study and design of an electro-optic frequency...
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STUDY AND DESIGN OF AN ELECTRO-OPTIC FREQUENCY SHIFTER FORDYNAMIC AND CONTINUOUS TUNING OF SHORT PULSES OF LIGHT
BY
DARIO FARIAS
INGEN., Universidad de los Andes, 1997.M.S., University of Illinois at Urbana-Champaign, 2000.
DISSERTATION
Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics
in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2003
Urbana, Illinois
c© Copyright by Darıo Farıas, 2003
STUDY AND DESIGN OF AN ELECTRO-OPTIC FREQUENCY SHIFTER FOR
DYNAMIC AND CONTINUOUS TUNING OF SHORT PULSES OF LIGHT
Darıo Farıas, Ph.D.Department of Physics
University of Illinois at Urbana-Champaign, 2003James N. Eckstein, Advisor
In this work, a novel all-optical device to achieve frequency shifting of short light
pulses is presented. A simple classical model will be shown to explain the basics of
the physical principles used for the device operation.
A theoretical study of the device operation is performed by using a semiclassical
wave equation analysis. This method uses a cavity coupled mode expansion formalism
to study the effect of a microwave field on a train of short optical pulses co-propagating
in a traveling-wave modulator structure. It is demonstrated that the device can
homogeneously and continuously shift the frequency spectrum of the light pulses,
as opposed to other frequency shifting methods, such as those involving four wave
mixing, cascaded second order nonlinearities, and semiconductor optical amplifiers,
in which there are residuals of the original signal that have to be filtered out.
As part of this work, a study of the effect of the applied field on the spectrum of
the light, the dependence on the relation between microwave wavelength and pulse
length, and the effect of changing the interaction length or the intensity of the ap-
plied microwave field, has been done. Computer simulations of this model have been
performed to predict and study the operation of the device.
Experimental measurements made on these kinds of devices have been performed
and compared to the theoretical predictions showing good agreement. Some exper-
imental techniques to improve the device operation and increase the frequency shift
obtained with controlled spectrum distortion levels were proposed and tested, showing
promising results.
iii
To my parents, Darıo Enrique and Marıa Teresa,
and to my brothers and sisters.
Thank you for your support and for so much love.
iv
Acknowledgments
First of all, I want to express my deepest gratitude to Professor James N. Eckstein
for his guidance and support throughout this project. The supportive environment in
his group, his generosity, and his enthusiasm for science and knowledge have helped
make my graduate school experience an enjoyable and fruitful journey. I want to
thank him for his support in my pursuit of a better understanding of the area of
non-linear optics and electro-optic devices. I also want to thank him for the positive
input and encouragement in pursuing the great satisfaction given by honest and hard
work. Under his supervision I have learned a lot about the true meaning of research
and its importance to science. The diversity of projects and research in our group has
helped me to learn about basic principles in other areas of physics and the importance
of multidisciplinary efforts when getting involved in research activities.
I would like to thank Dr. Jeffrey White and Ray Strange from the MRL Laser Lab
facility. I have learned a lot from their expertise and strong background in different
areas of optics and laser physics. Their insight, discussions, and feedback regarding
the experimental setups has been of invaluable help during these years of work. I
would also like to thank Professor Lance Cooper for his collaboration with comments
and feedback during the preparation and edition of this manuscript.
I would like to thank all the members of Professor Eckstein’s group for making
daily life in the laboratory environment enjoyable. Thanks for the patience and useful
discussions in many situation. I think we have all learned very important things from
each other.
During these years in Urbana-Champaign there has been a constant element in
my life. That has been the love and support of my parents, brothers, and sisters.
Even though there are thousands of miles between us, that has not stopped me from
feeling like a part of them. From them, I have learned the great value of a family, the
importance of pursuing a good education, and the love for learning and working to
accomplish new things everyday.
Finally, I would like to thank all the friends I have made in Champaign-Urbana
for the fond memories that I take with me. High-quality dancing at The Regent (with
v
the best of my music), a lot of good soccer games, and many other things that would
fill dozens of pages. Thanks to the Colombian Students Association for many great
experiences and events that kept “the roots” alive. I would also like to thank the rest
of my family and friends in Colombia from my high-school and undergraduate years.
Special thoughts to “la horda” for so many great memories.
And last, but certainly not least, I want to acknowledge the funding for this
work, which was supplied by the National Science Foundation under grant NO.EIA00-
81437, and the Physics Department with Teaching Assistantships during a couple of
semesters.
vi
Contents
List of Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Physical Model of the Electro-optic Frequency Shifter . . . . . . . . . . . . 7
2.1 Principles of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Coupled-Mode Analysis of the Electro-optic Frequency Shifter . . . . 13
2.2.1 Optical Photon Number Conservation . . . . . . . . . . . . . . 14
2.2.2 Optical Frequency Shifting . . . . . . . . . . . . . . . . . . . . 15
2.3 Simulation of a Simplified Model . . . . . . . . . . . . . . . . . . . . 18
3 Design and Fabrication Processes for the Electro-optic Frequency
Shifter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1 Optical Waveguide Processing . . . . . . . . . . . . . . . . . . . . . . 24
3.1.1 Proton Exchange . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.1.2 End-facet Polishing . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Microwave Structure Design . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.1 Basic Electro-optic Modulation and Bandwidth
Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.2 Broadband Traveling Wave Modulators . . . . . . . . . . . . . 39
4 Experimental Setup and Operation of the EOFS . . . . . . . . . . . . . . . . . . 46
4.1 Experimental Setup and Measurements . . . . . . . . . . . . . . . . . 47
4.1.1 Power and Pulse Length Dependence of the EOFS
Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.1.2 EOFS Cascading and Interaction Length Dependence . . . . . 53
vii
4.2 Microwave Techniques For Improved EOFS
Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1 Experimental Introduction of the Second Harmonic
Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 The EOFS and the Single Photon Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.1 Quantum Mechanics of the EOFS Operation . . . . . . . . . . . . . . 73
5.2 The EOFS in Quantum Information Science . . . . . . . . . . . . . . 77
6 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Appendices
A A Review of Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.2 Coupled-mode Wave Equation Formalism for Mode Evolution . . . . 86
A.2.1 Manley-Rowe Relations . . . . . . . . . . . . . . . . . . . . . . 88
B The Electro-optic Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.1 The Electro-optic Coefficient and the Effective Second Order Suscep-
tibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
viii
List of Figures
1.1 Optoelectronic frequency conversion scheme. . . . . . . . . . . . . . . 3
1.2 Difference frequency generation and Four wave mixing in nonlinear
optical material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Cross gain modulation and cross phase modulation conversion techniques. 5
2.1 (a) Schematic view of the EOFS. (b) Frequency up-shifting configuration. 8
2.2 Experimental set-up for spectral measurement of the light pulses. . . 11
2.3 Optical pulse spectra in down-shifting and up-shifting configurations
in Riaziat et al. (1993) . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Light spectrum shift with different interaction lengths and fixed mi-
crowave power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Light spectrum shift at different power levels and fixed length. . . . . 19
2.6 Frequency upshifting for different pulse lengths and fixed power and
interaction length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7 Relative bandwidth change as a function of the microwave power and
the ratio between pulse length and microwave length. . . . . . . . . . 22
3.1 Proton exchange reaction. . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Proton exchange setup. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Lapping tool and sample holder. . . . . . . . . . . . . . . . . . . . . . 31
3.4 Polishing setup showing the three aluminum oxide manual steps and
final lapping wheel step. . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5 Lithium niobate crystal polishing for waveguide coupling. . . . . . . . 33
3.6 Electro-optic Frequency Shifter (EOFS) structure. . . . . . . . . . . . 35
3.7 Broad-band traveling wave phase electro-optic modulator proposed by
Miyamoto et al. (1991). . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.8 Electro-optic traveling wave amplitude modulator with a ridge wave-
guide structure proposed by Mitomi et al. (1995). . . . . . . . . . . . 43
4.1 First experimental setup for spectral measurement of the pulses. . . . 48
ix
4.2 Experimental setup for spectral measurement of the pulses using the
phase-locked oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 Optical spectra of the up-shifting configuration for different power lev-
els. a) 8 ps pulses at 945 nm, b) 13 ps pulses at 927 nm, and c) 20 ps
pulses at 927 nm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Light spectrum shift with different power levels applied to the EOFS
in both up-shifting and down-shifting configurations. . . . . . . . . . 52
4.5 Numerical simulations showing the effect of pulse length and power on
the distortion of the pulses after frequency shifting. . . . . . . . . . . 53
4.6 Experimental setup for cascading of two EOFSs. . . . . . . . . . . . . 54
4.7 Spectra of the frequency shifting obtained when cascading two EOFSs. 55
4.8 Introduction of a second harmonic field component to extend the con-
stant gradient region in the refractive index wave. . . . . . . . . . . . 58
4.9 Introduction of a second harmonic field correction with relative weight
a = 1/8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.10 Simulations for the operation of the EOFS with a second harmonic
field component with relative amplitude a = 1/8. . . . . . . . . . . . 62
4.11 Experimental setup for the introduction of a second harmonic compo-
nent correction to the microwave field. . . . . . . . . . . . . . . . . . 64
4.12 Frequency shifting of 18 ps pulses with the second harmonic correction
to the microwave field. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.13 Frequency shifting of 29 ps pulses with the second harmonic correction
to the microwave field. . . . . . . . . . . . . . . . . . . . . . . . . . . 67
A.1 Sum frequency generation. . . . . . . . . . . . . . . . . . . . . . . . . 87
x
List of Abbreviations
APE Annealed Proton Exchange
CW Continuous Wave
DFG Difference Frequency Generation
DI Deionized (Water)
DWDM Dense Wavelength Division Multiplexing
EOFS Electro-optic Frequency Shifter
FWHM Full Width at Half Maximum
FWM Four Wave Mixing
GHz Gigahertz
IPA Isopropyl Alcohol
IR Infrared
LNO Lithium Niobate
MHz Megahertz
Nd:YLF Neodymium:Yttrium-Lithium-Fluoride
OSA Optical Spectrum Analyzer
PMF Polarization Maintaining Fiber
PLO Phase-Locked Oscillator
QCo Quantum Computing
QCr Quantum Cryptography
xi
QIS Quantum Information Science
QO Quantum Optics
RZ Return to Zero
SOA Semiconductor Optical Amplifier
SFG Sum Frequency Generation
SHG Second Harmonic Generation
TDM Time Division Multiplexing
WDM Wavelength Division Multiplexing
TE Transverse Electric
TM Transverse Magnetic
THz Terahertz
TDM Time Division Multiplexing
TWT Travelling Wave Tube (Amplifier)
W Watts
XGM Cross-Gain Modulation
XPM Cross-Phase Modulation
YIG Yttrium-Iron-Garnet
xii
Chapter 1
Introduction
1.1 Motivation
During the last ten years there has been an increasing need for research in the area of
optical communication systems. Current computer networks, Internet users, and data
transfer networks require higher bandwidth and faster data transmission rates every
day. Wavelength Division Multiplexing (WDM) emerged as an efficient response to
these needs by transmitting various channels using different wavelengths through the
same optical medium. With time, it has become clear that it is important to achieve
fast and efficient conversion of channels for flexible data transmission and reliable
routing in these WDM systems. Time Division Multiplexing (TDM), based on en-
hancing the channel capacity by a careful time window definition and synchronization
of various signals through the same channel, was an area of active research for some
time. It was abandoned to a degree because of the technical difficulty involved with
precise timing and phase control of the systems. WDM became the dominant ap-
proach to attack the bandwidth problem. However, in the last few years TDM has
gained more interest, particularly considering the progress in phase and timing con-
trol electronics and the more restrictive needs of modern systems in terms of data
synchronization. The creation of a hybrid system with both TDM and WDM tech-
niques would be a possible answer to the need for faster data communication rates.
A hybrid TDM-WDM system would most likely involve the use of time synchronized
and phase stable short pulse signals, as well as different wavelengths to increase the
channel capacity.
Another problem in data communication systems is that of switching and routing.
Efficient routing is a major interest in the future of distributed computing. Advanced
communication networks must be able to support data traffic among a large number
1
of distributed sites. High bandwidth and dynamically responsive optical networks
will enable parallel processing in multiprocessor arrays and communication between
different processors and computers. Such an optical network will also need to be
dynamically responsive to the global and local state of the computation or task being
performed.
In this work, an all-optical device using a nonlinear optical material is presented as
a possible solution to these problems: the Electro-optic Frequency Shifter (EOFS).
This device is able to achieve very fast routing and switching of the optical data
between different channels in real time. Accurately routing a particular bit stream
to a particular channel, with a reaction time fast enough to act in the inter-bit space
(typically 100 ps), would greatly extend the capabilities of distributed processing.
The device needs to have the ability to select the particular channel to which we
want to switch a data packet. The optimum goal would be to control the frequency
of a particular bit or pulse in the transit time through the device. The device that is
proposed could be used in Return to Zero (RZ) encoding systems. RZ data format is
preferred for reliable operation of high bandwidth WDM telecommunication systems
(Breuer et al. (1996); Caspar et al. (1999)). This device makes use of the linear
electro-optic effect that is found in some anisotropic materials. Particularly, we will
be interested in Lithium Niobate (LNO), which is widely used in the field. The
device is able to produce fast changes in the frequency of short pulses of light with a
long range in wavelength, within a large bandwidth WDM system. Currently, WDM
systems require channel separations on the order of 50 to 100 GHz and a bandwidth
per channel in the 20 to 40 GHz range. This shows that a 20 channel WDM system
can easily span the range from 1 to 2 THz. This is the range of switching frequencies
we should expect the EOFS to be capable of achieving.
The instantaneous (really limited by the material response time which is related to
optical phonon frequencies) nature of its conversion rate, in addition to its ability to
reduce stabilization problems associated with second laser sources or optoelectronic
conversion stages, makes the EOFS very flexible compared with other devices. The
frequency shifting obtained is simply related to the power that is applied to the de-
vice. It is this ability to control a microwave integrated circuit with reliability and
a fast response time which gives the EOFS advantages over other systems that are
only able to change the wavelength between prearranged values, as well as systems
that use subsequent steps of detection, electronic conversion, amplification, and re-
emission. These latter systems make up a large part of the group of opto-electronic
conversion schemes. Currently, these are the most widely used in optical commu-
2
PIN/Receiver
Amplifier/Electronics
Laser
External Modulator
Figure 1.1: Optoelectronic frequency conversion scheme.
nication networks because of their relatively easy fabrication process and low costs.
A typical opto-electronic frequency conversion device (see figure 1.1) consists of a
detection stage (usually a photodiode), an electronics and amplification stage that
will generate a modulation signal, and second laser source which is either modulated
directly or externally by an electro-optic modulator. The external modulation scheme
has generally proven to be faster, but it also requires the external modulator design,
which increases fabrication steps and costs.
Another advantage of the EOFS over other currently used schemes such as Sum (or
Difference) Frequency Generation (SFG/DFG), Four Wave Mixing (FWM) (Breuer
et al. (1996); Caspar et al. (1999)), and cross phase or cross gain modulation (XPM,
XGM) in semiconductor optical amplifiers (SOA) (Stubkjaer et al. (1996); Wang and
Tucker (1997); Durhuss et al. (1994)), is that the EOFS does not require a second
optical source. The DFG or FWM techniques, shown in figure 1.2, rely on wave
mixing processes that use a second optical wave or generate harmonics to produce the
desired output. They require good source control,their efficiency is not 100%, and they
usually require strong pumps as a second source to be able to have a relatively efficient
wave mixing in the nonlinear material. Once the signal and pump wavelengths are
set, the idler (output) wavelengths are then fixed. The possibility of wavelength
conversion to other values is limited by the availability of tunable sources and by
3
λi
pump
Nonlinear optical element λc
ωi
ωp
2ωp-ωi
FWM
ωi
ωp/2
ωp-ωi
DFG
ωp
Figure 1.2: Difference frequency generation and Four wave mixing in nonlinearoptical material.
velocity matching conditions between the optical waves, both of which depend on the
optical wavelengths and the crystal properties and orientation. The velocity matching
or quasi-phase matching is crucial in order to attain efficient wavelength conversion,
and not just a back and forth power flow between different frequency modes due to
the nonlinear interaction.
Now we can briefly describe the frequency conversion techniques that make use
of SOAs. These are illustrated in figure 1.3. The XGM uses optical gate switching
(controlling the gain saturation of the SOAs with an optical signal). The XPM uses an
interferometric scheme based on the refractive index modulation in the OSAs driven
by the original message or data stream. With this effect, a modulation of the second
source (target source) is produced and a “copy” of the data is created in the target
wavelength. The XGM devices usually work in an inversion mode (i.e., a high input
level signal will produce gain saturation and increase the absorption of the medium
blocking the second source). It could be possible to obtain tunability of the switching
range in these methods by having a controllable source. This light source must be
tuned within very tight requirements. In addition, the tuning capability to different
channels in all of these devices depends on the stability and rate at with which these
light sources can be changed. To change the emission wavelength of a second source
4
λi
CW
SOA Optically controled gate
λc (output)
Cross Gain Modulation
λi
CWλc
SOA1
SOA2
(λc)
(output)
Cross Phase Modulation
(λc)
Figure 1.3: Cross gain modulation and cross phase modulation conversion tech-niques.
to a particular value would require a response time closely related to the speed with
which the gain or electronic characteristics of the source change. Instability problems
related to bias level, temperature, and line width of the source could also occur. The
EOFS has the advantage that its switching capability depends solely on the control
electronics for the microwave excitation. There are microwave electronics circuits with
bandwidths above 20 GHz, which means that switching the power level output of the
circuit faster than the inter-bit time is an achievable goal. Another advantage is that
the conversion efficiency of the bit streams is practically 100 % (excluding insertion
losses), so there is no residue of the original signal that must be filtered out before
the output is routed to the new destination, and there is no need for an amplifier
stage. Another important advantage is that the fabrication process of the EOFS is
considerably simpler than the fabrication of XGM or XPM devices, in which there is
the issue of the fabrication of the SOAs, the integration of the second laser source,
and more complex filtering and interferometric schemes in addition to the waveguide
structures.
5
1.2 Thesis Overview
In chapter 2, basic physics of the operation of the Electro-optic Frequency Shifter
(EOFS) are discussed with a classical model. Then, the operation of the EOFS is
studied introducing a semiclassical approach via a coupled mode analysis. It is shown
how both analyses lead to the same conclusions. Numerical simulations are shown on
the expected operation performance of an EOFS and operational parameters will be
discussed to obtain an efficient operation of the device. The model is used to study
the effect of interaction length, microwave power, and pulse length on the device
operation.
In chapter 3, the techniques and procedures used for the device design and fabri-
cation are described. A review of how these techniques and processes have been used
before for other devices is presented, as well as how these techniques have developed
over the last few years. The application of these techniques to the EOFS will be
discussed as well.
In chapter 4, the laboratory experimental set-up and the results of the measure-
ments performed with the device are presented. Different experimental set-ups were
used to study the operation of one or two devices, as well as the power and pulse
length dependences. Techniques for improving the frequency range (under low dis-
tortion limits) are discussed and the respective experiments to test these ideas will
be shown.
In chapter 5, an attempt to describe the interaction of the EOFS in the quan-
tum limit of a single photon is shown. The problem of photon localization and wave
function description is introduced briefly. A single photon case is treated in a quan-
tized optical field approach (even though the microwave is considered to be a classical
field), using a strongly attenuated beam approximation. The results show that it is
possible to perform an analysis analogous to the semiclassical approach in chapter
2. A unitary operator that can approximately describe the action of the EOFS on
a single photon state is defined here. This is of potential interest for the fields of
quantum communication and quantum cryptography using photons.
Finally, chapter 6 includes a brief summary of the work, discussions about the
findings, and possible future work. The significance of these types of devices for
possible future telecommunication networks or distributed computing systems is also
discussed here.
6
Chapter 2
Physical Model of the Electro-optic
Frequency Shifter
2.1 Principles of Operation
The principle of operation of this device consists of using the linear electro-optic effect
in a travelling wave modulator structure (Riaziat et al. (1993)). An optical pulse
travels through a waveguide fabricated in a non-linear optical crystal, such as lithium
niobate (LNO). A microwave waveguide is patterned on the crystal surface, so that
an electric field can be applied to the material. The microwave signal operates at a
single stable frequency, synchronous with or at a small multiple of the bit stream rate.
The microwave co-propagates in the same direction in the crystal as the optical pulse
by using the electrode structure deposited on the LNO substrate (see figure 2.1(a)).
The optical and coplanar microwave electrode structures for a y propagating, z-cut
substrate case are shown here. Part (b) of this figure shows the relative position of the
optical pulse and the index of refraction wave required to accomplish frequency up-
shifting. The leading edge of the pulse travels slower than the trailing edge, resulting
in pulse and wavelength compression. This gradient in the refractive index seen by
the light is due to the modulation produced by the microwave electric field via the
electro-optic effect (n = n0 + αE). When the pulse of light is co-propagating with
the microwave signal, having an electric field given by
Em = E0 sin (kz − ωt), (2.1)
the index of refraction experiences a dynamic change due to the linear electro-optic
effect
αE = δn = −1
2n3
0rcEm. (2.2)
7
y y
index of refraction
optical pulse
n(y,t)-n0
Eopt(y,t)
direction of propagation
b)
y
Light pulse microwave
z
x
a)
Figure 2.1: (a) Schematic view of the EOFS. (b) Frequency up-shifting configura-tion.
Here, n0 is the index of refraction of the material and rc is the effective linear electro-
optic coefficient. From figure 2.1(b), we see that if the pulse is located at a position
with respect to the microwave (index of refraction modulation wave), such that the
front of the pulse travels slower than the back of the pulse, the pulse will be com-
pressed. Pulse stretching may also be produced by changing the relative position
of the pulse and the microwave, so that the front of the pulse is the one travelling
faster. Pulse compression or stretching with a fixed number of optical cycles leads to
frequency (wavelength) conversion. To calculate the shifting effect, two points near
the center of the pulse, which are spatially separated by one wavelength, are chosen.
The difference in their propagation speed is ∆u. After a propagation time, dt, the
change in wavelength is
dλ ' dt ·∆u = (n0/c)dz · (−c/n20)∆n, (2.3)
8
where
∆n ' ∂n
∂zλ =
∂δn
∂zλ. (2.4)
Substituting equation (2.4) into equation (2.3), the expression is transformed to
dλ ' −λ1
n0
∂δn
∂zdz. (2.5)
Now using equation (2.2) for the index variation, an equation for the fractional wave-
length change is obtained
dλ = −1
2λn2
0rc∂Em
∂zdz. (2.6)
This equation may be solved and integrated over an interaction length L to arrive at
an expression for the maximum wavelength change as
λ = λ0 exp[∓πLeff
λm
n20rcE0], (2.7)
where λm is the wavelength of the microwave in the material. The sign of the shift de-
pends on the gradient chosen by the relative position of the pulse within a microwave
period. Leff is an effective length of interaction due to the attenuation of the field
because of microwave losses. The field amplitude can be expressed as
E0(z) = E0(0)exp[−z/Z0], (2.8)
where Z0 is the microwave attenuation length. Going over the integration solution
in equation (2.6), the relation between L and Leff is found to be given by Leff =
Z0(1 − exp[−L/Z0]). Since the exponential argument in equation (2.7) is small, a
linear approximation is used to arrive at a frequency shift similar to the one given by
Riaziat et al. (1993)
∆f = ±f0πLeff
λm
n20rcE0. (2.9)
This shift calculation is done considering a fixed relative phase condition between
the optical and the microwave fields. Generally, the index of refraction of the optical
and microwave fields will be different unless there is a velocity matching condition.
This can be achieved by structure engineering or by other quasi-phase matching meth-
ods (Feng-Zeng et al. (1997); Mitomi et al. (1995); Noguchi et al. (1995); Miyamoto
et al. (1991)). Typical values for the index of refraction for LNO are nopt = 2.25
for the optical wave and nmic =√
εeff for the microwave. Here, εeff is approximated
as an average of the material and air dielectric constants for quasi-planar electrode
9
structures. For LNO in an x or y propagating mode in these planar electrode struc-
tures, the effective dielectric constant is close to 18, which makes the refractive index
4.25. The index mismatch produces a gradual phase shift between the microwave
and optical wave. If this continues so that the phase shift reaches 2π then, over
this period, the upshifting and downshifting phase configurations appear to be equal,
and the net effect is cancelled. This problem is solved by structure engineering or
other quasi-phase matching methods, such as microwave polarity inversion along the
microwave guide to produce a 180-degree shift to periodically reposition the waves
(Riaziat et al. (1993)). Due to this periodic re-synchronization, in a first estimate of
the device operation, a factor of 1/2 is introduced in equation (2.9). Typical values
for the variables in this equation are f0 = 2 ·1014 Hz (1500nm), nmic =√
18, n0 = 2.2,
and Leff = 4 cm. This effective length depends on the microwave structure, length
(in this case between 5 and 6 cm), and the operating frequency. Based on microwave
attenuation measurements previously done at 16 GHz (Riaziat et al. (1993)), 4 cm is
a reasonable value for the effective length. Other relevant values are rc = 30 · 10−12
m/V, λm = 0.442 cm (at 16 GHz), E0 '√
2PZ0/d, P = power in Watts, Z0 = 50 Ω,
and d = 10 µm. Using these values in the corrected version of equation (2.9), we
arrive at an expression for the shift in GHz:
∆f ' 412.83√
P (Watts). (2.10)
A similar EOFS, with a half-coplanar structure, has already been built and tested
by Eckstein and coworkers at Varian (Riaziat et al. (1993)). They used a 3.5 cm long
monolithic half co-planar microwave circuit deposited onto an x-cut LNO wafer. This
structure used a periodical electrode inversion structure to achieve the quasi-phase
matching. The pitch for the required inversion is obtained by calculating the required
length for the microwave phase to drift by 180 degrees at the pulse position. The
microwave phase is given by φµ = (ωt − ky), where ω and k are the microwave’s
angular frequency and wave-vector, respectively, and where y and t indicate position
and time. When the pulse has propagated for a distance l, its time of travel is given
by t = lnopt/c. The phase of the microwave is then
φµ = ωt− kl = 2πfµlnopt
c− 2π
λµ
l (2.11)
where λµ = cfµnµ
and nµ ' √εeff . Solving for l for the case in which the phase reaches
the −π value, the pitch needed for the periodic electrode reversal is given by (Riaziat
et al. (1993)
10
Figure 2.2: Experimental set-up for spectral measurement of the light pulses.
lπ =c
2fµ(√
εeff − nopt). (2.12)
The experimental set-up used in this previous experiment is shown in figure 2.2.
Light from a mode-locked Nd:YLF laser was passed through a pulse compressor and
then coupled to the devices in the LNO wafer using a polarization-maintaining fiber.
The laser was synchronized with the microwave source so that the frequency of the
microwave applied to the devices matched a multiple of the pulse repetition rate. This
was done by means of an RF synthesizer that provided the signal for the acousto-
optic modulator in the Nd:YLF laser and a 10 Mhz reference signal for the microwave
synthesizer. The microwaves were then passed through a power amplifier and applied
to the LNO devices. The optical pulse length was measured using an autocorrelator.
The light spectrum was measured using an optical spectrum analyzer (OSA) which
had a free spectral range of 8 THz and a finesse of 10000. Riaziat and Eckstein
measured frequency shifts of up to ±350 GHz with 16 Watts of power. The results
of this experiment are shown in figure 2.3. The left graph shows the operation of the
device in the down-shifting configuration. The right graph shows the results in the up-
shifting configuration. A 350 GHz span was observed by applying a microwave power
of 16 W. The difference between upshift and downshift configurations was obtained
by adjusting the relative phases between the microwaves and the optical pulse with a
phase shifter placed downstream of the power amplifier, as shown in figure 2.2. The
11
Down-shifting phase Up-shifting phase
on off off on∆f = -350GHz ∆f = +350GHz
P(f)
f
Figure 2.3: Optical pulse spectra in down-shifting and up-shifting configurations inRiaziat et al. (1993)
.
results demonstrate a variation in the spectrum shape after the shifting occurred. The
shifted pulse has a tail on the lower frequency side for the upshifting configuration.
Similarly, the tail for the downshifting configuration appears on the higher frequency
side. In the next section, a model that qualitatively and quantitatively explains this
effect is presented. We worked on an improvement of this design using a coplanar
microwave structure and a periodic poling quasi-phase matching technique, such as
the one used for SHG experiments in LNO (Jundt et al. (1991); Yamamoto et al.
(1991)). Using this technique, it may be possible to improve the optical-electric field
coupling, and thus the performance of the device, without having to rely entirely on
the application of microwave power or device length. The other option studied was
the use of a velocity matched structure which is engineered to speed up the phase
velocity of the electric field to match the optical phase velocity. This type of structure
is the one analyzed in the next section.
12
2.2 Coupled-Mode Analysis of the Electro-optic
Frequency Shifter
In this section, the frequency shifting of a short light pulse traveling along the device
is studied theoretically using a mode expansion formalism similar to the one used
in photon-by-photon power flow in three wave mixing (Boyd (1992)). While it is
possible to study the frequency shifting of a single optical pulse interacting with a
sinusoidal microwave field, it is computationally simpler to consider a system in which
the microwave and optical fields have the same periodicity. An optical pulse train
is created by “placing” pulses which are replicas of the original optical pulse in each
microwave period at the same phase. Using the nonlinear wave equation to introduce
the nonlinear interaction due to the material results in coupled differential equations
for the discrete mode spectrum.
With this periodic boundary condition, a light pulse at z = 0 and t = 0 becomes
part of a pulse train with pulse spacing of τ seconds, which has a spectrum of modes
separated by 1/τ Hz:
Eopt =∑m
Em(z, t) =∑m
Am(z)ei(kmz−ωmt) + c.c., (2.13)
where Am(z) is a slowly-varying amplitude for the mth mode, as a function of the
position z, which can also be directly related to time by the pulse propagation. The
pulse is originally centered at z = 0 at t = 0. These Am(z) amplitudes are centered
around a peak mode such that mpeak = Int[ωopt/ω1], where ωopt represents the central
frequency of the light pulses, km = mk1, and ωm = mω1. Here k1 and ω1 are the
wave vector and frequency of the microwave signal respectively. According to the
light spectrum distribution, the mode amplitudes (representing the pulse train) are
centered around mpeak, and gradually become smaller away from it. Without loss of
generality, they are chosen to be real, initially. Similarly, the synchronous microwave
signal is written as a superposition of modes in the following expression.
Eµ =s∑
l=1
−ial(z)ei(klz−ωlt) + c.c., (2.14)
where kl = lk1 and ωl = lω1 = (c/n0)kl. While the coefficients al(z) and Am(z) are
mode amplitudes, the difference between them is the range of frequencies which they
describe. al(z) corresponds to the lower (microwave) frequencies and Am(z) to the
much higher (optical) frequencies. Setting the initial conditions of Am(0) and a1(0) to
13
be real and positive in equation (2.14) corresponds to having the light pulse located
in the increasing index gradient portion of the microwave, which is the situation that
produces frequency upshifting. The other al amplitudes are taken to be zero at z = 0,
since a well-defined single microwave frequency is applied to the device.
The method used to model the device is based on a multi-mode generalization
of a the three wave mixing phenomena in non-linear optics. From this we get some
Manley-Rowe power flow type of relations which explain the power flow between the
different optical modes and the microwave field mode. For a short review on the
nonlinear optics theory involved in this model, see Appendix A. In the next sections
we will show how the action of the EOFS conserves the number of photons, and how
this semiclassical treatment can also explain the frequency shifting effect, obtaining
the same predictions as the simpler model presented in section 2.1.
2.2.1 Optical Photon Number Conservation
The microwave and optical waves are coupled by the non-linear response of the ma-
terial, leading to a non-linear polarization component that acts as a source term in
the homogeneous wave equation for the field components (Boyd (1992))
−∇2Em +ε
c2
∂2Em
∂t2= −4π
c2
∂2PNLm
∂t2, (2.15)
where Em = Em(z)e−iωmt + c.c.. The non-linear polarization term is given by
PNLm = PNL
m (z)e−iωmt + c.c., (2.16)
This nonlinear polarization term is related to the fields by the second order suscepti-
bility as
PNL(z, t) = 2deff(Eµ + Eopt)(Eµ + Eopt). (2.17)
Here deff is the effective second order susceptibility, which in this case is given by
deff = n40rc/16π. This relationship between the second order susceptibility and the
commonly measured electro-optic coefficient is explained in Appendix B. Substituting
the expressions in equations (2.13) and (2.14) into the equation (2.17), we get the
nonlinear polarization term expansion that can then be used in the wave equation
(2.15). Solving for the different frequency components (eiωmt) separately, the following
expression is obtained:
14
∂Am
∂z=
1
2kmn2
0rc
∑
l
alAm−l − a∗l Am+l, (2.18)
where all the al and Al coefficients are functions of z. Similarly, focusing on the
low frequency amplitudes (microwave harmonic components), equations for the al
coefficients are obtained:
∂al
∂z= −1
2kln
20rc
∑m
AmA∗m−l. (2.19)
In this expression, the contributions of the terms involving products of the microwave
coefficients aia∗i−l are neglected, because a1 is the only non-negligible term in these
products. We are thus left only with the AmA∗m−l contributions. Equation (2.18) is
now used to calculate the change in photon number per mode and the rate of change
of the total photon number in the optical pulse. The mode intensity is given by (Boyd
(1992))
Im =nmc
2πAmA∗
m. (2.20)
The number of photons per mode is then Nm = Im/~ωm. Using equations (2.18) and
(2.20), the following expression for the photon number change is obtained.
d
dzNopt ≡ d
dz
∑m
Im
~ωm
=n4
0rc
4π~∑
m,l
[al(Am−lA
∗m − AmA∗
m+l) +
a∗l (AmA∗m−l − Am+lA
∗m)
]. (2.21)
Evaluating the sum over m in the previous equation, it is clear that with changes of
variable of the form m → m + l in the first and third terms of the summation, the
terms inside the parentheses cancel each other out, showing that the total photon
number does not change. The photon number in each mode varies due to its coupling
with the adjacent modes’ amplitudes. However, the total number of photons remains
invariant. This shows that optical photons are transferred from one mode to another
by absorbing or emitting microwave quanta.
2.2.2 Optical Frequency Shifting
Equation (2.18) can also be used to derive an expression for optical frequency shifting.
For this purpose, we define the average frequency of the optical pulse train in the
15
device as
〈ω〉 ≡∑
m Nmωm∑m Nm
=1~∑
m Im∑m Nm
. (2.22)
This expression is now differentiated with respect to the propagation distance z. The
denominator is constant since the number of photons is conserved. The derivative of
the term Im is evaluated using equations (2.20) and (2.18).
dIm
dz=
n30ckmrc
4π
∑
l
al(Am−lA∗m − AmA∗
m+l) + a∗l (AmA∗m−l − Am+lA
∗m). (2.23)
Introducing an extra sum over m in the previous equation to take into account the
modes contribution leads to the expression
∑m
dIm
dz=
n30crc
4π
∑m
mk1
∑
l
al(Am−lA∗m − AmA∗
m+l) +
a∗l (AmA∗m−l − Am+lA
∗m). (2.24)
The sums over the elements that accompany the al coefficient can be made by intro-
ducing a change of variable from m to (m− l) as
n30ck1rc
4π∑m
m∑
l
al(Am−lA∗m) −
∑m
(m− l)∑
l
al(Am−lA∗m) =
n30ck1rc
4π
∑
m,l
lal(Am−lA∗m). (2.25)
The terms with a∗l can be grouped similarly and simplified so that the total sum over
m in equation (2.24) gives
∑m
dIm
dz=
n30ck1rc
4π
∑
m,l
lalAm−lA∗m + la∗l AmA∗
m−l. (2.26)
Now consider the complex phase that the amplitudes Am and al may acquire. All
of these, by definition, are real at z = 0. Notice that the terms in the differential
equations (2.18) and (2.19) are all real. Thus the coefficients will remain real and the
complex conjugation in equation (2.26) may be neglected. This leads to the simpler
equation
∑m
dIm
dz=
n30ck1rc
2π
∑
m,l
lalAmAm−l. (2.27)
16
To analyze this expression, consider the case where l only equals 1. That is,
when the microwave field is only composed of a1 and the other ai are negligible. The
variation of the factor a1 is negligible if we consider a lossless situation. It may be
taken out of the bracket, leaving a sum over all modes of the product of neighboring
mode amplitudes Am−1Am, and transforming the previous equation to
∑m
dIm
dz=
n30ck1rca1
2π
∑m
AmAm−1. (2.28)
If the optical pulses are sufficiently short, compared to the microwave wavelength,
the sum over m can be approximated by an integral over a continuous variable A(k)
such that Am = A(km). Then Am−1 ' Am − (∂A/∂k)∆k, where ∆k is the mode
spacing. Substituting the expression for Am−1 back into equation (2.28), the right
hand side can then written in terms of the total optical intensity plus a k-space sum.
This sum can be transformed into an integral of a perfect differential (d(A2m)/dk).
This integral is equal to zero, since the optical spectrum has a finite bandwidth and
the mode amplitudes go to zero in the integral limits:
∑m
dIm
dz=
n30ck1rca1
2π
∑m
AmAm = n20ck1rca1
∑m
Im. (2.29)
Going back to equation (2.22), the derivative with respect to z is simply
d〈ω〉dz
=1~∑
mdIm
dz∑m Nm
, (2.30)
which can then be used to obtain the following differential equation
1
〈ω〉d〈ω〉dz
=
∑m
dIm
dz∑m Im
= n20k1rca1. (2.31)
Using this expression, and taking into account that 2a1 = E0, an equation for the
evolution of the average photon energy in the pulse as it propagates is obtained:
d
dz〈ω〉 =
πn20rcE0
λm
〈ω〉. (2.32)
Integrating this equation leads to an expression for frequency shifting that is identical
to the one obtained for the wavelength in equation (2.7) using the simpler kinematic
model. The shift in energy depends exponentially on the distance over which the
interaction occurs, and for small shifts this can be approximated linearly as in equa-
tion (2.9). An equation describing the rate of change of the field amplitude a1 is
derived similarly. The energy gain/loss of each of the optical photons is provided by
the absorption/emission of the photons at the microwave frequency.
17
2.3 Simulation of a Simplified Model
The set of coupled equations that come from equations (2.18) and (2.19) can be solved
in a first approximation by including all of the microwave power in the fundamental
mode. This means aj ≈ 0 for all values of j greater than 1. Then the sums over
l in equation (2.18) are simplified to one term. A first order approximation for the
variation of the aj coefficients is obtained. The resulting equations are
∂Am
∂z=
1
2kmn2
0rc(a1Am−1 − a∗1Am+1),
∂a1
∂z= −1
2k1n
20rc
∑m
AmAm−1,
∂aj
∂z= −1
2kjn
20rc
∑m
AmAm−j. (2.33)
A fifth order Runge-Kutta algorithm was used to solve the system of equations.
Equations for the aj coefficients were included up to j = 3, since for higher numbers
their variation is negligible. The case of light output from a Nd:YLF laser with 10
ps long pulses, and a central wavelength at 1047 nm was considered, in order to
use values close to the previous experiment. For a first approximation, a Transform
Limited Gaussian pulse was considered, so that the frequency spectrum can be written
as
Am ∝ e−(m−m0)2/2σ2
. (2.34)
For this simulation the microwave field was chosen to operate at a frequency of 16
GHz. With this frequency and the central frequency of the laser, m0 was determined.
The index of refraction and electro-optic coefficient for the LNO were taken from
known values. The initial values of the coefficients Am and aj are obtained from
the intensity of the microwave and optical waves by considering the mode intensity
contributions Ij = (njc/2π)AjA∗j . The optical intensity is given as a sum of the mode
intensities, and the microwave intensity is related simply to a1.
To study the operation of the device, the set of equations (2.33) with j = 1, 2,
and 3 was numerically solved. Initially, the light power of the mode-locked laser was
chosen to be 50 mW, and the microwave power was chosen as 0.5 W. The overall shift
dependence on interaction length and microwave power was studied. Figure 2.4 shows
the dependence of frequency shifting with the length of interaction for a fixed power
at 0.5 W. Part a) shows the spectrum after different interaction lengths z = 0, 1, 2,
18
0
0.5
1
1.5
2
2.5
3
3.5
4
0 200 400 600 800
z=0 cm
z=1 cm
z=2 cm
z=3 cm
Relative Frequency (GHz)
0
0.05
0.1
0.15
100 200 300 400 500
0
100
200
300
400
500
0 0.5 1 1.5 2 2.5 3 3.5
z (cm)
a)
b)
Figure 2.4: Light spectrum shift with different interaction lengths and fixed mi-crowave power.
a)
b)
0
0.5
1
1.5
2
2.5
3
3.5
4
0 200 400 600 800
01 W4 W9 W16 W
Relative Frequency (GHz)
0
0.05
0.1
0.15
0.2
100 200 300 400 500 600
0
100
200
300
400
500
0 1 2 3 4 5
Power1/2 (W1/2)
Figure 2.5: Light spectrum shift at different power levels and fixed length.
19
and 3 cm. respectively. The inset shows the tail developed in the lower frequency side
after shifting for the z = 2 cm case. Part b) of the figure shows that the relationship
between the frequency shift and the interaction length is linear. The results for the
dependence of the frequency shifting with power level are shown in figure 2.5. In part
a), the spectrum is shown for power levels of 0, 1, 4, 9, and 16 W, respectively. The
length was fixed at 0.5 cm. The inset of this graph shows a distortion tail developing
in the low frequency side for the 9 W case. Part b) shows that the relationship
between the frequency shift and the square root of the power is linear. This agrees
with the shifting expressions obtained from both the simple physical model and the
semiclassical model as in equation (2.9).
From the results of these simulations, it is observed that the frequency shift scales
linearly with length and as the square root of the power. Also, as the shift increases
(in both cases), the spectrum develops an overall broadening and a small tail begins
to appear on the lower frequency side. This is for an upshifting configuration of
the EOFS. For the downshifting configuration, the result is symmetric on the high
frequency side. This demonstrates that the pulse begins to distort when either the
interaction length or the power applied is increased. The appearance of this distortion
tail on the lower (or higher, if in down-shifting configuration) side of the pulse spec-
trum qualitatively agrees with the results observed in previous experiments (Noguchi
et al. (1995) ). This distortion comes from the fact that the refractive index gradient
is not a constant over all the parts of the pulse, so that locally the shifting effect in the
different portions of the pulse might vary. The longer the interaction or the higher the
power, the more visible the effect will appear. This can potentially create a problem
when either the tail is too large or the bandwidth spread increases above inter-channel
spacing requirements in a WDM system, because this can produce inter-channel in-
terference. However, as long as this spread is either controlled within the bandwidth
requirements of the channel, or filtered out, the interference can be minimized.
The appearance of the distortion after the frequency shifting is performed suggests
that the use of shorter pulse lengths can improve the operation, since a shorter pulse
would be located in a smaller portion of the index of refraction wave. In this way,
the gradient that the pulse “observes” can be kept closer to a constant value over
an extended region. Simulations of the operation of the EOFS were performed to
study the dependence of the pulse length for fixed microwave frequency, power and
interaction length. The results of this modelling are shown in figure 2.6. Part a) of
this figure shows the unshifted spectra of transform limited gaussian pulses of FWHM
duration of 5, 10, 15, 20, 25, and 30 ps. Other operation variables were set to 930
20
0
1
2
3
4
5
322.65 322.7 322.75 322.8 322.85 322.9
5 ps
10 ps
15 ps
20 ps
25 ps
30 ps
Pow
er (
a.u)
Freq. (THz)
0
3
6
9
12
322.45 322.5 322.55 322.6 322.65 322.7
Pow
er (
a.u)
Freq. (THz)
a)
b)
Figure 2.6: Frequency upshifting for different pulse lengths and fixed power andinteraction length.
nm wavelength, 1 W of microwave power, 2.5 cm effective length, and 6.056 GHz
microwave frequency. Part b) shows the shifted spectra for the different pulses. From
this figure it can be seen how the overall shifting is almost the same for the different
pulses (approx. 220 GHz). However, the peaks of the spectra tend to drift to slightly
lower values for the longer pulses. This is also accompanied by a stronger and more
extended lower frequency band distortion. As the figure shows for the longer pulses,
this effect is considerably bigger and is an important factor to keep in mind when
thinking about the fidelity of the shifted signal and the problems of distortion and
inter-channel interference that could arise in WDM system, where it is important
that adjacent channels can be distinguished from each other.
Since the longer the pulse length is, the narrower is its spectrum, but the more
distortion there is after shifting. Consequently, there is a need for experimenting
and finding optimum parameters such as pulse-length, microwave frequency, pulse
repetition rate, and inter-channel spacing. Because of this, the distortion of the
pulse as a function of the ratio between the lengths of the pulse and the microwave
wavelength was also studied. For an efficient device operation, the pulse should
experience a constant gradient of the index of refraction wave. This can be achieved
by causing the gradient of the index of refraction wave to have nearly a constant
21
0.0001
0.001
0.01
0.1
1
10
0 0.2 0.4 0.6 0.8
Tpulse/Tmicrowave
( σσ σσ- σσ σσ
0)/ σσ σσ
0
0.1 W
0.5 W
1 W
4 W
ref= 0.25
Figure 2.7: Relative bandwidth change as a function of the microwave power andthe ratio between pulse length and microwave length.
value over the spatial region that contains the pulse. From figure 2.7, it is seen
that as the ratio between Tpulse and Tmicrowave increases, the resulting distortion of
the pulse in terms of bandwidth (rms measurement) spread also increases. This is
expected, since spatially this means that the optical pulse spans a larger portion
of the microwave period, and consequently not all sections of the pulse experience
the same index of refraction gradient. However, when the pulse is sufficiently short
compared to the microwave, the distortion is small and the frequency shift of the
pulse can be bigger compared to the longer pulse case. Increasing the power applied
to the microwave distorts the pulse spectrum in addition to increasing the frequency
shifting. The horizontal line in figure 2.7 outlines the relative 10% change in the
bandwidth, which could be considered as a design criterion for this kind of device.
Generally, when the ratio Tpulse/Tmicrowave is less than 1/8, the distortion can be kept
at low values even when high microwave powers are used. For ratios greater than 1/8,
the pulse distortion can become important, especially when high power (for longer
range frequency conversion) is used.
22
Chapter 3
Design and Fabrication Processes
for the Electro-optic Frequency
Shifter
In this chapter, the general design and characteristics of improved versions of the
EOFS device will be discussed. The processes for the fabrication of the EOFS, in-
cluding microwave structure design, optical waveguide diffusion, techniques for crystal
cutting and end fire polishing, will be described briefly. Most of these processes de-
scribed and tested here, have been previously used by other researchers and are found
in the literature. They have been adapted for the particular needs of the EOFS.
The first objective of the development of the device was the need to improve the
coupling of the electrical and optical fields, in order to increase the efficiency of the
frequency conversion and to reduce the power consumption. Traveling wave electro-
optic modulators have been built in z-cut LNO by using engineered microwave struc-
tures that achieve velocity matching of the microwave to the optical wave (Miyamoto
et al. (1991); Noguchi et al. (1995); Mitomi et al. (1995)). These types of structures
have proven to have better bandwidth performance for the modulation than previous
electro-optic modulators. Most of the modulators for high speed telecommunication
systems are currently based on this type of velocity matched travelling wave Mach-
Zehnder modulator structure. There have also been previous proposals and work
in velocity matching techniques for optical switches and traveling wave modulators.
Some of them are based on periodic phase reversal or intermittent interaction (Alfer-
ness et al. (1984); Nazarathy et al. (1987)). These approaches are similar to the
velocity matching technique with electrode polarity inversion (via air bridges) used
for the pulse frequency conversion in (Riaziat et al. (1993)). This type of structure,
23
however, is more complicated to fabricate than standard coplanar microwave struc-
tures. In addition, the optical and microwave field coupling is not optimized in these
periodic reversal structures. To generate an improved design of the device that was
previously demonstrated, switching to a design with a coplanar microwave structure
in a z-cut substrate was considered. This change can lead to better coupling between
the optical and electrical fields. In the new design, the phase matching problem can
be addressed in two ways. The first one is by the use of velocity phase matched
structures that rely on the dielectric and electrode engineering as mentioned above.
The second way is by introducing a quasi-periodic phase matching based on periodic
poling. Periodic poling is a technique widely used in second harmonic generation
(SHG) and four wave mixing (FWM) experiments. In our case, the wafers can be
periodically poled by the vendor (Crystal Technologies Inc.), so we did not need to
develop a procedure for this process. The pitch needed for SHG usually is on the
order of microns, but the periodic poling pitch for this traveling wave phase mod-
ulator structure is in the millimeter range. There could be a possible advantage of
this periodically poled structure over the velocity matched structure in terms of the
strength of the field coupling between the microwave and the optical wave that can
be obtained. Usually the velocity matching is achieved by introducing changes in the
geometry of the electrodes and dielectric insulating layers (such as thicker electrodes
to ensure that a larger portion of the field is in the air), to increase the microwave
phase velocity. However this has the effect of reducing the electrical field strength
in the region of the optical waveguide. There are some techniques, such as ridge
waveguide construction, that attenuate this effect.
One of the crucial steps in the fabrication of this type of structure is the optical
waveguide fabrication. It is important to achieve low loss waveguides that preserve
the nonlinear properties of the material. In section 3.1.1, the optical waveguide
fabrication process used in the lab will be discussed.
3.1 Optical Waveguide Processing
3.1.1 Proton Exchange
Optical waveguides can be created in a material by inducing an index of refraction
channel in such a way that the channel index is slightly larger than the surrounding
material. This index characteristic enables the guiding of light through the channel
because the light tends to be confined in the channel via total internal reflection.
24
For many years, different techniques have been studied for achieving a local index
of refraction change that is useful for guiding optical waves in non-linear crystals.
A very popular technique used previously was titanium diffusion. However, in the
1980’s, proton exchange in benzoic acid and in other similar melts started to be de-
veloped for similar purposes and proved to have better results in terms of resistance
to optical damage, defect concentration and optical losses, and fabrication simplicity
(Jackel et al. (1982); Wong (1985)). Another advantage of the proton exchange re-
action is that this process is perfectly compatible with a previous process of periodic
poling of the nonlinear crystal. The proton exchange reaction will not destroy or
alter the ferro-electric domains induced in the periodic poling because it is done at
relatively low temperatures (with respect to the Curie temperature), as compared to
the titanium diffusion method. The Curie temperature of LNO is 1142.3oC and the
titanium diffusion temperature is around 1100oC. With this in mind, it is likely that
the titanium diffusion approach can generate or alter the poling of the ferroelectric
domains in the crystal. This can have adverse effects when done on a sample in which
a periodic poling of the domains of the crystal is desired (i.e., quasi-phase matched
structures for SHG and frequency conversion).
The index of refraction change in proton exchange is, however, not isotropic.
The extraordinary index is modified considerably, while there is a small decrease
in the ordinary index. This leads to a constraint in the polarization of light that
can be coupled efficiently into the waveguide. This means that for x− and y− cut
substrates, the proton exchanged waveguides will only guide the TE modes (E field
parallel to substrate surface), and in the z− cut substrates, only the TM modes (E
field perpendicular to surface) will propagate. This can be a limitation for devices
if polarization independence of the coupling is needed. Also, in systems where the
polarization control is difficult, there will necessarily be losses in the light coupling.
This, on the other hand, can be used as an advantage if a device capable polarization
discrimination is needed.
The solution for polarization independent behavior of devices has been addressed
by using polarization discrimination with different waveguides depending on the po-
larization that is received by the device. This does not seem very efficient, but in
the past few years there have been very promising advances in the fabrication of an-
nealed proton exchange waveguides with ridge structures, which can sustain both TE
and TM polarization propagation and thus are used in devices that have polarization
independent performance (Nishida et al. (2003); Asobe et al. (2002)). Zinc doping
of LNO is one of the methods used. It has been found that zinc doping increases
25
the index of refraction of the TE and TM modes in the LNO waveguides. These
researchers grew Zn:LiNbO3 films on Mg:LiNbO3 substrates by liquid phase epitaxy.
After this, electric periodic poling was performed on the substrates, and finally, the
ridge structures were patterned by ion etching. The near field profiles for both TE
and TM modes were very similar, indicating that both polarizations can be guided by
these waveguides. The Mg doping of the LNO substrates has been shown to help in-
crease the waveguides’ resistance to photorefractive damage. This is very important,
especially since most of these waveguides are used for SHG, DFG, or SFG experi-
ments in which the pump power is usually high (60-100 mW). In these experiments
the polarization independent conversion was achieved by a double pass through the
waveguide with the use of a mirror and a quarter wave-plate. The double pass of the
light through the quarter wave plate rotates the polarization by 90 degrees in such
a way that the TE/TM mode components are interchanged after the reflection. The
wavelength converter itself works in the TM mode (locally), so for this device the
TM mode component is converted in the forward direction and the TE component is
converted in the backward propagation after the reflection and polarization rotation.
Previous polarization independent conversion had been demonstrated, but with the
use of two individual waveguides (Brener et al. (2000)). The other method to create
these polarization insensitive waveguides, reported by Nishida et al. (2003), is based
on the direct bonding (in a 500oC clean atmosphere) of a non-doped periodically poled
LNO wafer to a MgO doped wafer. Similarly, a ridge waveguide structure was pat-
terned by micro-lapping and polishing techniques. These waveguides demonstrated a
very good resistance to photorefractive damage at room temperature as compared to
normal annealed proton exchange (APE) waveguides.
In the proton exchange reaction, the LNO wafer is submerged in a hot benzoic acid
bath at temperatures usually ranging between 200 and 235oC. This temperature range
is good in order to guarantee that the diffusion of the ions is fast enough to produce
the desired exchange, while keeping the temperature in a region where the control
is simple and the reaction is stable. The melting and boiling points of benzoic acid
are 122 and 249oC respectively. There have been more recent reports of alternative
techniques for proton exchanged waveguides based on vapor phase exchange (Masalkar
et al. (1997)), where the index profile is graded and the index step is approximately
0.01-0.02. We are not aware of the electro-optic behavior and performance of this type
of waveguide. Another interesting technique is based on high-temperature proton
exchange in stearic acid in a sealed environment (Korkishko et al. (2000)). This
method seems to be able to produce low loss waveguides with good electro-optic
26
Exchange regionAluminum coating
LiNbO3
xxxx COOLiHCNbOHLiCOOHHCLiNbO )()( 5631563 +→+ −
Figure 3.1: Proton exchange reaction.
behavior with the advantage of using short fabrication times.
For the proton exchange reaction we tried, the wafer is coated with a protective
aluminum layer which has openings only at the places where the reaction is desired,
as shown in figure 3.1. The aluminum coatings were evaporated, and the thicknesses
used were between 200 and 500 nm. Standard lithographic processes were used to
pattern the aluminum film and etch the channels for the proton exchange reaction to
take place. The reaction induces ion exchange between the lithium ions in the wafer
and the hydrogen ions in the acid, according to the process
LiNbO3 + (C6H5COOH)x → Li1−xHxNbO3 + (C6H5COOLi)x (3.1)
This reaction creates a crystalline change in the substrate, which will then produce
an index of refraction change in the exchanged region. The index change is associated
with the loss of lithium ions from, and the resulting formation of hydroxyl groups in,
the substrate. The reaction is enabled by both the high hydrogen ion concentration
and the high temperature in the melt. Other types of proton sources, such as py-
rophosphoric acid and buffered benzoic acid solutions (lithium benzoate in benzoic
acid), have been used and reported in the literature (Nikolopoulos and Yip (1989,
1990); Loni et al. (1989); Ziling et al. (1993)). The typical step index obtained by
this exchange reaction is roughly 0.12 (index increases from 2.2 to 2.32). This index
step can be used to guide the light in a manner similar to that in a dielectric slab
waveguide. However, this reaction is known to deteriorate the electro-optic coefficient
in both lithium niobate and lithium tantalate (Wong (1985); Li et al. (1989); Canali
27
et al. (1986)). The values of the electro-optic coefficient after the exchange have been
observed to degrade up to 10% of the original value.
In an attempt to control the extent of the electro-optic degradation, researchers
started investigating different acid baths and techniques, such as the diluted melts
and post annealing of the substrates after the exchange (Rottschalk et al. (1987);
Loni et al. (1989); Yip and Nikolopoulos (1991)). Of all the techniques tried, the one
that gave the most encouraging results was the post annealing procedure. Annealing
procedures with different reaction times, temperatures, and atmospheric conditions
have been tried by several groups.1 It is known that the degradation of the electro-
optic effect, which is observed after the exchange reaction, can be reduced with a
subsequent annealing procedure, which is done usually between 300 and 400oC. Dur-
ing annealing, the crystalline structure relaxes slightly, producing an electro-optic
coefficient recovery and also a diffusion of the exchanged region deeper into the sub-
strate. This is accompanied by a decrease in the index step (from about 0.12 to 0.02
or 0.015) and a relaxation of the sharp refractive index step profile to a more graded
index profile (Savatinova et al. (1996)). The fact that the waveguide characteristics
are dependent on the exchange time and temperatures, as well as on the annealing
time, temperatures and atmosphere, has given researchers a lot of flexibility towards
obtaining reduced losses and greater resistance to optical damage, single mode op-
eration, and recovered electro-optic behavior. In previous literature, it has generally
been seen that relatively short exchange reactions (under 1 or 2 hours), followed by
longer annealing times, produce good results in terms of losses, resistance to damage,
and single mode behavior. The single mode behavior is desirable in applications where
there is a concern about mode dispersion or conversion that can occur. Our goal for
the annealing step was to get the electro-optic recovery as high as possible, and an
index step reduction to approximately 0.01. This will give a numerical aperture for
the waveguides of about 0.15− 0.2, which is very good for light coupling to and from
an optical fiber.
Using previously reported studies by Clark et al. (1983), Canali et al. (1986) and
Nikolopoulos and Yip (1990), we found a starting point to estimate the diffusion
coefficient and the behavior of the proton exchange reaction in benzoic acid for z-cut
LNO substrates. From these previous experiments, the temperature dependence of
the diffusion coefficient for the reaction can be estimated using
1In fact, it would be very difficult to mention all the wide variety of work and variations thatdifferent researchers have tried based on this annealing procedure. A good review of the earlier workin this technique can be found in Wong (1988).
28
Figure 3.2: Proton exchange setup.
D(T ) = D0 exp(−Q/RT ), (3.2)
where D0 is the diffusion constant to be determined, R is the universal gas constant,
Q is the activation energy for the process, and T is the absolute temperature. The
following numbers were found by the researchers after the experimental data fit:
Q = 94 kJ/mol, and D0 = 1.84× 109 µm2/h. With these values, the diffusion depth
can be calculated using the relationship
d = 2√
t×D(T ) = 8.58× 104t1/2 exp(−5.65× 103/T ) µm, (3.3)
First, it is important to have a temperature of exchange which is high enough
to have a fast diffusion rate, but not too high, so that the temperature control is
easier and the reaction progresses in a controlled way. This is done to avoid excessive
structural damage in the crystals. A pyrex container with lid and sample holder
were designed and built for this purpose. They can be seen in figure 3.2. The
lid has two small holes to introduce the sample holder arm and the temperature
probe. The temperature probe provides the feedback to the hotplate to keep the
bath temperature constant within ±0.5 degrees. The plate/acid bath setup was
surrounded with various layers of thick aluminum foil during the heating of the bath
and during the exchange reaction to help keep a stable temperature. The LNO
29
sample holder has many small holes that help the benzoic acid melt flow to guarantee
a more homogeneous temperature and concentration of the melt around the LNO
sample. The exchange temperature was set at 200o C. The diffusion coefficient at this
temperature, obtained from equation (3.2), is 0.077. The exchange times typically
ranged from 1/2 to 2 hours. For a 2 hour exchange at this temperature, the depth
of the first diffusion region is approximately 0.79 µm. The LNO substrates are very
susceptible to thermal shock and pyroelectric effects that can produce cracks in the
substrate. Hence, the process has to be done in a very careful way. Before dipping
the sample holder, it is lowered slowly (at a rate of about 2 cm/minute) to just above
the melt and left there for about 5 minutes. The hot atmosphere starts to heat up the
holder and sample. Finally, the sample is introduced in the acid bath very slowly. The
exchange time was recorded starting at this point. After the 2 hours of the exchange
process, the sample was also lifted at a rate of about 2 cm/ minute and then left on
the holder to allow it to cool down for 5 minutes. The samples were then dipped in
de-ionized (DI) water at 80− 90o C to avoid thermal shock. Slowly the temperature
was lowered by adding more room temperature DI water. The LNO samples were
then cleaned using methanol to get rid of benzoic acid residues. A final thorough
cleaning was performed using acetone, isopropyl alcohol (IPA) and DI water. Finally
the samples were blow dried with nitrogen.
The annealing procedure used was performed in a wet oxygen atmosphere. This
was obtained by bubbling oxygen through a 10 cm water column at a rate between
200 and 300 ml/min. The furnace was pre-baked with the oxygen flow on at 350oC
for at least one hour. The samples were loaded slowly (approximately 2 minutes
to move 0.5 m from the end of the furnace to the center). Then the samples were
annealed for times ranging from 1 to 6 hours. At the end of the anneal, the sample
was moved to the end of the furnace tube and the furnace turned off. The sample was
allowed to cool in the presence of the flowing oxygen. After the annealing process was
finished, the waveguides were checked for light coupling and mode behavior. This was
done with He-Ne (632 nm) light as a fast, first diagnostic tool, because the visible
wavelength makes the optical alignment and coupling easier. Eventually, the coupling
and mode behavior depends on the operation wavelength so that the parameters
should be optimized for the use of the light at one micron in our experiments. The
best results were obtained with annealing times between 2−4 hours. Longer annealing
times cause the waveguides to exhibit undesirable multimode coupling behavior. Our
results agree with previous experimental work where single mode waveguides have
been produced in x-cut LNO (Suchoski et al. (1988)). In this earlier work, exchange
30
Lapping Tool
Individual sample holder
Figure 3.3: Lapping tool and sample holder.
and anneal temperatures of 200 and 350oC respectively, have been used to create
single mode waveguides. They were exchanged for 10 min and annealed for 2 hours
for 0.8 µm operation, and exchanged for 30 min and annealed for 4 hours for the 1.55
µm case. We must keep in mind that the diffusion constant for the x direction is not
the same as for the z direction. In fact, the diffusion constant for the x-cut diffusion
is approximately 1.75 times that for the z-cut process (Canali et al. (1986)).
3.1.2 End-facet Polishing
To ensure good light coupling efficiency, the end faces of the crystals must be very
well polished, and rounded edges in the waveguide ends must be avoided. The defect
size at the end faces and corner rounding needs to be small compared to the light
wavelength in order to avoid scattering centers and reflections that can be an obstacle
for good coupling. We designed a polishing procedure in our lab with which we can
obtain the flatness required for good light coupling to the LNO devices. 2
The LNO wafers (0.5 mm, z-cut) were diced after the aluminum coating lithogra-
2The basic technique was adapted from an existing procedure used by the optoelectronics groupat NIST in Boulder, CO. We would like to acknowledge the help of Dr. Norman Sanford and Dr.David Funk for their help in this matter.
31
Figure 3.4: Polishing setup showing the three aluminum oxide manual steps andfinal lapping wheel step.
phy for the optical waveguides definition. The aluminum mask includes markings for
the cutting as well. To perform the cut, a DISCO dicing saw was used with resinoid
blades from Dicing Technologies. After the cut was done, it was clear that the rough-
ness and chipping during the cutting process was big enough to make the crystals
useless for optical coupling. Thus, the need for the end-facet polishing step. After
the proton exchange reaction was performed on the diced samples, they were mounted
on a 1 mm microscope glass slide. The surfaces were glued using Crystal Bond in
such a way that the aluminum film, and thus the surface waveguides, were kept in
the inside faces of the glued substrates. The glass/LNO crystal pair was diced again
in such a way that the end face cut coincided with the waveguide inputs and outputs,
and was perpendicular to the waveguide direction. Next, the samples were mounted
in special holders that can maintain the proper substrate orientation to guarantee a
polishing surface as perpendicular to the waveguides as possible. These holders were
then mounted in a lapping tool designed in our lab. See figure 3.3.
The manual polishing was done in three consecutive steps using three different
aluminum oxide grits (25 µm, 9 µm, and 3 µm). See figure 3.4. Pyrex glass flats
were used as the base surfaces for each of these polishing steps. These manual steps
were done for about 15 to 30 minutes, each. The fourth and most important step
32
1. After Cut 2. Intermediate AluminumOxide Step
3 . a) Final After Colloidal Silica b) waveguide end view Scale
100 µm
Figure 3.5: Lithium niobate crystal polishing for waveguide coupling.
was done using a lapping wheel with a motorized lapping tool holder. The lapping
wheel, which has a soft polishing cloth surface, turns in one direction while the tool
holder turns in the other. This guarantees a more even and flat final polishing. It also
helps to reduce corner rounding and a wedge like finish that will reduce the optical
coupling efficiency. In this step a colloidal silica solution (sub-micron particle size)
was used. The polishing time ranged from 45 minutes to 1.5 hours. Different rotation
speeds in the sample holder (40-100 r.p.m.) were used. The pressure applied on the
sample holder could be adjusted slightly. Faster spinning speeds and higher pressures
increase the polishing rate. However, care must be used with this to avoid polishing
too fast or with too much pressure since this can produce sample chipping. This step
was the most important for achieving an optical grade finish to ensure good light
coupling. In figure 3.5, different stages of the polishing can be seen. The first image
shows the edge of the crystal after the diamond dicing saw cut. The second and
third images show the sample after finishing the colloidal solution polishing. All the
pictures have been taken with the same magnifying power. As a help for the viewer,
a scale with 100 µm and 10 µm divisions is shown in this figure too.
33
3.2 Microwave Structure Design
Aside from the optical waveguide processing, there is a need to work on the microwave
structure design for the device. In this part of the process, many issues become
important, such as microwave losses, impedance matching, microwave phase velocity
(related to the velocity matching problem), and power handling techniques.
The first structure envisioned for the device is shown in figure 3.6. This is a
typical half coplanar microwave transmission line which serves as the waveguide for
the electric field travelling wave that modulates the index of refraction characteristics
of the nonlinear material. The signal electrode is the middle conductor and it has a
width w. There are two ground planes, one at each side of the signal electrode. The
distance between the ground planes is denoted a and their width is large compared
to the signal electrode width. Thus, the gap from the signal electrode to the ground
plane is given by (a − w)/2. In this case there is a thin dielectric layer between
the optical waveguide and the center signal electrode of the coplanar waveguide.
This dielectric helps prevent problems with optical losses caused by proximity to the
metal overlayer, therefore improving the spatial isolation of the optical mode from
the metal conductor. On the other hand, the dielectric should be sufficiently thin in
order to avoid reducing the effective field in the waveguide region, so that the efficient
optical/electrical field coupling needed for the device operation is maintained.
Using figure 3.6 conventions, the line impedance of one of these coplanar structures
with thin electrodes can be modelled by the following expressions.
Z =377π
4√
εeff
[ln
(21 +
√w/a
1−√
w/a
)]−1
, for 0.173 < w/a < 1, or (3.4)
Z =377
π√
εeff
ln(2√
a/w), for 0 < w/a < 0.173.
A review of the microwave structure theory and modelling that leads to the previous
approximations can be found in Ramo et al. (1994). It is important to consider the
line impedance for the design of the microwave structure because of the impedance
matching conditions. The standard impedance for most sources, loads, connectors,
and cable assemblies for microwave applications is 50 Ohms. This means that, in
order to guarantee a maximum power transfer from the source to the line and loads,
the line impedance of the design must be very close to 50 Ohms. In this way losses
related to power reflections because of the mismatching can be reduced. However, at
the same time that the impedance matching of the line to a 50 Ohm design reduces the
34
a
w
z
x
Figure 3.6: Electro-optic Frequency Shifter (EOFS) structure.
microwave power reflection losses, a higher impedance can increase the effective field
given the same input power. This could be done using thinner central electrodes, but
at the same time this can also increase the microwave power resistive losses. Generally,
designers of electro-optic modulators and similar devices use computational tools to
look for some optimal solution for this multiple parameter problem to guarantee good
field coupling and efficient power transfer.
Another problem that needs to be taken into account during the design process
is that of the velocity matching between the microwave field and optical field. The
optical index depends mainly on the material characteristics, while the microwave
index depends both on material characteristics and microwave structure geometry.
For this first case, the effective dielectric constant in equations (3.4) results from
the average of the dielectric constant in the substrate and in the dielectric above it
(air), because of the spatial symmetry of the problem. Thus εeff ' (εair +εdie)/2. The
dielectric constant in the nonlinear material for a y-propagating wave in a z-cut crystal
is given by εdie ' √εxεz. For the case of lithium niobate the values of the dielectric
constants are εx = 42, and εz = 28. This makes the effective dielectric constant for y
axis propagation be εdie ≈ 34. The index of refraction for the microwave propagation
is then
35
nµ ≈ √εeff =
√(εair + εdie)/2 =
√(1 + 34)/2 ≈ 4.2 (3.5)
Clearly this situation leads to a problem of velocity matching in this kind of
structure, because the optical index of refraction for the wavelengths of interest in
this material (between 0.8 and 1.5 µm commonly used in optical communication
systems) is close to 2.2. Velocity matching has been a problem studied in nonlinear
optics for many years. The interaction of different fields in a nonlinear crystal has
constraints given by the conservation of energy and the conservation of momentum.
The conservation of momentum is related to the propagation vectors and therefore to
the optical indices of refraction. In the case where the interacting fields are optical and
electrical, as in the velocity matching for travelling wave structures for modulators
and switches, various different techniques to achieve a better velocity matching have
been studied (Alferness et al. (1984)). They are usually divided into two big groups.
The first group achieves velocity matching by increasing the microwave velocity. This
is done by using an electrode structure that makes a larger portion of the microwave
field to travel in a material of lower index than the substrate. This kind of design will
be described further in the sub-section 3.2.1 The second group obtains the velocity
matching based on artificial or quasi-phase matching structures that involve some
kind of electrode polarity reversal or meandering of the electrodes.
One example of a solution among the second group is the previous work by Ri-
aziat et al. (1993). In this work, a meandering type of half-coplanar line (y axis
propagation in an x-cut substrate) was designed to alternate the polarity of the field
with respect to the crystal orientation. The continuity of the ground planes was
guaranteed by introducing air bridges joining the otherwise disconnected portions of
the ground plane. This structure fabrication process requires extra steps because of
the air-bridges needed, as compared to a simple planar electrode structure. Another
possible solution in this group of quasi-phase matched solutions, is the introduction
of periodically poled domains in the substrate so that the same polarity reversal ef-
fect is obtained by flipping the crystalline domain orientation instead of the electric
field. Periodic poling techniques based on proton exchange reaction, ion implantation
and high voltage application have been studied by many researchers (Kintaka et al.
(1996); Lim et al. (1989); Yamada et al. (1993)). This type of structure with the
periodically poled z-cut substrate can be advantageous because propagation losses
are lower compared to the structures with the meandering line and air-bridges. It
also allows the signal electrode to be deposited directly on top of the optical waveg-
uide, so that the electric field is stronger in the region where the light propagates.
36
Therefore, it helps to improve the optical and electrical field coupling, as compared
to the structures fabricated on x or y-cut substrates.
The periodically poled pitch required for this structure can be calculated from
equation (2.12). Using a value for the effective dielectric constant εeff = 35/2, as
expressed in equation (3.5), an optical index nopt = 2.2, and a microwave frequency
of 6 GHz, the value of the pitch period obtained is
lπ =c
2fµ(√
εeff − nopt)= 1.26 cm. (3.6)
This pitch length will depend clearly on the microwave frequency that is chosen
for the operation. One important thing to note is that the length of this pitch is
large compared to other periodically poled pitches normally used for SHG and FWM
frequency conversion experiments. As mentioned before, the typical pitch used in
these is on the order of microns. The current techniques for poling are very well
developed and reliable, so a pitch on the order of millimeters or centimeters is not
a technological challenge at this point. The periodically poled substrates can be
obtained commercially from several vendors.
The next section will describe some of the basic theory for the electrode structure
and design for the first group of velocity matched structures. An EOFS derived from
this kind of structure has been used for most of the experimental work that will be
shown in chapter 4.
3.2.1 Basic Electro-optic Modulation and Bandwidth
Considerations
Since the early stages of research into using electro-optic modulators as a means of
encoding and modulating laser light for communication systems, there have been
two consistent goals: High speed (operation bandwidth for switching) and low power
consumption. A very good review of electro-optic modulation of laser beams can be
found in the book by Yariv (1997). The discussion in this subsection follows the ideas
and derivations presented in there. In this book, the reader can find a general descrip-
tion of the electro-optic effect. The description includes applications for electro-optic
retardation making use of the birefringence of some non-linear materials, amplitude
and phase modulation of light, and longitudinal or transverse modulation for low
frequency electric fields. There is also a discussion of the limitations of modulation
at high frequencies, which of course is of great importance for high speed systems.
Consider the situation in which a sinusoidal modulation electric field is applied
37
to the crystal by means of a parallel plate capacitor structure. The frequency of
modulation applied by this field is f0 = ω0/2π. The system can be modelled by a
parallel RLC circuit in which Rs is the internal resistance of the source and C is
the parallel plate capacitance. If Rs > 1/(ω0C), most of the voltage drop will be in
the source resistance and thus not used for the modulation field. This can be solved
by resonating the capacitor with an inductance L where L = (ω20C)−1. A shunting
resistance RL is used so that the impedance of the resonant circuit is non-zero at the
resonant frequency. The idea is to choose this resistance so that RL > Rs so that most
of the voltage drop is used for the electro-optic modulation. As in a normal electronic
RLC circuit, this has a maximum bandwidth that is given by ∆ω/2π ≈ 1/2πRLC.
This will then be the maximum modulation bandwidth of the modulation signal
around ω0. This can be related to the peak retardation that can be achieved in the
specific modulator and then to the modulation voltage required for the modulator.
In a typical modulator the electro-optic retardation due to an applied field can be
written as
Γ = aEl, (3.7)
where a is a constant related to the optical frequency, the index of refraction, and
the electro-optic characteristics of the crystal. E is the applied electric field and l is
the length of interaction or optical path in the crystal. This expression is valid for a
DC or low frequency variation field. However, if the field varies appreciably during
the transit time of the light through the modulator (τd = nl/c), then the retardation
expression has to be modified to
Γ(t) = a
∫ l
0
e(z)dz = ac
n
∫ t
t−τd
e(t′)dt′. (3.8)
The change of variable from position to time has been made so that the light exits
the crystal at time t. The instantaneous field is given by e(t′) = Emeiωmt′ . Using this
expression in (3.8) and performing the integration, the retardation can be expressed
as
Γ(t) = Γ0
[1− eiωmτd
iωmτd
]eiωmt, (3.9)
where Γ0 = a(c/n)τdEm = alEm is the peak retardation. The factor in the brackets
is close to 1 when the transit time is negligible. Clearly, if the frequency-transit time
product is too high, then the retardation factor will decrease and the modulation will
not be effective. Having this in mind, a limit for a good reduction factor can usually
38
be defined so that the factor in brackets is not too far from 1. With this, a maximum
modulation frequency can be obtained. This modulation frequency will typically be
of the form (νm)max ∝ c/nl, which shows that the longer the path is, the smaller the
useful modulation frequency gets.
One technique that has been widely used to improve the modulator performance
and increase the bandwidth is to reduce the transit-time limitation by applying an
electric field signal that travels through the electrode structure along with the optical
signal. In this configuration, the electric field is perpendicular to the propagation
direction (transverse electro-optic modulation). The idea behind this is that if the
optical and modulation field phase velocities are the same, then a portion of the optical
field will always experience the same instantaneous field throughout the propagation.
In Yariv (1997) it is shown that with this type of traveling wave modulator structure
the retardation is given by
Γ(t) = Γ0
[eiωmτd(1−c/ncm) − 1
iωmτd(1− c/ncm)
]eiωmt, (3.10)
where cm is the microwave phase velocity. This shows an improvement in the fre-
quency limit or useful crystal length by a factor of (1 − c/ncm)−1. Also it shows
that for a perfect velocity matching case, the maximum retardation possible can be
obtained independently of the crystal length. In practice many modulators that are
built are not perfectly matched, or the mismatch will define a limit for the length
that is useful for modulation.
In the late 1980’s and early 1990’s, extensive research was done in the develop-
ment of broadband travelling wave modulator structures for amplitude and phase
modulators. The type of structure of interest to this work (basis of the EOFS), and
that is similar to one of the designs described in the next section, is a broad-band
travelling wave optical phase modulator.
3.2.2 Broadband Traveling Wave Modulators
It has already being shown in equation 3.10 how the bandwidth of a traveling wave
modulator is limited by the velocity mismatch between the optical wave and the
microwave. There are also other factors such as the microwave losses related to
impedance mismatch and electrode losses that further limit the bandwidth. In this
section, a technique based on modifying the microwave electrode structure geometry
will be discussed. Generally, the microwave phase velocity in coplanar or half copla-
nar modulators is slower than the optical phase velocity. The goal of the electrode
39
geometry modification is to decrease the microwave index (in other words, increase
microwave speed) in order to make it match the optical index as closely as possible.
This is basically done by introducing thicker electrode structures and dielectric layers
or ridge waveguides for the optical signal in such a way that a bigger portion of the
microwave field will travel in a lower index material (air and dielectrics). The overall
wave speed will be increased by the fact that a greater percentage of the microwave
field will travel in the “fast” medium.
The first step in the design of one of these velocity matched modulator structures
is then to define the geometry for the index matching, taking into account that the
impedance matching for the microwave circuitry must be achieved to optimize the
power transfer to the microwave structure. It has been shown previously that the
characteristic impedance and refractive index of a transmission line for a modulator
operating in a transverse electric mode are given by (Miyamoto et al. (1991))
nm =√
C/C0, (3.11)
and
Z0 = 1/c√
CC0, (3.12)
where c is the speed of light, C0 is the capacitance per unit length between the elec-
trodes if they are in free space, and C is the capacitance with the substrate and any
buffer or dielectric layer that exists in the structure. This type of approximation is
based in a quasistatic analysis that is considered valid up to frequencies of several tens
of gigahertz, in which the electrode dimensions (cross sectional) are small compared
to the microwave wavelength. The transverse mode behavior of the field is also a
very good approximation for the same reason. Another thing that makes the qua-
sistatic approximation possible is the fact that the material response time scales are
smaller compared to the time scale for the microwave field variation (which depends
on the microwave frequency). In other words, the material response is so fast that, for
practical purposes, it can be assumed that there is no delay between the microwave
change and the electro-optic effect modulation. From these expressions in equations
(3.11) and (3.12) the values of the capacitances can be calculated in terms of the mi-
crowave index and line impedance. Since, in a LNO, z-cut, y propagating modulator
using light wavelengths between 1 and 1.5 µm, the index is close to nm = 2.1 and the
microwave transmission line can be made to have an impedance close to Z0 = 50 Ω,
the typical values of capacitances can be calculated as (Miyamoto et al. (1991))
C0 = 1/Z0nmc = 3.1× 10−11 F/m, (3.13)
40
Waveguide
Electrodes
SiO2(or dielectric)
z-cut LNO
Figure 3.7: Broad-band traveling wave phase electro-optic modulator proposed byMiyamoto et al. (1991).
and
C = nm/Z0c = 1.4× 10−10 F/m. (3.14)
Using this basic idea, different types of electrode geometry designs and techniques
ranging from some analytical approximations and mathematical methods such as
the method of moments and finite element analysis, have been studied by many re-
searchers. For other examples of these kinds of broadband modulators, see the work
by Noguchi et al. (1995) and Mitomi et al. (1995). The basic structure proposed in
Miyamoto et al. (1991) can be seen in figure 3.7. Using as variable parameters the
metal layer and dielectric thicknesses, as well as signal electrode width and gap, the
design can be optimized using numerical methods (in this particular case, finite ele-
ment analysis was used) to obtain values that will provide the required line impedance
and microwave phase velocity.
The results obtained by Miyamoto et al. (1991) show generally that the greater
the gap is, the thicker the electrodes must be to keep the vacuum capacitance in the
required range. Increasing the buffer layer thickness decreases the line capacitance
with the dielectrics and helps to tune the matching conditions. The velocity matching
can be obtained by making thicker electrodes and buffer layers, but with the drawback
41
that the electric field underneath the electrodes is reduced compared to a planar
microwave structure. Thus, the good effect of velocity matching is compromised by
the inferior field coupling.
The structures proposed by Mitomi et al. (1995) and Noguchi et al. (1995) are
very similar to the previous one, but try to address the field coupling problem in a
better way. These structures introduce a type of ridge waveguide that is constructed
by etching the substrate. The effect of this is to make the buffer layer thinner so
that the optical wave will be closer to the electrodes. This increases the coupling
between the optical and microwave fields so that the modulation can be more efficient.
The device then becomes more efficient with respect to required voltage and power
consumption. The basic structure is shown in figure 3.8. The structure described here
is the basis for an electro-optic Mach-Zehnder modulator. In this type of structure,
the light is coupled into the modulator entry waveguide and then split equally into two
waveguides that act as the arms of an interferometer structure. The actual “optical
length” of each of the modulator arms is modulated by the application of the electrical
field to change the index of refraction. The light is then recombined and interference
effects are observed depending on the optical path difference that is experienced
between the two paths. This is a typical design used for an amplitude modulator and
is very widely used in optical telecommunication networks as an external modulator
for the laser sources in these systems. A very good description of light coherence
properties and many types of interferometer structures such as this one can be found
in the book by Mandel and Wolf (1995). It is important to note that the structure of
interest in this work is not a Mach-Zehnder interferometer like the one shown in figure
3.8. However, the idea behind the velocity matching techniques and the importance
of good coupling between the fields is the same. This is eventually reflected in the
frequency conversion range vs power efficiency of the device.
When studying the frequency response of these types of modulator structures,
there is also a need to study the microwave attenuation properties in order to have an
idea of which electrode geometry is best, what length to use, and what kind of power
handling the device will have. In typical microwave application integrated circuits,
the microwave attenuation coefficient is related to the skin depth effects. It is known
that the loss dependence with frequency is generally given by (Ramo et al. (1994))
α = α0
√f, (3.15)
where f is the microwave frequency and α0 is the microwave electrode loss in dB/cm.
It has been mentioned before how there is a bandwidth limit due to velocity matching
42
tm
tr
tb
W G
Figure 3.8: Electro-optic traveling wave amplitude modulator with a ridge wave-guide structure proposed by Mitomi et al. (1995).
and index matching problems. However, the attenuation also will restrict the optical
−3dB modulation bandwidth as shown in the work done by Noguchi et al. (1995).
∆f ≈ 193
(α0 · L)2GHz. (3.16)
In this expression L is the interaction length. The calculations performed by this
group of investigators provide a very good insight into the effects of electrode thick-
ness and ridge height on such microwave properties as index, losses, line impedance,
and the product of driving voltage (Vπ, which is the voltage required for a π phase
retardation) and the interaction length (L). This last parameter is usually used as an
indication of the performance and the modulation capability of the modulator. This
product is obtained by calculating the overlap integral between the optical and mi-
crowave fields, similar to what is done when calculating the accumulated retardation.
From the results of this work, it was observed that the electrode thickness has a small
effect on VπL. On the other hand, this product has a strong dependence on the ridge
height. It decreases generally with the ridge height, but after an optimal point, it
increases again. This is a very good indication that the microwave field coupling to
the optical wave is enhanced by the ridge. The microwave field is more concentrated
43
in the ridge since the distance to the metal electrode is shortened and the attenuation
of the field in the buffer layer is also reduced. Another advantage observed in these
calculations is that the electric field is more confined to the z-axis direction inside
the ridge compared to the confinement of the electric field in the presence of just a
buffer layer, so that the effective electro-optic interaction (due to the r33 coefficient)
is thus enhanced.
The ridge height has effects on other characteristics as well. The microwave losses
α0 depend strongly on the electrode thickness (thicker electrodes are less lossy than
thin electrodes). The ridge height introduces a further reduction in the microwave
losses, probably because it helps to increase the electrode section involved in the
electric field flow by providing additional field confinement. It was also observed
that increasing the metal thickness reduces the microwave index (which speeds up
the microwave by causing a greater portion of the microwave to travel in the air),
and that an increase in the ridge height reduces the microwave index further. The
reason for this is basically the same, since increasing the ridge height makes a bigger
portion of the microwave field travel through the buffer dielectric which has a lower
index of refraction than the LNO substrate. The increase in metal electrode thickness
makes the free space capacitance higher so the microwave index and line impedances
are reduced. The increased ridge height increases the line impedance, which helps
balance the reduction produced by the thicker electrodes, and keeps the design close
to a 50 Ohm impedance matching that will be compatible with the microwave power
supply and electronics.
The effects of the thickness of the buffer layer were also studied in these type of
structure. The results of the calculations in Mitomi et al. (1995) show that increasing
the thickness of the buffer layer for a given electrode configuration and thickness has
the effect of reducing the effective dielectric constant and, thus, the microwave index.
This can clearly be used to speed up the microwave for the velocity matching goal.
At the same time, and similar to the effect of increasing the ridge height, an increase
in the buffer layer thickness produces an increase in the characteristic line impedance
that helps to counteract the effect of electrode thickness. Also, the microwave losses
decrease as the buffer layer thickness is increased. On the other hand, the VπL
product experiences a gradual increment with increasing buffer layer thickness, while
increasing ridge depth resulted in an initial decrement and the presence of an optimal
point.
All these results show clearly that the ridge type of structures are advantageous
as compared to the planar type without any substrate etching in terms of being able
44
to achieve better bandwidth and lower power consumption characteristics. These
researchers studied similar structures, one with coplanar waveguide geometry (con-
sisting of a signal electrode with two ground planes around it), and another with
an asymmetric coplanar strip line geometry (with just one ground plane). Both de-
signs appear suitable for the creation of ultrahigh speed modulators by tuning the
electrode thickness, ridge height, and buffer layer thickness to be able to produce
structures with very good velocity and impedance matching characteristics. How-
ever, the coplanar waveguide structures seem to be preferable because they can be
built with thinner buffer and electrode layers, so their fabrication is easier. They also
show slightly lower driving voltage/length product characteristics. This means that
the electric field magnitude, and thus the coupling to the optical wave, is stronger
than in the asymmetric structures. This is one of the key factors of interest for the
particular device application of the EOFS.
45
Chapter 4
Experimental Setup and Operation
of the EOFS
In this chapter, the experimental setup used for the testing of the operation of the
EOFS will be presented. The results shown in this chapter have been obtained using
two devices with a traveling wave modulator structure with a ridge waveguide, as
described in chapter 3. The devices consist of a fiber pigtailed velocity matched
phase modulator structure supplied by Lucent Technologies. The modulators were
built on a z-cut lithium-niobate crystal using a velocity matched electrode structure
with a line impedance of approximately 47 Ohms. This makes the reflection coefficient
approximately −15 dB at the frequency range studied (5-12 GHz). The modulator
structures are properly terminated for power dissipation. The laser light was coupled
to the devices via polarization-maintaining fiber . Experimental evidence of frequency
shifting of a train of mode-locked pulses of light up to +/- 400 GHz will be shown.
Dynamic, continuous, and accurate shifting of the optical pulses was achieved by
controlling the phase and power of a microwave in this EOFS device. The microwave
structure is the means by which the electric field is applied to the crystal, and the
nonlinearity of the material is what provides the coupling to the optical field. The
operation of the device under different microwave powers and optical pulse lengths was
investigated. Some discussion is also presented in this chapter about future device
work, techniques to improve the device performance, as well as some preliminary
results on these future directions.
In the experiments performed, the effect of the EOFS on a train of mode-locked
pulses was studied. This system mimics the operation of a TDM system with a
return to zero (RZ) encoding format (Caspar et al. (1999)). The frequency shifting
obtained is simply related to the amount of power applied to the device, and the
46
conversion is practically instantaneous, so the device avoids the bottle neck problems
present in other types of optoelectronic conversion schemes. Up-shifting or down-
shifting operation can be achieved by adjusting the phase of the microwave field
applied to the device. The switching performance of the device ultimately depends
on the speed of the control electronics for the microwave excitation. The EOFS has
the capability of converting practically all the light to the new frequency without
using a second optical source or filtering stages to get rid of the original signal. This
makes it a very flexible device with advantages compared to other frequency shifting
schemes, such as those based on optoelectronic solutions, sum-frequency generation
(SFG), difference-frequency generation (DFG) , four wave mixing (FWM), and devices
based in semiconductor optical amplifiers (SOAs) that use cross-phase or cross-gain
modulation (XPM, XGM).
4.1 Experimental Setup and Measurements
Figure 4.1 shows the experimental setup used for the device testing. Optical pulses
(30-35 ps. long) from a doubled, mode-locked Nd:YLF laser were produced at a 75.7
MHz repetition rate and a wavelength of 527 nm. These pulses where used to pump a
cavity length matched dye laser to produce a train of mode-locked optical pulses in the
900-980 nm range. A Styryl 13 (LDS 925) dye supplied by Exciton, Inc. was used as
the gain medium for the dye laser. One of the reasons for using this particular dye was
to obtain light with a wavelength closer to the ones used in short haul communication
links (around 800 nm). Another reason for this was that the original Optical Spectrum
Analyzer (OSA) that was to be used required an operational wavelength around 1µm.
This original OSA consisted of a scanning Fabry-Perot cavity with a finesse of about
10,000. However, because of a loss of resolution due to transverse mode coupling into
this cavity, the OSA was later changed to a better OSA (ANDO AQ6317B), typically
used in optical telecommunication systems. Unfortunately, the wavelengths of 1.3
and 1.5µm typically used for long haul systems are not easily available using a dye
laser system. In any case, the operation of the device is scalable and ultimately the
wavelength choice will only be important for the device geometry definition and not
for the basic physics of the EOFS.
In addition to providing a means of confirming that the operation of the device
doesn’t rely heavily on the wavelength, another reason to use the dye laser system
is that it can more readily produce pulses of different lengths than other systems.
The length of the pulses was modified by introducing intra-cavity glass etalons with
47
RF Synthesizer
Microwave Synthesizer
Power Amp.
Phase shifter
Autocorrelator
Mode-locked Nd:YLF Laser
EOFS
PMF
OSA
Dye Laser Coupling optics
Figure 4.1: First experimental setup for spectral measurement of the pulses.
thicknesses ranging from 0.2 to 0.5 mm, to obtain pulses in the range from 7 to 25 ps
Full Width at Half Maximum (FWHM) length. The etalon’s filtering action serves
both to lengthen and smooth out the pulses so that their frequency and time domain
characteristics are closer to those of a Transform Limited Gaussian pulse. For this
kind of pulse the product of the FWHM of the time and frequency power distributions
is 0.441. This was used as a basic test of the pulse performance characteristics. The
output of the dye laser is coupled to the EOFS via a Polarization Maintaining Fiber
(PMF). The coupling efficiency is then optimized with translation and tilt adjustment
of the coupling optics. The typical optical loss measured at the fiber pigtail output was
7-8 dB. The pulse length and spectrum were monitored by using the autocorrelator
and the OSA as shown in the figure. A microwave phase shifter after the amplifier
was used to control the relative phase between the optical pulses and the microwave.
As explained in chapter 2, the microwave frequency is chosen to be an integer
multiple of the pulse repetition rate, to ensure that the phase relationship between
the microwave and the optical pulse remains constant when each one of the pulses goes
through the device. It is crucial to guarantee that this synchronization has a good
phase stability so that the frequency shifting effect of the microwave on the pulses is
homogeneous. Two schemes of synchronization of the microwaves to the optical pulse
repetition rate were tried. The first scheme uses the 10 MHz reference output of the
RF source that drives the mode-locker of the Nd:YLF laser as a frequency reference
base for an HP8341B microwave synthesizer. The synthesizer is set to generate the
microwave signal at a harmonic of the pulse repetition rate, in order to maintain a
48
Phase-locked YIG oscillator
Power Amp.
Phase shifter
Autocorrelator
Mode-locked Nd:YLF Laser
EOFS
PMF
ANDO-OSA
Dye Laser
PD LPF
Amp
Coupling optics
Figure 4.2: Experimental setup for spectral measurement of the pulses using thephase-locked oscillator.
fixed phase relationship between the different optical pulses and the microwaves. The
microwave signal is then amplified by a 20 W Traveling Wave Tube (TWT) amplifier
and passed through the microwave phase shifter before being applied to the EOFS.
The phase shifter permits the adjustment of the relative phase between the microwave
and the optical pulse to control whether the device is operated in an up-shifting or a
down-shifting configuration. The microwave power is adjusted to control the extent
of the optical frequency shift desired. The fiber output of the EOFS is then coupled
to an ANDO AQ6317B optical spectrum analyzer with a 2 GHz resolution.
The second synchronization scheme used replaces the microwave synthesizer with
a Phase-Locked YIG Oscillator (PLO) custom made by MicroLambda Wireless. This
is the second experimental setup that was used for the measurements, and it is shown
in figure 4.2. The phase locked oscillator uses a 75.7 MHz reference signal to generate
harmonics of this reference in the range of 10 to 12 GHz. The harmonic generated
is controlled by a 12 bit word supplied externally to the device. The reference signal
is obtained by using a fast photodetector to sample the IR leakage from the high
reflector mirror at the end of the cavity of the Nd:YLF laser. This signal is then
filtered by a 80 MHz 8-pole low-pass filter and then amplified to a -3 to 3 dBm level,
required by the PLO. The microwave output of the PLO was amplified with the TWT
amplifier in the same way as the first scheme. This second scheme proved to have
better phase stability and noise characteristics.
49
4.1.1 Power and Pulse Length Dependence of the EOFS
Operation
The first experiments performed studied the power dependence of the frequency shift-
ing of the EOFS. The results of these measurements are shown in figure 4.3. Parts
a, b, and c of this figure show the results using pulse lengths of 8, 13, and 20 ps,
respectively. The 8 ps pulses were obtained at 945 nm and the 13 and 20 ps pulses
at 927 nm. The microwave frequency used was 6.056 GHz (80th harmonic of the
pulse repetition rate). These pulses were close to Gaussian Transform Limited. The
microwave power levels supplied to the EOFS (after taking into account cable losses
of up to 3dB) were chosen as square integer multiples (0, 1, 4, 9, 16, and 25) of a
reference power level of 0.15 W. In these figures, only the up-shifting configuration
is shown, since the downshifting behavior was very similar. The results show how
the overall shifting is practically the same for the three cases once the power was set.
In all the cases a distortion in the lower frequency side of the spectrum is observed
after shifting. It is important to note that the pulse distortion behavior is different
depending on the pulse length. This distortion is more noticeable in the longer pulse
cases. These observations agree with the results expected from theory and modelling
of the EOFS operation. A reduced distortion in the frequency spectrum is expected
for shorter pulses, where the smaller ratio between the pulse length and the microwave
period helps to localize the optical pulse in a region where it experiences an almost
constant index of refraction gradient. This observed result agrees with previous ex-
perimental observations and calculations presented in chapter 2 (Riaziat et al. (1993);
Farias and Eckstein (2003)).
In these previous works it has also been shown how the frequency shifting pro-
duced by the EOFS depends on the square root of the power. Figure 4.4 shows the
frequency shifting vs. power dependence behavior of the EOFS operation taken from
the observations in the same experiments as in figure 4.3. Both up-shift and down-
shift configuration shifting results are shown here, showing the almost symmetric
behavior of the operation. Shifting of up to 170 GHz with 3.75 W of applied power
was observed. For the sake of the representation, the negative values in the horizontal
axis relate to the downshifting configuration. The results show how the scaling of the
shifting with the square root of the power agrees very well with the linear behavior
expected.
To compare the observed experimental results with the developed model for the
EOFS, numerical simulations were performed. For this purpose, the differential equa-
50
0
0.004
0.008
0.012
0.016
313.2 313.3 313.4 313.5
Pow
er (
a.u)
Freq (THz)
0
0.03
0.06
0.09
323.5 323.6 323.7 323.8 323.9
Pow
er (
a.u)
Freq (THz)
0
0.03
0.06
0.09
323.5 323.6 323.7 323.8 323.9
Freq (THz)
Pow
er (
a.u)
a)
c)
b)
orig
0.15 W
0.6 W1.35 W
2.4 W3.75 W
orig
0.15 W0.6 W 1.35 W
2.4 W3.75 W
orig0.15 W
0.6 W1.35 W 2.4 W
3.75 W
Figure 4.3: Optical spectra of the up-shifting configuration for different power levels.a) 8 ps pulses at 945 nm, b) 13 ps pulses at 927 nm, and c) 20 ps pulses at 927 nm.
51
-200
-150
-100
-50
0
50
100
150
200
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Shi
ft (G
Hz)
Sign*(Power)1/2 (W)1/2
Figure 4.4: Light spectrum shift with different power levels applied to the EOFS inboth up-shifting and down-shifting configurations.
tions for the evolution of the field mode amplitudes were solved using the procedure
developed in section 2.3. As a first approximation to the problem, the pulses were
considered to be Gaussian Transform Limited when generating the initial condition
for the power spectrum of the optical field. The spectrum was matched to represent
the optical pulse lengths of 8, 13 and 20 ps, respectively. Another approximation
in the numerical simulation is that a perfect velocity matching between the optical
pulse and the microwave was assumed and effects of dispersion and mode coupling
were neglected. Figure 4.5 shows the results of numerical simulations for frequency
up-shifting of 13, 20 and 25 ps pulses at 927 nm using a microwave frequency of
6.056 GHz. The spectrum distortion that appears at the lower frequency side is more
evident for the longer pulse case, which agrees qualitatively with the experimental
data. It can be seen how the range of the frequency shifting depends on the power
applied. The low frequency distortion effect for the upshifting configuration is shown
in this simulation, and was also observed in the experiment. The distortion in the
experiments is, however, more noticeable than in the simulated model. This indicates
that the quality of the microwave field that is supplied to the microwave structures
is not as good as expected and the portion of the optical pulse that experiences an
almost constant index of refraction gradient is not as large as in the model. This
can be related to the quality of the microwave source, amplification problems, and
velocity mismatch effects. Some of the velocity mismatch effects can be produced by
small changes of the index of refraction due to the change of temperature that can be
52
0
1
2
3
323.5 323.6 323.7 323.8 323.9
13ps-sim
Pow
er (
a.u)
Freq (THz)
0
3
6
9
323.5 323.6 323.7 323.8 323.9
20ps-sim
Pow
er (
a.u)
Freq (THz)
0
1.5
3
4.5
6
323.5 323.6 323.7 323.8 323.9
25ps-sim
Pow
er (
a.u)
Freq (THz)
a)
c)
b)
orig0.15W 0.6W 1.35W 2.4W
3.75W
orig 0.15W0.6W
1.35W2.4W
3.75W
orig0.15 W
0.6 W1.35 W
2.4 W3.75 W
Figure 4.5: Numerical simulations showing the effect of pulse length and power onthe distortion of the pulses after frequency shifting.
generated by ohmic heating in the proximity of the microwave electrodes. There is
also the possibility that in reality the pulses produced by the dye laser may be closer
to the typical single side exponential form that is normal in this kind of laser (Schafer
et al. (1977)), and the intra-cavity etalons are not able to fully clean the spectrum
to generate the more desired gaussian pulse shape. It is important to point out that
microwave attenuation, impedance and velocity matching considerations have not
been included in the model and could eventually be incorporated for a more complete
description of the problem.
4.1.2 EOFS Cascading and Interaction Length Dependence
It has been shown in the model developed in chapter 2 how the frequency shifting
obtained scales linearly with the interaction length and the square root of the power.
53
Phase-locked YIG oscillator
TWT Amp.
Phase Shifter 1
Autocorrelator
Mode-locked Nd:YLF Laser
EOFS1
PMF
OSA
Dye Laser
PD LPF
Amp
Coupling optics
EOFS2
Phase Shifter 2
3dB coupler
Figure 4.6: Experimental setup for cascading of two EOFSs.
This result is seen in equation (2.9). Extending the length of interaction or cascading
two or more EOFS one after the other is advantageous because the shifting will be
increased linearly as compared to a square root law increment when only power is used
as a frequency shifting range control. Following this line of argument, it can be seen
that fixing the total amount of power available and dividing it equally into n EOFSs
of equal length, the overall shifting can be improved by a factor of√
n, as compared
to having only one device and applying all the power to it. The application of lower
power to a bigger number of modulators reduces the heating and loss problems, and
might reduce both higher order nonlinear effects and effects related to thermal heating
of the material. This same idea would work similarly with longer devices, because
it is easier to gain from increased length of interaction than from power. On the
other hand, fabrication of a very long device can have other problems ranging from
increased power losses in the materials to alignment difficulties in the fabrication.
Cascading various devices gets rid of the problem of device fabrication but has the
drawback that, in order to obtain an accumulative shifting effect, phase control must
be done separately in each device to keep the microwaves and optical pulses in the
same phase relationship.
A modification to the experimental setup was then introduced to be able to cascade
two EOFSs. In this setup, shown in figure 4.6, a microwave signal at 10.598 GHz
54
0
0.001
0.002
0.003
0.004
0.005
316.8 317 317.2 317.4 317.6 317.8
Pow
er (
a.u)
Freq (THz)
orig
u-0.15W
u-0.6W
u-1.35W
u-2.4W u-3.25W
d-0.15W
d-0.6Wd-1.35Wd-2.4W
d-3.25W
Figure 4.7: Spectra of the frequency shifting obtained when cascading two EOFSs.
was generated with the second scheme that includes the PLO. The microwave was
amplified by the TWT and the output was split in two using a 3dB coupler. Each of
the outputs was then run into a phase shifter and then to two different EOFS devices.
The phase shifters were used to separately control the relative position between the
optical pulses and the microwave field in each EOFS. This was done to ensure that
after some phase is fixed in one of them, the other device is set similarly to avoid
cancelling out the effects. The phase synchronization can also be done with the use
of an optical delay line and only one microwave phase shifter. The introduction of
an optical delay line before the first EOFS can be used to adjust the relative phase
between the microwaves and the pulse and then the phase shifter can be used to adjust
the relative phase of the microwave that is supplied to the second EOFS so that the
correct phase is achieved. The fiber output of the first EOFS was coupled with an
FC adapter to the second EOFS input, and then the second output was taken to the
OSA. The optical coupling losses of this system increased to 17 dB (as compared to
8dB with one device) and the signal output level was approx 0.5 mW, which still
guaranteed a good operation of the OSA. Pulses that were close to transform limited
at a wavelength of 945 nm, a repetition rate of 75.7 MHz, and a length of 9 ps.
The experimental results are shown in figure 4.7. Both up-shifting and down-
55
shifting configurations are shown here. The plots are labelled with the power that
was supplied by the amplifier to each modulator, so the actual power supplied by
the TWT is twice these numbers. The figure shows a frequency shifting range up to
±450 GHz. With this kind of shifting available, it would be possible to address 16
channels with a 50 GHz inter-channel spacing. As power and shifting are increased,
the pulse distortion is more noticeable. As expected, the distortion for the up-shifting
configuration appears in the low frequency side of the spectrum, while it appears
on the high frequency side for the down-shifting case. The operation of the device
is limited by the pulse distortion because of the spectrum broadening that would
induce inter-channel interference. The spectra of the shifted channels can be post
processed by using band pass filters, to ensure channel differentiation. However this
can reduce the optical power efficiency of the device since some of the power will be
lost in the filtering. This can be handled within certain requirements without much
problem, as long as the frequency content is within the detector range, and there
is the possibility of some amplification to boost the signal if needed. The problem
of distortion is produced by the fact that the gradient of the index of refraction
is not really constant over the pulse extent. This has been confirmed in both the
experiment and the mathematical modelling. With this in mind, the next step is to
work towards obtaining more convenient microwave sources and techniques for the
microwave generation. Some of these ideas will be described in the next section.
4.2 Microwave Techniques For Improved EOFS
Operation
The experimental results obtained in sections 4.1.1 and 4.1.2 show that the problem of
not having a constant gradient in the index of refraction wave in the region where the
pulses are located can lead to significant distortion and broadening after the frequency
shifting is performed. It is also clear that this distortion increases as the shifting is
increased, so a limit on the shifting range that can be obtained will result from this
effect. On the other hand, if the optical pulse can be placed in a position where
the index gradient does not vary considerably, then the distortion can be reduced.
Therefore, an ideal waveform for the microwaves will be one with a long linear portion,
such as a triangular or saw-tooth wave. Using these kinds of waveforms would allow
more freedom on the position or phase that the pulse could have with respect to
the microwave, and the distortion would be reduced because the region of constant
gradient in the index of refraction is extended. The down side of this proposal is
56
that the generation and amplification of these waveforms makes the problem more
complicated. A perfect saw-tooth waveform has a high harmonic content, and thus
the limitation for this kind of approach is the availability of a wideband amplifier so
that all the frequency components can be enhanced and used for the application of the
microwave field to the EOFS structure. Also, more complex microwave engineering
designs that take into account the frequency dependent properties of the microwave
circuitry need to be studied so that after the amplification process the waveform has
the needed linear characteristics and can propagate properly along the microwave
structure. This seems to be a very challenging problem but some simplified solutions
can be considered.
As a first approach to try to extend the linear portion of the microwave field to be
used in the frequency shifting effect, the introduction of a second microwave frequency
component at twice the frequency of the original microwave is considered. This will
create an effective microwave field of the form
E(y, t) = E0sin(k0y − ωt) + aE0sin(2k0y − 2ωt), (4.1)
where a is a constant that will determine the relative amplitude of the second har-
monic field component with respect to the original field. The difference between the
original field and a corrected field with a second harmonic component can be shown
in figure 4.8 a). In this figure, the corrected field is obtained by adding the second
harmonic of a sinusoidal wave with different weights given by a = 1/2, a = 1/4,
a = 1/8, and a = 1/16. For simplicity, the figure just shows a simple sinusoidal
wave and the corrections with the phase parameter varying from 0 to 2π, to show one
oscillation period. In part b) of the figure, the gradient of the original waveform and
the corrected versions is also presented. It can be seen how the effect of the applied
second harmonic field can extend the “linear” behavior region and this is more clearly
seen in the gradients figure, where this region is shown by an approximately constant
value of the gradient. The desired effect is to obtain an extended region where the
gradient in the corrected waves is approximately constant. In the real situation, the
corrected plot represents the applied microwave field, so this is directly related (pro-
portional) to the refractive index variation induced in the nonlinear material. It is
now important to investigate which second harmonic weight or proportion makes the
field correction better for the purpose of driving the EOFS. In other words, what is
the best second harmonic weight to achieve the best “constant gradient region” for
the driving field?
Without loss of generality, the field can be viewed as the field at time t=0, since
57
-1.5
-1
-0.5
0
0.5
1
1 2 3 4 5
x (rad)
Ind
ex V
aria
tion
Gra
die
nt (
a.u
.)
orig w/o shca=1/2a=1/4a=1/8a=1/16
-1.25
-0.75
-0.25
0.25
0.75
1.25
0 1 2 3 4 5 6
x (rad)
Ind
ex V
aria
tion
(a.u
)
orig w/o shc
a=1/2
a=1/4
a=1/8
a=1/16
a)
b)
Figure 4.8: Introduction of a second harmonic field component to extend the con-stant gradient region in the refractive index wave.
58
a velocity matched structure is being considered. The spatial gradient of the index is
given by
∂
∂yE(y, 0) = k0E0cos(k0y) + 2ak0E0cos(2k0y). (4.2)
Since the objective is to try to get a longer linear portion of the wave at the k0y = π
point (to have an approximately constant gradient), the condition needed around this
point is (∂2/∂z2)E(y = π/k0, 0) ≈ 0. This then leads to the equation
∂2
∂y2E(y + ∆y, 0) = −k2
0E0sin[k0(y + ∆y)]− 4ak20E0sin[2k0(y + ∆y)] ≈ 0. (4.3)
Expanding the sine terms around the k0y = π and 2k0y = 2π values, the condition
is simply reduced to a = 1/8 (relative amplitude of second harmonic component).
This suggests that introducing a second harmonic component of 1/64 of the power
(approximately -18 dB) of the original microwave, the constant portion of the gradient
of the electric field can be extended so that the pulse distortion that occurs with the
shifting can be reduced. This is more clearly shown in figure 4.9 , where the field
wave and its gradient are shown with and without this second harmonic component
correction (a = 1/8). The part of the gradient where the spatial variation is small
or close to zero is practically doubled, as can be seen by the marked segments that
indicate the portion of the graph where the gradient varies by less than 10%. This
means that in the operation of the EOFS, either longer pulses could be used, or longer
shifting can be achieved without the distortion penalty.
Going back the results obtained in the physical and semiclassical models presented
in chapter 2, a basic relationship for the frequency shifting obtained in one of these
EOFS structures in terms of the microwave field applied is
∆f ∝ f0 · ∂
∂y(Ez). (4.4)
In the corrected versions of the fields presented in figures 4.8 and 4.9 the place
of the microwave field where the pulse is going to be located is when the phase of
the field equals π. To obtain the same effect for the situation that was studied in
chapter 2, where the pulse is located at the origin in t=0 (phase =0), the sign of the
added second harmonic component needs to be changed. Clearly the effect of the
second harmonic component is that it would reduce the value of the gradient of the
index field in magnitude but extend the region of linearity of the field to obtain the
performance improvement. In other words if the two waves are studied as separate
59
-1.5
-1
-0.5
0
0.5
1
1.5
0 1 2 3 4 5 6
x (rad)
Sin(x)
Sin(x)+ (1/8) Sin(2x)
Index of Refraction Wave
a) b)
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
2 2.5 3 3.5 4 4.5
Index Gradient
x (rad)
Figure 4.9: Introduction of a second harmonic field correction with relative weighta = 1/8.
excitations, the original wave phase is set for an up-shifting (or down-shifting) phase
configuration, while the smaller second harmonic wave is set for the opposite down-
shifting (or up-shifting) phase configuration respectively. If the total field is applied
using the optimal a = 1/8 factor obtained before, then the total field is given by an
expression of the form
Ez(y, t) = a1sin(k1y − ω1t)− 1
8a1sin(2k1y − 2ω1t), (4.5)
where the factor a1 is directly related to the applied power and ω and 2ω correspond
to the original single and second harmonic frequencies respectively. The field (index)
gradient is given by
∂
∂yEz(y, t) = k1a1cos(k1y − ω1t)− 1
82k1a1cos(2k1y − 2ω1t). (4.6)
The point of interest is the place where the maximum gradient is obtained when
t = y = 0. This gives a value of
∣∣ ∂
∂yEz
∣∣max
=3
4k1a1, (4.7)
where it is seen that the overall effect shifting is diminished (in the original case
without the second frequency component, the relationship is∣∣ ∂∂y
Ez
∣∣max
= k1a1). This
is intuitively obvious, because the second field phase is adjusted so that it is in an
opposite configuration (and consequently in the opposite down/up-shifting phase) as
60
compared to the original field. This results in some range of shifting being sacrificed in
the search of reduced distortion. If the same overall shifting range as in the individual
field source case is desired, then a power re-scaling must be done in order to keep the
relative amplitudes of the two frequency components fixed and the overall shifting
increased to the same level obtained when using only the single frequency component.
Equation (4.7) shows that the introduction of the second harmonic correction will
reduce the effective gradient to 3/4 of the single frequency value, and consequently
the differential frequency shifting would be reduced proportionally. In order to obtain
the same frequency shifting as before, the amplitude of the field (a1 coefficient) should
be multiplied by a factor of 4/3, and the second harmonic component fixed at 1/8
of this value. This means that, the power of the first component must be increased
as P ′1 = 16
9P1 = 1.78P1 and the second harmonic power set to P2 = 1
64P ′
1. It is
important to note that there are effects that are not taken into account in this rough
approximation that can make things more complex, such as the inter-mixing of the
field components that could occur in the nonlinear material. Also, considering the
fact that the P1 component is considerably stronger, there might be processes related
to other nonlinear phenomena that may have a greater effect on the first frequency
component than on the second component. Also, this “linearized” version of the
approach can be inappropriate, especially when the magnitude of the first harmonic
field is taken to high levels.
It is possible to include this second harmonic component in the semiclassical model
that was introduced in chapter 2. The evolution of the mode amplitudes can be
obtained from equations (2.18) and (2.19)
∂Am
∂z=
1
2kmn2
0rc
∑
l
alAm−l − a∗l Am+l, (4.8)
and
∂al
∂z= −1
2kln
20rc
∑m
AmA∗m−l. (4.9)
In a manner similar to that seen in the simplified solution model in section 2.3, it
is seen that the microwave field mode amplitudes that have an initial non-zero value
are a1 and a2. They correspond to the main and the second harmonic frequency
components respectively. Also, since the variation of the other harmonic components
of the microwave field will be very small, the approximation aj ≈ 0 for j > 2 is done.
With this in mind, the set of equations (2.33) is used in this case leading to
61
a)
b)
Orig
0.26 W1.07 W
2.4 W 4.27 W6.67 W
Orig
0.15 W
0.6 W
1.35 W2.40 W
3.75 W
0
2
4
6
8
10
322.5 322.6 322.7 322.8 322.9
Pow
er (
a.u.
)
Frequency (GHz)
0
2
4
6
8
10
322.5 322.6 322.7 322.8 322.9
Pow
er (
a.u.
)Frequency (GHz)
Figure 4.10: Simulations for the operation of the EOFS with a second harmonicfield component with relative amplitude a = 1/8.
∂Am
∂z=
1
2kmn2
0rc(a1Am−l − a∗1Am+1 + a2Am−2 − a∗2Am+2),
∂a1
∂z= −1
2k1n
20rc
∑m
AmAm−1,
∂a2
∂z= −1
2k2n
20rc
∑m
AmAm−2. (4.10)
This set of equations was solved numerically, in a similar to what was done in
chapter 2 to study the operation of the EOFS under these conditions. The results
are shown in figure 4.10. The simulated data corresponds to the operation of the
EOFS using a train of 24 ps long pulses (Gaussian Transform Limited) at 930 nm,
in a 1.5 cm effective length device. The original and second harmonic frequency
components were set at 6 and 12 GHz respectively. The plots show the pulse spectra
for an up-shifting configuration using different power levels supplied to the device.
Part a) of the figure shows the different spectra for the EOFS operation without
the second harmonic correction. Part b) shows the plots for the operation with the
second harmonic correction. The different plots are labelled with the power level in
the first harmonic component, i.e., P1 in part a) and P ′1 in part b). The power level
62
used for the second harmonic component in this simulation was set to P2 = 1/64P ′1.
As mentioned above, to be able to compare the operation of the device obtaining
similar shifts in the spectra, the powers have to be modified in order to compensate
for the fact that the introduction of the second harmonic correction would decrease
the amount of shifting obtained.
From the figure, it is clear that the introduction of the second harmonic correction
component for the microwave source can have a very beneficial effect on distortion
reduction. Figure 4.10 shows how the spectra in part b) has a considerable reduction
in the low frequency distortion tails that appear after shifting. The spectral widths
in the case of the harmonic component correction remain closer to the original (i.e.,
their bandwidth is not increased so much as the bandwidth of the pulses when the
correction is not used.) The cost of this reduction in the distortion is an increase in
the power consumption of the device, because the original component must be made
larger to compensate for the fact that the second harmonic correction introduces the
opposite shifting effect.
4.2.1 Experimental Introduction of the Second Harmonic
Correction
As discussed above, the idea of linearizing a portion of the index of refraction wave
so that the gradient (in the region where the pulse is located) is closer to a constant,
led to the introduction of higher harmonic components to the microwave source. For
experimental purposes there is a need to have an amplifier with a wide enough gain
bandwidth so that the different harmonic components can be amplified. This same
reason makes the approach only realizable when only a few harmonics are considered.
In the case of this experiment, the TWT amplifier had a gain bandwidth from 4.5 to
18 GHz. To extend the setup previously used, it was decided to use the microwave
frequency synthesizer as one source and the PLO oscillator as the second source
(for the second harmonic). The constraint was given by the PLO, which in this
case can operate between 10 and 12 GHz, approximately. The HP synthesizer was
set for a frequency of 5.9046 GHz and the PLO was set to generate a frequency
of 11.8092 GHz (the 78th and 156th harmonics of the 75.7 MHz pulse repetition
rate respectively). Lower frequency choices for the synthesizer were avoided due
to bandwidth constraints related to some of the microwave components, such as the
3dB coupler, whose bandwidth was limited from 6 to 18 GHz.
The experimental setup is shown in figure 4.11. The synthesizer signal is locked
63
RF Synthesizer
Mode-locked Nd:YLF Laser
Microwave Synthesizer
Phase-locked YIG oscillator
TWT Amp.
PD LPF
Amp
Phase shifter
3dB coupler
Phase shifter
Autocorrelator
Dye Laser
ANDO-OSA
Att.
Att.
Att. EOFS
Coupling optics
Figure 4.11: Experimental setup for the introduction of a second harmonic compo-nent correction to the microwave field.
via the RF synthesizer, while the PLO signal is phase-locked using the laser pulse
sampling technique explained in section 4.1. The two microwave signals are then
passed through different phase shifters so that each of their phases may be controlled
separately. The microwaves are then combined using a “reversed” 3dB coupler so that
the two microwaves are introduced in the coupler output ports, and exit the coupler
via the normal input port. There is an approximate 3dB loss for each microwave at
the output. To minimize problems with reflected power, the fourth terminal of the
3dB coupler was properly terminated with a 50 Ohm load. The microwave signal
was then introduced to the TWT amplifier before finally being applied to the EOFS.
Some attenuation pads (1 dB, 3dB, 6 db, 12 dB and 15 dB) were used between the
phase shifters and the 3dB coupler in order to get the relative power between the
two microwaves adjusted to the needed values (relative power differences between the
components). The continuous output power control from the HP synthesizer was also
used at the same time to help with this task. The rest of the Nd:YLF, dye laser, EOFS,
autocorrelator and OSA setup is the same one used for the experiments studying
the power dependence of the operation of a single EOFS device. The experimental
measurement technique was very similar to the previous experiment. The pulse length
and wavelength were controlled by the dye laser setup (using its birefringent filter to
choose wavelength, and the glass intra-cavity etalons as bandwidth filters).
To obtain the power configuration required, each of the sources was then turned on
individually and amplified to check the power level output by monitoring the -40dB
64
0
0.015
0.03
0.045
0.06
311.45 311.5 311.55 311.6
orig
f-2f
f
f+2fPow
er (
a.u)
Freq. (THz)
0
0.01
0.02
0.03
0.04
311.45 311.5 311.55 311.6
Pow
er (
a.u)
Freq. (THz)
0
0.015
0.03
0.045
0.06
311.45 311.5 311.55 311.6
Pow
er (
a.u)
Freq. (THz)
a)
b)
c)
10 dB
13 dB
16 dB
Figure 4.12: Frequency shifting of 18 ps pulses with the second harmonic correctionto the microwave field.
output of the TWT. According to these measurements, the attenuation pads in the
transmission lines or the output power of the HP synthesizer were adjusted to obtain
the desired relative power levels between the microwaves. The phase of each wave was
adjusted individually to obtain the upshifting or downshifting configuration needed.
Several configurations were tried: (i) both microwaves in the upshifting phase; (ii)
one microwave in the upshifting phase and the other in the downshifting phase; and
(iii) both microwaves in the downshifting phase. It is important to point out that the
situation of most interest was the one in which the waves were in the opposite shifting
configurations, in order to check the possibility of the reduction in the spectrum
distortion after the shifting. After the relative phases of the waves were adjusted,
both waves were amplified together and the phases were fine tuned to achieve the
desired configuration. The spectrum of the light was measured using the OSA as in
the previous experiments.
The first experimental results in these trials were obtained using 18 ps pulses,
at an approximate wavelength of 964 nm. The objective was to test the frequency
up-shifting operation. The frequency shifting was tried with the single frequency
component alone and with the second harmonic both in phase and out of phase with
65
respect to the first (up-shifting and down-shifting configurations respectively). The
reference point used was a total power output of the TWT of 1 Watt in the situation
where the second harmonic was set for downshifting to counteract the first frequency
component. It is very important to point out that there was a 3dB pad attenuation
and a 1 dB loss in the cable before this power was supplied to the EOFS, so the actual
power is -4dB below the 1W level (this means the effective power was at a 0.4 W level
instead.) This was done to test whether or not this configuration offered advantages
over others with respect to the distortion obtained. For the single frequency and
the “in-phase” configurations, the overall gain of the TWT amplifier was adjusted in
order to obtain a very similar overall shifting of the frequency spectra. In this way we
were able to compare the relative distortion using the same “normalization” idea used
for the mathematical simulations for this type of operation. The main reason for this
is that, according to the previous experimental results and calculations, the amount
of distortion increases with the amount of relative shifting. In order to compare
distortions in different spectra in a significant way, it is necessary to make the overall
shifting of the spectra very close to each other. The results of these measurements are
shown in figure 4.12. Parts a), b) and c), show measurements for the cases when the
single frequency component is set for frequency up-shifting and the second harmonic
correction is set 10 dB, 13 dB, and 16 dB below the first component power level.
There are four traces labelled in each graph: orig shows the unshifted spectra, f-2f
shows the case when the second harmonic correction is set for downshifting, f shows
the effect of only using the first component, and f+2f shows the effect of adding the
second component with an up-shifting phase. The trace used to monitor the power
level supplied was the f-2f case, which was set to 1 W of combined power. After this,
the power was controlled with the TWT gain to be able to obtain similar overall shifts
in the f and f+2f configurations. This explains the reason why the overall average
shifting increases from part a) to part c), since the power difference between the two
components is smallest in part c). In this way we see that the relative effect of the
second harmonic correction to downshift the spectrum would be more noticeable in
part a) than in c).
Similar measurements were done for 29 ps long pulses. These are shown in figure
4.13. The settings were similar to the ones for the 18 ps group. The wavelength was
chosen as 965 nm. Each graph shows the same four traces corresponding to the orig-
inal, f-2f, f, and f+2f spectra, with the same rule of thumb of setting the power level
to 1 W output of the TWT in the f-2f configuration. The results in both of these
sets agree with each other in various points. The first thing that is noticed is that
66
0
0.04
0.08
0.12
0.16
311.2 311.25 311.3 311.35
orig-n
f-2f
f
f+2f
Pow
er (
a.u)
Freq. (THz)
0
0.04
0.08
0.12
0.16
311.2 311.25 311.3 311.35
Pow
er (
a.u)
Freq. (THz)
0
0.04
0.08
0.12
0.16
311.2 311.25 311.3 311.35
Pow
er (
a.u)
Freq. (THz)
a)
b)
c)
10 dB
13 dB
16 dB
Figure 4.13: Frequency shifting of 29 ps pulses with the second harmonic correctionto the microwave field.
the spectrum distortion leading to a bandwidth spread after the frequency shifting
is weaker for the f-2f configuration. This agrees with the results obtained from the
mathematical simulations and the idea that introducing the second harmonic compo-
nent in an opposite phase configuration would reduce the magnitude of the gradient
of the index of refraction, but would extend the region where this gradient can be
considered a constant. Thus it is clear that this configuration reduces the bandwidth
distortion as compared to the f traces. With the same argument, it is possible to
explain why the traces corresponding to the f+2f configurations show a larger band-
width distortion as compared to the single frequency case. It is shown in figures 4.12
and 4.13 how the bandwidth distortion correction effect is more noticeable in the 10
dB difference sets than in the 13 dB or 16 dB sets. The correction is almost negligible
in the 16 dB difference set. This is not in very good agreement with the mathematical
modelling results, where it was shown that the best power ratio between the first and
second harmonic components is around 64, which means an 18dB difference. During
the measurements we were expecting to get better distortion correction around the
16 dB to 18 dB sets, however the effects for these experimental sets were negligible
and the best results were obtained around the 10 dB difference sets (even from 7
67
dB to 13 dB). This situation is probably due to the fact that the second harmonic
power level is very low compared to the single frequency power. Consequently, the
distortion effects caused by white noise amplification and other effects that distort
the spectra are stronger than the correction that can be achieved by the introduction
of the second harmonic component. Another factor is that the microwave losses are
frequency dependent and are higher at higher frequencies. Thus a stronger second
harmonic field should be applied to try to compensate for the losses. Another possi-
bility is that the microwave cables and connectors introduce small variations due to
modulations in the transfer characteristics from various sources: resonances caused
by cable length; impedance matching of connectors; and other related effects that
can cause the power transfer to the EOFS to be different for the different frequencies,
thereby causing this disagreement.
68
Chapter 5
The EOFS and the Single Photon
Limit
The fields of Quantum Optics (QO) and Quantum Information Science (QIS) have
been very active during the last few years. There is a great interest in applications for
Quantum Cryptography (QCr), Quantum Computing (QCo), and also in the study
of the basic principles of quantum mechanics. The quantum properties of many two-
level systems, such as spin-like systems and polarization states in photons have been
used among many others, for the idea of constructing the famous “qubits”, or units
of quantum information analogous to the classical bits. The polarization states of
photons have been used to encode 0’s and 1’s, like in classical bits, but in this case,
the quantum nature of photons (and all these two level systems) allows the possibility
of having states that are a superposition of the 0’s and 1’s as
|Ψ〉 = α|H〉+ β|V 〉, (5.1)
where |H〉 and |V 〉, denote orthogonal polarization states (horizontal and vertical
polarizations in this case, even though the important thing is to have a well defined
orthogonal basis) that can be associated with the 0 and 1 states.
Extensive theoretical work and experiments with photon based qubits and “en-
tangled” photon qubits have been made during the last decade. Many of these ex-
periments simulate the quantum limit of single photon operation by using strongly
attenuated beams. The aim of this chapter is to explore the possibility of finding a
way of using the EOFS with a single photon and to express its effect over a single
photon state. It known that is not possible to define a position operator for a single
photon (Mandel and Wolf (1995); Newton and Wigner (1949)), so that the localiza-
tion of the photon can be only made approximately. On the experimental side, the
69
technological challenges of creating or handling a single photon state at the present
time are really enormous. Nevertheless, there is a lot of interest in developing the so
called “single photon sources”. How to deal with a system like this, having a frag-
ile state that is very hard to isolate from interactions with the external world (like
absorption and losses in the materials and propagating media), is a hard problem.
In these experiments with strongly attenuated beams there is not exactly a single
photon state but really a collection of states where the average number of photons
per state is very low. This suggests that the model previously described in chapter
2 could be used as a starting point for studying the physics of the EOFS operation
when going to this single photon limit.
In quantum optics, the quantization of the field leads to the annihilation and
creation operators a and a†, and the number operator n. The number operator
denotes the total number of photons in all space. However, the practical situation in
quantum optics experiments deals with photo-detection processes and defined optical
paths that imply that we could think about some approximate localization of the
photon state. The fact that the photons are detected means there should be some
idea of photon localization, a wave function, or wave packet for detection (Mandel and
Wolf (1995); Scully and Zubairy (1997)). Following the ideas given in these books,
where the field intensity is related to a space number operator, some approximate
localization over a region of space may be defined. A photon absorption operator can
be introduced as
V (r, t) =1
L3/2
∑
k,saks
εksei(k · r−ωt), (5.2)
where L represents the dimension of a box containing the modes and the sum is made
over the set of modes to which the detector is sensitive to. Each mode has also a
polarization vector εksassociated with it. With this, the configuration space number
operator is defined as
n(V , t) =
∫
VV y(r, t) · V (r, t)d3r, (5.3)
where V is the region in which the photon or photon states are defined (the box of
side L). This is a good approximation if we guarantee that the dimension L is very
large compared to the optical wavelengths in question. In these previous equations,
the quantum mechanical creation and annihilation operators have been introduced.
The creation operator can be used to generate any Fock (number) state of the field by
repeated operation over the vacuum state. This is shown from the following general
70
properties of these operators:
a|vac〉 = 0,
a|n〉 =√
n|n− 1〉, for n = 1, 2, 3, ...,
a†|n〉 =√
n + 1|n + 1〉, for n = 0, 1, 2, 3, ... . (5.4)
The creation and annihilation operators are usually labelled by wave numbers (or
frequency) and by polarization states. In this way, they span a complete basis to
describe any state of the field. The Fock states are called number states because
they are eigenvalues of the number operator, which is related to the annihilation and
creation operators by
nks= a†
ksaks
. (5.5)
The boldface type in the previous equations and throughout this chapter denotes
vector quantities. There are also very particular commutation relationships that these
operators follow such us (Mandel and Wolf (1995); Cohen-Tannoudji et al. (1977))
[aks, a†
k0s′ ] = δ3
kk0δss′ , (5.6)
[aks, ak0s′ ] = 0, (5.7)
[a†ks
, a†k0s′ ] = 0. (5.8)
From these commutation relations, more general commutation rules can be obtained
as
[aks, nk0s′ ] = [aks
, a†k0s′ ]ak0s′
= aksδ3
kk0δss′ , (5.9)
and
[a†ks
, nk0s′ ] = −a†k0s′δ
3
kk0δss′ . (5.10)
The problem of representing a one photon state by the action of a creation operator
on the vacuum state is that a†ks|vac〉 corresponds to a photon that is not localized in
space because it is associated with a definite momentum or wavenumber k. In order
71
to define a single photon state that is to some degree localized there is a need to think
about a superposition of these modes. The rest of this introductory section follows
the procedure by Mandel and Wolf (1995) to define a position space wave function
and an energy density wave function. There is a way to express a one photon state
that is partially localized in space
|φ〉 =1
L3/2
∑
ks
φ(k, s)a†ks|vac〉, (5.11)
where the function φ(k, s) gives the amplitudes of the different modes and has a
normalization condition given by
〈φ|φ〉 =1
L3
∑
ks
∣∣φ(k, s)∣∣2 = 1. (5.12)
This guarantees that the photon is partially localized in a region of space, with the
disadvantage that the superposition of different momentum states makes the photon
energy (momentum) be not definite. The vector function of this partially localized
photon is
φ(r, t) =1
L3
∑
ks
φ(k, s)εksei(k · r−ωt). (5.13)
This represents the position space wave function of a photon in state |φ〉 and can be
used to obtain the photon probability density |φ(r, t)|2, as seen in equation (5.15),
which gives a more clear idea of the localization of the photon.
Now, if the local detection operator defined in equation (5.2) acts on the state |φ〉the result is
V (r, t)|φ〉 =1
L3
∑
k,s
∑
k0,s′φ(k0, s′)aks
a†k0s′|vac〉εks
ei(k · r−ωt)
=1
L3
∑
k,s
φ(k, s)εksei(k · r−ωt)|vac〉
= φ(r, t)|vac〉. (5.14)
Using equation (5.3), the expected value of the configuration space number operator
can be calculated, leading to a relationship that is the probability of finding a particle
with wave function φ(r, t) inside a volume V .
〈φ|n(V , t)|φ〉 =
∫
V|φ(r, t)|2d3r. (5.15)
72
In a similar manner, it is possible to introduce a definition for the average photon
energy and an energy wave function. The average photon energy is given by
〈φ|H|φ〉 =∑
k,s
~ω〈φ|nk,s|φ〉
=1
L3
∑
ks
~ω∣∣φ(k, s)
∣∣2, (5.16)
where the last equation is obtained by introducing the definition of |φ〉 in equation
(5.11) and using the creation and annihilation operator properties. The energy wave
function is given by
ψ(r, t) =1
L3
∑
ks
(~ω)1/2φ(k, s)εksei(k · r−ωt). (5.17)
In in a similar way to equation (5.15), the expected value of the energy is expressed
as
〈φ|H|φ〉 =
∫|ψ(r, t)|2d3r, (5.18)
so that |ψ(r, t)|2 is the energy density. However, this energy density is not locally
defined as the photon density over a volume V , but it is defined over all space.
5.1 Quantum Mechanics of the EOFS Operation
As shown in the beginning of this chapter, a single photon state can be approximately
represented by a superposition of modes with different amplitudes in a way similar
to the expansion for the fields that was done in the wave-mixing treatment in section
2.2. The interaction of the optical field with the electrical field in the EOFS can be
treated as a three wave mixing process in which a photon of one mode of the electric
field is combined with a photon of the optical field to generate a third photon. This
process is mediated by the nonlinearity of the material. Using the partially localized
wave function for a single photon introduced previously, and reducing the problem to
one polarization and propagation direction resembling the EOFS experimental set-
up, the mode amplitudes can labelled by m instead of by k and s. The quantized
hamiltonian of the system can be written as a superposition of many of these three
photon processes:
73
H =∑m
~ωm(nm + 1/2) + hg[a†m+1ama1 + h.c.]. (5.19)
In this hamiltonian, g is a mode coupling constant that depends on the nonlinear
susceptibility of the material by which the photon interaction is mediated. Also, the
approximation of the basis modes used is similar to the three wave mixing model
case in which the m-th mode frequency corresponds to the m-th multiple of a basis
frequency which is given by the microwave source. Since the microwave field is intense
compared to the mode amplitudes of the photon states (i.e., 〈nm〉 ¿ |a1|2), it can be
treated classically. Instead of using an operator for the field, it can be represented
by a field of complex amplitude a1 = a10e−iω1t. Here, ω1 = 2πf1, where f1 is the
microwave frequency. With this, the hamiltonian becomes
H =∑m
~ωm(nm + 1/2) + hg[a†m+1ama10e−iω1t + h.c.]. (5.20)
The evolution of the an operator is calculated using the Heisenberg equation of
motion
˙an =1
i~[an, H]. (5.21)
With the definition of the hamiltonian, this can be written as
˙an =∑m
−iωm[an, a†mam]−ig[an, a†m+1am]a10e
−iω1t−ig[an, a†mam+1]a∗10e
iω1t
(5.22)
Using the operator definitions and commutation relations in equations (5.6) to (5.10),
the expression for the time evolution of the an operator can be reduced to a more
simple expression given by
˙an = −iωnan − igan−1a10e
−iω1t + an+1a∗10e
iω1t. (5.23)
At this point, it is convenient to introduce the slowly varying amplitude approxima-
tion where the operators have a plane wave component and an amplitude component
that gives the slow evolution behavior of the modes. This is done by introducing a
new set of operators in similar way as in the interaction picture representation:
An = aneiωnt. (5.24)
Using these new operators, equation (5.23) can be written as
74
˙Ane−iωnt − iωnAne−iωnt = −iωnAne
−iωnt − igAn−1e
−iωn−1ta10e−iω1t
+An+1e−iωn+1ta∗10e
iω1t. (5.25)
Noting that the second term in the left and first term in the right cancel each other,
and that the exponential terms after grouping are the same (using the fact that ωn =
nω1), the expression is reduced to a simple evolution equation for the A operators
given by
˙An = −igAn−1a10 + An+1a
∗10
. (5.26)
The evolution of the A operators can now be calculated provided that their initial
conditions are set and that the microwave field is also known. For the case where
the microwave field is set to provide an upshifting configuration as in chapter 2, the
complex amplitude can be expressed as a10 = −i|a10|. This will transform equation
(5.26) to
˙An = −gAn−1|a10| − An+1|a10|
. (5.27)
It is clear how this evolution equation resembles the differential equation for the
evolution of the Am amplitudes that was found in section 2.3 using the fact that the
a1 amplitude was set initially as a real value.
∂Am
∂z=
1
2kmn2
0rca1(Am−1 − Am+1). (5.28)
One of these equations is related to a time evolution and the other to a space evolution.
However, a relationship between time and position can be used to connect these
equations (i.e. z = ct/n0, where n0 is the refractive index, and c the speed of light).
The results show that, in the limit of a strongly attenuated beam, the single photon
can be approximated by a superposition of Fock states where the average number per
state is much less than one in a similar way as the optical pulse was modelled by a
superposition of travelling wave modes. The structure of the evolution of the field
operators is mathematically equivalent to the evolution of the mode amplitudes that
was studied in this work and, thus, a mathematical approach to this problem is
possible in a similar way.
The next question is how to express the action of the EOFS as a unitary evolution
operator acting over the single photon state. From the experimental results in this
work it was observed that, for moderate lengths of interaction and low power levels,
the frequency shifting produced by the EOFS was homogeneous. In the low frequency
75
shifting regime, there was not a strong distortion and the mode amplitudes where
shifted homogeneously to another region of the spectrum. A single photon state
coming from a pulsed type of source will have a very similar spectral content to what
was observed in the experiments, so we can expect the behavior of the device to be
very similar. This leads to the idea of representing the action of the EOFS as an
operator that acts over the one photon state.
Writing a position space wave function for the initial state of the photon as
φbefore(r) =1
L3
∑
ks
φ(k, s)εksei(k · r), (5.29)
and introducing the operator U = ei(∆k·r), the state after the action of the EOFS is
represented as
φafter(r) = ei(∆k·r)φbefore(r)
=1
L3
∑
ks
φ(k, s)εksei(∆k·r)ei(k · r)
=1
L3
∑
ks
φ(k, s)εksei(k+∆k)·r. (5.30)
Looking at the previous expression, it is seen how the action of the operator has
shifted the mode amplitudes φ(k, s) from the k number to the k + ∆k number in a
continuous way. This represents the position space wave function for a single photon
state in which the mode amplitudes are the same but have homogeneously moved
from one region of the spectrum to another. This means the whole spectrum has
been shifted by an amount ∆k so it is natural to talk about displacing the single
photon state to a new value of the expected value or average of momentum (energy)
〈k + ∆k〉. Using the relation ω = ck, and the results in chapter 2 given by
d
dz〈ω〉 =
πn20rcE0
λm
〈ω〉, (5.31)
a linear approximation of the exponential behavior of the solution for the frequency
shifting can be used again to obtain
∆k ' πn20rcE0Leff
λm
k. (5.32)
This ∆k is then used to express the unitary operator as
U = ei(πn20rcE0Leff /λm)k·r. (5.33)
76
5.2 The EOFS in Quantum Information Science
One of the reasons for the interest in the single photon limit of the EOFS operation
is the possibility of incorporating this kind of device in QIS systems. The possibility
of introducing arbitrary shifts to a single photon opens the field for many possibilities
in the study of the quantum mechanics of these systems. There have been previous
studies of two-photon entanglement using pulsed sources (from femtosecond lasers),
where the observed results in the correlations and interference measurements show
considerable differences from those in experiments using parametric down conver-
sion pairs generated with CW light (Keller and Rubin (1997); Kim et al. (2000a,b)).
The correlations are inherently different, because in the photons coming from pulsed
sources the time uncertainty for the pair generation is reduced. On the other hand,
the spectrum of short pulses is considerably wider than that of CW sources, so there
is a new component of uncertainty about the frequency characteristics of the gen-
erated pair. The energy of the photons is correlated by energy conservation, but as
with CW sources where there is a big uncertainty about the time of emission, in these
pulsed sources there is a big uncertainty about frequency of the photon. Usually, in
the experiments with pairs generated from pulsed sources, the introduction of nar-
row bandpass filters is necessary to ensure that the frequency uncertainty is reduced.
In many of the first experiments performed, this produced problems such as having
weaker signals and less visibility in the correlation measurements (Grice and Walm-
sley (1997)). However, some techniques and improvements to the entangled photon
sources have been implemented to reduce this problem (Kim et al. (2000a)).
There have been other reported proposals and experiments in the field of quantum
cryptography. A proposal for quantum cryptography and detection of eavesdropping
based on the frequency modulation of weak light pulses was described by Klyshko
(1997). A very similar idea was then experimentally tested by Merolla et al. (1999).
This experiment consisted in the implementation of quantum cryptography using
a 1540 nm source (typical optical telecommunications wavelength) using a pair of
phase modulators in the emitter and receiving ends to create interference between the
phase modulated bands of a CW source. This group of researchers also described a
possible protocol for quantum key distribution using this setup. Another experimental
proposal for an interferometric technique for long-distance cryptography based in
the frequency division of two waves using acousto-optic modulators (AOMs) was
developed by Sun et al. (1995). This opens the possibility of introducing the EOFS
for such systems in which frequency shifting can be part of the cryptographic encoding
77
or key distribution protocol. One advantage of the EOFS over AOMs would be the
fact that the frequency shifting can be controlled dynamically by the power and phase
of the driving microwave. In addition, the range of the shifting doesn’t depend mainly
on the modulation frequency but on the power. Experiments with entangled photon
pairs in which this dynamic possibility can be incorporated into a dynamic type of
entanglement seems possible. It is important to note that this idea goes together
with the fact that the signals used should be pulsed. In this way, the condition of
the pulse localization for a convenient operation of the EOFS is still valid. In a CW
regime, the EOFS should have effects very similar to the AOM, but maybe with the
possibility of higher modulation rates (frequencies) that can be on the order of tens
of GHz, while with the AOM the working ranges maybe go up to the MHz range.
The introduction of frequency shifting encoding for quantum cryptography and
quantum communications is very appealing because of the fact that frequency en-
coded signals are more robust when dealing with long distance information transfer.
Maintaining polarization encoded states through miles of fiber or free space links is
a very difficult task, and it has been clearly shown that the performance of these
communication channels is severely compromised by distance, and even climate con-
ditions. A frequency encoded message can help solve the problem of the polarization
stability, since the signals can be transmitted for longer distance without them drift-
ing in frequency. With polarization encoded states there is much more instability in
this communication process, and thus the reliability of the system is hindered. The
EOFS could be, for example, part of an “entanglement transfer” mechanism in the
emitting and receiving ends of a quantum information channel. In this quantum com-
munication channel, the entanglement could be transferred from polarization encoded
states to frequency encoded states. In this respect, the polarization selectivity of the
EOFS could be advantageous, since only light in a specific polarization (i.e., z polar-
ization in a z-cut device) would be shifted, while the orthogonal component would
remain the same. Typical LNO EOFS devices only transmit one type of polarization,
so a scheme that incorporates one EOFS for each different state polarization should
be implemented depending on the frequency shift requirements. Recently, there has
a been great advance in developing LNO devices that transmit both polarizations
(Asobe et al. (2002)), while only one polarization is the one that experiences the
strong electro-optic effect. An EOFS made with this type of optical waveguide tech-
nology would be very useful because it would transmit all photons, only introducing
the frequency shift for the ones in a specific polarization state.
78
Chapter 6
Summary and Discussion
In this work, a novel all-optical device for frequency shifting short optical pulse trains,
similar to the ones used in Return to Zero encoding formats in Wavelength Division
Multiplexing and Dense Wavelength Division Multiplexing systems, has been pre-
sented. The basic operation principle of the Electro-optic Frequency Shifter (EOFS)
has been described with a simple physical model of the electro-optic effect. After
this, a coupled mode analysis based on a semiclassical three wave mixing formalism
was used to study the nonlinear interaction between the light and the microwave
field. From this model, the evolution of the pulse spectrum as it propagates through
the device was calculated numerically. Predictions for the expected frequency shift
caused by the EOFS operation were obtained from these numerical simulations. In
addition, spectrum distortion effects, which can appear due to the increment of power
input, interaction length, and the relative ratio between the pulse and the microwave
length, were studied with this model. From the simulation results, it was shown how
a good design criterion appears to be that the length of the optical pulse needs to be
less than 1/8 of the microwave wavelength.
The next step in the investigation was to perform experimental measurements to
compare with the predictions of the semiclassical model. The results of these exper-
iments were shown in chapter 4. The simulated and experimental data show that
the overall frequency shifting achieved in these EOFS structures is related linearly to
the microwave field amplitude that is applied (which means it is proportional to the
square root of the power). This makes the EOFS considerably different from other
modulation or frequency conversion schemes, in which the shifted frequencies or side-
bands depend on the microwave frequency. In the EOFS, the microwave power is the
knob that can be used to control the amount of shifting that is obtained. Looking
at the model and the expressions found in chapter 2, it is shown how the frequency
79
shifting of the pulse is also proportional to the original frequency (i.e., central pulse
frequency). However, an effect on shifting due to different central optical frequencies
(wavelengths) would only be clearly noticed if the frequencies (wavelengths) of the
different pulses under consideration differ by a large amount. In other words, the
relative frequency shift is a small fraction of the central frequency (hundreds of gi-
gahertz compared to hundreds of terahertz). This shows that, within a small range
of wavelengths or frequencies in a system, the frequency shifting of different nearby
channels will be homogeneous, for practical purposes. This is of great importance for
applications where there could be a need to shift various different signals by the same
amount.
It was also evident that some distortion in the frequency spectra is also present
when the shifting is performed, and that this distortion became more noticeable when
the range of the shifting was increased, as the distortion accumulated. This led to
an investigation of the distortion effect related to both microwave power and pulse
length characteristics in the model and in the experiment. The experimental obser-
vations indicated that the spectrum distortion, shown in the pulse after the shifting
operation, is strongly dependent on the pulse shape and length characteristics. Pulses
that are too long and span a considerable fraction of a microwave period have the
problem of different portions of the pulse experiencing different index gradients. This
problem can be even worse when there is a velocity mismatch added to the situation.
The experiments also showed a weak temperature dependence of the operation of
the EOFS. However, this distortion was not quantified. The ohmic heating, due to
the amplification of the microwaves required to produce a sufficiently strong electric
field, produces a change in the index of refraction of the nonlinear material which
is in contact with the termination load and also on the material near the microwave
electrodes. When the device was operated for extended periods of time and the tem-
perature change was noticeable, some distortion effects were noted, especially when
injecting high levels of power to the device. This effect was not quantified specifically,
even though it could be possible to study with the adaptation of a good temperature
control and measurement system and a feedback loop for temperature stability. It
would be interesting to study the effects of an overall crystal temperature change,
and to see if these effects are related to more local heating effects near the optical
waveguide of the microwave structure.
Having looked at the distortion effects from the experiments, the next step was to
identify ways to better control the extent of the distortion, and to thereby improve
the long range shifting capability of the device. One possibility was to obtain ma-
80
terials with a higher electro-optic coefficient so that lower fields can be used. This
would improve the performance of the device by reducing either power consumption
or the typical interaction length needed to achieve the required frequency shifting.
This is one of the reasons the current research on novel and improved non-linear
materials is so important for the field of electro-optic devices and their applications
for optical telecommunication systems, optical switching, and distributed comput-
ing networks. Devices with reduced length minimize fabrication processing time and
errors that lead to misalignment, velocity mismatch, coupling losses, and material
losses. Reduced dimension devices can make integration with other elements easier
when thinking about network operation. This is also good for reduction of costs in
any manufacturing process. However, the search for new materials was out of the
scope of this investigation, so we looked into other types of solutions.
The next potential candidate for the improvement of the device operation was the
design of the device itself. In the literature review of previous work, and in the tech-
niques studied during the development of this research, we have seen a great variety in
the design of electro-optic devices and electro-optic modulators. Some of these works
deal with the microwave electrode structure design and the optical waveguide design
and processing. Some of the problems addressed in these designs are the velocity and
impedance matching techniques, improvement of the optical and microwave fields’
coupling to reduce the power requirements, and the search for increased modulation
bandwidths. A few of these works are related proposals for microwave excitation
techniques. Integrated microwave engineering is a field that has had a very strong
development in the past decade, in great part due to the increasing demand of the
new wireless systems and military or defense applications. This intense research and
development of very high speed integrated circuits, including detectors, high stability
oscillators, filters, amplifiers, and other state of the art devices, has permitted great
advances in today’s new technology. With this in mind, the design of improved mi-
crowave driving circuits to obtain better microwave excitation waveforms is a more
realistic goal. As discussed in chapters 2 and 4, it is conceivable to have a constant
gradient of the index of refraction wave in a larger portion of one period (i.e., using
triangular wave forms instead of sinusoidal), so that this can reduce the constraints
on the pulse lengths that can be used. This microwave supply can, in principle, be
merged with a phase or a sampling clock recovery subsystem that will serve as a
mechanism to achieve the phase-locking needed to synchronize the operation of the
microwaves to the bit rate of the system. The main challenge of this idea is the
generation of waveforms with enough voltage swing to be able to get the high fields
81
required. Low amplitude signals with this type of waveform can be easily synthesized
in commercially available microwave synthesizers. However, the addition of large
bandwidth amplifiers to amplify the high harmonic content of these waveforms is
difficult. Problems that may exist in this stage, such as a non-flat gain profile of the
amplifier or the different behavior of the microwave structure at different frequencies
(microwave structure S11-reflected power- and losses behavior are frequency depen-
dent), can be compensated on the microwave source end so that the overall result of
the applied field is the one desired.
Another possibility for improving the device operation, and increasing the volt-
age/field capacity that can be supplied to the device, is to amplify only the microwave
during the period of time when the pulse is present. A low duty cycle of microwave
power can be used so that high fields can be applied without running into the prob-
lems of overheating the EOFS structure. The resulting temperature problems would
cause increased aging and deterioration of the devices or index matching problems as
mentioned before. This can be done by an amplitude modulation of the microwave
signal before the amplification. The microwave can be enhanced for a fraction of the
time when the pulses are propagating through the device. For the typical pulses used
in the experiments in this work (10 to 20 ps long), and using the fact that the repeti-
tion rate is 76 MHz (about 13 ns between pulses), it is seen that the microwave field
really needs to be on about 0.15% of the time. This gating of the microwave field,
provided that there is a good phase-locking stability, can result in a great advantage
when dealing with power consumption. On the other hand, this duty cycle ratio will
decrease eventually as the bit rate is increased, so this option would not be as effective
in controlling the heating when going to higher bit rates. For example, a 10 Gbit/s
rate would have an inter-pulse spacing of 100 ps, so that with 10 ps long pulses, the
microwave power needs to be on for more than 1/10 of the time.
In the experiments performed, it was also observed that the power supplied to the
EOFS could produce heating effects leading to spectrum broadening and distortion.
This is basically due to velocity mismatch induced by the thermal effect on the index
of refraction. This problem was more noticeable under certain conditions, especially
when dealing with high power and extended time operation. A simple technique for
trying to deal with this problem is to introduce a bias DC field to the modulator.
This is simply done by the adding a Bias Tee that couples the amplified microwave
power to a DC voltage before injecting it into the EOFS. The effect of the DC field is
to shift the index of refraction experienced by the optical pulse by means of the same
electro-optic effect in the nonlinear material. This is done in such a way that the
82
velocity matching between the optical pulse and the oscillatory part of the electric
field (coming from the amplified microwave) is achieved or at least improved. As
the temperature of the crystal changes due to the ohmic heating of the microwave
structure and load resistance,the DC bias can be adjusted to maintain the velocity
matching, and thus reduce the pulse spectrum distortion. Some simple feedback loop
can be added to this DC bias control to make the control automatic.
One idea that was identified for improving the frequency shifting operation was
to cascade the EOFSs one after the other. Since the frequency shifting scales linearly
with the length of the device (in this case related to the number of EOFSs), but only
with the square root of the power, it is preferable to cascade a couple of EOFSs,
and apply half-power to each, rather than to use only one with the total power. This
approach introduced the extra difficulty of having to control the microwave phase-shift
for the two EOFSs independently. By cascading two devices, shifts of over 400 GHz
were obtained, showing that the scalability of this device in order to reach the THz
regime is a very realistic goal. Improvements that can be made to achieve this goal
include increasing the length of one device, using a better nonlinear material (with
higher effective electro-optic coefficients such as the recent interest in new electro-
optic polymer materials), improving the microwave structure design to increase the
effective coupling between the optical and microwave fields, and reducing the velocity
mismatch (pulse distortion reduction) and the impedance mismatch (to reduce power
losses). Reducing the pulse distortion related to the velocity mismatching and the
pulse length to microwave wavelength ratio is crucial. This will permit an increase of
the frequency shifting obtained from a single EOFS by increasing the driving power.
Reasons for not using high powers for extended periods of time during our experiments
were the increased distortion effects induced both by the strong driving and the ohmic
heating. However, these kinds of effects related to high voltage application could be
minimized by using longer devices or cascading a couple of devices. If, on top of
that, a correction of the microwave field to achieve a better index gradient profile is
included, then higher powers could be used in a pulsed or gated manner to improve
the frequency shifting range that can be obtained.
The EOFS simplicity, both in fabrication and operation, makes it a very inter-
esting device for applications in optical telecommunication networks and distributed
computing systems. The ability to introduce an arbitrary (within some limits) fre-
quency shift to a pulse in a very fast, accurate, and dynamic way, makes this device
very flexible. Since the process is all optical, the shift can be performed in the transit
time through the device without a bottle neck as in other optoelectronic conversion
83
schemes. It also has the advantage of not requiring second optical sources that would
have to be integrated onto the device, introducing unavoidable problems with sta-
bility and noise. If the driving circuitry that controls the microwave can be made
fast enough, different pulses from the same train could ultimately be shifted in dif-
ferent amounts as part of a very efficient routing mechanism. A great part of today’s
optical communication networks use WDM and DWDM systems, because in past
years TDM technology was not so easily implemented. However, recent advances in
high speed integrated circuits (including phase and clock recovery subsystems, am-
plifiers, filters, and other relevant devices) and the improvement of new solid state
laser sources, modulators, and detectors, may prove significant for the development
of hybrid TDM-WDM type of networks. In these hybrid systems, the advantages
of short pulses can be used along with the multiple wavelength channel capacity to
make these networks faster without compromising their reliability.
One last topic mentioned in this work is the one of Quantum Information Science.
The introduction of the EOFS to systems for quantum encryption or quantum com-
munication channels could help in the creation of new encoding protocols, improving
the performance and/or reliability of links, and in helping to study of the effects of
dynamic transformations in entangled photon states. Some work has been done both
theoretically and experimentally in this area, as was shown in chapter 5. The idea of
controlling the frequency in single photon states may prove to be a very useful tool.
Since the operation of the EOFS is related to the idea of being able to “localize” the
photons in such a way as to control the phase relationship with the microwave field,
the use of the EOFS along with entangled photons from pulsed sources seems possible
in a near future.
84
Appendix A
A Review of Nonlinear Optics
A.1 Basic Definitions
In classical (linear) optics, the polarization or dipole moment per unit volume is
related to the applied optical field by the simple equation presented by Boyd (1992)
as
P (t) = χ(1)E(t), (A.1)
where χ is known as the linear susceptibility. In order to include nonlinear effects,
this equation is generalized by expressing the polarization as a power series expansion
with the intensity of the field as
P (t) = χ(1)E(t) + χ(2)E2(t) + χ(3)E3(t) + ..., (A.2)
where χ(2) is the second order non-linear optical susceptibility. Similarly, higher order
susceptibilities may be defined. The polarization is the key element for nonlinear
optical interactions because when it is a time varying component it becomes a source
for the wave equation in the nonlinear optical media. This is a inhomogeneous wave
equation in which the polarization will drive the electric field. This, in another way,
can be seen as the existence of charges that are being accelerated in the material that
consequently will generate radiation (Boyd (1992)).
Boyd also introduces a more formal definition of the nonlinear susceptibility for
materials with dispersion and loss, representing the full vector field and nonlinear
susceptibility as complex quantities. In Chapter 2 of Boyd (1992) we find a general
form of the wave equation in nonlinear optics
∇×∇× E +1
c2
∂2
∂t2E =
−4π
c2
∂2P
∂t2, (A.3)
85
where the tilde is used to denote a quantity that has a fast variation in time. On the
other hand, the polarization and displacement field terms can be divided into their
linear and nonlinear parts and the equation is transformed to
∇×∇× E +1
c2
∂2
∂t2D(1) =
−4π
c2
∂2PNL
∂t2. (A.4)
Here P = P (1) + PNL, D = D(1) + 4πPNL, and D(1) = E(1) + 4πP (1). Boyd shows
that in a dispersive medium the electric, polarization and displacement fields can
be expressed in terms of different frequency components. Neglecting dissipation, the
relationship between the displacement vector and electric field can be expressed in
terms of a real dielectric tensor. With this, a wave equation that is valid for each of
the frequency components of the field is obtained from (A.4)
∇×∇× Em +ε(1)(ωm)
c2
∂2Em
∂t2=−4π
c2
∂2PNLm
∂t2. (A.5)
Now a very well known vector identity, ∇×∇× E = ∇(∇ · E)−∇2E, can be used.
In the majority of cases the first term in the right hand side of this equation can
be replaced by −∇2Em because the term ∇ · E vanishes or can be shown to be very
small. This leads to the equation (2.15) which is the basis of the semiclassical analysis
of the Electro-optic Frequency Shifter presented in section 2.2.
A.2 Coupled-mode Wave Equation Formalism for
Mode Evolution
Consider a lossless medium, where monochromatic input beams are incident normally
for simplicity. The wave equation (A.5) must hold for all the frequency components of
the light. In particular we can describe the sum-frequency generation process where
two monochromatic input waves are combined together to produce a new frequency
by the nonlinear interaction in the material. This process is illustrated in Fig. A.1.
The waves at ω1 and ω2 enter the nonlinear material and a new wave at ω3 = ω1 +ω2
is generated. This new field can be expressed as a plane wave when the nonlinear
source term is relatively small.
E3(z, t) = A3ei(k3z−ω3t) + c.c., (A.6)
where k3 = n3ω3/c, and n3 = [ε(1)(ω3)]1/2. If the nonlinear interaction is not present
the A3 coefficient will be constant. For this small nonlinear term case, the solution
86
ω1
ω2
ω3 = ω1 + ω2
L
d = ½ χ(2)
Figure A.1: Sum frequency generation.
can still be of the plane wave form, but allowing the field amplitude to be a slowly
varying function of z as it propagates. The nonlinear polarization induced by the
material interaction with the waves will act as a source term. It can be expressed as
P3(z, t) = P3e−iω3t + c.c., (A.7)
where the polarization coefficient is related to the fields by the effective second order
susceptibility as P3 = 4deffE1E2 (Boyd (1992)). The applied fields are similarly
expressed as the sum frequency field above, but we separate the time and space
variation parts for calculation purposes.
E1(z, t) = A1ei(k1z−ω1t) + c.c. = E1e
−iω1t + c.c.. (A.8)
The expression for E2 is similar. Now the expression for the polarization term and the
field components are substituted into equation (A.5). Using the fact that for this case
∇×∇× ≈ −∇2, and the fields only depend spatially on z (so that∇ ' ∂/∂z ' d/dz),
a differential equation for the evolution of the mode amplitude A3 can be obtained:
d2A3
dz2+ 2ik3
dA3
dz=−16π
c2deffω2
3A1A2ei(k1+k2−k3)z. (A.9)
To simplify this equation, the slowly-varying-amplitude approximation is used. This
can be done in this case because d2A3
dz2 ¿ k3dA3
dz, so the second order derivative term
negligible with respect to the first order derivative term. The equation is simplified
to
87
dA3
dz=
8πideffω23
k3c2A1A2e
i(k1+k2−k3)z. (A.10)
Similarly, equations for the evolution of the amplitudes of the other modes can
be obtained (Zernike and Midwinter (1973); Boyd (1992)). The introduction of the
quantity ∆k = k1 +k2−k3, simplifies the expressions for the mode evolutions, leading
to the expressions
dA1
dz=
8πideffω21
k1c2A3A
∗2e−i∆kz, (A.11)
and
dA2
dz=
8πideffω22
k2c2A3A
∗1e−i∆kz. (A.12)
A.2.1 Manley-Rowe Relations
The intensity of the optical fields at different frequency components is given by
Im =nmc
2π|Ai|2, (A.13)
where m can be 1, 2 or 3 depending on the frequency component we refer to. Differ-
entiation of the previous equation with respect to z leads to
Im
dz=
nmc
2π
(A∗
i
dAi
dz+ Ai
dA∗i
dz
). (A.14)
The equations (A.10-A.12) and (A.14) are then used to obtain the spatial variation
of the intensities of different frequency waves as done in Boyd (1992), leading to
dI1
dz= −8deffω1Im(A3A
∗1A
∗2e−i∆kz), (A.15)
dI2
dz= −8deffω2Im(A3A
∗1A
∗2e−i∆kz), (A.16)
and
dI3
dz= 8deffω3Im(A3A
∗1A
∗2e−i∆kz). (A.17)
The total intensity variation is given by the sum of the three intensity variation
components. It is clear that the sum of these three quantities will be a constant
factor multiplied by (−ω1 − ω2 + ω3), which is equal to zero from the energy conser-
vation condition. This means that dI/dz = 0 for the entire system. From this same
conditions, the Manley-Rowe relations are obtained
88
d
dz
( I1
ω1
)=
d
dz
( I2
ω2
)= − d
dz
( I3
ω3
)(A.18)
These relationships show that for the case of the sum frequency generation the
power flows from one mode to another without being lost. More information about
the quantum nature of the process is obtained as the above equations imply that the
rate at which photons of the first mode (ω1) are created/destroyed is equal to the rate
for the second mode (ω2), and it is equal to the rate at which the photons for the third
mode (ω3) are destroyed/created respectively. This three wave mixing process thus
implies that there exists a photon number conservation relationship because for each
photon created at ω3, there is a need to destroy one photon at ω1 and one photon at
ω2. Similarly the annihilation of a photon at ω3 requires the creation of the other two
photons. The process of Difference Frequency Generation can be modelled similarly
and the corresponding power flow relationships can be obtained with similar results.
This process corresponds to the creation of a photon of frequency ω3 = ω2 − ω1.
The process described above is the type of photon creation/annhilation process
that is used to model the operation of the EOFS in chapter 2. The EOFS in principle
can be modelled by many of these three-wave mixing events, one after the other.
The frequencies of the allowed optical modes are the frequency components of the
mode locked pulse spectrum. The three wave modes for each process are related
to two neighbor modes of the optical spectrum and the third mode is related to
the microwave field. In this way, photons in the optical spectrum are shifted from
one mode to the other by the addition or substraction of a photon from the electric
field (sum frequency or difference frequency generation), depending on whether the
configuration is for up-shifting or down-shifting. This process is repeated various
times as the optical pulse is travelling through the device, between all the different
modes in the optical spectrum whose frequency difference matches the microwave
photons energy. The combined effect of many of these transitions leads to the final
frequency shifting of the spectrum in a continuously tunable, fast and efficient way.
89
Appendix B
The Electro-optic Effect
Most of the material in this appendix follows the treatment and notation used by
Boyd (1992), Yariv (1975), and Zernike and Midwinter (1973).
The electro-optic effect is the physical phenomenon in which an index of refraction
change is induced in a material by the application of a DC or low frequency field (with
respect to the characteristic material response time). For the particular case of the
EOFS the refractive index of the material depends linearly on the applied field. This
is called the linear electro-optic effect or Pockels effect. Similarly to what has been
done in Appendix A the nonlinear polarization can be expressed as
P(r, t) =∑
n
P(ωn)e−iωnt, . (B.1)
With this in mind the, linear electro-optic effect can be related to the components
of the second order susceptibility tensor using them as the constants that relate the
polarization to the electrical field amplitudes in the following way (Boyd (1992);
Zernike and Midwinter (1973))
Pi(ω) = 2∑
jk
χ(2)ijk(ω = ω + 0)Ej(ω)Ek(0). (B.2)
This means that the effect can only be seen in non-centrosymmetric materials because
they have non-vanishing second order susceptibility as contrary to centrosymmetric
media. In the latter, the lowest order change in the index of refraction would depend
quadratically on the field (quadratic electro-optic or Kerr effect).
Generally, the electro-optic effect is studied from the formalism that gives the ten-
sor relationship between the electric and displacement fields in anisotropic materials.
The displacement vector is given by
90
Di =∑
j
εijEj, (B.3)
where the dielectric tensor is a real symmetric matrix for the case of lossless and non-
optically active materials. This tensor has a matrix representation that, in principle,
can be diagonalized in a new principal axis system. The energy density in this new
system is given by
U =1
8πD · E =
1
8π
[ D2X
εXX
+D2
Y
εY Y
+D2
Z
εZZ
]. (B.4)
With this expression, the surfaces of constant energy density are clearly ellipsoids.
With a change of variable X =(
18πU
)1/2
DX (and similarly for the Y and Z axes) the
widely known index ellipsoid is defined (Yariv (1975); Boyd (1992))
1 =X2
εXX
+Y 2
εY Y
+Z2
εZZ
(B.5)
The index ellipsoid can be described in any other coordinate system where the form
will be slightly more elaborate and will contain coefficients that describe the optical
indicatrix of the material as follows
( 1
n2
)1x2 +
( 1
n2
)2y2 +
( 1
n2
)3z2 + 2
( 1
n2
)4yz
+ 2( 1
n2
)5xz + 2
( 1
n2
)6xy = 1
(B.6)
Using a procedure given by Born and Wolf (1975), the description of the optical
properties of an anisotropic material is simplified greatly. Using a given propagation
within the crystal, the plane perpendicular to it and passing through the ellipsoid
center is drawn. The semi axes of the ellipse that results from the intersection of
this plane with the ellipsoid, will define the indices of refraction for the two allowed
polarizations in that given direction of propagation. The orientations of these axes
also coincide with the displacement vector components. The optical indicatrix coef-
ficients are important to study the electro-optic effect when the crystal is placed in
a constant or low frequency field. For this purpose the impermeability tensor, which
corresponds to the inverse of the dielectric tensor, is defined
Ei =∑
j
ηijDj (B.7)
With this definitions, an expression for U in terms of D and the new tensor can be
derived. This will lead to a new form of the index ellipsoid. Comparing the coefficients
one by one it is shown that
91
( 1
n2
)1
= η11,( 1
n2
)2
= η22,( 1
n2
)3
= η33,( 1
n2
)4
= η23 = η32, (B.8)( 1
n2
)5
= η13 = η31,( 1
n2
)6
= η12 = η21.
The components of the impermeability tensor can be expressed as power series in
the electric field components as in Boyd (1992)
ηij = η(0)ij +
∑
k
rijkEk +∑
kl
sijklEkEl + ..., (B.9)
where the tensor rijk represents the linear electro-optic effect. Clearly, the sym-
metry of the impermeability tensor will make the electro-optic tensor symmetric
in the first two indices. This has led to a widely used convention of using con-
tracted notation for the electro-optic tensor using a two dimensional matrix rhk,
where the indices h = 1, 2, 3, 4, 5, 6 have a correspondence with the pairs ij =
11, 22, 33, 23 or 32, 13 or 31, 12 or 21. With this notation in mind the lowest (first
order) correction of the optical indicatrix constants, which is related to the linear
electro-optic effect can be written
∆( 1
n2
)i=
∑j
rijEj. (B.10)
For the case of Lithium Niobate, the crystal is of class 3m and the electro-optic
tensor is given by
rij =
0 −r22 r13
0 r22 r13
0 0 r33
0 r42 0
r42 0 0
−r22 0 0
(B.11)
where r13 = 9.6, r22 = 6.8, r33 = 30.9, and r42 = 32.6 in units of (10−12 m/V).
B.1 The Electro-optic Coefficient and the Effec-
tive Second Order Susceptibility
The electro-optic effect in the development of the coupled mode analysis is studied
from the point of view of a frequency mixing interaction between an optical field and
92
a low frequency component electric field. Thus, it is important to know the rela-
tionship between the electro-optic coefficient (which is what is commonly known and
measured) and the second order susceptibility that creates the wave mixing interac-
tion. In a first approximation the quasi-static case can be considered (i.e. considering
the low frequency field to be at zero frequency), which is what is typically done to
study the electro-optic effect in modulators. In the case of the EOFS, the structure
is really a traveling wave phase modulator but for practical purposes the quasi-static
approximation is good enough.
Consider the sum frequency analysis in appendix A. Taking A1 as the optical
wave, A2 as the applied electric field and A3 as the generated wave amplitudes, and
considering a velocity matching condition, the three-wave mixing (SFG) formalism
can be generalized to write the evolution of the A3 amplitude from equation (A.10)
as
dA3
dz=
8πideffω23
k3c2A1A2. (B.12)
Introducing the frequency dependence and considering a crystal of length L, and
slowly varying amplitudes, this equation can be integrated and transformed to
A3e−i(ωt−φ) =
8πiLdeffω23
k3c2A1A2e
−iωt, (B.13)
where the phase factor φ accounts for the dephasing between the incoming and the
generated wave. Using the relationships ω = ck and k = 2πn/λvac, where λvac is the
vacuum wavelength, this equation can be written as
A3e−i(ωt−φ) =
16π2L
nλdeffA1A2e
−i(ωt−π/2). (B.14)
From this equation it is clear that the nonlinear interaction via the second order
susceptibility creates the wave amplitude A3 and and introduces an extra 90 degree
phase delay. On the other hand, equation (B.10) can be used to obtain the index
variation in the electro-optic effect leading to
∆n =−n3rA2
2(B.15)
If we are in a crystal of length L, this index of refraction difference will produce a
phase change ∆φ = −πn3rA2L/λ. The phase change produced by the electro-optic
effect and its relation to electro-optic modulation has been studied previously by
Yariv (1997). This phase change should be the same as the one between the A3 and
A1 components in equation (B.14). Since A3 ¿ A1, then
93
∆φ ' |A3||A1| =
16π2L
nλdeffA2 =
−πn3rA2
λ, (B.16)
which leads us to the expression relating the electro-optic coefficient and the effective
second order susceptibility used in section 2.2.
r = −16π
n4deff , (B.17)
94
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Vita
Darıo Farıas was born on September 14, 1973, in Bogota, Colombia. He receivedB. Sc. degrees in Physics and Electrical Engineering from Universidad de Los An-des (Bogota, Colombia) in 1997, with a thesis entitled “Study of nonlinear opticalphenomena in the light-matter interaction using Floquet Theory”, done under theguidance of Prof. Ferney R. Rodrıguez. In 1997 he joined the graduate program inPhysics at the University of Illinois at Urbana-Champaign, where he completed theMasters degree in Physics in May 2000, under the supervision of Professor JamesN. Eckstein. His research has been related to non-linear optics, optoelectronics, andelectro-optic frequency conversion techniques. In January of 2004 he will join thePortland Technology Development Group at Intel Corporation, where he will workon the development of the lithographic techniques for the next generation processors.
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