active modelocking of an open-cavity helium-neon … modelocking of an open-cavity helium-neon laser...

Active Modelocking of an Open-Cavity Helium-Neon Laser

A Thesis

Presented to

The Division of Mathematics and Natural Sciences

Reed College

In Partial Fulfillment

of the Requirements for the Degree

Bachelor of Arts

Sam W. Spencer

May 2010

Approved for the Division(Physics)

Advisor L. Illing

Acknowledgements

I want to begin by thanking Lucas Illing for his continual guidance, clarifications,arbitration and reassurance during the chaotic and humbling thesis process, and forhelping me remember that not everything needs explanation. There are so manyothers who deserve acknowledgement: David Griffiths, for showing me that clarity ofdescription begets clarity of understanding; Nicholas Wheeler, for teaching me thatphysics can be communicated eloquently; Joellyn Johnson, for teaching me how tomove on; Robert Knapp, for letting me realize that an essay is not a proof; MaryJames, for introducing me to wonderful truths; Chris Thoen, for helping me feelconfident; and the 2006 MacNaughtonites and the 2010 physics students for end-less friendliness. I want to thank my parents and sister for their conversations andsupport. Most importantly, I thank Elisabeth Hawks for making me happy and forunconditional love.

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter 1: Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 Laser Resonator Stability . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Helium-Neon Gain Medium . . . . . . . . . . . . . . . . . . . . . . . 111.3 Acoustooptic Modulator . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.1 Standing vs. Traveling Acoustic Wave AOMs . . . . . . . . . 131.3.2 Diffraction Regimes and Traveling Wave AOM Theory . . . . 14

1.4 Time-domain modeling: Pulse Propagation . . . . . . . . . . . . . . . 191.4.1 Gaussian Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . 191.4.2 Pulse Propagation . . . . . . . . . . . . . . . . . . . . . . . . 20

1.5 Injection Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.5.1 Locked Regime: ωD ≤ ε . . . . . . . . . . . . . . . . . . . . . 261.5.2 Unlocked Regime: ωD > ε . . . . . . . . . . . . . . . . . . . . 261.5.3 Consequences of Injection Locking . . . . . . . . . . . . . . . . 26

Chapter 2: Experimental Design . . . . . . . . . . . . . . . . . . . . . . . 29

Chapter 3: Results and Discussion . . . . . . . . . . . . . . . . . . . . . 353.1 Unmodulated Operation . . . . . . . . . . . . . . . . . . . . . . . . . 353.2 Approaching Modelocking . . . . . . . . . . . . . . . . . . . . . . . . 413.3 Modelocking Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Abstract

In this thesis, we investigate the modelocking of a laser, which refers to the simulta-neous excitation of multiple modes of light within the laser cavity, and the establish-ment of a constant uniform phase difference between pairs of these modes, resultingin the formation of laser pulses. We discuss the physical mechanisms that lead tomodelocking, beginning with descriptions of the necessary laser components and pre-senting a mathematical model for propagating a Gaussian laser pulse through thesecomponents. We also discuss the nonlinear phenomenon of injection locking and therole it plays in making modelocking an experimental reality. We then experimen-tally demonstrate modelocking using a Helium-Neon laser tube with external cavitymirrors and an acoustooptic modulator. The laser is found to produce pulses of ap-proximately 2 nanoseconds in duration, at a repetition rate of about 160 MHz andan average output intensity on the order of 110 nanowatts. We present results of theexperiment, including light-intensity timeseries, optical spectra, and radio frequencyspectra of the beat frequencies between modes, to characterize modelocking behaviorand the injection locking process leading to modelocking.

Introduction

Generating short pulses of light from lasers has been an area of intense interest sincethe early 1960s, soon after the laser was invented. Short, intense laser pulses haveacquired hundreds of applications in both industry and research roles. Examples inindustry include laser ablation and micromachining [Clark-MXR, 2009], refractiveindex variation including lens aberration correction in the human eye [Ding et al.,2009], high speed fiber optic telecommunications [Wada, 2004], and the generation ofterahertz radiation for security screening [Federici et al., 2005]. Short laser pulses arealso essential in research projects like nonlinear optics studies [Boyd, 2008], inertialconfinement fusion [Pfalzner, 2006], high-harmonic generation [Ferray et al., 1988],and the time-resolved spectroscopy of chemical reactions [Manescu, 2004].

Many different methods have been developed for extracting short pulses from alaser, including Q-switching, gain switching, and cavity dumping, each of which relieson electronic modulation to quickly regulate the output power of the laser [Siegman,1986]. These techniques typically generate pulses on nanosecond (10−9 s) or picosec-ond (10−12 s) timescales and rely on the speed of variation of an electrical signalto form pulses. There is another method, called modelocking, that takes advantageof the fact that a laser cavity can naturally support multiple frequencies of light si-multaneously, and these frequencies, if correctly superimposed, can beat together toform short pulses. First discussed by Lamb [1964] and first implemented by Hargroveet al. [1964], modelocked lasers have become the premier choice for the generation ofshort, intense laser pulses. Modelocked lasers usually generate ‘ultrafast’ pulses onthe timescale of femtoseconds (10−15 s), with the current record held at pulses under10 femtoseconds.

This thesis documents an experimental attempt to construct a modelocked laserusing a Helium-Neon laser tube and free-space optical components. We will begin byintroducing the theory of modelocking and give theoretical details about the compo-nents with which we will assemble our laser. We will then motivate the design of theexperiment itself. Finally, we will present data obtained while operating the laser,and analyze our results.

Chapter 1

Theory

This chapter gives an outline of the theory of laser modelocking and an introductionto the experimental apparatus, including the mathematical description of its com-ponents. Thus we will begin with a general description of lasers and the physicalprinciples behind modelocking, and outline the two broad types of modelocking.

Gain Medium

Mirror Cavity

Pump Process

Figure 1.1: The schematic for a generic laser, including the three main components and asingle standing-wave beam of light.

Every laser requires three main components: a gain medium, an energy pumpingprocess, and an optical cavity. A gain medium is a substance which is quantum-mechanically excited into higher energy levels. This energy is released as light, eitherrandomly in the process of spontaneous emission, or by stimulated emission due tothe passing of light through the substance. Stimulated emission generates photonsthat have both the same phase and same frequency as the stimulating photons. Anoptical cavity, also called a laser resonator, is a set of mirrors that surrounds the gainmedium and reflects light back into it. If this reflected light encounters the excitedportion of the substance, it can stimulate the emission of more light and generatea coherent beam. However, if the light instead encounters an unexcited portion ofthe substance, the photons may simply be reabsorbed. In order to ensure that morelight is spontaneously emitted than absorbed, an energy pumping process is used toexcite as much of the substance in the gain medium as possible.

Most lasers operate in the ‘continuous wave’ (CW) regime, where operation in the

4 Chapter 1. Theory

Powe

r

Frequency

Powe

r

Frequency

Laser GainCavity Loss

Cavity longitudinal mode structure

Powe

r

Frequency

Output Spectrum

(a)

(b)

(c)

Figure 1.2: (a) The laser cavity supports an infinite Dirac delta frequency comb. (b) Thegain of the laser gain medium (solid curve) has a Lorentzian or Gaussian shape, and mustovercome the cavity losses (dashed curve) for lasing to occur. (c) Overlaying the regionwhere gain is greater than loss onto the Dirac comb, we end up with a series of narrowemission bands bounded by a wider envelope.

steady state implies generating a beam with constant frequency and intensity. Thelight emitted by a CW laser is ideally monochromatic. The frequency range over whichmost gain media amplify, called the gain bandwidth, is usually large enough suchthat multiple closely-spaced frequencies may oscillate simultaneously. Imagine a lasercavity of optical path length L, formed by two inward-facing highly reflective mirrors.This cavity will support standing light waves of certain evenly-spaced wavelengths,given by L = mλ

2, where m is a positive integer. The frequency corresponding to

this standing wave is ν = mc2L

, implying that the difference in frequency between theadjacent m and m+ 1 modes is given by

∆ν =c

2L. (1.1)

This forms an evenly-spaced comb of Dirac delta peaks in frequency space, represent-ing ‘axial modes’ that will be supported by the cavity. We thus call ∆ν the ‘axialmode spacing.’ If the cavity is long enough (and thus the allowed modes close enoughin frequency), then multiple axial modes will lie within the laser’s gain bandwidth.

5

Some of these modes will experience enough gain to overcome the losses that theyexperience when circulating in the cavity. See Fig. 1.2. In some lasers, called ‘inhomo-geneously broadened’ lasers1, these axial modes can lase simultaneously. Conversely,in ‘homogeneously broadened’ lasers2, normal circumstances will allow only one axialmode to lase at a time, even if multiple modes fall within the region where the gain isgreater than cavity losses. In either case, if multiple axial modes can be forced to lasesimultaneously, and if certain phase relationships between the modes are maintained,then the modes will beat together to form laser pulses.

Suppose a laser has been forced to lase in multiple axial modes simultaneously.If each neighboring pair of axial modes has the same relative phase shift, then allmodes will beat together to form pulses. If the mode pairs continue to have the samerelative phase shift over time, then the pulses will remain. We can call this condition‘constant uniform mode pair phase difference,’ or we can simply say that the laseris ‘modelocked’. Maintaining a constant uniform mode pair phase difference is theprimary challenge of modelocked laser design. This challenge arises because the phasesof a free-running laser’s modes are typically not linked in any meaningful way. Therelative phases of the modes may fluctuate randomly with respect to one another overtime, a phenomenon called phase noise, which is due to quantum noise, mechanicalvibrations, and temperature fluctuations. Experimentalists cannot hope to eliminatephase noise completely, but if the noise for each mode is correlated, then pulses willstill form, and we can still consider this state modelocking (see Fig. 1.7) [Wollenhauptet al., 2007].

In a practical laser, part of the power of the circulating pulses will be coupled outof the cavity. The frequency at which pulses are emitted from the cavity, which wecall the ‘round-trip frequency,’ is

νRT =c

2L. (1.2)

Note that this is equal to the axial mode spacing, ∆ν. Thus lasers with long cavities,and therefore close axial modes, will rapidly emit pulses.

Modelocked lasers fall into two primary classifications, based on the technique usedto bring the axial modes into phase. Some modelocked lasers employ an electronically-driven modulator to periodically change the cavity loss or frequency of light; thistechnique is called active modelocking, because the modulator actively shifts opticalpower between axial modes. Loss modulators achieve modelocking by varying thecavity loss at the round-trip frequency, so that the wings of the pulse experience losswhile the pulse peak does not, and thus the pulse is shortened. Frequency modulators,on the other hand, the kind used in this thesis, shift part of the light passing through

1This means that different atoms in the gain medium experience different frequency-shiftingeffects, so that different atoms have gain spectra that are slightly shifted relative to one another.In the HeNe and other gas lasers, this is mostly due to Doppler shifts that arise from the velocitydistribution of the atoms [Siegman, 1986, Section 12.2, pp. 462-465; Section 30.3, pp. 1185-1186].

2Homogeneous broadening means that all atoms in the gain medium experience the samefrequency-shifting effects, so the gain spectra of the atoms lie on top of one another. In homo-geneously broadened lasers, like those with solid-state gain media, the lasing of one mode ‘saturates’the gain of other modes. We will discuss gain saturation in more detail later.

6 Chapter 1. Theory

Ν

PSD

Figure 1.3: Power spectral density(PSD) of a multimode laser. Simulatedas a Dirac delta comb with a Gaussian en-velope. The following figures show resultingwaveform for different phase relationships.

0

t

Pow

er,E

Figure 1.4: All modes in phase. Ideally,with a constant mode pair phase difference,the laser generates a Gaussian pulse train.The thick dashed line represents power, andthe thin solid line represents electric field.

0

t

Pow

er,E

Figure 1.5: Random phases, no noise.If phases are random but fixed in time,then the waveform is deformed but peri-odic. Mode pair phase differences are con-stant but nonuniform.

0

t

Pow

er,E

Figure 1.6: Same phase noise. Althoughthe phase of the electric field is noisy, pulsesare still formed in the power waveform andwe have modelocking.

0

t

Pow

er,E

Figure 1.7: Correlated phase noise.Pulses will form, but the waveform will notbe clean. This is the best that experimen-talists can accomplish under normal circum-stances.

0

t

Pow

er

Figure 1.8: Independent phase noise.Waveform will be noisy. This is the casein a multifrequency continuous wave laser.

1.1. Laser Resonator Stability 7

them to a higher or lower frequency. This shifted light can excite other axial modes,and, by a process we will discuss later, can even establish the fixed mode pair phasedifference necessary for modelocking.

The other class of modelocked lasers, called passively modelocked lasers, involvea device that modulates light without being driven. Passive modelocking usuallyimplies the use of a ‘saturable absorber’: a substance that will absorb light up to acertain intensity, and transmit more intense light. Like an active loss modulator, thishas the effect of absorbing the wings of a pulse yet letting the pulse peak through.The advantage of the saturable absorber is that it can react very quickly to intensityvariations, and thus can produce pulses of shorter duration than active loss modula-tors, which are limited by their driving frequency. It is passively modelocked lasersthat hold the records for shortest pulse duration; Kerr-lens passive modelocking ofTi:sapphire lasers have produced pulses of under 6 femtoseconds, spanning less thantwo optical cycles [Sutter et al., 1999]. Despite the advantages of passive modelocking,actively modelocked lasers are arguably conceptually simpler, and, with most of theresources available on-hand in the Reed physics department, the actively modelockedlaser was chosen for this experiment.

In the following sections of this chapter, we will describe in turn the major com-ponents of the laser assembled in this thesis. We will begin by examining the lasercavity and the conditions necessary for the support of a laser beam between thecavity mirrors. We will then turn to our gain medium, the Helium-Neon dischargetube, and describe how the medium amplifies light by stimulated emission. With anunderstanding of how the gain medium generates coherent light, we will move on tothe active modulator employed in the laser, an acoustooptic modulator, and explainhow the device works as a frequency modulator to bring the axial modes into phase.With the theory of the components laid out, we will then model the operation ofthe modelocked laser by propagating a Gaussian electromagnetic pulse through eachlaser component. This argument will make some idealizing assumptions, includingthe assumptions that our frequency modulator can be tuned perfectly and that phasenoise is negligible. However, we will see that because the laser system is not perfectlylinear, the reality of imperfect tuning and the presence of phase noise still allow formodelocking to occur.

1.1 Laser Resonator Stability

Laser resonators, or laser cavities, come in a variety of styles. The simplest case ofa linear optical cavity consists of two spherical mirrors: a ‘high reflector’ (HR) thatmaximizes the reflected light, and an ‘output coupler’ (OC) that transmits a smallfraction of the light incident on it. Such a cavity can be designed to support a beamthat is Gaussian in geometry (with transverse intensity described by the Gaussianfunction). If a Gaussian beam can be reflected in on itself perfectly at both ends ofthe cavity, essentially none of the light in the beam will escape out of the sides of the

8 Chapter 1. Theory

R1 R2

L

Z2Z1

Beam Waist

Z = 0

Figure 1.9: A linear laser cavity with two spherical mirrors, stably supporting a Gaussianbeam. The cavity length, mirror radii, beam waist and distances to mirrors are all indicated.

R1 R2L < R1

L < R2

L

L < R1

L > R2

Stable

Unstable

L > R1

L > R2 StableL < R1+R2

StableL = R1+R2

UnstableL > R1+R2

Figure 1.10: Various conditions under which a Gaussian beam may or may not be supportedby a cavity.

1.1. Laser Resonator Stability 9

cavity. Such a cavity is said to stably support the Gaussian beam.3 Whether a cavityis stable, and thus able to support a Gaussian beam, depends on its geometry: thelength of the cavity and the curvatures of the mirrors. Consider a cavity composedof two concave mirrors, with radii of curvature R1 and R2 respectively. We place thez axis along the optical axis, with z = 0 at the beam waist, the narrowest portionof the beam. The length of the cavity can be expressed as the sum of the distancefrom the R1 mirror to the beam waist, called z1, and from the beam waist to the R2

mirror, called z2:

L = z2 − z1. (1.3)

Note that z1 is defined to be negative, because it runs from the beam waist inside thecavity to the mirror R1, opposite to the direction of the optical axis [Siegman, 1986].This causes L to always be a positive quantity. See Fig. 1.9.

Suppose that we are given two mirrors, of curvature R1 and R2, and we want tofind the cavity lengths that will stably support a Gaussian beam of wavelength λ.To make sure that the beam is stable within the cavity, we need the curvature ofthe wavefronts to match the curvature of the mirrors, so that the mirrors reflect thewavefronts perfectly. Following Chang [2005, p.50], we first write down, for the caseat hand, formulae relating the Gaussian beam curvature to z,

−R1 = z1 +z2R

z1

, (1.4)

R2 = z2 +z2R

z2

, (1.5)

where zR =πw2

0

λis the Rayleigh range, and w0 is the radius of the beam waist. The

minus sign in Eq. 1.4 is needed to settle a discrepancy in conventions. Gaussianwavefront curvature is taken to be negative for a converging beam, traveling in thedirection of the optical axis. Mirror curvatures, on the other hand, are positive forconcave mirrors (looking out from the inside of the cavity). The minus sign allowsus to relate a negative wavefront curvature to a positive mirror curvature [Siegman,1986].

We can solve Eq. 1.4 and 1.5 for z1 and z2,

z1 =−R1

2± 1

2

√R2

1 − 4z2R, (1.6)

z2 =R2

2± 1

2

√R2

2 − 4z2R. (1.7)

and plug them into Eq. 1.3, yielding a formula for the cavity length:

L =R2

2± 1

2

(√R2

2 − 4z2R −

√R2

1 − 4z2R

). (1.8)

3Note that so-called unstable cavities can still lase. Such laser cavities are usually designed sothat light will leak out around one of the cavity mirrors. See Siegman [1986] for more details.

10 Chapter 1. Theory

20 40 60 80 100L

0.2

0.4

0.6

0.8

1.0g1g2

Figure 1.11: Stability ranges of L. A graph of the product g1g2 for a cavity with a60 cm high reflector and a 45 cm output coupler. The upper stability limit is shown as adashed line. The cavity will be stable if 0 < L < R1 or R2 < L < R1 +R2, so in this case,0 < L < 45 cm or 60 cm < L < 105 cm.

Some algebra and squaring leads to the following expression for zR:

z2R =

L (L−R1) (L−R2) (R1 +R2 − L)

(−2L+R1 +R2)2 . (1.9)

The Rayleigh range is positive real by definition, so the right hand side of Eq. 1.9must be positive as well. The denominator is always positive, by the reality of thevariables involved. The numerator must also be positive, which places restrictions onthe values of R1 and R2. Either two of the parenthesized terms in the numerator arenegative, or none of them are. This implies two possible inequalities, which representtwo separate cases:

1. R1 +R2 ≥ L ≥ max (R1, R2),

2. min (R1, R2) ≥ L > 0.

Physically, this means that a cavity will be stable in two cases: either the centers ofcurvature of both mirrors lie within the cavity, or both centers lie beyond the cavity.4

See Fig. 1.10 for an illustration of these conditions.A single, useful inequality that applies for both cases can be derived algebraically.

Define g1 =(

1− LR1

)and g2 =

(1− L

R2

). In case 1, the quantities g1 and g2 are

both negative, so their product must be positive. Likewise, in case 2, g1 and g2 areboth positive, and the product must again be positive:

g1g2 > 0. (1.10)

Also, note that the length L and curvatures R are always positive, so g1 and g2 musteach be less than 1, and thus so must their product:

1 > g1g2. (1.11)

4In the case that the curvatures are equal, it is possible for R1 = R2 = L, so both cases aresatisfied simultaneously.

1.2. Helium-Neon Gain Medium 11

Together, Eqs. 1.10 and 1.11 form the ‘stability equation’ for laser resonators:

1 > g1g2 > 0. (1.12)

With the mirror curvaturesR1 andR2 in hand, we can use Eq. 1.12 to place boundarieson the length of the cavity. Graphing g1g2 as a function of L shows that Eq. 1.12 isequivalent to cases 1 and 2 above: the cavity is stable up to the radius of curvatureof the first mirror, and again from the raius of the second mirror to the sum of theradii. See Fig. 1.11 for a graph with the mirror curvatures used in this thesis.

1.2 Helium-Neon Gain Medium

This thesis utilizes the common helium-neon gas discharge tube as a gain medium. Asexplained in Siegman [1986], the tube is filled with a low-pressure mixture of heliumand neon gas, with about ten times as much helium as neon. The potential differencebetween anode and cathode at opposite ends of the tube is over 1000 volts. Thisvoltage accelerates electrons that collide with the helium atoms, exciting them to thelong-lived 21S state. If an excited helium atom collides inelastically with a groundstate neon atom, the neon atom can be excited to the 3s state, which has a 632.8 nmlaser transition to a 2p state. Photons of this frequency that interact with excitedneon atoms can stimulate the emission of additional photons of identical frequencyand phase. After many round-trips through the cavity, the photons begin to form acoherent beam.

Consider the light circulating in a laser cavity as it passes by the gain medium.The amount by which light is amplified depends critically on frequency, so we expressgain mathematically in terms of frequency. Transforming the electric field of thelight, E(t),to the frequency domain, E(ω), we can approximate the effect of the gainmedium by multiplying E(ω) by the transfer function g(ω):5

g(ω) = exp

[αp

2 + 4i(ω − ω0)/∆ω0

]. (1.13)

Here α is the laser gain coefficient, which is less than 1, p is the gain mediumlength, ω0 is the peak laser transition frequency, and ∆ω0 is the full-width half-maximum (FWHM) gain bandwidth. For the 632.8 nm HeNe transition, values are0.02 cm−1 < α < 0.1 cm−1 (typically), ω0 ≈ 4.737 × 1014 Hz, and ∆ω0 ≈ 1.56 × 109

Hz (as found by Niebauer et al. [1988]). The real part of Eq. 1.13 is

Re(g(ω)) = exp

[pα∆ω2

0

2 (∆ω20 + 4 (ω − ω0) 2)

], (1.14)

5This expression reproduced from Siegman [1986, p.1062], though a factor of 1/2 is introduced asSiegman’s expression describes an entire round trip, and thus propagation through twice the cavitylength L. Note that atomic collisions and Doppler broadening are ignored in this formula for thesake of simplicity; these phenomena cause broadening of the gain lineshape.


0Ω - Ω0

1

ReHgHΩLL

Figure 1.12: Small-signal gain vs. detuning from resonance. Dependence of lasergain on the detuning of the incident field from a resonance of the gain medium. Ignoringline broadening, the lineshape is Lorentzian.

representing the small-signal gain, or the maximum gain of light passing throughthe gain medium at frequency ω. A representative plot of this function for an arbi-trary gain medium can be seen in Fig. 1.12. This shows that only a narrow band offrequencies are amplified by the medium.

The gain values given by Re(g(ω)) are those experienced only by very small signals,and in general, the gain that any particular laser mode experiences is not constant;otherwise, the mode would continue to gain power forever. Instead, as the modeincreases in intensity, the gain ‘saturates,’ decreasing until it equals the cavity losses,at which point the mode intensity equals its steady-state value. This saturation isdue to the fact that the atoms in the gain medium need a finite amount of timeto become re-excited. Consider a single mode of light at an intensity greater thanthe steady-state intensity. The presence of this light stimulates emission from thegain atoms more quickly than the atoms can become re-excited by the pump process.Meanwhile, losses continue at a constant rate and the mode intensity decreases, so atthis instant the gain is less than the losses. As the mode loses intensity, the gain willincrease until it exactly equals the cavity losses. This shows that lasing in the steadystate is an equilibrium point in the dynamics of a laser.

The imaginary part of Eq. 1.13 is

Im(g(ω)) = exp

[− pα∆ω0 (ω − ω0)

∆ω20 + 4 (ω − ω0) 2

], (1.15)

representing the phase shift introduced to each frequency. The gain medium is thusdispersive, and we will see in a later section that a light pulse will be broadened uponinteracting with the gain medium. A representative plot is given in Fig. 1.13.

The bandwidth of the gain medium is relevant to modelocked laser design, sincethe bandwidth limits the minimum duration of pulses produced by the laser. Theminimum pulse duration is approximately equal to the inverse gain bandwidth. Forthe HeNe gain medium, with a gain bandwidth of ∆ω0 = 1.56 GHz, this estimationyields a minimum pulse duration of about 640 picoseconds. The current standardfor research-grade ultrafast pulse generation is the titanium-doped sapphire crystal.

1.3. Acoustooptic Modulator 13

0Ω - Ω0

1

ImHgHΩLL

Figure 1.13: Phase shift vs. detuning from resonance. Phase shift induced by a gainmedium as a function of the detuning of the incident field from a resonance of the gainmedium.

With a gain bandwidth of about 128 GHz, it can generate pulses lasting under 10 fem-toseconds. Comparatively, the HeNe is a relatively mediocre choice of gain medium,but it is much cheaper and easier to set up.

1.3 Acoustooptic Modulator

This section is intended to describe the variety of acoustooptic modulators (AOMs)used in active modelocking, and to present a theory of how the AOM in this thesisinteracts with laser light.

An acoustooptic modulator is a mechanical device that consists of a piezoelectrictransducer attached to a transparent crystal. The transducer sends radio-frequencyacoustic vibrations into the crystal, and the compression waves induce changes inindex of refraction to form a ‘phase grating’ off of which laser light can diffract.Depending on various operational and fabrication parameters, this device can acteither like a standard diffraction grating that produces many diffracted beams, oras a Bragg scatterer that produces a single diffracted beam at a specified angle, orthe behavior can lie somewhere in between. The distinction between these regimeswill be discussed in a following subsection. In either case, light from the principlebeam is diffracted into one or more higher-order beams. A second and arguably moreimportant distinction between classes of AOMs is whether the acoustic wave is astanding wave or a traveling wave.

1.3.1 Standing vs. Traveling Acoustic Wave AOMs

The standing wave AOM creates a stationary phase grating that oscillates in intensity.Sending a laser through this type of AOM causes the beam to undergo diffractionperiodic in time. Such an AOM can be placed inside the laser cavity and used as aloss modulator. At zero amplitude of the stationary wave, the beam passes throughunimpeded, but, as the amplitude increases, some of the incident optical power is


redirected into the diffracted beams, which escape out of the sides of the cavity,resulting in a loss. If this periodic loss modulation occurs at the round-trip frequency,a short pulse can form that coincides with low-loss periods. This is because everytime the pulse reaches the AOM, the pulse wings are subject to more loss than thepulse peak, shortening the pulse. This technique is called modelocking by amplitudemodulation.

The traveling wave AOM, the type employed in the experimental section of thisthesis, creates a moving phase grating. A laser beam passing through this AOM willconstantly be diffracted by the grating, but the traveling nature of the grating causeshigher and lower diffraction orders to become frequency-shifted by various amounts.This can be thought of, as will be explained in a following subsection, as the addition(or subtraction) of the energy and momentum of the acoustic waves to the opticalwave. The frequency of each successive diffraction order is shifted by another multipleof the acoustic wave frequency. The upshot is that these frequency-shifted beams aregenerated in phase with the incident beam. If the acoustic frequency is set to equalthe round-trip frequency, then diffracted beams that are reflected back into the lasercavity will resonate within it as axial modes, and proper alignment can ensure thatthe modes stay in phase as light circulates within the cavity. Thus, a traveling waveAOM can be used to bring the axial modes of a laser into phase. This technique iscalled modelocking by frequency modulation. With this application in mind, we willnow turn to focus on the theory of operation of a traveling wave AOM and the useof this device for shifting the frequency of light.

1.3.2 Diffraction Regimes and Traveling Wave AOM Theory

Based on how an AOM is constructed and employed, the device can operate in twodiffraction regimes: the Bragg regime and the Raman-Nath regime. In the Braggregime, light passing through a thick AOM crystal interacts with many acousticwavefronts to form a single, intense diffracted beam that only appears at certainangles of incidence. This is analogous to the diffraction of X-rays off of crystal layers;thus the reference to Bragg. In the Raman-Nath regime, a relatively thin AOM actsas a standard diffraction grating, generating many higher-order beams regardless ofincident angle. Although the AOM used in this thesis operates somewhere betweenthe regimes, it operates closer to the Raman-Nath (diffraction grating) regime, so thetheory of that regime will be discussed in detail.6

Under most circumstances, the regime in which an AOM operates is determinedby geometrical factors7. For λd

Λ2 1, the AOM is in the Bragg regime, and for λdΛ2 < 1,

we are in the Raman-Nath regime. Here λ is the optical wavelength of incident light,Λ is the wavelength of the acoustic wave, and d is the width of the crystal.

To understand the origin of this condition, consider first the Bragg diffraction

6This theory follows Boyd [2008, Section 8.4], who gives detailed descriptions of both diffractionregimes. The Bragg regime is described in Section 8.4.1, while the Raman-Nath regime is discussedin Section 8.4.2.

7For a more accurate method for distinguishing between regimes, the degree of modulation of theindex of refraction must be taken into account. See Moharam and Young [1978].


ΛL L

θ

Figure 1.14: Bragg diffraction. A plane wave scatters off of the layers in a crystal. If theincident angle is correct for constructive interference, the ray diffracted by the lower crystalface will travel an extra distance 2L = 2Λ sin θ = mλ, where m is an integer, so that therays scattered by the upper and lower planes are in phase.

regime, shown in Fig. 1.14: interference between reflections from multiple acousticwaves generates a diffracted beam, the angle of which is specified by the relativewavelengths of the optical and acoustic waves:

sin θ =mλ

2Λ, (1.16)

where θ is the half-angle of the diffracted beam away from the acoustic plane, andm is the diffraction order. Note that the efficiency of diffraction should increase ifthe laser beam encounters more acoustic wavefronts before exiting the AOM. Thusthe intensity of the diffracted beam will be greatest if the transverse distance coveredby the beam inside the AOM is much larger than the acoustic wavelength. Thetransverse distance, z is found geometrically by examining Fig. 1.15:

z = d tan θ. (1.17)

Encountering multiple wavefronts requires that z Λ.Here d is the width of the AOM. In most cases, θ is small, so we can say that

sin θ ≈ tan θ ≈ θ. In this approximation we can combine Eqs. 1.16 and 1.17 andsimplify the result to find the condition that must hold for Bragg scattering to occur:

λd

Λ2 1. (1.18)

If instead the AOM is thin enough that the beam only encounters one acousticwavefront within the AOM, we will have Raman-Nath scattering:

z = d tan θ < Λ. (1.19)

This is shown in Fig. 1.15.


θz = d tan θΛ

d

Figure 1.15: Requirements for Raman-Nath diffraction in a traveling-wave AOM.The wavelength of the acoustic waves, Λ, must be greater than the transverse distancecovered by the light beam through the AOM, d tan θ. This ensures that the beam does notdiffract off of multiple wavefronts.

It will be shown that Raman-Nath scattering generates diffracted beams at anglessimilar to a diffraction grating. Assuming for now that this is the case, we can combinethe above inequality with the diffraction grating equation,

sin θ =mλ

Λ, (1.20)

to obtain, for the first order diffracted beam (m = 1), again in the small-angleapproximation,

λd

Λ2< 1. (1.21)

Thus the relative dimensions of the AOM and the wavelengths involved give an in-dication as to the type of scattering the acoustooptic modulation will produce. TheAOM used in this thesis consists of a tellurium dioxide crystal, for which the ratioλd/Λ2 ≈ 5, placing the AOM between regimes. For the sake of brevity, only theRaman-Nath theory is presented here; this theory has the advantage of being moreclosely related to expressions about general frequency modulators than the Braggtheory.

We will now present the theory of Raman-Nath scattering. We first note that thepiezoelectric transducer induces a sinusoidal density variation

∆ρ = ∆ρ sin(qz − Ωt). (1.22)

This will generate a refractive index variation

∆n = ∆n sin(qz − Ωt). (1.23)

To find the amplitude ∆n of the refractive index variation in terms of the known∆ρ, we can employ the relationship between relative permittivity change and density


change, which we assume is small enough as to be linear:

∆ε =∂ε

∂ρ∆ρ = γe∆ρ/ρ0, (1.24)

where γe is the electrostrictive constant and ρ0 is the bulk density, both of which aremeasurable. Now note that the total refractive index, n = n0 + ∆n, is

n =√ε0 + ∆ε, (1.25)

as long as we assume that the relative permeability is 1 (a good approximation formost dielectrics). Since the relative permittivity variation ∆ε is small, we can makethe approximation n ≈ √ε0 + ∆ε/(2

√ε0), which results in the relation n0 =

√ε0, and

∆n ≈ ∆ε

2n0

. (1.26)

Using Eq. 1.24 we can finally relate the variation of the refractive index to densityvariations of the AOM crystal:

∆n =γe∆ρ

2n0ρ0

. (1.27)

Now we can figure out what happens to an incident light wave as it passes throughthe AOM. Sinusoidal voltage signals applied to the AOM-transducer result in densityvariations of the crystal, which lead to variation of the refractive index. This, inturn, will cause a phase shift. The exact shift depends on both position and time.To calculate the relative phase shift between different parts of the light wave aftertraveling across the AOM, we can use a varying wavenumber, call it ∆κ:

φ = ∆κd =ω

c∆nd =

ω

c∆nd sin(qz − Ωt) (1.28)

≡ δ sin(qz − Ωt), (1.29)

which serves to define δ. If we write the incident field in complex form as E(x, t),then the transmitted field E ′(x, t) is simply the incident field multiplied by a complextransmission function eiφ:

E ′(x, t) = E(x, t)eiδ sin(qz−Ωt). (1.30)

We can put this result into a more revealing form by using a variation of the Jacobi-Anger expansion:

eiδ sin θ =∞∑

m=−∞

Jm(δ)eimθ, (1.31)

which employs Bessel functions of order m acting on δ. Applying this expansion tothe transmitted field gives

E ′(x, t) = E(x, t)∞∑

m=−∞

Jm(δ)eim(qz−Ωt). (1.32)


Consider the effect that the modulation has on a plane wave of frequency ω andwave vector ~k = kx. We can assume that the beam covers a large enough transverse(z-direction) distance to interact with multiple wavelengths of the acoustic wave.Representing the plane wave in complex notation, we get

E ′(x, t) = A[ei(kx−ωt) + e−i(kx−ωt)]∞∑

m=−∞

Jm(δ)eim(qz−Ωt) (1.33)

= A

∞∑m=−∞

Jm(δ)[ei[(kx+mqz)−(ω+mΩ)t] + e−i[(kx+mqz)−(ω+mΩ)t]] (1.34)

= A∞∑

m=−∞

Jm(δ) sin[(kx+mqz)− (ω +mΩ)t]. (1.35)

This reveals that the transmitted field is in fact a superposition of plane wavesof varying magnitude, direction and frequency. The diffracted waves are essentiallyformed by adding or subtracting the momentum and energy of the acoustic waveto the incident optical wave. Higher-order diffracted beams vary in frequency inincrements of Ω, the acoustic wave frequency, and vary in wavevector in incrementsof ~q, the acoustic wavevector. Thus varying the AOM-transducer frequency providescontrol over the frequency of diffracted beams. Since the speed of sound in thecrystal is independent of frequency, changing the transducer frequency also changesthe wave number and thus the deflection angle. (Traveling wave AOMs are often usedas beam deflectors that can precisely alter the direction of diffracted orders withoutmuch dependence on incident beam angle). Imagine the wavevector of the m-thorder output from the AOM as the hypotenuse of a right triangle, with the opticalwavevector ~k as one leg, and an integer number of acoustic wavevectors m~q as theother leg. Then the angle of the diffracted mode away from the optical wavevector ~kis given by the trigonometry of the triangle,

tan θm =mq

k=mλ

Λ, (1.36)

which at small angles is identical to the diffraction grating equation.When using a traveling wave AOM for a modelocking experiment, one would,

ideally, capture all diffracted beams and redirect them so that they recirculate insidethe laser cavity. Doing so would eliminate the z-dependence of the phase shift, so thetotal phase shift equation would read

φ = δ sin(Ωt). (1.37)

This idealization will be useful shortly when modeling the behavior of a modelockedlaser. In practice, it is possible to capture just one diffracted beam, typically thefirst-order beam, and redirect it so that it circulates in the laser cavity, allowing aone-directional transfer of power between frequencies in the cavity.

1.4. Time-domain modeling: Pulse Propagation 19

1.4 Time-domain modeling: Pulse Propagation

Having described the components of the modelocked laser and described how eachaffects passing light, we can introduce a theory of optical pulses and propagate themthrough each component. Following the argument of Siegman [1986, Section 27.3],this section is intended to demonstrate how an optical pulse can maintain a Gaussianshape throughout a round-trip through the laser. Unfortunately, some of the ap-proximations used are invalid in the experimental setup described later in this thesis,but the idealized argument presented here is nonetheless useful for understanding thegeneration of Gaussian pulses in a general modelocked laser system.

We will begin by developing the mathematics for the Gaussian pulse, move onto find the effect each optical element has on the pulse, and finally find a stableround-trip pulse shape.

1.4.1 Gaussian Pulses

We can approximate the pulses that circulate in the laser as having Gaussian intensityprofiles. In the time domain, we can describe a Gaussian pulse in the form

E(t) = e−at2

ei(ωct+bt2) = exp(−Γt2 + iωct), (1.38)

where a is the envelope parameter, ωc is the carrier frequency, and Γ ≡ a− ib is the‘Gaussian pulse parameter’. If b is nonzero, then the quadratic dependence of opticalphase on time implies that the instantaneous frequency varies linearly along the pulse.This linear frequency variation is called ‘chirp,’ and is a result of the shifting of partsof the pulse spectrum. The instantaneous intensity of the pulse is

I(t) = e−2at2 = e−(4 log 2)(t/τ)2 , (1.39)

where τ =√

2 log 2/a is the full width at half-maximum (FWHM) of the intensityprofile. Thus a large a parameter implies a short pulse, while a small a implies a longpulse.

Propagation of the pulse through most optical components is more easily ac-complished in the frequency domain than in the time domain. Taking the Fouriertransform of the field results in

E(ω) =

∫ ∞−∞

E(t)e−iωtdt (1.40)

=1√2π

∫ ∞−∞

e−Γt2eiωcte−iωtdt (1.41)

=1√2Γ

exp

[−(ω − ωc) 2

4Γ

]. (1.42)

With this result in hand, we can turn to propagating the pulse through the laser.


1.4.2 Pulse Propagation

We can propagate our pulse through a simple laser setup by multiplying the fieldby either a transmission function in the time domain or a transfer function in thefrequency domain. Consider a linear laser consisting of generic laser gain medium(with a Lorentzian lineshape) and a generalized frequency modulator within a two-mirror cavity. Suppose the cavity has length L. We begin with the standard Gaussianpulse, in frequency representation, and assume that the carrier frequency is equal tothe peak frequency in the gain medium:

E0(ω) =1√2Γ0

exp

[−(ω − ω0) 2

4Γ0

]. (1.43)

First let us consider the effect the gain medium has on the pulse during one round-trip. We will multiply the field by the transfer function g(ω), from the section ongain media, but first we can make an approximation that will be useful in analysislater8:

g(ω) = exp

[αp

2 + 4i(ω − ω0)/∆ω0

](1.44)

≈ exp

[αp

(2− 4i

ω − ω0

∆ω0

− 8

∆ω20

(ω − ω0)2

)]. (1.45)

Here we have made a Taylor expansion approximation based on the assumption thatthe spectrum of E(ω) is much narrower than the total gain bandwidth of the lasertransition.9 Multiplying the Gaussian field by g(ω) yields a new field

E ′(ω) =1√2Γ0

exp

[2αp− 4iαp

(ω − ω0)

∆ω0

−(

1

4Γ0

+8αp

∆ω20

)(ω − ω0) 2

]. (1.46)

The three terms in the exponent represent different physical effects that the gainmedium has on the field. The first term represents the amplification afforded to eachfrequency, which is independent of frequency (according to the approximation wemade above, that the bandwidth of the pulse is much narrower than the FWHM ofthe gain medium). The second term is imaginary, and thus represents a phase shiftin frequency space, corresponding to a delay of the field in time. We will considerthis term later. The last term is real and quadratic in ω, and corresponds to the newwaveform of the field. This term is, like the exponent of Eq. 1.42, proportional to−(ω − ω0)2, so it is again a Gaussian waveform. We can describe this pulse with anew Gaussian pulse parameter, Γ′:

1

4Γ′=

1

4Γ0

+8αp

∆ω20

(1.47)

=⇒ Γ′ =Γ0∆ω2

0

32αpΓ0 + ∆ω20

. (1.48)

8Note that we are passing through the gain medium twice during the round-trip.9This assumption is invalid for the experimental setup of this thesis, because at practical cavity

lengths, the HeNe’s narrow gain bandwidth covers only a few intermode frequency spacings.


Decomposing this result into real and imaginary parts, Γ′ = a′ − ib′, we have

a′ =∆ω2

0 (32 (a2 + b2) pα + a∆ω20)

1024b2p2α2 + (32apα + ∆ω20)

2 , (1.49)

b′ =b∆ω4

0

1024b2p2α2 + (32apα + ∆ω20)

2 . (1.50)

These are complicated expressions, and the behavior of a and b over many roundtrips depends on the initial values of the parameters. However, we can run a simplesimulation to show that a pulse circulating in a cavity containing only a gain mediumwill broaden over time, returning to a sine wave. This is done by choosing valuesfor the constants and initial values a0 and b0, and then iterating Equations 1.49 and1.50. A representative plot of the evolution of a and b after each round-trip canbe seen in Fig. 1.16. Both a and b decrease slowly over time, meaning that thepulse width increases to infinity and the chirp decreases to zero, approaching a sinewave. However, if the initial pulse has a high chirp (large b), then there will be atransient effect where the pulse width decreases before the long-term increase begins.This suggests that a chirped pulse, and thus a pulse with phase-shifted modes, willbe narrowed in time when passing through the gain medium. However, withoutmodulation, the gain medium will eventually cause the pulse to broaden, and thechirp to decrease, due to a narrowing of the pulse spectrum.

Roundtrips

a, b

Figure 1.16: Evolution of pulse parameters in a cavity with gain. Calculated iter-atively over many round trips. The thick line represents a, and the thin line represents b.Both a and b decrease slowly over time. The dashed line shows the initial value of a.

Let us now consider the phase shift component of the exponential in Eq. 1.46,4iαp (ω − ω0) /∆ω0. Taking the inverse Fourier transform of this part of the pulsespectrum reveals that the constant term, −4iαpω0/∆ω0, simply shifts the phase ofthe pulse in the time-domain. The frequency-dependent term, 4iαpω/∆ω0, on theother hand, actually translates the pulse in the time domain, so that the pulse isdelayed. This small delay, due to the gain medium, means that the round-trip timeis not exactly 2L/c, and the axial mode spacing is not exactly c/2L. However, theproduct αp and the difference ω − ω0 are usually quite small compared to ∆ω0.


While the perturbation is theoretically nontrivial, its magnitude is usually within theuncertainty of an experimentalist’s tools for measuring the cavity length or round-trip frequency. Thus experimentalists usually ignore this perturbation and estimatethe axial mode spacing to be c/2L, as they can tune cavity length and modulationfrequency empirically for best modelocking.

Finally, we consider the effect of the frequency modulator on the pulse. Wedeveloped the modulator theory in the time domain, so we use the time-domainGaussian pulse. The gain and phase-shift terms induced by the gain medium areinconsequential in the steady state10, so we focus on the term involving the pulseparameter:

E(t) ∝ exp[−Γ′t2], (1.51)

The transmission function from Eq. 1.37, this time replacing the sine function witha cosine function, is given by T (t) = eiδ cos(Ωt). We can make a simplifying approx-imation of the cosine function by Taylor expansion, but we first need to figure outat which point in time the pulse will cross the modulator. Suppose the pulse crossesthe modulator at t = nπ/2 for odd integer values of n; that is, when cosine sweepsthrough zero and the phase changes most rapidly. This would yield a transmissionfunction approximately proportional to exp[iδΩt]. This would change the instanta-neous frequency of the pulse E(t), bringing the central pulse frequency outside of thebandwidth of the gain medium after many round-trips. Thus this pulse will not beamplified and will quickly die off due to cavity losses. To form pulses that survive inthe steady-state, then, we want the pulse to avoid periods of rapidly-changing phase.If the pulse passes through the modulator when the phase is nearly constant (at thecrest or trough of the cosine function), then we can avoid terms in the transfer func-tion that change linearly with time, and thus avoid shifting the central frequency ofthe pulse. If we assume the pulse duration to be much shorter than the modulationperiod11, we can make the following quadratic approximation for the transmissionfunction:

T (t) ≈ exp[iδ(1− Ω2t2/2

)]. (1.52)

After passing through the modulator, then, the field is

E ′(t) ∝ exp[−Γ′t2 + iδ

(1− Ω2t2/2

)](1.53)

= exp[iδ −

(Γ′ + iδΩ2/2

)t2]. (1.54)

The iδ term is a small constant phase shift. The coefficient of t2 is the new Gaussianpulse parameter: Γ′′ = Γ′+iδΩ2/2. This shows that |b′′| > |b′|, so the chirp of the pulseis increased during modulation, but a′′ = a′, so modulation does not directly compress

10The gain per round trip depends on the degree to which the gain medium is saturated, and inthe steady state, it is exactly cancelled by the cavity losses. The phase shift accumulates over everyround trip, so emitted pulses generally differ in optical phase.

11This is another approximation that fails in our experimental case, but is true for modelockedlasers employing broadband gain media.


the pulse. However, increasing the chirp of the pulse does broaden the spectrum, andlooking back at Eq. 1.49, we see that the when the pulse passes through the gainmedium again, the parameter a increases if the magnitude of b had increased duringthe previous round-trip. Thus upon exiting the gain medium, the pulse is narrowedslightly. This indirect pulse narrowing causes the spectrum to broaden and the pulseduration to decrease over many round trips, until a steady state is reached.

When we include the modulation in our simulation, as in Fig. 1.17, we see thatboth a and b stabilize to reach approximately the same value. This stable value isindependent of the initial pulse parameters; the stable value is reached even whena0 = b0 = 0.

Roundtrips

a, b

Figure 1.17: Evolution of pulse parameters with both gain and modulation. Thethick line represents a, and the thin line represents b. Whenever the lines cross, one ofthe values reaches a maximum or a minimum. The dashed and dotted lines represent thetheoretical steady-state values of a and b respectively, as obtained from Eq. 1.58.

Finally, we can solve our formulae analytically for the steady-state pulse parame-ter. In the steady state,

Γ′′ = Γ0 (1.55)

=⇒ Γ =2ipαδΩ2 +

√−4p2α2δ2Ω4 + ipαδΩ2∆ω2

0

8pα(1.56)

(we choose the root such that a is positive). Expanded into real and imaginary parts,we have

Γ =iδΩ2

4+

√δΩ (16p2α2δ2Ω4 + ∆ω4

0) 1/4

8√pα

(Cos[ψ] + iSin[ψ]), (1.57)

with ψ =1

2argument

pαδΩ2

(−4pαδΩ2 + i∆ω2

0

). (1.58)

This result is complicated. Numerically, we find that b is slightly higher than a forthe values used in the iterative simulation, as seen in the Fig. 1.17.

In this section, we have shown that a steady-state Gaussian pulse solution exists,where an initial Gaussian pulse maps back onto itself after propagating through the


gain medium and intracavtiy frequency modulator. We have also provided evidencethat this solution is stable in the sense that any initial Gaussian evolves toward thissolution.

1.5 Injection Locking

There are two practical problems with modelocked laser experiments that we implic-itly ignored in the preceding theory. The first is the fact that we cannot hope toperfectly tune the modulation frequency Ω to line up with the axial mode spacing∆ν. The frequency-shifted light will, in every practical case, be detuned slightly fromthe axial mode frequencies that the laser cavity supports. We expect that the spectraof both the allowed cavity modes and the frequency-shifted light will be broadenedby noise, but the overlap of these broadened peaks will not be complete, resulting inlow gain of the frequency-shifted light. Thus there would be little power in any ofthe modes except the principle mode, forming broad output pulses at best.

The second, and perhaps more important, problem we ignored was that of phasenoise. When we began our discussion of modelocking, we emphasized that a goal ofthe technique is to minimize phase noise differences between mode pairs. The pulse-propagation modeling we presented in the previous section showed that the spectrumof a pulse will be broadened inside our laser (spectral broadening being the other maingoal of modelocking), but it assumed that the pulse was noiseless, and that the opticalcomponents did not contribute any noise. In reality, every component contributesnoise to the system, whether it be from spurious vibrations of the mirrors or AOM,thermal deformations of those components, or variations of index of refraction in theair between components.

Thankfully, it turns out that modelocked lasers can operate effectively in thepresence of both detuning of the modulation frequency and standard experimentalnoise. This is due to the nonlinearity of the system. Suppose two axial modes, at ω1

and ω2 = ω1 + ∆ν, are oscillating simultaneously in a laser that includes a frequencymodulator that is currently turned off. The two modes have different phase noise.Now suppose that the frequency modulator is turned on. All we want this modulatorto do is take part of the incident light and increase its frequency by Ω, where Ω isclose but not equal to the axial mode spacing ∆ν. Considering only the ω1 mode fornow, the modulator will therefore output two frequencies of light, ω1 and ω∗ = ω1+Ω,where the ω∗ light has some phase relationship with the ω1 signal. We say that thelight at ω∗, which does not lie precisely at any axial mode, has been ‘injected’ into thecavity. If the laser were a completely linear system, we would have three frequenciesof light superimposed in the cavity: ω1, ω2 and ω∗. However, because of nonlineareffects, such as cross-gain saturation, the presence of light at ω∗ affects the generationof light at ω2. In fact, if the frequency ω∗ is close enough to ω2, then the frequency ofthe ω2 mode will be ‘pulled’ to the frequency ω∗, and the phase of this pulled modewill follow the phase of the ω∗ signal. Thus by turning on the modulator, we haveeffectively replaced the mode at ω2 with a mode at ω∗, and because the ω∗ mode wasgenerated from ω1, the two modes in the cavity have a constant phase relationship.

1.5. Injection Locking 25

This synchronization phenomenon, called ‘injection locking,’ occurs in almost anynonlinear oscillatory system in which a periodic signal is injected near resonance. Itwas described mathematically in a paper by Adler [1973], and this description wasapplied to laser oscillators by Kurokawa [1973].

To show how injection locking works, we will consider the behavior of a general self-sustained oscillator when injected with a periodic signal (a more in-depth treatmentof the injection locking of lasers is given in Siegman [1986, Chapter 29]). The systemis described by the so-called Adler’s equation12,

dψ

dt= −ωD + ε sinψ, (1.59)

where ψ is the phase difference between the oscillator and the injected signal, ωDis the detuning of the injected signal’s frequency from the natural frequency of theoscillator, and ε will be shown to be the injection locking range, which has the unitsof frequency. We can solve this equation for ψ(t), which tells us how the phase of theoscillator relates to the phase of the injected signal over time:

ψ(t) = 2 arctan

[ε

ωD−√ω2D − ε2ωD

tan

[√ω2D − ε22

t

]](1.60)

There are, then, two general regimes of solutions, which depend critically on therelative magnitudes of ωD and ε.

0 T 2 T 3 Tt

Ψ0

Ψ f

Π

ΨHtL

Figure 1.18: Evolution of phase difference, ωD < ε. Solution to Adler’s equation forsmall detuning. The phase of the oscillator approaches a fixed phase difference, ψf , withrespect to the injected signal, so the frequencies are identical and injection locking occurs.The dashed line shows the final phase difference.

12The equation in this form was adapted from Pikovsky et al. [2001], which derives Adler’s equationand provides a generalization that can be applied to many situations.


1.5.1 Locked Regime: ωD ≤ ε

If the the detuning ωD is less than the parameter ε, then the radical factor becomesimaginary: √

ω2D − ε2 = ih, (1.61)

which serves to define h. The solution (1.60) can now be written

ψ(t) = 2 arctan

[ε

ωD− ih

ωDtan

[ih

2t

]]. (1.62)

We can solve the problem of the tangent function taking an imaginary argument bymaking the replacement tan(iθ) = i tanh(θ):

ψ(t) = 2 arctan

[ε

ωD+

h

ωDtanh

[h

2t

]]. (1.63)

We plot this function in Fig. 1.18. We can see that the phase difference moves from

the initial value ψ0 to a final constant value ψf = 2 arctan

(ε+√ε2−ω2

D

ωD

), indicating

that the oscillator is oscillating at the same frequency as the injected signal but witha shifted phase. Here T is the natural period of the oscillator when undisturbed. Thisis the injection locked regime.

Note that if we had set ωD = ε, then the solution would be ψ(t) = 2 arctan(1) =π/2, so the phase difference would still be constant and we can still consider thissituation injection locking.

1.5.2 Unlocked Regime: ωD > ε

If the detuning ωD is greater than the parameter ε, then the radical factor is real, andwe can call Eq. 1.60 our solution. This is plotted in Fig. 1.19 for ωD slightly larger thanε (1.2:1). What is happening here is that, as long as ψ evolves slowly, the oscillationis basically at the injected frequency, but the phase periodically ‘slips,’ changing theinstantaneous frequency of oscillation. If we increase ωD to be much greater than ε(2:1), shown in Fig. 1.20, then the phase slips become more frequent, and the periodof the phase slips approaches the period of the undisturbed oscillator. Thus, for largedetunings, the oscillator essentially oscillates at the undisturbed frequency.

1.5.3 Consequences of Injection Locking

In the context of our laser, injection locking helps overcome the two difficulties pre-sented in the beginning of this section: modulator detuning and phase noise. As longas the detuning of the modulator is small enough, injected frequency-shifted modeswill replace the naturally occurring axial modes. Also, the phase noise of the injectedmode replaces the noise of the natural axial mode, essentially coupling the noise of

1.5. Injection Locking 27

T 2 T 3 Tt

Ψ0

Π

-Π

ΨHtL

Figure 1.19: Evolution of phase difference, ωD & ε. Solution to Adler’s equation fordetuning slightly larger than ε. The frequency of the oscillator is usually close to that of theinjected signal, but the phase ‘slips’ periodically, though less frequenty than the undisturbedperiod T . The dashed line shows the initial phase, to give a sense of the phase slip period.

T 2 T 3 Tt

Ψ0

Π

-Π

ΨHtL

Figure 1.20: Evolution of phase difference, ωD ε. Solution to Adler’s equation fordetuning far larger than ε. The period of phase slips approach the period of the undisturbedoscillator, so the oscillator usually oscillates near the undisturbed frequency.

each mode in the cavity. But how small does the detuning have to be? That is, whatcan we expect ε to be, and how can we increase it? Siegman [1986, Eq. 29.30] gives

ε = γ

√Iinj

I0

, (1.64)

where Iinj is the intensity of the injected signal, I0 is the intensity of the undisturbedmode, both measured inside the cavity, and γ is the ‘cavity decay rate,’ which isapproximately equal to δ

TRT, where δ is the percentage of light that is emitted from

the output coupler mirror, and TRT is the round-trip time through the cavity, which isjust 1/νRT . What all this means is that we can improve our chances of modelockingby increasing the intensity of the light we inject into the laser, by increasing theaxial mode spacing (making our cavity shorter), and increasing the amount of lighttransmitted through our output coupler mirror. Thus the laser will be more difficult


to injection-lock if we use a long cavity to reduce the duration of our pulses, soinjecting as much intensity into the cavity is a priority.

In this section, we have seen that the two problems we feared would ruin ourexperimental attempts at modelocking, being modulator detuning and phase noise,are mitigated by injection locking, which results from the inherent nonlinearities ofthe laser. On this positive note, we make the assumption that the theory presentedin this chapter will be sufficient for guiding us through the construction and analysisof a modelocked laser, and turn now to the experimental design of our laser.

Chapter 2

Experimental Design

In this chapter, we will describe and motivate the experimental setup used to achievemodelocking. First we will consider a simple design that employs a traveling-waveAOM for modelocking. We will then describe the practical problems with this designand introduce the actual experimental setup.

AOM

0th order

1st order

L0

HeNe tube

HR

HROC

L0

Figure 2.1: Theoretically simple split cavity for modelocking. Each leg of the cavity is thesame length, so a circulating pulse that splits at the AOM will be recombined at the AOMwith no relative delay between the legs. There are two high-reflecting (HR) mirrors and oneoutput-coupling mirror (OC). In this configuration, each mirror can ideally have the sameradius, equal to L.

The design of our modelocked laser is centered around the modulator available,a traveling-wave acoustooptic modulator (AOM). As explained in Chapter 1, thetraveling-wave AOM diffracts incident light into multiple beams, each of which isfrequency-shifted by a certain amount. A modelocked laser can be constructed bycapturing both the 0th order beam, which is not frequency shifted, along with one ofthe higher-order frequency shifted beams, such as the 1st order beam, and allowingboth to beat together inside of an optical cavity. A theoretically simple setup thatwill contain both 0th and 1st order beams within an optical cavity involves three

30 Chapter 2. Experimental Design

AOM

Reflected0th order

0,0

0,-1

0,1

Figure 2.2: After reflecting off of the out-put coupler, the 0th order is reflected ex-actly back in on itself and diffracted onceagain in the AOM. The 0th order of thissecond diffraction, labelled (0,0), is con-tained by the cavity, while the other or-ders are lost.

AOM1,1 Reflected

1st order

1,01,-1

Figure 2.3: Like the 0th order, the 1storder is reflected exactly back in on itselfand diffracted once again in the AOM.This time the 1st order of this seconddiffraction, labelled (1,1), continues tothe HeNe tube and is contained by thecavity, while the other orders are lost.

mirrors and is shown in Fig. 2.1. In this setup, an AOM and a third mirror haveessentially been placed inside of a linear two-mirror laser cavity. The third mirrorreflects the 1st order back into the AOM, so both 0th and 1st orders are reflectedback into the AOM. These reflected beams also diffract. Examining the diffractionof the reflected 0th order beam in Fig. 2.2, we see that only the unshifted part ofthis diffraction pattern (beam 0,0) remains inside the cavity. The diffraction of thereflected 1st order beam is seen in Fig. 2.3; in this case the first shifted beam ofthis diffraction pattern (beam 1,1) remains inside the cavity. Thus the 1,1 beam isfrequency-shifted by the AOM twice: once propagating left and once propagatingright, so we set the AOM modulation frequency to half of the round-trip frequency.This twice-shifted beam injection locks the modes circulating between the HeNe tubeand the OC, modelocking the laser.

It is important in this configuration that the distance from the AOM to eachmirror on the right-hand side of the the setup is the same, so that the 1st and 0thorder beams traverse the same distance before recombining. If we think of a pulsepropagating rightward in the time domain through this laser, we can see that it willsplit at the AOM, the split pulses will traverse each leg separately, and then recombineagain at the AOM. To make sure that the recombined pulses constructively interfere,each leg of the cavity must be the same length. This length is labeled as L0 onFig. 2.1.

This split cavity is theoretically sound, but difficult to construct for a numberof practical reasons. Most importantly, the losses that the beam incurs through theAOM are much greater than the gain from the HeNe tube during each round trip, sothere is no hope that enough photons will circulate in the cavity simultaneously forlasing to occur. In the standard two-mirror continuous wave configuration, the HeNetube used in this thesis has such a low gain that even a glass coverslip, if introducedinto the cavity as a beamsplitter, presents too much loss. If a thin coverslip cannotbe placed into the cavity, then a thicker crystal will definitely not be supported.

The solution to the problem of too much loss is to place the AOM outside of

31

the laser cavity, allowing unimpeded amplification inside the cavity, and then couplethe frequency-shifted light back into this cavity. Figure 2.4 shows the experimentalsetup used in this thesis. The setup consists of a ‘primary cavity’ configured like astandard continuous-wave laser cavity, and a ‘secondary cavity’ formed by using athird spherical mirror to couple light back into the primary cavity. Modelocking withan external modulator was first accomplished by Foster et al. [1965], using a tank ofwater instead of an acoustooptic device. The setup used in this lab follows the layoutin the graduate lab of Jones [2009] at the College of Optical Sciences of the Universityof Arizona.

AOM

Primary Cavity

R = 1 mHR

EdgeMirror

FastPhotodiode

FPIScope

ESA

R = 60 cm HR R = 45 cm OC

0th order

1st order

Secondary CavityL

L

Figure 2.4: Experimental setup. The 0th order diffracted from the AOM is reflected by anedge mirror into either a Fabry-Perot interferometer or a fast photodiode.

The entirety of the optical setup was built on a Newport RS 300 floating opticaltable. The primary cavity consists of a HeNe laser tube and an output coupler (OC)mirror through which the laser beam is emitted. The HeNe tube is a Melles Griot 05-LHB-570 powered by the power supply of a Spectra Physics 132 laser. The right end ofthe HeNe tube is terminated in a glass window that lies perpendicular to the Brewsterangle of 632.8 nm as it transitions from air to glass. This setup, called a Brewsterwindow, minimizes the reflectivity of the glass-air interface for light polarized parallelto the window. The Brewster window thus helps polarize the laser beam. The otherend of the HeNe tube is terminated in a high-reflecting (HR) mirror with a radiusof 60 cm. The spherical output coupler mirror, on the right beyond the Brewsterwindow in Fig. 2.4, has a radius of curvature of 45 cm, but the manufacturer of the


mirror is unknown. The output coupler is mounted on a micrometer stage for fineadjustment of the cavity length. According to Section 1.1 of the Theory chapter, thispair of mirrors can theoretically support Gaussian beams at cavity lengths from 0 to45 cm, and from 60 to 105 cm. Maximizing the cavity length is desirable, becausethis decreases the axial mode spacing, which increases the number of modes that canlase simultaneously and thus minimize the duration of our pulses. At 105 cm, theround-trip frequency is νRT = 157.5 MHz, and the round-trip time is approximately7 ns. The longest cavity achieved during experimentation was approximately 94 cmlong (νRT ≈ 141 MHz; t ≈ 6.3 ns), at which point the intensity began to fluctuateand further alignment became difficult.

Light emitted from the output coupler travels a short distance before encounteringa NEOS 46110-1-LTD acoustooptic beam deflector, driven in the range of 80 MHz to130 MHz by a NEOS 21110-1ASVCO AOM driver. The drive frequency is controlledby an external voltage, supplied by a Tektronix PS280 power supply. The travelingacoustic wave in the AOM causes the incident light to diffract into a number of beams;only the 0th and 1st order beams are of interest to us. The 0th order beam is notfrequency-shifted, so it can be measured for an accurate description of the field insidethe primary cavity. The 0th order beam is deflected by an edge mirror towards eithera photodiode or Fabry-Perot interferometer. A Thorlabs DET10A fast photodiodeis used for measurement of the intensity timeseries and the radio-frequency beatsbetween laser modes. This detector was chosen for its risetime of 1ns and reasonableresponsivity. The detector is connected to either a Tektronix TDS 620B oscilloscopeor an Agilent E4411B electronic spectrum analyzer.

Alternatively, the 0th order beam can be deflected into a Burleigh SAPlus LaserSpectrum Analyzer1. This is a Fabry-Perot interferometer (FPI), which consists of anoptical cavity of two partially-transmitting confocal mirrors in front of a photodiode.Like the laser cavity, this Fabry-Perot cavity only supports distinct, evenly-spacedfrequencies. Supported frequencies are transmitted through the cavity to the pho-todiode2, allowing the measurement of the intensity of a very narrow portion of theoptical spectrum. The distance between mirrors is varied by a piezoelectric trans-ducer over time, allowing the measurement of a range of optical frequencies. Byconnecting the photodiode in the interferometer to our scope, we can measure therelative intensities of different optical frequencies in our laser signal. The mirror setused in the cavity reflects light in the range 450-700 nm.

An important property of the FPI cavity is the axial mode spacing3, also calledthe ‘free spectral range’ in the case of interferometers. This is important becauseit limits the frequency range over which the interferometer can measure. Supposelight at a frequency of ν, and thus wavelength λ = c/ν, enters the FPI as the cavity

1The interferometer is equipped with a SA-98-B1 mirror set and driven by a Burleigh RG-91ramp generator.

2The photodiode signal runs through a Burleigh DA-100 amplifier.3Because of the confocal geometry of the Fabry-Perot interferometer used, it is possible for light

to be injected into the cavity off of the optical axis. In the paraxial approximation, light then travelsfour cavity lengths per round-trip, so the formula for the axial mode spacing is νRT = c/4L, exceptin the case that the input beam is aligned exactly on the optical axis, in which case νRT = c/2L.

33

length sweeps from L1 = nλ/2 to L2 = (n + 1)λ/2. Then the light will resonatein the cavity at both L1 and L2, so light will be transmitted to the photodiode atboth lengths, and the monochromatic light will cause the FPI to measure a signal atwhat appear to be two different frequencies, when in actuality the spectrum is justrepeated. The apparent frequency difference between these transmission peaks is thefree spectral range. The Fabry-Perot cavity we use in this experiment is very short,with a free spectral range of 8 GHz4, so the spectrum from a narrow-bandwidth laserlike the HeNe is sure to lie within this range, preventing any part of the spectrumfrom overlapping with other parts.

While the 0th order beam emanating from the AOM is used for measurement,the 1st order beam is used to modelock the laser. This is done by retroreflectingthe 1st order beam back into the AOM with a third spherical mirror, a Newport10DC1000ER enhanced aluminum mirror with a radius of 1 m. As in the split-cavityconfiguration, the 1,1 beam travels along the same path as the light incident on theAOM, but in the opposite direction. This 1,1 beam is coupled into the primarycavity. As in the split cavity case, the AOM drive frequency is half of the round-trip frequency of the primary cavity, so that the twice-shifted beam that is coupledback into the primary cavity has the same frequency as one of the primary cavity’saxial modes. The shifted beam then injection-locks the modes inside of the primarycavity, modelocking the laser. As labeled in Fig. 2.4, the distance between the outputcoupler and the third mirror should be as close as possible to the length of theprimary cavity. This is because a pulse circulating in the laser is split at the outputcoupler; one part traveling out to the secondary cavity, and the other returning tothe primary cavity. After one round-trip time, these pulses will once again meet atthe output coupler, and if there is any delay between them, the two split pulses willnot maximally constructively interfere. This is similar to the requirement that eachof the legs of the split cavity be the same length, but in this case the splitting at theoutput coupler is important, while the splitting at the AOM is not (the 0th orderbeam is measured, not reflected back into the cavity). Thus there are three variablesthat must be in relatively good agreement for modelocking to occur: the round-tripfrequency of the primary cavity, the round-trip frequency of the secondary cavity,and the doubled modulation frequency. Thankfully, due to the miracle of injectionlocking, these three frequencies do not have to be precisely the same for modelockingto occur, as we will see in the Results chapter.

Note that, although the AOM operates closer to Raman-Nath regime than theBragg regime, generating multiple diffracted beams, the angle of the AOM still affectsthe relative intensities of the beams. Specifically, the intensity of a given diffractionorder is maximized when the angle between the acoustic waves and the incident lightis at one of the Bragg angles. For this experiment, the AOM was rotated so thatthe intensity of the first diffracted order was maximized, to ensure that sufficientpower was fed back into the laser for modelocking to occur (recall that increasingthe intensity of the injected light can increase the range of modulation frequency

4The range depends on the cavity length, which changes over time, but we assume that thesechanges are very small compared to the total length.


detunings over which injection locking can occur). While this necessarily decreasedthe intensity of the zeroth order beam, and thus decreased the percentage of opticalpower redirected to the photodiode, a signal was still resolvable above noise whenanalyzed by our measuring equipment.

Chapter 3

Results and Discussion

Having provided a description of the experimental setup in the previous chapter, wecan turn to the results of the experiment and attempt to analyze them. First we willdescribe the modelocking procedure and examine the behavior of the laser during thisprocess.

The first, and most time-consuming, step in the modelocking procedure was toalign the primary cavity for lasing. This was increasingly difficult at longer cavitylengths. The laser was aligned at a cavity length of about 91 cm, yielding a theo-retical axial mode spacing of about ∆ν ≈ 165 MHz. (A small axial mode spacingwas necessary to maximize the number of modes within the modelocked frequencyspectrum.) Once the primary cavity was lasing, the AOM was aligned such that the1st order mode was most intense. The third mirror, reflecting the 1st order modeback into the AOM, was then set up at a distance L− 3.5 cm away from the outputcoupler, where the correction is necessary because of the high index of refraction ofthe AOM crystal.

3.1 Unmodulated Operation

With the components laid out, data collection began. The unmodulated output ofthe laser, with the AOM turned off, was measured first. The laser was found tooperate in only one axial mode for most of the time, whereas two-mode operationwas observed only once in a while. We will call these modes ‘primary modes,’ todistinguish them from the ‘shifted modes’ that will arise later due to modulation.Fig. 3.1 shows timeseries data of the intensity waveform for one-mode operation, takenwith the fast photodiode. This is essentially continuous-wave output. The meanvoltage was approximately 0.425 mV, while the RMS voltage was 0.450 mV. Anoptical power meter, set up just outside of the output coupler mirror, measured theaverage power to be about 110 nW, far weaker than the average laser pointer (1-5mW). We believe that the noise in the output was a combination of electronic noise(measured to have an RMS voltage of 0.220 mV) as well as fluctuations in laser powerdue to changing cavity losses.

Evidence that this output was due to single-mode operation comes from Figs. 3.2

36 Chapter 3. Results and Discussion

-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

Time HnsL

Inte

nsity

HmV

L

Figure 3.1: Timeseries waveform: AOM off, one mode. Single-mode output at 91cm. Mean voltage is 0.425 mV, RMS 0.450 mV.

0 5 10 15

0

5

10

15

20

-1 0 1 2 3 4 5 6 7 8 9

Scan time HmsL

Opt

ical

spec

trum

HmV

L

Relative frequency HMHzL

Figure 3.2: Free spectral range: AOM off, one mode. Power spectral density dataof single-mode laser operation from the FPI, covering the free spectral range (8 GHz) ofthe device. The upper horizontal axis gives relative frequency spacing, while the lower axisgives sweep time. The sweep rate is calculated to be 523.56 MHz per ms.

3.1. Unmodulated Operation 37

-2 -1 0 1 2

0

10

20

30

40-1000 -500 0 500 1000

Scan time HmsL

Opt

ical

spec

trum

HmV

L


Figure 3.3: FPI spectrum: AOM off, one mode. Detail of single-mode laser operation.The sweep rate is the same as for Fig. 3.2. The peak appears to have a FWHM of about100 MHz.

-20 -10 0 10 200.0

0.2

0.4

0.6

0.8

1.0

1.2

Time HnsL

Inte

nsity

HmV

L

Figure 3.4: Timeseries waveform: AOM off, two modes. The period of the beats isabout 6 ns, close to the expected period.


and 3.3, which show a single mode in the optical spectrum taken from the Fabry-Perot Interferometer (FPI). Figure 3.2 shows the output of the FPI when the rangeof frequencies swept out by the interferometer’s cavity is greater than the free spectralrange of 8 GHz. As a result, the spectrum repeats after 8 GHz. The mirrors in theFPI oscillate to sweep across a frequency range, so the output of the FPI is actuallya timeseries. We used the free spectral range sweep presented in Fig. 3.2 to find theratio between sweep time and frequency spacing. All of the FPI data presented inthis thesis have the same sweep rate as in Fig. 3.2, which is ∼523.56 MHz per ms.Knowing the sweep rate allows us then to zoom in on small frequency ranges for abetter spectral resolution. Looking in detail at the single mode output by the FPI, asin Fig. 3.3, we see that the optical mode had a FWHM of about 100 MHz, althoughthis may be artificially broadened by the combination of the limited risetime of theFPI’s photodiode and the FPI’s fast sweep rate. The FPI cannot directly measureabsolute frequency, but we can assume that the single lasing mode is close to thestandard HeNe 632.8 nm transition (at a frequency of 474.1 THz).

It is also possible for the laser to switch briefly to two-mode operation. Thiswas generally an unstable state, and would survive for only a few seconds at most.A sample timeseries waveform can be seen in Fig. 3.4. The corresponding radio-frequency beat signal on the electronic spectrum analyzer (ESA) is shown in Fig. 3.5,and a typical FPI spectrum is shown in Fig. 3.6.

In Fig. 3.4 we can see the rather clear beating of the modes, with a periodicity ofabout 6 ns, which is close to the round-trip frequency associated with a cavity of 91cm, T = 6.25 ns. The beat frequency is also visible on the ESA, as in Fig. 3.5, andgives a far more accurate measure of cavity length than can be achieved with a ruler.In this case, the cavity appears to have a length of 91.64 cm. The broadened natureof this peak is due to noise. The apparent width of the peak is deceptive becauseFig. 3.5 is a semi-logarithmic plot; plotting the same data linearly reveals that theFWHM of the peak is only about 2.42 kHz. The clearest evidence that two modescan coexist in the cavity is seen in the FPI optical spectrum, as in Fig. 3.6. Thefrequency labeled “∆ν” in the upper horizontal axis marks the axial mode spacingcorresponding to the beat frequency that we see on the ESA in Fig. 3.5, at about163.684 MHz. The distribution of power between the modes was seen to fluctuateover time, but the mode spacing remained constant, until the spectrum collapsed intoa single mode. Note that the vertical scale is this case is the same as in the plot ofone-mode operation (Fig. 3.3), so the total power in the spectrum is roughly the samein both cases.

When we discussed laser spectra in the Theory chapter, we predicted that, if theaxial mode spacing was small enough, multiple axial modes would fall within thelaser’s gain bandwidth, and these modes could lase simultaneously, even without themodulation necessary for modelocking. Why, then, is the presence of multiple modesso rare? We expect that there is simply too much loss in the cavity for more thanone mode to stably lase at once. That is, only the highest portion of the gain curveis greater than the cavity losses, and only the modes that lie within this region havea chance to lase. See Fig. 3.7.

A possible explanation for the relatively instability of multi-mode operation is a

3.1. Unmodulated Operation 39

163.675 163.680 163.685 163.690

-12

-11

-10

-9

-8

-7

-6

Frequency HMHzL

RF

spec

trum

HdB

L

Figure 3.5: ESA spectrum: AOM off, two modes. Radio-frequency beat signal visibleon ESA, indicating the presence of two modes in the cavity.

-2 -1 0 1 2

0

10

20

30

40-1000 -500 0 DΝ 500 1000

Scan time HmsL

Opt

ical

spec

trum

HmV

L


Figure 3.6: PSI spectrum: AOM off, two modes. Detail of two-mode laser operation.The sweep rate is the same as for Fig. 3.2. The mode spacing ∆ν was found from the ESAplot in Fig. 3.5.


Cavity Loss

Laser GainOnly 1-2

modes lase

Powe

r

Frequency

Figure 3.7: Possible relative position of gain curve to axial modes, with large cavity lossespreventing lasing in more than two modes.

-2 -1 0 1 2 3

0

10

20

30

40

-1000 -500 0 500 1000 1500

Scan time HmsL

Opt

ical

spec

trum

HmV

L


Figure 3.8: FPI spectrum: AOM on, significant detuning. Typical spectrum arisingwhen AOM is turned on and aligned correctly. Each shifted mode is shifted by the samefrequency. The shifted modes are weak because they are not supported by the primarycavity.

3.2. Approaching Modelocking 41

nonlinear phenomenon called ‘cross-saturation.’ In the Theory chapter we discussedthe general phenomenon of gain saturation, where the gain experienced by any modedecreases to the level of cavity losses as the mode intensity increases to its steady-state value. This prevents mode intensities from increasing indefinitely. In some gainmedia, stimulated emission by one mode not only saturates the gain of that mode, butalso contributes to the saturation of other modes nearby. This effect, called cross-saturation, can cause one particularly intense mode to reduce the gain of nearbymodes below the cavity loss line, so that the other modes die out. Cross-saturationis usually not stated to be a significant cause for concern in gas lasers like the HeNe[Siegman, 1986, Section 12.2, pp. 462-465]. However, because our laser has such asmall axial mode spacing of 160 MHz, and because such a small portion of the gaincurve seems to be available above the cavity loss line, we suspect that cross-saturationcould be responsible for the domination of one mode in the spectrum, as well as theinstability observed during two-mode operation.

3.2 Approaching Modelocking

With the laser operational and the third mirror correctly aligned, the final step inthe modelocking procedure was the tuning of the AOM modulation frequency Ω tocoincide with half of the axial mode spacing ∆ν. In this section, we will describethe effect of injecting frequency-shifted light into the primary cavity for a range ofdetunings. When the AOM was turned on, part of the light emitted by the primarycavity as primary modes was frequency-shifted by the modulation, becoming shiftedmodes. These shifted modes were visible not only in the optical spectrum as seenby the FPI, but also created beat frequencies by interfering with the primary modes.These beat frequencies were visible on the ESA.

When the modulation frequency was significantly detuned, shifted modes becamevisible on the FPI spectrum (Fig. 3.8). The intense mode at 0 MHz is the primarymode, and the weaker modes to the right are shifted modes. The shifted modeswere weak because they did not line up with axial mode frequencies supported bythe primary cavity. The beating of these shifted modes and the single cavity mode,shown in Fig. 3.9, was visible on the ESA. The central beat frequency was at twice themodulation frequency, 2Ω. The sidebands at approximately ±60 kHz are of unknownorigin; the spacing and magnitude of these sidebands appeared to be independent of Ωand cavity length, so it is conceivable that the AOM itself has an unintended resonanceat this frequency. Fortunately, the sidebands had little effect on laser operation, sincethey were about 15 decibels less intense than the central beat frequency. The entiretyof the beat structure could also be seen at higher multiples of 2Ω, corresponding tothe beating of the higher-order shifted modes with the primary mode and with oneanother. These higher-frequency beat signals we much weaker than the structure at2Ω, because the higher-order shifted modes themselves were much weaker than theprimary mode.

Important to modulation was the structure of the central peak of the beat signal.Unfortunately, the magnitude of the peak was quite noisy. We expect that this noise


-0.05 0.00 0.05

-120

-110

-100

-90


RF

spec

trum

HdB

L

Figure 3.9: ESA spectrum: AOM-shifted signal. Beat frequencies arising when AOMis turned on and aligned correctly. The central peak lies near twice the modulation frequency(2Ω ≈ 180 MHz in this case). Relative frequencies are given since the structure is virtuallyindependent of Ω.

175.175 175.180 175.185 175.190 175.195 175.2000

2

4

6

8

10

12

Frequency HMHzL

RF

spec

trum

HmV

L

Figure 3.10: ESA spectrum: Detail of AOM-shifted signal. Plot averaged over eightacquisitions. The linear scale helps emphasize the importance of the central peak comparedto the rest of the beat spectrum generated by the AOM.

3.2. Approaching Modelocking 43

163.4 163.5 163.6 163.7 163.8 163.9 164.0

!110

!100

!90

!80

!70

!60

Frequency !MHz"

RFspectrum!dB"

163.4 163.5 163.6 163.7 163.8 163.9 164.0

!110

!100

!90

!80

!70

!60

RFspectrum!dB"

163.4 163.5 163.6 163.7 163.8 163.9 164.0!120

!110

!100

!90

!80

!70

!60

RFspectrum!dB"

163.4 163.5 163.6 163.7 163.8 163.9 164.0

!110

!100

!90

!80

!70

!60

RFspectrum!dB"

163.0 163.5 164.0 164.5 165.0 165.5 166.0

!110

!100

!90

!80

!70

!60

Frequency !MHz"

RFspectrum!dB"

a)

b)

c)

d)

e)

Primary

Primary

Primary& Shifted

(2Ω)Second Primary

& ShiftedPrimary

& Shifted(2Ω)

Figure 3.11: Beat frequencies visible on the ESA as the detuning of the modulation fre-quency from the axial mode spacing is reduced. (a) Beat signals when two primary modeslase simultaneously. Each primary mode beats with the first shifted mode. (b) Detail ofbeat between primary modes alone. Dashed line used to indicate ∆ν in following figures.(c) Beat signal as shifted modes approach axial mode spacings. Dotted line indicates 2Ω.(d) Noise due to rapid switching in and out of modelocking. (e) Shifted modes withinmodelocking region (2Ω ≈ ∆ν).


was from the vibrations of the AOM itself. Fig. 3.10 shows the average of eight plotsof the structure of the central peak, taken over about 30 seconds. Note that thisfigure is vertically linear, whereas the other ESA plots have been logarithmic, so weare focusing on only the most intense beat frequencies. The upshot of averaging thenoisy signal over time is that it allows us to estimate an upper bound on the FHWMof the central AOM beat frequency. In this case, the FWHM is about 3.2 kHz, whichwas on the same order of magnitude as the FWHM of the primary-primary beat.This suggests that the noise in the modulation would not a have a huge impact onthe modelocking of the laser.

With the AOM turned on and detuned, it was still possible (although uncommon)for two primary modes to lase simultaneously, just as we saw in the previous sectionon unmodulated operation. In this case, there were three significant beat frequencies:one between the two primary modes, one between the first shifted mode and the firstprimary mode, and one between the second primary mode and first shifted mode. Theoccurrence of three beat signals can be seen in Fig. 3.11 (a). This gave us an indicationabout what the axial mode spacing (the primary-primary beat frequency, Fig. 3.11(b)) was at that point, and how far we would have to tune the modulation frequencyso that the shifted modes lined up with axial modes. Fig. 3.11 (c) through (e) showthe structure of the beat signals as the modulation frequency Ω was decreased. Thedashed line represents the axial mode spacing frequency ∆ν and the dotted linerepresents 2Ω. Part (c) shows the beat structure that arose at small detuning. Theentirety of the shifted beat signal was amplified, including the ±60 kHz sidebandsflanking the peak. Part (d) shows the behavior on the edge of the modelockingregion, with detuning around 40 kHz: significant noise appears near 2Ω as the laserrapidly switches in and out of modelocking. In part (e), the detuning is small enough(∼20 kHz) that the laser is modelocked. The noise is reduced and the peak of thebeat signal is amplified significantly.

3.3 Modelocking Behavior

When the beat signal was tuned to within about 20 kHz of the axial mode spac-ing, modelocking occurred. Fig. 3.12 shows the modelocked intensity waveform, withpulses separated by about 6 ns, with a FWHM of about 2 ns. The laser maintainedthis output as long as the modulation frequency did not drift outside of the lockingrange. The average output signal measured by the photodiode during this acquisitionwas 0.689 mV, which was higher than the unmodulated case of 0.425 mV, proba-bly because modelocking fills more of the gain medium’s bandwidth. For reference,Fig. 3.13 shows the beat signal for this modelocked data, showing a structure verysimilar to Fig. 3.9.

Figure 3.14 shows the optical spectrum corresponding to Fig. 3.12. We can seethat at least four modes were populated. This figure has the same vertical scale asin Figs. 3.3 and 3.6, so we can see that the most intense mode has about half theintensity of the single mode that arises in the unmodulated case, but considering allfour modes, there is a bit more total power in the spectrum now than there was in

3.3. Modelocking Behavior 45

-20 -10 0 10 200.0

0.5

1.0

1.5

Time HnsL

Inte

nsity

HmV

L

Figure 3.12: Timeseries waveform: Modelocked. Modelocked output at 91 cm. FWHMof pulses is about 2 ns.

163.5 163.6 163.7 163.8

-120

-110

-100

-90

-80

-70

-60

Frequency HMHzL

RF

spec

trum

HdB

L

Figure 3.13: ESA spectrum: Modelocked. Beat signal corresponding to Fig. 3.12. Thisdata was used to find 2Ω ≈ 163.698 MHz. Dashed line shows ∆ν ≈ 163.684 MHz.


-1 0 1 2 3

0

10

20

30

40-500 0 2W 4W 500 1000 1500

Scan time HmsL

Opt

ical

spec

trum

HmV

L

Relative frequency HHzL

Figure 3.14: FPI spectrum: Modelocked. Here 2Ω ≈ 163.698 MHz. Note that verticalscale is same as in Figs. 3.3 and 3.6.

the unmodulated case.The selection of the zero-point on the horizontal axis of Fig. 3.14 is not arbitrary.

In fact, there is a very weak mode centered at approximately this point, and this modelines up with the single primary mode that is visible when the laser is unmodulated.This is shown in Fig. 3.15, where a modelocked spectrum is shown as a solid line,and the corresponding unmodulated spectrum is shown as a dashed line, which wasacquired by turning off the AOM. In order to show the detailed structure of themodelocked spectrum, the FPI’s amplification of the spectrum was increased tenfoldabove the other FPI plots in this thesis. In order to show the (far more intense)unmodulated signal on the same plot, both sets of data were individually rescaled,so the scales of the two lines should not be considered accurate. What is important,though, is that the unmodulated mode overlaps with the the first, very weak mode inthe modelocked spectrum. This may seem counterintuitive: modelocking has causedhigher-frequency modes to become more intense than the central primary mode. Weexpect that this behavior can be attributed to two facts. The first is the fact that,because our experimental set up only utilized light that was up-shifted in frequency,the flow of optical power due to modulation is unidirectional. This creates a Gaussian-shaped envelope of mode intensities which becomes wider and is pushed rightwardsin frequency-space as time progresses. The second fact we must consider is the effectof gain, gain saturation, and cavity losses. While power is being transferred to higherfrequencies, lower frequencies (closer to the gain peak) experiences more gain thanhigher frequencies, and this increases the intensity of lower-frequency modes overtime. Uniform cavity losses prevent modes far from the gain peak from becomingvery intense. By considering all of these effects together, it is plausible that themodelocked laser can achieve the steady states indicated in Figs. 3.14 and 3.15.

3.3. Modelocking Behavior 47

-0.5 0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0-250 0 2W 4W 500 750 1000

Scan time HmsL

Opt

ical

spec

trum

Ha.u

.L


Figure 3.15: FPI spectrum: Modelocked vs. unmodulated. Amplified modelockedspectrum (solid line) and unmodulated spectrum (dashed line), showing relative frequenciesof modulated and unmodulated modes. Shows distribution of modelocked mode powerrelative to primary mode. Note that the two plots are scaled for simpler comparison.

It is important to reiterate that the existence of a ‘modelocking range,’ a rangeof detunings over which the laser stays modelocked, is a direct result of injectionlocking. If the laser system were completely linear, no injection locking would occur,and modelocking would require a perfectly tuned modulation frequency as well as astabilization equipment to reduce phase noise. It is difficult to show the frequencypulling associated with injection locking directly, because the FPI signal is noisy andthe horizontal scale is too coarse to notice small frequency shifts. Frequency pullingcan, however, be seen as a shift of the beat frequency between primary modes onthe ESA, as in Fig. 3.16. The thin line in this figure shows the primary-primary beatfrequency when the AOM is turned off, while the thicker line shows the modelockedbeat signal when the AOM is turned on and correctly tuned. Readers will have totrust the author that the unmodulated primary-primary beat frequency was neverseen as long as the laser was modelocked, because the injection of frequency-shiftedlight pulled the frequencies of the primary cavity modes to coincide with the injectedmodes. The best evidence that the thicker line represents modelocking is the factthat the modulated beat frequency is about 9 decibels greater in magnitude thanthe unmodulated beat, indicating that there are more modes, and that the phasedifferences between mode pairs is more uniform. This coupling of mode phases wasanother result of injection locking.

Assuming that the modelocking range for our laser was about 20 kHz above andbelow ∆ν, we can use the injection locking range equation 1.64,

ε = γ

√Iinj

I0

≈ 20 kHz, (3.1)


163.60 163.65 163.70 163.75 163.80

-120

-110

-100

-90

-80

-70

-60

Frequency HMHzL

RF

spec

trum

HdB

L

Figure 3.16: ESA spectrum: Modelocked vs. unmodulated. Unmodulated beatsignal (thin line) compared to modelocked beat signal (thick line). The displacement of thepeaks indicates that the primary modes were frequency-shifted by about 17.4 kHz in thepresence of the injected modes. This is evidence of the frequency-pulling associated withinjection locking.

to learn more about how our laser operated. Here where Iinj is the intensity of theinjected signal, I0 is the intensity of the undisturbed mode, both measured insidethe cavity. The cavity decay rate γ ≈ δ

TRT, where δ is the percentage of light that

is emitted from the output coupler mirror, and TRT ≈ 6.25 ns is the round-triptime through the cavity. Estimating δ ≈ 1%, the ratio of injected mode intensityto undisturbed mode intensity was about 0.02%. This means that very little light,after being split by the AOM, reflecting off the third mirror, being split by the AOMa second time, and transmitting through the ouput coupler, made it back into theprimary cavity. However, this was enough light to broaden the injection locking rangeto a point where modelocking could be achieved without too much difficulty.

Conclusion

In this thesis, we have introduced the theoretical components necessary for discussingmodelocking, utilized the theory to motivate the design of a particular modelockedlaser, and presented experimental evidence that such a laser operates as intended.Although the depth of theoretical material presented may be exhausting, it is hardlyexhaustive. A more general and rigorous form of pulse-propagation modeling can beaccomplished by using Haus’ ‘Master equation of modelocking’ [Haus et al., 1991],which is a partial differential equation that can accommodate active and passivemodelocking, and take into account complicated effects like intracavity dispersionand nonlinearities of the gain medium that arise for high peak intensities. Thisequation also allows for modeling of pulse shapes other than Gaussian pulses. Anothertype of modeling in the frequency domain, using ‘coupled-mode’ equations, presentsanalytical solutions that track how power is transferred between modes over time inan actively-modelocked laser. This frequency-domain modeling can easily account fordetuning of the modulation frequency. If we wanted a more accurate model of how weshould expect our laser to operate, we could have applied both the Master equationand the coupled-mode equations to our particular setup.

Our laser succeeded in producing pulses that were about 2 nanoseconds long,although the low intensity of the laser suggests that we could not expect to useits output for industrial or research applications. There are many other designs ofmodelocked lasers that could have been studied and assembled, the limiting factorsbeing time available to the student and funds available from the institution. Otherlaser designs could have been implemented that probably would have yielded shorter,more intense pulses, such as an erbium-doped fiber ring laser, or a diode laser with ACpumping at the round-trip frequency. If appreciable output powers could be obtainedwith one of these laser designs, a second-harmonic crystal could be used in opticalautocorrelation measurements to discover the shape of pulses that are too fast forphotodiode detectors to measure.

References

Acoustooptics Database: Deflectors, 2001. URL http://acoustooptics.phys.msu.

su/deflectors.asp. MSU, Faculty of Physics, Oscillations Department.

Robert Adler. A study of locking phenomena in oscillators. Proceedings of the IEEE,61(10):1380–1385, October 1973.

R. W. Boyd. Nonlinear Optics. Academic Press, 3 edition, 2008.

William S. C. Chang. PRINCIPLES OF LASERS AND OPTICS. Cambridge Uni-versity Press, 2005.

Clark-MXR. Micromachining Handbook. Clark-MXR, Inc., 2009. URL http://www.

cmxr.com/mmhandbook.html. Electronically available publication.

Li Ding, Lana J. Nagy, Lisen Xu, Jack C-H. Chang, Jennifer Swanton, Wayne H.Knox, and Krystel R. Huxlin. Enhancement of intra-tissue refractive index shap-ing (iris) of the cornea by two-photon absorption. In Conference on Lasers andElectro-Optics/International Quantum Electronics Conference, page CWE4. Opti-cal Society of America, 2009.

John F Federici, Brian Schulkin, Feng Huang, Dale Gary, Robert Barat, FilipeOliveira, and David Zimdars. Thz imaging and sensing for security application-sexplosives, weapons and drugs. Semiconductor Science and Technology, 20(7):S266, 2005.

M Ferray, A L’Huillier, X F Li, L A Lompre, G Mainfray, and C Manus. Multiple-harmonic conversion of 1064 nm radiation in rare gases. Journal of Physics B, 21:L31, 1988.

L. Curtis Foster, M. D. Ewy, and C. Burton Crumly. Laser Mode Locking By AnExternal Doppler Cell. Applied Physics Letters, 6:6–8, 1965.

L. E. Hargrove, R. L. Fork, and M. A. Pollack. Locking of he-ne laser modes inducedby synchronous intracavity modulation. Appl. Phys. Lett., 5:4–5, 1964.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen. Structures for additive pulse modelocking. J. Opt. Soc. Am. B, 8(10):2068–2076, 1991.

52 References

H.A. Haus. Mode-locking of lasers. IEEE Journal of Selected Topics in QuantumElectronics, 6:1173–1185, Nov/Dec 2000.

R. J. Jones. Active Modelocking of a Helium-Neon Laser, 2009. URL http://www.

optics.arizona.edu/opti511l/2009_files/modelocking_Fall2009.pdf.

K Kurokawa. Injection locking of microwave solid-state oscillators. Proceedings of theIEEE, 61(10):1386–1410, October 1973.

Willis E. Lamb. Theory of an optical maser. Phys. Rev., 134(6A):A1429–A1450, Jun1964.

Corneliu Manescu. Controlling and probing atoms and molecules with ultrafast laserpulses. PhD thesis, University of Florida, 2004.

M. G. Moharam and L. Young. Criterion for Bragg and Raman-Nath diffractionregimes. Appl. Opt., 17(11):1757–1759, 1978.

T. M. Niebauer, James E. Faller, H. M. Godwin, John L. Hall, and R. L. Barger.Frequency stability measurements on polarization-stabilized He-Ne lasers. Appl.Opt., 27(7):1285–1289, 1988.

Susanne Pfalzner. An introduction to inertial confinement fusion. Taylor & FrancisGroup, Boca Rotan, FL, 2006.

A. Pikovsky, M. Rosenblum, and J. Kurths. Synchronization: A universal concept innonlinear sciences. Number 12 in Cambridge Nonlinear Science Series. CambridgeUniversity Press, 2001.

Anthony E. Siegman. Lasers. University Science Books, 1986.

D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud,U. Keller, V. Scheuer, G. Angelow, and T. Tschudi. Semiconductor saturable-absorber mirror assisted Kerr-lens mode-locked Ti:sapphire laser producing pulsesin the two-cycle regime. Opt. Lett., 24(9):631–633, 1999.

Osamu Wada. Femtosecond all-optical devices for ultrafast communication and signalprocessing. New Journal of Physics, 6(1):183, 2004.

M. Wollenhaupt, A. Assion, and T. Baumert. Section 12.2: Generation of Femtosec-ond Laser Pulses via Mode Locking. In Frank Trager, editor, Springer Handbookof Lasers and Optics, chapter C.12. Springer, 2007.

active modelocking of an open-cavity helium-neon … modelocking of an open-cavity helium-neon laser...

Documents