RICE UNIVERSITY

461nm Laser For Studies In Ultracold Neutral Strontium

by

Aaron D Saenz

A Thesis Submitted

in Partial Fulfillment of the

Requirements for the Degree

Master of Science

Approved, Thesis Committee:

Thomas C. Killian, ChairmanAssistant Professor of Physics andAstronomy

Randall G. HuletFayez Sarofim Professor of Physics andAstronomy

Stanley A. DoddsAssociate Professor of Physics andAstronomy

Houston, Texas

July, 2005

ABSTRACT

461nm Laser For Studies In Ultracold Neutral Strontium

by

Aaron D Saenz

A 461 nm laser was constructed for the purposes of studying ultracold neutral

strontium. The dipole-allowed 1S0 → 1P1 transition at 460.862 nm can be used in

laser cooling and trapping, optical imaging, Zeeman slowing, and photoassociative

spectroscopy. We produce light at this wavelength by converting infrared light at

922nm from various IR sources, notably a Ti:Sapphire laser, via second harmonic

generation in a frequency doubling cavity using a potassium niobate crystal. This

thesis will discuss the motivation, optical resonator, locking electronics, and charac-

terization of a 461 nm laser.

Acknowledgments

I would like to thank my committee: S.A. Dodds, R.G. Hulet, for their time and

patience, and especially my advisor T.C. Killian for his guidance and assistance both

in the thesis and my work in general.

Many thanks go to my colleagues at the Killian Lab: S. Laha, P. Gupta, P.G.

Mickelson, Y.N. Martinez, S.B. Nagel, and C. Simien who gave good advice and

valuable suggestions. Y.C. Chen was sorely missed but continued to serve as the

apotheosis of post-doctoral excellence.

I give my thanks to my family and friends whose support is vital and appreciated.

G.D. Wiley, it would not have been fun without you.

Finally: Thanks Mom and Dad, for everything.

Contents

Abstract ii

Acknowledgments iii

List of Figures vi

1 Introduction 1

1.1 Ultracold Neutral Strontium . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Laser Cooling and Trapping . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Absorption Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Photoassociative Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Frequency Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2 Optics and Beam Coupling 14

2.1 Frequency Doubling Cavity . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Spatial Modes of the Cavity . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Beam Coupling and Alignment . . . . . . . . . . . . . . . . . . . . . 20

2.4 Output Beam Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Electronics 29

3.1 Error Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

v

3.2 Locking The Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Characterization of Electronics . . . . . . . . . . . . . . . . . . . . . 34

4 Characterization of the 461nm Laser 42

4.1 Efficiency of Frequency Conversion . . . . . . . . . . . . . . . . . . . 45

4.2 Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 Conclusion 52

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Improvements and Future Work . . . . . . . . . . . . . . . . . . . . . 52

A Computational Analysis of Beam Profiles 54

B Computational Modelling of Beam Profiles 60

References 64

List of Figures

1.1 Strontium Energy Diagram . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Laser Cooling on Strontium . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Absorption Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Photoassociative Spectroscopy Energy Diagram . . . . . . . . . . . . 8

1.5 Photoassociative Spectroscopy Results . . . . . . . . . . . . . . . . . 10

2.1 Optical Elements of Doubling Cavity . . . . . . . . . . . . . . . . . . 17

2.2 Close-up of Potassium Niobate Crystal . . . . . . . . . . . . . . . . . 17

2.3 Beam Waists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Beam Waists Inside Resonator . . . . . . . . . . . . . . . . . . . . . . 21

2.5 Infrared Beam Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6 461 nm Beam Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Logic Flow of Feedback Loop . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Schematic of Locking Circuit . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Coincidence of IR Transmission and Blue Output Modes . . . . . . . 35

3.4 Transmission Modes of Cavity . . . . . . . . . . . . . . . . . . . . . . 36

3.5 Close-up of Transmission Modes . . . . . . . . . . . . . . . . . . . . . 36

3.6 Output Modes of Cavity . . . . . . . . . . . . . . . . . . . . . . . . . 37

vii

3.7 Close-up of Output Modes . . . . . . . . . . . . . . . . . . . . . . . . 37

3.8 Switching Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.9 Comparison of Sweep Versus Lock Power in Blue . . . . . . . . . . . 40

4.1 Power Out vs. Temperature . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Efficiency of Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Temperature Controller Calibration . . . . . . . . . . . . . . . . . . . 48

4.4 Optimum Temperature for Input Power . . . . . . . . . . . . . . . . . 50

A.1 Matlab1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

A.2 Matlab2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.3 Matlab3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

A.4 Matlab4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.5 Matlab5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

B.1 MathematicaCode1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

B.2 MathematicaCode2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

B.3 MathematicaCode3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

B.4 MathematicaCode4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

B.5 MathematicaCode5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B.6 MathematicaCode6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Chapter 1

Introduction

This thesis describes the construction of a 461 nm laser in the form of a frequency

doubling cavity that facilitates various experiments on ultracold neutral strontium.

This laser excites an allowed dipole transition from the 1S0 to the 1P1 states, in Sr and

is the basis through which optical trapping and cooling, fluorescence and absorption

imaging, and photoassociative spectroscopy (PAS) can all be performed.

The following chapters will describe the physics and construction of the laser as

well as characterize its performance. Designed to compensate for a gap in available

commercial laser sources in the blue spectral range, the laser utilizes second harmonic

generation (SHG) to frequency double near infrared light (IR) from 922 nm to 461

nm. We will first review the various experimental uses of the 461 nm laser, and touch

upon the theory behind SHG before detailing the operation of the laser itself.

1.1 Ultracold Neutral Strontium

Ultracold neutral strontium atoms provide unique opportunities for research with

narrow intercombination line transitions that may lead to all optical means of ob-

taining quantum degeneracy [2][3][4], and/or may be utilized for optical frequency

standards [5] [6] [7] [8]. There are several available bosonic isotopes for quantum

degeneracy including the most abundant 88Sr. Strontium displays interesting atomic

2

Figure 1.1: Strontium Energy Diagram. Strontium energy levels for commonlyused transitions. Selected decay rates (1/s) and excitation wavelengths are shown*Taken from [1].

properties due to the absence of nuclear spin and the subsequent lack of hyperfine

structure. Strontium atoms also approach the ideal two level theoretical systems

used to commonly describe atomic laser cooling and trapping [9]. In such cases, the

modelled atomic transition is in a J=0 → J=1 system. In strontium, the two valence

electrons may couple in parallel or anti-parallel, corresponding to triplet or singlet

states that approximate the model [10]. Figure 1.1 shows a partial energy diagram of

88Sr, with emphasis given to the transitions used within our laboratory.

1.2 Laser Cooling and Trapping

We recently completed construction of a new apparatus for studying laser cooled

strontium. The doubling cavity I built generates the 460.862 nm photons for laser

cooling and trapping of strontium along the 1S0 → 1P1 allowed dipole transition.

3

Magneto-Optical Traps (MOT), thoroughly discussed in Metcalf and van der Straten’s

Laser Cooling and Trapping, are now commonly used tools in many atomic physics

laboratories [11]. In such traps, three orthogonal counter propagating beams cool and

trap atoms inside a quadrupole magnetic field. In our setup, such atoms are provided

by heating solid strontium to create an atomic beam that is collimated and Zeeman

slowed using the same 1S0 → 1P1 transition.

Optical cooling is performed by the well known Doppler technique, cooling stron-

tium atoms to sub-Kelvin temperatures, and obeying the well known Doppler limit

given by:

kBTdoppler =~Γ2

(1.1)

Where Tdoppler ≈ 760 µK is the Doppler limit temperature and Γ is the transition

rate. For the 1S0 → 1P1 transition the Doppler limit is many time greater than the

limit set by photon recoil:

kBTrecoil =~2k2

M(1.2)

Where k is the wavenumber of the light, and M the mass of the atom, and Trecoil ≈

1 µK.

In order to trap, orthogonal circular polarization is chosen for each propagat-

ing/counter propagating beam, and anti-Helmholtz coil pairs provide magnetic field

gradients to generate Zeeman shifts in the atoms traveling away from the intersection

of the three MOT beams. The MOT cooling and trapping provides us with cold,

4

dense, and spatial constrained samples aptly suited for atomic experiments. Typical

results for our 461 nm MOT are < 7 mK, > 1017 atoms/m3, and sizes ∼ 1 mm3.

Figure 1.2 highlights the MOT setup in our laboratory including the doubling

cavities which are the heart of the 461 nm laser described in this thesis. Not pictured

is the red, 689 nm, intercombination line MOT that follows the 1S0 → 3P1 transition

from Figure 1.1, and is the subject of S. Nagel’s master’s thesis [10]. Notice that the

setup also includes absorption imaging which is further discussed below.

1.3 Absorption Imaging

We are able to probe cooled and trapped strontium atoms through purely optical

means utilizing absorption imaging. My doubling cavity is used to provide an imaging

beam of 461 nm light. That image beam is split off the main MOT beam using

an acouto-optic modulator (AOM), making it slightly detuned from resonance (∼40

MHz), and passed through the atoms inside the MOT which absorb photons along the

1S0 → 1P1 transition. The absorbed photons are emitted in a randomized manner such

that relatively few travel along the same k vector as the imaging beam. Effectively,

the atoms cast a shadow in the path of the beam, and by placing a CCD camera

behind the atoms we can capture an image of that shadow.

The shadow contains a plethora of valuable information about the atomic cloud.

Not only can the camera capture information about the spatial outline and movement

of the cloud, we can use the intensity of the absorption to determine the number of

5

Magneto-Optical Trap

Strontium Reservoir

Zeeman-Slowed Atomic Beam

Frequency Doubling

Imaging Camera

To Zeeman Slower

To MOT

461 nm

Ti-Sapphire Laser

922 nm

Zeeman Beam

Imaging Beam

Magnetic Coils

Figure 1.2: Laser Cooling on Strontium Diagram of our experimental setup forstudies on ultracold neutral strontium.

6

CameraAtomsImage Beam

Figure 1.3: Absorption Imaging A beam slightly detuned from the 1S0 → 1P1

transition is incident upon the trapped atoms. The image of the cast shadow isrecorded by a CCD camera and the corresponding Optical Depth is calculated.

7

atoms and approximate density. Using Beer’s Law, we can relate the optical density

(OD) of the MOT to the image intensity with (Iatoms) and without (Ibackground) the

atomic cloud:

OD(x, y) = ln[Ibackground(x, y)/Iatoms(x, y)] (1.3)

OD(x, y) = α(ν)

∞∫

−∞

ni(x, y, z) dz (1.4)

OD(x, y) =α(ν)n0√

2πσz

e−x2

2σ2x

+−y2

2σ2y (1.5)

Where α(ν) is the aborption cross section at the image beam frequency ν, n0 is the

peak atom density, and we assume a Gaussian distribution of atoms consistent with

our MOT [12]. We image along the z axis, perpendicular to the magnetic coils. For σz

we must infer a value, typically ∼ √σxσy where sizes in the x and y axis are typically

similar to each other within a factor of two. In Figure 1.3, we see an overview of the

imaging process as well as an example of an OD distribution for our MOT.

Absorption imaging along the 1S0 → 1P1 transition is the most commonly used

and definitive diagnostic for ultracold neutral strontium in our lab. By varying the

time between capture and camera exposure, we can watch our MOT spatially expand,

yielding information on temperature and lifetime of our system. It is through this

simple but powerful technique that we can analyze the results of the various other

experiments we perform.

8

461 nm

Ene

rgy

Internuclear separation

1S0 + 1S

0

1S0 + 1P

1ν′=12

34....

1Σ+u

1Σ+g

1Πg

1Πu

1Σ+g

SC

RE

Figure 1.4: Photoassociative Spectroscopy Energy Diagram Abbreviations:State Changing Collision (SC) and Radiative Emission (RE) We only concern our-selves with the 1Σ+

u potential *Taken from [13].

1.4 Photoassociative Spectroscopy

Before we built the new apparatus, the doubling cavity was used to generate light

to perform photoassociative spectroscopy (PAS), which provides valuable insight into

molecular potentials and excited state lifetimes as well as scattering lengths of the 88Sr

and 86Sr isotopes. Several papers have been recently written on PAS at both short

and long range [13] [2] [14]. Figure 1.4 gives the atomic and molecular potentials

as functions of internuclear separation. Notice the levels in the excited molecular

potential, ν ′. During PAS, atoms in close proximity to one another can be optically

driven to combine into molecular states by excitation into these levels. This light

9

assisted combination is the foundation of photoassociation. Atoms paired into the

steep molecular potential quickly gain energy as they move along the curve. Many will

transition back to the atomic state and gain enough kinetic energy to leave the trap -

called radiative emission (RE). Even more kinetic energy is gained if the paired atoms

go through a state changing collision (SC) wherein the molecular state changes to a

lower-lying electronic configuration of free atoms [13]. In either case, photoassociated

atoms no longer remain inside the atomic MOT. This absence of atoms that have

been photoassociated allows us to set the number of atoms in the MOT as a direct

indicator of how effectively atoms are being coupled into the molecular state.

During our PAS experiments, a 461 nm beam was detuned from the 1S0 → 1P1

resonance and made incident upon 88Sr or 86Sr atoms that were trapped and cooled in

the intercombination line MOT briefly alluded to earlier in this section. By varying

that detuning, the PAS beam scanned the molecular potential. When on resonance

with a quantum level, the number of atoms in the MOT would be reduced, often by

more than 50%. When these experiments were performed at small detunings (0-2

GHz) corresponding to large internuclear spacings (∼ 20− 30 nm) they are known as

PAS studies at long range and gave insight into long range parameters of the excited

state potential. Figure 1.5 shows typical results for our PAS studies at long range

(88Sr only), which were normalized for intensity and duration of the beam. The level

spacings allowed us to determine the 1P1 lifetime at 5.22± .03 ns [2].

10

−2270 −2235 −2200 −2165 −2130 −2095 −2060

1.8

2

2.2

2.4

x 106

(a)

detuning (MHz)

num

ber

of a

tom

s

(a)

datafit

−2450 −2150 −1850 −1550 −1250 −950 −650

0.4

0.6

0.8

1

detuning (MHz)

frac

tion

rem

aini

ng (b)

|61

|60

|59

|58

|57

|56

|55

|54

|53

|52

|51

|50

|49

|48

61 60 59 58 57 56 55 54 53 52 51 50 49 48

0

10

20

quantum number

obs.

−ca

lc. (

MH

z) (c)

Figure 1.5: Photoassociative Spectroscopy Results Typical spectra taken forlong range PAS studies on 88Sr (a)The signal from a single quantum level has beenfit using a lorentzian curve (b)Various quantum levels (ν ′ from Figure 1.4) at longrange show the decrease in signal strength nearing the atomic resonance (c) Resultscorresponded well with theoretical calculations *Taken from [13].

PAS studies at short range occured at larger detunings (2-2000 GHz) correspond-

ing to smaller internuclear spacing (< 4 nm). These studies allowed us to probe the

molecular potential further, determining the ground state wavefunction, and giving

values for the scattering lengths of 88Sr and 86Sr that look promising for achieving

quantum degeneracy [2] [4] [3].

11

1.5 Frequency Doubling

Second harmonic generation of light is a prime example of non-linear optical phe-

nomena in which the frequency of light is doubled through polarization waves in a

medium. The polarization of the medium can be expressed as:

P = ε0χ1E1 + ε0χ2E

2 + ε0χ3E3... (1.6)

Where E is the electric field imposed on the medium and χi is the ith order suscepti-

bility of the medium. If an electromagnetic wave of frequency ω is incident on such

a medium with a non-trivial χ2 then a corresponding E-M wave will propagate with

frequency 2ω. The power of that wave will follow the relation:

Pω2 = [2η3

oω21d

2effL

2

A]P 2

ω1(sin ∆kL

2∆kL

2

)2 = ξnLP 2ω1

(sin ∆kL

2∆kL

2

)2 (1.7)

∆k =2ω1(n1 − n2)

c(1.8)

Where A is the area, k is the wavevector, L the length of the medium, c is the speed

of light in vacuum, n1 and n2 are indices of refraction for each frequency, ξnL is the

nonlinear conversion efficiency, deff is the nonlinear coefficient of the doubling crystal,

and η0 = 377/n1 [15].

Typically, ∆k is non zero, and the power of the frequency doubled wave is small

[15]. If the two waves can be phase-matched, however, ∆k goes to zero and the power

12

can be optimized. This matching can only occur if the indices of refraction for each

frequency are identical.

Potassium Niobate (KNbO3) is aptly suited for frequency doubling in the blue-

green portion of the visible spectra. KNbO3 is transparent in both the visible and

near infrared regimes, and has a large deff suitable for efficient doubling as seen

in equation 1.7. Potassium Niobate is often used in frequency conversions to the

blue-green spectra with reported efficiencies > 80% [16].

If the fundamental and second harmonic have orthogonal polarizations, the crys-

tal’s birefringence may make it possible for n2 to equal n1. The equality is fine tuned

by varying incident angles of light with respect to the axes of the crystal, or through

varying the temperature of the crystal. The first technique is utilized, in our case, as

the crystal is being manufactured: the crystal is cut at an angle, with respect to its

optical axes, to phase-match the 922 nm and 461 nm light near room temperature, as

discussed in [17] [18] [19]. We also use the second technique: the temperature suscep-

tibility of the crystal is such that the indices of refraction in the two perpendicular

axis, which correspond to the two polarizations of IR and blue light, are widely tun-

able [19] [20]. This allows us to find and fine-tune near-room temperature for which

n2 is equal to n1. For further information on the indices of refraction, the reader is

referred to Figure 7 in [20]

13

1.6 Outline

The thesis will describe the process through which the 461 nm laser was built

and operates. The motivation for the 461 nm laser, as well as the theory of SHG, is

discussed earlier in this introductory chapter. Chapter 2 will discuss the pertinent

optical resonator beam parameters of the laser, as well as detailing some of its optical

characteristics. Chapter 3 concerns itself with electronic feedback and stabilization

and relates the electronic parameters of the lock-loop setup with the transmitted and

output modes of the laser’s doubling cavity. Characterization of the laser, including

overall efficiency and temperature response is given in chapter 4, along with compar-

isons to similar systems from within and outside our laboratory. The final chapter

gives a brief conclusion to this thesis and discusses possible improvements and av-

enues for future work. Appendices describing the computational techniques involved

in modelling the beam profiles of the cavity are included.

Chapter 2

Optics and Beam Coupling

In order to create the 461 nm visible laser light, an infrared 922 nm beam couples

into a Potassium Niobate crystal within a resonant optical cavity. That 922 nm

source has varied over the history of the 461 nm laser. The ideal source would be

commercially available, easily tuned in frequency, able to be locked to an atomic

reference, of sufficient power and intensity, and stable over long periods of time.

These requirements suggest either a diode or Ti:Sapphire laser.

Diode lasers are easily tunable by means of optical feedback. Using a diffrac-

tion grating in Littrow to form an extended cavity setup, they can be continuously

tuned in frequency by 10s of GHz as the grating is adjusted via a piezo-electric trans-

ducer (PZT). Utilizing temperature and current control, the diffraction grating can

be adjusted even further for discontinuous tuning over 1000s of GHz. This high

range in tunability was ideal for the short range photoassociative spectroscopy stud-

ies described in section 1.4. For those experiments, the doubling cavity was fed by

a TEC100 diode laser from Sacher Lasertechnik with output of 60 mW and able to

be continually tuned without mode hops by > 8 GHz using compensating current as

a PZT adjusted grating angle. Later, IR power was boosted to 125 mW using the

more powerful TEC120. Laser linewidth was measured using a Fabry Perot etalon

15

and was approximately 85 MHz. The diode lasers, though aptly suited for the PAS

studies, do not provide sufficient power for laser cooling and trapping as detailed in

section 1.2. For the PAS experiments, relative measurements to a locked source using

a Burleigh WA-1000 wavemeter were required.

Coherent’s MBR-110 is a Verdi-10 pumped Ti:Sapphire infrared laser that pro-

vides > 1 W of IR power for 10 W of pumped power. Using a standard bow-tie

configuration and with a variably angled Fabry-Perot etalon, it can be easily tuned

over the near infrared spectrum (670 to 1090 nm), and can be continuously scanned

over ∼10 GHz while the etalon locks its frequency [15]. Two other 461 nm lasers built

previous to the one characterized in this thesis are already supplied by the Ti:Saph.

laser in our laboratory. A saturated absorption cell is pumped using one of those

lasers, and the corresponding signal locks the Ti:Saph. to the 1S0 → 1P1 transition.

The large available power makes the Ti:Saph laser ideal for the 461 nm laser for use

in the MOT or Zeeman beams. Long range studies in PAS were also accomplished by

passing the 922 nm beam through acousto-optic modulators (AOM) before entering

the doubling cavity, and again after exit [13]. In this case the detuning in the IR

beam was doubled as it was converted to blue light. This means that we were able

to obtain detunings > 2 GHz using AOMs with maximum detunings < 350 MHz a

piece.

For the setup described in this thesis, the IR source is always the Ti:Saph. laser

16

tuned to 922 nm and passed through a fiber optic cable. This fiber is designed for

transmission in the near IR, and has a high coupling efficiency > 75%. The optical

fiber passes the IR beam onto another optical table where it is coupled into the

doubling cavity as described below.

2.1 Frequency Doubling Cavity

At its heart, the frequency doubling cavity is a simple optical resonator formed

by two mirrors. Such a resonator, often referred to as a Fabry-Perot etalon, has two

helpful qualities for frequency doubling. First, for high reflectivity of the mirrors, light

circulates many times in the resonator before it escapes, creating higher intensities of

light inside the resonator. Frequency conversion of infrared to blue light is directly

proportional to the square of incident power in the crystal, as seen in equation 1.7.

Accordingly, increases in the IR intensity within the crystal are desirable to maximize

IR to blue light conversion. Second, destructive interference of the circulating light

limits the possible wavelengths of light allowed inside the resonator to λ = 2LN

Where

N is any integer, and L is the length of the cavity. For a given wavelength of our

IR source, the cavity will not generally allow the light inside. We scan L in order

to match λ and see a resonant peak in transmission out of the back of the cavity

when the two are in sync. Keeping the cavity on resonance is accomplished using a

servo-lock described in section 3.2.

For the 461 nm laser, the first mirror of the optical resonator is formed by the input

17

Input Coupler

Fast Photo-Diode

Potassium Niobate Crystal

Mirror

Dichroic Mirror

IR Fiber Coupler

Mode Matching Lens

Collimation Lens

Cylindrical Lens

Figure 2.1: Optical Elements ofDoubling Cavity.

PZTInput

Coupler

KNbO3

n~2.28

Thermo-electric cooler

R1=.975 @922nm

T1>.85@461nm

R2>.999 @922nm

R2>.999 @461nm

R=.08 @922nm

R=.036@461nm

Figure 2.2: Close-up of PotassiumNiobate Crystal.

coupler, and the second mirror by the back surface of the KNbO3 crystal. Figure

2.1 diagrams our setup and Figure 2.2 provides an enlarged look at the resonator.

Typically, optical resonators are characterized by three properties: the free spectral

range (FSR), finesse (F ), and full width half maximum of a resonance (FWHM). For

a cavity without loss, that is a cavity where the crystal does not frequency double

but merely acts as a transparent medium, relations for each are given by:

FSR =c

2Leff

(2.1)

F =π√

R

1−R(2.2)

FWHM =FSR

F(2.3)

With

Leff = Dair + Dcrystal ∗ ncrystal (2.4)

R =√

R1R2 (2.5)

Where c is the speed of light in vacuum, Dair ≈ 15 mm is the distance in air in

the resonator, Dcrystal ≈ 5 mm is the crystal length, ncrystal ≈ 2.28 is the index of

18

refraction for the crystal. Using these values for our doubling cavity, we can expect

FSR ≈ 5.7 GHz, Finesse ≈ 239, and FWHM ≈ 23.7 MHz.

Our doubling cavity, however, loses IR power due to conversion to blue light, power

lost out the input coupler, and inherit losses of the system (Lsys). For a system with

losses, we consider the follwing relations [21]:

Pω2 =16T 2

1 ξnLP 2ω1

[2−√R1(2− Lsys −√

εξnLPω1)]4

(2.6)

Pω2 ≈T 2

1 CPω1

[1− 2√

R1(2− Lsys −√

εC)](2.7)

Where T1 = 1 − R1 is the transmittance of the input coupler, ε ≡ Pω2

Pω1is the overall

conversion efficiency of IR to blue, C = ξnLPω1 is the infrared to blue conversion per

pass, and other variables are as in equation 1.7. If we maximize equation 2.7 with

regards to T1 [21]:

T optimized1 =

Lsys

2+

√L2

sys

4+ C (2.8)

As we will always have some losses, and hopefully quite a bit of conversion to blue

light, T optimized1 will be greater than 0. Thus the optimized input coupler would have

reflectance less than 1, as is our case. Finding the right T for a an optical resonator

with losses is known as impedance matching, and is analogous to the concept in

electronics [21].

19

Losses also affect the finesse of our cavity [22]:

F ≈ 2π

Lsys + C + ln 1R

(2.9)

Which does approach equation 2.2 as Lsys and C go to 0. For our doubling cavity,

this means that F 6= 239 and FWHM 6= 23.7 MHz as we calculated before. We

continue this discussion in section 3.3.

2.2 Spatial Modes of the Cavity

In order to successfully couple into the resonator the infrared light must match

specific spatial modes. These modes are determined by the design of the optical

resonator which has a preferred transverse Gaussian waist at a preferred position.

This preferred beam profile is stable within the resonator and by coupling into it, we

insure that circulating power within the resonator is maximized.

The lowest order transverse spatial mode, TEM00, is circular in pattern and be-

haves simply as it focuses, making it relatively easy to model within the crystal. It is

Gaussian in both transverse axes and is the preferred spatial profile for most optical

beams in atomic physics. Gaussian beam behavior is generally understood and is dis-

cussed in Lasers and Electro Optics by C.C. Davis as well as in many other references

[23].

The optical cavity is described by a discrete set of resonance frequencies which

correspond to integer number, N, of half wavelengths of the incident light such that

20

λ2N = L. These longitudinal modes are set by cavity spacing, L, and for a cavity

with sufficiently high finesse, can be very narrow. For our setup impedance matching

requires a relatively low finesse and broad linewidth. Thus, with a relatively narrow

IR source, it is the position, and not width of these resonances that will concern us. In

our cavity, a piezo-electric transducer (PZT) attached to the input coupler allows us

to vary cavity spacing and resonances. Typically, longitudinal modes are not a large

concern during alignment, but become critical when we discuss locking the cavity to

a particular resonance, as in section 3.2.

2.3 Beam Coupling and Alignment

Maximizing power within the cavity requires exact alignment of the infrared light

into the resonator. This alignment is dependent on incident angle and position, as

well as beam waist position and size. Before attempting to couple the IR source into

the doubling cavity, we first model the spatial modes of both resonator and IR beam

and determine what beam shaping must occur to match the two.

If the incoming mode of the infrared light does not match the fundamental mode

of the cavity, light will be coupled into various transverse spatial modes, limiting the

conversion efficiency of the doubling cavity. We can avoid this loss of efficiency by

passing the IR beam through a lens before it enters into the doubling cavity. This

mode matching lens will be chosen such that the new waist of the IR beam will

coincide with the natural waist of the cavity.

21

Input Coupler

Natural Cavity Waist

Virtual Cavity Waist

IR Waist out of Fiber

Mode-Matching

Lens

Figure 2.3: Beam Waists This diagram is not to scale.

0.59 0.6 0.61 0.62Position HmL

0.0001

0.0002

0.0003

0.0004

tsiaW

eziS

HmL Virtual Waist

Natural Cavity Waist

Input Coupler (Lens)

Crystal

Figure 2.4: Beam Waists Inside Resonator Using a Mathematica script (seeAppendix B), we can model the natural cavity waist and the virtual waist used formode matching. The dashed line is the natural cavity waist propagated out of thecavity ignoring the input coupler. The solid line to the left of the input coupler is thebeam profile for both the virtual beam and the natural cavity waist propagated outof the cavity with the input coupler acting as a lens.

The natural waist of the cavity, however, is calculated as if light originates in

the resonator and stays there [23], allowing us to consider the input coupler as a

focusing mirror. In such a case the natural waist is typically on the back mirrored

surface of the crystal as depicted in Figure 2.3. We can calculate the exact size of

this waist using standard ABCD matrix formulation, as described in [23]. In such

formulation we consider the beam to travel through a repeating unit cell consisting of

the crystal medium, open air, the reflective surface of the input coupler as a focusing

22

lens, open air again, and the crystal again. Once the ABCD matrix for this unit cell

is calculated, we can use the equation found in [15]:

w20 =

2λ0B

nπ√

4− (A + D)2(2.10)

Where w0 is size of the natural cavity waist, and A,B, and D refer to the corresponding

values of the matrix. Notice that the formula contains λ0

nwhich is the wavelength of

light in vacuum over the index of refraction of the medium. For a single media unit

cell as discussed in [15] this formula is sufficient, but for our case, with two media

of different n, we must alter the technique. We include the effect of the change of

indices of refraction in the ABCD matrices themselves. We are then free to use a

modified equation:

w20 =

2λ0B

π√

4− (A + D)2(2.11)

Where the direct dependence on n has been moved to within the values of A,B, and

D.

If we wish to model light as it enters into the resonator, we should consider how

that natural waist ‘looks’ from outside. Here we introduce the concept of a virtual

waist that mimics the position and size of the natural cavity waist as seen from outside

the cavity. In order to model the virtual waist, we propagate the natural waist out

of the cavity and towards the IR source. We then retrace the beam back towards the

crystal, but ignore the input coupler. The radius of curvature of the input coupler is

23

≈ 25 mm, and has a focal length ≈ −50 mm for beams passing through. This gives

us a virtual waist that is smaller and further from the back surface of the crystal than

the natural waist as seen in Figure 2.4. We use the mode matching lens to match the

IR beam to the size and position of this new waist. Determination of the virtual waist

given cavity spacing and input coupler radius of curvature is excellently discussed in

C. Simien’s masters thesis [24]. We determine our virtual waist to be 39.6 µm, 3.85

mm from the front surface of the crystal.

Correctly matching the real and virtual beam waists is simplified using computer

modelling. Measurements of the IR beam using a Beam Master beam profiler are run

through a Matlab script which then fits suitable Gaussian parameters to them using

the equation:

w2(z, w0) = w20[1 + (

λ(z − z0)

πw20

)2] (2.12)

Where the beam waist, w, can be determined at any position, z, given the initial beam

waist w0 and position z0. Equations for modelling Gaussian beams are taken from

[23] and entered into a Mathematica notebook which takes the Gaussian waists and

positions determined by Matlab and virtual waists from Mathematica, see Appendix

B, and plots them over distance. Modelling the two profiles as they pass through

various simulated thin-lenses allows us to mode match the IR beam into the cavity.

Figure 2.5 shows the results of such a program. The horizontal and vertical profiles

of the incoming IR light travel from left to right and are overlapped with the virtual

24

beam that propagates right to left out of the cavity. Sharp turns in the profiles

correspond to the passage through a lens, as detailed in the caption. Although we

could model a perfect mode-matching lens, we are limited in reality by available focal

lengths lenses, and millimeter spatial resolution in positioning. The model in Figure

2.5 uses a single commercially made lens and we allow for some astigmatism in our IR

beam. Appendix B provides a more thorough look into the beam coupling programs.

The reader will notice in Figure 2.5 that the horizontal and vertical beam profiles

out of the IR fiber vary widely from the virtual beam profile. This is NOT ideal.

After completion of this thesis and many hours successful use of the 461 nm laser, an

error was detected in the calculation of the natural cavity waist, and virtual waist.

When corrected, the virtual waist was about half the previous size, causing the virtual

profile outside of the cavity to be proportionally larger. The parameters for the virtual

waist we listed above are the corrected values. Figure 2.5 shows this correct virtual

waist, and the real beam profiles out of the IR fiber as currently used in our setup.

Obviously, our current mode matching lens, while sufficient, is not ideal, and likely

leads to coupling into higher order spatial modes, as seen in Figure 3.4 and discussed

in Chapter 4. Future improvements in the 461 nm laser will include a better mode

matching lens.

Once the mode matching lens has been chosen and placed, alignment into the

cavity can begin. After passing through the mode matching lens, the IR beam is

25

0 0.1 0.2 0.3 0.4 0.5 0.6Position HmL

0.0005

0.001

0.0015

0.002

0.0025

0.003

tsiaW

eziS

HmL

Vertical Waist

Horizontal Waist

Virtual Waist

Mode Matching

Lens

Input Coupler (Lens)

Figure 2.5: Infrared Beam Profile The horizontal and vertical beams propagatingfrom the left begin at z=0, the face of the IR fiber coupler. They then encounterthe f=200 mm lens at z=0.37 m. The IR beams focus down onto the input coupler(effectively a f=-49 mm lens) at z=0.595 m and end at the back surface of the crystallocated at z=0.62 m. The virtual beam propagates from right to left seeing the sameelements in reverse.

centered onto the input coupler and the reflection is aligned back onto the incoming

beam. The crystal is then adjusted in the transverse axes such that the beam falls

roughly along its longitudinal axis. Crystal angular alignment is then adjusted as

cavity spacing is scanned, until transmission modes appear out the back end of the

crystal. A high speed photodiode, see Figure 2.1, carries those transmission modes

onto an oscilloscope where they can be maximized using the various IR turning mirrors

and input coupler angle. Gross cavity spacing can be adjusted by moving the crystal

itself, and the angle of the crystal may sometimes be altered to maximize transmission.

A dichroic mirror at an angle close to 45o from the incident beam, separates the

incoming IR beam from the outgoing blue visible beam.

26

In order to maximize the efficiency of the cavity both the first and second harmonic

frequencies of light must be phased matched within the crystal using temperature,

as discussed in section 1.5. The fundamental and second harmonic are polarized at

different axes, and indices of refraction in different axes have different temperature

dependences. Temperature tuning matches the index of refraction of the fundamental

frequency with that of the second harmonic. Coatings on the back surface of the

crystal are used to provide the corresponding high reflectivity of the resonator mirror

for both frequencies. The coatings must be of proper thickness such that the nodes

of the fundamental and second harmonic coincide on the back surface of the crystal

[25]. Small variations in coating thickness will lead to slightly different node positions

causing destructive interference inside the optical resonator. Second harmonic losses

due to coating thickness are non-trivial and may explain low conversion efficiencies

discussed in section 4.1 [25] [26].

2.4 Output Beam Profile

The second harmonic generation of light provides a relatively ideal beam out of

the doubling cavity. Figure 2.6 shows the beam profile of the 461 nm laser as it comes

out of the doubling cavity and passes through a collimating lens, and two cylindrical

lenses for beam shaping. In our current setup, the 461 nm laser provides light for a

MOT, image beam, and 2D collimator (version of optical molasses).

There are several excellent discussions of losses in SHG from the near IR to the

27

0 1 2 3 4 5Position HmL

0.0002

0.0004

0.0006

0.0008

0.001

tsiaW

eziS

HmL

Vertical Waist

Horizontal Waist

Figure 2.6: 461 nm Beam Profile Horizontal and vertical waists propagate fromthe left starting at z=0.62 m - the outer most edge of the input coupler. A f=200 mmspherical lens at z=0.815 m collimates the horizontal waist over long distances. Twocylindrical lenses (f=500mm and f=-400 mm) at z=1.215 m and z=1.3 m respectivelyhelp match the vertical waist to the horizontal.

high end visible spectra [27][16][24][28][29]. Common optical culprits of loss include

thermal lensing and blue light induced infrared absorption (BLIIRA). Thermal lensing

occurs as temperature gradients form in the longitudinal and transverse axis due to

heating from the circulating IR light. These temperature gradients change the indices

of refraction of the KNbO3 crystal and alter the position and typically increase the

size of the resonator waist. Shifts in waist position and size affect the beam coupling

into the cavity and adversely affect SHG [27]. More importantly, increases in beam

waist lower intensity within the crystal and thus conversion efficiency. See section 3.3

and 4.2 for further discussion of thermal issues.

BLIIRA is not completely understood, though extensive studies have been per-

28

formed to model its behavior and determine a coefficient for the process [28] [29]. It

is typically modelled as blue light depopulating low-level traps inside the crystal lat-

tice through photo-ionization, permitting increased IR absorption [27][29]. Infrared

photons are normally absorbed into the crystal at a small yet measurable rate, but

as blue photons are incident upon the crystal, that rate dramatically increases [28].

This absorption is significant with as little as 10−3 W/cm2 of blue light, begins to

increase exponentially near 100 W/cm2 and continues so well past 104 W/cm2 (for

reference our laser intensity ∼ 1300 W/cm2) [28] [29]. BLIIRA, combined with other

loss mechanisms, helps explain why KNbO3 crystals fail to maintain a quadratic de-

pendence on incident power as suggested by theory, and instead enter into a linear

regime and fixed efficiency [16]. Although BLIIRA can be reduced at longer wave-

lengths or higher crystal temperatures [28], this does not match our phase-matching

criteria.

Chapter 3

Electronics

Feedback electronics prove necessary in order to compensate for fluctuations in

cavity spacing and crystal properties. Acoustic vibrations provide the greatest inter-

ference in cavity stability but thermal drifts may also contribute. By monitoring the

transmission through the doubling cavity and correspondingly varying cavity spacing

through a piezo-electric transducer (PZT), feedback electronics can lock the cavity

on resonance and compensate for the fluctuations.

An overview of the feedback process is outlined in Figure 3.1. Demonstrative

samples of various signals have been provided for clarification. Important to cavity

stability but not pictured in the figure are crystal temperature, high voltage amplifi-

cation for the PZT, voltage offset on the PZT, and beam alignment into the cavity.

Alignment has been discussed in the previous section and will be considered to al-

ready have been maximized in discussion of electronic feedback - likewise with crystal

temperature, discussed in the next section. High voltage amplification and voltage

offset will briefly be discussed with the ramp signal.

3.1 Error Signal

A modified Pound-Drever-Hall method generates an electronic error signal to pro-

vide feedback into the cavity. In a standard P-D-H setup, the incoming laser light is

30

Use VCO & EOM to place sidebands on

IR light

Ramp cavity to vary resonance

frequency

Mix Transmission with VCO to form Error

Signal

Transmission>

Set?

Feed back Error into

cavity(Locking)YES

NO

-40 -30 -20 -10 0 10 20 30 40

0

2

4

6

8

10

12

14

16

18

20

ν -β 0 ν +β 0

ν 0

0 10 20 30 40 500

1

2

3

4

5

6

Time (ms)

Inte

nsity

(A

rb.)

Ramp SignalTransmission Modes

0 10 20 30 40 500

2

4

6

8

Time (ms)

Inte

nsity

(A

rb.)

Error SignalTransmission Mode

5 10 150

1

2

3

4

5

6

Time (ms)

Inte

nsity

(A

rb.)

Set PointTransmission Mode

0 10 20 30 40 500

1

2

3

4

5

6

7

8

Time (ms)

Inte

nsity

(A

rb.)

Error SignalLocked Transmission

A B C D E

Figure 3.1: Logic Flow of Feedback Loop. A) Frequency sidebands are placedon either side of the IR laser center frequency. An Idealized example below where βis ∼15 MHz. B) A triangular wave ramp is sent to the cavity which passes throughresonances with the IR laser. C) An error signal is generated by demodulating themixed sideband and IR center frequency signals. This error signal is anti-symmetricabout peaks in transmission. D) A comparison between IR transmission level and amanually determined set point allow the locking mechanism to distinguish betweenoff and on resonance conditions. E) The anti-symmetric error signal is fed back intothe cavity PZT, causing cavity spacing to follow transmission peaks thus locking thecavity to the IR resonance.

modulated and the reflected beam off the cavity is phase detected at the modulation

frequency using an electronic mixer to produce a demodulated signal [30]. We have

modified this setup to use the transmitted rather than reflected signal. This requires

that the transmitted and outgoing light have coincident peaks, but this will always

be the case for our setup as shown later in section 3.3.

Infrared light from the Ti:Sapphire pumping laser passes through an Electro-Optic

Modulator (EOM), acquiring frequency sidebands. These sidebands occur at roughly

31

15 MHz on each side of the IR laser frequency similar to Figure 3.1A, and are driven

by a voltage controlled oscillator (VCO). Actual traces of our sidebands cannot be

seen as the resolution of our cavity is greater than 15 MHz, but we cite the EOM

technology’s reliability and the successful production of an error signal as sufficient

proof of their existence. After passing through the EOM, the IR beam is steered into

an optical fiber that carries it onto the Neutral Atoms table, where our strontium

studies occur, and from that fiber through a mode-matching lens and into the cavity.

A simple triangular wave signal ramps the cavity PZT and scans the etalon’s

transmission frequency. Transmission peaks occur when the cavity spacing is on

resonance with the input beam. The ramp varies 15 volts peak to peak at a typical

rate of > 10 Hz and inputs directly into the feedback electronics circuit. From there,

the ramp can pass into a high voltage amplifier whose input gain is ∼ 5. The amplifier

also provides a DC offset to the PZT through an amplified battery signal. This offset

is largely unimportant to the locking process as long as at least one transmission

signal occurs during a ramp cycle. In order to guarantee the transmission peak, the

offset is manually set such that peaks occur roughly in the center leg of the ramp

signal, and the laser is then locked. The 461 nm laser can operate for many hours

before DC offset drift causes it to unlock.

One generates the error signal by mixing the transmission signal with the original

VCO frequency. The signal is anti-symmetric: negative on one side of the transmission

32

peak, zero at resonance, and positive at the other side. Scope traces of the signal can

be seen in Figure 3.1C and can be analyzed as in [31]. The anti-symmetry forms the

backbone of the feedback signal and allows the locking circuit to center itself on a

transmission mode.

3.2 Locking The Cavity

Figure 3.2 provides a schematic of the locking circuit, which consists of a switching

circuit and a single-path servo-lock. The servo-lock is a standard element in laser

control, allowing the error signal derived previously to be amplified and integrated

and fed back into the cavity. The switching circuit allows the locking system to

determine for itself if the cavity is on or off resonance and respond accordingly.

A simple circuit determines if the cavity is near resonance by comparing the

transmission signal with a manually controlled set voltage. This set voltage is high

enough that smaller, non-desirable modes are not considered to be on resonance.

When the transmission signal is less than the set voltage, the switching circuit is

sent a low signal. At low input, the switching circuit shorts the integrator and the

boost of the locking circuit, resetting the capacitors and preventing the error signal

from locking the cavity. Also at low, the switching circuit passes the ramp signal

to the cavity causing it to continue scanning. Eventually, the ramped cavity should

hit a transmission peak and cause the comparator to send out a high signal. The

switching circuit then opens the capacitors to allow integration and stops sending

33

Figure 3.2: Schematic of Locking Circuit. The single path lock-loop and switch-ing circuit used to the lock the cavity including relevant passive element values.

the ramp signal to the cavity. There is a finite non-trivial time between the cavity

nearing resonance and the locking circuit being able to integrate sufficiently to lock

the cavity. To compensate for this time a small capacitor has been placed at the ramp

signal output from the switching circuit, causing the ramp to come down slowly and

assisting the integrator. This measure in of itself is helpful but not sufficient, and the

set voltage must be considerably lower than the peak of the transmission to allow for

extra integrator time. Thus there is some finesse involved in placing the set point

at the correct voltage - high enough to exclude undesirable peaks but low enough to

compensate for integrator lag.

34

The lock-loop is essentially an integrator with variable gain and boost to maintain

a secure lock. Integration follows the standard relation:

Vout =1

RC

∫Vin dt. (3.1)

Where Vout is output voltage, Vin is input voltage and R and C stand for the input

resistor and feedback capacitor respectively. During operation, the cavity will be

ramped until it nears a transmission peak and integration begins. The integrated

error signal will be fed back into the PZT and control the resonance of the cavity.

Variable gain allows for the error signal to adequately shift the PZT without oscillation

no matter what the absolute transmission, and thus error signal, strength. Boost gives

more gain at low frequencies to provide a more stable lock.

Normal procedure for locking the cavity is straightforward. Once cavity align-

ment and temperature have been optimized, the DC offset is placed appropriately.

The locking circuit is then activated, the cavity begins to lock, and gain is adjusted

manually to optimize 461 nm output. Fine-tuning of temperature may be required

but the system does approach a “single switch” setup allowing the user a minimum

number of tasks before maximum output of the cavity is achieved.

3.3 Characterization of Electronics

Typical operation of the feedback electronics provides a robust lock to the IR

transmission peaks, and generation of 461 nm light can be characterized in parallel to

35

0 10 20 30 40 50−1

0

1

2

3

4

5

6

7

8

Time (ms)

Vol

ts

Transmission ModesBlue Output Modes

Figure 3.3: Coincidence of IR Transmission and Blue Output modes. Trans-mission modes are displaced +1 Volt and Blue Output modes -1 Volt for clarity.Amplitudes are to scale.

electronic features. The two best diagnostics of cavity behavior are the transmitted

(IR) and output (visible) modes of the cavity. The transmitted modes are a direct

ingredient in the creation of the error signal and the output modes are our desired

461 nm laser output. The feedback and locking electronics do have an effect on the

optical properties of that laser and the entire electronics system is characterized in

those terms below.

As we proposed, transmitted and output modes occur in coincidence with each

other as seen in Figure 3.3. Figures 3.4 and 3.6 show the transmitted and output

modes independently of each other and with the error signal as reference. Secondary

36

0 10 20 30 40 500

2

4

6

8

10

Time (ms)

Vol

tsError SignalLocked TransmissionTransmission Modes

Figure 3.4: Transmission Modesof Cavity. Error signal is offset by+6 Volts and amplified (x10) for clar-ity. The smaller transmission modesare due to coupling into non-TEM00

transverse spatial modes.

0.5 1 1.5 20

2

4

6

8

10

Time (ms)

Vol

ts

Error SignalTransmission Mode

Figure 3.5: Close-up of Transmis-sion Modes. Error signal is offsetby +6 Volts and amplified (x10) forclarity.

peaks in the transmission arise from coupling into non-TEM00 transverse modes of

the cavity. These higher order modes limit conversion power of the doubling cavity

as discussed in section 2.2. Notice the anti-symmetry of the error signal needed to

lock the cavity as we described earlier. A corresponding match between 461 nm light

generation and the error signal is demonstrated as well. Noise that occurs during

the error signal is problematic during startup when the cavity occasionally locks to a

smaller transmission peak. Once locked to the correct mode, however, the switching

circuit keeps the cavity from seeing extraneous error signals. The locked versions of

each signal are provided as reference and to demonstrate that the transmission falls

as the cavity locks while the opposite occurs with the visible light. Explanations and

effects will be discussed later in this section.

37

0 10 20 30 40 500

1

2

3

4

5

6

Time (ms)

Vol

tsError SignalLocked OutputBlue Output Modes

Figure 3.6: Output Modes of Cav-ity. Error signal is offset by +3.5 voltsand amplified (x5) for clarity.

1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

Time (ms)

Vol

ts

Error SignalOutput Modes

Figure 3.7: Close-up of OutputModes. Error signal is offset by +3.5volts and amplified (x5) for clarity.

The free spectral range (FSR) of the cavity corresponds in time to the separation

between peaks. A simple method for experimentally measuring the FSR is taken

from [24]. The 922 nm source (Ti:Saph. laser) is manually scanned with part of the

beam coupled into the 461 nm and part aligned onto a Burleigh WA-1000 wavemeter.

When cavity is on resonance with the laser, the wavemeter reading is recorded, and the

distance between resonances is the FSR, measured to be 7.7± 0.1 GHZ. This process

is akin to our normal sweeping configuration, only we are scanning the laser and not

the cavity. Even this simple measurement gets us fairly close to our calculations in

section 2.1 of 5.7 GHz. Taking our experimental value for the free spectral range

we can determine a constant conversion factor of 316.4 MHz/s for the given ramping

of the cavity. Maintaining that same ramp and examining the inset of each figure

gives an experimentally determined full width half maximum of each peak as ∼ 50

38

MHz for the transmitted and ∼ 30 MHz for the output. If we use the experimentally

derived values for FSR and FWHM in the IR, we can calculate a finesse for our cavity

which includes our losses at 110 mW input power. Using equation 2.3 we find that

our finesse is ≈ 154, and using equation 2.9 we can say that Lsys + C ≈ .0145 at

110 mW. Likewise, one can determine that the error signal scans the cavity ∼ 0.36

MHz/mV. These error signals have been optimized for the parameters of the laser at

the time (110 mW input power, and 26.53oC control temperature) and vary as those

parameters change.

0 10 20 30 40 50−1

0

1

2

3

4

5

6

7

8

Time (ms)

Vol

ts

Transmission ModesBlue Output Modes

Figure 3.8: Switching Noise. The switching circuit of the lock-loop provides un-wanted feedback onto the cavity PZT as it is being swept. This noise is manifestedas multiple modes occurring after the primary mode, and can be seen on both trans-mitted and reflected signals. Transmission modes are displaced +1 Volt and BlueOutput modes -1 Volt for clarity. Amplitudes are to scale.

39

While close to ideal, operation of the locking-circuit generates a few problematic

effects on the 461 nm light. The first and most trivial problem is demonstrated in

Figure 3.8 as compared to Figures 3.4 and 3.6. Notice the extra transmission peaks

or fuzziness next to the main peak. This effect is caused by erroneous feedback from

the switching circuit as it compares the set voltage to transmission signal. Though

it has no direct path to the PZT while ramping, the switching circuit can provide

an additional path to ground for the ramp signal through a capacitor. This path

is necessary and cannot be excluded. It is thought that the switching circuit tries

to activate as normal when a transmission peak occurs and causes the ramp signal

to fluctuate as the cavity passes through resonance. During locking operation, this

behavior is not seen because the switching circuit is directly in control of PZT voltage.

Thus, the effect has minimal impact on the cavity and only needs to be eliminated

when trying to take characteristic scope traces of the electronics. The effect can be

removed by disconnecting the transmission signal from the comparator.

Another effect of the locking circuit is seen in Figure 3.9 where power in the blue

output is shown. During normal locking procedure, the output light increases as

the cavity stops sweeping and locks onto resonance. This increase is expected as the

cavity is optimized in temperature for locked output. If instead we optimize the cavity

as it sweeps, we notice there is little power difference between the two signals. The

temperature difference is non-trivial for our cavity: 0.025 volts on the temperature

40

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

Time (ms)

Vol

ts

Swept SignalLocked Signal

Figure 3.9: Comparison of Sweep Versus Lock Power in Blue. Optimizingthe crystal temperature for cavity output during swept operation shows that there islittle to no power lost during locked operation.

controller or ∼ 0.6oC. We know that phase-matching criteria set the temperature

in the crystal such that n2 is equal to n1 at all times [20][19], so we can surmise

that the temperature difference occurs as the temperature controller tries to keep

the crystal temperature in the beam path constant. This means that as circulating

power increases as we lock the cavity, thermal gradients are formed on the order of

∼ 0.4oC/mm inside the crystal (we assume linear heating and target temperatures

occurring at the center of the 3 mm tall crystal) which is small compared to ranges

seen in [28] and [20]. Additionally, Figures 3.4 and 3.6 show no signs of thermal locking

as described in [28] and [24]. In such cases, as a cavity sweeps from low to high and

41

off of the peak, the crystal length changes keeping the cavity near resonance [28].

The absence of thermal locking again suggests that thermal gradients are relatively

small. Thus, thermal lensing, or other thermal effects which are discussed in sections

2.4 and 4.2 may be negligible in our system.

Non-ideal effects of the feedback electronics on the production of 461 nm light

suggest areas where improvement is possible. Power is lost due to thermal and/or

absorption effects in the crystal (see section 2.4 as just discussed) [28] [29]. In routine

operation, the set voltage is not changed even if input and thus transmission strength

is varied, and occasionally the cavity will lock to undesired modes as mentioned

earlier in this section. Though this is easily diagnosed by low laser output, and easily

remedied by re-locking the cavity, it is non-ideal operation.

Overall, the locking electronics on the cavity work remarkably well. The cavity

can lock for many hours without need of adjustment. More impressively, however, the

cavity recovers from large perturbations without need of manual resetting. Acoustic

noise, which plagues the cavity’s optical stability, may unlock the cavity briefly, but

due to the switching circuit, the electronics will relock the laser quickly and repeatedly

as necessary.

Chapter 4

Characterization of the 461nm Laser

Once the 461 nm laser became operational, care was taken to optimize all avail-

able parameters such as beam coupling, beam alignment, crystal temperature, and

electronic feedback. We will discuss those four parameters briefly before continuing

on to the general characterization of the laser.

Beam coupling of the infrared light into the frequency doubling cavity changed as

the source of that infrared light changed from the Ti:Sapphire laser, to various diode

lasers, and finally back to the Ti:Saph. laser now passed through an optical fiber.

The basic assumptions and means of modeling/determining that coupling, however,

stayed the same through out. The system for deciding the focal length and position

of the mode matching lens, as described in section 2.3, yielded coupling efficiencies

∼ 80% in our current setup (as calculated below). Our characterization of the laser

in the current setup then should carry over as the pumping source for the 461 nm

laser changes again in the future.

Beam alignment into the cavity will always be able to be improved upon. Realisti-

cally the alignment depends on the stability of at least half a dozen optical elements,

all of which have more than one relevant axis of rotation/translation. Great means

could be taken to secure each of these elements to the utmost of scientific ability and

43

coupling could approach unity. As it is, Figure 3.4 shows that there is still some

power lost to higher order spatial modes due to poor alignment and mode matching

(see section 2.3). Comparing the heights of those secondary peaks to the primary

suggest that the output power may be improved by as much as 21% if all the power

in the secondary peaks were instead in the primary (increasing efficiency to 40%).We

are satisfied with characterizing the laser at the current beam alignment because the

current setup is robust enough to differ very little over the lifetime of several weeks.

Please refer to the improvements heading in the conclusion of this thesis for more on

maximizing the beam alignment in the future.

Optimization of the Potassium Niobate crystal temperature is accomplished by

adjusting a temperature controller whose thermo-electric cooling device (TEC) sits as

a heat sink under the crystal housing. This setup is discussed further below. Figure

4.1 shows output power versus temperature over a range of input powers. Though

not pictured in the figure, care was taken to determine that the peaks of each curve

correspond to global and not local maxima of the system. Each peak then coincides

with the optimization of the locked cavity, the normal operating mode of the laser, and

is satisfactory to characterize the laser’s temperature dependence for our purposes.

Electronic feedback is optimized by adjusting the error signal gain on the locking

circuit as described in Section 3.1. Maximum output of the cavity occurs when the

gain is as large as possible without causing strong oscillations in the error signal.

44

0

20

40

60

80

100

120

140

0.9 0.92 0.94 0.96Temperature (V)

Ou

tpu

t P

ow

er (

mW

)

26.72727.327.627.9Temperature (C)

23

52

75

104

123

150

178

205

227

253

276

305

324

349

372

407

Input Power(mW)

Figure 4.1: Power Out vs. Temperature. Given various input powers, outputpower was traced as a function of the crystal’s temperature.

45

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

Power In (mW)

Pow

er O

ut (

mW

)

Linear RegimeQuadratic RegimeQuadratic FitLinear Fit

Figure 4.2: Efficiency of Cavity. Optimized output power versus input power isshown along with fits to the quadratic and linear regimes.

There is a local, much less stable, maxima at lower gain than the optimal one and

future users of the system are encouraged to scan thoroughly in gain before determin-

ing output power is maximized. For this characterization of the laser care was taken

to use only the stable global maximum.

4.1 Efficiency of Frequency Conversion

Optimum output for any given input power of the IR pump laser can be seen in

Figure 4.2. Typical output power of the Potassium Niobate crystal follows a quadratic

increase from zero and later shifts to a linear regime [16]. Accordingly, we have fit

our data with quadratic and linear curves as seen, and R2 values near unity attest

46

to the accuracy of those fits. Overall efficiency in the linear regime is ∼ 33% and

holds steady out to the maximum input achievable in our setup. Though thermal

considerations would theoretically start to lower efficiency at higher powers, it is

clear that we have not yet entered that regime [16].

Potassium Niobate crystals are the standard for frequency doubling in the high

end visible spectrum, but results can vary widely based on individual quality of the

specimen. [28] and [27] cite efficiencies of ∼ 5% and > 75% respectively and at

wavelengths of 423 nm and 461 nm. [16] is the highest we found in our search at

> 81% at 471 nm. In context, our results seem to be acceptable but not exceptional.

Within our own lab, crystals from the same manufacturer and operating at the same

wavelength from the same source yield efficiencies close to 40%. It is important

to note, however, that in [27], the reported efficiency is for blue light created, not

emitted, from the cavity. Greater than 15% of the claimed efficiency is supposed

by accounting for losses due to BLIIRA and non-ideal optical elements. The high

efficiency in [16] also compensated for optical elements, though not for BLIIRA.

By observing the transmission modes of the cavity as well as the reflected (blue)

modes, we can make modest approximations on the limiting factors of our efficiency

as discussed in the opening of this section. The 21% gain from beam alignment would

give a predicted efficiency of 40%. This would be an improvement but would not ap-

proach the results of [16]. Even if operating at other wavelengths was desirable, crystal

47

properties would not necessarily yield greater efficiencies at nearby wavelengths such

as those used in the references above. It is hard to quantify the losses due to phase

mis-matching as discussed in section 2.3 and 1.5 but this may considered along with

other thermal effects and losses in the next section.

Even if this particular Potassium Niobate crystal, or some systematic portion of

our 461 nm laser setup is less than ideal, it is important to note that our results are

more than satisfactory for our purposes. Output powers in excess of 125 mW are

more than sufficient for the creation of stable Magneto Optical Trap, 2-D collima-

tion, absorption imaging, or photoassociative spectroscopy beams as discussed in the

introduction of this thesis.

4.2 Thermal Effects

Temperature regulation of the Potassium Niobate (KNbO3) crystal was maintained

using a simple HTC temperature controller from Wavelength Electronics. Figure 4.3

shows the conversion curve from monitored voltage on the temperature controller to

degrees Celsius. Monitoring of the temperature is accomplished through a 10 kΩ

thermistor positioned at the base of the housing of the crystal. This was also the

location of the thermo-electric cooling device (TEC) which acts as a heat sink on the

crystal and is the means through which the HTC regulates the crystal temperature.

The housing of the crystal is made of aluminum. As seen in Figure 4.1, variations as

small as 0.1oC yield large differences in the output power of the laser. The crystal can

48

-10

0

10

20

30

40

50

0 1 2 3 4 5

Monitor Voltage (V)

Tem

per

atu

re (

C)

y = -24.379x + 49.355R2 = 0.9995

25

25.5

26

26.5

27

27.5

28

0.86 0.91 0.96

Figure 4.3: Temperature Controller Calibration. The HTC 3000 temperaturecontroller from Wavelength Electronics utilizes a 10 kΩ thermocouple to monitorcrystal temperature. INSET: Routine operation of the crystal occurs over a relativelysmaller portion in temperature space. A linear fit reasonably approximates the voltageto temperature calibration in this regime.

49

take several minutes to fully stabilize at a new temperature, depending on the speed

at which the temperature controller stabilized, and on the amount the temperature

was changed. Furthermore, there is some oscillation in overall monitor voltage on the

order of ∼ 3 mV (0.075oC) over a period of a few days.

As discussed in sections 1.5 and 2.3, phase matching plays a critical role in the

infrared to blue light frequency conversion. The indices of refraction along different

axis in the KNbO3 crystal have differing dependences on temperature, allowing us

to temperature tune the 461 nm polarization axis to match the 922nm axis [19].

See the discussion on frequency doubling in the introduction of this thesis for more

information. Optimum temperature of the TEC decreases as input power increases

as seen in Figure 4.4. Temperature at the crystal near the axis of the incoming

beam should remain constant to maintain phase-matching [20][19], suggesting that

the difference in TEC temperature likely creates a temperature gradient across the

crystal (as we mentioned in section 3.3.

Along the longitudinal axis of the crystal, where variation in beam waist size can

lead to uneven heating, non-ideal temperature gradients result in a spatial dependence

on the index of refraction of the crystal. This effect is sometimes referred to as

thermal lensing. Thermal lensing causes distortion in the beam waist as the variation

in index of refraction affects the beam profile passing through the medium. Not only

does thermal lensing affect spatial mode matching, it can also distort phase matching

50

0.91

0.915

0.92

0.925

0.93

0.935

0.94

0.945

0 100 200 300 400 500

Power In (mW)

Tem

per

atu

re (

V)

27.03

27.1485

27.267

27.3855

27.504

27.6225

27.741

27.8595

Tem

per

atu

re (

C)

Figure 4.4: Optimum Temperature for Input Power.

51

and leads to an overall reduction in the efficiency of the cavity. See also section 2.4.

Large differences in the optimum temperature for a locked versus swept cavity (see

Figure 3.9 and section 3.3) suggest that the locked cavity is heated by the incident

and circulating IR light. This requires the temperature controller to cool the crystal

significantly in order to phase match. A larger beam waist inside the crystal should

lessen the heating but this will lead to lower intensities and power conversions. The

thermal gradient calculated in section 3.3, however, is small compared to other studies

as we mentioned [28] [20].

Thermal expansion of the crystal causes a condition called thermal (self) locking

as a cavity is scanned over frequency [28]. The absence of thermal locking in our

system, see section 3.3, suggests that thermal gradients along the beam axis are a

small source of loss in our system.

That thermal losses are small is best seen in Figure 3.9 where, as we mentioned,

there are no significant differences in blue power out even as circulating power in-

creases as the cavity locks. Thus despite what temperature gradients we may induce,

thermal considerations seem to have minimal effects on cavity efficiency.

Chapter 5

Conclusion

5.1 Summary

We have detailed the necessary steps in the creation and characterization of a

460.862 nm laser. We have discussed the relevant concepts in optics, electronics, and

atomic physics as pertaining to the construction and use of this device. During normal

operation we run the 461 nm laser at output power > 100 mW (125 mW typical),

corresponding to conversion efficiencies near 33%. About 30 mW is used in a magneto-

optical trap, < 1 mW for an imaging beam, and alternating between 70 mW and 100

mW in a 2D collimator (2D optical molasses), all along the dipole allowed 1S0 → 1P1

transition, and all on the new Neutral Atoms setup in our laboratory. The 461 nm

has proven itself an essential tool in the cooling, trapping, and study of Strontium.

5.2 Improvements and Future Work

Characterization of our doubling cavity suggests that we have not yet reached

input powers sufficient to destabilize our 461 nm laser. We can then increase the

input power of the cavity and supply the Zeeman beam (∼ 70 mW required power in

the blue) for the new setup as well. Alternatively, using the process detailed in this

thesis, we can create additional 461 nm lasers using alternative IR sources similar to

the diode lasers utilized during PAS studies at long range. Plans have already been

53

considered to use tapered diode amplifiers to provide IR power > 300 mW to various

doubling cavities. These sources would be less expensive than the Ti:Saph laser.

Beam coupling into the cavity may be improved with greater care in aligning opti-

cal mounts. Improved coupling will improve efficiency as we discussed in section 4.1.

Drawings have already been made for a new crystal mount which will be more stable

than the current one and may increase coupling efficiency. Acoustic noise, which has

a devastating but short-term effect on output efficiency, can also be improved upon

through standard isolation techniques.

Studies in ultracold neutral strontium are reaching a critical point in our labora-

tory. Results from our studies in photoassociative spectroscopy suggest that 86Sr may

readily be brought into quantum degenerate conditions using purely optical means.

Along with the intercombination-line MOT, and a new optical dipole trap, the blue

MOT along the 1S0 → 1P1 transition will provide an essential ingredient in that pro-

cess. Quantum degeneracy in 86Sr will open our laboratory to studies not yet explored

in atomic physics.

Appendix A

Computational Analysis of Beam Profiles

This Matlab script is used to fit beam profiles and extract the critical beam

parameters (beam waist position and size). Results from this program are typically

used in conjunction with the Mathematica modelling programs described in Appendix

B. Script begins on next page, with comments in text.

55

Figure A.1:

56

Figure A.2:

57

Figure A.3:

58

Figure A.4:

59

Figure A.5:

Appendix B

Computational Modelling of Beam Profiles

The following is Mathematica notebook code we used to model beam profiles as

they are shaped through lenses and other optical elements. This is not the exact

script used to calculate the models in Chapter 2, but rather the elements of code that

can be used to do such calculations. Once the reader understands this portion of the

code, he/she can generalize the process to model any desired profile.

Figure B.1: The expressions for waist position and size, and wavelength. We havegiven typically values, in meters, for the IR waist out of the fiber coupler.

Figure B.2: The complex gaussian beam relations found in [23] and many othertexts.

61

Figure B.3: These functions are designed such that the output of freespace and lensare inputs for any function. In freespace z is the distance the beam travels in openair, and in lens f is the focal length of the lens in meters.

Figure B.4: We demonstrate how the functions work by plotting waist as a functionof z. Notice that the function waist calls freespace. Plotting the waist function isequivalent to plotting the beam waist. Here we have used the IR beam out of theoptical fiber.

62

Figure B.5: Including a thin lens is done by using the lens function between twofreespace functions which represent the open air on either side of the lens. Noticethat functions are called in reverse order - this is an inherit necessity due tothe design of the functions, and will always be the method used. This plot showsmultiple beam profiles: the IR beam without the lens, and the IR beam with a lens.Notice that the two plots overlap one another and the user has to remember whichone is to be used where.

63

Figure B.6: We can solve the overlap issue by defining plot outputs as variables overthe relevant portions of z, and using a function called Show to plot each portion onthe same graph. Here we have also used the same formatting as in the Figures ofChapter 2 to demonstrate the necessary Mathematica code.

64

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