Phase Shift Spectroscopy in the Study of the 6Pl12
-+
6P312
M1
Transition in a Thallium Atomic Beam
by
Colin D. Bruzewicz
A thesis submitted in partial fulfillment
of the requirements for the
Degree of Bachelor of Arts with Honors
in Physics
WILLIAMS COLLEGE
Williamstown, Massachusetts
May 20, 2005
Abstract
A phase sensitive method of high precision atomic spectroscopy is presented. Due to the intrinsic weakness of the M1
transition, previous spectroscopic techniques based on the direct absorption of laser light are untenable in the reduced atomic density thallium beam unit. A feasibility study and results of bench top testing of a newly constructed high finesse Fabry-Perot cavity capable of resolving atomically induced phase shifts are described here. The future high precision measurements afforded by this technique will provide general tests of atomic wavefunction calculations of relevance to the refine-ment of the Standard Model of Electroweak Interactions.
Acknowledgments
Endless thanks are owed to all those who helped make this project a success.
To those who came before, your tireless efforts in the laboratory have provided the equipment and knowledge base that make this type of work possible.
To those in the present, I am best able to thank you personally. George Walther and Larry Mattison, your craftsmanship and friendliness never cease to amaze me. Emile Ouellette, your electrical expertise is unparalleled by anything I have ever seen before. Ralph Uhl, you have my eternal gratitude for showing me that there exists a time between being a physics undergrad-uate and a professor. Joe Kerckhoff, Bronfman basement seems a little less like no-man's-land with you around. Professor Sarah Bolton,
thank you for your insight and for appropriately cracking down on my lyrical license. Pro-fessor Tiku Majumder, your lab churns out hardened physics veterans who would like nothing more than to spend another year under your guidance. My crew, I swear the thesis is really done now. My family, thanks for never having a moment's doubt.
To those in the future, this work depends on you, so let me thank you in advance.
Contents
1 Introduction 5
1.1 Mission Statement ........................ 5
1.2 Precision Measurement ...................... 6
1.2.1 Signals ........................... 6
1.3 Existing Studies .......................... 7
1.3.1 Stark Shift Measurements ................ 8
1.3.2 The Forbidden (E2/Ml)
Transition at 1283 nm .... 8
1.3.3 Stark Shift Measurements of the Forbidden (E2/M1)
Transition at 1283 nm .................. 8
1.4 The Stark Shift .......................... 9
1.5 Outline of the Thesis ....................... 12
2 Atomic Physics 13
2.1 Atomic Transitions in Thallium ................. 13
2.1.1 Selection Rules ...................... 13
2.1.2 The Complex Index of Refraction ............ 15
2.2 Measurement Feasibility ..................... 18
2.2.1 Fabry-Perot Theory ....................18
2.2.2 Signal Size .........................20
2.2.3 Signal Resolution ..................... 21
3 Experimental Apparatus 25
3.1 Optical Probe ........................... 25
3.1.1 Fabry-Perot Cavity .................... 25
3.1.2 Diode Laser ........................ 28
3.1.3 Acousto-Optical Modulation ............... 30
3.2 Spectroscopic Medium ...................... 32
3.3 Signal Processing .........................33
3.3.1 Photodiodes ........................ 33
3.3.2 Lock-In Detection ..................... 34
4 Experimental Tests and Results 36
4.1 Cavity Tests ............................ 36
4.2 Phase Shift vs .Transmission Change .............. 37
4.3 The Expected Signal ....................... 38
4.4 Verified Theory .......................... 42
4.5 Further Reduction in Cavity Drift ................ 42
5 Future Study 44
5.1 Remaining Bench Testing .................... 44
5.2 "Forbidden" Transition Spectroscopy .............. 44
5.3 Two-step Spectroscopy ......................45
A The PZT Powered Mirror Mount 48
B Electronics 50
List of Figures
The Low-Lying Energy Levels of Atomic Thallium [Maj05]
. . 7
Optical Depths in the Vapor Cell and Atomic Beam for the
1283 nm Transition [Hol03]
. . . . . . . . . . . . . . . . . . . . 9
The Stark Shift in the 1283 nm Transition (separations greatly
exaggerated for clarity) . . . . . . . . . . . . . . . . . . . . . . 11
El Transitions in the 1283 nm Transition Caused by PNC
Stark-Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
The Frequency Dependence of the Real (left) and Imaginary (right) Parts of the Refractive Index . . . . . . . . . . . . . . 17 Simulated(1eft)
and Experimental(right)
Fabry-Perot Trans-mission Curves . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Light Trajectory in the Confocal Fabry-Perot [Adapted from [KRYO3]].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Slow Scan of a Single Fabry-Perot Peak. Notice the residual jitter associated with frequency uncertainty. . . . . . . . . . . 24
Optical and Signal Paths for the Proposed Experiment . . . . 26 Photograph of the Existing Fabry-Perot Cavity . . . . . . . . 27 Frequency Drift in a Standard Diode Laser [KBU05]
. . . . . . 29 Frequency Dependence of Dispersive Phase Shift and Faraday Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Acousto-Optical Modulation [SpeOO] . . . . . . . . . . . . . . 31 Cross Section of the Atomic
Beam Unit [Updated from [Nic98]]
32 The Two Methods of Collimation in the Atomic Beam Unit. Left: Pre-Collimation in the Nozzle [SpeOO] . Right: Knife-Edge Collimation [Nic98]
. . . . . . . . . . . . . . . . . . . . . 33 Measurements Confirming Estimated Experimental Cavity Lim-its. Note from the scales that the entire width of the graph on the left is equal to one of the divisions on tlie
graph to the right Transmission Change in a Fabry-Perot Cavity. Left: In the Presence of Absorption. Right: Subject to a Frequency Change From the set point marked X, the transmission change associ-ated with the absorption is indistinguishable from the trans-mission change caused by the frequency shift .......... Simulated phase shift spectrum of the F=l portion of the tran-sition, appropriately scaled in amplitude, Doppler width, and relative splitting .......................... Left: Predicted Transmission Change from Actual Phase Shifts and Absorptive Effects. Right: Combined Transmission Ef-
fects (above) Alter the Original Dispersion Curve (below) ... Predicted Stark Shift in the Dispersive Phase Shift ......
Proposed Method of Double-Passing Light Through the AOM Transitions of Interest in the Proposed Two-Step Measurement
Photographs of the PZT Mirror Mount. Left: The PZTs
(one pictured in lower left corner of the plate) are placed in between the two plates of the mirror mount to adjust the separation distance between the two mirrors. Right: The fully functional PZT mirror mount ........................
B.l Circuit Diagram for the Cavity Length Locking Circuit [Wie98]
51
List of Tables
2.1 Selection rules for atomic multipole transitions. [Ho103]
. . . . 14
Chapter 1
Introduction
Mission Statement
The historical development of physics has been a constant interplay between experiment and theory. Having observed some phenomenon, the first natural philosophers sought to describe it under some self-consistent mathematical framework. Often times these mathematical descriptions would suggest other observable processes that had yet to be seen, thus thrusting the onus of further discovery back on to the experimentalists. These searches would then uncover confirmations of certain aspects of the physical theory and counterexamples to others, and all the while finding new phenomena that had yet to be explained. This push-and-pull mechanism between theory and experiment continues to dominate physical research to this day.
At present, the theory of primary interest to this work, known as the Standard Model of Electroweak Interactions, is complicated by constraints unknown to earlier studies of classical physics. Analytic solutions to the governing equations of quantum mechanics simply do not exist, except in the very simplest examples. Thus theorists rely on approximate methods, which, although impressively accurate in their own right, beg for refinement. Through very careful measurements of exceedingly small physical effects, experimentalists are able to shed light on the accuracy of the competing theoretical methods of the day. It is to this cause that this work has devoted itself.
Precision Measurement
In 1974, Bouchiat and Bouchiat suggested the possibility of measuring effects due to the weak force through parity-violating processes in heavy atoms. At that time, however, spectroscopic techniques were not able to resolve the intrinsically small signals associated with those effects, and the authors con-cluded their paper with the following challenge: "The need for more sensi-tive experiments is evident" [BB74].
Since then a significant amount of work has been done to honor this request, especially in the element suggested by Bouchiat and Bouchiat: cesium [See TW881.
Its high atomic number and hy-
drogenic nature make it an ideal choice for measuring parity-violating effects; such effects scale with Z3, where Z
is the atomic number, and theoretical approximations are most accurate for hydrogenic atoms.
This work, however, is based on another, slightly more complex atom: thallium, whose low lying energy levels are given in Figure 1.1. Although possessing a higher atomic number than cesium (ZT1
= 81, ZCs = 55))
thal-lium is not quite hydrogenic, as its lone valence electron lies in its p rather than s, orbital. This complicates calculations of its wavefunctions, but suc-cessful measurements of parity-violating effects in thallium as well as cesium provide more general tests for all heavy atoms.
1.2.1 Signals
In the optical atomic physics laboratory, parity-violating effects manifest themselves through measurable changes in laser light due to interactions with atoms. Given the weakness of these interactions, it is useful to examine the magnitude of the changes in the transmitted light in order to determine the feasibility of measurements based on these effects. For the determination of signal size in this context, the relevant measure is the optical depth, a dimensionless parameter given by the following:
where n is the number of atoms per unit volume, a(w)
is the absorption cross-section, and 1 is the length of interaction. The absorption cross-section is, in effect, the probability of photon absorption as a function of photon frequency. The effect on transmitted intensity is then given by:
h
=378
nr El Only
If the optical depth is reduced below a certain level, any change in the laser light caused by interactions with the atoms will be imperceptible, for a given noise level, in the laboratory. Given that the optical depth is a product of three very different parameters, it is possible to compensate for the reduction of one of the variables with an appropriate increase in another. This tactic has been utilized in this lab in multiple forms and has made the measurement of several of thallium's properties possible.
1.3 Existing Studies
This experiment, aimed at measuring the Stark shift in the transition at 1283 nm, is far from being the first of its kind undertaken in this lab. In fact, this particular measurement splices together the two primary branches of research that have taken place over the past 10 years. Specifically, work has been done both to investigate the El transition at 378 nm with higher standards of precision and also to examine the spectrum of the much weaker (E2/M1)
transition at 1283 nm.
1.3.1 Stark Shift Measurements
In 2002, through the use of an atomic beam unit capable of Doppler width-reduced spectroscopy, Doret et al. improved the precision of existing Stark shift measurements for the strongly allowed El transition at 378 nm by a factor of 15 over previously published results [DFS02].
The high precision afforded by the atomic beam, however, comes at a price. The number of atoms available to interact with the light is reduced by about 4 orders of magnitude. Encouraged by the success of the 378 nm experiment, though, we have used similar techniques in further studies. In extending this work, one must remember that all atomic transitions are not created equal, and the specific characteristics of the 1283 nm transition offer a variety of exper-imental difficulties that must be overcome.
1.3.2 The Forbidden (E2/M1)
Transition at 1283 nm
Throughout the years of the creation of the atomic beam unit, a parallel strain of research was also being pursued in this lab. Through the use of a high atomic density vapor cell, spectroscopic measurements of the elec-tric quadrupole amplitude were made by Tsai in 1999 [MT99].
These ex-periments were complicated by the relative weakness of this electric dipole forbidden transition, as manifested in a much lower value of the absorption cross-section, which is approximately 10,000 times smaller than that of the 378 nm transition. In the vapor cell, however, the light interacted with ap-proximately 1,000 times the number of atoms, and the optical depth returned to an acceptable level, which led to detectable signals.
1.3.3 Stark Shift Measurements of the Forbidden (E2/M1)
Transition at 1283 nm
The goal of this experiment is to combine these two seemingly incompatible modes of research. Despite greatly reducing both the number of interacting atoms by using the atomic beam unit as well as the absorptive cross-section by studying the magnetic dipole transition at 1283 nm, it is believed that a resolvable signal can still be achieved. The reduction in signal size is offset slightly by the much longer lifetime of this state and the subsequent nar-rowness of the absorption peaks, which recovers one of these lost orders of magnitude. In sum, however, this reduction in optical depth necessarily im-
Optical Depth Vs. Temperature
plies a much smaller signal. Accordingly this experiment requires a different method of measurement based on newly constructed equipment, but given the theoretical calculations and feasibility studies that will follow, this tech-nique will likely prove useful in the high-precision study of this and other weak atomic transitions.
1.4 The Stark Shift
In the presence of a strong, static electric field, atomic energy levels undergo a well-studied change, known as the Stark shift. The shifts for a given field can be calculated directly from non-degenerate perturbation theory. The appropriate calculations can be found in most standard quantum physics textbooks and have been done in previous theses; the results have been sum-marized here (See [SpeOO]).
First order energy shifts due to an electric field along the ?-direction,
with the interaction Hamiltonian, H' = -8.
2,
where d is the dipole moment, are given by:
We know, however, that any such contribution is equal to zero, as the odd parity of the operator in conjunction with the even parity of the combined wavefunctions implies an odd parity function integrated over all space, which is, of course, identically zero. Moving on to second-order corrections, we have:
where an
is the scalar polarizability and encapsulates the rather complicated unperturbed wavefunction dependence for each level. It is possible, however, to gain some qualitative understanding of the relative the behavior of a for some values of n. For example, in the case of a6pIi2,
it can be inferred that its value is less than zero, as all contributions to a6pl12
come from higher energy states such that [A0)
-[g)
< 0. Similarly, we can determine that a6p3,,
is also less than zero, as all of its non-zero contributions come from higher energy states; the contribution from the ground state is zero, as that transition is electric
dipole forbidden. Further,
it is possible to compare the magnitudes of these two scalar polarizabilities. Given that the 6P3/2
state is closer in energy than the 6Pl12
state to all of the states that contribute to the infinite sum, we can conclude /[g3,2
-tg)
1
< l[i$l/2
-[;I/.
Therefore the absolute value of every contribution to a6p3/,
is greater than the corresponding contribution to U'6pIl2.
Thus we conclude that the downward Stark shift in energy is greater in the 6P3/2
state than in the 6Pl12
state. These results are illustrated in Figure 1.3.
For this experiment, however, it is most useful, and indeed only possible, to measure the shift in the energy of a transition between states. In this case, one is interested in the difference in the energy shifts between the ground state and the excited state at 1283 nm. Thus the value of interest is given
It is most convenient to express this in terms of the shift in frequency rather than energy, so recalling that A[
= hAv,
one concludes:
where kstark
is a theoretical parameter whose calculated value can be com-pared to the experimentally measured value. Although lcstarlc
comes from the difference of two infinite sums of radial integrals, its value can be estimated by truncating those sums and using approximate integral values from, for example, [NC77].
Static electric fields have the additional effect of mixing states of opposite parity. Thus the formerly unperturbed p-states are given some character of s and d-states. This allows for the possibility of strongly allowed El transitions at the 1283 nm wavelength, as shown in Figure 1.4. The degree to which this mixing occurs is determined by the atomic wavefunctions, and its quantification will further improve future theoretical calculations. This parity non-conservation is a hallmark of the weak force, and study of these Stark-induced effects provides independent tests of its influence.
Conceptually, it is probably most convenient to think of these amplitudes as two-step processes wherein either the ground or excited state is perturbed from its p-state to an intermediate s or d-state by the static electric field and then the laser light drives a the transition between the opposite parity states. The transition amplitude for the 6Pl12
--+
6P312
transition is then given by
Mixed S
and D States
Mixed Sand D States
where I?
is the static electric field and 5
is the electric field of the light. Rapoport estimated the size of thesc effects to be roughly 100 times smaller than the size of the unperturbed rates, but these should still be visible with this scheme [Rap97].
1.5 Outline of the Thesis
The remainder of this thesis is comprised of a complete proposal of a phase-sensitive method of measuring the Stark shift in the 6P1/2
+ 6P3/2
transition at 1283 nm in atomic thallium. The next chapter involves a detailed deriva-tion that predicts the feasibility of such a measurement. The third chapter is devoted to the experimental apparatus, while the fourth provides experi-mental evidence of the efficacy
of the method. The final chapter details the future steps to be taken in the execution of the experiment and other avenues of research opened by this technique.
Chapter 2
Atomic Physics
2.1 Atomic Tkansitions in Thallium
Optical atomic excitation is a direct result of the interaction of atoms with the oscillating electric and magnetic fields of light. These interactions con-sist primarily of the forced oscillation of charged particles and the subsequent imparting of a multipole moment, which implies a polarization or magneti-zation in the material. This process alters the light as it propagates through the material and is described by the complex index of refraction. The degree to which the light is altered depends on the particular induced multipole moment, each of which is characterized both by its magnitude and the con-ditions, otherwise known as selection rules, under which it manifests itself.
2. 1.
P
Selection Rules
Given the mathematical formalism of quantum mechanics, one can calculate the probability of exciting an atom to another quantum state armed with only the wavefunctions of the initial and final states and a mathematical description of the excitation interaction. One simply calculates the matrix element for the excitation operator, 0,given by:
and squares the result. Luckily for the experimentalists of the world, it often suffices to know only certain characteristics of the quantum states in order to determine the likelihood of absorption. The general protocols of
]
Rule Electric Dipole Magnetic Dipole Electric Quadrupole
An Unrestricted 0 Unrestricted
A1
f 1
0 0, f2
Aml
0,kl
0,
fl 0,
f 1,
f2
Am, 0
0,
fl 0
Aj
0,
fl
i.1
0, f
1,
k2
AF 0,
fl 0, fl
0, f1,
f2
AmF
0, kl
0, fl 0, f
1,
k2
Other F+Ff>2
Table 2.1: Selection rules for atomic multipole transitions. [Ho103]
determination have long since been codified in the form of selection rule tables for each of the primary methods of excitation. One such table has been reproduced in Table 2.1.
Upon first glance, one might be at a loss to find a combination of pa-rameters that does not actually allow for a transition, given the wide variety of mechanisms. In other words, even though a transition like that at 1283 nm is not allowed under the rules of electric dipole radiation, it is allowed under the protocols given for both magnetic dipole and electric quadrupole radiation. Although multiple paths of excitation do exist, this reasoning fails to account for the readily apparent differences in the strengths of the transi-tions. Specifically, a photon with a wavelength of 378 nm is far more likely to be absorbed by way of an electric dipole interaction than a photon with a wavelength of 1283 nm is to be by way of a magnetic dipole or electric quadrupole interaction. This manifests itself in the relative sizes of the ob-served absorption cross-sections; a highly probable transition will give a large signal while a less probable transition yields a smaller signal. In fact, the strength of electric dipole transitions is so much greater than that of mag-netic dipole or electric quadrupole transitions that the latter two are usually ignored in the presence of the former. To get an approximate comparison of these effects, consider the ratio of the probabilities of the two dipole in-teractions. The matrix element for the operator in the El case is given by (yif
1
-d 4
. E 4
[go)
and for M1
by (gf
I
-,Z
. Elgo).
Approximating d as eao
and p
as pg,
where e is the electron charge, a0
is the Bohr radius, and p~
is the
+
Bohr magneton, and recalling that BI
= e,
one concludes:
where a is the fine structure constant (Z
&).
Thus any optical effect caused by the interaction between the light and the atoms is from 1,000 to 10,000 times larger for an electric dipole transition than the corresponding effect in a magnetic dipole or electric quadrupole transition.
2.1.2 The Complex Index of Refraction
Equipped with this understanding of the relative probability of absorption and the subsequent expected signal size, it is necessary to translate this probability into an observable parameter. When examining the interaction of light with matter, the property of primary interest is the refractive index. Its derivation comes from a straightforward extrapolation of classical physics to quantum mechanics. In the classical regime, a charged particle in an atom under the influence of an oscillating electric field, say Eoe-""t,
will behave just as a damped, driven oscillator and will obey Equation 2.3, given by Newton's Second Law:
where r
= blm
and W:
= klrn.
This model allows for the calculation of the induced dipole moment, given classically by 5
= qr'.
With a value of the dipole moment, it is possible to determine the polarization and dielectric constant of the entire material, which finally leads to the index of refraction from the equation:
n=&
(2.5)
where n is the refractive index and 6
is the dielectric constant [Dem98].
This derivation, however, is only an approximation. First, it is predicated on the faulty belief that an atom can be represented as a classical point dipole. Second, this ignores the fact that light is an oscillating electromagnetic wave rather than a purely electric oscillation. As a result, one must incorporate the more subtle natures of both the thallium atom and the laser light.
To add the next terms of the correction to the complex index of refraction, one must first write the oscillating electric and magnetic fields of light so as to account for the finite size of the atom. This yields:
In the dipole approximation, it is assumed that z.
Fm
0,
as 7
is on the size scale of the Bohr radius (m
10-'Om)
while the characteristic length of the electromagnetic wave is that of a wavelength of light (m
lop6
m) [LDK96].
z
r'
can be simplified by the following Taylor expansion:
This correction is then put into the equations for the electric and magnetic fields and gives rise to the quadrupole moment. Given that the magnetic moments are so much weaker than their electric analogs, at this level of cor-rection, one need only consider the magnetic dipole, which is of comparable size to the electric quadrupole, as measured in [Tsa98].
The remaining step in the refinement of this approximation comes from quantum mechanical considerations. The atomic size scales require that rather than considering the classical multipole moments that one instead calculate the alialagous
matrix elements of the multipole moment operators given by ($f/O$o),
where 0 is the appropriate operator.
This yields a final expression for the complex index of refraction:
where Aj
is a scaling factor proportional to the square of the matrix element for each transition, j,
and is equal to zero for transitions forbidden under that particular interaction. This can be resolved into real and imaginary parts, yielding:
These can be further simplified with the assumption that for frequencies near the 1283 nm atomic resonance, the only ones of interest in this exper-iment, that only the A1283nm
term contributes substantially to the sum and further that Iwo
-wl
<
wo.
This yields:
A wo
-w
n, (w) =
1
-t-
4~~rnw~
(w0
-w)~
+ (1'/2)2
The index of refraction represents an experimental parameter whose ef-fects can be measured directly. To illustrate the interaction, examine the oscillating electric field component of the laser light. This can be done with-out loss of generality as the electric and magnetic fields undergo identical changes in this context. In complex notation, the electric field in the absence of a medium
can be expressed as in Equation 2.6. With the inclusion of matter, the index of refraction is introduced into the field to yield:
It is important to note, however, that the index of refraction is a frequency-
dependent, complex quantity with real and imaginary parts of the form n(
f) = n,(
f) +
zn,(
f ). Substituting this back into equation 2.14, one con-cludes:
= Eoe
z(nr(f)6?--wt)
e-n,(f)g.~
(2.15)
E(7,
t,f)
One can clearly see that the index of refraction has two separate effects: the first, an induced phase shift, and the second, a direct absorption. Work has been done to calculate the exact value of the refractive index, but for the purposes of this stage of the derivation, it suffices to note the indices' frequency dependence (See [Vet95]).
From Figure 2.1, for the direct ab-sorption, one finds the ubiquitous Voigt profile, which is the convolution of the homogeneous Lorentzian linewidth associated with the natural width of the absorption, and the inhomogeneous Gaussian linewidth associated with Doppler-broadening. The phase shift is described by a slightly more subtle dispersion relation. In the next section, the expected size of the phase shift will be calculated.
Measurement Feasibility
Although these arguments have done much to suggest the existence of a physical method of determining the resonant frequencies using the atomically induced phase shift, the more pressing task of the experimental physicist is to determine if such a method could ever possibly work in the laboratory setting.
2.2.1 Fabry-Perot Theory
Measuring the phase shift of light requires more complicated methodology than the more straightforward measurement of intensity, which requires little more than a photodiode. The Fabry-Perot interferometer has long been an optical laboratory standard, given its simplicity, versatility, and effectiveness. By separating two highly-reflective mirrors by a fixed distance, L, one is able to translate the emitted intensity into a measure of the frequency or phase of the light. Further, theoretical calculations for this setup abound, allowing for relatively simple feasibility calculations. The transmission of a Fabry-Perot cavity is given by the Airy function, which has the form:
where the FSR is the free spectral range and F
is the finesse. This has the effect of generating Lorentzian peaks at integral multiples of the free spectral range, which, for a confocal geometry, is given by c/4L,
where L is the radius
Transmission
Figure 2.2: Simulated(1eft)
and Experimental(right)
Fabry-Perot Transmis-sion Curves
of curvature-15 cm in this case-and c is the speed of light. As can be seen from Figure 2.3, the factor of 4L in the free spectral range calculation comes from the fact that light traverses the cavity at integral multiples of 4 times before returning to its original trajectory. These peaks are characterized by their maximum values and their full-widths-at-half-maximum, or FWHM. The latter is given explicitly by:
Free Spectral Range FWHM =
.F
Maximum transmission in this configuration implies the generation of a standing wave in the cavity that has nodes at the constrained endpoints oc-cupied by
the mirrors. Thus for a fixed frequency and cavity length, with the accumulation of phase in the light, one generates a transmission curve identi-cal to that in Figure 2.2, except that instead of being separated by multiples of the free spectral range, the peaks are separated by integral multiples of 27r.
This implies:
Af
4
(2.18)
FSR 27r
Equipped with this relation and assuming that all phase shifts of interest in this project are much less than 27r,
it is now possible to calculate changes in
Figure 2.3: Light Trajectory in the Confocal Fabry-Perot [Adapted from [KRYO3]]
the phase of the light based on easily measured changes in the transmitted intensity.
2.2.2 Signal
Size
Having established the cxistence
of a method of measuring phase shifts, it is now necessary to calculate the size of the expected shift induced by the thallium atoms. Using standard optical calculations, the phase shift is given bv:
This, however, is based on the real part of the index of refraction, whose amplitude is not directly measurable nor particularly easy to calculate. It is more experimentally useful to recast this equation in terms of the directly measurable optical depth. This requires the following relation:
which im~lies:
The components of the index of refraction are functions of frequency, and therefore so is their ratio, but that ratio can be estimated by examining its value at the top of the dispersion curve. It will be shown in the next section that this point is given by w E
wo
-0.3r.
Substituting this value for w into Equations 2.12 and 2.13, one concludes:
A 16eomwo
0.3r
(WO
+
(r/2)2
_
w)~
(2.23)
77
nr
n, 4comwo
A (wo
w)~
r/2n
-+ (17/2)2
which can be conservatively estimated as being of order unity over the ma-jority of the resonance center. Then one can estimate the phase shift as:
It should be noted that the constant 1 in the real part of the index of re-fraction has been neglected. This constant phase term is always present, independent of the light's frequency. In this experiment, liowever, we are in-terested only in the frequency dependent phase shift. In fact, the Fabry-Perot cavity is only capable of measuring these changes in phase, rather than some absolute phase. Thus the constant 1 is not measurable and can therefore safely be ignored in this derivation.
Equation 2.24, however, is the calculation for a only single pass through the thallium vapor, but given that the light is in a Fabry-Perot etalon and propagates through the cavity a number of times equal to the mirrors' finesse, the total phase shift is given by:
where the finesse is estimated to be about 50 and the optical depth at roughly
both of which have been experimentally verified.
2.2.3 Signal Resolution
Given that this lab has shown itself capable of measuring optical rotations corresponding to phase shifts many of orders of magnitude smaller than those given above, one might be tempted to accept this proposal without further delay. Unfortunately, in the world of experiment a1
physics, the relevant mea-sure of feasibility is the ratio of signal to noise, rather than the absolute signal size. It will be shown, however, that this ratio is well within experimentally appropriate limits.
To justify this claim, consider a single transmission peak from a Fabry-Perot etalon. In order to suggest plausibility, it is necessary to show that given an uncertainty in the laser frequency, A
f,
that the corresponding un-certainty in the measured signal divided by the absolute signal size is small. Suppose then that one examines the point on this peak corresponding to the greatest optical phase shift induced by interactions with thallium atoms. This shape has the analytical form of a standard Lorentzian curve given below:
where A is scaling constant with units of voltage divided by frequency squared and defined such that at f
= fo,
4A/r2
= V,,,
and l?
is the FWHM. With this mathematical relation, one can find the point of greatest concavity, otherwise known as the inflection point, by taking the second derivative of the curve with respect to frequency and setting that expression equal to zero. This yields inflection points given by:
which implies:
One now examines the effect of an uncertainty in frequency, Af,
at the inflection point. In other words, given a change in frequency, one wishes to know the corresponding change in signal. This is simply given by the slope of the Lorentzian curve at the inflection point, which can be calculated directly by evaluating the first derivative at that point. This yields:
dV
-2A(f
-fo)
df
((f
-fol2
+ ((r/2)2)2
The second equation comes from substituting for A and the assumption that the Lorentzian is linear over this small interval, which is commonplace in this type of calculation. This derivation has thus far related the easily ex-perimentally measured fractional signal uncertainty to a given uncertainty in frequency, but given that the experiment is interested in measuring opti-cal phase shift, it is more useful to state the fractional signal uncertainty in terms of an uncertainty in the phase of the periodic Fabry-Perot transmis-sion. Recalling Equation 2.18 and substituting those values, one arrives at the conclusion:
AV
~&(FsR)&
Vmax
8nr
nv
8~
a4
Vmax
3&~
These calculations are the fundament a1
relations motivating this experiment, but so far they fail to reflect experimental realities, that is, these equations hold independent of the instrumentation used. It is now necessary to plug in expected values of certain parameters to justify the feasibility in this lab. Returning to equation 2.30, it is necessary to show that the fractional uncer-tainty in voltage is on the order of a few percent, as any associated noise that is comparable to the size of the signal will preclude an effective measurement. This so called Titter" on the side of a Fabry-Perot fringe increases with the reflectivity of the mirrors but so too does the size of the phase shift signal. Thus one must balance these two competing factors to achieve a large enough signal that can still be resolved from the noise. Previous experiments have stabilized laser frequency to within 0.3 MHz, and measurements of the Fabry-Perot transmission curve show the finesse of the cavity to be roughly 50, which implies an FWHM of 10 MHz for the nominal free spectral range of 500 MHz [KBU05].
Plugging these values into Equation 2.30, one concludes that the fractional uncertainty in voltage is on the order of a few percent (-
4%),
as hoped. Now it is necessary to recast these parameters into the terms of the expected phase shift. By the relation in Equation 2.18, one finds that the uncertainty in laser frequency implies a phase uncertainty of 3.8 mrad. This can be compared to the expected phase shift calculated in Equation 2.25. This suggests that the uncertainty in phase is comparable to the expected signal, which would seem to discredit the plausibility of the experiment. As will be shown in the next chapter, for the purposes of extracting the atomically induced phase shift from the associated noise, one can use very powerful lock-in detection techniques to pick out the component of the signal that comes from the thallium atoms.
Frequency Scan of a Slngle
Fabry-Perot Peak
Frequency
Figure 2.4: Slow Scan of a Single Fabry-Perot Peak. Notice the residual jitter associated with frequency uncertainty.
Chapter 3
Experimental Apparatus
Figure 3.1 shows a schematic of the optical and signal paths involved in this experiment. This form of spectroscopy requires three general systems: an optical probe, a spectroscopic medium, and a method of signal processing. The important facets of each are discussed below.
Optical Probe
3.1.1
Fabry-Perot Cavity
The item of primary interest to this thesis is the newly constructed Fabry-Perot cavity that is to be integrated into the atomic beam unit shown in Figure 3.2. It is with this device that one can measure the induced phase shift and perform the desired spectroscopy. The fundamental design of a confocal Fabry-Perot cavity is very simple; one merely separates two highly reflective curved mirrors by their radius of curvature. Given that this particular cavity is fated to perform Stark shift measurements in the rather trying environment of the atomic beam unit, other engineering concerns have been taken into consideration.
At its simplest level, this cavity consists of two Macor ceramic plates attached by three 15 cm Invar rods. Small optical mirror mounts (slightly altered versions of the Thorlabs KS05)
are screwed directly onto these plates. Also the existing electric field plates used in previous experiments have been integrated into the system and are attached directly to the Invar rods. The optics involved in this cavity are two CVI custom-built spherical mirrors.
Optical
1283 "111
Laser
lrolatar
They are 0.5" in diameter and have been designed to have 99.5 %
reflectivity with a radius of curvature of 15 cm. The flat faces have been anti-reflection coated for 1283 nm. The cavity has free spectral range of c/4L,
which for L = 15 cm, is 500 MHz.
Given that the phase shifts to be measured in this experiment represent small fractions of a wavelength of light, the accuracy and precision of the separation of the two mirrors are of paramount importance. Although the low thermal expansion coefficient of Invar
C) does much to minimize the instability of this separation due to temperature drift, this experiment requires an even higher level of certainty. To achieve this goal, a locking system has been designed to constantly adjust the length of the cavity to compensate for external changes. The mechanics of this system were inspired by the larger Thorlabs KC-PZ1
system. The size of this commercial device precluded its use in our experiment, but its use of piezoelectric transducers, PZTs, placed directly under the adjustment screws of the mirror mount can be seen in one of our cavity's mirror mounts. By applying a parallel voltage to these PZTs, the separation between the two plates of the mirror mount, and therefore the two mirrors, increases or decreases according to the minute changes in the widths of the PZTs. For more information on the PZT powered mirror mount, see Appendix A.
This length stabilization method exploits the well-studied phenomenon of Fabry-Perot cavity signals. Ignoring for the time being any uncertainty in
the frequency of the laser, one can choose an arbitrary set frequency point along the very high slope of a single Fabry-Perot peak to act as a standard marker for a specific separation of the mirrors. As the length of the cavity changes as a result of thermal fluctuations, even on the submicron level, the transmission signal associated with the set frequency point changes dramat-ically, given the high signal change to frequency change relation implied by the steep slope of the Fabry-Perot peak. These differences in the transmis-sion signal are then sent to the feedback element as an error signal, which is converted into a correction signal that is sent to the PZTs
to actively compensate for the changing cavity length. The magnitude and direction of the change necessary to achieve stability are calculated and delivered via an electronic feedback circuit designed by the Wieman group at the Joint Insti-tute for Laboratory Astrophysics [Wie98].
Its correction signal is generated in several steps. First, the voltage corresponding to the measured intensity is compared to a set point voltage within the circuit. The difference between the two serves as an error signal that undergoes two simultaneous processes. One, known as a feed-forward system, inverts and magnifies the signal so as generate a correction component that will act in the opposite direction of the error signal. With this process alone, though, the correction signal will consistently overshoot the target voltage, leading to oscillation about the set point. To counteract that phenomenon, the signal also undergoes a feedback process that integrates the error signal over some time. The resulting cor-rection component is then added to the feed-forward portion to generate a complete correction signal that will stabilize the measured intensity to the set point. The relative strengths of feed-forward and feedback as well as the time constant for integration are adjusted through the values of electronic components used in the circuit and can be tuned to achieve critical damping of deviations from the set point.
3.1.2 Diode Laser
Returning now to the subject of laser uncertainty, it is plain that drift in the frequency of the diode laser will always exist. Unavoidable thermal fluc-tuations and current drifts give rise to changes in the cavity length, which alter the frequency of the laser light. Without a stabilization method, the uncertainty of laser frequency can be of order 10 MHz over a time scale of a few seconds, as shown in Figure 3.3. Much study, however, has gone into the means by which such uncertainty can be minimized. The general premise
Diode Laser Frequency Drift
behind extended cavity diode laser stabilization is to constantly compare the laser frequency to some fixed frequency and compensate for changes by ad-justing the PZT or injection current in the laser. Two separate methods of laser frequency stabilization exist in this laboratory. The first, based on the Orozco group's method of comparison of the diode laser frequency to that of a stabilized HeNe
laser, has been used in years past [ZE098].
Although wave-length non-specific, this technique requires the physical scanning of another Fabry-Perot cavity, which limits its bandwidth to roughly 20 Hz.
The second, more novel, approach employs laser correction signals based on low magnetic field Faraday rotations in atomic thallium at the 1283nm
transition. In the presence of a modest (N
10 G) magnetic field parallel to the direction of laser propagation, thallium vapor preferentially absorbs one of the two helicities of circularly polarized light, giving rise to small rotations in the polarization axis of linearly polarized light. Through very precise polarimetry, it is possible to measure these rotations. As one scans through a resonance, one finds a Faraday rotation curve essentially proportional to the derivative of the dispersion curve that describes the phase shift induced by the real part of the index of refraction. To lock the laser, one sits on the point of zero polarization axis rotation and adjusts the laser cavity PZT according to deviations from that point with an electronic servo circuit. This zero
Dispersive Phase Shift
I
Frequency Detuning
1
1
Faraday Rotation
Figure 3.4: Frequency Dependence of Dispersive Phase Shift and Faraday Rot ations
crossing point is ideal for this experiment; given that the Faraday rotation curve is proportional to the derivative of the dispersion curve, the lock point is at the same frequency as the point of maximum phase shift, the peak of the dispersion curve, as seen in Figure 3.4.
The uncertainty in the diode laser frequency can be limited to roughly
0.3 MHz. For more information on this system, refer to [KBU05].
The laser used in this experiment is a Sacher
Lasertechnik external cavity, semiconductor diode laser in a Littrow
configuration running at 1283 nm. It is quoted by Sacher
as capable of delivering a peak optical power of 20 mW,
but has been found experimentally to be closer to 10 mW.
This laser is fairly robust in terms of tunability, but is susceptible to problems related to optical feedback, especially when operating with a high-finesse Fabry-Perot cavity. Strong optical isolators have been placed early in the beam path and when properly aligned have largely eliminated feedback problems.
3.1.3 Acousto-Optical Modulation
To study the absorption spectrum at the 1283 nm transition, we measure the changes in cavity transmission over a range of frequencies of roughly 50 MHz. As stated earlier, however, laser frequency uncertainty presents a large potential source of signal noise. Thus one must combine the seemingly paradoxical ideals of frequency stability and tunability with a single laser. This is achieved through a two-step process involving a device known as an acousto-optical modulator, or AOM. The raw, unmodulated light that emerges from the laser is split into two paths, one that leads to the laser locking systems described above and another that leads to the AOM. The locking system fixes the laser's frequency at an arbitrary point while the light in the AOM is subjected to a frequency shift induced through the interaction of the laser light with a photoelastic crystal subjected to a strong RF signal. The acoustic RF wave produces a periodic strain in the crystal that alters the index of refraction in the material, which results in optical diffraction. When the Bragg criterion is satisfied, a significant portion (3
60%) of the
light is Doppler-shifted by the moving planes of altered refractive index by an amount equal to the frequency of the acoustic wave [ISO93].
This shift is illustrated in Figure 3.5. The light that emerges from the AOM is thus frequency shifted according to the RF signal generated by an HP signal synthesizer without affecting the stability of the locked laser frequency. In this way, the pitfalls associated with frequency instability are avoided without losing tunability.
The AOM is not
without its flaws. In the process of altering the light's frequency, the AOM diffracts the light as it exits the AOM crystal. Early tests have shown the cavity to be able to withstand this level of steering, but should this later be found to be unacceptable in the atomic beam unit, it is possible to double-pass the laser beam through the AOM to engender no net steering. This approach reduces the light's intensity, making it more difficult to see the beam after it exits the cavity. This makes alignment into the photodiode more onerous, but with patience, it will still be possible.
(ialc
Valve rcir
3.2
Spectroscopic Medium
The material of interest to this experiment is thallium vapor. Given that thallium is a solid at room temperature, it is necessary to heat the sample to temperatures upwards of 800°C in a vacuum to achieve an atomic den-sity capable of generating a detectable signal. At these high temperatures, spectroscopic signals are Doppler-broadened due to the Maxwellian velocity distribution of the individual atoms, that is, moving atoms are capable of absorbing a wide range of light frequencies, as the photons are shifted ac-cording to their relative motion parallel to the atoms. In order to measure the Stark shift as precisely as possible, this broadening must be reduced, but enough atoms must be produced to generate a detectable signal.
The device capable of balancing these two needs, and the ultimate desti-nation of the Fabry-Perot cavity, is the atomic beam unit, shown in Figure
3.6. This sophisticated vacuum system capable of generating a thin ribbon of thallium atoms has undergone minor changes over the course of this ex-periment, specifically the replacement of a destroyed diffusion pump and the redesign of the atomic chopping wheel, but the majority of its functional-ity can be found in previous theses (See for example, [Nic98][FriOl]).
The method of its operation is to heat the thallium to the same high temperatures, pre-collimate the resultant atoms through the design of the oven nozzle, and then further collimate the beam by blocking those atoms whose velocity is not perpendicular to the laser light. It should be noted that one must allow for a certain amount of non-orthogonality in the beams,
as a perfectly per-pendicular atomic beam would contain only a negligible number of atoms. Regardless, this collimation system shown in Figure 3.7 reduces Doppler-broadening by roughly a,
factor of 15-from ~500
MHz to ~30
MHz-but does so at the reduction of the number of atoms available to interact with the light. We believe that this reduction of optical depth has been the primary cause of previous failures to measure the Stark shift at 1283 nm with spectroscopic methods based on direct absorption. The method in this experiment relies instead on the dispersive phase shift of laser light, as represented in the real part of the thallium vapor's index of refraction, to overcome these problems.
3.3 Signal Processing
3.3.1 Photodiodes
This experiment differs from others recently conducted in this lab in that the amount of transmitted light to be measured is actually quite large and in a frequency range easily reached by available detectors. Furthermore the signal modulation takes place at a frequency on the order of 100 Hz, which is easily resolved with the widely-available v-Focus 2011 photodetector.
3.3.2 Lock-In Detection
In the field of laser spectroscopy, few will argue against the power of the lock-in detector. With a clever combination of electronic engineering and trigonometric identities, the signal to noise ratio made possible with this device makes it nearly as ubiquitous as the oscilloscope. In order to tap its power, however, one must modulate the desired signal at some known
frequency. Two methods of modulation have been used with our atomic beam unit to date. The first employs a standard light chopping wheel that is placed directly in front of the laser. One can then feed the output of the photodiode directly into the lock-in amplifier and extract the modulated signal. This simple method is unfortunately untenable in this experiment. Recalling the necessity of locking the length of the Fabry-Perot cavity in order to achieve full transmission, repeated interruption of the incoming laser light would preclude the use of the feedback system described above.
The second method indirectly modulates the signal nestled into the laser light without physically disrupting the beam through the system. By plac-ing a chopping wheel in front of the atomic beam itself, one modulates the phase shift incurred by the light interacting with the atoms without blocking the light. To put this another way, in the absence of thallium atoms, there will be a correction signal sent to the PZTs in order to offset the changes in cavity length
caused by thermal drift. In the presence of thallium atoms, there will be an additional correction signal sent to the PZTs to account for the phase shift incurred by the light as it interacts with the atoms. Given that the atoms are either blocked (causing no additional phase shift) or un-
blocked(causing
an additional phase shift), this results in a step function in the correction signal tliat
is combined with the baseline level correction. The modulated correction signal, that is the portion of the signal caused by the thallium atoms, is neatly picked out of the combined baseline and induced signal by the lock-in detector. The beauty of this design is that the Fabry-
Perot cavity length is always locked. One simply feeds the correction signal into the lock-in amplifier to detect the presence of atomic phase shifts.
There are, however, problems associated with the use of the atomic chop-ping wheel. The mechanical vibrations associated with wheel have the po-tential to unlock the Fabry-Perot cavity. To combat this problem, a thicker wheel that runs more true has been installed into the atomic beam unit. There is still the expectation that noise
at the frequency of the wheel's ro-tation will persist. It should be noted, though, that because the chopping wheel has six holes through which atoms may travel, rather than one, that the signal is modulated at six times the wheel rotation frequency. Provided that the wheel can rotate smoothly without destroying the cavity length lock at frequencies of about 20 Hz, one has reason to hope that the associated noise will be outside the bandwidth of the lock-in detector and will not affect the measurement.
Chapter 4
Experimental Tests and Results
Although the timeframe allotted for this experiment prevented an actual measurement of the Stark shift at the 1283nm
transition, much work was done to confirm the estimated experimental constraints used in the justi-fication of this method. Furthermore, simulations of the experiment have been conducted outside of the atomic beam unit so as to gauge the system's sensitivity as well as to predict the ultimate measured signal.
4.1 Cavity Tests
In the derivation of the expected signal resolution, certain levels of stability were assumed based on previous experiments. Tests were run to check the accuracy of these assumptions. By scanning the frequency of the laser over several Fabry-Perot peaks, it was possible to measure the FWHM of the peaks and infer the finesse of the cavity.
Further, measurements were made to test the locking ability of the cav-ity in order to determine the locked phase uncertainty. The fractional un-certainty was found to be less than 2%. This combined with a measured FWHM of 9 MHz implies a phase uncertainty of roughly 1.5 mrad. With the use of lock-in detection, this uncertainty is reduced by a factor of the square root of the bandwidth. Assuming a bandwidth of 100 Hz, this suggests that the cavity has a nominal uncertainty of 0.15 mrad in one second, which is significantly lower than the expected signal phase shift.
Locked Noise Signal Experimental Fabry-Perot Trsnamiaaion
Frequency
Uetunlng
(MHz1
Time
liecl
Figure 4.1: Measurements Confirming Estimated Experimental Cavity Lim-its. Note from the scales that the entire width of the graph on the left is equal to one of the divisions on the graph to the right
4.2 Phase Shift vs. Transmission Change
The purpose of the Fabry-Perot cavity in this context is to translate an easily measured change in transmitted intensity into a measure of the phase shift of the laser light. There are, however, multiple possible sources of changes in the intensity. Thus in the analysis of this simulated experiment, it is useful to differentiate between
atomically induced changes in the phase of the oscillating electric and magnetic fields and other external effects causing changes in the transmission through the cavity that are thus indistinguishable from actual phase shifts. Specifically, the signal measured in this experiment is that of the voltage applied to the PZTs
on the mirror mount necessary to keep the measured intensity of light coming out of the Fabry-Perot cavity constant. Thus any change in the measured output intensity, regardless of its origin, will be seen as part of the signal. Two such transmission changes are shown in Figure 4.2.
Laser frequency drift and changes in the length of the cavity cause the transmitted intensity output to move along the Airy function given by Equa-tion 2.16. These effects are actively compensated for by the locking circuit and present a baseline uncertainty in the phase of the light that can be
Transmission Transmission
Frequency
Frequency
Figure 4.2: Transmission Change in a Fabry-Perot Cavity. Left: In the Presence of Absorption. Right: Subject to a Frequency Change
calculated as shown earlier. It is important to note that these effects are independent of the interaction with atoms and therefore can be measured outside of the atomic beam apparatus. Further, such non-atomic effects do not occur at the frequency at which the thallium beam is modulated, and as a result are largely excluded from the output of the lock-in amplifier. This minimizes the effect of such uncertainties on the final measurement.
There is, however, an atomic effect that does occur at the modulated frequency that will persist through the lock-in amplifier: the direct absorp-tion of light due to the imaginary
part of the index of refraction. Although largely ignored in this discussion due to its small size, the effect of the di-rect absorption of light can be recast as a shift in the light's phase. It is fairly obvious that such absorption will reduce the transmitted intensity, but in the vernacular of Fabry-Perot theory, the understanding of this behavior can be refined. The process of absorption in the Fabry-Perot etalon effec-tively lowers the reflectivity and therefore the finesse of the confocal mirrors. This causes the transmission peaks to decrease in height and broaden while keeping the same center frequencies. From the perspective of the measured signal, however, the change in output intensity from the view of the set point is indistinguishable from the phase shift shown in Figure 4.3. Thus in any simulation of the expected output signal, it will be necessary to combine the results expected from both parts of the complex index of refraction.
The Expected Signal
Given the extensive theoretical knowledge surrounding the interaction of light with atoms, it is possible to predict the frequency dependence of the transmis-sion change induced in the interaction, as is done in Figure 4.4. Considering
Transmission
Induced
Phase
Shift
Frequency Detuning (GHz)
Transmission Change Transmission Change
only the effects of the phase shift that comes from the real part of the index of refraction, one would expect a dispersive curve centered on the atomic resonance. To this simulation one must also add the symmetric contribution from the transmission change associated with the absorption caused by the imaginary part of the refractive index, which is also centered at resonance. From
the derivation in Chapter 2, it is possible to quantify this result. Re-calling Equation 2.32, one can calculate the equivalent phase shift associated with the absorptive change in transmission. For example, an absorption of 0.1% implies a phase shift of N
100 prad
at the center of the resonance peak, for the measured finesse of 50. These results, exaggerated for clarity, can be seen in Figure 4.5.
Examining this simulation, one notices that the zero crossing has been shifted with the inclusion of the absorptive component. Rather than inducing no phase shift at the center of the resonance peak, a net zero phase shift is incurred at a slightly different frequency. In reality this type of shift would be much smaller, as the phase shift due to absorption is very small. Still, such a contribution would make the determination of the center of the resonance line difficult, as it would be necessary to somehow extract the absorptive portion of the phase shift from the total shift. Fortunately, the high precision experiments conducted in this lab are more concerned with relative shifts in frequencies, rather than absolute positions, as seen not only in this experiment but also in the hyperfine and isotopic splitting measurements conducted by Lyman [RLM99].
Thus for this purpose, one can assume that with the inclusion of the perturbing electric field, one will find a lineshape identical to that of the unperturbed system that is merely shifted
Phase
Shift
30 MHz Stark Shift
in frequency. Thus one still has a standard marker in the measurement of the Stark shift even though it is no longer the center of the resonance peak.
The width of the dispersion curve is determined primarily by the resid-ual Doppler width caused by the non-orthogonality of the laser and atomic beams. This can be estimated to be roughly 30 MHz. One can also esti-mate the magnitude of the Stark shift at an electric field of 30 kV/cm
to be approximately 30 MHz. Thus the curve should be displaced by roughly its width, which is more than enough to be resolved, as seen in Figure 4.6.
The actual measuring process will most likely involve the transmission change method used in the 2002 experiment. This technique locks the laser frequency to a certain point on the dispersion curve and then alters the frequency of the light going through the Fabry-Perot cavity by a precisely known amount by way of the AOM. This would have the effect of changing the transmitted intensity of the light coming out of the cavity substantially. Now a precisely measured electric field could be turned on that would return the signal to the original set point. Fkom
the known frequency shift and electric field, it would then be trivial to calculate kStark.
The 2002 experi-ment also used another technique that took data at several different points about the absorption peak and fit a theoretical curve to find the center of the resonance. A similar technique could be used in this context, given an appropriate line-fitting algorithm for this altered dispersion curve. It would also be possible to compare frequency scans in the presence and absence of the electric field to determine the Stark shift. All of these supplementary measurement methods are useful in corroborating the value ascertained from the transmission change method.
4.4 Verified Theory
In the theoretical calculations earlier, it was asserted that the system would be able to resolve phase shifts on the order of 0.1 mrad. To test this hy-pothesis, a modulation in the laser frequency, which has been shown to be indistinguishable from a phase shift, was introduced into the cavity. This was accomplished through the use of the acousto-optical modulator described earlier. With a modulation of 100 kHz, a signal could be seen with lock-in detection. Using the relation comparing shifts in frequency and phase, this can be seen to be equivalent to resolving a phase shift of
radians, which is significantly smaller than the expected atomically induced shift. This is a very encouraging result that further supports the feasibility of this experi-ment.
This result is, however, predicated on the use of lock-in detection, and to use such techniques, one must modulate the signal at a known frequency. This is to be done with the chopping wheel inside the atomic beam unit. Should the wheel be found to introduce an intolerable level of noise, another possible technique remains. In the successful 2002 experiment at 378 nm, a stepper motor that stably moved the chopping wheel 1/12 of a turn at a frequency of 1 Hz was used. This very low level of modulation can be used to essentially "lock-in detect by hand," allowing measurements to be read directly from the oscilloscope. This runs the risk of introducing slow drifts into the signal, as the modulation is so slow, but given the relatively high bandwidth of the locking circuit, such drifts should be at a tolerable level.
4.5 Further Reduction in Cavity Drift
In the discussion of potential sources of systematic error, this experiment benefits from the success of the 2002 experiment [Dor02].
A number of possible sources were investigated and their probable sizes quantified. The majority are related to the function of the atomic beam unit and should therefore be easily applied to the new system. The remainder are primarily associated with the electric field plate system, and given the proximity of additional metal pieces associated with the cavity should probably be re-examined.
Beyond previously calculated sources of systematic error, the new Fabry-
Perot cavity also introduces new sources that were investigated. The most pressing concern is that of temperature stability. Although the cavity is made of Invar, temperature fluctuations in the lab may be at the root of unstable readings that have been measured thus far. Temperature changes in the cavity lead to linear thermal expansion given by the following:
where a is the coefficient of thermal expansion. Then for a temperature in-stability of say O.l°C,
there exists a frequency instability of about 20MHz.
The PZT system in the cavity is able to correct for this instability to main-tain lock, but the correction signal may be flooded by this instability, which may be the cause of slowly oscillating lock-in detector readings. In place of the constant lock-in detector signal expected for a sinusoidal frequency mod-ulation at 50 Hz, early tests have shown non-zero signals with fairly periodic drifts of about 30% of the maximum signal size with periods on the scale of about one second that can extend into slower drifts with periods on the scale of roughly 10 seconds that bring the signal from its maximum value to an equally large negative value and back. In the evacuated environment of the atomic beam unit, temperature instability is reduced, but measures such as passive and active temperature regulation are conceivable should this insta-bility be deemed excessive. Temperatures can be locked in much the same way as laser frequency, in that deviations from the set point temperature can trigger correction signals that can be sent to the heating unit to achieve a stable temperature.
To get an estimate of the certainty necessary, consider trying to resolve A#=l
mrad. From Equation 2.18, this suggests a frequency stability on the order of 100 kHz, which is achievable through time averaging with the present locking system. In terms of length and temperature stability, one can combine Equations 2.18 and 4.1 to calculate a necessary temperature stability on the order 1
mK
and length stability on the order of 0.05 nm to resolve a 1mrad phase shift. These calculated values neglect the use of lock-in detection to pick out the atomically induced phase shift but are very useful in estimating the basic levels of parameter stability necessary to carry out this experiment.
Chapter 5
Future Study
As is the curse of dl
undergraduate theses, the author has but limited time to devote to this experiment, leaving many avenues of inquiry unexplored. The success of the Fabry-Perot cavity built in this time, however, does much to suggest future discovery in this lab.
5.1 Remaining Bench Testing
The most immediate concern in this experiment is the oscillation observed in the lock-in detector readings of the correction signal. Temperature stabiliza-tion measures will likely be necessary in the very near future. A consistent, constant lock-in signal will do much to prove the utility of this phase-sensitive detection method. In the near future, it will also probably become necessary to double-pass the laser light through the AOM to remove all possibility of steering problems in the laser beam. This will also serve to show that the transmission change that is responsible for the correction signal is due to the changes in laser frequency and not the steering of the beam.
Figure 5.1 shows one possible configuration that allows for double-pass.
5.2 "Forbidden" Transition Spectroscopy
This thesis hopes to present a complete examination and test of the instru-ments and methods to be used in the measurement of the Stark shift at the 1283 nm transition. Having firmly established the feasibility, both the-oretically and experimentally, of this technique, the time has come for the
Polarizing
Beam Splitter
Incoming
Quarter
Wave Plate
v=v,+
26v
Figure 5.1:
Proposed Method of Double-Passing Light Through the AOM
integration of the Fabry-Perot cavity into the atomic
beam unit and the search for the signal.
This technique also has lasting ramifications as it is essentially indepen-dent of wavelength. Thus it presents a method of investigation of intrinsically weak transitions that requires only the alteration of the laser wavelength and the coating of the reflective mirrors to that wavelength. The convenience presented by this flexibility makes this method useful in the examination of any number of transitions. Its usefulness can also
be extended to measure elements whose isotopic abundances would otherwise preclude detailed mea-surement. There exist isotopes whose transitions, although strongly allowed under electric dipole selection rules, are very small due to very low abun-
dances. The sensitivity of this method would compensate for the diminished optical depth.
5.3 Two-Step Spectroscopy
The Fabry-Perot cavity constructed for the 1283
nm transition also presents an opportunity to revisit the two-step 6Pllz
-+
7SIl2
-+
7Pl12
transition discussed by Burkhardt [Bur04].
In this experiment, a frequency-doubled 755 nm to 378 nm beam (N
1pW)
of UV light excited thallium atoms via an electric dipole transition. From
this excited state, it was hoped that an overlapping beam at 1301
nm, a wavelength that can be reached with the 1283
nm laser, would further excite the atoms via another electric dipole transition to the 7Pllz
state. Due to experimental constraints on the power of the UV light and the difficulty
of overlapping the two beams, no signal was ever resolved from this experiment. Further, the special Brewster-angle
vapor cell designed by Holmes in 2003 for this experiment was shown to be unable to withstand the high temperatures needed to produce an appropriate atomic density [Ho103].
With the replacement of the mirrors currently employed in the Fabry-Perot with similar mirrors coated for the 378 nm transition, it may be possible to amplify the power of the UV laser to achieve a detectable signal.
Figure 5.2: Transitions of Interest in the Proposed Two-Step Measurement
Given the high reflectivity of the Fabry-Perot mirrors at 378 nm, the laser beam traverses the cavity several times before emerging through the second mirror. Consider then the travel of a length of the beam equal to the length of the cavity. On average, the photons in this beam will traverse the cavity a number of times equal to the finesse of the mirrors, which can be coiiservatively estimated at about 50. Thus the amount of laser light in the cavity at any given time is 50 times what it would be for a single pass. Given the strength of the electric dipole transition, this has the effect of multiplying the population of atoms in the excited state by essentially the same factor. The 1301 nm probe beam, whose power (N
5mW) is more than enough to excite all of the atoms in the 7Slp
state it encounters, could then be passed through the cavity once, as it is almost entirely transmitted through the mirrors coated for 378 nm, and its transmission would be measured. This experiment would return to reliance on the imaginary part of the index of refraction, as the small absorption size of M1
transitions are irrelevant to these two El transitions. Assuming that the 378 nm transition could be saturated by resonating the UV light, the electric dipole amplitude of the 1301 nm transition would cause a very easily resolved direct absorption.
Preliminary experiments might first investigate the Stark shift in this transition in the atomic beam unit. It would also be possible to carry out high precision hyperfine and isotopic splitting measurements in this environment, much like those done by Lyman [RLM99].
The atomic beam unit fails to exploit the hole burning described by Burkhardt but does allow for Stark shift measurements that would be precluded by a vapor cell. Such experiments shall likely prove fruitful in the near future.
Appendix A
The PZT
Powered Mirror Mount
Optical mirror mounts are designed with two primary goals in mind: stability and tunability. Standard mounts usually achieve these ends through the use of two separate plates connected with springs. One of the plates is fixed in space through attachment to a large optical table while the other, which actually holds the optic, is afforded limited tunability by way of adjustable screws that alter the distance and angle between the plates. In the design of this mount, similar needs were considered, but given that there would be no optical table to which the mirror could be fixed and the need for piezoelectric transducers, other methods are used.
In place of an optical table, the source of stability is the relatively massive body of the Fabry-Perot cavity itself. By attaching one of the mirror mount plates directly to one of the cavity plates, one essentially fixes that part of the rnount in space. The second plate, which holds the confocal mirror, is then still allowed to move by virtue of adjustable screws. In the first attempt to integrate the PZTs, the standard slot that ordinarily holds the mirror was ignored and the
PZTs were glued directly to the body of the second plate. The mirror was then glued to the PZTs. Thus the second mirror could still be adjusted with the screws and the PZTs. However, the shear stress applied to the epoxy from the gravitational pull on the mirror was found to cause intolerable instability. The second, and currently used, system involves the replacement of the moving plate. Inspired by an existing Thorlabs design, the PZTs are placed directly under small pieces of stainless steel that hold the adjustable screws of the fixed plate in place. The PZTs can still expand Figure A.l: Photographs of the PZT Mirror Mount. Left: The PZTs (one pictured in lower left corner of the plate) are placed in between the two plates of the mirror mount to adjust the separation distance between the two mirrors. Right: The fully functional PZT mirror mount
or contract according to the voltage applied to them, allowing the distance between the two plates to change, but the mirror can be placed into its socket on the plate and fixed wit,h
a set screw. Besides vastly improving stability, this has the additional benefit of mediating the change in the separation distance with the linear restoring force of the springs attaching the two plates. The movement of PZTs is notoriously non-linear, but this problem should be reduced by the springs.
The design of this new plate is essentially a duplication of the existing plate with the inclusion of small holes into which one can put the PZTs, channels through which the electric wires can emerge, and small stainless steel pieces analogous to those in the existing Thorlabs design. This piece was custom made by the Machine Shop at Wesleyan University.
Appendix B
Electronics
The only piece of electronics worthy of note to this experiment is the locking circuit designed by the Wieman group at JILA. Its schematics are reproduced below:
Figure B.l: Circuit Diagram for the Cavity Length Locking Circuit [Wie98]
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