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A TECHNIQUE FOR PASSIVELY COMPENSATING THERMALLY INDUCED
MODAL DISTORTIONS IN FARADAY ISOLATORS FOR GRAVITATIONAL
WAVE DETECTOR INPUT OPTICS
RUPAL S. AMIN
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
Rupal S. Amin
This thesis is dedicated to my parents and family.
They were ah--iv there and never let me back down.
Acknowledgments are usually a section in a thesis, dissertation, or book that
thanks various instructors for their contributions. This section also seems to be
the most homogeneous in all books. Well to those people whom I will alv- i- call
professor, I deliver the following as my version of the acknowledgments.
(Tune of Yakko's N II i's of the world.")
Tanner and Reitze, showed some light, see, although it was infrared,
had me help correct a laser, and to Guido's pleasure, never found me dead,
optical alignment, wave confinement, all stored in my head,
Gouy shifts, higher order rifts, won't let me sleep in bed,
high work ethics, influenced my kinetics, when talking to Guido (whew),
and Coldwell kept smiling, 'cause code was compiling, on my CPU,
Malik was frantic, a CalTech semantic, and helped in afternoons,
but only in theory, and with a big query, why I never salute,
Good sir Klauder, your applause get louder, as I add you to my list,
you showed me tremendously, mainly mathematically, things I had missed,
Lundock and Guagliardo, the dynamic duo, pushing the physics each di-,
never complained in lab, lined up a slab, and helped me every i-.
Dr. Jim Simpson, a good physics denizen, taught my first physics class,
then Dr. Raffalle, got another advisee, from whom physics would pass,
Drs. Rassoul and Moldwin, won the gold in, geospace science alas,
they saw I was recruited, and to them I proved, CDVT was not crass,
Mantovani and Wood, time permitting they should, take Rupal avoidance
but when I had questions, they would never mention, how I bored
them and laughed,
Sawyer, Jin, and Blatt, at a distance they sat, e--: '1 me on to do well,
how they showed me, a bit of arrogancy, does little else but smell,
Gering I'm -Ip iii._ from this poem l ini]1 --ii- teaches undergrads who
Paul, TJ, and Nic, whose last names we'll omit, I am sorry for the big scares,
In high school, there was...Covington, Cloniger, Campbell and Wells, Asher,
all those teachers who casted their spells,
and helped me become who I am.
Also thank you NSF for making this thesis possible.
TABLE OF CONTENTS
ACKNOWLEDGMENTS ................... ...... iv
ABSTRACT ...................... ............ viii
1 INTRODUCTION ........................... 1
1.1 H history . . . . 1
1.2 Gravity Waves ............ .................. 3
1.3 Detection of Gravity Waves ......... ........ .... 6
1.4 Faraday Isolators ................... .... 12
1.5 M otivation .................. ............ .. 14
2 THEORY ................... ... ... ........ 16
2.1 Why Do Materials Absorb EM Energy? .............. .. 16
2.2 Birefringence: Innate and Thermal ................ .. 20
2.3 Magneto-Optics and the Fa-rd-vi Effect . 23
2.4 Faraday Isolator .................. ......... .. 28
2.5 Thermal Lens .................. .......... .. 30
2.6 Toy Model and Practice .................. .. 34
2.7 Wavefront Model .................. ........ .. 36
3 EXPERIMENT AND ANALYSIS .................. ..... 41
3.1 Design ................. . .... 41
3.2 How to Analyze a Thermal Lens .................. .. 42
3.2.1 Scanning Fabry-Perot Cavity ................ .. 42
3.2.2 Beam Profiling .................. ..... .. 52
3.3 Layout of Table .................. ......... .. 53
3.3.1 Table Overview .................. .. .. .. 53
3.3.2 Components .................. ....... .. 54
3.3.3 Beam Paths .................. ....... .. 64
3.4 Procedures .............. . ...... 66
3.4.1 MMT Alignment Techniques ................ .. 66
3.4.2 Overlap Alignment .................. ..... 67
3.4.3 Data Collection .................. .. .. .. 68
3.5 Absorption Coefficients .................. ... .. 70
3.6 Analysis .................. ............. .. 71
4 CONCLUSION ...................... ........ 77
4.1 Results ................... ..... ....... 77
4.2 Future Experiments .................. ....... .. 77
A COMPLEX-Q AND THE ABCD LAW ......... .......... 79
A. 1 Derivation of q ............................ 79
A.2 ABCD Laws for Gaussian Laser Propagation ....... ..... 82
B JONES MATRICES .................. .......... .. 85
C UF THERMAL LENSING CODE. .................. .... 87
REFERENCES .................. ................ .. 105
BIOGRAPHICAL SKETCH .................. ......... 108
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
A TECHNIQUE FOR PASSIVELY COMPENSATING THERMALLY INDUCED
MODAL DISTORTIONS IN FARADAY ISOLATORS FOR GRAVITATIONAL
WAVE DETECTOR INPUT OPTICS
Rupal S. Amin
C('! Ii: David B. Tanner
Major Department: Physics
A technique to compensate thermal lenses in next generation gravity wave
detector input optics has been studied. For long optics, such as Faraday isolators,
the thermal dependence of the index of refraction can be exploited to recover
94 percent of the original TER1 ,, mode at 50 W of incident light. Theoretical
calculations describe the possibility of recovering 95 percent TE1,,,, light under
200 W of incident radiation when an optimized compensator is used. This level
of compensation is demanded since next generation interferometers have stringent
specification for input laser mode quality and input amplitude.
For nearly three centuries this equation was regarded as absolutely accurate.
Withstanding the tests of time, Isaac Newton's "Law of Gravity" proved itself in
numerous earthly experiments and observations. From Galileo's proposition that
bodies fall at the same rate regardless of mass to Kepler's conclusion that planetary
orbits were ellipses in pre-Newtonian times to Hooke and Halley's debate over
planetary orbits, the law of gravity remained unchanged . It was considered
infallible, to the point that it was given the title of "The Law of Gravity."
However strong this model was, problems with the predictions started to
surface when people looked more deeply into the sky. The most prominent problem
dealt with one of the least visible planets. Mercury's orbit was found not to be
closed. Newton and Kepler predicted and calculated its orbit to be an unchanging
ellipse with the Sun at one of the foci [1, 2]. Instead the perihelion was shifting
by better than five thousand arcs seconds each century. This posed a dilemma for
physics developed in the mid-17th century.
The early twentieth century brought about a change in the model of this
force. In an effort to corrroborate Michelson and Morley's null ether experiment,
Albert Einstein along with several other physicists constructed a model based on
geometrized deformation of space-time . Since this theory handled the world
in what would be called "flat space," it became known as "special relativity."'
The application of Einstein's theory to general space-time coordinates resulted in
the ;, ii, i theory of relativity."b Nonetheless, the 1916 publication effectively
described the interplay between energy and the surrounding space-time [2, 3]. This
is what we call gravity.
Nonetheless, the model did not stop at updating Newton's laws of gravity.
The other motivation of GR was to raise the description of gravity to the regime of
electromagnetics. Completion of the famed Michelson-Morley experiment showed
that the electromagnetic equations written down in the mid-1800s were invariant
in the regime of relativistic mechanics [2, 3]. Gravity demanded a model with
appropriate relativistic corrections.
Once the foundations of general relativity were laid, investigations into aspects
of the gravitational force could begin. Predictions made by GR necessitated a set
of realizable experiments. Therefore a series of classical tests both astronomical and
Earth-based were devised.
The first of these tests calculated Mercury's perihelion shift. GR's exploitation
of geometric deviations obtained the correct perihelion shift missed by various
parameterized Newtonian theories [1, 2, 3].
Eddington's famed trek into Principe Island, Africa came second . Ac-
cording to GR, light flows along paths called geodesics. These paths which are
ordinarily straight lines in flat space can be curved by massive .,-I i ..!ir, -i- 1 ob-
jects. During 1919's eclipse, Sir Arthur Eddington made observations of the Hyades
cluster which was positioned near the solar disk [2, 3]. Predictions and observations
a The abbreviation SR will be used for special relativity.
b The abbreviation GR will be used for general relativity.
disagreed by one degree . This bolstered Einstein's theory at the November
meeting of the Royal Society .
Following the advent of the nuclear age, the third test used little more than
radioisotopes and a convenient five story building. Gamma ray photons climbing
out of the Earth's gravitational well di-ph ,i, ,1 a red shift of their frequency. Since
the radioactive mass was not in motion, the frequency change was not caused by
a Doppler effect. Therefore, the only remaining explanation for the photons losing
energy was through gravitational interaction. GR's predictions fell along empirical
data as with the previous two experiments .
1.2 Gravity Waves
Another prediction made by general relativity is the existence of gravity
waves [2, 3]. This effect is analogous to what happens when one shakes an electric
charge. The surrounding field oscillates and carries energy away from the source
in the form of light. Similarly binary orbiting masses, ..i-mmetric supernova,
and even the birth of the universe release gravity waves. The prediction can be
reconstructed through the following means.
First we assume that an observer is standing in a region of flat space-time.
Flat space-time can be represented by a 4 x 4 tensor that appears as
-1 0 0 0
0 1 0 0
TIV = (1.2)
0 0 1 0
0 0 0 1
We add a weak field perturbation, h,, to this flat space-time,
Sh,,, < 1,
gv = rTpv + hpv. (1.3)
Here gy, is the metric tensor that expresses the new space-time. We next apply the
( 1 2 167G) 7rG
( c9PV r -Tp (t.4)
C 2 a 2 y 4
where G is the gravitational constant, T, is the stress-energy tensor, and c is the
speed of light in a vacuum [2, 3, 4]. Originally, the statement called for a nearly
flat space-time. This means that we are not near any of the masses represented on
the righthand side of the equation. Therefore, we may set the righthand side of the
above equation to zero,
c 2 2 2 ) ,v 0. (1.5)
Since the flat space metric, Tr, is a constant, a derivative operating on it will yield
zero. This leaves only the perturbation metric upon which the operator may act,
c2 2 +V2 hP 0. (1.6)
The solution to this harmonic differential equation is a harmonic function. Through
separation of variables, we find that the harmonic solution takes the form of
h, A, e-(-wt), (1.7)
where A., is the amplitude of the space-time fluctuation, r is VX2 + y2 + 2, and
S= xx + /,/ + zz [2, 4]. The reader will notice the shift to spherical coordinates.
A., can be acquired by solving the differential equation inside the source and
applying the boundary conditions.
Since the Einstein equations are gauge independent, we can change basis to
a transverse traceless (TT) gauge. This provides an additional restriction in the
' The d'Alembertian is a four space derivative.
number of freedoms used to obtain A, [2, 3]. The TT gauge condition implies that
a solution for the z-axis would be
0 0 0 0
0 A11 Ay 0
0 Ay -A1 0
0 0 0 0
This results in a final TT solution for h,. being
hTTpO 0, = 0, 1,2, 3, (1.9)
h TTik t i, k = 1, 2, 3, (1.10)
Or dt2 C
where QTTik is the traceless transverse quadrupole moment of the source. From
inspecting the A matrix we find that two polarizations exist for gravity waves.
These are the "x" and "+" polarizations. This is due to gravity's monopole
nature; there is no "anti-mass." Since the wave is quadrupole in the lowest
order approximation, an .,-vmmetric source is required to produce gravitational
waves [2, 3].
Should these waves be detectable then the sources of these waves could be
observed. A new field in astronomy based on gravitational waves would extend
knowledge of the universe beyond that of the electromagnetic (EM) and particle
spectra. Unlike EM waves and cosmic rays, gravitational waves can travel through
the universe unimpeded by matter.
A single gravitational wave based telescope would in theory be able to "see" to
physical processes that conventional telescopes cannot begin to observe. Processes
such as binary star and black hole-black hole mergers, events hidden by nebulae,
and events that occurred prior to the decoupling of the microwave background
during the birth of our universe would become visible and not left to conjecture.
0 n 371
Figure 1-1: The effect of a gravitational wave on a circle. The wave is moving axi-
ally to the circle. The circle shows no net displacement. Its perimeter
returns to its original shape after 7r radians.
A single telescope could theoretically see to the edge of our universe, but such
a device would be omni directional and would not be able to distinguish events
let alone direction. The solution to this problem is the same solution used for
particle based telescopes, an array of detectors. An array provides the possibility
of coincidence. This scheme first permits verification of the wave and second allows
for some directionality.
1.3 Detection of Gravity Waves
Now we must examine whether or not these waves are detectable by current
technology and physical knowledge. A gravitational wave produces no net move-
ment. Rather it exerts a strain on its medium, the local space-time. As it passes
through an object, the object is stretched in one direction and compressed in the
The ratio of the change in an object's length to its original length is strain 
and is written as
Assuming that the detector is tuned to binary star mergers in the Virgo cluster
(20 million parsecs away) , one may expect to observe strains of magnitude
h ~ 10-20. In practical terms this is a displacement of one hydrogen atom's
Figure 1-2: A Fabry-Perot interferometer. R1 is reflectivity of mirror 1; R2 is re-
flectivity of mirror 2; L is the length; O.A. is the optical axis; wo is the
beam radius. Light enters the device through the left mirror. When
on resonance, an interferometer with mirrors of the same reflectivity
will transmit all light through the right mirror. In order to sit on the
fundamental resonance, the two mirrors must be an integral number
of wavelengths apart and possess the same radius of curvature as the
impacting light wave.
length over a distance from the Sun to Saturn. However, well-designed optical
interferometers can detect displacements of one atomic width on laboratory scales.
This ability rises from light's wave characteristic. A set of coherent light waves
out of phase with each other will generate destructive interference reducing the
overall output amplitude. Coherent light waves that are in phase will constructively
interfere or increase in overall output amplitude.
An estimation shows whether or not interferometers would be a practical
solution for strains of this magnitude. We begin with set of estimates for a Fabry-
Table 1-1: Hypothetical Fabry-Perot cavity characteristics
Arm length Reflectivity Wave Number Strain
1 m 0.999999 2x107 m-1 10-20
d An interferometer composed of two partially reflective mirrors. These mirrors
reflect the light along the optical axis. This results in multiple superimposed beams
interfering with each other.
The effective length of the interferometer is leff ~ 106 m. This provides a
phase shift of
6 ~_ 10-7.
This is a detectable phase shift .
Being a quadrupole wave, gravity waves can be readily detected by a Michel-
son interferometer. This interferometer is capable of detecting the stretch and
compression in both orthogonal directions simultaneously for one polarization.
The Michelson functions like the Fabry-Perot through phase differences in wave
A simple Michelson interferometer. BS = beam splitter; Ml, M2 =
mirrors. Light enters this interferometer and is directed on a beam
splitter. The beam splitter creates two beams that reflect off their re-
spective arm mirrors and return to the beam splitter. If the two beams
return with their waves in phase the full input power leaves the system
through the anti-symmetric port. If the two beams are slightly out of
phase, anti-symmetric (output) port will be dimmer than the afore-
mentioned resonant state. When the two beams are 1800 out of phase,
no light leaves through the anti-symmetric port. Instead it exits the
instrument through the input port.
propagating between the end mirrors and the beam splitter. Waves that arrive at
the beam splitter after reflection off the end mirror destructively interfere with each
other. Waves that return in phase add in constructive interference. Therefore, a
Michelson with immovable end mirrors and beamsplitter will be able to observe
Output= O 0 O 0
0 C 3
Figure 1-4: A diagram indicating Michelson interferometer output during gravita-
tional wave passage.
the gravity wave as it passes. Making the three interferometer components immov-
able restricts the number of items that can change the arm lengths to that of the
Being a quadrupole wave, the strain amplitude reaches maximum at one-
quarter cycle, 7/2 and returns to zero displacement at half-cycles, r. For a large
scale detector the light must sample this part of the wave. Sampling beyond one-
quarter cycle will only average over the detected signal . This will reduce the
effectiveness of the detector. Assume that an in-spiraling binary star system were
to have a frequency of 300 Hz. We can find the length of the detector arms.
where fg is the gravitational wave frequency, c is the speed of light, and I is the
interaction length. The Michelson's arms should be no more than 250 km!
A practical detector with 250 km arm lengths would not be feasible on Earth.
However, the phase shift calculation referred to beam folding in a cavity with a
much smaller length. Through highly reflective mirrors a 1 m Fabry-Perot may
act like a 1 Mm interferometer. Therefore it would be logical to introduce the
Fabry-Perot resonance cavities into the arms of a Michelson interferometer.
Another method of beam folding is the delay line. A delay line does not
reflect light along the optical axis as does the Fabry-Perot. Rather a single beam
Figure 1-5: This is a simplified schematic of LIGO. PR = power recycling mir-
ror; FP = Fabry-Perot cavity; BS = beamsplitter. Modifications to
the Michelson interferometer include a power recycling mirror and a
Fabry-Perot cavity in each arm. The power recycling mirror impedance
matches the laser light to the interferometer.
bounces a distinct number of times between the two end mirrors before leaving the
device . The delay line is not an interferometer by itself; it is a means of using
mirrors to increase the total optical path length.
Using these strategies several detectors around the world are being or have
been built. Virgo (Italy-France)  and GEOe (Germany-Britain)  are two
European detectors with 3 km and 600 m arm lengths respectively. GEO is
operational. The US effort named LIGO is composed of three detectors [11,
12]. Two detectors have arm lengths of 4 km and sit in Livingston, Louisiana,
and Hanford, Washington. A third detector with 2 km arms also shares the
e GEO is the only delay line based interferometer.
A simplified schematic of Germany's GEO detector, a delay line
setup.PR = power recycling mirror; BS = beam splitter; Ml, M3 =
retroreflectors; M2, M4 = turning mirrors. Light enters GEO's beam-
splitter and is broken into two beams. These beams travel down their
respective arms and reflect off the turning and retroreflector mirrors.
The beams then re-converge on the beamsplitter where the beams in-
terfere. Notice that the total arm length exceeds the length from the
beam splitter to the turning mirrors.
Hanford site. Other prototypes are found in Japan (TAMA300)  and Australia
Detecting such small phase shifts demands extremely clean light sources,
lasers, and extremely accurate mirror positioning to reduce any background noise.
These tasks have been broken into several parts at LIGO, the pre-stabilized laser,
the input optics, core optics-the interferometer itself, suspensions, and data
analysis.g The pre-stabilized laser comes from a LightWave neodymium doped
ytterbium aluminum garnet (Nd:YAG) laser. This laser is cleaned in terms of axial
f ACIGA's Gin Gin site is under construction.
g The last two will not be discussed.
frequency and transverse mode to attempt to insure no source induced phase shifts.
The laser then enters the input optics.
Input optics' table performs three addition functions beyond beam pointing.
Phase modulated sidebands are injected onto the laser, now called the "carrier
field." This permits accurate mirror positioning using the Pound-Drever-Hall
scheme. Next the laser is cleaned further through the aid of a triangle ring cavity.
This cavity permits only the TE1 I,, spatial mode to pass. All other modes are
ejected out of the beamline. The core optics are decoupled from the pre-stabilized
laser by a Faraday isolator. Finally, the laser enters a mode matching telescope
that accurately focuses (mode matches) to the interferometer.
Once the laser enters the core optics, a correctly mode matched laser will
resonate within the 4 km Fabry-Perot arm cavities until the cavities are pushed
off their resonant states via a gravitational wave. Since the Michelson component
of the LIGO interferometer functions on the dark fringe, completely destructive
interference, a passing gravitational wave will generate constructive interference and
the output port will brighten.
This last point about the working point of the interferometer is why isolating
the core optics from the pre-stabilized laser is important. An interferometer whose
output port is dark reflects its optical power upstream. Reflected laser light can
easily destabilize an interferometer's sensitive laser source. Effectively the lasing
medium will see two laser fields competing for power and thereby generating beats
in the output laser light . Fa- idii isolators guard against this.
1.4 Far-id-v Isolators
Faraday isolators are based around magneto-optic crystals. The current
industry standard is terbium gallium garnet (Tb3Ga5O12 or TGG). When used
in combination with two polarizers and a half-wave plate, an isolator behaves like
Figure 1-7: A schematic of UF Input Optics table for LIGO II. MMT = mode
matching telescope; EOM = electro-optic modulator; PSL = pre-
stabilized laser; MC = mode cleaner; IFO = interferometer. Notice the
positioning of the Far,-"I w isolator immediately upstream of the mode
* Phase modulation Optical Isolation
* Mode cleaning Mode Matching
TFP1 FR HWP TFP2
Figure 1-8: A diagram of the Far-d-li isolator's internal components. FR = Fara-
d-i- rotator; TFP = polarizer; HWP = half-wave plate
an electrical diode. Light travelling downstream passes through the device and
leaves with the same polarization. Light propagating upstream, however has its
polarizations altered and is deflected out of the beamline.
Although Far,-id1i isolators protect upstream components from core optic's
reflections their position in the beamline becomes problematic for future gravita-
tional detectors. In order to reduce shot noise' and extend the observable range
of LIGO, Advanced LIGO designs call for laser power to be increased from 10 W
to 200 W. This will heat the magneto-optic i -lI I1 A thermal gradient centered
on the laser will generate a I I111, ii, lens" [15, 16, 17, 18, 19, 20]. This lens acts
like a gradient-index (GRIN) lens . The accuracy of mode matching into the
core optics will degrade, and with less light resonating within the Fabry-Perot arm
cavities the LIGO will lose fringe contrast [15, 16, 17, 18, 19, 20].
Therefore, methods must be found to repair this problem of thermal lensing
in FaridIi isolators. A few techniques are already under consideration. One
tactic involves heating the TGG element via a nichrome heater. Materials that
are thermally homogeneous possess no thermal gradient and therefore no thermal
h The terms ,l.-' ii. ,1i and "downstil i11 refer to the direction of designed
A noise term due to statical counting errors.
-3 -2 -1 0 1 2 3
Radial Distance [mm]
Figure 1-9: A diagram of intensity distortion after a laser passes through a ther-
mal lens. A Gaussian input beam is shown in blue. The orange beam
indicates the intensity profile that generated when the Gaussian beam
encounters a thermal lens.
lens. A second tactic would use adaptive optical components for the final mode
matching telescope. The mirrors of this telescope could be deformed by a second
laser whose frequency is absorbed by the mirror substrate. A computer program
would determine how to write a deformation into the mirror. Both methods involve
active components and active monitoring of the laser's wavefront before delivery to
the core optics.
This thesis describes a method for a passive solution that would not require
monitoring or adjustment through the full power range of the Advanced LIGO
laser. This strategy is similar to that used with Nd:YAG rods with undoped
endcaps . The solution presented in this thesis resolves the thermal lens
through the thermal dependence of the index of refraction exclusively.
2.1 Why Do Materials Absorb EM Energy?
When light strikes a medium, one of two outcomes are possible. The light can
be scattered or absorbed. The case in which light scatters back towards the source
is referred to as reflection. Similarly the case when light scatters into the medium
is called transmission.
On a hot summer d4 -, touch a windowpane. It feels warm if it has had
sunlight falling upon it. This seems a rather trivial fact, until one asks why this
pane of glass should be warmed by the sun. Afterall, the window is clear; in order
to heat up, the pane needs to absorb light. Yet, apparently it transmits almost
all sunlight perfectly and we can see the outside environment without loss of
This question can be answered by using the Maxwell's equations and a little
approximation to the behavior of charges in a solid. Let us begin by considering
a non-metallic media first as it applies to this thesis. Once again the Maxwell
equations in a material are
V x H = + (2.2)
.D= p, (2.3)
= 0, (2.4)
B (fH + M (2.5)
=co + P, (2.6)
where P is polarization and M is magnetization, B is the magnetic field flux
density, D is the electric displacement.
Polarization is defined as the electric dipole moment per unit volume. More
simply it is the intramaterial field induced when an external electric field is
imposed on a dielectric substance
P =- oE. (2.7)
X is the susceptibility and when multiplied by the permittivity of free space, Co,
gives the difference between a dielectric's permittivity and that of vacuum. It can
be written as
Magnetization in non-conductive media is not a great factor so the magnetic
field is pB IoH. C'!i rges are restricted in movement; we can set V- = 0 since
our non-conductors are electrically neutral. Therefore, we shall not carry it into the
following analysis. We begin with the dynamic equation for the electric field,
Vx E= -o (2.9)
We take the curl of the above equation,
Vx (Vx) oVx t
Sx ( x -/.-V x H. (2.10)
Substituting V x H into the above equation we obtain
Sending through the a and replacing copo with c-2, we get
1 a 22E 2P of
V(V ) E V2 a- Po P0 (2.12)
If we move the E-field terms to one side, the left-hand side becomes the
familiar wave equation. The right-hand side contains source terms
1 _2E 82P f
V(V )- V2E + a o -o a. (2.13)
C2 a22 2 a
Why are we interested in terms on the right? The polarization term accounts
for indices of refraction, birefringence, modulation, and a few other physical
nuances. The single time derivative on current density accounts for a metal's
ability to absorb energy from light fields that attempt to scatter through it. Since
we shall not be considering the case of metallic free charges or currents traversing
our non-conductor, we assign f 0 and V E = 0.
We assume in our model that the electric field modulates harmonically and is
not so strong as to cause non-linear effects in the media. The weak field case allows
us to model the response of the outer electrons by Hooke's law, F '. Writing
down a well known second order differential equation with linear damping we have
mt + md Y + m 7d = -eE. (2.14)
Here K is our Hooke's constant. 7 refers to the dampening coefficient for bound
oscillating charges. As usual we assume that E and F are harmonic,
Sroe-iwt, = Ee-iwt
This gives us
2- m im Te- iwt e- = eEe-it. (2.15)
mwL) roe 7m0wC +efoe t(2.1t5)
Solving for ro gives us
ro = (2.16)
When we factor out the m we can rename K/m as w2. This is the effective reso-
nance for the bound electron states ,
ro = 2 i. (2.17)
Now recall that polarization is P = Poe-t = -Ne. We substitute this into
the equation for polarization and get
P= m (2.18)
UQo a) IUJ7
This equation alv--, remains less than oo, but, at w= o, polarization becomes
purely imaginary. The consequences of this will become apparent momentarily.
Replace the polarization term in the wave equation and regroup the derivatives,
2E p oNe2 -1 9 2'
( Ne2 1 2 a^E
P/oco 1+ o- o-2 L- I O t2
1 Ne2 12E
S(1 + (2.19)
c2 oM a i a t2
This differential equation is solvable through the separation of variables,
E(r, t) -R(rT(t), (2.20)
V2f= -K2/, (2.21)
= KT. (2.22)
The solution of this is
S= oe-i(-K.r ) (2.23)
2 2(t NC2 t
K2 + (2.24)
C2 om co i_ wL
The plane wave solution is one of only two popular solutions that can be quickly
obtained. For the moment, we shall stick to the plane wave solution for this
discussion. Focusing on the separation constant, we rewrite it more clearly,
K = k' + ik", (2.25)
where k" = a/2. We see that this is little more than a complex wavenumber.
What does this mean in terms of the electromagnetic wave? The exponential now
possesses a decaying quality e-r" that is being multiplied times the free-space term,
e-i". Upon defining the intensity, I= 2, a gains a special name, the absorption
coefficient . This will be seen later in the model and experiment. Multiply by
cw/c, K becomes the complex index of refraction,
N =- K. (2.26)
As before we can separate this value into it's real and imaginary part,
N = n ilC. (2.27)
Just as before the existence of the imaginary index implies lossy media .
2.2 Birefringence: Innate and Thermal
We have now seen how to characterize and model the absorption of EM-waves
in lossy media. This simple evaluation gives us an insight into how materials heat
up when irradiated. The right hand portion of equation (2.13) also indicated that
electrically neutral materials can interact with electromagnetic fields. In a moment
we shall see how dielectrics can alter electromagnetic waves through birefringence
and multiple indices of refraction. These phenomena too can be affected by
thermalization of host media. In order to understand how heating can generate
-1' i, ,!!y induced birefringence," a short adventure into native birefringence is
This jaunt starts with a redeclaration of the susceptibility. In real materials,
it is a 3 x 3 tensor, not a simple scalar . Why has physics built a model like
this? Simply because on the microscopic scale, the
not spherically symmetric. Rather, it follows "fl- '. paths imposed by the lattice
that the nuclei support. Therefore, < iv--I I- with non-symmetric potentials will
di-pl-iv multiple susceptibilities. This also points to anisotropies in phase velocities
as will be seen. Incidentally, the only two lattice structures that yield isotropic
susceptibility (and polarization) are cubic and triagonal [22, 23]. Crystals like
sodium chloride, fluorite, and even diamond are isotropic. Should one not wish to
work with a
exhibits optical isotropy.
P xcoE (2.28)
This implies that a modification to the displacement field may be made via
C = Emn = Co (1 + Xmn) (2.29)
The new permittivity's modifier in parenthesis will become a bit more transparent
later. Now, we write the Maxwell wave equation (again),
7 X V t 82t 1 t2E
Vx Vx + C2 mn (2.30)
In Fourier space, this transforms into
( ) 02 2
Ex EX ( + 2 2 mn
(f E) kE + E --X2mn. (2.31)
If we assume that Xmn is diagonal, we can expand this easily. The three equations
for components along the x-, y-, and z-axes,
x : -k k + 0 E + kPkyE + kzEz = XIE, (2.32)
y : k( + 2 Ey + kxkyEx + kxkzE, X2Ey, (2.33)
z: -k( + 2 E, + kxkyE + kxkEy 2 x33Ez. (2.34)
So we have an array of equations. An arbitrary yet contrived selection in propaga-
tion is now made. We set the wave vector, k, along the x-axis,
k / 0, ky = k, = 0, E = 0. (2.35)
Note that Ex = 0 because E-fields are perpendicular to the wave vector. In turn,
the above array yields
x : 0 = 0, (2.36)
y -k + 2 E = 2 22E, (2.37)
z: -k + W2E -2 33.E (2.38)
These two equations leave us with two phase velocities and two indices of refraction
in the y and z direction,
k 02 L)2
ky 2 222 (2.39)
ky = L)l + X22, (2.40)
C 2 3 (2.41)
Sk = -a + X33. (2.42)
The square root is the index of refraction for the y- and z-axes . The x-axis'
index of refraction can be obtained by inputting the Xmn appropriate for the x-axis,
X11. Although these susceptibilities appear to be material parameters, we will see
that they have a dependence on temperature that effects birefringence.
2.3 Magneto-Optics and the Far-id1-v Effect
In the previous section, the author mentioned "optically ,. I ,., media.
Optically active media entails any media that can adversely effect light as it
passes through the material. "Ad.'-. i effects include the rotation of the plane
of polarization, phase modulation, and large scale frequency shifting (such as
frequency doubling). The generation of such phenomena sometimes requires a
static or dynamic electromagnetic field. In the case of frequency doubling, there is
no need for extra external fields. For the application of Far,-i1di rotators, magneto-
optic activity is of importance.
An isotropic substance will not affect propagating light waves with the
exception of increasing the optical path length.a As discussed before, this means
simple cubic lattice crystals and glasses. However, when the dielectric is immersed
a Birefringence can be induced in isotropic materials through localized stress
gradients. In this text we are concerned with thermally induced stress.
in a magnetic field, it becomes optically active. This effect was first encountered
by Michael Far,-id1i in 1845; therefore, magneto-optic rotators are referred to as
Fa,-rid-i rotators .
One's first impulse to explain the rotation of polarization, the Far,-id- i effect,
is to include the full Lorentz force in the damped differential equation seen in
m 2+ m7Yd + K -E e x Bo, (2.43)
where Bo is the external magnetic field flux density . Again, we assume that
the electric field and position vector are harmonic,
mw2 i -eE + iewx B0o. (2.45)
Next -Ne is multiplied through to convert position vectors into polarization
-Ne (-mw2F imn$+ Kr) Ne ( + iew X (2.46)
S(-m 2 _imTw7 + K2) P Ne2 + iewP x B0. (2.47)
This expands to...
PI E, (PyBoz PzBoy)
-mw2 ima + K) = N2 Ey + iew (PBo PBoz) .(2.48)
P Ez (P1Boy PyBo,)
C'!... -iig the magnetic field to be in the z direction simplifies the problem and
somewhat simulates what a magneto-optically active (< ,-l I1 sees in a Farid-iv
(-mmw2 im2 ^
Solving for P,, P,
P, E P,
+K) = #N2 Ey + iew -P,
P E j
, and Pz, we get a set of complex solutions,
Ne2(_ B,Eye + Em(iy + w))
(Be2 + 2(y iw)2)
SNe2(iB Eye + E~m(iy + w))
(B2e2 + m2(7y )2)'
In their full from (dampening included) one can see that the after collecting
electric field terms that the susceptibility matrix does not have a real diagonal.
This is completely due to the dampening introduced. However, this is a rather
difficult expression to utilize. If we disregard dampening, complex components
can be packed into the susceptibility's off diagonal terms. This leads into a rather
useful phenomenon as we shall see.
where X12 is imaginary.
Without dampening, the components are as follows:
Ne2 2 2
vXII L(2 2) (2.55)
meo (a 0 U)2
iNe2 [ jc ,
X12 7 -7 t- 2 (2.57)
meo (o' U2) 2c2 'c
Here w~ and o J- .
As with section (1), X,,, and it's multipliers to and becomes the righthand
side of the inhomogeneous wave equation.
kx EQx = (1 + Xn) (2.58)
A pair of wave numbers are produced this time. However, unlike section 1,
these wave numbers differ for the two circular polarizations-left and right.
k A 1 + xiL (12) (2.59)
The two indices of refraction that come from this are,
right = + Vii + ((112) (2.60)
nieft = + Vii 2(X12) (2.61)
We can find the specific rotary power for a polarized light wave by first starting
with circularly polarized light,
1 ik +1 i
U ei + eikLz. (2.62)
vl) \ -l
where U is a normalized field vector and kL and kR are the wave vectors for left
and right circularly polarized light respectively [22, 14]. Continuing we obtain
1 ikRz + ikLz
2 (eik eikLz)
1 (kR-kL)z e-(kR-kL)z
2 l(e (k-kL)-z -_ (kR-kL)z)
cos (k (k-)z
S e(kR+kLZ cos k kL) (2.63)
sin (kR- kL)z
Since k = 2= we see that this gives us specific rotary power,
6 = (night neft)A. (2.64)
Finally, multiplying through Ji 1 length, 1, reveals the angle by which the plane
of polarization is rotated,
3 = rightt nteft) A (2.65)
This can be written as a function of magnetic field,
S= VBJl. (2.66)
where V is the Verdet constant, B, is the magnetic field in the direction of propa-
gation, and I again is the length of the media .
One will notice that the rotation angle does not depend on propagation
direction with respect to the magnetic field. This is why optically active media can
be exploited for use in FarE,-idi isolators.
TFPFR HWP TFP
A schematic describing the operation of a Faraday isolator. TFP
thin film polarizer; FR =Far,-idl isolator; HWP = half-wave plate.
The direction of light propagation is indicated by the large arrow head.
Additional arrows indicate the angle of light polarization with respect
to the optical axis. Circles indicate light with polarization vectors
perpendicular to the page. Arrows at 45' indicate light whose polariza-
tion has been rotated by 45' with respect preceding polarization angle.
Vertical arrows describe polarization vectors that lie in the page. All
polarization vectors in this schematic are perpendicular to the optical
2.4 Faraday Isolator
Physically, a Faraday isolator is a simple apparatus that serves the same
purpose as an electrical diodeb for optical systems. The layout shown in Figure 2.1
consists of two polarizers, the Far,-idii rotator," and a half-wave plate.
The first polarizer, "TFP 1," selects the polarization of light to be passed.
Light polarized perpendicularly to the polarizer reflects off "TFP 1" and out of
the beamline.d The light that reaches the Far,-idi rotator has its polarization
altered by angle 3, as stated in the preceding discussion. Typically this angle is
b A device based on a pn-junction that ideally permits current to flow in one
c A magneto-optically active medium surrounded by a strong magnet encased in
an optical housing.
d "Beamline" refers to the primary or intended path of the laser light.
45. The half-wave plate then rotates the light back to its original polarization
thereby allowing the second polarizer, "TFP 2", to transmit the laser downstream.
Upstream propagating light initially observes the second polarizer. Supposing
the light passes into the Faraday isolator, it meets the half-wave plate which
imparts a 450 rotation on the electric field. Next the Faraday rotator completes
the polarization's angular displacement by another 45. The upstream propagating
light is now reflected out of the beamline by "TFP 1." The Jones matrices for the
separate components are as follows:
MFR cos -si(2.68)
Half wave plate
( cos2a sin2a
MHWP = ; (2.69)
where 3 equals r/4 and a equals 8 .
Multiplying the components in the downstream propagating solution we get
1 ( cos3 + sinj3 0 (
MFI = sin (2.70)
V2 K cos3 sin/3 0
For any fields propagating upstream, the solution becomes
1 ( cos3 sin/3 0 (
2 = (2.71)
cosp + sinp 0
Notice that these calculations do not include the final polarizer observed by
the beam. By setting f to 450, we see that the matrices are orthogonal. This
represents the isolator's diode-like behavior. Therefore, placing an isolator between
two optical components in series will decouple the upstream component from the
2.5 Thermal Lens
The preceding sections have introduced how seemingly transparent materials
can absorb light energy and heat up. The author assumed that the medium was
being irradiated evenly via plane waves; furthermore, the author assumed that none
of the material's optical parameters were thermally dependent (i.e., X / x(T)).
If we allow the medium's parameters to be thermally dependent, we find that the
optical path length through the medium changes in three manners . Explicitly,
they are written out as
AOPL = AOPLinnate + AOPLthermal + AOPLstress + AOPLexpansion. (2.72)
The initial term, AOPLinnat = noL, refers to the increase in optical path
length as a medium is placed into a beamline. The term no is the material's index
of refraction at a constant temperature [22, 15].
Second in line stands the thermal response of the index of refraction. When
we let Xmn be thermally dependent, this implies that the index of refraction varies
with respect to temperature,
AOPLthermal(' d= LAT(r, (2.73)
where dn/dT is the derivative of refractive index with respect to temperature and
AT(r) is the difference in temperature between r = 0 and a radial point, r,[15, 21].
The third component of the equation is the stress term. Its approximation is
AOPLstress( -= pl0 12LAT(r. (2.74)
Here P12 is the photo-elastic coefficient, and a is the thermal expansion coefficient.
This effect hails from the elasto-optic effect
S 6 3
2)~ Yp +I >1 (2.75)
The change in the inverse square index of refraction is directly proportional to the
photo-optic coefficient, pij, the strain tensor, Sij, radial direction,rij, and electric
field, Eij. A spatial deformation in a piezo ( i--1 I1 generates a voltage by changing
charge distributions within the i iv- i1 .
Equally, an oscillating electric field or acoustic wave will cause changes in the
index of refraction through charge coupling. However, since the frequency of light
tends to be greater than the charge response time, the photo-elastic term is not
usually included in the OPL calculation .
Fourth in this list is expansion. Possibly the best known of the AOPL terms,
this term tracks surface deformation due to local temperature differences and total
on-axis length increase.
AOPLepansion 2anwAT(rl (2.76)
where a is the coefficient of thermal expansion, n is the index of refraction, and
w is the Gaussian-beam waist size in the medium. We can neglect bulk expansion
due to the boundary conditions imposed on the medium sitting in the beamline.
If the optic is far larger than the beam's diameter, we can assume that the colder
region clamps the expansion of the heated region. Thus, expansion in directions
transverse to the beam will be held to a minimum. Expansion only occurs at the
end faces of the media [15, 16, 17, 18, 19, 25]. According to Strain et al  we can
approximate the expansion of the surface-also called the change in sagitta-by
assuming that the volume entertaining expansion extends a waist length into the
material. Therefore, the above equation is sound in its approximation.
One will note that each of the three terms contains a single power of the
temperature difference. This temperature difference can be calculated using as-
sumptions made in the AOPLexpansion term plus a set of implicit assumptions .
Stated again for clarity they are
the medium does not excessively attenuate the beam;
the dimensions of the optical medium far exceeds that of the laser's Gaussian
heat flow is exclusively radial;
the absorption throughout the media is uniform.
We first begin with the heat diffusion equation.
V2T+ q 1 T (2.77)
kth Dth Ot
where T is temperature, q is the heat source distribution (Wm-2), kth is the
thermal conductivity (Wm-iK-1), and Dth is the thermal difTfIL-iv.il i (m2s-1). Since
we are not interested in transient solutions, the right hand side equals zero .
V2T+ = 0 (2.78)
Since LIGO depends on the TE1,,,, mode, we shall use this as our heat source,
q(r)= e (2.79)
SHello and Vinet studied transients for large mirrors to be used in gravitational
wave detectors .
We can now continue.
Sd dd 1 2aP -2
--r-T ( e "26 w)
r dr dr kthm w0
dT 2aP 2r2
r 2. re o dr (2.80)
dr r 1'.,, ,',,I
Rewriting the exponent as a sum we get...
oo (2,2r )2
dT 2aP 2f 2)2
dT- 2a -B ^n 0 dr,
2 r ,(-1) dv
dr 2kthw n!
2P T, (-1 2( )2 2)
T(r) (,0 t + f (2) n + Aln(r) + B,
wkthw nO 2(n + 1)n! 2
D (_-)"+l( 2 )n+1
T(r) th ( 1)2,! (r2)n+1 + Aln(r) + B.
4kth (n + 1)2n!
If we change the summation by setting j = n + 1 this solution becomes...
ap o("_j2,r2 ")j
T(r) 4 h .J + Aln(r) + B,
4kth j j.
A = 0,
The first coefficient was set to zero to prevent the equation from diverging as
r --- oo, and B's value is then obtained by setting r = 0 . The final result for
the temperature difference profile is
aP a o (_1)j(2'2 )
AT(r) 4t (2.81)
Let's evaluate this expression. We observe that the function is zero at r = 0.
As we plot the profile we see for several waist lengths the temperature difference
falls as expected. However, the exponent on r causes the function to go to positive
or negative infinity depending on whether the j factor is even or odd, respectively.
Since the laser effectively does not exist outside 3 waist lengths, the laser never
Figure 2-2: The toy model visualized. fl = positive lens focal length; f2 negative
lens focal length; d = separation distance. A positive thin lens can be
compensated by a negative thin lens. Should the focal lengths, fl, f2
of the two lenses greatly exceed the separation distance, d, the effect of
the first lens can be negated by the second lens if the magnitude of the
two focal lengths are equal.
observes this part of the model . Therefore, within physical beam dimensions
this model of temperature difference is valid.
Homogeneous heating does not result in a lens being generated; it only yields
an increased OPL. This is why a temperature gradient is necessary. Its existence
causes phase differences in the EM wave.
This rather innocuous looking solution permits us to proceed with the develop-
ment of an analysis scheme to simulate the thermal lens and its compensation.
2.6 Toy Model and Practice
We have now introduced the concept of thermal lensing as it applies to flat
face media. A lens of any sort simply alters the phase of the imposed field. This
results in the field iid. -i-ik i :-; at the focus of the "lens." A physical lens imposes
the same type of phase distortions. However the phase difference enters through
the differences in physical path length of the media. Taking this into account, we
can develop a toy model of the requirements for the eventual solution. The focal
length of a two lens telescope is
1 1 1 d
t f- + f2 f (2.82)
ftot fl f2 1f21
where d is the distance between the two lenses. Assuming that the two lenses' focal
lengths exceed d, d < fl, f2, we can simplify this the lensmaker's equation to the
1 1 1
S- + 1 (2.83)
ftot fl f2
Effectively this happens in the real world because the focal length change in
a low-absorbing media is small. Nevertheless, let fl be a positive lens of strength
"f." It is simple to see that a negative lens of equal strength will be required to set
ftot = oo. This is equivalent to stating that no lenses are in the beamline.
However, compensation of a thermal lens is not as simple as indicated by
the above equations. The lensmaker's equations are built around spherical non-
deformable lenses. Thermal lenses are not spherical nor are they unchanging. We
can take advantage of AOPL through the AOPLthermal . Strictly writing the
dn/dT lensing terms for two materials yields
AOPL -= LATi(r) + L2AT2(r). (2.84)
We see that when AOPL is set to zero, the two OPL terms have opposite signs,
LATi(r) L2AT2(r). (2.85)
We can factor out the power in the thermal difference terms since the target
medium and its compensator material will observe nearly the same laser intensity,
dnl al dn2 a2(26)
dT kthl dT kth2
Since lengths, thermal conductivity,and absorption coefficient are alv--i positive,
the dn/dT of the compensator material must be opposite that of the target
Although a compensation material may have an appropriate dn/dT, a parame-
ter must be available to properly compensate the thermal lens in the first medium.
Length of the medium 1, dni/dT, dn2/dT, and thermal profile are fixed for both
materials. Only length of the second (compensation) medium remains as a variable
in repairing thermal lenses.
One should also note that this linearized model implies that an optimum
length of compensation material exists for each thermally lensing material. Further
contemplation indicates that the compensation strategy is independent of incident
power. The latter conclusion should not be accepted as truth. This model is
based on a linear expansion of thermally induced effects in materials and does not
consider the possibility of nonlinear effects such as melting, frequency conversion,
or other effects that occur beyond the envelope of assumptions. This does not make
the model incorrect as a first approximation to correction of thermal lenses .
2.7 Wavefront Model
Solutions that track the paraxial wave profile require numerical solutions.
Since we are only interested in axial motions of the beam and not pointing
deflections, we can restrict the propagation to the Laguerre-Gauss mode repre-
sentationf [15, 17, 20]. We begin the algorithm by declaring the TE1,,,, mode's
UOO= 2_e (2.87)
To match the toy model's thin lens type assumptions, we place this field in the
center of the first Ji i 1 TGG. The thermal lens' phase shift is then applied,
Uaber = U006 A (2.88)
f See section 1, chapter 3
A is the laser wavelength in a vacuum. The aberrated beam is written in the
Uaber = CSnOL ( o o0 (2.89)
Using Laguerre-Gauss overlap integrals, we can obtain the coefficients,
Cn0,uncomp = 27 r U*berUnordr. (2.90)
We next propagate the wavefront to the position of the compensator material,
FK51 Schott glass. This is performed by applying the Gouy phase shift. The Gouy
phase denotes how the wavefront's shape changes as it propagates through free
space. It acquires its phase from the ratio of propagated distance versus half the
collimation length. Implemented the new field becomes
uaber = CnOLno -2) (2.91)
Explicitly cGouy arctan( ). ZR is half the collimation distance, the Rayleigh
At the center of the glass, the counter-thermal lens is applied,
Ucomp = Uabere A (2.93)
Notice the AOPL is negative in this case and will subtract from the original
thermally induced phase shift. Again, we expand this field in the local Laguerre-
The amplitudes once again are found through overlap integrals,
Cno,comp = 27 j oumpUntordr. (2.95)
We now have the amplitudes of the modes that compose the two uncom-
pensated and compensated laser fields. Since we are interested in the power
retention of the TE1 ,,,, mode, we need only find the magnitude squared of the first
Laguerre-Gauss coefficient The resulting intensities are normalized to one at zero
0,uncomp C00,uncomp 2, (2.96)
1o0,comp IC0O,comp 2. (2.97)
Notice that the representation in each basis calls for an infinite sum. We must
be careful here during the construction of the model and use enough of the basis to
regenerate the smooth transitions that a real wavefront would observe. This lack
of precision encouraged the use of twenty Laguerre-Gauss modes to represent the
We need a third solution to compare the strategy of thermal compensation
against focus compensation. Focus compensation uses the tactic of finding an
optimum spherical lens to refocus the fundamental mode. A solution can be found
by searching for a fundamental Gaussian mode that generates the largest overlap
lopt =27r tu$gryoptrcd (2.98)
The constants used in this analysis are in Table 2.1
Table 2-1: A table of material constants.
Material a(cm-1) kth(WmK-1) dn/dT (10-6K-1) L(mm)
TGG 6.3 x 10-3 7.4 20 35
FK51 9.2 x 10-3 0.9 -6 67
Deletion of two Gaussian modes
0 1 2 3 4
Figure 2-3: This is a demonstration of simulation precision. The green line shows
plots a Gaussian curve based on the analytical solution. The blue
curve approximates the Gaussian curve using nineteen Hermite-Gauss
modes. The black curve approximates the Gaussian with only eigh-
teen Hermite-Gauss modes. Notice the over estimation and artifactual
ripples in the approximations especially in the wings.
One should note that the absorption coefficient for TGG was measured in-
house. The FK51's a term was fitted following data acquisition. Schott's internal
transmittance report for FK51 gave a standardized term of 8 x 10-3cm-1. Thermal
conductivity was obtained for TGG from Mansell et al.  and for FK51 from
Schott's data sheetg . The absorption coefficient will become especially
important in chapter 3's analysis section.
The results of these intensity terms are plotted below with respect to incident
power. The rapidly falling curve is the uncompensated TE1 ,,,, mode. Without
g Schott Glass data sheet 487845.
A plot of simulated normalized intensity remaining in the TE 1,,, mode
versus incident laser power. The uncompensated, black, curve drops
rapidly due to the thermal lens shifting laser light from the fundamen-
tal spatial mode into higher order TEM modes. The focal compen-
sation curve is generated by determining the best fixed lens basis on
the radius of curvature of the wavefront in the spherical approximation
for each power. The red curve represents ideal thermal compensation.
At 180 W, the retained intensity in the TE1,,,, mode drops to 95 per-
cent. The rolls off is due to the separation between the lensing and
compensating materials in the beamline.
compensation, the amount of power in the TE1,,,, mode drops to 5 percent at 180
W. Focus compensation appears to retain 60 percent of the original modal power.
This still does not meet Advanced LIGO requirements of greater than 95 percent
power retention. An ideal compensator, however, would keep 97.5 percent light
power in the fundamental mode at 180 W. At 200 W, the ideal compensator drops
to 94.5 percent. Unlike the previous section's toy model, the separation distance
between the two materials cannot be ignored. The Gouy phase shift removes the
possibility of perfect compensation.
50 100 150
EXPERIMENT AND ANALYSIS
The following chapter discusses the experiment in detail. This chapter con-
cerns itself with the methods and techniques used in observing thermal lenses. We
begin with a section regarding the detectors and design of the tabletop. Following
this will be a briefing of the equipment and any peculiarities found. Then, comes
a subsequent description of the experimental setup. Next, measurement methods
of optical absorption coefficients will ensue. A description of procedures, data
collection, and analysis will conclude this chapter.
In chapter two, we learned that thermal lenses can alter the spatial mode
structure of a laser. In order to observe how much light is moved from the funda-
mental TE.1I,, mode to higher order modes, one needs an instrument to separate
the different spatial modes. This device is a simple non-degenerate scanning Fabry-
Perot cavity. All other optical components found on the table are based around
redirecting, mode matching, or thermalizing materials that interact with light
entering the cavity.
Also, the use of a Fabry-Perot cavity flows logically from the design of the
large gravitational wave detectors. Since all optical gravity wave detectors utilize
Fabry-Perot interferometers whose working point is the TEA1,,,, mode, a scheme
that incorporates a cavity would offer insight into how badly thermal lenses
depreciate signal contrast and how well compensation recovers this signal contrast.
3.2 How to Analyze a Thermal Lens
3.2.1 Scanning Fabry-Perot Cavity
The scanning non-degenerate Fabry-Perot cavity forms the heart of this
experiment's detection scheme. So we shall spend a little time in the theory of
its characteristics and operation. The Fabry-Perot cavity is a physically simple
device. It consists of two facing mirrors. These mirrors are aligned in the beamline
in order to temporarily trap light passing through them. This results in light waves
bouncing off the mirrors multiple times. Numerous contained counter-propagating
waves generate interference. Therefore, the Fabry-Perot cavity can be referred to as
an interferometer or an optical resonator a [14, 22, 27].
To understand the nature of the cavity we investigate how fields act within the
cavity. The analysis starts by writing down the Collins integral ,
-i e-it E(xi, yl)eB (AxB+Dx -2xlX2+AyI+Dy-2ylY2)dxldyl. (3.1)
This equation is generalized Kirchhoff integral written in terms of the ABCD
matrixb used in ray and wave optics. Here A is the wavelength of the light field,
and u is the angular frequency of the light field. The subscripts 1 and 2 denote two
different analysis planes perpendicular to the z or optical axis. One should note
that this equation assumes the light beam is cylindrically symmetric.
A solution that retains the field distribution yet allows the beam's amplitude
and beam diameter change as the beam propagates through an optical system is a
a The term resonator will be used interchangeably with Fabry-Perot cavity.
b see Appendix A
beam with a Gaussian field distribution,
El(xl, y) = Eoe (3.2)
where w is the laser beam's diameter. This type of beam can also be written in the
Ei(x., y) =Eoe 2qjjl (3.3)
q is a wavefront's complex radius of curvature," and k is the wavenumber.
Within a resonator, a wavefront regenerates itself upon each round trip
amplitude and lateral extent included. This stipulation requires an eigensolution of
the Collins integral be found,
ic Iik L I\ (G(X2+y2+X2+ )-2(xiX2-yly2)
7Ei(x2, Y2) L j Ei(x, y)e2L92Ao((YI+ i -2(1 12)dxidyl. (3.4)
The round trip ray transfer matrix,
M 2L (3.5)
has been applied to the integral thereby simulating the periodicity of the optics.
The term gj equals
1 L (3.6)
where L is the optical path length of the cavity, and pj is the radius of curvature of
the jth mirror. G is 2g1g2 1. 7, the eigenvalue in the Collins equation takes note
of any diffractive losses (scattering, absorption by cavity medium).
c see Appendix A
Implementing the Gaussian field a family of solutions to the cavity field may
be found. Expressed in cylindrical and rectangular coordinates, the fields on mirror
j are as follows :
Circular symmetry, Laguerre-Gauss modes (TE ,:)
r 2r2) 2_ COS(1)
Ell (r, E) -- Eo L 2e
P. \j/ sin(l4)
7 exp 2L (2L p + + 1) arccos g/ g12)
Rectangular symmetry, Hermite-Gauss modes (TEMmn)
E(x,y) = Eoe Hm Wj Hm )
7 exp ik (2L- (m + n + 1)arccos /1g-g2
Conditions for resonance demands 7 is real. However, setting 7
of finding the resonance frequencies is simplified,
i k(2L- (2p+1+1) arccos /g2)
1.0 eik(2L- (2p+1+1) arccos 12)
Resonance frequencies may now be found by utilizing 7's exponent.
2L (2p + + 1) arccos g1g2
(2p + 1 + 1) arccos /gig2
2L p (2P +1 + 1) arccos gvg2
2L (2p + 1 + 1) arccos Vgig
2L (2p + 1 + 1) arccos 1g-2.
1 the effort
These are the frequencies for the transverse modes [14, 22, 27]. Axial modes can be
added by superposition to obtain the final result,
c 2p+l+ 1g -
uV = 2L + arccos .g (3.11)
An identical treatment applies to Hermite-Gauss modes,
c m+n+l 1
Vnq 2- ( +L n + arccos (3.12)
V 2L \ 2
Let us look carefully at these results. If we set each g = 0 through p =
L, we find that the resonant frequencies are all spaced L apart. Also, laser
frequencies that exhibit higher order spatial modes resonate simultaneously with
fundamental resonance modes. Optical resonators with these characteristics are
called "confocal" cavities . These are not applicable to this experiment.
If we construct a cavity with 9192 between 0 and 1, we see that the no two
spatial modes resonate at the same frequency unless their indices yield coincidence.
For example the Hermite-Gauss TEM11d and the Laguerre-Gauss TEMloe modes
share the same frequency. Such cavities are nondegenerate and can be used to sift
through various optical modes. This is the type of cavity necessary for successfully
analyzing modal content of laser radiation.
In order to observe the separate spatial modes of the laser radiation entering
the cavity, we must change the length of the cavity. Although there lies the
possibility of altering the radius of curvature of the end mirrors or changing the
axial frequency of the laser, these are both less practical methods of moving the
Fabry-Perot through the free spectral range. A brief discussion of the free spectral
range will follow. C'!i ,ihiig the length of the cavity will change the cavity's
d The TEMf11 mode is often called the "clo., i,. !I mode.
e Referred to as the "bull's eye" mode.
Cui 1 -
A graphic of this experiment's Fabry-Perot transmission curves over
the full free spectral range. The data di-p-,1- DC voltage versus time.
Implicit on the horizontal axis is mirror position. The two largest peaks
are the fundamental resonances, TE31,,, The second highest peak is
the Laguerre-Gauss TE. 1,, "bull's eye" mode followed by the Hermite-
Gauss TE.113,, tilt mode. One can see the number of modes passing
through is limited to several higher order modes. One should also
notice the ability of this nondegenerate cavity to separate the probe
laser's spatial modes.
resonance frequency. In this experiment, a piezoelectric crystal altered the length of
the cavity by approximately 1 micrometer.
A few other characteristics of Fabry-Perot cavities are worth mentioning since
they affect experimental results. These are free spectral range, bandwidth, finesse,
and maximum circulating power [14, 27]. Most of these attributes, based on mirror
reflectivity, determine cavity quality.
NE... .e dAL-: 7//02
CS]. .01%7/=V no'
TB A: 200 .a T.: CH4M.V PT: 25
FPI Near Mode Matched
A transmission curve showing the cavity nearly aligned. This figure
spans the free spectral range of the cavity. Notice the suppression of
higher order modes. The tallest three peaks are the TE1L,,, mode in-
tensity. The next highest peaks are the Hermite-Gauss TEMlo tilt
mode in transmission. Height discrepancies are due to the cavity not
full achieving resonance.
Free spectral range (abbreviated FSR) is defined by
This number informs one how far apart two resonances of the same spatial fre-
quency sit from one another in frequency space [14, 22]. For example, if one had a
1 meter long cavity on TE31 ,,, resonance, one would have to slew through nearly
150 MHz of frequency space before the TE31,,, was observed again.
Transmission Curve for Fabry-Perot Cavity
.1 -i I I II
Figure 3-3: This transmission curve is calculated for the Fabry-Perot cavity on
fundamental resonance. Normalized intensity is plotted versus phase
number. k = 27/A; L = mirror separation length. This indicates in
order to sweep through the free spectral range, one must either alter
the input laser wavelength or change the mirror separation length.
Peak fall off is due to computer precision. All peak maxima achieve 100
percent transmission on resonance.
The bandwidth or full width at half maximum of the transmission peaks is
given by the equations,
6= n(7 1 R2I) c- (3.14)
Here Rj is the amplitude reflectivity of the jth mirror modulus squared. One
empirical method of determining the bandwidth is to scan the frequency of the
light entering a fixed length Fabry-Perot. A plot of transmission intensity versus
frequency can be created. A cavity sitting on resonance will transmit any light
wave whose frequency falls within the bandwidth.
Most resonant electronic circuits have a measure of quality [14, 27]. Res-
onators are no different. The quality factor's expression is not altered by the idea
that light waves possess frequencies of hundreds of terahertz (1014 Hz),
The 7 in the first equation is the decay time for circulating energy in a cavity to
fall by a factor of e-1 ,
c Iln(7y12V R 2)
v in the second equation is the frequency at which Q is being analyzed. A second,
more often quoted equation is finesse [14, 27],
AT iir' books, such as Siegman  and Davis , also document finesse as
F = vI 2 (3.18)
Both terms yield information regarding how long light will circulate within a
cavity. Greater circulation times generate greater contrast between resonant and
nonresonant cavity states. Also it is interesting to notice
At their resonance frequencies, resonant electrical circuits store large quantities of
energy and power. This too occurs with the Fabry-Perot cavity. Energy density
inside the cavity can be acquired via
lo 1 R2
PE, maz R )2' (3.19)
C (1 1122/NIR2)
where I is the incident intensity. Circulating power can be acquired from p by
multiplying with lightspeed and cross-sectional area of the beam,
Pmax = Po | R )' (3.20)
(1 |H|2 Ri)2
where Po is the incident laser power. The amount of circulating optical power
becomes important when trying to hold a Fabry-Perot cavity on resonance.
Although the input laser power can be negligible, circulating powers can result in
physical expansion of resonator housing material.
The above equations for this experiment yield the following results:
The device itself is reflection coated for 1064 nm laser light. The operating
wavelength for this cavity is 1064 nm. The "g" product lies comfortably between 0
and 1. Reflectivities of approximately 0.99 percent deliver high finesse that resulted
in large contrast. Calculating the difference between the "bull's cye mode and the
TE3 ,,, mode, we obtain 744 MHz. The free spectral range exceeds this value by
193 MHz. Therefore the two modes of interest should be clearly distinguishable
from one another. This was confirmed during the data runs; the two modes well
The single drawback of high finesse lay in the cavity's ability to remain on
resonance. Due to the large number of bounces observed within the cavity, power
in the cavity increased by a factor of 199. This level of power provided a heat
source that pushed the cavity off resonance. This problem was remedied by placing
a second polarizing beamsplitter and halfwave plate immediately behind the
interaction region to reduce the amount of cavity input light.
Table 3-1: Analyzer cavity characteristics
speed of light
Radii of curvature
Free Spectral Range
Input laser power
299792458 m s-1
3.2.2 Beam Profiling
A second and less precise method of observing thermal lenses is beam profiling.
This method involves acquiring cross-sections of the laser at multiple distances. A
X2 fit is then performed to determine the laser waist size and position as incident
power on target is varied. The X2 fit is based on the waist propagation equation,
w(z) found from the complex-q, f
q(z) = z+qo,
q(z) z + i
1 1 iA
q(z) 2 + 2
1 1 iA
q(z) R(z) Tw(z)2'
Sw() = wo l + (3.21)
f The complex-q is a fundamental part of the eigensolution to the Fresnel-
Kirchoff equation; see Appendix A.
R(z) is the radius of curvature of the wave front. The term ZR is the Rayleigh
range. This is the half-distance that a wave front with infinite radius of curvature
and a finite waist will remain collimatedd" . At the Rayleigh distance, the
beam waist expands to a factor /2 larger than the waist [14, 27].
Examining this equation we find that the waist equation is hyperbolic.
Beyond the Rayleigh range the beam more closely follows the .-i-~I',il. This
implies that for accurate waist measurements one needs to obtain data outside the
Rayleigh range. Obtaining data within the Rayleigh range often results in poor X2
fits. This often occurs when there exists a large beam waist. Large beam waists
cause long collimation distances and therefore slow radii changes. When this is
combined with errors of approximately ten percent, X2 analyses can result in poor
fits or multiple fit solutions.
The beam profiler instrument functions by timing how long light falls on a
pyroelectric detector as a slit rotates past. A computer program then interprets
this time as beam width. Using a rapid response detector constructed for use in
the near infrared, an intensity profile can be plotted. The detector used in this
experiment was obtained from Photon-Inc. and will here on be referred to by its
model name, "BeamScan."
3.3 Layout of Table
3.3.1 Table Overview
The overview of the table will give a structured look at the tabletop experi-
ment's components. We shall start with the two lasers followed by a note on all
components. The probe beamline will then be discussed through the cavity and
tandem photodetectors. The high power laser's beamline will be explained before
continuing into procedures.
For any probe laser to stably resonate in an Fabry-Perot ,i .i-. r cavity, the
laser itself must be stable. The probe laser must not be powerful enough to induce
a thermal lens in any of the media with which it interacts. The laser incorporated
into this experiment was a LightWave Non-Planar Ring Oscillator (NPRO) model
126. This laser delivers 450mW of 1064 nm radiation from a neodymium doped
yttrium aluminum garnet (Nd:YAG) monolithic iv-l I1 that is pumped by laser
diodes. The NPRO was chosen for its stability and reliability. A turnkey device,
this laser delivers 94.6 percent of its power in the TE~ ,,,, mode.g M2 for this laser
was approximately 1.1. All other light fields lie in higher-order modes.
Heat for our optical crystal and glass comes from a neodymium doped yttrium
lithium fluoride (Nd:YLF) crystal laserh built by Photon Industries, Inc. This
laser delivered 52 Watts at 93.4 percent output capacity. Similar to the probe laser,
this laser also had an M2 of 1.1 verifying a nearly pure TE1I,,,, mode. Unlike the
probe laser, Photon Industries utilized a linear resonator whose transverse modes
were limited by a mechanical aperture. Although M2 measurements confirmed
the linear resonator's transverse modes were restricted, the design still permitted
the Nd:YLF to "mode hop." Resonators that "mode hop" have several axial
frequencies that compete for energy simultaneously. Therefore these lasers exhibit
frequency modulation. Modulation of the high power laser light was an advantage.
Frequency instability prohibited much light from entering and passing through
the analyzer cavity thereby protecting downstream sensors. Light that eventually
seeped through increased the noise floor on the photodetectors.
g M2 measurements verified the NPRO's operational quality.
h This laser will be referred to as the pump or YLF after this point.
Figure 3-4: Graphed above are the X2 fits of the Nd:YAG NPRO laser's vertical
and horizontal axes. The vertical axis radius exceeds the horizontal
axis radius indicating a slight ellipticity. This is inherent in solid state
lasers. Error bars are approximated by the datum size.
r"./ -*K. 1 .1 "'1
,i Ri *'i H r 1 m *iL
- w. u r i
i .. h M
WRI 1131 S I M? 1<
Figure 3-5: This screenshot of BeamScan's proprietary computer program illus-
trates the Gaussian nature of the pump laser used.
Figure 3-6: This figure is the X2 fit of the pump laser. Similar to the probe laser,
this laser possesses a slight .,-Jrmmetry. Data variation is due to the
BeamScan experiencing difficulty when observing full laser power.
Error bars are approximated by the datum size.
The experiment was assembled on a 4 ft. x 8 ft. x 1 ft. Newport Research Series
Plus table. These tables are filled with silicon oil to aid in intra-table vibration
dampening. The floor mounts were XL-A series. Isolation from vibration was
not deemed critically important. The YLF laser's water cooler/power supply
transmitted some vibration through the water lines directly into the table. The
top of the table had a breadboard configuration with bolt points spaced 1 in. on
center. The boltholes accepted 0.25" x 20 bolts. The experiment often exploited the
rectangular arrangement of the boltholes. This permitted a more precise method of
obtaining beam profilimetry.
The centerpieces of this experiment were obtained from two manufacturers. A
w rotator was obtained from Litton AirTronics. The compensation material
Figure 3c6u This figure is the blt he pump laser. Similar to the probe laser,
this laser possesses a slight .trimmetry. Data variation is due to the
BeamScan experiencing difficulty when observing full laser power.
not deemed critically important. The YLF laser's water cooler/power supply
Fa-ndiv?-li rotator was obtained from Litton AirTronics. The compensation material,
FK51 glass wedges, were supplied by Schott Glass company. The rotator was based
on terbium gallium garnet crystal technology and housed in a neodymium based
magnet housing. This magnet generates approximately 0.6 Tesla in its bore. The
AirTronics rotator also had a large aperture, 1 cm in diameter. The TGG (< i-I Il's
dimensions were 1 cmx3 cm. In order for the rotator to function properly, the
< i -1 I1 was required to be centered within the magnet.
Two FK51 Schott glass wedges were ordered with the dimensions 3.65 cm by
9.67 cm by 2.2 cm. Both wedges were anti-reflection coated for 1053 nm and 1064
nm. They were placed closely together to prevent multiple inter-wedge scattering.
Low Power F,-gvidv Isolator
A Faraday isolator was used to retain the stability of the probe laser and
to further protect the laser from any high power radiation that might propagate
upstream. The Far- idiv isolator chosen for the probe laser was obtained from
Electro-Optics Technology (EO-Tech). EO-Tech's isolator was based on terbium
gallium garnet technology similar to the Far 'idiv rotator used for the thermal
All four waveplates were acquired from CVI Laser Corporation.' All three
half-wave plates were based on air gap separated quartz plates. As quartz is
already birefringent at room temperature, these < i v-I i1 do not impose added
polarizations. The short physical length of the wave plates also minimized the
possibility of thermal lensing, especially important for pump laser power control.
The fourth plate was a quarter-wave plate and was deploi-, 1 only during overlap
* CVI serves as CVI Laser Corporation's unofficial abbreviation.
Three polarizers were bought from CVI. Two rectangular thin film polarizers
(TFPs) were used to create the test Faraday isolator and limit the region of
pump laser interaction. These polarizers functioned on Brewster's angle (53).
They were designed for 30 dB suppression of "s" polarized light at high incident
laser powers. BK7 glass formed the substrate and did not appear to develop a
substantial thermal lens when tested using beam profilimetry. The third polarizer
was a polarizing beam splitter cube (PBSC). Its substrate was also BK7, and it
also yielded a 30 dB extinction ratio. This particular type of beam splitter was
not incorporated into the high power region since it di-pl~ i poor thermal lens
performance. The poor performance was caused by an overlap between incident
and reflected radiation within the incident prism. This performance was observed
in all PBSCs and was dimension independent. Therefore, the PBSC was used to
enhance the extinction of any pump radiation that escaped towards the analyzer
The fourth polarizer was bought from New Focus. A Glan-Thompson 5525
calcite based polarizer provided 50 dB extinction. This polarizer was used with a
CVI half-wave plate and CVI PBSC during absorption coefficient measurements for
TGG and FK51 (see section 3.5). This polarizer was not intended to withstand a
high power laser; it was never used in the thermal lensing setup.
One-inch steering mirrors for this experiment came from two sources, New
Focus and CVI. No mirror in this experiment except for the analyzer cavity was
set for zero incidence. Nonetheless, New Focus mirrors were designed for zero
incidence and included a silvered back decal to prevent light leakage. These mirrors
comprised the pump laser's aiming periscopes. CVI mirrors designed for 100
percent reflection for s-polarized light J at 45 degrees incidence were used in the
probe laser periscopes. Although these mirrors were designed to operate at two
different incidence angles, we found the mirrors to be interchangeable.
Mirrors placed in the analyzer cavity were manufactured by REO. Indicated in
the preceding section, both mirrors had reflectivities of approximately 0.99 for 1064
nm radiation. The input coupler and output coupler mirrors had radii of curvatures
of 0.3 m and 0.5 m respectively. Unlike the steering mirrors, the cavity mirrors'
diameters were 1 cm. The thickness of the mirrors was also 1 cm. REO formed the
substrates from BK7 glass.
Newport fabricated lenses for mode matching telescopes 1, 2, and 3. A
substrate of fused silica was coated with an anti-reflective coating for the one
micron infrared regime. Fused silica possesses half the dn/dT of TGG. The short
path length of the silica and preshaped curvature suppressed thermal lensing or
Cavity and Associated Electronics
The cavity itself was machined out of stainless steel. Steels such as Invar and
i v-I i1- such as fused quartz display less susceptibility to thermal expansion. How-
ever, an analyzer cavity's performance does not rely on remaining on resonance.
Rather, the device sweeps through the resonance state to obtain peak values.
Therefore, this material was deemed suitable. Also, a screw end design allowed the
input coupler mirror to be manually moved through 7 cm of travel. This permitted
the cavity's eigenmodes to be altered should the experiment necessitate it. The
J Mirrors had high reflective coatings for 1064 nm light. The bandwidth included
mirrors rested on end caps that were attached to the tube via three screws. Viton-
O ring spacers between the end cap and the cavity tube permitted the mirrors to
be aligned independently. Situated between the output coupler mirror and the end
cap was a piezoelectric crystal tube (PZT). This i-lI I1 when stressed by 200
V moved the output coupler through approximately one micron, two free spectral
The piezocrystal driver was assembled in-house. The design for both the
power supply and PZT driver come from Gerhard Heinzel at the Laser Zentrum
in Hannover (LZH), Germany. The driver primary components consisted of two
high voltage operational amplifiers, Apex's model OP-87. These op-amps received
amplitude fluctuation signals from two smaller OP-27 op-amps. The latter devices
r(l iv ,1 signal data from an external function generator that was set to a triangle
waveform. A DC offset allowed for manual tuning of the output coupler position
in the cavity. However, excessive heat in the OP-87s due to everyd- i operation
and poor heat sink quality, resulted in the driver's voltage slowly wandering. The
direction of voltage change was monotonic, predictable, and therefore did not
influence the experiment.
A Thurlby Thandar Instruments, TTI, model TG120 function generator sent
a 0.7 Hz triangle wave to the PZT driver. This function generator was chosen for
its simplicity of operation. It also came with an output attenuator which when
switched off increased the output amplitude by factor 10.
A ThorLabs Inc. PDA255 photodiode was chosen for this experiment. Pow-
ered by a AC/DC transformer-rectifier, this photodiode observed voltage into the
102 microvolt range. This design incorporated a BNC coaxial cable connector to
E!taute daLt: 7/5/502
*: .QIJ3V/DWV fC q A: 2GG 'I*a OT. : CSU I4JU1 PT: 2S
Figure 3-7: Oscilloscope voltage versus time data set showing the noise background
for the PDA225. Note, no lasers were active during this data acquisi-
deliver current signals. The small diode (1 mm x 1 mm) was silicon based. Unfor-
tunately the AC power source resulted in output signal fluctuations in the tens of
A ThorLabs Inc. DET110 12 V battery powered fast response diode aided
in absorption coefficient measurements. DC battery power provided a quieter
baseline from which to operate. Linearity tests indicate that the device remained
predictable from low incident light levels to near saturation.
In addition to the photodiodes, a dichromatic CCD and monitor were em-
ploy, 1 to observe modes resonating within the cavity. Tandy Corporation assem-
bled the CCD and monitor. This strategy of utilizing a camera to see light in the
cavity permitted for a faster alignment as well as immediate lower order mode
A Digital Hameg Model 407 oscilloscope was chosen to capture the photo-
diode signal. Similar to many other digital oscilloscopes, the Hameg utilizes a
- 41M I l 13A h I PP
Figure 3-8: Oscilloscope data set showing the noise background for the DET110.
Notice the reduction in background noise.
megaohm input impedance. ThorLabs designed their photodiode for a 50 Ohm
output impedance. Therefore, a 50 Ohm terminator and a coax tee were added to
impedance match the signal. This oscilloscope also possessed a serial port permit-
ting computer control through a PC running Hameg's proprietary software. This
software was used to validate data being taken from cursor bars on the scope's
Attenuators and Mounts
Two 5215 New Focus attenuators sat in front of the photodetectors (photo-
diode and CCD). These neutral density filters (Inconel coated) prevented both
detectors from saturating. Also a high filter setting allowed the photodiode to
function in the linear range.
Lens mounts, posts, and bases for this experiment came from commercial
suppliers. These lens holders did not generate observable fluctuations in the data
set through vibrations. The mount for the cavity was obtained from Lees. This
MkIauI daL: 7T/S/D2
0l2l .DMl/EXIV Ino
O A-I 200 .. TR: C3H4AD PT: 25
mount was utilized in a previous Advanced LIGO prototype experiment. This
preceding experiment demanded stable cavity pointing for extended periods of
time. The researchers implementing the mount found no observable motion in the
device once it was set. This mount was deploy, '1 in this experiment.
Lens mounts for the probe laser's second mode matching telescope (\ \ T 2)
were placed on New Focus 9801 five freedom stages with single axis New Focus
translation stages on top. The five-freedom stage was used to align the final MMT
lenses' optical axis to the beam. A translation stage allowed large axial translation
(>2 cm) while keeping the lens centered on the beamline. Micrometers on the
translation stages and fine adjust screws on the five-freedom stages held the lenses
without inadvertent sliding.
Power Meters and Laser Protection
Two power meters were used in the experiment. ScienTech's S310 with an
Ultra series pyroelectric head was used to measure the output of the high power
pump laser. A highly defocussing lens was required to diffuse the beam to prevent
detector damage. ScienTech's detector head had a 0 to 100 W range with a
precision of 0.1 W. The second power meter was a Coherent 200 power meter
rated for a maximum of 10 W. Both power meters showed d.1 iv times in their
readout. ScienTech detector's time to lock was approximately 40 s to a couple of
minutes depending on power differences. Coherent's detector locked onto a power
reading within forty seconds.
Finally, steel plates (16 gauge) were bolted to the perimeter of the table.
These plates served as a physical barrier to the pump laser should an optic
misdirect the beam off the table. High temperature flat black paint minimized
backscatter. Two anodized plates served as beam blocks in the event the pump or
probe laser needed to be blocked without closing the shutter.
3.3.3 Beam Paths
After leaving the NPRO laser housing, the probe laser passed through EO-
Tech's Far,-Iiv isolator. This isolator's input and output polarizers were positioned
for maximum transmission. Next in the beamline was a vertical periscope that
increased the probe beam height from 9 cm to 16.5 cm. This was the initial height
of the pump laser; all equipment except EO-Tech's isolator was centered at 16.5
cm. A half-wave plate in a CVI rotational mount rotated the polarization for
transmission through the interaction region's TFPs. The first mode matching
telescope (\ \ T 1) followed this waveplate. Composed of a 250 mm lens and a
200 mm lens, this MMT developed a 1 mm waist within the center of the TGG
Si -I ,1 A second pair of 45 degree incidence mirrors directed the beam through
the interaction region, the test Faraday isolator. TFP 1 indicated the beginning
of the interaction region. Incorrectly polarized probe beam deflected off TFP
1 into ScienTech's detector. The probe beam that flowed through the polarizer
next encountered the Litton Faraday rotator and its TGG < i --I 1 A 45 degree
monolithic quartz rotator sat downstream of the TGG. Behind the 45 degree quartz
rotator was the second TFP, TFP 2. The probe laser continued into a second
half-wave plate and the polarizing beam splitter cube (PBSC). This suppressed any
depolarized high power pump radiation that may have egressed on the probe laser's
beamline. A third pair of mirrors redirected the probe beam into the second MMT.
The second MMT (\I\ T 2) mode matched the probe beam into the cavity and
was comprised of a 150 mm lens and a 125 mm lens. A fourth pair of mirrors sent
the beam into the Fabry-Perot cavity for modal i -i-i Modes that resonated
were transmitted to a mirror. This mirror launched the probe beam through the
New Focus attenuators, through a final focusing lens, and onto the ThorLabs
photodiode. Light that transmitted through the mirror passed through an identical
set of final optics before landing on the CCD camera.
50 W Nd:YLF Pump laser
Cavity with PZT
IE- ^t "
M1 & M2
L 'I 450mW
The pump laser beamline was less complicated. The YLF laser first inter-
cepted a half-wave plate. The half-wave plate and TFP 1 were used in combination
to regulate the amount of high power radiation entering the interaction region.
A split MMT (\ I\T 3) formed a waist coplanar with the probe laser within the
TGG. One hundred millimeter and one hundred fifty millimeter lenses comprised
the pump's MMT. However, a mirror was required to redirect the pump beam
within the MMT due to design constraints. Following MMT 3 were a fifth pair of
mirrors. These mirrors were used to direct the beam on TFP 1 at Brewster's angle
and to align the pump coaxially with the probe beamline. Pump light that was
not deflected into the interaction region entered the ScienTech power meter. The
ScienTech detector head was placed here since the FK51 glass scattered some of the
pump light. Placing the detector head near TFP 2 would have given false powers.
Pump radiation that was deflected into the interaction region passed through the
TGG i I I1 and quartz rotator. TFP 2 then deflected the pump laser into the
BeamScan which was being used in coaxial alignment and beam quality control.
Any depolarized pump light escaping towards the cavity was suppressed using the
aforementioned half-wave plate and PBSC.
3.4.1 MMT Alignment Techniques
Setting the MMT lenses in the correct positions was simplified by utilizing
the beam profiler's centroid tracking subroutine. This tracking program provided
a reference point indicating the laser's original direction. A lens could then be
positioned in front of BeamScan and translated until the laser fell on the reference
point. It was possible that the beam could appear to travel through the reference
point and be directed off the beamline due to beam width. To thwart this the
BeamScan reference points at two positions on the beamline were acquired. When
the laser propagated directly though the lens' optical center, centroid data at two
points along the beamline would show no lateral displacement on the centroid
graphic user interface (GUI).
With the lenses installed in their appropriate positions, beam profilimetry
and a X2 fit provided information on propagated waist position and size. Should
the beam be converging incorrectly the MMT was manually modified and another
profile taken. Since the probe beam was elliptical, the mode matching calculations
were based on the horizontal waist.
3.4.2 Overlap Alignment
Alignment of the pump laser on the probe laser was critical since the entirety
of the thermal lens needed to be observed without excitation of the HG10 mode.k
Mirrors M7, M8, and the cavity were rotated to suppress the HGio mode into the
noise floor (approximately 2 percent of the TE.1,,, mode). To correctly establish
pointing, two working points' and a reasonably large working length were
required. The experiments two working points were the BeamScan, intercepting
bounce light from TFP 2, and the CCD camera. This provided nearly 2.5 m of
working distance or a theoretical 1 mrad pointing accuracy."
A quarter-wave plate was set behind the quartz rotator allowing 10 W
of pump laser to flow towards the cavity and 240 mW of probe light to enter
BeamScan. The pump laser was then blocked without closing the shutter. The
BeamScan's pointing feature was then centered on the probe beam indicating the
k The excitation of the HGlo mode indicated a pointing deflection generated by
sensing a noncentered thermal lens.
1 Two points are needed geometrically to form a straight line.
m Longer working lengths provide greater pointing resolution when handling
n This calculation was based on the BeamScan's best ability to detect lateral
short reference point. The probe light spot's center on the monitor noted the long
The probe beam was then stopped by a second block plate and the pump
beam opened. Mirror M10 was used to walk the beam beyond the short reference
point's center. Mirror M11 was then used to walk the beam through the circular
symmetric modes to a clear laterally symmetric mode. The beam walking process
was repeated until the pump laser's position indicator fell on the BeamScan's cen-
ter crosshairs, and the shape of the pump laser through the cavity was circularized.
This insured that the two beams were sharing the same volume of space. The
quarter-wave plate was finally removed from the beamline.
3.4.3 Data Collection
Once the two beams were correctly aligned, the pump laser was set at 3.0
W and the pump's half-wave plate rotated to transmit all possible light through
TFP 1. The pump laser was then brought to running capacity (52.0 W) and left
to stabilize over 15 minutes. The probe laser and all electronics were allowed to
thermally stabilize over 10 to 15 minutes. Thermal stabilization for the YLF was
especially important as its power level would fluctuate 0.5 W until cooling water
and lasing crystals reached an equilibrium temperature.
The statement all possible light implies that some light was polarized ran-
domly with respect to the laser field. A couple hundred milliwatts of errant
radiation entered the interaction region. This did not affect measurements at zero
power. A measurement of TE 1,,,, magnitude with and without the pump engaged
showed no visible changes in the mode of the probe laser.
The TTi function generator was not needed in the data collection, only during
alignment. Therefore, the generator was removed from the PZT. This provided
operators to manually slew the cavity and to find the best resonance points.
Differences between maximum power and laser power on the ScienTech
detector head indicated the amount of power entering the interaction region.
Pump laser power levels were increased by 8 W increments until 52 W flowed
through the interaction region. A wait time of 5 minutes was imposed to insure the
thermalization of the TGG and to allow the ScienTech power meter to stabilize.
Following each power increase, the amount of power entering and leaving the
interaction region (probe beamline) was checked using the Coherent 200 detector
on the 0.3 W and later the 1.0 W setting. This check would disclose any power
loss or gain in the probe laser. Power fluctuations in the probe would have resulted
in false lensing data. The check also provided data regarding thermally induced
birefringence and TFP performance. Furthermore, the check provided information
of the noise floor as pump power was increased.
Next the intensity of each visible mode was recorded. The CCD allowed the
mode to be recognized on the display screen. Here the PZT driver's monotonic
drifting characteristic became useful. The PZT driver would automatically drift
through a resonant mode of when the DC offset positioned the output coupler at
roughly 10 percent resonance. This method of drifting proved to be more effective
than manually adjusting the DC offset to hold resonance. Manipulating the DC
offset and watching for maximum intensity values on the oscilloscope resulted in
overcompensation through human error.
One should note that as the cavity drifted through resonance the peak was not
symmetric. As indicated in the first section of this chapter, circulating powers in
the cavity forced the cavity off resonance rather quickly. For example, drifting into
resonance often took tens of seconds. The "kick out" of resonance happened within
a second. This occasionally gave lower intensity readings. However, several drift
measurements remedied this problem.
The sum of all observable modes provided confirmation that the experiment
was not losing or gaining light through depolarization or nonlinear effects. Nev-
ertheless, the primary modes of interest were the fundamental TE 1,,,, mode, the
Laguerre-Gauss LGlo mode, and the Hermite-Gauss HGoo mode.
Once the control data run (TGG only) was complete, 6.8 cm of FK51 Schott
glass was inserted behind the quartz rotator. A gap of 1 millimeter remained
between the wedges. Mirrors M5 and M6 were adjusted to compensate for the
beam displacement. The above procedures were repeated with the FK51 sitting in
3.5 Absorption Coefficients
Absorption coefficients were acquired through a four-point method. This
method involved observing incident, transmitted, front surface, and back surface
reflected light levels. A focused probe beam, w = 100U pm, entered the target
(i -I 1 A non-normal incidence angle provided the means to separate the reflected
beams. To obtain the absorption coefficient, the transmitted and reflected beams
It,r = Itransmitted + Ifrontrefl + Ibackrefl (3.22)
The difference between the sum and the incident radiation divided by the incident
radiation gave the absorption factor. Dividing by the length, 1, of the material
yielded the absorption coefficient,
a1 incident t,r
To obtain light intensities necessary for this computation, the ThorLabs
battery powered photodiode was used to directly intercept the light. A Glan-
Thompson polarizer, half-wave plate, and a PBSC prevented laser light from
saturating the photodiode. A block plate aided in differentiating between the
two reflections. An MMT consisting of a 19.0 mm and 60 mm lens focused the
probe beam. Absorption coefficient results are noted in chapter 2, section 7: The
The results for the both uncompensated and compensated measurements are
plotted below superimposed on the theoretical data curves seen in the chapter
2. One will note that the experimental uncompensated data points verify that
laser power shifts dramatically due to the thermal lens in the TGG. XN .inl ii. 1 ,1
intensity in the TE11,,,, mode falls from 1, perfect mode matching, to 60 percent at
50.2 W. The radius of the circles indicates approximately one sigma. With respect
to the theoretical calculation and the empirical data, no fitting parameters were
used during plotting. To obtain a correct fit, it was necessary to establish the
absorption coefficients of the TGG and FK51. Absorption coefficients for TGG,
QTGG, indicated that the Litton AirTronics sample exceeded a factors found in
literature by a factor of 3. This factor became important during the compensation
tests with FK51.
Compensated data however told a slightly different story than what the
theory pointed towards. Initial data runs showed the fundamental mode dropped
qualitatively more slowly than the uncompensated data. At 50 W, the fundamental
mode dropped to 87 percent of the TE11,,,, intensity value observed at zero incident
pump power. However, HGo0 exceeded 5.5 percent of the TE1,,,, intensity at 34.3
W. At full pump power, both HGo0 and LGo1 modes exceeded 9.5 percent of the
TE1 [,,, intensity.
The second compensation data run was performed by modifying the proce-
dures. Following each increment in pump power, the probe beam was i, idlusted
to minimize the HGio mode. This realigned the probe beam and permitted an
Focus Comp. (Theory)
100% Thermal Comp. (Theory)
Thermal Comp. (Exper.)
Thermal Uncomp. (Theory)
Thermal Uncomp. (Exper.)
0 50 100 150 200
A plot of experimental data superimposed on Chapter 2's theoret-
ical results. The uncompensated thermal lensing data follows the
predicted uncompensated line's rapid fall off well. The compensated
thermal lensing data however appear to track the focus compensated
curve instead of the red thermal compensation curve. This mismatch
was determined to result from the lack of FK51 glass interacting with
the distorted laser beam. No fitting parameters were applied; datum
point size is representative of the error bars.
increased visibility of the axial mode shift, LGIo. Results using the modified pro-
cedures indicated a retention of TE13 ,,, mode. At 50 W the normalized intensity
TE3L,,, intensity falls to 93.6 percent. The HGlo to TE31L,,, and LGlo to TE3L,,,
ratios crested at 1.3 percent and 8.7 percent at 50 W. Since the amount of higher
order modes showed no significant increase beyond the initial FK51 data run, we
conclude that the HGlo excitation was caused by the glass's wedge geometry and
nonuniform thermal boundary conditions. Furthermore, the change in LGlo is
minimal. Therefore, this implies that the second compensated data set would have
been obtained in the initial data run if the wedges had not displaced the probe
0 50 100 150 200
A plot of experimental data superimposed on Chapter 2's theoretical
results. The green line represents 68 percent compensation superim-
posed on the experimental compensation data. The focus compen-
sation curve has been removed. No fitting parameters were applied;
datum point size is representative of the error bars.
beam. According to theoretical compensation curves state that fundamental mode
intensity should remain above 99 percent at approximately 50 W and fall to 95
percent at 200 W.
The reason for the lack of optimum compensation arises in the aforementioned
aTGG. Being a factor 3 greater than what literature records, the data indicates that
the length of FK51 glass used in the experiment was not enough to perform the
calculated task. A modification to the computer model's glass length shows that
this experiment had achieved 68 percent of first order compensation limit. In terms
of length, 3 cm more glass would have compensated the thermal lens in the TGG.
100% Thermal Comp. (Theory)
68% Thermal Comp. (Theory)
m Thermal Comp. (Exper.)
Thermal Uncomp. (Theory)
Thermal Uncomp. (Exper.)
e =,I ,I, ,
0 10 20 30 40 50 60
A zoom-in of the data plot shows that the thermally compensated
data in this experiment follow a 68 percent compensation curve. The
solitary black point was obtained through refinement of the final
mode matching telescope, lenses L3 and L4. This refinement increased
modal restoration to 97 percent.
To verify that this experiment did not simply perform a focus compensation,0
MMT 2 was altered to improve mode matching with the cavity following FK51
data acquisition. At maximum incident power the new point increased to 97
percent. This was 3.4 percent greater than the 50 W FK51 data point and 4
percent greater than the focus compensated data point. All datum error bars span
approximately 1 percent. Therefore, it can be concluded that this maximization
was attributed primarily to spatial mode compensation.
o Focus compensation implies that higher order transverse modes are still intact.
- Focus Comp. (Theory)
- 100% Thermal Comp. (Theory)
68% Thermal Comp. (Theory)
m Thermal Comp. (Exper.)
e Thermal Comp. + Focus Comp. (Exper.)
I I I I I I I I I
Focus Comp. (Theory)
0.2 100% Thermal Comp. (Theory)
68% Thermal Comp. (Theory)
Thermal Uncomp. (Theory)
0 50 100 150 200
Figure 3-13: An overlay of the 68 percent compensation curve and the focal com-
pensation curve. One can see that the two curves track each other
until 60 W. After this point compensation bows to focal compensation
The model notes that for optimum compensation, a point solution must be
found for each TGG crystal. However, MMT 2's ability to improve compensation
would allow an error margin to exist for designers of mode critical systems.
Throughout the experiment tilt modes did not exceed 3.4 percent for the
TGG data run. The FK51 faired better with a maximum tilt of 1.3 percent. This
pointed out that the alignment of the pump on probe was concentric to within 5
percent of the probe beam radius.
Power data taken immediately upstream and downstream of the interaction
region disclosed two intriguing results regarding depolarization. Although the
probe laser light did not fluctuate more than 3.5 percent, pump light leaking
through TFP 2 increased by 20 percent when only the TGG sat in the beamline.
The FK51 increased the pump leakage by nearly 3-fold. This depolarization of
the light field was attributed to thermally induced birefringence. At mid-range
pump powers, a clear "clover 1. i mode was visible on frequency doubling ceramic
sensors. This was the definite verification of the stress induced depolarization.
However, it was not entirely clear why probe light was not as greatly influenced.
This experiment has demonstrated a viable passive method for counteracting
thermal lenses in Faridiv isolators. By exploiting the thermal dependence of the
index of refraction, we are able to experimentally recover more than 93 percent
of the original laser mode at 50 W pump power. It must be emphasized that this
is a point solution for each TGG i -I I1 An optimized solution would permit
future gravitational wave detectors to have numerous highly effective alignment
and working points. During alignment at 20 W, Advanced LIGO would observe
nearly 100 percent mode matching. Following the increase to full running power,
Advanced LIGO's core optics would retain 97.5 percent of the light in the TE1,,,,
mode. However, an MMT downstream of the compensated Faraday isolator can
improve mode matching should the glass not be of appropriate length.
4.2 Future Experiments
Future studies into passive compensation should include applications to other
long input optics such as those found in electro-optic modulators. Also thermally
induced birefringence must be studied as a function of incident light intensity. As
indicated by the power data taken before and after the interaction region, pump
light was depolarized and proceeded towards the cavity. Although it may not
resonate in the core optics, this process does remove light from the fundamental
mode and may effect final results.
Data acquired in April 2002 indicated that FK51 has a depolarization coef-
ficient of the same magnitude of TGG. Due to FK51's glass characteristic, axial
rotation will have no effect in counteracting depolarization as it has in IAP Faraday
rotators' Most important, as an optical diode, thermally induced birefringence
can readily defeat an isolator's ability to decouple subsystems. Finally, temperature
and the varying of boundary conditions of the FK51 should be considered. This
experiment relied on large pieces of freestanding glass. These pieces were not uni-
formly cooled since one side conducted heat to its pedestal and the other four sides
released heat through convection. Ultimately designs for compensated Fa,-d iv
isolators may demand smaller rod shapes to minimize the cost of glass, and thermal
control of the glass could provide a negative focus adaptive optic.
Another possibility for compensating the thermal lens generated by TGG
dn/dT's that are two orders of magnitude greater than the FK51 glass tested. This
implies that if applicable, these compensation materials could be placed closer
to the crystal of interest (TGG). Placing the compensating medium closer to the
thermal lens reduces the Gouy phase shift. This increases the incident laser power
that the isolator can tolerate without the losing an excessive amount of TE [,,,,
mode. However, this assumes that the dn/dT term will remain the dominant term
in the AOPL sum. The introduction of gels removes the possibility of thermally
induced birefringence. These gels also show large expansion coefficients. This may
increase the complexity of optimum compensation.
a IAP Russia at Nihzny Novogrod has compensated thermally induced birefrin-
gence in TGG by rotating one (i i axially by 45 degrees with respect to it coun-
COMPLEX-Q AND THE ABCD LAW
A.1 Derivation of q
C'! ipter 3.2.1 started to demonstrate the difficulties of full wavefront prop-
agation. The scheme of using the full Fresnel-Kirchoff integral equation in an
unconfined region becomes increasingly difficult. A less precise but more convenient
method of field propagation utilizes components of the phase factor obtainable
through the wave equation [28, 29]. This term is referred to as the "complex-q."
We assume that the laser phase front shape changes slowly with respect to
the z-axis (the direction of propagation). We also assume that the laser's intensity
profile is Gaussian and that the paraxial approximation applies [28, 29]. With these
assumptions we can begin our derivation with the wave equation,
(V2 -2 ) U(x, y, ,t) 0 (A.1)
Using separation of variables we can break the equation into a time and space
U(x, y, z, t) =u(x, y, z)T(t). (A.2)
The space term becomes
V2u(x, y, z)- k2U(x, y, z) =0. (A.3)
The time terms becomes
c2 t2T(t) k2T(t) = 0. (A.4)
C 2 Ot2
The time component has the familiar harmonic solution of
T(t) = Toe-wt. (A.5)
This part of the solution behaves as a simple multiplier to the space term. There-
fore the time component will not be carried through the rest of this solution.
To solve the space differential equation we assume a solution that looks like
u(x, y, z) = (x, y, z)e-iz. (A.6)
Expanding the differential equation we obtain
2 + y2 -i2k = 0. (A.7)
(x2 9y2 ) Oz
We can assume a solution of the following form
) = e-i(P(z)+ kr2)8
2q' j (A.8)
where P(z) is a complex phase shift, k = + y2, and q q(z).
Let's focus on the second phase shift,
S= e- r2. (A.9)
Since this portion of the model is to represent the laser field's Gaussian amplitude
profile q must be imaginary . We also know that the Gaussian profile's char-
acteristic length is the waist length, wo . Therefore, the form of q at the laser
o = (A.10)
By sending the solution through the differential equation we can determine
the what the phase parameters are. Written below is the full expansion of the
kc-i(p()+ k 2+ ,2) k~ 2 (P(z)+ ,2)
C-e 2 -2
q -i( -. 2+- ) ) 2qi(P(+)+ +
2+- :) + 2g,(x2+ 2)e -(P(z)+ 2 -
If we separate this equation through powers of k we obtain the following:
22 2 2
+ q(x2 + 2)e-i(p(z)+ xk2+
k _i(P(Z)+ k: 2) k _i(pZ)+k 2- 2)
-2kP'(z)e-(P(z)+ ) q 0.
1. q' =1;
2. P'(z) i
Integrate item 1 with respect to z,
q(z) = qo + z.
This simple addition law indicates how one can propagate a complex-q factor in
free space [29, 28]. The logic for this will become apparent in the following section.
Inserting this into item 2 we acquire
P(z) In z + ) + C', (A.16)
P(z) In 1+ i arctan (A.17)
The first term represents the phase difference between a plane wave and a Gaus-
sian phase front. The imaginary coefficient is the Gouy phase shift . This
component models the phase shift as the laser's wavefront travels through a waist.
A.2 ABCD Laws for Gaussian Laser Propagation
Gaussian lasers are propagated through the ABCD law,
() Aqo(z) + B (A.
Cqo(z) + D
This formalism acquires its four terms (A, B, C, and D) from 2 x 2 matrices
implemented in ray optics. The derivation for the ABCD law in Gaussian optics is
In the thin lens approximation, a lens of focal length f effects a spherical
wavefront's radius of curvature in the following form
1 1 1
R'(z) R(z) f
This is applicable for Gaussian laser phasefronts provided the lens is thin enough
that it does not greatly alter the transverse radius of the laser beam,
1 1 1
q1(z) qo(z) f
This indicates that the complex-q is the phasefront's radius of curvature in the
complex plane [28, 29].
Combining this with free space propagation (item 1 of section A.1),
q,(z) = qo(z) + z,
and then propagating the beam again, we acquire
(1 )qo(z) + (z+ 2)
q(z) = -T) -
q (z) ( )qo + (1 )
Looking at the coefficients of qo(z) in the form of the i
the coefficients are the same as the four elements in the 2 >
optics to move a ray through a thin lens,
ABCD law, we find that
S2 matrix utilized by ray
Mtot = M2 M. Mi,
(1- 2) (-I + Z2 (A.z
1 ( (1- ) )
A list of ABCD matrices can be found in Siegman, Fowles, Hodgson and Weber,
and many other optics book.
This coincidence occurs because both ray optics and paraxial wave analysis
track the phasefront's radius of curvature utilizing the thin lens approximation.
(r' A B ( (A.
/' C D 0)
The radius of curvature for a spherical wavefront in ray optics becomes
r' Ar + B
0' Cr + D
We can now see that the complex-q is merely the complex radius of curvature of
the laser's i,. fi l' .i
a Walther  presents a similar but more generalized approach in his book.
The polarization of an electromagnetic field represents the direction that the
plane in which the electric field is directed. This polarization can be represented by
a 2 x 1 vector. C'!i ,;, in polarization angle due to interactions with various media
(dust, interfaces, magneto-optics, etc.) can be modeled through a change of basis
using a 2 x 2 matrix called Jones matrices [14, 22].
The formalism for this interaction is as follows
P, M 1 V11 ] P t
( y) K( 11 [) ( ) (B.1)
where P' and P' are the new electric field polarization magnitudes in the or-
thogonal directions, the matrix M is the change in direction of polarization due
to the interaction with some set of media, and the final P vector is the original
component magnitudes of the electric field [14, 22].
For example, let's start with a electromagnetic wave completely polarized in
the x-axis. This wave interacts first with a quarter-wave plate, QWP, that has
been rotated by 450 with respect to the incoming light's polarization and then
a polarizer set to pass y polarized light. To find the final vector amplitude, we
multiply the matrices right to left in the order that our electromagnetic field "sees"
the optical components,
I .,wp ,l 1 (B.2)
MPol o (B.3)
M final = Mpoi. iWP
( 1) (B .4)
i o (B.5)
The solution obtained indicates that the resulting light is right circularly polarized.
This rather trivial exercise demonstrates that the Jones matrices for optical
components can be multiplied in the order of observation. Futhermore, it is
relatively simple to analyze the resulting polarization direction through the
component magnitudes. Further information and a full list of Jones matrices
for optical components can be found in nearly any modern optics textbook (for
example, Hodgson and Weber, Fowles, or Siegman).
UF THERMAL LENSING CODE
%Master.m %Written for UF-LIGO
%Author Guido Mueller 28 October 2001
%Comments by Rupal S. Amin // MatLAB-Octave Code
% All lengths in millimeters
lambda = 1.064e-3; %Wavelength of laser
P_max=200; %Maximum power of laser [W]
L = 30; %The distance between the two slabs
%Load crystal and glass parameters
crystal_l_params; %Input TGG crystal parameters.
crystal_2_params; %Input FK51 glass parameters.
w_in = 1; %Laser waist size in TGG
invR_in = 0; %Radius of curvature of wavefront at the waist
c_l=zeros(19,1); %Initialize constant array.
%input q after pushed upstream through the slab.
q_in = i*pi*w_in^2/lambda L_1/(2*n_l);
%Propagate the input q into the center of the TGG
q_l = q_in+L_l/(2*n_l);
%Obtain the first 201 terms to approximate the infinite sum in
%eq. 2 of the Amaldi paper.
m = 201;
%Iterate for 51 steps / tranverse data 1-D grid density.
%Lambda = delta OPL; a 51 point array denoting the
%delta OPL phase shifts on the x-axis.
%Our "crystal" is 8 units wide.
delta = 4.0;
%Radial position on transverse axis
r = delta/num_step;
c_out = zeros(19,1);
%Initialize the position and optical pathlength distortion arrays
Lambda_l = Lambda;
Lmabda_2 = Lambda;
%out = zeros(2,num_step);
%Calculate the sum for each of the 201 points on the (transverse) x-axis.
%Octave does not have a "factorial(n)" function.
fac = fac*l;
sum = sum+(-l)^l*(2*r^2/w_in^2)^l/(l*fac);
Lambda(step) = sum;
pos(step) = r; %Keep track of transverse position
%Increment to the next position on the x-axis.
r=r + delta/num_step;
pow_step = 31; %The laser power will be increased in 30 steps.
dP=P_max/(pow_step-l); %Determine power step size
P=0; %Initialize laser power variable
P_out = zeros(4,pow_step); %Initialize mode intensity matrix
%Start solving for Laguerre-Gauss mode series constants through 30 power
P = P+dP; % Step up the power!
%The thermal lens's delta OPL coefficients.
coef_l = a_l*P/(4*pi*k_l) dn_dT_1 L_1;
coef_2 = a_2*P/(4*pi*k_2) dn_dT_2 L_2;
%Multiply the above coefficients by the optical pathlength distortion
Lambda_l = coef_l Lambda;
%The second coefficient is set to be negative the first coefficient.
%This enables us to obtain optimum compensation.
coef_2 = -coef_l;
Lambda_2 = coef_2 Lambda;
%Call the "radial_lens" subroutine.
= radial_lens(q_l, num_step, delta, Lambda_l, lambda);
%Call the "focus_comp" subroutine.
[P_out(3,k), invR_eff, invR_app]
= focus_comp(q_l, num_step, delta, Lambda_l,lambda);
%Calculate the absolute intensity output for all modes.
%Detemine the optical distance from the TGG to FK51 glass.
d_dif = L_1/(2*n_l)L+L_2/(2*n_2);
%Determine the complex q coming out of the TGG
p_2 = (invR_eff+i*imag(1/q_l))^(-1)+d_dif;
w_2 = sqrt(-lambda/(pi*imag(1/p_2)));
q_2 = q_l+d_dif;
%Call "radial_gouy" to propagate the wavefront to the FK51 glass.
c_3 = radial_gouy(c_l,q_2);
w_ave = (w_in+w_2)/2;
% Calculate the coefficient of the FK51 glass
%(removed for optimum comp)
%coef_2 = a_2*P/(4*pi*k_2) dn_dT_2 L_2;
coef_2 = -coef_l;%*0.68;
Lambda_2 = coef_2 Lambda;
%Call the "radial_correc" subroutine.
c_4 = radial_correc(c_3,q_in, num_step, delta, Lambda_2,lambda);
%Detemine the intensity of the compensated modes.
P_out(4,k) = abs(c_4)^2;
%Input incident power into "power" matrix
P_out(l,k) = P;
%P_out = P_out; %one dimensional analysis,
%second dim gives additional losses
%Plot and write data to a file.
fid = fopen('result','w');
formatstring = ['%.6e','\t%.6e' ,'\t%.6e' ,'\t%.6e' ,'\n'];
fprintf(fid, format_string P_out);
%Thermal lensing subroutine implementing the phase shift due to the
% first crystal.
%Input variables:q_l, num_step, delta, Lambda_l, lambda
%Output variables:c_l: Laguerre-Gauss amplitude coefficients
% P_un: power in phase front
% invR_eff: 1/(effective radius of curvature ) used in focus