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A THEORETICAL AND EXPERIMENTAL STUDY OF THE POLARIZA'MON STATES OF AN~~+:YAG LASER
Waam Robert Daigliesh
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy
Graduate Depariment of Physics University of Toronto
O Copyright by William Robert Dalgliesh 1998
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Abstract
A Theoretical and Emerimental Study of the Polarization States of a N~~+:YAG Laser
Waam Robert Dalgliesh
PhD. 1998
Department of Physics
University of Toronto
In this work a model descniing the polarization states of a N~~+:YAG, single
frequency laser is developed. Based on the symmetry of the ~ d ~ + site in the cryd, and the
selection d e s for electric dipole transitions in Kramers degenerate states, the phase
relationships between the wmponents of the electric ùipole moment are found, without the
use of the double group. Sumxning over the six sites of the crystal, the response of the gain
material to an applied electric field is found, and the polarization behavior is determined to all
orders in the e l e d c field through a vector extension of the mean field model, similar to work
done in gas lasers. The result is 16 coupled non-linear differential equations describing the
polarization state of the laser.
The predictions of the model are compared with results in the iiteraîure. Whde the
model cm explain most of these results, it is found that the effective output mirror model used
to describe polarized feedback is inappropriate for a microchip laser with a long feedback
path.
New experiments are performed in the low birefXngence regime, and for the case of
eliipticdy polarized pump light. The predictions of the mode1 are in semi-quantitative
agreement with the low birefiingence experiments, while for the case of circularly polarized
pump fields, it is found that some of the approximations of the model become inappropriate.
New experimental results on the bistability of the Nd:YAG microchip laser with
polarized feedback are presented and bnefly discussed. Suggestions for future experirnental
work and refinements to the rnodel are presented.
Table of Contents
Chapter 1 -- Introduction ...................... .... ................................................. 1
Part 1
Chapter 2 œ Theory ..................................................................................................... 6
.................................................................................................................................. htroduction -6
................................................................................................ Gamet Stnrcture ............,................ 6 *- ......................................................................................................... Spcîmscopy of Nd in YAG -9
3+ SeIection niles for Nd in YAG .......................................... .. ......... 10 ................................................................................................... Derivation of phase relationships 12
Theory of the polanzation stats d a N~*:YAG laser .................................................................... 15
...................................................................................................................................... Pump rate 16
............................................................................................................... Laser transition ............ ., 22
...............*......*.......... .... Transformation to laboratory fiame ... -26 ............................................................................................................................... Field equations 30
Methods used to fhd the stationaxy solutions ................................................................................. 39
Integration in tirne .................................................................................................................. ..39 . . ............................................... ............................ S tability analysis .... -40
............................................................................................ ...................... Algebraic solution , -42
Chapter 3 -- Cornparison with results in literature ...................................................... 44
........................................................................................................................ Control parameters -44
....................................................................................... RÊsults in the iiteraîure with no feedback 54
............................................................................................ Resdts in the literaîure with feedback 61
Part II
.................................................................................. Chapter 4- Experimental sehip 71
........................................................................................................ Introduction ........ ..,. .... 71
.............................................................................................................. Apparatus and techniques 71
.................................................................................................................. Nd:YAG microchip -72
Pump source .............................................................................................................................. 73
Soleil-Babinet compensator ....................................................................................................... 74
b h l i t 0 ~ g SyStem ............. .,. .................................................................................................... 75
Feedbadr ................................................................................................................................... 78
Characterizhg the system ............................................................................................................. -79
Spatial mode characterization .................................................................................................... 79
Longitudinwl made characterization... ........................................................................................ 80
ûther considerations .................................................................................................................. 81
Chapter 5 -- Experimental Redts and Analysis ....................................................... 83
Elliptical pump p o M o n ........................................................................................................... 98
Hysteresis experiment .................................................................................................................. 103
Chapter 6 n Summary and Recornmendations for Future Work .............................. 109
References ......................... , ................................................................................... 113
List of Tables
Table 3.1 . Model parameters required for CaIculations ................... ,.., ................................................... 44
Table 3 2 - Parameters reqnired for calculaiions ciffigwe 3.2 and figure 3.3. ...................................... ., S O
Table 3.3 - Paranieters required for calculations of figure 3.2 and figure 3.3. ......................................... 56
Table 3 -4 - Parameters requUed for caiculati011~ of figure 3.7 and figure 3.8 ......................................... 57
Table 3 -5 - Parameîers required for d&ons of figure 3 -2 and fi- 3.3. ............................... ...,. ... ..6û Table 3 -6 - Parameters required for c a l d o n offigures 3.10 to 3.15 ............................................ ....... 63
Table 5.1 O Parameters required for calculations in figures 5.3 to 5.10 ................................................... -86
Table 5.2 - Parameters reqirired for caldations of figure 5.11 and figure 5.12. ..................................... 95
List of Figures
................................................. Figure 2.1 w Diagram o f the tocahions of the constituent ions in Y3&ûi2 7
Figure 2.2 - Diagram showing the unit c d of the cubic YAG Mce ....................................................... 7
.................... Figure 2.3 - Diagram showing the nearest neighbors of a ~ d * ion in the hm YAG lattice- 8
Figure 2.4 - Diagram showing the orientation of the symmetry axes uf the subunit ceiI ........................... 9
......................................... Figure 2.5 - Diagram showing the Iower lying enagy levels a f ~ d * in YAG 10
Figure 2.6 - The poIarization dection niles for transitions beîween Kramers pairs of srites .................. 11 Figure 2-7 - The four level laser system ................................................................................................. 16
. . .................................................................................... Figure 2.8 - Diagram af the polanzanon eiii pse. 38
Figure 3.1 a The dative strengths of fluorescence of the R to Y levek ................................................... 47
.............................................................. Figure 3 2 - Intensity of the laser output for an isobropic cavity 49
....................................... Figure 3 -3 - Orientation of the laser output poIarization for an isotropie cavity 50
Figure 3.4 - Orientation of the symmetry axes of the ~ d * sites in YAG ................................................. 52
Figure 3 -5 - OrientaDion of the symmetry axes of the ~ d * sites in YAG ............................................. -53
Figure 3 -6 - Orientation of the Iaser output for piimping dong stress axis .............................................. 56
Figure 3.7 - Output ellipticity for pumping at 45 degrees to the axïs of the stress ................................... 57
Figure 3 -8 - Output orientation for pumping at 45 degrees to the axis of the stress ................................. 58
Figure 3.9 - Output intensity as a fiinction of the pump for a Iarge birefnngen ce.. .................................. 60 ........................................................ Figure 3.10 - Largest reai part ofsmbility exponents with feedback 63
Figure 3.11 - Largest reai part of stabiiïty exponents with Fp=l/2. Fr=l ................................................. 65
.............................................. Figure 3.12 - Largest r*tl part of nability wrponents with F p 1, FL= 112 -65
................................................... Figure 3.13 - Largest real part d stability exponents with Fp= FL=1/2 66
Figure 3.14 - Largest real part of stability exponents with Fp= FL=1/2 and reduced feedback ................. 67
Figure 3.15 - h g h q part of stability exponent with Fp=FL=1/2 ....................... .... ......................... 68
Figure 4.1 œ Schematic drawing of the experimental apparatus ............................................................... 72
Figure 4.2 - Caliiration of Soleil-Babinet compensator ......................................................................... 37 Figure 4.3 - Experhentai setllp including polarized feedback .............................................................. -78
.................................................... Figure 4.4 - Intensity profile in the x and y directions of laser output -80
........................................................... Figure 4.5 - The gain p r d e s of the R2-Y3 and RI -Y2 transitions 81
Figure 5 . L O Measured output orientation relative to the pump orientation ............................................ 84
Figure 5.2 - Measured absalute value of ellipticity vs . pwnp orientation ............................................ 2
.................... Figure 5.3 - Calculateci orientation for two s a l e =ors af the dipole moment anisotropies 87
Figure 5.4 - Calcuiated ellipticity for two scde &tors of the dipole moment anisotropies ................... ..87 Figure 5-5 - Calcuiated orientation for three crystai angles ............................................... ,.., ............. 89
Figure 5.6 - Caldated ellipticity for three crystal angles .........................................~...........~................ 89
Figure 5.7 - Calculateci orientation for three operating fiequencies ......................................................... 91
........................................................... Figure 5.8 - Caiculated ellipticity for three operating fkquencies 91
Figure 5.9 - Calculateci orientation with and without a linear dichroism ................................................. 93
Figure 5.10 - Calculated eilipticity wiîh and without a linear dichroism ...................~~~~..~~~.~~~..~.............. 93
Figure 5.11 - CaiÇulated vs . experimentai output orientation .................................................................. 94
Figure 5.12 - Caiculated vs . experimental eliipticity ......... ..,. ..... ..... ..................................................... 94
Figure 5.13 - Calculatecl vs . experimental ellipticity for three pump powers ......................................... 97
Figure 5.14 - Measured absolute value of ellipticity vs . ellipticity of pump beam, for &+, =4S0 ............. 99
Figure 5.15 - Measured absolute value of eilipticity vs . elIipticity of pump beam, for &+=9 O ............... 99
Figure 5- 16 - Calculated eiiipticity vs . phase retardation of pump beam for three values of yk .............. 101 Figure 5.17 O Polarization switching with 40 cm feedback path ............................................................. 105
Figure 5.18 . Poiarization switching with 3 m feedback pa th. .............................................................. 105
Figure 5.19 Largest reai part of stability exponents ............................................................................... 107
............................................................................... Figure 5.20 Largest real part of stability exponents 107
vii
Chapter 1 - Introduction
The basic principles of a laser, Le., oscillation at opticai nequencies through
amplincation by stimuiated emission, plus feedback nom a mirror system or cavity etc.,
are easy to express. They are not considered complicated, probably because the laser has
a weU known analogue: the electronic oscillator. On the other han& lasers are often
technicdy ditncult and difncuit Eom a theoretical or fundamentai point of view. Lasers
are ditticult to describe theoreticdy because they are multivariable, nonlinear devices in
space and time with a large number of control parameters. Given the daunting nature of
the theoretid problem, it is understandable that no single comprehensive theory of the
laser exists. There are only theones tailoreci to describe specific aspects of one laser or a
generic type of lasers. We refer to nich theones as "models".
The best known rnodel is the mean-field theory of Lamb [l]. Although it was
origindy developed as a rnodel for low-gain gas lasers, it was sufaciently generai that it
could be applied to many other lasers. It was intended to capture the threshold and
saturation aspects of lasers. The main assumptions or approximations and some of the
rationale behind them are as foliows:
(i) The problem of determining the transverse spatial variation of the opticd field of a red
laser was circumvented by making a plane wave approximation.
(ii) The longitudinal variation of the optical field of a real laser was ignored by assuming
the field was a uniform (pure) standing wave.
Introduction
(ii)Havbg made the mean field approximation (i and ü), it was necessary to distriiute the
point losses that occur at the rnirrors of a real laser, so the local loss codd balance the
local gain, as is appropriate for a d o m field.
(iv)The response of the gain medium to the optical field was calculated to third order in
the field. This is the lowest order possible if the process of saturation is to be captured
by the model
(v) The vector nature of the laser field was ignored. Such a scaiar theory is an appropriate
model for lasers containhg opticai components nich as Brewster angle windows that
strongly discriminate against one component of the field.
(vi)In order to keep the mathematics tractable, the original model assumed that only a
single longitudinal mode could oscillate.
The model of Lamb was extremely successfil. It was soon extended to include
other longitudinal modes. Today there ex& many phenomenological variants of the
original scalar model applicable to a large range of lasers. However, it is clear that none
of these models is capable of describing a laser with weakly anisotropic optical
components. Such lasers are capable of operathg on different polarization modes and
require as a minimum a vector extension of Lamb's mean field model.
The extension of Lamb's theory to include the polarization aspects of so caiied
quasi-isotropie lasers can be traced back to de Lang [2]. In the onpinal scdar theory there
are two factors contributing to the rate of change of the amplihide of the opticai field.
The fkst is the gain due to the ampEQing medium and the second is the loss ber round
trip) due to the cavity. The conceptuai jump to the vector case is simple; replace the
round trip change in the amplitude by the round trip change in the vector field. The cavity
contribution to the change can then be easily formdated in ternis of Jones matrices. The
mathematicai jump is not simple.
Introduction 3
For gas lasers, the mathematicai problem is more one of bookkeeping than
anything else. The response of the gain medium to polarized Light can only be treated
correctly by including the degeneracy of the states. In a gas each state is (2J+1)-fold
degenerate, and keeping track of ail elechic dipole connections between the sub-levels, up
to third order in the field, is prone to erron. There is no diflFiculty in the case of fke
atoms or moleniles with deteminhg the polarization dependence of relative amplitudes
and phase relationships of the matrix elements of the dipole moments between the sub-
Iwels. Al1 of this idonnation is contained in the Wgner-Eckart theorem [3].
Today the theory of gas lasers is weil advanced. One now understands the
polarkation characteristics of the saturated gain medium, not ody in its dependence on the
intensity of the saturating field, but also on the polarization state of the field. One aiso
understands the polarization competition between the saturated gain medium and the
anisotropies of the cavity, and the nature of both the stable and unstable polarization
modes [4]. The mode1 has been tested and found to be in remarkable agreement with
experirnents, without the use of ad hoc adjustable parameters.
To a large extent, the work on polarization states of gas lasers is academic, since
there is a declining commercial interest in these types of lasers. On the other haad, there is
a growing commercial interest in semiconductor lasers and diode pumped solid date
lasers. 1t is worthwhile examining the polarization behavior of these, both £tom a practical
and a theoretical point of view. If one understands and can control the polarization state
of a compact and inexpensive laser, then the potential exists to use nich a device in
cornputers and as switches in conununication systerns. Intuitively, polarization switching
shouid be fàster than intensity switching since there is no need to 'push around" the
populations of levels. In this thesis we concern ourselves with the polarization behavior of
the most cornmon solid state laser, N~~':YAG.
The experimental literature on the polarization characteristics of quasi-isotropic
N~~':YAG lasers is very limited [5 to 251. Most of these papers contain ody a passing
comment on the polarization state of the output. There are only four papers [9, 10, 12
and 251 devoted, at least in part, to a study of the parameters that control the polarization
state. These revealed that the polarization behavior was very sensitive to a number of
experimental parameters, (more than originally expected, since many are not even
rnentioned in the other 17 eariy papers), and that the spectroscopic properties of a haK
integral spin system in 4 symmetry are important but not undentood. The experimemal
and theoretical situation can best be summarized by quoting the last paragraphs in two of
the papers.
From Esherick [9] we quote:
'Udortunately, the spectroscopy of the ~ d ~ ' ion, with its three 4f electrons, is not as easily treated with group theory because the odd number of electrons requires the use of double groups. h spite of this, if one hypothesizes that the local 4 symrnetry of the YAG crystal field forces some f o m of polarization selection mie on the FTd3' ion, then projection of the laboratory axis system ont0 the six ~ d ~ + ionic lanice sites indicates that a polarization effect can be observeci, even dong a 1 11 axis. Clearly a more complete theoretical study of this effect, using appropnate crystal-field-split wavefiinctions to calculate relevant matrix elements, is in order."
The Iast paragraph in Besnard et ai. [25] reads:
'On the experimental side much rernains to be done to characterize the system better. The present observations represent only the first sep in understanding the polarkation behavior of rnicrochip solid-state lasers. On the theoretical side, while we may be able to constmct a mode1 that mimics the experimentd results, a theory with a solid base must await a complete caiculation including the distribution and symmetry of the Nd sites, nonadiabatic effects and a dynamic control parameter. This promises to be a nontrivial problem."
One would have thought that the spectroscopie properties of N~~':YAG were weU
understood, and indeed many are. The location and site symmetry of aü the ions are well
known fiom X-ray studies. The identification and theoretical calculations of all of the
hundred or so levels of the (403 configuration of ~ d ~ ' using ciystal field theory is also
well advanced. On the other han& very Little is known about the key properties of the
transition dipole moments. Levels formed purely f?om the (40' configuration have odd
Introduction 5
parïty and wodd have zero electric dipole transition moments between them. However,
as is weU understood, the odd crystal field components mLr in a portion of higher-lying
levels coming âom co&gurations with even parity. This is the physics of 'Forced'' dipole
emission in ~ d ~ + . One expects components of the dipole moments along the x, y, and z 9
axes. The key information relevant to laser action that was lacking at the beginnllig of this
work was measurement or caiculation of the p :, :, and : , polarization s e l d o n d e s
and phase relationships, as are contained (for free atoms) in the Wigner-Eckari theorem.
In 4 symmetry aU degeneracy is removed except that associated with motion
reversal. For ha-integral spin systems this means aU states are doubly degenerate. The
states are referred to as Kramers pairs. The symrnetry is described by that obscure subject
called the theory of double groups. We were not able to fhd any discussion of
polarization / phase relationships for double groups in the lîterature. In this thesis we
develop them f?om the spatial syrnrnetry @3 and motion reversal properties of the
system, without a knowledge of the double group. We were able to locate some
unpublished estimates of the three reduced matrix elements P:, p$, and p: for many of
the transitions in ~ d ~ + in YAG.
We have used the properties of the matrix elements of the dipole moments to
construct the first microscopie theory of the polarkation states of a single fiequency
N~~':YAG laser. The development is very much along the lines of the theory developed
at Toronto for the polarization properties of quasi-isotropie gas lasers. Here however we
are able to extend the theory to all orders in the laser field strength. This theory is
compared with published results on the polarization behavior of microchip and short-rod
N~~':YAG lasers, (Le., single frequency lasers). AU the eariy experimental work involved
optical pumping of the laser with linearly polarized iight. Here we report new
measurements using an ellipticaliy polarized pump field. These red ts are also compared
with the predictions of our model. F i y a new measurement of polarization fiips
induced by optical feedback is presented and discussed.
Chapter 2 - Theory
Introduction
This part of the thesis is divided into two chapters. Chapter 2 outlines a
microscopie theory of the polarization states of a single fkequency Nd:YAG laser. It
begins with an outline of the crystal structure of YAG, and the known spectroscopy of
~ d ~ ' in the host lattice. This is foiIowed by a section containhg new material concerning
the polarization seleaion rules and phase relationships among the components of the
matrk elements of the electnc dipole moments. AU of the above is needed in developing
the subsequent (and main) part of this chapter, a theory of the polarization states of the
N~~*:YAG laser.
The final section of this chapter outlines the methods used to solve the 16 non-
linear, coupled equations that govem the populations of the levels and the output of the
laser. Chapter 3 is concerned with showing that the theory is capable of giWig a
reasonable explmation of experimental results that can be found in the existing literature,
on the polarization states of single fkequency N~.":YAG lasers.
Gamet Structure
Figure 2.1 shows the formula unit of the Nd:YAG (Y&O i2) crystal. Although
33 atoms are shown in figure 2.1, the formula unit contains only 20 atoms since those on
the corner positions should be counted as 1/8 whüe those on the faces should be counted
as 1/2 due to the sharing with other formula units.
octal dumlnum sttes
yttdum or neodymlurn sftes
oxygen sites
fdral aluminurn des
Figure 2.1- Diagram showing the locations of the constituent ions in Y3AlsOi2. The sides of the unit shown are p d e l to the sides of the unit ceil.
The unit ceii of gamet, shown in figure 2.2, is made up of eight of these formula
units grouped together in four distinct orientations. Figure 2.1 showed the formula unit
labeled 1, while units 2, 3 and 4 are those obtained through rotations of unit 1 by 180
degrees about the y, z, and x axes respectively. The unit ceU has cubic symmetry, and
contains 160 atoms, 24 of which are yttrium. The latter can be substituted by neodymium.
Figure 2.2- Diagram showing the orientation of the eight subunits of figure 2.1 in a unit ceii of the cubic YAG lanice.
The active ion in the laser is triply ionized neodymium which replaces yttrium in
the YAG crystal. Yttnum (neodymiw) in the crystal has a coordination number of8, and
is located at the center of a dodecahedron f o n d by the oxygen atoms around it. Figure
2.3 shows the positions of the nearest neighbor oxygen atoms around a possible
neodymium site (the oxygen sites are numbered to uidicate how the two halves job
together). It can be difncult to visualize this orientation of the atoms. Oxygen sites 1 to 4
are CO-planai, fonning a rectangle, 5 and 6 are on a lhe parailel to and above the rectangle
(figure on the left), and 7 and 8 are on another line parallel to and below the rectangle
(figure on the nght). AU are equidistant from the ~ d ~ ' site at the center. There are three
orthogonal two-fold symmetry axes. The 'long" two-fold axis is pardel to the long side
(1-3) of the reztangle formed by the sites 1-2-34 The 'Intexmediate" two-fold axis is
pardel to the side (1-2). The 'Short" two-fold axis is perpendicular to the page in figure
2.3,
3
4 Figure 2.3- Diagram showing the dodecahedron formed by the eight oxygen ions tbat are the nearest
neïghbors of a ~ d * ion in the hast YAG Iattice, and the 3 two-fold symmetry axes of the site.
B is clear fiom figure 2.3 that the neodymium site has three two-fold symmetry
axes, but no center of inversion and thus the site symmetq is Di. However, there are 6
distinct ways that the dodecahedron cm be oriented in the gamet crystal. These six sites
are arranged such that overail there is cubic symmetry (see figure 2.4). The fact that the
c r y d as a whole has cubic syrnmetry, while the individual sites have only Dt syrnmetry
has important implications for the use of Nd:YAG as a laser material. The main point of
this seaion is that the geornetry of the Nd:YAG crystai is complicated, but known [26, 27
and 281.
Figure 2.4- Diagram showing the orientation of the "short" symmeay axis of the dodecahedron of the subunit ceil. The "long" and "intermediate" axes Lie dong the fàœ diagonais in one of the two posm'bIe
directions; a x rotation about the short axk carries one into the other.
Spectroscopy of ~ d * in YAG
The energy levels of FJd3' in YAG are dso weli known [29 and 301. ~ d ~ ' has
three 4f valence electrons outside a xenon shell. All of the energy levels involved in the
laser operation are foxmed fkom this configuration and have odd parity. The next nearest
even-parity states are formed f?om the 4P5d and 4f6s configurations. They lie some
50 000 m-1 above the levels formed by the 4f electrons.
In the fiee ion, the 17 multiplets of the 4? configuration form a total of 41 energy
levels. Laser action involves transitions between members of die 4~ and 4~ multiplets. The
lower laser level, coming as it does f?om the 4~1m ion state, is one level of six, labeled
accordmg to their energies as Yi to Ys. The upper ievel, f?om the %= ion state, is one of
two labeled 4 and IL2. The amal laser transition is between Rz and Yj, producing
radiation at 1.0641 pm. Another transition near this fiequency occurs between RI and Y2
at 1.0645 ~III In the crystal field, d degeneracy except the two-fold Kramers degeneracy
is removed (see figure 2.5).
laser transition 1.0641 p
/'6
relevant free ion Crystal field levels rnumpleis
Figure 2.5- Diagram showing a number of the lower lying multiplets of the fkee ion and the crysml field Ieveis relevant to the 1.0641 pm iaser lasntion of ~ d * in YAG.
Kraxners' degeneracy is the remit of invariance under motion reversal, and occurs
only for systems with an odd number of eiectrons. In the Hamiltonian there are kinetic
energy ternis and spin-orbit interactions ( s -p ) which are invariant under the
aansformation s t s , and p+p. To break the invariance under the motion reversai
operation, a magnetic field perturbation is required. The crystal field interaction is purely
electrostatic in nature, thus ail leveis are two fold degenerate for ~ d ~ ' ions in sites of D2
Selection rules for IYd* in YAG
In a system with an odd number of eIectrons, the angular momentum will take on
half-integer values. Motion reversal will transfom a state with angular momentum - d 2
to d 2 . Thus one member of a Krarners pair will involve a mixture of fiee ion States with
Jz= ...5/2, 112, -312, -7/2,... and these will trawform under the motion reversal operator K,
to the orthogonal member of the Kramers pair formed fiom a mixture of f?ee ion states
with Jz= -712, 3/2, - 1/2, -5/2,... We ditrarily label these by 1 and 2. Tt follows therefore
fiom the polarization selection d e s for dipole transitions in kee atoms that dipole
transitions between levels of opposite members of Krarners pairs, i.e., states with mixtures
of different Jz components, are a transitions, whiie transitions between lwels formed nom
mixtures of States with the same I, components are x transitions. This d o w s us to
constnict figure 2.6 showing the polarization seleciion d e s for dipole transitions between
Kramers pairs. One c m visualùe a state as consisting of one of the two components of a
pseudo spin 1/2 system
1 a,> Kla,>=lw Figure 2.6- Figure showing the polanzation seLeciion niles for dipoIe transitions between Kramers p a h
of States.
If the R and Y levels were pure odd panty States, no electric dipole transitions
would be allowed between them. The transitions are made possible by the rnixing in of
components of the higher lying even parity configurations, by the odd components of the
crystai field interaction. This process is known as a forced elearic dipole transition.
Màtrk elements of the dipole moment are formeci between the large odd panty part of one
state, and the small component of even parity in the other state.
While the principle of forced transitions is weli known, there has in fact been linle
numericd work done for Bld3'. We were able to locate unpubiished theoretical
cdculations of the transition moments (p)2, &)', and (p)2 [3 11. There are no rneasured
vaiues, nor were there any published phase relations for transitions between the degenerate
levels. Such information is central to understanding the polarization behavior of the gain
medium. For fkee ions this information is contained in the Wigner-Eckart theorem. We
need to h o w what replaces the Wigner-Eckart theorem for ions in sites of D2 symmetry.
We develop the needed reiationships f?om the symmetry of the crystd field, and the
properties of the motion reversai operator. To the best of our knowledge, this is the first
derivation of phase relations based solely on symmetry arguments. We develop the phase
reiationships without the use of the double group.
Derivation of phase reiationships
The motion reversai operator, K, transforms the states 1 to the other member of
the Rramers pair 2, i.e. :
1%) = (2- 1)
K, being a unitaryy antilinear operator bas the foilowing general propdes [32]:
Kc = c g K (2-5)
and specifïc to K we have for a system with an odd number ofelectrons:
The K operator commutes with the âipole operator er, Le. it commutes with the
components of the dipole moment, A in Cartesian f o m In a circdar basis however,
equation 2.5 shows that
Appl-g these results to transitions between aates of the same member of the Krarners
pair, we have:
Since transitions between similar Kramen states (Il>+ 1> or 12>+(2>) involve x
transitions, the only non-zero component of the dipole moment in this case will be the z
component. Further, since the dipole moment is a Hermitian operator, equation 2.8 can be
reduced to:
(b* IPzla*) = (a1 IPzlbl) (2-9)
Transitions between dissimilar Kramers States (1 P+(D or 12>+11>) invoive only
the x and y components of the dipole moments. For these cases we have:
The important results f?om this development are:
(b2 I lda*) = (a, l ~ z l b l ) -- (2.1 1)
(b2 I ~ x l a t ) = -(a, 1p.l bi) (2.12)
(b2 IP&J = -(a, IPJ h) (2.13)
For a fiee ion, a single reduced matrix element is required. Not unexpectedly, in
& sites three quantities are required. However, we stiu Iack the phase relationships
equivalent to those contained in the coupling coefficients for the fiee ion dipole
transitions. We can deduce the phase relationships by considering the linear response to
an applied field D, oscillating at the transition frequency ad. It is easy to show (and it wiii
be shown later) that the dipole density induced by a driving field cm be written as:
where DqyJ represent the components of the applied field in the coordinate h e of an
individual ~ d ~ ' site, ih represents a Lorentzian line shape, and p, is the population of the
ground state. Our argument is based on the fàct that a cubic crystal is opticaiiy isotropie
in linear spectroscopy. For linear spectroscopy, p, in equation 2.14 cannot be field O dependent, Le. it m u t be the field fiee or thermal equilibrium value p, . The denvation of
equation 2.14 also depends upon the assumption that the initial states of a m e r s pair
are e q d y populated pih~=p-. R d the pseudo halfhtegral spin picture of the
Kramers pair. If paIa1#p-, the ion would be oriented, a property which is inconsistent
with D2 symmetry. Thus the ground states are equaüy occupied.
Accepting equation 2.14, we can now proceed to deduce the phase relationships.
Fkst we show that the last two lines of equation 2.14 (those involving a mixed product of
dipole components) must be zero by symmetry. Being an optically isotropic crystd, P must be invariant under a rotation of the crystal by x radians about any axis. If the ion
were rotated by x radians about a & site x axis for example, the wave fùnctions do not
change, but the dipole components would transform as x+q y+, z +z. Since the last
two hes are the only ones to involve a rnix of the components of the dipole moment, they
are the only terms to change sign, and must vanish if Ï? is to remain invariant.
Using equations 2.12 and 2.13 it is easy to show that
2.14 are sums of cornplex conjugates. Therefore, for the ternis to vanish, the products
must be pure imaginary numbers. This is enough to deduce the relative phase between
the x and y components of the dipole moments in Our system. We must have:
where p, and p, are real constants, 4 is an arbitrary phase and n is an arbitrary integer. The
choice of making the integer n odd or even determines which of the degenerate m e r s
levels is arbitrarily labeled as 1 or 2. Using the relationships 2.11 to 2.13, we cm mite the
phase relationships for ali of the relevant dipole transition moments between Kramers pairs
as :
Theory of the polarization states of a N~*:YAG laser
We are now in a position to develop a theory of a single eequency N~~':YAG
laser. It is apparent fiom the description of the crystal geome- ion energy Ievels and
dipole transition moments that this is a complicated system. A complete theory addresshg
every aspect of the operation of a microchip laser would be extremely difncult, if not
impossible. Our goal is to tailor a theory to explain the polarization characteristics of the
steady state operation and dynamicd behavior of a microchip Nd:YAG laser.
Our approach to modeling this laser system is a vectorial extension of Lamb's
scalar theory where the fields are treated classically, and the gain materiai is treated
quantum mechanically. It closely paraüels the veaoriai treatment of gas lasers [4, 33 and
341. For each of the six Nd sites, we calculate the response to two opticai fields: the
pump field and the laser field. By summing over the sites, we then determine the
material's dipole density P at the laser fiequency. From the dipole moment density, we
caldate the laser field fkom MaxweU's equations and demand a result consistent with the
imagineci starting field.
Figure 2.7 shows the four level system. The pump levels are labeled a and b, with
a being the ground state, and the laser levels are labeled c and d. The subsctipts 1 or 2
indicate, as above, the separate members of a pair of degenerate Ievels.
The pump field plays a passive role, so the nrst step is to establish the steady state
pump rate to the upper pump lwel and relate this to the pumping rate to the upper laser
Figure 2.7 - A figure illustrating the four lwel laser system. P indicates the pump transition, L the Iaser
transition, and R indicates crystai relaxation processes.
Pump rate
The tirne development of the system is detemined using density matrix formalism.
The starting equation is:
where p refers to the ~ d ~ ' ions, and r represents relaxation processes associated with the
bath of other ions in the crystal. Looking at individual matruc: elements we have:
where H = HO - F-Ë is the Hamiltonian, is the Hamiltonian of the
of extemal optical fields, Ë, at this stage, is the pump field, and r is a
ion in the absence
p henomenologid
reiaxation ma& nie diagonal elements of the density ma& represent populations,
whiie the o f f ' o n a l elements represent coherences between the different possiile
eigenstates. We consider the pump field to resonate only with the pump transitions
( a 4 ) . With this in mind, we have for the population of the upper pump level:
Here we have omitted a subscript 1 on the relaxation rate y, anticipahg that it is the same
for both members of the Kramers pair. Note that the population terms on the right hand
side of the equation cancel, and we have the standard result, Le., the rate of change of
populations arises fiom fields dnving optical coherences. It is convenient at this tirne to
introduce a new symbol W to represent matrix elements (lp -E/) and to suppress A h m
now on by absorbing it into the W; and y.
the upper pump Ievels can now be m e n :
The dynamic equations for the populations of
+ ~ b i i i K l b 1 + ~ b 1 a 2 K 2 b l - i~ b ~ b l b l (2.25)
For the optical coherences we have
From the form of these equations, it is apparent that the optical coherences wilI oscillate at
the pump eequency. FoUowing what is now a standard procedure, we write these t e m
as a slowly varying amplitude and an oscillating part:
and using equatim 2.3 0, we have:
or:
For the pump field we will wrïte Ê = Dexp(-ia ,t) t cc. The time dependence is
removed nom W = (lp E]) by d e m g a new symbol W = p + D , where D is the
amplitude of the pump field, including its polarization. Making use of the rotating wave
approximation to drop the tems oscillating with a fiequency of 2a, we are left with:
For ~ d ~ ' in YAG, it is known that the relaxation rate &om the upper pump level to
the upper laser level is vexy fast (<OS ns [35]). Thus we can assume negligible population
b d d up in the upper pump levels (pblbl=@2,2=û). It is a h known that N~~':YAG lasers
cm be pumped to very high powers. This implies that the ground state is rapidly
replenished. Consequently, we can take P.l.l=pm=p,o, the thermal population of the
ground state. With constant populations, the amphdes of the optical coherences become
time independent (@& = 0 ) and equation 2.35 reduces to:
O 'dl32 = ' h W ~ b 2 ~ u
where the cornplex LorentPan h e shape is given by:
and &, = -&.
We can now insert the steady state coherences into the equations for the pumphg
rates to the upper pump level. We find:
and similarly
where the Lorentrian line shape L is given by
Recalling that transitions w i t h a Krarners set are x transitions while those between
conjugate sets are o transitions, the population equations can be written as:
b h l D Z ( P Z ) ~ ~ ~ ~ ~ ; + [ ( p x ) b h l ~ x +('Y)bL2'Y] - Y b P b l b l
+ (pY)a2blD;] (2.44)
In steady state, the pumping rate to the upper level (the tems in the curly
brackets) is equal to the decay rate. Because of the fast crystal relaxation nom states Ib>
to the upper laser level Id>, we take the pumping rate to id> as proportional to the steady
state pumping rate to the upper pump level.
Using the phase relationships developed in equations 2.17 to 2.21 we can write the purnp
rates in terms of the magnitudes of the dipole moments.
For Iinearly polarized pump light, there is no phase clifference between the x and y
components of the fieid so the last term on the right hand side of the equation vanishes. In
this case, Ri is equal to Rz. For elliptically polarized pump fields, the two rates are not
equaL Physically the purnp field is favoring one Ebmers level over the other, Le. it is
orienthg the excited ion. Thus equations 2.48 and 2.49 contain those aspects of
polarization spectroscopy relevant to optical pumping ofNd:YAG lasers. Of course, at
this stage we are still restricting the presentation to a single ion site (the field wmponents
are in ternis of the local site axes, not the crystal or labontory axes). In the foiiowing, we
assume that equations 2.48 and 2.49 aiso determine the pumping rates to the upper laser
lwels [dl> and Id+, i.e. we assume any orientation cre;rted by the pumping with elliptidy
polarized light is preserved when the ion relaxes f?om states Ib to Id>. Later, we will
dow for destruction of this orientation by including population relaxation within the
Kramers pair of the upper laser level.
Laser transition
For the laser transition, we now apply equation 2.22 to levels Id> and Ic>. For the
populations of Idi> we indude the relaxation rate y , out of the levei, the relaxation rate y*
of the population Merence between opposite members of a Kramers pair and the pump
rate RL into the level. The dynamic equation for pdldl becomes:
before, the time dependence will be removed fiom X by defïning X = (&-q) where
- F = Ëexp(-io ,t) + cc . The optical coherences will again be written as a slowly varying
amplitude and a term oscillating at the laser fiequency. Using the rotating wave
approximation, as was done for the pump transition, we find that the rates of change of
the populations of the laser levels are given by:
Proceedurg in a simila. rnanner, we find the equations for the optical coherences are given
by:
At this point, it is possible to use some weU known properties of Nd:YAG to
simplify the system of equations for the laser levels. The optical coherences of the laser
transition are known to relax on a time scale on the order of picoseconds (the low signal
gain bandwidth is on the order of 4 cm-'=120 GHz). The lifetime of the lower laser level
(c), is on the order of 30 nanoseconds. In the earlier experiments [25], we observed
polarization dynamics on the scale of rnicroseconds, so the coherences, and the population
of the lower laser level can be adiabatically eliminated fkom the system of equations.
Mathematically the adiabatic approximation means setting the time denvatives on the left
hand side of the relevant equation equai to zero. This does not mean however that the
variable being eiiminated nom the equations is constant. The same variable appears on the
nght hand side of the equation, and its tirne dependence is then determined by the other
quantities in the equation One says the variable being eliminated (the fast one) is 'Slaved"
to the others (the slow ones). We cannot make the same approximation for the upper
laser levei, whose Iifetime is on the order of 1/4 millisecond.
W e can make M e r approximations. Since Nd:YAG lasers can be made to
operate at hi& power levels, it is reasonable to assume that population build up in the
lower laser levei is not an important process. We thus assume zero population for the
lower Iaser level.
Opticd pumping by the pump field does not create coherences between the laser
levels. The laser field creates optical coherences between the states Id> and Ic> diiectly.
Howwer, any coherence between a Kramers pair of states for the laser levels builds up
nom the repumping of the upper laser level f?om the lower laser level. Given the short
lifetime of the lower laser levei, and the shorter Kramen coherence time of 5 r),
we will assume that the Kramers coherences can dso be neglected, and wiu be set to zero.
Wlth the approximations made above, we can determine the amplitudes of the
optical coherences fiom
Ttius we have
'dkl = - idcXdlcl~dldl
= -2 dc ' X dlc2Pdldl
'd2d = -idcXd2cl~d2d2
p d 2 ~ 2 = - b X d z c 2 ~ d 2 d z
where h, the complex Lorentzian line shape is given by:
Substituthg these into the equations for the populations @es
Definhg the Loren* line shape, Law through
the dynamic equation for p d i d i becomes:
- i ~ d ~ d l d l - i~ ic (pdtdl - ~ c U d 2 ) +
Findy, applying the electric dipole relationships developed earlier, yields the site specific
dynamic equations for the populations of the upper laser level.
The ternis in square brackets represent orientations of the ion, and vanish when the
laser field is hearly polarized. The next step in the calculations is to find the material
dipole density. The dipole den* P = ~ r ( @ ) is found by substituthg the optical
coherences Eorn equations 2.61 to 2.64, and ushg the phase reiationships 2.17 to 2.21.
The r d t for the amphde of the part of P that varies as eUDt is:
We have ahos t compieted our mode1 for the laser material. The equations so fiir
developed are specinc to each site of the active ion, and expressed in coordinates of the
symmefxy ofthe site. To reflect this, we add a superscript i to ail site specific variables,
and write the site specinc coordinates axes as q, r, and s (note that these are determined by
the geometxy of the sites). It is also convenient to write the populations as a sum and
difference of the populations of the Krarners pair. The gain matenal equations for a single
site are thus M e n :
Trruisformation to Iaboratory fmme
To be ofany use, these equations need to be transformed to a comrnon, laboratory
based coordinate system Most Nd:YAG q s t a l s are cut nich that the laser axis is dong
the (1.1.1) direction of the unit celi. We chose the z axis of the laboratory fhme to Iie
h g this direction There is no restriction however on the x and y axes. Defining the
rotation f?om the unit ceil hime to the laboratory m e in terms of Euler angles [3q,
requires a rotation of 4 5 " about the z axis, a second rotation of -54.735651' about the
resulting x axis, and the final rotation about the resulting z axis is determined by the crystal
angle &. The rotation matruc fkom the unit cell fiame to the laboratory fhne is thus:
For longitudii pumping, the pump and laser fields will have components only
dong the x and y directions. For the pump field, we have:
o; = $ .D, D" q = [i' - (D,~+D~~)I$ -(D;? +D;?)] (2.78)
where the x, y, and z coordinates are now in the common b e . The pump rate can be
written:
site 5 ([O, 1, O], CI/&, O, Jfi], [VA, O, - 1/&]} (2.89)
The constants üsted in equations 2.81 to 2.84 can be detefmined by applying the
rotation T to îhe coordinates of the individuai sites listecl above. With the constants
dehed, the pump rates can be written as:
RI = A c': D: +c': D: +CI; (D.DL + D,D:) - ici (D,D: - D,D;)] (2.91) [ = A[c'; D: +c'; D: +CI; (D,D; +D,D:)+~C': (D,D; -D,D;)] (2.92)
Writing the pump field with explicit real and imginary parts
The equations for the popdations of the upper laser level wiU involve the same
constants, with the dipole moments from the laser transition instead of the pump
transition We imply this by removing the prime fiom the constants Ci4 and the field
variables. The resulting dynamic equations for the populations are
and
In tmosfonning the material dipole moment derisity into the laboratov fiame, there
wiii be terrns in each of the x, y and z directions, even though the field components were
only dong x and y. In spite of the crystal havhg cubic synunetry, we h d in summing the
polarization of the sites that the crystal becomes biaxiai. ûwing to the lower symmetry of
the Nd sites pz) , each site has unequal dipole moments in the three directions. A linearly
polarized purnp beam wiIi not popdate the six sites equaiiy, and so, the polarizability made
up of the sum of each site is no longer isotropie. For a low signai gain on the order of 1%
however, we found that the bireningence induced is one or two orders smaüer than the
typical strain birefigence acpected in the microchip. For this reason, the propagation
problem brought about by the induced anisotropy of the crystal is ignored. We will treat
the laser field as a plane wave propagating in the z direction, and simply ignore the
polarization terms induced in the z direction. The result of the transfomation to
iaboratoiy coordinates becomes:
We rernind the reader that the constants Ci' to C l contain the geometry of the
orientation of each site relative to the common laboratory name, in addition to the
magnitudes of the relevant dipole moments.
Fidd equations
The finai step in developing the mathematid mode1 is to derive a dynamical
equation for the complex amplitude of the laser field. MaxweWs equations in ME23 units
are:
We wÏli assume a form of the electric field where:
This is an ùinnite plane wave, with E. and E, representing slowly varyùig complex
amplitudes. What we have concealed fiom the reader is that a complete treatment of the
dipole density P yields a result in the same fom as 2.103 provided we neglect the spatial
harmonic aspect of hole bumùig. This is a common approximation of no consequence for
singie fiequency lasers. As it is wrinen, the field is a sum of a forward and a backward
traveling wave, and the total field is zero at the origin and at the end of the cavity if we
nn: demand that k = - (where n is the order of the longitudinal mode) as required nom the
L
boundary conditions of the laser. For our infinite plane wave, we can reduce Maxwell's
equations to
Takhg the curl of equation (2.105) we get
Finally, inserthg equations (2.104) and (2.107) we arrive at the wave equation
Looking at the x component oniy, the derivatives of the electric field are given by:
= -(e.p[-i(at - kz)] -e*p[-ifot + b)) + &E,(exp[-i(or - hl]- exp(-i(wt + la)^ (2.1 10) a2
- ($1 = ,(..p[-i(at - i-11-eq[-i(at +b@ -2io %(exp[-i(a -ia)] --[-(rot + g ~ (2.113) x
-a'~~(exp[-i(at - k)] -exp[-i(ut +)a)])
Inserthg these into the wave equation, (with similar redts for the polarization) gives:
exp[-i(ot - kz)] - exp[-i(o t + kz)])
Dropping the second-order derivatives, the common exponential terms, the spatial
derivative af the field amplitude, and the derivative in the polarization, we are lefi with:
Rearrangïng the rernaining t e m and absorbing a &or of EO into the dipole moment
density, we have:
with an quivalent equation for the y component of the field. n i e two equations may be
combined into a single matrix equation
where E and P are 1 by 2 column matrices of complex amplitudes, Le.
So fàr we have worked out how the dipole density of the gain materid drives the
laser fieid. In a mean fieId mode1 the localized losses or changes in the field due to mirrors
etc., are distributed throughout the cavity. To fix the cavity contribution to the rate of
change of the laser field, we caiculate the round trip change using Jones matrices and
divide by the round trip tirne C/2L. Writing the Jones matrices for the round trip of the
where U is the unit 2 by 2 matrix. Inserting the material dipole moment densities into this
equation laves:
where we have writîen the complex Lorentzian & as
We are interested in solving diEerential equations for the relative amphdes and
relative phases of the field. We &te the slowly varying comptex amplitudes of the fields
in terms of real amplitudes and phases.
E x = ex arp[i(4 + 40/2)] (2.120)
Er = &, ex$(+ - 60/2)] (2.121)
We substitute these relationships into 2.118, then we separate the matrut equation into its
two component equations and f indy separate each of these into its real and im3ginary
parts. Mer some tedious algebra, the finai form o f the equations in the field variables are:
a&, - = (ar - l)&, t &[bey exp[-i4,]] ôt
;!(ci f p i + . ]] + pic: ~-e[2ev[-i4~ l]]
The last steps toward writing the equations in a form suitable for computation, is
to rescde the populations and the fields, and to define some new constants. The faaor
O wiU be absorbed into the intensities, and will be absorbed into the
ydh0
populations. We wiU also recast the Lorentzian line shape, and the operating eequency as
1 = Y 3 0 - , *
old and Aanm =- . Y cd
It is also convenient to make the fdowing definitions:
9, = ~e(be-*) @, = ~ e ( c e ' * ~ ) a3 = Im(be-'%) a, = h(ceih )
Noting the fact that:
meory 37
the fina equations goverring the dynamics of the popiilations and the laser field variables
are:
Equations 2.126 to 2.13 1 together with the definitions of the constants CI u, 4 in
equations 2.81 to 2.84, represent the 16 coupled differential equations for our mode1 of
the polarkation States of a single fiequency Nd:YAG laser. There are sixteen equations
because there is one equation for p. and pd for each of the six quivalent sites of Nd in
YAG.
The variables involved in those equations, (Cx, Cy, ,,O), are not quantities one can
measure directly in an experiment. It is convenient to relate these to more easily
recognized quantities, such as the intensity (I) of the laser field, the ellipticity (11) of the
polarization, and the angle (a) of the major axis of the polarization ellipse (see figure 2.8).
minor axis
major axk
JI Figure 2.8- Locus of the tip of the reai electric field vector (polarkation ellipse).
The intensity is simply the sum of the squares of the laser field in the x and y
directions, i. e.
2 2 I=E,+E,. (2.132)
The efipticity of the polarization is reiated to amplitudes and relative phase by:
whüe the angle of the major axis is given by:
The reader will certainly appreciate the staternent that the 16 coupled non-linear
equations are indeed compiicated. In the last section of this chapter, we discuss the
numerical methods used to detennine their steady state solutions and the stability of the
solutions.
Methods used to find the stationary solutions
The mathematicai complexity of the problem appears to d e out the possibility of
finding aaalytic solutions to these equations. The most straightforward numencd rnethod
of finding the steady state solutions is to integrate the variables in tirne. Starting fiom
some arbitrary initiai conditions, the variables are aiiowed to evolve, ushg the tirne
derivatives multipiied by the cavity round trip t h e as the changes in the variables at each
integration step. The cavity round trip time (on the order of IO*" seconds), is short
enough compared to the time scales relevant to the polarization dynarnics of the problem,
that there is no ditncuity in using this as the integration step sue. The process continues
untii the changes in al l of the variables per iteration become smailer than some ahitrary
value. The variables wiu thus nahirally tend towards any stable stationary solution. There
are several points that need to be addressed when using such a procedure.
Equation 2.131, for the rate of change of the cormnon phase, is completeiy
determined by the other variabks in the problem, and the cornmon phase does not appear
in aay of the other equations. Stationary solutions to the equations can thus be fouad by
imegrating the remaining 15 equations, assuming that the operating frequency of the iaser
is Iaiown. The last equaîion can then be used to determine the appropriate length of the
laser cavity corresponding to this fiequency.
Since the atomic decay time of the upper laser level is much slower than the cavity
decay tirne, the system WU undergo spiking, creating large fluctuations in the populations
and the fields as the solutions are intepteci. To avoid this numericd problem, and to
speed up the temporal evolution towards a solution, the populations can be effeaively
removed nom the integration If the field variables are assumed to be known, equations
2.126 and 2.127 for the population sums and merences can be written in the form,
6; = + bipd +ci and = dipi + bipf + ei (2.13 5 )
where the a, b, c, d are fùnctions of the field variables. In the aeady state these then
be solved analytically yielding
These can be substituted into the three field equations and the t h e integration carrîed out.
A weakness of this method of solution is that it will not h d more than one
solution fiom a given çtarting point for the integration. To guarantee that aIi possible .
solutions have been found, it would be necessary to pedionn the integration fkom all
possible initiai conditions. Furthemore we divided by the field amplitudes at one step in
the derivation of the h a 1 equation. Consequently when the electric field dong one of the
axes is zero, the integration fails. To avoid this problem, the arbitrary coordinate axes
were rotated whenever the solutions approached a linear polarization dong one of the
origina axes.
The solutions found by this method will be statiomy, though not necessarily stable
since the dynamics of the populations were removed fiom the caldation A fidl linear
stability analysis was d e d out. The stabiiity ofthe staîionary solutions to the equations
is determineci by examining the wolution of s m d perturbations of the variables. Ail of the
dx- eqyations are of the fonn 2 = fi (x,, x,, .. .) , and c m be expandeci in a Taylor series at about a stationary solution xi, xi,. . . xi,) . We have: {
Keeping only the first terms fiom the Taylor series, the resdt is a hear system of
equations in the perturbations of the variables.
Equations 2.138 can be written in matrix form and solved as an eigenvalue problem The
eigenvecîoa of the system represent the new variables v., which wiil diagonalize the
av ma& In this form, the derivatives of the perturbations can be d e n as =
a Xmvm J
with solutions va = c, exp(h,t) . The real part of the eigemralues will determine if the
perturbations will grow or be damped in time. If the real parts of al1 of the eigenvdues are
negative, the solution must be stable. The imaginary parts of the eigenvalues represent
relaxation oscillation kequencies.
The reai parts of the eigenvalues indicate how quickly the fields will relax to a
solution (for negative values), or how quickly perturbations wiU grow (positive values).
Near an instability, when A.', = O , the solutions are slow to converge. and numerid
integration slows to a point where it is no longer feasible to use this method. This
situation arises when polarized feedback is included in the system; stationary solutions to
the equations musî then be found algebraicdy. With a l l of the derivatives set quai to
zero, equations 2.126 to 2.13 1 form a system with 16 variables. The populations can be
h t t e n once again in ternis of the laser field, reducing the system to three non-linear
equations, and a fourth equation providing a constraint on the iength of the cavity. The
solutions can be found by a three dimensional bisection method.
Algetjrajc solution
The bisection method works by horning in on a bracketed solution. At least one
solution must exist inside an interval over which a hc t ion changes sign. Here there are
three hctions, the denvatives of the two field components and the relative phase between
them. Evaluaîing the fùnction at the midpoint of an interval which brackets a solution, and
replacing the one of the two end points at which the hct ion had the same sign, reduces
the interval by half at each step. This process continues until the bracketing interval is
s d e r than some arbitrary @ut s d l ) value.
This method is guaranteed to find stationary solutions, whether or not
these solutions are stable. Because the bisection is in three dimensions, it is very time
consuming to search ail of the parameter space for solutions. This method is most usefid
ifa solution is known or suspected to exist in some region. In a feedback experiment, the
solution fa f?om the flip point can be found using the tirne integration method desnibed
earlier. The solution can then be tracked through the critical point and beyond into the
region where the particular solution is unstable using the bisection method. In this
manner, exhaustive searches of panuneter space are avoided.
In writing eqpatiom 2.126 to 2.131, the operating firequency of the laser is
assumed. Once the solutions are found, equation 2.13 1 is used to determine the
correspondhg length of the cavity. For experiments hvolving polarized feedback the
solutions depend on the phase of the feedback Thus, laiowledge of the operating
fiequency is critical since a srnail change in the fkequenqdwavenumber can cause a
sigdicanî change in the phase of the feedback if the distance to the feedback element is
large. To ensure that d ofthe solutions are for the same length of laser cavity, the length
of the cavity is continuously monitored in the search for solutions, and the operating
fkequency is adjusted accordingiy.
The laser equations contain many factors, such as relaxation rates, strength of
feedback etc. These are referred to as control parameters, and the full set as parameter
space. It is well known that coupled nonlinear equations exhibit wideiy different solutions
in different regions of parameter space. Herein Iies a trap for experimentalists: if all the
control parameters are not known, one can often fïnd a region of parameter space that
mimics certain experimentai results, even if the theory is wrong. The tme test of a theory
is to make an independent measurement of the control panuneters and thus predict the
behavior of the laser. As an alternative, one can make measurements of the laser over a
wide region of parameter space and see if the fitted constants remain consistent as some
other control parameters are varied. One's trust in the model is increased if the fitted
parameters seem reasonable, based on other experience or knowledge. Finally one can use
the laser itself to measure certain parameters, provided the experiments are generic in
nature, i. e. the behavior is not model-specinc.
In spite of the fkct that YAG lasers have found extensive commercial use, there
still remah a lack of knowiedge of many of the rnicroscopic parameters. This poses a
dalnculty in testing our mode!. In the foIlowing chapter, we use a combination of the
ploys above to overcome the McuIty, and use our model to "explain" results found in the
existing literature on the polarkation behavior of Nd:YAG lasers.
Chapter 3 - A Cornparison with Results in the Literature
There have been a number of observations or cornments on the polarization states
of single mode N~~':YAG lasers [5 to 251. Only a few of these are devoted to a study of
the polarization state and its dependence on some control parameter such as the
polarization of the pump, stress on the aystal etc. [9, 10, 12 and 251. In order to compare
our theory to these experim-, there are many more parameters which need to be
specined. Table 3.1 k a list of ai i parameters requked to perform the calculations
descriied in the theov chapter. To this list must be added aii the parameters that
determine the optical feedback, if present.
Table 3.1 - Mode1 parameters required for a caladation of the polarization states of a single frrquency N~»:YAG laser.
I Symbol 1 Definition
of the population of the upper laser level
1 Decay rate of the optical coherence of the laser transition
Relaxation rate of the population ciifference between members of Kramers pairs
in the upper laser level
strength of the components of the dipole moments for the pump
transition in local site coordinates
p , p , p, Relative strength of the components of the dipole moments for the laser
transition in local site coordinates
A Cornparison wifh Results in the Literature
The operating fiequency of the laser measured f?om line center
Length of the laser
hteasity of the pump field
Isotropie loss of the laser cavity
EUipticity of the pump field
Orientation of the axis for the linear birefiingence
$P
+C
Orientation of the axis for the linear dichroisrn
Orientation of the polarkation ellipse of the pump field
Orientation of the crystal about the laser axis
Linear birefikgence of the laser cavity
Linear dichroism of the laser cavity
Very few* of these parameters have been measured or were specined in a particular
acperiment. The only way it is possible to make a meanin- cornparison between theory
and previous experiments is either to make reasonable estimates of the unlaiown
quantities, or to show that the results are insensitive to the values of the parameter. The
cornparison is weakened if we are forced to consider any of the input parameters as fke
fitting parameters. In the foliowing paragraphs we discuss the values of each of the input
parameters.
The decay rates yd, ycd, and yk are characteristics of ~ d ~ ' in YAG. The Metime of 3 1 the upper laser level is 23 O ps [37], wtiich yields yd4.3 x10 s- . At room temperature the
laser transition is dominated by homogeneous broadening and has a haif width at half 7 -1 maximum of y&= 2 cm-' = 6 x10 s . There are no measurements of yk. Fortunate1y, for
A Campafison with Results in the Literature 46
lineariy polarized pumping, the results are totally insensitive to this relaxation rate- For
convenience, in the calculations we have set yk equal to yd.
We have estimâtes of the relative dipole moments for many transitions, includhg
both the pump and laser transitions [3 11. In that work, a point-charge mode1 was used to
parametrize the coetnaents of the crystd field for ~ d ~ ' sites in YAG. The &ecfive
charges on the ions are determitleci by fitting the even components of the crystal field to
the measured energy levels of ?Jd3' in YAG. Given the eEectve charges, the odd
components of the crystal field can then be found and used to calculate the dipole
moments of the transitions.
As a check on the reliability of these estimates, we have compared the relative
Uitensiw of the fluorescence for the transitions between the R* and Yi to Ys levels
measured experirnentally 1381, with the values calcuiated using the theoretical values of
the dipole moments. These are shown in figure 3.1. In general, the agreement is w i t h a
faaor of 2. Lacking any other Uiformation, we will assume that the estimated relative
dipole moments have about the same level of uncertainty.
The fiequency o s e t fiom line center Am has never been reported in the literature.
However, maximum output occurs at line center and it is hard to conceive that Aa is large
compared to the homogeneous width of the gain m e .
The length of the laser is aot a very important parameter in the equations. As was
noted in the theory chapter, the length of the laser is comected to the nequency. On a
large scale the length detemiines the round trip time. On a srnail scde (4 pm), a
variation in the length determines the operating fkequency. The only concem we have for
the length is to make sure that solutions for different polarkations are calculated for the
same length laser.
A Cornpananson with Resufts in the Literature 47
Figure 3 -1- The relative strengîhs of fluorescence of the R2 to Y level transitions are compared to values calculated with the theoreticai values of the dipole moments. Laser action at 1.0641 pm occurs on the
strongest line shown.
The Mensity of the pump field and the loss in the laser cavity are related in our
equations, in that we use arbitraxy units for intensities. The intensity of the pump field is
written as a ratio of the pump power to the threshold pump power, and this includes the
loss in the cavity. The arbitrary units thus conceal our ignorance of the exact doping
concentration, the absoIute strength of the dipoles, and the fiaction of the pumping rate to
the upper pump level which is transferred to the upper laser level. The major source of
the loss is the transmission of the output rnirror. The refledvity at the operating
wavelength is generdy reported in the fiterature.
The pump field is heariy polarized in most of the experiments in the literaîure.
The orientation of the pump polarization is usudy given with respect to the axis of the
linear birefihgence (at least for experiments involving stressed crystals).
A Cornparison with Resuits in the Literature 48
The orientation of the crystal with respect to the Iaboratory axes is unknown in aii
of the experiments. The experiments in the literahire invoive crystals grown dong the
[l, 1,1] direction of the cubic axes (and cut pexpendicular to it), so the laser (2) axis is weli
defined, but the x and y axes are not specined.
The linear buefikgence in rnany experiments can be determineci by the b a t
fkequency between the two orthogonal polarization modes, when they operate on the same
spatial mode. Typicai beat fiequencies [7,20, 21 and 251 range fkom 1 MHz to 300
corresponding to values of&O.00002 to 0.006 radians per pass (depending of course on
the length of the laser cavity). The orientation of the axis is known if the buekgence
anses fiom stress applied to the crystal, but varies fiorn point to point on the microchip for
unstressed crystais. Yoshino [20] mea~u~ed the magnitude of the bireningence and its
spatiai variation by measuring the beat fkequency for an unstressed c r y d as a h c t i o n of
position on the crystal. It varied by some 30% over a distance of 100 pm, a distance
comparable to the diameter of the lowest order mode.
The hear dichroism in the cavity is not stated in the experiments, and we have no
fum knowtedge of its size. It is presumably smali, particulariy for the monolithic
microchip laser. The same situation exists with respect to circular dichroism and Ncular
birefhgence Paraday rotation). In the caldations descriid below we have set to zero
the linear and circular dichroism, and the Faraday rotation. Thus the Jones matrix
representing the laser cavïty is characterized by a birefiingence, and the isotropic loss.
From the discussion above, the most important experirnental parameter about
which we have no information is the orientation of the x and y crystai axes. Thus the est question we must face is whether or not the polarkation behavior of N~~':YAG lasers is
sensitive to the crystai orientation.
As a measure of this sensitivity, the output of the Iaser was calculated for the case
of a completely isotropic cavity, as a hc t ion of the crystal onentation We believe this to
be the most sensitive test possible since the output is completely d e t e d e d by the gain
materiai in an isotropic cavity. 11 was found that the laser output ('IL) is always linearly
polarued (m). On the other han4 the intensity and the orientation (h) of the lineariy
polarized output showed a weak modulation with the orientation of the crystaI. Figure 3.2
shows the intensity of the output of the laser, with respect to the orientation of the pump
field polarization (For cornenience, in the caldation we vary the orientation of the
pump rather than the crystai. In an isotropic cavity, it is ody the relative orientation of the
two that maffers). We see from figure 3.2 that the calcuiated output intensity is iosensitive
to the orientation ofthe crystal axis.
Figure 3.2- htensity of the laser output as a hct ion of +&. The control parameters are listed in table 3 -2.
A Cornpananson with Resuits in the Literature
O 90 180 270 360
@.+c ( d e m e s ) Figure 3.3- Orientation of the laser output polarization as a fiinction of the orientation of the crystai. The
control parameters are Listed in table 3.2,
Ta
Parameter
Yd
Y&
Yk
pq : Pr : ps
1 . 0
Pq:Pr:Ps
A ~ L
L
lp
le 3.2 - Parameten required for calcuiations of figure 3.2 and figure 3.3.
Value Parameter Value
A Cornparison with Results in the Literature 51
Figure 3.3 shows the orientation of the hear polarization output relative to the
orientation of the linearly polarized pump field. W e see that for the same set of control
parameters there is a smaii variation in the orientation of the output polarization of the
laser (on the order of4O with respect to the direaion ofthe linearly polarized pump field).
We conclude that the polarization behavior of the laser is not sensitive to the orientation
of the crystal. However we also see, in the case of an isotropie cavity, that the
polarkation of the laser is determhed by the pump, being (nearly) parallel to it.
The reader may have noticed that both the intensity and the orientation show a
periodicity of 60'. This cm be understood from the geometry of the crystal. We reiterate
that the rotation of the orientation of the pump field with respect to the crystal is
equivalent to a rotation of the crystal about its z axis (the [1,1, 11 axis of the unit ceil).
Since the pump and laser fields are propagating in the z direction, the output intensity and
orientation depend on the components of the dipole moments of each PJd3' site, rnapped
ont0 the x-y plane of the laser. While the 60" rotation does not leave the crystal invariant,
it does, as we shall see, maintain the orientations of the components of the dipole moments
in the x-y plane.
Figure 3 -4 shows the orientations of the symmetry axes of the ~ d ~ ' sites in YAG.
For clarity, oniy three of the six sites are shown. The omitted sites (lY,2',3') are found by
rotations of those shown by 90 degrees about their short axis (solid line). Figure 3.5
shows the same sites, but with the perspective changed so that the laser axis cornes alrnost
straight out of the page. This figure is essentiaily the projection ont0 the laser x-y plane.
It is obvious f?om the second picture (rotated 60' about the laser axis fkom the first) that
the projection is invariant under a 60° rotation about the laser axis. M e r the rotation, the
projections of the sites 1, 2, 3 are equivalent to the projections of the primed sites l', 2',
3' before the rotation. Thus we understand the periodicity displayed in figures 3.2 and
A Compananson Wth Results in the Literature
\ t \
Figure 3.4- Orientarion of the symmetry axes of three of the six ~ d * sites in YAG. The three C2 axes of each site are dinerentiated by their sizes and their fill (soiid, dashed or dotted). The other three sites are f o d îhmugh rotations of these sites by 90° about the short axk (solid he). In the figure the dashed
lines lie dong the fhce üiagonals of the cubic crysrai axes, whiie the soiid h e s are paralle1 to one of the axes.
3.3. We have verifïed that the modulation approaches zero as the ratio of the dipole
components approaches 1 : 1 : 1. Given the smaii dependence on crystd orientation as
determined by calculaiion, it is reasonable to neglect its effect on experimental resuits. We
can choose an ahitrary orientation of the crystai axes when solving our 16 coupled non-
linear equations.
A Cornparison with Results in the iiterature
Figure 3.5- Orientation of the symmetry axes of- of the six ~ d * sites in YAG looking d o m dong the laser axis. The prime coordinates are! rotated by 60° murid the laser axis fiom the unprimed cwrdiBates.
In summary, we have some a priori knowledge of aii input parameters (that may or
may not be relevant) for calculating the polarkation states of single mode N~~':YAG
lasers that are longitudinaily pumped with linearIy polarized light. In the theory we have
assumed that the cavity does not contain components that are strongly anisotropic and we
have in the following caldations taken the Faraday rotation, the linear dichroism and the
circuiar dichroism as zero. We are now in a position to make, what we hope will be,
meaningful cornparisons beîween the polarkation predicted by our mode1 and the
polarkation results found in the literature.
A Cornpananson with Results in the îiterature
Resuits in the Iiterature with no feedback
Zhou et al. [SI noted that the polarkation was "indeterminate" uniess the laser
was transversely stressed. When stressed, the output was linearly polarized parallel to the
applied stress. Esherick and Owyoung [q noted that the output is lineariy polarized, and
that in most cases this happens spontaneously. Occasionaily it was necessary to apply a
stress to cause operation in a iinearly polarized mode. In another publication [9] using a
stressed rod, they noted that the output polarization is parallel to the stress axes, and the
intensity was a bc t ion of the pump wavelength and the orientation of the linearly
polarized pump (paralle1 or perpendicular to the stress axis). Zayhowski and Mooradian
noted that the output was "randomly" oriented (presumably 60m laser to laser rather than
wth time), when not stresseci, but paralle1 to the stress axis othenvise.
One cannot avoid jumping to the conclusion that the bireningence, either residual
in the crystd, induced by the mount, or deliberately created by the application of a stress,
plays a major role in determiring the polarization behavior of the laser. Above, we
predicted for an isotropie cavity that the polarization of the output was nearly parallel to
the pump. This is not consistent with the experimentd observations. In the following
section we examine theoreticdy the role that birefiingence plays in determinhg the output
polarization
We consider the ideal case when there is no residual birefihgence in the crystal,
and there is a stress-induced birefigence dong the x or y axis. In our nomenclature, a
positive ei means n a y where the n's represent the indices of rehction for light polarized
dong the lab x and y axes. A negative E' means n&. As discussed in Owyoung [6],
stress dong y makes n a y i.e. corresponds to a positive si in Our nomenclature.
Conversely, a negative si impiies stress dong the x axis.
We consider &st two cases where the pump is iinearly polarized dong x. For
positive ei (case 1) this means the pump is perpendicular to the axis of stress. For negative
A Cornpananson with Results in the Merature 55
ci (case 2) the pump is parallel to the stress. In these cases, the predicted output remains
heariy polarized (q~=0). However the orientation with respect to the x axis, depends
upon the magnitude and sign of the birehgence. Figure 3.6 shows a plot of the
orientation versus si. For case 1, (si=) we see for large si that the orientation (m is d 2
(= 1.6 radians), Le. ünearly polarized along the y axis or parallel to the stress. For smaü E',
approaches zero, i.e. linearly polarized along x, or paralle1 to the pump. For case 2,
(&O), remains equal to zero, i.e. paralle1 to the pump and the stress axis.
Quaiîtatively, this explains many of the experimental observations. If the residual
birefigence or the stress induced bûeniagence exceeds about 0.0005 radiandpass then
the output is linearly polarized dong the axis of maximum index. This will be random in
an unstressed crystal, or along the stress axis otherwise. Quantitatively, it is reassurïng
that 0.0005 radiandpass is compatible with values estimated above fkom the mode beating
and fiequency jump experiments. Of course, if the birehgence is due to residual strain in
the crystal, there is no guarantee that the pump light is polarized parallel or perpendicular
to the axis of the residual strain. In the foiiowing paragraphs, we examine the extreme
case, when the angle between the two directions is 45'.
Figures 3.7 shows the ellipticity of the output calculated as a finction of E' for the
case of a linearly polarized pump field, aligned at 4S0 to the x axis (OP=x/4). For a
birehgence near zero, the solutions are hearly polarized (m). As the biremgence is
increased (adding a stress along the x or y axis for negative or positive Ej, the solutions
quickly become circular (qL=kl), and slowly retum to linear for large birefringence.
Linear polarizations thus ocnir when there is no competition between the pump field and
the birehgence, as we saw above when the axes of the two are aligned or, as we see
here, when one of the two dominates the other. The application of a stress to the
microchip serves to create a birefihgence in the cavity large enough to force the operation
into a linearly polarized mode. The relative strengths of the iduences of the pump field
A Cornparison with Results in the Literature
Figure 3.6 Orientation ufthe hearly poiarized output with respect to the x a i s as a fiindon of the birefiingence îi
Table 3 3 - Parameters required for caiculations
Parameter Value Parameter Value
varied I
A CornpanSon with Results in the Li'terature
Figure 3.7 Output ellipticity for the case of a linearly poiarized pump field aligned at 45 degrees to the axisofthestress
Table 3.4 - Parameters required for ca
Parameter Value
dations of figure 3.7 and figure 3.8 . Parameter Value
A CornpanSon with Results in the iiterature 58
and the birefiingence depend on the merence in the dipole moments in each direction at
the PJd3+ sites. If the differences in the dipole moments are increased, the a m e s Aden,
indicating that it takes a Iarger stress to force Iinearly polarized operation. Once again we
see that linear polarization is expected in the usual case where the residual birefhgence is
larger than 0.002 radians per pass. (It is worth noting that the ellipticity scale can be
misleadhg For 1yû.4 the ratio of the intensities dong the axes of the polarkation ellipse
is 25 to 1, Le. the light is stiu nearly linearly polarized).
Figure 3.8 shows the orientation of the output relative to the stress axis caldated
for the same situation (the stress axis switches fiom the x axis to the y suris as si inmeases
and passes through zero). Near zero birefhgence, the solutions are aligned with the
pump field (h+4S0). For large birefigence the major axis of the polarization ellipse is
-6 -5 -4 -3 -2 1 O 1 2 3 4 5 6 x 1 0 ~ d (radiandpass)
Figure 3.8 Output orientation for the case of a linearly polarized pump field aiigned at 45 degrees to the axis ofthe stress
aligned with the stress axïs.
40
n 20
U) a,
E + g -20 -40
- 4
- - - -
0 - -
- -
- * -
I I - l I I I I I i I I l I
A Cc~npari$on with Results ri, the Literature 59
We conclude from ail of this that our mode1 of a quasi-isotropic single fiequency
N~~':YAG laser gives a plausible account of the experimentally observed polarization with
and without an appüed stress. Even in the case where no stress is appiied, typical residual
strain birehgences are dcient iy large to cause the laser to operate in the high si
regions of figures 3.6 to 3.8
Another set ofobservations, made by two groups [9 and 121, concemed the output
polarization and intensity as a hc t ion of the pump fiequency and orientation. For
stressed ciystals, they investigated the dependence of the intensiv of the output on the
Eequency of the pump source, and the orientation of the iinearly polarized pump field.
Since the experiments were peiformed in the region where the stress induced birefringence
dominated the output polarization of the laser, we are not surpnsed that the output was
always linearly polarized. It was found that the intensity of the laser variecl when pumped
pardel or perpendicular to the applied stress, and this variation depended strongly on the
fkequency of the pump source. At the peak of the 809 nm absorption, (which corresponds
to almost aii of the other experiments reported), they observed approximately a 25%
increase in the intensity in the case of pumping perpendicular vs. that for pumping pardel
to the stress axk.
Figure 3.9 shows the intensity of the Iaser, calculated as a fùnction of the angle
between the linear pump polarization and the stress axis, for the case of a large
birefiingence. The solutions remained linearly polarized throughout, and aligned with the
stress axis. The predicted output intensity at 90' is 25% higher than at OO. That the
caiculated and observed ratios are in close agreement is coincidence. The variation of the
calculated intensiw with the pump orientation arises fiom the merence in the components
of the dipole moments, for both the pump transition and the laser transition. If either of
these sets were isotropie, there wouid be no variation in the output intensity of the laser
with the pump orientation Lt is easy to appreciate that the variation will therefore depend
on the tiequency of the pump field. At dinerent pump frequencies, different transitions
with Merent sets of dipole moments will determine the size of the variation. The
A Campan" with Resuits Iir the bterature
Angle of pump with respect to stress axis (degrees) Figure 3.9 Output intensity as a function of the angle ktween the pump piarization and the stress axis,
for the case d a large birefiingence
Table 3.5 - Parameters required for caiculations of figure 3.9
Value Parameter
yd
Value
4.3 ~ 1 0 ~ S-l
Parameter
I
A Cornparison with Results in the Literature 61
magnitude of the variation aiso depends on the pump power to threshold pump power
ratio. For higher pump powerj, the variation is less pronounceci.
Results in the literature with feedback
In 1993, Besnard et ai. [25] reported a number of polarization observations where
polarized optical feedback was used to cause switching between two orthogonal linearly
polarized modes. They used a microchip Nd:YAG laser longitudindy pumped at 590 nrn
with a iinearly polarized beam. The eee-ninnllrg laser was 98% linearly polarhed, and the
output was independent of the orientation of the heady polarized pump b e a a Clearly
this corresponds to a large residual bireaingence. Polarized feedback was applied pardel
or perpendicuiar to the linearly polarized output of the Eee-nuuiuig laser. The experiment
consisted of modulating the distance to the feedback rnirror, and obseMng the output
either p d e l or perpendicular to the direction of the polarized feedback The main
r e d t s fiom the experiment were: (i) the laser switched between operation in two
orthogonal linearly polarized modes, Ci) there were polarkation oscillations accompanyhg
the switchiq, (fi) no hysteresis was observed between the forward and backward scans of
the feedback mirror, (iv) the period of the oscillations was always about 15 to 20 p,
independent of the pump power, and findy, (v) no polarkation switching was observed
for reduced feedback
Two auxiliary results were also reported by Besnard et al. The nrst was that the
fkequency of the mode jumped an amount between 1 and 100 MHz when the laser
changed its polarization state, with a typical value being close to 100 MHz. This ailows us
to estimate the residual birefkhgence as &2x10-~ radiandround trip. The second
obsewation was that the intensity in the mode pardel to the direction of the polarized
feedback varied some 15% as the phase of the feedback was varied, and the laser was
operated 25% above threshold. Recaii that polarized feedback aligned perpendicular to
the polarization of the fkee-nuuiing laser can be modeled as a change in the reflectivity of
the output mirmr. When the light retuming to the laser cavity is in phase or M O 0 out of
A Cornpanson with Results in the Lirerature 62
phase, the output mirror acts as ifit has a Iower or higher loss in one direction. When the
iight has a phase shift of 90°, the output mirror appears to be buefiingent. Using these
ideas and the usual third order scalar calculation of the output intensity, Besnard et al.
eshateci the sîrength of the feedback as Er= 1 . 1 ~ 1 0 ~ . Consequentiy we can take the
feedback-induced effective anisotropy of the cavQ as E&o*~ + i sin&) where & is the
phase of the feedback This must be added to (or subtracted nom) the residual
birefiingence of the cavity E'.
Of ali the early experiments, that of Besnard et al. is unique in that we are able to
speci@ or give rûasonable estimates of all of the input parameters to which the model is
sensitive. Consequentiy, the observations reported were expected to provide a &cal test
of the model,
With or without feedback, our mode1 predids a linearly polarized mode, either
pardel or perpendicular to the polarization axes of the feedbaclq for any value of the
phase of the fedback What Merentiates the solutions is the stability of the two modes.
Recd fiom the discussion of a linear stability analysis that one need ody examine the
largest real part of the stability exponents hr for any stationary solution to determine if the
mode is stable or unstable. Figure 3.10 shows hr- for both polarization modes as a
fiindon of the phase of the feedback for two cases: EFO (no feedback) and ~ f - 1 . 1 ~ 1 0 ~ .
We see that one mode is always stable (Ar-<O), and the other is dways unstable
( 0 ) This is in conûict with the experimentai results.
A CornpanSon with Results in the Literature
Phase of feedback (radians)
Figure 3-10 Largest real part of smbility exponents. Solid ines, stability parameters (3Lr-) for the two orthogonal modes as a fiinction of the phase of the feedback The stability parameter in the absence of
f h k is shown as a dotted h e - The control patametexs are listed in table 3.6, below.
Table 3.6 - Parameters required for calcu 1 I I
Parameter Value Parameter 1
Value
A Cornparison with Results in the Merature
The question arises as to whether the theory is wrong or incomplete, or that some
reasonable adjustment of one or more of the input parameters wili produce a si&cant
change in the predicted output. We have only estimates âom [3 11 of the ratios of the
components of the dipole moments px:py:%, and we h o w that the polarization behavior of
the system wodd change dramaticdy if R:P>.:Pz had the values 1 : 1 : 1 (Le. the sites wodd
have cubic instead of& symmetry). Thus it is nahiral to explore the sensitivity of the
predicted polarizatioa behavior to variations in the ratios px:hpz. In order to keep the
number of adjustable parameters to a minimum, we have chosen to scale only the
anisotropic part of the components. We write the magnitude of the component of the
dipole moments in the-form
where the factor F is used to adjust the magnitude of the diiference between the
components. F is defined such that setting F equal to 1 r e m s the components to
Momson's original esthates of the dipole components. Setting F equal to zero makes aii
components equai.
Figure 3.11 shows the ~tabîïity diagram, for the case of setting F=1/2 for the pump
transition, and F=l for the laser transition. We see that one mode is always unstable, and
the other mode is stable for a Lunited range of the phase of the feedback This is not
compatible with the experimentai results.
Figure 3.12 shows the stability diagram for F=l for the pump transition, and F=1/2
for the laser transition. We see that this case is also incompatible with the experirnent,
predicting only one stable mode.
A Cornparison with Resuits in the titeralure
O 1 2 3 4 5 6 7
Phase of feedback (radians)
Figure 3.11 Largest real part of stability exponents for our two stationaq lineariy polarized solutions. Control parameters are as tisted in table 3.6 except Fp=t/2, FL=L
4E7
OEO
O 1 2 3 4 5 6 7
Phase of feedback (dians)
Figure 3.12 Largest real part of stabiliiy exponents for our two stationary linearly polarized solutions. Conml parameters are as listed in table 3.6 except Fp=l, FL=1/2.
A Cornparison with Resuits in the Literature 66
Figure 3.13 shows a plot of the real part of the largest stabiIity exponent for F=1/2
for both the pump and laser transitions. The model now predicts polarization flips near
w2 and e 3 7 d 2 , Le. almost equal duration of orthogonal modes over a complete cycle
of the phase of the feedback This is consistent with the srperiment of Besnard et ai.
However, we must consider how reasonable it is to appiy the same F to both transitions,
and whether or not the predicted stability behavior is simply fortuitous.
3E7
2E7
in e tn Y
I - OEO
-A €7
-2E7
-3E7 O 1 2 3 4 5 6 7
Phase of feedback (radians)
Figure 3.13 Largest real part ofstability exponents. Conml parameters are as lisred in table 3.6 exœpt Fp= FL=I/2.
Recaii that the matrix elements of the components of the dipole moments were
calculateci using a point charge model. Clearly ali transition moments scale with the
crystal field potential, and thus one expects errors in the model to affect aIi transitions in a
similar manner. Thus it is reasonable to scale the relative components for diierent
transitions in a similar hhion.
A Cornparison with Resuits in the Liierature 67
Was there anything "rnagic" in the choice of F=l/2? The answer is no, because the
experiment was performed by adjusting the strength of the feedback to approach equal
duration of the two modes. Thus one can interpret the experiment as one in which one
uses the strength of the feedback to determine the correct value of F. This hterpretation
is strengthened by a second result reported in [25]. Figure 5 in [25], shows that with the
strength of the feedback reduced, there are odations in the polarization, but no f ip to
the orthogonal mode. Figure 3.14 shows our calculateci stabilie diagram for ~f reduced by
50%. We see that oniy 1 mode is wer stable, and that over approxhately 1/4 of a cycle
this mode is unstable. This is consistent with figure 5 in [25].
1E7
CI L Y
I -4 OEO
-1E7 .
Phase of feedback (radians)
Figure 3.14 Largest reai part of stabiliîy exponents. Control parameters are as listeci in table 3.6 except Fp= FL=lf2 and the feedback intensity is reduced by 50%-
As mentioned above, Besnard et ai. reported that ail oscillations appeared to be
oscillations in polarization and not intensity. From the stability analysis we find the
eigenmodes of the decay near a stationary solution are intensity and polarization, and that
A Cornparison with Resuits in the titerature 68
it is the reai part of the eigenvalues associated with the polarization that change sign near a
fiip point. While this is in agreement with the experiment, we must remember that for a
noise fiee theory, the trajectory of the solutions for a non-stationary situation is
detennined by the time dependence of the cuntrol parameter. WWithout solving the time
dependent w o n s , we do not know for certain ifa periodic variation in the phase of the
fdback wiIl excite polarization oscillations. However it probably does do this since
feedback dters the relative phase dong the two orthogoaat directions, and it is the relative
phase of the two components that changes the polarization state of light.
At uUs stage we have discussed and appea. to have scplained al of the
observations of Besnard except the Eequency of the polarkation osciuations. Near a flip
point this is given by the imaginary part of the stability exponent. Figure 3.15 shows a
plot of hi versus phase of the feedback, when xi is converted to a fiequency in MHz We
Phase of feedback (radians)
Figure 3.15 Imaginary part ofstabiliîy exponent Control parameters are as üsttd in table 3.6 except FP=FL=l/ 2.
A Compafison with Results in the Literature 69
see that it ranains almost constant near 100 MHz, Le., just the fiequency shift between the
two stationary solutions. The period of oscillation given in [25] is typicaliy 20 ps
(1120 MHz) and totally incompatible with figure 3.15.
We now s p d a t e on what determines the period of the polarization oscillation.
nie theory is based on a mean-field approximation and is valid on t h e scales long
comparai to cavity decay rates. (A cavity has three decay rates, one for intensity, and two
for the polarization). For polarization the tirne constants s are on the order of the round
trip t h e in the cavity divided by the anisotropy. For the residud birefbgence e'=0.002
and a I/2 mm cavity7 r,~ is on the order of (10-~')/0.002=10~ sec. The microchip cavity
can respond faster than the observed period of 20 ps. However the extemal feedbadc has
&'=do4 and was some 15 cm long. In this case sd10~9/104=10 ps, or just the penod
obsetved experimentally. On this fime scale we cannot use a mean field approximation, or
what is quivalent, we cannot replace the output mirror with an effective &or to mimic
polarized feedback Clearly the effective mirror technique can be used for tirne scales
longer that sa7 Le. it is valid for stationary solutions or slower dynamics.
The ievel of agreement between resuits found in the literature and caldations
based on our modei, while satisfying, do not represent a strong confirmation of the theory.
There are too many uncertainties and "loose ends" to make such a ciaim.
We saw above that it was possible to have no stable solutions in the presence of
feedback (see figure 3.11). However our limited search in parameter space has located
only two stationary solutions. Even in the simpler case of gas lasers, six statiomy
solutions are known. Thus other stationary solutions must exist; we simply have not yet
found them. Finding ail solutions must wait until a more efficient method of searching
solution space is developed.
In the calculations, the h e m dichroism was set to zero. The isotropic loss in the
case of Besnard was sr=0.003. Later we wiii see that an anisotropy of the order of 10"
Le. a 1% variation in the isotropic loss can and does alter the predicted polarization
A Comparison M~ Results in the Merature 70
behavior. Dichroism is capable of stabilting solutions that are unstable in orientation of
the Iineady polarized output. However we have not explored parameter space (sr&)
lookiag for other stable solutions for a number of rasons. F h and foremost, the main
thnist of thk work is to test the theory, not fit it. Exploring parameter space is also an
overwhelming task, if not impossible, simply because of the large number of input or
control parameters. F i y , cataloguing, understanding and interpreting numerid
solutions of the mode1 is beyond the endmce of any graduate student. What we have
done, ùistead, is to examine experimentally regions of parameter space that have been
ignored in the early experiments. Part II of this thesis describes this work-
Chapter 4 - Experimental Setup
Introduction
In part I, we presented a theov of the polarization behavior of a single mode
quasi-isotropie N~~+:YAG laser and compared it with existing experirnental results found
in the literature. AU previous experirnents involved longitudinal pumping with linearly
polarized light. The computations required certain input parameters or conditions and in
none of the reported experiments were these completely specined. This is understandable
as the relevant parameters or conditions were not known at the time. However, now that
we have a theory, new experiments can be designed and one can attempt to control the
conditions andlor masure the parameters.
This chapter deals with a description of the experimental setup and a set of
measurements aimed at characterizing the Nd:YAG rnicrochip laser. Chapter 5 presents
new results and a cornparison with theory.
Apparatus and techniques
Figure 4.1 shows the optical setup of the Nd:YAG microchip laser. The rnicrochip
is longitudinally pumped with a diode laser operating at 809 nm. A lens and an
anamorphic prism pair are mounted at the output of the diode to collimate the pump beaq
whiie a polarizer ensures that the light is heariy polarized. Two aluminurn coated rnirrors
are used to steer the pump beam above and then towards the microchip. A Soleil-Babinet
compensator, iaserted into the beam, is used to control the polarization state of the pump
beam, whüe a final lens is used to focus the pump beam ont0 the microchip laser. The
output nom the microchip at 1064 nm is sent through a polarizer, and then through a cut-
off filter to remove the pump radiation. The Nd:YAG beam is nnally examineci with a
beam profiler, a Fabry-Perot interferorneter and a photodiode detecîor. The whole
experiment is mounted on a bread board, set on a 1 ton slab of granite, mounted on rubber
blocks to reduce acoustic noise and mechanical viirations.
Figure 4. I - Schematic drawing of the experimental apparatus
Nd:YAG microchip
The Nd:YAG microchip is a 560 pm thick, 4x4 mm square. The crystal was grown
by the Czochrdski method, and contains 1 Atomic % Nd. The rnicrochip has plane
paraUeI surfaces, with dielectric snirrors directly deposited on them These are coated to
have 80% transmission at 809 nm and 99.9% reflectMty at 1064 m on the input face,
with 93% reflectivity at 810 nrn and 0.5% transmission at 1064 m on the output d e .
Gcpeninental Setup 73
The length of the laser cavity was chosen to ensure that the laser can operate on
only one longitucfinal mode. A 560 thick microchip has a longinidinal mode
separation of 4.9 cm"; the gain bandwidth at room temperature is 4 cm-'. N o d y with
cwo plane parallei mirrors, the resonator wouid be unstable. However, the radiiy
symmetric pump beam not only favors the formation of a symmetric low order mode, it
also stabitws the cavity. The pump beam deposits heat h o the gain material. The
resulting radial thermal distribution creates a waveguide since the gain material has a
positive index-of-refiaction themial coefficient. Furthemore, the pump beam will induce
a m a t u r e of the end mirrors, which also contributes to the stabilization of the transverse
mode. AU of these scalar effects have been treated in the literature 139 to 461.
The usuai experimental test of the theory is to Vary some control parameter in a
known manner and to see if the measured changes in the output polarization coincide with
those calculated. The orientation of the crystal axes, of the stress axes (residual or
applied) and orientation of the direction of the polarkation of the pump beam are the
control parameters that can be easily varied in an experirnent. To rninimize stress induced
birehgence, the crystal is simply dowed to rest in a shallow well. The holder is made of
aiuminum and mounted on a heatsink to cany heat away from the microchip.
If we were to chose to rotate the crystal about the axis normal to the microchip
d a c e (in order to change the orientation of the purnp polarization), the question arises as
to whether or not the other operating conditions would be unwittingiy altered. We find
that the alignment of the microchip in the pump b m is aitical to determinhg the spatial
mode of the output. Smail changes in the orientation of the microchip in the pump beam
can alter the shape of the lowest-order tranmerse mode, and cm cause higher order
transverse modes to appear. We aiso found that translating the microchip by as Little as
100 p could cause large changes in the output intensity. This spatial inhomogeneity is
consistent with the observations of Yoshuio [20]. As reported in part I, he measured the
beat fiequency between two orthogonal polarization modes as a fûnction of crystai
position. Since the fkequency separation is caused by the buefiringence in the cavity, a
EXpenhentai Setup 74
variation in the beat fiequency is a masure of the variation in the residud birefiingence.
They found variations on the order of 30% in the birefbgence of the unstresseci crystal
moving over a distance of about 50 to 100 p. On this sale, it is not surprising then that
we were unable to make a mount for the microchip laser that would d o w us to rotate the
crystal about the axïs dehed by the pump beam, without also changing the operathg
mode. Thus the c r y d was mounted instead in a five point @bal mount, to provide fine
adjustment of the angle and position of the aystal, while the orientation of the polarization
of the pump was rotated using a Soleil-Babinet compensator.
Pump source
A diode laser (SDL-542281) was used as the pump source for the experiments.
This diode laser can operate at up to 150 m W of continuous wave output power. It has a
1 p by 3 p emitting surface, and operates in a single transverse mode. The polarization
ratio of the diode is 20:l. It has a thermoelectric (TE) cooler, and a monitor photodiode
mounted in its package, and is controiied with an SDL-800 laser diode driver. The diode
is tunable over a range of 4 m about 809 nm, so by setting its operating temperature with
the TE cooler, it can be tuned to optimke the absorption of the pump beam by the
Nd:YAG microchip.
Because of the emitting dimensions, the diode has a divergence of 9" in the broad
cavity direction and 30' in the nmow one. Since the pump beam geometry is responsible
for the mode formation in the microchip laser, it is important to have a collimateci, radially
symmetric output Eom the pump laser. To collimate the beam, and to correct for this
asymmetrk radiation pattern, we use a lens mounted with an anamorphic prism pair
(Meiles-Griot 06 GLC 002, and 06 GPA 004). The prisms are an identical set, mounted
to provide an expansion of the beam by three times dong one meridian only.
The diode is mounted with its polarization aligned parallel to the table, and a
polarizer on the output removes whatever component remains in the orthogonal direction.
mpen'mental Setup 75
The pump beam is completely S polarized with respect to the mirmrs used to s t e a it, so
the light arriving at the compensator is linearly polarized.
Because the pump b a r n establishes the spatial mode of the laser, it is necessary to
focus the pump radiation to a cirdar spot, at least as small as the radius of the operating
mode in the microchip. Focusing the pump beam however creates a dficuIty in
maintaining the stability of the pump radiation absorbed by the microchip. It is usual to
have a high reflectivity at the pump fkequency on the output face of the microchip, as this
ailows more of the pump radiation to be absorbed. However this l a d s to an interférence
pattem inside the microchip, analogous to Newton's rings. Here the Mgs are caused by
interfering wavefionts of ciiffirent cunrature reflecting off of the flat surfaces of the
microchip. If the interference pattern changes, this changes the spatial distribution of
energy deposition in the aystal and as a consequence the mode must change. In
particular, if a short focal length lens is used, we find that the laser output becomes highly
dependent on the position of the rnicrochip with respect to the beam waist of the pump.
When operating near the threshold condition of the laser, variations in chip position and
orientation are saCient to cause significant fluctuations in the output power. If a longer
focal length lens is use& there is less variation in the interference pattern. Increasing the
focal length increases the confocal parameter and now near the bearn waist, the in t e r f e~g
beams have the sarne (zero) mature . In the experiments we used a 2 1 cm lens to focus
the pump ont0 the microchip. With carefbi alignment, variation in the output power of the
microchip laser cm be kept to below 4%. With a 2 1 cm lens, the overaii distance fiom the
diode laser to the microchip becomes nearly 2 m. There is an advantage to having a large
separation between the diode and the microchip. It helps to minimize feedback into the
diode, which makes its output unstable.
Soleil-Babinet compensator
To control the orientation of the pump polarization, we used a Soleil-Babi.net
compensator (Special Optics mode1 8-400), set as a haKwave plate. The compensator
acts iike a zero order waveplate with a variable retardation Mode1 8-400 can provide up
to one wavelength retardation over a spectrai range of 200 to 2700 nm The effect of a
half wave retardation on lineariy polarized iight is to rotate the orientation by twice the
angle between the waveplate crystal axîs and the original polarkation Once the
retardation of the Soleil-Babinet is calibrateci for the operating Iiequency of the diode
laser, we can adjust the orientation of the pump polarkation simply by rotating the entire
compensator,
Ody linearly polarized pump beams have been used in previous experiments. With
our setup it is a simple matter to investigate the effect of having a linear, elliptical or
circularly polarized pump beam. To calibrate the Soleil-Babinet, we k s t measued the
poIarization of the pump beam as a fûnction of the retardation. Figure 4.2 shows the
measured ellipticity of the pump beam as a fünction of the displacement of the quartz
plate. The heariy polarized input beam was oriented at 45" with respect to the axes of
the cornpensator plates. The solid h e in the figure is the theoretical shusoidal
dependence of the absolute value of the eUipticity on the displacement of the compensator
plates. The compensator is a quarter wave plate at 809 nm for a displacement near
5.5 mm, and a halfwave plate at 10.5 mm
To ver@ that rotating the compensator did not alter the position of the purnp
beam on the microclip, the chip was replaced with a 100 pm pinhole. No variation in the
transmitted intensity was detected when the compensator was rotated.
Experimental Setup
-1 0 1 2 3 4 5 6 7 8 9 1 0 1 1
Displacement (in rnm)
Figure 4.2 -- Ellipticity of the pump polarization as a bction of plate displacement in a Soleil Babinet compensator.
Monitoring System
A polarizer and detector were used to monitor the output of the Nd:YAG laser.
The polarizer was placed before any mirror, to avoid distortion of the state of polarization
Following the polarizer is a cut-off filter which attenuates the radiation at 809 nm, while
dowing the 1.0641 prn radiation to pass through. The intensity of the idhued light was
measured as a function ofthe rotation of the polarizer. From the maximum and minimum
intensities and the orientation of the polarizer, one can determine the orientation and the
absolute value of the ellipticity of the polarization ellipse of the laser field.
Throughout the experiment, the output was monitored with a confocal Fabry-
Perot interferometer, (TecOptics SA-7.5 spectrum analyzer) to ensure that the laser
merimental Setup 78
operated on a single longitudinal mode. The interferorneter has a free spectral range of
7.5 GEEz and a finesse of 175.
The spatial extent of the beam was rnonitored using a Merchantek Pc-Beamscope.
This is a dual-siît scannhg beam profiler, with a tramlatable probe style detector. When
the profiler is not scanning it does not block the path of the Light.
Feedback
For experiments requiring polarized feedback into the microchip laser, some
modifications to the apparatus were required. Figure 4.3 shows the second setup.
Figure 4.3 - Experimental setup including polarized feedback
The Soleil-Babinet compensator was not used in the experiments involving
feedback. A polarization insensitive beam splitter was added in the output section of the
apparatus to create a feedback path into the laser. There are two polarizers in the
feedback paîh. The fkst poiarizer is used to control the polarkation of the feedback itself
The second polarizer is used to control the intensity of the feedback by adjusting the angle
between the two polarizers. The feedback lem is placed at its focal distance (8 cm) fiom
the microchip to couple as much light as possible back into the laser. The phase of the
feedback is controlled by adjusting the length of the feedback path by moving the feedback
mirror, whïch is mounted on an electromagnetic transducer @MT).
The theoretical mode1 developed in Part 1 is a plane wave, single ftequency model.
The closest we can corne to realiPng this situation experimentally is to use a laser that
operates in a single longitudinal mode and the lowest order transverse mode. The
following paragraphs summarize our observations of the modal characteristics of Our
microchip laser.
Spaanl mode characterizatioo
Figure 4.4 shows the intensity profile taken with a beam profiler, of the output of
the microchip laser, in two orthogonal directions. The output is symmetric in the x and y
directions at a distance of 20 cm nom the crystai. The output was found to have a half
width of 740 pn, indicating that the size of the beam waist at the crystal must be on the
order of 90 p. The profiles of the beam indicate that the laser is operating in the
fundamental or (0,O) transverse mode.
Figure 4.4 - hiensity p r d e of the output of the laser at 1.064 un, measured in the x and y directions
Longitudinal mode characterization
The operating fiequency of the rnicrochip laser was monitored with a Fabry-Perot
interferorneter. It was found that for low purnp powers, (cl -3 times threshold power), the
laser operated hearly polarized at a single frequency. At higher pump power levels, a
second mode appeared. Both modes were iinearly polarized, and the second mode could
appear oriented either in the same direction, or orthogonal to the original mode.
For a 560 jun long crystai, the separation of longitudinal modes is 0.55 nm at
1.06pm. Using a grating monochromator, it was deterrnined that the wavelengths of the
two operating modes were 1 .O642 pm for the stronger (original) mode, and 1.06475 pm
for the other. Taking the Merence of the measured wavelengths shows that the laser was
operating on consecutive longitudinal modes of the cavity. It is uniikely that the two
modes corne nom the sarne optical transition. Figure 4.5 shows the low signal gain
profiles of the Rz to Y3, and Ri to Y2 transitions at room temperature [471, together with
the measured wavelengths of the two modes of our microchip laser. Clearly a laser
operating at a single fkequency will have maximum output at the peak of the gain curve,
and generally will be operated in that region. This does little to saturate the gain coming
-&entai Setup 81
eom the Ri to YI transition. Consequently, at higher pump power, a longinidinai mode
operating at a longer waveiength can and does corne above threshold.
Wavelength (microns) Figure 4.5 - The gain profIIes of the R A and Rt-Y2 transitions are shown together with the
wavelengths of the two operating modes. The dotted line shows the total gain profile.
The presence of the second operating mode imposes a lirnit on the pump power of
the laser. AU experirnents were performed at low pump powers to avoid multi-
longitudinal mode operation-
Other considerations
We saw earlier that many of the experiments reported in the iiterature applied a
stress to the crystal and consequently operated in a regime where the birefiingence and not
the polarization of the pump laser d e t e d e d the output characteristics of the microchip
laser. We wished to examine the relatively unexplored region of low birehgence. This
&pmenrnentai Setup 82
is the reason the microchip was simply allowed to rest in a shallow weiI. The orientation
of the birefringence axis, while fixed for a parti& experiment, was not diredy
masurable. Howcver we were, foilowing Besnard et al., able to measure the magnitude
of the birehgence at several spots on our microchip. Using our confocal interferorneter
we observeci the Eequency jump when the laser was forced to switch polarization modes
with the application of hearly polarized feedbadc We found the residual birehgence to
be between 2x1 O-' and radiaadpass depending on the position in the cxystal.
The orientation of the c x y d axes must be specified before a caldation can be
made- While the crystal is cut with the face perpendidar to the [1,1,1] direction, we do
not know the orientation of the other axes (of course they could have been detennined
using Xray scattering). Fortunately, as we saw eariier, the output polarization
characteristics according to our theory, are not very sensitive to the orientation of the
crystal axes.
The first sehip described above was used to study the polarization states of the
microchip operating at 1.0641 pn as a fiindon of the polarization aate of the pump. The
second setup was used to study the effect of feedback for two feedback distances and a
linearly polarized purnp. The results and analysis is presented in the foIiowing chapter.
Chapter 5 - Experimental Results and Analysis
Using a lineariy polarized pump beam and the experimentai setup given in figure
4.1, we examined the output polarization of the microchip as a fiindon of the orientation
of the pump polarization with respect to the aystal axes. The purnp power was 1.14
times the threshold pump power. Both the measued eilipticity (TL) and the orientation
(h) of the output depend on the orientation of the pump field. Figure 5.1 shows the
measured orientation of the polarization ellipse of the output of the laser with respect to
the polarization direction of the pump. It can be seen from the figure, that the output of
the laser followed the orientation of the pump, staying within 8" on either side, as the
pump fieid was rotated with respect to the crystal axes. This is a clear indication that this
system is operating in the small bireningence region. (See figure 3.8, where for small E'
the calculated orientation of the output is determineci by the orientation of the pump).
Figure 5.1 iuustrates that working in the s m d E' region is more dEcult than in the
large si region. The measurements should repeat themselves for a rotation of 180". In
figure 5.1, the O0 and 180° positions are disparate. The moa likely source of the problem
is the compensator. The compensator may be changing the spot position on the chip,
enough to alter the mode, but not its intensity. At this stage we do not concem ourseives
with this technical dficulty. As we s h d see below, the quaiity of the data is stdi more
than d c i e n t to reveal more serious diffidties. First, however, we report the measured
absolute values of the eilipticities Iqrl.
fipementai Resulfs and Analysis
Figure 5.1 - Output orientation relative to the pump orientation & versus &,. The zero position on the horizontal scaie is arbitrary
Figure 5.2 - Absolute value ofellipticity Iqd, vs. pump orientation &. The origin on the horizontal scale was arbitrarily chosen to make &=9û0 for the measured value oflqw, near the center of the &, axis.
Gcpennlental Resuks and Analysis 85
Figure 5.2 shows the measured elliptic* of the output under the same conditions
as figure 5.1. Since the apparatus used for this experiment cannot distinguish right from
left cirdar polarizations, it is the absolute values of the ellipticities that are mea~u~ed and
ploned. It is likely that the output beam has opposite helicity on either side of the points
with linear polarizations (qL=û). It cm be seen that the eilipticity varies with the
orientation of the pump polarkation, wÏth values ranging f?om O to 0.4. The ellipticity, as
defined in chapter 2, varies quickly near iinear polarizations, and more slowly around
ciradar polarisations. The maximum elIipticity rneasured (0.4) corresponds to a ratio of
intensities of about 1125 between the minor and major axes of the polarization ellipse.
A cornparison of the resuits shown in figures S. 1 and 5.2 with curves caldateci
with input parameters similar to those in part 1 (except the magnitude of ai), reveals what
looks, at fxst sight, k e a difnculty with our model. There is significant disagreement
between theory and experiment. We hope to convince the reader that the problem lies not
with the model but rather with the values chosen for certain control pararneters. In part I,
the experiments with which the calculations were compared were di done in the large
birefigence regime. In that case, the choice of other control parameters had little or no
effêct on the calculation and could be left fixed at some arbitrary value. In part 1 we set
Ao>=O, er=O and kept the orientation of the crystal axis Oc fixed. However, we were
obliged to impose a constraint on the input parameters to explain the eady experiments of
Besnard. There, in the presence of feedback we concluded that the estimate of the
anisotropy of the dipole moments estimated by Momson [3 11 was perhaps too large. We
used a factor F=1/2 to reduce the anisotropy by a factor of 2.
The usual method of tracking down an inadequacy in a calculation Like the present,
is to sift through parameter space looking for a signature in the output that cm be tied to a
variation in one or perhaps two control parameters. To be effective this must be done
methodically, starting always fiom the satne reference set of control pararneters. Our
reference set is that of section I, where we let si '%oat" w i t h a range consistent with
Gpeninental Results and Analysis
other measurements. Here the range of ci is 2xl0-' to 1.3x10-~, and was determined by the
fiequency jumps &om 3 1 to 60 MHz reported in the experimental section.
The fxst slice of parameter space exploreci was the anisotropy of the dipole
moments. Figures 5.4 and 5.3 show the computed orientation and ellipticity for two
cases: F=l and F=1/2. Clearly the resdts for the case of a linearly polarized pump beam
are insensitive to the value of F. niey also show much more symmetry thaa the
qerimental results shown in figures 5.1 and 5.2. We therefore fix F at 112 to be
consistent with what we found in part I, albeit for a difEerent pump transition.
Table 5.1 - Puameters required for caldations io figures 5.3 to 5.10 escept where othelwise noted.
Parameter
Yd
ydc
Yk
p,:p,:p,
. P .
Pq:Pr:Ps
A ~ L
L
h
Value
4.3 X ~ O ~ s-'
8 1 1.2 ~ 1 0 S-
4.3 X I O ~ s-'
4.0:1.3:4.1
4.5 : 3.4 : 1.8
O
560 prn
1.3
Parameter
1
TP
4~
4~
Ob
o d
gi
sr
Value 1
0.003
0.0
varied
15"
O.*
N/A
O.
2.8~10"
Btpenmental Results and Anaiysis
Figure 5.3 - Cornparison of calculateci output orientation relative to the pump orientation & versus &+, for two d e factors ofthe dipole moment aaisotropies. Other parameeters are as Listeci in table 5.1.
Figure 5.4 - Cornparison of dculated absolute value of ellipticity vs. pump orientation %+, for two sale factors of the dipole moment anisotropies. ûther parameters are as listeci in table 5.1.
Experimental Resuits and Analysis 88
The next "slice" of parameter space explored is the variation with the orientation
ofthe crystal axes with respect to the axes of the residual birefringence. Earlier, for large
si we found the results insensitive to gC+. That is not the case here. Figure 5.6 shows
the ellipticity is insensitive to the orientation of the crystal axes. However the orientation
of the output polarization ellipse (figure 5.5) varies with the orientation of the crystal axes
relative to the axes of the birefigence. Most important, note that the symmetry about
b+=O changes with &+. It is symmetric for &+=15", most positive for &&,=30°
and most negative for t$c+=OO. This asymmetry may be considerably accentuated by
increasing F from 1/2 to 1.
The asymmetry about h+=O appears to be a unique signature ofthe orientation
of the crystal axes with respect to the axes of birefhgence. It varies from a minimum to a
maximum as one changes 4 . 6 &om O0 to 30°, and returns to a minimum for +,+ equal
to 60". This variation in b+, is symmetric about zero for &+=lSO or 45 O... etc. Since
figure 5.1 appears nearly symmetric we claim to have determined the orientation of the
crystal axes with respect to the axes of the birefigence (4c+,=150, 45O, ...). We fix it at
15' in al l subsequent calculations. In so doing we may be "painting ourselves into a
corner", but it is the best we can do at this stage.
Gtpen'mmtal Results and Analpis
Figure 5.5 - Cornparison ofcalcuiated output orientation dative to the pump orientation & versus &+, for t h crystal an@. Other parameters are as iisted in table 5.1.
Figure 5.6 - Cornparison of caiculated absolute value of ellipticity [qd, vs. a+,, for three qstai angies. m e r parameters are as liste- in table 5.1.
menmental Resuits and Analysis 90
We now examine the sensitivity of the caldateci orientation and ellipticity of the
output beam to the operating eequency ofthe laser. Previous calcuiations were for hd,
Le. near line center. Figures 5.7 and 5.8 show the calculated values of $L-& and lqLl as a
bction of 4,+,, the angle between the pump beam and the axis of the bir-gence, for
three cases: A a 4 , A a i U O GHz. Here there are noticeable merences between the
m e s . The Aa=+30 GHz case has some of the characteristics of the experimental w e s
in figures 5.1 and 5.2. Notably in the plots of Iqd, the peaks are displaced "outwards" for
Aa positive, a feaîure evident in figure 5.2. The calculated orientation shows a major and
a minor positive swing. While the experimental results for h+p are somewhat in doubt,
there appears to be only one large positive swing as +p+ is varied through 180°. Of
course the origin for the horizontal axis of the experirnentd m e s is unknown. It appears
as ifwe have corne close to the comect value by selecting the zero angle to correspond to
one of the two linearly polarized output positions.
As we shail see below, introducing a linear dichroism into the caldation has
signatures in the b-4, and lqLI curves very similar to those that are displayed in figures 5.7
and 5.8. where a AmeO GHz was explored. Therefore, we would not be justifiai in
fixing the operating fiequency at Aa=30 GHz, without fûrther evidence. There is
howwer direct evidence. The wavelength measured diredy was 1.0642 Pm. The peak of
the gain curve occurs at 1 .O641 p, and this ciifference amounts to h = 3 0 GHz (recall
Aa=ao-o). It seems reasonable then that many of the asyrnrnetries in the measured m e s
for b-4, and IqLl occur because the laser is operating off line center. Later we will
speculate on the physicai reason for this behavior.
In al l of the calculations we have assumed that the linear dichroism of the cavity
was zero. A linear dichroism is characterized by its magnitude and the orientation of the
axes, which we chose to specm as behg with respect to the axis of the birehgence. At
one point, we had considered this as the only source of the asyrnmetnes in figures 5.1 and
5.2. We had determineci that the values er=l .7x10-' and $d+=90 gave results close to the
Eicpetimental Results and Anaiysis
Figure 5.7 - Comparison af calcuiated output orientation reIative to the pump onentation versus &+,, for three operating fiequencies. Other parameters are as listed in tabIe 5.1,
Figure 5.8 - Comparison of caiculated absolute value af ellipticity lqLJ, vs. &+, for three operating fbquencies. Other parameters are as listed in table 5.1.
Gcperïhental Results and Analyss 92
experimentd curves for b-& and Iqrl. T o show the sensitivity of the output polarization
to a linear dichroism, we compare in figures 5.9 and 5.10, caldations with and without
this hear dichoism. We chose to have A d dso, in order to compare with figures 5.7
and 5.8. Upon inspection we see that the a m e s generated by introducing a dichroism are
similar to the m e s computed for Aa=30 GHz. There is one noticeable difference
however introducing a dicbroisrn aiters the relative heights of the two "bumps" in the plot
of htl. This is seen experimentally in figure 5.2.
Given that our measurements of the fiequency showed the that the experiments
were perfomed off h e center and given that the %umpsY' in the experknentai results nom
h l are not the same heighf we computed new a m e s with Aa=30 GHz, 4 d 4 4 0 , and
adjusted er and si to rnimic the experimental results. Figures 5.11 and 5.12 illustrate that
our mathematical mode1 is capable of explaining these new measurements. The value of&'
used was 1x10'~ and is small with respect to the isotropie loss due to the mirrors of ~ x I o * ~ .
Dielectric coatings are known to be weakly anisotropic. However we do not know if the
anisotropy in the reflection is in amplihide or phase.
Bpen'mental Results and Anatysis
Figure 5.9 - Cornparison afcalcuiated output orientation reiatnte to the pump orientation & versus
Figure 5.10 - Cornparison of calculated absolute value of eiiipticity ImJ, vs. &+,, with and without a Linear dichmism, Other parameters are as listed in table 5.1.
Eq~en'menttal Results and Anaiysr's
Figure 5.1 1 - Cornparison afcaicuiat.ed output orientation relative to the pump orientation versus &h with experimental data Control parameters are as iisted in table 5.2.
Figure 5.12 - Cornparison of calcuiated absolu& MLue of ellipticity IqLJ, VS. &+,, with experimenoil data Control parameters are as listed in table 5.2.
E3peninental Results and Analyss
Table 5.2 - Parameters mpired for cal b
Parameter Value I
dations of figure 5.11 and figure 5.12.
Parameter Value
1 varied
Let us pause to summarize the situation at this point. The general approach we
have adopted to test the theory developed in section I, is to show that theoty and
experirnent are in agreement for values of the input parameters that are within reasonable
bounds. By reasonable bounds we mean that the parameters should be close to values
measured or estimated by independent means. The eady experiments were sensitive to the
birehgence and the ratio of the components of the dipole moments. The experiments
muid be explained by assuming &0.002 and F d 2 , both of which we judge to be
reasonable. In this section two new messurements were presented so far. These are in the
small zi region were the predicted polarization behavior is sensitive to the values of other
control parameters. The additional control parameters are the orientation of the crystal
axes, the operathg f?equency of the laser and the magnitude and orientation of the Iinear
dichroism. In spite of the obvious difndty with the measurement of the orientation of the
major a>o's of the polarkation ellipse of the laser, measured relative to the orientation of
fipeninental Results and Analysis 96
the lineariy polarized pump field, we have been able to find qualitative or semiquantitative
agreement between the caldateci and obsewed vdues of h+p and lqL1 as a hct ion of
&+. The final input control parameters are listed in table 5.2. We need comment, at this
stage, only upon si and &. The value of ei used in the caiculation, 2.8x10", is weil within
the range determineci by measurllig the Eequency jump in a separate experiment. The
value of E~ was a fhctor 3x10.~ smaller than the isotropic loss due to the mirron. We
would have doubted a ratio close to 1 but can hardly feel uncornfortable with an
anisotropic loss less than 1% of the isotropic loss.
While theory and experiment appear to agree, up to this point we have in fact
examined a vev Limiteci region of parameter space. We have oniy varied $p in the present
experiment. We now examine the eIlipticity of the output, under the same conditions as
above, but as a hc t ion of the pump power. Above the pump power was 1.14 times the
threshold value. Figure 5.13 shows the rneasured eilipticity as a f'unction of the orientation
of the hearly polarized pump beam, for the values of the pump intensity, 1.06, 1.14 and
1.22 times the threshold value. Counter to our intuition, the efipticity rises with
increasing pump power. The theoretical c w e s foilow the same trend as the experirnental
points. The control parameters for the 1.14 times threstiold case were as determined
(estimated) above. No additional parameters were introduced to predict the curves for the
other two pump powers. The agreement between theory and experiment is satisfactory.
I3perimentaî Results and Anaiysis 97
'> - II- *.--
expaxntai daîa for three pump powers. Control parameters are as listed in table 5.2. The squares and the dotted h e represent pumping at 1.22 times the threshold pump power, the CircIes and the solid line are for 1.14 tMes the threshold pump gower, and the triangIes and dashed line are for L.06 times the
threshold pwnp power.
Varying the pump power amounts to v@g a control parameter in the same
region of parameter space explored above, theoretically and experimentaiiy. A more
severe test of the mode1 would be provided if we could examine another region of
panuneter space e~perimentally~ The only remaining control parameter that we have at
our disposai is the eiiipticity of the pump field. Up to this point, the effect of having an
elliptically or even circularly polarized pump beam has not been treated in the literature.
In the following paragraphs, we present new measurements made by pumping the
Nd:YAG microchip with eiiipticaliy polarized light, and cornparisons of the results with
our mode1 of the laser.
&peninental Results and Anaiysis
Ellipticai pump poIarization
Measurernents were made of the ellipticity of the output as a fhction of the
ellipticity of the pump beam, for two orientations of the polarkation ellipse of the pump
beam. In order to generate a circularly polarized pump beam, the linearly polarized input
to the Soleil-Babinet compensator must be alignai at 45O with respect to the crystal axes
of the compensator. It is not a coincidence that this orientation dso corresponds to
&,+4S0. The original orientation of the crystal was chosen by rotating it to the position
when the laser output was hearly polarized without the compensator in place. (Actually,
it is theory that dows us to claim that 4p-&rl+l= $p+)-
With &,+, fked at 4S0, Le. near the first peak of the lqLl n w e in figure 5.12, the
only slice of parameter space we can explore, without rotating the c r y d is the eilipticity
of the pump beam. This is accomplished by changing the phase retardation of the
compeasator. The output of the compensator wiU be ellipticaiiy polarized with a fixed
axis of the polarization ellipse. As the retardation is varieci nom O to 27c, the polarization
of the pump beam wiU change nom Linear, paralle1 to the original pump, to for instance left
handed ciradar (d2 retardation), to linear perpendicular to the original pump (x
retardation), to right circular ( 3 d 2 retardation), and back to the original state (27c
retardation). We report two experiments. The first is as we have just discussed. In the
second, we explored a sirnilar slice of parameter space except the orientation of the pump
polarization ellipse was set at 9' with respect to the birefkgence ais . The crystal was
rotated for this measurement and thus there was the chance that a new section of cxystal
was being used as the gain medium.
Figure 5.14 shows the value of the measured absolute ellipticity of the laser as a
hct ion of the ellipticity of the pump beam for 4 p 4 b 4 5 0 . For clarity, the phase
retardation is shown dong the horizontal axis, with qp on the upper axis. It is clear that
tbere is a sudden change in the ebpticity near qp=+0.8. We could interpret this as a
f3pen'mentai Results and Analyss
Pump field phase retardation (degrees) Figure 5.14 - Measured absolute value of ellipticity vs. eilipticity of pump beam, for +41&5~.
Pump field phase retardation (degrees) Figure 5.15 - Measured absolute d u e of eiiipticity vs. ellipticity of pump beam, for &+,=9*.
Bcpefhnental Results and Anaiysk 100
change in the sign of the ellipticity for qfl.8 and qp<-0.8, or that there is an instabiüty in
the mode at these points. In the experiment where the crystal was rotated to &+,=9O,
there was no indication that the eiiiptiaty q~ had c h g e d sign over the range of qp
exploreci. Figure 5.15 shows the r d t s for cbp+,=900.
We first disaiss figure 5.14. Shce the crystal was not moved for these
measurements the resdts should be calculable using the sarne input parameters as above,
(table 5.2), except for varying qp. The predicted curve is in strong disagreement with the
measurements. Howwer, being in a new region of parameter space, we will not be
surprised if the solutions are sensitive to control parameters to which the previous
experiments were not. The earlier experiments were insensitive to yb the relaxation of the
dBerence in population between the Kramers pair of states of the upper laser level. The
present experiments are very sensitive to this parameter. Above we had taken y m . We
now explore the effect of changing yk on the solutions to the model equatiow.
Figure 5.16 shows a plot of the computed eilipticity q~ as a function of the
eilipticity of the pump for &+4S0. The parameters (aside nom y 9 were as in table 5.2
for figure 5.12. This fixes the calcuiated value for q,=û at or near the experimental value
shown in figure 5.12. All of the computed values ofqL pass through this point, iiiustrating
as claimed above, that the cdculations are insensitive to the value of yk for linearly
polarized pump light.
We see immediately fkom figure 5.16 that the calculated ellipticity of the output,
q ~ , depends critically upon yb for highiy elliptical pump Light. We cm understand the
physics behind this behavior. If the crystal is pumped with circularly polarized Light, then
in our model, one member of a Kramers pair is preferentidly populated (recali figure 2.6).
If there is negügible relaxation between the states of a pair, then the gain for one ellipticity
of the laser wili be much higher than for the other, thus driving the laser towards a circuiar
mode. If the relaxation rate between the states is rapid (yrc>>"Id), this preferentiai gain is
Gtperimental Results and Anaiysis 101
1 0 s Now it is the geometry of the sites and the concornital pumping of each site that
piays a role.
-200 -1 00 O IO0 200
Pump field phase retardation (degrees) Figure 5.16 - CaiIculated ellipticity vs. phase retardation of pump beam for three values of yk. ûther
parameters are as listed in table 5.2,
If the pump is circularly polarized, it follows that the unsaturated gain medium
must exhibit cyiindrical symmetry. The low signal gain is thus independent of orientation
for lineady polarized iight. However the low signal gain may, and usually does exhibit
cirdar dichroism and cirdar birefkingence Faraday rotation). These symmetry
propertîes reveai themselves in Our mode4 in the population of the sublevels of the upper
laser level, and the variation fiom site to site. For a linearly polarized probe of the
populations of the laser level, the cylindrical symmetry arises from the euuality of the total
population of the matching pairs of sites (see figure 2.4). The Faraday rotation exists if
the Kramers conjugate sublevels are not equdy populated. This reflects the orientation of
the ions that arises fkom pumping with circularly polarized light. In the case of fast
Gpeninental Resuïts and Analysis 102
relaxation (large y& the orientation is lest and we simply have equal population sublevel
by sublevel and site by site. m e low signal gain now appears isotropie. Since hearly
polarized light cannot create ciradar anisotropies, the laser field must be linearly polarized
in a cavity for which the cavity modes are Iuiearly polariseci. That is very nearly the case
for our system and thus the laser mode is nearly hearly polarized Thus we understand
the large changes predicted by the model as one changes yc in the case of a circularly
poiarized pump bearn
There is a second and important feature of the caldated curves of qL. For a slow
relaxation of the difrence in population between the Kramers pairs, the predicted q~
passes through zero for small q,. For large yb varies slowly as qp is changed fkorn
t@. It is clear fkom the experimental results that remains nearly constant for smaii qp.
We cannot escape the conclusion that y3)Yd. This opens up a Pandora's box. In our
model we have assumed that sublevel population ratios, generated by the pump are
presewed in the relaxation to the laser level. If Our model requires yk for the laser level to
be fast to be in accord with experiments for low qp, we must have severe resenrations
about our assumption of preservation of population ratios in the pumping process. It
would not be dif]Eicult to build into the mode4 a Rramers relaxation in the pumping
process. However that would introduce another parameter, besides yk for the upper laser
level. Already we have essentially zero knowledge about the value of yk. Introducing
adjustable fitting parameters into nonlinear coupled equations moves a model fiom the
realm of physics, to the realm of pure mathematics. The experiments with elliptically
polarized pump beams show us that there is a need to measure yk and the pumping process
in N~~':YAG.
While the observation near qP=û show that a fast relaxation ( y p y d ) is necessary
and therefore that our model of the pumping process is suspect, it does not establish that
the model is wrong. There is some evidence that it is the fast Krarners relaxation in the
upper laser level that plays a sipnincant role. We find in the calculations, when the
merimental Results and Analysis 103
predicted changes rapidly with q, that it is difiïcult to find stable solutions. This is
usually an indication, in the time integration method use& that we are in a region of
parameter space where there are instabiIities. T'us the sudden change seen near q,=ih0.8
in figure 5.14 may weil represent a jump to another mode.
In the case of quasi-isotropic gas lasers, the mode1 has been rehed and the
cornputer programs developed to the point that all stationary modes (stable and unstable)
may be found. This has greatly fiiciiitated a physicai understanding of the modes. Here
we are in the very edy stages of developing a mode1 of the N~~':YAG laser, and in the
present case, we are only able to h d a single solution For certain values of q, we are not
yet absolutely certain we have found a stable solution.
Having at least d e t e b e d for this laser, the orientation of the ions either by the
pump beam or through saturation by the laser field when elliptically polarized, it is evident
we have made an error in applying the model to the 1.0641 p transition. There is a
second transition, (Ri to Y*) very close by (see figure 4.5). If the RI levels are oriented by
a cirdarly polarized pump then there is effectively a large background Faraday rotation at
the laser wavelength. This couid have a dramatic effect on the polarization behavior of the
laser. The message is clear. Either modo@ the model and measure the properties of the RI
Ievel when opticaily pumped, or measure the polarization properties of ~ d ~ ' : YAG,
operating on an isolated transition.
W e now lave the measurements and discussion of the polarization properties of
the quasi-isotropie YAG laser, under elliptical pumping conditions and retum to the case
of pumping with linearly polarized light. We consider the case where such a system is
subject t O polarized optical feedback
Hysteresis experiment
The purpose of this experiment was to determine if it was possible to make the
microchip exhibit hysteresis in polarization switching by adjusting the length of the
E k p e M t a l Results and Analysr's 104
feedback path into the laser. The experimenta apparatus was set up as in figure 4.3 to
provide optical feedback into the laser. The orientation of the Iinearly polarized pump was
adjusted to produce a hearly polarized output, oriented parailel to that of the pump field.
Eariier we showed that this condition could be interpreted as setting &-+.
In the absence of feedback the laser operates in a well defined linearIy polarized
state. Polarized feedback was provided perpendicular to the output poIarization. The
output intensity was measured through a polarizer oriented perpendicular to the
polarkation direction of the Eee nuining laser. Two measurements were made, one for a
long feedback path, (3 m), the other for a short total distance (40 cm). The polarization
E p points were measured for both increasiag and decreasing feedback distance, Le. for
changing the phase of the feedback For a contbous change in bf, the strength of the
feedback was adjusted to yield nearly equal operating times in the two observed linearly
polarized orthogody oriented modes.
Figures 5.17 and 5.18 show the intensity measured as the feedback mirror was
scatl~led at a rate of about 1 @S. nie solid and dashed curves represent one forward and
one backward pass of the feedback mirror respectively. The zero signai points coincide
with the laser operating in the polarization mode orthogonal to the passlig direction of the
polarizer in front of the detector. These cuves show polarization nips as the mirror is
translated. The curves for the two directions of scan were superimposed by assuming
e q d voltages on the EhlT (see figure 4.3) correspond to equal feedback distance.
Eperimental Results and Anaiysis
2 0 22 2 4 2 6 2 8 3.0 3 2 3 A 3.6 3.8 4.0
tirne (seconds)
Figure 5.17 - Poiarization switching with 40 cm f-ck path. The solid line represents a forward scan (positive voltage on EMT) and the dashed line represent a backwards scan. The two are made to overlap
by equating the EMT voltages of each scan
0.0 0 2 0.4 0.6 0.8 1 .O 1.2 1 -4 1.6 1.8 2.0
time in seconds Figure 5.18 - Polarization switching with 3 m feeâback pat6.
Gcpen'mentai Results and Analysr's 106
It is clear fkom the figures that the laser with feedback is bistable. We annbue the
fluctuations in the output to a variation in the feedback caused by mechanid vibrations.
In figure 5.17 there is no evidence of hysteresis, the measured polarization fips ocnilring
at the same time or same phase of the feedback In figure 5.18, there is a large hysteresis.
In part 1 we showed that the absence of hysteresis in the experiments of Besnard
et. al. codd be expiaineci simply in terms of the modulation of the real part of the larges
stability exponent by the feedback (see figure 3.13). We now show that same argument
carmot be used to explain the present obsemations. For convenience we repeat the m e s
of figure 3.13 in figure 5.19. The horizontal axis is the phase of the feedback In the
figure it was tacitiy assumed that the phase of the feedback was the sarne for each mode
for a given feedback distance. This is not correct. The two modes have slightly diierent
fiequencies because of the residual birefkgence of the crystal / cavity. For short
feedback distance and typical birefiingeace, the correction to the phase of the feedback is
smajl- In our case, with a frequency shift of 5-50 MH& and a feedback path of 3 m, the
relative shift of the phase of the feedback axes is 0.3 to 3 radians. This cannot account for
the observed polarization flips. In figure 5.20, we show plots of the two stability
exponents fiom figure 3.13, shifted dong the horizontal âxis by a measurable arnount.
Marked on the diagram are regions when the laser would be monostable, bistable and
unstable. The hysteresis of stabiiity for forward and backward scans of $f predicted by this
diagram is inconsistent with our experirnentai results.
In part 1 we pointed out that polarization oscillations, in the presence of feedback
couid be determined in part by the round trip time in the feedback section, and thus one
could not use a mean field approximation in this case. Here however, our time scale is
much longer, the scan over a 2x change in +f taking 1 s, not microseconds as in the
experiments of Besnard. nius we must look to other shortcornings in our model.
Gpeninentd Resuits and Analysr's
O 1 2 3 4 5 6 7 P b of feed back (radians)
Figure 5-19 Largest reai part of stability exponents- Control parameters are as Iisted in table 3.6 except Fp= FL=1/2.
O 1 2 3 4 5 6 7 Phase of feedbadc (radians)
Figure 5.20 k g e s t real part of scability exponents. Control parameters are as listeci in table 3.6 except Fp= FL= 112.
fipen'mental Results and Analysis 108
We have identifieci one possble source of the discrepancy between theory and
experiment. This is a new problem and it has to do with the stability analysis itself In the
usual mean field treatrnent of lasers, the fiequency is tnated as constant (adiabatically
elimiaated). As a consequence, the mean phase of the laser field acquires neutral stability.
c;ia Here however the rate of change of frequency with polarization (orientation) - is on 3,
the order of 10' Wradian, and it is qyestionable if the usual treatment of the fiequency of
the laser and thus also the stability of polarization modes is acauate. The old problem,
encountered above, of not being able to find ail the stationary solutions and thus having an
incomplete anaiysis of the possible modes and their stability* exists here also. An
understanding of the polarkation behavior of the laser in the presence of feedback is thus
lacking at this tirne-
Chapter 6 - Summary and Recomrnendaüons for Future Work
This thesis is dMded into two parts. In part I, the fist and ody microscopie
theory of the polarization States of a single kequency N~~':YAG laser is developed. While
there k s t s a general procedure for constructing a mean field vector model of lasers,
models for specific lasers must be tailored to include the hown properties of the gain
medium. Thus our model is rather specinc to YAG. The model was first tested by
cornparison with existing experimental data.
For N~~':YAG, not all of the relevant control parameters are known. Typically
any unknown parameter is regarded as a parameter to be fit by comparing the theoretical
predictions of the model with experimentai observations. For nonlinear coupled equations
this is not an acceptable method of testing a model. We have avoided nich a process by
showing, in some cases, that certain obsenrations are insensitive to specifïc control
pararneters. We were able to provide a plausible explanation of all of the relevant
experimental results found in the Iiterature, there had been no pnor explanations. Thus
we established that the main features of the model are probably correct. However, in the
case of polarized feedback, we concluded that the usual mean field trick of includùig the
e f f i of feedback as a pseudo anisotropy of the output mirror was inappropriate in the
case of a short cavity and a long feedback path. Here time delayed equations should be
The main difECU1ty in developing the theory was connected with the spectroscopy
of BTd3' at a crystal site ofD2 symmetry. From the point of view of the polarization states
Summary and Recommendatims for Future Work 110
of lasers, the most important aspect of the spectroscopy is the polarization selection des
and the phase relationships between the matrix eiements of the components of the dipole
moment. We were able to detennine these relationships without a knowledge of the
double group. As fat as we know, this is a new red t .
The mathematical modei, valid to dl orders in the laser field, consists of sixteen
coupled nonfinea. equations. For quasi-isotropie gas lasers, the mean field vector theory,
valid to third order in the field, consists of only three coupled nonlinear equations. Even
in that case, it took some ten years to simplify the formulation and develop the
mathematical tools to find ail of the six stationary solutions. Here the problem is much
more cornpiex and it is not surprising that considerable effort was spent in wrïting code
and applying laiown numencal solution techniques to fhd stationary solutions. In many
cases only one solution was found; in the rest, only two were found. This difliculty, on
top of the usud problem of interpreting n&cd resdts considerably slowed our
progress in understanding or physically interpreting the behavior predicted by the model.
AU of the intuitive interpretations presented in this thesis are based on the very recently
gamered insights into quasi-isotropie gas lasers.
In part II of this thesis, new measurements are reported on the polarization
behavior of a ~ d ~ ' microchip YAG laser. These differed frorn ail previous experiments in
that they involved longitudinal pumphg of the laser with elliptically polarized light. A
cornparison of the measurements with the behavior predicted by the model deepened our
understanding of the N~~':YAG laser and of the limitations to our mean field model.
We learned that orientation of the ~ d ~ ' ion either by elliptically polarized pump
radiation or through sahiraton by an elliptically polarized laser field must be included in
any model. Our mode1, as presented, assumes the pump induced polarization is preserved
in the relaxation process from the upper pump level to the upper laser levei- It would
require only a trivial modification to the program to relax this constraint: insert a factor f
(O S fa 1) in front of the pump tem that drives the ciifference in population in equations
2.48 and 2.49 in chapter 2. Of course, it is desirable to have an independent measurement
Summary and Recommendations for Future Work 111
of f The model as presented already dowed for the destruction of the orientation in the
upper laser levei, through the inclusion of the Kramers relaxation rate yk. The
measuements strongly suggest that y* is perhaps a factor of 100 or more greater than the
population deçay rate yd. Here too, it is desirable to have an independent masure of this
quantity.
Having exposed that optically induced orientation of the sites may play a
siBnificant role in determjning the polarization behavior of the N~~+:YAG laser, it was then
recognized that the nearby transition Ri+& cannot be ignored. A clear test of the model
requires an isolated laser he.
One of the main characteristics of the experimentai conditions in part II was the
use of a stress fiee mount for the microchip. This pennitted us to examine N~~+:YAG
lasers with a s m d residuai biremgence. We were able to show that in the case of a
hearly polarized pump beam that the output was mostly linearly polarized and tracked the
pump polarkation as it was rotated with respect to the crystal axis. Here theory and
experiment were in agreement, serni-quantitatively. ( R e d that the value yk is almost
irrelevant in such a situation). Caiculations revealed that the variation of the relative
orientation of the polarization ellipse of the purnp and laser beams is sensitive to the
orientation of the crystal axis with respect to the axis of the residual birefhgence. In
spite of technical diEcu1ties presumed to be associated with the displacement of the pump
beam upon rotations of the Soleil-Babinet compensator, we were able to fk the cxystal
axes at 15' with respect to the birefiingence. Clearly, future experiments cd for the use
of a crystal, the axes ofwhich have been predetermined by Xray studies.
One feature specinc to microchip lasers that was not stressed in the thesis, is that a
single chip is not one laser but rather a continuum of lasers. The laser being studied
depends both on how the chip is üiuminated and upon the exact spot on the chip being
used. The spatial uniformity is such that we were not able to produce repeatable results if
the chip was rotated, reoriented or translated during a set of measuremenrs. We strongly
Summary and Recommendations for Future Work 112
suggest for future experiments that YAG rods with w e d mirrors be used. These would
at lemt have stable spatial modes and WU remain singie fiequency at Iow pump powers.
In this nunmary, we have added suggestions for fuhue experiments at the
appropriate place in the text. AU of the experirnents reported here used longitudtial
pumping at 809 nm. However a marked dependence on the pump wavelength has been
reported in the literature. Our treatment of the spectroscopy, in partidar the character of
the dipole transition moments and polarization selection des, should fom a basis for
understanding such experhnents. Cleariy, observations of the dependence of the
p o b t i o n characteristics of the laser on the wavelength of the pump should be carried
out for isolated pump transitions.
We have suggested a minor modification to the mode1 to d o w for the loss of ion
orientation during the relaxation process from the upper pump level to the upper laser
level. VJe have also indicated that Our mean field approximation would remain vaiid in the
presence of feedbaclg provided the feedback distance is short. Our inabiiity to predict the
hysteresis observed with a long feedback distance forced us to reconsider the implicit
adiabatic elimination of the fiequency as a variable in the mathematical model. If the
fkequency is to be retained as a variable, the second derivatives of the mean phase must be
retained in MaxweWs wave equation. However, we believe more progress can be made in
understanding the polarization behavior of quasi-isotropic single fiequency YAG lasers if
the model is tmcated at third order in the laser field. The problem could then be
refonnulated in t e m of Stokes parameters. This should overcome ail the difEcuIties
encountered here in finding the stationary solutions. If the experience with the problem of
quasi-isotropic gas lasers is any guide, it would also permit a clear identification of the role
the polarization state of the laser plays in establishing the saturated optical properties of
this important gain medium.
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