Gain enhancement in a XeC1 pumped Raman amplifier
by Jeffrey Rifkin
A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in
Physics
Montana State University
© Copyright by Jeffrey Rifkin (1987)
Abstract:
Gain enhancement in an excimer-pumped Raman amplifier consisting of molecular hydrogen is
experimentally measured. Gain enhancement is measured both as a function of optical delay and input
pump intensity, and in the two limits when the laser mode spacing is large, and comparable to the
Iinewidth associated with the Raman medium. Experimental results are compared with the predictions
of a transient multimode theory. This theory is based on the coupled equations of Raman scattering
which describe the transient growth of the Stokes field and coherence of the Stokes transition. A sum
over longitudinal modes with fixed and totally random phases is assumed for both the pump laser and
Stokes fields.
The theory compares well with experiment in the regime where the Raman linewidth is large compared
to the modespacing of the laser. The theory also makes the interesting prediction that in this limit, the
broadband gain will be larger than the gain for a narrowband laser. .GAIN ENHANCEMENT IN A XeCl PUMPED
RAMAN AMPLIFIER
by
Jeffrey Rifkin
A thesis submitted in partial fulfillment
of the requirements for the degree
of
Master of Science
in
Ph y s ic s
MONTANA STATE UNIVERSITY
Bozeman, Montana
December 1987
ii
APPROVAL
of a thesis submitted by
Jeffrey Rifkin
This thesis has been read by each member of the thesis
committee and has been found to be satisfactory regarding
content, English us a g e , format, citations, bibliographic
style, and consistency, and is ready for submission to the
College of Graduate Studies.
' I-Lr-"-(T 7
Date
IA rpe s o n , G r a d u a t e Committee
Cha^rper
Approved for the Major Department
i T , - 4 - «1
Date
epartment
Approved
/ - / / -y7
Date
for the College
of Graduate Studies
Graduate Dean
iii
STATEMENT OF PERMISSION TO USE
In presenting this thesis in partial
requirements
University,
for a master's
I agree
fulfillment
degree at Montana
of the
State
that the Library shall make it available
to borrowers under rules of the Library.
Brief quotations
from this thesis are allowable without special permission
provided
that accurate acknowledgment of source is made.
Permission for extensive quotation from or reproduction
of this thesis may be granted by my major professor,
his absen ce , by the Director of Libraries when,
opinion of either,
scholarly purposes.
this thesis
Date
in the
the proposed use of the material
is for
Any copying or use of the material
for financial
my written permission.
Signa ture
or in
in
gain shall not be allowed without
iv
ACKNOWLEDGEMENTS
The author wishes
entire Secretarial
to extend his deepest gratitude
staff at the Physics Department
Montana State University.
would never have
Without
their help,
to the
of
this document
seen print.
The author would also like to thank his committee
members Dr. R . T . Robiscoe and Dr.
J . E . DrumheIler who read
carefully through this thesis and offered many suggestions
about how it might be improv ed .
deserves thanks.
Dave MacPhe r son also
His patient advice,
helped unravel mysteries,
J . L . Carlsten,
and a model
Dr.
insight
too many in number to count.
The author is especially grateful
Dr.
and physical
to his advisor.
who was a constant source of inspiration
of excellent
leadership and management.
Carlsten provided an environment where even the weakest
of students could inevitably excel
himself fortunate
and the author considers
indeed for the opportunity to participate.
V
TABLE OF CONTENTS
Page
1. INTRODUCTION................................
1
2. DERIVATION OF THE RAMAN SCATTERING EQUAT IO NS ........
5
Int roduction .........................................
Response of the Me d i um .. .. .........
Response of the F i e l d ........................
5
8
12
3. AC CO UTE RME NTS ..........................
Equipment and T e c hn i q u e ............................
The L a s e r ..........................................
The Me d i u m ..............
4. T H E O R Y ... .....................................
Ph y s i c s .................. ................... -........
Computer M o d e l .......................................
5. EX PER IM ENT S........
Gain Enh anc e me nt .......................
Measurement of Beam P r o f i l e s . ......
Measurement of Zero of Optical D e l a y ..............
6 . EXPERIMENTAL RE S U L T S ....................
General Re s u l t s ................................... . .
Gain Enhancement at 100 Atmospheres..... .........
Zero of Optical D e l a y ...........................
Gain Enhancement at 10 At mospheres................
7. COMPARISON WITH T H E O R Y .................
Introd uct ion .............
100 Atmos ph ere s. ... ......
10 At mos phe res .......................................
8 . CONCLUSI ON S......
REFERENCES CI T E D ........................................
17
17
21
31
37
37
39
45
46
48
49
51
51
54
55
57
61
61
64
67
71
73
vi
TABLE OF CONTENTS— Continued
Page
A P P E N D I C E S ...............................................
78
Appendix A
- Relative
Size ofA L ................
Appendix B - Transformation to the Travelling
Reference F r a m e .......................
Appendix C
- Computer
P r o gr am s. ................
79
81
84
vii
LIST OF FIGURES
Figure
Page
1.
Energy level diagram for stimulated Raman
scat ter ing .................................
6
2.
Self consistency di ag ra m ...........................
7
3.
Schematic of experiment to measure
magnification of IOcm lens. ............... .........
19
Digitized plot of the results of the
magnification
experiment...................
20
(a) schematic of the XeCl laser and (b ) an
energy level diagram of the XeCl b o n d ............
22
Digitized plot of a spectrograph showing the
spectral lines of the XeCl
laser. ........
24
7.
Temporal profile of a typical
laser p u l s e ........
26
8.
Schematic of the Mach— Zehnder interferometer
used to measure the laser and Stokes
I in ewidths...................... ......... ...........
27
Interferogram made with the Mach Zehnder
interferometer and a least squares fit to
the d a t a .................................
29
Autocorrelation function for (a) the Stokes,
(b) the pump and an exponential fit to the data. .
30
Schematic of experiment to check for diffraction
limited operation of the XeCl laser ...............
32
Superposition of the theoretical profile for a
diffraction limited beam at focus and the
spatial profile of the focused XeCl b e a m .........
33
13.
Energy level diagram for H g . . . ....................
35
14.
Comparison ofexperimental and theoretical
values for gain enhancement as a function of
optical de l a y ................... .....................
41
4.
5.
6.
9.
10.
11.
12.
viii
LIST OF FIGURES— CONTINUED
Figure
15.
Page
Comparison of experimental and theoretical
values for correlated and uncorrelated
gain enhancement as a function of input
pump energy. ... ; .... ...............................
42
16.
Flow chart for the program Mode ............ .......
43
17.
Schematic of the gain enhancement ex pe riment....
47
18.
Schematic of experiment to measure zero of
optical d e l a y ........................................
50
Profile of Stokes beam through the amplifier
c e l l ....... ..........................................
52
Profile of pump beam through the amplifier
ce l l ........................ ................... ......
53
Experimental data for the autocorrelation
function used to determine the point of zero
delay from the zero of optical delay
exp eriment ...........................................
56
Gain enhancement profile used to determine the
location of the enhancement peak from the
experiment to measure zero of optical
d e l a y ........ ........................................
58
Experimental and theoretical results for gain
enhancement as a function of input pump energy
at 10 atmosphere s .... ........... ...................
60
Theoretical comparison showing the relative
size and spacing of the laser modes and the
Raman I inewidth......................................
62
Theoretical comparison of size of the laser
bandwidth and theRaman I inewidth... ..............
63
Plot of the intensity of the theoretically
calculated laser field resulting from the
superposition of 64 0 m o d e s .........................
65
19.
20.
21.
22.
23.
24.
25.
26.
ix
LIST OF FIGURES— CONTINUED
Figure
27.
Page
Superposition of the traveling and laboratoryreference fr ame s............
82
28.
Computer program L f i e l d s ........................
85
29.
Computer program M o d e .....................
89
X
ABSTRACT
Gain enhancement in an excimer-pumped Raman amplifier
consisting of molecular hydrogen is experimental Iy measured.
Gain enhancement is measured both as a function of optical
delay and input pump intensity, and in the two limits when
the laser mode spacing is large, and comparable to the
Iinewidth associated with the Raman medium.
,
Experimental results are compared with the predictions
of a transient multimode theory.
This theory is based on
the coupled equations of Raman scattering which describe the
transient growth of the Stokes field and coherence of the
Stokes transition.
A
sum over longitudinal modes with
fixed and totally random phases is assumed for both the pump
laser and Stokes fields.
The theory compares well with experiment in the regime
where the Raman linewidth is large compared to the
modespacing of the laser.
The theory also makes the
interesting prediction that in this limit, the broadband
gain will be larger than the gain for a narrowband laser;
I
CHAPTER ONE
INTRODUCTION
Raman scattering was first observed in 1928 by C.V.
Raman*.
A Raman medium is defined as a medium capable of
coupling optical
rotational,
frequencies which, differ by a vibrational,
or electronic frequency.
Raman scattered light
is referred to as either Stokes or anti-Stokes radiation 2
depending upon whether the coupling downshifts
or upshifts
the frequency of the scattered light.
The
laser or pump used for Raman scattering
categorized in one of two ways,
(monochromatic)
describe
or broadb an d.
the broadband pump.
phase diffusion,
fluctuations
either narrowband
Numerous models exist to
The three most common are the
chaotic, and multimode models.
models all describe
can be
These
the pump as a broadband wave with either
in amplitude
and/or fluctuations
phase diffusion mode I^ assumes
in phase.
Thp
constant amplitude with
randomly fluctuating phases , e.g. E( t )=Eco s ((i)t-kz+<t> (t) ) ,
where E is the amplitude and <(> is the p h a s e .
The chaotic
mode I^ assumes both randomly fluctuating amplitudes and
phases , e.g. E( t )=E (t ) co s (<ot-k z+<(>(t ) ) , while
the multimode
mode I^ uses a distribution of widely separated modes,
in both amplitude and phase
fixed
so that B( t )=^En Cos ((on t-kn z+<j)n ) .
2
.When the pump is of low enough intensity,
spontaneous Raman scattering
will
take p l a c e .
(spontaneous
emission)
As pump intensity is increased,
Raman scattering or SRS will occur.
two categories:
only
or SE
stimulated
SRS can be divided into
Raman generation and Raman amplification.
The Raman generator has been studied by numerous
authors4-14.
The result
from these studies relevant to this
work is the prediction that in certain cases,
Stokes radiation,
the generated
or field, may be correlated with the pump
during its growth from SE in the
generator.
By correlation,
one means that fluctuations in the Stokes field correspond
to fluctuations
in the pump field.
This is predicted to be
the case for the I) phase diffusion model and 2 ) both the
chaotic and multimode models in the limit when the laser
bandwidth,
is larger than the Raman Iinewidth,
Raman amplification,
Fr .
although studied by an equally
large number of authors15~ 34 has been mostly theoretical,
rather than experimental
in n a t u r e .
Numerous
applications
of. Raman amplification however have been exploited.
of spatially poor quality pump
conversionl5-19,
lasers through Raman
and access to high energy beams
multiplexing of several pump
Cleanup
through the
lasers with a single Stokes
beam in a Raman amplification m ed iu m18-22 are two of the
most
common applications.
Early work on Raman amplif ication 3 >23-29 using models
derived from the phase diffusing and multimode models
showed
3
that the amplified Stokes spectrum was always narrower than
the pump spectrum below a critical pump intensity.
this critical
even equaled
intensity,
Above
the Stokes spectrum approached or
the pump spectrum in width.
It was also
discovered that the gain for broadband amplification was
nonlinear and always
less than the narrowband
critical pump intensity.
These
gain below the
two effects were attributed
to dispersion in the Raman amplification medium and hence
related to the spectral width of the pump.
intensity was increased above the critical
As pump
intensity,
.
broadband gain became independent of both pump bandwidth and
dispersion,
value.
and consequently approached the narrowband
Raymer et a I .3 , using the phase diffusion model,
predicted
theoretically that the pump and Stokes fields may
become
correlated in ph a s e , which leads to monochromatic
gain.
Lombardi
et a I .3 0 f later showed experimental Iy that
the Stokes does indeed acquire
the same phase as the p u m p .
Vokhnik et a 1 .3 1 and Stappaerts
authors to investigate
the effects of correlation between
the pump and injected Stokes
These authors
correlated,
et a I .3 2 were the first
seed on gain
en hancement.
showed that when the two fields were perfectly
the gain enhancement is a maximum and approaches
the narrowband value.
Both Agarwal33 and George s34 using
the chaotic model made
the interesting prediction that when
the pump linewidth is narrower than the Raman I inewidth,
when the two beams are well
correlated,
gains which are
and
4
larger than the monochromatic
It appears
gain are to be expected.
that no single model
will describe every experimental
for Raman scattering
situation and each
researcher must decide which model will work for any given
laser and each particular experiment.
a X e C I laser
(described in chapter three)
Raman scattering.
at describing
In Dr.
is used to do
This thesis results from a first attempt
the radiation from a XeCl
interaction with the Raman medium.
random-phase model
Carls t e n 1s lab
is developed,
effects of correlation,
on the broadband gain are studied.
theoretical predictions
A multimode-fixed-
and using this model,
linewidth,
performed and experimental
laser and it's
the
and input pump intensity
Experiments are
results are compared with the
of this model.
5
CHAPTER TWO
DERIVATION OF THE RAMAN SCATTERING EQUATIONS
Introduc t ion
A Raman medium is one capable of coupling optical
frequencies which differ by a vibrat io na l, rotational, or
electronic
frequency.
The body of the work presented in
this thesis will be concerned with vibrational Raman
scattering;
thus a brief description is appropriate.
Raman scattering
is a two photon processes.
Physical
insight into vibrational Raman scattering is obtainable with
a simple model.
Consider the three level molecule
in Fig.I.
When a pump photon of frequency u>p (also referred to as u>L )
is absorbed by the molecule,
is emitted and the molecule
difference
frequency
a Stokes photon of frequency 0»s
is left vibrating at the
, where
3 is the energy
difference between the molecular levels one and three.
difference
This
frequency is referred to as the Raman shift.
A theoretical
description of Raman scattering
is
considerably more complicated since both Sc hr od i n g e r 's
equation and Maxwell's
equations must be used.
A feel for
the theoretical approach may be obtained by referring to the
block diagram in Fig.2.
and Stokes fields
The interaction between the pump
in the medi urn is described by
6
F i g u r e I.
E n e r g y level d i a g r i m for s t i m u l a t e d Raman
scattering.
A three level molecule interacts with a pump
photon of frequency u»p and a Stokes photon of frequency ui. .
The molecule is left vibrating at the difference frequency
toIS-
7
Punp
Laser
S c h ro d in g e r's
eq u atio n
Nonlinear
P o la riz a tio n
S to k e s
Field
\
\
Maxwell's
eq u a tio n s
Figure 2.
Self consistency d i a g r a m for s t i m u l a t e d Ra m a n
scattering.
The p u m p and Stokes fields interact via
S c h r o d i n g e r 1s equation
and
produce
a non-linear
pelariz ation in the me di um .
The non— linear polarization
then drives the Stokes field via Marvell's equation.
8
S ch ro din ge r's equation.
medium results
A nonlinear polarization in the
from this interaction and becomes the driving
term in.Maxwell's wave equation.
completely describes
The nonlinear polarization
the macroscopic response of the medium
to the incident fields, while Maxwell's wave equation
completely describes
medium.
the response of the fields
to the
The remainder of this section will be devoted to
the details of this process.
Response of the Medium
One begins by writing out the wave function for a system
of three
level molecules
(described in chapter three)
interacting with a laser field of frequency H m l and a Stokes
field of frequency hwg.
An energy level diagram for the
system is shown in Fig .I
9 = aiexp
Following Raymer e t a T^ we let
1^ l U 1 + B2 O x p -1 ^ l + 5 L.) u 2 + a3 ex p ~ 1 (51+ 5 L - $ S *U 3 . (I )
The a^ are the amplitudes of level
and the
is equal
to w^t-k^z
are the stationary states having energies iTwi.
Since we are free to choose
convenient to choose
phase,
i,
the zero of energy it is
it to be the ground state with the
$i=0 .
The Hamiltonian for the system is
H = H0 + H'.
(2)
H q is the Hamiltonian for the unperturbed system having
eigenvalues
describes
while H',
the perturbing Hamiltonian
the interaction of light with the system and is
9
given by the dipole
interactions 5
H 1 = |t•E = -ex~( Be xp i ® + c.c.).
(3)
We let the system evolve by putting # into
Schrodinger's
equation,
Ilty=Hijr.
I K a l 5J1 + [a2 -ia 2 toL3exp
This yields
LD 2
L 5 s ^Ug)
+ I-a3~ia3 ((o l - cos) I exp
(4 )
- V a 1 U 1 + (Iw 2 +V)a 2 exp
^^U 2
+ (Itog+v)agexp 1 (toL - O S )Ip3 ^
Perturbing
the system with H'
finding populations
creates the probability of
in excited states.
populating excited states
The amplitude for
is determined by evaluating
iKty I ty> = <ty I H I ty> .
Collecting terms proportional
(S)
to U 1 we obtain
(6 )
111I - Vllal + VllaZexp-i^k + V13a3exp- 1 ( $ L - $ S , •
The dipole interaction only couples adjacent
states,
therefore V 11 = Y-^g = Q, and E q n .6 simplifies to,
iaI = -Q i 2 a2
where
(7)
the Rotating Wave Approximation
(RWA)SS has been used
and Q 1J =eXj[ j E/21 is the Rabi f r e q u e n c y ^ *
Similarly,
we
find
ia2 = Aj_,a2 - Q 2 1 ai - $223 aS
( 8)
1 ag = agAg - Qg 2 a2 •
(9)
and
describes how far the laser is tuned from the
resonant 1-2 transition and is assumed to be very large for
the purposes of this derivation
(see appendix A), while
10
A S-<I)3- ^m L -m S ^ is the detuning for the Raman transition and
assumed to be negligibly small.
Determination of the population
requires
the simultaneous
equations 7, 8 , and 9.
first.
Iai I2 of level
i
solution of the system of
It is convenient to solve E q .8
Integrating both sides of this expression yields
a2 (t) = / G ( t ')explAL ( t ' t)dt'
0
(1 0 )
where G (t ')=i (Q i 2 E 1 + Qg 2 Sg).
Integrating by parts
gives
G(t) - G(O) exp i^ L t
G (t ) - G(O) expa2 (t) = --------------- ---------------iAL
U A l )2
(1 1 )
G (t ) — G(O) exp ^a L *
U A l )3
The second and third terms
in this expression will be small
compared to the first term for large A l and can be ignored.
At t=0,
S 1 (O)=I and S 2 (O)=Sg(O)=O so that G(O)=- IQ 12 S1 .
Thus
(Ql2Si + Qg2Sg)
Q 12S1 Bxp iAL t
a 2 (t) --------------------- ---------.
AL
(12 )
aL
For large Al , the second term will be oscillating very fast
compared to the first term and can be ignored so that
(Q12aI + Qg2 S3 )
a 2 (t ) * :
---------------- .
AL
This
is tantamount
conveniently reduce
to a 2 « 0 .
(13).
These approximations
the three level problem to a two level
11
problem;
the system can be considered to be oscillating
between levels I and 3 only.
Rewriting equations 7 and 9 with the expression for a2
given by E q .13 we obtain
aI =
„
and
i(|Gl 2 l2 a i + GI 2 O 32 S3 )
---------------AL
(14)
■•
i (^32^12al — I ® 3 2 I ^a3^
a3 = --- — ------------------ •
AL
Again,
equations
(15)
rather than solving these two simultaneous
it is expedient
to introduce
the density matrix
formulation^® by defining
-
Q
*
= 2 aIa 3
(16)
where Q* is related to the off diagonal
elements
and has the physical
density matrix
interpretation of a coherence
between the states a^ and a2 .
Additionally,
with this
formulation it is possible to include the effects of
collisions between molecules
collisions
in the m e d i u m .
The effect of
is to introduce dephasing between the ground and
excited states which in turn destroys the coherence between
these st a te s.
The time evolution of the coherence
is given
by
9Q
_
\
»
i2Q32fi12
»
- 2 (a^a 3 + S^a3 ) - TQ
« -------- - FQ
3t
(17)
aL
or
6 Q*
—
= -FQ* + iklE SEL
at
with k
d12d23
2 a L1j2
(18)
12
where terms proportional
neglected
, collisional
to ag have been assumed small and
dephasing
is accounted for by the
inclusion of the damping term P, and the d ij are the matrix
elements e
Stokes
|X I Uj >.
E q .I 8 tells us that the pump and
fields create
coherence
between molecules destroy it.
microscopic response
The macroscopic
in the medium and collisions
Eq .18,
then describes the
of the medium to the applied fields.
response of the medium is manifest in
the polarization which is given by
P iJ = N< 9 i Id'XlVj > + c .c .
(19)
where d •X is the dipole moment per unit volume and N is the
number of dipoles contained within the v o l u m e .
component of the polarization which drives
The
the Stokes field
is given by
P23 = N<^2 l d * Xl
With the wave
> + (:.«;.
( 20 )
function given by E q. I along with Eq.13 and
E q .1 6 the polarization can be written in terms of the
coherence Q * , yielding
P2 3 = N d 2 3 (0 2 1ci<’exP i$S + Q 2 lQ*exP-i$S ^ •
So, using
the pump and Stokes
fields
(2 1 )
in Schrodinge r 1s
equation we have created a polarization in the medium,
thus completely described the macroscopic
and
response of the
medium to the applied fields.
Response of the Field
The polarization drives the Stokes field via Maxwell's
13
wave equation whose derivation follows.
Maxwell's
We begin with
equations^?
dB
I. V x E = - —
II. V x E = J
at
3D
+ —
III. V ' D = p
at
and O h m 1s law
IV. J = <rE
along with the relations
V. B =
VI. D = E q E + P
(iH
and the vector identity
(V-A) -V 2A.
VII. V = V = A =
Substituting V. into I . and taking the curl, of both sides
results
in
a (V=H)
V = V = E = -p —
at
Applying
the identity VII.
and substituting IV. and VI.
BE
B2E
B2P
(V" H) - V 2E = - (io — - p e q —
- p— .
at
at 2
(2 2 )
at 2
The polarization is now separated into linear and nonlinear
p a r t s , that is, P=P l + P ^ ^ .
by37 P jj= E q X l E, s o
The linear polarization is given
that Eq.22 can be rewritten
BE
B2E
3 2 Pn l
V 2E - (V* B) - per— - pe—
= p—
at
where
(23)
at2
e=(l+xL )eQ is the permittivity^^ of the medium having
a linear susceptibility Xl media,
at 2
For wave propagation in gaseous
there are no free charges which means
conductivity a , and the divergence
that both the
of E can be taken to be
14
zero.
Then,
writing
fie as I/v 2 , Eq-. 2 3 simplifies
I 92ES
4.jr52p23
V 2®S - — —
- ---v 2 dt 2
c 2 d t2
where
(24)
the Stokes field Eg Jias been substituted for E, the
polarization from E q .21 has been substituted
for
aQ(j a
transformation to Gaus si an unit $38 has been m a d e .
Equations 18 and 24 are sufficient to describe
stimulated Raman scattering,
though E q .24 is more commonly
written as a first order differential
slightly different form.
equation and in a
This transformation will be
derived next.
Considering propagation in the z direction only,
the
Stokes field can be written
Eg = -(EgeXp
2
where $g=wgt-kgz
phase.
+ c .c . )
(25)
and Eg is complex with a slowly varying
Substituting
this field into the left side of Eq.24
yields
-ikggxp
where
i$ 0 —
k DEg + c .c.
(26)
the Slowly Varying Envelope Approximations 6 (SVEA) has
been evoked, and D is the differential operator defined by
9
I a
D = _ + ----az
v at
Writing out the complex exponential
term,
and letting a=DE and U ft=DE*,
(27)
as a real
and imaginary
the expression given by
15
26 can be rewritten
-ik s [(a-a )cos$s + i (a+a* )sin$gJ .
Similarly,
(28)
the right side of.E q .24 can be written
2 nNd 2 3 wS
[ (p+P*)cos$s + i(p-p*)sin$s ]
(29)
C6A1
where p fi^2 Q » P
note,
an<l SVEA was again invoked.
But now
for any complex number y=x+iy,
.
Y+y
x = Eey = ---
*
Y-Y
and
y = Imy
(30)
i2
so that
_
_
—N2ntogVgd23
ImDEcos$g + ReDEsin$g =
--- -------- (R e Q ^ Q ^ c o s ^ g
C^A 1
(31)
- ImQj 2 Q sin@g ).
Using
the results
of Eq.30,
E q .31 can be written as
NnQtogvgd12 A^g
-ik2Q Ej^
with k,
(32)
Ic 2 A 1
E q .32 is the complete wave
field.
equation for the Stokes
It tells us that spatial
and temporal
changes in the
propagation of the Stokes field are driven both by the
coherence and the pump laser.
It should be noted that an
equivalent expression to E q .3 2 can also be written for the
pump field,
but this is of no interest in the present work,
which is concerned only with the evolution of the Stokes
field.
In summary then,
the two equations,
complete description of the problem.
18 and 32,
give a
They describe,
as
16
promised,
the response of the medium to. the applied fields
and the response
of the fields to the medium.
17
CHAPTER THREE
ACCOUTERMENTS
Eqnipment and Technique
Spatial profiles and interference patterns,
the next
section of this chapter,
discussed in
were digitized using an
Reticon 11G " series self— scanning linear photodiode
array.
The array uses 1024 photodiodes on 25pm centers.
The arrays are typically encased by a glass window.
glass strongly attenuates
Since
the UV and adds the additional
complication of interference
from its two surfaces,
the
arrays used in these experiments were specially ordered
without w i n d o w s .
The array collects
a scan.
Each scan is triggered by a read pulse
4ms to complete.
external
.5 H z .
light continuously and is cleared by
In practice,
and takes
the array is triggered by an
source with a repetition rate of approximately
To reduce
the background arising from stray room
light, the array is read twice.
At the onset of the read
pulse
the array scans once which clears
wait,
scans again.
it and after a .2 ms
The second scan is read either with a
storage oscilloscope or with digitizing electronics which
send the information directly to a computer for storage.
When using the photodiode
array to measure
spatial profiles
18
of the excimer laser,
a 4.2ms delay is introduced between
the trigger and the laser to insure
begins after the arrival
Usually,
that the second scan
of the laser pulse.
the spatial profiles of interest are
inordinately small
in diameter and in utterly inaccessible
locations e.g.
inside of a Raman cell.
special
an imaging technique developed by Dr.
cases,
Carlsten was u s e d .
in the beam,
In these not so
An f= 1 0 cm focal length lens is inserted
12 cm from the location of interest and the
photodiode
array is located at the image distance,
60cm from
the lens.
The photodiode array "sees" a magnified image of
the beam 72cm from the point of interest.
An experimental measurement of the magnification was
made using the apparatus
shown in Fig.3.
A Imm aperture was
placed in the attenuated
laser beam which had been clipped
"upstream" with a 4.5mm aperture to obtain a diffraction
limited beam,
as discussed in the next section.
The
photodiode array was initially placed immediately in front
of the aperture and a measurement was made to confirm that
the diameter of the aperture was I m m .
Next,
the f=1Ocm lens
was placed in the system and the aperture was imaged onto
the array where another measurement was made.
A digitized
plot of the measured results are shown in figures 4 a for the
actual
aperture and 4b for the image of the aperture.
One
calculates the magnification by simply taking the ratio of
the image to object distance,
which for this case is 5.
The
19
Attenuator
a)
4/
-> - L Photodiode
Laser
-> b f
Array
'—
Aperture
b)
Laser
f=10cn
->
->U
Photodiode
Array
K— X -------------------*
12cn
72cm
F i g u r e 3.
Schemetic
of e x p e r i m e n t
to m e e e n r e
the
magnification of the f*10cm lens.
The laser was attenuated
with an appropriate dielectric filter and then clipped with
* Imm aperture.
The aperture was first measured (a) with a
photodiode array, and then (b) the image of the aperture was
projected onto the array.
Relative Intensity
20
Figure 4.
Digitized plot of (•) the aperture measured with
the photodiode array and (b) the image of the aperture
measured with the array.
21
experimentally measured value was 5.13.
Energy measurements were made with a Laser Precision Rj72 00 energy r atiome te r, Rj P-7 00 energy pr o b e , and Rj I serial
to parallel
interface.
An HP— 85 computer was interfaced to
the forementioned electronics which were used in combination
to collect
the data.
In the next section,
shown.
These
several plots of laser lines are
lines were resolved with a Bausch and Lomb
spectrograph which had a second order dispersion of 7.5 A/mm
and second order spectral
photodiode
range of 185 0-3 7 00 angstroms.
A
array was placed at the output of the
spectrograph and monitored with a Tektronix 2230 100MHz
digital
storage oscil lo sco pe .
The spectrograph,
inadequate as a device to accurately measure
though
the width of
each laser line, was immensely useful as a monitor of
changes
in I inewidth during
the course of an experiment.
The Laser
The laser used in this thesis work was an injection
locked Lambda Physik EMG-150ET, which is a pulsed,
discharge
Fig.5a.
laser.
The
gas
A schematic of the laser is shown in
laser uses an oscillator-amplifier
configuration.
The oscillator light,
after passing through
a number of intra-cavity Brewsters prisms becomes
significantly narrowed and polarized to better than 90% in
the horizontal.
The
light,
after leaving the oscillator.
Energy
22
In te r n u c le a r Distance
Figaro 5.
(a) schematic for the
level diagram for the Xe Cl bo n d .
Xe Cl
laser
and
(b)
energy
23
passes
through two Fresnel
rhombs which rotate the
polarization from the horizontal
the light is injected
unstable resonator.
the same vertical
oscillator
The
into the amplifier,
and then
a three pass
The amplifier light becomes
locked into
state of polarization as the injected
light with an efficiency of better than 80%.
laser's active medium,
halogen gas,
is inert,
into the vertical
a gas,
chlorine and the noble
consists of the
gas,
xenon.
Since xenon
the ground state for the laser is repulsive and
the molecules bond only in the excited state forming
excimers39
state).
(molecules which exist only in the excited '
As is shown in Fig.5b,
lasing only takes place on
the transition between bound and unbound states with the
ground state always empty.
Thus,
the excimer laser is
considered to be very efficient.
According to the Lambda Thysik catalog40,
spectrum consists of four
and 308.15nm,
the laser
lines, two strong lines at
and two weak lines
307.9nm
at 307.6nm and 308.56nm,
along with a weak subsidiary line at 308.2nm.
Shown in
Fig.6 is a plot of the digitized output from the
spectrograph described in the previous section.
in Fig.6a,
we typically see only the 307.9nm,
308.2nm lines.
As is shown
308. Inm and
By tuning
in the horizontal the rear
mirror
of the oscillator cavity,
we are able to select to a
certain
degree which frequency light returns to the Brewster prisms
at the appropriate angle for maximum transmission.
Thus we
CO
I^
oq
o)
M
O
n
O
m
o
n
q
M
o
r>
o
CO
CO
ro
o
n
CM
00
O
n
m
CO
o
n
308.6
Relative Intensity
24
Wavelength X
Fignre 6.
Digitized plot from the spectrograph shoving (a)
the three lines of the IeCl laser and (b) the result of
tuning the laser on the 307.9nm line.
25
are able to tune between the three
the laser on the 307.9
doing so,
lines.
We chose to tune
line and as is shown in Fig.6b,
in
suppress the other two.
The temporal profile of the laser was measured with a
Hamamatsu R1193U-02 phototube and a Tectronix 7834 500MHz
storage oscilloscope.
profile
A digitized plot of the temporal
is shown in Fig.7.
By approximating
the pulse as
rectangular with a height equal to the peak power and an
area equal
to the total energy,
the pulse width was
calculated and equal to 15.3ns.
A measurement was also made of the laser Iinewidth.
make
this measurement,
To
a Mach— Zehnder interferometer shown
schematically in Fi g. 8 was constructed.
The path length of
one beam was varied relative to the other using a three
mirror corner-cube mounted on a linear motor drive which was
originally constructed for the gain enhancement
experiment
discussed in chapter 6.
The output of the interferometer,
was a series of
interference profiles dr interfere grams.
was for a different path length difference
with a linear photodiode array.
Each interferogram
and was recorded
An example of an
interferogram is shown by the solid line in Fi g . 9.
The interference
fringes are described by the irradiance
fun c tion41 which is defined as
I(x) = I1 + J2 + 2(Iil2)1/2Rey(x).
Y (x ) is the normalized autocorrelation function and is
(33)
Relative Intensity
26
Time
Fi gare 7
Temporal
profile
of a typical
laser pul se
27
DELAY
P H O TO D IO D E
ARRAY
Figure 8.
Schematic of the M a ch-Zehnder interferometer
to measure the I inew idt h a of the input Stokes and pump.
used
28
Diode N um ber
Figure 9.
Interference profile g e n e r a t e d with the HachZehnder interferometer.
The solid line is the actual data
and the dashed line is a least squares fit using eqn.33.
For this data Iy I was .91.
29
given by
<
(x )E 2 (x +Ax ) >
(IlI2 )172
so that
R e y i 2 (X) = Iyi2 Icos(ax-*)
(34)
where a is the periodicity of the irradiance
function I (x )
and * is the phase.
A least squares fit was made
profile using
equations 32 and 33 with
the fitting parameters.
dashed line
to each interference
in Fi g. 9.
A typical
Iy(x)I, a, and * as
fit is shown as the
A plot was then made of the magnitude
of y as a function of path length difference
the X 's in Fig.10b.
The auto correlation function and the
power spectrum form a Fourier transform pair,
Iyl with an exponential
Fig.11b,
results
and is given by
thus fitting
as shown by the solid line
in
in a Lorentzian spectral distribution.
exponential
fit had a half width at 1/e2 of 6.9mm which
corresponds
to .47cm""1 HWHM for the Lorentzi an frequency
distribution.
The coherence
The
length for the laser can also
be obtained from Fig.IOb and is given by the HWHM at 1/e
which is 3.5mm or 23 .5 p s .
The length of the oscillator cavity of the laser was
measured to be 1.43m which corresponds
mode
spacing 8, of
longitudinal
.0035 cm- -*-.
to a longitudinal
Thus there are 134
laser modes out to the point at which the
Lorentzian frequency distribution has fallen in intensity by
30
-10
0
10
Position X(mm)
Figure 10.
M a g n i t u d e of the n o r m a l i z e d au to correlation
function y , for (a) the Stokes beam and (b ) the laser b e a m .
The X 1s re p r e s e n t spe cific v a l u e s of Iyl for di f f e r e n t
interferometer path length differences and were generated as
p a r a m e t e r s from fitting a series of interference profiles
simlar to Fi g. 9.
The solid line is an exponential fit to
31
5 0%
In all of the experimental work performed
thesis,
the laser was
for this
clipped with a 4.5mm aperture
resulting in a flat tophat spatial distribution.
The laser
operating in this configuration is near diffraction limited,
where a diffraction limited beam is defined as one which is
a plane wave at fo c u s •
A simple check for diffraction
limited operation can be made by making use of the fact that
the image focal plane for a lens is the Fourier transform of
the object
focal plane^^.
To determine diffraction limited o p e r a t i o n ^ t the
apparatus
in Fig.11 was u s e d .
A lens, LI,
of focal length
198.6 cm was inserted 3 cm from the aperture.
The profile at
focus was imaged with a second lens, L 2 , onto a linear
photodiode array.
The resulting pattern is shown by the
solid line in Fig.12.
The Fourier transform of a flat
tophat spatial distribution is given by the Airy p a t t e r n ^
Kr)
= I0
where ?-2nr/Xf and I q is the peak intensity at fo c u s .
dashed line in F i g »12 is the calculated profile.
The
The two
compare well which shows that the beam is near diffraction
limited.
The Medium
The medium used for scattering in this thesis work was
32
LI
E M G - 150
INJECTION LOCKED
XeCI L A S E R
Fl
LINEAR
— A -fh PHOTODIODE
— AMJA| I*
191.6 cm
ARRAY
^
L2
F2
F i g u r e 11.
Bxperimentil
apparatus
u s e d to m e a s u r e
d i f f r a c t i o n li mi te d o p e r a t i o n of the Xecl laser.
The
incident beam was clipped by aperture Al and focused with a
191.6cm focal length lens LI.
A second lens L2 was used for
imagi ng the p r of i le at focus onto a p h o t o d i o d e a r r a y .
Filters Fl and F2 were used to attenuate the beam.
Relative Intensity
33
—
0.8
-
0.4
Distance r (m m )
Figure 12.
laser beam
theoretical
The solid line is the experimentally m e a s u r e d
at focus and the dashed line is the calculated
profile at focus for a diffraction limited beam.
34
molecular Hydrogen,
H2 .
Molecular hydrogen is a diatomic
homonuclear molecule44 and has energy levels divisible into
three groups or modes of motion4 ^.
levels are electronic
states,
The primary energy
with each electronic
state
divisible into approximately equal groups of energy levels
corresponding to vibrational
st ates.
Each vibrational
in turn exhibits a fine structure attributable
states4 6 .
level
to rotational
An energy level diagram for the H 2 is shown in
Fig.13.
The quantum numbers associated with these
levels are n ,
v , J , and m which are identified respectively as electronic
and vibrational
quantum numbers > and rotational
angular
momentum and it's projection along an internuclear axis^^.
The
selection rules for two photon Raman scattering are
A v = O , AJ=±2,
or Av=±l and AJ=O,±2.
The branches of the AJ=-
2,0,2 are labeled 0, Q, and S , respectively4 ^.
The
selection rules for v come from perturbation theory for
dipole radiation while the selection rules for J arise from
the fact
that the H2 absorbs two photons which each have a
spin of ±1.
At room temperature the separation between vibrational
levels is large
compared to kT so that all of the molecules
exist in the vibrational
states however are
ground state.
The rotational
separated by energies which are
comparable to kT and the molecules have a distribution of
rotational energies with approximately 66% of the population
35
First
Excited
Electronic
State
> .0=;
Ground
Electronic
State
Vibrational
levels E u'
Rotational
levels Er"
F i g u r e 13.
En erg y level d i a g r a m
Eisberg and Resnick, p.430).
for
H,
(a da pt ed
from
36
in the J=I
level^?.
The scattering for this thesis work is from the
vibrational
rotational
while
transition.
level
the initial
the subscripts.
With this notation the initial
is given in the superscripted parenthesis,
and final vibrational
levels are given by
37
CHAPTER FOUR
THEORY
Physic s
-
In this section the equations appropriate
multimode,
described.
fixed— random—phase,
This model
for a
broadband laser are
is a straightforward extension of the
coupled equations of transient stimulated Raman scattering
derived in chapter 2.
travelling
frame
These equations, written in the
(see appendix B) are
d/dz Eg = -ik2Q <,EL
(35)
d/dr Q6 = -FQ* + Ik1E^Eg.
These expressions,
as previously mentioned,
describe the
transient growth of the Stokes field and coherence of the
Raman transition for a medium of three level molecules
interacting with two fields,
a pump,
El , and a Stokes,
Eg,
which differ in frequency by the Raman shift of the m e d i u m .
The retarded time r measures
time
in a reference
frame
traveling with the laser pulse and is given by r=t-z/c,
where
z is a measure of position in the Raman m e d i u m .
Q* is
related to the off diagonal density matrix elements and can
be interpreted as the coherence of the transition between
the initial
and final le ve ls .
T is the Raman linewidth
38
(HffHM) arising from col lisional dephasing in the medium,
and k 2 are constants
For simplicity,
and
given by equations 18 and 32.
it is assume
depletion can be ignored.
that dispersion and pump
Additionally,
we include a delay
?D which corresponds physically to an optical delay between
the Stokes and pump fields.
The solution of equations 35 is
given by3
•n
Es (z,T+Tn ) = E s (0,T+Tn ) + (k1k«z)l / 2 f
exp-1'^ T-'c *)
------------- x
1 2
i [p(T)-p(T')]i/:
,
(
3
6
)
I1 ({4k1k2 z [ p ( T ) p ( T ,)]}1 / 2 )EL (T)EL (T')E g (O ,T 1+rD )dr 1.
The
constant
(k]_k2 z ) ^ 2 is related to the exponential
gain
G , for a narrow band laser with G=ZE^ £ ^ ^ 2 z/F and will be
discussed in the next section.
p(r)
laser pulse up to the time t , and
is the energy in the
is the modified Bessel
function of order one
To extend the theory using a multimode,
phase model,
fixed-random-
the input pump and input Stokes fields are
taken as a sum over longitudinal modes with fixed but random
phases.
The fields are given by
E^( t ) = ^ELn exp^ ^n ^T+<*>Ln^
n
and
(37)
B g (O ,r ) = ]>Eg m (0)expi(,n8T+<t>S m (0) }
where
the input field amplitudes E^n and E g m (O) are fixed
and real.
identical
It is assumed that the mode spacing,
8, is
for both the pump and the Stokes fields,
the distributions
of the initial
random phases,
as are
and
39
^S m (O ) •
The appropriateness of these assumptions requires
consideration but for the purposes
of this work,
their
validity is justified by the success of the model.
Computer Model
The computer programs are printed in their entirety in
appendix C .
The first program Lfields calculates
field and input Stokes field,
the laser
integrated laser energy p (r ),
and the difference p (T )- p (r 1).
The code
is very straightforward.
parameters are loaded e.g.
frequency distribution,
time,
First,
the relevant
the ha Ifwidth of the Lorentzian
pulse length,
and number of modes.
Next,
pulse resolution in
each mode is assigned a
random phase, weighted with a Lorentzian distribution in
intensity,
and normalized.
Though it is not obvious from the code , the following
actually accomplished.
and one narrowband,
Consider
two fields,
is
one broadband
and each normalized to the same energy.
The amplitude of each broadband mode can be written as some
function of the narrowband amplitude,
so that
hhA2
i = nbA 2 (B?)
(38)
where nbA is the amplitude of a narrowband mode
a constant and
and equal to
gives the Lorentzian distribution with
B
(39)
<k Q—fcj>2
(I
+
1L
1/2
40
where C is a normalization constant and T l is the bandwidth
HWHM of the laser.
phase
Each laser mode has an amplitude and a
so that E i (t )=l5bA i e xpi
total field
(i.e.
, where $i = w i t-kiz+*i .
The
the superposition of all the m o d e s ) then,
can be written
EU)
The multimode
= ^ E i (t) = nbA ^ B iBxpl5I .
(40)
laser field is thus written in terms of the
monochromatic field.
Finally,
Lfie l d s
calculates
P (t ) at each instant of time
the integrated
laser energy
(with the input time
resolution)
over the pulselength along with the difference
p (t )- p (T ).
A relatively short pulse length of 4.685ns was
chosen to minimize
the calculation time.
The program Elode calculates
function of input pump en e r g y .
profile
gain enhancement as a
It will calculate a delay
for a fixed input pump energy as shown by the dashed
line in Fig.14, or a table of correlated and uncorelated
gains for varying pump energy as shown by the
Fig.15.
solid line in
Though it is essentially nothing more than a
straightforward numerical
trapezoidal
integration of E q .3 6 using the
integration m e t h o d ^ ,
loops over many different
optimize the algorithm,
involves many nested
array elements.
In an effort to
much of the code became necessarily
ob sc ure .
A simplified flow chart for the program showing critical
decisions and paths
is shown in Fig.16.
Mode begins by
Energy (/nJ)
41
Total Optical Delay (p s)
Figure 14.
Comparison of the experimental (solid line) and
theoretical (dashed line) values of the gain enhancement as
a function of optical delay.
The en ergy of the i n j e c t e d
Stokes and input pump was .43pJ and .42mJ , respectively.
42
EL(mJ)
Figure 15.
Correlated and u n correlated gain enhancement as
a function of input pump energy.
Superimposed on the data
(the X's) is the theoreti ca l result (solid line).
Also
shown is the monochromatic gain (dashed line).
43
Figure 16.
Flow c h a r t for the p r o g r a m M o d e us e d to
calculate the theoretical predictions for gain enhancement
as a function of optical delay.
44
reading the previously calculated values of the laser field
E l (t ), input Stokes
p(r),
field E g (r ), integrated laser energy
and the difference p (t )- p (t ').
Mode also reads the results of another program in which
the Bessel
function I1 O
is calculated using a series
approximation given in reference 48.
over a range appropriate
Values
are calculated
for solving the integral
in MODE
and the results are stored in an array called bdata.
This
program also calculates the slope between the elements of
bdata and stores
these values
in an array called bdif.
the function call F b e s s e l , MODE uses these results,
method of linear interpolation,
bdif to calculate Bessel
the array elements
and the slopes
With
the
stored in
function values which lie between
of bdata.
Next Mode sets the flag IflagZ which is later used to
determine whether a delay profile or gain enhancement
is in use.
Mode then proceeds
gain enhancement experiment,
into the gain loop.
this by varying
In the
gain was increased by
increasing the laser intensity.
accomplishes
loop
The computer program
the constant Cl.
To see how
this is done we rewrite Eq.36 using Eq.40
Eg(z,T+TD ) = Eg(0,T; + TD )
exp-r (? -t ')
+ (k1k 2 z|nbE|2 )1/2EL (r)|
[p(T)-p(-C 1) I1 /2
(41)
r1 ({4k1k 2 z|nbE|2 [p-(T)p(T ')] }1 / 2 )E*(r ')Es (0,r'+TD )dT.
where we have used the fact that nbA 2=InbEl2 .
The constant
45
Cl is related to the gain for a narrowband
laser by
GF
C1 = (— 5)1/2 _ (k1k2 z|nbE I2 )1^2 .
By varying
, we are able to vary the gain.
The optical delay is implemented by shifting the Stokes
field with respect
to the pump field.
is measured relative
Note
that in Eq.41,
This means that time
to the beginning of the pump pulse.
the Stokes field at time t + t d depends on
the laser field during the time interval 0 to x .
Physically,
this means that during the time interval 0<t<TD ,
when the laser pulse precedes
the Stokes pulse,
the medium
"sees" no Stokes field and we have no Raman scattering,
Eg=O.
Conversely,
follows
for •c-'Cj)<t<T, when the laser pulse
the Stokes pulse,
again, no Stokes field is present,
there can be no Raman scattering and E g = O .
place
i.e.
The shift takes
in the do loop to 2 7 00 where !DELAY corresponds to Tjq .
Zzap is the subroutine which contains
calculating
calculates
the integral,
the integrand.
subroutine Bbambi
exponential
while Bbambi
The variable
the algorithm for
is a subroutine which
Iflagl
in the
is used to halt the iteration when the
exp-*'* becomes inconsequentially small.
46
CHAPTER FIVE
EXPERIMENTS
Gain. Enhancement
The experimental
enhancement
apparatus used to collect
data is shown in Fi g . 17.
in chapter three.
The
the gain
laser is described
Two Raman cells were used and both were
filled with H 2 ini.tially at 100 atmospheres and later for a
second experiment at 10 atmospheres.
One was 50cm in length
and was used as a Raman generator and the other was I OOcm in
length and was used as a Raman amplifier.
Raman generator s t u d i e s ^ ,
As in previous
the laser beam was clipped with a
4.5mm aperture to obtain a diffraction limited beam (see
chapter three).
beams.
A beam splitter was used to obtain two
One was brought to focus
in the center of the Raman
generator cell with an f=Im lens.
The Stokes emission from
the generator cell was collimated with another f=Im lens,
passed through an optical
delay (a three mirror corner-cube
mounted on a linear motor drive),
combined with the other
other beam,
I inewidth,
attenuated,
and finally
(pump) beam at a beam combiner.
The
the p u m p , was also used to monitor the laser
power,
and energy.
The pump intensity was varied
/
with dielectric
Stokes beam.
attenuators before
it was combined with the
After the beam combiner,
the two beams were
47
POWER
DETECTOR
SPECTROGRAPH
RAMAN
GENERATOR
XeCI
LASER
STOKES
DELAY
RAMAN AMPLIFIER
ENERGY
DETECTOR
Figure 17.
Schematic
STOKES
DETECTOR
of the gain enhancement
experiment
48
made col linear by centering each beam in a 1.5mm aperture on
each side of the Raman amplifier cell.
To check that the
beams were centered in the ap ertures, we examined the image
of each beam in the respective
was inserted to reduce
aperture.
A beam contractor
the diameters of the two beams.
f=SOOmm focal length lens was used to reduce
diameter and an f= -25 Omm focal
recollimate
An
the beam
length lens was used to
it, which is to say the optics were adjusted to
give as uniform a profile as possible through the amplifier,
cell.
The two beams were then injected into the amplifier.
A measurement of the Stokes
the same Mach-Zehnder
linewidth was also made with
interferometer described in chapter 3.
The optics were exchanged for optics coated at the
appropriate wavelength for the Stokes but in all other
respects
the apparatus was identical
as were
the method of
data collection and analysis.
Measurement of Beam Profiles
The spatial profiles of both the Stokes and pump were
measured at the amplifier cell entrance, center and exit.
The cell was first removed and then an f= 1 Ocm focal length
lens was used to image the desired location onto a linear
photodiode
chapter 3.
array,
using the imaging technique described in
Measurement
of Zero of Optical Delay
An experiment was performed to measure exactly,
the
point at which the two legs of the gain enhancement
apparatus were
identical
in length.
The results
experiment are discussed in chapter 6.
experiment
is shown in Fi g . 18.
of this
A schematic of the
As can be seen from this
figure there were no changes made from the gain enhancement
experiment discussed ab ov e .
As it h a p pe ns , there is
sufficient pump reflection from the surfaces of optics
coated at the Stokes wavelength that the experimental
apparatus could be used,
interferometer.
unaltered,
as another Mach-Zehnder
The pump traverses both legs of the
apparatus and is combined at the beam combiner where the
recombined beams were examined with a linear photodiode
array 90 degrees from the previous direction of propagation.
A measurement
of the polarization through the corner-cube
revealed that the polarization rotates by approximately 10%.
A Glan-Thompson polarizer was
inserted between the
recombined beams and the photodiode array and was used in
conjunction with attenuators
well matched in intensity.
allowing the beams
An attenuator was also inserted
between the recombined beams and the photodiode
block the Stokes radiation.
to be fairly
array to
50
P h o todio de
A rra y
— A2
O ptical
Del ay
Laser
Ranan Cell
Figure 18.
The gain enhancement experimental apparatus was
used as a Mach-Zehnder interferometer to measure the zero of
optical delay.
The filters A2 were used to remove all light
at 3 5 1 n m .
After the beams w e r e re c o m b i n e d ,
they were
projected with lens L2 onto a photodiode array.
A polarizer
P2 was used to vary the intensity of the two beams relative
to one another.
51
CHAPTER SIX
EXPERIMENTAL RESULTS
General Results
Both the Stokes and pump beam profiles were measured
with a linear photodiode
entranc e, center,
are
array at the amplifier cell
and .exit.
The measured spatial profiles
shown in figures 19 and 20.
vary by ^20% through the cell.
The waists were
found to
The profiles at cell center,
figures 19b and 2 Ob were fit to a gaussian profile,
beam waists
(half width at 1/e2 ) of
giving
.28mm for the Stokes and
.33mm for the p u m p .
A pump energy of
corresponds
.42mJ and a pulselength of 15.3ns
to an average power of 27.5kW.
But the average
power is also equal to the peak power density
over the cross-sectional area of the beam,
integrated
that is
OO
2
I
P ave = 1O-^exP 2 (r/^o) 2nrdr = - I q Cjiwq ).
0
2
This expression can be solved for the peak power density I q ,
which at cell
center for a beam waist of .33mm,
corresponds
to 8.OMW/cm2 .
The linewidth of both the Stokes and pump were measured
as described
in chapter 3.
In Fi g . 1 0a,
the autocorrelation
52
-
0.6
-
0.2 0 0.2
0.6
Transverse Distance r ( m m )
t the
Figure 19.
Spatial profile of the input Stokes beam
amplifier (a) cell entrance , (b ) cell center, and (c
cell
exit.
53
-
0.2 0 0.2
Transverse Distance r ( m m )
Figure 20.
Spitisl profile of the input pump beam at the
amplifier (a) cell entrance, (b ) cell center, and (c) cell
exit.
54
function for the Stokes and it's exponential
The Stokes distribution had a HWHM of
the pump was
.47cm- 1 .
beam is not surprising
fit are sh o w n .
.5 8 cm- * while that of
The larger I inewidth for the Stokes
since the Raman Iinewidth at 100
atmospheres of hydrogen^® is 0.08cm-
Gain Enhancement at 100 Atmospheres
Fig . 14 is a plot of the amplified Stokes energy (solid
line)
as a function of optical
is varied,
delay.
As the optical delay
the pump and Stokes beams become correlated and
we begin to see gain enhan cement.
The gain enhancement
continues
a maximum when the pump
to increase,
experiences
and injected Stokes beams become highly correlated,
falls
and then
as the two beams once again become unc orr elated.
the data
For
in Fi g . 8 the incident pump energy was 0.42mJ and
the injected Stokes energy was 0.4 3 pJ .
enhancement
The correlated gain
is approximately 13 times the uncorrelated
v a l u e , and the profile has a temporal width
corresponding to 4.3mm of optical
(FWHM)
of 14.3ps
delay in path length.
A measurement of the correlated and uncorrelated gain
for various pump intensities was m a d e , thus documenting the
dependence
of the gain enhancement on pump energy.
was obtained from a number of plots
different pump energies.
Fig.15,
The data
similar to Fig.14 but at
The results,
(the X's)
shown in
are plotted on a log linear scale so that the usual
exponential
growth will be a linear function of pump energy.
55
The correlated Stokes gain increases linearly from the value
of the injected Stokes and eventualIy saturates due to pump
depletion.
The uncorrelated Stokes gain initially grows
much slower than the correlated gain, but eventually
develops
before
a slope comparable
to that of the correlated gain
saturating.
Zero of Optical Delay
Note
that in Fig .14 on page 41,
gain is the point of zero delay.
An unsuccessful
point
the point of maximum
This result
is artificial.
attempt was made to determine
the zero delay
interferometrica 11y with the experiment
described in chapter 5.
apparatus
A number of interference profiles
similar to those obtained in the Iinewidth experiment and
shown in Fig.10 were obtained.
These,
too, were
fit with
E q .3 3 to obtain the normalized autocorrelation function.
A
plot of the normalized autocorrelation function for this
experiment
is shown in Fig.21.
Though it was
impossible to obtain a fit to this data
directly it seems reasonable
to assign the location of the
visibility peak at 22.6mm with
.5mm error bars.
path at the Stokes
frequency is different
pump.
for this difference
Corrections
the peak by
.702mm.
The optical
from that for the
in optical path shift
The new location for the peak is at
2 1 .9mm ± .5m m .
With this identical
configuration of quartz,
a run was made
56
1.0
0.8
IIIIIIIIIIIII I
I
t
IIII I
IIIII
I
-
—
3^
—
x
0.6 -
JD
X
-
X
-
CO 0.4
>
—
-
0.2
-
-
-
-
X
X
X
X
n
0.0p I » I i » I «i It ii «I «I II II It II I tI Ii II tI II II Ii II li lI lIl i I» II II II II II II I tI II i II I II II II »I I I I » I I I I t § iI lI lIf Il i
' 16 17
18 19 20 21
22 23 24 25 26 27 28
X
I
Position (m m )
Fi gur e 21.
E x p e r i m e n t a l data for the a u t o c o r r e l a t i o n
function showing approximately the point of zero delay: the
point at wh ic h both paths through the gain e n h a n c e m e n t
apparatus are optically equivalent.
57
to obtain a gain enhancement profile which is shown in
Fig.22.
The enhancement peak is located at 19.8mm ± .lmm,
occurring
2.1mm ± 1.5mm sooner than the interferometrica IIy
determined zero of optical
5ps
in time.
Theoretical
from physical
chapter,
that the zero of optical
this measurement
On theoretical
delay is expected
Even assuming the
grounds then, we are compelled to
error bars was too optimistic,
experiment
in the next
indicates that the converse
reject this experimental measurement.
careful
to 7ps ±
along with arguments
both discussed
the gain enhancement peak.
largest error,
is true.
This corresponds
calculations,
considerations,
indicate
to precede
delay.
Perhaps the choice of
but in any case,
another more
is needed either to confirm or refute
this result.
Gain Enhancement at 10 Atmospheres
The gain enhancement experiment discussed
previous
section was performed again in an identical
except that the pressure
The motivation for
this experiment was to study gain enhancement
when the Raman linewidth becomes
spacing.
fashion
in the Raman generator and
amplifier was reduced to 10 atmospheres.
mode
in the
in the limit
comparable to the laser
Since the Raman linewidth in H 2 at room
temperature
depends
linearly on pressure
atmospheres
to greater than 100 atmospheres^?,
pressure by a factor of ten,
in effect,
in the range from 5
reduces
reducing the
the
Total Optical Delay (m m )
Figure 22.
Gain enhancement
the enhancement peak.
profile
showing
the location of
Iinewidth by a factor of ten.
The results of this measurement are given by the X's in
Fig.23.
Both the correlated and uncorrelated gain again
grow from the value of the energy of the injected Stokes
beam,
but in a manner quite different
atmosphere data.
uncorrelated
For this case,
then the 100
both the correlated and
gain seem to be restrained from growth until
some minimum pump energy is reached
correlated
gain).
After this,
comparable
slopes until saturation,
converge.
In the region where
(».3mJ for the
both grow linearly with quite
at which point they both
the correlated and
uncorrelated gains are both linear,
the correlated gain is
approximately one and a half orders of magnitude
then the corresponding uncorrelated gain.
greater
Log Es (AiJ)
60
El (m J)
Figure 23.
Correlated and u n correlated gain enhancement as
a fun ct io n of input pump e n e r g y at 10 atmospheres. The
experimental data are given by the I 1s while the solid line
shows the theoretical prediction.
61
CHAPTER SEVEN
COMPARISON WITH THEORY
Introduc t ion
In this section a comparison is made between the
experimental
multimode,
results of chapter 6 and the predictions
fixed-random-phase
and 37 in chapter 4.
of the
theory given by equations 36
For simplicity,
the input Stokes field
was taken to have the same Lorentzian lineshape, mode
spacing,
and phase
appropriateness
but were not,
distribution as the p u m p .
The
of these assumptions merit consideration,
for lack of time,
explored in this w o r k .
The mode distribution was artificially chopped at 640
modes to make computation tractable.
For a laser
modespacing of .0035 cm- -*-, the magnitude of the
the cutoff point
is about
.14 Ijnax.
This mode
intensity at
distribution
along with the Raman I inewidth at 100 atmospheres
for
comparison is shown on two different scales in figures 24
and 25.
overall
The. individual modes are shown in F i g .24 and the
envelope
is shown in Fig.25.
graphically from these figures,
As one can see
our calculations
at 100
atmospheres are for a Raman Iinewidth large compared to the
mode spacing of the pump and Stokes
compared to their I inewidths.
fields,
but
small
Intensity
62
0.0
Frequency cj(cm ')
Figure 24.
Distance between laser modes (vertical lines)
separated by .0035 ce-1 is small compared to the Raman
linewidth of .OScm-I (solid line) .
Intensity
63
O O ------ '
-
1.5
-
1.0
1— ----1
—
-
0.5
0.0
Frequency
Figure 25.
Lorentzian frequency distribution of the laser
given by the dashed line was chopped at 1.12ca~l and is
large compared to the Raman linewidth of .08 cm- 1 plotted as
the solid line.
64
A random number generator was used to give each mode a
random phase.
For a 4.685ns pulse,
these modes results
Using
modes,
the superposition of
in the spiky envelope
these values
shown in Fig.26.
for the linewidth and number of
and using gain and optical
delay as parameters,
Eq.36
was numerically integrated to give the Stokes field at the
amplifier cell exit.
100 Atmospheres
The dashed line in Fig. 14 on page 41 is the predicted
Stokes energy plotted as a function of optical
delay.
The
profile has a temporal width of 21ps corresponding to 6.3mm
of optical path delay,
and a correlated to uncorrelated gain
ratio of approximately 13,
experimental
result.
comparing well with the
The gain coefficient^!
for this
calculation was 6.8xl0~® cm/W giving a theoretical power
density equal
density.
to 71% of the experimentally measured power
This is reasonable
since the on axis power density
for the pump varies by approximately a factor of two across
the Im amplifier cell.
An interesting
a 2ps asymmetry
indicates
precedes
feature of the theoretical prediction is
in the gain enhancement profile,
a larger gain enhancement when the Stokes field
the pump field.
This effect
transiency of the Raman transition.
precedes
which
the pump pulse,
that
is related to the
When the Stokes pulse
is, when a spike
in the Stokes
Relative Intensity
65
Time (n s)
Figure 26.
Plot of the theoretically calculated intensity
envelope for the laser.
The laser field was given a 4 .6 5 ps
pulse length and results from the superposition of 6 40
longitudinal modes with fixed but random phases.
66
field precedes the corresponding spike in the pump field
(see Fig.26) by some time Tjj which is short compared to the
coherence
time of both the medinm and the pump,
then a
strong Q* with the proper phase will be present in the
medium as the pump spike arrives.
The strong Q 6 and pump
fields are then maximally effective in driving Stokes
amplification.
In these calculations the coherence time for
the medium was 60ps while that of the pump was approximately
14ps so that a 2ps asymmetry is not unreasonable.
The solid line in Fig.15 on page 42 is the theoretical
prediction for the gain as a function of input pump energy
for the correlated and uncorrelated cases.
the data well
The theory fits
in the region before pump depletion.
Note
that in the region in which both the correlated and
uncorr elated gains are linear,
the theory predicts that the
correlated and uncorrelated gains grow at slightly different
rates.
Another interesting prediction of this theory is a
correlated gain w h i ch is larger than the monochromatic
The monochromatic
gain is plotted as the dashed
Fig.15 for comparison.
random,
Since
gain.
line in
the phases of the modes are
this enhancement is not the result of modelocking,
as one might naively expect.
Instead,
it comes from
amplitude modulation in the pump field due to beating of the
various random-phase modes,
and results in peaks
in
intensity which are larger than the intensity of a
67
mono chroma tic field of the same average
intensity.
Since
the gain depends exponentialIy on the pump intensity,
the
correlated gain for a multimode pump is larger than the
monochromatic
gain.
Gains which are higher than
monochromatic have also been predicted for the chaotic
model33'34
The multimode,
fixed— random— phase model
expected to approach the chaotic model
mode
is
in the limit when the
spacing of the pump is small compared to the Raman
Iinewidth.
The results presented here are consistent with
that prediction.
10 Atmospheres
At 10 atmospheres,
the value
for the Raman I inewidth is
smaller than the value at 100 atmospheres by a factor of
ten.
In this regime,
the laser mode
spacing
is no longer
small compared to the Raman !inewidth and the correlated
gain is expected to fall.
The solid line in Fig.23 on page 60 is the theoretical
prediction for gain enhancement at 10 atmospheres.
can readily see,
experimental
exponential
the theory fits the data poorly.
way.
The
data seems to experience a nonlinear
growth below some critical
input pump value for
both the correlated and uncorrelated cases, while
predicts
As one
the theory
that only the uncorrelated gain should grow this
The correlated gain is predicted to grow linearly from
the value of the input Stokes energy in a manner
identical
68
to the 100 atmosphere case.
predicts
where
Though the theory correctly
the slope for the uncorrelated
it grows linearly,
correlated
gain.
gain in the region
it fails to do so for the
Also, whereas
the experimental
data
indicates that the correlated gain should be approximately
one and a half orders of magnitude
uncorrelated
gain,
greater than the
the theory indicates that this value
should be closer to three orders of magnitude,
experimental
twice the
result.
In trying to discover the discrepancy between the
theoretical
and experimental
results,
several
been ruled out as possible explanations.
dispersion.
Because
of dispersion,
factors have
The first is
different frequency
components of the pump and Stokes pulses should accumulate
relative phase
shifts as they propagate
dispersive medium of the experiment.
pump frequencies will be different
Stokes frequencies,
fall because
through the
Since the index at the
than the index at the
one would expect the correlated gain to
the two pulses spread out differently, which is
to say that the two pulses will no longer be well
correlated.
over AX,
In fact though,
dn/dX is approximately constant
the bandwidth of the pump or Stokes pulse and
consequently the change in phase across AX is negligible.
The effect of dispersion will only be to shift the envelope,
of one pulse relative to the other and this can be
compensated for with the optical delay T g .
69
Another possibility is a change
in the spatial overlap
of the input pump and Stokes pulses
in the amplifier cell.
We found the gain enhancement
function of the spatial
overlap or alignment
beams in the amplifier cell
during an experimental
to be a very sensitive
of the two
so that vibration of the optics
run would displace
the two beams
relative to one another causing the gain to fall.
The slope
of the correlated gain however depends not on the alignment
of the two beams,
but instead on the constant k^ given by
E q .1 8 and the input pump intensity.
The input pump
intensity does vary from shot to shot experimentally,
but
time averages to a constant value during an experimental
run.
So,
it would not be surprising to see the experimental
value
for the correlated gain falling below the theoretical
prediction but it should n one-the-1 ess grow with the same
slo pe.
One possibility which merits
serious investigation is
the laser Iinewidth which was also found to vary during the
course of an experiment.
The spectrograph used to monitor
the laser linewidth reveals only gross characteristics
i.e.
whether one line or many lines were present in the laser
spectrum.
laser line.
It was insensitive to the width of the actual
Since
100 atmosphere
several months had elapsed between the
and 10 atmosphere
experiments,
very possibly
the operating characteristics of the laser had changed.
It
may be that the second experiment was run with a different
70
laser linewidth than the first.
case,
Should this have been the
then the discrepancy between the experimental
results
and theoretical predictions would come as no surprise.
Before
anything more
can be learned,
a more
monitor of the laser linewidth is needed.
experiments
measure
sensitive
Future
are planned using a Fabry-Perot to accurately
and monitor the laser linewidth.
71
CHAPTER EIGHT
CONCLUSIONS
Conclusions for this thesis have already been given in
previous
chapters ; only a brief summary will be given here.
A multimode
fixed— random—phase model works well
gain enhancement with a XeCl
atmospheres.
enhancement
The model
to describe
laser when pumping H 2 at 100
successfully describes
gain
as a function of optical delay (correlation
between the pump and Stokes fields)
a function of input pump intensity.
and gain enhancement as
The model
also predicts
that the gain enhancement will be larger than the
monochromatic
gain.
Larger than monochromatic
also been predicted when a chaotic model
pumping has been used33 ,34 #
gains have
for broadband
It is expected that the
multimode fixed-random-phase model becomes approximately
equivalent to the chaotic model
in the limit when the laser
modespacing becomes much smaller than the Raman linewidth.
At 100 atmospheres of H 2 the Raman linewidth is small
compared to the laser linewidth but large compared to the
laser mode spacing
support
thus the results presented in this work
that prediction.
The model
is much less successful
enhancement at 10 atmospheres
at predicting gain
of H 2 when the Raman linewidth
72
becomes
comparable
to tb e 'mode spacing of the laser.
be that the theoretical prediction differs
experimental
It may
from the
results because the value used for the laser
linewidth in the calculations was
incorrect.
The bandwidth
of the laser was found to vary dramatically with time so
that the linewidth of the laser measured during the 100
atmosphere experiment- and used for all calculations in this
thesis- may have been different
than the linewidth of the
laser during the 10 atmosphere ex pe riment.
experiments are planned which will
include
Future
a Fabry-Perot
interferometer to monitor the linewidth in real time.
The multimode fixed-random-phase
theory predicts an
asymmetry in the profile of gain enhancement
as a function
of optical delay.
this asymmetry
The experiment to measure
yielded a curious and likely erroneous result:
was present
in the experimental data,
in the wrong direction.
clever measurement
Whether,
but unfortunately went
A more careful or perhaps more
is still n e e d e d .
ultimately,
the multimode
fixed-random-phase
model will prove to be the correct model
is yet u nk no w n.
exist.
optimism
Many uncertainties
fo_r the XeCl laser
about the model still
The fact that so many of the predictions
the model
an asymmetry
given by
are either correct or close gives one cause for
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74
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A.Z. Grasyuk, I.G. Znbarev, N.V. Suyazov, "Influence of
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in stimulated scattering," ZhETF Pis. Red. 16,
237(1972).
25 .
F.A. Korolev, O.M. Vokhnik, V.I. Odintsov, "Signal gain
in stimulated scattering with broadband pumping," S o v .
Tech. P h y s . Lett. 2, (1976) .
26 .
G.P. Dzhotyan, Y.E. D'yakov, I.G. Zubarev, A.B.
Mironov, S .I . Mikhailov, "Amplification during
stimulated Raman scattering in a nonmonochromatic pump
field," Sov. P h y s . JETP 46, 3(1977).
27 .
W.R. Trutna, Ir., Y.K. Park, R.L. Byer, "The dependence
of Raman gain on pump laser bandwidth," IEEE J .Q.E. 15,
648(1979).
28.
J. Eggleston, R.L. Byer, "Steady-state stimulated Raman,
scattering by a multimode laser," IEEE J .Q.E. 16,
850(19 80) .
29 .
G.P. Djotyan, L.L. Mi na sy an i "On the theory of the
simulated Raman scattering of nonmonochromatic optical
radiation," Op. Comm. 58, 27 8( 1 9 8 6 ) .
30.
G.G. Lombardi, H. Injeyan, "Phase correlation in a
Raman amplifier" JOSA B 3, 1461(1986).
31.
O.M. Vokhnik, I.V. Nikitin, V.I. Odintsov, "Delay of
broadband-pumped stimulated Raman gain," O p t . Spectr.
(USSR) 45, 1( 19 78) .
32 .
E.A. Stappaerts, W.H.Long, Jr., H. Komine, "Gain
enhancement in Raman amplifiers with broadband
pumping," Op. Lett. 5, 4 ( 1 9 8 0 ) .
33 .
G.S. Ag a r w a l , "Stimulated Raman scattering
fields, " Op. Comm. 35, 2 6 7( 19 80 ).
34 .
A.T. Georges, "Theory of stimulated Raman scattering
a chaotic incoherent, pump field," Op. Comm. 41,
61( 1 98 2) .
35 .
A. Yariv, Quantum Ele ct ro ni cs .
and Son s, 1975.
36.
Y.R. Shen, The Principles Of Nonlinear O p t i c s .
York:
John Wiley and Sons, 1984.
37 .
P. Lorra in, D . Corson,
Waves.
San Francisco:
New York:
in chaotic
in
John Wiley
New
Electromagnetic Fields And
W.H.Freeman and Company, 1970.
77 .
38.
.J .D . Jackson, Classical Electrodynamics.
John Wiley and Sons, 1975.
New York:
39.
0. Svel to. Principles Of L a s e r s .
Pr ess, 1976.
40.
Lambda Physi k , Suppiement-EMG 1 5 0 .
41.
E. H e c h t , A. Zajac,
Wesley, 1974.
42.
J .W . Goodman, Introduction To Fourier O p t i c s .
Francisco:
McGraw-Hill, 1968.
43.
J .L . Carlsten, J , R i fk i n , D .C . MacPhe r son, "Spatial
mode structure of stimulated Stokes emission from a
Raman generator, " J . Opt. S o c . Am. B 3 , 1986.
44.
S . Gasiorowic z , Quantum P h y s i c s .
and So ns , 1974
45.
G . Herzbe rg, Spectra Of Diatomic Mo l e c u l e s .
Van No st rand Reinhold, 1950.
46.
R . Eisberg, R . Resnick, Quantum Physics Of Atoms,
Molecules, Solids, Nuclei, And Pa rticles. New York:
John Wiley and Sons, 1985.
47.
R .W . Mi n c k , E .E . Hagenlocker, W.G. R a d o , "Consideration
and evaluation of factors influencing the stimulated
optical scattering in gases," Ford Motor Company, 1966.
48.
M . Abram ow itz , I.A. Stegun. Handbook of Mathematical
Func t ion s. National Bureau of Standards, 1972 .
49.
S . Kopnin, Computational P h y s i c s . Menlo Park:
Benjamin Cummings Publishing Company, 1986.
50.
0.W . Young, "High-resolution cw stimulated Raman
spectroscopy in molecular hydrogen," Opt. Lett. 2 ,
91(1978).
51.
W .K . B i sc h el , M.J. Dyer, "Wavelength dependence of the
absolute Raman gain coefficient for the Q(I) transition
in h 2," J. Out. S o c . Am. B 3 , 677(1986).
Optic s .
New York:
California:
Plenum
Addison-
New York:
San
John Wiley
New Y o r k :
The
78
APPENDICES
79
APPENDIX A
RELATIVE SIZE OF A l
80
APPENDIX A
RELATIVE SIZE OF A l
The energy of the first electronically excited state of
hydrogen is approximately 1 0.2 eV* ® .
approximately 82,2 00 cm- -*-.
The laser,
This corresponds to
at a wavelength of
3 08 n m , and Stokes at 35I n m , corresponds
28,5 OOcm ^ respectively.
to 32,45 Ocm- ^ and
Al , which is the difference
between the first electronically excited state of hydrogen
and the laser energy is 4 9,75 9cm""*.
81
APPENDIX B
TRANSFORMATION TO THE TRAVELING REFERENCE FRAME
82
APPENDIX B
TRANSFORMATION TO THE TRAVELING REFERENCE FRAME
Fig. 27 shows both the traveling and laboratory reference
frames.
Both z and Z measure distances
that z (t ,z)=Z.
in the medium so
T on the other hand measures
with respect to a point on the pulse;
time at rest
therefore t (t ,Z)=t-
Z/Vg, where V g is the velocity of the pulse in the m e d i u m .
Defining
the differential
operator D as in chapter 2, we
simply apply the chain rule
33 zD
3 3t D
3z
Br
3 3z
3 zd Z
13
3z
V g S z d t
3 dr
Iddr
3
3 r 3Z
V g S r d t
3z
since
3z
3z
3r
I
— = I, -T- = 0, — = — ■
3Z
31
3Z
vg
3r
and —
3t
=1.
83
Laboratory
Medium
—
T = t - v / c —>|
Figaro 27. The traveling frame is at rest with respect to
the wave.
Points on the wave are located relative to an
arbitrary origin, here chosen at the boundary of the medium
with z-Z-O.
84
APPENDIX C
COMPUTER PROGRAMS
85
Figure 28.
Computer program L f i e l d s .
C
C *o*#**oo**o*o&**o**####o*#oo*#o#$*$$#$*$$***##ao$$*###
C
L F I E L D S .FOR
7-08-87
C *#**#o#**#******#**###******#<t************************
C
C
INITIALIZATION
REAL P(0:5000) ,P 2 (5000) ,R N( 640),E L R ,E L I ,P H I L (640) ,
* PHIS( 640) ,
» A M P L (640),A M P S (640),REESO(0:5000)
C
COMPLEX EL( 0:5000), ESO (0: 5000). ,ES (0:5000),,
* LASER,STOKES
C
INTEGERftA ID
C
OPEN
OPEN
OPEN
OPEN
OPEN
(I,FILE='FIELDS.DAT',S T A T U S = 'O L D ')
(2,F I L E = 'EL.DAT',STATUS='OLD')
(3,F ILE ='ES0.DAT',STATUS='OLD')
(4,F I L E = 'REES0.DAT',STATUS='OLD')
(5,F ILE='P.DAT',STATUS='OLD')
C
R E A D (I ,ft) D E L T A ,HW I D T H ,T M A X ,TST EP ,I R I ,M O D E S ,!RANDOM
C
o o n o
C=3.E-2
P I = 3 .1415926
C 2 = 2 .ftP I ftC ftDELTA
40
50
LOAD RANDOM NUMBER ARRAY AND CALCULATE INITIAL PHASES
IF (!RANDOM.NE.I) GOTO 5 0
M=3 2 7 5 3
„ DO 40 1=1,MODES
J R I = I R A N D d R I , M)
IRT=JRI
RN(I)=IRIft2 .ftPIZM
PHIL(I)=RN(I)
PHIS(I)=RN(I)
CONTINUE
GOTO 70
DO 60 K = I ,MODES
PHIL(K)=O
PHIS(K)=O
CONTINUE
60
C
C
#***#****##**#*****o*#*####**o**#$*#*#***#**o**
C CALCULATE THE LASER AND STOKES FIELDS AND THE LASER ENERGY
86
Figure 28.
Continued.
C CALCULATE WEIGHTED AMPLITUDES USING A LORENTZIAN LINESHAPE
C
70
J O= MODES/2
ASUM=O
DO 200,1=1,MODES
A M P L ( J ) = S Q R T d / (HWIDTH**2+(DELTA*(J-JO))**2) )
c
OPEN (9,F I L E = 1A M P L .D A T ' ,S T A TU S= 'OLD 1)
c
WRITE (9,*) J,AMPL(J)
A SUM=A SU M+ AM PL (J )* *2
200
CONTINUE
SCALE=I/SQRT(ASUM)
DO 300,J=O,MODES
A M P L (J )=SC AL E* AM PL (J)
AMPS(J)=AMP l (J)
300
CONTINUE
Pl = O.
P SU M= O.
C
C CALCULATE THE FIRST LATTICE POINT OF P ,THE FIRST ELEMENT C
OF LASER THE LASER ENERGY, AND SIMULTANEOUSLY CALCULATE
C THE INPUT STOKES FIELD.
W R I T E (*,*) 'CALCULATING THE LASER E N E R G Y '
T=O.
I=O
CALL FIELD(LASER,STOKES,A M P L ,A M P S ,C 2 ,M O D E S ,T,
PHIL,PHIS)
EL(I)=LASER
Pl=EL(I)*CONJG(EL(I))
ESO(I) = STOKES
P(I)=O.
C
W R I T E (2,*) EL(I)
W R I T E (3,*) ESO(I)
W R I T E (5,*) P(I)
C
R EE SO( I)= ESO (I)*CONJG (ES0(I))
W R I T E (4,*) T,REESO(I)
c
SO=REESO(I)
C
C
CALCULATE THE REMAINING ELEMENTS OF P AND LASER
I=I
DO 1200,T=TSTEP,TMAX-TSTEP,TSTEP
C
CALL FIELD (LASER,STOKES,AMPL,A M P S ,C 2 ,MODES,T,
* PHIL,PHIS)
EL(I)=LASER
87
Figure
28..
Continued.
P 2 (I )= E L (I )*CONJG(E L ( I ) )
P( D = P S U M + (P1 + P2 (I) ) *TSTEP/2 .
PSUM=P(I)
P 1 = P 2 (I)
ESO(I)=STOKES
W R I T E ( 2 , *) EL(I)
W R I T E O , * ) ESO(I)
W R I T E (5,*) P(I)
C
C
C
1200
C
REES0(I)=ES0(I)*CONJG(ESO (I))
TIME=T*.001
WRITE (4, *) TIME,REESO(I)
SO=SO+REES O (I )
1= 1 + 1
CONTINUE
END
C
Q
*###*#*#**#**#oo#****#*#oo*
SUBROUTINE FIELD(LASER,STOKES ,A M P L ,A M P S ,C2,MODES,T,
* PHIL,PHIS)
C
200
COMPLEX LASER,STOKES
REAL P H I L (640),PHIS(640),AMPL(640),AMPS(640)
ELR=O.
ELI=O.
ESR=O.
ESI=O.
DO 200,M=l,MODES
C3=C2*M*T
U L = C 3+PHIL(M)
ELR=ELR+AMPL(M) *CO S(UL)
ELI= E L I + A M P L (M ) *SIN(U L )
U S = C 3 + P H I S (M)
E SR=E SR+AMPS(M)*COS(US)
ESI=ESI+AMPS(M)*SIN(US)
CONTINUE
L A S E R = C M P L X (ELR,ELI)
s t o k e s = c m p l x (e s r ,ESI)
RETURN
END
C
INTEGER FUNCTION I R A N D (I R I ,M )
C
INTEGER*4 ID
F igure
28 .
C o n t inued .
IA = 2 0 1
IC= 3 67 9
ID=IA*IRI+IC
IRAND=MO d (ID,M)
END
89
noon
Figure 29.
Computer program Mode.
MODE.FOR
2-23-87
a#*#********##*####***##*##**##**##**$**###**0***$$$*#
INITIALIZATION
o
REAL F E E S SEL,BESSEL,B D A T A (850),BDIF(850),ELR,ELI,
.* P( 0:200 0) ,REES( 0:2000) ,ENERGYY (500)
n n
n
o
, o n
COMPLEX L ( 0 : 2 00 0);ES0(0:2000),E S (0:2000),
* E S I (0:2000),
* B A M B I ,ZAP,CS
200
C
INTEGE r 0A i d
$ $ $ $ $ $ # $ $ $ $ $ # $ $ * $ # $ $ $ $ $ $ $ * $ # $ $ $ $ $ # $ $ $ * $ $ $ $ $
OPEN (I ,F I L E = 'M O D E .D A T ',S T A T U S = 'O L D ')
R E A D (I,0 ) D E L T A ,G A M M A ,T M A X ,S T E P ,D M I N ,D M A X ,D S T E P ,
° GMIN.GMAX,GSTEP
$ $ $ $ $ » $ $ $ $ * * $ $ $ $ $ $ $ $ * $ $ $ $ $ $ $ $ $ $ $ $ $ $ # $ $ $ $ » * $ $ *
C = 3 .E-2
P I = 3 .1415926
NOW CONVERT GAMMA FROM WAVENUMBERS TO TIME
G A M M A = 2 .0P I 0C 0GAMMA
C 2 = 2 .0P I 0C 0DELTA
N I = (TMAX-STEP)/ STEP
0000
READ THE INPUT FIELDS INTO ARRAYS
WRITE (°,°) 'TRANSFERRING FIELDS NOW...'
OPEN (I,F I L E = 'ES0.DAT',STATUS='OLD')
OPEN (2,F I L E = 'EL.DAT',S T A T U S = 'O L D ')
DO 200 K=O , NI
READ(1,«) ESO(K)
R E A D (2,0 ) EL(K)
" "CONTINUE
$ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ * $ $ # $ $ $ $ $ $ $ $ $ $
OPEN (I,F I L E = 'P.DAT',STATUS='OLD')
DO 250 L=O,NI
READ (I,0 ) P(L)
250
CONTINUE
C
$$$$$$$$$
READ BESSEL FUNCTION INTO AN ARRAY
W R I T E f 0 ,0 ) 'READIN THE OLD BESSEL FUNCTION....'
OPEN (I, F ILE='BDATA.DAT',S T A T U S = 'O L D ')
OPEN (2,F I L E = 'B D I F .D A T ',S T A T U S = 'O L D ')
DO 300 1=1,800
READ(1,°) BDATA(I)
R E A D (2,0 ) BDIF(I)
300
CONTINUE
C $$*$$$$$$$$$$*$*$$$$$$$$$$$$$*$$$$$$$»$$$#*»»#*#»#####
90
Figure 29.
Continued.
C
Q
CALCULATE THE STOKES FIELD
#**##*#**#**$*&*$$*$#*0*$**$$$$3*0#**0$****&$#***$#0**
WRITE**,#) 'CALCULATING THE STOKES FIELD....'
OPEN (2,F I L E = 1R E E S .D A T ',S T A T U S = 'O L D ')
OPEN (3,F I L E = 'G A I N l .D A T ',S T A T U S = 'O L D ')
OPEN (4,FILE='G A I N 2 . D A T ' ,STATUS='OLD')
C UCG IS THE UNCORRELATED GAIN. CG IS THE CORRELATED GAIN.
ES(O)=ESO(O)
C CHECK TO SEE IF GAIN LOOP IS BEING USED
IF (G M I N .E Q .G M A X ) THEN
IFLAG2=!
ELSE
IFLAG2=0
END IF
DO 8000 GL=GMIN,GMAX,GSTEP
N=0+IFLAG2
400
C l = S Q R T (G L #GAMMA/2.)
DO 7000 D=DMIN,DMAX,DSTEP
N=N+1
Id e l a y = N I N T (d / s t e p )
c
C
2700
C
R E E S (0)= E S (0)# CONJ G (ES(0))
WRITE (2,*) 0, REES(O)
E N E R G Y Y (N)=REES(0)
DO 2700, L=O , NI
K=L+IDELAY
IF ( (K.LT.O).OR.(K.GT.N1)) THEN
E S l ( L ) = C M P L X (0.,0.)
ELSE
ESl(L)=ESO(K)
END IF
CONTINUE
J= I
DO 3000,
TO=STEP,TMAX-STEP, STEP
C 3 = C 1 # E L (J )
CALL ZZAP(ZAP,STEP,TO,GAMMA,EL,J ,C l ,P, BDATA ,
* B D I F ,E S I ,A R G )
E S ( J ) = E S l (J)+C3#ZAP
C
C
REES (J)=ES(J)*CONJG(ES( J ) )
R E E S 0 =ES0(J)8C O N J G (ESO(J))
REES(J) = A M I N 1 (10000.8R E E SO,REES(J))
T I ME=.O O l 8TO
WRITE (2,8 ) TIME, REES(J)
E N E R G Y Y (N)= E N E R G Y Y (N)+ R E E S (J )
91
Figure 29.
Continued.
J=J+1
CONTINUE
WRITE (*,*)
WRITE (*,*) 'DELAY IS ',D,'ps
GAIN IS ',GL
ENERGY=ENERGYY(N)/T M A X ftSTEP
WRITE (ft,ft) 'AVG OUTPUT ENERGY= ',ENERGY
WRITE (ft,ft)
IF (I F L A G 2 .EQ.I ) THEN
C THE VARIABLE DELAY IS USED FOR PLOTTING ONLY ( => (X,Y)
DELAY=Dft.001
WRITE (3,ft) DELAY,ENERGY
GOTO 7000
ELSE
IF (N .N E .I ) GOTO 4000
UCG=ENERGY
END IF
4000
CG=ENERGY
7000
CONTINUE
W R I T E (4,ft) G L ,U C G ,CG
W R I T E (I ,ft) G L ,ARG
8000
CONTINUE
C
END
3000
C
C
C
SUBROUTINES
SUBROUTINE Z Z AP(ZAP,STEP,TO,GAMMA,EL,J ,C l ,P,BDATA,
ft B D I F ,E S I ,A R G )
ZZAP IS THE INTEGRAL HAVING THE INTEGRAND BAMBI
COMPLEX E L ( 0:2000),S U M ,SUMO,SUMTO,B A M B I ,ZAP,
» ESI (0:200 0)
REAL B D A T A (850),BDIF(850),P(0:2000),ARG
C
SUM=CMPL x (0.,0.)
C
C
C
C
CALCULATE FIRST AND LAST ELEMENTS OF INTEGRAND
I=O
T=O
CALL B B A M B I (TO,T,GAMMA,E L ,1,1,Cl,P,B D A T A ,B D I F ,
ft E S I ,I F L A G l ,BAMBI.ARG)
SUMO= BAMBI
T=TO
SUMTO= Cl ftC O N J G ( E L U ) )ftESl (J)
IF(J.EQ.l) GOTO 3210
)
92
Figure 29.
C
Continued.
LOOP TO CALCULATE ES(Z,TAU)
T=J-I
DO 3200,T=TO-STEP,STEP,-STEP
C
CALL B B A M B I (T O ,T,GAMMA,E L ,I ,J ,C l ,P,BDATA,B D I F ,
E S I ,I F L A G l ,B A M B I ,A R G )
*
C
IF (I F L A G l .E Q .I ) GOTO 3205
SUM=SUM+BAMBI
C
I= I-I
CONTINUE
3200
C
3205
3210
C
SUM= 2 . *SUM
ZAP= STEP/2.*(SUMO +SUM+SUMTO)
RETURN
END
C
C
*
C
C
SUBROUTINE B B A M B I (TO,T,GAMMA,EL,I ,J ,C l ,P,BDATA,
BDIF,E S I ,I F L A G l ,B A M B I ,A R G )
BAMBI IS THE INTEGRAND
COMPLEX BAMBI
COMPLEX E L (0:2000),ESI(0:2000)
REAL B D A T A (850),BDIF(850),P(0:2000),
6
F B E S S E L ,ARG
C
4900
C
5000
. X=TO-T
PDIF=SQRT(P(J)-Pd) )
ARG=2*C1*PDIF
BESSEL=FBESSEL(ARG,BDATA,BDIF)
GX=GAMMA0X
I F (G X .G T .80) GOTO 4900
E B P = E X P (- G X + A R G )0BESSEL/PDIF
IF (E B P .L E .I .E - 4 ) THEN
IFLAGl=I
ELSE
IFLAGl=O
END IF
BAMBI=EBP°CONJG(EL(I))°ES1(I)
GOTO 5000
BAMBI=O.
RETURN
END
93
Figure 29.
Continued.
non
C
Q ****#*****o*******#********o********* * $ *<*#**#**#*#***#
C
FUNCTIONS
Q *#**#**##*#*#******###**#** 0 $#*********#***##**###****
C
CALCULATE THE BESSEL FUNCTION
C
FBESSEL CALCULATES THE BESSEL FUNCTION OF AN ARGUMENT
C
BY INTERPOLATING BETWEEN PREVIOUSLY CALCULATED
C
RESULTS (PDATA).
C
THE METHOD OF LINEAR IN T E R POLATON IS USED:
C
C
Y = Y l + (Y 2 - Y 1 )/ (X2-X1) <-(X-Xl)
C
C
BDATA ARE THE BESSEL FUNCTION ARRAY ELEMENTS WHICH
C
HAVE ALREADY BEEN CALCULATED AND READ INTO PROGRAM.
C
BDIF IS AN ARRAY CONTAINING THE SLOPES BETWEEN ANY
C
TWO POINTS OF THE BDATA A R R A Y . ARG IS THE ARGUMENT
C
OF THE BESSEL F U N T I O N . BARG PUTS ARG WITHIN THE
RANGE OF THE BDATA ARRAY.
IARG AND JARG ARE THE
NUMBERS OF THE BDATA ARRAY ELEMENTS SURROUNDING B A R G .
o
REAL FUNCTION F B E S S E L (A R G ,BDATA,BDIF)
o o
REAL B D A T A (850)* B D I F (850)
CALCULATE THE ARGUMENT OF THE BESSEL FUNCTION
B A R G = A R G / .05
IBARG=BARG
J B A R G = IBARG+I
BESSELl= B D A T A (IB A R G )+ B D I F (JB A R G )*(ARG- I B A R G *.05)
FBESSEL=BESSELl
C
END