Non-Forward Stimulated Raman Scattering from a
Phase-Modulated Pump Beam
David Eimerl, Ph.D.
EIMEX Software and Consulting
4042 Camrose Avenue
Livermore, CA 94550
(925)-371-5364
Abstract
We present a new theory of non-forward stimulated Raman scattering in the field of a
phase modulated pump, where the pump phase modulation is an arbitrary function of time. The
theory is accurate enough to be determine the performance limitations on laser systems where SRS
is a potentially controlling parasitic. Calculations are described for S:FAP amplifier media where
the laser beam has a pure RF modulation typical of those attainable in the laboratory. The
predictions of this theory are compared with the oft-used D’yakov approximation.
(1) Introduction and Background:
Stimulated Raman Scattering (SRS) is a well-known phenomenon wherein a laser beam
scatters inelastically off a medium whose polarizability is enhanced by the scattering process
itself.1-13. In SRS the medium polarizability oscillates at the frequency of a material excitation,
and so generates sidebands of any light present in the medium. The separation of the sidebands is
just the frequency of the material excitation. Thus an intense laser beam acquires at least one
sideband down-shifted in frequency by the material excitation frequency . While there are many
theoretical treatments involving the quantum theory of the scattering medium, the process itself is
essentially a classical process. It is described by a simple classical Hamiltonian containing the free
electromagnetic field E, a simple harmonic oscillator representing the medium excitation Q, and
the interaction term
H int   | E | 2 ,
where the polarizability depends on the medium excitation, Q
  0   'Q
It is the dependence of  on Q that encapsulates the stimulated Raman effect9. The
electromagnetic field contains at least two waves, one at the laser frequency and another at the
down-shifted frequency, (the Stokes frequency). The resulting dynamics is well-known for laser
beams that have benign temporal and spatial structure generating a co-propagating Stokes wave.
The Stokes wave experiences exponential gain. SRS can also generate non-forward Stokes waves,
which also experience the same exponential gain for benign laser beams.
In reality, the incident laser beam is often not benign, having both spatial and temporal
structure.
For a laser beam with a general spatial and temporal structure, the dynamics can only be modeled
using numerical techniques, due to the presence of three different velocities (pump, Stokes and
medium excitation) for the waves. This type of calculation is prone to numerical errors arising
from the need to interpolate at least one of the three waves in the problem at each computational
step. In general, the reliability of numerical techniques remains open to question at this time. Some
theoretical work has been reported for laser pumps that are essentially Gaussian noise sources2,5,6,7,
but there is some question about their validity1. In any case results for noisy beams would hardly
apply to a laser beam, which has a high degree of coherence in general.
In situations where the Stokes wave is seen as an undesirable parasitic, a means of
suppressing its production is desired. There is a general belief that laser bandwidth can assist in
suppressing SRS, but there are few, if any, detailed calculations for real laser beam structures in
the literature.
Page 2
Sometimes the suppression has been estimated using the theoretical results for noisy
Gaussian beams10. However, this approach is prone to error associated with selecting the correct
equivalent noise bandwidth to use. For example, it is not known what noise bandwidth would
correspond to a pure RF-modulated laser beam, yet this type of bandwidth (RF) is relatively
straightforward to implement in the laboratory. The accuracy of any model for the equivalent noise
bandwidth for real laser temporal structures can only be determined by comparing with the exact
solution for the actual laser beams being used. The lack of such accurate results for laser beams
with structure precludes the testing and evaluation of these approaches.
The equivalent noise bandwidth concept and similar approximate methods are useful for
determining if SRS is likely to be present in a given laser system, but they are too vague to
provide accurate limitations on the performance or output of a laser system where SRS is the prime
parasitic. For this purpose, an accurate numerical or analytic model for SRS pumped by the actual
laser beam is required.
This paper reports a new mathematical model of SRS in the presence of a laser beam with
an arbitrary phase as a function of time. The theory applies to plane wave laser beams with a
general phase modulation, and to Stokes beams propagating in any direction. It therefore applies to
forward, tranverse, and backward SRS. The model is accurate enough to evaluate the suppression
of SRS using phase-modulation techniques. These results are sufficiently accurate to predict limits
on the performance of a laser system where SRS is a potentially controlling parasitic.
(2) The Equations of SRS: A Review
The Stokes wave is presumed to be a plane wave propagating along the z-axis towards z
= ∞. Its amplitude and phase are not constant, but may vary in an arbitrary fashion, so long as the
rate of variation is slow compared to the material dispersion. Thus in general
E  E ( z, t ) exp( i( s [t  ns z / c] ) )
(1)
The pump wave is also presumed to be a plane wave, but propagating at an angle  to the z-axis,
in the (x,z) plane. In general, its amplitude and phase may vary in an arbitrary fashion, except that
the pump wave must be a solution of Maxwell’s equations, that is, the pump wave is a fixed
structure propagating at velocity c/np.
E p  aA( ) exp( i p )
where   t  (n p / c) ( z cos   x sin  ) (2)
The constant “a” has been introduced so that the intensity of the pump wave is just |A|2 with no
other factors. The medium excitation that is created has a carrier phase that is simply the beat
between the pump and the Stokes waves:
Q  Q( z, t ) exp(  i p  i s [t  n s z / c] )
Page 3
(3)
Without any loss in generality, the carrier waves at the pump and Stokes’ frequencies are
presumed matched to the Raman shift.
 p  R  s
(4)
With these definitions, the equations of SRS, in the laboratory frame, are as follows:
.
  ns  
 
 E s  1 / 2AQ *

z
c

t


(5)


*
    Q   AE s

t


Here the slowly varying amplitude approximation has been made to remove the second-order
derivatives of Maxwell’s equations. The symbols have their usual meaning. Note that A is a
function of the running time of the pump,, and that the dependence of both E and Q on z and t is
arbitrary, except for the condition that is be slow enough for the slowly-varying amplitude
approximation to be valid. The next step is to transform these equations from the laboratory frame
to a frame that is co-moving with the pump along the z-axis.
T  t  (n p / c) z cos 
Z z
(6)
Then
  n s  n p cos   

 E s  1 / 2AQ *

c
T 
 Z
 

*
 
 Q   AE s
T 

Now the pump field A is a function of T and x:
  T  (n p / c) x sin 
(8)
We then shift the time origin as follows:
T '  T  (n p / c) x sin 
to obtain the equations often quoted for non-forward SRS.
Page 4
(9)
(7)
  n s  n p cos   

 E s  1 / 2A(T ' )Q *

c
T ' 
 Z
 

*
 
 Q   A(T ' ) E s

T
'


(10)
It appears that the x-dependence has been removed from the equations. However, we made
the assumption that the Stokes wave is a plane wave propagating along the positive z-axis. It can
therefore have no x-dependence whatsoever. If we solve equn (10) and substitute back to find the
Stokes amplitude, the x-dependence in the transformation from T’ back to t will appear in the
Stokes amplitude. The Stokes amplitude will therefore acquire an x-dependence, which is
inconsistent with the assumption that it is spatially a plane wave propagation along the z-axis. The
shift in time origin is therefore inconsistent with the original assumption that the Stokes wave is
spatially a single plane wave.
To resolve this apparent discrepancy, recall that the pump (complex) amplitude is a
constant along the pump wavefront, which is not parallel to the Stokes wavefront. Now consider
various locations on the Stokes wavefront, which in fact is an (x,y) plane. As we allow the xposition to vary, we necessarily cross from one pump wavefront to another, because the
wavefronts are not parallel. So the driving term has a phase that varies in x. The effect is
essentially a time delay – each point on the Stokes wavefront experiences the same driving terms,
but as we move in x, the driving terms are delayed by an amount proportional to the distance that
the pump wavefront has separated from the Stokes wavefront. To some extent this will result in a
non-flat wavefront for the Stokes wave.
If the pump is noisy, and fluctuating about a mean value, then one might expect the Stokes
wave-front to fluctuate also. If the fluctuations are rapid enough that diffractive effects do not
occur as the Stokes wave propagates, one might expect that the Stokes wavefront will fluctuate
insignificantly about a flat wavefront. Also, to the extent that the noise is steady state, the gain
should not depend on the initial conditions, or equivalently on any time delay in applying the
driving term to the Stokes wave. So while mathematically the x-dependence is undeniable, it is
reasonable to suppose that it will make no difference to the problem we are concerned with.
Thus we are justified in taking equn (10) as appropriate and ignoring the shift in time origin
(9).
Page 5
(3) SRS gain in the field of a phase modulated pump beam
The starting point for our theory of non-forward SRS is equ. (10), neglecting the issues of
the implied x-dependence.
 1 
*
 
 E s  1 / 2A(t )Q
 z V t 


*
    Q   A(t ) E s
t 

(11)
where 1/V is the relative fugacity, or the velocity dispersion.
1 / V  n s / c  (n p / c) cos 
(12)
Note that V is not the relative velocity between the pump and Stokes waves. Rather 1/V is the
difference in the inverse of their velocities. Note also that for some angle, (1/V) may vanish. The
SRS model is singular at this point.
The pump is assumed to have a pure phase modulation:
A(t ) 
A0 exp( i (t ) )
(13)
where A0 is a constant, and is an arbitrary function of time, except that its bandwidth is small
compared to the laser frequency. Note that in this coordinate frame, the pump has no dependence
on z. Taking the spatial Fourier transform of (11)
d

i ( t ) *
 ikV   es  1 / 2V . A0 e q
dt


d

i ( t ) *
    q   A0 e e s
dt


(14)
where corresponding Fourier transforms are written as lower case. Define a state function Z as the
combination
q * e i
Z
es A0
(15)
Then
ikV 
d
ln es  1 / 2 g 0V . Z
dt
d
  ln q *
dt
  (1 / Z )
Page 6
(16)
where g0 = |A0|2 = I0 is the monochromatic gain (units: cm-1). Taking the time averages of these
equations immediately gives conditions that the time average <Z> must satisfy:
Z 
 ik /( g 0 / 2)
 1/ Z 
1
(17)
In steady state, Z = 1, and the spatial growth rate of the field is just the real part of (ik). If this turns
out to be true for the modulated case also, then (17) suggests that we should identify <Z> as the
mean reduction in the gain in the presence of bandwidth.
This is indeed the case. From (16), the Stokes field is
t
ln es (k , t )  ln es (k ,) 
 dt ' ( 1 / 2 g V . Z (k , t )  ikV )
(18)
0

and
E s ( z, t ) 
dk
 2
e s (k , t ) e ikz
(19)
Write Z = <Z> + Z where Z represents the fluctuations in Z, so <Z> = 0. Insert this into (18)
and take the limit as t becomes large. This gives
ln es (k , t )  ln es (k ,) 

(1 / 2) g 0Vt. (  Z  ik /( g 0 / 2) )
0
t

0
 dt ' ( 1 / 2 g 0V . Z (k , t )  ikV )  ( 1 / 2 g 0V ). dt ' Z (k , t )
(20)
The first integral represents transient effects that can be incorporated into the “source” field
es(k,-∞). In any case it is a bounded quantity – effectively an additive constant. In the second
integral the fluctuations average to zero over time, making this term also bounded on average. This
leaves the second term, which is proportional to Vt, as the dominant term. Now, the field must
always remain finite, so the coeffiecient of Vt must be zero. Thus the gain reduction is
g / g0
 Re  Z 
(21)
It remains to develop the equation for the state function Z. In equs(16), subtract these two
equations to obtain a first order equation for Z:
dZ / dt  1 / 2 g 0V Z (Z  p)  (1  Z )  iZd / dt
(22)
Here p is the (complex) constant ik/(g0/2). Dividing (22) by Z and taking the time average, we find
1 / 2 g 0V (  Z   p )  (  1 / Z  1 )
Page 7
(23)
As p is varied, equ(23) is rigorously satisfied by the solutions of (22), but the solution is only
consistent with (17) for one particular value for p, namely the value of p which makes both sides of
(23) vanish. This special value for p is the desired solution for the gain reduction due to bandwidth,
which is just Re(p). In this way the gain emerges as type of eigenvalue. Equations (17) and (22)
are the central equations of this theory. They do not have an analytic solution for arbitrary phase
modulation φ(t), but will require a numerical solution.
In general the eigenvalue of problems such as this one must be obtained by an iterative
technique of some type. However, in the present problem, the procedure can be streamlined
somewhat, as described in the next section.
(4) Solving for Re(<Z>)
Writing Z = X + iY, and p = u+iv, we find
dX / dt  1 / 2 g 0V (( X  u ) X  (Y  v)Y )  (1  X )  Yd / dt
dY / dt  1 / 2 g 0V (( X  u )Y  (Y  v) X )  (  Y )  Xd / dt
(24)
In principle this can be solved by iterating over u and v, but this is a very slow process. It is more
efficient to continually update the values of u and v as information is obtained on Z(t). Thus we
add to (24) two other equations:
du / dt   ( X  u )
dv / dt   (Y  v)
(25)
where  is chosen to be small compared to any of the rates in the problem. These equations can be
integrated using a standard integration method such a fourth order Runge-Kutta technique. The
calculation proceeds until u and v approach their steady state asymptotic values. The output of the
calculation is the asymptotic value of u, as t →∞.
While the use of (25) speeds up the calculation, there are some subtleties in this approach.
In the absence of bandwidth, u = 1 and v = 0. Bandwidth drives u below 1, and induces
fluctuations in both u and v. The magnitude of the fluctuations depends on the rates in (24) and the
value of . These fluctuations are not completely unphysical in that the desired solution is the
asymptotic value of u and v, and the physical fluctuations only disappear in the asymptotic limit.
However, using (25) the fluctuations do not disappear numerically. The test for reaching the
asymptotic limit is typically one where the change in u or v since the last time it was interrogated
is smaller than some desired accuracy. If the numerical fluctuations are larger than the desired
accuracy, the procedure will not converge. For some situations, especially the case where V is very
large (at low scattering angle) the required value of is so small that convergence is very slow and
ultimately numerical accuracy is compromised by the accumulation of errors in the integration
procedure.
Page 8
Note that this model and its equations are singular for 1/V = 0, which usually occurs for
nearly forward scattering. Thus the entire model will fail close to forward scattering. This is not a
problem in principle, because we already know the result for forward scattering; this solution is u
= 1, v = 0. But as this singularity (1/V=0) is approached the convergence and stability of the
numerical procedures will degrade. So it is necessary to have a robust algorithm for those angles
close to forward where the gain reduction is not exactly 1, say u =0.99 or below.
These algorithms have been tested using a fourth order Runge-Kutta integrator. Phase
functions corresponding to RF, square wave and gaussian noise have been used as test phase
functions. It was found that for forward angles and material parameters appropriate to S:FAP, an
algorithm based on (24) and (25) is not adequate. Convergence is poor for near forward angles and
low gain, and there can be significant errors in the gain reduction factor. It is necessary to reduce
the fluctuations even further, perhaps by forcibly eliminating the higher frequencies. One
technique which appears to work is to implement extra integration variables, as follows:
du 3 / dt   ( X  u )
dv3 / dt   (Y  v)
du / dt   ( X  u 3 )
(26)
dv / dt   (Y  v3 )
Algorithms based on (24) and (26) appear to converge gracefully for parameters appropriate to
typical solid state Raman media.
Page 9
(5)
Results for RF bandwidth in S:FAP Amplifiers
The model has been used to predict the gain for a phase-modulated pump using parameters
appropriate to S:FAP. The phase function is
 (t )   sin( t )
(27)
where the modulation frequency is taken to be 10.384 Gcycles/sec (2 10.384 rads/sec) and the
amplitude is taken to be 14.4 radians. These parameters are readily achievable in the laboratory.
For S:FAP the material constants are g0 = 1.2 cm/GW, and  = 84 GHz. The results of the
calculations are presented in Fig 1.
Fig1: SRS Gain Suppression in S:FAP using RF pump bandwidth
Suppression of SRS in S:FAP by RF Bandwidth
1.2
RF Frequency = 10.384 GHz
Beta = 14.4 radians
g_zero = 1.2 cm/GW
1 <= Intensity <= 8 GW/cm2
Suppression Factor
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
35
40
45
50
Scattering Angle (degrees)
(6)
Comparison with the D’yakov Approximation:
Previous approaches to estimating SRS generation with RF modulated beams have been
based on the D’yakov approximation taken from the Russian literature10. Unfortunately, the
mathematical basis of this approximation is obscure, and it has certain inconsistencies that raise
Page 10
questions about its general validity. However, in the absence of an accurate model, it has been
used to estimate the effect of bandwidth on non-forward SRS10. In terms of the FWHM linewidths, expressed in wave-numbers, the D’yakov result is
 g

 g0
2 c( R   p )  g



  1 
g 0V


 g 0
2
 2 c R
 
0
g 0V

(28)
where  R   /  c is the FWHM of the Raman line, (which is Lorenzian), and  p is the FWHM
of the (presumed Gaussian noise) pump spectrum. The appropriate solution is the larger of the two
roots.
We note that for the phase modulation of equ (27) the frequency spectrum is a set of Bessel
functions, and is not even remotely like a Gaussian. Using the new theory we are in a position to
determine how well equn(28) approximates the true solution for RF bandwidth. The RMS
bandwidth of (27) is  RMS   / 2 . We would therefore expect that a reasonable choice for
the equivalent noise bandwidth might be
 eff   . / 2
(29)
where ~1 is an adjustable parameterhe D’yakov approximation for the cases presented in Fig
1 is presented in Fig 2, using an effective noise bandwidth corresponding to ~1

Fig2: The D’yakov Approximation
Dyakov Approximation for SRS Suppression in S:FAP
1.2
Effective Noise
Bandwidth
Parameter = 1
Suppression Factor
1
0.8
0.6
0.4
0.2
0
0
5
10
15
20
25
30
Scattering Angle (degrees)
Page 11
35
40
45
50
A comparison of Figs 1 and 2 reveals several differences. The most important is that the
shape of the curves is not the same, so a simple scaling by adjusting the choice of effective noise
bandwidth () will not make the models coincident. In particular as the scattering angle is
increased away from the forward direction, the curves for RF say close to the forward result (unity)
and then drop rapidly at a critical angle, whereas the D’yakov approximation falls away more
rapidly from the forward value as the angle is increased. Secondly even if were chosen to fine
tune the location of the curves in angle, it appears that RF modulation causes a greater reduction in
the SRS gain at high angles than the D’yakov approximation predicts.
In summary, the D’yakov approximation gives reasonable guidance as to the general
features of the SRS suppression of SRS, but is not accurate enough to predict the limitations on
laser system performance when SRS is a potentially controlling parasitic. For this the model
described herein is required.
(7)
Conclusions
We have described a theory for non-forward SRS gain in the presence of arbitrary phase
modulation on the pump laser. The theory is accurate enough to predict performance limits on
laser systems where SRS is a potentially controlling parasitic. The results of this theory for RF
modulation have been compared to previous estimates, by comparing with the D’yakov
approximation for SRS in the presence of a pump laser with a gaussian noise (diffusing) phase.
The theory predicts that the forward gain persists to higher scattering angles compared to the
D’yakov approximation, but that the asymptotic value of the gain at high scattering angles is
lower.
Page 12
Appendix: Further Mathematical Details: A summary of the mathematical proof
that the intensity gain reduction factor is Re<Z> in the presence of spontaneous
scattering source terms
The model presented above is analogous to the model of monochromatic SRS gain where
the gain is evaluated without any reference to the spontaneous scattering source terms. Indeed,
taking the steady state limit in equ(10) immediately gives exponential growth for the Stokes field
regardless of the source terms. And a cursory review of the derivation of equ(21) indicates that it
is based essentially on the asymptotic (i.e. long time average) behavior of the gain coefficient.
The approach leaves some questions:
(a) The model does not include the source terms associated with spontaneous scattering..
Indeed, from eqn(20) the model describes the gain experienced by an injected Stokes pulse,
without any spontaneous scattering. This may not be appropriate for a study of the limitations on
SRS as a potentially controlling parasitic, as we have in Mercury. Note that the transient terms in
eun (20) (the first integral) represent the evolution of the phase of the injected Stokes pulse to the
phase it has under conditions of gain, so the calculated (asymptotic) gain is probably correct for a
spontaeneous scattering source also. This is true for monochromatic pump beams, but it needs
confirmation for a phase-modulated pump beam.
(b) The time average of the Stokes field is not the same as the time average of the Stokes
intensity. It is possible that time average of the electric field can be zero, even if the intensity is
finite. The question arises whether the gain we found is in fact the gain for the Stokes intensity.
To answer these questions it necessary to construct the Stokes intensity in terms of the spontaneous
source terms, and show that the gain is indeed correct. The proof that this is so is presented here.
We first modify equn(14) to include a source term for the Stokes wave, s(k,t).
d

i ( t ) *
 ikV   es  1 / 2V . A0 e q
dt


d

i ( t ) *
    q   A0 e es
dt 

 s (k , t )
Then defining
t2
(k , t1 , t 2 )  1 / 2 g 0V  dtZ (k , t )
t1
the Stokes field is
t
es (k , t )   dt ' e ( k ,t ,t ')  ikV ( t  t ') s (k , t ' )

Page 13
in the absence of an injected Stokes pulse. In these expressions, Z(k,t) is the state function
evaluated using (22) with a general value of p=2ik/g0. Taking the inverse Fourier transform, the
Stokes wave at (z,t) is obtained.
t
z

0
E ( z, t )   dt '  dz 'G( z  z ' , t , t ' ) S ( z, t ' )
where the Green’s function is
G ( z, t , t ' )  
dk  ( k ,t ,t ')ikV (t t ' z / V )
e
2
and S(z,t) is source for the Stokes field. It is delta-correlated.
 S ( z, t ) S * ( z ' , t ' )   I N
 ( z  z ' ) (t  t ' )
The Stokes intensity is just I= |E|2 (modulo fundamental constants):
t
z

0
I ( z, t )  I N  dt '  dz ' | G( z  z ' , t , t ' ) |2
Now write the Green’s function to make the k-integral explicit:
G( z, t , t ' )  
dk
Ke ikz
2
where K is defined by this equation. Then the z-integral in the expression for Is eliminates one of
the two k-integrals in Is (by limiting the region of support of the integrals to k≈k’) and we obtain
Is  IN
 dk dt e
p
where
t
p  g 0V Re  dt ' ' Z (k , t" )  ikV (t  t ' z / V )
t'
Now we replace Z by <Z> + Z, and find
p  g 0V (t  t ' ) Z R (k )  ikV (t  t ' z / V )  fluctuations
where ZR(k) is the Re<Z> evaluated for a general k-vector k. In the limit where t is large, the
fluctuation terms become insignificant.
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We do not know in general the k-dependence of the state function <Z>. However, it is not
necessary to know this in order to prove the result. The procedure is to do the k-integral and the tintegral in the expression for Is by successive applications of the method of steepest descents.
For the k integral, the saddle point is located at ks, where ks(t) is the solution of the saddle point
equation (∂p/∂k = 0):
p / k

g 0V (t  t ' )dZ R (k ) / dk  iV (t  t ' z / V )
 0
At this saddle point, the integrand is
p  p1  p(k s (t ), t )  g 0V (t  t ' ) Z R (k s )  ik sV (t  t ' z / V )
Repeat this for the t-integral(∂p1/∂t = 0):
p / k s .dk s / dt
 p / t
 0
The term in ∂ks/∂t vanishes, because ∂p/∂ks is zero at the double saddle point, so this is just
g 0VZ R (k s )  ik sV  0
At the second saddle point the integrand is
p
 g 0V (t  t ' ) Z R (k s )  ik sV (t  t ' z / V )
 ik s z
 ( g 0 z)Z R
Thus the intensity gain for the Stokes noise is just the monochromatic gain reduced by Re<Z>.
The variation of <Z> with p (or 2ik/g0) is slow so the method of the previous section should give a
precise estimate of the gain reduction.
Page 15
References:
1. D.Eimerl, upublished results on the D’yakov result for non-forward SRS gain,
2. S.A.Akhmanov,Yu.E.D’yakov,and L.I.Pavlov,”Statistical Phenomena in Raman Scattering
stimulated by a broad-band pump”, Sov.Phys.JETP, 39,249 (1974) Translated from: Zh.
Eksp. Teor. Fiz. 66,520 (February 1974
3. D.Eimerl, “Theory of Temporal Stochasticity in Stimulated Raman Scattering in Dispersionless
Media”, D. Eimerl, J. Math. Phys. 20, 1811 (1979)
4. D.Eimerl, unpublished original work, EIMEX IP.
5. D.Eimerl, “Inhomogeneously Pumped Stimulated Raman Scattering”, International Conference on
Lasers, Orlando FL, (1978), Proceedings, STS press, 333 (1979)
6. William.R.Truntna Jr, Yong Kwan Park, and Robert L Byer,”The Dependence of Raman Gain
on Pump Laser Bandwidth”, IEEE J.Qu.Electron., QE-15,648 (1979)
7. John Eggleston and Robert L.Byer, “Steady-State Stimulated Raman Scattering by a
Multimode Laser”,IEEE J.Qu.Electron., QE-16,850 (1980)
8. George C.Valley,”A Review of Stimulate Brillouin Scattering Excited with a Broad-Band
Laser”,IEEE.J.Qu.Electron., QE-22,704 (1986)
9. John F. Reintjes, “Stimulated Raman Scattering”, CRC Handbook of Laser Science and
Technology, Supplement 2: Optical Materials, M.J.Weber, Editor, Chapter 8.3 , CRC Press,
Boca Raton, FL (1995)
10. J.R.Murray, J.Ray Smith, R.B.Ehrlich,D.T.Kyrakis, C.E.Thompson, T.L.Weiland and
R.B.Wilcox,”Experimental Observation and Supprression of transverse stimulated Brillouin
Scattering in Large Optical Components”, J.Opt.Soc Am. B, 6, 2402,(1989)
11. Andy J. Bayramian, Camille Bibeau, Raymond J. Beach, Christopher D. Marshall, and Stephen
A. Payne,” Consideration of stimulated Raman Scattering in Yb:Sr5(PO4)3 laser amplifiers”,
Applied Optics, 39,3746 (2000)
12. H.Haken, “Laser Theory”, (Springer, Berlin, Heidelberg 1984)
13. H. Ritsch and M.A.M Marte, “Quantum Noise in Raman Amplifiers : Effect of Pump
Bandwidth and Super- and Sub-Poisson Pumping”. Phys.Rev A47, 2354 (1993)
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