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The Semi-Classical Approach to Stimulated Raman Scattering
Term Project
ELEC342 - Photonics and Optical Communications
Lieuwe Leene
08562315
Abstract:
This report presents the fundamental understating of Stimulated Raman Scattering by the use of
semi-classical G. Placzek model that is based on the bond-polarizability theory. The theory will
be developed in the context of optoelectronics with particular consideration toward the Raman
effect in optical fibers. The subsequent discussion will then use the basis to discuss the
implications of the Raman effect on lightwave systems.
I. Introduction
Raman scattering was first properly identified
by Sir C.V.Raman in 19281 as the phenomenon of
inelastic scattering as we know it today which was
already several year after one of his students mistook
the effect as a „weak fluorescence‟ in 1925. The
sensational finding bewildered the researches for the
following two decades that resulted in some of the
most profound consequences in the area of non
linear optics and quantum chemistry known to man
that have now almost become commonplace in the
area of spectroscopy and optoelectronics.
One must emphasize that even though this
phenomenon was predicted by A. Smekel2 in the
field of quantum mechanics one year before Sir
Raman‟s discovery the full potential of Raman
Scattering was not reviled until G. Placzek‟s
redefining theoretical contribution in 19343. At the
time of Raman‟s discovery, the quantum mechanical
explanation required the knowledge of all eigen
states of a particular scattering system in order to
even consider the calculation of the first and second
derivative of the molecular polarizability.
Unfortunately, Quantum chemistry had not yet
developed into the state that would allow the
measurement of such empirical data, which is where
the importance of G. Placzek contribution came into
play. As Placzek reintroduced the understanding of
Raman Scattering with a semi-classical model that
orients itself around the bond-polarizability theory
that became of much use to both physicists as well as
chemists due to the elegance in formulation as well
as its ease to application. On the basis of Placzek‟s
theory all details of interest on the molecular Raman
effect have quantitatively been accounted for and is
now found as a basis for some of the higher order
proceedings of the Raman effect such as Stimulated,
Hyper, and inverse Raman scattering.
The Raman Scattering process was first
thought of as a fluorescent type of scattering because
the scattering was inelastic as and the scattered light
photons had a frequency shift that was dependant on
the medium of occurrence. However, it was quickly
determined that this is not entirely the case due to the
fact that Raman Scattering can occur at any given
frequency and as a result the process it not a directly
resonant effect the same way fluorescence is. This
allowed Sir Raman to make the fundamental
distinction between fluorescence and Raman
Scattering, being that the phenomena is an inelastic
scattering process that radiates coherent light.
Fig 1: Experimental setup R.H. Stolen‟s observation of the
Raman effect in a fiber4
Stimulated Raman Scattering was a
introduced almost simutanously with the
introduction of the Raman Effect. R. H. Stolen was
one of the first to experimentally observe SRS using
one of early Corning single mode fibers in 1971
when SRS had already been well developed
theoretically. Even though SRS can generate a very
respectable gain over a long distance the process has
a high dependence on the fiber attenuation which
was one of the reasons SRS was not empirically
shown in fiber optics until almost 50 years after its
theoretical prediction. This was the beginning of 25
years of intensive experimentation that lead to the
Raman fiber amplifiers and lasers as we know them
today.
Recent Wavelength Division Multiplexing
(WDM) systems have developed particular interest
in the field of Raman amplification by SRS. Because
the Raman Effect allows for a very wide band of
amplification, approximately on the order of ,
at any specified frequency band it is no wonder that
there is potential for both very efficient amplification
and simultaneously amplification over a multiple
wavelength channels. We see the potential benefits
but we also see the risk of potentially unwanted
Raman scattering as the channel densities increase
resulting in intensities possibly approaching or
exceeding the Raman threshold. This includes seeing
the nonlinear effects taking a increasingly more
significant effect on the propagating signals.
II. Light Scattering and the Placzek Basis
In practice, light intensities are high and the
medium may have strong excitation – which is
considered to be the „high quanta limit‟ making the
semi-classical approach to the topic of stimulated
Raman scattering very appropriate as the
macroscopic behavior fits the classical wave theory
well. We will first introduce the classical picture that
represents the basis of light scattering and following
that is precursor to the Placzek model that was
developed with the notion of classical mechanics to
explain spontaneous Raman scattering (SRS).
The classical approach that studies the nature
of light scattering is based on revising the interaction
of light and matter on a non linear basis and treats
the interaction as a parametric amplifier. In attempt
to explain the observed Raman scattering we
consider the induced electric dipole moment of a
molecule as the source of electromagnetic radiation.
It is well demonstrated in electro magnetism text
books that the intensity radiated by an oscillating
dipole at the angle θ to the axis of oscillation is
(1)
Where is the amplitude of the induced dipole at
the frequency , , and .The latter being
constants for the speed of light and material
permittivity respectively.
Let us now introduce the time dependant
electric field E, the electric vector incident of the
radiation, as a plane wave oscillating at the
frequency ωi . We may then make the nonlinear
consideration with regard to the polarization vector.
The molecular time dependant induced electric
dipole moment may be considered as a superposition
of constituent dipole moment vectors as follows;
(2)
Where is assumed to be a rapidly converging
series, as . Having the
relationship with respect to E in the following form;
(3)
(4)
(5)
Here, one observes that is linear with
respect to E whereas higher elements of the series
are nonlinear with the field E. As expected is the
second-rank polarizability tensor what in this case is
time dependant. β is a third-rank tensor that
corresponds to the hyperpolarizability and similarly
γ a fourth rank tensor that corresponds to the second
hyperpolarizability tensor. Our current interest lies
with since it corresponds to the Raman scattering
phenomena as we shall see later. While Raman
scattering only occurs only at high light intensities
the hyper-Rayleigh & hyper-Raman scattering
requires even higher intensities before the processes
takes significant effect such we may simply mention
but ignore the hyperpolazirability tensors for our
purposes. We may now rewrite the polarization
vector in general orientation as;
(6)
Now, making use of this relationship we may
adopt the Placzek Model for further consideration.
We shall assume a molecule that only experiences
non-rotational modes of vibration about the point of
equilibrium such that we allow ourselves to use a
Taylor expansion with respect to normal coordinates
of vibration for each component of of the second
order polarizability tensor at the equilibrium
configuration.
(7)
Where is the equilibrium value of and
are the normal coordinates associated with
the vibrational modes at the frequencies as
a summation over all coordinates. Accordingly, we
make the electrical harmonic approximation which is
analogous to the mechanical harmonic
approximation that allows us to first consider the
response of one normal mode of vibration .
Mechanical harmonicity in a molecular vibration
indicates that the restoring force on the molecule is
proportional to the first power of the displacement
which can easily be interoperated a single spring
fixation to the equilibrium position. This analogy
allows us to picture the dipole response as illustrated
in fig 2.
Fig 2: Illustration of the Placzek Model.5
And also allows us to greatly simplify our
polarizability tensor to fundamental form that still
allows for the correct interpretation of Raman
scattering.
(9)
We can now consider the response of the
microscopic polarization vector to an incident
electromagnetic disturbance characterized by the
plane wave E. Given that we are considering a non-
absorbing linear (isotropic) medium (e.g optical
fiber) and assumptions as stated above, the time
dependence of is as follows
(10)
Where is the thermal amplitude of the vibration
such that the expression for and is
(11)
(12)
From the trigonometric identity one will arrive at a
function for the polarization vector as a function of
linear polarization response with the addition of two
additional frequency components that can be
identified as follows.
(13)
(14)
This shows induced the linear induced electric dipole
has three frequency components; the
component that induces radiation at the frequency ωi
and corresponds to Rayleigh scattering. The other
two frequency components are the stokes and anti
stokes frequencies that induce radiation at and
that correspond to Raman scattering. The
nomenclature of these particular frequency
components is not of explicit significance has a
historical origin that is related to whether the
radiation satisfies Stoke‟s Law of florescence.
Fig 3: Illustration of the Raman effect in frequency domain
6
Fig 4: Illustration of the Raman effect in time domain 6
Using the classical interpretation of Raman
scattering we have identified the correct frequency
dependence of the observed stokes and anti-stokes
frequency components, namely the molecular modes
of vibration. By substituting the expression (12) back
into the expression for radiation intensity (1);
(15)
We observe that just light Rayleigh scattering,
Raman scattering is also proportional to inverse
wavelength due to the similarity in nature of the
scattering.
III. Dynamics of Raman Scattering
Now that some of the fundamental aspects of
Raman scattering have been identified we can start
the considering more involved topics that are of
interest to us. One of these topics being Raman gain
since we hope to evaluate the implications of Raman
scattering on fiber optic communications.
The derivation of the Raman gain coefficient
will require us to consider two wave propagating in
the direction z with frequencies and which
correspond in our context to the pump and signal
frequencies where > .. The familiar wave
equation with the plane wave approximation with the
associated polarization and displacement vectors is
as follows.
(16)
(17)
(18)
Where we are considering N independent molecules
per unit volume allowing us to summarize this result
as
(19)
Now, has modal amplitudes proportional to both
and which are the two signal and pump waves
contained in the field and are expressed formally
as
(20)
(21)
We may also represent the molecular vibration as
(22)
Now, we need to obtain a relation between the
vibrational amplitude with respect to the fields
and in order to begin to consider the Raman
gain coefficient. In an earlier treatment on stimulated
Raman scattering Shen and Bloembergen presented7
us with a method that allows for an elegant
expression for that summarizes the dynamics in a
Lagrangian system. The Lagrangian density is given
by
(23)
Since our plane wave asserts that
is constant such that we can equate the derivatives
other two Lagrangian components that presents a
coupling of energy between the light waves and
vibrational waves. We may determine by
considering a molecule in the system that has the
kinetic energy , potential energy
, and let
.
The Lagrangian now has the form
(24)
Note that here we assumed the pump intensity as a
constant and neglected the anti stokes term.
(25)
Equating the Lagrangian component derivatives
and with respect to conserved momentum of
gives
(26)
Phenomenologically the damping term
was added to the expression resulting in the form of
a harmonic oscillator function with respect to the
vibrational amplitude which is a cogent result if we
relate it to the intuitive model of vibrating molecule
as stable oscillatory system.
We know from our previous analysis that the
resonant oscillation occurs when and from our
experience with the 2nd
order harmonic oscillator
differential equation we find that the peak response
is
(27)
If we focus our attention to the term as the
forward propagating wave of interest that we would
like to see amplified, we may reconsider the
condition for the propagating wave (19) by
substituting the co propagating waves , and
the wave appropriately. We have particular
interest in the gain in over a small segment on
the z axis so before we continue let us quickly
abstract the relations (21) and (22) so that we can
express the gain directly
Clearly in is is the stokes gain term and
in is is the anti-stokes gain term. For
consistency we shall assume two things to simplify
our derivation, the anti stokes are negligible and the
pump intensity remains constant (non depleting).
This allows us to formally relate the gain of as
follows
(28)
We can clearly see that we require both the
frequency matching of as well as the κ
vector matching of for the process of
spontaneous Raman scattering to take place and in
steady state ( ) we can identify from
(26) & (28)
(29)
This result allows the Raman gain coefficient to be
summarized as
(30)
(31)
As seen from the derivation the gain has a significant
dependence on the vibration frequency which can
be considered as the frequency shift from the pump
wave to signal wave denoted as which it is also
the parameter used when practical considerations
made with regard to the Raman gain over the
spectrum of frequency shift, .
Fig 5: illustration of the spontaneous Raman gain spectrum in a
fused silica fiber at pump wavelength 1.0µm 8
In a series of experiments the Raman gain
coefficient of a silica-core fiber was measured and
the spontaneous Raman cross-section measurements
were compared to the known standard benzene
spectral response which quickly introduced two new
concepts effective area & effective length that apply
to almost all nonlinear processes.
A useful method for evaluating the effective
area9 is by setting up the gain in terms of optical
power and effective area instead of directly
inspecting the radial variation of the electric field.
Let us first consider the wave function that describes
the distribution of intensity as result of the
waveguide modes for both and as
(32)
(33)
Then substitution for equation (30) gives our
coupling equation
(34)
For convenience let
(35)
such that when we multiply both sides of (34) by
and integrate over and θ we obtain
(36)
We can now formulate this expression in terms of
optical intensities by introducing the optical intensity
„P‟ and the it‟s square root „F‟ as
(37)
Substitution for (36) gives
(38)
(39)
Let and
Generalizing our previous expression as
(40)
This last generalization turns out to be useful since
the effective cross sectional gain is entirely
specified by the waveguide‟s material parameters,
and two source frequencies and .
Using the previous result we will proceed
with the two paraxial wave equations for the rest of
our discussion which allow the coupling of pump
power and signal (stokes) power as follows
(41)
We recall that throughout our derivation we
have assumed the pump wave to be a constant.
However we realize that the pump will also
experience attenuation as it propagates through a
medium. For this reason we introduce the concept of
effective length that will to some extent compensate
for this shortcoming. Manipulation of (41) with the
inclusion of the loss term over length L gives
(42)
(43)
Note that here we assumed the pump intensity in
much greater than the signal intensity such that the
energy lost due to amplification over distance L is
negligible. We can then identify and approximate in
the limit
(44)
If the pump and signal intensities are comparable
however we must use the coupled equations (41) and
(45) that result in significantly more difficult
computations.
(45)
The Manley-Rowe relations allow us to
identify one of the main motivations for SRS
amplification. Set aside the quantum mechanical
basis, these relations assume a lossless system and
allow us to relate the photon flux of both intensities.
The photon fluxes are as follows
, (46)
And with , (42) and (46) easily show that
the Manley-Rowe relations for SRS are as follows
(47)
(48)
(49)
This shows that it is a “photon for a photon” process
and that amplification by Raman scattering
intrinsically has an internal quantum efficiency of
100%. Given the low loss fibers used in modern
systems can theoretically allow for very high
efficiencies.
IV. The Stimulated Distinction & Threshold
Since we have just shown that we may express
propagation equation for the Raman photon number
by considering a segment dz then the output photon
number is 1 + gain – loss, formally formulated as
(52)
This allows us to identify both spontaneous and
stimulated components of the rate equation. By
integration gives the gain over length
The spontaneous scattering limit ( ) results in
(51)
The Stimulated scattering limit ( ) results in
(52)
And the observed threshold condition with the same
assumption as (44)
(53)
V. The Quantum Mechanical Side Note
The assumption however makes this threshold value
(53) fruitless as we intend to have energy couple
from the pump wave to the signal wave. R.H Stolen
has shown us in his treatment for the fundamentals
of Raman Amplifiers in Fibers that we can
incorporate the associated quantum mechanical basis
and obtain a more meaningful expression for the
SRS threshold.
So lets us first develop a brief understanding
of the quantum mechanical mechanism associated
with spontaneous Raman scattering. We must initiate
our thinking by considering two particles of interest
and the electronic „excited‟ states in a segment of
medium. The spontaneous process is fairly strait
forward and consists of a photon with energy
exciting the medium from ground state into higher
state of excitation whereby either the spontaneous
process occurs and at some statistical instant of time
the medium will emit a photon with energy or a
incident phonon (quasi particle representing
quantized vibrational energy) can stimulate the
process downwards also emitting a photon with the
energy .
Fig 6: illustration of the (left) special process and
(right) state transition process 4
The above process describes the generation of a
stokes photon where as the anti stokes photon also
has a possibility to scatter but this process only
occurs when a incident phonon is present. However
given that the molecule is in thermal equilibrium the
population of phonons is given by the Bose-Einstein
distribution
(54)
And formulating rate equations of the two processes
described above
(55)
(56)
Where S is the associated rate constant and N
indicates the associated particle population.
Combining these results shows that radiation of anti-
stokes likely to be magnitudes less at low
temperatures or even zero at while stokes photos
would still have a positive rate.
Consideration of the SRS process will
conclude that we must add the population as a
incident photon with energy will stimulate
another photon of energy downwards. From our
experience with similar laser rate equations we know
that the additional stimulated photon will be in
coherent and in phase with the incident photon.
Conclusively SRS is summarized by rate equation
(57)
It is important to point out that the fact that this
process develops gain from a stimulated photon tells
us that it is polarization dependant. This result makes
amplification in a fiber a slightly more challenging
that it seemed at first sight however this overcome
by either inducing SRS at multiple polarizations
which is energy inefficient or polarizing the two
light sources in the same direction which requires
more advanced waveguiding techniques and thus
more expensive fiber.
In comparison with our classical analysis, we are
interested in the stimulated cross-sectional gain per
small length segment. We may do so by introducing
a generalized gain parameter that is includes the
wave velocity
(58)
(59)
Where is the refractive index at . With a very
similar approach as with regard to the derivation of
(50) we can assert that when consider the stimulated
gain to be dominant the change in photon number is
as follows
(60)
by considering the stimulated scattering limit
we will integrate the total equivalent photon
power over the entire Raman gain curve for the total
pump power necessary to couple energy over length
L
(61)
Assugested by R. H. Stolen we should approximate
the Raman gain curve as a Lorentzian function with
peak gain and curve width at half gain of . One
may then approximate the integral even further by
only considering the most dominant terms of the
series expansion
(62)
Where the resulting integral will give
(63)
(64)
Where Stolen then presents us the formal expression
for stimulated Raman threshold power
(65)
Fortunately, the Raman threshold power varies
slowly with the fiber parameters and can typically be
approximated a much simpler condition for the
threshold power
(66)
Overly conservative
This last result finalizes the relevant mathematical
basis that would allow for proper analysis of optical
systems based on the SRS rate equations.
VI. Discussion
Even a basic analysis of the current optical
fiber communication systems will allow the
identification of a crucial fact. We must find a way
to deal optical nonlinearities instead of instead of
avoiding them if we wish to improve out
communication networks. This fact is perfectly
illustrated by W.P. Urquhart 10
. When we consider
the linear basis for communications were a limited
by two fundamental constraints. One limitation is
detector sensitivity that requires a finite minimum
power at the end of the transmission fiber, which is a
result of statistical white noise associated with any
real system (zone 1). The second being the minimum
intrinsic loss associated with a propagating fiber the
disallows the production of fibers with lower losses,
which is a result of light scattering process becoming
the dominant term in fiber losses (zone 2). The non-
linear interaction of propagating light and the fiber‟s
molecular structure results in effects like RS and
FWM and creates a third constraint to
communication systems (zone 3). The latter however
is variable and in some sense technology has already
allows us to shift the curve of zone 3 by
technological innovation (e.g. dispersion
compensators). The illustration also demonstrates
that WDM systems that supposedly holds the future
for communication systems, experience these
limitations in the largest magnitude as they push the
limits of maximum channels over maximum lengths
Fig 7: Illustration of the relative restriction zones of light transmission quantified with respect to fiber loss and pump
power 9
Approaching the non linear domain with
equivalent innovation is a far greater challenge that
requires a much deeper physical understanding than
the basis that was presented. As there are still some
aspects of the Raman effect left completely
unexplained. One of them being associated with the
fact that in the classical analysis we have taken the
Raman gain spectrum for granted and as a result we
unable to explain why backwards SRS has a
amplification is smaller than forward SRS and has a
saturation a few magnitudes smaller than the pump
intensity. A more crucial addition to this is that the
simplicity of the coupled wave equations breakdown
for light pulses shorter than 100 fs, requiring a
modern „spacetime‟ approach to SRS. In addition we
have not properly considered the phonon buildup
under SRS. Relating back to our rate equations () we
see that phonons can contribute to both gain but
clearly also the reverse SRS process resulting in
absorption. Fortunately, empirical evidence shows11
that this approximation was appropriate.
A important consideration of the Raman
effect is it implication on wavelength channel cross
talk. Shown by Fig 5 the Raman gain is dependant
and proportional to frequency shift up to 4 THz. By
using the threshold approximation and by making
several general estimations to channel band
separation and minimum allowable cross talk A.
Chraplyvy was able to present a very quantifying
result relating the number of wavelength channels
and the maximum allowable power per channel as
shown in fig 8.
Fig 8: Illustration correlating number of WDM wavelength
channels with the maximum power per channel 7
Now that we have discussed some of the
concequences of SRS we should introduce the
Raman Amplifiers as the outstanding boon from the
Raman effect discovery. The basis for Raman
amplifiers was the basis for out coupled paraxial
equations that have shown both a exponential gain
relation as well as the photon conservation
expression. Both of these are very beneficial since
they sum up as efficient gain over any large band of
frequencies with the fiber as transmission medium.
Recent developments12
showed that not only do
these amplifiers allow efficient amplification but
they also allow amplification of 45 dB with a pump
power of 2W and show high saturation output
powers of 20 dBm with noise levels of -50dB. It is
pointed out that the Raman amplifier requires a
significantly higher pump power due to the low
spontaneous Raman scattering cross section that
needs to initially overcome the overall attenuation
before it can initiate the stimulated process. More
advanced material enhancement of fibers such as Ge-
doped fibers allow for a higher Raman scattering
cross section and as a result lower the pump power
to 0.5W or allow for discreet amplification that could
potentially reduce noise levels.
It is important to realize that both classical
and quantum mechanical theories many of the
theoretical formulation of SRS depends on empirical
results that can easily disdain some of the subtle
effects that only surface in particular cases. It is very
likely that the future development in this field lies
very much with material science and quantum
chemistry as the Raman effect has a fundamentally a
material basis either specified by the classical
polarizability tensor or the Hamiltonian that describe
transition coefficients.
Conclusion
We have successfully introduced a brief basis
of understanding of the Raman effect and its
stimulated variant applicable in both areas of
Classical mechanics and Quantum mechanics.
Including the mathematical formulation we have
introduced several approximations that allow a
straightforward explanation and formulation of SRS
in terms of rate equations. We have also introduced
concepts such as effective length and effective area
that are commonplace in the studies of non linear
optics in the application of fiber communications.
Our discussion has pointed out several of the
undermining shortcomings of the presented semi
classical understanding of the Raman effect and
presented the topics for further study. In addition, the
discussion briefly identified the main implications on
lightwave systems that identified the general
challenge that results from nonlinearities as well as a
quantifying signal intensity constraint in WDM
systems. Conclusively, the Raman amplifier was
presented with in association with its recent
developments and future challenges.
1 C. V. Raman and K. S. Krishnan, ‘Optical analog of the Compton effect’. Nature London 121, 711, 1928
C. V. Raman and K. S. Krishnan, ‘A new type of secondary radiation.’ Nature London 121, 501, 31 March 1928 (16 February).
2 A.Smekel,Zurquantentheoriederdispersion,
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Theory of Raman Scattering by Molecules, Wiley (2002). 7 Y.R.Shen and N.Bloembergen,“Theory of Stimulated
Brillioun and Raman Scattering*”, Physical Review, Vol 137, Num 6A, 15 March 1963. 8 Andrew r. Chraplyvy, "Limitations on Lightwave Communications Imposed by Optical-Fiber Nonlinearities," Oct. 1990, Journal of Lightwave Technology, vol. 8, No. 10. 9 W.P. Urquhart, P.J. Laybourn, “Effective core area for
stimulated Raman scattering in single-mode optical fibers”, Proc. Inst. Elect. Eng., Vol. 132, pp. 201-204, 1985. 10 W.P. Urquhart and P.J.R. Laybourn, “Stimulated Raman scattering in optical fibres: the design of distortion-free transmission”, IEE PROCEEDINGS, Vol. 133, Pt. J, No. 5, 1986. 11 G.C.Fralick and R.T.Deck, “Reassesment of the theory on Stimulated Raman Scattering”, Physical Review B, 3rd Series, vol. 32, Nov. 15, 1985. 12 E.M. Dianov, “Advances in Raman Amplifiers” - Journal of Lightwave Technology, 2002 Additional References: 12
R.W. Hellwarth, “Theory of Stimulated Raman Scattering”, Physical Review, Vol 130, Num 5, June 1, 1963 13 S.P.Singh,R.Gangwar,and N.Singh, “non linear scattering effects in optical fibers”, Progress In Electromagnetics Research ,PIER74 , 379–405 , 2007 14
C.S. Wang,“Theory of Stimulated Raman Scattering”, Physical Review, Vol 182, Num 2, 10 June 1969. 15
J.Bromage,“Raman Amplification for Fiber Communication Systems”, journal of lightwave technology, vol.22, no.1, January 2004.