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,

Naturwissenschaften 11:873–875,1923. 3 G.Placzek, Hand buchder Radiologie VI Leipzig: Akademische Verlagsgellschaft, Teil II 205,1934. 4 R.H.Stolen and E.P.Ippen ,Raman gain in glass optical waveguides, Appl. Phys. Lett. , 22:276–278,1973. 5 R.H. Stolen, Fundamentals of Raman amplification in fibers, Raman Amplifiers for Telecommunications vol. 1, Springer, New York (2003) chap. 2. 6 D. A. Long, The Raman Effect, A Unified Treatment of the

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.