and W. KAISER - Science Direct

Pro#. Quant. Electr., Vol. 6, pp. 55-140 © Pergamon Press Ltd., Printed in Great Britain

0079~o727 '79/0701~055S05.00/0

HIGH INTENSITY RAMAN INTERACTIONS A. PF.NZKOFER,*A. LAtYaEREAU,~"and W. KAISER Physik Department der Technischen Universit/it Miinchen, M/inchen, Germany

CONTENTS 1. Introduction

56

2. General Theory of Raman Scattering 2.1. Quantum-mechanical treatment of Raman scattering 2.1.1. Spontaneous Raman scattering 2.1.2. Stimulated Raman scattering 2.1.3. Raman polarizability 2.2. Semiclassical theory of stimulated Raman scattering 2.3. Nonlinear susceptibility of the third order 2.4. Frequency dependence of X~3~and formal solutions by Fourier integrals

57 57 60 60 61 61 64 64

3. Raman Processes with Two Light Waves 3.1. Stimulated Raman Stokes emission. 3.1.1. Quasi-stationary case 3.1.1.1. Stimulated Stokes scattering in the forward direction 3.1.1.2. Stimulated Stokes scattering in the backward direction 3.1.2. Transient stimulated Stokes scattering 3.1.2.1. Transient scattering with small conversion 3.1.2.2. Transient process with pump depletion 3.1.2.3. Outlook: new coherent phenomena 3.1.3. Stimulated process with pump pulses of broad frequency bandwidth 3.1.3.1. Quasi-stationary regime with small dispersion (Toot> T2, 7"3) 3.1.3.2. Quasi-stationary regime with large dispersion (T3 > Zcor> 7"2) 3.1.3.3. Transient regime in dispersive media (T2 > Zcor) 3.2. Anti-Stokes Raman scattering 3.2.1. Quantum-mechanical treatment of anti-Stokes Raman scattering 3.2.1.1. Spontaneous anti-Stokes Raman scattering 3.2.1.2. Incoherent Raman probe scattering. Determination of population lifetime Tt 3.2.2. Anti-Stokes Raman process at high laser intensities 3.2.2.1. The inverse Raman effect 3.2.2.2. Anti-Stokes amplification in inverted systems

68 68 70 71 81 85 86 89 91 92 93 93 94 94 94 95 96 97 99 99

4. Raman Processes with Three and More Light Waves 4.1. Stokes-anti-Stokes coupling 4.1.1. Theory 4.1.2. Stokes-anti-Stokes coupling at a Raman resonance 4.1.3. Parametric four-photon interaction outside Raman resonances 4.2. Coherent spectroscopy (active spectroscopy) 4.2.1. Quasi-stationary case 4.2.1.1. Coherent anti-Stokes Raman spectroscopy--CARS 4.2.1.2. Coherent Stokes spectroscopy. The Raman induced Kerr effect. 4.2.1.3. Coherent spectroscopy with three input waves 4.2.2. Transient regime 4.2.2.1. Coherent anti-Stokes probe scattering 4.2.2.2. Coherent Stokes probe scattering 4.3. Secondary processes 4.3.1. Higher order Stokes and anti-Stokes generation 4.3.2. Interaction and competition with other nonlinear larocesses 5. Special Raman Type Processes 5.1. Raman scattering in crystals 5.2. Electronic Raman scattering 5.3. Spin-flip Raman scattering

99 100 100 103 106 10~ 110 110 113 114 115 115 118 122 122 125 127 128 130 131

6. Concluding Remarks

134

References

135

Appendix

140

* Present address: Fachbereich Physik tier Universit//t Regensburg, Regensburg, Germany. ~"Present address: Physikalisches Institut der Universit/it Bayreuth, Bayreuth, Germany. 55 .'.P.Q.E.6/2--^

56

A. PENZKOFER,A. LAUBEREAUand W. KAISER

1. I N T R O D U C T I O N For several decades, inelastic light scattering has proved to be a valuable tool in the study of elementary material excitations. The frequency shifts, the line widths, the intensities, and the degrees of polarization of the scattered spectra give detailed information which may be correlated to a microscopic picture of the medium. As a classical example we point to the extensive Raman and infrared data accumulated in the thirties which laid the foundation for our understanding of the normal vibrational modes of polyatomic molecules. With the advent of the laser, conventional Raman spectroscopy has experienced a genuine renaissance. The high intensity, the favorable monochromaticity and the ideal directionality of the laser beam have eased the taking of data and improved their quality tremendously. Numerous new investigations have been made possible with this new light source. In 1962 it was found accidentally that at very high light intensities a new phenomenon sets in. (1~ Light waves at a Stokes shifted frequency were amplified by many orders of magnitude in the presence of a strong laser beam. Similar to the light generation in a laser system one observes exponential amplification. While the laser gain is determined by the inversion of the active medium, one finds the intensity of the pump laser in the gain factor of the Stokes amplification. During the past 15 years, this stimulated Raman process has received considerable attention. Numerous new aspects never anticipated at the very beginning were discovered. Quantitative understanding of the various phenomena has been achieved with (frequently) good agreement between theory and experiment. The field has now reached such a degree of maturity that a review article appears to be timely. In Section 2 we present a general theory of Raman scattering. A fully quantummechanical treatment shows quite clearly the connection between the spontaneous and stimulated scattering processes. In addition, the equivalent noise input for stimulated scattering is readily deduced from these arguments. The semiclassical calculation discussed in the following section gives a good insight into the coherent properties of the interaction process and allows the investigation, the excitation, and the relaxation of the medium. The semiclassical approach will be used in the main part of this article. We begin the discussion of the various topics of high intensity inelastic light scattering with the important stimulated Stokes process (3.1). Different investigations of generator, amplifier and oscillator systems are presented. In the backward direction one has found a situation of especially strong coupling between the incoming and backward travelling wave. Short and intense Stokes pulses are generated in this process. Transient stimulated Raman scattering in liquids and gases at high pressure requires light pulses of several picoseconds duration since the relevant time constant of the stimulated Raman process, the dephasing time Tz, is of this order of magnitude. Transient processes allow the study of dynamical properties of the medium. Two aspects of the excitation of the medium are of interest: the coherent excitation of the resonant mode and the excess population of the upper excited level. The respective time constants, the dephasing time and the population lifetime, are measured by interrogating pulses. In Section 3.1.3 we discuss briefly the complex situation in which the input laser is not monochromatic and the medium is dispersive. Under certain conditions the nonlinear Stokes generation is strongly reduced. Anti-Stokes Raman scattering is treated separately to see clear differences from the Stokes process (3.2). At high light intensity one finds loss at the anti-Stokes side, an effect used in the inverse Raman process. The anti-Stokes Raman gain in inverted systems is documented by a recent experiment. The Stokes-anti-Stokes interaction is analyzed in Section 4. Of practical interest is the situation where two input laser beams of tunable frequency difference interact with the medium and generate anti-Stokes radiation. The interpretation of the observed coherent anti-Stokes Raman scattering (CARS) is discussed in detail. Of interest is the transient case of coherent Stokes and anti-Stokes probe scattering. It allows direct observation of the dephasing time of homogeneously broadened transitions and opens the possibility of obtaining information on inhomogeneously broadened systems.

High intensity Raman interactions

57

Under certain experimental conditions (e.g. tight focusing) higher order Stokes and antiStokes waves are observed (4.3). In most cases the input intensities are exceedingly high and the interaction with other nonlinear processes becomes significant. Of particular importance is the self-focusing of the laser beam with a resulting rapid increase in light intensity. When working with light pulses of several ns duration the competition of stimulated BriUouin scattering has to be considered in the individual experimental situation. Most of the examples given in this article are concerned with vibrational modes of polyatomic molecules. In Section 5 we discuss Raman scattering in crystals, the electronic Raman effect in metal vapours, and spin-flip Raman scattering of free electrons in semiconductors. The article closes with various concluding remarks. The equations are presented in electrostatic units (esu). Several relevant conversion factors are listed in the appendix. 2. GENERAL THEORY OF RAMAN SCATTERING This section is devoted to a general discussion of Raman scattering. Treating the nonresonant coupling of light with an atomic or molecular transition, we derive several important features of the Raman process. Few assumptions will be made concerning the specific properties of the physical system which generates the scattering effect. For clarity, we refer frequently to the interaction with molecular vibrations. We shall first outline a quantum mechanical description. This concept comprises both spontarieous and stimulated Raman scattering; it is usoful for the understanding of the onset of stimulated scattering from quantum noise. An occupation number formalism will be used, yielding rate equations for the photon modes of the incident and scattered fields. Phase effects of the light fields and coherency properties of the stimulated emission are not contained in the present quantum mechanical picture. For intense lightwaves the classical description of the optical fields in terms of Maxwelrs equations is justified, while the physical system is still treated quantum mechanically. This semiclassical approach leads to a set of differential equations which accounts for numerous features of different stimulated scattering processes. The coherent aspect of the material excitation and the phase relationship which builds up between the individual molecules become quite apparent from the semiclassical theory. It will be shown that the response of the physical system to the perturbation of an intense external light field is analogous to a driven oscillator when population changes are small and may be neglected. The individual Raman processes will be discussed in later sections; they are presented as special solutions of the semiclassical equations. 2.1. Quantum Mechanical Treatment of Raman Scattering In this section we treat both the radiation field and the physical system quantummechanically. The approach is appropriate for the weak scattered fields which occur in spontaneous Raman scattering and at the beginning of the stimulated amplification process. There are numerous theoretical papers in the literature on this subject. (2-5) The present treatment is intended to reveal the relationship between the spontaneous and the stimulated scattering phenomena. We follow a recent discussion given by Wang. (5) The total physical system is described by the Hamiltonian function H which consists of terms representing the molecular system Hv, the electromagnetic radiation field Hem, and the interaction between vibrations and electric fields Hint. H = Hv + Hem + Hint.

(1)

The Hamiltonian Ho contains the kinetic energy and the potential energy of the uncoupled system. 1 H~ = ~ ~m {t~i2 + o92q/2} •

(2)

The index i runs over all molecules, m is the reduced mass and q is the relevant local coordinate. For the ease of a molecular vibration, q is the normal mode amplitude.

58


COo represents the transition frequency of the molecules considered. The Hamiltonian He,, comprises the electric and the magnetic energy of the electromagnetic field.

Hem: ~;(DE.-~ HB)dV: lf(F, E2--[-1B2)dV.

(3)

E is the electric field strength, D = eE is the dielectric displacement, H represents the magnetic field strength and B = / ] H is the magnetic inductance, e is the relative permittivity and /~ the relative permeability (here /] _~ 1). The integration is carried out over the interaction volume. The interaction term responsible for Raman scattering is treated in the electric dipole approximation, where Hint = -.f~ ~E' dE'. Following Placzes,-(6) o n e expands the molecular polarizibility ~ as a function of the local coordinate q. c~= So + ~qqq + ' " .

(4)

The term linear in q is important for the interaction Hamiltonian : Hint

--

2

q(Ri) E(R~)E(R~).

(5a)

Ri denotes the position of the ith molecule. The sum in equation (5a) extends over all molecules in the interaction volume. The electric field E(Ri) in equation (5a) consists of the incident laser field, and of the scattered Stokes and anti-Stokes components. Here, we restrict ourselves to the laser and the first Stokes field E = EL + Es. The relevant interaction Hamiltonian for Stokes light generation reduces to Hint = - ~ (O~-~q)q(R,)EL(R,)Es(R,).

(5b)

The Hamiltonian Hint provides a non-resonant coupling between an incident light field of arbitrary frequency with the molecular transition. Equation (5b) is quite different from the direct electric dipole interaction, which is only effective for frequencies of the incident field close to the transition frequency COoof the physical system. For the quantum mechanical calculation we introduce the corresponding operators. Second quantization is applied to the field variables using the following transformations: (ll = (mNo)- 1/2 ~ e x p (ikvRi)i(b~ - b_ko)(hCOo/2)l/2 ;

(6)

kv

qi = (mNo)- x/2 ~ e x p (ikvR~)(bkv + b +ko)(h/2COo) x/2 ;

(7)

E = ~ (2rchCOz/eV) 1/2 ezi {a~ exp ( - ikaai) - ak~ exp (ikxRi)} •

(8)

kv

No is the number of molecules in the interaction volume V. e~ is a unit vector representing the light polarization, bko and ak~ are the annihilation operators for the quantized material excitation and for the electromagnetic field, respectively. With the help of equations (6-8), the Hamiltonians Hv, Hem, and Hint are written in the form : H. = ~, tUoo(b~bko + i/2);

(9)

k~

(10)

film = ~ hcok,~(a~ ak,~ + 7/2);

ka

~l..tL Vm

ks,kL\

COO /

+ a ~akL + (eLe s){bk~

+ bkoak~a~} 6(kL -- ks - kv).

(11)


59

Equations (9) and (10) show that the molecular system and the radiation field are represented by ensembles of harmonic oscillators. Energy quanta of the material excitation, equation (9), will be called phonons in the following. The subscripts L and S refer to the laser and Stokes photons, respectively. The creation and annihilation operators obey the usual commutator rules. # denotes the refractive index. 7 is the identity operator. The sum in equation (11) has to be carried out over the laser and Stokes modes. Only the resonant terms which fulfill the condition of energy conservation are retained in /~,t (rotating wave approximation). In order to calculate the steady-state transition rate W for the Raman process we apply Fermi's golden rule: W = (2~/h) l ]2p(hCOf)

(12)

p(hcoy) denotes the density of final states I f ) which are connected to the initial state Ii> by energy conservation. For simplicity we confine ourselves to discrete energy levels, p = h-lfi(coy _ c o i _ COo).It is convenient to use in equation (12) the occupation number representation and define the initial state Ii> in terms of the occupation numbers ns, no, and nL of the Stokes modes, molecular system, and laser modes: Ii> = Ins, nL>. Due to the well-known selection rules of the creation and annihilation operators of harmonic oscillators, the matrix elements in equation (12) are non-vanishing for the two final states If~) = Ins + 1, no + 1, n L 1) a n d If2) = Ins - 1, no - 1, nL + 1). The transition to the final state fl produces Stokes photons and material excitation at the expense of incident laser photons (hCOL~ hCOs+ hCOo) while the final state f2 represents the inverse process, loss of Stokes photons and generation of laser photons (hcos + hco0 ---,hCOL).The total rate dns/dt of Stokes photon generation is the difference of the two processes. Using the matrix elements -

-

(ns + 1,no + 1, n L -

11/-Sin,Ins, no, nL> = const [(ns +

1)(nv + 1)nL]1/2

and (ns - 1, no - 1, nz + 1 ]Hi.t Ins, no, nL) = const Insnv(nL + 1)31/2 we arrive at the following result: dns d t = N (\~qJ t~2

pg # L4~3 ~mclL

ms (eLes)2_1 ( + ns + no)fi(cos -- COL+ COo). COO

(13)

N = N o / V is the number density of molecules. The laser intensity IL = (hcoLC//~L V ) E k L/1L was introduced in equation (13). According to equation (13), the rate of Stokes light generation is determined by three terms (a fourth one proportional to nsno has been neglected since ns, no ~ tiE). The first term (the 1 in the bracket) represents spontaneous Raman scattering. I t describes the scattering of laser light off the quantum fluctuations, when the population of Stokes modes is negligible, ns ~ 1. The second term (proportional to ns) describes the rate of Stokes light generation resulting from the amount of Stokes photons present. For a large Stokes population, ns > 1, the scattering increases rapidly. This term is responsible for the stimulated Raman effect. The third term (proportional to nv) corresponds to parametric coupling between the Stokes light field and the material excitation. In general, no is small and approximately equal to the value at thermal equilibrium nv = (exp (hcoo/kT) - 1)- 1 ; no.may be neglected in many cases. Even in the stimulated Raman process, where a large number of Stokes photons and a corresponding number of material transitions (phonons) are generated, no is negligible due to the stronger damping of the material excitation as compared to the Stokes photons, i.e. no and If > are the initial and final state of the Raman transition. Insertion of equation (21c) into (21b) leads to = \h3~s J

,-

coi-

tel

tel

"

Equation (21d) shows the dependence of (c%~/&l)on the level system of the medium. A resonant enhancement of Raman scattering is expected when transition frequencies co, - co, or tOy - cot are near to the laser frequency (resonance Raman scattering ~25a,254'269)). For most papers discussed in this article the electronic transition frequencies are considerably higher than the frequencies of the incident light fields. In these case (O~/dq) is a constant. 2.2. Semiclassical Theory of Stimulated Raman Scatterino The quantum-mechanical approach of Section 2.1 has shown that stimulated Raman scattering occurs for large occupation numbers of the Stokes mode. At the corresponding intensities the light fields may be treated classically. The theory discussed after equation (12)

62

A. PENZKOFER, A. LAUBEREAUand W. KAISER

is valid in the steady-state case; it does not include transient effects, and gives no information on the phases of the individual atoms or molecules. The semiclassical theory to be presented now gives a detailed picture of the physical situation. The molecular system will be treated quantum-mechanically. The time evolution of the coherence of the medium is well accounted for by the theory. On the other hand, the coherence of the scattered radiation is anticipated to some extent by the ansatz of classical electromagnetic waves. This point is supported by experimental findings of a highly monochromatic stimulated Stokes emission. The (molecular) transition involved in the scattering process is described by a two-level model. Transitions to other energy levels are assumed to be off-resonance and are omitted. Molecular vibrations are notably anharmonic, i.e. the energy levels are not equally spaced. As a result, transitions between higher excited vibrational states give only negligible contributions to the (stimulated) scattering process. E

"fitoo/2

1o >

0 -~tOo/2

Ib>

FIG. 1. Two-level system of Raman transition.

The two-level system is depicted in Fig. 1. The eigenstates of the unperturbed molecule are the upper state la> and lower state Ib>. The energy separation is AE = ho0. The Hamiltonian is written as the sum of the free system and the Raman interaction; the latter has been discussed in equation (5a): ,-, ~-~q) [ a a ~ q",E,E, ffI = I-Iv + fflint = f-Iv -- ~1 ~i " ".

(22)

The wave function of an individual molecule i is written as

[q/i) = ai(t)la) + bi(t)lb ).

(23a)

The density operator of the molecular system is given by 1

L

(23b)

The time evolution is expressed by the equation of motion: 0/5 i ^ ~- - ~ [p,/1].

(24)

Taking into account the properties of the four matrix elements of ~, p~a, Pab, Pb~, Pbb = (b I P [ b), the analysis of equation (24) shows that three real quantities fully describe the time dependence of the molecular system. This fact is used in the pseudo-spin vector model (v)Here, we introduce two dynamic variables, the expectation value of the vibrational coordinate (q) = Tr(~q) = (a [~qla) + ( b l ~ q l b ) and the population probability n = p~, of the upper state. (q) is related to the off-diagonal elements of the density matrix: ( q ) = qab(Pab + Pba)

(25)

and will be called collective amplitude, q~ denotes the transition matrix element (alq[b), which is taken to be real by proper choice of the basis l a) and Ib). The closure relation [a) (a] + Ib> (bl = 1 and the assumptions q~a = qbb = 0 have been used in equation (25). The application of equation (24) to 8 ( q ) / d t = qab(dpab/Ot + Opba/Ot) and to dn/dt = dp~o/dt gives :~s-a3)


63

a(q> 1 g-----i-+ ~ ( q> = - iWoqab(Pab -- P~a) an 1 i act 2 O t + - ~ l ( n - ~)= ---~qE2h q~(P~'

P~°)"

(26a) (26b)

On the left hand side of equations (26a) and (b), damping terms with two time constants Tt and T2 have been introduced in analogy to the Bloch equations for a spin system. (9'14~ These terms account for relaxation processes due to the interaction between, and within the molecules which were not included in the Hamiltonian of equation (21). T2 denotes the dephasing time which is connected with the off-diagonal dements of the density matrix. This time constant is sensitive to the phase relation between the molecules of the system. The second relaxation time T1, connected to the diagonal elements of ~3, represents the population lifetime of the excited state. Since there may be processes which effect the phase of the collective excitation without population decay we expect T2 < 2T1 in general. The factor 2 results from the definition of the time constants. T2 is related to the relaxation of an amplitude while T1 gives the decay of the stored energy. The damping terms in equations (26a) and (b) lead to an exponential decay of the freely relaxing system. This time dependence is expected for sufficiently long values of T1 and T2, i.e. if a large number of random events constitute the relaxation times. 114~~ denotes the thermal equilibrium value of n. Elimination ofp~b - Pb~in equations (26) (with the help of equation (24) and the condition po~ + Pbb = 1) leads to second order differential equations for the collective amplitude (q> :ts,9) a2(q) 2a(q) tooq2 (O~) at 2 + T2 a--T- + w 2 ( q ) = h -~q E E ( 1 - 2n)

(27)

and for the excited state population n: dn 1 1 /'~a~ d(q) dt + 11-~-(n- n ) = 2-~mo~q) E2 dt _.

(28)

On the left hand side of equation (27), a term ( q ) l T 2 was omitted since we have small damping, l/T2 ~ O~o, in general. In equation (28), a term proportional to ( q ) / T 2 has been neglected in comparison with a ( q ) / ~ t for the same reason. It is interesting to see from equation (27) that the collective amplitude of the molecular ensemble obeys the equation of motion of a classical damped oscillator. The driving term on the right hand side depends upon the population of the upper state. There is no coupling with an equally populated system, n = ½. A change of sign occurs for population inversion, n > ½. The coupling between the applied light field and the molecular transition is determined by the Raman polarizability a~/dq and the transition matrix element q,,. Having in mind an anharmonic vibration, we use for qob as an approximation the result q~ = (h/2mogo)l/2. (8) The propagation of the light field is governed by the classical electromagnetic theory. The following wave equation is obtained from Maxwell's equations :tls> V2E-

(~)2 ~02- E -

Y-~#a 41r 82 pNL c -~E = ~ - ~ - ~ •

(29)

The third term has been introduced to account for linear losses. V denotes the absorption coefficient. The total polarization of the system is split into two parts in equation (29); the linear component containing the refractive index # is placed on the left hand side and the nonlinear contribution pN~, acts as a source term on the right hand side. The latter part is responsible for the stimulated Raman process. The nonlinear polarization has two contributions: A first term due to the interaction of the light field with the two-levd systems,

64


and a second part, PNR, resulting from all the other off-resonance transitions of the moleculesJ 16) pNL =

N(~-~) ( q ) E + PNR.

(30)

It should be noted that for the case of condensed matter a local field correction is included in the Raman polarizability Oct/Oq of equation (30) (and of equations (27 and 28)). N is the number density of molecules. Since Oct/8q is treated as a scalar quantity, equation (30) refers to highly polarized Raman lines. The scattered light, polarized perpendicular to the incident laser light is not included in the present calculation. Equations (27) to (30) form a complete set of differential equations describing stimulated Raman scattering by the semiclassical approach. In many practical cases, simplifying assumptions make the system tractable. The following chapters present solutions for various special experimental situations. Equations (27) to (30) constitute the nonlinear coupling between photons and phonons as well as between photons themselves. Various names have appeared in the literature emphasizing different aspects for the light-matter interaction. The present paper will discuss both the specific conditions and the physical context of the individual processes in order to clarify the current terminology. Some authors have introduced new names for known and previously explored scattering effects. 2.3. Nonlinear Susceptibility of the Third Order For the theoretical description of various nonlinear optical phenomena, a power expansion of the induced macroscopic polarization as a function of the applied electromagnetic field is generally usedJ ls-2°) This approach is convenient for parametric processes and scattering effects. For the present paper, the term of interest is proportional to E 3" PNt~t, X) = ~3}(t)EEE.

(31a)

The proportionality factor in equation (31a) is the third .order nonlinear susceptibility ~t3) in the time representation. Equation (31a) denotes a macroscopic relationship, i.e. the fourth rank nonlinear susceptibility tensor is a macroscopic material parameter just like the linear susceptibility. A special component of the nonlinear polarization has the form ~1s~

;_o

P~L(t, x) =

dr1

m,rl,o

= 1

dz3

¢~

~3

x Z},m,n,o,~3' ~ It; zl, zz, z3)Em(t- Zl)En(t- r2)Eo(t- r3). (31b) The subscripts 1, m, n, 0 denote the vector components in the x, y and z directions. Under stationary and quasi-stationary conditions P't(t) may be described by the electric fields at time t: 3

P?L(t,x) =

~13~.... Em(t)E.(t)Eo(t).

~ m,n,o

(SIc)

= 1

Equations (31) emphasize the point of view of a parametric light-light interaction. Certain microscopic aspects of the light-matter interaction, e.g. the excited-state population, are not evident. On account of its usefulness, the Z{3}formalism has found wide application. In the discussion of Raman scattering processes it is convenient to describe the nonlinear polarization and the electric field by its frequency components. For a set of discrete frequencies con we write: 1

El(t) = ~ ~p { Et(cop) exp ( - icopt) + c.c. } ; = 21 Z,

{P)L(co")exp(-ico"t) + c.c.};

(32a) (32b)

High intensityRaman interactions

65

yielding with the aid of equation (31c) ~,'L

1

Pi (t) = ~ -..,,.-~-o{ZI'a~'"'°(- tOa; tO,, tOa,tO~,.)E,.(tO~,)E,(tOa)Eo(tOr).

(33a)

,,B,7

In the last equation we used the relation tO~ = tO, + tOa + tOy- From equations (32b) and (33a) we find for the frequency component P~t(tO6):

pZt(w,0 = ,,,~,oD ~13g,,.,o(_tO,;o,~,,tOa,tO~,)E,,.(tO,,)E,,(tOa)Eo(tO:,).

(33b)

D denotes the number of terms Em(~o,)E.(tOa)Eo(tOv)that contribute to p~L(tO~). We have D = 6 when all three frequencies tO,, toa, and tO~ are different; if two frequencies are equal one finds D = 3, while D = 1 holds for three identical frequencies. The nonlinear susceptibility Zc3) is usually defined as 3)

D

"3)

Xl.m..,o(- tO~; tO,, tOa,tO~) = ~- ~}..... o(- ~o~;tO~,tO~,cot).

(34a)

The ordering of the electric field components E(co,) to E(tO~)in equations (33) is insignificant. Thus the frequencies may be interchanged in the argument of :tc3) c16.i s) ;to,, tOp, tOy) =

tO,tO,,

tO )

= )~lao~ ,,( _ tO6; tOy,tO~,tO~).

(34b)

Owing to the energy conservation of the inherent four-photon interaction, the sum of the frequencies tO~ to toy has to be zero. The frequency values of the photons which are converted in the nonlinear process conventionally enter the argument of)~(3) with positive signs; and the generated frequencies enter the argument of Zca) with negative signs. As shown below, the Stokes Raman scattering (toE --' COs+ toO)is described in the Z13) formalism by )Ct3)(- COs; COL, --(-DE, toS). For the anti-Stokes scattering (toe + toO--' tOA) we use XC3~(--tOA ; tOE, --COL,toA) and for the Stokes-anti-Stokes interaction (tOe + tOE ~ (-DS "Jr- tOA) the responsible susceptibility is ~((3)(_COs; tOE,COL,--a~). The following points concerning the spatial symmetry of ;tca) are of value. The large number of independent elements of the fourth rank tensor is often reduced by the symmetry properties of the material. In a medium with cubic symmetry four independent elements Ayyyy, ,,(3) Ayyzz, ,,ca) ]~yzyz, ~t3) ~yzzy .t3) exist. Here, the light is assumed to propagate along the x-axis, and the transverse fields E and p~L have components in the y and z directions. In isotropic media, e.g. gases and liquids, the number of independent tensor elements is three on account of the additional isotropy condition Zc3) .ca) + A~,ca) . yea) yyyy zy,~, (34c) y z y z -- A y z z y "

2.4. Frequency Dependence of z ~3~and Formal Solutions by Fourier Integrals The frequency dependence of XI~,,,o is rather complicated and has been discussed extensively in the literature, cla-2°) The magnitude of X13) is resonantly enhanced if one frequency or combinations of the frequencies to, to to~ coincide with eigenfrequencies of the molecular system. For the purpose of this paper it suffices to distinguish between two terms of the nonlinear susceptibility: (i) a resonant contribution Zres 13) due to the molecular transition at frequency tOo. 7.~3~ is a complex quantity and becomes important when frequency differences, approach the resonance value o f tOo, e.g. when Ito~ - to~.l = ~00. In the resonance region ^re~"~3)is strongly frequency dependent: (ii) The contributions of all the other molecular transitions (assumed to be far off-resonance) are collected by the nonresonant susceptibility )C}v3~.The value of X}v3~is only weakly frequency dependent. Taking these terms together, we write )~13) = X~,~ (3) + X~3~.

(35a)

66


Accordingly, the polarization pNL in equations (33) has two terms, a resonant and a nonresonant contribution : pNL = Pr~ + PNR

(35b)

where Pres = Z~3]EEE and

PNR = Zk3"t~EEE.

(35c)

For the Stokes Raman scattering (toL ---'toS + toO)we now derive an explicit expression of Equations (30) and (31) give two different relations for the nonlinear polarization pXL. The first equation describes pNL in terms of the coherent amplitude (q), while the latter relates pNL to the nonlinear susceptibility X~3).The comparison gives

~(3) res.

Z~3~EEE.

Pres = N ~qq ( q ) E =

(35d)

In the following we neglect the tensor character ofx (3) which relates light fields of different polarization, and consider the total light field to be linearly polarized. This approach is equivalent to the treatment of the Raman polarizability ~/3q as a scalar. The calculation is performed in the frequency domain. Accordingly, we introduce the Fourier transforms of the total light field, the nonlinear polarization, and the coherent amplitude and write :¢2o)

E(x, t) = ~

dto[bSL(x,to) + Es(x, 09)] e-"'*; -oo

pNL(x, t) = ~

de) pNL(x, ~0)e -i,.,,;

(36a)

-oo .

(q(x, t)) = ~

dto Q(x, co)e-"". -oo

Propagation of the light waves in the x-direction is assumed in equation (36a). The total light field consists of two non-overlapping components ~L(Og) and Es(to) which are concentrated around a central laser and Stokes frequency, respectively. Since the physical quantities E, pNr, and (q) are real, we have

~:L~(-- to) = ~,S(to) ~L(__to) = p,'L,(~0)

(36b)

~2(-- to) = ~2" (to)" Introducing equations (36a and b) in our material, equation (27) yields a solution for Q(x, to). Use of equations (30) and (35) give the nonlinear polarization

pN%~)=

f?~

dto~ dto2

[ (O°~/Oq)2N(12n)/4m

0,2 _ (to~ _ to)2 + 2i(to~ - to)/T2 + xk°~

X EL((D1)E~(oJ2)Es((D

] - - co 1 -{- (.D2).

(37a)

In equation (37a) we have tacitly assumed that the time dependence of the excited state population n may be neglected. Similarly, equations (33) (generalized for a distribution of frequency components) and (36) lead to a second equation for the nonlinear polarization: pNL(to) =

f

oo

dot dto2zta)(-to;tol,

-(-02, to + tot - to2)EL(tot)E~(to2)Es(to - col + ~02).

-oo

(37b)


67

Comparison of equations (37a and b) readily gives the frequency dependence of the nonlinear susceptibility

1 [~2 X(3)( -- CO; 0)1, -- CO2, CO -- CO1 "l- CO2) = ~

k~"~q]

N(1-

2n)

CO2 -- (0)1 __ 0))2 ..1_ 2i(co1 -

CO)/T2+ x~3~" (38a)

The first part on the right hand side equation (38a)denotes the resonant part of the nonlinear susceptibility introduced in equation (35a). Separating X(3) into real and imaginary parts, X(3) = X' - ix", we find for monochromatic laser and Stokes fields : ±

,020 - (co -

Xt(--O.)S;(Z)L, --COL, (I)8)= 4m \dq/ (1 - 2n)[o92_ (O)L- (OS)2] 2 + 4(COL-- COS)2/T22+ x~3~ = X; + g~3~

(38b)

and 1

-~m N

Xt'(--(.OS ; COL,--COL, (/~S) :

0/~_~/2 (1 -- 2r/) [coo2

(COL -- O)S)/T2 (COL -C~S')~-]:I'+--4(-O~L_ COs)2/T22 = Xs"

(38c) "% 3

3 3

Na ! w

i

0

f

x e., .m .m

X (3)" : X s

¢~

t~o Frequency w v = W[. - W S

FIG. 2. Resonance structure of nonlinear susceptibility for Stokes light generation.

The frequency dependence of X,

in IS

lOUt 'S

¢ O

"_.u

~

W

~= io z

15 6 =

•

IU

0

,..I"'

I

I °ut

'S

,.J*'J

.--;"

¢:

- 10"8

r...f..r "I"'" r..d

Ig;

e..d

--- 161°

--'------"''"

Sx .i"

"'"

0 I,,,'r~ 0

I /Time

- Sx16"

I 8 t rinunits

I

I

I0

12 16 of r o u n d trip t i m e ]

FIG. 10. Stokes light generation in Raman oscillator versus time. Input laser has shape of a stepfunction. I~", Stokes intensity inside the oscillator; I~°', Stokes intensity outside the oscillator. Solid lines, R = 0.9 ; dotted line, R = 0.2.

frequency, i.e. TMI(COL)= TM2((--OL)"~-- 1. At the Stokes frequency the reflectivity of the mirrors are RM1 = 1 and RM2 = R. The length of the active medium I is taken to be equal to the length of the resonator. For easy computation the pump pulse is assumed to be square. Our model calculation is restricted to the stimulated amplification of quantum noise starting at one end of the cavity. After the first passage through the medium at time tl = l#s/C, the emitted Stokes signal is given by equation (66). I~Ut(1) = ISN [exp(gS/L/) -- 1](I -- R).

(66)

Since we started from quantum noise ISN, we used equation (51). The back-reflected Stokes component is amplified on its way back and forth through the oscillator (stimulated amplification in the backward direction will be discussed in detail below). After a time t2 = 311~s/C the Stokes signal has the intensity I~Ut(2) -- ISN [exp(gslLl) -- 1] (1 --R){1 + R exp (2#slL/)}.

(67)

After N - 1 round trips, i.e. after a time tN = (2N -- 1)ll~s/c, the Stokes output has the form: ffs"l(N) = ISN [ e x p ( g s l L l ) -- 1] (1 --

R] RN exp (2N gslLl) -- 1

, ~ ~

i

"

(68)

There is a kind of threshold for the Raman oscillator. For R exp (2gslLl) > 1

(69)

the Stokes output increases exponentially with N, until depletion of the laser pump limits the Stokes generation. According to equation (69) exponential amplification occurs for Gs = gslLl > ½In(I/R). For R = 0.9 we calculate gslLl > 0.05. With a small net gain of

80

A. PENZKOFER, A. LAUBEREAUand W. KAISER 1021

~

I

I

I

I

t

r~

E .c I0

®

4~ 0

J2 0

0

E

--2

0

~

3

® t0" L_ OJ,

0.5 0.6 0.7 Dye Laser Wavelength ~L ClJ'm'l

0.8

FIG. 11. Infrared and ultraviolet wavelengths generated by Stokes and anti-Stokes Raman scattering. Pump source is a dye laser of variable wavelength zt. Raman medium, gaseous hydrogen. (a) Stokes side 1, 2, 3, 4 stands for first, second third, and fourth Stokes component. (b) AntiStokes side. The numbers 1, 2, 3, 4 indicate the higher order anti-Stokes components (after Ref. 60).

Gs = 0.2 one obtains high conversion of laser into Stokes light after seventy transits. This result should be compared to the single pass Raman generator where 9 s l L l ~ 25 is necessary for efficient ('-~1%) Stokes generation. This estimate shows that Raman oscillators may be operated at intensities down by a factor of 100 compared to Raman generators. In Fig. 10, the rapid growth of the Stokes intensity inside, I~n, and outside, Igu', the resonator is depicted as a function of time (number of round trips). The following parameters were chosen in the calculation : Ie = 108 W/cm z, 9s = 10- 9 cm/W, l = 10 cm, and IsN = 10- 3 W / c m 2. R = 0.9 and R = 0.2 corresponds to the solid and broken curves, respectively. Saturation for R = 0.9 occurs after approximately N = 15 round trips. In the saturation range, the Stokes output reaches a value of I~ u' = (¢Os/COe)Ie similar to the case of the Raman generator or amplifier. We note that the Stokes intensity inside the cavity may exceed the laser pump intensity with I~"(N) = IgL't(N)/(1 -- R) (see Fig. 10). We. now turn to the discussion of various experimental investigations of Raman oscillators. The emitted Raman beam is determined by the pumped volume (or by the aperture of the mirrors) and by the number of transits inside the oscillator. (11) Diffraction limited Stokes output has been achieved. The brightness [W/cm 2 sr] of the Stokes radiation may exceed the brightness of the pump substantially. (55'59) A tunable Raman oscillator was designed by pumping a single mode fused silica fibre with an argon laser. (27'57'256~ The 100m long fibre had a core diameter of 3.3~m and a loss of 17db/km at 5145A. ~57~ The Raman oscillator was tuned by an external prism over 3oocm-1 with a linewidth of ,-~7 cm-1. Stolen et al. (27'2s6) believe that fibre Raman oscillators are potential rivals to dye lasers and parametric oscillator. Quasi-CW Raman oscillators, where high pressure hydrogen gas as an active medium is pumped by tunable dye lasers, gives a versatile coherent light source with a frequency range from the visible to the far infrared ~56'6°) (see Fig. 11). Schmidt et alJ 6°~ used special mirrors with high reflectivity at the higher Stokes components in order to increase the conversion efficiency for the far infrared frequencies. In another experiment, a tunable Q-switched Nd-glass laser was applied to pump an oscillator wave guide system containing liquid nitrogen and hydrogen as Raman-active material. Tunable infrared radiation of high conversion efficiency was reported. ~61'62~


I

10.3

I

I

I/

p : 95 bar

/

81

~ •

/

/ /

•

o

~, 167

b

/0 • e_ •

c

p I

s

g

~o Io.9

!x 10s

/, x I0 e

6x I0e

8xlOs

Input Peak Intensity ]oL rWIcm2"l FIG. 12. Effects of diffuse scattering on stimulated Raman interaction. Raman medium, hydrogen at 95atm and room temperature. Sample length 1 = 30cm. The pulse durations are given in the figure. Curve (a), calculated Stokes output for Raman oscillator with R = 10- s and gs = 1.8 x 10- 9 cm/W. Open circles, experimental points. Curve (b), calculated Stokes output according to Raman generator theory. Curve (c), Stokes output from (sli[ghtly) transient Raman generator. Closed circles, experimental points (after Ref. 63).

The reduced gain requirements of Raman oscillators as compared with Raman generators make such systems sensitive to the small optical feed-back Of optical components. Glass surfaces of standard polish exhibit backward scattering of the order of 10 -5, a value large enough to see Raman oscillation. Experimental investigations to demonstrate this effect are presented in Fig. 12. (63) Firstly, relatively long laser pulses of ~ 13 ns duration passed through a pressure cell filled with Hz-gas at 95 arm. The cell windows were tilted by several degrees to avoid specular reflection, but the diffuse reflectivity of R ~ 10-5 gave small but significant optical feedback. The very rapid rise of the Stokes output with input peak intensity (see Fig. 12, left) is fully accounted for by the resonator theory. Next, the same H2-cell was pumped with a laser pulse of tp = 1.7 ns, which was too short to sustain several transits in the Raman system. As a result, Raman oscillation did not occur and the Stokes output rose according to the single pass generator theory (see Fig. 12, right). It should he noted that in the latter experiment one is just on the borderline of quasi-stationary and transient behaviour. For He-gas at 95 atm, the Raman linewidth is A¢ = 0.14 cm- ~ which gives T2 = (21rcA¢)- ~ -~ 4 x 10-1 ~ s, i.e. this value is a factor of 40 smaller than the pulse duration tp. The gain coefficient deduced from Fig. 12 curve c comes out to be L65 x 10 - 9 c m / W , somewhat smaller than the calculated steady-state value of O = 1.8 x 10- 9 cm/W. Of interest is an experiment where the stimulated Raman effect was measured over eleven orders of magnitude in liquid N2 and 02. ~36~In Fig. 13, the Stokes output power is plotted versus input laser power for liquid N2. Startingat low Stokes generation, one readily sees the linear range of spontaneous Raman scattering (a),"the exponential build-up of Stokes power (b), and the saturation range resulting from depletion of the laser (d). The rapid rise of Stokes power (c) is due to optical feedback. Grun et al. explain their data by feedback resulting from Rayleigh scattering, t36) The small feedback due to the optical components was not considered. 3.1.1.2. STIMULATED STOKES SCATTERING IN THE BACKWARD DIRECTION. Starting from equations (42) or (45) we obtain the differential equations of the light fields for the case of backward Raman scattering when Q, Es, T2, and ks are replaced by Qa, EsB, T~ and

82


I

I

I

I

I

d

I

o

--

o

rn I0 2

N2

I/I c

•~

~-

e

LJ

~ 166_ 0 O_

m ide_

o

i0I0

/-

0

I

I

I

I

I

0.2 03 Input Loser Power PL EMW.] FIG. 13. Stimulated Stokes generation in liquid nitrogen. (a) Spontaneous scattering ; (b) exponential amplification of spontaneous Stokes light ; (c) rise caused by feedback of Stokes light; (d) saturation region (after Ref. 36). 0.I

- k ~ =-Ogs#s/C, respectively. In solids, liquids, and gases at high pressure, the spontaneous Raman linewidth is independent of the direction of observation, i.e. A03,. = A03ff= 2(T2) 1 = 2 ( T ~ - 1 . For gasesat low pressure, however, the Raman linewidth is Doppler broadened and A03v is larger in the backward than in the forward direction. (47'65'66) In the case of T2B = T2 one has ~((3)B = X(3). Following the same lines as for the derivation of equations (45a and b) we arrive at: 8Eft ~.x

6~EL

#s 0Eft c dt

7s ~ = _ i2zr COsZ(3)B( ~s - 03s ; 03L, -- COL,03S)IEL [ZEft #sC

]-/L ~EL ~L . 2 r e COL 03 ,lEO 2 c3x ~- o C~t F ~ - E L = t /~L - c ~((3)n*(--COS;03L,-- L, 03S)I s EL.

(70a)

(70b)

Equations (70) hold for the quasi-stationary situation. It has been shown above that the interaction length l between pump and Stokes pulse represents an important system parameter. In the forward direction, l is determined by the length of the Raman-active medium. In the backward scattering process, the interaction length equals the cell length if the pump pulse is sufficiently long, i.e. if tp > 21lt/c. For shorter pulses the effective interaction length in the backward direction is /elf "~ tpc/2/~ since pump and scattered pulse have opposite directions. For very short pulses the values of/elf are small, and stimulated backward Raman emission does not occur (e.g. tp = 5 ps gives lCtf = 5 x 10-2cm). (i) Backward process with small gain. We first discuss the situation where the laser pump is neither depleted by the backward Raman process nor by simultaneous scattering in


83

t h e forward direction, EL = EL(X- vt). v denotes the pulse velocity. In this case, stimulated scattering in both directions is completely de¢oupled and can b e treated separately. For simplicity we assume 7L = 7S = 0. Introducing the retarded time frame for propagation in the - x direction, 7 = t + x/v and x' = x, we obtain: t~EsB (x,P t) c~x' -•

i

2n e,'s Z'3's(--ms, O.)L, --(DL, (DS)lULls E~. #sC

(71)

The Stokes intensity is determined by the expression : disB (x,t t) t3x'

gB(OgS) IL I~"

(72)

The gain coefficient gB is given by equation (47b) replacing Z(3) Z(3}B. The solution of equation (72) has the form :

IsB (x,t t) = I~(I, t) exp [gn(cOS)IL(I -- X')].

(73)

The backward travelling stimulated Stokes pulse has at the beginning of the medium x = 0 the following intensity :

•

I~(0, 7) = Ig(l, 7) exp [gn(eJs) ~ [ L / ' J "

(73a)

The input Stokes signal In(l, 7) at the e n d o f the sample x = I may result from a n external Stokes pulse at a frequency COsor from a reflection of Stokes light generated in the forward direction: Without such input Signals, the equivalent quantum noise level Isz. starts the stimulated backward Stokes wave and equation (73a) has to be replaced by equation (74):

I~(0) = IsN [exp (gSBIL l) -- 1]

(74)

gsB is the gain coefficient at resonance, COs= COL-- ~00. In liquids or solids, the forward and backward Stokes waves have the same intensity, since g~ = gs and Z 1 3 ) e ' ' = Zt3)''. Gases, on the other hand, display different Doppler line broadening in the forward and backward directions, so that g~ < gs at low pressure. A backward Raman amplifier is depicted schematically in Fig. 14. The pump laser at frequency COLenters the Raman amplifier RA from the left. The Stokes input at COs is generated in RL which stands for a Raman oscillator or generator, or for a second tunable laser. According to equation (73a), the intensity IL and the ratio I~(O)/I~(l) determine the Raman gain coefficient gB(~). Early experiments t46) were made with H2-gas in RA and RL. More recently, the gain profiles of inhomogeneously broadened Raman lines of liquids were investigated where the input Stokes source RL consisted of a tunable dye laser. (67) Gain measurements in the backward direction have certain experimental advantages, for instance, the easy separation of the intense laser pump beam and the much weaker Stokes signal. it

I~{0} I~(l} FIG. 14. Schematic of backward Raman amplifier. PL, pump laser; RL, Stokes light source (laser, generator, or oscillator); RA, Raman amplifier sarhple.

(ii) Backward scattering with pump depletion. For a large Raman gain, i.e. for G¢ = g¢ILl ~ 20, the Stokes conversion becomes substantial and depletion of the laser

84


pump has to be taken into account. Frequently, stimulated backward scattering is accompanied by a strong forward emission. In this case, the equations connecting the three light intensities IL, Is, and Isn have to be considered. Similar to equation (61), one finds (6s) (~)L = 7S = 0 ) : _ _

C~IL+ Ox

OIL - -

1

. n I LIB~ 0) (ySILIs -I- ~S SJ--

v Ot

L

tOs

OIs 10Is 0~ + v - ~ = 9SIL Is

OIff Ox

1 OIsB -v Ot

B

(75a) (75b)

B

(75C)

ffs ILIs

where dispersion of the group velocity v is neglected. Numerical solutions of equations (75) were obtained by Johnson and Marburger. (6s) Under certain conditions the forward and backward Raman processes compete with each other, resulting in the periodic emission of Stokes pulses in the forward and backward direction. Figure 15 should illustrate the situation. A square laser pulse enters the Ramanactive medium from the left, generating intense forward and backward Raman signals within an interaction length ld or a time td = Id #/C. If the backward travelling Stokes wave depletes the laser strongly, the Stokes generation terminates after a time t = 2td. During the time 2td < t < 3td new laser radiation enters the medium, repeating the Stokes generation process. In this way a series of Stokes pulses is emitted in the forward and backward direction. Experimentally, the break-up of longer laser pulses and the appearance of trains of short Stokes pulses in the forward and backward direction have been reported in gases at very high pressure. (34'69'7°) medium

Raman J I

I

J

I

t

-4

-3

-2

-I Length

o

I

I

2Xtd

J

r, .... "1',

I

i

I

t

i

1(aS ,

,r .... '

i

,r .... I I

t

I

ixtd i

I

,,

3xl d

,

~ ....

]

:

: I

I I

t

I

, l'xtd

I

2

3

4

!

I [ i n units of t d c/JJ, "1

FIG. 15. Schematic pulsation of stimulated forward and backward Raman scattering; td depletion time.

The situation is quite different when an input Stokes signal travelling in the opposite direction to the laser pulse is presenL Considering the Stokes input to be significantly larger than the noise signal (which serves as the initial condition for simultaneous forward scattering) the backward signal reaches the saturation region much more quickly. As a result, the backward process depletes the laser independently of the occurrence of weaker forward scattering. In this case, only the second term on the right hand side of equation (75a) need to be retained, which gives : OIL 1 OIL ~Br I B O)L 0X "q- /3 0 t ~-----~SILZS (DS--"

(76)


85

b tJ

O_

0 0.

!

25 SO Time t r n s 3 FIG, 16, Oscilloscope traces obtained in backward stimulated Raman process. (a) Input laser pulse; (b) total backward travelling radiation (Raman and Brillouin component); (c) backward scattered Raman signal ; (d) transmitted laser pulse (after Ref. 9). 0

The coup!ed differential equations (75c and 76) have to be integrated for the initial conditions 1L(x = O, t) = ILo(t) and I~(l, t) = l~(t). Analytical and .numerical solutions have been discussed for a variety of input pulse shapes. (9~ Of special interest is the fact that a backward travelling pulse continuously encounters undepleted pump light, and the leading edge can be amplified to a value far in excess of the pump intensity. Accompanying the high amplification of the pulse is a strong steepening and sharpening of the Stokes pulse. Maier et al. t9'71) reported detailed investigations of the backward stimulated Raman process in liquid CS2. The backward travelling signal was initiated by self-focusing of the laser beam at the end of the cell. The peak power of the Stokes pulse was amplified to a level twenty times the instantaneous laser pump power and the pulse duration was found to be 30 ps. Several oscilloscope traces obtained in these experiments are redrawn in Fig. 16. The incident laser pulse of approximately 15 ns duration is shown in (a). The total backward-emitted light consisting of Raman Stokes light followed by intense Brillouin light is seen in (b). It should be noted that the Raman-Stokes emission (b) and (c) occurs several nanoseconds after the appearance of the laser pulse, in the form of a sharp light burst shorter than 0.3 ns, the time resolution of the detection system. The transmitted laser signal (d) shows a sharp break, demonstrating the depletion of the laser power during the interaction with the backward travelling Stokes pulse (c) (length of the cell 30 cm). The backward Raman process with pump depletion was subsequently investigated in high pressure H2 gas. ~72~ The input Stokes signal was generated in the region of high intensity of a focused laser beam. Pulse shortening was also observed by stimulated backward Raman scattering in CH4 pumped with a KrF-laser3257) Back~vard Raman interaction is able to convert efficiently a long pulse to a shorter o n e of higher peak intensity. This technique is being investigated for application in laser fusion experiments. 3.1.2. Transient stimulated Stokes scattering. The transient stimulated Raman process deserves special attention since it enables us to determine directly important material parameters of the molecular transition under investigation. (23-2s'73-vs) In particular, experimental techniques have been devised to study the dephasing time T2 and the energy

86


relaxation time Tx in the liquid phase, t79) While T2 may be deduced from the spectral linewidth in special cases, the population lifetime 7"1 has not been measured previously by any other experimental method. Transient excitation of a molecular transition occurs when the duration of the laser pump becomes comparable to or shorter than T2. Numerical analysis indicates that transient effects have to be taken into account for tp < 2 0 T 2 .(24) (See Fig. 17). The dephasing time of liquids and gases (at 300 K) is smaller than 10- ~~ s and 10- 9 s, respectively. With a gain of approximately twenty-five we estimate the appearance of transient effects for pulse durations of tp < 10- x0 s in liquids and tp < 10- 8 s in gases. We concentrate on the Stokes generation at resonance, o9~ = ~Oo, and neglect linear losses and group velocity dispersion for simplicity. From equations (42) we obtain with X(N3/~"~ 0 :

aQ(x, t) 1 i (3~ ~-+ ~22Q = 4~-m~o(~q) EL E , (1-- 2n)

(77)

dEs(x, t) 1 OEs .rtCOs{Oc~) c~ + - t N EL Q* v Ot C#s \~qJ c3x

On(x,t) 8~

+

+ -v ~

n-h

=

(78)

-- i C---~L Oqq IV/ZS~d

. 1 [Oc¢'X E

Le

Q* - E

(79) EsQ).

(80)

Equations (77)-(80) correlate the transient Stokes process with the dynamic material response. In the following sections the time dependence of Es, Q, and n will be discussed in detail. The semiclassical approach in the time domain is used here. In many investigations, especially for liquids, the generated excess population is small, i.e. n ~- h, and the pump laser is not depleted. This situation will be discussed in the next section. The case of very high Stokes conversion will be treated afterwards. 3.1.2.1. TRANSIENT SCATTERING WITH SMALL CONVERSION. W e again use the retarded time frame, t ' = t - x/v, x ' = x. With the approximation n - ~ and neglecting pump depletion so that EL = EL(f) we rewrite equations (77) and (78) t3Q*(x', t') 1 Q, = _ iKeE, E s ~3t' +

aEs(x', t') t3x'

1 /8~'~ 1

(81a)

(81b)

-- i K s E L Q *

~O~sN{8~']

KQ = 4--m-~m~qq)( - 2ti); Ks = ~

\Oq,]

(81c)

Solutions of equations (81a) and (b) have been published by Carman et al. 124) and Akhmanov et al. t2s) containing modified Bessel functions Io and I1 of zeroth and first order. For the Stokes field Es and the coherent vibrational amplitude Q one obtains :

Es(x', t') = Eso(t') + (KQKsXt) 1/2 EL(t')~" .I

and

,,=i oL{exp ( , ,) e~(t")Eso(t")1o(\/Z[KsKex'(W(t')

W(t"))])}dr"

(83)


87

where the abbreviation W(t') = J'E ~o dt"lEL(t") 12 is a measure of the pump energy up to the time t'. The solutions (82) and (83) are derived for negligible group delay, i.e. the Stokes and laser pulse travel with equal velocity in the medium. We note that in equation (82) the amplification starts from an initial Stokes field Es(x' = 0, t') = Eso(t') and the initial vibrational amplitude Q(x', t' -~ - oo) is assumed to be zero. Before discussing equations (82) and (83), we give a solution of equation (80) for the occupation number n

n(x', t') = h + exp(-t/T1)~-~

~q

-~ (ELE~Q* -- E*EsQ)exp(t"/T1)dt".

(84)

We see from equation (84) that n decays with the time constant T1. n may be calculated after Es(x', t') and Q(x', t') have been obtained from equations (82) and (83). In the limit of strong transient interaction, i.e. for pump pulses of duration tp ~ T2 and large gain gsILl ,> 1, one derives from equation (82) for the Stokes gain (23'24)

I Edt,,) 12 dt,,)l/2.

In (I Es(x', t')/Es(O, t')l e) ~- 4(Xs~eX'

(85)

This result indicates that for a highly transient situation the exponential amplification is proportional to the square root of the pulse energy. Equation (85) should be compared with the quasi-stationary gain where we have found: In (I Es(x', t)/Es(O, t)[2)

2Ks~:ex, l EL(t, ) 12 T2 = OsILX'

~--

= G,

(86)

i.e. the gain is proportional to the pulse intensity under stationary conditions. We have calculated the energy conversion efficiency r/s of laser into Stokes light for Gaussian input pulses. (14) In Fig. 17, the required gain #slLX for Stokes efficiencies of r/s = 0.1% and 1% is plotted as a function of the ratio tp/T2. For decreasing values of rifT2 the transient character of the stimulated Raman process becomes more dominant. Figure 17 shows quite clearly that larger values of the product ffslLX are necessary for shorter pulses in order to achieve the same conversion efficiency r/s. The two curves for r/s = 0.1% and 1% are very close together, reflecting the high nonlinearity of the stimulated Raman process. The following example should illustrate the situation: we calculate the required peak laser intensity for pulses of t o = 3 ps and for a sample length of x = 2 cm; a conversion efficiency of 1% is desired for a CH-stretching mode in CHaCCI3 10 3

I

II

I

I

I

i

i ~1 I0

J

i

J t I0 z

%•11

o

.c"

I

_!0 z

"0

.

_

n.,

i01

i

t el

I0 "~

~ I

Ratio

of

Time

t

Conslants

tp/T 2

FIG. 17. Required Stokes gain versus pulse duration tp for two "energy conversion etticiencies ~/s = 10 .2 and 10 -3. T2 denotes the dephasing time of the interacting transition (after Ref. 14). J.P.Q.E, 6 / 2 ~

88


E u

-, 10-1 c

tU 0

• [1-2 E

10-3

0

1 2 Time t/tp FIG. 18. Generated Stokes pulse versus time for a moderately transient, tp/Tz = 4.0, and a highly transient, tp/T2 = 0.25, situation. The Gaussian incident laser pulse of duration tp is indicated by the broken curve: r/s = 10-2 (after Ref. 14).

where T2 -~ 2 ps (A~v - 5 c m - 1). According to Fig. 17, for tp/T2 = 1.5 we need a gain of G = gslLX = 95 and estimate with gs ~-5 x 10 - 9 cm/r~/ a pump peak intensity of IL --~ 9 X 109 W / c m 2. This value is considerably larger than the steady-state intensity IL "~ 2.5 × 10 9 W / c m 2 which is calculated from G = 23 (see right hand side of Fig. 17). Equation (82) allows the calculation of the time dependence of the generated Stokes pulse. In Fig. 18, the normalized Stokes intensity is plotted for two parameters tp/T2 = 0.25 and 4. The Stokes pulses grow very rapidly on account of the highly nonlinear stimulated Raman process; the intensity rises by several orders of magnitude when the pump intensity changes by a factor of approximately two. We point to the pulse shortening and the shift of the Stokes pulse compared to the Gaussian input pulse (broken curve). Detailed calculations of Stokes pulses for various laser pulse shapes have been given in the literature. (24'25) The generation of Stokes light under transient condition was studied by several authors. (sl - 90) Figure 19 shows an experimental photograph of a streak camera with the laser and Stokes pulse on the same time scale. The shortening and the delay of the Stoke pulse is clearly visible. The picture was taken with H2-gas at lObar as Ramanactive medium where T2 ~ 200 ps is relatively long and transient effects are seen with laser pulses of approximately 1 ns in duration/s6) Of special interest is the excitation of the Raman medium. There are two aspects to the material excitation. The laser and Stokes waves generate: (i) a coherent excitation with collective amplitude Q ; and (ii) an increase of the population of the first excited state with occupation number n. ~s) (i) In Fig. 20, the coherent excitation [Q[2 is plotted as a function of time for three parameters tp/Tz. The ordinate is in units of h/mtoo where m and COo denote the reduced mass and the frequency of the Raman transition, respectively. The calculations of Fig. 20 are made for typical experimental situations of x = 1 cm, ~/s = 1%, and Gaussian laser pulses. As indicated in the figure, the coherent excitation grows very fast on account of the rapidly rising Stokes field. We recall that the Stokes field together with the laser field drives the coherent excitation. Of more interest is the behaviour of [ Q [ 2 for longer times.

High intensity R a m a n interactions

89

- - - - 500

--

0

ps

500

L

S

FIG. 19. Streak camera trace of laser and Stokes pulse. R a m a n medium, H2 gas at 10 bar (after Ref. 86).

When the pumping process has terminated, the vibrational excitation relaxes freely. The exponential decay of the ]Q 12-curves directly gives the relaxation time T2/2 (remember: T2 is the dephasing time of the amplitude). The question may be raised, how can we observe the coherent excitation IQI27 It will be shown below, that coherent Stokes or anti-Stokes scattering of delayed probe pulses allows us to measure directly the relaxation of ]Q]2 (Section 4.2.2). (ii) The excess population n ( t ) - ~ of the upper excited vibrational state (v = 1) is presented as a function of time in Fig. 21. We recall that for each Stokes photon generated one molecule is excited from the ground to the first vibrational state. The curves are calculated for the same conditions given above: x = 1 cm, r/s = 1~, and Gaussian laser pulses. We note that the generated excess population in condensed media remains small, less than 10-3. The parameter T1/T2, i.e. the population lifetime determines the time behaviour ofn for long times. In fact, the exponential decay of n(t) - ~ enables us to determine directly the important time constant T1. The observation of n(t) as a function of time is made by incoherent anti-Stokes scattering of delayed probe pulses as discussed in Section 3.2.1.2. 3.1.2.2. TRANSIENTPROCESSWITH PUMPDEPLETION.For large conversion of Stokes light, the laser becomes strongly depleted,t91) If the dephasing time T2 is sufficiently long and the coherent amplitude Q remains excited, the nonlinear polarization p~L oc QEs (see equation (41b)) generates a new laser field reversing the initial energy flow. A second laser pulse develops at the expense of the Stokes field, quickly depleting the latter. The repeated exchange of energy leads to a modulation of the laser and Stokes pulses. The observation of pulse break-up on account of severe pulse depletion was reported by Lowdermilk and Kachen for H2-gas at 30 atm where Tz = 240 ps.!92) The results are shown in Fig. 22. With a laser pulse of 1 ns duration, two maxima of the laser pulse and

90

A. PENZKOFER,A. LAUBEREAUand W. KAISER 10 "3

I

I

I

I

tplT2= 0,25

i,o-i 0

1

2

Time tltp FIG. 20. Coherent vibrational excitation (in units of h/mcoo) versus time for the values of the parameter tp/T2 (pulse duration tp to dephasing time T2). The exponential decay of the freely relaxing system with time constant T2/2 should be noted. Calculations are made for an energy conversion efficiency ~/s = 1% and a sample length of 1 cm. m and COo denote the reduced mass and the vibrational frequency, respectively.

10-3

I

I

tplT 2 =/.

C

10"~

C

tp/T 2 =/. Tl / T2 = 2

.9

0 D.

lo-S u x UJ

tp/T z = I " T1/ T2 =0.S

\ tp/Tz= i. lo-6

1 Time

2 tltp

FIG. 21. Excess population n - ~ of the first excited vibrational state versus normalized time. After the excitation, n decays exponentially with the population lifetime 7"1. The parameters tp/T2 and T1/T2 contain the pulse duration tp and the dephasing time T2. Calculations are made for ~/s= 1% a n d l = l c m .

High intensity Raman interactions I

I

I

I

I

I

I

I

91 I

1.0--

:" 0.6 -i cal

c

0.6

-

l.xp

'

•

N

A/,

lea,

I Z

0.2

0

-.-- ,

,

,.U I L~t...N,~

0

1.0 Time

I 2.0

t Cns3

FIG. 22. Transient stimulated Stokes process in the saturation range. Solid curves, calculated; dashed curves, experimental. Raman sample, H2 gas at 30 bar; T2 = 0.24 ns (after Ref. 92).

two time-delayed maxima of the Stokes emission were observed. It was stated that only one molecule in 104 was vibrationally excited, i.e. the experiments were performed with n~l. 3.1.2.3. OUTLOOK: NEW COHERENT PHENOMENA.The transient effects discussed so far result from the coherent interaction of light with the medium. The coherent excitation is represented by the collective amplitude Q. In other words, the off-diagonal elements of the density matrix play an important role in the coherent scattering process. Up to the present time, the problem of coherent optical pulse propagation in context with the Raman interaction Hamiltonian (equation (5)) has been only slightly touched. A few theoretical papers have addressed this field. (93) For electric dipole interactions--and even more for magnetic dipole interactions--the situation is quite different. A wealth of publications exists on these t w o topics. (94'95) A short comparison of coherent Raman pulse propagation with the well-established electric dipole case should be of value here; it reveals several interesting prospects of this field. Coherent effects of electric dipole transitions, e.g. self-induced transparency, free induction decay, etc. are readily visualized in terms of the pseudo-spin vector model. This model describes the instantaneous state of the molecular ensemble by a vector, the first two components of which are related to the off-diagonal elements of the density matrix while the third one represents the population (i.e. diagonal elements of the density matrix~ The model is also suitable for coherent Raman processes. (14'52) A n important dynamic variable is the direction of the pseudo-vector. The precession angle 0 depends upon the so-called "area under the electric field" which corresponds to the time integral over the field amplitude. The various propagation effects may be discussed according to the evolution of the angle 0. A first kind of phenomena is connected with small angles 0 ~ n; i.e. with small area pulses. A transient transparency effect occurs for this situation while the molecular system essentially remains in the ground state during the passage of the light p u l s e . (14'96'25s) It is interesting to see that the stimulated Stokes scattering discussed so far (n ,~ 1) represents the analogous case of small area pulses for the Raman type interaction. The condition of small scattering efficiency discussed in Section 3.1.2.1 establishes the limiting case of a short propagation length so that the propagation loss of the laser pulse may be neglected. More interesting is the situation with pump depletion. The modulation of the laser amplitude pointed out in Section 3.1.2.2 has its direct analogue in the electric dipole case where break-up of small area pulses has been found in several theoretical and experimental investigations. (97) We also note that the exponential decay of the coherent

92


excitation discussed in context with Fig. 21 corresponds to free induction decay in the dipole case. A second class of effects occurs for intense pulses, 0 ~ re, where the population of the upper state is substantially changed during the interaction. Under these conditions, striking phenomena, e.g. self-induced transparency and photon echoes were found for electric dipole transitions. The corresponding effects for Raman interactions have been observed in only a few cases. The major difficulty in these investigations is the weakness of the Raman coupling as a second order process in comparison with electric dipole interaction which occurs in the first order perturbation theory. ~98~The observation of a Raman echo was recently reported for a spin-flip transition in CdS at 1.6 K on a nanosecond time-scale by Hu, Geschwind and Jedju. ~17°~ Gaseous systems and solids at low temperature have good prospects in the future; they have long dephasing times T2 allowing the application of long excitation pulses with large amounts of incident photons. Furthermore, resonance enhancement of the Raman coupling by electronic transition might also be of help in studying these coherent light-matter interactions in the Raman case. 3.1.3. Stimulated process with pump pulses of broad frequency bandwidth. In the previous sections we have made a series of assumptions which may be fulfilled in certain experimental situations, but may not hold for others. We have tacitly assumed the laser pump source to be highly monochromatic with bandwidth-limited pulses. This point is connected with the slowly varying amplitude approximation where we have neglected all second order time derivatives. The transient stimulated process was discussed in the time domain, neglecting dispersion. Now we wish to comment on the stimulated Raman process with stochastic or phase modulated pump pulses. (24'25'76'99-1°2) The propagation in dispersive media will be considered. The efficiency of nonlinear optical processes such as stimulated Raman scattering is strongly affected by the coherence properties of the intense pump laser. In addition, the spectral properties of the Stokes radiation and of the coherent vibrational amplitude Q reflect the quality of the laser source. Three time constants are important for the following discussion :(99) (i) In many practical cases the pump laser does not emit a single monochromatic wave train. The observed frequency bandwidth A~OLmay be related to a correlation time rcor of the stochastic pump emission (87)

r c o r = I/ACOL.

Of course, the correlation time does not exceed the duration of the pulse, Zcor __~tp. (ii) As pointed out above, the coherent (phase) properties of the material excitation are characterized by the dephasing time T2. Various features of the transient stimulated Raman process depends upon this important time constant (see preceding section). (iii) Different group velocities of the laser and Stokes pulse lead to a separation of the two pulses in space and time. For a material of length l one estimates a time delay of (88)

T3 = (~L ~- las)l/c .

The minus and plus sign corresponds to Stokes pulses travelling in the forward and backward direction, respectively. The phase relationship between laser and Stokes radiation becomes destroyed in dispersive media and a reduced Stokes gain results. Stokes generation occurs effectively over a coherence length Lcoh,where Lcoh ----"Tcor

= ~33 l -

•

(89)

A comparison of L¢oh with the sample length l (for low gain) or with the saturation length Ln (for high gain) allows an estimate of the effectiveness of the Stokes generation. The saturation length is defined by equation (90) : L o ~- 30/0slL,

(90)


93

where gs and IL are the gain coefficient and the laser peak intensity, respectively. It will be shown in the following that the parameters %or, 7-2, T3, Lcoh, and LD are relevant in dispersive media for the Stokes generation by laser pulses with a broad frequency bandwidth. A detailed mathematical treatment is given by Akhmmaov et al. (99) 3.1.3.1. QUASI-STATIONARY REGIME WITH SMALL DISPERSION (Tcor > T2, T3). For "~eor > T2 transient effects may be disregarded. The material excitation follows the laser fluctuations. The relation %or > T3 indicates small dispersion without effect on the stimulated Raman process. The theory of the preceding Section 3.1.1. applies in this case to a good approximation. We note in passing that the Stokes gain of amplitude modulated pulses may appear surprisingly high compared to temporally smooth, bandwidth limited pulses, when an average intensity and not the true peak value of fluctuating pulse is considered. Intensity peaks give high Stokes conversion.

3.1.3.2. QUASI-STATIONARY REGIME WITH LARGE DISPERSION (T3 > "Ceor> T2). For > T2 the coherent material excitation follows the phase changes of the pump laser without transient effects. The second condition, T3 > Zcor, indicates reduced overlap between the fluctuations of the pump and the generated Stokes radiation. The coherence l e n g t h Lcoh : (%or/T3)l is shorter than the length of the sample, and effective Stokes generation occurs only over part of the Raman-active medium. The situation is different, however, for very large gain, where saturation of the Stokes generation occurs after a length Lo. For increasing laser intensities, the saturation length Lo "" 30/gslL becomes smaller. One expects a strong increase of the Stokes conversion when LD approaches Lcoh. When LD is even shorter than Leo, the Stokes generation becomes as effective as for bandwidth limited pulses. %or

U"

c

&

&

.c

t

0

o

2

! 0 tO

/

0 I

0

I

I

2 ~ 6 Input Intensity 1L [IdW/cm23

Fro. 23. Influenceof pump laser bandwidth on Stokes amplification.Stokes gain versus input intensity.Ramanmedium;liquidnitrogenwith A~ = 0.067cm- t. Curve 1, forward(opentriangles) and backward(closedtriangles)amplificationfor A~L.% 0slL) :

Es(x', t') = Eso exp (½gslL X').

(122)

For the anti-Stokes wave we obtain from equation (118b) the essential terms

E](x', t') = E]o exp ( - ½gslLX') +

exp (iAkA x')gs IL AkA Eso exp (½#slLX). 1 ,

(123)

The first factor describes inverse Raman scattering. The second term represents anti-Stokes generation by parametric Stokes-anti-Stokes coupling, which is considerably smaller than the Stokes production of equation (122) ([AkA I >> gslL). The dependence of Stokes-anti-Stokes coupling on AkA was investigated in detail by Shen and Bloembergen. (123) In Fig. 26, the normalized Raman amplification for Stokes and anti-Stokes radiation is calculated as a function of phase mismatch in units of gsl L. The calculations are made for ~3~ = 0.1 I~a)"[. Zero amplification for AkA = 0 and a superior gain of the Stokes intensity at AkA ~ 0 is readily seen from Fig. 26. For a phase mismatch Aka ~> 2gslL one finds the familiar Stokes generation Is(O/Is(O) = exp(gslL/) while the anti-Stokes generation amounts to less than 4~o of the Stokes intensity. It has been shown above that Stokes conversion of several per cent is possible for gslLl ~-- 25, while the anti-Stokes production is negligible. It is important to realize that the required phase mismatch of AkA ~ 50/1 is readily fulfilled in a collinear geometry in liquids and solids. In many practical cases one has AkA > 10 2 cm-1 and a cell length of several cm to tens of cm. These numbers indicate that stimulated Stokes generation is the dominant process in the direction of the laser beam in liquids and solids. Under a certain small angle to the beam axis of the laser where AkA-,0, the Stokes and anti-Stokes gain decreases strongly. Experimentally, this result has been eonfirrned' qualitatively. Around the laser direction dark rings in the Stokes and anti-Stokes emission were observed at a definite angle. This phase-matching angle was precalculated from the normal dispersion of the Raman-active medium.( t 24-126)

106


In Fig. 26 the pump laser was assumed to be constant. In the saturation region (depletion of pump laser) the anti-Stokes light generation may approach the amount of Stokes light generation. Two final remarks should be made. First, in gases of low dispersion the contribution of Stokes-anti-Stokes coupling has to be estimated for each individual case. Second, for the backward process the phase mismatch AkA is very large and backward Stokes generation is predominant. 4.1.3. Parametric four-photon interaction outside Raman resonances. In the general parametric four-photon process, COL+ OgL~ OgA+ O9S, tWO input laser photons at ~OL are converted into one signal photon at frequency OgAand one idler photon at COs.~5A6'22~ We identify the signal with the anti-Stokes, and the idler with the Stokes wave and discuss the situation where o9A and COsare off a Raman resonance. We will distinguish three situations: (i) The frequency differences ogA- 09L and O9L- COS are assumed to be only several linewidths outside the Raman resonance at o90. For this case we neglect the imaginary part of )~3~ and retain only the real part. We recall that ~3)"decays faster (with (COo- o9,)- 2) than the real part Z~3)' which is proportional to (COo- o9~)-1 (see equation (38)); (ii) The frequencies OgA and COs are situated far away from any resonance. The nonlinear susceptibility reduces to the nonresonant part in this case; (iii) The resonance behaviour is studied when COsapproaches a molecular excitation frequency, i.e. a single frequency resonance of Zt3~. The initial conditions for parametric light generation starting from quantum noise are Eso = 0 for Stokes (idler) light generation and E]o = 0 for anti-Stokes (signal) light generation.(t 26) The general solutions (118) reduce to (see also equation (114)): Es (x', t') = exp [-(~ - iAka/2)x'] ~-~ E,~o sinh (fix').

(124a)

E,~(x', t') = exp [(~ + iAkA/2)x'] ~XAS - Eso sinh (fx').

(124b)

The corresponding expressions for the Stokes and anti-Stokes intensities are

ls(x', t') = exp [(~ + ~*)x'-] IxsA ill212IAN [sinh (fx')[ z

(125a)

IA(X', t') = exp[(~ + ~*)x'] I

(125b)

Asl

zsN [sinh(fx')[

2.

ISN and IAN represent the equivalent intensity level of quantum noise at frequency COsand O9A,respectively (see equation (48)). (i) For the discussion of the parametric four-photon interaction outside but close to a Raman resonance we neglect Z~3)" and ~3~. Equation (114) simplifies to i ~2rc

2{o9s

co___Sa~ Z~3; + [-4n2 ]ELI,(Ogs

+ Ak] +

O9aA~2Z~3,,2

(COS OgA~Z~3,.]t'2~ JELl 2 akA ~ s + a ~ / _1 ;

(126)

Inspection of equation (126) shows that the four-photon interaction depends on the mismatch Aka. For AkA --0, ct is purely imaginary; i.e. no amplification takes place. The situation is different, when Ak,a # 0. For a certain phase mismatch

-T

(127a)


107

optimum parametric four-photon interaction occurs with the amplification factors ~,,2 .opt

=

_nCIELI2Z~~v i

-~o~ ~ 2

k/,tsga /

d

(127b)

This optimum situation will be considered in more detail. In the notation of equation (114) we have (128a) and (128b)

c \ #S~A /

Here, 9P denotes the optimum gain factor of the four-photon process :

32n2 (0-~ (-OA~t/2 Z~3V"

(128c)

Rewriting (11 lb) and (112b) yields XSA = i 2_~ntOS Z~3y E2L c

#s

and

KAS = i -2~ - ~coA Z~3~' E~. 2 c #a

and equations (125) have the new form: Is(X', t', Ak~pt) = 1AN/~AtoSsinh 2 (½gdLX')

(129)

IA(x', t', Ak~ t) = IsN/.tStoA sinh 2 (lgt,ILX,). #AtoS

(13o)

#StoA

and

The derived equations show that significant Stokes and anti-Stokes amplification occurs for Ak7 't. At tot- - Os = too ___F, gp has its maximum value (gp = 08/2). For tot. - COs= too _+ ~F (ct ~> 3) it is ge = gs/~. In Raman generator experiments the stimulated Raman process dominates since gs is at least a factor of two larger than ge. The situation may change for transient scattering (see below). Amplification of input light at tos near Raman resonances is very effective due to the large #e values (see following section 4.2). (ii) and iii) In our previous treatment, the nonlinear susceptibility was split into a resonant part X~s due to a Raman resonance while all other contributions were collected by the second term XNR.When the Stokes frequency varies over a large range the frequency dependence of the second term has to be considered. Of particular interest here is the situation when tos approaches a single frequency resonance. We now take the contribution of the single frequency resonance explicitly into account by writing

~((3)(--toS ;toL, --toL, toS) = X~3)"F"~ ) + ~/3~

(131a)

Z(3)(-tos ;COL,toL,--toA) = Z~3) + X~3)+ ] ~

(131b)

Z(3)(-toA ;tOE, --toL, toa) = X~3~* + ~

(131C)

Z(3)(-- toA ;COL,toL, COs)= ~3). + X~3). + ] ~3~.

(131d)

X~s and Xt denote the contribution of (infrared) single frequency resonances for the Stokes process and for the Stokes-anti-Stokes coupling, respectively, Far from Raman and other frequency resonances X~3~,X~, and X~3~ may be neglected and X~3) reduces to X~. In this limiting case, equations (114) simplifies to:


108

I,2=i [ELI

(ms

~AAA )

1 [-9rC2

7- EL 4

msm2:.sV

--(AkA+3~nclELI2(~s+~)2~a~)211/2.

(132a)

Again there exists an optimum phase mismatch Ak~pt = - (3n/c) [ EL[ 2 (ms/Ps + WA/pA)~t~ where the parametric four-photon interaction is maximum : 3x [i(ms ~]P~z= ~ cIELI2 Z~v3Z Ps

m a ) ( ~ s O g a ~ 1/21 pA -7- \~---~AJ "

(132b)

Equations (129) and (130), which are also valid for the situation discussed here, predict growth of the parametric light with the maximum gain factor : 24~Z2//COsmA'~a/2 -13'

9v = C - - ~ L ~ - ~ a )

(132c)

Z~.

2 ~ is approximately a factor of 100 less than the Raman susceptibility Z~3) (mr. - ms = mo) of Raman active media, e.g. 2 ~ = 10-14 cm3/erg in condensed matter. In parametric four photon processes with incident pump and signal waves (e.g. mL + mL --* ms + ma or mL + oL + ms ~ w with l(ms, x = 0) 4:0) light generation by Z ~ may be significant. Far off the Raman resonance but for ms close to the single frequency resonance the single frequency terms X~ and X}3) dominate and Z~3) and 2~v3~may be neglected in equations (131). For this situation we rewrite equations (114) yielding: V47z2

E '4

~I,2= Ic 1.7~ELI2~SSZ~3,.~_L7 I L' (L)S('OAz2/.~S] "IA --

(

+ 27~

AkA

c

IELI2~ )~}31)J • r,x..

~2-]1/2

(133a)

Outside the single frequency resonance we may neglect the imaginary parts setting Z~3r' = Z~3Y' = 0. Optimum parametric interaction occurs for Ak~p' =

:[

ms 3 ' EL [2 _ _ Z}s )

Ps

where

=clELl i~ssZ~3)'T-2Z}3)'~--~,/

A"

(133b)

Correspondingly, the opUmum gain coefficient is given by 327c-~2(ms w----~a~ 1/2 Z~3Y.

(133c)

gP = C2~L \ ~ S ~ A /

Near single frequency resonances )C}t3)' and the Raman susceptibility Xs have similar values and 0P and Os are of comparable magnitude. The parametric four-photon process WL + C0L-+ COS+ ma may effectively compete with the Raman Stokes processes 09 L ~ COs + inv.

The. solutions derived here give us the following information. According to equations (128c), (132c) and (133c), the amplification by off-resonance four-photon interaction is determined by the real parts of the nonlinear susceptibility Z~3). This result is substantially different from our previous findings for the resonant process 2WL --' ms + mA and for stimulated Stokes or anti-Stokes scattering. In the stimulated Stokes case, the value of the imaginary part Zt3r' alone determines the magnitude of the gain coefficient (see equations (17)). The amplification according to equations (129) and (130) exhibits an exponential dependence on pump intensity IL and distance x in the large gain limit (avltx ,> 1), similar to the pure Stokes process. Small differences of the value of the exponent decide which process dominates.

109


D

x

®

9 a~ e, a-!

x 0

1.._

®

xCt8 0 _mE . ---

1"

-

X

0,

131

-

6

8

a

~

-0.~ -8

-6

-t.

-2

0

Difference

Z

t,

((J~v-l,t)o) / T2

FIG. 27. Nonlinear susceptibility in Stokes-anti-Stokes coupling. (a) Frequency dependence of Ix, ,l relevant for coherent anti-Stokes spectroscopy. (b) Frequency dependence of the real and imaginary part of Z(3). The nonresonant contribution X~3~is real and nearly frequency independent.

We stress the influence of the k-mismatch on the gain factor. No amplification occurs at AkA = 0, while gain is found for Aka ~: 0 with a maximum for AkA "-" --½gilL. The frequency dependence of g(a) around a Raman resonance is depicted in Fig. 27b ; it should be compared with Fig. 2. In the off-resonant process one has to consider the real part of :t(3) which is made up of a resonant term g~3)' and a nonresonant contribution X~. We note that for frequencies outside the Raman band, O~o_ F, the real part of X[3) is larger than the imaginary part. We expect from Fig. 27b an asymmetry of the four-photon interaction around the Raman resonance. For small positive values of ogv - O~o, i.e. for larger Stokes shifts, the value of X(3)' may be quite small and the four-photon process is negligible. According to Fig. 27b, the maximum of ;((3)' is smaller than that of X(3)''. As a result, a collimated input laser pulse generates Raman Stokes light with larger gain, and Stokes generation is the dominant process. The situation is different for the transient case, i.e. for very short input pulses. The gain for stimulated Stokes generation may be substantially reduced (see 3.1.2) while the off-resonance Stokes-anti-Stokes interaction is less effected. We note that X~] has major contributions from electronic processes with characteristic time constants of the order of 10-15 s. Furthermore, pump light with broad bandwidth and focused beams reduce the Stokes gain and favour the four-photon interaction. Experimentally, broad stimulated Raman lines and the simultaneous occurrence of light continua was reported by a number of authors. (aS' 127,240J 4.2. CoherentSpectroscopy (Active Spectroscopy) In the previous part of this paper we were mainly concerned with the question what happens when an intense coherent electromagnetic wave traverses a Raman active medium. The magnitude of the gain coefficient was of particular importance, because it determines the frequency of the electromagnetic wave which is generated. We have seen that the gain has a maximum when the difference between the pump and the Stokes frequency is equal

110


to the Raman resonance of the material. The intense monochromatic Stokes radiation is a manifestation of the strong interaction with the medium. The spectroscopist is interested in data measured over a frequency range as wide as possible. With two incident light waves, where one is tunable, frequency dependent information may be obtained. As a first example we have discussed the gain measurements at the Stokes side with parallel polarization between the two light beams (see Fig. 8). More general, radiation at the Stokes and anti-Stokes side with different polarizations may be studied. With very short pulse excitation, and ultrashort probe pulses it is possible to determine coherent time constants of Raman active transitions. Various examples of investigated systems will be discussed in the following. 4.2.1. Quasi-stationary Case 4.2.1.1. COHERENT ANTI-STOKESRAMAN SPECTROSCOPY--CARS. There exists an extensive literature on the observation of anti-Stokes radiation when two intense light pulses of frequencies COLand COswith COL-- COS"- COOenter the Raman active medium. ~128-130,159,263) The first experiments date back to 1963 where several discrete frequencies at ~os were used to obtain frequency dependent information on the anti-Stokes side. t132,133) Renewed interest in anti-Stokes spectroscopy began when tunable frequencies from dye lasers or from parametric oscillators became readily available, t134-162) In some experiments two dye lasers are used, where one is fixed at COL while the second one is tuned at frequency COs on the Stokes side of the spectrum. An experimental system is depicted schematically in Fig. 28. For liquids and solids kmatching between the three beams requires a well defined phase-matching angle 0, which has to be readjusted when the input frequency Cos is varied. The wave vector triangle is shown in the lower part of Fig. 28. In gases, the colour dispersion is very small and the different beams travel collinearly through the sample for all frequencies.

\

F

FIG. 28. Coherent anti-Stokes scattering (CARS) (schematic). Two laser beams at frequencies COLand tas traverse the Raman active sample, RS. Anti-Stokes radiation at ~0Ais observed. The phase-matched wave vectors are depicted in the lower part of the figure. In condensed phases the phase-matching angle is wavelength dependent; fixed laser FL, tuned laser TL. Most experiments reported so far on coherent anti-Stokes scattering were carried out with nanosecond pulses or even with CW lasers. The quasi-stationary theory applies to these investigations. The following conditions strongly simplify the calculations: The input intensities IL and Is should be sufficiently small in order to avoid effective stimulated Stokes scattering. More quantitatively, one should choose gslLl < 1 and gslTl < 1. Under these conditions the input pulses are not strongly affected by the interaction and one has 1L(X, t') = IL(t') and Is(x, t') = Is(f). In nonabsorbing media equation (108b) has the form:

OEa (x', t') 0x'

.2xtoa { [Z~3~, + 6 ~v3~] [EL [2 Ea + ['Z~3~* + ~ ~v3~] E~E~ exp ( - iAka x')}. (133) #a c

We neglect the first term on the right hand side of equation (133), since EA is small


111

compared to the input field Es. Under phase-matching conditions, equation (133) reduces to ~EA(X', t') = i 2rrCOA [~3~. + ~ ~3~] E2 E~. c~x' #a c

(134)

Integration over the effective interaction length/eft leads to Ea(l, t') = i

2/~OJA

ktAC

[Z~3~, + 3 ~3]-] E2LE~/err.

(135a)

The generated anti-Stokes intensity has the form 2561t4c°2 IX[3~, + 3 ~v3~]212(t,)is(t ,) 12ff. I a(l, t') = pS].LAI.12LC4

(135b)

In condensed phases/eft is limited by the spatial overlap of the noncollinear (phase-matched) beams of frequency f-OL and COs.In gases with collinear geometries left is given by the length of the medium. We see from equation (135b) that the coherent anti-Stokes signal is proportional to the square of the absolute nonlinear susceptibility Zt3)(-O~A; O)L, O3L, --(-,OS)(see equation (131)). This result differs strikingly from the stimulated Stokes process where the Stokes field growth exponentially with the imaginary part of Xt3~(--Ogs;COL,--O~L,COS)(see equations (47b) and (50). We note that the Stokes input signal gives optimum anti-Stokes generation at exact phase-matching AkA = 0; in the previous section, 4.1, a certain phase-mismatch was necessary for Stokes and anti-Stokes light generation starting from quantum noise. The relevant nonlinear susceptibility around a resonance frequency o90 of the medium is of special importance for spectroscopic investigations. We have 1~((3)(-- O)A ; (DL, (DL, -- (L)S) I 2 =

FN~ao~2 =

I 3,' + ix~ 3,'' + ~

]2

(co~-og~)(1-2h) +

l' +

-

+ 4( 0 /T2)2_1

(136) where o9~ = tnL -- a,'S. In Fig. 27a, the frequency dependence of equation (136) is depicted. We note that the maximum of lzt3>12 is not located exactly at the resonance frequency. Outside the half width of the resonance, the anti-Stokes signal is essentially determined by the real parts of Z~3).Since Z[3r changes sign and g~3] always remains positive, one has a cancellation of the real parts at a certain frequency beyond the resonance position. The resulting minimum in the anti-Stokes signal gives rise to a strongly asymmetric frequency dependence around O9o.We emphasize that the frequency profiles measured by coherent anti-Stokes scattering may differ substantially from the familiar spontaneous Stokes or anti-Stokes Raman lines. We recall that--in contrast--the Stokes gain (equation (47)) and the antiStokes loss (equation (103); the inverse Raman effect) have the same frequency dependence as the spontaneous Raman lines. The complex frequency dependence of the CARS spectra poses a basic problem. The observed frequencies of the maxima do not coincide with the resonance frequencies and the measured line shapes do not provide the linewidths of the transitions of interest. For quantitative studies it is necessary to calculate the frequency dependence of I and compare these spectra with the observed data. Such calculations require the resonance frequencies and the knowledge of a series of other material parameters. In the past, these data were taken from previous spontaneous measurements. On the other hand, the coherent anti-Stokes spectroscopy has experimental advantages compared to the common spontaneous Raman technique : (i) The coherent anti-Stokes radiation is generated within a small beam divergence in the phase-matched direction while the spontaneous Raman emission occurs in a solid

112


angle of 4n. The background noise is drastically reduced by accepting light only within the small angle of the coherent anti-Stokes emission. (ii) Measurements on the anti-Stokes side avoid the frequently occurring fluorescence which might be induced by the incident light waves at ~OLand COs. (iii) Even with moderate pump powers for PL and Ps one obtains a coherent antiStokes emission IA which is several orders of magnitude more intense than the conventional spontaneous Stokes intensity l~p. With the help of equations (47, 50) and (135b) one estimates around resonance for the ratio l,~/I~p:

IA/l~p " gSILI(Is/IsN).

(137)

The gain gslLl is of the order of unity and IsN was calculated above to 10 W/cm 2 for a solid angle of 1 sr. For an input of I s - 100 W/cm 2 within 10 -4 sr one estimates Ia/Isp ,,, 105, a number quite favourable for coherent anti-Stokes spectroscopy. The high sensitivity of the coherent anti-Stokes process allows the investigation of components in strongly diluted liquid and gaseous systems, tls 2,153~ Strong anti-Stokes signals were observed at gas pressures as low as ! mtorr315 t~ It should be noted, however, that the coherent anti-Stokes emission is proportional to N 2 since Zt3~ is proportional to N The square dependence of the particle concentration N is a serious limitation to extend the technique to even lower densities. In Fig. 29 the coherent anti-Stokes Raman signal of benzene is presented when the frequency O9s is tuned through the vibration at Ogo/2rrc = 992 cm- 1.~146~We point to the log-scale of the ordinate and the asymmetry of the curve extending over more than 100 cm- 1, while the spontaneous Raman linewidth is only 2.1 c m - 1. Another example of a CARS spectrum is depicted in Fig. 30. The coherent anti-Stokes emission of D2 at 48 torr was measured in the centre of an electric discharge. (,UoZC/8n/~2)(a/l)2

This intensity value has to be compared with the laser intensity I s needed for efficient Raman Stokes generation I s > 25/gsl. When IL > IL: one finds self-focusing of the beam and strong stimulated Raman emission in spite of the fact that 1L < I s. In this case, the stimulated Raman radiation is generated by the high light intensity in the focused beam. (182,19°-19s) It should be mentioned that the limiting beam diameter found in self-focusing is suggested to be determined by effective Raman conversion. (196) Figure 38 shows a typical picture of Stokes emission versus input power as frequently found in media (mainly liquids) with large nonlinear index of refraction (small selffocusing length Xf). (182) A t a certain input power, self-focusing sets in (xy = l) and the Stokes emission rises suddenly over many orders of ten. Even higher order Stokes components are found at the same critical power level. Stimulated Stokes scattering occurs without self-focusing when I~ > IL > I s. We point to the different dependence on material length and on beam diameter of I s and I/~. For a = const., shorter samples are more favourable for the stimulated Raman process: i.e. for sufficiently large input power, self-focusing can be avoided. The intensity dependent index of refraction may also lead to changes in frequency of the incident electromagnetic wave. The spectral broadening is estimated to be :(t 85) (153)

A09 ~- tOLlt2 E2 x / c A t

where At represents the time during which the field changes by EL. According to equation (153), spectral broadening may become substantial when large changes of the electroto s

m

/

10~t

i~o102 ,o

I-

~63J 0

t 2 Loser Input Power

3 PL CMW3

g

FIG. 38. StimulatedRaman scattering with self-focusing.Raman medium, nitrobenzene; sample length, 10 cm. At PL --~3 X 105 W self-focusingsets in and the Stokes output rises suddenly. Second Stokesgenerationoccurs at nearly the same input power (after Ref. 182).


127

magnetic field occur rapidly. In fact, short light pulses and rapid self-focusing generate frequency widths of many cm-1. We recall that pump waves with broad frequency spectra show smaller Raman gain (see 3.1.3); i.e. frequency broadening reduces the stimulated Raman emission. Self-phase modulation due to ~t2 (which is determined by ~3)) is closely related to the more general four-photon process COL+ COL--* CO3+ CO4where signal (CO3)and idler (CO4) frequencies are generated. In certain cases, the four-photon interaction is resonantly enhanced: a broad frequency spectrum is rapidly generated suppressing stimulated Raman interactions. As an example to this point we quote investigations on water performed with light pulses at 1.06 ~tm of 6 ps duration. ~127'197) Emission spectra extending over several thousand wave numbers were observed without indication of significant stimulated Raman scattering. We have seen in previous parts of this paper that the dephasing time Tz is the relevant time constant for the stimulated Raman process. When the duration of excitation (e.g. the pulse duration) becomes smaller than Tz, one finds a reduced interaction and higher values of the product OILl are required for a certain value of the conversion of laser into Stokes light (see Fig. 17). Similarly, other nonlinear optical phenomena have their relevant time constants which may differ by many orders of magnitude31°~ For the stimulated Brillouin process--where light waves couple to acoustic phonons--the phonon lifetime T n represents the characteristic response time. In liquids at room temperature one finds values of T Bof the order of 10-9 s, while solids, especially at low temperature, have much longer time values. Reduced transient Brillouin amplification is observed for pulse durations shorter than T ~. An interesting example was reported by vonder Linde e t al. ~33~ (see Fig. 39). The conversion efficiency of stimulated Raman and Brillouin scattering is plotted versus the duration of the input pulses. For long ( ~ 14 ns) laser pulses the Brillouin process dominates on account of its substantially larger steady-state gain coefficient. The Brillouin efficiency decreases drastically when tp < 3 ns since we are now in the transient regime for the Brillouin process with T B = 3 ns. Figure 39 shows convincingly the growing Raman efficiency for shorter pulses. After the disappearance of the competing Brillouin process stimulated Raman scattering takes over. Finally, for very short pulses with t o < T2, the transient amplification gives a reduced Raman emission. 5. SPECIAL RAMAN TYPE PROCESSES In the preceding part of this article we mainly considered Raman interactions with vibrational transition. We had in mind the numerous Raman active normal modes of molecules in liquids or gases. In this section we wish to comment on other Raman o.o ~1.0

I

I

ii

/

1"

/ u

" ' 0.5 c o

c o ¢J

_

~ "

!

/

\./

i _

0

//

\

"---~'

.......

g Pulse Ourotion

"1

I0

.O---15

tpCnS3

FIG. 39. Competition between stimulated Raman and Brillouin scattering in CS2. For tp ~ 3 ns Brillouin scattering dominates. For tp < 2.5 ns Raman scattering is the most intense process. With picosecond light pulses the Raman conversion is reduced by transient interaction. Closed symbols correspond to stimulated Brillouin scattering; open symbols to stimulated Raman scattering (after Ref. 33). J,V'.Q.E. 6 / 2 - - r

128


processes. First, we briefly discuss the situation in solids. Then, inelastic light scattering from electronic transitions (bound electrons) and from quasi-free electrons in semiconductors will be treated. The basic equations derived above hold for all Raman type processes, but large differences in scattering cross sections are observed. 5.1 Raman Scatterin9 in Crystals The interaction of light with lattice vibrations follows well established selection rules. "99-2°2) In the classical nonpolar crystals diamond, silicon and germanium the optical phonon is Raman active. On the other hand, the fundamental lattice vibration in NaCl-type crystals is Raman forbidden (only a second order Raman effect is observed). In polar crystals without inversion centres certain lattice vibrations are Raman and/or infrared active. In the second part of this section we shall comment on light scattering by polaritons, which requires some extension of the picture discussed so far. The following difference between liquids and solids should be noted. In gases and liquids we consider individual molecules which are characterized by a two-level system. Raman excitation leads to a moderate excess population n of the first vibrational state, v = 1. In solids we study optical phonons, a collective excitation of the crystal lattice which extends over many atomic sites. Optical phonons have harmonic oscillator levels. With a highly directional incident light beam one excites a small number of lattice modes. High phonon occupation numbers (large number of quanta per mode) may be produced by stimulated Raman excitation. Inelastic scattering of light from lattice vibrations follows closely the theory outlined above for molecular vibrations, tS) In equations (27)-(30) the parameters q and ~o~ denote the amplitude of the lattice vibration and the phonon frequency, respectively; the phonon density is nN where N stands for the number of atoms per cm 3. At high light intensities stimulated Raman scattering has been observed in a number of different crystals. We name just a few here, together with the frequency of the excited lattice vibration: Si (521 cm-1),t2o3) CaWO4 (911 cm-1), K D P (915 cm-1),~2o4) CaCO3 (1086 cm-t),ts,124,2os,2o6) and diamond (1332 cm-1).(206,207) Of interest is the possibility of getting direct information on the phonon lifetime. In a series of experiments one specific lattice mode was excited via transient stimulated Raman scattering, and delayed probe pulses allowed the observation of the decay of the coherent phonon excitation. So far, the fundamental optical phonon of diamond, ~172) the symmetric CO~- mode in CaCO3 t173'2°8) and a polariton mode in GaP t4a) were investigated. In Fig. 40a we show for diamond the scattered probe signal as a function of delay time to. ~:2) A time constant of 2.9 ps (300 K) and 3.4 ps (77 K) may be inferred from the exponential decay of the measured data. The probable decay process is depicted schematically in Fig. 40b. By stimulated Raman scattering we excite TO phonons at ~ = 1332 cm-1 very near to the centre of the Brillouin zone. These phonons decay via anharmonic coupling into TA and LA acoustic phonons. Energy and momentum are conserved in the decay channels indicated in Fig. 40b. In Table 1 we list several crystals together with the frequencies of the lattice vibrations which were excited by the stimulated Raman process. The stated gain factors are either experimentally determined or calculated from known material parameters. We see in Table 1 that the gain factors of solids do not differ drastically from those of liquids. Amorphous solids, in particular glasses, have very broad Raman linewidths and correspondingly small gain factors. In spite of this, the extreme interaction lengths (100 m) possible with optical fibres allowed the generation of CW stimulated Raman scattering e.g. in fused quartz. Polariton scattering was investigated in a number of polar crystals such as GaP, tl 7o, 2o9,210) LilO3 ~211) or LiNbOa32 x2- 2x6) The coupling between transverse optical lattice modes and the electromagnetic fields give rise to an excitation of mixed phononphoton character, which is called the polariton. The frequency of the polariton is strongly wave-vector dependent. The dispersion curve is directly investigated by spontaneous Raman scattering at small angles close to the forward direction. At high input intensities,

High intensity,Ramaninteractions |

I

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and W. KAISER - Science Direct

and W. KAISER - Science Direct

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