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14-Stimulated Raman Scattering

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14-Stimulated Raman Scattering

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     14

     Stimulated Raman Scattering

     Raman scattering is a two-photon process [1]. As illustrated in Fig. 1a, an incident photon is annihilated and a new photon of a different frequency is created. This implies that the scattering is inelastic, and the medium is left in a different energy state. When the scattered photon is lower in energy, or frequency, it is said to be Stokes shifted. The shift in frequency of the Stokes photon is related to a characteristic frequency of the medium. This is illustrated in Fig. 2a. Calling this characteristic material excitation frequency vv, the Stokes frequency is then vS ? vL 2 vv ; where vL is the incident laser frequency. Thus, Raman (Stokes) scattering results in an excitation from the ground state of the medium to an excited state mediated by a two-photon inelastic scattering. The internal material excitation can involve the creation of an excited electronic state, an excited vibrational ?C rotational state, a lattice vibration, a spin ?ip in semiconductors, or electron waves in plasmas [2]. The phenomenon occurs in gases, liquids, solids, and plasmas. If the material is already in an excited state (e.g., by thermal excitation), as illustrated in Fig. 2b, then the scattered photon can be higher in energy, or frequency, and is called an anti-Stokes photon. The probability of anti-Stokes scattering is smaller than that of Stokes scattering by a Boltzmann factor expe2"vv =kB TT; where kB is Boltzmann??s constant and T is the temperature. Since typically "vv q kB T; spontaneous Raman scattering is dominated by Stokes emission. Stimulated Raman scattering (SRS) involves the incidence of both Stokes and laser photons on a medium, as illustrated in Fig. 1b. The result is the stimulation of an additional Stokes photon coherent with the incident Stokes photons. Thus the Stokes photon ?eld experiences gain. As in spontaneous Raman scattering, SRS leaves the medium in an excited state.

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

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     Figure 1 Schematic diagram of (a) spontaneous Raman scattering and (b) stimulated Raman scattering.

     SRS is strictly a Stokes generation process. However, in practice, coherent anti-Stokes photons are also produced, sometimes in nearly equal amounts to the Stokes photons. This is in stark contrast to the spontaneous Raman scattering case. The anti-Stokes generation is not readily described in the photon picture, but it can be understood in

    terms of third order nonlinear optics as a four-wave mixing process. SRS is of considerable scienti?c and technological signi?cance. It is used in such diverse areas as spectroscopy and coherent frequency conversion. Other uses include signal ampli?cation, beam combining, and beam cleanup (i.e., removal of phase as well as amplitude distortions in optical beams). In the sections that follow, some of the important formulas used in the description of SRS are given. The last section brie?y describes some of the applications of SRS. Several of the parameters that appear in formulas related to SRS are given in Table 1, including both SI and cgs units. Sometimes units are mixed, such as

     Figure 2 Energy level diagrams for (a) Stokes Raman scattering and (b) anti-Stokes Raman scattering.

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

     Stimulated Raman Scattering

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     Table 1 Physical Units of Important Physical Parameters Utilized in Formulas Related to Stimulated Raman Scattering Physical parameter Differential Raman scattering cross-section Differential polarizability Mass Boltzmann??s constant Temperature Molecular number density Third order polarization Electric ?eld amplitude Third order Raman susceptibility Raman gain coef?cient Raman gain intensity factor Vacuum permittivity Distance, length Speed of light Wave vector Circular frequency Raman line width Planck??s constant Intensity Power SI ?ds=dV? ? m2 =strad ??a=?q? ? C-m=V [m ] ? kg [kB] ? J/K [T ] ? K [N ] ? m23 [P (3)] ? C/m2 [A ] ? V/m e3T ?xR ? ? m2 =V2 [G(vS)] ? m21 [gS] ? m/W [10] ? F/m ? C/V-m [z ] ? [L ] ? m [c ] ? m/s [k ] ? m21 [v] ? Hz ? s21 [G] ? Hz ? s21 ?"? ? ?h=2p? ? J-s [I ] ? W/m2 [P] ? W cgs ?ds=dV? ? cm2 =strad ??a=?q? ? cm2 [m ] ? g [kB] ? erg/K [T ] ? K [N ] ? cm23 [P (3)] ? sV/cm ? (erg/cm3)1/2 [A ] ? sV/cm ? (erg/cm3)1/2 e3T ?xR ? ? cm2 =sV2 ? cm3/erg ? esu [G(vS)] ? cm21 [gS] ? cm-s/erg ?ª [z ] ? [L ] ? cm [c ] ? cm/s [k ] ? cm21 [v] ? Hz ? s21 [G] ? Hz ? s21 ?"? ? ?h=2p? ? erg-s [I ] ? erg/s-cm2 [P? ? erg=s

     when the Raman gain intensity factor is quoted in cm/W (centimeters per watt). Table 2 gives conversion factors between parameters in SI and cgs or mixed units.

     I.

     SPONTANEOUS RAMAN SCATTERING

     The incoherent spontaneous Raman scattering from an incident laser beam is depicted schematically in Fig. 3. The transmitted laser radiation is attenuated by the conversion of laser photons to scattered photons. This attenuation is described in terms of the total Raman scattering cross-section de?ned by [3] dI L ? 2sR NI L dz e1T

     where IL is the laser intensity, N is the number density of Raman

scatterers

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

     796 Table 2 Conversion Formulas Between the SI and cgs Systems of Units for Several Physical Parameters Utilized in Stimulated Raman Scattering

     ds eSIT ? 1024 dV ecgsT eSIT ? e1T2 ?ê 10213 ?a ecgsT ?q 3 m eSIT ? 1023 m ecgsT ecgsT N eSIT ? 106 N P e3T eSIT ? 1 ?ê 1025 P e3T ecgsT 3 A eSIT ? 3 ?ê 104 A ecgsT e3T xe3T eSIT ? 4p ?ê 1028 xR ecgsT R 32 2 G eSIT ? 10 G ecgsT gS eSIT ? 105 gS ecgsT eSIT ? 107 gS ecgsT gS ecm=WT ? 102 gS L eSIT ? 1022 L ecgsT c eSIT ? 1022 c ecgsT k eSIT ? 102 k ecgsT ecgsT I eSIT ? 1023 I 2 IeW=cm T ? 1024 I eSIT ? 1027 I ecgsT P eSIT ? 1027 P ecgsT

     Chapter 14

     a ?q

     ds dV

     (e.g., molecular density), and sR is the Raman scattering cross-section (m2 or cm2) for laser photons scattered into all Raman modes (v, k). In the quantum mechanical picture, the laser and Stokes ?elds are described in the terms of the laser and Stokes photon numbers, mL and mS, respectively. The generation of Stokes photons is given by the relation [3] dmS c dmS ? / mL emS t 1T n dz dt e2T

     where c/n is the speed of light in the medium. The vacuum state is characterized by mS ? 0; and the unity term in Eq. (2) yields spontaneous Raman scattering. The spontaneous scattering regime is thus characterized by mS p 1; and

     Figure 3

     Incoherent Raman scattering and laser attenuation in a Raman active medium.

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

     Stimulated Raman Scattering

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     mL . constant over distances small compared to (sRN )21. Then, for scattering into a single Stokes mode, mS / mL z: Experimentally, photon numbers can be related to optical power. The experimental geometry for measuring Stokes emission is illustrated in Fig. 4. The angles u and f represent the scattering direction from a volume element of the material of length dz. The measurement yields the Stokes power emitted in a small solid angle DV and is given by [4] dP S ? N ds eu; fTDVdzP L dV e3T

     The quantity ds/dV, which is a function of the angles u and f, is called the differential Raman scattering cross-section. It is measurable since the other quantities in Eq. (3) are either measurable or established by the experimental conditions. This quantity is then related to the total Raman cross-section by [4] Z vL ds eu; fTdV e4T

    sR ? vS 4p dV For dipole scattering [4], ds ds eu; fT ? sin2 u dV dV u?908 and the total cross-section is thus 8p vL ds sR ? 3 vS dV 908

     e5T

     e6T

     The important physical quantity in SRS formulas is the differential scattering cross-section ds/dV

     Figure 4

     Raman scattering measurement geometry.

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

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     II.

     STOKES GAIN

     When the Stokes photon number is large emS @ 1T; the unity term in Eq. (2) can be ignored. Thus if the laser photon number is approximately constant, mS will experience exponential gain. In the classical picture, the photon numbers relate to power (or intensity), and the Stokes intensity experiences gain as it propagates a distance z through the Raman gain medium: I S ezT ? I S e0Texp?GevS Tz? e7T

     The Raman gain coef?cient G(vS) is given in Table 3, where it is related to the differential scattering cross-section. It is also seen to be proportional to the normalized Raman lineshape function g(vS), where gevS T , T 2 for vL 2 vS . vv : T2 is the relaxation time of the Raman mode of excitation [2]. The Raman gain coef?cient can also be expressed in terms of the Raman gain intensity factor

     Table 3 Formulas Related to Stimulated Raman Scattering

     c GevS T ? 8p2 "vN 3 3n

     S S 3 2

     Quantum approach Raman gain coef?cient Raman gain intensity factor

     ?? ds ??

     G

     dV 908 {1

     2

     2 exp?"evL 2 vS T=kB T?}I L gevS T

     gevS T ? ?v

     v 2evL 2vS T?

     tG2

     gS ? GevS T IL Nonlinear optics approach Raman susceptibility xe3T e2vS ; vL ; 2vL ; vS T ? 61 R

     Ne?a=?qT2 0 2 2 0 m?vv 2evL 2vS T tiGevL 2vS T?

     eSIT

     Raman gain coef?cient

     xe3T e2vS ; vL ; 2vL ; vS T ? ecgsT R v eSIT GevS T ? 26 nSSc Im?xe3T e2vS ; vL ; 2vL ; vS T?jAL j2 R v GevS T ? 224p nSSc Im?xe3T e2vS ;

vL ; 2vL ; vS T?jAL j2 ecgsT R

     S 0 GevS T . 212 mc 2 nL nS vv ?v 0

     Ne?a=?qT2 0 6m?v2 2evL 2vS T2 tiGevL 2vS T? v

     v Ne?a=?qT2

     2

     G=2

     v 2evL 2vS T? 2

     I teG=2T2 L

     ; gS I L ; gS I L

     eSIT ecgsT

     GevS T . Stokes-pump coupling outputs

     16p vS Ne?a=?qT2 0 mc 2 nL nS vv

     G=2 I ?vv 2evL 2vS T?2 teG=2T2 L

     =vL TI L e0T?I S e0TexpeuT I S eLT ? ?I S e0TtevLSTI L e0TtI S e0TexpeuT evS =v =vS TI S e0T?I L e0Texpe2uT I L eLT ? ?I L e0TtevSLTI S e0TtI L e0Texpe2uT evL =v

     u ? ?evL =vS TI S e0T t I L e0T?gS

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

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     gS, where gS ? GevS T IL e8T

     It is possible to relate SRS to third order nonlinear optics. As noted by Shen [1], the rate of Stokes photon generation, in the limit of large photon numbers, is related classically to the polarization through the energy theorem, i.e., "vS dmS 2 2 / Re?eivS TE S ??P* ? / mL mS / jE L j jE S j S dt e9T

     The last relation implies that the Stokes polarization is third order in the ?eld, i.e., P S / jEL j2 E S : Thus a third order Raman susceptibility can be de?ned through X e10T Pe3T evS T ? 610 xe3T e2vS ; vL ; 2vL ; vS TEj evL TE* evL TEl evS T i k ijkl

     jkl

     With the ?eld amplitude written as E ? A expeik??rT; the form of Eq. (10) shows that the SRS process is automatically phase matched. The Raman gain coef?cient is related to the imaginary part of the third order susceptibility, which can be seen through the relation given by Eq. (9). The most common form of SRS is by molecular vibrational Raman scattering in a macroscopically isotropic medium. A simple one-dimensional molecular vibrator model has been developed to describe this phenomenon [5]. The central idea of the model is that the linear polarizability of the molecule changes as the molecule vibrates. To ?rst order, the polarizability is related to a normal mode of vibration q by the relation ?a a ? a0 t q e11T ?q 0 where a0 is the ordinary polarizability, governing molecular absorption and index of refraction, typically in the infrared, and (?a/?q )0 represents the ?rst term in

    a Taylor series expansion of the polarizability in terms of the displacement of the vibrator about its equilibrium position and is called the differential polarizability. The physics of SRS can be understood in terms of the simple model. The normal mode q oscillates at the characteristic material excitation frequency vv (e.g., the vibrational frequency of the molecule). This modulation of the molecular polarizability, through Eq. (11), causes the molecule to radiate at the sideband frequency vS ? vL 2 vv as well as at the incident laser frequency vL. The superposition of the waves at vL and vS produces a beat wave at vL 2 vS that coherently drives the normal mode vibration. This reinforced molecular vibration then coherently ampli?es the wave at vS. (Classically, there is also a sideband generated at vA ? vL t vv ; which is the anti-Stokes frequency.

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

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     Chapter 14

     However, the phase of this wave is p radians out of phase with the time derivative of the polarization driving it. By the energy theorem, it is thus attenuated, as discussed further below.) The Raman susceptibility for Stokes emission is found by solving the equation of motion for the normal mode, assuming the presence of both Stokes and laser ?elds, and relating the polarization to the induced dipole moment [3,4]. The Raman susceptibility given by this model is shown in Table 3. A damping force term proportional to 2 G(dq/dt ), where G is the damping coef?cient, has been assumed to be acting on the molecular vibrator. G is also known as the Raman line width, as it measures the full width at half maximum of the spectral response of the susceptibility. As noted above, the Raman gain coef?cient, as well as the Raman gain intensity factor, is proportional to the imaginary part of the Raman susceptibility. This is shown by the two relations for G(vS) given in Table 3. The last relation, which gives the Raman line shape explicitly, is approximately true and holds well in the typical case of a narrow Raman line such that G ,, vv : (Note that the real part of the Raman susceptibility plays an important role in nonlinear refraction and leads to the Raman induced Kerr effect that was discussed in Chapter 6.) The Raman susceptibility governs the interaction of the Stokes and laser waves and thus is strong for waves such that vL 2 vS . vv : For this reason, it is called a resonant contribution to the third order susceptibility. It should be noted that for the third order interaction of waves in any medium there is always also a background nonresonant susceptibility, due to, for example, electron orbital distortion.

     III.

     STOKES AMPLIFICATION

     For ef?cient Stokes ampli?cation, the depletion of the pump laser cannot be ignored. To describe the coupling between the Stokes and pump waves, the nonlinear polarizations for both waves, within the simple one-dimensional model, are introduced: Pe3T ? 610 xe3T e2vS ; vL ; 2vL ; vS TjEL j ES R S

     2

     e12T e13T

     Pe3T ? 610 xe3T e2vL ; vS ; 2vS ; vL TjES j EL L R

     2

     The Stokes and laser susceptibilities are related by [1]

     e3T xe3T e2vL ; vS ; 2vS ; vL T ? xR * e2vS ; vL ; 2vL ; vS T R

     e14T

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

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     Note again by the form of the nonlinear polarizations that the Stokes?C laser coupling is automatically phase matched. In terms of the complex wave amplitudes E ? A expeik??rT; the coupling of the waves is described through the differential equations dAS 3vS e3T 2 ?i x evS TjAL j AS dz nS c R dAL 3vL e3T 2 ?i x evL TjAS j AL dz nL c R e15T

     e16T

     Again, for jALj . constant, Eq. (15) predicts gain for jASj, with the gain factor proportional to Imexe3T T. Note also that since the gain is a function of jALj2, phase R distortions in the laser beam are not directly transferred to the Stokes wave. This is the basis (partially) for amplifying a ??clean?? Stokes wave with a ??dirty?? pump wave. The phase of the Stokes wave will depend on the Reexe3T T. R Differential equations describing the spatial evolution of the ?eld modulus for both laser and Stokes beams follow directly from Eqs. (15) and (16): nS djAS j 6 2 2 ? 2 Im?xe3T evS T?jAL j jAS j R c vS dz nL djAL j 6 2 2 ? 2 Im?xe3T evL T?jAS j jAL j R c vL dz

     2 2

     e17T

     e18T

     Since Im?xe3T evS T? , 0; it is obvious by Eqs. (14), (17), and (18) that the Stokes R wave experiences gain at the expense of the pump, which experiences loss. Equations (17) and (18) can be combined to show that the following invariant exists for Stokes?C pump coupling: I S ezT I L ezT I S e0T I L e0T t ? t ? constant vS vL vS vL e19T

     This equation states that the total number of photons in the two-photon scattering process remains a constant. It also allows the Stokes and laser outputs at the end of a medium of length L to be expressed in terms of the input intensities. These expressions are given in Table 3. The behavior of the Stokes and pump intensities as functions of the

    Raman gain ?C length product is shown in Fig. 5 for three values of the input Stokes-topump photon ?ux ratio, r ? evL =vS TI S e0T=I L e0T: Note that the growth of the Stokes photon ?ux is initially exponential but begins to saturate as the pump is depleted. The saturation is more apparent when the input Stokes photon ?ux is larger.

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

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     Chapter 14

     Figure 5 Laser and Stokes photon ?ux as functions of Raman gain at the output of a medium of thickness L.

     In many cases, I L e0T q evL =vS TI S e0T; particularly when the Stokes mode grows from quantum noise (i.e., spontaneous Raman scattering). Then, as gS I L e0TL ! 1; I S eLT ! evS =vL TI L e0T; i.e., pump photons are entirely converted to Stokes photons, with the energy difference showing up as material excitation. In reality, before this occurs, higher-order Stokes and anti-Stokes modes appear and drain off energy. Since there is no phase matching condition imposed on Stokes ampli?cation, Stokes generation from noise can occur in both the forward and backward directions along the pump beam. However, this symmetry is broken in some cases. For a broadband-pump source of line width GL, where GL , G; the Raman line width, the forward Stokes gain is proportional to G21, while the backward gain is proportional to eGL t GT21 [1]. Thus the forward gain can be much stronger. Backward SRS is generally not observed when picosecond laser pump pulses are used, since this pulse duration limits the effective interaction length for backward Stokes ampli?cation [2]. The Stokes photon can serve as an input to generate a new photon (by Raman scattering) at frequency vS 2 vv ? vL 2 2vv : This is called the second Stokes wave. As the incident pump power is increased, the ?rst Stokes wave grows from noise and then saturates as the pump is depleted. The second Stokes

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

     Stimulated Raman Scattering

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     wave is then generated and eventually saturates as the ?rst Stokes wave is depleted. This process continues to higher Stokes generation as the pump power is continually increased. This process has been demonstrated experimentally using picosecond pulses to avoid backward Stokes generation. The Gaussian nature of the beams causes a gradual depletion of the pump as well as a gradual growth of the higher-order Stokes beams as illustrated in Fig. 6[1, 6].

     IV.

     ANTI-STOKES GENERATION

     As mentioned in Section II, the vibration of the normal coordinate q produces a sideband at the anti-Stokes frequency vA ? vL t vv : The

    phase of this wave is such that it experiences loss instead of gain, in contrast to the Stokes wave. This can be shown by considering the normal mode excitation produced by the beating of the anti-Stokes and laser pump waves and calculating the resultant Raman susceptibility for the anti-Stokes wave [1, 3]. The result is an expression nearly identical to the Stokes susceptibility, shown in Table 3, with vL 2 vS replaced by vA 2 vL (which are both equal to vv); the major exception is that the imaginary part of the resonant frequency denominator expression has the opposite sign. Thus Im?xe3T e2vA ; vL ; 2vL ; vA T? . 0; which implies loss instead of gain. R It turns out, however, that there is another way to generate the anti-Stokes wave. This is the four-wave mixing process governed by the Raman susceptibility, and also possibly by the background nonresonant susceptibility, which produces a nonlinear polarization oscillating at the frequency vA ? vL t vL 2 vS ? vL t vL 2 evL 2 vv T ? vL t vv : The quantum mechanical picture of this interaction is shown schematically in Fig. 7. Note that anti-Stokes

     Figure 6 Normalized laser (circles), ?rst Stokes (squares), and second Stokes (diamonds) power as functions of incident laser intensity. (From Ref. 6.)

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

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     Chapter 14

     Figure 7

     Anti-Stokes generation by four-wave mixing of laser and Stokes waves.

     generation by four-wave mixing is a parametric process, leaving the material in its ground state, in contrast to SRS, which leaves the material in an excited state. The nonlinear polarization produced by the third order interaction of the laser and Stokes waves is given by Pe3T evA T ? 310 xe3T e2vA ; vL ; vL ; 2vS TAL AL A* exp?ie2k L 2 k S T??r? S A e20T

     Two things should be noticed in this expression. First, the anti-Stokes polarization does not depend on the anti-Stokes ?eld amplitude. Thus the antiStokes wave cannot directly achieve exponential gain. Second, the process is not automatically phase matched. Ef?cient anti-Stokes generation can only be achieved for k A . 2k L 2 k S : In fact, in a normally dispersive isotropic medium, the anti-Stokes wave cannot be ef?ciently generated along the direction of the pump [4]. The usual phase matching condition is sketched in Fig. 8. It is noted that the anti-Stokes wave must couple to a Stokes wave generated off-axis. Thus the anti-Stokes wave is emitted in a conical shell about the pump direction. In general, the anti-Stokes wave will be coupled to the Stokes wave through the four-wave mixing process and will affect the

    ampli?cation of the Stokes wave. This problem has been treated within the nondepleted pump approximation [1,3]. The coupled mode equations are given by [3] dAS ? 2G0S AS t kS A* expeiDkzT A dz and dAA ? 2G0A AA t kA A* expeiDkzT S dz e22T e21T

     Complex Raman gain coef?cients Gj0 eGj ? Re?G0j ?; with j ? S; AT and coupling

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

     Stimulated Raman Scattering

     805

     Figure 8

     Phase matching diagram for anti-Stokes generation by four-wave mixing.

     coef?cients kj / xe3T evj T have been introduced, and z Dk ? e2k L 2 k S 2 k A T??^ e23T

     is the phase mismatch along the propagation direction (z-axis). The phase matching condition is illustrated in Fig. 8. Equations (21) and (22) allow two modes each for the Stokes and antiStokes waves. These waves have the same eigenvalues for the wave propagation factors b (i.e., Aj , e bz ), which for the two modes (?? t ?? and ?? 2 ??) are given by 1 0 1 0 * * b^ ? 2 eG0S t GA T ^ ?eG0S 2 GA t iDkT2 t 4kS k* ?1=2 A 2 2 e24T

     The ratio of the normal mode amplitudes for the anti-Stokes and Stokes waves is given as " #1=2 A^ * k* eb^ t G0S t iDk=2T A A ? e25T 0 * A^ kS eb^ t GA 2 iDk=2T S The following relationships apply: nS vA 0 G0A ? 2 G * ; kS ? 2G0S expei2wL T; nA v S S nS v A 0 GS* expei2wL T; kA ? nA vS

     e26T

     with w L the phase of the laser ?eld, AL ? jAL jexpeiwL T: Normally nS vA =nA vS . 1; and Eq. (24) reduces to the approximate form [3] Dk 4G0S 1=2 12i e27T b^ . ^i 2 Dk Three propagation regimes can be identi?ed.

     Copyright ? 2003 by Marcel Dekker, Inc. All Rights Reserved.

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     Chapter 14

     Strongly phase mismatched regime. In this regime jDkj q jG0S j; and the following approximations hold: iDk 0 b^ . ^ G S t e28T 2 At * Dk A t .i 0 GS AS A2 * A ?0 A2 S e29T e30T

     These equations imply that the anti-Stokes and Stokes waves are effectively decoupled, with one mode (t ) primarily anti-Stokes experiencing loss and the other mode (2 ) primarily Stokes experiencing gain, since ReeG0S T , 0: Strongly phase matched regime. Here Dk ? 0; and nS vA 0 b2 ? 2GS 1 2 .0 nA vS A^ * A ? 21 A^ S bt ? 0 e31T e32T e33T

     In this regime, the Stokes and anti-Stokes waves are strongly coupled so that each eigenmode is an equal combination of both Stokes and

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