Материал: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo

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Ref. p. 232]	4.3 Stimulated scattering	229

Table 4.3.3. Stimulated Brillouin scattering in backward direction: frequency shift, linewidth, and gain factor.

Medium	Stokes shift Linewidth		Gain	Gain factor	Gain factor	Ref.
	ν0/c	δν	coe cient	gBe	gBe
	[cm−1]	[MHz]	gBa /α	(calculated)	(measured)
			(calculated)	[cm / MW]	[cm / MW]
			[cm2 / MW]
Acetic acid	0.152					[67Wig]
Acetone	0.154	180	0.022	0.017	0.020	[70Poh]
Aniline	0.259					[67Wig]
Benzaldehyde	0.224					[67Wig]
Benzene	0.211		0.024	0.024	0.018	[68Den]
Bromobenzene	0.188					[67Wig]
Carbondisulﬁde	0.194	75	0.213	0.197	0.068	[97Jo]
Carbontetrachloride	0.146	650	0.0134	0.0084	0.006	[68Den, 70Poh]
Chloroform	0.148					[67Wig]
Cyclohexane	0.180			0.007	0.0068	[68Den]
p-Dichlorobenzene	0.184					[67Wig]
Ethanol	0.152		0.010	0.012		[72Mai]
Fluorinert FC 72					0.006	[97Yos]
Fluorinert FC 75					0.005	[97Yos]
Glass BSC-2	0.866					[67Wig]
Glass DF-3	0.638					[67Wig]
Glycerol	0.386					[67Wig]
n-Hexane		220		0.027	0.026	[68Den, 70Poh]
InSb						[95Lim]
Methanol	0.142		0.013	0.013	0.013	[68Den, 70Poh]
Methylodide	0.166					[67Wig]
Nitrobenzene	0.228					[67Wig]
m-Nitrotoluene	0.229					[67Wig]
n-Nitrotoluene	0.217					[67Wig]
Octanol	0.194					[67Wig]
Pyridine	0.226					[67Wig]
Quartz	1.16				0.005	[89Agr]
Sulfurhexaﬂuoride					0.0015	[93Fil, 97Jo]
(20 atm)
Tetrabromomethane	0.173					[67Wig]
Toluene	0.193	480		0.013	0.013	[68Den, 70Poh]
Water	0.197		0.0008	0.0066	0.0048	[68Den, 77Rys,
						94Yos]
p-Xylene	0.199					[67Wig]

1. and 2. The peak value of the stimulated gain for SRLS is given by:


ge	=	(∂ε/∂ρ)T2 ωS ρo (γ − 1)		,	(4.3.18)
RL		4 c2 nS2 v2
		4 c2 nS2 v2
where	γ = Cp/CV . It is interesting to notice that gRLe				does not depend on scattering angle
(ωS = ωL ). A ﬁnite optical absorption coe cient α of the medium gives rise to a second con-
tribution with peak value:
ga	=	α (∂ε/∂ρ)T ωS βT	.		(4.3.19)
ga	=		.		(4.3.19)
RL		2 c nS Cp ΓRL
		2 c nS Cp ΓRL

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230	4.3.3 Individual scattering processes	[Ref. p. 232

Table 4.3.4. Stimulated Rayleigh scattering: gain factors and linewidth values.

Medium	Gain factor
	gRLe
	[cm/MW]

Gain coe cient	Linewidth	Reference
gRLa (max.)/α	δν
[cm2/MW]	[MHz]

Acetone	2	0.47	21	[70Rot]
Benzene	2.2	0.57	24	[70Rot]
Carbondisulﬁde	6	0.62	36	[70Rot]
Ethanol		0.38	18	[70Rot]
Methanol	8.4	0.32	20	[70Rot]
Tetrachloromethane	2.6 × 10−4	0.82	17	[70Rot]
Water	0.02	0.019	27.5	[70Rot]

Here ΓRL = π δν is the halfwidth (HWHM) of the spontaneous Rayleigh line in circular frequency units that originates from the damping of entropy changes via thermal conductivity Λ :

ΓRL =	4 kL2	Λ sin2 (θ/2)		(4.3.20)
			.

		ρo Cp

Via ΓRL the gain constant gRLa strongly depends on scattering angle. The frequency dependencies of SRLS and STRS have opposite sign, leading to the total gain factor:

g(ωS) =	(gBe − gBa ) 2 ΓRL(ωL − ωS)	.	(4.3.21)
	(ωL − ωS)2 + ΓRL2

For dominant coupling via electrostriction positive gain occurs on the Stokes side, ωS < ωL . Small absorption values, α > 10−3 cm−1 , can be su cient for dominant STRS that produces gain on the anti-Stokes side. For zero frequency shift, ωS = ωL, the gain factor vanishes in the steady state, (4.3.21), but not in the transient case. Stimulated scattering in forward scattering is particularly delicate since ΓRL → 0, (4.3.20), and Stokes–anti-Stokes coupling has to be included. Values for g and δν are listed in Table 4.3.4.

3. Stimulated Rayleigh wing scattering is connected with the overdamped rotational motion of liquid molecules in combination with an anisotropic polarizability tensor. The latter is also involved in the optical Kerr e ect enhancing the nonlinear refractive index of the medium (optical selffocusing, self-phase modulation). The maximum steady-state gain factor for SRWS is given by:

gRW =	16 π2 N ωS (α − α )2	.	(4.3.22)
	45 kB To c2 nS2

The di erence of the molecular polarizability parallel and perpendicular to the (assumed) molecular axis of rotational symmetry is denoted by α − α . kB is the Boltzmann constant, To the sample temperature. The frequency dependence of the gain factor is analogous to the previous cases:

g(ωS) =	gRW 2 ΓRW (ωL − ωS)	.	(4.3.23)
	(ωL − ωS)2 + ΓRW2

Maximum gain of SRWS occurs for ωS = ωL − ΓRW . The halfwidth ΓRW of the Rayleigh wing line may be taken from spontaneous scattering observations or from the reorientational time τor . The latter can be derived from spontaneous Raman spectroscopy, NMR, or time-resolved spectroscopy, e.g. transient optical Kerr e ect observations: τor = T2,RW = 1/ΓRW . An estimate of the halfwidth may be computed from shear viscosity and the size of the molecules using the Debye theory:

ΓRW =	3 kB To	.	(4.3.24)

	8 π R η S

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Ref. p. 232]	4.3 Stimulated scattering	231

Here R denotes an e ective mean radius of the molecule. The proportionality ΓRW η S was demonstrated experimentally for numerous examples.

Equations (4.3.6), (4.3.22), and (4.3.23) hold for large scattering angles where Stokes–anti- Stokes coupling can be neglected. Close to forward direction simultaneous anti-Stokes scattering enhances the stimulated Stokes scattering, in contrast to SRS. Including the Stokes–anti-Stokes coupling maximum gain is predicted for an optimum scattering angle

θopt =	2 gRW IL c	2
θopt =	nL ωL	.	(4.3.25)

The corresponding gain factor for stimulated ampliﬁcation in the scattering direction θopt without frequency shift, ωS = ωA = ωL , amounts to:

gopt = 2 gRW ,

(4.3.26)

where gRW is given by (4.3.22). For more general cases the reader is referred to the literature, e.g. [72Mai]. Frequency shift and gain factor numbers are compiled in Table 4.3.5.

Table 4.3.5. Frequency shift and gain factor of stimulated Rayleigh wing scattering.

Medium	Frequency shift × c−1	Gain factor GRW	Reference
	[cm−1]	[1012 m/W]
Azoxybenzene	0.036		[68Fol]
Benzene		6	[72Mai]
Benzonitrol	0.198		[68Fol]
Benzoylchloride	0.184		[68Fol]
Benzylidenaniline	0.065		[68Fol]
Bromobenzene		14	[72Mai]
1-Bromonaphthalene	0.076		[68Fol]
Carbondisulﬁde		30	[72Mai]
Chlorobenzene		10	[72Mai]
Chloronaphthalene	0.100		[68Fol]
1,4-Dimethylnitrobenzene	0.090		[68Fol]
m-Dinitrobenzene	0.116		[68Fol]
2,4-Dinitrotoluene	0.098		[68Fol]
Naphthalene	0.5		[68Fol]
Nitroacetophenone	0.105		[68Fol]
o-Nitroaniline	0.107		[68Fol]
p-Nitroanisol	0.075		[68Fol]
Nitrobenzaldehyde	0.101		[68Fol]
Nitrobenzene	0.111	76	[72Mai]
o-Nitrophenol	0.078		[68Fol]
m-Nitrotoluene	0.097		[68Fol]
o-Nitrotoluene	0.133		[68Fol]
p-Nitrotoluene	0.145		[68Fol]
Styrene	0.4		[68Fol]
Toluene		20	[72Mai]

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232	References for 4.3

References for 4.3

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66Eck	Eckhardt, G.: IEEE J. Quantum Electron. 2 (1966) 1.
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72Lau	Laubereau, A., von der Linde, D., Kaiser, W.: Phys. Rev. Lett. 28 (1972) 1162.
72Mai	Maier, M. Kaiser, W., in: Laser Handbook, Arecchi, F.T., Schulz-Dubois, E.O. (eds.),
	Vol. 2, Amsterdam: North Holland, 1972, p. 1077.
73Lau	Laubereau, A., von der Linde, D., Kaiser, W.: Opt. Commun. 1 (1973) 173.
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75Cha	Chatelet, M., Ogsengorn, B.: Chem. Phys. Lett. 36 (1975) 73.
75Lau	Laubereau, A.: Unpublished data, 1975.
76Ber	Berne, B.J., Pecora, R.: Dynamic light scattering, New York: Wiley, 1976, Chap. 10.5.
76Mai	Maier, M.: Appl. Phys. 11 (1976) 209.
77Rys	Rysakov, V.M., Korotkov, V.I.: Sov. J. Quantum Electron. (English Transl.) 7 (1977)
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