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

ωA = ωL + ωo . The anti-Stokes process is depicted in Fig. 4.3.2a (dashed arrows) and “consumes” material excitation, so that (4.3.6) is not applicable. The corresponding four-wave interaction via χ(3)(−ωA; ωL, ωL, −ωS) is termed Stokes–anti-Stokes coupling and depicted in Fig. 4.3.2c. The significance of the process is determined by its wave vector mismatch ∆kA , depicted in the lower part of Fig. 4.3.2c, and the initial intensity ratio IA(0)/IS(0) (IA : anti-Stokes intensity). ∆kA is governed by the scattering angle and the color dispersion of the refractive index n(ω) of the medium since

For a collinear geometry we simply have ∆kA = kA + kS − 2kL . For ∆kA = 0 and IA/IS = 1 , the inverse process of anti-Stokes scattering fully inhibits stimulated Stokes scattering. An example in this context is exact forward scattering in gases, where ∆kA is small, so that the observed weakness of SRS in exact forward direction is explained in this way. For a large mismatch, |∆kA| > 3 g IL , on the other hand, the Stokes–anti-Stokes coupling is negligible. This condition is always fulfilled for backward scattering so that simultaneous anti-Stokes scattering cannot perturb the stimulated Stokes process notably. For IA IS , the perturbation of Stokes scattering by antiStokes production is negligible, too. In this case the process of Fig. 4.3.2c is also called Coherent Anti-Stokes Raman Scattering, CARS, an important nonlinear spectroscopy (preferentially applied

for phase-matching geometries, ∆k 0 ).

A =

Outside Raman resonances the properties of Stokes–anti-Stokes coupling di er notably from the near-resonant case considered here.

4.3.2.3.3 Higher-order Stokes and anti-Stokes emission

For high conversion e ciency of the stimulated scattering the Stokes intensity IS becomes comparable to the incident radiation IL , and the material excitation is significant. As a consequence secondary processes show up, generating a cascade of higher-order Stokes and anti-Stokes lines with relative frequency shift ωo and decreasing intensity levels. Two mechanisms are relevant here:

1.stimulated Stokes scattering where the intense first-order Stokes component serves as the pump radiation for generating the second-order line and so forth;

2.coherent Stokes or anti-Stokes scattering o the material excitation generated by the primary Stokes scattering producing new frequency-shifted lines. The mechanism is e ected by wavevector mismatches of the individual processes.

The Stokes–anti-Stokes coupling discussed above is responsible for the generation of the firstorder anti-Stokes component. Higher-order Stokes scattering limits the energy conversion e ciency of first-order Stokes production. The higher-order stimulated scattering should be distinguished from higher-order spontaneous scattering since only a fundamental material transition is involved in the former case.

4.3.2.4 Transient stimulated scattering

The build-up of a material excitation in stimulated scattering involves the response time T2 (dephasing time) of the medium. When the pulse duration tp of the incident laser is comparable to or smaller than T2 , the interaction becomes less e cient and the actual gain of the stimulated Stokes

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process is smaller than in the steady state. Equation (4.3.6) for the stationary case is not valid for tp/T2 < 10 . The smaller transient gain for a given input situation may be overcome experimentally by increased pump intensities. For details the reader is referred to the literature [78Lau]. Here only three remarks are given:

1.For homogeneous broadening of the material transition ωo involved in the stimulated scattering the relaxation time can be simply derived from the linewidth δν (FWHM)

For inhomogeneous broadening (4.3.9) may be also used to estimate an e ective T2 from the line broadening that may be su cient for a semi-quantitative discussion of the transient scattering. For the competition among di erent Raman transitions in transient SRS both gain factor g and dephasing time T2 are relevant.

2. For frequency-modulated laser pulses the temporal behavior is not fully described by the duration tp of the pulse envelope. Because of intensity fluctuations the e ective duration of the

should be fulfilled.

3.Choice of a short tp may allow to suppress stimulated scattering of transitions with longer T2 that would have to occur in a less favorable transient situation. An example is SRS in liquids in forward direction with picosecond pulses that is observed in spite of the larger stationary gain factor of SBS. Here the di erent interaction lengths of forward (SRS) and backward scattering (SBS) also play a role.

4.3.3 Individual scattering processes

4.3.3.1 Stimulated Raman scattering (SRS)

The gain constant for stimulated amplification of the first Stokes component (4.3.6) at resonance, ωS = ωL − ωo is given by

Here N denotes the molecular number density. A highly polarized vibrational Raman line with halfwidth Γ (HWHM, isotropic scattering component) is considered. (∂α/∂q) is the isotropic part of the Raman polarizability (derivative of the molecular polarizability with respect to the vibrational coordinate q of transition ωo) . m represents the reduced mass of the molecular vibration. ni (i = L, S) is the refractive index at frequency ωi. (∂α/∂q) is connected to the Raman scattering cross section by the relation:

dσ (∂α/∂q)2 ω4 h nS

dΩ = 4 π c4 m ωS n . (4.3.12)

o L

The frequency dependence of the gain factor is given by:

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A Lorentzian lineshape is assumed in (4.3.13) that holds well in gases at su ciently high pressure, weakly associated liquids and solids. SRS of notably depolarized Raman lines is discussed in [78Lau]. Frequency shifts observed for SRS in the generator setup are compiled in Table 4.3.1. A list of gain factors gS and other parameters is presented in Table 4.3.2. The relaxation time T2 in condensed matter is in the range 10−12 to 10−10 s.

Table 4.3.1. Frequency shifts (in wavenumber units) observed in stimulated Raman scattering of various materials.

(a) Liquids

(continued)

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(continued)

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