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

Here we have replaced in the argument of the exponential the product “const. nL t” by a more familiar term with laser intensity IL, the gain factor g for stimulated Stokes scattering, and the interaction length z. Equation (4.3.6) indicates exponential amplification of an initial signal IS(0) that may be supplied by spontaneous scattering or by an additional input beam. The exponential growth of the scattered light is only limited by the energy conservation of (4.3.2), since for every scattered photon one incident laser photon has to be annihilated. The corresponding laser depletion leads to gain saturation not included in (4.3.6). Conversion e ciencies above 50 % have been observed for stimulated scattering in a number of cases. Equation (4.3.6) refers to steady state.

The gain factor g is an important material parameter for stimulated scattering. The dependence of g on the frequency shift of the scattering is indicated in Fig. 4.3.1b. Maximum gain occurs in the center of the down-shifted Brillouin and Raman lines (Stokes process). For stimulated Rayleigh scattering the peak gain occurs for a Stokes shift equal to half of the full width, δν/2, of the respective line. The negative gain values in Fig. 4.3.1b indicate loss via stimulated scattering on the anti-Stokes side.

Typical values of the peak gain factors are listed in Tables 4.3.2–4.3.5. Under steady-state conditions stimulated Brillouin scattering often represents the dominant interaction. In absorbing media additional mechanisms occur. The corresponding processes, stimulated thermal Brillouin and stimulated thermal Rayleigh scattering, are discussed below.

4.3.2.2 Experimental observation

Stimulated scattering was studied using the following three di erent experimental approaches:

1.generator setup,

2.oscillator setup,

3.stimulated amplification setup.

4.3.2.2.1 Generator setup

Here only an intense laser beam is directed into the sample. The kind of stimulated scattering is

selected by the material and laser beam properties. As a general rule, a large gain of g I z 30 is

L =

required under steady-state conditions for the traveling-wave situation with a single pass through the medium (length z), in order to observe the respective stimulated process. The scattering occurs in forward and/or backward direction because of a simple geometrical argument (maximum interaction length in these directions). The process builds up from an equivalent noise input IS(0) , see (4.3.6), that can be estimated from zero point fluctuations of the electromagnetic field [79Pen]. The growth of the Stokes component is finally limited by the simultaneous decrease of incident laser radiation. The observations are di cult to analyze because of the competition of nonlinear interactions including optical self-focusing. The latter is often involved in liquid media. The observed frequency shift of the stimulated process may slightly deviate from the value known from spontaneous scattering (up to a few cm−1 in SRS) because of simultaneous self-phase modulation in the medium.

4.3.2.2.2 Oscillator setup

An optical resonator made up by mirrors or reflecting surfaces can provide feedback of the stimulated Stokes radiation so that the e ective interaction length is increased by multiple passes

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through the medium. As a result the laser intensity requirements are lowered. The scattering angle is controlled by the cavity axis, so that o -axis emission is possible relative to the laser beam. The frequency-dependent feedback and the lower intensity level of the setup can be su cient to select a specific stimulated scattering process. Among di erent Raman transitions only the one with largest gain factor g shows up in SRS in general.

4.3.2.2.3 Stimulated amplification setup

Two well defined beams representing the laser component and the incident Stokes radiation are directed into the scattering medium. Scattering angle and mechanism are determined by the direction and frequency shift of the incident Stokes beam. A second tunable laser is used for the latter in general. The pump intensity IL is smaller by one or more orders of magnitude compared to the generator case, so that self-focusing and other competing e ects including secondary scattering processes can be avoided. Quantitative information on the amplitude and/or frequency dependence of the gain factor g(νS) may be deduced from careful measurements of the amplification factor.

An example for the technique is Raman gain spectroscopy that is often applied in the lowintensity limit g IL z 1 . An alternative is Raman loss spectroscopy of the transmitted laser component, since the production of Stokes photons corresponds to the annihilation of the same number of laser photons.

4.3.2.3 Four-wave interactions

4.3.2.3.1 Third-order nonlinear susceptibility

Stimulated Stokes scattering can be treated as a four-photon (or four-wave) interaction involving the third-order nonlinear susceptibility χ(3)(−ωS; ωL, −ωL, ωS) . The interaction is illustrated by the energy level scheme of Fig. 4.3.2b. The two waves are resonantly coupled via a di erence frequency resonance, ωL − ωS = ωo, to the relevant material excitation. The latter is enhanced by the scattering thus increasing the coupling strength. The photons at frequencies ωL and ωS enter the process twice (see Fig. 4.3.2a). Stimulated amplification is provided in the resonant case by the imaginary part χ3 of χ(3), while the real part leads to frequency modulation. The gain factor is related to the imaginary part by:

Outside di erence frequency resonances the real part of χ(3) is also important for stimulated amplification. The general case of stimulated 4-photon amplification is treated in [79Pen]. The (fourth-rank) tensor character of χ(3) is omitted here for brevity considering only parallel polarization of the light field components.

The corresponding wave-vector diagram is shown in the lower part of Fig. 4.3.2b. The general case with o -axis geometry is considered. The scattering couples to a material excitation with wave vector ko. The e ective scattering angle is strongly influenced by interaction-length arguments. Because of the maximum interaction length, geometries with approximate forward and backward scattering are most important. In cases where the corresponding frequency shift ωo vanishes, e.g. SBS, stimulated scattering exactly in forward direction is not possible. For backward scattering of short pulses, e.g. SRS of a picosecond laser, the interaction length may be governed by the duration tp (FWHM of intensity envelope) of the incident laser pulse setting an upper limit of= tp/2 vg (vg : group velocity). In forward direction a less stringent limitation is set by group velocity dispersion between laser and Stokes pulses, = tp ∆(1/vg) . As a result SRS of picosecond pulses preferentially occurs in forward direction.

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