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

4.3Stimulated scattering

A. Laubereau

4.3.1 Introduction

The first example of stimulated scattering was incidentally discovered in 1962 as a “new laser line in the emission of a ruby laser” [62Woo]. The phenomenon occurred when the laser was equipped with a nitrobenzene cell for Q-switching operation. The emitted frequency component was identified as an amazingly intense Raman line [62Eck] due to stimulated Raman scattering predicted theoretically in 1931 [31Goe]. Hundreds of papers appeared since then on the novel phenomenon. Compared to the wealth of experimental evidence full quantitative information about the individual scattering processes however is rather scarce since many publications confine themselves to reported frequency shifts. A quantitative analysis is also often impeded by competing nonlinear e ects and by the not too well known properties of the applied laser pulses. Three cases were investigated in detail: Stimulated Raman Scattering (SRS), Stimulated Brillouin Scattering (SBS), and stimulated Rayleigh scattering.

This chapter, Chap. 4.3, follows the discussions given by Maier and Kaiser [72Mai], by Maier [76Mai], and by Penzkofer et al. [79Pen]. Circular frequencies are denoted in the following by ωi while the corresponding frequency values are represented by νi = ωi/2 π . The term “circular” is often omitted in context with the ωi’s.

4.3.1.1 Spontaneous scattering processes

Fluctuations of the molecular polarizability and of the number density of atoms or molecules give rise to various scattering processes when light passes a transparent medium. The scattering is characterized by the frequency νsc of the scattered light relative to the incident laser frequency νL , the linewidth δν , its polarization properties, and the scattering intensity. Here we introduce the scattering cross section dσ/dΩ relating the power Psc of the light scattered into a solid angle ∆Ω to the incident laser power PL :

with number density N of the (quasi-)particles generating the scattering. The interaction length is denoted by . dσ/dΩ is the di erential cross section with respect to solid angle but integrated over the spectral lineshape. The spectrum of four scattering processes is depicted schematically in Fig. 4.3.1a. Two unshifted components are indicated: the narrow Rayleigh line scattered from nonpropagating entropy (temperature) fluctuations and the broader Rayleigh-wing line due to orientation fluctuations of anisotropic molecules. The lines are accompanied by the Brillouin doublet representing scattering from propagating isentropic density fluctuations. In the quantum-mechanical approach the Brillouin lines are related to the annihilation (frequency up-shifted anti-Stokes component) and creation (down-shifted Stokes line) of acoustic phonons with conservation of quantum energy and (pseudo-)momentum:

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with Planck’s constant h and wavevector k, = h/2 π . Subscript “o” refers to the material excitation, i.e. acoustic phonons. The positive sign in (4.3.2) and (4.3.3) corresponds to the Stokes process (sc = S), while the negative sign applies for anti-Stokes scattering (sc = A). Due to the dispersion relation of acoustic phonons (phase velocity v of sound waves) the frequency shift is given by

Here c/n denotes the speed of light in the medium; θ is the scattering angle between wave vectors

kL and ksc (ksc kL , since νo νL ). Equation (4.3.4) refers to isotropic media, e.g. gases and

=

liquids. For anisotropic solids three Brillouin doublets occur in the general case and (4.3.4) has to be modified according to the considered transverse or longitudinal acoustic phonon branch and the respective orientation-dependent sound velocity in the crystal. As a consequence of (4.3.3), ko ≤ 2 kL , so that only acoustic phonons close to the center of the first Brillouin zone are involved (note kL 105 cm−1 ).

Figure 4.3.1a also schematically shows the Stokes and anti-Stokes line of Raman scattering o a molecular vibration or o an optical phonon branch, displaying a larger frequency shift. As before, only phonons of relatively small ko are involved. Polyatomic molecules display a variety of such vibrational Raman lines. In gases many vibration-rotation Raman lines occur in addition and also rotational lines with small frequency shifts. In ionic crystals the relevant material excitation is of mixed phonon-photon character and termed polariton. Since the excited states of molecular vibrations and optical phonons are weakly populated, the anti-Stokes line intensity is also small compared to the corresponding Stokes line.

A further unshifted scattering component in liquids, the Mountain line [76Ber], is only mentioned here since it was not yet observed in stimulated scattering because of its weakness and broad width. Typical values for the frequency shift νo/c and the linewidth δν/c (FWHM) in wavenumber units of the various processes are given in Tables 4.3.1–4.3.5. Some scattering cross sections for the Raman interaction are listed in Table 4.3.2. The scattered light intensity is small. Even for the large

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number density of condensed matter of 1022 cm−3 a small fraction < 10−5 of the incident light is distributed into the whole solid angle 4 π per cm interaction length by spontaneous scattering.

4.3.1.2Relationship between stimulated Stokes scattering and spontaneous scattering

The elementary interaction for Stokes scattering is illustrated in Fig. 4.3.2a (solid arrows). The process involves a transition from an initial to a final energy level of the medium (horizontal lines). The relationship between the stimulated and the spontaneous process is close and originates from the Boson character of photons, i.e. the analogy of the eigenmodes of the electromagnetic field with the harmonic oscillator, the transition probability of which increases with occupation number. As a result the rate of photons scattered into an eigenmode of the Stokes field (subscript “S”) depends on the occupation number nS of this mode. Under steady-state conditions we have:

The first term in the bracket on the right-hand side of (4.3.5) represents spontaneous scattering depending linearly on incident photon number nL or laser power, compare (4.3.1), as long as nS 1, i.e. a negligible number of scattered photons per mode of the radiation field is present. The second term on the right-hand side of (4.3.5) describes stimulated scattering that dominates for nS > 1 and requires su ciently high laser intensities. In this regime an avalanche build-up of scattered photons can occur.

4.3.2 General properties of stimulated scattering

4.3.2.1 Exponential gain by stimulated Stokes scattering

Integration of (4.3.5) yields exponential growth of Stokes-scattered photons, nS exp (const. nL t) , or equivalently for forward scattering in the z-direction:

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