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

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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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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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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