- •Preface
- •Contents
- •1.1 Fundamentals of the semiclassical laser theory
- •1.1.1 The laser oscillator
- •1.1.2.2 Homogeneous, isotropic, linear dielectrics
- •1.1.2.2.1 The plane wave
- •1.1.2.2.2 The spherical wave
- •1.1.2.2.3 The slowly varying envelope (SVE) approximation
- •1.1.2.3 Propagation in doped media
- •1.1.3 Interaction with two-level systems
- •1.1.3.1 The two-level system
- •1.1.3.2 The dipole approximation
- •1.1.3.2.1 Inversion density and polarization
- •1.1.3.3.1 Decay time T1 of the upper level (energy relaxation)
- •1.1.3.3.1.1 Spontaneous emission
- •1.1.3.3.1.2 Interaction with the host material
- •1.1.3.3.1.3 Pumping process
- •1.1.3.3.2 Decay time T2 of the polarization (entropy relaxation)
- •1.1.4 Steady-state solutions
- •1.1.4.1 Inversion density and polarization
- •1.1.4.2 Small-signal solutions
- •1.1.4.3 Strong-signal solutions
- •1.1.5 Adiabatic equations
- •1.1.5.1 Rate equations
- •1.1.5.2 Thermodynamic considerations
- •1.1.5.3 Pumping schemes and complete rate equations
- •1.1.5.3.1 The three-level system
- •1.1.5.3.2 The four-level system
- •1.1.5.5 Rate equations for steady-state laser oscillators
- •1.1.6 Line shape and line broadening
- •1.1.6.1 Normalized shape functions
- •1.1.6.1.1 Lorentzian line shape
- •1.1.6.1.2 Gaussian line shape
- •1.1.6.1.3 Normalization of line shapes
- •1.1.6.2 Mechanisms of line broadening
- •1.1.6.2.1 Spontaneous emission
- •1.1.6.2.2 Doppler broadening
- •1.1.6.2.3 Collision or pressure broadening
- •1.1.6.2.4 Saturation broadening
- •1.1.6.3 Types of broadening
- •1.1.6.3.1 Homogeneous broadening
- •1.1.6.3.2 Inhomogeneous broadening
- •1.1.6.4 Time constants
- •1.1.7 Coherent interaction
- •1.1.7.1 The Feynman representation of interaction
- •1.1.7.3 Propagation of resonant coherent pulses
- •1.1.7.3.2 Superradiance
- •1.1.8 Notations
- •References for 1.1
- •2.1.1 Introduction
- •2.1.3 Radiometric standards
- •2.1.3.1 Primary standards
- •2.1.3.2 Secondary standards
- •References for 2.1
- •2.2 Beam characterization
- •2.2.1 Introduction
- •2.2.2 The Wigner distribution
- •2.2.3 The second-order moments of the Wigner distribution
- •2.2.4 The second-order moments and related physical properties
- •2.2.4.3 Phase paraboloid and twist
- •2.2.4.4 Invariants
- •2.2.4.5 Propagation of beam widths and beam propagation ratios
- •2.2.5.1 Stigmatic beams
- •2.2.5.2 Simple astigmatic beams
- •2.2.5.3 General astigmatic beams
- •2.2.5.4 Pseudo-symmetric beams
- •2.2.5.5 Intrinsic astigmatism and beam conversion
- •2.2.6 Measurement procedures
- •2.2.7 Beam positional stability
- •References for 2.2
- •3 Linear optics
- •3.1 Linear optics
- •3.1.1 Wave equations
- •3.1.2 Polarization
- •3.1.3 Solutions of the wave equation in free space
- •3.1.3.1 Wave equation
- •3.1.3.1.1 Monochromatic plane wave
- •3.1.3.1.2 Cylindrical vector wave
- •3.1.3.1.3 Spherical vector wave
- •3.1.3.2 Helmholtz equation
- •3.1.3.2.1 Plane wave
- •3.1.3.2.2 Cylindrical wave
- •3.1.3.2.3 Spherical wave
- •3.1.3.2.4.2 Real Bessel beams
- •3.1.3.2.4.3 Vectorial Bessel beams
- •3.1.3.3 Solutions of the slowly varying envelope equation
- •3.1.3.3.1 Gauss-Hermite beams (rectangular symmetry)
- •3.1.3.3.2 Gauss-Laguerre beams (circular symmetry)
- •3.1.3.3.3 Cross-sectional shapes of the Gaussian modes
- •3.1.4.4.2 Circular aperture with radius a
- •3.1.4.4.2.1 Applications
- •3.1.4.4.3 Gratings
- •3.1.5 Optical materials
- •3.1.5.1 Dielectric media
- •3.1.5.2 Optical glasses
- •3.1.5.3 Dispersion characteristics for short-pulse propagation
- •3.1.5.4 Optics of metals and semiconductors
- •3.1.5.6 Special cases of refraction
- •3.1.5.6.2 Variation of the angle of incidence
- •3.1.5.7 Crystal optics
- •3.1.5.7.2 Birefringence (example: uniaxial crystals)
- •3.1.5.8 Photonic crystals
- •3.1.5.9 Negative-refractive-index materials
- •3.1.5.10 References to data of linear optics
- •3.1.6 Geometrical optics
- •3.1.6.1 Gaussian imaging (paraxial range)
- •3.1.6.1.1 Single spherical interface
- •3.1.6.1.2 Imaging with a thick lens
- •3.1.6.2.1 Simple interfaces and optical elements with rotational symmetry
- •3.1.6.2.2 Non-symmetrical optical systems
- •3.1.6.2.3 Properties of a system
- •3.1.6.2.4 General parabolic systems without rotational symmetry
- •3.1.6.2.5 General astigmatic system
- •3.1.6.2.6 Symplectic optical system
- •3.1.6.2.7 Misalignments
- •3.1.6.3 Lens aberrations
- •3.1.7 Beam propagation in optical systems
- •3.1.7.2.1 Stigmatic and simple astigmatic beams
- •3.1.7.2.1.1 Fundamental Mode
- •3.1.7.2.1.2 Higher-order Hermite-Gaussian beams in simple astigmatic beams
- •3.1.7.2.2 General astigmatic beam
- •3.1.7.3 Waist transformation
- •3.1.7.3.1 General system (fundamental mode)
- •3.1.7.3.2 Thin lens (fundamental mode)
- •3.1.7.4 Collins integral
- •3.1.7.4.1 Two-dimensional propagation
- •3.1.7.4.2 Three-dimensional propagation
- •3.1.7.5 Gaussian beams in optical systems with stops, aberrations, and waveguide coupling
- •3.1.7.5.1 Field distributions in the waist region of Gaussian beams including stops and wave aberrations by optical system
- •3.1.7.5.2 Mode matching for beam coupling into waveguides
- •3.1.7.5.3 Free-space coupling of Gaussian modes
- •References for 3.1
- •4.1 Frequency conversion in crystals
- •4.1.1 Introduction
- •4.1.1.1 Symbols and abbreviations
- •4.1.1.1.1 Symbols
- •4.1.1.1.2 Abbreviations
- •4.1.1.1.3 Crystals
- •4.1.1.2 Historical layout
- •4.1.2 Fundamentals
- •4.1.2.1 Three-wave interactions
- •4.1.2.2 Uniaxial crystals
- •4.1.2.3 Biaxial crystals
- •4.1.2.5.1 General approach
- •4.1.3 Selection of data
- •4.1.5 Sum frequency generation
- •4.1.7 Optical parametric oscillation
- •4.1.8 Picosecond continuum generation
- •References for 4.1
- •4.2 Frequency conversion in gases and liquids
- •4.2.1 Fundamentals of nonlinear optics in gases and liquids
- •4.2.1.1 Linear and nonlinear susceptibilities
- •4.2.1.2 Third-order nonlinear susceptibilities
- •4.2.1.3 Fundamental equations of nonlinear optics
- •4.2.1.4 Small-signal limit
- •4.2.1.5 Phase-matching condition
- •4.2.2 Frequency conversion in gases
- •4.2.2.1 Metal-vapor inert gas mixtures
- •4.2.2.3 Mixtures of gaseous media
- •References for 4.2
- •4.3 Stimulated scattering
- •4.3.1 Introduction
- •4.3.1.1 Spontaneous scattering processes
- •4.3.1.2 Relationship between stimulated Stokes scattering and spontaneous scattering
- •4.3.2 General properties of stimulated scattering
- •4.3.2.1 Exponential gain by stimulated Stokes scattering
- •4.3.2.2 Experimental observation
- •4.3.2.2.1 Generator setup
- •4.3.2.2.2 Oscillator setup
- •4.3.2.3 Four-wave interactions
- •4.3.2.3.1 Third-order nonlinear susceptibility
- •4.3.2.3.3 Higher-order Stokes and anti-Stokes emission
- •4.3.2.4 Transient stimulated scattering
- •4.3.3 Individual scattering processes
- •4.3.3.1 Stimulated Raman scattering (SRS)
- •4.3.3.2 Stimulated Brillouin scattering (SBS) and stimulated thermal Brillouin scattering (STBS)
- •4.3.3.3 Stimulated Rayleigh scattering processes, SRLS, STRS, and SRWS
- •References for 4.3
- •4.4 Phase conjugation
- •4.4.1 Introduction
- •4.4.2 Basic mathematical description
- •4.4.3 Phase conjugation by degenerate four-wave mixing
- •4.4.4 Self-pumped phase conjugation
- •4.4.5 Applications of SBS phase conjugation
- •4.4.6 Photorefraction
- •References for 4.4
Ref. p. 232] |
4.3 Stimulated scattering |
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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 :
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(4.3.1) |
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:
Landolt-B¨ornstein
New Series VIII/1A1
218 |
4.3.1 Introduction |
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h νL = h νsc ± h νo ,
kL = ksc ± ko
Fig. 4.3.1. (a) Schematic of the spectral intensity distribution of spontaneous light scattering in condensed matter with unshifted Rayleigh and Rayleigh-wing lines (quasi-elastic scattering) as well as Stokesand anti-Stokes-shifted Brillouin and Raman lines (inelastic light scattering). (b) Frequency dependence of the corresponding gain factors of stimulated scattering (see text).
(4.3.2)
(4.3.3)
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
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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
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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
Landolt-B¨ornstein
New Series VIII/1A1