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

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Contents

XIII

Part 4 Nonlinear optics

4.1Frequency conversion in crystals

G.G. Gurzadyan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.1.1	Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 141
4.1.1.1	Symbols and abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 141
4.1.1.1.1	Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 141
4.1.1.1.2	Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 142
4.1.1.1.3	Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 142
4.1.1.2	Historical layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 143
4.1.2	Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 144
4.1.2.1	Three-wave interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 144
4.1.2.2	Uniaxial crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 145
4.1.2.3	Biaxial crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 145
4.1.2.4	E ective nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 147
4.1.2.5	Frequency conversion e ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 151
4.1.2.5.1	General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 151
4.1.2.5.2	Plane-wave ﬁxed-ﬁeld approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 152
4.1.2.5.3	SHG in “nonlinear regime” (fundamental wave depletion) . . . . . . . . . . . . . . . . . .	. . 154
4.1.3	Selection of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 154
4.1.4	Harmonic generation (second, third, fourth, ﬁfth, and sixth) . . . . . . . . . . . . . . . .	. . 156
4.1.5	Sum frequency generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 167
4.1.6	Di erence frequency generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 172
4.1.7	Optical parametric oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 176
4.1.8	Picosecond continuum generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 186
	References for 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 187
4.2	Frequency conversion in gases and liquids
	C.R. Vidal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 205
4.2.1	Fundamentals of nonlinear optics in gases and liquids . . . . . . . . . . . . . . . . . . . . . .	. . 205
4.2.1.1	Linear and nonlinear susceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 205
4.2.1.2	Third-order nonlinear susceptibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 206
4.2.1.3	Fundamental equations of nonlinear optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 207
4.2.1.4	Small-signal limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 207
4.2.1.5	Phase-matching condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 208
4.2.2	Frequency conversion in gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 209
4.2.2.1	Metal-vapor inert gas mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 209
4.2.2.2	Mixtures of di erent metal vapors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 209
4.2.2.3	Mixtures of gaseous media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 209
	References for 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 212
4.3	Stimulated scattering
	A. Laubereau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 217
4.3.1	Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 217
4.3.1.1	Spontaneous scattering processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	. . 217
4.3.1.2	Relationship between stimulated Stokes scattering and spontaneous scattering	. . 219

XIV

Contents

4.3.2 General properties of stimulated scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 4.3.2.1 Exponential gain by stimulated Stokes scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 4.3.2.2 Experimental observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.3.2.2.1 Generator setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.3.2.2.2 Oscillator setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.3.2.2.3 Stimulated ampliﬁcation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 4.3.2.3 Four-wave interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 4.3.2.3.1 Third-order nonlinear susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 4.3.2.3.2 Stokes–anti-Stokes coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.3.2.3.3 Higher-order Stokes and anti-Stokes emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.3.2.4 Transient stimulated scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

4.3.3 Individual scattering processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 4.3.3.1 Stimulated Raman scattering (SRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

4.3.3.2Stimulated Brillouin scattering (SBS) and stimulated thermal Brillouin

scattering (STBS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 4.3.3.3 Stimulated Rayleigh scattering processes, SRLS, STRS, and SRWS . . . . . . . . . . . . 228

References for 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

4.4Phase conjugation

H.J. Eichler, A. Hermerschmidt, O. Mehl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4.4.2 Basic mathematical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

4.4.3 Phase conjugation by degenerate four-wave mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 236

4.4.4 Self-pumped phase conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

4.4.5 Applications of SBS phase conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

4.4.6 Photorefraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

References for 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Ref. p. 40]	1.1 Fundamentals of the semiclassical laser theory	3

1.1 Fundamentals of the semiclassical laser theory

V.A. Lopota, H. Weber

A rigorous description of light–matter interaction requires a fully quantized system of ﬁeld equations, which is the content of quantum optics [70Hak, 95Wal, 97Scu, 95Man, 01Vog]. This theory is well developed and the results are conﬁrmed perfectly by many experiments (see Chap. 5.1). But most problems of laser design and laser technology can be solved in a satisfactory way by applying the semiclassical theory. This means a non-relativistic quantum-mechanical approach for the electronic system and a non-quantized, classical electromagnetic ﬁeld.

Non-relativistic means that the velocity of the interacting electrons is small compared with the velocity of light. This holds for the outer shell electrons of the atoms and molecules, which are relevant in laser physics. It is not true for the free-electron laser and for the interaction of strong ﬁelds with plasmas, which demand a relativistic treatment.

A non-quantized electromagnetic ﬁeld implies that the photon is neglected. In laser technology the photon ﬂux in most applications is extremely high and the granulation of light beams is of no importance. It is of signiﬁcance for metrology, where the lower limit of detectability is partly given by photon statistics. There are some other e ects, which are not covered by the semiclassical theory:

–Planck’s law, related to photon statistics,

–squeezed states,

–entangled photons,

–zero-point energy e ects,

–spontaneous emission,

and some spectral line shifts (Lamb-shift [47Lam]), of minor importance for laser technology, although of great experimental interest for the conﬁrmation of the fundamental theory. The spontaneous emission of excited atoms/molecules is responsible for the lower limit of laser line width [74Sar, 95Man] and for the on-set of laser oscillation. Therefore, spontaneous emission has to be included in the semiclassical theory by a phenomenological term as shown in Fig. 1.1.1.

It is the intention of this chapter to compile the relevant relations of laser dynamics, their application in laser design and to discuss the limitations and approximations. The mathematical derivations can be taken from the references.

1.1.1 The laser oscillator

The laser oscillator is based on the principle of the feed-back ampliﬁer, a principle invented by A. Meissner 1913 and patented 1919 [19Mei]. All coherent electromagnetic waves are generated by such self-sustained oscillators, from radio frequencies to microwaves and ﬁnally lasers. Basov and Prokhorov published 1954 a theoretical paper on masers [54Bas], Schawlow and Townes in 1958 [58Sch] a theoretical paper discussing the possibility of masers in the visible range of the spectrum, and Maiman realized 1960 the ﬁrst laser [60Mai].

Landolt-B¨ornstein

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4	1.1.1 The laser oscillator	[Ref. p. 40

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Fig. 1.1.1. The semiclassical laser theory (SVE-approximation: Slowly Varying Envelope approximation, see Sect. 1.1.2.2.3).

≈

Fig. 1.1.2. Schematic set-up of a laser oscillator.

The principle set-up of a laser oscillator is plotted in Fig. 1.1.2. Light is ampliﬁed by induced emission in an active medium (gas discharge, doped crystals or liquids, pn-transitions). The active medium is characterized by an intensityand frequency-dependent gain factor G(J ) (with J : intensity). The beam bounces forth and back between the two mirrors of an optical resonator. On-set of laser oscillation requires a gain factor exceeding the total losses per round trip:

G0RV > 1 (threshold condition)

(1.1.1)

with

G0: small-signal gain factor for the intensities,

√

R = R1R2: average reﬂection factor of the mirrors, V : internal loss factor of the resonator.

With increasing intensity J the gain decreases due to saturation of the ampliﬁer

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 40]	1.1 Fundamentals of the semiclassical laser theory	5

G(J ) ≤ G0 .
In steady state the gain has to compensate the losses:
G(J )RV = 1	(steady-state condition) .	(1.1.2)

If the relation G(J ) is known, depending on the speciﬁc ampliﬁer, (1.1.2) gives the internal intensity of the laser system in steady state.

The wavelength of the ﬁeld is determined by the resonance condition. After one round trip the

phase shift ∆ϕ of the ﬁeld must be
∆ϕ = 2π p , p = 1, 2, 3, . . . (resonance condition) ,	(1.1.3)

otherwise the ﬁeld would be reduced by destructive interference. The resonator is mainly responsible for the mode structure of the output ﬁeld and can be described by a non-quantized ﬁeld. Details are given in Chap. 8.1. For the interaction ﬁeld–ampliﬁer a plane wave is assumed and di raction is neglected.

1.1.2 The electromagnetic ﬁeld

Light is a special case of propagating electromagnetic waves, as was predicted by Maxwell 1856 [54Max] and conﬁrmed experimentally by Hertz [88Her]. The electromagnetic ﬁeld is characterized by the electric/magnetic vector ﬁelds E, H. In this section the propagation of quasi-monochromatic waves with frequency ω and wavelength λ is investigated. The wavelength range from the infrared (λ ≈ some 10 µm) to the UV (λ ≈ 0.1 µm) is normally called light.

1.1.2.1 Maxwell’s equations

The electromagnetic ﬁeld is used in the classical representation, neglecting the quantization. The materials equations, based on quantum mechanics, are introduced phenomenologically. The ﬁnal result is a wave equation, describing the propagation of electromagnetic waves.


The classical electromagnetic ﬁeld is completely described by Maxwell’s equations:
	∂B
curl E = −				,	(1.1.4)
curl E = −		∂t		,	(1.1.4)
curl H =	∂D		+ j ,		(1.1.5)
curl H =	∂t		+ j ,		(1.1.5)
	∂t
div D = ρ ,					(1.1.6)
div B = 0					(1.1.7)
with