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

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

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Ref. p. 212]	4.2 Frequency conversion in gases and liquids	205

4.2 Frequency conversion in gases and liquids

C.R. Vidal

This chapter covers the properties of a nonlinear medium having spherical symmetry like gases and liquids. They therefore clearly di er from the properties of most solids (see Chap. 4.1).

Lasers have become so powerful these days that one can easily generate various kinds of optical overtones


ωo = ni · ωi ± kq · ωres,q > 0 ,	(4.2.1)

i,q

where ni and kq are some integer (including ni = 0 or kq = 0), using a suitable nonlinear medium with eigenfrequencies ωres,q and an incident laser frequency ωi (conservation of energy).

In case of frequency conversion in gases one generally has kq = 0 and deals with sum or di erence frequency mixing

ωs = ωi ± ωj > 0 (4.2.2)

which may be enhanced by exploiting suitable resonances of the atomic or molecular gas.

In case of stimulated scattering one generally has ni = 1. Then ωres,q is a suitable manifold of atomic or molecular (rotational or vibrational) resonances of the gaseous or liquid scattering medium numbered by the index q. Like in classical spectroscopy the plus sign stands for Stokes processes, whereas the minus sign is responsible for Anti-Stokes processes.

Linear and nonlinear susceptibilities are discussed in [87Vid].

The complex linear susceptibility is given by

and the dipole moment matrix elements µag between the states |a and |g . The nonlinear polarization is

P	∞	(4.2.5)
P	NL = P (n) .	(4.2.5)

n=2

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206		4.2.1 Fundamentals of nonlinear optics in gases and liquids		[Ref. p. 212

The deﬁnition of the electric ﬁeld amplitude is given by
E(r, t) =	1	j	ej Eˆ(r, ωj ) exp(i kj r − i ωj t) + c.c. ,	(4.2.6)

	2

resulting in the deﬁnition of the total polarization

P (r, t) = 2 ej P (r, ωj ) exp(−i ωj t) + c.c. , (4.2.7)

where the n th-order polarization is given by

(n)		n!N	(n)
			α1 n
P αs	(r, ωs) = 2 n−1 0		α1 n		(−ωs; ω1	. . . ωn) Eα1 (r, ω1) . . . Eαn (r, ωn) .	(4.2.8)
P αs	(r, ωs) = 2 n−1 0		χαsα1	...αn	(−ωs; ω1	. . . ωn) Eα1 (r, ω1) . . . Eαn (r, ωn) .	(4.2.8)
			...α

The αs are the unit vectors of the spatial coordinates, which may be cartesian, cylindrical, or spherical. The polarization can be expressed in terms of the density matrix [71Han]


P (t) = N Tr [ρ(t) µ] = N ρmn(t) µmn ,	(4.2.9)

whose elements are given by i ρ˙mn = [H, ρ]mn , where the Hamiltonian H = H 0 + H H = −µE(t) . From a perturbation approach one obtains

(n)

(−ωs; ω1

. . . ωn) =

χα1,α2...αn

gb1

ρ(g)

g|esµ|b1 b1|e1µ|b2 . . . bn|enµ|g

−

(Ωb1g

ω1

− · · · −

ωn)(Ωb2g

−

ω2

− · · · −

ωn) . . . . . . (Ωbng

ωn)

...b

contains

(4.2.10)

These processes are responsible for the lowest-order frequency conversion in gases such as sum or di erence frequency mixing, stimulated scattering processes and photorefraction. For the degenerate case the dominant terms in a system of spherical symmetry are [71Han]:

χ(3)(	3 ω; ω, ω, ω) = χ(3)(3 ω) = −3		g\|esµ\|a a\|e1µ\|b b\|e2µ\|c c\|e3µ\|g							,	(4.2.11)
−	T		(Ωag	−	ω)(Ωbg	−	2 ω)(Ωcg	−	3 ω)
		abc

where the index T stands for the third harmonic generation.

For the nondegenerate case we have the general third-order nonlinear susceptibility [62Arm, 71Han]

(3)

, ω2, ω3) =

χα1,α2,α3,αs (−ωs; ω1

ρ(g)

g|esµ|a a|e1µ|b b|e2µ|c c|e3µ|g

(4.2.12)

−

6 3

−

ω1

−

ω2

−

ω3)(Ωbg

−

ω2

−

ω3)(Ωcg

ω3)

(Ωag

gabc

obeying the conservation of energy

ωs = ω1 + ω2 + ω3 .

(4.2.13)

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χ(1) = χ¯(1) + i χ˜(1) =

with the complex transition frequency

Ωag = ωag − i Γag




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