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

4.2 Frequency conversion in gases and liquids

C.R. Vidal

4.2.1 Fundamentals of nonlinear optics in gases and liquids

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)

j

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.

4.2.1.1 Linear and nonlinear susceptibilities

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

The complex linear susceptibility is given by

χ(1) = χ¯(1) + i χ˜(1) =	1		|µag |2			(4.2.3)
			(Ωag	−	ω)
		a
with the complex transition frequency
Ωag = ωag − i Γag						(4.2.4)

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

P	∞	(4.2.5)
	NL = P (n) .

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

1

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

j

where the n th-order polarization is given by

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

mn

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

1

gb1

ρ(g)

g|esµ|b1 b1|e1µ|b2 . . . bn|enµ|g

.

−

−

n!

n

(Ωb1g

ω1

− · · · −

ωn)(Ωb2g

−

ω2

− · · · −

ωn) . . . . . . (Ωbng

ωn)

n

...b

contains

(4.2.10)

4.2.1.2 Third-order nonlinear susceptibilities

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

1

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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Ref. p. 212]	4.2 Frequency conversion in gases and liquids	207

4.2.1.3 Fundamental equations of nonlinear optics

Maxwell’s equations in SI units [62Jac, 87Vid] are given by (1.1.4)–(1.1.7) and the material equations (1.1.8) and (1.1.9), see Chap. 1.1.

With the following three approximations

1.magnetization M = 0 : µ0 H = B → µ = 1 ,

2.source-free medium: ρ = 0 ,

3.currentless medium: j = 0

we get the simplified Maxwell equations

∂H

(4.2.14)

× E = −µ0

∂t

∂E

∂P

× H = 0

+

(4.2.15)

∂t

∂t

resulting in the wave equation

1 ∂2E

1

∂2P

∆E −

=

(4.2.16)

c2

∂t2

0c2

∂t2

with the polarization P = P L + P NL . This gives the driven wave equation

n2 ∂2E

0 ∂2E

1 ∂2P NL

∆E −

− i

=

.

(4.2.17)

c2

∂t2

c2

∂t2

0c2

∂t2

ˆ

ˆ

(z, ω) and the slow-amplitude approximation

With the plane-wave approximation E

(r, ω) = E

ˆ

ˆ

ˆ

ˆ

∂Ej

∂Ej

(4.2.18)

ω Ej ,

k Ej ,

∂t

∂z

we get the fundamental equations of nonlinear optics

ˆ

ωj

NL

κj ˆ

d Ej

= i

Pj

exp(−i kj z) −

Ej

,

(4.2.19)

d z

2 0 c nj

2

where κ is the absorption coe cient and where the total derivative is given by the partial derivatives

ˆ			ˆ		nj		ˆ
d Ej	=	∂Ej		+			∂Ej	,	(4.2.20)
d z			∂z		c ∂t

ˆ

and Ej is a slowly-varying-envelope function in space and time.

4.2.1.4 Small-signal limit

In this case the only nonlinear polarization for a medium of density N is given by

3

(3)

Ps(3)(ωs) =

0

N χT

(−ωs; ω1, ω2, ω3)E1E2E3 .

(4.2.21)

2

Within the plane-wave approximation one obtains

ˆ

= i c ns

N χT

E10E20E30 exp −

κ1

2

+ κ3

− i ∆ k z

,

(4.2.22)

d z

d Es

3 π ωs

(3)

+ κ2

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208		4.2.1 Fundamentals of nonlinear optics in gases and liquids	[Ref. p. 212
where the wave-vector mismatch is given by the conservation of momenta
∆ k = ks − k1 − k2 − k3 .			(4.2.23)
The wave vector kj of the j th wave is given by the refractive index nj
kj =	ωj nj	.	(4.2.24)
	c

With the optical depth τj = κj L = σj(1)(ωj ) N L and the length L of the nonlinear medium we have

Eˆs(L) = i 3 π ωs N L 0χT(3) E10 E20 E30				τ −		τs			τs − τ0
			τ			2		exp			i ∆ k L 1 ,
			exp							−	−
	c ns		s − 0		−	i ∆ k L			2
			2

(4.2.25)

where the total optical depth τ0 = τ1 + τ2 + τ3 . With the intensity

Φj = 0 nj c |Ej |2

2

the intensity conversion is given by

Φs

=

6 π ωs

(3)

(−ωs; ω1, ω2, ω3)

2

Φ10Φ20Φ30

F (∆ k L, τ0, τs) ,

N L χT

ns

c2ns

n1n2n3

containing the general phase-matching factor

s cos (∆ k L)

exp (−τ0) + exp (−τs) − 2 exp −

τ0

2

τ

+

F (∆ k L, τ0, τs) =

< 1 .

2

2

τs − τ0

+ (∆ k L)2

(4.2.26)

(4.2.27)

(4.2.28)

4.2.1.5 Phase-matching condition

Maximum conversion e ciency is achieved for conservation of momenta kj where ∆ k = 0
ω1 n1 + ω2 n2 + ω3 n3 = ωs ns .	(4.2.29)

In case of the third harmonic generation this gives n1 = ns . Frequency mixing in a two-component system results in

				3
Na		ωs χ¯b(1)	(ωs) −	=1ωj χ¯b(1)	(ωj )
	=		j		.	(4.2.30)
		3
Nb		j	(1)	(1)
		=1ωj χ¯a (ωj ) − ωs χ¯a (ωs)

For the third harmonic generation in a two-component system we have:

Na = χ¯(1)b (3 ω) − χ¯(1)b (ω) .

Nb χ¯(1)a (ω) − χ¯(1)a (3 ω)

The frequency mixing in a one-component system is given by:

3

ωs χ¯(1)(ωs) = ωj χ¯(1)(ωj ) . j=1

(4.2.31)

(4.2.32)

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