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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

resulting in the wave equation

1 ∂2E

∂2P

∆E −

(4.2.16)

∂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)

∂t2

0c2

∂t2

(z, ω) and the slow-amplitude approximation

With the plane-wave approximation E

(r, ω) = E

∂Ej

(4.2.18)

ω Ej ,

k Ej ,

∂t

∂z

we get the fundamental equations of nonlinear optics

ωj

κj ˆ

d Ej

= i

exp(−i kj z) −

(4.2.19)

d z

2 0 c nj

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

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)

Ps(3)(ωs) =

N χT

(−ωs; ω1, ω2, ω3)E1E2E3 .

(4.2.21)

Within the plane-wave approximation one obtains

= i c ns

N χT

E10E20E30 exp −

κ1

+ κ3

− i ∆ k z

(4.2.22)

d z

d Es

3 π ωs

(3)

+ κ2

Landolt-B¨ornstein

New Series VIII/1A1

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

the intensity conversion is given by

Φs

6 π ωs

(3)

(−ωs; ω1, ω2, ω3)

Φ10Φ20Φ30

F (∆ k L, τ0, τs) ,

N L χT

c2ns

n1n2n3

containing the general phase-matching factor

s cos (∆ k L)

exp (−τ0) + exp (−τs) − 2 exp −

τ0

F (∆ k L, τ0, τs) =

< 1 .

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 )
Na	=		j		.	(4.2.30)
	=	3	j		.	(4.2.30)
Nb		j	(1)	(1)
		j
		=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:

ωs χ¯(1)(ωs) = ωj χ¯(1)(ωj ) . j=1

(4.2.31)

(4.2.32)

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 212]	4.2 Frequency conversion in gases and liquids	209

4.2.2 Frequency conversion in gases

The following conditions have to be met for large conversion e ciencies:

1.a large nonlinear susceptibility χ(3)T which may be enhanced by a proper two-photon resonance,

2.large column densities with a proper phase matching,

3.small optical depths for the incident and generated waves to avoid reabsorption.

4.2.2.1 Metal-vapor inert gas mixtures

Metal-vapor inert gas mixtures are generally generated in concentric heat pipes because for e cient frequency mixing the phase matching can be accurately and independently adjusted through the partial pressures in the heat pipe [71Vid, 87Vid, 96Vid].

Tables of the multi-wave mixing experiments in di erent gaseous nonlinear media are arranged according to the elements, Table 4.2.1. For every element the wavelength is given together with the method of generation. The method of generation is indicated where (ω1 + ω1)Res + ω2 , for example, indicates a two-photon resonance of ω1 in the particular atomic or molecular medium and the additional wave ω2 can make the resulting radiation tunable.

4.2.2.2 Mixtures of di erent metal vapors

The modiﬁed concentric heat pipe [71Vid, 87Vid, 96Vid] is used for phase matching with small partial pressures avoiding strong homogeneous broadening.

In Table 4.2.2 mixtures of di erent metal vapors are listed.

4.2.2.3 Mixtures of gaseous media

For λ > 106 nm one prefers gas cells with lithium ﬂuoride windows [87Vid]. For λ < 106 nm one should use either pulsed nozzle beams [87Bet] without windows or gas cells with a fast shutter [85Bon].

In Table 4.2.3 mixtures of gaseous media are given.

Landolt-B¨ornstein

New Series VIII/1A1

210	4.2.2 Frequency conversion in gases	[Ref. p. 212

Table 4.2.1. Metal-vapor inert gas mixtures.

Vapor	Wavelength [nm]	Method	Ref.

Na	354.7	3 · ω1	[75Blo1, 75Blo2, 76Oha]
Na	330.5	(ω1 + ω1)Res + ω2	[74Blo]
Na	268 (cw)	(ω1 + ω2)Res + ω3	[84Bol]
Na	231	(ω1 + ω2)Res + ω3	[76Bjo]
Na	151.4	7 · ω1	[77Gro2, 79Mit]
Na	117.7	9 · ω1	[77Gro1, 79Mit]
Rb	354.7	3 · ω1	[71You, 75Blo1, 76Pue,
			76Oha]
Cs	213.4	(ω1 + ω1)Res + ω1	[74Leu, 75War]
Be	121–123	(ω1 + ω1)Res + ω2	[79Mah]
Mg	173.5	2 · ω1 + ω1 + ω1	[85Hut]
Mg	140–160	(ω1 + ω1)Res + ω2	[76Wal, 80Jun, 96Ste]
Mg	143.6 (cw)	(ω1 + ω1)Res + ω1	[83Tim]
Mg	115, 121.2, 127	(ω1 + ω1)Res + ω2	[81Car, 85Car]
Mg	121–129	(ω1 + ω1)Res + ω2	[78McK]
Mg+	123.6	(ω1 + ω1)Res + ω2	[85Leb]
Ca+	200	3 · ω1	[76Fer]
Ca	127.8	(ω1 + ω1)Res + ω2	[75Sor]
Zn	106–140	(ω1 + ω1)Res + ω2	[82Jam]
Cd	118.2, 152, 177.3	(ω1 + ω1)Res + ω2; 3 · ω1	[72Kun]
Cd	128.7–135.3	(ω1 + ω1)Res + ω2	[84Miy]
Cd	145.3–171.1	(ω1 + ω1)Res − ω2	[86Miy]
Sr	155.3, 166.7, 169.7, 173.5, 183.5 (cw)	(ω1 + ω2)Res + ω3	[90Nol1]
Sr	184.6, 185.7, 192.0, 195.8, 208.8, 217.9 (cw)	(ω1 + ω2)Res + ω3	[90Nol1]
Sr	171.2, 168.3, 169.7, 190 (cw)	(ω1 + ω2)Res + ω3	[77Bjo, 78Fre]
		(ω1 + ω1)Res + ω2	[90Nol2]
Ca	153.0, 159.5, 161.3, 163.3, 167.0 (cw)	(ω1 + ω2)Res + ω3	[90Nol1]
Ca	169.7, 170.9, 172.3, 173.1, 176.9 (cw)	(ω1 + ω2)Res + ω3	[90Nol1]
Zn	134.5–141.6 (cw)	(ω1 + ω2)Res + ω3	[90Nol1]
Cd	138.1–140.3	(ω1 + ω1)Res + ω2	[88Sch]
Sr	177.8–195.7	(ω1 + ω1)Res + ω2	[74Hod, 75Sor, 76Sor]
Sr	165–166	(ω1 + ω2)Res + ω3	[78Eco]
Sr	192.3	(ω1 + ω1)Res + ω1	[80Pue, 81Egg]
Sr	171.2	(ω1 + ω1)Res + ω2	[80Eco]
Ba	190–200	(ω1 + ω1)Res + ω2	[80Hei]
Hg	109–196	(ω1 + ω1)Res ± ω2	[83Hil2]
Hg	184.9, 143.5, 140.1, 130.7, 125.9, 125.0	(ω1 + ω1)Res ± ω2	[81Bok]
Hg	125.1, 183.3, 208.5	(ω1 + ω1)Res ± ω2	[81Tom]
Hg	124.7–125.5, 122.8–123.5, 117.4–122	(ω1 + ω1)Res + ω2	[82Mah, 82Tom]
Hg	120.3	(ω1 + ω1)Res + ω2	[76Hsu]
Hg	87.5–105, 99.1–126.8	(ω1 + ω1)Res + ω2	[85Her, 83Hil2]
Hg	89.6	(ω1 + ω1)Res + ω1	[78Sla]
Hg	132–185	(ω1 + ω1)Res + ω2	[86Hil2]
Tl	195.1	(ω1 + ω1)Res + ω1	[75Wan]
Eu, Yb	185.5, 194	(ω1 + ω1)Res + ω2	[75Sor, 76Sor]

Table 4.2.2. Mixtures of di erent metal vapors.

Mixture	Wavelength [nm]	Method	Ref.

Na + K	2–25 µm	(ω1 − ω2)Res − ω3	[74Wyn]
Na + Mg	354.7	3 · ω1	[75Blo2]

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 212]	4.2 Frequency conversion in gases and liquids					211

Table 4.2.3. Mixtures of gaseous media.

Gas	Wavelength [nm]	Method			Ref.

He, Ne, Ar, Kr, Xe	231.4	3	· ω1		[67New, 69War]
He	53.2	5	· ω1		[76Rei, 77Rei,	77She,
			· ω1		78Rei2, 78Rei1]
He	38	7	· ω1	; 5 · ω1; 7 · ω1	[77Rei, 78Rei2]
He	82.8, 50, 35.5	3	· ω1	; 5 · ω1; 7 · ω1	[83Bok]
He, Ne	106.4	(ω1 + ω1)Res + ω2			[78Rei2]
He, Ne	88.7	3	· ω1		[78Rei2, 78Rei1]
He, Ne	76, 70.9, 62.6, 59.1	4	· ω1 ± ω2		[77She, 78Rei2]
He, Xe	49.7, 35.5	5	· ω1; 7 · ω1		[83Bok]
Ne	53.2, 118.2	5	· ω1; (ω1 + ω1)Res + ω2		[76Rei, 77She, 78Rei2]
Ne	72.05–73.58, 74.3–74.36	3	· ω1		[84Hil]
Ar	120.4	(ω1 + ω1)Res + 3 · ω1			[80Din]
Ar	85.7–87.0, 97.4–104.8	3	· ω1		[83Hil3, 83Mar]
Ar	106.7	(ω1 + ω1)Res + ω1			[82Mil]
Ar	102.6–102.8	3	· ω1		[79Rei, 80Rei]
Ar, Kr	61.6, 53.2	5	· ω1		[78Rei2, 81Rei]
Ar	57	(ω1 + ω1)Res + ω1			[76Hut]
Kr	131.2, 92.3, 92.8, 94.2	(ω1 + ω1)Res ± ω2			[85Bon]
Kr	121.6	3	· ω1		[81Hil, 80Lan, 81Bat]
Kr	123.6	(ω1 + ω1)Res + ω1			[82Mil]
Kr	112.4, 120.3–123.6	(ω1 + ω1)Res + ω2			[79Cot1, 79Cot2]
Kr	131.2	(ω1 + ω1)Res − ω2			[85Bon]
Kr	72.5–83.5; 127–180	(ω1 + ω1)Res ± ω2			[87Hil]
Kr	121–200	(ω1 + ω1)Res − ω2			[90Mar]
Kr, Xe	71–92	(ω1 + ω1)Res + ω2			[89Miy]
Kr, Xe	110–210	(ω1 + ω1)Res ± ω2			[82Hil]
Xe	84.6–109.5, 155–220	(ω1 + ω1)Res ± ω2			[82Hil, 82Hag, 83Hil1]
Xe	155	(ω1 + ω1)Res − ω2			[83Hut]
Xe	147	(ω1 + ω1)Res + ω1			[82Mil]
Xe	140.3–146.9	3	· ω1; (ω1 + ω1)Res + ω2		[81Hil, 83Val]
Xe	74.8, 75, 75.2	(ω1 + ω1)Res + ω1			[82Mui]
Xe	125.4, 125.9, 126.1	(ω1 + ω1)Res − ω2			[82Mui]
Xe	101.5, 101.8, 13.0	(ω1 + ω1)Res + ω2			[82Mui]
Xe	163.1–194.6	(ω1 + ω1)Res ± ω2			[74Kun]
Xe	118.2	3	· ω1		[83Kun, 76Kun, 82Gan,
			· ω1		83Bok]
Ar, Kr, Xe, CO, N2	72; 90.4–102.5	3	· ω1		[87Pag]
Ne, Ar, Kr, Xe, Hg	60–200	(ω1 + ω1)Res ± ω2			[86Hil1]
Ar, Xe, CO	74.2, 80.4, 95.1, 98.2, 100.1,	ω1 + ω1 + ω2; 3 · ω1			[89Cro]
	116.5, 117.8, 123.6