Ref. p. 187]		4.1 Frequency conversion in crystals					151
Table 4.1.4. Units and conversion factors.
Nonlinear coe cient	MKS or SI units			CGS or electrostatic units
χ(1)	1	(SI, dimensionless)	=		1	(esu, dimensionless)
ij					4 π
dij or χ(2)	1	V−1m	=		3 × 104		(erg−1 cm3) 21
ijk					4 π
					4 π		1
δij	1	C−1m2	=				(erg−1 cm3) 2
					3 × 105
Note that in SI units P (n)	=	ε0 χ(n)E n (with P (n)	expressed in C m−2 ), whereas in CGS or esu
units P (n) = χ(n)E n (with P (n)		expressed in esu).

4.1.2.5 Frequency conversion e ciency

4.1.2.5.1 General approach

The conversion e ciency of a three-wave interaction process for the case of square nonlinearity

P nl = ε0 χ(2)E2

(4.1.14)

can be determined from the wave equation derived from Maxwell’s equations [64Akh, 65Blo, 73Zer, 99Dmi], see also (1.1.4)–(1.1.7),

× × E +	(1 + χ(1)) ∂2E			= −	1	∂2P nl		(4.1.15)
	c2		∂t2		ε0 c2		∂t2

with the initial and boundary conditions for the electric field E .

An exact calculation of the nonlinear conversion e ciency for SHG, SFG, and DFG generally requires a numerical calculation. In some simple cases analytical expressions are available. In order to choose the proper method, the contribution of di erent e ects in the nonlinear mixing process should be determined. For this purpose the following approach is introduced [99Dmi]:

–Consider the e ective lengths of the interaction process:

1.Aperture length La:

La = d0 ρ−1 ,	(4.1.16)
where d0 is the beam diameter.
2. Quasistatic interaction length Lqs:
Lqs = τ ν−1 ,	(4.1.17)

where τ is the radiation pulse width and ν is the mismatch of reverse group velocities. For SHG

	ν = uω−1 − u2−ω1 ,		(4.1.18)
	where uω and u2ω are the group velocities of the corresponding waves ω and 2ω .
3.	Di raction length Ldif :
	L	= k d2 .	(4.1.19)
	dif	0
4.	Dispersion-spreading length Lds:
	Lds = τ 2g−1 ,		(4.1.20)

Landolt-B¨ornstein

New Series VIII/1A1

152				4.1.2 Fundamentals	[Ref. p. 187
where g is the dispersion-spreading coe cient
1			∂ 2k
g =				.	(4.1.21)
	2		∂ ω2
5. Nonlinear interaction length Lnl:
Lnl = (σ a0)−1 .					(4.1.22)
Here σ is the nonlinear coupling coe cient:
σ1,2 = 4 π k1,2 n1−,22 de ,					(4.1.23)
σ3 = 2 π k3 n3−2 de ,					(4.1.24)
and
a0 = a12			1
			(0) + a22 (0) + a32 (0) 2 ,		(4.1.25)
where an(0) are the wave amplitudes of interacting waves λ1 , λ2 , and λ3					at the input
surface of the crystal.

–The length of the crystal L should be compared with Le from above equations. If L < Le the respective e ect can be neglected.

4.1.2.5.2 Plane-wave fixed-field approximation

When the conditions L < Lnl and L < Le are fulfilled, the so-called fixed-field approximation is realized. For SHG, ω + ω = 2ω and ∆k = 2kω − k2ω , the conversion e ciency η is determined by the equation:

	η = P2ω /Pω =	2 π2de2 L2Pω		sinc2	|∆k| L	.
		ε0 c nω2 n2 ω λ22 A			2

For SFG, ω1 + ω2 = ω3 and ∆k = k1 + k2 − k3 , the conversion e ciency η is:

	η = P3/P1 =	8 π2de2 L2P2	sinc2	|∆k| L	.
		ε0 c n1 n2 n3 λ32 A		2

For DFG, ω1 = ω3 − ω2 and ∆k = k1 + k2 − k3 , the conversion e ciency η is:

	η = P1/P3 =	8 π2de2 L2P2	sinc2	|∆k| L	.
		ε0 c n1 n2 n3 λ12 A		2

(4.1.26)

(4.1.27)

(4.1.28)

Note that all the above equations are for the SI system, i.e. [de ] = m/V ; [P ] = W ; [L] = m ; [λ] = m ; [A] = m2 ; ε0 = 8.854 × 10−12 A s/ (V m) ; c = 3 × 108 m/s .

When the powers of the mixing waves are almost equal, the conversion e ciency is for THG,

ω + 2 ω = 3ω :
P3ω	1 ;	(4.1.29)
η =

(P2 ω Pω ) 2

Landolt-B¨ornstein

New Series VIII/1A1

Ref. p. 187]	4.1 Frequency conversion in crystals	153

for FOHG in the case of ω + 3ω = 4ω :

P4ω

η = 1 ,

(P3ω Pω ) 2

or for 2 ω + 2 ω = 4ω :

η= P4ω ;

P2 ω

for SFG, ω1 + ω2 = ω3 :

P3

η = 1 ; (P1P2) 2

for DFG, ω1 = ω3 − ω2 :

P1

η = 1 .

(P2P3) 2

(4.1.30)

(4.1.31)

(4.1.32)

(4.1.33)

In some cases (mentioned additionally) the conversion e ciency is calculated from the power (energy) of fundamental radiation, e.g. for fifth harmonic generation, ω + 4ω = 5ω :

η =	P5ω	.	(4.1.34)
	Pω

Corresponding equations are valid for energy conversion e ciencies by substituting the pulse energy instead of power in the above equations.

The e ciency η in the case of OPO is calculated by the equation

η =	EOPO	,	(4.1.35)
	E0

where EOPO is the total OPO radiation energy (signal + idler) and E0 is the energy of the pump radiation. Conversion e ciency can also be determined in terms of pump depletion:

Eunc		(4.1.36)
η = 1 − Epump	,

where Eunc is the energy of unconverted pumping beam after the OPO crystals. Pump depletions are usually significantly greater than the ordinary η values.

The quantum conversion e ciency (for the ratio of converted and mixing quanta) in the case of exact phase-matching (∆ k = 0) for sum-frequency generation, ω1 + ω2 = ω3 , is determined by the following equation (SI system):

η = P1		λ1	= sin2	2 π de L					,	(4.1.37)
							ε0 c n1 n2 n3 λ1 λ3 A
	P3	λ3					2P2
and for di erence-frequency generation, ω1 = ω3 − ω2 :
η = P3		λ3	= sin2	2 π de L				.		(4.1.38)
						ε0 c n1 n2 n3 λ1 λ3 A
	P1	λ1					2P2

In the presence of linear absorption all the above equations for conversion e ciencies should be multiplied by the factor

exp (−αL) ≈ 1 − αL ,

(4.1.39)

where α is the linear absorption coe cient of the crystal.

Landolt-B¨ornstein

New Series VIII/1A1