144	4.1.2 Fundamentals	[Ref. p. 187

4.1.2 Fundamentals

4.1.2.1 Three-wave interactions

Dielectric polarization P (dipole moment of unit volume of the substance) is related to the field E by the material equation of the medium [64Akh, 65Blo] (Chap. 1.1):

P (E) = ε0 (χ(1) E + χ(2) E2 + χ(3) E3 + . . . )

(4.1.1)

with

ε0 = 8.854 × 10−12 CV−1m−1 : dielectric permittivity of free space,

χ(1) = n2 − 1 : the linear, and χ(2) , χ(3) etc.: the nonlinear dielectric susceptibilities.

In the present chapter, Chap. 4.1, we consider only three-wave interactions in crystals with square nonlinearity (χ(2) = 0). The following nonlinear frequency conversion processes are considered:

Second Harmonic Generation (SHG):

ω + ω = 2 ω ,

(4.1.2)

Sum-Frequency Generation (SFG) or up-conversion:

ω1 + ω2 = ω3 ,

(4.1.3)

Di erence-Frequency Generation (DFG) or down-conversion:

ω3	− ω2 = ω1 ,	(4.1.4)
Optical Parametric Oscillation (OPO):
ω3	= ω2 + ω1 .	(4.1.5)
For e cient frequency conversion phase matching should be fulfilled:
k1	+ k2 = k3	(4.1.6)
with

ki : the wave vectors for ω1 , ω2 , ω3 , respectively.

Two types of phase matching are introduced:

type I: o + o → e or e + e → o , type II: o + e → e or o + e → o ,

or with shortened notations:

ooe: o + o → e or e → o + o , eeo: e + e → o or o → e + e , eoe: e + o → e or e → e + o , oeo: o + e → o or o → e + o .

In the shortened notation (ooe, eoe, . . . ) applies: ω1 < ω2 < ω3, i.e. the first symbol refers to the longest-wavelength radiation, and the latter to the shortest-wavelength radiation. Here, o-beam, or ordinary beam, is the beam with polarization normal to the principal plane of the crystal, i.e.

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Ref. p. 187]	4.1 Frequency conversion in crystals	145

the plane containing the wave vector k and crystallophysical axis Z (or optical axis, for uniaxial crystals). The e-beam, or extraordinary beam, is the beam with polarization in the principal plane.

The methods of angular and temperature phase-matching tuning are used in frequency converters. Angular tuning is rather simple and more rapid than temperature tuning. Temperature tuning is generally used in the case of 90 ◦ phase matching, i.e., when the birefringence angle is zero. This method is mainly used in crystals with a strong temperature dependence of phase matching: LiNbO3, LBO, KNbO3, and Ba2NaNb5O15.

4.1.2.2 Uniaxial crystals

For uniaxial crystals the di erence between the refractive indices of the ordinary and extraordinary beams, birefringence ∆n , is zero along the optical axis (crystallophysical axis Z) and maximum in the normal direction. The refractive index of the ordinary beam does not depend on the direction of propagation, however, the refractive index of the extraordinary beam ne(θ) is a function of the polar angle θ between the Z axis and the vector k (but not of the azimuthal angle ϕ) (Fig. 4.1.1):

								1
ne (θ) = no		1 +	1 + tan2 θ			θ		2	(4.1.7)
			no 2 tan2					,
			ne

where no and ne are the refractive indices of the ordinary and extraordinary beams in the plane normal to the Z axis and termed as corresponding principal values. Note that if no > ne , the crystal is negative, and if no < ne , it is positive. For an o-beam the indicatrix of the refractive indices is a sphere with radius no , and an ellipsoid of rotation with semiaxes no and ne for an e-beam (Fig. 4.1.2). In the crystal the beam, in general, is divided into two beams with orthogonal polarizations; the angle between these beams ρ is the birefringence (or walk-o ) angle.

Equations for calculating phase-matching angles in uniaxial crystals are given in Table 4.1.1 [86Nik, 99Dmi].

Z

k

Y

X

Fig. 4.1.1. Polar coordinate system for description of refraction properties of uniaxial crystals (k is the light propagation direction, Z is the optic axis, θ and ϕ are the coordinate angles).

4.1.2.3 Biaxial crystals

For biaxial crystals the optical indicatrix has a bilayer surface with four points of interlayer contact which correspond to the directions of two optical axis. In the simple case of light propagation in

Landolt-B¨ornstein

New Series VIII/1A1

146			4.1.2 Fundamentals		[Ref. p. 187
	Z		Z
no > ne	no	z	ne > no
			no	ne ( )
		ne ( )
					z
	0		0
	ne	no X (Y )		no	ne X (Y )

a

b

Fig. 4.1.2. Dependence of refractive index on light propagation direction and polarization (index surface) in uniaxial crystals: (a) negative: no > ne and (b) positive: ne > no .

Table 4.1.1. Equations for calculating phase-matching angles in uniaxial crystals [86Nik, 99Dmi].

Negative uniaxial crystals

Positive uniaxial crystals

tan2

θpmooe = (1

−

U )/(W

−

1)

tan2 θpmeeo

≈

(1

−

U )/(U

−

S)

2

eoe

(1

U )/(W

R)

tan

2

oeo

V )/(V

Y )

tan

θpm

≈

−

−

2

θpm

= (1

−

−

2

oee

tan

eoo

= (1

tan

θpm

≈ (1 − U )/(W − Q)

θpm

− T )/(T − Z)

Notations:

U = (A + B)2/C2 ; W = (A + B)2/F 2 ; R = (A + B)2/(D + B)2 ;

Q = (A + B)2

E)2 ; S = (A + B)2/(D + E)2 ; V = B2/(C

−

A)2 ;

2

/E

2

/(A +2

2

2

/D

2

;

Y = B

; T = A

/(C − B) ; Z

= A

A = no1/λ1 ; B = no2/λ2 ; C = no3/λ3 ;

D = ne1/λ1 ; E = ne2/λ2 ; F = ne3/λ3 .

These expressions can

be generalized

to

noncollinear phase matching. In this

case, for example,

the phase-matching angle θpmooe is determined from the above presented equation using the new coe cients U and W :

U = (A2 + B2 + 2AB cos γ)/C2 , W = (A2 + B2 + 2AB cos γ)/F 2 ,

where γ is the angle between the wave vectors k1 and k2 .

the principal planes XY , Y Z, and XZ the dependencies of refractive indices on the direction of light propagation represent a combination of an ellipse and a circle (Fig. 4.1.3). Thus in the principal planes a biaxial crystal can be considered as a uniaxial crystal, e.g. a biaxial crystal with nZ > nY > nX in the XY plane is similar to a negative uniaxial crystal with no = nZ and

ne (ϕ) = nY 1 + (nY /nX )2		tan2	1	(4.1.8)
			ϕ .
	1 + tan2	ϕ	2

The angle VZ between the optical axis and Z axis for the case nZ > nY > nX can be found from:

nY				nZ2	− nX2	1
sin VZ =	nZ			nY2	− nX2	2		(4.1.9)
and for the case nX > nY > nZ :
		nY		nX2	− nZ2	1
cos VZ =		nX		nY2	− nZ2		2 .	(4.1.10)

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New Series VIII/1A1

Ref. p. 187]				4.1 Frequency conversion in crystals					147
Z				Optic axis		Z		nX
		nY
								nY
Optic axis		nX
		nX	nZ Y					nZ	nX Y
nY						nZ
				n	Y
nZ
				X
X
a				b

Fig. 4.1.3. Dependence of refractive index on light propagation direction and polarization (index surface) in biaxial crystals: (a) nX < nY < nZ , (b) nX > nY > nZ .

For a positive biaxial crystal the bisectrix of the acute angle between optical axes coincides with nmax and for a negative one the bisectrix coincides with nmin .

Equations for calculating phase-matching angles upon propagation in principal planes of biaxial crystals are given in Table 4.1.2 [87Nik, 99Dmi].

4.1.2.4 E ective nonlinearity

Miller delta formulation [64Mil]:
ε0Ei(ω3) = δijk Pj (ω1)Pk (ω2) ,						(4.1.11)
where the Miller coe cient,
1			χijk(2) (ω3)
δijk =					,	(4.1.12)
	2ε0		χii(1)(ω1) χjj(1)(ω2) χkk(1)
				(ω3)

has small dispersion and is almost constant for a wide range of crystals.

For anisotropic media the coe cients χ(1) and χ(2) are, in general, the secondand third-rank tensors, respectively. In practice, the tensor

dijk =	1	χijk	(4.1.13)
	2

is used instead of χijk . Usually, the “plane” representation of dijk in the form dil is used, the relation between l and jk is:

jk		l
11	↔	1
22	↔	2
33	↔	3
23 or 32	↔	4
31 or 13	↔	5
12 or 21	↔	6

Landolt-B¨ornstein

New Series VIII/1A1

B¨ornstein-Landolt

VIII/1A1 Series New

Table 4.1.2. Equations for calculating phase-matching angles in biaxial crystals upon light propagation in the principal planes [87Nik, 99Dmi].

(a) nX < nY < nZ

Principal

Type of

Equations

Notations

plane

interaction

1 − U

U =

A + B

2

; W =

A + B

2

ooe

tan2 ϕ =

; A =

nZ1

; B =

nZ2

; C =

nY 3

; F =

nX3

W − 1

C

F

λ1

λ2

λ3

λ3

XY

≈

W − R

U =

C

2

; W =

F

2

D + B

2

; B =

λ2

; C = λ3

; D = λ1

; F =

λ3

; R =

; A = λ1

eoe

tan2

ϕ

1 − U

A + B

2

A + B

2

A

+ B

2

nY

1

nZ2

nY 3

nX1

nX3

≈

W − Q

U =

C

; W =

F

; A = λ1

; B = λ2

; C = λ3

; E = λ2

; F =

λ3

; Q =

A + E

oee

tan2

ϕ

1 − U

A + B

A + B

A

+ B

nZ1

nY 2

nY 3

nX2

nX3

≈

U − S

U =

C

2

2

λ1

; B =

λ2

; C =

λ3

; D =

λ1

; E =

λ2

; S = D + E

; A =

eeo

tan2

θ

1 − U

A + B

A

+ B

nY 1

nY 2

nX3

nZ1

nZ2

Y Z

V − Y

V =

C − A

2

; Y =

E

2

; B = λ2

; C = λ3

; E =

λ2

; A = λ1

oeo

tan2 θ =

1 − V

B

2

B

2

nX1

nY

2

nX3

nZ2

T − Z

T =

; Z = D

; B = λ2

; C = λ3

; D = λ1

C − B

; A = λ1

eoo

tan2 θ =

1 − T

A

A

nY 1

nX2

nX3

nZ1

XZ

W − 1

U =

C

2

; W =

F

2

λ1

; B =

λ2

; C =

λ3

; F =

λ3

; A =

ooe

tan2 θ =

1 − U

A + B

A + B

nY 1

nY 2

nX3

nZ3

θ < VZ

≈

W − R

U =

C

2

; W =

F

2

D + B

2

; B =

λ2

; C = λ3

; D = λ1

; F =

λ3

; R =

; A = λ1

eoe

tan2

θ

1 − U

A + B

2

A + B

2

A

+ B

2

nX1

nY 2

nX3

nZ1

nZ3

≈

W − Q

U =

C

; W =

F

; A = λ1

; B = λ2

; C = λ3

; E = λ2

; F =

λ3

; Q =

A + E

oee

tan2

θ

1 − U

A + B

A + B

A

+ B

nY 1

nX2

nX3

nZ2

nZ3

XZ

≈

U − S

U =

C

2

2

λ1

; B =

λ2

; C =

λ3

; D =

λ3

; E =

λ2

; S = D + E

; A =

eeo

tan2

θ

1 − U

A + B

A

+ B

nX1

nX2

nY 3

nZ1

nZ2

θ > VZ

V − Y

V =

C − A

2

; Y =

E

2

; B = λ2

; C = λ3

; E =

λ2

; A = λ1

oeo

tan2 θ =

1 − V

B

2

B

2

nY 1

nX2

nY 3

nZ2

T − Z

T =

; Z = D

; B = λ2

; C = λ3

; D =

λ1

C − B

; A = λ1

eoo

tan2 θ =

1 − T

A

A

nX1

nY 2

nY 3

nZ1

(continued)

148

Fundamentals 2.1.4

187 .p .[Ref

B¨ornstein-Landolt

VIII/1A1 Series New

Table 4.1.2 continued.

(b) nX > nY > nZ

Principal

Type of

Equations

Notations

plane

interaction

≈

U − S

U =

C

2

2

λ1

; B =

λ2

; C =

λ3

; D =

λ1

; E =

λ2

; S = D + E

; A =

eeo

tan2

ϕ

1

− U

A + B

A

+ B

nY 1

nY 2

nZ3

nX1

nX2

XY

V − Y

V =

C − A

2

; Y =

E

2

; B = λ2

; C = λ3

; E =

λ2

; A = λ1

oeo

tan2

ϕ =

1

− V

B

B

nZ1

nY 2

nZ3

nX2

2

2

T − Z

T =

; Z = D

; B = λ2

; C = λ3

; D =

λ1

C − B

; A = λ1

eoo

tan2

ϕ =

1

− T

A

A

nY 1

nZ2

nZ3

nX1

W − 1

U =

C

2

; W =

F

2

λ1

; B =

λ2

; C = λ3

; F = λ3

; A =

ooe

tan2

θ =

1

− U

A + B

A + B

nX1

nX2

nY 3

nZ3

2

2

2

Y Z

≈

W − R

U =

C

; W =

F

; A = λ1

; B =

λ2

; C =

λ3

; D = λ1

; F =

λ3

; R =

D + B

eoe

tan2

θ

1

− U

A + B

A + B

A

+ B

nY 1

nX2

nY 3

nZ1

nZ3

2

2

2

≈

W − Q

U =

C

; W =

F

; A = λ1

; B =

λ2

; C =

λ3

; E = λ2

; F =

λ3

; Q =

A + E

oee

tan2

θ

1

− U

A + B

A + B

A

+ B

nX1

nY 2

nY 3

nZ2

nZ3

XZ

≈

U − S

U =

C

2

2

λ1

; B =

λ2

; C =

λ3

; D =

λ1

; E =

λ2

; S = D + E

; A =

eeo

tan2

θ

1

− U

A + B

A

+ B

nX1

nX2

nY 3

nZ1

nZ2

θ < VZ

V − Y

V =

C − A

2

; Y =

E

2

; B = λ2

; C = λ3

; E =

λ2

; A = λ1

oeo

tan2

θ =

1

− V

B

B

nY 1

nX2

nY 3

nZ2

2

2

T − Z

T =

; Z = D

; B = λ2

; C = λ3

; D =

λ1

C − B

; A = λ1

eoo

tan2

θ =

1

− T

A

A

nX1

nY

2

nY 3

nZ1

XZ

W − 1

U =

C

2

; W =

F

2

λ1

; B =

λ2

; C =

λ3

; F = λ3

; A =

ooe

tan2

θ =

1

− U

A + B

A + B

nY 1

nY 2

nX3

nZ3

θ > VZ

≈

W − R

U =

C

2

; W =

F

2

2

; A = λ1

; B =

λ2

; C =

λ3

; D = λ1

; F =

λ3

; R =

D + B

eoe

tan2

θ

1

− U

A + B

A + B

A

+ B

nX1

nY 2

nX3

nZ1

nZ3

≈

W − Q

U =

C

2

; W =

F

2

2

; A = λ1

; B =

λ2

; C =

λ3

; E = λ2

; F =

λ3

; Q =

A + E

oee

tan2

θ

1

− U

A + B

A + B

A

+ B

nY 1

nX2

nX3

nZ2

nZ3

187] .p .Ref

crystals in conversion Frequency 1.4

149

150 4.1.2 Fundamentals [Ref. p. 187

Kleinman symmetry conditions [62Kle]: d21 = d16 , d24 = d32 , d31 = d15 , d13 = d35 , d14 = d36 , d25 = d12 = d26 , d32 = d24 are valid in the case of non-dispersion of electron nonlinear polarizability. The equations for calculating the conversion e ciency include the e ective nonlinear coe cients de , which comprise all summation operations along the polarization directions of the interacting waves and thus reduce the calculation to one dimension. E ective nonlinearities de for di erent crystal point groups under valid Kleinman symmetry conditions are presented in Table 4.1.3.

The conversion factors for SI and CGS-esu systems are given in Table 4.1.4.

Table 4.1.3. Expressions for de in nonlinear crystals when Kleinman symmetry relations are valid.

(a) Uniaxial crystals

Point group

Type of interaction

ooe, oeo, eoo

eeo, eoe, oee

4, 4mm

d15 sin θ

0

6, 6mm

d15 sin θ

0

¯

d 22

cos θ sin (3ϕ)

d22 cos

2

θ cos ϕ

6m2

3m

d 15

sin θ − d22 cos θ sin (3ϕ)

d22 cos2 θ cos (3ϕ)

2

¯

(d11 cos (3ϕ) − d22 sin (3ϕ)) cos θ

(d11 sin (3ϕ) + d22 cos (3ϕ)) cos2

θ

6

sin θ

3

(d

11

cos (3ϕ)

−

d

22

sin (3ϕ)) cos θ + d

(d11

sin (3ϕ) + d

22

cos (3ϕ)) cos

θ

32

15

2

d 11

cos θ cos (3ϕ)

d11 cos

θ sin (3ϕ)

¯

(d14 sin (2ϕ) + d15 cos (2ϕ)) sin θ

(d14 cos (2ϕ) − d15 sin (2ϕ)) sin (2θ)

4

¯

d 36

sin θ sin (2ϕ)

d36 sin (2θ) cos (2ϕ)

42m

(b) Biaxial crystals (assignments of crystallophysical and crystallographic axes: for mm2 and 222 point groups: X, Y, Z → a, b, c ; for 2 and m point groups: Y → b )

Point	Principal	Type of interaction
group	plane	ooe, oeo, eoo	eeo, eoe, oee
2	XY	d23 cos ϕ	d36 sin (2ϕ)
	Y Z	d21 cos θ	d36 sin (2θ)
	XZ	0	d21 cos2 θ + d23 sin2 θ − d36 sin (2θ)
m	XY	d13 sin ϕ	d31 sin2 ϕ + d32 cos2 ϕ
	Y Z	d31 sin θ	d13 sin2 θ + d12 cos2 θ
	XZ	d12 cos θ − d32 sin θ	0
mm2	XY	0	d31 sin2 ϕ + d32 cos2 ϕ
	Y Z	d31 sin θ	0
	XZ	d32 sin θ	0
222	XY	0	d36 sin (2ϕ)
	Y Z	0	d36 sin (2θ)
	XZ	0	d36 sin (2θ)

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New Series VIII/1A1