Investigation of the nonsymmetry of effective nonlinear optical coefficient expressions for low-symmetry crystals

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Xin et al. Vol. 22, No. 10 /October 2005 /J. Opt. Soc. Am. B 2185

Investigation of the nonsymmetry of effectivenonlinear optical coefficient expressions

for low-symmetry crystals

Yin Xin, Zhang Shaojun, and Tian Zhaobing

State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, Shandong, China

Received February 28, 2005; revised manuscript received May 1, 2005; accepted May 3, 2005

For low-symmetry crystals that belong to 2 and m point groups, two kinds of nonlinear optical coefficientsmatrix were derived after the positive directions of the optical coordinate axes were defined with the aid of thepiezoelectric coordinate axes. These two kinds of nonlinear optical coefficient matrix are the cause of the non-symmetry of the effective nonlinear optical coefficients about the x, y, and z optical coordinate planes. For thecorresponding wave vector k�� ,�� in the Nth quadrant, the effective nonlinear optical coefficient expressionsare deff

I=aie2�dijk��3 ,�2 ,�1��N+2aj

e1ake1 (Type I) and deff

II=aie2�dijk��3 ,�2 ,�1��N+1aj

e1ake2 (Type II), where N is the

number of the quadrant. Only one kind of nonlinear optical coefficient matrix is the maximum effective non-linear optical coefficient. It is necessary to find out which quadrant, in which the wave vector k�� ,�� is located,corresponds to the maximum effective nonlinear optical coefficient. © 2005 Optical Society of America

OCIS codes: 190.4360, 190.4400.

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. INTRODUCTIONn experiments with harmonic-wave generation of nonlin-ar optics, the harmonic wave intensity is expressed asollows1:

I2� � � �2l2deff2

n1�n2

�n22�� sin2��k��,��l/2�

��k��,��l/2�2 ��p�2

A2 � , �1�

here � is the frequency of the fundamental wave; l is theength of the crystal through which the light passes; n1

�,2

�, and n22� are the refractive indices of fundamental

ave e1�, fundamental wave e2

�, and harmonic wave e22�,

espectively; deff is the effective nonlinear optical (NLO)oefficient; k�� ,�� is the wave vector; p� is the power ofhe fundamental wave; A is the area of the light spot; andp�

2 /A2� is the power density.When �k�� ,��=0,

sin2��k��,��l/2�

��k��,��l/2�2 = 1.

t this time, the Type I or Type II phase-matching condi-ion is satisfied.2 That is,

n22� = n1

� �Type I phase match�, �2�

n22� = 1/2�n1

� + n2�� Type II phase match�. �3�

Based on refractive-index surface equation2

sin2� cos �

n−2 − nx−2 +

sin2� cos �

n−2 − ny−2 +

cos2

n−2 − nz−2 = 0, �4�

he phase-matched surfaces of biaxial crystals can be de-ived; � is the angle from the z axis to wave vector k�� ,��,nd � is the angle from the x axis to the projection of waveector k�� ,�� in the x–y coordinate plane.

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According to an arrangement with the value of the re-ractive indices of fundamental wave nx

�, ny�, and nz

� andarmonic waves nx

2�, ny2�, and nz

2�, 14 types of phase-atched surface were calculated by Hobden.2

If � and � are all from 0° to 90°, the phase-matched sur-ace in eight quadrants of an optical coordinate systemre symmetric about the x, y, and z coordinate planes.The effective NLO coefficients in relation (1) can be ex-

ressed as follows3,4:

deffI = ai

e2dijkaje1ak

e1 �Type I�, �5�

deffII = ai

e2dijkaje1ak

e2 �Type II�, �6�

here

aie2 = �− cos � cos � sin � − sin � cos �

− cos � sin � sin � + cos � cos �

sin � sin �� , �7�

ake1 = �cos � cos � cos � − sin � sin �

cos � sin � cos � + cos � sin �

− sin � cos �� , �8�

cot 2� =cot2 � cos2 � − cos2 � cos2 � + sin2 �

cos � sin 2�, �9�

nd � is the angle between the z axis and the optic axis inhe z–x plane. Equations (7) and (8) are usually calledrojection factors.If the refractive indices of the fundamental wave and

he harmonic wave satisfy the phase-matching conditionsf Eqs. (2) and (3), the values of deff

I and deffII can be cal-

ulated by substitution of � and � of corresponding waveector k�� ,�� into Eqs. (5)–(9).

005 Optical Society of America

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2186 J. Opt. Soc. Am. B/Vol. 22, No. 10 /October 2005 Xin et al.

We know that the phase-matched surfaces of all crys-als are symmetric about the x, y, and z optical-coordinatelanes, as are the effective NLO coefficients of uniaxialrystals and some biaxial crystals that belong to pointroups 222 and mm2, but some low-symmetry crystalshat belong to point groups 2, m, and 1 are notymmetric.5–7 Why are the effective NLO coefficients ofhese low-symmetry crystals nonsymmetric about the x,, and z optical-coordinate planes? Now we discuss thisopic from the perspective of the optical coordinate axes.

. DEFINITION OF THEPTICAL-COORDINATE SYSTEM

he definition of the optical-coordinate system for biaxialrystals tells us that optical-coordinate axes x, y, and zre the refractive indices (nx, ny, and nz) of the principalxes and that the principal refractive indices (nz, ny, andx) obey the arrangement nz�ny�nx.

8 In this definitionhe x, y, and z coordinate axes have been given, but theirositive directions have not been given. In the definitionf the piezoelectric coordinate system, not only the piezo-lectric coordinate axes but also their positive directionsave been given.9 In the measurement of electro-optic co-fficients by some scientists, the positive directions of theptical coordinate axes are defined with the aid of the pi-zoelectric coordinate axes, such as KTiOPO4 crystal.10

Because they are the same as piezoelectric coefficientsnd electro-optic coefficients, the NLO coefficients arelso third-order tensors, and this fact enlightens us in de-ning the positive directions of the optical coordinatexes. In determining whether NLO coefficients are posi-ive or negative, one must first define the positive direc-ions of the optical coordinate axes. We can do so if wenow the positive directions of the piezoelectric coordi-ate axes.For uniaxial crystals and biaxial crystals in 222 andm2 point groups, the optical coordinate axes coincideith the a, b, and c axes in a crystallographic coordinate

ystem. For 222 and mm2 point groups, which axis is par-llel to the crystallographic c axis, x, y, or z? The answers supplied by the principal refractive indices nz, ny, andx of crystals such as KTiOPO4, which belongs to them2 point group. Its optical coordinate axes x, y, and z

re parallel to its crystallographic coordinate axes a, b,nd c and its piezoelectric coordinate axes X, Y, and Z, re-pectively. The positive direction of the optical coordinatexes of KTiOPO4 crystal can be determined from the IRE

Fig. 1. Right-hand rotation ru

tandards on piezoelectric coordinate axes.9 From the IREtandard for the mm2 point group, we know that its pi-zoelectric coordinate Z axis is parallel to its crystallo-raphic c axis (twofold symmetric axis) and that the posi-ive direction of the Z axis is determined by the positiveiezoelectric coefficients d33. The X axis and the Y axisbey the right-hand rotation rule for rectangular coordi-ate systems, as Fig. 1 shows. Positive piezoelectric coef-cients d33 can be determined by use of a quasi-static pi-zoelectric d33 measurement meter.

KTiOPO4 is a good crystal for us to use to describe theffective NLO coefficients of uniaxial crystals, and theseiaxial crystals belong to the 222 and mm2 point groupsith symmetry about the x, y, and z coordinate planes.TiOPO4 has five NLO coefficients, d311, d322, d333, d113,nd d223, where the subscripts of rectangular coordinateigns 1, 2, and 3 represent the coordinate axes x, y, and z,espectively.

For convenience of description, we have divided eightuadrants into x, y, and z coordinate planes according theraditional method. The signs of the coordinate axes inach quadrant are listed in Table 1. Table 1 lists the posi-ive signs or negative signs (called simply “signs” in whatollows) of the coordinate axes in quadrant I as

x → + , y → + , z → + .

o the subscripts of the NLO coefficients dijk are

1 → + , 2 → + , 3 → + .

or the mm2 point group, the NLO coefficients in quad-ant I are

d311, d322, d333, d113, d223.

In quadrant III the signs of the coordinate axes are

x → − , y → − , z → + .

o the NLO coefficients are

d311� → d3�−1��−1� → d311, d322� → d3�−2��−2� → d322,

d333� → d333, d113� → d�−1��−1�3 → d113,

d223� → d�−2��−2�3 → d223.

he signs of the NLO coefficients in quadrant III are theame as those in quadrant II. From the above transforma-ion it can be seen that the signs of the NLO coefficientsave symmetry about the twofold axis.

rectangular coordinate system.
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In quadrant II which is near quadrant I, the signs ofhe coordinate axes are

x → − , y → + , z → + ,

nd the NLO coefficients are

d311� → d3�−1��−1� → d311, d322� → d322,

d333� → d333, d113� → d�−1��−1�3 → d113, d223� → d223.

he signs of the NLO coefficients in these two near quad-ants are the same. It also can be derived that the signs ofhe NLO coefficients in quadrants IV and II are the sameor the symmetry of the twofold axis.

In quadrant, V near quadrant, the signs of the coordi-ate axes are

x → + , y → + , z → − ,


d311� → d11�−3� → − d311, d322� → d�−3�22 → − d322,

d333� → d�−3��−3��−3� → − d333, d113� → d11�−3� → − d113,

d223� → d22�−3� → − d223.

It can be seen that the absolute signs of the NLO coef-cients in quadrant V are all the inverse of those in quad-ant I, and the relative signs of the NLO coefficients inhese quadrants are the same. By using the same trans-ormation, we can derived that the absolute signs of theLO coefficients in quadrants II, III, and IV are positivend in quadrants VI, VII, and VIII are negative, but theirelative signs in each quadrant are all the same.

For point group mm2 the relative signs of the NLO co-fficients in all the quadrants of the optical-coordinateystem are the same, and the phase-matched surfaces inhe eight quadrants are symmetric about the x, y, and zoordinate planes. So the absolute value of the effectiveLO coefficients deff

I and deffII expressed in Eqs. (5) and

6) are also symmetric about the x, y, and z coordinatelanes. The positive or negative signs of the effective NLOoefficients do not influence the harmonic-wave intensityxpressed in relation (1).

For all point groups of uniaxial crystals and of those bi-xial crystals that belong to the 222 point group, it can beerived by the same method that the absolute values ofhe effective NLO coefficients deff

I and deffII are symmetric

bout the x, y, and z coordinate planes, except for theoint groups in which symmetry centers exist.Now we discuss the signs of the NLO coefficients in

ach quadrant for low-symmetry crystals that belong to

Table 1. Signs of Coordina

Axis I II III

x

y

z

oint groups 2 and m to determine the cause of nonsym-etry of the effective NLO coefficients deff

I and deffII of

hose crystals.

. SIGNS OF NONLINEAR OPTICSOEFFICIENTS IN LOW-SYMMETRYRYSTALShe so-called low-symmetry crystals contain those crys-

als that belong to three point groups: 2, m, and 1. We be-in by discussing the signs of NLO coefficients for pointroup 2.

. Point Group 2ccording to IRE standards for point group 2,iezoelectric-ordinate axis Y is parallel to the b axis (two-old symmetry axis) in the crystallographic coordinateystem, Z is parallel to c, and X is perpendicular to Y and.9 When four fingers rotate from the c axis to the a axist an obtuse angle, the direction of the thumb is the posi-ive direction of the Y axis, which can be determined byhe positive or negative sign of piezoelectric coefficient22.For the crystals that belong to point group 2, one of the

ptical coordinate axes is parallel to crystallographic axisor piezoelectric axis Y; the other two are located in

lane c–a of a crystallographic coordinate system (i.e.,he Z–X plane of the piezoelectric-coordinate system),nd they are perpendicular to each other.8 Which axis isarallel to the b axis? The answer is decided by the prin-ipal refractive indices nz, ny, and nz of different crystals.hen the positive direction of the optical coordinate axis

arallel to the b axis is determined by use of a quasi-staticiezoelectric d33 measurement meter, the positive direc-ions of the other two axes obey the right-hand rotationule of a rectangular coordinate system, as Fig. 1 shows.

If crystallographic axis b is parallel to optical-oordinate axes y, z, and x, the matrix forms of the NLOoefficients are, respectively,

0 0 0 d14 0 d16

d21 d22 d23 0 d25 0

0 0 0 d34 0 d36 �y � b�,

0 0 0 d14 d15 0

0 0 0 d24 d25 0

d31 d32 d33 0 0 d36 �z � b�,

es in the Eight Quadrants

Quadrant

V VI VII VIII

te Ax

IV

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d11 d12 d13 d14 0 0

0 0 0 0 d25 d26

0 0 0 0 d35 d36 �x � b�,

here the simple subscripts 1–6 are adapted; they are

11 → 1, 22 → 2, 33 → 3, 11 → 1,

23 → 4, 31 → 5, 12 → 6.

Like the results derived for point group mm2, the signsf the NLO coefficients for point group 2 are also symmet-ic about the twofold axis (i.e., the b axis). So the signs ofhe NLO coefficients in quadrant VI are the same as thosen quadrant I. If the y axis is parallel to the b axis, theoefficients are

d211, d222, d233, d231,

d123, d112, d323, d312.

he signs of the coordinate axes in quadrant V (nearuadrant I) are

x → + , y → + , z → − ,

nd the signs of the NLO coefficients are

d211� → d211, d222� → d222,

d233� → d2�−��−3� → d233, d231� → d2�−3�1 → − d231,

d123� → d12�−3� → − d123, d112 → d112,

d323� → d�−3�2�−3� → d323, d312� → d�−3�12 → − d312.

t can be seen that the signs of the NLO coefficients inuadrant V are not the same as those in quadrant I.It can be derived that the signs of the NLO coefficients

n quadrant II are the same as those in quadrant V for theymmetry of the twofold axis.

Now we discuss the signs of the NLO coefficients inuadrants IV, III, VIII, and VII. In quadrant IV the signsf the optical-coordinate axes are

x → + , y → − , z → + ,


d211� → d�−2�11 → − d211, d222� → d�−2��−2��−2� → − d222,

d233� → d�−2�33 → − d233,

d231� → d�−2�31 → − d231, d123� → d1�−2�3 → − d123,

d112� → d11�−2� → − d112, d323� → d3�−2�3 → − d323,

d312� → d31�−2� → − d312.

t can be seen that the absolute signs of the NLO coeffi-ients in quadrant IV are the inverse of those in quadrant

, although their relative signs in this quadrant are same.By the same method, it can be derived that the absolute

igns of the NLO coefficients in quadrants II, V, and VIre the inverse of those in quadrants III, VIII, and VII,espectively.

The NLO coefficient matrix �Y �b� in each quadrant foroint group 2 is listed in Table 2.The inverse signs of the NLO coefficients matrices in

wo quadrants do not influence the absolute values of theffective NLO coefficients deff

I and deffII expressed in Eqs.

5) and (6) and do not influence the harmonic-wave inten-ity expressed in relation (1). So they can be consideredhe same kind of sign.

From Table 2 it can be seen that there are two kinds ofelative sign of NLO coefficients matrices in eight quad-ants.

If the crystallographic b axis is parallel to the optical zxis or x axis, two kinds of relative sign of the NLO coef-cients matrices in eight quadrants can also be derived byhe same method.

Table 2. Matrices of NLO Coefficients in EightQuadrants for Point Group 2 „y ¸b…

Quadrant x, y, z NLO Coefficient Matrix

I � 0 0 0 d14 0 d16

d21 d22 d23 0 d25 00 0 0 d34 0 d36

�

II � 0 0 0 −d14 0 d16

d21 d22 d23 0 −d25 00 0 0 d34 0 −d36

�

III

−� 0 0 0 −d14 0 d16

d21 d22 d23 0 −d25 00 0 0 d34 0 −d36

�

IV � 0 0 0 d14 0 d16

d21 d22 d23 0 d25 00 0 0 d34 0 d36

�

V � 0 0 0 −d14 0 d16

d21 d22 d23 0 −d25 00 0 0 d34 0 −d36

�

VI � 0 0 0 d14 0 d16

d21 d22 d23 0 d25 00 0 0 d34 0 d36

�

VII

−� 0 0 0 d14 0 d16

d21 d22 d23 0 d25 00 0 0 d34 0 d36

�

VIII

−� 0 0 0 −d14 0 d16

d21 d22 d23 0 −d25 00 0 0 d34 0 −d36

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. Point Group mor point group m, one of the optical-coordinate axes isarallel to the crystallographic b axis (i.e., perpendicularo the m plane); the other two are located in the crystal-ographic c–a plane, and they are perpendicular to eachther. The positive directions of these two axes can be de-ermined with the aid of the positive signs of piezoelectricoefficients d in the directions of these two optical axes.he positive direction of another axis parallel to the bxis obeys the right-hand rotation rule of the rectangularoordinate system; When four fingers rotate from theositive direction of one axis to that of the other in the–a plane, the direction of the thumb is the positive di-ection of another axis (parallel to the b axis), as Fig. 1hows.

As for point group 2, which axis is parallel to the b axiss decided by the principal refractive indices nz, ny, and nxf different crystals. If the crystallographic b axis is par-llel to optical axes y, z, and x, the matrix forms of theonlinear NLO coefficients are

d11 d12 d13 0 d15 0

0 0 0 d24 0 d26

d31 d32 d33 0 d35 0 �y � b�,

d11 d12 d13 0 0 d16

d21 d22 d23 0 0 d26

0 0 0 d34 d35 0 �z � b�,

0 0 0 0 d15 d16

d21 d22 d23 d24 0 0

d31 d32 d33 d34 0 0 �x � b�,

espectively. If the optical y axis is parallel to the crystal-ographic b axis, the signs of the coordinate axes in quad-ant I are

x → + , y → + , z → + ,


d111, d122, d133, d131, d223,

d212, d311, d322, d333, d331.

The signs of the coordinate axes in quadrant VI are

x → − , y → + , z → − ,


d111� → d�−1��−1��−1� → − d111, d122� → d�−1�22 → − d122,

d133� → d�−1�33 → − d133, d131� → d�−1��−3��−1� → − d131,

d223� → d22�−3� → − d223, d212� → d2�−1�2 → − d212,

d311� → d�−3��−1��−1� → − d311, d322� → d�−3�22 → − d322,

d333 → d�−3��−3��−3� → − d333, d331� → d�−3��−3�1 → − d311.

It can be seen that the absolute signs of the NLO coef-cients in quadrant VI are the inverse of those in quad-ant I, although their relative signs in this quadrant areame.

In quadrant V (near quadrant I), the signs of the coor-inate axes are

x → + , y → + , z → − ,

nd the nonlinear optical coefficients are

d111� → d111, d122� → d122, d133� → d1�−3��−3� → d133,

d131� → d1�−3�1 → − d131, d223� → d22�−3� → − d223,

d212� → d212 → d212,

d311� → d�−3�11 → − d311, d322� → d�−3�22 → − d322,

d333� → d�−3��−3��−3� → − d333,

d331� → d�−3��−3�1 → − d331.

he signs of the NLO coefficients in quadrant V are notame as those in quadrant I.

By the same method as for quadrant VI, it can be de-ived that the absolute signs of the NLO coefficients inuadrant II are the inverse of those in quadrant V, al-hough the relative signs are same.

Now we discuss the signs of the NLO coefficients inuadrants IV, III, VIII, and VII. In quadrant IV the signsf the optical-coordinate axes are

x → + , y → − , z → + .

he signs of the NLO coefficients in quadrant IV are

d111� → d111, d122� → d1�−2��−2� → d122,

d133� → d133, d131� → d131,

d223� → d�−2��−2�3 → d223,

d212� → d�−2�1�−2� → d212, d311� → d311,

ig. 2. Phase-matched quadrants that correspond to wave vec-or k�� ,��.

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d322� → d3�−2��−2� → d322,

d333� → d333, d331� → d331.

he signs of the NLO coefficients in quadrant IV are asame as those in quadrant I.

By the same method, it can be derived that the absoluteigns of the NLO coefficients in quadrants III, VIII, andII are as same as those in quadrants II, V, and VI, re-pectively.

It can be seen that the signs of the NLO coefficients areymmetric about the m symmetry plane. The NLO coeffi-ients matrices �Y �b� in each quadrant for point group mre listed in Table 3.From Table 3 it can be seen that there are four kinds of

bsolute sign of NLO coefficient matrices for point group. As described above for point group 2, the inverse signs

f NLO coefficient matrices in two quadrants have no in-uence on the absolute value of the effective NLO coeffi-ients, so they can be considered one kind.

Table 3. Matrices of NLO coefficients in EightQuadrants for Point Group m „y ¸b…

Quadrant x, y, z NLO Coefficent Matrix

1 �d11 d12 d13 0 d15 00 0 0 d24 0 d26

d31 d32 d33 0 d35 0 �

2

−� d11 d12 d13 0 −d15 00 0 0 −d24 0 d26

−d31 −d32 −d33 0 d35 0 �

3

−� d11 d12 d13 0 −d15 00 0 0 −d24 0 d26

−d31 −d32 −d33 0 d35 0 �

4 �d11 d12 d13 0 d15 00 0 0 d24 0 d26

d31 −d32 d33 0 d35 0 �

5 � d11 d12 d13 0 −d15 00 0 0 −d24 0 d26

−d31 −d32 −d33 0 d35 0 �

6

−�d11 d12 d13 0 d15 00 0 0 d24 0 d26

d31 d32 d33 0 d35 0 �

7

−�d11 d12 d13 0 d15 00 0 0 d24 0 d26

d31 d32 d33 0 d35 0 �

8 � d11 d12 d13 0 −d15 00 0 0 −d24 0 d26

−d31 −d32 −d33 0 d35 0 �

If the z axis or the x axis is parallel to the b axis foroint groups 2 and m, the signs of the NLO coefficient ma-rices in each quadrant can be derived by the sameethod.For point group 1 there are 27 NLO coefficients; to de-

ive their signs in each quadrant is too complicated for us.o we do not discuss that point group in this paper.

. EFFECTIVE NONLINEAR OPTICALOEFFICIENT EXPRESSIONSN LOW-SYMMETRY CRYSTALSrom the above discussion it is clear that the nonsymme-

ry of the effective NLO coefficients about the x, y, and zoordinate plane are caused by the different NLO coeffi-ient matrices in eight quadrants of the optical-coordinateystem. Except for their optical-coordinate axes, theirositive directions must be stipulated before measure-ent of the NLO coefficients of the low-symmetry crys-

als. It would be best if the values and the signs of NLOoefficients were given in quadrant I.

For wave vector k�� ,�� in quadrant I (i.e., � from 0° to0° and � from 0° to 90°), the effective NLO coefficient ex-ressions are

deffI = ai

e2�dijk��3,�2,�1��IIIaje1ak

e1, �10�

deffII = ai

e2�dijk��3,�2,�1��IIaje1ak

e2, �11�

here the subscripts III and II are the quadrant num-ers, as Fig. 2 shows.For wave vector k�� ,�� in quadrant N, the effective

LO coefficients expressions are

deffI = ai

e2�dijk��3,�2,�1��N+2aje1ak

e1, �12�

deffII = ai

e2�dijk��3,�2,�1��N+1aje1ak

e2 �13�

or subscripts N=I,II,III,IV,I . . . or NV,VI,VII,VIII,V . . ..For point groups 2 and m there are two kinds of NLO

oefficient matrices in eight quadrants, but only one kindorresponds to the maximum effective NLO coefficient,hatever the value of deff

I or deffII. So it is necessary to

nd in which quadrant wave vector k�� ,�� is, to ensurehat the corresponding effective NLO coefficient is maxi-um.

Y. Xin’s e-mail address is [email protected].

EFERENCES1. H. Rabin and C. L. Tang, Nonlinear Optics, Vol. 1 of

Quantum Electronics (Academic, 1975), pp. 209–281.2. M. V. Hobden, “Phase matched second-harmonic generation

in biaxial crystals,” J. Appl. Phys. 38, 4365–4372 (1967).3. H. Ito, H. Naito, and H. Inaba, “New phase-matchable

nonlinear optical crystals,” IEEE J. Quantum Electron.QE-10, 247–252 (1974).

4. H. Ito, “Generalized study on angular dependence ofinduced second-order nonlinear optical polarizations and

1


phase matching in biaxial crystals,” J. Appl. Phys. 49,3992–3998 (1975).

5. G. Aka, A. Kahn-harari, F. Mougel, and D. Vivien, “Linearand nonlinear-optical properties of a new gadoliniumcalcium oxoborate crystal, Ca4GDO�BO3�3,” J. Opt. Soc.Am. B 14, 2238–2247 (1997).

6. M. Iwai, T. Kobayashi, H. Furuya, Y. Mori, and T. Sasaki,“Crystal growth and optical characterization of rare-earth(Re) calcium oxyborate ReCa4O�BO3�3 (Re=Y or Gd) as newnonlinear optical material,” Jpn. J. Appl. Phys. Part 1 36,L276–L279 (1997).

7. Z. Wang, G. Xu, J. Liu, D. Hu, X. Xu, J. Wang,and Z. Shaso, “Nonlinear second-harmonic generation ofBiB3O6,” J. Opt. Soc. Am. B 21, 1348–1353 (2004).

8. J. Minghua, Crystal Physics (Shandong Science andTechnology Press, 1982; in Chinese), pp. 28–36.

9. “Standards on piezoelectric crystals,” Proc. IRE 37,1378–1390 (1949).

0. J. D. Biertein and C. B. Arweller, “Electro-optic anddielectric properties of KTiOPO4,” Appl. Phys. Lett. 49,917–919 (1986).

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Investigation of the nonsymmetry of effective nonlinear optical coefficient expressions for low-symmetry crystals