Calculation of nonlinear-susceptibility tensor components in ferroelectrics: cubic, tetragonal, and rhombohedral symmetries

Murgan et al. Vol. 19, No. 9 /September 2002 /J. Opt. Soc. Am. B 2007

Calculation of nonlinear-susceptibility tensorcomponents in ferroelectrics: cubic,

tetragonal, and rhombohedral symmetries

Rajan Murgan and David R. Tilley

School of Physics, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia

Yoshihiro Ishibashi

Faculty of Communication, Aichi Shukutoku University, Nagakute-cho, Aichi Prefecture 480-1197, Japan

Jeff F. Webb

National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 21009, China

Junaidah Osman

School of Physics, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia

Received November 5, 2001; revised manuscript received May 3, 2002

We present the formalism for the calculation of all second- and third-order nonlinear susceptibility coefficientsbased on the Landau–Devonshire free-energy expansion for cubic symmetry in the high-temperature paraelec-tric phase and the Landau–Khalatnikov dynamical equations. Second-order phase transition and single-frequency input waves are considered. Detailed results are given for all nonvanishing tensor elements of thesecond- and third-order nonlinear optical effects in the paraelectric and the tetragonal and rhombohedralferroelectric phases. © 2002 Optical Society of America

OCIS codes: 190.0190, 160.2260, 190.4400, 190.4720, 190.4160, 190.3270.

1. INTRODUCTIONThe Landau–Devonshire expansion of the free energy Fin terms of the electric polarization vector P can be usedto describe phenomenologically the properties of ferroelec-tric (FE) materials. Based on this theory, the expres-sions for the static dielectric constant above and below theCurie temperature Tc have been derived.1,2 The formulafor the nonlinear (NL) static dielectric response has alsobeen obtained.2

With the use of the Landau–Khalatnikov (LK) dynami-cal equation, Ishibashi and Orihara3 have extended thetheory to give expressions for the NL dynamic dielectricresponse, that is, the nonlinear optic (NLO) coefficients,and for the third-order NL susceptibility coefficients inthe paraelectric (PE) phase above Tc . Subsequently,based on the simplest form of the theory in which F is afunction of a scalar P, the calculation of the NLO coeffi-cients has been extended into the FE phase. In this lat-ter work Osman et al.4 have demonstrated that allsecond-order (x (2)) processes vanish naturally in theinversion-symmetry PE phase and that they are nonzeroin the FE phase in which the presence of a spontaneouspolarization P0 breaks the inversion symmetry. Expres-sions for the third-order coefficients x (3) [namely third-harmonic generation (THG)] and for the intensity-dependent refractive index are also given in Ref. 4.

0740-3224/2002/092007-15$15.00 ©

However, the work of Ref. 4 is restricted in two ways.First, the derivation is applicable only to second-orderphase transitions in FE materials. Second, P has beentreated as a scalar, and therefore the complete range oftensor elements x ijk

(2) and x ijkl(3) cannot be derived. Ref. 4,

in fact, gives expressions only for xzzz(2) and xzzzz

(3) , wherethe z axis has been chosen as the direction of P0 .

Since it is possible to find the most general expansion ofthe free energy in terms of the polarization by making useonly of the space group of the PE state,5 that is, by writingF formally as an expansion in terms of invariants of thehigh-temperature PE phase, Osman et al.6 have pre-sented a full account of the derivation of all the tensor el-ements for both first- and second- order transitions in thePE and FE phases. They assume that the material is iso-tropic in the PE phase with the symmetry group denotedby ` for T > Tc . For T < Tc the symmetry group is`m, where the material is assumed to have continuousrotational symmetry about z with the z axis taken as thedirection of P0 . A complete range of expressions for thesecond-order NL coefficients in the FE phase and thethird-order coefficients in both phases is tabulated in Ref.6.

Successive phase transitions occur in many ferroelec-trics crystals, one example of which is barium titanate(BaTiO3). BaTiO3 has a perovskite-type structure and

2002 Optical Society of America

2008 J. Opt. Soc. Am. B/Vol. 19, No. 9 /September 2002 Murgan et al.

belongs to the space group Oh1 at high temperatures. It

undergoes a phase transition from the PE to the FE phaseat 120 °C; subsequently at 0 °C and 270 °C, the directionof the spontaneous polarization changes from [100] to[110] and then to [111], respectively.7 Thus the crystalsymmetry changes from cubic Oh

1 to tetragonal, C4V1 ,

orthorhombic, C2V14 , and rhombohedral, C3V

5 , with de-creasing temperatures.7 These successive phase transi-tions can be described well using the Devonshire theory.8

Quite often, the isotropic free-energy expansion used inRef. 6 is applicable to a description of BaTiO3 crystal.However, it is only an approximation since it omits termsthat are manifested in real crystals. For a cubic pointgroup, for example, Px

4 1 Py4 1 Pz

4 and Py2Pz

2 1 Pz2Px

2

1 Px2Py

2 are separately invariant. Fujita and Ishibashi9

have given an account of the successive phase transitionsin BaTiO3 by means of a free-energy expression in whichthe above terms are included.7 In the spirit of the tech-nique employed in Ref. 6, it would be of interest to extendtheir calculations to a system like BaTiO3 . This is ourintention here.

In this paper, we begin with the free-energy expressionfor cubic symmetry in the high-temperature PE phase.We follow the formalism presented in Ref. 6 and thereforerepeat here for clarity some of the important equationsappearing there. First, we show that in the PE phase allsecond-order processes (thus all the second-order NL sus-ceptibility tensor elements) vanish since the cubic pointgroup has inversion symmetry, and then derive all nonva-nishing tensor elements for the third-order NL processes.Second, based on the cubic energy expression, as is donein Ref. 6, we derive all nonvanishing tensor elements forthe tetragonal and rhombohedral symmetries in the FEphase. This is possible since Haas has shown5 that thephase transition from cubic symmetry to the various sym-metries of the low-temperature phase can be treated assecond order, provided certain conditions are fulfilled forthe lower symmetry groups. All these conditions (de-tailed in Ref. 5,) are satisfied by the groups C4V

1 and C3V5 .

Most ferroelectrics, especially oxide ferroelectrics, exhibita first-order transition from the PE cubic phase to thelower-symmetry FE phases. However, in the FE phase,especially at temperatures much lower than the transi-tion temperature, the type of transition is immaterial fora discussion of their physical properties. Moreover, sincewe are only focusing on principles, it is reasonable to con-centrate only on second-order phase transition. In addi-tion to this, we consider only single-frequency inputwaves.

In subsequent sections, we discuss in general terms allthe resonant features of the nonvanishing tensor-elementcomponents, give a nonresonant numerical estimate ofone of the coefficients, and illustrate numerically one spe-cific coefficient for second-harmonic generation (SHG) andone for third-harmonic generation (THG) in the tetrago-nal FE phase. As an application, we look at SHG in bulkferroelectrics having tetragonal symmetry. We concludethe paper with a discussion touching on the subject ofmorphotropic phase boundary in ferroelectric materials inwhich we predict that in the vicinity of this phase bound-ary some linear susceptibility coefficients in the FE phasebecome very large.10

2. PARAELECTRIC PHASE: CUBICSYMMETRYWe begin by considering a bulk ferroelectric material suchas BaTiO3 in the PE phase at a temperature above theCurie temperature Tc . Under equilibrium conditions,the free energy per unit volume for cubic symmetry hasthe following expression in terms of the polarization com-ponents truncated at the fourth order10:

F 5a

2e0Pi Pi 1

b1

4e02 ~Pi Pi!

2

1b2

2e02 ~Px

2Py2 1 Py

2Pz2 1 Pz

2Px2!, (1)

where a 5 a(T 2 T0) and, for second-order phase transi-tion, Tc 5 T0 . The usual summation convention is em-ployed in Eq. (1). As in Ref. 6, we include the factor e0 inEq. (1) so that b1 and b2 have mechanical dimensions,and a is dimensionless with a as the inverse of the Curieconstant. Based on the published data for BaTiO3 ,a ; 6 3 1026 deg21.6 b1 and b2 are material-dependentparameters whose numerical estimates are given in Ap-pendix A.

Consider now exposing the bulk FE material to high-intensity incident IR radiation having a single frequencyv. Under this condition, we add a field term to the freeenergy

FE 5 F 2 E • P, (2)

where E is an applied field due to the incident IR radia-tion. The LK equation of motion for P is

OPi 5 2]F/]Pi 1 Ei 5 fi~P! 1 Ei , (3)

where the operator O is

O 5 g]

]t~relaxational dynamics!, (4a)

5 md2

dt2 1 gd

dt~oscillatory dynamics!. (4b)

Equation (3) defines the function fi(P); it is the dynami-cal equation from which the NLO coefficients are derived.We follow the conventional route and invert Eq. (3) to findthe response equation that is linear in P and nonlinear inE, thus deriving expressions for the usual NLO tensor el-ements of x (2) and x (3).

In the PE phase the equilibrium value is P 5 0. TheNLO coefficients are the coefficients of the Taylor seriesfor P as a function of E, which may be compactly writtenas

Pi 5 Pi~1 ! 1 Pi

~2 ! 1 Pi~3 !

5 e0x il~1 !El 1 e0x ilm

~2 ! ElEm 1 e0x ilmn~3 ! EiEmEn , (5)

where Pi(n) is the nth-order polarization induced by Ei(t),

which is

Ei~t ! 51

2@Eoi exp~2ivt ! 1 Eoi* exp~ivt !#, (6)


written in terms of complex amplitudes Eoi . We definethe corresponding response function by

~Pi~n !!vs

5 e0K~2vs ;v1 , ¯vn!x ia1a2 ..an

~n ! ~2vs ;v1 , ¯vn!

3 ~Ea1!v1~Ea2

!v2¯~Ean!vn

, (7)

where vs 5 v1 1 v2 1 ¯ 1 vn .6,11 The definitiongiven by Eq. (7) leads to x (n) being invariant under allpermutations of (a1v1),(a2v2) ,..., (anvn). This is re-ferred to as the intrinsic permutation symmetry.11 Thefactor K in Eq. (7), defined explicitly in Refs. 6 and 11, is aproduct of a combinatorial factor arising from the symme-trization of x (n).12

In order to find the NLO coefficients, we start from theTaylor expansion of fi(P), which begins with the linearterm since the equilibrium condition yields, from Eq. (3),fi(0) 5 ]F/]Pi 5 0:

fi~P ! 5 fil~0 !Pl 11

2film~0 !PlPm 1

1

6filmn~0 !PlPmPn ,

(8)

where fil(0) 5 ]fi /]Pl evaluated at P 5 0, and so on; thesummation convention is used here. Using Eqs. (5) and(8), the LK equation of motion now becomes

O~Pi~1 ! 1 Pi

~2 ! 1 Pi~3 !!

5 Ei 1 fil~0 !~Pl~1 ! 1 Pl

~2 ! 1 Pl~3 !!

11

6filmn~0 !~Pi

~1 ! 1 Pl~2 ! 1 Pl

~3 !!

3 ~Pm~1 ! 1 Pm

~2 ! 1 Pm~3 !!~Pn

~1 ! 1 Pn~2 ! 1 Pn

~3 !!, (9)

where explicitly,

fil~0 ! 5 2a

e0d il , (10)

C ilmn 5 2filmn~0 !

5 F6b1

e02 d ind ild im 1

2b2

e02 ~d jnd jmd il 1 dkndkmd il

1 d imd jnd jl 1 d ind jmd jl 1 dkmd indkl

1 dknd imdkl!G , (11)

and film(0) 5 0. To obtain the linear response, we takeonly terms in Eq. (9) that are linear in E. The solution is

Pi~1 ! 5

1

2@r1~v!Eoi exp~2ivt ! 1 r1* ~v!Eoi* exp~ivt !#,

(12)

where the linear response function is

r1~v! 5 S O 1a

e0D 21

5 S 2ivg 1a

e0D 21

~relaxational dynamics!,

5 S 2mv2 2 ivg 1a

e0D 21

~oscillatory dynamics!.(13)

The solution given by Eq. (12) yields the dielectric tensorthat is diagonal, which is consistent with a cubic-symmetry isotropic medium. As in Ref. 6, the linearresponse function for oscillatory dynamics can bewritten in terms of the soft-mode frequency, mvT

2

5 a/e0 5 (a/e0)@T 2 T0#, as

r1~v! 5 @m~vT2 2 v2 2 ivg/m !#21. (14)

Since film(0) 5 0, this ensures that all second-order NLeffects vanish, which is consistent with the symmetry re-sult. To determine the third-order coefficients, we selectonly the third-order terms in E from Eq. (9). This yields

S O 1a

e0DPi

~3 ! 5 21

6C ilmnPl

~1 !Pm~1 !Pn

~1 ! . (15)

By using Eq. (12), Eq. (15) can be written in terms of com-plex amplitudes as

S O 1a

e0DPi

~3 !

5 21

6C ilmn

1

2@r1Eol exp~2ivt ! 1 r1* Eol* exp~ivt !#

31

2@r1Eom exp~2ivt ! 1 r1* Eom* exp~ivt !#

31

2@r1Eon exp~2ivt ! 1 r1* Eon* exp~ivt !#, (16)

where each response function r1 in Eq. (16) is evaluatedat the frequency of the corresponding E. The solution ofEq. (16) is

Pi~3 ! 5 2

1

48C ilmn@~r3r1

3EolEomEon!exp~23ivt !

1 ~r1r12r1* EolEomEon* !exp~2ivt ! 1 c.c.#. (17)

If we ignore the presence of a dc field in the incident ra-diation, then Pi

(3) is the third-order NL polarization due tothe contributions of third-order NL processes, namely theTHG [the first term on the right-hand side of Eq. (17)] andthe intensity-dependent refractive index (the secondterm). Thus we find

x ilmn~3 ! ~23v; v, v, v! 5 2

1

6e0C ilmnr3r1

3, (18)

for the NL susceptibility for THG, and


x ilmn~3 ! ~2v; 2v, v, v! 5 2

1

6eoC ilmnur1u2r1

2, (19)

for the intensity-dependent refractive index. We simplifythe notation in Eqs. (18) and (19) by defining rn5 r(nv). As reflected by Eqs. (18) and (19) and by theform of C ilmn , only two independent nonvanishing com-ponents of x ilmn exist, which is in agreement with Ref. 13,where the nonvanishing elements of x (n) and thesymmetry-required relations between those nonvanishingelements are known and tabulated. All the nonvanish-ing NLO tensor elements for the third-order NL processesare tabulated in Table 1. The values of K as defined inEq. (7) are 1/4 and 3/4, respectively, for THG andintensity-dependent refractive index, and there is a totalof 21 nonvanishing elements for each. The susceptibili-ties contain frequency dependences that are standard forresonant NLO and temperature dependences that arespecific to FE and which result from the softening of thedynamics near T0 . These have been discussed in ourearlier work6 and will not be repeated here.

3. FERROELECTRIC PHASEA. Tetragonal SymmetryWe now apply the formalism to the FE phase by lookingat the tetragonal symmetry first, where, for BaTiO3 , thestructural phase transition into this symmetry grouphappens at 0 °C. The presence of spontaneous polariza-tion in the tetragonal-symmetry FE phase along the te-

Table 1. Cubic Symmetry: Nonvanishing NLOTensor Elements in the Paraelectric Phase, TÌTC

Process, K Susceptibility x ilmn(3)

Third-harmonic generation xiijj~3! 5 xijji

~3! 5 2b2

3e03 r3r1

3

(i Þ j)K 5 1/4x ilmn

(3) (23v; v, v, v)

Symmetric on permutation of (lmn) xiiii~3! 5 2

b1

e03 r3r1

3

Intensity-dependent refractive index xiijj~3! 5 xijji

~3! 5 2b2

3e03 ur1u2r1

2

(i Þ j)K 5 3/4x ilmn

(3) (2v; v, v, 2v)

Symmetric on interchange of (lm) xiiii~3! 5 2

b1

e03 ur1u2r1

2

dc Kerr effect xiijj~3! 5 xijji

~3! 5 28b2

3e03 r0

2r12

(i Þ j)K 5 3x ilmn

(3) (2v; 0, 0, v)

Symmetric on interchange of (lm) xiiii~3! 5 2

8b1

3e03 r0

2r12

tragonal axis (taken to be the z axis here) leads to Px5 Qx , Py 5 Qy , Pz 5 P0 1 Qz . Following Ref. 6, wewrite the high-temperature free-energy expression of thecubic symmetry for the tetragonal symmetry as

f 5 f0 1a

2e0~Qx

2 1 Qy2 1 Qz

2! 1b1

4e02 ~Qx

4 1 Qy4 1 Qz

4!

1b2

2e02 ~Qx

2Qy2 1 Qy

2Qz2 1 Qz

2Qx2!

1a

2e0~P0

2 1 2QzPo!

1b1

4e02 ~6Qz

2P02 1 4Qz

3P0 1 4QzP03 1 P0

4!

1b2

2e02 @~Qz

2 1 2QzP0 1 P02!~Qx

2 1 Qy2!# 2 EP.

(20)

Equation (20) may be written simply in the form f 5 f01 Df, where f0 is the cubic expression [Eq. (1)] and Dfconsists of terms related to the tetragonal symmetry.Note that as we approach the transition point from thelow- to the high-temperature phase, f → f0 since P0→ 0. Notice also that the highest power of Qz in Df is

the third order; this suggests that none of the fourth-order derivatives would come from the tetragonal symme-try expression.

We proceed, as in the cubic-symmetry formalism, by ob-taining the derivatives of free energy f in Eq. (20) with re-spect to polarization evaluated at P0 . The relevant de-rivatives are

fil~P0! 5 2F a

e0d il 1 3

b1

e02 d izdzid ilP0

2 1b2

e02 P0

2

3 ~d ildzkdkz 1 2dzidzkdkl!G , (21a)

film~P0! 5 26b1

e02 P0dzid imd il 1 2

b2

e02 P0~d ildzkdkm

1 d imdzkdkl 1 dzidkmdkl!, (21b)

filmn~P0! 5 2F6b1

e02 d ind ild im 1

2b2

e02 ~d jnd jmd il

1 dkndkmd il

1 d imd jnd jl 1 d ind jmd jl 1 dkmd indkl

1 dknd imdkl!G . (21c)

The magnitude of P0 is given by the condition for mini-mum f, ]f/]Pz 5 0 evaluated at Pz 5 P0 , which is

a 1b1

e0P0

2 5 0. (22)

Expansion of fi(P) about P0 gives


fi~P ! 5 fi~P0! 1 fil~P0!Ql 11

2film~P0!QlQm

11

6filmn~P0!QlQmQn . (23)

Writing Qi as Qi 5 Qi(1) 1 Qi

(2) 1 Qi(3) , the LK equation

of motion for the tetragonal case takes the form

O~Qi~1 ! 1 Qi

~2 ! 1 Qi~3 !!

5 fil~P0!~Ql~1 ! 1 Ql

~2 ! 1 Ql~3 !! 1

1

2film~P0!

3 ~Ql~1 ! 1 Ql

~2 ! 1 Ql~3 !!~Qm

~1 ! 1 Qm~2 ! 1 Qm

~3 !!

11

6filmn~P0!~Ql

~1 ! 1 Ql~2 ! 1 Ql

~3 !!

3 ~Qm~1 ! 1 Qm

~2 ! 1 Qm~3 !!~Qn

~1 ! 1 Qn~2 ! 1 Qn

~3 !! 1 Ei .

(24)

Similar definition as given by Eq. (5) follows for Qi(n) .

Proceeding along the lines of the previous section, the lin-ear response is

@O 2 fil~P0!d il#Ql~1 ! 5 El , (25)

where the derivatives explicitly are

fxx 5 fyy 5 2S a

e01

b2

e02 P0

2D , (26)

fzz 5 22b1

e02 P0

2. (27)

Using Eq. (6) and Eqs. (25)–(27), and by writing Qi(1) as

Qi~1 ! 5

1

2@Bii~v!Eoi exp~2ivt ! 1 Bii* ~v!Eoi* exp~ivt !#,

(28)

we obtain the expressions for Bxx , Byy , and Bzz as

Bxx~v! 5 Byy~v! [ s~v!, (29)

Bzz~v! [ s~v!, (30)

where

s~v! 51

Q~v! 1a

e01

b2

e02 P0

2

, (31)

s~v! 51

Q~v! 1a

e01

3b1

e02 P0

2

51

Q~v! 22a

e0. (32)

Q(v) in Eqs. (31) and (32) is

Q~v! 5 2igv ~relaxational dynamics!, (33a)

5 2igv 2 mv2 ~oscillatory dynamics!. (33b)

Notice that Eqs. (29) and (30) have the uniaxial propertyof tetragonal symmetry.

We now focus on the second-order terms. From Eq.(24), we have

OQi~2 ! 5 fil~P0!Ql

~2 ! 11

2film~P0!Ql

~1 !Qm~1 ! . (34)

Substitution of the expression for Qi(1) brings Eq. (34) into

the form

@O 2 fil~P0!d il#Qi~2 !

51

2film~P0!

1

2@Bll~v!Eol exp~2ivt !

1 Bll* ~v!Eol* exp~ivt !#1

2@Bmm~v!Eom exp~2ivt !

1 Bmm* ~v!Eom* exp~ivt !. (35)

Calculation of the nonvanishing NLO tensor elementsfrom the above follows along the lines presented in Sec-tion 2. Each tensor element contains a product of threeresponse functions, one coming from the inversion of@O 2 fil(P0)d il# and two arising from the factors such asBllBmm . The first is s(vNL) for i 5 x or y and s(vNL)for i 5 z, where vNL is the frequency of the NL processin question.

Table 2 gives the results of the explicit evaluation forthe tensor elements of two second-order NL processes,namely SHG and optical rectification. For SHG, K 5

12

Table 2. Tetragonal Symmetry: NonvanishingSecond-Order, x(2), NLO Tensor Elements

in the Ferroelectric Phase, TËTC

Process, K Susceptibility x ilm(2)

Second-harmonicgenerationK 5 1/2

xzzz~2! 5 2

3

e03 b1P0s~2v!s2~v!

x ilm(2) (22v; v, v) xzyy

~2! 5 xzxx~2! 5 2

b2

e03 P0s~2v!s 2~v!

Symmetric onexchange of (l, m)

xxzx~2! 5 xyzy

~2! 5 xxxz~2! 5 xyyz

~2!

5 2b2

e03 P0s~2v!s~v!s~v!

OpticalrectificationK 5 1/2

xzzz~2! 5 2

3

e03 b1P0s~0!s* ~v!s~v!

x ilm(2) (0; 2v, v) xzyy

~2! 5 xzxx~2! 5 2

b2

e03 P0s~0!s* ~v!s~v!

xxzx~2! 5 xyzy

~2! 5 2b2

e03 P0s~0!s~v!s* ~v!

xxxz~2! 5 xyyz

~2! 5 2b2

e03 P0s~0!s* ~v!s~v!


as defined in Eq. (7) and there are three independent el-ements and a total of seven nonvanishing tensor ele-ments, which is in agreement with Ref. 13. The elementsare symmetric on exchange of the indices lm. For opticalrectification, K 5

12 , the number of independent elements

is four and the number of nonvanishing elements isseven, which again agrees with Ref. 13.

We consider next the third-order terms. Collectingterms with third order in Q from Eq. (24) yields

@O 2 fil~P0!d il#Qi~3 ! 5

1

2film~P0!~Ql

~1 !Qm~2 ! 1 Qm

~2 !Ql~1 !!

11

6filmn~P0!~Ql

~1 !Qm~1 !Qn

~1 !!.

(36)

We may regard the form of Ql(3) above as having two con-

tributions C1 and C2. Focusing first on C1, we have

C1 51

2film~P0!@O 2 fil~P0!d il#

21~Ql~1 !Qm

~2 ! 1 Qm~2 !Ql

~1 !!.

(37)

Since the values of film(P0) derived previously show sym-metric properties on permutation of indices lm, C1 maybe rewritten as

C1 5 filr~P0!@O 2 fil~P0!d il#21Ql

~1 !Qr~2 ! . (38)

Here we use the index r instead of m to account for crossterms that arise later in the expressions. SubstitutingEqs. (28) and (34), we have

Qi1~3 ! 5

1

16filr~P0!frmn~P0!@Bll~v!Eol exp~2ivt !

1 Bll* ~v!Eol* exp~ivt !#

3 @Brr~2v!Bmm~v!Bnn~v!EomEon exp~2ivt !

1 Brr~0 !Bmm* ~v!Bnn~v!Eom* Eon 1 c.c.#. (39)

The second contribution C2 is

C2 51

6filmn~P0!@O 2 fil~P0!d il#

21Ql~1 !Qm

~1 !Qn~1 ! .

(40)

Using Eq. (28) in the above, we have

Qi2~3 ! 5

1

48filmn~P0!$@Bll~v!Bmm~v!Bnn~v!EolEomEon

3 exp~23ivt !#

1 @Bll~v!Bmm* ~v!Bnn~v!EolEom* Eon

3 exp~2ivt ! 1 c.c.#%. (41)

Adding the two contributions together yields Qi(3) 5 Qi1

(3)

1 Qi2(3) , and with the definition of Eq. (7) and the nonva-

nishing derivatives of Eq. (21), the general expressions forthe third-order NLO-susceptibility tensors are calculatedas before. In calculating the explicit expressions forthese tensors, care has to be taken that all the intrinsic-permutation-symmetry properties are satisfied.6 For ex-

ample, the coefficient for THG, x ilmn(3) (23v; v, v, v), must

be symmetric on permutation of the indices lmn, and thecoefficient for the dc Kerr effect, x ilmn

(3) (2v; 0, 0, v), mustbe symmetric on interchange of lm. Following Ref. 6,this is taken care of by symmetrizing the susceptibilitiesas follows:

x iill~3 ! ~23v;v, v, v! 5

1

3~x iill 1 x ilil 1 x illi!, (42)

x ilil~3 ! ~2v; 0, 0, v! 5

1

2~x ilil 1 x iill!, (43)

and so on.The x (3) expressions involving single-frequency inci-

dent fields can be calculated from Eq. (42), and the resultsare presented in Table 3. The list of the nonvanishing el-ements and the relations between them are in agreementwith the symmetry predictions for the tetragonal group.13

There are respectively five and nine independent andnonvanishing tensor elements for THG. The intensity-dependent refractive index has, respectively, eight and 15independent and nonvanishing elements, and for dc Kerreffect, there are six and 13, respectively.

It can be seen from Table 3 that the x (3) matrix ele-ments include the x (2)-related functions s(2v) and s(2v),the former of which has a soft-mode resonance. Theseare an intrinsic part of the x (3) response and they do notresult from cascade processes in which two second-orderprocesses follow one another in time. For example, a cas-cade THG could arise from SHG followed by sum-frequency generation, or, symbolically, v 1 v → 2v, thenv 1 2v → 3v. In a quantum-mechanical analogy, cas-cade is a real process, whereas the terms in Table 3 andsubsequent tables come from virtual processes.

B. Rhombohedral SymmetryWe now derive all nonvanishing, nonlinear-susceptibilitytensor elements for rhombohedral symmetry, where thespontaneous polarization P0 is taken to be along the (1, 1,1) direction. We therefore write the high-temperaturefree-energy expression [Eq. (1)] for the rhombohedralsymmetry by replacing (Px , Py , Pz) with Px 5 P01 Qx , Py 5 P0 1 Qy , Pz 5 P0 1 Qz . Proceeding asbefore, we determine the derivatives of the free energywith respect to polarization [evaluated now at(P0 , P0 , P0)], which are

f il0 5 2F S a

e0d il 1

3b1

e02 P0

2d il 12b2

e02 P0

2d ilD1

2b2

e02 P0

2~d jld ij 1 dkld ik!G , (44a)

f ilm0 5 2F6b1

e02 P0id ild im 1

2b2

e02 ~P0jd jmd il

1 Pokdkmd il 1 Pojd imd jl 1 Poid jmd jl

1 Poidkmdkl 1 Pokd imdkl!G , (44b)


Table 3. Tetragonal Symmetry: Nonvanishing Third-Order, x(3), NLO Tensor Elements in theFerroelectric Phase, TËTC

Process, K Susceptibility, x ilmn(3)

Third-harmonic generationK 5 1/4 xxxxx 5 xyyyy 5

s~3v!s 3~v!

e03 F2b2

2P02s~2v!

e02 2 b1G

x ilmn(3) (23v; v, v, v)

Symmetric on permutation of(lmn)

xzzzz 5b1s~3v!s3~v!

e03 F18P0

2b1s~2v!

e02 2 1G

xyxxy 5 xxxyy 5b2

3e03 s~3v!s 3~v!F2b2P0

2s~2v!

e02 2 1G

xxxzz 5 xyyzz 5b2s~3v!s~v!s2~v!

3e03 F4b2P0

2s~2v! 1 6P02b1s~2v!

e02 2 1G

xzyyz 5 xzxxz 5b2s~3v!s 2~v!s~v!

3e03 F4b2P0

2s~2v! 1 6P02b1s~2v!

e02 2 1G

dc Kerr effectK 5 3 xxxxx 5 xyyyy 5

4s 2~v!s~0!

e03 F4b2

2P02s~0!s~v!

e02 2 b1s~0!G

x ilmn(3) (2v; 0, 0, v)

Symmetric on interchange of(lm)

xzzzz 5b1s

2~v!s2~0!

e03 F144b1P0

2s~v!

e02 2 4G

xyxxy 5 xxyyx 5 22b2s 2~v!s 2~0 !

3e03

xyxyx 5 xxxyy 5s 2~v!s 2~0 !b2

e03 F8b2P0

2s~v!

e02 2

14

9 Gxyzzy 5 xxzzx 5

4s 2~v!s2~0 !b2

e03 F4b2P0

2s~v!

e02 2

1

3Gxyyzz 5 xxxzz 5 xzzyy 5 xzzxx

54b2s~v!s~v!s~0 !s~0 !

e03 H F6b1P0

2s~v!

e02 1

2P02b2s~v!

e02 G 2 1J

xzyyz 5 xzxxz 54b2s 2~0 !s2~v!

e03 F4b2P0

2s~v!

e02 2

1

3GIntensity-dependentrefractive indexK 5 3/4

xxxxx 5 xyyyy 5s 3~v!s* ~v!

e03 F2b2

2P02s~0!

e02 2 b1G

x ilmn(3) (2v; 2v, v, v)

Symmetric oninterchange of (lm)

xzzzz 5b1s

3~v!s* ~v!

e03 F18b1P0

2s~0 !

e02 2 1G

xyxxy 5 xxyyx 5 2b2s 3~v!s* ~v!

3e03

xyxyx 5 xxxyy 5s 3~v!s* ~v!b2

2e03 F2b2P0

2s~v!

e02 2

2

3Gxyzzy 5 xxzzx 5

b2s 2~v!s~v!s* ~v!

e03 F3b2P0

2s~0 !

e02 2

1

3Gxyyzz 5 xxxzz 5

b2s~v!s~v!

2e03 H F6b1P0

2s~v!s* ~v!s~0 ! 1 2P02b2s~v!s* ~v!s~0 !

e02 G

21

3@ s~v!s* ~v! 1 s~v!s* ~v!#J

xzyyz 5 xzxxz 5b2s2~v!s~v!s* ~v!

e03 F2b2P0

2s~0 !

e02 2

1

3Gxzzyy 5 xzzxx 5

b2s~v!s~v!

2e03 H F6b1P0

2s* ~v!s~v!s~0 ! 1 2P02b2s* ~v!s~v!s~0 !

e02 G

21

3@ s* ~v!s~v! 1 s* ~v!s~v!#J


f ilmn0 5 2F6b1

e02 d ind ild im 1

2b2

e0 02 ~d jnd jmd il

1 dkndkmd il 1 d imd jnd jl 1 d ind jmd jl

1 dkmd indkl 1 d imdkldkn!G , (44c)

with the magnitude of P0 given by

a 11

e0~b1 1 2b2!P0

2 5 0. (45)

It is worth pointing out that because of the symmetry ofthe P0 direction all the diagonal derivatives fii are iden-tical, as can be seen from Eq. (44a). There is also an-other group of nondiagonal derivatives fij . This leadsagain to two response functions for both second- andthird-order NL susceptibility coefficients, which are

s~v! 5~O1 2 q !

e0@O1~O1 2 q ! 2 2q2#, (46a)

s~v! 5q

e0@O1~O1 2 q ! 2 2q2#, (46b)

where

On 5 Q~nv! 2 p, (46c)

p 5 fii 5a

e01

P02

e02 ~3b1 1 2b2!,

(46d)

q 5 fij 52b2

e02 P0

2. (46e)

Q(v) is given by Eq. (33). Note that the response func-tions s(v) and s(v) as given in Eqs. (46a) and (46b) aredifferent from Eqs. (31) and (32) for the tetragonal sym-metry. The rest of the formalism follows along the samelines of the tetragonal-symmetry derivation; the detailsare therefore omitted in this section.

Table 4 gives all nonvanishing, NL susceptibility coef-ficients for the second-order NL processes, SHG, and op-tical rectification. Results for the third-order NL pro-cesses are tabulated in Table 5. There are fiveindependent elements for THG, nine for the intensity-dependent refractive indices, and eight for the dc Kerr ef-fect. We have fewer independent components than givenby Popov et al.,13 and this is as expected since they used adifferent axis orientation for the rhombohedral symmetry.For reasons of compactness and clarity, we define herenew quantities used in tabulating the explicit expressionsfor second- and third-order processes, which are

u 5 fiij 5 fiji 5 fjii 52b2

e02 P0 , (47a)

v 5 fiii 56b1

e02 P0 , (47b)

Table 4. Rhombohedral Symmetry: Nonvanishing Second-Order, x(2), NLO Tensor Elements in theFerroelectric Phase, TËTC

Process, K Susceptibility, x ilm(2)

Second-harmonicgenerationK 5 1/2

xzzz~2! 5 xxxx

~2! 5 xyyy~2! 5 x1

SHG 5e0@~O2 2 q !~vs2 1 2uh1! 1 2q~vs 2 1 uh2!#

8~O2 1 q !~O2 2 2q !

x ilm(2) (22v; v, v)

Symmetric onexchange of (l, m)

xxxy~2! 5 xxxz

~2! 5 xyyx~2! 5 xyyz

~2! 5 xzzx~2! 5 xzzy

~2! 5 x3SHG 5

e0@O2~2vss 1 2uh2! 1 q~2vs 2 1 2uh3!#

8~O2 1 q !~O2 2 2q !

xxyy~2 ! 5 xxzz

~2 ! 5 xyxx~2 ! 5 xyzz

~2 ! 5 xzxx~2 ! 5 xzyy

~2 ! 5 x2SHG 5

e0@O2~vs 2 1 uh2! 1 q~vs2 1 2uh1!#

8~O2 1 q !~O2 2 2q !

OpticalrectificationK 5 1/2x ilm

(2) (0;2v,v)

xzzz~2! 5 xxxx

~2! 5 xyyy~2! 5 x1

OR 5e0@~ p 1 q !~vusu2 2 2ug1! 2 2q~vu su2 1 ug2!#

8~q 2 p !~ p 1 2q !

xxxy~2 ! 5 xxxz

~2 ! 5 xyyx~2 ! 5 xyyz

~2 ! 5 xzzx~2 ! 5 xzzy

~2 ! 5 x3OR

5e0@~ p 1 q !~vss* 1 ug2! 2 q~vs* s 1 ug2 1 vu su2 1 2ug1!#

8~q 2 p !~ p 1 2q !

xxyx~2 ! 5 xxzx

~2 ! 5 xyxy~2 ! 5 xyzy

~2 ! 5 xzxz~2 ! 5 xzyz

~2 ! 5 x4OR

5e0@~ p 1 q !~vs* s 1 ug2! 2 q~vss* 1 ug2 1 vu su2 1 2ug1!#

8~q 2 p !~ p 1 2q !

xxyy~2 ! 5 xxzz

~2 ! 5 xyxx~2 ! 5 xyzz

~2 ! 5 xzxx~2 ! 5 xzyy

~2 ! 5 x2OR 5

e0@ p~vu su2 1 ug2! 2 q~vusu2 1 2ug1!#

8~q 2 p !~ p 1 2q !


Table 5. Rhombohedral Symmetry: Nonvanishing Third-Order, x(3), NLO Tensor Elementsin the Ferroelectric Phase, TËTC

Process, K Susceptibility, x ilmn(3)

Third-harmonic generationK 5 1/4 xxxxx 5 xyyyy 5 xzzzz 5

~O3 2 q!t1 1 2qt2

e0~O3 1 q!~O3 2 2q!

x ilmn(3) (23v; v, v, v)

Symmetric on permutationof (lmn)

xyxxx 5 xzxxx 5 xxyyy 5 xzyyy 5 xxzzz 5 xyzzz 5~O3 2 q!t2 1 q~t1 1 t2!

e0~O3 1 q!~O3 2 2q!

xxxyy 5 xyyxx 5 xyyzz 5 xzzyy 5 xzzxx 5 xxxzz 5~O3 2 q!t4 1 qt3

e0~O3 1 q!~O3 2 2q!

xxxxy 5 xxxxz 5 xyyyx 5 xyyyz 5 xzzzx 5 xzzzy 5~O3 2 q!t3 1 qt4

e0~O3 1 q!~O3 2 2q!

xxyzz 5 xzxyy 5 xyzxx 5 xxzyy 5 xyxzz 5 xzyxx 5q~t3 1 t4!

e0~O3 1 q!~O3 2 2q!

Intensity-dependentrefractive indexK 5 3/4

xxxxx 5 xyyyy 5 xzzzz 5~O1 2 q!r1 1 2qr2

e0~O1 1 q!~O1 2 2q!

x ilmn(3) (2v; 2v, v, v)

Symmetric on interchangeof (lm)

xxxxy 5 xxxxz 5 xyyyx 5 xyyyz 5 xzzzx 5 xzzzy 5~O1 2 q!r3 1 q~r7 1 r8!

e0~O1 1 q!~O1 2 2q!

xyxxx 5 xzxxx 5 xxyyy 5 xzyyy 5 xxzzz 5 xyzzz 5~O1 2 q!r2 1 q~r1 1 r2!

e0~O1 1 q!~O1 2 2q!

xxyyx 5 xxzzx 5 xyxxy 5 xyzzy 5 xzxxz 5 xzyyz 5~O1 2 q!r7 1 q~r3 1 r8!

e0~O1 1 q!~O1 2 2q!

xxzyy 5 xxyzz 5 xyxzz 5 xyzxx 5 xzyxx 5 xzxyy 5~O1 2 q!r9 1 q~r6 1 r4!

e0~O1 1 q!~O1 2 2q!

xxxyy 5 xyyzz 5 xzzxx 5 xyyxx 5 xzzyy 5 xxxzz 5~O1 2 q!r4 1 q~r6 1 r9!

e0~O1 1 q!~O1 2 2q!

xxyxx 5 xxzxx 5 xyxyy 5 xyzyy 5 xzxzz 5 xzyzz 5~O1 2 q!r6 1 q~r4 1 r9!

e0~O1 1 q!~O1 2 2q!

xxyyz 5 xxzzy 5 xyxxz 5 xyzzx 5 xzxxy 5 xzyyx 5~O1 2 q!r8 1 q~r3 1 r7!

e0~O1 1 q!~O1 2 2q!

xxxyz 5 xyyxz 5 xzzyx 5 xxzxy 5 xyzyx 5 xzxzy 5~O1 1 q!r5

e0~O1 1 q!~O1 2 2q!

dc Kerr effectK 5 3 xxxxx 5 xyyyy 5 xzzzz 5

~ p 1 q!~c1 1 d1 1 d2! 1 2q~c2 1 d3 1 d4 1 d5!

~ p 1 2q!~q 2 p!

x ilmn(3) (2v; 0, 0, v)

Symmetric on interchangeof (lm)

xxxxy 5 xxxxz 5 xyyyx 5 xyyyz 5 xzzzx 5 xzzzy 5~ p 1 q!~c3 1 c4 1 d6 1 d7!

~ p 1 2q!~q 2 p!

xyxxx 5 xzxxx 5 xxyyy 5 xzyyy 5 xxzzz 5 xyzzz 5q~c1 1 d1 1 d2! 2 p~c2 1 d3 1 d4 1 d5!

~ p 1 2q!~ p 2 q!

xxxyz 5 xyyzx 5 xzzxy 5~ p 1 q!~c6 1 c8!

~q 2 p!~ p 1 2q!

xyzxx 5 xzyxx 5 xzxyy 5 xxzyy 5 xxyzz 5 xyxzz 5q~c6 1 c5!

~ p 2 q!~ p 1 2q!

xxxyy 5 xyyzz 5 xzzxx 5 xyyxx 5 xzzyy 5 xxxzz 5~ p 1 q!~c6 1 c5!

~q 2 p!~ p 1 2q!

xxyyz 5 xxzzy 5 xyzzx 5 xyxxz 5 xzxxy 5 xzyyx 5q~c3 1 c4 1 d6 1 d7!

~ p 2 q!~ p 1 2q!

xxyzx 5 xyxzy 5 xzxyz 5q~c6 1 c8!

~ p 2 q!~ p 1 2q!


h 5 fiiii 56b1

e02 , (47c)

l 5 fiijj 5 fijij 5 fijji 52b2

e02 , (47d)

h1 5 s 2 1 2ss, (48a)

h2 5 s2 1 3s 2 1 2ss, (48b)

h3 5 s 2 1 4ss, (48c)

g1 5 u su2 1 ss* 1 s* s, (48d)

g2 5 3u su2 1 usu2 1 ss* 1 s* s, (48e)

t1 51

4ve0

2sx1SHG 1

1

4ue0

2@2sx1SHG 1 2~s 1 s!x2

SHG#

11

48he0

3s3 11

8le0

2ss2, (49a)

t2 51

4ve0

2sx2SHG 1

1

4ue0

2@~s 1 s!x1SHG

1 ~s 1 3s!x2SHG# 1

1

48he0

3s 3

11

16le0

2~ ss2 1 s 3!, (49b)

t3 51

4ve0

2sx3SHG 1

1

4ue0

2~s 1 3s!x3SHG 1

1

16he0

33s2s

11

8le0

3~ ss2 1 s 3 1 ss 2!, (49c)

t4 51

4ve0

2sx3SHG 1

1

4ue0

2~s 1 3s!x3SHG 1

1

16he0

3ss 2

11

16le0

3~3ss 2 1 2s 3 1 s3!, (49d)

r1 51

8ve0

2s* x1SHG 1

1

8ue0

2@2s* x1SHG

1 2~s* 1 s* !x2SHG# 1

1

48he0

3susu2

11

48le0

3~2s* s 2 1 4su su2!, (50a)

r2 51

8ve0

2s* x2SHG 1

1

8ue0

2@~ s* 1 s* !x1SHG

1 ~s* 1 3s* !x2SHG# 1

1

48he0

3su su2

11

48le0

3~ s* s2 1 2susu2 1 3su su2!, (50b)

r3 51

8ve0

2s* x3SHG 1

1

8ue0

2~3s* 1 s* !x3SHG

11

48he0

3susu2

11

48le0

3~ s* s2 1 susu2 1 2su su2 1 su su2

1 s* s 2!, (50c)

r4 51

48he0

3su su2 11

48le0

3~susu2 1 2su su2 1 2su su2

1 s* s 2!, (50d)

r5 51

48he0

3su su2 11

48le0

3~su su2 1 2su su2 1 susu2

1 s* s 2 1 s2s* !, (50e)

r6 51

48he0

3susu2 11

48le0

3~su su2 1 2su su2 1 susu2

1 s* s 2 1 s2s* !, (50f )

r7 51

8ve0

2s* x2SHG 1

1

8ue0

2~2s* 1 2s* !x3SHG

11

48he0

3s 2s*

11

48le0

3~2su su2 1 2su su2 1 s* s 2 1 susu2!,

(50g)

r8 51

8ue0

2~3s* 1 s* !x3SHG 1

1

48he0

3su su2

11

48le0

3~ s* s2 1 susu2 1 2su su2 1 su su2

1 s* s 2!, (50h)

r9 51

48he0

2su su2 11

48le0

3~3su su2 1 su su2 1 susu2

1 s* s 2!, (50i)

c1 51

48he0

3usu2s 11

48le0

3~2s* s 2 1 4su su2!, (51a)

c2 51

48he0

3u su2s 11

48le0

3~s* s 2 1 2susu2 1 3u su2s!,

(51b)

c3 51

48he0

3s2s* 11

48le0

3~2usu2s 1 2su su2

1 2u su2s!, (51c)


c4 51

48he0

3usu2s 11

48le0

3~ usu2s 1 su su2 1 2u su2s

1 s2s* 1 s* s 2!, (51d)

c5 51

48he0

3u su2s 11

48le0

3~ usu2s 1 2su su2 1 2u su2s

1 s* s 2!, (51e)

c6 51

48he0

3s 2s* 11

48le0

3~ usu2s 1 2su su2 1 2u su2s

1 s* s 2!, (51f)

c7 51

48he0

3s 2s* 11

48le0

3~2usu2s 1 2su su2

1 2u su2s!, (51g)

c8 51

48he0

3u su2s 11

48le0

3~s2s* 1 susu2

1 2u su2s 1 su su2 1 s* s2!, (51h)

d1 51

8ve0

2sx1OR 1

1

8ue0

2@2sx1OR 1 ~s 1 2s!x2

OR#,

(52a)

d2 51

8ve0

2sx1OR 1

1

8ue0

2@sx1OR 1 ~s 1 3s!x2

OR#,

(52b)

d3 51

8ve0

2sx2OR 1

1

8ue0

2@~s 1 s!x1OR 1 ~s 1 2s!x2

OR#,

(52c)

d4 51

8ve0

2sx2OR 1

1

8ue0

2@sx1OR 1 ~s 1 3s!x2

OR#,

(52d)

d5 51

8ve0

2sx2OR 1

1

8ue0

2@~s 1 s!x2OR 1 3s!x2

OR],

(52e)

d6 51

8ve0

2sx3OR 1

1

8ue0

2@ sx3OR 1 ~s 1 s!x4

OR#, (52f )

d7 51

8ve0

2sx3OR 1

1

8ue0

2@2sx3OR 1 ~s 1 s!x4

OR#.

(52g)

In the expressions given above x1SHG , x2

SHG , x3SHG , x1

OR ,x2

OR , x3OR are as stated in Table 4.

4. RESONANCE FEATURES ANDNUMERICAL ILLUSTRATIONSWe now discuss the general resonance features of the non-vanishing tensor elements derived in the previous sec-tion, and focus on characteristics of these coefficients foroscillatory dynamical systems.

We begin by looking at the second-order NL suscepti-bility coefficients for the case of tetragonal symmetry asdisplayed by Table 2. Notice that the frequency depen-dence of the various x ilm

(2) is governed by the linear re-sponse functions s(v) and s(v) [Eqs. (31) and (32)], andthe temperature dependences of the coefficients arisepartly from the derivatives film(P0) [Eq. (21b)] and partlyfrom the presence of the soft-mode terms fxx , fyy , and fzz[Eqs. (26) and (27)]. We see from Table 2 that each coef-ficient contains a temperature-dependent term b1P0 orb2P0 , and we observe that P0 → 0 as T → T0 . There-

Fig. 1. Plot of scaled xxzx(2) (22v; v, v). Solid curve, real;

dashed curve, imaginary. Versus reduced frequency f at reducedtemperature (a) t 5 0.76, (b) t 5 0.69. (c) Versus reduced tem-perature t at reduced frequency f 5 0.15.


fore, for second-order transition, the tensors for SHG andoptical rectification all have the same property: x ilm

(2)

→ 0 as T → T0 . This is not surprising, since in thehigh-temperature PE phase, x ilm

(2) is zero because of the in-version property of the cubic phase.

The nature of these resonances is illustrated in Figs.1(a) to 1(c) for the case of xxzx

(2) (22v; v, v). In these fig-ures, xxzx

(2) is scaled by a factor e03/b2 and frequencies v by

v0 where v02 5 aT0 /e0m. We use the estimated values

of b1 and b2 given in Appendix A to calculate the scaledcoefficient. Figures 1(a) and 1(b) display scaled xxzx

(2) ver-sus f at two different reduced temperatures t 5 T/T05 0.76 and 0.69. As mentioned above, resonances areobserved at the fundamental and the SHG frequencies,and there is a shift to higher resonant frequencies atlower temperatures owing to the temperature dependenceof the resonant frequency. Figure 1(c) shows a plot ofscaled xxzx

(2) versus t at f 5 0.15 where a resonance in tem-perature resulting from soft-mode dynamics is observedat t ' 0.97. As expected, the coefficient approaches zeroas t → 1.

For the third-order NL susceptibility coefficients, thereare resonances at both input and output frequencies as inthe isotropic free-energy case6; the THG coefficients, forexample, have resonances at frequencies v and 3v. Inaddition there are resonances at 2v owing to the presenceof the terms s(v) and s(v) for THG, and at zero frequencyowing to s(0) and s(0) for the other two effects. As forthe temperature dependences, we observe that in most co-efficients the dependences come directly from P0 [see Eq.(22)] and indirectly from the responses s(v) and s(v)where, as can be seen from Eqs. (31) and (32), thesefrequency-dependent functions are also sensitive to tem-perature through the presence of a and P0 terms in theirexpression.

Figures 2(a) to 2(c) illustrate these complex resonancesfor the case of xxxxx

(3) (23v; v, v, v). In these figures thecoefficient is scaled by a factor e0

3/b1 and the same re-duced frequency and temperature as for Fig. 1 are used.As discussed above, xxxxx

(3) (23v; v, v, v) is resonant atthe fundamental and the THG frequencies as shown byFigs. 2(a) and (b) at two different temperatures t5 T/T0 5 0.76 and 0.69. Again it is observed that thereis an upward shift in these frequencies at lower tempera-tures. The temperature dependence of the coefficient isillustrated by Fig. 2(c) for f 5 0.15. As expected, it hasthe same characteristic as in the case of the SHG coeffi-cient since this feature is a result of the presence of thesoft-mode term in the expression.

From the results of Tables 4 and 5 for the rhombohe-dral symmetry, observation of the explicit expressions re-veals that in general the nature of the resonances and di-vergences are similar to those of the tensor elements inthe case of tetragonal symmetry. Also as in the tetrago-nal case, each element is strongly dependent on the ma-terial parameters b1 and b2 .

As an example of a numerical estimate, we presenthere, based on our formalism, a nonresonant SHG valueof xzzz

(2) (22v; v, v) for the case of tetragonal symmetry.By using the experimental values of BaTiO3 from Ref. 14,the soft-mode frequency is calculated to be vT > 3

3 1012 Hz. In addition, by using our calculated valuesof b1 and b2 in Appendix A, we obtain the magnitude ofxzzz

(2) (22v; v, v) as >1.36 3 10211 mV21 at a frequency of4.75 3 1013 Hz and temperature of 273 K for g 5 3.323 1025. These values are reasonably acceptable. Tocompare the magnitude, the experimental value obtainedfrom Ref. 15 is (1.32 6 1.00) 3 10211 mV21 at the inputfrequency of 2.835 3 1014 Hz and room temperature.

In Section 5 we look at SHG in a bulk FE material asan application of our formalism.

Fig. 2. Plot of scaled xxxxx(3) (23v; v, v, v). Solid curve, real;

dashed curve, imaginary. Versus reduced frequency f at reducedtemperature (a) t 5 0.76, (b) t 5 0.69. (c) Versus reduced tem-perature t at reduced frequency f 5 0.15.


5. SECOND-HARMONIC GENERATION INBULK FERROELECTRIC WITHTETRAGONAL SYMMETRYThe coefficients that have been derived in Sections 3 and4 can be used for detailed calculations by standard meth-ods in nonlinear optics.11,15,16 As an example, we givehere a calculation for SHG in bulk ferroelectric materialassumed to be in the FE phase having tetragonal symme-try. The example given here is SHG with no depletion ofthe pump beam. Suppose an incident far IR beam ispropagating with a single frequency v1 in the y direction,and, as in the formalism, the spontaneous polarization P0is assumed to be along the z axis. With these assump-tions, the driving E field in general is in the x-z plane.The relevant NL wave equation to be solved for the SHGfield is

]2Ei2v1

]y2 5 m0

]2

]t2 @e0Ei2v1 1 Pi

L 1 PiNL#, i 5 x, z,

(53)

where PL and PNL are the linear and nonlinear contribu-tion, respectively, to polarization in the bulk FE, and Ei

2v1

is the ith-component-generated field propagating at fre-quency 2v1 . For the z component, for example, this fieldmay be written as

Ez2v1 5

1

2@Ez

2v1~ y !exp~ik2zy !exp~2i2v1t ! 1 c.c.#.

(54)

k2z 5 e2z(2v1)2/c2 is the wave number of the SHG beam,where e2z is evaluated with Eq. (30) at frequency 2v1 .From Eqs. (29) and (30) and the SHG nonvanishing coef-ficients of Table 2, we have

PL 5 e0@xxx~1 !~22v; v!Ex

2v1x 1 xzz~1 !~22v; v!Ez

2v1z#,(55)

for the linear polarization and

PNL 5 e0$@xxxz~2 ! ~22v; v, v!Ex

v1Ezv1

1 xxzx~2 ! ~22v; v, v!Ez

v1Exv1#x

1 @xzxx~2 ! ~22v; v, v!Ex

v1Exv1

1 xzzz~2 ! ~22v; v, v!Ez

v1Ezv1#z%, (56)

for the NL polarization, where Eiv1 is the ith component of

the driving field and for the x component it is

Exv1 5

1

2@Ex

v1 exp~ik1xy !exp~2iv1t ! 1 c.c.#, (57)

in which Exv1 is a constant. Equations (55) and (56) sug-

gest that three configurations of driving fields may be ar-ranged: (1) x-z plane polarized, (2) purely x polarized,and (3) purely z polarized. For simplicity, we concentrateon the case where the driving field or the incident beam ispurely x polarized. This leads to the generation of PNL

and thus the SHG beam in the z direction as depicted byEq. (56). Based on the slowly varying amplitudeapproximation11,15,16 in which the envelope Ez

2v1 is as-

sumed to have a slow spatial variation compared with theexponential in Eq. (54), Eq. (53) reduces to

2ik2z

dEz2v1

dy5 2

2v12

c2 xzxx~2 ! ~22v; v, v!~Ex

v1!2exp~iDkx y !,

(58)

where

Dkx 5 2k1x 2 k2z (59)

is the phase-mismatch factor. Equation (58) is easilysolved for the SHG amplitude Ez

2v1. The main propertyof the solution is that Ez

2v1 attains its maximum value af-ter a propagation length L } 1/Dkx , and this maximumvalue is proportional to L. An important practical con-sideration is therefore to minimize the impedance mis-match Dkx . The possibility of achieving phase match-ing, that is Dkx 5 0, is illustrated by Fig. 3 where a plotof uDkxu versus frequency v/2p for BaTiO3 is given basedon our estimated values of b1 and b2 shown in AppendixA. As shown by Fig. 3, a close to phase-matching condi-tion is possible at around 6.0 3 1013 Hz. For example, at;6.2 3 1013 Hz we calculate that Dkx ' 26.7 m21 andL ' 23.3 cm. The large value for L is not surprisingsince as Dkx → 0, L → `.

In this section, we have given a detailed discussion ofthe simplest example of a complete calculation using theNL coefficients that were derived in Sections 2, 3, and 4.One physical point of difference between ferroelectrics inthe far IR and standard NLO crystals in the visible andnear IR is that the dielectric function eJ (linear response)of the former is very large and may contain a resonantfrequency interval. It may be asked, what are the prac-tical consequences of the large and highly dispersive eJ ?One part of the answer is that, as we have seen, phasematching can be achieved. A complete assessment wouldinvolve calculations for various other NLO aspects.These might include SHG in a Fabry–Perot etalon or asuperlattice (x (2)), multistable transmission throughthese structures (x (3)), self-guiding of surface polaritons(x (3)), and bistability arising from the Kerr-like tensors.

6. CONCLUSIONSFollowing the formalism introduced by Ref. 6, we havepresented the results of a systematic expansion, based on

Fig. 3. Graph of uDkxu versus frequency v/2p for SHG underphase-mismatching condition.


the high-temperature, cubic-symmetry free energy as ex-pressed in Eq. (1), to obtain the second- and third-orderNLO tensor coefficients in the cubic PE phase and in thetetragonal and rhombohedral FE phases. We have giventhe results for the second-order phase transition in ferro-electrics involving only a single input frequency. Exten-sion to the first-order phase transition and to multiple in-put frequencies can easily be made, and we hope to dothese for future publication.

The results we obtain for the case of cubic symmetryare similar to the results of Ref. 6. This is as expectedsince the PE phase is isotropic. However, for complete-ness we have presented all the nonvanishing third-ordertensor elements for the case of cubic symmetry in Table 1.The second- and third-order NL coefficients for the case oftetragonal symmetry are presented in Tables 2 and 3.The general features of the frequency and temperaturedependences of the coefficients for oscillatory dynamicalsystems are discussed, and specific tensor elements forSHG and THG processes are numerically illustrated.The results for the case of rhombohedral symmetry aredisplayed in Tables 4 and 5. In general, the resonancefeatures of the nonvanishing NL tensor elements aresimilar to those for the case of tetragonal symmetry.

The objective here has been to evaluate the nonvanish-ing, NL-susceptibility tensor elements for the cubic PEphase and the tetragonal and rhombohedral FE phases ofbulk FE materials. Since our formalism uses group sym-metries of real crystals, we would expect the results weobtained could be used as the model for experimentalmeasurements on NL-susceptibility coefficients of variousperovskite FE materials. The expressions that we havederived can also be applied to calculations of many NLOeffects, such as harmonic generation and effects resultingfrom an intensity-dependent linear susceptibility.

In addition to the above, another possible application isin the study of morphotropic phase boundary (MPB) insolid solution systems of perovskite-type oxideFE.10,11,17–19 It is known that oxide FEs belonging to theBaTiO3 family often form systems of complete solid solu-tion, and some of them show several different phases ac-cording to the change in concentration of the endmembers.10 One such system is PbZrO3 –PbTiO3 , whichexhibits an MPB in the temperature–composition phasediagram, that is, at the boundary between rhombohedraland tetragonal FE phases. If the free energy of such sys-tems is described by Eq. (1), the MPB corresponds to thevertical line b1 /b2 5 1 in the a-b1 /b2 phase diagram.10

It is found that the transversal linear dielectric suscepti-bilities for the tetragonal and rhombohedral FE phasesdiverge when b1 5 b2 for all temperatures.10 Since theresponse functions we have derived here—s(v) for thecase of tetragonal and r(v) for the case of rhombohedralsymmetry—contain the b1 and b2 dependence, we expectsimilar types of divergence may also occur in the NL-susceptibility coefficients. Since the parameters b1 andb2 may depend on composition, the MPB must be a func-tion of the composition. This therefore can be used asone of the general guiding principles in the search for ma-terials with large NL dielectric susceptibility coefficients.We are currently investigating along this line and hope topublish the results in the near future.

APPENDIX ABased on available experimental values of BaTiO3 ,14 wegive here numerical estimates of b1 and b2 in the tetrag-onal phase. The value of b1 is calculated from Eq. (22),i.e., P0

2 5 2(e0a/b1), and b2 by use of the expression10

exx 52b1

b2 2 b1ezz , (A1)

where exx and ezz are components of the static dielectricconstants for the tetragonal symmetry. Since for mostFE materials x ii @ 1, we take e ii > x ii , and thereforefrom Eqs. (31) and (32) we have

ezz 51

uau 13b1P0

2

e02

, exx 51

uau 1b2P0

2

e02

. (A2)

To calculate b1 , we use the experimental value of uau> 5.22 3 1024 at room temperature,2 and the slope ofthe P0

2-versus-T graph, which is calculated to be approxi-mately 2.57 3 1024 C2 m24 deg21.20 This yields b1> 1.913 3 10213 m3 J21.

To obtain an estimation of b2 , we first calculate ezz andthen use both exx expressions given above. By using theestimated value of b1 and the room temperature valueP0 > 0.1945 Cm22,14 we obtain ezz > 336. This is thensubstituted into Eq. (A1). By equating both exx expres-sions in Eq. (A1) and (A2), we find b2 > 5.74 3 10213

m3 J21.With these calculated values the ratio of b2 /b1 is found

to be approximately 3, which is acceptable for the tetrag-onal phase where b2 /b1 . 1.10

ACKNOWLEDGMENTSJ. Osman thanks the Nippon Sheet Glass Foundation andUniversiti Sains Malaysia for financial support.

J. Osman may be reached by e-mail at [email protected].

REFERENCES1. C. Kittel, Introduction to Solid State Physics (Wiley, New

York, 1986).2. M. E. Lines and A. M. Glass, Principles and Applications of

Ferroelectrics and Related Materials (Clarendon, Oxford,UK, 1977).

3. Y. Ishibashi and H. Orihara, ‘‘A phenomenological theory ofnonlinear dielectric response,’’ Ferroelectrics 156, 185–188(1994).

4. J. Osman, Y. Ishibashi, S.-C. Lim, and D. R. Tilley, ‘‘Nonlin-ear optic coefficients in the ferroelectric phase,’’ J. KoreanPhys. Soc. 32, S446–S449 (1998).

5. C. Haas, ‘‘Phase transitions in ferroelectric and antiferro-electric crytals,’’ Phys. Rev. 140, A863–A868 (1965).

6. J. Osman, Y. Ishibashi, and D. R. Tilley, ‘‘Calculation ofnonlinear susceptibility tensor components in ferroelec-trics,’’ Jpn. J. Appl. Phys. 37, 4887–4893 (1998).

7. J. Grindlay, An Introduction to the PhenomenologicalTheory of Ferroelectricity (Pergamon, New York, 1970).

8. A. F. Devonshire, ‘‘Theory of barium titanate (Part 1),’’ Phi-los. Mag. 40, 1040–1063 (1949).


9. K. Fujita and Y. Ishibashi, ‘‘Roles of the higher orderaniso-tropic terms in successive structural phasetransitions: The method of determination of phenomeno-logical parameters,’’ Jpn. J. Appl. Phys., 36, 254–259(1997).

10. Y. Ishibashi and M. Iwata, ‘‘Morphotropic phase boundaryin solid solution systems of perovskite-type oxide ferroelec-trics,’’ Jpn. J. Appl. Phys. 37, L985–L987 (1998).

11. P. N. Butcher and D. Cotter, The Elements of Nonlinear Op-tics (Cambridge University, Cambridge, UK 1990).

12. H. Orihara and Y. Ishibashi, ‘‘A phenomenological theory ofnonlinear dielectric response II. Miller’s rule and nonlin-ear response in nonferroelectrics,’’ J. Phys. Soc. Jpn. 66,242–246 (1997).

13. S. V. Popov, Yu. P. Svirko, and N. I. Zheludev, SusceptibilityTensors for Nonlinear Optics (Institute of Physics, London,1995).

14. Y. G. Wang, W. L. Zhong, and P. L. Zhang, ‘‘Size effects on

the Curie temperature of ferroelectric particles,’’ Solid StateCommun. 92, 519–523 (1994).

15. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, NewYork, 1984).

16. D. L. Mills, Nonlinear Optics (Springer-Verlag, Berlin,1991).

17. Y. Ishibashi and M. Iwata, ‘‘A theory of morphotropic phaseboundary in solid-solution systems of perovskite-type oxideferroelectrics,’’ Jpn. J. Appl. Phys. 38, 800–804 (1999).

18. M. Iwata and Y. Ishibashi, ‘‘Theory of morphotropic phaseboundary in solid solution systems of perovskite-type oxideferroelectrics: engineered domain configurations,’’ Jpn. J.Appl. Phys. 39, 5156–5163 (2000).

19. L. Gag, J. Osman, and D. R. Tilley, ‘‘Effective-mediumtheory of dielectric-constant anomalies in ferroelectric com-posites,’’ Ferroelectrics 255, 59–72 (2001).

20. E. Fatuzzo and W. J. Merz, Ferroelectricity (North-Holland,Amsterdam, 1967).

Documents

Calculation of nonlinear-susceptibility tensor components in ferroelectrics: cubic, tetragonal, and rhombohedral symmetries