Zuo-Guang. Ye Advanced Dielectric Piezoelectric and Ferroelectric Materials: Synthesis, Characterisation and Applications

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Handbook of dielectric, piezoelectric and ferroelectric materials940

practical applications is to find a technique that can mass-produce modulated

domain materials at low cost. Yamada and his co-workers (Yamada et al.,

1993) realized a significant breakthrough along this direction. They successfully

fabricated a periodic domain grating in a ~0.2mm thick LN thin wafer by

applying a pulsed field at room temperature. The period of the domain structures

was defined by the lithographically fabricated metal electrode. They also

confirmed that the periodic electrode should be fabricated on the positive z

face of LN, for the reversed domain nucleates more easily on the +z face

than on the –z face. By using this technique, great progress has been made

towards fabricating thicker samples as well as other ferroelectrics, such as

LT, KTP and SBN. The details for field poling on LN, LT, KTP and SBN

crystals at room temperature were described by Miller (1998), Zhu S.N. et

al. (1995a,b), Rick and Lau (1996) and Zhu et al. (1998). In this method, the

location of the reversed domains is defined by the lithographically fabricated

electrode and the domain duty cycle is controlled by the spacing ratio of

electrode and switch time t

. The t

should be selected according to the

expression

Q idt kP A

= =

∫

where Q is the total delivered charge, k a coefficient around 2.2–2.5 from

experiential, and P

and A are the spontaneous polarization and total area of

reversed domain, respectively. If the period of a superlattice is Λ and the

average width of the reversed domains is d, the duty cycle ρ = d/Λ. Miller

(1998) proposed that domain reversal under periodic electrodes could be

divided into several stages. First, domain nucleates along each strip electrode

edges. Then, domain apex propagates toward the opposite face. Once the

apex reaches the –z face, it extends rapidly and coalesces under the electrode

and extends out of the area covered by electrode strips. The duty cycle of the

domain is controlled by the electrode width, and amplitude and duration of

the applied field. For small-period patterns (Λ < 10µm), the width of the

electrode is generally designed not to exceed Λ/4 to avoid domain merging

prior to coalescence and the field amplitude is set at the field with the value

of highest nucleation site density. In conventional poling, in order to prevent

the back-switching effect (Fatuzzo and Merz, 1967), the external field is

ramped to zero over a duration of ~several tens of milliseconds to stabilize

the reversed domains, instead of removing it abruptly.

By far, most of the LN and LT superlattices are made of congruent

composition crystals, because they are easy to grow and are available

commercially with high quality and low price. Recent progress in the growth

techniques makes it possible to grow LN and LT with stoichiometric properties.

Gopalan et al. (1998) and Kitamura et al. (1998) found that the electric field

for domain reversal in the stoichiometric crystals was much lower than that

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Novel physical effects in dielectric superlattices 941

in the congruent crystals, and the values are about one-fifth for LN and one-

thirteenth for LT, respectively. However, the spontaneous polarizations P

and the Curie temperature were relatively insensitive to the non-stoichiometry.

The internal fields in the congruent crystals, which were calculated from the

asymmetry in the P

versus electric field hysteresis, disappeared in

stoichiometric crystals. These results further verify that the origin of the

internal field and large changes in the poling fields of LN and LT appear to

be largely dependent on the ratios of [Li]/[Li+Nb] and [Li]/[Li+Ta], therefore,

on non-stoichiometric point defects in these two crystals, respectively. Interest

has been aroused about the stoichiometric crystals because lower poling

field and better poling characteristic make them candidates for poling thicker

samples (thicker than a few millimeters) to fabricate bulk devices for nonlinear

optical application. Recently, Batchko et al. (1999) further improved the

electric field poling technique that incorporates domain back-switching as a

means for realizing high-fidelity short-period domain pattern essential for

SHG of blue and ultra-violet (UV) light. A high-quality LT superlattice with

period as short as 1.7 µm was prepared for UV SHG (Mizuuchi et al., 1997).

LN (Burr et al., 1997), SBN (Zhu Y.Y. et al., 1997a,b) and KTP (Wang et al.

1998) superlattices with 1 mm thickness were fabricated successfully.

The superlattices with various domain gratings, such as chirped period

(Loza et al., 1999), quasi-period (Zhu S.N. et al., 1997a,b; Zhu Y.Y. et al.,

1998), Thue-Morse structure and domain lens and prism array (Yamada et

al., 1996, Chiu et al., 1996) were also fabricated using the above method.

The poling properties of various doped LN, LT, SBN and KTP crystals have

also been studied at room temperature and at low temperatures (~170 K)

(Rosenman et al., 1998). These studies enable significant optimization of the

process parameters.

31.3 Preparation of DSL by the photorefractive

effect

Usually domain inversion does not change the dielectric constants of the

material. In order to modulate the dielectric constants of DSL, other methods

should be used, such as volume holographic record and thermal fixation. In

this case, 1D, 2D or 3D DSL (or, say, photonic crystals) can be realized. In

our experiments, we have used an oxidized Fe-doped LN (0.1 wt.% Fe)

single domain crystal for the holographic recording. A beam of the blue line

of an argon-ion laser is split into three nearly equal-intensity ones, S

, S

, and

, which are recombined and interfered to cause spatial variations of the

refractive index. Thus a 2D square periodic grating can be recorded into the

crystal as shown in Fig. 31.4(a). The oxidized Fe : LN sample exhibits high

resistance to relax and the grating remains stable after the writing beams are

removed. If a beam of an argon-ion laser is split into four equal-intensity

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Handbook of dielectric, piezoelectric and ferroelectric materials942

ones and an S

is applied parallel to the normal of the sheet of Fig. 31.4(b),

a 3D cubic DSL may be formed (Xu and Ming, 1993a; Wang et al.,

1996a, b).

31.4 Application of DSLs in nonlinear parametric

interactions

The concept of superlattice was initiated by Esaki and Tsu (1970). Since

then, the study of superlattices has experienced three stages according to the

symmetry of the superlattice. The first stage was began in the 1970s, which

was devoted to the investigation of periodic superlattices. The second stage

of the study of quasi-periodic superlattices was triggered by the discovery of

quasi-crystals, with the periodic one still remaining a subject. The quasi-

periodic structure is of increasing interest because it is intermediate between

periodic and random. Recently, aperiodic and disorder structures have attracted

considerable attention. Like semiconductor and metal superlattices, the structure

of dielectric superlattices may be periodic, quasi-periodic, aperiodic and

disorder, 1D or 2D, which show different physical properties.

31.4.1 Wave vector conservation

Usually wave vector conservation plays an important role in interactions

between electromagnetic waves and media, no matter whether the interactions

are linear or nonlinear. One best known example is the Bragg condition in X-

ray diffraction, where the wave vector conservation between the incident

and diffracted X-rays is fulfilled with a reciprocal vector provided by the

-axis

Fe : LiNbO

–h

(a)

(b)

31.4

(a) Optical geometry for recording a 2D refractive index grating

into a Fe:LN single domain crystal. (b) Schematic diagram of four-

wave diffraction.

is the incident wave satisfying the exact Bragg

condition of four-wave diffraction.

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Novel physical effects in dielectric superlattices 943

crystal lattice. In optical range, more interesting phenomena exist related to

the wave vector conservation.

In the linear interaction regime, when the light transmission in 1D DSL is

considered, the Bragg condition is important for constructive interference to

take place. The treatment can be extended to light transmission in 2D DSL.

In that case, more than one Bragg condition, for example, two Bragg conditions

can be fulfilled (Feng and Ming, 1989). It is noted that an X-ray can hardly

satisfy two Bragg conditions at the same time in a crystal because the lattice

periods are non-adjustable; whereas in 2D DSL, the reciprocal vectors can

be adjusted to make the incident waves satisfy the two conditions

simultaneously.

Among the nonlinear interaction regime, the most studied are optical

frequency conversion and Kerr-form nonlinearity. The former one is related

to the second-order nonlinear process, while the latter one is related to the

third-order nonlinear process.

In the optical frequency conversion process, the wave vector conservation

involves light waves with different wavelengths. In that case, the wave vector

conservation is just the so-called phase-matching condition. In the SHG

process, the phase mismatch between the fundamental and the second harmonic

can be compensated either using the birefringence of the crystal (birefringence

phase-matching (PM) method) or using the reciprocal vectors in DSL (QPM

method). When optical bistability is discussed in 1D and 2D DSLs with

Kerr-form nonlinearity, the situation becomes quite different. In 1D DSL, it

is the change of the phase mismatch between the propagating wave vector

and the reciprocal vector provided by the periodic structure that brings the

incident wave from a forbidden transmission state to an allowed state, or

from a low transmission state to a high transmission state. The bistability in

1D DSL is thus attributed to the phase-mismatch mechanism. Recent research

work by present authors has revealed that a novel bistable mechanism, that

is, the refractive index modulation (RIM) mechanism, which is characteristically

different from the phase mismatch mechanism, exists in 2D DSL containing

Kerr-form nonlinearity. It is well known that in 1D DSL the light transmission

in the exact phase-matched condition (the incident wave vector satisfied the

exact Bragg condition) corresponds to the forbidden state. However, in 2D

DSLs, when the transmitting light satisfies the exact Bragg condition of

multi-wave diffraction, whether the transmission state is located in the forbidden

band or in the allowed band is determined by the value of a set of parameters

which are defined as the RIM strengths of the 2D periodic structure. If the

values of these parameters are arranged suitably, the light will propagate in

the allowed band. Changes of these values will lead to high or low transmission

states for each diffracted wave, since in the allowed band in the exact Bragg

condition, the intensities of the multi-diffracted waves are oscillation functions

of these parameters.

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Handbook of dielectric, piezoelectric and ferroelectric materials944

31.4.2 Nonlinear optical frequency conversion and optical

parametric oscillation in 1D periodic DSL

Nonlinear optical frequency converters are attractive sources of coherent

radiation in applications for which laser sources are unavailable or for which

wide tunability is needed. One promising technique is QPM in periodic

DSLs. A significant advantage of QPM is that any interaction within the

transparency range of the material can be non-critically phase matched at

any temperature, even the interactions for which birefringence phase matching

is impossible (for example, LT crystals). Another benefit is that the interacting

waves can be chosen so that coupling occurs through the largest element of

the χ

(2)

tensor. In LN, QPM with all waves polarized parallel to the z-axis

yields a gain enhancement over the birefringence phase-matched process of

(2d

/πd

)

. Much work has been done on LN, LT, KTP, SBN and RbTiOAsO

(RTA) because of their large nonlinear optical coefficients. As early as 1980, we

prepared for the first time a periodic DSL LN by the Czochralski method (Feng

et al., 1980). Later with the same method, we fabricated the first periodic

DSL LT (Wang et al., 1986). With these periodic DSLs, we verified the QPM

theory proposed by Bloembergen et al. in 1962 (Armstrong et al., 1962).

In 1985, Feisst and Koidl (1985) performed the experiment with LN DSL

of less periods, prepared through the application of an alternating electric

current during the growth process. Later, Magel et al. (1990) realized the

blue light SHG in an LN fiber DSL. In 1990s, the QPM technique, spurred

by the need for blue light laser sources for data storage, compact disk players,

etc., has made great progress (Feng et al., 1980; Yamada et al., 1993; Miller

et al., 1997; Mizuuchi et al., 1997; Zhu S.N. et al., 1995a, b, 1997a, b; Byer

1994; Webjorn et al., 1994; Chen and Risk, 1994; Myers et al., 1995; Ito et

al., 1995; Karlsson et al., 1996; Zheng et al., 1998; Bierlein et al., 1990; Arie

et al., 1998; Lu Y.Z. et al., 1991, 1994, 1996a, b, c, d; Englander et al., 1997;

Myers et al., 1996; Burr et al., 1997; Missey et al., 1998; Kartaloglu et al.,

1998; Reid et al., 1997; Wang et al., 1998; Karlsson et al., 1996; Mizuuchi

and Yamamoto, 1992; Hadi et al., 1997; Arbore and Fejer, 1997; Serkland et

al., 1997; Kintaka et al., 1997; Eger et al., 1997; Arbore et al., 1997; Reid

et al., 1998; Chou et al., 1998; Landry and Maldonado, 1997; Qin et al.,

1998). A highly efficient QPM SHG has been demonstrated in LN DSL and

KTP DSL in both continuous wave (CW) and pulsed regimes. For example,

single-pass CW and quasi-CW SHG with efficiencies high as 42% (Miller et

al., 1997) and 65% (Pruneri et al., 1996) were realized in LN DSLs, respectively.

An internal conversion efficiency of 64% was achieved using a KTP DSL for

single-pass SHG of high-repetition-rate, low-energy, diode-pumped lasers

(Englander et al., 1997). By placing a KTP DSL in an external resonant

cavity, a conversion efficiency of 55% was obtained for a CW Nd : YAG

laser. When used for optical parametric oscillator (OPO), DSLs show

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Novel physical effects in dielectric superlattices 945

advantages such as high gain, low threshold and engineerability of domain

structures, thus making it possible to develop a robust, all solid-state, diode-

pumped, miniaturized OPO (Myers et al., 1996; Burr et al., 1997; Missey et

al., 1998; Kartaloglu et al., 1998; Reid et al., 1997). For example, Batchko

et al., (1998) demonstrated a 532 nm CW pumped single-resonant OPO

based on a 5.2cm long first-order QPM LN sample with a 64% quantum

efficiency and a threshold less than 1 W. With all these achievements, it is

possible, by intracavity frequency doubling the outputs (signal and idler

waves), to generate blue and red light for display applications. For more

detailed discussions on the SHG and OPO with the DSL, interested readers

are referred to a the review article by Byer (1997).

Usually, DSL LN and LT are limited to periods that are not very short and

to thicknesses ≤1 mm because of the side growth of inverted domains. This

inhibits the use of the DSL LN and LT in UV SHG and high-power pulsed

OPOs. However, with the improvement of the poling technique and the use

of new materials, this situation has been changed gradually. High-quality DSL

LT with period as short as 1.7µm was prepared for UV SHG (Mizuuchi et al.,

1997). DSL LN (Burr et al., 1997), SBN (Zhu Y.Y. et al., 1997a, b) and KTP

(Wang et al., 1998) with 1mm thickness were fabricated successfully. The

low coercive field of RTA, which allows the poling of thick crystals (3mm)

(Karlsson et al., 1996), together with its high damage threshold, low temperature

dependence and high resistance to photorefractive damage, makes this material

more suitable for high-power applications. Thus far in our laboratory, efficient

pulsed SHG (including ns, ps and fs) has been obtained with periodic DSL

LN, LT, SBN samples prepared by the room temperature poling technique

(RTPT) or Czochralski method. We obtained experimental results for several

samples by using a picosecond automatic tunable OPO as a fundamental

source (Lu Y. L. et al., 1996a). The CW 0.35mW, 405nm blue light and

CW1.34 mW, 489nm blue light generations by directly doubling an 810nm

GaAlAs laser diode and a 978nm InGaAs laser diode, respectively, have been

reported (Lu Y. L. et al., 1996a, b). Periodic DSL Nd : MgO : LN has been

grown for self-frequency-doubling (Lu Y. L. et al., 1996b). Visible dual-

wavelength light generation through upconversion and QPM frequency doubling

has been realized in erbium-doped periodic DSL LN (Zheng et al., 1998).

31.4.3 Nonlinear optical frequency conversion in 1D

quasi-periodic DSLs

Before 1984, research work on superlattices was mainly focused on periodic

structures. The discovery of quasi-crystals has opened a new field in condensed

matter physics and therefore attracted much attention (Steinhardt and Ostlund,

1997; Janot, 1992). Can this kind of material be of any practical use? It is

well known that the key to QPM is to construct a 1D periodic structure that

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Handbook of dielectric, piezoelectric and ferroelectric materials946

provides a reciprocal vector to compensate the mismatch of wave vectors

due to the dispersive effect of the refractive index. This 1D periodic structure

can provide a series of reciprocal vectors, each of which is an integer times

a primitive vector. In other words, the reciprocal vectors of a periodic DSL

are determined by an integer and one structural parameter. It is the reciprocal

vectors that make the optical parametric processes in the material phase

matched. Compared with the periodic structure, a 1D quasi-periodic dielectric

superlattice (QPDSL) has low space group symmetry, but its symmetry is

higher than that of an aperiodic structure. Its reciprocal vectors are governed

by two integers and two structural parameters rather than by one integer and

one structural parameter as in the case of the periodic one. This fact makes

the QPDSL more flexible in structure design. Therefore, a QPDSL can provide

more reciprocal vectors to the QPM optical parametric process. Because of

this, not only the QPM multi-wavelength SHG but also some coupled

parametric processes, such as the third-harmonic generation (THG) and fourth-

harmonic generation, can be realized with high efficiency (Feng et al., 1990;

Zhu and Ming, 1990, Zho Y.Y. et al., 1997a, b, 1998; Qin et al., 1999). This

is an example of possible applications of quasiperiodic structure materials in

nonlinear optics.

The construction of QPDSL

The QPDSL with Fibonacci sequence is constructed as follows. We first

define two fundamental blocks A and B, which are arranged according to the

production rule S

= S

j–1

j–2

with j > 3, S

= A and S

= AB. Each block is

composed of one positive and one negative ferroelectric domain, so that the

neighboring domains are interrelated by a dyad axis in the x direction. The

thickness of the positive ferroelectric domain in blocks A and B are l

and

, respectively. The thickness of the negative ferroelectric domain in block

A is l

and that in block B is l

. Here, l

is chosen to be equal to l

, that

is l

= l

. The details of the structure parameters can be seen in Fig.

31.5(a). The sequence of the blocks, ABAABABA . . . produces a QPDSL

(see Fig. 31.5(b)).

The sample was fabricated by poling a z-cut LT single domain wafer at

room temperature. In the sample of QPDSL for experiment, blocks A and B

consist nominally of 11, 13 µm and 11, 65µm, respectively. The sample is

the 13 generation, 377 A and B blocks, with a total length of ∼8 mm and a

thickness of ∼0.5mm.

Theoretical treatment of the nonlinear optical processes in QPDSLs

Here, we consider a case in which a single laser beam with ω

= ω is incident

from the left onto the surface of a QPDSL made from, for example, LN,

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Novel physical effects in dielectric superlattices 947

propagates along the x-axis with its polarization along the z-axis in order to

use the largest nonlinear optical coefficient d

as shown in Fig. 31.5. Through

the nonlinear optical effect, the SHG and THG exist simultaneously in the

QPDSL. Here these three electric fields, E

, E

and E

must be taken into

account, which satisfy the wave equation:

∇

∂

= + E

µε µ

31.1

where P = P

+ P

. The parametric SHG and THG processes can be analyzed

by solving the coupled nonlinear wave equations that describe the interaction

of the three fields in the 1D QPDSL structure:

ifxAA ikxifxAA ikx

2122

= – 2 ( ) exp(– ) – ( ) exp(– )

′′

κκ∆∆

ifxA ikxifxAA ikx

12 1

= – ( ) exp(– ) – ( ) exp(– )

′′

κκ∆∆

ifxAA ikx

212 2

= – ( ) exp( )

′

∆

31.2

where f (x) is a quasiperiodic function in positive and negative domain with

fx() =

–1







AAA

(a)

(b)

31.5

A QPDSL made from a LT single crystal. (a) Two building blocks,

A and B, each composed of one positive and one negative

ferroelectric domain. (b) Schematic diagram of a QPDSL with

Fibonacci sequence.

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Handbook of dielectric, piezoelectric and ferroelectric materials948

′

ωω

′

ωωω

33 1 2 3

123

cnnn

∆k

= k

2ω

– 2k

∆k

= k

3ω

– k

2ω

– k

Here f (x) can be Fourier transformed to (Birch et al., 1990)

fx f iG x( ) = exp( )

m,n

m,n m,n

31.3

with the reciprocal vector

mpn

m,n

2[ + ()]πγ

31.4

and

pl l

Gpl

m,n

2[1 + ( )]

() +

sin

()

sin

()

sin ( )

31.5

Here m and n are two integers and D = γ (p)l

+ l

. In fact, only the Fourier

component that is phase matched contributes significantly to the parametric

interaction. If the non-phase-matched components are ignored, the coupled

equations become:

iAA ikxiAA iKx

2122

= – 2 exp(– ) – exp(– )κκ∆∆

iA ikxiAA iKx

121

= – exp( ) – exp(– )κκ∆∆

iAA iKx

212 2

= – exp ( )κ∆

31.6

with

∆K

= k

2ω

– 2k

– G

m,n

∆K

= k

3ω

– k

2ω

– k

– G

m′,n′

31.7

κκ

ωω

m,n

= =

′

κκ

ωωω

m,n

123

= = f

cnnn

′′

′

31.8

The equation can be solved analytically for the boundary conditions of A

(0)

= 0 and A

(0) = 0 under a small signal approximation:

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Novel physical effects in dielectric superlattices 949

kk A

CikkAL

=–

++4

exp –

(– +4 )

∆∆

[]

–

–+4

exp –

(+ +4 )

∆∆

kk A

CikkAL

[]

–

exp( )

kk A

ikL

∆

∆∆

∆

31.9

AC i k k AL

31 2

= exp

(+ +4 )∆∆

[]

+ exp

(– +4 )

Cikk AL∆∆κ

[]

–

exp( )

κκ

kk A

ikL

∆∆

∆

31.10

with

kk A

kk k A

= –

–

++ +4

2+4

κκ

∆∆

∆∆ ∆

∆

and

kk A

kk k A

–

+– +4

2+4

κκ

∆∆

∆∆ ∆

∆

∆k

= ∆k

+ ∆k

where L is the length of crystal.

Under the QPM conditions

∆K

= k

2ω

– 2k

– G

m,n

= 0 31.11

∆K

= k

3ω

– k

2ω

– k

– G

m′,n′

= 0

Equations (9) and (10) can be simplified to

AiALcAL

= sin ( )κκ

AALcAL

312

32 2

= – 2 sin

κκ κ

()

31.12

Using the following relation, we can get the harmonic intensity:

31.13

If the κ

L is very small, then we can get the conversion efficiency:

ελ

m,n

nnc

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