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

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Handbook of dielectric, piezoelectric and ferroelectric materials950

ελ

m,n

144

ddL

nnnc

′

31.14

Usually under the small signal approximation, the second harmonic generated

through the second-order nonlinear process and the third harmonic generated

through the third-order nonlinear process are proportional to L

Here because

of the coupling effect between the SHG and sum frequency generation (SFG)

processes, the third harmonic generated through the second-order nonlinear

process is proportional to L

. Thus it is possible to obtain THG in a QPDSL

with high efficiency. In addition, it can be seen clearly that the effective

nonlinear optical coefficients play an important role in nonlinear parametric

processes. Thus it is helpful to derive the expression for the effective nonlinear

optical coefficients.

Effective nonlinear optical coefficients

For QPM SHG in a periodic DSL, the conversion efficiency is proportional

to the square of the effective nonlinear optical coefficient, which is represented

sin

()

with l the length of the reverse domain (Fejer et al., 1992). n = 1 corresponds

to the first-order QPM with the largest conversion efficiency. For QPM THG

in a QPDSL, the situation becomes much more complex. In order to see the

relationship, it is needed to obtain the expression of the effective nonlinear

optical coefficient for QPDSL.

For the QPDSL, the modulated nonlinear coefficient d(x) = d

f (x). Then

the effective nonlinear coefficient is

ddfd

pl l

Gpl

m,n

= =

2[1 + ( )]

() +

sin

()

sin

()

sin ( )

31.15

where X

m,n

= πD

–1

[1 + γ(p)](ml

– nl

). Obviously the value of d

m,n

depends

strongly on the adjustable structure parameters such as l and the ratio l

Note that when p = 1, the result is just one for the Fibonacci sequence. When

= l

, the QPDSL turns back to a periodic DSL and d

m,n

reduces to

sin

()

with m = n, the effective nonlinear coefficient for the

periodic DSL.

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Novel physical effects in dielectric superlattices 951

QPM multi wavelength SHG (Zhu and Ming, 1990; Zhu S. N. et al.,

1997b; Qin et al., 1999)

To verify the theoretical predictions, two types of QPDSL have been fabricated:

one with p = 1, the so-called Fibonacci type, and the other one a generalized

QPDSL with p = 2. The sample was fabricated by poling a z-cut LT single

domain wafer at room temperature (Zhu S. N. et al., 1995a,b). The SHG

spectrum of the QPDSL LT was measured with the fundamental tuned in the

infrared. With the QPDSL samples, we obtained QPM second harmonic

blue, green, red and infrared light output with conversion efficiencies up to

∼5–20%. The measured and calculated results are in good agreement.

It is theoretically predicted and experimentally observed that the X-ray

diffraction and Raman spectra of quasi-periodic superlattice exhibit self-

similarity. However in the SHG spectra of the QPDSL, careful analysis of

the measured spectrum show that the self-similarity no longer holds. This is

due to the dispersive effect of the refractive index, although the reciprocal

vector does in reciprocal space.

As pointed out, here the QPM multi-wavelength SHGs are wholly

determined by the distribution of the reciprocal vectors of the QPDSL. Therefore

with the aid of Fourier transformation, arbitrary wavelength-response functions

can be obtained by design of appropriate DSL (Chou et al., 1999). The QPM

structures with multiple phase-matching wavelengths can be used for

wavelength-division-multiplexed wavelength conversion. Using the SH spectra

obtained in QPDSL either with p = 1 or p = 2, the dispersion relationship of

the refractive index has been deduced (Zhu S. N. et al., 1997b; Qin et al.,

1999).

Direct third-harmonic generation (Feng et al., 1990; Zhu S. N. et al.,

1997a; Zhu Y. Y. et al., 1998; Qin et al., 1999)

THG has a wide application as a means to extend coherent light sources to

short wavelengths. The creation of the third harmonic directly from a third-

order nonlinear process is of little practical importance because of the intrinsic

low third-order optical nonlinearity. Conventionally, an efficient THG was

achieved by a two-step process. Two nonlinear optical crystals are needed:

the first for SHG and the second for SFG. In this regard, QPDSL has some

advantages over the conventional method. Here only one crystal is needed

and the harmonic generation can be realized using the largest nonlinear

optical coefficient over the entire transparency range of the material with

high efficiency. For the QPDSL, the QPM conditions for THG in a collinear

interaction are given by equation 11. THG was tested with a tunable optical

parametric oscillator. Several THGs have been detected; however, only one

TH has a high conversion efficiency (>20%). Others are very weak. Theoretical

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Handbook of dielectric, piezoelectric and ferroelectric materials952

analysis has shown that the THG can be generated through the nearly QPM

SFG with the SHG either QPM or mismatched. However, an efficient third

harmonic can be generated only when both the SHG and SFG processes are

quasi-phase matched. Recently, QPM THG has been demonstrated using a

simple silica structure of six modulation periods through cubic nonlinearity

(χ

ijkl

) (Williams et al., 1998). CW frequency tripling by simultaneous three-

wave mixings has been realized in an LN crystal (Pfister et al., 1997). In an

LN waveguide, ultraviolet THG of 355nm has been observed (Kintaka et al.,

1997).

With the aid of RTPT, we have performed a systematic study on

QPDSL SBN related to SHG and THG (Zhu S. N. et al., 1997a, b; Zhu Y. Y.

et al., 1998).

31.4.4 Nonlinear optical frequency conversion

in 2D DSLs

In 1998, Berger (Berger, 1998) extended the QPM study from 1D to 2D, and

proposed the concept of an χ

(2)

photonic crystal in order to contrast and

compare it with a regular photonic crystal having a periodic linear susceptibility.

In a configuration proposed by Berger, a crystal has a 2D periodic modulation

of nonlinear susceptibility, and a linear susceptibility that is a function of the

frequency, but constant in space. It is an in-plane generalization of 1D QPM

structures and can be realized in periodic poled LN or in GaAs. An interesting

property of these structures is that new phase-matching processes appear in

the 2D plane as compared with the 1D case. It is shown that these in-plane

phase-matching resonances are given by a nonlinear Bragg law (Berger,

1998):

1 – + 4 sin









31.16

where λ

2ω

is the SH wavelength inside the material and 2θ the walk-off

angle between k

2ω

and k

. More generally, this equation gives the direction

of coherent radiation at the wavelength λ

2ω

for a phased array of nonlinear

dipoles having a phase relation fixed by the propagation of the pump. Nonlinear

Bragg law is a generalization for nonlinear optics of the Bragg law. It gives

the direction of resonant scattering at the wavelength λ

2ω

of a plane wave

with vector k

by a set of nonlinear dipoles. Since then, the concept of a χ

(2)

photonic crystal has been gradually accepted in the nonlinear optics community.

Actually χ

(2)

photonic crystals can also be called 2D DSLs when χ

(2)

modulation

is obtained by ferroelectric modulation in the 2D system. Recently several

novel classes of nonlinear optical effects in 2D DSLs were discovered by

researchers, such as second-order SHG in 2D square poled LN, tunable SHG

in a tetragonal poled LN, and higher-order harmonic generation with cascaded

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Novel physical effects in dielectric superlattices 953

process in a hexagonally poled LN. In addition, in 2D DSLs, generation of

a flat-top second-harmonic beam and nonlinear optical diffraction with

transversely patterned QPM gratings have be realized by Fejer’s group (Kurz

et al., 2002). Baudrier-Raybaut et al. (2004). investigated continuous SHG

in 2D random domain structures.

In our laboratory, Xu et al. (2004) reported a novel nonlinear optical

phenomenon – a conical second-harmonic (SH) beam, generated from 2D

DSLs – a hexagonally poled LiTaO

(HPLT) crystal. The conical SH beam

emerged when the HPLT was illuminated with a z-polarized fundamental

beam with no special requirements. The conical beams were visualized as

rings when projected onto a screen behind the crystal, and were presented in

axial symmetry or mirror symmetry when the fundamental beam propagated

along a symmetrical axis of the hexagonal structure. This reveals the presence

of another type of nonlinear interaction – a scattering involved optical

parametric generation. The interaction is greatly enhanced by a QPM process

in the 2D DSLs; thus, the infrared scattering signal (ω) is converted to the

visible band (2ω) through mixing with the incident wave (ω). Further study

confirms that the conical beam records the spatial distribution of the scattering

signal and reveals the structure information of the 2D DSLs.

The QPM approach can enhance not only the elastic scattering signal

intensity but also the inelastic scattering signal intensity in a nonlinear medium.

Raman scattering, as a sort of inelastic scattering, is always accompanied by

frequency change. Raman scattering can be mediated by many different

types of elementary excitations in a medium, such as transverse optical

phonons, phonon polaritons or plasmons. Compared with elastic scattering,

Raman scattering is much weaker, which is possibly enhanced by the QPM

process in a χ

(2)

modulated medium. Xu et al. (2005) reported the experimental

studies on QPM enhanced phonon–polariton (P–P) Raman scattering in artificial

microstructure – an HPLT crystal. The QPM approach was first operated to

generate two laser beams to drive an intense coherent (P–P) field in the

HPLT crystal and was then used to amplify the generated Raman signals by

parametric amplification, which resulted in the generation of a stimulated P–

P Raman scattering. The anti-Stokes and Stokes spectrum with very low

Raman shift down to 2cm

–1

and very high scattering order up to the 11th

rank was detected from the medium. The resulting spectrum exhibits a comb-

shaped structure with an equal frequency interval tunable by changing the

frequency of excited polariton. The QPM approach was successfully introduced

into the 2D DSLs medium for enhancement of inelastic scattering.

31.5 Application of DSLs in acoustics

It was well known that for an unpoled ferroelectric crystal, the resonances at

low frequencies, which are related to the geometry and dimensions of the

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Handbook of dielectric, piezoelectric and ferroelectric materials954

measured sample, are absent. Only the resonance at a very high frequency,

which is related to the domain structure in the sample, is observed. The

position of the resonant peak and its bandwidth depend on the sizes of

domains and their size distribution. For a single domain ferroelectric crystal,

only the low frequency resonance related to the geometric sizes of the sample

and its high-order harmonics can be detected, whereas high frequency resonance

is absent. All resonances in a ferroelectric material, whether poled or unpoled,

originate from the domains and are excited through piezoelectric effect. In a

ferroelectric superlattice, piezoelectric coefficient h

ijk

, as a third-rank tensor,

is a periodic or quasi-periodic function of spatial coordinates, depending on

the array of domains. The modulation of piezoelectric coefficient can result

in some novel acoustic effects. It is the reason why ferroelectric superlattice

sparks so much interest in the ultrasonic field. Since 1988 Zhu et al. (1988)

have systematically studied the excitation and propagation of elastic waves

in DSL and successfully fabricated various DSL devices (Zhu et al., 1996).

Owing to piezoelectricity, the discontinuity of the piezoelectric stresses at

the domain walls may be produced under the action of an external electric

field. The stress must be balanced by a strain S(u

), where u

(m = 1, 2, …)

represent the positions of the domain walls. If the external field is an alternating

field, the strain can propagate as an elastic wave:

S(u) = S(u

) cos(ωt – ku) 31.17

where ω, k and t are the angle frequency, wave vector and time, respectively.

Every domain wall can be viewed as a δ sound source. All domain walls in

a DSL are arranged in a certain sequence forming an array. The elastic waves

excited by this δ sound source array will interfere with each other when

certain frequency conditions are satisfied. Those satisfying the condition for

constructive interference will appear as resonances. This is the physical

basis for the ultrasonic excitation in DSL.

31.5.1 High-frequency resonance in DSLs

As an example, considering the case of Fig. 31.6 (b), an alternating voltage

is applied on the z faces of the superlattice, thus a longitudinal planar wave

propagating along the z-axis will be excited inside the sample. This situation

is described by the wave equation:

∂









∂













33 3

–

( – )

31.18

where u

represents the particle displacement along the z direction, v is the

velocity of sound, h

and

are the piezoelectric and elastic

coefficients,

respectively, and D

is the component of the electric displacement along the

z-axis.

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Novel physical effects in dielectric superlattices 955

For a DSL with periodic domain structure, by using the Green’s function

method to solve the elastic wave equation, the electric impedance of the DSL

can be derived, and then the resonance frequency can be obtained as follows:

( + )

⋅

, n = 1, 2, 3, … 31.19

where υ is the velocity of the longitudinal wave propagating along the z-

axis, a and b are the thickness of positive and negative domains, respectively,

and a + b is the period Λ of the DSL. It is obvious that the resonance

frequency is determined only by the period of the DSL, i.e. a + b, not by the

total thickness of the wafer. The thinner the laminar domains, the higher the

resonance frequencies. As we know, the thickness of a resonator working at

several hundred MHz is about several micrometers. An ordinary material

with such a thickness is too thin to be fabricated by current mechanical

processing methods and too thick to be deposited by film growth techniques.

However, it is easy to grow by the Czochralski method (Zhu and Ming et al.,

1992a) or fabricated by poling (Chen et al., 1997). In practice, a domain

period with several micrometers has been achieved by using these two methods,

which corresponds to resonance frequencies of hundreds MHz to several

GHz (Zhu and Ming, 1992b).

For a DSL with domain configuration like Fig. 31.6 (a), acoustic waves

can be excited through two different schemes. One is an in-line scheme with

the acoustic propagation vector parallel to the applied electric field. The

other is a cross-field scheme, which is characterized by an electric field

perpendicular to the propagation vector. A schematic diagram of the two

kinds of schemes is shown in Fig. 31.7. Zhu and Ming (1992b) studied the

two kinds of acoustic excitations in a DSL theoretically and Chen et al.

(1997) experimentally confirmed their theory in the DSLs based on LT

superlattices and fabricated a set of resonators operating in the range of

several hundreds MHz. The devices made of DSLs can be divided into two

(a)

(b)

31.6

Periodic superlattice with the spontaneous polarization

modulated along (a) the

-axis and (b) the

-axis.

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Handbook of dielectric, piezoelectric and ferroelectric materials956

categories, resonator and transducer types, depending on their boundary

conditions. For resonator, both electrode faces of DSL are free, whereas for

transducer, one face is fully matched to a transmission medium. The insertion

loss of transducer is an important parameter for an acoustic device. For

transducers made of homogeneous piezoelectric materials, such as a single-

domain LN wafer, the static capacitance is the main part of the impedance at

the resonance frequency under high-frequency operation. As a result, the

insertion loss of the transducer is very high. In DSL case, the impedance of

transducer may be adjusted by choosing the number of periods N and the

area of the electrode face A. As a example, an insertion loss of near 0 dB

at 555MHz was achieved in a 50Ω measurement system (Zhu et al.,

1988).

Acoustic excitation and propagation in a quasi-periodic superlattice have

been studied both theoretically and experimentally (Zhu et al., 1989). The

excitation spectrum in a Fibonacci superlattice is expressed by the following

equation:

H(k) ∝ sin (kL/2) ∑ (sin X

m,n

) δ{k – 2π(m + nτ)/D} 31.20

where X

= πτ(mτ – n)/(1 + τ

) and τ = (1+√5)/2 the golden mean. The self-

similarity of the Fibonacci sequence in the reciprocal space was experimentally

confirmed by the acoustic excitation spectrum.

–

cut

–

cut

–

cut

31.7

A schematic diagram of the ultrasonic excitation in an acoustic

superlattice made of a LT crystal in which the domains are arranged

periodically along the

-axis, and the spontaneous polarization

directions of these domains are parallel to the

-axis. The excitation

by the electrodes coated on the

faces (

–

cut) corresponds to the

in-line scheme, while electrodes coated on the

faces (

–

cut) and

the

faces (

–

cut) excite acoustic waves by the cross-field scheme

(after Chen

et al.,

1997).

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Novel physical effects in dielectric superlattices 957

31.5.2 Polariton excitation in DSLs

In a real crystal, coupling exists among the motions of electrons, photons

and phonons. For example, infrared absorption and polariton excitation result

from the coupling between lattice vibrations and electromagnetic waves in

an ionic crystal. It is expected that the same coupling may occur in artificial

materials with modulated domain structures. Lu Y. Q. et al. (1999b) considered

the case of a ferroelectric superlattice with the ‘head to head’ configuration

(Fig. 31.6(b)). This structure is similar to a 1D diatom chain with positive

and negative ‘ions’ connected periodically. They called the superlattice an

ionic-type phononic crystal (ITPC) and expected that a polariton excitation

might occur in it.

In a real ionic crystal, the polariton excitation originates from the coupling

of electromagnetic wave and the lattice vibration of crystal, appearing within

the infrared region. In an ITPC, this excitation may originate from the coupling

between the superlattice vibration and electromagnetic wave. The piezoelectric

effect leads to the vibration of superlattice when there is external

electromagnetic field. The vibration frequency depends on the period and

material constants of superlattice, so the polariton excitation in such a

ferroelectric superlattice is expected to occur in the microwave region.

To verify the prediction, Lu Y. Q. et al. (1999b) prepared a LN superlattice

with a period of 7.2µm and a ‘head to head’ configuration by the growth

striation method, and calculated its polariton dispersion curve. The measurement

of dielectric spectrum confirmed that there was a gap between the calculated

and f

where ε < 0 (Fig. 31.8), where an incident electromagnetic wave

would be strongly reflected. The measured reflection coefficient as a function

of frequency showed that there was an absorption peak (~26 dB) at 502

MHz. The results verified there was a polariton mode in the measured sample

indeed. According to the above results, one can expect that other long-

wavelength optical properties (such as Raman and Brillouin scattering) in a

real ionic crystal also exist in such a ferroelectric superlattice. The only

difference is that they occur in different frequencies, one in the infrared

region (THz) and the other one in the microwave region (GHz).

31.6 Application of DSLs in electro-optic

technology

In the linear optics regime, physics effects in a ferroelectric superlattice

mainly involved in modulated electro-optic coefficient r

ijk

. As a third-rank

tensor, its elements have opposite signs in a positive and a negative domain.

In the presence of an external field along some axis of the crystal, the

modulation of electro-optic coefficient will accordingly lead to the modulation

of refractive index, or the alternating rotation of the principle axis due to the

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Handbook of dielectric, piezoelectric and ferroelectric materials958

deformation of the refractive index ellipsoid in the superlattice. Lu et al.

(2000) and Zhu and Ming (1992a) studied the electro-optic effects and the

transmission spectra in a periodic and a quasi-periodic LN superlattice,

respectively. Figure 31.9 shows the electro-optic effect in a LN superlattice.

The electrodes are coated on the y surfaces of the superlattice. In the absence

of an external electric field, the principal axes of the positive domains overlap

with those of the negative domains and the dielectric tensor has only diagonal

components with respect to the principal axes. The superlattice is homogeneous

to the propagation of light in the linear optics regime. There is no refractive

index modulation accompanying the domain modulation. In the presence of

an external electric field, however, the dielectric tensor is perturbed because

of the electro-optic effect, which results in a small dielectric modulation

along the propagation direction of light. The new dielectric principal axes

may no longer overlap because new off-diagonal components appear in their

31.8

(a) Calculated polariton dispersion curve of an ionic-type

phononic crystal with the period of 7.2 mm without consideration of

damping. There is a frequency gap between

and

where no EM

waves are permitted to propagate in the sample. (b) The measured

reflection coefficient of the sample in the microwave band. The

minimum of the reflection coefficient indicates that there is a strong

microwave absorption peak at 502 MHz (after Lu

et al.

, 1999).

100 200 300 400 500 600 700 800 900

Frequency (MHz)

(b)

Frequency (MHz)

1200

1000

800

600

400

200

0 50 100 150 200 250

–1

)

Reflection coefficient (dB)

–5

–10

–15

–20

–25

–30

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Novel physical effects in dielectric superlattices 959

original dielectric tensors. The new principal axes rotate from the original

principal axes by an angle with opposite signs in positive and negative

domains, respectively. The angles depend on the applied external field. In

Fig. 31.9, the initial condition at x = 0 which is determined by the polarizer

is given by E

(0) = 0 and E

(0) = 1, where E

and E

are the mode amplitudes

for y-polarized and z-polarized light, respectively. According to the coupled-

mode theory, Lu et al. (2000) and Zhu and Ming (1992a) considered the

coupling effect of light beams with orthogonal polarization in a periodic and

a quasi-periodic superlattice, respectively. They found that the energy could

be transferred back and forth between these two orthogonal modes in this

electro-optic system. At the analyzer (y-polarized), i.e. x = L (which is directly

related to the number of domain blocks), E

is extinguished, and the

transmission of the y-component is wavelength dependent and is controllable

by an applied electric field. Their results verify that the periodic and quasi-

periodic superlattices are similar to adjustable Solc filters. Moreover, Lu et

al. (2000) proposed an electro-optic tuning scheme to tune the output frequency

of a QPM OPO. Compared to temperature tuning, the electro-optic tuning

provides a faster response. The tuning rate was expected to excess 3 nm/kV/

mm. Recently Lu et al. (2000) proposed a high-frequency traveling-wave

integrated electro-optic modulator based on a periodically poled LN. The

traveling velocity of the optical wave and the electrical wave velocity in the

waveguide can be quasi-matched due to the periodic structure. Using this

design, a wide-bandwidth electro-optic modulator with several hundred GHz

can be realized.

Analyzer

Polarizer

Input

31.9

Schematic diagram of an DSL electro-optic system.

X, Y, Z

denote the principal axes of the unperturbed dielectric tensor, and

, Z

and

, Z

are the principal axes of the perturbed dielectric

tensor of negative

and

positive domains, respectively.

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