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    I . C . Chang Aurora Associates Santa Clara , California

    1 2 . 1 GLOSSARY

    a acoustic attenuation

    d θ o , d θ a divergence : optical , acoustic

    D Bm impermeability tensor

    D f , D F bandwidth , normalized bandwidth

    D n birefringence

    D θ deflection angle

    l o optical wavelength (in vacuum)

    f , F acoustic frequency , normalized acoustic frequency

    L acoustic wavelength

    r density

    τ acoustic transit time

    c phase mismatch function

    a divergence ratio

    D optical aperture

    E i , E d electric field : incident , dif fracted light acoustic wave

    k i , k d , k a wavevector : incident , dif fracted light , acoustic wave

    L , , interaction length , normalized interaction length

    L o characteristic length

    M figure of merit

    n o , n e refractive index : ordinary , extraordinary

    12 .1


    P a , P d acoustic power , acoustic power density

    p , p m n , p i j k l elasto-optic coef ficient

    S , S i j strain , strain tensor components

    t r rise time

    T scan time

    V acoustic velocity

    1 2 . 2 INTRODUCTION

    When an acoustic wave propagates in an optically transparent medium , it produces a periodic modulation of the index of refraction via the elasto-optical ef fect . This provides a moving phase grating which may dif fract portions of an incident light into one or more directions . This phenomenon , known as the acousto-optic (AO) dif fraction , has led to a variety of optical devices that perform spatial , temporal , and spectral modulations of light . These devices have been used in optical systems for light-beam control and signal- processing applications .

    Historically , the dif fraction of light by acoustic waves was first predicted by Brillouin 1 in 1922 . Ten years later , Debye and Sears 2 and Lucas and Biquard 3 experimentally observed the ef fect . In contrast to Brillouin’s prediction of a single dif fraction order , a large number of dif fraction orders were observed . This discrepancy was later explained by the theoretical work of Raman and Nath . 4 They derived a set of coupled wave equations that fully described the AO dif fraction in unbounded isotropic media . The theory predicts two dif fraction regimes ; the Raman-Nath regime , characterized by the multiple of dif fraction orders , and the Bragg regime , characterized by a single dif fraction order . Discussion of the early work on AO dif fraction can be found in Refs . 5 and 6 .

    Although the basic theory of AO dif fraction in isotropic media was well understood , there had been relatively few practical applications prior to the invention of the laser . It was the need of optical devices for laser beam control that stimulated extensive research on the theory and practice of acousto-optics . Significant progress of the AO devices has been made during the past two decades , due primarily to the development of superior AO materials and ef ficient broadband transducers . By now , acousto-optics has developed into a mature technology and is deployed in a wide range of optical system applications .

    It is the purpose of this chapter to review the theory and practice of bulkwave acousto-optic devices and their applications . The review emphasizes design and implemen- tation of AO devices . It also reports the status of most recent developments . Previous review of acousto-optics may be found in references 7 – 11 .

    In addition to bulkwave acousto-optics , there have also been studies on the interaction of optical guided waves and surface acoustic waves (SAW) . However , the ef fort has remained primarily at the research stage and has not yet resulted in practical applications . As such , the subject of guided-wave acousto-optics will not be discussed here . The interested reader may refer to a recent review article . 1 2

    This chapter is organized as follows : The next section discusses the theory of acousto-optic interaction . It provides the necessary background for the design of acousto-optic devices . The important subject of acousto-optic materials is discussed in the section following . Then a detailed discussion on three basic types of acousto-optic devices is presented . Included in the discussion are the topics of deflectors , modulators , and tunable filters . The last section discusses the use of AO devices for optical beam control and signal processing applications .



    Elasto-optic Ef fect

    The elasto-optic ef fect is the basic mechanism responsible for the AO interaction . It describes the change of refractive index of an optical medium due to the presence of an acoustic wave . To describe the ef fect in crystals , we need to introduce the elasto-optic tensor based on Pockels’ phenomenological theory . 1

    An elastic wave propagating in a crystalline medium is generally described by the strain tensor S , which is defined as the symmetric part of the deformation gradient

    S i j 5 S ­ u i ­ x j

    1 ­ u j ­ x i

    D Y 2 i ,j 5 1 to 3 (1) where u i is the displacement . Since the strain tensor is symmetric , there are only six independent components . It is customary to express the strain tensor in the contracted notation

    S 1 5 S 1 1 , S 2 5 S 2 2 , S 3 5 S 3 3 , S 4 5 S 2 3 , S 5 5 S 1 3 , S 6 5 S 1 2 (2)

    The conventional elasto-optic ef fect introduced by Pockels states that the change of the impermeability tensor , D B i j , is linearly proportional to the symmetric strain tensor .

    D B i j 5 p i j k l S k l (3)

    where p ijkl is the elasto-optic tensor . In the contracted notation ,

    D B m 5 p m n S n m ,n , 5 1 to 6 (4)

    Most generally , there are 36 components . For the more common crystals of higher symmetry , only a few of the elasto-optic tensor components are non-zero .

    In the above classical Pockels’ theory , the elasto-optic ef fect is defined in terms of the change of the impermeability tensor D B i j . In the more recent theoretical work on AO interactions , analysis of the elasto-optic ef fect has been more convenient in terms of the nonlinear polarization resulting from the change of dielectric tensor D » i j . We need to derive the proper relationship that connects the two formulations .

    Given the inverse relationship of » i j and B i j in a principal axis system D » i j is :

    D » i j 5 2 » i i D B i j » j j 5 2 n 2 i n

    2 j D B i j (5)

    where n i is the refractive index . Substituting Eq . (3) into Eq . (5) , we can write :

    D » i j 5 χ ijk , S k , (6)

    where we have introduced the elasto-optic susceptibility tensor ,

    χ ijk , 5 2 n 2 i n 2 j p ijk , (7)


    For completeness , two additional modifications of the basic elasto-optic ef fect are discussed as follows .

    Roto - optic Ef fect . Nelson and Lax 1 4 discovered that the classical formulation of elasto-optic was inadequate for birefringent crystals . They pointed out that there exists an additional roto-optic susceptibility due to the antisymmetric rotation part of the deforma- tion gradient .

    D B 9 ij 5 p 9 ijkl R k l (8)

    where R i j 5 ( S i j 2 S j i ) / 2 . It turns out that the roto-optic tensor components can be predicted analytically . The

    coef ficient of p ijkl is antisymmetric in kl and vanishes except for shear waves in birefringent crystals . In a uniaxial crystal the only nonvanishing components are p 2 3 2 3 5 p 2 3 1 3 5 ( n 2 2 o 2 n

    2 2 e ) / 2 , where n o and n e are the principal refractive indices for the ordinary and

    extraordinary wave , respectively . Thus , the roto-optic ef fect can be ignored except when the birefringence is large .

    Indirect Elasto - optic Ef fect . In the piezoelectric crystal , an indirect elasto-optic ef fect occurs as the result of the piezoelectric ef fect and electro-optic ef fect in succession . The ef fective elasto-optic tensor for the indirect elasto-optic ef fect is given by 1 5

    p * ij 5 p i j 2 r i m S m e j n S n » m n S m S n


    where p i j is the direct elasto-optic tensor , r i m is the electro-optic tensor , e j n is the piezoelectric tensor , » m n is the dielectric tensor , and S m is the unit acoustic wave vector . The ef fective elasto-optic tensor thus depends on the direction of the acoustic mode . In most crystals the indirect ef fect is negligible . A notable exception is LiNbO 3 . For instance , along the z axis , r 3 3 5 31 3 10 2 1 2 m / v , e 3 3 5 1 . 3 c / m 2 , E s 33 5 29 , thus p * 5 0 . 088 , which dif fers notably from the contribution p 3 3 5 0 . 248 .

    Plane Wave Analysis of Acousto-optic Interaction

    We now consider the dif fraction of light by acoustic waves in an optically transparent medium in which the acoustic wave is excited . An optical beam is incident onto the cell and travels through the acoustic beam . Via the elasto-optical ef fect , the traveling acoustic wave sets up a spatial modulation of the refractive index which , under proper conditions , will