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1、 PART 3 MODULATORS This page intentionally left blank ACOUSTO-OPTIC DEVICES I-Cheng Chang Accord Optics Sunnyvale, California 6.3 6 6.1 GLOSSARY dqo, dqadivergence: optical, acoustic Bmimpermeability tensor f, Fbandwidth, normalized bandwidth nbirefringence q defl ection angle lo, l optical waveleng
2、th (in vacuum/medium) acoustic wavelength r density t acoustic transit time y phase mismatch function A optical to acoustic divergence ratio a optical to acoustic wavelength ratio D optical aperture Ei, Ed electric fi eld, incident, diffracted light f, F acoustic frequency, normalized acoustic frequ
3、ency Hacoustic beam height ki, kd, kawavevector: incident, diffracted light, acoustic wave L, linteraction length, normalized interaction length Lo characteristic length M fi gure of merit no, nerefractive index: ordinary, extraordinary Pa, Pdacoustic power, acoustic power density p, pmn, pijkl elas
4、to-optic coeffi cient 6.4 MODULATORS S, SIstrain, strain tensor components tr,Trise time scan time V acoustic velocity W bandpass function 6.2 INTRODUCTION When an acoustic wave propagates in an optically transparent medium, it produces a periodic modu- lation of the index of refraction via the elas
5、to-optical effect. This provides a moving phase grating which may diffract portions of an incident light into one or more directions. This phenomenon, known as the acousto-optic (AO) diffraction, has led to a variety of optical devices that can be broadly grouped into AO deflectors, modulators, and
6、tunable filters to perform spatial, temporal, and spectral modulations of light. These devices have been used in optical systems for light-beam control, optical signal processing, and optical spectrometry applications. Historically, the diffraction of light by acoustic waves was first predicted by B
7、rillouin1 in 1921. Nearly a decade later, Debye and Sears2 and Lucas and Biquard3 experimentally observed the effect. In contrast to Brillouins prediction of a single diffraction order, a large number of diffraction orders were observed. This discrepancy was later explained by the theoretical work o
8、f Raman and Nath.4 They derived a set of coupled wave equations that fully described the AO diffraction in unbounded isotropic media. The theory predicts two diffraction regimes; the Raman-Nath regime, characterized by the multiple of diffraction orders, and the Bragg regime, characterized by a sing
9、le diffraction order. Discussion of the early work on AO diffraction can be found in Ref. 5. The earlier theoretically work tend to treat AO diffraction from a mathematical point of view, and for decades, solving the multiple-order Raman-Nath diffraction has been the primary interest on acousto-opti
10、cs research. As such, the early development did not lead to any AO devices for practical applications prior to the invention of the laser. It was the need of optical devices for laser beam modu- lation and deflection that stimulated extensive research on the theory and practice of AO devices. Signif
11、icant progress has been made in the decade from 1966 to 1976, due to the development of supe- rior AO materials and efficient broadband ultrasonic transducers. During this period several impor- tant research results of AO devices and techniques were reported. These include the works of Gordon6 on th
12、e theory of AO diffraction in finite interaction geometry, by Korpel et al. on the use of acoustic beam steering,7 the study of AO interaction in anisotropic media by Dixon;8 and the invention of AO tunable filter by Harris and Wallace9 and Chang.10 As a result of these basic theoretical works, vari
13、ous AO devices were developed and demonstrated its use for laser beam control and optical spectrometer applications. Several review papers during this period are listed in Refs. 11 to 14. Intensive research programs in the 1980s and early 1990s further advanced the AO technology in order to explore
14、the unique potential as real-time spatial light modulators (SLMs) for optical signal processing and remote sensing applications. By 1995, the technology had matured, and a wide range of high performance AO devices operating from UV to IR spectral regions had become commercially available. These AO d
15、evices have been integrated with other photonic components and deployed into optical systems with electronic technology in diverse applications. It is the purpose of this chapter to review the theory and practice of bulk-wave AO devices and their applications. In addition to bulk AO, there have also
16、 been studies based on the interaction of optical guided waves and surface acoustic waves (SAW). Since the basic AO interaction structure and fabrication process is significantly different from that of the bulk acousto-optics, this subject is treated separately in Chap. 7. This chapter is organized
17、as follows: Section 6.3 discusses the theory of acousto-optic interac- tion. It provides the necessary background for the design of acousto-optic devices. The subject of acousto-optic materials is discussed in Sec. 6.4. The next three sections deal with the three basic types of acousto-optic devices
18、. Detailed discussion of AO deflectors, modulators, and tunable filters are presented in Section 6.5, 6.6, and 6.7, respectively. ACOUSTO-OPTIC DEVICES 6.5 6.3 THEORY OF ACOUSTO-OPTIC INTERACTION Elasto-Optic Effect The elasto-optic effect is the basic mechanism responsible for the AO interaction. I
19、t describes the change of refractive index of an optical medium due to the presence of an acoustic wave. To describe the effect in crystals, we need to introduce the elasto-optic tensor based on Pockels phenomenological theory.15 An elastic wave propagating in a crystalline medium is generally descr
20、ibed by the strain tensor S, which is defined as the symmetric part of the deformation gradient S u x u x i j ij i j j i = + =213,to (1) where ui 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
21、contracted notation SSSSSSSSSSSS 111222333423513612 = (2) The conventional elasto-optic effect introduced by Pockels states that the change of the imperme- ability tensor Bij is linearly proportional to the symmetric strain tensor. BpS ijijklkl = (3) where pijkl is the elasto-optic tensor. In the co
22、ntracted notation B pSm n mmn n =,16to (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 nonzero. In the above classical Pockels theory, the elasto-optic effect is defined in terms of the change of the i
23、mpermeability tensor Bij. In the more recent theoretical work on AO interactions, analysis of the elasto-optic effect has been more convenient in terms of the nonlinear polarization resulting from the change of dielectric tensor eij. We need to derive the proper relationship that connects the two fo
24、rmulations. Given the inverse relationship of eij and Bij in a principal axis system eij is ijiiijjjijij Bn nB= 22 (5) where ni is the refractive index. Substituting Eq. (3) into Eq. (5), we can write ijijklkl S= (6) where we have introduced the elasto-optic susceptibility tensor ijkl ijijkl n n p=
25、22 (7) For completeness, two additional modifications of the basic elasto-optic effect are discussed as follows. 6.6 MODULATORS Roto-Optic Effect Nelson and Lax16 discovered that the classical formulation of elasto-optic effect was inadequate for birefringent crystals. They pointed out that there ex
26、ists an additional roto-optic susceptibility due to the antisymmetric rotation part of the deformation gradient. =BpR ijijklkl (8) where Rij = (Sij Sji)/2. It turns out that the roto-optic tensor components can be predicted analytically. The coefficient of pijkl is antisymmetric in kl and vanishes e
27、xcept for shear waves in birefringent crystals. In a uniaxial crystal the only nonvanishing components are ppnn oe23232313 22 2= ()/ , where no and ne are the principal refractive indices for the ordinary and extraordinary wave, respectively. Thus, the roto-optic effect can be ignored except when th
28、e birefringence is large. Indirect Elasto-Optic Effect In the piezoelectric crystal, an indirect elasto-optic effect occurs as the result of the piezoelectric effect and electro-optic effect in succession. The effective elasto-optic tensor for the indirect elasto-optic effect is given by17 pp r S e
29、S S S ijij im m jn n mn m n = (9) where pij is the direct elasto-optic tensor, rim is the electro-optic tensor, ejn is the piezoelectric tensor, emn is the dielectric tensor, and Sm is the unit acoustic wavevector. The effective elasto-optic tensor thus depends on the direction of the acoustic mode.
30、 In most crystals the indirect effect is negligible. A notable exception is LiNbO3. For instance, along the z axis, r33 = 31 1012 m/v, e33 = 1.3 c/m2, Es 33 29=, thus p = 0.088, which differs notably from the contribution p33 = 0.248. Plane Wave Analysis of Acousto-Optic Interaction We now consider
31、the diffraction of light by acoustic waves in an optically transparent medium. As pointed out before, in the early development, the AO diffraction in isotropic media was described by a set of coupled wave equations known as the Raman-Nath equations.4 In this model, the incident light is assumed to b
32、e a plane wave of infinite extent. It is diffracted by a rectangular sound column into a number of plane waves propagating along different directions. Solution of the Raman-Nath equations gives the amplitudes of these various orders of diffracted optical waves. In general, the Raman-Nath equations c
33、an be solved only numerically and judicious approxi- mations are required to obtain analytic solutions. Using a numerical procedure computation Klein and Cook18 calculated the diffracted light intensities for various diffraction orders in this regime. Depending on the interaction length L relative t
34、o a characteristic length Lo = n2/lo, where n is the refractive index and and lo are wavelengths of the acoustic and optical waves, respectively, solutions of the Raman-Nath equations can be classified into three different regimes. In the Raman-Nath regime, where L Lo, the AO diffraction appears as
35、a predominant first order and is said to be in the Bragg regime. The effect is called Bragg diffraction since it is similar to that of the x-ray diffraction in crystals. In the Bragg regime the acoustic column is essentially a plane wave of infinite extent. An important feature of the Bragg diffract
36、ion is that the maximum first-order diffraction efficiency obtainable is 100 percent. Therefore, practically all of todays AO devices are designed to operate in the Bragg regime. ACOUSTO-OPTIC DEVICES 6.7 In the immediate case, L Lo, the AO diffraction appears as a few dominant orders. This region i
37、s referred as the near Bragg region since the solutions can be explained based on the near field effect of the finite length and height of the acoustic transducer. Many modern AO devices are based on the light diffraction in anisotropic media. The Raman- Nath equations are no longer adequate and a n
38、ew formulation is required. We have previously pre- sented a plane wave analysis of AO interaction in anisotropic media.13 The analysis was patterned after that of Klienman19 used in the theory of nonlinear optics. Unlike the Raman-Nath equations wherein the optical plane waves are diffracted by an
39、acoustic column, the analysis assumes that the acoustic wave is also a plane wave of infinite extent. Results of the plane wave AO interaction in aniso- tropic media are summarized as follows. The AO interaction can be viewed as a parametric process where the incident optical plane wave mixes with t
40、he acoustic wave to generate a number of polarization waves, which in turn generate new optical plane waves at various diffraction orders. Let the angular frequency and optical wavevector of the incident optical wave be denoted by wm and ? ko, respectively, and those of the acoustic waves by wa and
41、? ka. The polarization waves and the diffracted optical waves consist of waves with angular frequencies wm = wo + mwa and wavevectors ? Kkmk moa =+ (m 1, 2, ). The diffracted optical waves are new normal modes of the interaction medium with the angular frequencies wm = wo + mwa and wavevectors ? km
42、making angles qm with the z axis. The total electric field of the incident and diffracted light be expanded in plane waves as ? ? E r te Ezjtkr mmmm ( , ) ( )exp (). .=+ 1 2 c c (10) whereemis a unit vector of the electric field of the mth wave, Em is the slowly varying amplitude of the electric fie
43、ld and c.c. stands for the complex conjugate. The electric field of the optical wave sat- isfies the wave equation, + = ? ? ? E c E t P t o 1 2 2 2 2 2 (11) where ? is the relative dielectric tensor and ? P is the acoustically induced polarization. Based on Pockels theory of the elasto-optic effect,
44、 ? ? ? P r tS r t E r t o ( , )( , ) ( , )= (12) where ? is the elasto-optical susceptibility tensor defined in Eq. (7). ? S r t( , ) is the strain of the acoustic wave ? ? ? S r tsSej t k r aa ( , )/ (. ) () =+ 1 2 c c (13) where s is a unit strain tensor of the acoustic wave and S is the acoustic
45、wave amplitude. Substituting Eqs. (10), (12), and (13) into Eq. (11) and neglecting the second-order derivatives of electric-field amplitudes, we obtain the coupled wave equations for AO Bragg diffraction. dE dz jc k SEe mo mm mm j k r m m =+ (/ ) cos ( 2 1 4 ? ? + + + 11 1 S Ee m j k r m ? ?) (14)
46、where m m mm n np= 2 1 2 , pep s e mmm = , ? ? 1 m is the angle between ? km and the z axis, and ? kKkkmkk mmmoam =+ is the momentum mismatch between the optical polarization waves and mth-order normal modes of the medium. Equation (14) is the coupled wave equation describ- ing the AO interaction in
47、 an anisotropic medium. Solution of the equation gives the field of the optical waves in various diffraction orders. 6.8 MODULATORS Two-Wave AO Interaction In the Bragg limit, the coupled wave equation reduces to the two-wave interaction between the incident and the first-order diffracted light (m =
48、 0, 1): dE dz j n n p SE e dide oo i j k z = 2 2cos ? (15) dE dz j n n p S E e idie oo d j k z = 2 2cos ? (16) where ni and nd are the refractive indices for the incident and diffracted light, pep s e edi = is the effective elasto-optic constant for the particular mode of AO interaction, go is the a
49、ngle between the z axis and the median of incident and diffracted light and, ? k zkz= is the component of the momentum mismatch ? k along the z axis, and ? k is the momentum mismatch between the polar- ization wave ? Kd and the free wave ? kd of the diffracted light. ? kKkkkk ddiad =+ (17) Equations (15) and (16) admit simple analytic solutions. At