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Spatial spectrum of leaky waves generated by scattering of a guided wave from an acoustic wave
Volume 60, number 5
OPTICS C O M M U N I C A T I O N S
1 December 1986
S P A T I A L S P E C T R U M O F LEAKY WAVES G E N E R A T E D BY S C A T T E R I N G OF A G U I D E D WAVE F R O M AN A C O U S T I C WAVE E.A. KOLOSOVSKY, D.V. PETROV and I.B. Y A K O V K I N Institute of Semiconductor Physics, Siberian Branch of the USSR Academy of Sciences, 630090 No~,osibirsk, USSR Received 6 December 1985; revised manuscript received 29 July 1986
The behavior of the radiation m o d e spectrum has been studied for acoustooptic conversion of a guided m o d e o f an anisotropic waveguide to a leaky wave over a wide range of lengths of the interaction region and of acoustic power densities.
1. Introduction
The diffraction of optical guided modes by a surface acoustic wave (SAW) has been studied since the seventies (see, e.g., refs. [1,2]). Nevertheless, there is a certain type of acoustooptic interaction (AOI) with interesting properties of the wave dynamics in the interaction region which, to our mind, has not been clearly understood. It is the acoustooptic conversion of the guided mode to a leaky wave (LW) in an anisotropic optical waveguide. LW in the graded index anisotropic waveguides (e.g. Ti : LiNbO3) were studied in refs. [3-5]. The conversion of the guided mode to a LW by a SAW, or by a diffraction grating induced by the electrooptic effect, was first theoretically and experimentally studied in refs. [6-8]. The geometries of the Bragg AOI in anisotropic waveguides have also been studied already for the case when one participant in the interaction is a leaky wave [1,9], although the latter circumstance was disregarded. The theoretical description of such AOI is difficult because LW, unlike guided modes, have complex propagation constants. Therefore, LW are not eigenwaves of a waveguide, because their amplitudes grow indefinitely with increasing transverse distance from the waveguide, so that the LW cannot be normalized. AOI in Ti : LiNbO 3 waveguides has been analysed in ref. [8]. A LW field representation by means of coupled modes as a sum of the fields of collinear guided TE-mode and an integral of TM-radiation modes was used. However, this approach is approxi280
mate since the system of eigenwaves lacks an important property of waves of the anisotropic waveguide i.e. hybrid polarization. Besides, the radiated wave field is calculated with the assumption that there is no strong dependence of the coupling coefficients on the value of the propagation constant of the radiation modes. In the present paper we consider the spatial spectrum of waves radiated into the substrate under noncollinear AO conversion to LW. The problem is solved by means of a field expansion in the interaction region in terms of eigenwaves of the anisotropic waveguide i.e. guided modes and radiation modes. The interaction efficiency and the behavior of optical waves in the interaction region are expressed in terms of dimensionless variables, specific waveguide parameters do not appear in the theory.
2. Coupled wave equations for non-collinear interaction in an anisotropic waveguide The conversion of a guided wave into a leaky wave is considered as a conversion of a guided mode into a spectrum of radiation modes (fig. la). The system of coupled mode equations describing such an interaction has usually been considered for an anisotropic waveguide with coUinear interaction along a crystallographic axis [11,12] and also for non-collinear interaction of guided modes without radiation modes [ 12, 13]. Let they-axis be directed along the group (coin0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 60, number 5
OPTICS COMMUNICATIONS
1 December 1986
ky
(a)
(b)
~o Fig. 1. (a) The condition of phase matching for non-collinear AOI. The section of the wave vector surface of the TM-guided mode and TE-LW(solid lines), and the ordinary and extraordinary waves in substrates (dashed lines). (b) AOI geometry.
ciding with the phase) velocity of the SAW with an aperture L, the x-axis is directed along the normal to the plane of the waveguide (fig. lb). Unlike scattering from diffraction gratings with finite dimensions along b o t h y - and z-coordinates (see, e.g., refs. [13,14]), the acoustooptic problem can be significantly simplified owing to the unlimited dimensions of the interaction region along y. That is, the wave amplitudes in the interaction region depend only on the zcoordinate. The whole field of the optical waves is expressed in terms of their components E t and H t transverse to the z-axis. The fields in the interaction region are written as a sum of the guided and radiation mode fields
t(r, t)J
X kHE"t(flt~y',x)j u([t3uyF x)l
waveguide. The first term in (1) includes integrating along the dispersion curve of the guided mode tz. {3uy is the y-component of PU and acts here as a parameter. In the first term of (1) there is integration over one component of [~u only, because for the guided modes, (3uy and (3uzare connected by the dispersion equation. The integration enables us to take into account the possible generation of guided modes with the same polarization and mode number due to the coupling with the continuous spectrum of radiation modes. In the second term there is a double integral because for radiation modes the components flRy and ORz are inde pendent variables. The transverse distributions of guided eigen modes Eut(fl~y, x) and radiation modes Et(PR, x) are normalized as follows 4-0o
exp [i(6o.t -
P = f ez • [E~t(13uy,x) × Hl~t~.y,X )
t~uyY)l
--0o
+E~t(fllay, x) X tt~t(fluy , x)] d x , +ff
d~ R aR(~R , z)
x k Ht(~ R , x ) J exp [i(co R t-/3RyY)],
(2)
(guided modes)
(1)
f e z • [Et(p~,
x) × Ht(PR, x)
+Et(]~ R , x ) X H t*~ R' , X ) ] dx = P 6 ( I [ ~ I - IPRi) where Pu and PR are the longitudinal wave vectors of the guided mode and radiation mode. The guided modes have a continuous spectrum because they can travel in a continuum of directions in the plane of the
(radiation modes). This normalization results in different dimensions for the guided and the radiation mode fields. The dis28!
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OPTICS COMMUNICATIONS
tribution of hybrid polarization radiation mode fields with arbitrary propagation directions in an anisotropic, graded index waveguide were obtained using the methods described in ref. [15 ]. Analytical relations for the field distributions in the anisotropic substrate were used as initial conditions for numerical solution of the Maxwell equations at any point of the waveguide including the interface with vacuum. Boundary conditions enable us to express the amplitude of the fields in terms of the field amplitude A 0 in the substrate. The normalization condition makes it possible to express A 0 in terms of the normalization coefficient P. For this purpose we employed the method of normalization for continuous spectrum wave functions [16]. The TM radiation mode field normalized by the condition (2) must assume the asymptotic form of the Ey-component in the depth of the substrate
Ey = 2A 0 cos (70 x + ~ ) , 2 2 _ fl2)1/2 is the transverse wave number,' (70 = (kono the phase) with the constant A 0 connected to P by the relation
P = 4A27r(70/ko)[1 +((kono/70)tanO)2],
(3)
where 0 is the angle between the proPagation direction and the optic axis, k 0 is the wave number in vacuum. The expression (3) is general and holds true for graded index anisotropic waveguides with an arbitrary distribution of the dielectric permittivity of the waveguide layer. A similar result has been obtained earlier only for the case of an isotropic film waveguide using a well-known analytical field representation [17,18]. The tensor of SAW-induced dielectric permittivity perturbation is written as
A~(x,y, t) : ½A~(x) exp [i(~2 t - KsY)] + c.c.,
(4)
where ~2 is the frequency, K S is the SAW wavenumber. The tensor A~(x) will be expressed in terms of the acoustic power density Pac/L (Pac is the total power carried iny-direction). For this purpose we used the expression from the theory of SAW Pac/L = M~2 U 2, where U± is the component of elastic displacement on the interface with vacuum, normal to the boundary, M is a dimensional coefficient. Then
A~(x) = (Pac/L)I/2(~2/MV2)I/Z d(x), 282
(5)
! December 1986
where d(x) is the normalized distribution of the tensor of the dielectric permittivity perturbation, VS =
a/K s . By applying a well-known procedure for deriving coupled mode equations [11 ~17], we obtain a system of equations for the slowly varying amplitudes of interacting waves
dAO3uy, z)/dz = --i f B(J3Ry, ~ + [3uz, z) × KuR(flRy, ~ + [3uz) exp(--i~z) d~ ,
dB(flRy , ~ + 13~z, z)/dz = -iA(fl#y, z) * ,~+ [3uz) exp (i~z) × ~:uR(flRy
(6)
where ~ = 13Rz -- 13uz is the mismatch of the longitudinal components of the wave vectors of the interacting waves. The coupling coefficient is defined as K.R(flRy, ~ +flt~z) = (Pac/L)I/2(Q/MV2)I/2I(~),
1(0 = Xo fe
.y, z)
× dt(X)ER(flRy, ~ +[3uz,x)dx.
(7)
When the SAW frequency is given, (6) describes the relationship between the guided mode, with longitudinal wave vector Pu' and the continuous spectrum of radiation modes with the same y-component of the longitudinal wave vector [3Ry = [ivy + K s (fig. lb). According to (6), the anisotropic properties and specific properties of the waveguide layer are combined in a single characteristic i.e. the coupling coefficient ~pR"
3. Solution of coupled waves equations There exist different solutions of the system of integro-differential equations (6) depending on the conditions at the input of the interaction region at z = 0. The initial conditions A([3uy , z = O) = 1, B([3Ry, ~ + /3uz , z = 0) = 0 for all the ~Ry and ~ conform to the conversion of the guided mode to radiation modes including the set which forms LW (the "direct" problem). Expressing B([3Ry , ~ + t3uz , z) by means of a Laplace transformation, using [19], we obtain
VoLume 60, number 5
1 f A(fluY'Z)=2~I
o o + i~o
OPTICS COMMUNICATIONS
Dt..(O12 1
exp(pz)dp p +T '
ko I ol 2
oo-i°° with
300
T= f IKuR(flRy-'~ +--3uz)I2 d~ p+i~
a
(8)
'
where p is the integration variable and a 0 > 0. When the coupling coefficient I~uR(} ) i2 is given, the expression (8) is the general solution of (6). To evaluate the solution analytically, we must approximate the }KuR(~)i2 curve by a specific function. For this purpose, the calculations have been carried out for a waveguide in lithium niobate. The profiles of the ordinary and extraordinary refractive indices of the waveguide layer were described by gaussian functions. The data on SAW structure have been used with the SAW propagating on the surface of Y-cut crystal in X-direction. Here X and Y refer to crystallographic axes and they coincide with y and x axes on fig. lb respectively. For given guided mode angles of incidence and SAW frequency the calculated functions IKuR(~)I2 in all cases had a bell-shaped form (fig. 2). The position of the maximum was observed at values corresponding to the radiation mode with the effective refractive index I~R [/ko, equal to the real part o f the effective LW refractive index (the circle in fig. 1b). The Lorentz curve is well suited to approximate K~R IKuR(})I 2 = 2 a2b 7r (~ _ ~0)2 + (2b)2 '
(9)
/2rrPac
~
~\1/2
-~=\-~---E-Mv 2 II(~o)]Z)
,
(10)
i.e. for given b it is proportional to the square root o f the acoustic power density. The equation (8) with the coupling coefficient (9) has an analytic form
A([3uy, z)
= [cos (uz) + (c/u) sin (uz)] e x p ( - c z ) ,
B(flRy,~+[3uz,z)=l (b)2
1 (~b -- ~0/b) 2 + 4
X Ig(~, u, z) + g(~, --u, z)l 2 ,
200
100
-is
-1o
(11)
-5
o
5
1o ~/k/,105
Fig. 2. Dependence of coupling coefficient of mismatch for experiment [8]. For TMo-TE o conversions 260 MHz (1), 160 MHz (3), 120 MHz (4); for TM1-TM 1 conversion 124 MHz (2). Dots represent the calculated values, solid lines are approximations according to (9). where u = (a 2 - c2)1/2, c = b + i~0/2,
ic/u) X smc [g (~/b + u/b + ic/b)bz] , ( s i n c x - sinx/x).
g(~, u, z) = i(bz) (1 •
where ~0 characterizes the position of the curve's maximum, a and b are parameters characterizing its height and width. The a over b ratio is determined by a
1 December 1986
1
Now the expressions for M t2 and IBI2b are functions of four dimensionless variables G/b, ~o/b, a/b, zb only. In the limiting case b = 0, as consequence of the wellknown relation for 6-function 6 (t) = limx_,ox/~(t 2+x 2), we obtain an expression [1,2], describing the change of the amplitudes IA [2 and tBI 2 in AOI of two guided modes. In the earlier analysis of the system (6) only the solution for the amplitude of guided wave IA 12 has been found. For instance, in [8,10] the solution has been obtained as an approximation for a smooth dependence of the coupling coefficient on ~. The behavior of different components of the spectrum of the 283
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OPTICS COMMUNICATIONS
radiation field, produced by the interaction, has not been studied comprehensively until now. Eq. (8) can be analyzed (when b is given) in different ways corresponding to various conditions of the experiment. We may, for instance, change the length of interaction region, leaving the a / b ratio unchanged. We may also leave the interaction region unchanged but let a / b vary. Fig. 3 illustrates the dependence of the intensity of the guided wave and the spectral components on ( a / b ) 2 (i.e. acoustic power density) for a number of b z values, ~0 is assumed to be zero. This implies that the condition of the phase synchronism is satisfied for the radiation mode with an effective refractive index equal to the real part o f the effective refractive
1 December 1986
index of the LW. As a / b increases, e.g. with increasing SAW power, the energy transfer can be either monotonic ( a / b < 1) or oscillatory (a/b > i). This can be made clearer in the following manner. For small SAW power densities a negligible part of the guided mode power is converted to the radiation mode spectrum. In spite of the increasing length o f the interaction, the inverse conversion remains insignificant. This conversion takes place in a layer near the surface o f the crystal whose thickness is o f the order of the SAW wavelength where the dielectric permittivity perturbations are localized. With such a / b , the value o f the dielectric permittivity perturbations is insufficient for noticeable conversion over any length of the interac-
f!s
d)
t ¢xll,'~z
~__-..,~._~:~_~._a~_~x~_~ . . k~l~h "u
f)
el
i
#oo
"~--. 0.8
t
----.
3,0
(b.z}
Fig. 3. Spatial spectrum of radiation mode s for different values of z b, 0.25 (a); 0.5 (b); 1.0 (c); 2.5 (d); zb = 0.5 and Go/b = - 4 (e) ; (f) shows the intensity of guided mode versus the length of interaction. 284
Volume 60, number 5
OPTICS C O M M U N I C A T I O N S
tion region. The spectrum of the radiation modes is narrowing as the length of interaction increases;for zb ~ oo it is a 6-function. In this limit, for small SAW powers, the approach of [8,20] becomes rigorous. An increase of the SAW power density (a/b > I) results in the qualitative change of the structure of the field radiated into the substrate (fig. 3). For small zb the spectrum approximates the case a/b < 1, i.e. the maximum of the radiation modes spectrum is located near the real part of the effective refractive index of the LW. This maximum is observed for a given length of the interaction region for a/b values for which complete conversion takes place from a guided mode to diffracted radiation modes. A further increase of the length of interaction results in power conversion predominantly to two side maxima. Experimentally, it implies the following conditions. Let us choose a sufficiently long interaction region. For small densities of the acoustic power a narrow spectrum of radiation modes is to be expected. With increasing SAW power density the spectrum broadens and two spectral maxima appear. Fig. 3e illustrates the behavior of the radiation mode spectrum with an initial deviation from the phase matching conditions of the guided mode and LW. When the SAW power densities are small, the power is transferred to the spectral component with the maximum coupling coefficient. Further increase ofa/b shifts the highest radiation intensity from the spectral component with = 0 (the phase matching condition is satisfied for it quite accurately), to the spectral component with = - 6 b , i.e. at a still greater mismatch. Note also that zb = 1 defines a characteristic boundary. When zb values are less or larger we observe a "symmetrical" structure of the radiated field spectrum.
4. Formulation of the "inverse" problem
1 December 1986
gram of the wave vector surface the LW is shown as a plane wave spectrum whose centre is situated in the vicinity of the longitudinal wave vector of the LW. Each/th spectral component interacts with SAW at the Bragg condition and is coupled with a specific guided mode with am P litude A(J)(a(D ~ " R y + K s l z ) as , shown in fig. 4. Thus, for eachfth pair of an incident and diffracted waves we have the system of equations (6). The inverse process results not in one but in a spectrum of guided modes, each diffracted guided mode being coupled with a spectrum of radiation modes with a definite value of the y-component of the wave vector. The initial conditions are
A (D(a(j) '.~Ry + K s , z = O) = 0 V-'Ry,~'t~ z =
B(J) ~p.z '
-
~Rz
(13)
for all]. The use of the delta-function is necessary because eachjth pair of the interacting waves must obey the law of energy conservation. After integration of the second equation of (6) and substitution of its solution into the first equation, we obtain z
dA (j)ta(/') ~ ") + K s , z') WRy + K s , z)/dz = - fjA ( / ) ( B.,Ky 0
x( f
-I-¢o
Dc~D(~)12exp[-i~(z-z')]d~)dz '
--oo
- iBjKiuR(~j) exp (--i~jz).
(14)
ky kono
kone
There is an "inverse" problem, a conversion of a LW to a guide wave. Let us represent the field of a leaky wave as a sum (z ~< 0)
Y',n Ft't I¢)'" T
.
.uj L//t~O.), x) J exp [1(~R t
g _
~'R.v~" VRz z)],(12)
where B] is the amplitude describing radiated plane waves from the set forming a LW. Thus, on the dia-
keno
kz
Fig. 4. The phase matching condition for the "inverse" problem.
285
Volume 60, number 5
OPTICS COMMUNICATIONS
Applying the approximation of the coupling coefficient (9) and a Laplace transformation we have a system o f equations which enable us to describe the spectral change of the incident LW and the amplitude variations of each spectral component of the guided mode. The results of these calculations will be published later.
References [ 1] C.S. Tsai, IEEE Tran. on Circuits and Systems, CAS-26 (1979) 1072. [2] I.B. Yakovkin and D.V. Petrov, Diffraction of light on acoustic surface waves (Nauka, Novosibirsk, 1979). [3] K. Yamanouchi, T. Kamiya and K. Shibayama, IEEE Trans., MTT-26 (1978) 298. [4] J. Ctyroky and M. Cada, Optics Comm. 27 (1978) 353. [5] S.K. Sheem, W.K. Burns and A.F. Milton, Optics Lett. 3 (1978) 76. [6] D.V. Petrov, Pis'ma v JTF (USSR) 9 (1983) 1120. [7] D.V. Petrov, Optics Comm. 50 (1984) 300.
286
1 December 1986
[ 8] D.V. Petrov and J. Ctyroky, Kvantovaya Elektronika (USSR) 12 (1985) 987. [9] B. Kim and C.S. Tsai, IEEE J. Quantum Electron. QE15 (1979) 642. [10] S. Yamamoto and Y. Okamura, J. Appl. Phys. 50 (1979) 2555. [ 11 ] D. Marcuse, Bell Sys. Techn. J. 54 (1975) 985. [ 12] D.V. Petrov, Kvantovaya Elektronika (USSR) 11 (1984) 1403. [13] K.K. Svidzinsky, Kvantovaya Electronika (USSR) 7 (1980) 1914. [ 14] R.P. Kennan, IEEE J. Quantum Electron., QE-14 (1978) 924. [15] E.A. Kolosovsky, D.V. Petrov, A.V. Tzarev and I.B. Yakovkin, Optics Comm. 43 (1982) 21. [16] L.D. Landau and E.M. Lifshiz, Quantum Mechanics (Nauka, 1974). [17] V.V. Shevchenko, Continuous transitions in open waveguides (Golem, Boulder, 1974). [18] H. Kogelnik, Theory of dielectric waveguides in integrated optics, ed. T. Tamir, Topics in Applied Physics, Vol. 7 (Springer Berlin, 1975). [19] W.H. Louisell, Radiation and noise in quantum electics (McGraw-Hill, 1964). [20] H.F. Taylor and A. Yariv, Proc. IEEE 62 (1974) 1044.
OPTICS C O M M U N I C A T I O N S
1 December 1986
S P A T I A L S P E C T R U M O F LEAKY WAVES G E N E R A T E D BY S C A T T E R I N G OF A G U I D E D WAVE F R O M AN A C O U S T I C WAVE E.A. KOLOSOVSKY, D.V. PETROV and I.B. Y A K O V K I N Institute of Semiconductor Physics, Siberian Branch of the USSR Academy of Sciences, 630090 No~,osibirsk, USSR Received 6 December 1985; revised manuscript received 29 July 1986
The behavior of the radiation m o d e spectrum has been studied for acoustooptic conversion of a guided m o d e o f an anisotropic waveguide to a leaky wave over a wide range of lengths of the interaction region and of acoustic power densities.
1. Introduction
The diffraction of optical guided modes by a surface acoustic wave (SAW) has been studied since the seventies (see, e.g., refs. [1,2]). Nevertheless, there is a certain type of acoustooptic interaction (AOI) with interesting properties of the wave dynamics in the interaction region which, to our mind, has not been clearly understood. It is the acoustooptic conversion of the guided mode to a leaky wave (LW) in an anisotropic optical waveguide. LW in the graded index anisotropic waveguides (e.g. Ti : LiNbO3) were studied in refs. [3-5]. The conversion of the guided mode to a LW by a SAW, or by a diffraction grating induced by the electrooptic effect, was first theoretically and experimentally studied in refs. [6-8]. The geometries of the Bragg AOI in anisotropic waveguides have also been studied already for the case when one participant in the interaction is a leaky wave [1,9], although the latter circumstance was disregarded. The theoretical description of such AOI is difficult because LW, unlike guided modes, have complex propagation constants. Therefore, LW are not eigenwaves of a waveguide, because their amplitudes grow indefinitely with increasing transverse distance from the waveguide, so that the LW cannot be normalized. AOI in Ti : LiNbO 3 waveguides has been analysed in ref. [8]. A LW field representation by means of coupled modes as a sum of the fields of collinear guided TE-mode and an integral of TM-radiation modes was used. However, this approach is approxi280
mate since the system of eigenwaves lacks an important property of waves of the anisotropic waveguide i.e. hybrid polarization. Besides, the radiated wave field is calculated with the assumption that there is no strong dependence of the coupling coefficients on the value of the propagation constant of the radiation modes. In the present paper we consider the spatial spectrum of waves radiated into the substrate under noncollinear AO conversion to LW. The problem is solved by means of a field expansion in the interaction region in terms of eigenwaves of the anisotropic waveguide i.e. guided modes and radiation modes. The interaction efficiency and the behavior of optical waves in the interaction region are expressed in terms of dimensionless variables, specific waveguide parameters do not appear in the theory.
2. Coupled wave equations for non-collinear interaction in an anisotropic waveguide The conversion of a guided wave into a leaky wave is considered as a conversion of a guided mode into a spectrum of radiation modes (fig. la). The system of coupled mode equations describing such an interaction has usually been considered for an anisotropic waveguide with coUinear interaction along a crystallographic axis [11,12] and also for non-collinear interaction of guided modes without radiation modes [ 12, 13]. Let they-axis be directed along the group (coin0 030-401/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 60, number 5
OPTICS COMMUNICATIONS
1 December 1986
ky
(a)
(b)
~o Fig. 1. (a) The condition of phase matching for non-collinear AOI. The section of the wave vector surface of the TM-guided mode and TE-LW(solid lines), and the ordinary and extraordinary waves in substrates (dashed lines). (b) AOI geometry.
ciding with the phase) velocity of the SAW with an aperture L, the x-axis is directed along the normal to the plane of the waveguide (fig. lb). Unlike scattering from diffraction gratings with finite dimensions along b o t h y - and z-coordinates (see, e.g., refs. [13,14]), the acoustooptic problem can be significantly simplified owing to the unlimited dimensions of the interaction region along y. That is, the wave amplitudes in the interaction region depend only on the zcoordinate. The whole field of the optical waves is expressed in terms of their components E t and H t transverse to the z-axis. The fields in the interaction region are written as a sum of the guided and radiation mode fields
t(r, t)J
X kHE"t(flt~y',x)j u([t3uyF x)l
waveguide. The first term in (1) includes integrating along the dispersion curve of the guided mode tz. {3uy is the y-component of PU and acts here as a parameter. In the first term of (1) there is integration over one component of [~u only, because for the guided modes, (3uy and (3uzare connected by the dispersion equation. The integration enables us to take into account the possible generation of guided modes with the same polarization and mode number due to the coupling with the continuous spectrum of radiation modes. In the second term there is a double integral because for radiation modes the components flRy and ORz are inde pendent variables. The transverse distributions of guided eigen modes Eut(fl~y, x) and radiation modes Et(PR, x) are normalized as follows 4-0o
exp [i(6o.t -
P = f ez • [E~t(13uy,x) × Hl~t~.y,X )
t~uyY)l
--0o
+E~t(fllay, x) X tt~t(fluy , x)] d x , +ff
d~ R aR(~R , z)
x k Ht(~ R , x ) J exp [i(co R t-/3RyY)],
(2)
(guided modes)
(1)
f e z • [Et(p~,
x) × Ht(PR, x)
+Et(]~ R , x ) X H t*~ R' , X ) ] dx = P 6 ( I [ ~ I - IPRi) where Pu and PR are the longitudinal wave vectors of the guided mode and radiation mode. The guided modes have a continuous spectrum because they can travel in a continuum of directions in the plane of the
(radiation modes). This normalization results in different dimensions for the guided and the radiation mode fields. The dis28!
Volume 60, number 5
OPTICS COMMUNICATIONS
tribution of hybrid polarization radiation mode fields with arbitrary propagation directions in an anisotropic, graded index waveguide were obtained using the methods described in ref. [15 ]. Analytical relations for the field distributions in the anisotropic substrate were used as initial conditions for numerical solution of the Maxwell equations at any point of the waveguide including the interface with vacuum. Boundary conditions enable us to express the amplitude of the fields in terms of the field amplitude A 0 in the substrate. The normalization condition makes it possible to express A 0 in terms of the normalization coefficient P. For this purpose we employed the method of normalization for continuous spectrum wave functions [16]. The TM radiation mode field normalized by the condition (2) must assume the asymptotic form of the Ey-component in the depth of the substrate
Ey = 2A 0 cos (70 x + ~ ) , 2 2 _ fl2)1/2 is the transverse wave number,' (70 = (kono the phase) with the constant A 0 connected to P by the relation
P = 4A27r(70/ko)[1 +((kono/70)tanO)2],
(3)
where 0 is the angle between the proPagation direction and the optic axis, k 0 is the wave number in vacuum. The expression (3) is general and holds true for graded index anisotropic waveguides with an arbitrary distribution of the dielectric permittivity of the waveguide layer. A similar result has been obtained earlier only for the case of an isotropic film waveguide using a well-known analytical field representation [17,18]. The tensor of SAW-induced dielectric permittivity perturbation is written as
A~(x,y, t) : ½A~(x) exp [i(~2 t - KsY)] + c.c.,
(4)
where ~2 is the frequency, K S is the SAW wavenumber. The tensor A~(x) will be expressed in terms of the acoustic power density Pac/L (Pac is the total power carried iny-direction). For this purpose we used the expression from the theory of SAW Pac/L = M~2 U 2, where U± is the component of elastic displacement on the interface with vacuum, normal to the boundary, M is a dimensional coefficient. Then
A~(x) = (Pac/L)I/2(~2/MV2)I/Z d(x), 282
(5)
! December 1986
where d(x) is the normalized distribution of the tensor of the dielectric permittivity perturbation, VS =
a/K s . By applying a well-known procedure for deriving coupled mode equations [11 ~17], we obtain a system of equations for the slowly varying amplitudes of interacting waves
dAO3uy, z)/dz = --i f B(J3Ry, ~ + [3uz, z) × KuR(flRy, ~ + [3uz) exp(--i~z) d~ ,
dB(flRy , ~ + 13~z, z)/dz = -iA(fl#y, z) * ,~+ [3uz) exp (i~z) × ~:uR(flRy
(6)
where ~ = 13Rz -- 13uz is the mismatch of the longitudinal components of the wave vectors of the interacting waves. The coupling coefficient is defined as K.R(flRy, ~ +flt~z) = (Pac/L)I/2(Q/MV2)I/2I(~),
1(0 = Xo fe
.y, z)
× dt(X)ER(flRy, ~ +[3uz,x)dx.
(7)
When the SAW frequency is given, (6) describes the relationship between the guided mode, with longitudinal wave vector Pu' and the continuous spectrum of radiation modes with the same y-component of the longitudinal wave vector [3Ry = [ivy + K s (fig. lb). According to (6), the anisotropic properties and specific properties of the waveguide layer are combined in a single characteristic i.e. the coupling coefficient ~pR"
3. Solution of coupled waves equations There exist different solutions of the system of integro-differential equations (6) depending on the conditions at the input of the interaction region at z = 0. The initial conditions A([3uy , z = O) = 1, B([3Ry, ~ + /3uz , z = 0) = 0 for all the ~Ry and ~ conform to the conversion of the guided mode to radiation modes including the set which forms LW (the "direct" problem). Expressing B([3Ry , ~ + t3uz , z) by means of a Laplace transformation, using [19], we obtain
VoLume 60, number 5
1 f A(fluY'Z)=2~I
o o + i~o
OPTICS COMMUNICATIONS
Dt..(O12 1
exp(pz)dp p +T '
ko I ol 2
oo-i°° with
300
T= f IKuR(flRy-'~ +--3uz)I2 d~ p+i~
a
(8)
'
where p is the integration variable and a 0 > 0. When the coupling coefficient I~uR(} ) i2 is given, the expression (8) is the general solution of (6). To evaluate the solution analytically, we must approximate the }KuR(~)i2 curve by a specific function. For this purpose, the calculations have been carried out for a waveguide in lithium niobate. The profiles of the ordinary and extraordinary refractive indices of the waveguide layer were described by gaussian functions. The data on SAW structure have been used with the SAW propagating on the surface of Y-cut crystal in X-direction. Here X and Y refer to crystallographic axes and they coincide with y and x axes on fig. lb respectively. For given guided mode angles of incidence and SAW frequency the calculated functions IKuR(~)I2 in all cases had a bell-shaped form (fig. 2). The position of the maximum was observed at values corresponding to the radiation mode with the effective refractive index I~R [/ko, equal to the real part o f the effective LW refractive index (the circle in fig. 1b). The Lorentz curve is well suited to approximate K~R IKuR(})I 2 = 2 a2b 7r (~ _ ~0)2 + (2b)2 '
(9)
/2rrPac
~
~\1/2
-~=\-~---E-Mv 2 II(~o)]Z)
,
(10)
i.e. for given b it is proportional to the square root o f the acoustic power density. The equation (8) with the coupling coefficient (9) has an analytic form
A([3uy, z)
= [cos (uz) + (c/u) sin (uz)] e x p ( - c z ) ,
B(flRy,~+[3uz,z)=l (b)2
1 (~b -- ~0/b) 2 + 4
X Ig(~, u, z) + g(~, --u, z)l 2 ,
200
100
-is
-1o
(11)
-5
o
5
1o ~/k/,105
Fig. 2. Dependence of coupling coefficient of mismatch for experiment [8]. For TMo-TE o conversions 260 MHz (1), 160 MHz (3), 120 MHz (4); for TM1-TM 1 conversion 124 MHz (2). Dots represent the calculated values, solid lines are approximations according to (9). where u = (a 2 - c2)1/2, c = b + i~0/2,
ic/u) X smc [g (~/b + u/b + ic/b)bz] , ( s i n c x - sinx/x).
g(~, u, z) = i(bz) (1 •
where ~0 characterizes the position of the curve's maximum, a and b are parameters characterizing its height and width. The a over b ratio is determined by a
1 December 1986
1
Now the expressions for M t2 and IBI2b are functions of four dimensionless variables G/b, ~o/b, a/b, zb only. In the limiting case b = 0, as consequence of the wellknown relation for 6-function 6 (t) = limx_,ox/~(t 2+x 2), we obtain an expression [1,2], describing the change of the amplitudes IA [2 and tBI 2 in AOI of two guided modes. In the earlier analysis of the system (6) only the solution for the amplitude of guided wave IA 12 has been found. For instance, in [8,10] the solution has been obtained as an approximation for a smooth dependence of the coupling coefficient on ~. The behavior of different components of the spectrum of the 283
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OPTICS COMMUNICATIONS
radiation field, produced by the interaction, has not been studied comprehensively until now. Eq. (8) can be analyzed (when b is given) in different ways corresponding to various conditions of the experiment. We may, for instance, change the length of interaction region, leaving the a / b ratio unchanged. We may also leave the interaction region unchanged but let a / b vary. Fig. 3 illustrates the dependence of the intensity of the guided wave and the spectral components on ( a / b ) 2 (i.e. acoustic power density) for a number of b z values, ~0 is assumed to be zero. This implies that the condition of the phase synchronism is satisfied for the radiation mode with an effective refractive index equal to the real part o f the effective refractive
1 December 1986
index of the LW. As a / b increases, e.g. with increasing SAW power, the energy transfer can be either monotonic ( a / b < 1) or oscillatory (a/b > i). This can be made clearer in the following manner. For small SAW power densities a negligible part of the guided mode power is converted to the radiation mode spectrum. In spite of the increasing length o f the interaction, the inverse conversion remains insignificant. This conversion takes place in a layer near the surface o f the crystal whose thickness is o f the order of the SAW wavelength where the dielectric permittivity perturbations are localized. With such a / b , the value o f the dielectric permittivity perturbations is insufficient for noticeable conversion over any length of the interac-
f!s
d)
t ¢xll,'~z
~__-..,~._~:~_~._a~_~x~_~ . . k~l~h "u
f)
el
i
#oo
"~--. 0.8
t
----.
3,0
(b.z}
Fig. 3. Spatial spectrum of radiation mode s for different values of z b, 0.25 (a); 0.5 (b); 1.0 (c); 2.5 (d); zb = 0.5 and Go/b = - 4 (e) ; (f) shows the intensity of guided mode versus the length of interaction. 284
Volume 60, number 5
OPTICS C O M M U N I C A T I O N S
tion region. The spectrum of the radiation modes is narrowing as the length of interaction increases;for zb ~ oo it is a 6-function. In this limit, for small SAW powers, the approach of [8,20] becomes rigorous. An increase of the SAW power density (a/b > I) results in the qualitative change of the structure of the field radiated into the substrate (fig. 3). For small zb the spectrum approximates the case a/b < 1, i.e. the maximum of the radiation modes spectrum is located near the real part of the effective refractive index of the LW. This maximum is observed for a given length of the interaction region for a/b values for which complete conversion takes place from a guided mode to diffracted radiation modes. A further increase of the length of interaction results in power conversion predominantly to two side maxima. Experimentally, it implies the following conditions. Let us choose a sufficiently long interaction region. For small densities of the acoustic power a narrow spectrum of radiation modes is to be expected. With increasing SAW power density the spectrum broadens and two spectral maxima appear. Fig. 3e illustrates the behavior of the radiation mode spectrum with an initial deviation from the phase matching conditions of the guided mode and LW. When the SAW power densities are small, the power is transferred to the spectral component with the maximum coupling coefficient. Further increase ofa/b shifts the highest radiation intensity from the spectral component with = 0 (the phase matching condition is satisfied for it quite accurately), to the spectral component with = - 6 b , i.e. at a still greater mismatch. Note also that zb = 1 defines a characteristic boundary. When zb values are less or larger we observe a "symmetrical" structure of the radiated field spectrum.
4. Formulation of the "inverse" problem
1 December 1986
gram of the wave vector surface the LW is shown as a plane wave spectrum whose centre is situated in the vicinity of the longitudinal wave vector of the LW. Each/th spectral component interacts with SAW at the Bragg condition and is coupled with a specific guided mode with am P litude A(J)(a(D ~ " R y + K s l z ) as , shown in fig. 4. Thus, for eachfth pair of an incident and diffracted waves we have the system of equations (6). The inverse process results not in one but in a spectrum of guided modes, each diffracted guided mode being coupled with a spectrum of radiation modes with a definite value of the y-component of the wave vector. The initial conditions are
A (D(a(j) '.~Ry + K s , z = O) = 0 V-'Ry,~'t~ z =
B(J) ~p.z '
-
~Rz
(13)
for all]. The use of the delta-function is necessary because eachjth pair of the interacting waves must obey the law of energy conservation. After integration of the second equation of (6) and substitution of its solution into the first equation, we obtain z
dA (j)ta(/') ~ ") + K s , z') WRy + K s , z)/dz = - fjA ( / ) ( B.,Ky 0
x( f
-I-¢o
Dc~D(~)12exp[-i~(z-z')]d~)dz '
--oo
- iBjKiuR(~j) exp (--i~jz).
(14)
ky kono
kone
There is an "inverse" problem, a conversion of a LW to a guide wave. Let us represent the field of a leaky wave as a sum (z ~< 0)
Y',n Ft't I¢)'" T
.
.uj L//t~O.), x) J exp [1(~R t
g _
~'R.v~" VRz z)],(12)
where B] is the amplitude describing radiated plane waves from the set forming a LW. Thus, on the dia-
keno
kz
Fig. 4. The phase matching condition for the "inverse" problem.
285
Volume 60, number 5
OPTICS COMMUNICATIONS
Applying the approximation of the coupling coefficient (9) and a Laplace transformation we have a system o f equations which enable us to describe the spectral change of the incident LW and the amplitude variations of each spectral component of the guided mode. The results of these calculations will be published later.
References [ 1] C.S. Tsai, IEEE Tran. on Circuits and Systems, CAS-26 (1979) 1072. [2] I.B. Yakovkin and D.V. Petrov, Diffraction of light on acoustic surface waves (Nauka, Novosibirsk, 1979). [3] K. Yamanouchi, T. Kamiya and K. Shibayama, IEEE Trans., MTT-26 (1978) 298. [4] J. Ctyroky and M. Cada, Optics Comm. 27 (1978) 353. [5] S.K. Sheem, W.K. Burns and A.F. Milton, Optics Lett. 3 (1978) 76. [6] D.V. Petrov, Pis'ma v JTF (USSR) 9 (1983) 1120. [7] D.V. Petrov, Optics Comm. 50 (1984) 300.
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1 December 1986
[ 8] D.V. Petrov and J. Ctyroky, Kvantovaya Elektronika (USSR) 12 (1985) 987. [9] B. Kim and C.S. Tsai, IEEE J. Quantum Electron. QE15 (1979) 642. [10] S. Yamamoto and Y. Okamura, J. Appl. Phys. 50 (1979) 2555. [ 11 ] D. Marcuse, Bell Sys. Techn. J. 54 (1975) 985. [ 12] D.V. Petrov, Kvantovaya Elektronika (USSR) 11 (1984) 1403. [13] K.K. Svidzinsky, Kvantovaya Electronika (USSR) 7 (1980) 1914. [ 14] R.P. Kennan, IEEE J. Quantum Electron., QE-14 (1978) 924. [15] E.A. Kolosovsky, D.V. Petrov, A.V. Tzarev and I.B. Yakovkin, Optics Comm. 43 (1982) 21. [16] L.D. Landau and E.M. Lifshiz, Quantum Mechanics (Nauka, 1974). [17] V.V. Shevchenko, Continuous transitions in open waveguides (Golem, Boulder, 1974). [18] H. Kogelnik, Theory of dielectric waveguides in integrated optics, ed. T. Tamir, Topics in Applied Physics, Vol. 7 (Springer Berlin, 1975). [19] W.H. Louisell, Radiation and noise in quantum electics (McGraw-Hill, 1964). [20] H.F. Taylor and A. Yariv, Proc. IEEE 62 (1974) 1044.