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Resonant scattering of electromagnetic waves by a lossy periodic dielectric waveguide
MATHEMATICAL COMPUTER MODELLING
Mathematicaland Computer Modelling32 (2000) 1059-1070
PERGAMON
www.elsevier.nl/locate/mcm
Resonant Scattering of Electromagnetic Waves by a Lossy Periodic Dielectric Waveguide A.
A.
BIKOV, V. Yu. POPOV AND A. G. SVESHNIKOV MSU, Physical Faculty, Moscow, Russia M. KLIBANOV University of North Carolina at Charlotte Charlotte, NC 28223, U.S.A. I. 0. VOLKOVA AND A. V. TIKHONRAVOV MSU, Research Computer Center, Moscow, Russia
Abstract-The problem of the excitation of guided modes in planar dielectric waveguides with periodically modulated surfaces is considered. In particular, the form of the radiation pattern of a lossy waveguide, the width and the position of the central angle of the radiation pattern as a function of the surface modulation depth are studied. A special numerical-analytical method for the calculation of the intensities of guided modes has been developed. In the case of a small modulation depth, this method makes use the expansion of the solutions of Maxwell equations into the Laurent series near the scattering matrix pole. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Scattering,
Periodic waveguide,Directionalradiationpattern.
1. INTRODUCTION In this work, we consider the resonant scattering of electromagnetic
waves by the modulated
surface of a planar lossy dielectric waveguide. The resonant nature of the scattering is displayed in the significant increase of the intensities of scattered fields and waveguide modes near the waveguide boundary as compared to the intensity of the incident plane wave. It is well known that there are several mechanisms causing a resonant scattering of electromagnetic waves by the periodically modulated surface. A resonant scattering may be connected with the Wood anomaly [l-3] or with the excitation of waveguide modes [4,5]. Methods for the calculation of the amplitudes of induced waveguide modes and scattered fields have been developed by several authors [6-81. In spite of the variety of the approaches having been used, some important aspects of the plane wave scattering by modulated dielectric waveguides were not investigated in reasonable detail. One of these aspects is a detailed analysis of the radiation pattern of the dielectric waveguide in the receiving regime. The main goal of our paper is the calculation and analysis of the main parameters of the radiation pattern, such as the central angle, the width, and the amplitude of the central lobe. To calculate the intensities of scattered fields and waveguide modes, we have developed a special method combining both the numerical and analytical approaches. In the’case of a small amplitude of the surface modulation, the scattering This work was partly supported
by Russian Fund of Fundamental
0895-7177/00/$ - see front matter PII: SO895-7177(00)00190-4
Researches,
Grant 95Ol-01284a.
@ 2000 Elsevier Science Ltd. All rights reserved.
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by &&TD
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matrix is expanded into a series with respect to the modulation depth. The peculiarity of the problem under consideration is that this series turns to be a Laurent series. To calculate the coefficients of the Laurent series, the Rayleigh hypothesis is used [9,10]. The first terms of the scattering matrix series, being responsible for the effect of resonant increase of the scattered field, are calculated analytically. In a general case for the correct calculation of the matrix coefficients, a special numerical method has been developed. This method is similar to the method of a halfinversion [11,12]. To contrast with the traditional coupled-modes method [13,14], our method allows us to analyze waveguide structures of an arbitrary depth and form. In the case of big amplitudes of the surface modulation, the series representation of the scattering matrix is not valid. In this case, to obtain the scattering matrix, Maxwell equations are reduced to a system of ordinary differential equations by the Gale&n-type method 1151. To develop a stable numerical algorithm, we have applied a special method for the solution of the boundary value problems for ordinary differential equations [16]. On the whole, the combination of the numerical and analytical methods allows us to develop a high performance computer code. This code enables one to calculate the intensities of fields without any restrictions on the parameters of waveguides.
2. SCATTERING WAVEGUIDE
OF THE PLANE WAVE ON A THREE-LAYER WITH THE MODULATED BOUNDARY
Consider the scattering of the plane electromagnetic wave by a three-layer planar dielectric waveguide with the modulated boundary between the waveguide and the ambient medium. The refractive indices of the waveguide, the substrate, and the ambient medium are equal to n2, nr , n3, respectively, where Re n2 > Re ni,
Re nz > Re ns,
Im 72s= 0,
Im It2 and Im ni are small values. The shape of the boundary between the waveguide and the ambient medium is given by the periodic function y = f(z) with the period a : f(z + a) = f(z). The boundary between the waveguide and the substrate, is the plain y = -b, where 6 is the mean thickness of the waveguide. Let us consider the excitation of the waveguide by the TE-polarized plane wave, propagating under the angle cp to the y axis (see Figure 1). The wave vector belongs to the sy-plane, the only nonzero component of the electric field is given by E, = E,-,a-“2 exp(-ians(ssincp
+ ycoscp)).
Here Eo is a given field amplitude, K = 27r/X is the wavenumber, X is the wavelength in vacuum. Here and thereafter, the time factor exp( -iwt) is omitted. All nonzero field components, E,, H,, H,, depend on the coordinates x and y and do not depend on z. Any component U of the electrical and magnetic fields satisfies the Floquet condition U(z f or Y) = exp(f+)U(z,
y),
(1)
where p = --an3 sin cp
(2)
is Floquet parameter. It follows from the Maxwell equations and Floquet condition that electromagnetic fields in the substrate, in the uniform part of the waveguide and in the ambient medium may be presented as a superposition of uniform and nonuniform plane waves. The complex amplitudes of these waves are determined from the boundary conditions for the fields.
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Loeey Periodic Dielectric Waveguide
-b , Figure 1. A diagram of the dielectric waveguide with the periodically modulated boundary between the waveguide and the ambient medium. The rehactive indices are nr = 1.5, 122= 1.8, ne = 1.0, period a = 1, mean thickness b = 1, wavelength in vacuum X = 2, effective refractive index n,s = 1.756558. The central angle of the radiation pattern cpo = -49.161712O.
For the brevity of equations, we will use the matrix designations. To describe the dependence of the fields on 2, the following vector satisfying equation (1) is used: X(z) = {. . . ,X_m(Z),
. . . ,X*(z),
. . . ,Xm(2),
. . . },
where X,(z)
pm = p + T.
= a-‘/2exp(ip,z),
To describe the dependence of the fields on y coordinate in the substrate, the waveguide, and the ambient medium, we use the infinite matrices l?i, l?2, I’S with the diagonal elements equal to 72’ = (K’$
j = 1,2,3,
- &l/21
and all other elements equal to zero: l?j =diag
(
. . . . r!A ,...,
$‘,.,.,
r,!$,.,.).
We designate complex amplitudes of waves propagating upwards and downwards as Aim;“),Bkm;“) (in the ambient media), Aimm),Bim) (in waveguide), A(;Z),B(;R) (in the substrate), respectively. Let us In fact, Aj”) = 0 for all m because there are no upward waves in the substrate. write these amplitudes as the column vectors As,As, and Bi,Bs,Bs. For example, A3 = A(O) Atrn) }T where “T” designates a transposition. The vector Bs is { . . . ) Ah-“) ,*.*, 3 3.“) 3 ,*** , specified by the incident field. It has only one nonzero component Br) = Eo, where EO is the amplitude of the incident plane wave. Assume that the Rayleigh hypothesis is fulfilled. Then it is possible to present the electrical field U(z, y) = Ez(z, y) as a superposition of plane waves. This field can be written down as follows: U(Z, 3) = X(s)(exp(iIk+s + exp( -GY)Bs)
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in the region y > f(z), U(Z, y) = X(z)(exp(il?zy)Az
+ exp(-if’zY)Bz)
in the region -b < y < f(s),
V(G Y) = X(z) exp(-irl (Y + b))Bl in the region y < -b. Scattered fields in the substrate and the ambient covering medium are specified by the vectors A3 and Br, connected with the vector B3 by the linear equations A3 = R31 B3,
BI
= 5f-31B3r
where R3r and T3r are the reflection and transmission matrices. The amplitudes of waveguide modes are described by the vectors AZ, and B3, that are also determined by the vector B3. For example, A2 = S32B3,
where S32 is the scattering matrix. The method for the calculation of the matrices R31, T31, and section.
S32
is outlined in the following
3. A METHOD OF CALCULATION OF THE REFLECTION, TRANSMISSION, AND SCATTERING MATRICES Let R32 and T32 be the reflection and transmission matrices for the boundary between the ambient medium and the substrate in the case of the incidence of the plane wave from the ambient medium. Let R3r and T3r be the analogous matrices for the flat boundary between the waveguide and the substrate in the case of the excitation from the waveguide region (Figure 1). The vectors of the complex amplitudes of the fields in the waveguide and in the ambient medium are connected by the relations A3
B2 = T32 I33 + R23 A2.
=R32B3+T23A2,
Similar relations are valid at the boundary between the substrate A2
= R2lB2,
and the waveguide
B1 = 5721B2,
where x2 = exp(-il?&)A3,
63 = exp(+iT2b)B2.
These relations allow us to write down the following expression for the matrices R31, T31, S32: R31 = R32 + T23s32,
(3)
T31 = Tzl exP(ihb)(%
f R23S32)r
(4)
Sexp(ir2b)R21
exp(iI’2b)T32,
(5)
S32 =
where
(I is the unity matrix).
S = H-l,
(6)
H = I - exp(ir’2b)R21 exp(il?2b)R23,
(7)
Lossy Periodic DielectricWaveguide Equations (5)-(7)
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are applicable if the matrix H is nonsingular.
In the case of an arbitrary surface modulation, it is convenient to use the incomplete Galerkin method [15] to calculate the partial scattering matrices R32, T23, T32. This method reduces the boundary value problem for the Maxwell equations to the boundary value problem for a system of linear differential equations. This last problem can then be solved by a special numerical method [16]. It is essential that the method from reference [16] does not impose any restrictions on structure parameters (the thickness of the waveguide, the ratio between the wavelength, and metrical parameters of the waveguide). At the same time, in general, this method requires a lot of computational
work. This situation stimulated us to develop an analytical approach allowing
us to study the problem at least for a small depth of the surface modulation.
Here we present
the sketch of this approach. Let us denote the depth of the surface modulation as h and expand the matrices R32 and T32 in Taylor series T32 = Tii) + hTik) + h2T,(i) + . . . .
Note that here
(O' and R,,
T,(,o)are usual reflection and transmission matrices for the flat surface.
To calculate the Taylor coefficients R&j and TiF), n = 1,2, . . . , the method from reference [lo] is used. We assume that the waveguide surface is modulated by the sine law f(2)
= hsin
T (
, >
To avoid overloading the description of the approach, we do not present the expressions for R$
T!$ . We would like only to note that Rg’ and T.$
are diagonal matrices with the nonzero
main diagonal, or OStdiagonal, and matrices Ri’,) and T$:) have nonzero elements only on two diagonals (+lSt and -lSt diagonals), Rg’ and T3(:) are three-diagonal matrices with the nonzero -2nd, main, and +2’ld diagonals. Note that in the case of more complicated modulation laws, the other coefficients may be also nonzero. It follows from equations (7),(g), that the matrix H is also expanded in the Taylor series H =
2
hn@-‘).
n=O
The inverse matrix S = H-l
may have the pole in the series expansion S =
2
h”S@).
(9)
n=--p
The order of the pole depends on the presence of nonzero elements on the main diagonal of the matrix N(O). If H(O) m,m # 0 for all numbers m, S is a regular matrix. In this case, the amplitudes of all guided modes tend to zero when h tends to zero and in the limit, the scattered fields are described by Fresnel formulas. If H(O) m,m = 0 )
(10)
for some m = M, then S has a pole of a second order [17]. The exact formulas for the matrices Stn) are not essential for our discussion. To analyze the dependence of scattered fields on the depth of the surface modulation, we ought to note that the main term of the Laurent series expansion S ( - 2, has one nonzero element with the indices M, M, S(-l) has four nonzero elements with the indices M f 1, M f 1. It is easy to show that condition (10) is equivalent to the characteristic equation for a planar dielectric waveguide with flat boundaries (41. It may be exactly fulfilled for the real value of the
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Floquet parameter only in the absence of losses in the waveguide material. condition is the existence of the such index M, that
The other necessary
(11) In other words, the plane wave corresponding to the index M must be locked in the guide region. If equation (11) is fulfilled, it is possible to write down equation (10) in a standard form [4]: tan wY2)
where 72 = (ni equation (12) are The connection given by the link
72b1+ P3) = - CPlP3 _ y> I
(12)
S2/~2)1/2, pj = (/?2/~2 - n?)1/2, j = 1,3. Those P values that are the roots of longitudinal wave numbers’of guided modes. of equation (10) and the characteristics equation of the planar guide (12) is equation P=L’M.
(13)
It is possible to explain the link relation (13) in terms of the phase synchronism condition. Consider the analogous waveguide with the thickness b and the extremely small amplitude of the surface modulation h 0. The source of the emitted plane waves must be the guided mode propagating along the waveguide. Under the condition h 0, the radiation losses are negligible and the amplitude of the guided mode may be treated as a constant over the large region in comparison with the waveguide thickness and the wavelength. The scattered field may be represented as a set of the plane waves generated on the low periodical relief. Consider the full set of the direction angles of the emitted plane waves {cpn}. Under condition (13), the angle of the incidence of the exciting plane wave cp is equal to one of the (Pi. In this paper, we consider the case of the presence of only one zero element on the main diagonal of the matrix H,f$,,,. Under this condition, the S has the pole of the second order (p = 2). Note that even in the relatively simple case of the TEpolarized electromagnetic waves the Hf$m may have two elements on the main diagonal. It may happen if the corresponding value of the Floquet parameter p,,, = p + (2nm)/a is equal to zero. In other words, in this case, one of the above-mentioned plane waves radiated by the analogous waveguide with the extremely small corrugation amplitude has the directional angle directed along the normal to the waveguide plane. In the general case of the mixed polarization, the number of zero diagonal elements may be two, three, and greater. The order of the pole of the S matrix may be greater than two. Let us consider the dependence of the scattered fields on the corrugation depth h. The main factor determining the scattered fields appearance is the presence of the pole of the S matrix. The maximal possible rate of the growth of the fields when the h tends to zero is hp2. Note that then only the nonzero element of Ste2) matrix is the element in the position with the indices M, M. To realize the possibility of the maximal growth rate of the scattered fields, it is necessary to excite the waveguide by the external EM wave represented by the Bs vector with the nonzero element (B~)M. The third condition of equation (11) shows that this element of the Bs vector represents the amplitude of the nonuniform plane wave in the ambient media. The nonuniform plane wave is increasing by the exponential law under the condition that the y coordinate is increasing. Such appearance of the exciting field may be obtained with the special type of the input device, for example, the prism input element [4,5]. The excitation of the periodic waveguide by the plane wave cannot provide the exciting field of the necessary type. The next term in the series representation of the scattering matrix SC-‘) has four nonzero elements. Their indices are M f 1, M f 1. To obtain the increase rate of the scattered fields proportionally, h-l one must use the exciting field determined by the B3 vector with the indices of the nonzero elements M - 1 or M + 1.
Lossy Periodic
Dielectric Waveguide
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Let us consider some computational aspects of the numerical simulation electromagnetic
waves
scattering by the periodic guide. There exist the analytical expressions for scattering matrix (6). If h ---+0, the same elements of the scattering matrix tend to infinity. This is the result of the inversion of the ill-conditioned matrix H. The inversion of this matrix with the standard numerical algorithms is not correct and leads to great growth in computer round-off errors. The numerical-analytical method was developed for the correct matrix inversion. Consider the pole of the scattering matrix S. The main system matrix H has the second-order zero and may be expressed as the product of three matrices: H = H1H2H3, where HI and Hz matrices both have the first-order zero. These matrices are simple and may be inverted analytically. The HZ matrix is regular and it has the general form. This matrix may be inverted numerically with high precision. As a result, the S matrix presented in the form S = (H3)-1(H$-1(H1)-1. The main (unlimited) part of the inverse matrix
S =
5
hn#*)
(14)
n=-p
may be calculated analytically. The exact expressions for the terms mentioned in (14) are not essential, so they are omitted in this paper. The q value depends on the order of the pole of the scattering matrix. If the second-order pole exists, the optimal value is q = 0. At the same time, expression (14) is applicable only for the small values of h. For large h, it is necessary to use the exact expression (6). In this case, the inversion of the H matrix is correct. To exclude the two different expressions for the scattering matrix in the different ranges of the parameter h, the numerical-analytical
method was developed.
The main part of the scattering
matrix is calculated by expression (14), correction is fulfilled numerically. As a result, the inversion of the ill-conditioned matrix is excluded. The additional advantage of this approach consists in the uniformity of the formulas used in the all range of the h parameter. This may be essential in the optimization problems.
4. ANALYSIS OF THE RADIATION PATTERN OF THE LOSSY DIELECTRIC WAVEGUIDE In practice, it is useful to analyze the radiation pattern of the periodical dielectric waveguide. To introduce the radiation pattern, it is necessary to interpret the field induced in the waveguide as the superposition of the fields of the primary plane wave, the guided mode, and the scattered homogeneous and inhomogeneous plane waves. These fields are calculated under the conditions of the external excitation of the above-mentioned guide by the plane wave falling from the ambient medium. The field of the guided mode is the superposition of the direct and the inverse locked plane-wave modes in the guide region (-b < y < 0). The locked field decreases exponentially in the substrate and in the ambient medium. The conditions of the locking are represented by equations (11). It is possible to associate the complex value of the amplitude with the field of the induced guided mode. We will denote this value by the A, symbol. This field is similar to the field of the guided mode of the planar dielectric waveguide without the corrugation. We will briefly designate this value as the amplitude of the induced mode. Let us designate the angular dependence of the amplitude of the induced mode as the amplitude radiation pattern. Note that the amplitude of
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the guided mode in the case of the excitation of the dielectric waveguide with the flat boundaries is zero. The main practice interest consists in the investigation of the dependence of the amplitude radiation pattern on the corrugation depth h and the waveguide parameters, such as the extinction in the guide material. The simplest explanation of the radiation pattern appearance may be obtained with the introduction of the complex variables, Let us consider the angle of the incidence of the initial plane wave 4 and the depth of the corrugation h as complex parameters. As we had seen earlier, the point h = 0 is the pole of the scattering matrix if the same conditions are fulfilled. In particular, the angle of incidence 4 must be equal to the same directional angle $0 described earlier. Under the conditions of the phase synchronism, the point r$ = $0 on the real axis of the complex plane C$= 4’ + Q!J”is the pole of the guided mode amplitude. For the lossy waveguide (Im nz > 0), the pole of the guided mode amplitude changes to the complex plane. The appearance of the radiation pattern in this case will be illustrated later in this section. It is possible to introduce the intensity of the induced mode I9 along with the amplitude of the induced mode A,. The intensity of the induced mode is determined as the flow of the Pointing vector calculated for the electromagnetic field of the induced mode across the total waveguide cross-section including the parts of the substrate and the ambient medium where the induced fields are nonzero. Really, the integration will be carried out over the region z = const, -oo < y < oo. It is essential that only the field of induced mode is taken into account. The flow of the Pointing vector per the unit of the length for the incident plane wave is taken to be the unit. It is possible to show that the I9 and the A, values are connected by the usual relation 1, = PIA,?,
05)
where p is the longitudinal wavenumber of the induced guided mode [4,5]. Hence, it is possible to consider only one of the two sets of the characteristics-the complex induced amplitude or the intensity and phase. We will consider the intensity because there exist precision experimental methods of the measurement of this parameter in the range of the optical wavelengths. The dependence of the intensity of the induced mode on the angle of incidence of the initial plane wave is named the intensity radiation pattern of the dielectric waveguide in receiving regime. The most significant results may be obtained by consideration of the amplitude radiation pattern including the module and phase dependence. In this paper, we restrict ourselves by the only module and consider the intensity radiation pattern. The more obviousness results will be obtained by the simultaneous consideration of the set of the intensity directional patterns for the waveguides with the various h values. Totally, they form the two-dimensional function of the cp and h variables. The intensity radiation pattern possesses the characteristic features connected with location of the poles of the scattering matrix on the complex plane. The numerical results presented later may be interpreted as the results of the motion of the pole of the scattering matrix on the complex plane of the incident angle because of the variation of the corrugation depth. The results of the numerical calculations for the various parameters of the waveguide are presented in Figures 2-5. At first, the waveguide without the losses is considered. The common appearance of the radiation pattern depends mainly on such structural peculiarities of the waveguide under consideration as the order of the pole. The specific values of the waveguide parameters such as the refractive indices and thickness do not affect on the main features. The waveguide parameters for Figure 1 are ns = 1,
in the covering media,
12s= 1.8,
in the waveguide,
121= 1.5,
in the substrate.
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log h Figure 2. The set of the curves of the constant value of the induced mode intensity in the waveguide without losses: Im ns = 0. The value of log(h) is put aside the horizontal axis. The value of h is varying in the range between 10m4 (left end of the axis) and 10m2 (right end). The value of (0 is put aside the vertical axis. The range of cp is between -49.161688O (top end) and -49.16172OO (bottom end). --T-
log h Figure 3. The results of the lossy waveguide excitation: Im ns = 6.9 x 10es. All parameters are the same as in Figure 2 with the exception of the range of ‘p variable is between -49.161592” (top end) and -49.161752O (bottom end).
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The dimensionless spatial variables. the sine law with wavelength in the The main mode
A. A. BIKOV et
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variables are used for the values of the period, the thickness, and the other The boundary between the guide and the ambient media is modulated by the period a = 1, the average thickness of the waveguide is equal to 1. The vacuum X = 2. of the waveguide with the flat surfaces has the effective refractive index %tr =def p = 1 756558. K. *
The main guided mode of the analogous planar guide with negligible small corrugation depth radiates the plane wave under the angle cps = -49.161712” (the angles are counted from the vertical direction as is shown in Figure 1). The phase synchronism conditions requires that the angle of the incidence of the primary plane wave must be near the mentioned value, The dependence of the guided mode intensity on the variables h and cp is shown in Figure 2. The logarithm of the amplitude of the corrugation h is put aside the horizontal axis. The value of the h variable is varying in the range between 10B4 and 10a2. The angle of the incidence of the primary plane wave is put aside the vertical axis. The range of the cp variable is between -49.161688” and -49.161720”. The lines of the constant value of the induced mode intensity are shown in Figure 2. The decimal logarithmic scale is chosen for representation of the mode intensity. For example, the line marked by the 4 symbol corresponds to the value of the intensity equal to 104. The maximal value of the induced mode intensity in the shown range of the parameters is near 106. This value tends to infinity if the corrugation depth tends to zero. More exactly, if h --+ 0, the maximal intensity over the mentioned range of the initial angle C$is proportional to h-2. The infinite increasing of the mode intensity in the case of h 0 is accompanied by the displacement of the central angle of the radiation pattern. The central angle is the function of the h variable. It is determined as the angle corresponding to the maximal value of the mode intensity. If h 0, the central angle tends to the $0 value. The deflection of the central angle from the 40 value is proportional to h2. The next significant characteristic of the radiation pattern is the width on the specified level equal to half of the maximal level. It is possible to show (Figure 2) that the width of the radiation pattern is proportional to h2, also. The displacement of the central angle of the radiation pattern and the dependence of the central angle on the h both are the consequence of the displacement of the pole. The pole tends to the 0. limit position on the real axis in the case of h The fact that the width of the radiation pattern is proportional to h2 is typical for the planar periodic guides. The other peculiarity of Figure 2 is the absence of the term proportional to h’ in the power series for the dependence of the central angle of the radiation pattern on h. This is due to the specific choice of the function f(z) which is symmetric in the same manner with respect to the replacement h to -h. After this replacement, the function f(x) preserves its form and changes only the phase. Let us pass to the investigation of the lossy waveguide. Naturally, in this case, the guided mode 0. because of the coupling coefficient between the intensity does not tend to the infinity if h plane wave and the guided mode tends to zero. From the same value of h, the intensity begins to decrease if h ---+ 0. The function I,(+, h) must have the point of the maximum in the finite value of the h variable. The one example of such appearance of the radiation pattern for the lossy waveguide is shown in Figure 3. The imaginary part of the refractive index of the guide media is Im n;! = 6.9 x lo-‘. The range of tp variable in Figure 3 is five times wider than in Figure 2: -49.161592’ < 4 < -49.161752”.
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Lossy Periodic Dielectric Waveguide
log h Figure 4. The results for the loasy waveguide with Im 722= 5.6 x lo-‘. All parameters are the same ss in Figure 2 with the exception of the range of cp variable is between -49.161112’ and -49.161912’.
Figure 5. The sum of the intensities of all plane waves reflected to the ambient medium for the lossy waveguide identical to the one shown in Figure 3: Im 7~2= 6.9 x lo+.
It is interesting to compare the radiation patterns for the different extinction in the guide material. Figure 4 is analogous to Figure 3, but the imaginary part of the refractive index of the guide media is Im n2 = 5.6 x lo- 7. The range of the angle of incidence of the primary plane This is again five times wider than in Figure 3. The wave cp is -49.161112” < 4 < -49.161912’. maximal value of the mode intensity is smaller and achieves at the greater value of h.
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The comparison of Figures 2-4 shows that the appearance of the radiation pattern for the problem of the planar periodic waveguide excitation is in the same manner analogous to the resonance curve of the lossy oscillator. This analogy is not full because the coupling coefficient between the plane wave and the guided mode tends to zero if h -+ 0. Figure 5 shows some of the consequences of this peculiarity. Here the total intensity of all the plane waves reflected upwards is shown. All parameters of the guide are the same as in Figure 3. The reflection for the lossy waveguide in the csse of h - 0 tends to the analogous value for the planar dielectric slab and may be calculated by Freshnel formulas. The general appearance of the radiation pattern of the waveguide with the modulated surface for TE polarization for the very wide range of the parameters is similar to that shown in Figures 2-5. The main result of this work is the elaboration of the numerical-analytical algorithm for the calculation of the scattering of the electromagnetic waves by the planar dielectric waveguide with the modulated surface. The second result is the demonstration of the effect of the displacement of the central angle of the radiation pattern for the lossy periodical planar dielectric waveguide.
REFERENCES 1. A. Hessel and A.A. Oliner, A new theory of Wood’s anomalies on optical gratings, Applied Optics 4 (lo), 1275-1297, (1965). 2. S.A. Akhmanov, V.N. Seminogov and V.I. Sokolov, The diffraction of light by a rough surface with a deep arbitrary profile, Zh. Eksp. Tear. Fiz. 93 (5), 1654-1669, (1987). 3. V.N. Seminogov and V.I. Sokolov, The diffraction of,limited light beams by a surface with arbitrary relief under the condition of Wood’s anomalies, Kvantowaya Elektronika 16 (7), 1481-1493, (1989). 4. H.-G. Unger, Planar Optical Waveguides and Fibres, Mir, Moscow, (1980). 5. M. Barnosky, Editor, Introduction to Integral Optics, Mir, Moscow, (1978). 6. J.M. Elson, L.F. DeSandre and J.L. Stanford, Analysis of anomalous resonance effects in multilayer-overcoated, low efficiency gratings, J. Optical Sot. of America, Part A 6 (l), 74-88, (1988). 7. S.S. Wang and Il.. Magnusson, Theory and applications of guided-mode resonance filters, Applied Optics 32 (lo), 2606-2613, (1993). 8. M.G. Moharam and T.K. Gaylord, Rigorous coupled-wave analysis of metallic surface-relief gratings, J. Optical Sot. of America, Part A 3 (ll), 1780-1787, (1986). 9. R. Petit, Editor, EEectromugnetic Theory of Arrays, Springer-Verlag, Berlin, (1980). 10. S. Solimeno, B. Crosignali and P. DiPorto, Guiding, Di&uction and Confinement of Optical Radiation, Chapt. 6.10, Academic Press, New York, (1986). 11. V.P. Shestopalov and Yu.K. Sirenko, Dynamic Theory of Graitings, Nauk. Dumka, Kiev, (1989). 12. V.P. Shestopalov, L.N. iitvinenko, S.A. Masalov and V.G. Sologub, Dimction of Waves by Graitings, KKar’kov, (1973). 13. A. Yariv, Coupled-mode theory for guided wave optics, IEEE J. Quantum Electronic QE-9 (9), 919-931, (1973). 14. A. Yariv and P. Yell, Optical Waves in Crystal&, Mir, Moscow, (1987). 15. A.G. Sveshnikov, The incomplete Galerkin method, Dokl. Akad. Nauk. SSSR 238 (5), 1076-1079, (1977). 16. A.A. Bikov, The use of the run method to solve stiff boundary value problems for systems of linear difference equations, Zh. Vychisl. Mat. Mat. Fiz. 29 (3), 355-370, (1989). 17. A.A. Bikov, V.Yu. Popov and A.G. Sveshnikov, The use of the singular perturbations method to calculate the electromagnetic-wave-scattering operator in open periodic waveguide structure, Z/x. Vychisl. Mat. Mat. F&z. 34 (7), 1038-1052, (1994).
Mathematicaland Computer Modelling32 (2000) 1059-1070
PERGAMON
www.elsevier.nl/locate/mcm
Resonant Scattering of Electromagnetic Waves by a Lossy Periodic Dielectric Waveguide A.
A.
BIKOV, V. Yu. POPOV AND A. G. SVESHNIKOV MSU, Physical Faculty, Moscow, Russia M. KLIBANOV University of North Carolina at Charlotte Charlotte, NC 28223, U.S.A. I. 0. VOLKOVA AND A. V. TIKHONRAVOV MSU, Research Computer Center, Moscow, Russia
Abstract-The problem of the excitation of guided modes in planar dielectric waveguides with periodically modulated surfaces is considered. In particular, the form of the radiation pattern of a lossy waveguide, the width and the position of the central angle of the radiation pattern as a function of the surface modulation depth are studied. A special numerical-analytical method for the calculation of the intensities of guided modes has been developed. In the case of a small modulation depth, this method makes use the expansion of the solutions of Maxwell equations into the Laurent series near the scattering matrix pole. @ 2000 Elsevier Science Ltd. All rights reserved. Keywords-Scattering,
Periodic waveguide,Directionalradiationpattern.
1. INTRODUCTION In this work, we consider the resonant scattering of electromagnetic
waves by the modulated
surface of a planar lossy dielectric waveguide. The resonant nature of the scattering is displayed in the significant increase of the intensities of scattered fields and waveguide modes near the waveguide boundary as compared to the intensity of the incident plane wave. It is well known that there are several mechanisms causing a resonant scattering of electromagnetic waves by the periodically modulated surface. A resonant scattering may be connected with the Wood anomaly [l-3] or with the excitation of waveguide modes [4,5]. Methods for the calculation of the amplitudes of induced waveguide modes and scattered fields have been developed by several authors [6-81. In spite of the variety of the approaches having been used, some important aspects of the plane wave scattering by modulated dielectric waveguides were not investigated in reasonable detail. One of these aspects is a detailed analysis of the radiation pattern of the dielectric waveguide in the receiving regime. The main goal of our paper is the calculation and analysis of the main parameters of the radiation pattern, such as the central angle, the width, and the amplitude of the central lobe. To calculate the intensities of scattered fields and waveguide modes, we have developed a special method combining both the numerical and analytical approaches. In the’case of a small amplitude of the surface modulation, the scattering This work was partly supported
by Russian Fund of Fundamental
0895-7177/00/$ - see front matter PII: SO895-7177(00)00190-4
Researches,
Grant 95Ol-01284a.
@ 2000 Elsevier Science Ltd. All rights reserved.
Typeset
by &&TD
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matrix is expanded into a series with respect to the modulation depth. The peculiarity of the problem under consideration is that this series turns to be a Laurent series. To calculate the coefficients of the Laurent series, the Rayleigh hypothesis is used [9,10]. The first terms of the scattering matrix series, being responsible for the effect of resonant increase of the scattered field, are calculated analytically. In a general case for the correct calculation of the matrix coefficients, a special numerical method has been developed. This method is similar to the method of a halfinversion [11,12]. To contrast with the traditional coupled-modes method [13,14], our method allows us to analyze waveguide structures of an arbitrary depth and form. In the case of big amplitudes of the surface modulation, the series representation of the scattering matrix is not valid. In this case, to obtain the scattering matrix, Maxwell equations are reduced to a system of ordinary differential equations by the Gale&n-type method 1151. To develop a stable numerical algorithm, we have applied a special method for the solution of the boundary value problems for ordinary differential equations [16]. On the whole, the combination of the numerical and analytical methods allows us to develop a high performance computer code. This code enables one to calculate the intensities of fields without any restrictions on the parameters of waveguides.
2. SCATTERING WAVEGUIDE
OF THE PLANE WAVE ON A THREE-LAYER WITH THE MODULATED BOUNDARY
Consider the scattering of the plane electromagnetic wave by a three-layer planar dielectric waveguide with the modulated boundary between the waveguide and the ambient medium. The refractive indices of the waveguide, the substrate, and the ambient medium are equal to n2, nr , n3, respectively, where Re n2 > Re ni,
Re nz > Re ns,
Im 72s= 0,
Im It2 and Im ni are small values. The shape of the boundary between the waveguide and the ambient medium is given by the periodic function y = f(z) with the period a : f(z + a) = f(z). The boundary between the waveguide and the substrate, is the plain y = -b, where 6 is the mean thickness of the waveguide. Let us consider the excitation of the waveguide by the TE-polarized plane wave, propagating under the angle cp to the y axis (see Figure 1). The wave vector belongs to the sy-plane, the only nonzero component of the electric field is given by E, = E,-,a-“2 exp(-ians(ssincp
+ ycoscp)).
Here Eo is a given field amplitude, K = 27r/X is the wavenumber, X is the wavelength in vacuum. Here and thereafter, the time factor exp( -iwt) is omitted. All nonzero field components, E,, H,, H,, depend on the coordinates x and y and do not depend on z. Any component U of the electrical and magnetic fields satisfies the Floquet condition U(z f or Y) = exp(f+)U(z,
y),
(1)
where p = --an3 sin cp
(2)
is Floquet parameter. It follows from the Maxwell equations and Floquet condition that electromagnetic fields in the substrate, in the uniform part of the waveguide and in the ambient medium may be presented as a superposition of uniform and nonuniform plane waves. The complex amplitudes of these waves are determined from the boundary conditions for the fields.
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Loeey Periodic Dielectric Waveguide
-b , Figure 1. A diagram of the dielectric waveguide with the periodically modulated boundary between the waveguide and the ambient medium. The rehactive indices are nr = 1.5, 122= 1.8, ne = 1.0, period a = 1, mean thickness b = 1, wavelength in vacuum X = 2, effective refractive index n,s = 1.756558. The central angle of the radiation pattern cpo = -49.161712O.
For the brevity of equations, we will use the matrix designations. To describe the dependence of the fields on 2, the following vector satisfying equation (1) is used: X(z) = {. . . ,X_m(Z),
. . . ,X*(z),
. . . ,Xm(2),
. . . },
where X,(z)
pm = p + T.
= a-‘/2exp(ip,z),
To describe the dependence of the fields on y coordinate in the substrate, the waveguide, and the ambient medium, we use the infinite matrices l?i, l?2, I’S with the diagonal elements equal to 72’ = (K’$
j = 1,2,3,
- &l/21
and all other elements equal to zero: l?j =diag
(
. . . . r!A ,...,
$‘,.,.,
r,!$,.,.).
We designate complex amplitudes of waves propagating upwards and downwards as Aim;“),Bkm;“) (in the ambient media), Aimm),Bim) (in waveguide), A(;Z),B(;R) (in the substrate), respectively. Let us In fact, Aj”) = 0 for all m because there are no upward waves in the substrate. write these amplitudes as the column vectors As,As, and Bi,Bs,Bs. For example, A3 = A(O) Atrn) }T where “T” designates a transposition. The vector Bs is { . . . ) Ah-“) ,*.*, 3 3.“) 3 ,*** , specified by the incident field. It has only one nonzero component Br) = Eo, where EO is the amplitude of the incident plane wave. Assume that the Rayleigh hypothesis is fulfilled. Then it is possible to present the electrical field U(z, y) = Ez(z, y) as a superposition of plane waves. This field can be written down as follows: U(Z, 3) = X(s)(exp(iIk+s + exp( -GY)Bs)
A. A. BIKOV et al.
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in the region y > f(z), U(Z, y) = X(z)(exp(il?zy)Az
+ exp(-if’zY)Bz)
in the region -b < y < f(s),
V(G Y) = X(z) exp(-irl (Y + b))Bl in the region y < -b. Scattered fields in the substrate and the ambient covering medium are specified by the vectors A3 and Br, connected with the vector B3 by the linear equations A3 = R31 B3,
BI
= 5f-31B3r
where R3r and T3r are the reflection and transmission matrices. The amplitudes of waveguide modes are described by the vectors AZ, and B3, that are also determined by the vector B3. For example, A2 = S32B3,
where S32 is the scattering matrix. The method for the calculation of the matrices R31, T31, and section.
S32
is outlined in the following
3. A METHOD OF CALCULATION OF THE REFLECTION, TRANSMISSION, AND SCATTERING MATRICES Let R32 and T32 be the reflection and transmission matrices for the boundary between the ambient medium and the substrate in the case of the incidence of the plane wave from the ambient medium. Let R3r and T3r be the analogous matrices for the flat boundary between the waveguide and the substrate in the case of the excitation from the waveguide region (Figure 1). The vectors of the complex amplitudes of the fields in the waveguide and in the ambient medium are connected by the relations A3
B2 = T32 I33 + R23 A2.
=R32B3+T23A2,
Similar relations are valid at the boundary between the substrate A2
= R2lB2,
and the waveguide
B1 = 5721B2,
where x2 = exp(-il?&)A3,
63 = exp(+iT2b)B2.
These relations allow us to write down the following expression for the matrices R31, T31, S32: R31 = R32 + T23s32,
(3)
T31 = Tzl exP(ihb)(%
f R23S32)r
(4)
Sexp(ir2b)R21
exp(iI’2b)T32,
(5)
S32 =
where
(I is the unity matrix).
S = H-l,
(6)
H = I - exp(ir’2b)R21 exp(il?2b)R23,
(7)
Lossy Periodic DielectricWaveguide Equations (5)-(7)
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are applicable if the matrix H is nonsingular.
In the case of an arbitrary surface modulation, it is convenient to use the incomplete Galerkin method [15] to calculate the partial scattering matrices R32, T23, T32. This method reduces the boundary value problem for the Maxwell equations to the boundary value problem for a system of linear differential equations. This last problem can then be solved by a special numerical method [16]. It is essential that the method from reference [16] does not impose any restrictions on structure parameters (the thickness of the waveguide, the ratio between the wavelength, and metrical parameters of the waveguide). At the same time, in general, this method requires a lot of computational
work. This situation stimulated us to develop an analytical approach allowing
us to study the problem at least for a small depth of the surface modulation.
Here we present
the sketch of this approach. Let us denote the depth of the surface modulation as h and expand the matrices R32 and T32 in Taylor series T32 = Tii) + hTik) + h2T,(i) + . . . .
Note that here
(O' and R,,
T,(,o)are usual reflection and transmission matrices for the flat surface.
To calculate the Taylor coefficients R&j and TiF), n = 1,2, . . . , the method from reference [lo] is used. We assume that the waveguide surface is modulated by the sine law f(2)
= hsin
T (
, >
To avoid overloading the description of the approach, we do not present the expressions for R$
T!$ . We would like only to note that Rg’ and T.$
are diagonal matrices with the nonzero
main diagonal, or OStdiagonal, and matrices Ri’,) and T$:) have nonzero elements only on two diagonals (+lSt and -lSt diagonals), Rg’ and T3(:) are three-diagonal matrices with the nonzero -2nd, main, and +2’ld diagonals. Note that in the case of more complicated modulation laws, the other coefficients may be also nonzero. It follows from equations (7),(g), that the matrix H is also expanded in the Taylor series H =
2
hn@-‘).
n=O
The inverse matrix S = H-l
may have the pole in the series expansion S =
2
h”S@).
(9)
n=--p
The order of the pole depends on the presence of nonzero elements on the main diagonal of the matrix N(O). If H(O) m,m # 0 for all numbers m, S is a regular matrix. In this case, the amplitudes of all guided modes tend to zero when h tends to zero and in the limit, the scattered fields are described by Fresnel formulas. If H(O) m,m = 0 )
(10)
for some m = M, then S has a pole of a second order [17]. The exact formulas for the matrices Stn) are not essential for our discussion. To analyze the dependence of scattered fields on the depth of the surface modulation, we ought to note that the main term of the Laurent series expansion S ( - 2, has one nonzero element with the indices M, M, S(-l) has four nonzero elements with the indices M f 1, M f 1. It is easy to show that condition (10) is equivalent to the characteristic equation for a planar dielectric waveguide with flat boundaries (41. It may be exactly fulfilled for the real value of the
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A. A. BIKOV et al.
Floquet parameter only in the absence of losses in the waveguide material. condition is the existence of the such index M, that
The other necessary
(11) In other words, the plane wave corresponding to the index M must be locked in the guide region. If equation (11) is fulfilled, it is possible to write down equation (10) in a standard form [4]: tan wY2)
where 72 = (ni equation (12) are The connection given by the link
72b1+ P3) = - CPlP3 _ y> I
(12)
S2/~2)1/2, pj = (/?2/~2 - n?)1/2, j = 1,3. Those P values that are the roots of longitudinal wave numbers’of guided modes. of equation (10) and the characteristics equation of the planar guide (12) is equation P=L’M.
(13)
It is possible to explain the link relation (13) in terms of the phase synchronism condition. Consider the analogous waveguide with the thickness b and the extremely small amplitude of the surface modulation h 0. The source of the emitted plane waves must be the guided mode propagating along the waveguide. Under the condition h 0, the radiation losses are negligible and the amplitude of the guided mode may be treated as a constant over the large region in comparison with the waveguide thickness and the wavelength. The scattered field may be represented as a set of the plane waves generated on the low periodical relief. Consider the full set of the direction angles of the emitted plane waves {cpn}. Under condition (13), the angle of the incidence of the exciting plane wave cp is equal to one of the (Pi. In this paper, we consider the case of the presence of only one zero element on the main diagonal of the matrix H,f$,,,. Under this condition, the S has the pole of the second order (p = 2). Note that even in the relatively simple case of the TEpolarized electromagnetic waves the Hf$m may have two elements on the main diagonal. It may happen if the corresponding value of the Floquet parameter p,,, = p + (2nm)/a is equal to zero. In other words, in this case, one of the above-mentioned plane waves radiated by the analogous waveguide with the extremely small corrugation amplitude has the directional angle directed along the normal to the waveguide plane. In the general case of the mixed polarization, the number of zero diagonal elements may be two, three, and greater. The order of the pole of the S matrix may be greater than two. Let us consider the dependence of the scattered fields on the corrugation depth h. The main factor determining the scattered fields appearance is the presence of the pole of the S matrix. The maximal possible rate of the growth of the fields when the h tends to zero is hp2. Note that then only the nonzero element of Ste2) matrix is the element in the position with the indices M, M. To realize the possibility of the maximal growth rate of the scattered fields, it is necessary to excite the waveguide by the external EM wave represented by the Bs vector with the nonzero element (B~)M. The third condition of equation (11) shows that this element of the Bs vector represents the amplitude of the nonuniform plane wave in the ambient media. The nonuniform plane wave is increasing by the exponential law under the condition that the y coordinate is increasing. Such appearance of the exciting field may be obtained with the special type of the input device, for example, the prism input element [4,5]. The excitation of the periodic waveguide by the plane wave cannot provide the exciting field of the necessary type. The next term in the series representation of the scattering matrix SC-‘) has four nonzero elements. Their indices are M f 1, M f 1. To obtain the increase rate of the scattered fields proportionally, h-l one must use the exciting field determined by the B3 vector with the indices of the nonzero elements M - 1 or M + 1.
Lossy Periodic
Dielectric Waveguide
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Let us consider some computational aspects of the numerical simulation electromagnetic
waves
scattering by the periodic guide. There exist the analytical expressions for scattering matrix (6). If h ---+0, the same elements of the scattering matrix tend to infinity. This is the result of the inversion of the ill-conditioned matrix H. The inversion of this matrix with the standard numerical algorithms is not correct and leads to great growth in computer round-off errors. The numerical-analytical method was developed for the correct matrix inversion. Consider the pole of the scattering matrix S. The main system matrix H has the second-order zero and may be expressed as the product of three matrices: H = H1H2H3, where HI and Hz matrices both have the first-order zero. These matrices are simple and may be inverted analytically. The HZ matrix is regular and it has the general form. This matrix may be inverted numerically with high precision. As a result, the S matrix presented in the form S = (H3)-1(H$-1(H1)-1. The main (unlimited) part of the inverse matrix
S =
5
hn#*)
(14)
n=-p
may be calculated analytically. The exact expressions for the terms mentioned in (14) are not essential, so they are omitted in this paper. The q value depends on the order of the pole of the scattering matrix. If the second-order pole exists, the optimal value is q = 0. At the same time, expression (14) is applicable only for the small values of h. For large h, it is necessary to use the exact expression (6). In this case, the inversion of the H matrix is correct. To exclude the two different expressions for the scattering matrix in the different ranges of the parameter h, the numerical-analytical
method was developed.
The main part of the scattering
matrix is calculated by expression (14), correction is fulfilled numerically. As a result, the inversion of the ill-conditioned matrix is excluded. The additional advantage of this approach consists in the uniformity of the formulas used in the all range of the h parameter. This may be essential in the optimization problems.
4. ANALYSIS OF THE RADIATION PATTERN OF THE LOSSY DIELECTRIC WAVEGUIDE In practice, it is useful to analyze the radiation pattern of the periodical dielectric waveguide. To introduce the radiation pattern, it is necessary to interpret the field induced in the waveguide as the superposition of the fields of the primary plane wave, the guided mode, and the scattered homogeneous and inhomogeneous plane waves. These fields are calculated under the conditions of the external excitation of the above-mentioned guide by the plane wave falling from the ambient medium. The field of the guided mode is the superposition of the direct and the inverse locked plane-wave modes in the guide region (-b < y < 0). The locked field decreases exponentially in the substrate and in the ambient medium. The conditions of the locking are represented by equations (11). It is possible to associate the complex value of the amplitude with the field of the induced guided mode. We will denote this value by the A, symbol. This field is similar to the field of the guided mode of the planar dielectric waveguide without the corrugation. We will briefly designate this value as the amplitude of the induced mode. Let us designate the angular dependence of the amplitude of the induced mode as the amplitude radiation pattern. Note that the amplitude of
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BIKOV et al.
the guided mode in the case of the excitation of the dielectric waveguide with the flat boundaries is zero. The main practice interest consists in the investigation of the dependence of the amplitude radiation pattern on the corrugation depth h and the waveguide parameters, such as the extinction in the guide material. The simplest explanation of the radiation pattern appearance may be obtained with the introduction of the complex variables, Let us consider the angle of the incidence of the initial plane wave 4 and the depth of the corrugation h as complex parameters. As we had seen earlier, the point h = 0 is the pole of the scattering matrix if the same conditions are fulfilled. In particular, the angle of incidence 4 must be equal to the same directional angle $0 described earlier. Under the conditions of the phase synchronism, the point r$ = $0 on the real axis of the complex plane C$= 4’ + Q!J”is the pole of the guided mode amplitude. For the lossy waveguide (Im nz > 0), the pole of the guided mode amplitude changes to the complex plane. The appearance of the radiation pattern in this case will be illustrated later in this section. It is possible to introduce the intensity of the induced mode I9 along with the amplitude of the induced mode A,. The intensity of the induced mode is determined as the flow of the Pointing vector calculated for the electromagnetic field of the induced mode across the total waveguide cross-section including the parts of the substrate and the ambient medium where the induced fields are nonzero. Really, the integration will be carried out over the region z = const, -oo < y < oo. It is essential that only the field of induced mode is taken into account. The flow of the Pointing vector per the unit of the length for the incident plane wave is taken to be the unit. It is possible to show that the I9 and the A, values are connected by the usual relation 1, = PIA,?,
05)
where p is the longitudinal wavenumber of the induced guided mode [4,5]. Hence, it is possible to consider only one of the two sets of the characteristics-the complex induced amplitude or the intensity and phase. We will consider the intensity because there exist precision experimental methods of the measurement of this parameter in the range of the optical wavelengths. The dependence of the intensity of the induced mode on the angle of incidence of the initial plane wave is named the intensity radiation pattern of the dielectric waveguide in receiving regime. The most significant results may be obtained by consideration of the amplitude radiation pattern including the module and phase dependence. In this paper, we restrict ourselves by the only module and consider the intensity radiation pattern. The more obviousness results will be obtained by the simultaneous consideration of the set of the intensity directional patterns for the waveguides with the various h values. Totally, they form the two-dimensional function of the cp and h variables. The intensity radiation pattern possesses the characteristic features connected with location of the poles of the scattering matrix on the complex plane. The numerical results presented later may be interpreted as the results of the motion of the pole of the scattering matrix on the complex plane of the incident angle because of the variation of the corrugation depth. The results of the numerical calculations for the various parameters of the waveguide are presented in Figures 2-5. At first, the waveguide without the losses is considered. The common appearance of the radiation pattern depends mainly on such structural peculiarities of the waveguide under consideration as the order of the pole. The specific values of the waveguide parameters such as the refractive indices and thickness do not affect on the main features. The waveguide parameters for Figure 1 are ns = 1,
in the covering media,
12s= 1.8,
in the waveguide,
121= 1.5,
in the substrate.
bossy Periodic Dielectric Waveguide
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log h Figure 2. The set of the curves of the constant value of the induced mode intensity in the waveguide without losses: Im ns = 0. The value of log(h) is put aside the horizontal axis. The value of h is varying in the range between 10m4 (left end of the axis) and 10m2 (right end). The value of (0 is put aside the vertical axis. The range of cp is between -49.161688O (top end) and -49.16172OO (bottom end). --T-
log h Figure 3. The results of the lossy waveguide excitation: Im ns = 6.9 x 10es. All parameters are the same as in Figure 2 with the exception of the range of ‘p variable is between -49.161592” (top end) and -49.161752O (bottom end).
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The dimensionless spatial variables. the sine law with wavelength in the The main mode
A. A. BIKOV et
al.
variables are used for the values of the period, the thickness, and the other The boundary between the guide and the ambient media is modulated by the period a = 1, the average thickness of the waveguide is equal to 1. The vacuum X = 2. of the waveguide with the flat surfaces has the effective refractive index %tr =def p = 1 756558. K. *
The main guided mode of the analogous planar guide with negligible small corrugation depth radiates the plane wave under the angle cps = -49.161712” (the angles are counted from the vertical direction as is shown in Figure 1). The phase synchronism conditions requires that the angle of the incidence of the primary plane wave must be near the mentioned value, The dependence of the guided mode intensity on the variables h and cp is shown in Figure 2. The logarithm of the amplitude of the corrugation h is put aside the horizontal axis. The value of the h variable is varying in the range between 10B4 and 10a2. The angle of the incidence of the primary plane wave is put aside the vertical axis. The range of the cp variable is between -49.161688” and -49.161720”. The lines of the constant value of the induced mode intensity are shown in Figure 2. The decimal logarithmic scale is chosen for representation of the mode intensity. For example, the line marked by the 4 symbol corresponds to the value of the intensity equal to 104. The maximal value of the induced mode intensity in the shown range of the parameters is near 106. This value tends to infinity if the corrugation depth tends to zero. More exactly, if h --+ 0, the maximal intensity over the mentioned range of the initial angle C$is proportional to h-2. The infinite increasing of the mode intensity in the case of h 0 is accompanied by the displacement of the central angle of the radiation pattern. The central angle is the function of the h variable. It is determined as the angle corresponding to the maximal value of the mode intensity. If h 0, the central angle tends to the $0 value. The deflection of the central angle from the 40 value is proportional to h2. The next significant characteristic of the radiation pattern is the width on the specified level equal to half of the maximal level. It is possible to show (Figure 2) that the width of the radiation pattern is proportional to h2, also. The displacement of the central angle of the radiation pattern and the dependence of the central angle on the h both are the consequence of the displacement of the pole. The pole tends to the 0. limit position on the real axis in the case of h The fact that the width of the radiation pattern is proportional to h2 is typical for the planar periodic guides. The other peculiarity of Figure 2 is the absence of the term proportional to h’ in the power series for the dependence of the central angle of the radiation pattern on h. This is due to the specific choice of the function f(z) which is symmetric in the same manner with respect to the replacement h to -h. After this replacement, the function f(x) preserves its form and changes only the phase. Let us pass to the investigation of the lossy waveguide. Naturally, in this case, the guided mode 0. because of the coupling coefficient between the intensity does not tend to the infinity if h plane wave and the guided mode tends to zero. From the same value of h, the intensity begins to decrease if h ---+ 0. The function I,(+, h) must have the point of the maximum in the finite value of the h variable. The one example of such appearance of the radiation pattern for the lossy waveguide is shown in Figure 3. The imaginary part of the refractive index of the guide media is Im n;! = 6.9 x lo-‘. The range of tp variable in Figure 3 is five times wider than in Figure 2: -49.161592’ < 4 < -49.161752”.
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Lossy Periodic Dielectric Waveguide
log h Figure 4. The results for the loasy waveguide with Im 722= 5.6 x lo-‘. All parameters are the same ss in Figure 2 with the exception of the range of cp variable is between -49.161112’ and -49.161912’.
Figure 5. The sum of the intensities of all plane waves reflected to the ambient medium for the lossy waveguide identical to the one shown in Figure 3: Im 7~2= 6.9 x lo+.
It is interesting to compare the radiation patterns for the different extinction in the guide material. Figure 4 is analogous to Figure 3, but the imaginary part of the refractive index of the guide media is Im n2 = 5.6 x lo- 7. The range of the angle of incidence of the primary plane This is again five times wider than in Figure 3. The wave cp is -49.161112” < 4 < -49.161912’. maximal value of the mode intensity is smaller and achieves at the greater value of h.
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A. A.
BIKOV
et al.
The comparison of Figures 2-4 shows that the appearance of the radiation pattern for the problem of the planar periodic waveguide excitation is in the same manner analogous to the resonance curve of the lossy oscillator. This analogy is not full because the coupling coefficient between the plane wave and the guided mode tends to zero if h -+ 0. Figure 5 shows some of the consequences of this peculiarity. Here the total intensity of all the plane waves reflected upwards is shown. All parameters of the guide are the same as in Figure 3. The reflection for the lossy waveguide in the csse of h - 0 tends to the analogous value for the planar dielectric slab and may be calculated by Freshnel formulas. The general appearance of the radiation pattern of the waveguide with the modulated surface for TE polarization for the very wide range of the parameters is similar to that shown in Figures 2-5. The main result of this work is the elaboration of the numerical-analytical algorithm for the calculation of the scattering of the electromagnetic waves by the planar dielectric waveguide with the modulated surface. The second result is the demonstration of the effect of the displacement of the central angle of the radiation pattern for the lossy periodical planar dielectric waveguide.
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