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Solutions of the thin phase grating diffraction equation
Volume 25, number 2
OPTICS COMMUNICATIONS
May 1978
SOLUTIONS OF THE THIN PHASE GRATING DIFFRACTION EQUATION R. MAGNUSSON and T.K. GAYLORD
School of Electrical Engineering, Georgia Institute of Technology, A tlan ta, Georgia 30332, USA Received 21 December 1977 Revised manuscript received 27 February 1978
An integral solution and a Bessel function solution of the thin phase grating difference-differential equation that describes diffraction by a thin arbitrarily-shaped periodic grating are presented and discussed.
1. Introduction Wave diffraction by thin periodic structures is of great importance in many areas of physics. Thin grating devices, for example, are used in the deflection, guidance, coupling, and modulation of acoustic and electromagnetic waves. In this paper, solutions of the general system of difference-differential equations that govern wave diffraction by thin phase gratings are presented. The phase grating (for example, a spatial modulation of the refractive index) is assumed to have an arbitrary periodic form. Thus, the total grating phase modulation, An(x), may be expanded in a Fourier series as
,~a(x) = ~
h=l
(nch cos hKx + nsh sin hKx)
(1)
where K is the magnitude of the grating vector, nch and nsh are the amplitudes of the hth cosine and sine Fourier components of the grating, and x is the coordinate in the grating vector direction as shown in fig. 1. A wave incident upon a thin grating (of thickness d) is diffracted into multiple waves as illustrated in fig. 1. The complex amplitudes, Si(z), of these waves are given by the (infinite) system of difference-differential equations [1 ] oo
dSi/dz + (3 h~=l (nhSi_ h - n~Si+h) = O,
(2)
¢
where z is the normalized thickness coordinate (0 ~< z ~< 1), i and h are integers, * denotes complex conjugate,/3 is a constant (~ = 7rd/X cosO, where X is the free-space wavelength of the incident wave and 0 is the angle of incidence), and n h represents the complex amplitude of each of the hth gratings and is given by
n h =nsh +jnch ,
(3)
where j = ( - 1 ) 1/2. The incident wave is normalized to unity amplitude. On the incidence side of the periodic structure (z ~< 0) there are no diffracted components of the wave. Thus the boundary conditions are S0(0)-- 1,
Si(O)=Ofori=/=O.
(4,5)
For a single-spatial-frequency grating (having, for example, a phase modulation of An (x) = nsl sin Kx), the solution is the well-known result [2] Si(z ) = (-1)iJi(2nsl [3z), where Ji is an integer-order Bessel function of the first kind. 129
Volume 25, number 2
OPTICS COMMUNICATIONS
May 1978
X
f S-2
S
1
S0
Z
14g. 1. The geometry of a thin phase grating with an incident plane wave and multiple diffracted waves. The spatial phase modulation is indicated by the line pattern. It is assumed, for simplicity, that the grating has the same average refractive index as its surroundings so that no reflections occur. Solutions for two superposed sine gratings have been obtained by Nath [3], Mertens [4], and others [5]. Zankel and Hiedemann [6] have treated the case of a sine-series grating and have thereby obtained a Bessel function solution for gratings possessing odd symmetry. Solutions for a grating having any arbitrary functional form within a grating period are presented here.
2. Solutions
2.1. Integral solution An integral solution of eq. (2) subject to the boundary conditions, eqs. (4) and (5), is 27
Si(z ) = (27)-1 f
exp I-j2t3zAn (~)l exp (-ji~) d~,
(6)
0 where ~ = Kx and An (~) = An (~ + 2n). This solution may be verified by direct substitution into eq. (2). Thus a specific solution, Si(z), may be obtained by substituting the corresponding periodic function An(~) into eq. (6) and integrating. If An (~) is of suitably elementary form (e.g. a sinusoid, square, sawtooth, triangle, rectangle etc. within one grating period), then simple analytical solutions may be obtained upon direct integration [i.e. without Fourier expansion of 2m(~)]. For more general functions, An (~), numerical integration may be used to evaluate Si(z ) from eq. (6).
2.2. Bessel function solution Eq. (6) may be integrated for an arbitrary periodic function, An(~), by expanding An(~) in a Fourier series as 130
Volume 25, number 2
OPTICS COMMUNICATIONS
May 1978
given by eq. (1) and using the identity exp(-jb sin
a) z ~ i=
(-1)iJi(b)exp(jia)
(7)
oo
The factor exp [-j213zA~n(~)] in the integrand of eq. (6) thus becomes 00
co
e~
~I
~I
~
~
m=l n=l icm=-00 isn=-~
(-j)icmJic m (acm)exp(jicmm~)(-1)isnJisn(asn)exp(Jisnn~),
(8)
where acm = 2~Zncm, asn= 2~Znsn, IJm = 1 and Iln= 1 denote products with an infinite number of factors, and m, n, iem, and isn are integers. Combining eqs. (8) and (6) leads to fi
St(z)=
00
~ H (-j)icm(I1)isnJicm(acm)Jisn(asn), icm,isn m =1 n =1
(9)
where the symbol ~icm, isn represents the summation over values of the integers icm and isn subject to the constraint
m=l
icmm +
n=l
isnn =i .
(10)
The constraint (10) represents the necessary condition for nontrivial (S i q: 0) solutions. Eq. (9) is a direct solution of the thin phase grating difference-differential equation, eq. (2), subject to the boundary conditions, eqs. (4) and (5). It does not require an integration as does eq. (6). Eqs. (9) and (10)may be combined into a single expression for S i by solving eq. (10) for one of the integers icm or isn and substituting it into eq. (9). For example, solving for icl yields the solution 00
/C2 =-°°
ic3 =-00
00
iC4 =-~
X ( - j ) i - 2 i c 2 - 3 i c 3 - 4 i c 4 - ' ' ' -isl
iSl =--~
iS2 =-°°
is3 =-00
2is2-3is3-...
XJi-2ic2-3ic3-4ic4 - .." -is1-2is2-3is3- ...(acl ) X (-j)ic2Jic 2 (ac2)(-j)ic3Jic 3 (ac3)(-j)ic4Jic 4 (ac4)... X (-1)islJis I (asl)(-- 1) is2Jis 2 (as2)(- 1)is3Jis3 ( a s 3 ) . . .
(11)
The constraint of eq. (10) is now included in the solution. This result is similar to that found by Zankel and Hiedemann [6] by considering an equation similar to eq. (2) but with nch= 0. Eq. (1 1) is simple to program for computer calculations. A multitude of grating shapes (including sinusoid, square, sawtooth, triangle, and rectangle) and modulation amplitudes n 1, where n 1 = (n21 + n21 )1/2, have been tested using eq. (1 I) and the results compared to results obtained from the integral solution, eq. (6). Even though eq. (1 1) contains an infinite number of terms each with an infinite number of factors, convergence is obtained for practical physical cases by retaining only a few factors and a small number of terms. 131
Volume 25, number 2
OPTICS COMMUNICATIONS
May 1978
3. Conclusions The thin phase grating difference-differential equation has been solved for general grating profiles. An integral solution and a Bessel function solution have been presented. The integral solution, for elementary grating profiles, may be integrated directly to give simple analytical expressions for the diffracted amplitudes. The Bessel function solution is an infinite series, each term of which contains an infinite number of factors. It is found to converge rapidly.
Acknowledgement This work was sponsored by the National Science Foundation and by the Army Research Office.
References [1] [2] [3] [4] [5] [6]
132
R. Magnusson and T.K. Gaylord, J. Opt. Soc. Am. 67 (1977) 1165. C.V. Raman and N.S.N. Nath, Proc. Indian Acad. Sci. A2 (1935) 406. N.S.N. Nath, Proc. Cambr. Phil. Soc. 34 (1938) 213. R. Mertens, Proc. Indian Aead. Sci. A48 (1958) 288. Also see numerous references cited in ref. [4]. K.L. Zandel and E.A. Hiedemann, J. Acoust. Soc. Am. 31 (1959) 44.
OPTICS COMMUNICATIONS
May 1978
SOLUTIONS OF THE THIN PHASE GRATING DIFFRACTION EQUATION R. MAGNUSSON and T.K. GAYLORD
School of Electrical Engineering, Georgia Institute of Technology, A tlan ta, Georgia 30332, USA Received 21 December 1977 Revised manuscript received 27 February 1978
An integral solution and a Bessel function solution of the thin phase grating difference-differential equation that describes diffraction by a thin arbitrarily-shaped periodic grating are presented and discussed.
1. Introduction Wave diffraction by thin periodic structures is of great importance in many areas of physics. Thin grating devices, for example, are used in the deflection, guidance, coupling, and modulation of acoustic and electromagnetic waves. In this paper, solutions of the general system of difference-differential equations that govern wave diffraction by thin phase gratings are presented. The phase grating (for example, a spatial modulation of the refractive index) is assumed to have an arbitrary periodic form. Thus, the total grating phase modulation, An(x), may be expanded in a Fourier series as
,~a(x) = ~
h=l
(nch cos hKx + nsh sin hKx)
(1)
where K is the magnitude of the grating vector, nch and nsh are the amplitudes of the hth cosine and sine Fourier components of the grating, and x is the coordinate in the grating vector direction as shown in fig. 1. A wave incident upon a thin grating (of thickness d) is diffracted into multiple waves as illustrated in fig. 1. The complex amplitudes, Si(z), of these waves are given by the (infinite) system of difference-differential equations [1 ] oo
dSi/dz + (3 h~=l (nhSi_ h - n~Si+h) = O,
(2)
¢
where z is the normalized thickness coordinate (0 ~< z ~< 1), i and h are integers, * denotes complex conjugate,/3 is a constant (~ = 7rd/X cosO, where X is the free-space wavelength of the incident wave and 0 is the angle of incidence), and n h represents the complex amplitude of each of the hth gratings and is given by
n h =nsh +jnch ,
(3)
where j = ( - 1 ) 1/2. The incident wave is normalized to unity amplitude. On the incidence side of the periodic structure (z ~< 0) there are no diffracted components of the wave. Thus the boundary conditions are S0(0)-- 1,
Si(O)=Ofori=/=O.
(4,5)
For a single-spatial-frequency grating (having, for example, a phase modulation of An (x) = nsl sin Kx), the solution is the well-known result [2] Si(z ) = (-1)iJi(2nsl [3z), where Ji is an integer-order Bessel function of the first kind. 129
Volume 25, number 2
OPTICS COMMUNICATIONS
May 1978
X
f S-2
S
1
S0
Z
14g. 1. The geometry of a thin phase grating with an incident plane wave and multiple diffracted waves. The spatial phase modulation is indicated by the line pattern. It is assumed, for simplicity, that the grating has the same average refractive index as its surroundings so that no reflections occur. Solutions for two superposed sine gratings have been obtained by Nath [3], Mertens [4], and others [5]. Zankel and Hiedemann [6] have treated the case of a sine-series grating and have thereby obtained a Bessel function solution for gratings possessing odd symmetry. Solutions for a grating having any arbitrary functional form within a grating period are presented here.
2. Solutions
2.1. Integral solution An integral solution of eq. (2) subject to the boundary conditions, eqs. (4) and (5), is 27
Si(z ) = (27)-1 f
exp I-j2t3zAn (~)l exp (-ji~) d~,
(6)
0 where ~ = Kx and An (~) = An (~ + 2n). This solution may be verified by direct substitution into eq. (2). Thus a specific solution, Si(z), may be obtained by substituting the corresponding periodic function An(~) into eq. (6) and integrating. If An (~) is of suitably elementary form (e.g. a sinusoid, square, sawtooth, triangle, rectangle etc. within one grating period), then simple analytical solutions may be obtained upon direct integration [i.e. without Fourier expansion of 2m(~)]. For more general functions, An (~), numerical integration may be used to evaluate Si(z ) from eq. (6).
2.2. Bessel function solution Eq. (6) may be integrated for an arbitrary periodic function, An(~), by expanding An(~) in a Fourier series as 130
Volume 25, number 2
OPTICS COMMUNICATIONS
May 1978
given by eq. (1) and using the identity exp(-jb sin
a) z ~ i=
(-1)iJi(b)exp(jia)
(7)
oo
The factor exp [-j213zA~n(~)] in the integrand of eq. (6) thus becomes 00
co
e~
~I
~I
~
~
m=l n=l icm=-00 isn=-~
(-j)icmJic m (acm)exp(jicmm~)(-1)isnJisn(asn)exp(Jisnn~),
(8)
where acm = 2~Zncm, asn= 2~Znsn, IJm = 1 and Iln= 1 denote products with an infinite number of factors, and m, n, iem, and isn are integers. Combining eqs. (8) and (6) leads to fi
St(z)=
00
~ H (-j)icm(I1)isnJicm(acm)Jisn(asn), icm,isn m =1 n =1
(9)
where the symbol ~icm, isn represents the summation over values of the integers icm and isn subject to the constraint
m=l
icmm +
n=l
isnn =i .
(10)
The constraint (10) represents the necessary condition for nontrivial (S i q: 0) solutions. Eq. (9) is a direct solution of the thin phase grating difference-differential equation, eq. (2), subject to the boundary conditions, eqs. (4) and (5). It does not require an integration as does eq. (6). Eqs. (9) and (10)may be combined into a single expression for S i by solving eq. (10) for one of the integers icm or isn and substituting it into eq. (9). For example, solving for icl yields the solution 00
/C2 =-°°
ic3 =-00
00
iC4 =-~
X ( - j ) i - 2 i c 2 - 3 i c 3 - 4 i c 4 - ' ' ' -isl
iSl =--~
iS2 =-°°
is3 =-00
2is2-3is3-...
XJi-2ic2-3ic3-4ic4 - .." -is1-2is2-3is3- ...(acl ) X (-j)ic2Jic 2 (ac2)(-j)ic3Jic 3 (ac3)(-j)ic4Jic 4 (ac4)... X (-1)islJis I (asl)(-- 1) is2Jis 2 (as2)(- 1)is3Jis3 ( a s 3 ) . . .
(11)
The constraint of eq. (10) is now included in the solution. This result is similar to that found by Zankel and Hiedemann [6] by considering an equation similar to eq. (2) but with nch= 0. Eq. (1 1) is simple to program for computer calculations. A multitude of grating shapes (including sinusoid, square, sawtooth, triangle, and rectangle) and modulation amplitudes n 1, where n 1 = (n21 + n21 )1/2, have been tested using eq. (1 I) and the results compared to results obtained from the integral solution, eq. (6). Even though eq. (1 1) contains an infinite number of terms each with an infinite number of factors, convergence is obtained for practical physical cases by retaining only a few factors and a small number of terms. 131
Volume 25, number 2
OPTICS COMMUNICATIONS
May 1978
3. Conclusions The thin phase grating difference-differential equation has been solved for general grating profiles. An integral solution and a Bessel function solution have been presented. The integral solution, for elementary grating profiles, may be integrated directly to give simple analytical expressions for the diffracted amplitudes. The Bessel function solution is an infinite series, each term of which contains an infinite number of factors. It is found to converge rapidly.
Acknowledgement This work was sponsored by the National Science Foundation and by the Army Research Office.
References [1] [2] [3] [4] [5] [6]
132
R. Magnusson and T.K. Gaylord, J. Opt. Soc. Am. 67 (1977) 1165. C.V. Raman and N.S.N. Nath, Proc. Indian Acad. Sci. A2 (1935) 406. N.S.N. Nath, Proc. Cambr. Phil. Soc. 34 (1938) 213. R. Mertens, Proc. Indian Aead. Sci. A48 (1958) 288. Also see numerous references cited in ref. [4]. K.L. Zandel and E.A. Hiedemann, J. Acoust. Soc. Am. 31 (1959) 44.