- Email: [email protected]
Double grating coupler on a grounded dielectric slab waveguide
Optics Communications92 ( 1992) 35-39 North-Holland
OPTICS COMMUNICATIONS
Double grating coupler on a grounded dielectric slab waveguide A. Alphones Faculty of Engineering and Design, Kyoto Institute of Technology, Matsugasaki, Kyoto 606, Japan Received l0 September 1991;revisedmanuscript received27 November 1991
The input and output couplingcharacteristicsof the grounded dielectricslab waveguidecorrugatedboth at ground and filmcover interfacesare investigatedusingsingular perturbation procedurebased on multiplescales. Design parameterssuch as grating length for maximum input and output couplingefficienciesare comparedwith single grating structuresat conductorsurface or at film-coverinterface.
1. Introduction Grating structures are commonly employed in optics as beam couplers and as frequency selective filters. Transmission grating is widely used in integrated optics in which permittivity of the substrate is very close to that of the film, but the cover is usually free space. When the transmission grating is excited by an incident wave in the cover, reradiation can occur in the substrate and there is an unavoidable leakage of power. If the substrate is replaced by a grounding conductor, the structure becomes a reflection grating. For optical waves, the leakage of power in the conducting substrate through absorption is not more than that for a dielectric substrate. Periodic dielectric waveguides can be analysed by many analytical and numerical procedures. One of the methods is an asymptotic technique of singular perturbation procedure based on multiple scales [ 1 ], in which grating on the conducting surface was compared with the grating on the film-cover interface. This perturbation procedure is a systematic approach of obtaining coupled mode canonical equations and the coupling coefficients can be expressed in terms of physical parameters. In the range of most practical interest, doubly periodic structure are inherently leaky and recently a rigorous analysis of guided waves in such structures has been reported [2 ], in which specific examples are not considered to estimate the effect of energy
leakage. In the present paper, a singular perturbation procedure is used to analyse the coupling characteristics on grating structures consisting of peroiodic layers both at conducting surface and at film-cover interface having same period.
2. Formulationof the problem Consider a grounded dielectric waveguide that is periodically corrugated both at ground surface and film-cover interface with the same period, as shown in fig. 1. Only the transverse electric mode (having components Ey, Hx and Hz) is considered here with no variation in the y-direction and time dependence of exp ( - itot). Normalisation is used by taking the speed of light in vaccuum to be unity. The relative
b,/ eceO~t~0
x=d ~
x L-~g / X : 0 ( Z ) (
EfCO'PO
x
=
O
~
) l
"Z
Fig. 1. Configurationof double periodic structure at conductor surfaceand at film-coverinterfacewith same period.
0030-4018/92/$05.00 © 1992 ElsevierSciencePublishers B.V. All rights reserved.
35
Volume92, number 1,2,3
OPTICS COMMUNICATIONS
15 August 1992
EyE +dth cos(Kzo +01) (O/Ox)Eyl
depth of the surface corrugation is a small parameter ~. By Fourier expansion, the periodic functions at conducting surface O(z) and at film-cover interface d(z) are expressed as
and
O( z ) =d[~ql cos( Kz + Ol)
( O/Ox )Ey2 + dql cos(Kzo +01 ) ( O2/OxZ)gyl
"~(~2/~2 cos(2Kz+O2)
+...] ,
( 1)
d(z) = d [ 1 +~/, cos(Kz+OI ) +~2q2 cos(2Kz+O2) +...] ,
(2)
where q~ is the amplitude of the fundamental harmonic, K = (2n/A) is~the grating vector and A is the corrugation period. Here t h e analysis is followed along the lines originally doneby Park and Seshadri [ 1 ]. The perturbation is carried out upto the order ~2 in z-direction and Zo=Z, z2=O2z with O/Oz~=0 and the expansion of Ey is
..1_~" 1 r12~2 '1,
(02/Ox2)Eyo
cos2( Kzo + 0~) ( 03/ Ox3 ) Eeo
(7)
are continuous at x = d. The zeroth order problem correspond to the solution of uniform slab waveguide, where the fields in the film and cover are given by
Eyoc=Nsas(z2) e x p [ - O t c i ( x - d ) ] exp(ifliZo), sin(kix)
(8)
.
EY°f=Nsag(z2) sin(kid) exp0fliZo),
(9)
where ot~i = (f12_c02~¢) , ki2 = (o.)2(:f_fli 2)
Er~x, z) =E~o(x, zo, z2) ÷,~E~(x, Zo, z2) +02Er2(x, Zo, z2),
+ ½dZrl~cosZ(Kzo -~-01)
(3)
and the dispersion relation is given by - k i cot(/qd).
(10)
and the differential equation satisfying Ey is given by
aci =
(02/0x2 + 02/Oz~ + c02~.)Ey. = T.~Eyo ,
The normalisation constant Ns is evaluated by assuming the power carried by the guided wave in the z-direction as Iagl 2. In the first order problem, the solutions with wave numbers (fli-K) and (Pi+K) are considered with the assumption that only ( ~ i - K) lie in the fast wave region O<...(fli-K)<...ogx/~c and all other Floquet modes as slow waves.
(4)
where n = f (for film), c (for cover); s=0, 1, 2 and Tfo = Tf, = / ' c o = Tc~ = 0 ,
Tr2 = T~2 = - 202/0Zo0z2. By expanding Ey in a Taylor series about x = 0 and x=d, the equivalent boundary conditions for each order are t~°: Ero = 0
Eylc = Nr exp ( - i01 ) {bi exp [ik_ ~c( x - d) ] atx=0,
+ai exp[ -ik_~c(x-d) ]} exp [i(/~i -K)z0]
Eyo and (O/Ox)Eyo
(5)
are continuous at x = d O~:Ey~ +drh cos(Kzo+Ot) (O/OX)Eyo=O a t x = 0 ,
Eyl and ~xE),I Jl-d~l c o s ( K z o ' ¥ 0 1 )
+F~cexp[-oqc(x-d)]exp[i(fli+K)zo], Eylf=(F_lfs!n!k-lfX! +G_ cos(k_|fx)'~ \ slntK-lfa} lfcos(k_lfd)] × exp[i(fli - K ) z o ]
02 0--~ Ero
(6)
are continuous at x = d
+ ½d2q2 cos2(Kzo +0~) (02/Ox2)E~o = 0
(w sin(k, fx) ,.~ cos(k, fx)~ ~ ' " f sin(klfd) * trlf cos(k, fd)] ×exp[i(fl~ +K)zo] ,
02:Ey2 +dqt cos(Kzo + 01) (O/OX)Ey~
(12)
where atx=0,
k2_,~ =co2~c(#~ - K ) 2 , k21f=o)2(~f - ( f l i - K ) 2 ,
36
(11)
a~¢ = (#i + K ) 2 - t o 2 ~ c , k2f = (fli "l"K)2-(,02(~f •
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
15 August ! 992
The normalisation constant Nr is evaluated by considering the Poynting vector of incident and radiated powers as Iall 2 and Ibil 2 respectively, and is given by
where
N~ =2og/k_,¢ .
P2 = ( k ~ t a c O r _ lf . ,---77-~,,
el =
k
~2 see(kid)
- Ift-t ci S ~ )
.
(13)
The application of first order boundary conditions at x = 0 and at x = d yield the relations between F_ if, G_tf, Fif, G~f and F~. Elimination of these amplitude constants leads to a canonical equation relating incident (ai), guided (a,) and radiated (bi) waves.
2 --
cos(kid)
slnl, t¢_ if a)
--Ot2i k - If cot(k_ Ifd) ,
/'3 =ik_ ~cOtci[aci +k_ if cot(k_ ifd) ] , /'4 = k _ l f cot(k_ lfd)-ik_,c,
Ps mklfO~2i sec(/qd) + (k 2 _[_O~¢2i)aei..[_k2faei
(14)
sin(kifd)
where C~ is the coupling coefficient and C~ the reflection coefficient for the double grating coupler and are given by
P6 = (gi - t - o / c i ) K l f ~
bi = Crga, + frrai ,
ot¢i
/
k_ lf cot ( k_ lfd) + ik_ l¢ C~r=- k , lfCOt(k_lfd)-ik_l¢ "
n=f, c.
cos(kid)
(16)
(17)
( 18 )
id2q2 k 2 C ~ - 4fli( l +acid)
dthk-'¢k~ fli( l +a¢id)
X (Otei +k_,fcos(kid)[sin(k_,fd)]-1.) k_ lrcot ( k_ lfd) - ik_ l¢
+
P8
2'
(19)
(20)
(20) where Cu is the extinction coefficient and its real part gives the leakage coefficient. The imaginary part of the extinction coefficient affects the optimum angle of incidence in the case of an input coupler, and from reciprocity it also affects the radiation angle of output coupler. The output coupling efficiency Qo defined as the ratio of the total power radiated from the grating to the guided wave power incident at z 2= 0 is expressed as L Q0=(!
'bil2de2)(lag[2-o)-I
= 1- e x p ( 2 C u r L )
(21)
and the change in the radiation angle due to Cm is given by 0r ~-~tan - ' (j~i ? K ~ C g g i ) .
/'4
- o / 2 i k l f cot(klfd) ,
(15)
Substituting this in eq. (4) and by solving, a complete solution for second order problemis obtained with arbitrary amplitude constants. Application of second order boundary conditions at x = 0 and at x = d followed by the elimination of the arbitrary constants results in an amplitude transport equation. Equations (14) and (18) constitute a pair of coupled canonical equations.
da,/ dz2 = Cggag + Cgrai,
2--
1'7 = -alca¢i[a¢i +k~f cot (k~fd) ] ,
C~-
The second order solutions are consideredto obtain a solvability condition, which relates the interaction between the principal guided wave and first order incident and radiated waves.
Ey2,=¢~(x)exp(iPizo),
--2.
Ps =k,f cot ( ktfd) + al¢ ,
C ~ - dill N~ a¢i 2 Nr k _ l f c o t ( k _ l f d ) - i k _ l c k _ l f s i n ( k , ltd) +
+ (k~ + O/2i)O~ci + k 2 ifO~ci .
(22)
The input coupling efficiency is defined as the ratio of power of the principal guided wave at z2 = L to the 37
Volume 92, number 1,2,3
Q0
OPTICS COMMUNICATIONS
l(e {
o
i
3
Q,
I
2
3
4
5
n~L Fig. 2. (a) Output (b) input coupling efficienciesagainst the grating length, (a) grating on the conductor surface, (b) grating on the film-coverinterface, (c) double grating.
total incident power on the grating can be expressed as L
-I
0
-- - 21 e x p ( C ~ L ) - 1 I2/C~L.
(23)
3. Numerical results The numerical calculations have been carried out for the same parameters as in ref. [ 1 ] for the purpose of comparison. The guide is considered as lossless which corresponds to p = 0 case of ref. [ 1 ]. All the data are normalised quantities by taking the speed
15 August 1992
of light in vaccum to be unity. For co= 1.0, ¢c= 1.0, ~f=13.0, d=1.1336, ~h=0.1 and 0r=45 °, the dispersion equation for the grounded slab dielectric waveguide has been solved with the condition of fast wave requirement and other required data are evaluated, fli=2.8771, K=2.17, otci=2.6977, ki=2.1731, k_~c=0.7071, optimum length of CurL= - 1.2565. Figure 2a shows the output coupling efficiency against the grating length. It is found that the double corrugation structure has better coupling efficiency with short grating length or in otherway weak modulation of corrugation is sufficient to get desired coupling in a sufficient length. This can be explained phenomenologically. The guided waves which experience the scattering say at first on the film-cover interface generate multiple space harmonics and further as they propagate downward encounter another periodic surface which generate a set of harmonics to the each previously genearted harmonic. Strong couplings may involve two or more space harmonics that may come from more than one mode. But choosing K such that only ( ~ - K ) lie in the fast wave region 0~< (//i-K) ~
Table 1 Comparisons of design parameters for different grating couplers for ~h = 0.1 and 0= 45 (leg all data are normalised). Design parameters
Double grating
Grating on the conducting surface
Grating on the film-cover interface
Cu
-0.0170805 +i0.035355 43.608 73.563
-0.00560 +i0.000883 44.96 224.375
-0.003122 -i0.000688 45.03 402.406
~(deg) L
38
Volume92, number 1,2,3
OPTICS COMMUNICATIONS
15 August 1992
which shifts the radiation angle for optimum coupling. The grating length with doubly corrugated structure requires one third of that Of grating on the conductor surface. Hence both input and output coupling efficiencies could be enhanced by weak modulated doubly periodic structures with sufficient grating length.
structure have been investigated using singular perturbation method based on multiple scales. It is found that both input and output coupling parameters show better performance compared with single grating structures. A study is lbeing carried out with asynchronous and different modulation indices of doubly periodic structures.
4. Conclusions
References
The coupling characteristics of doubly corrugated
[ 1] W.S.Park and S.R. Seshadri,IEE Proc. H 132 (1985) 149. [2] S.T.Peng,J. Opt. Soc.Am. A 7 (1990) 1448.
39
OPTICS COMMUNICATIONS
Double grating coupler on a grounded dielectric slab waveguide A. Alphones Faculty of Engineering and Design, Kyoto Institute of Technology, Matsugasaki, Kyoto 606, Japan Received l0 September 1991;revisedmanuscript received27 November 1991
The input and output couplingcharacteristicsof the grounded dielectricslab waveguidecorrugatedboth at ground and filmcover interfacesare investigatedusingsingular perturbation procedurebased on multiplescales. Design parameterssuch as grating length for maximum input and output couplingefficienciesare comparedwith single grating structuresat conductorsurface or at film-coverinterface.
1. Introduction Grating structures are commonly employed in optics as beam couplers and as frequency selective filters. Transmission grating is widely used in integrated optics in which permittivity of the substrate is very close to that of the film, but the cover is usually free space. When the transmission grating is excited by an incident wave in the cover, reradiation can occur in the substrate and there is an unavoidable leakage of power. If the substrate is replaced by a grounding conductor, the structure becomes a reflection grating. For optical waves, the leakage of power in the conducting substrate through absorption is not more than that for a dielectric substrate. Periodic dielectric waveguides can be analysed by many analytical and numerical procedures. One of the methods is an asymptotic technique of singular perturbation procedure based on multiple scales [ 1 ], in which grating on the conducting surface was compared with the grating on the film-cover interface. This perturbation procedure is a systematic approach of obtaining coupled mode canonical equations and the coupling coefficients can be expressed in terms of physical parameters. In the range of most practical interest, doubly periodic structure are inherently leaky and recently a rigorous analysis of guided waves in such structures has been reported [2 ], in which specific examples are not considered to estimate the effect of energy
leakage. In the present paper, a singular perturbation procedure is used to analyse the coupling characteristics on grating structures consisting of peroiodic layers both at conducting surface and at film-cover interface having same period.
2. Formulationof the problem Consider a grounded dielectric waveguide that is periodically corrugated both at ground surface and film-cover interface with the same period, as shown in fig. 1. Only the transverse electric mode (having components Ey, Hx and Hz) is considered here with no variation in the y-direction and time dependence of exp ( - itot). Normalisation is used by taking the speed of light in vaccuum to be unity. The relative
b,/ eceO~t~0
x=d ~
x L-~g / X : 0 ( Z ) (
EfCO'PO
x
=
O
~
) l
"Z
Fig. 1. Configurationof double periodic structure at conductor surfaceand at film-coverinterfacewith same period.
0030-4018/92/$05.00 © 1992 ElsevierSciencePublishers B.V. All rights reserved.
35
Volume92, number 1,2,3
OPTICS COMMUNICATIONS
15 August 1992
EyE +dth cos(Kzo +01) (O/Ox)Eyl
depth of the surface corrugation is a small parameter ~. By Fourier expansion, the periodic functions at conducting surface O(z) and at film-cover interface d(z) are expressed as
and
O( z ) =d[~ql cos( Kz + Ol)
( O/Ox )Ey2 + dql cos(Kzo +01 ) ( O2/OxZ)gyl
"~(~2/~2 cos(2Kz+O2)
+...] ,
( 1)
d(z) = d [ 1 +~/, cos(Kz+OI ) +~2q2 cos(2Kz+O2) +...] ,
(2)
where q~ is the amplitude of the fundamental harmonic, K = (2n/A) is~the grating vector and A is the corrugation period. Here t h e analysis is followed along the lines originally doneby Park and Seshadri [ 1 ]. The perturbation is carried out upto the order ~2 in z-direction and Zo=Z, z2=O2z with O/Oz~=0 and the expansion of Ey is
..1_~" 1 r12~2 '1,
(02/Ox2)Eyo
cos2( Kzo + 0~) ( 03/ Ox3 ) Eeo
(7)
are continuous at x = d. The zeroth order problem correspond to the solution of uniform slab waveguide, where the fields in the film and cover are given by
Eyoc=Nsas(z2) e x p [ - O t c i ( x - d ) ] exp(ifliZo), sin(kix)
(8)
.
EY°f=Nsag(z2) sin(kid) exp0fliZo),
(9)
where ot~i = (f12_c02~¢) , ki2 = (o.)2(:f_fli 2)
Er~x, z) =E~o(x, zo, z2) ÷,~E~(x, Zo, z2) +02Er2(x, Zo, z2),
+ ½dZrl~cosZ(Kzo -~-01)
(3)
and the dispersion relation is given by - k i cot(/qd).
(10)
and the differential equation satisfying Ey is given by
aci =
(02/0x2 + 02/Oz~ + c02~.)Ey. = T.~Eyo ,
The normalisation constant Ns is evaluated by assuming the power carried by the guided wave in the z-direction as Iagl 2. In the first order problem, the solutions with wave numbers (fli-K) and (Pi+K) are considered with the assumption that only ( ~ i - K) lie in the fast wave region O<...(fli-K)<...ogx/~c and all other Floquet modes as slow waves.
(4)
where n = f (for film), c (for cover); s=0, 1, 2 and Tfo = Tf, = / ' c o = Tc~ = 0 ,
Tr2 = T~2 = - 202/0Zo0z2. By expanding Ey in a Taylor series about x = 0 and x=d, the equivalent boundary conditions for each order are t~°: Ero = 0
Eylc = Nr exp ( - i01 ) {bi exp [ik_ ~c( x - d) ] atx=0,
+ai exp[ -ik_~c(x-d) ]} exp [i(/~i -K)z0]
Eyo and (O/Ox)Eyo
(5)
are continuous at x = d O~:Ey~ +drh cos(Kzo+Ot) (O/OX)Eyo=O a t x = 0 ,
Eyl and ~xE),I Jl-d~l c o s ( K z o ' ¥ 0 1 )
+F~cexp[-oqc(x-d)]exp[i(fli+K)zo], Eylf=(F_lfs!n!k-lfX! +G_ cos(k_|fx)'~ \ slntK-lfa} lfcos(k_lfd)] × exp[i(fli - K ) z o ]
02 0--~ Ero
(6)
are continuous at x = d
+ ½d2q2 cos2(Kzo +0~) (02/Ox2)E~o = 0
(w sin(k, fx) ,.~ cos(k, fx)~ ~ ' " f sin(klfd) * trlf cos(k, fd)] ×exp[i(fl~ +K)zo] ,
02:Ey2 +dqt cos(Kzo + 01) (O/OX)Ey~
(12)
where atx=0,
k2_,~ =co2~c(#~ - K ) 2 , k21f=o)2(~f - ( f l i - K ) 2 ,
36
(11)
a~¢ = (#i + K ) 2 - t o 2 ~ c , k2f = (fli "l"K)2-(,02(~f •
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
15 August ! 992
The normalisation constant Nr is evaluated by considering the Poynting vector of incident and radiated powers as Iall 2 and Ibil 2 respectively, and is given by
where
N~ =2og/k_,¢ .
P2 = ( k ~ t a c O r _ lf . ,---77-~,,
el =
k
~2 see(kid)
- Ift-t ci S ~ )
.
(13)
The application of first order boundary conditions at x = 0 and at x = d yield the relations between F_ if, G_tf, Fif, G~f and F~. Elimination of these amplitude constants leads to a canonical equation relating incident (ai), guided (a,) and radiated (bi) waves.
2 --
cos(kid)
slnl, t¢_ if a)
--Ot2i k - If cot(k_ Ifd) ,
/'3 =ik_ ~cOtci[aci +k_ if cot(k_ ifd) ] , /'4 = k _ l f cot(k_ lfd)-ik_,c,
Ps mklfO~2i sec(/qd) + (k 2 _[_O~¢2i)aei..[_k2faei
(14)
sin(kifd)
where C~ is the coupling coefficient and C~ the reflection coefficient for the double grating coupler and are given by
P6 = (gi - t - o / c i ) K l f ~
bi = Crga, + frrai ,
ot¢i
/
k_ lf cot ( k_ lfd) + ik_ l¢ C~r=- k , lfCOt(k_lfd)-ik_l¢ "
n=f, c.
cos(kid)
(16)
(17)
( 18 )
id2q2 k 2 C ~ - 4fli( l +acid)
dthk-'¢k~ fli( l +a¢id)
X (Otei +k_,fcos(kid)[sin(k_,fd)]-1.) k_ lrcot ( k_ lfd) - ik_ l¢
+
P8
2'
(19)
(20)
(20) where Cu is the extinction coefficient and its real part gives the leakage coefficient. The imaginary part of the extinction coefficient affects the optimum angle of incidence in the case of an input coupler, and from reciprocity it also affects the radiation angle of output coupler. The output coupling efficiency Qo defined as the ratio of the total power radiated from the grating to the guided wave power incident at z 2= 0 is expressed as L Q0=(!
'bil2de2)(lag[2-o)-I
= 1- e x p ( 2 C u r L )
(21)
and the change in the radiation angle due to Cm is given by 0r ~-~tan - ' (j~i ? K ~ C g g i ) .
/'4
- o / 2 i k l f cot(klfd) ,
(15)
Substituting this in eq. (4) and by solving, a complete solution for second order problemis obtained with arbitrary amplitude constants. Application of second order boundary conditions at x = 0 and at x = d followed by the elimination of the arbitrary constants results in an amplitude transport equation. Equations (14) and (18) constitute a pair of coupled canonical equations.
da,/ dz2 = Cggag + Cgrai,
2--
1'7 = -alca¢i[a¢i +k~f cot (k~fd) ] ,
C~-
The second order solutions are consideredto obtain a solvability condition, which relates the interaction between the principal guided wave and first order incident and radiated waves.
Ey2,=¢~(x)exp(iPizo),
--2.
Ps =k,f cot ( ktfd) + al¢ ,
C ~ - dill N~ a¢i 2 Nr k _ l f c o t ( k _ l f d ) - i k _ l c k _ l f s i n ( k , ltd) +
+ (k~ + O/2i)O~ci + k 2 ifO~ci .
(22)
The input coupling efficiency is defined as the ratio of power of the principal guided wave at z2 = L to the 37
Volume 92, number 1,2,3
Q0
OPTICS COMMUNICATIONS
l(e {
o
i
3
Q,
I
2
3
4
5
n~L Fig. 2. (a) Output (b) input coupling efficienciesagainst the grating length, (a) grating on the conductor surface, (b) grating on the film-coverinterface, (c) double grating.
total incident power on the grating can be expressed as L
-I
0
-- - 21 e x p ( C ~ L ) - 1 I2/C~L.
(23)
3. Numerical results The numerical calculations have been carried out for the same parameters as in ref. [ 1 ] for the purpose of comparison. The guide is considered as lossless which corresponds to p = 0 case of ref. [ 1 ]. All the data are normalised quantities by taking the speed
15 August 1992
of light in vaccum to be unity. For co= 1.0, ¢c= 1.0, ~f=13.0, d=1.1336, ~h=0.1 and 0r=45 °, the dispersion equation for the grounded slab dielectric waveguide has been solved with the condition of fast wave requirement and other required data are evaluated, fli=2.8771, K=2.17, otci=2.6977, ki=2.1731, k_~c=0.7071, optimum length of CurL= - 1.2565. Figure 2a shows the output coupling efficiency against the grating length. It is found that the double corrugation structure has better coupling efficiency with short grating length or in otherway weak modulation of corrugation is sufficient to get desired coupling in a sufficient length. This can be explained phenomenologically. The guided waves which experience the scattering say at first on the film-cover interface generate multiple space harmonics and further as they propagate downward encounter another periodic surface which generate a set of harmonics to the each previously genearted harmonic. Strong couplings may involve two or more space harmonics that may come from more than one mode. But choosing K such that only ( ~ - K ) lie in the fast wave region 0~< (//i-K) ~
Table 1 Comparisons of design parameters for different grating couplers for ~h = 0.1 and 0= 45 (leg all data are normalised). Design parameters
Double grating
Grating on the conducting surface
Grating on the film-cover interface
Cu
-0.0170805 +i0.035355 43.608 73.563
-0.00560 +i0.000883 44.96 224.375
-0.003122 -i0.000688 45.03 402.406
~(deg) L
38
Volume92, number 1,2,3
OPTICS COMMUNICATIONS
15 August 1992
which shifts the radiation angle for optimum coupling. The grating length with doubly corrugated structure requires one third of that Of grating on the conductor surface. Hence both input and output coupling efficiencies could be enhanced by weak modulated doubly periodic structures with sufficient grating length.
structure have been investigated using singular perturbation method based on multiple scales. It is found that both input and output coupling parameters show better performance compared with single grating structures. A study is lbeing carried out with asynchronous and different modulation indices of doubly periodic structures.
4. Conclusions
References
The coupling characteristics of doubly corrugated
[ 1] W.S.Park and S.R. Seshadri,IEE Proc. H 132 (1985) 149. [2] S.T.Peng,J. Opt. Soc.Am. A 7 (1990) 1448.
39