Abstract

A study of coupling between finite number of general-shaped grooves in electromagnetic plane-wave scattering problem is presented. The formulation for a single groove [1] is extended to two grooves. The importance of inclusion of coupling interaction between two grooves in scattering analysis is presented and its dependence on the grooves separation distance and the angle of incident of the electromagnetic field is demonstrated quantitatively. For larger angle of incident and smaller separation distance between grooves indicate larger discrepancy between between simulation results with and without inclusion of the coupling effects. Although the results presented here considers two grooves, the formulation can be extended to arbitrary number of grooves.

©2009 Optical Society of America

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Rigorous formulation for electromagnetic plane-wave scattering from a general-shaped groove in a perfectly conducting plane

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J. Opt. Soc. Am. A 24(6) 1647-1655 (2007)

References

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  1. M. A. Basha, S. Chaudhuri, S. Safavi-Naeini, and H. J. Eom, “Rigorous formulation for electromagnetic plane wave scattering from a general shaped-groove in a perfectly conducting plane,” J. Opt. Soc. Am. A. 24, 1647–1655 (2007).
    [Crossref]
  2. Z. Ma and E. Yamashita, “Modal analysis of open groove guide with arbitrary groove profile,rdquo; IEEE Microwave Guided Wave Lett. 2, 364–366 (2007).
    [Crossref]
  3. M. A. Basha, “Optical MEMS Switches: Theory, Design, and Fabrication of a New Architecture,” PhD Thesis, Electrical and Computer Engineering, University of Waterloo (2007), (http://uwspace.uwaterloo.ca/handle/10012/3116).
  4. H.J. Eom, “Electromagnetic Wave Theory for Boundary-Value Problems,” (chapter 7),(Springer-Verlag, 2004).
  5. H.J. Eom, “Wave Scattering Theory: A series Approach Baesd on The Fourier Transform,” (chapter 1) (Springer-Verlag, 2001).
  6. P. M. Morse and H. Feshbach. newblock Method of theoretical Physics, volume I. McGraw-Hill Book Company (1953).
  7. J. A. Kong, “Electromagnetic Wave Theory,” (New York: John Wiley & Sons, c1990).

2007 (2)

M. A. Basha, S. Chaudhuri, S. Safavi-Naeini, and H. J. Eom, “Rigorous formulation for electromagnetic plane wave scattering from a general shaped-groove in a perfectly conducting plane,” J. Opt. Soc. Am. A. 24, 1647–1655 (2007).
[Crossref]

Z. Ma and E. Yamashita, “Modal analysis of open groove guide with arbitrary groove profile,rdquo; IEEE Microwave Guided Wave Lett. 2, 364–366 (2007).
[Crossref]

Basha, M. A.

M. A. Basha, S. Chaudhuri, S. Safavi-Naeini, and H. J. Eom, “Rigorous formulation for electromagnetic plane wave scattering from a general shaped-groove in a perfectly conducting plane,” J. Opt. Soc. Am. A. 24, 1647–1655 (2007).
[Crossref]

M. A. Basha, “Optical MEMS Switches: Theory, Design, and Fabrication of a New Architecture,” PhD Thesis, Electrical and Computer Engineering, University of Waterloo (2007), (http://uwspace.uwaterloo.ca/handle/10012/3116).

Chaudhuri, S.

M. A. Basha, S. Chaudhuri, S. Safavi-Naeini, and H. J. Eom, “Rigorous formulation for electromagnetic plane wave scattering from a general shaped-groove in a perfectly conducting plane,” J. Opt. Soc. Am. A. 24, 1647–1655 (2007).
[Crossref]

Eom, H. J.

M. A. Basha, S. Chaudhuri, S. Safavi-Naeini, and H. J. Eom, “Rigorous formulation for electromagnetic plane wave scattering from a general shaped-groove in a perfectly conducting plane,” J. Opt. Soc. Am. A. 24, 1647–1655 (2007).
[Crossref]

Eom, H.J.

H.J. Eom, “Electromagnetic Wave Theory for Boundary-Value Problems,” (chapter 7),(Springer-Verlag, 2004).

H.J. Eom, “Wave Scattering Theory: A series Approach Baesd on The Fourier Transform,” (chapter 1) (Springer-Verlag, 2001).

Feshbach, H.

P. M. Morse and H. Feshbach. newblock Method of theoretical Physics, volume I. McGraw-Hill Book Company (1953).

Kong, J. A.

J. A. Kong, “Electromagnetic Wave Theory,” (New York: John Wiley & Sons, c1990).

Ma, Z.

Z. Ma and E. Yamashita, “Modal analysis of open groove guide with arbitrary groove profile,rdquo; IEEE Microwave Guided Wave Lett. 2, 364–366 (2007).
[Crossref]

Morse, P. M.

P. M. Morse and H. Feshbach. newblock Method of theoretical Physics, volume I. McGraw-Hill Book Company (1953).

Safavi-Naeini, S.

M. A. Basha, S. Chaudhuri, S. Safavi-Naeini, and H. J. Eom, “Rigorous formulation for electromagnetic plane wave scattering from a general shaped-groove in a perfectly conducting plane,” J. Opt. Soc. Am. A. 24, 1647–1655 (2007).
[Crossref]

Yamashita, E.

Z. Ma and E. Yamashita, “Modal analysis of open groove guide with arbitrary groove profile,rdquo; IEEE Microwave Guided Wave Lett. 2, 364–366 (2007).
[Crossref]

IEEE Microwave Guided Wave Lett. (1)

Z. Ma and E. Yamashita, “Modal analysis of open groove guide with arbitrary groove profile,rdquo; IEEE Microwave Guided Wave Lett. 2, 364–366 (2007).
[Crossref]

J. Opt. Soc. Am. A. (1)

M. A. Basha, S. Chaudhuri, S. Safavi-Naeini, and H. J. Eom, “Rigorous formulation for electromagnetic plane wave scattering from a general shaped-groove in a perfectly conducting plane,” J. Opt. Soc. Am. A. 24, 1647–1655 (2007).
[Crossref]

Other (5)

M. A. Basha, “Optical MEMS Switches: Theory, Design, and Fabrication of a New Architecture,” PhD Thesis, Electrical and Computer Engineering, University of Waterloo (2007), (http://uwspace.uwaterloo.ca/handle/10012/3116).

H.J. Eom, “Electromagnetic Wave Theory for Boundary-Value Problems,” (chapter 7),(Springer-Verlag, 2004).

H.J. Eom, “Wave Scattering Theory: A series Approach Baesd on The Fourier Transform,” (chapter 1) (Springer-Verlag, 2001).

P. M. Morse and H. Feshbach. newblock Method of theoretical Physics, volume I. McGraw-Hill Book Company (1953).

J. A. Kong, “Electromagnetic Wave Theory,” (New York: John Wiley & Sons, c1990).

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Figures (5)

Fig. 1.
Fig. 1. Schematic diagram of plane wave scattering from two general-shaped grooves
Fig. 2.
Fig. 2. Comparison between the scattered near-field using the superposition and the exact formulations of two grooves for different incident angle θinc and period T where (a) T = 1.3λ, (b) T = 5λ, and (c) T = 13λ.
Fig. 3.
Fig. 3. Comparison between phases of scattered near-field using the superposition and the exact formulations of two grooves for different incident angle θinc and period, where T (a) T = 1.3λ, (b) T = 5λ, and (c) T = 13λ.
Fig. 4.
Fig. 4. Comparison between the scattered far-field using the superposition and the exact formulations of two grooves for different incident angle θinc and period T, where (a) T = 1.3λ, (b) T = 5λ, and (c) T = 13λ.
Fig. 5.
Fig. 5. Contour path in the complex Γ-plane [5]

Equations (22)

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E y inc x z = exp ( jk x x + jk z z ) ,
E y r x z = exp ( jk x x jk z z ) ,
E y s x z = 1 2 π E ˜ y s ( ζ ) exp ( jζx j κ 0 z ) ,
E y p , 1 x z = n = 1 sin β n 1 ( x + a pT ) × ( A n p , 1 e j ζ n 1 z + B n p , 1 e j ζ n 1 ( z + d ) ) , x pT a
2 a 2 k z e jqT k x G m ( k x a ) = 1 2 π p = 0 1 n = 1 a 2 β n 1 R m , n q , p ( k 0 ) A n p , 1 + m 1 β m 1 μ r δ m , n δ p , q A m q , 1
+ 1 2 π p = 0 1 n = 1 a 2 β n 1 e j ζ n 1 d R m , n q , p ( k 0 ) B n p , 1 m 1 β m 1 μ r e j ζ m 1 d δ m , n δ p , q B m q , 1 ,
R m , n q , p ( k 0 ) = a 4 G n ( ζa ) G m ( ζa ) e j ( p q ) κ 0 .
G n ( ζ ) = e ( 1 ) n e ζ 2 ( β n 1 a ) 2 .
[ C 1 C 2 ] = [ CA 0,0 CA 1,0 CA 0,1 CA 1,1 ] [ A 1,0 A 1,1 ] + [ CB 0,0 CB 1,0 CB 0,1 CB 1,1 ] [ B 0,1 B 1,1 ]
C q ( m ) = 2 a 2 k z e jq Tk x G m ( k x a ) ,
CA q , p m n = 1 2 π a 2 β n R m , n q , p ( k 0 ) + a ζ m β m μ r δ m , n δ p , q ,
CB q , p m n = 1 2 π a 2 β n e j ζ n d R m , n q , p ( k 0 ) m β m μ r e j ζ m d δ m , n δ p , q
R m , n q , p ( k 0 ) = a 2 G n ( ζa ) G m ( ζa ) e j ( p q ) κ o 2 .
f ( ζ ) = ( ( 1 ) m + n + 1 ) e jsTζ ( 1 ) m e j ( sT + 2 a ) ζ ( 1 ) n e j ( sT 2 a ) ζ ( ζ 2 β n 2 ) ( ζ 2 β m 2 ) κ o 2
R m , n s ( k 0 ) = 2 πa β m 2 β k 0 2 β m 2 δ mn δ qp 2 I
f ( ζ ) = ( 1 ) m + 1 e j ( sT + 2 a ) ζ e j ( sT 2 a ) ζ ( ζ 2 β n 2 ) ( ζ 2 β m 2 ) κ o 2
I = j ( 1 ) m + 1 k 0 2 0 e ( jk 0 vk 0 ) ( sT + 2 a ) e ( jk 0 vk 0 ) ( sT 2 a ) ζ ( ( 1 + jv ) 2 β 2 ) ( ( 1 + jv ) 2 α 2 ) v ( j 2 + v ) dv = R m , n s ( k 0 )
f ( ζ ) = ( 1 ) m e j ( sT + 2 a ) ζ e j ( sT 2 a ) ζ ( ζ 2 β n 2 ) ( ζ 2 β m 2 ) κ o 2
I = j ( 1 ) m + 1 k 0 2 0 e ( jk 0 vk 0 ) ( sT + 2 a ) e ( jk 0 vk 0 ) ( sT 2 a ) ζ ( ( 1 + jv ) 2 β 2 ) ( ( 1 + jv ) 2 α 2 ) v ( j 2 + v ) dv = R m , n s ( k 0 )
R m , n s = ( ( 1 ) m + n + 1 ) e j sT ζ + ( 1 ) m + 1 e j sT + 2 a ζ + ( 1 ) n + 1 e j sT 2 a ζ ( ζ 2 β n 2 ) ( ζ 2 β m 2 ) κ o 2 dv
I = ( 1 + ( 1 ) m + n ) I o ( sT ) + ( 1 ) m + 1 ) I o ( sT + 2 a ) + ( 1 ) n + 1 ) I o ( sT 2 a )
I 0 ( x ) = 0 je j k 0 x e k 0 x v v ( j 2 + v ) k 0 2 ( ( 1 + jv ) 2 β 2 ) ( ( 1 + jv ) 2 α 2 ) dv

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