Abstract

This work presents a novel finite-element solution to the problem of scattering from multiple two-dimensional cavities in infinite metallic walls. The technique presented here is highly efficient in terms of computing resources and is versatile and accurate in comparison with previously published methods. The formulation is based on using the surface integral equation with the free-space Green’s function as the boundary constraint. The solution space is divided into local bounded frames containing each cavity. The finite-element formulation is applied inside each frame to derive a linear system of equations associated with nodal field values. The surface integral equation is then applied at the opening of the cavities to truncate the computational domain and to connect the matrix subsystem generated from each cavity. The near and far fields are generated for different single and multiple cavity examples. The results are in close agreement with methods published earlier.

© 2009 Optical Society of America

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References

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  1. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998).
    [CrossRef]
  2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
    [CrossRef] [PubMed]
  3. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).
  4. R. F. Harrington and J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870-873 (1976).
    [CrossRef]
  5. D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE Case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
    [CrossRef]
  6. K. Barkeshli and J. L. Volakis, “TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
    [CrossRef]
  7. K. Barkeshli and J. L. Volakis, “Scattering from narrow rectangular filled grooves,” IEEE Trans. Antennas Propag. 39, 804-810 (1991).
    [CrossRef]
  8. Y. Shifman and Y. Leviatan, “Scattering by a groove in a conducting plane a PO-MoM hybrid formulation and wavelet analysis,” IEEE Trans. Antennas Propag. 49, 1807-1811 (2001).
    [CrossRef]
  9. Y.-L. Kok, “Boundary-value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am. A 9, 302-311 (1992).
    [CrossRef]
  10. T. J. Park, H. J. Eom, and K. Yoshitomi, “An analytic solution for transverse-magnetic scattering from a rectangular channel in a conducting plane,” J. Appl. Phys. 73, 3571-3573 (1993).
    [CrossRef]
  11. T. J. Park, H. J. Eom, and K. Yoshitomi, “An analysis of transverse electric scattering from a rectangular channel in a conducting plane,” Radio Sci. 28, 663-673 (1993).
    [CrossRef]
  12. T. J. Park, H. J. Eom, and K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905-911 (1993).
    [CrossRef]
  13. R. A. Depine and D. C. Skigin, “Scattering from metallic surfaces having a finite number of rectangular grooves,” J. Opt. Soc. Am. A 11, 2844-2850 (1994).
    [CrossRef]
  14. M. A. Basha, S. K. 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]
  15. M. A. Basha, S. K. Chaudhuri, and S. Safavi-Naeini, “Electromagnetic scattering from multiple arbitrary shape grooves: A generalized formulation,” IEEE/MTT-S International Microwave Symposium (IEEE, 2007), pp. 1935-1938.
    [CrossRef]
  16. J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
    [CrossRef]
  17. J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
    [CrossRef]
  18. “Finite elements-boundary integral methods,” in The Finite Element Method in Electromagnetics, J.M.Jin, 2nd ed. (Wiley, 2002).
  19. B. H. McDonald and A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theory Tech. 20, 841-847 (1972).
    [CrossRef]
  20. O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
    [CrossRef]
  21. Ansoft HFSS Version 10.1., “Ansoft Corporation,” http://www.ansoft.com. (July 2009).

2007 (1)

2002 (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

2001 (1)

Y. Shifman and Y. Leviatan, “Scattering by a groove in a conducting plane a PO-MoM hybrid formulation and wavelet analysis,” IEEE Trans. Antennas Propag. 49, 1807-1811 (2001).
[CrossRef]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[CrossRef]

1994 (1)

1993 (3)

T. J. Park, H. J. Eom, and K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905-911 (1993).
[CrossRef]

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analytic solution for transverse-magnetic scattering from a rectangular channel in a conducting plane,” J. Appl. Phys. 73, 3571-3573 (1993).
[CrossRef]

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analysis of transverse electric scattering from a rectangular channel in a conducting plane,” Radio Sci. 28, 663-673 (1993).
[CrossRef]

1992 (1)

1991 (2)

K. Barkeshli and J. L. Volakis, “Scattering from narrow rectangular filled grooves,” IEEE Trans. Antennas Propag. 39, 804-810 (1991).
[CrossRef]

O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
[CrossRef]

1990 (3)

K. Barkeshli and J. L. Volakis, “TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
[CrossRef]

J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
[CrossRef]

J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
[CrossRef]

1978 (1)

D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE Case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
[CrossRef]

1976 (1)

R. F. Harrington and J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870-873 (1976).
[CrossRef]

1972 (1)

B. H. McDonald and A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theory Tech. 20, 841-847 (1972).
[CrossRef]

Auckland, D. T.

D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE Case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
[CrossRef]

Barkeshli, K.

K. Barkeshli and J. L. Volakis, “Scattering from narrow rectangular filled grooves,” IEEE Trans. Antennas Propag. 39, 804-810 (1991).
[CrossRef]

K. Barkeshli and J. L. Volakis, “TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
[CrossRef]

Basha, M. A.

M. A. Basha, S. K. 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, S. K. Chaudhuri, and S. Safavi-Naeini, “Electromagnetic scattering from multiple arbitrary shape grooves: A generalized formulation,” IEEE/MTT-S International Microwave Symposium (IEEE, 2007), pp. 1935-1938.
[CrossRef]

Chaudhuri, S. K.

M. A. Basha, S. K. 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, S. K. Chaudhuri, and S. Safavi-Naeini, “Electromagnetic scattering from multiple arbitrary shape grooves: A generalized formulation,” IEEE/MTT-S International Microwave Symposium (IEEE, 2007), pp. 1935-1938.
[CrossRef]

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Depine, R. A.

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Ebbesen, T. W.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[CrossRef]

Eom, H. J.

M. A. Basha, S. K. 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]

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analytic solution for transverse-magnetic scattering from a rectangular channel in a conducting plane,” J. Appl. Phys. 73, 3571-3573 (1993).
[CrossRef]

T. J. Park, H. J. Eom, and K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905-911 (1993).
[CrossRef]

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analysis of transverse electric scattering from a rectangular channel in a conducting plane,” Radio Sci. 28, 663-673 (1993).
[CrossRef]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[CrossRef]

Harrington, R. F.

D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE Case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
[CrossRef]

R. F. Harrington and J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870-873 (1976).
[CrossRef]

Jin, J. M.

J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
[CrossRef]

J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
[CrossRef]

Kok, Y.-L.

Leviatan, Y.

Y. Shifman and Y. Leviatan, “Scattering by a groove in a conducting plane a PO-MoM hybrid formulation and wavelet analysis,” IEEE Trans. Antennas Propag. 49, 1807-1811 (2001).
[CrossRef]

Lezec, H. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[CrossRef]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Mautz, J. R.

R. F. Harrington and J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870-873 (1976).
[CrossRef]

McDonald, B. H.

B. H. McDonald and A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theory Tech. 20, 841-847 (1972).
[CrossRef]

Mittra, R.

O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
[CrossRef]

Park, T. J.

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analysis of transverse electric scattering from a rectangular channel in a conducting plane,” Radio Sci. 28, 663-673 (1993).
[CrossRef]

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analytic solution for transverse-magnetic scattering from a rectangular channel in a conducting plane,” J. Appl. Phys. 73, 3571-3573 (1993).
[CrossRef]

T. J. Park, H. J. Eom, and K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905-911 (1993).
[CrossRef]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

Ramahi, O. M.

O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
[CrossRef]

Safavi-Naeini, S.

M. A. Basha, S. K. 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, S. K. Chaudhuri, and S. Safavi-Naeini, “Electromagnetic scattering from multiple arbitrary shape grooves: A generalized formulation,” IEEE/MTT-S International Microwave Symposium (IEEE, 2007), pp. 1935-1938.
[CrossRef]

Shifman, Y.

Y. Shifman and Y. Leviatan, “Scattering by a groove in a conducting plane a PO-MoM hybrid formulation and wavelet analysis,” IEEE Trans. Antennas Propag. 49, 1807-1811 (2001).
[CrossRef]

Skigin, D. C.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[CrossRef]

Volakis, J. L.

K. Barkeshli and J. L. Volakis, “Scattering from narrow rectangular filled grooves,” IEEE Trans. Antennas Propag. 39, 804-810 (1991).
[CrossRef]

J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
[CrossRef]

K. Barkeshli and J. L. Volakis, “TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
[CrossRef]

J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
[CrossRef]

Wexler, A.

B. H. McDonald and A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theory Tech. 20, 841-847 (1972).
[CrossRef]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[CrossRef]

Yoshitomi, K.

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analytic solution for transverse-magnetic scattering from a rectangular channel in a conducting plane,” J. Appl. Phys. 73, 3571-3573 (1993).
[CrossRef]

T. J. Park, H. J. Eom, and K. Yoshitomi, “Analysis of TM scattering from finite rectangular grooves in a conducting plane,” J. Opt. Soc. Am. A 10, 905-911 (1993).
[CrossRef]

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analysis of transverse electric scattering from a rectangular channel in a conducting plane,” Radio Sci. 28, 663-673 (1993).
[CrossRef]

IEE Proc., Part H: Microwaves, Antennas Propag. (1)

J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (6)

J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled aperture in a thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
[CrossRef]

O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
[CrossRef]

R. F. Harrington and J. R. Mautz, “A generalized network formulation for aperture problems,” IEEE Trans. Antennas Propag. 24, 870-873 (1976).
[CrossRef]

K. Barkeshli and J. L. Volakis, “TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions,” IEEE Trans. Antennas Propag. 38, 1421-1428 (1990).
[CrossRef]

K. Barkeshli and J. L. Volakis, “Scattering from narrow rectangular filled grooves,” IEEE Trans. Antennas Propag. 39, 804-810 (1991).
[CrossRef]

Y. Shifman and Y. Leviatan, “Scattering by a groove in a conducting plane a PO-MoM hybrid formulation and wavelet analysis,” IEEE Trans. Antennas Propag. 49, 1807-1811 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE Case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
[CrossRef]

B. H. McDonald and A. Wexler, “Finite-element solution of unbounded field problems,” IEEE Trans. Microwave Theory Tech. 20, 841-847 (1972).
[CrossRef]

J. Appl. Phys. (1)

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analytic solution for transverse-magnetic scattering from a rectangular channel in a conducting plane,” J. Appl. Phys. 73, 3571-3573 (1993).
[CrossRef]

J. Opt. Soc. Am. A (4)

Nature (London) (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[CrossRef]

Radio Sci. (1)

T. J. Park, H. J. Eom, and K. Yoshitomi, “An analysis of transverse electric scattering from a rectangular channel in a conducting plane,” Radio Sci. 28, 663-673 (1993).
[CrossRef]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Other (4)

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).

M. A. Basha, S. K. Chaudhuri, and S. Safavi-Naeini, “Electromagnetic scattering from multiple arbitrary shape grooves: A generalized formulation,” IEEE/MTT-S International Microwave Symposium (IEEE, 2007), pp. 1935-1938.
[CrossRef]

Ansoft HFSS Version 10.1., “Ansoft Corporation,” http://www.ansoft.com. (July 2009).

“Finite elements-boundary integral methods,” in The Finite Element Method in Electromagnetics, J.M.Jin, 2nd ed. (Wiley, 2002).

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Figures (16)

Fig. 1
Fig. 1

Schematic of the scattering problem from a cavity with arbitrary shape in an infinite PEC surface. An ABC or PML is used to truncate the computational domain.

Fig. 2
Fig. 2

Schematic of the scattering problem from a cavity in a PEC surface. The dashed line represents the bounded region that contains all sources, inhomogeneities, and anisotropies.

Fig. 3
Fig. 3

Schematic of surface integral contour in the half-space above cavity.

Fig. 4
Fig. 4

Schematic showing the extension of the surface integral method to multiple cavities.

Fig. 5
Fig. 5

Amplitude of total E-field at the cavity opening for a 1 λ × 1.5 λ air-filled rectangular cavity, TM case, θ = 0 , calculated using the method introduced in this work (FEM-TFSIE), the mode matching method [14], and HFSS (FEM-HFSS).

Fig. 6
Fig. 6

Amplitude of total E-field at the cavity opening for a 1 λ × 1.5 λ air-filled rectangular cavity, TM case, θ = 45 , calculated using the method introduced in this work (FEM-TFSIE), the mode matching method [14], and HFSS (FEM-HFSS).

Fig. 7
Fig. 7

Amplitude of total H-field at the cavity opening for a 0.6 λ × 0.4 λ air-filled rectangular cavity, TE case, θ = 45 , calculated using the method introduced in this work (FEM-TFSIE), the FEM used in [18], and HFSS (FEM-HFSS).

Fig. 8
Fig. 8

Amplitude of total E-field at the cavity opening for an isosceles right triangle with aperture size of 1 λ with no filling, TM case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE) and HFSS (FEM-HFSS).

Fig. 9
Fig. 9

Amplitude of total E-field at the cavity opening for a 0.7 λ × 0.35 λ silicon-filled ( ϵ r = 11.9 ) rectangular cavity, TM case, θ = 0 , calculated using the method introduced in this work (FEM-TFSIE). A 0.42 λ × 0.07 λ PEC strip is positioned at the geometric center of the cavity.

Fig. 10
Fig. 10

Amplitude of total E-field at the cavities’ opening for two identical 1 λ × 0.5 λ air-filled rectangular cavities, TM case, θ = 0 , calculated using the method introduced in this work (FEM-TFSIE) and HFSS (FEM-HFSS). The cavities are separated by 0.2 λ .

Fig. 11
Fig. 11

Amplitude of total E-field at the cavities’ opening for two identical 1 λ × 0.5 λ air-filled rectangular cavities, TM case, θ = 45 , calculated using the method introduced in this work (FEM-TFSIE) and HFSS (FEM-HFSS). The cavities are separated by 0.2 λ .

Fig. 12
Fig. 12

Amplitude of total H-field at the cavities opening for two identical 0.4 λ × 0.2 λ air-filled rectangular cavities, TE case, θ = 0 , calculated using the method introduced in this work (FEM-TFSIE) and HFSS (FEM-HFSS). The cavities are separated by 0.2 λ .

Fig. 13
Fig. 13

Magnitude of surface current on the PEC surface between two 0.8 λ × 0.4 λ air-filled rectangular cavities, TM case, θ = 0 , calculated using the method introduced in this work (FEM-TFSIE).

Fig. 14
Fig. 14

Magnitude of surface current on the PEC surface between two 0.8 λ × 0.4 λ air-filled rectangular cavities, TM case, θ = 45 , calculated using the method introduced in this work (FEM-TFSIE).

Fig. 15
Fig. 15

Amplitude of the total E-field at the cavities’ opening for six identical 0.8 λ × 0.4 λ air-filled rectangular cavities, TM case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), the mode matching method used in [15], and HFSS (FEM-HFSS). The cavities are separated by 0.8 λ .

Fig. 16
Fig. 16

Amplitude of the far field for six identical 0.8 λ × 0.4 λ air-filled rectangular cavities, TM case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE) and the mode matching method used in [15]. The cavities are separated by 0.8 λ .

Equations (61)

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[ 1 p ( x , y ) u t ] + k 0 2 q ( x , y ) u t = g ,
r = ( 1 p ( x , y ) u t ) + k 0 2 q ( x , y ) u t g
R i = Ω in w i { [ 1 p ( x , y ) u t ] + k 0 2 q ( x , y ) u t g } d Ω = 0 .
R i = Ω in [ 1 p ( x , y ) w i u t k 0 2 q ( x , y ) w i u t + g w i ] d Ω + w i p ( x , y ) u t d Γ = 0 .
R i = Ω in [ 1 p ( x , y ) w i u t k 0 2 q ( x , y ) w i u t + g w i ] d Ω = 0 .
u t = i = 1 m u i t α i ( x , y ) ,
R = [ M ] [ U ] [ F ] = 0 ,
M i j = Element [ 1 p ( x , y ) α i ( x , y ) α j ( x , y ) k 0 2 q ( x , y ) α i ( x , y ) α j ( x , y ) ] d Ω ,
F i = Element g α i ( x , y ) d Ω .
[ M i i M i b 0 M b i M b b M b o 0 M o b M o o ] [ u i u b u o ] = [ F i F b F o ]
2 E z ( ρ ) + k 0 2 E z ( ρ ) = j ω μ J z ( ρ ) , ρ Ω ,
2 G e ( ρ , ρ ) + k 0 2 G e ( ρ , ρ ) = δ ( ρ ρ ) , ρ , ρ Ω .
G e ( ρ , ρ ) = j 4 H 0 2 ( | ρ ρ source | ) + j 4 H 0 2 ( k | ρ ρ image source | ) .
Ω G e ( ρ , ρ ) [ 2 E z ( ρ ) + k 0 2 E z ( ρ ) ] d Ω = j ω μ Ω J z ( ρ ) G e ( ρ , ρ ) d Ω ;
Ω ( E z 2 G e G e 2 E z ) d Ω = Γ + Γ ( E z G e n G e E z n ) d Γ ,
Ω E z ( ρ ) ( 2 G e + k 0 2 G e ) d Ω = j ω μ Ω J z ( ρ ) G e d Ω + Γ + Γ ( E z G e n G e E z n ) d Γ .
E z ( ρ ) = j ω μ Ω J z ( ρ ) G e ( ρ , ρ ) d Ω Γ + Γ [ E z ( ρ ) G e ( ρ , ρ ) n G e ( ρ , ρ ) E z ( ρ ) n ] d Γ .
E z ( ρ ) = j ω μ Ω J z ( ρ ) G e ( ρ , ρ ) d Ω Aperture E z ( ρ ) G e ( ρ , ρ ) n d Γ .
E z ( ρ ) = E z inc ( ρ ) + E z ref ( ρ ) Aperture E z ( ρ ) G e ( ρ , ρ ) n d Γ .
E z inc = exp [ j k ( x sin θ y cos θ ) ] ,
E z ref = exp [ j k ( x sin θ + y cos θ ) ] ,
E z ( ρ ) = j = 1 n E z j k = 1 2 ψ j k ( x j ) ,
ψ j k ( x j ) = { x j Δ x , k = 1 1 x j Δ x , k = 2 } .
[ u o ] = [ T ] + [ S ] [ u b ] ,
S i j = x j Δ x x j ψ j 1 ( x j ) G e ( x i , y , x j , y ) y d x + x j x j + Δ x ψ j 2 ( x j ) G e ( x i , y , x j , y ) y d x .
[ M i i M i b M b i M b b + M b o S ] [ u i u b ] = [ F i F b M b o T ] .
2 H z ( ρ ) + k 0 2 H z ( ρ ) = j ω ϵ M z ( ρ ) , ρ Ω ,
2 G h ( ρ , ρ ) + k 0 2 G h ( ρ , ρ ) = δ ( ρ ρ ) , ρ , ρ Ω .
| G h ( ρ , ρ ) y | y = 0 = 0 .
G h ( ρ , ρ ) = j 4 H 0 2 ( | ρ ρ source | ) j 4 H 0 2 ( k | ρ ρ image source | ) .
H z ( ρ ) = j ω ϵ Ω M z ( ρ ) G h ( ρ , ρ ) d Ω Γ + Γ [ H z ( ρ ) G h ( ρ , ρ ) n G h ( ρ , ρ ) H z ( ρ ) n ] d Γ .
H z ( ρ ) = j ω ϵ Ω M z ( ρ ) G h ( ρ , ρ ) d Ω + Aperture G h ( ρ , ρ ) H z ( ρ ) n d Γ .
H z ( ρ ) = H z inc ( ρ ) + H z ref ( ρ ) + Aperture G h ( ρ , ρ ) H z ( ρ ) n d Γ .
H z inc = exp [ j k ( x sin θ y cos θ ) ] ,
H z ref = exp [ j k ( x sin θ + y cos θ ) ] .
H z ( ρ ) n = H z ( x = x , y ) H z ( x , y ) y y
H z = j = 1 n H z j ψ j ( x j ) ,
ψ j ( x j ) = { 1 , x j Δ x j 2 < x j < x j + Δ x j 2 0 , elsewhere } ,
S i j = x j Δ x j 2 x j + Δ x j 2 G h ( x i , y , x j , y ) ψ j ( x j ) y y d x ,
[ u o ] = [ T ] [ S ] { [ u o ] [ u b ] } ,
[ u o ] = { [ 1 ] + [ S ] } 1 [ T ] + { [ 1 ] + [ S ] } 1 [ S ] [ u b ] .
[ M i i M i b M b i M b b + M b o ( 1 + S ) 1 S ] [ u i u b ] = [ F i F b M b o ( 1 + S ) 1 T ] .
[ M ] ( 1 ) [ u ] ( 1 ) = [ F ] ( 1 ) ,
[ M ] ( 2 ) [ u ] ( 2 ) = [ F ] ( 2 ) ,
[ [ M ] ( 1 ) 0 0 [ M ] ( 2 ) ] [ [ u ] ( 1 ) [ u ] ( 2 ) ] = [ [ F ] ( 1 ) [ F ] ( 2 ) ] .
[ [ u o ] ( 1 ) [ u o ] ( 2 ) ] = [ [ T ] ( 1 ) [ T ] ( 2 ) ] + [ [ S ] ( 11 ) [ S ] ( 12 ) [ S ] ( 21 ) [ S ] ( 22 ) ] [ [ u b ] ( 1 ) [ u b ] ( 2 ) ] ,
[ [ M i i ] ( 1 ) [ M i b ] ( 1 ) 0 0 [ M b i ] ( 1 ) [ M b b ] ( 1 ) + [ M b o ] ( 1 ) [ S ] ( 11 ) 0 [ M b o ] ( 1 ) [ S ] ( 12 ) 0 0 [ M i i ] ( 2 ) [ M i b ] ( 2 ) 0 [ M b o ] ( 2 ) [ S ] ( 21 ) [ M b i ] ( 2 ) [ M b b ] ( 2 ) + [ M b o ] ( 2 ) [ S ] ( 22 ) ] [ [ u i ] ( 1 ) [ u b ] ( 1 ) [ u i ] ( 2 ) [ u b ] ( 2 ) ] = [ [ F i ] ( 1 ) [ F b ] ( 1 ) [ M b o ] ( 1 ) [ T ] ( 1 ) [ F i ] ( 2 ) [ F b ] ( 2 ) [ M b o ] ( 2 ) [ T ] ( 2 ) ] .
[ [ M ] ( 1 ) [ C ] ( 12 ) [ C ] ( 21 ) [ M ] ( 2 ) ] [ [ u ] ( 1 ) [ u ] ( 2 ) ] = [ [ F ] ( 1 ) [ F ] ( 2 ) ] ,
[ C ] ( 12 ) = [ 0 0 0 [ M b o ] ( 1 ) [ S ] ( 12 ) ] ,
[ C ] ( 21 ) = [ 0 0 0 [ M b o ] ( 2 ) [ S ] ( 21 ) ] ,
[ M ] ( 1 ) = [ [ M i i ] ( 1 ) [ M i b ] ( 1 ) [ M b i ] ( 1 ) [ M b b ] ( 1 ) + [ M b o ] ( 1 ) [ S ] ( 11 ) ] ,
[ M ] ( 2 ) = [ [ M i i ] ( 2 ) [ M i b ] ( 2 ) [ M b i ] ( 2 ) [ M b b ] ( 2 ) + [ M b o ] ( 2 ) [ S ] ( 22 ) ] ,
[ F ] ( 1 ) = [ [ F i ] ( 1 ) [ F b ] ( 1 ) [ M b o ] ( 1 ) [ T ] ( 1 ) ] ,
[ F ] ( 2 ) = [ [ F i ] ( 2 ) [ F b ] ( 2 ) [ M b o ] ( 2 ) [ T ] ( 2 ) ] ,
[ u ] ( k ) = [ [ u i ] ( k ) [ u b ] ( k ) ] , ( k = 1 , 2 ) .
[ [ M ] ( 1 ) [ C ] ( 12 ) [ C ] ( 1 N ) [ C ] ( 21 ) [ M ] ( 2 ) [ C ] ( 2 N ) [ C ] ( N 1 ) [ C ] ( N 2 ) [ M ] ( N ) ] [ [ u ] ( 1 ) [ u ] ( 2 ) [ u ] ( N ) ] = [ [ F ] ( 1 ) [ F ] ( 2 ) [ F ] ( N ) ] .
J ( x ) = z ̂ 1 j ω μ E z y
J ( x ) = y ̂ × H z = x ̂ H z
E ( x , y ) = 2 × Aperture M ( x , y ) G ( x , y , x , y ) d x ,
E ( x , y ) = z ̂ Aperture | 2 E z ( x , y ) | y = 0 G y d x .
E ( x , y ) = 2 j ω ϵ ( x ̂ Aperture | H z ( x , y ) y | y = 0 G y d x y ̂ Aperture | H z ( x , y ) y | y = 0 G x d x ) .

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