Abstract

A rigorous formulation is used to calculate the transmission properties of a thin, perfectly conducting biperiodic capacitive mesh on a dielectric boundary. The formulation is analogous to the well-known modal method used for inductive meshes, with the modal electric fields replaced by modal currents. Measurements made at submillimeter wavelengths are presented for square capacitive meshes on a crystal quartz substrate (n = 2.1). These measurements are shown to be in good agreement with the theory. The applicability of simple equivalent circuit models is investigated and the variation of the equivalent circuit parameters with the refractive index of the substrate is discussed. A modified expression of Babinet’s principle is presented which is valid in the nondiffracting region for thin meshes on a dielectric interface.

© 1989 Optical Society of America

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References

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  1. C. C. Chen, “Transmission Through a Conducting Screen Perforated Periodically with Apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627–632 (1970).
    [CrossRef]
  2. R. C. McPhedran, D. Maystre, “On the Theory and Solar Application of Inductive Grids,” Appl. Phys. 14, 1–20 (1977).
    [CrossRef]
  3. L. C. Botten, R. C. McPhedran, J. M. Lamarre, “Inductive Grids in the Resonant Region: Theory and Experiment,” Int. J. Infrared Millimeter Waves 6, 511–575 (1985).
    [CrossRef]
  4. P. J. Bliek, L. C. Botten, R. Deleuil, R. C. McPhedran, D. Maystre, “Inductive Grids in the Region of Diffraction Anomalies: Theory, Experiment, and Applications,” IEEE Trans. Microwave Theory Tech. MTT-28, 1119–1125 (1980).
    [CrossRef]
  5. C. C. Chen, “Scattering by a Two-Dimensional Periodic Array of Conducting Plates,” IEEE Trans. Antennas Propag. AP-18, 660–665 (1970).
    [CrossRef]
  6. J. P. Montgomery, “Scattering by an Infinite Periodic Array of Thin Conductors on a Dielectric Sheet,” IEEE Trans. Antennas Propag. AP-23, 70–75 (1975).
    [CrossRef]
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 559, 560.
  8. R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Strip Gratings at a Dielectric Interface and Application of Babinet’s Principle,” Appl. Opt. 23, 3236–3242 (1984).
    [CrossRef] [PubMed]
  9. R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, D. B. Rutledge, “Bow-Tie Antennas on a Dielectric Half-Space: Theory and Experiment,” IEEE Trans. Antennas Propag. AP-35, 622–631 (1987).
    [CrossRef]
  10. L. C. Botten, “Theories of Singly and Doubly Periodic Diffraction Gratings” Ph.D. Thesis, U. Tasmania (1977), unpublished.
  11. Electroformed inductive meshes obtained from Buckbee-Mears Co., P.O. Box 43210, St. Paul, MN 55164.
  12. J. A. How, L. B. Whitbourn, “A Two-Scanning Mirror Fourier Transform Interferometer for Plasma Diagnostics,” Int. J. Infrared Millimeter Waves 4, 335–342 (1983).
    [CrossRef]
  13. D. Véron, L. B. Whitbourn, “Strip Gratings on Dielectric Substrates as Output Couplers for Submillimeter Lasers,” Appl. Opt. 25, 619–628 (1986).
    [CrossRef] [PubMed]
  14. E. E. Russell, E. E. Bell, “Measurement of the Optical Constants of Crystal Quartz in the Far Infrared with the Asymmetric Fourier-Transform Method,” J. Opt. Soc. Am. 57, 341–348 (1967).
    [CrossRef]
  15. E. V. Loewenstein, D. R. Smith, R. L. Morgan, “Optical Constants of Far Infrared Materials. 2: Crystalline Solids,” Appl. Opt. 22, 398–406 (1973).
    [CrossRef]
  16. R. Ulrich, “Far-Infrared Properties of Metallic Mesh and its Complimentary Structure,” Infrared Phys. 7, 37–55 (1967).
    [CrossRef]
  17. L. B. Whitbourn, R. C. Compton, “Equivalent-Circuit Formulas for Metal Grid Reflectors at a Dielectric Boundary,” Appl. Opt. 24, 217–220 (1985).
    [CrossRef] [PubMed]
  18. S. W. Lee, G. Zarrillo, C. L. Law, “Simple Formulas for Transmission Through Periodic Metal Grids or Plates,” IEEE Trans. Antennas Propag. AP-30, 904–910 (1982).
  19. N. Marcuvitz, Waveguide Handbook (McGraw-Hill, New York, 1951), pp. 56–60, 280–289.
  20. N. Amitay, V. Galindo, “On the Scalar Product of Certain Circular and Cartesian Wave Functions,” IEEE Trans. Microwave Theory Tech. MTT-16, 265–266 (1968).
    [CrossRef]

1987 (1)

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, D. B. Rutledge, “Bow-Tie Antennas on a Dielectric Half-Space: Theory and Experiment,” IEEE Trans. Antennas Propag. AP-35, 622–631 (1987).
[CrossRef]

1986 (1)

1985 (2)

L. B. Whitbourn, R. C. Compton, “Equivalent-Circuit Formulas for Metal Grid Reflectors at a Dielectric Boundary,” Appl. Opt. 24, 217–220 (1985).
[CrossRef] [PubMed]

L. C. Botten, R. C. McPhedran, J. M. Lamarre, “Inductive Grids in the Resonant Region: Theory and Experiment,” Int. J. Infrared Millimeter Waves 6, 511–575 (1985).
[CrossRef]

1984 (1)

1983 (1)

J. A. How, L. B. Whitbourn, “A Two-Scanning Mirror Fourier Transform Interferometer for Plasma Diagnostics,” Int. J. Infrared Millimeter Waves 4, 335–342 (1983).
[CrossRef]

1982 (1)

S. W. Lee, G. Zarrillo, C. L. Law, “Simple Formulas for Transmission Through Periodic Metal Grids or Plates,” IEEE Trans. Antennas Propag. AP-30, 904–910 (1982).

1980 (1)

P. J. Bliek, L. C. Botten, R. Deleuil, R. C. McPhedran, D. Maystre, “Inductive Grids in the Region of Diffraction Anomalies: Theory, Experiment, and Applications,” IEEE Trans. Microwave Theory Tech. MTT-28, 1119–1125 (1980).
[CrossRef]

1977 (1)

R. C. McPhedran, D. Maystre, “On the Theory and Solar Application of Inductive Grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

1975 (1)

J. P. Montgomery, “Scattering by an Infinite Periodic Array of Thin Conductors on a Dielectric Sheet,” IEEE Trans. Antennas Propag. AP-23, 70–75 (1975).
[CrossRef]

1973 (1)

E. V. Loewenstein, D. R. Smith, R. L. Morgan, “Optical Constants of Far Infrared Materials. 2: Crystalline Solids,” Appl. Opt. 22, 398–406 (1973).
[CrossRef]

1970 (2)

C. C. Chen, “Transmission Through a Conducting Screen Perforated Periodically with Apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627–632 (1970).
[CrossRef]

C. C. Chen, “Scattering by a Two-Dimensional Periodic Array of Conducting Plates,” IEEE Trans. Antennas Propag. AP-18, 660–665 (1970).
[CrossRef]

1968 (1)

N. Amitay, V. Galindo, “On the Scalar Product of Certain Circular and Cartesian Wave Functions,” IEEE Trans. Microwave Theory Tech. MTT-16, 265–266 (1968).
[CrossRef]

1967 (2)

Amitay, N.

N. Amitay, V. Galindo, “On the Scalar Product of Certain Circular and Cartesian Wave Functions,” IEEE Trans. Microwave Theory Tech. MTT-16, 265–266 (1968).
[CrossRef]

Bell, E. E.

Bliek, P. J.

P. J. Bliek, L. C. Botten, R. Deleuil, R. C. McPhedran, D. Maystre, “Inductive Grids in the Region of Diffraction Anomalies: Theory, Experiment, and Applications,” IEEE Trans. Microwave Theory Tech. MTT-28, 1119–1125 (1980).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 559, 560.

Botten, L. C.

L. C. Botten, R. C. McPhedran, J. M. Lamarre, “Inductive Grids in the Resonant Region: Theory and Experiment,” Int. J. Infrared Millimeter Waves 6, 511–575 (1985).
[CrossRef]

P. J. Bliek, L. C. Botten, R. Deleuil, R. C. McPhedran, D. Maystre, “Inductive Grids in the Region of Diffraction Anomalies: Theory, Experiment, and Applications,” IEEE Trans. Microwave Theory Tech. MTT-28, 1119–1125 (1980).
[CrossRef]

L. C. Botten, “Theories of Singly and Doubly Periodic Diffraction Gratings” Ph.D. Thesis, U. Tasmania (1977), unpublished.

Chen, C. C.

C. C. Chen, “Scattering by a Two-Dimensional Periodic Array of Conducting Plates,” IEEE Trans. Antennas Propag. AP-18, 660–665 (1970).
[CrossRef]

C. C. Chen, “Transmission Through a Conducting Screen Perforated Periodically with Apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627–632 (1970).
[CrossRef]

Compton, R. C.

Deleuil, R.

P. J. Bliek, L. C. Botten, R. Deleuil, R. C. McPhedran, D. Maystre, “Inductive Grids in the Region of Diffraction Anomalies: Theory, Experiment, and Applications,” IEEE Trans. Microwave Theory Tech. MTT-28, 1119–1125 (1980).
[CrossRef]

Galindo, V.

N. Amitay, V. Galindo, “On the Scalar Product of Certain Circular and Cartesian Wave Functions,” IEEE Trans. Microwave Theory Tech. MTT-16, 265–266 (1968).
[CrossRef]

How, J. A.

J. A. How, L. B. Whitbourn, “A Two-Scanning Mirror Fourier Transform Interferometer for Plasma Diagnostics,” Int. J. Infrared Millimeter Waves 4, 335–342 (1983).
[CrossRef]

Lamarre, J. M.

L. C. Botten, R. C. McPhedran, J. M. Lamarre, “Inductive Grids in the Resonant Region: Theory and Experiment,” Int. J. Infrared Millimeter Waves 6, 511–575 (1985).
[CrossRef]

Law, C. L.

S. W. Lee, G. Zarrillo, C. L. Law, “Simple Formulas for Transmission Through Periodic Metal Grids or Plates,” IEEE Trans. Antennas Propag. AP-30, 904–910 (1982).

Lee, S. W.

S. W. Lee, G. Zarrillo, C. L. Law, “Simple Formulas for Transmission Through Periodic Metal Grids or Plates,” IEEE Trans. Antennas Propag. AP-30, 904–910 (1982).

Loewenstein, E. V.

E. V. Loewenstein, D. R. Smith, R. L. Morgan, “Optical Constants of Far Infrared Materials. 2: Crystalline Solids,” Appl. Opt. 22, 398–406 (1973).
[CrossRef]

Marcuvitz, N.

N. Marcuvitz, Waveguide Handbook (McGraw-Hill, New York, 1951), pp. 56–60, 280–289.

Maystre, D.

P. J. Bliek, L. C. Botten, R. Deleuil, R. C. McPhedran, D. Maystre, “Inductive Grids in the Region of Diffraction Anomalies: Theory, Experiment, and Applications,” IEEE Trans. Microwave Theory Tech. MTT-28, 1119–1125 (1980).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the Theory and Solar Application of Inductive Grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

McPhedran, R. C.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, D. B. Rutledge, “Bow-Tie Antennas on a Dielectric Half-Space: Theory and Experiment,” IEEE Trans. Antennas Propag. AP-35, 622–631 (1987).
[CrossRef]

L. C. Botten, R. C. McPhedran, J. M. Lamarre, “Inductive Grids in the Resonant Region: Theory and Experiment,” Int. J. Infrared Millimeter Waves 6, 511–575 (1985).
[CrossRef]

R. C. Compton, L. B. Whitbourn, R. C. McPhedran, “Strip Gratings at a Dielectric Interface and Application of Babinet’s Principle,” Appl. Opt. 23, 3236–3242 (1984).
[CrossRef] [PubMed]

P. J. Bliek, L. C. Botten, R. Deleuil, R. C. McPhedran, D. Maystre, “Inductive Grids in the Region of Diffraction Anomalies: Theory, Experiment, and Applications,” IEEE Trans. Microwave Theory Tech. MTT-28, 1119–1125 (1980).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the Theory and Solar Application of Inductive Grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

Montgomery, J. P.

J. P. Montgomery, “Scattering by an Infinite Periodic Array of Thin Conductors on a Dielectric Sheet,” IEEE Trans. Antennas Propag. AP-23, 70–75 (1975).
[CrossRef]

Morgan, R. L.

E. V. Loewenstein, D. R. Smith, R. L. Morgan, “Optical Constants of Far Infrared Materials. 2: Crystalline Solids,” Appl. Opt. 22, 398–406 (1973).
[CrossRef]

Popovic, Z.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, D. B. Rutledge, “Bow-Tie Antennas on a Dielectric Half-Space: Theory and Experiment,” IEEE Trans. Antennas Propag. AP-35, 622–631 (1987).
[CrossRef]

Rebeiz, G. M.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, D. B. Rutledge, “Bow-Tie Antennas on a Dielectric Half-Space: Theory and Experiment,” IEEE Trans. Antennas Propag. AP-35, 622–631 (1987).
[CrossRef]

Russell, E. E.

Rutledge, D. B.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, D. B. Rutledge, “Bow-Tie Antennas on a Dielectric Half-Space: Theory and Experiment,” IEEE Trans. Antennas Propag. AP-35, 622–631 (1987).
[CrossRef]

Smith, D. R.

E. V. Loewenstein, D. R. Smith, R. L. Morgan, “Optical Constants of Far Infrared Materials. 2: Crystalline Solids,” Appl. Opt. 22, 398–406 (1973).
[CrossRef]

Tong, P. P.

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, D. B. Rutledge, “Bow-Tie Antennas on a Dielectric Half-Space: Theory and Experiment,” IEEE Trans. Antennas Propag. AP-35, 622–631 (1987).
[CrossRef]

Ulrich, R.

R. Ulrich, “Far-Infrared Properties of Metallic Mesh and its Complimentary Structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

Véron, D.

Whitbourn, L. B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 559, 560.

Zarrillo, G.

S. W. Lee, G. Zarrillo, C. L. Law, “Simple Formulas for Transmission Through Periodic Metal Grids or Plates,” IEEE Trans. Antennas Propag. AP-30, 904–910 (1982).

Appl. Opt. (4)

Appl. Phys. (1)

R. C. McPhedran, D. Maystre, “On the Theory and Solar Application of Inductive Grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

IEEE Trans. Antennas Propag. (4)

R. C. Compton, R. C. McPhedran, Z. Popovic, G. M. Rebeiz, P. P. Tong, D. B. Rutledge, “Bow-Tie Antennas on a Dielectric Half-Space: Theory and Experiment,” IEEE Trans. Antennas Propag. AP-35, 622–631 (1987).
[CrossRef]

S. W. Lee, G. Zarrillo, C. L. Law, “Simple Formulas for Transmission Through Periodic Metal Grids or Plates,” IEEE Trans. Antennas Propag. AP-30, 904–910 (1982).

C. C. Chen, “Scattering by a Two-Dimensional Periodic Array of Conducting Plates,” IEEE Trans. Antennas Propag. AP-18, 660–665 (1970).
[CrossRef]

J. P. Montgomery, “Scattering by an Infinite Periodic Array of Thin Conductors on a Dielectric Sheet,” IEEE Trans. Antennas Propag. AP-23, 70–75 (1975).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

C. C. Chen, “Transmission Through a Conducting Screen Perforated Periodically with Apertures,” IEEE Trans. Microwave Theory Tech. MTT-18, 627–632 (1970).
[CrossRef]

P. J. Bliek, L. C. Botten, R. Deleuil, R. C. McPhedran, D. Maystre, “Inductive Grids in the Region of Diffraction Anomalies: Theory, Experiment, and Applications,” IEEE Trans. Microwave Theory Tech. MTT-28, 1119–1125 (1980).
[CrossRef]

N. Amitay, V. Galindo, “On the Scalar Product of Certain Circular and Cartesian Wave Functions,” IEEE Trans. Microwave Theory Tech. MTT-16, 265–266 (1968).
[CrossRef]

Infrared Phys. (1)

R. Ulrich, “Far-Infrared Properties of Metallic Mesh and its Complimentary Structure,” Infrared Phys. 7, 37–55 (1967).
[CrossRef]

Int. J. Infrared Millimeter Waves (2)

L. C. Botten, R. C. McPhedran, J. M. Lamarre, “Inductive Grids in the Resonant Region: Theory and Experiment,” Int. J. Infrared Millimeter Waves 6, 511–575 (1985).
[CrossRef]

J. A. How, L. B. Whitbourn, “A Two-Scanning Mirror Fourier Transform Interferometer for Plasma Diagnostics,” Int. J. Infrared Millimeter Waves 4, 335–342 (1983).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (4)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975), pp. 559, 560.

N. Marcuvitz, Waveguide Handbook (McGraw-Hill, New York, 1951), pp. 56–60, 280–289.

L. C. Botten, “Theories of Singly and Doubly Periodic Diffraction Gratings” Ph.D. Thesis, U. Tasmania (1977), unpublished.

Electroformed inductive meshes obtained from Buckbee-Mears Co., P.O. Box 43210, St. Paul, MN 55164.

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Figures (12)

Fig. 1
Fig. 1

Geometric parameters associated with a mesh. (a) Coordinate axes and mesh parameters for a capacitive mesh with rectangular conducting elements. Note that the theory discussed here only deals with the case where τ = 0. (b) Parameters describing the direction and polarization of a plane wave incident on a mesh.

Fig. 2
Fig. 2

Transmittance results as a function of normalized frequency, calculated for a square capacitive mesh with c/d = 0.724 for a normally incident plane wave. Curves are shown for a range of substrate refractive indices. The vertical broken lines mark the position of the normalized Rayleigh frequency (ωR = 2π/n) for each curve.

Fig. 3
Fig. 3

Transmittance results as a function of normalized frequency calculated for capacitive meshes on a substrate with refractive index of 2.1: (a) square meshes and (b) circular meshes. Curves are shown for a range of mesh parameters (c/d,a/d).

Fig. 4
Fig. 4

Comparison of transmittance as a function of frequency according to theory (broken line) and experiment (two solid lines) for mesh 1.

Fig. 5
Fig. 5

Measured (solid line) and calculated (broken line) transmittance as a function of frequency for a 10-mm thick crystal quartz substrate with no mesh.

Fig. 6
Fig. 6

Comparison of transmittance as a function of frequency according to theory (broken line) and experiment (two solid lines) for mesh 2.

Fig. 7
Fig. 7

Measured transmittance as a function of normalized frequency for meshes 1–3.

Fig. 8
Fig. 8

Theoretical transmittance as a function of normalized frequency for meshes 1–3.

Fig. 9
Fig. 9

Schematic of the equivalent circuit model of meshes on dielectric substrates: (a) capacitive mesh and (b) inductive mesh. Z1 = Z0 is the impedance of free space, and Z2 = Z0/n is the impedance of the dielectric.

Fig. 10
Fig. 10

Comparison of transmittance as a function of normalized frequency according to the equivalent circuit model (broken lines) and our rigorous calculations for square capacitive meshes (solid lines). The mesh parameter is c/d = 0.724 and curves are shown for n = 2.1, 1.5, 1.0 (from left to right).

Fig. 11
Fig. 11

Comparison of calculated equivalent circuit transmittance as a function of normalized frequency for different methods of choosing the equivalent circuit parameters. This diagram compares the methods of Ulrich,16 Lee et al.,18 and our own equivalent circuit fits with the results from our square capacitive mesh program. These curves are for a square capacitive mesh with c/d = 0.724 and n = 1.

Fig. 12
Fig. 12

Comparison of transmittance of a capacitive mesh as a function of normalized frequency using the modified form of Babinet’s principle applied to results calculated by our square inductive mesh program (broken lines) and results calculated by our square capacitive mesh program (solid lines). Curves are shown for n = 3.5, 2.1, 1.5, 1.2 (from left to right).

Tables (4)

Tables Icon

Table I Reciprocity Check for a Square Capacitive Mesh with c/d = c′/d = 0.724, n = 2, and kd = 10

Tables Icon

Table II Comparison of Results Produced by our Square Capacitive Mesh Program with Those Produced by an Earlier Square Inductive Mesh Program,2,10 for a Mesh with c/d = c′/d = 0.724 in Free Space and a Normally incident Plane Wave with λ/d = 0.9

Tables Icon

Table III Physical Dimensions of Capacitive Meshes

Tables Icon

Table IV Equivalent Circuit Parameters for Square Capacitive and Inductive Meshes with c/d = 0.724

Equations (88)

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f ( x + p d , y + q d , z ) = f ( x , y , z ) exp [ j ( p α d + q β d ) ] .
R p q 1 ( i ) ± = 1 ξ p q ( β p q x ˆ α p y ˆ ) R p q ( i ) ± ,
R p q 2 ( i ) ± = 1 ξ p q ( α p x ˆ + β p q y ˆ ) R p q ( i ) ± ,
R p q ( i ) ± = 1 d d exp [ j ( α p x + β p q y ± γ p q ( i ) z ) ] .
E i T = υ A i υ R 00 υ ( a ) ,
E T ( a ) = p q υ A p q υ ( a ) R p q υ ( a ) + δ p 0 δ q 0 A i υ R p q υ ( a ) ,
z ˆ × H T ( a ) = p q υ A p q υ ( a ) Y p q υ ( a ) R p q υ ( a ) + δ p 0 δ q 0 A i υ Y p q υ ( a ) R p q υ ( a ) ,
E T ( d ) = p q υ A p q υ ( d ) R p q υ ( d ) ,
z ˆ × H T ( d ) = p q υ A p q υ ( d ) Y p q υ ( d ) R p q υ ( d ) .
Y p q 1 ( i ) = γ p q ( i ) Z 0 k , Y p q 2 ( i ) = ( i ) k Z 0 γ p q ( i ) ,
J = n m u C n m u J n m u ,
J n m u = z ˆ × E n m u .
E T ( a ) = E T ( d ) = 0 on metal , E T ( a ) = E T ( d ) elsewhere ,
z ˆ × [ H T ( a ) H T ( d ) ] = J on metal , = 0 elsewhere .
A p q υ ( d ) = A p q υ ( a ) + δ p 0 δ q 0 A i υ .
p q υ { [ Y p q υ ( a ) + Y p q υ ( d ) ] A p q υ R p q υ + 2 δ p 0 δ q 0 A i υ Y p q υ ( a ) R p q υ } = n m u C n m u J n m u on metal , = 0 elsewhere .
A p q υ = Z ρ q υ [ n m u ( C n m u I p q υ n m u ) 2 δ p 0 δ q 0 A i υ Y p q υ ( a ) ] ,
Z p q υ = 1 Y p q υ ( a ) + Y p q υ ( d ) ,
I p q υ n m u = metal R p q υ * · J n m u d 2 r .
n m u C n m u ( p q υ Z p q υ I p q υ n m u I p q υ N M U * ) = 2 υ A i υ Y 00 υ ( a ) Z 00 υ I 00 υ N M U * .
υ A i υ Y 00 υ ( a ) A P Q υ ( i ) = υ A i υ Y P Q υ ( i ) A P Q υ ( i ) ,
ω = k d = 2 π d / λ ,
T = T 1 T 2 F 1 + R 1 R 2 F 2 2 R 1 R 2 F cos ( 2 ϑ + φ 21 R ) ,
T av = T 1 T 2 F 1 F 2 ( 1 T 1 ) ( 1 T 2 ) .
T = 4 n X 2 1 + ( 1 + n ) 2 X 2 .
X I = ω L I 1 ω 2 L I C I ,
X C = ω 2 L C C C 1 ω C C .
L = L MKS μ 0 d , C = C MKS 0 d .
T C = 4 n ( 1 + n ) 2 [ 1 ω 2 C C 2 ( 1 + n ) 2 ] .
T I = 4 n ω 2 L I 2 .
C C q s = lim ω 0 ( 1 + n ) 2 ω 2 [ 1 ( 1 + n ) 2 4 n T C ] ,
L I q s = lim ω 0 T I 4 n ω 2 .
C ( n ) = 1 + n 2 2 C ( 1 ) , L ( n ) = L ( 1 )
X C = 1 ω C C + ω L C ,
C C = ( 1 + n 2 ) | I 001 n m u | 2 [ p q ( | I p q 2 n m u | 2 d ξ p q ) ] 1 ,
L C = 1 2 | I 001 n m u | 2 [ p q | I p q 1 n m u | 2 d ξ p q + 1 + n 4 ( 1 + n 2 ) 2 p q | I p q 2 n m u | 2 d ξ p q ] .
X C ( n , ω ) = 1 C C ( n ) 1 ω + ω L C 0 ( n ) f ( n , ω ) ,
1 X I ( n , ω ) = 1 L I ( n ) 1 ω ω C I 0 ( n ) g ( n , ω ) ,
f ( n , ω ) = a 0 ( n ) + a 1 ( n ) ω 2 + a 2 ( n ) ω 4 + ,
g ( n , ω ) = b 0 ( n ) + b 1 ( n ) ω 2 + b 2 ( n ) ω 4 + .
T C ( 1 , ω ) + T I ( 1 , ω ) = 1 .
C I ( 1 , ω ) = 4 L C ( 1 , ω ) , L I ( 1 , ω ) = C C ( 1 , ω ) / 4 .
X C ( n , ω ) = 2 ω C ( 1 + n 2 ) + ω L ( ω ) f ( n , ω ) ,
1 X I ( n , ω ) = 4 ω C 2 ω ( 1 + n 2 ) L ( ω ) g ( n , ω ) .
X C ( n , ω ) X I ( n , ω ) = 1 2 ( 1 + n 2 ) .
T I ( n ) = 16 n 2 4 n ( 1 + n ) 2 T C ( n ) 4 n ( 1 + n ) 2 + [ 4 ( 1 + n 2 ) 2 ( 1 + n ) 4 ] T C ( n ) .
E n m 1 ( x , y ) = g n m [ m c cos ( n π x c ) sin ( m π y c ) x ˆ n c sin ( n π x c ) cos ( m π y c ) y ˆ ]
E n m 2 ( x , y ) = g n m [ n c cos ( n π x c ) sin ( m π y c ) x ˆ + m c sin ( n π x c ) cos ( m π y c ) y ˆ ]
g n m = n m n 2 c c + m 2 c c ,
J n m 1 ( x , y ) = g n m [ n c sin ( n π x c ) cos ( m π y c ) x ˆ + m c cos ( n π x c ) sin ( m π y c ) y ˆ ] ,
J n m 2 ( x , y ) = g n m [ m c sin ( n π x c ) cos ( m π y c ) x ˆ n c cos ( n π x c ) sin ( m π y c ) y ˆ ] .
I p q υ n m u = 0 c d x 0 c d y R p q υ * ( x , y ) · J n m u ( x , y ) .
I p q 1 n m 1 = j π g n m ξ p q d d [ ( n β p q c ) 2 ( m α p c ) 2 ] I p q n m ,
I p q 2 n m 1 = j π g n m α p β p q ξ p q d d [ ( n c ) 2 + ( m c ) 2 ] I p q n m ,
I p q 1 n m 2 = j π g n m n m ξ p q c c d d I p q n m ,
I p q 2 n m 2 = 0 ,
I p q n m = [ ( 1 ) n exp ( j α p c ) 1 α p 2 ( n π / c ) 2 ] [ ( 1 ) m exp ( j β p q c ) 1 β p q 2 ( m π / c ) 2 ] .
E n m 11 ( r , θ ) = g n m [ n a χ n m r J n ( χ n m r a ) cos ( n θ ) r ˆ J n ( χ n m r a ) sin ( n θ ) θ ˆ ] ,
E n m 21 ( r , θ ) = g n m [ n a χ n m r J n ( χ n m r a ) sin ( n θ ) r ˆ + J n ( χ n m r a ) cos ( n θ ) θ ˆ ] ,
E n m 12 ( r , θ ) = h n m [ J n ( χ n m r a ) cos ( n θ ) r ˆ n a χ n m r J n ( χ n m r a ) sin ( n θ ) θ ˆ ] ,
E n m 22 ( r , θ ) = h n m [ J n ( χ n m r a ) sin ( n θ ) r ˆ + n a χ n m r J n ( χ n m r a ) cos ( n θ ) θ ˆ ] ,
g n m = n π χ n m a J n ( χ n m ) χ n m 2 n 2 ,
h n m = n π 1 a J n 1 ( χ n n ) .
J n m 11 ( r , θ ) = g n m [ J n ( χ n m r a ) sin ( n θ ) r ˆ + n a χ n m r J n ( χ n m r a ) cos ( n θ ) θ ˆ ] ,
J n m 21 ( r , θ ) = g n m [ J n ( χ n m r a ) cos ( n θ ) r ˆ + n a χ n m r J n ( χ n m r a ) sin ( n θ ) θ ˆ ] ,
J n m 12 ( r , θ ) = h n m [ n a χ n m r J n ( χ n m r a ) sin ( n θ ) r ˆ + J n ( χ n m r a ) cos ( n θ ) θ ˆ ] ,
J n m 22 ( r , θ ) = h n m [ n a χ n m r J n ( χ n m r a ) cos ( n θ ) r ˆ + J n ( χ n m r a ) sin ( n θ ) θ ˆ ] .
I p q υ n m l u = 0 a d r 0 2 π d θ r R p q υ * ( r , θ ) · J n m l u ( r , θ ) .
I p q 1 n m 11 = 4 π n d d ( j ) n 1 n cos ( n θ p q ) J n ( ξ p q a ) ξ p q χ n m 2 n 2 ,
I p q 1 n m 21 = tan ( n θ p q ) I p q 1 n m 11 ,
I p q 2 n m 11 = 4 π n d d ( j ) n 1 a sin ( n θ p q ) J n ( ξ p q a ) [ 1 ( ξ p q a / χ n m ) 2 ] χ n m 2 n 2 ,
I p q 2 n m 21 = cot ( n θ p q ) I p q 2 n m 11 ,
I p q 1 n m 12 = 4 π n d d ( j ) n 1 ξ p q cos ( n θ p q ) J n ( ξ p q a ) ξ p q 2 ( χ n m / a ) 2 ,
I p q 1 n m 22 = tan ( n θ p q ) I p q 1 n m 12 ,
I p q 2 n m 12 = I p q 2 n m 22 = 0 ,
C n m u = 2 A i Y 001 ( a ) Z 001 I 001 n m u * p q υ Z ρ q υ | I p q υ n m u | 2 .
A 001 = 2 A i Z 001 Y 001 ( a ) { 1 Z 001 | I 001 n m u | 2 p q υ Z p q υ | I p q υ n m u | 2 } = 2 A i Z 001 Y 001 ( a ) { p q υ Z ρ q υ | I p q υ n m u | 2 Z 001 | I 001 n m u | 2 + p q υ | I p q υ n m u | 2 } ,
A 001 = 2 p q υ Z p q υ | I p q υ n m u | 2 / ( Z 0 | I 001 n m u | 2 ) 1 + [ 1 + ( d ) ] p q υ Z p q υ | I p q υ n m u | 2 / ( Z 0 | I 001 n m u | 2 ) .
γ p q ( i ) = j ξ p q [ 1 ( i ) 2 ( k ξ p q ) 2 ] .
Z p q 1 = k Z 0 2 j ξ p q [ 1 + 1 + ( d ) 4 ( k ξ p q ) 2 ] = k Z 0 2 j ξ p q ,
Z p q 2 = Z 0 j ξ p q k [ 1 + ( d ) ] [ 1 1 + ( d ) 2 2 [ 1 + ( d ) ] ( k ξ p q ) 2 ]
T = 4 n X 2 1 + ( 1 + n ) 2 X 2 ,
X = k d L 1 k d C ,
C = ( 1 + n 2 ) | I 001 n m u | 2 [ p q ( | I p q 2 n m u | 2 ξ p q d ) ] 1 ,
L = 1 2 | I 001 n m u | 2 [ p q | I p q 1 n m u | 2 ξ p q d + 1 + n 4 ( 1 + n 2 ) 2 p q | I p q 2 n m u | 2 ξ p q d ] .
1 X = k d C + 1 k d L ,
L = | I 001 n m u | 2 2 ( p q | I p q 1 n m u | 2 ξ p q d ) 1 ,
C = 1 + n 2 2 | I 001 n m u | 2 p q ( 2 | I p q 2 n m u | 2 + | I p q 1 n m u | 2 ξ p q d ) .

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