Abstract

The transmitting characteristics of high-transparency double-layer metallic meshes with submillimeter period were analyzed using an analytical model, which was established using angular spectrum propagation theory and verified through experiments. It was found through analysis that rotating misalignment has significant effect on the distribution of diffraction spot intensity. Large period and small linewidth can be used to obtain high transmittance and low levels of stray light. Substrate thickness has little effect on transmitting characteristics of mesh, and so it is a variable free to choose in optimizing shielding characteristics of mesh. We think, together with other ways and means of optimizing shielding characteristics of mesh, the model can also be used for the optimization of a high-pass mesh filter.

© 2008 Optical Society of America

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2007

2005

H. Y. Sang, Z. Y. Li, and B. Y. Gu, “Photonic states deep into the waveguide cutoff frequency of metallic mesh photonic crystal filters,” J. Appl. Phys. 97, 033102 (2005).
[CrossRef]

H. Wen, B. Hou, Y. Leng, and W. Wen, “Resonance-induced wave penetration through electromagnetic opaque object,” Opt. Express 13, 7005-7010 (2005).
[CrossRef] [PubMed]

2004

S. Govindaswamy, J. East, F. Terry, E. Topsakal, J. L. Volakis, and G. I. Haddad, “Dual-frequency-selective surfaces for near-infrared bandpass filters,” Microw. Opt. Tech. Lett. 43, 95-98 (2004).
[CrossRef]

2003

H. A. Smith, M. Rebbert, and O. Sternberg, “Designer infrared filters using stacked metal lattices,” Appl. Phys. Lett. 82, 3605-3607 (2003).
[CrossRef]

2002

R. Sauleau, G. L. Ray, and P. Coquet, “Parametric study and synthesis of 60 GHz Fabry-Perot resonators,” Microw. Opt. Tech. Lett. 34, 247-252 (2002).
[CrossRef]

K. D. Möller, O. Sternberg, H. Grebel, and K. P. Stewart, “Near-field effects in multilayer inductive metal meshes,” Appl. Opt. 41, 1942-1948 (2002).
[CrossRef] [PubMed]

2000

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

1998

A. Kao, K. A. McIntosh, O. B. McMahon, R. Atkins, and S. Verghese, “Calculated and measured transmittance of metallodielectric photonic crystals incorporating flat metal elements,” Appl. Phys. Lett. 73, 145-147 (1998).
[CrossRef]

G. Jin, Y. Yan, and M. Wu, Binary Optics (National Defence Industry, 1998).

1997

S. Gupta, G. Tuttle, M. Sigalas, and K.-M. Ho, “Infrared filters using metallic photonic band gap structures on flexible substrates,” Appl. Phys. Lett. 71, 2412-2414 (1997).
[CrossRef]

1996

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, and C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759-2761 (1996).
[CrossRef]

1994

1993

M. Kohin, S. J. Wein, J. D. Traylor, R. C. Chase, and J. E. Chapman, “Analysis and design of transparent conductive coatings and filters,” Opt. Eng. 32, 911-925 (1993).
[CrossRef]

1968

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill1968).

Atkins, R.

A. Kao, K. A. McIntosh, O. B. McMahon, R. Atkins, and S. Verghese, “Calculated and measured transmittance of metallodielectric photonic crystals incorporating flat metal elements,” Appl. Phys. Lett. 73, 145-147 (1998).
[CrossRef]

Brueck, S. R.

Brueck, S. R. J.

Cai, W.

Chapman, J. E.

M. Kohin, S. J. Wein, J. D. Traylor, R. C. Chase, and J. E. Chapman, “Analysis and design of transparent conductive coatings and filters,” Opt. Eng. 32, 911-925 (1993).
[CrossRef]

Chase, R. C.

M. Kohin, S. J. Wein, J. D. Traylor, R. C. Chase, and J. E. Chapman, “Analysis and design of transparent conductive coatings and filters,” Opt. Eng. 32, 911-925 (1993).
[CrossRef]

Chettiar, U. K.

Coquet, P.

R. Sauleau, G. L. Ray, and P. Coquet, “Parametric study and synthesis of 60 GHz Fabry-Perot resonators,” Microw. Opt. Tech. Lett. 34, 247-252 (2002).
[CrossRef]

Drachev, V. P.

East, J.

S. Govindaswamy, J. East, F. Terry, E. Topsakal, J. L. Volakis, and G. I. Haddad, “Dual-frequency-selective surfaces for near-infrared bandpass filters,” Microw. Opt. Tech. Lett. 43, 95-98 (2004).
[CrossRef]

Fischer, J.

Frauenglass, A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill1968).

Govindaswamy, S.

S. Govindaswamy, J. East, F. Terry, E. Topsakal, J. L. Volakis, and G. I. Haddad, “Dual-frequency-selective surfaces for near-infrared bandpass filters,” Microw. Opt. Tech. Lett. 43, 95-98 (2004).
[CrossRef]

Grebel, H.

Greenhouse, M. A.

Grossman, J.

Gu, B. Y.

H. Y. Sang, Z. Y. Li, and B. Y. Gu, “Photonic states deep into the waveguide cutoff frequency of metallic mesh photonic crystal filters,” J. Appl. Phys. 97, 033102 (2005).
[CrossRef]

Gupta, S.

S. Gupta, G. Tuttle, M. Sigalas, and K.-M. Ho, “Infrared filters using metallic photonic band gap structures on flexible substrates,” Appl. Phys. Lett. 71, 2412-2414 (1997).
[CrossRef]

Haddad, G. I.

S. Govindaswamy, J. East, F. Terry, E. Topsakal, J. L. Volakis, and G. I. Haddad, “Dual-frequency-selective surfaces for near-infrared bandpass filters,” Microw. Opt. Tech. Lett. 43, 95-98 (2004).
[CrossRef]

Hains, C.

Ho, K.-M.

S. Gupta, G. Tuttle, M. Sigalas, and K.-M. Ho, “Infrared filters using metallic photonic band gap structures on flexible substrates,” Appl. Phys. Lett. 71, 2412-2414 (1997).
[CrossRef]

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, and C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759-2761 (1996).
[CrossRef]

Hou, B.

Isaacson, P.

Jin, G.

G. Jin, Y. Yan, and M. Wu, Binary Optics (National Defence Industry, 1998).

Jin, P.

J. Tan, Z. Lu, J. Liu, P. Jin, and Y. Wang, “Analysis of Fraunhofer diffractive characteristics of tilted metallic mesh for its effect on optical measurement,” Meas. Sci. Technol. 18, 1703-1709 (2007).
[CrossRef]

Kao, A.

A. Kao, K. A. McIntosh, O. B. McMahon, R. Atkins, and S. Verghese, “Calculated and measured transmittance of metallodielectric photonic crystals incorporating flat metal elements,” Appl. Phys. Lett. 73, 145-147 (1998).
[CrossRef]

Kildishev, A. V.

Kohin, M.

M. Kohin, S. J. Wein, J. D. Traylor, R. C. Chase, and J. E. Chapman, “Analysis and design of transparent conductive coatings and filters,” Opt. Eng. 32, 911-925 (1993).
[CrossRef]

Kong, J. A.

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

Ku, Z.

Leng, Y.

Li, D.

Li, Z. Y.

H. Y. Sang, Z. Y. Li, and B. Y. Gu, “Photonic states deep into the waveguide cutoff frequency of metallic mesh photonic crystal filters,” J. Appl. Phys. 97, 033102 (2005).
[CrossRef]

Liu, J.

J. Tan, Z. Lu, J. Liu, P. Jin, and Y. Wang, “Analysis of Fraunhofer diffractive characteristics of tilted metallic mesh for its effect on optical measurement,” Meas. Sci. Technol. 18, 1703-1709 (2007).
[CrossRef]

Lu, Z.

J. Tan, Z. Lu, J. Liu, P. Jin, and Y. Wang, “Analysis of Fraunhofer diffractive characteristics of tilted metallic mesh for its effect on optical measurement,” Meas. Sci. Technol. 18, 1703-1709 (2007).
[CrossRef]

J. Tan and Z. Lu, “Contiguous metallic rings: an inductive mesh with high transmissivity, strong electromagnetic shielding, and uniformly distributed stray light,” Opt. Express 15, 790-796 (2007).
[CrossRef] [PubMed]

Mahoney, L.

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

McCalmont, J. S.

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, and C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759-2761 (1996).
[CrossRef]

McIntosh, K. A.

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

A. Kao, K. A. McIntosh, O. B. McMahon, R. Atkins, and S. Verghese, “Calculated and measured transmittance of metallodielectric photonic crystals incorporating flat metal elements,” Appl. Phys. Lett. 73, 145-147 (1998).
[CrossRef]

McMahon, O. B.

A. Kao, K. A. McIntosh, O. B. McMahon, R. Atkins, and S. Verghese, “Calculated and measured transmittance of metallodielectric photonic crystals incorporating flat metal elements,” Appl. Phys. Lett. 73, 145-147 (1998).
[CrossRef]

Möller, K. D.

Neumann, A.

Oswald, J. A.

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

Peckerar, M.

Ray, G. L.

R. Sauleau, G. L. Ray, and P. Coquet, “Parametric study and synthesis of 60 GHz Fabry-Perot resonators,” Microw. Opt. Tech. Lett. 34, 247-252 (2002).
[CrossRef]

Rebbert, M.

Sang, H. Y.

H. Y. Sang, Z. Y. Li, and B. Y. Gu, “Photonic states deep into the waveguide cutoff frequency of metallic mesh photonic crystal filters,” J. Appl. Phys. 97, 033102 (2005).
[CrossRef]

Sauleau, R.

R. Sauleau, G. L. Ray, and P. Coquet, “Parametric study and synthesis of 60 GHz Fabry-Perot resonators,” Microw. Opt. Tech. Lett. 34, 247-252 (2002).
[CrossRef]

Shalaev, V. M.

Sigalas, M.

S. Gupta, G. Tuttle, M. Sigalas, and K.-M. Ho, “Infrared filters using metallic photonic band gap structures on flexible substrates,” Appl. Phys. Lett. 71, 2412-2414 (1997).
[CrossRef]

Sigalas, M. M.

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, and C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759-2761 (1996).
[CrossRef]

Smith, H. A.

Soukolis, C. M.

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, and C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759-2761 (1996).
[CrossRef]

Sternberg, O.

H. A. Smith, M. Rebbert, and O. Sternberg, “Designer infrared filters using stacked metal lattices,” Appl. Phys. Lett. 82, 3605-3607 (2003).
[CrossRef]

K. D. Möller, O. Sternberg, H. Grebel, and K. P. Stewart, “Near-field effects in multilayer inductive metal meshes,” Appl. Opt. 41, 1942-1948 (2002).
[CrossRef] [PubMed]

Stewart, K. P.

Tan, J.

J. Tan, Z. Lu, J. Liu, P. Jin, and Y. Wang, “Analysis of Fraunhofer diffractive characteristics of tilted metallic mesh for its effect on optical measurement,” Meas. Sci. Technol. 18, 1703-1709 (2007).
[CrossRef]

J. Tan and Z. Lu, “Contiguous metallic rings: an inductive mesh with high transmissivity, strong electromagnetic shielding, and uniformly distributed stray light,” Opt. Express 15, 790-796 (2007).
[CrossRef] [PubMed]

Terry, F.

S. Govindaswamy, J. East, F. Terry, E. Topsakal, J. L. Volakis, and G. I. Haddad, “Dual-frequency-selective surfaces for near-infrared bandpass filters,” Microw. Opt. Tech. Lett. 43, 95-98 (2004).
[CrossRef]

Topsakal, E.

S. Govindaswamy, J. East, F. Terry, E. Topsakal, J. L. Volakis, and G. I. Haddad, “Dual-frequency-selective surfaces for near-infrared bandpass filters,” Microw. Opt. Tech. Lett. 43, 95-98 (2004).
[CrossRef]

Traylor, J. D.

M. Kohin, S. J. Wein, J. D. Traylor, R. C. Chase, and J. E. Chapman, “Analysis and design of transparent conductive coatings and filters,” Opt. Eng. 32, 911-925 (1993).
[CrossRef]

Tuttle, G.

S. Gupta, G. Tuttle, M. Sigalas, and K.-M. Ho, “Infrared filters using metallic photonic band gap structures on flexible substrates,” Appl. Phys. Lett. 71, 2412-2414 (1997).
[CrossRef]

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, and C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759-2761 (1996).
[CrossRef]

Verghese, S.

A. Kao, K. A. McIntosh, O. B. McMahon, R. Atkins, and S. Verghese, “Calculated and measured transmittance of metallodielectric photonic crystals incorporating flat metal elements,” Appl. Phys. Lett. 73, 145-147 (1998).
[CrossRef]

Verghesee, S.

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

Volakis, J. L.

S. Govindaswamy, J. East, F. Terry, E. Topsakal, J. L. Volakis, and G. I. Haddad, “Dual-frequency-selective surfaces for near-infrared bandpass filters,” Microw. Opt. Tech. Lett. 43, 95-98 (2004).
[CrossRef]

Wang, Y.

J. Tan, Z. Lu, J. Liu, P. Jin, and Y. Wang, “Analysis of Fraunhofer diffractive characteristics of tilted metallic mesh for its effect on optical measurement,” Meas. Sci. Technol. 18, 1703-1709 (2007).
[CrossRef]

Wang, Z.-B.

Wein, S. J.

M. Kohin, S. J. Wein, J. D. Traylor, R. C. Chase, and J. E. Chapman, “Analysis and design of transparent conductive coatings and filters,” Opt. Eng. 32, 911-925 (1993).
[CrossRef]

Wen, H.

Wen, W.

Wu, B. I.

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

Wu, M.

G. Jin, Y. Yan, and M. Wu, Binary Optics (National Defence Industry, 1998).

Xiao, S.

Yan, D.

Yan, Y.

G. Jin, Y. Yan, and M. Wu, Binary Optics (National Defence Industry, 1998).

Yang, E.

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

Ye, Y.-H.

Yuan, H.-K.

Zhang, J.

Zhang, J.-Y.

Zhang, S.

Appl. Opt.

Appl. Phys. Lett.

A. Kao, K. A. McIntosh, O. B. McMahon, R. Atkins, and S. Verghese, “Calculated and measured transmittance of metallodielectric photonic crystals incorporating flat metal elements,” Appl. Phys. Lett. 73, 145-147 (1998).
[CrossRef]

H. A. Smith, M. Rebbert, and O. Sternberg, “Designer infrared filters using stacked metal lattices,” Appl. Phys. Lett. 82, 3605-3607 (2003).
[CrossRef]

J. S. McCalmont, M. M. Sigalas, G. Tuttle, K.-M. Ho, and C. M. Soukolis, “A layer-by-layer metallic photonic band-gap structure,” Appl. Phys. Lett. 68, 2759-2761 (1996).
[CrossRef]

S. Gupta, G. Tuttle, M. Sigalas, and K.-M. Ho, “Infrared filters using metallic photonic band gap structures on flexible substrates,” Appl. Phys. Lett. 71, 2412-2414 (1997).
[CrossRef]

J. Appl. Phys.

H. Y. Sang, Z. Y. Li, and B. Y. Gu, “Photonic states deep into the waveguide cutoff frequency of metallic mesh photonic crystal filters,” J. Appl. Phys. 97, 033102 (2005).
[CrossRef]

Meas. Sci. Technol.

J. Tan, Z. Lu, J. Liu, P. Jin, and Y. Wang, “Analysis of Fraunhofer diffractive characteristics of tilted metallic mesh for its effect on optical measurement,” Meas. Sci. Technol. 18, 1703-1709 (2007).
[CrossRef]

Microw. Opt. Tech. Lett.

R. Sauleau, G. L. Ray, and P. Coquet, “Parametric study and synthesis of 60 GHz Fabry-Perot resonators,” Microw. Opt. Tech. Lett. 34, 247-252 (2002).
[CrossRef]

S. Govindaswamy, J. East, F. Terry, E. Topsakal, J. L. Volakis, and G. I. Haddad, “Dual-frequency-selective surfaces for near-infrared bandpass filters,” Microw. Opt. Tech. Lett. 43, 95-98 (2004).
[CrossRef]

B. I. Wu, E. Yang, J. A. Kong, J. A. Oswald, K. A. McIntosh, L. Mahoney, and S. Verghesee, “Analysis of photonic crystal filters by the finite-difference time-domain technique,” Microw. Opt. Tech. Lett. 27, 81-87 (2000).
[CrossRef]

Opt. Eng.

M. Kohin, S. J. Wein, J. D. Traylor, R. C. Chase, and J. E. Chapman, “Analysis and design of transparent conductive coatings and filters,” Opt. Eng. 32, 911-925 (1993).
[CrossRef]

Opt. Express

Opt. Lett.

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill1968).

G. Jin, Y. Yan, and M. Wu, Binary Optics (National Defence Industry, 1998).

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Figures (12)

Fig. 1
Fig. 1

Structures of double-layer metallic meshes.

Fig. 2
Fig. 2

Schematic of a double-layer metallic mesh diffraction.

Fig. 3
Fig. 3

Phase difference of angular spectrums with the same direction of propagation.

Fig. 4
Fig. 4

Comparison of transmittances measured with a UV-3101PC spectrophotometer for double-layer mesh with simulation results obtained using Eq. (18).

Fig. 5
Fig. 5

Diffraction spot intensity distribution of double-layer mesh obtained using Eq. (17).

Fig. 6
Fig. 6

Experimental observation diffraction spot intensity distribution of double-layer mesh.

Fig. 7
Fig. 7

Misalignments between two meshes of a double-layer metallic mesh.

Fig. 8
Fig. 8

Distribution of diffraction spot intensity of a double-layer square metallic mesh with 10 ° rotating misalignment using Eq. (19) (mesh period, 320 μm ; mesh line width, 2.31 μm ; and substrate thickness, 1 mm ).

Fig. 9
Fig. 9

Normalized transmittance of double-layer square metallic mesh at different periods using Eq. (18).

Fig. 10
Fig. 10

Normalized transmittance of double-layer square metallic mesh at different linewidths using Eq. (18).

Fig. 11
Fig. 11

Normalized transmittance of double-layer square metallic mesh at different substrate thicknesses using Eq. (18).

Fig. 12
Fig. 12

Normalized distribution of diffraction spot intensity along axis x of double-layer square metallic meshes using Eq. (17).

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

U 0 ( f x , f y ) = u ( x , y , 0 ) exp [ i 2 π ( f x x + f y y ) ] d x d y ,
u ( x , y , 0 ) = U 0 ( f x , f y ) exp [ i 2 π ( f x x + f y y ) ] d f x d f y .
u ( x , y , 0 ) = m = n = U 0 ( f m , f n ) exp [ i 2 π ( f m x + f n y ) ] .
D image = m = n = [ p = q = D ( m , n ) ( p , q ) ] .
D image = m = n = ( D ( 0 , 0 ) ( m , n ) + D ( m , n ) ( 0 , 0 ) ) D ( 0 , 0 ) ( 0 , 0 ) ,
Δ φ = A B × n s × k = 2 π λ h ( 1 γ 1 ) × n s ,
E ( m , n ) ( 0 , 0 ) = a ( m , n ) ( 0 , 0 ) exp [ i ( φ ω t ) ] ,
E ( 0 , 0 ) ( m , n ) = a ( 0 , 0 ) ( m , n ) exp [ i ( φ Δ φ ω t ) ] ,
A m n 2 = a ( 0 , 0 ) ( m , n ) 2 + a ( m , n ) ( 0 , 0 ) 2 + 2 a ( 0 , 0 ) ( m , n ) a ( m , n ) ( 0 , 0 ) cos ( Δ φ ) .
a square   ( m , n ) = r 2 sinc ( r n ) sinc ( r m ) ,
a ring   ( m , n ) = sinc ( n ) sinc ( m ) J 1 ( π n 2 + m 2 ) 2 n 2 + m 2 + r J 1 ( r π n 2 + m 2 ) 2 n 2 + m 2 ,
a ( m , n ) ( 0 , 0 ) = a square   ( m , n ) × a ring   ( 0 , 0 ) ,
a ( 0 , 0 ) ( m , n ) = a square   ( 0 , 0 ) × a ring   ( m , n ) .
D image = m = n = A m n × F aperture a ( 0 , 0 ) ( 0 , 0 ) F aperture ,
F aperture = 2 J 1 [ π N g ( ξ n / g ) 2 + ( η m / g ) 2 ] π N g ( ξ n / g ) 2 + ( η m / g ) 2 .
I image = | D image | 2 = | m = n = A m n × F aperture a ( 0 , 0 ) ( 0 , 0 ) F aperture | 2 .
I image = | D image | 2 = m = n = ( A m n × F aperture ) 2 3 ( a ( 0 , 0 ) ( 0 , 0 ) F aperture ) 2 .
T d = m = n = ( A m n ) 2 3 ( a ( 0 , 0 ) ( 0 , 0 ) ) 2 .
D image = [ m = n = D ( m , n ) ( 0 , 0 ) × F aperture D ( 0 , 0 ) ( 0 , 0 ) × F aperture ] ( x , y ) + Tran ( x 1 x , y 1 y ) { [ m = n = D ( 0 , 0 ) ( m , n ) × F aperture ] ( x 1 , y 1 ) } ,
Tran ( x 1 x , y 1 y ) = { x = x 1 cos θ y 1 sin θ y = y 1 cos θ + x 1 sin θ .
T drm = m = n = ( a ( m , n ) ( 0 , 0 ) ) 2 + m = n = ( a ( 0 , 0 ) ( m , n ) ) 2 ( a ( 0 , 0 ) ( 0 , 0 ) ) 2 .

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