Abstract

We present the first general theoretical description of speckle suppression efficiency based on an active diffractive optical element (DOE). The approach is based on spectral analysis of diffracted beams and a coherent matrix. Analytical formulae are obtained for the dispersion of speckle suppression efficiency using different DOE structures and different DOE activation methods. We show that a one-sided 2D DOE structure has smaller speckle suppression range than a two-sided 1D DOE structure. Both DOE structures have sufficient speckle suppression range to suppress low-order speckles in the entire visible range, but only the two-sided 1D DOE can suppress higher-order speckles. We also show that a linear shift 2D DOE in a laser projector with a large numerical aperture has higher effective speckle suppression efficiency than the method using switching or step-wise shift DOE structures. The generalized theoretical models elucidate the mechanism and practical realization of speckle suppression.

© 2017 Optical Society of America

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References

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  9. Q. Ma, C.-Q. Xu, A. Kitai, and D. Stadler, “Speckle reduction by optimized multimode fiber combined with dielectric elastomer actuator and lightpipe homogenizer,” J. Disp. Technol. 12(10), 1162–1167 (2016).
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  21. W. Thomas and C. Middlebrook, “Non-moving Hadamard matrix diffusers for speckle reduction in laser pico-projectors,” J. Mod. Opt. 61(sup1Supp 1), S74–S80 (2014).
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    [Crossref]
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2017 (2)

A. Lapchuk, G. Pashkevich, O. Prygun, I. Kosyak, M. Fu, Z. Le, and A. Kryuchyn, “Very efficient speckle suppression in the entire visible range by one two-sided diffractive optical element,” Appl. Opt. 56(5), 1481–1488 (2017).
[Crossref]

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

2016 (3)

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

T.-T.-K. Tran, Ø. Svensen, X. Chen, and M. N. Akram, “Speckle reduction in laser projection displays through angle and wavelength diversity,” Appl. Opt. 55(6), 1267–1274 (2016).
[Crossref] [PubMed]

Q. Ma, C.-Q. Xu, A. Kitai, and D. Stadler, “Speckle reduction by optimized multimode fiber combined with dielectric elastomer actuator and lightpipe homogenizer,” J. Disp. Technol. 12(10), 1162–1167 (2016).
[Crossref]

2015 (2)

2014 (2)

W. Thomas and C. Middlebrook, “Non-moving Hadamard matrix diffusers for speckle reduction in laser pico-projectors,” J. Mod. Opt. 61(sup1Supp 1), S74–S80 (2014).
[Crossref] [PubMed]

Z. Tong and X. Chen, “Principle, design and fabrication of a passive binary micro-mirror array (BMMA) for speckle reduction in grating light valve (GLV) based laser projection displays,” Sens. Actuators A Phys. 210(1), 209–216 (2014).
[Crossref]

2013 (3)

2012 (1)

W. Gao, Z. Tong, V. Kartashov, M. N. Akram, and X. Chen, “Replacing two-dimensional binary phase matrix by a pair of one-dimensional dynamic phase matrices for laser speckle reduction,” J. Disp. Technol. 8(5), 291–295 (2012).
[Crossref]

2011 (1)

2010 (2)

2009 (1)

2005 (2)

2004 (2)

J. I. Trisnadi, C. B. Carlisle, and V. Monteverde, “Overview and applications of grating light valve TM based optical write engines for high-speed digital imaging,” Micromach. Microfabr. 5348, 52–64 (2004).

J. I. Trisnadi, “Hadamard speckle contrast reduction,” Opt. Lett. 29(1), 11–13 (2004).
[Crossref] [PubMed]

2002 (1)

S. R. Kubota, “The Grating Light Valve Projector,” Opt. Photonics News 13(9), 50–53 (2002).
[Crossref]

1998 (1)

1976 (1)

Akram, M. N.

An, S.

Barnett, S. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

Basu, C.

C. Basu, M. Meinhardt-Wollweber, and B. Roth, “Lighting with laser diodes,” Adv. Opt. Technol. 2(4), 313–321 (2013).

Bogdan, O.

Bogdan, O. V.

Borodin, Y.

Buck, F.

Carlisle, C. B.

J. I. Trisnadi, C. B. Carlisle, and V. Monteverde, “Overview and applications of grating light valve TM based optical write engines for high-speed digital imaging,” Micromach. Microfabr. 5348, 52–64 (2004).

Chellappan, K. V.

Chen, X.

T.-T.-K. Tran, Ø. Svensen, X. Chen, and M. N. Akram, “Speckle reduction in laser projection displays through angle and wavelength diversity,” Appl. Opt. 55(6), 1267–1274 (2016).
[Crossref] [PubMed]

Z. Tong and X. Chen, “Principle, design and fabrication of a passive binary micro-mirror array (BMMA) for speckle reduction in grating light valve (GLV) based laser projection displays,” Sens. Actuators A Phys. 210(1), 209–216 (2014).
[Crossref]

W. Gao, Z. Tong, V. Kartashov, M. N. Akram, and X. Chen, “Replacing two-dimensional binary phase matrix by a pair of one-dimensional dynamic phase matrices for laser speckle reduction,” J. Disp. Technol. 8(5), 291–295 (2012).
[Crossref]

Edgar, M. P.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Erden, E.

Fu, M.

Gao, W.

W. Gao, Z. Tong, V. Kartashov, M. N. Akram, and X. Chen, “Replacing two-dimensional binary phase matrix by a pair of one-dimensional dynamic phase matrices for laser speckle reduction,” J. Disp. Technol. 8(5), 291–295 (2012).
[Crossref]

Gibson, G. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Goodman, J. W.

Halldórsson, T.

Hsu, W.-F.

Jang, J.

Karpoltsev, S.

Kartashov, V.

W. Gao, Z. Tong, V. Kartashov, M. N. Akram, and X. Chen, “Replacing two-dimensional binary phase matrix by a pair of one-dimensional dynamic phase matrices for laser speckle reduction,” J. Disp. Technol. 8(5), 291–295 (2012).
[Crossref]

M. N. Akram, V. Kartashov, and Z. Tong, “Speckle reduction in line-scan laser projectors using binary phase codes,” Opt. Lett. 35(3), 444–446 (2010).
[Crossref] [PubMed]

Kim, D.-W.

Kitai, A.

Q. Ma, C.-Q. Xu, A. Kitai, and D. Stadler, “Speckle reduction by optimized multimode fiber combined with dielectric elastomer actuator and lightpipe homogenizer,” J. Disp. Technol. 12(10), 1162–1167 (2016).
[Crossref]

Klymenko, A.

Klymenko, V.

Kononov, A.

Korchovyi, A. A.

Kosyak, I.

Kryuchyn, A.

Kubota, S. R.

S. R. Kubota, “The Grating Light Valve Projector,” Opt. Photonics News 13(9), 50–53 (2002).
[Crossref]

Lamb, R.

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Lapchuk, A.

A. Lapchuk, G. Pashkevich, O. Prygun, I. Kosyak, M. Fu, Z. Le, and A. Kryuchyn, “Very efficient speckle suppression in the entire visible range by one two-sided diffractive optical element,” Appl. Opt. 56(5), 1481–1488 (2017).
[Crossref]

A. Lapchuk, G. A. Pashkevich, O. V. Prygun, V. Yurlov, Y. Borodin, A. Kryuchyn, A. A. Korchovyi, and S. Shylo, “Experiment evaluation of speckle suppression efficiency of 2D quasi-spiral M-sequence-based diffractive optical element,” Appl. Opt. 54(28), E47–E54 (2015).
[Crossref] [PubMed]

A. Lapchuk, V. Yurlov, A. Kryuchyn, G. A. Pashkevich, V. Klymenko, and O. Bogdan, “Impact of speed, direction, and accuracy of diffractive optical element shift on efficiency of speckle suppression,” Appl. Opt. 54(13), 4070–4076 (2015).
[Crossref]

A. Lapchuk, A. Kryuchyn, V. Petrov, O. V. Shyhovets, G. A. Pashkevich, O. V. Bogdan, A. Kononov, and A. Klymenko, “Optical schemes for speckle suppression by Barker code diffractive optical elements,” J. Opt. Soc. Am. A 30(9), 1760–1767 (2013).
[Crossref] [PubMed]

A. Lapchuk, A. Kryuchyn, V. Petrov, V. Yurlov, and V. Klymenko, “Full speckle suppression in laser projectors using two Barker code-type diffractive optical elements,” J. Opt. Soc. Am. A 30(1), 22–31 (2013).
[Crossref] [PubMed]

S. An, A. Lapchuk, V. Yurlov, J. Song, H. Park, J. Jang, W. Shin, S. Karpoltsev, and S.-K. Yun, “Speckle suppression in laser display using several partially coherent beams,” Opt. Express 17(1), 92–103 (2009).
[Crossref] [PubMed]

Le, Z.

Lee, J.-H.

Lee, S.-D.

Ma, Q.

Q. Ma, C.-Q. Xu, A. Kitai, and D. Stadler, “Speckle reduction by optimized multimode fiber combined with dielectric elastomer actuator and lightpipe homogenizer,” J. Disp. Technol. 12(10), 1162–1167 (2016).
[Crossref]

Meinhardt-Wollweber, M.

C. Basu, M. Meinhardt-Wollweber, and B. Roth, “Lighting with laser diodes,” Adv. Opt. Technol. 2(4), 313–321 (2013).

Middlebrook, C.

W. Thomas and C. Middlebrook, “Non-moving Hadamard matrix diffusers for speckle reduction in laser pico-projectors,” J. Mod. Opt. 61(sup1Supp 1), S74–S80 (2014).
[Crossref] [PubMed]

Monteverde, V.

J. I. Trisnadi, C. B. Carlisle, and V. Monteverde, “Overview and applications of grating light valve TM based optical write engines for high-speed digital imaging,” Micromach. Microfabr. 5348, 52–64 (2004).

Padgett, M. J.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Park, H.

Pashkevich, G.

Pashkevich, G. A.

Petrov, V.

Pétursson, P. R.

Phillips, D. B.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

Prygun, O.

Prygun, O. V.

Radwell, N.

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Roth, B.

C. Basu, M. Meinhardt-Wollweber, and B. Roth, “Lighting with laser diodes,” Adv. Opt. Technol. 2(4), 313–321 (2013).

Scheffold, F.

Shin, W.

Shyhovets, O. V.

Shylo, S.

Song, J.

Stadler, D.

Q. Ma, C.-Q. Xu, A. Kitai, and D. Stadler, “Speckle reduction by optimized multimode fiber combined with dielectric elastomer actuator and lightpipe homogenizer,” J. Disp. Technol. 12(10), 1162–1167 (2016).
[Crossref]

Sun, B.

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Sun, M.-J.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Svensen, Ø.

Taylor, J. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

Thomas, W.

W. Thomas and C. Middlebrook, “Non-moving Hadamard matrix diffusers for speckle reduction in laser pico-projectors,” J. Mod. Opt. 61(sup1Supp 1), S74–S80 (2014).
[Crossref] [PubMed]

Tong, Z.

Z. Tong and X. Chen, “Principle, design and fabrication of a passive binary micro-mirror array (BMMA) for speckle reduction in grating light valve (GLV) based laser projection displays,” Sens. Actuators A Phys. 210(1), 209–216 (2014).
[Crossref]

W. Gao, Z. Tong, V. Kartashov, M. N. Akram, and X. Chen, “Replacing two-dimensional binary phase matrix by a pair of one-dimensional dynamic phase matrices for laser speckle reduction,” J. Disp. Technol. 8(5), 291–295 (2012).
[Crossref]

M. N. Akram, V. Kartashov, and Z. Tong, “Speckle reduction in line-scan laser projectors using binary phase codes,” Opt. Lett. 35(3), 444–446 (2010).
[Crossref] [PubMed]

Tran, T.-T.-K.

Trisnadi, J. I.

J. I. Trisnadi, C. B. Carlisle, and V. Monteverde, “Overview and applications of grating light valve TM based optical write engines for high-speed digital imaging,” Micromach. Microfabr. 5348, 52–64 (2004).

J. I. Trisnadi, “Hadamard speckle contrast reduction,” Opt. Lett. 29(1), 11–13 (2004).
[Crossref] [PubMed]

Tschudi, T.

Urey, H.

Völker, A.

Wang, L.

Weber, B.

Wu, S.-T.

Wu, Y.-H.

Xu, C.-Q.

Q. Ma, C.-Q. Xu, A. Kitai, and D. Stadler, “Speckle reduction by optimized multimode fiber combined with dielectric elastomer actuator and lightpipe homogenizer,” J. Disp. Technol. 12(10), 1162–1167 (2016).
[Crossref]

Yeh, C.-F.

Yu, C.-J.

Yun, S.-K.

Yurlov, V.

Zakharov, P.

Adv. Opt. Technol. (1)

C. Basu, M. Meinhardt-Wollweber, and B. Roth, “Lighting with laser diodes,” Adv. Opt. Technol. 2(4), 313–321 (2013).

Appl. Opt. (7)

K. V. Chellappan, E. Erden, and H. Urey, “Laser-based displays: a review,” Appl. Opt. 49(25), F79–F98 (2010).
[Crossref] [PubMed]

W.-F. Hsu and C.-F. Yeh, “Speckle suppression in holographic projection displays using temporal integration of speckle images from diffractive optical elements,” Appl. Opt. 50(34), H50–H55 (2011).
[Crossref] [PubMed]

A. Lapchuk, V. Yurlov, A. Kryuchyn, G. A. Pashkevich, V. Klymenko, and O. Bogdan, “Impact of speed, direction, and accuracy of diffractive optical element shift on efficiency of speckle suppression,” Appl. Opt. 54(13), 4070–4076 (2015).
[Crossref]

A. Lapchuk, G. A. Pashkevich, O. V. Prygun, V. Yurlov, Y. Borodin, A. Kryuchyn, A. A. Korchovyi, and S. Shylo, “Experiment evaluation of speckle suppression efficiency of 2D quasi-spiral M-sequence-based diffractive optical element,” Appl. Opt. 54(28), E47–E54 (2015).
[Crossref] [PubMed]

T.-T.-K. Tran, Ø. Svensen, X. Chen, and M. N. Akram, “Speckle reduction in laser projection displays through angle and wavelength diversity,” Appl. Opt. 55(6), 1267–1274 (2016).
[Crossref] [PubMed]

A. Lapchuk, G. Pashkevich, O. Prygun, I. Kosyak, M. Fu, Z. Le, and A. Kryuchyn, “Very efficient speckle suppression in the entire visible range by one two-sided diffractive optical element,” Appl. Opt. 56(5), 1481–1488 (2017).
[Crossref]

L. Wang, T. Tschudi, T. Halldórsson, and P. R. Pétursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37(10), 1770–1775 (1998).
[Crossref] [PubMed]

J. Disp. Technol. (2)

Q. Ma, C.-Q. Xu, A. Kitai, and D. Stadler, “Speckle reduction by optimized multimode fiber combined with dielectric elastomer actuator and lightpipe homogenizer,” J. Disp. Technol. 12(10), 1162–1167 (2016).
[Crossref]

W. Gao, Z. Tong, V. Kartashov, M. N. Akram, and X. Chen, “Replacing two-dimensional binary phase matrix by a pair of one-dimensional dynamic phase matrices for laser speckle reduction,” J. Disp. Technol. 8(5), 291–295 (2012).
[Crossref]

J. Mod. Opt. (1)

W. Thomas and C. Middlebrook, “Non-moving Hadamard matrix diffusers for speckle reduction in laser pico-projectors,” J. Mod. Opt. 61(sup1Supp 1), S74–S80 (2014).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Micromach. Microfabr. (1)

J. I. Trisnadi, C. B. Carlisle, and V. Monteverde, “Overview and applications of grating light valve TM based optical write engines for high-speed digital imaging,” Micromach. Microfabr. 5348, 52–64 (2004).

Nat. Commun. (1)

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Opt. Express (3)

Opt. Lett. (2)

Opt. Photonics News (1)

S. R. Kubota, “The Grating Light Valve Projector,” Opt. Photonics News 13(9), 50–53 (2002).
[Crossref]

Sci. Adv. (1)

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

Sens. Actuators A Phys. (1)

Z. Tong and X. Chen, “Principle, design and fabrication of a passive binary micro-mirror array (BMMA) for speckle reduction in grating light valve (GLV) based laser projection displays,” Sens. Actuators A Phys. 210(1), 209–216 (2014).
[Crossref]

Other (4)

B. Ismay, Semiconductor Laser Diode Technology and Applications (InTech, 1999), p. 376.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts & Company, 2007).

K. J. Horadam, Hadamard matrices and their applications (Princeton University Press, 2007), p. 278.

Wikipedia, “Determinant,” https://en.wikipedia.org/wiki/Determinant .

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Figures (8)

Fig. 1
Fig. 1 Scheme of optical system for speckle suppression by an active DOE.
Fig. 2
Fig. 2 The pseudorandom sequence based OSDOE and TSDOE structures.
Fig. 3
Fig. 3 Sequence of sixteen DOE structures based on Sylvester-Hadamard matrix [10].
Fig. 4
Fig. 4 Liquid crystal realization of OSDOE (a) and TSDOE (b) structure based on pseudorandom binary sequences with N = 7.
Fig. 5
Fig. 5 Dispersion of speckle contrast for the method using DOE shift for speckle suppression with different number of diffraction orders.
Fig. 6
Fig. 6 Dispersion of speckle contrast for the method using DOE based on Sylvester-Hadamard matrix with different matrix order.
Fig. 7
Fig. 7 Intensity of diffraction order of DOE based on M-sequence of length N = 15 and spectrum of rectangular function with the width of T.
Fig. 8
Fig. 8 Photograph of diffraction orders (a) and cross-section of the intensity distribution along the white line (b) for 2D Barker code DOE with code length N = 13. The bold dash-dot line in (b) is the curve of the function sinc2(πx/T0).

Tables (1)

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Table 1 Dependence of effective diffraction orders on input numerical aperture of a laser projector

Equations (74)

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SC=ΔI/ I ¯ .
k=S C 0 /SC.
SC= n=1 N I n 2 / n=1 N I n .
SC= n=1 N I n 2 / n=1 N I n =1/ N ef 1D .
SC= n=0,m=0 N 0 , N 0 I nm 2 / n=0,m=0 N 0 , N 0 I nm =1/sqrt( N ef N ef )=1/ N ef .
exp( i 2πn T 0 ( x+vt ) )exp( i 2πm T 0 ( y+vttan( α ) ) ) exp( i 2π n 1 T 0 ( vt+s ) )exp( i 2π m 1 T 0 ( y+vttan( α ) ) ) = 1 mN T 0 ' 0 mN T 0 exp( i2πn( x+s )/ T 0 )exp( i2πm( y+stan( α ) )/ T 0 ) exp( i2π n 1 ( x+s )/ T 0 )exp( ( i2π m 1 ( y+stan( α ) )/ T 0 ) ) ds 1 mN l=1 mN exp( i( 2πm( y+Tltan( α )/m )/ T 0 ) )exp( i( 2π m 1 ( y+Tltan( α )/m )/ T 0 ) ) 0 T 0 exp( i( 2πn( x+s )/ T 0 ) )exp( i( 2π n 1 ( x+s )/ T 0 ) )ds 1 T 0 2 0 T 0 exp( i( 2πn( y+s' ) )/ T 0 )exp( i( 2π n 1 ( x+s' )/ T 0 ) )ds ' 0 T 0 exp( i( 2πn( x+s )/ T 0 ) )exp( i( 2π n 1 ( x+s )/ T 0 ) )ds = δ mm 1 δ nn 1 .
I(x) _ = n= N 1 N 1 m= N 1 N 1 | a nm | 2 .
I(x) = E * ( x,y )E( x,y ) = 1 N 2 l=0 N1 j=1 N1 E * ( x+lT,y+jT )E( x+lT,y+jT ) .
exp( i( 2πn NT ( x+vt ) ) )exp( i( 2πm NT ( x+vt ) ) ) = 1 2N1 l=0 N exp( i( 2πn NT ( x'+Tl ) ) )exp( i( 2πm NT ( x''+Tl ) ) ) = 1 2N1 exp( i( 2πn NT ( x'x'' ) ) ) 1exp( i2π( nm ) ) 1exp( i 2π N ( nm ) ) .
1 2N1 1exp( i2π( nm ) ) 1exp( i 2π N ( ( nm ) ) ) ={ nm=Nl; is any integer 0; all other case .
I( x' ) = n=0 N | f n (x') | 2 , f n (x)= l= l 0 l 0 a n+lN exp( i2π( n+lN )x/ T 0 ) .
I( x,y ) = n=0 N1 m=0 N1 ( l= l 0 l 0 j= l 0 l 0 a n+lN,m+jN exp( i2π( n+lN )x/ T 0 )exp( i2π( m+jN )y/ T 0 ) ) ( l= l 0 l 0 j= l 0 l 0 a n+l( N+1 ),m+j( N+1 ) * exp( i2π( n+lN )x/ T 0 )exp( i2π( m+jN )y/ T 0 ) ) = n=0 N1 m=0 N1 | f nm (x,y) | 2 .
f nm (x,y)= l= l 0 l 0 j= l 0 l 0 a n+lN,m+jN exp( i2π(lN+n)x/ T 0 )exp( i2π(m+jN)y/ T 0 ) .
E(x,y,t)= n=0;m=0 N;N F nm (x,y)Π( ( t( nN+m )Δt )/Δt ) .
A 00 = a 00 cos( φ ).
A ij = a ij sin( φ ).
I 00 = I t cos 2 ( φ ).
I ij = I t sin 2 ( φ )/( N ef 2 1 ).
SC= I t 2 cos 4 ( φ )+ I t 2 sin 4 ( φ )/( N ef 2 1 ) / I t = cos 4 ( φ )+ sin 4 ( φ )/( N 0 1 ) .
I 0 = I t cos 2 ( φ ).
I i = I t sin 2 ( φ )/( N ef 1).
I 00 = I t cos 4 ( φ ).
I 0i = I i0 = I t cos 2 ( φ ) sin 2 ( φ )/( N ef 1 ).
I ij = I t sin 4 ( φ )/ ( N ef 1 ) 2 .
SC= cos 8 ( φ )+2 cos 4 ( φ ) sin 4 ( φ )/( N ef 1 )+ sin 8 ( φ )/ ( N ef 1 ) 2 .
A 00 = n + + n exp( i2φ ).
I 00 =[ ( n + n ) 2 +4 n + n cos 2 ( φ ) ]/ N 4 , I nm =4 n + n sin 2 ( φ )/[ N 4 ( N ef 2 1 ) ].
SC= { [ ( n + n ) 2 +4 n + n cos 2 ( φ ) ] 2 +16 n + 2 n 2 sin 4 ( φ )/( N ef 2 1 ) }/ N 8 .
I 0 = I t ( [ ( N1 )+exp( i2φ ) ] N [ ( N1 )+exp( i2φ ) ] N )= I t ( N2 ) 2 +4( N1 ) cos 2 ( φ ) N 2 .
I n =4( N1 ) sin 2 ( φ )/[ N 2 ( N ef 1 ) ].
I 00 = I t [ ( N2 ) 2 +4( N1 ) cos 2 ( φ ) ] 2 / N 4 .
I 0n = I n0 =4 I t sin 2 ( φ )( N1 )[ ( N2 ) 2 +4( N1 ) cos 2 ( φ ) ]/[ N 4 ( N ef 1 ) ].
I nm =16 I t ( N1 ) 2 sin 4 ( φ )/[ N 4 ( N ef 1 ) 2 ].
SC= 1 N 4 ( ( N2 ) 2 +4( N1 ) cos 2 ( φ ) ) 4 +256 ( N1 ) 4 sin 8 ( φ )/ ( N ef 1 ) 2 + 32 sin 4 ( φ ) ( N1 ) 2 [ ( N2 ) 2 +4( N1 ) cos 2 ( φ ) ] 2 /( N ef 1 ) .
F nm s (x,y)=exp(i φ mn ) F nm (x,y)sin( φ )+ F 00 cos( φ ).
M= ( F s ) T F s* .
M=( 1 b b b b b 1 b 2 b 2 b 2 b b 2 1 b 2 b 2 b b b 2 b 2 b 2 1 b 2 b 2 b 2 1 ).
det( 1ε b b b b b 1ε b 2 b 2 b 2 b b b 2 b 2 1ε b 2 b 2 1ε b 2 b 2 b b 2 b 2 b 2 1ε )=0.
ε 1,2 =[ 2+( N 0 2 ) b 2 ± ( N 0 2 ) 2 b 4 +4( N 0 1 ) b 2 ]/2, ε 3 = ε 4 == ε N 0 =1 b 2 .
SC= 1/ N 0 +2( N 0 1) cos 2 ( φ )/ N 0 2 + ( N 0 2 ) 2 cos 4 ( φ ) / N 0 2 .
M=( 1 b 2 b 2 b 2 b 2 1 b 2 b 2 b 2 b 2 1 b 2 b 2 b 2 b 2 1 ).
ε 1 =1+( N 0 1 ) b 2 , ε 2 = ε 3 == ε N =1 b 2 .
SC= ( 1+( N 0 1 ) b 2 ) 2 +( N 0 1 ) ( 1 b 2 ) 2 / N 0 = b 4 ( N 0 1 )/ N 0 +1/ N 0 .
E nx =exp( i ψ n )sin( φ ) S nx ( x,z )+cos( φ ).
E nm = E nx E my =( exp( i ψ n )sin( φ ) S nx ( x,z )+cos( φ ) )( exp( i ψ m )sin( φ ) S * my ( y,z )+cosφ ) =exp( i( ψ n ψ m ) ) sin 2 ( φ ) S nx ( x,z ) S * my ( y,z )+ sin( φ )cos( φ )( S nx ( x,z )exp( i ψ n )+ S * my ( y,z )exp( i ψ m ) )+ cos 2 ( φ ).
E nm E ij * ={ 1; n=i, m=j cos 4 ( φ )+ sin 2 ( φ ) cos 2 ( φ )( δ ni + δ mj ); ni or mj .
M=( A B B B B B A B B B B B A B B B B B A B B B B B A ).
A=( 1 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 1 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 1 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 1 ).
B=( b 4 + s 2 b 2 b 4 b 4 b 4 b 4 b 4 + s 2 b 2 b 4 b 4 b 4 b 4 b 4 + s 2 b 2 b 4 b 4 b 4 b 4 b 4 + s 2 b 2 ).
ε 1 =1( b 4 + 2 2 b 2 ) is for (N1)(N1) times degenerate, ε 2 =1 b 4 +(N2) s 2 b 2 is for 2(N1) time degenerate , ε 3 =1+2( N1 ) s 2 b 2 +( N1 )( N+1 ) b 4 .
SC= 1/ N 2 +2 s 2 b 2 ( N1 ) 2 / N 4 +2 s 4 b 4 ( N1 )( 5N6 )/ N 4 +4 s 2 b 6 ( N1 )/ N 2 + b 8 ( N 2 1 )/ N 2 .
I n =sin c 2 (nπ/N).
det( M )=det( u b b b b u b 2 b 2 b b 2 u b 2 b b 2 b 2 u )=0.
det( u b b b b bub u b 2 0 0 0 bub 0 u b 2 0 0 bub 0 0 u b 2 0 bub 0 0 0 u b 2 ) = ( u b 2 ) N2 [ u( u b 2 )( N 0 1 )b( bub ) ]=0.
ε 1,2 =[ 2+( N 0 2 ) b 2 ± ( N 0 2 ) 2 b 4 +4( N 0 1 ) b 2 ]/2, ε 3 = ε 4 == ε N e0 =1 b 2 .
det( M )=det( u b 2 b 2 b 2 b 2 u b 2 b 2 b 2 b 2 u b 2 b 2 b 2 b 2 u b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 b 2 u b 2 b 2 b 2 u b 2 b 2 b 2 u ).
det( M )=u ( u b 2 ) N 0 1 +( N 0 1 ) b 2 ( u b 2 ) N 0 1 =0.
ε 1 =1+( N 0 1 ) b 2 , ε 2 = ε 3 == ε N =1 b 2 .
det( M )=det( A B B B A B B B A B B B B B B B B B B B B A B B A ).
A=( 1ε b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 1ε b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 1ε b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 b 4 + s 2 b 2 1ε ).
B=( b 4 + s 2 b 2 b 4 b 4 b 4 b 4 b 4 + s 2 b 2 b 4 b 4 b 4 b 4 b 4 + s 2 b 2 b 4 b 4 b 4 b 4 b 4 + s 2 b 2 ).
det| M |=det| D C C D |=det| D+C |det| DC |.
D=( A B B B A B B B A ), C=( B B B B B B B B B ).
DC=( AB 0 0 0 0 AB 0 0 0 0 AB 0 0 0 0 AB ), D+C=( A 1 B 1 B 1 B 1 B 1 A 1 B 1 B 1 B 1 B 1 A 1 B 1 B 1 B 1 B 1 A 1 ).
AB=( 1 b 4 s 2 b 2 -ε s 2 b 2 s 2 b 2 s 2 b 2 s 2 b 2 1 b 4 s 2 b 2 -ε s 2 b 2 s 2 b 2 s 2 b 2 s 2 b 2 1 b 4 s 2 b 2 -ε s 2 b 2 s 2 b 2 s 2 b 2 s 2 b 2 1 b 4 s 2 b 2 -ε ).
A 1 =A+B=( 1+ b 4 + s 2 b 2 -ε 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 1+ b 4 + s 2 b 2 -ε 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 1+ b 4 + s 2 b 2 -ε 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 2 b 4 + s 2 b 2 1+ b 4 + s 2 b 2 -ε ).
B 1 =2B=( 2( b 4 + s 2 b 2 ) 2 b 4 2 b 4 2 b 4 2 b 4 2( b 4 + s 2 b 2 ) 2 b 4 2 b 4 2 b 4 2 b 4 2( b 4 + s 2 b 2 ) 2 b 4 2 b 4 2 b 4 2 b 4 2( b 4 + s 2 b 2 ) ).
det|DC|=det|AB | N/2 det| D+C |.
ε 1 = ε 2 == ε N1 =1( b 4 +2 s 2 b 2 ), ε N =1 b 4 +( N2 ) s 2 b 2 .
det| D 1 C 1 |=det|AB | N/4 det| D 1 + C 1 |.
det|M|= ( det|AB| ) N1 det| A m |.
D m + C m = A m =( 1+( N1 )( b 4 + s 2 b 2 )ε N b 4 + s 2 b 2 N b 4 + s 2 b 2 N b 4 + s 2 b 2 1+( N1 )( b 4 + s 2 b 2 )ε N b 4 + s 2 b 2 N b 4 + s 2 b 2 N b 4 + s 2 b 2 1+( N1 )( b 4 + s 2 b 2 )ε ).
ε 1 = ε 2 == ε N1 =1 b 4 +( N2 ) s 2 b 2 , ε N =1+2( N1 ) s 2 b 2 +( N1 )( N+1 ) b 4 .
ε 1 =1( b 4 +2 s 2 b 2 ) is for (N1)(N1) times degenerate , ε 2 =1 b 4 +( N2 ) s 2 b 2 is for 2(N1) times degenerate , ε 3 =1+2( N1 ) s 2 b 2 +( N1 )( N+1 ) b 4 .

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