Abstract

For a rough diffuser that fully depolarizes a linearly polarized laser, the speckle contrast of a fully developed speckle image can be reduced from 1 to 0.5 by polarization diversity. This reduction is achieved by rotating an x-polarized laser to the y-polarized orientation to form four independent speckle patterns with equal intensities during the detector exposure time. For the case of arbitrary rotation of the polarization, we derived a generalized speckle contrast formula for the superposed speckle patterns. This formula completes the theory of speckle contrast for the sum of correlated speckle intensities as it relates to polarization diversity.

© 2012 Optical Society of America

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References

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  1. A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
    [CrossRef]
  2. S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
    [CrossRef]
  3. Z. Tong, X. Chen, M. N. Akram, and A. Aksnes, “Compound speckle characterization method and reduction by optical design,” J. Disp. Technol. 8, 132–137 (2012).
    [CrossRef]
  4. G. Ouyang, Z. Tong, M. N. Akram, K. Wang, V. Kartashov, X. Yan, and X. Chen, “Speckle reduction using a motionless diffractive optical element,” Opt. Lett. 35, 2852–2854 (2010).
    [CrossRef]
  5. V. Kartashov and M. N. Akram, “Speckle suppression in projection displays by using a motionless changing diffuser,” J. Opt. Soc. Am. A 27, 2593–2601 (2010).
    [CrossRef]
  6. S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
    [CrossRef]
  7. Z. Tong and X. Chen, “Speckle reduction by angle diversity using a translucent spatial light modulator,” Opt. Appl. (to be published).
  8. Z. Tong, M. N. Akram, and X. Chen, “Dynamic interference fringes generated by optical interferometer for laser speckle reduction,” Speckle (to be published).
  9. J. I. Trisnadi, “Hadamard speckle contrast reduction,” Opt. Lett. 29, 11–13 (2004).
    [CrossRef]
  10. Z. Tong, M. N. Akram, and X. Chen, “Speckle reduction using orthogonal arrays in laser projectors,” Appl. Opt. 49, 6425–6429 (2010).
    [CrossRef]
  11. 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, 291–295 (2012).
    [CrossRef]
  12. M. N. Akram, Z. Tong, G. Ouyang, X. Chen, and V. Kartashov, “Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror,” Appl. Opt. 49, 3297–3304 (2010).
    [CrossRef]
  13. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).
  14. N. George, A. Jain, and R. D. S. Melville, “Speckle, diffusers, and depolarization,” Appl. Phys. 6, 65–70 (1975).
    [CrossRef]
  15. F. Yang, L. Ruan, S. A. Jewell, and J. R. Sambles, “Polarization rotator using a hybrid aligned nematic liquid crystal cell,” Opt. Express 15, 4192–4197 (2007).
    [CrossRef]
  16. B. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).

2012 (2)

Z. Tong, X. Chen, M. N. Akram, and A. Aksnes, “Compound speckle characterization method and reduction by optical design,” J. Disp. Technol. 8, 132–137 (2012).
[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, 291–295 (2012).
[CrossRef]

2011 (1)

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

2010 (4)

2008 (1)

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

2007 (1)

2006 (1)

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

2004 (1)

1975 (1)

N. George, A. Jain, and R. D. S. Melville, “Speckle, diffusers, and depolarization,” Appl. Phys. 6, 65–70 (1975).
[CrossRef]

Akram, M. N.

Z. Tong, X. Chen, M. N. Akram, and A. Aksnes, “Compound speckle characterization method and reduction by optical design,” J. Disp. Technol. 8, 132–137 (2012).
[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, 291–295 (2012).
[CrossRef]

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Z. Tong, M. N. Akram, and X. Chen, “Speckle reduction using orthogonal arrays in laser projectors,” Appl. Opt. 49, 6425–6429 (2010).
[CrossRef]

M. N. Akram, Z. Tong, G. Ouyang, X. Chen, and V. Kartashov, “Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror,” Appl. Opt. 49, 3297–3304 (2010).
[CrossRef]

G. Ouyang, Z. Tong, M. N. Akram, K. Wang, V. Kartashov, X. Yan, and X. Chen, “Speckle reduction using a motionless diffractive optical element,” Opt. Lett. 35, 2852–2854 (2010).
[CrossRef]

V. Kartashov and M. N. Akram, “Speckle suppression in projection displays by using a motionless changing diffuser,” J. Opt. Soc. Am. A 27, 2593–2601 (2010).
[CrossRef]

Z. Tong, M. N. Akram, and X. Chen, “Dynamic interference fringes generated by optical interferometer for laser speckle reduction,” Speckle (to be published).

Aksnes, A.

Z. Tong, X. Chen, M. N. Akram, and A. Aksnes, “Compound speckle characterization method and reduction by optical design,” J. Disp. Technol. 8, 132–137 (2012).
[CrossRef]

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Chen, X.

Z. Tong, X. Chen, M. N. Akram, and A. Aksnes, “Compound speckle characterization method and reduction by optical design,” J. Disp. Technol. 8, 132–137 (2012).
[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, 291–295 (2012).
[CrossRef]

G. Ouyang, Z. Tong, M. N. Akram, K. Wang, V. Kartashov, X. Yan, and X. Chen, “Speckle reduction using a motionless diffractive optical element,” Opt. Lett. 35, 2852–2854 (2010).
[CrossRef]

Z. Tong, M. N. Akram, and X. Chen, “Speckle reduction using orthogonal arrays in laser projectors,” Appl. Opt. 49, 6425–6429 (2010).
[CrossRef]

M. N. Akram, Z. Tong, G. Ouyang, X. Chen, and V. Kartashov, “Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror,” Appl. Opt. 49, 3297–3304 (2010).
[CrossRef]

Z. Tong, M. N. Akram, and X. Chen, “Dynamic interference fringes generated by optical interferometer for laser speckle reduction,” Speckle (to be published).

Z. Tong and X. Chen, “Speckle reduction by angle diversity using a translucent spatial light modulator,” Opt. Appl. (to be published).

Egge, S. V.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Furukawa, A.

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

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, 291–295 (2012).
[CrossRef]

George, N.

N. George, A. Jain, and R. D. S. Melville, “Speckle, diffusers, and depolarization,” Appl. Phys. 6, 65–70 (1975).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

Hirata, S.

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

Imanishi, D.

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

Ito, S.

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

Jain, A.

N. George, A. Jain, and R. D. S. Melville, “Speckle, diffusers, and depolarization,” Appl. Phys. 6, 65–70 (1975).
[CrossRef]

Jewell, S. A.

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, 291–295 (2012).
[CrossRef]

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

M. N. Akram, Z. Tong, G. Ouyang, X. Chen, and V. Kartashov, “Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror,” Appl. Opt. 49, 3297–3304 (2010).
[CrossRef]

G. Ouyang, Z. Tong, M. N. Akram, K. Wang, V. Kartashov, X. Yan, and X. Chen, “Speckle reduction using a motionless diffractive optical element,” Opt. Lett. 35, 2852–2854 (2010).
[CrossRef]

V. Kartashov and M. N. Akram, “Speckle suppression in projection displays by using a motionless changing diffuser,” J. Opt. Soc. Am. A 27, 2593–2601 (2010).
[CrossRef]

Kwon, J. W.

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

Lee, S.

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

Lee, S. Y.

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

Melville, R. D. S.

N. George, A. Jain, and R. D. S. Melville, “Speckle, diffusers, and depolarization,” Appl. Phys. 6, 65–70 (1975).
[CrossRef]

Ohse, N.

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

Osterberg, U.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Ouyang, G.

Park, C.

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

Park, S.

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

Ruan, L.

Saleh, B. A.

B. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).

Sambles, J. R.

Sato, Y.

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

Shin, S. C.

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

Tamamura, K.

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

Teich, M. C.

B. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).

Tong, Z.

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, 291–295 (2012).
[CrossRef]

Z. Tong, X. Chen, M. N. Akram, and A. Aksnes, “Compound speckle characterization method and reduction by optical design,” J. Disp. Technol. 8, 132–137 (2012).
[CrossRef]

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Z. Tong, M. N. Akram, and X. Chen, “Speckle reduction using orthogonal arrays in laser projectors,” Appl. Opt. 49, 6425–6429 (2010).
[CrossRef]

M. N. Akram, Z. Tong, G. Ouyang, X. Chen, and V. Kartashov, “Laser speckle reduction due to spatial and angular diversity introduced by fast scanning micromirror,” Appl. Opt. 49, 3297–3304 (2010).
[CrossRef]

G. Ouyang, Z. Tong, M. N. Akram, K. Wang, V. Kartashov, X. Yan, and X. Chen, “Speckle reduction using a motionless diffractive optical element,” Opt. Lett. 35, 2852–2854 (2010).
[CrossRef]

Z. Tong and X. Chen, “Speckle reduction by angle diversity using a translucent spatial light modulator,” Opt. Appl. (to be published).

Z. Tong, M. N. Akram, and X. Chen, “Dynamic interference fringes generated by optical interferometer for laser speckle reduction,” Speckle (to be published).

Trisnadi, J. I.

Wakabayashi, K.

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

Wang, K.

Welde, K.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Yan, X.

Yang, F.

Yoo, S. S.

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. (1)

N. George, A. Jain, and R. D. S. Melville, “Speckle, diffusers, and depolarization,” Appl. Phys. 6, 65–70 (1975).
[CrossRef]

Displays (1)

S. C. Shin, S. S. Yoo, S. Y. Lee, C. Park, S. Park, J. W. Kwon, and S. Lee, “Removal of hot spot speckle on laser projection screen using both the running screen and the rotating diffuser,” Displays 27, 91–96 (2006).
[CrossRef]

J. Disp. Technol. (2)

Z. Tong, X. Chen, M. N. Akram, and A. Aksnes, “Compound speckle characterization method and reduction by optical design,” J. Disp. Technol. 8, 132–137 (2012).
[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, 291–295 (2012).
[CrossRef]

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

Opt. Eng. (1)

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Osterberg, and A. Aksnes, “Sinusoidal rotating grating for speckle reduction in laser projectors: feasibility study,” Opt. Eng. 50, 083202 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (1)

A. Furukawa, N. Ohse, Y. Sato, D. Imanishi, K. Wakabayashi, S. Ito, K. Tamamura, and S. Hirata, “Effective speckle reduction in laser projection displays,” Proc. SPIE 6911, 69110T(2008).
[CrossRef]

Other (4)

Z. Tong and X. Chen, “Speckle reduction by angle diversity using a translucent spatial light modulator,” Opt. Appl. (to be published).

Z. Tong, M. N. Akram, and X. Chen, “Dynamic interference fringes generated by optical interferometer for laser speckle reduction,” Speckle (to be published).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2006).

B. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 1991).

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Figures (5)

Fig. 1.
Fig. 1.

Polarized field Ep when rotating orientation of the laser polarization, where Ep has two orthogonal components Epx and Epy.

Fig. 2.
Fig. 2.

Speckle contrast C when the rotation steps of the polarizer N are modified at a particular value of θ: black squares, θ=π/4; red circles, θ=π/2; green inverted triangles, θ=π; blue triangles, θ=3π/2; magenta stars, θ=2π.

Fig. 3.
Fig. 3.

Setup for the experiment investigating the polarization diversity for the characterization of the speckle intensity coherency matrix. S, laser source; P1 and P2, linear polarizers; QWP, quarter-wave plate; BE, 20× beam expander; PS, projection screen; CCD, CCD detector.

Fig. 4.
Fig. 4.

Plot of the experimentally obtained 31×31 speckle intensity coherency matrix, where the polarization orientation of the laser beam is modified temporally within the range of θ=340°. The dashed, solid, and dotted black lines represent the correlation coefficients for the 7th, 14th, and 22nd speckle intensities, respectively. The solid gray lines indicate to the correlation coefficients for the other speckle intensities. The solid green line represents Eq. (13a) for n=14, θ=340°, and N=31. The solid red line represents the cosine function fit given by Eq. (14).

Fig. 5.
Fig. 5.

7th, 14th, and 22nd speckle images and the summed images of each possible pair and the light intensity distribution on each CCD pixel caused by picking the 240th row of multiple speckle images. (a) The 7th (I7), 14th (I14), and 22nd (I22) speckle images when rotating the laser beam polarization and the summed images of each pair (I7+22, I7+14, and I14+22 are the sums of I7 with I22, I7 with I14, and I14 with I22); (b) Light intensity distributions for all 640 of the CCD column pixels, created by selecting the 240th row of the 7th (solid black line), 14th (solid red line), and 22nd (solid blue line) speckle images.

Tables (1)

Tables Icon

Table 1. Statistical Properties of the Exampled Speckle Images and the Sums of Every Pair Shown in Fig. 5(a)

Equations (22)

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

CN=n=1NI¯n2n=1NI¯n,
CN=n=1Nλn4n=1Nλn2.
ImIn¯=σmσnρm,n+I¯mI¯n,
I¯N=n=1NI¯n,
I¯N2=n=1NI¯n2+2n=1Nm=n+1NInIm¯=n=1NI¯n2+2n=1Nm=n+1N(σmσnρm,n+I¯mI¯n),
σN2=I¯N2I¯N2=n=1NI¯n2+2n=1Nm=n+1N(σmσnρm,n+I¯mI¯n)n=1NI¯n22n=1Nm=n+1NI¯mI¯n=n=1Nσn2+2n=1Nm=n+1Nσmσnρm,n,
CN=σNI¯N=n=1Nσn2+2n=1Nm=n+1Nσmσnρm,nn=1NI¯n.
Cm+n=σm2+σn2+2σmσnρm,nI¯m+I¯n,
ρm,n=(I¯m+I¯n)2Cm+n2σm2σn22σmσn.
CN=Nσn2+2σn2n=1Nm=n+1Nρm,nNI¯n=1N+2N2n=1Nm=n+1Nρm,n.
[Esn,xEsn,y]=[tn,xxtn,xytn,yxtn,yy][Epn,xEpn,y].
Epn=Epn,x+Epn,y,Esn=Esn,x+Esn,y,
μm,n=EsmEsn*¯|Esm|2¯|Esn|2¯=TmEpm(TnEpn)*¯|TmEpm|2¯|TnEpn|2¯=TmTn*¯|Tm|2¯|Tn|2¯EpmEpn*¯|Epm|2¯|Epn|2¯=ηm,nγm,n.
γm,n=cos[(mn)Δθ].
ρm,n=cos2[(mn)θN1],
CN=1NN1+2n=1Nm=n+1Ncos2[(mn)θN1]N2N1,
ρexp=cos2(0.2m2.8).
CN=12N1+{sin[Nθ/(N1)][Nθ/(N1)]sin[θ/(N1)]θ/(N1)}22N1=12N1+sin2θ2θ2N1.
CN=n=1Nm=1Ncos2[(mn)θN1]N2N1=N2+n=1Nm=1Ncos[2(mn)θN1]2N2N1=12N1+n=1Ncos(2nθN1)m=1Ncos(2mθN1)+n=1Nsin(2nθN1)m=1Nsin(2mθN1)2N2N1.
n=1Nsin(2nθN1)=sinNθN1sinN+1N1θsinθN1,
n=1Ncos(2nθN1)=sinNθN1cosN+1N1θsinθN1.
CN=12N1+{sin[Nθ/(N1)]Nsin[θ/(N1)]}22N1,

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