Abstract

Speckle suppression in projection displays with a laser light source can be achieved by imaging a changing diffuser with random phase cells onto the screen. Theoretical expressions for the speckle contrast in this method have been earlier obtained in the case when different realizations of the phase diffuser produced statistically independent patterns of the light field on the screen. In the present paper, these expressions are generalized in the case when different realizations of the phase diffuser produce partly correlated speckle patterns. The possible structure of a motionless changing diffuser is presented. It includes a dynamic diffractive optical element (DDOE) and a light homogenizer. The DDOE can be based on the electrically controlled spatial light modulator (SLM) with a deformable polymer layer. This type of SLM can handle high light power and, therefore, can be used in projection displays with powerful laser beams.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).
  2. S. Lowenthal and D. Joyeux, “Speckle removal by a slowly moving diffuser associated with a motionless diffuser,” J. Opt. Soc. Am. 61, 847–851 (1971).
    [CrossRef]
  3. L. Wang, T. Tschudi, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
    [CrossRef]
  4. L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
    [CrossRef]
  5. N. George and A. Jain, “Speckle reduction using multiple tones of illumination,” Appl. Opt. 12, 1202–1212 (1973).
    [CrossRef] [PubMed]
  6. K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys., Part 1 43, 5904–5906 (2004).
    [CrossRef]
  7. V. Yurlov, A. Lapchuk, S. Yun, J. Song, and H. Yang, “Speckle suppression in scanning laser display,” Appl. Opt. 47, 179–187 (2008).
    [CrossRef] [PubMed]
  8. V. Yurlov, A. Lapchuk, S. Yun, J. Song, I. Yeo, H. Yang, and S. An, “Speckle suppression in scanning laser displays: aberration and defocusing of the projection system,” Appl. Opt. 48, 80–90 (2009).
    [CrossRef]
  9. M. N. Akram, V. Kartashov, and Z. Tong, “Speckle reduction in line-scan laser projectors using binary phase codes,” Opt. Lett. 35, 444–446 (2010).
    [CrossRef] [PubMed]
  10. S. An, A. Lapchuk, V. Yurlov, J. Song, H. Park, J. Jang, W. Shin, S. Kargapoltsev, and S. Yun, “Speckle suppression in scanning laser display using several partially coherent beams,” Opt. Express 17, 92–103 (2009).
    [CrossRef] [PubMed]
  11. J. I. Trisnadi, “Hadamard speckle contrast reduction,” Opt. Lett. 29, 11–13 (2004).
    [CrossRef] [PubMed]
  12. J. I. Trisnadi, “Speckle contrast reduction in laser projection displays,” Proc. SPIE 4657, 131–137 (2002).
    [CrossRef]
  13. I. Peled, M. Zenou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” presented at CLEO/IQEC-2009, Baltimore, Maryland, USA, 31 May–5 June 2009.
  14. 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 micro-mirror,” Appl. Opt. 49, 3297–3304 (2010).
    [CrossRef] [PubMed]
  15. C.-d. Liao, and J.-c. Tsai, “The evolution of MEMS displays,” IEEE Trans. Ind. Electron. 56, 1057–1065 (2009).
    [CrossRef]
  16. F. P. Shevlin, O. Squalli, and N. Descharmes, “An optical system and method,” International patent WO2009077198, 25 June 2009.
  17. M. Kamm and O. Ripoll, “Image generation unit and method to use an image generation unit,” U.S. patent application US2008/0204847, 28 August 2008.
  18. J. W. Goodman, Introduction to Fourier Optics (Roberts, 1996).
  19. V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
    [CrossRef]
  20. A. Malthe-Sørrensen, E. Zimmer, T. Naterstad, and B. Jacobsen, “Method and device for variable optical attenuator,” U.S. patent 6,897,995, 24 May 2005.
  21. C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport, 2005).
  22. G. Ouyang, Z. Tong, M. N. Akram, K. Wang, V. Kartashov, and X. Chen, “Speckle reduction using a motionless diffractive optical element,” Opt. Lett. 35, 2852–2854 (2010).
    [CrossRef] [PubMed]
  23. S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Speckle reduction using a sinusoidal rotating grating,” presented at the Norwegian Electrooptics Meeting, Ålesund, Norway, 7–9 April 2010.

2010 (3)

2009 (4)

V. Yurlov, A. Lapchuk, S. Yun, J. Song, I. Yeo, H. Yang, and S. An, “Speckle suppression in scanning laser displays: aberration and defocusing of the projection system,” Appl. Opt. 48, 80–90 (2009).
[CrossRef]

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

C.-d. Liao, and J.-c. Tsai, “The evolution of MEMS displays,” IEEE Trans. Ind. Electron. 56, 1057–1065 (2009).
[CrossRef]

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

2008 (1)

2007 (1)

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

2005 (1)

C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport, 2005).

2004 (2)

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys., Part 1 43, 5904–5906 (2004).
[CrossRef]

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

2002 (1)

J. I. Trisnadi, “Speckle contrast reduction in laser projection displays,” Proc. SPIE 4657, 131–137 (2002).
[CrossRef]

2000 (1)

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
[CrossRef]

1998 (1)

1996 (1)

J. W. Goodman, Introduction to Fourier Optics (Roberts, 1996).

1973 (1)

1971 (1)

Akram, M. N.

Aksnes, A.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Speckle reduction using a sinusoidal rotating grating,” presented at the Norwegian Electrooptics Meeting, Ålesund, Norway, 7–9 April 2010.

An, S.

Berglind, R.

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

Boeddinghaus, M.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
[CrossRef]

Chen, X.

Descharmes, N.

F. P. Shevlin, O. Squalli, and N. Descharmes, “An optical system and method,” International patent WO2009077198, 25 June 2009.

Egge, S. V.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Speckle reduction using a sinusoidal rotating grating,” presented at the Norwegian Electrooptics Meeting, Ålesund, Norway, 7–9 April 2010.

Elbert, A.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
[CrossRef]

George, N.

Goodman, J. W.

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

J. W. Goodman, Introduction to Fourier Optics (Roberts, 1996).

Greenberg, B.

I. Peled, M. Zenou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” presented at CLEO/IQEC-2009, Baltimore, Maryland, USA, 31 May–5 June 2009.

Halldorsson, T.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

Hedin, G.

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

Henriksen, L.

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

Jacobsen, B.

A. Malthe-Sørrensen, E. Zimmer, T. Naterstad, and B. Jacobsen, “Method and device for variable optical attenuator,” U.S. patent 6,897,995, 24 May 2005.

Jain, A.

Jang, J.

Joyeux, D.

Kamm, M.

M. Kamm and O. Ripoll, “Image generation unit and method to use an image generation unit,” U.S. patent application US2008/0204847, 28 August 2008.

Kargapoltsev, S.

Kartashov, V.

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

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 micro-mirror,” Appl. Opt. 49, 3297–3304 (2010).
[CrossRef] [PubMed]

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

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Speckle reduction using a sinusoidal rotating grating,” presented at the Norwegian Electrooptics Meeting, Ålesund, Norway, 7–9 April 2010.

Kasazumi, K.

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys., Part 1 43, 5904–5906 (2004).
[CrossRef]

Kitaoka, Y.

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys., Part 1 43, 5904–5906 (2004).
[CrossRef]

Kotler, Z.

I. Peled, M. Zenou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” presented at CLEO/IQEC-2009, Baltimore, Maryland, USA, 31 May–5 June 2009.

Lapchuk, A.

Liao, C. -d.

C.-d. Liao, and J.-c. Tsai, “The evolution of MEMS displays,” IEEE Trans. Ind. Electron. 56, 1057–1065 (2009).
[CrossRef]

Loewen, E.

C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport, 2005).

Lowenthal, S.

Malthe-Sørrensen, A.

A. Malthe-Sørrensen, E. Zimmer, T. Naterstad, and B. Jacobsen, “Method and device for variable optical attenuator,” U.S. patent 6,897,995, 24 May 2005.

Mizuuchi, K.

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys., Part 1 43, 5904–5906 (2004).
[CrossRef]

Naterstad, T.

A. Malthe-Sørrensen, E. Zimmer, T. Naterstad, and B. Jacobsen, “Method and device for variable optical attenuator,” U.S. patent 6,897,995, 24 May 2005.

Österberg, U.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Speckle reduction using a sinusoidal rotating grating,” presented at the Norwegian Electrooptics Meeting, Ålesund, Norway, 7–9 April 2010.

Ouyang, G.

Palmer, C.

C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport, 2005).

Park, H.

Peled, I.

I. Peled, M. Zenou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” presented at CLEO/IQEC-2009, Baltimore, Maryland, USA, 31 May–5 June 2009.

Petursson, P.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

Ripoll, O.

M. Kamm and O. Ripoll, “Image generation unit and method to use an image generation unit,” U.S. patent application US2008/0204847, 28 August 2008.

Shevlin, F. P.

F. P. Shevlin, O. Squalli, and N. Descharmes, “An optical system and method,” International patent WO2009077198, 25 June 2009.

Shin, W.

Song, J.

Squalli, O.

F. P. Shevlin, O. Squalli, and N. Descharmes, “An optical system and method,” International patent WO2009077198, 25 June 2009.

Svardal, B.

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

Svortdal, T.

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

Tong, Z.

Trisnadi, J. I.

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

J. I. Trisnadi, “Speckle contrast reduction in laser projection displays,” Proc. SPIE 4657, 131–137 (2002).
[CrossRef]

Tsai, J. -c.

C.-d. Liao, and J.-c. Tsai, “The evolution of MEMS displays,” IEEE Trans. Ind. Electron. 56, 1057–1065 (2009).
[CrossRef]

Tschudi, T.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

Ulvensøen, J. H.

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

Wang, K.

Wang, L.

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
[CrossRef]

L. Wang, T. Tschudi, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projection systems by diffractive optical elements,” Appl. Opt. 37, 1770–1775 (1998).
[CrossRef]

Welde, K.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Speckle reduction using a sinusoidal rotating grating,” presented at the Norwegian Electrooptics Meeting, Ålesund, Norway, 7–9 April 2010.

Yamamoto, K.

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys., Part 1 43, 5904–5906 (2004).
[CrossRef]

Yang, H.

Yeo, I.

Yun, S.

Yurlov, V.

Zenou, M.

I. Peled, M. Zenou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” presented at CLEO/IQEC-2009, Baltimore, Maryland, USA, 31 May–5 June 2009.

Zimmer, E.

A. Malthe-Sørrensen, E. Zimmer, T. Naterstad, and B. Jacobsen, “Method and device for variable optical attenuator,” U.S. patent 6,897,995, 24 May 2005.

Appl. Opt. (5)

IEEE Trans. Ind. Electron. (1)

C.-d. Liao, and J.-c. Tsai, “The evolution of MEMS displays,” IEEE Trans. Ind. Electron. 56, 1057–1065 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Soc. Inf. Disp. (1)

V. Kartashov, L. Henriksen, J. H. Ulvensøen, B. Svardal, T. Svortdal, R. Berglind, and G. Hedin, “Image improvement in the laser projection display with the spatial light modulator with deformable polymer,” J. Soc. Inf. Disp. 17, 581–587 (2009).
[CrossRef]

Jpn. J. Appl. Phys., Part 1 (1)

K. Kasazumi, Y. Kitaoka, K. Mizuuchi, and K. Yamamoto, “A practical laser projector with new illumination optics for reduction of speckle noise,” Jpn. J. Appl. Phys., Part 1 43, 5904–5906 (2004).
[CrossRef]

Opt. Eng. (Bellingham) (1)

L. Wang, T. Tschudi, M. Boeddinghaus, A. Elbert, T. Halldorsson, and P. Petursson, “Speckle reduction in laser projections with ultrasonic waves,” Opt. Eng. (Bellingham) 39, 1659–1664 (2000).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Proc. SPIE (1)

J. I. Trisnadi, “Speckle contrast reduction in laser projection displays,” Proc. SPIE 4657, 131–137 (2002).
[CrossRef]

Other (8)

I. Peled, M. Zenou, B. Greenberg, and Z. Kotler, “MEMS based speckle reduction obtained by angle diversity for fast imaging,” presented at CLEO/IQEC-2009, Baltimore, Maryland, USA, 31 May–5 June 2009.

S. V. Egge, M. N. Akram, V. Kartashov, K. Welde, Z. Tong, U. Österberg, and A. Aksnes, “Speckle reduction using a sinusoidal rotating grating,” presented at the Norwegian Electrooptics Meeting, Ålesund, Norway, 7–9 April 2010.

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

A. Malthe-Sørrensen, E. Zimmer, T. Naterstad, and B. Jacobsen, “Method and device for variable optical attenuator,” U.S. patent 6,897,995, 24 May 2005.

C. Palmer and E. Loewen, Diffraction Grating Handbook (Newport, 2005).

F. P. Shevlin, O. Squalli, and N. Descharmes, “An optical system and method,” International patent WO2009077198, 25 June 2009.

M. Kamm and O. Ripoll, “Image generation unit and method to use an image generation unit,” U.S. patent application US2008/0204847, 28 August 2008.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 1996).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

General structure of the projection display with the laser light source and the motionless changing diffuser for the speckle suppression.

Fig. 2
Fig. 2

Distribution of light intensity in the plane of diffuser 1. Filled and empty circles show two diffraction patterns with intensities I diffuser 1 ( m 1 ) ( x , y ) and I diffuser 1 ( m 2 ) ( x , y ) . The central diffraction order is fully attenuated in both patterns. The diffuser is divided into w = 1 , , W elementary cells with area Δ s .

Fig. 3
Fig. 3

(a) DDOE consists of three sinusoidal phase gratings parallel to each other with the same spatial period Λ and different orientation angles 0°, 60°, and 90°. (b) Speckle contrast versus amplitude of phase gratings calculated by Eq. (21) with the DDOE from (a).

Fig. 4
Fig. 4

TSLM for use in the DDOE. Typical practical parameters: gel thickness of 15 20 μ m , air gap of 5 10 μ m , spatial periods of gratings Λ = 30 80 μ m [16]. Different periods and orientation angles of gratings are shown symbolically by parallel lines inside yellow rectangles.

Equations (51)

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

C = M + K ± 1 M K ,
I = m = 1 M I m = m = 1 M | k = 1 K A k B k ( m ) | 2 = m = 1 M k = 1 K l = 1 K A k A l B k ( m ) B l ( m ) ,
B k ( m ) = w = 1 W b k w ( m ) .
b k w ( m ) = σ w ( m ) ( r k w + i j k w ) ,
r k w ¯ = 0 ,     j k w ¯ = 0 ,     r k w j l v ¯ = 0     for   any   k , l , w ,   and   v ,
r k w 2 ¯ = 1 ,     j k w 2 ¯ = 1 ,     r k w 4 ¯ = 3 ( r k w 2 ¯ ) 2 = 3 , j k w 4 ¯ = 3.
r k w r k v ¯ = j k w j k v ¯ = { 1 if   w = v 0 if   w v . }
r k w r l w ¯ = j k w j l w ¯ = { 1 if   k = l 0 if   k l . }
| B k ( m ) | 2 ¯ = ( w = 1 W σ w ( m ) r k w ) 2 + ( w = 1 W σ w ( m ) j k w ) 2 ¯ = w = 1 W ( σ w ( m ) ) 2 r k w 2 ¯ + w = 1 W ( σ w ( m ) ) 2 j k w 2 ¯ = 2 w = 1 W ( σ w ( m ) ) 2 = J B ,
| B k ( m ) | 4 ¯ = ( w = 1 W σ w ( m ) r k w ) 4 + 2 ( w = 1 W σ w ( m ) r k w ) 2 ( w = 1 W σ w ( m ) j k w ) 2 + ( w = 1 W σ w ( m ) j k w ) 4 ¯ = 3 [ ( w = 1 W σ w ( m ) r k w ) 2 ¯ ] 2 + 2 ( w = 1 W σ w ( m ) r k w ) 2 ¯ ( w = 1 W σ w ( m ) j k w ) 2 ¯ + 3 [ ( w = 1 W σ w ( m ) j k w ) 2 ¯ ] 2 = 8 [ w = 1 W ( σ w ( m ) ) 2 ] 2 = 2 J B 2 .
| b k w ( m ) | 2 = ( σ w ( m ) ) 2 ( r k w 2 ¯ + j k w 2 ¯ ) = 2 ( σ w ( m ) ) 2 = Q I diffuser 1 , w ( m ) Δ s ,
I screen , k ( m ) ¯ = B k ( m ) B k ( m ) ¯ = | B k ( m ) | 2 ¯ = 2 w = 1 W ( σ w ( m ) ) 2 = w = 1 W Q I diffuser 1 , w ( m ) Δ s = J B .
Q = J B / P ,
C = σ I I ¯ = I 2 ¯ ( I ¯ ) 2 I ¯ ,
E [ I A ] = m = 1 M k = 1 K l = 1 K A k A l B k ( m ) B l ( m ) ¯ .
B k ( m ) B l ( m ) ¯ = [ w = 1 W σ w ( m ) r k w + i w = 1 W σ w ( m ) j k w ] [ w = 1 W σ w ( m ) r l w i w = 1 W σ w ( m ) j l w ] ¯ .
E [ I A ] = m = 1 M k = 1 K | A k | 2 | B k ( m ) | 2 ¯ = J B M k = 1 K | A k | 2 .
I ¯ = M K J A J B ,
I 2 ¯ = m = 1 M n = 1 M k = 1 K l = 1 K p = 1 K q = 1 K A k A l A p A q B k ( m ) B l ( m ) B p ( n ) B q ( n ) ¯ .
I 2 ¯ = J B 2 [ K | A | 4 ¯ + ( K 2 K ) J A 2 ] ( M 2 + M + Ψ P 2 ) ,
Ψ = m = 1 M n = 1 , n m M [ S diffuser 1 I diffuser 1 ( m ) ( x , y ) I diffuser 1 ( n ) ( x , y ) d x d y ] 2
σ I 2 = I 2 ¯ ( I ¯ ) 2 = J A 2 J B 2 M K ( M + K + 1 ) + J A 2 J B 2 ( K 2 + K ) Ψ P 2 .
C = σ I I ¯ = M + K + 1 M K + K 2 + K M 2 K 2 P 2 Ψ .
I diffuser 1 ( 0 ) ( x , y ) = H [ 2 J 1 ( φ 0 ( x , y ) ) φ 0 ( x , y ) ] 2 ,
where   φ 0 ( x , y ) = π d λ z x 2 + y 2 .
I diffuser 1 ( m ) ( x , y ) = H q = J q 2 ( Θ 2 ) [ 2 J 1 ( φ m , q ( x , y ) ) φ m , q ( x , y ) ] 2 ,
m = 1 , 2 , 3 ,
I diffuser 1 ( m ) ( x , y ) = H s = q = J s 2 ( Θ 2 ) J q 2 ( Θ 2 ) [ 2 J 1 ( φ s , q , a , b ( x , y ) ) φ s , q , a , b ( x , y ) ] 2     for   m = 4 , 5 , 6 ,
I diffuser 1 ( 7 ) ( x , y ) = H r = s = q = J r 2 ( Θ 2 ) J s 2 ( Θ 2 ) J q 2 ( Θ 2 ) [ 2 J 1 ( φ r , s , q ( x , y ) ) φ r , s , q ( x , y ) ] 2 ,
E [ I 2 A ] = m = 1 M n = 1 M k = 1 K l = 1 K p = 1 K q = 1 K A k A l A p A q B k ( m ) B l ( m ) B p ( n ) B q ( n ) ¯ .
m = 1 M k = 1 K | A k | 4 | B k ( m ) | 4 ¯ = 2 J B 2 M k = 1 K | A k | 4 .
m = 1 M k = 1 K p = 1 , p k K A k A k A p A p | B k ( m ) | 2 | B p ( m ) | 2 ¯ = J B 2 M k = 1 K p = 1 , p k K | A k | 2 | A p | 2 .
E [ I 2 A ] m = n = 2 J B 2 M [ k = 1 K | A k | 4 + k = 1 K p = 1 , p k K | A k | 2 | A p | 2 ] .
m = 1 M n = 1 , n m M k = 1 K | A k | 4 B k ( m ) B k ( m ) B k ( n ) B k ( n ) ¯ .
B k ( m ) B k ( m ) B k ( n ) B k ( n ) ¯ = [ ( w = 1 W σ w ( m ) r k w ) 2 + ( w = 1 W σ w ( m ) j k w ) 2 ] [ ( w = 1 W σ w ( n ) r k w ) 2 + ( w = 1 W σ w ( n ) j k w ) 2 ] ¯ .
B k ( m ) B k ( m ) B k ( n ) B k ( n ) ¯ = w = 1 W ( σ w ( m ) ) 2 ( σ w ( n ) ) 2 r k w 4 ¯ + w = 1 W x = 1 , x w W ( σ w ( m ) ) 2 ( σ x ( n ) ) 2 r k w 2 r k x 2 ¯ + Σ r x Σ r y ¯ + w = 1 W ( σ w ( m ) ) 2 ( σ w ( n ) ) 2 r k w 2 ¯ j k w 2 ¯ + w = 1 W x = 1 , x w W ( σ w ( m ) ) 2 ( σ x ( n ) ) 2 r k w 2 j k x 2 ¯ + Σ r x Σ j y ¯ + w = 1 W ( σ w ( m ) ) 2 ( σ w ( n ) ) 2 j k w 2 ¯ r k w 2 ¯ + w = 1 W x = 1 , x w W ( σ w ( m ) ) 2 ( σ x ( n ) ) 2 j k w 2 r k x 2 ¯ + Σ j x Σ r y ¯ + w = 1 W ( σ w ( m ) ) 2 ( σ w ( n ) ) 2 j k w 4 ¯ + w = 1 W x = 1 , x w W ( σ w ( m ) ) 2 ( σ x ( n ) ) 2 j k w 2 j k x 2 ¯ + Σ j x Σ j y ¯ ,
Σ r x = w = 1 W x = 1 , x w W σ w ( m ) σ x ( m ) r k w r k x ,     Σ r y = y = 1 W z = 1 , z y W σ y ( m ) σ z ( m ) r k y r k z ,
Σ j x = w = 1 W x = 1 , x w W σ w ( m ) σ x ( m ) j k w j k x ,     Σ j y = y = 1 W z = 1 , z y W σ y ( m ) σ z ( m ) j k y j k z .
r k w 2 r k x 2 ¯ = { r k w 2 ¯ r k x 2 ¯ = 1 1 = 1 for   w x r k w 4 ¯ = 3 ( r k w 2 ¯ ) 2 = 3 for   w = x , }
j k w 2 j k x 2 ¯ = { 1 for   w x 3 for   w = x , }
r k w 2 j k x 2 ¯ = r k w 2 ¯ j k x 2 ¯ = 1 1 = 1     for   any   w   and   x .
B k ( m ) B k ( m ) B k ( n ) B k ( n ) ¯ = 8 w = 1 W ( σ w ( m ) ) 2 ( σ w ( n ) ) 2 + 4 w = 1 W x = 1 , x w W ( σ w ( m ) ) 2 ( σ x ( n ) ) 2 + 4 w = 1 W x = 1 , x w W σ w ( m ) σ x ( m ) σ w ( n ) σ x ( n ) = 4 [ w = 1 W ( σ w ( m ) ) 2 ] [ w = 1 W ( σ w ( n ) ) 2 ] + 4 ( w = 1 W σ w ( m ) σ w ( n ) ) 2 .
B k ( m ) B k ( m ) B k ( n ) B k ( n ) ¯ = 4 J B 2 J B 2 + Q 2 ( w = 1 W I diffuser 1 , w ( m ) I diffuser 1 , w ( n ) Δ s ) 2 = J B 2 + Q 2 Θ m , n 2 ,
m = 1 M n = 1 , n m M k = 1 K | A k | 4 B k ( m ) B k ( m ) B k ( n ) B k ( n ) ¯ = m = 1 M n = 1 , n m M k = 1 K | A k | 4 { J B 2 + J B 2 P 2 Θ m , n 2 } = J B 2 ( M 2 M ) k = 1 K | A k | 4 + J B 2 P 2 m = 1 M n = 1 , n m M k = 1 K | A k | 4 Θ m , n 2 .
m = 1 M n = 1 , n m M k = 1 K p = 1 , p k K | A k | 2 | A p | 2 | B k ( m ) | 2 | B p ( n ) | 2 ¯ = J B 2 ( M 2 M ) k = 1 K p = 1 , p k K | A k | 2 | A p | 2 .
m = 1 M n = 1 , n m M k = 1 K l = 1 , l k K | A k | 2 | A l | 2 B k ( m ) B l ( m ) B l ( n ) B k ( n ) ¯ .
B k ( m ) B l ( m ) B l ( n ) B k ( n ) ¯ = [ w = 1 W σ w ( m ) r k w + i w = 1 W σ w ( m ) j k w ] [ w = 1 W σ w ( m ) r l w i w = 1 W σ w ( m ) j l w ] ¯ × [ w = 1 W σ w ( n ) r l w + i w = 1 W σ w ( n ) j l w ] [ w = 1 W σ w ( n ) r k w i w = 1 W σ w ( n ) j k w ] ¯ .
B k ( m ) B l ( m ) B l ( n ) B k ( n ) ¯ = 4 ( w = 1 W σ w ( m ) σ w ( n ) ) 2 = Q 2 Θ m , n 2 ,
m = 1 M n = 1 , n m M k = 1 K l = 1 , l k K | A k | 2 | A l | 2 B k ( m ) B l ( m ) B l ( n ) B k ( n ) ¯ = J B 2 P 2 m = 1 M n = 1 , n m M k = 1 K l = 1 , l k K | A k | 2 | A l | 2 Θ m , n 2 .
E [ I 2 A ] = J B 2 ( M 2 + M ) [ k = 1 K | A k | 4 + k = 1 K p = 1 , p k K | A k | 2 | A p | 2 ] + J B 2 P 2 [ m = 1 M n = 1 , n m M k = 1 K | A k | 4 Θ m , n 2 + m = 1 M n = 1 , n m M k = 1 K l = 1 , l k K | A k | 2 | A l | 2 Θ m , n 2 ] .
I 2 ¯ = J B 2 [ K | A | 4 ¯ + ( K 2 K ) J A 2 ] ( M 2 + M + Ψ P 2 ) ,

Metrics