Abstract

A cascade of a thick grating and a thin diffuser is shown to scatter radiation efficiently and uniformly over a wide angle. Cascading the grating with a diffuser causes the single-beam power spectrum of the diffuser to be replicated at each diffraction angle of the grating. The grating period is chosen so that the first diffraction order falls near the one-half point of the power-spectrum peak of the diffuser. The relative strengths of the diffraction orders are optimized to obtain uniformity of the resulting intensity distribution in the plane of the diffraction orders. The intensity distribution in the perpendicular plane is governed solely by the diffuser. Such a cylindrical system is considered on the basis of the requirements for projection TV’s of a large horizontal span (100°) and a narrower vertical span (∼15°). Broadband illumination is studied by consideration of three simultaneously illuminating wavelengths. Experimental results are given for a cascade of a grating formed in photoresist and an etched-glass diffuser.

© 1999 Optical Society of America

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References

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  1. C. N. Kurtz, “Transmittance characteristics of surface diffusers and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am 62, 982–989 (1972).
    [CrossRef]
  2. C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am 63, 1080–1092 (1973).
    [CrossRef]
  3. N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
    [CrossRef]
  4. N. George, A. Jain, R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
    [CrossRef]
  5. K. J. Allardyce, N. George, “Diffraction analysis of rough reflective surfaces,” Appl. Opt. 26, 2364–2375 (1987).
    [CrossRef] [PubMed]
  6. L. G. Shirley, N. George, “Diffuser radiation patterns over a large dynamic range. 1: strong diffusers,” Appl. Opt. 27, 1850–1861 (1988).
    [CrossRef] [PubMed]
  7. N. George, “Speckle at various planes in an optical system,” Opt. Eng. 25, 754–764 (1986).
    [CrossRef]
  8. L. G. Shirley, N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am A 4, 734–745 (1987).
    [CrossRef]
  9. E. W. Marchand, “Diffraction effects with lenticular projection screens,” J. Opt. Soc. Am 65, 139–145 (1975).
    [CrossRef]
  10. M. D. Kirkpatrick, G. Mihalakis, “Projection screens for high definition television,” in Large-Screen-Projection, Avionic and Helmet-Mounted Displays, H. M. Assenheim, R. A. Flasck, T. M. Lippert, J. Bentz, eds., Proc. SPIE1456, 40–47 (1991).
    [CrossRef]
  11. R. J. Bradley, J. F. Goldenberg, T. S. McKechnie, “Ultra-wide viewing angle rear projection television screen,” IEEE Trans. Consum. Electron. CE-31, 185–193 (1985).
    [CrossRef]
  12. K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
    [CrossRef]
  13. H. P. Baltes, A. M. J. Huiser, “Coherent and incoherent grating reconstruction,” J. Opt. Soc. Am. A 3, 1268–1275 (1986).
    [CrossRef]
  14. D. Newman, J. C. Dainty, “Detection of gratings hidden by diffusers using intensity interferometry,” J. Opt. Soc. Am. A 1, 403–411 (1984).
    [CrossRef]
  15. H. P. Baltes, “Speckle correlation and the detection of phase gratings hidden by diffusers,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 223–226 (1985).
    [CrossRef]
  16. E. Simova, M. Kavehrad, “Light shaping diffusers for indoor wireless infrared communications via a holographic approach,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 284–291 (1996).
    [CrossRef]
  17. N. George, D. J. Schertler, ″Optical system for diffusing light,″ U.S. patent application UR-0160 (1998).
  18. P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).
  19. We derived these forms from Eqs. (36) and (34), respectively, of Ref. 6 by letting the secondary rms phase delay S2 go to zero.
  20. M. J. Beesley, J. G. Castledine, “The use of photoresist as a holographic recording medium,” Appl. Opt. 9, 2720–2724 (1970).
    [CrossRef] [PubMed]
  21. R. A. Bartolini, “Characteristics of relief phase holograms recorded in photoresists,” Appl. Opt. 13, 129–139 (1974).
    [CrossRef] [PubMed]
  22. S. Austin, F. T. Stone, “Fabrication of thin periodic structures in photoresist: a model,” Appl. Opt. 15, 1071–1074 (1976).
    [CrossRef] [PubMed]
  23. R. C. Enger, S. K. Case, “High-frequency holographic transmission gratings in photoresist,” J. Opt. Soc. Am. 73, 1113–1118 (1983).
    [CrossRef]
  24. S. Lindau, “Controlling the groove depth of holographic gratings,” in Optical System Design, Analysis and Production, P. J. Rogers, R. E. Fischer, eds., Proc. SPIE399, 323–328 (1983).
    [CrossRef]
  25. M. Miler, “Photoresist as a recording material for holographic elements,” in International Conference on Holography, Correlation Optics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 2–9 (1993).
    [CrossRef]
  26. L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1988).
  27. Allied Chemical, Allied Corp., Morristown, N.J., 07960.

1988 (1)

1987 (2)

L. G. Shirley, N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am A 4, 734–745 (1987).
[CrossRef]

K. J. Allardyce, N. George, “Diffraction analysis of rough reflective surfaces,” Appl. Opt. 26, 2364–2375 (1987).
[CrossRef] [PubMed]

1986 (2)

N. George, “Speckle at various planes in an optical system,” Opt. Eng. 25, 754–764 (1986).
[CrossRef]

H. P. Baltes, A. M. J. Huiser, “Coherent and incoherent grating reconstruction,” J. Opt. Soc. Am. A 3, 1268–1275 (1986).
[CrossRef]

1985 (1)

R. J. Bradley, J. F. Goldenberg, T. S. McKechnie, “Ultra-wide viewing angle rear projection television screen,” IEEE Trans. Consum. Electron. CE-31, 185–193 (1985).
[CrossRef]

1984 (1)

1983 (1)

1981 (1)

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

1976 (1)

1975 (2)

E. W. Marchand, “Diffraction effects with lenticular projection screens,” J. Opt. Soc. Am 65, 139–145 (1975).
[CrossRef]

N. George, A. Jain, R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

1974 (2)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

R. A. Bartolini, “Characteristics of relief phase holograms recorded in photoresists,” Appl. Opt. 13, 129–139 (1974).
[CrossRef] [PubMed]

1973 (1)

C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am 63, 1080–1092 (1973).
[CrossRef]

1972 (1)

C. N. Kurtz, “Transmittance characteristics of surface diffusers and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am 62, 982–989 (1972).
[CrossRef]

1970 (1)

Allardyce, K. J.

Austin, S.

Baltes, H. P.

H. P. Baltes, A. M. J. Huiser, “Coherent and incoherent grating reconstruction,” J. Opt. Soc. Am. A 3, 1268–1275 (1986).
[CrossRef]

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

H. P. Baltes, “Speckle correlation and the detection of phase gratings hidden by diffusers,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 223–226 (1985).
[CrossRef]

Bartolini, R. A.

Beesley, M. J.

Bradley, R. J.

R. J. Bradley, J. F. Goldenberg, T. S. McKechnie, “Ultra-wide viewing angle rear projection television screen,” IEEE Trans. Consum. Electron. CE-31, 185–193 (1985).
[CrossRef]

Case, S. K.

Castledine, J. G.

Clemmow, P. C.

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

Dainty, J. C.

DePalma, J. J.

C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am 63, 1080–1092 (1973).
[CrossRef]

Enger, R. C.

George, N.

L. G. Shirley, N. George, “Diffuser radiation patterns over a large dynamic range. 1: strong diffusers,” Appl. Opt. 27, 1850–1861 (1988).
[CrossRef] [PubMed]

L. G. Shirley, N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am A 4, 734–745 (1987).
[CrossRef]

K. J. Allardyce, N. George, “Diffraction analysis of rough reflective surfaces,” Appl. Opt. 26, 2364–2375 (1987).
[CrossRef] [PubMed]

N. George, “Speckle at various planes in an optical system,” Opt. Eng. 25, 754–764 (1986).
[CrossRef]

N. George, A. Jain, R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

N. George, D. J. Schertler, ″Optical system for diffusing light,″ U.S. patent application UR-0160 (1998).

Goldenberg, J. F.

R. J. Bradley, J. F. Goldenberg, T. S. McKechnie, “Ultra-wide viewing angle rear projection television screen,” IEEE Trans. Consum. Electron. CE-31, 185–193 (1985).
[CrossRef]

Hoadley, H. O.

C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am 63, 1080–1092 (1973).
[CrossRef]

Huiser, A. M. J.

Jain, A.

N. George, A. Jain, R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

Jauch, K. M.

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

Kavehrad, M.

E. Simova, M. Kavehrad, “Light shaping diffusers for indoor wireless infrared communications via a holographic approach,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 284–291 (1996).
[CrossRef]

Kirkpatrick, M. D.

M. D. Kirkpatrick, G. Mihalakis, “Projection screens for high definition television,” in Large-Screen-Projection, Avionic and Helmet-Mounted Displays, H. M. Assenheim, R. A. Flasck, T. M. Lippert, J. Bentz, eds., Proc. SPIE1456, 40–47 (1991).
[CrossRef]

Kurtz, C. N.

C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am 63, 1080–1092 (1973).
[CrossRef]

C. N. Kurtz, “Transmittance characteristics of surface diffusers and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am 62, 982–989 (1972).
[CrossRef]

Lindau, S.

S. Lindau, “Controlling the groove depth of holographic gratings,” in Optical System Design, Analysis and Production, P. J. Rogers, R. E. Fischer, eds., Proc. SPIE399, 323–328 (1983).
[CrossRef]

Marchand, E. W.

E. W. Marchand, “Diffraction effects with lenticular projection screens,” J. Opt. Soc. Am 65, 139–145 (1975).
[CrossRef]

McKechnie, T. S.

R. J. Bradley, J. F. Goldenberg, T. S. McKechnie, “Ultra-wide viewing angle rear projection television screen,” IEEE Trans. Consum. Electron. CE-31, 185–193 (1985).
[CrossRef]

Melville, R. D. S.

N. George, A. Jain, R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

Mihalakis, G.

M. D. Kirkpatrick, G. Mihalakis, “Projection screens for high definition television,” in Large-Screen-Projection, Avionic and Helmet-Mounted Displays, H. M. Assenheim, R. A. Flasck, T. M. Lippert, J. Bentz, eds., Proc. SPIE1456, 40–47 (1991).
[CrossRef]

Miler, M.

M. Miler, “Photoresist as a recording material for holographic elements,” in International Conference on Holography, Correlation Optics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 2–9 (1993).
[CrossRef]

Newman, D.

Schertler, D. J.

N. George, D. J. Schertler, ″Optical system for diffusing light,″ U.S. patent application UR-0160 (1998).

Shirley, L. G.

L. G. Shirley, N. George, “Diffuser radiation patterns over a large dynamic range. 1: strong diffusers,” Appl. Opt. 27, 1850–1861 (1988).
[CrossRef] [PubMed]

L. G. Shirley, N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am A 4, 734–745 (1987).
[CrossRef]

L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1988).

Simova, E.

E. Simova, M. Kavehrad, “Light shaping diffusers for indoor wireless infrared communications via a holographic approach,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 284–291 (1996).
[CrossRef]

Stone, F. T.

Appl. Opt. (5)

Appl. Phys. (2)

N. George, A. Jain, “Space and wavelength dependence of speckle intensity,” Appl. Phys. 4, 201–212 (1974).
[CrossRef]

N. George, A. Jain, R. D. S. Melville, “Experiments on the space and wavelength dependence of speckle,” Appl. Phys. 7, 157–169 (1975).
[CrossRef]

IEEE Trans. Consum. Electron. (1)

R. J. Bradley, J. F. Goldenberg, T. S. McKechnie, “Ultra-wide viewing angle rear projection television screen,” IEEE Trans. Consum. Electron. CE-31, 185–193 (1985).
[CrossRef]

J. Opt. Soc. Am (3)

C. N. Kurtz, “Transmittance characteristics of surface diffusers and the design of nearly band-limited binary diffusers,” J. Opt. Soc. Am 62, 982–989 (1972).
[CrossRef]

C. N. Kurtz, H. O. Hoadley, J. J. DePalma, “Design and synthesis of random phase diffusers,” J. Opt. Soc. Am 63, 1080–1092 (1973).
[CrossRef]

E. W. Marchand, “Diffraction effects with lenticular projection screens,” J. Opt. Soc. Am 65, 139–145 (1975).
[CrossRef]

J. Opt. Soc. Am A (1)

L. G. Shirley, N. George, “Wide-angle diffuser transmission functions and far-zone speckle,” J. Opt. Soc. Am A 4, 734–745 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

K. M. Jauch, H. P. Baltes, “Coherence of radiation scattered by gratings covered by a diffuser. Experimental evidence,” Opt. Acta 28, 1013–1015 (1981).
[CrossRef]

Opt. Eng. (1)

N. George, “Speckle at various planes in an optical system,” Opt. Eng. 25, 754–764 (1986).
[CrossRef]

Other (10)

M. D. Kirkpatrick, G. Mihalakis, “Projection screens for high definition television,” in Large-Screen-Projection, Avionic and Helmet-Mounted Displays, H. M. Assenheim, R. A. Flasck, T. M. Lippert, J. Bentz, eds., Proc. SPIE1456, 40–47 (1991).
[CrossRef]

H. P. Baltes, “Speckle correlation and the detection of phase gratings hidden by diffusers,” in International Conference on Speckle, H. H. Arsenault, ed., Proc. SPIE556, 223–226 (1985).
[CrossRef]

E. Simova, M. Kavehrad, “Light shaping diffusers for indoor wireless infrared communications via a holographic approach,” in Diffractive and Holographic Optics Technology III, I. Cindrich, S. H. Lee, eds., Proc. SPIE2689, 284–291 (1996).
[CrossRef]

N. George, D. J. Schertler, ″Optical system for diffusing light,″ U.S. patent application UR-0160 (1998).

P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields (Pergamon, New York, 1966).

We derived these forms from Eqs. (36) and (34), respectively, of Ref. 6 by letting the secondary rms phase delay S2 go to zero.

S. Lindau, “Controlling the groove depth of holographic gratings,” in Optical System Design, Analysis and Production, P. J. Rogers, R. E. Fischer, eds., Proc. SPIE399, 323–328 (1983).
[CrossRef]

M. Miler, “Photoresist as a recording material for holographic elements,” in International Conference on Holography, Correlation Optics, and Recording Materials, O. V. Angelsky, ed., Proc. SPIE2108, 2–9 (1993).
[CrossRef]

L. G. Shirley, “Laser speckle from thin and cascaded diffusers,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1988).

Allied Chemical, Allied Corp., Morristown, N.J., 07960.

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Figures (13)

Fig. 1
Fig. 1

System diagram showing the scattering angles.

Fig. 2
Fig. 2

Total scattered power from a parabolic and a conical surface correlation as functions of the rms phase delay S and a fixed correlation length.

Fig. 3
Fig. 3

Geometry of the grating–diffuser cascade for (a) the grating surface nearest the diffuser, (b) the grating surface facing the illumination, and (c) the composite grating and diffuser on one substrate.

Fig. 4
Fig. 4

(a) Parabolic diffuser scattering patterns for l c /(λS) = 1 and for five illumination angles of 0°, ±20°, and ±43.2° and the sum. (b) Diffuser patterns scaled by the normalized coefficients of Table 1 and the pattern resulting from the diffuser–grating cascade.

Fig. 5
Fig. 5

(a) Parabolic diffuser scattering patterns for l c /(λS) = 4 and for illumination angles corresponding to the diffraction angles of a 5° grating and their sum. (b) The scaled diffuser patterns and the optimized sum.

Fig. 6
Fig. 6

Wavelength variation of the cascade of Fig. 5(b).

Fig. 7
Fig. 7

Scanning electron micrograph of a 5° grating identical to that used in the experiments.

Fig. 8
Fig. 8

Relative intensity pattern scattered from the diffuser for single-beam illumination.

Fig. 9
Fig. 9

Relative intensity pattern for the cascade of a 5° grating and a diffuser measured in the plane of the grating diffraction orders.

Fig. 10
Fig. 10

Relative intensity pattern from a commercially available microfiche view screen.

Fig. 11
Fig. 11

Relative intensity patterns for the cascades of a 5° and a 15° grating and a diffuser measured in the plane of the grating diffraction orders.

Fig. 12
Fig. 12

Full lobe width versus the rms phase delay S for the parabolic and the conical surfaces for l c /λ = 10. The efficiency is noted by the percentages.

Fig. 13
Fig. 13

Small-angle scattering patterns of four surfaces: The diffuser used in the cascade, 3M Magic Tape type 810, a frosted-glass microscope slide, and glass etched with Armour Etch. The efficiency is noted by the percentages.

Tables (5)

Tables Icon

Table 1 Peak-Angle Location of the Diffuser Patterns for the 20° Grating and the Grating Coefficients for a Uniform Power Distribution

Tables Icon

Table 2 Peak-Angle Locations of the Diffuser Patterns for the 5° Grating and the Grating Coefficients for a Uniform Power Distribution

Tables Icon

Table 3 Grating Coefficients at Three Wavelengths for Optimization at 0.55 µm

Tables Icon

Table 4 Ideal Grating Coefficients for Optimization at Three Wavelengths

Tables Icon

Table 5 Diffraction-Order Strengths at Three Wavelengths for the 5° Grating at 0.457 µm Used in the Experiments as a Fraction of the Incident Power

Equations (37)

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

vx, y, z=i exp-ikR0λR0cos θ  dxdyvIIx, y×expi2πλR0xx+yy,
Vfx, fy; z; ν= dxdyvx, y, z; ν×exp-i2πxfx+yfy,
vx, y, z; ν= dfxdfyVfx, fy; z; ν×expi2πxfx+yfy.
fx=-αλ0=-xλ0R0=-sin θ cos ϕλ0, fy=-βλ0=-yλ0R0=-sin θ sin ϕλ0.
dvx, y, z; ν=VIfx, fy; z; νexpi2πxfx+yfy×tx, y; -λ0fx, -λ0fydfxdfy,
VIfx, fy; 0; ν=j Bjδfx+αjλ0δfy+βjλ0,
vIIx, y=j Bj exp-i 2πλ0αjx+βjy×tDx, y; αj, βj,
tDx, y; α, β=exp-ik0Hn cos θ1+hx, y×n cos θ1-cos θ2,
I=R02A vx, y, zv*x, y, z.
I=cos θ 2πλ2  dρρJ02π ρλsin2 θ-2 sin θ sin θ0 cosϕ-ϕ0+sin2 θ01/2×exp-S21-r12ρ,
S=2π σλn cos θ1-cos θ0.
r12ρ1-ρlc2,  ρlc,
r12ρ1-|ρ|lc,  |ρ|lc.
IP=π cos θlcλS2exp-πlcλS2sin2 θ-2 sin θ sin θ0 cosϕ-ϕ0+sin2 θ0
IC=2π cos θlcλS221+2πlcλS22sin2 θ-2 sin θ sin θ0 cosϕ-ϕ0+sin2 θ0-3/2
tGx, y=j Bj exp-i2π jxΛ,
tGx, y=δfyj=-MM Bjδfx+jΛ,
vIIIx, y, 0; ν= VIIfx, fy; 0; νexpi2πxfx+yfytDx, y; -λ0fx, -λ0fydfxdfy,
vIIIx, y, 0; ν=j Bj  dfxdfyδfyδfx+jΛ×exp-ikhDx, yn cos θ1j-cos θjexp-ikL cos θj+nH cos θ1jexpi2πxfx+yfy.
fx=-sin θj cos ϕjλ, fy=-sin θj sin ϕjλ.
vIIIx, y, 0; ν=j Bj exp-ikhDx, y×n cos θ1j-cos θj×exp-ikL cos θj+nH cos θ1j×exp-i2π jxΛ
ϕj=0, sin θj=jλΛ, n sin θ1j=sin θj.
vx, y, z=i exp-ikR0λR0cos θ j=- Bj×exp-ikL cos θj+nH cos θ1j× dxdy exp-ikhDx, yn cos θ1j-cos θjexpikxR0-jλΛx+yR0 y.
I=cos θAλ2j=-l=- BjBl exp-ikLcos θj-cos θl+nHcos θ1j-cos θ1l dxdydxdy×F2n cos θ1j-cos θjλ, -n cos θ1l-cos θlλ; r12×expi2πλxR0x-x-λΛjx-lx+yR0y-y.
F2η1, η2; r12ρ=exp-i2πη1hDx, y+η2hDx, y,
I=cos θAλ2j=- Bj2  dxdydxdy×F2n cos θ1j-cos θjλ, -n cos θ1j-cos θjλ; r12×expi2πλxR0-jλΛx-x+yR0y-y+cos θ 1A2λ2j=-lj BjBl exp-ikLcos θj-cos θl+nHcos θ1j-cos θ1l× dxdydxdy×F2n cos θ1j-cos θjλ, -n cos θ1l-cos θlλ; r12×expi2πλxR0x-x-λΛjx-lx+yR0y-y.
 =14  dudvF2n cos θ1j-cos θjλ, ×-n cos θ1l-cos θlλ; r12×expi2πλxR0 u+yR0 v×exp-i2πΛu2j+l  dsdw×exp-i2πΛs2j-l.
IP=π cos θ j=-MM Bj2lcλSj2 exp-πlcλSj2×sin2 θ-2jλΛsin θ cos ϕ+jλΛ2,
IC=2π cos θ j=-MM Bj2lcλSj221+2πlcλSj22×sin2 θ-2jλΛsin θ cos ϕ+jλΛ2-3/2,
Sj=2πσλn2-jλΛ21/2-1-jλΛ21/2.
Iθm=j=-MM Bj2Ijθm=B02I0θm+j=1M Bj2Ijθm+I-jθm,
AB=-I,
A=I1θ0+I-1θ0IMθ0+I-Mθ01I1θ1+I-1θ1IMθ1+I-Mθ11I1θM+I-1θMIMθM+I-MθM1, B=B12BM2-C, I=I0θ0I0θ1I0θM.
tGx, y=exp-ik0Lx,
tGx, y=j=- Bj exp-i2π jxΛ.
ϕ0=-k0Lx=Phase j=- Bj exp-i2π jxΛ.
ϕ=kk0 ϕ0.

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