Abstract

We propose and test experimentally a new scheme to implement spatially multiplexed multiwavelength holographic memory. An electro-optically modulated phase grating array on LiNbO3 substrate is used as a guided-wave interconnect to activate the reconfigurable reference beam. The object beam is provided by free-space interconnect. An electro-optic modulation efficiency of 18 ± 2.5% is achieved with an applied voltage of 100 V. The reference beams with different diffraction angles can implement the angle-multiplexing holographic recording. We believe this is the first report of the implementation of guided-wave electro-optic interconnect together with free-space interconnect in holographic memory applications.

© 1997 Optical Society of America

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References

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  1. R. T. Chen, H. Lu, D. Robinson, T. Jannson, “Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing,” Appl. Phys. Lett. 59, 1144–1146 (1991).
    [CrossRef]
  2. A. A. Rizvi, M. S. Zubairy, “Implementation of associative memory using grating structures,” Appl. Opt. 33, 3642–3646 (1994).
    [CrossRef] [PubMed]
  3. R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
    [CrossRef]
  4. E. Arons, D. Dilworth, “Analysis of Fourier synthesis holography for imaging through scattering materials,” Appl. Opt. 34, 1841–1847 (1995).
    [CrossRef] [PubMed]
  5. G. A. Rakuljic, V. Leyva, A. Yariv, “Optical data storage by using orthogonal wavelength-multiplexed volume holograms,” Opt. Lett. 17, 1471–1473 (1992).
    [CrossRef]
  6. S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
    [CrossRef]
  7. H. Zhou, F. Zhao, F. S. T. Yu, “Effects of recording-erasure dynamics of storage capacity of a wavelength-multiplexed reflection-type photorefractive hologram,” Appl. Opt. 33, 4339–4344 (1994).
    [CrossRef] [PubMed]
  8. S. Campbell, X. Yi, P. Yeh, “Hybrid sparse-wavelength angularly multiplexed optical data storage system,” Opt. Lett. 19, 2161–2163 (1994).
    [CrossRef] [PubMed]
  9. S. Campbell, P. Yeh, “Sparse-wavelength angle-multiplexed volume holographic memory system: analysis and advances,” Appl. Opt. 35, 2380–2388 (1996).
    [CrossRef] [PubMed]
  10. S. Tao, Z. H. Song, D. R. Selviah, J. E. Midwinter, “Spatioangular-multiplexing scheme for dense holographic storage,” Appl. Opt. 34, 6729–6737 (1995).
    [CrossRef] [PubMed]
  11. F. H. Mok, “Angle-multiplexed storage of 5000 holograms in lithium niobate,” Opt. Lett. 18, 915–917 (1993).
    [CrossRef] [PubMed]
  12. D. Psaltis, M. Levene, A. Pu, G. Barbastathis, K. Curtis, “Holographic storage using shift multiplexing,” Opt. Lett. 20, 782–784 (1995).
    [CrossRef] [PubMed]
  13. G. Barbastathis, M. Levene, D. Psaltis, “Shift multiplexing with spherical reference waves,” Appl. Opt. 35, 2403–2417 (1996).
    [CrossRef] [PubMed]
  14. C. Denz, G. Pauliat, G. Roosen, T. Tschudi, “Volume holographic multiplexing using a deterministic phase encoding technique,” Opt. Commun. 85, 171–176 (1991).
    [CrossRef]
  15. C. Alves, G. Pauliat, G. Roosen, “Dynamic phase-encoding storage of 64 images in BaTiO3 photorefractive crystal,” Opt. Lett. 19, 1894–1896 (1994).
    [CrossRef]
  16. K. Curtis, D. Psaltis, “Cross talk for angle- and wavelength-multiplexed image plane holograms,” Opt. Lett. 19, 1774–1776 (1994).
    [CrossRef] [PubMed]
  17. K.-H. Tu, T. Tamir, H. Lee, “Multiple-scattering theory of wave diffraction by superposed volume gratings,” J. Opt. Soc. Am. A 7, 1421–1435 (1990).
    [CrossRef]
  18. A. Belendez, I. Pascual, A. F. Fimia, “Model for analyzing the effects of processing on recording material in thick holograms,” J. Opt. Soc. Am. A 9, 1214–1223 (1992).
    [CrossRef]
  19. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  20. V. Minier, J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
    [CrossRef]
  21. T. Kubota, M. Takeda, “Array illuminator using grating couplers,” Opt. Lett. 14, 651–652 (1989).
    [CrossRef] [PubMed]
  22. R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
    [CrossRef]
  23. A. Pu, D. Psaltis, “High-density recording in photopolymer-based holographic three-dimensional disk,” Appl. Opt. 35, 2389–2398 (1996).
    [CrossRef] [PubMed]
  24. E. N. Glytsis, T. K. Gaylord, “Anisotropic guided-wave diffraction by interdigitated electrode-induced phase gratings,” Appl. Opt. 27, 5031–5050 (1988).
    [CrossRef] [PubMed]
  25. D. G. Sun, C. Zhao, R. T. Chen, “Intraplane to interplane optical interconnects with a high diffraction efficiency electro-optic grating,” Appl. Opt. 36, 629–634 (1997).
    [CrossRef] [PubMed]
  26. M. C. Hutley, Diffraction Gratings (Academic, New York, 1982).
  27. A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

1997 (1)

1996 (3)

1995 (3)

1994 (5)

1993 (3)

F. H. Mok, “Angle-multiplexed storage of 5000 holograms in lithium niobate,” Opt. Lett. 18, 915–917 (1993).
[CrossRef] [PubMed]

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

V. Minier, J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
[CrossRef]

1992 (4)

R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
[CrossRef]

R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
[CrossRef]

G. A. Rakuljic, V. Leyva, A. Yariv, “Optical data storage by using orthogonal wavelength-multiplexed volume holograms,” Opt. Lett. 17, 1471–1473 (1992).
[CrossRef]

A. Belendez, I. Pascual, A. F. Fimia, “Model for analyzing the effects of processing on recording material in thick holograms,” J. Opt. Soc. Am. A 9, 1214–1223 (1992).
[CrossRef]

1991 (2)

C. Denz, G. Pauliat, G. Roosen, T. Tschudi, “Volume holographic multiplexing using a deterministic phase encoding technique,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

R. T. Chen, H. Lu, D. Robinson, T. Jannson, “Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing,” Appl. Phys. Lett. 59, 1144–1146 (1991).
[CrossRef]

1990 (1)

1989 (1)

1988 (1)

1983 (1)

Alves, C.

Arons, E.

Barbastathis, G.

Baumbick, R.

R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
[CrossRef]

Belendez, A.

Campbell, S.

Chen, R. T.

D. G. Sun, C. Zhao, R. T. Chen, “Intraplane to interplane optical interconnects with a high diffraction efficiency electro-optic grating,” Appl. Opt. 36, 629–634 (1997).
[CrossRef] [PubMed]

R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
[CrossRef]

R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
[CrossRef]

R. T. Chen, H. Lu, D. Robinson, T. Jannson, “Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing,” Appl. Phys. Lett. 59, 1144–1146 (1991).
[CrossRef]

Curtis, K.

Denz, C.

C. Denz, G. Pauliat, G. Roosen, T. Tschudi, “Volume holographic multiplexing using a deterministic phase encoding technique,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Dilworth, D.

Fimia, A. F.

Gaylord, T. K.

Glytsis, E. N.

Hutley, M. C.

M. C. Hutley, Diffraction Gratings (Academic, New York, 1982).

Jannson, T.

R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
[CrossRef]

R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
[CrossRef]

R. T. Chen, H. Lu, D. Robinson, T. Jannson, “Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing,” Appl. Phys. Lett. 59, 1144–1146 (1991).
[CrossRef]

Kubota, T.

Lee, H.

Levene, M.

Leyva, V.

Lu, H.

R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
[CrossRef]

R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
[CrossRef]

R. T. Chen, H. Lu, D. Robinson, T. Jannson, “Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing,” Appl. Phys. Lett. 59, 1144–1146 (1991).
[CrossRef]

Midwinter, J. E.

Minier, V.

V. Minier, J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
[CrossRef]

Moharam, M. G.

Mok, F. H.

Pascual, I.

Pauliat, G.

C. Alves, G. Pauliat, G. Roosen, “Dynamic phase-encoding storage of 64 images in BaTiO3 photorefractive crystal,” Opt. Lett. 19, 1894–1896 (1994).
[CrossRef]

C. Denz, G. Pauliat, G. Roosen, T. Tschudi, “Volume holographic multiplexing using a deterministic phase encoding technique,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Psaltis, D.

Pu, A.

Rakuljic, G. A.

Rizvi, A. A.

Robinson, D.

R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
[CrossRef]

R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
[CrossRef]

R. T. Chen, H. Lu, D. Robinson, T. Jannson, “Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing,” Appl. Phys. Lett. 59, 1144–1146 (1991).
[CrossRef]

Roosen, G.

C. Alves, G. Pauliat, G. Roosen, “Dynamic phase-encoding storage of 64 images in BaTiO3 photorefractive crystal,” Opt. Lett. 19, 1894–1896 (1994).
[CrossRef]

C. Denz, G. Pauliat, G. Roosen, T. Tschudi, “Volume holographic multiplexing using a deterministic phase encoding technique,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Savant, G.

R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
[CrossRef]

Selviah, D. R.

Song, Z. H.

Sun, D. G.

Takeda, M.

Tamir, T.

Tao, S.

Tschudi, T.

C. Denz, G. Pauliat, G. Roosen, T. Tschudi, “Volume holographic multiplexing using a deterministic phase encoding technique,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Tu, K.-H.

Wang, M.

R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
[CrossRef]

Wang, M. R.

R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
[CrossRef]

Wen, M.

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Xu, J. M.

V. Minier, J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
[CrossRef]

Yang, Z.

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Yariv, A.

Yeh, P.

Yi, X.

Yin, S.

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Yu, F. S. T

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Yu, F. S. T.

Zhang, J.

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Zhao, C.

Zhao, F.

H. Zhou, F. Zhao, F. S. T. Yu, “Effects of recording-erasure dynamics of storage capacity of a wavelength-multiplexed reflection-type photorefractive hologram,” Appl. Opt. 33, 4339–4344 (1994).
[CrossRef] [PubMed]

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Zhou, H.

H. Zhou, F. Zhao, F. S. T. Yu, “Effects of recording-erasure dynamics of storage capacity of a wavelength-multiplexed reflection-type photorefractive hologram,” Appl. Opt. 33, 4339–4344 (1994).
[CrossRef] [PubMed]

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

Zubairy, M. S.

Appl. Opt. (9)

A. A. Rizvi, M. S. Zubairy, “Implementation of associative memory using grating structures,” Appl. Opt. 33, 3642–3646 (1994).
[CrossRef] [PubMed]

E. Arons, D. Dilworth, “Analysis of Fourier synthesis holography for imaging through scattering materials,” Appl. Opt. 34, 1841–1847 (1995).
[CrossRef] [PubMed]

H. Zhou, F. Zhao, F. S. T. Yu, “Effects of recording-erasure dynamics of storage capacity of a wavelength-multiplexed reflection-type photorefractive hologram,” Appl. Opt. 33, 4339–4344 (1994).
[CrossRef] [PubMed]

S. Campbell, P. Yeh, “Sparse-wavelength angle-multiplexed volume holographic memory system: analysis and advances,” Appl. Opt. 35, 2380–2388 (1996).
[CrossRef] [PubMed]

S. Tao, Z. H. Song, D. R. Selviah, J. E. Midwinter, “Spatioangular-multiplexing scheme for dense holographic storage,” Appl. Opt. 34, 6729–6737 (1995).
[CrossRef] [PubMed]

G. Barbastathis, M. Levene, D. Psaltis, “Shift multiplexing with spherical reference waves,” Appl. Opt. 35, 2403–2417 (1996).
[CrossRef] [PubMed]

A. Pu, D. Psaltis, “High-density recording in photopolymer-based holographic three-dimensional disk,” Appl. Opt. 35, 2389–2398 (1996).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Anisotropic guided-wave diffraction by interdigitated electrode-induced phase gratings,” Appl. Opt. 27, 5031–5050 (1988).
[CrossRef] [PubMed]

D. G. Sun, C. Zhao, R. T. Chen, “Intraplane to interplane optical interconnects with a high diffraction efficiency electro-optic grating,” Appl. Opt. 36, 629–634 (1997).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

R. T. Chen, H. Lu, D. Robinson, T. Jannson, “Highly multiplexed graded-index polymer waveguide hologram for near-infrared eight-channel wavelength division demultiplexing,” Appl. Phys. Lett. 59, 1144–1146 (1991).
[CrossRef]

IEEE J. Lightwave Technol. (1)

R. T. Chen, H. Lu, D. Robinson, M. Wang, G. Savant, T. Jannson, “Guided-wave planar optical interconnects using highly multiplexed polymer waveguide holograms,” IEEE J. Lightwave Technol. 10, 888–897 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

S. Yin, H. Zhou, F. Zhao, M. Wen, Z. Yang, J. Zhang, F. S. T Yu, “Wavelength multiplexed holographic storage in a sensitive photorefractive crystal using a visible-light tunable diode laser,” Opt. Commun. 101, 317–321 (1993).
[CrossRef]

C. Denz, G. Pauliat, G. Roosen, T. Tschudi, “Volume holographic multiplexing using a deterministic phase encoding technique,” Opt. Commun. 85, 171–176 (1991).
[CrossRef]

Opt. Eng. (2)

V. Minier, J. M. Xu, “Coupled-mode analysis of superimposed phase grating guided-wave structures and integrating coupling effects,” Opt. Eng. 32, 2054–2063 (1993).
[CrossRef]

R. T. Chen, D. Robinson, H. Lu, M. R. Wang, T. Jannson, R. Baumbick, “Reconfigurable optical interconnection network for multimode optical fiber sensor arrays,” Opt. Eng. 31, 1098–1106 (1992).
[CrossRef]

Opt. Lett. (7)

Other (2)

M. C. Hutley, Diffraction Gratings (Academic, New York, 1982).

A. Yariv, P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984).

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Figures (9)

Fig. 1
Fig. 1

Diffraction schematic of modulated thin phase grating; a = 2 µm and b = 6 µm.

Fig. 2
Fig. 2

Distribution curve of index modulation as a function of x.

Fig. 3
Fig. 3

Distribution pattern of modulated effective optical path difference along z axis.

Fig. 4
Fig. 4

Fabricated LiNbO3 EO grating array.

Fig. 6
Fig. 6

Relationship between θm and λ of three different diffraction orders: 1, the first order; 2, the fifth order; 3, the tenth order.

Fig. 5
Fig. 5

Angular distribution of multiple diffraction orders coupled by EO modulated grating: (a) far-field pattern of multiple diffraction orders, (b) measured result, (c) theoretical result; θ m represents the output angle associated with the mth diffraction order.

Fig. 7
Fig. 7

Experimental setup for the evaluation of the EO thin phase grating array: (a) the setup, (b) modulation transfer curve represents 18 ± 2.5% (f = 60 Hz).

Fig. 8
Fig. 8

Proposed configuration for hologram recording: (a) schematic for hologram recording, (b) schematic for hologram reading, (c) picture of a real device.

Fig. 9
Fig. 9

Experiments at λ = 0.633 µm: (a) input image, (b) reconstructed holographic image with the second diffraction order as the reference beam, (c) reconstructed holographic image with the fifth diffraction order as the reference beam.

Equations (18)

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

Δneθ1=ne3 sin2 θ1Δno+no3 cos2 θ1Δneno2 cos2 θ1+ne2 sin2 θ13/2,
Δne=-12ne3γ33Ez,
Δno-12no3γ13Ez,
Δneffd=-de0 Δneθ1, xdx,
Δneffd=00<zaΔneffda<za+b0a+b<z2a+b-Δneffd2a+b<za+2b.
E1=ziΛ1Zi0ziexpikΔLizdz,
ΔLiz=rθmz+δ1k0<zarθmz+δ2ka<za+brθmz+δ1ka+b<z2a+brθmz-δ2k2a+b<z2a+2b,
rθm=ns sin θ1+n2 sin θmk,
δ1=kn1 cos θ1,
δ2=kΔneθ1d/cos θ1.
E1θm=1a0aexpirθmz+δ1dz+1baa+bexpirθmz+δ2dz+1aa+b2a+bexpirθmz+δ1dz+1b2a+b2a+2bexpirθmz+δ2dz=expiarθm-1expiδ1iarθm+expibrθm-1expiarθm+δ2ibrθm+expiarθm-1expia+brθm+δ1iarθm+expibrθm-1expi2a+brθm-δ2ibrθm=4arsina2rθmcosa+b2rθm×expi2a+b2rθm+δ1+4brsinb2rθmcosa+b2rθm-δ2×expi3a+2b2rθm.
I1θm=E1·E1*=2 sina2rθmcosa+b2rθma2rθm2+2 sinb2rθmcosa+b2rθm-δ2b2rθm2+2 cosa+b2rθm-δ1×2 sina2rθmcosa+b2rθma2rθm×2 sinb2rθmcosa+b2rθm-δ2b2rθm.
I2θm=sinNΛrθm2sinΛrθm22,
Iθm=mI1θm·I2θm.
Λ·rθm=2mπ.
θm=sin-11n2mλΛ-ns sin θ1.
ηm=PmnPtn-Pln,
D/2f=tan 30° or f=3/2D.

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