Abstract

To optimize the functions of the surface-relief diffractive optical elements for optical limiters and other applications, we present a systematic design and analysis with numerical illustrations of the transmission properties of different surface-relief phase gratings and Fabry–Perot elements. The spectral response and the tolerance of fabrication errors have been included. An analysis shows that the blazed grating and the echelon grating, a multilevel element as a substitute of the blazed grating, can make the on-axis transmittance very low (less than 1% with one grating) over the broad visible band with a large tolerance of fabrication errors. The results are highly significant for the design of optical-limiting devices.

© 1999 Optical Society of America

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References

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  1. H. K. Liu, C. Li, R. Wang, “Adaptive grating optical limiting devices,” U.S. patent application 98,244 (16June1998).
  2. Y. Z. Liang, D. Z. Zhao, H. K. Liu, “Multifocus dichromated gelatin hololens,” Appl. Opt. 22, 3451–3459 (1983).
    [CrossRef] [PubMed]
  3. H. K. Liu, S. M. Zhou, “Reconfigurable optical interconnections via dynamic computer-generated holograms,” U.S. patent5,539,543 (23July1996).
  4. H. K. Liu, Y. Jin, N. I. Marzwell, “Advanced ultra-high-capacity optical random access memory and pattern recognition techniques,” Opt. Eng. 37, 779–788 (1998).
    [CrossRef]
  5. N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
    [CrossRef]
  6. P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode array by continuous surface-relief elements,” J. Mod. Opt. 40, 723–732 (1993).
    [CrossRef]
  7. J. Saarinen, E. Noponen, J. Turunen, T. Suharan, H. Nishihara, “Asymmetric beam deflection by doubly grooved binary gratings,” Appl. Opt. 33, 2401–2405 (1995).
    [CrossRef]
  8. J. Cederquist, A. M. Tai, “Computer-generated hologram for geometric transformation,” Appl. Opt. 23, 3099–3104 (1984).
    [CrossRef]
  9. Y. Ishii, T. Kubota, “Wavelength demultiplexer in multimode fiber that uses optimized holographic optical elements,” Appl. Opt. 32, 4415–4422 (1993).
    [CrossRef] [PubMed]
  10. K. S. Urguhart, P. Marchand, Y. Fainman, S. H. Lee, “Diffractive optics applied to free-space optical interconnects,” Appl. Opt. 33, 3670–3682 (1994).
    [CrossRef]
  11. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Arm. 72, 1385–1392 (1982).
    [CrossRef]
  12. A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
    [CrossRef] [PubMed]
  13. E. K. Popov, L. V. Tsonev, E. G. Loen, “Scalar theory of transmission relief grating,” Opt. Commun. 80, 307–311 (1991).
    [CrossRef]
  14. K. Yokomori, “Dielectric surface-relief gratings with high diffraction efficiency,” Appl. Opt. 23, 2303–2310 (1984).
    [CrossRef] [PubMed]
  15. S. K. Case, “Properties of optical elements with ultra-high spatial frequency surface corrugations,” in Applications of Holography, L. Huff, ed., Proc. SPIE523, 269–276 (1985).
    [CrossRef]
  16. S. J. Walker, J. Jahus, L. Li, W. M. Manfield, P. Mulgrew, D. M. Tennant, C. W. Roberts, L. C. West, N. K. Ailawadi, “Design and fabrication of high-efficiency beam splitters and beam defectors for integrated planar micro-optic system,” Appl. Opt. 32, 2494–2501 (1993).
    [CrossRef] [PubMed]
  17. Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wavefront shaping,” Opt. Eng. 35, 2556–2565 (1996).
    [CrossRef]
  18. M. Shiek-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
    [CrossRef]
  19. D. J. Harter, M. L. Shand, Y. B. Band, “Power/energy limiter using reverse saturable absorption,” J. Appl. Phys. 56, 865–868 (1984).
    [CrossRef]
  20. K. M. Nashold, D. P. Walter, “Investigations of optical limiting mechanisms in carbon particle suspensions and fullerence solutions,” J. Opt. Soc. Am. B 12, 1228–1237 (1995).
    [CrossRef]
  21. C. Li, G. Fang, L. Cai, H. K. Liu, “Optical gratings with wavelength and subwavelength structures for optical limiting,” in Far- and Near-Field Optics: Physics and Information Processing. S. Jutamulia, T. Asakura, eds., Proc. SPIE3467, 186–198 (1998).
    [CrossRef]
  22. L. Cai, C. Li, H. K. Liu, “Optical elements for elimination of on-axis visible transmission,” presented at the SPIE Topical Meeting on Diffractive Optics and Micro-Optics, Kailua-Kona, Haw., 8–11 June 1998.
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  24. L. Z. Cai, C. Y. Wang, Optics (Shandong University Press, Jinan, China, 1992).
  25. T. Fujuta, H. Mishihara, J. Koyama, “Fabrication of microlenses using electron beam lithography,” Opt. Lett. 6, 613–615 (1981).
    [CrossRef]
  26. M. Ekberg, F. Nikolajeff, M. Larsson, S. Hard, “Proximity-compensated blazed transmission grating manufacture with direct-writing electron-beam lithography,” Appl. Opt. 33, 103–107 (1994).
    [CrossRef] [PubMed]
  27. P. Blair, M. R. Taghizadeh, W. Parker, C. D. W. Wilkinson, “High-efficiency binary fanout grating by modulation of a high-frequency carrier grating,” Appl. Opt. 34, 2406–2413 (1995).
    [CrossRef] [PubMed]

1998 (1)

H. K. Liu, Y. Jin, N. I. Marzwell, “Advanced ultra-high-capacity optical random access memory and pattern recognition techniques,” Opt. Eng. 37, 779–788 (1998).
[CrossRef]

1996 (1)

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wavefront shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

1995 (3)

1994 (2)

1993 (3)

1992 (1)

1991 (2)

E. K. Popov, L. V. Tsonev, E. G. Loen, “Scalar theory of transmission relief grating,” Opt. Commun. 80, 307–311 (1991).
[CrossRef]

M. Shiek-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

1989 (1)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

1984 (3)

1983 (1)

1982 (1)

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Arm. 72, 1385–1392 (1982).
[CrossRef]

1981 (1)

Ailawadi, N. K.

Band, Y. B.

D. J. Harter, M. L. Shand, Y. B. Band, “Power/energy limiter using reverse saturable absorption,” J. Appl. Phys. 56, 865–868 (1984).
[CrossRef]

Blair, P.

Cai, L.

C. Li, G. Fang, L. Cai, H. K. Liu, “Optical gratings with wavelength and subwavelength structures for optical limiting,” in Far- and Near-Field Optics: Physics and Information Processing. S. Jutamulia, T. Asakura, eds., Proc. SPIE3467, 186–198 (1998).
[CrossRef]

L. Cai, C. Li, H. K. Liu, “Optical elements for elimination of on-axis visible transmission,” presented at the SPIE Topical Meeting on Diffractive Optics and Micro-Optics, Kailua-Kona, Haw., 8–11 June 1998.

Cai, L. Z.

L. Z. Cai, C. Y. Wang, Optics (Shandong University Press, Jinan, China, 1992).

Case, S. K.

S. K. Case, “Properties of optical elements with ultra-high spatial frequency surface corrugations,” in Applications of Holography, L. Huff, ed., Proc. SPIE523, 269–276 (1985).
[CrossRef]

Cederquist, J.

Dandliker, R.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode array by continuous surface-relief elements,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Danziger, Y.

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wavefront shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

Ehbets, P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode array by continuous surface-relief elements,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Ekberg, M.

Fainman, Y.

Fang, G.

C. Li, G. Fang, L. Cai, H. K. Liu, “Optical gratings with wavelength and subwavelength structures for optical limiting,” in Far- and Near-Field Optics: Physics and Information Processing. S. Jutamulia, T. Asakura, eds., Proc. SPIE3467, 186–198 (1998).
[CrossRef]

Friesem, A. A.

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wavefront shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

Fujuta, T.

Gaylord, T. K.

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Arm. 72, 1385–1392 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hagan, D. J.

M. Shiek-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Hard, S.

Harter, D. J.

D. J. Harter, M. L. Shand, Y. B. Band, “Power/energy limiter using reverse saturable absorption,” J. Appl. Phys. 56, 865–868 (1984).
[CrossRef]

Hasman, E.

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wavefront shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

Herzig, H. P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode array by continuous surface-relief elements,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Ichikawa, H.

Ishii, Y.

Jaakkola, T.

Jahus, J.

Jin, Y.

H. K. Liu, Y. Jin, N. I. Marzwell, “Advanced ultra-high-capacity optical random access memory and pattern recognition techniques,” Opt. Eng. 37, 779–788 (1998).
[CrossRef]

Kjelberg, I.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode array by continuous surface-relief elements,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Koyama, J.

Kubota, T.

Kuisma, S.

Larsson, M.

Lee, S. H.

Li, C.

H. K. Liu, C. Li, R. Wang, “Adaptive grating optical limiting devices,” U.S. patent application 98,244 (16June1998).

L. Cai, C. Li, H. K. Liu, “Optical elements for elimination of on-axis visible transmission,” presented at the SPIE Topical Meeting on Diffractive Optics and Micro-Optics, Kailua-Kona, Haw., 8–11 June 1998.

C. Li, G. Fang, L. Cai, H. K. Liu, “Optical gratings with wavelength and subwavelength structures for optical limiting,” in Far- and Near-Field Optics: Physics and Information Processing. S. Jutamulia, T. Asakura, eds., Proc. SPIE3467, 186–198 (1998).
[CrossRef]

Li, L.

Liang, Y. Z.

Liu, H. K.

H. K. Liu, Y. Jin, N. I. Marzwell, “Advanced ultra-high-capacity optical random access memory and pattern recognition techniques,” Opt. Eng. 37, 779–788 (1998).
[CrossRef]

Y. Z. Liang, D. Z. Zhao, H. K. Liu, “Multifocus dichromated gelatin hololens,” Appl. Opt. 22, 3451–3459 (1983).
[CrossRef] [PubMed]

H. K. Liu, S. M. Zhou, “Reconfigurable optical interconnections via dynamic computer-generated holograms,” U.S. patent5,539,543 (23July1996).

H. K. Liu, C. Li, R. Wang, “Adaptive grating optical limiting devices,” U.S. patent application 98,244 (16June1998).

L. Cai, C. Li, H. K. Liu, “Optical elements for elimination of on-axis visible transmission,” presented at the SPIE Topical Meeting on Diffractive Optics and Micro-Optics, Kailua-Kona, Haw., 8–11 June 1998.

C. Li, G. Fang, L. Cai, H. K. Liu, “Optical gratings with wavelength and subwavelength structures for optical limiting,” in Far- and Near-Field Optics: Physics and Information Processing. S. Jutamulia, T. Asakura, eds., Proc. SPIE3467, 186–198 (1998).
[CrossRef]

Loen, E. G.

E. K. Popov, L. V. Tsonev, E. G. Loen, “Scalar theory of transmission relief grating,” Opt. Commun. 80, 307–311 (1991).
[CrossRef]

Lohmann, A. W.

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wavefront shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

Manfield, W. M.

Marchand, P.

Marzwell, N. I.

H. K. Liu, Y. Jin, N. I. Marzwell, “Advanced ultra-high-capacity optical random access memory and pattern recognition techniques,” Opt. Eng. 37, 779–788 (1998).
[CrossRef]

Miller, J. M.

Mishihara, H.

Moharam, M. G.

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Arm. 72, 1385–1392 (1982).
[CrossRef]

Mulgrew, P.

Nashold, K. M.

Nikolajeff, F.

Nishihara, H.

J. Saarinen, E. Noponen, J. Turunen, T. Suharan, H. Nishihara, “Asymmetric beam deflection by doubly grooved binary gratings,” Appl. Opt. 33, 2401–2405 (1995).
[CrossRef]

Noponen, E.

Parker, W.

Popov, E. K.

E. K. Popov, L. V. Tsonev, E. G. Loen, “Scalar theory of transmission relief grating,” Opt. Commun. 80, 307–311 (1991).
[CrossRef]

Regnaul, P.

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode array by continuous surface-relief elements,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Roberts, C. W.

Saarinen, J.

J. Saarinen, E. Noponen, J. Turunen, T. Suharan, H. Nishihara, “Asymmetric beam deflection by doubly grooved binary gratings,” Appl. Opt. 33, 2401–2405 (1995).
[CrossRef]

Said, A. A.

M. Shiek-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Shand, M. L.

D. J. Harter, M. L. Shand, Y. B. Band, “Power/energy limiter using reverse saturable absorption,” J. Appl. Phys. 56, 865–868 (1984).
[CrossRef]

Shiek-Bahae, M.

M. Shiek-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Soileau, M. J.

M. Shiek-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Streibl, N.

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

Suharan, T.

J. Saarinen, E. Noponen, J. Turunen, T. Suharan, H. Nishihara, “Asymmetric beam deflection by doubly grooved binary gratings,” Appl. Opt. 33, 2401–2405 (1995).
[CrossRef]

Taghizadeh, M. R.

Tai, A. M.

Tennant, D. M.

Tsonev, L. V.

E. K. Popov, L. V. Tsonev, E. G. Loen, “Scalar theory of transmission relief grating,” Opt. Commun. 80, 307–311 (1991).
[CrossRef]

Turunen, J.

Urguhart, K. S.

Van Stryland, E. W.

M. Shiek-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

Vasara, A.

Walker, S. J.

Walter, D. P.

Wang, C. Y.

L. Z. Cai, C. Y. Wang, Optics (Shandong University Press, Jinan, China, 1992).

Wang, R.

H. K. Liu, C. Li, R. Wang, “Adaptive grating optical limiting devices,” U.S. patent application 98,244 (16June1998).

West, L. C.

Westerholm, J.

Wilkinson, C. D. W.

Yokomori, K.

Zhao, D. Z.

Zhou, S. M.

H. K. Liu, S. M. Zhou, “Reconfigurable optical interconnections via dynamic computer-generated holograms,” U.S. patent5,539,543 (23July1996).

Appl. Opt. (10)

Y. Z. Liang, D. Z. Zhao, H. K. Liu, “Multifocus dichromated gelatin hololens,” Appl. Opt. 22, 3451–3459 (1983).
[CrossRef] [PubMed]

J. Saarinen, E. Noponen, J. Turunen, T. Suharan, H. Nishihara, “Asymmetric beam deflection by doubly grooved binary gratings,” Appl. Opt. 33, 2401–2405 (1995).
[CrossRef]

J. Cederquist, A. M. Tai, “Computer-generated hologram for geometric transformation,” Appl. Opt. 23, 3099–3104 (1984).
[CrossRef]

Y. Ishii, T. Kubota, “Wavelength demultiplexer in multimode fiber that uses optimized holographic optical elements,” Appl. Opt. 32, 4415–4422 (1993).
[CrossRef] [PubMed]

K. S. Urguhart, P. Marchand, Y. Fainman, S. H. Lee, “Diffractive optics applied to free-space optical interconnects,” Appl. Opt. 33, 3670–3682 (1994).
[CrossRef]

A. Vasara, M. R. Taghizadeh, J. Turunen, J. Westerholm, E. Noponen, H. Ichikawa, J. M. Miller, T. Jaakkola, S. Kuisma, “Binary surface-relief gratings for array illumination in digital optics,” Appl. Opt. 31, 3320–3336 (1992).
[CrossRef] [PubMed]

K. Yokomori, “Dielectric surface-relief gratings with high diffraction efficiency,” Appl. Opt. 23, 2303–2310 (1984).
[CrossRef] [PubMed]

S. J. Walker, J. Jahus, L. Li, W. M. Manfield, P. Mulgrew, D. M. Tennant, C. W. Roberts, L. C. West, N. K. Ailawadi, “Design and fabrication of high-efficiency beam splitters and beam defectors for integrated planar micro-optic system,” Appl. Opt. 32, 2494–2501 (1993).
[CrossRef] [PubMed]

M. Ekberg, F. Nikolajeff, M. Larsson, S. Hard, “Proximity-compensated blazed transmission grating manufacture with direct-writing electron-beam lithography,” Appl. Opt. 33, 103–107 (1994).
[CrossRef] [PubMed]

P. Blair, M. R. Taghizadeh, W. Parker, C. D. W. Wilkinson, “High-efficiency binary fanout grating by modulation of a high-frequency carrier grating,” Appl. Opt. 34, 2406–2413 (1995).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

D. J. Harter, M. L. Shand, Y. B. Band, “Power/energy limiter using reverse saturable absorption,” J. Appl. Phys. 56, 865–868 (1984).
[CrossRef]

J. Mod. Opt. (2)

N. Streibl, “Beam shaping with optical array generators,” J. Mod. Opt. 36, 1559–1573 (1989).
[CrossRef]

P. Ehbets, H. P. Herzig, R. Dandliker, P. Regnaul, I. Kjelberg, “Beam shaping of high-power laser diode array by continuous surface-relief elements,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

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

J. Opt. Soc. Arm. (1)

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Arm. 72, 1385–1392 (1982).
[CrossRef]

Opt. Commun. (1)

E. K. Popov, L. V. Tsonev, E. G. Loen, “Scalar theory of transmission relief grating,” Opt. Commun. 80, 307–311 (1991).
[CrossRef]

Opt. Eng. (3)

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wavefront shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

M. Shiek-Bahae, A. A. Said, D. J. Hagan, M. J. Soileau, E. W. Van Stryland, “Nonlinear refraction and optical limiting in thick media,” Opt. Eng. 30, 1228–1235 (1991).
[CrossRef]

H. K. Liu, Y. Jin, N. I. Marzwell, “Advanced ultra-high-capacity optical random access memory and pattern recognition techniques,” Opt. Eng. 37, 779–788 (1998).
[CrossRef]

Opt. Lett. (1)

Other (7)

C. Li, G. Fang, L. Cai, H. K. Liu, “Optical gratings with wavelength and subwavelength structures for optical limiting,” in Far- and Near-Field Optics: Physics and Information Processing. S. Jutamulia, T. Asakura, eds., Proc. SPIE3467, 186–198 (1998).
[CrossRef]

L. Cai, C. Li, H. K. Liu, “Optical elements for elimination of on-axis visible transmission,” presented at the SPIE Topical Meeting on Diffractive Optics and Micro-Optics, Kailua-Kona, Haw., 8–11 June 1998.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

L. Z. Cai, C. Y. Wang, Optics (Shandong University Press, Jinan, China, 1992).

H. K. Liu, C. Li, R. Wang, “Adaptive grating optical limiting devices,” U.S. patent application 98,244 (16June1998).

H. K. Liu, S. M. Zhou, “Reconfigurable optical interconnections via dynamic computer-generated holograms,” U.S. patent5,539,543 (23July1996).

S. K. Case, “Properties of optical elements with ultra-high spatial frequency surface corrugations,” in Applications of Holography, L. Huff, ed., Proc. SPIE523, 269–276 (1985).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Geometric configuration of diffraction phase element: (a) rectangular grating; (b), (c) blazed gratings; (d) sinusoidal grating; (e) Fabry–Perot element.

Fig. 2
Fig. 2

Spectral OAET of the rectangular grating with different modulation parameter m at the extinction wavelength, λ0 = 532 nm.

Fig. 3
Fig. 3

Spectral dependence of the OAET of the blazed grating.

Fig. 4
Fig. 4

Spectral dependence of the OAET of the sinusoidal grating.

Fig. 5
Fig. 5

Spectral dependence of the OAET of the F–P element with R = 0.8.

Fig. 6
Fig. 6

OAET of the rectangular grating versus wavelength λ and depth error δh: (a) m = 0, (b) m = 1, (c) m = 2.

Fig. 7
Fig. 7

OAET of the blazed grating versus λ and δh: (a) m = 3, (b) m = 4, (c) m = 5, (d) m = 10.

Fig. 8
Fig. 8

OAET of the sinusoidal grating versus λ and δh: (a) m = 1, (b) m = 5, (c) m = 10, (d) m = 20.

Fig. 9
Fig. 9

OAET of the F–P with R = 0.8 versus λ and δh: (a) m = 0, (b) m = 1, (c) m = 2.

Fig. 10
Fig. 10

Profile of an echelon grating as a multilevel diffraction element.

Fig. 11
Fig. 11

Dependence of the OAET of an echelon grating on λ and δh: (a) m = 2, M = 4; (b) m = 4, M = 8; (c) m = 8, M = 16; (d) m = 16, M = 32.

Fig. 12
Fig. 12

Diffraction of an echelon grating at a paraxial region over the visible band with a = 1.064 µm, m = 8, M = 16, L = 3.40 mm.

Fig. 13
Fig. 13

Angular distribution of OAET of an echelon grating with the same parameters as those in Fig. 12, assuming monochromatic incident light with λ0 = 532 nm.

Tables (1)

Tables Icon

Table 1 Zeros of the Function J0(u)

Equations (38)

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

tx=rectx-a2a+exp-i 2πλn-n0h×rectx-a-b2b * n=0N-1 δx-nd,
Iθ=|FTtx|2=a2 sin c2a sin θλ+b2×sin c2b sin θλ+2ab sin ca sin θλ×sin cb sin θλcosπλ2n-n0h+b sin θ* sin2Nπd sin θλsin2πd sin θλ,
I0=N2a2+b2+2ab cos2πλn-n0h.
a=b,
n-n0h=m+12λ, m=0, 1, 2,,
I0=2N2a21+cos2πλn-n0h.
T=I04N2a2=121+cos2πλn-n0h.
Tλ=121+cos2m+1π λ0λ, m=0, 1, 2.
Tλ, δh=121+cosπλ2m+1λ0+2n-n0δh.
tx=rectx-d2dexpi 2πhλdn-n0x* n=0N-1 δx-n+12d.
Iθ=d2 sinc2d sin θ-n-n0hλ×sin2Nπ dλsin θsin2π dλsin θ.
I0=N2d2 sinc2n-n0hλ,
T=I0N2d2=sinc2n-n0hλ.
n-n0h=mλ,  m=1, 2, 3.
Tλ=sinc2mλ0λ,  m=1, 2, 3,,
Tλ, δh=sinc2mλ0+n-n0δhλ.
hx=h0+h cos2π xd,
tx=expiu cos2π xdrectxNd,
u=2πλn-n0h,
expiu sin υ=- Jnuexpinυ
T=|J0u|2,
2πλ0n-n0h=Um,  m=1, 2, 3,.
Tλ=J0λ0λ Um2,
Tλ, δh=J0λ0λ Um+2πλn-n0δh2.
T=11+4R1-R2sin2δ2,
δ=4πnhλ
δ=4πnhλ=2m+1π,  m=0, 1, 2,,
Tmin=1-R21+R2.
Tλ=11+4R1-R2sin2m+12π λ0λ
Tλ, δh=11+4R1-R2sin2m+12π λ0λ+2πnδhλ
Tθ=1MNsin2πaλsin θsinMπλ ΔsinNπλ Δπaλsin θ sinπλ Δsinπλ Δ2,
Δ=n-n0 cos θh+n0a sin θ
Δ=n0d sin θ
T=sinM πλn-n0hM sinπλn-n0h2.
h=mλMn-n0,  m=1, 2, 3,.
Tλ, δh=sinmλ0+Mn-n0δhπλM sinmM λ0+n-n0δhπλ2.
Tλ, θ=sinπaλsin θMN πλ a sin θsinM πλn-n0 cos θMn-n0 mλ0+n0a sin θsinπλn-n0 cos θMn-n0 mλ0+n0a sin θ×sinMN πλ a sin θsinM πλ a sin θ2.
Tλ, θ=sin2π λ0λsin θsinMπ λ0λmn-n0 cos θMn-n0+2n0 sin θsin2NMπ λ0λsin θMN2π λ0λsin θ sinπ λ0λmn-n0 cos θMn-n0+2n0 sin θsin2Mπ λ0λsin θ2.

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