Abstract

We address the problem of shaping the radiant intensity distribution of a highly nonparaxial coherent field by means of a diffractive element located in the plane of the beam waist. To be capable of wide-angle energy redistribution the element must necessarily contain wavelength-scale transverse features, and consequently it must be designed on the basis of rigorous diffraction theory. We consider, in particular, wide-angle Gaussian to flat-top beam shaping in one dimension. Scalar designs are provided and their validity is evaluated by rigorous diffraction theory, which is also used for optimization deep inside the nonparaxial domain, where the scalar designs fail. Experimental verification is provided by means of electron-beam lithography.

© 1997 Optical Society of America

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References

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  1. P. W. Rhodes, D. L. Shealy, “Refractive optical systems for irradiance distribution of collimated radiation: their design and analysis,” Appl. Opt. 19, 3545–3553 (1980).
    [CrossRef] [PubMed]
  2. D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
    [CrossRef]
  3. N. C. Roberts, “Beam shaping by holographic filters,” Appl. Opt. 28, 31–32 (1989).
    [CrossRef] [PubMed]
  4. F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 31, 529–536 (1991).
    [CrossRef]
  5. C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam into a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
    [CrossRef]
  6. W.-H. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 469–471 (1981).
    [CrossRef]
  7. W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–356 (1982).
    [CrossRef] [PubMed]
  8. C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  9. M. T. Eismann, A. M. Tai, J. N. Cederquist, “Iterative design of a holographic beamformer,” Appl. Opt. 28, 2641–2650 (1989).
    [CrossRef] [PubMed]
  10. Y. Ohtsuka, A. Tanone, “Acousto-optic modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
    [CrossRef]
  11. E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
    [CrossRef]
  12. H. Aagedal, S. Teiwes, F. Wyrowski, “Consequense of illumination wave on optical function of non-periodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
    [CrossRef]
  13. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
    [CrossRef]
  14. J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
    [CrossRef]
  15. J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
    [CrossRef]
  16. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 5.
    [CrossRef]
  17. J. Huttunen, A. T. Friberg, J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
    [CrossRef]
  18. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  19. J. Turunen, “Diffraction theory of microrelief gratings,” in Microoptics, H. P. Herzig, ed. (Taylor & Francis, London, 1996).
  20. M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980), Sect. 5.5.
  21. M. Kuittinen, J. Turunen, “Exact-eigenmode model for index-modulated apertures,” J. Opt. Soc. Am. A 13, 2014–2020 (1996).
    [CrossRef]
  22. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
    [CrossRef]
  23. R. Bräuer, O. Bryngdahl, “Design strategy of diffractive elements with prescribed diffraction angles in non-paraxial region,” Opt. Commun. 115, 411–416 (1995).
    [CrossRef]
  24. R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980), Sect. 7.
  25. R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  26. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]

1996 (1)

1995 (2)

J. Huttunen, A. T. Friberg, J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

R. Bräuer, O. Bryngdahl, “Design strategy of diffractive elements with prescribed diffraction angles in non-paraxial region,” Opt. Commun. 115, 411–416 (1995).
[CrossRef]

1994 (2)

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequense of illumination wave on optical function of non-periodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

1993 (3)

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

1992 (1)

1991 (4)

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 31, 529–536 (1991).
[CrossRef]

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam into a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

1989 (2)

1983 (1)

1982 (2)

W. B. Veldkamp, C. J. Kastner, “Beam profile shaping for laser radars that use detector arrays,” Appl. Opt. 21, 345–356 (1982).
[CrossRef] [PubMed]

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

1981 (2)

W.-H. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

Y. Ohtsuka, A. Tanone, “Acousto-optic modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
[CrossRef]

1980 (1)

1978 (1)

Aagedal, H.

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequense of illumination wave on optical function of non-periodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam into a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Botten, L. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980), Sect. 7.

Bräuer, R.

R. Bräuer, O. Bryngdahl, “Design strategy of diffractive elements with prescribed diffraction angles in non-paraxial region,” Opt. Commun. 115, 411–416 (1995).
[CrossRef]

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bryngdahl, O.

R. Bräuer, O. Bryngdahl, “Design strategy of diffractive elements with prescribed diffraction angles in non-paraxial region,” Opt. Commun. 115, 411–416 (1995).
[CrossRef]

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Cederquist, J. N.

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980), Sect. 7.

Eismann, M. T.

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam into a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Friberg, A. T.

J. Huttunen, A. T. Friberg, J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Han, C.-Y.

Huttunen, J.

J. Huttunen, A. T. Friberg, J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Ichikawa, H.

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

Ishii, Y.

Kastner, C. J.

Knop, K.

Kuittinen, M.

Lee, W.-H.

W.-H. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 5.
[CrossRef]

McPhedran, R. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980), Sect. 7.

Miller, J. M.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

Murata, K.

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam into a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Neviere, M.

M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980), Sect. 5.5.

Noponen, E.

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

Ohtsuka, Y.

Y. Ohtsuka, A. Tanone, “Acousto-optic modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
[CrossRef]

Rhodes, P. W.

Roberts, N. C.

Ross, N.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Roux, F. S.

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 31, 529–536 (1991).
[CrossRef]

Shafer, D.

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Shealy, D. L.

Taghizadeh, M. R.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

Tai, A. M.

Tanone, A.

Y. Ohtsuka, A. Tanone, “Acousto-optic modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
[CrossRef]

Teiwes, S.

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequense of illumination wave on optical function of non-periodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

Tervonen, E.

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

Turunen, J.

M. Kuittinen, J. Turunen, “Exact-eigenmode model for index-modulated apertures,” J. Opt. Soc. Am. A 13, 2014–2020 (1996).
[CrossRef]

J. Huttunen, A. T. Friberg, J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

J. Turunen, “Diffraction theory of microrelief gratings,” in Microoptics, H. P. Herzig, ed. (Taylor & Francis, London, 1996).

Vasara, A.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

Veldkamp, W. B.

Westerholm, J.

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 5.
[CrossRef]

Wyrowski, F.

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequense of illumination wave on optical function of non-periodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

Appl. Opt. (5)

J. Mod. Opt. (2)

E. Tervonen, A. T. Friberg, J. Turunen, “Acousto-optic conversion of laser beams into flat-top beams,” J. Mod. Opt. 40, 625–635 (1993).
[CrossRef]

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (7)

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Y. Ohtsuka, A. Tanone, “Acousto-optic modification of a Gaussian laser beam,” Opt. Commun. 39, 70–74 (1981).
[CrossRef]

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequense of illumination wave on optical function of non-periodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Rigorous diffraction theory of Dammann gratings,” Opt. Commun. 81, 337–342 (1991).
[CrossRef]

J. Turunen, A. Vasara, H. Ichikawa, E. Noponen, J. Westerholm, M. R. Taghizadeh, J. M. Miller, “Storage of multiple images in a thin synthetic Fourier hologram,” Opt. Commun. 84, 383–392 (1991).
[CrossRef]

W.-H. Lee, “Method for converting a Gaussian laser beam into a uniform beam,” Opt. Commun. 36, 469–471 (1981).
[CrossRef]

R. Bräuer, O. Bryngdahl, “Design strategy of diffractive elements with prescribed diffraction angles in non-paraxial region,” Opt. Commun. 115, 411–416 (1995).
[CrossRef]

Opt. Eng. (2)

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 31, 529–536 (1991).
[CrossRef]

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam into a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Opt. Laser Technol. (1)

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14, 159–160 (1982).
[CrossRef]

Phys. Rev. E (1)

J. Huttunen, A. T. Friberg, J. Turunen, “Diffraction of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Other (4)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 5.
[CrossRef]

J. Turunen, “Diffraction theory of microrelief gratings,” in Microoptics, H. P. Herzig, ed. (Taylor & Francis, London, 1996).

M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980), Sect. 5.5.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer, Berlin, 1980), Sect. 7.

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Figures (8)

Fig. 1
Fig. 1

(a) Experimental geometry for wide-angle beam shaping: DE, diffractive element; PZT, piezoelectric transducer. (b) Geometry of surface profile h(x).

Fig. 2
Fig. 2

Scalar designs and dashed curves denote a binary solution with a single groove in a period and a multilevel solution, respectively. Vertical lines denote the signal window. (a) Radiant intensity distributions. (b) Cumulative efficiency curves.

Fig. 3
Fig. 3

Analysis of the binary scalar design: profile quality as a function of the angular full width Ω = 2θ1 of the signal window. The solid, dashed, and dotted curves are for Ω = 47.2°, Ω = 23.0°, and Ω = 15.4°, respectively.

Fig. 4
Fig. 4

Same as Fig. 3, but for the multilevel scalar design.

Fig. 5
Fig. 5

Same as Fig. 2, but rigorous electromagnetic designs obtained by refining the scalar designs are considered.

Fig. 6
Fig. 6

Error analysis for the single-groove binary design, showing the effect of (a) error in beam width w, which decreases in steps of 0.05λ from the top to bottom curve, (b) lateral positioning error (step size 0.02λ), (c) error in profile depth h (step size 0.02λ).

Fig. 7
Fig. 7

Same as Fig. 6, but for the multilevel design.

Fig. 8
Fig. 8

Experimental result. The solid curve denotes the radiant intensity distribution for a single-groove solution illuminated by a Gaussian beam; the dashed curve denotes the theoretical distribution for the same structure.

Equations (10)

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

Eyx=C exp-x-x¯2/w2,
Jθ=Jin0cos2 θTk sin θ2  Jin0Tkθ2,
Tα=m=-TmE˜yα-2πm/d,
E˜yα=-Eyxexp-iαxdx
Tm=1d0dtxexp-i2πmx/ddx
tx=expik0hnzdz,
E˜yα=Cπw exp-wα/22
w2 ln 2πd.
αM/d,
ηθ=-θθ Jθdθ-π/2π/2Jinθdθ

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