Abstract

A method is developed for the quantitative assessment of microlens deficiencies caused by inaccuracies in the multistep mesa fabrication or by incomplete mass-transport smoothing. Analytical expressions are obtained and tolerable imperfections are deduced for the practical effort-saving fabrication of high-performance micro-optical elements.

© 1994 Optical Society of America

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References

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  1. M. Oikawa, K. Iga, “Distributed-index planar microlens,” Appl. Opt. 21, 1052–1056 (1982).
    [Crossref] [PubMed]
  2. O. Wada, “Ion-beam etching of InP and its application to the fabrication of high-radiance InGaAsP/InP light-emitting diodes,” J. Electrochem. Soc. 131, 2373–2380 (1984).
    [Crossref]
  3. N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” Appl. Opt. 24, 2520–2525 (1985).
    [Crossref] [PubMed]
  4. Z. D. Popovic, R. A. Sprague, G. A. N. Connell, “Techniques for monolithic fabrication of microlens arrays,” Appl. Opt. 27, 1281–1284 (1988).
    [Crossref] [PubMed]
  5. T. Tatsumi, T. Saheki, T. Takei, K. Nukui, “High-performance micro-Fresnel lens fabricated by UV lithography,” Appl. Opt. 23, 1742–1744 (1984).
    [Crossref] [PubMed]
  6. T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
    [Crossref] [PubMed]
  7. J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
    [Crossref]
  8. H. Hosokawa, T. Yamashita, “ZnS micro-Fresnel lens and its uses,” Appl. Opt. 29, 5106–5110 (1990).
    [Crossref] [PubMed]
  9. B. Stegmüller, H. Westermeier, W. Thulke, G. Franz, D. Sacher, “Surface-emitting InGaAsP/InP distributed feedback laser diode at 1.53 μm with monolithic integrated microlens,” IEEE Photon. Technol. Lett. 3, 776–778 (1991).
    [Crossref]
  10. H. W. Lau, N. Davies, M. McCormick, “Microlens array fabricated in surface relief with high numerical aperture,” in Miniature and Micro-Optics: Fabrication and System Applications, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1544, 178–188 (1991).
  11. Z. L. Liau, V. Diadiuk, J. N. Walpole, D. E. Mull, “Large-numerical-aperture InP lenslets by mass transport,” Appl. Phys. Lett. 52, 1859–1861 (1988); “Gallium-phosphide microlens by mass transport,” 55, Appl. Phys. Lett.97–99 (1989).
    [Crossref] [PubMed]
  12. V. Diadiuk, Z. L. Liau, J. N. Walpole, J. W. Caunt, R. C. Williamson, “External-cavity coherent operation of InGaAsP buried-heterostructure laser array,” Appl. Phys. Lett. 55, 2161–2163 (1989).
    [Crossref]
  13. Z. L. Liau, J. N. Walpole, L. J. Missaggia, D. E. Mull, “GaInAsP/InP buried-heterostructure surface-emitting diode laser with monolithic integrated bifocal microlens,” Appl. Phys. Lett. 56, 1219–1221 (1990).
    [Crossref]
  14. Ar+ ion milling has been used to etch the multilevel mesa structure, and crystallographic orientational effects of wet chemical etching (observed in Refs. 11–13) have been eliminated.
  15. W. W. Mullins, “Flattening of a nearly plane solid surface due to capillarity,” J. Appl. Phys. 30, 77–83 (1959).
    [Crossref]
  16. P. S. Maiya, J. M. Blakely, “Surface self-diffusion and surface energy of nickel,” J. Appl. Phys. 38, 698–704 (1967).
    [Crossref]
  17. H. Nagai, Y. Noguchi, T. Matsuoka, “Thermal deformation of surface corrugations on InGaAsP crystals,” J. Cryst. Growth 71, 225–231 (1985).
    [Crossref]
  18. Z. L. Liau, H. J. Zeiger, “Surface-energy-induced mass-transport phenomenon in annealing of etched compound semiconductor structures: theoretical modeling and experimental confirmation,” J. Appl. Phys. 67, 2434–2440 (1990).
    [Crossref]
  19. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 463.
  20. H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, New York, 1961).

1991 (1)

B. Stegmüller, H. Westermeier, W. Thulke, G. Franz, D. Sacher, “Surface-emitting InGaAsP/InP distributed feedback laser diode at 1.53 μm with monolithic integrated microlens,” IEEE Photon. Technol. Lett. 3, 776–778 (1991).
[Crossref]

1990 (3)

Z. L. Liau, J. N. Walpole, L. J. Missaggia, D. E. Mull, “GaInAsP/InP buried-heterostructure surface-emitting diode laser with monolithic integrated bifocal microlens,” Appl. Phys. Lett. 56, 1219–1221 (1990).
[Crossref]

Z. L. Liau, H. J. Zeiger, “Surface-energy-induced mass-transport phenomenon in annealing of etched compound semiconductor structures: theoretical modeling and experimental confirmation,” J. Appl. Phys. 67, 2434–2440 (1990).
[Crossref]

H. Hosokawa, T. Yamashita, “ZnS micro-Fresnel lens and its uses,” Appl. Opt. 29, 5106–5110 (1990).
[Crossref] [PubMed]

1989 (1)

V. Diadiuk, Z. L. Liau, J. N. Walpole, J. W. Caunt, R. C. Williamson, “External-cavity coherent operation of InGaAsP buried-heterostructure laser array,” Appl. Phys. Lett. 55, 2161–2163 (1989).
[Crossref]

1988 (3)

Z. L. Liau, V. Diadiuk, J. N. Walpole, D. E. Mull, “Large-numerical-aperture InP lenslets by mass transport,” Appl. Phys. Lett. 52, 1859–1861 (1988); “Gallium-phosphide microlens by mass transport,” 55, Appl. Phys. Lett.97–99 (1989).
[Crossref] [PubMed]

Z. D. Popovic, R. A. Sprague, G. A. N. Connell, “Techniques for monolithic fabrication of microlens arrays,” Appl. Opt. 27, 1281–1284 (1988).
[Crossref] [PubMed]

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

1987 (1)

T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
[Crossref] [PubMed]

1985 (2)

H. Nagai, Y. Noguchi, T. Matsuoka, “Thermal deformation of surface corrugations on InGaAsP crystals,” J. Cryst. Growth 71, 225–231 (1985).
[Crossref]

N. F. Borrelli, D. L. Morse, R. H. Bellman, W. L. Morgan, “Photolytic technique for producing microlenses in photosensitive glass,” Appl. Opt. 24, 2520–2525 (1985).
[Crossref] [PubMed]

1984 (2)

T. Tatsumi, T. Saheki, T. Takei, K. Nukui, “High-performance micro-Fresnel lens fabricated by UV lithography,” Appl. Opt. 23, 1742–1744 (1984).
[Crossref] [PubMed]

O. Wada, “Ion-beam etching of InP and its application to the fabrication of high-radiance InGaAsP/InP light-emitting diodes,” J. Electrochem. Soc. 131, 2373–2380 (1984).
[Crossref]

1982 (1)

1967 (1)

P. S. Maiya, J. M. Blakely, “Surface self-diffusion and surface energy of nickel,” J. Appl. Phys. 38, 698–704 (1967).
[Crossref]

1959 (1)

W. W. Mullins, “Flattening of a nearly plane solid surface due to capillarity,” J. Appl. Phys. 30, 77–83 (1959).
[Crossref]

Bellman, R. H.

Blakely, J. M.

P. S. Maiya, J. M. Blakely, “Surface self-diffusion and surface energy of nickel,” J. Appl. Phys. 38, 698–704 (1967).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 463.

Borrelli, N. F.

Caunt, J. W.

V. Diadiuk, Z. L. Liau, J. N. Walpole, J. W. Caunt, R. C. Williamson, “External-cavity coherent operation of InGaAsP buried-heterostructure laser array,” Appl. Phys. Lett. 55, 2161–2163 (1989).
[Crossref]

Connell, G. A. N.

Davies, N.

H. W. Lau, N. Davies, M. McCormick, “Microlens array fabricated in surface relief with high numerical aperture,” in Miniature and Micro-Optics: Fabrication and System Applications, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1544, 178–188 (1991).

Diadiuk, V.

V. Diadiuk, Z. L. Liau, J. N. Walpole, J. W. Caunt, R. C. Williamson, “External-cavity coherent operation of InGaAsP buried-heterostructure laser array,” Appl. Phys. Lett. 55, 2161–2163 (1989).
[Crossref]

Z. L. Liau, V. Diadiuk, J. N. Walpole, D. E. Mull, “Large-numerical-aperture InP lenslets by mass transport,” Appl. Phys. Lett. 52, 1859–1861 (1988); “Gallium-phosphide microlens by mass transport,” 55, Appl. Phys. Lett.97–99 (1989).
[Crossref] [PubMed]

Dwight, H. B.

H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, New York, 1961).

Franz, G.

B. Stegmüller, H. Westermeier, W. Thulke, G. Franz, D. Sacher, “Surface-emitting InGaAsP/InP distributed feedback laser diode at 1.53 μm with monolithic integrated microlens,” IEEE Photon. Technol. Lett. 3, 776–778 (1991).
[Crossref]

Hosokawa, H.

Iga, K.

Lau, H. W.

H. W. Lau, N. Davies, M. McCormick, “Microlens array fabricated in surface relief with high numerical aperture,” in Miniature and Micro-Optics: Fabrication and System Applications, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1544, 178–188 (1991).

Leger, J. R.

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

Liau, Z. L.

Z. L. Liau, H. J. Zeiger, “Surface-energy-induced mass-transport phenomenon in annealing of etched compound semiconductor structures: theoretical modeling and experimental confirmation,” J. Appl. Phys. 67, 2434–2440 (1990).
[Crossref]

Z. L. Liau, J. N. Walpole, L. J. Missaggia, D. E. Mull, “GaInAsP/InP buried-heterostructure surface-emitting diode laser with monolithic integrated bifocal microlens,” Appl. Phys. Lett. 56, 1219–1221 (1990).
[Crossref]

V. Diadiuk, Z. L. Liau, J. N. Walpole, J. W. Caunt, R. C. Williamson, “External-cavity coherent operation of InGaAsP buried-heterostructure laser array,” Appl. Phys. Lett. 55, 2161–2163 (1989).
[Crossref]

Z. L. Liau, V. Diadiuk, J. N. Walpole, D. E. Mull, “Large-numerical-aperture InP lenslets by mass transport,” Appl. Phys. Lett. 52, 1859–1861 (1988); “Gallium-phosphide microlens by mass transport,” 55, Appl. Phys. Lett.97–99 (1989).
[Crossref] [PubMed]

Maiya, P. S.

P. S. Maiya, J. M. Blakely, “Surface self-diffusion and surface energy of nickel,” J. Appl. Phys. 38, 698–704 (1967).
[Crossref]

Matsuoka, T.

H. Nagai, Y. Noguchi, T. Matsuoka, “Thermal deformation of surface corrugations on InGaAsP crystals,” J. Cryst. Growth 71, 225–231 (1985).
[Crossref]

McCormick, M.

H. W. Lau, N. Davies, M. McCormick, “Microlens array fabricated in surface relief with high numerical aperture,” in Miniature and Micro-Optics: Fabrication and System Applications, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1544, 178–188 (1991).

Missaggia, L. J.

Z. L. Liau, J. N. Walpole, L. J. Missaggia, D. E. Mull, “GaInAsP/InP buried-heterostructure surface-emitting diode laser with monolithic integrated bifocal microlens,” Appl. Phys. Lett. 56, 1219–1221 (1990).
[Crossref]

Morgan, W. L.

Morse, D. L.

Mull, D. E.

Z. L. Liau, J. N. Walpole, L. J. Missaggia, D. E. Mull, “GaInAsP/InP buried-heterostructure surface-emitting diode laser with monolithic integrated bifocal microlens,” Appl. Phys. Lett. 56, 1219–1221 (1990).
[Crossref]

Z. L. Liau, V. Diadiuk, J. N. Walpole, D. E. Mull, “Large-numerical-aperture InP lenslets by mass transport,” Appl. Phys. Lett. 52, 1859–1861 (1988); “Gallium-phosphide microlens by mass transport,” 55, Appl. Phys. Lett.97–99 (1989).
[Crossref] [PubMed]

Mullins, W. W.

W. W. Mullins, “Flattening of a nearly plane solid surface due to capillarity,” J. Appl. Phys. 30, 77–83 (1959).
[Crossref]

Nagai, H.

H. Nagai, Y. Noguchi, T. Matsuoka, “Thermal deformation of surface corrugations on InGaAsP crystals,” J. Cryst. Growth 71, 225–231 (1985).
[Crossref]

Noguchi, Y.

H. Nagai, Y. Noguchi, T. Matsuoka, “Thermal deformation of surface corrugations on InGaAsP crystals,” J. Cryst. Growth 71, 225–231 (1985).
[Crossref]

Nukui, K.

Oikawa, M.

Popovic, Z. D.

Sacher, D.

B. Stegmüller, H. Westermeier, W. Thulke, G. Franz, D. Sacher, “Surface-emitting InGaAsP/InP distributed feedback laser diode at 1.53 μm with monolithic integrated microlens,” IEEE Photon. Technol. Lett. 3, 776–778 (1991).
[Crossref]

Saheki, T.

Scott, M. L.

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

Setsune, K.

T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
[Crossref] [PubMed]

Shiono, T.

T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
[Crossref] [PubMed]

Sprague, R. A.

Stegmüller, B.

B. Stegmüller, H. Westermeier, W. Thulke, G. Franz, D. Sacher, “Surface-emitting InGaAsP/InP distributed feedback laser diode at 1.53 μm with monolithic integrated microlens,” IEEE Photon. Technol. Lett. 3, 776–778 (1991).
[Crossref]

Takei, T.

Tatsumi, T.

Thulke, W.

B. Stegmüller, H. Westermeier, W. Thulke, G. Franz, D. Sacher, “Surface-emitting InGaAsP/InP distributed feedback laser diode at 1.53 μm with monolithic integrated microlens,” IEEE Photon. Technol. Lett. 3, 776–778 (1991).
[Crossref]

Veldkamp, W. B.

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

Wada, O.

O. Wada, “Ion-beam etching of InP and its application to the fabrication of high-radiance InGaAsP/InP light-emitting diodes,” J. Electrochem. Soc. 131, 2373–2380 (1984).
[Crossref]

Walpole, J. N.

Z. L. Liau, J. N. Walpole, L. J. Missaggia, D. E. Mull, “GaInAsP/InP buried-heterostructure surface-emitting diode laser with monolithic integrated bifocal microlens,” Appl. Phys. Lett. 56, 1219–1221 (1990).
[Crossref]

V. Diadiuk, Z. L. Liau, J. N. Walpole, J. W. Caunt, R. C. Williamson, “External-cavity coherent operation of InGaAsP buried-heterostructure laser array,” Appl. Phys. Lett. 55, 2161–2163 (1989).
[Crossref]

Z. L. Liau, V. Diadiuk, J. N. Walpole, D. E. Mull, “Large-numerical-aperture InP lenslets by mass transport,” Appl. Phys. Lett. 52, 1859–1861 (1988); “Gallium-phosphide microlens by mass transport,” 55, Appl. Phys. Lett.97–99 (1989).
[Crossref] [PubMed]

Wasa, K.

T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
[Crossref] [PubMed]

Westermeier, H.

B. Stegmüller, H. Westermeier, W. Thulke, G. Franz, D. Sacher, “Surface-emitting InGaAsP/InP distributed feedback laser diode at 1.53 μm with monolithic integrated microlens,” IEEE Photon. Technol. Lett. 3, 776–778 (1991).
[Crossref]

Williamson, R. C.

V. Diadiuk, Z. L. Liau, J. N. Walpole, J. W. Caunt, R. C. Williamson, “External-cavity coherent operation of InGaAsP buried-heterostructure laser array,” Appl. Phys. Lett. 55, 2161–2163 (1989).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 463.

Yamashita, T.

Yamazaki, O.

T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
[Crossref] [PubMed]

Zeiger, H. J.

Z. L. Liau, H. J. Zeiger, “Surface-energy-induced mass-transport phenomenon in annealing of etched compound semiconductor structures: theoretical modeling and experimental confirmation,” J. Appl. Phys. 67, 2434–2440 (1990).
[Crossref]

Appl. Opt. (1)

T. Shiono, K. Setsune, O. Yamazaki, K. Wasa, “Rectangular-apertured micro-Fresnel lens arrays fabricated by electron-beam lithography,” Appl. Opt. 26, 587–591 (1987).
[Crossref] [PubMed]

Appl. Opt. (5)

Appl. Phys. Lett. (2)

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[Crossref]

V. Diadiuk, Z. L. Liau, J. N. Walpole, J. W. Caunt, R. C. Williamson, “External-cavity coherent operation of InGaAsP buried-heterostructure laser array,” Appl. Phys. Lett. 55, 2161–2163 (1989).
[Crossref]

Appl. Phys. Lett. (2)

Z. L. Liau, J. N. Walpole, L. J. Missaggia, D. E. Mull, “GaInAsP/InP buried-heterostructure surface-emitting diode laser with monolithic integrated bifocal microlens,” Appl. Phys. Lett. 56, 1219–1221 (1990).
[Crossref]

Z. L. Liau, V. Diadiuk, J. N. Walpole, D. E. Mull, “Large-numerical-aperture InP lenslets by mass transport,” Appl. Phys. Lett. 52, 1859–1861 (1988); “Gallium-phosphide microlens by mass transport,” 55, Appl. Phys. Lett.97–99 (1989).
[Crossref] [PubMed]

IEEE Photon. Technol. Lett. (1)

B. Stegmüller, H. Westermeier, W. Thulke, G. Franz, D. Sacher, “Surface-emitting InGaAsP/InP distributed feedback laser diode at 1.53 μm with monolithic integrated microlens,” IEEE Photon. Technol. Lett. 3, 776–778 (1991).
[Crossref]

J. Appl. Phys. (3)

Z. L. Liau, H. J. Zeiger, “Surface-energy-induced mass-transport phenomenon in annealing of etched compound semiconductor structures: theoretical modeling and experimental confirmation,” J. Appl. Phys. 67, 2434–2440 (1990).
[Crossref]

W. W. Mullins, “Flattening of a nearly plane solid surface due to capillarity,” J. Appl. Phys. 30, 77–83 (1959).
[Crossref]

P. S. Maiya, J. M. Blakely, “Surface self-diffusion and surface energy of nickel,” J. Appl. Phys. 38, 698–704 (1967).
[Crossref]

J. Cryst. Growth (1)

H. Nagai, Y. Noguchi, T. Matsuoka, “Thermal deformation of surface corrugations on InGaAsP crystals,” J. Cryst. Growth 71, 225–231 (1985).
[Crossref]

J. Electrochem. Soc. (1)

O. Wada, “Ion-beam etching of InP and its application to the fabrication of high-radiance InGaAsP/InP light-emitting diodes,” J. Electrochem. Soc. 131, 2373–2380 (1984).
[Crossref]

Other (4)

H. W. Lau, N. Davies, M. McCormick, “Microlens array fabricated in surface relief with high numerical aperture,” in Miniature and Micro-Optics: Fabrication and System Applications, C. Roychoudhuri, W. B. Veldkamp, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1544, 178–188 (1991).

Ar+ ion milling has been used to etch the multilevel mesa structure, and crystallographic orientational effects of wet chemical etching (observed in Refs. 11–13) have been eliminated.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), p. 463.

H. B. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed. (Macmillan, New York, 1961).

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Figures (6)

Fig. 1
Fig. 1

(a) Microlens fabrication by mass-transport smoothing of etched multistep mesa structure; (b) residual ripples in an incompletely smoothed lens, which results in wave-front deformation and light scattering.

Fig. 2
Fig. 2

Illustrations of (a) a misaligned mesa step, (b) the resulting hump and depression in an otherwise ideal lens profile after mass-transport smoothing, (c) the net distortions that can be used in the calculation of wave-front deformation and light scattering.

Fig. 3
Fig. 3

Optical micrographs, top views, of (a) an etched five-step mesa structure in a GaP substrate, (b) the spherical lens formed after mass-transport smoothing at approximately 1100 °C for 270 h, (c) an image formed for a character placed near the microscope light source. The structure in (a) has an overall height (perpendicular to the paper) of 9.7 μm. The encircled region in (b) is used for interferometric surface profiling in Fig. 4.

Fig. 4
Fig. 4

Optical interferometric profiling of a portion of a spherical lens surface [cf. the encircled region in Fig. 3(b)]. A spherical profile with radius of curvature of 349 μm has been subtracted. The residual ripples can be seen with a ring pattern (portion) corresponding to the original etched structure.

Fig. 5
Fig. 5

Calculated far field, E(θ), of a cylindrical microlens with residual ripples: (a) with lens uniformly illuminated, (b) with a Gaussian beam illumination.

Fig. 6
Fig. 6

Calculated profile of an extra mass after mass-transport smoothing and its comparison with a Gaussian profile of the same height and width. Scaling parameter Δ is the original mesa step width, and δA is the cross-sectional area of the extra mass.

Equations (24)

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

E ( θ ) = E 0 D - D / 2 D / 2 exp { 2 π i λ × [ x sin θ + ( n - 1 ) x δ D sin 2 π x Δ ] } d x ,
exp [ 2 π i λ ( n - 1 ) x δ D sin 2 π x Δ ] 1 - 1 4 [ 2 π ( n - 1 ) x δ λ D ] 2 + 1 4 [ 2 π ( n - 1 ) x δ λ D ] 2 cos 4 π x Δ + i [ 2 π ( n - 1 ) x δ λ D ] sin 2 π x Δ .
E ( θ ) E 0 sin ξ ξ - E 0 [ π ( n - 1 ) δ 2 λ ] 2 × 2 ξ cos ξ + ( ξ 2 - 2 ) sin ξ ξ 3 + i E 0 π ( n - 1 ) δ 2 λ × ( sin ξ - ξ cos ξ ξ 2 - sin ξ - ξ cos ξ ξ 2 ) ,
E ( 0 ) E 0 { 1 - 1 3 [ π ( n - 1 ) δ 2 λ ] 2 } - i E 0 [ ( n - 1 ) δ 2 N λ ] .
- 1 3 [ π ( n - 1 ) δ 2 λ ] 2 .
1 2 [ 2 π λ ( n - 1 ) x δ D sin 2 π x Δ ] 2 ,
E ( θ ) = E 0 - D / 2 D / 2 exp ( - x 2 w 2 ) exp { 2 π i λ [ x sin θ + ( n - 1 ) x δ D sin 2 π x Δ ] } d x - D / 2 D / 2 exp ( - x 2 w 2 ) d x .
E ( θ ) E 0 exp ( - ζ 2 ) - E 0 2 [ π ( n - 1 ) δ 2 λ ] 2 ( 2 w D ) ×     ( 1 - 2 ζ 2 ) exp ( - ζ 2 ) - i E 0 [ π ( n - 1 ) δ 2 λ ] ( 2 w D ) ×     [ ζ exp ( - ζ 2 ) - ζ exp ( - ζ 2 ) ] ,
E ( 0 ) E 0 { 1 - 1 2 [ π ( n - 1 ) δ 2 λ ] 2 ( 2 w D ) 2 } .
δ z ( x , 0 ) δ A δ ( x ) = δ A π 0 cos k x d k ,
δ z ( x , t ) = δ A π 0 exp ( - γ t k 4 ) cos k x d k ,
ξ x / ( γ t ) 1 / 4 ,
κ k ( γ t ) 1 / 4 ,
δ z ( x , t ) = δ A π ( γ t ) 1 / 4 0 exp ( - κ 4 ) cos κ ξ d κ .
( γ t ) 1 / 4 = Γ ( 1 / 4 ) Δ 4 π .
δ z ( x ) = 4 δ A Γ ( 1 / 4 ) Δ 0 exp ( - κ 4 ) cos κ ξ d κ ,
ξ = 4 π x Γ ( 1 / 4 ) Δ .
0 exp ( - κ 4 ) cos κ ξ d κ = 0 exp ( - κ 4 ) n = 0 ( - 1 ) n ( κ ξ ) 2 n ( 2 n ) ! d κ = n = 0 ( - 1 ) n ξ 2 n ( 2 n ) ! 0 κ 2 n exp ( - κ 4 ) d κ .
0 κ 2 n exp ( - κ 4 ) d κ = 1 4 0 u ( n / 2 ) + ( 1 / 4 ) - 1 exp ( - u ) d u = 1 4 Γ ( 2 n + 1 4 ) .
δ z ( x ) = δ A Δ n = 0 ( - 1 ) n Γ [ ( 2 n + 1 ) / 4 ] Γ ( 1 / 4 ) Γ ( 2 n + 1 ) ξ 2 n .
δ z ( x ) δ A Δ exp { - x 2 / [ ( 2 / 3 ) Δ ] 2 } ,
δ I I = - D / 2 D / 2 [ 2 π ( n - 1 ) δ z / λ ] 2 E ( x ) d x - D / 2 D / 2 E ( x ) d x ,
δ I I = 2 - [ 2 π ( n - 1 ) δ z / λ ] 2 d x D ,
δ I I ( 5 / 6 ) [ 2 π ( n - 1 ) ( δ A / Δ ) / λ ] 2 N ,

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