Abstract

A new method for fabricating plastic spherical microlenses was developed, which allowed self-alignment of lenses and self-organized formation of a spherical shape. First a low-surface-energy fluoropolymer thin film was deposited and patterned as a stencil. Then photosensitive phenol resin was patterned on it as the lens material. Finally the resin was annealed in an oil bath to form a sphere. The molten phenol resin spontaneously formed a sphere and positioned itself in the center of the fluoropolymer ring pattern as a result of the difference of surface free energy and the equivalently zero-gravity condition in the oil bath. When a light-emitting-diode printer head was loaded with spherical microlenses, its optical output increased by 1 order of magnitude.

© 2003 Optical Society of America

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References

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  1. J. L. Jewell, S. L. McCall, Y.-H. Lee, A. Scherer, A. C. Gossard, J. H. English, “Optical computing and related micro-optic devices,” Appl. Opt. 29, 5050–5053 (1990).
    [CrossRef] [PubMed]
  2. N. C. Craft, A. Y. Feldblum, “Optical interconnects based on arrays of surface-emitting lasers and lenslets,” Appl. Opt. 31, 1735–1739 (1992).
    [CrossRef] [PubMed]
  3. H. Honmou, M. Itoh, “Optical coupling of laser diode array to singlemode-fibre array with heat-treated hemispherical microlens,” Electron. Lett. 31, 793–794 (1995).
    [CrossRef]
  4. H. Saeki, M. Ikeno, S. Suzuki, H. Kawashima, S. Uematsu, “Effect of microlens array for MOS color imager,” IEEE Trans. Consum. Electron. 31, 88–95 (1985).
    [CrossRef]
  5. C. Van Berkel, B. P. McGarvey, J. A. Clarke, “Microlens arrays for 2D large area image sensors,” Pure Appl. Opt. 3, 177–182 (1994).
    [CrossRef]
  6. F. Yamasaki, M. Deguchi, T. Kakuda, T. Shiaki, T. Nakashima, “High efficiency optical system design for a liquid crystal projector,” IEEE Trans. Consum. Electron. 46, 851–856 (2000).
    [CrossRef]
  7. S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
    [CrossRef]
  8. O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
    [CrossRef]
  9. B. P. Keyworth, D. J. Corazza, J. N. McMullin, L. Mabbott, “Single-step fabrication of refractive microlens arrays,” Appl. Opt. 36, 2198–2202 (1997).
    [CrossRef] [PubMed]
  10. Y. Lu, Y. Yin, Y. Xia, “A self-assembly approach to the fabrication of patterned, two-dimensional arrays of microlenses of organic polymers,” Adv. Mater. 13, 34–37 (2001).
    [CrossRef]
  11. D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
    [CrossRef]
  12. M. Tamura, “Bonding method for micro beads of photoelectric element,” Japan patent unexamined 2000-353824 (19December2000).
  13. M. Tamura, “Method for arranging microbead onto LED substrate,” Japan patent unexamined 2001-18447 (23January2001).
  14. D. M. Hartmann, O. Kibar, S. C. Esener, “Optimization and theoretical modeling of polymer microlens arrays fabricated with the hydrophobic effect,” Appl. Opt. 40, 2736–2746 (2001).
    [CrossRef]
  15. H. Usui, H. Tamura, H. Hiramoto, “Method for preparing plastic microlens,” Japan patent application unexamined, 2002-353511 (6December2002).
  16. H. Usui, H. Koshikawa, K. Tanaka, “Effect of substrate temperature on the deposition of polytetrafluoroethylene by an ionization-assisted evaporation method,” J. Vac. Sci. Technol. A 13, 2318–2324 (1995).
    [CrossRef]
  17. H. Usui, H. Koshikawa, K. Tanaka, “Characteristics of polyetetrafluoroethylene thin films prepared by ionization-assisted deposition,” IEICE Trans. Electron. E-81-C, 1083–1089 (1998).
  18. H. Tamura, “LED printer head,” Japan patent unexamined 2001-18448 (23January2001).

2001 (2)

Y. Lu, Y. Yin, Y. Xia, “A self-assembly approach to the fabrication of patterned, two-dimensional arrays of microlenses of organic polymers,” Adv. Mater. 13, 34–37 (2001).
[CrossRef]

D. M. Hartmann, O. Kibar, S. C. Esener, “Optimization and theoretical modeling of polymer microlens arrays fabricated with the hydrophobic effect,” Appl. Opt. 40, 2736–2746 (2001).
[CrossRef]

2000 (1)

F. Yamasaki, M. Deguchi, T. Kakuda, T. Shiaki, T. Nakashima, “High efficiency optical system design for a liquid crystal projector,” IEEE Trans. Consum. Electron. 46, 851–856 (2000).
[CrossRef]

1998 (1)

H. Usui, H. Koshikawa, K. Tanaka, “Characteristics of polyetetrafluoroethylene thin films prepared by ionization-assisted deposition,” IEICE Trans. Electron. E-81-C, 1083–1089 (1998).

1997 (1)

1995 (3)

H. Usui, H. Koshikawa, K. Tanaka, “Effect of substrate temperature on the deposition of polytetrafluoroethylene by an ionization-assisted evaporation method,” J. Vac. Sci. Technol. A 13, 2318–2324 (1995).
[CrossRef]

H. Honmou, M. Itoh, “Optical coupling of laser diode array to singlemode-fibre array with heat-treated hemispherical microlens,” Electron. Lett. 31, 793–794 (1995).
[CrossRef]

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

1994 (3)

C. Van Berkel, B. P. McGarvey, J. A. Clarke, “Microlens arrays for 2D large area image sensors,” Pure Appl. Opt. 3, 177–182 (1994).
[CrossRef]

D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
[CrossRef]

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

1992 (1)

1990 (1)

1985 (1)

H. Saeki, M. Ikeno, S. Suzuki, H. Kawashima, S. Uematsu, “Effect of microlens array for MOS color imager,” IEEE Trans. Consum. Electron. 31, 88–95 (1985).
[CrossRef]

Blum, O.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

Carson, R.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

Chen, T.

D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
[CrossRef]

Clarke, J. A.

C. Van Berkel, B. P. McGarvey, J. A. Clarke, “Microlens arrays for 2D large area image sensors,” Pure Appl. Opt. 3, 177–182 (1994).
[CrossRef]

Corazza, D. J.

Cox, W. R.

D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
[CrossRef]

Craft, N. C.

Deguchi, M.

F. Yamasaki, M. Deguchi, T. Kakuda, T. Shiaki, T. Nakashima, “High efficiency optical system design for a liquid crystal projector,” IEEE Trans. Consum. Electron. 46, 851–856 (2000).
[CrossRef]

Du, T. C.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

English, J. H.

Esener, S. C.

Feldblum, A. Y.

Gossard, A. C.

Hartmann, D. M.

Hayes, D. J.

D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
[CrossRef]

Hiramoto, H.

H. Usui, H. Tamura, H. Hiramoto, “Method for preparing plastic microlens,” Japan patent application unexamined, 2002-353511 (6December2002).

Honmou, H.

H. Honmou, M. Itoh, “Optical coupling of laser diode array to singlemode-fibre array with heat-treated hemispherical microlens,” Electron. Lett. 31, 793–794 (1995).
[CrossRef]

Ikeno, M.

H. Saeki, M. Ikeno, S. Suzuki, H. Kawashima, S. Uematsu, “Effect of microlens array for MOS color imager,” IEEE Trans. Consum. Electron. 31, 88–95 (1985).
[CrossRef]

Ishida, J.

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

Itoh, M.

H. Honmou, M. Itoh, “Optical coupling of laser diode array to singlemode-fibre array with heat-treated hemispherical microlens,” Electron. Lett. 31, 793–794 (1995).
[CrossRef]

Jewell, J. L.

Kakuda, T.

F. Yamasaki, M. Deguchi, T. Kakuda, T. Shiaki, T. Nakashima, “High efficiency optical system design for a liquid crystal projector,” IEEE Trans. Consum. Electron. 46, 851–856 (2000).
[CrossRef]

Kawashima, H.

H. Saeki, M. Ikeno, S. Suzuki, H. Kawashima, S. Uematsu, “Effect of microlens array for MOS color imager,” IEEE Trans. Consum. Electron. 31, 88–95 (1985).
[CrossRef]

Keyworth, B. P.

Kibar, O.

Kilcoyne, S. P.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

Koshikawa, H.

H. Usui, H. Koshikawa, K. Tanaka, “Characteristics of polyetetrafluoroethylene thin films prepared by ionization-assisted deposition,” IEICE Trans. Electron. E-81-C, 1083–1089 (1998).

H. Usui, H. Koshikawa, K. Tanaka, “Effect of substrate temperature on the deposition of polytetrafluoroethylene by an ionization-assisted evaporation method,” J. Vac. Sci. Technol. A 13, 2318–2324 (1995).
[CrossRef]

Lear, K. L.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

Lee, Y.-H.

Lu, Y.

Y. Lu, Y. Yin, Y. Xia, “A self-assembly approach to the fabrication of patterned, two-dimensional arrays of microlenses of organic polymers,” Adv. Mater. 13, 34–37 (2001).
[CrossRef]

Mabbott, L.

MacFarlane, D. L.

D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
[CrossRef]

McCall, S. L.

McGarvey, B. P.

C. Van Berkel, B. P. McGarvey, J. A. Clarke, “Microlens arrays for 2D large area image sensors,” Pure Appl. Opt. 3, 177–182 (1994).
[CrossRef]

McMullin, J. N.

Nakashima, T.

F. Yamasaki, M. Deguchi, T. Kakuda, T. Shiaki, T. Nakashima, “High efficiency optical system design for a liquid crystal projector,” IEEE Trans. Consum. Electron. 46, 851–856 (2000).
[CrossRef]

Narayan, V.

D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
[CrossRef]

Ogata, S.

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

Peters, F. H.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

Robinson, G.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

Saeki, H.

H. Saeki, M. Ikeno, S. Suzuki, H. Kawashima, S. Uematsu, “Effect of microlens array for MOS color imager,” IEEE Trans. Consum. Electron. 31, 88–95 (1985).
[CrossRef]

Sasano, T.

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

Scherer, A.

Schneider, R. P.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

Shiaki, T.

F. Yamasaki, M. Deguchi, T. Kakuda, T. Shiaki, T. Nakashima, “High efficiency optical system design for a liquid crystal projector,” IEEE Trans. Consum. Electron. 46, 851–856 (2000).
[CrossRef]

Suzuki, S.

H. Saeki, M. Ikeno, S. Suzuki, H. Kawashima, S. Uematsu, “Effect of microlens array for MOS color imager,” IEEE Trans. Consum. Electron. 31, 88–95 (1985).
[CrossRef]

Tamura, H.

H. Usui, H. Tamura, H. Hiramoto, “Method for preparing plastic microlens,” Japan patent application unexamined, 2002-353511 (6December2002).

H. Tamura, “LED printer head,” Japan patent unexamined 2001-18448 (23January2001).

Tamura, M.

M. Tamura, “Method for arranging microbead onto LED substrate,” Japan patent unexamined 2001-18447 (23January2001).

M. Tamura, “Bonding method for micro beads of photoelectric element,” Japan patent unexamined 2000-353824 (19December2000).

Tanaka, K.

H. Usui, H. Koshikawa, K. Tanaka, “Characteristics of polyetetrafluoroethylene thin films prepared by ionization-assisted deposition,” IEICE Trans. Electron. E-81-C, 1083–1089 (1998).

H. Usui, H. Koshikawa, K. Tanaka, “Effect of substrate temperature on the deposition of polytetrafluoroethylene by an ionization-assisted evaporation method,” J. Vac. Sci. Technol. A 13, 2318–2324 (1995).
[CrossRef]

Tatum, J. A.

D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
[CrossRef]

Uematsu, S.

H. Saeki, M. Ikeno, S. Suzuki, H. Kawashima, S. Uematsu, “Effect of microlens array for MOS color imager,” IEEE Trans. Consum. Electron. 31, 88–95 (1985).
[CrossRef]

Usui, H.

H. Usui, H. Koshikawa, K. Tanaka, “Characteristics of polyetetrafluoroethylene thin films prepared by ionization-assisted deposition,” IEICE Trans. Electron. E-81-C, 1083–1089 (1998).

H. Usui, H. Koshikawa, K. Tanaka, “Effect of substrate temperature on the deposition of polytetrafluoroethylene by an ionization-assisted evaporation method,” J. Vac. Sci. Technol. A 13, 2318–2324 (1995).
[CrossRef]

H. Usui, H. Tamura, H. Hiramoto, “Method for preparing plastic microlens,” Japan patent application unexamined, 2002-353511 (6December2002).

Van Berkel, C.

C. Van Berkel, B. P. McGarvey, J. A. Clarke, “Microlens arrays for 2D large area image sensors,” Pure Appl. Opt. 3, 177–182 (1994).
[CrossRef]

Warren, M. E.

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

Xia, Y.

Y. Lu, Y. Yin, Y. Xia, “A self-assembly approach to the fabrication of patterned, two-dimensional arrays of microlenses of organic polymers,” Adv. Mater. 13, 34–37 (2001).
[CrossRef]

Yamasaki, F.

F. Yamasaki, M. Deguchi, T. Kakuda, T. Shiaki, T. Nakashima, “High efficiency optical system design for a liquid crystal projector,” IEEE Trans. Consum. Electron. 46, 851–856 (2000).
[CrossRef]

Yin, Y.

Y. Lu, Y. Yin, Y. Xia, “A self-assembly approach to the fabrication of patterned, two-dimensional arrays of microlenses of organic polymers,” Adv. Mater. 13, 34–37 (2001).
[CrossRef]

Adv. Mater. (1)

Y. Lu, Y. Yin, Y. Xia, “A self-assembly approach to the fabrication of patterned, two-dimensional arrays of microlenses of organic polymers,” Adv. Mater. 13, 34–37 (2001).
[CrossRef]

Appl. Opt. (4)

Electron. Lett. (2)

O. Blum, S. P. Kilcoyne, M. E. Warren, T. C. Du, K. L. Lear, R. P. Schneider, R. Carson, G. Robinson, F. H. Peters, “Vertical-cavity surface-emitting lasers with integrated refractive microlenses,” Electron. Lett. 31, 44–45 (1995).
[CrossRef]

H. Honmou, M. Itoh, “Optical coupling of laser diode array to singlemode-fibre array with heat-treated hemispherical microlens,” Electron. Lett. 31, 793–794 (1995).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

D. L. MacFarlane, V. Narayan, J. A. Tatum, W. R. Cox, T. Chen, D. J. Hayes, “Microjet fabrication of microlens arrays,” IEEE Photon. Technol. Lett. 6, 1112–1114 (1994).
[CrossRef]

IEEE Trans. Consum. Electron. (2)

H. Saeki, M. Ikeno, S. Suzuki, H. Kawashima, S. Uematsu, “Effect of microlens array for MOS color imager,” IEEE Trans. Consum. Electron. 31, 88–95 (1985).
[CrossRef]

F. Yamasaki, M. Deguchi, T. Kakuda, T. Shiaki, T. Nakashima, “High efficiency optical system design for a liquid crystal projector,” IEEE Trans. Consum. Electron. 46, 851–856 (2000).
[CrossRef]

IEICE Trans. Electron. (1)

H. Usui, H. Koshikawa, K. Tanaka, “Characteristics of polyetetrafluoroethylene thin films prepared by ionization-assisted deposition,” IEICE Trans. Electron. E-81-C, 1083–1089 (1998).

J. Vac. Sci. Technol. A (1)

H. Usui, H. Koshikawa, K. Tanaka, “Effect of substrate temperature on the deposition of polytetrafluoroethylene by an ionization-assisted evaporation method,” J. Vac. Sci. Technol. A 13, 2318–2324 (1995).
[CrossRef]

Opt. Eng. (1)

S. Ogata, J. Ishida, T. Sasano, “Optical sensor array in an artificial compound eye,” Opt. Eng. 33, 3649–3655 (1994).
[CrossRef]

Pure Appl. Opt. (1)

C. Van Berkel, B. P. McGarvey, J. A. Clarke, “Microlens arrays for 2D large area image sensors,” Pure Appl. Opt. 3, 177–182 (1994).
[CrossRef]

Other (4)

M. Tamura, “Bonding method for micro beads of photoelectric element,” Japan patent unexamined 2000-353824 (19December2000).

M. Tamura, “Method for arranging microbead onto LED substrate,” Japan patent unexamined 2001-18447 (23January2001).

H. Usui, H. Tamura, H. Hiramoto, “Method for preparing plastic microlens,” Japan patent application unexamined, 2002-353511 (6December2002).

H. Tamura, “LED printer head,” Japan patent unexamined 2001-18448 (23January2001).

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Figures (13)

Fig. 1
Fig. 1

Schematic procedure for forming the spherical microlenses. Step 4 in the text is shown in (a). A PTFE layer was deposited on the photoresist by lift-off to define a stencil pattern. Step 6 in the text is shown in (b). Phenol resin was photolithographically patterned on the PTFE pattern. A plan view is given in (b’). Step 8 in the text is shown in (c). The substrate was annealed in an oil bath upside down to cause the phenol resin to form a sphere.

Fig. 2
Fig. 2

SEM image of the spherical microlens. The diameter is ∼10 μm.

Fig. 3
Fig. 3

SEM image of a microlens formed without the PTFE stencil layer.

Fig. 4
Fig. 4

Meniscus at the rim of the microlens.

Fig. 5
Fig. 5

(a) Plan and (b) slant views of a 10-μm microlens formed on a LED chip.

Fig. 6
Fig. 6

(Top) Cross-sectional and (bottom) plan views of microlenses (a) before and (b) after a displacement of δ from the position coaxially fit to the opening of the PTFE stencil.

Fig. 7
Fig. 7

Optical micrograph of an array of spherical microlenses formed on a Si substrate. The lens spacing corresponds to 1200 dpi.

Fig. 8
Fig. 8

Optical micrograph of a microbridge connecting the spheres.

Fig. 9
Fig. 9

Infrared absorption spectra of the microlens, phenol resin, and PTFE films.

Fig. 10
Fig. 10

Configuration of the LED loaded with the spherical microlens for measuring and calculating the near-field emission pattern.

Fig. 11
Fig. 11

Simulated power distribution of the light emitted from the LED loaded with the microlens at a distance of 200 μm from the lens-device interface. The arrows represent the length of 200 μm.

Fig. 12
Fig. 12

Line profile of the simulated power distribution along the same diameter as in Fig. 11. The vertical broken lines represents the experimentally observed beam diameter. The dotted curve gives the least-squares fit to the power distribution by a double Gaussian function.

Fig. 13
Fig. 13

Model of the LED printer head.

Equations (12)

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

σOP=σLS+σLO cos θ.
θ=cos-1σOP-σLSσLO.
ΔE=SσLP-σOP+σOS-σLS,
σLP=σL+σP-2σLdσPd,
σOP=σO+σP-2σOdσPd,
σOS=σO+σS-2σOdσSd.
σLS=σL+σS-2σLdσSd-ILSe
ΔES=2σLd-σOdσSd-σPd+ILSe.
ILθ=I1a1 exp-b1θ2+a2 exp-b2θ2,
0π/2 ILθ2π sin θdθ=0.125I1.
IBθ=I0 cosθθ0π2,
0θ0 IBθ2π sin θdθ=1.28I0.

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