Abstract

To improve the coupling efficiency between a laser diode and a single-mode fiber, we propose a two-microlens coupling scheme that uses two solgel spherical microlenses for high coupling efficiency. The conventional reflow technique was employed and extended to the inorganic–organic hybrid SiO2/ZrO2 solgel material to form the microlenses. Preliminary results show that the coupling efficiency was increased to −1.28 dB (74.5%) by the proposed scheme, compared with a coupling efficiency of −10.13 dB (9.7%) by the butt-joint method. The proposed fabrication technique demonstrates that use of a reflowed solgel spherical microlens is a cost-effective mass-production approach to application of micro-optical elements in optical communication.

© 2005 Optical Society of America

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References

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  1. W. Unziker, “Low-cost packaging of semiconductor laser arrays,” Circ. Devices 13, 19–25 (1997).
    [CrossRef]
  2. J. Yamashita, A. Adachi, S. Kaneko, T. Hashimoto, “High efficient and small sized coupling optics for monolithic array LD module,” IECE Trans. Electron. E77-C, 1776–1780 (1994).
  3. K. Kawano, “Coupling characteristics of lens systems for laser diode modules using single-mode fiber,” Appl. Opt. 25, 2600–2605 (1986).
    [CrossRef] [PubMed]
  4. H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
    [CrossRef]
  5. P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. 39, 387–392 (2000).
    [CrossRef]
  6. P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
    [CrossRef]
  7. F. A. Rahman, K. Takahashi, C. H. Teik, “A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber,” Opt. Commun. 208, 103–110 (2002).
    [CrossRef]
  8. H. L. An, “Theoretical investigation on the effective coupling from laser diode to tapered lensed single-mode optical fiber,” Opt. Commun. 181, 89–95 (2000).
    [CrossRef]
  9. K. Shiraishi, “A fiber with a long working distance for integrated coupling between laser diodes and single-mode fibers,” J. Lightwave Technol. 13, 1736–1744 (1995).
    [CrossRef]
  10. H. Sakaguchi, N. Seki, S. Yamamoto, “Power coupling from laser diodes into single-mode fibres quadrangular pyramid-shaped hemiellipsodal ends,” Electron. Lett. 17, 425–426 (1981).
    [CrossRef]
  11. V. S. Shah, L. Curtis, R. S. Vodhabel, D. P. Bour, W. C. Young, “Effective power coupling from a 980 nm, broad-area laser to a single-mode fiber using a wedge-shaped fiber end-face,” J. Lightwave Technol. 8, 1313–1318 (1990).
    [CrossRef]
  12. D. Blac, S. Pelissier, K. Saravanamuttu, S. I. Najafi, M. P. Andrews, “Self-processing of surface-relief gratings in photosensitive sol-gel glasses,” Adv. Mater. 11, 1508–1511 (1999).
    [CrossRef]
  13. J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
    [CrossRef]
  14. K. Kintaka, J. Nishii, N. Tohge, “Diffraction gratings of photosensitive ZrO2gel films fabricated with the two-ultraviolet-beam interference method,” Appl. Opt. 39, 489–493 (2000).
    [CrossRef]
  15. M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Reflow technique for fabrication of an elliptical microlens array in sol-gel material,” Appl. Opt. 42, 7174–7178 (2003).
    [CrossRef]
  16. M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Simple technique for fabrication of cylindrical sol-gel MLA by reflowing technique,” Opt. Eng. 42, 2180–2181 (2003).
    [CrossRef]

2003 (2)

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Reflow technique for fabrication of an elliptical microlens array in sol-gel material,” Appl. Opt. 42, 7174–7178 (2003).
[CrossRef]

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Simple technique for fabrication of cylindrical sol-gel MLA by reflowing technique,” Opt. Eng. 42, 2180–2181 (2003).
[CrossRef]

2002 (1)

F. A. Rahman, K. Takahashi, C. H. Teik, “A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber,” Opt. Commun. 208, 103–110 (2002).
[CrossRef]

2000 (3)

H. L. An, “Theoretical investigation on the effective coupling from laser diode to tapered lensed single-mode optical fiber,” Opt. Commun. 181, 89–95 (2000).
[CrossRef]

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. 39, 387–392 (2000).
[CrossRef]

K. Kintaka, J. Nishii, N. Tohge, “Diffraction gratings of photosensitive ZrO2gel films fabricated with the two-ultraviolet-beam interference method,” Appl. Opt. 39, 489–493 (2000).
[CrossRef]

1999 (2)

D. Blac, S. Pelissier, K. Saravanamuttu, S. I. Najafi, M. P. Andrews, “Self-processing of surface-relief gratings in photosensitive sol-gel glasses,” Adv. Mater. 11, 1508–1511 (1999).
[CrossRef]

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
[CrossRef]

1998 (1)

J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
[CrossRef]

1997 (1)

W. Unziker, “Low-cost packaging of semiconductor laser arrays,” Circ. Devices 13, 19–25 (1997).
[CrossRef]

1995 (1)

K. Shiraishi, “A fiber with a long working distance for integrated coupling between laser diodes and single-mode fibers,” J. Lightwave Technol. 13, 1736–1744 (1995).
[CrossRef]

1994 (1)

J. Yamashita, A. Adachi, S. Kaneko, T. Hashimoto, “High efficient and small sized coupling optics for monolithic array LD module,” IECE Trans. Electron. E77-C, 1776–1780 (1994).

1990 (1)

V. S. Shah, L. Curtis, R. S. Vodhabel, D. P. Bour, W. C. Young, “Effective power coupling from a 980 nm, broad-area laser to a single-mode fiber using a wedge-shaped fiber end-face,” J. Lightwave Technol. 8, 1313–1318 (1990).
[CrossRef]

1988 (1)

H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

1986 (1)

1981 (1)

H. Sakaguchi, N. Seki, S. Yamamoto, “Power coupling from laser diodes into single-mode fibres quadrangular pyramid-shaped hemiellipsodal ends,” Electron. Lett. 17, 425–426 (1981).
[CrossRef]

Adachi, A.

J. Yamashita, A. Adachi, S. Kaneko, T. Hashimoto, “High efficient and small sized coupling optics for monolithic array LD module,” IECE Trans. Electron. E77-C, 1776–1780 (1994).

An, H. L.

H. L. An, “Theoretical investigation on the effective coupling from laser diode to tapered lensed single-mode optical fiber,” Opt. Commun. 181, 89–95 (2000).
[CrossRef]

Andrews, M. P.

D. Blac, S. Pelissier, K. Saravanamuttu, S. I. Najafi, M. P. Andrews, “Self-processing of surface-relief gratings in photosensitive sol-gel glasses,” Adv. Mater. 11, 1508–1511 (1999).
[CrossRef]

Blac, D.

D. Blac, S. Pelissier, K. Saravanamuttu, S. I. Najafi, M. P. Andrews, “Self-processing of surface-relief gratings in photosensitive sol-gel glasses,” Adv. Mater. 11, 1508–1511 (1999).
[CrossRef]

Bour, D. P.

V. S. Shah, L. Curtis, R. S. Vodhabel, D. P. Bour, W. C. Young, “Effective power coupling from a 980 nm, broad-area laser to a single-mode fiber using a wedge-shaped fiber end-face,” J. Lightwave Technol. 8, 1313–1318 (1990).
[CrossRef]

Bu, J.

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Reflow technique for fabrication of an elliptical microlens array in sol-gel material,” Appl. Opt. 42, 7174–7178 (2003).
[CrossRef]

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Simple technique for fabrication of cylindrical sol-gel MLA by reflowing technique,” Opt. Eng. 42, 2180–2181 (2003).
[CrossRef]

Chanclou, P.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. 39, 387–392 (2000).
[CrossRef]

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
[CrossRef]

Cheong, W. C.

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Simple technique for fabrication of cylindrical sol-gel MLA by reflowing technique,” Opt. Eng. 42, 2180–2181 (2003).
[CrossRef]

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Reflow technique for fabrication of an elliptical microlens array in sol-gel material,” Appl. Opt. 42, 7174–7178 (2003).
[CrossRef]

Curtis, L.

V. S. Shah, L. Curtis, R. S. Vodhabel, D. P. Bour, W. C. Young, “Effective power coupling from a 980 nm, broad-area laser to a single-mode fiber using a wedge-shaped fiber end-face,” J. Lightwave Technol. 8, 1313–1318 (1990).
[CrossRef]

Gadonna, M.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. 39, 387–392 (2000).
[CrossRef]

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
[CrossRef]

Hashimoto, T.

J. Yamashita, A. Adachi, S. Kaneko, T. Hashimoto, “High efficient and small sized coupling optics for monolithic array LD module,” IECE Trans. Electron. E77-C, 1776–1780 (1994).

He, M.

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Reflow technique for fabrication of an elliptical microlens array in sol-gel material,” Appl. Opt. 42, 7174–7178 (2003).
[CrossRef]

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Simple technique for fabrication of cylindrical sol-gel MLA by reflowing technique,” Opt. Eng. 42, 2180–2181 (2003).
[CrossRef]

Honkannen, S.

J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
[CrossRef]

Kaneko, S.

J. Yamashita, A. Adachi, S. Kaneko, T. Hashimoto, “High efficient and small sized coupling optics for monolithic array LD module,” IECE Trans. Electron. E77-C, 1776–1780 (1994).

Karstensen, H.

H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

Kawano, K.

Kintaka, K.

Lostec, J.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. 39, 387–392 (2000).
[CrossRef]

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
[CrossRef]

Najafi, S. I.

D. Blac, S. Pelissier, K. Saravanamuttu, S. I. Najafi, M. P. Andrews, “Self-processing of surface-relief gratings in photosensitive sol-gel glasses,” Adv. Mater. 11, 1508–1511 (1999).
[CrossRef]

Ngo, N. Q.

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Simple technique for fabrication of cylindrical sol-gel MLA by reflowing technique,” Opt. Eng. 42, 2180–2181 (2003).
[CrossRef]

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Reflow technique for fabrication of an elliptical microlens array in sol-gel material,” Appl. Opt. 42, 7174–7178 (2003).
[CrossRef]

Nishii, J.

Nordman, N.

J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
[CrossRef]

Nordman, O.

J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
[CrossRef]

Pavy, D.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. 39, 387–392 (2000).
[CrossRef]

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
[CrossRef]

Pelissier, S.

D. Blac, S. Pelissier, K. Saravanamuttu, S. I. Najafi, M. P. Andrews, “Self-processing of surface-relief gratings in photosensitive sol-gel glasses,” Adv. Mater. 11, 1508–1511 (1999).
[CrossRef]

Peyghambarian, N.

J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
[CrossRef]

Poudoulec, A.

Rahman, F. A.

F. A. Rahman, K. Takahashi, C. H. Teik, “A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber,” Opt. Commun. 208, 103–110 (2002).
[CrossRef]

Rantala, J. T.

J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
[CrossRef]

Sakaguchi, H.

H. Sakaguchi, N. Seki, S. Yamamoto, “Power coupling from laser diodes into single-mode fibres quadrangular pyramid-shaped hemiellipsodal ends,” Electron. Lett. 17, 425–426 (1981).
[CrossRef]

Saravanamuttu, K.

D. Blac, S. Pelissier, K. Saravanamuttu, S. I. Najafi, M. P. Andrews, “Self-processing of surface-relief gratings in photosensitive sol-gel glasses,” Adv. Mater. 11, 1508–1511 (1999).
[CrossRef]

Seki, N.

H. Sakaguchi, N. Seki, S. Yamamoto, “Power coupling from laser diodes into single-mode fibres quadrangular pyramid-shaped hemiellipsodal ends,” Electron. Lett. 17, 425–426 (1981).
[CrossRef]

Shah, V. S.

V. S. Shah, L. Curtis, R. S. Vodhabel, D. P. Bour, W. C. Young, “Effective power coupling from a 980 nm, broad-area laser to a single-mode fiber using a wedge-shaped fiber end-face,” J. Lightwave Technol. 8, 1313–1318 (1990).
[CrossRef]

Shiraishi, K.

K. Shiraishi, “A fiber with a long working distance for integrated coupling between laser diodes and single-mode fibers,” J. Lightwave Technol. 13, 1736–1744 (1995).
[CrossRef]

Takahashi, K.

F. A. Rahman, K. Takahashi, C. H. Teik, “A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber,” Opt. Commun. 208, 103–110 (2002).
[CrossRef]

Teik, C. H.

F. A. Rahman, K. Takahashi, C. H. Teik, “A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber,” Opt. Commun. 208, 103–110 (2002).
[CrossRef]

Thual, M.

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. 39, 387–392 (2000).
[CrossRef]

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
[CrossRef]

Tohge, N.

Unziker, W.

W. Unziker, “Low-cost packaging of semiconductor laser arrays,” Circ. Devices 13, 19–25 (1997).
[CrossRef]

Vahakangas, J.

J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
[CrossRef]

Vodhabel, R. S.

V. S. Shah, L. Curtis, R. S. Vodhabel, D. P. Bour, W. C. Young, “Effective power coupling from a 980 nm, broad-area laser to a single-mode fiber using a wedge-shaped fiber end-face,” J. Lightwave Technol. 8, 1313–1318 (1990).
[CrossRef]

Yamamoto, S.

H. Sakaguchi, N. Seki, S. Yamamoto, “Power coupling from laser diodes into single-mode fibres quadrangular pyramid-shaped hemiellipsodal ends,” Electron. Lett. 17, 425–426 (1981).
[CrossRef]

Yamashita, J.

J. Yamashita, A. Adachi, S. Kaneko, T. Hashimoto, “High efficient and small sized coupling optics for monolithic array LD module,” IECE Trans. Electron. E77-C, 1776–1780 (1994).

Young, W. C.

V. S. Shah, L. Curtis, R. S. Vodhabel, D. P. Bour, W. C. Young, “Effective power coupling from a 980 nm, broad-area laser to a single-mode fiber using a wedge-shaped fiber end-face,” J. Lightwave Technol. 8, 1313–1318 (1990).
[CrossRef]

Yuan, X.-C.

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Reflow technique for fabrication of an elliptical microlens array in sol-gel material,” Appl. Opt. 42, 7174–7178 (2003).
[CrossRef]

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Simple technique for fabrication of cylindrical sol-gel MLA by reflowing technique,” Opt. Eng. 42, 2180–2181 (2003).
[CrossRef]

Adv. Mater. (1)

D. Blac, S. Pelissier, K. Saravanamuttu, S. I. Najafi, M. P. Andrews, “Self-processing of surface-relief gratings in photosensitive sol-gel glasses,” Adv. Mater. 11, 1508–1511 (1999).
[CrossRef]

Appl. Opt. (3)

Circ. Devices (1)

W. Unziker, “Low-cost packaging of semiconductor laser arrays,” Circ. Devices 13, 19–25 (1997).
[CrossRef]

Electron. Lett. (2)

H. Sakaguchi, N. Seki, S. Yamamoto, “Power coupling from laser diodes into single-mode fibres quadrangular pyramid-shaped hemiellipsodal ends,” Electron. Lett. 17, 425–426 (1981).
[CrossRef]

J. T. Rantala, N. Nordman, O. Nordman, J. Vahakangas, S. Honkannen, N. Peyghambarian, “Sol-gel hybrid glass diffractive element by electron-beam exposure,” Electron. Lett. 34, 455–456 (1998).
[CrossRef]

IECE Trans. Electron. (1)

J. Yamashita, A. Adachi, S. Kaneko, T. Hashimoto, “High efficient and small sized coupling optics for monolithic array LD module,” IECE Trans. Electron. E77-C, 1776–1780 (1994).

J. Lightwave Technol. (3)

V. S. Shah, L. Curtis, R. S. Vodhabel, D. P. Bour, W. C. Young, “Effective power coupling from a 980 nm, broad-area laser to a single-mode fiber using a wedge-shaped fiber end-face,” J. Lightwave Technol. 8, 1313–1318 (1990).
[CrossRef]

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
[CrossRef]

K. Shiraishi, “A fiber with a long working distance for integrated coupling between laser diodes and single-mode fibers,” J. Lightwave Technol. 13, 1736–1744 (1995).
[CrossRef]

J. Opt. Commun. (1)

H. Karstensen, “Laser diode to single-mode fiber coupling with ball lenses,” J. Opt. Commun. 9, 42–49 (1988).
[CrossRef]

Opt. Commun. (2)

F. A. Rahman, K. Takahashi, C. H. Teik, “A scheme to improve the coupling efficiency and working distance between laser diode and single mode fiber,” Opt. Commun. 208, 103–110 (2002).
[CrossRef]

H. L. An, “Theoretical investigation on the effective coupling from laser diode to tapered lensed single-mode optical fiber,” Opt. Commun. 181, 89–95 (2000).
[CrossRef]

Opt. Eng. (2)

P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, “Focusing and coupling properties of collective micro-optics on fiber ribbons,” Opt. Eng. 39, 387–392 (2000).
[CrossRef]

M. He, X.-C. Yuan, N. Q. Ngo, W. C. Cheong, J. Bu, “Simple technique for fabrication of cylindrical sol-gel MLA by reflowing technique,” Opt. Eng. 42, 2180–2181 (2003).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of the proposed two-microlens coupling scheme.

Fig. 2
Fig. 2

Measured SEM image of solgel spherical MLA 2.

Fig. 3
Fig. 3

Measured surface profile of the solgel MLA 2 along the x direction.

Fig. 4
Fig. 4

Discrepancy in surface profile of the measured surface and the designed shape of MLA 2.

Fig. 5
Fig. 5

AFM image of a segment of the surface of MLA 2.

Fig. 6
Fig. 6

Light-intensity distribution on the confocal plane of MLA 2. (The grid scale is 20 μm × 20 μm.)

Fig. 7
Fig. 7

Light output power P coupled into a SMF as a function of LD drive current I.

Equations (4)

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

f = D e 2 N . A . ,
R = ( n - 1 ) f ,
h = R - ( R 2 - D 2 4 ) 1 / 2 .
H = 4 h 2 ( 3 R - h ) 3 η D 2 .

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