Abstract

Light focused from air into a spherical microlens is affected by diffraction at the lens surface as its diameter approaches the wavelength of light. Through an extension of Mie theory, we show that a converging wave that is incident upon a Si microlens with a diameter less than approximately 4 λ creates a spot as much as 25% smaller than predicted with vector diffraction theory. Si microlenses only a wavelength in diameter are shown to be virtually insensitive to variations in the maximum illumination angle, and changes in index of refraction are not found to cause the proportional changes in spot size that would be expected from vector diffraction theory.

© 2001 Optical Society of America

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  1. B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
    [CrossRef]
  2. S. M. Mansfield and G. S. Kino, Appl. Phys. Lett. 57, 2615 (1990).
    [CrossRef]
  3. M. Yoshita, M. Baba, S. Koshiba, H. Sakaki, and H. Akiyama, Appl. Phys. Lett. 73, 2965 (1998).
    [CrossRef]
  4. M. Yoshita, T. Sasaki, M. Baba, and H. Akiyama, Appl. Phys. Lett. 73, 635 (1998).
    [CrossRef]
  5. C. D. Poweleit, A. Gunther, S. Goodnick, and J. Menendez, Appl. Phys. Lett. 73, 2275 (1998).
    [CrossRef]
  6. L. P. Ghislain and V. B. Elings, Appl. Phys. Lett. 72, 2779 (1998).
    [CrossRef]
  7. Q. Wu, G. D. Feke, R. D. Grober, and L. P. Ghislain, Appl. Phys. Lett. 75, 4064 (1999).
    [CrossRef]
  8. Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, Phys. Rev. Lett. 83, 2652 (1999).
    [CrossRef]
  9. B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, Appl. Phys. Lett. 65, 388 (1994).
    [CrossRef]
  10. S. M. Mansfield, W. R. Studenmund, G. S. Kino, and K. Osato, Opt. Lett. 18, 305 (1993).
    [CrossRef]
  11. L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
    [CrossRef]
  12. D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
    [CrossRef]
  13. M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1980).
  14. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  15. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
    [CrossRef]

2000 (1)

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

1999 (3)

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

Q. Wu, G. D. Feke, R. D. Grober, and L. P. Ghislain, Appl. Phys. Lett. 75, 4064 (1999).
[CrossRef]

Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, Phys. Rev. Lett. 83, 2652 (1999).
[CrossRef]

1998 (4)

M. Yoshita, M. Baba, S. Koshiba, H. Sakaki, and H. Akiyama, Appl. Phys. Lett. 73, 2965 (1998).
[CrossRef]

M. Yoshita, T. Sasaki, M. Baba, and H. Akiyama, Appl. Phys. Lett. 73, 635 (1998).
[CrossRef]

C. D. Poweleit, A. Gunther, S. Goodnick, and J. Menendez, Appl. Phys. Lett. 73, 2275 (1998).
[CrossRef]

L. P. Ghislain and V. B. Elings, Appl. Phys. Lett. 72, 2779 (1998).
[CrossRef]

1994 (1)

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, Appl. Phys. Lett. 65, 388 (1994).
[CrossRef]

1993 (1)

1990 (1)

S. M. Mansfield and G. S. Kino, Appl. Phys. Lett. 57, 2615 (1990).
[CrossRef]

1959 (1)

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

Akiyama, H.

M. Yoshita, M. Baba, S. Koshiba, H. Sakaki, and H. Akiyama, Appl. Phys. Lett. 73, 2965 (1998).
[CrossRef]

M. Yoshita, T. Sasaki, M. Baba, and H. Akiyama, Appl. Phys. Lett. 73, 635 (1998).
[CrossRef]

Baba, M.

M. Yoshita, T. Sasaki, M. Baba, and H. Akiyama, Appl. Phys. Lett. 73, 635 (1998).
[CrossRef]

M. Yoshita, M. Baba, S. Koshiba, H. Sakaki, and H. Akiyama, Appl. Phys. Lett. 73, 2965 (1998).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1980).

Crozier, K. B.

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

Elings, V. B.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

L. P. Ghislain and V. B. Elings, Appl. Phys. Lett. 72, 2779 (1998).
[CrossRef]

Feke, G. D.

Q. Wu, G. D. Feke, R. D. Grober, and L. P. Ghislain, Appl. Phys. Lett. 75, 4064 (1999).
[CrossRef]

Fletcher, D. A.

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

Gammon, D.

Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, Phys. Rev. Lett. 83, 2652 (1999).
[CrossRef]

Ghislain, L. P.

Q. Wu, G. D. Feke, R. D. Grober, and L. P. Ghislain, Appl. Phys. Lett. 75, 4064 (1999).
[CrossRef]

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

L. P. Ghislain and V. B. Elings, Appl. Phys. Lett. 72, 2779 (1998).
[CrossRef]

Goodnick, S.

C. D. Poweleit, A. Gunther, S. Goodnick, and J. Menendez, Appl. Phys. Lett. 73, 2275 (1998).
[CrossRef]

Goodson, K. E.

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

Grober, R. D.

Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, Phys. Rev. Lett. 83, 2652 (1999).
[CrossRef]

Q. Wu, G. D. Feke, R. D. Grober, and L. P. Ghislain, Appl. Phys. Lett. 75, 4064 (1999).
[CrossRef]

Gunther, A.

C. D. Poweleit, A. Gunther, S. Goodnick, and J. Menendez, Appl. Phys. Lett. 73, 2275 (1998).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

Katzer, D. S.

Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, Phys. Rev. Lett. 83, 2652 (1999).
[CrossRef]

Kino, G. S.

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, Appl. Phys. Lett. 65, 388 (1994).
[CrossRef]

S. M. Mansfield, W. R. Studenmund, G. S. Kino, and K. Osato, Opt. Lett. 18, 305 (1993).
[CrossRef]

S. M. Mansfield and G. S. Kino, Appl. Phys. Lett. 57, 2615 (1990).
[CrossRef]

Koshiba, S.

M. Yoshita, M. Baba, S. Koshiba, H. Sakaki, and H. Akiyama, Appl. Phys. Lett. 73, 2965 (1998).
[CrossRef]

Mamin, H. J.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, Appl. Phys. Lett. 65, 388 (1994).
[CrossRef]

Manalis, S. R.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

Mansfield, S. M.

Menendez, J.

C. D. Poweleit, A. Gunther, S. Goodnick, and J. Menendez, Appl. Phys. Lett. 73, 2275 (1998).
[CrossRef]

Minne, S. C.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

Osato, K.

Palanker, D. V.

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

Poweleit, C. D.

C. D. Poweleit, A. Gunther, S. Goodnick, and J. Menendez, Appl. Phys. Lett. 73, 2275 (1998).
[CrossRef]

Quate, C. F.

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

Rugar, D.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, Appl. Phys. Lett. 65, 388 (1994).
[CrossRef]

Sakaki, H.

M. Yoshita, M. Baba, S. Koshiba, H. Sakaki, and H. Akiyama, Appl. Phys. Lett. 73, 2965 (1998).
[CrossRef]

Sasaki, T.

M. Yoshita, T. Sasaki, M. Baba, and H. Akiyama, Appl. Phys. Lett. 73, 635 (1998).
[CrossRef]

Simanovskii, D.

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

Studenmund, W. R.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, Appl. Phys. Lett. 65, 388 (1994).
[CrossRef]

S. M. Mansfield, W. R. Studenmund, G. S. Kino, and K. Osato, Opt. Lett. 18, 305 (1993).
[CrossRef]

Terris, B. D.

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, Appl. Phys. Lett. 65, 388 (1994).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wilder, K.

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1980).

Wu, Q.

Q. Wu, G. D. Feke, R. D. Grober, and L. P. Ghislain, Appl. Phys. Lett. 75, 4064 (1999).
[CrossRef]

Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, Phys. Rev. Lett. 83, 2652 (1999).
[CrossRef]

Yoshita, M.

M. Yoshita, M. Baba, S. Koshiba, H. Sakaki, and H. Akiyama, Appl. Phys. Lett. 73, 2965 (1998).
[CrossRef]

M. Yoshita, T. Sasaki, M. Baba, and H. Akiyama, Appl. Phys. Lett. 73, 635 (1998).
[CrossRef]

Appl. Phys. Lett. (9)

S. M. Mansfield and G. S. Kino, Appl. Phys. Lett. 57, 2615 (1990).
[CrossRef]

M. Yoshita, M. Baba, S. Koshiba, H. Sakaki, and H. Akiyama, Appl. Phys. Lett. 73, 2965 (1998).
[CrossRef]

M. Yoshita, T. Sasaki, M. Baba, and H. Akiyama, Appl. Phys. Lett. 73, 635 (1998).
[CrossRef]

C. D. Poweleit, A. Gunther, S. Goodnick, and J. Menendez, Appl. Phys. Lett. 73, 2275 (1998).
[CrossRef]

L. P. Ghislain and V. B. Elings, Appl. Phys. Lett. 72, 2779 (1998).
[CrossRef]

Q. Wu, G. D. Feke, R. D. Grober, and L. P. Ghislain, Appl. Phys. Lett. 75, 4064 (1999).
[CrossRef]

L. P. Ghislain, V. B. Elings, K. B. Crozier, S. R. Manalis, S. C. Minne, K. Wilder, G. S. Kino, and C. F. Quate, Appl. Phys. Lett. 74, 501 (1999).
[CrossRef]

D. A. Fletcher, K. B. Crozier, G. S. Kino, C. F. Quate, K. E. Goodson, D. Simanovskii, and D. V. Palanker, Appl. Phys. Lett. 77, 2109 (2000).
[CrossRef]

B. D. Terris, H. J. Mamin, D. Rugar, W. R. Studenmund, and G. S. Kino, Appl. Phys. Lett. 65, 388 (1994).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

Q. Wu, R. D. Grober, D. Gammon, and D. S. Katzer, Phys. Rev. Lett. 83, 2652 (1999).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1980).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Schematic of a spherical microlens of diameter d and index n illuminated by a converging wave of wavelength λ at a maximum angle θ.

Fig. 2
Fig. 2

Variation in spot size with Si microlens diameter for d=4 λ, 2 λ, λ for a NA of 0.8 at ϕ=π/2. The dotted curve shows the spot expected from vector diffraction theory.1

Fig. 3
Fig. 3

Variation in spot size with illumination NA’s of 0.8, 0.6, 0.4, and 0.2 in a Si microlens of diameter λ at ϕ=π/2. The dotted curve shows the spot expected from Mie theory.

Fig. 4
Fig. 4

Variation in spot size with index of refraction n=3.4, 1.7 in a Si microlens of diameter 2 λ for a NA of 0.8 at ϕ=π/2. The solid curves show results from the converging wave model, and the dashed curves show the spots expected from vector diffraction theory.

Equations (2)

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

E0=m=0n=mAmnMmn+BmnNmn,
Amn=02π0πE0 · Mmnsinθdθdϕ02π0πMmn2sinθdθdϕ

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