Abstract

Focusing characteristics of hemispherical microlenses formed on the end of single-mode fibers are investigated. The Gaussian beam passing through a hemispherical microlens formed on the end of a single-mode fiber is always aberrated and truncated due to its spherical aberration and aperturing. However, numerical computations show that for the majority of the microlenses whose truncations and aberrations are small, the truncated or aberrated Gaussian beam can be assumed as a Gaussian beam. The shifts of the size and position of minimum 1/e spot size and the shift of the maximum intensity position due to the spherical aberration and the finite size of a microlens are also discussed. To analyze these shifts, Fresnel diffraction integrals are used in the intermediate field region to the hemispherical microlens. Minimum 1/e spot sizes for hemispherical microlenses are measured and compared with the theoretical values of minimum 1/e spot sizes derived with diffraction theory as well as with those derived using the paraxial theory.

© 1986 Optical Society of America

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References

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  1. K. S. Lee, F. S. Barnes, “Microlenses on the End of Single-Mode Optical Fibers for Laser Applications,” Appl. Opt. 24, 3134 (1985).
    [CrossRef] [PubMed]
  2. H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).
  3. H. W. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  4. L. D. Dickson, “Characteristics of Propagating Gaussian Beam,” Appl. Opt. 9, 1854 (1970).
    [CrossRef] [PubMed]
  5. P. Belland, J. P. Crenn, “Changes in the Characteristics of a Gaussian Beam Weakly Diffracted by a Circular Aperture,” Appl. Opt. 21, 522 (1982).
    [CrossRef] [PubMed]
  6. J. T. Hunt, P. A. Renard, R. G. Nelson, “Focusing Properties of an Aberrated Laser Beam,” Appl. Opt. 15, 1458 (1976).
    [CrossRef] [PubMed]
  7. A. Yoshida, “Spherical Aberration in Beam Optical Systems,” Appl. Opt. 21, 1812 (1982).
    [CrossRef] [PubMed]
  8. V. N. Mahajan, “Uniform Versus Gaussian Beams: a Comparison of the Effects of Diffraction, Obscuration, and Aberrations,” J. Opt. Soc. Am. A 3, 470 (1986).
    [CrossRef]
  9. S. Nemoto, T. Makimoto, “Generalized Spot Size for a Higher-Order Beam Mode,” J. Opt. Soc. Am. 69, 578 (1979).
    [CrossRef]
  10. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 3.
  11. D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric Study of Apertured Focused Gaussian Beams,” Appl. Opt. 11, 565 (1972).
    [CrossRef] [PubMed]
  12. G. Goubau, edited by E. C. Jordan, Electromagnetic Theory and Antennas (Pergamon Press, Oxford, U.K., 1963), pp. 907–918.
  13. Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211 (1981).
    [CrossRef]
  14. W. H. Carter, “Focal Shift and Concept of Effective Fresnel Number for a Gaussian Laser Beam,” Appl. Opt. 21, 1989 (1982).
    [CrossRef] [PubMed]
  15. A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  17. J. P. Crenn, “Optical Theory of Gaussian Beam Transmission Through a Hollow Circular Dielectric Waveguide,” Appl. Opt. 21, 4533 (1982).
    [CrossRef] [PubMed]
  18. S. D. Conte, C. de Boor, Elementary Numerical Analysis (McGraw-Hill, New York, 1980).
  19. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 5.
  20. A. H. Firester, M. E. Heller, P. Sheng, “Knife-Edge Scanning Measurements of Subwavelength Focused Light Beams,” Appl. Opt. 16, 1971 (1977).
    [CrossRef] [PubMed]
  21. V. N. Mahajan, “Axial Irradiance and Optimum Focusing of Laser Beams,” Appl. Opt. 22, 3042 (1983).
    [CrossRef] [PubMed]

1986 (1)

1985 (1)

1983 (1)

1982 (4)

1981 (1)

Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211 (1981).
[CrossRef]

1979 (1)

1977 (1)

1976 (1)

1972 (1)

1970 (1)

1965 (2)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

H. W. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 3.

Avizonis, P. V.

Barnes, F. S.

Belland, P.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 5.

Carter, W. H.

Conte, S. D.

S. D. Conte, C. de Boor, Elementary Numerical Analysis (McGraw-Hill, New York, 1980).

Crenn, J. P.

de Boor, C.

S. D. Conte, C. de Boor, Elementary Numerical Analysis (McGraw-Hill, New York, 1980).

Dickson, L. D.

Firester, A. H.

Ghatak, A. K.

A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Goubau, G.

G. Goubau, edited by E. C. Jordan, Electromagnetic Theory and Antennas (Pergamon Press, Oxford, U.K., 1963), pp. 907–918.

Heller, M. E.

Holmes, D. A.

Hunt, J. T.

Kogelnik, H.

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

Kogelnik, H. W.

Korka, J. E.

Lee, K. S.

Li, Y.

Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211 (1981).
[CrossRef]

Mahajan, V. N.

Makimoto, T.

Nelson, R. G.

Nemoto, S.

Renard, P. A.

Sheng, P.

Thyagarajan, K.

A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

Wolf, E.

Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211 (1981).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 5.

Yoshida, A.

Appl. Opt. (11)

H. W. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562 (1965).
[CrossRef]

L. D. Dickson, “Characteristics of Propagating Gaussian Beam,” Appl. Opt. 9, 1854 (1970).
[CrossRef] [PubMed]

D. A. Holmes, J. E. Korka, P. V. Avizonis, “Parametric Study of Apertured Focused Gaussian Beams,” Appl. Opt. 11, 565 (1972).
[CrossRef] [PubMed]

J. T. Hunt, P. A. Renard, R. G. Nelson, “Focusing Properties of an Aberrated Laser Beam,” Appl. Opt. 15, 1458 (1976).
[CrossRef] [PubMed]

A. H. Firester, M. E. Heller, P. Sheng, “Knife-Edge Scanning Measurements of Subwavelength Focused Light Beams,” Appl. Opt. 16, 1971 (1977).
[CrossRef] [PubMed]

P. Belland, J. P. Crenn, “Changes in the Characteristics of a Gaussian Beam Weakly Diffracted by a Circular Aperture,” Appl. Opt. 21, 522 (1982).
[CrossRef] [PubMed]

A. Yoshida, “Spherical Aberration in Beam Optical Systems,” Appl. Opt. 21, 1812 (1982).
[CrossRef] [PubMed]

W. H. Carter, “Focal Shift and Concept of Effective Fresnel Number for a Gaussian Laser Beam,” Appl. Opt. 21, 1989 (1982).
[CrossRef] [PubMed]

J. P. Crenn, “Optical Theory of Gaussian Beam Transmission Through a Hollow Circular Dielectric Waveguide,” Appl. Opt. 21, 4533 (1982).
[CrossRef] [PubMed]

V. N. Mahajan, “Axial Irradiance and Optimum Focusing of Laser Beams,” Appl. Opt. 22, 3042 (1983).
[CrossRef] [PubMed]

K. S. Lee, F. S. Barnes, “Microlenses on the End of Single-Mode Optical Fibers for Laser Applications,” Appl. Opt. 24, 3134 (1985).
[CrossRef] [PubMed]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Imaging of Optical Modes—Resonators with Internal Lenses,” Bell Syst. Tech. J. 44, 455 (1965).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

Y. Li, E. Wolf, “Focal Shifts in Diffracted Converging Spherical Waves,” Opt. Commun. 39, 211 (1981).
[CrossRef]

Other (6)

S. D. Conte, C. de Boor, Elementary Numerical Analysis (McGraw-Hill, New York, 1980).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), Chap. 5.

A. K. Ghatak, K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), p. 3.

G. Goubau, edited by E. C. Jordan, Electromagnetic Theory and Antennas (Pergamon Press, Oxford, U.K., 1963), pp. 907–918.

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Figures (6)

Fig. 1
Fig. 1

Intensity distributions at the maximum intensity position Zp for the truncated Gaussian beam passing through microlenses whose sizes are limited. Incident beam size ω1 = 3 μm, wavelength of the laser beam λ = 0.83 μm, radius curvature of lens R = 7.5 μm, and refractive index of lens n = 1.55. The truncated Gaussian beam (—) is compared with the (untruncated) Gaussian beam (- - - -) having the same 1/e spot size ωi: (a) aperture size of lens a = 6 μm, ωi = 0.793 m; (b) a = 4.8 μm, ωi = 0.784 μm; (c) a = 3.6 μm; ωi = 0.751 μm; (d) a = 3 μm, ωi = 0.708 mμ.

Fig. 2
Fig. 2

Intensity distributions at Zp for the aberrated Gaussian beam passing through microlenses which possess different amounts of spherical aberration: ω1 = 3 μm, λ = 0.83 μm; R = 6 μm, n = 1.55, a = 9 μm. The aberrated Gaussian beam (—) is compared with the (unaberrated) Gaussian beam (- - - -) having the same 1/e spot size ωi: (a) coefficient of spherical aberration a2 = −0.0001λ μm−4, ωi = 0.696 μm; (b) a2 = −0.0005λ μm−4, ωi = 0.829 μm; (c) a2 = −0.001λ μm−4, ωi = 0.931 μm; (d) a2 = −0.005λ μm−4, ωi = 1.11 μm.

Fig. 3
Fig. 3

Maximum intensity position (Zp1) and minimum 1/e spot size position (Z21) as a function of degree of aperturing (a/ω1). Radius of curvature of lens R = 7.5 μm, ω1 = 2.5 μm, refractive index of lens n = 1.55, wavelength = 632.8 nm.

Fig. 4
Fig. 4

Maximum intensity position (Zp2) and minimum 1/e spot size position (Z22) in the presence of spherical aberration: ω1 = 2.5 μm, R = 7.5 μm, λ = 0.6328 μm, a = 9 μm, and n = 1.55).

Fig. 5
Fig. 5

1/e spot sizes at the maximum intensity position and at Z21 as a function of degree of aperturing (a/ω1). Radius of curvature of lens R = 7.5 μm, ω1 = 2.5 μm, n = 1.55, and λ = 632.8 nm.

Fig. 6
Fig. 6

1/e spot sizes at the maximum intensity position and at Z22 as a function of spherical aberration coefficient a2(λ). R = 7.5 μm, ω1 = 2.5 μm, a = 9 μm, n = 1.55, and λ = 632.8 nm.

Tables (1)

Tables Icon

Table I Comparison Between Experimental Values and Theoretical Values of the Minimum 1/e Spot Size (λ = 0.83 μm, ω1 = 3.0 μm, n = 1.55)

Equations (10)

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Δ f = - f / ( 1 + π 2 N G 2 ) ,
N G = ω 1 2 / λ f .
ψ ( r , z ) = i A 1 exp ( - i k z - i k 2 z r 2 ) λ z × 0 a ρ J 0 ( k ρ z r ) exp { - ρ 2 [ 1 ω 1 2 + i k 2 f + i k 2 z - i k ϕ ( ρ ) ρ 2 ] } d ρ ,
ϕ ( ρ ) = a 2 ρ 4 ,
I ( r , z ) = ψ ( r , z ) 2 .
I ( o , z ) = | A 1 λ z 0 a exp ( - ρ 2 { 1 w 1 2 + i k 2 [ 1 Z - 1 f - 2 ϕ ( ρ ) ρ 2 ] } ) d ρ | 2
W ( o ) = ω 1 [ A 2 + ( B / Z 01 ) 2 ] 1 / 2 A D - C B ,
R ( o ) = A 2 + ( B / Z 01 ) 2 A C + B D / Z 01 2 ,
ϕ ( ρ ) = - 1 8 n 2 ( n - 1 ) R 3 ρ 4 .
a 2 = - 1 8 n 2 ( n - 1 ) R 3 ,

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