Abstract

The optical properties of plano-convex refractive microlenses with low Fresnel Number (typically FN <10) are investigated. It turns out that diffraction effects at the lens aperture limit the range of the effective focal length. The upper limit of the focal length is determined by the diffraction pattern of a pinhole with equal diameter. In addition achromatic microlenses can be realized because refraction and diffraction have opposing effects on the focal length. Gaussian beam propagation method has been used for simulation. The presented results are of relevance for applications, where microlenses with small apertures and long focal lengths are used, for example, Shack Hartmann wavefront sensors or confocal microscopes.

© 2006 Optical Society of America

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References

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  1. D. Malacara, and Z. Malacara, Handbook of lens design, (Dekker, New York, 1994).
  2. Y. Li, and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39,211 (1981).
    [CrossRef]
  3. Y. Li, and H. Platzer, "An experimental investigation of diffraction patterns in low Fresnel-number focusing systems," Optica Acta 30,1621 (1983).
    [CrossRef]
  4. W. Wang, A. T. Friberg, E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A. 12,1947 (1995).
    [CrossRef]
  5. J. Arnaud, "Representation of Gaussian beams by complex rays," Appl. Opt. 24, (1985).
    [CrossRef] [PubMed]
  6. U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, "Unconventional treatment of focal shift," Opt. Commun. 157,218-224 (1998).
    [CrossRef]
  7. J. W. Goodman, Introduction to Fourier Optics, 2nd ed., (MacGraw-Hill, New York, 1968), Chap. 4, pp. 63-69.

1998 (1)

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, "Unconventional treatment of focal shift," Opt. Commun. 157,218-224 (1998).
[CrossRef]

1995 (1)

W. Wang, A. T. Friberg, E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A. 12,1947 (1995).
[CrossRef]

1985 (1)

J. Arnaud, "Representation of Gaussian beams by complex rays," Appl. Opt. 24, (1985).
[CrossRef] [PubMed]

1983 (1)

Y. Li, and H. Platzer, "An experimental investigation of diffraction patterns in low Fresnel-number focusing systems," Optica Acta 30,1621 (1983).
[CrossRef]

1981 (1)

Y. Li, and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39,211 (1981).
[CrossRef]

Arnaud, J.

J. Arnaud, "Representation of Gaussian beams by complex rays," Appl. Opt. 24, (1985).
[CrossRef] [PubMed]

Blattner, P.

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, "Unconventional treatment of focal shift," Opt. Commun. 157,218-224 (1998).
[CrossRef]

Dändliker, R.

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, "Unconventional treatment of focal shift," Opt. Commun. 157,218-224 (1998).
[CrossRef]

Friberg, A. T.

W. Wang, A. T. Friberg, E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A. 12,1947 (1995).
[CrossRef]

Herzig, H. P.

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, "Unconventional treatment of focal shift," Opt. Commun. 157,218-224 (1998).
[CrossRef]

Li, Y.

Y. Li, and H. Platzer, "An experimental investigation of diffraction patterns in low Fresnel-number focusing systems," Optica Acta 30,1621 (1983).
[CrossRef]

Y. Li, and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39,211 (1981).
[CrossRef]

Platzer, H.

Y. Li, and H. Platzer, "An experimental investigation of diffraction patterns in low Fresnel-number focusing systems," Optica Acta 30,1621 (1983).
[CrossRef]

Vokinger, U.

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, "Unconventional treatment of focal shift," Opt. Commun. 157,218-224 (1998).
[CrossRef]

Wang, W.

W. Wang, A. T. Friberg, E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A. 12,1947 (1995).
[CrossRef]

Wolf, E.

W. Wang, A. T. Friberg, E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A. 12,1947 (1995).
[CrossRef]

Y. Li, and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39,211 (1981).
[CrossRef]

Appl. Opt. (1)

J. Arnaud, "Representation of Gaussian beams by complex rays," Appl. Opt. 24, (1985).
[CrossRef] [PubMed]

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

W. Wang, A. T. Friberg, E. Wolf, "Structure of focused fields in systems with large Fresnel numbers," J. Opt. Soc. Am. A. 12,1947 (1995).
[CrossRef]

Opt. Commun. (2)

Y. Li, and E. Wolf, "Focal shifts in diffracted converging spherical waves," Opt. Commun. 39,211 (1981).
[CrossRef]

U. Vokinger, R. Dändliker, P. Blattner, and H. P. Herzig, "Unconventional treatment of focal shift," Opt. Commun. 157,218-224 (1998).
[CrossRef]

Optica Acta (1)

Y. Li, and H. Platzer, "An experimental investigation of diffraction patterns in low Fresnel-number focusing systems," Optica Acta 30,1621 (1983).
[CrossRef]

Other (2)

D. Malacara, and Z. Malacara, Handbook of lens design, (Dekker, New York, 1994).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed., (MacGraw-Hill, New York, 1968), Chap. 4, pp. 63-69.

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Figures (9)

Fig. 1.
Fig. 1.

Model of a plano-convex microlens illuminated by a plane wave.

Fig. 2.
Fig. 2.

Intensity (a.u.) distribution behind a microlens of Ø=635 µm, ROC=2.03 mm, illuminated by a plane wave at 633 nm. The lens stands in the x,y plan while z corresponds to the propagating axis.

Fig. 3.
Fig. 3.

Intensity (a.u.) distribution behind a microlens of Ø=635 µm and ROC=25.2 mm illuminated by a plane wave at 633 nm. The lens stands in the x,y plan while z corresponds to the propagating axis.

Fig. 4.
Fig. 4.

Intensity (a.u.) distribution behind a microlens of Ø=635 µm, ROC=252 mm, illuminated by a plane wave at 633 nm. The lens stands in the x,y plan while z corresponds to the propagating axis.

Fig. 5.
Fig. 5.

Intensity (a.u.) distribution behind an aperture of Ø=635 µm illuminated by a plane wave at 633 nm. The lens stands in the x,y plan while z corresponds to the propagating axis.

Fig. 6.
Fig. 6.

ROC [mm] versus position of peak irradiance for six diameters of microlenses illuminated by a plane wave at 633nm.

Fig. 7.
Fig. 7.

ROC versus position of peak irradiance Zp for microlenses of Ø=635 µm illuminated by a plane wave at two different wavelengths.

Fig. 8.
Fig. 8.

ROC versus position of peak irradiance Zp for a microlens of Ø=635 µm, illuminated by a plane wave at four different wavelengths. The crossings between two curves corresponding to achromatic microlenses are shown with circles.

Fig. 9.
Fig. 9.

Relative variations on Zp calibrated at 550 nm for five microlenses of Ø=635 µm for five ROC, illuminated by a plane wave at different wavelengths.

Equations (9)

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f E = ROC ( n 1 ) .
FN = ρ 2 λ f E .
δ = Z p f E .
U ( r , z ) = 1 i λ Σ A 0 ( r 0 ) exp ( ikR ) R cos ( θ ) dS ,
U ( r = 0 , z ) = A 0 i λ φ = 0 2 π r 0 = 0 ρ 0 exp ( ik z 2 + r 2 0 ) z 2 + r 2 0 r 0 dr 0 d ϕ ,
I ( r = 0 , z ) = ( 2 A 0 sin ( k ρ 2 0 4 z ) ) 2 ,
dI ( r = 0 , z ) dz = A 0 k ρ 0 2 sin ( k ρ 2 0 2 z ) 1 z 2 = 0 ,
Z p = ρ 0 2 λ .
n = C 0 + C 1 λ 2 + C 2 λ 4 ,

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