Abstract

In this paper, we calculate the transverse spherical aberration TA of a thin lens and defines a normalized aberration Y equal to TA divided by the theoretical resolution limit. As a rule of thumb, (a) a thin lens that suffers only from spherical aberration may be considered effectively diffraction-limited as long as Y < 1.6. Similarly, (b) the coupling efficiency of a Gaussian beam to a single-mode fiber may be high even when Y > 1.6, and, specifically, (c) the lens need be diffraction-limited only over a radius approximately equal to the radius (to the 1/e-point) of the Gaussian beam.

© 2000 Optical Society of America

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References

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  1. F.A. Jenkins, H.E. White, Fundamentals of Optics, 4th ed., McGraw-Hill, New York, 1976.
  2. LC Martin, Technical Optics, 1, Pitman, London, 1959, p. 114.
  3. Table 1 differs from the tables that I have distributed in annual short courses during the past 20 years. Here, I use image distance l′ and effective F-number l′/D, whereas, in the short courses, I used focal length f′ and F-number f′/D, even for imaging at unit magnification.
  4. Warren M. Smith, Modern Optical Engineering, 2nd ed., McGraw-Hill, New York, 1990.

Jenkins, F.A.

F.A. Jenkins, H.E. White, Fundamentals of Optics, 4th ed., McGraw-Hill, New York, 1976.

Martin, LC

LC Martin, Technical Optics, 1, Pitman, London, 1959, p. 114.

Smith, Warren M.

Warren M. Smith, Modern Optical Engineering, 2nd ed., McGraw-Hill, New York, 1990.

White, H.E.

F.A. Jenkins, H.E. White, Fundamentals of Optics, 4th ed., McGraw-Hill, New York, 1976.

Other (4)

F.A. Jenkins, H.E. White, Fundamentals of Optics, 4th ed., McGraw-Hill, New York, 1976.

LC Martin, Technical Optics, 1, Pitman, London, 1959, p. 114.

Table 1 differs from the tables that I have distributed in annual short courses during the past 20 years. Here, I use image distance l′ and effective F-number l′/D, whereas, in the short courses, I used focal length f′ and F-number f′/D, even for imaging at unit magnification.

Warren M. Smith, Modern Optical Engineering, 2nd ed., McGraw-Hill, New York, 1990.

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Figures (3)

Figure 1
Figure 1

A planoconvex lens displaying spherical aberration. The vertical line to the right of the lens represents the paraxial image plane. The longitudinal aberration LA is assumed to be very much smaller than the image distance l. The transverse aberration is TA, and the radius at which the ray hits the lens is h. n is the index of refraction of the lens, and λ is the wavelength of the light.

Figure 2
Figure 2

The normalized aberration Y of a 1 cm (focal length) lens as a function of effective F-number or numerical aperture for the cases corresponding to Table 1(a) and (d), that is, to the optimum configurations for unit magnification and for one infinite conjugate.

Figure 3
Figure 3

The point spread function of a diffraction-limited Gaussian beam compared with that of a Gaussian beam that is focused by a thin, planoconvex lens with significant spherical aberration. The numerical aperture of the lens is 1.5 times that of the beam. The focal length of the lens is 12.5 mm. and the wavelength of the light is 1.3 µm. The Image has been optimized by defocusing.

Tables (2)

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Table 1 Normalized aberration Y for various thin lenses.

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Table 2 Coupling efficiency of Gaussian beam into step-index fiber.a

Equations (9)

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ta=lhLS
LS=h28f31nn1n+2n1q2+4n+1pq+3n+2n1p2+n3n1Ah2
ah=Ah4/4, oraθ=Al4θ4/4,
Y  f/λ · Φ4,
dθ=TA θ24 NA,
Uθ=exp2πipθ/λ · expθ/θ02,
T=2π0NAexp2θ/θ02θdθ/πθ02/22
ur=0NA2θUθJ0krθ
c=T0urg*rrdr0|ur|2rdr · 0|gr|2rdr½

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