Abstract

To maximize the diffraction efficiency of cylinder lenses with high numerical apertures (such as F/0.5 lenses) we use an iterative algorithm to determine the optimum field distribution in the lens plane. The algorithm simulates the free-space propagation between the lens and the focal plane applying the angular spectrum of plane waves. We show that the optimum field distribution in the lens plane is the phase distribution of a converging cylindrical wave-front and an amplitude distribution with Gaussian-profile. The computed results are verified by rigorous calculations, simulating a F/0.5 lens with subwavelength structures.

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References

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  1. E. Noponen, J. Turunen and A. Vasara, Electromagnetic theory and design of diffractive-lens arrays, J. Opt. Soc. Am. A 10,434-443(1993)
    [CrossRef]
  2. K. Hirayama, E. N. Glytsis and T. K. Gaylord, Riogorous electromagnetic analysis of diffractive cylinder lenses, J. Opt. Soc. Am. A 13, 2219-2231(1996)
    [CrossRef]
  3. M. Schmitz and O. Bryngdahl, Rigorous concept for the design of diffractive microlenses with high numerical apertures, J. Opt. Soc. Am. A 14, 901-906(1997)
    [CrossRef]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968)
  5. C. B. Burckhardt, Diffraction of a plane wave at a sinusoidally stratified dielectric grating, J. Opt. Soc. Am. 56, 1502-157(1966)
    [CrossRef]
  6. F. G. Kaspar, Diffraction by thick, periodically stratified gratings with complex dielectric constant, J. Opt. Soc. Am. 63, 37-45(1973)
    [CrossRef]
  7. K. Knop, Rigorous diffraction theory ofor transmissionphase gratings with deep rectangular grooves, J. Opt. Soc. Am. 68, 1206-1210(1978)
    [CrossRef]
  8. M. Schmitz, R. Bruer and O. Bryngdahl, Comment on numerical stability of rigorous differential methods of diffraction, Opt. Commun. 124, 1-8(1996)
    [CrossRef]
  9. Lifeng Li, Use of Fourier series in the analysis of discontinuous periodic structures, J. Opt. Soc. Am. A 13, 1870-1876(1996)
    [CrossRef]

Other (9)

E. Noponen, J. Turunen and A. Vasara, Electromagnetic theory and design of diffractive-lens arrays, J. Opt. Soc. Am. A 10,434-443(1993)
[CrossRef]

K. Hirayama, E. N. Glytsis and T. K. Gaylord, Riogorous electromagnetic analysis of diffractive cylinder lenses, J. Opt. Soc. Am. A 13, 2219-2231(1996)
[CrossRef]

M. Schmitz and O. Bryngdahl, Rigorous concept for the design of diffractive microlenses with high numerical apertures, J. Opt. Soc. Am. A 14, 901-906(1997)
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 1968)

C. B. Burckhardt, Diffraction of a plane wave at a sinusoidally stratified dielectric grating, J. Opt. Soc. Am. 56, 1502-157(1966)
[CrossRef]

F. G. Kaspar, Diffraction by thick, periodically stratified gratings with complex dielectric constant, J. Opt. Soc. Am. 63, 37-45(1973)
[CrossRef]

K. Knop, Rigorous diffraction theory ofor transmissionphase gratings with deep rectangular grooves, J. Opt. Soc. Am. 68, 1206-1210(1978)
[CrossRef]

M. Schmitz, R. Bruer and O. Bryngdahl, Comment on numerical stability of rigorous differential methods of diffraction, Opt. Commun. 124, 1-8(1996)
[CrossRef]

Lifeng Li, Use of Fourier series in the analysis of discontinuous periodic structures, J. Opt. Soc. Am. A 13, 1870-1876(1996)
[CrossRef]

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Figures (6)

Figure 1:
Figure 1:

Iterative algorithm to evaluate the propagation of light between the lens and the focal plane.

Figure 2:
Figure 2:

a) Diffractive binary lens and b) the ideal phase distribution of the transmitted electrical field.

Figure 3:
Figure 3:

a) Amplitude - and b) phase distribution of the transmitted electrical field assuming a normal incident plane wave. Solid curves : calculated field, dashed curves : ideal field.

Figure 4:
Figure 4:

a) Amplitude- and b) phase distribution of the transmitted electrical field assuming an incident Gaussian-beam. Solid curves : calculated field, dashed curves : ideal field.

Figure 5:
Figure 5:

Energie distribution in the focal plane assuming an incident plane wave (solid curve) and Gaussian-beam illumination (dashed curve).

Figure 6:
Figure 6:

Diffraction efficiency η as a function of the half-width ξ of the incident Gaussian-beam. The dashed curve indicates the diffraction efficiency with x0 being the location of the first minimum. The solid curve indicates the diffraction efficiency with x0 = 0.675λ.

Equations (10)

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φ ( x ) = k 0 n ( F F 2 + x 2 ) ,
E y ( x , z = 0 ) = E y ( x , z = 0 ) exp ( i φ ( x ) ) .
E y ( x , z = 0 ) = 1 2 π ψ ( k x ) exp ( i k x x ) d k x .
E y ( x , z = F ) = 1 2 π ψ ( k x ) exp ( i k z ( F ) ) exp ( i k x x ) d k x
k z = k 0 2 k x 2 .
O ( x , ν ) = { ν x x 0 0 ν 1 1 x < x 0 ,
E y ( x , z = 0 ) = exp ( 4 ln ( 2 ) x 2 ξ 2 ) ,
S z = 1 2 RE ( E y H x * ) ,
S ¯ z ( x , z ) = S z ( x , z ) x 20 λ dx S z ( x , z = λ ) inc ,
η = x x 0 dx S ¯ z ( x , z = F ) .

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