Abstract

We modify the concept of superresolution for achieving arbitrarily extended focal depth with a finite aperture. The Legendre polynomials are used for designing a novel apodizer that produces arbitrarily high depth of focus. This kind of apodizer can also arbitrarily reduce the sensitivity to spherical aberration. Since these benefits are achieved at the expense of light throughput, we report on a formula for evaluating light throughput of this kind of filter.

© 1988 Optical Society of America

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References

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1986

1985

J. Ojeda Castañeda, L. R. Berriel-Valdos, E. Montes, Opt. Lett. 10, 520 (1985).
[CrossRef]

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985).
[CrossRef]

1984

1983

1979

M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
[CrossRef]

1976

1971

1969

B. R. Frieden, Opt. Acta 16, 795 (1969).
[CrossRef]

1952

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[CrossRef]

Andres, P.

Antosiewicz, H. A.

H. A. Antosiewicz, “Bessel functions of fractional order,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (Dover, New York, 1972), p. 440.

Bai, H.

Berriel-Valdos, L. R.

Biswas, S. C.

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985).
[CrossRef]

Boivin, A.

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985).
[CrossRef]

Boyer, G. R.

Calvo, F.

M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
[CrossRef]

Diaz, A.

Frieden, B. R.

B. R. Frieden, Opt. Acta 16, 795 (1969).
[CrossRef]

Indebetouw, G.

Mills, J. P.

Mino, M.

Montes, E.

Ojeda Castañeda, J.

Ojeda-Castañeda, J.

Okano, Y.

Thompson, B. J.

Toraldo di Francia, G.

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[CrossRef]

Villeneuve, J. E.

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985).
[CrossRef]

Yzuel, M. J.

M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
[CrossRef]

Appl. Opt.

Can. J. Phys.

J. E. Villeneuve, A. Boivin, S. C. Biswas, Can. J. Phys. 63, 287 (1985).
[CrossRef]

J. Opt. Soc. Am. A

Nuovo Cimento Suppl.

G. Toraldo di Francia, Nuovo Cimento Suppl. 9, 426 (1952).
[CrossRef]

Opt. Acta

B. R. Frieden, Opt. Acta 16, 795 (1969).
[CrossRef]

M. J. Yzuel, F. Calvo, Opt. Acta 26, 1397 (1979).
[CrossRef]

Opt. Lett.

Other

H. A. Antosiewicz, “Bessel functions of fractional order,” in Handbook of Mathematical Functions, M. Abramowitz, I. A. Stegun, eds. (Dover, New York, 1972), p. 440.

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Figures (4)

Fig. 1
Fig. 1

Normalized amplitude transmittance of the apodizers in the 1-D interval: −½ ≦ ζ ≦ ½.

Fig. 2
Fig. 2

Normalized amplitude distribution along the optical axis for the apodizers in Fig. 1.

Fig. 3
Fig. 3

Strehl ratio or normalized on-axis irradiance at the best focal plane for variable spherical aberration, for the apodizers in Fig. 1.

Fig. 4
Fig. 4

Light throughput for the apodizers in Fig. 1.

Equations (19)

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p ( r , W 20 ) = 2 π 0 Ω p ˜ ( ρ ) exp [ i 2 π W 20 ( ρ / Ω ) 2 ] J 0 ( 2 πrρ ) ρ d ρ .
q ( W 20 ) = p ( r = 0 , W 20 ) = [ π Ω 2 exp ( W 20 ) ] q ˜ ( ζ ) exp ( i 2 π W 20 ζ ) d ζ ,
ζ = ( ρ / Ω ) 2 0 . 5 , q ˜ ( ζ ) = p ˜ ( ρ ) .
δ ( ζ ζ ) = n = 0 ϕ n * ( ζ ) ϕ n ( ζ ) .
q ˜ ( ζ ) = n = 0 N ϕ n * ( 0 ) ϕ n ( ζ ) .
exp  ( i 2 π W 20 ζ ) = n = 0 ( i ) n ( 2 n + 1 ) j n ( π W 20 ) P n ( 2 ζ ) ,
δ ( ζ ) = exp ( i 2 π W 20 ζ ) d W 20 = n = 0 a n P n ( 2 ζ ) ,
a n = ( i ) n ( 2 n + 1 ) j n ( π W 20 ) d W 20 .
q ˜ ( ζ ) = n = 0 N a n P n ( 2 ζ )
1 = n = 0 [ ( i ) n ( 2 n + 1 ) P n ( 0 ) ] j n ( π W 20 ) .
q ( W 20 ) = n = 0 N [ ( i ) n ( 2 n + 1 ) P n ( 0 ) ] j n ( π W 20 ) ,
P 2 n + 1 ( 0 ) = 0 and P 2 n ( 0 ) = ( 1 ) n ( 2 n ) ! / [ 2 2 n ( n ! ) 2 ] .
q ( W 20 ) = n = 0 N [ ( 2 n ) ! ( 4 n + 1 ) / 2 2 n ( n ! ) 2 ] i 2 n ( π W 20 ) .
q ˜ ( ζ ) = n = 0 N [ ( 1 ) n ( 2 n ) ! ( 4 n + 1 ) / 2 2 n ( n ! ) 2 ] P 2 n ( 2 ζ ) .
q ˜ ( ζ ) = n = 0 N ( 4 n + 1 ) P 2 n ( 0 ) P 2 n ( 2 ζ ) .
q ˜ ( ζ = 0 ) = n = 0 N ( 4 n + 1 ) P 2 n 2 ( 0 ) .
T = | q ˜ ( ζ ) / q ˜ ( 0 ) | 2 d ζ = | q ˜ ( 0 ) | 2 m = 0 N n = 0 N ( 4 m + 1 ) ( 4 n + 1 ) P 2 m ( 0 ) P 2 n ( 0 ) × P 2 m ( 2 ζ ) P 2 n ( 2 ζ ) d ζ .
T = | q ˜ ( 0 ) | 2 m = 0 N ( 4 m + 1 ) P 2 m 2 ( 0 ) .
T = | q ˜ ( 0 ) | 1 .

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