Abstract

The purpose of this work was to develop formulas and accurate numerical techniques for computation of the optical transfer function (OTF) in the general case of unrestricted aberration and with the following different forms of nonuniform real amplitude: (1) when the real amplitude is described by a polynomial, (2) a Gaussian distribution of real amplitude, and (3) a pupil with a central obstruction. The resulting computer program has been carefully tested and used to study the influence of nonuniform amplitude on the OTF in typical cases, for which detailed numerical results are given.

© 1989 Optical Society of America

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References

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  1. J. Ojeda-Castaneda, L. R. Berriel-Valdos, E. L. Montes, “Bessel Annular Apodizers: Imaging Characteristics,” Appl. Opt. 26, 2770 (1987).
    [CrossRef] [PubMed]
  2. L. N. Hazra, A. Guha, “Far-Field Diffraction Properties of Radial Walsh Filters,” J. Opt. Soc. Am. A 3, 843 (1986).
    [CrossRef]
  3. K. Tanaka, O. Kanzaki, “Focus of a Diffracted Gaussian Beam Through a Finite Aperture Lens: Experimental and Numerical Investigation,” Appl. Opt. 26, 390 (1987).
    [CrossRef] [PubMed]
  4. P. Kuttner, “Image Quality of Optical Systems for Truncated Gaussian Laser Beams,” Opt. Eng. 25, 189 (1986).
    [CrossRef]
  5. R. Barakat, “Diffracted Electromagnetic Fields in the Neighborhood of the Focus of Paraboloidal Mirror Having a Central Obscuration,” Appl. Opt. 26, 3790 (1987).
    [CrossRef] [PubMed]
  6. H. H. Hopkins, “Calculation of the Aberrations and Image Assessment for a General Optical System,” Opt. Acta 28, 667 (1981).
    [CrossRef]
  7. H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
    [CrossRef]
  8. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).
  9. E. L. O’Neil, “Transfer Function for an Annular Aperture,” J. Opt. Soc. Am. 46, 285 (1956).
    [CrossRef]
  10. M. Mino, Y. Okano, “Improvement in the OTF of a Defocused Optical System Through the Use of Shaded Apertures,” Appl. Opt. 10, 2219 (1971).
    [CrossRef] [PubMed]

1987

1986

P. Kuttner, “Image Quality of Optical Systems for Truncated Gaussian Laser Beams,” Opt. Eng. 25, 189 (1986).
[CrossRef]

L. N. Hazra, A. Guha, “Far-Field Diffraction Properties of Radial Walsh Filters,” J. Opt. Soc. Am. A 3, 843 (1986).
[CrossRef]

1981

H. H. Hopkins, “Calculation of the Aberrations and Image Assessment for a General Optical System,” Opt. Acta 28, 667 (1981).
[CrossRef]

1971

1970

H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
[CrossRef]

1956

Barakat, R.

Berriel-Valdos, L. R.

Guha, A.

Hazra, L. N.

Hopkins, H. H.

H. H. Hopkins, “Calculation of the Aberrations and Image Assessment for a General Optical System,” Opt. Acta 28, 667 (1981).
[CrossRef]

H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
[CrossRef]

Kanzaki, O.

Kuttner, P.

P. Kuttner, “Image Quality of Optical Systems for Truncated Gaussian Laser Beams,” Opt. Eng. 25, 189 (1986).
[CrossRef]

Mino, M.

Montes, E. L.

O’Neil, E. L.

Ojeda-Castaneda, J.

Okano, Y.

Tanaka, K.

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

H. H. Hopkins, “Calculation of the Aberrations and Image Assessment for a General Optical System,” Opt. Acta 28, 667 (1981).
[CrossRef]

H. H. Hopkins, M. J. Yzuel, “The Computation of Diffraction Patterns in the Presence of Aberrations,” Opt. Acta 17, 157 (1970).
[CrossRef]

Opt. Eng.

P. Kuttner, “Image Quality of Optical Systems for Truncated Gaussian Laser Beams,” Opt. Eng. 25, 189 (1986).
[CrossRef]

Other

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1972).

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Figures (10)

Fig. 1
Fig. 1

Region of integration for the OTF.

Fig. 2
Fig. 2

Integration along the line y = yk.

Fig. 3
Fig. 3

Amplitude variation, group I.

Fig. 4
Fig. 4

Amplitude variation, group II.

Fig. 5
Fig. 5

Pupil with a central obstruction, group III(b).

Fig. 6
Fig. 6

Optical transfer functions of diffraction-limited systems with nonuniform amplitudes in (i) group I, (ii) group II, (iii) group III.

Fig. 7
Fig. 7

Optical transfer functions in the presence of spherical aberration W40 = 3λ in different focal planes with nonuniform amplitudes in group I: (i) W20 = −2.4λ, (ii) W20 = −3.0λ, (iii) W20= −3.6λ.

Fig. 8
Fig. 8

Optical transfer functions in the presence of spherical aberration W40 = 3λ in different focal planes with nonuniform amplitudes in group II: (i) W20 = −2.4λ, (ii) W20 = −3.0λ, (iii) W20 = −3.6λ.

Fig. 9
Fig. 9

Optical transfer functions in the presence of spherical aberration W40 = 3λ in different focal planes with nonuniform amplitudes in group III: (i) W20 = −2.4λ, (ii) W20 = −3.0λ, (iii) W20 = −3.6λ.

Fig. 10
Fig. 10

Optical transfer functions in the presence of coma W31 = 3λ in the azimuth ψ = 0 with nonuniform amplitudes in (i) group I, (ii) group II, (iii) group III.

Equations (29)

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D 0 ( s 0 , t 0 ) = ( 1 / A ) S f 0 ( x 0 + 1 2 s 0 , y 0 + 1 2 t 0 ) × f * ( x 0 ½ s 0 , y 0 ½ t 0 ) d x 0 d y 0 ,
A = A | f 0 ( x 0 , y 0 ) | 2 d x 0 d y 0 ,
D ( s , ψ ) = ( 1 / A ) S f ( x + 1 2 s , y ) f * ( x 1 2 s , y ) d x d y ,
x 0 = x cos ψ y sin ψ , y 0 = y cos ψ + x sin ψ , A = A | f ( x , y ) | 2 d x d y
f ( x ± ½ 2 s , y ) = f 0 { ( x ± ½ s ) cos ψ y sin ψ , y cos ψ + ( x ± ½ s ) sin ψ }
f 0 ( x 0 , y 0 ) = τ 0 ( x 0 , y 0 ) exp { i 2 π W 0 ( x 0 y 0 ) } ( x 0 , y 0 ) A = 0 ( x 0 , y 0 ) A ,
f ( x , y ) = τ ( x , y ) exp { i 2 π W ( x , y ) } ( x , y ) A = 0 ( x , y ) A ,
τ ( x , y ) = τ 0 ( x cos ψ y sin ψ , y cos ψ + x sin ψ ) , W ( x , y ) = W 0 ( x cos ψ y sin ψ , y cos ψ + x sin ψ )
τ 0 ( x 0 , y 0 ) = a 00 + a 11 y 0 + a 20 ( x 0 2 + y 0 2 ) + a 22 y 0 2 + a 31 ( x 0 2 + y 0 2 ) y 0 + a 33 y 0 3 + a 40 ( x 0 2 + y 0 2 ) 2 + a 42 ( x 0 2 + y 0 2 ) y 0 2 + a 44 y 0 4 ,
τ 0 ( x , y ) = exp { ( x 0 2 + y 0 2 ) / σ 2 } ,
τ 0 ( x 0 , y 0 ) = 0 [ x 0 / a ] 2 + [ ( y 0 c ) / b ] 2 < 1 = 1 1 [ x 0 / a ] 2 + [ ( y 0 c ) / b ] 2 [ x 0 2 + y 0 2 ] = 0 [ x 0 2 + y 0 2 ] > 1 ,
W 0 ( x 0 , y 0 ) = m p W m p ( x 0 2 + y 0 2 ) m p / 2 y 0 P ,
x = [ ( 1 y 2 ) ½ s ] to x = [ ( 1 y 2 ) ½ s ]  ,
D ( x , ψ ) = ( 1 A ) [ 1 ( ½ s ) 2 ] + [ 1 ( ½ s ) 2 ] d y [ ( 1 y 2 ) ½ s ] , + [ ( 1 y 2 ) ½ s ] f ( x + 1 2 s , y ) × f * ( x ½ s , y ) d x
I ( s , ψ ; y k ) = [ ( 1 y k 2 ) ½ s ] + [ ( 1 y k 2 ¯ ) ½ s ] τ ( x + 1 2 s , y k ) τ ( x 1 2 s , y k ) × exp { i 2 π [ W ( x + ½ s , y k ) W ( x ½ s , y k ) ] } d x ,
( 1 / ɛ 0 ) [ ( 1 y k 2 ) ½ s ] = J k η ,
ɛ k = [ ( 1 y k 2 ) ½ s ] / J k
x ± j = ± ( j ½ ) ɛ k ( j = 1 , 2 , , J k ) , y = y k ,
I ( s , ψ ; y k ) = j = J k ( j 0 ) + J k  τ ( x j + 1 2 s , y k ) τ ( x j 1 2 s , y k ) x j ½ ε k x j + ½ ε k × exp { i 2 π W ˜ ( x j , y k ; s ) } d x ,
W ˜ ( x , y ; s ) = W ( x + ½ s , y ) W ( x ½ s , y )
I ( s , ψ ; y k ) = ε k j = J k ( j 0 ) + J k τ ( x j + 1 2 s , y k ) τ ( x j 1 2 s , y k ) exp { i 2 π W ˜ j k } × sinc { π / 2 [ W ˜ j + 1 , k W ˜ j 1 , k ] } ,
W ˜ j k = W ( x j + ½ s ; s ) W ( x j ½ s ; s )
W ˜ ( x , y ; s ) x = ( W ˜ j + 1 , k W ˜ j 1 , k ) / ε k
D ( x , ψ ) = ( 1 / A ) [ 1 ( 1 2 s ) 2 ] k = 1 n ω k ε k j = J k ( j 0 ) J k τ ( x j + 1 2 s , y k ) × τ ( x j ½ s , y k ) exp { i 2 π W ˜ j k } sin c { ½ ( W ˜ j + 1 , k W ˜ j 1 , k ) } ,
D ( x , ψ ) = R ( s , ψ ) exp { i θ ( s , ψ ) } ,
T ( s , ψ ) = + [ R ( s , ψ ) ] 2 + [ I ( s , ψ ) ] 2 ,
θ ( s , ψ ) = arg { R ( x , ψ ) + i I ( s , ψ ) }
D ( s , ψ ) = 1 π [ 1 ( ½ s ) 2 ] + [ 1 ( ½ s ) 2 ] [ ( 1 y 2 ) ½ s ] + [ ( 1 y 2 ) ½ s ] { a 00 + a 20 [ x 2 + x s + 1 4 s 2 + y 2 ] } × { a 00 + a 20 [ x 2 x s + ¼ s 2 ] } exp ( i 4 π W 20 s x ) d x d y ,
D ( x , ψ ) = ( 1 π ) exp ( s 2 2 σ 2 ) [ 1 ( ½ s ) 2 ] + [ 1 ( ½ s ) 2 ] exp ( 2 y 2 σ 2 ) d y × [ ( 1 y 2 ) ½ s ] + [ ( 1 y 2 ) ½ s ] exp ( 2 x 2 σ 2 ) d x ,

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