Abstract

The line spread function and edge spread function for incoherent illumination are expressed in terms of the transfer function in the presence of rotationally nonsymmetric aberrations. A sampling theorem is derived which expresses the edge spread function in terms of sampled values of the line spread function. Typical numerical results are presented for third-order coma.

© 1965 Optical Society of America

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References

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  1. C. Andre, Ann. Ecole Normale Suppl. No. 5 (1876).
  2. H. Struve, Wied. Ann. 27, 1008 (1882).
    [Crossref]
  3. G. Toraldo di Francia, La Diffrazione della Luce (Torino, Edizione Scientifiche Einaudi, 1958), p. 252. This reference contains a convenient summary of Struve’s work.
  4. Rayleigh Lord, “Wave Theory of Light” in Collected Papers (Cambridge University Press, Cambridge, England, 1902; Dover1965), Vol. 3.
  5. R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 (1964).
    [Crossref]
  6. W. Weinstein, J. Opt. Soc. Am. 44, 610 (1954).
    [Crossref]
  7. F. Dixon, Proc. Phys. Soc. (London) 75, 713 (1960).
    [Crossref]
  8. The dimensionless parameter υ is given by υ=(πD/λ)z= πz/λ, where F is the f/number of the optical system, λ is the wavelength of the incident light, and z is the lateral distance in the image plane measured from the optical axis. Both z and λ are to be measured in the same units (usually microns).
  9. A. Papoulis, The Fourier Integral and Its Applications (Mc-Graw-Hill Book Co., Inc., New York, 1962), p. 38.
  10. The dimensionless spatial frequency ω is related to the physical parameters of the system by the equation Ω=ω/2λF, where Ω, is the dimensional spatial frequency expressed in lines/mm, λ is the wavelength of light in mm, and F is the f number of the system. The physical cutoff of the system occurs at ω= 2.
  11. H. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed., p. 129.
  12. H. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Co., Inc., New York, 1954).
  13. The normalization of the line spread function varies with the investigator. We have consistently normalized the line spread function in our calculations so that 0⩽τ⩽1; the value unity occurring when the system is aberration free.
  14. R. Barakat, J. Opt. Soc. Am. 54, 920 (1964).
    [Crossref]
  15. R. Barakat and A. Houston, J. Opt. Soc. Am. 55, 881 (1965).
    [Crossref]

1965 (1)

1964 (2)

1960 (1)

F. Dixon, Proc. Phys. Soc. (London) 75, 713 (1960).
[Crossref]

1954 (1)

1882 (1)

H. Struve, Wied. Ann. 27, 1008 (1882).
[Crossref]

1876 (1)

C. Andre, Ann. Ecole Normale Suppl. No. 5 (1876).

Andre, C.

C. Andre, Ann. Ecole Normale Suppl. No. 5 (1876).

Barakat, R.

Bode, H.

H. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Co., Inc., New York, 1954).

Carslaw, H.

H. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed., p. 129.

Dixon, F.

F. Dixon, Proc. Phys. Soc. (London) 75, 713 (1960).
[Crossref]

Houston, A.

Lord, Rayleigh

Rayleigh Lord, “Wave Theory of Light” in Collected Papers (Cambridge University Press, Cambridge, England, 1902; Dover1965), Vol. 3.

Papoulis, A.

A. Papoulis, The Fourier Integral and Its Applications (Mc-Graw-Hill Book Co., Inc., New York, 1962), p. 38.

Struve, H.

H. Struve, Wied. Ann. 27, 1008 (1882).
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, La Diffrazione della Luce (Torino, Edizione Scientifiche Einaudi, 1958), p. 252. This reference contains a convenient summary of Struve’s work.

Weinstein, W.

Ann. Ecole Normale Suppl. No. 5 (1)

C. Andre, Ann. Ecole Normale Suppl. No. 5 (1876).

J. Opt. Soc. Am. (4)

Proc. Phys. Soc. (London) (1)

F. Dixon, Proc. Phys. Soc. (London) 75, 713 (1960).
[Crossref]

Wied. Ann. (1)

H. Struve, Wied. Ann. 27, 1008 (1882).
[Crossref]

Other (8)

G. Toraldo di Francia, La Diffrazione della Luce (Torino, Edizione Scientifiche Einaudi, 1958), p. 252. This reference contains a convenient summary of Struve’s work.

Rayleigh Lord, “Wave Theory of Light” in Collected Papers (Cambridge University Press, Cambridge, England, 1902; Dover1965), Vol. 3.

The dimensionless parameter υ is given by υ=(πD/λ)z= πz/λ, where F is the f/number of the optical system, λ is the wavelength of the incident light, and z is the lateral distance in the image plane measured from the optical axis. Both z and λ are to be measured in the same units (usually microns).

A. Papoulis, The Fourier Integral and Its Applications (Mc-Graw-Hill Book Co., Inc., New York, 1962), p. 38.

The dimensionless spatial frequency ω is related to the physical parameters of the system by the equation Ω=ω/2λF, where Ω, is the dimensional spatial frequency expressed in lines/mm, λ is the wavelength of light in mm, and F is the f number of the system. The physical cutoff of the system occurs at ω= 2.

H. Carslaw, Introduction to the Theory of Fourier’s Series and Integrals (Dover Publications, Inc., New York, 1956), 3rd ed., p. 129.

H. Bode, Network Analysis and Feedback Amplifier Design (D. Van Nostrand Co., Inc., New York, 1954).

The normalization of the line spread function varies with the investigator. We have consistently normalized the line spread function in our calculations so that 0⩽τ⩽1; the value unity occurring when the system is aberration free.

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Figures (8)

F. 1
F. 1

Edge spread function E(υ,ϕ) in the presence of third-order coma W131 = 0.5λ in the paraxial plane W2 = 0: (A) ϕ = 0°, (B) ϕ=45°, (C) ϕ=90°.

F. 2
F. 2

Line spread function τ (υ,ϕ) in the presence of third-order coma W131=0.5λ in the paraxial plane W2 = 0: —— ϕ = 0°,– – – ϕ = 45°,– · – ϕ = 90°.

F. 3
F. 3

Edge spread function E(υ,ϕ) in the presence of third-order coma W131 = 1.0λ in the paraxial plane W2=0: (A) ϕ = 0°, (B) ϕ=45°, (C) ϕ = 90°.

F. 4
F. 4

Line spread function τ(υ,ϕ) in the presence of third-order coma W131 = 1.0λ in the paraxial plane W2 = 0: —— ϕ = 0°, – – – ϕ = 45°, – · – ϕ = 90°.

F. 5
F. 5

Edge spread function E(υ,ϕ) in the presence of third-order coma W131 = 1.5λ in the paraxial plane W2 = 0: (A) ϕ = 0°, (B) ϕ = 45°, (C) ϕ = 90°.

F. 6
F. 6

Line spread function τ(υ,ϕ) in the presence of third-order coma W131 = 1.5λ in the paraxial plane W2 = 0: —— ϕ = 0°, – – – ϕ = 45°,– · – ϕ = 90°.

F. 7
F. 7

Edge spread function E(υ,ϕ) at the geometric edge υ=0 in the paraxial plane W2 = 0 as a function of third-order coma W131 in the azimuths: —— ϕ = 90°, – – – ϕ = 45°. In the azimuth ϕ = 0°, E(0,ϕ) = 0.5 irrespective of the amount of third-order coma.

F. 8
F. 8

Maximum illuminance of the line spread function in the paraxial plane W2 = 0 as a function of third-order coma W131 in the azimuths: ––– ϕ = 0°, – – – ϕ = 45°, – · – ϕ = 90°.

Equations (20)

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0 ( υ ) = { 0 υ < 0 1 υ > 0 .
0 ( ω ) = π δ ( ω ) + ( i ω ) 1 ,
i ( υ , ϕ ) = 2 2 [ π δ ( ω ) + 1 i ω ] T ( ω , ϕ ) e i υ ω d ω = π + 1 i 2 2 T ( ω , ϕ ) ω e i υ ω d ω .
lim υ f ( x ) sin υ x x d x = π f ( 0 ) ,
E ( υ , ϕ ) = 1 2 + 1 2 π i 2 2 T ( ω , ϕ ) ω e i υ ω d ω .
E ( υ , ϕ ) = 1 2 + 1 π 0 2 T r ( ω , ϕ ) sin υ ω ω d ω + 1 π 0 2 T i ( ω , ϕ ) cos υ ω ω d ω .
E ( υ ) = 1 2 + 1 π 0 2 T ( ω ) sin υ ω ω d ω .
L i ( υ ) = 2 π 0 2 T ( ω ) sin υ ω ω d ω , ( υ > 0 ) .
E ( υ ) = 1 2 [ 1 + L i ( υ ) ] , 0 υ < , = 1 2 [ 1 L i ( υ ) ] , < υ 0 .
E ( 0 , ϕ ) = 1 2 + 1 π 0 2 T i ( ω , ϕ ) ω d ω .
[ E ( 0 , ϕ ) E ( , ϕ ) ] [ E ( + , ϕ ) E ( 0 , ϕ ) ] = 2 π 0 2 T i ( ω , ϕ ) ω d ω .
τ ( υ , ϕ ) = 1 2 π 2 2 T ( ω , ϕ ) e i υ ω d ω .
T ( ω , ϕ ) = n = B n ( ϕ ) e i ( n π / 2 ) ω , ( 2 ω 2 )
B n ( ϕ ) = 1 4 2 2 T ( ω ) e i ( n π / 2 ) ω d ω = π 2 τ ( n π 2 , ϕ ) ,
T ( ω , ϕ ) = π 2 n = τ ( n π 2 , ϕ ) e i ( n π / 2 ) π .
2 π T r ( ω , ϕ ) = τ ( 0 , ϕ ) + n = 1 [ τ ( n π 2 , ϕ ) + τ ( n π 2 , ϕ ) ] cos 2 π 2 ω ,
2 π T i ( ω , ϕ ) = n = 1 [ τ ( n π 2 , ϕ ) τ ( n π 2 , ϕ ) ] sin n π 2 ω .
E ( υ , ϕ ) = 1 2 + 1 2 n = τ ( n π 2 , ϕ ) Si ( 2 υ + n π ) ,
E ( 0 , ϕ ) = 1 2 + 1 2 n = 1 [ τ ( n π 2 ) τ ( n π 2 ) ] Si ( n π ) ,
2 π 0 2 T i ( ω , ϕ ) ω d ω = n = 1 [ τ ( n π 2 ) τ ( n π 2 ) ] Si ( n π ) ,