Abstract

The effect of random longitudinal motion of an image is discussed. Numerical results are given for systems with circular and annular pupils. It is shown that, for a given value of the image motion, the time-averaged Strehl ratio is larger for a larger value of the obscuration ratio of an annular pupil. We also show that the time-averaged image irradiance distribution near its center and the corresponding encircled power can be obtained from their aberration-free values by multiplying them by the Strehl ratio.

© 2007 Optical Society of America

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References

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  1. W. B. Wetherell, "Image quality criteria for the large space telescope," in Space Optics, Proceedings of the Ninth International Congress of the International Commission for Optics, B. J. Thompson and R. R. Shannon, eds. (National Academy of Sciences, 1974), pp. 55-103.
    [PubMed]
  2. V. N. Mahajan, "Degradation of an image due to Gaussian motion," Appl. Opt. 17, 3329-3334 (1978).
    [CrossRef] [PubMed]
  3. W. B. Wetherell, "The calculation of image quality," in Applied Optics and Optical Engineering, R.R.Shannon and J.C.Wyant, eds. (Academic, 1992), Vol. 8.
  4. V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2004).
  5. M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).
  6. W. T. Welford, "Use of annular apertures to increase focal depth," J. Opt. Soc. Am. 50, 749-753 (1960).
    [CrossRef]

1978 (1)

1960 (1)

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

Mahajan, V. N.

V. N. Mahajan, "Degradation of an image due to Gaussian motion," Appl. Opt. 17, 3329-3334 (1978).
[CrossRef] [PubMed]

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2004).

Welford, W. T.

Wetherell, W. B.

W. B. Wetherell, "The calculation of image quality," in Applied Optics and Optical Engineering, R.R.Shannon and J.C.Wyant, eds. (Academic, 1992), Vol. 8.

W. B. Wetherell, "Image quality criteria for the large space telescope," in Space Optics, Proceedings of the Ninth International Congress of the International Commission for Optics, B. J. Thompson and R. R. Shannon, eds. (National Academy of Sciences, 1974), pp. 55-103.
[PubMed]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Other (4)

W. B. Wetherell, "The calculation of image quality," in Applied Optics and Optical Engineering, R.R.Shannon and J.C.Wyant, eds. (Academic, 1992), Vol. 8.

V. N. Mahajan, Optical Imaging and Aberrations, Part II: Wave Diffraction Optics, 2nd ed. (SPIE, 2004).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Oxford, 1999).

W. B. Wetherell, "Image quality criteria for the large space telescope," in Space Optics, Proceedings of the Ninth International Congress of the International Commission for Optics, B. J. Thompson and R. R. Shannon, eds. (National Academy of Sciences, 1974), pp. 55-103.
[PubMed]

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Figures (7)

Fig. 1
Fig. 1

Strehl ratio S ( σ ; ϵ ) as a function of the standard deviation σ of the longitudinal image motion. ϵ is the obscuration ratio of the annular pupil. σ is in units of 8 λ F 2 , and its numerical value represents the peak defocus wave aberration in units of wavelength.

Fig. 2
Fig. 2

Time-averaged PSF I ( r ; σ ) as a function of r and encircled power P ( r c ; σ ) as a function of r c for various values of σ for a system with a circular pupil. Both r and r c are in units of λ F . S = 0.1868 when σ = 2 .

Fig. 3
Fig. 3

Time-averaged PSF I ( r ; σ ; ϵ ) as a function of r and encircled power P ( r c ; σ ; ϵ ) as a function of r c for various values of σ for an annular pupil with ϵ = 0.25. S = 0.1983 when σ = 2 .

Fig. 4
Fig. 4

Same as Fig. 3, except that ϵ = 0.5. S = 0.2434 when σ = 2 .

Fig. 5
Fig. 5

Same as Fig. 3, except that ϵ = 0.75. S = 0.3897 when σ = 2 .

Fig. 6
Fig. 6

Same as Fig. 3, except that ϵ = 0.9 and larger values of σ are used. S = 0.3638 when σ = 5 .

Fig. 7
Fig. 7

Same as Fig. 6, except that the PSFs are not normalized to unity at the center.

Tables (2)

Tables Icon

Table 1 Time-Averaged Strehl Ratio 〈 S (σ; ϵ)〉 for Annular Pupils with an Obscuration Ratio ϵ for Various Values of the Standard Deviation σ of the Longitudinal Image Motion in Units of 8λ F 2

Tables Icon

Table 2 Table 1. (Continued)

Equations (13)

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I ( r ; z ; ϵ ) = [ 2 R z ( 1 ϵ 2 ) ] 2 | ϵ 1 exp ( i B d ρ 2 ) J 0 ( π r ρ R / z ) ρ d ρ | 2 ,
B d ( z ) = π N ( R / z 1 )
N = a 2 / λ R
I ( r ; R ; ϵ ) = 1 ( 1 ϵ 2 ) 2 [ 2 J 1 ( π r ) π r ϵ 2 2 J 1 ( π ϵ r ) π ϵ r ] 2 .
I ( r ) = [ 2 J 1 ( π r ) / π r ] 2 .
S ( B d ) = { sin [ B d ( 1 ϵ 2 ) / 2 ] B d ( 1 ϵ 2 ) / 2 } 2 .
B d = π Δ / 4 λ F 2 ,
I ( r ; Δ ; ϵ ) = ( 2 1 ϵ 2 ) 2 | ϵ 1 exp ( 2 π i Δ ρ 2 ) J 0 ( π r ρ ) ρ d ρ | 2 ,
p ( Δ ) = 1 2 π σ exp ( Δ 2 / 2 σ 2 ) ,
I ( r ; σ ; ϵ ) = 1 2 π σ I ( r ; Δ ; ϵ ) exp ( Δ 2 / 2 σ 2 ) d Δ .
S ( σ ; ϵ ) = 1 2 π σ { sin [ π Δ ( 1 ϵ 2 ) ] π Δ ( 1 ϵ 2 ) } 2 × exp ( Δ 2 / 2 σ 2 ) d Δ .
P ( r c ; σ ; ϵ ) = π 2 2 ( 1 ϵ 2 ) 0 r c I ( r ; σ ; ϵ ) r d r ,
P ( r c ) = 1 J 0 2 ( π r c ) J 1 2 ( π r c ) ,

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