Abstract

The line of sight of an aberrated optical system is defined in terms of the centroid of its point-spread function (PSF). It is also expressed in terms of its optical transfer function as well as its pupil function. Whereas the PSF’s obtained according to wave-diffraction optics and ray geometrical optics are quite different from each other, they have the same centroid. Although different amplitude distributions across an aberration-free pupil give the same centroid location, in the case of an aberrated pupil not only the phase but also the amplitude distribution affects the centroid location. If the amplitude across a pupil is uniform, then the centroid may be obtained from the aberration only along the perimeter of the pupil, without regard for the aberration across its interior irrespective of its shape. Next, an optical system with aberrated but uniformly illuminated annular pupil is considered. The aberration function is expanded in terms of Zernike annular polynomials. It is shown that only those aberrations that vary with angle as cos θ or sin θ contribute to the line of sight. A simple expression is obtained for the line of sight in terms of the Zernike aberration coefficients. Similar results are obtained for annular pupils with radially symmetric illumination. Finally, as numerical examples, some specific results are discussed for annular pupils aberrated by classical primary and secondary coma. As an example of a radially symmetric illumination, we obtain numerical results for Gaussian illumination of aberrated annular pupils. It is emphasized that the centroid and the peak of the PSF’s aberrated by coma are not coincident. Moreover, as the amount of the aberration increases, the separation of the centroid and peak locations also increases.

© 1985 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.
  2. e.g., see B. Tatian, “Asymptotic expansions for correcting truncation error in transfer-function calculations,” J. Opt. Soc. Am. 61, 1214–1224 (1971);V. N. Mahajan, “Asymptotic behavior of diffraction images,” Can. J. Phys. 57, 1426–1431 (1979).
    [CrossRef]
  3. It is not difficult to show that Eq. (10) may also be obtained by substituting Eqs. (1) into Eqs. (2);see V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 287.
  4. See M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 206.see, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), Sec.6.2.
  5. R. V. Shack, “Interaction of an optical system with the incoming wavefront in the presence of atmospheric turbulence,” Tech. Rep. No. 19 (Optical Sciences Center, University of Arizona, Tucson, Ariz., August1967).
  6. C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
    [CrossRef]
  7. V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981);J. Opt. Soc. Am. 71, 1408 (1981);J. Opt. Soc. Am. A1, 685 (1984).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 469.
  9. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 416.
  10. V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am, 72, 1258–1266 (1982).
    [CrossRef]
  11. R. Barakat, A. Houston, “Diffraction effects of coma,” J. Opt. Soc. Am. 54, 1084–1088 (1964).
    [CrossRef]
  12. V. N. Mahajan, “Luneburg apodization problem I,” Opt. Lett. 5, 267–269 (1980).
    [CrossRef] [PubMed]
  13. V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A 1, 1316 (A) (1984).

1984 (1)

V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A 1, 1316 (A) (1984).

1982 (1)

V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am, 72, 1258–1266 (1982).
[CrossRef]

1981 (1)

1980 (1)

1976 (1)

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

1971 (1)

1964 (1)

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 416.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

See M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 206.see, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), Sec.6.2.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 469.

Butts, R. R.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

Hogge, C. B.

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

Houston, A.

Mahajan, V. N.

V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A 1, 1316 (A) (1984).

V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am, 72, 1258–1266 (1982).
[CrossRef]

V. N. Mahajan, “Zernike annular polynomials for imaging systems with annular pupils,” J. Opt. Soc. Am. 71, 75–85 (1981);J. Opt. Soc. Am. 71, 1408 (1981);J. Opt. Soc. Am. A1, 685 (1984).
[CrossRef]

V. N. Mahajan, “Luneburg apodization problem I,” Opt. Lett. 5, 267–269 (1980).
[CrossRef] [PubMed]

Shack, R. V.

R. V. Shack, “Interaction of an optical system with the incoming wavefront in the presence of atmospheric turbulence,” Tech. Rep. No. 19 (Optical Sciences Center, University of Arizona, Tucson, Ariz., August1967).

Tatarski, V. I.

It is not difficult to show that Eq. (10) may also be obtained by substituting Eqs. (1) into Eqs. (2);see V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 287.

Tatian, B.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 416.

See M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 206.see, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), Sec.6.2.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 469.

IEEE Trans. Antennas Propag. (1)

C. B. Hogge, R. R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. AP-24, 144–154 (1976).
[CrossRef]

J. Opt. Soc. Am (1)

V. N. Mahajan, “Strehl ratio for primary aberrations: some analytical results for circular and annular pupils,” J. Opt. Soc. Am, 72, 1258–1266 (1982).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

V. N. Mahajan, “Line of sight of an aberrated optical system,” J. Opt. Soc. Am. A 1, 1316 (A) (1984).

Opt. Lett. (1)

Other (6)

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 482.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 469.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 416.

It is not difficult to show that Eq. (10) may also be obtained by substituting Eqs. (1) into Eqs. (2);see V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971), p. 287.

See M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, New York, 1975), p. 206.see, for example, W. T. Welford, Aberrations of the Symmetrical Optical System (Academic, New York, 1974), Sec.6.2.

R. V. Shack, “Interaction of an optical system with the incoming wavefront in the presence of atmospheric turbulence,” Tech. Rep. No. 19 (Optical Sciences Center, University of Arizona, Tucson, Ariz., August1967).

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Figures (7)

Fig. 1
Fig. 1

PSF I(xs; ∊) for several typical values of primary-coma aberration W3 in units of λ. The amplitude A(u, υ) across the pupil is uniform. The PSF’s are normalized by the aberration-free central value given by Eq. (41). xs represents x in units of λF. (a) = 0,(b) 2 = 0.5.

Fig. 2
Fig. 2

Variation of Im, Ip, and Ic with W3 where the irradiances are in units of the aberration-free central irradiance and W3 is in units of λ. (a) = 0, (b) 2 = 0.5.

Fig. 3
Fig. 3

Variation of xm, xp, and 〈x〉 with W3. The x values are in units of λF and W3 is in units of λ. (a) = 0, (b) 2 = 0.5.

Fig. 4
Fig. 4

Same as Fig. 1 except that the aberration is secondary coma W5.

Fig. 5
Fig. 5

Same as Fig. 1 except that the aberration is a combination of primary and secondary coma given by Eqs. (57). Note that in this figure the horizontal coordinate is xsxm.

Fig. 6
Fig. 6

Same as Fig. 1 except that the amplitude across the pupil is Gaussian, given by Eqs. (61) with γ = 1. The PSF’s are normalized by the aberration-free central irradiance I(0; 1, ) given by Eq. (63).

Fig. 7
Fig. 7

Same as Fig. 6, except that the aberration is secondary coma W5.

Tables (5)

Tables Icon

Table 1 Typical Values of xm, xp, and 〈x〉 in Units of λF and the Corresponding Irradiances Im, Ip, and Ic in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by Primary Comaa

Tables Icon

Table 2 Typical Values of xm, xp, and 〈x〉 in Units of λ F and Corresponding Irradiances Im, Ip, and Ic in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by Secondary Coma

Tables Icon

Table 3 Typical Values of xm, xp, and 〈x〉 in Units of λF and Corresponding Irradiances Im, Ip, and Ic in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by a Combination of Primary and Secondary Comaa

Tables Icon

Table 4 Typical Values of xm, xp, and 〈x〉 in Units of λF and Corresponding Irradiances Im, Ip, and Ic in Units of the Aberration-Free Central Irradiance of PSF’s for Gaussian Pupils with γ = 1 Aberrated by Primary Comaa

Tables Icon

Table 5 Typical Values of xp and 〈 x〉 in Units of λF and Corresponding Irradiances Ip and Ic in Units of Aberration-Free Central Irradiance of PSF’s for Gaussian Pupils Aberrated by Secondary Comaa

Equations (99)

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I ( x , y ) = ( 1 / λ 2 R 2 ) | S P ( u , υ ) exp [ 2 π i ( x u + y υ ) / λ R ] d u d υ | 2 ,
P ( u , υ ) = A ( u , υ ) exp [ 2 π i W ( u , υ ) / λ ] inside the pupil , = 0 outside the pupil .
x = E 1 x I ( x , y ) d x d y
y = E 1 y I ( x , y ) d x d y ,
E = I ( x , y ) d x d y
E = S | P ( u , υ ) | 2 d u d υ = S I ( u , υ ) d u d υ ,
I ( u , υ ) = A 2 ( u , υ )
α = x / R
β = y / R ,
τ ( ξ , η ) = E 1 I ( x , y ) exp [ 2 π i ( ξ x + η y ) ] d x d y .
x = 1 2 π i ( τ ξ ) ξ = η = 0 .
y = 1 2 π i ( τ η ) ξ = η = 0 .
x = 1 2 π ( Im τ ξ ) ξ = η = 0
y = 1 2 π ( Im τ η ) ξ = η = 0 .
τ ( ξ , η ) = E 1 Σ P ( u , υ ) P * ( u λ R ξ , υ λ R η ) ] d u d υ ,
x = ( λ R / 2 π E ) S Im [ P ( u , υ ) P * ( u , υ ) u ] d u d υ
x = ( R / E ) S I ( u , υ ) W ( u , υ ) u d u d υ .
y = ( R / E ) S I ( u , υ ) W ( u , υ ) u d u d υ .
f ( u , υ ) = P ( u , υ ) exp [ 2 π i ( x u + y υ ) / λ R ] ,
I ( x , y ) I ( 0 , 0 ) = | S f ( u , υ ) d u d υ | 2 S | f ( u , υ ) | 2 d u d υ 1 .
A ( u , υ ) = A 0
I ( u , υ ) = A 0 2 = I 0 ,
E = S I 0 ,
x = ( R / S ) S W ( u , υ ) u d u d υ
y = ( R / S ) S W ( u , υ ) u d u d υ .
x = ( R / S ) W ( u , υ ) u ̂ d s
y = ( R / S ) W ( u , υ ) υ ̂ d s ,
S = π ( 1 2 ) a 2 .
u = h cos θ
υ = h sin θ ,
θ = tan 1 ( υ / u )
h = ( u 2 + υ 2 ) 1 / 2 .
d s = ( u d θ , υ d θ ) .
x = [ R / π ( 1 2 ) a ] 0 2 π [ W ( a , θ ; ) W ( a , θ ; ) ] × cos θ d θ
y = [ R / π ( 1 2 ) a ] 0 2 π [ W ( a , θ ; ) W ( a , θ ; ) ] × sin θ d θ .
ρ = h / a .
W ( h , θ ; ) = n = 0 m = 0 n m 2 ( n + 1 ) R n m ( ρ ; ) × ( c n m cos m θ + s n m sin ) m θ ,
m = 1 / 2 m = 0 , = 1 , m 0 ,
s n 0 = 0 .
1 R n m ( ρ ; ) R n m ( ρ ; ) ρ d ρ = ( 1 2 ) 2 ( n + 1 ) δ n n ,
x = [ R / ( 1 2 ) a ] n = 1 2 ( n + 1 ) × [ R n 1 ( 1 ; ) R n 1 ( ; ) ] c n 1
y = [ R / ( 1 2 ) a ] n = 1 2 ( n + 1 ) × [ R n 1 ( 1 ; ) R n 1 ( ; ) ] s n 1 ,
R n 1 ( 1 ; 0 ) = 1 .
x = [ R / a ] n = 1 2 ( n + 1 ) c n 1
y = [ R / a ] n = 1 2 ( n + 1 ) s n 1 .
I ( h ) = A 2 ( h ) .
x = ( R / E ) a a 0 2 π I ( h ) [ cos θ W ( h , θ ; ) h sin θ h W ( h , θ ; ) θ ] h d h d θ
y = ( R / E ) a a 0 2 π I ( h ) [ sin θ W ( h , θ ; ) h + cos θ h W ( h , θ ; ) θ ] h d h d θ ,
E = 2 π a a I ( h ) h d h d θ .
W ( h , θ ; ) = n = 0 m = 0 n m 2 ( n + 1 ) S n m ( ρ ; ) × ( c n m cos m θ + s n m sin m θ ) ,
S n m ( ρ ; ) = M n m [ R n m ( ρ ; ) i 1 ( n m ) / 2 ( n 2 i + 1 ) × R n m ( ρ ; ) S n 2 i m ( ρ ; ) S n 2 i m ( ρ ; ) ]
R n m ( ρ ; ) S n 2 i m ( ρ ; ) = 1 0 2 π R n m ( ρ ; ) × S n 2 i m ( ρ ; ) A ( a ρ ) ρ d ρ / 1 0 2 π A ( a ρ ) ρ d ρ ,
1 S n m ( ρ ; ) S n m ( ρ ; ) A ( ρ ) ρ d ρ / 1 A ( ρ ) ρ d ρ = [ 1 / ( n + 1 ) ] δ n n .
x = ( π a R ) / ( E ) n = 1 2 ( n + 1 ) c n 1 × 1 I ( a ρ ) ρ [ ρ S n 1 ( ρ ; ) ] d ρ
y = ( π a R ) n = 1 2 ( n + 1 ) s n 1 × 1 I ( a ρ ) ρ [ ρ S n 1 ( ρ ; ) ] d ρ .
x = r cos ϕ
y = r sin ϕ ,
I ( r ; ) = [ 1 / ( 1 2 ) 2 ] [ 2 J 1 ( π r s ) π r s 2 2 J 1 ( π r s ) π r s ] 2 I ( 0 ; ) .
r = ( x 2 + y 2 ) 1 / 2
r s = r / λ F
F = R / 2 a
I ( 0 ; ) = E S / λ 2 R 2 ,
W ( h , θ ) = W n ( h / a ) n cos θ , a h a ,
= W n ρ n cos θ , ρ 1 ,
x = 2 W n F i = 0 ( n 1 ) / 2 2 i .
α = 2 ( W n / D ) i = 0 ( n 1 ) / 2 2 i ,
D = 2 a
x = 2 W 1 F ,
α = 2 W 1 / D .
x = 2 W 3 F ( 1 + 2 ) .
R 3 1 ( ρ ; ) = 3 ( 1 + 2 ) ρ 3 2 ( 1 + 2 + 4 ) ρ ( 1 2 ) [ ( 1 + 2 ) ( 1 + 4 2 + 4 ) ] 1 / 2 ,
x m = 4 W 3 F ( 1 + 2 + 4 ) / 3 ( 1 + 2 ) ,
I ( x ; ) = [ I ( 0 ; ) / ( 1 2 ) 2 ] [ 2 1 J 0 ( π B ) d t ] 2 ,
B = ( 2 t W 3 x s ) t 1 / 2 ,
x s = x / λ F ,
x = 2 W 5 F ( 1 + 2 + 4 ) .
x m = W 5 F ( 1 + 2 + 4 + 6 ) / ( 1 + 2 ) .
B = ( 2 t 2 W 5 x s ) t 1 / 2
R 5 1 ( ρ ; ) = 10 ( 1 + 4 2 + 4 ) ρ 5 12 ( 1 + 4 2 + 4 4 + 6 ) ρ 3 + 3 ( 1 + 4 2 + 10 4 + 4 6 + 8 ) ρ ( 1 2 ) 2 [ ( 1 + 4 2 + 4 ) ( 1 + 9 2 + 9 4 + 6 ) ] 1 / 2 .
W ( h , θ ) = ( W 5 ρ 5 + W 3 ρ 3 ) cos θ ,
W 3 = 1.2 W 5 ( 1 + 4 2 + 4 4 + 6 ) / ( 1 + 4 2 + 4 ) .
x m = 0.6 W 5 F ( 1 + 4 2 + 10 4 + 4 6 + 8 ) / ( 1 + 4 2 + 4 ) .
x = W 5 F ( 0.4 + 2 2 + 7.2 4 + 2 6 + 0.4 8 ) / ( 1 + 4 2 + 4 ) .
B = ( 2 t 2 W 5 + 2 t W 3 x s ) t 1 / 2 .
A ( h ) = A 0 exp [ γ ( h / a ) 2 ]
= A 0 exp ( γ ρ 2 ) , ρ 1 ,
I ( r ; γ ; ) = [ γ / ( e γ e γ 2 ) ] 2 I ( 0 ; γ ; ) × [ 2 1 exp ( γ t ) J 0 ( π r s t 1 / 2 ) d t ] 2 ,
I ( 0 ; γ ; ) = ( π a 2 A 0 2 / λ R ) 2 [ e γ e γ 2 ) / γ ] 2
I ( x ; γ ; ) = [ γ / ( e γ e γ 2 ) ] 2 I ( 0 ; γ ; ) × [ 2 1 exp ( γ t ) J 0 ( π B ) d t ] 2 ,
x = 4 W 3 F [ 1 2 γ + 2 exp ( 2 γ 2 ) exp ( 2 γ ) exp ( 2 γ 2 ) exp ( 2 γ ) ] .
x m = 2 W 3 F [ ( 2 / γ ) + γ [ 4 e γ 2 e γ e γ 2 ( 1 + γ 2 ) e γ ( 1 + γ ) ] .
x = 12 W 5 F exp ( 2 γ 2 ) ( 4 + 2 / γ + 1 / 2 γ 2 ) exp ( 2 γ ) ( 1 + 1 / γ + 1 / 2 γ 2 ) exp ( 2 γ 2 ) exp ( 2 γ ) .
W ( h , θ ) = n ρ n ( W n cos θ + W n sin θ ) ,
I ( r , ϕ ; ) = ( a 2 / λ R ) 2 | 1 0 2 π A ( h ) exp { π i [ 2 W ( h , θ ) r s ρ cos ( θ ϕ ) ] } ρ d ρ d θ | 2 ,
2 W ( h , θ ) r s ρ cos ( θ ϕ ) = B cos ( θ ψ ) ,
B 2 = ( n 2 W n ρ n r s ρ cos ϕ ) 2 + ( n 2 W n ρ n r s ρ sin ϕ ) 2
tan ψ = ( n 2 W n ρ n r s sin ϕ ) / ( n 2 W n ρ n r s cos ϕ ) .
I ( r , ϕ ; ) = ( π a 2 / λ R ) 2 [ 2 1 A ( h ) J 0 ( π B ) d t ] 2 ,
t = ρ 2 .

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