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.

Virendra N. Mahajan and Guang-ming Dai J. Opt. Soc. Am. A 24(9) 2994-3016 (2007)

References

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Typical Values of x_{m}, x_{p}, and 〈x〉 in Units of λF and the Corresponding Irradiances I_{m}, I_{p}, and I_{c} in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by Primary Coma^{a}

W_{3}

x_{m}

x_{p}

〈x〉

I_{m}

I_{p}

I_{c}

I(0)

0

0

0

0

1.0000

1.0000

1.0000

1.0000

(0)

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

(1.0000)

0.5

0.67

0.66

1.00

0.8712

0.8712

0.6535

0.3175

(0.78)

(0.78)

(1.50)

(0.9283)

(0.9283)

(0.0524)

(0.0403)

1.0

1.33

1.30

2.00

0.5708

0.5717

0.1445

0.0791

(1.56)

(1.55)

(3.00)

(0.7410)

(0.7412)

(0.1357)

(0.0319)

1.5

2.00

1.80

3.00

0.2715

0.2844

0.0004

0.0618

(2.33)

(2.32)

(4.50)

(0.5063)

(0.5064)

(0.0139)

(0.0010)

2.0

2.67

1.57

4.00

0.0864

0.1978

0.0061

0.0341

(3.11)

(3.07)

(6.00)

(0.2936)

(0.2946)

(0.0160)

(0.0000)

W(h, 0) = W_{3}ρ^{3} cos θ. The units of W_{3} are λ. The aberrated central irradiance I(0) is also given here. The numbers without parentheses are for a circular pupil (∊ = 0), and those with parentheses are for an annular pupil with ∊^{2} = 0.5.

Table 2

Typical Values of x_{m}, x_{p}, and 〈x〉 in Units of λ F and Corresponding Irradiances I_{m}, I_{p}, and I_{c} in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by Secondary Coma

W_{5}

x_{m}

x_{p}

〈x〉

I_{m}

I_{p}

I_{c}

I(0)

0

0

0

0

1.0000

1.0000

1.0000

1.0000

(0)

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

(1.0000)

0.5

0.50

0.49

1.00

0.8150

0.8153

0.4114

0.4955

(0.63)

(0.62)

(1.75)

(0.8400)

(0.8402)

(0.0760)

(0.1768)

1.0

1.00

0.83

2.00

0.4464

0.4664

0.0025

0.2332

(1.25)

(1.21)

(3.50)

(0.4948)

(0.4966)

(0.0282)

(0.0002)

1.5

1.50

0.81

3.00

0.1685

0.3237

0.0098

0.1873

(1.88)

(1.74)

(5.25)

(0.2003)

(0.2196)

(0.0130)

(0.0009)

2.0

2.00

1.11

4.00

0.0420

0.2523

0.0073

0.1389

(2.50)

(1.71)

(7.00)

(0.0573)

(0.1478)

(0.0074)

(0.0065)

Same as Table 1, except that the aberration is secondary coma, W(h, θ) = W_{5}ρ^{5} cos θ, where W_{5} is in units of λ.

Table 3

Typical Values of x_{m}, x_{p}, and 〈x〉 in Units of λF and Corresponding Irradiances I_{m}, I_{p}, and I_{c} in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by a Combination of Primary and Secondary Coma^{a}

W_{5}

x_{m}

x_{p}

〈x〉

I_{m}

I_{p}

I_{c}

I(0)

0

0

0

0

1.0000

1.0000

1.0000

1.0000

(0)

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

(1.0000)

1.0

−0.60

−0.59

−0.40

0.9676

0.9682

0.8763

0.3721

(5.0)

(−5.60)

(−5.60)

(−5.35)

(0.8832)

(0.8832)

(0.0005)

(0.0039)

2.0

−1.20

−1.18

−0.80

0.8765

0.8784

0.5870

0.0030

(10.0)

(−11.19)

(−11.23)

(−10.69)

(0.6101)

(0.6128)

(0.0000)

(0.0014)

3.0

−1.80

−1.77

−1.20

0.7429

0.7459

0.2981

0.0014

(15.0)

(−16.79)

(−16.94)

(−16.04)

(0.3353)

(0.3558)

(0.0000)

(0.0000)

4.0

−2.40

−2.37

−1.60

0.5886

0.5914

0.1173

0.0465

(20.0)

(−22.38)

(−22.80)

(−21.38)

(0.1493)

(0.2296)

(0.0000)

(0.0008)

Same as Table 1, except that the aberration is a combination of primary and secondary coma given by Eqs. (57).

Table 4

Typical Values of x_{m}, x_{p}, and 〈x〉 in Units of λF and Corresponding Irradiances I_{m}, I_{p}, and I_{c} in Units of the Aberration-Free Central Irradiance of PSF’s for Gaussian Pupils with γ = 1 Aberrated by Primary Coma^{a}

W_{5}

x_{m}

x_{p}

〈x〉

I_{m}

I_{p}

I_{c}

I(0)

0

0

0

0

1.0000

1.0000

1.0000

1.0000

(0)

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

(1.0000)

0.50

0.61

0.60

0.69

0.8805

0.8806

0.8670

0.4567

(0.76)

(0.76)

(1.42)

(0.9288)

(0.9288)

(0.1126)

(0.0602)

1.00

1.22

1.15

1.37

0.6013

0.6062

0.5590

0.1708

(1.51)

(1.51)

(2.84)

(0.7435)

(0.7435)

(0.1273)

(0.0348)

1.50

1.82

1.40

2.06

0.3205

0.3672

0.2479

0.1199

(2.27)

(2.24)

(4.25)

(0.5112)

(0.5122)

(0.0014)

(0.0033)

2.00

2.43

1.46

2.75

0.1305

0.2947

0.0624

0.0773

(3.03)

(2.93)

(5.67)

(0.3005)

(0.3065)

(0.0399)

(0.0000)

Same as Table 1, except that the irradiances given here are normalized by the aberration-free central irradiance I(0; 1;∊) given by Eq. (63).

Table 5

Typical Values of x_{p} and 〈 x〉 in Units of λF and Corresponding Irradiances I_{p} and I_{c} in Units of Aberration-Free Central Irradiance of PSF’s for Gaussian Pupils Aberrated by Secondary Coma^{a}

W_{5}

x_{p}

〈x〉

I_{p}

I_{c}

I(0)

0

0

0

1.0000

1.0000

1.0000

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

0.50

0.41

0.56

0.8452

0.8105

0.6322

(0.59)

(1.57)

(0.8451)

(0.0123)

(0.2253)

1.00

0.64

1.12

0.5659

0.4161

0.3793

(1.14)

(3.14)

(0.5144)

(0.0026)

(0.0025)

1.50

0.63

1.68

0.4541

0.1147

0.3083

(1.49)

(4.70)

(0.2595)

(0.0075)

(0.0011)

2.00

0.74

2.24

0.3824

0.0075

0.2476

(1.67)

(6.27)

(0.1892)

(0.0043)

(0.0084)

Same as Table 2, except that the irradiances given here are normalized by the aberration-free central irradiance I(0; 1; ∊) given by Eq. (63).

Tables (5)

Table 1

Typical Values of x_{m}, x_{p}, and 〈x〉 in Units of λF and the Corresponding Irradiances I_{m}, I_{p}, and I_{c} in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by Primary Coma^{a}

W_{3}

x_{m}

x_{p}

〈x〉

I_{m}

I_{p}

I_{c}

I(0)

0

0

0

0

1.0000

1.0000

1.0000

1.0000

(0)

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

(1.0000)

0.5

0.67

0.66

1.00

0.8712

0.8712

0.6535

0.3175

(0.78)

(0.78)

(1.50)

(0.9283)

(0.9283)

(0.0524)

(0.0403)

1.0

1.33

1.30

2.00

0.5708

0.5717

0.1445

0.0791

(1.56)

(1.55)

(3.00)

(0.7410)

(0.7412)

(0.1357)

(0.0319)

1.5

2.00

1.80

3.00

0.2715

0.2844

0.0004

0.0618

(2.33)

(2.32)

(4.50)

(0.5063)

(0.5064)

(0.0139)

(0.0010)

2.0

2.67

1.57

4.00

0.0864

0.1978

0.0061

0.0341

(3.11)

(3.07)

(6.00)

(0.2936)

(0.2946)

(0.0160)

(0.0000)

W(h, 0) = W_{3}ρ^{3} cos θ. The units of W_{3} are λ. The aberrated central irradiance I(0) is also given here. The numbers without parentheses are for a circular pupil (∊ = 0), and those with parentheses are for an annular pupil with ∊^{2} = 0.5.

Table 2

Typical Values of x_{m}, x_{p}, and 〈x〉 in Units of λ F and Corresponding Irradiances I_{m}, I_{p}, and I_{c} in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by Secondary Coma

W_{5}

x_{m}

x_{p}

〈x〉

I_{m}

I_{p}

I_{c}

I(0)

0

0

0

0

1.0000

1.0000

1.0000

1.0000

(0)

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

(1.0000)

0.5

0.50

0.49

1.00

0.8150

0.8153

0.4114

0.4955

(0.63)

(0.62)

(1.75)

(0.8400)

(0.8402)

(0.0760)

(0.1768)

1.0

1.00

0.83

2.00

0.4464

0.4664

0.0025

0.2332

(1.25)

(1.21)

(3.50)

(0.4948)

(0.4966)

(0.0282)

(0.0002)

1.5

1.50

0.81

3.00

0.1685

0.3237

0.0098

0.1873

(1.88)

(1.74)

(5.25)

(0.2003)

(0.2196)

(0.0130)

(0.0009)

2.0

2.00

1.11

4.00

0.0420

0.2523

0.0073

0.1389

(2.50)

(1.71)

(7.00)

(0.0573)

(0.1478)

(0.0074)

(0.0065)

Same as Table 1, except that the aberration is secondary coma, W(h, θ) = W_{5}ρ^{5} cos θ, where W_{5} is in units of λ.

Table 3

Typical Values of x_{m}, x_{p}, and 〈x〉 in Units of λF and Corresponding Irradiances I_{m}, I_{p}, and I_{c} in Units of the Aberration-Free Central Irradiance of PSF’s for Uniformly Illuminated Pupils Aberrated by a Combination of Primary and Secondary Coma^{a}

W_{5}

x_{m}

x_{p}

〈x〉

I_{m}

I_{p}

I_{c}

I(0)

0

0

0

0

1.0000

1.0000

1.0000

1.0000

(0)

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

(1.0000)

1.0

−0.60

−0.59

−0.40

0.9676

0.9682

0.8763

0.3721

(5.0)

(−5.60)

(−5.60)

(−5.35)

(0.8832)

(0.8832)

(0.0005)

(0.0039)

2.0

−1.20

−1.18

−0.80

0.8765

0.8784

0.5870

0.0030

(10.0)

(−11.19)

(−11.23)

(−10.69)

(0.6101)

(0.6128)

(0.0000)

(0.0014)

3.0

−1.80

−1.77

−1.20

0.7429

0.7459

0.2981

0.0014

(15.0)

(−16.79)

(−16.94)

(−16.04)

(0.3353)

(0.3558)

(0.0000)

(0.0000)

4.0

−2.40

−2.37

−1.60

0.5886

0.5914

0.1173

0.0465

(20.0)

(−22.38)

(−22.80)

(−21.38)

(0.1493)

(0.2296)

(0.0000)

(0.0008)

Same as Table 1, except that the aberration is a combination of primary and secondary coma given by Eqs. (57).

Table 4

Typical Values of x_{m}, x_{p}, and 〈x〉 in Units of λF and Corresponding Irradiances I_{m}, I_{p}, and I_{c} in Units of the Aberration-Free Central Irradiance of PSF’s for Gaussian Pupils with γ = 1 Aberrated by Primary Coma^{a}

W_{5}

x_{m}

x_{p}

〈x〉

I_{m}

I_{p}

I_{c}

I(0)

0

0

0

0

1.0000

1.0000

1.0000

1.0000

(0)

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

(1.0000)

0.50

0.61

0.60

0.69

0.8805

0.8806

0.8670

0.4567

(0.76)

(0.76)

(1.42)

(0.9288)

(0.9288)

(0.1126)

(0.0602)

1.00

1.22

1.15

1.37

0.6013

0.6062

0.5590

0.1708

(1.51)

(1.51)

(2.84)

(0.7435)

(0.7435)

(0.1273)

(0.0348)

1.50

1.82

1.40

2.06

0.3205

0.3672

0.2479

0.1199

(2.27)

(2.24)

(4.25)

(0.5112)

(0.5122)

(0.0014)

(0.0033)

2.00

2.43

1.46

2.75

0.1305

0.2947

0.0624

0.0773

(3.03)

(2.93)

(5.67)

(0.3005)

(0.3065)

(0.0399)

(0.0000)

Same as Table 1, except that the irradiances given here are normalized by the aberration-free central irradiance I(0; 1;∊) given by Eq. (63).

Table 5

Typical Values of x_{p} and 〈 x〉 in Units of λF and Corresponding Irradiances I_{p} and I_{c} in Units of Aberration-Free Central Irradiance of PSF’s for Gaussian Pupils Aberrated by Secondary Coma^{a}

W_{5}

x_{p}

〈x〉

I_{p}

I_{c}

I(0)

0

0

0

1.0000

1.0000

1.0000

(0)

(0)

(1.0000)

(1.0000)

(1.0000)

0.50

0.41

0.56

0.8452

0.8105

0.6322

(0.59)

(1.57)

(0.8451)

(0.0123)

(0.2253)

1.00

0.64

1.12

0.5659

0.4161

0.3793

(1.14)

(3.14)

(0.5144)

(0.0026)

(0.0025)

1.50

0.63

1.68

0.4541

0.1147

0.3083

(1.49)

(4.70)

(0.2595)

(0.0075)

(0.0011)

2.00

0.74

2.24

0.3824

0.0075

0.2476

(1.67)

(6.27)

(0.1892)

(0.0043)

(0.0084)

Same as Table 2, except that the irradiances given here are normalized by the aberration-free central irradiance I(0; 1; ∊) given by Eq. (63).