Zernike-Gauss Polynomials and Optical Aberrations of Systems with Gaussian Pupils

Virendra N. Mahajan

doi:10.1364/AO.34.008057

Applied Optics
Vol. 34,
Issue 34,
pp. 8057-8059
(1995)
•https://doi.org/10.1364/AO.34.008057

Zernike-Gauss Polynomials and Optical Aberrations of Systems with Gaussian Pupils

Virendra N. Mahajan

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Virendra N. Mahajan, "Zernike-Gauss Polynomials and Optical Aberrations of Systems with Gaussian Pupils," Appl. Opt. 34, 8057-8059 (1995)

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Abstract

In the first two Notes of this series,1,2 we discussed Zernike circle and annular polynomials that represent optimally balanced classical aberrations of systems with uniform circular or annular pupils, respectively. Here we discuss Zernike-Gauss polynomials which are the corresponding polynomials for systems with Gaussian circular or annular pupils.3–5 Such pupils, called apodized pupils, are used in optical imaging to reduce the secondary rings of the point-spread functions of uniform pupils.6 Propagation of Gaussian laser beams also involves such pupils.

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Equations (13)

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Aberration	Radial Polynomial	Uniform	Gaussian	Weakly Truncated Gaussian ∊ = O
Piston	$R_{0}^{0}$	1	1	1
Distortion (tilt)	$R_{1}^{1}$	ρ/(1 + ∊ ²)^1/2	$a_{1}^{1} ρ$	$\sqrt{γ / 2 ρ}$
Field curvature (defocus)	$R_{2}^{0}$	(2ρ ² − 1 ∊ ²)/(l − ∊ ²)	$a_{2}^{0} ρ^{2} + b_{2}^{0}$	$(γ ρ^{2} - 1) / \sqrt{3}$
Astigmatism	$R_{2}^{2}$	ρ ²/(1 + ∊ ² + ∊ ⁴)^1/2	$a_{2}^{2} ρ^{2}$	$(γ / \sqrt{6}) ρ^{2}$
Coma	$R_{3}^{1}$	$\frac{3 (1 + ∊^{2}) ρ^{3} - 2 (1 + ∊^{2} + ∊^{4}) ρ}{(1 - ∊^{2}) {[(1 + ∊^{2}) (1 + 4 ∊^{2} + ∊^{4})]}^{1 / 2}}$	$a_{3}^{1} ρ^{3} + b_{3}^{1} ρ$	$\sqrt{γ / 2} (\frac{γ}{2} ρ^{3} - ρ)$
Spherical aberration	$R_{4}^{0}$	[6ρ ⁴ − 6(1 + ∊ ²)ρ ² + (1 + 4∊ ² + ∊ ⁴)]/(1 − ∊ ²)²	$a_{4}^{0} ρ^{4} + b_{4}^{0} ρ^{2} + c_{4}^{0}$	$(γ^{2} ρ^{4} - 4 γ ρ^{2} + 2) / 2 \sqrt{5}$

∊	$a_{1}^{1}$	$a_{2}^{0}$	$b_{2}^{0}$	$a_{2}^{2}$	$a_{3}^{1}$	$b_{3}^{1}$	$a_{4}^{0}$	$b_{4}^{0}$	$c_{4}^{0}$
0.00	1.09367 (1.00000)	2.04989 (2.00000)	−0.85690 (−1.00000)	1.14541 (1.00000)	3.11213 (3.00000)	−1.89152 (−2.00000)	6.12902 (6.00000)	−5.71948 (−6.00000)	0.83368 (1.00000)
0.25	1.04364 (0.97014)	2.18012 (2.13333)	−1.00080 (−1.13333)	1.08940 (0.96836)	3.01573 (2.94566)	− 1.84513 (−1.97099)	6.95563 (6.82667)	− 6.98197 (−7.25333)	1.25153 (1.42667)
0.50	0.92963 (0.89443)	2,70412 (2.66667)	−1.56449 (−1.66667)	0.93620 (0.87287)	3.14319 (3.11400)	− 2.06618 (−2.17980)	10.79549 (10.66667)	−13.08900 (−13.33333)	3.46706 (3.66667)
0.75	0.80827 (0.80000)	4.59329 (4.57143)	−3.51548 (−3.57143)	0.74439 (0.72954)	4.55179 (4.53877)	−3.57767 (−3.63858)	31.47559 (31.34694)	− 48.77879 (−48.97959)	18.39840 (18,63265)
0.90	0.74453 (0.74329)	10.53581 (10.52632)	−9.50324 (−9.52632)	0.63890 (0.63679)	9.60573 (9.60023)	−8.69629 (−8.72012)	166.33357 (166,20499)	−300.66342 (−300.83102)	135.36926 (135.62604)

Aberration	Radial Polynomial	Uniform	Gaussian	Weakly Truncated Gaussian ∊ = O
Piston	$R_{0}^{0}$	1	1	1
Distortion (tilt)	$R_{1}^{1}$	ρ/(1 + ∊ ²)^1/2	$a_{1}^{1} ρ$	$\sqrt{γ / 2 ρ}$
Field curvature (defocus)	$R_{2}^{0}$	(2ρ ² − 1 ∊ ²)/(l − ∊ ²)	$a_{2}^{0} ρ^{2} + b_{2}^{0}$	$(γ ρ^{2} - 1) / \sqrt{3}$
Astigmatism	$R_{2}^{2}$	ρ ²/(1 + ∊ ² + ∊ ⁴)^1/2	$a_{2}^{2} ρ^{2}$	$(γ / \sqrt{6}) ρ^{2}$
Coma	$R_{3}^{1}$	$\frac{3 (1 + ∊^{2}) ρ^{3} - 2 (1 + ∊^{2} + ∊^{4}) ρ}{(1 - ∊^{2}) {[(1 + ∊^{2}) (1 + 4 ∊^{2} + ∊^{4})]}^{1 / 2}}$	$a_{3}^{1} ρ^{3} + b_{3}^{1} ρ$	$\sqrt{γ / 2} (\frac{γ}{2} ρ^{3} - ρ)$
Spherical aberration	$R_{4}^{0}$	[6ρ ⁴ − 6(1 + ∊ ²)ρ ² + (1 + 4∊ ² + ∊ ⁴)]/(1 − ∊ ²)²	$a_{4}^{0} ρ^{4} + b_{4}^{0} ρ^{2} + c_{4}^{0}$	$(γ^{2} ρ^{4} - 4 γ ρ^{2} + 2) / 2 \sqrt{5}$

∊	$a_{1}^{1}$	$a_{2}^{0}$	$b_{2}^{0}$	$a_{2}^{2}$	$a_{3}^{1}$	$b_{3}^{1}$	$a_{4}^{0}$	$b_{4}^{0}$	$c_{4}^{0}$
0.00	1.09367 (1.00000)	2.04989 (2.00000)	−0.85690 (−1.00000)	1.14541 (1.00000)	3.11213 (3.00000)	−1.89152 (−2.00000)	6.12902 (6.00000)	−5.71948 (−6.00000)	0.83368 (1.00000)
0.25	1.04364 (0.97014)	2.18012 (2.13333)	−1.00080 (−1.13333)	1.08940 (0.96836)	3.01573 (2.94566)	− 1.84513 (−1.97099)	6.95563 (6.82667)	− 6.98197 (−7.25333)	1.25153 (1.42667)
0.50	0.92963 (0.89443)	2,70412 (2.66667)	−1.56449 (−1.66667)	0.93620 (0.87287)	3.14319 (3.11400)	− 2.06618 (−2.17980)	10.79549 (10.66667)	−13.08900 (−13.33333)	3.46706 (3.66667)
0.75	0.80827 (0.80000)	4.59329 (4.57143)	−3.51548 (−3.57143)	0.74439 (0.72954)	4.55179 (4.53877)	−3.57767 (−3.63858)	31.47559 (31.34694)	− 48.77879 (−48.97959)	18.39840 (18,63265)
0.90	0.74453 (0.74329)	10.53581 (10.52632)	−9.50324 (−9.52632)	0.63890 (0.63679)	9.60573 (9.60023)	−8.69629 (−8.72012)	166.33357 (166,20499)	−300.66342 (−300.83102)	135.36926 (135.62604)

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Applied Optics