Effect of rotation and translation on the expected benefit of an ideal method to correct the eye’s higher-order aberrations

Antonio Guirao; David R. Williams; Ian G. Cox

doi:10.1364/JOSAA.18.001003

Journal of the Optical Society of America A
Vol. 18,
Issue 5,
pp. 1003-1015
(2001)
•https://doi.org/10.1364/JOSAA.18.001003

Effect of rotation and translation on the expected benefit of an ideal method to correct the eye’s higher-order aberrations

Antonio Guirao, David R. Williams, and Ian G. Cox

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Antonio Guirao, David R. Williams, and Ian G. Cox, "Effect of rotation and translation on the expected benefit of an ideal method to correct the eye’s higher-order aberrations," J. Opt. Soc. Am. A 18, 1003-1015 (2001)

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Abstract

An ideal correcting method, such as a customized contact lens, laser refractive surgery, or adaptive optics, that corrects higher-order aberrations as well as defocus and astigmatism could improve vision. The benefit achieved with this ideal method will be limited by decentration. To estimate the significance of this potential limitation we studied the effect on image quality expected when an ideal correcting method translates or rotates with respect to the eye’s pupil. Actual wave aberrations were obtained from ten human eyes for a 7.3-mm pupil with a Shack–Hartmann sensor. We computed the residual aberrations that appear as a result of translation or rotation of an otherwise ideal correction. The model is valid for adaptive optics, contact lenses, and phase plates, but it constitutes only a first approximation to the laser refractive surgery case where tissue removal occurs. Calculations suggest that the typical decentrations will reduce only slightly the optical benefits expected from an ideal correcting method. For typical decentrations the ideal correcting method offers a benefit in modulation 2–4 times higher (1.5–2 times in white light) than with a standard correction of defocus and astigmatism. We obtained analytical expressions that show the impact of translation and rotation on individual Zernike terms. These calculations also reveal which aberrations are most beneficial to correct. We provided practical rules to implement a selective correction depending on the amount of decentration. An experimental study was performed with an aberrated artificial eye corrected with an adaptive optics system, validating the theoretical predictions. The results in a keratoconic subject, also corrected with adaptive optics, showed that important benefits are obtained despite decentrations in highly aberrated eyes.

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n	$\pm m$	Zernike Polynomials	Monomial Representation
0	0	1	1
1	1	$2 ρ cos θ$	x
1	-1	$2 ρ sin θ$	y
2	0	$\sqrt{3} (2 ρ^{2} - 1)$	$- 1 + 2 (x^{2} + y^{2})$
	2	$\sqrt{6} ρ^{2} cos 2 θ$	$x^{2} - y^{2}$
	-2	$\sqrt{6} ρ^{2} sin 2 θ$	$2 xy$
3	1	$\sqrt{8} (3 ρ^{3} - 2 ρ) cos θ$	$x [3 (x^{2} + y^{2}) - 2]$
	-1	$\sqrt{8} (3 ρ^{3} - 2 ρ) sin θ$	$y [3 (x^{2} + y^{2}) - 2]$
	3	$\sqrt{8} ρ^{3} cos 3 θ$	$x (x^{2} - 3 y^{2})$
	-3	$\sqrt{8} ρ^{3} sin 3 θ$	$y (3 x^{2} - y^{2})$
4	0	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	$6 (x^{2} + y^{2})^{2} - 6 (x^{2} + y^{2}) + 1$
	2	$\sqrt{10} (4 ρ^{4} - 3 ρ^{2}) cos 2 θ$	$(x^{2} - y^{2}) [4 (x^{2} + y^{2}) - 3]$
	-2	$\sqrt{10} (4 ρ^{4} - 3 ρ^{4}) sin 2 θ$	$2 xy [4 (x^{2} + y^{2}) - 3]$
	4	$\sqrt{10} ρ^{4} cos 4 θ$	$x^{4} - 6 x^{2} y^{2} + y^{4}$
	-4	$\sqrt{10} ρ^{4} sin 4 θ$	$4 xy (x^{2} - y^{2})$
5	1	$\sqrt{12} (10 ρ^{5} - 12 ρ^{3} + 3 ρ) cos θ$	$x [10 (x^{2} + y^{2})^{2} - 12 (x^{2} + y^{2}) + 3]$
	-1	$\sqrt{12} (10 ρ^{5} - 12 ρ^{3} + 3 ρ) sin θ$	$y [10 (x^{2} + y^{2})^{2} - 12 (x^{2} + y^{2}) + 3]$
	3	$\sqrt{12} (5 ρ^{5} - 4 ρ^{3}) cos 3 θ$	$x (x^{2} - 3 y^{2}) [5 (x^{2} + y^{2}) - 4]$
	-3	$\sqrt{12} (5 ρ^{5} - 4 ρ^{3}) sin 3 θ$	$y (3 x^{2} - y^{2}) [5 (x^{2} + y^{2}) - 4]$
	5	$\sqrt{12} ρ^{5} cos 5 θ$	$16 x^{5} - 20 x^{3} (x^{2} + y^{2}) + 5 x (x^{2} + y^{2})^{2}$
	-5	$\sqrt{12} ρ^{4} sin 5 θ$	$16 y^{5} - 20 y^{3} (x^{2} + y^{2}) + 5 y (x^{2} + y^{2})^{2}$
6	0	$\sqrt{7} (20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1)$	$20 (x^{2} + y^{2})^{3} - 30 (x^{2} + y^{2})^{2} + 12 (x^{2} + y^{2}) - 1$
	2	$\sqrt{14} (15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) cos 2 θ$	$(x^{2} - y^{2}) [15 (x^{2} + y^{2})^{2} - 20 (x^{2} + y^{2}) + 6]$
	-2	$\sqrt{14} (15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) sin 2 θ$	$2 xy [15 (x^{2} + y^{2})^{2} - 20 (x^{2} + y^{2}) + 6]$
	4	$\sqrt{14} (6 ρ^{6} - 5 ρ^{4}) cos 4 θ$	$(x^{4} + y^{4} - 6 x^{2} y^{2}) [6 (x^{2} + y^{2}) - 5]$
	-4	$\sqrt{14} (6 ρ^{6} - 5 ρ^{4}) sin 4 θ$	$4 xy (x^{2} - y^{2}) [6 (x^{2} + y^{2}) - 5]$
	6	$\sqrt{14} ρ^{6} cos 6 θ$	$32 x^{6} - 48 x^{4} (x^{2} + y^{2}) + 18 x^{2} (x^{2} + y^{2})^{2} - 1$
	-6	$\sqrt{14} ρ^{6} sin 6 θ$	$2 xy [16 x^{4} - 16 x^{2} (x^{2} + y^{2}) + 3 (x^{2} + y^{2})^{2}]$

n	m	F
n	m	Fixed Translation	Gaussian Translation
3	1	36	48
3	3	12	24
4	0	40	80
	2	40	80
	4	20	40
5	1	144	192
	3	78	156
	5	30	60
6	0	140	280
	2	140	280
	4	112	224
	6	42	84

n	m	No Benefit				50% Reduction
		$σ_{t} = 0$	$σ_{r} = 0 °$	$σ_{r} = 5 °$	$σ_{r} = 10 °$	50% Reduction

		$σ_{r}$	$σ_{t} / r_{0}$	$σ_{t} / r_{0}$	$σ_{t} / r_{0}$	$σ_{r}$	$σ_{t} / r_{0}$
2	0	—	—	—	—	—	—
2	2	34°	—	—	—	15°	—
3	1	67°	0.144	0.144	0.142	30°	0.072
3	3	22°	0.204	0.197	0.176	10°	0.102
4	0	—	0.112	0.112	0.112	—	0.056
	2	34°	0.112	0.110	0.105	15°	0.056
	4	17°	0.158	0.148	0.119	7°	0.079
5	1	67°	0.072	0.072	0.071	30°	0.036
	3	23°	0.080	0.077	0.069	10°	0.040
	5	13°	0.129	0.117	0.078	6°	0.065
6	0	—	0.060	0.060	0.060	—	0.030
	2	34°	0.060	0.060	0.056	15°	0.030
	4	17°	0.067	0.063	0.050	7°	0.034
	6	11°	0.109	0.094	0.043	5°	0.055

n	$\pm m$	Zernike Polynomials	Monomial Representation
0	0	1	1
1	1	$2 ρ cos θ$	x
1	-1	$2 ρ sin θ$	y
2	0	$\sqrt{3} (2 ρ^{2} - 1)$	$- 1 + 2 (x^{2} + y^{2})$
	2	$\sqrt{6} ρ^{2} cos 2 θ$	$x^{2} - y^{2}$
	-2	$\sqrt{6} ρ^{2} sin 2 θ$	$2 xy$
3	1	$\sqrt{8} (3 ρ^{3} - 2 ρ) cos θ$	$x [3 (x^{2} + y^{2}) - 2]$
	-1	$\sqrt{8} (3 ρ^{3} - 2 ρ) sin θ$	$y [3 (x^{2} + y^{2}) - 2]$
	3	$\sqrt{8} ρ^{3} cos 3 θ$	$x (x^{2} - 3 y^{2})$
	-3	$\sqrt{8} ρ^{3} sin 3 θ$	$y (3 x^{2} - y^{2})$
4	0	$\sqrt{5} (6 ρ^{4} - 6 ρ^{2} + 1)$	$6 (x^{2} + y^{2})^{2} - 6 (x^{2} + y^{2}) + 1$
	2	$\sqrt{10} (4 ρ^{4} - 3 ρ^{2}) cos 2 θ$	$(x^{2} - y^{2}) [4 (x^{2} + y^{2}) - 3]$
	-2	$\sqrt{10} (4 ρ^{4} - 3 ρ^{4}) sin 2 θ$	$2 xy [4 (x^{2} + y^{2}) - 3]$
	4	$\sqrt{10} ρ^{4} cos 4 θ$	$x^{4} - 6 x^{2} y^{2} + y^{4}$
	-4	$\sqrt{10} ρ^{4} sin 4 θ$	$4 xy (x^{2} - y^{2})$
5	1	$\sqrt{12} (10 ρ^{5} - 12 ρ^{3} + 3 ρ) cos θ$	$x [10 (x^{2} + y^{2})^{2} - 12 (x^{2} + y^{2}) + 3]$
	-1	$\sqrt{12} (10 ρ^{5} - 12 ρ^{3} + 3 ρ) sin θ$	$y [10 (x^{2} + y^{2})^{2} - 12 (x^{2} + y^{2}) + 3]$
	3	$\sqrt{12} (5 ρ^{5} - 4 ρ^{3}) cos 3 θ$	$x (x^{2} - 3 y^{2}) [5 (x^{2} + y^{2}) - 4]$
	-3	$\sqrt{12} (5 ρ^{5} - 4 ρ^{3}) sin 3 θ$	$y (3 x^{2} - y^{2}) [5 (x^{2} + y^{2}) - 4]$
	5	$\sqrt{12} ρ^{5} cos 5 θ$	$16 x^{5} - 20 x^{3} (x^{2} + y^{2}) + 5 x (x^{2} + y^{2})^{2}$
	-5	$\sqrt{12} ρ^{4} sin 5 θ$	$16 y^{5} - 20 y^{3} (x^{2} + y^{2}) + 5 y (x^{2} + y^{2})^{2}$
6	0	$\sqrt{7} (20 ρ^{6} - 30 ρ^{4} + 12 ρ^{2} - 1)$	$20 (x^{2} + y^{2})^{3} - 30 (x^{2} + y^{2})^{2} + 12 (x^{2} + y^{2}) - 1$
	2	$\sqrt{14} (15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) cos 2 θ$	$(x^{2} - y^{2}) [15 (x^{2} + y^{2})^{2} - 20 (x^{2} + y^{2}) + 6]$
	-2	$\sqrt{14} (15 ρ^{6} - 20 ρ^{4} + 6 ρ^{2}) sin 2 θ$	$2 xy [15 (x^{2} + y^{2})^{2} - 20 (x^{2} + y^{2}) + 6]$
	4	$\sqrt{14} (6 ρ^{6} - 5 ρ^{4}) cos 4 θ$	$(x^{4} + y^{4} - 6 x^{2} y^{2}) [6 (x^{2} + y^{2}) - 5]$
	-4	$\sqrt{14} (6 ρ^{6} - 5 ρ^{4}) sin 4 θ$	$4 xy (x^{2} - y^{2}) [6 (x^{2} + y^{2}) - 5]$
	6	$\sqrt{14} ρ^{6} cos 6 θ$	$32 x^{6} - 48 x^{4} (x^{2} + y^{2}) + 18 x^{2} (x^{2} + y^{2})^{2} - 1$
	-6	$\sqrt{14} ρ^{6} sin 6 θ$	$2 xy [16 x^{4} - 16 x^{2} (x^{2} + y^{2}) + 3 (x^{2} + y^{2})^{2}]$

n	m	F
n	m	Fixed Translation	Gaussian Translation
3	1	36	48
3	3	12	24
4	0	40	80
	2	40	80
	4	20	40
5	1	144	192
	3	78	156
	5	30	60
6	0	140	280
	2	140	280
	4	112	224
	6	42	84

Abstract

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Equations (16)

Journal of the Optical Society of America A