Geometrical optical transfer function: is it worth calculating?

José A. Díaz; Virendra N. Mahajan

doi:10.1364/AO.56.007998

Applied Optics
Vol. 56,
Issue 28,
pp. 7998-8004
(2017)
•https://doi.org/10.1364/AO.56.007998

Geometrical optical transfer function: is it worth calculating?

José A. Díaz and Virendra N. Mahajan

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José A. Díaz and Virendra N. Mahajan, "Geometrical optical transfer function: is it worth calculating?," Appl. Opt. 56, 7998-8004 (2017)

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Abstract

In this paper, we explore the merit of calculating the geometrical optical transfer function (GOTF) in optical design by comparing the time to calculate it with the time to calculate the diffraction optical transfer function (DOTF). We determine the DOTF by numerical integration of the pupil function autocorrelation (that reduces to an integration of a complex exponential of the aberration difference function), 2D digital autocorrelation of the pupil function, and the Fourier transform (FT) of the point-spread function (PSF); and we determine the GOTF by the FT of the geometrical PSF (that reduces to an integration over the pupil plane of a complex exponential that is a scalar product of the spatial frequency and transverse ray aberration vectors) and the FT of the spot diagram. Our starting point for calculating the DOTF is the wave aberrations of the system in its pupil plane, and the transverse ray aberrations in the image plane for the GOTF. Numerical results for primary aberrations and some typical imaging systems show that the direct numerical integrations are slow, but the GOTF calculation by a FT of the spot diagram is two or even three times slower than the DOTF calculation by an FT of the PSF, depending on the aberration. We conclude that the calculation of GOTF is, at best, an approximation of the DOTF and only for large aberrations; GOTF does not offer any advantage in the optical design process, and hence negates its utility.

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Aberration	Value
Defocus: $B_{d} ρ^{2}$	$B_{d} = 1.4$
Astigmatism: $A_{a} ρ^{2} (\cos^{2} θ - 1 / 2)$
$ϕ = 0$ , $π / 2$	$A_{a} = 2.8$
$ϕ = π / 4$	$A_{a} = 1$
Coma: $A_{c} ρ^{3} \cos θ$
$ϕ = 0$	$A_{c} = 1$
$ϕ = π / 2$	$A_{c} = 2$
$ϕ = π / 4$	$A_{c} = 0.7$
Spherical aberration
Minimum $σ_{w}$ : $A_{s} (ρ^{4} - ρ^{2})$	$A_{s} = 2.3$
Minimum $σ_{s}$ : $A_{s} [ρ^{4} - (4 / 3) ρ^{2}]$	$A_{s} = 1.2$

	Imaging System
	#1		#2		#3
Zernike coefficients	On- Axis	Off- Axis	On- Axis	Off- Axis	On- Axis	Off- Axis
$a_{1}$	−1.15	0.71	0.19	0.47	−1.16	−0.29
$a_{2}$	0	0.07	0	0.22	0	1.06
$a_{4}$	−0.29	0.49	0.26	0.36	0.11	0.06
$a_{6}$	0	0.48	0	−0.09	0	−0.19
$a_{8}$	0	0.03	0	0.04	0	0
$a_{10}$	0	0	0	−0.01	0	0.79
$a_{11}$	0.27	0.04	0.10	0.05	0.27	−0.24
$a_{12}$	0	0.11	0	−0.02	0	−0.18
$a_{16}$	0	0	0	−0.03	0	−0.21
$a_{22}$	−0.01	−0.02	0	0	−0.30	−0.37
$a_{37}$	0	0	0	0	−0.02	−0.02

	DOTF			GOTF
Aberration	NI	DA	FT	NI	FT
Defocus	0.26	0.040	0.033	0.47	0.030
Spherical aberration
Minimum $σ_{w}$	0.27	0.046	0.030	0.55	0.074
Minimum $σ_{s}$	0.17	0.053	0.029	0.53	0.079
Astigmatism
$ϕ = 0$ , $π / 2$	0.16	0.060	0.026	0.53	0.028
$ϕ = π / 4$	0.17	0.072	0.046	0.57	0.047
Coma
$ϕ = 0$	0.14	0.040	0.033	0.50	0.080
$ϕ = π / 2$	0.23	0.037	0.030	0.48	0.048
$ϕ = π / 4$	0.19	0.041	0.026	0.57	0.066

		DOTF			GOTF
Imaging System		NI	DA	FT	NI	FT
#1	On–axis: $σ_{s} = 4.7$ ; $σ_{w} = 0.4$ ; $P - V = 1.5$	0.34	0.060	0.055	0.50	0.08
	Off–axis: $σ_{s} = 7.0$ ; $σ_{w} = 0.7$ ; $P - V = 3.2$
	$ϕ = 0$	0.33	0.061	0.054	0.56	0.12
	$ϕ = π / 2$	0.27	0.061	0.054	0.53	0.09
#2	On–axis: $σ_{s} = 4.2$ ; $σ_{w} = 0.3$ ; $P - V = 1$	0.25	0.068	0.051	0.61	0.09
	Off–axis: $σ_{s} = 4.6$ ; $σ_{w} = 0.4$ ; $P - V = 1.9$
	$ϕ = 0$	0.54	0.062	0.049	0.61	0.10
	$ϕ = π / 2$	0.45	0.062	0.049	0.70	0.14
#3	On–axis: $σ_{s} = 8.5$ ; $σ_{w} = 0.4$ ; $P - V = 1.7$	0.58	0.063	0.042	0.68	0.09
	Off–axis: $σ_{s} = 21.5$ ; $σ_{w} = 1.4$ ; $P - V = 7.8$
	$ϕ = 0$	0.74	0.068	0.049	0.61	0.10
	$ϕ = π / 2$	0.75	0.068	0.049	0.70	0.14

Aberration	Value
Defocus: $B_{d} ρ^{2}$	$B_{d} = 1.4$
Astigmatism: $A_{a} ρ^{2} (\cos^{2} θ - 1 / 2)$
$ϕ = 0$ , $π / 2$	$A_{a} = 2.8$
$ϕ = π / 4$	$A_{a} = 1$
Coma: $A_{c} ρ^{3} \cos θ$
$ϕ = 0$	$A_{c} = 1$
$ϕ = π / 2$	$A_{c} = 2$
$ϕ = π / 4$	$A_{c} = 0.7$
Spherical aberration
Minimum $σ_{w}$ : $A_{s} (ρ^{4} - ρ^{2})$	$A_{s} = 2.3$
Minimum $σ_{s}$ : $A_{s} [ρ^{4} - (4 / 3) ρ^{2}]$	$A_{s} = 1.2$

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Equations (10)

Applied Optics