Comparison of geometrical and diffraction imaging in the space and frequency domains

Virendra N. Mahajan; José A. Díaz

doi:10.1364/AO.55.003241

Applied Optics
Vol. 55,
Issue 12,
pp. 3241-3250
(2016)
•https://doi.org/10.1364/AO.55.003241

Comparison of geometrical and diffraction imaging in the space and frequency domains

Virendra N. Mahajan and José A. Díaz

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Virendra N. Mahajan and José A. Díaz, "Comparison of geometrical and diffraction imaging in the space and frequency domains," Appl. Opt. 55, 3241-3250 (2016)

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Abstract

The geometrical and diffraction point-spread functions of an optical imaging system have been reviewed and compared in the past [Proc. SPIE 3729, 434 (1999) [CrossRef] ]. In this paper, we review and compare corresponding optical transfer functions. While the truth lies with the diffraction optical-transfer functions (OTF), it is considered easier and quicker to calculate the geometrical OTF, especially for large aberrations. We describe the theory of the two OTFs and explore the range of spatial frequencies and the magnitude of the primary aberrations over which the geometrical OTF may provide a reasonable approximation of the diffraction OTF. Moreover, balancing of spherical aberration with defocus for optimum diffraction OTF is studied as a function of both the aberration value as well as the spatial frequency. How to gauge the progress of an optical design in the frequency domain based on the geometrical OTF is outlined as the ray spot size is used in the space domain.

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Aberration	$r_{s}$	$σ_{s}$
Defocus $B_{d} ρ^{2}$	$4 B_{d}$	$2 \sqrt{2} B_{d}$
Balanced Astigmatism
$A_{a} ρ^{2} (\cos^{2} θ - 1 / 2)$	$4 A_{a}$	$2 \sqrt{2} A_{a}$
Coma
$A_{c} ρ^{3} \cos θ$	$3 A_{c}$	$2 \sqrt{2 / 3} A_{c}$
Balanced Spherical
$A_{s} [ρ^{4} - (4 / 3) ρ^{2}]$	$(8 / 3) A_{s}$	$(4 / 3) A_{s}$
$A_{s} (ρ^{4} - ρ^{2})$	$4 A_{s}$	$2 \sqrt{2 / 3} A_{s}$

Aberration	$σ_{w}$
Defocus $B_{d} ρ^{2}$	$B_{d} / 2 \sqrt{3}$
Balanced Astigmatism
$A_{a} ρ^{2} (\cos^{2} θ - 1 / 2)$	$A_{a} / 2 \sqrt{6}$
Coma
$A_{c} ρ^{3} \cos θ$	$A_{c} / 6 \sqrt{2}$
Balanced Spherical
$A_{s} (ρ^{4} - ρ^{2})$	$A_{s} / 6 \sqrt{5}$

Aberration	10%		5%
Defocus	$B_{d} = 1.4$	$ν \leq 0.11$	$B_{d} = 2.7$	$ν \leq 0.054$
Astigmatism
$ϕ = 0, π / 2$	$A_{a} = 2.8$	$ν \leq 0.11$	$A_{a} = 5.3$	$ν \leq 0.054$
$ϕ = π / 4$	$A_{a} = 1.1$	$ν \leq 0.28$	$A_{a} = 2.1$	$ν \leq 0.13$
Coma
$ϕ = 0$	$A_{c} = 1$	$ν \leq 0.75$	$A_{c} = 2$	$ν \leq 0.21$
$ϕ = π / 2$	$A_{c} = 2$	$ν \leq 0.32$	$A_{c} = 4.1$	$ν \leq 0.13$
$ϕ = π / 4$	$A_{c} = 0.7$	$ν \leq 0.79$	$A_{c} = 1.1$	$ν \leq 0.77$
Spherical
Minimum $σ_{w}$	$A_{s} = 2.3$	$ν \leq 0.28$	$A_{s} = 3$	$ν \leq 0.17$
Minimum $σ_{s}$	$A_{s} = 1.2$	$ν \leq 0.33$	$A_{s} = 5.2$	$ν \leq 0.068$

	10%		5%
Aberration	$r_{s}$	$σ_{s}$	$r_{s}$	$σ_{s}$
Defocus	5.6	4	10.8	7.6
Astigmatism
$ϕ = 0, π / 2$	5.6	4	10.6	7.6
$ϕ = π / 4$	2	1.4	4.2	3
Coma
$ϕ = 0$	3	1.6	6	3.3
$ϕ = π / 2$	2.1	1.1	3.3	1.8
$ϕ = π / 4$	6	3.2	12.3	6.7
Spherical
Minimum $σ_{w}$	9.2	3.8	12	4.9
Minimum $σ_{s}$	3.2	1.6	13.9	6.9

Aberration	$r_{s}$	$σ_{s}$
Defocus $B_{d} ρ^{2}$	$4 B_{d}$	$2 \sqrt{2} B_{d}$
Balanced Astigmatism
$A_{a} ρ^{2} (\cos^{2} θ - 1 / 2)$	$4 A_{a}$	$2 \sqrt{2} A_{a}$
Coma
$A_{c} ρ^{3} \cos θ$	$3 A_{c}$	$2 \sqrt{2 / 3} A_{c}$
Balanced Spherical
$A_{s} [ρ^{4} - (4 / 3) ρ^{2}]$	$(8 / 3) A_{s}$	$(4 / 3) A_{s}$
$A_{s} (ρ^{4} - ρ^{2})$	$4 A_{s}$	$2 \sqrt{2 / 3} A_{s}$

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Applied Optics