Three-dimensional phase optical transfer function in axially symmetric microscopic quantitative phase imaging

Jianhui Huang; Jianhui Huang; Yijun Bao; Thomas K. Gaylord

doi:10.1364/JOSAA.403861

Journal of the Optical Society of America A
Vol. 37,
Issue 12,
pp. 1857-1872
(2020)
•https://doi.org/10.1364/JOSAA.403861

Three-dimensional phase optical transfer function in axially symmetric microscopic quantitative phase imaging

Jianhui Huang, Yijun Bao, and Thomas K. Gaylord

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Jianhui Huang, Yijun Bao, and Thomas K. Gaylord, "Three-dimensional phase optical transfer function in axially symmetric microscopic quantitative phase imaging," J. Opt. Soc. Am. A 37, 1857-1872 (2020)

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- Biomedical Optics and Imaging
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About this Article
History
- Original Manuscript: July 27, 2020
- Revised Manuscript: September 28, 2020
- Manuscript Accepted: September 29, 2020
- Published: November 3, 2020

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Abstract

Three-dimensional quantitative phase imaging (3D QPI) is widely recognized as a potentially high-impact microscopic modality. Central to determining the resolution capability of 3D QPI is the phase optical transfer function (POTF). The magnitude of the POTF over its spatial frequency coverage (SFC) specifies the intensity of the response for each allowed spatial frequency. In this paper, a detailed analysis of the POTF for an axially symmetric optical configuration is presented. First, a useful geometric interpretation of the SFC, which enables its visualization, is presented. Second, a closed-form 1D integral expression is derived for the POTF in the general nonparaxial case, which enables rapid calculation of the POTF. Third, this formulation is applied to disk, annular, multi-annuli, and Gaussian illuminations as well as to an annular objective. Taken together, these contributions enable the visualization and simplified calculation of the 3D axially symmetric POTF and provide a basis for optimizing QPI in a wide range of applications.

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${N A}_{O}$ and ${N A}_{OI}$	(Outer) and inner numerical aperture of the objective lens
${N A}_{C}$ and ${N A}_{CI}$	(Outer) and inner numerical aperture of the condenser lens
$n (r)$	3D refractive index distribution of the tested object
$n_{0}$	Refractive index of the immersion material of the objective
$n_{o b}$	Average refractive index of the object (and surrounding RI-matching material)
$λ$	Wavelength of the illumination light
$ρ_{_{O}}$ and $ρ_{_{O I}}$	(Outer) and inner objective NA normalized by $2 π n_{0} / λ$
$ρ_{C}$ and $ρ_{_{C I}}$	(Outer) and inner condenser NA normalized by $2 π n_{0} / λ$
$O (k)$	Function that defines the intensity distribution of light collected by the objective
$S (k)$	Function that defines the intensity distribution of source (incident) light
$σ$	Gaussian illumination width coefficient
$(k_{x}, k_{y}, k_{z})$	Normalized source wave vector space, $k_{y} = (1 - k_{x}^{2} - k_{z}^{2})^{1 / 2}$
${\bar{ρ}}_{S}$ and $ρ_{S}$	Source wave vector and its normalized version by dividing by $2 π n_{0} / λ$
${\bar{ρ}}_{D}$ and $ρ_{D}$	Diffracted wave vector and its normalized version by dividing by $2 π n_{0} / λ$
$\bar{K}$ and $K$	grating vector in the object and its normalized version by dividing by $2 π n_{0} / λ$ , $K = (η, μ)$
$(η, μ)$	normalized object’s spatial-frequency space
$μ_{x}$ and $μ_{z}$	normalized lateral spatial frequency of the object in $x$ and $z$ directions
$η$	normalized axial spatial frequency of the object in $y$ direction
$μ$ and $μ$	normalized lateral spatial frequency vector and its modulus, $μ = (μ_{x}, μ_{z}), μ = \| μ \|$
$k$ and $k$	normalized lateral wave vector of the source light and its modulus, $k = (k_{x}, k_{z}), k = \| k \|$
$k^{'}$	normalized shifted lateral wave vector of the source light, $k^{'} = k + μ / 2$
$η_{s e l}$ and $μ_{s e l}$	selected detectable normalized axial- and lateral-spatial frequency of the object
$T_{3 D} (η, μ)$	3D phase optical transfer function (POTF)
$T_{2 D} (η, μ)$	2D phase optical transfer function (POTF), which indicates the axially symmetric $T_{3 D} (η, μ)$
$T_{D} (η, μ)$	diffracted-light-ellipse integral
$T_{S} (η, μ)$	source-light-ellipse integral
$l_{D}$ and $w_{D}$	integration limit and integrand for calculating $T_{D} (η, μ)$
$l_{S}$ and $w_{S}$	integration limit and integrand for calculating $T_{S} (η, μ)$

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Journal of the Optical Society of America A