Two-point resolution in bright background under partially coherent light
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Two‑point resolution in bright background under partially coherent light Kimiaki Yamamoto1 Received: 11 June 2020 / Accepted: 1 September 2020 © The Optical Society of Japan 2020
Abstract Two-point resolution for ordinary microscopy was studied in a bright background under partially coherent light. The basic equation was derived from the unified theory of image intensity for two point-like objects for bright and dark backgrounds. For an extended Rayleigh’s criterion, it is shown that the two-point resolution in a bright background is noticeably different from that in a dark background. The resolution for annular illumination is also dealt with, and its characteristics are elucidated. Keywords Two-point resolution in bright background · Two-point resolution · Resolution · Microscopy
1 Introduction Many studies have been conducted on two-point resolution under partially coherent light in ordinary microscopy, and its characteristics are well known [1–12]. However, almost all studies have been performed for dark backgrounds, and there is little information about the resolution for a bright background. We have previously reported a few findings pertaining to the resolution for a bright background [13–16]. In particular, to the best of the author’s knowledge, no study has provided a theory unifying both cases, namely, dark and bright backgrounds, and deals with the resolution evaluated on the same criterion. This paper presents a unified theory, and equations to calculate the resolution for both cases. As a calculated result, it elucidates the difference in the resolution characteristics of the two cases. Furthermore, the resolution for annular sources is also dealt with.
2 Theory 2.1 Imaging equations for a general optical system The optical system shown in Fig. 1 is considered. Let us assume that the optical system satisfies the sine condition. In * Kimiaki Yamamoto [email protected] 1
KM OptLab, 1706‑12 Kanoya‑cho, Hachioji City, Tokyo 193‑0815, Japan
Fig. 1, ns, nc, no, and ni are refractive indices of the respective spaces; h0 and hi are the radii of the entrance and exit pupils of an objective; and hs , hc are the radii of an illumination system (a condenser) in the entrance and exit pupil planes corresponding to the ho of the objective. The coordinate system in each plane is expressed by a vector designated by the same symbol as the plane. For convenience in the formulation of the imaging equations, we introduce the following reduced coordinate systems [16–21]:
ni sin ||𝜃i || V, 𝜆 ns sin ||𝜃s || p(px , py ) = P 𝜆
v(vx , vy ) =
x(x, y) =
XO X = i , hO hi
u(ux , uy ) =
q(qx , qy ) =
no sin ||𝜃o || U 𝜆
QS Q = C hS hC
(1)
(2)
,
where the vectors represented by capital letters are actual geometrical coordinates in the respective planes. In these reduced coordinates, the intensity I(v) of an image can be expressed as [16–18]
I(v) =
|2 | S(p)|| ĝ (x,p)P(x) exp(−i2𝜋x ⋅ v)dx|| dp , ∫ | |∫
ĝ (x,p) =
∫
t(q) â (x − p) exp(i2𝜋p ⋅ q)dq,
(3)
(4)
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