Microsc. Microanal. 23, 1116–1120, 2017 doi:10.1017/S1431927617012624
© MICROSCOPY SOCIETY OF AMERICA 2017
Bright-Field Microscopy of Transparent Objects: A Ray Tracing Approach
Anatoly K. Khitrin,1,✠ Jonathan C. Petruccelli,2 and Michael A. Model3,*
1Department of Chemistry and Biochemistry, Kent State University, Kent, OH 44242, USA 2Department of Physics, State University of New York Albany, Albany, NY 12222, USA 3Department of Biological Sciences, Kent State University, Kent, OH 44242, USA
Abstract: The formation of a bright-field microscopic image of a transparent phase object is described in terms of elementary geometrical optics. Our approach is based on the premise that the image replicates the intensity distribution (real or virtual) at the front focal plane of the objective. The task is therefore reduced to finding the change in intensity at the focal plane caused by the object. This can be done by ray tracing complemented with the requirement of energy conservation. Despite major simplifications involved in such an analysis, it reproduces some results from the paraxial wave theory. In addition, our analysis suggests two ways of extracting quantitative phase information from bright-field images: by vertically shifting the focal plane (the approach used in the transport-of-intensity analysis) or by varying the angle of illumination. In principle, information thus obtained should allow reconstruction of the object morphology.
Key words: Optical theory, transport-of-intensity equation, transmission microscopy, quantitative phase imaging
INTRODUCTION
The diffraction theory of image formation developed by Ernst Abbe in the 19th century remains central to under- standing transmission microscopy (Born and Wolf, 1970). It has been less appreciated that certain effects in transmission imaging can be adequately described by geometrical, or ray optics. In particular, the geometrical approach is valid when one is interested in features significantly larger than the wavelength. Examples of geometrical descriptions include the explanation of Becke lines at the boundary of two media with different refractive indices (Faust, 1955) or the axial scaling effect (Visser et al., 1992). In this work, we show that a model based on geometrical optics can be used to describe the formation of a bright-field transmission image of a refractive (phase) specimen. Our approach is based on the notion that in an infinite
tube length microscope, an image replicates the real or virtual intensity distribution at the front focal plane of the objective. Thus, the effect of a refractive object can be analyzed by examining the pattern formed by extending the incoming rays back to the focal plane. Figure 1 illustrates the concept. In the absence of a specimen, the intensity distribution at the focal plane is uniform, and no image is
Received May 19, 2017; accepted September 25, 2017
* Corresponding author.
mmodel@kent.edu ✠The premature death of Anatoly Khitrin (1955–2017) took away from us a scientist
of rare knowledge and imagination. His specialty was quantum physics and magnetic resonance, but he also had a unique ability to quickly get to the root of the most diverse problems, whether it was membrane biophysics, climate change, or trouble with a home furnace. He made original and useful inventions but never cared to patent them. Anatoly was a modest and gentle person, not seeking any special recognition, not jumping on bandwagons and “hot” topics. He thought about deep questions, achieved remarkable insights, helped colleagues and taught students – in other words, did what scientists are supposed to be doing.
formed. A phase specimen causes refraction of the rays, which is produces lighter and darker areas at the focal plane. These darker and lighter areas are translated into the image.
THEORY AND DISCUSSION
Next, we present the above model of image formation in quantitative terms. Consider a typical situation in light microscopy (Fig. 2), where an object (e.g., a biological cell) is attached to the coverglass on the side of the objective. It is illuminated by light coming from the condenser on the opposite side. The cell has a slightly higher refractive index than the surrounding liquid (typically by ~2–3%) and is assumed to have a homogeneous structure. The focal plane of a coverglass-corrected objective is positioned approximately on the level of the cells but can be shifted up or down by moving either the sample or the objective. Figure 3 gives a more detailed view of the ray path
through the sample. A single refraction at the interface of the cell and its aqueous environment is assumed. It causes a change in the distribution of intensity at the focal plane of the objective, which, in turn, determines the intensity distribution at the image plane. It is possible, of course, to have a situation when the focal plane is belowthe cell, in which case the intensity would also be affected by a second refraction at the cell-coverglass boundary. This case will not be considered. Figure 3 introduces the main parameters used in the
model. From the law of refraction, we have: sinα sinβ = n2
n1 =n; (1)
where n is the relative refractive index. To simplify the following derivations, we assume that n−1≪1, which is
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