Bright-Field Microscopy of Transparent Objects 1119
used in the calculations, the nominal shift must be multiplied by the ratio of n1 (or n2, as they are nearly equal) to the refractive index of the immersion medium of the objective (Carlsson, 1991; Visser et al., 1992). Second, the focal plane must remain within the sample or the medium, but not within the coverglass, as that would introduce a second refraction which is not accounted for by Equation (14). For example, if the first image is focused on the coverglass surface, the second or subsequent images should be focused further into the sample.
(2) Varying the illumination angle The other way of creating an interpretable contrast is
to vary the illumination angle, for example, by using an off-center condenser diaphragm. Variable angle illumination has been used in differential phase contrast microscopy (Hamilton and Sheppard, 1984; Tian and Waller, 2015; Chen et al, 2016) and computer tomography (Sung et al, 2009). Here we show that quantitative data can, in principle, be extracted from the above ray model. If γx and γy are small tilt angles in the xz- and yz-planes, one can use two pairs of images, taken at ±γ. For the angle γ, Equation (3) is modified as:
dα= n - 1 ðÞ= n - 1 n - 1
ðÞ1 - γtanα :
can be expressed as: I -1
+γx x0; y0ðÞ- I -1 I -1 +γy x0; y0ðÞ- I -1
ðÞtan α + γ ðÞ1 - tan α tan γ tan α + γ
tan α + tan γ (16) At γtanα≪1, the difference between each pair of images -γx x0; y0ðÞ=4γxðn - 1Þ -γy x0; y0ðÞ=4γyðn - 1Þ
∂h ∂x 1 + ∂h
∂h ∂y 1 + ∂h
These are algebraic equations for ∂h "#
2 ∂x
"#
2 ∂y
derivatives are not too large, the contrast essentially represents the slope of the profile h(x,y) along the corresponding direction. After the derivatives are found, the profile h(x,y)can be obtained by simple integration. However, one complication might arise if one strives for a higher resolution. The images forming each pair in Equation (17) are misregistered by the amount γh(x,y). This would make the task of numerical reconstruction of the profile h(x,y)at high resolution less straightforward. In summary, we have presented a simple theory of
∂x and ∂h ∂y.When the
transmission image formation based on ray tracing. The theory relates directly to the quantity of interest—the object profile. If the profile of a cell is known from independent measurements, one should be able to find the average refractive index, as well as related quantities—water and protein concentration. Local protein/water variations are usually less important than the integral values over the entire cell volume, and thus the geometrical description is appro- priate. Although ray tracing is a very simplified description of light propagation, our results are equivalent to those based
; : ð17Þ
on paraxial wave theory (Teague, 1983; Streibl, 1984). The other finding is the possibility of extracting quantitative phase information from variable illumination angle, which leads to simpler equations. Future work will test the practicality of this approach.
ACKNOWLEDGMENTS
The research was supported by the ACS PetroleumResearch Fund 58813-ND6 to A.K.K. and the University Research Council Grant to M.A.M.
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