Analysis of three-dimensional objects in quantitative phase contrast microscopy: a validity study of the planar approximation for spherical particles (Conference Presentation)

Imaging transparent or semi-transparent refractive objects is often required in various fields such as biology, medicine, and fluid dynamics. With conventional bright field microscopy, transparent refractive objects are difficult to observe in focal position, since they mainly affect the phase of the optical field, i.e., the shape of the wavefront, but not the intensity [1]. Microscopy with phase imaging capabilities has been developed, solving this issue: dark field microscopy [1], Schlieren-based methods [1,2], Zernike phase contrast microscopy [1], Nomarski differential interference contrast microscopy [3], and Shack–Hartmann-based techniques [4] are commonly used nowadays. Whereas those technologies typically allow a qualitative description of the phase, digital image processing has enabled the quantitative phase retrieval in microscopy, e.g., by using transport equation of intensity [5], digital holography [6,7], or ptychography [8]. The knowledge of both phase and intensity in a plane, giving together the complex amplitude, offers a whole description of the optical field, assuming a monochromatic, scalar, and coherent case. In digital holographic microscopy, the complex amplitude is used to implement the holographic propagation in depth, by numerical computations [9]. This presentation will discuss the validity and limitation of assessing the thickness of an object by using the phase profile in the plane back-propagated to the object. Because this assessment would be perfect if the object was two-dimensional, perpendicular to the optical axis, which is only an approximation with three-dimensional objects, it is referred here as the planar approximation. The presented study deals with simulated homogeneous spherical particles and uses Mie scattering for computing the optical field behind the particle. Results will be shown for various parameters, such as the particle size and its refractive index. Assuming a plane wave illumination, possibly after some numerical compensation, the quantitative value of the phase phi(x,y) is often considered as proportional to the optical thickness e_opt(x,y) of the observed object: phi(x,y) - phi_0 = (2pi/lambda) * e_opt(x,y), (1) where phi_0 is a constant, lambda denotes the wavelength and e_opt(x,y) is given by e_opt(x,y) = (n_obj - n_ext) * e(x,y), (2) with n_obj and n_ext, the refractive indices of the object and of the external medium, respectively, and e(x,y), the physical thickness of the object. Equation (1) results from the hypothesis of straight-line propagation of optical rays across the object. Indeed, under this assumption, each ray passing through the object is delayed, in comparison to the propagation in the external medium, by t_obj - t_ext = e / (c/n_obj)) - e / (c/n_ext)) = e_opt / c, (3) where t_obj and t_ext represent the time for propagating by the distance e inside and outside the object, respectively; and c is the speed of light in vacuum. Equation (3) straightforwardly leads to Eq. (1). However, this hypothesis is limited, as objects are essentially three-dimensional, and light is subject to refraction and reflection at interfaces. From a physical optical point of view, a three-dimensional object distorts the wavefront of the incident wave. Therefore, the quantitative measure of the phase, back-propagated in the median plane of the object, generally differs from the thickness profile. Assessing these differences is highly valuable since it allows to estimate the validity of the planar approximation, i.e., the validity of considering that the optical thickness of the observed object is proportional to the phase image. This is addressed in the presented study. For the parameters with which this planar approximation is not valid, parametric inverse problem approach using Mie model can be used [10], since it intrinsically takes into account the three-dimensional features of the object, assuming a perfectly spherical shape.