An accurate physical model is one of the main contributing factors towards the quality of images that can be formed with an optical tomographic microscope. Our model in [1] was based on the beam propagation method, which splits the sample into layers, and computationally propagates the light layer-by-layer from the source all the way to the detector. The starting point for our derivations was the scalar Helmholtz equation

where denotes a spatial coordinate, is what we called the total light-field at , is the Laplacian, is the identity operator, and is the wavenumber of the field at . The spatial dependence of the wavenumber is due to variations of the speed of light induced by the inhomogeneous nature of the sample under consideration. Once we have the Helmholtz equation, we can repeat the process described in [1], and obtain an algorithm for computationally forming an image from the measured data.

However, how does the Helmholtz equation above relate to the actual Maxwell's equations?

*Maxwell's equations* provide the full description of the optical waves, and there are four equations to consider. The first equation is often called the *Faraday's law of induction*

where and are electric and magnetic fields, respectively, and is the curl operator. The second equation is sometimes called the *Ampère's circuital law*

where is the is the total current density, is a magnetic field, and is the electric displacement field. The total current density is related to the electric field as , where is the distribution of condictivity in the sample. The magnetic field is related to as , where is the distribution of permeability in space. Similarly, the electric displacement field is related to as , where is the distribution of permittivity in space. The third equation is the Gauss's law

where is divergence operator and is the distribution of charge in the sample. Finally, the fourth equation is the Gauss's law for magnetism

To obtain the scalar Helmhotz equation, we assume that the sample is charge free and non-magnetic, *i.e.*, and . We also limit ourselves to the electric fields that have transverse magnetic (TM) polarizations, i.e., and , and that are thus divergence free . Then by applying the identity , and limiting ourselves to the scalar , we obtain

The usual Helmholtz equation is then obtained by taking a temporal Fourier transform form both sides, ignoring , and rearranging the terms

where . Typically, it is more convenient to write , where corresponds to the wave vector in the air, and has been modified to incorporate , and is thus complex.

**References:**

[1] U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Optical tomographic image reconstruction based on beam propagation and sparse regularization,” *IEEE Trans. Comp. Imag.*, vol. 2, no. 1, 2016,