A new approach for microstructure imaging

A recurring issue with microstructure studies is specimen lighting. In particular, microscope lighting must be deployed in such a way as to highlight biological elements without enhancing caustic effects and diffraction. We describe here a high frequency technique due to address this lighting issue. First, an extensive study is undertaken concerning asymptotic equations in order to identify the most promising algorithm for 3D microstructure analysis. Ultimately, models based on virtual light rays are discarded in favor of a model that considers the joint computation of phase and irradiance. This paper maintains the essential goal of the study concerning biological microstructures but offers several supplementary notes on computational details which provide perspectives on analyses of the arrangements of numerous objects in biological tissues.


Introduction
Solving the amplitude simultaneously with the phase is a consistent issue in physics. For example, the same mathematical computations can be applied to solve the Schrödinger or the Helzmoltz equation 1 . In this way, the eikonal and amplitude equations (supplementary note 1) can estimate bright effects in microstructures.

Rewording equations for scalar electric field
The eikonal and transport equation are rewritten to the differential system (1) where ϕ is ϕ=k 0 δ 0 with k 0 the modulus of the wave vector in the vacuum and δ 0 the eikonal; n is the refractive index, and E 0 is the amplitude of the scalar electric field.
The solution is given in Cartesian coordinates ( x , y , z ) whose z acts as the propagation time. Therefore, the first equation (1) successively becomes the resolving equation (2).
The second equation (1) evolves to times: the gradient part (3) where a is given by a= and μ is given by μ=E 0 2 ∂δ 0 ∂ z and then the divergent part (4).
It should be noted that a , b and I 0 =E 0 2 depend on the eikonal δ 0 , and a unique solution exists for this differential system of the two equations (2) and (4) for very general initial boundaries 1 .

Numerical schema of Lax-Friedrichs type
The differential system obtained from the two equations (2) and (4) can be computed with equations of differences. Regular sampled grids are drawn in a three-dimensional space: the x and y coordinates are sampled with the same step Δ xy giving sampled plans [ y j , y j +1 ] , and the z coordinates are also sampled with the step Δ z giving the sampled axis ] . Therefore, each point in three-dimensional space is shown by ( x i , y j , z k )=(i Δ xy , j Δ xy , k Δ z ) and belongs to a three-dimensional matrix.
Numerical schema for the eikonal equation (2) resolves from the equation of the differences (5).
The numerical Lax-Friedrichs 2 type schema substitutes each sample by centering with its nearest neighbors (7).
It should be noted the ratio Δ z Δ xy keeps a fixed value even if the differences becomes very small.
Numerical schema for the transport equation (4) resolve from the equation of the differences (8).
where the sample v i , j k is v i , j k =μ ( x i , y j , z k ) and the functions f (aμ ) and g(bμ ) are f (aμ )=aμ and g(bμ )=bμ , respectively. The numerical Lax-Friedrichs type schema allow for computing the relationship (8). Herein, these computations require coupling for each propagation time z k with the computation of the eikonal equation. The coefficients a and b and the irradiance I 0 use the partial differential equations The numerical Lax-Friedrichs 2 type schema are renewed for these computations.
The numerical Lax-Friedrichs type schema are used in order to warrant better numerical convergence. This approach is feasible and efficient in several practical situations 1 .

Smooth wedge simulations
Smooth wedges are used to illustrate results of the numerical schema (Fig. S1). The 2D numerical schema computes the phase and the irradiance (Fig. S2). The 3D numerical schema allows to build images in a xy plane to study the irradiance (Fig. S3).