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RP Coating – Advanced Software for
Designing Optical Multilayer Structures

Demo File: Reflection of a Focused Beam

In the context of reflection of light at dielectric mirrors, one often only considers plane waves. However, tightly focused laser beams exhibit a substantial beam divergence. Therefore, it may not be accurate to model their reflection with plane waves only, neglecting the substantial angular spread.

RP Coating provides functions for calculating the reflection or transmission of plane waves only. However, we can use a few lines of script code to do more comprehensive calculations, where we take into account the angular spread. This is demonstrated in the following. For simplicity, we assume focusing only in one dimension, but the treatment could quite easily be generalized to two dimensions.

It is assumed that we deal with a Gaussian beam; however, we want to reserve the capability to treat beams with arbitrary spatial profiles. Therefore, we start by defining the electric field profile corresponding to the beam, keeping the option to arbitrarily change that later on:

lambda := 1180 nm
w0 := 2 um { beam radius }
theta := lambda / (pi * w0)
alpha := 20 deg { angle of incidence }
show "R0:       ", 100 * R_p(lambda, alpha):f3:"%"
w0_eff := w0 / cos(alpha)
k_x := (2pi / lambda) * sin(alpha)
E0%(x) := exp(-(x / w0_eff)^2) * expi(k_x * x) { electric field in arbitrary units }

This is the beam profile in the plane of the mirror surface, i.e., not in a plane perpendicular to the beam. It is assumed that the beam focus is on the mirror surface – we have no wavefront curvature there. However, that could easily be changed if necessary.

We now numerically calculate the angular spread with a Fast Fourier Transform (FFT), simply obtain the reflected field in the Fourier domain, and can transform back to the spatial domain:

x_m := 10 * w0 { max. x value }
N_x := 2^12 { number of points for numerical grid }
dx := 2 * x_m / N_x { spatial step size for grid ranging from -x_m to +x_m - dx }
defarray E0%[0, 2 * x_m - dx, dx] (periodic)
calc
  begin
    for x := -x_m to +x_m - dx step dx do
      E0%[x] := E0%(x); { incident field on numerical grid }
    P_in := arraysumsqr(E0%[]); { incident power }
    FFT(E0%[], E_f%[], +1);
    { Array E_f%[] now contains the incident field in the spatial Fourier domain.
      Now calculate the corresponding data for the reflected beam,
      using the function r_p%() for p polarization: } 
    processarray_e(E_f%[], '*', if abs(lambda * fx) < 1 then r_p%(lambda, arcsin(lambda * fx)), 'fx');
      { Note that the spatial frequency is fx = sin(alpha) / lambda. }
    { Now transform back to the spatial domain to obtain E_refl%[]: }
    FFT(E_f%[], E_refl%[], -1);
    P_refl := arraysumsqr(E_refl%[]); { reflected power }
    R_eff := P_refl / P_in; { effective reflectivity }
  end
show "R_eff:    ", 100 * R_eff:f3:"%"

Results

Here, we don't show the code for defining the dielectric mirror structure, which is basically arbitrary; it is assumed that we have a simple Bragg mirror (for 1000 nm at perpendicular incidence), and the wavelength and incidence angle of the beam are chosen such that we operate the mirror in the region where the reflectivity sharply drops, see the black circle in the following diagram:

Using the data as calculated above, we can plot the angular profile of the reflected beam:

We see that it is not centered around 20°, the angle of incidence; the maximum is somewhat shifted towards smaller angles because the mirror reflectivity is higher there.

Further, we can inspect the near field profile, which is also significantly deformed; for comparison, see the fine dashed curve for the incident beam profile:

The effective reflectance is found to be 50.7%, which is significantly lower than the value of 57.0% for a plane wave.

Note that we obtain quite pronounced effects, but only by assuming a rather tightly focused light beam and also an operation point concerning angle and wavelength where the effects could be expected to be relatively strong. For common practical situations, where laser beams are reflected on such mirrors, such effects are in most cases negligibly weak.

Anyway, you can see that RP Coating is flexible enough to study such things, even though the provided functions deal with plane waves only. If required, one could easily make user-defined functions, e.g. for obtaining the effective reflectivity of a Gaussian beam with a given beam waist radius. Also, one could create all sorts of other diagrams, for example color diagrams for illustrating resulting beam profiles.

If you find the implementation of such tricks complicated, keep in mind that you can obtain such additional code within the technical support. Also, also, for some cases you can find such code in the demo scripts delivered with this software.