Simulation of spherical wave propagation through focus
Here are movies showing simulations of spherical wave propagation through
a circular or a spidered annular aperture.
Spherically converging wavefront - circular aperture, linear stretch
Spherically converging wavefront - circular aperture, clipped log stretch
Spherically converging wavefront - spidered annular aperture, linear stretch
Spherically converging wavefront - spidered annular aperture, clipped log stretch
This simulation was performed by applying the aperture to a
spherically converging wave. The diameter of the aperture was 30
millimeters. For the spidered annular aperture, the inner diameter
was 6.15 mm and three spiders were chosen. The pixel scale was .235
millimeters, and the electromagnetic wavelength was 1 micron. The
converging wave had a radius of curvature of 30 cm at the aperture.
Thus the f number of the converging beam was 10.
The movies show the evolution of the wave amplitude as it propagates
from the location of the aperture at z = -30 cm through focus at z = 0
cm and back out to z = 30 cm on the other side of focus. The first
movie shows the wavefront amplitude on a linear scale, while the
second shows the wavefront amplitude on a logarithmic scale, clipped
at about 17.5 percent of the peak amplitude at the focal plane. The
location of the wavefront is shown in the upper left corner, along
with the Fresnel number. Note that the propagation distance is not
evenly sampled - many more samples are taken near focus, so that the
movie spends most of its time within about a centimeter of the focal
position. The wave amplitude increases as the wavefront approaches
focus, and so to maintain the dynamic range of the movie the maximum
value changes with propagation distance. This maximum is shown in the
upper left corner - the manner in which it was chosen is described in
more detail below. Finally the size of the entire image is shown at
the bottom of the movie.
Each frame of the movie was created by propagating a wavefront from
the aperture to the distance corresponding to the location of the
frame. One could have accomplished the same thing by propagating the
same wavefront through a series of small steps, but numerical error
tends to accumulate this way. There are two lengths of importance in
this problem. The first is the the transition between near and far
field propagation. For this simulation, this transition was chosen to
occur at a Fresnel number of 20, which for the parameters used in this
problem was about +/-.2 cm from focus. Recall that the Fresnel
number F is defined as
where a is the aperture diameter, R_c is the radius of curvature, and
lambda is the wavelength. To propagate the wavefront from the
aperture location at z = -30 cm, different propagators were used
depending on the final location of the wavefront. For locations
between -30 cm and -.2 cm, a near field propagator was used. For
propagation distances between -.2 cm and .2 cm a far field propagator
was used. For propagation distances beyond .2 cm, two far field
propagations were performed - the first to the focal plane, and the
second to distances beyond z = .2 cm.
The second length is related to the wavefront sampling interval. As
the spherical wave converges towards focus, the sampling interval is
chosen so that the image size shrinks linearly with propagation
distance. Conversely, as the wave diverges past focus, the image size
increases linearly. At a distance corresponding to the depth of
focus, the images switch over to a fixed sampling interval, so the
image size remains constant near focus. The depth of
focus is defined as
For the parameters above, the depth of focus is .4 millimeters.
The electromagnetic field of a converging wavefront passing through a
circular aperture may be represented near focus in terms of Lommel
functions (Born and Wolf, Section 8.8.1). The on-axis value of the
wavefront amplitude, |E(z)|, is given by
where |E_ap| is the amplitude of the wavefront at the aperture.
The images in the movie were scaled so that the maximum tracked the
envelope of the electric field amplitude. Below is a log plot of the
envelope of the above equation, together with the predicted on-axis
values and those taken from the simulated images. The plot shows the
range of propgation distances between -.5 cm and .5 cm. Beyond about
half a centimeter the approximations required to express the wavefront
in terms of Lommel functions begins to break down.
