3D Optical System/Propagation Modeling:
3D optical lens system ray trace analysis.
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3D farfield modeling of truncated Gaussian optical
beams
It is often useful to know the farfield radiation pattern of specific
types of optical beams. The farfield pattern for an arbitrary beam
may be numerically calculated if the beam intensity and phase is
assumed to be defined over a flat, rectangular, ‘entrance’
aperture (located at z=0). The beam intensity is assumed to be zero
everywhere outside this entrance aperture (the aperture is zerobaffled).
Figure 1 illustrates the geometry of the problem.
Figure 1. Assumed geometry of
the farfield extrapolation problem
Given the complex aperture function E_{in}(x,y),
the farfield pattern E_{out}(k) is a function of
the direction angles Φ,Θ as well as the wavelength
λ. Specifically, E_{out}(k) is a 2D Fourier
transform of the aperture function E_{in}(x,y),
where C_{0} is an overall complex
amplitude factor that we may ignore if we seek only relative amplitude
and/or phase information over the output screen. Also, k is
the propagation vector oriented in the direction of the farfield
point being computed.
Eq. 1 is based on Fraunhofer diffraction theory which
assumes that the farfield pattern is projected sufficiently far
away from the input aperture. Taking the aperture diameter to be
D, and the distance between the aperture and the output screen
to be R, the Fraunhofer criterion is that R >> D^{2}/(8λ).
For engineering purposes, this criterion is often simplified to
R > D^{2}/λ. For example, assuming D
= 100 μm for a collimated fiber optic beam, and taking λ = 1.5
μm, then the Fraunhofer criterion is approximately R > 0.7 cm.
It is obviously possible to compute the transform
integral (Eq. 1) using 2D fast Fourier transforms. An alternative
approach is to model the function E_{in}(x,y) over the input aperture
with continuous polynomials. In the examples below, the real and
imaginary parts of E_{in}(x,y) were modeled with bicubic splines. The
real part of E_{in}(x,y), for example, was modeled over the input aperture
as:
(2)
Plugging this expression (along with the imaginary
part) into Eq. 1, the integral can then be computed to third order
in closed form over each x_{i},y_{j} element of the input aperture. This
elementbyelement summation is repeated for each farfield direction
in which the farfield solution is sought.
The accuracy of this method depends on how well the
input grid samples the aperture function. I have found that the
results are generally quite accurate even for sparsely sampled input
functions. The smoothness of the aperture function over the input
grid determines how wellsampled the function is. If the successive
derivatives of the interpolating splines decrease rapidly in magnitude,
then truncating the integration at third order produces very small
errors.
The following examples illustrate computed farfield
patterns of a Gaussian input beam blocked by a knifeedge occlusion
to various extents. Decibel plots of the input aperture and the
resulting farfield distributions are shown for 0%, 25%, and 75%
blocked Gaussian beams. Amplitudes in these plots have been truncated
at a decibel ‘floor’ of –60 dB (values below –60
dB are plotted as –60 dB). Note that the coordinates of the
input aperture plots are in units of (nondimensionalized) distance
(grid distance divided by wavelength), while the coordinates of
the farfield distribution plots are in degrees (angles).
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