3D Optical System/Propagation Modeling:
3D optical lens system ray trace analysis.
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3D far-field modeling of truncated Gaussian optical
It is often useful to know the far-field radiation pattern of specific
types of optical beams. The far-field 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 zero-baffled).
Figure 1 illustrates the geometry of the problem.
Figure 1. Assumed geometry of
the far-field extrapolation problem
Given the complex aperture function Ein(x,y),
the far-field pattern Eout(k) is a function of
the direction angles Φ,Θ as well as the wavelength
λ. Specifically, Eout(k) is a 2D Fourier
transform of the aperture function Ein(x,y),
where C0 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 far-field
point being computed.
Eq. 1 is based on Fraunhofer diffraction theory which
assumes that the far-field 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 >> D2/(8λ).
For engineering purposes, this criterion is often simplified to
R > D2/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 Ein(x,y) over the input aperture
with continuous polynomials. In the examples below, the real and
imaginary parts of Ein(x,y) were modeled with bicubic splines. The
real part of Ein(x,y), for example, was modeled over the input aperture
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 xi,yj element of the input aperture. This
element-by-element summation is repeated for each far-field direction
in which the far-field 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 well-sampled 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
The following examples illustrate computed far-field
patterns of a Gaussian input beam blocked by a knife-edge occlusion
to various extents. Decibel plots of the input aperture and the
resulting far-field 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 (non-dimensionalized) distance
(grid distance divided by wavelength), while the coordinates of
the far-field distribution plots are in degrees (angles).
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