Terahertz System Analysis Using Zemax Prepared by Andrew Mueller for Ted Stinson and other members of the Basov Infrared Laboratory
I. Introduction The commercial optical design software Zemax was used to model and evaluate the performance of a THz Near Field microscope. In this
system, geometric optics approximations were not valid due to the long wavelength of Thz radiation. So by using the Physical Optics
Propagation (POP) tool build into the engineering version of Zemax, it was possible to model the beam using a paraxial approximation.
The initial goals of this venture are listed below. For the most part, all were at least partially fulfilled, though some conclusions were
realized late in the process of building the experiment, or did not easily suggest a revised experiment design.
Goals of the THz system analysis using Zemax:
Determine the intensity and power of Thz radiation that could make it from the emitter and be focused onto the AFM
tip.
Learn what simple modifications to the system’s geometry could yield even better focusing performance at
the tip
Discover if the paraxial Gaussian beam approximation of Thz radiation would show a situation where the light would
not be focused as desired.
Develop a working knowledge of Zemax and its applicability to projects of this type, since it may be useful again in this
lab.
Basic Layout
This image shows the basic geometry
of the system. A broadband THz signal
with wavelengths ranging from
100um to 3000um is focused onto an
AFM tip using parabolic mirrors. Some
lengths, like length A and B as shown
in the image, were not set to specific
values in the Zemax file. The intention
was to leave these lengths as
variables so that changing them might
improve the focus on the AFM tip.
However, using the system with a
broadband signal showed that it
could not be optimized for specific
paraxial beam waist locations. What
worked best for one wavelength
would not translate to others. After
working with the system for some
time, it appeared the best lengths for
A and B were as short as possible, as
permitted by hardware and spatial
constraints. But the lengths still did not have a large impact on the system's performance. For example, increasing A from 30mm to
130mm caused the radius of the beam at the tip to increase by less than 2% both for 100um and 3000um light. Keeping A and B
short has more of an effect on ratio of power that makes it to the tip, because of clipping at the various reflective surfaces. Increasing A
from 30mm to 130mm decreased the power transmitted to the tip by less than 5% for wavelengths shorter than 1000um.
Objective of this Report
This report is focused on the modeling of an optical instrument and covers insights gained using Zemax with significance for the project.
Though not a general guide on how to use Zemax, it does outline techniques that would be useful for modeling of similar systems. This
report would be useful for any projects that involve off-axis layouts, parabolic mirrors, or Gaussian beam models of long wavelength light.
II. System Layout and Geometric Optics
A. System-Specific Layout
1. Sequential vs Non Sequential
Zemax has two modes with different features and advantages for building and analyzing off axis systems: Sequential and Non-
Sequential. The mode is chosen under the File menu. For sequential mode, the order that surfaces are listed in the Lens Data
Editor matters. Rays are only traced from surface 1 to surface 2 and so on. This means rays will not interact with a surface twice
and will pass through a surface if it is not the next in the list. Using sequential mode requires a good understanding of exactly
how the rays are supposed to propagate. It was mainly used during this investigation because physical optics propagation is
only supported for a set of surfaces in this mode.
For Non-Sequential mode, the order of surfaces in the Lens Data Editor does not matter, and rays will interact with any surface
they encounter. Also in this mode, the incident rays can be defined in many more ways. As described in a later section, a point
source of radiation can be modeled which was used for studying how light would propagate from the tip to the THz detector.
Also, more arbitrary 3D surface shapes can be modeled or imported for use in Non-Sequential mode.
2. Aperture Type – Incidence Angle
For the sake of viewing the experiment setup in the Zemax 3D layout viewer, correctly tracing some rays through the system
can be helpful. However, there is not a strong
correlation between how a beam is defined for
raytracing and for Physical Optics Propagation.
Defining the incident beam in the Zemax general
settings only applies to the tracing of rays. Still, we
modeled the incident beam geometrically using the
Object Cone Angle Feature. This was useful because
we did not know or care yet what the stop or
smallest aperture was in our system. But we did know, from specifications of the Thz emitter, that most of the beam was inside
a cone angle of 12.1 degrees.
3. Using Coordinate Breaks
A three dimensional system of mirrors and lenses can be built and analyzed in Zemax while still using Sequential Mode. But
because rays are propagated through surfaces in a specific order, the location of surfaces is defined in the context of this order.
The location and orientation of a surface is not defined by a single universal coordinate system, but instead by the surface that
preceded it. This way of building in 3D can be confusing for those with previous 3d modeling experience.
Coordinate breaks are ‘pseudo-surfaces’ that are added in the Lens Data editor. They are not true surfaces because they have
no effect on rays propagated through the system. They only work to modify the coordinates system for the following surface in
the Lend Data Editor list. To add a coordinate break, add another surface in the Lens Data editor with the Insert key. Double
click the first column entry of this surface to bring up the surface properties. Inside the Surface Type dropdown menu the
Coordinate Break option is found. Coordinate Breaks have six parameters as shown here in the Lens data Editor:
When the order parameter is set to zero, coordinate break applies the five
transformations in this order: Decenters x and y, tilts about the local Z, tilts
about the new Y, and then about the new x. When the order is set to any
other number, then the transformations are applied in the opposite order.
This can be useful when one needs one coordinate break to put a surface in
an arbitrary configuration, and one coordinate break to bring the coordinate
system back to what it was before that surface. Finally, the thickness of the
coordinate break acts the same way as it does for any other surface. It is the
length between the current surface and the next surface (or added after the
transformations of a coordinate break). The image shows a simplified 2D
representation of the order in which a coordinate break applies
transformations.
4. Making custom shape fold mirror
The aperture of a mirror or lens is the two dimensional shape that is
projected over the surface of the element. The plane of the projection is
normal to the (local) optical axis. Rays inside the shape are modified by
the lens or mirror as normal; rays outside the shape are clipped or
ignored. Essentially the lens or mirror is this shape when viewed along the
optical axis. Custom shaped apertures for lenses and mirrors can be
defined using a .UDA file which is selected for a particular surface on the
Aperture tab of the surface properties menu.
Starting on page 78, the Zemax manual provides an excellent explanation
on how to define a custom aperture using a list of User Defined Aperture
Entities. Through this method, fold mirrors with one semicircular edge and
three straight edges were added to the system.
5. Making an Off Axis Parabolic Mirror
Zemax fully supports mirror surfaces, but the technique for making an Off Axis
Parabolic Mirror (OAP) is rather involved. The procedure is to make the full
parabolic mirror that the final OAP is part of. Then the parts of the mirror that
are not needed are ‘cropped’ away.
First, add a Mirror type surface. Enter double the desired mirror focal length as
the radius of curvature value. The focal length of a parabolic mirror is the
closest distance from the mirror surface to the point of focus.
For a 90 degree off axis parabolic mirror, the distance from the focus to the
center of the mirror is double the focal length of the ‘parent’ parabola. So if
you know this distance, shown as the Reflected Focal Length in the image, then
it can simply be entered in the radius of curvature value to build the parent
parabola. In the Thz system, the first collimating OAP has a Parent Focal Length
of 25.4mm and a Reflected Focal Length of 50.8mm. -50.8 was used as the
Radius value for this surface. The radius is negative just so the mirror is
oriented correctly in the system.
Second, enter -1 for the Conic value of the mirror surface in the Lens Data Editor. A parabola has a conic constant of -1. The
Semi-Diameter of the mirror is its radius as measured from its optical axis. Make sure the Semi-Diameter is large enough so this
‘parent’ parabola includes the surface of the OAP. For example, if you wish to model a 90 degree OAP with a 100mm reflected
focal length and a 30mm Diameter (see image above), then the parent parabola would need to have a Semi-Diameter of at least
115mm .
Third, under the Aperture tab of the surface properties, choose Circular Aperture. Enter the radius of the OAP as the Max
Radius. For a 90 degree OAP, the reflected focal length can be entered for either the Aperture X-decenter or Aperture Y-
decenter, depending on where the OAP should be located. This is the distance from the center axis of the parent OAP to the
center of the OAP aperture.
B. Angles of beams incident on AFM tip A collimated Thz beam is reflected off of the wide parabolic mirror surrounding the AFM tip. The resulting beam is a cone that
focuses on the tip. Angles that describe the orientation of the central ray of this cone can be found. These angles were useful
for constructing the entire Thz system layout in Zemax. In most of the Zemax project files used, a mirror was used where the
AFM tip would be in the physical system. The mirror worked to reflect incident rays symmetrically so that the rays propagated
after this mirror would represent the path of light radiated by the tip and collected by the Thz detector. The orientation of this
mirror depended on a number or geometrical factors and was found using the method shown here.
(
) Estimates:
The estimate values in blue are used in a recent Solidworks file of the Thz experiment. They were also used in the Zemax files.
(
) (
) (
)
True focal length of this OAP:
(
(
) ( )
) (
(
) ( )
)
(
(
)( )
)
This angle is used by the coordinate breaks above and below the TipMirror surface in most project files. It is the angle between
the normal of the mirror surface and the YZ plane of the 3D view.
III. Physical Optics Propagation
A. Gaussian beam equations and parameters
1. Rayleigh Range,
The Rayleigh range is the distance from the waist
to where the beam radius has increased by √ .
For a circular beam, it is the distance from the
waist to where the area has doubled.
(1.1) ( )
(1.2) ( ) (
)
2. Waist, w
(2.1) ( )
(2.2) ( ) √
3. Divergence Angle
(3.1) ( )
Valid for
B. Power transmitted to tip
1. Reevaluation of incident divergence angle
The incident beam was initially modeled in
Zemax using the values found in this image
provided by the manufacturer of the Thz
emitter:
At first, the divergence angle was set to the
12.1 angle from the image. But this led
to clipping of a significant portion of the
beam at the first OAP because the divergence
angle as defined in Zemax is the the 1/e2
size
of the beam. That is, the radius of a circle that traces 86% of the beam's total power. It was then decided that the 12.1 degree
angle shown in the image is not the divergence angle, but instead an angle that traces the propagation of 99% of the beam's
power. The A.E. Seigman book on lasers has a formula [2]:
(
)
This implies that the angle tracing 99% of the beams power is related to the true divergence angle by:
(
) ( ) ( )
So the true divergence angle theta is:
((
) ( ))
This image of a Gaussian beam visualizes most parameters in the equations.
A divergence angle of 7.77 was used in the Physical Optics Propagating settings of the Thz system.
2. Separate X, Y and Polarization
Turning on Separate X, Y in the POP settings makes Zemax define
the beam using x and y coordinates rather than a radial
coordinate. This setting allows the program to work better with
non-radially symmetric beams. Since the entire Thz system
incorporates coordinate breaks and other non-radially symmetric
features, it is probably best to leave this setting on when
simulating the most accurate propagation.
According to the Zemax manual, turning on the Use Polarization feature will "permit the modeling of effects of optical coatings
on the phase and amplitude of the transmitted or reflected beam". Since all the reflective surfaces in the Thz system are gold
which reflects very well in the Thz range, the effect of optical coatings on beam propagation was not thoroughly investigated.
By keeping the Use Polarization feature turned off, all mirrors reflect perfectly and this assumption was adequate for this
project. However, Zemax has the tools to model a system with more
physically accurate mirrors if the added accuracy is ever needed. One
would make a custom coating.dat file with the parameters specific to
that mirror surface.
As shown by data later in this section, turning on the Use Polarization
feature in the POP settings with no coating specified on the mirror
surfaces will decrease transmitted intensity by a few percent per
reflecting surface. This is because Zemax assumes mirrors to be bare
aluminum with refractive index 0.7 + 7i – a non-ideal reflecting surface
[3]. (A true aluminum surface would have a varying refractive index
with wavelength, but in this case Zemax uses these fixed values which
translate to about 94% reflectivity at normal incidence1).
3. Power transmitted to Tip – Initial Data
Now with the system set up so Thz radiation is transmitted to the AFM tip through 2 OAP's and one fold mirror, Zemax can
predict how much of the source's power will make it to the tip.
The following tables show data at the tip for several wavelengths emitted by the Thz source. The values of greatest interest are
probably the Peak Irradiance and Size X,Y. The power initially emitted by the source in our simulation is exactly 1 Watt. This
table included data updated since the
discovery of the software glitch described in
Section C of this chapter. The tilt angle used
for the first OAP was 89.999 degrees.
The first column is data defining the
Gaussian beam at the tip focus point with
both settings described above turned off.
Zemax is approximating the beam as radially
symmetric throughout the system. Values
for Waist X, Size X and Distance X are
actually singular radial measurements for
this column. The image shows what a non-
radial Gaussian beam might look like and
visualizes a few of the parameters in the data. However, the beam can also have different values for the Distance parameter in
X and Y, which is not shown in the image.
1 It may seem odd that the default no-coating setting would assume a non-ideal reflecting surface. But a Zemax staff member has explained on their forums that this was done to keep users from naively expecting 100% reflection. [4]
The default settings for coating in the properties menu of a
mirror. The mirror will have refractive index 0.7 + 7.0i
100um Off Separate XY Both
Power (W) 9.98E-01 9.98E-01 7.90E-01 Peak Irradiance (W/mm
2)
2.67E+01 2.67E+01 2.06E+01
Waist X (mm) 1.48E-01 1.49E-01 1.49E-01 Waist Y (mm) 1.48E-01 1.48E-01 Size X (mm) 1.49E-01 1.49E-01 1.49E-01 Size Y (mm) 1.48E-01 1.48E-01 Distance X (mm) -4.20E-02 -3.08E-02 -3.08E-02 Distance Y (mm) -5.31E-02 -5.31E-02
1000um Off Separate XY Both
Power (W) 9.89E-01 9.89E-01 7.82E-01 Peak Irradiance (W/mm
2)
2.78E-01 2.78E-01 2.14E-01
Waist X (mm) 1.26E+00 1.37E+00 1.37E+00 Waist Y (mm) 1.16E+00 1.16E+00 Size X (mm) 1.43E+00 1.49E+00 1.49E+00 Size Y (mm) 1.41E+00 1.41E+00 Distance X (mm) -2.72E+00 -2.57E+00 -2.57E+00 Distance Y (mm) -2.94E+00 -2.94E+00 Power (W) 9.89E-01 9.89E-01 7.82E-01
3000um Off Separate XY Both
Power (W) 7.71E-01 7.64E-01 6.00E-01 Peak Irradiance (W/mm
2)
3.98E-02 4.00E-02 3.10E-02
Waist X (mm) 2.55E+00 2.73E+00 2.73E+00 Waist Y (mm) 1.99E+00 1.99E+00 Size X (mm) 3.55E+00 4.50E+00 4.50E+00 Size Y (mm) 3.21E+00 3.21E+00 Distance X (mm) -6.59E+00 -1.02E+01 -1.02E+01 Distance Y (mm) -5.26E+00 -5.26E+00
C. Long wavelength problems
1. Introduction
Once the Thz system was set up in Zemax, it was used to calculate the parameters of the Gaussian beam that was
projected onto the AFM tip’s location. The system worked as expected for wavelengths below 2000um or so. The
majority of the beam’s power would be collected by the first OAP, collimated, and then refocused on the tip. But for
longer wavelengths, the power available near the tip dropped by multiple orders of magnitude. It was soon
discovered the reason for this decrease was a very poorly collimated beam between the first OAP and fold mirror. The
profile of the beam in this ‘collimated zone’ often looked similar to the following image. The beam was far from
circular and had a very short Rayleigh range in at least one axis (when using the Separate X,Y feature). A well
collimated beam does not change significantly along the direction of travel. For Gaussian beams, this is the case when
the Rayleigh range is long compared with the propagation distance [7]. From early on, potential reasons for the poor
collimation were categorized as follows:
Profile of the beam in the ‘collimated region’ with a 3000um signal. It
often had this characteristic ‘butterfly’ shape.
a) A real-world phenomena
Nothing was wrong with Zemax or
its calculations and it was giving a
valid prediction of the physical
Thz system’s performance.
b) An incorrectly set up project
file
Perhaps the Physical Optics
Propagation tool was not being
used correctly, or Zemax was
reporting relevant errors which
were wrongly ignored.
c) An error in Zemax, or a limit of
its programming
This seemed more likely if the
problem appeared and disappeared
upon changing what should be inconsequential ways. A computational error also seemed more likely if we
were using Zemax in especially unique ways that the original programmers would not have considered.
2. Clipping
At first, it seemed one explanation for the collimation problems found for long wavelengths may have involved one of
the mirrors clipping the beam. As long as the wavelength is not too large though, it was calculated before that the 1st
OAP reflected 99% or more of the incident beam because the 99% angle of 12.1 degrees would project a circle of
radius 10.9mm at the distance of 1 focal length (50.8mm). The 1st OAP has a radius of 12.7mm. However, for long
wavelengths, the Rayleigh range is longer and the size of the beam does not seem to increase linearly with the
divergence angle. The tangent of the 12.1 degrees times the distance from the waist is no longer a good
approximation for the size of the beam.
If the dashed lines represent the 12.1 degree 99% angle and the solid black lines the actual beam profile, then one
can see how the beam will clip even if the circle traced by the 12.1 degree angle at 1 focal length is within the OAP
aperture.
For lambda=100um, the 1st
OAP reflects 99.8% of
the incident beam. For lambda=3000um, it
reflects 96.5% of the beam. The image shows the
incident signal falling on the OAP at
lambda=3000um. Notice the faint outline of the
projection of the mirror.
So yes, less than 99% of the beam is reflected for
long wavelengths, but 96.5% is still quite good. It
is unlikely that diffraction from this small clipping
is the cause of behavior seen in the collimated
region for long wavelengths.
3. Zemax chooses the incorrect propagator
Zemax employs two distinct mathematical methods or propagators for defining the phase of a beam’s electric field
and commuting how it changes with distance. The Zemax manual has extensive information on how each propagator
works, and when one should be used over the other. Before getting into how they are explicitly used by the program,
the manual explains in what theoretical cases one propagator should work better than the other, and these cases
depend on the dimensionless Fresnel number:
Where is the radius of of the beam at the first surface, is the distance between the surfaces, and is the
wavelength.
The Angular Spectrum Propagator is valid when the Fresnel number between surfaces is large. This is the case when
the surfaces are close together. But the propagator also works well over long distances and if the beam does not
change size significantly. [1,603]
The Fresnel Diffraction Propagator is valid when the Fresnel number is small. It is most appropriate when the beam
changes size significantly between one surface and the other. The Zemax manual provides some information on the
mathematical methods that these propagators use.
In the case of the Thz system, all regions the beam
passes through fulfill the requirements of the Angular
Spectrum Propagator reasonably well. Zemax has an
option on the Physical Optics tab of a surface’s
properties to force use of the Angular Spectrum
Propagator:
Checking this option for the first OAP in the Thz
system does change results of the POP tool,
suggesting Zemax was using the Fresnel Diffraction
propagator in the collimated zone. But unfortunately
the odd behavior persists while only using Angular
Spectrum. For long wavelength light, the beam has a
large divergence angle and strange asymmetrical
cross section in the collimated zone. So the problem
is not related to or not only caused by Zemax automatically choosing an inappropriate propagator.
4. Varying waist vs. varying divergence angle
Zemax gives the option of specifying the waist or divergence angle when defining an incident Gaussian beam. If the
waist is specified, then the divergence angle of the beam is proportional to its wavelength as shown by equation (3.1)
If the divergence angle is specified, then the waist will vary with wavelength. After some deliberation, we finally came
to the conclusion that fixing the divergence angle and letting the waist vary would best approximate the way in which
the Thz source functioned.
5. Unexpected performance for zR f
We tried to change parameters of our project file and discover in what specific situations the error would exist. We
observed that if the beam was reflected by the 1st OAP and had a Rayleigh range very close to the focal length of the
OAP, then collimation in the 'collimated zone' after the 1st OAP was quite poor. The behavior could be observed while
propagating any wavelength of light with any Rayleigh range so long as the focal length of the mirror was adjusted
accordingly. This graph shows how the Rayleigh range in the collimate zone after the first OAP changes with a varying
mirror focal length. Data was collected in this way:
a) A wavelength and divergence angle for the incident Gaussian beam were chosen. The divergence angle for all
data was constant: 7.77 degrees just like in the Thz system.
b) An 90 degree off axis parabolic mirror was added so that it reflected the beam. The reflected focal length of the
mirror and distance between the mirror and the beam waist were varied together so that the reflected light
would always be perfectly collimated if we were only tracing geometric rays.
c) While the reflected focal length was varied over a range of values (from 80% to 120% of the beam’s incident
Rayleigh range), the Rayleigh range after the mirror (in the collimated zone) was recorded.
d) The process was repeated for several wavelengths of light.
The graph shows how the Rayleigh range in the collimated zone decreases by several orders of magnitude if the distance from the incident waist to the OAP is about the Rayleigh range of the incident beam (and the OAP is set up to collimated geometric rays correctly – waist to OAP is equal to reflected focal length). Given this information, it is highly probable the long wavelength issue has some relation to the Rayleigh range of the incident beam. Therefore, it helps to restate our hypotheses as such:
i) Rayleigh range has real-world significance
Zemax is correctly modeling a real-world phenomenon related to the Rayleigh range. We could
expect to see the results of the graph above in a physical experiment.
ii) Zemax computation methods use the concept of the Rayleigh range
If Zemax algorithms calculate and use the Rayleigh range, then the error is due to a sort of
discontinuity of processing methods. This is a computational error.
a) Real-World significance of the Rayleigh Range
The radius of curvature of a beam is minimized one Rayleigh range from its waist, and is equal to twice the Rayleigh
range. At the waist, the wavefront is planar, while very far from the waist the wavefront has a radius equal to its
distance from the waist (like that of a point source). This is one fact that supports the physical significance of the
Rayleigh range. But it does not lend any useful insight into why reflecting the beam near this point would cause the
effects seen in Zemax.
The math underlying propagation of a Gaussian beam
was investigated. The Rayleigh range of a beam after it
has passed through a thin on-axis lens can be found in
terms of its original Rayleigh range[8].
( )
z is the distance between the incident waist and the
lens; f is the focal length of the lens. We are
investigating the case of z = f.
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
80% 85% 90% 95% 100% 105% 110% 115% 120%
Ray
leig
h R
ange
in C
olli
amte
d Z
on
e
Focal Length set to Percentage of Incident Rayleigh Range
Rayleigh Range in Colliamted Zone
100 um
200 um
500 um
1000 um
3000 um
Using the definition of r and M from the image, the equation simplifies to:
Which would simply trace a parabola on the graph above. Clearly this on-axis theoretical model does not describe the
unusual performance of Zemax with the OAP.
b) The Rayleigh range in Zemax computations
As described above, Zemax will choose to the Angular Spectrum Propagator or the Fresnel Diffraction Propagator
based on properties of the beam. The Zemax manual describes how the program chooses a propagator based on the
properties of a pilot beam. The pilot beam is a simple Gaussian beam constructed by fitting Gaussian beam
parameters to the initial (more complex) distribution at the first surface before a region. Zemax calculates the
location of the Rayleigh range of this simple beam. It then uses this information to determine where the two
propagators should be used for propagating the real beam and in what order [1,605]. The process is quite a bit more
complicated and well described in the manual, but the only relevant knowledge gained is that certain Rayleigh ranges
are found and used for intermediate calculations.
6. Fix
Two ways to avoid the problem were found. First, under the Physical Optics tab of a surface’s properties, there is an
option to Use Rays to Propagate to Next Surface. The problem is not apparent if Zemax traces rays from the mirror to
the next surface, and then starts using Angular spectrum propagation after. It is not very clear if using this option will
oversimplify the situation and not show us some physically significant result. But if the surface after the 1st
OAP is
very close to OAP, then the distance over which rays are used is short and possibly inconsequential.
When discussing the ray propagation feature, the Zemax manual states this:
….This is a very desirable property, because geometrical optics may be used to propagate through whole optical components that would be difficult to model with physical optics propagation. These include highly tilted surfaces and gradient index lenses, to name a few. [1,609]
Our 90 degree off axis parabolic mirror may qualify as a highly tilted surface.
Second, changing the tilt of the first coordinate break in the system from -90 to an acute but very close angle, like
-89.999, will also cause the problem to disappear. This is persuasive evidence that the phenomena in the collimated
zone with long wavelengths was a computational error all along. This is because we would not expect such a minute
change of angle to cause such a significant change in the real reflected beam.
References [1] Zemax Development Corporation, "ZEMAX User's Guide" (2009)
[2] A. Siegman, Lasers (University Science Books, Mill Valley, New Edition, 1986), pp.669
[3] Nicholson, M. How is a MIRROR Without a Coating Handled?. Zemax (2007). at <https://www.zemax.com/support/knowledgebase/how-is-a-
mirror-without-a-coating-handled>
[4] Zemax Staff,. MIRROR reflectivity in NS mode. Zemax Forums (2015). at <http://forum.zemax.com/Topic2763.aspx>
[5] Mathar, R. Solid Angle of a Rectangular Plate. (Max-Planck Institute of Astronomy, 2014). at
<http://www.mpia.de/~mathar/public/mathar20051002.pdf>
[6] Tocci, M. Demystifying the Off-Axis Parabola Mirror. Zemax (2006). at https://www.zemax.com/support/knowledgebase/demystifying-the-off-
axis-parabola-mirror
[7] Paschotta, R. Collimated Beams. RP Photonics Encyclopedia. at http://www.rp-photonics.com/collimated_beams.html
[8]Saleh, B. Beam Optics. Fundamentals of Photonics (1991). at http://gautier.moreau.free.fr/cours_optique/chapter03.pdf