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