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Tutorial Texts Series
Matrix Methods for Optical Layout, Gerhard Kloos, Vol. TT77
Fundamentals of Infrared Detector Materials, Michael A. Kinch, Vol. TT76
Practical Applications of Infrared Thermal Sensing and Imaging Equipment, Third Edition, HerbertKaplan, Vol. TT75
Bioluminescence for Food and Environmental Microbiological Safety, Lubov Y. Brovko, Vol. TT74
Introduction to Image Stabilization, Scott W. Teare, Sergio R. Restaino, Vol. TT73
Logic-based Nonlinear Image Processing, Stephen Marshall, Vol. TT72
The Physics and Engineering of Solid State Lasers, Yehoshua Kalisky, Vol. TT71
Thermal Infrared Characterization of Ground Targets and Backgrounds, Second Edition, Pieter A. Jacobs,Vol. TT70
Introduction to Confocal Fluorescence Microscopy, Michiel Mller, Vol. TT69
Artificial Neural Networks An Introduction, Kevin L. Priddy and Paul E. Keller, Vol. TT68
Basics of Code Division Multiple Access (CDMA), Raghuveer Rao and Sohail Dianat, Vol. TT67
Optical Imaging in Projection Microlithography, Alfred Kwok-Kit Wong, Vol. TT66
Metrics for High-Quality Specular Surfaces, Lionel R. Baker, Vol. TT65
Field Mathematics for Electromagnetics,Photonics,and Materials Science, Bernard Maxum, Vol. TT64
High-Fidelity Medical Imaging Displays, Aldo Badano, Michael J. Flynn, and Jerzy Kanicki, Vol. TT63
Diffractive OpticsDesign,Fabrication,and Test,Donald C. OShea, Thomas J. Suleski, Alan D.Kathman, and Dennis W. Prather, Vol. TT62
Fourier-Transform Spectroscopy Instrumentation Engineering, Vidi Saptari, Vol. TT61
The Power- and Energy-Handling Capability of Optical Materials,Components,and Systems,Roger M.Wood, Vol. TT60
Hands-on Morphological Image Processing,Edward R. Dougherty, Roberto A. Lotufo, Vol. TT59
Integrated Optomechanical Analysis,Keith B. Doyle, Victor L. Genberg, Gregory J. Michels,Vol. TT58
Thin-Film Design Modulated Thickness and Other Stopband Design Methods,Bruce Perilloux, Vol. TT57
Optische Grundlagen fr Infrarotsysteme,Max J. Riedl, Vol. TT56
An Engineering Introduction to Biotechnology, J. Patrick Fitch, Vol. TT55
Image Performance in CRT Displays, Kenneth Compton, Vol. TT54
Introduction to Laser Diode-Pumped Solid State Lasers, Richard Scheps, Vol. TT53
Modulation Transfer Function in Optical and Electro-Optical Systems, Glenn D. Boreman, Vol. TT52
Uncooled Thermal Imaging Arrays,Systems,and Applications, Paul W. Kruse, Vol. TT51
Fundamentals of Antennas, Christos G. Christodoulou and Parveen Wahid, Vol. TT50
Basics of Spectroscopy, David W. Ball, Vol. TT49
Optical Design Fundamentals for Infrared Systems, Second Edition, Max J. Riedl, Vol. TT48
Resolution Enhancement Techniques in Optical Lithography, Alfred Kwok-Kit Wong, Vol. TT47 Copper Interconnect Technology, Christoph Steinbrchel and Barry L. Chin, Vol. TT46
Optical Design for Visual Systems, Bruce H. Walker, Vol. TT45
Fundamentals of Contamination Control, Alan C. Tribble, Vol. TT44
Evolutionary Computation Principles and Practice for Signal Processing, David Fogel, Vol. TT43
Infrared Optics and Zoom Lenses,Allen Mann, Vol. TT42
Introduction to Adaptive Optics,Robert K. Tyson, Vol. TT41
Fractal and Wavelet Image Compression Techniques,Stephen Welstead, Vol. TT40
Analysis of Sampled Imaging Systems,R. H. Vollmerhausen and R. G. Driggers, Vol. TT39
Tissue Optics Light Scattering Methods and Instruments for Medical Diagnosis, Valery Tuchin, Vol. TT38
Fundamentos de Electro-ptica para Ingenieros, Glenn D. Boreman, translated by Javier Alda, Vol. TT37
Infrared Design Examples,William L. Wolfe, Vol. TT36
Sensor and Data Fusion Concepts and Applications,Second Edition, L. A. Klein, Vol. TT35
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Bellingham, Washington USA
Tutorial Texts in Optical Engineering
Volume TT77
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Library of Congress Cataloging-in-Publication Data
Kloos, Gerhard.Matrix methods for optical layout / Gerhard Kloos.
p. cm. -- (Tutorial texts series ; TT 77)
ISBN 978-0-8194-6780-51. Optics--Mathematics. 2. Matrices. 3. Optical instruments--Design and construction. I. Title.
QC355.3.K56 2007
681'.4--dc222007025587
Published by
SPIEP.O. Box 10
Bellingham, Washington 98227-0010 USA
Phone: +1 360 676 3290Fax: +1 360 647 1445
Email: [email protected]
Web: spie.org
Copyright 2007 Society for Photo-optical Instrumentation Engineers
All rights reserved. No part of this publication may be reproduced or distributedin any form or by any means without written permission of the publisher.
The content of this book reflects the work and thought of the author(s).Every effort has been made to publish reliable and accurate information herein,but the publisher is not responsible for the validity of the information or for any
outcomes resulting from reliance thereon.
Printed in the United States of America.
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Introduction to the Series
Since its conception in 1989, the Tutorial Texts series has grown to more than 70titles covering many diverse fields of science and engineering. When the series
was started, the goal of the series was to provide a way to make the material
presented in SPIE short courses available to those who could not attend, and to
provide a reference text for those who could. Many of the texts in this series are
generated from notes that were presented during these short courses. But as
stand-alone documents, short course notes do not generally serve the student or
reader well. Short course notes typically are developed on the assumption that
supporting material will be presented verbally to complement the notes, which
are generally written in summary form to highlight key technical topics and
therefore are not intended as stand-alone documents. Additionally, the figures,tables, and other graphically formatted information accompanying the notes
require the further explanation given during the instructors lecture. Thus, by
adding the appropriate detail presented during the lecture, the course material can
be read and used independently in a tutorial fashion.
What separates the books in this series from other technical monographs andtextbooks is the way in which the material is presented. To keep in line with the
tutorial nature of the series, many of the topics presented in these texts are
followed by detailed examples that further explain the concepts presented. Many
pictures and illustrations are included with each text and, where appropriate,tabular reference data are also included.
The topics within the series have grown from the initial areas of geometrical
optics, optical detectors, and image processing to include the emerging fields of
nanotechnology, biomedical optics, and micromachining. When a proposal for a
text is received, each proposal is evaluated to determine the relevance of the
proposed topic. This initial reviewing process has been very helpful to authors in
identifying, early in the writing process, the need for additional material or other
changes in approach that would serve to strengthen the text. Once a manuscript is
completed, it is peer reviewed to ensure that chapters communicate accurately theessential ingredients of the processes and technologies under discussion.
It is my goal to maintain the style and quality of books in the series, and to
further expand the topic areas to include new emerging fields as they become of
interest to our reading audience.
Arthur R. Weeks, Jr.
University of Central Florida
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Contents
Preface xi
1 An Introduction to Tools and Concepts 1
1.1 Matrix Method 1
1.2 Basic Elements 21.2.1 Propagation in a homogeneous medium 2
1.2.2 Refraction at the boundary of two media 3
1.2.3 Reflection at a surface 5
1.3 Comparison of Matrix Representations Used in the Literature 5
1.4 Building up a Lens 6
1.5 Cardinal Elements 7
1.6 Using Matrices for Optical-Layout Purposes 10
1.7 Lens Doublet 12
1.8 Decomposition of Matrices and System Synthesis 131.9 Central Theorem of First-Order Ray Tracing 14
1.10 Aperture Stop and Field Stop 16
1.11 Lagrange Invariant 18
1.11.1 Derivation using the matrix method 18
1.11.2 Application to optical design 18
1.12 Petzval Radius 19
1.13 Delano Diagram 19
1.14 Phase Space 20
1.15 An Alternative Paraxial Calculation Method 211.16 Gaussian Brackets 22
2 Optical Components 25
2.1 Components Based on Reflection 25
2.1.1 Plane mirror 25
2.1.2 Retroreflector 26
2.1.3 Phase-conjugate mirror 26
2.1.4 Cats-eye retroreflector 27
2.1.5 Roof mirror 28
2.2 Components Based on Refraction 292.2.1 Plane-parallel plate 29
2.2.2 Prisms 31
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viii
2.2.3 Axicon devices 47
2.3 Components Based on Reflection and Refraction 49
2.3.1 Integrating rod 492.3.2 Triple mirror 52
3 Sensitivities and Tolerances 53
3.1 Cascading Misaligned Systems 55
3.2 Axial Misalignment 56
3.3 Beam Pointing Error 57
4 Anamorphic Optics 59
4.1 Two Alternative Matrix Representations 59
4.2 Orthogonal and Nonorthogonal Anamorphic Descriptions 604.3 Cascading 60
4.4 Rotation of an Anamorphic Component with Respect to the
Optical Axis 61
4.4.1 Rotation of an orthogonal system 62
4.4.2 Rotation of a nonorthogonal system 65
4.5 Examples 65
4.5.1 Rotated anamorphic thin lens 65
4.5.2 Rotated thin cylindrical lens 66
4.5.3 Cascading two rotated thin cylindrical lenses 674.5.4 Cascading two rotated thin anamorphic lenses 68
4.5.5 Quadrupole lens 69
4.5.6 Telescope built by cylindrical lenses 72
4.5.7 Anamorphic collimation lens 72
4.6 Imaging Condition 73
4.7 Incorporating Sensitivities and Tolerances in the Analysis 75
5 Optical Systems 77
5.1 Single-Pass Optics 77
5.1.1 Triplet synthesis 77
5.1.2 Fourier transform objectives and 4farrangements 79
5.1.3 Telecentric lenses 80
5.1.4 Concatenated matrices for systems of n lenses 81
5.1.5 Dyson optics 82
5.1.6 Variable single-pass optics 84
5.2 Double-Pass Optics 92
5.2.1 Autocollimator 92
5.3 Multiple-Pass Optics 95
5.4 Systems with a Divided Optical Path 995.4.1 Fizeau interferometer 99
5.4.2 Michelson interferometer 101
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ix
5.4.3 Dyson interferometer 103
5.5 Nested Ray Tracing 107
6 Outlook 111
Bibliography 113
Index 119
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Preface
This book is intended to familiarize the reader with the method of Gaussian matri-
ces and some related tools of optical design. The matrix method provides a means
to study an optical system in the paraxial approximation.
In optical design, the method is used to find a solution to a given optical task,which can then be refined by optical-design software or analytical methods of aber-
ration balancing. In some cases, the method can be helpful to demonstrate that
there is no solution possible under the given boundary conditions. Quite often it is
of practical importance and theoretical interest to get an overview on the solution
space of a problem. The paraxial approach might then serve as a guideline during
optimization in a similar way as a map does in an unknown landscape.
Once a solution has been found, it can be analyzed under different points of
view using the matrix method. This approach gives insight on how degrees of
freedom couple in an optical device. The analysis of sensitivities and tolerances is
common practice in optical engineering, because it serves to make optical devices
or instruments more robust. The matrix method allows one to do this analysis in a
first order of approximation. With these results, it is then possible to plan and to
interpret refined numerical simulations.
In many cases, the matrix description gives useful classification schemes of
optical phenomena or instruments. This can provide insight and might in addition
be considered as a mnemonic aid.
An aspect that should not be underestimated is that the matrix description re-
presents a useful means of communicating among people designing optical instru-
ments, because it gives a kind of shorthand description of main features of an opti-cal instrument.
The book contains an introductory first chapter and four more specialized chap-
ters that are based on this first chapter. Sections 1.11.14 are intended to provide a
self-contained introduction into the method of Gaussian transfer matrices in parax-
ial optics. The remaining sections of the chapter contain additional material on how
this approach compares to other paraxial methods.
The emphasis of Chapters 3 and 4 is on refining and expanding the method of
analysis to additional degrees of freedom and to optical systems of lower symmetry.
The last part of Chapter 4 can be skipped at first reading.To my knowledge, the text contains new results such as theorems on the design
of variable optics, on integrating rods, on the optical layout of prism devices, etc.
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xii Preface
I tried to derive the results in a step-by-step way so that the reader might apply
the methods presented here to her/his design problems with ease. I also tried to
organize the book in a way that might facilitate looking up results and the ways ofhow to obtain them.
It would be a pleasure for me if the reader might find some of the material
presented in this text useful for her/his own engineering work.
Gerhard Kloos
June 2007
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Chapter 1
An Introduction to Tools and
Concepts
1.1 Matrix Method
Ray-transfer matrices is one of the possibile methods to describe optical systems
in the paraxial approximation. It is widely used for first-order layout and for the
purpose of analyzing optical systems (Gerrard and Burch, 1975). The reason why
the paraxial approximation is often used in the first phase of a design or of an optical
analysis becomes obvious if we have a look at the law of refraction in vectorial form
as follows:
n1 a N=n2 b N , (1.1)
wherea is the vector of the ray incident on the interface with the normal N. Thisinterface separates two homogeneous media with indices of refraction n1 andn2.
The refracted ray is described by the vector b. For optical-layout purposes, we needan explicit expression of this ray in terms of the other quantities because we have
to trace the ray through the optical system. Using vector algebra, Eq. (1.1) can be
rewritten in the following way:
b= n1
n2
a n1
n2
N a 1 n1
n2
2
1( N a)2 N . (1.2)The form obtained like this is complicated and it is difficult to trace the ray without
making use of a computer. Therefore, a linearized form of this law would be helpful
for thinking about the optical system, and this is the motivation for starting with a
paraxial layout.
It would be a precious tool for analyzing optical instruments if the approxi-
mated description would also allow for cascading subsystems to describe a com-
pound system. The method of ray-transfer matrices provides this advantage and
cascading of subsystems is performed by matrix multiplication.
Another aspect, which might be sometimes underestimated, is that paraxialdescriptions, and especially the matrix method, provide a convenient shorthand
notation to communicate and discuss ideas to other optical designers. In a way, this
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2 Chapter 1
branch of optics is axiomatic like thermodynamics, for example. The framework
of the underlying theory can be reduced to a limited number of basic rules and
elements. But combining these rules and elements allows one to study a greatvariety of optical systems.
1.2 Basic Elements
We will now look for linearized relations that describe three situations, namely,
propagation of a ray, and its refraction and reflection. The matrices obtained in this
way serve as building blocks of the matrix description.
1.2.1 Propagation in a homogeneous medium
Let us first consider the propagation of a ray in a homogeneous medium. We as-
sume that the ray propagates in the y z plane and choose the z-axis as the optical
axis. In any plane perpendicular to the optical axis, the ray can now be described
by its distance from the optical axis, y, and by the angle , which it has with a
line parallel to the optical axis. As the ray propagates along the optical axis, these
coordinates may change and take different values in different planes perpendicular
to the optical axis. We now choose two reference planes separated by a distancet
inside a homogeneous medium (Fig. 1.1) and determine the inputoutput relation-
ship. The ray starts with the coordinates[y(1), (1)]. Due to the propagation alonga rectilinear line, the angle remains unchanged,
(2) = (1). (1.3)
The height in the second reference plane depends on the distance traveled and on
the starting angle,
y(2) =y (1) +ttan (1). (1.4)
Figure 1.1 Propagation in a homogeneous medium. The two reference planes are at a
distancet.
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An Introduction to Tools and Concepts 3
Under the assumption that the paraxial approximation is valid, i.e., for small an-
gles (1), we can linearize the trigonometric function in Eq. (1.4) as
y(2) =y (1) +t (1). (1.5)
Equations (1.3) and (1.5) can now be combined and written as a matrix relation,y(2)
(2)
=
1 t
0 1
y(1)
(1)
. (1.6)
The matrix depends on the distance of the two reference planes. We will later refer
to it as the translation matrixT, defined as
T = 1 t0 1
. (1.7)1.2.2 Refraction at the boundary of two media
Now, we will try to obtain a linearized expression for the refraction of a ray at a
spherical surface described by the radius R. This surface separates two homoge-
Figure 1.2 Refraction at a spherical surface. The spherical surface separates two media
with refractive indices n1 andn2.
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4 Chapter 1
neous media of refractive indices n1 and n2. Let us first draw a line representing
the ray as it hits the spherical surface in a reference plane (Fig. 1.2). We consider
how the input and output variables are changed in this single reference plane wherethe refraction takes place. The distance from the optical axis remains unchanged
for the ray leaving the reference plane, i.e.,
y(2) =y (1). (1.8)
The change in angle is described by the law of refraction,
n1sin i(1) =n2sin i
(2), (1.9)
where the anglesi (1) andi (2) refer to the normal vector that is perpendicular to the
surface.
Assuming that the paraxial approximation is valid, Eq. (1.9) can be linearizedas
n1i(1) =n2i
(2). (1.10)
But we need expressions in terms of the angles (1) and (2) that are measured
with respect to a line parallel to the optical axis. To obtain relations between these
angles and the angles appearing in Eq. (1.9), we have a closer look at the triangles
in Fig. 1.2. Applying the exterior angle theorem for triangles twice, we have
i(1) = (1) +, (1.11)
i(2) = (2) +. (1.12)
Substituting these equations into Eq. (1.10), we find
(2) = n1
n2(1) +
n1n2
n2. (1.13)
Neglecting the small distance between the intersection of the spherical surface with
the optical axis and the reference plane, we approximate the angle appearing in
Eq. (1.13) as
tan = y(1)
R. (1.14)
Linearizing the trigonometric function for small angles (tan = ), Eq. (1.14) issubstituted into Eq. (1.13) and we have
(2) = n1
n2(1) +
n1n2
n2Ry(1). (1.15)
This is the linearized inputoutput relation we were looking for. In combination
with Eq. (1.8), we can write it in matrix form asy(2)
(2)
=
1 0
n1n2n2R
n1n2
y(1)
(1)
. (1.16)
The corresponding matrix will be used later as the refraction matrix R , defined as
R=
1 0
n1n2n2R
n1n2
. (1.17)
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An Introduction to Tools and Concepts 5
Figure 1.3The unfolding of a spherical mirror.
1.2.3 Reflection at a surface
A geometrical consideration quite similar to the one that led to Eq. (1.17) can also
be used to find the matrix for a spherical concave mirror. In this case, the output
ray remains on the same side of the reference plane.
It is interesting to note that we can formally obtain the matrix of an unfolded
spherical concave mirror by settingn1=1 andn2= 1 in Eq. (1.17), i.e.,
S= 1 0
2R
1 . (1.18)Unfolding refers to the symmetry operation (or coordinate break) depicted in
Fig. 1.3. This can be helpful in finding the matrix chain of a compound optical
system. Please note that some signs might change in the system matrix with re-
spect to the starting system because reference is made to an optical axis with a
different direction after the coordinate break.
1.3 Comparison of Matrix Representations Used in theLiterature
In the literature, different notations used to write the ray-transfer matrices can be
found. Many authors use coordinates that have nas the second coordinate, where
n is the index of refraction (Guillemin and Sternberg, 1984). An advantage of
this notation is that the determinant value of the ray-transfer matrices is always 1.
This provides a useful check during calculations and can also simplify theoretical
arguments based on the determinant. In the description used here, the determinant
of the ray-transfer matrixA has the value
det A= n1
n2
, (1.19)
with n1 as the refractive index of the medium at the entrance reference plane and
n2 as the refractive index of the medium at the exit reference plane.
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6 Chapter 1
The second coordinatencan also be introduced as a modified ray slope (Sieg-
man, 1986) as
r (z)=n(z)dr(z)
dz . (1.20)
The interpretation of this coordinate in terms of slope can be fruitful in some cir-
cumstances.
1.4 Building up a Lens
With the prerequisite of Eqs. (1.7) and (1.17), we can determine the matrix of a
spherical lens. The refraction at the first surface is expressed by the matrixR(a).
The ray is then propagated through the lens using the translation matrix T and
finally refracted at the second surface of the lens. To describe this refraction, thematrix R(b) is used. The combined effect is calculated as the product of these
matrices,
S=R (b)T R(a). (1.21)
More explicitly, this equation reads as
S=
1 0
n2n3n3R2
n2n3
1 t
0 1
1 0
n1n2n2R1
n1n2
, (1.22)
where t is the thickness of the lens and R1 and R2 are the radii of the first andthe second surfaces of the lens, respectively. Because the lens is in air, we can
specialize the set of refractive indices as n1 = 1, n2 = n, andn3 = 1. Therefore,we have
S=
1 n1
R1
tn
tn
n1R1
1nR2
+ n1R1
1nR2
tn
1 1nR2
tn
. (1.23)
This might suggest the following abbreviations:
P1= n1
R1, (1.24)
P2= 1n
R2. (1.25)
With these abbreviations, Eq. (1.23) then takes the form
S=
1P1
tn
tn
P1P2+P1P2tn
1P2tn
. (1.26)
The so-called thin lens is obtained by letting the lens thickness t tend to zero in
Eq. (1.26),
S=
1 0
P1 P2 1
. (1.27)
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An Introduction to Tools and Concepts 7
1.5 Cardinal Elements
To identify the lower-left entry in the matrix of the thin lens, we first look at a lens
described by a more general matrix of the form
A=
a11 a12
a21 a22
. (1.28)
Its focal plane can be found by letting a ray parallel to the optical axis pass through
the lens and determine the distance b from the exit reference plane to the plane
where it intersects the optical axis. Expressing this in matrix notation, we have
0
out = 1 b0 1
a11 a12a21 a22
yin
0 , (1.29)
or 0
out
=
a11+ ba21 a12+ ba22
a21 a22
y in
0
. (1.30)
This implies that
0= (a11+ba21)yin. (1.31)
This equation should hold for all values ofy in. Therefore, it follows that
a11+ ba21=0. (1.32)
The position of the second focal plane of the lens described by the matrix A is
therefore determined by
b= a11
a21, (1.33)
and we can identify b as the focal length fof the lens.
Applying this result to the thin lens of Eq. (1.27), for which a11 = 1 holds, wesee that the lower-left entry represents the negative inverse of its focal length, i.e.,
the matrix of the thin lens is
F =
1 0 1
f 1
. (1.34)
The second focal plane is one of the cardinal elements of a lens. The position of the
first focal plane is calculated on the same footing, but by letting a parallel ray enter
from the other side into the system or by finding the distance for which the light
from a point source in front of the lens is collimated. In both ways, the following
result is obtained for the position of the first focal plane:
a = a22
a21
. (1.35)
A straightforward way to obtain other cardinal elements is by direct comparison
with the thin lens. We are interested in finding the positions of the planes with
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8 Chapter 1
respect to which a lens given by the matrix Acould be described similar to a thin
lens. To this end, we take the following approach:1 h2
0 1
a11 a12
a21 a22
1 h1
0 1
=
1 0
1f
1
, (1.36)
whereh1 andh2 are the distances that have to be determined. The corresponding
planes are called principal planes and, together with the focal planes, they are the
cardinal elements of a lens. After performing the matrix multiplication on the left-
hand side, we have
a11+a21h1 a12+a11h1+a22h2+a21h1h2
a21
a22
+a12
h2
= 1 0
1f 1
. (1.37)The position of the first principal plane is therefore given by
h1= 1a11
a21, (1.38)
measured with respect to the first reference plane of the lens. The position of the
second principal plane is at
h2= 1a22
a21, (1.39)
measured with respect to the second reference plane of the lens.
A beautiful illustration of the principal planes concept is given by Lipson et
al. (1997). We can trace typical rays through the lens and draw this on a piece
of paper. If we now fold this paper along the lines that represent the principal
planes, we can hold it in such a way that the part between the principal planes
is perpendicular to the other parts. These other parts are combined to represent a
simplified arrangement (Fig. 1.4), which corresponds to a thin lens.
This is in complete analogy to Eq. (1.36). The results on the cardinal elements
are collected in Fig. 1.5.
With these prerequisites, we can state the cardinal elements of the thick lensgiven by Eq. (1.23). The equation for the focal lengthfof the lens is
1
f=
n1
R1+
1n
R2
(n1)(1n)t
nR1R2. (1.40)
Its principal planes are at
h1= fn1
R1
t
n, (1.41)
h2= f1n
R2
t
n. (1.42)
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An Introduction to Tools and Concepts 9
Figure 1.4 Principal planes visualized by folding. The optical system is described by the
matrixA. It has the focal points F1 andF2 and its principal planes are at h1 andh2, respec-
tively.
Figure 1.5 Cardinal elements. The focal pointsF1 and F2 and the positions h1 and h2 of
the principal planes serve to characterize the optical system given by the matrix A.
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10 Chapter 1
1.6 Using Matrices for Optical-Layout Purposes
In the derivation of the position of the second focal plane, we considered the optical
arrangement formed by a lens, which was given by the matrix A, and a translationmatrixT, i.e.,
S=T A. (1.43)
On this combined optical arrangement, the condition s12 = 0 was then imposed toensure that the ray height in the output plane was independent of the ray angle in
the input plane. We used this condition because in the paraxial approximation, it
characterizes a point in the second focal plane. This way of reasoning can also be
applied to other situations.
Its application to the first focal plane is convenient; to this end, we consider the
combined arrangement given by the matrix product,
S=AT . (1.44)
We then impose a condition on the combined matrix Sthat expresses (in the linear
approximation) that a bundle of rays at a given ray height yin but with different
angles in in the entrance plane ofSwill be transformed into a parallel beam, i.e.,
a bundle of rays with the same angle, at the exit plane. The general inputoutput
relation is youtout
=
s11 s12s21 s22
y in in
. (1.45)
To ensure that out has a single value for a given ray height y in, it has to be inde-
pendent of in. A look at the inputoutput relation suggests that this condition is
met if we choose
s22=0. (1.46)
This choice determines the distance contained in the translation matrix and thereby
the position of the first focal plane, which corresponds to the matrixA.
At this point, we have conditions for the first focal plane (s22 = 0)and for thesecond focal plane (s11 = 0), and we might ask: what is the characteristic featureof a ray-transfer matrixSthat describes imaging? The rays leaving at a point at y in
in the object plane with different angles in intersect in a point at y out in the image
plane. If the matrixA describes a lens, we have to add two spacings on both sides
to model imaging, so we have
S=BAG, (1.47)
with B =
1 b0 1
andG =
1 g0 1
. Considering the inputoutput relation again, we
find thaty out is independent of in if
s12=0. (1.48)
This is the characteristic feature of a matrix Sthat represents imaging.
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An Introduction to Tools and Concepts 11
We can apply this condition immediately to find the imaging relation for a thin
lens. The corresponding matrix chain is
S=
1 b
0 1
1 0
1f
1
1 g
0 1
=
1 bf g+b bgf
1f
1 gf
. (1.49)
Usings12=0, we have the well-known imaging condition
1
g+
1
b=
1
f, (1.50)
which expresses thatb varies in a hyperbolic way as a function ofg and vice versa.
The signs of the distances are positive in Eq. (1.50) because the direction of the
distances is chosen as the direction of the optical axis.To find a relation for the first focal plane, we asked under which conditions par-
allel rays leaving the system might be independent of the input angle. Alternatively,
we can consider the situation where the rays leaving the system are independent of
the ray height in the entrance plane. This is the case if a collimated input beam is
transformed into a collimated output beam. Making reference to the inputoutput
relation forS, we see that setting
s21= 0 (1.51)
ensures that out does not depend on the ray heighty in in the input reference plane.
Because collimated rays are considered, no additional translation matrices have to
be introduced here and therefore S = A. Earlier, we related the matrix entry a12to the negative inverse of the focal length of an optical system via Eq. (1.33). This
matrix entry takes the value of zero now, which corresponds to the case of an afocal
system.
Typical examples for such systems are telescopes. In the paraxial approxima-
tion, we might model a telescopic arrangement using thin lenses. We choose two
lenses with focal lengthsf1andf2, separated by a distanced. Concatenation of the
corresponding matrices gives us the system matrix
S=
1 0
1f2
1
1 d
0 1
1 0
1f1
1
=
1 d
f1d
1f1
1f2
+ df1f2
1 df2
. (1.52)
Now, we impose the condition thats21=0 should hold. This implies that
1
f1
1
f2+
d
f1f2=0. (1.53)
The setting ofd=f1+f2solves this equation and we have
S=
f2
f1f1+ f2
0 f1
f2
(1.54)
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12 Chapter 1
Figure 1.6The significance of zero-matrix entries.
for the system matrix of the telescopic arrangement. It represents a Newtonian tele-
scope if both focal lengths are positive. If the focal length of the first lens is nega-
tive, the matrix describes a Galilean telescope, which is composed of a concave and
a convex lens. Optical arrangements of this type also serve as transmissive beamexpanders (Das, 1991) and intracavity telescopes (Siegman, 1986). The results on
the significance of special matrix entries are summarized in Fig. 1.6.
1.7 Lens Doublet
We encountered telescopic arrangements as the first examples of a lens doublet and
we now have a closer look at optical systems composed of two lenses. The matrix
that describes two lenses separated by a distance dforms the starting point of our
discussion:
S=
1 d
f1d
1f1
1f2
+ df1f2
1 df2
. (1.55)
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An Introduction to Tools and Concepts 13
The terms21 is related to the focal length of the doublet (measured with respect to
its second principal plane).
1
f= 1
f1+ 1
f2 d
f1f2. (1.56)
[As shown before, this principal plane is at a distance z = (1s11)/(s21)from thesecond reference plane of the system.] To facilitate the discussion, it is convenient
to reference the intermediate distance to the second focal plane of the first lens and
to the first focal plane of the second lens by setting
d=f1+E +f2. (1.57)
With this setting, the equation for the focal length of the doublet reduces to
f = f1f2
E. (1.58)
It can now be discussed in terms of the signs of the three parameters that intervene.
Depending on whether f1 < 0 orf1 > 0,f2 < 0 orf2 > 0, orE < 0, E = 0, orE > 0, twelve cases can be distinguished. The case where f1 > 0 andf2 > 0 and
E=0, for example, represents the Galilean telescope.At this point, it is near at hand to make a distinction between divergent (f 0)doublets in terms of their three parameters. A compound
microscope represented as a doublet is characterized by f1 > 0 and f2 > 0 and
E > 0, and it is interesting to note that it is an example of a divergent system
(Prez, 1996)
1.8 Decomposition of Matrices and System Synthesis
In the layout of a new optical system, it is advantageous to know how the ray-
transfer matrix of a given optical system can be factorized. Let us consider the
design of an optical device with given properties and that some of these features
can be expressed in terms of a system matrix. To realize the device, it is now of
interest to systematically explore in which ways a device with the given features
can be realized. To this end, it is useful to divide the device into subsystems,
the combination of which would create the desired functionality. In the matrix
description, this is equivalent to considering matrix products of the target matrix,
and this is where factorizing the system matrix comes into play. The problem of
a synthesis of optical systems using this approach has been studied in depth by
Casperson (1981).
In what follows, we will consider optical systems that have both their object
and image planes in air. Therefore, n1 = 1 andn2 = 1 and the determinant of the
system matrixScan be written as
det S= n1
n2=1. (1.59)
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14 Chapter 1
Therefore, the condition
s11s22s12s21= 1 (1.60)
is contained implicitly in Eqs. (1.61) and (1.62). A generalization is possible andcan, for example, be found in the work of Casperson (1981).
The appropriate factorization depends on the matrix entries. If we consider a
nonimaging problem, we can assume s12=0 for the system matrix. Such a matrixcan be factorized as
S=
s11 s12
s21 s22
=
1 0s221
s121
1 s12
0 1
1 0s111
s121
. (1.61)
If the lower-left entry of the system matrix can be assumed to be nonzero (s21=0),
i.e., if we do not look for an afocal system, the following matrix decompositionis appropriate:
S=
s11 s12
s21 s22
=
1 s111
s21
0 1
1 0
s21 1
1 s221
s21
0 1
. (1.62)
What is left are the cases in which both s12 = 0 and s21 = 0. These cases corre-spond to optical systems that are imaging and afocal devices. In the above-cited
work, four possibilities for a decomposition of this diagonal matrix are given. The
system matrix is either decomposed in a product of matrices A andB witha21= 0
andb21= 0 as
S=
s11 0
0 s22
=
1 t
0 1
s11 ts22
0 s22
, (1.63)
S=
s11 0
0 s22
=
s11 t s11
0 s22
1 t
0 1
, (1.64)
or a product of matrices witha12=0 andb12=0 as
S= s11 0
0 s22
= 1 0
1f 1 s11 0
s11f
s22
, (1.65)
S=
s11 0
0 s22
=
s11 0s22f
s22
1 0
1f
1
. (1.66)
Depending on the application, the matrices appearing in the product can then be
further decomposed by applying the same set of rules.
1.9 Central Theorem of First-Order Ray Tracing
We will now turn to a theorem that is of prime importance to ray tracing using
the matrix method. It can be applied to different sets of rays. Its main content isthat the number of rays necessary to characterize an optical system in the linear
approximation is rather small.
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An Introduction to Tools and Concepts 15
Let us consider two rays labeled a and b that are traced through the optical
system described by the matrixA. Each ray vector entering the system is mapped
onto an output ray vector as follows:ya
a
youta
outa
, (1.67)
yb
b
youtb
outb
. (1.68)
The mapping is given by the system matrixA. Therefore, we have
youta
outa= A ya
a , (1.69)
youtb
outb
= A
yb
b
. (1.70)
We assume that we can completely determine the four ray coordinates and that
we want to use this information to determine the system matrix A. Its entries are
therefore the unknown variables of the problem, and we can state it by rewriting
the above equations as the following system of linear equations:
ya a 0 0
yb b 0 0
0 0 ya a
0 0 yb b
a11
a12
a21
a22
=
youta
youtb
outa
outb
. (1.71)
Because the matrix is partitioned, two sets of linear equations can be solved inde-
pendently. If the determinant
D=det
ya a
yb b
=0, (1.72)
the problem has a unique solution, namely,
a11=det
youta ayoutb b
D
, (1.73)
a12=det
ya youtayb y
outb
D
, (1.74)
a21=det
outa a
outb b D
, (1.75)
a12=det
ya outayb
outb
D
. (1.76)
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16 Chapter 1
D = 0 is equivalent to the condition that the input ray vectors are linearly in-dependent. We can therefore conclude that the ray-transfer matrix is completely
determined if we know a set of two linearly independent input ray vectors and thecorresponding output ray vectors. In the linear approximation, the passage of any
other third ray through the system is then known because we can trace it through
the system using the matrix A. Putting it in different words, the theorem states that
in the approximation used, the inputoutput relation is completely characterized
once the input and output data of two linearly independent rays are known.
This gives the theoretical basis of why an optical system can be characterized
to such an extent by just tracing the principal ray and the axial ray.
1.10 Aperture Stop and Field Stop
Theaperture stopis defined as the opening of an optical system that limits the input
angle at zero height in the object plane. A ray with these coordinates can be trans-
ported through the system. If the input angle of a ray is slightly greater than this
critical angle, the ray is blocked. We might have several candidates in the system to
cause this blockage, and which of them forms the aperture stop can be determined
in the following way using the matrix method. We label the free diameters of the
candidates as y(k). To every candidate now corresponds a matrix P(k) that maps
the start ray into the reference plane at z(k),
y(k)
(k)
= P(k)
0(k)
. (1.77)
This implies that
(k) = y(k)
p(k)12
. (1.78)
The aperture stop is at the position z(k) for which (k) takes the minimum value of
all the candidates. It has the heighty as =y(k) if(k )is the label for that minimum.
The axial ray is the ray that starts at zero height in the object plane and that
passes through the aperture stop at the maximum possible height. If we supposethat the matrix Pdescribes the mapping of the ray from the object plane to the
aperture plane, we can trace this ray to that plane using
P
0
=
yas
. (1.79)
Its start coordinates in the object plane are
y in
in =
0yas
p12 ,
and this ray can now be traced through the complete optical system. We describe
the second part of the system, i.e., the part between the aperture plane and the image
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An Introduction to Tools and Concepts 17
plane, byQ. Therefore, the system matrix is
S=QP , (1.80)
and the coordinates of the axial ray as its leaves the system areyout
out
=
y as
p12
s12
s22
. (1.81)
While the axial ray starts at zero height in the object plane and passes through the
aperture stop at its margin, the principal ray starts at the marginal height of the
object (if this corresponds to the field stop) and passes through the aperture stop
at zero height. In the matrix description, we can express this relation by using the
matrix P, which describes the mapping from the object plane to the plane of theaperture stop, as
P
yfield
=
0
. (1.82)
To be able to trace the principal ray through a complete system, we need its input
angle, which we can calculate from the following equation:
= p11
p12yfield. (1.83)
Therefore, the input coordinates of the principal ray are given byy in
in
=
yfield
p11p12
yfield
, (1.84)
and the output coordinates after passage through the whole system areyout
out
= y fieldS
1
p11p12
. (1.85)
It is interesting to note the followingsymmetrythat exists between the axial ray and
the principal ray:
P
0
=
y as
for the axial ray, (1.86)
yfield
= P1
0
for the principal ray, (1.87)
where the corresponding inverse matrix has been used. Das (1991) expressed this
symmetry relation by writing . . . the field stop is nothing but the new aperture
stop, when the object is placed at the center of the actual aperture stop.
(Please note that in writing the symmetry relation it was assumed that the ex-tension of the object can be identified with the extension of the field stop. This is
quite often the case, but more intricate situations are possible.)
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18 Chapter 1
1.11 Lagrange Invariant
1.11.1 Derivation using the matrix method
We know that
det A= n1
n2(1.88)
holds for a ray-transfer matrix. During the derivation of the central theorem of first-
order ray tracing, the following result was obtained for two linearly independent
rays that pass through the system described by this matrix:
A=
a11 a12
a21 a22
=
1
D
youta b y
outb a yay
outb yby
outa
outa b outb a ya
outb yb
outa
. (1.89)
If we use this result in Eq. (1.88), we have
(youta b youtb a)(ya
outb yb
outa )(
outa b
outb a)(yay
outb yby
outa )
(ya b yba)2 =
n1
n2.
(1.90)
Performing the multiplications, we find
youta outb
outa y
outb
yab a yb=
n1
n2. (1.91)
This equation can now be rearranged slightly, to separate input and output quanti-
ties, as
n1(ya b a yb)= n2
youta outb
outa y
outb
. (1.92)
We can therefore conclude that the following quantity is conserved during the pas-
sage through the system:
L= n(yab ayb). (1.93)
This is the Lagrange invariant.
1.11.2 Application to optical design
We now turn to an application of the Lagrange invariant that is useful when design-
ing imaging systems. Let us consider the invariance condition given by Eq. (1.92)
and specialize it for the case where ray a is the axial ray and ray b is the principal
ray. From the discussion earlier, we know that these are two linearly independent
rays and are therefore suitable to characterize the system. These rays pass through
an imaging system. As reference planes, the object plane and the image plane are
natural choices, so we have
n(yar pr ar ypr )= n(y ar
pr
ar y
pr ). (1.94)
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An Introduction to Tools and Concepts 19
In both these planes, the height of the axial ray is equal to zero. Therefore, the
above equation reduces to
n(ar ypr )= n( ar y
pr ). (1.95)
The quantitiesar andar can be identified with the aperture of the system in the
object plane and the image plane, respectively, and yar andyar are the heights of
the object and the image. In this way, the Lagrange invariant allows one to establish
a direct relation between these quantities. This relation is, for example, of use to
study the magnification and angular magnification of an optical instrument.
1.12 Petzval Radius
The Petzval radius is the reciprocal of the curvature of the image field. This is a
concept of aberration theory, and it is surprising that we can determine its value
from paraxial quantities. The quantities to be considered are quite similar to those
that we encountered building up a lens, namely,
P1= n1
R1, P2=
1n
R2.
We state the relation for the Petzval radiusRp of a system made up ofN refractive
surfaces, without proof, as
1
Rp
=
Nk=1
Dk
nk, (1.96)
where Dk = nk/Rk is the refractive power of the kth surface, nk is the dif-ference in refractive indices, and Rk is the radius of curvature of the kth surface.
A proof can be found in textbooks on optics (Born, 1933; Born and Wolf, 1980)
The Petzval radius should tend to infinity to have a flat image field. This corre-
sponds to finding a combination for the right-hand side of the equation that brings
its value close to zero. An example for an optical device that minimizes this quan-
tity is Dysons system (Dyson, 1959). This optical arrangement will be consideredin more detail in Section 5.1.
1.13 Delano Diagram
The Delano diagram or y y diagram is a visual tool of paraxial analysis (Delano,
1963; Shack, 1973; Besenmatter, 1980). It is created by tracing a principal ray (y)
and an axial ray (y) through an optical system and by drawing the corresponding
ray heights (y , y ) in a single diagram. The position on the optical axis does not
appear explicitly in this diagram and this representation is therefore somewhat ab-
stract compared to a usual ray trace. But in many cases it can give an overview thatis like a kind of shorthand notation for the system. Figure 1.7 shows four Delano
diagrams that represent the different cases of matrices with zero entries.
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20 Chapter 1
Figure 1.7Delano diagram, four cases.
1.14 Phase Space
Looking at the set of coordinates used in matrix optics, it is near at hand to try a
representation in a plane using the ray height as one coordinate, and the ray angle as
the other coordinate. A set of rays in a given reference plane can then be represented
as a surface in this abstract plane.
To familiarize ourselves with this concept, let us choose a set of rays whose
coordinates are represented by a rectangle in this so-called phase space. We would
like to see what might be the effect of the translation matrixTon this set of rays. 0
0
0
0
+
t0
0
, (1.97)
y0
0
y0
0
+
t0
0
, (1.98)
0
0
0
0
, (1.99)
y00 y0
0 . (1.100)
Graphically, this can be expressed as in Fig. 1.8. The corresponding mapping for
the refraction matrix is shown in Fig. 1.9.
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An Introduction to Tools and Concepts 21
Figure 1.8 Effect of the translation matrix in phase space. The spatial coordinate is repre-sented along they -axis and the angular coordinate is represented along the -axis.
Figure 1.9 Effect of the refraction matrix in phase space. As in Fig. 1.8, the spatial coor-
dinate is represented along the y -axis and the angular coordinate is represented along the
-axis.
It is an important feature of this phase space that the volume (or surface inthe two-dimensional case considered here) is conserved if we consider mappings
between input and output planes where the refractive indices are 1. This conserva-
tion is a consequence of Eq. (1.19).
From the theoretical point of view, it is more convenient to use coordinates
(y, n). The corresponding volume (or surface) is then always conserved. The
phase-space approach is common in laser technology (Hodgson and Weber, 1997).
1.15 An Alternative Paraxial Calculation Method
An alternative paraxial method (Berek, 1930) uses distances sk measured along theoptical axis and ray heights hk measured perpendicular to it as a set of coordinates
for the description of a ray. For readers who use this method, it might be interesting
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22 Chapter 1
to see how both methods are connected. They will be familiar with the so-called
transition equations. These equations state how a ray is transformed that leaves
a lens surface labeled with k and
and that reaches another surface (labeled withk+1) situated at a distance e k as follows:
sk+1= sk e
k, (1.101)
sk+1= hk+1
uk+1, (1.102)
s k = hk
uk, (1.103)
whereuk+1andu
k
are angles. We can combine these equations to obtain
hk+1
uk=
hk
ukek. (1.104)
Settinguk+1=uk, we can write the linear relation,
hk+1
uk+1
=
1 ek0 1
hk
uk
, (1.105)
which is quite similar to the relation for the translation matrix T.
The other method also makes use of the relation
nk
1
Rk
1
sk
= nk
1
Rk
1
s k
(1.106)
for the two sides of a refracting surface with radius Rk. Usinguk = hk/sk for theparaxial angle, we can write this equation as follows:
uk = hk
Rk 1nk
n
k +
nk
n
k
uk. (1.107)
Assuminghk =hk in addition, we have an augmented linear relation of the formhk
uk
=
1 0
nk nkRk n
k
nknk
hk
uk
. (1.108)
This relation corresponds to the refraction matrix R of the matrix description.
1.16 Gaussian Brackets
While cascading matrices in order to determine a system matrix, the recursive char-acter of the problem became clear. But at this point, we were unable to state the
underlying recursion law that would allow us to express the ray that finally leaves
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An Introduction to Tools and Concepts 23
the system, and that is described by the product matrix, as a function of the in-
put ray without performing the matrix multiplication. The use of the mathematical
concept of Gaussian brackets makes it possible to state this recursion formula.An introduction to the algebra of Gaussian brackets and the recursion law for
cascaded linearized optical systems was given by Herzberger (1943). In this text,
the Gaussian brackets are defined by the following recursion formula:
[a1, . . . , ak] =a1[a2, . . . , ak] + [a3, . . . , ak], (1.109)
with[ ] =1. This implies
[a1] =a1, (1.110)
[a1, a2] =a1a2+1, (1.111)[a1, a2, a3] =a1a2a3+a1+ a3, (1.112)
[a1, a2, a3, a4] =a1a2a3a4+a1a2+a1a4+a3a4+1. (1.113)
From Herzbergers article, we take a description of a lens using his symbols, but
arrange the linear transformation as a matrix as follows:x2
2
=
1 d12
n121
d12n12
1+2 d12n12
12 1 d12n12
2
x1
01
. (1.114)
wherex is the distance between intersections of the optical axis and the ray withthe reference plane andis the inclination angle of the ray. The subscripts indicate
the corresponding surfaces. k is the refractive power of the kth surface, dk,k+1 is
the distance between the k th and the (k+ 1)st surface, andnk,k+1 is the refractiveindex between the kth and the(k+ 1)st surface. Using Eqs. (1.110)(1.112), thiscan be recast in the following way:
x2
2
=
1,
d12n12
d12
n12
1, d12n12 , 2 d12n12 , 2
x1
01
. (1.115)
For the general case, the recursion law reads as follows (Herzberger, 1943):
x
=
1,
d12n12
, . . . , dk1,knk1,k
d12
12, 2, . . . ,
dk1,knk1,k
1, . . . , dk1,knk1,k
, k
d12n12
, . . . , k
x1
01
. (1.116)
This equation establishes a link between the input coordinates and the output coor-
dinates of an optical system. The optical properties of the system are described by
the matrix entries, which are Gaussian brackets. In order to state these entries ex-
plicitly in terms of refractive powers, distances, and refractive indices, the recursive
relations have to be evaluated using Eq. (1.109) in a kind of backtracking proce-dure. The approach of the matrix method is advantageous, because it allows us to
obtain the system matrix by concatenating matrices, i.e., by matrix multiplication.
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24 Chapter 1
In cases, where it is useful to state inputoutput relations in a recursive way, the
method of Gaussian brackets provides an alternative approach.
In the framework of the matrix method presented in this text, I found no way tostate the recursion law with a similar conciseness. Therefore, I would like to draw
the readers attention to the Gaussian-brackets method that allows such a formula-
tion. It is beyond the scope of this book to derive the method here, and interested
readers are referred to Herzbergers work. In optical design, Gaussian brackets
were applied to the layout of zoom systems (Pegis and Peck, 1962; Tanaka, 1979,
1982).
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Chapter 2
Optical Components
2.1 Components Based on Reflection
Mirrors are a key component of many optical devices. The matrix for the reflection
at a spherical mirror was considered in the introductory chapter. We will now
discuss reflectors more generally.
2.1.1 Plane mirror
The matrix describing the reflection at a plane mirror can be obtained by taking the
matrix for reflection at a spherical reflector and letting the radius of the spherical
mirror tend to infinity. In this way, the unity matrix is obtained as
A=
1 0
0 1
. (2.1)
The signs that appear in this matrix are surprising at first, and it is instructive to
derive the matrix also in an alternative way. In Fig. 2.1, the reflection of a ray at a
plane mirror, which is perpendicular to the optical axis, is depicted. In the matrix
representation used here, we unfold the ray using the reference plane of the mirror
as the plane of the coordinate break. Figure 2.2 shows the result of this unfolding.
It is this coordinate break that causes the positive signs in the matrix of the plane
reflector.
Figure 2.1Plane mirror.
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26 Chapter 2
Figure 2.2Unfolded plane mirror.
2.1.2 Retroreflector
Unlike a plane mirror, a retroreflector redirects the beam back into the same direc-
tion from where it came. In the matrix description, this is expressed by a negative
sign of the matrix entry a22. Reflection at a retroreflector is combined with a change
in the height of the beam. The height of the incident beam is changed from yin
toy out = yin. This corresponds to a parallel shift and is expressed by a negative
sign of the matrix entrya11. The complete matrix of the retroreflector reads as
A=
1 0
0 1
. (2.2)
More generally, a retroreflector has the following matrix:
A=
1 a12
0 1
. (2.3)
2.1.3 Phase-conjugate mirror
An element that redirects an incident beam into itself without any shift in ray height
would be described by the following matrix (Lam and Brown, 1980):
A= 1 00
1 . (2.4)
There are devices that exploit nonlinear optical effects and that are able to operate in
such a way on an incident laser beam. To describe these so-called phase-conjugate
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Optical Components 27
mirrors in the paraxial approximation, the matrix stated above can be used. The
nonlinear effect itself is far beyond the scope of this approach. Retroreflectors, on
the other hand, can be advantageously modeled using the matrix method.
2.1.4 Cats-eye retroreflector
A prominent example of this type of optics is the cats-eye retroreflector. It ba-
sically consists of a lens and a plane mirror (Fig. 2.3). The distance between the
lens and the mirror is chosen as the focal length of the lens (with respect to the
backward principal plane of the lens). The lens will be approximated as a thin lens.
To describe the plane mirror, we can use the matrix given earlier [Eq. (2.1)]. The
distances involved are first designated by g and b. Following the ray through the
unfolded arrangement, we find the following matrix chain:
S=
1 g
0 1
1 0
1f
1
1 b
0 1
1 0
0 1
1 b
0 1
1 0
1f
1
1 g
0 1
. (2.5)
After performing the matrix multiplications, we have
S= 1+2 b
f g
f 1 2 g
f 2g +2b4
bg
f +2
g2
f b
f 1
2f
bf
1
2g
f
bf
1
+12 bf
. (2.6)
We know that lettingb =fmakes the arrangement work as a cats-eye retroreflec-
tor. It is instructive to see what happens if we choose the entrance and exit planes
of the system either at the position of the lens or at a distance f in front of it. The
first alternative corresponds to settingg = 0 and the second one to letting g = f.
In this way, we find
S(g =0, b =f )= 1 2(f g)
0 1
(2.7)
Figure 2.3Cats-eye arrangement.
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28 Chapter 2
and
S(g =f, b =f )= 1 0
0 1 . (2.8)The form of the second matrix is equal to the matrix of a retroreflector stated in
Eq. (2.2).
2.1.5 Roof mirror
A roof mirror is formed by two plane mirrors meeting at a right angle. This optical
arrangement is also being designated as a double mirror (DeWeerd and Hill, 2004).
We will consider the plane that is perpendicular to the roof edge. In this plane,
the mirror can be unfolded as shown in Fig. 2.4 using an xy plane as the plane
of symmetry. The coordinates of the rays that leave the mirror after reflection canbe found by tracing a rectilinear line through the unfolded arrangement. If we first
ignore the coordinate break caused by the unfolding operation, the figure shows the
passage of a ray through a distance 2t. This propagation can be described by the
following matrix:
T =
1 2t
0 1
. (2.9)
Additionally, it has to be taken into account that the orientation of the reference
axis changes due to the unfolding operation. The corresponding sign change of the
coordinates in the new coordinate system can be seen in Fig. 2.4 and expressed bythe following matrix:
1 0
0 1
. (2.10)
Combining, this gives the component matrix,
S=
1 0
0 1
1 2t
0 1
=
1 2t
0 1
. (2.11)
Therefore, a roof reflector acts like a retroreflector [Eq. (2.3)] in the plane that is
perpendicular to the roof edge.
Figure 2.4Unfolding the roof mirror.
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Optical Components 29
Figure 2.5Arrangement with roof mirror and plane mirror.
It is interesting to see how a combination of this roof mirror and a plane mirror
as depicted in Fig. 2.5 would act on a ray. To this end, it is advantageous to unfold
the optical arrangement with respect to the reference plane of the plane mirror. This
leads to the following matrix chain:
S=
1 2t
0 1
1 b
0 1
1 0
0 1
1 b
0 1
1 2t
0 1
. (2.12)
The evaluation of this matrix product gives the system matrix of the arrangement
of Fig. 2.5 as follows:
S=
1 4t+2b
0 1
. (2.13)
This equation has some similarity to the equation of a plane mirror, but is of the
following more general form:
A = 1 a120 1
. (2.14)
2.2 Components Based on Refraction
Lenses are of course very prominent examples of optical components based on
refraction. Because their description using the matrix method are treated in other
chapters, other components based on refraction are considered here.
2.2.1 Plane-parallel plate
The plane-parallel plate is a component that is often encountered in optical setups.It can be a simple glass plate used for path-length compensation or a component
that is used to influence the polarization and has the form of a plate. Its system
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30 Chapter 2
matrix can be concluded as a special case of the system matrix of a thick lens
[Eq. (1.23)] in air by letting both radii tend to infinity. In this way, we find
S=
1 t
n
0 1
, (2.15)
wheretis the thickness of the plane-parallel plate andn is its refractive index.
To recall this equation during design work, we may also apply the linearized
law of refraction twice. In matrix notation, this reads as y(1)
(1)
=
1 0
0 1n
yin
in
(2.16)
for the rays entering the plate and yout
out
=
1 0
0 n
y(2)
(2)
(2.17)
for those leaving it. In between, they pass through a homogeneous medium of
refractive indexn. Putting this together, we have
S=
1 0
0 n
1 t
0 1
1 0
0 1n
=
1 t
n
0 1
. (2.18)
This matrix can be used to determine the shift z depicted in Fig. 2.6, which occurs
if a plane-parallel plate is introduced into a convergent beam. For comparison, we
first describe the situation without a plane-parallel plate in the following way: yout,1
out,1
=
1 z10 1
1 t
0 1
y in
in
=
1 t+z10 1
y in
in
. (2.19)
At the intersection with the optical axis, we have y out,1 =0. This implies
z1 = y in
in t . (2.20)
Figure 2.6Shift caused by a plane-parallel plate.
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Optical Components 31
The situation with a plane-parallel plate can be described by
yout,2out,2
=
1 z2
0 1 1 t
n0 1
yin in
=
1 tn
+z2
0 1 yin
in
. (2.21)
Usingy out,2 =0, we find
z2 = yin
in
t
n. (2.22)
The difference of Eqs. (2.22) and (2.20) gives the shift zwe are looking for in the
paraxial approximation,
z = z2 z1 = t1
n 1
. (2.23)
Because the term in brackets is generally negative, the shift z takes a positive
value as is expected from Fig. 2.6.
2.2.2 Prisms
Prisms have a vast range of applications in several fields of optics. In optical spec-
troscopy, for example, an important branch of science, prisms serve both as disper-
sive means (Demtrder, 1999) and as components for beam shaping. In optical data
storage technology, prisms are also used for beam shaping purposes, namely, to cir-cularize the asymmetric beam emitted by a semiconductor laser
(Marchant, 1990).
2.2.2.1 Two types of prisms
It is useful to divide prisms into two distinct groups that have different properties
with respect to dispersion. A convenient graphical tool to make this distinction
is the so-called tunnel diagram (Yoder, 1985). To draw this diagram, the prism
has to be optically unfolded as if it has a surface from which the beam of light
is reflected back into the prism. Figures 2.72.10 show two examples that are
representative for the two groups. The tunnel diagram in Fig. 2.8 shows that the
Dove prism (Fig. 2.7) is equivalent to a plane-parallel plate. There is a variety of
prisms that can be understood in terms of this basic component. The second prism
(Fig. 2.9), on the other hand, can be reduced to a prism with an apex angle using
the tunnel diagram (Fig. 2.10).
Figure 2.7 Dove prism.
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32 Chapter 2
Figure 2.8Tunnel diagram of the Dove prism.
Figure 2.9Prism with one internal reflection.
Figure 2.10Tunnel diagram of the prism with one internal reflection.
We can conclude from these examples that there are the following:
1. Prisms that are reducible to a plane-parallel plate.
2. Prisms that are not reducible to a plane-parallel plate.
Another way of stating this important difference is talking of nondispersive (Wolfe,
1995) and dispersive prisms (Zissis, 1995). The first case can be treated with the
matrix of a plane-parallel plate, which has already been derived. We will therefore
have a closer look at the other type of prisms.
Thin-prism approximation for dispersive prisms If both the prism angle and the
incidence angle measured against the normal of the first face of the prism are small,
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Optical Components 33
Snells law can be linearized and the approximated expression for the deviation
angle is as follows (Heavens and Ditchburn, 1991):
=(n1). (2.24)
It is interesting to observe that the angle of incidence does not appear in this ap-
proximated expression.
In terms of the matrix method, the thin prism can be written in the following
way: y
=
1 0
0 1
y
+
0
(n1)
. (2.25)
We will apply this later to get an approximated expression for an axicon.
Trigonometric description of dispersive prisms In many cases, the linear approxi-
mation is not sufficient and it is necessary to perform the trigonometric calculations
to trace a ray through a prism.
Using beam shaping, especially laser beam circularization, is a representative
example of how prism arrangements can be analyzed with trigonometric transfer
functions. Semiconductor lasers emit with a high beam divergence perpendicular
to the junction and with a low beam divergence parallel to it. In several applica-
tions, it is of importance to transform this elliptical Gaussian beam into a circular
Gaussian beam or, more generally speaking, to adapt the elliptical Gaussian beamto a given optical system by expanding or compressing it along an axis. There are
also applications in which the intensity profile that results in a plane perpendicular
to the optical axis differs from a circular one after the Gaussian beam is shaped.
Only beam shaping of collimated light is considered here, i.e., prisms that can be
used to perform the expansion or compression after the beam emitted by the laser
diode has passed a collimating lens are described.
Of course, prism arrangements are not the only way to realize beam shaping
in an optical system. The reader who would like to know more about alternative
techniques is referred to Dickey and Holswade (2000).
2.2.2.2 Brewster condition
To minimize losses, the so-called Brewster condition important. Reflective losses
at a surface are minimal if a polarized beam is parallel to the plane of incidence,
i.e., a p-polarized beam, is incident at Brewsters angle (Young, 1997; Marchant,
1990). This angle depends on the refractive index in the following way:
0 =arctan(n). (2.26)
The Brewster condition is also often expressed by saying that the ray reflected by
the surface and the ray refracted into the medium are perpendicular with respect toeach other, i.e.,
0 =90 deg 1, (2.27)
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34 Chapter 2
where1 is the angle that the refracted ray forms with the surface normal pointing
into the medium. This implies that the following equation is an alternative expres-
sion of the Brewster condition in terms of the angle1:
1 =arccot(n). (2.28)
2.2.2.3 Refracting prism
Relation of incident and exit angle in the case of a single prism To derive a rela-
tion between the angle of incidence 0and the exit angle3of the refracting prism
characterized by its apex angle and the refractive index n, the law of refraction
has to be applied twice. This leads to
3 =arcsin(n sin 2), (2.29)
1 =arcsin
1
nsin 0
, (2.30)
where 1 and 2 are the corresponding angles inside of the prism as depicted in
Fig. 2.11. In a prism, these angles are related by
1+ 2 =. (2.31)
If these equations are put together, the following relation is obtained:
3 =arcsin
n sin
arcsin
1
nsin 0
. (2.32)
In an analogous way, the corresponding equation for the inverse relation can be
derived as
0 =arcsin
n sin
arcsin
1
nsin 3
. (2.33)
Figure 2.11 Refracting prism.
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Optical Components 35
From Eq. (2.32), a relation can be derived that links the incidence angle and
the apex angle for the case where the beam exiting the prism is perpendicular to
the second surface of the prism, i.e., 3 = 0. Using the trigonometric identityarcsin x =90 deg arccos x, Eq. (2.32) can be reformulated as follows:
3 =arcsin
n sin( 90 deg + 0)
. (2.34)
For3 =0, this equation implies that
90 deg + 0 =0. (2.35)
This relation is used later.
The transfer function for the beamwidth altered by a single prism To study the
beam-shaping effect of a prism on a collimated beam, it is appropriate to consider
first how a single refracting surface changes an incident beam of width w0. The
angle of incidence of this beam, with respect to the surface normal is, designated
by 0. The angle 1, with respect to this normal inside a medium of refractive index
nis, determined by Snells law as
1 =arcsin
1
nsin 0
. (2.36)
If0is not zero, the beamwidth is increased after transition from a medium of lower
refractive index to a medium of higher refractive index. Figure 2.12 shows a cutthrough a surface that forms the boundary between air and the prism material. The
beamwidth after refraction is calledw1. From the figure, the following relations for
the cosines of the two angles are obvious:
cos 0 =w0
h, (2.37)
cos 1 =w1
h. (2.38)
Figure 2.12 Change of beamwidth at a refracting surface.
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36 Chapter 2
If they are combined, a relation follows that is an expression for the beamwidths as
a function of incident and exiting angle, i.e.,
w1
w0=
cos 1
cos 0. (2.39)
Together with Snells law [Eq. (2.36)], this leads to the beamwidth transfer function
for a refracting surface,
w1
w0=
cos{arcsin[(1/n) sin 0]}
cos 0. (2.40)
The corresponding equation for the other refracting surface of the prism, i.e., the
transition from a medium with refractive index n to air, is
w3
w1=
cos[arcsin(n sin 2)]
cos 2. (2.41)
In a prism of apex angle , the link between the known angle 1 and the angle2is given by =1+ 2.
In this way, one finds
w3
w1
=cos[arcsin(nsin{ arcsin[(1/n) sin 0]})]
cos{
arcsin[(1/n) sin 0
]}. (2.42)
Substituting for w1 from Eq. (2.40), the beamwidth transfer function for a prism
with the apex angle is obtained as
w3
w0=
cos[arcsin(nsin{arcsin[(1/n) sin 0]})]
cos{ arcsin[(1/n) sin 0]}
cos{arcsin[(1/n) sin 0]}
cos 0.
(2.43)
This equation shows that the transfer functions of the refracting surfaces can be
multiplied to obtain the transfer function of the prism. This is a general property of
the beamwidth transfer functions.
A prism for expansion along one axis It is of interest to consider the situation
where the light beam is incident on the prism in accordance with the Brewster
condition [Eq. (2.26)], i.e.,sin 0
cos 0=n. (2.44)
To avoid reshaping at the other surface of the prism, i.e., compression in the direc-
tion of the axis that has been expanded before, it is convenient to have the beam
exiting the prism perpendicular to its rear surface(3 =0). Combining this condi-
tion [Eq. (2.35)] with the Brewster condition leads to the equation,
=90 deg + arctan(n). (2.45)
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Optical Components 37
Figure 2.13Expanding prism.
This implies that the following relation between the apex angle of the prism and its
refractive index should hold:
=arccot(n). (2.46)
In this situation of practical importance (Fig. 2.13), the relation for the beamwidths
is especially simple, i.e.,
w3w0
=n. (2.47)
This formula can be concluded from Eq. (2.33) using the Brewster condition and
the additional condition expressed in Eq. (2.46). To this end, it is helpful to consider
the argument arcsin(sin 0/n) first. Using Snells law and the trigonometric
identity arcsin x =90 arccos x, one has
arcsin
sin 0
n
= 90 deg + 0. (2.48)
Combining this equation with Eq. (2.46) and the Brewster condition, it follows that
arcsin
sin 0
n
= 0. (2.49)
This implies that the first factor in Eq. (2.43) is equal to 1. Using Snells law and
the trigonometric identity stated before, the second factor can be expressed as
cos[arcsin(sin 0/n)]
cos 0
=sin 0
cos 0
. (2.50)
If this is combined with the Brewster condition (tan 0 = n), one finds that the
second factor is equal to the refractive index n.
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38 Chapter 2
It seems worthwhile to present a shortcut to derive Eq. (2.47) for the case where
it can already be assumed that the second surface of the prism leaves the beamwidth
unaltered, i.e.,3 =0
w3
w1=1. (2.51)
In this case, it is sufficient to consider
w1
w0=
cos 1
cos 0. (2.52)
Using the Brewster condition (1/ cos 0 =n/ sin 0), it is concluded that
w1
w0=
cos 1
sin 0n. (2.53)
The sine term can now be expressed by Snells law (sin 0 =n sin 1)as
w1
w0=
cos 1
sin 1=cot 1. (2.54)
At this point, it is convenient to make use of the alternative expression of the Brew-
ster condition(cos 1 =n)in order to obtain Eq. (2.47) and we have
w3
w0=
w3
w1
w1
w0=n.
The type of prism considered here has numerous applications in optical devices.
In the literature, the abbreviationM =w1/w0for magnification can sometimes be
encountered in conjunction with the following equation (Hanna et al., 1975):
M =1
n
n2 sin2 0
1sin2 0. (2.55)
Its equivalence with Eq. (2.52) can directly be seen using Snells law (sin 0 =
n sin 1).
A prism for compression along one axis An analogous relation holds for a prism
that reduces the beamwidth along an axis instead of expanding it, namely,
w3
w0=
1
n. (2.56)
Such a prism can be realized by letting a beam pass undeviated through the first
surface, so that the beam shaping occurs at the exit surface (Fig. 2.14). In contrast
to the beam-shaping effect described earlier, in this case a transition from a mediumwith a higher refractive index to a medium with a lower index of refraction is of
importance(n sin 2 =sin 3).
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Optical Components 39
Figure 2.14Compressive prism.
Equation (2.56) can be derived in a way very similar to the derivation of
Eq. (2.47) if it is assumed that w2/w0 =1 holds, i.e., that perpendicular incidence
on the first surface is realized. Additionally, it has to be assumed that
tan 3 =n cot 2 =n. (2.57)
Combined with Snells law, the two equivalent equations lead to
w3
w2=
cos 3
cos 2=
1
cot 2=
1
n, (2.58)
and combined with the assumption of perpendicular incidence finally gives
w3
w0=
w2
w0
w3
w2=
1
n. (2.59)
Tolerancing For the purpose of tolerancing, it is advantageous to have explicit
formulas for the quantities of interest as functions of physical quantities that are
controlled by adjustment (the incidence angle0) or by manufacturing (the refrac-
tive indexn, the apex angle). Equations (2.32) and (2.43) are appropriate starting
points for such a sensitivity analysis.
An important point is the question of how the exiting angle changes if the input
angle0 is not well adjusted. In Fig. 2.15, the output angle is plotted as a function
of the deviation = 0 opt
0 from the optimum angle opt
0 = arctan(n) for
an expanding prism (Fig. 2.15) made of the standard glass BK 7. The wavelength
considered is = 405 nm. This determines the refractive index used in Eq. (2.32),which is n = 1.53024. In Fig. 2.16, the change of beamwidth calculated from
Eq. (2.43) is shown as a function of the deviation for the same prism.
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40 Chapter 2
Figure 2.15Exit angle versus adjustment error for the prism of Fig. 2.13.
Figure 2.16Beamwidth versus adjustment error for the prism of Fig. 2.13.
Figure 2.17 gives a representation of the change of the exiting angle with dis-
adjustment for a compressive prism (Fig. 2.14). To allow for comparison with
Fig. 2.9, the same material and wavelength as before are chosen in this example.
Figure 2.18 shows the dependence of the beamwidth on a deviation from the opti-
mum input angle.Being a function of wavelength and temperature, the refractive index that has to
be considered in the analysis might change in practice, depending on the conditions
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Optical Components 41
Figure 2.17Exit angle versus adjustment error for the prism of Fig. 2.14.
Figure 2.18Beamwidth versus adjustment error for the prism of Fig. 2.14.
under which the laser is operated. Figure 2.19 shows how the exiting angle changes
in the case of an expanding prism (Fig. 2.13) if there is a deviation n =nnopt
from the optimum refractive index. The apex angle is considered to be a fixed
value. The same holds for the angle of incidence. Again, Eq. (2.32) can be usedfor a simple analysis. Figure 2.20 shows the same dependence for a compressive
prism (Fig. 2.14).
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Optical Components 43
Figure 2.21Arrangement of two prisms to increase dispersion.
Figure 2.22Arrangement of two prisms to decrease dispersion.
in optical spectroscopy to increase dispersion, while the second case is of impor-
tance here. Arrangements of two prisms in which the second case is realized are
common in devices for optical recording (Okuda et al., 1995) and also have been
used in dye laser systems (Niefer and Atkinson, 1988).
Relation of incident and exit angle in the case of a two-prism arrangement Hav-
ing a look at the result for a single prism, the following equations can be written
immediately:
7 =arcsin
nII sin
II arcsin
sin 4
nII
, (2.60)
3 =arcsin
nI sin
I arcsin
sin 0
nI
, (2.61)
where the angles are designated as depicted in Fig. 2.22 and II is the apex angleof the second prism andnII is its index of refraction, while the variablesI andnI
describe the first prism. In the equation for 4, the relative orientation of the two
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44 Chapter 2
prisms intervenes. It is expressed by the variable =3 4 as
4
=arcsin
nIsin
I arcsin
sin 0nI
. (2.62)
If this expression is substituted into the equation for 7, the output angle of the
two-prism arrangement is found as a function of the input angle, i.e.,
7 =arcsin
nII sin
II arcsin
1
nIIsin
arcsin
nI sin
I
arcsin
sin 0
nI
. (2.63)
The transfer function for the beamwidth altered by two prisms Exploiting the factstated earlier that the beamwidth transfer function can be composed as the prod-
uct of the transfer function of the refracting surfaces that intervene, the transfer
function for the two prisms can now be written directly as
w7
w0=
cos(arcsin{nII sin[II arcsin(sin 4/nII)]})
cos[II arcsin(sin 4
nII )]
cos[arcsin(sin 4/nII)]
cos 4
cos(arcsin{nI sin[I arcsin(sin 0/n
I)]})
cos[I arcsin(sin 0/n)]
cos[arcsin(sin 0/nI)]
cos 0.
(2.64)
The link bet