lens design approach to optical relays by...
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Lens Design Approach to Optical Relays
Item Type text; Electronic Thesis
Authors OShea, Kevin
Publisher The University of Arizona.
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Download date 28/04/2018 06:59:59
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1
LENS DESIGN APPROACH TO OPTICAL RELAYS
By
Kevin O’Shea
A Thesis Submitted to the Faculty of the
COLLEGE OF OPTICAL SCIENCES (GRADUATE)
In Partial Fulfillment of the Requirements For the Degree of
MASTER OF SCIENCE
In the Graduate College
THE UNIVERSITY OF ARIZONA
2005
2
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be
made available to borrowers under rules of the Library.
Brief quotations from this master’s thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be
granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In
all other instances, however, permission must be obtained from the author.
SIGNED: _________________________________ Kevin P. O’Shea
APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below:
__________________________________ 11/28/05Dr. José M. Sasián Date Professor of Optical Sciences
3
ACKNOWLEDGEMENTS
I would like to thank José Sasián for being my advisor and giving me a topic
which allowed me to investigate an area of great interest to me.
To Rick Juergens, for being a member of my thesis committee and for always
being an understanding friend.
To John Greivenkamp, for being a member of my thesis committee and for
reading the draft of my thesis.
To Bob Pierce and Frank Grochocki, for reading various parts of the draft of this
thesis and providing feedback.
Lastly, to Optical Research Associates for providing me with an academic
software license which allowed me to do the computer aided design and analysis work for
this thesis.
4
TABLE OF CONTENTS
TABLE OF ILLUSTRATIONS ..................................................................................... 5 LIST OF TABLES.......................................................................................................... 8 ABSTRACT.................................................................................................................... 9
Chapter 1 INTRODUCTION............................................................................................ 10 Concatenation ............................................................................................................... 11
First Order Considerations........................................................................................ 13 Aberration Theory and its effect on concatenation orientation ................................ 14
Telecentricity ................................................................................................................ 19 Pupil Aberrations .......................................................................................................... 23 Scope of Validation....................................................................................................... 29
Chapter 2 INFINITE CONJUGATE EXTERNAL STOP MIRROR SYSTEMS............ 32 Imager Mirror systems.................................................................................................. 33
All Spherical Mirror Systems ................................................................................... 33 All Conic Mirror Systems......................................................................................... 41 Imaging Lens Performance Comparison Summary.................................................. 48
Collimator Mirror systems............................................................................................ 50 Imaging Mirror System Reversal.............................................................................. 50
Pupil Aberrations and Telecentricity ............................................................................ 52 Chapter 3 IMAGING RELAY MIRROR SYSTEMS...................................................... 54
Validation of Aberration Cancellation Upon Concatenation........................................ 57 Performance comparisons............................................................................................. 71
Chapter 4 CONCLUSION ................................................................................................ 75 APPENDIX A................................................................................................................... 77
Pupil Relay.................................................................................................................... 77 APPENDIX B ................................................................................................................... 81 REFERENCES ................................................................................................................. 83
5
TABLE OF ILLUSTRATIONS Figure 1. An example of an imaging mirror system which results from the concatenation of a collimator and an imager. The dashed line shows the separation of the two components. The global coordinate system shown is for later reference. ........................ 12 Figure 2. Example of an imaging mirror system. ............................................................ 12 Figure 3. Example of an collimator mirror system. The path of the mirror system in Figure 2 was reversed to generate this mirror system....................................................... 12 Figure 4. Relay mirror system with symmetry about the stop. The mirror systems in Figures 2 and 3 were concatenated to form this relay. ..................................................... 16 Figure 5. Relay mirror system with symmetry about its center. The mirror system in Figures 2 and 3 were concatenated to form this relay. ..................................................... 16 Figure 6. Illustration of obliquity of OAR with respect to the image surface due to image tilt. The principal ray for the off axis field point (blue) is parallel to the OAR (i.e. the lens is telecentric). ............................................................................................................ 20 Figure 7. Gradient of constant astigmatism showing the introduction of anamorphic deviation into the pattern of principal rays. ...................................................................... 22 Figure 8. Gradient of constant coma showing the introduction of quadratic error into the pattern of principal rays. ................................................................................................... 22 Figure 9. Numbered blue diamonds indicate normalized field points used for spot size optimization and subsequent analysis. The normalization factor in the x direction is 2.29 degrees; the normalization factor in the y direction is 1.29 degrees................................. 32 Figure 10. Single mirror path folded to prevent interference upon concatenation. ......... 33 Figure 11. Ray fan plots for one mirror spherical system with 250 mm focal length. Scale is mm. ...................................................................................................................... 34 Figure 12. Off axis section of a Schwarzchild system with infinite conjugate object (250 mm focal length). The stop is shown in the location of the virtual front focal plane. ..... 36 Figure 13. A 5 mirror pupil relay. The lens is afocal and has good imaging from the aperture stop to the exit pupil............................................................................................ 36 Figure 14. Schwarzchild system with pupil relay. ........................................................... 37 Figure 15. Ray fan plots for two mirror spherical system with 250 mm focal length. Scale is mm. ...................................................................................................................... 38 Figure 16. Three mirror spherical system with infinite conjugate object (250 mm EFL). Fold mirror is introduced to allow clearance for the stop................................................. 39 Figure 17. Ray fan plots for three mirror spherical system with 250 mm focal length. Scale is mm. ...................................................................................................................... 40 Figure 18. One mirror conic system with infinite conjugate object (250 mm focal length)............................................................................................................................................ 41
6
Figure 19. Ray fan plots for one mirror conic system with 250 mm focal length. Scale is mm. ................................................................................................................................... 42 Figure 20. Off axis section of a Gregorian telescope with infinite conjugate object (250 mm focal length). .............................................................................................................. 43 Figure 21. Folded view of the Gregorian telescope with infinite conjugate object (250 mm focal length). .............................................................................................................. 44 Figure 22. Ray fan plots for two mirror conic system with 250 mm focal length. Scale is mm. ................................................................................................................................... 45 Figure 23. Three mirror Anastigmat, all conic surfaces. ................................................. 46 Figure 24. Ray fan plots for three mirror conic system (TMA) with 250 mm focal length. Scale is mm. ...................................................................................................................... 47 Figure 25. Composite RMS spot size comparison........................................................... 48 Figure 26. Illustration of image/object decenter. Green line is axis of symmetry of parent system. The OAR is decentered relative to this axis. ........................................... 51 Figure 27. Example of a collimator mirror system. ......................................................... 51 Figure 28. Wave front aberration map for pupil reimaging............................................. 52 Figure 29. Telecentricity error for TMA showing an anamorphic and a small quadratic component. Scale is radians. ............................................................................................. 53 Figure 30. System 22C5XH............................................................................................. 55 Figure 31. System 22S1XR. Two afocal pupil relays are introduced to ensure an accessible aperture stop..................................................................................................... 55 Figure 32. System 23S1XR. An example where the Schwarzchild pupil relay is not necessary. .......................................................................................................................... 56 Figure 33. System 33C2XR ............................................................................................. 56 Figure 34. Illustration of the OAR image plane obliquity (image tilt) for the three mirror spherical system. ............................................................................................................... 57 Figure 35. Illustration of the cancellation of field tilt upon concatenation in the orientation yielding symmetry about the stop................................................................... 58 Figure 36. Plot of full field astigmatism for the one mirror conic system showing components of constant and linear astigmatism. .............................................................. 59 Figure 37. Plot of full field astigmatism after concatenation of system 11C1XR........... 60 Figure 38. Plot of full field coma for the one mirror conic imager showing components of constant and linear coma. ............................................................................................. 61 Figure 39. Plot of full field coma after concatenation of system 11C1XR...................... 62 Figure 40. System 11C1XH............................................................................................. 64 Figure 41. Coma residual for system 11C1XH................................................................ 65 Figure 42. Distortion plot for one mirror conic. .............................................................. 66 Figure 43. Distortion plot for system 11C1XH after concatenation................................ 67 Figure 44. Two OAPs used to illustrate an issue with the PSF. ...................................... 69
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Figure 45. PSF for the on axis case (angle of incidence equals 0 degrees) (a) and the case for which the sum of the AOI and AOR is 90 degrees (b). .............................................. 69 Figure 46. Log scale PSF at (a) 15 (b) 30 (c) 45 (d) 60 (e) 75 and (f) 90 degree sum of OAR AOI and AOR on the mirrors. Size of each box is 6.3 microns on a side.............. 70 Figure 47. Spot Size comparison for Spherical Relays, symmetry about the stop. .......... 71 Figure 48. Spot Size comparison for Conic Relays, symmetry about the stop................. 72 Figure 49. Spot Size comparison for Spherical Relays, symmetry about the center. ....... 72 Figure 50. Spot Size comparison for Conic Relays, symmetry about the center.............. 73 Figure 51. Source tilt vs. magnification, symmetry about the stop. ................................ 73
8
LIST OF TABLES Table 1. Aberration Groups ............................................................................................. 15 Table 2. Dependence of aberrations of axial symmetry on magnification. ..................... 17 Table 3. Dependence of aberrations of planar symmetry on magnification in the concatenation orientation which yields symmetry about the aperture stop. ..................... 18 Table 4. Dependence of aberrations of planar symmetry on magnification in the concatenation orientation which yields symmetry about the center of the stop. .............. 18 Table 5 Summary of Pupil and Imagery Aberration Equalities........................................ 29 Table 6. W13100 for the One Mirror Imagers .................................................................... 64 Table 7. W13100 for the 2 mirror conic relays. ................................................................... 64 Table 8. Approximate Packaging Dimensions for 1X Spherical Systems with symmetry about the stop. ................................................................................................................... 74 Table 9. Approximate Packaging Dimensions for 1X Conic Systems with symmetry about the stop. ................................................................................................................... 74
9
ABSTRACT
A process to design a relay lens is presented. The process is to concatenate a
collimator lens and an imaging lens. For this study the imager and collimator are
required to have an external or remote stop in collimated space to prevent interference
upon concatenation. The relay is created by concatenating the collimator and imager at
the external or remote stop. This process allows the use of optimized infinite conjugate
imagers to develop a relay lens. A collimator lens can be created by reversing the path of
an imager. Magnification is achieved by scaling the focal length of the imager while
keeping the focal length of the collimator constant. Computer design software is used to
develop examples of relays designed using the process. A discussion of the aberration
theory governing the integration of the collimator and imager to create a relay is also
presented.
10
Chapter 1 INTRODUCTION
Relay lenses image a finite conjugate object to a finite conjugate image. They are
used in a variety of applications, including lithography and machine vision. An approach
to design a relay lens is to first design a collimator lens and an infinity corrected imager
lens, both with external pupils located in collimated space. The collimator lens and the
imaging lens are concatenated at their respective external pupils to create a relay lens,
meaning that the two lenses are connected in series. This straight forward technique
enables the creation of a relay lens using two infinite conjugate lenses optimized for the
same entrance pupil size and shape and the same field of view. The collimator lens can
be created by reversing the path of an imager lens. While previous studies of relay lenses
have yielded first order solutions which are good starting points for optimization2-5, the
concatenation process presented can yield a relay system which does not require further
optimization when given the proper inputs.
This study explores the described approach to design a relay system using
examples of un-obscured reflective telecentric relay systems. Consequently, the
discussion will be tailored to these types of systems. The third order theory which
governs the linking of the collimator and imager mirror systems will be discussed,
establishing a reference which gives the lens designer information to help understand the
subtleties of relay systems and a concrete means to approach the design of a relay lens.
Although telecentricity is not required for the concatenation to be successful, it is a
desirable characteristic because the magnification of the resulting system will be
11
insensitive to changes in focus. Since the systems are designed to be telecentric, the
connection between pupil aberrations and telecentricity is also studied in detail. For
completeness, the connection between aberrations of imagery of the pupil and imagery of
the system is also discussed.
Concatenation
The method proposed to design a relay lens is the concatenation of a collimator
lens and a lens which is focused for an infinite object conjugate at an external pupil. An
example of such a system is shown in Figure 1. For this study, a planar object and image
are of interest. In this case, the concatenation requires that the collimator lens and the
imaging lens both have a pupil located in the collimated space external to the lens, and
that this external pupil is coincident with the aperture stop; successful addition of the
lenses relies on proper mapping of the field of view in the collimated space as well as
proper pupil matching. The pupils must be matched and the collimator lens must provide
a collimated field of view with angular extent equivalent to the field of view of the
imaging lens. Both the collimator lens and the imaging lens can be designed as infinite
conjugate external stop lenses. Reversing the path of an imaging lens creates a collimator
with an external stop. Examples of a collimator mirror system and an imaging mirror
system are shown in Figures 2 and 3. After concatenation, the aperture stop of the relay
lens will be located between the collimator lens and imaging lens.
12
Figure 1. An example of an imaging mirror system which results from the concatenation of a collimator and an imager. The dashed line shows the separation of the two components. The global coordinate system
shown is for later reference.
Figure 2. Example of an imaging mirror system.
Figure 3. Example of a collimator mirror system. The path of the mirror system in Figure 2 was reversed to generate this mirror system.
+Y
+Z
13
First Order Considerations
For two lenses to be useable in the concatenation, the lenses must obey a few
rules. First, the imaging properties of the lenses must be optimized for an object at
infinity. Second, the entrance pupil for each lens must be coincident with the aperture
stop and located in collimated space. Third, the entrance pupil for the lenses must be the
same shape and size. Finally, each lens must be designed for the same field of view in
object space.
The path of one of the lenses from a pair which adheres to the guidelines above
can be reversed to form the collimator lens; the resulting collimator lens and the imager
lens will form a relay lens when they are connected at their respective aperture stops.
The relay lens will have a magnification equal to the ratio of the focal length of the
imager lens to the focal length of the collimator lens. Since the collimator and imager
components of the relay will stay in focus if separated, the components of a set of relay
lenses are interchangeable. This allows multiple magnifications from a set of infinite
conjugate lenses with multiple focal lengths. The approach that will be employed in this
study is to fix the focal length of the collimator lens and vary the focal length of the
imager lens to generate various discrete magnifications.
While not required for the concatenation to be successful, telecentricity is a
desirable characteristic because the magnification of a telecentric system is constant with
changes in focus and the illumination on the image plane will be more uniform. The cos4
irradiance variation typically seen in imaging lenses reduces to the relative decrease in
irradiance due to the obliquity of field angles on the aperture stop in a telecentric lens.
14
Aberration Theory and its effect on concatenation orientation
The possibilities of concatenation orientation of collimator and imager mirror
systems are limited to those which yield bilateral or planar symmetry, meaning that the
relay lenses will be symmetric about the yz plane (see Figure 1). Studies of the
aberrations of systems with this type of symmetry can be found in several sources6-10. In
particular, Sasian studied the imagery of bilateral symmetric systems9 by developing a
wave aberration function organized by order and symmetry. The first three aberration
groups are presented below (from table 1 in Sasian’s paper). H is the magnitude of the
field vector, ρ is the magnitude of the aperture vector, φ is the angle between the aperture
and field vector, α is the angle between the field vector and the plane of symmetry and βis the angle between the aperture vector and the plane of symmetry. Within the third
order group, the aberrations are organized in the subgroups of double plane symmetry,
plane symmetry and axial symmetry.
This wave aberration function indicates a dependence of the aberrations on the
orientation of concatenation. Figures 4 and 5 show the two possible orientations of the
imager relative to the collimator that will result in bilateral symmetry. Since both
orientations will have symmetry about the aperture stop (figure 5 has an inverted
symmetry) the odd aberrations in the subgroup corresponding to axial symmetry will
cancel independent of concatenation orientation. Both orientations will also have plane
symmetry, so a subset of the odd aberrations in the subgroup corresponding to plane
symmetry will cancel after concatenation. The choice of orientation affects which
aberrations cancel. The orientation illustrated in Figure 4 results in pure symmetry about
15
the stop. In this orientation the system has a negative magnification. The result is that
the aberrations in the plane symmetry subgroup which have an odd dependence on H will
cancel. In this orientation linear coma, cubic distortion, linear astigmatism, field tilt and
cubic piston will cancel.
Table 1. Aberration Groups
Scalar Form Name
First Group
W00000 Constant Piston
Second Group
W01001 ρ cos(β) Field Displacement W10010 H ρ cos(α) Linear Piston
W02000 ρ2 Defocus W11100 H ρ cos(φ) Magnification W20000 H2 Quadratic Piston
Third Group
W02002 ρ2 cos2(β) Constant Astigmatism W11011 H ρ cos(α)cos(β) Anamorphism W20020 H2 cos2(α) Quadratic Piston
W03001 ρ3 cos(β) Constant Coma W12101 H ρ2 cos(φ) cos(β) Linear Astigmatism W12010 H ρ2 cos(α) Field Tilt W21001 H2 ρ cos(β) Quadratic Distortion I W21110 H2 ρ cos(φ) cos(α) Quadratic Distortion II W30010 H3 cos(α) Cubic Piston
W04000 ρ4 Spherical Aberration W13100 H ρ3 cos(φ) Linear Coma W22200 H2 ρ2 cos2(φ) Quadratic Astigmatism W22000 H2 ρ2 Field Curvature W31100 H3 ρ cos(φ) Cubic Distortion W40000 H4 Quartic Piston
16
Figure 4. Relay mirror system with symmetry about the stop. The mirror systems in Figures 2 and 3 were concatenated to form this relay.
The alternate orientation results in symmetry about a center24, meaning that there
is a center point about which the system is symmetric. The center point is the center of
the aperture stop. This orientation is illustrated in figure 5. The aberrations in the plane
symmetry subgroup which cancel in this orientation are those which contain an odd order
of ρ. So the aberrations linear coma, cubic distortion, constant coma and quadratic
distortion cancel.
Figure 5. Relay mirror system with symmetry about its center. The mirror system in Figures 2 and 3 were concatenated to form this relay.
In order to determine the dependence of the aberrations on magnifications greater
than unity, we start with a collimator of focal length f. If the collimator is duplicated and
its path is reversed, we generate an imager lens. Scaling the imager lens by a factor of M
17
yields a focal length of Mf. The aberrations of the lens scale by the factor of M. The
scaling yields an imager with a stop diameter equal to M times that of the collimator.
When the imager lens is stopped down to an aperture diameter equal to that of the
collimator lens, the aberrations scale as the inverse of Mn, where n is the power of ρ for a
given aberration. The factor M is the magnification of the resulting relay. Equations 1
and 2 show the dependence of the relay aberrations on magnification and the collimator
aberration. Plus is for aberrations which add and minus is for aberrations which balance
or cancel. Table 2 shows the dependence for the specific aberrations.
±= nCOLL
COLLRELAY MMWWW (1)
COLLnRELAY WMW
±= −1||11 (2)
Table 2. Dependence of aberrations of axial symmetry on magnification. Aberration Power of ρ Dependence
on magnification*
Spherical 4
+ 304000
11M
W
Linear Coma 3
− 213100
11M
W
Quadratic Astigmatism 2
+ MW 1122200
Field Curvature 2
+ MW 1122000
Cubic Distortion 1 0 * - Coefficient is that of the collimator.
18
Table 3. Dependence of aberrations of planar symmetry on magnification in the concatenation orientation which yields symmetry about the aperture stop.
Aberration Power of ρ Dependence on magnification*
Constant Coma 3
+ 203001
11M
W
Linear Astigmatism 2
− MW 1112101
Field Tilt 2
− MW 1112010
Quadratic Distortion I 1 210012WQuadratic Distortion II 1 211102W
* - Coefficient is that of the collimator.
Table 4. Dependence of aberrations of planar symmetry on magnification in the concatenation orientation which yields symmetry about the center of the stop.
Aberration Power of ρ Dependence on magnification*
Constant Coma 3
− 203001
11M
W
Linear Astigmatism 2
+ MW 1112101
Field Tilt 2
+ MW 1112010
Quadratic Distortion I 1 0 Quadratic Distortion II 1 0
* - Coefficient is that of the collimator.
The choice of which concatenation orientation to use is driven by the
requirements of the application. For instance, it may be desirable to have a minimum
optical axis ray (OAR - the zero field point ray which passes through the center of the
19
aperture stop and the hence the pupils) angle of obliquity with respect to the object and
image plane. In this case the orientation with anti-parallel H vectors is preferable since
the field tilt will cancel. In another application minimization of distortion may be
required, so the orientation with anti-parallel ρ vectors is preferable. As a consequence,
the aperture stop for each lens is required to have enough clearance to allow for
concatenation in either orientation. This requirement may be met by folding the optical
path with planar mirrors or by introducing a pupil relay lens which images the aperture
stop or a pupil to an alternate location. Finally, regardless of orientation, the systems in
this study will be designed to be telecentric, a property described in the next section.
Telecentricity
An optical system is telecentric if the aperture stop of the system is imaged to an
infinite conjugate. The system is telecentric in object space if the entrance pupil is at
infinity; it is telecentric in image space if the exit pupil is at infinity. Telecentricity in
object space is achieved by placing the aperture stop at the rear focal plane. Placing the
aperture stop at the front focal plane yields a system which is telecentric in image space.
A telecentric system has the property that to first order all the rays which pass though the
center of the stop for all field points are parallel in image and/or object space, depending
on the type of telecentricity. While not necessarily a requirement for finite conjugate
relays, this property is desirable because the magnification is constant versus defocus and
the illumination across the image plane is more uniform.
20
In an axially symmetric telecentric optical system, the principal rays are
perpendicular to the image plane and the exit pupil. This removes the cos4 dependence of
illumination on field. In a bilaterally symmetric optical system, a system may be
telecentric and it may suffer from field tilt. The result is that the improvement in
uniformity over a system which is not telecentric is maintained, but the OAR is not
perpendicular to the image plane. This is illustrated in figure 6. To first order, the
principal rays have a constant finite angle of incidence across the focal plane equivalent
to the angle of incidence of the OAR at the focal plane. There will be a decrease of
irradiance due to the cosine projection, but the decrease will be constant over field.
Figure 6. Illustration of obliquity of OAR with respect to the image surface due to image tilt. The principal ray for the off axis field point (blue) is parallel to the OAR (i.e. the lens is telecentric).
The angular deviation of real rays from the angle of the OAR is governed by
higher order effects, since a telecentric imaging lens acts as a collimator for the pupil.
The collimation error of this path is the error in telecentricity. To further investigate this,
a discussion of pupil aberrations is necessary.
21
The infinite conjugate solutions utilized in this study image an external or remote
stop to infinity. This is how telecentricity is achieved. This imagery is not perfect, so
there will be a higher order telecentricity error in the form of an angular deviation from
the angle of the OAR. The principal ray for each field passes through the center of the
stop, so the axial aberrations of the pupil reimaging will be a measure of the deviation
from pure telecentricity. Since the derivative of wave front is ray slope, the deviation
from pure telecentricity is determined by computing the gradient of the relevant pupil
aberration. For a system lacking rotational symmetry, spherical aberration is no longer
the sole possible third order aberration present at the center of the pupil. In the case of
bilaterally symmetric systems, constant astigmatism and constant coma may also be
present. Constant astigmatism has the functional form ρ2cos2(β). The gradient of this
function, plotted in Figure 7, shows that constant astigmatism in the pupil imaging yields
anamorphism in telecentricity. If the on axis pupil imaging suffers from constant coma,
which has the functional form ρ3cos(β), the telecentricity error has the pattern shown in
figure 8. The vectors show the direction cosines of the principal rays at the focal plane.
22
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Figure 7. Gradient of constant astigmatism showing the introduction of anamorphic deviation into the pattern of principal rays.
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Figure 8. Gradient of constant coma showing the introduction of quadratic error into the pattern of principal rays.
23
Pupil Aberrations
In the previous section, studying the derivative of the pupil aberrations showed
the patterns that telecentricity errors will manifest depending on which pupil aberrations
are present. In this section, we further analyze the pupil aberrations in the context of their
relation to the aberrations of the system imagery. This analysis was performed for axially
symmetric systems by Longhurst21, and was further developed by Wynne22. Here it is
extended to systems with bilateral symmetry, so the aberrations of concern are those in
the first two sub-groups of group three in table 1.
In the following derivations, ( )∆ is the Abbe difference operator, u is the
marginal ray angle, n is the index of refraction, u is the chief ray angle, Ψ is the
Lagrange invariant, I is the angle of incidence of the OAR, x is the marginal ray height,
x is the chief ray height and j is the number of mirror surfaces. The aberration
coefficients are as defined in table 1; barred coefficients correspond to aberrations of the
pupil imagery. The quantity
∆ 21n , which appears in many of the derivations, is zero
for reflective systems. Since the derivations are tailored for reflective systems, any terms
containing this quantity as a factor will be dropped from the derivation. The structural
parameters, functional form of the aberration coefficients and identities used can be
found in Appendix B.
24
1. Constant astigmatism of the pupil
{ }∑=
=j
iiIJW
102002 (3)
( )∑=
∆−=
j
i ixxxn
uIn1
22 sin21 (4)
( )∑=
Ψ∆+
∆
−=
j
i inxnu
xxIn
12
22 1sin21 (5)
( )∑=
∆−=
j
i ixxxn
uIn1
222 sin2
1 (6)
2002002002 WW = (7)
2. Anamorphism of the pupil
∑=
=
j
i iIJx
xW1
11011 2 (8)
( )∑=
∆−=
j
i ixn
uIn1
22 sin (9)
( )∑=
Ψ∆−
∆−=
j
i inxnuIn
12
22 1sin (10)
( )∑=
∆−=
j
i ixxxn
uIn1
22 sin (11)
∑=
=
j
i iIJx
x1
2 (12)
1101111011 WW = (13)
25
3. Quadratic Piston of the pupil
i
j
iIJx
xW ∑=
=
1
2
20020 (14)
( )i
j
ixn
uInxx∑
=
∆−
=
1
222
sin21 (15)
( )i
j
ixn
uInxx∑
=
∆−
=
1
22 sin21 (16)
( )i
j
ixn
uIn∑=
∆−=
1
22 sin21 (17)
0200220020 WW = (18)
4. Constant Coma of the Pupil
{ }∑=
=j
iiIIJW
103001 (19)
( )∑=
∆−=
j
i ixn
uAIn1
sin21 (20)
( )∑=
∆−=
j
i ixxxn
uAIn1
sin21 (21)
∑=
+
+
+
=
j
i iIIIVIIIV x
xJxxJx
xJJxx
1
2(22)
3001003001 WW = (23)
26
5. Linear Astigmatism of the pupil
i
j
iIIIII Jx
xJW ∑=
+
=
112101 2 (24)
( ) ( )∑=
Ψ∆+
∆−=
j
i inuInxn
uAIn1
sinsin (25)
( ) ( )i
j
i nuInxn
uAIn∑=
Ψ∆+
∆−=
1sinsin (26)
( ) ( )∑=
Ψ+
∆−=j
i ixx
nuInxn
uAIn1
sinsin (27)
( )∑=
−
∆−=
j
i iIII x
xJxnuAIn
1sin (28)
∑=
−
+
=
j
i iIIIIIIII x
xJxxJx
xJ1
22 (29)
( )i
j
iIIIIVIVIIIII x
xJJxxJJx
xxxJ∑
=
−
−+
+
=1
2222 (30)
∑=
+
−=
j
i iIIIIV Jx
xJxxW
121110 2 (31)
2111012101 WW = (32)
27
6. Field Tilt of the pupil
i
j
iIVII Jx
xJW ∑=
+
=
112010 (33)
( ) ( )∑=
Ψ∆+
∆−=
j
i ixnR
InxnuAIn
1
1sin21sin2
1 (34)
( )i
j
iIIIVIIIV xnR
InxxJx
xJxxJJ∑
=
Ψ∆+
+
+
+=
1
2 1sin21 (35)
2100112010 WW = (36)
7. Quadratic Distortion I of the pupil
i
j
iVIIIII JJx
xJxxW ∑
=
++
=
1
2
21001 (37)
( )i
j
iIVJx
xxnuAIn∑
=
−
∆−=
1sin2
1 (38)
( ) ( )i
j
ixnR
Inxxxn
uAIn∑=
Ψ∆−
∆−=
1
1sin21sin2
1 (39)
i
j
iIVII JJx
x∑=
+
=
1(40)
1201021001 WW = (41)
28
8. Quadratic Distortion II of the pupil
( )i
j
iIVIIIII JJx
xJxxW ∑
=
++
=
1
2
21110 22 (42)
( ) ( )∑=
∆Ψ−
∆−=
j
i ixxxnR
InxnuAIn
1
1sinsin (43)
( ) ( )∑=
Ψ∆+
∆
−=
j
i ixnR
InxnuAInx
x1
1sinsin (44)
( )∑=
−
∆
−=
j
i iIVJxn
uAInxx
12sin (45)
( )∑=
+−+
=
j
i iIIIIVIIIII JJJJx
x1
22 (46)
1210121110 WW = (47)
9. Cubic Piston of the pupil
( )∑=
++
+
=
j
i iVIVIIIII Jx
xJJxxJx
xW1
23
30010 (48)
( )∑=
++
+
=
j
i iVIVIIIII JJJx
xJxx
xx
1
2(49)
( )∑=
∆−
=
j
i ixn
uAInxx
1sin2
1 (50)
{ }∑=
=j
iiIIJ
1(51)
0300130010 WW = (52)
29
Table 5 Summary of Pupil and Imagery Aberration Equalities Pupil
Aberration Description Image
Aberration Description
02002W Constant Astigmatism 20020W Quadratic Piston 11011W Anamorphism 11011W Anamorphism 20020W Quadratic Piston 02002W Constant Astigmatism 03001W Constant Coma 30010W Cubic Piston 12101W Linear Astigmatism 21110W Quadratic Distortion II 12010W Field Tilt 21001W Quadratic Distortion I 21001W Quadratic Distortion I 12010W Field Tilt 21110W Quadratic Distortion II 12101W Linear Astigmatism 30010W Cubic Piston 03001W Constant Coma
Scope of Validation
To validate the process described to yield a relay lens, it is necessary to show that
the lenses behave as described after the components have been integrated. To do this a
set of infinite conjugate remote pupil telecentric mirror systems will be designed and
concatenated as discussed. To determine the necessary solutions, a set of requirements
for the relays must be determined. These requirements then flow requirements to the
infinite conjugate mirror systems.
Specifications for the imaging relay mirror systems are magnifications of 1X, 2X
and 5X (note that magnifications less than 1 can be achieved by reversing the systems
with magnification greater than 1, the trade off is that the cone of light collected from the
source will decrease – such systems are not considered here), a planar object with a 20
30
mm width and 16:9 width to height aspect ratio, light collection from the object of f/5, no
chromatic dependence, no obscuration of rays and minimum spot size. For comparison, a
set of spherical mirror and a set of conic mirror systems ranging in complexity from 2 to
6 mirrors are required. The relay mirror systems for this study are required to be
telecentric in both object and image space.
The specifications on the relay mirror systems yield specifications on the set of
the infinite conjugate mirror systems. Solutions chosen will be un-obscured external
pupil telecentric reflective mirror systems, optimized for spot size at focal lengths of 250
mm, 500 mm and 1250 mm. The flexibility of the orientation of concatenation described
previously requires that aperture stop have a minimum clearance of 25 mm with respect
to all critical surfaces in the z direction. The shortest focal length mirror systems will be
used to create the collimator mirror systems, so the necessary field of view (FOV) is
calculated using the shortest focal length. The resulting half FOV is 1.29 degrees in the
direction of the aperture decenter (i.e. in the yz plane) and 2.29 degrees in the direction
perpendicular to the aperture decenter (the xz plane), derived from the first order relation
y=f*tan(θ), where y is the half image height f is the focal length and θ is the half field
angle. The stop diameter will be 50 mm, yielding focal ratios of f/5, and f/10 and f/25 for
the 250 mm, 500 mm and 1250 mm focal lengths respectively. The infinite conjugate
mirror systems will range in complexity from one to three mirrors and will be divided
into spherical and conic surface categories.
Recent studies have provided insight into the development of useful starting
points for optimization4,5,11,12. The starting design forms used for this study are common
31
solutions that are documented in several literature sources and patents13-16. Chapter 2
documents the details of the solutions and the relevant performance parameters for the
design forms chosen for the creation of the suite of imaging relays.
Chapter 3 describes the resulting imaging relay mirror systems, validates the
aberration theory discussed earlier in this chapter and shows performance comparisons of
the various possible relays. Finally, Chapter 4 summarizes and concludes the work
performed.
32
Chapter 2 INFINITE CONJUGATE EXTERNAL STOP MIRROR SYSTEMS
In order to validate the concatenation process, it is necessary to design a set of
infinite conjugate mirror systems which will be utilized to construct the relay mirror
systems. This chapter details these mirror systems. Each design form is optimized for
spot size at focal lengths of 250 mm, 500 mm and 1250 mm at the 6 field points
illustrated in Figure 9. A layout of each system is given for the 250 mm focal length
and ray intercept plots are provided for each design at this focal length (for brevity of
presentation field point 4 will be omitted from the ray intercept plots – 5 fields are the
maximum plotted on a single ray intercept plot for the software used).
Figure 9. Numbered blue diamonds indicate normalized field points used for spot size optimization and
subsequent analysis. The normalization factor in the x direction is 2.29 degrees; the normalization factor in the y direction is 1.29 degrees.
-1
-0.5
0
0.5
1
0 0.5 1
1
2
3
5
6
4
33
Imager Mirror systems All Spherical Mirror Systems Single Mirror System
The solution for the one mirror case is straight forward. The layout of the 250
mm focal length one mirror configuration is shown in Figure 10. It is simply a single
mirror with the aperture stop placed at its front focal plane. A fold mirror is introduced
into the path to meet the stop clearance requirement.
Figure 10. Single mirror path folded to prevent interference upon concatenation.
34
-0.25
0.25
1
-0.25
0.25
,0.00 0.00RELATIVE FIELD( , )0.00O 0.00O
-0.25
0.25
2
-0.25
0.25
,0.00 1.00RELATIVE FIELD( , )0.00O 1.29O
-0.25
0.25
3
-0.25
0.25
,0.00 -1.00RELATIVE FIELD( , )0.00O -1.29 O
-0.25
0.25
4
-0.25
0.25
,1.00 1.00RELATIVE FIELD( , )2.29O 1.29O
-0.25
0.25
5
-0.25
0.25
( X , Y )Y-FAN ,1.00 -1.00
RELATIVE FIELD( , )2.29O -1.29 O
X-FAN
Figure 11. Ray fan plots for one mirror spherical system with 250 mm focal length. Scale is mm.
35
Two Mirror System
The two mirror solution utilizes an off axis section of a Schwarzschild system13, a
system comprised of two concentric spherical mirrors. Decentering the aperture by an
adequate distance yields an un-obscured spherical two mirror solution, shown in Figure
12. The degrees of freedom employed in the optimization are the distance from the
aperture stop to the primary mirror, the mirror curvatures, the spacing between the
mirrors, the back focal distance and the tilts and decenters of the mirrors relative to their
respective parent axes. This solution cannot satisfy the requirements of telecentricity and
stop accessibility simultaneously. The concave secondary has a focal length which is
shorter than the separation of the two mirrors. As a consequence, the front focal plane is
located behind the primary. Due to its superior performance relative to other two mirror
spherical solutions, this form is chosen and a pupil relay18 is added when necessary to
yield an accessible stop. The necessity of the pupil relay is dictated by whether the
mating lens has sufficient stop clearance to access the pupil in the Schwarzschild system.
The pertinent properties and performance of the pupil relay are discussed in
Appendix A, but it is presented here to show its application. It employs an afocal relay
which also reimages the aperture stop to a new location as illustrated in Figure 13. When
placed in tandem with the Schwarzschild system, the pupil relay maps the aperture stop
to an accessible location as indicated in figure 14. Note that the pupil relay has good
performance on its own. It does not aid in the correction of the spot size of the
Schwarzschild system. Its sole purpose is to allow the Schwarzschild system and imager
to simultaneously meet the requirements of telecentricity and aperture stop accessibility.
36
125.00 MM
Figure 12. Off axis section of a Schwarzschild system with infinite conjugate object (250 mm focal length). The stop is shown in the location of the virtual front focal plane.
80.00 MM
Figure 13. A 5 mirror pupil relay. The lens is afocal and has good imaging from the aperture stop to the exit pupil.
37
150.00 MM
Figure 14. Schwarzschild system with pupil relay.
38
-0.025
0.025
1
-0.025
0.025
,0.00 0.00RELATIVE FIELD( , )0.00O 0.00O
-0.025
0.025
2
-0.025
0.025
,0.00 1.00RELATIVE FIELD( , )0.00O 1.29O
-0.025
0.025
3
-0.025
0.025
,0.00 -1.00RELATIVE FIELD( , )0.00O -1.29 O
-0.025
0.025
4
-0.025
0.025
,1.00 1.00RELATIVE FIELD( , )2.29O 1.29O
-0.025
0.025
5
-0.025
0.025
( X , Y )Y-FAN ,1.00 -1.00
RELATIVE FIELD( , )2.29O -1.29 O
X-FAN
Figure 15. Ray fan plots for two mirror spherical system with 250 mm focal length. Scale is mm.
39
Three Mirror System
The three mirror solution is illustrated in Figure 16. The solution is constructed
by placing a positive power mirror in front of a finite conjugate mirror doublet similar to
the Schwarzschild system. The degrees of freedom utilized are the spacing from the
aperture stop to the mirror, the curvatures of the mirrors, the spacings between the
mirrors, the back focal distance and the tilts and decenters of the mirrors relative to their
respective parent axes. A fold mirror is placed between the tertiary and image plane to
allow for clearance of the aperture stop in the z direction. This fold mirror has the effect
of decreasing a relatively large package size.
350.00 MM
Figure 16. Three mirror spherical system with infinite conjugate object (250 mm EFL). Fold mirror is introduced to allow clearance for the stop.
40
Figure 17. Ray fan plots for three mirror spherical system with 250 mm focal length. Scale is mm.
-0.0125
0.0125
1
-0.0125
0.0125
,0.00 0.00RELATIVE FIELD( , )0.00O 0.00O
-0.0125
0.0125
2
-0.0125
0.0125
,0.00 1.00RELATIVE FIELD( , )0.00O 1.29O
-0.0125
0.0125
3
-0.0125
0.0125
,0.00 -1.00RELATIVE FIELD( , )0.00O -1.29 O
-0.0125
0.0125
4
-0.0125
0.0125
,1.00 1.00RELATIVE FIELD( , )2.29O 1.29O
-0.0125
0.0125
5
-0.0125
0.0125
( X , Y )Y-FAN ,1.00 -1.00
RELATIVE FIELD( , )2.29O -1.29 O
X-FAN
41
All Conic Mirror Systems One Mirror System
The solution for the one mirror case differs from the spherical solution by a conic
constant. The conic constant is a degree of freedom utilized to improve performance
over the single mirror spherical case. The layout of the 250 mm focal length one mirror
configuration is shown in Figure 18. This configuration is folded in the same way that
the spherical one mirror is folded in order to prevent interference between surfaces upon
concatenation.
30.00 MM
Figure 18. One mirror conic system with infinite conjugate object (250 mm focal length).
42
Figure 19. Ray fan plots for one mirror conic system with 250 mm focal length. Scale is mm.
-0.25
0.25
1
-0.25
0.25
,0.00 0.00RELATIVE FIELD( , )0.00O 0.00O
-0.25
0.25
2
-0.25
0.25
,0.00 1.00RELATIVE FIELD( , )0.00O 1.29O
-0.25
0.25
3
-0.25
0.25
,0.00 -1.00RELATIVE FIELD( , )0.00O -1.29O
-0.25
0.25
4
-0.25
0.25
,1.00 1.00RELATIVE FIELD( , )2.29O 1.29O
-0.25
0.25
5
-0.25
0.25
( X , Y )Y-FAN ,1.00 -1.00
RELATIVE FIELD( , )2.29O -1.29O
X-FAN
43
Two Mirror System
The two mirror conic solution is an off axis section of a Gregorian telescope. The
Gregorian telescope is a two conic mirror system with an intermediate focus.
Decentering the aperture in a Gregorian telescope yields an un-obscured conic two mirror
solution13, shown in Figure 20. Unlike the two mirror spherical system, this
configuration has an accessible front focal plane, so it can be made telecentric without the
aid of auxiliary optics. The degrees of freedom employed in the optimization are the
distance from the aperture stop to the primary mirror, the mirror curvatures, the mirror
conic constants, the spacing between the mirrors, the back focal distance and the tilts and
decenters of the mirrors relative to their respective parent axes.
100.00 MM
Figure 20. Off axis section of a Gregorian telescope with infinite conjugate object (250 mm focal length).
44
Figure 21. Folded view of the Gregorian telescope with infinite conjugate object (250 mm focal length).
45
-0.025
0.025
1
-0.025
0.025
,0.00 0.00RELATIVE FIELD( , )0.00O 0.00O
-0.025
0.025
2
-0.025
0.025
,0.00 1.00RELATIVE FIELD( , )0.00O 1.29O
-0.025
0.025
3
-0.025
0.025
,0.00 -1.00RELATIVE FIELD( , )0.00O -1.29O
-0.025
0.025
4
-0.025
0.025
,1.00 1.00RELATIVE FIELD( , )2.29O 1.29O
-0.025
0.025
5
-0.025
0.025
( X , Y )Y-FAN ,1.00 -1.00
RELATIVE FIELD( , )2.29O -1.29O
X-FAN
Figure 22. Ray fan plots for two mirror conic system with 250 mm focal length. Scale is mm.
46
Three Mirror System
The three mirror conic solution is a Three Mirror Anastigmat (TMA)16, illustrated
in Figure 23, an often used configuration due to its superior performance and flexibility
to constraints. The degrees of freedom employed in the optimization are the distance
from the aperture stop to the primary mirror, the mirror curvatures, the mirror conic
constants, the spacing between the mirrors, the back focal distance and the tilts and
decenters of the mirrors relative to their respective parent axes.
150.00 MM
Figure 23. Three mirror Anastigmat, all conic surfaces.
This design form has excellent performance and size relative to all of the previous
forms. The constraint of telecentricity drives the package size larger by requiring a large
spacing between the aperture stop and the primary mirror, however it should be noted
that even with this large spacing, the TMA has a significantly smaller size than the three
mirror spherical design without the aid of a fold mirror.
47
-0.0125
0.0125
1
-0.0125
0.0125
,0.00 0.00RELATIVE FIELD( , )0.00O 0.00O
-0.0125
0.0125
2
-0.0125
0.0125
,0.00 1.00RELATIVE FIELD( , )0.00O 1.29O
-0.0125
0.0125
3
-0.0125
0.0125
,0.00 -1.00RELATIVE FIELD( , )0.00O -1.29 O
-0.0125
0.0125
4
-0.0125
0.0125
,1.00 1.00RELATIVE FIELD( , )2.29O 1.29O
-0.0125
0.0125
5
-0.0125
0.0125
( X , Y )Y-FAN ,1.00 -1.00
RELATIVE FIELD( , )2.29O -1.29 O
X-FAN
Figure 24. Ray fan plots for three mirror conic system (TMA) with 250 mm focal length. Scale is mm.
48
Imaging Lens Performance Comparison Summary
The designs presented can be compared directly using the composite RMS spot
size for the field points described at the beginning of the chapter. Composite RMS spot
size is mathematically defined as ∑
i
inx where x is the RMS spot size of the ith field
and n is the number of fields. Figure 25 shows the comparison of this metric for all the
systems.
Figure 25. Composite RMS spot size comparison.
As expected, performance improves with number of mirrors. Also of note is the
fact that adding a third spherical mirror yields a telecentric system with an accessible
aperture stop, while adding a third conic mirror yields better performance and stop
Composite RMS Spot Size (microns) vs Number of Mirrors
0
20
40
60
80
100
120
140
160
1 2 3
Spherical 250 mm focal lengthSpherical 500 mm focal lengthSpherical 1250 mm focal lengthConic 250 mm focal lengthConic 500 mm focal lengthConic 1250 mm focal length
49
accessibility without the use of a fold mirror. The advantage of the conic surfaces is not
quite as apparent, though. In the one mirror case, the 250 and 500 mm focal length
mirrors do receive a performance increase from the conic, however the 1250 mm focal
length mirror does not. This is largely due to the fact that the value of the conic constant
after optimization is close to zero in the 1250 mm case. The aperture decenter can
actually be varied to make the conic constant zero. In the case of the two mirror systems,
there is not a substantial difference in performance between the spherical and conic
systems. Recall, however, that the two mirror spherical systems may require a pupil
relay lens in order to achieve stop accessibility. The accessibility of the pupil with only a
fold mirror is what the conics buy in this case. For the three mirror systems, the size of
the three mirror spherical system was allowed to increase so the performance of the
spherical and conic three mirror systems would be approximately equivalent. So, here
the conic buys a substantial decrease in package size, as illustrated by Figures 16 and 23.
If the spherical three mirror system was constrained in package size, the TMA would
show superior performance.
These systems have been designed to be used to show the efficacy of the
concatenation process. If the concatenation process proposed and the aberration theory
are correct, then the concatenation of these systems should prove their validity. The 250
mm focal length solutions can be used to generate the necessary collimator mirror
systems, and the 250, 500 and 1250 mm focal length mirror systems can be used as
imagers to generate concatenated systems with magnifications of 1X, 2X and 5X. Next
the process of reversal to create a collimator lens is briefly discussed.
50
Collimator Mirror systems
In the preceding section, a set of infinite conjugate imagers was presented. Those
mirror systems can be used as collimator mirror systems if their path is reversed. This
section discusses the reversal of such a lens to generate a collimator lens.
Imaging Mirror System Reversal
The path of an infinite conjugate object imaging lens can be reversed to create a
collimator lens, such as the one illustrated in Figure 27. The process of creating such a
lens is primarily comprised of rotating the imager about the y axis. The remainder of the
task is to ensure that the source is in the correct position to create a collimated beam
which is parallel to the global z axis. During optimization of the infinite conjugate mirror
systems, the decenter of the image plane is allowed to vary in order to keep the OAR
centered on the image plane. When the system is rotated around the y axis to create a
collimator, the object point from which the optical axis ray originates in the collimator
lens must be specified with an equivalent decenter. This is illustrated in Figure 26. The
collimators used in this study are created from the 250 mm focal length imagers.
51
Figure 26. Illustration of image/object decenter. Green line is axis of symmetry of parent system. The OAR is decentered relative to this axis.
Figure 27. Example of a collimator mirror system.
125.00 MM
52
Pupil Aberrations and Telecentricity
Waves
-3.759
0.5213
-1.619
WAVEFRONT ABERRATION
Field = ( 0.000, 0.000) DegreesWavelength = 587.6 nmDefocusing = 0.000000 DIOPTERS
Figure 28. Wave front aberration map for pupil reimaging.
The wave front plot in figure 28 corresponds to the on axis pupil imaging of the
250 mm focal length three mirror spherical system and shows both coma and astigmatism
for the pupil imaging on axis. Figure 29 is the telecentricity plot for the same system. It
shows both anamorphism and quadratic deviation, validating the pupil aberration theory
presented. The difference in the lengths of the vectors at the (0,1) and (1,0) normalized
fields shows anamorphic telecentricity. The difference in the x location of the tips of the
vectors as a function of y field and the bowing of the pattern show the quadratic
components. The quadratic components are small, but they are present. This follows
from the fact that astigmatism dominates the wavefront, and the residual coma is small.
53
0.0019
-12-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
X Image Plane Location - mm
YIm
age
Plane
Loca
tion
-mm
TELECENTRICITY ERRORVS FOCAL PLANE POSITION
0.0019
-12-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
X Image Plane Location - mm
YIm
age
Plane
Loca
tion
-mm
TELECENTRICITY ERRORVS FOCAL PLANE POSITION
Figure 29. Telecentricity error for TMA showing an anamorphic and a small quadratic component. Scale is
radians.
54
Chapter 3 IMAGING RELAY MIRROR SYSTEMS
This chapter explores the concatenation of the collimator and imager mirror
systems developed in the previous chapters. First, layouts with ray traces for a sampling
of systems are shown. This is followed by a validation of the aberration theory presented
in chapter 2. Finally, a comparison of the designs will be presented based on
performance and packaging.
As previously mentioned, the relay mirror systems will range in complexity from
2 to 6 mirrors with surface types of spherical and conic and magnifications of 1, 2 and 5.
In order to organize the prescriptions and results, the following alphanumeric sequence
will be used as a naming convention: the number of mirrors in the collimator lens, the
number of mirrors in the imager lens, S for spherical or C for conic, the magnification
and a designator for the orientation of concatenation (H for the orientation which yields
symmetry about the stop, R for the orientation which yields symmetry about the center).
For example a 2X 5 mirror conic system with 2 mirrors in the collimator and 3 mirrors in
the imager with symmetry about the stop will be system 23C2XR.
The collimator mirror systems will have a 250 mm focal length in all cases. All
systems will be analyzed using an object size of 20mm width by 11.25 mm height (this
yields a 16:9 aspect ratio). A variety of systems is illustrated in Figures 30-33.
55
550.00 MM
Figure 30. System 22C5XH.
250.00 MM
Figure 31. System 22S1XR. Two afocal pupil relays are introduced to ensure an accessible aperture stop.
56
400.00 MM
Figure 32. System 23S1XR. An example where the Schwarzschild pupil relay is not necessary.
400.00 MM
Figure 33. System 33C2XR
57
Validation of Aberration Cancellation Upon Concatenation
As described in Chapter 2, certain aberrations will cancel upon concatenation.
An analysis of some 1X systems demonstrates the cancellation.
For the case of symmetry about the stop, aberrations which are dependent on an
odd order of H cancel. These are field tilt, linear astigmatism, linear coma, cubic piston
and cubic distortion. The cancellation of field tilt is demonstrated by comparing the
angle of the OAR with respect to the image plane and object plane of the imager before
and after concatenation. For the case prior to concatenation, we can look at the imager.
Since the collimator is simply a reversal of the imager lens the OAR will have an equal
magnitude obliquity. The three mirror spherical system is chosen since the imager lens
has a relatively large OAR obliquity. Before concatenation the angle of incidence of the
imager’s OAR is 11.6 degrees. The obliquity is illustrated in Figure 34.
Figure 34. Illustration of the OAR image plane obliquity (image tilt) for the three mirror spherical system.
425.00 MM
58
When the collimator and imager mirror systems are integrated, the tilts of the object and
image plane are allowed to vary. A cancellation of the image plane tilt is observed in the
1X system, as illustrated in Figure 35.
Figure 35. Illustration of the cancellation of field tilt upon concatenation in the orientation yielding
symmetry about the stop.
To demonstrate the cancellation of linear astigmatism and linear coma
System11C1XR, the 1X system with a 1 mirror conic collimator and imager is employed.
A full field plot of each aberration illustrates the cancellation. The astigmatism plot
shows the magnitude and direction of the astigmatic spot. Linear astigmatism is the
rotation of the lines vs. the H vector. Figure 36 shows the magnitude and direction of the
astigmatism for the 250 mm focal length single mirror conic.
650.00 MM
59
Figure 36. Plot of full field astigmatism for the one mirror conic system showing components of constant
and linear astigmatism.
17-Sep-05
ASTIGMATIC LINE IMAGEvs
FIELD ANGLE IN OBJECT SPACE
2.2mm
-3 -2 -1 0 1 2 3
X Field Angle in Object Space - degrees
-3
-2
-1
0
1
2
3Y
Fiel
dAn
gle
inOb
ject
Spac
e-
degr
ees
60
Figure 37. Plot of full field astigmatism after concatenation of system 11C1XR.
17-Sep-05
ASTIGMATIC LINE IMAGEvs
OBJECT HEIGHT
0.81mm
-15 -10 -5 0 5 10 15
X Object Height - mm
-15
-10
-5
0
5
10
15Y
Obje
ctHe
ight
-mm
61
17-Sep-05
FRINGE ZERNIKE PAIR Z7 AND Z8vs
FIELD ANGLE IN OBJECT SPACE
7.8waves
-3 -2 -1 0 1 2 3
X Field Angle in Object Space - degrees
-3
-2
-1
0
1
2
3Y
Fiel
dAn
gle
inOb
ject
Spac
e-
degr
ees
Figure 38. Plot of full field coma for the one mirror conic imager showing components of constant and linear coma.
62
Figure 39. Plot of full field coma after concatenation of system 11C1XR.
17-Sep-05
FRINGE ZERNIKE PAIR Z7 AND Z8vs
OBJECT HEIGHT
5.3waves
-15 -10 -5 0 5 10 15
X Object Height - mm
-15
-10
-5
0
5
10
15Y
Obje
ctHe
ight
-mm
63
In Figure 37, there is a residual constant astigmatism, but the linear astigmatism
shown in the previous figure has cancelled. In the case of coma, a full field map of the
combination of the coefficients of the fringe Zernike polynomials 7 and 8 show the
magnitude and orientation of coma over the field of view (Figure 38). For the single
mirror imager the field suffers from both linear and constant coma. After concatenation
that the linear term does in fact cancel (Figure 39), since the residual error is constant
coma.
This shows that the aberrations mentioned do in fact cancel at 1X magnification.
It remains to verify the dependence of the residual on magnification. To do this we look
at linear coma and assume that other aberrations follow accordingly. The claim is that
upon concatenation linear coma cancels according to COLLnRELAY WMW
±= −1||11 . We
verify this by using the data from the full field plots and assuming that if constant coma is
subtracted from the field, the remaining aberration is linear coma (i.e. we neglect higher
order effects). So the magnitude of W03001 is the value of coma at the center of the field.
The linear coma coefficient W13100 is calculated by determining the H value of the node
in the coma field and dividing W03001 by it. For ease of computation, the normalization
value of H will be x=10 mm for the relay. For consistency, the normalization value for
the imagers will be 2.29 degrees. This calculation is done for the infinite conjugate
single mirrors of all three focal lengths and for the corresponding 2X and 5X cases. The
results are tabulated in Tables 6 and 7.
64
Table 6. W13100 for the One Mirror Imagers focal length (mm) W03001 H W13100
250 0.133 0.310 -0.43 500 0.070 0.852 -0.08 1250 0.015 1.97 -0.01
Table 7. W13100 for the 2 mirror conic relays. Magnification W03001 H W13100 (1-1/M2)Wcoll
2X 0.065 .195 -0.33 -0.32 5X 0.119 .288 -0.41 -0.41
We see from tables 6 and 7 that the linear coma does in fact cancel and that the
dependence on magnification follows equation 2.
For the orientation yielding symmetry about the center, linear and constant coma,
and quadratic and cubic distortion cancel. System 11C1XH is analyzed. Figure 41
illustrates the residual coma after concatenation. Constant and linear coma cancel and a
small higher order residual remains.
50.00 MM
Figure 40. System 11C1XH.
65
New lens from CVMACRO:cvnewlens.seq
18-Sep-05
FRINGE ZERNIKE PAIR Z7 AND Z8vs
OBJECT HEIGHT
0.16waves
-15 -10 -5 0 5 10 15
X Object Height - mm
-15
-10
-5
0
5
10
15
YObject
Height
-mm
Figure 41. Coma residual for system 11C1XH.
66
18-Sep-05
IMAGE DISTORTIONvs
FIELD ANGLE IN OBJECT SPACE
0.58mm
-3 -2 -1 0 1 2 3
X Field Angle in Object Space - degrees
-3
-2
-1
0
1
2
3YField
Anglein
Object
Space-degr
ees
Figure 42. Distortion plot for one mirror conic.
67
Figure 43. Distortion plot for system 11C1XH after concatenation.
19-Sep-05
0.25
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
IMAGE DISTORTIONvs
OBJECT HEIGHT
X Object Height
YOb
ject
Heig
ht-mm
68
The cancellation of distortion is shown in Figures 42 and 43 above. There is quadratic
distortion of type I and II in the imager and the collimator which are the bowing and the
keystone like effects in Figure 42. The vector plot is a measure of the shift in the real
chief ray from the paraxial position. The tips of the vectors form the distortion pattern.
After concatenation we see that the residual distortion is only anamorphism. Note that
this is equivalent to the cancellation of the pupil aberration constant coma.
Considerations for the Point Spread Function
Thus far the discussion has involved the geometric aspects of the concatenation of
the relays. For completeness, a brief discussion of an issue related to the physical optics
in a system like the ones chosen is in order20.
Figure 44 shows two off axis parabolas (OAP) concatenated in the orientation
yielding symmetry about the center point. A parabola is a not an isoplanatic imager, so
the angular spacing of rays in converging space is non-uniform for a uniform grid of rays
in the collimated space. Since the systems considered here may behave similarly, we
show the effects of the off axis distance on the point spread function (PSF) of this system.
This is not analyzed in detail for any of the systems in this study. Rather, it is shown as a
consideration for these types of systems. The purpose is to show this effect as a function
of the OAR angle of incidence on the mirrors for an f/1 beam. This effect is negligible
when the systems are symmetric about the stop.
69
50.00 MM
Figure 44. Two OAPs used to illustrate an issue with the PSF.
(a) (b)
Figure 45. PSF for the on axis case (angle of incidence equals 0 degrees) (a) and the case for which the sum of the AOI and AOR is 90 degrees (b).
As a function of angle of incidence, we see a smearing of the point spread
function in the direction of aperture decenter. The PSFs are plotted for 15 degree
increments of the sum of the angle of incidence (AOI) and angle of reflection (AOR) of
the OAR on a log scale in Figure 46.
0.001016 mm
25
0.001016 mm
25
70
Figure 46. Log scale PSF at (a) 15 (b) 30 (c) 45 (d) 60 (e) 75 and (f) 90 degree sum of OAR AOI and AOR
on the mirrors. Size of each box is 6.3 microns on a side.
dB
-50.00
0.0000
-25.00
(a)
(c)
(e)
(b)
(d)
(f)
71
Performance comparisons
For brevity of presentation, the composite spot size of the six field points
described in the previous chapter will be the metric for relative performance comparison.
Also compared will be package volumes corresponding to the smallest box which will fit
around each relay lens, as well as the obliquity of the OAR relative to the object plane vs.
magnification in the case of a system symmetric about the stop. Recall that the
orientation of concatenation only minimizes this, and for magnifications other than unity
it is non-zero.
The comparisons of performance are shown in Figures 47-50.
Composite RMS Spot Size (microns) vs Magnification
0
100
200
300
400
500
600
700
800
1 2 3 4 5
111213212223313233
Figure 47. Spot Size comparison for Spherical Relays, symmetry about the stop.
72
Composite RMS Spot Size (microns) vs Magnification
0
100
200
300
400
500
600
700
1 2 3 4 5
111213212223313233
Figure 48. Spot Size comparison for Conic Relays, symmetry about the stop.
Composite RMS Spot Size (microns) vs Magnification
0
100
200
300
400
500
600
700
800
1 2 3 4 5
111213212223313233
Figure 49. Spot Size comparison for Spherical Relays, symmetry about the center.
73
Composite RMS Spot Size (microns) vs Magnification
0
100
200
300
400
500
600
700
1 2 3 4 5
111213212223313233
Figure 50. Spot Size comparison for Conic Relays, symmetry about the center.
Source Tilt vs Magnification
-15
-10
-5
0
5
10
15
20
1 2 3 4 5
Magnification
Sour
ceTil
t(de
gree
s)
111213212223313233
Figure 51. Source tilt vs. magnification, symmetry about the stop.
74
When the collimator and imager mirror systems are integrated, the angle of
incidence on the image plane is constrained to be zero. For the case of the systems
symmetrical about the stop, Figure 51 shows the resulting tilt of the object plane relative
to the OAR. This plot shows that it is possible to cancel field tilt for magnifications other
than one with proper choice of input systems.
Table 8. Approximate Packaging Dimensions for 1X Spherical Systems with symmetry about the stop.
System x (mm) y (mm) z (mm) Volume (m^3)
11 34.94 46.18 501.60 8.09E-04 12 103.44 355.83 1053.70 3.88E-02 13 398.47 1277.36 3409.13 1.74E+00 21 103.57 355.76 1053.92 3.88E-02 22 115.70 355.42 2385.33 9.81E-02 23 406.54 1278.43 3661.11 1.90E+00 31 406.68 1277.42 3409.13 1.77E+00 32 406.57 1277.82 3660.88 1.90E+00 33 406.60 1277.86 6316.66 3.28E+00
Table 9. Approximate Packaging Dimensions for 1X Conic Systems with symmetry about the stop.
System x (mm) y (mm) z (mm) Volume (m^3)
11 34.88 56.03 496.22 9.70E-04 12 99.86 310.88 1000.00 3.10E-02 13 64.26 288.95 1588.97 2.95E-02 21 99.73 310.70 1000.37 3.10E-02 22 1000.00 310.91 594.80 1.85E-01 23 99.90 311.12 1700.93 5.29E-02 31 72.64 288.63 1588.97 3.33E-02 32 99.80 310.98 1700.56 5.28E-02 33 72.50 289.05 2681.73 5.62E-02
Tables 4 and 5 primarily show that the three mirror conic solution creates much
smaller packages than its spherical counterpart. The other solutions have comparable
package sizes.
75
Chapter 4 CONCLUSION
A process to design a relay lens utilizing the concatenation of an imager and
collimator lens has been presented and demonstrated. The aberration theory involved in
the integration of the components of the relay has been validated and a connection
between pupil aberrations and telecentricity error in a bilaterally symmetric optical
system has been made as well as the connection between aberrations of the system
imagery and aberration of the pupil imagery. The process yields a method of developing
relay lens solutions utilizing infinite conjugate solutions as inputs.
The lens design examples presented validate the aberration theory discussed, and
provided insight into the required complexity of a relay to yield a particular performance
level or to meet particular constraints. In particular the six mirror conic solution showed
near diffraction limited performance in a significantly smaller package than the
equivalent spherical system. Placing a packaging requirement on such a system makes
the choice of solution clear.
We can conclude from this study that viable relay solutions can be obtained by the
process described. The resulting output is in fact predictable based on the inputs. The
possibilities of solutions are limited only by the inputs and constraints imposed on the
resulting relay.
Future work on this topic should include the extension of this process to include
afocal relay systems. These systems exist for the specific case that the object and image
are located at infinity. In these systems the pupils and image planes reverse roles, so the
76
concatenation takes place at an image plane rather than at a pupil. A brief introduction to
these types of systems was contained in the use of the pupil reimaging lens. Extending
this process to include such systems will only augment the value in this process.
77
APPENDIX A Pupil Relay
The pupil relay used to gain access to the Schwarzschild collimator pupil must be
briefly discussed. The design used is a reoptimization of a design by Cook18. The result
has smaller field but a larger stop diameter than the patented form to conform to the
parameters of the relays in this study. It is an extremely compact design with excellent
wave front and pupil reimaging performance.
80.00 MM
Figure A1. Afocal pupil reimager layout.
The aperture stop is 50 mm and the half field of the collimated beam is 2.29
degrees by 1.29 degrees to match the field of view of the mirror systems described in
chapter 2. The performance parameters of interest are the wavefront error induced by the
78
afocal and the quality of the reimaging of the aperture stop. These performance metrics
are shown in Figures 57 and 58.
The technique employed to simultaneously optimize the collimated beam and the
pupil reimaging quality is the creation of effective point sources on the stop sources using
rays from different field points. Each field has a ray which passes through a particular
point on the stop. If an assortment of rays from several field points which pass through
this location is used a point source is effectively created with a numerical aperture
equivalent to the sine of the field of view. Constraining these rays to converge to the
corresponding point on the image of the stop allows for simultaneous optimization of the
collimated wave front and the pupil reimaging.
79
Figure A2. Wave front error plots for afocal pupil reimager.
24-Sep-05
New lens from CVMACRO:cvnewlens.seq
OPTICAL PATH DIFFERENCE (WAVES)587.5618 NM
-0.5
0.5
1
-0.5
0.5
,0.00 0.00RELATIVE FIELD( , )0.00O 0.00O
-0.5
0.5
2
-0.5
0.5
,0.00 1.00RELATIVE FIELD( , )0.00O 1.29O
-0.5
0.5
3
-0.5
0.5
,0.00 -1.00RELATIVE FIELD( , )0.00O -1.29 O
-0.5
0.5
4
-0.5
0.5
,1.00 1.00RELATIVE FIELD( , )2.29O 1.29O
-0.5
0.5
5
-0.5
0.5
( X , Y )Y-FAN ,1.00 -1.00
RELATIVE FIELD( , )2.29O -1.29 O
X-FAN
80
Figure A3. Ray intercept plots for pupil reimaging. Scale is mm.
-0.2
0.2
1
-0.2
0.2
,0.00 0.00RELATIVE FIELD( , )0.00O 0.00O
-0.2
0.2
2
-0.2
0.2
,0.00 1.00RELATIVE FIELD( , )0.00O 0.00O
-0.2
0.2
3
-0.2
0.2
,0.00 -1.00RELATIVE FIELD( , )0.00O -0.00O
-0.2
0.2
4
-0.2
0.2
,1.00 0.71RELATIVE FIELD( , )-0.00 O -0.00O
-0.2
0.2
5
-0.2
0.2
( X , Y )Y-FAN ,1.00 -0.71
RELATIVE FIELD( , )-0.00 O -0.00O
X-FAN
81
APPENDIX B
The parameters and identities used to derive the relations between pupil
aberrations and aberrations of imagery are presented here for reference.
( ) xnuInJ I
∆−= 22 sin21 (B1)
( ) xnuAInJ II
∆−= sin21 (B2)
( ) xnuInJ III
Ψ∆−= sin (B3)
( ) xnRInJ IV
Ψ∆−= 1sin
21 (B4)
( ) xnInJV11sin2
12
2
∆Ψ−= (B5)
niA = (B6)
xnuxunxAxA −=−=Ψ (B7)
{ }∑=
=j
iiIJW
102002 Constant Astigmatism (B8)
∑=
=
j
i iIJx
xW1
11011 2 Anamorphism (B9)
∑=
=
j
i iIJx
xW1
2
20020 Quadratic Piston (B10)
{ }∑=
=j
iiIIJW
103001 Constant Coma (B11)
∑=
+
=
j
i iIIIII JJx
xW1
12101 2 Linear Astigmatism (B12)
82
∑=
+
=
j
i iIVII JJx
xW1
12010 Field Tilt (B13)
i
j
iVIIIII JJx
xJxxW ∑
=
++
=
1
2
21001 Quadratic Distortion I (B14)
( )i
j
iIVIIIII JJx
xJxxW ∑
=
++
=
1
2
21110 22 Quadratic Distortion II (B15)
( )∑=
++
+
=
j
i iVIVIIIII Jx
xJJxxJx
xW1
23
30010 (B16)
( ) xnuAInx
xJxxJx
xJJ IIIVIIIV
∆−=
+
+
+ sin212
(B17)
( ) xnuAInx
xJxxJ IIIII
∆−=
+
sin2
2(B18)
022 =+=
+
IVIIIIVIII JJx
xJxxJ (B19)
Ψ∆+
∆=
∆ 21nxn
uxnu (B20)
83
REFERENCES
1. Abe Offner, “Unit Power Imaging Catoptric Anastigmat”, U.S. Patent 3,748,015 (1973). 2. F. Bociort, M.F. Bal and J.J.M. Braat, “Systematic analysis of unobscured mirror systems for microlithography”. 3. Joseph M. Howard and Bryan D. Stone, “Imaging a point with two spherical mirrors”. 4. Joseph M. Howard and Bryan D. Stone, “Imaging with Three Spherical Mirrors” Applied Optics, Vol. 39 Issue 19 Page 3216 (2000). 5. Joseph M. Howard and Bryan D. Stone, “Imaging with Four Spherical Mirrors” Applied Optics, Vol. 39 Issue 19 Page 3232 (2000). 6. José M. Sasian “Imagery of the Bilateral Symmetrical Optical System” Ph. D. Dissertation (1988). 7. Kevin P. Thompson, “Aberration fields in tilted and decentered optical systems” Ph.D. Dissertation (1980). 8. John Rogers, “Aberrations of Unobscured Reflective Optical Systems,” Ph.D. Dissertation (1983). 9. José M. Sasian, “How to approach the design of a bilateral symmetric optical system” Optical Engineering Vol. 33 No. 6 Pages 2045-2061 (1994). 10. John Rogers, “Vector aberration theory and the design of off axis systems” SPIE Vol. 554 IODC(1985). 11. Andrew Rakich, “Four families of flat-field three-mirror anastigmatic telescopes with only one mirror aspherized” Proc. SPIE Int. Soc. Opt. Eng. 4768, 32 (2002) . 12. Andrew Rakich and Norman Rumsey “Method for deriving the complete solution set for three-mirror anastigmatic telescopes with two spherical mirrors” , JOSA A, Vol. 19 Issue 7 Page 1398 (2002). 13. J. Michael Rodgers, “Un-obscured mirror designs” Proc. SPIE Int. Soc. Opt. Eng. 4832, 33 (2002). 14. Rudolph Kingslake, Lens Design Fundamentals, Academic Press, New York (1978).
84
15. Dietrich Korsch, “Reflective Optics”, Academic Press Inc., San Diego (1991). 16. Lacy Cook, “Three Mirror Anastigmatic Optical System”, U.S. Patent 4,265,510 (1979). 17. Robert Shannon, “The Art and Science of Optical Design”, Cambridge University Press, New York (1997). 18. Lacy Cook, “Compact Afocal Reimaging and Image Derotation Device,” U.S. Patent 5,078,502 (1992). 19. CodeV is a product of Optical Research Associates. 20. Pantazis Mouroulis, “Optical Design and Engineering: Lessons Learned”, Proc. SPIE Vol. 5865 (2005) 21. R.S Longhurst, “A Note on the Calculation of Principal Ray Aberration”, Proc. Phys. Soc. B 65 116-117 (1952) 22. C.G. Wynne, “Primary Aberrations and Conjugate Change”, Proc. Phys. Soc. LXV (1952) 23. The identities used can be found in the following paper: José M. Sasián, “Aberrations from a prism and a grating”, Applied Optics Volume 39, No. 1 (2000) 24. A detailed explanation of center of symmetry can be found on page 6 in: Elizabeth Wood, “Crystals and Light, and Introduction to Optical Crystallography”, D. Van Nostrand Company, Princeton, NJ (1964).