LENS DESIGN APPROACH TO OPTICAL RELAYS By...

Lens Design Approach to Optical Relays

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Authors OShea, Kevin

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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

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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.

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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


-0.25

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


-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


-0.025

0.025

2

-0.025

0.025


-0.025

0.025

3

-0.025

0.025


-0.025

0.025

4

-0.025

0.025


-0.025

0.025

5

-0.025

0.025

( X , Y )Y-FAN ,1.00 -1.00


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


-0.0125

0.0125

2

-0.0125

0.0125


-0.0125

0.0125

3

-0.0125

0.0125


-0.0125

0.0125

4

-0.0125

0.0125


-0.0125

0.0125

5

-0.0125

0.0125

( X , Y )Y-FAN ,1.00 -1.00


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


-0.25

0.25

2

-0.25

0.25


-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


-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


-0.025

0.025

2

-0.025

0.025


-0.025

0.025

3

-0.025

0.025


-0.025

0.025

4

-0.025

0.025


-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


-0.0125

0.0125

2

-0.0125

0.0125


-0.0125

0.0125

3

-0.0125

0.0125


-0.0125

0.0125

4

-0.0125

0.0125


-0.0125

0.0125

5

-0.0125

0.0125

( X , Y )Y-FAN ,1.00 -1.00


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


7.8waves

-3 -2 -1 0 1 2 3


-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


OBJECT HEIGHT

5.3waves

-15 -10 -5 0 5 10 15


-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


OBJECT HEIGHT

0.16waves

-15 -10 -5 0 5 10 15


-15

-10

-5

0

5

10

15

YObject

Height

-mm

Figure 41. Coma residual for system 11C1XH.

66

18-Sep-05

IMAGE DISTORTIONvs


0.58mm

-3 -2 -1 0 1 2 3


-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


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.


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


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


-0.5

0.5

2

-0.5

0.5


-0.5

0.5

3

-0.5

0.5


-0.5

0.5

4

-0.5

0.5


-0.5

0.5

5

-0.5

0.5

( X , Y )Y-FAN ,1.00 -1.00


X-FAN

80

Figure A3. Ray intercept plots for pupil reimaging. Scale is mm.

-0.2

0.2

1

-0.2

0.2


-0.2

0.2

2

-0.2

0.2


-0.2

0.2

3

-0.2

0.2


-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).

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