)l_ 3-Lb NASA-£R-191270

LfS

Polarization Analysis of a

Balloon-Borne Solar Magnetograph

Final Report for

NASA contract NAS8-38609 D.O. 01

(NASA-CR-191270) POLARIZATION

ANALYSIS OF A BALLOON-BORNE SOLAR

MAGNETOGRAPH Final Report (Alabama

Univ.) 43 p

Daniel J. Reiley

Russell A. Chipman (PI)

Physics Department

School of Science

University of Alabama in Huntsville

Huntsville, AL 35899

(205) 895 -6417x318

N93-16458

Unclas

G3/35 013169_
Table of Contents

Body of the report

The main text of the report contains the particular results of our

research which relate directly to the EXVM (Experimental Vector

Magnetograph) and the (Balloon-borne Vector Magnetograph) BVM.

I. Polarization 2

A brief overview of which elements in the EXVM and BVM are

relevant to this polarization analysis.

II. Polarization Specification 2

A qualitative discussion of the possible meaning of the 1O- s

polarization specification for the BVM. A recommendation of which

polarization specification is most relevant for the BVM. A diattenuation

budget for the various surfaces in the BVM which will allow the

polarization specification to be met.

III. Specifying the Coatings 5

An explanation of the various coating specifications which are

recommended.

IV. Optical Design of the EXVM 8

V. Coating specification sheets for the BVM 22

Appendices

The appendices of this report contain the more general results of

our research on the general topic of polarization aberrations.

Appendix I. 27

A general discussion of polarization aberration theory, in terms of

the SAMEX solar magnetograph,

Appendix II. 28

Rigorous derivations for the Mueller matrices of optical systems.
I. Polarization

The mirrors and lensesin optical systemscanchangelight's polarization state. Since

solar magnetographsoperate by measuringthe polarization state of the solar disc,the

effect of the mirrors and lenseson the polarization state must bewell-characterized.

One way to characterizemirrors' and lensespolarization properties is to use

polarization aberration theory. Polarization aberration theory is applied to a previous solar

magnetographdesignin the paper "Polarization analysisof the SAMEX solar

magnetograph,"which is included in this report asAppendix I.

Sincethe polarimeter usedin this systemwill bea rotating-retarder polarimeter, the

polarization state exiting the polarimeter will be constant. Therefore, the only optical

elementswhich will affect the accuracyof the Solar magneticfield measurementswill be

the polarization that iscausedbethe optics in front of the polarimeter.

II. Polarization specification

The polarization for this systemwas specifiedto be below 10-s This specification

canhaveseveralmeaningsin terms of the polarization aberration coefficients that are

associatedwith the system.Theseerrors are derived in Appendix II, and are only

summarizedin this section.

We haveenumeratedthree polarization errors which are possiblein the solar

magnetograph:

Polarization Error # 1: The first polarization error is that unpolarized light can couple into

polarized light. The first effect of this polarization error is that magnetic field is measured
when no magneticfield is present. The secondeffect of this polarization error is that the

orientation of linearly polarized light is measuredincorrectly, causingthe orientation of the

magneticfield to bemeasuredincorrectly. The symptomof this polarization error is a

spurious,radially oriented linear polarization with magnitude that increasesquadratically

with field coordinate. Mathematically, this polarization error is2 I Re (P 120o) I< 1 0- s

For the BVM with an articulating secondary mirror and the coatings specified in Section V,

the value for this polarization error is 2 I Re(P _2oo) I< .9@. 10 -s . Since this value

depends only on the chief ray, this value is the same for either the 60 cm telescope or the

30 cm. telescope.

Polarization Error #2: The second polarization error is that polarized light couples into

the wrong polarization state. The first effect of this polarization error is that the

orientation of linearly polarized light is measured incorrectly, causing the orientation of the

magnetic field to be measured incorrectly. The second effect of this polarization error is an

error in the measured degree of linear polarization, also causing the orientation of the

magnetic field to be measured incorrectly. The magnitude of this polarization error is

2 I P _oo I< 1 0-5. For the BVM with an articulating secondary mirror and the coatings

specified in Section V, the value for this polarization error is 2 LRe( P _2oo) I< - 1.8. 10 -'_

The value is approximate because the specifications in Section V are for diattenuation only.

Since this value depends only on the chief ray, this value is the same for either the 60 cm

telescope or the 30 cm. telescope.

Polarization Error #3: The third polarization error is that polarized light couples into

unpolarized light. The effect of this polarization error is that the magnitude of the

measured field is underestimated, raising the threshold of the measurable magnetic field.

The magnitude of this polarization error is represented by the magnitude of all the

polarization aberration coefficients. For the BVM with an articulating secondary mirror, a

30 cm. telescope, and the coatings specified in Section V, the value for this polarization
error is D < -3. 10 -a The value is approximate because the specifications in Section V

are for diattenuation along the chief ray only. This value is approximately quadratic in

f-number, so the value for a 60 cm. telescope is D < - 12. 10 - _

We believe that polarization error # 1 is the most relevant to this problem, The 1 0-5

polarization specification would then require that 2 R e P t2oo < 1 O- s.

The polarization aberration coefficients depend on the angles of incidence for the

chief and marginal rays as well as the coatings on the various surfaces. With a tilted

secondary mirror, the angles of incidence on the relevant surfaces are listed in Table 1.

prefilter

primary mirror

secondary mirror

front surface of lens

inner surface of lens

back surface of lens

Table 1

Angles of Incidence

(with secondary tilted .3 ° )

marginal ray chief ray

0 ° .16 °

3.6 ° .16 °

4.5 ° .3 °

2.3 ° 2.2 °

9.3 ° 4.7 °

0,5 o 3.2 °

Polarization tilt diattenuation is small even with no coatings on any of the surfaces,

Re P _2oo = 8.3. 10-5 The distribution of polarization diattenuation piston when the

surfaces are uncoated are listed in Table 2.
Surface

Table 2

Polarization Piston Diattenuation

all surfaces uncoated

(with secondary tilted .3 °

RO P l;_m)

Primary mirror

Secondary mirror

tilted .15"

front surface of lens

inner surface of lens

back surface of lens

1.6" 10 -s

.8' 10 -'_

6" 10 - 's

SUM 8.3" 10 -s

The negative sign on the coefficients in Table 2 for the mirrors indicates that the elements

are used in reflection rather than transmission. The .3" tilt on the secondary mirror

corresponds to a 10 arcminute change in the center of the field of view.

The values in Table 2 were calculated using the following equation:

.2 _2

Rp(Oc,,)-R_(Oc.,)=t p-l_.

=(1 -lip)2-( I -]1,,) 2

=2(_(0c,)-r,,(o_,))

= 2Ra(P 12oo),
where R_ and Rp are the s- and p- intensity reflection coefficients, r, and r,, are the sand p- amplitude reflection coefficients, q s and q _ are the small deviations of r, and r_

from one, and Oc., is the chief ray angle of incidence on surface _.

III. Specifying the coatings

The coatings should be specified so that the diattenuation along the chief ray path is

less than 1 0-5 We suggest the following d/attenuation budget listed in Table 3.

6
Prefilter

0<Rp-Rs <.1 10 -s

Primary mirror

.05 10 -s < Rp-R_

Secondary mirror

.05 lO -_ < Rp-R s

Doublet (front)

0 < Rp-R_ < .1 lO -s

Doublet (internal surface)

Rp-R_ = 1.510 -s (fixed)

Doublet (back)

0 ° <0 < .16 °

< .5 10 -s 0° <0 <.16 °

< .5 10 -u 0 ° <0 <.18 °

0 ° <0<.8 °

O<Rp-R, <.110 -5 00<0<3.3 °

This diattenuation budget will give a polarization aberration of 2 R e (P_2oo) < .9. 1 0-5.

We also suggest that the mirrors have high reflectivity, > 98% over the entire range of

relevant incident angles and apertures.

Several characteristics of this diattenuation budget may not be immediately obvious.

The first characteristic is that the diattenuation of the internal surface of the doublet is the

limiting factor. To balance the diattenuation of the internal surface of the doublet, we

place upper and lower bounds on the diattenuation of the telescope mirrors. Although the

specified differences in reflectivity are all of the same sign, the diattenuations balance

because the prefilter and the lens are used in transmission. The antirefiection coating on

the back side of the doublet will be the hardest to make because of the relatively large

angles of incidence on that surface.

Notice that no coatings are required if the polarimeter is placed in front of the first

doublet.

Since the required diattenuation values are close to the diattenuation values for the
uncoatedinterfaces, coatingvendorsshould havelittle problem designingeconomical

coatingswhich meet this specification. We choosenot to provide coating designswith this

report becausethe most economicalway to meet thesespecificationsvarieswidely from

vendor to vendor. Most vendorshavestockcoating designswhich will meet these

specifications,and will not chargefor coatingdesign.

A more seriousproblem than devisingcoatingdesignswhich meetthis polarization

specification is the problem of verifying that thesespecificationswere actually met. These

low polarization coatingswould require very preciseellipsometry to verify that the

polarization specificationsare met. One possibleway to accomplishthis is to take

advantageof the fact that diattenuation is quadratic in angleof incidence. This quadratic

behavioris what determines the polarization specificationfor the 45" angleof incidenceon

the coating specification sheetsin Appendix IV. Simpler, bessprecise, ellipsometric

techniquescanbe usedto find the relatively largediattenuation at the large angleof

incidence.

The coatingsshouldbespecifiedto degradethe surfacefigure by lessthan X/10

RMS over the entire clear aperture, to prevent degradation of image quality. Surface

quality should be specified to meet MIL-SPEC 60-40 scratch-dig specifications; this is a

typical surface quality specification for scientific-grade optics. Surface durability should be

specified to meet the MIL-SPEC scotch tape test and eraser test; this is a fairly stringent

requirement, which we feel is justified because this instrument will be used outdoors.

Although the polarization specifications for these coatings are for the real part of

polarization piston, we can use them to estimate what the other aberration terms would be,

and therefore estimate the magnitude of other polarization errors. First, since we know the

angles of incidence for the marginal ray, and since we have specified the diattenuation for

the chief ray, we can use the quadratic behavior of diattenuation to estimate the real parts

of polarization tilt, Re( ,° L___), and polarization defocus,/e(,( f" _o_:J) • Since, for many
coatings,the imaginary parts of the polarization aberration coefficients are closeto the real

parts,Re (P x_x_ ) _- l _ ( P x,,, ), we can estimate the other polarization effects.

One effect of polarization error #2 listed above is the coupling of circular

polarization states into linear ones. This coupling is represented by the M 43 element of

the matrix in eq. 2 of Appendix II.

M 4_ = 2h2 pooooq 12oo

For radially symmetric systems, this term is simply a quadratic term from the center of the

pupil to the edge of the pupil. For tilted and decentered systems, this quadratic variation is

moved from the center of the field to another point in the field. This coupling term is

plotted in Figures 3 & 4. For a magnetograph with a tilting secondary mirror and the

coatings outlined above, the magnitude of this error should be M 43 < .9 1 0-5. This value

was calculated by assuming the transmission of the system is close to one, P oooo _ 1, and

the real part of polarization piston is about equal to the imaginary part of polarization

piston, Ro( P 1o22) -z /I_( P Lo_:_).

Polarization error #3 can also be estimated. The depolarization of the system is

estimated in Appendix III to be

! 2 1 2 2

D= _P1022+_ ]-t PIIll.

Assuming a quadratic variation in diattenuation and retardance on each surface, and

assuming that the real and imaginary parts of the polarization aberrations are equal, the

magnitude of polarization piston will be

IPLlil I _ 1.4' 10 -'_,

and the magnitude of polarization defocus will be

[ Pl_ll I----29' I0 -s

The depolarization of the system will then be D _- 3. 10- 8 This is the maximum

depolarization at the edge of the field of view for a system with a tilted secondary mirror.
The following two pages describe the polarization errors graphically.

10
Diattenuation Vectors

In the image plane, the main polarization defect introduced by the coatings is the

diattenuation caused by the real part of P1200

Foa a centered system (no tilted secondary), the diattenuation vectors are radially

oriented in the image plane, with quadratically increasing amplitude..

For the specified coatings, the magnitude of the largest line shown here is .7E-5.

For a tilted secondary, the diattenuation vectors have the same pattern, but are not

centered in the center of the field. The exact location of the center depends on the

coatings and amouont of tilt, but this figure is reasonable.

For the specified coatings, the magnitude of the largest line shown here is 1E-5.

==Effect of retardance along the chief ray

In the image plane, the next most important effect of polarization aberrations is the
Nasa2

retardance caused by the imaginary part of P1200. These values will have the same

orientation and approximately the same values as the diattenuation plots shown above

• Effect of other polarization aberrations

In the image plane, the effect of the other polarization aberrations is depolarization.

For a radially symmetric system, this depolarization increases quadratically

from the center of the field of view.

-0.5

-I

-i -0.5 0 0.5 1

With a tilted secondary, this depolarization increases quadratically

from some other point in the field of view.

The plot shown here is rotated 90 degrees with respect to the diattenuation plots.

-i

-i -0.5 0 0.5 1

For the specified coatings, the maximum contour line shown here would be

less than 3E-8.

2
IV. Optical design of the EXVM

While working on the polarization analysis of the NASA-designed instrument, we

noticed that it could be fabricated much more economically if catalog lenses were used

instead of custom lenses. After discussing the possibility with Mona Hagyard, Alan Gary,

and Ed West of NASA, we decided to redesign the optical system to take advantage of

these lenses.

Catalog lenses are lenses which are made for common optical tasks such as

collimating and focussing laser beams. Companies such as Melles-Griot, Spindler &

Hoyer, JML, Newport, Edmund, and CVI have dedicated large amounts of capital to

produce large numbers of commonly-requested focal lengths and diameters. Because these

lenses are made on a regular basis, they are much less expensive than custom lenses of

similar quality.

lens.

Table 4 lists the optical prescription for this system. Each of the lenses is a catalog

Figure 1 is a schematic of the system.

Table 4

Optical prescription for the EXVM

RDY THI GLA

> OBJ: INHN1TY INFINITY

1: INFINITY 0.001000 BK7 SCHOTT

2: INFINITY 650.000000

STO: -2394.73740 -877.239800 REFL

K: -I.000000

4: -858.24060 1069.000000 REFL

K: -2.82538

5: INFINITY 7.000000 BK7 SCHOTT

6: INFINITY 5.000000

7: INFINITY 50.000000 BK7 SCHOTT

8: INFINITY 5.000000

9: INFINITY 6.000000 BK7 SCHOTT

10: INFINITY 5.000000

11: INFINITY 7.00(/000 BK7 SCHOTT

12: INFINITY 5.000000

13: INFINITY 3.000000 BK7 SCHOTT

14: INFINITY 315.0(0)010)

15: 718.39000 4.000000 SF8 SCHOTT

16: 92.73000 6.600000 SSK'4 SCHOTT

17: -128.08000 105.00000/)

Clear Aperture

306.653

306.653

304.8

83.9923

24.1539

23.8961

23.6162

21.7747

21.4949

21.2739

20.994

20.7362

20.4563

20.3458

27.4918

27.6017

27.8829

13
18: INFINITY 300.000000

19: INFINITY 33.500000

20: 210.75000 5.000000

21: -81.29000 4.400000

22: -515.63000 75.000000

23: INFINITY - 128.880000

ADE: 45

24: 141.25000 -4.800000

25: 47.31500 -3.000000

26: -61.74800 -15.0000(X)

27: INFINITY 0.010000

28: INFINITY -48.000000

29: INFINITY -4.000000

30: INFINITY -48.000000

31: INFINITY 0.010000

32: INFINITY -112.017000

33: - 188.36000 - 12.500000

34: 139.24000 -6.000000

35: 415.67000 -55.000000

36: INFINITY 165.233309

ADE: 45

37: INFINITY 4.000000

38: INFINITY 12.000000

39: INFINITY 21.000000

40: INFINITY 0.000275

41: INFINITY 18.000000

42: INFINITY 0.000000

43: INFINITY 21.000000

44: INFINITY 12.000000

45" INFINITY 4.000000

46: INFINITY 80.000000

47: 673.17000 6.000000

48: 222.27000 10.000000

49: -302.87000 439.000000

50: 121.71195 3.800000

51: -89.71796 2.500000

52: -268.15906 131.029506

53: 32.16000 4.460000

54: -22.47000 1.500000

55: -89.36000 34.925611

56: INFINITY - 120.845313

ADE: -15

57: - 182.72000 -2.000000

58: -44.88000 -4.800000

59: 64.04000 -72.000000

60: INFINITY 257.078523

ADE: 15

61: INFINITY -72.000000

ADE: 45

62: INFINITY 108.0(X1000

ADE: -45

IMG: INFINITY 0.000000

SPECIFICATION DATA

EPD 304.80000

DIM MM

WL 656.27

YAN 0.00000

632.80

0.05717

BK7 SCHO'I_F

BAK4 SCHOTT

F3 SCHOTT

REFL

SF5 SCHOTT

BKT SCHOTT

BK7 SCHOTT

BK7 SCHOTT

BK7 SCHOTT

BK7 SCHOTT

SF5 -SCHOTT

REFL

BK7 SCHOTT

BK7 SCHOTT

BK7 SCHOTT

BK7 SCHOTT

BK7 SCHOTT

SF5 SCHOTT

BKT SCHOTT

BK7 SCHOTT

SF5 -tdCHOTT

SKll SCHOTT

SF5 gCHOTT

REFL

SF5 SCHOTT

SKI_ SCHOTT

REFL

REFL

REFL

525.02

0.08167

20.9224

18.1876

20.4624

20.5015

20.5668

20.5172

20.4319

20.7112

20.7653

23.7989

23.7975

33.5O49

34.0372

43.7446

43.7432

66.3972

66.5534

66.8955

66.142

63.8781

63.842

63.6776

63.713

63.713

63.8744

63.8744

64.0626

64.2261

64.2619

65.3518

65.1645

65.1296

13.7199

13.2806

13.0547

17.9977

17.4991

17.4214

9.74839

16.8009

16.9879

17.6003

20.7726

32.0994

35.2717

40.0302

14
REFRACTIVE INDICES

GLASS CODE

SSK4 SCHO'I'q"

SF8 S'CHOTI"

BAK4 SCHOTT

F3 SCI-IOTT

BK7 SCHOTT

SF5 -SCHOTT

S K 1"1 SCHOTT

INFINITE CONJUGATES

EFL -14063.1395

BFL 109.3923

FFL -0.3236E+ 06

FNO -46.1389

IMG DIS 108.0000

OAL 2301.9661

PARAXIAL IMAGE

HT 20.0458

ANG 0.0817

EXIT PUPIL

DIA 13.2190

THI -500.5187

656.27

1.614266

1.682505

1.565761

1.608063

1.514323

1.666612

1.561011

632.80

1.615305

1.684452

1.566704

1.609545

1.515089

1.668457

1.561883

525.02

1.621923

1.697362

1.572695

1.619244

1.519867

1.680666

1.567374

15
Figure 1: Schematic of the EXVM

E_XVM - Sote_" ivl,_gnetc.,g_.xph

/ 1

Lyot ¢_t,r

/ / .....

sH3eeeT_

J _ ,,

16
Table 5 lists the first-order propertiesof the system. HMY is the height of the

marginal ray, UMY is the slopeof the marginal ray, HCY is the height of the chief ray, and

UCY is the slope of the chief ray. A secondrepresentationof the system'sfirst-order

properties is in Figure 2, the y-y bar diagramof the system.

Table 5

First-Order properties of the EXVM system

HMY UMY HCY UCY

EP 152.400000 0.000000 0.000000 0.001425

1 152.400000 0.000000 -0.926518 0.000938

2 152.400000 0.000000 -0.926517 0.001425

STO 152.400000 0.127279 0.000000 -0.001425

4 40.745715 -0.032327 1.250428 0.004339

5 6.187768 -0.021270 5.889186 0.002855

6 6.038879 -0.032327 5.909172 0.004339

7 5.877242 -0.021270 5.930868 0.002855

8 4.813749 -0.032327 6.073623 0.004339

9 4.652112 -0.021270 6.095319 0.002855

10 4.524493 -0.032327 6.112450 0.004339

11 4.362856 -0.021270 6.134146 0.002855

12 4.213967 -0.032327 6.154132 0.004339

13 4.052330 -0.021270 6.175829 0.002855

14 3.988520 -0.032327 6.184394 0.004339

15 -6.194598 -0.015503 7.551287 -0.001762

16 -6.256610 -0.019362 7.544239 0.001940

17 -6.384400 -0.000403 7.557043 -0.033548

18 -6.426724 -0.000265 4.034458 -0.022073

19 -6.506285 -0.000403 -2.587524 -0.033548

20 -6.519789 0.011009 -3.711396 -0.014919

21 -6.464743 0.0(184(16 -3.785991 -0.015829

22 -6.427755 0.021331 -3.855639 -0.021001

23 -4.827902 -0.021331 -5.430683 0.021001

24 -2.078715 -0.006732 -8.137238 0.035827

25 -2.046401 -0.012020 -8.309207 0.021038

26 -2.010341 -0.001344 -8.372319 0.102462

27 -1.990187 -0.000884 -9.909252 0.067415

28 - 1.990196 -0.001344 -9.908578 0.102462

29 - 1.925704 -0.000884 - 14.826762 0.067415

30 - 1.922168 -0.001344 - 15.096423 0.102462

31 -1.857677 -0.000884 -20.014606 0.067415

32 -1.857685 -0.0(11344 -20.013932 0.102462

33 -1.707182 -0.003984 -31.491437 0.010229

34 -1.657381 -0.002464 -31.619301 0.030977

35 -1.642596 -0.006831 -31.805163 -0.0013019

36 -1.266885 0.006831 -31.804096 0.000019

37 -0.138158 0.004495 -31.800888 0.000013

38 -0.120180 0.006831 -31.800837 0.000019

39 -0.038206 0.004495 -31.800604 0.000013

40 0.056179 0.006831 -31.800336 0.000019

41 0.056181 0.004495 -31.800336 0.000013

42 0.137083 0.0(16831 -31.8001 (16 0.000019

43 0.137083 0.(/04495 -31.800106 0.000013

44 0.231468 0.006831 -31.799838 0.000019

45 0.313442 0.004495 -31.799605 0.000013

46 0.331420 0.006831 -31.799554 0.000019

47 0.877909 0.003536 -31.798001 0.019142

48 0.899127 0.004338 -31. 683148 0.006086

49 0.942511 0.004976 -31.622283 0.063529

17
50

51

52

53

54

55

56

57

58

59

60

61

62

IMG

3.127019

3.106067

3.101882

1.701589

1.585757

1.556562

0.000000

-5.385824

-5.414985

-5.531890

-4.751638

-1.965719

-1.185466

-0.015088

-0.005514

-0.001674

-0.010687

-0.025971

-O.019463

-0.044568

0.044568

0.014580

0.024355

-0.010837

0.010837

-0.010837

0.010837

0.010837

-3.732943

-3.534241

-3.425446

7.297245

7.163775

7.154148

4.874194

-3.014621

-3.078942

-3.268270

-5.634672

-14.083994

-16.450397

-20.000000

0.052290

0.043518

0.081834

-0.029926

-0.006418

-0.065280

0.065280

0.032160

0.039443

0.032867

-0.032867

0.032867

-0.032867

-0.032867

18
Figure 2: y- y bar diagram of the EXVM.

- 160

140

120

- 100

- 80

- 60

40

1 "\\

_-J-_--_-3o20 <--i_:_Z__,o

1

y bar

19
Table 6 lists the surface-by-surfacethird-order aberrations of the EXVM. SA stand

for spherical aberration, TCO stands for tangential coma, TAS stands for tangential

astigmatism, SAG stands for saggital astigmatism, PTB stands for Petzval blur, and DST

stands for distortion. All aberrations are in millimeters.

Table 6

Third-order aberrations of the EXVM

SA TCO TAS SAG PTB DST

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000012

2 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000012

STO 3.624628 -0.243555 0.003637 0.000000 -0,001818 0.000000

-3.624628 0.000000 0.000000 0.00(3000 0.000(300

4 - 1.136824 0.123181 0.000625 0.003591 0.005074 -0.000130

1.136776 0.104658 0.003212 0.001071 0.000033

5 -0.005470 0.002203 -0.000296 -0.000099 0.000000 0.000013

6 0.005338 -0.002150 0.000289 0,000096 0.000000 -0.000013

7 -0.005195 0.002092 -0.00028 1 -0.000094 0.000000 0.00(3013

8 0.004255 -0,001714 0.000230 0.000077 0.00(3000 -0.000010

9 -0.004112 0.001656 -0.000222 -0.000074 0.000000 0.000010

10 0.004000 -0.001611 0.000216 0,000072 0.000000 -0.000010

11 -0.003857 0.001553 -0.000208 -0.000069 0.000000 0.000009

12 0.003725 -0.001500 0.000201 0.000067 0.000000 -0.000009

13 -0.003582 0.001443 -0.000194 -0.000065 0.000000 0.000009

14 0.003526 43.001420 0.000191 0.000064 0.000000 -0.000009

15 0.011116 -0.012094 0.005631 0.002707 0,001245 -0.000982

16 -0.016057 0.046209 -0.044970 -0.015419 -0.000643 0.014791

17 0.008307 -0.046646 0.093831 0.035623 0.006518 -0.066680

18 0,000000 0.000003 0,000229 0.000076 0.000000 0,006349

19 0.000000 -0.000003 -0.000232 -0.000077 0,000000 -0.006428

20 0.002187 0.010711 0.021248 0.009591 0.003762 0.015656

21 -0.010937 -0.011472 -0.004501 -0.001827 -0.000490 -0.000639

22 0.005467 -0.006563 0.004241 0.002490 0,00 ] 615 -0.000996

23 0.000000 0.000000 0.000000 0.000000 0.0(X)(X)O 0.000000

24 -0.002159 -0.006579 -0.012924 -0.008470 -0.006243 -0.008601

25 0.002601 0.021819 0.063919 0.023238 0.002897 0.064989

26 -0.000593 -0.013574 -0.115584 -0.046569 -0.012061 -0.355156

27 0.000000 0.000029 -0.002204 -0.000735 0.000000 0.056016

28 0.00(3000 -0.000029 0.002204 0.000735 0,000000 -0.056016

29 0.00000(3 0.000028 -0.002132 -0.000711 0.000000 0.054201

30 0.000000 -0,000028 0,002128 0.000709 0.000000 -0,054101

31 0.000000 0.000027 -0.002057 -0.000686 0.000000 0.052286

32 0.000000 -0.000027 0.002057 0.000686 0.000000 -0,052286

33 0.000006 0.000629 0.025908 0.011272 0,003954 0.393726

34 -0.000052 -0.002109 -0.029773 -0.010581 -0.000984 -0.144424

35 0.000047 0.001007 0.009266 0.004503 0.002121 0.031962

36 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

37 -0.000001 0.000000 0.000000 0.000000 0.000000 0.000000

38 0.000001 0.000000 0.000000 0.000000 0.0000(0) 0.000(300

39 0.000000 0.000000 0,000000 0.000000 0.000000 0.000000

40 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

41 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

42 -0.000001 0.000000 0.000000 0.000000 0.000000 0.000000

43 0.000001 0.000000 0.000000 0.000000 0,000000 0.000000

44 -0.000002 0.00(3000 0,000000 0.000000 0.000000 0.000000

45 0.000003 0.000000 0,000000 0.000000 0.000000 0.000000

46 -0.000003 0.000000 0.000000 0.000000 0.000000 0.000000

47 0.000013 -0.000221 0.002591 0.001737 0.001310 -0.010080

48 -0.000005 0.000247 -0.004633 -0.001956 -0.0006 l 7 0.031830

2O
49

50

51

52

53

54

55

56

57

58

59

60

61

62

SUM

0.000030

0.001168

-0.001403

0.000687

0.000823

-0.008358

0.009103

0.000(_

0.048899

-0.088585

0.063628

0.000000

0.000000

0.000000

0.024479

-0.000087

0.003753

0.009618

-0.008760

0.018063

-0.090578

0.064030

0.000000

0.162022

-0.198013

0.035672

0.000000

0.000000

0.000000

-0.038080

-0.005347

0.010140

-0.023498

0.040530

0.156585

-0.331361

0.159999

0.000000

0.183774

-0.149627

0.018974

0.000000

0.000000

0.000000

0.081811

-0.000143

0.007459

-0.008851

0.015702

0.068533

-0.113232

0.059912

0.000000

0.064475

-0.051266

0.014529

0.000000

0.000000

0.000000

0.068093

0.002459

0.006119

-0.001528

0.003288

0.024507

-0.004167

0.009868

0.000000

0.004826

-0.002086

0.012307

0.000000

0.000000

0.000000

0.061233

-0.012892

0.007992

0.020220

-0.066755

0.501119

-0.409025

0.140474

0.000000

0.071211

-0.038199

0.002715

0.000000

0.000000

0.000000

0.182185

21
Figure 3: Wavefront aberrations of the EXVM in the final image plane.

REILEY

10;83;01

Z-FAN 100 RELATIVE X-FAN

FIELD HEIGHT

I00 _ 0817 )D 1,00

- ld0

i OO

u hi _'!L AP] /P

F IELb HEb]H7

c 0572 )m

- I.UU

IUO

_-- -= 7- -. 5-__

ilflO ;'7 L A _ I_,'[

r [ELI1 HEIGHT

' OOOO _ o

CIo_;<" ,_kjLt t' _t , , :,

}

OPTICAL PATH DIFFERFNLE _/AVE_-, _

15-DEC-92

!l i-_

1.00

-I00

IU(J

] IIII

..... ,/%,i NM

..... b388 NM

- - 65E,}I NM

22
Section V - Coating specification sheets

Prefilter

diagram of blank

Average intensity transmittance

> % over entire diameter, incident angles 0° - .16 °

X. = 525nm

< % outside of nm bandpass, centered on)_ =525nm

over entire diameter, incident angles 0 ° - .16 °

Polarization

R_ - s-polarization intensity reflectivity

Rp - p-polarization intensity reflectivity

0 < R_, -/i'_ < .1"10 -5

over entire clear aperture, incident angles 0 ° - .16 °

= 525nm

Surface figure

surface figure must be degraded less than _/10 at _ =633 nm

Surface quality

must meet MIL-SPEC 60-40 scratch-dig specification over entire surface

Surface durability

must meet MIL-SPEC scotch tape and eraser tests

23
Primary mirror

diagram of mirror

Average intensity reflectivity

> 98% over entire diameter, incident angles 0° - 3.8 °

X = 525nm

Polarization

R s - s-polarization intensity reflectivity

R p - p-polarization intensity reflectivity

.05"10 -s < Rs-R_, <.5"10 -_

over entire clear aperture, incident angles 0° - .16 °

X = 525nm

< .4 for incident angle = 45 °

Surface figure

surface figure must be degraded less than X/10 at X =633 nm

Surface quality

must meet MIL-SPEC 60-40 scratch-dig specification over entire surface

Surface durability

must meet MIL-SPEC scotch tape and eraser tests

24
Secondary mirror

diagram of mirror

Average intensity reflectivity

> 98% over entire diameter, incident angles 0° - 4.6 °

X_ = 525nm

Polarization

R s - s-polarization intensity reflectivity

R p - p-polarization intensity reflectivity

.05" IO -2 < Rs-Rp <.5' 10 -'s

over entire clear aperture, incident angles 0 ° - .8 °

= 525nm

R _ - R _ < for incident angle = 45"

X. = 525nm

Surface figure

surface figure must be degraded less than k/10 at )_ =633 nm

Surface quality

must meet MIL-SPEC 60-40 scratch-dig specification over entire surface

Surface durability

must meet MIL-SPEC scotch tape and eraser tests

25
Front doublet surface

diagramof lens

Average intensity transmittance

< 98% over entire diameter, incident angles0° -

= 525nm

O

Polarization

/?s - s-polarization intensity reflectivity

R _ - p-polarization intensity reflectivity

0< R o -Rs < .1"10 -'_

over entire clear aperture, incident angles 0° - .8 °

= 525nm

R, - R p < .03 for incident angle = 45"

)_ = 525nm

Surface figure

surface figure must be degraded less than X/10 at _, =633 nm

Surface quality

must meet MIL-SPEC 60-40 scratch-dig specification over entire surface

Surface durability

must meet MIL-SPEC scotch tape and eraser tests

26
Back doublet surface

diagramof lens

Averageintensity transmittance

>98% over entire diameter, incident angles0° - 5.1°

X. = 525nm

Polarization

R s - s-polarization intensity reflectivity

R p - p-polarization intensity reflectivity

0 < R, -R_ <.1.10 -5

over entire clear aperture, incident angles 0 ° - 3.3 °

;_ = 525nm

/i'_ -R,, < 2.1 0 .4 for incident angle = 45"

= 525nm

Surface figure

surface figure must be degraded less than )x/10 at X =633 nm

Surface quality

must meet MIL-SPEC 60-40 scratch-dig specification over entire surface

Surface durability

must meet MIL-SPEC scotch tape and eraser tests

27
Appendix I. The paper "Polarization Analysis of the SAMEX Solar

Magnetograph" contains both a good summary of polarization aberration

theory and a good demonstration of polarization aberration analysis.

28
Appendix II. Polarization Aberrations of unresolved point spread functions

This section explicitly provides the mathematics behind the polarization specification

which was presented in Section III. First, we present the reasoning for the derivation of the

Mueller matrix averaged over the point spread function. Next, we derive the average

Mueller matrix for rotationally symmetric systems. Then, we use this Mueller matrix to

explore the three possible explanations of the 1 O- 5 polarization specification in

rotationally symmetric systems. Finally, we explain how the result from rotationally

symmetric systems can be generalized into a result for slightly decentered systems, such as

a vector magnetograph with an actuating secondary mirror.

In many imaging situations, the point spread function that is formed by the system's

optics is smaller than the resolution of the system's readout. The EXVM is a good example

of this type of system; the pixels on the CCD array are slightly larger than the point spread

function.

For this situation, the Stokes vector averaged over the point spread function is equal

to the pupil-averaged Stokes vector. This equality can be understood by recognizing that

the intensity in the pupil of any two orthogonal polarization states is the same as the

intensity of those polarization states in the image. For example, if a 90% of the intensity in

the pupil is y-polarized and 10% is x-polarized, the intensity distribution in the image plane

will also be 90% y-polarized and 10% x-polarized. Just as the polarization state changes

across the pupil, the polarization state will also change across the point spread function, but

the average Stokes vector will be the same.

This equality can also be derived. The Jones vector in the exit pupil is

E= E "

29
The Stokesvector in the exit pupil is defined in terms of sumsand differencesin intensity

measurements,

-S= -- 145"I1'3s°1 t 2RO(t:':/:'_) /'

]

where 1, Q, u, v are the Stokes vector elements, I h is the intensity of the light which

would pass through a horizontal polarizer, 1 _ is the intensity of the light which would pass

through a vertical polarizer, 14_. is the intensity of the light which would pass through a

45 ° polarizer, / _3so is the intensity of the light which would pass through a 135 ° polarizer,

1 R is the intensity of light which would pass through a right circular polarizer, and l L is the

intensity which would pass through a left circular polarizer.

The pupil-averaged stokes vector is

21m(E'_Ey) y ,..,,. <21m(l::l:.)>,,,,,,,.,/ 21m(<['_t:.> .) J

where the brackets represent

<f >,,,,,,,,= f f a

where p is the pupil coordinate. The units are chosen such that the area of the pupil is

unity, f _ E _. is the intensity of the x-polarized light in the exit pupil. Since there are no

optical elements between the exit pupil and the image, there will be the same amount of

x-polarized light in the image plane. Therefore,

3O
<E_E_>_p.= <E'_E'_> t,

where the primed quantities refer to the quantities in the image plane. Clearly, the same

relationship holds true for the y-polarized light:

<EyEy>p._u= <E" E" >Y psi "

Since there is the same amount of light polarized both horizontally and vertically in the exit

pupil and in the image, the average of the first two elements in the Stokes vector are equal

in the pupil and the point spread function,

<l> p.mt = </>psi

<Q> m,mt= <Q> p_!

Since the choice of horizontal and vertical is arbitrary, the same relationship holds true for

the third element in the Stokes vector,

<g!>p.p_l= <U>p_f.

Since the first three elements of the Stokes vector are equal, the fourth must be equal as

well,

< V > pupi I = < V > ps],

These equalities can also be derived rigorously using properties of the Fourier

transform. The relationship between the Jones vector in the pupil plane, _, and the Jones

vector in the image plane, E ", is a Fourier transform,

Instead of explicitly using the x- and y- components of the electric field, we will derive the

more general case, for E _ E _, where both c_ and [3 can be either x or y. Multiplying the

Fourier transform for both components of the electric field yields

E"E'_, = fz::e -' 2"_; _'v-,_o_-r_,__,o'2"_'';d_"

--=f f E_t_l_o d d

31
Averaging over the point spreadfunction yields

<E'_,E _> t= f E',E _cl-_

Rearranging the order of integration yields

= f f'-' {_F { E'_E_} }d'o

= f E'_E,ct'p

= < E ;/_-0 > oap, _.

Therefore,

Since a and [3 can both be either v or y, this proof explicitly shows that the Stokes vector

averaged over the exit pupil is equal to the Stokes vector averaged over the point spread

function.

Because the average Stokes vector in the point spread function is the same as the

average Stokes vector in the image, the average Mueller matrix in the point spread function

is the same as the average point spread function in the pupil. Qualitatively, this equality is

even easier to understand than the equality for the Stokes vector. Since there are no

polarizers, retarders, or depolarizers between the exit pupil and the image, the Mueller

matrix for propagation from the exit pupil to the image must be the identity matrix.

32
This relationship can also be proven rigorously. Assuming a uniform input

polarization state, we can distribute the averages among the matrices in the following way

when we average over the point spread function.

< S > es F = < M S ln > psF

= <M >psFSin

A similar relationship holds when we average over the pupil.

< S > pup_l = < M Sin > puptl

= <M>pupilSin

Since we have already proven that the Stokes vector averaged over the pupil is equal to the

Stokes vector over the point spread function (< S > ,),,,,,_ = < S > P.sF ), we can equate the

previous two equations

< M > PSF S i_ = < S > PSF = < S > p,.,p,_ = < A'! > p,,_,_ S ,,,.

Therefore, the Mueller matrix averaged over the point spread function is equal to the

Mueller matrix averaged over the exit pupil,

< _V_ > PSI: = < _ > pupLl •

m

The easiest way to find the pupil-averaged Mueller matrix, M (h), is to begin with the

polarization aberration Jones matrix, J (p. ¢. h ). This matrix will have different forms

depending on the symmetries of the system. To convert this Jones matrix into a Mueller

matrix, use the following relationships:

33
2Mll(P

2MI2(P

2Mla(P

2Ml4(p,¢

2MzI(P

2Mzz(P

2M2a(P

2M2,(P,qb

2Ma_(p

2M32(P

2M3a(P

2Ma4(p,¢ h

2M41(p,¢ It

2M42(P,¢ h

2M4a(P,¢ h

2M44(P,_,h

¢,h)

¢,h)

¢,h)=J[t

i d _l + d'2_ d 2_+ J ]2J _2+ ./'22J =

d 11 + d21 d21 - J 12 d 12 -- J22J22

J l_ + J2l J22 + J 12J L_+ Jz2J2_

J _2+ J21J22- J _2J li - J22J2_)

J l_ + J _2J 12- J2_ J21 - J 22J22

J _l + J22J22- J2_ J2L - J _2J 12

O,h) =J 12J II + J llJ _z-J2_J2_ - J_Jz2

h) = J(J _ J 12+ J22J_l - J 21J_2- J l_ J 12)

¢,h) =J_lJ_ + J_J _l + J _.-/_2 + J_J I_

_ , h ) = d __d __+ J 21J __- J _2,-l _.'_- .] "e'_.] l_

¢, h) =J _ J___ + J'e_ J _ + J _J2_ + J_'eJ ,_

) = J(J _J_+ J2_.J _- J _,J_ - J_2J l_)

) = j(J2_ J_ + J l_J_2 - J ll J21 - J22 J 12)

J ,c represents the element from row v and column c in the Jones matrix. M rc represents

the element from row t and column c in the Mueller matrix. This transformation from the

34
Jones calculus into the Mueller calculus can be found in several references: E.L. O'Neil,

Introduction to Statistical Optics, (Addison Wesley, 1963), A. Gerrard and J.M. Burch,

Introduction to Matrix Methods in Optics, (Wiley, 1975).

A second, more succinct, method of transforming a Jones matrix into a Mueller

matrix uses the Kroneker product:

M=U(J®J*)U -j

U is defined as

LJ _---

1 0 0 1 1

1 0 0 -1

0 1 1 0 "

0 i -i 0

The Kroneker product of two 2X2 matrices is

a_ b I ®

c_ d I c 2

C-t I Ct2 (3-102 b_az bibz 1

b2"_=[alc2 aid2 blc2 blcl2

dzJ _c_ct2 clb2 djct2 ct_b2

I

I

_,C1C2 old2 dlc2 did2

This formalism can be found in "Obtainment of the polarizing and retardation parameters

of a non-depolarizing optical system from the polar decomposition of its Mueller matrix," J.

Gil, E. Bernabeu, Optik, vol. 76, no. 2, 1987.

Converting the polarization aberration Jones matrix into a Mueller matrix yields a

matrix which is too complicated to even write down. However, by looking at the resulting

Mueller matrix for the various aberration terms, we can see the form of the entire Mueller

matrix. In the following analysis, we will use the polarization aberration expansion for a

35
rotationally symmetricsystem.The resultsfrom other typesof systems,suchasa

Cassegraintelescopewith an articulating secondarymirror, canbe inferred from our

results. We choosenot to work with the polarization aberration expansionfor a

decenteredsystembecausethe math becomessounwieldy that the benefitsof the added

generality are more than outweighedby the difficulties of the addedcomplexity. The

polarization aberration expansionwhich wewill beginwith is the Jonesmatrix for

rotationally symmetricsystems:

J(p, ¢,h) =PooooOo + P lzooO _h 2

P)lll/zp(cos¢o_ - sin ¢_2)

P io22P2(COS 2¢cY l - sin 2¢02),

The polarization aberration terms have both real and imaginary parts

P l xxx -= V l xxx + i q l xxx

The Mueller matrix for a system with only polarization defocus is

I 2 2 4

Poooo + P tozzP

2P oooo r iozz9 Zoos2##

- 2 P 0000/" 1022 [k 2 Slll 2 ##

0

2Poooor 1o221:)2 COS2_ )

2Poo0oq IozzP z s) t_ 2##

- 2Poooor lozzP Zsin 2d_

P_o2z9 4 sin 4_

)? ) 2 -I

/ oooo-/ _o_z9 cos _

2 P o0oo q iozzP z cos 2 ##

o )

- 2Pooooq _ozzP z sin 2?0

_ 2Pooooq _oz292 cos2## '

2 4

Poooo- P IozzP

As expected, this is the Mueller matrix for a retarder and diattenuator with orientation that

rotates at twice the angular pupil coordinate. The Mueller matrix for a system with only

polarization tilt is

36
I P_oooo+Pllllh2p2

2Poooor lltihpcost#

- 2Poooor ollhp sine _

2Poooor llllhpcosO

Poooo + PllllhZp2cos2O

- PZ_lilh Zp Zsin2¢_

2Pooooq i_lL hp s)n

- 2Poooo r l lllhpsin

-P_lllh2p2sin2#

Poooo- 2 P_l_l h2p2ces2@

2PooooqlLllhpcos¢_ o) --2Pooooqtlllhpmn_

- 2Pooooqllllhpcos_

Poooo 2 + Pllkl h2p2

As expected, this is the Mueller matrix for a retarder and diattenuator oriented radially.

The Mueller matrix for a system with only polarization piston is

,flV_ plslon

I p 2oooo + h 4 2

P 1200

2h 2 P oooo r 12oo

0

o

2h2 Poooo r 12oo 0

/o2 4 2

oooo +h Pi2oo 0

0 2 4p2

PO000 -- /_ " 1200

0 2h2pooooq 12oo

0 )

0

- 2h2pooooq l_oo

2 4 2

Poooo-h Pf2oo

As expected, this is the Mueller matrix for a vertical or horizontal diattenuator and

retarder.

Integrate over p and q) to obtain the pupil-averaged Mueller matrix.

1 2_

_(h)= f f M(p,C,/-/)Odpd¢

o o

For rotationally symmetric systems, the pupil-averaged Mueller matrix is

2 i 2 4 2

fPoooo";P,o_'_+1⁄2haP_,,, +h P,2::o 2h2poo_:,r,:, .... 0

2h:_P oooor ,_.,, t -' i lh Ill' * l_ 4 p "] . 0

0 0 t '"oooo- h" P f_.,o .,,

\ 0 0 2 h' P ._,,_,>q,, .... o )

0

, -2h'P><,o<q,2o<,

? :i)Oi3,, - _P, .,.. - : h : P "L_ LI - h _[_:L2o_

37
I PZ 2h2Poooo r 12oo 0 0 1

0000

2

2hZPoooo r 12oo Poooo 0 0

_=

0 0 pZoooo - 2 h 2p ooooq 1200

0 0 2h2Pooooq l_oo Po2ooo

2

This is the Mueller matrix which would be measured in the image plane if the system were

placed in a polarimeter. The approximation in eq. 2 is the usual approximation used in

aberration theory; terms of high order in h and terms on the order P 2_x_x are neglected.

Making this approximation has a slight drawback; the resulting Mueller matrix is slightly

unphysical. If linearly polarized light at 45 ° is incident on the Mueller matrix,

the pupil-average stokes vector is

/" Po oooP °°° )

= [2h2Poooo c 12oo

Sour |

k_ Rh 2 p ooooq 120o

The degree of polarization for this Stokes vector is

DOP = _/Q2 + U2 + v2 / !

-_ 1 + 2h4p 21200

This Stokes vector is unphysical because the degree of polarization is greater than one.

38
This is not a seriousproblem, however,because,to be consistentwith our previous

approximations,we shoulddrop terms of order P 2l,,xx. In this case, the degree of

polarization for this Stokes vector is unity,

DOP _ 1 .

Now that we have a pupil-averaged Mueller matrix, we can rigorously define the

three polarization aberration criteria which we enumerated in Section II.

light.

Polarization error # 1 was that unpolarized light should not couple into polarized

The Stokes vector for unpolarized light is

When this unpolarized light is incident on the system, she Stokes vector averaged over the

exit pupil is

S

P oooo

2h2Poooo r 12oo .

0

0

This is the polarization error which we believe is the most important, and should be kept

below 1 0-5 Since P oooo, the transmission for the system, is approximately equal to one,

the polarization specification for this system should be

39
2r12oo< 10-s.

This is the polarization specificationwhich we recommend. This specification is essentially

a limit on the diattenuation alongthe chief ray.

Polarization error #2 is that polarized light should maintain its polarization state.

This specification would limit all off-diagonal elements in the pupil-averaged Mueller

matrix. One way to quantify this specification would be

21P12ool < 10 -s.

This specification is essentially a limit on the diattenuation and retardance along the chief

ray path. We do not recommend this specification because we believe that it is too

stringent. Furthermore, all of the typical, low diattenuation coatings which we have

analyzed have low retardance as well. Therefore, any set of coatings which would meet the

first specification would also have a small value for the second specification.

Polarization error #3 is that polarized light should not couple into unpolarized light.

Determining the amount of depolarization in a Mueller matrix is an active area of

research, but we can determine some properties of the pupil-averaged Mueller matrix by

examining the Mueller matrix in eq. 1 on a term-by-term basis. For a system with only

polarization defocus, the pupil-averaged Mueller matrix is

\

0 0 0 '_)

/`2 0 0 0000

)2 0 f oooo 0

2 1 2

0 0 Poooo- _ P !022

2 + LpZ

0000 3 102Z

0

M de/ocu s =

0

0

For a system with only polarization tilt, the pupil-averaged Mueller matrix is

40
ttIt

I 2 1

Poooo+_h2P_Lll 0 0

0 P 2oooo 0

0 0 Doooo2

0 0 0

0)

0

0

2 I 2

Poooo- _/z P_lll

The pupil-averaged Mueller matrices for a system with only tilt or with only defocus is the

Mueller matrix for a depolarizer. The first element in the Stokes vector will be increased,

but the other three will remain the same or decrease. For a system with only polarization

piston, the pupil-averaged Mueller matrix is

2 4 2

Poooo +h P12oo

M p,,to,_= / 2 tz2 p oooor 120o

\ o

2 h. 2 p oooo r _2oo 0

p_oooo +/_4 Pl_oo 2 0

0 pZoooo-h 4 P_zoo 2

0 - 2h _ Pooooq 12oo

0 )

0

2h-2 Pooooq _2oo

I::)2 _ /-)4D2

0000 * _" t 1200

The defocus term and the tilt term are clearly depolarizers. The piston term, perhaps not

so clearly, is a diattenuator and a retarder. The depolarization of the entire Mueller matrix

is

1 2 l 2 2

D=sPlo22+Sh Pl111

This depolarization is clearly a small effect; all of the aberration terms are on the order of

pZ] XXX "

For slightly decentered systems, such as a Cassegrain telescope with a tilted

secondary mirror, the polarization aberration expansion in Jones matrix form becomes

41
much longer, and the corresponding pupil-averaged Mueller matrix becomes

correspondingly larger. Rather than writing the equations, it is more instructive to think

through the problem.

The Mueller matrix in eq. 2 is the Mueller matrix which represents a diattenuator.

The magnitude of the diattenuation increases quadratically from the center of the field to

the edge. For the Mueller matrix for the decentered system will look much more

complicated, but the meaning will be very similar. The orientation of the diattenuation will

be oriented radially about some non-axial image point, and the magnitude of the

diattenuation will increase quadratically from the same, non axial image point. Therefore,

we specified the coatings to meet the 1 O- _ polarization specification at the most extreme

point in the image plane with a tilted secondary mirror.

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