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Polarization aberrations. I. Rotationally symmetric
optical systems
James P. MCGuire, Jr.
Russell A. Chipman
May 21, 1990
(NASA-CR-I8421_) POLARIZATION ABERRATIONS.
I: ROTATIONALLY SYMMETRIC OPTICAL SYSTEMS
Final Report (Alabama Univ.) 32 p CSCL 20F
G3174
N92-IO63B
Unclas
0303964
4,4 D
1Introduction
The analysis of the polarization characteristics displayed by optical systems can
be. divided'into two categories: geometrical and physical. Geometrical analysis
calculaees the change in polarization of a wavefront between pupils in an opti-
cal instrument [1,2,3,4]. Physical analysis propagates the polarized fields wherever
the geometrical analysis is not valid, i.e. near the edges of stops, near images, in
anisotropic media, etc. [1,5,6,7,8]. The changes, geometrical and physical, polariza-
tion causes in the performance of lens and mirror systems are readily calculated by
several commercial computer codes [9,10,11,12]. The inverse problem of designing
a system with specified polarization characteristics is more difficult. Examples in
the literature include a polarization compensated polarizing microscope[13] and a
telescope with ultra-low polarization for a solar polarimeter [14]. Design requires a
fundamental understanding of the origin of polarization aberrations and how they
change with both the optical and coating prescriptions for a system. Polarization
aberration theory provides a starting point for geometrical design and facilitates
subsequent optimization.
Chipman has derived several polarization aberration expansions similar to the
classical wavefront expansion for rotationally symmetric systems valid for weak sec-
ond order aberrations [15,16,2]. Reference [17] explores polarization aberrations
graphically using a "symmetrized" second order expansion valid for slrong aberra-
tions but does not describe a method to calculate the aberration coefficients. In
this paper, we calculate and discuss an exponential expansion of the polarization
aberrations valid for strong polarization aberrations through fourth order 1. The
results are applied to the interpretation of polarization raytracing results.
The polarization aberrations described in this paper arise from differences in the
transmitted (or reflected) amplitudes and phases at interfa,:es. In contrast, classical
Wavefront aberrations arise from differences in optical path i_.ngth as rays propagate
'between interfaces [18]. Figure I shows the calculation of the optical path length
Wand the optical path length and the polarization J along a ray. Repetition of
the calculation depicted in Figure I (a) for multiple rays and wavelengths samples
the wavefront aberration function. Repetition of the calculation 1 (b) for multiple
rays and wavelengths samples the wavefront aberration function and the polariza-
tion aberration matrix (PAM) of the system. The PAM ,__scribes the variation in
polarization with object coordin, pupil coordinates, and wavelength. This paper
calculates the PAM for isotropic rotationally symmetric _ystems through fourth
order and includes the interface phase, amplitude, linear ,Jiattenuation (defined
in Table 1), and linear retardance aberrations. Polarization aberrations resulting
from propagation through anisotropic media such as crystals are not considered in
this paper. For propagation through anisotropic crystals, the propagation terms
exp[jWn,n] in Figure [ (b) would be replaced by Jones mat rices J,,n.
The order of an aberration term referred to in this paper is the order of the
wavefront representation (n), not the order of the transverse aberrations (n - 1).
Thus, defocus and tilt are second order aberrations, while spherical aberration,
coma, and astigmatism are fourth order aberrations (not third order aberrations).
Section 2 discusses the exponential form of Jones matrices used in this paper.
Section 3 introduces the PAM in Jones matrix form. In Section 4, the exact cal-
culation of polarization aberrations through polarization raytracing is described.
Section 5 presents the coordinate system used in this paper. Section 6discusses the
paraxial approximation including: thd'p_ki_l PAM for a single surface, paraxial
angle of incidence, and the paraxial orientation of the plgne _of incidence. In Sec-
tion 7, a Taylor series simplifies coating dependence of the single surface paraxial
1The preliminary version of tkis paper used did not use the exponential form which resulted in
increaJled complexity of computation and interpretation,
-_t ,qh _
Table 1: Polarization terminology
Term Definition
Diattenuation The property of having an intensity transmittance
which is dependent on the incident polarization
state.
The property of altering the polarization state of
light. Polarization includes the subsets of diattenu-
ation and retardance.
An optical element which transmits a fixed polariza-
tion state independent of the incident polarization
state. Examples include dichroic sheets (Polaroid)
and Glan-Thompson prisms.
Any optical element showing polarization. Exam-
ples include retarders, polarizers, and metallic in-
terfaces.
The property of having a phase or optical path
length which is dependent on the incident polariza-
tion state.
Polarization
Polarizer
Polarizationelement
Retardance
Note: There is a distinction between polarizer and polarization element. Chipman
provides a more detailed discussion [2].
PAM. Section 8 calculates the paraxial PAM for a system of isotropic rotationally
symmetric elements through fourth order. A general discussion of the terms is
contained in Section 9. Section 10 contains a detailed discussion of the vector aber-
rations (defined later) comparing and contrasting them with classical scalar phase
aberrations. Section 11 discusses interpretation of polarization raytracing in the
"context of the aberration theory results. Appendix A conr:dns paraxial expressions
for polarization basis vectors. Appendix B examines the polarization by uncoated
interfaces. Appendix C lists the polarization aberration coefficients for a system of
isotropic rotationally symmetric elements.
2 Jones Matrices
In this section, we present the Jones matrb¢ formalism[1920] for the analysis of
polarization as used in this paper.
The Jones vector U(t) is
G(t) = (_(t)try(t) ) (t)
where _._(t) and lYe(t) ar Mthe projections of the electromagnetic field on any two
orthogonal basis states qand r. The Jones matrix Jrelates the incident Uand
transmitted fields _,
0'=J0
and isa 2x2 matrix wi_h complex element#
.....J= (jll j12
(2)
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f
Table 3: Physical significanceof the exponentialpolarizationcoefficients
Coefficient Matrix Physical significance
ao o'0 _(a0):
_(a0):
al o.1 _(al):
_(al):
a2 o'2 _(a2):
e(a2):
a3 o'3 _(a3):
_(a3),:
Polarization independent amplitude
Polarization independent phase
Linear diattenuation along the coordinate
axes
Linear retardance along the coordinate
axes
Linear diattenuation at 45 degrees to the
coordinate axes
Linear retardance at 45 degrees to the co-
ordinate axes
Circular diattenuation
Circular retardance
where the Jones matrix is
(5)
j,y,,_l (fl, ,,_A) j,v,,22(h, £ A)
=exp[a,y,,oo.o -b a,v,,lo.1 -4-a,), :or2 -4-a,_,,3a'3] ((3)
and his the object coordinate, ff is the pupil coordinate. Rnd A is the wavelength.
It is convenient to separate the Jones matrix for an imaging system into a polar-
ization aberration matrix (PAM) which describes the aberrations and a "quadratic
phase" characteristic of ideal imaging systems
J,_,(h,ff,A) =J(h,:,A) exp 2f
where J(h,_, A) is the PAM, fis focal length, and k= 2_/A is the wavenumber.
Both the wavefront aberrations and the finite extent of lens are described by the
elements of J (the finite size of the lens is an "aberration" which reduces resolution
from that predicted by geometrical optics.) In the limit of ,_.non-polarizing optical
system, the PAM has the form
1 0
J(h,],A)= P(h,p]exp[jkW(h,],A)] (0 1 ) (8)
where P(h,p") is the pupil function which describes the amplitude transmittance
of the pupil and W(h,_,A) is the wavefront aberration function. In succeeding
sections, quasi-monochromatic light is assumed and the explicit wavelength depen-
dence A is suppressed.
4 Exact Polarization Raytracing
In this section, we describe the procedure for calculating the changein polariza-
tion along a ray through a system of iSofropic surfaces." The technique is called
polarization raytracing [23,24]. , .::. _ _--.,,: .... :_ _,:: ....... •_;_-..1
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Jq(iq,0q) -" Jq(i_)R(0q)
(a,,(i,)cosO, a,,(i,)sinO, ) (19)
"=-avq(iq)sinO q arq(iq)eosOq
Equation (19) is exact Jones matrix for an isotropicinterface.Isotropicinterfaces
do not display circularpolarizationor circularretardance. If a ray Rl isincident
on a system of surfaces,the polarizationalong 1_Iisfound by cascading the effects
from each surface
1
J(R1) = ]'I aq(iq,O,) . (20)
q=Q
The set of all Jones matrices for a system for each possible ray path ]_t and
wavelength is the PAM, as described in Section 3. Polarization raytracing codes
[10,11,12] sample the PAM by calculating (20) for selected rays and wavelengths.
Reference [9]uses Mueller matrices to sample polarizationaberrations. The rest
of this paper examines a fourthorder approximation to the PAM for rotationally
symmetric systems.
5 Global Coordinates
For polarization aberrations expansions, an object and pupil coordinate system is
required. We denote the object height by Hand normalize it to one at the edge
of the field of view. The object is located along the global y-axis without loss of
generality because of the rotational symmetry of the sy,rem. The entrance pupil
coordinates are denoted by either (z, y, z) or (p, _b,z) in Cartesian and cylindrical
coordinates. In the paraxial approximation the pupils at, _ _]at, so the zdependence
is dropped for convenience in much of this paper. The pv.pi[ coordinates z, y, and
pare normalized to one at the edge of the entrance pupil. The polar angle ¢ is
defined so that
= -psin ¢ (21)
y = pcos¢ . (22)
Figure 2 illustrates the normalized coordinate system.
6 Paraxial Polarization Aberration Matrices
In this section, we determine the paraxial PAM for a rotationally symmetric sur-
face. Appendix Aprovides paraxial expressions for the surface normal and local
(/_9, Sq, Pq) basis vectors. The single surface Jones matrix of Section 4 is simplified
to first the paraxial Jones matrix and then to the para_xial PAM. The accuracy of
the paraxial approximation for the angle of incidence and orientation of the plane
of incidence are examined.
Consider the polarization of a ray for a system of surfaces. Because the paraxial
fields remain in the x-y plane [see (A-9) and (A-10)], we choose x-y Jones basis
vectors at each surface. Then, the Jones matrix for each surface consists of a
rotation into s-p coordinates, the Jones matrix in s-p coordinates, and finally a
rotation back to the x-y coordinates
Z PRECED!r.'G PAGE BLANK NOT FILMED
dependence of the paraxial PAM is suppressed for notational convenience. The
pataxial angle of incidence is obtained by expanding the angle of incidence (17) in
a Taylor series
1
= +2Hpcos, +p,i , (2s)
where the term linearin object and pupil coordinates was retained. The parax-
ial orientation of the angle of incidence (24) in terms of cos[28¢(H,p,¢)] and
sin [20q (H, p, ¢)] is
cos[2Oq(H,p,¢)] -H2i_q + 2Hpc°sOicqime + p2i2mqc°s2@ (29)
sin[2Oq(H,p,¢)] =-2Hpsin¢icqimq-p2i_¢sin2@ (30)
2
|q
Equations (28) through (30) define the paraxial system geometry.
Next, consider the accuracy of the paraxial orientation of the plane of incidence
and angle of incidence at a spherical interface. The paraxial orientation of the
plane of incidence assumes that the ,_q and /5q basis vectors are in the x-y plane.
A measure of the accuracy of this approximation is the ratio of the intensity of the
field in the x-y plane to the total intensity
r_,(O) =Ib'=12 + IUwI!
[U[2 =cos 2 0(31)
where 0is the angle the ray makes with the z-axis. When I_ v_ 1, most of the
light is polarized in the x-y plane and the paraxial approximation is good. When
I_ m 0, most of the light is polarized in either the ×-z or y-z planes and the paraxial
approximation is not valid. Now consider the angle of incidence. The surface normal
for a sphere from the definition (A-2) is
-Rt' Rt'R_ _- (z_ +v:) (32)
where R_ is the radius of curvature of the sphere. If the axial plane waves /_ =
(0, 0, 1) are incident then
= R-S g _ + T, + " (33)
where p= v/_ + y_. Figure 3 shows a comparison between several approximatmns
to the angle of incidence for a spherical mirror. The high degree of linearity of
the exact curve, even with angles of incidence as large as 30*, permits the para.xial
approximation to the geometry to be used for many systems.
0PRECEDINGPAGE BLANK NOT FILMED
I
ro2 =r.2/ro (39)
I
ra4 -- ra4/ro--0.5(ra2/ro) 2 . (40)
Finally, substituting the Fresnel expansion coefficients into (25) gives the paraxial
PAM e_pansion
.2 + ao4q i_ + ...)0.0+ "_(aoo_ +ao._q ,_ -4 .)(cos 2t_0.t +sin 20_0._) j (41)
•2 +a14q +..
Jq(H, p, _) = exp (at2q z_ zq
where if =i_(H, p, ¢) and 0_ =0q(H, p, ¢) are defined in (28) through (30) and the
coating coefficients ao,2ka and al,2k,q are
!
a0._,q =[r'.,2_,.+ rp,_,q + j(¢.,_,. + ¢.,_,_)]/2 (42)
The subscripts of coefficients a0,2_,q and at,2k,q are assigned as follows. The first
subscript assumes the value 0 for the polarization independent contribution and 1
for the linear polarization along the s-p axis. The second subscript, 2k, is the order
of the coefficient. The last subscript, q, designates surface number.
8 Polarization Aberration Matrices for Systems
In this section, we obtain the paraxial PAM for an optical system of isotropic
rotationally symmetric elements through fourth order. Sections 9 and 10 discuss
the system PAM.
To begin, we introduce the Baker-Campbell-HansdorIf 'iBCH) identity [25]..
Baker-Campbell-Hausdorff 1If Aand Bare matr_'cs, or certain other non-
commuting operators, then
exp A exp B = exp [A +B + C2 +C3 + -"-] (44)
where Ck is a linear combination of k-fold commutators o/A and B, in particular
1[A, B] (45)
c2=_ 1
C3 = 1"2[[A,[A,B]] - [B,[A,B)i] (46)
The BCH identity consistently carries out operator products, retaining terms to a
given order.
Now, the PAM of an optical system with Qsurfaces is the product of the PAMs
of the individual surfaces
J = JqJq-1 ""J2JI (47)
which simplifies to
J - exp [ao0.o "b a10.1. "_ (120"2 h- 430"3] (48)
using the paraxial surface PAM (41), the BCH identity, and t,he commutation rela-
tions in Table 4. The system "PAPYricoeffiCienis.lto fourth order are
J
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J(H,p, ¢) "-"exp
Poooo_o +
Po2ooii20.o+
PoIIIHpcos¢ 0.0+
Poo2_p_0.o +
Poo4op%"o+
Po131Hp 3cos¢ 0.0 +
Po;22H2p 2cos 2¢ o'o +
Po311H3pcos4, 0.0+
Po4ooH40.o +
Pz2ooH_o'I +
Pz111Hp(cos ¢o't -sin
Plo2_p2(co_ 2¢ _l -sin
Pt4ooH4o'l +
constant piston
quadratic piston
tilt
defocus
spherical aberration
coma
astigmatism and field curvature
distortion
quartic piston
¢o'_) +
2¢ o'2) +
P1311H3p(2 cos 4, 0.1 -sin 4, o'2) +
PL_2H2p _cos 4, (cos 4, o'I -sin 4, o"2) +
e11_p3 cos4,[(I+ cos2¢ )0._- sin 24,o._]+
Pt042p4(cos 2¢ O'l --sin 24, o.2) +
P3131HP 3sin ¢ o.3 + circular polarization coma
P_tl H3p sin 4, 0"3 + circular polarization distortion
sixth order terms
or in Cartesian coordinates (53)
Pooooo.o+ constant p}s_on
Po2ooH= 0"o + quadratic pi_on
Pollt Hy0"o +tilt
P00_2(z 2+y2)a'0 +defocus
P0040(z _ + Y_)_0"0 +spherical ab_:rration
Po131Hy(z 2 + y2)0"o +coma
Po_22H2(y 2 -x2)o'o + astigmatism and field curvature
Po3,1H 3Y0.0 +distortion
Po4ooH4 0"o +quartic piston
PL2ooH2o'[ +
J(H,z,y) =exp PIlllH(y_l +z0"2) +
P,o22[(:- + +
P14ooH4o'1 +
P,_,,H3(2y0"1 +z0"2) +
PI2;2H_Y (Y0"1 +20"2) +
•') 0",+(: + +
P31_,Hx (z 2 + y_) o'3 -eircular polarization coma
P331tH3zo'3 +circular polarization distortion
sixth order terms
54)
where the terms are grouped based on their H, p, ¢ dependencies and P=u_,_ are
the polarization aberration coefficients. The polarization aberration coefficients are
sums over interface contributions as given in Appendix Cand assigned subscripts
using the convention: tdenotes the type of polarization behavior[2], udenotes the
order of the Hdependence, vdenotes the order of the pdependence, and wdenotes
the order of the ¢ depender/ce. /.=:! .... "_" ._
The fourth order paraxial PAM is p_axial in system geometry and fourth order
in coating response to changes in:_:,gle o'f {nc{cl'en_ee)_._s_tbtl¢ con_e*C_uence is that
[3 PRECEDI!_G PAGE BLA_K NOT FILMED
thin film coatings (particularly those with many layers), they can be significant.
Our equations only calculate the system wavefront aberration contributions arising
from coatings. The geometrical portion of the classical wavefront aberrations (those
arising from optical path length differences) are calculated by the paraxial ray trace
and coffventional aberration calculations [18].
The second class of aberrations contains the amplitude or apodisation aberra-
tions which are characterized by the real parts of the o'0 coefficients, _(Pouvw=).
The amplitude aberrations are variations of the amplitude of the electromagnetic
field across the exit pupil which are independent of the incident polarization state.
They do not describe the shape of the transmitted wavefront, only its amplitude.
This apodisation is due to the optical system, not to intensity variations in the
incident light, such as the Gaussian profile of a laser beam. The amplitude aber-
rations describe the average of the coating amplitude transmittance of the s and
p light (the polarization terms describe the difference). Amplitude aberrations are
scalar aberrations and have the same functional dependence on object and pupil
coordinates as the classical wavefront aberrations. Contours of constant apodisa-
tioa aberrations through fourth order are shown in Figure 4. Since the functional
form is the same, the generic names of the functions have been retained with the
prefix amplitude added: amplitude tilt, amplitude coma, amplitude spherical aber-
ration, etc. For example, the term _(Pa040)p 4 is amplitude spherical aberration.
If _(P0040) is negative the center of the pupil is brighter and the pupil becomes
dimmer quartically with pupil radius. For _(P0uvwz) positive, the pupil is brighter
at the edge. The interpretation of all of the amplitude aberration follows in the
same manner as amplitude spherical aberration. Intentional apodisation (versus
apodksation aberrations) is discussed in Reference [26].
The third and fourth classes of aberrations contain linear diattenuatioa and lin-
ear retardance aberrations and are characterized by the real and imaginary parts
of the coefficients of _i and o'_ respectively. These two :!asses of aberrations are
characterized by the real and imaginary parts of the coefficients of a't and 0"2 re-
spectively. These two classes of aberration will be treated :ogether under the name
vector aberrations. Vector aberrations are conceptually different from the scalar
aberrations since both a magnitude and orientation must be specified. The parax-
ial vector aberration patterns through fourth order are illustrated in Figure 5. The
length of the lines denotes the strength of the linear polarization element. The ori-
entation of the line denotes the orientation of the linear p,:,larization element. The
patterns are the same for both linear diattenuation and linear retardance aberrations
since both are vector aberrations. Adetailed discussion of the vector aberrations is
found in the next section.
The effect of the vector polarization aberrations on p,:larized incident light is
the same as a linear polarization element with spatially varying strength and ori-
entation. Figures 6and 7show the effect of the three second order aberrations on
linearly polarized light. The magnitudes of the aberrations depicted are not typi-
cal, but have been chosen to clearly display the effect of the aberrations. Figures
8and 9 depict the effect of the three second order aberrations on circularly polar-
ized light. The type and orientation of ellipse indicates the type and orientation of
the polarization state. The position and direction of the arrow denotes phase and
handedness of the polarization state.
The last class of aberrations contain the circular aberrations. Figure 4 shows con-
tours of constant circular aberrations through fourth order. The real and imaginary
parts of the coefficient of a3 correspond to the circular diattenuation and circular
retardance aberrations respectively. The circular aberrations are variations of the
circular diattenuation and circular retardance across the exit pupil. The imaginary
part of the term P3t3t [P331t] produces a wavefront with _(P313t) coma [_(P33tt)
distortion] when right circularly polarized, t_Kht is incident and -_('Pa131) coma
15 PRECEDII'_G PAGE BLAi",_K NOT FILMED
Second order
_ ._ /\
--/f..-_K\ / "" \
.... /,° -, %
-- /,,._ _.x\ I ' :, I
--,.-- • I .I
/,,, \ \ Q/
Fourth order
I i B • • i • •
_ _ [I \ _ -I I I _I _
C_oo P.. C:_2
/-- \/.. \
/\/, , \
I, -,1, " ,
•_' #," . I
I l
Ii i i
II #
,, /\ ',". .'/
\-- /\-.- /
_m Co_
Figure 5: Paraxial vector polarization aberration patterns through fourth order
-._j.
17
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Linearly polarized light incident P1o22 = jTr/2
(a) (b)
P11_1=jTr/2 P12oo = jTr/2
(c) (d)
Figure 7: Effect of linear retardance aberrations on linearly polarized light. Part
(a) shows uniform linearly polarized light in the entrance pupil. Parts (b), (c),
and (d) show the polarization state across the exit pupil if the system has only the
aberration P10=2 = j,'r/2, Pllll =jTr/2, and Pt2oo =j_r/2.
19 PRECEDING PAGE BLANK NOT FILMED
0 ©
C' 0 © C,._G
coc
00000 0_0_0
OoO o_'c?
0 0 0
0 0
Circularly polarized light incident
(a)
D ° O© 0
/\ / (...r O© ©
o to--o ooooo
0 \ CC ©0°
© © ©
PlzlI =iT2 P12oo =jz/2
(c) (d)
Figure 9: Effect of linear retardance aberrations on circularly polarized light. Part
(a) shows uniform circularly polarized light in the entrance pupil. Parts (b), (c),
and (d) show the polarization state across the exit pupil if the system has only the
aberration P10._2 = j,'r/2, PlllZ =j_r/2, and P12oo =jTr/2.
21 PRECEDING PAGE BLANK NOT FILMED
f---_ /-\
ll-,,\ I I I I
l t i I \\-I/
\ ,,- ,, i
I I
\\-il
H=0 H =H o H=2H o
Figure i0: Para_xialangle of incidenceat a sphericalsurface for three fieldan-
gles.The length ofthe linesdenotes the magnitude of the angle of incidence.The
orientationof the linesdenotes the orientationof the plane ofincidence.
/..-.. \I.-. \/..\
/" " \ / " "\/' " \
m
ll' 'Ill' '1 [' '[
\""-" I\ !\ I
\ • /,.... ,, .. ,
\...__./ \...i \.../
_ ........ i
e_o22 e_ _o62
Figure ii: The firstthree principalvector aberration pa_erns for isotropicrota-
tionallysymmetric interfaces.The PI022 pattern variesquadraticallywith radius.
The PI042 pattern variesquarticallywith radius. Finally,the Pz0s_ pattern varies
with radiusto the sixthpower.
moves off-a_cis.The non-principalvector aberration patterns add to the principal
pattern to give a decentered view of the principalpatterns The aberrations change
asthe paraxialangleof incidencefunctionchanges foroi_'-_isobject points.[n fact,
the set of nth order vector aberrationpatternsiscomplete when the nth order prin-
cipalpatterncan be decentered with a linearcombination o( the nth order patterns.
Again, these principalpatternsare analogous to sphericalaberration. In classical
aberration theory when the object ison-a_ds,arotationallysymmetric interface
only has sphericalaberration (but many differentorders of sphericalaberration).
As the object moves off-a0ds,the other aberrationsare introduced.
The completeness of the set of second order vector aberrationpatterns can be
addressed by constructing a shiftedsecond order principalpattern from alinear
combination of the second order vector aberrations. Figure 13 shows the second
order patterns Pz022,Pi200,and P11zt adding to givea decentered view of the prin-
cipalpattern. Figure 13 (a)shows the superpositionof the Pz_00and Pi022 patterns.
The vector aberrationpatterns(arraysof weak lineardiattenuatorsand/or arrays
ofweak linearretarders)add as PAMs, not as vectors!Two orthogonal weak linear
diattenuatorsof the same magnitude,',addt'ozer0"di_tt_n_atiork_g.dreduced ampli-
tude (transmission).Two orthogonal weak linearretaxde_'swith equal retardance
add to zero net retardance. Figure 13 (b) shows the result of adding Pz20o and
23 PRECEDING PAGE BLANK NOT FILMED
•,41. E
m
+ -,- __ __ +
o
m
m
I_1200+P1022
/
m m
mm
m
/
m
÷__ I
I
\
\
P12oo+PlO22 +Pllll P12oo+PllO22÷l_llll
Figure 13: Addition of second order aberration patterns to give a decentered view
of the second order principal aberration pattern
• o. _ .
25 PRECEDING PAGE BLANK NOT FILMED
Table 6: Symmetry of pupil aberration sections of rotationally symmetric systems
On Axis Objects Off Axis Objects
Tangential Sabdttal
Aberration type Linear Linear Elliptical
Aberration orientation Radial or Horizontal or Arbitrary
tangential vertical
Symmetry Rotational None Odd about
the y-axis
Note: An aberration has odd symmetry if the pattern is mirror symmetric about
some line.
Tangential Sagittal
/
/
I\\
I%
| •
%I
•,p
\\
, I
/
/
m
/\
/, , \
%I
\-. ,/
\/
(a) _b)
I ' ' I I ' ' I
(c) (d)
Figure 15: Example pupil aberration section for on axis object. Exit pupil aber-
ration maps with the tangential and sagittal sections highlighted are shown in (a)
and (b), respectively. The tangential and sagittal sections are shown in (c) and (d),
respectively. The sagittal section is rotated 90 degrees for a more compact display.
The aberration is P102_ -P1042 +P3311.
27 PRECEDING PAGE BLANK NOT FILMED
Field height Tangential section
-o.z I Jl' ))1
H--o I, ' , , I
Sagittal section
II , , I
Figure 17: Tangential and sagittal pupil aberrations plot for a system with nth
order principal aberrations and piston
can be either diattenuating or retarding giving polarization dependent apodisation
or phase, respectively. The circular aberrations described the circular retardance
and circular diattenuation introduced by an optical system. For systems of isotropic
rotationally symmetric surfaces, the circular aberrations appeared first at fourth
order. Vector aberrations were conceptually different t'r,,m classical aberrations.
They described the linear retardaace and linear diatteau._tion variation introduced
by an optical system. These polarization aberrations were decomposed into a set
of vector patterns which were each attached a weight or at_ aberration coefficient.
The theory described in this paper applies to many optical systems built today.
Thin films have a much deeper role in optical system design than merely changing
the transmittance of a system. Thin films induce polarization 'aberrations or, if the
designer is clever, control the polarization aberrations. This extension of aberration
theory was made possible by including a Taylor series ,×pansion of the Fresnel
coefficients for each interface in the optical system. The resulting aberration theory
allows the integration of thin film design and optical design for polarization critical
optics.
13 Acknowledgments
The authors wish to acknowledge the support of the Jet Propulsion Laboratory
and both the Solar Physics and Optics Branches at Marshall Space Flight Center.
Dr. James Breckinridge of the Jet Propulsion Laboratory initially suggested and
supported research in the area of instrumental polarization.
A Paraxial Basis Vectors and Fields
In this appendLx, we derive the paraxial quantities necessary for the paraxial PAM
in Section 6. First, ge___[:ex, pre,s_p.ns_,for the surface n_r'n!al and local basis vectors
_¢re presented. Next, t[a'e par_ial'appr0ximation [§"applle_i'to.these'quantities. The
paraxial field is shown to lie in the x-y plane.
29 PRECEDING PAGE BLANK NOT FILMED
where
Icq -- Rq Riq
•y,nq + r_.!% (A-12)
and i_q and i,,_q are the angle of incidences for the chief and marginal rays respec-
tively. This follows directly from linear nature of the paraxial raytrace [16]. The
paraxial system geometry is entirely contained in ic and ira. It is not necessary to
work with the radii of the individual lenses, entrance pupils, etc. The asphericity
and wavefront aberrations have a third order effect on the surface normal and local
(/_,,_,q, 15) basis vectors. Both 2_, and /_, are in the x-y plane. There is no z
component of the electric field.
B Polarization at Uncoated Interfaces
This appendix examines the polarization on reflection and transmission from an
uncoated interface. We determine the interface coefficients of Chapter 7 with a
Taylor series expansion of the Fresnel coefficients. The accuracy is discussed tbr
reflection by gold at A - 10.6 pm.
The Fresnel coefficients expanded in the angle of incidence are
sin(/- i')
/'s --" sin(/+ i')
N-l( 1. 2 N2-6N-3i4+... )
-N+I 1-_ + 12Na
tan(/- i')
rp - tan(i + i')
- 1 .2 ._N -6N+9i4
NII+ + +...
-X+l iS.W
2 cos isin(/')
t s sin(i + i')
2N (N-li2 3N3 +3N_-TN+ 1i4 +,,, )
-N+I I+--_ + 24
sin(/- i')
tp =sin(i + i')
(B-l)
(B-2)
(B-3)
-N(N - 1)(9N 2 - 6N + 5) i4 ]
2N N(N 1)i_ + + ... (B-4)
-N+ I I+ 2 24
where rj and rp are reflection coefficients, t, and tp are reflection coefficients, iis the
angle of incidence in radians, the 'denotes quantities after the interface, N-- n'/n
is the ratio of the refractive indices, and Snail's law
nsin i = n' sin i' (B - 5)
was used [16,15]. The intetfa'¢.e'coefficients .(B_I) and (B-2) inexponential form axe
1
r,(i) =exp[ro+r2i 2+...] (B-6)
rp(i) = exp[r0- r:i 2 +'"] (B-7)
31 PRECEDING PAGE BLANK NOT FILMED
0.98
i 0.96
0.94
0.92
0.9
<0.88
175
---, 150
_0
_ 125
_ 100
e,,
_ 75
_ 50
25
Angle of Incidence (deg)
(a)
' ' ' 8"0
20 40 60
Angle of Incidence (deg)
(b)
Figure 18: The amplitude reflectance (a) apd phase changer(b) on reflection from
a gold coating at 106 #sn- Curves-are "shown for both the s and p components of
the incident light. Val_ "_were computed using a refractive index, n= 8.8 - 64j.
33 PRECEDING PAGE BLANK NOT FILMED
Q
/%o40 = _ ao,_i_,, (C-9)
q=l
Q(c-10)
q=1
q
Pun = 2_._an_i_qimq (c-11)
q=l
Q
Pto22 =__._at2q "_ (C-12)
lmq
q=I
Q
P_oo = _ a_,qi_, (C-131
q----I
Q
P1311 2_'] .3• (c-14)-: al4qlcgZrng
q=l
Q
Pn22 3Z .2 .2 (C-15)
-: al4q|cqZrnq
q=I
Q
P_ 2_ " '_ (c-161
--_ Gl4q Zcq Srn q
q=l
Q
P_o,,= Z a",iL (c-tn
q=l
_3131
P3331
Z- <-18/-- j al2qZmq al2picpi,.np -- al2ql,: _ 7,nq al2plmp
q=l \ p=l p=l
E"Z"<-io):j al2qtcqZmq al2plcp -- at2q_q al2picpirnp
q=l p=l p=t
References
[1] S. Inou4. Studies on depolarization of fight at microscope lens surfaces I: The
origin of stray light by rotation at lens surfaces. Ezper. Cell ges., 3:I99--208,
1951.
[2] R. A. Chipman. Polarization analysis of optical syste::'_s. Opt. Eng., 28:90-99,
1989.
[3]E. W. Hansen. Overcoming polarizationaberrationsin microscopy. In R. A.
Chipman, editor, SPIE Prec. 891: Polarization ConsLderation for Optical Sys-
tems, pages 190-197, SPIE, Bellinghazn WA, 1988.
[4] R. A. Chipman. Polarization'analysis of Optical systems. In R. A. Chipman,
editor, SPIE Prec. 1166." Polarization Considerations for Optical Systems II,
SPIE, Bellingham WA, 1989 ....
[5] J. D. Mangus an_ _. Alonso. Image senslUvlty anomalies of glancing incidence
telescopes. In P. W. Sanford, editor, Proceedings of the X.ray Optics Sympo-
sium, pages 244-275, Mullard Space Sciences Laboratory of University College,
London, April 1973.
35 PRECEDINGPAGE BLANK NOT FILMED
[23] R. A. Chipman. Polarization raytracing. In C. LondcxSn and R. E. Fischer,
editors, SPIE Proc. 766: Recent Trends in Optical System Design; Computer
Lens Design Workshop, pages 61-68, SPIE, Bellingham WA, 1987.
[24] E. AValuschka. Polarization ray trace. Opt. Eng., 28:86-89, 1989.
[25] J. S_.nchez Mandragdn and K. B. Wolf, editors. Lie Methods in Optics. Lecture
Notes in Ph_/sics, Springer-Verlag, Berlin, 1986.
[26] P. Jacquinot and Mme B. Roizen-Dossier. Apodisation. In E. Wolf, editor,
Progress in Optics, North Holland, Amsterdam, 1964.
[27] C. Whitney. Pauli-algebraic operators in polarization optics. J. Opt. Soc. Am.,
61:1207-1213, 1971.
_,_ _ _ 2.,.
37 PRECEDING PAGE BLANK NOT FILMED
Polarizationaberrations.If.Tiltedand decentered
opticalsystems
James P. MCGuire Jr.
Russell A. Chipman
May 21, 1990
Many optical systems built have tilted or decentered elements. These include
unob_ured systems and systems with fold flats. Because of typically larger angles
of incidence, polarization aberrations can be significant in these systems. Two types
of tilted _mci decentered systems composed of rotationally symmetric elements are
examined. One is systems with collinear centers of curvatures but with decentered
pupils. Symmetry in such systems allows the analysis to proceed along lines very
similar to those in Paper [. The other is systems with arbitrary tilts and decenters.
In these systems, the field dependences of the aberrations from each surface are not
concentric. The extension is made by using a polarization aberration matrix with
vector, instead of scalar, arguments.
The extension to tilted and decentered systems used in this paper is based on
the principle that each surface has an axis of symmetry; and these aberrations
can be found in the conventional fashion. Buchroeder used this principle to design
systems composed of tilted and decentered elements [1]. Thompson used vector
algebra to combine the aberration contributions from tilted and decentered elements
[2]. P_gers extended the vector techniques to explore anamorphic and keystone
distortion due to the tilt of the object relative to the elements in the optical system
[3].Section i briefly introduces the basis of aberration theory for tilted and decen-
tered systems. Section 2 outlines the coordinate system and some vector operations.
In Section 3 we present the PAM with vector arguments which is the basis of the
calculations in this paper. Section 4 examines systems with decentered pupils. Ex-
ample calculations for an infrared LIDAR beam expander are given in Section 5.
Section 6 explores aberrations in systems with arbitrary tilts and decenters. The
second order PAM of Section 3 is manipulated into a form convenient for summing a
system of tilted and decentered elements. The accuracy of this method is discussed.
A simple IR scan mirror assembly is analyzed in Section 7References to equations
and sections in Paper I are preceded I.
1Overview
Each spherical surface and its entrance pupil form a rotationally symmetric system.
Thus, each optical surface introduces aberrations of the same form, whether used
in a rotationally symmetric or unsymmetric system. The ,'enters of the aberration
contributions of each surface are displaced due to the tilt. F_gure 1 shows a spherical
surface, pupil, and object. The axis of symmetry is called tile local azis and connects
the center of the pupil and the center of curvature for the _urface. The central ray
is defined by the center of the object and the center of the pupil for the surface.
Both the object field and the pupils for each surface are fou_,,J by imaging the object
and the entrance pupil through each of the surfaces prior t,_ the surface in question.
In rotationally symmetric systems, the local axis and the ,:entral ray coincide. [n
syster_ with decentered pupils, the vector aberration expansion is made about the
line connecting the centers of curvatures of the elements [n arbitrarily tilted and
decentered systems, the vector aberration expansion for tilted optical systems is
made about the local axis of each element.
When the center of the pupil coincides, or nearly coincides, with the center of
curvature of the mirror (fie. in a Schrnidt system), the local axis is either ill-defined
or so oblique that aberration expansions about it are impractical. In these cases,
the expansion should be made about the line {onnecting the object and the center
of curvature." The opt{cal system _ rotatmnally symmetric about this hne with the
pupil decentered and tilted instead of the object.
ORIGINAL p_c IS
OF POOR QUALITY
e=(p,t) (3)
where pand ¢ are the polar pupilcoordinates.The pupil vector isnormalized so
that p = [ at the edge of a circularpupil
Vector multiplicationintroduced by Thompson [2]isanalogous to complex num-
ber multiplication
Hf- (Hp, 0+¢) (4)
and gives a vectorresult.The square root of a vector
e _.) (5)
isa vectorand followsdirectlyfrom the definitionof vectormultiplication.The dot
product
=Hpcos@
gives a scalar result. The following vector identities
(6)
C0 =
are useful in the manipulation of vector aberrations.
converting to trigonometric form.
All are easily verified by
3 Polarization Aberration Matrices with Vector
Arguments
In this section, we discuss PAMs for a single surface with vector arguments. The
PAM with scalar arguments is rewritten first for arbitrar_ object orientations and
then with vectornotation.
Firstconsiderthe second order terms in the PAM derived in Section1.9
J(H, p,¢) - exp
PooooO'o+constant piston
Po:ooH_o + quadratic piston
PoliiHpcos4) O'o + tilt
Poo2op2o'o + defocus
Pl_ooH2_'l +sm 4)_2)_r"P_,llH p ( cos.¢ _'i - _ I -","
(10)
Equation (10) describes the aberrations for objects of height Hlocated along the y-
axis. The pupil dependence is described by polar coordinates (p, 4)) with 4) measured
from the object (which lies on the y-axis). If the object is located 0from the y-axis,
then the PAM is
3PRECEDING PAGE BLANK NOT FILMED
Fold mirror
-_Primary
(a)
_Primary
Fold mirror
--_Secondar¥
(b)
Figure 3: IR LIDAR off-axis beam expander (a) and the equivalent rotationally
symmetric system (b). In systems with decentered pupits such as (a), the aber-
rations are easily calculated by analyzing the equivalent -otationally symmetric
system (b) and then decentering the pupil.
PRECEOING PAGE BLA_K NOT FILMED
Table 1: Optical design of the LIDAR off-axis beam expander and the associated
chief and marginal angles of incidence
Overall
F-number 1
Expansion ratio 16: i
Field of view 50 prad
Diameter 110 cm
Coatings Gold
Wavelength 10.6 #m
Parabolic primary
Focal length
Diameter
Off axis section diameter
Beam displacement from optical axis
110 cm
110 cm
30 cm
50 cm
Parabolic secondary
Focal length
Diameter
Beam displacement from optical axis
Paraxial angles of incidence
Chief
At primary mirror (icl)
At fold mirror (i¢2)
At secondary mirror (ic3)
Marginal
At primary mirror (i,_1)
At fold mirror (im2)
At secondary mirror (i,=3)
6.875 cm
7.5 cm
2.5 cm
-50 #tad
50 _rad
424 prad
750mrad
-500mrad
72.7mrad
Table 2: Aberration coefficients for the LIDAR beam expander
P00o0 = -0.012 - 0.092/
P020o = 0
Po_11 = 0
Po02o = 0
Pl:0o: = 3.5 x 10,1° 4- 2.8 x 10-gj
. ,: ,.... .x 10=, _1,4x lO-_j
P_o,_2 = 0.002 + O.O13j
7PRECEDii_G PAGE BLANK NOT FILMED
PooooqCro +constant p_ton
r_,,Po,.,._,_H -h,).,_o-o + til_
., -expz,Poo o , .+ dofoc s
_q P12ooqI/I 2. _. +
_q P1o_2q/_
which may be rewritten 15)
where the polarization independent aberrations P0u_ were dropped and the aber-
ration coefficientsare
P_:00 =)-_P_2oo_ (17)
q
I
_r.oo =_ _ Pl_,ooq/q (IS)
q
_1200 -2 1 _
-- at2oo- p-_:_'_:_oZP12oo_.,i (19)
q
Pllu = _Ptltlq (20)
q
1 ZPllllq_ q(21)
fftln - Ptnl q
Pro2:,' = ZPt°22q -(22)
q
Equation (16) describes the polarization dependent aberrations through second or-
der for systems of tilted and decentered elements. The interpretation of each aber-
ration is discussed below.
The first aberration to consider is polarization de.focus
Pto::jY2".4 • (23)
Since each contributionto the defocus terms isindependent of/_ and thereforeh,
the aberration is independent of the tilts or decenters of the elements.
The next term to consider is polarization ldt
_:,',,-_ "_,.;V .... - Eltlt)ff:._,. (24)
which shows the aberration resulting from a sum of tilts is of the same form as
a single surface tilt. Figure 6 shows the pupil aberration for the tilt aberration
in (a) rotationally symmetric system and (b) tilted and decentered system. The
aberration is centered in the field at 5lt_l- The strength of the aberration is given
PRECEDING PAGE BLANK NOT FILMED
,, m
J
JJj
JC'f"
,,/ //
_, \ \ p,
I1\\
(a)
$
\\1\
i J.,'l/'
Z..l/
(b)
Figure 6: The pupil aberration map for the tilt aberration. Examples for (a) ro-
tationally symmetric systems and (b) tilted or decentered systems are shown. The
object vectors are superimposed over the pupil to show the effect of the tilts and
decenters on the orientation of the aberra!ion paCtern.
11 PRECEDING PAGE BLANK NOT FILMED
(a)
Iii I I, I i (_-_'.:_
/j'/:///l/_
I I'Ill Ii_-':_'-''_°
(b)
• . = -, _,_. _,'*_tl_ .
F_gure 8: The _upd aberra_t6n map for quadrat,c pz_ton aberration. Examples for
(a) rotatlonally symmetric systems and (b) tilted or decentered systems are shown.
The object vectors are superimposed over the pupil to show the effect of the tilts
and decenters on the orientation of the aberratioa pattern.
13 PRECEDING PAGE BLANK NOT FILMED
Compensating
m ___
Scan mirror Pupil
(a)
Pupil
(b)
Compensating//, /
mirror ----
Scan mirror Pupil
Figure 10: Infrared scan mirror system pointed at (a) 22.5 a,(b) 45 ° ,and (c) 67.5 °.
The fixed pupil of the optical system is shown.
15 PRECEDING PAGE BLANK NOT FILMED
Polarization Analysis of LIDARs
Dr. James P. MCGuire, Jr. Dr. Russell A. Chipman
Physics Department
University of Alabama in Huntsville
Huntsville, Alabama 35899
June 8, 1990
I Introduction
The objective of LIDAR systems is to accurately measure atmospheric winds. The
measurement proceeds as follows. A circularly polarized beam is sent by the LIDAR
into the atmosphere. Particulates backscatter some of the light back into the LIDAR
and change the handedness of the circularly polarized fight. The return signal is
combined with the local oscillator at the heterodyne receiver using a polarizing
beamsplitter. Doppler shifts and trip time of the return beam provide velocity and
ranging information.
Any difference in the polarization state of the return beam from the expected
circular polarization state, decreases the fraction of the return signal combined
with the local oscillator. Thus changes in polarization due to the optics before the
beamsplitter, reduce signal and instrumental accuracy. Thi_ loss of signal due to
instrumental polarization can be minimized during the design phase, if polarization
is understood.
All of the mirrors in LIDAR systems change the polarization state of the light
because the rays strike at non-normal incidence. The polarization change depends
on field position, object position, and incident polarization state. Polarization aber-
rations couple some portion of the incident polarization state into the orthogonal po-
larization state. Polarization analysis of optical systems is reviewed by Chipman[1].
2Results
Significant contributions in two areas of polarization analysis were made: aberration
theory and polarization raytracing. The results are described in two papers which
were completed with funding from this contract[2,3] and have been submitted to
Optical Engineering for publication. These papers are the final report for this
contract. This contract also provided partial support for the completion of Dr.
MCGuire's dissertation [4]. Some important aspects of this research are highlighted
below.
Polarization aberration theory describes the low order polarization in an optical
system with a Taylor series approximation. This approximation is particularly good
in reflective IR systems because of the high indices of IR materials. Section II.5
uses polarization aberration theory to analyze a NASA IR LIDAR beam-expander.
Polarization aberrations were found to couple less than 1.0_ of the light into the
orthogonal polarization state.
Polarization raytracing is the analogue of conventional raytracing with the po-
larization modifying properties of the system calculated instead of optical path
differences. Current commercial polarization raytracing codes [5,6] are good at cal-
ORIGINAL PAGE IS
OF POOR QUALITY
culating the polarization of an optical system. However the results are not presented
in a form which leads to a thorough understanding of the polarization aberrations
in asystem. Section 1.8 discusses several alternative methods of display which
would make the design of LIDAR or any polarization critical system far easier. Dis-
cussions about incorporatating these new techniques have been initiated with lens
design software developers. Improved polarization analysis software should result.
References
[1] R. A. Chipman. Polarization analysis of optical systems. Opt. Eng., 28:90-99,
1989.
[2] J. P. MCGuire, Jr. and R. A. Chipman. Polarization aberrations in optical
systems. In R. E. Fisher and W. J. Smith, editors, SPIE Proc. 818: Current
Developments in Optical Engineering II, pages 240-257, SPIE, Bellingham WA,
1987. A revised version of this paper, "Polarization aberrations. I. Rotation°
ally symmetric optical systems," including exponential form and several other
refinements has been submitted to Opt. Eng. 1990.
[3] J. P. MCGuire, Jr. and R. A.. Chipman. Polarization aberrations. II. Tilted and
decentered optical systems. 1990. This paper has been submitted to Opt. Eng.
[4] J. P. MCGuire, Jr. Image formation and analysis in optical systems with polar-
ization aberrations. PhD thesis, University of Alabama in Huntsville, 1990.
[5] CODE V. Optical Research Associates, Pasadena CA, 1989.
[6] SYIVOPSYS. Eastboothbay MA, 1988.
_alo'_al _=_nau_c.s an_
_ace _In',n_raO_
i. Report No.
Final Draft
4. "ritle and Subtit_
Report Documentation Page
2. Government Accessi0n No. 3. Recipient's Catalog No.
5, Report Date
Polarization Effects
(Tasks 1 & 2)
7, Author(s)
McGuire/Chipman
9. Performing Orgeniz_ltion Name and Address
University of Alabama in Huntsville
Center for Applied Optics
Huntsville, AL 35899
12. Sponsoring Agency Name and Address
NASA/MSFC
8116/90
6. Performing Orgenization Code
5-32186
8. Performing Organization Report No,
Draft Final
10. Work Unit No.
il. Contract or Grant No.
NAS8-36955, D.O. 33
13. Ty_ Of Report and Period Covered
Final Draft
14. Sponsoring Agency Code
EB-23
15. Supplementa_ Notes
16. Abstract
1"i. Key Words°'(Suggested by Author(s)]
19. Security Classif. (of this report)
18. Distribution Statement
20. Security Classif. (of this page) 21. No. of pages 22. Price
/
/
NASA FORM I(126 OCT 86
