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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 61 "Analytical Study of Optic
al Wavefront Aberrations Using Maple" }}{PARA 19 "" 0 "" {TEXT -1 15 "
Marc A. Murison" }}{PARA 256 "" 0 "" {TEXT -1 103 "Astronomical Applic
ations Dept.\nU.S. Naval Observatory\n3450 Massachusetts Ave., NW\nWas
hington, DC 20392" }}{PARA 258 "" 0 "" {TEXT -1 60 "murison@riemann.us
no.navy.mil\nhttp://aa.usno.navy.mil/AESOP/" }}{PARA 257 "" 0 "" 
{TEXT -1 10 "1 May 1996" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}
{PARA 259 "" 0 "" {TEXT 377 8 "ABSTRACT" }}{PARA 260 "" 0 "" {TEXT -1 
35 "This paper describes a package for " }{TEXT 383 10 "analytical" }
{TEXT -1 643 " ray tracing of relatively simple optical systems.  AESO
P (An Extensible Symbolic Optics Package) enables analysis of the effe
cts of small optical element misalignments or other perturbations.  (I
t is possible to include two or more simultaneous independent perturba
tions.)  Wavefront aberrations and optical path variations can be stud
ied as functions of the perturbation parameters.  The power of this ap
proach lies in the fact that the results can be manipulated algebraica
lly, allowing determination of misalignment tolerances as well as deve
loping physical intuition, especially in the picometer regime of optic
al path length variations. " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "1
. Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 304 "Modern high-pre
cision optical systems, such as space astrometric interferometers (e.g
. FAME: Johnston et al. 1997; POINTS: Reasenberg et al. 1996; GAIA: Lo
iseau and Malbet 1996, Loiseau and Shaklan 1996; Lindegren and Perryma
n 1996; SIM: SIM97), can require optical path tolerances in the sub-na
nometer (" }{XPPEDIT 18 0 "1*nm = 10^(-9) *m" "/*&\"\"\"\"\"\"%#nmGF%*
&)\"#5,$\"\"*!\"\"F%%\"mGF%" }{TEXT -1 16 ") to picometer (" }
{XPPEDIT 18 0 "1* pm = 10^(-12) * m" "/*&\"\"\"\"\"\"%#pmGF%*&)\"#5,$
\"#7!\"\"F%%\"mGF%" }{TEXT -1 50 ") regimes over total path lengths on
 the order 10 " }{TEXT 257 1 "m" }{TEXT -1 503 ".  Such tolerances pla
ce extreme requirements on optical analysis programs.  Two questions a
re of paramount importance:  1) to which specific perturbations is a s
ystem most sensitive? and 2) are there couplings between different per
turbations that produce high sensitivities (i.e., are there strong cor
relations between perturbation parameters)?  AESOP can be used to answ
er these questions (Murison, 1993), as well as to develop physical int
uition in the picometer optical path difference (OPD) regime." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "A
 common optical subsystem employed in astronomical interferometers is \+
a " }{TEXT 365 15 "beam compressor" }{TEXT -1 86 ", used to convert a \+
large aperture input beam (starlight) to a narrow output beam (~1 " }
{TEXT 364 2 "cm" }{TEXT -1 583 ") suitable for combining with another \+
such beam to produce interference fringes of sufficient visibility.  A
 typical beam compressor consists of a pair of confocal paraboloidal m
irrors, as sketched in Figure 1.  If perfectly aligned, a flat input w
avefront results in a radially compressed flat output wavefront.  Misa
lignment analysis of even such a simple system as this generally requi
res resorting to numerical programs.  Usually, such programs are ill-s
uited for studies involving both misalignment parameter variation and \+
aperture-averaged OPD determination, especially in the " }{TEXT 363 2 
"pm" }{MPLTEXT 0 21 0 "" }{TEXT -1 277 " regime.  The need for picomet
er OPD tolerances is a relatively recent development, driven by ever m
ore demanding sciece objectives.  Such tolerance requirements will lik
ely become more common, and the lack of adequate analysis tools will c
orrespondingly be felt more strongly." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "To develop a physical underst
anding of alignment sensitivities, one would much prefer an " }{TEXT 
258 10 "analytical" }{TEXT -1 786 " rather than a numerical descriptio
n of the output wavefront as a function of the misalignment parameters
.  Unfortunately, an analytical wavefront description of misaligned op
tical systems as simple as a beam compressor, or even a single focussi
ng optic, can be too complex to attempt by hand in the kind of detail \+
required for sensitivity studies (see, e.g., Noecker et al. 1993).  Ho
wever, computer algebra systems such as Maple have advanced to such a \+
state of capability and sophistication that, coupled with the processi
ng power of modern desktop computers, complete analytical descriptions
 are, as we shall see, now becoming possible.  This paper describes AE
SOP (An Extensible Symbolic Optics Package), an analytical ray tracing
 package written in the Maple programming language." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 480 "AESOP was deve
loped to support the analysis effort involved in determining critical \+
sensitivities to optic misalignments in a proposed dual interferometri
c astrometric telescope, POINTS (Reasenberg et al. 1988, 1995a, 1995b,
 1996).  POINTS consists of a pair of independent Michelson stellar in
terferometers and a laser metrology system that measures both the crit
ical starlight paths and the angle between the two interferometer base
lines.  The nominal design has baselines of 2 " }{TEXT 317 1 "m" }
{TEXT -1 28 ", telescope apertures of 35 " }{TEXT 318 2 "cm" }{TEXT 
-1 557 ", and observes target stars separated by roughly 90 degrees.  \+
One of the distinguishing features of POINTS is that it employs hologr
aphic optical elements (HOEs) to accomplish picometer metrology over t
he full aperture of the starlight optical path.  See the Reasenberg et
 al. references for a full description of the instrument, its capabili
ties, and the astrophysical, astrometric, and planet-finding science t
hat POINTS would significantly impact.  Also, information can be found
 on the POINTS web pages at http://www-cfa.harvard.edu/~reasen/points.
html." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 392 "A key analysis problem regarding the POINTS interferomet
ers is the determination of optical path length errors as a function o
f various optical element misalignments. The path length error budget \+
in a precision system such as this is only several tens of picometers.
  With such a tight error budget, it is imperative to determine which \+
perturbations lead to large path length errors.  At the " }{TEXT 259 
2 "pm" }{TEXT -1 670 " level, often we cannot trust our optical intuit
ion in determining misalignment sensitivities.  In such cases, we must
 rely on numerical analysis to an uncomfortable degree, lacking reliab
le independent checks on the numerical results.  AESOP was created in \+
part to fill this niche.  In the case of POINTS, a numerical program c
alled RayTrace (see Murison (1993) for a description) was written spec
ifically to perform ultra-high precision, sub-picometer OPD variation \+
analyses.  AESOP was developed in parallel with RayTrace.  The two ana
lysis approaches -- numerical and analytic -- are completely independe
nt and therefore serve as excellent checks upon one another.  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 257 "
AESOP traces an input ray through a misaligned optical system and prod
uces an analytic description of the output ray as a function of the sy
stem parameters, the misalignment parameters, and the input ray positi
on and direction.  A crucial diagnostic is the " }{TEXT 319 17 "apertu
re-averaged" }{TEXT -1 819 " OPD variation.  The physical principles i
nvolved are quite simple, since AESOP takes a classical geometric opti
cs approach.  At a given reflecting surface, all that is required is t
o calculate the reflected ray direction and the accumulated optical pa
th up to that intersection point.  Similarly, at a refracting surface \+
we use Snell's law to calculate the refracted ray direction.  If a hol
ographic optical element (HOE) is encountered, it is a similarly simpl
e process to calculate the output ray direction and the change in opti
cal phase across the element (Murison and Noecker, 1993).  The POINTS \+
optical subsystems involve all three types of optical surfaces.  AESOP
 currently handles reflecting and holographic optical elements.  Refra
cting surfaces will be easy to add, due to the extensible structure of
 AESOP." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 632 "As simple as the physics is, such a ray tracing process \+
is impossible to do analytically by hand (especially aperture-averaged
 effects and wavefront analysis).  The inhibiting factor is the rapidl
y increasing (with each successive optical surface) complexity of the \+
intermediate expressions that must be algebraically manipulated.  This
 kind of repetitive manipulation of unwieldy objects is precisely what
 computers can do well.  Hence, a programmable computer algebra system
 like Maple is well suited in principle to analyzing misaligned optica
l systems, at least simple ones involving relatively few focussing opt
ical surfaces.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 1022 "In section 2, a few mathematical topics relevant
 to this kind of ray tracing in general and to AESOP in particular are
 briefly reviewed.  Most optical surfaces are conicoids, so quadric su
rfaces and their normal vectors are introduced in section 2.1.  Sectio
n 2.2 introduces the concepts of optical rays, ray bundles, and wavefr
onts.  Sections 2.3 and 2.4 explain the methods used by AESOP to deter
mine the intersection of a ray with an optical surface and calculate t
he exit ray direction.  Section 2.5 covers the averaging of wavefront \+
error over the aperture of a centrally obstructed optical system (comm
on in astronomical systems).  Section 2.6 introduces the expansion of \+
a perturbed wavefront in a Zernike series, the low-order components of
 which correspond to the classical wavefront aberrations such as coma,
 astigmatism, etc.  The Zernike series representation of a wavefront i
s extremely convenient, instructive, and a helpful aid in the analysis
 of perturbed optical systems (as we shall see in Appendix B).  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 857 "
Assuming the background material in section 2, section 3 covers the AE
SOP design approach.  Optical systems and geometric ray tracing lend t
hemselves naturally to an object-oriented design, which AESOP takes fu
ll advantage of.  Indeed, many design simplifications result, helping \+
to make analytical ray tracing not only feasible but extensible as wel
l.  In practice, it has proven easy to extend AESOP capabilities as ne
w ones are needed, though one does have to be careful to enforce the e
ncapsulation structure of the AESOP objects, since the Maple language \+
is not itself object oriented.  Finally, section 3.2 gives an overview
 of the AESOP ray tracing process.  Appendix A contains a calculation \+
of the unit normal vector for quadric surfaces.  Appendix B includes e
xcerpts of a Maple session in which a misaligned beam compressor is an
alyzed by AESOP. " }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "2. Mathema
tical Considerations" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 20 "2.1 Quadr
ic Surfaces" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 22 "2.1.1 Surface Fami
lies" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "A general conicoid, or qua
dric surface, has the form" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "    \+
 " }{XPPEDIT 18 0 "s(rho) = rho^2/( 2*f + sqrt(4*f^2 - epsilon*rho^2 )
" "/-%\"sG6#%$rhoG*&F&\"\"#,&*&\"\"#\"\"\"%\"fGF,F,-%%sqrtG6#,&*&\"\"%
F,*$F-\"\"#F,F,*&%(epsilonGF,*$F&\"\"#F,!\"\"F,F:" }{TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "rho" "I$rho
G6\"" }{TEXT -1 58 " is the perpendicular distance from the axis of sy
mmetry, " }{XPPEDIT 18 0 "f" "I\"fG6\"" }{TEXT -1 40 " is the focal le
ngth at the vertex, and " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }
{TEXT -1 68 " is the conic constant.  The quadric surface types as a f
unction of " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 5 " ar
e:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "epsil
on" "I(epsilonG6\"" }{TEXT -1 40 " = 0         paraboloid of focal len
gth " }{XPPEDIT 18 0 "f" "I\"fG6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 5 "     " }{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 32 " =
 1         spheroid of radius " }{XPPEDIT 18 0 "2*f" "*&\"\"#\"\"\"%\"
fGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "eps
ilon" "I(epsilonG6\"" }{TEXT -1 24 " < 0         hyperboloid" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "     0 < " }{XPPEDIT 18 0 "epsilon
" "I(epsilonG6\"" }{TEXT -1 23 " < 1   prolate spheroid" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "epsilon" "I(epsilon
G6\"" }{TEXT -1 28 " > 1         oblate spheroid" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 42 "The eccentricity of a prolate spheroid is " }
{XPPEDIT 18 0 "e = sqrt( (1 - epsilon)/epsilon )" "/%\"eG-%%sqrtG6#*&,
&\"\"\"\"\"\"%(epsilonG!\"\"F*F+F," }{TEXT -1 36 ", and that of an obl
ate spheroid is " }{XPPEDIT 18 0 "e = sqrt( (epsilon - 1)/epsilon )" "
/%\"eG-%%sqrtG6#*&,&%(epsilonG\"\"\"\"\"\"!\"\"F*F)F," }{TEXT -1 46 ".
  Commonly, the conic constant is denoted by " }{XPPEDIT 18 0 "k = eps
ilon - 1" "/%\"kG,&%(epsilonG\"\"\"\"\"\"!\"\"" }{TEXT -1 9 " so that \+
" }{XPPEDIT 18 0 "k = 0" "/%\"kG\"\"!" }{TEXT -1 203 " refers to a sph
eroid.  The quantity used here is more convenient for our purposes, si
nce the paraboloid is the most frequently encountered focussing surfac
e in astronomical optics.  For small values of " }{XPPEDIT 18 0 "rho" 
"I$rhoG6\"" }{TEXT -1 8 " we have" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 
1 0 12 "assume(f>0):" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 51 "sag :
= rho^2 / (2*f + sqrt(4*f^2 - epsilon*rho^2)):" }}}{EXCHG {PARA 0 "" 
0 "" {MPLTEXT 1 0 31 "s(rho) = series( sag, rho, 7 );" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"sG6#%$rhoG++F',$*$%#f|irG!\"\"#\"\"\"\"\"%\"
\"#,$*&%(epsilonGF.F+!\"$#F.\"#k\"\"%,$*&F3\"\"#F+!\"&#F.\"$7&\"\"'-%
\"OG6#F.\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 409 "The leading ter
m we recognize as the sagitta of a paraboloid.  All conicoids are para
boloidal at second order.  Hence, all conicoids exhibit arbitrarily go
od focussing of a sufficiently narrow, on-axis input wavefront.  The f
irst aberrational term is spherical aberration, which enters in with a
 linear term at fourth order in radius.  We see that wavefront aberrat
ions are a function of conicoid family type." }}}}{SECT 0 {PARA 5 "" 
0 "" {TEXT -1 31 "2.1.2 The Surface Normal Vector" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 11 "The vector " }{XPPEDIT 18 0 "grad s" "*&%%gradG\"
\"\"%\"sGF$" }{TEXT -1 38 " is the projection onto the xy plane (" }
{XPPEDIT 18 0 "rho = sqrt( x^2 + y^2)" "/%$rhoG-%%sqrtG6#,&*$%\"xG\"\"
#\"\"\"*$%\"yG\"\"#F+" }{TEXT -1 37 ") of a vector which is normal to \+
the " }{TEXT 320 6 "convex" }{TEXT -1 53 " side of the surface s.  Def
ine the surface function " }{XPPEDIT 18 0 "g(x,y,z) = z - s(x,y)" "/-%
\"gG6%%\"xG%\"yG%\"zG,&F(\"\"\"-%\"sG6$F&F'!\"\"" }{TEXT -1 7 ".  Then
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "N = gra
d(g)/abs(grad(g)) " "/%\"NG*&-%%gradG6#%\"gG\"\"\"-%$absG6#-F&6#F(!\"
\"" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "[-diff(s,x),-diff(s,y),1]/abs(
grad(g)) " "*&7%,$-%%diffG6$%\"sG%\"xG!\"\",$-F&6$F(%\"yGF*\"\"\"\"\"
\"-%$absG6#-%%gradG6#%\"gGF*" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 46 "is a unit vector normal to the surface on the " }{TEXT 
321 7 "concave" }{TEXT -1 220 " side.  Calculation of the surface norm
al vector is the crucial step in determining the output direction of a
 ray incident on an optical surface.  As shown in Appendix A, we have \+
a relatively simple result for conicoids:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "    " }{XPPEDIT 18 0 "N = VECTOR( [ -x/sqrt((1-epsilon)*rh
o^2+4*f^2), -y/sqrt((1-epsilon)*rho^2+4*f^2), sqrt(4*f^2-epsilon*rho^2
)/sqrt((1-epsilon)*rho^2+4*f^2) ] )" "/%\"NG-%'VECTORG6#7%,$*&%\"xG\"
\"\"-%%sqrtG6#,&*&,&\"\"\"F+%(epsilonG!\"\"F+*$%$rhoG\"\"#F+F+*&\"\"%F
+*$%\"fG\"\"#F+F+F4F4,$*&%\"yGF+-F-6#,&*&,&\"\"\"F+F3F4F+*$F6\"\"#F+F+
*&\"\"%F+*$F;\"\"#F+F+F4F4*&-F-6#,&*&\"\"%F+*$F;\"\"#F+F+*&F3F+*$F6\"
\"#F+F4F+-F-6#,&*&,&\"\"\"F+F3F4F+*$F6\"\"#F+F+*&\"\"%F+*$F;\"\"#F+F+F
4" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{SECT 0 {PARA 
4 "" 0 "" {TEXT -1 37 "2.2 Rays, Ray Bundles, and Wavefronts" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "In the geometric optics regime, we
 may develop the concept of a wavefront " }{TEXT 267 1 "W" }{TEXT -1 
53 " as follows.  Consider an infinitely narrow beam, or " }{TEXT 260 
4 "ray," }{TEXT -1 1 " " }{TEXT 261 1 "r" }{TEXT -1 16 ", defined by a
n " }{TEXT 262 6 "anchor" }{TEXT -1 1 " " }{TEXT 263 5 "point" }{TEXT 
-1 1 " " }{TEXT 264 1 "p" }{TEXT -1 74 ", a point in space from which \+
the ray originates; a propagation direction " }{TEXT 265 1 "v" }{TEXT 
-1 98 ", conveniently but not necessarily represented as a unit vector
; and an optical path length (OPL) " }{TEXT 266 1 "t" }{TEXT -1 37 ", \+
defined as the index of refraction " }{TEXT 373 1 "n" }{TEXT -1 64 " o
f the propagation medium integrated over a geometric distance " }
{TEXT 374 1 "L" }{TEXT -1 6 " from " }{TEXT 375 1 "p" }{TEXT -1 7 " al
ong " }{TEXT 376 1 "v" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "t = int( n(l),
 l=0..L" "/%\"tG-%$intG6$-%\"nG6#%\"lG/F*;\"\"!%\"LG" }{TEXT -1 23 ". \+
 Hence, we may write " }{XPPEDIT 18 0 "r = p + t*v" "/%\"rG,&%\"pG\"\"
\"*&%\"tGF&%\"vGF&F&" }{TEXT -1 3 ".  " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 44 "A wavefront may be viewed as a surface W in " }{XPPEDIT 
18 0 "R^3" "*$%\"RG\"\"$" }{TEXT -1 124 " of constant optical phase pr
opagating through space (or through an optical medium).  An infinitesi
mally small neighborhood " }{XPPEDIT 18 0 "U epsilon W" "*(%\"UG\"\"\"
%(epsilonGF$%\"WGF$" }{TEXT -1 15 " of each point " }{XPPEDIT 18 0 "p \+
epsilon W" "*(%\"pG\"\"\"%(epsilonGF$%\"WGF$" }{TEXT -1 27 " propagate
s in a direction " }{XPPEDIT 18 0 "v(p)" "-%\"vG6#%\"pG" }{TEXT -1 19 
" that is normal to " }{TEXT 268 1 "W" }{TEXT -1 4 " at " }{TEXT 269 
1 "p" }{TEXT -1 71 ".  We can therefore associate a ray with each poin
t of W.  We define a " }{TEXT 270 3 "ray" }{TEXT -1 1 " " }{TEXT 271 
6 "bundle" }{TEXT -1 33 " as the set of rays belonging to " }{TEXT 
272 1 "W" }{TEXT -1 141 ".  A concave wavefront (or portion of a wavef
ront) produces a converging ray bundle, and a convex wavefront produce
s a diverging ray bundle. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 667 "In an analytical ray tracing proc
edure, one can consider a single ray that is transformed by passage th
rough an optical system.  The resulting output ray then strikes a dete
ctor surface.  The position on the detector of the output ray intersec
tion point, the output ray direction, and the total OPL (the optical p
ath from the incident ray anchor point to the detector surface) are al
l functions of the input ray anchor point and direction.  Taking the i
nput ray anchor point position as lying on an incident wavefront (a fu
nction we can represent analytically, usually a plane), we can constru
ct a corresponding output wavefront from the ray trace of the input ra
y.  " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "2.3 Surface Intersectio
n Point" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 290 "Given a ray propagatin
g toward an optical surface, we must find the intersection point of th
e ray with that surface.  Define the surface local coordinate frame wi
th origin at the surface vertex and Z axis along the vertex normal.  A
n equation for an input ray parameterized by optical path " }{TEXT 
273 1 "t" }{TEXT -1 3 " is" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "    \+
 " }{XPPEDIT 18 0 "r(t)=r[0] + t*v" "/-%\"rG6#%\"tG,&&F$6#\"\"!\"\"\"*
&F&F+%\"vGF+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }
{XPPEDIT 18 0 "r[0]=[x[0],y[0],z[0]]" "/&%\"rG6#\"\"!7%&%\"xG6#F&&%\"y
G6#F&&%\"zG6#F&" }{TEXT -1 29 " is the ray anchor point and " }{TEXT 
274 1 "v" }{TEXT -1 29 " is the unit direction vector" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "v = [ sin psi * cos
 lambda, sin psi * sin lambda, cos psi ]" "/%\"vG7%**%$sinG\"\"\"%$psi
GF'%$cosGF'%'lambdaGF'**F&F'F(F'F&F'F*F'*&F)F'F(F'" }{TEXT -1 3 " = " 
}{XPPEDIT 18 0 " [ v[x], v[y], v[z] ]" "7%&%\"vG6#%\"xG&F$6#%\"yG&F$6#
%\"zG" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "where (" 
}{XPPEDIT 18 0 "psi,lambda" "6$%$psiG%'lambdaG" }{TEXT -1 239 ") are t
he polar and azimuthal angles with respect to the +Z axis and counterc
lockwise from the +X axis in the local coordinate frame, respectively.
  The equation of the surface -- the sagitta -- in the local coordinat
e frame is of the form" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }
{XPPEDIT 18 0 "g(x,y,z) = z - s(x.y)" "/-%\"gG6%%\"xG%\"yG%\"zG,&F(\"
\"\"-%\"sG6#(F&F'!\"\"" }{TEXT -1 4 " = 0" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 25 "We can find the value of " }{TEXT 277 1 "t" }{TEXT -1 68 
" which corresponds to the intersection point (x,y,z) by substituting
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "x = x[0
] + t*sin psi * cos lambda" "/%\"xG,&&F#6#\"\"!\"\"\"*,%\"tGF(%$sinGF(
%$psiGF(%$cosGF(%'lambdaGF(F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "x[0] \+
+ t*v[x]" ",&&%\"xG6#\"\"!\"\"\"*&%\"tGF'&%\"vG6#F$F'F'" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "y = y[0] + t*sin ps
i * sin lambda" "/%\"yG,&&F#6#\"\"!\"\"\"*,%\"tGF(%$sinGF(%$psiGF(F+F(
%'lambdaGF(F(" }{TEXT -1 5 "   = " }{XPPEDIT 18 0 "y[0] + t*v[y]" ",&&
%\"yG6#\"\"!\"\"\"*&%\"tGF'&%\"vG6#F$F'F'" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 5 "     " }{XPPEDIT 18 0 "z = z[0] + t*cos psi" "/%\"zG,&&F#6
#\"\"!\"\"\"*(%\"tGF(%$cosGF(%$psiGF(F(" }{TEXT -1 12 "          = " }
{XPPEDIT 18 0 "z[0] + t*v[z]" ",&&%\"zG6#\"\"!\"\"\"*&%\"tGF'&%\"vG6#F
$F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "into the scalar equation
 " }{XPPEDIT 18 0 "g(x,y,z)" "-%\"gG6%%\"xG%\"yG%\"zG" }{TEXT -1 17 " \+
and solving for " }{TEXT 275 1 "t" }{TEXT -1 173 ".  For a (perhaps pe
rturbed) quadric surface there will in general be two solutions.  AESO
P automatically chooses the correct solution.  Then we substitute the \+
solution for " }{TEXT 276 1 "t" }{TEXT -1 37 " back into the equations
 for the ray," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 
18 0 "r(t) = r[0] + t*v" "/-%\"rG6#%\"tG,&&F$6#\"\"!\"\"\"*&F&F+%\"vGF
+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "to determine the x, y, and
 z values of the intersection point.  " }}}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 22 "2.4 Exit Ray Direction" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 388 "Upon encountering an optical surface, a ray must know how to i
nteract with that surface and choose an output direction.  After the i
ntersection point is found, we determine the unit normal vector at tha
t point, thus providing a local reference for measuring input and outp
ut angles.  The normal vector is easily found from the gradient of the
 surface equation at the intersection point.  " }}}{SECT 0 {PARA 5 "" 
0 "" {TEXT -1 16 "2.4.1 Reflection" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
348 "For a reflecting surface, the incident and reflecting angles are \+
equal.  The reflected beam lies in the plane defined by the incident b
eam and the normal to the surface at the intersection point.  Thus, we
 can write the reflected ray direction as being equal to the incident \+
ray direction plus a component along the surface normal vector directi
on," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "v[r]
 = v[i] + alpha*N" "/&%\"vG6#%\"rG,&&F$6#%\"iG\"\"\"*&%&alphaGF+%\"NGF
+F+" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "where subs
cripts " }{TEXT 279 1 "r" }{TEXT -1 5 " and " }{TEXT 278 1 "i" }{TEXT 
-1 62 " correspond to the reflected and incident rays, respectively, \+
" }{TEXT 280 1 "N" }{TEXT -1 61 " is the normal to the surface at the \+
intersection point, and " }{XPPEDIT 18 0 "alpha" "I&alphaG6\"" }{TEXT 
-1 103 " is a scale factor that must be determined.  Since the inciden
t and reflected angles are equal, we have" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 5 "     " }{XPPEDIT 18 0 "dot( N, v[r] ) / (abs(N)*abs(v[r]) \+
) = -dot( N, v[i] ) / (abs(N)*abs(v[i])) " "/*&-%$dotG6$%\"NG&%\"vG6#%
\"rG\"\"\"*&-%$absG6#F'F,-F/6#&F)6#F+F,!\"\",$*&-F%6$F'&F)6#%\"iGF,*&-
F/6#F'F,-F/6#&F)6#F<F,F5F5" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 17 "Combine this with" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "
     " }{XPPEDIT 18 0 "dot( N, v[r] ) = dot( N, v[i] ) + alpha*abs(N)^
2" "/-%$dotG6$%\"NG&%\"vG6#%\"rG,&-F$6$F&&F(6#%\"iG\"\"\"*&%&alphaGF1*
$-%$absG6#F&\"\"#F1F1" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 133 "and the further condition that the magnitude of the refl
ected beam is equal to the magnitude of the incident beam (certainly t
rue if " }{XPPEDIT 18 0 "v[i]" "&%\"vG6#%\"iG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "v[r]" "&%\"vG6#%\"rG" }{TEXT -1 41 " are unit vectors),
 and we can solve for " }{XPPEDIT 18 0 "alpha" "I&alphaG6\"" }{TEXT 
-1 1 ":" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "
alpha = -2 * dot( N, v[i] ) / abs(N)^2" "/%&alphaG,$*(\"\"#\"\"\"-%$do
tG6$%\"NG&%\"vG6#%\"iGF'*$-%$absG6#F+\"\"#!\"\"F5" }{TEXT -1 1 " " }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "After " }{XPPEDIT 18 0 "v[r]" "&%\"
vG6#%\"rG" }{TEXT -1 137 " is determined, the result is then transform
ed back to the global reference frame for propagation of the ray to th
e next optical surface." }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 16 "2.4.
2 Refraction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 160 "For a refracting \+
surface, we must use Snell's law, which leads to more complications th
an the simple law of reflection.  As in the reflecting case, we may wr
ite" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "v[r]
 = v[i] + alpha*N" "/&%\"vG6#%\"rG,&&F$6#%\"iG\"\"\"*&%&alphaGF+%\"NGF
+F+" }{TEXT -1 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "where the
 subscript " }{TEXT 285 1 "r" }{TEXT -1 69 " is associated with the re
fracted ray.  But now we have the condition" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 5 "     " }{XPPEDIT 18 0 "n[i] * sin theta[i] = n[r] * sin \+
theta[r]" "/*(&%\"nG6#%\"iG\"\"\"%$sinGF(&%&thetaG6#F'F(*(&F%6#%\"rGF(
F)F(&F+6#F0F(" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "w
here " }{XPPEDIT 18 0 "n[i]" "&%\"nG6#%\"iG" }{TEXT -1 5 " and " }
{XPPEDIT 18 0 "n[r]" "&%\"nG6#%\"rG" }{TEXT -1 36 " are the indices of
 refraction and  " }{XPPEDIT 18 0 "theta[i]" "&%&thetaG6#%\"iG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta[r]" "&%&thetaG6#%\"rG" }
{TEXT -1 66 " are the corresponding angles of incidence and refraction
.  Using " }{XPPEDIT 18 0 "dot( N, v[r] ) = dot( N, v[i] ) + alpha*abs
(N)^2" "/-%$dotG6$%\"NG&%\"vG6#%\"rG,&-F$6$F&&F(6#%\"iG\"\"\"*&%&alpha
GF1*$-%$absG6#F&\"\"#F1F1" }{TEXT -1 17 " , the condition " }{XPPEDIT 
18 0 "abs(v[r]) = abs(v[i])" "/-%$absG6#&%\"vG6#%\"rG-F$6#&F'6#%\"iG" 
}{TEXT -1 53 " , and Snell's law, we find that the scale factor is " }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "alpha = ab
s(v[i])*( sqrt(1-(n[i]/n[r])^2*sin^2 theta[i] ) - cos theta[i] )/abs(N
)" "/%&alphaG*(-%$absG6#&%\"vG6#%\"iG\"\"\",&-%%sqrtG6#,&\"\"\"F,*(*&&
%\"nG6#F+F,&F66#%\"rG!\"\"\"\"#%$sinG\"\"#&%&thetaG6#F+F,F;F,*&%$cosGF
,&F@6#F+F,F;F,-F&6#%\"NGF;" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "theta[i]" "&%&thetaG6#%\"iG" }
{TEXT -1 51 " is determined from the incident beam direction via" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "cos theta[i]
 = dot(N,v[i])/(abs(N)*abs(v[i]))" "/*&%$cosG\"\"\"&%&thetaG6#%\"iGF%*
&-%$dotG6$%\"NG&%\"vG6#F)F%*&-%$absG6#F.F%-F46#&F06#F)F%!\"\"" }{TEXT 
-1 1 " " }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 21 "2.4.3 HOE Diffractio
n" }}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 39 "2.4.3.1 Direction of the Di
ffracted Ray" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 536 "An adequate descr
iption of ray tracing across a holographic optical element (HOE) is be
yond the scope of this paper.  However, it is part of AESOP's current \+
capabilities, so I present the relevant equations here, without motiva
tion or proof.  HOE ray tracing is mostly neglected in the standard op
tics texts, with the exception of Welford (1986).  Even in the latter,
 the account is incomplete, cursory, and potentially misleading.  The \+
reader is referred instead to Murison and Noecker (1993) for a complet
e and accurate development.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Define the quantity " }{XPPEDIT 
18 0 "T = +1" "/%\"TG\"\"\"" }{TEXT -1 71 " for transmission (i.e., th
e incident ray passes through the HOE), and " }{XPPEDIT 18 0 "T = -1" 
"/%\"TG,$\"\"\"!\"\"" }{TEXT -1 105 " for reflection (for example a di
ffraction grating ruled on the surface of a mirror).  Then let us defi
ne" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "S = T
*sign( dot(N,v[i]) )" "/%\"SG*&%\"TG\"\"\"-%%signG6#-%$dotG6$%\"NG&%\"
vG6#%\"iGF&" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "whe
re " }{TEXT 282 1 "N" }{TEXT -1 12 " is now the " }{TEXT 281 4 "unit" 
}{TEXT -1 54 " normal vector at the surface intersection point, and " 
}{XPPEDIT 18 0 "v[i]" "&%\"vG6#%\"iG" }{TEXT -1 130 " is the incident \+
beam direction of propagation (also now required to be a unit vector).
  Additionally, define the auxiliary vector" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 5 "     " }{XPPEDIT 18 0 "u = cross(N,v[i]) - (m*lambda)/la
mbda[c] * cross( N, k[c[1]]-k[c[2]]" "/%\"uG,&-%&crossG6$%\"NG&%\"vG6#
%\"iG\"\"\"**%\"mGF-%'lambdaGF-&F06#%\"cG!\"\"-F&6$F(,&&%\"kG6#&F36#\"
\"\"F-&F96#&F36#\"\"#F4F-F4" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 6 "where " }{TEXT 283 1 "m" }{TEXT -1 40 " is the diffracti
on order (an integer), " }{XPPEDIT 18 0 "lambda" "I'lambdaG6\"" }
{TEXT -1 73 " is the readout wavelength (i.e., wavelength of the incid
ent wavefront), " }{XPPEDIT 18 0 "lambda[c]" "&%'lambdaG6#%\"cG" }
{TEXT -1 41 " is the HOE construction wavelength, and " }{XPPEDIT 18 
0 "k[c[1]]" "&%\"kG6#&%\"cG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 
0 "k[c[2]]" "&%\"kG6#&%\"cG6#\"\"#" }{TEXT -1 114 " are related to the
 unit vectors directed from the HOE construction points to the surface
 intersection point.  If " }{XPPEDIT 18 0 "v[1],v[2]" "6$&%\"vG6#\"\"
\"&F$6#\"\"#" }{TEXT -1 30 " are those unit vectors, then " }{XPPEDIT 
18 0 "k[c[1]] = V[1]*v[1]" "/&%\"kG6#&%\"cG6#\"\"\"*&&%\"VG6#\"\"\"\"
\"\"&%\"vG6#\"\"\"F/" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "k[c[2]] = V[
2]*v[2]" "/&%\"kG6#&%\"cG6#\"\"#*&&%\"VG6#\"\"#\"\"\"&%\"vG6#\"\"#F/" 
}{TEXT -1 8 ", where " }{XPPEDIT 18 0 "V[k] = +1" "/&%\"VG6#%\"kG\"\"
\"" }{TEXT -1 35 " (k=1,2) if the construction point " }{XPPEDIT 18 0 
"C[k]" "&%\"CG6#%\"kG" }{TEXT -1 21 " is a real focus and " }{XPPEDIT 
18 0 "V[k] = -1" "/&%\"VG6#%\"kG,$\"\"\"!\"\"" }{TEXT -1 27 " if the c
onstruction point " }{XPPEDIT 18 0 "C[k]" "&%\"CG6#%\"kG" }{TEXT -1 
83 " is a virtual focus.  Then the diffracted ray direction, a unit ve
ctor, is given by" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }
{XPPEDIT 18 0 "v[r] = S*sqrt(1-u^2) * N - cross(N,u)" "/&%\"vG6#%\"rG,
&*(%\"SG\"\"\"-%%sqrtG6#,&\"\"\"F**$%\"uG\"\"#!\"\"F*%\"NGF*F*-%&cross
G6$F4F1F3" }{TEXT -1 1 " " }}}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 31 "2
.4.3.2 Optical Path Correction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 218 
"In geometrical ray tracing of a HOE, a correction must be added to th
e optical path upon traversing the HOE surface.  Again, refer to Muris
on and Noecker (1993) for a detailed development. The corrected optica
l path is" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 
0 "L = L[0](p) - Delta L(p)" "/%\"LG,&-&F#6#\"\"!6#%\"pG\"\"\"*&%&Delt
aGF+-F#6#F*F+!\"\"" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 6 "where " }{XPPEDIT 18 0 "L[0](p)" "-&%\"LG6#\"\"!6#%\"pG" }{TEXT 
-1 55 " is the optical path to the surface intersection point " }
{TEXT 284 1 "p" }{TEXT -1 47 " as calculated in the normal geometric w
ay, and" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "
Delta L(p) = m*lambda/lambda[c] * (D(p) - D(p[0] )" "/*&%&DeltaG\"\"\"
-%\"LG6#%\"pGF%**%\"mGF%%'lambdaGF%&F,6#%\"cG!\"\",&-%\"DG6#F)F%-F36#&
F)6#\"\"!F0F%" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "
is the phase correction.  " }{XPPEDIT 18 0 "p[0]" "&%\"pG6#\"\"!" }
{TEXT -1 122 " is an arbitrary reference point; for a conicoid, a good
 choice is the location of the surface vertex.  [The location of  " }
{XPPEDIT 18 0 "p[0]" "&%\"pG6#\"\"!" }{TEXT -1 117 " can be arbitrary \+
since it introduces a constant offset in the optical path.  We are onl
y interested in optical path " }{TEXT 322 11 "differences" }{TEXT -1 
119 " (or variations).]  The distance function D is a function of the \+
HOE construction points and is given by the expression" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "D(p) = abs( V[1]*v[
1](p) - V[2]*v[2](p) )" "/-%\"DG6#%\"pG-%$absG6#,&*&&%\"VG6#\"\"\"\"\"
\"-&%\"vG6#\"\"\"6#F&F0F0*&&F-6#\"\"#F0-&F36#\"\"#6#F&F0!\"\"" }{TEXT 
-1 1 " " }}}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 41 "2.5 OPD Averaged o
ver an Annular Aperture" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "One ma
y define the optical path difference (OPD) as the difference in total \+
optical path through a system, starting from an initial ray anchor pos
ition (" }{XPPEDIT 18 0 "rho,phi" "6$%$rhoG%$phiG" }{TEXT -1 435 "), m
inus the total optical path of an axial ray through the unperturbed sy
stem (the fiducial, or chief, ray).  The output wavefront is then conv
eniently represented by the OPD.  Frequently, we have need of the OPD \+
averaged over the beam aperture.  Generally, there is a central (usual
ly circular) obscuration, for example the secondary mirror in a telesc
ope.  Hence the OPD averaged over an annular input beam of inner and o
uter radii " }{TEXT 286 1 "a" }{TEXT -1 5 " and " }{TEXT 287 1 "b" }
{TEXT -1 3 " is" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }
{XPPEDIT 18 0 "OPD[avg]  = int( int( rho*OPD(rho,phi), phi=0..2*pi ), \+
rho=a..b ) / (pi*(b^2-a^2))" "/&%$OPDG6#%$avgG*&-%$intG6$-F)6$*&%$rhoG
\"\"\"-F$6$F.%$phiGF//F2;\"\"!*&\"\"#F/%#piGF//F.;%\"aG%\"bGF/*&F8F/,&
*$F<\"\"#F/*$F;\"\"#!\"\"F/FC" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 170 "Once the OPD is determined by tracing a ray through t
he system (and subtracting the fiducial ray optical path), the averagi
ng integral is easy to perform.  The procedure " }{TEXT 323 17 "annula
r_average()" }{TEXT -1 38 " is the AESOP function that does this." }}}
}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 50 "2.6 Expansion of the Wavefront i
n a Zernike Series" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Zernike cir
cle polynomials are a complete orthogonal set over the interior of the
 unit circle.  Hence an arbitrary function " }{XPPEDIT 18 0 "W(rho,phi
)" "-%\"WG6$%$rhoG%$phiG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "rho" 
"I$rhoG6\"" }{TEXT -1 124 " is restricted to the range [0,1], may be c
ompletely represented by an infinite series of Zernike polynomials.  W
e may write" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 
18 0 "W(rho,phi) =  sum( sum( A[n,m]*U[n,m](rho,phi) + B[n,m]*V[n,m](r
ho,phi), m=0..infinity ), n=0..infinity )" "/-%\"WG6$%$rhoG%$phiG-%$su
mG6$-F)6$,&*&&%\"AG6$%\"nG%\"mG\"\"\"-&%\"UG6$F2F36$F&F'F4F4*&&%\"BG6$
F2F3F4-&%\"VG6$F2F36$F&F'F4F4/F3;\"\"!%)infinityG/F2;FEFF" }{TEXT -1 
1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "where the values of " }
{TEXT 292 1 "m" }{TEXT -1 19 " are restricted to " }{XPPEDIT 18 0 "n-m
=even" "/,&%\"nG\"\"\"%\"mG!\"\"%%evenG" }{TEXT -1 2 ", " }{TEXT 288 
1 "A" }{TEXT -1 5 " and " }{TEXT 289 1 "B" }{TEXT -1 23 " are coeffici
ents, and " }{TEXT 290 1 "U" }{TEXT -1 5 " and " }{TEXT 291 1 "V" }
{TEXT -1 13 " are given by" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "    \+
 " }{XPPEDIT 18 0 "U[n,m](rho,phi) = R[n,abs(m)](rho) * cos(m*phi)" "/
-&%\"UG6$%\"nG%\"mG6$%$rhoG%$phiG*&-&%\"RG6$F'-%$absG6#F(6#F*\"\"\"-%$
cosG6#*&F(F5F+F5F5" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 5 "     " }{XPPEDIT 18 0 "V[n,m](rho,phi) = R[n,abs(m)](rho) * sin(
m*phi)" "/-&%\"VG6$%\"nG%\"mG6$%$rhoG%$phiG*&-&%\"RG6$F'-%$absG6#F(6#F
*\"\"\"-%$sinG6#*&F(F5F+F5F5" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 29 "where the radial polynomials " }{TEXT 293 1 "R" }{TEXT 
-1 13 " are given by" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }
{XPPEDIT 18 0 "R[n,m](rho) = sum( (-1)^k * (n-k)! / ( k! * ((n+m)/2-k)
! * ((n-m)/2-k)! ) * rho^(n-2*k), k=0..(n-m)/2" "/-&%\"RG6$%\"nG%\"mG6
#%$rhoG-%$sumG6$**),$\"\"\"!\"\"%\"kG\"\"\"-%*factorialG6#,&F'F4F3F2F4
*(-F66#F3F4-F66#,&*&,&F'F4F(F4F4\"\"#F2F4F3F2F4-F66#,&*&,&F'F4F(F2F4\"
\"#F2F4F3F2F4F2)F*,&F'F4*&\"\"#F4F3F4F2F4/F3;\"\"!*&,&F'F4F(F2F4\"\"#F
2" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 205 "See Murison
 (1995) for a discussion, including determination of the coefficients \+
and an example using AESOP.  [See also Born and Wolf (1980) and Zernik
e (1934) for more information on Zernike polynomials.]" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 354 "The Zernike
 series representation is useful for providing explicit expressions fo
r the well-known low-order wavefront aberrations such as coma, astigma
tism, defocus, and so on.  This turns out to be an appealing way of co
nverting the often large and inscrutable AESOP wavefront expressions i
nto tidy, intuitively understandable results.  In general, the " }
{XPPEDIT 18 0 "m=1" "/%\"mG\"\"\"" }{TEXT -1 35 " terms correspond to \+
coma, and the " }{XPPEDIT 18 0 "m=2" "/%\"mG\"\"#" }{TEXT -1 39 " term
s correspond to astigmatism, with " }{TEXT 294 1 "n" }{TEXT -1 70 " de
grees of radial \"rippliness\".  Hence, the classical aberrations are
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "     n=1, m=1     wavefront t
ilt" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "     n=2, m=0     defocus
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "     n=2, m=2     astigmatism
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "     n=3, m=1     coma" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "     n=4, m=0     spherical aberra
tion" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 392 "Another low-order, but n
on-classical, aberration that is sometimes important is the \"trefoil
\" term n=3, m=3.  Zernike components of the wavefront are illustrated
 in the example shown in Appendix B.  Another advantage of a Zernike s
eries representation is that each Zernike term affects the variance in
dependently.  Hence, the Zernike polynomials \nminimize the wavefront \+
variance term by term." }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 30 "3. A
ESOP Design Considerations" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "3.1
 An Object Oriented Approach" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "G
eometrical optics lends itself very naturally to an object-oriented ap
proach when creating computer programs, either numerical or algebraic.
  AESOP takes advantage of this by defining useful " }{TEXT 324 7 "obj
ects" }{TEXT -1 60 " as Maple table structures.  These Maple tables co
ntain, or " }{TEXT 295 11 "encapsulate" }{TEXT -1 311 ", all of the in
formation relevant to the corresponding objects.  The Maple procedures
 that constitute AESOP, and which the user uses to create a Maple proc
edure which can analyze an optical system, manipulate these AESOP obje
cts.  Following is a list of the most important AESOP objects with bri
ef descriptions." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 17 "3.2 AESOP O
bjects" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 36 "3.2.1 Optical Surface D
ata Structure" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 622 "The optical syst
em is comprised of AESOP optical elements.  Each element type has a co
rresponding procedure which, given certain information, creates the op
tical element object.  All AESOP optical elements share a common table
 structure.  The table element [eqn] contains the equation describing \+
the optical element's surface shape (usually, but not necessarily, a c
onicoid) in the surface local coordinate frame.  The local frame origi
n is located at the surface vertex, and the positive Z axis is coincid
ent with the vertex normal vector.  The [dir] and [pos] elements conta
in, respectively, a Maple vector and an AESOP " }{TEXT 339 5 "point" }
{TEXT -1 207 " which describe the surface vertex normal vector directi
on and the vertex position, both in the global reference frame.  The [
type] element is the object identifier.  Finally, the [coord] table el
ement is a " }{TEXT 340 5 "point" }{TEXT -1 21 " which contains the (
" }{XPPEDIT 18 0 "x,y,z" "6%%\"xG%\"yG%\"zG" }{TEXT -1 109 ") coordina
te labels that the user wishes to appear in the [eqn] expression.  An \+
example will make this clear:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 
17 "read `objects.p`:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 76 "sphe
roid( f, point([a,b,c]), \n          vector([d[u],d[v],d[w]]), [u,v,w]
 );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7)/%&coordG-%'VECTOR
G6#7%%\"uG%\"vG%\"wG/%%typeG%'mirrorG/%$eqnG,(F/\"\"\"%#f|irG!\"#*$,(*
$F7\"\"#\"\"%*$F-F<!\"\"*$F.F<F?#F6F<F6/%%flenGF7/%$posG-F*6#7%%\"aG%
\"bG%\"cG/%$dirG-F*6#7%&%\"dG6#F-&FR6#F.&FR6#F//%#ccGF6" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 19 "Usually, one uses (" }{XPPEDIT 18 0 "x,y,
z" "6%%\"xG%\"yG%\"zG" }{TEXT -1 28 ") for the coordinate labels." }}}
}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 35 "3.2.2 AESOP Optical Surface Obje
cts" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 307 "The current AESOP optical \+
surface types are as follows.  They all have the same Maple table stru
cture and, for the most part, differ only in the form of the equation \+
describing the surface.  This assortment represents the surface types \+
needed in analyzing POINTS; other types are easy to add as need arises
." }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 20 "3.2.2.1 optical_flat" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 343 12 "optical_flat" 
}{TEXT -1 77 " is a mirror with infinite focal length.  It is the simp
lest optical surface." }}}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 16 "3.2.2
.2 conicoid" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 344 8 "
conicoid" }{TEXT -1 97 " is a reflecting surface with a general conico
id shape which is a function of the conic constant." }}}{EXCHG {PARA 
0 "" 0 "" {MPLTEXT 1 0 24 "pos := point([a,b,c]):  " }}}{EXCHG {PARA 
0 "" 0 "" {MPLTEXT 1 0 32 "dir := vector([v[x],v[y],v[z]]):" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 42 "conicoid( f, epsilon, pos, dir
, [x,y,z] );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TABLEG6#7)/%&coordG
-%'VECTORG6#7%%\"xG%\"yG%\"zG/%%typeG%'mirrorG/%$eqnG,&F/\"\"\"*&,&*$F
-\"\"#F6*$F.F:F6F6,&%#f|irGF:*$,(*$F=F:\"\"%*&%(epsilonGF6F-F:!\"\"*&F
CF6F.F:FD#F6F:F6FDFD/%%flenGF=/%$posG-F*6#7%%\"aG%\"bG%\"cG/%$dirG-F*6
#7%&%\"vG6#F-&FW6#F.&FW6#F//%#ccGFC" }}}}{SECT 0 {PARA 20 "" 0 "" 
{TEXT -1 28 "3.2.2.3 spheroid, paraboloid" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 100 "Because the equation for a quadric surface simplifies so
mewhat for the special conic constant value " }{XPPEDIT 18 0 "epsilon \+
= 1" "/%(epsilonG\"\"\"" }{TEXT -1 13 ", a separate " }{TEXT 345 8 "sp
heroid" }{TEXT -1 39 " surface is available.  Similarly, the " }{TEXT 
346 10 "paraboloid" }{TEXT -1 38 " is a conicoid with the special valu
e " }{XPPEDIT 18 0 "epsilon=0" "/%(epsilonG\"\"!" }{TEXT -1 2 ". " }}}
}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 15 "3.2.2.4 asphere" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 347 7 "asphere" }{TEXT -1 
134 " object is one whose conicoidal surface is perturbed by a series \+
of radial ripples.  AESOP employs a general asphere model of the form
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "s(rho) \+
= rho^2/( 2*f + sqrt(4*f^2 - epsilon*rho^2 ) ) + sum( A[k]*rho^(k), k=
1..infinity )" "/-%\"sG6#%$rhoG,&*&F&\"\"#,&*&\"\"#\"\"\"%\"fGF-F--%%s
qrtG6#,&*&\"\"%F-*$F.\"\"#F-F-*&%(epsilonGF-*$F&\"\"#F-!\"\"F-F;F--%$s
umG6$*&&%\"AG6#%\"kGF-)F&FCF-/FC;\"\"\"%)infinityGF-" }{TEXT -1 1 " " 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "which is a conicoid plus a radi
al power series.  The AESOP generating function for this is " }{TEXT 
349 9 "asphere()" }{TEXT -1 35 ".  Here is an illustrative example:" }
}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 33 "clist := [ seq( A[i], i=1..4
 ) ]:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 48 "asphere( f, epsilon,
 clist, pos, dir, [x,y,z] );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%&TAB
LEG6#7)/%&coordG-%'VECTORG6#7%%\"xG%\"yG%\"zG/%%typeG%'mirrorG/%$eqnG,
.F/\"\"\"*&,&*$F-\"\"#F6*$F.F:F6F6,&%#f|irGF:*$,(*$F=F:\"\"%*&%(epsilo
nGF6F-F:!\"\"*&FCF6F.F:FD#F6F:F6FDFD*&&%\"AG6#F6F6F8FFF6*&&FI6#F:F6F8F
6F6*&&FI6#\"\"$F6F8#FQF:F6*&&FI6#FAF6F8F:F6/%%flenGF=/%$posG-F*6#7%%\"
aG%\"bG%\"cG/%$dirG-F*6#7%&%\"vG6#F-&F`o6#F.&F`o6#F//%#ccGFC" }}}}
{SECT 0 {PARA 20 "" 0 "" {TEXT -1 12 "3.2.2.5 pHOE" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 14 "The procedure " }{TEXT 350 6 "pHOE()" }{TEXT -1 
73 " creates a simple focussing HOE on a paraboloidal mirror of focal \+
length " }{TEXT 348 1 "f" }{TEXT -1 58 ".  It is assumed that one of t
he construction points, say " }{XPPEDIT 18 0 "C[2]" "&%\"CG6#\"\"#" }
{TEXT -1 73 ", is virtual, so that a beam starting from the other cons
truction point, " }{XPPEDIT 18 0 "C[1]" "&%\"CG6#\"\"\"" }{TEXT -1 30 
", will diffract to a focus at " }{XPPEDIT 18 0 "C[2]" "&%\"CG6#\"\"#
" }{TEXT -1 55 ".  Further restrictions are that the diffraction order
 " }{XPPEDIT 18 0 "m = 1" "/%\"mG\"\"\"" }{TEXT -1 71 ", and the reado
ut wavelength is equal to the construction wavelength.  " }{XPPEDIT 
18 0 "C[1]" "&%\"CG6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "C[2]
" "&%\"CG6#\"\"#" }{TEXT -1 46 " must be specified in the surface loca
l frame." }}}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 21 "3.2.2.6 beam_split
ter" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "This is identical to the " 
}{TEXT 341 12 "optical_flat" }{TEXT -1 48 " object except for the iden
tification tag.  The " }{TEXT 342 13 "beam_splitter" }{TEXT -1 68 " ob
ject exists solely for human convenience and program readability." }}}
}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 12 "3.2.2.7 lens" }}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 11 "Currently, " }{TEXT 351 4 "lens" }{TEXT -1 40 "es
 are unimplemented.  Declaration of a " }{TEXT 352 4 "lens" }{TEXT -1 
33 " will produce an error message.  " }}}}}{SECT 0 {PARA 5 "" 0 "" 
{TEXT -1 27 "3.2.3 Miscellaneous Objects" }}{SECT 0 {PARA 20 "" 0 "" 
{TEXT -1 5 "point" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "An AESOP " }
{TEXT 353 5 "point" }{TEXT -1 181 " is identical in most respects to a
 Maple vector.  It's main purpose is to support the conceptual distinc
tion between a direction vector and a position (point) in space.  The \+
AESOP " }{TEXT 354 5 "point" }{TEXT -1 285 " object is restricted to t
hree components.  Otherwise, it is equivalent to a Maple vector with t
hree elements and is recognized by Maple as such.  There is a correspo
nding `type/point` function so that the Maple type procedures (such as
 is(), hastype(), etc.) will recognize the AESOP " }{TEXT 355 5 "point
" }{TEXT -1 3 ".  " }}}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 4 "beam" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 356 4 "beam" }{TEXT -1 
93 " object is a Maple table that has four elements.  The first two el
ements consist of an AESOP " }{TEXT 357 5 "point" }{TEXT -1 27 " [pos]
, which contains the " }{TEXT 358 4 "beam" }{TEXT -1 225 " (or ray) an
chor position, and a Maple vector [dir], which contains the ray propag
ation direction vector.  Next is a scalar element [path] for storing t
he expression corresponding to the accumulated optical path.  Finally,
 a " }{TEXT 359 4 "beam" }{TEXT -1 101 " contains an identifier ['type
'] := 'beam', which procedures may query to check that the object is a
 " }{TEXT 360 4 "beam" }{TEXT -1 74 ".  An associated type() function \+
makes the Maple type procedures aware of " }{TEXT 361 4 "beam" }{TEXT 
-1 12 "s.  It is a " }{TEXT 362 4 "beam" }{TEXT -1 83 " object which s
erves as the optical ray being propagated through an optical system." 
}}}}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 30 "3.3 How AESOP Does Ray Trac
ing" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "An overview of the ray trac
ing process using AESOP is as follows.  " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 425 "(1) In a Maple procedure \+
that the user writes, the user first defines the various optical eleme
nts comprising the optical system.  These surfaces are assembled into \+
a Maple list which AESOP routines will make use of.  Perturbations (mi
salignments) are applied in the form of rotations and/or translations \+
of specified optical elements.  AESOP provides object rotation and tra
nslation procedures to make this a simple process." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "(2) The user the
n defines the input ray, which is subsequently launched into the optic
al system by calling the AESOP procedure " }{TEXT 325 10 "raytrace()" 
}{TEXT -1 555 ".  AESOP then automatically traces the ray to each succ
essive optical element, performing series expansions on the perturbati
on parameter(s) as necessary and simplifying the cumbersome expression
s as as much as possible, until finally an output ray is produced at t
he detector.  Progress during this process is communicated via informa
tional messages and key intermediate expressions to the monitor screen
.  If nothing else, there is plenty of stuff the user can peruse while
 waiting for the ray trace to finish, since AESOP is intentionally a b
it chatty." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 117 "(3) The OPD is then calculated from the output ray expre
ssions, followed by calculation of the aperture-averaged OPD." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 394 "
(4) Optionally, the Zernike components of the OPD are determined next,
 either at the Maple prompt or from within the user's driver procedure
.  The resulting Zernike coefficients may then be combined to produce \+
wavefront aberration plots.  The aberrated wavefronts are represented \+
by 3D Maple surface plots.  Maple procedures are supplied for making t
he wavefront plots in the Maple worksheet.  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "An illustrative example
 of this entire process for a simple beam compressor is shown in Appen
dix B." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 658 "In practice, the essential step for useful analytical ra
y tracing is to make series expansions at each intersection of a ray w
ith an optical surface.  (The original insight for this trick is due t
o R.D. Reasenberg.)  This reduces the \"equation bloat\" considerably.
  Even so, it is still rather easy to cause the intermediate expressio
ns to mushroom in size so that they overwhelm the available machine re
sources.  The equation bloat seems to go as some power of the number o
f focussing optical elements in a system.  Flat surfaces certainly con
tribute to increasing equation complexity, but at a rate that pales in
 comparison to that of focussing surfaces.  " }}{PARA 0 "" 0 "" {TEXT 
-1 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 381 "Since we are interes
ted in analyzing optical systems whose elements are slightly misaligne
d, the small parameters to perform the series expansion on are natural
ly the misalignment perturbations.  Hence, AESOP is not meant to analy
ze the very interesting properties of ideal, perfectly aligned optical
 systems.  It requires at least one misalignment or other perturbation
 parameter." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 1071 "For a given optical system, a certain amount (sometim
es a great amount) of tinkering on the part of the user is required to
 hit upon the best ways of simplifying the cumbersome expressions so t
hat their size is managable.  Great care has been taken in the types o
f simplification taking place in the AESOP ray tracing routines.  Howe
ver, they are no doubt optimized for the particular systems the author
 has analyzed and will therefore perhaps be less than optimum for othe
r kinds of optical systems.  Hence, AESOP is nowhere near the \"black \+
box\" stage, where a user can provide necessary input, crank the handl
e, and magically produce an answer without caring overmuch about the i
nternals of the black box.  Nonetheless, AESOP can be quite useful and
 represents a significant advance in capability for analyzing perturbe
d optical systems and performing misalignment sensitivity studies.  It
 also serves as an invaluable check on numerical programs as well as a
n essential aid to developing reliable insight into the arcane and bea
utiful world of high-precision optics.  " }}}}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 15 "4. Availability" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 
"The AESOP source code, help files, examples, background papers, and o
ther information may be found at the web site " }{TEXT 378 30 "http://
aa.usno.navy.mil/AESOP/" }{TEXT -1 3 ".  " }}}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 13 "5. References" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "N
ote: copies of relevant Smithsonian Astrophysical Observatory (SAO) Te
chnical Memoranda may be obtained from the author.  They are also avai
lable at http://aa.usno.navy.mil/AESOP/." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Born, M., and Wolf, E. (19
80).  " }{TEXT 370 20 "Principles of Optics" }{TEXT -1 288 ", sixth ed
ition, Pergammon, section 9.2.\n\nJohnston, K.J., Seidelmann, P.K., Ge
rmain, M.E., Monet, D.M., Murison, M.A., Urban, S., Davinic, N., Ricka
rd, L.J., Weiler, K., Shao, M., Vaughan, A., Fansen, J., and Norris, D
. (1997).  \"Fizeau Astrometric Mapping Explorer (FAME)\", submitted t
o " }{TEXT 379 20 "Astronomical Journal" }{TEXT -1 108 ".\n\nLindegren
, L., and Perryman, M.A.C. (1996).  \"GAIA: Global Astrometric Interfe
rometer for Astrophysics\", " }{TEXT 366 28 "Astron. and Astrophys. Su
pp." }{TEXT -1 1 " " }{TEXT 367 3 "116" }{TEXT -1 75 ", 579.\n\nLoisea
u, S., and Malbet, F. (1996).  \"Global Astrometry with OSI\", " }
{TEXT 369 28 "Astron. and Astrophys. Supp." }{TEXT 368 4 " 116" }
{TEXT -1 142 ", 373.\n\nLoiseau, S., and Shaklan, S. (1996).  \"Optica
l Design, Modelling and Tolerancing of a Fizeau Interferometer Dedicat
ed to Astrometry\", " }{TEXT 381 28 "Astron. and Astrophys. Supp." }
{TEXT 380 4 " 117" }{TEXT -1 91 ", 167.\n\nMurison, M.A. (1993).  \"Ra
y Trace Analyses of Selected POINTS Optical Subsystems\", " }{TEXT 
298 24 "SAO Technical Memorandum" }{TEXT -1 112 " TM93-08.\n\nMurison,
 M.A. (1995).  \"Expansion of Wavefront Errors in an Infinite Series o
f Zernike Polynomials\", " }{TEXT 299 24 "SAO Technical Memorandum" }
{TEXT -1 125 " TM95-04.\n\nMurison, M.A., and Noecker, M.C. (1993).  \+
\"Ray Tracing of and Optical Path across a Holographic Optical Element
\", " }{TEXT 300 24 "SAO Technical Memorandum" }{TEXT -1 135 " TM93-04
.\n\nNoecker, M.C., Murison, M.A., and Reasenberg, R.D. (1993).  \"Opt
ic-misalignment tolerances for the POINTS interferometers\", " }{TEXT 
302 75 "Proceedings of the SPIE - The International Society for Optica
l Engineering" }{TEXT -1 2 ", " }{TEXT 301 4 "1947" }{TEXT -1 331 ", 2
18.\n\nReasenberg, R.D., Babcock, R.W., Chandler, J.F., Gorenstein, M.
V., Huchra, J.P., Pearlman, M.R., Shapiro, I.I., Taylor, R.S., Bender,
 P., Buffington, A., Carney, B., Hughes, J.A., Johnston, K.J., Jones, \+
B.F., and Matson, L.E. (1988).  \"Microarcsecond optical astrometry - \+
An instrument and its astrophysical applications\", " }{TEXT 297 10 "A
stron. J." }{TEXT -1 1 " " }{TEXT 296 2 "96" }{TEXT -1 159 ", 1731.\n
\nReasenberg, R.D., Babcock, R.W., Murison, M.A., Noecker, M.C., Phill
ips, J.D., and Schumaker, B.L. (1995a). \"POINTS: The Instrument and i
ts Mission,\" " }{TEXT 306 24 "Proceedings of the SPIE," }{TEXT -1 
258 " Conference #2477 on Spaceborne Interferometry II (Orlando, FL, 1
7-20 April 1995).\n\nReasenberg, R.D., Babcock, R.W., Murison, M.A., N
oecker, M.C., Phillips, J.D., Schumaker, B.L., Ulvestad, J.S., McKinle
y, W., Zielinski, R.J., and Lillie, C.F. (1995b).  \"\", " }{TEXT 304 
27 "Bull. American Astron. Soc." }{TEXT -1 2 ", " }{TEXT 305 3 "187" }
{TEXT -1 243 ", #71.04.  \n\nR.D. Reasenberg, R.W. Babcock, M.A. Muris
on, M.C. Noecker, J.D. Phillips, B.L. Schumaker, J.S. Ulvestad, W. McK
inley, R.J. Zielinski, and C.F. Lillie (1996).  \"POINTS: High Astrome
tric Capacity at Modest Cost via Focused Design,\" " }{TEXT 303 24 "Pr
oceedings of the SPIE," }{TEXT -1 111 " Conference #2807 on Space Tele
scopes and Instrumentation IV (Denver, CO, 6-7 August 1996), in press.
\n\nSIM97.  " }{TEXT 382 29 "http://huey.jpl.nasa.gov/sim/" }{TEXT -1 
25 "\n\nWelford, W.T. (1986).  " }{TEXT 307 30 "Aberrations of Optical
 Systems" }{TEXT -1 45 ", Adam Hilger, Boston.\n\nZernike, F. (1934). \+
 " }{TEXT 371 7 "Physica" }{TEXT -1 1 " " }{TEXT 372 1 "1" }{TEXT -1 
6 ", 689." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "6. Biography" }}
{EXCHG {PARA 0 "" 0 "" {TEXT 308 15 "Marc A. Murison" }{TEXT -1 796 " \+
is an Astronomer in the Astronomical Applications Department of the U.
S. Naval Observatory, in Washington, DC.  He obtained his Ph.D. in Ast
ronomy at the University of Wisconsin-Madison in 1988.  He then held a
 postdoctoral position with the Hubble Space Telescope Wide Field/Plan
etary Camera team.  Subsequently, he was a Physicist at the Smithsonia
n Astrophysical Observatory where, among other things, he was part of \+
the effort in analyzing the optical systems of the proposed astrometri
c interferometer satellite, POINTS.  He currently is part of an effort
 by the Naval Observatory to build its own astrometric satellite.  His
 research interests include the chaotic dynamics of the asteroid belt,
 high-precision solar system ephemerides, and ultra-high precision ana
lysis of optical systems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 
0 {PARA 3 "" 0 "" {TEXT -1 55 "Appendix A: The Unit Normal Vector of a
 Quadric Surface" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The sagitta is
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 4 "sag;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&%$rhoG\"\"#,&%#f|irGF%*$,&*$F'F%\"\"%*&%(epsilonG\"\"
\"F$F%!\"\"#F.F%F.F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Convert t
he sagitta to a function of (x,y)." }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 37 "sagxy := subs(rho=sqrt(x^2+y^2),sag):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "Then take the gradient and collect on " }
{XPPEDIT 18 0 "epsilon" "I(epsilonG6\"" }{TEXT -1 1 "." }}}{EXCHG 
{PARA 0 "" 0 "" {MPLTEXT 1 0 47 "map(collect, grad(z-sagxy,[x,y,z]), [
epsilon]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Now replace x and y
 with " }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 57 ", where conveni
ent.  We get the resulting gradient vector" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 52 "g := map( collect, subs(x^2+y^2=rho^2,\"), [x,y,z] );
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"gG-%'VECTORG6#7%*&,&*$,&%#f|ir
G\"\"#*$,&*$F-F.\"\"%*&%(epsilonG\"\"\"%$rhoGF.!\"\"#F5F.F5F7!\"#**F6F
.F,F9F0#F7F.F4F5F7F5%\"xGF5*&F*F5%\"yGF5F5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 64 "Now that we have the gradient term, form the unit normal \+
vector." }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 52 "v := array( [ -dif
f(sagxy,x), -diff(sagxy,y), 1 ] ):" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 43 "subs( x^2+y^2=rho^2, evalm( v / mag(g) ) ):" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 38 "N := map( collect, \", [epsilo
n,x,y] );" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"NG-%'VECTORG6#7%,&*.,
(\"\"\"F,*&,&*$,&%#f|irG\"\"#*$,&*$F1F2\"\"%*&%(epsilonGF,%$rhoGF2!\"
\"#F,F2F,F:!\"#**F9F2F0F<F4#F:F2F8F,F:F2%\"xGF2F,*&F.F2%\"yGF2F,F>F9F2
F0F<F4F>F?F,F8F,F:*(F+F>F?F,F0F:F<,&*.F+F>F9F2F0F<F4F>FAF,F8F,F:*(F+F>
FAF,F0F:F<*$F+F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }
{TEXT 326 5 "mag()" }{TEXT -1 127 " is a function which returns the ma
gnitude of a vector.  This expression can be simplified considerably. \+
 Let's get rid of the " }{XPPEDIT 18 0 "x^2" "*$%\"xG\"\"#" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "y^2" "*$%\"yG\"\"#" }{TEXT -1 22 " terms and
 substitute " }{XPPEDIT 18 0 "Q^2" "*$%\"QG\"\"#" }{TEXT -1 5 " for " 
}{XPPEDIT 18 0 "4*f^2-epsilon*rho^2" ",&*&\"\"%\"\"\"*$%\"fG\"\"#F%F%*
&%(epsilonGF%*$%$rhoG\"\"#F%!\"\"" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 
"" 0 "" {MPLTEXT 1 0 12 "assume(Q>0):" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 72 "map( simplify, N, \{ x^2+y^2=rho^2, 4*f^2-epsilon*rho
^2=Q^2 \}, [x,y,f] ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Simplify
ing again, we get the pleasing result" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 19 "map( simplify, \" );" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%'VECTORG6#7%,$*&%\"xG\"\"\",&*$%$rhoG\"\"#F**$%#Q|irGF.F*#!\"\"F.
F2,$*&%\"yGF*F+F1F2*&F0F*F+F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "
That is, we have" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 44 "N := subs
( Q=sqrt(4*f^2-epsilon*rho^2), \" ):" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 33 "N := rootfunc( N, collect, rho );" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"NG-%'VECTORG6#7%,$*&%\"xG\"\"\",&*&,&F,F,%(epsilo
nG!\"\"F,%$rhoG\"\"#F,*$%#f|irGF3\"\"%#F1F3F1,$*&%\"yGF,F-F7F1*&,&F4F6
*&F0F,F2F3F1#F,F3F-F7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "(The pro
cedure " }{TEXT 327 10 "rootfunc()" }{TEXT -1 320 " applies an arbitra
ry procedure to an expression which contains subexpressions raised to \+
fractional powers.  Now, the Maple procedure collect() allows one to c
ollect on a set of variables and apply a function to the coefficients \+
of those variables, but it unfortunately ignores expressions raised to
 fractional powers.  " }{TEXT 328 10 "rootfunc()" }{TEXT -1 144 " also
 automatically processes the components of vectors and lists.  It is a
 handy function for easy manipulation of expressions under radicals.)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 30 "Appendix B: A Sample AESOP Run" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 19 "B.1 OPD Calculation" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
159 "Following is a Maple session in which AESOP is used to calculate \+
the aberrated wavefront of a misaligned beam compressor.  The primary \+
mirror has focal length " }{TEXT 309 1 "f" }{TEXT -1 47 ", and the bea
m compression ratio is denoted by " }{TEXT 310 1 "C" }{TEXT -1 94 ".  \+
The perturbation consists of a rotation of the primary mirror about it
s vertex by an angle " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 
56 ".  The detector surface (a plane) is located a distance " }{TEXT 
311 1 "d" }{TEXT -1 137 " below the primary mirror surface.  The drive
 procedure, in which the optical system is defined and AESOP ray traci
ng invoked, is called " }{TEXT 329 10 "BeamComp()" }{TEXT -1 25 ".  Th
e first argument of " }{TEXT 330 10 "BeamComp()" }{TEXT -1 136 " is th
e perturbation type, in this case the primary mirror rotation.  The th
ird and fourth arguments are the order of the expansions in " }
{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 17 " and in radius.  " }
}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 34 "read `d:/optics/BeamCompress
or.p`;" }}{PARA 6 "" 1 "" {TEXT -1 505 "#=============================
==============#\n#                   AESOP                   #\n# (An \+
Extensible Symbolic Optics Processor) #\n#             version 96.10.0
2              #\n#===========================================#\n#    \+
          Marc A. Murison              #\n#          U.S. Naval Observ
atory           #\n#      Astronomical Applications Dept.      #\n#   \+
    murison@riemann.usno.navy.mil       #\n#      http://aa.usno.navy.
mil/AESOP/       #\n#===========================================#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"~G" }}{PARA 6 "" 1 "" {TEXT -1 
1016 "Use:\n  BeamComp( pert_type1::integer, \011#first perturbation t
ype\n            eps1::name,           #name of first pert. variable\n
            pert_type2::integer,  #second perturbation type\n         \+
   eps2::name,           #name of second pert. variable\n            o
rder_eps1::integer   #expansion order of first pert.\n            orde
r_eps2::integ     #expansion order of second pert.\n            order_
r::integer      #radial expansion order\n  )\n\nPerturbation Types:\n \+
 TRANS_PRIMARY      translation of primary mirror in X\n  ROT_PRIMARY \+
       rotation of primary mirror around Y axis\n  TRANS_SECONDARY    \+
translation of secondary mirror in X\n  ROT_SECONDARY      rotation of
 secondary mirror around Y axis\n  FIELD              rotation of the \+
input ray around X axis\n  NOTHING            do nothing\n\nExample:\n
  verbosity := 1:\n  assume( Delta >= 0, psi >= 0 ):\n  BeamComp( TRAN
S_PRIMARY, Delta, ROT_SECONDARY, psi, 2, 10 );\n\nNOTE: you MUST assum
e(eps1 >= 0): and assume(eps2 >= 0):\n      before calling BeamComp()
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 85 "verbosity := 1:\nassume( \+
theta >= 0 ):\nBeamComp(ROT_PRIMARY,theta,NOTHING,dummy,3,12);" }}
{PARA 0 "" 0 "" {TEXT -1 27 "[...much output deleted...]" }}{PARA 6 "
" 1 "" {TEXT -1 20 "BeamComp[518]: Done!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 330 "This run took 518 seconds on a 100 MHz Pentium machine w
ith 32 MB of RAM available.  Peak memory usage, as reported by the Map
le status bar, was 10.4 MB.  The wavefront and the aperture-averaged w
avefront, which are the main results (and which are too bulky to repro
duce here),  are stored in the global variables OPD and OPD_avg." }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 49 "B.2 Zernike Series Decomposition \+
of the Wavefront" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "In this secti
on we perform a Zernike series analysis of the wavefront (OPD) just ca
lculated.  First, we set the compression ratio to 10, the distance to \+
the detector to 20 " }{TEXT 331 2 "cm" }{TEXT -1 16 ", and normalize \+
" }{XPPEDIT 18 0 "rho" "I$rhoG6\"" }{TEXT -1 45 " so that it spans the
 interval [0,1] and now " }{TEXT 314 1 "R" }{TEXT -1 64 " represents t
he radius of the input beam.  The OPD simplifies to" }}}{EXCHG {PARA 
0 "" 0 "" {MPLTEXT 1 0 108 "opd := subs( rho=R*rho, C=10, d=20, \n    \+
         collect( OPD, [theta,cos(phi),sin(phi),rho], simplify ) );" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$opdG,(*&,&*&,,**,&\")++%4\"\"\"\"%
#f|irG\"'ToYF-F.!#6%\"RG\"#6%%rho|irGF2#F-\"%C5**,&\"(+?6&F-F.\"'*p>#F
-F.!\"*F1\"\"*F3F;#!\"\"\"$c#**,&\"(+S.\"F-F.\"&R[%F-F.!\"(F1\"\"(F3FD
#F-\"#K**,&\"'+SLF-F.\"&fY\"F-F.!\"&F1\"\"&F3FL#F=\"\")**,&\"'++9F-F.
\"%ziF-F.!\"$F1\"\"$F3FT#F-\"\"%F--%$cosG6#%$phiGFTF-*&,.**,&\")+!=0\"
F-F.\"'uzVF-F.F0F1F2F3F2#F-\"%Wh**,&\")+g*Q\"F-F.\"'X;eF-F.F:F1F;F3F;#
F=\"%sI**,&\"(+gE%F-F.\"'\"zz\"F-F.FCF1FDF3FD#F-\"$%Q**,&\"(+Sq%F-F.\"
'f,?F-F.FKF1FLF3FL#F=\"$#>**,&\"(+g3\"F-F.\"&@m%F-F.FSF1FTF3FT#F-\"#C*
*,&\"'+!)>F-F.\"%@))F-F.F=F1F-F3F-#F=FTF-FWF-F-F-%'theta|irGFTF-*&,4*&
,.**,&\"&+?'F-F.\"%-FF-F.!#7F1\"#7F3FdqF4**,&\"&++)F-F.\"%BNF-F.!#5F1
\"#5F3Fjq#F=\"$7&**,&\"&+S#F-F.\"%r5F-F.!\")F1FNF3FN#F-\"#k**,&\"&+g#F
-F.\"%\"=\"F-F.!\"'F1\"\"'F3Fir#F=FF**,&\"$+'F-F.\"#GF-F.!\"%F1FVF3FV#
FL\"\"#**,&\"%+SF-F.\"$*>F-F.!\"#F1FasF3Fas#F=FasF-FWFasF-**,&\"%+=F-F
.\"#uF-F.FcqF1FdqF3Fdq#FL\"%'4%**,&\"$+)F-F.\"#LF-F.FiqF1FjqF3Fjq#FKF
\\r**,&\"%+9F-F.\"#eF-F.FarF1FNF3FN#FLF>**,&F]sF-F.\"#DF-F.FhrF1FirF3F
ir#FKFF**,&\"%+5F-F.\"#UF-F.F_sF1FVF3FV#FL\"#;**,&FdsF-F.\"$h\"F-F.Ffs
F1FasF3Fas#F=FVF.\"$)>FdsF-F-FjpFasF-*(,.*(F.FiqF1F2F3F2#F=F5*(F.FarF1
F;F3F;#F-F>*(F.FhrF1FDF3FD#F=Fcr*(F.F_sF1FLF3FL#F-Fau*(F.FfsF1FTF3FTFe
u*&F1F-F3F-FasF-FWF-FjpF-F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 136 "Now we perform the Zernike series expan
sions on the OPD to the same order in radius that the ray trace calcul
ation used.  The procedure " }{TEXT 332 15 "ZernikeSeries()" }{TEXT 
-1 26 " automatically calculates " }{TEXT 315 3 "all" }{TEXT -1 134 " \+
nonzero Zernike series coefficients, up to and including the order spe
cified.  I have clipped the output, due to space constraints.  " }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 41 "ZernikeSeries(opd,12,[theta],O
N,'c','S');" }}{PARA 0 "" 0 "" {TEXT -1 22 "[...output deleted...]" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "The Zernike series coefficients a
re now in a globally accessible table called " }{TEXT 312 1 "c" }
{TEXT -1 76 ", and the Zernike wavefront representation is stored in t
he global variable " }{TEXT 313 1 "S" }{TEXT -1 233 ".  Determination \+
of the Zernike series coefficients is a time-consuming process (this p
articular example took ~11 minutes), but it is not memory intensive li
ke the ray tracing process.  To show an example, the classical coma te
rm is " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 19 "ZernikeTerm(3,1,c);
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*(,&**%\"RG\"\"$,6*&F&\"\"#%#f|irG
\"\"(!+'*od%\\#*&F&\"\"%F+\"\"&\"+'HB!o:*&F&F*F+\"\"'!,+g@4w&*&F&F3F+F
'!*?&f)=)*$F&\"\")\"+++?i*)*&F&F/F+F/\",+g`\"[O*&F&F3F+F*!,++Gp\">*$F+
F8\",++W,:'*&F&F8F+\"\"\"\"*vP\"3Q*$F+\"\"*\"+gPd$o#FAF+!#6%'theta|irG
F'#FA\"(![!e#**F&F'F+!#5,,F7\"#vF>\"&?z\"F:\"%W8F<!$?$F2!%w`FAFGFA#!\"
\"\"'S]@FA,&*$%%rho|irGF'F'FW!\"#FA-%$cosG6#%$phiGFA" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 41 "We see that it has a first-order term in " }
{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 45 " (other aberrations \+
begin at second order in " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }
{TEXT -1 428 ").  Hence, we expect coma to be an important aberration \+
resulting from rotational misalignment of the primary mirror.  (In fac
t, for this particular example, it turns out that all of the coma term
s [3,1], [5,1], [7,1], etc. are first order in the perturbation, while
 the other aberrations are second or third order in the perturbation. \+
 Hence, coma is by far the dominant aberration after wavefront tilt, a
s we shall see below.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 116 "We now subtract the original OPD from th
e Zernike series representation to check that the two are indeed equiv
alent." }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 24 "factor( expand(S-op
d) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 201 "This is reassuring!  We see by calculating the \"co
st\" of each that the Zernike series representation contains quite a f
ew more terms than the heavily simplified (in the Maple sense) express
ion for OPD." }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 8 "cost(S);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%*additionsG\"$^#%0multiplicationsG
\"%LC%*divisionsG\"#I%*functionsG\"#<" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 10 "cost(opd);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%*add
itionsG\"#`%0multiplicationsG\"$*f%*divisionsG\"#G%*functionsG\"\"%" }
}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 41 "B.3 Analysis of the Wavefront \+
Aberrations" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 87 "Now comes the fun part.  We will read in, among others, t
he Zernike plotting function, " }{TEXT 333 22 "PlotZernikeWavefront()
" }{TEXT -1 70 ".  This Maple procedure produces a color 3D Maple surf
ace plot of the " }{TEXT 316 18 "residual wavefront" }{TEXT -1 707 " a
fter the specified Zernike terms have been subtracted.  Hence, it is a
 useful visual and quantitative tool for determining what are the impo
rtant aberrations for the particular system and misalignment under stu
dy.  Typically, one takes a look at a plot of the wavefront, from whic
h the aberration with the largest magnitude is usually apparent.  One \+
then subtracts the Zernike term(s) corresponding to that aberration an
d views the resulting residual wavefront, from which the next most imp
ortant aberration is now apparent.  This process is repeated as desire
d, resulting in a series of 3D plots showing all of the important aber
rations.  In our example here, we use a primary mirror focal length of
 100 " }{TEXT 334 2 "cm" }{TEXT -1 32 " and an input beam radius of 10
 " }{TEXT 335 2 "cm" }{TEXT -1 41 ", which sets the output beam radius
 at 1 " }{TEXT 336 2 "cm" }{TEXT -1 381 " since we have previously tak
en the compression ratio to be 10.  For this session, a perturbation m
agnitude of 0.2 arc second (~1 microradian)  of primary mirror rotatio
n is specified.  Changing the perturbation magnitude only changes the \+
scale of the aberrations and not their relative importance, as long as
 we remain in the regime that is valid within the expansion orders use
d." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 91 "For the first plot, we subtract the average wavefront.  T
he resulting wavefront residual is" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 15 "verbosity := 0:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 
1 0 117 "PlotZernikeWavefront( opd, theta, c, [[0,0]], \n    `microns`
, evalf(0.2*arcsec), 100.0, 10.0, orientation=[-70,65] );" }}{PARA 13 
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17060 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 263 "(The fifth argument is a string which sets the physica
l units of the plot, in this case microns.)  The plot represents a map
ping between the input beam coordinates (XY plane) and the output wave
front (OPD).  Divide the horizontal scale by the compression factor (
" }{XPPEDIT 18 0 "C = 10" "/%\"CG\"#5" }{TEXT -1 229 " in this case) t
o get output beam coordinates.  We recognize that the primary aberrati
on is, as expected, wavefront tilt, to the tune of about 1.5 microns a
t the edge of the beam.  Hence, let us additionally remove the tilt te
rm:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 101 "PlotZernikeWavefront(
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14010 14091 14086 14104 14112 15040 15080 20020 17010 17060 0 0 0 0 0 
0 0 1 1 0 0 0 0 3 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "
The primary aberration is now ~0.2 microns of coma.  All of the coma t
erms are first order in the rotation angle " }{XPPEDIT 18 0 "theta" "I
&thetaG6\"" }{TEXT -1 80 ", and therefore dominant.  Let us then addit
ionally subtract all orders of coma:" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 131 "PlotZernikeWavefront( opd, theta, c,\n    [[0,0],[1,
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110.000000 14010 14091 14086 14104 14112 15040 15080 20020 17010 
17060 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 36 "The residuals now consist of a ~100 " }{TEXT 338 4 "pic
o" }{TEXT -1 114 "meter mix of defocus and astigmatism, which both are
 quadratic in radius.  By removing all of the first order (in " }
{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 495 ") terms, we see tha
t the second order terms result in aberrations that are over 2,000 tim
es smaller.  Similar to the series of coma terms, whose higher-order (
in radius) components were of similar magnitude to the classical third
 order coma term [3,1], the defocus and astigmatism terms of various r
adial orders contribute comparable amounts to the aberrations as the c
lassical components [2,0] and [2,2], respectively.  Hence, let us addi
tionally remove all defocus and astigmatism terms to get" }}}{EXCHG 
{PARA 0 "" 0 "" {MPLTEXT 1 0 212 "PlotZernikeWavefront( opd, theta, c,
\n    [[0,0],[1,1],[3,1],[5,1],[7,1],[9,1],[11,1],[2,0],[2,2],[4,0],\n
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`fm`, evalf(0.2*arcsec), 100.0, 10.0 );" }}{PARA 13 "" 1 "" {INLPLOT "
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NTLIGHTG6%$\"\"%!\"\"Fai\\lFai\\l-%+PROJECTIONG6%$\"$5\"F)$\"#lF)\"\"
\"" 3 645 563 563 2 0 1 0 2 4 2 3 2 1.000000 65.000000 110.000000 
14010 14091 14086 14104 14112 15040 15080 20020 17010 17060 0 0 0 0 0 
0 0 1 1 0 0 0 24062 25823 0 0 0 0 0 1 }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 39 "Now we recognize exceedingly small 0.2 " }{TEXT 337 2 "fm" }
{TEXT -1 145 " residuals due to trefoil ([3,3], [5,3], etc.) aberratio
ns.  These aberrations happen in this system to be third order in the \+
perturbation angle " }{XPPEDIT 18 0 "theta" "I&thetaG6\"" }{TEXT -1 
16 ".  For example, " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 19 "Zerni
keTerm(3,3,c);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*.%\"RG\"\"$%'thet
a|irGF&,6*&F%\"\")%#f|irG\"\"\"\")0$=!\\*$F%F*\"+++q[6*&F%\"\"'F+F&!*?
bX0\"*&F%F1F+\"\"#!+++w`C*&F%\"\"%F+\"\"&\"*?(y3?*&F%F7F+F7\"+++KKY*&F
%F4F+\"\"(!*OrA:$*&F%F4F+F1!++gL#=(*$F+\"\"*\"*S!fvL*$F+F*\"+++SEvF,F+
!#6%%rho|irGF&-%$cosG6#,$%$phiGF&F,#F,\"'g,')" }}}}{SECT 0 {PARA 4 "" 
0 "" {TEXT -1 14 "B.4 Conclusion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
320 "We have firmly (and simply!) established that this particular wav
efront's dominant aberrations are tilt and coma, followed at a much lo
wer level by defocus and astigmatism.  Even more important, we have th
e dependence of each aberration type (as well as of the aperture-avera
ged wavefront) on the perturbation parameter " }{XPPEDIT 18 0 "theta" 
"I&thetaG6\"" }{TEXT -1 255 ", as well as the optical system parameter
s, allowing us to determine the sensitivity of these aberrations to (i
n this case) rotation of the primary mirror.  Hence, given an error bu
dget, one can easily determine tolerances for the perturbation magnitu
des." }}}}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {MPLTEXT 1 0 76 "save OPD, OPD_avg, opd, S, c, `d:/optics/AESO
P/MapleTech/AESOP_article.eqs`;" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 
0 0 "" }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 0 0 1 1 1803 }