Variance of the wave-aberration of the optical System
with small décentration
Maciej Rafałowski
Warsaw University of Technology, Institute of Design of Precise and Optical Instruments, ul. Karola Chodkiewicza 8, 02-525 Warszawa, Poland.
The variance of the wave-aberration of the composed optical system with small décentration of single element is derived using the vector form of wave-aberration equation of phase distorter with décentration. The possibilities of optimization of aberrations of centred system from the standpoint of minimization of the décentration sensitivity of the selected single element are discussed. The conditions for minimization of coma of décentration and their practical aspects are presented.
1. Introduction
In the p reviou s article [ 1 ] it was show n that the w a ve-a b erra tio n o f the o p tica l system w ith sm all d écen tration can be described in the v e c to r fo rm w ith the a b erra tio n coefficien ts o f centred system in trod u ced b y
H
opkins
[2].
It m akes possible to in trod u ce the phase d istorter d escription o f décen tration o f any lens in the o p tica l system. U n d er lim itation s o f the Fresnel a p p ro x im a tio n and in the isop lan a tic regio n o f the ob ject the d istorter o f an y type can be transferred th rou gh focu sin g elem ents to the object space o f the system w ith ou t a lterin g its influence on the im a gin g process [3 ]. I t is also possible to determ in e the p ertu rb ation o f the w a ve fro n t in the exit pupil. M o r e o v e r , the m eth ods o f im age assessment based on the d iffra ctio n th e o ry o f im a gin g can be applied. T h e y enable, a m o n g others, the varia n ce calcu lation fo r the w a vefro n t in the exit pupil plane (under M a réch a l a p p ro xim a tio n ). T h is can be d on e con ven ien tly b y num erical m ethods. In sim ple cases the an alytical ap p roach is possible as well. A s an exa m p le the varian ce fo r the system w ith spherical ab erration and defocu sin g w as discussed in [1 ]. In the present p a p er the varia n ce o f the w a ve ab erration fo r the system w ith all p rim a ry aberrations an d single d écen tration w ill be presented. T h e p ossib ility o f m in im iza tio n o f the d écen tration sensitivity o f the selected elem ent b y o p tim iza tio n o f aberrations o f cen tred system w ill be discussed, and the con d itio n s fo r m in im iza tio n o f com a o f d écen tration w ill be form ulated.2. Analysis
I t was stated in [ 1 ] that the op tica l system w ith d ecen tration can be expressed as a set o f:
i) P u re ly qu ad ratic phase correctors that describe focusing p roperties o f all elem ents.
ii) D istorters due to aberration s o f the centred system, w hich can be described as
____ N
<P(A
q=
0 ) = £ [ w20i ë2û2 + w40i é4+ ( w1152 + w3 1ë 2) t ë ' â ) + w22t ë - â ) 2] . ( i ) i= 1iii) D isto rters due to the décen tration s o f the first order. T h e y can be expressed as
0 {A
q # 0 ) = £\w 20i[2(Q ■ A g jâ 2 + 2{à-Aa^Q2]
i
=i C
+ w40i4{
q-A
qJ
q2
+ W ju
[(Ü · d p f) â2 + (ë ·A ajâ2
+ 2 (â · d a f) (ë · fl)] + w 31i[ (a ■Agt) g
2 + 2(
q · dg,·) (ë ·a) + (
q ·A aJ
ë 2] + w22i [2(â ·A
qù(
q · fl) ■+ 2 (ë * d a 4) (ë · fl)]J
(2) w h ere: w mni - coefficients o f p rim a ry ab erration s o f i-th elem en t o f the system,ë - lo ca liza tio n v ec to r o f a p o in t in the aperture plane,
à
- lo ca liza tio n v e c to r o f the p o in t in the im age plane,Agt -
the shift o f the a b erra tio n a l fu n ction in the exit p u pil o f i-th elem ent,Aat -
the shift o f the im a ge p o in t in the im a ge plane o f i-th element, ë, â,Agit
d a , are n o rm a lized to the m a xim al p u p il and im age heights respectively in the im a ge space o f i-th elem ent. E q u a tio n (2) shou ld be taken in to account twice. F irstly, fo r d escribin g the effect o f o w n aberration s o f i-th elem ent and, secondly, fo r describin g the effect o f ab erration s o f the w a ve fro n t incident on the i-th elem ent o f the system.F o r the case o f a system o f
N
elem ents w ith the single décen tration o fk-th
elem ent Eqs. (1) and (2) can be rew ritten as<p =
4>(dc = 0) + # ( d c # 0) (3) w here:0(Ac = 0) = c20ë2 fl2 + c40ë4 + (c11à2 + c31ë2) (ë'd) + c22(ë-fl)2,
i3·1)
<P{Ac # 0) = 2b20[pâ2(ë-dc) + ië 2(fl*dc)]
+ 4bAOpg2 {g-Ac)
+ bl l [p(à-Ac)â2
+ 1
(ë ·Ac)â2
-1-2i (â ·Ac)
(ë ·à)
] + *>3i [P (â ‘Ac)62
+ 2p (ë * d e ) (ë · fl)+ 2d20(qa2{Q-Ac) + tQ2 (a-Ac)]
+ 44 40qQ2(Q-Ac)
+ d l l [q{a· Ac)à2
+ 1 (
q ·Ac)a2
+ 2i (a ·Ac)
(
q · â )] + ¿3 j[_q (â
·A c)
q2 +
2q {
q · d c )(
q · â) + i ( ê d c ) ê 2] + 2d22 [fl (â · d e ) (ê · à )^+ 1 (
q · J e ) (ë · â )] (3.2) w h ere:Ac —
v ec to r o f lateral shift o f centre o f curvature o f decen tred fc-th surface(o r shift o f focus o f the thin lens); the index
k
is o m itted fo r sim plicity,N
cmn
= Z vvmm- - sum o f the coefficients o f p rim a ry ab erration fo r the w h o lei=1 system,
bmn
= w milk - coefficien t o f ab erration o fk-
th surface o r thin lens o f the system,* — 1
dmn
= Z wm/i« - sum o f “ a b erra tio n coefficients o f p receed in g part o f the i=1 system fo r the w a ve fro n t in cident on fc-th elem ent o f thesystem,
p
= —M A, q =
1—M A, t
= 1— M 0, M A, M 0
— pu pil and im a ge m a gn ifi cations o fk-
th elem ent o f the system under con sidérations.In the con sideration s o n ly a sm all décen tration due to the p ro d u ctio n tolerances o f fa irly w ell-corrected centered system are taken in to account. In this case it is useful to e m p lo y the varian ce
E
o f the w a ve fro n t as a desing param eter.M
aréchal
[ 4 ] sh ow ed that fo r sm all aberration, i.e.,SC
^ 0.8 w e haveSC
^ [1 — l/ 2 k 2 £ ] 2. (4)O b vio u s ly , the variance m in im iza tio n m axim izes the Strehl criterion . U n d er M a ré ch a l a p p ro xim a tio n , fo r vectors described in the p o la r-co o rd in a te system, the varia n ce o f the w a ve fro n t is given fo r a circu lar apertu re by
E
=
- f f <P2QdQd<x — \
[ J J
dg dot]2.
( 5 )
n 0 0
71
0 0I f the ab erration fu n ction is g iven on the p o la r-c o o rd in a te system the in tegration s are sim ple. Because o f the extent o f analysis o n ly the final result is q u o ted in Eqs. (6). W e assume the vectors q
(
q cos a, g sin a), a (a c o s y ,a s in y ),Ac(Accosp,Acsinfi)
and, fo r a sim plification o f Eqs. (6), w e in trod u ce the param eterA
= cos/i cosy + sin/?siny = cos cosine o f the an gle betw een the azim u th o f a chosen im age p oin t and the décen tration azim uth o f
k
-th elem ent. F r o m Eqs. (5) and (3) w e haveE = E^-l· E2 + E2-\- E
a+ E 5
+ £ 6w here
(
6
)
E ,
= c 20« 2^ ^ 2 0 + ^ 40 + i a 2c 22J + c40^ c 40 + ^ a 2c22J+ c 1i a 4 ^ a 2c 11+ i c 31J + c i 1i a 2 + c i 2^ a 4,
E 2 =
AcaA
| (c 20a 2 + c40) j j i & 2o + ^ P &3i + £ ř i,22J+ Cj j
a2
j^a2pb20
+^pbA0
+ ^ a2tb t x
+^ tb 31
+ u2 ph22 J + c31 [ j fl2Pb
20 +Pb
40 + a 2tb
11+ i i 6 31+ ? a 2 pfr22 J + C 2 2 « 2 ^ ^ 2 0 + ^ 3 1 + ^ ^ 2 2] J » £ 3 = dcav4 | (c20a2 + c 40) j j t á 20+ ^ 3 i + ^ i d 22J + Ci 1 a2 V 20+ 5^ 4 0+\ a 2tdx 1
+^ td 31
+ a2 $d22 J f 3 1 2 ~| + C3 1 2 fl2^ 2 0 + ^ 4 0 + a 2 t d 1 1 + - i d 3 1 + - a 2 ^ 22 + c 22a 2^ ř d 20 + ^ d 3i + ^ ^ 2 2 j j » £ 4 = (d c )2 | a 2h20^ p 2a 2 + i ř 2^ 2^ h 20 + ^ p 2h40 + ( l + 2 ^ 2) a 2 p íb 11 + ^ (1 + >12) přh31 + ^ 2 a 2p 2 + ^ í 2^ 2fe22 J + b4o ^ 2 p 2b4o + ^ ( l + 2 ^ 2) a 2 p ř b u + p ř b 3i + ^ 2^ 2p 2h22J+ a 2fe11^ ( l + 8yl2) a 2t 26 11+ i ( l + 2 ^ 2) t 2fe31 + 3 a M 2 pih 22J
(
6
.1
)(
6
.2
)+ b3 i^ ( l + 2 A 2)a2p2b}, + l- t % 1+ j ( i + 3 A 2)a2ptb22^
[,v+I
+ a2bl
(2 + A 2)t
(6.4)E ,
=(Ac)2^a2d20^ q 2a2+ j t 2A ^ jd 20+ ^ q 2d40+ (l+ 2 A 2)a2qtdi t + ^ { l+ A 2)qtd3,
+ ( f ‘1‘l1+ ^ t2j A 2d22^ + dM^2q2dM+ ~ a + 2 A 2)a2qcdu +qtd3i+ ^ a2AIq2d2^j
+ a2^
, | j ( l +SA2)a2t2d n
+ ^ ( 1 +2A2) t % , +
J+d
3i[ ¿ ( 1+2A2)a2q2d31
+ ^ 3 1 + ^ 1 + 3 ^ V « td 22]+ a2dl
-(2 + A 2)t
( ,
v
+
ïï
'
E6
= 2(Ac)2 ^a2b20d20(^pqa2 + ^ t 2A 2^ + ^ a 2pq(b20d40b40d20) + a * ( ^ + A 2^J
x i p t b ^ d ^ + q tb ^ d20)
+ ^a2
[(p + ^ 2) i h20d 31+ (g + p,42) i ¿ 31 ¿ 20]+ n 2^ p g a 2 + ^ i 2^ (¿2o ¿ 22 +
b22d22)
+ 2
pqb40d40+ ^ a 2( \ + 2A2) (ptb4Qd
1
4
+ qtbi l d40) + -(p tb 40d3 l+ qtb3 ld40) + - a 2pq(b40d22 + b22d40)
(6.5) + i a 4 ( l + 8^ ) ^dll + l- a 2(
1 + 2A2) t2(h11 d31 + h31 d11) + ^ a^ 2 ( ^ 11 d22 + p i6 22d x l) + ^ a 2 ( l + 2 4 2)p4 + i f2J¿ 31d3i
+ ^ a2[(/42 + 1) (p ih31 d22 +qtb22 d3l)-\-2A2 (qtb31 d22+ ptb22 d
3 l)] + a 2 ^ a 2/42 + ^ i 2 (>12 + 2) J ¿ 22 <*22·· (6.6)£a g iven by Eq. (6.1) is the varian ce due to the aberrations o f the w h o le centered system. T h e part o f the varian ce g iven b y Eqs. (6.2) and (6.3) contains the first ord er term s o f décentration. T h e y v a ry w ith im age height and the im age azim uth. £ 4,
Es,
E6
com b in e qu ad ratic term s o fAc. E 2
andE4
are the functions o f ab erration coefficients o f fc-th elem ent on ly, andE 3, E s
o f aberrations o f the w a vefro n t incident on fc-th elem ent o f the system.E6
is a com bin ed term.3. Discussion
T h e varian ce expression g iven enables to determ ine the influence o f décen tration on the im a ge q u a lity w ith reduced am ou n t o f data. F o r p rim a ry aberrations o n ly five coefficients, p ara xia l aperture and im a ge m agn ification s fo r e very op tica l elem ent are required. Since the terms
E 2, E 3
and £4»E6
are m a th em atically sim ilar(E3
reduced toE 2, E s
toE4
andE6
to 2E4
fo rp = q, bmn
=dmn)
the num erical a lg oryth m can be very simple.A t this p o in t w e can m in im ize the variance o r m in im ize the d écen tration sensitivity o f
k
-th elem ent o f the system b y a p p ro p ria te choices o f the selected ab erration coefficients. A s p oin ted ou t b yB
arakat
andH
ouston
[5]
this p rocedu re is n o t gen era lly possible in practice. E ven w h en w e solve the m ath em atical p ro b lem and o b ta in the d erived fu n ctional relations betw een the a b erra tio n coefficients, it is ra rely possible to ach ieve these relation s exa ctly in the system design. T h e practical b alan cin g p ro b le m reduces to ach ievin g the p a rtia l balancing. Secondly, the variance term sE2
—E6
are functions o fA
(an gle betw een azim uths o f the décen tration and im a ge point). T h e im age qu a lity changes n o t o n ly w ith the im age height but also w ith the azim uth. T h is is n o t true fo ra =
0. In such a case the varian ce is influenced o n ly b y spherical ab erration o f the w h o le system and c om a o f décentration. E q u a tio n (6) sim plifies toE
= ^ c 40 + (d c )2 ^ 2 p 2h io + Pt&3i&4o + ^ 2H i J+
(Ac)2^2 q2d l0 + qtd3í d40
+ it2d2
3l
+ 2
(Ac
)21^2pqb40 d40
+ i(ptb40 d3l + qtb3l d40)
+ ^t2b3
ťd
31 j .. (7)C o m a o f décen tration is also caused b y spherical a b erra tio n and com a o f the decentred elem en t and the same aberration s o f the w a ve fro n t incident o f this elem ent only. T h e b alan cin g p ro b lem seems v e ry sim ple in this case. F o r
ÔE
n
ÔE
„
JÔ T
db4
0 = ° ’ X T ' = ° ’od4Q
w e have the con d ition s* 4 0 - - — * » . W
< 4 0 - - J ^ » · <8b»
I f these con d ition s are fu lfilled the décen tration o f the elem ent under con sideration can n ot influence the valu e o f the varia n ce fo r the w h o le system. A s m en tion ed above,
it is n o t alw ays possible to fu lfil these con d ition s exactly. H o w e v e r, th ey can be used as o p tim iza tio n criteria fo r m in im izin g the c o m a o f décen tration sensitivity fo r a selected elem ent o f the o p tica l system.
T h e com a o f d écen tration is the one o f all d écen tration aberrations w hich is observed in a sim ple w a y in the alignm en t process in produ ction . T h is ab erration d eteriorates the im a ge q u a lity u n ifo rm ly in the w h o le field o f v iew o f the op tica l systems. I t is discussed very often w hen a n a lyzin g the décen tration effects on the im age qu ality. T h e possibility o f m in im iza tio n o f this aberration (b y o p tim iza tio n o f w a vefro n t aberrations fo r the selected elem ent inside the optica l system ) in the desing process seems to be a v ery interesting one. I t can be used fo r the desing o f the elem ents o f the o p tica l system, w hich are difficu lt fo r practical adjustm ent and assem bly as w e ll as fo r the desing o f these elem ents o f the system which are used fo r com pen sation fo r the c om a o f décentration. M ic ro s c o p ic objectives can serve as an exam ple. T h e sensitivity to décen tration o f these elem ents can be o p tim ize d during the desing fro m the p o in t o f v iew o f requirem ents o f the adjustm ent.
T h e m o re d etailed analysis o f the p ossibility o f im p lem en ta tion o f th e con d ition s given b y Eqs. (8a) and (8b), and o f the o p tim iza tio n o f oth er décen tration aberrations w ill be given in the fo llo w in g w ork .
References
[1] Rafalowski M., Opt. Appl. 17 (1987), 3.
[2] Hopkins H. H., vVave Theory o f Aberrations, Clarendon Press, Oxford 1950. [3] Jóžw ick i R., Opt. Acta 31 (1984), 169.
[4] Maréchal A., Rev. Opt. 29 (1950), 1.
[5] Barakat R., Houston A., J. Opt. Soc. Am. 55 (1965), 1142.
Received June 1, 1987 Варианция волновой аберрации оптической систем ы с м а ло й децентрировкой Выведена формула для варианции волновой аберрации сложной оптической системы с децентри ровкой одного элемента на базе векторного уравнения волновой аберрации интерпретируемой как фазовый дистортер. Рассмотрены возможности оптимизации аберрации центральной оптической системы для минимализации чувствительности выбранного элемента системы на децентрировку. Представлены условия для минимализации комы децентрировки и практические возможности их использоания.
