1976 lens & mirror design via the principal surface - applied optics 15, p. 2579

7/30/2019 1976 Lens & Mirror Design via the Principal Surface - Applied Optics 15, p. 2579

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a of paper intended foror Since changes be made

preprint IFnot be reproduced

the of the

LABORATORY

LENS AND MIRROR DESIGN V I A THE PRINCIPAL SURFACE*

Anne GreenbaurnAlexander J . Glass

John B. Trenholme

J ul y 14, 1975

UCRL - 76639PREPRINT

This paper was presented a t th e In te rn at io na l Lens DesignConference i n Haverford, Pennsylvania, on June 1975and prepared f o r s u b mi t t a l t o A p p l i e d Op t i c s


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LENS AND MIRROR DESIGN V I A THE PRINCIPAL SURFACE*

Anne Greenbaum, Alexander J. Glass, and John B . Trenholme

Lawrence L ivermore Labora to ry , Un iv er s i t y o f Ca l i fo rn ia

L ivermore , Ca l i fo rn ia 94550

ABSTRACT

For many l a s e r a p p l i c at i o n s , i t i s des i r ed t o f ocus a co l l i m a ted beam

The trans-w i t h a s p e ci f i ed t ra n sf o rm a ti o n o f t h e i n t e n s i t y d i s t r i b u t i o n ,

f o r m a t i o n p r o p e r t i e s o f a l e n s o r m i r r o r s yst em can b e s p e c i f i e d i n terms

o f t h e p r i n c i p a l s ur fa ce , whi ch maps t he he i gh t o f t h e i n c i de n t

p a r a l l e l r a y o nt o a gi v e n an g l e a t t h e f o cu s.

a t t h e foc us i s t he n gi v e n b y t he r e l a t i o n =

a s ph e ri c s u r fa c e i n a n o p t i c a l system i s s u f f i c i e n t t o y i e l d d i f f r a c t i o n

l i m i t e d f oc u si n g. By means o f two asphe r i c su r faces, d i f f r a c t i o n l i m i t e d

performance wi th a spec i f i e d p r i nc i p a l su r f ace can be achi eved .

,

The i n t e n s i t y d i s t r i b u t i o n

One

,-

The p ro bl em o f o p t i c a l d e si g n i s s t a t e d as f o l l o w s : G iv en a p r i n c i p a l

su r f ace and a maximum focal angle f i n d t h e p a i r o f o p t i c a l su r-

f ac es f o r wh ic h d i f f r a c t i o n l i m i t e d f oc u si ng i s a ch ie ve d.

s p e c i f i c a t i o n o f and un i que l y de te rm i nes t he l ens des i gn t o wi th in

a s c al e f ac t or , g i v en t h e r e f r a c t i v e in de x o f t h e l e n s . I t i s f u r t h e r

shown t ha t one s t r a ig h t f o rw ard Runge-Kut ta i n t e g r a t i o n r ou t i ne generates

b o t h s u rf a ce s f or e i t h e r a l e n s o r a p a i r o f m i r r o r s ur fa ce s.

I t i s shown t h a t

The complete fam i l y o f ap l ana t i c lenses w i l l be descr ibed . Dev ia t ion

f r o m s p h e r i c i t y w i l l be discussed , as w i l l t he p o s s i b i l i t y o f r e a l i z i n g

the spec i f i ed l ens desi gns .

i n t e n s i t y i n t o i l l u m i n a t i o n a b o u t t h e f o c u s w i l l a lso be descr ibed .

Ex tens ion of t he method t o o f f - a x i s a b e r r a t i o n s w i l l be considered.

The fam i l y o f l enses wh ich map un i fo rm in c i de n t

work was performed of t h e EnergyResearch and Development Administrat ion.


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INTRODUCTION

I n t h e d e si gn o f f o c u si n q o p t i c s f o r l a s e r f u s i o n e xp er im en ts ,

spec ia l requ irements a r i s e wh i ch a re d i f f e r e n t f r o m t h e

cons ide ra t i ons wh i ch t h e d e s iq n o f t y p i c a l i ma gi ng o p t i c s . I n

t h i s paper, we sh a l l o u t l i n e t h e s p e ci a l f e at u r es o f l a s e r fo c u si n g

o p t i c s w hi ch d i f f e r e n t i a t e t h e i r d e s ig n f r o m o t h e r s ystems, and r e p o r t

a phi losophy o f l e n s d es ig n p a r t i c u l a r l y s u i t e d t o l a s e r f oc u si ng o p t i c s

I n p a r t i c u l a r , we s h a l l d e v el op t h e f o rma l is m f o r t h e d es i qn lenses

f r o m t h e p r i n c i p a l su r face , wh ich rep resen ts t h e mapp inq o f ra y he i gh t

i n t h e e nt ra nc e p u p i l o n t o r a y a n g le a t t h e f o c a l p o i n t .

f o rm u la t io n f o r r e f l e c t i n g o p t i cs w i l l be q iven. Sp ec i f i c examples o f

f a m i li e s o f l e n s s u rf ac e s w i l l be presented.

The equ i v a len t

I n l a s e r f u s i o n e xp e ri me n ts , one i l l u m i n a t e s a s p h e r i c a l t a r g e t as

u n i fo r m ly as p o s s ib l e , o v e r i t s e n t i r e s u rf ac e , k ee pi ng t h e l i g h t as n e ar

t o normal in c iden ce as dual requirement o f near-normal

i nc i de n ce and u ni f or m i l l u m i n a t i o n a r i s e s f r om t h e d e s i r a b i l i t y o f

c r e a t i n g a h e a t i ng o f t h e p lasma o v e r t h e e n t i r e sur face.

The l i q h t i s g en er at ed as t h e o u t p u t o f a l a r g e , s h o r t- p u l s e l a s e r ,

u s u a l l y a l a se r , r a d i a t i n q a t The beam p r o f i l e i s

g e n e r a ll y a f u n c t i o n o f ray h e i g h t i n t h e en tr a nc e p u p i l o n l y , and i s

g i v e n by t h e o p e r a t i n g c o n s t r a i n t s o f t h e l a s e r s ystem.

I n th e d e si gn o f a l a r g e , s h o r t- p u l s e g l a ss l a s e r , t h e c r u c i a l

parameter i s t he t o t a l i n t e g r a l o f t h e l a s e r i n t e n s i t y al on g t h a t p a r t

o f th e o p t i c a l p a th t h a t l i e s i n g la ss , e i t h e r l a s e r gl as s o r o p t i c a l

Thi s parameter, th e so - c a l l e d "B- I n t e g ra l " , m us t b e k e p t t o a

minimum, i n o rd e r t o p re v e n t t h e g ro wt h o f h i g h s p a t i a l f r e qu e n ci e s

( r i p p l e s ) o n t h e beam, due t o t h e n o n l i n e a r c o u p l i n g o f t h e i n t e n s e


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l i q h t t o t h e o p t i c a l medium.

the t o t a l t h i c kness o f t he l enses mus t be kep t t o a minimum. I n add i t i on ,

s i nce op t i c a l sur faces a re mos t vu l ne rab l e t o damaqe, a t h i gh i n t e ns i t i e s ,

t h e number of surfaces must be kep t t o a minimum.

m i l i t a t e a g a i n s t t h e use o f a l a r g e number o f e lem en ts i n t h e f o c u s i ng

o p t i c s .

i n any q iven l ens .

Thus i n d e s ig n i ng l a s e r f o c u s i n q o p t i c s ,

These considerat ions

I n g e ne r al , a number of no q r e a t e r t h an t wo i s d e s i r a b l e

L as er f u s i o n o p t i c s f a l l i n t h e c a te go ry o f e ne rgy d e l i v e r y systems,

r a t h e r t ha n sys tems. The q u a l i t y o f the image, i n f oca l

p l an e o f t h e l e n s, i s l e s s i m p or t a nt th an t h e p a t t e r n o f i l l u m i n a t i o n

genera ted on the ta rae t su r face .

f e a tu r e s o f l a s e r i l l u m i n a t i o n , t h e r e ar e fe we r a d d i t i o n a l c o n s t r a i n t s

on t he l ens des i gn .

c o l l i m a te d , and i n c i d e n t p a r a l l e l t o t h e a x i s o f t h e system, o f f - a x i s

aber ra t ions can be neg lec ted . A l so , s i nce t he l as e r i s monoch roma ti c,

ch roma t ic abe r ra t i on i s o f no concern. Thus th e me r i t of a p a r t i c u l a r

d es ig n i s e n t i r e l y s p e c i f i e d by how w e l l t h e i l l u m i n a t i o n re qu ir em en ts

are met .

I n a d di t i o n , due t o t h e s p ec i a l

For example, s in ce th e la se r beam i s q enera l l y

C l e ar l y , t o e f f e c t a g i ve n t r a ns f o rm a t io n o f r a y h e i g h t on t o f o c al

Recent advancesang le w i th a few e lements requ i re s a spher i c sur f aces .

i n a s p h e r i c f a b r i c a t i o n s t r o n g l y s u p p o r t t h i s a p p r o a c h .

i n c r e a s i n g l y f e a s i b l e t o f a b r i c a t e s t e e p an d co mp l ic a te d a s p he r i c s ur f a ce s

a t r easonab le cos t . It must be understood, however, t h a t these a r e

k i nd op t i ca l des igns , o f wh i ch on l y a f ew cop i es w i l l be made. The i n i t i a l

c o st s o f t o o l i n g , and t e s t s et up must be d i s t r i b u t e d o v e r

these few copies.

b e p re pa re d t o b e ar t h i s i n c re a s ed c o st , as p a r t o f t h e p r i c e o f d e ve lo pi ng

a s o p h i s t i c a t e d a nd s p e c i a l i z e d i n d u s t r i a l base t o s u p p o rt t h e i r needs.

I t i s

The l a r g e l a b o r a t o r i e s w o r k in g i n l a s e r f u s i o n must


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I t i s hoped t h a t i n t he l ong run, t he e n t i r e o p t i c a l i n d us t r y w i l l b e n e f i t

f rom t he advances made and pa id f o r under such spe ci al iz ed proqrams as

l a s e r f u s io n .

EXPLICIT METHODS OF LENS DESIGN

S i nce we a re cons i de r i ng ve r y spec i a l i z ed op t i ca l systems cons i s t i nq

on ly o f a few e lements, th e cus tomary i n t e r ac t i ve methods o f l e ns design,

such as are embodied i n gener a l l y a va i l ab le lens des ign p rograms, a re

n o t t h e most e f f i c i e n t methods a v a i l a b l e f o r d e s ig n .

t o d es iq n th e l e ns su rfa ces e x p l i c i t l y i n o r de r t o o b t a i n e x a c t l y t h e

d e s ir e d t r a n s f o rm a t i o n p r o p e r t i e s . We s h a l l r e f e r t o t h i s approach as

t h e u se o f " e x p l i c i t methods" , i n s te a d o f i t e r a t i v e a dj us tm en t o f s u r fa c e

parameters to ob ta in an op t ima l approx imat ion to the des i red per fo rmance.

I t i s more e f f i c i e n t

I n q eo me tr ic al o p t i c s , t h e o b j e c t i v e o f d es ig n i s to de te rmine a

s e t o f r e f r a c t i v e o r r e f l e c t i n q s u rf a ce s w hi ch map a s e t o f r ay s, s p e c i f i e d

i n t erm s o f r a y h e i q h t and a n ql e t o t h e

object space, on to a s e t o f rays on a p lane i n image space, w i t h

p resc r i bed r ay he i gh

be expressed as

where R and

and ang l e t o t he ax i s . Th i s t r ans fo rma t i on can

t h e a x i s r e s p e c ti v e l y .

I n t h i s d i scuss i on, t he sys tem i s assumed t o be cy l i nd r i ca l l y symmetr i ca l ,

and skew rays are not considered.

if t h e mapping i s d e f i n e d i n t erms o f r a y p o s i t i o n a t t h e f oc us , then a

s yst em f r e e o f s p h e r i c a l a b e r r a t i o n c an be c o n st r u ct e d b y t h e s p e c i f i c a t i o n

o f a s i n g l e a s p he r ic s u r fa c e . T hi s i s c a r r i e d o u t by e ns u ri ng t h a t t h e

o p t i c a l p a t h f r om e v e ry p o i n t on t h e o b j e c t p l an e t o t h e f o c a l p o in t i s

t he same. We sh a l l r e f e r t o t h i s as the equal pa th cond i t i on . The

As i s p o i nt e d o u t i n Lun ebu rg' s t r e a t i s e ,


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mapping can be represented in the form

where F denotes the focal plane, and the angle between the ray and the

axis a t the focal poi nt. In th is case, th e system is fre e of on-axis

aberrations, b u t the intensity distribution about the focal point is

completely determined.

A single aspheric surface can also be used t o obtain a desired

intensi ty dis t r ibut ion, b u t in this case, the system is afocal,

all rays do n o t pass t h rough a single focal point.

is given by

The

b u t the ray angle a t the image plane i s completely determined.

intensity in the image plane i s given by

The

=

where i s the incident intensity distr ibution in the object plane.

This method has been used t o design laser illumination systems, and i s

discussed i n a separate paper.

In order t o satisfy the equal p a t h condition, and, simultaneously,

t o obtain the desired intensity mapp ing , the use of two aspheric surfaces

i n the opt ical system is requ ired. Since the equal p a t h condition is

sa t i s f i ed in t h i s case, we can define the transformation in terms of ray

angle a t the focal point (where all ray heights are identically zero).

We shal l assume in the discussion t h a t the in cident beam i s

parallel t o the axis , a l t h o u q h the method readily generalizes

t o converginq o r diverging incident l i g h t . Under this assumption, the

desired mappinq is function where


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i s t h e r a y h e i g h t i n t h e o b j e c t p lan e, and 01 i s t h e f o c a l an ql e.

f u n c t i o n d e f i ne s a s u r f ac e i n space, r e f e r r e d t o as t h e p r i n c i p a l s ur fa ce ,

s i n c e i t i s t a n ge n t t o t h e s ec on da ry p r i n c i p a l p l a ne a t t h e a x i s .

p r i n c i p a l s u r f a ce i s sometimes c a l l e d t h e e q ui v a l e n t r e f r a c t i n g s u r f ac e,

s i n c e i t r e pr e se n ts t h e i n t e r s e c t i o n between t h e i n c i d e n t r a y s i n t h e

ob je c t space, and th e focused rays i n th e image space.

a knowledge o f the p r in c i pa l su r face comp le te ly de te rmines two aspher ic

s ur fa ce s, t o w i t h i n a s c a l e f a c t o r , q i v en e i t h e r t h e f oc a l a n gl e of t h e

marg ina l ray , o r t h e r a t i o o f t h e back f o ca l l e n gt h t o t 5 e l e ns

th ickness .

CALCULATION OF THE LENS SURFACES

T h is

The

We s h a l l s ee t h a t

I n o rd e r t o s i mp l i f y t h e d is c u ss io n , we s h a l l c o n sid e r a s ys te m o f

two aspheric sur faces.

i n t r o d u c i n g an number o f s p h e r i c a l s u r fac e s i n t h e system, b u t

t o do so r e nd e rs t h e e x p o s i t i o n l e s s s t r a i g h t f o r w a r d .

fab r ica t ion , one would gener a l l y use two elements ( f ou r su r faces ) w i t h

one aspheric surface on each element.

T he re i s n o d i f f i c u l t y i n

For ease o f

We cons ider th e two aspher ic sur fac es as shown i n F iq . 1 . The

p r i n c i p a l s u r f a c e i s d e f i n e d by th e fu n c t i o n where the ray he igh t

exte nds t o a maximum val ue correspo nding t o a maximum value o f t he

foca l ang le ,

h e i g h t i n t e r s e c t s t h e p r i n c i p a l s u rf a ce a t t h e p o i n t a t wh ic h t h e two

o p t i c a l s u r fa c e s c ro s s . By i n s p e c t i o n , we see t h a t t h e p r i n c i p a l s u r fa c e

always passes through the i n t e r s e c t i o n o f t h e t wo l e n s s u r fa c es .

The marg ina l ray which i s i nc i den t on the sys tem a t

R e fe r r i n g t o F ig . 1, we wa nt t o i n t e q ra t e t h e e q ua t i o n f o r t h e d i s -

placement of t h e f i r s t s ur fa ce f rom t h e l e ns as a funct ion


8/19

of the focal angle where i s known.

the slope of the f i r s t surface in terms of the angle of deflection of the

ray entering the optical medium, as

From Snells law, we can write

s i ncos - 1

where the minus sign arises from the definition of

index of the optical medium.

object space fromthe f i r s t ref rac ting sur face t o the principal surface,

the distance as the distance i n image space from the principal surface t o

the second refracting surface, and the distance as the distance traveledt h rough the optical medium by the actual ray. From the law of sines, we

have

Here i s the ref rac t ive

We define the distance P as the distance

We want to express entirely in terms of the angle a, and the distance

Simple geometry yields the r e su l t

while the equal p a t h condition takes the form

Z - = P ( n s in a - sin -

Equating two expressions for P yie lds the resul t ,

- -- a)= =s in - s i n


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Equat ion 9 c a n b e w r i t t e n as

s i n a cos + - cos a ] s i n n s i n , (10)

and so lved fo r i n any o f se veral ways (see Appendix I ) .

Eq. we can w r i t e

Combining wi th

n cos -n s i n

which can be i n teg ra ted us ing Runge-K u t t a i n t e g r a t i o n .

i n Eq . ( 9 ) i s i nde te rmina te a t t he ve r tex , where R = R

r u l e m ust be a p p l i e d t o s t a r t t h e i n t e g r a t i o n .

t h e f u n c t i o n i s determined. Since i s known, the f i r s t surface

i s t hus spec i f i ed .

S ince the func t i on

andm

A t each step,

The coord ina tes o f the second sur f ace are then g iven as

R ' = R - Y s i n a s i n (a-and

= s i n cos - -

where Z ' i s measured from t he ve rt ex , as shown i n 1.

I f we examine th e ra y alo ng th e ax is , we express th e equal pa th

c o n d i t i o n a s

Z t ( n -

where t i s t h e a x i a l t h i c k n e s s o f the ens. Cancel ing th e i n Eq.

o b t a i n a genera l r e l a t i o n among lens th ickness, index, and marg ina l

ra y parameters, namely,

( n - l ) t = tan

I n the case i n which t he back foca l l eng th , and lens th ickness,

t, ar e spec i f i ed , we de f ine Z as the d i s tance f rom the foca l p lane

t o t h e i n t e r s e c t i o n o f t he ray w i t h t h e f i r s t sur face.

o f t h e p r e vi o u s f o rm a li sm c a r r i e s o ve r i n t a c t , w i t h t h e e x ce p t io n

Then a l l


10/19

that in Eq. 9 and the function now takes the form

= - ( n - 1 ) t - a)

and that the coordinate of the second surface, measured from the focal

plane, i s given by

For reflecting surfaces, exactly the same considerations apply.

Referring t o Fig. 2, we now define the angle as the angle of deflection

of the incident ray. The distance P is the distance in object space from

the f i r s t ref lect ing surface t o the principal surface, which the distance

i s the distance in image space from the second reflecting surface

t o the principal surface. Using these def ini tio ns, and taking we find

that Eqs. 10-13 carry over without change.

pond ing t o the lens vertex i s again the point of inter secti on of the

two re fl ec ti ng surfa ces. This point, which is , of course, never rea liz ed

in practical system, corresponds t o the deflection point of a ray

which i s tang ent t o the f i r s t re f lec t ing sur face .

EXAMPLES OF THE APPLICATION OF THE METHOD

We note that the point corres-

To i l lus t r a te the app licat ion of the method, we have computed the

family of apla nat ic lens es, fo r which the principal surface i s given

by R = sin and th e family of lenses which map equal beam

areas on to equal angles a t th e focus, fo r which t he princip al

surface i s given by R

obtained by requiring t h a t R = s in da.

height a t th e marginal ray i s taken t o be 1 .

lens profiles for aplanatics of increasing numerical aperture.

expected, the of both surfaces increases dramatically with

This l a t t e r mapping i s

In bo t h cases, the ray

In Fig. 3 , see

As i s


11/19

i n c r e a s i n g N.A.

pro f i l es fo r equa l a rea mapp ing a re shown fo r i nc reas ing N.A.

these cases were computed f o r n 1.5.

i s l i m i t e d by t o t a l i n t e r n a l r e f l e c t i o n i n t h e le n s. The desi gns were

computed on a CDC 7600 computer, and each design, embodying 100 po i nt s

across the lens (100 rays) took approximately

SUMMARY

The same i s seen t o b e t r u e i n F i g. where the lens

A l l o f

The achiev able val ue o f N.A.

We have shown t ha t fo r l a se r focus ing op t i cs , wh ich cons is t s i n

general o f a few, aspheri c l ens su rfaces , e x p l i c i t des ign a r e

advantageous. We have p resented the gene ral so lu t i on o f t he two- sur faceproblem, which


12/19

APPENDIX I

An equation i n t he f orm o f E q . 10, namely

a cos s i n c

must be so lve d w i t h some care , t o a m b i gu i ti es i n

to in t roduce complex no ta t ion , and wr i te

It i s c on ve ni en t

( a = r

Equat ion A-1 then takes the form

- = c / r ,

t he so l u t i on o f wh ich can be w r i t t e n

=

(A-3)

(A-4)

w i t h r t

i n terms o f t h e c o e f f i c i e n t s and c, by use o f s tandard t r i gonomet r i c

i d e n t i t i e s .

The sine and cosine o f can eas i l y be ob ta i ned


13/19

REFERENCES:

1 . C .E . Thomas, App l . Optics 1267 (1975).

2 . J . Nuckolls, L . Wood, A. Thiessen, and G. Nature 239, 139

3. Trenholrne, Laser Program Annual Report, UCRL Director'sOf fi ce , Lawrence Livermore Laboratory, Livermore, Ca li fo rn ia 94550.

4. V.I. Bespalov and V.I. Talanov, JETP Lett. 307 (1966).

5 . R. Theory of Optics [University of California

6. J .S. JOSA 64, 55 (1974).

7. A. Greenbaurn and A.J. Glass, "Optical Methods i n Energy Conversion",

(1972).

Press

SPIE Meeting, Rochester, New York (June 1975).


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FIGURE CAPTIONS

Fig. 1 Definition of ray parameters fo r refracting surfaces.

Fiq. 2 Definition of ray parameters for surfaces.

Fig. 3 Lens profiles for aplanatic lenses of varying N .A . The

do t t e d l ine indicates the principal surface. n =

Fig. 4 Lens pr of i le s fo r equal area mappina. The dot ted l i n e

indicates the principal surface. n = 1.5.


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FIGURE 1


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1976 lens & mirror design via the principal surface - applied optics 15, p. 2579

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