光学设计实战：Zemax计算机辅助应用

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18 Chapter 4.3

L¡ght and lmaging Systems

or flux

density)

varies directly as the square of the

image NA. and iuversely as the

sqr-rale of

the eflective

inragefTnurnber.

A.3.7

Marginal and Chief Rays

In the analysis of

optical

systems,

thele are two

rays

that are especially use-

ful. These are

the marginal ray and

the chief ray. These ale shown schernatically

in Figure A.3.2. The

rnalginal ray originates

at the object

point

on the axis,

and

goes

to the edge

(or

niargin) of the defining

aperture ol' stop of the system. Fron]

thele, the rnarginal ray is focused

back to the axis. Thus,

for the rnarginal ray, it

is center-edge-center.

The chief ray, also callecl

the

principal

ray, originates

ât the object

point

the edge of the field of

view,

ancl

goes

to the

center of the defining aperture or

stop of the systern. At

the stop the chief l'ay

crosses the axis. Fror¡ the

center of

the stop, the chief

ray continues to the

edge

the iruage field. Tlius,

for the chief

ray, it is edge-center'-edge.

The axial height

(the

transverse distance

away frorn the axis) of the rnarginal

ray is zero

at the

object

and at all irnages of

tlie object. At these locations,

the axial

height of

the chief ray cletermines

the

size

(semi-diameter)

the

object

and its

images. Similarly,

the axial height of the

chief ray is zero at the stop and

at all

irnages

of the stop

(pupils).

Atthese locations, the axial

height of the marginal ray

detennines the size of

the

stop

opening and its images.

Throughout lens design,

the marginal and chief rays

âre usecl as convenient reference rn¡'s.

Actually,

lay

frorn any off-axi" object

point that

passes

through the

center'

the stop is a chief lay. Simiiarly,

cuÐ'ray ftom au on-axis

ol off-axis object

point that passes through the edge of the stop opening

is zr rnarginal ray. But

when

lens

designers s¡reak of ¡åe chief

ray, they usually meau the

chief ray

fi'orn

the ex-

trenre edge of the fieìd. And the

rrarginal ray usually means the on-axis

rnarginal

ray. How the tel'lns chief ray

and marginal ray ate being

used car-r be detenniued

from

the context.

A.3.8

Perfect lmagery

Maxwell, in acldition to studying

electromagnetisrn. also studied

imagery.

ploposecl tlrat, independent of the colìstructiou¿ìl

details, any axially centered

optical

system

must

satisfy the following

three cliteria in older to

achieve

^erre¡r

imagery:

1. Flonr any ob.ject

point, atl rays that

pass

thlor-rgh

the system lnust converge to

single

irnage

point.

points

are iniaged

âs poirìts;

2. If the ob-ject surfàce is a

plane perpendicular to the optical axis. then

the irl-

ages of

any and all

object

points

must also lie on a

¡rlane

perpencliculal to tlre

axis,

or the system has

a flat fietd;

Section

4.3.9: Causes

of lmage Quality

Degradation 19

OBJECT

Figure

4.3,2, Marginal

ancl Chief Ra¡'s.

The marginal

ray

origittatas

ot tlrc centur oJ tlrc object

fie

(on

the

axis) and rra,eLs

first

to the edge

(or

mdrgin) of

the

,stop

apciltrre. A.lter

pttssin¡¡

through

the lens, the

t ¿r\i

tt'ot)els lo the centcr of

lhe innge

field

(again

on î\rc a¡ìs).

Tlrc chic.f

(or

prittt i¡tal) rav oritirtttlt's

ot tlu'

edge

of the

object

.field

and tntvels

JirsÍ

to tllc certcr of llrc sto¡t

operturc. Alter

crossirtg tltc axis

ttt the sto¡t,

the chief

ra,t,

t:onliuucs

on to Ilrc edge

of rhe imaga.field.

TltLts, thc

ntargittul

trt-v.r/tott,s

(h?

si¿a o.Í

lha

sto!)

apeilure,

and tlrc chief ra¡' sltows

the size of the.fiald

rtJ viev.

3. Stlaight

lines on the object

plane must be

intagetl as stlaight

lines

on the iru-

age

plane,

01'the

system has

S:g-melrical

distorti,on.

Except for

plane mirrors, no

t'eal optical systenl

cornpletely

satisfies

Max-

well's

criteria, although

some come

close. The

concellt of

perfect itnagely

can

also be

extended to

general,

non-axial

systetns.

4.3.9

Causes

of lmage

Quality

Degradation

There

ale six main

causes of

irnage

quality degraclation.

Fout'

are related

image

sharpness.

the four, the first is

design error

residuals, or

geonletrical alrellatious.

Ex-

cept in

a very few highly

restricted special

cases,

the lens designer

uever

able

to cornpletely

elirninate ail

aberlations in

real

system.

The

geornetlical

light fays

are

(almost)

never

perfectly

leunited.

If image

quality is limited by

aberrations,

the systeln is said

to be aberration

limited.

The second

cause of

pool

images

is ellors

in optical

and urechanical

fabl'i-

cation and

alignruent. Tliis

area is the

plovince of

the optics sliop,

the

machine

sho¡t, ar-rd the

glass lnanufacturer'.

Predicting,

artal5,7i¡1o,

r'estlictiug

and

cotnpen-

sating for

these l'eal-world

erlors is done

duriug tolelancing.

In addition.

rttarty

optical

systenls

degrade somervlrat

with time:

for examl:le. reflectitlg

telescopes

ì-\

Ø44c>\

à"è---

LENS

(sroP)

20 Chapter

4.3

Light and lmaging Systems

nlust occâsionally

be recollirnated.

Itl sotre lalge

and cotnplex systelns.

active op-

tics

can be

used to rnaintain alignrnent.

The

thircl cause of irnperfect

irnages is

difflaction of

ligltt t'ry apertut'e

and

obscuration

edges in the systeni.

Diffraction

is a cotrsequence

of the

wave

nattlre

of liglit

and

is thus impossible

to elirninate.

Difflaction spleads

out the light in

the

irnages of

point

objects and t'ules out

the

possibility of a

pelfect

point

image of

point object. In other

rvords, difffaction

creates the

fundametttal

limit to image

quality

beyond

which it is inrpossible

go.

In soltle

systetns,

the effects of dif-

fraction

may be smail

relative

to other sources

of inrage

degladation. But

in highly

collected

systerrs,

diffraction

is the r¡ain soulce

of irnage degladation,

and

these

systerns

ale

said

to be diff|action

lilnited.

The

foulth

cause of unsharp

images is atrnospheric

tut'bttlence,

what as-

tronomers

call

attnosphelic seeing.

An extreme

case of seeing

is tlie sliinimering

tliat

is apltalent

rvhen

looking

at a distânt object

in the

desert iti

the

sutnlner

heat.

Seeing

always

importattt

effect for

ground-based telescopes

that attempt

achieve

high angular

resolution

while viewing thlough

the dynantic

and inlrorno-

geneous air. Note

that seeing is

lelatecl to bLrt

not the saure

as the scintillation

twinkling of

stals. In

very

special

systems,

adaptive optics

can

paltially

relrove

the efTects

of seeing

in real time. Optical

systems

whose itnage

quality

is limited

atnrosphel'ic

turbulence are said

to be seeing

limited.

4.3.10

The

Point Spread

Function

Becanse it

is impossible

to image a

point

object

as a mathernatically

perfect

point

image,

the concept of

point

spt'eacl

function, or PSF,

has been

introduced.

The

PSF

gives the

physically cotrect light clistribution

in the ilnage

of a

point ob-

.ject

due to the

efïects of abelrations

and difftaction.

Neither fabrication

and alignrnent

errors

nor atrnospheric seeing

are includ-

ed in a stanclard

PSF

analysis. These erl'ols

are handled separately

because

they

ale of a st¿rtistical

rather

than a detelministic

r-r¿rture. Althougli

these sources

ellor are often

important,

the PSF due only

to aben'ations

and difiraction

is very

useful for the

lens designel because

provides a rneasttre of

the

best

image

qual-

ity a

nourinal design is

capable of achieving.

Most letts

clesign coutputel

progralns

can

calcul¿rte the

PSF and clisltlay

tlre results either nuurerically

grapliically.

For

a two-dimensional extended

object,

the light distribution

in the

irnage is

derived

nìathelnatically

by a tivo-dirnensional

convolution

of the olrject

(ol

its

pert'ect

irnage)

with the PSF. Inrage

processiug and

Foulier

optics

ale based on

this lelationship,

and they are

career specializations

in their orvn

right.

4.3.11

lmage Motion

Tlie

fif th cause of inrage

quality

degr:adation

is image r.notion.

Iura-qe

utotiott

cau

produced

a lineal translatic¡n ol

an angulal rotation

ofan optical

systeltl.

Section 4.3.12: Stray Light 21

The

classic example is camera vibration,

althor.rgli thele ale rnany othel sources.

addition,

sometirnes

the

object

itself is in motion.

In eithel

case. unless precaLr-

tions

are taken, the

result

is image

snrealing. AlthoLrgh

irnage

motion

often

irl-

portant

the

design and use of optical systenrs, it is only r:alely a

factol

in lens

design.

A.3.12 Stray

Light

Stlay

light, tlie sixtli cause

irnage

quality

degladation, is caused b),

r-'n-

wanted

light scattered and/or ref'lected by optical ancl rrech¿rnical sulfaces

u,ithin

an optical systeln. Stlay light can take two folrns. The f ilst folur

is a loughly uni-

form

veiling glare

across the irnage that reduces contrast. The second

form is

gl-rost

images. Ghost irrrages can also take two forms. The

filst is a complete du-

plicate

irnage, often invelted

ancl

in failly

goocl

focus. The seconcl consists of dis-

crete flare

spots, often highly aberrated, corresponding

bright

ligttt

sources

within or close to the fielci of view.

Excessive stlay light

can

disqualify

a potential optical design as readily

excessive

aberrations.

Stlay

light

particularly

difficult

plobleur

for

complex

systems

that either have

rnany lens

surfaces ol that have lalge rlecharnical

distanc-

es between surfaces

(including

the inrage sulface).

It is im¡roltant for the optical

designer to be aware of stlay-light

considerations

fi'om

the beginning of the

cle-

sign

process.

Waiting

until

a design is nearly finished and

then trying to correct

stta¡,-lioþ1

problems is

usually

doomecl to failure.

Since

about 1

940,

there have been two advances

that have revolLrtionized

the

control of stray light. The first

aclvance

thin-film anti-r'eflection

lens coatings.

Urrcoated

glass

leflects about 4Vo of the incident light

per

sulface,

with higher'-in-

clex

glasses

reflecting somewhat lnol'e than lower-index

glasses.

Fot a urulti-ele-

ment lens

with r.r.rany air-to-glass

surfaces,

the total of

this

reflected

light cau be

substantial. The big

problem is not

the

loss of ir.r.rage light; the big

problern is the

increase in stray light. This leflected liglit must

somewhele,

irncl rnuch of it

in-

variably

encls up on the irnage surface

stl'ay

light. A coated

Ieus

surface,

how-

ever, typically reflects less than 1Vo, and often rnuch

less. Thus, stlay

light is

reduced. Coated lenses nrake

plactical rnany corrplex systenrs

that otherwise

would

r¡seless.

Tlre second advance is the development of

collrprìtel'

prograllìs that analyze

stray

light,

especially light

reflected by

inteliof mechanical surl'aces

such as baf-

fles.

The results

frorn this capability liave

pointed

the

way towarcl

gt'eat

itnpt'ove-

nrents in high-pelforrrance optical systenrs such

olbiting

telescopes.

At the

present time, the nrajol

stray-light

cornputel

prograrns

ale APART,

ASAP,

ancl

GUERAP. Note tlìat these

prograrns

anerlyze

stray light, but they

clo noI clesign

baffles. Baffles must still be designed by hand. Thin-fil¡

coatings

and stray-light

supplession are each caleer specializations in

their

own

ri-ght.

Chapter

4.3 Light and lmaging

Systems

Figure 4.3.3. Coordinate and Sign

Convcntio¡rs. The

loctl

coorditLaÍe.\ ot a

given

lens surface.fonn u rect-

trttgular, Carlesicut, ot'.rvt .s¡,.slem.

Coctrditttút

l¡lts are bt' rota¡ion

ttrtl¡las a,

þ,

curcl

'¡

ubotl tltc x,

t',

ttntl

axas,

res¡tectit'elt,. Tlrc tilt.ç

nu.rt be ap¡tlied sequetúiall\J. Far

CODE

\t, all til¡ angla.s are

slto¡'n

ltt¡sitit

For ZEÌvIAX,

¡|rc

a untl

tilts are shovtt n(.ß¿ttit'¿.

4.3.13

Focal and Afocal

Systems

either the

object 01'the image is located

at a finite distance from the lens,

then the lens is called

focal. Thele are three cliffelent cases

focal

systerns. In

the filst case, the

object is at infinity and the irnage

at a

finite distance. An ex-

ample

alì ordinary camera lens

focused on a distant object. In the

second

case,

the object is at a

finite distance and the

image is at infinity. Exarnples are a hand

magnifier and a cornpound

micloscope, both focused for the

eye. In the third case,

both the object and irnage ale

at finite dist¿rnces. An exan-rple

is a ntacro car.ìlera

lens focusecl up close.

However, if

both

the

object and the irnage

are at infinity, then the system is

called afocal.

The classic example

of an afocal system is a telescope with an eye-

piece

focused for the

eye. A laser

bear.n

expander

is also afocal.

4.3.14 Fast

and Slow Lenses

and

Detectors

The tenn fast applied

to an optical

systern oliginated historically to clescribe

lens

giving

a relatively bright

(high

in'adiance or illuminance)

ilnage that

woulcl

expose a

photographic

piate quickly.

A fast lens

illulninates an iurage

point

with

a large solid angle

liglit;

tliat is, it is a lens with a large

image numerical aper'-

tule or a

srnall

focal

ratio or.f/nunrber.

Sirnilarly, a fàst

photodetector,

such

plrotoglaphic

film, is

one requiling little exposule to record

tlie light. As rniglit be

expectecl.

slow rneans the

opposite of fast, and the

speed of atl optical systenl or

detecto| r'efèr's to how

fast or slow it is.

Section 4.3.15: Coordinate Systems

and Sign Conventions 23

4.3.15

Coordinate

Systems and

Sign Conventions

Coordinate systems and sign conventions

ale subjects about which

there is

universal

agreelnent. Different lens design progranls

may use different sys-

tems,

and ttris

lack of standardization rnay lead

to confusion.

palticular-.

CODE

V and ZEMAX

do not use the salne

systerÌ1,

although the

two

systelÌls

are

sirnilar.

Thns, first the

CODE V system

will

be described,

followecl by the ZEI\4-

AX system. For other

progl'ams,

the Llsel'

must deterrnine his orvn

specif.ic couveu-

tions.

Figure

A.3.3

illustrates the

CODE

systern. The

coorclinate

axes cornplise

a rectangular, Cartesian, or x¡,¿ coordinate systeln.

The oligin is at the

vertex

whatever surface is under consideration; tliat is,

these al'e local coordinates, l'tot

global cooldinates. The

axis is the optical axis

(or

sometimes

just

a urechanical

leference axis) with

positive values

the right of the origin. Light oliginates

frorn somewhere aiong the negative

axis, and initialty

moves ir.r

the

direction of

the

positive

axis fi'om left to

right. A reflection reverses this direction of light

travel but not tlìe orientation of the coordir-rate

system.

The

1,¿¡¡i.

is orthogonal to

tlie

axis and runs

and

down, with

positive

valnes of

.1,

above the origin. Thus

the

¡r¿

plane

is the

plane

of tl.re paper.

The ¡

axis

orthogonal

to botli the

and

axes

and

tuns in and out of the

paper,

with

positive

values

,r into the

paper.

This

sign

convention

creates a right-handed

x},¿ system.

Because local coordinates

are used, tlie origin of the coordinate systeln

lnoves successively fi'om the vertex

of one surface

to the vertex

o1'the

next as

yor,r

progress

through a lens. As many as

seven

parametels govern

this cooldinate

transfbrmation.

One is the thickness associated

with

the

present

sulface. Others

al'e the decenters and tilts associated rvith the

surface.

to three decenters

and thlee tilts can be specified for each surface.

The order in which these parameters are applied duling the transfonnation is

impoltant;

changing

the order rnay

give

different results. First the coordinate sys-

tern is

rnoved

down

the optical axis by the arnount specified by

the thickness

pa-

rameter. Then the decenters associated with tlie new surface are applied in

the

order of x, then

and then

Actually, the

order so

far does not rì1atter. Also, note

that the

thickness

parameter

and the

decenter

paraÍneter

have the sarre

effect.

Finally,

the tilts associated with the

new surface

are appliecl, and lrele the order

does indeed

matter.

The three rotation

tilt angles are

cx.,

and

CODE

V, a

positive

ct tilt

is a

left-handed rotation ofcooldinates about the

positive

¡-axis. A

positive

tilt

is a left-handed

rotation of coordinates about the

positive

),-axis.

Ancl

positive

tilt

is a light-handed

rotatiorì of

coordinates âbout the

positive

¿-axis.

ThLrs, the

CODE V

coordinate

system

is not cornpletely right-l.randed. For this sign conven-

tion,

retèr

again to Figule 4.3.3., whele the rotation angles are all shown

positive.

To apply the tilts,

first the ct tilt is applied

give

a new intennediate coor'-

dinate

systeur. Then in this intelnrediate coordiuate systerÌ1,

the

tilt is applied to

24 Chapler 4.3

Light and lmaging Systems

give

a second

new intel'ìnecliate

cooldinate systellì.

Lastly, in the second intenne-

diate coordinate system,

the y tilt is applied to

give the final coordinate systenl.

course,

sorne

these

seven

pal'anìeters can

zero. In

particular,

avoicl

confusion, specifying only one nolìz-el'o

tilt angle at

a time is recotnurended

(use

rnultiple single-angle tlanstbnnations

as needed). In

addition, note carefully

that

yoLr

wish to re\/erse ol

unclo

a coorclinate

transformatior'ì.

yotì

cannot melely

l'e\/erse the

signs of the

palarnetefs: you

must

also corlpletely

re\/erse the older'

that the

paralneters

are applied. CODE

V lias a command

that allo'"vs this to be

done easily.

In ZEMAX,

coolclinates, clecenters, and

tilts are handled the sarne

r'vay

rvith

three exceptions.

First, in ZEMAX

there is no

decenter;

this function nrust be

hanclled by

the tl.rickness

pararreter.

Secor.td,

positive

clt.

tilt is a

rlglrt-lianded

t'o-

tation

about the

positive.r-axis. Ancl third, a

positive

tilt is a llglrt-hanclecl

rota-

tion about the

positive

l'-axis.

Thus, the

ZEMA'X system is

cornpletely

right-handed,

and the

and

angles shown

in Figure 4.3.3.

now

become

nega-

tive.

There are rnore

sign conventions. If the

lens sut'face is curved,

a positive cur-

vature has the center of

curvature

the right

of the surface,

atrd a negative cttr-

vature

has

the center of curvatule

to the left of the surface.

For a lens as a rvhole,

positive

lens causes

rays to increase convelgence,

while a negative

leus causes

rays

to decrease convergence

(or

inclease divergence). Convex

lettses are

positive

and concave

lenses ale uegative, bnt

concave mirrors ale

positive.

Still

rnore sign

conventions will be introduced

later'.

Finally,

r'ecall that for lotationally syrlrnetlic,

axially

centered systetns, any

plane

containing the optical

axis is a rneridional

plane, ar.rd all rneridional

planes

are equivalent.

Thus,

rvith

no loss of

generality,

the¡'z rneridional

plane is usttalll,

selected.

and

objects

and images at'e located

along the

),-axis.

4.3.16 Optical

Prescriptions

oldel

to be able to enter

lens

into a col'r'rputer for optimization

or evalu-

ation, the optical

prescription is lequired. An optical

prescription contaius

all the

inforrration

needed to specify the nominal

lens configuration. Tolelances

are ustì-

ally not included

ancl are hanclled separately.

Because

most lenses consist of collections

of surfaces through

which the

light

¡rasses

seqLrentially, the

nrain palt of a conventional

plescliption is a table

listing the snrf'ace

paramete[s.

The sulfaces

are numbered

and tabulatecl in orclel

stalting with the object sulface

(sulface

ancl ending

with

the irnage surface. A

rninirrurn o1'three sulfaces is required:

the

ob.ject sulface,

the stop sut'face.

atrd

the image sul'face. The sto¡r cannot be on the object ol

irnage surfaces. The

details

of how an optical

prescription is plesented depend very tnuch on

your

particular

lens desi

gn pl'ograril.

Section 4.3.16: Optical Prescript¡ons 25

For eacl-t

sulfãce, the

parameters

incÌude: sulface nnrnber'. raclius

culva-

tule,

thickness

to the uext

surface,

glass

type in the next r-neclinln.

any

asplielic

data,

any apertul'es and

obscuratious, arly tilt and clecenter

data, ¿rtrd auy solves.

One

sulf¿ice is designated tlìe

stop surface.

Illstead

of ladius of cuLvatule,

the leciplocal of'r¿rclir.rs

of cnrvature. knolvrr

sirnply

as tl'ìe curvature.

is often substituted.

CnrvatuLe has the acli'antages that a

greatel'absolute value denotes

gl'eater

effect and that a flat

surface does not need

to be descl'ibed

with

an infinite

quantity.

However, cLrrvature cloes requile

the

in-

convenient

use

of tiny nuurbel's

(of

course,

(he

coulplrter

cloes uot nrind)

Thickness is vertex

(not

edge) sepal'¿rtior'ì along the optical axis l'rorn

one sul'-

face

to the

next. Glass type can

any

optical

glass

well

as a crystal,

wittel',

aif,

vacuull1,

and so forth.

A special

glass

clesignation

n-rillol-, whel'e

tlre Iight is le-

flected

back into the

plevious

rnedium.

Solves

are specizrl

opelations that acl.just

lens

palarretefs

to satisfy

conditions; solves aLe cliscussecl in

Cliapter

4.13.

multiple

configulation

lens

is ¿r special

situation. Hele thele are two or

rnore

discrete lens configulatiol'rs,

with one or tìlore lens

¡talametels

chaneing

be-

tween configurations.

One

of the

most cornulou nrultiple configuration lens types

is a zoom lens. In a zoorn lens, the lens elelnents

lnove longitudinally relative to

each other

to vary the focal

length. But rnultiple configuration lenses are not lirn-

ited to zoorn lenses. Otherexarnples ir-rclude

lenses

witli

scanning rniuors,

rnulti-

ple wavelength

passbands,

beam splittels, and so forth. In

Chapter

8.4 the lens is

given

multiple entrance

pupil

diar.neters.

lens

prescription,

the

palameters

that change

betrveen

configurations

are listed separately along with their

values

in each configr-rration. Altelnatively,

more than one lens tallle is

given.

Anothel special

situation is unusual characteristics, sucl'r as sulfaces

u,itlr

difflactive

properties

gradierrt

index rnedia. Again, these palametels are often

listed separately.

Three other

pal'alneters

groups

parameters

ale also

part

of the optical

prescription.

The first is the

set of wavelengths used. The set nray consist of only

one waveleugth fol'a rnonochromatic or an all-r'eflecting system.

For

a color'-cor'-

Iected refracting

systern, two or rnore wavelengths are necessary. The extferne

wavelengths

define the

wavelength

passband, and usually a centlal rvavelength is

clesignated as the reference, plincipal,

prirnary

w¿rvelength for calculating

first-order

properties.

The second

pararneter gloup

specifies

tlle

positions

of the object

points

in the

field of

view.

Either angulal position

as seen

frorn

the entrallce

pupil

lineal po-

sition on the object surface can I¡e

used. Three ol

foLrr

object

points ale commonly

specified.

One object

point is

usLrally on the optical axis, and one is

usually at the

edge

of the field.

Half-field

angles or distances relative to the optical axis are

giv-

en;

full-field

40"

corresponds to a half-fielcl of

*20".

For asymnretric or'

off-axis

systems, many object

points

allayed in two climensions across the obiect

surface

rnay be needed.

26 Chapter

4.3 Light and

lmaging Systems

The

third

paratneter

contfols

the diameter

of the on-axis

beam

transrnitted

the system.

Mgs.t_9o-B!lolly,

entrauce

pup{9tqT_.tg(ElÐfgspecified.

Alterna-

tively, the

dtg]]]glgl-9|l¡€--ep9llrlre-rrn

the slgplulgg

rarl-Þ9_s!9c-ifieçl,

Another

alternative

is to specify

tbe-Éûlumbg,L.or

nttmerical

aperture of

the irnage.

For fi-

nite

conjugate lenses,

the nutnerical

apertut'e

of the object

is often

used. Other'

variations are also

possible, but the idea

is the sarne.

Finally,

an optical

prescription includes

a title,

any apodization

(nonuniforn

partial transmission

across

the etitrance

pupil), any vignetting

factol's

way of

specifying

the sizes

of off-axis

beams),

any special

controls

for the

program

(such

as ray airning),

and the

units used. Millimeters

are becorning

the

world

standard

for optics;

use inches

you rnust; avoid

centimeters.

For

convenience,

when listing

prescription,

your

softwale

tlay

append sev-

eral derivecl

quantities.

These

rnay include:

focal length,

focal

ratio, back

focal

length,

overall length,

maxirnnul

irnage

lieight, entlance

and

exit

pu¡til diarnetels

and locations,

clear

apertures,

and so forth.

Be sut'e

to differe¡tiate

between spec-

ified

and delived

quantities.

The

derivecl

quantities are technically

not

part

of the

prescription.

A.3.17

Aspheric

Surfaces

Optical

surfaces

are most often

spherically

shaped, but

general conics,

poly-

nomials,

and

other shapes

can also be used.

For the more

comrnon rotationally

symmetric

surfaces

(including

spheres),

the

departure of

a surface frorn

plane'

also

called the sag of

a surface

parallel to the optical

axis, is

given

by:

whele:

lt2

),2

is the axial height

on the surface

(transvelse

distance

froni

the axis),

is the curvatule

of the surface

at its

vettex,

k is

the conic constant of

the sulface,

and

D, ate the

4th-,6th-, 8th-, and

l0ttr-order

polynomial surface

deforma-

tion

coefficients, respectively.

For

the conic constarìt,

| 1 a t)

| I

+ klc'h'l-

Section

4.3.18: Thin Lenses 27

-1

gives

paraboloid,

-1

gives

a liyperboloid,

and

gives

an oblate ellipsoid

(like

a dool knob or the top o{'a spinning

planet.

such

as the Earth

especially Jupiter).

Note

that

definition, arz_r' surface that

is not a

pute

sphere

is arr asphet'e.

pleviously mentioned, a

plane

surface

is considered a sphere

rvith

an infinite

ladius of

curvature. Spheles are

distinguished from all othel sulface shapes

be-

cause

spheres

(and

flats) are the only surfaces

that can be tnade relatively easily

in the optical shop.

Any asphere is hard

to make.

4.3.18

Thin Lenses

Fol a real singlet

lens with a nouzero

thickness, the

palaxial

fbcal

length is

,.ì

, ,,1

I I

lt1

-tltl

j=rtt-,,[[-;-,\\)

where/is

the focal length,

is the

lefractive index of

the

glass,

r, atrd rr al'e

respec-

tively

the front

and

lear ladii

of curvature, and ¡

is the axial thickness.

If the thickness

of the lens

small

relative to its other

dimensions, then

this equation reduces

to the

lensrnaker's equation :

ch-

+ Aha +

Bh6

ChB

+ Dltto

(4.3.2)

where

¿ is the conic

eccentlicity ft'orl] analytical

geouretry, ancl:

k=0givesasphere,

-l

gives

prolate ellipsoid

(like

the

pointecl end of

a football),

The

concept of a thin lens is

encounteled again

ancl agaiti in lens

design.

thin lens has snch snrall

culvatures and centel

and edge thicknesses

that

all of its

refractive

power

is effectively concentrated

in a

plane.

course, truly

thin lenses

not exist in the real

wor-ld, and the concept

is an approxirnaticln.

Nevertheless,

for rnany

applications, the errors

are srnall and

can

neglected,

at least initially.

Note that at the

center

a thin lens,

the two lens surfaces

ar-e

parallel to each

othel and are effectively

in contact. Thus,

a ray

passing through the centel

o1'a

thin

lens is

uncleviated.

Many useful

optical formulas

are thin-lens fo¡:uiulas.

Perhaps

tlìe most use-

ful is the thin-lens

eouation:

r,-r)lr-11

./ \/'r

12)

rvhere/is the focal length. s is the object

distauce,

anci

sign conventions

ale as

given

above.

Note that the focal

length of

tliin

lens is equal

to the iniage

when viewing a distant object.

Itl

--+-

.r .r'

(A.3.3)

,ç' is the

irnage distancc.

The

to the distance

frorn the

lelis

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