Chapter 4.Matrix Methods in paraxial optics

Chapter 4.Matrix Methods in paraxial optics

• Development of systematic methods of analyzing optical systems with numerous elements

• Matrices developed in the paraxial (small angle) approximation• Matrices for analyzing the translation, refraction, and reflection of

optical rays• Matrices for thick and thin lenses, optical systems with numerous

elements

Let’s start with definition of cardinal points (planes) : focal (F), principal (H), and nodal (N) points (planes)

Complex optical systemsComplex optical systems

Thick lenses, combinations of lenses etc..Thick lenses, combinations of lenses etc..

tt

nnLL

nn nn’’

Consider case where t is not Consider case where t is not negligible. negligible.

We would like to maintain our We would like to maintain our Gaussian imaging relationGaussian imaging relation

Psn

sn

=+''

But where do we measure s, sBut where do we measure s, s’’ ; f, f; f, f’’from? How do we determine P?from? How do we determine P?

We try to develop a formalism that We try to develop a formalism that can be used with any system!!can be used with any system!!

Cardinal points and planes:Focal (F) points

Cardinal points and planes:Focal (F) points

. . .

F'

1 k (마지막 면)

u =01

F'

1 k

u =01

제2초점(second focal point, image side focal point) : F' -무한대에 있는 축상 물체점의 상점-광축과 평행하게 입사한 광선이 모이는 점(실상) 또는 모이는 것처럼 보이는 점(허상)

제1초점(first focal point, object side focal point): F -무한대에 상이 생기는 축상 물체점-상측에서 광축과 평행하게 입사한 광선이 모이는 점또는 모이는 것처럼 보이는 점.

F

F

u'k=0 u'

k=0

11 kk

... ...

Cardinal points and planes:2nd principal planes (PP) and points

Cardinal points and planes:2nd principal planes (PP) and points

nnLLnn nn’’

HH22

ƒ’ƒ’

FF22

PPPP22

제 2 주요면 (상측 주요면) : PP2-물체측에서 광축과 평행하게 입사한 광선을 상측에서 보아 굴절되는 것처럼 보이는 가상면.

제2 주요점 (상측 주요점 ): H2 - 제2 주요면과 광축의 교점.

Cardinal points and planes:1st Principal planes (PP) and points

Cardinal points and planes:1st Principal planes (PP) and points

nnLLnn nn’’

HH11

ƒƒ

FF11

PPPP11제 1 주요면 (물체측 주요면) : PP1

-상측에서 광축과 평행하게 입사한 광선을 물체측에서 보아 굴절되는 것처럼 보이는 가상면.

제1 주요점 (물체측 주요점 ): H1 – 제1 주요면과 광축의 교점.

Objective distance, image distanceObjective distance, image distance

l l'

1 k

H H'

h h

P P'

u1

u'k

o o'

n1

n'1

h1

A1= H

1= H'

1

u'1

u1

l1

l'1

면의 물체거리

면의 상거리

o

1 2

물체거리(object distance)

: 제1 주요면에서 물체까지의 거리

l =HO

상거리(image distance)

: 제2 주요면에서 상면까지의 거리

l’ = H’O’

Focal lengthFocal length

1 k......

u1=0

h1

hk

F'u'k

Ak

H'HA1

bfl

f'b

efl, f'

fb

f

F u1

h1

hk

u'k=0

․유효 초점거리(effective focal length, efl):f' - 제2 주요점에서 제2초점까지의 거리

efl = f’ = H’F’ = h1/u’k , u1 = 0

․후 초점거리(back focal length, bfl): f‘b-광학계의 마지막 면의 정점에서 제2 초점까지의 거리

bfl = f’b = AkF’ = hk/u’k , u1 = 0

․제2 주요면의 위치 = bfl - efl

․앞 초점거리(front focal length): fb= 작동거리(working distance) - 제1면의 정점에서 제1초점까지의 거리

fb = A1F =h1/u1 , u’k = 0

․물체측 초점거리(Object side focal length):f- 제1 주요점에서 제1 초점까지의 거리

f = HF =hk/u1 , u’k = 0

․ 제1 주요면의 위치 : fb - f

Utility of principal planesUtility of principal planes

HH22

ƒ’ƒ’

FF22

PPPP22

HH11

ƒƒ

FF11

PPPP11

s s’

nnLLnn nn’’

hh

hh’’

Suppose s, sSuppose s, s’’, f, f, f, f’’ all measured from Hall measured from H11 and Hand H22 ……

Cardinal points and planes:Nodal (N) points and planes

Cardinal points and planes:Nodal (N) points and planes

nn nn’’

NN22

NPNP22

NN11

NPNP11

nnLL

Nodal point and optical centerNodal point and optical center

N' Nu

u' c

1 2

절점(nodal point : N, N’)-광학계는 입사각과 출사각이 같은 광선이 1개 존재. -제1절점:이 광선을 물체측에서 보아 입사하는 것처럼 보이는 점. -제2절점:이 광선을 상측에서 보아 출사하는 것처럼 보이는 점.

광심(optical center : C) : 이 광선이 실제로 광축과 교차하는 점.

i) 제1절점으로 입사한 광선은 제2절점에서 출사 (제1절점 - 광심 - 제2절점) ii) (제1)절점으로 입사한 광선은 입사각과 출사각이 같다. iii) 상측 매질의 굴절률과 물체측 매질의 굴절률이 같으면 절점과 주요점은 같다.

N = H , N' = H' iv) 제2절점을 기준으로 광학계를 회전시키면 상의 위치는 변화하지 않는다.

F'

F'

Cardinal planes of simple systems1. Thin lens

Cardinal planes of simple systems1. Thin lens

Psn

sn

=+''

Principal planes, nodal planes, Principal planes, nodal planes,

coincide at centercoincide at center

VV

H, HH, H’’

VV’’

VV’’ and V coincide andand V coincide and

is obeyed.is obeyed.

Cardinal planes of simple systems1. Spherical refracting surface

Cardinal planes of simple systems1. Spherical refracting surface

nn nn’’

Gaussian imaging formula Gaussian imaging formula obeyed, with all distances obeyed, with all distances measured from Vmeasured from V

VV

Psn

sn

=+''

Conjugate Planes – where y’=yConjugate Planes – where y’=y

HH22

ƒ’ƒ’

FF22

PPPP22

HH11

ƒƒ

FF11

PPPP11

s s’

nnLLnn nn’’

yy

yy’’

Combination of two systems: e.g. two spherical interfaces, two thin lenses …

Combination of two systems: e.g. two spherical interfaces, two thin lenses …

nn22nn nn’’HH11’’HH11

HH22 HH22’’

HH’’

yyYY

dd

ƒ’ƒ’

ƒƒ11’’

FF’’ FF11’’

1. Consider F1. Consider F’’ and Fand F11’’

hh’’

Find hFind h’’

Combination of two systems:Combination of two systems:

nn22nn nn’’

HH11’’HH11

HH22 HH22’’HH

yyYY

ddƒƒ

1. Consider F and F1. Consider F and F22

FF22

ƒƒ22

hh

FF

Find hFind h

SummarySummary

HH11’’HH11 HH22 HH22’’HH HH’’

ƒƒ ƒ’ƒ’hh hh’’

FF FF’’

dd

SummarySummary

2

2121

211

2

2

2

12

1

2

21

2

,''

''

'''

''''''

nPPdPPP

orfn

ffdn

fn

fn

fn

hnn

PPdHH

ffdh

hnn

PPdHH

ffdh

−+=

=−+=

=⎟⎟⎠

⎞⎜⎜⎝

⎛−==−=

=⎟⎟⎠

⎞⎜⎜⎝

⎛===

Thick LensThick Lens

nn22

RR11 RR22

HH11,H,H11’’ HH22,H,H22’’

In air n = nIn air n = n’’ =1=1

Lens, nLens, n22 = 1.5= 1.5

RR11 = = -- RR22 = 10 cm= 10 cm

d = 3 cmd = 3 cm

Find Find ƒƒ11,,ƒƒ22,,ƒƒ, h and h, h and h’’

Construct the Construct the principal planes, H, principal planes, H, HH’’ of the entire of the entire systemsystem

nn nn’’

Principal planes for thick lens (n2=1.5) in airPrincipal planes for thick lens (n2=1.5) in air

EquiEqui--convex or convex or equiequi--concave and moderately thick concave and moderately thick ⇒⇒ PP11 = P= P22 ≈≈ P/2P/2

3' dhh =−=

12

22

'ff

ndh

ff

ndh

•−=

•=HH HH’’ HH HH’’

Principal planes for thick lens (n2=1.5) in airPrincipal planes for thick lens (n2=1.5) in air

PlanoPlano--convex or convex or planoplano--concave lens with Rconcave lens with R22 = = ∞∞

⇒⇒ PP22 = 0= 0

dh

h

32'

0

−≅

=

12

22

'ff

ndh

ff

ndh

•−=

•=

HH HH’’ HH HH’’

Principal planes for thick lens (n=1.5) in airPrincipal planes for thick lens (n=1.5) in air

For meniscus lenses, the principal planes move For meniscus lenses, the principal planes move outside the lensoutside the lens

RR22 = 3R= 3R11 (H(H’’ reaches the first surface)reaches the first surface)

P Same for all lensesSame for all lenses

12

22

'ff

ndh

ff

ndh

•−=

•=

HH HH’’ HH HH’’ HH HH’’HH HH’’

Examples: Two thin lenses in airExamples: Two thin lenses in air

2

2

ffd

PPdh ==

ƒƒ11 ƒƒ22

dd

HH11’’HH11 HH22 HH22’’

n = nn = n2 2 = n= n’’ = 1= 1

Want to replace HWant to replace Hii, H, Hii’’ with H, Hwith H, H’’

1

1'ffd

PPdh −=−=

hh hh’’

HH HH’’

Examples: Two thin lenses in airExamples: Two thin lenses in air

ƒƒ11 ƒƒ22

dd

n = nn = n2 2 = n= n’’ = 1= 1

2121

2

2121

111,

ffd

fff

ornPPdPPP

−+=

−+=

HH HH’’

FF FF’’

ƒƒ ƒ’ƒ’ fss1

'11=+

ss’’ss

Two separated lenses in airTwo separated lenses in air

ff11’’=2f=2f22’’

d = 0.5 d = 0.5 ff22’’

HHHH’’

FF’’FF

ff’’

d = d = ff22’’

HHHH’’

FF’’FF

ff’’

Two separated lenses in airTwo separated lenses in air

ff11’’=2f=2f22’’

d = 2d = 2ff22’’

HHHH’’

FF’’FF

ff’’

d = 3d = 3ff22’’

Principal points at Principal points at ∞∞

e.g. Astronomical telescopee.g. Astronomical telescope

Two separated lenses in airTwo separated lenses in air

ff11’’=2f=2f22’’

d = 5d = 5ff22’’

ff’’

e.g. Compound microscopee.g. Compound microscopeHH

FF’’FF

HH’’

Two separated lenses in airTwo separated lenses in air

ff11’’==--2f2f22’’

d = d = --ff22’’

e.g. Galilean telescopee.g. Galilean telescope

Principal points at Principal points at ∞∞

Two separated lenses in airTwo separated lenses in air

ff11’’==--2f2f22’’

d = d = --1.51.5ff22’’e.g. Telephoto lense.g. Telephoto lens

HH HH’’

FF’’

ff’’

FF

What is the ray-transfer matrix What is the ray-transfer matrix

How to use the ray-transfer matrices How to use the ray-transfer matrices

How to use the ray-transfer matrices How to use the ray-transfer matrices

Translation Matrix Translation Matrix

( ) ( )( ) ( )

1 0 1 0 0 0 0

1 0 0

1 0 0

0 01 1 0

0 01

tan

1

0 1

1 10 1 0 1

y y L y L

y y L

y

y yy L x x

α α α α

α

α α

α αα

= = + ≅ +

= +

= +

−⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Refraction Matrix Refraction Matrix

' :

1 1

yR

yR

yR

Paraxial Snell s Law n n

y n y n y y n nyR n R n R R R n n

α θ φ θ

α θ φ θ

θ α

θ θ

α θ θ α α

′ ′ ′= − = −

= − = −

= +

′ ′=

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞′ ′= − = − = + − = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟′ ′ ′ ′⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠

( ) ( )1 0

1 0: 0

1 1 : 0

y y

y y Concave surface Rn n Convex surface R

R n n

α

α α

′ = +

⎡ ⎤′ <⎡ ⎤ ⎡ ⎤⎢ ⎥= ⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥⎢ ⎥′ − >⎣ ⎦ ⎣ ⎦⎜ ⎟ ⎜ ⎟′ ′⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Reflection Matrix Reflection Matrix

( ) ( )

( )

:

2

1 0

2 1

1 02 1

y y yR R R

Law of Reflection

y y yR R R

y y

yR

y y

R

α θ φ θ α θ φ θ θ α

θ θ

α θ θ α

α

α α

α α

′ ′ ′= − = − = + = + = +− −

′=

′ ′= + = + = +

′ = +

⎛ ⎞′ = +⎜ ⎟⎝ ⎠

⎡ ⎤′⎡ ⎤ ⎡ ⎤⎢ ⎥=⎢ ⎥ ⎢ ⎥′ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

Thick Lens Matrix IThick Lens Matrix I

0 011

0 011

2 1 12

2 1 1

3

3

1 0:

11 2 :

0 1

:

L

L L

y yyRefraction at first surface Mn n n

n R n

y y ytTranslation from st surface to nd surface M

yRefraction at second surface

α αα

α α α

α

⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥= =− ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎡ ⎤⎢⎣

2 2

2 22

1 0

L L

y yMn n n

n R nα α

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥= =′−⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎦ ′ ′⎢ ⎥⎣ ⎦

Thick Lens Matrix IIThick Lens Matrix II

( )

( )

( )

3 2 1

2 1

1

21

1

2 1

:1 0 1 0

10 1

:

11 0

1

1

L L L

L L

L

L LL L

L

L L

L

L L

LL

L

Thick lens matrix M M M M

tM n n n n n n

n R n n R n

Assuming n n

t n n t nn R n

M n n nn n nn R nn R n

t n n t nn R n

t n nn nn R n R

=

⎡ ⎤ ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥′= − −⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦′ ′⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

′=

⎡ − ⎤+⎡ ⎤ ⎢ ⎥

⎢ ⎥ ⎢ ⎥= −⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥⎣ ⎦

⎣ ⎦−

+

=−⎡ ⎤−

+ +⎢ ⎥⎣ ⎦ 1 2

1L L

L

n n n n tn R n R

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥− −⎢ ⎥+⎢ ⎥⎣ ⎦

Thin Lens MatrixThin Lens Matrix

2 1

1 2

:1 0

1 1 1

1 1 1

1 01 1

L

L

Thin lens matrix

M n nn R R

n nbutf n R R

Mf

⎡ ⎤⎢ ⎥= ⎛ ⎞−⎢ ⎥−⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎛ ⎞−= −⎜ ⎟

⎝ ⎠

⎡ ⎤⎢ ⎥= ⎢ ⎥−⎢ ⎥⎣ ⎦

The thin lens matrix is found by setting t = 0:

nL

Summary of Matrix MethodsSummary of Matrix Methods

Summary of Matrix MethodsSummary of Matrix Methods

System Ray-Transfer Matrix System Ray-Transfer Matrix

Introduction to Matrix Methods in Optics, A. Gerrard and J. M. Burch

1

1

yα⎡ ⎤⎢ ⎥⎣ ⎦

2 2

2 2

n

n

yα

+

+

⎡ ⎤⎢ ⎥⎣ ⎦

System Ray-Transfer Matrix System Ray-Transfer Matrix Any paraxial optical system, no matter how complicated, can be represented by a 2x2 optical matrix. This matrix M is usually denoted

: system matrixA B

MC D⎡ ⎤

= ⎢ ⎥⎣ ⎦

A useful property of this matrix is that

0Detf

nM AD BCn

= − =

where n0 and nf are the refractive indices of the initial and final media of the optical system. Usually, the medium will be air on both sides of the optical system and

0Det 1f

nM AD BCn

= − = =

Significance of system matrix elements

Significance of system matrix elements

The matrix elements of the system matrix can be analyzed to determine the cardinal points and planes of an optical system.

0

0

f

f

y yA BC Dα α

⎡ ⎤ ⎡ ⎤⎡ ⎤= ⇒⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

Let’s examine the implications when any of the four elements of the system matrix is equal to zero.

0 0

0 0

f

f

y Ay B

Cy D

α

α α

= +

= +

D=0 : input plane = first focal planeA=0 : output plane = second focal planeB=0 : input and output planes correspond to conjugate planesC=0 : telescopic system

System Matrix with D=0System Matrix with D=0Let’s see what happens when D = 0.

0

0

0 0

0

0f

f

f

f

y yA BC

y Ay B

Cy

α α

α

α

⎡ ⎤ ⎡ ⎤⎡ ⎤= ⇒⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

= +

=

When D = 0, the input plane for the optical system is the input focal plane.

Ex) Two-Lens System Ex) Two-Lens System f1 = +50 mm f2 = +30 mm

q = 100 mmr s

InputPlane

OutputPlane

F1 F2F1 F2

T1 R1 R2 T3T2

03 2 2 1 1

02 1

1

2 1 1 2

1 0 1 01 1 1

1 11 10 1 0 1 0 1

11 0 1 1 01 1 1

1 1 11 1 10 1 0 1 0 1

f

f

y y s q rM M T R T R T

f f

q q rr qrf fs q s

M rf f f f

α α

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦⎣ ⎦ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

− + −⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − − + −⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

1

1 1

1 1rf f

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥− − +⎢ ⎥⎣ ⎦

1 13 2 2 1 1

2 1 1 2 1 1

1 2 1 1 2 2 1 1

2 1 1 2 1 1

110 1 1 1 11 1

1 1 1

1 1 11 1

q q rr qf fs

M T R T R Tq q r rr q

f f f f f f

q s s q q r r q q r rr q sf f f f f f f f

q q r rr qf f f f f f

⎡ ⎤− + −⎢ ⎥⎡ ⎤ ⎢ ⎥= = = ⎢ ⎥ ⎢ ⎥⎛ ⎞ ⎛ ⎞⎣ ⎦ − − − − + − − +⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎛ ⎞ ⎛ ⎞+ +− − − + − − − − +⎢ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎢= ⎢ ⎛ ⎞ ⎛ ⎞⎢ − − − − + − − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎣

⎤⎥⎥⎥⎥

⎢ ⎥⎦

( ) ( ) ( )( )

2 1 1

2 1 1

1 2

1 1 0

30 50 100 50175

100 50 30

q r rD r qf f f

f f q frq f f

r mm

⎛ ⎞= − + − − + =⎜ ⎟

⎝ ⎠− +

⇒ =− −

− += =

− −

ƒƒ11 ƒƒ22

dd

HH HH’’

FF FF’’

ƒƒ ƒ’ƒ’

ss’’ss

hh

rr

1 2

1 2 1 2 1 2

1 1 1 f fd ff f f f f f f d= + − → =

+ −2

2

ffd

PPdh ==

2 1 2 1

2 1 2

f d f f f dr f h ff f f d

⎛ ⎞− −= − = =⎜ ⎟ + −⎝ ⎠

< check! >

System Matrix with A=0, C=0System Matrix with A=0, C=0

0

0

0

0 0

0f

f

f

f

y yBC D

y B

Cy D

α α

α

α α

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

=

= +

When A = 0, the output plane for the optical system is the output focal plane.

When C = 0, collimated light at the input plane is collimated light at theexit plane but the angle with the optical axis is different. This is a telescopic arrangement, with a magnification of D = αf/α0.

0

0

0 0

0

0f

f

f

f

y yA BD

y Ay B

D

α α

α

α α

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

= +

=

0

0

0

0 0

0

0f

f

f

f

f

y yAC D

y Ay

Cy D

ym A

y

α α

α α

⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

=

= +

= =

When B = 0, the input and output planes are object and image planes, respectively, and the transverse magnification of the system m = A.

System Matrix with B=0System Matrix with B=0

Ex) Two-Lens System with B=0Ex) Two-Lens System with B=0

f1 = +50 mm f2 = +30 mm

q = 100 mmr s

ObjectPlane

ImagePlane

F1 F2F1 F2

T1 R1 R2 T3T2

( )( )

( )( )

1

1 2 2 1 1

2 2 1 1

1 2 2 1 2 2 1 2

1 1 2 2 1 2 1 1 2

1 2 1

1 01

1 1

q rr qq r r q q r r fB r q s s r q q r rf f f f f

f f f ff f r q f qr r f f f q f f q

f r q q r f f f r r f q f f q f f

q s s qm Af f f

+ −⎛ ⎞+

= + − − − − + = ⇒ =⎜ ⎟ +⎝ ⎠ − − +

+ − − += =

+ − + − − + + −

⎛ ⎞+= = − − −⎜ ⎟

⎝ ⎠

Location of Cardinal Points (Planes)for an Optical System

Location of Cardinal Points (Planes)for an Optical System

Distances measured to the right of the respective reference plane are positive, distances measured to the left are negative. As shown:

p < 0 q > 0 f1 < 0 f2 > 0r > 0 s < 0v > 0 w < 0

Ex) Thick Lens AnalysisEx) Thick Lens Analysis

R1 = +30 mm R2 = +45 mm

Input plane(RP1 )

V1 V2

t = 50 mm

nL = 1.8n0 = 1.0 n0 = 1.0

Find for the lens:

(a) Principal Points(b) Focal Points(c) Focal Length(d) Nodal Points

output plane(RP2 )

( )

( )

( )

( )

1

2 1 1 2

, :

1

1 1

50* 0.8 50*1.011.8*30 1.8

50* 0.80.8 0.8 0.8*501 145 1.8*30 30 1.8*45

L

L L

LL L L

L L

Thick lens matrix assuming n n

t n n t nn R nA B

MC D t n nn n n n n n t

n R n R n R n R

′=

⎡ − ⎤+⎢ ⎥

⎡ ⎤ ⎢ ⎥= = =⎢ ⎥ ⎢ ⎥−⎡ ⎤− − −⎣ ⎦ ⎢ ⎥+ + +⎢ ⎥⎢ ⎥⎣ ⎦⎣ ⎦−⎡ ⎤

+⎢ ⎥⎢ ⎥=⎢ ⎥−⎡ ⎤ −

+ + +⎢ ⎥⎢ ⎥⎢ ⎣ ⎦⎣ ⎦

: det 1Check M AD BC

=

⎥

= − =

Thick Lens AnalysisThick Lens Analysis

10.2593 27.77 0.02206 1.494A B mm C mm D−= = = − =

Thick Lens AnalysisThick Lens Analysis

1

2

1.494 67.720.02206

0.2593 11.750.02206

1.494 1 22.390.02206

1 0.2593 33.580.02206

1.494 1v 22.390.02206

1 0.2593 33.580.02206

45.33

45.33

p mm

q mm

r mm

s mm

mm

w mm

f p r mm

f q s mm

= = −−

= − = +−

−= = −−−

= = −−

−= = −−−

= = −−

= − = −

= − = +

Thick Lens AnalysisThick Lens Analysis

RP1 RP2

t = 50 mm

PP1

F1

F2

PP2

H1H2

si = +86.7 mmso = -95 mm

In general, for any optical system:

1 2

0 0

1 20 0

1

1 1 1:

i

i

i

i

n sf f ms s n s

sfor n n f f f ms s f s

+ = =′

′= − = = − + = = −

y 축

x 축

z 축A

2)

1 2

A 2A 1 c1c2

(+)r1(-)r2

3)

k+1

k-1 k1 2 3 4

o o'

d1 d2

y1

z1z k-1

yk-1

y2

z 2

물체면 상면

Ray tracingRay tracing부호에 관한 규약

1) 광선은 최초에 좌 → 우로 진행좌 → 우 : 순방향 우 → 좌 : 역방향

2) 우수 직각 좌표계를 사용. 좌표계의 원점은 면의 정점에 둔다. 광축은 z-축

3) 곡률 반경의 부호는 곡률 중심의 위치에 따른다.

4) 여러개의 면이 있으면 각 면의 정점을 원점으로 하는 좌표계를 사용.

※ 면 번호(Surface Number) : 광선이 만나는 순서대로 붙인 면의 번호

Ray tracingRay tracing부호에 관한 규약

5) 굴절률의 부호 ---- 순방향 : + 역방향 : -

n=1

1

2

n =11

n '=12

순방향일때 굴절률도 (+)역방향일때 굴절률도 (-)

n'1=n

2=-1

Paraxial ray tracingParaxial ray tracing

C o'o

u u'

n n'i

i'

h r

A

α

sin 'sin ' ' ' ( , ' 1)' - ' , -' '- ' -' ' ( ' )

sin ~ /'' ' , : power of refraction

n i n i ni n i i ii u i un u n nu nn u nu n nh r r h r

n nn u nu hK kr

α αα α

αα α α

= ⇒ = <<= =

== + −

= ⇒ =−

∴ = + =

Q

근축 광선 추적(Paraxial Ray Tracing) = Gaussian ray tracing (GRT)- 근축 광학적 근사를 통하여 계산된 광선의 진행경로.

굴절 방정식(refraction equation)

l l’

결상 방정식(imaging equation)

' ' '' , , or, ' ' '

h h n n n n n nu u K Kl l l l l l r

−= = ⇒ = + − = =

면 불변량 (surface invariant) = (Abbe’s Zero invariant)

From Snell's law, ' ' '

1 1 1 1' , , = '' '

n u n nu n

h h hu u Q n nl l r l r l r

α α

α

− = −

⎛ ⎞ ⎛ ⎞= = ⇒ = − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Paraxial invarianceParaxial invariance

sin 'sin 'n i n i=

면 불변량 (surface invariant) = (Abbe’s Zero invariant)

불변량 (invariant) : 굴절 전후에 변화하지 않는 물리량

굴절 불변량 (snell’s law) :

From Snell's law, ' ' '

1 1 1 1' , , = '' '

n u n nu n

h h hu u Q n nl l r l r l r

α α

α

− = −

⎛ ⎞ ⎛ ⎞= = ⇒ = − = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

' 'ni n i=근축 굴절불변량 (snell’s law) :

Lagrange 불변량

'' ' '

, ' , ''

H= ' ' '

n n n nl l

h h h hu u l ll l u u

nu n u

η ηβ β

η η

= → − = −

= = → = =

∴ =

n n'

u'

h

uo o'

l l'

o'1

o1

A

ββ'

η

η'