< Cardinal points (planes) >

Chapter 18.

Matrix Methods in paraxial optics Chapter 18.

Matrix Methods in paraxial optics

< Cardinal points (planes) >

Focal (F) points (planes)

• EFL: Effective Focal Length (or simply “focal length”)

• FFL: Front Focal Length

• BFL: Back Focal Length Principal (H) surface (planes) Nodal (N) points (planes)

< Matrix methods to find the cardinal points in paraxial optics >

Ray-transfer matrix

• Translation Matrix

• Reflection Matrix

• Refraction Matrix

• Lens Matrix

• Mirror Matrix

Complex optical systems Complex optical systems

Thick lenses, combinations of lenses etc..

Thick lenses, combinations of lenses etc..

tt

n

n

n n n n ’ ’

Consider case where

Consider case where tt is not negligible. is not negligible.

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

imaging relation

R P n R

s n n s

n

L

 



 



 







2 1

1 ) 1

' ' (

'

But where do we measure

But where do we measure s, s

s, s

’

; f, f

’

from? How do we determine

P

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

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

We need to define cardinal points (planes) :

 focal (F), principal (H), and nodal (N) points (planes)

Example: Focal Lengths & Principal Planes

generalized optical system (e.g. thick lens, multi-element system)

EFL: Effective Focal Length (or simply “focal length”) FFL: Front Focal Length

BFL: Back Focal Length

FP: Focal Point/Plane

PS: Principal Surface/Plane

The significance of principal planes

object

multi-element optical system

lateral hold, where f= (EFL)

Utility of principal planes Utility of principal planes

H

H

ƒ’ƒ’

FF₂₂ H

H

ƒƒ FF₁₁

s s’

n

n

n n n n ’ ’

hh

h’h’

Suppose

Suppose s, s s, s’ ’, f, f , f, f’ ’ all measured from H all measured from H

and H and H

… …

' 1 '

f f

s s

  

' '

f f

n   n 1 1 1

if '

' n n

s  s  f 

Cardinal points and planes:

Cardinal points and planes:

제 1 초점 (first focal point, object side focal point): F

 무한대에 상이 생기는 축 상 물체 점

 상 측에서 광 축과 평행하게 입사한 광선이 모이는 점, 또는, 모이는 것처럼 보이는 점.

u'_k=0

1 k

...

F₁

u'_k=0

1 k

...

F₁

Cardinal points and planes:

Cardinal points and planes:

. . .

F'

1 k (마지막 면)

u =0₁

F'

1 k

u =0

제 2 초점 (second focal point, image side focal point) : F

 무한대에 있는 축 상 물체 점의 상점

 광축과 평행하게 입사한 광선이 모이는 점(실상), 또는, 모이는 것처럼 보이는 점(허상)

F₂

F₂

1st Principal planes (PP

) and points(H

) 1st Principal planes (PP

) and points(H

)

n n

n n n’ n ’

H

H

ƒƒ FF₁₁

PPPP₁₁

제 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.

2nd principal planes (PP

) and points (H

) 2nd principal planes (PP

) and points (H

)

n n

n n n’ n ’

H

H

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.

Objective distance, image distance Objective distance, image distance

물체거리 (object distance)

: 제1 주요면에서 물체까지의 거리 s = HO

상거리 (image distance)

: 제2 주요면에서 상면까지의 거리 s’ = H’O’

l l'

1 k

H H'

h h

P P'

u₁ u'_k

o o'

s s’

n₁ n'₁

h₁ A₁= H₁= H'₁

u'₁ u₁

l₁

l'₁ 면의 물체거리

면의 상거리 o

1 2

* 면에서 물체까지의 거리 l₁, l₂ 등과는 다름.

l₁

l₂

Effective focal length & back focal length Effective focal length & back focal length

1 k...

u₁=0

h₁

h_k

F' u'_k

A_k H'

A₁ H

bfl f'_b

efl, f' f_b

f

F u₁

h₁

h_k

u'_k=0

유효 초점거리(effective focal length, efl): f'

- 제2 주요점에서 제2초점까지의 거리 : efl = f’ = H’F’

후 초점거리(back focal length, bfl): f‘

-광학계의 마지막 면의 정점에서 제2 초점까지의 거리 : bfl = f’_b = A_kF’

제2 주요면의 위치 = bfl - efl

efl, f’

bfl, f’_b

Front focal length = Working distance of a lens Front focal length = Working distance of a lens

물체측 초점거리(Object side focal length): f

- 제1 주요점에서 제1 초점까지의 거리 : f = HF

앞 초점거리(front focal length): f

- 제1면의 정점에서 제1초점까지의 거리 : f_b = A₁F

제1 주요면의 위치 : f

1 k...

u₁=0

h₁

h_k

F' u'_k

A_k H'

A₁ H

bfl f'_b

efl, f' f_b

f

F u₁

h₁

h_k

u'_k=0

efl, f’

bfl, f’_b

f f_b

Utility of principal planes Utility of principal planes

H

H

ƒ’ƒ’

FF₂₂ H

H

ƒƒ FF₁₁

s s’

n

n

n n n n ’ ’

hh

h’h’

Suppose

Suppose s, s s, s’ ’, f, f , f, f’ ’ all measured from H all measured from H

and H and H

… …

' 1 '

f f

s s

  

' '

f f

n   n 1 1 1

if '

' n n

s  s  f 

Nodal points (N

, N

) and Nodal planes (NP

, NP

) Nodal points (N

, N

) and Nodal planes (NP

, NP

)

n n n’ n ’

N N

NPNP₂₂

N N

NPNP₁₁

n n

절점(nodal point : 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.

Nodal point (N) and optical center (c) Nodal point (N) and optical center (c)

N' u N

u' c

1 2

광심(optical center : C) :

절점(nodal point)을 정의하는 하나의 광선 (입사각과 출사각이 같은 광선)이 실제로 광 축과 교차하는 점.

Nodal point 의 성질

i) 제1절점으로 입사한 광선은 제2절점에서 출사 (제1절점 - 광심 (C) - 제2절점) ii) 제1절점으로 입사한 광선은 입사각과 출사각이 같다.

iii) 상측 매질의 굴절률과 물체측 매질의 굴절률이 같으면 절점과 주요점은 같다.

N₁ = H₁ , N₂ = H₂

iv) 제2절점을 기준으로 광학계를 회전시키면 상의 위치는 변화하지 않는다.

F'

N₂ N₁ F'

C

Cardinal planes of simple systems Thin lens

Cardinal planes of simple systems Thin lens

s P n s

n   '

'

Principal planes, nodal planes, Principal planes, nodal planes, coincide at center

coincide at center VV

H, H H, H’’ V’V’

V’ V ’ and V coincide and and V coincide and

is obeyed.

is obeyed.

Cardinal planes of simple systems Spherical refracting surface

Cardinal planes of simple systems Spherical refracting surface

n n n n ’ ’

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

measured from V VV

s P n s

n   '

'

Combination of two systems:

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

e.g. two spherical interfaces, two thin lenses …

n n

n n

H

H

’ n n ’ ’

H

H

’

H

H

’

y y

Y Y

d d

ƒ ƒ

’ ’ ƒ ƒ

’ ’

F

F

’

F

’

1. Consider F

1. Consider F’’ and Fand F₁₁’’

h h ’ ’

Find

Find h h’ ’

Combination of two systems:

Combination of two systems:

n n

n n n’ n ’

H

H

’

H

H

H

H

’

H

y y Y Y

d d ƒ ƒ

1. Consider F and F 1. Consider F and F₂₂

F

F

ƒ ƒ

h h

F

F

Find

Find h h

Summary Summary

H

H

’

H

H

H

H

’

H

H

ƒ ƒ h h h h ’ ’ ƒ’ ƒ’

F

F

F

’

d d

n n

n n n n ’ ’

Summary Summary

2 2 1 2

1

2 1

1 2 2

2 1

2 1

2 2

1 2

,

' '

' '

' '

'

' ' '

' '

n P d P

P P

P or

f n f

f dn f

n f

n f

n

n n P

d P H

f H d f

h

n n P

d P H

f H d f

h

















 



 











 



 





Hecht, 6.1, p.214

Thick Lens Thick Lens

n n

R R

R R

H

H

,H

’

H

,H

’

In air n = n

In air n = n ’ ’ =1 =1 Lens, n

Lens, n

= 1.5 = 1.5

R R

= - = - R R

= 10 cm = 10 cm d = 3 cm

d = 3 cm Find

Find ƒ ƒ

, , ƒ ƒ

, , ƒ ƒ , h and h , h and h ’ ’ Construct the

Construct the

principal planes, H, principal planes, H, H H ’ ’ of the entire of the entire

system system

n n n n ’ ’

Principal planes for thick lens (n ₂ =1.5) in air Principal planes for thick lens (n ₂ =1.5) in air

Equi Equi - - convex or convex or equi equi - - concave and moderately thick concave and moderately thick

  P P

= P = P

≈ ≈ P/2 P/2

' d 3 h

h   

1 2

2 2

' f

f n

h d

f f n

h d











H H H’ H ’ H H H’ H ’

Principal planes for thick lens (n ₂ =1.5) in air Principal planes for thick lens (n ₂ =1.5) in air

Plano

Plano - - convex or convex or plano plano - - concave lens with R concave lens with R

= =  

  P P

= 0 = 0

d h

h

3 ' 2

0







1 2

2 2

' f

f n

h d

f f n

h d











H H H’ H ’ H H H H ’ ’

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

R R

= 3R = 3R

(H (H ’ ’ reaches the first surface) reaches the first surface)

P Same for all lenses Same for all lenses

1 2

2 2

' f

f n

h d

f f n

h d











H H H’ H ’ H H H H ’ ’ H H H’ H ’

H H H’ H ’

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

2 2

f d f P

d P

h  

ƒ ƒ

ƒ ƒ

d d

H

H

’

H

H

H

H

’

n = n

n = n

= n’ = n ’ = 1 = 1 Want to replace H

Want to replace H

, H , H

’ ’ with H, H’ with H, H ’

1

'

f d f P

d P

h    

h h h h ’ ’

H H H’ H ’

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

ƒ ƒ

ƒ ƒ

d d

n = n

n = n

= n’ = n ’ = 1 = 1

2 1 2

1

2 2 1 2

1

1 1

1 ,

f f

d f

f f

or

n P d P

P P

P













H H H’ H ’

F F F F ’ ’

ƒ ƒ ƒ’ ƒ’ s s f

1 '

1 1  

s’ s ’

s s

Two separated lenses in air Two separated lenses in air

If f

If f

’ ’ =2f =2f

’ ’

d = 0.5 d = 0.5 f f

’ ’

H H H’ H ’

F’ F ’ F F

f’ f ’

d = d = f f

’ ’

H H H H ’ ’

F’ F ’ F F

f’ f ’

Two separated lenses in air Two separated lenses in air

f f

’ ’ =2f =2f

’ ’

d = 2 d = 2 f f

’ ’

H H H’ H ’

F’ F ’ F F

f’ f ’

d = 3 d = 3f f

’ ’

Principal points at Principal points at

e.g. Astronomical telescope e.g. Astronomical telescope

Two separated lenses in air Two separated lenses in air

f f

’ ’ =2f =2f

’ ’

d = 5 d = 5 f f

’ ’

f’ f ’

e.g. Compound microscope e.g. Compound microscope

H H

F’ F ’ F F

H’ H ’

Two separated lenses in air Two separated lenses in air

f f

’ ’ = = - - 2f 2f

’ ’

d = d = - - f f

’ ’

e.g. Galilean telescope e.g. Galilean telescope

Principal points at

Principal points at  

Two separated lenses in air Two separated lenses in air

f f

’ ’ = = - - 2f 2f

’ ’

d = d = - - 1.5 1.5 f f

’ ’

e.g. Telephoto lens e.g. Telephoto lens

H H H H ’ ’

F’ F ’

f’ f ’

F F

우와아~ !!! 복잡하다.

우와아 우와아 ~ !!! ~ !!! 복잡하다 복잡하다 . .

HH₁₁’’

HH₁₁ HH₂₂ HH₂₂’’

HH H’H’

ƒ

ƒ ƒ’ƒ’

h

h hh’’

FF FF’’

d d nn₂₂

nn n’n’

렌즈 렌즈 2 2 개가 개가 있는 있는 경우도 경우도 힘들다 힘들다 . .

좋은 좋은 방법이 방법이 없을까 없을까 ? ?

H H₁₁’’ H

H₁₁ HH₂₂ HH₂₂’’ HH H’H’

ƒ

ƒ ƒ’ƒ’

hh h’h’

FF F’F’

dd n n₂₂ n

n n’n’

H H₁₁’’ H

H₁₁ HH₂₂ HH₂₂’’ H

H H’H’

ƒƒ hh hh’’ ƒ’ƒ’

FF FF’’

dd n n₂₂ n

n nn’’

3 3 개 개 이상 이상 있으면 있으면 ? ? 못하겠다 못하겠다 . .

포기 포기 ? ?

Matrix Methods in paraxial optics Matrix Methods in paraxial optics

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

• Matrices for analyzing the translation, refraction, and reflection of optical rays

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

Matrix Method Matrix Method

 

 



 



 



 

 

 





1 1 2

2





y D

C

B A

y

1 1

2

1 1

2







D Cy

B Ay

y









ABCD Matrix

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





  tan 

sin

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 refraction translation translation

Translation Matrix Translation Matrix

0 0

1 1 0

0 0

1

1 1

0 1 0 1

y y

y L x x

 



    

     

     

     

   

     

( y_o, _o)

( y₁, ₁ )

L

1 0

y

y

L tan

y

L

        

   

   

1 0 0

1 0 0

1

0 1

y y L

y



 

 

 

Refraction Matrix Refraction Matrix

' :

1 1

Paraxial Snell s Law n n

y n y n y y n n

R n R n R R R n y n

 

    

  

        

                      

 

 

1 0

: 0

1 1 : 0

y y

y y Concave surface R

n n

Convex surface R

R n n



 

  

 

 

        

      

           

y=y’

y

^   ^  ^ R

y y

R R

         

n n'

Reflection Matrix Reflection Matrix

   

 

:

2

1 0

2 1

1 0 2 1 Law of Reflection

y y

R R R y

y y

R y

y y

R

 

   



 

 

 

      

  

      

 

     

    

    

y=y’

y y y

R R R

^   ^  ^         

 

Thick Lens Matrix I Thick Lens Matrix I

0 0

1

1

0 0

1

1

1 0

: _L

L L

y y

Refraction at first surface y n n n M

n R n  



 

   

         

   

      

2 1 1

2

2 1 1

1 2 : 1

0 1

y t y y

Translation from st surface to nd surface M

  

      

 

      

 

     

3 2 2

3

3 2 2

2

1 0

: y _{L L} y y

Refraction at second surface n n n M

n R n

  

 

          

         

    

Thick Lens Matrix II Thick Lens Matrix II

 

 

 

1

2

1

1

2 1 1 2

: 1

1 0

1

1 1

L

L L

L L

L

L L

L

L L

L L L L

L L

Assuming n n

t n n t n

n R n

M n n n

n n n

n R n

n R n

t n n t n

n R n

t n n

n n n n n n

n R n R n R n R t

 

  

    

   

      

   

  

  

  

 

           

   

 

2 1

1 0 1 0

1

L L 0 1 L

L L

M n n n t n n n

n R n n R n

   

 

    

       

3 2 1

:

Thick lens matrix M  M M M

Thin Lens Matrix Thin Lens Matrix

2 1

1 2

:

1 0

1 1

1

1 1 1

1 0

1 1

L

L

Thin lens matrix

M n n

n R R

n n

but f n R R

M

f

 

 

              

 

    

 

 

 

   

 

 

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

n_L

Summary of Matrix Methods Summary of Matrix Methods

0

System Ray-Transfer Matrix System Ray-Transfer Matrix

Introduction to Matrix Methods in Optics, A.

Gerrard and J.

M. Burch

1 1

y



  

 

2 2

2 2

n n

y







 

 

 

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 matrix A B

M C D

 

  

 

A useful property of this matrix is that

Det

f

M AD BC n

   n

where n₀ and n_f are the refractive indices of the initial and final media of the optical system

Det

1

f

M AD BC n

   n 

Usually, the medium will be air on both sides of the optical system and

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

0 0 f

f

y A B y

C D

 

     

 

     

   

 

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



 

 

 

D=0 : input plane = first focal plane

A=0 : output plane = second focal plane

B=0 : input and output planes correspond to conjugate planes C=0 : telescopic system

D=0 A=0

B=0 C=0

System Matrix with D=0 System Matrix with D=0

Let’s see what happens when D = 0.

0 0

0 0

0

0

f f

f f

y A B y

C

y Ay B Cy

 





     

 

     

   

 

 



When D = 0, the input plane for the optical system is the first (object) focal plane.

Ex) Two-Lens System : find the 1

focal plane Ex) Two-Lens System : find the 1

focal plane

f₁ = +50 mm f₂ = +30 mm

q = 100 mm

r s

Input

Plane Output

Plane

F₁ F₁ F₂ F₂

T₁ R₁ T₂ R₂ T₃

1 1

2 1 1 2

1 1

1 0 1 1 0 1

1 1 1

1 1 1

1 1 1

0 1 0 1 0 1 1

1

q q r

r q

r f f

s q s

M r

f f f f r

f f

    

       

          

                 

 

0

3 2 2 1 1

0

2 1

1 0 1 0

1 1 1

1 1

1 1

0 1 0 1 0 1

f f

y y s q r

M M T R T R T

f f

 

   

            

          

     

3 2 2 1 1

1 1

2 1 1 2 1 1

1 2 1 1 2 2 1 1

2 1 1 2 1 1

1 1

0 1 1 1 1

1 1

1 1 1

1 1 1

1 1

M T R T R T

q q r

r q

f f

s

q q r r

r q

f f f f f f

q s s q q r r q q r r

r q s

f f f f f f f f

q q r r

r q

f f f f f f



    

 

   

              

     

 

               

    

   

           







     

2 1 1

2 1 1

1 2

1 1 0

30 50 100 50 100 50 30 175

q r r

D r q

f f f

f f q f

r q f f

r mm

 

       

 

 

   

 

 

 

ƒ

ƒ₁₁ ƒƒ₂₂

d d HH H’H’

FF F’F’

ƒ

ƒ ƒ’ƒ’

s’s’ ss

h h rr

1 2

1 2 1 2 1 2

1 1 1

f f

d f

f  f  f  f f   f f d

 

2 2

f d f P d P

h 

2 1 2 1

2 1 2

f d f f f d

r f h f

f f f d

   

       

< check! >

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

0 0 0

0 0

f

0

f f

f

y B y

C D

y B

Cy D

 



 

     

      

   

 



 

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 the exit plane but the angle with the optical axis is different. This is a

telescopic arrangement, with a magnification of D = _f/₀.

0 0

0 0

0

0

f f f

f

y A B y

D y Ay B

D

 



 

     

      

   

 

 



0 0

0

0 0

0 f

0

f

f f

f

y A y

C D

y Ay

Cy D

m y A

y

 

 

     

      

   

 



 

 

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=0 System Matrix with B=0

Conjugate planes