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• Chapter 4_L6

 Ray optics (Geometrical optics)

 Wave reflection and transmission

 Fabry-Perot interferometer

89

• Ray optics 90

 Ray optics: light is described by rays that travel in different optical media in accordance

with a set of geometrical rules. Ray optics is concerned with the location and direction of

light rays. The wavelength of light is assumed to be zero (much smaller than the objects).

 Paraxial ray approximation: The ray travels close to the optical axis at small inclinations.

 Matrix optics: an efficient tool to study the ray’s position and inclination.

 The ray at the interface is described by

the position r and the direction as 

( , )r 1 2

1r 2 r

z

r

Optical system

Optical axis

In p

u t

p la

n e

O u

tp u

t p

la n

e 1 1( , )r  2 2

( , )r 

 If the ray is rotated clockwise to

the optical axis, or else the angle is

negative

0 

1 20; 0  sin tan     The optical axis is usually the normal of a surface, if the incident ray is on the optical axis, the output ray will be on it as well .

• Marix optics 91

 In the paraxial approximation, the relation is generally

2 1 1

2 1 1

r Ar B

Cr D

 

 

 

2 1

2 1

is the ray transfer matrix

r rA B

C D

A B M

C D

 

              

      

 Planar interface

n1 n2 1 1 2 2sin sinn n 

1 2

1 0

0 / M

n n

      

 Free space propagation

n1 n2 2 1 1 1 2 /r r Ln n 

1 21 /

0 1

n L n M

      

n1

L

Snell’s law

• Matrix optics 92

 Thin lens

1/ 1/ 1/p q f 

1 0

1/ 1 M

f

    

  p q

 Spherical interface

n1 n2

2 1 1

2 2

1 0

1M n n n

n R n

         

R

• Matrix optics 93

 For reflection on mirrors, angle is positive for incident ray clockwise,

while for reflected ray anticlockwise

 Planar mirror

2 1 

1 0

0 1 M

      

/ 2f R

1 0

2 / 1 M

R

    

 

R

 Spherical mirror

 R is positive for concave mirror, while negative for

convex mirror

• Matrix optics 94

 For successive optical elements in a system

2 1

1

2 1

3 2

2

3 2

3 1

2 1

3 1

r r M

r r M

r r M M

 

 

 

       

   

        

  

       

  

M1 M2 MN

2 1...NM M M M

• Propagation of a spherical wave 95

 The transfer matrix is useful to calculate the propagation (radius) of a spherical wave

2 1

2 1 2 1

1

0 1

( )

z z M

R R z z

      

  

2 1

1 0

1 1

1 1 1

M

f

R R f

        

 

1 1

1

2 2

2

r R

r R

1 2

1

AR B R

CR D

 

• Chapter 4_L6

 Ray optics (Geometrical optics)

 Wave reflection and transmission

 Fabry-Perot interferometer

96

• Wave reflection and transmission 97

 The Electric field E of the optical wave, can be divided into a p-polarized wave Ep (in the plane of incidence) and a s-polarized wave Es (orthogonal to the plane of incidence).  Plane of incidence is the plane that formed by the incident wave & the normal of the dielectric interface.

s wave p wave

• Wave reflection and transmission 98

 Reflection and transmission of electric field (not power)

 Fresnel equations

2 1 1 2 1 1

2 1 1 2 2 1 1 2

1 1 2 2 1 2

1 1 2 2 1 1 2 2

cos cos 2 cos

cos cos cos cos

cos cos 2 cos

cos cos cos cos

p p

s s

n n n r t

n n n n

n n n r t

n n n n

  

   

  

   

  

 

  

 

 For the electric field (p wave):

rp is the reflectivity,

tp is the transmissivity

For the power/intensity

Rp=rp 2, Tp=tp

2, Rp+Tp=1

1

2

 When 1 0 

2 1 1

2 1 2 1

2 p p

n n n r t

n n n n

  

 

If rp R=0.31

• 99 Wave reflection and transmission

• Special angles 100

 When the reflectivity Rp is zero, the

incident angle is named Brewster angle.

1 2

1 1 2 2

2 1 1 2

/ 2

sin sin

cos = cos

0

B B

B B

p

n n

n n

r

  

 

 

 

 

2 1

1

tan B n

n  

 When n1>n2, it is possible to have total

reflection, that is, the transmission is zero.

1 2 1

2

2 1

1

sin sin 1

The critical angle: sin =c

n

n

n

n

 

  

1

2

Brewster polarizer

Optical fiber

• Chapter 4_L6

 Ray optics (Geometrical optics)

 Wave reflection and transmission

 Fabry-Perot interferometer

101

• Fabry-Perot interferometer/etalon 102

 A F-P eltalon is usually made of two parallel highly

reflecting mirrors. The power reflectivity of each mirror is R,

and the transmissivity is T. (Note: R+T=1)

At point b, the transmitted electric field is

Ein

0 exp cos

in

l E E T jk

    

 

At point c, the transmitted electric field is

'

1

3 exp

cos in

l E E RT jk

    

 

At the wave front, the phase difference between wave E1 and wave E0 is

0 0

0 0

0 0

2

cos

2 tan sin

sin sin

2 cos

l k k l

l l

n n

kl

 

 

 

 

  

  

 

 

 

1 0

0

0 1

exp

exp

...

1 exp

cos 1 exp

m

m

out m

in

E E R j

E E R jm

E E E E

l E T jk

R j

 

 

 

  

    

  

nwave number k=2 / 

• Fabry-Perot interferometer 103

 Then, the total power transmission TFP is

 

 

2

2

2

2

/

1 exp

1 2 cos

FP out in

FP

T E E

T

R j

T T

R R

  

    

2 coskl  

 The maxima occur at , therefore,

the corresponding frequencies are

2m  

2 cos m

c v m

nl  

 The free spectral range (FSR) is

2 cos FSR

c v

nl   

The maximum transmission is:

,max 1FPT 

 The minimum transmission is 2

,min 2(1 ) FP

T T

R 

• Fabry-Perot interferometer 104

 

2

21 2 cos FP

T T

R R 

  

 The width of the transmission peak

at half maximum:

  2 2

2 2 2

1 2 cos

2

1 2 1

2 2

R T

R

R T

R

T

R

   

     

   

2 coskl  

2 cos FWHM

c T v

nl R   

 The finesse describes how sharp the transmission peak with respecti

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