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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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