laser and non -linear optics by imran aziz by azizimran33

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									Introduction to Nonlinear Optics


          MOHAMMAD IMRAN AZIZ
             Assistant Professor
           PHYSICS DEPARTMENT
   SHIBLI NATIONAL COLLEGE, AZAMGARH
                   (India).


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    How to make a laser in three
           easy steps …

   • Pick a medium that has the potential for optical gain – i.e., an
   amplifying medium.
   • Select a means of putting energy into that medium – i.e., an
   excitation system.
   • Construct an optical feedback system for stimulating further
   emission, i.e., an optical resonator.




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            Introduction

Question:
 Is it possible to change the
 color of a monochromatic
 light?



                                          NLO sample
                          input                        output


Answer:
 Not without a laser light

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    Stimulated emission, The MASER
             and The LASER
   (1916) The concept of stimulated emission Albert
    Einstein
   (1928) Observation of negative absorption or stimulated
    emission near to resonant wavelengths, Rudolf
    Walther Ladenburg
   (1930) There is no need for a physical system to always
    be in thermal equilibrium, Artur L. Schawlow




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                  E2                                   E2
h                                                h

                  E1                         E1
     Absorption                   Spontaneous
                                   Emission

                                    E2
        h                    h             h

                                    E1
                  Stimulated
                  Emission
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Light (Microwave) Amplification
               by
           Stimulated
     Emission of Radiation




       LASER
      (MASER)
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             The Maser

Two groups were working on Maser in 50s

 Alexander M. Prokhorov and Nikolai
  G. Bassov (Lebedev institute of
  Moscow)
 Charles H. Townes, James P. Gordon
  and Herbert J. Zeiger (Colombia
  University)
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Left to right: Prokhorov, Townes and Basov at the Lebede
 institute (1964 Nobel prize in Physics for developing the
 “Maser-Laser principle”)
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Townes (left) and
Gordon (right) and
the ammonia maser
they had built at
Colombia University




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                         The LASER
   (1951) V. A. Fabrikant “A method for the application of
    electromagnetic radiation (ultraviolet, visible, infrared, and radio
    waves)” patented in Soviet Union.
   (1958) Townes and Arthur L. Schawlow, “Infrared and Optical
    Masers,” Physical Review
   (1958) Gordon Gould definition of “Laser” as “Light Amplification
    by Stimulated Emission of Radiation”
   (1960) Schawlow and Townes
    U. S. Patent No. 2,929,922
   (1960) Theodore Maiman Invention of the first Ruby Laser
   (1960) Ali Javan The first He-Ne Laser


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Maiman
and the
first ruby
laser




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Ali Javan and
the first He-Ne
Laser




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    Properties of Laser Beam

A laser beam
 Is intense

 Is Coherent

 Has a very low divergence

 Can be compressed in time up to few
  femto second


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         Applications of Laser

   (1960s) “A solution looking for a
    problem”

   (Present time) Medicine, Research,
    Supermarkets, Entertainment,
    Industry, Military, Communication,
    Art, Information technology, …
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      Start of Nonlinear Optics
Nonlinear optics started
  by the discovery of
  Second Harmonic
  generation shortly
  after demonstration
  of the first laser.
(Peter Franken et al
  1961)


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2. The Essence of Nonlinear Optics

When the intensity
 of the incident light



                               Output
 to a material
 system increases
 the response of
 medium is no
 longer linear
                                             Input intensity


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 Response of an optical Medium

The response of an
  optical medium to               h           
  the incident
                                               h
  electro magnetic          h                             
  field is the                                    



  induced dipole                          h
                                                       
  moments inside
  the medium

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    Nonlinear Susceptibility
Dipole moment per unit volume or polarization

                  Pi  Pi   ij E j
                             0



The general form of polarization

 Pi  Pi  χ E j  χ
       0    (1)
            ij
                          (2)
                          ijk   E j Ek  χ E j Ek El  
                                            (3)
                                            ijkl




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          Nonlinear Polarization
   Permanent
    Polarization
    First order
                                      P   Ej
                                        1    (1)
    polarization:                       i     ij
   Second order
    Polarization                      Pi   E j Ek
                                         2     ( 2)
                                               ijk
   Third Order
    Polarization                      Pi   E j Ek El
                                         3    ( 3)
                                              ijkl

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   How does optical nonlinearity
             appear
The strength of the
electric field of the light
                                                    e
wave should be in the
range of atomic fields                    h            a0

                                                         N
    Eat  e / a   2
                  0


   a0   / me
            2         2


                      7
   Eat  2 10 esu
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    Nonlinear Optical Interactions
   The E-field of a laser beam
                         ~
                         E (t )  Eeit  C.C.
   2nd order nonlinear polarization
          ~ ( 2)
          P (t )  2  ( 2) EE*  (  ( 2) E 2e 2it  C.C.)


                                           2
                            ( 2)

                                         
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          2nd Order Nonlinearities
   The incident optical field
        ~             i1t        i 2t
        E (t )  E1e         E2e          C.C.
   Nonlinear polarization contains the following terms
        P(21 )   E ( 2)
                             1
                              2
                                                  (SHG)
        P(2 2 )   ( 2 ) E2
                            2
                                                  (SHG)
        P(1   2 )  2  E1 E2  ( 2)
                                                  (SFG)
        P(1   2 )  2  ( 2 ) E1 E2
                                     *
                                                  (DFG)
        P(0)  2  ( 2 ) ( E1 E1*  E2 E2 ) (OR)
                                        *

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 Sum Frequency Generation
         2                    2
                     ( 2)                   3  1   2
       1                     1

Application:                                  2
Tunable radiation in the                              3
UV Spectral region.                           1

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        Difference Frequency
              Generation
         2          2
                          ( 2)            3  1   2
         1                       1
Application:
The low frequency
photon, 2 amplifies in
                                                    2
                                           1
the presence of high
frequency beam  . This
                                                    3
                   1
is known as parametric
amplification.   aziz_muhd33@yahoo.co.in
           Phase Matching


                            ( 2)

                                                 2
    •Since the optical (NLO) media are dispersive,
     The fundamental and the harmonic signals have
     different propagation speeds inside the media.

    •The harmonic signals generated at different points
     interfere destructively with each other.
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                SHG Experiments




   We can use a
    resonator to increase
    the efficiency of SHG.

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        Third Order Nonlinearities
   When the general form of the incident electric field is in
    the following form,
          ~            i1t       i 2t       i3t
          E (t )  E1e        E2e         E3e
    The third order polarization will have 22 components
    which their frequency dependent are

         i ,3 i , (i   j   k ), (i   j   k )
        (2 i   j ), (2 i   j ), i, j, k  1,2,3
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           The Intensity Dependent
               Refractive Index
    The incident optical field
                  ~                it
                  E (t )  E ( )e  C.C.

    Third order nonlinear polarization


P ( )  3 (       ) | E ( ) | E ( )
    ( 3)          ( 3)                             2



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    The total polarization can be written as

P   TOT
          ( )   E ( )  3 (       ) | E ( ) | E ( )
                  (1)            ( 3)                          2



    One can define an effective susceptibility

                 eff    4 | E ( ) | 
                          (1)                       2   ( 3)



    The refractive index can be defined as usual


                        n  1  4eff
                         2


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By definition


                n  n0  n2 I
where
                   n0c
                I     | E ( ) | 2

                   2

                    12 2 ( 3)
                n2  2 
                     n0 c
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    Typical values of nonlinear refractive index

Mechanism                 n2 (cm2/W)    1111
                                         ( 3)
                                                (esu)   Response time (sec)
Electronic Polarization      10-16          10-14              10-15
Molecular Orientation        10-14          10-12              10-12
Electrostriction             10-14          10-12               10-9
Saturated Atomic
                             10-10           10-8               10-8
Absorption
Thermal effects              10-6            10-4               10-3
Photorefractive Effect       large          large       Intensity dependent



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Third order nonlinear susceptibility of some material

                                                  Response
      Material                  1111
                                                  time

      Air                          1.2×10-17

      CO2                          1.9×10-12        2 Ps

      GaAs (bulk room
                                    6.5×10-4        20 ns
      temperature)
      CdSxSe1-x doped
                                      10-8          30 ps
      glass

      GaAs/GaAlAs (MQW)               0.04          20 ns

      Optical glass             (1-100)×10-14      Very fast

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     Processes due to intensity
     dependent refractive index
1. Self focusing and self defocusing
2. Wave mixing
3. Degenerate four wave mixing
   and optical phase conjugation




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    Self focusing and self defocusing
   The laser beam has Gaussian intensity
    profile. It can induce a Gaussian refractive
    index profile inside the NLO sample.




                            ( 3)

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Wave mixing




                                     
                           
                                2n0Sin( /2)

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       Optical Phase Conjugation
   Phase conjugation mirror


                                           PCM
                  M




                                 s
                                           PCM

                   M


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Aberration correction by PCM


            Aberrating
             medium
                                   PCM




 s           Aberrating
              medium
                                   PCM



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What is the phase conjugation
The signal wave

~                 it              Es  ε s As e
                                         ˆ          iks .r
Es (r , t )  Es e  C.C.


The phase conjugated wave

           ~               * it
           Ec (r , t )  rEs e  C.C.

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Degenerate Four Wave Mixing
         A1                                         A2

                             ( 3)

                                                    A3

                                               A4
 •All of the three incoming beams A1, A2 and A3 should be originated
  from a coherent source.
 •The fourth beam A4, will have the same Phase, Polarization, and
   Path as A3.
 •It is possible that the intensity of A4 be more than that of A3
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              Mathematical Basis
    The four interacting waves

                     ~                    i ( ki .r t )
                     Ei (r.t )  Ai (r )e                  C.C.

The nonlinear polarization

                                                        * i (( k1  k 2  k3 ).r t )
P   NL
          6  E1E2 E  6  A1 A2 A e
              ( 3)            *
                              3
                                          ( 3)
                                                        3



               The same form as the phase
                     conjugate of A3

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Origin of Nonlinearities in Optics

   The fast response of media to an
    electromagnetic wave in visible and near
    IR is caused by a displacement of
    electrons, both free ones in metals and
    bound ones in dielectrics.




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         Origin of Nonlinearities in
                     Optics
  The fast response of media to an
  electromagnetic wave in visible and
  near IR is caused by a displacement
  of electrons, both free ones in metals
  and bound ones in dielectrics.


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                        1. Free electrons
      The motion of electron in the field of a light wave:
                            
                                          
                    E(t )  E0 exp i(t  kr )        
                                                     
                                                   (1
                    H (t )  H0 exp i(t  kr )
is described by an equation:
                               
                                              
                                          e  1   
                                                      
                          2
                        d r
                                     E  V  H                   (2)
                         dt    2
                                         m        c          
           
                                                  
Becaus V  E , the vector product  H is proportional E 2 .
  e
                                                  V
                                                              to
The solution of (2) can be found in a
  form:                                               
                   r   E   EE   EEE  ...
                             (1)          ( 2)       ( 3)
                                                                     (3)
wher  (1) is         ( 2 ) ,  ( 3) ... are nonlinear polarisabilities.
                                                                 
  e           linear                                               
The induced electrical dipole moment isd  er (4)
              ,
  equal to
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                         2. Bound electrons
For the case of bound electron the equation has the following
                            2
                                   form: e 
                                    
                       
                 2 r   r  F   E (t )
               r                                  (5)
                                     NL
                                         m
                  
where       the FNL takes into account real anharmonisity of
  term             the 
oscillator:FNL  ar r  br r r  ...
               
Considerin FNL as a small term the solution of (5) can be
presented as:
                                          
                      r   E   EE   EEE  ...
                           (1)   ( 2)    ( 3)
                                                     (3)




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               3. Macroscopic characteristics
To describe the media response for the electromagnetic field
                                                
one must calculate a polarization vector       P          , which is a dipole
moment of a unit volume.
                                         
                            P  Nd  Ner                               (6
Where N is the concentration of electrons.                              )
                                                                       
If a nonlinear dependenced of E on                           takes place the
                                                                        d
 
P vectors and can be presented in the form:
                                                    
         d  d ( E)  d L  d NL   E   EE   EEE  ...
                                     (1)      ( 2)    ( 3)              (7
                                                              )
         P  P( E)  PL  PNL   E   EE   EEE  ...
                                     (1)      ( 2)      ( 3)           (8)
where (1) ,  (1) are tensors of 2 rank,  ( 2 )
                                              (2) ,           are tensors of
    3 rank
 ( n ) so 
and  ( non.2)
                   are nonlinear susceptibilities
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                         4. Local field factor
In a microscopic model of nonlinearity (we presented two
                                                        (n)
such models) it is important to describe correctly microscopic
            (n)
and macroscopic       values. For crystals of cubic symmetry:
                                            n 2 ( )  2 
                        (1) ( )  N (1)                                   (9)
                                                   3     
where the term in brackets is so-called Lorentz factor (local
field factor). For nonlinear susceptibility in particular for
quadratic nonlinearity:
                                             n 2 (1  2 )  2   n 2 (1 )  2 
    ( 2 ) (1  2 )  N ( 2 ) (1  2 )                                    
                                                      3                 3       
    n 2 (2 )  2 
                                                                        (10)
          3       
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            5. How high is the nonlinearity
If the response of the media is caused by electrons in
         (n)
nonresonant case for          the following ratio is valid:
                         ( n 1)   1
                                                           (11)
                           (n)
                                    EA
where E A is an interatomic field. For hydrogen E A  10 V cm .
                                                             9


One can see from this that appreciable nonlinear effects can
be observed at relatively high light intensities, which are the
features of pulse lasers. The nonlinear optics experiments
became real after innovation of Q-switched laser with pulse
duration of 10-8 s and intensities of 1010-1011 W/cm2. Now
femtosecond lasers became available, which generate pulses
with duration of 6-30·10-15 s at the intensity up to 1017-1020
W/cm2. In this case the electric field in the light wave exceeds
the value of EA. It opens completely new branch of optics:
physics of superstrong fields.
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Besides the above electronic nature of nonlinear response a
strong nonlinearity can be caused by an anharmonisity of
atomic oscillation in molecules, orientation of polar molecules
in an electric field, heating of medium. The slower is a
mechanism responsible for nonlinearity the stronger is the
nonlinearity.
Let us present the values of characteristic time constants and
the values of  ( n ) for different mechanism of nonlinear
polarization.
                     nonresonant        resonant     orientation in
     Mechanism
                      electronic       electronic   liquid crystals
  Time constant, s       10-14          10-7-10-8        1-10-1
       (2), esu        10-9            10-6-10-8

       (3), esu     10-14-10-15           10-10      10-1-10-2

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              III. Optical Harmonic
                    Generation

  The high intensity light wave induces
  the nonlinear polarization in a
  medium. The wave of polarization is
  a source for new electromagnetic
  waves.

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              1. Second-harmonic generation
First of all we should notice that the tensor  ( 2 ) , for
centrosymmetric media is equal to zero.
                           ( 2)         
                         PNL   EE ( 2)                        (12)
The operation of symmetry transforms the terms from (12) in
the following way:
                             ( 2)   ( 2)
                                   
                            E  E                               (13)
                                   
                            P  P
         ( 2)                           ( 2)
 Then PNL   EE   (E)(E)  PNL
               ( 2)     ( 2)
                                                        , that can not take
 place under nonzero  ( 2 ) .
 The same is valid for all even order        ( n ) , n  2m .

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For a simplicity we assume that the medium is isotropic.
Then the polarization:
 P  P (1)  P ( 2 )  P (3)  ...   (1) E   ( 2) E 2   (3) E 3  ... (14)
The incident waves propagating in z-direction can be
presented as:             E1  E10 cos(1t  k1 z )
                          E2  E20 cos(2t  k 2 z )                        (15)
 PNL )   ( 2) [ E10 cos(1t  k1 z )  E20 cos(2t  k 2 z )]2 
  (2


   ( 2) [ E10 cos2 (1t  k1 z )  E20 cos2 (2t  k 2 z ) 
              2                       2


  2 E10 E20 cos(1t  k1 z ) cos(2t  k 2 z )] 
   ( 2) {0.5E10 [1  cos 2(1t  k1 z )]  0.5E20 [1  cos 2(2t  k 2 z )] 
                2                                2


               
  E10 E20 cos[( 1  2 )t  (k1  k 2 ) z ] 
               
  E10 E20 cos[( 1  2 )t  (k1  k 2 ) z ]}                             (16)
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   A spectrum of polarization waves contains new
   frequencies: 2 , 2 ,    ,    ,   0 .
                  1    2   2   1   2   1

        E1 , E2




               0        ω1      ω2                 ω
               P




                0
                      ω2-ω 1   2ω1   ω2+ω1   2ω2   ω
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                2. Third-harmonic generation
If the medium possesses cubic nonlinearity, under the action
of two monochromatic waves1  2 and P ( 3) the polarization
would contain the components with frequencies:
31 , 32 , 21  2 , 22  1 .




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       IV. Wave Nonlinear Optics
As the optical harmonic generation takes
place both induced waves of polarization and
free running electromagnetic waves of
harmonics are propagating in the medium. If
the dimensions of the medium are much
larger than pumping wavelength the phase
matching determines the efficiency of the
energy transfer from the pumping wave to
harmonics. Let us consider the phase
matching conditions for the case of second
harmonic generation.
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                       1. Maxwell equations
The propagation of the light in the medium is described by
Maxwell equations:
                                  wher
          1 B                   e
                                          
 rot E                           B  H
           c t
                                              

rot H  4   1 D
                                   D  E  4P            (18)
              j           (17)              (1)      NL
           c      c t              E  4P  4P
      
div B  0                         For         optical
                                 range
div D  4
                                      1,   0, j  0    (19)
Combining first and second equations from (17) one may
obtain so-called wave equation:
                                       
                         1  E 4  D
                              2       2
                      E  2 2                             (20)
                          c t    c t  2


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Inserting (18) into (20) we are getting:
                                          
              1  E 4  P
                      2        2 (1)
                                       4  P
                                           2 NL
            E  2 2                                              (21)
                  c t     c t  2
                                        c t 2
The nonlinear polarization term in the right hand side of (21)
plays a role of a source of electromagnetic waves
                            2. Phase mismatch
For quadratic media(  ( 2)  0)and relatively low nonlinearity the
plane wave solution of (21) for the intensity of the second
harmonic looks like:
                                              2    (n  n2 ) z
                                             sin
                          [  ( 2 ) I  ]2              c
                 I 2                                             (22)
                                                (n  n2 ) 
                                  2                             2
                              n2
                                              
                                                     c       
                                                              
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                                           2
  Phase                 k  2k  k2       (n  n2 )
  mismatch                                  c
                                              max
                                       I2/I2
                                   1




                                                            Δkz/2
                 -2π       -π          0         π      2π
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For the case of the exact phase matching the energy of the
pumping wave can be completely transferred into second
harmonic
              I
                                             I2




                                             I
              0
                             L                2L z
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                      3. Phase matching
How the condition  k  2k  k  0 or n  n  0 can be
                              2             2
realized? In an isotropic medium with normal dispersion 2 > n
                                                       n
and  k never equals to zero
                                        Directions of
                                       phase matching
But in birefringent uniaxial
crystal there are two beams
ordinary and extraordinary. For
                                                               o
so-called negative crystal no>ne.                            nω
If pumping wave is ordinary one
and     second    harmonic     is                                  e
extraordinary one the material                                 nω
dispersion ( n2 > n ) can be                                 n 2ω
                                                                   o

compensate for the difference in
refractive indices for o and e                           e
beams: n  n2
           o    e
                                                        n 2ω
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For the process of third-harmonic generation the condition of
phase matching looks the same: k  0
As it was mentioned already ( 2 ) and  ( 3) values for the fast
nonresonant electronic polarization do not much differ for
many materials and the only way to enhance the efficiency of
nonlinear energy transformation is to phase match the
interacting waves.




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       V. Other Nonlinear Effects
           1. Modulation of a refractive index
Cubic nonlinearity causes not only wave generation with new
frequency but also appearance of a wave of nonlinear
polarization with the frequency of pumping wave:
                         1  1  1  1                      (23)
            PNL (1 )   (1; 1 , 1 ,1 ) E (1 ) E (1 )
                         ( 3)                        2



As a result of such selfaction a nonlinear refractive index n2I
appears at the frequency :1
          n  n0  n2 I          n2   (3) (1 ; 1 , 1 ,1 ) (24)
For the fast nonresonant nonlinearity n2 is relatively small:
n2~10-13 cm2/kW.
For slower mechanisms of the nonlinearity n2 can be much
larger in particular for liquid crystals: n2~0.1 cm2/kW.
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                      2. Selffocusing
If the intensity of a laser beam is high enough instead of
diffraction an opposite effect of selffocusing takes place.
Phase velocity depends on the intensity through nonlinear
refractive index:
                         Vph=c/n0+n2I                 (25)
If n2 > 0 the phase velocity at the axis of the beam is lower
and nonlinear medium is working as a lens.




                                          Z




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                VI. Nonlinear Optical
                     Diagnostics
Nonlinear susceptibilities  ( 2 ) and  ( 3) are tensors and they
inherit the symmetry properties of the crystalline medium. It
means that nonlinear optical effects are structure sensitive. It
can be employed to study different structure transformations.
A lot of such experiments were done. I will mention just one
related with laser induced melting of semiconductors.
                       R    melting                 Ge, Si
                             point
                       RL
                                                    R(t)
                       RS
                                      laser pulse


                                                           t
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       1. Nonlinear optical diagnostics of phase
                      transitions
                                                                     Semiconductor            in liquid
                  Idea              of                               stateMetal in            liquid
                  experiment beam
                     Powerful laser
                                                                      R   state
                                                                              melting               2
                                                                                                    3
                                                                                                    AB
                                                                                                        6
                                                                                                        5

                      which melts the surface                                 point
                                                                      NL
                     (Ruby, Nd:YAG , Eximer)                         RS
                                                                                               nonlinear
    Pr
      ob                                                              NL
                                                                                               reflection
        in                                                            NL
             gH                                                n     RL
                                                                      L
                                                          c tio
                  e-                                 le                L
                    Ne                            ef                 RL
Pr                     be                      arr                                               linear
   ob                     am                ne
las ing N                             L   i                   ar )       L                     reflection
   er
      be a d: YA                                        nl ine (2ω   RS
          m                                         No ctio n
            (ω G                                       le
               )                                   re f                               laser pulse
                         Semiconductor
                                                                                                            t


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           VIII. Conclusions
1.Nonlinear optics is an attractive and fast
    developing part of modern optics.
2.Nonlinear effects are structure sensitive in
    their nature. It can be used for time-
    resolved        monitoring  of    structural
    transformation (up to femtosecond time
    resolution).
3.Artificial photonic media on the base of
    porous semiconductors open new exciting
    possibilities for the control of nonlinear
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