# Optical nonlinear processes

Where P is the vector representing the electric dipole moment per unit volume induced by the external electric field E, 0 is vacuum permittivity and i...

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Preparatory School to the Winter College on Optics and the Winter College on Optics: Advances in Nano-Optics and Plasmonics

6 - 17 February, 2012

Optical nonlinear processes

M. Bertolotti Dept. SBAI, University Roma "La Sapienza" Roma Italy

Winter College on Optics Advances in Nano-Optics and Plasmonics (6-17 February 2012)

Nonlinear Optics M.Bertolotti Universita’ di Roma, “La Sapienza”, Roma, Italy Dipartimento SBAI Via Antonio Scarpa 16 00161 Roma 1

OUTLINE Linear and Nonlinear Systems Linear susceptibility: dispersion Nonlinear Susceptibility Nonlinear Optical Interactions Second harmonic Other nonlinear processes 2

Linear and Nonlinear Systems A linear system is defined as one which has a response proportional to external influence and has a well-known property, i.e. if influences, F1, F2 …., Fn are applied simultaneously, the response produced is the sum of the responses that would be produced if the influences were applied separately. A nonlinear system is one in which the response is not strictly proportional to the influence and the transfer of energy from one influence to another can occur. 3

If the influences are periodic in time, the response of a nonlinear system can contain frequencies different from those present in the influences. However, the point to emphasize here is that, as well as the generation of new frequencies, nonlinear optics provides the ability to control light with light and so to transfer information directly from one beam to another without the need to resort to electronics. Traditionally, nonlinear optics has received a phenomenological approach in terms of the effect of an electric field on the polarization within a material. 4

MAXWELL EQUATIONS Electromagnetic processes are described by Maxwell’s equations which constitute a set of linear equations. In SI units: D  

div D  

B E   t B  0

B rot E   t div B  0

H 

D J t

rot H 

D J t

where E and B are the electric and magnetic fields. The displacement fields D and H arise from the external charge and conduction current densities  and J. In most cases of interest in nonlinear optics,  5 = 0 and J = 0.

‘Constitutive relations’ connect the charge and current distributions within the medium and the displacement fields to the electric and magnetic fields.

D  P   0 E  E

B  H

where P is the induced polarization in the medium resulting from the field E,  is defined as a dielectric constant and 0 is the permittivity of free space (8.85 × 10-12 F m-1 in MKS units). Optical materials are mostly non magnetic  = r 0 = 0. 6

LINEAR THEORY Usually one assumes a linear response of a dielectric material to an external field

P  0E Where P is the vector representing the electric dipole moment per unit volume induced by the external electric field E, 0 is vacuum permittivity and  is a quantity characteristics of the considered material with no dimensions, called electric susceptibility. 7

In general  is a tensor  xx  xy  xz Ex Px Py  0  yx  yy  yz E y Pz

zx

zy

zz Ez

The symmetry properties of the material indicate which ones of the ij coefficients are zero. Alternatively

Dx  xx  xy  xz Ex Dy   yx  yy  yz E y Dz

zx

zy

zz Ez 8

HOMOGENEOUS MATERIALS  not depending on space div E   rot E  

If

div E 

B t

 

rot E  

B t

div B  0

div B  0

 B   E  rot     j   t      

 E  rot B    j   t  

0

j0 div E  0

div B  0

B rot E   t

E rot B   t

9

WAVE EQUATION  div E  0

Starting from Maxwell’s Eqs

   B rot E    t     0 div B 0div B     D   rot B  0 t 

rot rot E  2 E  grad div E    rot B t

 D  2D  E  grad div E  0  0 2 t t t 2

 2 E 1  2E  E   0 2  2 2 t v t 2

  2 Ex  2 E x  2 Ex  i  E   2 2 2  y z   x 2

  Ey  Ey  Ey    j  2 2 2   y z   x 2

2

2

  2 Ez  2 Ez  2 Ez    k  2 2 2  y z   x

v

c c  r n

c

1 00

refractive index n  r 10

DISPERSION Experimentally the refractive index is a function of wavelength (frequency)

n( )  r ( )

 r (  )  1  (  )

This phenomenon is called DISPERSION. The polarization in a material medium can be explained considering the electrons tied to the atoms as harmonic oscillators. Nucleus: ~2000 electron mass, i.e., infinite mass 11

DISPERSION i t   mx   x  kx  eE 0e Damping Restoring Force

(one-dimensional model)

Driving Force

From the solution: eE 0ei t x m(02   2  i )

 

 m

02 

k m

the induced moment is calculated:

E

e2 it p  ex   E e 0 2 2 m 0    i

e-

x

12

For N oscillators per volume unit, the polarization is: N  e2 it PNp E e 0 m  02   2  i

Calling 

e2  atomic polarizability  2 2 m(0    i ) E  E 0ei t

P  N E   0  E



N

0

where  is the electric susceptibility.  N   0 r   0 (1   )   0  1    0   N n2  1    1  n  r

0

13

2 Ne n2  1  2 2 m(   0 0   i )

If the second term is lower than 1 (as it happens in gases): Ne2 n  1 2 0 m(02   2  i )

In the expression n comes out to be a complex number. 14

ABSORPTION The term i is responsible for absorption. The complex index can be written as 2 2 2 2 Ne (   ) Ne   0 n  n  ik  1  i    2 2 2 2 2 2 2 2     2 2  2 0 m 0   2 0 m 0      

If we consider a plane wave 1.0

E  A exp  i( t  kz) 

k    

 c

n

2

-1

( a. u. )

where

0.5

0

n

-0.5 -3.0

-1.5

0

1.5

3.0

0

15

we see that, substituting the complex refractive index, one has 2  k (n  ik)  which gives

 2 k  2 n     E  A exp i   t  z   exp   z       

The last exponential represents a term of attenuation. The attenuation coefficient may be defined from: 1 dI 2   I(z)  E  I (0)e  z I dz By comparison with the previous equation  2 k  2 n     E  A exp i   t  z   exp   z       

4   k

16

It can be noticed that a small value of k leads to an elevated attenuation. k  0.0001 and   0.5m gives   25 cm 1. 1.0

( a. u. )

k n - 1 0.5

0

-0.5 -3.0

-1.5

0

1.5

3.0

0

Both n and k are functions of the frequency.

17

Substance Air (1 atm)

n (for yellow light)

r

(static value)

1.0002926

1.0002925

CO (1 atm)

1.00045

1.0005

Polistyren

1.59

1.6

Glass

1.5  1.7

2  3

Fused quartz

1.46

1.94

Water

1.33

9

Ethanol

1.36

5

Table I Values of n and r for some materials

n as a function of  for some materials

18

normal dispersion

n  1 for   0

At resonance ( = 0) the slope of

n is negative anomalous dispersion

Absolute refraction index at 20°C for the line D of Sodium (=5890 Å) n

 n

Gas

k n - 1

n

1.528

Acetone

1.359

Carbon dioxide

1.000448

Calcite

1.658

Water

1.333

Air

1.000292

Dispersive Crown

1.520

Ethanol

1.361

Nitrogen

1.000296

Heavy Flint

1.650

Benzene

1.502

Helium

1.000036

Amorphous quartz

1.458

Etere etilico

2.352

Hydrogen

1.000132

Heavy glass

1.970

Solfuro di Carbonio

1.627

Oxygen

1.000271

( a. u. )

Solids

 Liquids

1.0

0.5

0

-0.5 -3.0

-1.5

0

1.5

3.0

0 19

METALS In a metal the electrons are free and they do not oscillate around the atoms. Therefore k = 0 and 0 = 0. In the equation for n2 it is sufficient to put 0 = 0. 2 Ne n2  1   0 m( 2  i )

N  density of electrons

If  <<  2

n 1

p 2 2

p 2

For Al, Cu, Au, Ag

Ne2  0 m

Frequency of plasma

N ~ 1023 cm-3 and P~ 2.1016 s-1. 20

For  > P

n is real and the waves propagate freely.

For  P n is pure imaginary and the field is exponentially attenuated with the distance from the surface. Therefore the radiation is reflected from the surface.  Therefore, for visible radiation and infrared  < P and n is imaginary. In general, n is complex because there is : 2 2 2   i    Ne Ne Ne   i 1  n2   2 2 2 2 0m    i      i   0m      0m     

p 2

 

2

i

 

p 2   2

  21

NONLINEAR SUSCEPTIBILITY Dipole moment per unit volume or polarization in the linear case

The general form of polarization in a nonlinear medium is

22

NONLINEAR SUSCEPTIBILITY In some cases also the magnetic field is important and quadrupole terms In these cases the general form of polarization in a nonlinear medium is

+ χijk(2) Ej Bk + χijkl(2)(·E)E + ..

23

JUSTIFICATION OF THE PRESENCE OF A NONLINEAR RESPONSE If the force exercised by the electric field of the wave becomes comparable with the Coulomb’s force between the electron and the nucleus, the oscillator is perturbed (anharmonic oscillator) and, at the lower level of the perturbation, we can write: x(t)  x(t)    20 x(t)  Dx 2 (t)  (e / m)E(t) E

(4)

x 24

The solution of eq.(4) can express as the sum of two terms (1) (2) x(t)  x (t)  x

(t)

(5)

in which x(1)(t) is obtained solving eq.(4) without the anharmonic term, whereas x(2)(t) is considered a small correction of the solution at the first order x(1)(t) and is obtained utilizing x(1)(t) in the anharmonic term  x

(2)

(t)  x

(2)

(t)  02 x (2) (t)

2 eE(t) (1)   D  x (t)  . m

(6)

In this way, considering the case in which the forcing electric field is formed by the sum of two fields at different frequencies 1 E(t)  E1 cos 1t  E 2 cos 2 t   E1e  j1t  E 2e  j2 t  c.c. 2

(7) 25

We have the solution at the first order 1 x (1) (t)   x (1) (1 )e  j1t  x (1) ( 2 )e  j2 t  c.c. 2

(8)

and subsequently the solution at the second order, solving eq.(6) with the use of (8) is x

(2)

1 (2) (t)  [x  1  2  e  j 1 2  t  x (2)  1  2  e j 1 2  t  2  x (2)  21  e j2 1t  x (2)  22  e  j2 2 t  c.c] (9)

in which x

(2)

x

(2)

1 D(e / m) 2 E1E 2   1  2    2 02  12  j1 02  22  j2  02   1  2 2  j  1  2    



1 D(e / m) 2  E 2k (2k )   ; k  1,2. 2 2  2  j 2 2  42  j 0 k k 0 k k



(10) 26

Therefore the solution of the second order brings to the generation of oscillations at a frequency different from the ones of the forcing field. In particular, it is possible to have frequencies equal to the sum or to the difference of the field frequencies or to the double (second harmonic). Moreover, we emphasize that the previous formulas remain valid also if just a single forcing field  is present. In this case x(2)(t) will be the sum of a second harmonic term (2) with a null pulsation term (term of optical rectification). 27

Now, remembering the expression polarization of the medium, we can write P(t)   Ne  x (1) (t)  x (2) (t) 

for

the

(11)

where N is the number of dipoles for volume unit; that is P(t)  PL (t)  PNL (t)

(12)

Which, compared with (10) x

(2)

x

(2)

1 D(e / m)2 E1E 2   1  2    2 02  12  j1 02  22  j2  02   1  2 2  j  1  2    



1 D(e / m) 2  E 2k (2k )   ; k  1,2. 2 2 2  2  j 02  42k  jk 0 k k

permits to write



(10)

PL  0(1) E PNL  (2) E  E.

(13)

28

SECOND HARMONIC PRODUCTION The nonlinear properties in the optical region have been demonstrated for the first time in 1961 by Franken et al. during an experiment of second harmonic generation. Sending red light of a ruby laser ( = 6.943 Å) onto a crystal of quartz, they observed ultraviolet light.

29

To describe the phenomenon, it is necessary to observe that in many crystal materials the nonlinear polarization depends on the direction of propagation, on the polarization of the electric field and on the orientation of the optical axis of the crystal. Since in such materials the vectors P and E are not necessarily parallel, the coefficient  is a tensor. The second order polarization can be written as Pi(2)   d ijk E jE k (14) j,k

where i, j, k represent the coordinates x, y, z. The main part of the coefficients dijk, however, are usually zero and so only a few of them must be considered. 30

Only the non-centrosymmetric crystals can have a non null tensor dijk. In facts, let us consider an isotropic crystal. In this case dijk is independent from the direction and therefore it is constant. If now we invert the direction of the electric field, also the polarization must change sign, that is  Pi(2)   d ijk (  E j )(  E k )   d ijk E jE k   Pi(2) .

It is clear that, not being able to be  Pi(2)   Pi(2) , dijk must be null. Moreover, in materials for which d ≠ 0, since no physical meaning can be assigned to an exchange of Ej with Ek, it must be dijk = dikj. 31

Now if we consider the Maxwell equations writing we have

D  0 E  P

(15)

D E P rot B  j    j  0  t t t B rot E   . t

(16)

The polarization can be written as the sum of a linear term plus a nonlinear one P  0  L E  PNL (17) where, in case of materials with second order nonlinearity is, f.e.

 PNL i   d ijk E jE k .

(18)

32

So eq.(16) can be written, assuming j = 0 PNL E rot B    t t

(19)

from which 2 2  E  PNL 2 .  E   2   2 t t

(20)

If we consider the unidimensional case of propagation along a direction z, we have  2 (PNL )i  2 Ei  2 Ei   2   . 2 2 z t t

(21) 33

Let us consider now three monochromatic fields with frequencies 1, 2, 3 using the complex notation 1  E1i (z)e j 1t  k1z   c.c.  2 1 j  t k ( 2 ) E k (z, t)  E 2k (z)e  2 2 z   c.c.  2 1 j 3 t  k 3z  ( 3 )  c.c. E j (z, t)  E3 j (z)e  2

E (i 1 ) (z, t)

(22)

where the indices i, j, k represent the components x or y. (18)  PNL i   d ijk E jE k .

The polarization at frequency 1 = 3 - 2, for example, from (18) and from (22) results ( 1 ) Pi

1 j  3 2  t   k 3  k 2 z     d ijk E3 j (z)E 2k (z)e  c.c. 2 j,k

(23) 34

Substituting eqs.(22) into (21) for the component E1i, it is necessary to calculate  2 E ( 1 ) 1  2  E (z)e  1t  k1z   c.c.. 2 2  1i 2 z z

If we assume

dE1i d 2 E1i k1  dz dz 2

(24)

(25)

we have  2 E(i 1 ) 1 2 dE1i (z)  j 1t  k1z     k1 E1i (z)  2 jk1 e  c.c. 2  2 dz  z

(26)

with similar expressions for  2 E(j2 ) z

2

and

 2 E(k3 ) . 2 z

35

Finally, substituting (26) and (23) into (21) we have dE1i (z)  0   j k 3  k 2  k1 z  j 1 d E E  c.c.  ijk 3 j 2k e dz 2 1

(27)

and in analogous way dE2k j2 0   j k1  k 3  k 2  z   c.c. d E E  ijk 1i 3 je dz 2 2 dE 3 j

3 0  j k1  k 2  k 3 z d E E e  j  c.c..  ijk 1i 2k dz 2 3

(28)

The second harmonic generation is obtained immediately from (27) and (28) for the case of 1 = 2 and 3 = 21. Therefore it is enough to consider only, 36 f.e., (27) and the last one of (28).

To further simplify the analysis we can assume that the power lost by the frequency 1 (fundamental) is negligible, and therefore dE1i dz

 0.

(29)

So it is sufficient to consider just the last one of (28) dE2k j2 0   j k1  k 3  k 2  z  d E E  c.c.  ijk 1i 3 je dz 2 2 dE 3 j dz

 j

3 0  j k1  k 2  k 3 z d E E e  c.c..  ijk 1i 2k 2 3 dE 3 j

where and

0 d jik E1i E1k e jkz  dz  3   1  2   j

k  k (3j)  k1(i)  k1(k) .

(31)

(28) (30)

37

In eq.(31) k1(i) is the constant of propagation of the beam at 1 polarized in the direction i. The solution of (30) for E3j(0) = 0 for a crystal of length L is or

0 e j k  L  1 E 3 j (L)   j d jik E1i E1k  j k 

0 2 I(L)  E 3 j (L)    2

 d jik E1i E1k

2

2 sen  k  L / 2  2 L . 2  k  L / 2 

(32)

According to (32) a requirement for an efficient second harmonic generation is that k = 0, that is from (31) with 3 = 2, 1 = 2 =  k  k (3j)  k1(i)  k1(k) . 2 k    2k ( ) .

(31) (33)

38

If k ≠ 0, the second harmonic wave generated at a generic plane z1 which propagates until another plane z2 is not in phase with that generated in z2. This produces an interference described by the factor sen 2  k  L / 2 

 k  L / 2  in (32).

2

 I(L)  E 3j (L)  0 2  2

 d jik E1i E1k

2

2

L

sen 2  k  L / 2 

.

(32)

 k  L / 2  The condition (33) is never practically satisfied because, due to dispersion, the refractive index depends on . k  2   2k ( ) .

2

(33) 39

Therefore, we have k  k

(2 )

being k ( )

and therefore

 2k

( )

2 (2 ) n   n ( ) c

n ( )  c

(34)

(35)

k  0.

40

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. 41

Coherence length We have no more second harmonic production when Sin(∆kL/2)/∆kL/2 = 0 This is achieved when ∆kL/2 =π Which means L = 2π/∆k

that is named coherence length 42

However, it is possible to make k = 0 (phasematching condition) using various skills; the most used of which takes advantage from the natural birefringence of the anisotropic crystals. From (34) we can see that k = 0 implies n

(2 )

n

( )

(2 )  2k ( )  (36) k  k

2 (2 ) n  n ( ) c

(34)

so that the refractive indices of second harmonic and of fundamental frequency have to be equal. In the materials with normal dispersion, the index of the ordinary and extraordinary wave along a direction increase with , as it is shown in the table. 43

, m

Index no (ordinary beam)

ns (extraordinary beam)

0,2000

1,622630

1,563913

0,3000

1,545570

1,498153

0,4000

1,524481

1,480244

0,5000

1,5144928

1,472486

0,6000

1,509274

1,468267

0,7000

1,505235

1,465601

0,8000

1,501924

1,463708

0,9000

1,498930

1,462234

1,0000

1,496044

1,460993

1,1000

1,493147

1,459884

1,2000

1,490169

1,458845

1,3000

1,487064

1,457838

1,4000

1,483803

1,456838

1,5000

1,480363

1,455829

1,6000

1,476729

1,454797

1,7000

1,472890

1,453735

1,8000

1,468834

1,452636

1,9000

1,464555

1,451495

2,0000

1,460044

1,450308

44

This makes it possible to satisfy eq.(36) when both the beams are of the same kind (that is both extraordinary or ordinary). Or (36) can be satisfied, in some cases, using an ordinary and an extraordinary wave. n (2 )  n ( ) (36) In order to illustrate this point we can consider the dependence of the refractive index of the extraordinary wave in a uniaxial crystal, from the angle  between the direction of propagation and the optical axis (z) of the crystal.

1 cos2  sen 2   2 . 2 2 ns () n0 ns

(37) 45

If ns(2 )  n (0) an angle n exists for which ns(2 ) (n )  n 0( ) . In this case if the fundamental beam (frequency ) is propagated along n as a ordinary beam, the second harmonic beam will be generated along the same direction as an extraordinary beam. This situation is shown in the figure.

46

The angle n is determined by the intersection between the sphere (shown as a circle in the figure) which corresponds to the index surface of the ordinary beam to  with the index ellipsoid of the extraordinary beam. The angle n, for negative uniaxial crystals (that is for crystals for which ns(2 )  n 0( ) is given by cos2 n

2 (2 )   n0   

sen 2n

2 (2 )  ns   

1

(38)

2 ( )   n0   

that is sen 2 

2 2 (  ) (2  ) n0   n0      2 2 (2 ) (2 )   ns   n0     

.

(39) 47

According to (32), if we deviate from the matching condition, for a fixed length L of the nonlinear crystal, we have a reduction of the second harmonic power generated by the factor (2 )

P  (2 ) Pmax

0 2 I(L)  E 3 j (L)    2

sen 2  k  L / 2 

 k  L / 2 

 d jik E1i E1k

2

2

L

2

.

(40)

sen 2  k  L / 2 

 k  L / 2 

2

.

(32)

48

This relation can be easily verified varying the angle  =  - n between the direction of index matching and the propagation direction. A diagram of the second harmonic power according to  is shown in the figure (where the theoretical curve (sen x/x)2 is also shown).

49

In the case of nanostructures, phase matching is not so important because in any case the radiation propagates over lengths which are comparable with the wavelength and so decoherence is negligible

50

Optical conservation laws Conservation of momentum k(2ω) = 2k(ω) Conservation of energy 2ω = ω + ω 2ħω = ħω +ħω 51

Nonlinear Optical Interactions • The E-field of a laser beam

~ it E(t )  Ee  C.C.

• 2nd order nonlinear polarization

~( 2) P (t )  2 ( 2) EE*  ( (2) E 2e2it  C.C.)

2

( 2)

52

2nd Order Nonlinearities • The incident optical field

~ E (t )  E1e

 i1t

 E2 e

 i 2 t

 C.C.

• Nonlinear polarization contains the following terms P(21)  (2)E12

(SHG)

P(22)  (2)E22

(SHG)

P(1 2)  2(2)E1 E2

(SFG)

summfrequency generation

P(1 2)  2(2)E1 E2*

(DFG)

difference frequency generation

P(0)  2(2) (E1 E1*  E2E2*) (OR)

53

Sum Frequency Generation

2

2 

( 2)

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

1

 3  1   2 2 1

3 54

Difference Frequency Generation

2

1

2 

( 2)

1

Application: The low frequency photon, 2 amplifies in the presence of high frequency beam  . This 1 is known as parametric amplification.

 3  1   2

1

2

3 55

Third Order Nonlinearities When the general form of the incident electric field is in the following form,

~  i1t  i 2 t  i 3 t E (t )  E1e  E2 e  E3e

The third order polarization will have 22 components

i ,3i ,(i j k ),(i j k ) (2i j ),(2i j ),i, j, k 1,2,3 56

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

57

The total polarization can be written as

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

(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

58

By definition

n  n0  n2 I where

n0 c I | E ( ) |2 2

12 2 ( 3 ) n2  2  n0 c 59

Typical values of nonlinear refractive index Mechanism

n2 (cm2/W)

( 3) (esu) 1111

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 Absorption

10-10

10-8

10-8

Thermal effects

10-6

10-4

10-3

Photorefractive Effect

large

large

Intensity dependent

60

Third order nonlinear susceptibility of some material Material



Response time

Air

1.2×10-17

CO2

1.9×10-12

2 Ps

GaAs (bulk room temperature)

6.5×10-4

20 ns

CdSxSe1-x doped glass

10-8

30 ps

GaAs/GaAlAs (MQW)

0.04

20 ns

Optical glass

(1-100)×10-14

Very fast 61