1. Introduction to Nonlinear Optics
MOHAMMAD IMRAN AZIZ
Assistant Professor
PHYSICS DEPARTMENT
SHIBLI NATIONAL COLLEGE, AZAMGARH (India).

2. 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.

3. Introduction Question:
Is it possible to change the color of a monochromatic light?
Answer:
Not without a laser light
output
NLO sample
input

4. 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

6. E2 E2 E1 E1 Absorption Spontaneous E2 E1 Stimulated Emission Emission

7. Light (Microwave) Amplificationby
Stimulated
Emission of Radiation
LASER
(MASER)

8. 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)

9. Left to right: Prokhorov, Townes and Basov at the Lebede institute (1964 Nobel prize in Physics for developing the “Maser-Laser principle”)

10. Townes (left) and Gordon (right) and the ammonia maser they had built at Colombia University

11. 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

12. Maiman and the first ruby laser

13. Ali Javan and the first He-Ne Laser

15. 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

16. Applications of Laser (1960s) “A solution looking for a problem”
(Present time) Medicine, Research, Supermarkets, Entertainment, Industry, Military, Communication, Art, Information technology, …

17. 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)

18. 2. The Essence of Nonlinear Optics
When the intensity of the incident light to a material system increases the response of medium is no longer linear
Input intensity
Output

19. Response of an optical Medium
The response of an optical medium to the incident electro magnetic field is the induced dipole moments inside the medium

20. Nonlinear Susceptibility
Dipole moment per unit volume or polarization
The general form of polarization

21. Nonlinear Polarization
Permanent Polarization
First order polarization:
Second order Polarization
Third Order Polarization

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

23. Nonlinear Optical Interactions
The E-field of a laser beam
2nd order nonlinear polarization

24. 2nd Order Nonlinearities
The incident optical field
Nonlinear polarization contains the following terms

25. Sum Frequency Generation
Application:
Tunable radiation in the
UV Spectral region.

26. Difference Frequency Generation
Application:
The low frequency photon, amplifies in the presence of high frequency beam This is known as parametric amplification.

27. Phase Matching 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.

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

30. Third Order Nonlinearities
When the general form of the incident electric field is in the following form,
The third order polarization will have 22 components which their frequency dependent are

31. The Intensity Dependent Refractive Index
The incident optical field
Third order nonlinear polarization

32. The total polarization can be written as
One can define an effective susceptibility
The refractive index can be defined as usual

33. By definition
where

34. Typical values of nonlinear refractive index
Mechanism
n2 (cm2/W)
(esu)
Response time (sec)
Electronic Polarization
10-16
10-14
10-15
Molecular Orientation
10-12
Electrostriction
10-9
Saturated Atomic Absorption
10-10
10-8
Thermal effects
10-6
10-4
10-3
Photorefractive Effect
large
Intensity dependent

35. Third order nonlinear susceptibility of some material
 1111
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
Optical glass
(1-100)×10-14
Very fast

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

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

38. Wave mixing

39. Optical Phase Conjugation
Phase conjugation mirror M
PCM
PCM s M

40. Aberration correction by PCM
Aberrating
medium
PCM s
Aberrating
medium

41. What is the phase conjugation
The signal wave
The phase conjugated wave

42. Degenerate Four Wave Mixing
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

43. Mathematical Basis The four interacting waves
The nonlinear polarization
The same form as the phase conjugate of A3

44. 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.

45. 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.

46. The motion of electron in the field of a light wave:
1. Free electrons
The motion of electron in the field of a light wave: (1
is described by an equation:
(2)
Because
, the vector product
is proportional to
The solution of (2) can be found in a form:
(3)
where
is linear,
are nonlinear polarisabilities.
The induced electrical dipole moment is equal to
(4)

47. For the case of bound electron the equation has the following form:
2. Bound electrons
For the case of bound electron the equation has the following form:
(5)
where the term
takes into account real anharmonisity of the
oscillator:
Considerin
as a small term the solution of (5) can be
presented as:
(3)

48. 3. Macroscopic characteristics
To describe the media response for the electromagnetic field one must calculate a polarization vector , which is a dipole moment of a unit volume.
(6)
Where N is the concentration of electrons.
If a nonlinear dependence of on takes place the vectors and can be presented in the form:
(7)
(8)
where are tensors of 2 rank, are tensors of 3 rank
and so on.
are nonlinear susceptibilities

49. 4. Local field factor
In a microscopic model of nonlinearity (we presented two such models) it is important to describe correctly microscopic and macroscopic values. For crystals of cubic symmetry:
(9)
where the term in brackets is so-called Lorentz factor (local field factor). For nonlinear susceptibility in particular for quadratic nonlinearity:
(10)

50. 5. How high is the nonlinearity
If the response of the media is caused by electrons in nonresonant case for the following ratio is valid:
(11)
where is an interatomic field. For hydrogen
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 W/cm2. Now femtosecond lasers became available, which generate pulses with duration of 6-30·10-15 s at the intensity up to 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.

51. 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 for different mechanism of nonlinear polarization.
Mechanism
nonresonant
electronic
resonant
orientation in
liquid crystals
Time constant, s
10-14
1-10-1
 (2), esu
10-9
 (3), esu
10-10

52. 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.

53. 1. Second-harmonic generation
First of all we should notice that the tensor , for centrosymmetric media is equal to zero.
(12)
The operation of symmetry transforms the terms from (12) in the following way:
(13)
Then , that can not take place under nonzero
The same is valid for all even order

54. The incident waves propagating in z-direction can be presented as:
For a simplicity we assume that the medium is isotropic. Then the polarization:
(14)
The incident waves propagating in z-direction can be presented as:
(15)
(16)

55. A spectrum of polarization waves contains new frequencies:

56. 2. Third-harmonic generation
If the medium possesses cubic nonlinearity, under the action of two monochromatic waves and the polarization would contain the components with frequencies:

57. 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.

58. 1. Maxwell equations
The propagation of the light in the medium is described by Maxwell equations:
where
(18)
(17)
For optical range
(19)
Combining first and second equations from (17) one may obtain so-called wave equation:
(20)

59. 2. Phase mismatch Inserting (18) into (20) we are getting: (21)
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 and relatively low nonlinearity the plane wave solution of (21) for the intensity of the second harmonic looks like:
(22)

60. Phase mismatch

61. For the case of the exact phase matching the energy of the pumping wave can be completely transferred into second harmonic

62. 3. Phase matching
How the condition or can be realized? In an isotropic medium with normal dispersion > and
never equals to zero
But in birefringent uniaxial crystal there are two beams ordinary and extraordinary. For so-called negative crystal no>ne. If pumping wave is ordinary one and second harmonic is extraordinary one the material dispersion ( > ) can be compensate for the difference in refractive indices for o and e beams:

63. For the process of third-harmonic generation the condition of phase matching looks the same:
As it was mentioned already and 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.

64. 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:
(23)
As a result of such selfaction a nonlinear refractive index n2I appears at the frequency :
(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.

65. 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.

66. VI. Nonlinear Optical Diagnostics
Nonlinear susceptibilities and 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.

67. 1. Nonlinear optical diagnostics of phase transitions
Semiconductor in liquid state
Idea of experiment
Metal in liquid state

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