1. Chapter 4. The Intensity-Dependent Refractive Index - Third order nonlinear effect - Mathematical description of the nonlinear refractive index - Physical processes that give rise to this effect Reference : R.W. Boyd, “Nonlinear Optics”, Academic Press, INC.

2. 4.1 Description of the Intensity-Dependent Refractive Index Refractive index of many materials can be described by where, : weak-field refractive index : 2nd order index of refraction : optical Kerr effect (3rd order nonlinear effect)

3. Polarization : In Gaussian units,

4. Appendix A. Systems of Units in Nonlinear Optics (Gaussian units and MKS units) 1) Gaussian units ; The units of nonlinear susceptibilities are not usually stated explicitly ; rather one simply states that the value is given in “esu” (electrostatic units).

5. 2) MKS units ; : MKS 1 : MKS 2 In MKS 1, In MKS 2,

6. 3) Conversion among the systems (Gaussian) (MKS)

7. Alternative way of defining the intensity-dependent refractive index (4.1.4) Example)

9. Physical processes producing the nonlinear change in the refractive index 1) Electronic polarization : Electronic charge redistribution 2) Molecular orientation : Molecular alignment due to the induced dipole 3) Electrostriction : Density change by optical field 4) Saturated absorption : Intensity-dependent absorption 5) Thermal effect : Temperature change due to the optical field 6) Photorefractive effect : Induced redistribution of electrons and holes  Refractive index change due to the local field inside the medium

11. 4.2 Tensor Nature of the 3rd Susceptibility Centrosymmetric media Equation of motion : Solution : Perturbation expansion method ;

12. 3rd order polarization : 4th-rank tensor : 81 elements Element with even number of index where, D : Degeneracy factor (The number of distinct permutations of the frequency wm, wn, wp) Let’s consider the 3rd order susceptibility for the case of an isotropic material. 21 nonzero elements : and, (Report) (Report)

13. Expression for the nonlinear susceptibility in the compact form : Example) Third-harmonic generation : Example) Intensity-dependent refractive index :

14. Nonlinear polarization for Intensity-dependent refractive index in vector form Defining the coefficients, A and B as (Maker and Terhune’s notation)

15. In some purpose, it is useful to describe the nonlinear polarization by in terms of an effective linear susceptibility, as where, Physical mechanisms ;

16. 4.3 Nonresonant Electronic Nonlinearities # The most fast response : [a0(Bohr radius)~0.5x10-8cm, v(electron velocity)~c/137] Classical, Anharmonic Oscillator Model of Electronic Nonlinearities Approximated Potential : (1.4.52) where, According to the notation of Maker and Terhune,

17. Far off-resonant case, Typical value of

18. 4.4 Nonlinearities due to Molecular Orientation The torque exerted on the molecule when an electric field is applied : induced dipole moment

19. Second order index of refraction Change of potential energy : Optical field (orientational relaxation time ~ ps order) : 1) With no local-field correction Mean polarization :

20. Defining intensity parameter, i) J  0 : linear refractive index

21. ii) J 0

22. Second-order index of refraction :

23. 2) With local-field correction and : Lorentz-Lorenz law

24. 8.2 Electrostriction : Tendency of materials to become compressed in the presence of an electric field Molecular potential energy : density change Force acting on the molecule :

25. Increase in electric permittivity due to the density change of the material : Field energy density change in the material = Work density performed in compressing the material So, electrostrictive pressure : : electrostrictive constant where,

26. Density change : where, : compressibility For optical field,

27. <Classification according to the Maker and Terhune’s notation> Nonlinear polarization :

28. : For dilute gas : For condensed matter (Lorentz-Lorenz law) Example) Cs2, C ~10-10 cm2/dyne, ge ~1  c(3) ~ 2x10-13 esu Ideal gas, C ~10-6 cm2/dyne (1 atm), ge =n2-1 ~ 6x10-4 c(3) ~ 1x10-15 esu