Nonlinear Optics Lab. Hanyang Univ. 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.

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

Nonlinear Optics Lab. Hanyang Univ. Polarization : In Gaussian units,

Nonlinear Optics Lab. Hanyang Univ. 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).

Nonlinear Optics Lab. Hanyang Univ. 2) MKS units ; : MKS 1 : MKS 2 In MKS 1, In MKS 2,

Nonlinear Optics Lab. Hanyang Univ. 3) Conversion among the systems (Gaussian) (MKS)

Nonlinear Optics Lab. Hanyang Univ. Alternative way of defining the intensity-dependent refractive index (4.1.4) Example)

Nonlinear Optics Lab. Hanyang Univ.

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

Nonlinear Optics Lab. Hanyang Univ.

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

Nonlinear Optics Lab. Hanyang Univ. 3 rd order polarization : where, D : Degeneracy factor (The number of distinct permutations of the frequency  m,  n,  p ) 4 th -rank tensor : 81 elements Let’s consider the 3 rd order susceptibility for the case of an isotropic material. 21 nonzero elements : and, (Report) Element with even number of index

Nonlinear Optics Lab. Hanyang Univ. Expression for the nonlinear susceptibility in the compact form : Example) Third-harmonic generation : Example) Intensity-dependent refractive index :

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

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

Nonlinear Optics Lab. Hanyang Univ. 4.3 Nonresonant Electronic Nonlinearities # The most fast response : [a 0 (Bohr radius)~0.5x10 -8 cm, 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,

Nonlinear Optics Lab. Hanyang Univ. Far off-resonant case, Typical value of

Nonlinear Optics Lab. Hanyang Univ. 4.4 Nonlinearities due to Molecular Orientation The torque exerted on the molecule when an electric field is applied : induced dipole moment

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

Nonlinear Optics Lab. Hanyang Univ. Defining intensity parameter, i) J  0 : linear refractive index

Nonlinear Optics Lab. Hanyang Univ. ii) J 0

Nonlinear Optics Lab. Hanyang Univ. Second-order index of refraction :

Nonlinear Optics Lab. Hanyang Univ. 2) With local-field correction and : Lorentz-Lorenz law

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

Nonlinear Optics Lab. Hanyang Univ. 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 : where, : electrostrictive constant

Nonlinear Optics Lab. Hanyang Univ. Density change : where, : compressibility For optical field,

Nonlinear Optics Lab. Hanyang Univ. Nonlinear polarization :

Nonlinear Optics Lab. Hanyang Univ. : For dilute gas : For condensed matter (Lorentz-Lorenz law) Example) Cs 2, C ~ cm 2 /dyne,  e ~1   (3) ~ 2x esu Ideal gas, C ~10 -6 cm 2 /dyne (1 atm),  e =n 2 -1 ~ 6x10 -4   (3) ~ 1x esu