This is a summary of the topics in linear optics we have covered in modules 8 and 9.

We have studied second and third order nonlinear effects.  When the magnitude of external electric field is large, then the induced polarization has a nonlinear dependence on the electric field and can be expressed as a power series with respect to the electric field.

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P_i(t) can be expressed in terms of its Fourier transform P_i(w).

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If the incident field is a contains two frequency components,

E(t) = [½(E₁exp(-iw₁t) + E₂exp(-iw₂t)) + ½(E₁*exp(iw₁t) + E₂*exp(iw₂t))],

then w_n and w_m can take on the values ±w₁ and ±w₂, and E'(w_n) is the field strength associated with w_n.

P_i(w’) is the source for an electric field oscillating with frequency w’.  We find that the nonlinear dependence of the polarization on the electric field introduces source terms at frequencies other than the incident frequencies.

Second-order nonlinear optical interactions are described by the c⁽²⁾ term.  This term is present only in media without inversion symmetry.  Second-order nonlinear optical interactions are responsible for the linear electro-optic effect, optical rectification, second harmonic generation, parametric up-conversion and parametric amplification.

The linear electro-optic effect modifies the dielectric tensor e in the presence if an external field with components E_k.

e_ij ® e_ij + e₀ c_ijk⁽²⁾ E_k.

An external field can therefore be used to turn a uniaxial crystal into a biaxial crystal.   Both polarization components of a laser beam propagate with the same speed along the extraordinary axis of a uniaxial crystal.  But after passage through a biaxial crystal of length L one polarization component will be retarded with respect to the other.  The intensity transmitted by a polarization analyzer varies with the strength of the external field.  This is the basis for electro-optic amplitude modulation. 

Parametric up-conversion in a crystal can be used to convert a signal from a low frequency w₁ to a high frequency w' by mixing it with a strong laser beam of frequency w₂.  The signal (w₁) and pump (w₂) photons are annihilated while simultaneously a photon of frequency w' is created. 

Parametric amplification involves an input signal at at w₁ together with an intense pump beam at w₂, w₂ > w₁.  A photon of the pump beam with energy hw₂ interacts with a photon of energy hw₁ and splits into two photons, one with energy h(w₂-w₁) and one with energy hw₁

Second-harmonic generation produces electromagnetic radiation with frequency 2w, when an electromagnetic wave with frequency w moves through the anisotropic, nonlinear medium.

 When EM waves with different frequencies are mixed in a nonlinear material, color dispersion generally gives rise to a momentum mismatch Dk.  This limits the active volume of emission to a layer of thickness ~IDkl^-1.  The problem of destructive interference can be overcome by making use of optical birefringence in anisotropic crystals.

Third-order nonlinear optical interactions are described by the c⁽³⁾ term.  Third-order nonlinear optical interactions are responsible for four-wave mixing, the nonlinear index of refraction and the optical limiting phenomenon.  The nonlinear index of refraction is responsible for self-focusing and soliton propagation in fibers. 

The quadratic electro-optic effect (QEO) or Kerr effect is also a third order nonlinear optical process.  The quadratic electro-optic effect modifies the dielectric tensor e in the presence if an external field.

e_ij ® e_ij + e₀ c_ijko⁽³⁾ E_k E_o,

where E_k and E_o are components of the external field.  Unlike the linear electro-optic effect, the quadratic electro-optic effect does not require a material without inversion symmetry but can be generated in materials with any molecular orientation.