Maxwell’s equations in nonlinear media, frequency conversion techniques (SHG, SFG, OPO), stimulated scattering, phase conjugation, wave-guide optics, nonlinear crystals.

In this twenty first century, even milliwatt semiconductor lasers can produce nonlinear effects in materials. This course deals with the fundamentals and applications of the nonlinear interaction of radiation with matter. Its goal is to give a student a working knowledge of nonlinear effects, nonlinear materials and the applications of nonlinear optics in various technologies.

**Course prerequisites:**

**Textbook:**

- “Nonlinear Optics, phenomena, materials and devices,” George & Robert Stegeman, Wiley Series in Pure and Applied Optics, Glenn Boreman, Series editor, 2012.

**Suggested reading:**

- “Nonlinear Optics” 2
^{nd}edition, Robert Boyd (Academic Press) - “Principles of Nonlinear Optics” Ron Shen (Wiley)

**List of topics** Introduction to NLO

- any change in optical properties or generation of new frequencies
- classical expansion in powers of the field

Electron on a Spring – Ï‡^{(2)} Processes

- electron on a spring, equation
- induced polarization
- linear susceptibility
- non-parabolic potential well
- nonlinear driving equation and perturbative solution
- structure of Ï‡
^{(2)} - SHG, sum and difference frequency
- Miller’s rule
- permutation, Kleinman symmetry, centrosymmetric versus non-centrosymmetric
- cascade processes giving Ï‡
^{(3)}

Coupled Mode Equations

- meaning of Ï‡
^{(2)}in frequency domain - integral formulation
- nonlinear polarizations
- Maxwell’s equations in terms of total polarization
- simplifications, slowly varying phase and amplitude approximation
- coupled mode equations for SHG and for three wave interactions

Phase-matching

- Type I and Type II phase-matching
- uniaxial crystals phase-matching directions, e.g. KDP, special case of 90
^{o}phase-matching - QPM, advantages, domain inversion in ferroelectric media
- walk-off, acceptance angle, temperature acceptance
- non-collinear phase-matching

Susceptibility Tensor

- d-tensor
- connection between crystal class and number of independent elements
- projections and deff for different crystal classes

SHG & Three Wave Mixing

- low conversion efficiency
- phase mismatch & coherence length
- large conversion on phase match
- SHG eigenmodes
- Manley-Rowe relations
- focused beams, pulsed beams, nonlinear (cascaded) phase shift

Optical Parametric Amplifiers and Oscillators

- coupled mode equations, solutions
- parametric gain length, typical numbers
- bandwidth
- tuning curves in real materials
- cavities and cavity modes, singly resonant cavity, self-consistent analysis
- doubly resonant cavity
- modern examples

Materials

- typical materials including semiconductors, dielectrics, organics, LB films
- measurement techniques, Kurtz method, Maker fringes
- molecular hyperpolarizabilities
- electric field induced SHG
- optical rectification

Perturbation Theory of Susceptibilities

- review of quantum mechanics, 1
^{st}order perturbation theory - linear susceptibility
- resonances
- 2
^{nd}order susceptibility - 3
^{rd}order susceptibility

Third Order Susceptibilities

- symmetries etc. of Ï‡
^{(3)}, Ï‡^{(3)}phenomena - single frequency inputs
- degeneracy factors for self-action and THG
- confusion- units, self-action + SVEA

Nonlinear Refraction and Absorption

- real and imaginary parts of Ï‡
^{(3)} - nonlinear refraction and absorption
- TPA, non-degenerate TPA
- pump-probe
- intensity-dependent refractive index
- self-focusing and defocusing
- self-phase modulation
- bistability

Solitons

- self-trapping spatial solitons, temporal solitons, modern developments

Four-Wave Mixing

- degenerate FWM, gain, reflectivity, measurement of Ï‡
^{(3)} - non-degenerate FWM, grating picture
- Raman Nath versus Bragg
- diffuse nonlinearities (carriers)

Mechanisms for NLR & NLA

- thermal; molecular re-orientation, CS
_{2} - concept of saturation
- electrostriction
- 2-level system
- conjugated polymers
- carrier and ultrafast NLO in semiconductors
- Kerr effect

Nonlinear Spectroscopy

- Third Harmonic Generation
- Z-scan
- Raman active nonlinearities
- TPA resonances
- phase shifts
- RIKES, CARS, CSRS

Stimulated Scattering

- stimulated Raman, Raman gain
- Stokes and anti-Stokes
- Coherent Anti-stokes Raman
- Stimulated Brillouin, Brillouin gain

*Measured evolution of the output spatial distribution of a 1.064um pulse (left) and its second harmonic signal (right) after propagation through a periodically poled KTP crystal, showing mutual self-focusing and the development of a spatial soliton (mutually self-trapped beam) as the pulse power is increased.*