Lectures on nonlinear optics

This is the full set of twelve lectures in Nonlinear Optics which I in the spring of 2003 gave at the Royal Institute of Technology, Department of Laser Physics and Quantum Optics, Stockholm, Sweden. The course was aimed towards advanced undergraduate and doctoral students and gave 5 points, corresponding to five weeks of full-time study. The lecture notes available below are a product from the continuous preparation for the lectures.

Throughout the course, the convention on nonlinear susceptibilities and degeneracy factors follows that of Butcher and Cotter's The Elements of Nonlinear Optics (Cambridge University Press, 1990), for which a separate errata and summary of conventions can be found.

These lecture notes have also been published electronically by the Royal Institute of Technology, as Lecture Notes on Nonlinear Optics, ISBN 91-7283-517-6, TRITA-FYS 2003:26 (Transactions of the Royal Institute of Technology, Stockholm, 2003).
URL: http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-9154.

The complete series of lectures

nlo2003.pdf [2.96 MB] The entire set of lecture notes, supplementary notes and home assignments in a single PDF document. [ download ]

nlo2003.djvu [1567 kB] The entire set of lecture notes, supplementary notes and home assignments in a single DejaVu document. [ download ]

nlo2003.ps [10561 kB] The entire set of lecture notes, supplementary notes and home assignments in a single PostScript document. [ download ]

Plug-in viewers of the PDF and DejaVu formats can be downloaded for free from the sites of Adobe and LizardTech, respectively. In order to display PostScript files on screen, you need a viewer like GhostView, which can be downloaded for free from the web site of the University of Wisconsin.


Additional resources

errata.pdf [107 kB] Errata for Butcher and Cotter's The Elements of Nonlinear Optics (Cambridge University Press, 1990). Also available online. [ download ]

bcconven.pdf [102 kB] Summary of the "Butcher and Cotter convention" on nonlinear susceptibilities, adopted from The Elements of Nonlinear Optics (Cambridge University Press, 1990). TeX source for the document is available here. [ download ]


Lecture 1 - Introduction to nonlinear optics

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Contents

  1. The contents of the course
  2. Examples of applications of nonlinear optics
  3. A brief history of nonlinear optics
  4. Outline for calculations of polarization densities
    1. Metals and plasmas
    2. Dielectrics
  5. Introduction to nonlinear dynamical systems
  6. The anharmonic oscillator

Lecture 2 - The nonlinear susceptibilities and their symmetries

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Contents

  1. Nonlinear polarization density
  2. Symmetries in nonlinear optics
    1. Intrinsic permutation symmetry
    2. Overall permutation symmetry
    3. Kleinman symmetry
    4. Spatial symmetries
  3. Conditions for observing nonlinear optical interactions
  4. Phenomenological description of the susceptibility tensors
  5. Linear polarization response function
  6. Quadratic polarization response function
  7. Higher order polarization response functions

Lecture 3 - Quasi-monochromatic fields and the degeneracy factor in nonlinear optics

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Contents

  1. Susceptibility tensors in the frequency domain
  2. First order susceptibility tensor
  3. Second order susceptibility tensor
  4. Higher order susceptibility tensors
  5. Monochromatic fields
  6. Convention for description of nonlinear optical polarization
    1. Note on the complex representation of the optical field
    2. Example: Optical Kerr-effect

Lecture 4 - Quantum mechanics I: Formulation of linear optical interactions

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Contents

  1. The Truth of polarization densitites
  2. Outline
  3. Quantum mechanics
  4. Perturbation analysis of the density operator
  5. The interaction picture
  6. The first order polarization density

Lecture 5 - Quantum mechanics II: Formulation of nonlinear optical interactions

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Contents

  1. The second order polarization density
  2. Higher order polarization densities

Lecture 6 - Quantum mechanics III: Linking the microcscopic to the macroscopic

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Contents

  1. Assembly of independent molecules
  2. First order electric susceptibility
  3. Second order electric susceptibility
  4. Overall permutation symmetry of second order susceptibility

Lecture 7 - Spatial symmetries in nonlinear optics

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Contents

  1. Motivation for analysis of susceptibilities in rotated coordinate systems
  2. Optical properties in rotated coordinate frames
    1. First order polarization density in rotated coordinate frames
    2. Second order polarization density in rotated coordinate frames
    3. Higher order polarization density in rotated coordinate frames
  3. Crystallographic point symmetry groups
  4. Schönflies notation for the non-cubic crystallographic point groups
  5. Neumann's principle
  6. Inversion properties
  7. Euler angles
  8. Example of the direct inspection technique applied to tetragonal media
    1. Does the 422 point symmetry group possess inversion symmetry?
    2. Step one - Point symmetry under twofold rotation around the x1-axis
    3. Step two - Point symmetry under fourfold rotation around the x3-axis

Lecture 8 - The nonlinear electromagnetic wave equation

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Contents

  1. Wave propagation in nonlinear media
    1. Maxwell's equations
    2. Constitutive relations
  2. Two frequent assumptions in nonlinear optics
  3. The wave equation
  4. The wave equation in frequency domain (optional)
  5. Quasimonochromatic light - Time dependent problems
  6. Three practical approximations
  7. Monochromatic light
    1. Monochromatic optical field
    2. Polarization density induced by monochromatic optical field
  8. Monochromatic light - Time independent problems
  9. Example I: Optical Kerr-effect - Time independent case
  10. Example II: Optical Kerr-effect - Time dependent case

Lecture 9

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Contents

  1. General process for solving problems in nonlinear optics
  2. Formulation of the two case studies in this lecture
    1. Case study I: Second harmonic generation in negative uniaxial media
    2. Case study II: Optical Kerr-effect - continuous wave case
  3. Second harmonic generation
    1. The optical interaction
    2. Symmetries of the medium
    3. Additional symmetries
    4. The polarization density
    5. The wave equation
    6. Boundary conditions
    7. Solving the wave equation
  4. Optical Kerr-effect - Field corrected refractive index
    1. The optical interaction
    2. Symmetries of the medium
    3. Additional symmetries
    4. The polarization density
    5. The wave equation - Time independent case
    6. Boundary conditions - Time independent case
    7. Solving the wave equation - Time independent case
  5. References

Lecture 10

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Contents

  1. What are solitons?
  2. Classes of solitons
    1. Bright temporal envelope solitons
    2. Dark temporal envelope solitons
    3. Spatial solitons
  3. The normalized nonlinear Schrödinger equation for temporal solitons
    1. The effect of dispersion
      1. β > 0, negative group velocity dispersion
      2. β < 0, positive group velocity dispersion
    2. The effect of a nonlinear refractive index
    3. The basic idea behind temporal solitons
    4. Normalization of the nonlinear Schrödinger equation
  4. Spatial solitons
  5. Mathematical equivalence between temporal and spatial solitons
  6. Soliton solutions
  7. General travelling wave solutions
  8. Soliton interactions
  9. Dependence on initial conditions
  10. References

Lecture 11

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Contents

  1. Singularities of non-resonant susceptibilities
  2. Modification of the Hamiltonian for resonant interaction
  3. Phenomenological representation of relaxation processes
  4. Perturbation analysis of weakly resonant interactions
  5. Validity of perturbation analysis of the polarization density
  6. The two-level system
    1. Terms involving the thermal equilibrium Hamiltonian
    2. Terms involving the interaction Hamiltonian
    3. Terms involving relaxation processes
  7. The rotating-wave approximation
  8. The Bloch equations
  9. The resulting electric polarization density of the medium

Lecture 12

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Contents

  1. Recapitulation of the Bloch equations for two-level systems
  2. The resulting electric polarization density of the medium
  3. The vector model of the Bloch equations
  4. Transient build-up at exact resonance as the optical field is switched on
    1. The case T1 >> T2 - Longitudinal relaxation slower than transverse relaxation
    2. The case T1 ≈ T2 - Longitudinal relaxation approximately equal to transverse relaxation
  5. Transient build-up at off-resonance as the optical field is switched on
  6. Transient decay for a process tuned to exact resonance
    1. The case T1 >> T2 - Longitudinal relaxation slower than transverse relaxation
    2. The case T1 ≈ T2 - Longitudinal relaxation approximately equal to transverse relaxation
  7. Transient decay for a slightly off-resonant process
    1. The case T1 >> T2 - Longitudinal relaxation slower than transverse relaxation
    2. The case T1 ≈ T2 - Longitudinal relaxation approximately equal to transverse relaxation
  8. Transient decay for a far off-resonant process
    1. The case T1 >> T2 - Longitudinal relaxation slower than transverse relaxation
    2. The case T1 ≈ T2 - Longitudinal relaxation approximately equal to transverse relaxation
    3. The case T1 << T2 - Longitudinal relaxation faster than transverse relaxation
  9. The connection between the Bloch equations and the susceptibility
    1. The intensity-dependent refractive index in the susceptibility formalism
    2. The intensity-dependent refractive index in the Bloch-vector formalism
  10. Summary of the Bloch and susceptibility polarization densities
  11. Appendix: Notes on the numerical solution to the Bloch equations

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Last modified Wednesday 16 Nov 2011