OPTI 546 Physical Optics (last updated 4/25/11)
Unit 1: Paraxial ray optics
•
•
•
•
•
•
Optical path length and Fermat’s principle, calculus of variations, Euler-Lagrange
equations, ray equations of geometrical optics.
Paraxial ray optics for first-order optical systems, examples including
homogeneous dielectric media, thin lenses, curved mirrors, and GRIN media.
ABCD matrix approach to paraxial ray optics, analysis of first-order optical
systems, examples and special cases, synthesis of ABCD systems.
Optical path length through an ABCD system.
Periodic optical systems, eigenray analysis.
Geometric optics analysis of optical resonators, stable and unstable resonators,
examples and special cases.
Unit 2: Diffraction and propagation
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Review of Maxwell’s equations and the constitutive relations for linear media.
Maxwell’s time-dependent wave equation for a simple isotropic dielectric
medium, Helmholtz equation for a monochromatic field.
Special solutions of the Helmholtz equation, plane-waves and Fourier transforms,
spherical waves, and Bessel beams solutions.
Green’s functions for Maxwell’s wave equation and the Helmholtz equation, role
of boundary conditions.
Historical perspective of the development of Scalar Diffraction Theory.
Rayleigh-Sommerfeld diffraction formula, Fresnel and Fraunhofer
approximations.
Diffraction by a planar object, Babinet’s principle, diffraction by an aperture,
diffraction by an opaque disk, the Spot of Arago.
The paraxial wave equation for beam propagation, Gaussian beam solution and
the complex beam parameter in free-space, the Guoy phase-shift.
Higher-order transverse modes, Hermite-Gaussian and Laguerre-Gaussian modes.
Non-diffracting beams, generation of Bessel beams.
Green’s function for the paraxial wave equation, relation between paraxial ray and
wave optics via the path integral.
Generalized Huygen’s integral for propagation through first order optical systems.
ABCD law for Gaussian beams in first-order optical systems.
Special examples, focusing of a Gaussian beam, Fourier transforming system,
mode converter, imaging systems, coherent versus incoherent illumination.
Unit 3: Interferometers and resonators
•
•
•
•
•
Basics of interference, Young’s double-slit experiment, theory of partial
coherence, mutual coherence and visibility, modification of interference fringes
due to partial temporal or spatial coherence.
Specific interferometers, division of wavefront versus division of amplitude,
Michelson interferometer, basic ideas of fringe formation for simple wavefronts,
Twyman-Green, and Mach-Zender.
Plane-parallel Fabry-Perot, transmission characteristics for normal incidence and
the Airy function, free-spectral range, frequency bandwidth, and finesse,
longitudinal modes, Gaussian mode of a curved mirror Fabry-Perot resonator and
mode-matching, higher-order transverse modes, transverse mode resonances.
Optical resonators with curved mirrors, equivalent periodic systems, transverse
modes of stable resonators, transverse mode frequencies.
Fox & Li method for obtaining optical resonator modes, stable and unstable
resonators, example resonator with a Gaussian reflectivity mirror.
Unit 4: Optical interactions
Non-relativistic classical electrodynamics
•
•
•
Maxwell-Lorentz equations: Microscopic Maxwell equations for a system of
point charges, Newton-Lorentz equations for the motion of the charges,
conservation laws for energy and momentum for the Maxwell-Lorentz system.
Scalar and vector potentials: Vector and scalar potentials and their relation to the
physical fields, wave equations for the potentials, nonuniqueness of potentials and
gauge transformations, Lorentz gauge and Coulomb (or radiation) gauge and
associated wave equations for the potentials, separation into transverse and
longitudinal fields in the Coulomb gauge.
Classical electrodynamics in the Coulomb gauge: Electric versus magnetic field
contributions to the Lorentz force, Coulomb interaction energy, transverse optical
polarization, equations of classical electrodynamics in the Coulomb gauge.
Classical theory of dispersion
•
Classical model for dielectrics: Classical atoms for a charge neutral system, gas of
atoms as a model for a dielectric medium, center-of-mass (COM) motion and
relative motion for a single classical atom treated in the Coulomb gauge, classical
bound states of the relative motion and the failure of classical theory to predict
stable atoms, Lorentz electron oscillator model based on the electron cloud model.
•
•
•
Newton-Lorentz equations for an atom in an applied light field in the Coulomb
gauge, dipole approximation for the relative motion of a bound electron-ion pair,
electronic polarizability.
Optical polarization for Lorentz model: Macroscopic spatial averages for the
electromagnetic field and polarization, wave equation for the electric field and
oscillator equation for the polarization, solution for the polarization in the time
domain, linear optical susceptibility and optical properties, optical dispersion for
the Lorentz oscillator model.
Propagation equations: The scalar Helmholtz equation, paraxial wave equation for
propagation in a dilute medium.
Periodic structures
•
•
Propagation in periodic structures: Scalar Helmholtz equation for a spatially
inhomogeneous but isotropic medium, Green’s function solution of the Helmholtz
equation with spatially dependent refractive-index, first-Born approximation, 3D
diffraction in far-field approximation for an incident plane-wave, phase-matching
and the Bragg condition for a periodic structure, Raman-Nath versus Bragg
diffraction regions, forward scattering versus backscattering.
Acousto-optics: Interaction of light and sound, scalar paraxial wave equation for
acousto-optic diffraction, coupled wave analysis in the Bragg regime, AO
modulators and deflectors, Bragg versus Raman-Nath regions, exact solution for
Debye-Sears diffraction in the Raman-Nath region.
Magneto- and Electro-optics
•
•
•
Magneto-optics: Lorentz electron oscillator model in the presence of an applied
static magnetic field, circular electron motion and the Larmor frequency, circular
polarizations as medium eigenvectors, Zeeman effect and the splitting of spectral
absorption lines, propagation in the transparency region, magneto-optical Faraday
effect, Faraday rotation, application to optical isolators, description in terms of a
dielectric tensor, comparison with optical activity.
Electrooptics (Lite): Lorentz electron oscillator model with an applied static
electric field, anharmonic potential for large electron displacements, simplified
model for the electro-optic Pockel and Kerr effects, EO modulation.
Anisotropic Lorentz model: Periodic arrangement of atoms in crystals and the unit
cell, anisotropic Lorentz electron oscillator model for electrons in the anisotropic
environment of the unit cell, force constant tensor and symmetry properties,
tensor notation for the Lorentz model, crystal axes system that diagonalizes the
force constant tensor.
•
•
Optical Polarization in crystals: Macroscopic polarization for a dielectric crystal,
linear susceptibility tensor, principal axes system, examples of an isotropic
medium and the susceptibility tensor in the principal axes, general properties of
the linear susceptibility tensor for transparent crystals, optically isotropic crystals,
uniaxial crystals, principal axes and the ordinary and extraordinary indices of
refraction, negative and positive uniaxials, biaxial media, dielectric tensor and the
displacement vector, longitudinal fields in crystals, the impermeability tensor.
Crystal optics: The index ellipsoid, isotropic, uniaixal, and biaxial media,
modification of the impermeability tensor due to an applied DC electric field, the
electrooptic coefficients, modified index ellipsoid for KDP as an example, new
principal axes system, the linear EO or Pockel’s effect, EO modulation.
Introduction to Nonlinear Optics
•
•
Introduction to nonlinear optics: Simplified Lorentz model for nonlinear optics
(NLO), anharmonic electron motions, the first- and second-optical polarizations,
second- and higher-order NLO, second-harmonic generation (SHG), optical
rectification, sum and difference frequency generation, generalization of the firstand second-order polarizations, Helmholtz equations, solution for SHG, phasematching for efficient SHG and the coherence length, Maker fringes, index phasematching in uniaxial crystals, quasi phase-matching.
Third-order nonlinear optics: Third-harmonic generation, nonlinear refractiveindex and Kerr-type nonlinearities, paraxial wave equation, self-phase modulation
and spectral broadening in fibers, external self-focusing and defocusing, selffocusing and self-trapping, critical power for self-focusing.