Past exam of the nonlinear Optics and Quantum States of Light course of the University of Cambridge / 2012 / Paper 1 / 3

For stationary photodetection, the normalized correlation in second-order optical coherence is $g^{(2)}(\tau )=⟨E^{-}(t)E^{-}(t+\tau )E^{+}(t+\tau )E^{+}(t)⟩/⟨E^{-}{E}^{+}{⟩}^2$. Normal ordering expresses the absorption of two photons. In a single mode,A coherent state has $g^{(2)}(0)=1$. A single-mode thermal field has $g^{(2)}(0)=2$ and bunches, while a number state with $n>0$ gives $1-1/n$; a single-photon state has zero ideal coincidence probability at zero delay. Antibunching, a zero-delay suppression relative to nearby delays, is distinct from simply reporting an intensity variance. Multimode averaging and finite detector response reduce observed contrast.

The Hanbury Brown and Twiss effect is measured by splitting the light onto two detectors and histogramming coincidences versus delay. Background counts, detector timing, dead time and afterpulsing need calibration. Two detectors avoid confusing the single detector's inability to register arbitrarily close counts with real antibunching. First-order interference and second-order correlations contain different information: weak first-order coherence does not imply an absence of intensity bunching.^[ref]

For a second-order response, $P_i^{(2)}({\omega }_3)={ϵ}_0{\sum }_{jk}{\chi }_{ijk}^{(2)}({\omega }_3;{\omega }_1,{\omega }_2)E_j({\omega }_1)E_k({\omega }_2)$, with the appropriate frequency conservation and negative-frequency convention for difference-frequency processes. Interchanging the two input field-frequency pairs gives intrinsic permutation symmetry of nonlinear susceptibility:For second-harmonic generation the input frequencies agree, so the last two Cartesian indices are symmetric and can be contracted into six input-pair labels. This reduces the unrestricted 27 Cartesian entries to at most 18 before crystal symmetries are imposed.

Crystal point-group transformations must leave the material tensor invariant and can force entries to vanish or become related. Under spatial inversion both electric field and polarization reverse, whereas the field product is even. A centrosymmetric bulk therefore has zero electric-dipole ${\chi }^{(2)}$. Surfaces and interfaces can break inversion symmetry, and higher-multipole processes are outside that simple prohibition.

Kleinman symmetry is a stronger approximate permutation symmetry used sufficiently far from resonances when dispersion can be neglected over the participating frequencies. It is not justified merely because two input frequencies are equal; resonant or appreciably dispersive media need the full frequency-dependent relations. These distinctions determine which polarization and propagation geometries have nonzero effective coupling, in addition to whether they satisfy phase matching.^[ref]

Frequency-resolved optical gating characterizes an ultrashort pulse using a nonlinear gate and a measured two-dimensional spectrogram. In second-harmonic frequency-resolved optical gating, split the pulse into two replicas, vary their relative delay, overlap them noncollinearly in a thin second-harmonic crystal, and spectrally resolve the cross-generated beam. Its propagation direction separates it from the individual beams' self-generated harmonics. For complex pulse envelope $E(t)$, the ideal trace iswhere $\Omega$ denotes the output frequency offset in the envelope convention. An iterative phase retrieval algorithm seeks a pulse consistent with both this measured Fourier magnitude and the nonlinear product relation. The delay axis provides timing structure, while the resolved spectrum adds information missing from a plain intensity autocorrelation.^[ref]

The measured trace does not fix absolute optical phase or absolute time origin. SHG FROG also has a conjugate time-reversal ambiguity: $E(t)$ and $E^{\ast }(-t)$ produce the same trace. Extra information, such as a known dispersion change and a second measurement, can resolve that direction-of-time ambiguity. Finite phase-matching bandwidth, group-velocity mismatch, spatial chirp, spectrometer calibration, delay error, noise and instability over the scan must be controlled. A thin gate helps avoid temporal blurring, but trades against signal. FROG therefore reconstructs a pulse under an explicit instrument model rather than reading optical phase directly from a single intensity image.