Optical autocorrelation

Optical autocorrelation is the correlation of an optical field, its intensity, or a nonlinear signal with a time-delayed replica of itself. In ultrafast optics, the term most commonly denotes a measurement used to characterize the temporal duration and coherence of an ultrashort pulse. Related correlation functions also describe stationary optical fields and photon-counting statistics, although these quantities differ in their mathematical definitions and physical interpretations.

An autocorrelation trace is invariant under an overall displacement in time. Most forms are also unchanged by temporal reversal, and intensity autocorrelation discards the optical phase entirely. Consequently, an autocorrelation measurement constrains pulse duration and temporal structure without generally determining a unique electric field.

Mathematical formulation

For a finite-energy complex electric-field envelope (E(t)), the first-order field autocorrelation is

[ \Gamma^{(1)}(\tau) = \int_{-\infty}^{\infty} E^*(t)E(t+\tau),dt , ]

where (\tau) is the relative delay and the asterisk denotes complex conjugation. For a statistically stationary field, the corresponding ensemble-averaged expression is

[ \Gamma^{(1)}(\tau) = \left\langle E^*(t)E(t+\tau)\right\rangle . ]

Normalization by (\Gamma^{(1)}(0)) gives the first-order degree of coherence (g^{(1)}(\tau)), which governs the visibility of interference between delayed copies of the field. Its characteristic width defines the field's coherence time.

The Wiener–Khinchin theorem relates the field autocorrelation to the optical power spectrum:

[ \Gamma^{(1)}(\tau) = \frac{1}{2\pi} \int_{-\infty}^{\infty} |\widetilde{E}(\omega)|^2 e^{-i\omega\tau},d\omega . ]

Thus, field autocorrelation and spectral intensity form a Fourier transform pair. Norbert Wiener established the continuous-time harmonic-analysis framework underlying this relation, while Aleksandr Khinchin developed its formulation for stationary stochastic processes. Because the spectrum in this expression contains only (|\widetilde{E}(\omega)|^2), neither quantity determines the spectral phase.

For a pulse intensity

[ I(t)=|E(t)|^2, ]

the second-order intensity autocorrelation is

[ A_I(\tau) = \int_{-\infty}^{\infty} I(t)I(t-\tau),dt . ]

This function is real, nonnegative, and even:

[ A_I(-\tau)=A_I(\tau). ]

Its maximum occurs at zero delay, and its total area equals the square of the pulse energy:

[ \int_{-\infty}^{\infty}A_I(\tau),d\tau

\left( \int_{-\infty}^{\infty}I(t),dt \right)^2 . ]

The even symmetry eliminates any distinction between a pulse and its time-reversed counterpart. More generally, different intensity profiles can possess the same autocorrelation, so inversion of (A_I(\tau)) is not unique.

Experimental realization

Optical intensity autocorrelation usually employs a variable-delay interferometer that divides a pulse into two replicas. After one replica acquires a delay (\tau), the beams overlap in a nonlinear medium. A time-integrating detector records the nonlinear signal as a function of delay.

In a background-free second-harmonic generation autocorrelator, the beams enter a nonlinear crystal along distinct directions. Phase matching spatially separates the cross term from second-harmonic light generated by either beam alone. Under the assumptions of instantaneous nonlinear response, negligible propagation distortion, and complete spatial overlap, the measured signal is proportional to

[ S_{\mathrm{SHG}}(\tau) \propto \int_{-\infty}^{\infty} I(t)I(t-\tau),dt . ]

The delay scan converts a temporal interval shorter than the response time of an ordinary photodetector into a slowly varying trace. Temporal resolution is therefore established by optical overlap and the nonlinear interaction rather than by the electronic bandwidth of the detector.

During the early development of picosecond diagnostics, Erich P. Ippen, Charles V. Shank, and You Watanabe established the response treatment for noncollinear nonlinear autocorrelators. Ippen and Shank related delay-dependent frequency conversion to the underlying pulse envelope, while Watanabe quantified the effects of finite crystal length and group-velocity mismatch on the measured correlation width. This work placed instrumental broadening within the same convolution formalism used for detector response and dispersive propagation.

Other nonlinear processes can produce an intensity-dependent correlation signal. Two-photon absorption generates a delay-dependent response inside a suitable detector, while two-photon fluorescence records the integrated emission following simultaneous absorption. These implementations measure the same ideal intensity autocorrelation only when their nonlinear response is effectively instantaneous across the temporal and spectral extent of the pulse.

Pulse-width interpretation

The width of an intensity autocorrelation exceeds the width of the pulse that produces it. Their ratio depends on the assumed temporal profile because autocorrelation combines every point in the pulse with every delayed point.

For a Gaussian intensity profile with full width at half maximum (\Delta t),

[ I(t) = I_0 \exp\left( -\frac{4\ln 2,t^2}{\Delta t^2} \right), ]

the autocorrelation is also Gaussian and has width

[ \Delta \tau_{\mathrm{AC}} = \sqrt{2},\Delta t . ]

For an intensity profile proportional to (\operatorname{sech}^2(t/T)), the autocorrelation does not retain exactly the same functional form. Its full width at half maximum is approximately (1.543) times the intensity width. These conversion factors express model-dependent inference rather than direct reconstruction of the pulse.

A broad pedestal, a satellite pulse, or a temporally asymmetric envelope can alter the trace without producing a unique corresponding intensity profile. The measured width can also include broadening from group-velocity dispersion, imperfect beam overlap, a nonuniform delay calibration, and finite nonlinear-crystal response. Each contribution modifies the trace through a distinct physical transfer function rather than through a universal correction factor.

Interferometric autocorrelation

A collinear arrangement preserves interference between the two delayed fields. For second-harmonic detection, the interferometric autocorrelation is proportional to

[ S_{\mathrm{IAC}}(\tau)

\int_{-\infty}^{\infty} \left| \left[E(t)+E(t-\tau)\right]^2 \right|^2 dt . ]

The trace contains fringes at the optical carrier frequency, modulated by a broader envelope associated with temporal overlap. For identical coherent pulses in the ideal limit, the zero-delay signal is eight times the signal measured at delays where the pulses no longer overlap. This (8{:}1) ratio follows from coherent field addition and subsequent quadratic frequency conversion.

Interferometric autocorrelation is sensitive to coherence and to certain phase-dependent features that do not appear in an intensity autocorrelation. It nevertheless remains many-to-one: distinct electric fields can produce identical interferometric traces. The presence of carrier fringes therefore does not convert the measurement into a complete determination of temporal amplitude and phase.

Relation to optical coherence

Autocorrelation in deterministic pulse measurement is distinct from the normalized second-order coherence function used in quantum optics:

[ g^{(2)}(\tau)

\frac{ \left\langle {:}I(t)I(t+\tau){:} \right\rangle }{ \langle I(t)\rangle^2 }, ]

where normal ordering is indicated by colons. Roy J. Glauber incorporated such correlation functions into the quantum theory of optical coherence. Robert Hanbury Brown and Richard Q. Twiss demonstrated intensity-correlation measurements in astronomy, establishing the operational connection between second-order coherence and photon statistics.

A pulse autocorrelator integrates a deterministic nonlinear response across laboratory time, whereas a photon-correlation experiment estimates an ensemble average from detection events. The two measurements can contain mathematically similar products of intensities, but their normalization, statistical meaning, and instrumental realizations are different.

For chaotic light, the Siegert relation connects first-order and second-order coherence under Gaussian statistical assumptions:

[ g^{(2)}(\tau)

1+\left|g^{(1)}(\tau)\right|^2 . ]

No corresponding universal relation converts an arbitrary ultrashort-pulse intensity autocorrelation into its complex electric field.

Information content and reconstruction limits

An optical spectrum determines the field autocorrelation through Fourier transformation, while an intensity autocorrelation supplies constraints on the pulse envelope. Neither measurement contains sufficient information to recover an arbitrary spectral phase. Even their combination can retain ambiguities because phase retrieval depends on the structure of the measured nonlinear signal.

Complete pulse-characterization methods introduce an additional independent coordinate or a known reference interaction. Frequency-resolved optical gating records the spectrum of a nonlinear signal as a function of delay, producing a two-dimensional spectrogram. Spectral phase interferometry for direct electric-field reconstruction encodes phase differences between spectrally sheared replicas. These methods extend the information content beyond that of a one-dimensional autocorrelation trace.

Optical autocorrelation consequently occupies a defined role in temporal metrology: it measures correlation width and reveals coarse temporal structure, while its symmetries and loss of phase prevent unique waveform reconstruction.

See also

• Cross-correlation, which compares two distinct fields or signals rather than a waveform with itself.
• Optical coherence, which describes statistical correlations between optical fields at separated positions or times.
• Michelson interferometer, whose divided-path geometry forms the basis of many variable-delay correlators.
• Nonlinear optics, which provides the frequency-conversion and multiphoton processes used for ultrafast temporal gating.
• Pulse compression, which changes pulse duration through control of spectral phase.
• Phase retrieval, which concerns reconstruction from measurements lacking directly observed phase information.
• Photon correlation spectroscopy, which uses temporal intensity fluctuations to characterize dynamical processes.