Group Delay

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

The group delay (($T_\textrm{g}$)) of an optical element (e.g. a dielectric mirror or a piece of optical fiber) is essentially the time delay experienced by a short (but not too short and not too intense) light pulse for propagation through that element. That time delay generally depends on the optical frequency or wavelength. More precisely, the group delay is defined as the derivative of the change in spectral phase with respect to the angular optical frequency:

The group delay has the units of a time (e.g. picoseconds) and generally (in dispersive media) depends on the optical frequency (→ group delay dispersion, chromatic dispersion) and possibly on the polarization state (→ polarization mode dispersion) and the optical mode (→ intermodal dispersion) in the case of a waveguide.

As a simple example, for propagation over a distance ($d$) in vacuum we have ($\varphi = 2\pi \: d / \lambda = \omega \: d / c$), so that the resulting group delay is ($d / c$). In this case, it is independent of the optical frequency. In more complicated cases, e.g. with resonant structures, the phase delay has a more complicated dependency on ($\omega$), but one can calculate with analytical or numerical means the frequency derivative to obtain the group delay, which is then in general frequency-dependent, which implies that we have chromatic dispersion.

For linear propagation of a narrow-band optical pulse with a simple temporal and spectral shape, the group delay is the time delay which the pulse maximum (i.e., the maximum of the temporal intensity profile) experiences when propagating through the optical element. For broadband optical pulses, and particularly in situations where nonlinearities affect the propagation, the situation can be more sophisticated, leading to a time delay which can deviate from the group delay. Inappropriate interpretations of the group delay can then cause confusion. There are even situations where the temporal pulse shape undergoes a complicated evolution, e.g. with multiple intensity peaks which can vary in strength such that the global pulse maximum can suddenly shift its temporal location.

The group velocity of light in a medium is the inverse of the group delay per unit length. For a piece of optical material with a certain length, the group delay is the length divided by the group velocity. Note that for common transparent optical materials (also for solid-state laser crystals and optical fibers), the group velocity can significantly deviate from the phase velocity. For example, one meter of fused silica bulk material causes a group delay of 4.879 ns at 1550 nm, whereas from the phase velocity one would (wrongly) calculate 4.817 ns. For a shorter wavelength like 400 nm, this discrepancy is larger: the group delay is 5.049 ns instead of 4.878 ns. In optical fibers, the group delay is further modified by dopants of the fiber core and by the effect of waveguide dispersion.

For an optical resonator, the group delay (and not the phase delay) for one round trip determines the resonator mode spacing, also called the free spectral range.

Measurement of Group Delay

The group delay of an optical element can be measured in various ways. A conceptually most direct method is based on measuring the arrival times of ultrashort pulses e.g. with fast photodetectors. Similarly, one can measure a phase change of intensity-modulated light, with its optical power detected before and after the optical element (phase shift method).

There are more sophisticated and far more powerful interferometric methods, e.g. based on white light interferometry, which allow the measurement of wavelength-resolved spectral phase changes and the group delay with a precision of a few femtoseconds.

Differential Group Delay

In some situations, a quantity of primary interest is a differential group delay, i.e., a difference of two different group delays. For example, in a birefringent optical medium there is generally a difference in group delay between two polarization directions. That can be a problem for nonlinear frequency conversion with ultrashort pulses in nonlinear crystals, where the differential group delay leads to a temporal walk-off of the interacting pulses. See also the article on polarization mode dispersion, which is also observed even in nominally not birefringent fibers. In some cases, a differential group delay is compensated by inserting a piece of birefringent material which provides the opposite differential group delay.

Another example is a differential group delay for two pulses or for two optical telecom signals with different central wavelengths propagating through an optical fiber. That kind of differential group delay arises from the group velocity dispersion of the fiber.

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