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# Spectral Phase

Author: the photonics expert

Definition: the phase of the electric field in the frequency domain

Formula symbol: <$\varphi$>

The electric field of an optical pulse may be described in the time domain or in the frequency domain. In the frequency domain, it can be of interest to know not only the power spectral density (i.e., the intensity spectrum) but also the spectral phase. This is defined as the phase of the electric field in the frequency domain, i.e., the complex phase of the function

$$E(\nu ) = \int\limits_{ - \infty }^{ + \infty } {\exp \left( { - i2\pi \nu t} \right)\;E(t)\;{\rm{d}}t}$$

Complete pulse characterization includes measuring not only the optical spectrum, i.e. the squared modulus of <$E(\nu )$>, but also the spectral phase, which contains additional information. This is possible e.g. with the methods of frequency-resolved optical gating (FROG) and spectral phase interferometry for direct electric-field reconstruction (SPIDER, → spectral phase interferometry).

Note that there are different sign conventions in wave optics; the equation above is for physicists' convention.

## Spectral Phase and Group Delay

The group delay for light in an optical component or setup can be defined as the derivative of the spectral phase delay with respect to angular optical frequency:

$${T_{\rm{g}}} = \frac{{\partial \varphi }}{{\partial \omega }}$$

That can be understood by considering a light pulse, where the peak intensity is found at a time where all spectral components are in phase. After passage through an optical component, leading to frequency-dependent phase changes, that condition is no longer fulfilled at the original time of the pulse peak, but at a later time, where the spectral components again acquire the same phase. That temporal shift of the pulse is determined by the group delay provided that the underlying linear approximation is valid – i.e., possibly not for broadband pulses experiencing more complex changes of spectral phase.

## Examples

It is instructive to consider the changes of spectral phase associated with certain operations:

• A constant change in temporal phase translates directly into the same change in the spectral phase (for time-dependent phase changes, the relation is much less obvious), and to no group delay.
• A time delay by <$T$> corresponds to a change in spectral phase which is <$2\pi \: \nu \: T$>, i.e. proportional to the optical frequency.
• Chromatic dispersion directly affects the spectral phase and also causes a group delay. For example, the effect of third-order dispersion corresponds to adding a term to the spectral phase which varies with the third power of the frequency offset.

When the spectral phase of a pulse is constant or depends linearly on the frequency, the pulse is unchirped, which implies that it is at the transform limit. A chirp in the time domain is associated with a nonlinear frequency dependence of the spectral phase. A dispersive pulse compressor basically has the task of applying spectral phase shifts so that the resulting spectral phase is constant (or changes only linearly with frequency). The deviations from a flat spectral phase are more informative measure of the quality of pulse compression than e.g. just the pulse duration achieved.

The spectral phase can be useful for understanding the phenomenon of spectral interference. For example, consider two identical pulses with a relative time delay <$T$>. The difference in spectral phase, which is linear in frequency (see above), causes a spectral modulation. See the article on spectral phase interferometry for more details.

## Modifying the Spectral Phase

There are pulse shapers which can be used to modify the spectral phase of pulses. Such a setup consists of, e.g., a first diffraction grating to separate different frequency components spatially, a liquid crystal modulator for applying position-dependent phase shifts, and a second diffraction grating to recombine the frequency components.

By properly adjusting all the phase values, it is possible e.g. to obtain transform-limited pulses, being as short as the given spectral width allows, or to form longer pulses with complicated temporal shapes. Conditions for such capabilities are that the full optical bandwidth can be processed, and that the spectral resolution (related to the maximum occurring group delay) is sufficiently high. On the other hand, a fast optical modulator is not required.

## More to Learn

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### Bibliography

 [1] J. P. Heritage et al., “Picosecond pulse shaping by spectral phase and amplitude manipulation”, Opt. Lett. 10 (12), 609 (1985); https://doi.org/10.1364/OL.10.000609 [2] I. A. Walmsley and V. Wong, “Characterization of the electric field of ultrashort optical pulses”, J. Opt. Soc. Am. B 13 (11), 2453 (1996); https://doi.org/10.1364/JOSAB.13.002453 [3] C. Iaconis and I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses”, Opt. Lett. 23 (10), 792 (1998); https://doi.org/10.1364/OL.23.000792