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Definition: the phase of the electric field in the frequency domain
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 spectral 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
Complete pulse characterization includes measuring not only the intensity spectrum (i.e., the squared modulus of E(ν)), 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 interferometry).
Food for Thought
Can you find out without doing a calculation, what the effect of a weak Kerr nonlinearity on the spectral phase of a sech2-shaped pulse is? As a hint, use the fact that the effects of group delay dispersion and Kerr nonlinearity can cancel each other in a fundamental soliton pulse, apart from a remaining constant phase shift.
Spectral Phase and Group Delay
The group delay can be defined as the derivative of the spectral phase with respect to angular frequency. This means e.g. that the group delay of all spectral components is zero if the spectral phase is constant. If this is not the case, the group delay may be frequency-dependent: different frequency components can then be considered to arrive at different times. However, this kind of interpretation is somewhat problematic, as becomes apparent e.g. when considering that the instantaneous frequency may acquire a certain value more than once within the pulse duration, whereas the group delay for a particular frequency can have only one value. For simple pulse shapes, however, the group delay can be used to describe the position of the pulse maximum of the whole pulse, or of the pulse obtained after a bandpass filter.
Examples
It is instructive to consider the changes of spectral phase associated with certain operations:
- A constant change of temporal phase translates directly into the same change of the spectral phase. (For time-dependent phase changes, the relation is much less obvious.)
- A time delay by T corresponds to a change of spectral phase which is 2π ν T, i.e., proportional to the optical frequency. In cases where the temporal position is not of interest, the spectral phase can be made less arbitrary by temporally shifting the pulse so that the linear term in frequency vanishes.
- chromatic dispersion directly affects the spectral phase. 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 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 to apply 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 achieved pulse duration.
The spectral phase can be useful to understand the phenomenon of spectral interference. For example, consider two identical pulses with a relative time delay T. The difference of spectral phase, which is linear in frequency (see above), causes a spectral modulation. See the article on double pulses 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 e.g. of a first diffraction grating to spatially separate different frequency components, 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 somewhat 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.
Bibliography
| [1] | J. P. Heritage et al., "Picosecond pulse shaping by spectral phase and amplitude manipulation", Opt. Lett. 10 (12), 609 (1985) |
| [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) |
| [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) |
See also: chirp, transform limit, pulse characterization, spectral interferometry, pulse compression, double pulses
