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# Group Velocity Mismatch

Author: the photonics expert

Acronym: GVM

Definition: the phenomenon that pulses become temporally separated in a medium due to different group velocities, or (quantitatively) the difference of inverse group velocities

Units: s/m

When optical pulses with different optical frequencies propagate in a transparent medium (e.g. in an optical crystal), their group velocities will in general be different. As a result of this group velocity mismatch, two pulses which are initially overlapping in time can after some propagation distance no longer overlap.

Similarly, group velocity mismatch can occur for different polarization directions in a birefringent medium.

Quantitatively, the group velocity mismatch is defined as the difference of the inverse group velocities

$${\rm{GVM}} = \frac{1}{{{\upsilon _{{\rm{g1}}}}}} - \frac{1}{{{\upsilon _{{\rm{g2}}}}}} = {\left. {\frac{{\partial k}}{{\partial \omega }}} \right|_1} - {\left. {\frac{{\partial k}}{{\partial \omega }}} \right|_2}$$

and has units of seconds per meter.

## Common Consequences of Group Velocity Mismatch

The phenomenon of group velocity mismatch, also called temporal walk-off, is particularly relevant in the context of nonlinear frequency conversion with ultrashort pulses. Typical values of the GVM for nonlinear crystals and significantly different wavelengths in the visible or near-infrared spectral regions (e.g. 1064 nm and 532 nm) are of the order of 0.1 ps/mm, corresponding to 0.1 ns = 100 ps per meter. This shows that for a crystal length of e.g. 10 mm the group velocity mismatch can have significant effects for the frequency conversion of 5-ps pulses, and very strong effects for femtosecond pulses. For that reason, it is generally necessary to use shorter crystals for converting shorter pulses, which requires higher optical intensities for maintaining a high conversion efficiency. As the applicable intensities can be limited e.g. by optical damage, group velocity mismatch can effectively limit the achievable conversion efficiency for short pulses.

In a frequency-domain picture, the group velocity mismatch limits the width of the spectral range in which phase matching is achieved, also called the phase-matching bandwidth. This is not an additional effect, however, but only the same phenomenon as seen in the frequency domain.

Group velocity mismatch is also important, for example, for Raman amplification of ultrashort pulses in optical fibers. For example, picosecond pulses at 1064 nm may be used for amplifying pulses at 1110 nm. In a large mode area silica fiber, the group velocity mismatch is approximately the same as in bulk silica; for the given wavelengths, it amounts to 1.1 ps/m. This means that e.g. 1-ps pulses substantially lose their temporal overlap within 1 m of fiber, so that the effective Raman gain is reduced.

There are also many situations where the group velocity mismatch can be very useful. In some cases, it can increase the efficiency of nonlinear frequency conversion. In optical parametric oscillators, it can sometimes be exploited for wavelength tuning and for significant pulse shortening. In other cases, it is used for purposes of optical signal processing.

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