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Self-phase Modulation

Due to the Kerr effect, high optical intensity in a medium (e.g. an optical fiber) causes a nonlinear phase delay of the propagating light. That phase delay which has the same temporal shape as the optical intensity. In simple models, the effect is described as a nonlinear change in the refractive index:

with the nonlinear index <$n_2$> and the optical intensity <$I$>. In the context of self-phase modulation, the emphasis is on the temporal dependence of the phase shift, whereas the transverse dependence for some beam profile leads to the phenomenon of self-focusing.

Of course, the nonlinear phase delay comes in addition to the linear phase delay.

Note that the description of self-phase modulation with a time-dependent refractive index is somewhat simplified, and accurate only for not too short pulses. In more extreme situations, extended models are required, which also take into account the effect of self-steepening – a change of the temporal pulse shape even in the absence of chromatic dispersion. That may be interpreted as a nonlinear modification of the group velocity. There can also be higher-order nonlinear effects, leading to more complicated propagation phenomena.

Effects on Optical Pulses

Time-dependent Phase Shift

If an optical pulse is transmitted through a medium, the Kerr effect causes a time-dependent phase shift according to the time-dependent pulse intensity. In this way, an initial unchirped optical pulse acquires a so-called chirp, i.e., a temporally varying instantaneous frequency.

The central part of the pulse exhibits an up-chirp.

For a Gaussian beam with beam radius <$w$> in a homogeneous medium with length <$L$>, the on-axis phase change per unit optical power is described by the proportionality constant

(In some cases, it may be more convenient to omit the factor <$L$>, obtaining the phase change per unit optical power and unit length.)

Note that 2 times smaller coefficients sometimes occur in the literature, which can have two different reasons:

• An incorrect (2 times too low) equation for the peak intensity of a Gaussian beam is sometimes used. With that, one underestimates the on-axis phase change.
• One may calculate the overall nonlinear phase shift of a Gaussian-shaped waveguide mode, where one gets a kind of averaging between higher- and lower-intensity parts. That reduces the nonlinear phase shift by a factor of 2.

Spectral Changes

The time-dependent phase change caused by SPM is associated with a modification of the optical spectrum:

• If the pulse is initially unchirped or up-chirped, SPM leads to spectral broadening (an increase in optical bandwidth).
• Spectral compression can result if the initial pulse is downchirped (always assuming a positive nonlinear index).

For strong SPM, the optical spectrum can exhibit strong oscillations (see Figure 2). The reason for the oscillatory character is essentially that the instantaneous frequency undergoes strong excursions, so that in general there are contributions from two different times to the Fourier integral for a given frequency component. Depending on the exact frequency, these contributions may constructively add up or cancel each other (see also Ref. [7]).

The simulation has been done with the software RP Fiber Power.

Spatial Effects

For a propagating laser beam, the optical intensity is not only time-dependent, but also spatially dependent. The nonlinear phase changes are then also spatially dependent, and that leads to nonlinear self-focusing.

Self-phase Modulation in Optical Fibers

In optical fibers, SPM can be the dominant effect on an ultrashort pulse if the peak power is high (leading to strong SPM) while the chromatic dispersion is weak, so that the pulse duration remains approximately constant. Figure 3 shows an example case where that assumption is well fulfilled within the first 30 mm of fiber; here, the overall spectral width rises about linearly with the propagation distance. Thereafter, it grows faster because anomalous dispersion leads to pulse compression and thus to an increased peak power and an enhanced nonlinear interaction.

In optical fibers with anomalous chromatic dispersion, the chirp from self-phase modulation may be compensated by dispersion; this can lead to the formation of solitons. In the case of fundamental solitons in a lossless fiber, the spectral width of the pulses stays constant during propagation, despite the SPM effect.

In optical fibers with normal dispersion, a modulational instability can occur. That can also contribute to pulse break-up in supercontinuum generation.

Case Studies

Case Study: Nonlinear Pulse Compression in a Fiber

We explore how we can spectrally broaden light pulses by self-phase modulation in a fiber and subsequently compress the pulses using a dispersive element. A substantial reduction in pulse duration by more than an order of magnitude is easily achieved, while the pulse quality is often not ideal.

Self-phase Modulation in Semiconductors via Carrier Density Changes

The term self-phase modulation is occasionally used outside the context of the Kerr effect, when other effects cause intensity-dependent phase changes. In particular, this is the case in semiconductor lasers and semiconductor optical amplifiers, where a high signal intensity can reduce the carrier densities, which in turn lead to a modification of the refractive index and thus the phase change per unit length during propagation. Comparing that effect with SPM via the Kerr effect, there is an important difference: such carrier-related phase changes do not simply follow the temporal intensity profile because the carrier densities do not instantly adjust to modified intensities. That aspect is pronounced for pulse durations below the relaxation time of the carriers, which is typically in the range of picoseconds to a few nanoseconds.

Self-phase Modulation in Mode-locked Lasers

Self-phase modulation has important effects in mode-locked femtosecond lasers. It results mainly from the Kerr nonlinearity of the gain medium, although for very long laser resonators even the Kerr nonlinearity of air can be relevant [5]. Without chromatic dispersion, the nonlinear phase shifts can be so strong that stable operation is no longer possible: the laser would not reach a steady state with well-defined pulses. In that case, (quasi-)soliton mode locking [4] is a good solution, where a balance of self-phase modulation and dispersion is utilized, similar to the situation of solitons in fibers.

Self-phase Modulation via Cascaded Nonlinearities

Strong self-phase modulation can also arise from cascaded <$\chi^{(2)}$> nonlinearities. Basically this means that a not phase-matched nonlinear interaction leads to frequency doubling, but with subsequent backconversion. In effect, there is little power conversion to other wavelengths, but the phase changes on the original wave can be substantial. This effect may also be used to compensate self-phase modulation from other origins [6].

More to Learn

Case studies:

• Case Study: Nonlinear Pulse Compression in a Fiber

Encyclopedia articles:

• Kerr effect
• Kerr lens
• solitons
• cross-phase modulation
• self-steepening

Blog articles:

• The Photonics Spotlight 2015-07-01