Self-phase modulation (SPM)

Self-phase modulation is the phase shift that an optical pulse acquires from its own intensity as it travels through a medium with a Kerr nonlinearity. The peak of the pulse sees a slightly higher refractive index than its wings, so the phase varies across the pulse, the pulse becomes chirped, and its spectrum broadens while the temporal envelope, in the absence of dispersion, stays the same. In standard single-mode fiber at 1550 nm the nonlinear coefficient is $\gamma \approx 1.3$ W⁻¹km⁻¹: 1 W over 1 km, or 10 W over 100 m, gives a nonlinear phase of about 1.3 rad.

In a Kerr medium the index depends on intensity as $n=n_0+n_{2}I$, and the resulting phase is

$${\varphi }_{\text{SPM}}(t)=\frac{2\pi }{{\lambda }_0}{n}_{2}I(t)L_{\text{eff}}$$

In a waveguide it is more convenient to write this with power and the nonlinear coefficient $\gamma =2\pi n_2\mathrm{/}(\lambda A_{\text{eff}})$:

$${\varphi }_{\text{SPM}}(t)=\gamma P(t)L_{\text{eff}}$$

$L_{\text{eff}}$ is the effective interaction length, reduced from the geometric length by attenuation:

$$L_{\text{eff}}=\frac{1-e^{-\alpha L}}{\alpha }$$

Chirp and spectral broadening

The time-dependent phase produces an instantaneous frequency shift:

$$\mathrm{\Delta }\omega (t)=-\frac{d{\varphi }_{\text{SPM}}}{dt}\propto -\frac{dI}{dt}$$

For positive $n_2$, the leading edge ($dI\mathrm{/}dt>0$) is downshifted in frequency and the trailing edge ($dI\mathrm{/}dt<0$) is upshifted. Near the pulse center the chirp is approximately linear and positive, which is what pulse compressors exploit.

For a Gaussian pulse the maximum frequency excursion is about $0.86\mathrm{\Delta }{\omega }_0{\varphi }_{\text{max}}$, where $\mathrm{\Delta }{\omega }_0$ is the initial spectral half-width at $1\mathrm{/}e$ intensity and ${\varphi }_{\text{max}}$ is the peak nonlinear phase. Because each frequency is generated at two instants on opposite sides of the peak, the two contributions interfere and the spectrum develops an oscillatory structure. The number of peaks is approximately

$$M\approx \frac{{\varphi }_{\text{max}}}{\pi }+\frac{1}{2}$$

so ${\varphi }_{\text{max}}=2.5\pi$ (7.9 rad) gives three peaks. Counting peaks, or fitting the broadened spectrum on an optical spectrum analyzer, is the standard laboratory measurement of SPM, and with a known peak power and effective area it yields $\gamma$ and $n_2$.

Interaction with dispersion

Combined with chromatic dispersion, SPM behaves differently depending on the sign of $D$:

The balance is quantified by two length scales: the nonlinear length $L_{\text{NL}}=1\mathrm{/}(\gamma P_0)$ and the dispersion length $L_D=T_0^2\mathrm{/}\mathrm{\mid }{\beta }_2\mathrm{\mid }$. For SMF at 1550 nm, $D=17$ ps/(nm·km) corresponds to ${\beta }_2\approx -21.7$ ps²/km. A pulse with $T_0=1$ ps has $L_D\approx 46$ m, and a fundamental soliton ($L_D=L_{\text{NL}}$) needs a peak power of $\mathrm{\mid }{\beta }_2\mathrm{\mid }\mathrm{/}(\gamma T_0^2)\approx 17$ W. When $L_{\text{NL}}\ll L_D$, SPM dominates; when $L_{\text{NL}}\gg L_D$, group-velocity dispersion dominates.

In optical communication

Typical parameters:

Worked example. For 10 mW (+10 dBm) per channel over 80 km at 0.2 dB/km: $\alpha =0.046$ km⁻¹, $L_{\text{eff}}\approx 21$ km, and $\varphi =\gamma P{L}_{\text{eff}}\approx 0.3$ rad per span. Because the amplifiers restore the power at every span, the phase adds across spans, so a ten-span link accumulates about 3 rad before dispersion is considered.

SPM is one of the main nonlinear impairments in long-haul fiber communication, alongside cross-phase modulation in WDM systems. It can be partially compensated digitally using nonlinear backpropagation in coherent receivers, which is exact only in the absence of noise and with full knowledge of the link. The practical approach is to operate near an optimum launch power, below which amplifier noise dominates and above which nonlinear distortion does.

Other uses

• Supercontinuum generation: high-peak-power pulses in fiber produce very wide spectra, with SPM as the first stage before soliton fission and four-wave mixing
• Pulse compression: SPM in normal-dispersion fiber adds a near-linear chirp that a grating or prism pair removes, shortening the pulse
• Spectral compression: a negatively chirped pulse narrows spectrally under SPM
• Nonlinear measurement: SPM-based measurements infer $n_2$ and effective area
• Integrated waveguides: in silicon, two-photon absorption and free carriers limit the achievable SPM at 1550 nm; silicon nitride does not have this limit

Pitfalls

The nonlinear phase follows peak power; a 1 mW average train with 1% duty cycle has 100 mW peak. Evaluate $A_{\text{eff}}$ at the operating wavelength, since it grows with wavelength in step-index fiber. In amplified links, $L_{\text{eff}}$ applies per span, and each span must be counted.

Common questions

What is the difference between SPM and XPM?

SPM is the phase a wave imposes on itself; cross-phase modulation is the phase one wave imposes on another. For co-polarized waves XPM is twice as strong per unit intensity, but in WDM systems it is reduced by walk-off between channels.

Does SPM change the pulse duration?

SPM alone changes only the phase and spectrum. Any change in duration comes from dispersion acting on the new spectral content.

How much spectral broadening does SPM produce?

For a Gaussian pulse the spectral width grows by roughly $0.86{\varphi }_{\text{max}}$ times its initial value once ${\varphi }_{\text{max}}$ exceeds a few radians; at ${\varphi }_{\text{max}}=1$ rad the broadening is modest.

References: Agrawal, Nonlinear Fiber Optics 6th ed. 2019; Boyd, Nonlinear Optics 4th ed. 2020; Saleh & Teich, Fundamentals of Photonics 3rd ed. 2019.