Categories: Fiber optics, Nonlinear optics, Optics, Physics.

Optical wave breaking

In fiber optics, optical wave breaking (OWB) is a nonlinear effect caused by interaction between group velocity dispersion (GVD) and self-phase modulation (SPM). It only happens in the normal dispersion regime (β2>0\beta_2 > 0) for pulses meeting a certain criterium, as we will see.

SPM creates low frequencies at the front of the pulse, and high ones at the back, and if β2>0\beta_2 > 0, GVD lets low frequencies travel faster than high ones. When those effects interact, the pulse gets temporally stretched in a surprisingly sophisticated way.

To illustrate this, the instantaneous frequency ωi(z,t)=ϕ/t\omega_i(z, t) = -\ipdv{\phi}{t} has been plotted below for a theoretical Gaussian input pulse experiencing OWB, with settings T0=100fsT_0 = 100\:\mathrm{fs}, P0=5kWP_0 = 5\:\mathrm{kW}, β2=2ps2/m\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m} and γ=0.1/W/m\gamma = 0.1/\mathrm{W}/\mathrm{m}.

In the left panel, we see the typical S-shape caused by SPM, and the arrows indicate the direction that GVD is pushing the curve in. This leads to steepening at the edges, i.e. the S gradually turns into a Z. Shortly before the slope would become infinite, small waves start “falling off” the edge of the pulse, hence the name wave breaking:

Instantaneous frequency profile evolution

Several interesting things happen around this moment. To demonstrate this, spectrograms of the same simulation have been plotted below, together with pulse profiles in both the tt-domain and ω\omega-domain on an arbitrary linear scale (click the image to get a better look).

Initially, the spectrum broadens due to SPM in the usual way, but shortly after OWB, this process is stopped by the appearance of so-called sidelobes in the ω\omega-domain on either side of the pulse. In the meantime, in the time domain, the pulse steepens at the edges, but flattens at the peak. After OWB, a train of small waves falls off the edges, which eventually melt together, leading to a trapezoid shape in the tt-domain. Dispersive broadening then continues normally:

Spectrograms of pulse shape evolution

We call the distance at which the wave breaks LWBL_\mathrm{WB}, and want to predict it analytically. We do this using the instantaneous frequency ωi\omega_i, by estimating when the SPM fluctuations overtake their own base, as was illustrated earlier.

To get ωi\omega_i of a Gaussian pulse experiencing both GVD and SPM, it is a reasonable approximation, for small zz, to simply add up the instantaneous frequencies for these separate effects:

ωi(z,t)ωGVD(z,t)+ωSPM(z,t)=tzT02(β2/T021+β22z2/T04+2γP0exp ⁣( ⁣ ⁣t2T02))\begin{aligned} \omega_i(z,t) &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t) = \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) \end{aligned}

Assuming that zz is small enough such that z20z^2 \approx 0, this expression can be reduced to:

ωi(z,t)β2tzT04(1+2γP0T02β2exp ⁣( ⁣ ⁣t2T02))=β2tzT04(1+2Nsol2exp ⁣( ⁣ ⁣t2T02))\begin{aligned} \omega_i(z,t) \approx \frac{\beta_2 tz}{T_0^4} \bigg( 1 + 2\frac{\gamma P_0 T_0^2}{\beta_2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) = \frac{\beta_2 t z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) \end{aligned}

Where we have assumed β2>0\beta_2 > 0, and NsolN_\mathrm{sol} is the soliton number, which is defined as:

Nsol2LDLN=γP0T02β2\begin{aligned} N_\mathrm{sol}^2 \equiv \frac{L_D}{L_N} = \frac{\gamma P_0 T_0^2}{|\beta_2|} \end{aligned}

This quantity is very important in anomalous dispersion, but even in normal dispersion, it is still a useful measure of the relative strengths of GVD and SPM. As was illustrated earlier, ωi\omega_i overtakes itself at the edges, so OWB occurs when ωi\omega_i oscillates there, which starts when its tt-derivative, the instantaneous chirpyness ξi\xi_i, has two real roots for t2t^2:

0=ξi(z,t)=ωit=β2zT04(1+2Nsol2(12t2T02)exp ⁣( ⁣ ⁣t2T02))β2zT04f(t2T02)\begin{aligned} 0 = \xi_i(z,t) = \pdv{\omega_i}{t} &= \frac{\beta_2 z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg) \equiv \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big) \end{aligned}

Where the function f(x)f(x) has been defined for convenience. As it turns out, this equation can be solved analytically using the Lambert WW function, leading to the following exact minimum value Nmin2N_\mathrm{min}^2 for Nsol2N_\mathrm{sol}^2, such that OWB can only occur when Nsol2>Nmin2N_\mathrm{sol}^2 > N_\mathrm{min}^2:

Nmin2=14exp ⁣(32)1.12\begin{aligned} \boxed{ N_\mathrm{min}^2 = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big) \approx 1.12 } \end{aligned}

If this condition Nsol2>Nmin2N_\mathrm{sol}^2 > N_\mathrm{min}^2 is not satisfied, ξi\xi_i cannot have two roots for t2t^2, meaning ωi\omega_i cannot overtake itself. GVD is unable to keep up with SPM, so OWB will not occur.

Next, consider two points at t1t_1 and t2t_2 in the pulse, separated by a small initial interval (t2t1)(t_2 - t_1). The frequency difference between these points due to ωi\omega_i will cause them to displace relative to each other after a short distance zz by some amount Δt\Delta t, estimated by:

ΔtzΔβ1Δβ1β1(ωi(z,t2))β1(ωi(z,t1))zβ2ΔωiΔωiωi(z,t2)ωi(z,t1)zβ2Δξi(t2t1)Δξiξi(z,t2)ξi(z,t1)\begin{aligned} \Delta t &\approx z \Delta\beta_1 \qquad &&\Delta\beta_1 \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1)) \\ &\approx z \beta_2 \Delta\omega_i \qquad &&\Delta\omega_i \equiv \omega_i(z,t_2) - \omega_i(z,t_1) \\ &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1) \qquad \quad &&\Delta\xi_i \equiv \xi_i(z,t_2) - \xi_i(z,t_1) \end{aligned}

Where β1(ω)\beta_1(\omega) is the inverse of the group velocity. For a certain choice of t1t_1 and t2t_2, OWB occurs when they catch up to each other, which is when Δt=(t2t1)-\Delta t = (t_2 - t_1). The distance LWBL_\mathrm{WB} at which this happens first must satisfy the following condition for some value of tt:

LWBβ2ξi(LWB,t)=1    LWB2=T04β22f(t2/T02)\begin{aligned} L_\mathrm{WB} \: \beta_2 \: \xi_i(L_\mathrm{WB}, t) = -1 \qquad \implies \qquad L_\mathrm{WB}^2 = - \frac{T_0^4}{\beta_2^2 \: f(t^2/T_0^2)} \end{aligned}

The time tt of OWB must be where ωi(t)\omega_i(t) has its steepest slope, which is at the minimum value of ξi(t)\xi_i(t), and by extension f(x)f(x). This turns out to be f(3/2)f(3/2):

fmin=f(3/2)=14Nsol2exp(3/2)=1Nsol2/Nmin2\begin{aligned} f_\mathrm{min} = f(3/2) = 1 - 4 N_\mathrm{sol}^2 \exp(-3/2) = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2 \end{aligned}

Clearly, fmin0f_\mathrm{min} \ge 0 when Nsol2Nmin2N_\mathrm{sol}^2 \le N_\mathrm{min}^2, which, when inserted above, leads to an imaginary LWBL_\mathrm{WB}, confirming that OWB cannot occur in that case. Otherwise, if Nsol2>Nmin2N_\mathrm{sol}^2 > N_\mathrm{min}^2, then:

LWB=T02β2fmin=LDNsol2/Nmin21\begin{aligned} \boxed{ L_\mathrm{WB} = \frac{T_0^2}{\beta_2 \, \sqrt{- f_\mathrm{min}}} = \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}} } \end{aligned}

This prediction for LWBL_\mathrm{WB} appears to agree well with the OWB observed in the simulation:

Optical wave breaking simulation results

Because all spectral broadening up to LWBL_\mathrm{WB} is caused by SPM, whose ω\omega-domain behaviour is known, it is in fact possible to draw some analytical conclusions about the achieved bandwidth when OWB sets in. Filling LWBL_\mathrm{WB} in into ωSPM\omega_\mathrm{SPM} gives:

ωSPM(LWB,t)=2γP0tβ24Nsol2exp(3/2)1exp ⁣( ⁣ ⁣t2T02)\begin{aligned} \omega_{\mathrm{SPM}}(L_\mathrm{WB},t) = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \end{aligned}

Assuming that Nsol2N_\mathrm{sol}^2 is large in the denominator, this can be approximately reduced to:

ωSPM(LWB,t)2γP0tβ2Nsolexp ⁣( ⁣ ⁣t2T02)=2γP0β2tT0exp ⁣( ⁣ ⁣t2T02)\begin{aligned} \omega_\mathrm{SPM}(L_\mathrm{WB}, t) \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \end{aligned}

The expression xexp(x2)x \exp(-x^2) has its global extrema ±1/2e\pm 1 / \sqrt{2 e} at x2=1/2x^2 = 1/2. The maximum SPM frequency shift achieved at LWBL_\mathrm{WB} is therefore given by:

ωmax=2γP0eβ2\begin{aligned} \omega_\mathrm{max} = \sqrt{\frac{2 \gamma P_0}{e \beta_2}} \end{aligned}

Interestingly, this expression does not contain T0T_0 at all, so the achieved spectrum when SPM is halted by OWB is independent of the pulse width, for sufficiently large NsolN_\mathrm{sol}.


  1. D. Anderson, M. Desaix, M. Lisak, M.L. Quiroga-Teixeiro, Wave breaking in nonlinear-optical fibers, 1992, Optical Society of America.
  2. A.M. Heidt, A. Hartung, H. Bartelt, Generation of ultrashort and coherent supercontinuum light pulses in all-normal dispersion fibers, 2016, Springer Media.