Frequency chirp

Frequency chirp is a variation of the instantaneous optical frequency of a signal over the duration of a pulse or symbol. A chirped pulse has its leading and trailing edges at different frequencies, and so a wider spectrum than its duration alone requires. In optical communications chirp is usually an unwanted by-product of modulation, most strongly of direct current modulation of a semiconductor laser; it matters because fiber chromatic dispersion sends different frequencies at different speeds, so the chirp decides whether dispersion spreads the pulse faster or, for a while, compresses it. Deliberate chirp is also a tool: FMCW lidar sweeps the frequency on purpose, and ultrafast lasers stretch and recompress pulses by chirping them.

The chirp parameter

The standard model is a Gaussian pulse whose phase is quadratic in time,

$$A(0,T)=A_0\exp \text{\hspace{0.17em}⁣}[-\frac{(1+iC)T^2}{2T_0^2}]$$

with $T$ the time in a frame moving with the pulse, $T_0$ the half-width at $1\mathrm{/}e$ intensity ($T_{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}}\approx 1.665T_0$), and $C$ the chirp parameter. The instantaneous frequency offset from the carrier is then linear across the pulse,

$$\delta \omega (T)=\frac{C}{T_0^2}T$$

so in this convention $C>0$ means the frequency rises from the leading edge to the trailing edge and $C<0$ means it falls. The spectrum is wider than that of an unchirped pulse of the same duration by the factor $\sqrt{1+C^2}$: 4.1 for $\mathrm{\mid }C\mathrm{\mid }=4$ and 6.1 for $\mathrm{\mid }C\mathrm{\mid }=6$. For a 50 ps FWHM pulse ($T_0=30.0$ ps) with $C=-4$ the frequency one half-width either side of the center is offset by $\pm 21.2$ GHz.

Chirp and dispersion

In a fiber with GVD parameter ${\beta }_2$, the width $T_1$ after a distance $z$ follows from

$$\frac{T_1^2}{T_0^2}={(1+C\xi )}^2+{\xi }^2$$ $$\xi =\frac{{\beta }_{2}z}{T_0^2}$$

as derived in the group velocity dispersion entry. The sign of $C{\beta }_2$ decides the outcome. When $C{\beta }_2>0$ the edges of the pulse are already at the frequencies that dispersion will move outwards, and the pulse broadens faster than an unchirped one. When $C{\beta }_2<0$ the fiber first undoes the chirp: the pulse compresses to a minimum width $T_0\mathrm{/}\sqrt{1+C^2}$ at the distance $z=L_D\mathrm{\mid }C\mathrm{\mid }\mathrm{/}(1+C^2)$, where $L_D=T_0^2\mathrm{/}\mathrm{\mid }{\beta }_2\mathrm{\mid }$, and then broadens.

For a 50 ps FWHM pulse in G.652 fiber at 1550 nm (${\beta }_2=-22.27$ ps²/km from the ITU-T formula with ${\lambda }_0=1310$ nm and $S_0=0.092$), the dispersion length is 40.5 km. After 20 km an unchirped pulse has broadened by a factor of 1.115; with $C=-4$ the factor is 3.02, and with $C=-6$ it is 3.99. With $C=+2$ the same pulse compresses to 22.4 ps FWHM at 16.2 km.

Chirp in directly modulated lasers

Changing the drive current of a semiconductor laser changes its carrier density, which changes both the gain and the refractive index of the active region; the ratio of the two effects is the linewidth enhancement factor $\alpha$, typically 2 to 5 in multiple-quantum-well telecom lasers. The index change shifts the lasing frequency, and in the rate-equation model the frequency deviation follows the output power $P(t)$ as

$$\mathrm{\Delta }\nu (t)=\frac{\alpha }{4\pi }[\frac{d}{dt}\ln P(t)+\kappa P(t)]$$

The first term is the transient chirp, present only while the power changes and ringing at the relaxation-oscillation frequency after a fast edge; the second is the adiabatic chirp, a steady frequency offset between the high and low power levels set by the gain-compression constant $\kappa$. On a rising edge the transient term pushes the frequency up and on a falling edge it pulls it down, so a directly modulated pulse starts blue and ends red: $C<0$ in the convention above, with $\mathrm{\mid }C\mathrm{\mid }$ of the order of $\alpha$.

In the C-band, where standard fiber has ${\beta }_2<0$, this gives $C{\beta }_2>0$, the fast-broadening case, and the blue leading edge runs ahead while the red trailing edge falls behind. This interaction, more than loss, limited 1550 nm directly modulated links at 2.5 and 10 Gb/s to short uncompensated reach and moved longer links to external modulation. A push-pull Mach-Zehnder modulator can be driven with no chirp, or with a chosen sign by unbalancing the arms; an electro-absorption modulator has a small chirp that depends on bias. In the O-band below the zero-dispersion wavelength, ${\beta }_2>0$ and the same negative laser chirp gives $C{\beta }_2<0$; together with the small dispersion there, a few ps/(nm·km), this keeps chirp a minor impairment for O-band directly modulated lasers. Chirp-managed lasers and dispersion-supported transmission turned the adiabatic term to advantage for a time by pairing it with an optical filter, and in coherent systems chirp from the transmitter is one more linear distortion for the DSP to equalize. In nonlinear fiber, self-phase modulation itself imposes a chirp with $C>0$ near the pulse center, which in anomalous dispersion is what allows an optical soliton to hold its shape.

The Dispersion and Pulse Broadening Calculator takes the initial chirp $C$ of a Gaussian pulse as an input and shows the output pulse, the width versus distance with and without chirp, and, when $C{\beta }_2<0$, the distance of minimum width and the width there.

References: G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Academic Press, 2019), chapter 3. G. P. Agrawal, Fiber-Optic Communication Systems, 5th ed. (Wiley, 2021), chapter 2. T. L. Koch and R. A. Linke, "Effect of nonlinear gain reduction on semiconductor laser wavelength chirping," Applied Physics Letters 48, 613 (1986). L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), chapter 5.