Sources, Modulators, and Detectors For Fiber-Optic Communication Systems Part 3

Relaxation Oscillations

An important characteristic of the output of any rapidly switched laser (not just semiconductor lasers) is the relaxation oscillations that can be observed in Fig. 6. These overshoots occur as the photon dynamics and carrier dynamics are coming into equilibrium. Such oscillations are characteristic of the nonlinear coupled laser rate equations and can be found by simple perturbation theory. These relaxation oscillations have a radian frequency QR given, to first order, by9:

where I is the current,is the current at threshold,is the photon lifetime in the cavity, given by

and

FIGURE 6 Experimental example of turn-on delay and relaxation oscillations in a laser diode when the operating current is suddenly switched from 6 mA below the threshold current of 177 mA to varying levels above threshold (from 2 to 10 mA). The GaAs laser diode was 50 |im long, with a SiO2 defined stripe 20 |im wide. Light output and current pulse are shown for each case.8

where aL is from Eq. (4). The factor % is the ratio of the unpumped absorption loss to the cavity loss. For semiconductor lasers, % is on the order of 1 to 3. It can also be shown that where Itr is the current at transparency.

When x ~ 1, at 1.5 times threshold current, wherethe time between successive relaxation oscillation maxima is approximately the geometric mean of the carrier and photon lifetimes:Typical numbers for semiconductor lasers are so at 1.5 times threshold current, the relaxation oscillation frequency isand the time between the relaxation oscillation peaks is 1 ns.

The decay rate of these relaxation oscillations yR is given by

and is roughly 2 ns at twice threshold for typical heterostructure lasers. At 1.5 times threshold, whenThe relaxation oscillations will last approximately as long as the spontaneous emission lifetime of the carriers.

This analysis employs several assumptions which do not seriously affect the relaxation oscillation frequency, but which will overestimate the time that relaxation oscillations will last. The analysis ignores gain saturation, which reduces gain with increased photon density P and is important at high optical powers. It also ignores the rate of spontaneous emission in the cavity Rsp, which is important at small optical powers. Finally, it ignores the impact of changing carrier density on spontaneous emission lifetime. A more exact formulation10 includes these effects:

where gT is the modal gain per unit time. This more exact analysis increases the rate of decay, since the sign ofis negative and alsoA more typical experimental decay rate for lasers atwavelength is

The number of relaxation oscillations (before they die out) in an LD at 1.5 times threshold is proportional toThe longer the carrier lifetime, the more relaxation oscillations will occur (because the carriers do not decay rapidly to steady state). Shorter carrier lifetimes also mean shorter turn-on times. Thus, achieving short carrier lifetimes by high carrier densities is important for high-speed semiconductor lasers. This can be achieved by using as small an active region as possible (such as quantum wells) and by reducing the reflectivity of the laser facets to raise the threshold carrier density.

The relaxation oscillations disappear if the current is just at threshold. However, we’ve also seen that under this situation the turn-on time becomes very long. It is more advantageous to turn the laser on fast, suffering the relaxation oscillations and using a laser designed to achieve a high decay rate, which means using the laser with the highest possible relaxation oscillation frequency.

Other useful forms for the relaxation oscillations are:

whereThese expressions can be found by inserting the following equations into Eq. (24):

where

is the time it takes light to bleed out the mirror and

so that

Note that the relaxation oscillation frequency increases as the photon density increases, showing that smaller laser dimensions are better.

Relaxation oscillations can be avoided by biasing the laser just below threshold, communication systems often operate with a prebiased laser. In digital and high-speed analog systems, relaxation oscillations may limit speed and performance.

Modulation Response and Gain Saturation

The modulation response describes the amplitude of the modulated optical output as a function of frequency under small-signal current modulation. There is a resonance in the modulation response at the relaxation oscillation frequency, as indicated by the experimental data in Fig. 7. It is more difficult to modulate the laser, above the relaxation oscillation frequency. Carrying out a small-signal expansion of the rate equations around photon density P, the modulation response (in terms of current density J) is12:

where is the fraction of spontaneous emission that radiates into the mode and, as before, Te is the spontaneous carrier lifetime, and

FIGURE 7 Measured small-signal modulation response of a high speed DFB laser at several bias levels. Zero-dB modulation response is defined in terms of the low-frequency modulation response, Eq. (32).11

This modulation response has the form of a second-order low-pass filter. Resonance occurs when (from Eq. (29), with negligible internal loss); that is, at the relaxation oscillation frequency.

The modulation response at a frequency well below the relaxation oscillation frequency can be expressed as the change in optical poweras a function of current I using the limit of Eq. (31) whenFromand relating output power to photon density throughthe low frequency modulation response becomes

which is expected from Eq. (8) when

The 3-dB modulation radian frequency bandwidthcan be expressed in terms of the relaxation oscillation parameters by13:

where the oscillation frequencyand damping rateare as previously described. The parameters are strongly power dependent and the bandwidth increases with optical power. Whenthe 3-dB bandwidthAt high optical powers the presence of gain saturation (reduced gain at high optical power densities) must be included; the modulation bandwidth saturates, and the limiting value depends on the way that the gain saturates with photon density. Using the following approximate expression for gain saturation:

where No is the equilibrium carrier density and Ps is the saturation photon density, a simple expression can be found for the limiting value of the modulation bandwidth at high optical powers:

Typical numbers for a 1.55-^m InGaAsP laser are 20 to 40 GHz.

Frequency Chirping

When the carrier density in the active region is rapidly changed, the refractive index also changes rapidly, causing a frequency shift proportional to dn/dt. This broadens the laser linewidth from its original width of -100 MHz into a double-peaked profile with a gigahertz linewidth, as shown in the experimental results of Fig. 8. The frequency spread is directly proportional to the dependence of the refractive index n on carrier density N. This is a complex function that depends on wavelength and degree of excitation, but for simplicity a Taylor expansion around the steady-state carrier density No can be assumed: n = no + n1(N – No), where n1 = 9n/9N.The (normalized) ratio of this slope to that of the gain per unit length gL is called the linewidth enhancement factor Pc.

 

 

FIGURE 8 Time-averaged power spectra of 1.3 |im InGaAsP laser under sinusoidal modulation at 100 MHz. Horizontal scale is 0.05 nm per division. Spectrum broadens with increase in modulation current due to frequency chirping.14

The magnitude of the frequency spread between the double lobes of a chirped pulse, 28mCH, can be estimated in the small-signal and large-signal regimes from analyzing the time dependence of a modulated pulse in terms of the sum of all frequency components.15

Small-Signal Modulation. For a modulation frequencythat is less than the relaxation oscillation frequency, and assuming thata small modulation currentwill cause a frequency chirp of magnitude

where(remembering thatis negative). The origin of chirp is the linewidth enhancement factor pc. It will be largest for gain-guided devices where pc is a maximum. The chirp will be smaller in lasers with am<<a,, such as will occur for long lasers, where mirror loss is amortized over a longer length, but such lasers will have a smaller differential quantum efficiency and smaller relaxation oscillation frequency. Typical numbers at 25-mA modulation current can vary from 0.2 nm for gain-guided lasers to 0.03 nm for ridge waveguide lasers.

Large-Signal Modulation. There is a transient frequency shift during large-signal modulation given by:

 

When a gaussian shape pulse is assumed,

the frequency shift becomes

The importance of the linewidth enhancement factor Pc is evident from this equation; its existence will inevitably broaden modulated laser linewidths.