Dynamic Effects

CHAPTER FIVE Dynamic Effects 5.1 INTRODUCTION In Chapter 2, the rate equations for both the carrier and the photon density of a given optical mode, in the active region were developed from simple intuitive arguments. The below-threshold and above-threshold limits to the steady-state solutions of the rate equations were considered to give a feel for the operating characteristics of the laser. We also took a first look at the small-signal intensity modulation response of the laser. In this chapter we wish to expand on these simplified discussions considerably, drawing on the results of Chapter 4 for the evaluation of the various generation and recombination rates. We will first summarize many of the results obtained in Chapter 2. This review will serve as a convenient reference to the relevant formulas. We will then move on to dynamic effects, starting with a differential analysis of the rate equations. We will derive the small-signal intensity and frequency modulation response of the laser as well as the small-signal transient response. We then consider large-signal solutions to the rate equations. The turn-on delay of the laser and the general relationship between frequency chirping and the modulated output power will also be treated here. Next we consider laser noise. Using the Langevin method, we will determine the laser’s relative intensity noise (RIN) and the frequency fluctuations of the laser’s output from which we can estimate the spectral linewidth of the laser. We will also discuss the role of the injection current noise in determining the RIN at high power and examine the conditions necessary for noise-free laser operation. Diode Lasers and Photonic Integrated Circuits, Second Edition. Larry A. Coldren, Scott W. Corzine, and Milan L. Masanoviˇ c.´ © 2012 John Wiley & Sons, Inc. Published 2012 by John Wiley & Sons, Inc. 247 248 DYNAMIC EFFECTS In the final two sections we consider dynamic effects in specific types of lasers. The first section considers carrier transport limitations in separate-confinement het- erostructure (SCH) quantum-well lasers. The second deals with the effects of weak external feedback in extended cavity lasers. 5.2 REVIEW OF CHAPTER 2 Figure 5.1 summarizes the reservoir model used to develop the rate equations in Chapter 2. To ensure particle conservation, each arrow in the flowchart represents the number of particles flowing per unit time. This is why the rates per unit volume and the densities are multiplied by the active-region volume V of the carrier reservoir or the mode volume Vp of the photon reservoir. Before deriving the rate equations, we will briefly follow the flowchart from input to output. Starting with the carrier reservoir, we have the rate of carrier injection into the laser I/q. Of these carriers only ηi I /q reach the active region, where ηi is the injection or internal efficiency of the laser introduced in Chapter 2. The rest recombine elsewhere in the device (this includes all carriers recombining at rates not directly tied to the carrier density in the active region). Once in the carrier reservoir, the carriers have a number of options. Some recombine via nonradiative recombination at the rate Rnr V . Others recombine spontaneously at the rate Rsp V , of which a certain fraction emit photons into the mode of interest at the rate Rsp V (or βsp Rsp V where βsp is the spontaneous emission factor). Other carriers recombine via stimulated emission into the mode of interest at the rate R21V . Although photons in other modes can also induce stimulated recombination of carriers, we limit ourselves initially to a single mode. Finally, photons in the photon reservoir I/q ( − ) 1 ηi I/q Heat & light ηi I/q RnrV RspV NV Heat P Carrier reservoir sp ′ RspV R12V R21V NpVp NpVp τ η0 τ p p NpVp P0 Photon reservoir N V ( − ) p p 1 η0 τ p Heat & light FIGURE 5.1: Model used in the rate equation analysis of semiconductor lasers. 5.2 REVIEW OF CHAPTER 2 249 can be absorbed, serving as an additional source of carriers, which are generated at the rate R12V . In Chapter 2, we also considered carriers that might leak out of the active region via lateral diffusion and/or thermionic emission at the rate Rl V . However, because Rl V is often negligible and furthermore plays a role identical to Rnr V in the rate equations, we will not include it explicitly in this chapter (if we wish to include carrier leakage at any time, we can set Rnr → Rnr + Rl ). In the photon reservoir, the stimulated and spontaneous emission rates into + the mode provide the necessary generation of photons at the rate R21V Rsp V . Stimulated absorption in the active region depletes photons at the rate R12V .All other photons leave the cavity at the rate Np Vp /τp . Of those leaving the cavity, only η0Np Vp /τp leave through the desired mirror to be collected as useful output power, P0, where η0 is the optical efficiency of the laser to be defined below. The rest of the photons exit the cavity through a different mirror or disappear through: (1) free carrier absorption in the active region (which does not increase the carrier density), (2) absorption in materials outside the active region, and/or (3) scattering at rough surfaces. 5.2.1 The Rate Equations By setting the time rate of change of the carriers and photons equal to the sum of rates into, minus the sum of rates out of the respective reservoirs, we immediately arrive at the carrier and photon number rate equations: dN η I V = i − (R + R )V − (R − R )V, (5.1) dt q sp nr 21 12 dN N V p = − − p p + Vp (R21 R12)V Rsp V . (5.2) dt τp Setting R21 − R12 = vg gNp using Eq. (4.37), dividing out the volumes, and using = V /Vp (by definition of Vp ), we obtain the density rate equations derived in Chapter 2 in slightly more general form: dN η I = i − (R + R ) − v gN , (5.3) dt qV sp nr g p dN p = − 1 + vg g Np Rsp . (5.4) dt τp It is a matter of preference to use either the density or the number rate equations in the analysis of lasers.1 Throughout this book we choose to use the density versions. This choice forces us to do a little more book-keeping by explicitly including V and Vp in various places (as we will see in Section 5.5) but in the end places results 1It is important to note that the gain term in the density versions has a in the photon density rate equation, but not in the carrier density rate equation. While in the number rate equations, the gain term is symmetric. This asymmetry in the density rate equations is often overlooked in the literature. 250 DYNAMIC EFFECTS in terms of the more familiar carrier density and related terms (e.g., the differential gain, the A, B, and C coefficients, etc.). To complete the description, the output power of the mode and total spontaneous power from all modes are given by Np Vp P0 = η0hν and Psp = hνRsp V, (5.5) τp where αm η0 = F . (5.6) αm +αi The optical efficiency defined here times the injection efficiency yields the differential quantum efficiency defined in Chapters 2 and 3, ηd = ηi η0. The prefactor F in Eq. (5.6) is the fraction of power not reflected back into the cavity that escapes as useful output from the output coupling mirror as derived in Chapter 3. The mirror loss can usually be defined as αm = (1/L) ln(1/r1r2), and αi is the spatial average over any internal losses present in the cavity. The photon lifetime is given by 1 ω = vg (αm +αi ) = , (5.7) τp Q where vg is the group velocity of the mode of interest including both material and waveguide dispersion. The more general definition of τp in terms of the cavity Q is useful in complex multisection lasers where αm can prove difficult to define (see the latter part of Appendix 5). In such cases, we can use the definition: Q ≡ ω (energy stored in cavity)/(total power lost). = We can use Eq. (4.67) to set Rsp vg gnsp /V in Eq. (5.4). However, a useful → approximation to Rsp is found by setting Rsp βsp Rsp and evaluating βsp at its threshold value: vg gnsp q nsp βsp ≈ βsp = = , (5.8) th ηi ηr I /q th Ith τp ηi ηr th where Eq. (4.82) was used to define βsp and ηr = Rsp /(Rsp + Rnr ). The latter equal- ity makes use of Eq. (5.11). The term in brackets at threshold is typically between 1.5 and 2 in short-wavelength materials, but it can be as large as 10 in long- wavelength materials due to low radiative efficiencies (high Auger recombination). 5.2.2 Steady-State Solutions The steady-state solutions of Eqs. (5.3) and (5.4) are found by setting the time derivatives to zero. Solving Eq. (5.4) for the steady-state photon density and 5.2 REVIEW OF CHAPTER 2 251 Eq. (5.3) for the DC current, we have Rsp (N ) Np (N ) = , (5.9) 1/τp − vg g(N ) qV I (N ) = (Rsp (N ) + Rnr (N ) + vg g(N )Np (N )). (5.10) ηi As the functionality implies, it is useful to think of N as the independent parameter of the system, which we can adjust to determine different values of both current and photon density.

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