Journal of Applied Spectroscopy, Vol. 77, No. 4, 2010
MODULATION RESPONSE OF DYNAMIC SINGLE-MODE QUANTUM-WELL HETEROLASERS IN THE 1.5 μm RANGE B. F. Kuntsevich* and V. K. Kononenko
UDC 621.375.826
Numerical modelling is used to study the effect of tuning the laser output over the gain bandwidth on the modulation response of GaInAs–GaInAsP quantum-well heterolasers for different modulation frequencies of the pump current. It is found that the maximum frequency bandwidth of the response band and the greatest feasibility of high speed modulation for transmission of signals in information systems are achieved in the center of the gain band. Raising the dc component of the pump current increases the response bandwidth. For typical parameters of this system (near 1.5 μm) the maximum response bandwidth can approach ≈40 GHz. For certain parameters, the amplitude-frequency characteristics of the heterolasers have two local maxima: one at low frequency corresponding usually to a resonance for the 1/2 subharmonic and one at high frequency, for the fundamental resonance. Keywords: semiconductor laser, pump current modulation, response bandwidth, amplitude-frequency characteristic, resonance frequency Introduction. Solving a number of scientific and practical problems requires single-mode (single frequency) semiconductor lasers with a narrow and stable emission line. Furthermore, in a number of cases the frequency of the laser radiation must be tuned over a certain range within the confines of the gain band. As a rule, single frequency operation is obtained using an external dispersive cavity. For example, narrow band cw lasing (Δλ ≤ 0.15 nm) at an output power of up to 20 mW tuneable over an interval of ≈10 nm around the central wavelength of 884 nm has been obtained [1] in a GaAs–AlGaAs heterolaser with an external dispersive cavity at room temperature. Single frequency emission from a laser with stressed active layers (λ = 975 nm) and a cw power of >100 mW has been obtained [2] with suppression of side longitudinal modes at a level of >30 dB. It has been pointed out [3] that with the right position for the fixed axis of rotation of an external mirror, an injection laser can essentially be tuned continuously over its entire gain band. An external waveguide grating mirror has been used to obtain a significant narrowing (to 0.1 nm) of the output spectrum and it was shown that the output wavelength of a semiconductor laser can be tuned smoothly over a range of 10–18 nm [4]. It has been noted [5] that one of the most important problems in fiber optic communications lines with channel spectral densification is the creation of laser sources with a narrow and stable spectrum (no more than 0.1 nm) that must be maintained in a dynamic regime with modulation of an information signal at frequencies up to 10 GHz. These requirements may correspond best to semiconductor lasers with external fiber Bragg gratings on single-mode fiber light guides. For GaInAsP–InP lasers the half width of the laser output spectrum with a fiber Bragg grating is no more than 0.1 nm with pulsed modulation of the pump current at a frequency of up to ≈5 GHz. Other features of the output parameters of single frequency semiconductor lasers are discussed in [6–11]. Methods of creating dynamic single-frequency semiconductor lasers that operate stably to generate a single longitudinal mode even with high speed direct modulation and are used in fiber-optic communications have been examined in [12, 13]. This brief discussion shows that, up to now, a number of papers have been published on dynamic single-frequency semiconductor lasers that can in some cases be discretely or continuously tuned within their gain bands. However, no systematic studies have yet been made of the maximum output parameters of semiconductor quantum-well ∗
To whom correspondence should be addressed.
B. I. Stepanov Institute of Physics, National Academy of Sciences of Belarus, prosp. Nezavisimosti 68, Minsk 220072, Belarus; e-mail:
[email protected]. Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 77, No. 4, pp. 583–590, July–August, 2010. Original article submitted February 19, 2010. 0021-9037/10/7704-0541 ©2010 Springer Science+Business Media, Inc.
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heterolasers with high frequency modulation of the current during tuning of the output frequency within their gain bands, which are used in fiber-optic data transmission lines (in the range of 1.3 to 1.5 μm). One exception is some recent studies [14–16] of the general behavior of the amplitude-frequency and amplitude-detuning characteristics of quantum-well heterolasers which emit in various regions. This paper is a study by numerical modelling of the amplitude-frequency characteristics of semiconductor lasers in the region of 1.5 μm, which serve as primary sources in fiber-optic information systems. Knowledge of the amplitude-frequency characteristics makes it possible to determine a parameter of great importance of fiber-optic information systems, the frequency width of the transmission band (response) of the laser source. Basic Equations and Parameters. For concreteness the calculations have been done for dynamic single-frequency quantum-well GaInAs–GaInAsP heterolasers with two quantum wells of width ≈5 nm each in their active regions. It is assume that lasing with pump current modulation takes place in a single longitudinal mode which can lie at any point in the gain band (by using a selective cavity). The dynamics of monochromatic lasing at frequency νg [14] is described by a system of standard rate equations for the density S of photons in the cavity and the concentration N of nonequilibrium current carriers: . S = vμ (Γk (νg) − k1) S + βNaRsp , (1) . η′j Rsp Γk (νg) N= − − vμ S, edNa ηsp Na
(2)
where v is the velocity of light in the active medium; μ is the space factor of the active medium in the cavity; Γ is the optical clipping parameter; k(νg) and k1 are the gain and loss coefficients; Rsp is the rate of spontaneous radiative recombination; ηsp is the luminescence quantum yield; β is a coefficient that determines the fraction of the contribution of spontaneous emission to the laser mode; j is the pump current density; η′ is the injection efficiency; Na is the number of quantum wells in the active region; d is the thickness of the gain layer (width of a quantum well); and, e is the elementary charge. Here the kinetic equations are extended to a system of identical quantum wells with uniform pumping. The gain spectrum k(νg) and rate of spontaneous radiative recombination Rsp are determined in terms of a model for the active medium in the case of optical transitions without selection rules for the wave vector of an electron between states of the fundamental subbands [15]: hνg − ΔF ζc − Ec1 ⎞ ⎛ ζh − Evh1 ⎞ ⎛ ⎛ ⎟ ⎜1 + exp ⎟ exp ⎜1 + exp ⎜ kT kT ⎠ ⎝ kT ⎠ ⎝ ⎜ mvht ln k (νg) = κ1Nc1 ⎜ m ⎜ e ⎛ hνg − ΔF + ζc − Ec1 ⎞ ⎛ hνg − ΔF − ζh − Evh1 ⎞ ⎜ ⎟ ⎜1 + exp ⎟ ⎜1 + exp ⎝ kT kT ⎠⎝ ⎠ ⎝ (3)
ζh − Evl1 ⎞ ζc − Ec1 ⎞ ⎛ hνg − ΔF ⎛ ⎟ ⎜1 + exp ⎟ exp ⎜1 + exp kT ⎠ ⎝ kT ⎠ kT ⎝
⎞ ⎟ mvlt ⎟ ln + ⎟, me hνg − ΔF + ζc − Ec1 ⎞ ⎛ hνg − ΔF − ζh − Evl1 ⎞ ⎟⎟ ⎛ ⎜1 + exp ⎟ ⎜1 + exp ⎟⎠ kT kT ⎝ ⎠⎝ ⎠
Rsp =
ζh − Evh1 ⎞ ⎛ ⎛ A1 ζc − Ec1 ⎞ ⎛ ⎛ ζh − Evl1 ⎞⎞ N ln ⎜1 + exp ⎟ ⎜N ln ⎜1 + exp ⎟ + Nvl1 ln ⎜1 + exp ⎟⎟ . d c1 ⎝ kT ⎠ ⎝ vh1 ⎝ kT ⎠ kT ⎠⎠ ⎝
(4)
Here κ1 = 4π(a0)2k0; k0 = r0 ⁄ νgρ(hνg)d; r0 = Acvme ⁄ πh− 2 ; Acv is the Einstein coefficient for direct interband transitions; a0 is the effective Bohr radius; A1 = 4π(a0)2Acv is the probability of transitions without selection rules; Nc1, Nvh1, and 542
Fig. 1. Time variation in S (1) and in the shape of the modulating signal (2) for νm = 0.03 (a), 1.8 (b), 4.2 (c), 5.7 (d), 8.6 (e), and 14.0 GHz (f); hνg = 857 meV; xb = 1.53; xm = 0.6. Nvl1 are the effective (two dimensional) densities of states in the electron and hole subbands; me, mvht, and mvlt are the effective masse of an electron and the transverse components of the effective masses of heavy and light holes; T is the temperature; ΔF is the difference in the Fermi quasilevels; and, Ec1, Evh1, and Evl1 are the initial levels for the electron and hole fundamental subbands. For this laser quantum well system we find the relationship between the chemical potential for the electrons ζc and holes ζh and the difference ΔF in the Fermi quasilevels in the following numerical form: ζc = 79.227 + 0.584(ΔF – Eg – 96.7) (meV), ζh = 17.473 + 0.416(ΔF – Eg – 96.7) (meV), where Eg is the band gap of the semiconductor (GaInAs). The relationship between The concentration N of nonequilibrium current carriers and ζc at each time t is found from the formula N = (Ncl/d) ln (1 + exp ((ζc – Ecl)/kT)), which ultimately allows us to find ΔF and ζh. In this case the gain spectrum has a bell shape, which, in fact, corresponds to taking the effect of spectral broadening into account [15, 17]. The radiation is assumed isotropic and including the polarization characteristics of the output radiation (TE- or TM-mode) is not difficult. We note also that the parameters ηsp and η′ are assumed close to unity and it is important to take their variations, with, for example, temperature, into account when considering heat transfer and diffusion behavior, and heating and Auger recombination effects. The modulation of the pump current has the form j = jb + jm sin (2πνmt), where jm and νm are the modulation depth and frequency and jb is the dc component of the current. Here we set jb = xbjth and jm = xmjb, where jth = edNaRsp-th/η′ηsp is the steady-state threshold, Rsp-th is the threshold rate of spontaneous recombination, and xb and xm are specified parameters. The threshold parameters were determined as follows. Using Eq. (3), a value of ΔFb was chosen such that within the confines of the gain band the maximum (weak signal) gain coefficient, which is attained at some frequency νgb, satisfies the condition Γk(νgb) ⁄ k1 = xth, where xth is a specified number of thresholds. Then, at a given frequency νgb the threshold value ΔFth was determined and with the known ΔFth Eq. (4) was used to calculate Rsp-th and then the threshold current density jth. It was then assumed that xb = xth, so it was possible to calculate jb and, for the given 543
xm, also to find a value for jm. These threshold values were taken to be fixed as the emission frequency was tuned over the gain band. The following parameters were specified in the calculations for the GaInAs–GaInAsP system [15]: d = 5 nm, 5 –1 23 –2 –1 –1 –1 6 k0 = 2.07⋅10 cm , r0 = 2.739⋅10 cm ⋅s ⋅eV , Acv = 6.547⋅108 s , a0 = 7.27 nm, κ1 = 1.375⋅10 cm, A1 = –3 2 –1 11 –2 11 –2 12 –2 4.35⋅10 cm ⋅s , T = 300 K, Eg = 0.718 eV, Nc1 = 4.43⋅10 cm , Nvh1 = 5.62⋅10 cm , Nvl1 = 1.30⋅10 cm , Ec1 –2 –5 –1 = 69.7 meV, Evh1 = 27.0 meV, Evl1 = 88.4 meV, Na = 2, Γ = 0.845⋅10 , βNa = 10 , μ = 1, and k1 = 15 cm . The parameters of the components of the heterostructure, effective masses, and, therefore, the subband levels were found on the basis of data given in [18]. In particular, we found the following: mc/me = 0.041, mvh/me = 0.31, mvl/me = 0.040, mvht/me = 0.052, and mvlt/me = 0.12. Then the initial energy of the output photons is hν1 = Eg + Ec1 + Evh1 = 0.8147 eV (λ1 = 1.52 μm). Data from [19, 20] can be used for estimating the parameters of quantum-well laser systems emitting in the region of 1.3 μm. We note that for the specified typical parameters of heterolasers used for fiber-optic information systems in 2 the 1.5 μm wavelength band, we have a threshold current density of ~400 A/cm . At twice the threshold and pump currents of ~40 mA (for a cavity length of l ≈ 500 μm and a strip contact width of w ≈ 10 μm) we obtain a cw output power of P = hνgdwlvk1S = hνgwlη′(j − jth) ⁄ e ≈ 15 mW (for a laser quantum yield of ηst = η′ ≈ 1). Computational Results and Discussion. Figure 1 shows some typical time variations in the density S of photons in the cavity for different modulation frequencies with a fixed value of hνg. Varying the modulation frequency νm clearly leads to a significant change in both the shape and duration of the output pulses. Figure 1a corresponds to cw lasing, when Γk(νg) ≈ k1. In this case pulses are emitted only if a threshold level is exceeded, with S proportional to the pump current. It is known [21] that in nonlinear systems secondary resonances are possible at νm = νrelm ⁄ n, where m and n are mutually prime numbers, in addition to the fundamental resonance. Here νrel is the frequency of the relaxation oscillations. A characteristic sign of a resonance for the 1/n subharmonic is fine structure in the trailing edges of the laser pulses with a period of ≈1/nνm. Thus, the curve in Fig. 1b corresponds to the 1/2 subharmonic. The fundamental resonance is characterized by the emission of relatively short symmetric pulses (Fig. 1c). Figures 1d and e corresponds to 2T (period doubling) and 3T (tripling) emission modes, where T = 1/νm is the modulation period. For modulation frequencies νm >> νrel pulses (Fig. 1f) with a low modulation depth are emitted (essentially in opposite phase to the modulating signal). It should be noted that the density of photons at the minima between the laser pulses does not go to zero, but decreases by five or six orders of magnitude (Fig. 1a–e). Further calculations showed that for a given value of νm, the parameters of the output pulses depend substantially on νg. Therefore, the temporal and energy characteristics of the output pulses are determined by a combination of the parameters of the laser system. One general and intuitive way of describing the behavior of a given emitter as the modulation frequency is varied over a wide range is its amplitude-frequency characteristics, which are shown in Fig. 2 for different values of the parameter xb and positions of the lasing frequency νg within the gain band. Here the ordinates are samples of the output pulse amplitudes Sm in terms of the modulation period T. The ranges of the modulation frequencies where the curves "split into two" correspond to emission modes that differ from the 1T (e.g., 2T). In Fig. 2 these segments mainly correspond to a doubled period. One of the major and most often encountered parameters determining the suitability of such lasers for use in fiber-optic information systems is the ν−3dB modulation frequency (frequency width of the transmission or response band), at which the response decreases by 3 dB (i.e., by a factor of two) compared to the initial value (at νm → 0). [22, 23]. In Fig. 2 the level corresponding to the –3dB response is indicated by a horizontal line. Note that in Fig. 2 the minimum modulation frequency νm was chosen to be 0.001 GHz in the calculations. Figure 2a–d, corresponds to a comparatively low dc component of the pump current: xb = 1.25. It can be seen that as νm is raised the response initially increases, reaches a maximum, and then falls off monotonically. The exception is the ranges of the modulation frequency where, because of the appearance of modes differing from 1T, the monotonic fall is disrupted. On the whole, this behavior of the amplitude-frequency characteristics is consistent with the previously known variations. Usually the modulation frequency corresponding to the maximum of the amplitudefrequency characteristic (νr) is referred to as the resonance frequency. In Fig. 1a–d, νr equals 0.7, 1.8, 1.6, and 1.1 GHz, while ν−3dB = 1.7, 9.3, 8.7, and 2.8 GHz. Therefore, when emission modes other than 1T are present, the ratio α = ν−3dB/νr is considerably higher. For example, under the conditions of Fig. 2a and d, we have α ≈ 2.5, and
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Fig. 2. Sm as a function of νm for xb (a–d), 1.53 (a′–d′), 1.93 (a′′–d′′) and hνg = 820 (a, a′, a′′), 830 (b, b′), 840 (b′′, c), 850 (c′, d), 860 (c′′), 870 (d′) and 880 meV (d′′); xm = 0.6.
for the conditions of Fig. 2b and c, we have α ≈ 5.2. This is important from the standpoint of using these lasers in fiber-optic information systems. Figure 2a′–d′, shows the amplitude-frequency characteristics for xb = 1.52. As before the minimum values of νr (and ν−3dB) occur at limiting points, i.e., on either the low- or high-frequency side of the absorption band. The presence of modes other than 1T leads to an increase in α. Figure 2a′′–d′′ corresponds to the maximum value xb = 1.93. In that case, only 1T modes are observed. The maximum values of νr (and ν−3dB) occur in the central region of the gain band. The following conclusions can be reached on the basis of this analysis and some additional calculations: 1. The magnitude and frequency width of the response, as well as the resonance frequency, are determined by a combination of the system parameters. νr (and ν−3dB), as well as α, approach a minimum at the limiting points for meeting the lasing threshold conditions, i.e., at the low- and high-frequency edges of the gain band. For specified values of xb and xm, by choosing νg it is possible to find a spectral range near the center of the gain band that corresponds to the maximum value of ν−3dB (and α). In our case, the maximum values of ν−3dB ≈ 40 GHz and α ≈ 11 are 545
Fig. 3. ν1r ⁄ 2 (1) and νr (2) as functions of hνg for xb = 1.01 (a), 1.25 (b), 1.53 (c) and 1.93 (d); xm = 0.6. attained for the conditions of Fig. 2b′′. As a comparison we note that a value of ≈15 GHz has been reported elsewhere [24] for the modulation response bandwidth of a GaInAs–GaInAsP quantum-well laser, although the effect of tuning νg on ν−3dB was not examined there. 2. In the region of these parameters for fixed xm, as xb is raised νr increases. This is consistent with published data [25]. At the same time, in the approximation of small deviations of the system parameters from the steady state, it was found that ν−3dB ≈ 1.55νr. This relation holds for negligibly small xm. Our calculations imply that when the nonlinear properties of real systems are taken into account, as xb is raised α also increases. Thus, for example, in Fig. 2a α ≈ 2.4, and for Fig. 2a′′ α ≈ 9.2. 3. For fixed xm, an increase in xb leads to a rise in the possible maximum values of νg at which the system response is still comparable to the response as νg → ν1. Thus, for example, comparable response magnitudes were obtained up to hνg ≈ 850 meV in Fig. 2a–d, and up to hνg ≈ 880 meV in Fig. 2a′′–d′′. Figure 2b′, a′′, and b′′ and some additional calculations imply that, for some parameters of these systems, GaInAs-GaInAsP quantum-well lasers have two local maxima in their amplitude-frequency characteristics. A more detailed analysis shows that these maxima are physically different. Usually the low frequency local maxima correspond to a resonance for the 1/2 subharmonic (see Fig. 1b and the explanations for it) and the high frequency maxima, to the fundamental resonance (Fig. 1c). Note that the maximum value in the amplitude-frequency characteristic (response) may appear in the regions of either low- or the high-frequency local maxima. A similar multiresonance structure in the nonlinear response has been observed previously in some cases for other semiconductor laser systems [14, 26, 27]. The shapes and durations of the pulses at these local maxima do differ and this is of some practical interest. Thus, we shall examine this question in more detail. Calculated resonance frequencies ν1r ⁄ 2 (curves 1) for the 1/2 subharmonic and the fundamental resonance frequency νr (curves 2) as the output frequency is tuned over the gain band are shown in Fig. 3. When the dc component of the pump current is relative small (Fig. 3a, xb ≈ 1.01) the response peaks correspond only to resonances for the 1/2 subharmonic. The maximum values of ν1r ⁄ 2 occur near the center of the gain band. With rising xb (Fig. 3b), there is an increase in ν1r ⁄ 2 and the peak shifts somewhat toward higher νg. In addition, a single point corresponding to the fundamental resonance showed up on the low-frequency side. With further increases in xb (Fig. 3c and d), two curves exist in some spectral ranges. This means that the amplitude-frequency characteristics contain two local maxima, i.e., have been transformed into "double humped" curves. It can be seen that the fundamental resonance corresponds to the high-frequency local maxima. In addition, the ν1r ⁄ 2 curve may be nonmonotonic (Fig. 3d). As xb increases, the maximum values of ν1r ⁄ 2 and νr both become larger. 546
We have also calculated the dependences of νr and ν−3dB on xm with a fixed value of xb = 1.53 for hνg = 840, and 860 meV. In the first two cases, resonance frequencies ν1r ⁄ 2 and νr appear within certain spectral ranges and nonmonotonic behavior is observed in the ν1r ⁄ 2(xm) curve. Here the ν−3dB(xm) curve is also nonmonotonic. For hνg = 860 meV, when xm is increased, ν1r ⁄ 2 rises monotonically from 1.0 to 1.7 GHz, and ν−3dB from 2.6 to 5.4 GHz. Conclusion. Numerical modelling calculations have shown that changes in the dc component of the pump current, modulation frequency, and position of the laser mode within the gain band have a relatively strong effect on the temporal and energy parameters of the output from GaInAs–GaInAsP quantum-well heterolasers. The frequency width of the transmission band (response) of the laser approaches a minimum on the low-frequency side as hνg → hν1 and on the high-frequency side near the maximum hνg, when the threshold lasing conditions are still met. The maximum response widths ν−3dB that correspond to possible high-speed modulation modes for information systems occur in the central region of the gain band. Here a rise in the dc component of the pump current leads to an increase in the response bandwidth. For the system parameters chosen here, ν−3dB reaches a maximum of up to ≈40 GHz. Including nonlinear recombination processes such as Auger recombination may cause a reduction in this maximum. For certain values of the system parameters, the amplitude-frequency characteristics of heterolasers contain two local maxima. The low-frequency maximum usually corresponds to a resonance for the 1/2 subharmonic and the highfrequency maximum, to the fundamental resonance. The behavior discovered here can be used for developing more efficient fiber-optic information systems using quantum-well heterolasers.
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