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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 3019–3036
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Simple form of multimode laser diode rate equations incorporating the band filling effect

Kenji Wada, Hiroyuki Yoshioka, Jiaxun Zhu, Tetsuya Matsuyama, and Hiromichi Horinaka  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 3019-3036 (2011)
http://dx.doi.org/10.1364/OE.19.003019


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Abstract

To derive a simple form of the multimode laser diode rate equations incorporating the band filling effect, the laser diode gain in the direct bandgap model is introduced into the conventional multimode laser diode rate equations. By numerically examining each modal gain under the gain-switching condition, it is found that both the differential gain coefficient and the carrier density at transparency show an approximately linear dependency on the oscillation frequency. As a result, it is possible to derive a simple form of the multimode laser diode rate equations with linearized gain, which can be used to simulate the behaviors of a gain-switched laser diode characterized by the band filling effect, in both the multimode and single-mode oscillation cases.

© 2011 Optical Society of America

1. Introduction

Rate equations have been widely used for analyzing the dynamic behavior of laser diodes [1

1. T. Ikegami, K. Kobayashi, and Y. Suematsu, “Transient behaviour of semiconductor injection lasers,” Electron. Commun. Jpn. 53B, 82–89 (1970).

22

22. M. Osinski, D. F. G. Gallagher, and I. H. White, “Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method,” Electron. Lett. 21, 981–982 (1985). [CrossRef]

]. For rigorous analysis of laser diode operation, the laser diode gain in the rate equations should be described with terms that take into account certain aspects of semiconductors, namely, the band structure and many-body effects [18

18. W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser Physics, (Springer-Verlag, 1994). [CrossRef]

,19

19. W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals, (Springer-Verlag, 1999).

]. In one such attempt, pioneering work by Osinski and Adams demonstrated that the asymmetric multimode power spectrum of a gain-switched pulse from an InGaAsP Fabry-Perot laser diode could be simulated by introducing a semiconductor gain term in the rate equations [20

20. M. Osinski and M. J. Adams, “Intrinsic manifestation of regular pulsations in time-averaged spectra of semiconductor lasers,” Electron. Lett. 20, 525–526 (1984). [CrossRef]

,21

21. M. Osinski and M. J. Adams, “Picosecond pulse analysis of gain-switched 1.55 μm InGaAsP lasers,” IEEE J. Quantum Electron. 21, 1929–1936 (1985). [CrossRef]

]. The asymmetry seemed to stem from the band filling effect in the laser diode, which is a phenomenon peculiar to a laser diode whereby the gain peak is blue-shifted with increasing carrier density. On the other hand, when a linear approximation of the relation between the laser diode gain and the carrier density holds, the gain term in the rate equations is reduced to a simple form G(N – N0) (G: differential gain coefficient; N, N0: carrier density and carrier density at transparency). This simple form of the linearized gain has been traditionally used in analyzing laser diode operation [1

1. T. Ikegami, K. Kobayashi, and Y. Suematsu, “Transient behaviour of semiconductor injection lasers,” Electron. Commun. Jpn. 53B, 82–89 (1970).

17

17. C. Chen, G. Ding, B. S. Ooi, L. F. Lester, A. Helmy, T. L. Koch, and J. C. M. Hwang, “Optical injection modulation of quantum-dash semiconductor lasers by intra-cavity stimulated Raman scattering,” Opt. Express18, 6211–6219 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6211. [CrossRef] [PubMed]

]. Even though the importance of introducing a realistic semiconductor gain has been noted, as mentioned above, the linearized gain form in the rate equations seems to remain valued by researchers in terms of its convenience. For GaAs-based laser diodes, at least, the numerical framework of the rate equations with the linearized gain provides foreseeable analytical solutions in steady-state or quasi steady-state analyses [8

8. R. Olshansky, P. Hill, V. Lanzisera, and W. Powazinik, “Frequency response of 1.3 μm InGaAsP high speed semiconductor lasers,” IEEE J. Quantum Electron. 23, 1410–1418 (1987). [CrossRef]

,9

9. K. Y. Lau, “Gain switching of semiconductor injection lasers,” Appl. Phys. Lett. 52, 257–259 (1988). [CrossRef]

] and also provides many good estimations in analyzing both single-mode and multimode operation in the presence of optical feedback fields [4

4. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron.16, 347–355 (1980). [CrossRef]

, 11

11. I. V. Koryukin and P. Mandel, “Dynamics of semiconductor lasers with optical feedback: Comparison of multimode models in the low-frequency fluctuation regime,” Phys. Rev. A 70, 053819 (2004). [CrossRef]

, 13

13. J. Ohtsubo, Semiconductor Lasers –Stability, Instability and Chaos, Second Ed., (Springer-Verlag, 2007). [PubMed]

] or under large-amplitude modulation conditions [2

2. P. M. Boers and M. Danielsen, “Dynamic behaviour of semiconductor lasers,” Electron. Lett. 15, 206–208 (1975). [CrossRef]

, 5

5. S. Tarucha and K. Otsuka, “Response of semiconductor laser to deep sinusoidal injection current modulation,” IEEE J. Quantum Electron. 17, 810–816 (1981). [CrossRef]

7

7. P. M. Downey, J. E. Bowers, R. S. Tucker, and E. Agyekum, “Picosecond dynamics of a gain-switched InGaAsP laser,” IEEE J. Quantum Electron. 23, 1039–1047 (1987). [CrossRef]

, 10

10. K. A. Corbett and M. W. Hamilton, “Comparison of the bifurcation scenarios predicted by the single-mode and multimode semiconductor laser rate equations,” Phys. Rev. E 62, 6487–6495 (2000). [CrossRef]

, 12

12. K. Wada, H. Sato, H. Yoshioka, T. Matsuyama, and H. Horinaka, “Suppression of side fringes in low-coherence interferometric measurements using gain- or loss modulated multimode laser diodes,” Jpn. J. Appl. Phys. 44, 8484–8490 (2005). [CrossRef]

, 14

14. C. -C Lin, H. -C Kuo, P. -C Peng, and G. -R Lin, “Chirp and error rate analyses of an optical-injection gain-switching VCSEL based all-optical NZR-to-PRZ converter,” Opt. Express16, 4838–4847 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4838. [CrossRef] [PubMed]

, 16

16. K. Wada, S. Takamatsu, H. Watanabe, T. Matsuyama, and H. Horinaka, “Pulse-shaping of gain-switched pulse from multimode laser diode using fiber Sagnac interferometer,” Opt. Express16, 19872–19881 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19872. [CrossRef] [PubMed]

, 17

17. C. Chen, G. Ding, B. S. Ooi, L. F. Lester, A. Helmy, T. L. Koch, and J. C. M. Hwang, “Optical injection modulation of quantum-dash semiconductor lasers by intra-cavity stimulated Raman scattering,” Opt. Express18, 6211–6219 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6211. [CrossRef] [PubMed]

]. Some typical aspects of laser diodes, namely, linewidth broadening [22

22. M. Osinski, D. F. G. Gallagher, and I. H. White, “Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method,” Electron. Lett. 21, 981–982 (1985). [CrossRef]

, 23

23. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]

], spontaneous emission [2

2. P. M. Boers and M. Danielsen, “Dynamic behaviour of semiconductor lasers,” Electron. Lett. 15, 206–208 (1975). [CrossRef]

], gain saturation [6

6. C. B. Su, V. Lanzisera, and R. Olshansky, “Measurement of nonlinear gain from FM modulation index of In-GaAsP lasers,” Electron. Lett. 21, 893–895 (1985). [CrossRef]

,7

7. P. M. Downey, J. E. Bowers, R. S. Tucker, and E. Agyekum, “Picosecond dynamics of a gain-switched InGaAsP laser,” IEEE J. Quantum Electron. 23, 1039–1047 (1987). [CrossRef]

], the carrier-density-dependent carrier lifetime [24

24. B. Sermage, J. P. Heritage, and N. K. Dutta, “Temperature dependence of carrier lifetime and Auger recombination in 1.3 μm InGaAsP,” J. Appl. Phys. 57, 5443–5449 (1985). [CrossRef]

], and band filling [5

5. S. Tarucha and K. Otsuka, “Response of semiconductor laser to deep sinusoidal injection current modulation,” IEEE J. Quantum Electron. 17, 810–816 (1981). [CrossRef]

,10

10. K. A. Corbett and M. W. Hamilton, “Comparison of the bifurcation scenarios predicted by the single-mode and multimode semiconductor laser rate equations,” Phys. Rev. E 62, 6487–6495 (2000). [CrossRef]

], are taken into account by introducing phenomenologically-derived parameters into the rate equations. As for the band filling effect, however, although its expression is elegant, the modified gain is given as a polynomial with respect to the carrier density, which conflicts with the definition of the differential gain coefficient. In this paper, therefore, using the simple form of the linearized gain G(NN0), we derive consistently formulated multimode laser diode rate equations incorporating the band filling effect.

2. Asymmetry in the power spectrum of a gain-switched pulse

Figure 1 shows an example of an experimentally observed multimode power spectrum of a gain-switched pulse from an 800 nm Fabry-Perot laser diode (Rohm, RLD-78PIT). The laser diode was dc-biased at nearly its threshold current and sinusoidally modulated with an amplitude of about four times the threshold current at a modulation frequency of 1 GHz. The power spectrum was observed with an optical spectrum analyzer with a resolution of 0.01 nm. Two spectral peaks seen in each mode [25

25. B. W. Hakki, “Optical and microwave instabilities in injection lasers,” J. Appl. Phys. 51, 68–73 (1980). [CrossRef]

], forming deformed trapezoidal shapes, are indicated by arrows in the magnified view. The ratio of the longer wavelength peak intensity to the shorter one (SPl/SPs) in the individual modes decreases gradually and continuously in the spectral range from the central mode at 802.5 nm to the shorter wavelength modes around 798 nm, through the peak ratio of nearly one in the 801 nm mode. In contrast, the peak ratio decreases continuously but slowly in the spectral range from the central mode to the longer wavelength modes around 805 nm, because of the existence of excessively high longer wavelength peaks. This leads to an asymmetric multimode power spectrum of a gain-switched pulse, as reported before [22

22. M. Osinski, D. F. G. Gallagher, and I. H. White, “Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method,” Electron. Lett. 21, 981–982 (1985). [CrossRef]

, 26

26. P. -L Liu, C. Lin, I. P. Kaminow, and J. J. Hsieh, “Picosecond pulse generation from InGaAsP lasers at 1.25 and 1.3 μm by electrical pulse pumping,” IEEE J. Quantum Electron. 17, 671–674 (1981). [CrossRef]

]. By introducing semiconductor gain into the multimode laser diode rate equations, the numerical simulation conducted by Osinski and Adams [20

20. M. Osinski and M. J. Adams, “Intrinsic manifestation of regular pulsations in time-averaged spectra of semiconductor lasers,” Electron. Lett. 20, 525–526 (1984). [CrossRef]

,21

21. M. Osinski and M. J. Adams, “Picosecond pulse analysis of gain-switched 1.55 μm InGaAsP lasers,” IEEE J. Quantum Electron. 21, 1929–1936 (1985). [CrossRef]

] demonstrated that the asymmetric spectral shape stemmed from the semiconductor gain aspect.

Fig. 1 Experimentally observed power spectrum of the gain-switched pulse from an 800 nm Fabry-Perot laser diode.

3. Dynamics of a gain-switched multimode laser diode

Table 1. Notation and values of parameters used in the rate equations

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For comparison, the conventional differential gain model incorporating the band filling effect is also shown below [5

5. S. Tarucha and K. Otsuka, “Response of semiconductor laser to deep sinusoidal injection current modulation,” IEEE J. Quantum Electron. 17, 810–816 (1981). [CrossRef]

,10

10. K. A. Corbett and M. W. Hamilton, “Comparison of the bifurcation scenarios predicted by the single-mode and multimode semiconductor laser rate equations,” Phys. Rev. E 62, 6487–6495 (2000). [CrossRef]

]; the differential gain coefficient is used, and the gain has a parabolic spectrum (parameter values used in the gain expression are listed in Table 2):
gn=G0N[1{2(λ(N)λn)Δλg}2]G0N0
(14)
λ(N)=λ0+k[NthNNth]
(15)
λn=λ0+nδλ=λ0+nλ022nrL
(16)
Nth=N0+1G0τp
(17)
where λn and δλ are the central wavelength of the n-th mode and the wavelength difference between adjacent modes, respectively.

Table 2. Notation and values of parameters for the conventional differential gain model

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The injected current I is defined here as the sum of a dc current and a sinusoidally modulated current, given by
I=Idc+Imwsin(2πfmt)
(18)
where Idc and Imw are the dc-bias current and the amplitude of the modulated current, respectively, and fm is the modulation frequency. These parameter values are set to Idc = 0.95Ith (Ith is the threshold current), Imw = 2.0Ith, and fm = 1.0GHz throughout the simulation. When the value of I is negative in Eq. (18), it must be replaced by zero according to the rectification property of diodes. Because the form of the above gain expressions is complicated, the threshold current Ith was estimated to be 54 mA for both gain models by numerically calculating the I–L characteristics (injected dc current vs. light output power). The number of longitudinal modes assumed in the calculation is 41 (n is varied from −20 to 20), and an initial small amplitude with a random phase is adopted for the optical field of each mode. The temporal evolutions of variables (En(t) and N(t)) in both of the above rate equations (the direct bandgap model and the conventional differential gain model) are then numerically evaluated by using the fourth-order Runge–Kutta method under the gain-switching condition of Eq. (18). Repeating the numerical integration produces a regular gain-switched pulse train within 30 cycles of the modulation period (1/fm). The Fourier transform of the temporal data of the optical fields thus obtained provides the power spectrum of the gain-switched pulse.

Figure 2 shows multimode power spectra of gain-switched pulses simulated when the laser diode gain is described with the direct bandgap model (a) and with the conventional differential gain model (b, c). The parameter k tied to the band filling effect is set to 0 nm in (b) (no band filling) and 20 nm in (c). By assuming an α parameter value of 5, all linewidths of the multimode spectra are broadened [22

22. M. Osinski, D. F. G. Gallagher, and I. H. White, “Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method,” Electron. Lett. 21, 981–982 (1985). [CrossRef]

], corresponding to the experimental result in Fig. 1. The above-mentioned spectral peak ratio (SPl/SPs) in the individual modes gradually varies in all of the multimode spectra in Fig. 2, whereas symmetry of the peak ratio variation about the central mode is seen only in (b). The other two power spectra contain asymmetry, corresponding to the experimental result in Fig. 1. From these results, the asymmetry in the spectral peak variation stems from the band filling effect.

Fig. 2 Simulated power spectra of gain-switched pulses from multimode laser diodes; (a) is calculated with the direct bandgap model, whereas (b) and (c) are calculated with the conventional differential gain model with (b) k=0 and (c) k=20 nm. The numbers in (a) represent the mode numbers.

To examine this in detail, the gain spectra for these cases are depicted in Fig. 3. By varying the value of the carrier density from 1.25 × 1024m−3 to 1.73 × 1024m−3 in 0.04 × 1024m−3 steps in Eq. (4) for (a) and in Eq. (14) for (b) and (c), thirteen gain spectra curves are plotted in each figure. In the case of Fig. 3(b), the gain spectra have a symmetric parabolic shape about the central mode at 800 nm, which leads to the symmetric power spectrum of the gain-switched pulse in Fig. 2(b). In Figs. 3(a) and 3(c), as a result of the decrease in the carrier density within the gain-switched pulse width, the gain remains in the longer wavelength region more than the shorter wavelength region. This phenomenon is based on the red shift of the gain peak with decreasing carrier density, which corresponds to the reverse process of the band filling effect. The residual gain produces the excessively high spectral peak in the individual modes in the longer wavelength region, causing the asymmetry in the power spectrum of the gain-switched pulse.

Fig. 3 Gain spectra when the carrier density is varied from 1.25 × 1024m−3 to 1.73 × 1024m−3 in 0.04 × 1024m−3 steps. (a) is calculated with the direct bandgap model, and (b) and (c) are calculated with the conventional differential gain model with (b) k=0 and (c) k=20 nm.

Fig. 4 The pulse intensity (blue) and the pulse width (red) of each pulse component of a gain-switched pulse with a multimode power spectrum.

Fig. 5 Temporal waveforms of pulse components at −7th, −5th, −3rd, 0th, 3th, 5th, and 9th modes in the multimode oscillation, and a composite pulse of all pulse components, labeled “all modes”.

In addition to the above considerations, the pulse shape functions of those pulse components are estimated in detail by applying the TBP–CBP method (Time Bandwidth Product –Coherence-time Bandwidth Product) which was proposed to estimate shape functions of the power spectra of light emitting diodes [32

32. K. Wada, J. Fujita, J. Yamada, T. Matsuyama, and H. Horinaka, “Simple method for estimating shape functions of optical spectra,” Opt. Commun. 281, 368–373 (2008). [CrossRef]

]. To obtain TBP and CBP values, the pulse complex spectrum should be calculated by Fourier-transforming the complex optical pulse fields while assuming that the inside phase is constant. The power spectrum is obtained by calculating squared absolute values of the complex pulse spectrum. The FWHM of the power spectrum Δf multiplied by the pulse width Δt equals the TBP value. Then, the coherence function C is calculated by substituting the complex optical pulse field E(t) and its replica E(t + τ) into the following equation assuming the inside phase is constant:
C(τ)=|E(t)+E(t+τ)|2dt2|E(t)|2dt1,
(19)
where τ is the time delay. The value of τ is varied in Eq. (19) until the coherence function is obtained. The FWHM of the coherence function Δτ (called coherence time) multiplied by the FWHM of the power spectrum Δf equals the CBP value. The combination values (CBP, TBP) thus calculated characterize the pulse shape function in terms of the relative ratio of the average height of inflection points between the leading and trailing edges of the pulse to the pulse peak (called “wing height” in [32

32. K. Wada, J. Fujita, J. Yamada, T. Matsuyama, and H. Horinaka, “Simple method for estimating shape functions of optical spectra,” Opt. Commun. 281, 368–373 (2008). [CrossRef]

]) and the relative ratio of the average width between the pulse leading and trailing edges to the pulse width (called “wing width” in [32

32. K. Wada, J. Fujita, J. Yamada, T. Matsuyama, and H. Horinaka, “Simple method for estimating shape functions of optical spectra,” Opt. Commun. 281, 368–373 (2008). [CrossRef]

]). For simplicity, they are called “the position of the pulse waist” and “the ratio of the pulse edge width”, respectively. A small TBP value indicates a large ratio of the pulse edge width under a constant pulse width regardless of the CBP value, and a large CBP value indicates a high position of the pulse waist under a constant TBP value. Figure 6 plots the combination values (CBP, TBP) of the pulse components for the corresponding mode numbers in Fig. 2(a). To intuitively estimate the pulse shape functions and the degree of variation of those pulse shapes, the combination values of typical functions, namely, Gaussian, hyperbolic secant-squared, and Lorentzian, are also indicated in Fig. 6. Those combination values of the pulse components, located in the vicinity of the combination value of a sech2-shape function (0.78, 0.315), form a semielliptical-like locus. The apex of the semiellipse, composed of the combination values around the central mode (−2nd, −1st, and 0th modes), has a closest approach to the combination value of the sech2-shape function. As the absolute mode number becomes large, the shape function is found to approach a Lorentz-like shape. It is found in Fig. 6 that the combination values of the pulse component at the 9th mode include approximately the same TBP value as that at the −5th mode and the same CBP value as that at the −7th mode. This indicates the following: Comparing the shape function of the pulse component at the 9th mode with that at the −5th mode, both have approximately the same ratio of the pulse edge width, and the difference is that the former has a lower pulse waist position. Then, comparing the shape function of the pulse component at the 9th mode with that at the −7th mode, the former has a smaller ratio of the pulse edge width and a higher pulse waist position. These estimations correspond to the results in Fig. 5. Thus, a gain-switched pulse from a multimode laser diode contains a distinctive variation in the shape function of inside pulse components.

Fig. 6 TBP–CBP plot for estimating pulse shape functions. Numbers in the figure correspond to the mode numbers in Fig. 2(a).

4. Derivation of multimode laser diode rate equations

As shown above, the direct bandgap model is found to be appropriate for describing the multimode laser diode gain. The good agreement between Figs. 2(a) and 2(c) implies that the conventional differential gain model is also appropriate for describing the multimode laser diode gain. However, the introduction of the term phenomenologically added in Eq. (14) to incorporate the band filling effect nullifies the definition of the differential gain coefficient, that is, dg/dNG0. Because of this, despite a positive differential gain coefficient, the gain decreases with increasing carrier density at higher values in the wavelength region from 804 nm to 806 nm in Fig. 3(c), shown as a gain curve crossing. Therefore, we returned to the estimation of the differential gain coefficient with the direct bandgap model.

Fig. 7 Examples of the modal gain (for −5th, 0th, and 5th modes) versus the carrier density in the direct bandgap model.

In the same way, the values of the differential gain coefficient and the carrier density at transparency are estimated for all 41 modes, as shown in Fig. 8. Those parameter values increase monotonically as a function of the mode number or the oscillation frequency. To our knowledge, the carrier density at transparency is treated as a constant for the conduction band of interest. The result in Fig. 8, however, makes it clear that the carrier density at transparency should be treated as a parameter that depends on the oscillation frequency.

Fig. 8 Estimated differential gain coefficient (red) and carrier density at transparency (blue) in each mode in the direct bandgap model.

By applying a linear approximation to the relation between those parameters and the mode number, the form of the laser diode gain expression is greatly simplified:
gn=G0n(NN0n)
(20)
where,
G0n=G00+nδG0
(21)
N0n=N00+nδN0
(22)
Here, G0n and N0n are the differential gain coefficient of the n-th mode and the carrier density at transparency of the n-th mode, respectively; G00 and N00 are the corresponding values of the central mode; and δG0 and δN0 are the differences in differential gain coefficients and carrier densities at transparency between adjacent modes, respectively. The parameter values in Eqs. (21) and (22) are listed in Table 3.

Table 3. Notation and values of parameters for the linearized laser diode gain model

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Figure 9 shows the simulated power spectrum of the gain-switched pulse using the linearized laser diode gain derived in Eqs. (20)(22). The same parameter values in Table 1 and the same pumping condition were used in the simulation, except for the gain term. The mode number n was varied from −20 to 20 in the same way as in the case of the direct bandgap model. The excellent agreement between Fig. 2(a) and Fig. 9 indicates that the linearized laser diode gain is equivalent to the gain in the direct bandgap model, and therefore, it incorporates the band filling effect despite its comparatively simple description.

Fig. 9 Simulated power spectrum of the gain-switched pulse from a multimode laser diode in the linearized laser diode gain model.

Figure 10 plots the gain spectra using Eq. (23) to compare it with the result in Fig. 3(a). Because the linear approximation between the gain and the carrier density was applied to the higher carrier densities, the gain spectra in Fig. 3(a) and Fig. 10 show good agreement at higher carrier densities, although the discrepancy between them is increased at lower carrier densities. This is an essential problem in the use of the linearized gain model. If necessary, saturation effects in the laser diode gain should be introduced. On the other hand, a disagreement seen in the longer wavelength region above 804 nm stems from the linear approximation applied to the result in Fig. 8. However, this does not have a serious effect on the validity of the model. When using the same numerical method with the more realistic GHLBT-SME model that takes a bandtail into account [33

33. F. Stern, “Band-tail model for optical absorption and for the mobility edge in amorphous silicon,” Phys. Rev. B 3, 2636–2645 (1971). [CrossRef]

], we confirmed that the calculated gain spectra showed good agreement with those in Fig. 10 in the longer wavelength region.

Fig. 10 Gain spectra using Eq. (23) when the carrier density is varied from 1.25×1024m−3 to 1.73 × 1024m−3 in 0.04 × 1024m−3 steps.

5. Dynamics of a gain-switched single-mode laser diode

In analyzing the dynamic behavior of a single-mode laser diode using the linearized laser diode gain, a set of the combination values (G0, N0) that satisfy the relations in Eqs. (20)(22) should be chosen. In this case, the mode number n should be treated not as an integer but as a real number. Consequently, this model is the same as the conventional gain written in the form G0(N − N0). However, although only two parameter values are chosen, their values characterize the particular spectral position of the laser diode gain while taking into account the band structure of the semiconductor. Therefore, the parameter values chosen reflect the band filling effect.

In the single-mode oscillation case, the threshold current Ith is analytically calculated by assuming ɛ, β = 0, as follows:
Ith=eV(C1Nth+C2Nth2)
(26)
Nth=N0+1G0τp
(27)
Figure 11 shows the variation of the threshold current as a function of the oscillation frequency for a single-mode oscillation case in the linearized laser diode gain model. For simplicity, the oscillation frequencies are chosen discretely corresponding to the modes in the multimode spectrum. The variation of the threshold current is found to be slightly parabolic but approximately flat, except for the longer wavelength region above 804 nm. A gain-switched pulse is then simulated at each mode under the pumping conditions Idc = 0.95Ith and Imw = 2.0Ith, whose absolute values are approximately constant in the wavelength region below 804 nm. The modulation frequency is set to 1 GHz. The fourth term on the right-hand side of Eq. (1) concerning the frequency detuning was omitted throughout the simulation for the single-mode oscillation case. Figure 12 plots the peak intensity and the pulse width of gain-switched pulses simulated when different combination values (G0, N0) are used. When a sufficient carrier density is supplied, the speed of the gain-switching becomes faster in the shorter wavelength region where the gain varies significantly in association with the band filling effect. However, because the amount of blue shift of the gain peak is limited under a given pumping condition, the speed of the gain-switching certainly has the fastest value at a particular spectral position in the shorter wavelength region, where the pulse must have the shortest pulse width. From Fig. 12, the particular spectral position seems to be located below 794 nm, and as the oscillation frequency becomes higher, the peak intensity increases and the pulse width decreases, concurrently showing saturation characteristics. The increase in the peak intensity seen in the longer wavelength region is based on the increase of the input energy accompanied with the increase of threshold current in the corresponding wavelength region.

Fig. 11 Variation of the threshold current as a function of the oscillation frequency for the single-mode case in the linearized laser diode gain model.
Fig. 12 Simulated peak intensity (blue) and pulse width (red) of gain-switched pulses from a single-mode laser diode at each oscillation frequency in the linearized laser diode gain model.

Figure 13 shows examples of simulated gain-switched pulses from a single-mode laser diode. As simulated in the 20th mode in Fig. 13, a gain-switched pulse followed by a lower and broader second pulse is characteristically generated in the shorter wavelength region corresponding to the mode numbers higher than 13. In the single-mode oscillation case, all gain-switched pulses have a normal pulse shape; that is, the pulses acquire a fast leading edge and a slower trailing edge, regardless of the oscillation frequency.

Fig. 13 Examples of simulated gain-switched pulse from a single-mode laser diode at different oscillation frequencies corresponding to the modes in the multimode oscillation in Fig. 2(a).

A detailed analysis using the TBP–CBP method in Fig. 14, however, gives rise to a difference in shape functions of the gain-switched pulse simulated in Figs. 12 and 13. Increasing the mode number from −20 to 20, an inverted S-shaped locus of the combination values (CBP, TBP) is exhibited. The increase in both the CBP and TBP values at mode numbers from −20 to −12 implies that the long pulse tail caused by the red shift of the gain peak becomes shorter, accompanied by a rising position of the pulse waist, as the oscillation frequency is increased. As a result, the pulse oscillating in the −12th mode (shown in the middle-left panel of Fig. 13) has a shape function approximately the same as the sech2-shape. Then, increasing the mode number from −12 to 13, both the CBP and TBP values decrease monotonically, which indicates that the pulse shape function varies from a nearly sech2-shaped to slightly Lorentzian. Such variation of the pulse shape function makes the position of the pulse waist lower and the ratio of the pulse edge width larger. The increase in the CBP value at mode numbers from 13 to 20 implies that the position of the pulse waist becomes high inversely because a part of the trailing edge is separated from the main body of the pulse due to a speeding-up of the gain-switching. This also saturates the decreasing trend of the TBP value. High-speed gain-switching enhanced by the band filling effect is thus found to cause not only a shortening of the pulse width but also a distinctive variation of the pulse shape function in the single-mode oscillation case.

Fig. 14 TBP–CBP plot for estimating shape functions of gain-switched pulses from a single-mode laser diode. Numbers in the figure correspond to the mode numbers in the multimode oscillation in Fig. 2(a).

6. Conclusions

Acknowledgments

This research was partially supported by the Ministry of Education, Culture, Sports, Science and Technology, under Grant-in Aid for Scientific Research (C), No. 21560044.

References and links

1.

T. Ikegami, K. Kobayashi, and Y. Suematsu, “Transient behaviour of semiconductor injection lasers,” Electron. Commun. Jpn. 53B, 82–89 (1970).

2.

P. M. Boers and M. Danielsen, “Dynamic behaviour of semiconductor lasers,” Electron. Lett. 15, 206–208 (1975). [CrossRef]

3.

D. J. Channin, “Effect of gain saturation on injection laser switching,” J. Appl. Phys. 50, 3858–3860 (1979). [CrossRef]

4.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron.16, 347–355 (1980). [CrossRef]

5.

S. Tarucha and K. Otsuka, “Response of semiconductor laser to deep sinusoidal injection current modulation,” IEEE J. Quantum Electron. 17, 810–816 (1981). [CrossRef]

6.

C. B. Su, V. Lanzisera, and R. Olshansky, “Measurement of nonlinear gain from FM modulation index of In-GaAsP lasers,” Electron. Lett. 21, 893–895 (1985). [CrossRef]

7.

P. M. Downey, J. E. Bowers, R. S. Tucker, and E. Agyekum, “Picosecond dynamics of a gain-switched InGaAsP laser,” IEEE J. Quantum Electron. 23, 1039–1047 (1987). [CrossRef]

8.

R. Olshansky, P. Hill, V. Lanzisera, and W. Powazinik, “Frequency response of 1.3 μm InGaAsP high speed semiconductor lasers,” IEEE J. Quantum Electron. 23, 1410–1418 (1987). [CrossRef]

9.

K. Y. Lau, “Gain switching of semiconductor injection lasers,” Appl. Phys. Lett. 52, 257–259 (1988). [CrossRef]

10.

K. A. Corbett and M. W. Hamilton, “Comparison of the bifurcation scenarios predicted by the single-mode and multimode semiconductor laser rate equations,” Phys. Rev. E 62, 6487–6495 (2000). [CrossRef]

11.

I. V. Koryukin and P. Mandel, “Dynamics of semiconductor lasers with optical feedback: Comparison of multimode models in the low-frequency fluctuation regime,” Phys. Rev. A 70, 053819 (2004). [CrossRef]

12.

K. Wada, H. Sato, H. Yoshioka, T. Matsuyama, and H. Horinaka, “Suppression of side fringes in low-coherence interferometric measurements using gain- or loss modulated multimode laser diodes,” Jpn. J. Appl. Phys. 44, 8484–8490 (2005). [CrossRef]

13.

J. Ohtsubo, Semiconductor Lasers –Stability, Instability and Chaos, Second Ed., (Springer-Verlag, 2007). [PubMed]

14.

C. -C Lin, H. -C Kuo, P. -C Peng, and G. -R Lin, “Chirp and error rate analyses of an optical-injection gain-switching VCSEL based all-optical NZR-to-PRZ converter,” Opt. Express16, 4838–4847 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4838. [CrossRef] [PubMed]

15.

T. Mayer, H. Braun, U. T. Schwarz, S. Tautz, M. Schillgalies, S. Lutgen, and U. Strauss, “Spectral dynamics of 405 nm (Al, In) GaN laser diodes grown on GaN and SiC substrate,” Opt. Express16, 6833–6845 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-6833. [CrossRef]

16.

K. Wada, S. Takamatsu, H. Watanabe, T. Matsuyama, and H. Horinaka, “Pulse-shaping of gain-switched pulse from multimode laser diode using fiber Sagnac interferometer,” Opt. Express16, 19872–19881 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19872. [CrossRef] [PubMed]

17.

C. Chen, G. Ding, B. S. Ooi, L. F. Lester, A. Helmy, T. L. Koch, and J. C. M. Hwang, “Optical injection modulation of quantum-dash semiconductor lasers by intra-cavity stimulated Raman scattering,” Opt. Express18, 6211–6219 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6211. [CrossRef] [PubMed]

18.

W. W. Chow, S. W. Koch, and M. Sargent III, Semiconductor-Laser Physics, (Springer-Verlag, 1994). [CrossRef]

19.

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals, (Springer-Verlag, 1999).

20.

M. Osinski and M. J. Adams, “Intrinsic manifestation of regular pulsations in time-averaged spectra of semiconductor lasers,” Electron. Lett. 20, 525–526 (1984). [CrossRef]

21.

M. Osinski and M. J. Adams, “Picosecond pulse analysis of gain-switched 1.55 μm InGaAsP lasers,” IEEE J. Quantum Electron. 21, 1929–1936 (1985). [CrossRef]

22.

M. Osinski, D. F. G. Gallagher, and I. H. White, “Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method,” Electron. Lett. 21, 981–982 (1985). [CrossRef]

23.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]

24.

B. Sermage, J. P. Heritage, and N. K. Dutta, “Temperature dependence of carrier lifetime and Auger recombination in 1.3 μm InGaAsP,” J. Appl. Phys. 57, 5443–5449 (1985). [CrossRef]

25.

B. W. Hakki, “Optical and microwave instabilities in injection lasers,” J. Appl. Phys. 51, 68–73 (1980). [CrossRef]

26.

P. -L Liu, C. Lin, I. P. Kaminow, and J. J. Hsieh, “Picosecond pulse generation from InGaAsP lasers at 1.25 and 1.3 μm by electrical pulse pumping,” IEEE J. Quantum Electron. 17, 671–674 (1981). [CrossRef]

27.

G. Lasher and F. Stern, “Spontaneous and stimulated recombination radiation in semiconductors,” Phys. Rev. 133, A553–A563 (1964). [CrossRef]

28.

T. Suhara, Semiconductor laser fundamentals (Kyoritsu, 1998), Chap. 3. in Japanese.

29.

W. B. Joyce and R. W. Dixon, “Analytic approximations for the Fermi energy of an ideal Fermi gas,” Appl. Phys. Lett. 31, 354–356 (1977). [CrossRef]

30.

A. E. Siegman, Lasers (University science books, 1986), Chap. 26.

31.

R. A. Linke, “Modulation induced transient chirping in single frequency lasers,” IEEE J. Quantum Electron. 21, 593–597 (1985). [CrossRef]

32.

K. Wada, J. Fujita, J. Yamada, T. Matsuyama, and H. Horinaka, “Simple method for estimating shape functions of optical spectra,” Opt. Commun. 281, 368–373 (2008). [CrossRef]

33.

F. Stern, “Band-tail model for optical absorption and for the mobility edge in amorphous silicon,” Phys. Rev. B 3, 2636–2645 (1971). [CrossRef]

OCIS Codes
(270.3430) Quantum optics : Laser theory
(300.6170) Spectroscopy : Spectra
(320.1590) Ultrafast optics : Chirping
(320.5390) Ultrafast optics : Picosecond phenomena
(320.5540) Ultrafast optics : Pulse shaping
(250.5960) Optoelectronics : Semiconductor lasers

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: November 29, 2010
Revised Manuscript: January 21, 2011
Manuscript Accepted: January 30, 2011
Published: February 2, 2011

Citation
Kenji Wada, Hiroyuki Yoshioka, Jiaxun Zhu, Tetsuya Matsuyama, and Hiromichi Horinaka, "Simple form of multimode laser diode rate equations incorporating the band filling effect," Opt. Express 19, 3019-3036 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3019


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References

  1. T. Ikegami, K. Kobayashi, and Y. Suematsu, "Transient behaviour of semiconductor injection lasers," Electron. Commun. Jpn. 53B, 82-89 (1970).
  2. P. M. Boers, and M. Danielsen, "Dynamic behaviour of semiconductor lasers," Electron. Lett. 15, 206-208 (1975). [CrossRef]
  3. D. J. Channin, "Effect of gain saturation on injection laser switching," J. Appl. Phys. 50, 3858-3860 (1979). [CrossRef]
  4. R. Lang, and K. Kobayashi, "External optical feedback effects on semiconductor injection laser properties," IEEE J. Quantum Electron. 16, 347-355 (1980). [CrossRef]
  5. S. Tarucha, and K. Otsuka, "Response of semiconductor laser to deep sinusoidal injection current modulation," IEEE J. Quantum Electron. 17, 810-816 (1981). [CrossRef]
  6. C. B. Su, V. Lanzisera, and R. Olshansky, "Measurement of nonlinear gain from FM modulation index of In-GaAsP lasers," Electron. Lett. 21, 893-895 (1985). [CrossRef]
  7. P. M. Downey, J. E. Bowers, R. S. Tucker, and E. Agyekum, "Picosecond dynamics of a gain-switched InGaAsP laser," IEEE J. Quantum Electron. 23, 1039-1047 (1987). [CrossRef]
  8. R. Olshansky, P. Hill, V. Lanzisera, and W. Powazinik, "Frequency response of 1.3 μm InGaAsP high speed semiconductor lasers," IEEE J. Quantum Electron. 23, 1410-1418 (1987). [CrossRef]
  9. K. Y. Lau, "Gain switching of semiconductor injection lasers," Appl. Phys. Lett. 52, 257-259 (1988). [CrossRef]
  10. K. A. Corbett, and M. W. Hamilton, "Comparison of the bifurcation scenarios predicted by the single-mode and multimode semiconductor laser rate equations," Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 62, 6487-6495 (2000). [CrossRef]
  11. I. V. Koryukin, and P. Mandel, "Dynamics of semiconductor lasers with optical feedback: Comparison of multimode models in the low-frequency fluctuation regime," Phys. Rev. A 70, 053819 (2004). [CrossRef]
  12. K. Wada, H. Sato, H. Yoshioka, T. Matsuyama, and H. Horinaka, "Suppression of side fringes in low-coherence interferometric measurements using gain- or loss modulated multimode laser diodes," Jpn. J. Appl. Phys. 44, 8484-8490 (2005). [CrossRef]
  13. J. Ohtsubo, Semiconductor Lasers -Stability, Instability and Chaos, Second Ed., (Springer-Verlag, 2007). [PubMed]
  14. C.-C. Lin, H.-C. Kuo, P.-C. Peng, and G.-R. Lin, "Chirp and error rate analyses of an optical-injection gain switching VCSEL based all-optical NZR-to-PRZ converter," Opt. Express 16, 4838-4847 (2008), http:// www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4838. [CrossRef] [PubMed]
  15. T. Mayer, H. Braun, U. T. Schwarz, S. Tautz, M. Schillgalies, S. Lutgen, and U. Strauss, "Spectral dynamics of 405 nm (Al, In) GaN laser diodes grown on GaN and SiC substrate," Opt. Express 16, 6833-6845 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-6833. [CrossRef]
  16. K. Wada, S. Takamatsu, H. Watanabe, T. Matsuyama, and H. Horinaka, "Pulse-shaping of gain-switched pulse from multimode laser diode using fiber Sagnac interferometer," Opt. Express 16, 19872-19881 (2008), http: //www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-19872. [CrossRef] [PubMed]
  17. C. Chen, G. Ding, B. S. Ooi, L. F. Lester, A. Helmy, T. L. Koch, and J. C. M. Hwang, "Optical injection modulation of quantum-dash semiconductor lasers by intra-cavity stimulated Raman scattering," Opt. Express 18, 6211-6219 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-6-6211. [CrossRef] [PubMed]
  18. W. W. Chow, S. W. Koch, and M. SargentIII, Semiconductor-Laser Physics, (Springer-Verlag, 1994). [CrossRef]
  19. W. W. Chow, and S. W. Koch, Semiconductor-Laser Fundamentals, (Springer-Verlag, 1999).
  20. M. Osinski, and M. J. Adams, "Intrinsic manifestation of regular pulsations in time-averaged spectra of semiconductor lasers," Electron. Lett. 20, 525-526 (1984). [CrossRef]
  21. M. Osinski, and M. J. Adams, "Picosecond pulse analysis of gain-switched 1.55 μm InGaAsP lasers," IEEE J. Quantum Electron. 21, 1929-1936 (1985). [CrossRef]
  22. M. Osinski, D. F. G. Gallagher, and I. H. White, "Measurement of linewidth broadening factor in gain-switched InGaAsP injection lasers by CHP method," Electron. Lett. 21, 981-982 (1985). [CrossRef]
  23. C. H. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J. Quantum Electron. 18, 259-264 (1982). [CrossRef]
  24. B. Sermage, J. P. Heritage, and N. K. Dutta, "Temperature dependence of carrier lifetime and Auger recombination in 1.3 μm InGaAsP," J. Appl. Phys. 57, 5443-5449 (1985). [CrossRef]
  25. B. W. Hakki, "Optical and microwave instabilities in injection lasers," J. Appl. Phys. 51, 68-73 (1980). [CrossRef]
  26. P.-L. Liu, C. Lin, I. P. Kaminow, and J. J. Hsieh, "Picosecond pulse generation from InGaAsP lasers at 1.25 and 1.3 μm by electrical pulse pumping," IEEE J. Quantum Electron. 17, 671-674 (1981). [CrossRef]
  27. G. Lasher, and F. Stern, "Spontaneous and stimulated recombination radiation in semiconductors," Phys. Rev. 133, A553-A563 (1964). [CrossRef]
  28. T. Suhara, Semiconductor laser fundamentals (Kyoritsu, 1998), Chap. 3. in Japanese.
  29. W. B. Joyce, and R. W. Dixon, "Analytic approximations for the Fermi energy of an ideal Fermi gas," Appl. Phys. Lett. 31, 354-356 (1977). [CrossRef]
  30. A. E. Siegman, Lasers (University science books, 1986), Chap. 26.
  31. R. A. Linke, "Modulation induced transient chirping in single frequency lasers," IEEE J. Quantum Electron. 21, 593-597 (1985). [CrossRef]
  32. K. Wada, J. Fujita, J. Yamada, T. Matsuyama, and H. Horinaka, "Simple method for estimating shape functions of optical spectra," Opt. Commun. 281, 368-373 (2008). [CrossRef]
  33. F. Stern, "Band-tail model for optical absorption and for the mobility edge in amorphous silicon," Phys. Rev. B 3, 2636-2645 (1971). [CrossRef]

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