1. Introduction

Random number generation (RNG) is widely used in many applications including cryptographycally secured communications, Monte Carlo simulation, industrial testing, quantitative finance, gambling, etc. [1,2]. Both pseudorandom number generators (PRNGs) based on deterministic algorithms, and random number generators based on hardware physical devices, have been used for the previous applications. Examples of physical processes used in these generators include: radioactive decay, Zener noise, Johnson’s noise [1,2], chaos noise [2,3] and quantum phenomena [1]. Quantum random number generators (QRNGs) are a particular case of hardware physical RNGs in which the data are obtained from quantum events. The main advantage of QRNGs is that their randomness is inherent to quantum mechanics making quantum systems a perfect source of entropy for random number generation [2].

Most of the existing QRNGs are based on quantum optics [2]. Light from single-photon sources, semiconductor lasers or light emitting diodes are used in these generators. QRNGs based on single-photon detection methods include: two-path splitting of single photons [4], generators measuring the time of arrival of photons [5], photon counting generators [6], and attenuated pulse generators [7]. The properties of some of these methods have been experimentally compared in [8]. Multiphoton QRNGs have also been proposed and experimentally and theoretically demonstrated. For instance, generators based on quantum vacuum fluctuations [9], on amplified spontaneous emission (ASE) signals [10], and on phase noise in continuous wave [11–13] and in pulsed semiconductor lasers [14–17] including full integration on an InP platform [18], have been demonstrated.

In these last generators the current applied to a single-mode semiconductor laser is periodically modulated from a well below threshold value to a value above threshold, in such a way that gain-switching operation appears, typically at Gbps rates [14–19]. While the laser is below threshold the optical phase becomes random due to the spontaneous emission noise. The laser emits then a series of pulses with a random phase. Phase fluctuations are converted into amplitude fluctuations by using an unbalanced Mach-Zehnder interferometer, with a delay matching the pulse repetition period. Detection and filtering of the amplitude provides the generation of random values with an arcsine distribution. Fast generation rates, up to 43 Gbps quantum random bit generation, have been experimentally shown [15]. There are several advantages in using QRNGs based on pulsed gain-switching of single-mode laser diodes. They are made of commercially available components and the high signal level permits the use of standard photodetectors. They are simple, fast, robust, have low cost, and operate with flexible clock frequencies.

The description and understanding of the physical processes underlying the operation of a QRNG are essential for their applications. For instance in cryptography, randomness is very often used for the generation of a secret key. QRNGs based on gain-switched laser diodes are appropriate for Quantum Key Distribution (QKD) systems because high speed random bit generation for state preparation is required [17,20]. For these type of QRNGs it is essential to have a good quantitative description of the phase diffusion process in which their fluctuations are characterized and measured.

Theoretical and experimental studies of the fluctuations of laser light began in the decade of the sixties [21–25]. Study of fluctuations of the light emitted by semiconductor lasers has also received a lot of attention [26–31]. A good theoretical description, deduced from first principles and valid for below and above threshold operation, is based on the Fokker-Planck equation or alternativelly on stochastic rate equations of Langevin’s type [23,32]. Good quantitative agreement between experiments and theory can only be achieved when the complete set of parameters of the stochastic rate equations is known for the specific laser diode. Having this knowledge permits a good quantitative description of the dependence of phase diffusion on the laser and modulation parameters.

Modelling of phase fluctuations in gain-switched semiconductor lasers for QRNG has been performed by numerical integration of the laser stochastic rate equations [15,17,18,33,34]. All these models have considered a linear dependence of the carrier recombination rate, the spontaneous emission rate, and the semiconductor material gain on the carrier density. Very recently, numerical simulations using a model in which the spontaneous emission rate has a quadratic dependence have been performed and compared with results from models using linear terms for that rate [35]. These numerical simulations use the complete set of parameters that have been extracted for a single-mode discrete mode laser (DML) [35,36]. Using these parameters a very good quantitative agreement between experiments and theory for describing optical frequency comb formation has been shown for a very wide range of gain-switching conditions [36,37].

In this paper we report an experimental and theoretical study of the phase diffusion in the gain-switched DML characterized in [35,36]. Our theoretical modelling includes nonlinear dependencies of the carrier recombination rate and the semiconductor material gain on the carrier number. We compare with the results obtained with linear models in order to know if these linear approximations are able to quantitatively describe the phase diffusion process. We focus our attention in the below threshold operation that is the regime where most of the phase diffusion occurs in QRNG. Using the simple analytical expressions for the laser linewidth derived from the Schawlow-Townes law, and the extracted parameters for our laser, we show that nonlinear dependence of both, carrier recombination rate and semiconductor material gain must be considered for a good quantitative description of the phase diffusion. In particular, good agreement between experiments and theory is obtained when a 2-parameter logarithmic material gain and a 3-parameter cubic carrier recombination are considered. We show that going beyond the usual linear dependencies is important to quantitatively describe the measured nonlinear dependence of the spectral linewidth, and hence of the phase diffusion coefficient, as a function of the bias current.

Our paper is organized as follows. In section 2, we present our theoretical model. Section 3 is devoted to the calculation of simple expressions of the spectral width and their comparison with our experimental results. In section 4, we analyze theoretically and experimentally the phase diffusion for below and above threshold operation. Finally, in section 5 we discuss the validity of the linear approximations and summarize our results.

2. Theoretical model

Gain-switched single-mode semiconductor laser dynamics can be modelled by using a set of stochastic rate-equations that read (in Ito’s sense) [26,35,36,38]

Numerical integration of the previous stochastic rate equations present instabilities when the photon number is very small because $P$ can become negative when the values of the current are smaller than the threshold current, $I_{th}$, a situation always found when using these lasers for QRNG. This problem can be solved by integrating the equivalent equations for the complex electrical field [35]. These equations read

Analysis of phase diffusion in gain-switched laser diodes has been done by using a rate-equation model in which the carrier recombination and the unsaturated gain are assumed to be linear [15,17,18,33,34] not only for QRNG but also for quantum key distribution using the same laser systems [20]. These models are based on a linearization of $R(N)$ around the carrier number at threshold, $N_{th}$ [26,40], where $N_{th}=N_t+1/(G_N\tau _p)$. This linear expansion reads $R(N)\sim R(N_{th})+(N-N_{th})/\tau _n$, where $\tau _n$ is the differential carrier lifetime at threshold that is given by $\tau _n^{-1}=\bigl (\frac {dR}{dN}\bigr )_{N_{th}}$. Substituting this expansion in Eq. (3) and using that the threshold current is given by $I_{th}=eR(N_{th})$ we obtain

3. Spectral width below threshold: comparison between theory and experiments

In this section we first focus on the calculation of the width of the optical spectrum. In the second part of this section we compare with experiments performed with a discrete mode semiconductor laser (DML). The spectral linewidth below threshold is given by the usual Schawlow-Townes law that is accurate for the amplified spontaneous emission (ASE) regime [26]. This law reads

Very simple expressions are obtained for the two considered unsaturated gain functions, $G_\textrm {lin}(N)$, and $G_\textrm {log}(N)$. These read

Our experiments are performed using a single-mode DML. Details on the device, measurement procedure and extraction of the parameters of the model are detailed in [35–37]. The parameters have been extracted using $G_\textrm {lin}(N)$ and their values are the following: $G_N=1.48\times 10^4$s$^{-1}$, $N_t=1.93\times 10^7$, $\epsilon = 7.73\times 10^{-8}$, $\tau _p = 2.17$ ps, $\alpha =3$, $\beta =5.3\times 10^{-6}$, $A=2.8\times 10^8$ s$^{-1}$, $B=9.8$ s$^{-1}$, and $C=3.84\times 10^{-7}$ s$^{-1}$ [35–37]. We note that measurements for extracting parameters have been performed for bias current above threshold, a condition for which $G_\textrm {lin}(N)$ works well. Measurements have been performed fixing the temperature at 25$^o$C. For this value $I_{th}$ is 14.14 mA, the emission wavelength is 1546.985 nm at $I$=30 mA, and the wavelength separation between consecutive longitudinal modes is 1.28 nm. $I_{th}$ is a parameter in our model that is used for calculating $N_{th}$ from $N_{th}=\bar {N}(I_{th})=5.045\times 10^7$. This value coincides to that obtained by using $G_\textrm {lin}(N)$, that is $N_t+1/(G_N\tau _p)$. We also note that the values of $\tau _n$ and $\tau _e$ are rather different: using Eq. (8) and Eq. (9) we obtain that $\tau _n=$0.24 ns and $\tau _e$= 0.57 ns.

The laser linewidth (FWHM) as a function of $I$ when $I<I_{th}$ is shown with squares in Fig. 1. Data close to threshold have been obtained with a high resolution (0.08 pm) optical spectrum analyzer. Data corresponding to currents below 13 mA have been obtained with another optical spectrum analyzer with worse wavelength resolution (0.06 nm). As the current decreases, $\Delta \nu$ has the nonlinear increase typically found in ASE operation. We first try to describe that increase by using Eq. (16), derived for $G_\textrm {lin}(N)$. Nonlinear behaviour is also expected for $\Delta \nu _\textrm {lin}$ because $\bar {N}$ does not depend linearly on $I$, as seen from Eq. (18). Figure 1 also shows the results obtained with Eq. (16). $\Delta \nu _\textrm {lin}$ describes well the experimental results close to $I_{th}$ as expected since $G_\textrm {lin}(N)$ works well close to threshold [40]. However it greatly underestimates the linewidth as $I$ decreases: $\Delta \nu _\textrm {lin}$ is 32 and 50 $\%$ smaller than the experimental values when $I=0.7 I_{th}$ and $I=0.5I_{th}$, respectively.

 

Fig. 1. Measured linewidth as a function of $I$ for bias currents below threshold. Theoretical results obtained with Eq. (16) and with Eq. (17) are plotted with black and red solid lines, respectively.

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In order to try to improve the theoretical description of the linewidth we now consider the logarithmic dependence of $G(N)$, $G_\textrm {log}(N)=G_0\ln (N/N_0)$. However we do not know the values of the $G_0$ and $N_0$ parameters. A very simple way of obtaining $G_0$ is to use Eq. (17) to fit the experimental results because $\Delta \nu _\textrm {log}$ only depends on $G_0$. We consider then the value of $G_0$ for which we obtain equal values in experiments and theory at an intermediate value of Fig. 1 ($I=9.75$ mA$\sim 0.7 I_{th}$), that is $G_0=1.05\times 10^{12} s^{-1}$. The value of $N_0$ can then be obtained from $G_\textrm {log}(N_{th})=1/\tau _p$, that is $N_0=N_{th}\exp (-1/(G_0\tau _p))=3.25\times 10^7$. These values are similar to those found in the literature. For instance typical $g_0$ and $n_t$ in [26] are 1800 cm $^{-1}$ and 2.2 $\times 10^{18}$ cm $^{-3}$, respectively. If we calculate the corresponding $g_0$ and $n_t$ for our device we get: $g_0=G_0/(\Gamma v_g)=2040$ cm $^{-1}$ and $n_t=N_0/V_{act}=2.13\times 10^{18}$ cm $^{-3}$, where we have used the confinement factor, $\Gamma$, group velocity, $v_g$, and active volume, $V_{act}$, corresponding to our device [36].

The description of the linewidth significantly improves if $G_\textrm {log}(N)$ is used. Figure 1 shows that $\Delta \nu _\textrm {log}$ describes much better than $\Delta \nu _\textrm {lin}$ the measured linewidth: $\Delta \nu _\textrm {log}$ is only 18 $\%$ smaller than the experimental values when $I=0.5I_{th}$. These results show that considering the logarithmic dependence of the unsaturated material gain on the carrier number is necessary for a good description of the linewidth below threshold.

Another confirmation of the good results obtained with the logarithmic gain is by plotting the optical spectrum calculated from the numerical simulation of our model. We numerically integrate Eqs. (5)–(6) with $G_\textrm {log}(N)$ using a 2.5 ps sampling time and a 81.92 ns temporal window. The optical spectrum is calculated by averaging over 1000 temporal windows. We compare in Fig. 2 the theoretical optical spectrum obtained with the corresponding experimental one obtained for a well-below threshold current (9 mA). A good agreement between experiment and theory is found.

 

Fig. 2. Optical spectrum for a constant bias current of 9 mA. Experimental and simulated results are plotted with black and red solid lines, respectively. Zero frequency corresponds to a 1546.54 nm wavelength.

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4. Phase diffusion

Studies on phase diffusion in gain-switched laser diodes for QRNG have been based on the numerical simulation of stochastic rate equations. Most of these analysis consider multiplicative noise for the three variables: $P, \phi$, and $N$, in a similar way to Eqs. (1)–(3). It has been shown that noise in the carrier equation is not important for the statistical properties of the phase [30,31,34]. Two different dependencies of the spontaneous emission rate on $N$, linear and quadratic, have been considered giving rise to different multiplicative noise terms in equations for $P$ and $\phi$. Comparison between results obtained with both dependencies has been recently performed [35] to show that the quadratic dependence is the best choice, in agreement with [26]. Since these numerical studies are aimed to study phase diffusion in QRNG the bias current is modulated, typically with a squared wave dependence like that described in Section 2.

Analytical studies of phase diffusion have not considered modulation of the bias current. They just consider a constant value of $I$ and additive noises in which the steady-state average values of the variables appear in the terms that multiply the Langevin terms, $F_P(t)$ and $F_\phi (t)$ [27,29]. In fact, these terms have been derived from first principles for a system where the matter and the radiation have reached equilibrium [22]. Taking this into account we consider the following equations

Taking into account Eq. (12) and that $\bar {N}$ and $\bar {P}$ obtained from Eqs. (19)–(20) and Eq. (3) are equal to those obtained from Eqs. (1)–(3) we get that

We now consider the phase statistics when $I>I_{th}$. The spectral linewidth above threshold is given by the "modified" Schawlow-Townes law [26,29]

When the steady-state corresponding to $I>I_{th}$ has been reached the phase diffuses in such a way that its variance, $\sigma _\phi ^2$, increases linearly with $t$ with diffusion coefficient, $D_\phi$ [27,29], that is

Taking into account Eq. (23) we get that

The phase diffusion coefficient $D_\phi$ is given by

In Fig. 3 we show $D_\phi$ as a function of the bias current, for values below and above $I_{th}$. In this figure we have considered the logarithmic gain dependence, that is $\Delta \nu$ given by Eq. (17). $D_\phi$ decreases with $I$ in such a way that around threshold it is dramatically reduced: 3 orders of magnitude in a narrow current range (from 13 to 16 mA). The agreement between experimental and theoretical results below threshold is good. We have not measured values above threshold because of the resolution limit of our equipment (10 MHz) although we should expect also good agreement between theory and experiments, as was shown in [40]. A gap in the theoretical results is apparent from Fig. 3. This corresponds to the transition around $I_{th}$ because Eq. (24) is not valid if $I=I_{th}$. However we note that the results obtained from that equation have been plotted for $I\geq$14.16 mA=1.001 $I_{th}$, a value very close to threshold.

 

Fig. 3. Phase diffusion coefficient as a function of the bias current. Measured and theoretical results are plotted with squares and solid line, respectively

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In Fig. 4 we show the experimental and theoretical values of the standard deviation of the phase after 1 ns (in $2\pi$ units), $\sigma _\phi (1\, \textrm {ns})/(2\pi )=\sqrt {D_\phi /2}/\pi$, as a function of $I$. $\sigma _\phi (1 \,\textrm {ns})$ is reduced several orders of magnitude when the current is increased above threshold: for instance it goes from 4$\pi$ at $I$=7 mA to 0.018$\pi$ at $I=$ 30 mA, that is a reduction in a factor of 222. Our analytical and experimental results are in agreement with those obtained from numerical simulation [33,41]. The system analyzed in this work is a true quantum entropy source when the standard deviation of the phase at the end of the period is greater than 2$\pi$ ($\sigma _\phi > 2\pi$) [34,42]. The good quantitative agreement between experiment and theory permits to assure that the previous condition is fulfilled after 1 ns time evolution if $I<11.6 mA\sim 0.8 I_{th}$.

 

Fig. 4. Standard deviation of the phase after 1 ns (in 2$\pi$ units) as a function of the bias current. Measured and theoretical results are plotted with squares and solid line, respectively

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5. Discussion and conclusions

Up to now we have discussed the importance of considering the logarithmic gain dependence in order to obtain a realistic quantitative description of the phase diffusion when the laser is below threshold, that is the regime where most of that diffusion occurs in QRNGs using laser diodes. This description has been made using nonlinear carrier recombination terms. We now discuss the validity of considering a linear carrier recombination, approximation that has been usually followed for modelling phase diffusion for QRNG and QKD using gain-switched semiconductor lasers [15,17,18,20,33,34]. In this approximation Eq. (11) describes the change of the carrier number. Using this equation the steady state value of $N$ corresponding to the below threshold operation depends linearly on $I$: $\bar {N}=\tau _e I/e$. Substitution of this expression in Eq. (16) and Eq. (17) gives

 

Fig. 5. Measured linewidth as a function of $I$ for bias currents below threshold. Theoretical results obtained with Eq. (28) and with Eq. (29) are plotted with black and red dashed lines, respectively. Theoretical results obtained with Eq. (16) and with Eq. (17) and nonlinear $R(N)$ have been also plotted with black and red solid lines, respectively.

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The equations that have been used to obtain the analytical expression of the phase diffusion coefficient, Eqs. (19) and (20), are not the same than Eqs. (1) and (2). Spontaneous emission noise terms in Eqs. (1) and (2) are good approximations to those of Eqs. (19) and (20), that correspond to the exact form that was deduced from first principles for a system where the matter and the radiation have reached equilibrium [22,31]. When the matter-radiation equilibrium has not yet been achieved, the exact form of the spontaneous emission strength is unknown [31]. In situations where a high-frequency modulation of the bias current is applied, like those found in QRNG, the laser is in a transient regime and the assumption of the validity of the spontaneous emission strength in Eqs. (1) and (2) cannot be justified in a rigorous way [31]. Nevertheless, noise terms like those in Eqs. (1) and (2) have been used for studying the statistics during the transient regime of laser diodes showing good agreement between experiments and theory [30,38,43–50]. A different set of rate equations in which additive noises have strengths given by the time dependent averages of the variables have been recently proposed [34]. Statistics obtained with these types of modelling could be different because of the different nature of noise terms. This comparison will be the subject of future work. However, a first estimation of the frequencies for which we would expect similar results when using Eqs. (1)–(2) and Eqs. (19)–(20) can be done by evaluating the time it takes the system to reach the steady state conditions after switching-off the current. Under typical modulation conditions used in QRNG this time is several hundreds of ps [35], so similar results would be expected if the modulation frequency is around 1 GHz or smaller. Extension of our analytical calculations to faster modulation frequencies can be done by modifying in an appropriate way the theory developed in [49].

Typical rate equations, like those used in this work, assume certain approximations. For instance $B$ is taken as a constant value although it is a parameter that depends on the carrier density [26,51]. It means that considering a single value for $B$, whether it is obtained with above or below threshold estimations, will not be enough to achieve perfect agreement between theoretical and experimental results. Therefore, using rate equations for analyzing the gain switched semiconductor laser for QRNG, in which the evolution above and below threshold is described with a single model, implies the consideration of these types of approximations.

Experimental results corresponding to parameter extraction of our laser were obtained in [35,36] but a few of them were assumed, in particular, $B$. The value of $B$ was chosen based on reported values of high speed 1.55 micron lasers. We chose the value $B=9.8 s^{-1}$ that corresponds to $B_d=1.5 \times 10^{-10} cm^3s^{-1}$ [52] (where $B_d$ is the corresponding recombination parameter written in equations for carrier and photon densities). The range of variation of $B_d$ is not very large. For instance, $B_d$ varies from $0.6 \times 10^{-10} cm^3s^{-1}$ to $2.4 \times 10^{-10} cm^3s^{-1}$, depending on different material composition, carrier density and temperature [26,51]. Our chosen value of $B$ is in the middle of the previous range. We can study the implications of assuming this value by evaluating how the theoretical linewidth changes from the value we calculated in Fig. 1 with respect to the values obtained at the extremes of the previous range. We have calculated $\Delta \nu _\textrm {log}$ for $B=3.92 s^{-1}, 9.8 s^{-1}$, and $15.68 s^{-1}$, that corresponds to $B_d=0.6\times 10^{-10} cm^3s^{-1}, 1.5\times 10^{-10} cm^3s^{-1}$ , and $2.4\times 10^{-10} cm^3s^{-1}$, respectively. The largest changes of the linewidth are obtained for the smallest values of the current and are only 4% and 9% for $B=3.92 s^{-1}$, and $15.68 s^{-1}$, respectively. So the implications of our assumption for the experimental validation performed in this work are small.

Our results have been obtained only with one laser. It would be desirable to have more experiments to support the generalization of our results. We think that our results could hold for other types of single-mode edge emitting lasers (DBR, DFB$\ldots$) because their dynamics and the way in which spontaneous emission noise affects that dynamics are similar. In fact, stochastic rate equations modelling is a common tool to describe the statistics of the emitted light in all those devices. Some recent results [53] have compared the gain-switching performance of the laser analyzed in this manuscript (Discrete Mode Laser) with DFB lasers, indicating that they are very similar: the driving conditions rather than the device properties determine the coherence of the emitted signal when generating optical frequency combs. Extending these experimental comparisons to the case of gain-switching for QRNG would be interesting in order to support the generalization of our results.

Summarizing, we have analyzed theoretically and experimentally the phase diffusion problem in a gain-switched single-mode semiconductor laser. We have shown the importance of a good knowledge of the values of the laser parameters and the need of considering logarithmic material gain and nonlinear recombination of carriers for achieving a good agreement between theory and experiments. This is essential for a good quantitative description of QRNGs based on gain-switched laser diodes.