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Abstract
An improved technique of continuous shaping current-injected waveforms based on the single-mode rate equations is proposed to suppress relaxation oscillations (ROs) from direct modulation of distributed feedback laser (DFB). The signal expression of shaping current is deduced theoretically from the dependence of DFB desired output waveforms in detail, and the specific parameters derivation of the different polynomial degree is also discussed necessarily. Furthermore, a polynomial p-function with inverse operation is adopted to construct the Fourier series corresponding to injection current waveform signal. The equivalent circuit model with DFB phenomenological description is injected into shaping current signal to verity the proposed validity by evaluating the static and dynamic characteristics. The simulation results of the optimized shaping signal show the good agreement with the desired output pulse including rising and falling edge and suppress the ROs amplitude dramatically at the two jump edges.
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1. Introduction
With the development of science and technology, higher requirements are put forward for the accuracy and efficiency of communication technology [1]. Semiconductor lasers are widely used as light sources in many optical communication transmission systems [2–4]. Direct modulation requires higher performance semiconductor lasers. As a kind of semiconductor lasers, distributed-feedback (DFB) lasers are in widespread use in the fields of modern optical communication [5], such as Wavelength Division Multiplexing (WDM) and Orthogonal Frequency Division Multiplexing (OFDM) transmission systems. The DFB laser is a type of laser diode whose active region distribute is periodically structured as a diffraction grating [6–8]. The diffraction grating provides optical feedback to selective wavelength. Because of the single longitudinal mode operation has been fully realized, DFB lasers are especially suitable for the light source of integrated optical circuit [9–11].
The application of NG-PON2 (Next-Generation Passive Optical Network Stage 2) multi-path multiplexing means the pursuit of high-speed, easy to implement and low-cost modulation technology. Generally, there are direct modulation DFB lasers and external modulation DFB lasers. The method of external modulation means that the modulator is located outside the resonator the laser. Therefore, the external modulation changes the parameters of the output laser. The advantages of this method are fast transmission rate and long transmission distance [12], but the disadvantages are complex external devices and high cost. The direct modulation is to change the parameters of oscillation by modulating signals in laser oscillation process, which is simple and low cost [13–15]. But this method can cause relaxation oscillations (ROs). In order to improve the disadvantages of directing modulation, the methods of suppress ROs include controlling the current flowing through the device, or injecting light source forms feedback [16], or using simple circuit shaping injection current [17] or analysis of signal current shaping to conduct modulation of the output of the semiconductor laser.
Roy Lang proposed a method to suppress relaxation oscillation which employed GaAs injection lasers both as external source of injected radiation and modulated laser [18]. Suematsu Y showed that the external circuit could suppress the relaxation oscillation of the output light of the injection laser.
The frequency response and transient characteristics of the direct modulation was analyzed by taking an electric resonant circuit connected to the laser diode and tuned to eh ROs as an example [19]. L. Bickers showed that the dynamic linewidth of 1.5$\mu m$ DFB semiconductor laser could be reduced by properly shaping the modulated current waveform [20].
N. Dokhane used the transient phase space technique to modulate the injection laser current to eliminate the relaxation oscillation [21]. Based on the relationship between the field intensity and the carrier density in two-dimensional space, the shape of the injection current switch was simply modified. Using piece-wise constant current to suppress relaxation oscillations may cause difficult discontinuities. Lucas Illing designed the current input waveforms to control the nonlinear internal degree of freedom of the laser moving along the specific state space trajectory to optimize the light intensity [22]. The photon intensity is not easy to measure, so it is difficult to optimize the photon intensity.
In order to obtain the relationship between the output photon density and the modulation current, Hoang Nam Nhat studied the rate equations of direct modulation semiconductor laser [23]. It was proved that the differential equations could be simplified to a special case at a certain time. Besides, he obtained the unique solution of the differential equation. LAN Yi et al. used Volterra series theory instead of Fourier series to establish the dynamic model of nonlinear laser. For the p-order inversion, he compensated the nonlinear misalignment of the laser, determined the injected waveforms and completed the current shaping [24].
Our research team proposed an improved and high-speed equivalent circuit model of DFB lasers on the basic of rate equations [25]. This model took into account the effects of parasitic parameters and non-radiative recombination compared with traditional equivalent circuit model, which further verified the availability of the model. The new equivalent circuit model provided an effective tool for directly modulating the characteristics of DFB lasers. In the same year, the technique and optimization requirements for shaping piecewise step current waveforms were demonstrated [26]. This technique was used to optimize the piecewise step current, and the relaxation oscillations of the laser were eliminated by injecting the piecewise step current.
The shaping current technique is simpler, cheaper and easier to implement than other directly modulated methods (Fig. 1). The relaxation oscillations can be suppressed by changing the current- injected without increasing the time delay. To overcome the problems associated with shaping current, there are two methods to modulate. One is step current shaping and the other is continuous current shaping. The former uses piecewise constant current, which would possibly present difficult discontinuities. The latter avoids the discontinuity of step current by using continuous current, but this method has not been applied to the semiconductor lasers actual circuit model at present. In this manuscript, a significant feature of our method is that a finite rise time forming, continuous, continuous shaping current is obtained by introducing a p-function. The optimized shaping current is injected into the equivalent circuit model of DFB laser to overcome the nonlinear distortion. The performance of the laser is reflected by the output power. It is verified that the continuous shaping current technique can be applied to the actual circuit model.
2. Derivation of injection current formula expression
2.1 Continuous current shaping based on rate equations
The rate equations are effective theories to study and design all kinds of optoelectronic devices, especially semiconductor lasers. The semiconductor laser is assumed to be a single-mode laser. The photonics process in the cavity can be expressed by two differential equations. The relationship between carrier density and photon density [22] is:
where $N$ and $S$ present carrier density and photon density, $G({S(t ),N(t )} )$ is the optical gain factor with nonlinear effect, ${J_0}$ and $J(t )$ are the bias current and modulated current. $\Gamma $ is the field limiting factor, $d$ represents the thickness of the laser active layer, ${\gamma _c}$ and ${\gamma _s}$ is the decay rate of photon density and carrier density.
In the case of no modulation ($J(t )= 0$), we select the static working point $({{S_0},{N_0}} )$. The nonlinear gain near this point is expanded to the first order.
The steady-state solution of the rate equation represents the variation of the $N$ and $S$ with the injection current. Under the condition of ${{dS} / {dt}} = {{dN} / {dt = 0}}$ and $({{S_0},{N_0}} )$, they can be expressed by:
where $J + {J_m} = {{({{J_0} + J(t )} )} / {({{\gamma_s}{N_0}ed} )}} - 1$ is defined current, $s(t )= {{S(t )} / {{S_0}}}$ and $n(t )= {{N(t )} / {{N_0} - 1}}$ are the dimensionless definitions of $S(t )$, $N(t )$. Dimensionless gain is $g = {G / {{G_0}}}$. Hence,
The dimensionless frequency and time corresponding to the relaxation oscillation angle is ${\omega _R} = \sqrt {{\gamma _c}{\gamma _n} + {\gamma _s}{\gamma _p}} $ and $\tau = t{\omega _R}$. First, the dimensionless photon density equation is described as follows because of ${{dS} / {dt}}$, $s = {S / {{S_0}}}$ and $\tau = t{\omega _R}$.
Then, the simplification gives $\frac{{ds}}{{d\tau }} = \frac{1}{{{\omega _R}}} \cdot [{ - {\gamma_c}s + \Gamma G({N,S} )s} ]$ with the condition of $s = {S / {{S_0}}}$. $\frac{{ds}}{{d\tau }} = \frac{1}{{{\omega _R}}} \cdot [{ - {\gamma_c}s + \Gamma g{G_0}s} ]$ is further simplified by $g\textrm{ = }{G / {{G_0}}}$. Due to Eq. (3) that $\Gamma {G_0} = {\gamma _c}$, the dimensionless photon density equation is expressed as
Similarly, the dimensionless carrier density equation ${{dn} / {d\tau }}$ can be derived as followed. The expression of ${{dn} / {d\tau }}$ is
On account of $S = {S_0}s$, $N(t) = {N_0}({n(t )+ 1} )$, $G = g{G_0}$ and ${G_0}{S_0} = \frac{{{J_0}}}{\alpha } - {\gamma _s}{N_0}$, the dimensionless carrier density equation is shown as
Furthermore, the defined current is rearranged to $1 - \frac{{{J_0}}}{{{\gamma _s}{N_0}\alpha }}\textrm{ = }1 - ({1 + J} )={-} J$ when $J(t )= 0$. Equation (8) can be replaced by:
Thus, the dimensionless differential equation can be written by
For computational ease, we introduce that the expression of desired output photon density waveforms is $s(\tau )= {e^{y(\tau )}}$. To solve for $n(\tau )$ using the ${{ds} / {d\tau }}$ equation, differentiate, and rearrange ${{dn} / {d\tau }}$ the equation to solve for ${J_m}(\tau )$. The first order differential equation of $s(\tau )$ is given by
Using Eq. (11) and Eq. (12), it follows that
Combined it with Eq. (9), the first derivative of (13) is solved to obtain the injection of the required output waveform as follows.
2.2 Selection of desired waveforms
The output waveform is selected as a rectangular pulse because of its perfect symmetry. It can divide the waveform into two symmetrical segments. More specifically, it will perform the ${s_{down}}$ and ${s_{up}}$. Since the idealized continuous rectangular wave is non-existent, it can only be as close to the rectangular pulse as possible. For a better discussion, the high-order Fourier series method is used to constrain the desired waveforms to achieve the effect of steadily approximation to rectangular pulse. The formula of the basis function $y(\tau )$ is as follows:
Here $x = \frac{\tau }{T}$. With the aim of minimizing the transition time between rising and falling, the formula of the f-function is given by $f(x )= {{({a + 1} )x} / {ax + 1}}$. For nonlinear systems, the convergence speed of state vector is faster than that of ordinary stable systems under the premise of exponential stability. Thus, $s(\tau )= {e^{y(\tau )}}$ is introduced to improve the convergence speed. The shaping current at the rising and falling edges can be represented by different $p(x)$.The p-function needs to satisfy several conditions in Table 1. $p^{\prime}(x )$, $p^{\prime\prime}(x )$ and $p^{\prime\prime\prime}(x )$ represent the first, second and third derivatives of the p-function at x respectively.
Table 1. The conditions of p-function
In addition, the derivative of p-function is zero to ensure that the final current waveforms ${J_m}$ has zero first derivative and is zero at $x = 1$ and 0. The Eq. (14) gives the relationship between shaping current-injected ${J_m}(\tau )$ and y-correlation function, including $y(\tau )$, ${y^{\prime}}(\tau )$ and ${y^{^{\prime\prime}}}(\tau )$. So, the expression of $J_m^{\prime}(\tau )$ is related to ${y^{\prime}}(\tau )$, ${y^{^{\prime\prime}}}(\tau )$ and ${y^{^{\prime\prime\prime}}}(\tau )$. Besides, the values of ${y^{\prime}}(\tau )$, ${y^{^{\prime\prime}}}(\tau )$ and ${y^{^{\prime\prime\prime}}}(\tau )$ should be constrained to be zero to ensure $J_m^{\prime}(\tau )= 0$ when $x = 1$ and 0. And the Eq. (15) shows the relationship between $y(\tau )$ and $p(x )$. Hence, the values of ${p^{\prime}}(x )$, ${p^{^{\prime\prime}}}(x )$ and ${p^{^{\prime\prime\prime}}}(x )$ should be constrained to be zero when $x = 1$ and 0. We furthermore require that the total input current $({0 < 3 + J + {J_m} < 7} )$ of the laser must be positive, which is a physically reasonable requirement.
According to the conditions in Table 1, p-function is discussed in two cases. The former describes the p-function needed by the rising edge, while the latter describes the p-function needed by the falling edge.
The Eq. (16) and Eq. (17) are taken into $s(\tau )= {e^{y(\tau )}}$ to get the rising edge and falling edge of the expected waveforms. Thus, the shaping current waveforms expression can be obtained after shaping. The square wave can be approximately obtained by up-down conversion of the two waveforms.
Table 2 lists the parameters we use, and the values of constants are the best results obtained through repeated experiments
Table 2. The meaning of parameters in equation
3. Equivalent circuit model DFB lasers
In the second section, the dimensionless rate equations are obtained by various transformations. Combining with the rate equations, the expression of continuous shaping current can be shown in this manuscript. Next, the shaping current need to be injected into an actual equivalent circuit model to observe the results.
3.1 Establishment of DFB laser’s equivalent circuit model
The starting point of establishing an equivalent circuit model of DFB lasers is the rate equations describing its electron-optic characteristics, as shown in Eq. (1). In the modulation response of DFB laser, there are many factors that produce response damping. These factors included carrier transverse diffusion, spontaneous emission, nonlinear absorption and so on. These nonlinear effects are considered by using a unified field dependent optical gain compression factor, which is introduced into the single-mode rate equations [26] and the following rate equations are obtained.
In Eq. (18), the first part on the right of the equation represents the increase of carrier density caused by current injection. ${I_j} = JWL$, $W$ and $L$ are the width and length of the active layer respectively. $\alpha = e{V_{act}}$, $e$ is an electron’s charge. ${V_{act}}$ is the volume of the active region which is composed of $dWL$. So, the first term can be roughly divided into $\frac{J}{{ed}}$. The second part ${R_n}(N )$ and third part ${R_r}(N )$ represent the rate of non-radiative recombination and the rate of radiative recombination, which are expressed as
In Eq. (19), The first part on the right of the equation represents the increase in photon density caused by light gain. The second part is the increase of photon density caused by the recombination coupling of spontaneous emission into lasing mode. The third part represents the decrease in photon density due to optical loss.
Using (18) and (19) to construct the equivalent circuit model of DFB laser, we need to know the Shockley relationship [27] between the injected carrier density and the junction voltage:
We substitute (20) into (18) and get the following result
where ${C_d}$ is the diffusion capacitance at junction, ${I_{st}}$, ${I_r}$ and ${I_n}$ represent currents in different recombination processes. Their specific expressions are given by ${C_d} = \frac{{\alpha {N_e}}}{{\eta {V_T}}}\exp \left( {\frac{{{V_j}}}{{\eta {V_T}}}} \right)$, ${I_n} = \alpha {R_n}(N)$, ${I_r} = \alpha {R_r}(N)$,and ${I_{st}} = \alpha GS$,where ${V_j}$ is the junction voltage.The Eq. (22) can be obtained by multiplying $e{V_{act}}$ on both sides of formula (14) and ordering ${V_{ph}} = s{V_{act}}{V_T}$, ${C_{ph}} = {e / {{V_T}}}$ and ${R_{ph}} = {{{V_T}{\tau _{ph}}} / e}$. Where ${V_T} = {{KT} / q}$. It can be seen that ${V_{ph}}$ is the light pressure in the active region. ${C_{ph}}$ has capacitance dimension and can be equivalent to a capacitance, and ${R_{ph}}$ has resistance dimension and can be equivalent to a resistance.
As shown in Fig. 2, ${R_s}$ and ${C_p}$ are the series resistance and parallel capacitance generated by the contact between the laser chip and the electrode respectively. The value of ${R_s}$ is 5Ω. Besides, the value of the equivalent resistance of current leakage in the active region ${R_d}$ is $1 \times {10^{15}}\Omega $[26].
Fig. 1. Comparison diagram of direct modulation and the result after shaping. The red dotted line of (a) demonstrates how a rectangular wave total input current gives rise to (b) relaxation oscillations of the DFB laser output power. The blue line in (a) and (b) show the shaped current and output power that desired.
Fig. 2. Diagram of the DFB laser’s equivalent circuit model [26]. It includes three parts: parasitical circuit, electrical circuit and optical circuit.
The equivalent circuit model of laser is of great significance for predicting and analyzing the working characteristics of laser. In fact, many equivalent circuit models have been proposed to analyze laser output behavior, including the small-signal equivalent-circuit model for photonic crystal Fano lasers [28], the equivalent-circuit model for dual-wavelength quantum cascade lasers [29], and the noise equivalent-circuit model for semiconductor lasers [30]. Under the perfect work condition of the DFB laser, the equivalent circuit model in Fig. 2 is established by our research team based on the single-mode rate equations. Furthermore, this model is simulated by PSpice soft. The soft is a general circuit analysis program for microcomputer series developed from SPICE (Simulation Program with Integrated Circuit Emphasis). Compared with other equivalent circuit models, this model considers the effects of parasitic parameters and nonradiative recombination. The simulation results and measured results obtained by PSpice can verify the effectiveness of the circuit model, and it has been proved that the circuit model can accurately predict the static and dynamic characteristics of DFB laser.
As shown in Fig. 3, the curves are flat at low frequency, and with the modulation frequency close to the relaxation oscillation frequency the modulation response distorts. The result shows that increasing the bias current without limit can lead to self-pulsation. For a bias current of 30 mA, the DFB laser can be modulated at frequencies of about 12 GHz.
Fig. 3. Small signal frequency response of the equivalent circuit model for the high-speed DFB laser at basis current: 30 mA (black line), 40 mA (red line) and 50 mA (blue line)
3.2 Relationship between photon density and output power
DFB laser is based on FP laser, using grating filter to make the device have only one longitudinal mode output (see Fig. 4). Fabry Perot interferometer is the first optical resonator in the history of laser technology, which is a parallel plane cavity. The characteristic of the cavity is that it can make full use of the active medium to make the beam oscillate in the whole volume of the active medium.
In practical terms, photon density is a microcosmic quantity that is not suitable for direct observation. For the laser, the output power of the laser is the best optical parameter of the model.
The output power of the left and right sides can be expressed by
where $\theta $ is the angle of emergence, and is related to the parameters of the resonator [27].The expression for the output power becomes $P = \theta {V_{ph}}$.With the help of definition ${V_{ph}} = s{V_{act}}{V_T}$, the relationship between photon density and output power is obtained
The relationship between ${J_m}$ and output power is written by
4. Optimization results and analysis
In Chapter III, the equivalent circuit model is established with nonlinear effect based on the rate equations of DFB laser. The rising part of a rectangular current waveform is injected into DFB’s equivalent circuit model and the results as shown in Fig. 5, Fig. 6. Obviously, the output power waveforms have nonlinear distortion. ROs is an inevitable process in the process of laser establishment. It is an effect produced by the transient process of unsteady image at the beginning. The relaxation oscillations will cause the instability of laser output, which is manifested as multiple spikes. The stronger the pump excitation, the higher the damping oscillation frequency and the faster the attenuation. Although there is such a spike, it will eventually tend to steady state.
Fig. 4. Structure of DFB laser. It includes F-P cavity and optical grating. ${R_L}$ and ${R_R}$ are the reflectivity of left and right end faces; ${P_L}$ and ${P_R}$ are the output power of the left and right sides respectively.
Fig. 5. Diagram of the output waveform of the rising edge. The blue line shows desired waveforms. The red dotted line shows the output power of the directly modulated DFB laser with the rising edge of the input rectangular wave, where ROs are clearly observed.
Fig. 6. Diagram of the output waveform of the falling edge. The blue line shows desired waveforms. The red dotted line demonstrates that the falling edge also has significant relaxation oscillations, but eventually tends to steady state.
Similarly, the falling part of the rectangular current waveform is injected into the equivalent circuit model of the DFB laser to observe the results.
In the upper right corner of Fig. 7, the values between blue dotted lines represents the relaxation oscillation resonance frequency in the on level, while the values between red dotted lines represents the relaxation oscillation resonance frequency in the off level. The relaxation oscillation resonance frequency in the on level and in the off level can be observed by frequency chirp. The frequency chirp of semiconductor laser means that the output light frequency will jump with time, that is, the output light phase will jump with time. The circuit model is also driven by an ideal rectangular wave current source. The expression of frequency chirp is given by $\Delta v(t )\cong \frac{\alpha }{{4\pi }}\left( {\frac{d}{{dt}}\ln P(t )+ \frac{{2\Gamma \varepsilon }}{{V\eta hv}}P(t )} \right)$.
Fig. 7. Frequency chirp versus time from the high-speed DFB laser’s equivalent circuit model at various bias currents 30 mA,40mA and 50 mA.
The specific work has been described in detail in previous manuscripts [25]. As shown in the Fig. 7, the relaxation oscillation resonance frequency in the on level and in the off level are about 10GHz.
For suppressing the ROs of laser output, the shaping current formula obtained in the third section is used. Firstly, the rising edge is verified. The rising part of the shaping current waveform is schematically shown in Fig. 8(a). The rising part of the shaping current is injected into the established equivalent circuit model. Considering the non-observable shape of photon density and the relationship between photon density and output power, the output power waveform is as shown in Fig. 8(b), which is the rising edge of desired output waveforms. Significantly, the ROs have been eliminated.
Fig. 8. The shaping of the rising edge. (a) The shaped current waveforms of the modulated rising edge. (b)The output power waveforms after laser injection.
In the same way, the (17) is substituted into the derived current to get the shaping current waveform of falling edge, as shown in Fig. 9(a). The modulated falling edge current waveforms is injected into the circuit model of DFB laser. The output power waveforms is observed as shown in Fig. 9(b). It is can be found that ROs are almost completely eliminated. By observing the results of Fig. 8 and Fig. 9, it can be said that the modulation of the rising edge and the falling edge of a rectangular pulse is successful. The shaping current technique is suitable for the equivalent circuit model of DFB laser with a band width of 12 GHz at basic current 30 mA. Finally, the rising and falling edges of the shaping current should be combined and injected into the DFB’s equivalent circuit model to validate its effect.
Fig. 9. The shaping of the falling edge. (a) The shaped current waveforms of the modulated falling edge. (b) The output power waveforms after injecting into the laser.
It is widely known that the actual input current waveform of laser is random and arbitrary. Thus, the non-periodic current waveform is injected to simulate the randomness of the actual input current in Fig. 10. The injection current waveforms are shown in Fig. 10(a). After the simulation of the laser, the output power waveforms are shown in Fig. 10(b). We directly inject the unmodulated waveforms into the DFB laser model, and the output power waveforms have obvious nonlinear distortion in Fig. 11. In contrast, the calculated waveforms are injected to obtain waveforms without ROs. The waveforms shown in Fig. 10 is in good agreement with our expected waveforms.
Fig. 10. Shaping current waveforms and output power waveforms. (a) Shaping injection current waveforms. (b) Waveforms of the corresponding output power.
Fig. 11. Ordinary directly modulation current waveforms and output power waveforms. (a) Injection current waveforms of ordinary directly modulated. (b) Waveforms of the corresponding output power.
Obviously, the shaping current waveform in Fig. 10(a) is obtained by the calculation software. A joint simulation is established between the calculation software and PSpice software, and the results are shown in Fig. 10 (b). However, the periodic input current in Fig. 11(a) is provided by PSpice circuit simulation software. The simulation results in Fig. 11(b) further verify the effectiveness of the equivalent circuit model established by PSpice. Besides, the ROs of Fig. 11(b) reflect the instability of the laser oscillation process, which is characterized by multiple spikes. Although the unstable state can eventually transition to the stable state, there will be inevitable interference in the process of information transmission. That is to say, the accuracy of information transmission cannot be guaranteed. Continuous shaping current technique can be reasonably used to improve this deficiency.
The selection of desired waveform in this manuscript is the rectangular pulse. Whether other complex waveforms can be obtained by the second section of the derivation theory remains to be studied. In addition, the time required for the rising edge and falling edge is the optimal result obtained through repeated experiments. We hope that we can take the rising or falling time as the optimization goal, and use the optimization software to continue to optimize desired waveforms. In this way, we can refer to the same method to output any desired waveforms. Besides, the time to reach the ideal waveforms can be as short as possible through optimization.
5. Conclusion
In this communication, the continuous shaping current technique is proposed and verified based on the equivalent circuit model of DFB laser. The expression of shaping current-injected is successfully obtained by analyzing the rate equations of semiconductor laser and desired waveforms. With a p-function derived by our team transforms on the desired output waveform and optimization, the expression of shaping current works well in injecting into the DFB laser's equivalent circuit model. The analysis on the resulted shows that the continuous shaping current technique provides powerful guiding ability for the suppression of relaxation oscillations. Besides, the comparison of the proposed continuous shaping current technique shows that this method is more efficient and low-cost. The proposed technique is applied in the circuit model which conforms to industry standards and can accurately predict the static and dynamic characteristics of DFB laser.
Finally, a potential future work is adopting the proposed technique to actual optical communication system, and our research team is working on it. Furthermore, our research team will continue to study the universal continuous shaping current technique in order to suppress the relaxation oscillations in any output signal. Therefore, continuous shaping current being a simple but effective modulation technique has a good prospect.
Funding
National Natural Science Foundation of China (No. 61471224, No. skr20-1-008, No. ZR2017MF070).
Acknowledgement
The authors would like to thank Professor Wei-ping Huang for his active guidance and valuable suggestions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this manuscript may be obtained from the authors upon reasonable request.
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