Flatnessbased precompensation of laser diodes
Abstract
A physical nonlinear dynamical model of a laser diode is considered. We propose a feedforward control scheme based on differential flatness for the design of inputcurrent modulations to compensate diode distortions. The goal is to transform without distortion a radiofrequency current modulation into a light modulation leaving the laserdiode and entering an optic fiber. We prove that standard physical dynamical models based on dynamical electron and photons balance are flat systems when the current is considered as control input, the flat output being the photon number (proportional to the light power). We prove that inputcurrent is an affine map of the flat output, its logarithm and their timederivatives up to order two. When the flat output is an almost harmonic signal with slowly varying amplitude and phase, these derivatives admit precise analytic approximations. It is then possible to design simple analogue electronic circuits to code approximations of the nonlinear computations required by our flatnessbased approach. Simulations with the parameters of a commercial diode illustrate the practical interest of this precompensation scheme and its robustness versus modelling and analogue implementation errors.
footnoteinfo]This work has been supported by the projet INRIA STICTunisie (2006/2007).
First]A. Abichou Second]S. El Asmi Third]P. Rouchon
onlinear systems, flatness, laser diode, precompensation, inversion.
1 Introduction
RadiooverFiber is a very attractive technique for wireless access network infrastructure, because it can transmit microwaves and millimeterwaves through optical fibers for a long distance. Therefore, the distribution of radio signals over optical fiber, which is of great interest for many applications such as broadband wireless access networks, sensor networks, radar and satellite communications, has been intensively studied. And there have been rapid advances in the techniques to generate and transport radio signals over optical fiber in recent years (see, e.g., [2]). When the wireless channel is in series with the optical link, nonlinear distortion due to the electrical/optical conversion is the biggest concern (see, e.g., [7]). This makes nonlinearity compensation an attractive solution to improve link performance.
The focus of this work is to investigate the standard physical model used for describing commercial laser diodes. This model is based on rate equation for the electron and photon populations. This allows us to propose a precompensation based on differential flatness for the design of the input current in order to cancel diode distortions. Simulations corresponding to a commercial laser diode illustrate the precompensation scheme. When the distortion due to the optic fiber is negligible, our flatnessbased precompensation scheme can be interesting to increase the bandwidth and transmission rate.
This paper is organized as follows. In section 2 we recall the standard diode model and show that it is structurally flat, with the photon number being the flat output . We show via elementary computations that the input current is a linear combination of , , , and with coefficients depending via explicit formula on the physical parameters (equation (3)). This relation is the starting point of the flatness based precompensation scheme addressed in section 3. In Section 4, we propose natural approximations when is quasiharmonic with slowly varying amplitude and phase. These approximations simplify considerably the computation burden in such a way that a nonlinear analogue circuit can be easily design to realize our flatnessbased precompensation scheme. In conclusion, we propose some hint to deal with phase influence that becomes important when fiber distortions cannot be negligible.
2 The flat physical nonlinear model
A standard nonlinear model relating the inputcurrent to the power of the emitted light is recalled in [5] where this model is used to analyze the nonlinear ”transfer” between inputcurrent modulation and the resulting output ) (see also [1] for a simple exposure on such basic models):
(1) 
where

and are photon and carrier lifetimes;

is the normalized carrier density;

is the threshold current and is the constant related to other physical parameters, .
Since the inputcurrent appears only in the second differential equation of (1) (the electron balance equation), the first state, , is a flat output. More precisely, this means that, if we consider (1) as a control system with input and output , we have a nonlinear input/output system such that its inverse admits no dynamics. This system is differentially flat (see [3, 4, 6, 8]) with as flat output: the input is a nonlinear function of , , . Notice that this property is independent of the precise form of when it depends effectively on . This is always the case for physical reasons.
Let us detail now the computations with the precise form of given here above. The first equation of (1) yields
where and are constant parameters. Thus we can derive this expression versus to get as a function of , and :
A linear combination of the two equations of (1) gives explicitly:
Thus we have the explicit formula relating to the derivatives of :
With the normalized input/ouput variables
(2) 
we have the following simple but nonlinear input/output relationships that is equivalent to the physical model (1):
(3) 
where the four parameters , , and are positive timescales. Remember that the normalized electron density is a combination of and :
3 Flatnessbased precompensation
We can now use (3) for deriving feedforward strategy in order to compensate the nonlinear distortion due to the diode. Assume that our goal is to transfer the amplitude and phase modulated signal from the current to the light leaving the diode. Here is the carriage pulsation, typically around a few Ghz for radio frequencies, and and are slowly varying amplitude and phase where the bits are encoded. This means that and . Without lost a generality, we can assume that . Assume finally that this modulated signal has to be converted in photons according to where is some positive constant. This means that our goal is to find the current leading to such . If one knows the diode parameters we can use (3) to compute .
We have tested in simulation this simple precompensation scheme for the commercial diode considered in [5]. For this diode, we have
We choose a modulation frequency of Ghz (), and
(4) 
The average normalized light power is corresponds physically to mW since mW. The simulations of figure 1 illustrate the interest of taking into account the derivative terms in (3) to modulate the input current .
4 Analogue implementation issues
For such high carrier frequency GHz, it is difficult and almost impossible to use realtime numerical computations. Let us now propose two approximations such that the above precompensation scheme can be applied in realtime via a specific electronic circuit.
The first approximation is relative to the computation of derivative of . Since we are looking for such that with and slowly varying quantities, we have
(5) 
The second approximation is relative to the nonlinearity attached to the terms in (3). Now we will use and propose an approximation up to second order terms:
(6) 
Plugging these approximations into (3) shows that can be expressed approximatively as a polynomial of degree in and . Such polynomial computations can be easily done by a nonlinear analogue circuit. Figure 2 shows that the diode response remains almost the same when such polynomial approximations are used. Other simulations displayed on figure 3 show that such precompensation scheme are also robust to parameters uncertainties.
5 Conclusion
We have shown that it is possible to compute, via flatnessbased motion planing techniques, adapted input current modulations in order to generate, in realtime and via analogue nonlinear circuits, a given modulation of the light entering the optic fiber. If the distortion along the fiber is not negligible (long fiber and/or refractive index strongly dependent versus optic frequency), we have to consider also the phase dynamic of the light emitted by the diode (see, e.g., [1] for more details about the spectrum widening due to phase/amplitude variations). Following [5] we have to complete (1) by
where is the phase of the emitted light and a positive constant. Thus the emitted light entering the optic fiber is represented by the classical electric field (complex notations)
where is the optical frequency (around Hz). It is interesting to notice that the diode dynamics (described by three states ) and one input ) is still flat with being the new flat output. This results from . This structural property could certainly be exploited in other precompensation schemes taking into account such additional phase effect.
Footnotes
 thanks: [
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