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9
Analysis of DFB Laser Diode Characteristics Based on Transmission-Line Laser Modelling (TLLM)
9.1 INTRODUCTION
In Chapter 8 we introduced transmission-line laser modelling (TMLM). In this chapter, TLLM will be modified to allow the study of dynamic behaviour of distributed feedback laser diodes, in particular the effects of multiple phase shifts on the overall DFB LD performance. We can easily model any arbitrary phase-shift value by inserting some phase-shifter stubs into the scattering matrices of TLLM. This helps to make the electric field distribution and hence light intensity of DFB LDs more uniform along the laser cavity and hence minimise the hole burning effect.
9.2 DFB LASER DIODES
As explained in Chapter 2, the feedback necessary for the lasing action in a DFB laser diode is distributed throughout the cavity length. This is achieved through the use of a grating etched in such a way that the thickness of one layer varies periodically along the cavity length. The resulting periodic perturbation of the refractive index provides feedback by means of Bragg diffraction rather than the usual cleaved mirrors in Fabry–Perot laser diodes [1–3]. Bragg diffraction is a phenomenon which couples the waves propagating in the forward and backward directions. Mode selectivity of the DFB mechanism results from the Bragg condition. When the period of the grating, , is equal to mB=2neff, where B is the Bragg wavelength, neff is the effective refractive index of the waveguide and m is an integer representing the order of Bragg diffraction induced by the grating, then only the mode near the Bragg wavelength is reflected constructively. Hence, this particular mode will lase whilst the other modes exhibiting higher losses are suppressed from oscillation. The coupling between the forward and backward waves is strongest when m ¼ 1 (i.e. first-order
Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1
232 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM
Bragg diffraction). By choosing appropriately, a device can be made to provide distributed feedback only at the selected wavelengths.
In recent years, DFB LDs have played an important role in the long-span and high-bit-rate optical fibre transmission systems because of their stronger capability of single longitudinal mode operation. To overcome the two modes’ degeneracy and achieve a pure single-mode operation, quarterly-wavelength-shifted (QWS) DFB lasers have been proposed [4]. However, in such QWS DFB lasers, spatial hole burning effects enhance the side modes when the coupling coefficient is large (i.e. L > 3). In order to combat this effect, multiple-Phase shift DFB lasers have been proposed [5–8]. It has been shown that side modes can be effectively suppressed and a stable and pure single-mode operation results. With the development of laser structures, efficient and relatively accurate simulation models are becoming more and more important for laser designs and operation optimisation due to the complication and expense involved in laser fabrication processes.
Distributed feedback semiconductor lasers have a greater mode selectivity than Fabry– Perot devices and so are preferred as sources for long-haul high-capacity-fibre systems. However, dynamic single-mode (DSM) operation is still difficult. Accurate multi-mode dynamic computer models could help in designing DSM DFB devices. Many DFB models calculate the individual mode threshold gains in an attempt to assess wavelength stability. However, these usually neglect the saturation and inhomogeneity of the gain which occurs at the onset of lasing. Dynamic models are available, but these assume a single oscillating mode, making the study of mode stability impossible.
The ideal semiconductor laser model would mimic the operation of the real devices in every detail, simulating all characteristics of the laser while accounting for all variations in device structure, processing, drive electronics and external optical components [9–10]. The model could be connected to other device models to form an optical system model. Such a model would improve the design of photonic devices, circuits and systems. It could also be used for detailed optimisation in particular applications.
Limitations in computing resources require that simplifications and assumptions have to be made before a model is developed. Many optoelectronic device models use rate equations to describe the interactions between the average electron and photon populations in the device [9–11]. Numerous adaptations of this technique have been proposed. For example, using a photon rate equation for each longitudinal laser mode gives the laser’s spectrum during modulation [12] and dynamic frequency shifting (chirping) may be estimated from the transient responses of both populations [13]. The laser rate equations may also describe saturation in laser amplifiers [14], the dynamic behaviour of model-locked lasers [15] and the transient response of cleaved-coupled-cavity lasers [16].
The limitation of using photon density as a variable is that it does not contain optical phase information. Optical phase is important when there is a set of coupled optical resonators such as in coupled-cavity lasers, external-cavity lasers, DFB lasers, or even Fabry–Perot lasers with unintentional feedback from external components. In these cases, the output wavelength of the devices and its current to light characteristics are determined by optical interference between the resonators. Although rate equations can be used in simple cases, by calculating effective reflection coefficients at discrete wavelengths [16], finding these wavelengths becomes difficult with multiple resonators exhibiting gain and variable refractive indices, such as in the DFB laser [17].
A development of the rate equation approach is to use a SPICE-compatible equivalent circuit of the laser diode. This may be used to find the time-varying photon density for a given drive current waveform or, alternatively, to find the frequency response of the devices
DFB LASER DIODES 233
[18]. This approach has an advantage in that it includes parasitic components in the laser chirp and mount and can be linked to models of the drive circuit for evaluation of the system’s response to modulation.
An alternative variable to photon density is optical field, which contains phase information and thus offers the possibility of dealing with multiple reflections. The optical field within a resonator system may be solved in the frequency domain or in the time domain. Frequency-domain models often use a transfer-matrix description of the laser that may be obtained by multiplying together the transfer matrices describing each individual reflection [19–20]. However, if the spectrum of a modulated laser is required, the multiplication has to be performed for each wavelength at each time step [17]. This is computationally inefficient.
Time-domain models using optical fields are better suited to modulated devices with multiple resonators than frequency-domain models because the former are simpler to develop and require less computation. Time-domain optical-field models are commonly based on scattering matrix descriptions of the individual reflections and of the gain medium. The scattering matrices may be connected by delays (transmission lines) so that reflected waves out of one scattering matrix can be connected to each adjacent matrices after the delay. The delays represent the optical propagation time along a portion of the waveguide. A solution for the network is found by iteration, each iteration representing an increase in time equal to the delay.
At high-frequency modulation (16–17 GHz) [21], the dynamic characteristics of lasers are important and design methods that can help to predict the chirp and modulation efficiency are needed. The dynamic response of lasers is generally studied by solving a set of rate equations that govern the interaction between the carriers and photons inside the active region of the laser cavity. In the earliest work, the equations are usually linearised to allow solutions to be found for small-signal oscillations. Although this gives insight to the important physical parameters, it has limited applicability. Large-signal dynamics with non-linear effects such as gain saturation, spatial hole burning and changes of electron and photon densities along the length of lasers are now essential in the study of DFB lasers where these effects are more significant than in Fabry–Perot lasers [22–23]. The transmission-line laser model based on the transmission-line modelling (TLM) method, is being developed to study many of the dynamic effects in lasers.
Transmission-line laser modelling, which was developed by Lowery, employs time-domain numerical algorithms for laser simulation [24–33]. This model splits the laser cavity longitudinally into a number of sections. In each section, TLLM uses a scattering matrix to represent the optical process, such as stimulation emission, spontaneous emission and attenuation. The matrices of these sections are then connected by transmission lines, which account for the propagation delays of the waves. From the iterations of scattering and connecting processes, the output electric field in the time domain can be obtained. Then, by applying a Fourier transform, we can easily obtain the laser output spectra. Large-signal dynamics with non-linear effects such as the changes of electron and photon densities along the length of the laser and spatial hole burning are key issues in the analysis of DFB laser diodes. These dynamic effects can be investigated easily by using transmission-line laser modelling.
TLLM models have been used to analyse QWS DFB LDs [32]. With the insertion of a zero-reflection interface (identity matrix) half way along the cavity, the effects of QWS on laser operation have been simulated successfully. However, using this method we can only analyse DFB laser structures with one =2 phase shift at the centre of the cavity. We cannot use this technique to analyse other phase shift values or multi-phase-shift (MPS) lasers.
234 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM
9.3 TLLM FOR DFB LASER DIODES
In general, two operations, scattering and connecting, are involved in transmission-line laser modelling. The scattering operation takes voltage pulses incident on the nodes, kAi, and scatters them to give voltage pulses reflected from the nodes, kAr. The reflected and incident voltage pulses are related together via the following scattering matrix, S which includes
stimulation, emission, spontaneous emission and attenuation processes. That is
kAr ¼ S kAi þ Is ð9:1Þ
where k is the iteration number and Is is the spontaneous wave. As discussed in Chapter 8, the scattering operation can be derived from a knowledge of the impedances of the transmission lines and associated components, such as resistors at the nodes. Equation (9.1) can be modified to include the source voltage pulses, kAs, so
kAr ¼ S kAi þkAs þ Is ð9:2Þ
The reflected pulses that propagate to the next scattering nodes become new incident pulses for the next scattering operation. This process can be expressed as
kþ1Ai ¼ C kAr ð9:3Þ
In eqn (9.3) C is the connection matrix that can be derived from the topology of the network. It should be noted that for all pulses to arrive at the nodes synchronously, the transmission lines must have equal delay times. Each delay time should also be equal to the iteration time step t. In the numerical calculation, we need to initialise the value of voltage Ai and then repeat eqns (9.1) and (9.2) to find the time evolution of the voltage Ai or Ar. In transmission-line laser models, the voltage pulses represent the optical fields along the cavity. A chain of transmission lines connects these fields from the laser rear facet via optical cavity to the laser front facet. The scattering matrices represent the optical processes of stimulated emission, spontaneous emission and attenuation. The local carrier density will be updated according to the rate equation model at each time step and the magnitudes of these processes at a particular matrix will also be re-calculated with the new information of the carrier density. It should be noted that the local carrier density should be updated at each time step ðtÞ accoding to the rate equation model. The updated carrier density will then be used to set the magnitude of the optical processes in the scattering matrix.
9.4 A DFB LASER DIODE MODEL WITH PHASE SHIFT
In a DFB laser diode, the forward and backward waves are coupled along the entire cavity length because of the refractive index modulation along the cavity. This coupling can be
Figure 9.1 The TLLM model for uniform DFB laser diodes.
A DFB LASER DIODE MODEL WITH PHASE SHIFT 235
Figure 9.2 The TLLM model representing a phase shift.
represented by impedance discontinuities placed between the model sections as shown in Fig. 9.1. However, a model for the phase shift is needed to model such DFB laser diodes. In doing so, phase stubs are employed and connected to the main transmission line. In this model circulators are used (see Fig. 9.2) to send the waves out of the stubs in the correct direction. For example, a forward wave will enter the first left-hand circulator (port 1) and is directed to the stub port (port 2). Since the stub presents an impedance mismatch, part of the wave will be reflected back into port 2. The circulator then directs this reflected wave to port 3, where it continues on as a forward wave. The remainder of the wave enters the stub to be delayed before returning to port 2 to be directed to port 3. Backward waves simply pass from port 3 to port 1 of this first circulator. A second set of three-port circulators is used to delay the backward waves.
The phase delay caused by a stub can be varied by altering its impedance. For example an infinite stub impedance gives a reflection with zero phase shift; a matched capacitive stub gives a phase shift of ð2ptfÞ radians; a zero impedance stub gives p radians; a matched inductive (shorted) stub gives ðÿ2pftÞ radians where f is the optical frequency. Other phase shifts are available over a limited bandwidth by using other reflection coefficients.
A complete DFB laser diode model with phase shift is shown in Fig. 9.3. Here, scattering matrices have been inserted between the circulators of each section. Also, alternate con-necting transmission lines have different impedances. This creates impedance mismatches at thesectionboundaries,whichcoupletheforwardandbackwardwaves[28].Eachsectionhas an associatedcarrierrateequationmodeltoenablethelocalgain,refractiveindexandspontaneous noise to be calculated from the injection current and the carrier recombination rates [24].
Figure 9.3 A complete DFB laser diode model with phase shift. p is a phase-shift stub, l and c are gain-filter stubs and i is the injection current.
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