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8 Circuit and Transmission-Line Laser Modelling (TLLM) Techniques 8.1 INTRODUCTION Although today microwave and optical engineering appear to be separate disciplines, there has been a tradition of interchange of ideas between them. In fact, many traditional microwave concepts have been adapted to yield optical counterparts. The laser, as an optical device that plays a key role in optoelectronics and fibre-optic communications, grew from the work of its microwave predecessor, the maser (microwave amplification by stimulated emission of radiation) [1]. The operating principle behind the laser is very similar to that of the microwave oscillator. In a semiconductor laser, the required feedback may either be provided by the cleaved facets of Fabry–Perot lasers or by a periodic grating in distributed feedback lasers. The optical technique of injection locking of lasers by external light [2] is an idea borrowed from the phenomenon of injection locking of microwave oscillators by an external electronic signal [3]. The close relationship between optical and microwave principles suggests that it may be advantageous to apply microwave circuit techniques in modelling of semiconductor lasers. Engineers work best when using tools they are familiar with. In particular, electrical and electronic engineers are familiar with well-established electrical circuit models as tools to aid themselves in the understanding and prediction of behaviour of electrical machines or electronic devices. Since the early days of radio frequency (RF) and microwave engineering, microwave circuit theory has allowed us to explore fundamental properties of electromagnetic waves by giving us an intuitive understanding of them without the need to invoke detailed and rigorous electromagnetic field theories [4–5]. In the same spirit, microwave circuit formulation of the semiconductor laser diode enhances our understanding of the device, which is otherwise obscured by hard-to-visualise mathematical formulations. Complex mathematical models are too sophisticated to be desirable for engineers, especially those who are not specialists in the field of laser physics but would like to have a quick-to-digest method of understanding and designing semiconductor laser devices. It is far more convenient to work in terms of voltages, currents and impedances. In fact, electromagnetic field theory and distributed-element circuits (transmission lines) give identical solutions Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1 196 CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES when we are dealing with transverse electromagnetic (TEM) fields, where voltages and currents in the transmission lines are uniquely related to the transverse electric and magnetic fields, respectively. The attractiveness of using equivalent circuit models for semiconductor laser devices stems from their ability to provide an analogy of laser theory in terms of microwave circuit principles. In addition, microwave circuit models of laser diodes are compatible with existing circuit models of microwave devices such as heterojunction bipolar transistors (HBTs) and field-effect transistors (FETs) – an attractive feature for optoelectronic integrated circuit (OEIC) design [6]. Equivalent circuit models have effectively helped many to understand, design and optimise integrated circuits (ICs) in the microelectronics industry and they have the potential to do the same for the optoelectronics industry. The main theme of this chapter is microwave circuit modelling techniques applied to semiconductor laser devices. Two types of microwave circuit model for semiconductor lasers have been investigated: the simple lumped-element model based on low-frequency circuit concepts and the more versatile distributed-element model based on transmission-line modelling. The former (lumped-element circuit model) is based on the simplifying assumption that the phase of current or voltage across the dimension of the components has little variation. This is true when considering only the modulated signal instead of the optical carrier signal. In this case, Kirchoff’s law can be applied, which is nothing more than a special case of Maxwell’s equations [7–8]. Strictly speaking, laser devices have dimensions in the order of the operating wavelength, thus lumped-element models may not be suitable in ultrafast applications where propagation plays an important role such as in active mode locking [9]. However, the lumped-element circuit is reasonably accurate for microwave applications if all the important processes and effects are modelled accordingly by the circuit on an equivalence basis. The latter of the two circuit modelling techniques (i.e. transmission-line modelling) is a more powerful circuit model that includes distributed effects, which will be discussed in detail in this chapter. It is worth pointing out that at microwave frequencies and above, voltmeters and ammeters for direct measurement of voltages and currents do not exist, so voltage and current waves are only introduced conceptually in the microwave circuit to make optimum use of the low-frequency circuit concepts. 8.2 THE TRANSMISSION-LINE MATRIX (TLM) METHOD The transmission-line matrix (TLM) was originally developed to model passive microwave cavities by using meshes of transmission lines [9–10]. The numerical processes involved in TLM resemble the mechanism of wave propagation but they are discretised in both time and space [10–11]. Much work has been carried out using the TLM method for analysis of passive microwave waveguide structures (see [12] and references therein). Most of the work done involved two-dimensional and three-dimensional TLMs, with the exception of the application to lumped networks [13–14], the heat diffusion problem [15] and semiconductor laser modelling [16]. Although the TLM is unconditionally stable when modelling passive devices, the semiconductor laser is an active device and therefore requires more careful consideration. The basics of the one-dimensional (1-D) TLM will be presented in the following section, which forms the basis of the transmission-line laser model (TLLM) [17]. THE TRANSMISSION-LINE MATRIX (TLM) METHOD 197 8.2.1 TLM Link Lines The TLM is a discrete-time model of wave propagation simulated by voltage pulses travelling along transmission lines. The medium of propagation is represented by the transmission lines – a general or lossy transmission line consists of series resistance, shunt admittance, series inductance and shunt capacitance per unit length, whereas an ideal or lossless transmission line has reactive elements only. The transmission line may be described by a set of telegraphist equations [7], which can be shown to be equivalent to Maxwell’s equations. There are two types of TLM element that can be used as the building blocks of a complete TLM network – they are the TLM stub lines and link lines [13]. For a lossless transmission line, the velocity of propagation is expressed by vp ¼ pffiffiffiffiffiffiffiffiffiffi ¼ l ð8:1Þ d d where Ld is inductance per unit length, Cd is capacitance per unit length, l is the unit section length, and t is the model time step. In Fig. 8.1, it is shown that a lumped series Figure 8.1 TLM link-lines. inductor (L) is equivalent to a transmission line with inductance per unit length of Ld, where [13] L ¼ Ldl The characteristic impedance ðZ0Þ of the transmission line can be found from rffiffiffiffiffiffi L 0 Cd t ð8:2Þ ð8:3Þ However, there is a small error associated with the shunt capacitance of the transmission line, which can be expressed as Ce ¼ ðtÞ2 ð8:4Þ 198 CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES Similarly, the lumped shunt capacitor (C) is equivalent to a transmission line with capacitance per unit length of Cd (Fig. 8.1) where C ¼ Cdl ð8:5Þ The characteristic impedance ðZ0Þ of the line can be expressed by [13] Z0 ¼ t ð8:6Þ and the associated error in the form of a series inductor is given by Le ¼ ðtÞ2 ð8:7Þ The errors Ce and Le are of the order of ðtÞ2 and can be reduced by using a smaller model time step. In practice, there is no component that is purely inductive nor purely capacitive. The parasitic errors can therefore be adjusted by changing the time step ðtÞ to model stray inductance or capacitance. If two adjacent reactive elements are required, then the parasitic error from one line can be ‘absorbed’ into its adjacent line so that the parasitic error may be eliminated for at least one of the lines. 8.2.2 TLM Stub Lines In the preceding section, we saw how lumped reactive elements can be simulated by TLM link lines. The lumped reactive elements may also be modelled by TLM stub lines, as shown in Fig. 8.2. The lumped inductor L is also equivalent to a short-circuit stub with a characteristic impedance of [13] 2L 0 t ð8:8Þ Figure 8.2 TLM stub lines. SCATTERING AND CONNECTING MATRICES 199 and has a parasitic capacitance expressed by ðtÞ2 e 4L ð8:9Þ On the other hand, the lumped capacitor is equivalent to an open-circuit stub with characteristic impedance of [13] t 0 2C ð8:10Þ and has a parasitic inductance expressed by ðtÞ2 e 4C ð8:11Þ For TLM stub lines, the length of the transmission line is chosen such that it takes half a model time step ðt=2Þ for the pulse to travel from one end to another (see Fig. 8.2). The reason is to allow the voltage pulses to propagate to the termination of the stub and back again at the scattering node in one complete time iteration ðtÞ. This way, all incident voltage pulses will arrive at their scattering nodes in exactly the same time, irrespective of stub lines or link lines, i.e. the voltage pulses are synchronised. 8.3 SCATTERING AND CONNECTING MATRICES The most basic algorithm of TLM involves two main processes: scattering and connecting. When the incident voltage pulses, Vi, arrive at the scattering node, they are operated by a scattering matrix and reflected voltage pulses, Vr, are produced. These reflected pulses then continue to propagate along the transmission lines and become incident pulses at adjacent scattering nodes – this process is described by the connecting matrix. Formally, the TLM algorithm may be expressed as kVrT ¼ SkViT kþ1ViT ¼ CkVrT ½ScatteringŠ ½ConnectingŠ ð8:12Þ The terms ViT and VrT are the transpose matrices of the incident and reflected pulses, respectively. The terms k and k þ1 denote the kth and ðk þ1Þth time iteration, respectively. The scattering and connecting matrices are denoted by S and C, respectively. As the matrices involved in eqn (8.12) depend on the type of TLM sub-network, a worked example based on the TLM sub-network of Fig. 8.3 follows. The TLM sub-network consists of three ‘branches’ of lossy transmission lines as shown in Fig. 8.3, where scattering and connecting of the voltage pulses are clearly described pictorially. The normalised impedances are unity for the two lines connected to adjacent ... - tailieumienphi.vn