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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 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]. 196 CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES 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 v p ¼ 1 ffiffiffiffiffiffiffiffiffiffi L d C d p ¼ Ál Át ð8:1Þ where L d is inductance per unit length, C d is capacitance per unit length, Ál is the unit section length, and Át is the model time step. In Fig. 8.1, it is shown that a lumped series inductor (L) is equivalent to a transmission line with inductance per unit length of L d , where [13] L ¼ L d Ál ð8:2Þ The characteristic impedance Z 0 ðÞof the transmission line can be found from Z 0 ¼ ffiffiffiffiffiffi L d C d r ¼ L Át ð8:3Þ However, there is a small error associated with the shunt capacitance of the transmission line, which can be expressed as C e ¼ ðÁtÞ 2 L ð8:4Þ Figure 8.1 TLM link-lines. THE TRANSMISSION-LINE MATRIX (TLM) METHOD 197 Similarly, the lumped shunt capacitor (C) is equivalent to a transmission line with capacitance per unit length of C d (Fig. 8.1) where C ¼ C d Ál ð8:5Þ The characteristic impedance Z 0 ðÞof the line can be expressed by [13] Z 0 ¼ Át C ð8:6Þ and the associated error in the form of a series inductor is given by L e ¼ ÁtðÞ 2 C ð8:7Þ The errors C e and L e are of the order of ÁtðÞ 2 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 ÁtðÞto 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] Z 0 ¼ 2L Át ð8:8Þ Figure 8.2 TLM stub lines. 198 CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES and has a parasitic capacitance expressed by C e ¼ ðÁtÞ 2 4L ð8:9Þ On the other hand, the lumped capacitor is equivalent to an open-circuit stub with characteristic impedance of [13] Z 0 ¼ Át 2C ð8:10Þ and has a parasitic inductance expressed by L e ¼ ðÁtÞ 2 4C ð8:11Þ For TLM stub lines, the length of the transmission line is chosen such that it takes half a model time step Át=2ðÞfor 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 ÁtðÞ. 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, V i , arrive at the scattering node, they are operated by a scattering matrix and reflected voltage pulses, V r , 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 k V rT ¼ S k V iT ½Scattering kþ1 V iT ¼ C k V rT ½Connecting ð8:12Þ The terms V iT and V rT are the transpose matrices of the incident and reflected pulses, respectively. The terms k and k þ 1 denote the kth and ðk þ 1Þth 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 SCATTERING AND CONNECTING MATRICES 199 Figure 8.3 The TLM stub line: (a) incident pulses arriving at scattering node; (b) incident pulses scattered into reflected pulses (scattering); (c) reflected pulses arriving at adjacent nodes (connecting). 200 CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES nodes and Z s for the remaining branch (open–circuit stub line). The associated normalised resistances are R and R s , respectively. The matrices V iT and V rT are given as V rT ¼ V 1 V 2 V 3 2 4 3 5 r V iT ¼ V 1 V 2 V 3 2 4 3 5 i ð8:13Þ where V 1 , V 2 and V 3 are the voltage pulses on ports 1, 2, and 3, respectively on each scattering node (see Fig. 8.3). The superscripts r and i denote reflected and incident pulses, respectively. It is convenient to break up the scattering matrix S and express it as [15–19] S ¼ p  q À r ð8:14Þ where the matrices p, q, and r are defined in the following. For the type of TLM sub-network in Fig. 8.3, we have q ¼ q 1 q 2 q 3 ½ where q i ¼ 2ðR s þ Z s Þ 1 þ R þ 2ðR s þ Z s Þ i ¼ 1; 2 2ð1 þ RÞ 1 þ R þ 2ðR s þ Z s Þ i ¼ 3 8 > > > < > > > : ð8:15Þ The matrix q may be found by replacing the sub-network by its Thevenin-equivalent circuit, which is shown in Fig. 8.4. By definition of the voltage pulse, V i x [5], its generator or source Figure 8.4 Thevenin equivalent circuit of the TLM sub-network. SCATTERING AND CONNECTING MATRICES 201 require a value of 2V i x , where x denotes the port number (1, 2 or 3). The nodal voltage vðÞ can be defined as v ¼ q k V iT ð8:16Þ This will be explained further by using two simple TLM sub-networks as examples later. The matrix p is found by applying the voltage division rule and is given as p ¼ p 1 p 2 p 3 ½ T where p i ¼ 1 1 þ R i ¼ 1; 2 Z s R s þ Z s i ¼ 3 8 > > > < > > > : ð8:17Þ and the matrix r is given by r ¼ r 11 00 0 r 22 0 00r 33 2 4 3 5 where r ii ¼ 1 À R 1 þ R i ¼ 1; 2 Z s À R s R s þ Z s i ¼ 3 8 > > > < > > > : ð8:18Þ If lossless transmission lines are used, we have R ¼ R s ¼ 0, and eqns (8.15), (8.17) and (8.18) become q ¼ q 1 q 2 q 3 ½ ð8:19aÞ q i ¼ 2Z s 1 þ 2Z s i ¼ 1; 2 2 1 þ 2Z s i ¼ 3 8 > > < > > : ð8:19bÞ p ¼ 1 1 1 2 6 4 3 7 5 ð8:19cÞ r ¼ 100 010 001 2 6 4 3 7 5 ð8:19dÞ 202 CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES The connecting matrix depends on how the transmission lines are connected, that is, which port(s) of one scattering node is/are connected to which port(s) of other adjacent scattering node(s). The matrix element in the connecting matrix is unity only if a connection allows a pulse to travel from port i of node m to port j of node n. In the example given in Fig. 8.3, the connecting matrix C may be expressed by 3  3 elements, where we have pulses travelling from 1. port 2 of node nÀ 1 to port 1 of node n 2. port 3 of node n to port 3 of node n (open-circuit stub) 3. port 1 of node n þ 1 to port 2 of node n Thus, the connecting matrices for node n and its adjacent nodes n À 1 and n þ 1 are given as C nÀðnÀ1Þ ¼ 000 100 000 2 4 3 5 C nÀn ¼ 000 000 001 2 4 3 5 C nÀðnþ1Þ ¼ 010 000 000 2 4 3 5 ð8:20Þ A simple example of another TLM sub-network is shown in Fig. 8.5(a), which consists of a resistor ‘sandwiched’ between two lossless transmission lines. In the Thevenin equivalent circuit shown in Fig. 8.5(b), each transmission line is replaced by its characteristic impedance in series with a voltage generator of twice the incident voltage pulse. The Figure 8.5 (a) A TLM sub-network and (b) its Thevenin equivalent circuit. SCATTERING AND CONNECTING MATRICES 203 incident voltage pulses are denoted as V i 1 and V i 2 , while the reflected pulses are V r 1 and V r 2 . The impedances of the lossless transmission lines are Z 1 and Z 2 . From Fig. 8.5(b), the nodal voltages (v 1 and v 2 ) may be expressed as v 1 ¼ i 1 Y 1 v 2 ¼ i 2 Y 2 ð8:21Þ where i 1 ¼ 2V i 1 Z 1 þ 2V i 2 R þ Z 2 ðÞ i 2 ¼ 2V i 1 R þ Z 1 ðÞ þ 2V i 2 Z 2 ð8:22Þ and Y 1 ¼ 1 Z 1 þ 1 R þ Z 2 Y 2 ¼ 1 Z 2 þ 1 R þ Z 1 ð8:23Þ By substituting eqns (8.22) and (8.23) into (8.21), we have v 1 ¼ 1 R þ Z 1 þ Z 2  2ðR þ Z 2 ÞV i 1 þ 2Z 1 V i 2 Âà v 2 ¼ 1 R þ Z 1 þ Z 2  2Z 2 V i 1 þ 2ðR þ Z 1 ÞV i 2 Âà ð8:24Þ By the original definition of the nodal voltage [5], we have the relationships v 1 ¼ V i 1 þ V r 1 v 2 ¼ V i 2 þ V r 2 ð8:25Þ Now we can write down the reflected voltage pulses in terms of the incident pulses as V r 1 ¼ v 1 À V i 1 V r 1 ¼ 1 R þ Z 1 þ Z 2  ðR À Z 1 þ Z 2 ÞV i 1 þ 2Z 1 V i 2 Âà ð8:26Þ and V r 2 ¼ v 2 À V i 2 V r 2 ¼ 1 R þ Z 1 þ Z 2  2Z 2 V i 1 þðRÀ Z 2 þ Z 1 ÞV i 2 Âà Finally, the complete scattering matrix of the TLM network in Fig. 8.5 at the kth iteration is expressed by [13] S k ¼ 1 R þ Z 1 þ Z 2  ðR À Z 1 þ Z 2 Þ 2Z 1 2Z 2 ðR À Z 2 þ Z 1 Þ ! ð8:27Þ 204 CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES [...]... carrier-induced refractive index change, and spontaneous emission noise The TLLM is very flexible and has successfully been used to model a wide range of laser devices including Fabry–Perot lasers [16], DFB (index-coupled and gain-coupled) and DBR lasers [21–23], quantum well (QW) lasers [24], cleavedcoupled-cavity (CCC) lasers [25] external cavity (EC) mode-locked lasers [26], fibre grating lasers [27] and laser. .. gratings [20] and shunt conductances can be included to model gain-coupled DFB lasers [21] A TLM sub-network which consists of a series resistor and two reactive stub lines will now be used to model the wavelength dependence of semiconductor laser gain in the transmission-line laser model 8.4 TRANSMISSION-LINE LASER MODELLING (TLLM) The transmission-line laser model is a wide-bandwidth dynamic laser. .. multimode picosecond dynamic laser chirp based on transmission line laser model, IEE Proceedings Pt J., 135(2), 126–132, 1988 41 Henry, C H., Theory of the Linewidth of Semiconductor Lasers, IEEE J Quantum Electron., QE-18(2), 259–264, 1982 42 Valle, A., Rodriguez, M and Mirasso, C R., Analytical calculation of timing jitter in single-mode semiconductor lasers under fast periodic modulation, Opt Lett., 17(21),... active mode locking based on the transmission line laser model, IEE Proceedings Pt J., 136(5), 264–272, 1989 27 Zhai, L., Lowery, A J and Ahmed, Z., Locking bandwidth of actively mode-locked semiconductor lasers using fiber-grating external cavities, IEEE J Quantum Electron., 31(11), 1998–2005, 1995 28 Lowery, A J., New inline wideband dynamic semiconductor laser amplifier model, IEE Proceedings Pt J., 135(3),... 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Microwave Engineering New York: McGraw-Hill, 1966 6 Agrawal, G P and Dutta, N K., Semiconductor Lasers, 2nd edition New York: Van Nostrand Reinhold, 1993 7 Ramo, S., Whinnery, J R and Van Duzer, T., Fields and Waves in Communication Electronics, 3rd edition New York: John Wiley & Sons, 1994 8 Vasil’Ev, P P., Ultrashort pulse generation in diode lasers, Opt Quant Electron., 24, 801–824, 1992 9 Johns, P B and... counter-propagating waves that occurs in DFB lasers can also be modelled At the laser facets, the reflections that provide feedback to achieve lasing action are simulated by unmatched terminal loads The model assumes single transverse mode operation of the laser diode, and that variations of the carrier and photon densities in the lateral and transverse dimensions are not significant, except for broad-area lasers... multimode chirp in DFB semiconductor lasers, IEE Proceedings Pt J., 137(5), 293–300, 1990 24 Nguyen, L V T., Lowery, A J., Gurney, P C R and Novak, D., A Time Domain Model for HighSpeed Quantum-Well Lasers Including Carrier Transport Effects, IEEE J Select Top Quantum Electron., 1(2), 494–504, 1995 25 Lowery, A J., New dynamic multimode model for external cavity semiconductor lasers, IEE Proceedings Pt... TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES where ZC is the impedance of the capacitive stub line When a short-circuit stub is used to simulate an inductive stub line, ZðlÞ is zero, and we have ZiL ¼ jZL tanðk0 lÞ ð8:49Þ where ZL is the impedance of the inductive stub line In the laser cavity, the free space wavenumber ðk0 Þ should be replaced by the propagation constant of the semiconductor material, . dependence of semiconductor laser gain in the transmission-line laser model. 8.4 TRANSMISSION-LINE LASER MODELLING (TLLM) The transmission-line laser model is. 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