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 m B =2n eff , where  B is the Bragg wavelength, n eff 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 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 232 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM [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. DFB LASER DIODES 233 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, k A i , and scatters them to give voltage pulses reflected from the nodes, k A r . 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 k A r ¼ S Á k A i þ I s ð9:1Þ where k is the iteration number and I s 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, k A s ,so k A r ¼ S Á k A i þ k A s þ I s 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 kþ1 A i ¼ C Á k A r 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 Át. In the numerical calculation, we need to initialise the value of voltage A i and then repeat eqns (9.1) and (9.2) to find the time evolution of the voltage A i or A r . 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 ðÁtÞ 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. 234 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM 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 ð2pÁtfÞ radians; a zero impedance stub gives p radians; a matched inductive (shorted) stub gives ðÀ2p f ÁtÞ 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 the section boundaries, which couple the forward and backward waves [28]. Each section has an associated carrier rate equation model to enable the local gain, refractive index and spontaneous noise to be calculated from the injection current and the carrier recombination rates [24]. Figure 9.2 The TLLM model representing a phase shift. 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. A DFB LASER DIODE MODEL WITH PHASE SHIFT 235 The single scattering matrix S shown in Fig. 9.3 represents a section of laser with length ÁL. This matrix operates on the forward- and backward-travelling incident waves to produce a set of reflected waves. These are then passed along the transmission lines ready to become new incident waves upon adjacent scattering nodes after one iteration time step. If two sections of the model were to be used to represent each period of the DFB grating on the real device, the number of sections and hence the computational task would be excessive. However, it is possible to represent an odd number of grating periods with a single pair of model sections without compromising the model’s accuracy [28]. This technique relies on the model having a square grating modulation. This can be decomposed into a number of sinusoidal gratings at harmonics of the grating period by Fourier techniques. One of these harmonics models the real device’s grating period. Note that the amplitude of each harmonic decreases with the harmonic number, that is, the fifth harmonic produces a coupling of one-fifth of the amplitude of the fundamental harmonic. For this example, the coupling of each period of the square grating has to be increased by a factor of five over the coupling of the real laser’s grating to compensate. A simpler and much neater rule is that the coupling  per unit length must be equal for model and real devices [28]. If a small number of sections is used, the optical field will be sampled less than once per wave period. This under-sampling is essential for realistic computer times. Under-sampling has been used in all TLLMs and does not compromise accuracy if the sampling rate (section length/group velocity) is more than twice the bandwidth of the optical wave [24]. The use of two sections per grating period ensures that the DFB’s spectrum always lies near the centre of the modelled spectrum. 9.5 ANALYSIS OF TLLM FOR DFB LASER DIODES Once the transmission-line representation of the device has been derived, an algorithm can be produced. One of the advantages of TLLM is that the algorithm is always an exact representation of the transmission-line model; no inaccuracies are introduced once the transmission-line representation has been formulated. This means that all approximations have physical meaning because they are associated with the parameters of the transmission lines. The terms in eqns (9.1) to (9.3) will now be derived for the DFB laser model. Note that the travelling optical electric fields are represented by voltage pulses A (forwards) and B (backwards) in the model. Thus, a unity constant m, with dimension of metres, is used to convert between electric field and voltage to maintain dimensional correctness. 9.5.1 Scattering Matrix for a Uniform DFB LD The scattering matrix can be split into two scattering matrices, one for each wave direction. This is possible as there is no cross coupling between the wave directions in the scattering operation. In a uniform DFB LD, the scattering process for the forward wave, with incident pulses from the previous section A i ðnÞ, the gain filter’s capacitive stub A i C ðnÞ and the gain filter’s inductive stub A i L ðnÞ, may be expressed as [27] k AðnÞ A C ðnÞ A L ðnÞ 2 4 3 5 r ¼ S u k AðnÞ A C ðnÞ A L ðnÞ 2 4 3 5 i þ k I s Z p =2 0 0 2 4 3 5 S ð9:4Þ 236 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM where S u ¼ 1 y ðg þ yÞ 2Y C 2Y L g 2Y C À y 2Y L g 2Y C 2Y L À y 2 6 4 3 7 5 ð9:5Þ I S ¼ ffiffiffiffiffiffiffiffi i 2 S  q ¼ mNðnÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2bLhfB=Z P p ð9:6Þ Z p ¼ 120pn g =n 2 eff ð9:7Þ y ¼ 1 þ Y C þ Y L ð9:8Þ Y L ¼ Y C tan 2 cpÁt=ðÞ ð9:9Þ Q ¼ ffiffiffiffiffiffiffiffiffiffiffi Y L Y C p ð9:10Þ Át ¼ " n eff ÁL=c ð9:11Þ g ¼ exp aÁL NðnÞÀN 0 ðÞ=2½À1 ð9:12Þ  ¼ exp À sc ÁL=2ðÞ ð9:13Þ where A i ðnÞ, A i C ðnÞ and A i L ðnÞ are the travelling waves in the main transmission line, the capacitive stub and the inductive stub, respectively. The parameters i and r denote incident and reflected pulses to and from the main scattering matrices, respectively. k is the iteration number, S u is the scattering matrix, I S the noise current representing spontaneous emission [26], Z p is the transverse wave impedance for a TE mode in the cavity [24], m is a unit constant with dimension of metres, L is the laser cavity length, hf is the photon energy, B is the radiative recombination coefficient, n g is the group refractive index, n eff is the effective mode refractive index, Y C is the capacitive admittance of the open-circuit stub, Y L is the inductive admittance of the short-circuit stub, Át is the time step, Q is the quality factor of a parallel RLC filter whose R value is unity [34] (see also Fig. 8.9), c and l are, respectively, the light velocity and wavelength in free space, g is the gain coefficient, a is the gain coefficient per unit carrier coefficient, ÁL the section length, À is the confinement factor, NðnÞ is the carrier density within the nth section and N 0 is the carrier density for transparency, g is the attenuation caused by free carrier absorption and scattering across a section and  sc is the power attenuation coefficient. It should be noted that, as mentioned in section 2.3.4, due to the dispersive properties of the semiconductor, the actual material gain g given in eqn (9.12) is also affected by the optical frequency f, and hence the wavelength l. So far, the gain has been assumed to be at the resonant frequency. However, if the optical frequency is tuned away from the resonant peak, the exact value of the gain becomes different from the peak value. On the basis of experimental observation, Westbrook [33] extended the linear peak gain model further so gðN Á h fÞ¼a 1 ðN À N 0 ÞÀa 2 ½h f ÀðE 0 þ a 3 ðN À N 0 ÞÞ 2 ð9:14Þ where h ¼ 6:626  10 À34 J.s is Planck’s constant, f ¼ c= is the optical frequency, a 1 is dg=dN at the gain curve peak a 1 ¼ 2:7  10 À16 cm 2 ÀÁ , N 0 is the transparency carrier density N 0 ¼ 9  10 17 cm À3 ðÞ, a 2 is the width parameter of the gain spectrum ANALYSIS OF TLLM FOR DFB LASER DIODES 237 a 2 ¼ 4  10 5 cm À1 eV À2 ÀÁ , E 0 is the gain peak energy at the transparency and a 3 is dE 0 =dN, the gain peak position carrier dependence a 3 ¼ 1:4  10 À20 eVcm 3 ÀÁ 9.5.2 Scattering Matrix for the DFB Laser Diode with Phase Shift For a DFB LD with phase shift, the scattering process for the forward wave, with incident pulses from the previous section A i ðnÞ, the gain filter’s capacitive stub A i C ðnÞ, the gain filter’s inductive stub A i L ðnÞ and the phase shifting stub A i P ðnÞ, is given by [30] k AðnÞ A C ðnÞ A L ðnÞ A P ðnÞ 2 6 6 6 4 3 7 7 7 5 r ¼ S p Á k AðnÞ A C ðnÞ A L ðnÞ A P ðnÞ 2 6 6 6 4 3 7 7 7 5 i þ k I S Z C =2 0 0 0 2 6 6 6 4 3 7 7 7 5 S ð9:15Þ where S p ¼ 1 yð1 þ Z s Þ ðg þ yÞðZ S À 1Þ 2Y C ðZ S À 1Þ 2Y L ðZ S À 1Þ 2y gðZ S þ 1Þð2Y C À yÞðZ S þ 1Þ 2Y L ðZ S þ 1Þ 0 gðZ S þ 1Þ 2Y C ðZ S þ 1Þð2Y L À yÞðZ S þ 1Þ 0 2ðg þ yÞZ S 4Y C Z S 4Y L Z S yð1 À Z S Þ 2 6 6 6 4 3 7 7 7 5 ð9:16Þ where Z S ¼ 1 tan p " n eff l          ð9:17Þ  ¼ ÀÁLNðnÞÀN p ÀÁ n g dn r dN ð9:18Þ In the above equations n is the number of sections, Z S is the phase-adjusting stub’s impedance normalised to the cavity wave impedance,  is the change in phase length across a section which is due to the dynamic change of the carrier density, " n eff is the guide’s group effective refractive index, N p is an arbitrary carrier density for zero phase shift and is usually set to the threshold carrier density [25], l is the light wavelength, n r is the refractive index and dn r =dN is the active region’s refractive index carrier dependence which is related to the Henry factor  H as [35] dn r dN ¼À  H 4p dg dN ¼À  H a 4p ð9:19Þ The scattering process for the backward wave can be obtained by using the above formula with all wave amplitudes A to be replaced by wave amplitudes B. It should be noted that all parameters in the above equations may vary from one section to another, hence they should have subscripts n, also some parameters are time dependent and vary with the iteration number k. 238 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM 9.6 CONNECTION MATRIX C The connection equations in TLLM DFB models which describe the cross coupling between the two wave directions occurring at the section interfaces can be expressed as [28] kþ1 Aðn þ 1Þ BðnÞ ! i ¼ 1 þ ÁL ÀÁL ÁL 1 À ÁL ! k AðnÞ Bðn þ 1Þ ! r ð9:20Þ for a low–high impedance boundary, and kþ1 Aðn þ 2Þ Bðn þ 1Þ ! i ¼ 1 À ÁL ÁL ÀÁL 1þ ÁL ! k Aðn þ 1Þ Bðn þ 2Þ ! r ð9:21Þ for a high–low impedance boundary. k is the iteration number and i denotes incident pulses to the main scattering matrices S,  is the coupling coefficient and r denotes reflected pulses from the main scattering matrices. Note that there is a one-iteration time step delay as the optical field samples move along the lines. For a standard DFB device, eqns (9.20) and (9.21) are applied alternately along the device length, i.e. n ¼ð1; 3; 5; 7; .Þ. For QWS grating devices [31] a zero-reflection interface (identity matrix) is inserted half way along the cavity. Figure 9.4 shows the schematic diagram of the TLLM for the QWS DFB laser diode. When a facet is placed at a low–high impedance boundary a simple resistive termination can be used giving: kþ1 B i ðnÞ¼ ffiffiffi R p k A r ðnÞ at the front facet and kþ1 A i ð1Þ¼ ffiffiffi R p k B r ð1Þ at the rear facet, where R is the facet power reflectivity and n is the number of sections [24,36]. 9.6.1 Connection Matrix C for the Stubs Within a Section There are also equations governing the reflections at the ends of the transmission line stubs. These are half a time step long to ensure that pulses arrive back at the originating scattering Figure 9.4 The TLLM model for a QWS DFB laser diode. CONNECTION MATRIX C 239 matrix after a delay of one time step. For the inductive stubs in each section, the reflection coefficient is negative, giving kþ1 A L ðnÞ B L ðnÞ ! i ¼ À10 0 À1 ! k A L ðnÞ B L ðnÞ ! r ð9:22Þ For the capacitive stubs in each section, the reflection coefficient is positive, giving kþ1 A C ðnÞ B C ðnÞ ! i ¼ 10 01 ! k A C ðnÞ B C ðnÞ ! r ð9:23Þ For the phase-adjusting stubs, which may be inductive or capacitive when the tangent in eqn (9.17) is negative we have kþ1 A p ðnÞ B p ðnÞ ! i ¼ À10 0 À1 ! k A p ðnÞ B p ðnÞ ! r ð9:24Þ and when it is positive we have kþ1 A p ðnÞ B p ðnÞ ! i ¼ 10 01 ! k A p ðnÞ B p ðnÞ ! r ð9:25Þ 9.7 CARRIER DENSITY RATE EQUATION Neglecting the effect of diffusion along the laser cavity we may apply the carrier density rate equation for each section of the model. However, this assumption may not be accurate for laser diodes with low facet reflectivities, including laser diode amplifiers. Electrons may be injected into the conduction band in the active region of the laser by sandwiching it between two higher-bandgap semiconductor layers and applying a forward bias across the structure. The electrons may leave the region into which they have been injected by diffusion to other regions. The electrons may also recombine to the valence band by stimulated or spontaneous recombination. Electrons involved in these processes may be accounted for using a carrier density rate equation given by dNðnÞ dt ¼À NðnÞ  s À ac " n eff ðNðnÞÀN 0 ÞSðnÞþ IðnÞ ewdL ð9:26Þ Where  s is the carrier lifetime, IðnÞ is the component of the injection current injected into section n, the laser diode width, thickness and length are, respectively, denoted by w, d and L. e is the electron charge and the photon density SðnÞ within a section is related to the incident waves from either side by SðnÞ¼ A i ðnÞ ÀÁ 2 þ B i ðnÞ ÀÁ 2 hi " n eff m 2 h f cZ p ð9:27Þ 240 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM [...]... J., Amplified spontaneous emission in semiconductor laser amplifiers: validity of the transmission-line laser model, IEE Proceedings Part J, 137(4), 241–247, 1990 30 Lowery, A J., New dynamic model for multimode chirp in DFB semiconductor lasers, IEE Proceedings Part J, 137(5), 293–300, 1990 31 Lowery, A J., Transmission-line laser modelling of semiconductor laser amplified optical communication systems,... time-domain model for spontaneous emission in semiconductor lasers and its use in predicting their transient response, Int J Num Model., 1, 153–164, 1988 27 Lowery, A J., Transmission-line modeling of semiconductor lasers: the transmission-line laser model, Int J Num Model., 2, 249–265, 1989 28 Lowery, A J., Dynamic modeling of distributed-feedback lasers using scattering matrices, Electron Lett., 25(19),... oscillating modes, Electron Lett., 11, 372–374, 1975 13 Osinski, N and Adams, M J., Transient time-averaged spectra of rapidly-modulated semiconductor lasers, IEE Proc Part J Optoelectronics, 132, 34–37, 1985 14 Lowery, A J., Modelling ultra-short pulses (less than the cavity transient time) in semiconductor laser amplifiers, Int J Optoelectronics, 3, 497–508, 1988 15 Demokan, M S., A model of a diode laser. .. DFB laser structures, IEEE J Quantum Electron., 28, 1277–1293, 1992 7 Zhou, P and Lee, G S., Chirped grating !=4 DFB laser with uniform longitudinal field distribution, Electron Lett., 26, 1660–1661, 1990 8 Hillmer, H., Magari, K and Suzuji, Y., Chirped grating for DFB laser diode using bent waveguides, IEEE Photonics Technol Lett., 5, 10–12, 1993 9 Lowery, A J., A two port bilateral model for semiconductor... Tucker, R S., High-speed modulation of semiconductor lasers, J Lightwave Technol., LT-3, 1180– 1192, 1985 19 Bjork, G and Nilsson, O., A new exact and efficient numerical matrix theory of complicated laser structures: Properties of asymmetric phase-shifted DFB lasers, J Lightwave Technol., LT-5, 140– 146, 1987 20 Zhang, L M and Carroll, J., Large-Signal dynamic model of the DFB laser, IEEE J Quantum Electron.,... Electron., 25, 20–30, 1989 24 Lowery, A J., A New dynamic semiconductor laser model based on the transmission-line modeling method, IEE proceedings Part J, 134(51), 281–289, 1987 25 Lowery, A J., A Model for multimode picosecond dynamic laser chirp based on transmission line laser model, IEE proceedings Part J, 135(2), 126–132, 1988 252 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM 26 Lowery, A... REFERENCES 1 Agrawal, G P and Dutta, N K., Semiconductor Lasers, 2nd Edition New York: Van Nostrand Reinhold Company Inc, 1993 2 Nosu, K and Iwashita, K., A consideration of factors affecting future coherent lightwave communication systems, J Lightwave Technol., 6, 686–694, 1988 REFERENCES 251 3 Kogelnik, H and Shank, V V., Coupled-wave theory of distributed feedback lasers, J Appl Phys., 43, 2327–2335,... Lett., 5, 10–12, 1993 9 Lowery, A J., A two port bilateral model for semiconductor lasers, IEEE J Quantum Electron., 28, 82–91, 1992 10 Buss, J., Principle of semiconductor laser modelling, IEE Proc J Optoelectronics, 132, 42–51, 1985 11 Boers, P M., Vlaardingerbroek, M T and Danielsen, M., Dynamic behavior of semiconductor lasers, Electron Lett., 11, 206–208, 1975 12 Hillbrand, H and Russer, P., Large... transmission-line laser model, IEE Proceedings Part J, 136, 264–272, 1989 36 Buss, J., Dynamic single-mode operation of DFB lasers with phase shifted gratings and reflecting mirrors, IEE Proceedings Part J, 133, 163–164, 1986 37 Zhang, L M., Yu, S F., Nowell, M C., Marcenac, D D., Carroll, J E and Plumb, R G S., Dynamic analysis of radiation and side-mode suppression in a second-order DFB laser using time-domain... p=3 All the electric field distributions have been normalised so that the intensity at the laser facets becomes unity It can be seen that the internal electric field distribution of the QWS DFB LDs is highly non-uniform This induces the spatial hole burning effect and thus affects the single-mode stability of the laser With three phase shifts incorporated into the cavity, the intensity distribution spreads . of DFB Laser Diode Characteristics Based on Transmission-Line Laser Modelling (TLLM) 9.1 INTRODUCTION In Chapter 8 we introduced transmission-line laser. rate equation for each longitudinal laser mode gives the laser s spectrum during modulation [12] and dynamic frequency shifting (chirping) may be estimated