Laser điốt được phân phối thông tin phản hồi và các bộ lọc du dương quang P4

4 Transfer Matrix Modelling in DFB Semiconductor Lasers 4.1 INTRODUCTION In Chapter 3, eigenvalue equations were derived by matching boundary conditions inside DFB laser cavities. From the eigenvalue problem, the lasing threshold characteristic of DFB lasers is determined. The single /2-phase-shifted (PS) DFB laser is fabricated with a phase discontinuity of /2 at or near the centre of the laser cavity. It is characterised by Bragg oscillation and a high gain margin value. On the other hand, the SLM deteriorates quickly when the optical power of the laser diode increases. This phenomenon, known as spatial hole burning, limits the maximum single-mode optical power and consequently the spectral linewidth. Using a multiple-phase-shift (MPS) DFB laser structure, the electric field distribution is flattened and hence the spatial hole burning is suppressed. In dealing with such a complicated DFB laser structure, it is tedious to match all the boundary conditions. A more flexible method which is capable of handling different types of DFB laser structures is necessary. In section 4.2, the transfer matrix method (TMM) [1–4] will be introduced and explored comprehensively. From the coupled wave equations, it is found that the field propagation inside a corrugated waveguide (e.g. the DFB laser cavity) can be represented by a transfer matrix. Provided that the electric fields at the input plane are known, the matrix acts as a transfer function so that electric fields at the output plane can be determined. Similarly, other structures like the active planar Fabry–Perot (FP) section, the passive corrugated distributed Bragg reflector (DBR) section and the passive planar waveguide (WG) section can also be expressed using the idea of a transfer matrix. By joining these transfer matrices as a building block, a general N-sectioned laser cavity model will be presented. Since the outputs from a transfer matrix automatically become the inputs of the following matrix, all boundary conditions inside the composite cavity are matched. The unsolved boundary conditions are those at the left and right facets. In section 4.3, the threshold equation of the N-sectioned laser cavity model will be determined and the use of TMM in other semiconductor laser devices will be discussed. An adequate treatment of the amplified spontaneous emission spectrum ðPNÞ is very important in the analysis of semiconductor lasers [5], optical amplifiers [6–8] and optical filters [9–10]. In semiconductor lasers, PN is important for both the estimation of linewidth [11] and the estimation of single-mode stability in DFB laser diodes [12]. In optical Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1 102 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS amplifiers and filters, PN has also been used to simulate the bandwidth, tunability and the signal gain characteristic. In section 4.4, the TMM formulation will be extended so as to include the below-threshold spontaneous emission spectrum of the N-sectioned DFB laser structure. Numerical results based on 3PS DFB LDs will be presented. 4.2 BRIEF REVIEW OF MATRIX METHODS By matching boundary conditions at the facets and the phase-shift position, the threshold condition of the single-phase-shifted DFB LD can be determined from the eigenvalue equation. However, this approach lacks the flexibility required in the structural design of DFB LDs. Whenever a new structural design is involved, a new eigenvalue equation has to be derived by matching all boundary conditions. For a laser with the MPS DFB structure, the formation of eigenvalue equation becomes tedious since it may involve a large number of boundary conditions. One possible approach to simplifying the analysis, whilst improving flexibility and robustness, is to employ matrix methods. Matrices have been used extensively in engineering problems which are highly numerical in nature. In microwave engineering [13], matrices are used to find the electric and magnetic fields inside various microwave waveguides and devices. One major advantage of matrix methods is their flexibility. Instead of repeatedly finding complicated analytical eigenvalue equations for each laser structure, a general matrix equation is derived. Threshold analysis of various laser structures including planar section, corrugated section or a combination of them can be analysed in a systematic way. Since they share the same matrix equation, the algorithm derived to solve the problem can be re-used easily for different laser structures. However, because of the numerical nature of matrix methods, they cannot be used to verify the existence of analytical expressions in a particular problem. In all matrix methods, the structures involved will first be divided into a number of smaller sections. In each section, all physical parameters like the injection current and material gain are assumed to be homogeneous. As a result, the total number of smaller sections used varies and mostly depends on the type of problem. For a problem like the analysis of transient responses in LDs [14], a fairly large number of sections are needed since a highly non-uniform process is involved. On the other hand, only a few sections are required for the threshold analysis of DFB lasers since a fairly uniform process is concerned. For an arbitrary one-dimensional laser structure as shown in Fig. 4.1, the wave propagation is modelled by a 2 2 matrix A such that any electric field leaving (i.e. ERðziþ1Þ Figure 4.1 Wave propagation in a general 1-D laser diode structure. BRIEF REVIEW OF MATRIX METHODS 103 Table 4.1 Name Different types of matrix method U V Domain Scattering matrix TLM TMM ERðziþ1Þ and ESðziÞ ERðziþ1Þ and ESðziþ1Þ ERðziþ1Þ and ESðziþ1Þ ERðziÞ and ESðziþ1Þ ERðziÞ and ESðziÞ ERðziÞ and ESðziÞ frequency time frequency and ESðziÞ) and those entering (i.e. ERðziÞ and ESðziþ1Þ that section are related to one another by U ¼ AV ð4:1Þ where U and V are two column matrices each containing two electric wave components. Depending on the type of matrix method, the contents of U and V may vary. In the scattering matrix method, matrix U includes all electric waves leaving the arbitrary section, whilst matrix V contains those entering the section. In both transmission line matrix (TLM) and transfer matrix methods (TMM), matrix U represents the electric wave components from one side of the section, whilst wave components from the other side are included in matrix V. For analysis of semiconductor laser devices, both TLM and TMM have been used. The difference between TLM and TMM lies in the domain of analysis. TLM is performed in the time domain, whereas TMM works extremely well in the frequency domain. Table 4.1 summarises the characteristics of matrix methods. Using the time-domain-based TLM, transient responses like switching in semiconductor laser devices can be analysed. Steady-state values may then be determined from the asymptotic approximation. However, it is difficult to use TLM to determine noise characteristics, and hence the spectral linewidth, of semiconductor lasers. Due to the fact that most noise-related phenomena are time-averaged stochastic processes, a very long sampling time will be necessary if TLM is used. In general, TLM is not suitable for the analysis of noise characteristics in semiconductor laser devices. In 1987, Yamada and Suematsu first proposed using the TMM for analysing the transmission and reflection gains of laser amplifiers with corrugated structures. This frequency-domain-based method works extremely well for both steady-state and noise analysis [6,9]. In the present study, we are interested in the steady-state and noise characteristics of DFB lasers. Hence, the use of TMM will be more appropriate. 4.2.1 Formulation of Transfer Matrices Based upon the coupled wave equations, one can derive the transfer matrix for a corrugated DFB laser section. From the solution of the coupled wave equations, one can express EðzÞ ¼ ERðzÞ þ ESðzÞ ¼ RðzÞeÿjb0z þ SðzÞejb0z ð4:2Þ where ERðzÞ and ESðzÞ are the complex electric fields of the wave solutions, RðzÞ and SðzÞ are two slow-varying complex amplitude terms and b0 is the Bragg propagation constant. From eqn (3.3), RðzÞ and SðzÞ have proposed solutions of the form RðzÞ ¼ R1egz þ R2eÿgz ð4:3aÞ SðzÞ ¼ S1egz þS2eÿgz ð4:3bÞ 104 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS where R1, R2, S1 and S2 are complex coefficients which are found to be related to one another by [15] S1 ¼ ejR1 ð4:4aÞ R2 ¼ eÿjS2 ð4:4bÞ where ¼ j=ð ÿj þgÞ and is the residue corrugation phase at the origin. By substituting eqn (4.4) into (4.3), one obtains RðzÞ ¼ R1egz þ S2eÿjeÿgz ð4:5aÞ SðzÞ ¼ R1ejegz þ S2eÿgz ð4:5bÞ Instead of four variables, the solution of the coupled wave equations is simplified to functions of two coefficients R1 and S2. Suppose the corrugation inside the DFB laser extends from z ¼ z1 to z ¼ z2 as shown in Fig. 4.2, the amplitude coefficients at the left and the right facets can then be written as Rðz1Þ ¼ R1egz1 þS2eÿjeÿgz1 ð4:6aÞ Sðz1Þ ¼ R1ejegz1 þ S2eÿgz1 ð4:6bÞ Rðz2Þ ¼ R1egz2 þS2eÿjeÿgz2 ð4:6cÞ Sðz2Þ ¼ R1ejegz2 þ S2eÿgz2 ð4:6dÞ From eqns (4.6a) and (4.6b), one can express R1 and S2 such that Sðz1Þeÿj ÿRðz1Þ 1 ð2 ÿ 1Þegz1 Rðz1Þej ÿ Sðz1Þ 2 ð2 ÿ 1Þeÿgz1 ð4:7aÞ ð4:7bÞ Figure 4.2 A simplified schematic diagram for a 1-D corrugated DFB laser diode section. BRIEF REVIEW OF MATRIX METHODS 105 By substituting the above equations back into eqns (4.6c) and (4.6d), one obtains Rðz2Þ ¼ E ÿ 2Eÿ1 Rðz1Þ ÿðE ÿ Eÿ1Þeÿj Sðz1Þ ð4:8aÞ Sðz2Þ ¼ ðE ÿEÿ1Þej Rðz1Þ ÿ 2E ÿEÿ1 Sðz1Þ ð4:8bÞ where E ¼ eðz2ÿz1Þ; Eÿ1 ¼ eÿðz2ÿz1Þ ð4:8cÞ From the above equations, it is clear that the electric fields at the output plane z2 can be expressed in terms of the electric waves at the input plane. By combining the above equations with eqn (4.2) we can relate the output and input electric fields through the following matrix equation [6] ERðz2Þ ERðz1Þ t11 ESðz2Þ ESðz1Þ t21 t12 ERðz1Þ t22 ESðz1Þ ð4:9Þ where matrix Tðz2 j z1Þ represents any wave propagation from z ¼ z1 to z ¼ z2 and its elements tijði; j ¼ 1; 2Þ are given as ðE ÿ 2Eÿ1Þ eÿjb0ðz2ÿz1Þ 11 ð1 ÿ 2Þ ÿðE ÿEÿ1Þ eÿjeÿjb0ðz2þz1Þ 12 ð1 ÿ 2Þ ðE ÿ Eÿ1Þ ejejb0ðz2þz1Þ 21 ð1 ÿ 2Þ ð2E ÿ Eÿ1Þ ejb0ðz2ÿz1Þ 22 ð1 ÿ 2Þ ð4:10aÞ ð4:10bÞ ð4:10cÞ ð4:10dÞ For convenience, the matrix written in this way is called the forward transfer matrix because the output plane at z ¼ z2 is located further away from the origin. Similarly, waves propagating inside the corrugated structure can also be expressed as the backward transfer matrix such that [16] ERðz1Þ ERðz2Þ u11 ESðz1Þ 1 2 ESðz2Þ u21 u12 ERðz2Þ u22 ESðz2Þ ð4:11Þ where matrix Uðz1 j z2Þ represents any field propagation inside the section from z ¼ z2 to z ¼ z1. By comparing eqn (4.9) with eqn (4.11), it is obvious that Uðz1 j z2Þ ¼ ½Tðz2 j z1ފÿ1 ð4:12Þ ... - tailieumienphi.vn