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 ðP N Þ is very important in the analysis of semiconductor lasers [5], optical amplifiers [6 –8] and optical filters [9–10]. In semiconductor lasers, P N 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 amplifiers and filters, P N 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. E R ðz iþ1 Þ Figure 4.1 Wave propagation in a general 1-D laser diode structure. 102 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS and E S ðz i Þ) and those entering (i.e. E R ðz i Þ and E S ðz iþ1 Þ 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 EðzÞ¼E R ðzÞþE S ðzÞ¼RðzÞe Àjb 0 z þ SðzÞe jb 0 z ð4:2Þ where E R ðzÞ and E S ðzÞ are the complex electric fields of the wave solutions, RðzÞ and SðzÞ are two slow-varying complex amplitude terms and b 0 is the Bragg propagation constant. From eqn (3.3), RðzÞ and SðzÞ have proposed solutions of the form RðzÞ¼R 1 e gz þ R 2 e Àgz ð4:3aÞ SðzÞ¼S 1 e gz þ S 2 e Àgz ð4:3bÞ Table 4.1 Different types of matrix method Name UVDomain Scattering matrix E R ðz iþ1 Þ and E S ðz i Þ E R ðz i Þ and E S ðz iþ1 Þ frequency TLM E R ðz iþ1 Þ and E S ðz iþ1 Þ E R ðz i Þ and E S ðz i Þ time TMM E R ðz iþ1 Þ and E S ðz iþ1 Þ E R ðz i Þ and E S ðz i Þ frequency BRIEF REVIEW OF MATRIX METHODS 103 where R 1 , R 2 , S 1 and S 2 are complex coefficients which are found to be related to one another by [15] S 1 ¼ e j R 1 ð4:4aÞ R 2 ¼ e Àj S 2 ð4:4bÞ where  ¼ j=  À j þ gðÞand  is the residue corrugation phase at the origin. By substituting eqn (4.4) into (4.3), one obtains RðzÞ¼R 1 e gz þ S 2 e Àj e Àgz ð4:5aÞ SðzÞ¼R 1 e j e gz þ S 2 e Àgz ð4:5bÞ Instead of four variables, the solution of the coupled wave equations is simplified to functions of two coefficients R 1 and S 2 . Suppose the corrugation inside the DFB laser extends from z ¼ z 1 to z ¼ z 2 as shown in Fig. 4.2, the amplitude coefficients at the left and the right facets can then be written as Rðz 1 Þ¼R 1 e gz 1 þ S 2 e Àj e Àgz 1 ð4:6aÞ Sðz 1 Þ¼R 1 e j e gz 1 þ S 2 e Àgz 1 ð4:6bÞ Rðz 2 Þ¼R 1 e gz 2 þ S 2 e Àj e Àgz 2 ð4:6cÞ Sðz 2 Þ¼R 1 e j e gz 2 þ S 2 e Àgz 2 ð4:6dÞ From eqns (4.6a) and (4.6b), one can express R 1 and S 2 such that R 1 ¼ Sðz 1 Þe Àj À Rðz 1 Þ  2 À 1ðÞe gz 1 ð4:7aÞ S 2 ¼ Rðz 1 Þe j À Sðz 1 Þ  2 À 1ðÞe Àgz 1 ð4:7bÞ Figure 4.2 A simplified schematic diagram for a 1-D corrugated DFB laser diode section. 104 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS By substituting the above equations back into eqns (4.6c) and (4.6d), one obtains Rðz 2 Þ¼ E À  2 E À1 1 À  2 Rðz 1 ÞÀ  E À E À1 ðÞe Àj 1 À  2 Sðz 1 Þð4:8aÞ Sðz 2 Þ¼  E À E À1 ðÞe j 1 À  2 Rðz 1 ÞÀ  2 E À E À1 1 À  2 Sðz 1 Þð4:8bÞ where E ¼ e ðz 2 Àz 1 Þ ; E À1 ¼ e Àðz 2 Àz 1 Þ ð4:8cÞ From the above equations, it is clear that the electric fields at the output plane z 2 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] E R ðz 2 Þ E S ðz 2 Þ ! ¼ T z 2 j z 1 ðÞÁ E R ðz 1 Þ E S ðz 1 Þ ! ¼ t 11 t 12 t 21 t 22 ! Á E R ðz 1 Þ E S ðz 1 Þ ! ð4:9Þ where matrix Tðz 2 j z 1 Þ represents any wave propagation from z ¼ z 1 to z ¼ z 2 and its elements t ij ði; j ¼ 1; 2Þ are given as t 11 ¼ ðE À  2 E À1 ÞÁe Àjb 0 ðz 2 Àz 1 Þ ð1 À  2 Þ ð4:10aÞ t 12 ¼ ÀðE À E À1 ÞÁe Àj e Àjb 0 ðz 2 þz 1 Þ ð1 À  2 Þ ð4:10bÞ t 21 ¼ ðE À E À1 ÞÁe j e jb 0 ðz 2 þz 1 Þ ð1 À  2 Þ ð4:10cÞ t 22 ¼À ð 2 E À E À1 ÞÁe jb 0 ðz 2 Àz 1 Þ ð1 À  2 Þ ð4:10dÞ For convenience, the matrix written in this way is called the forward transfer matrix because the output plane at z ¼ z 2 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] E R ðz 1 Þ E S ðz 1 Þ ! ¼ Uðz 1 j z 2 ÞÁ E R ðz 2 Þ E S ðz 2 Þ ! ¼ u 11 u 12 u 21 u 22 ! Á E R ðz 2 Þ E S ðz 2 Þ ! ð4:11Þ where matrix Uðz 1 j z 2 Þ represents any field propagation inside the section from z ¼ z 2 to z ¼ z 1 . By comparing eqn (4.9) with eqn (4.11), it is obvious that Uðz 1 j z 2 Þ¼ Tðz 2 j z 1 Þ½ À1 ð4:12Þ BRIEF REVIEW OF MATRIX METHODS 105 where the superscript À1 denotes the inverse of the matrix. Due to conservation of energy, both matrices Tðz 2 j z 1 Þ and Uðz 1 j z 2 Þ must satisfy the reciprocity rule such that their determinants always give unity value [4]. In other words, T jj ¼ t 11 t 22 À t 12 t 21 ¼ 1 U jj ¼ u 11 u 22 À u 12 u 21 ¼ 1 ð4:13Þ 4.2.2 Introduction of Phase Shift (or Phase Discontinuity) For a single PS DFB laser cavity as shown in Fig. 4.3, the phase shift at z ¼ z 2 divides the laser cavity into two sections. The field discontinuity is usually small along the plane of phase shift and any wave travelling across the phase shift is assumed to be continuous. As a result, the transfer matrices are linked up at the phase shift position as: E R ðz þ 2 Þ E S ðz þ 2 Þ ! ¼ P ð2Þ Á E R ðz À 2 Þ E S ðz À 2 Þ ! ¼ e j 2 0 0e Àj 2 ! Á E R ðz À 2 Þ E S ðz À 2 Þ ! ð4:14Þ where P ð2Þ is the phase-shift matrix at z ¼ z 2 ; z þ 2 and z À 2 are the greater and lesser values of z 2 , respectively, and  2 corresponds to the phase change experienced by the electric waves E R ðzÞ and E S ðzÞ. Alternatively, the physical phase shift of the corrugation may be used [9]. To avoid any confusion, we will use the phase shift of the electric wave hereafter. On combining eqn (4.14) with the transfer matrix shown earlier in eqn (4.9), the overall transfer matrix chain of a single-phase-shifted DFB laser becomes E R ðz 3 Þ E S ðz 3 Þ "# ¼ t ð2Þ 11 t ð2Þ 12 t ð2Þ 21 t ð2Þ 22 2 4 3 5 Á e j 2 0 0e Àj 2 "# Á t ð1Þ 11 t ð1Þ 12 t ð1Þ 21 t ð1Þ 22 2 4 3 5 Á E R ðz 1 Þ E S ðz 1 Þ "# ¼ T ð2Þ Á P ð2Þ Á T ð1Þ Á E R ðz 1 Þ E S ðz 1 Þ "# ð4:15Þ Figure 4.3 Schematic diagram showing a 1PS DFB laser diode section. 106 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS Without affecting the results of the above equations, one can multiply a unity matrix I after matrix T (1) . This matrix I behaves as if an imaginary phase shift of zero or a multiple of 2 has been introduced. As a result, the above matrix equation can be simplified such that E R ðz 3 Þ E S ðz 3 Þ ! ¼ Yðz 3 j z 1 ÞÁ E R ðz 1 Þ E S ðz 1 Þ ! ð4:16Þ where Yðz 3 j z 1 Þ¼ Y 1 m¼2 F ðmÞ ¼ y 11 z 3 j z 1 ðÞy 12 z 3 j z 1 ðÞ y 21 z 3 j z 1 ðÞy 22 z 3 j z 1 ðÞ ! ð4:17aÞ F ðmÞ ¼ T ðmÞ Á P ðmÞ ¼ f ðmÞ 11 f ðmÞ 12 f ðmÞ 21 f ðmÞ 22 "# ¼ t ðmÞ 11 e j m t ðmÞ 12 e Àj m t ðmÞ 21 e j m t ðmÞ 22 e Àj m "# ð4:17bÞ P ð1Þ ¼ I ¼ 10 01 ! ð4:17cÞ In the above equation, the overall matrix Yðz 3 j z 1 Þ comprises the characteristics of the field propagation inside the DFB laser cavity, whilst the corrugated matrix T ðmÞ and the phase- shift matrix P ðmÞ ðm ¼ 1; 2Þ are combined to form the matrix F ðmÞ . The use of the transfer matrix method is not restricted to the corrugated DFB laser structure. By modifying the values of  and  in the elements of the transfer matrix, other structures like the planar Fabry–Perot structure, the planar waveguide structure and the corrugated Distributed Bragg Reflector structure can also be represented using the transfer matrix. A DBR structure is different from the DFB structure because DBR structures have no underlying active region. The corrugated DBR structure simply acts as a partially reflecting mirror, the amount of reflection depending on the wavelength. The maximum reflection occurs near the central Bragg wavelength. Table 4.2 summarises all laser structures that can be represented by transfer matrices. The differences between them are also listed. When the grating height g reduces to zero and the grating period  approaches infinity, the feedback caused by the presence of corrugations becomes less important. At g ¼ 0,  becomes zero as does the variable . When  becomes infinite, the detuning coefficient  is reduced to the propagation constant 2n=. In this case, the DFB corrugated structure becomes a planar structure. Following eqns (4.9) and (4.10), the transfer matrix equation of Table 4.2 Laser structures that can be represented using the TMM Structure Active layer Corrugation Comments FP 38 ¼ 0 and >0 WG 88 ¼ 0 and  0 DFB 33finite  and >0 DBR 83finite  and  0 BRIEF REVIEW OF MATRIX METHODS 107 the planar structure becomes E R ðz 2 Þ E S ðz 2 Þ ! ¼ T ð1Þ Á E R ðz 1 Þ E S ðz 1 Þ ! ¼ t ð1Þ 11 t ð1Þ 12 t ð1Þ 21 t ð1Þ 22 "# Á E R ðz 1 Þ E S ðz 1 Þ ! ð4:18Þ where t ð1Þ 11 ¼ e ðz 2 Àz 1 Þ e Àjbðz 2 Àz 1 Þ t ð1Þ 12 ¼ t ð1Þ 21 ¼ 0 t ð1Þ 22 ¼ e Àðz 2 Àz 1 Þ e jbðz 2 Àz 1 Þ ð4:19Þ In the above equation, the amplitude gain term  decides the characteristics of the planar structure. For >0, the amplitude of the electric wave passing through will be amplified and the structure will behave as if it is a laser amplifier. For  0, the amplitude of the electric wave will either remain constant or be attenuated, as the planar structure becomes a passive waveguide. Similarly, the sign of  will decide whether a corrugated structure belongs to the DFB or DBR type. By joining these matrices together as building blocks, one can extend the idea further to form a general N-sectioned composite laser cavity as shown in Fig. 4.4. Laser structures that comprise different combinations of the sections shown in Table 4.2 can be modelled. By joining these matrices together appropriately, one ends up with E R z Nþ1 ðÞ E S z Nþ1 ðÞ ! ¼ F ðNÞ Á F ðNÀ1Þ ÁÁÁF ð2Þ Á F ð1Þ Á E R z 1 ðÞ E S z 1 ðÞ ! ¼ Yðz Nþ1 j z 1 ÞÁ E R z 1 ðÞ E S z 1 ðÞ ! ð4:20Þ Figure 4.4 Schematic diagram of a general N-section laser cavity. The phase shifts f 1 ; 2 ; .; N g are shown. Active regions along the laser cavity are shaded. 108 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS where matrix Yðz Nþ1 j z 1 Þ becomes the overall transfer matrix for the N-sectioned laser cavity. Using the backward transfer matrix together with eqns(4.11) and (4.14), one obtains E R z 1 ðÞ E S z 1 ðÞ ! ¼ Y N m¼1 G ðmÞ Á E R z Nþ1 ðÞ E S z Nþ1 ðÞ ! ¼ Zðz 1 j z Nþ1 ÞÁ E R z Nþ1 ðÞ E S z Nþ1 ðÞ ! ð4:21Þ where G ðmÞ ¼ P ðmÞ hi À1 ÁU ðmÞ ¼ g ðmÞ 11 g ðmÞ 12 g ðmÞ 21 g ðmÞ 22 "# ¼ u ðmÞ 11 e Àj m u ðmÞ 12 e Àj m u ðmÞ 21 e j m u ðmÞ 22 e j m "# ð4:22aÞ Zðz 1 j z Nþ1 Þ¼ z 11 ðz 1 j z Nþ1 Þ z 12 ðz 1 j z Nþ1 Þ z 21 ðz 1 j z Nþ1 Þ z 22 ðz 1 j z Nþ1 Þ ! ð4:22bÞ In the above equation, P ðmÞ Âà À1 is the inverse of the phase shift matrix P ðmÞ and Zðz 1 j z Nþ1 Þ is the overall backward transfer matrix. Comparing eqns (4.20) with (4.21), it is clear that matrices Yðz Nþ1 j z 1 Þ and Zðz 1 j z Nþ1 Þ are inverse to one another such that Zðz 1 j z Nþ1 Þ¼ Yðz Nþ1 j z 1 Þ½ À1 ð4:23Þ where the superscript À1 indicates the inverse of the matrix. From the property of the inverse of matrix products, individual transfer matrices G ðmÞ and F ðmÞ are related to one another. That is G ðmÞ ¼  F ðmÞ  À1 for m ¼ 1toN ð4:24Þ The above equation shows the equivalence between the forward and the backward transfer matrices in the general N-sectioned laser cavity. Unless stated otherwise, the forward transfer matrix is assumed hereafter. 4.2.3 Effects of Finite Facet Reflectivities It was discussed in Chapter 3 that the lasing characteristic of the DFB laser depends on the facet reflectivity. In this section, the facet reflectivity will be implemented using the TMM. Figure 4.5 Schematic diagram showing reflections at the laser facets of a DFB LD. BRIEF REVIEW OF MATRIX METHODS 109 In Fig. 4.5, a simplified schematic diagram for the reflections at the facets of the N-sectioned laser cavity is shown. In Fig. 4.5, ^ r 1 and ^ r 2 are the amplitude reflections at the left and right facets, respectively and medium 1 is the active region of the LD. In most practical cases, medium 2 is air. Due to the finite thickness of coating on the laser facets, any electric field passing through may suffer a phase change of  i ði ¼ 1; 2Þ. Depending on the direction of propagation, all the outgoing electric fields at the left facet (i.e. E R ðz þ 1 Þ and E S ðz À 1 Þ) can be expressed in terms of the incoming waves as E R ðz þ 1 Þ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 1 q e Àj 1 E R ðz À 1 Þþ ^ r 1 E S ðz þ 1 Þð4:25aÞ E S ðz À 1 Þ¼À ^ r 1 E R ðz À 1 Þþ ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 1 q e j 1 E S ðz þ 1 Þð4:25bÞ Rearranging the above equation for the electric fields at z ¼ z þ 1 , one obtains E R ðz þ 1 Þ¼ e Àj 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 1 p E R ðz À 1 Þþ ^ r 1 e Àj 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 1 p E S ðz À 1 Þð4:26aÞ E S ðz þ 1 Þ¼ ^ r 1 e Àj 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 1 p E R ðz À 1 Þþ e Àj 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 1 p E S ðz À 1 Þð4:26bÞ In matrix form, the above equations can be written as E R ðz þ 1 Þ E S ðz þ 1 Þ ! ¼ 1 e j 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 1 p Á 1 ^ r 1 ^ r 1 1 ! Á E R ðz À 1 Þ E S ðz À 1 Þ ! ð4:27aÞ Similarly, the reflection at the right facet can be written as E R ðz þ Nþ1 Þ E S ðz þ Nþ1 Þ "# ¼ 1 e j 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 2 p Á 1 À ^ r 2 À ^ r 2 1 ! Á E R ðz À Nþ1 Þ E S ðz À Nþ1 Þ ! ð4:27bÞ On combining the propagation matrix Yðz Nþ1 j z 1 Þ with the reflections at the laser facets, the overall transfer function of the N-sectioned DFB laser structure becomes E R ðz þ Nþ1 Þ E S ðz þ Nþ1 Þ "# ¼ 1 e j 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 2 p Á 1 À ^ r 2 À ^ r 2 1 ! Á Yðz Nþ1 j z 1 ÞÁ 1 e j 1 ffiffiffiffiffiffiffiffiffiffiffiffiffi 1 À ^ r 2 1 p Á 1 ^ r 1 ^ r 1 1 ! E R ðz À 1 Þ E S ðz À 1 Þ ! ð4:28Þ It will be easier to simplify the above matrix equation by an overall transfer matrix H such that E R ðz þ Nþ1 Þ E S ðz þ Nþ1 Þ "# ¼ H Á E R ðz À 1 Þ E S ðz À 1 Þ ! ¼ h 11 h 12 h 21 h 22 ! Á E R ðz À 1 Þ E S ðz À 1 Þ ! ð4:29Þ where h i;j ði; j ¼ 1; 2Þ are the elements of the overall transfer matrix H. 110 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS [...]... some popular semiconductor laser structures [27] 4.4 FORMULATION OF THE AMPLIFIED SPONTANEOUS EMISSION SPECTRUM USING THE TMM In the previous section, the threshold equation of the N-sectioned laser cavity was defined using the transfer matrix In fact, the TMM can also be applied to the below-threshold 112 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS Table 4.3 Semiconductor laser structures that... the TMM (After [27]) Laser structure FP lasers Conventional DFB LD Single PS DFB Multiple PS DFB Multiple electrode DFB –Non-uniform current injection Corrugation-pitch-modulated DFB –Different corrugation period in each section Linear chirped corrugation –continuous change in corrugation period Tapered corrugation –continuous change in corrugation depth g N-layer surface emitting laser Number of transfer... Lee, G S., Chirped grating !/4-shifted distributed feedback laser with uniform longitudinal field distribution, Electron Lett., 26, 1660–1661, 1990 25 Makino, T., Theoretical analysis of the spectral linewidth of a surface-emitting DFB semiconductor laser, Optics Comm., 81(2), 71–74, 1991 26 Makino, T., Transfer-matrix formulation of spontaneous emission noise of DFB semiconductor lasers, J Lightwave... identical at threshold 4.4.2 Determination of Below-Threshold Spontaneous Emission Power When a laser diode is biased in the below-threshold regime, there is finite optical power output due to spontaneous emission From the Poynting vector of the propagating field, the 116 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS spontaneous emission power PN ðzÞ within an angular frequency bandwidth Á! can be written... SEMICONDUCTOR LASERS Figure 4.6 Simplified schematic diagram showing a 3PS DFB laser diode structure of phase shifts 2 and 4 are allowed to move along the cavity Their relative positions are defined by a position parameter, , as ¼ It should be noticed that when 2L1 L ð4:61Þ ¼ 0 or 1, the structure becomes a single-phase-shifted laser Figure 4.7 Below-threshold spectra of various DFB semiconductor laser. .. mirrorless DFB laser cavity where ^1 ¼ ^2 ¼ 0, the above threshold equation is simplified such that r r y22 ðzNþ1 j z1 Þ ¼ 0 ð4:34Þ In fact, eqn (4.33) is a general expression that can be used to determine the lasing threshold characteristics of semiconductor laser devices These include FP lasers, conventional DFB lasers (both mirrorless and those having finite facet reflections), single-phase-shifted DFB laser. .. matrix has been introduced and explored Compared with the boundary matching approach, the TMM is more robust and flexible By converting the coupled wave equations into a matrix equation, the wave propagating characteristics of the corrugated DFB section can be represented using a transfer matrix The transfer matrix approach was extended to include phase discontinuity and the residue reflection at the... maintained along the laser cavity A different technique is required in the above-threshold condition when variables become longitudinally dependent 4.4.3 Numerical Results from Various DFB Laser Diodes In this section, the below-threshold PN of various DFB laser diodes will be presented Results obtained from a conventional, a QWS and a 3PS DFB LD will be compared For all these lasers, a laser cavity length... distributed feedback semiconductor laser diodes at threshold using the transfer-matrix method (TMM), Semi Sci and Technol., 8(5), 1126–1132, 1994 28 Henry, C H., Theory of spontaneous emission noise in open resonator and its application to lasers and optical amplifiers, J Lightwave Technol., LT-4(3), 288–297, 1986 29 Yamamoto, Y., Coherence, Amplification, and Quantum Effects in Semiconductor Lasers New York: Wiley,... discussed 4.6 REFERENCES 1 Makino, T., Transfer-matrix analysis of the intensity and phase noise of multisection DFB semiconductor lasers, IEEE J Quantum Electron., QE-27(11), 2404 –2415, 1991 2 Makino, T., Transfer-matrix formulation of spontaneous emission noise of DFB semiconductor lasers, J Lightwave Technol., LT-9(1), 84 –91, 1991 3 Yamada, M and Sakuda, K., Analysis of almost-periodic distributed . surface emitting N –– laser 112 TRANSFER MATRIX MODELLING IN DFB SEMICONDUCTOR LASERS spontaneous emission noise inside the semiconductor laser [28–29] 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