7 Above-Threshold Analysis of Various DFB Laser Structures Using the TMM 7.1 INTRODUCTION The above-threshold lasing characteristics of DFB semiconductor laser diodes were presented in the previous chapter using a modified transfer matrix. Instead of using an averaged carrier concentration, the inclusion of the actual carrier distribution allows phenomena such as the spatial hole burning effect and non-linear gain to be included. In the analysis, a parabolic gain model and high-order carrier recombination were assumed. Lasing mode characteristics such as the longitudinal distribution of carrier density, photon density, refractive index and the internal field intensity were shown for various laser structures. In this chapter, results obtained from the lasing mode characteristics will be used to determine the mode stability and noise characteristics of DFB LDs. For a coherent optical communication system, it is essential that the LD used oscillates at a stable single mode and that a narrow spectral linewidth is achieved. Using the information obtained for the lasing mode characteristics, a method derived from the above-threshold transfer matrix model will be introduced in section 7.2 which allows the gain margin to be evaluated. By introducing an imaginary wavelength into the transfer matrix equation, chara- cteristics of other non-lasing side modes can be evaluated and hence the single-mode stability can be obtained. Numerical results obtained using this method will be presented in section 7.3. In section 7.4, an alternative method which allows the theoretical prediction of the above- threshold spontaneous emission will be presented. Based on the Green’s function method, one can use transfer matrices to help determine the single-mode stability of a DFB laser structure by inspecting the spectral components of oscillating modes. The TMM also allows the noise characteristics of the DFB LD to be evaluated. In section 7.5, it will be shown that various contributions to the spectral linewidth can be determined using the information obtained from the above-threshold transfer matrices. In the analysis, the effective linewidth enhancement factor [1] has been used instead of the material-based linewidth enhancement [2]. Using a more realistic effective linewidth enhancement factor, impacts caused by structural changes can be investigated in a systematic way. Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1 7.2 SINGLE-MODE STABILITY IN DFB LDs Using the TMM-based above-threshold laser cavity model as presented in the previous chapter, distributions of the carrier density, photon density, refractive index and the normalised internal field intensity were obtained for various DFB laser structures. From the emitting photon density at the facet, the output optical power can be evaluated. Figure 7.1 summarises results obtained for QWS, 3PS and DCC þ QWS LDs with biasing current as a parameter. Within the range of biasing current that we are interested in, no ‘kink’ is observed in any of the three cases. Compared with the QWS structure, it seems that the introduction of multiple phase shifts along the 3PS laser cavity has increased the overall cavity loss. The figure also shows that the 3PS laser structure having the largest amplitude threshold gain has a relatively larger value of threshold current. Among the three cases, both the 3PS and the DCC þ QWS structures appear to have relatively larger output power under the same normalised biasing current. Semiconductor lasers having stable single longitudinal outputs and narrow spectral linewidths are indispensable in coherent optical communication systems. With a built-in wavelength selective corrugation, a DFB laser diode has a single longitudinal output. Other oscillation modes failing to reach the threshold condition become the non-lasing side modes. As the biasing current increases, the spatial hole burning effect becomes significant and mode competition between the lasing mode and the most probable non-lasing side mode may occur. Mode competition has been observed for a QWS DFB laser [3–5], which resulted Figure 7.1 Optical output power of three different types of DFB LD. 172 ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES in multiple mode oscillation as the biasing current increased. In this section, a numerical method that allows theoretical prediction of the above-threshold single-mode stability of DFB LDs will be presented. In the analysis, it is assumed that the detailed lasing mode characteristics have been obtained from the above-threshold transfer matrix model as discussed in the previous chapter. Single-mode stability implies the suppression of non-lasing side modes. There are two possible ways to demonstrate single-mode stability in DFB LDs. The first approach involves the evaluation of the normalised gain margin ÁL between the lasing mode and the probable non-lasing side modes. The single-mode stability is said to be threatened if the gain margin, Á, drops below 5 cm À1 for a 500 mm length laser cavity. An alternative method to check the stability of the device involves the measurement of the spectral characteristics. With the help of an optical spectrum analyser, the measured intensity difference between the lasing mode and the side modes will give single-mode stability. The second approach is often used to measure the single-mode stability of DFB LDs. In this section, we will concentrate on the first approach which leads to the evaluation of the above-threshold gain margin. From the numerical method discussed in the previous chapter, oscillation characteristics of the lasing mode were obtained at a fixed biasing current. By dividing the DFB laser into a large number of smaller sections, longitudinal distributions like the carrier and photon densities were determined. Since the laser cavity is now dominated by the lasing mode, the characteristics of other non-lasing side modes should be derived from the lasing mode. In order to evaluate the characteristics of other non-lasing side modes, the dominant lasing mode has to be suppressed in a mathematical way. In the analysis, an imaginary wavelength  i is introduced [6]. As a result, the complex wavelength  c of an unknown side mode becomes  c ¼  þ j i ð7:1Þ where  i takes into account the mathematical gain the side mode may need to reach its threshold value and  becomes the actual wavelength of the side mode. By changing values of both  and  i , wavelengths of other non-lasing side modes can be evaluated. The numerical procedure involved in determining the characteristics of other non-lasing side modes is summarised as follows: 1. A numerical procedure similar to the one discussed in section 6.3 is adopted. To initialise the iteration process, a 5  5 mathematical grid which consists of points (;  i ) is formed. Each of these (;  i ) points will be used as an initial guess. Since the most probable non-lasing side mode is usually found near the lasing mode, it is advisable to start with those wavelengths which are close to that of the lasing mode. 2. Lasing mode characteristics like the longitudinal distribution of the carrier, photon density and refractive index are retrieved from the data files obtained earlier. Matrix elements of each transfer matrix are then determined. 3. From the boundary condition at the left facet, electric fields E R ðz 1 Þ and E S ðz 1 Þ at the left facet are found which serve as the input electric field to the transfer matrix chain. 4. By passing the electric field through the transfer matrix chain, the output electric field at the right laser facet is determined. The discrepancy with the right facet boundary condition is then evaluated and stored. SINGLE-MODE STABILITY IN DFB LD 173 5. Steps (2) to (4) are then repeated with other pairs of (;  i ) obtained from the 5  5 mathematical grid. By comparing the discrepancy obtained from each of these (;  i ) pairs, the one showing the minimum discrepancy is then selected. Depending on the position of the selected (;  i ) on the mathematical grid, a new mathematical grid is formed ready for the next iteration. 6. Procedures shown above are then repeated until the discrepancy with the right facet boundary condition falls below 10 À14 . The final  obtained becomes the non-lasing side mode and distributions of the amplitude gain ðzÞ and the detuning coefficient ðzÞ are stored. 7. The average values of "  SM and "  SM associated with the side mode are evaluated from the corresponding longitudinal distribution as "  SM ¼ P N j¼1  j N ð7:2Þ "  SM ¼ P N j¼1  j N ð7:3Þ where  j and  j ð j ¼ 1toNÞ are the amplitude gain value and the detuning value obtained from each transfer matrix, and N is the total number of transfer matrices. 8. The whole numerical procedure can be repeated for other non-lasing side modes. All "  SM obtained are then sorted in increasing order. The one showing the smallest value becomes the most probable side mode. Characteristics of a new dominant lasing mode must be loaded every time a new injection current is used. From the result obtained, the gain margin between the lasing mode and the most probable non-lasing side mode can be evaluated as Á ¼ "  L À "  SM ð7:4Þ To maintain a stable single-mode oscillation, one must ensure that ÁL > 0:25 for a 500 mm length laser cavity. 7.3 NUMERICAL RESULTS ON THE GAIN MARGIN OF DFB LDs In this section, numerical results obtained for various DFB LDs, including the QWS, 3PS and the DCC þ QWS laser structures, will be presented. Figure 7.2 shows the characteristics of the lasing mode (0) and side modes (Æ1) in the " L; " L ÀÁ plane. In the analysis, an anti- reflection-coated QWS DFB structure with L ¼ 2 is assumed for a 500 mm length cavity. For each oscillating mode shown in Fig. 7.2, the cross and the black circle correspond to the oscillating mode at threshold and at 5I th , respectively. When the biasing current increases from the threshold value, an increase in the lasing mode amplitude gain and a corresponding reduction of gain margin between the lasing mode and the þ1 side mode can be seen. Such a phenomenon is well known to be induced by the spatial hole burning effect [5]. 174 ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES Figure 7.3 shows the normalised amplitude gain change of the QWS DFB LD with respect to the biasing current. The amplitude gain distributions for the lasing mode and the non- lasing side modes are shown. It is obvious that the " L value of each mode varies in a different way. When the biasing current increases from threshold, it is clear that the amplitude gain of the þ1 mode reduces remarkably and approaches that of the lasing mode. By contrast, the amplitude gain of the À1 mode becomes larger and hence becomes less significant. In Fig. 7.4, for the same QWS structure, the variation of the normalised detuning coefficient " L is shown. In such a strongly coupled LD, both the lasing mode and the þ1 side mode shift towards the shorter wavelength side (negative " L) with increasing biasing current. Among the different modes shown, the shift of the lasing mode is stronger since it is found closer to the gain peak. With multiple phase shifts introduced along the corrugation, the characteristics of the 3PS structure are shown in Fig. 7.5. In the analysis, the 3PS DFB is assumed to be anti-reflection coated. Phase shifts  2 ¼  3 ¼  4 ¼ =3 and PSP ¼ 0:5 are assumed for the 500 mm long cavity. Compared with the QWS structure, the 3PS structure shows a smaller shift in mode characteristics. This may be clearer when the variation of both amplitude gain and the detuning coefficient are shown as a function of normalised injection current. From Fig. 7.6 where the amplitude gain change is shown, the injection current alters the oscillating mode in a different way. It can be observed that the gain margin between the lasing mode and the Figure 7.2 Lasing characteristics of the QWS DFB laser structure in the ( " L, "L) plane showing two values of normalised injection current. NUMERICAL RESULTS ON THE GAIN MARGIN OF DFB LDS 175 Figure 7.3 Average amplitude gain " L of the QWS DFB laser structure versus the normalised injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown. Figure 7.4 Average detuning coefficient " L of the QWS DFB laser structure versus the normalised injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown. Figure 7.5 Lasing characteristics of the 3PS DFB laser structure in the ð " L; " LÞ plane showing two values of normalised injection current. Figure 7.6 Average amplitude gain " L of the 3PS DFB laser structure versus the normalised injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown. most probable side mode (þ1) shows little change. A similar situation can be seen in Fig. 7.7 where the variation of the detuning coefficient is demonstrated. The lasing mode shown has a milder shift with increasing biasing current. On combining results from both Figs. 7.6 and 7.7, it appears that the 3PS laser structure is not seriously affected by the spatial hole burning effect. No severe reduction of gain margin and a fairly mild shift in detuning coefficient are observed. Results obtained from a DCC þ QWS laser structure for the lasing characteristics, amplitude gain and the detuning coefficient are shown in Figs 7.8 to 7.10, respectively. A 500 mm length cavity is assumed. Other parameters used follow those adopted in the TMM- based laser model as discussed in the previous chapter. These parameters include  1 = 2 ¼ 1=3,  avg L ¼ 2 and CP ¼ 0:46. Due to effects of spatial hole burning, all figures appear to have a similar trend in the shape of their curves. On the other hand, the single- mode stability in the DCC þ QWS structure is improved due to the presence of the distributed coupling coefficient. As shown in Fig. 7.9, the amplitude gain difference between the lasing mode and that of the þ1 mode remains at a high value even at a high biasing current. Figure 7.11 shows the normalised gain margin (ÁL) between the lasing mode and the most probable side mode. Results obtained from QWS, 3PS and DCC þ QWS structures are shown. The gain margin of both the QWS and the DCC þ QWS structures reduces when the biasing current increases. From the above-threshold analysis, these structures are Figure 7.7 Average detuning coefficient " L of the 3PS DFB laser structure versus the normalised injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown. 178 ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES Figure 7.8 Lasing characteristics of the DCC þ QWS DFB laser structure in the ð " L; " LÞ plane showing two values of normalised injection current. Figure 7.9 Average amplitude gain " L of the DCC þ QWS DFB laser structure versus the nor- malised injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown. Figure 7.10 Average detuning coefficient " L of the DCC þ QWS DFB laser structure versus the normalised injection current. Results for both the lasing mode and non-lasing side modes (Æ1) are shown. Figure 7.11 The variation of gain margin with respect to changes in the injection current for different DFB LD structures. 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Analysis of Various DFB Laser Structures Using the TMM 7.1 INTRODUCTION The above-threshold lasing characteristics of DFB semiconductor laser diodes were presented. determine the single-mode stability of a DFB laser structure by inspecting the spectral components of oscillating modes. The TMM also allows the noise characteristics