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5 ThresholdAnalysisandOptimisation of Various DFB LDs Using the Transfer Matrix Method 5.1 INTRODUCTION In the previous chapter, the transfer matrix method (TMM) was introduced to solve the coupled wave equations in DFB laser structures. Its efficiency and flexibility in aiding the analysis of DFB semiconductor LDs has been explored theoretically. A general N-sectioned DFB laser model was built which comprised active/passive and corrugated/planar sections. In this chapter, the N-sectioned laser model will be used in the practical design of the DFB laser. The spatial hole burning effect (SHB) [1] has been known to limit the performance of DFB LDs. As the biasing current of a single quarterly-wavelength-shifted (QWS) DFB LD increases, the gain margin reduces. Therefore, the maximum single-mode output power of the QWS DFB LD is restricted to a relatively low power operation. The SHB phenomenon caused by the intense electric field leads to a local carrier depletion at the centre of the cavity. Such a change in carrier distribution alters the refractive index along the laser cavity and ultimately affects the lasing characteristics. By changing the structural parameters inside the DFB LD, an attempt will be made to reduce the effect of SHB. As a result, a larger single-mode power, and consequently a narrower spectral linewidth, may be achieved. A full structural optimisation will often involve the examination of all possible structural combinations in the above-threshold regime. On the other hand, the analysis of the structural design may be simplified, in terms of time and effort, by optimising the threshold gain margin and the field uniformity. The structural changes and their impacts on the characteristics of DFB LDs will now be presented. By introducing more phase shifts along the laser cavity, a three-phase-shift (3PS) DFB LD will be investigated in section 5.2. In particular, impacts due to the variation of both phase shifts and their positions on the lasing characteristics of the 3PS DFB LD will be discussed. To reduce the SHB effect, it is necessary to have a more uniform field distribution, whilst maintaining a large gain margin ðLÞ. The optimised structural design for the 3PS DFB laser based on the values of L and the flatness (F) of the field distribution will be discussed in section 5.3 [2]. Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1 124 THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS By changing the height of the corrugation and thus the coupling coefficient along a DFB laser cavity, a distributed coupling coefficient (DCC) DFB laser can be built. In section 5.4, the threshold characteristics of this structure will be shown. In particular, effects due to the variation of the coupling ratio and the position of the corrugation change will be investigated. To maintain a single-mode oscillation, a single phase shift is introduced at the centre of the cavity. By changing the value of the phase shift, the combined effect with the non-uniform coupling coefficient will be presented. Optimised structural combinations that satisfy both a high gain margin and a low value of flatness will be selected for later use in the above-threshold analysis. In section 5.5, the combined effect of both multiple phase shifts and non-uniform coupling coefficients will be investigated using a DCC þ3PS DFB laser structure. Finally, a summary will be presented at the end of this chapter. 5.2 THRESHOLD ANALYSIS OF THE THREE-PHASE-SHIFT (3PS) DFB LASER By introducing more phase shifts along the laser cavity, it has been shown [3–5] that the spatial hole burning effect can be reduced in a 3PS DFB LD which is characterised by a more uniform internal field distribution. Experimental measurement has been carried out [5] using a fixed value of phase shift. However, independent changes in thevalue of phase shift have not been fully explored. Using the TMM, it was shown in Table 4.1 of Chapter 4 that four transfer matrices are necessary to determine the threshold condition of 3PS DFB lasers. In Fig. 5.1, a Figure 5.1 Schematic diagram showing a 3PS DFB LD. schematic diagram of the 3PS DFB laser structure is shown. In the figure, 2, 3 and 4 represent phase shifts and the length of each smaller section is labelled Lj ðj ¼ 1;2Þ. In the analysis, zero facet reflection at the laser facets is assumed. Following the formulation of the transfer matrix method, the overall transfer matrix of the 3PS DFB laser becomes: Yðz j z Þ ¼ Fð4ÞFð3ÞFð2ÞFð1Þ ¼ y11ðz5 j z1Þ y12ðz5 j z1Þ ð5:1Þ 21 5 1 22 5 1 where FðjÞðj ¼ 1 to 4) corresponds to the transfer matrix of each smaller section. For a mirrorless cavity, the threshold condition can be found by solving the following equation y22ðz5 j z1Þ ¼ 0 ð5:2Þ THRESHOLD ANALYSIS OF THE THREE-PHASE-SHIFT (3PS) DFB LASER 125 Figure 5.2 Resonance modes of various DFBs that include: (a) a conventional DFB laser diode; (b) a single QWS DFB laser diode; (c) a three /2-phase-shifted DFB laser diode. Using a numerical approach such as Newton–Raphson’s method [6] for analytical complex equations, the threshold equation above may be solved. Figure 5.2 shows the resonance modes obtained from a symmetrical 3PS DFB laser where 2 ¼ 3 ¼ 4 ¼ =2 and L1 ¼ L2 are assumed. For comparison purposes, results obtained from a mirrorless conventional DFB laser and a single =2 phase-shifted DFB laser are also included. In all three cases, the coupling coefficient and the overall laser cavity length L are fixed at 40 cmÿ1 and 500 mm, respectively. Oscillation modes at the Bragg wavelength are found for both the single =2 and a 3PS DFB structure. However, the Bragg resonance mode of the 3PS DFB laser does not show the smallest amplitude threshold gain. Instead, degenerate oscillation occurs since it is shown that both the ÿ1 and þ1 modes share the same value of amplitude threshold gain. It is interesting to see how a single =2 phase shift enables SLM operation whilst multi-mode oscillation occurs in the case where there are three phase shifts, i.e. f=2;=2;=2g. The pair of braces f g used hereafter will indicate a phase combination in the 3PS structure, that is f2;3;4g. 5.2.1 Effects of Phase Shift on the Lasing Characteristics In order that stable SLM operation can be achieved in the 3PS DFB laser, one must change the value or the position of the phase shift. Figure 5.3 shows oscillation modes of various 3PS DFB laser structures. In the analysis, the values of the three phase shifts are assumed to be equal and the phase shift positions are the same as in Fig. 5.2. A shift of resonance mode can be seen when all phase shifts change from =2 to 2=5. The þ1 mode which demonstrates the smallest amplitude threshold gain will become the lasing mode after lasing 126 THRESHOLD ANALYSIS AND OPTIMISATION OF VARIOUS DFB LDS Figure 5.3 Resonance modes in various 3PS DFBs that include: (a) a f=2, =2, =2g 3PS DFB laser; (b) a f2=5;2=5;2=5g 3PS DFB laser; (c) a f3=5;3=5;3=5g laser. threshold is reached. On the other hand, the ÿ1 mode will become the lasing mode when the three phase shifts change from =2 to 3=5. With all three phase shifts displaced from the usual =2 values, SLM can be achieved in the 3PS DFB LD. 5.2.2 Effects of Phase Shift Position (PSP) on the Lasing Characteristics The 3PS DFB laser structure we have discussed so far is said to be symmetrical. For a cavity length of L, the position of phase shifts is assumed in such a way that L1 ¼ L2 ¼ L=4. To investigate the effect of the phase shift position (PSP) on the threshold characteristics, a position factor is introduced such that L1 2L1 L1 þL2 L ð5:3Þ where 3 is assumed to be located at the centre of the cavity. Using the above equation, it should be noted that both ¼ 0 and ¼ 1 correspond to a single-phase-shifted DFB laser structure. In Fig. 5.4, the variation of the amplitude threshold gain is shown with the position factor for different values of normalised coupling coefficient L. All the phase shifts are fixed at 2 ¼ 3 ¼ 4 ¼ =3. At a fixed value of , the figure shows a decrease in amplitude threshold gain as the L value increases. Along the curve L ¼ 1:0, discontinuities at ¼ 0:12 and ¼ 0:41 indicate possible changes in the oscillation mode. Figure 5.4 The change of amplitude gain with respect to the phase shift position for different values of coupling coefficient . Figure 5.5 The variation of detuning coefficient with respect to the phase shift position for coupling coefficient . ... - tailieumienphi.vn