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3 Structural Impacts on the Solutions of Coupled Wave Equations: An Overview 3.1 INTRODUCTION The introduction of semiconductor lasers has boosted the development of coherent optical communication systems. With the built-in wavelength selection mechanism, distributed feedback semiconductor laser diodes with a higher gain margin are superior to the Fabry– Perot laser in that a single longitudinal mode of lasing can be achieved. In this chapter, results obtained from the threshold analysis of conventional and single-phase-shifted DFB lasers will be investigated. In particular, structural impacts on the threshold characteristic will be discussed in a systematic way. The next two sections of this chapter present solutions of the coupled wave equations in DFB laser diode structures. In section 3.4 the concepts of mode discrimination and gain margin are discussed. The threshold analysis of a conventional DFB laser diode is studied in section 3.5, whilst the impact of corrugation phase at the DFB laser diode facets is discussed in section 3.6. By introducing a phase shift along the corrugations of DFB LDs, the degenerate oscillating characteristic of the conventional DFB LD can be removed. In section 3.7, structural impacts due to the phase shift and the corresponding phase shift position (PSP) will be considered. As mentioned earlier in Chapter 2, the introduction of the coupling coefficient into the coupled wave equations plays a vital role since it measures the strength of feedback provided by the corrugation. In section 3.7, the effect of the selection of corrugation shape on the magnitude of will be presented. With a =2 phase shift fabricated at the centre of the DFB cavity, the quarterly-wavelength-shifted (QWS) DFB LD oscillates at the Bragg wavelength. However, the deterioration of gain margin limits its use as the current injection increases. This phenomenon induced by the spatial hole burning effect, which is the major drawback of the QWS laser structure, will be examined at the end of this chapter. The limited application of the eigenvalue equation in solving the coupled wave equations will also be considered. Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1 80 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS 3.2 SOLUTIONS OF THE COUPLED WAVE EQUATIONS In Chapter 2 it was shown that the characteristics of DFB LDs can be described by a pair of coupled wave equations. The strength of the feedback induced by the perturbed refractive index or gain is measured by the coupling coefficient. Relationships between the forward and the backward coupling coefficients RS and SR were derived for purely index-coupled, mixed-coupled and purely gain-coupled structures. By assuming a zero phase difference between the index and the gain term, the complex coupling coefficient could be expressed as RS ¼ SR ¼ i þjg ¼ ð3:1Þ where becomes a complex coupling coefficient. According to eqn (2.98), the trial solution of the coupled wave equation can be expressed in terms of the Bragg propagation constant such that EðzÞ ¼ RðzÞeÿjb0z þ SðzÞejb0z ð3:2Þ where the coefficients RðzÞ and SðzÞ are given as [1] RðzÞ ¼ R1eðgzÞ þR2eðÿgzÞ ð3:3aÞ and SðzÞ ¼ S1eðgzÞ þS2eðÿgzÞ ð3:3bÞ In the above equations, R1, R2, S1 and S2 are complex coefficients and g is the complex propagation constant to be determined from the boundary conditions at the laser facets. Without loss of generality, one can assume ReðgÞ > 0. As a result, those terms with coefficients R1 and S2 become amplified as the waves propagate along the cavity. By contrast, those terms with coefficients R2 and S1 are attenuated. By combining the above equations with eqn (3.2), it can be shown easily that the propagation constant of the amplified waves becomes b0 ÿImðgÞ whilst the decaying waves propagate at b0 þ ImðgÞ. By substituting eqns (3.3a) and (3.3b) into the coupled wave equations, the following relations are obtained by collecting identical exponential terms [2] ÿR1 ¼ jeÿjS1 ð3:4aÞ ÿR2 ¼ jeÿjS2 ð3:4bÞ ÿS1 ¼ jejR1 ð3:4cÞ ÿS2 ¼ jejR2 ð3:4dÞ where ÿ ¼ s ÿjd ÿ g ð3:5aÞ ÿ ¼ s ÿjd þ g ð3:5bÞ By comparing eqns (3.4a) and (3.4c), a non-trivial solution exists if the following equation is satisfied ¼ j ¼ j ð3:6Þ SOLUTIONS OF THE COUPLED WAVE EQUATIONS 81 Based on the equation shown above, eqn (3.4) is simplified to become R1 ¼ 1eÿj S1 ð3:7aÞ R2 ¼ eÿj S2 ð3:7bÞ Similarly, by equating eqns (3.4a) and (3.4c), one obtains g2 ¼ ðs ÿjdÞ2 þ 2 ð3:8Þ It is important that the dispersion equation shown above is independent of the residue corrugation phase . With a finite laser cavity length L extending from z ¼ z1 to z ¼ z2 (where both z1 and z2 are assumed to be greater than zero), the boundary conditions at the terminating facets become Rðz1Þeÿjb0z1 ¼ r1 Sðz1Þejb0z1 ð3:9aÞ Sðz2Þejb0z2 ¼ r2 Rðz2Þeÿjb0z2 ð3:9bÞ where r1 and r2 are amplitude reflection coefficients at the laser facets z1 and z2, respectively. According to eqns (3.3) and (3.4), the above equations could be expanded in such a way that ð1 ÿ r1Þe2gz1 2 r1= ÿ1 1 ðr2 ÿÞe2gz2 2 1= ÿ r2 1 ð3:10aÞ ð3:10bÞ In the above equation, all RðzÞ and SðzÞ terms are expressed in terms of R1 and R2, whilst r1 and r2 are the complex field reflectivities of the left and the right facets, respectively such that r1 ¼ r1 e2jb0z1 ej ¼ r1 ej 1 r2 ¼ r2 eÿ2jb0z2 eÿj ¼ r2 ej 2 ð3:11aÞ ð3:11bÞ with 1 and 2 being the corresponding corrugation phases at the facets. Equations (3.10a) and (3.10b) are homogeneous in R1 and R2. In order to have non-trivial solutions, the following condition must be satisfied ð1 ÿ r1Þe2gz1 ðr2 ÿ Þe2gz2 r1 ÿ 1 ÿr2 ð3:12Þ Then the above equation can be solved for and 1= whilst employing the relation g ¼ ÿj ÿ 1 ð3:13Þ 82 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS derived from eqns (3.5a) and (3.5b). After some lengthy manipulation [2], one ends up with an eigenvalue equation gL ¼ ÿj sinhðLÞ nðr1 þ r2Þð1 ÿ r1 r2ÞcoshðgLÞ ð1 þ r1 r2Þ2o ð3:14Þ where ¼ ðr1 ÿ r2Þ2 sinh2ðgLÞþð1 ÿ r1 r2Þ2 D ¼ ð1 þ r1 r2Þ2 ÿ4r1r2 cosh2ðgLÞ r1 ¼ r1e2jb0z1ej ¼ r1ej 1 r2 ¼ r2eÿ2jb0z2eÿj ¼ r2ej 2 ð3:15aÞ ð3:15bÞ ð3:15cÞ ð3:15dÞ By squaring eqn (3.1), and after some simplification, one ends up with a transcendental function ðgLÞ2 þðLÞ2 sinh2ðgLÞð1 ÿ r2Þð1 ÿ r2Þ þ 2jLðr1 þ r2Þ2ð1 ÿ r1r2ÞgL sinhðgLÞcoshðgLÞ ¼ 0 ð3:16Þ In the above equation, there are four parameters which govern the threshold characteristics of DFB laser structures. These are the coupling coefficient , the laser cavity length L and the complex facet reflectivities r1 and r2. Due to the complex nature of the above equation, numerical methods like the Newton–Raphson iteration technique can be used, provided that the Cauchy–Riemann condition on complex analytical functions is satisfied. Before starting the Newton–Raphson iteration, an initial value of ð;Þini is chosen from a selected range of ð;Þ values. Usually, the first selected guess will not be a solution of the threshold equation and hence the iteration continues. At the end of the first iteration, a new pair of ð0;0Þ will be generated and checked to see if it satisfies the threshold equation. The iteration will continue until the newly generated ð0;0Þ pair satisfies the threshold equation within a reasonable range of error. Starting with different initial guesses of ð;Þini, other oscillating modes can be determined in a similar way. By collecting all ð ; Þ pairs that satisfy the threshold equation, the one showing the smallest amplitude gain will then become the lasing mode. The final value ð;Þfinal is then stored up for later use, in which the threshold current and the lasing wavelength of the LD are to be decided. In general, eqn (3.16) characterises all conventional DFB semiconductor LDs with continuous corrugations fabricated along the laser cavity. 3.3 SOLUTIONS OF COMPLEX TRANSCENDENTAL EQUATIONS USING THE NEWTON–RAPHSON APPROXIMATION All complex transcendental equations can be expressed in a general form such that WðzÞ ¼ UðzÞ þ jVðzÞ ¼ 0 ð3:17Þ SOLUTIONS OF COMPLEX TRANSCENDENTAL EQUATIONS 83 where z ¼ x þ jy is a complex number and UðzÞ and VðzÞ are, respectively, the real and imaginary parts of the complex function. From the above equation, one can deduce the following equality easily UðzÞ ¼ VðzÞ ¼ 0 ð3:18Þ By taking the first-order derivative of eqn (3.17) with respect to z, one can obtain @W @U @V @U @V @z @z @z @x @x ð3:19Þ The second equality sign can be obtained using the chain rule. Applying the Taylor series, the functions U(z) and V(z) can be approximated about the exact solution ðxreq, yreqÞ such that Uðxreq;yreqÞ ¼ Uðx;yÞ þ @x ðxreq ÿ xÞ þ @y ðyreq ÿ yÞ ð3:20Þ Vðxreq;yreqÞ ¼ Vðx;yÞ þ @x ðxreq ÿxÞ þ @y ðyreq ÿ yÞ ð3:21Þ where the values of (x, y) chosen are sufficiently close to the exact solutions. Other higher derivative terms from the Taylor series have been ignored. One then obtains the following equations for xreq and yreq from the above simultaneous equations [2] Vðx;yÞ @y ÿ Uðx;yÞ @y req Det yreq ¼ y þ Uðx;yÞ @x ÿ Vðx;yÞ @x ð3:22Þ ð3:23Þ where Det ¼ @U2þ@V2 ð3:24Þ Terms like @U=@x, @V=@x, @U=@y and @V=@y are the first derivatives of functions U(z) and V(z). For an analytical complex function W(z), the Cauchy–Riemann condition which states that @U @V @U @V @x @y @y @x ð3:25Þ must be satisfied [3]. ... - tailieumienphi.vn