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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 singlephase-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 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 ẳ SR ẳ i ỵj g 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 Ezị ẳ Rzịejb0 z ỵ Szịe jb0 z ð3:2Þ where the coefficients RðzÞ and SðzÞ are given as [1] Rzị ẳ R1 egzị ỵ R2 egzị 3:3aị Szị ẳ S1 egzị ỵ S2 egzị 3:3bị and 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 > 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 À ImðgÞ whilst the decaying waves propagate at b0 ỵ Imgị By substituting eqns (3.3a) and (3.3b) into the coupled wave equations, the following relations are obtained by collecting identical exponential terms [2] ^ À R1 ¼ j ej S1 3:4aị R2 ẳ j ej S2 3:4bị j S1 ẳ j e R1 ^ S2 ẳ j e j R2 3:4cị 3:4dị ^ ¼ s À jd À g ð3:5aÞ À ¼ s jd ỵ g 3:5bị where By comparing eqns (3.4a) and (3.4c), a non-trivial solution exists if the following equation is satised &ẳ ^ j ẳ j 3:6ị 81 SOLUTIONS OF THE COUPLED WAVE EQUATIONS Based on the equation shown above, eqn (3.4) is simplified to become R1 ẳ ej S1 & 3:7aị R2 ẳ &ej S2 ð3:7bÞ Similarly, by equating eqns (3.4a) and (3.4c), one obtains g2 ẳ s j dị2 ỵ 3:8ị 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 Rz1 ị ej b0 z1 ẳ ^1 Sz1 Þ e j b0 z1 r Sðz2 Þ e j b0 z ẳ ^2 Rz2 ị e r 3:9aị Àj b0 z2 ð3:9bÞ r where ^1 and ^2 are amplitude reflection coefficients at the laser facets z1 and z2 , respectively r According to eqns (3.3) and (3.4), the above equations could be expanded in such a way that ð1 À & r1 Þ e2g z1 Á R1 r1 =& À ðr2 À & Þ e2g z2 Á R1 R2 ẳ 1=& r2 R2 ẳ 3:10aị 3:10bị 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 r r r1 ¼ ^1 e2 j b0 z1 e j ¼ ^1 e j É r2 ¼ ^2 e r À2 j b0 z2 Àj e ¼ ^2 e r j É2 ð3:11aÞ ð3:11bÞ with and 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 À & r1 ị e2g z1 r2 &ị e2gz2 ẳ r1 À & À & r2 ð3:12Þ Then the above equation can be solved for & and 1=& whilst employing the relation Àj g¼   &À & 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 ẳ j o sinhLị n r1 ỵ r2 ị1 r1 r2 ị coshgLị ặ ỵ r1 r2 ị2 D 3:14ị where ẳ r1 r2 ị2 sinh2 gLị ỵ r1 r2 ị2 2 3:15aị D ẳ ỵ r1 r2 ị r1 r2 cosh g Lị 3:15bị r1 ẳ ^1 e2jb0 z1 e j ¼ ^1 e j r r 3:15cị r2 ẳ ^2 e r 2jb0 z2 j e ẳ ^2 e r j 3:15dị By squaring eqn (3.1), and after some simplification, one ends up with a transcendental function 2 gLị2 ỵ Lị2 sinh2 gLị r1 ị1 r2 ị ỵ 2j L r1 ỵ r2 ị2 r1 r2 Þ gL sinhðgLÞ coshðgLÞ ¼ 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 ; 0 Þ will be generated and checked to see if it satisfies the threshold equation The iteration will continue until the newly generated ð0 ; 0 Þ 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 ð0 ; 0 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 ẳ Uzị ỵ j Vzị ẳ 3:17ị SOLUTIONS OF COMPLEX TRANSCENDENTAL EQUATIONS 83 where z ẳ x ỵ j y 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 Uzị ẳ Vzị ẳ 3:18ị By taking the first-order derivative of eqn (3.17) with respect to z, one can obtain @W @U @V @U @V ẳ ỵj ẳ þj @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 , yreq Þ such that @U @U xreq xị ỵ yreq yị @x @y @V @V xreq xị ỵ yreq yị Vxreq ; yreq ị ẳ V x; yị ỵ @x @y Uxreq ; yreq ị ẳ U x; yị ỵ ð3:20Þ 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] @U @V À Uðx; yÞ @y @y ẳxỵ Det @V @U Vx; yị Ux; yị @x @x ẳyỵ Det Vx; yị xreq yreq 3:22ị 3:23ị where  2  2 @U @V Det ẳ ỵ @x @y 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 ¼ ; @x @y must be satisfied [3] @U @V ẳ @y @x 3:25ị 84 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS On replacing all the @=@y terms with @=@x using the above Cauchy–Riemann condition, eqns (3.22) and (3.24) can be simplified such that  @U Det ẳ @x 2 Vx; yị xreq ẳ x 3:26ị @V @U ỵ Ux; yị @x @x Det ð3:27Þ Here, only the first-order derivative terms @U=@x and @V=@x are used These can be determined from the complex function of eqn (3.19) Given an initial guess of ðx; yÞ, the numerical iteration process then starts A new guess is generated by following eqns (3.23), (3.26) and (3.27) Unless the new guess is sufficiently close to the exact solution (within 10À9 , let’s say), the new guess solution formed will become the initial guess of the next iteration The iteration process continues until approximate solutions of ðxreq , yreq Þ appear The advantages of this method are its speed and flexibility In addition, the derivative term @W=@z is found analytically first, before any numerical iteration is started Using this method, one can avoid any errors associated with other numerical methods such as numerical differentiation 3.4 CONCEPTS OF MODE DISCRIMINATION AND GAIN MARGIN At a fixed value of , pairs of ð; Þfinal can be determined following the method discussed in the previous section Each ð; Þfinal pair, which represents an oscillation mode, is plotted on the – plane Similarly, pairs of ð; Þfinal values can be obtained by changing the values of By plotting all ð; Þfinal points on the – plane, the mode spectrum of the DFB LD is formed A simplified – plot is shown in Fig 3.1 Different symbols shown represent various longitudinal modes obtained for various coupling coefficients whilst the solid curve shows how longitudinal modes join to form an oscillating mode When the biasing current increases, the longitudinal mode showing the smallest amplitude gain will reach the threshold condition first and begin to lase Other modes failing to reach the threshold condition will then be suppressed and become non-lasing side modes The – plane is split into two halves by the  ¼ line, or the Bragg wavelength As one moves along the positive -axis, any oscillation modes encountered will be denoted as the ỵ1, ỵ2 modes and so on Similarly, negative values such as À1, À2 are used for the modes found on the negative -axis The importance of the single longitudinal mode (SLM) in coherent optical communications has been discussed earlier in Chapter To measure the stability of the lasing spectrum, one needs to determine the amplitude gain difference between the lasing mode and the most probable side mode of the DFB laser ½4; 5Š A larger amplitude gain difference, better known as the gain margin ðÁÞ, implies a better mode discrimination In other words, the SLM oscillation in the DFB LD involved is said to be more stable In practice, the actual requirement of Á may vary from one system to another depending on the encoding format THRESHOLD ANALYSIS OF A CONVENTIONAL DFB LASER 85 Figure 3.1 A simplified – plot showing the mode spectrum and the oscillating mode of a DFB LD Different symbols are used to show longitudinal modes obtained from various values (return to zero, RZ, or non-return to zero, NRZ), transmission rate, the biasing condition of the laser sources, the length and characteristics of the single-mode fibre (SMF) used A simulation based on a 20 km dispersive SMF [6] indicated that a Á of cmÀ1 is required for a 2.4 Gb sÀ1 data transmission in order that a bit error rate, BER < 10À9 can be achieved A detailed analysis of the requirement of Á under different system configurations is clearly beyond the scope of the present analysis On the other hand, from the above data one can get some idea of the typical values of gain margin required in a coherent optical communication system The value of the gain margin, however, is difficult to measure directly from an experiment An alternative method is to measure the spontaneous emission spectrum For a stable SLM source, a minimum side mode suppression ratio (SMSR) of 25 dB [7] between the power of the lasing mode and the most probable side mode is necessary 3.5 THRESHOLD ANALYSIS OF A CONVENTIONAL DFB LASER For a conventional DFB laser having zero facet reflection, the threshold equation (3.16) becomes j g L ẳ ặ L sinhLị 3:28ị Using the NewtonRaphson iteration approach, the eigenvalue equation can be solved as a fixed coupling coefficient Results obtained for the above equation are shown in Fig 3.2 All 86 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS Figure 3.2 Relationship between the amplitude threshold gain and the detuning coefficient of a mirrorless index-coupled DFB LD parameters used have been normalised with respect to the overall cavity length L Discrete values of L have been selected between 0.25 and 10.0 As shown in the inset of Fig 3.2, solutions obtained from various L products are shown using different symbols Oscillation modes are then formed by joining the appropriate solutions together Solid lines have been used to represent the to ỵ4 modes From the figure, it is clear that oscillating modes distribute symmetrically with respect to the Bragg wavelength, whilst no oscillation is found at the Bragg wavelength Furthermore, it can be seen that the ỵ1 and modes although having different lasing wavelengths share the same amplitude gain As a result, degenerate oscillation occurs and these modes will have the same chance to lase once the lasing condition is reached Figure 3.2 also reveals that the amplitude of the threshold gain decreases with increasing values of L Since a larger value of implies a stronger optical feedback, a smaller threshold gain results Similarly, lasers having a long cavity length help to reduce the amplitude gain since a larger single pass gain can be achieved With no oscillation found at the Bragg wavelength, a stop band region is formed between the ỵ1 and À1 modes of the conventional mirrorless DFB LD From Fig 3.2, one can conclude that the normalised stop band width is a function of L Although the change in stop band width becomes less noticeable at lower values of L, the measurement of the stop band width has been used to determine the coupling coefficient of DFB LDs [8] Figure 3.3 shows the characteristic of a DFB LD having finite facet reflections It is shown in the figure that the mode distribution is no longer symmetrical and no oscillation is found at the Bragg wavelength The À1 mode, having the smallest amplitude gain, becomes the lasing mode IMPACT OF CORRUGATION PHASE AT LASER FACETS 87 Figure 3.3 Relationship between the amplitude threshold gain and the detuning coefficient of a DFB LD with finite reflectivities 3.6 IMPACT OF CORRUGATION PHASE AT LASER FACETS So far, symmetrical laser cavities sharing identical facet reflectivities have been used In order to understand the effects of the residue phases at facets ½2; 9Š, asymmetric cavities are now considered The threshold characteristic of one of these asymmetric DFB LDs is shown in Fig 3.4 The amplitude reflectivity ^1 ¼ 0:0343 is assumed whilst the other facet is r assumed to be naturally cleaved such that ^2 ¼ 0:535 Discrete values of L have been r chosen In the figure, the corrugation phase is fixed at % whilst changes in steps of %=2 Different symbol markers have been used to represent the different Solutions obtained from the same L product are joined together as usual to form the oscillation mode Consider L ¼ 1:0 as an example It can be seen that the lasing mode changes from the negative to the positive mode as the facet phase changes from À%=2 to % For L > 1:0, the amplitude gain at the Bragg wavelength remains so high that it never reaches the threshold condition The À1 mode showing the smallest amplitude gain becomes the dominant lasing mode If we replace the natural cleaved facet with a highly reflective surface such that ^2 ¼ 1:0, r we change the lasing characteristic to that shown in Fig 3.5 Various values of L have been used for comparison purposes In a similar way to Fig 3.4, the oscillation mode shifts from the to the ỵ1 mode when changes from À%=2 to % From both Figs 3.4 and 3.5, it is clear that SLM operation depends on both the facet reflectivity and the associated phase On the other hand, due to tolerances inherent during the process of fabrication, it is difficult to control the corrugation phase at the laser facets accurately [10] 88 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS Figure 3.4 The lasing characteristic of a DFB LD having asymmetric facet reflectivities The corrugation phase is fixed whilst is allowed to change Results obtained from various L products are compared Figure 3.5 The lasing characteristic of a DFB LD having asymmetric facet reflectivities The corrugation phase is fixed whilst is allowed to change Results obtained from various L products are compared THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY 89 Various methods have been proposed for adjusting the corrugation phase One such method is to use the ion beam etching technique ½11; 12Š A continuous flux of neutralised argon gas, which acts as an abrasive tool, is targeted at one laser facet By passing the facet slowly across the beam at a constant rate, a 20–50 nm depth can be etched away at the laser facet in a single process An annealing process is usually applied afterwards Experimental results ½11; 12Š show that the annealing process does not cause significant variation in the threshold and the external quantum efficiency in DFB lasers Apart from the extra annealing process required, the ion beam etching technique is effective in adjusting the position of facets and thus the associated corrugation phases Since the etching depth required may vary from one DFB laser to another, the ion beam etching technique is classified as a chip-by-chip optimisation method To improve the efficiency, other methods such as the phase control technique [13] can be used Basically, a multi-layer coating with precise refractive indices and thicknesses is applied to the laser facets so that the overall facet phase and the amplitude reflection can be controlled and determined easily 3.7 THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY In the previous section, the threshold analysis of conventional DFB lasers comprising uniform corrugations was presented SLM operation can be achieved when different values of facet reflectivity are employed On the other hand, due to the randomness of the corrugation phase at the laser facet, stable SLM oscillation is not guaranteed To improve the single-mode performance of DFB lasers, phase discontinuity or phase shift is introduced [14] along the corrugation As shown in Fig 3.6, phase shifts along the corrugation can be introduced by two methods Figure 3.6(a) shows a non-uniform active layer width, whilst the shape and the dimension of the corrugation remain constant ½15; 16Š In Fig 3.6(b), on the other hand, the corrugation shows a phase slip whilst the active layer dimensions remain uniform ½17; 18Š Using method (a), the actual phase shift depends on the length of the Figure 3.6 Phase shift or discontinuity fabricated along the corrugation of a DFB laser (a) Phase shift formed by uniform corrugation but non-uniform active layer width; (b) phase shift formed by uniform active layer dimension but discontinuous corrugation 90 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS Figure 3.7 Schematic representation of a single-phase-shifted (1PS) DFB LD The phase shift is represented by phase-adjustment region and the difference in strip width ðW2 À W1 Þ Precise control over the active layer width is required Using method (b), phase discontinuity is introduced during fabrication in which the slip is written directly along the corrugation In the analysis, we adopted the latter method in preparing phase shifts in a DFB laser Consider a single-phase-shifted (1PS) DFB laser as shown in Fig 3.7 A phase slip of 2 is fabricated along the corrugation at the z origin so that the cavity is subdivided into sections As one can see, these sections may have different lengths and each resembles a conventional DFB laser cavity with uniform corrugation In the analysis, zero facet reflectivity is assumed Following the argument presented earlier in Chapter 2, the refractive index of each section can be written as n1ị zị ẳ n0 ỵ n cos2b0 z ỵ 0ị 2ị n zị ẳ n0 ỵ n cos2b0 z 0Þ ð3:29aÞ ð3:29bÞ where superscripts (1) and (2) correspond to sections and 2, respectively In the above equations, it is assumed that the phase shift is equally split between sections and Based upon the coupled wave theory, counter-running waves are built up in each section such that the following equations can be derived for each section of the laser dR1ị ỵ  jịR1ị dz dS1ị ỵ  jịS1ị dz dR2ị ỵ  jịR2ị dz dS2ị ỵ  jịS2ị dz ẳ j S1ị ej0 3:30aị ẳ j R1ị e j0 3:30bị ẳ j S2ị e j0 3:30cị ẳ j R2ị ej0 3:30dị THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY 91 where Rð1Þ , Sð1Þ and Rð2Þ , Sð2Þ are the counter-running waves propagating in sections (1) and (2), respectively In both sections, the corrugation shape and grating depth are assumed to be equal As a result, the coupling coefficient remains constant throughout From eqn (2.98), the solution of the coupled wave equations can be written as Ekị zị ẳ Rkị zịejb0 z ỵ Skị zịe jb0 z 3:31ị where kị kị Rkị zị ẳ R1 eg z ỵ R2 e g z kị S zị ẳ kị S1 e g z ỵ 3:32aị kị S2 e g z 3:32bị kị kị kị kị and k ẳ and for sections (1) and (2), respectively Here, R1 , R2 , S1 and S2 are the complex coefficients associated with the particular section Since the discontinuity caused by the phase slip is assumed to be very small, the waves in the two sections can be considered to be continuous at z ẳ In other words, R1ị z ẳ 0ị ẳ R2ị z ẳ 0ị 3:33aị S1ị z ẳ 0ị ẳ S2ị z ẳ 0ị 3:33bị r By allowing ^1 and ^2 to be the respective amplitude facet reflection coefficients at the left r and right laser facets, the boundary conditions of the 1PS DFB laser become r Rð1Þ L1 ịe jb0 L1 ẳ ^1 S1ị L1 ị ejb0 L1 2ị S L2 ịe jb0 L2 2ị ẳ ^2 R ðL2 Þ e r Àjb0 L2 ð3:33cÞ ð3:33dÞ By matching all the boundary conditions, non-trivial solutions exist if and only if the following eigenvalue equation is satisfied [17] jÉ 1ị^1 ỵ 1ị jẫ 2ị^2 ỵ 2ị r r ; ẳ e j20 1ị 1ị j2ị^ ỵ T 2ị j ^1 ỵ T r r2 3:34ị where ^ ẫ kị ẳ ỵ e2gLkị ^ kị ẳ e2gLkị 3:35ị ^ T kị ẳ ỵ e2gLkị ^ ẳ  À j À  with k ¼ and 2, and L1ị ẳ L1 ; L2ị ẳ L2 3:36ị 92 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS ^1 and ^2 in eqn (3.34) can be expressed in terms of the following complex field reflectivities, r r r1 , and r2 , respectively r1 ¼ ^1 eÀjð2b0 L1 À 0ị r r2 ẳ ^2 ej2b0 L2 0ị r ð3:37Þ Compared with the conventional DFB laser, the boundary condition at the phase shift has to be matched for the mirrorless 1PS DFB laser structure Nevertheless, it was pointed out by Utaka et al [17] that the use of non-zero facet reflection may not be desirable This is because the random corrugation phases at the laser facets will cause extra difficulty in controlling the lasing characteristic Therefore, it is best to have AR coatings applied to both facets of the 1PS DFB laser For a mirrorless, symmetrical 1PS DFB laser cavity with L1 ¼ L2 ¼ L=2, the phase shift is located at the centre of the cavity As a result, eqn (3.34) can be simplified further [19] such that " #2 ^ À ð1 À egL Þ À Á ẳ e2j ỵ egL ^ 3:38ị 3.7.1 Effects of Phase Shift on the Lasing Characteristics of a 1PS DFB Laser Diode To investigate the effects of phase shifts on the lasing characteristic of 1PS DFB lasers, a symmetrical laser cavity is assumed with a single phase shift fabricated at the centre of the DFB laser Using a numerical method such as the Newton–Raphson method, the eigenvalue equation (3.38) can be solved numerically for the normalised amplitude threshold gain thL (amplitude gain of the lasing mode) and the lasing wavelength for specific values of and phase shift Figure 3.8 illustrates how the variation of phase shift value affects thL for the mirrorless 1PS DFB LD All parameters used are normalised with respect to the overall cavity length L Three different L values are plotted in the figure for comparison purposes All curves in Fig 3.8 are symmetrical and have a minimum amplitude threshold gain at ¼ 90 (or %=2 in radians) as can be seen This phase change corresponds to a quarter wavelength shift and so the name single !=4-shifted DFB, or quarterly-wavelength-shifted (QWS) DFB, laser is usually used to represent this laser structure When the phase shift approaches zero or %, the phase-shifted structure is reduced to the conventional, mirrorless DFB laser in which degenerate oscillation results Figure 3.9 shows the variation of the lasing wavelength with respect to the phase shift As in Fig 3.8, results of three sets of L products are shown and compared In this case, the Bragg wavelength !B is assumed to be 1330 nm and the actual wavelength is shown on the left y-axis The corresponding normalised detuning coefficient thL is shown on the righthand side At ¼ %=2, the lasing wavelengths of all three L values coincide at the Bragg wavelength This reveals an important characteristic of the symmetrical 1PS DFB laser A QWS DFB LD always oscillates at the Bragg wavelength irrespective of the L chosen THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY Figure 3.8 DFB LD 93 The variation of the normalised amplitude threshold gain with the phase shift of a 1PS Figure 3.9 The variation of the lasing wavelength with the phase shift of a 1PS DFB LD 94 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS When L increases, the range of lasing wavelengths also increases with varying phase shift At L ¼ 2:0, the range of wavelengths is found to be 10.8 nm whilst it is about 7.4 nm for L ¼ 0:50 So far, the phase shift has been assumed to be at the centre of the laser cavity In the next section, effects of the phase shift position on the lasing characteristics of 1PS DFB lasers will be discussed 3.7.2 Effects of Phase Shift Position (PSP) on the Lasing Characteristics of a 1PS DFB Laser Diode To investigate the effect of the location of the phase shift [20], a parameter known as the phase shift position (PSP) is introduced along the asymmetric laser cavity such that PSP ¼ L1 L ð3:39Þ The variation of the amplitude gain and gain margin obtained from a 500 mm long DFB laser cavity with ¼ 20 cmÀ1 (i.e L ¼ 1) are shown in Figs 3.10 and 3.11, respectively In the analysis, the Bragg wavelength is assumed to be at 1330 nm and the phase shift is fixed at %=2 Both Figs 3.10 and 3.11 show symmetrical distributions of curves at PSP ¼ 0:5, where the phase shift is found When the phase shift moves from the centre towards the laser facets, the effect of the phase shift becomes less influential Solutions obtained from the threshold equation indicate that degenerate oscillation begins to occur at PSP ¼ 0:26 and PSP ¼ 0:74 In this situation, the QWS DFB laser is reduced to the conventional one The dramatic fall of Figure 3.10 The variation of the amplitude threshold gain with the PSP of a 1PS DFB LD The phase shift is fixed at %/2 ADVANTAGES AND DISADVANTAGES OF QWS DFB LASER DIODES 95 Figure 3.11 The variation of the gain margin with the PSP of a 1PS DFB LD The phase shift is fixed at %/2 the gain margin shown in Fig 3.11 confirms the above argument When the position of the phase shift moves from the centre PSP ẳ 0:5ị to the laser facets, the gain margin drops from a peak value of 14.7 cmÀ1 to zero value at PSP ¼ 0:26 and PSP ¼ 0:74 As long as the phase shift is fabricated near the centre of the cavity, a QWS DFB laser can operate at the Bragg wavelength 3.8 ADVANTAGES AND DISADVANTAGES OF QWS DFB LASER DIODES By introducing a QWS at the centre of the DFB laser cavity, SLM operation at the Bragg wavelength can be achieved However, as first discussed by Soda et al [16], for a high L QWS DFB LD, the gain margin drops drastically with increasing biasing current Multimode oscillation at two distant wavelengths is observed when the optical output power increases Such a reduction in gain margin is thought to be induced by the longitudinal spatial hole burning effect [21] When DFB LDs are biased below threshold current, where spontaneous emission is still dominant, the longitudinal carrier and the field intensity distributions are relatively uniform However, when the bias current exceeds that of the threshold value, the optical field inside the laser cavity becomes intensified at places where corrugation reflections occur [22] For a QWS DFB LD, the field intensity is so intense at the phase shift position that the rate of spontaneous recombination increases near the phase shift In order to maintain a round-trip gain of unity, carriers located near to the phase shift will move to fill the carrier-depleted zone In Chapter 2, it was mentioned that the 96 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS refractive index of the semiconductor depends on the carrier injection Such a local variation of carrier concentration will result in a non-uniform distribution of refractive indices along the laser cavity The situation is made worse by the fact that the gain and refractive index are related to one another as a result of the Kramer–Kroenig relationship [23] When the biasing current changes, the gain of the lasing mode and other non-lasing side modes will change in such a way that the gain margin reduces and consequently, multi-mode oscillation occurs Due to the deterioration of single-mode stability, the longitudinal spatial hole burning effect limits the QWS DFB LD to a lower power of operation Although laser structures having a smaller L value are found to be less vulnerable to the longitudinal spatial hole burning effect, these structures are characterised by larger amplitude gain values and relatively large currents In order to suppress the spatial hole burning whilst improving the maximum single-mode output power available, it was proposed that a laser structure having a flatter field intensity may be used [24] To optimise the structure with respect to the intensity distribution, a parameter known as the atness (F) is dened Fẳ L Z Izị À Iavg Þ2 dz ð3:40Þ cavity where IðzÞ is the local field intensity and Iavg is the average field intensity In the QWS DFB laser, an optimum value of flatness is found when L ¼ 1:25 [16] In flattening the field intensity whilst improving the optimum L value that can be used in QWS DFB lasers, a three-electrode QWS DFB laser structure shown in Fig 3.12 was proposed [25] By passing a larger biasing current to the central electrode, carriers lost due to spatial hole burning are compensated for [26] An alternative approach which retains uniform current injection is also used By introducing more phase shifts along the DFB laser cavity, a multiple-phase-shift (MPS) structure can flatten the field distribution [27] Figure 3.13 shows a three-phase-shift (3PS) DFB LD Figure 3.12 A three-electrode QWS DFB LD (after [25]) SUMMARY 97 Figure 3.13 Schematic representation of a three-phase-shift (3PS) DFB LD (after [28]) AR: antireflection coating Longitudinal spatial hole burning must be considered when the LD operates at the abovethreshold condition To decide the above-threshold characteristic, one must take into account the local carrier concentration Using the perturbation method [27] or the quasi-uniform gain assumption [16], characteristics of QWS DFB LDs operating slightly above the threshold current are predicted However, these methods may not be appropriate when the biasing current becomes high and other non-linear effects such as the gain saturation [28] must be considered Throughout the analysis, the deviation of the eigenvalue equation becomes tedious as the laser structure becomes more and more complex The use of numerical analysis, such as the Newton–Raphson method, becomes impractical since the first-order derivative is required A new model that can cope with different designs of DFB LDs, such as the three-electrode QWS [29] and/or the 3PS DFB LD structures [30], while maintaining a wider range of current injection is necessary Such a model needs to be capable of considering any local variation and the gain saturation effect with increasing output power 3.9 SUMMARY In this chapter, the coupled wave equations have been solved under various structural configurations By matching all boundary conditions, eigenvalue equations were derived From the solutions of the eigenvalue equations, the threshold current and the lasing wavelength were determined Impacts due to the coupling coefficient, the laser cavity length, the facet reflectivities, the residue corrugation phases and phase discontinuities were discussed in a systematic way with regard to the lasing threshold characteristics With a single QWS fabricated at the centre of the DFB cavity, the QWS DFB LD oscillates at the Bragg wavelength Due to non-uniform field distribution, however, the single-mode stability 98 STRUCTURAL IMPACTS ON THE SOLUTIONS OF COUPLED WAVE EQUATIONS is threatened by the spatial hole burning effect To extend the analysis to the above-threshold operation, a new model is required such that all localised effects and other non-linear effects can be included in the analysis 3.10 REFERENCES Kogelnik, H and Shank, C V., Coupled-wave theory of distributed feedback 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Results obtained from various L products are compared THE EFFECTS OF PHASE DISCONTINUITY ALONG THE DFB LASER CAVITY 89 Various methods have been proposed for adjusting the corrugation phase One... other hand, due to the randomness of the corrugation phase at the laser facet, stable SLM oscillation is not guaranteed To improve the single-mode performance of DFB lasers, phase discontinuity or... discontinuity is introduced during fabrication in which the slip is written directly along the corrugation In the analysis, we adopted the latter method in preparing phase shifts in a DFB laser