With the built-in wavelength selection mechanism, distributedfeedback semiconductor laser diodes with a higher gain margin are superior to the Fabry–Perot laser in that a single longitud
Trang 1Structural Impacts on the Solutions
of Coupled Wave Equations:
An Overview
3.1 INTRODUCTION
The introduction of semiconductor lasers has boosted the development of coherent opticalcommunication systems With the built-in wavelength selection mechanism, distributedfeedback 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 phase-shifted DFB lasers will be investigated In particular, structural impacts on thethreshold characteristic will be discussed in a systematic way The next two sections of thischapter present solutions of the coupled wave equations in DFB laser diode structures Insection 3.4 the concepts of mode discrimination and gain margin are discussed Thethreshold analysis of a conventional DFB laser diode is studied in section 3.5, whilst theimpact of corrugation phase at the DFB laser diode facets is discussed in section 3.6 Byintroducing a phase shift along the corrugations of DFB LDs, the degenerate oscillatingcharacteristic of the conventional DFB LD can be removed In section 3.7, structuralimpacts due to the phase shift and the corresponding phase shift position (PSP) will beconsidered
single-As mentioned earlier in Chapter 2, the introduction of the coupling coefficient into thecoupled 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 themagnitude of will be presented With a =2 phase shift fabricated at the centre of the DFBcavity, 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 ofthe QWS laser structure, will be examined at the end of this chapter The limitedapplication of the eigenvalue equation in solving the coupled wave equations will also beconsidered
Distributed Feedback Laser Diodes and Optical Tunable Filters H Ghafouri–Shiraz
# 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1
Trang 23.2 SOLUTIONS OF THE COUPLED WAVE EQUATIONS
In Chapter 2 it was shown that the characteristics of DFB LDs can be described by a pair ofcoupled wave equations The strength of the feedback induced by the perturbed refractiveindex or gain is measured by the coupling coefficient Relationships between the forwardand the backward coupling coefficients RSand SRwere derived for purely index-coupled,mixed-coupled and purely gain-coupled structures By assuming a zero phase differencebetween the index and the gain term, the complex coupling coefficient could be expressed as
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 constantsuch that
where the coefficients RðzÞ and SðzÞ are given as [1]
By substituting eqns (3.3a) and (3.3b) into the coupled wave equations, the followingrelations are obtained by collecting identical exponential terms [2]
Trang 3Based on the equation shown above, eqn (3.4) is simplified to become
Rðz1Þ ej b0 z1 ¼ ^r1Sðz1Þ ej b0z1 ð3:9aÞSðz2Þ ej b0z2 ¼ ^r2Rðz2Þ ej b0 z2 ð3:9bÞ
where ^r1and ^r2are amplitude reflection coefficients at the laser facets z1and z2, respectively.According to eqns (3.3) and (3.4), the above equations could be expanded in such a way that
Trang 4derived from eqns (3.5a) and (3.5b) After some lengthy manipulation [2], one ends up with
an eigenvalue equation
gL¼j sinhðLÞ
D nðr1þ r2Þ 1 rð 1r2Þ coshðgLÞ 1 þ rð 1r2Þ1o
ð3:14Þwhere
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 andthe complex facet reflectivities r1and r2 Due to the complex nature of the above equation,numerical methods like the Newton–Raphson iteration technique can be used, provided thatthe Cauchy–Riemann condition on complex analytical functions is satisfied
Before starting the Newton–Raphson iteration, an initial value ofð ; Þiniis chosen from aselected range ofð ; Þ values Usually, the first selected guess will not be a solution of thethreshold equation and hence the iteration continues At the end of the first iteration, a newpair ofð 0; 0Þ will be generated and checked to see if it satisfies the threshold equation Theiteration will continue until the newly generatedð 0; 0Þ pair satisfies the threshold equationwithin a reasonable range of error Starting with different initial guesses ofð ; Þini, otheroscillating modes can be determined in a similar way By collecting allð 0; 0Þ pairs thatsatisfy the threshold equation, the one showing the smallest amplitude gain will then becomethe lasing mode The final value ð ; Þfinal is then stored up for later use, in which thethreshold current and the lasing wavelength of the LD are to be decided In general,eqn (3.16) characterises all conventional DFB semiconductor LDs with continuouscorrugations 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
Trang 5where z¼ x þ j y is a complex number and UðzÞ and VðzÞ are, respectively, the real andimaginary parts of the complex function From the above equation, one can deduce thefollowing equality easily
Uðxreq; yreqÞ ¼ U x; yð Þ þ@U
@xðxreq xÞ þ@U
Vðxreq; yreqÞ ¼ V x; yð Þ þ@V
@xðxreq xÞ þ@V
where the values of (x, y) chosen are sufficiently close to the exact solutions Other higherderivative terms from the Taylor series have been ignored One then obtains the followingequations for xreq and yreq from the above simultaneous equations [2]
Trang 6On replacing all the @=@y terms with @=@x using the above Cauchy–Riemann condition,eqns (3.22) and (3.24) can be simplified such that
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 thismethod, one can avoid any errors associated with other numerical methods such asnumerical differentiation
3.4 CONCEPTS OF MODE DISCRIMINATION
AND GAIN MARGIN
At a fixed value of , pairs ofð ; Þfinalcan be determined following the method discussed inthe previous section Eachð ; Þfinalpair, which represents an oscillation mode, is plotted onthe – plane Similarly, pairs ofð ; Þfinalvalues can be obtained by changing the values of
By plotting allð ; Þfinalpoints on the – plane, the mode spectrum of the DFB LD isformed A simplified – plot is shown in Fig 3.1 Different symbols shown representvarious longitudinal modes obtained for various coupling coefficients whilst the solid curveshows how longitudinal modes join to form an oscillating mode
When the biasing current increases, the longitudinal mode showing the smallest amplitudegain will reach the threshold condition first and begin to lase Other modes failing to reachthe threshold condition will then be suppressed and become non-lasing side modes The –plane is split into two halves by the ¼ 0 line, or the Bragg wavelength As one movesalong the positive -axis, any oscillation modes encountered will be denoted as theþ1, þ2modes and so on Similarly, negative values such as1, 2 are used for the modes found onthe negative -axis
The importance of the single longitudinal mode (SLM) in coherent optical tions has been discussed earlier in Chapter 1 To measure the stability of the lasing spectrum,one needs to determine the amplitude gain difference between the lasing mode and the mostprobable side mode of the DFB laser½4; 5 A larger amplitude gain difference, better known
communica-as the gain marginðÞ, implies a better mode discrimination In other words, the SLMoscillation in the DFB LD involved is said to be more stable In practice, the actualrequirement of may vary from one system to another depending on the encoding format
Trang 7(return to zero, RZ, or non-return to zero, NRZ), transmission rate, the biasing condition ofthe laser sources, the length and characteristics of the single-mode fibre (SMF) used Asimulation based on a 20 km dispersive SMF [6] indicated that a of 5 cm1 is requiredfor a 2.4 Gb s1data transmission in order that a bit error rate, BER < 109can be achieved.
A detailed analysis of the requirement of under different system configurations is clearlybeyond the scope of the present analysis On the other hand, from the above data one can getsome idea of the typical values of gain margin required in a coherent optical communicationsystem
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 SLMsource, a minimum side mode suppression ratio (SMSR) of 25 dB [7] between the power ofthe lasing mode and the most probable side mode is necessary
3.5 THRESHOLD ANALYSIS OF A CONVENTIONAL DFB LASERFor a conventional DFB laser having zero facet reflection, the threshold equation (3.16)becomes
Trang 8parameters used have been normalised with respect to the overall cavity length L Discretevalues 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 Oscillationmodes are then formed by joining the appropriate solutions together Solid lines have beenused to represent the 4 to þ4 modes From the figure, it is clear that oscillating modesdistribute symmetrically with respect to the Bragg wavelength, whilst no oscillation is found
at the Bragg wavelength Furthermore, it can be seen that theþ1 and 1 modes althoughhaving different lasing wavelengths share the same amplitude gain As a result, degenerateoscillation occurs and these modes will have the same chance to lase once the lasingcondition is reached Figure 3.2 also reveals that the amplitude of the threshold gaindecreases with increasing values of L Since a larger value of implies a stronger opticalfeedback, 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 betweenthe þ1 and 1 modes of the conventional mirrorless DFB LD From Fig 3.2, one canconclude that the normalised stop band width is a function of L Although the change instop band width becomes less noticeable at lower values of L, the measurement of the stopband width has been used to determine the coupling coefficient of DFB LDs [8] Figure 3.3shows the characteristic of a DFB LD having finite facet reflections It is shown in the figurethat the mode distribution is no longer symmetrical and no oscillation is found at theBragg wavelength The1 mode, having the smallest amplitude gain, becomes the lasingmode
mirrorless index-coupled DFB LD
Trang 93.6 IMPACT OF CORRUGATION PHASE AT LASER FACETS
So far, symmetrical laser cavities sharing identical facet reflectivities have been used Inorder to understand the effects of the residue phases at facets½2; 9, asymmetric cavities arenow considered The threshold characteristic of one of these asymmetric DFB LDs is shown
in Fig 3.4 The amplitude reflectivity ^r1¼ 0:0343 is assumed whilst the other facet isassumed to be naturally cleaved such that ^r2¼ 0:535 Discrete values of L have beenchosen In the figure, the corrugation phase 1is fixed at whilst 2changes in steps of =2.Different symbol markers have been used to represent the different 2 Solutions obtainedfrom the same L product are joined together as usual to form the oscillation mode.Consider L¼ 1:0 as an example It can be seen that the lasing mode changes from thenegative to the positive mode as the facet phase 2changes from=2 to For L > 1:0,the amplitude gain at the Bragg wavelength remains so high that it never reaches thethreshold condition The 1 mode showing the smallest amplitude gain becomes thedominant lasing mode
If we replace the natural cleaved facet with a highly reflective surface such that ^r2¼ 1:0,
we change the lasing characteristic to that shown in Fig 3.5 Various values of L have beenused for comparison purposes In a similar way to Fig 3.4, the oscillation mode shifts fromthe1 to the þ1 mode when 2changes from=2 to From both Figs 3.4 and 3.5, it isclear that SLM operation depends on both the facet reflectivity and the associated phase Onthe other hand, due to tolerances inherent during the process of fabrication, it is difficult tocontrol the corrugation phase at the laser facets accurately [10]
DFB LD with finite reflectivities
Trang 10Figure 3.4 The lasing characteristic of a DFB LD having asymmetric facet reflectivities Thecorrugation phase 1 is fixed whilst 2 is allowed to change Results obtained from various Lproducts are compared.
corrugation phase 1is fixed whilst 2is allowed to change Results obtained from various L productsare compared
Trang 11Various methods have been proposed for adjusting the corrugation phase One suchmethod is to use the ion beam etching technique½11; 12 A continuous flux of neutralisedargon gas, which acts as an abrasive tool, is targeted at one laser facet By passing the facetslowly across the beam at a constant rate, a 20–50 nm depth can be etched away at the laserfacet in a single process An annealing process is usually applied afterwards Experimentalresults½11; 12 show that the annealing process does not cause significant variation in thethreshold and the external quantum efficiency in DFB lasers Apart from the extra annealingprocess required, the ion beam etching technique is effective in adjusting the position offacets and thus the associated corrugation phases.
Since the etching depth required may vary from one DFB laser to another, the ion beametching technique is classified as a chip-by-chip optimisation method To improve theefficiency, other methods such as the phase control technique [13] can be used Basically, amulti-layer coating with precise refractive indices and thicknesses is applied to the laserfacets so that the overall facet phase and the amplitude reflection can be controlled anddetermined easily
3.7 THE EFFECTS OF PHASE DISCONTINUITY ALONG
THE DFB LASER CAVITY
In the previous section, the threshold analysis of conventional DFB lasers comprisinguniform 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 thecorrugation phase at the laser facet, stable SLM oscillation is not guaranteed To improve thesingle-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 beintroduced by two methods Figure 3.6(a) shows a non-uniform active layer width, whilstthe shape and the dimension of the corrugation remain constant½15; 16 In Fig 3.6(b), onthe other hand, the corrugation shows a phase slip whilst the active layer dimensions remainuniform ½17; 18 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 shiftformed by uniform corrugation but non-uniform active layer width; (b) phase shift formed by uniformactive layer dimension but discontinuous corrugation