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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 thespatial hole burning effect can be

Threshold Analysis and Optimisation

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 thecoupled wave equations in DFB laser structures Its efficiency and flexibility in aiding theanalysis of DFB semiconductor LDs has been explored theoretically A general N-sectionedDFB 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 DFBlaser

The spatial hole burning effect (SHB) [1] has been known to limit the performance ofDFB LDs As the biasing current of a single quarterly-wavelength-shifted (QWS) DFB LDincreases, the gain margin reduces Therefore, the maximum single-mode output power ofthe QWS DFB LD is restricted to a relatively low power operation The SHB phenomenoncaused by the intense electric field leads to a local carrier depletion at the centre of thecavity Such a change in carrier distribution alters the refractive index along the laser cavityand ultimately affects the lasing characteristics By changing the structural parameters insidethe DFB LD, an attempt will be made to reduce the effect of SHB As a result, a largersingle-mode power, and consequently a narrower spectral linewidth, may be achieved Afull structural optimisation will often involve the examination of all possible structuralcombinations in the above-threshold regime On the other hand, the analysis of the structuraldesign may be simplified, in terms of time and effort, by optimising the threshold gainmargin and the field uniformity

The structural changes and their impacts on the characteristics of DFB LDs will now bepresented 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 bothphase shifts and their positions on the lasing characteristics of the 3PS DFB LD will bediscussed To reduce the SHB effect, it is necessary to have a more uniform fielddistribution, whilst maintaining a large gain marginð

LÞ The optimised structural designfor the 3PS DFB laser based on the values of

L and the flatness (F) of the fielddistribution 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

By changing the height of the corrugation and thus the coupling coefficient along a DFBlaser 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 thevariation of the coupling ratio and the position of the corrugation change will beinvestigated 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 withthe non-uniform coupling coefficient will be presented Optimised structural combinationsthat 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 couplingcoefficients will be investigated using a DCCþ 3PS DFB laser structure Finally, a summarywill 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 thespatial hole burning effect can be reduced in a 3PS DFB LD which is characterised by a moreuniform internal field distribution Experimental measurement has been carried out [5] using

a fixed value of phase shift However, independent changes in the value of phase shift have notbeen fully explored Using the TMM, it was shown in Table 4.1 of Chapter 4 that four transfermatrices are necessary to determine the threshold condition of 3PS DFB lasers In Fig 5.1, a

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 Ljð j ¼ 1; 2Þ In theanalysis, zero facet reflection at the laser facets is assumed Following the formulation of thetransfer matrix method, the overall transfer matrix of the 3PS DFB laser becomes:

Using a numerical approach such as Newton–Raphson’s method [6] for analytical complexequations, the threshold equation above may be solved Figure 5.2 shows the resonancemodes obtained from a symmetrical 3PS DFB laser where 2¼ 3 ¼ 4¼ =2 and L1¼ L2

are assumed For comparison purposes, results obtained from a mirrorless conventional DFBlaser and a single =2 phase-shifted DFB laser are also included In all three cases, thecoupling coefficient and the overall laser cavity length L are fixed at 40 cm
1and 500 mm,respectively Oscillation modes at the Bragg wavelength are found for both the single =2and a 3PS DFB structure However, the Bragg resonance mode of the 3PS DFB laser doesnot show the smallest amplitude threshold gain Instead, degenerate oscillation occurs since

it is shown that both the
1 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 mode oscillation occurs in the case where there are three phase shifts, i.e.f=2; =2; =2g.The pair of bracesf g used hereafter will indicate a phase combination in the 3PS structure,that isf2; 3; 4g

multi-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 changethe value or the position of the phase shift Figure 5.3 shows oscillation modes of various3PS 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 modecan be seen when all phase shifts change from =2 to 2=5 The þ1 mode whichdemonstrates the smallest amplitude threshold gain will become the lasing mode after lasing

Figure 5.2 Resonance modes of various DFBs that include: (a) a conventional DFB laser diode; (b) asingle QWS DFB laser diode; (c) a three /2-phase-shifted DFB laser diode

threshold is reached On the other hand, the
1 mode will become the lasing mode when thethree phase shifts change from =2 to 3=5 With all three phase shifts displaced from theusual =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 cavitylength of L, the position of phase shifts is assumed in such a way that L1¼ L2¼ L=4 Toinvestigate the effect of the phase shift position (PSP) on the threshold characteristics, aposition factor is introduced such that

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

2¼ 3¼ 4 ¼ =3 At a fixed value of , the figure shows a decrease in amplitudethreshold gain as the L value increases Along the curve L¼ 1:0, discontinuities at

¼ 0:12 and ¼ 0:41 indicate possible changes in the oscillation mode

Figure 5.3 Resonance modes in various 3PS DFBs that include: (a) a f=2, =2, =2g 3PS DFBlaser; (b) af2=5; 2=5; 2=5g 3PS DFB laser; (c) a f3=5; 3=5; 3=5g laser

Figure 5.5 The variation of detuning coefficient with respect to the phase shift position for couplingcoefficient .

Figure 5.4 The change of amplitude gain with respect to the phase shift position for different values

of coupling coefficient 

Such a change in oscillation is confirmed when the relationship between the detuningcoefficient and the position factor is shown in Fig 5.5 Along L¼ 1:0, it can be seen thatthe
1 mode remains as the oscillation mode when increases from zero When ¼ 0:12 isreached, however, a sudden change of oscillation mode is observed Similar mode jumpingoccurs at ¼ 0:41 When the PSP shifts, there is a continuous change in the resonant cavityformed by the DFB laser such that the actual lasing mode may alter At ¼ 0:77, it isinteresting to see how all L values converge to the same lasing wavelength It appears that

at this particular phase shift position, the effect of the variation of L is irrelevant and thelasing characteristic depends on the presence of the =3 phase shifts

5.3 OPTIMUM DESIGN OF A 3PS DFB LASER STRUCTURE

A complete structural optimisation of MPS DFB lasers cannot be achieved without analysingthe above-threshold performances This involves solving the carrier rate equation, which is afairly complex process and needs intensive computation On the other hand, it is believedthat the complexity of the structural design in the 3PS DFB laser can be reduced byoptimising the threshold amplitude gain difference and the flatness of field distribution.Hence, we can simply concentrate on those structures satisfying these design criteria For ahigh-performance DFB LD, both a stable single-mode oscillation and a uniform fielddistribution are important to prevent LDs from being affected by the spatial hole burningeffect In our analysis, DFB laser structures having a high gain margin ð

LÞ areconsidered, whilst the spatial hole burning effect is included by analysing the correspondingeffects on field uniformity Reports by Kimura and Sugimura [3– 4] as well as Ogita et al [5]suggested that the lasing characteristics are strongly influenced by both and  To maintain

a stable SLM oscillation, and consequently improve the performance of the spectrallinewidth, these structural parameters need to be optimised

To achieve a stable laser source that oscillates at a single longitudinal mode, it is importantthat there is a gain margin

L > 0:25½1 In the analysis, we assumed the length of thelaser L to be 500 mm For a 3PS DFB LD, Fig 5.6 shows the relationship between the gainmargin and the phase shift  in a symmetrical structure for different values of L rangingfrom 1 to 3 The position factor ¼ 0:5 corresponds to the case where L1¼ L2¼ L=4 In allcases, the degenerate oscillations occur at ¼ 0, =2 and , and the distributions of gainmargins are symmetrical with respect to ¼ =2 It is also shown that the variation of Lhas little effect on the gain margin of the 3PS laser structure Along the line L¼ 1, it isfound that a stable laser having

L > 0:25 can be obtained provided that 47<  < 73or

107<  < 133

In Fig 5.7, a contour map is shown that relates the gain margin to the values of phaseshifts in three-phase-shift f 2; 3; 4g DFB LDs In the calculations, L ¼ 2 and ¼ 0:5 areassumed The phase shift 3introduced at the centre of the cavity is separated from the rest

so that its value can be selected independently Other phase shifts are assumed to be equal as

2¼ 4¼ side As stated earlier, to satisfy the requirement of

L > 0:25, sidemust either

be greater than 105or less than 80if  can be varied freely between 0 and  A maximum

Figure 5.6 Variation of the gain margin versus the phase shift for different coupling coefficients.

Figure 5.7 Relationship between the gain margin

L and phase shifts for a 3PS DFB laser diode

value of

L¼ 0:73 is obtained at f0; =2; 0g and f; =2; g which corresponds to asingle =2-phase-shifted DFB laser.

The variation of

L with respect to the position factor is shown in Fig 5.8 In thisfigure, the values of phase shifts are equal (i.e 2¼ 3¼ 4¼ ) and three different sets ofresults are calculated with ¼ =2, 2=5 and =3 By changing the values of the phaseshifts,

L also changes for each particular value of At a fixed phase shift ¼ =2 (solidline), it is shown that a non-zero value of gain margin is observed where < 0:13 and > 0:725 As approaches zero, the phase shifts 2 and 4 move towards the laser facetsand their contributions become less influential Also, as approaches unity, both 2and 4

move towards the central phase shift 3 In this case, the 3PS laser structure is reduced to asingle-phase-shifted structure and the lasing characteristic is described by an effective phaseshift of eff 2þ 3þ 4

Figure 5.9 shows the dependence of

L upon for different values of L In theanalysis, all phase shifts are assumed to be identical (i.e ¼ =3) From this figure, it isclear that L has little effect on

L in 3PS DFB lasers

5.3.2 Structural Impacts on the Uniformity of the Internal Field Distribution

In this section, the structural impact on the internal field distribution will be discussed Toquantify the uniformity of the field distribution, it was shown in Chapter 3 that the flatness

Figure 5.8 Variation of the gain margin versus for various 3PS DFB laser diode structures

(F) of the internal field of a general N-sectioned DFB laser cavity is defined as

F¼1L

In order to minimise the effects of longitudinal spatial hole burning, it has been shownexperimentally ½1; 7 that a DFB laser cavity with F < 0:05 is necessary for stable SLMoscillation To optimise the structural design of 3PS DFB lasers, F < 0:05 will be used asone of the design criteria In order to evaluate the flatness of the internal field distribution,the threshold equation of the 3PS DFB laser needs to be solved first The normalisedamplitude threshold gain
thL and the normalised detuning coefficient thL of the lasingmode are then used to determine the field distribution In our analysis, a 500 mm long DFBlaser is subdivided into a substantial number of small sections with equal length From theoutput of each transfer matrix, both the forward and the backward propagating electricfields can be determined, and the electric field intensity at an arbitrary position z0is found tobe

Iðz0Þ ¼ Ej ðz0Þj2þ Ej ðz0Þj2 ð5:5Þ

Figure 5.9 Variation of the gain margin versus for different coupling coefficients

In Fig 5.10, the internal field distributions of three different structures are shown Thesestructures include a conventional mirrorless, a single =2-phase-shifted and a three-phase-shiftf=3; =3; =3g DFB laser All the electric field distributions have been normalised sothat the intensity at the laser facets is unity It can be seen that the single =2-phase-shiftedDFB laser has a flatness value of F¼ 0:301 Such a high value of F (which means that thefield is highly non-uniform) induces a local carrier escalation near the centre of the cavityafter the laser threshold is reached, consequently affecting the single-mode stability of thelaser device With three phase shifts incorporated into the cavity, the intensity distributionspreads out and the overall distribution becomes more uniform (see dashed line with

F¼ 0:012Þ By optimising the values and the positions of the phase shifts with respect to theflatness, a 3PS DFB laser can maintain a uniform field distribution even at a high value of

L, which is necessary to reduce the spectral linewidth of the laser

The effect of on F is shown in Fig 5.11 for different combinations of L When smallvalues of Lð< 1:5Þ are used, the field intensity distribution becomes less uniform when thephase shifts 2 and 4 shift towards the laser facets (i.e as tends to 0) As the opticalfeedback becomes stronger with increasing L, the field intensity distribution becomes moreintense near the centre of the laser cavity where is found to be about 0.77

The contour map shown in Fig 5.12 can be used to optimise the value of phase shifts withrespect to F In a similar way to Fig 5.7, the central phase shift 3 is used as the x-axis andother phase shifts are represented in the y-axis In this figure, all phase combinations with

F< 0:05 form a ribbon shape stretching from the lower left-hand corner to the upper hand corner of the contour The worst case, which leads to the largest value of F, can be

right-Figure 5.10 Field distribution in various DFB laser diode structures

Figure 5.11 Variation of flatness versus for different coupling coefficients .

Figure 5.12 Relationship between the flatness and the phase shift for a 3PS DFB LD

found at phase combinations off0; =2; 0g and f; =2; g For F < 0:05, sidemust lie inthe range 67.5 < side< 112:5 for an unrestricted value of 3 By comparing the map inFig 5.12 with the one shown earlier in Fig 5.7, it can be seen that a trade-off exists

in selecting the appropriate phase shift value for the optimum values of

L and F On onehand, phases should be chosen such that the gain margin is large enough to avoid modehopping On the other hand, the corresponding combination of phase shifts will result in arelatively large value of F Due to the SHB effect, the associated single-mode stabilitydeteriorates with increasing output power As a result, a compromise has to be made inselecting the phase shifts in 3PS DFB lasers such that high performance LDs with high

ð> 0:05 cm
1Þ values and small Fð< 0:05Þ can be obtained Figure 5.13 is acombination of the region

L  0:25 in Fig 5.8 and the region F  0:05 fromFig 5.12 In the shaded area of this figure, it can be seen that a phase combination off=3; =3; =3g will satisfy both the design criteria of gain margin and flatness

So far, the value of phase shifts 2and 4 have been assumed to be identical By varying

2and 4 from 0to 180, the contour maps of both the gain margin and the flatness of the3PS DFB LD can be plotted as shown in Figs 5.14 and 5.15, respectively [8] In the analysis,the PSP is fixed at 0.3 and 3is fixed at =9 (or 20) Contours shown are for

L > 0:25and F < 0:05 As expected, both contours show symmetrical distributions along the linewhere 2¼ 4 From Figs 5.14 and 5.15, it can be seen that most of the region that satisfies

L > 0:25 does not match with the region for F < 0:05 The only area that matches bothselection criteria is found when 40 <  < 90 and 40 <  < 80

Figure 5.13 Variations of the gain margin and flatness with respect to phase shifts Shaded areacovers all phase combinations that satisfy both

L > 0:25 and F < 0:05 selection criteria for stablesingle-mode operation