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The average values of SMand SMassociated with the side mode are evaluated from thecorresponding longitudinal distribution as SM¼ PN j¼1j SM¼ PN j¼1j From the result obtained, the gain m

For a coherent optical communication system, it is essential that the LD used oscillates at

a stable single mode and that a narrow spectral linewidth is achieved Using the informationobtained for the lasing mode characteristics, a method derived from the above-thresholdtransfer matrix model will be introduced in section 7.2 which allows the gain margin to beevaluated By introducing an imaginary wavelength into the transfer matrix equation, chara-cteristics of other non-lasing side modes can be evaluated and hence the single-mode stabilitycan be obtained Numerical results obtained using this method will be presented in section 7.3

In section 7.4, an alternative method which allows the theoretical prediction of the threshold spontaneous emission will be presented Based on the Green’s function method,one can use transfer matrices to help determine the single-mode stability of a DFB laserstructure by inspecting the spectral components of oscillating modes

above-The TMM also allows the noise characteristics of the DFB LD to be evaluated Insection 7.5, it will be shown that various contributions to the spectral linewidth can bedetermined using the information obtained from the above-threshold transfer matrices Inthe analysis, the effective linewidth enhancement factor [1] has been used instead of thematerial-based linewidth enhancement [2] Using a more realistic effective linewidthenhancement factor, impacts caused by structural changes can be investigated in asystematic way

Distributed Feedback Laser Diodes and Optical Tunable Filters H Ghafouri–Shiraz

# 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

7.2 SINGLE-MODE STABILITY IN DFB LDs

Using the TMM-based above-threshold laser cavity model as presented in the previouschapter, distributions of the carrier density, photon density, refractive index and thenormalised internal field intensity were obtained for various DFB laser structures From theemitting photon density at the facet, the output optical power can be evaluated Figure 7.1summarises results obtained for QWS, 3PS and DCCþ QWS LDs with biasing current as aparameter Within the range of biasing current that we are interested in, no ‘kink’ is observed

in any of the three cases Compared with the QWS structure, it seems that the introduction ofmultiple phase shifts along the 3PS laser cavity has increased the overall cavity loss Thefigure also shows that the 3PS laser structure having the largest amplitude threshold gain has

a relatively larger value of threshold current Among the three cases, both the 3PS and theDCCþ QWS structures appear to have relatively larger output power under the samenormalised biasing current

Semiconductor lasers having stable single longitudinal outputs and narrow spectrallinewidths are indispensable in coherent optical communication systems With a built-inwavelength selective corrugation, a DFB laser diode has a single longitudinal output Otheroscillation modes failing to reach the threshold condition become the non-lasing side modes

As the biasing current increases, the spatial hole burning effect becomes significant andmode competition between the lasing mode and the most probable non-lasing side modemay occur Mode competition has been observed for a QWS DFB laser [3–5], which resulted

Figure 7.1 Optical output power of three different types of DFB LD

172 ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

in multiple mode oscillation as the biasing current increased In this section, a numericalmethod that allows theoretical prediction of the above-threshold single-mode stability ofDFB LDs will be presented In the analysis, it is assumed that the detailed lasing modecharacteristics have been obtained from the above-threshold transfer matrix model asdiscussed in the previous chapter.

Single-mode stability implies the suppression of non-lasing side modes There are twopossible ways to demonstrate single-mode stability in DFB LDs The first approach involvesthe evaluation of the normalised gain margin

L between the lasing mode and the probablenon-lasing side modes The single-mode stability is said to be threatened if the gain margin,

, drops below 5 cm
1for a 500 mm length laser cavity An alternative method to checkthe stability of the device involves the measurement of the spectral characteristics With thehelp of an optical spectrum analyser, the measured intensity difference between the lasingmode and the side modes will give single-mode stability The second approach is often used

to measure the single-mode stability of DFB LDs In this section, we will concentrate on thefirst approach which leads to the evaluation of the above-threshold gain margin

From the numerical method discussed in the previous chapter, oscillation characteristics

of the lasing mode were obtained at a fixed biasing current By dividing the DFB laser into alarge number of smaller sections, longitudinal distributions like the carrier and photondensities were determined Since the laser cavity is now dominated by the lasing mode, thecharacteristics of other non-lasing side modes should be derived from the lasing mode Inorder to evaluate the characteristics of other non-lasing side modes, the dominant lasingmode has to be suppressed in a mathematical way In the analysis, an imaginary wavelength

i is introduced [6] As a result, the complex wavelength c of an unknown side modebecomes

where i takes into account the mathematical gain the side mode may need to reach itsthreshold value and  becomes the actual wavelength of the side mode By changing values

of both  and i, wavelengths of other non-lasing side modes can be evaluated

The numerical procedure involved in determining the characteristics of other non-lasingside modes is summarised as follows:

1 A numerical procedure similar to the one discussed in section 6.3 is adopted Toinitialise the iteration process, a 5 5 mathematical grid which consists of points (; i)

is formed Each of these (; i) points will be used as an initial guess Since the mostprobable non-lasing side mode is usually found near the lasing mode, it is advisable tostart with those wavelengths which are close to that of the lasing mode

2 Lasing mode characteristics like the longitudinal distribution of the carrier, photondensity and refractive index are retrieved from the data files obtained earlier Matrixelements of each transfer matrix are then determined

3 From the boundary condition at the left facet, electric fields ERðz1Þ and ESðz1Þ at the leftfacet are found which serve as the input electric field to the transfer matrix chain

4 By passing the electric field through the transfer matrix chain, the output electric field atthe right laser facet is determined The discrepancy with the right facet boundarycondition is then evaluated and stored

SINGLE-MODE STABILITY IN DFB LD 173

5 Steps (2) to (4) are then repeated with other pairs of (; i) obtained from the 5 5mathematical grid By comparing the discrepancy obtained from each of these (; i)pairs, the one showing the minimum discrepancy is then selected Depending on theposition of the selected (; i) on the mathematical grid, a new mathematical grid isformed ready for the next iteration.

6 Procedures shown above are then repeated until the discrepancy with the right facetboundary condition falls below 10
14 The final  obtained becomes the non-lasing sidemode and distributions of the amplitude gain
ðzÞ and the detuning coefficient ðzÞ arestored

7 The average values of 
SMand SMassociated with the side mode are evaluated from thecorresponding longitudinal distribution as



SM¼

PN j¼1
j

SM¼

PN j¼1j

From the result obtained, the gain margin between the lasing mode and the most probablenon-lasing side mode can be evaluated as

of the lasing mode (0) and side modes (1) in the 
L; 
L

plane In the analysis, an reflection-coated QWS DFB structure with L¼ 2 is assumed for a 500 mm length cavity.For each oscillating mode shown in Fig 7.2, the cross and the black circle correspond to theoscillating mode at threshold and at 5Ith, respectively When the biasing current increasesfrom the threshold value, an increase in the lasing mode amplitude gain and a correspondingreduction of gain margin between the lasing mode and theþ1 side mode can be seen Such aphenomenon is well known to be induced by the spatial hole burning effect [5]

anti-174 ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

Figure 7.3 shows the normalised amplitude gain change of the QWS DFB LD with respect

to the biasing current The amplitude gain distributions for the lasing mode and the lasing side modes are shown It is obvious that the 
L value of each mode varies in adifferent way When the biasing current increases from threshold, it is clear that theamplitude gain of theþ1 mode reduces remarkably and approaches that of the lasing mode

non-By contrast, the amplitude gain of the
1 mode becomes larger and hence becomes lesssignificant In Fig 7.4, for the same QWS structure, the variation of the normalised detuningcoefficient L is shown In such a strongly coupled LD, both the lasing mode and theþ1 sidemode shift towards the shorter wavelength side (negative L) with increasing biasing current.Among the different modes shown, the shift of the lasing mode is stronger since it is foundcloser to the gain peak

With multiple phase shifts introduced along the corrugation, the characteristics of the 3PSstructure are shown in Fig 7.5 In the analysis, the 3PS DFB is assumed to be anti-reflectioncoated Phase shifts 2¼ 3¼ 4¼ =3 and PSP ¼ 0:5 are assumed for the 500 mmlong cavity Compared with the QWS structure, the 3PS structure shows a smaller shift inmode characteristics This may be clearer when the variation of both amplitude gain and thedetuning coefficient are shown as a function of normalised injection current From Fig 7.6where the amplitude gain change is shown, the injection current alters the oscillating mode

in a different way It can be observed that the gain margin between the lasing mode and the

Figure 7.2 Lasing characteristics of the QWS DFB laser structure in the ( L, 
L) plane showing twovalues of normalised injection current

NUMERICAL RESULTS ON THE GAIN MARGIN OF DFB LDS 175

Figure 7.3 Average amplitude gain 
L of the QWS DFB laser structure versus the normalisedinjection current Results for both the lasing mode and non-lasing side modes (1) are shown.

Figure 7.4 Average detuning coefficient L of the QWS DFB laser structure versus the normalisedinjection current Results for both the lasing mode and non-lasing side modes (1) are shown

Figure 7.5 Lasing characteristics of the 3PS DFB laser structure in theðL; 
LÞ plane showing twovalues of normalised injection current.

Figure 7.6 Average amplitude gain 
L of the 3PS DFB laser structure versus the normalised injectioncurrent Results for both the lasing mode and non-lasing side modes (1) are shown

most probable side mode (þ1) shows little change A similar situation can be seen in Fig 7.7where the variation of the detuning coefficient is demonstrated The lasing mode shown has amilder shift with increasing biasing current On combining results from both Figs 7.6and 7.7, it appears that the 3PS laser structure is not seriously affected by the spatial holeburning effect No severe reduction of gain margin and a fairly mild shift in detuningcoefficient are observed.

Results obtained from a DCCþ QWS laser structure for the lasing characteristics,amplitude gain and the detuning coefficient are shown in Figs 7.8 to 7.10, respectively A

500 mm length cavity is assumed Other parameters used follow those adopted in the based laser model as discussed in the previous chapter These parameters include

TMM-1=2¼ 1=3, avgL¼ 2 and CP ¼ 0:46 Due to effects of spatial hole burning, all figuresappear to have a similar trend in the shape of their curves On the other hand, the single-mode stability in the DCCþ QWS structure is improved due to the presence of thedistributed coupling coefficient As shown in Fig 7.9, the amplitude gain difference betweenthe lasing mode and that of the þ1 mode remains at a high value even at a high biasingcurrent

Figure 7.11 shows the normalised gain margin (

L) between the lasing mode and themost probable side mode Results obtained from QWS, 3PS and DCCþ QWS structures areshown The gain margin of both the QWS and the DCCþ QWS structures reduces whenthe biasing current increases From the above-threshold analysis, these structures are

Figure 7.7 Average detuning coefficient L of the 3PS DFB laser structure versus the normalisedinjection current Results for both the lasing mode and non-lasing side modes (1) are shown

178 ABOVE-THRESHOLD ANALYSIS OF VARIOUS DFB LASER STRUCTURES

Figure 7.8 Lasing characteristics of the DCCþ QWS DFB laser structure in the ðL; 
LÞ planeshowing two values of normalised injection current.

Figure 7.9 Average amplitude gain 
L of the DCCþ QWS DFB laser structure versus the malised injection current Results for both the lasing mode and non-lasing side modes (1) are shown

nor-Figure 7.10 Average detuning coefficient L of the DCCþ QWS DFB laser structure versus thenormalised injection current Results for both the lasing mode and non-lasing side modes (1) are shown.

Figure 7.11 The variation of gain margin with respect to changes in the injection current for differentDFB LD structures

characterised by an intense electric field located at the centre of the cavity, and hence areaffected by spatial hole burning Nevertheless, the DCCþ QWS laser structure can maintainthe gain margin at a sufficiently high level, even under large biasing conditions By contrast,the gain margin of the 3PS structure shows little change over the range of biasing current Inall three cases, the most dramatic change of gain margin occurs when the biasing current isstill close to that of the threshold value At this biasing condition, the photon density (asshown in the previous chapter) is still fairly uniform and the non-linear gain effect is still farfrom mature It is believed to be the dominant spatial hole burning effect which alters thecharacteristics of oscillating modes When biasing current increases, the average photondensity inside the laser cavity increases and so does its value found along the plane phase orcorrugation discontinuities Under such a high current injection, the non-linear gain effectbecomes dominant.

The change of lasing wavelength with respect to the injection current change is shown inFig 7.12 Results obtained from the QWS, 3PS and the DCCþ QWS structures are shown.Among them, the 3PS structure shows relatively minor changes (0.07 nm) with the biasingcurrent The introduction of multiple phase shifts along the corrugation has suppressed thespatial hole burning effect to such an extent that the injection current hardly changes therefractive index and hence a less gradual change in lasing wavelength is obtained With such

a stable output, it appears that the 3PS structure has much potential for use as an opticalcarrier On the contrary, structures like the DCCþ QWS structure show a larger dynamicchange of lasing wavelength A single-mode continuous tuning of about 0.16 nm is achieved

It appears that this structure has a potential application in the WDM optical network

Figure 7.12 The variation of the lasing wavelength with respect to the injection current for differentDFB LD structures

NUMERICAL RESULTS ON THE GAIN MARGIN OF DFB LDS 181

7.4 ABOVE-THRESHOLD SPONTANEOUS

EMISSION SPECTRUM

By measuring the mode intensity difference from the spectrum, single-mode stability can bedetermined A minimum side mode suppression ratio of 25 dB is necessary for a stablesingle mode [7] With the help of the method using Green’s function as discussed inChapter 4, the above-threshold spontaneous emission spectrum can be evaluated using thetransfer matrices From the output of an individual transfer matrix, the contribution due tothe distributed noise source is found

From eqn (4.42) in Chapter 4, the spontaneous emission power emitted for unitybandwidthð
! ¼ 1Þ at the right laser facet of an N-sectioned mirrorless DFB laser cavitycan be expressed in terms of the elements of the overall transfer matrix as

In this equation, y22ðzNþ1 j z1Þ is a matrix element obtained from the overall transfer matrix

Y zð Nþ1j z1Þ whilst y22ðzj z1Þ and y12ðzj z1Þ are elements of the matrix Y z j zð 1Þ at arbitrary

z In the above equation, nsp is the local population inversion factor which is usuallyapproximated as [8]

XN j¼1

Qð jÞ þ Qð j þ 1Þ

where Qð jÞ ¼ nspðzjÞgðzjÞ½jy22ðzjj z1Þj2 þ jy12ðzjj z1Þj2, N is the total number of transfermatrices used and
L¼ L=N is the length of each transfer matrix represented From thematrix multiplication, matrix elements y22 zjj z1

and y12 zjj z1

at an arbitrary matrixoutput plane of z¼ zj can always be determined from those at z¼ zj
1 as