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In practice, grating-embedded semiconductor wavelength tunable filters are among themost popular active optical filters since they are suitable for monolithic integration withother semic

In practice, grating-embedded semiconductor wavelength tunable filters are among themost popular active optical filters since they are suitable for monolithic integration withother semiconductor optical devices such as laser diodes, optical switches and photo-detectors [4] As a result,
=4-shifted DFB LDs can be used as semiconductor optical filterswhen biased below threshold [5–6] This is a grating-embedded semiconductor opticaldevice, which has the advantages of a high gain and a narrow bandwidth However, thedrawbacks are that the bandwidth and transmissivity will change with the wavelength tuning[5] Fortunately Magari et al have solved these problems by using a multi-electrode DFBfilter [7–8] in which a wavelength tuning range of 33.3 GHz (
0.25 nm) with constant gainand constant bandwidth has been obtained by controlling the injection current Since then,various DFB LD designs have been developed [9–11].

In this chapter, the wavelength selection mechanism is discussed in detail Subsequently,the idea of the transfer matrix method (TMM) is again thoroughly explored and the derivedsolutions from coupled wave equations are also discussed in detail By converting thecoupled wave equations into a matrix equation, these transfer matrices can represent thewave propagating characteristics of DFB structures Therefore, using this approach, variousaspects from different DFB optical filters to enhance the active filter functionality shall beinvestigated Finally, we shall compare some of the issues for DFB LDs with those fordistributed Bragg reflector (DBR) semiconductor optical filters

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

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

10.2 WAVELENGTH SELECTION

Figure 10.1 is a narrowband transmission filter which rejects unwanted channels If the filter

is tunable, the centre wavelength (frequency)
0(see Fig 10.1) can be shifted by changing,for example, the voltage or the current applied to the filter Tunable filters can be classifiedinto three categories: passive, active and tunable LD amplifiers, as shown in Table 10.1[12–14] The passive category is composed of those wavelength-selective components thatare basically passive and can be made tunable by varying some mechanical elements of thefilters, such as mirror position or etalon angle These include Fabry–Perot etalons, tunablefibre Fabry–Perot filters and tunable Mach–Zehnder (MZ) filters For Fabry–Perot filters, thenumber of resolvable wavelengths is related to the value of the finesse F of the filter One ofthe advantages of such filters is the very fine frequency resolution that can be achieved.The disadvantages are primarily their tuning speed and losses The Mach–Zehnderintegrated optic interferometer tunable filter is a waveguide device with log2ð Þ stages,N

Figure 10.1 Operation principle of wavelength selection

Table 10.1 A comparison of filtering technologies [12–14]

254 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

where N is the number of wavelengths This filter has been demonstrated with 100wavelengths separated by 10 GHz in optical frequency, and with thermal control of the exacttuning [15] The number of simultaneously resolvable wavelengths is limited by the number

of stages required and the loss incurred in each stage

In the active category, there are two filters based on wavelength-selective polarisationtransformation by either electro-optic or acousto-optic means In both cases, the orthogonalpolarisations of the waveguide are coupled together at a specific tunable wavelength In theelectro-optic case, the wavelength selected is tuned by changing the dc voltage on theelectrodes; in the acousto-optic case, the wavelength is tuned by changing the frequency ofthe acoustic drive A filter bandwidth in full width at half maximum (FWHM) of approxi-mately 1 nm has been achieved by both filters However, the acousto-optic tunable filter has

a much broader tuning range (the entire 1.3 to 1.55 mm range) than the electro-optic type.The third category of filter is LD amplifiers as tunable filters Operation of a resonant laserstructure, such as a DFB or DBR laser, below the threshold results in narrowbandamplification These types of filter offer the following important advantages: electronicallycontrolled narrow bandwidth, the possibility of electronic tuning of the central frequency,net gain (as opposed to loss in passive filters), small size, and integrability This type of filter

is becoming more attractive since only the desired lightwave signal will be passing throughthe cavity and being amplified simultaneously (thus it is also known as an amplifier filter)

We shall investigate the principles and performance of these filters in detail

10.3 SOLUTIONS OF THE COUPLED WAVE EQUATIONS

In Chapter 2, the derivation of coupled wave equations was discussed in detail Thecharacteristics of DFB filters can be described by using these coupled wave equations In thefollowing analysis, we have assumed a zero phase difference between the index and the gainterm, hence the complex coupling coefficient can be expressed as

^

in which sand  are the amplitude gain coefficient and detuning parameter, respectively If

we compare the equations (10.6) and (10.8), a non-trivial solutions exists if the followingequation is satisfied

¼ j¼j

It is vital to note that in the absence of any coupling effects, the propagation constant is just

s j With a finite laser cavity length L extending from z ¼ z1 to z¼ z2, the boundaryconditions at the terminating facets become

R zð Þe1 jb0z1¼ ^r1S zð Þe1 jb0z1 ð10:12aÞ

S zð Þe2 jb0z2¼ ^r2R zð Þe2 jb0 z2 ð10:12bÞwhere ^r1 and ^r2 are the amplitude reflection coefficients at the laser facets z1 and z2,respectively and 0 is the Bragg propagation constant The above equations could beexpanded in such a way that

After further simplification of eqn (10.15), the following eigenvalue equation can beobtained [17]

gL¼j sinh gLð Þ

D nðr1þ r2Þ 1  rð 1r2Þ cosh gLð Þ  1 þ rð 1r2 1=2o

ð10:16Þwhere

¼ rð1þ r2Þ2sinh2ðgLÞ þ 1  rð 1r2Þ2 ð10:17aÞ

D¼ 1 þ rð 1r2Þ2 4r1r2cosh2ðgLÞ ð10:17bÞEventually, we are left with four parameters that govern the threshold characteristics of DFBlaser structures – the coupling coefficient, , the laser cavity length, L and the complex facetreflectivities r1 and r2 We have studied the coupling coefficient Owing to the complexnature of the above equation, numerical methods like the Newton–Raphson iterationtechnique can be used, provided that the Cauchy–Riemann condition on complex analyticalfunctions is satisfied

10.3.1 The Dispersion Relationship and Stop Bands

As noted in Chapter 2, for a purely index-coupled DFB LD, ¼ i For such a case, thedispersion relation of eqn (10.11) is analysed graphically as depicted in Fig 10.2 At thedetuning parameter, ¼ 0 (Bragg wavelength), the complex propagation constant g is purelyimaginary when s<  or ð s= < 1Þ This indicates evanescent wave propagation in theregion known as the stop band [18] Within this band, any incident wave is reflectedefficiently By contrast, when s> ðor s= > 1Þ, the propagation constant g will thenbecome a purely real value As predicted, when s increases, the imaginary part of thepropagation constant g decreases appreciably while the real part increases significantly.Consequently, when the waves propagate away from the Bragg wavelength, the imaginarypart of the propagation constant g increases at a faster pace than the real part at a givenamplitude gain, s Physically, it means that the wave will be attenuated when it propagatesaway from the Bragg wavelength It is paramount to note that we have consideredReðgÞ > 0

10.3.2 Formulation of the Transfer Matrix

From eqns (10.4) to (10.9), we can simply relate the complex coefficients as [17]

Figure 10.2 Normalised dependence of (a) real and (b) imaginary parts of g on  and the amplitudegain sfor a purely index-coupled DFB LD.

258 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

As shown in Fig 10.3, the corrugation inside the DFB laser is assumed to extend from z¼ z1

Equations (10.24) can then be combined with the solution of the coupled wave equations, theoutput and input of the electric fields through the matrix approach can therefore be related as

in which the argument z¼ x þ jy is a complex number while U zð Þ and V zð Þ are the real andimaginary parts of the transcendental equations.

If W zð Þ ¼ 0, the real and imaginary parts will subsequently be zero values If the order derivative of eqn (10.31) with respect to z is taken as

Only the first-order derivatives @U=@x and @V=@x are used to solve eqn (10.32)

SOLUTIONS OF THE COUPLED WAVE EQUATIONS 261

Initially, a pair x; yð Þ is guessed in order to start the numerical iteration process A newpair x
approx; yapprox

is then generated until it is sufficiently close to the exact solution.Though there are many other numerical methods to solve transcendental equations, thismethod is used due to its flexibility and speed In addition, any errors associated with othernumerical methods, such as numerical differentiation, can be avoided However, thederivative term @W=@z must be solved analytically before any numerical iteration isundertaken Another numerical method in which the term @W=@z cannot be solvedanalytically for the case of tapered-structure DFB LDs shall now be discussed

10.4 THRESHOLD ANALYSIS OF DFB LASER DIODES

For a conventional DFB laser with a zero facet reflection, the threshold eigenvalueeqn (10.16) becomes

The above transcendental equation is then solved using the Newton–Raphson iterationapproach in which the coupling coefficient is given The results obtained are shown inFig 10.4

Figure 10.4 The normalised amplitude gain versus the normalised detuning coefficient of a uniformindex-coupled DFB LD

262 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES

Note that all parameters used have been normalised with respect to the overall cavitylength L Different values of normalised coupling coefficient L in the range 0.25 to 5.0 havebeen set As predicted mathematically, there exist two pairs of possible solutions for eachoscillation mode (complex conjugates) Thus, from the results, we can see that theoscillating modes distribute symmetrically with respect to the Bragg wavelength, where

L¼ 0 In addition, no oscillation can be found at the Bragg wavelength This regionbetween theþ1 and 1 modes is called the stop band as discussed in section 10.3.1 FromFig 10.4, it can also be seen that when the coupling strength increases, the normalisedamplitude gain will decrease, in other words, the threshold current will be decreasing This isbecause a larger value of L indicates a stronger optical feedback along the DFB cavity.Similarly, if the coupling strength is fixed, a longer cavity length will also reduce thethreshold gains since a larger single pass gain can be achieved easily

In laser operation, the main (fundamental) mode is large and the sub-modes aresufficiently suppressed because the coupling between the main mode and the sub-modes islarge and, as such, the gain concentrates on the main mode However, if DFB LDs are to beused as amplifier filters, the lasers will then be biased below the lasing threshold, thereforethe gain difference between the main mode and the sub-modes is always smaller than in laseroperation As a result, the wavelength tuning range for an optical amplifier filter is smallerthan that of a laser

10.4.1 Phase Discontinuities in DFB LDs

The analysis of phase-adjusted DFB LDs is rather similar to the conventional DFB LDsdescribed in the previous section The only difference is that the boundary conditions at thephase shift position (PSP) have to be matched Whenever a propagating wave travels past aphase discontinuity along the corrugation, it will experience a phase delay

As noted earlier, TMM is used since it can match the boundary conditions easily bycascading the matrices Thus, the phase discontinuity along the cavity of the DFB LDs can

be best explained by using a two-section DFB structure with a single phase shift at the centre

of the corrugation as depicted in Fig 10.5 zþ and z are assumed to be the slight deviationsfrom z

Figure 10.5 Schematic diagram of a single-phase-shifted DFB LD

THRESHOLD ANALYSIS OF DFB LASER DIODES 263

If the distance between z and z is infinitesimal, we can relate the electric fields at zþ and



ES
z

ð10:42Þ

where P is the phase discontinuity matrix, which causes the complex electric field delay

of at z¼ z ... tointroduced at the middle of the DFB cavity, the lowest mode threshold gain exists at theBragg wavelength (L¼ 0) This is an interesting feature in which SLM happens.

It is also interesting... modesrelative to the lasing mode This brings us to the idea of introducing three phase shifts alongthe cavity to reduce the discontinuity level in the electric field intensity.