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10 Wavelength Tunable Optical Filters Based on DFB Laser Structures 10.1 INTRODUCTION In recent years, advances in wavelength division multiplexing (WDM) and dense wavelength division multiplexing (DWDM) technology have enabled the deployment of systems that are capable of providing large amounts of bandwidth [1]. Wavelength tunable optical filters appear to be the key components in realising these WDM/DWDM lightwave systems. Optical filtering for the selection of channels separated by 2 nm is currently achievable, and narrower channel separations will be possible in the near future with improved technology [2–3]. This would give more than 100 broadband channels in the low-loss fibre transmission region of 1.3 mm and/or 1.55 mm wavelength bands, with each wavelength channel having a transmission bandwidth of several gigahertz. In practice, grating-embedded semiconductor wavelength tunable filters are among the most popular active optical filters since they are suitable for monolithic integration with other 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 filters when biased below threshold [5–6]. This is a grating-embedded semiconductor optical device, which has the advantages of a high gain and a narrow bandwidth. However, the drawbacks 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 DFB filter [7–8] in which a wavelength tuning range of 33.3 GHz (0.25 nm) with constant gain and 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 derived solutions from coupled wave equations are also discussed in detail. By converting the coupled wave equations into a matrix equation, these transfer matrices can represent the wave propagating characteristics of DFB structures. Therefore, using this approach, various aspects from different DFB optical filters to enhance the active filter functionality shall be investigated. Finally, we shall compare some of the issues for DFB LDs with those for distributed Bragg reflector (DBR) semiconductor optical filters. Distributed Feedback Laser Diodes and Optical Tunable Filters H. Ghafouri–Shiraz # 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1 254 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES Figure 10.1 Operation principle of wavelength selection. 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 classified into 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 that are basically passive and can be made tunable by varying some mechanical elements of the filters, such as mirror position or etalon angle. These include Fabry–Perot etalons, tunable fibre Fabry–Perot filters and tunable Mach–Zehnder (MZ) filters. For Fabry–Perot filters, the number of resolvable wavelengths is related to the value of the finesse F of the filter. One of the 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–Zehnder integrated optic interferometer tunable filter is a waveguide device with log2ðNÞ stages, Table 10.1 A comparison of filtering technologies [12–14] Passive Active Laser diode amplifiers Type Etalon (F200) Fibre Fabry–Perot Waveguide Mach–Zehnder Fibre Bragg Gratings (FBGs) Electro-optic TE/TM Acousto-optic TE/TM DFB amplifier 2-section DFB amplifier Phase-shift controlled DFB amplifier Resolution 0.38 A (5 GHz) 1 A–2 A 6 A 10 A 1–2 A 0.85 A 0.32 A (4 GHz) Range 45 A >50 nm 160 A 400 nm 4–5 A 6 A 9.5 A (120 GHz) No. of Tuning channels speed 30 ms 30 ms 128 ms 50 ms 10 ns 100 10 ms 2–3 1 ns 8 ns 18 ns SOLUTIONS OF THE COUPLED WAVE EQUATIONS 255 where N is the number of wavelengths. This filter has been demonstrated with 100 wavelengths separated by 10 GHz in optical frequency, and with thermal control of the exact tuning [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 polarisation transformation by either electro-optic or acousto-optic means. In both cases, the orthogonal polarisations of the waveguide are coupled together at a specific tunable wavelength. In the electro-optic case, the wavelength selected is tuned by changing the dc voltage on the electrodes; in the acousto-optic case, the wavelength is tuned by changing the frequency of the 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 laser structure, such as a DFB or DBR laser, below the threshold results in narrowband amplification. These types of filter offer the following important advantages: electronically controlled 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 through the 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. The characteristics of DFB filters can be described by using these coupled wave equations. In the following analysis, we have assumed a zero phase difference between the index and the gain term, hence the complex coupling coefficient can be expressed as RS ¼ SR ¼ i þjg ¼ ð10:1Þ where is the complex coupling coefficient. According to eqn (2.98), the complex ampli-tude terms of the forward, RðzÞ, and backward, SðzÞ, propagating waves can be written as [16] RðzÞ ¼ R1egz þ R2eÿgz ð10:2Þ SðzÞ ¼ S1egz þS2eÿgz ð10:3Þ where R1, R2, S1 and S2 are the complex coefficients and g, known as the complex pro-pagation constant, depends on the boundary conditions at the laser facets. By substituting eqns (10.2) and (10.3) into eqn (2.98), we have R1 ¼ jeÿjS1 ð10:4Þ R2 ¼ jeÿjS2 ð10:5Þ and S1 ¼ jejR1 ð10:6Þ S2 ¼ jejR2 ð10:7Þ 256 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES where ¼ s ÿj ÿg ð10:8Þ ¼ s ÿj þg ð10:9Þ in which s and 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 following equation is satisfied ¼ ¼ j ð10:10Þ Similarly, we can obtain the following dispersion equation, which is independent of the residue corrugation phase, . g2 ¼ ðs ÿ jÞ2þ2 ð10:11Þ It is vital to note that in the absence of any coupling effects, the propagation constant is just s ÿ j. With a finite laser cavity length L extending from z ¼ z1 to z ¼ z2, the boundary conditions at the terminating facets become Rðz1Þeÿjb0z1 ¼ r1Sðz1Þejb0z1 Sðz2Þejb0z2 ¼ r2Rðz2Þeÿjb0z2 ð10:12aÞ ð10:12bÞ where 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 be expanded in such a way that ð1 ÿr1Þe2gz1 2 r1= ÿ 1 1 or ðr2 ÿ Þe2gz2 2 1= ÿ r2 1 ð10:13aÞ ð10:13bÞ In the above equations, r1 and r2 are the complex field reflectivities of the left and right facets, respectively. such that r1 ¼ r1e2jb0z1ej ¼ r1ej 1 r2 ¼ r2eÿ2jb0z2eÿj ¼ r2eÿj 2 ð10:14aÞ ð10:14bÞ where 1 and 2 are the corresponding corrugation phases at the facets. Equations (10.13a) and (10.13b) are homogeneous in R1 and R2. Hence, in order to obtain a non-trivial solution, we must satisfy ð1 ÿ r1Þe2gz1 ðr2 ÿ Þe2gz2 r1 ÿ 1 ÿr2 ð10:15Þ SOLUTIONS OF THE COUPLED WAVE EQUATIONS 257 After further simplification of eqn (10.15), the following eigenvalue equation can be obtained [17] gL ¼ ÿjsinhðgLÞ nðr1 þ r2Þð1 ÿ r1r2ÞcoshðgLÞ ð1 þ r1r2Þ1=2o where ¼ ðr1 þ r2Þ2sinh2ðgLÞ þ ð1 ÿ r1r2Þ2 D ¼ ð1 þ r1r2Þ2ÿ4r1r2 cosh2ðgLÞ ð10:16Þ ð10:17aÞ ð10:17bÞ Eventually, we are left with four parameters that govern the threshold characteristics of DFB laser structures–the coupling coefficient, , the laser cavity length, L and the complex facet reflectivities r1 and r2. We have studied the coupling coefficient. Owing 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. 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, the dispersion relation of eqn (10.11) is analysed graphically as depicted in Fig. 10.2. At the detuning parameter, ¼ 0 (Bragg wavelength), the complex propagation constant g is purely imaginary when s < ðor s= < 1Þ. This indicates evanescent wave propagation in the region known as the stop band [18]. Within this band, any incident wave is reflected efficiently. By contrast, when s > ðor s= > 1Þ, the propagation constant g will then become a purely real value. As predicted, when s increases, the imaginary part of the propagation constant g decreases appreciably while the real part increases significantly. Consequently, when the waves propagate away from the Bragg wavelength, the imaginary part of the propagation constant g increases at a faster pace than the real part at a given amplitude gain, s. Physically, it means that the wave will be attenuated when it propagates away from the Bragg wavelength. It is paramount to note that we have considered ReðgÞ > 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] S1 ¼ ejR1 R2 ¼ eÿjS2 And thus eqns (10.2) and (10.3) become RðzÞ ¼ R1egz þS2eÿjeÿgz SðzÞ ¼ R1ejegz þ S2eÿgz ð10:18Þ ð10:19Þ ð10:20Þ ð10:21Þ ... - tailieumienphi.vn