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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 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 log 2 NðÞstages, Figure 10.1 Operation principle of wavelength selection. Table 10.1 A comparison of filtering technologies [12–14] No. of Tuning Type Resolution Range channels speed Passive Etalon (F $ 200) $30 ms Fibre Fabry–Perot $30 ms Waveguide Mach–Zehnder 0.38 A ˚ 45 A ˚ 128 ms (5 GHz) Active Fibre Bragg Gratings (FBGs) $1A ˚ –2 A ˚ >50 nm $50 ms Electro-optic TE/TM 6 A ˚ 160 A ˚ $10 ns Acousto-optic TE/TM 10 A ˚ 400 nm $100 $10 ms Laser diode DFB amplifier 1–2 A ˚ 4–5 A ˚ 2–3 1 ns amplifiers 2-section DFB amplifier 0.85 A ˚ 6A ˚ 8ns Phase-shift controlled 0.32 A ˚ 9.5 A ˚ 18 ns DFB amplifier (4 GHz) (120 GHz) 254 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES 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 þ j g ¼  ð10:1Þ where  is the complex coupling coefficient. According to eqn (2.98), the complex ampli- tude terms of the forward, RzðÞ, and backward, SzðÞ, propagating waves can be written as [16] RzðÞ¼R 1 e gz þ R 2 e Àgz ð10:2Þ SzðÞ¼S 1 e gz þ S 2 e Àgz ð10:3Þ where R 1 , R 2 , S 1 and S 2 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 R 1 ¼ je Àj S 1 ð10:4Þ ^ R 2 ¼ je Àj S 2 ð10:5Þ and ^ S 1 ¼ je j R 1 ð10:6Þ S 2 ¼ je j R 2 ð10:7Þ SOLUTIONS OF THE COUPLED WAVE EQUATIONS 255 where  ¼  s À j À g ð10:8Þ ^  ¼  s À j þ g 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  j ¼ j ^  ð10:10Þ Similarly, we can obtain the following dispersion equation, which is independent of the residue corrugation phase, . g 2 ¼  s À jðÞ 2 þ  2 ð10:11Þ It is vital to note that in the absence of any coupling effects, the propagation constant is just  s À j. With a finite laser cavity length L extending from z ¼ z 1 to z ¼ z 2 , the boundary conditions at the terminating facets become Rz 1 ðÞe Àjb 0 z 1 ¼ ^ r 1 Sz 1 ðÞe jb 0 z 1 ð10:12aÞ Sz 2 ðÞe jb 0 z 2 ¼ ^ r 2 Rz 2 ðÞe Àjb 0 z 2 ð10:12bÞ where ^ r 1 and ^ r 2 are the amplitude reflection coefficients at the laser facets z 1 and z 2 , respectively and  0 is the Bragg propagation constant. The above equations could be expanded in such a way that R 2 ¼ 1 À r 1 ðÞe 2gz 1 r 1 = À 1 Á R 1 ð10:13aÞ or R 2 ¼ r 2 À ðÞe 2gz 2 1= À r 2 Á R 1 ð10:13bÞ In the above equations, r 1 and r 2 are the complex field reflectivities of the left and right facets, respectively. such that r 1 ¼ ^ r 1 e 2jb 0 z 1 e j ¼ ^ r 1 e j 1 ð10:14aÞ r 2 ¼ ^ r 2 e À2jb 0 z 2 e Àj ¼ ^ r 2 e Àj 2 ð10:14bÞ where  1 and  2 are the corresponding corrugation phases at the facets. Equations (10.13a) and (10.13b) are homogeneous in R 1 and R 2 . Hence, in order to obtain a non-trivial solution, we must satisfy 1 À r 1 ðÞe 2gz 1 r 1 À  ¼ r 2 À ðÞe 2gz 2 1 À r 2 ð10:15Þ 256 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES After further simplification of eqn (10.15), the following eigenvalue equation can be obtained [17] gL ¼ Àj sinh gLðÞ D Á r 1 þ r 2 ðÞ1 À r 1 r 2 ðÞcosh gLðÞÆ1 þ r 1 r 2 ðÞ 1=2 no ð10:16Þ where  ¼ r 1 þ r 2 ðÞ 2 sinh 2 gLðÞþ1 À r 1 r 2 ðÞ 2 ð10:17aÞ D ¼ 1 þ r 1 r 2 ðÞ 2 À 4r 1 r 2 cosh 2 gLðÞ 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 r 1 and r 2 . 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 < or  s = < 1ðÞ. 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] S 1 ¼ e j R 1 ð10:18Þ R 2 ¼ e Àj S 2 ð10:19Þ And thus eqns (10.2) and (10.3) become RzðÞ¼R 1 e gz þ S 2 e Àj e Àgz ð10:20Þ SzðÞ¼R 1 e j e gz þ S 2 e Àgz ð10:21Þ SOLUTIONS OF THE COUPLED WAVE EQUATIONS 257 Figure 10.2 Normalised dependence of (a) real and (b) imaginary parts of g on  and the amplitude gain  s for 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 z ¼ z 1 to z ¼ z 2 . The amplitude coefficients at the left and right facets can then be written as Rz 1 ðÞ¼R 1 e gz 1 þ S 2 e Àj e Àgz 1 ð10:22aÞ Sz 1 ðÞ¼R 1 e j e gz 1 þ S 2 e Àgz 1 ð10:22bÞ Rz 2 ðÞ¼R 1 e gz 2 þ S 2 e Àj e Àgz 2 ð10:22cÞ Sz 2 ðÞ¼R 1 e j e gz 2 þ S 2 e Àgz 2 ð10:22dÞ From eqns (10.22a) and (10.22b), R 1 and S 2 can be expressed as R 1 ¼ Sz 1 ðÞe Àj À Rz 1 ðÞ  2 À 1ðÞe gz 1 ð10:23aÞ S 2 ¼ Rz 1 ðÞe j À Sz 1 ðÞ  2 À 1ðÞe Àgz 1 ð10:23bÞ Subsequently, by substituting the above equations into eqns (10.22c) and (10.22d), we have Rz 2 ðÞ¼ E À  2 E À1 1 À  2 Rz 1 ðÞÀ  E À E À1 ÀÁ e Àj 1 À  2 Sz 1 ðÞ ð10:24aÞ Sz 2 ðÞ¼  E À E À1 ÀÁ e j 1 À  2 Rz 1 ðÞÀ  2 E À E À1 1 À  2 Sz 1 ðÞ ð10:24bÞ where E ¼ e g z 2 Àz 1 ðÞ ; E À1 ¼ e Àg z 2 Àz 1 ðÞ ð10:24cÞ Note that the electric field at the output plane z 2 can be expressed in terms of the electric waves at the input plane. Given the solution of the coupled wave equations from eqn (2.98) EzðÞ¼RzðÞe Àjb 0 z þ SzðÞe jb 0 z ð10:25Þ Figure 10.3 A simplified schematic diagram for a one-dimensional corrugated DFB laser diode section. SOLUTIONS OF THE COUPLED WAVE EQUATIONS 259 Equations (10.24) can then be combined with the solution of the coupled wave equations, the output and input of the electric fields through the matrix approach can therefore be related as E R z 2 ðÞ E S z 2 ðÞ ! ¼ T z 2 j z 1 ðÞÁ E R z 1 ðÞ E S z 1 ðÞ ! ¼ t 11 t 12 t 21 t 22 ! Á E R z 1 ðÞ E S z 1 ðÞ ! ð10:26Þ where the matrix T z 2 j z 1 ðÞrepresents any wave propagation from z ¼ z 1 to z ¼ z 2 and its elements t ij i; j ¼ 1; 2ðÞare given as t 11 ¼ E À  2 E À1 ÀÁ Á e Àjb 0 z 2 Àz 1 ðÞ ð1 À  2 Þ ð10:27aÞ t 12 ¼  E À E À1 ÀÁ Á e Àj e Àjb 0 z 2 þz 1 ðÞ ð1 À  2 Þ ð10:27bÞ t 21 ¼ À E À E À1 ÀÁ Á e j e jb 0 z 2 þz 1 ðÞ ð1 À  2 Þ ð10:27cÞ t 22 ¼ À  2 E À E À1 ÀÁ Á e jb 0 z 2 Àz 1 ðÞ ð1 À  2 Þ ð10:27dÞ Or from eqn (10.24) in hyperbolic functions [7] Rz 2 ðÞ Sz 2 ðÞ ! ¼ F z 2 j z 1 ðÞÁ Rz 1 ðÞ Sz 1 ðÞ ! ¼ f 11 f 12 f 21 f 22 ! Á Rz 1 ðÞ Sz 1 ðÞ ! ð10:28Þ where f 11 ¼ cosh g z 2 À z 1 ðÞ½þ  À jðÞz 2 À z 1 ðÞ g z 2 À z 1 ðÞ sinh g z 2 À z 1 ðÞ½ð10:29aÞ f 12 ¼Àj  z 2 À z 1 ðÞ g z 2 À z 1 ðÞ sinh g z 2 À z 1 ðÞ½ ð10:29bÞ f 21 ¼ j  z 2 À z 1 ðÞ  z 2 À z 1 ðÞ sinh g z 2 À z 1 ðÞ½ ð10:29cÞ f 22 ¼ cosh g z 2 À z 1 ðÞ½À  À jðÞz 2 À z 1 ðÞ g z 2 À z 1 ðÞ sinh g z 2 À z 1 ðÞ½ð10:29dÞ Owing to conservation of energy, the determinant of the matrix T z 2 j z 1 ðÞmust always be unity [19–20]. That is t 11 t 22 À t 12 t 21 ¼ 1 ð10:30Þ 10.3.3 Solutions of Complex Transcendental Equations using the Newton–Raphson Approximation Transcendental equations will be formed in order to find the threshold gain of DFB LDs [21]. In general these equations can be expressed in complex form such that WzðÞ¼UzðÞþjVzðÞ¼0 ð10:31Þ 260 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES in which the argument z ¼ x þ jy is a complex number while UzðÞand VzðÞare the real and imaginary parts of the transcendental equations. If WzðÞ¼0, the real and imaginary parts will subsequently be zero values. If the first- order derivative of eqn (10.31) with respect to z is taken as @WzðÞ @z ¼ @UzðÞ @z þ j @VzðÞ @z ¼ @UzðÞ @x Á @x @z þ j @VzðÞ @x Á @x @z  ¼ @UzðÞ @x þ j @VzðÞ @x , @x @z ¼ 1 ð10:32Þ By using the Taylor series, the functions UzðÞand VzðÞcan be approximated about the exact solution x approx ; y approx ÀÁ such that Ux approx ; y approx ÀÁ ¼ Ux; yðÞþ @U @x x approx À x ÀÁ þ @U @y y approx À y ÀÁ ð10:33Þ Vx approx ; y approx ÀÁ ¼ Vx; yðÞþ @V @x x approx À x ÀÁ þ @V @y y approx À y ÀÁ ð10:34Þ where x; yðÞis the initial guess which is chosen to be sufficiently close to the exact solutions. The other higher-order derivative terms from the above Taylor series have been ignored. Thus, by solving the above simultaneous equations, we have x approx ¼ x þ Vx; yðÞ @U @y À Ux; yðÞ @V @y det ð10:35Þ y approx ¼ y þ Ux; yðÞ @V @x À Vx; yðÞ @U @x det ð10:36Þ where det ¼ @U @x  2 þ @V @y  2 ð10:37Þ For an analytical complex function WzðÞ, the Cauchy–Riemann condition must be satisfied [22] @U @x ¼ @V @y ; @U @y ¼À @V @x ð10:38Þ The partial differential with respect to y, @=@y will then be replaced with @=@x using the above Cauchy–Riemann condition det ¼ 2 @U @x  2 ð10:39Þ x approx ¼ x À Vx; yðÞ @V @x þ Ux; yðÞ @U @x det ð10:40Þ 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; yðÞis guessed in order to start the numerical iteration process. A new pair x approx ; y approx ÀÁ is then generated until it is sufficiently close to the exact solution. Though there are many other numerical methods to solve transcendental equations, this method is used due to its flexibility and speed. In addition, any errors associated with other numerical methods, such as numerical differentiation, can be avoided. However, the derivative term @W=@z must be solved analytically before any numerical iteration is undertaken. Another numerical method in which the term @W=@z cannot be solved analytically 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 eigenvalue eqn (10.16) becomes jgL ¼ÆL sinh gLðÞ ð10:41Þ The above transcendental equation is then solved using the Newton–Raphson iteration approach in which the coupling coefficient is given. The results obtained are shown in Fig. 10.4. Figure 10.4 The normalised amplitude gain versus the normalised detuning coefficient of a uniform index-coupled DFB LD. 262 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES [...]... LDS Distributed Bragg reflector laser diodes can be formed by replacing one or both of the discrete laser mirrors with a passive grating reflector as shown in Fig 10.21 By definition, the grating reflectors are formed along a passive waveguide section, so one of the issues is to make the transition between the active and passive waveguides without introducing unwanted discontinuities So far we have just... Mito, I., Tunable wavelength filters using !/4-shifted waveguide grating resonators, Appl Phys Lett., 53, 83–85, 1988 33 Kazovsky, L G., Stern, M., Menocal, S G and Zah, C E., DBR active optical filters: transfer function and noise characteristics, IEEE J Lightwave Technol., 8(10), 1441–1451, 1990 34 Agrawal, G P and Dutta, N K., Semiconductor Lasers, 2nd edition New York: Van Nostrand Reinhold, 1993 35 Komori,... boundary conditions at the phase shift position (PSP) have to be matched Whenever a propagating wave travels past a phase discontinuity along the corrugation, it will experience a phase delay As noted earlier, TMM is used since it can match the boundary conditions easily by cascading the matrices Thus, the phase discontinuity along the cavity of the DFB LDs can be best explained by using a two-section DFB... ON DFB LASER STRUCTURES zÀ  If the distance between z and zÆ is infinitesimal, we can relate the electric fields at zþ and   as follows " À Á# " # " À Á# E R zÀ E R zþ 0 e j  À þÁ ¼ À Á Á E S z E S zÀ 0 eÀj  " À Á# À ER z  À Á ¼ P Á ð10:42Þ ES z À  where P is the phase discontinuity matrix, which causes the complex electric field delay of  at z ¼ z By applying the phase discontinuity... not possible to achieve a single longitudinal mode (SLM) Nevertheless, if a single phase shift of %/2 is introduced at the middle of the DFB cavity, the lowest mode threshold gain exists at the Bragg wavelength (L ¼ 0) This is an interesting feature in which SLM happens It is also interesting to consider the electric field intensity in these structures By solving the threshold condition of eqn (10.16)... factors change the longitudinal mode intensity distribution and alter the gain suppression of the side modes relative to the lasing mode This brings us to the idea of introducing three phase shifts along the cavity to reduce the discontinuity level in the electric field intensity 10.4.2 Below-threshold Characteristics The amplification characteristics of conventional DFB LDs are calculated under low-input... ne j E m! ð10:55Þ where ' is the conductivity, n is the number of free electrons per unit volume, m and ! are the mass of an electron and the optical angular frequency, respectively E is the electric field Equation (10.55) clearly shows that the conductivity is a function of carrier density and optical frequency therefore changes in carrier density affects the conductivity This subsequently affects... length will also reduce the threshold gains since a larger single pass gain can be achieved easily In laser operation, the main (fundamental) mode is large and the sub-modes are sufficiently suppressed because the coupling between the main mode and the sub-modes is large and, as such, the gain concentrates on the main mode However, if DFB LDs are to be used as amplifier filters, the lasers will then be... passive DBR section, the refractive index changes due to the free carrier plasma effect (refer to section 10.5), and this in turn changes the effective corrugation period and the Bragg frequency Since the 280 WAVELENGTH TUNABLE OPTICAL FILTERS BASED ON DFB LASER STRUCTURES Figure 10.21 Schematic illustration of a three-section distributed Bragg reflector (DBR) laser diode carrier density is not clamped like... spontaneous emission of DFB semiconductor laser amplifiers, IEEE J Quantum Electron., QE-24(8), 1507–1518, 1988 21 O’Neil, P V., Advanced Engineering Mathematics, 4th edition, Part 6 London: Thomson, 1995 22 Arfken, G., Mathematical Methods for Physicists, 3rd edition New York: Academic Press, 1985 23 McCall, S L and Platzman, P M., An optimized p/2 distributed feedback laser, IEEE J Quantum Electron., . 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. than in laser operation. As a result, the wavelength tuning range for an optical amplifier filter is smaller than that of a laser. 10.4.1 Phase Discontinuities