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Microstubs resonators integrated to bent Y-branch waveguide
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Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31 www.elsevier.com/locate/photonics
Microstubs resonators integrated to bent Y-branch waveguide Y. Pennec a,*, M. Beaugeois b, B. Djafari-Rouhani a, R. Sainidou a, A. Akjouj a, J.O. Vasseur a, L. Dobrzynski a, E.H. El Boudouti a, J.-P. Vilcot a, M. Bouazaoui b, J.-P. Vigneron c a
Institut d’Electronique, de Microe´lectronique et de Nanotechnologie, UMR CNRS 8520, Universite´ de Lille1, BP 60069, 59652 Villeneuve d’Ascq Cedex, France b Laboratoire PHLAM, UMR CNRS 8523, Universite´ de Lille1, 59655 Villeneuve d’Ascq Cedex, France c Laboratoire de Physique du Solide, Faculte´s Universitaires Notre Dame de la Paix, Rue de Bruxelles 61, 5000 Namur, Belgium Received 13 June 2007; received in revised form 9 January 2008; accepted 14 January 2008 Available online 19 January 2008
Abstract We report numerical simulations, based on a finite difference time domain (FDTD) method, of light propagation in twodimensional semiconductor micro-optical waveguides coupled to one or several lateral stubs. It is shown that when the stub is covered with a perfectly metallic thin layer, the transmission spectrum contains several narrow dips. Such simulation of the metallic coating can be used in the far infrared frequency domain, far from the optical regime. We propose a selective filtering device based on the interaction between several stubs. Inserting an appropriate defect stub between a set of periodical stubs leads to a tunnelling transmission, with a narrow peak inside the gap. This filtering phenomenon is used to propose a demultiplexer based on a Y-shaped waveguide for separating signals with different frequencies. Finally, we show that the filtering effect of a stub can also be reproduced when the metal is described in the frame of a Drude model instead of being perfect, which makes plausible the realization of the above devices in the near optical regime. # 2008 Elsevier B.V. All rights reserved. PACS : 41.20.-q; 42.25.Bs; 42.79.Gn; 42.79.Sz; 42.82.Bq; 42.82.Et Keywords: Photonic waveguide; Resonator; Stub; FDTD; Filter; Demultiplexer
1. Introduction We report a theoretical study of microcavity resonator devices based on semiconductor waveguides with a very large lateral refractive index contrast (air– semiconductor–air), using 2D-FDTD numerical simulations. In a previous publication [1], we have shown that the transmission coefficient through a twodimensional semiconductor micro-waveguide coupled to a rectangular cavity grafted on each side of the guide * Corresponding author. Tel.: +33 3 20 43 68 07. E-mail address: [email protected] (Y. Pennec). 1569-4410/$ – see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2008.01.002
displays several narrow dips. Such a cavity, called a stub, was covered with a perfectly metallic thin layer to prevent the radiation to escape outside from the boundaries of the stub. Such a device could easily be used in the far infrared frequency domain (far from the plasma frequency) where the metal behaves as a perfect mirror. It is then believed that, owning to their small size comparatively to other resonators such as microdisks or microrings [2–4], microstubs could find original applications in future optoelectronic integrated circuits. In this paper, we report the study of a couple of functionalities of microstub resonator-based devices, already studied in the literature for different systems
Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31
such as fiber Bragg gratings [5], photonic band gap air bridges [6] and more recently 2D-photonic crystal materials [7]. One aim is to propose the filtering of specific wavelength from a large band input signal, either in rejection or in selection, by coupling the guide to a set of stubs. Then, we use this filtering process to investigate the possibility of a new device for wavelength demultiplexing based on a bent Y-shaped component. Furthermore, we show that the main features of the transmission spectrum through a waveguide coupled to a lateral stub remain valid if the metal covering the stub is not perfect, but described in the framework of the Drude model. This makes plausible the extension of the above applications to the near optical regime. 2. Results and discussion The geometry of the device is displayed in Fig. 1a, where w and l are, respectively, the width and the length of the stub and a is the thickness of the waveguide. The thickness of the metal is d = a/5. The waveguide and the
Fig. 1. (a) Schematic representation of a bus waveguide coupled to a stub covered with a thin metallic layer and (b) transmission spectrum through the above system when the coated metal is perfect and the dimensions of the stub are (w, l) = (a, 1.4a).
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stub are made of a semiconductor structure [8], which refractive index is n = 3.2429 and the embedding medium is air. This choice ensures a large contrast in the properties of the waveguide constituent material and the surrounding medium. The transmission through the waveguide is calculated by using the finite difference time domain (FDTD) method [9,10] with perfectly matching method as boundary conditions [11]. The incoming wave is created inside the taper region by an initial current source parallel to the x-axis, with a Gaussian profile along the x-direction. We choose to study the transverse magnetic polarization with the electric field ~ E belong~ along z. ing to the (x, y) plane and the magnetic field H All the transmission spectra, probed at the right end of the waveguide, are normalized with respect to the one corresponding to a perfect (without stub) waveguide. The transmission is reported in dB as a function of the dimensionless frequency V = va/(2pc) = a/l where l is the wavelength of light in vacuum. In Fig. 1b, we remind the calculated transmission for a stub grafted on the waveguide with the following parameters: w ¼ a and l = 1.4a [1]. The microstub is covered with a perfectly metallic thin layer that prevents the radiation to escape outside from the boundaries of the stub. The hypothesis of a perfect metal is valid in the far infrared frequency range, far from the plasma frequency of the metal. The results are given for V in the range 0.2–0.4 which ensures the waveguide to be in the monomode regime according to the (analytical) solutions of the dispersion curves for the straight waveguide. The transmission spectrum displays several dips with attenuation exceeding 20 dB. In Ref. [1], we have shown that for various lengths and widths of the stub, the frequencies of the dips can be favourably fitted with the eigenfrequencies Vg of a rectangular electromagnetic cavity given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi 1 a a m21 (1) þ m22 0 Vg ¼ 2n w l where n is the refractive index and m1 and m2 are two positive integers that define the order of the modes. While the width of the stub should be clearly w ¼ a in the analytical model, the length l0 remains somewhat undefined because the boundary conditions at the lower face of the cavity can be applied somewhere between the middle and the top of the waveguide; this means that l0 can be chosen between l0 = l and l0 = l + 0.5a. A good agreement was obtained between the numerical results and the eigenfrequencies of the cavity (see Table 1) by choosing l0 = 1.5a.
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Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31
Table 1 Comparison between the reduced resonance frequencies obtained by the FDTD simulation and by the eigenfrequencies calculated from Eq. (1) Vg (FDTD Fig. 1b)
Vg (Eq. (1))
(m1, m2)
0.256 0.322 0.329 0.362 0.415
0.257 0.325 0.344 0.370 0.430
(1, (2, (1, (2, (2,
2) 1) 3) 2) 3)
The numbers (m1, m2) give the order of the modes.
Furthermore, we also checked in a few examples that quite similar results are obtained in the case of a 3D rectangular waveguide coupled to a lateral stub, so we restricted the numerical simulations only to 2D structures for the rest of the paper. We first investigate the transmission coefficient through the waveguide when we increase the number N of the stubs from one to five (Fig. 2a), with a separation distance of 2a between the stubs. Increasing N leads to the widening of the dips and opening of band gaps. We concentrate here on the gap (Fig. 2b) occurring in the frequency range [0.240, 0.260] with an attenuation of the transmission exceeding 20 dB. We now change the
Fig. 2. (a) Schematic representation of the waveguide coupled to five stubs (w, l) = (a, 1.4a) where the length of the center stub is increased with respect to the others and (b) transmission spectra calculated for the above model for two different lengths (l = 1.5a and l = 1.54a) of the defect stub and showing the evolution of a narrow pass band inside the gap.
geometrical parameters of the third stub in order to make a ‘‘defect’’ inside the periodical chain of stubs (Fig. 2a). The defect stub could be created by a modification of the width or the length of the stub. For instance, we choose to increase the length of the defect stub from l = 1.4a to l = 1.5a which has the effect of decreasing its resonance frequency with respect to that associated with the other stubs in the chain, therefore ensuring that the resonance frequency falls inside the gap. The transmission spectrum of such a structure incorporating a defect (solid line in Fig. 2b) displays one narrow peak centred at frequency Vg = 0.2527 inside the gap. This corresponds to the tunnelling of the wave through the localized mode associated with the defect. As a result, such a device could be used as a selective filter. Furthermore, it is possible to displace the narrow frequency band inside the gap by adjusting the length or the width of the defect stub. In Fig. 4b, the transmission spectrum is presented for another length of the stub, l = 1.54a (dotted line in Fig. 2b) which shows a downward shift of the peak with a central frequency Vg = 0.2518. We have thus created a tunable selective filter at the reduced frequencies Vg = 0.2518 and Vg = 0.2527. The second purpose of the paper is to propose a Yshaped demultiplexer, based on microstubs cavities (Fig. 3a). The main goal is to select different frequencies on each branch of the Y-structure from a broad band signal. As explained below, the first step in the selective filter constitution is to associate microstubs in a periodic structure. The second step of the selective filter realisation is to place, on each branch, a ‘defect’ stub in the middle of the periodic stub arrangement as illustrated in Fig. 3a. In this model, we replace on the T1 (resp. T2) branch the third stub by a new one with the geometrical parameters (l, w) = (a, 1.54a) (resp. (l, w) = (a, 1.5a)). The calculated map of the magnetic field (Fig. 3b) at the monochromatic frequencies Vg = 0.2518 (resp. Vg = 0.2527) shows a strong enhancement of the field at the defect stub and its nearest neighbours into the T1 (T2) branch. Each selected frequency (Vg = 0.2518 and Vg = 0.2527) is then available at the end of the corresponding port (T1 and T2). At the same frequency, the regular periodic arrangement of the stubs acts as a stop band and no signal is available at port T2 for Vg = 0.2518 and at port T1 for Vg = 0.2527. As a result, Fig. 3 shows clearly that the large band input Gaussian signal T0 has been separated and directed towards the two branches of the Y-junction. It is also worth noticing that each branch of the Y-shaped waveguide will select its own narrow pass band. This example could also be extended to a multichannel device.
Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31
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Fig. 3. (a) Schematic representation of the Y-shaped waveguide in which the branches are coupled to five stubs where the length of the center stub is increased with respect to the others: l = 1.54a for T1 and l = 1.50a for T2 and (b) map of the magnetic field for two monochromatic incident radiations at the frequencies of the transmission zero Vg = 0.2527 and Vg = 0.2518.
The above applications were discussed by assuming that the stubs are covered with a perfect metal layer. The hypothesis of a perfect metal is well adapted to infrared frequencies but not to optical frequencies where the true behaviour of the metal cannot be omitted. Nevertheless, this hypothesis has the advantage of reducing considerably the simulation time, since no particular discretization is required in the metallic part of the structure. In the case of a real metal, the discretization should be performed at the scale of the skin depth, which means a much smaller mesh than in the case of the semiconductor waveguide. To show that the above applications should possibly be also feasible at the telecommunication wavelength, we have studied the transmission through a waveguide coupled to a single stub covered with a real metal whose dielectric function is described by the Drude model: eðvÞ ¼ e1
v2p
Fig. 4. (a) Transmission spectrum through the system of Fig. 1a when the metal is real and the dimensions of the stub are (w, l) = (a, a). The transmission curves are plotted for a value of the absorption coefficient g = 1014 Hz and (b) maps of the magnetic field (FDTD model) for three monochromatic incident radiations at frequencies corresponding to the deep peaks of (a) (g = 1014 Hz). The maps are obtained by averaging the field over one period of oscillation. The red (blue) colour corresponds to the highest (lowest) values of the magnetic field given in arbitrary units.
vðv þ igÞ
Here vp is the plasma frequency of the metal, g the absorption coefficient and e1 comes from the
contribution of the bound electrons to the polarizability. For metal Ag used here, vp = 1.36 1016 rad s1, g = 1.0 1014 Hz and e1 = 4.2 [12]. In Fig. 4a we show
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Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31
with solid line the transmission spectrum for a stub with the geometric parameters (w, l) = (a, a). The transmission spectrum displays three dips with attenuation less than 20 dB that confirm the interaction of the waveguide modes with the cavity resonances. The quality factor of the peaks may reach values of the order of 100. Fig. 4b displays the computation of the maps of the magnetic field at the frequencies of the dips, namely V = 0.2052, V = 0.2338 and V = 0.3357. One can notice that the incident field in the waveguide interacts with the stub and then is totally reflected back, the outgoing wave becoming vanishingly small. Depending on the frequency, one can observe inside the stub different number of oscillations in relation with the size of the cavity. In Fig. 4a, we have also investigated the effect of losses on the transmission spectrum by changing the parameter g in a broad range (g = 1013, 1014 and 1015 Hz). We find that for g below or of the order of 1014 Hz, there is no significant modification in the results. On the contrary, when g goes much above 1014 Hz, up to 1015 Hz, the peaks broaden and become less deep. This means that over a reasonable range of the absorption coefficient (g 1014 Hz), the effect of the metal to confine the waves in the stubs can mainly be attributed to the negative value of the real part of its dielectric constant. To give a better insight about the range of validity of the Drude model and its possible application to our structures, we have fitted the experimental values of the real and imaginary parts of the Ag dielectric constant [13] with the Drude model. A good agreement can be obtained for g ranging from 1013 to 1014 Hz when the wavelength varies from 2.5 to 0.5 mm, a range which is even broader than the one we have displayed in Fig. 4a. For example, for a common waveguide thickness a = 0.5 mm, the wavelength in Fig. 4a ranges between 3 and 1.25 mm. For wavelengths lower than 0.5 mm, it is no more possible to fit the dielectric constant, even by increasing the absorption coefficient. This is due to the proximity of the plasmon resonance where the Drude model fails to give an appropriate description of the metal. We have done the same analysis for two other metals which are commonly used in the infrared, namely gold and aluminium. To fit the experimental gold data [13], Drude model with parameters vp = 1.4 1016 rad s1, g = 1.0 1014 Hz and e1 = 7.0 [14] is suitable for wavelengths falling in the range [0.8, 2.5] mm, while it becomes invalid for shorter wavelengths due to the proximity of the resonance. To fit the experimental Al data [15], one can take the following parameters: vp = 2.3 1016 rad s1, e1 = 1.0 and an absorption coefficient g = 2.0 1014 Hz in the wavelength range [1.6, 2.5] mm. One
can notice that this latter material is less adapted for the wavelength range of Fig. 4a with a waveguide thickness of 0.5 mm. 3. Summary In conclusion, we have investigated theoretically the propagation of waves through semiconductor waveguides which refractive index is n = 3.2429 that ensures a large contrast in the properties of the waveguide constituent material and the surrounding medium. Such system, coupled to a microstub cavity covered with a perfect metal layer could be used as a rejective filter. The combination of a set of stubs widens the zeros of transmission into gaps in which a selective tunnelling transmission becomes possible through a defect stub. The rejective or selective frequency can be adjusted by choosing appropriately the width or the length of the stub. Moreover, we have discussed a model of demultiplexer based on a Yshaped waveguide. This system can be utilized for separating signals with different frequencies. The validity of such simulations of the metal as a perfect coating conductor is limited to the far infrared regime, far from the plasma frequency. Nevertheless, the physical trends have been shown to remain possible in the near optical regime by the modelization of the metallic coating in the framework of the Drude model. Acknowledgment This work is partly supported by ‘Le Fonds Europe´en de De´veloppement Re´gional (FEDER)’ under the Belgian-French INTERREG III programme PREMIO. References [1] Y. Pennec, B. Djafari-Rouhani, A. Akjouj, J.O. Vasseur, L. Dobrzynski, J.P. Vilcot, M. Beaugeois, M. Bouazaoui, R. Fikri, J.P. Vigneron, App. Phys. Lett. 89 (2006) 101113. [2] S.C. Hagness, D. Rafizadeh, S.T. Ho, A. Taflove, J. Lightwave Technol. 15 (1997) 2154–2165. [3] R. Grover, P.P. Absil, V. Van, J.V. Hryniewicz, B.E. Little, O. King, L.C. Calhoun, F.G. Johnson, P.T. Ho, Opt. Lett. 26 (2001) 506. [4] T.J. Johnson, M. Borselli, O. Painter, Opt. Express 14 (2006) 817. [5] G.P. Agrawal, S. Radic, IEEE Photon. Technol. Lett. 6 (1994) 995. [6] J.C. Chen, H.A. Haus, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, J. Lightwave Technol. 14 (1996) 2575. [7] K. Ogusu, K. Takayama, Opt. Lett. 32 (2007) 2185.
Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31 [8] A. Beaurain, S. Dupont, H.-W. Li, J.-P. Vilcot, C. Legrand, J. Harari, M. Constant, D. Decoster, Microwave Opt. Technol. Lett. 40 (2004) 216–218. [9] K.S. Yee, IEEE Trans. Antennas Propagat. AP-14 (1966) 302– 307. [10] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Norwood, MA, 1995.
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[11] J.P. Berenger, J. Comput. Phys. 114 (1994) 185. [12] H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, H. Yao, Opt. Express 13 (2005) 10795. [13] P.B. Johnson, R.W. Christy, Phys. Rev. B 6 (1972) 4370. [14] A.K. Sharma, B.D. Gupta, Nanotechnology 17 (2006) 124. [15] E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, 1997.
Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31 www.elsevier.com/locate/photonics
Microstubs resonators integrated to bent Y-branch waveguide Y. Pennec a,*, M. Beaugeois b, B. Djafari-Rouhani a, R. Sainidou a, A. Akjouj a, J.O. Vasseur a, L. Dobrzynski a, E.H. El Boudouti a, J.-P. Vilcot a, M. Bouazaoui b, J.-P. Vigneron c a
Institut d’Electronique, de Microe´lectronique et de Nanotechnologie, UMR CNRS 8520, Universite´ de Lille1, BP 60069, 59652 Villeneuve d’Ascq Cedex, France b Laboratoire PHLAM, UMR CNRS 8523, Universite´ de Lille1, 59655 Villeneuve d’Ascq Cedex, France c Laboratoire de Physique du Solide, Faculte´s Universitaires Notre Dame de la Paix, Rue de Bruxelles 61, 5000 Namur, Belgium Received 13 June 2007; received in revised form 9 January 2008; accepted 14 January 2008 Available online 19 January 2008
Abstract We report numerical simulations, based on a finite difference time domain (FDTD) method, of light propagation in twodimensional semiconductor micro-optical waveguides coupled to one or several lateral stubs. It is shown that when the stub is covered with a perfectly metallic thin layer, the transmission spectrum contains several narrow dips. Such simulation of the metallic coating can be used in the far infrared frequency domain, far from the optical regime. We propose a selective filtering device based on the interaction between several stubs. Inserting an appropriate defect stub between a set of periodical stubs leads to a tunnelling transmission, with a narrow peak inside the gap. This filtering phenomenon is used to propose a demultiplexer based on a Y-shaped waveguide for separating signals with different frequencies. Finally, we show that the filtering effect of a stub can also be reproduced when the metal is described in the frame of a Drude model instead of being perfect, which makes plausible the realization of the above devices in the near optical regime. # 2008 Elsevier B.V. All rights reserved. PACS : 41.20.-q; 42.25.Bs; 42.79.Gn; 42.79.Sz; 42.82.Bq; 42.82.Et Keywords: Photonic waveguide; Resonator; Stub; FDTD; Filter; Demultiplexer
1. Introduction We report a theoretical study of microcavity resonator devices based on semiconductor waveguides with a very large lateral refractive index contrast (air– semiconductor–air), using 2D-FDTD numerical simulations. In a previous publication [1], we have shown that the transmission coefficient through a twodimensional semiconductor micro-waveguide coupled to a rectangular cavity grafted on each side of the guide * Corresponding author. Tel.: +33 3 20 43 68 07. E-mail address: [email protected] (Y. Pennec). 1569-4410/$ – see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2008.01.002
displays several narrow dips. Such a cavity, called a stub, was covered with a perfectly metallic thin layer to prevent the radiation to escape outside from the boundaries of the stub. Such a device could easily be used in the far infrared frequency domain (far from the plasma frequency) where the metal behaves as a perfect mirror. It is then believed that, owning to their small size comparatively to other resonators such as microdisks or microrings [2–4], microstubs could find original applications in future optoelectronic integrated circuits. In this paper, we report the study of a couple of functionalities of microstub resonator-based devices, already studied in the literature for different systems
Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31
such as fiber Bragg gratings [5], photonic band gap air bridges [6] and more recently 2D-photonic crystal materials [7]. One aim is to propose the filtering of specific wavelength from a large band input signal, either in rejection or in selection, by coupling the guide to a set of stubs. Then, we use this filtering process to investigate the possibility of a new device for wavelength demultiplexing based on a bent Y-shaped component. Furthermore, we show that the main features of the transmission spectrum through a waveguide coupled to a lateral stub remain valid if the metal covering the stub is not perfect, but described in the framework of the Drude model. This makes plausible the extension of the above applications to the near optical regime. 2. Results and discussion The geometry of the device is displayed in Fig. 1a, where w and l are, respectively, the width and the length of the stub and a is the thickness of the waveguide. The thickness of the metal is d = a/5. The waveguide and the
Fig. 1. (a) Schematic representation of a bus waveguide coupled to a stub covered with a thin metallic layer and (b) transmission spectrum through the above system when the coated metal is perfect and the dimensions of the stub are (w, l) = (a, 1.4a).
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stub are made of a semiconductor structure [8], which refractive index is n = 3.2429 and the embedding medium is air. This choice ensures a large contrast in the properties of the waveguide constituent material and the surrounding medium. The transmission through the waveguide is calculated by using the finite difference time domain (FDTD) method [9,10] with perfectly matching method as boundary conditions [11]. The incoming wave is created inside the taper region by an initial current source parallel to the x-axis, with a Gaussian profile along the x-direction. We choose to study the transverse magnetic polarization with the electric field ~ E belong~ along z. ing to the (x, y) plane and the magnetic field H All the transmission spectra, probed at the right end of the waveguide, are normalized with respect to the one corresponding to a perfect (without stub) waveguide. The transmission is reported in dB as a function of the dimensionless frequency V = va/(2pc) = a/l where l is the wavelength of light in vacuum. In Fig. 1b, we remind the calculated transmission for a stub grafted on the waveguide with the following parameters: w ¼ a and l = 1.4a [1]. The microstub is covered with a perfectly metallic thin layer that prevents the radiation to escape outside from the boundaries of the stub. The hypothesis of a perfect metal is valid in the far infrared frequency range, far from the plasma frequency of the metal. The results are given for V in the range 0.2–0.4 which ensures the waveguide to be in the monomode regime according to the (analytical) solutions of the dispersion curves for the straight waveguide. The transmission spectrum displays several dips with attenuation exceeding 20 dB. In Ref. [1], we have shown that for various lengths and widths of the stub, the frequencies of the dips can be favourably fitted with the eigenfrequencies Vg of a rectangular electromagnetic cavity given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2ffi 1 a a m21 (1) þ m22 0 Vg ¼ 2n w l where n is the refractive index and m1 and m2 are two positive integers that define the order of the modes. While the width of the stub should be clearly w ¼ a in the analytical model, the length l0 remains somewhat undefined because the boundary conditions at the lower face of the cavity can be applied somewhere between the middle and the top of the waveguide; this means that l0 can be chosen between l0 = l and l0 = l + 0.5a. A good agreement was obtained between the numerical results and the eigenfrequencies of the cavity (see Table 1) by choosing l0 = 1.5a.
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Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31
Table 1 Comparison between the reduced resonance frequencies obtained by the FDTD simulation and by the eigenfrequencies calculated from Eq. (1) Vg (FDTD Fig. 1b)
Vg (Eq. (1))
(m1, m2)
0.256 0.322 0.329 0.362 0.415
0.257 0.325 0.344 0.370 0.430
(1, (2, (1, (2, (2,
2) 1) 3) 2) 3)
The numbers (m1, m2) give the order of the modes.
Furthermore, we also checked in a few examples that quite similar results are obtained in the case of a 3D rectangular waveguide coupled to a lateral stub, so we restricted the numerical simulations only to 2D structures for the rest of the paper. We first investigate the transmission coefficient through the waveguide when we increase the number N of the stubs from one to five (Fig. 2a), with a separation distance of 2a between the stubs. Increasing N leads to the widening of the dips and opening of band gaps. We concentrate here on the gap (Fig. 2b) occurring in the frequency range [0.240, 0.260] with an attenuation of the transmission exceeding 20 dB. We now change the
Fig. 2. (a) Schematic representation of the waveguide coupled to five stubs (w, l) = (a, 1.4a) where the length of the center stub is increased with respect to the others and (b) transmission spectra calculated for the above model for two different lengths (l = 1.5a and l = 1.54a) of the defect stub and showing the evolution of a narrow pass band inside the gap.
geometrical parameters of the third stub in order to make a ‘‘defect’’ inside the periodical chain of stubs (Fig. 2a). The defect stub could be created by a modification of the width or the length of the stub. For instance, we choose to increase the length of the defect stub from l = 1.4a to l = 1.5a which has the effect of decreasing its resonance frequency with respect to that associated with the other stubs in the chain, therefore ensuring that the resonance frequency falls inside the gap. The transmission spectrum of such a structure incorporating a defect (solid line in Fig. 2b) displays one narrow peak centred at frequency Vg = 0.2527 inside the gap. This corresponds to the tunnelling of the wave through the localized mode associated with the defect. As a result, such a device could be used as a selective filter. Furthermore, it is possible to displace the narrow frequency band inside the gap by adjusting the length or the width of the defect stub. In Fig. 4b, the transmission spectrum is presented for another length of the stub, l = 1.54a (dotted line in Fig. 2b) which shows a downward shift of the peak with a central frequency Vg = 0.2518. We have thus created a tunable selective filter at the reduced frequencies Vg = 0.2518 and Vg = 0.2527. The second purpose of the paper is to propose a Yshaped demultiplexer, based on microstubs cavities (Fig. 3a). The main goal is to select different frequencies on each branch of the Y-structure from a broad band signal. As explained below, the first step in the selective filter constitution is to associate microstubs in a periodic structure. The second step of the selective filter realisation is to place, on each branch, a ‘defect’ stub in the middle of the periodic stub arrangement as illustrated in Fig. 3a. In this model, we replace on the T1 (resp. T2) branch the third stub by a new one with the geometrical parameters (l, w) = (a, 1.54a) (resp. (l, w) = (a, 1.5a)). The calculated map of the magnetic field (Fig. 3b) at the monochromatic frequencies Vg = 0.2518 (resp. Vg = 0.2527) shows a strong enhancement of the field at the defect stub and its nearest neighbours into the T1 (T2) branch. Each selected frequency (Vg = 0.2518 and Vg = 0.2527) is then available at the end of the corresponding port (T1 and T2). At the same frequency, the regular periodic arrangement of the stubs acts as a stop band and no signal is available at port T2 for Vg = 0.2518 and at port T1 for Vg = 0.2527. As a result, Fig. 3 shows clearly that the large band input Gaussian signal T0 has been separated and directed towards the two branches of the Y-junction. It is also worth noticing that each branch of the Y-shaped waveguide will select its own narrow pass band. This example could also be extended to a multichannel device.
Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31
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Fig. 3. (a) Schematic representation of the Y-shaped waveguide in which the branches are coupled to five stubs where the length of the center stub is increased with respect to the others: l = 1.54a for T1 and l = 1.50a for T2 and (b) map of the magnetic field for two monochromatic incident radiations at the frequencies of the transmission zero Vg = 0.2527 and Vg = 0.2518.
The above applications were discussed by assuming that the stubs are covered with a perfect metal layer. The hypothesis of a perfect metal is well adapted to infrared frequencies but not to optical frequencies where the true behaviour of the metal cannot be omitted. Nevertheless, this hypothesis has the advantage of reducing considerably the simulation time, since no particular discretization is required in the metallic part of the structure. In the case of a real metal, the discretization should be performed at the scale of the skin depth, which means a much smaller mesh than in the case of the semiconductor waveguide. To show that the above applications should possibly be also feasible at the telecommunication wavelength, we have studied the transmission through a waveguide coupled to a single stub covered with a real metal whose dielectric function is described by the Drude model: eðvÞ ¼ e1
v2p
Fig. 4. (a) Transmission spectrum through the system of Fig. 1a when the metal is real and the dimensions of the stub are (w, l) = (a, a). The transmission curves are plotted for a value of the absorption coefficient g = 1014 Hz and (b) maps of the magnetic field (FDTD model) for three monochromatic incident radiations at frequencies corresponding to the deep peaks of (a) (g = 1014 Hz). The maps are obtained by averaging the field over one period of oscillation. The red (blue) colour corresponds to the highest (lowest) values of the magnetic field given in arbitrary units.
vðv þ igÞ
Here vp is the plasma frequency of the metal, g the absorption coefficient and e1 comes from the
contribution of the bound electrons to the polarizability. For metal Ag used here, vp = 1.36 1016 rad s1, g = 1.0 1014 Hz and e1 = 4.2 [12]. In Fig. 4a we show
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Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31
with solid line the transmission spectrum for a stub with the geometric parameters (w, l) = (a, a). The transmission spectrum displays three dips with attenuation less than 20 dB that confirm the interaction of the waveguide modes with the cavity resonances. The quality factor of the peaks may reach values of the order of 100. Fig. 4b displays the computation of the maps of the magnetic field at the frequencies of the dips, namely V = 0.2052, V = 0.2338 and V = 0.3357. One can notice that the incident field in the waveguide interacts with the stub and then is totally reflected back, the outgoing wave becoming vanishingly small. Depending on the frequency, one can observe inside the stub different number of oscillations in relation with the size of the cavity. In Fig. 4a, we have also investigated the effect of losses on the transmission spectrum by changing the parameter g in a broad range (g = 1013, 1014 and 1015 Hz). We find that for g below or of the order of 1014 Hz, there is no significant modification in the results. On the contrary, when g goes much above 1014 Hz, up to 1015 Hz, the peaks broaden and become less deep. This means that over a reasonable range of the absorption coefficient (g 1014 Hz), the effect of the metal to confine the waves in the stubs can mainly be attributed to the negative value of the real part of its dielectric constant. To give a better insight about the range of validity of the Drude model and its possible application to our structures, we have fitted the experimental values of the real and imaginary parts of the Ag dielectric constant [13] with the Drude model. A good agreement can be obtained for g ranging from 1013 to 1014 Hz when the wavelength varies from 2.5 to 0.5 mm, a range which is even broader than the one we have displayed in Fig. 4a. For example, for a common waveguide thickness a = 0.5 mm, the wavelength in Fig. 4a ranges between 3 and 1.25 mm. For wavelengths lower than 0.5 mm, it is no more possible to fit the dielectric constant, even by increasing the absorption coefficient. This is due to the proximity of the plasmon resonance where the Drude model fails to give an appropriate description of the metal. We have done the same analysis for two other metals which are commonly used in the infrared, namely gold and aluminium. To fit the experimental gold data [13], Drude model with parameters vp = 1.4 1016 rad s1, g = 1.0 1014 Hz and e1 = 7.0 [14] is suitable for wavelengths falling in the range [0.8, 2.5] mm, while it becomes invalid for shorter wavelengths due to the proximity of the resonance. To fit the experimental Al data [15], one can take the following parameters: vp = 2.3 1016 rad s1, e1 = 1.0 and an absorption coefficient g = 2.0 1014 Hz in the wavelength range [1.6, 2.5] mm. One
can notice that this latter material is less adapted for the wavelength range of Fig. 4a with a waveguide thickness of 0.5 mm. 3. Summary In conclusion, we have investigated theoretically the propagation of waves through semiconductor waveguides which refractive index is n = 3.2429 that ensures a large contrast in the properties of the waveguide constituent material and the surrounding medium. Such system, coupled to a microstub cavity covered with a perfect metal layer could be used as a rejective filter. The combination of a set of stubs widens the zeros of transmission into gaps in which a selective tunnelling transmission becomes possible through a defect stub. The rejective or selective frequency can be adjusted by choosing appropriately the width or the length of the stub. Moreover, we have discussed a model of demultiplexer based on a Yshaped waveguide. This system can be utilized for separating signals with different frequencies. The validity of such simulations of the metal as a perfect coating conductor is limited to the far infrared regime, far from the plasma frequency. Nevertheless, the physical trends have been shown to remain possible in the near optical regime by the modelization of the metallic coating in the framework of the Drude model. Acknowledgment This work is partly supported by ‘Le Fonds Europe´en de De´veloppement Re´gional (FEDER)’ under the Belgian-French INTERREG III programme PREMIO. References [1] Y. Pennec, B. Djafari-Rouhani, A. Akjouj, J.O. Vasseur, L. Dobrzynski, J.P. Vilcot, M. Beaugeois, M. Bouazaoui, R. Fikri, J.P. Vigneron, App. Phys. Lett. 89 (2006) 101113. [2] S.C. Hagness, D. Rafizadeh, S.T. Ho, A. Taflove, J. Lightwave Technol. 15 (1997) 2154–2165. [3] R. Grover, P.P. Absil, V. Van, J.V. Hryniewicz, B.E. Little, O. King, L.C. Calhoun, F.G. Johnson, P.T. Ho, Opt. Lett. 26 (2001) 506. [4] T.J. Johnson, M. Borselli, O. Painter, Opt. Express 14 (2006) 817. [5] G.P. Agrawal, S. Radic, IEEE Photon. Technol. Lett. 6 (1994) 995. [6] J.C. Chen, H.A. Haus, S. Fan, P.R. Villeneuve, J.D. Joannopoulos, J. Lightwave Technol. 14 (1996) 2575. [7] K. Ogusu, K. Takayama, Opt. Lett. 32 (2007) 2185.
Y. Pennec et al. / Photonics and Nanostructures – Fundamentals and Applications 6 (2008) 26–31 [8] A. Beaurain, S. Dupont, H.-W. Li, J.-P. Vilcot, C. Legrand, J. Harari, M. Constant, D. Decoster, Microwave Opt. Technol. Lett. 40 (2004) 216–218. [9] K.S. Yee, IEEE Trans. Antennas Propagat. AP-14 (1966) 302– 307. [10] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Norwood, MA, 1995.
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[11] J.P. Berenger, J. Comput. Phys. 114 (1994) 185. [12] H. Gao, H. Shi, C. Wang, C. Du, X. Luo, Q. Deng, Y. Lv, X. Lin, H. Yao, Opt. Express 13 (2005) 10795. [13] P.B. Johnson, R.W. Christy, Phys. Rev. B 6 (1972) 4370. [14] A.K. Sharma, B.D. Gupta, Nanotechnology 17 (2006) 124. [15] E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, 1997.