- Email: [email protected]
Electro-optical modulation in two-dimensional photonic crystal linear waveguides
Physica E 46 (2012) 177–181
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Electro-optical modulation in two-dimensional photonic crystal linear waveguides Cheng-Yang Liu n Department of Mechanical and Electro-Mechanical Engineering, Tamkang University, New Taipei City, Taiwan
H I G H L I G H T S c
c
c
We report the electro-optical modulation in a 2-D photonic crystal linear waveguide. Line waveguides can be obtained by reducing the radius of a line of rods. The switching mechanism is a change in the conductance in the waveguide region.
G R A P H I C A L
A B S T R A C T
We present the electro-optical modulation in two-dimensional photonic crystal linear waveguides.
a r t i c l e i n f o
abstract
Article history: Received 20 October 2011 Received in revised form 6 August 2012 Accepted 14 September 2012 Available online 24 September 2012
The electro-optical modulation in a two-dimensional photonic crystal linear waveguide is presented. In order to create a linear waveguide, the radius of a line of rods is reduced. The linear waveguide composed of a dielectric cylinder in air is studied by solving Maxwell’s equations using the plane wave expansion method and the finite-difference time–domain method. The switching mechanism is a change in the conductance in the waveguide region and hence modulating the guided modes and eventually switching is achieved. Such a mechanism of modulation should open up a new application for designing tunable components in photonic integrated circuits. & 2012 Elsevier B.V. All rights reserved.
1. Introduction Recently, there has been growing interest in studies of the propagation of electromagnetic waves in periodic dielectric structures [1]. These structures exhibit forbidden frequency regions where electromagnetic waves cannot propagate for both polarizations along any direction. This may bring peculiar physical phenomena to control lightwave and construct integrated optical devices. An important element of optical circuits is a linear waveguide to carry lightwave and photonic crystals (PCs) provide unique advantages for linear waveguides. The electromagnetic
n Correspondence address: No. 151, Ying-chuan Road, Tamsui District, New Taipei City, Taiwan. Tel.: þ886 2 26215656x2061; fax: þ 886 2 26209745. E-mail address: [email protected]
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.09.018
wave is strongly confined in the defect channel, allowing low loss in linear waveguides and sharp bends [2–5]. Such structures can be used to design highly efficient new optical devices. Optical waveguides in two-dimensional (2-D) PCs produced by insertion of linear defects into PC structures had been proposed [6] and experimentally proved [7]. It is important to obtain tunable PC waveguides for applications in optical devices. Tunable PC structures that utilize synthetic opals and inverse opals infiltrated with functional materials have been proposed [8–10]. The photonic bandgap of a 2-D PC is continuously varied using the temperature dependent refractive index of a liquid crystal (LC) [11]. The propagation of tunable lightwave in Y-shaped waveguides in 2-D PCs by use of LCs and semiconductors as linear defects is discussed [12,13]. A tunable PC waveguide coupler based on nematic LCs is presented by the authors [14], and then we have investigated the tunable bandgap
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C.-Y. Liu / Physica E 46 (2012) 177–181
in a 2-D PC modulated by a nematic LC [15]. These results can be used to obtain a tunable field-sensitive polarizer [16]. Electrooptical switches are the key components of such photonic integrated circuits. A PC channel waveguided 2 2 directional coupler switch that utilizes electrically or optically induced loss in the coupling region between coupled waveguides has been proposed and analyzed [17]. In this paper, we theoretically demonstrate the electro-optical modulation in a 2-D PC linear waveguide. To create a linear waveguide in the PC structure, the radius of a line of rods is reduced. Using a plane wave expansion method (PWE), we have solved Maxwell’s equations for the guided modes of electromagnetic waves in a periodic arrangement of dielectric cylinders. The guided modes can be controlled by changing the conductance in the waveguide region. The propagation properties of electromagnetic waves inside the linear waveguide are calculated by means of finite-difference time–domain simulations (FDTD). The switching mechanism is a change in the conductance in the waveguide region and modulating the guided modes. Such a mechanism of optical modulation should open up a new application for designing components in photonic integrated circuits. Details of the calculations and discussion of the results will be presented in the remainder of the paper.
2. Numerical method The PWE method is illustrated in several papers [2,18]. Here we summarize the theory very briefly. Following the discussion of previous literature, we can express the lightwave equation that is satisfied by the magnetic field in order to determine the photonic bandgaps of periodic structures as 2 1 o r rHðrÞ ¼ HðrÞ ð1Þ eðrÞ c where rHðrÞ ¼ 0. The dielectric tensor eðrÞ ¼ eðr þRÞ is periodic with respect to the lattice vectors R generated by primitive translation. The main numerical problem in obtaining the eigenvalue is the evaluation of the Fourier coefficients of the inverse dielectric tensors. The best method is to calculate the matrix of Fourier coefficients of real space tensors and take its inverse in order to obtain the required Fourier coefficients. This method (HCS) was shown by Ho et al. [18]. The eigenvalues computed with the HCS method for 512 plane waves are estimated to be in error less than 1%. In our calculations, the convergence is quite fast for lowenergy bands. We consider a square lattice of dielectric cylinders in air and examine transverse electric (TE) mode where electric field is parallel to the axis of cylinders. PWE method is used to obtain photonic band structures and guided modes. The electromagnetic wave propagation is simulated by using the FDTD method [19]. FDTD method is a powerful, accurate numerical method that permits computer-aided design and simulation of PC structures. The photonic device is laid out in the x–y plane. The propagation is along the x direction. The computational domain used is rectangular and is bounded by a perfectly matching layer to minimize back reflections. The space steps in the x and y directions are Dx and Dy, respectively. We assumed that Dx and Dy ¼0.05. The sampling in time is selected to ensure numerical stability of the algorithm. A more detailed treatment of the FDTD method is given in Ref. [19]. 3. Electro-optical modulation We consider that the 2-D square lattice PC is composed of dielectric cylinders surrounded by air. The material is
Fig. 1. Schematic view of the proposed 2-D photonic crystal linear waveguide. The linear waveguide is formed by reducing the radius of dielectric rods of a single row whose radius is ri o r.
homogeneous in the z direction, and periodic along x and y with lattice constant. The lattice constant is a and the radius of cylinders is r¼ 0.2a. The refractive index of the dielectric cylinders is n ¼3.4 (Si). This structure has a photonic bandgap for TE modes ranging from 0.2872 to 0.4204 in normalized frequency units (oa/ 2pc), where o is the angular frequency and c is the light velocity in the free space. In this structure, the waveguides made by reducing the radius of a line of rods are inherently single mode because total internal reflection modes are prevented to exist. Fig. 1 depicts the proposed 2-D photonic crystal linear waveguide. The linear waveguide is formed by reducing the radius of dielectric rods of a single row whose radius is ri or. The dispersion relation of the linear waveguide is obtained by using the PWE method. The guided modes in the linear waveguide are computed using a supercell approximation, which consists of placing a large crystal with a defect into a supercell and repeating it periodically in space. In the case below, the supercell contains a 1 7 crystal. In this paper, we propose the loss tangent of dielectric material in the waveguide region can be modified by external commands to spoil the guiding, thereby modulating the lightwave. This is a Da switch in which the change in optical absorption coefficient Da is employed. The change in conductance Ds is proportional to Da. The behavior is analogous to that discussed in Ref. [20] where electric absorption was assumed to reduce the Q factor of microring resonators coupled to strip channel waveguides. Fig. 2 shows the dispersion relation of the guided modes of the linear waveguide at ri ¼0.05a for different conductance s values. The inset of Fig. 2 shows the supercell for this linear waveguide. Shaded regions correspond to the projected band structures of the perfect PCs. It is observed that the guided modes arising from a single waveguide appear inside the photonic bandgap. Inside the photonic bandgap, there exist two kinds of dispersion relations at certain conductance values and there exist cutoff frequencies of guide modes. The cutoff frequencies at s ¼1 10 1 S/cm and 1 106 S/cm are 0.2948 and 0.3864, respectively. For the 1550 nm center wavelength, we assumed a 2-D PC of 217 nm diameter dielectric rods arrayed in a square lattice a¼542.5 nm in the air background. Therefore, the cutoff wavelength lcutoff at s ¼1 10 1 S/cm and 1 106 S/cm are 1840.23 nm and 1403.98 nm, respectively. In order to let the incident lightwave propagate in the linear waveguide, incident wavelength linc%lcutoff must be satisfied. Fig. 3 shows the normalized power response as a function of incident wavelength at ri ¼0.05a for conductivities s ¼1 10 1 S/cm and 1 106 S/cm. The transmissions are in agreement with dispersion relations in Fig. 2. At s ¼1 10 1 S/cm, the incident wavelength linc ¼1550 nm can propagate in the linear waveguide because linc%lcutoff is satisfied. At s ¼ 1 106 S/cm, the incident wavelength linc ¼1550 nm cannot propagate in the linear waveguide because linc%lcutoff is not satisfied. Hence, the cutoff frequencies
C.-Y. Liu / Physica E 46 (2012) 177–181
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Fig. 4. Normalized power response as a function of conductivity of the linear waveguide with ri ¼0.05a at input wavelength 1550 nm.
Fig. 2. Dispersion relation of the guided modes of the linear waveguide at ri ¼ 0.05a for different conductance s values: s ¼ 1 10 1 S/cm (solid line) and s ¼ 1 106 S/cm (dashed line). Shaded regions correspond to the projected band structures of the perfect photonic crystals. The inset shows the supercell for this linear waveguide.
Fig. 3. Normalized power response as a function of incident wavelength at ri ¼ 0.05a for different conductances: s ¼ 1 10 1 S/cm (solid line) and s ¼ 1 106 S/cm (dashed line).
of PC linear waveguides depend on the conductance in the waveguide region and lightwave propagation in linear waveguides can be controlled by changing the conductance. In this paper, we consider that the 2-D square lattice PC is composed of dielectric cylinders surrounded by air. This perfect PC has complete photonic bandgap for the TE modes, but not for the TM modes. Therefore, we cannot obtain the guided modes in the PC with square lattice for the TM modes. By combining our observations, we can design a PC that has photonic bandgaps for both polarizations. By adjusting the dimensions of the lattice, we can even arrange for the photonic bandgaps to overlap, resulting in an absolute bandgap for all polarizations [21]. If we can obtain the guided modes in an absolute bandgap, the guided modes can be controlled by changing the conductance for all polarizations. Fig. 4 depicts the normalized power response as a function of conductivity of the linear waveguide with ri ¼0.05a at input wavelength 1550 nm. For linc ¼1550 nm, the maximum of output
Fig. 5. Field patterns observed in frequency domain of the electro-optical linear waveguide at input wavelength l ¼ 1550 nm and ri ¼ 0.05a for conductance (a) s ¼1 10 1 S/cm, (b) s ¼5 102 S/cm and (c) s ¼ 1 106 S/cm.
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power is 3.42 dB at s ¼1 10 1 S/cm and the minimum of output power is 188.23 dB at s ¼1 106 S/cm. The results provide a nice illustration of how the linear waveguide can make it possible to control by changing the conductivity of the materials. The power response in Fig. 4 shows that there is a specific value of optical power at output port for a specific value of conductivity (s ¼5 102 S/cm), at this transient value the optical power launched at the input port will be absorbed gradually in the waveguide region and the device suffer high attenuation coefficient a in the waveguide region [22]. An increase or decrease in the conductivity will redirect the optical power to either
transmission or absorption respectively. The FDTD method is used to solve the light propagation in 2-D PC linear waveguide. Fig. 5 shows the field patterns observed in frequency domain of the electro-optical linear waveguide at input wavelength l ¼1550 nm and ri ¼0.05a for different conductance. Fig. 5(a) shows that lightwave can propagate in the linear waveguide at s ¼1 10 1 S/cm. Fig. 5(c) shows that lightwave cannot propagate in the linear waveguide at s ¼1 106 S/cm. These results would provide a basis for the novel application of switching devices in photonic circuits.
Fig. 6. Schematic view of the proposed 2-D photonic crystal waveguide branches. The linear waveguide is formed by reducing the radius of dielectric rods of a single row whose radius is ri o r.
Fig. 7. Normalized power response as a function of conductivity of the waveguide branches with ri ¼0.05a at input wavelength 1550 nm.
Fig. 8. Field patterns observed in frequency domain of the electro-optical waveguide branches at input wavelength l ¼1550 nm and ri ¼ 0.05a for conductance (a) s ¼ 1 10 1 S/cm, (b) s ¼ 5 102 S/cm and (c) s ¼1 106 S/cm.
C.-Y. Liu / Physica E 46 (2012) 177–181
To achieve optical modulation in 2-D PC waveguides for Si/air or Si/SiO2 alternating materials, the free carrier absorption loss of Si can be controlled by three ways [23]. The first way is carrier injection from forward biased PN junctions on the rods. The second way is depletion of doped rods with metal–oxide–semiconductor gates. The third way is generation of electrons and holes by above gap light shining upon the designated rods. If the PC waveguide is implemented in III–V semiconductor hetero-layers, the electric absorption effect could be used. The dependence of the conductance upon doping density is shown in [23]. The prior PC switching device relies upon a point defect resonator or two point defects located between two PC waveguides [24]. They assumed that the Q factor of cavities would be spoiled by loss induced electrically at the defects. The realization of a thermo-optically controlled Mach– Zehnder interferometer based on an AlGaAs/GaAs epitaxial waveguide operating at wavelength of 1550 nm is reported [25]. The fairly low switching power required for the tunable PC device has demonstrated its potential for use in future low cost and compact optical circuits.
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researched for enhancing transmission through waveguide region and reducing reflections.
5. Conclusion We have demonstrated numerically the electro-optical modulation in a 2-D PC linear waveguide with square lattice. In order to create a linear waveguide, the radius of a line of rods is reduced. The operation of the waveguides is controlled by optical loss induced in the dielectric rods in the linear waveguides. Using the FDTD method on a 1550 nm device, we predict low insertion loss in the different switching states although the required change in conductance is large in these non-optimized waveguides. Such an electro-optical mechanism should open up a new application for designing tunable components in photonic integrated circuits. Further theoretical investigations and experimental efforts are needed to bring the tunable linear waveguide into reality.
Acknowledgments 4. Tunable waveguide branches The tunable electro-optical waveguide branches designed by the proposed scheme are depicted in Fig. 6. The dielectric cylinders with the 2-D square lattice and a radius r ¼0.2a are surrounded by air. The waveguides are introduced by reducing the radius of a line of rods in the crystal. The FDTD method is also used to solve the lightwave propagation in 2-D PC waveguide branches. For the T-shaped waveguide shown in Fig. 6, the transmission spectrum thus obtained are shown in Fig. 7. The lightwave of wavelength 1550 nm is launched into input port and the temporal variation of the electromagnetic field in each output port is monitored. The transmission spectra are a function of the conductivity of the waveguide. At linc ¼1550 nm, the optical powers received in the two output ports are identical at s ¼1 10 1 S/cm and the transmittance is 6.96 dB per each output waveguide. When conductance is s ¼ 1 106 S/cm, the output power is almost null and the transmittance is 158.03 dB per each output waveguide. Fig. 8 shows field patterns observed in frequency domain of the electro-optical waveguide branches at input wavelength l ¼1550 nm and ri ¼0.05a for different conductances. Fig. 8(a) shows that the fields are completely confined within the waveguide regions and split equally into the output waveguides at s ¼1 10 1 S/cm. Fig. 8(c) shows that lightwave cannot propagate in the waveguide branches at s ¼1 106 S/cm. We can see that the proposed electro-optical waveguides can not only be a tunable linear waveguide but also be a tunable waveguide branch by controlling the conductivity of waveguides. The waveguides are intended to be interconnected and cascaded in the forward direction into an optical cross-connect network. In this case, further optimization to transmission can be achieved by minimizing the reflections at the waveguide region and waveguide bends. The broadband and narrowband techniques will be
This work was supported in part by the National Science Council of Taiwan under Grant NSC 101-2221-E-032-009. References [1] E. Yablonovitch, Physical Review Letters 58 (1987) 2059. [2] J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light, Princeton University Press, New Jersey, 1995. [3] H. Benisty, Journal of Applied Physics 79 (1996) 7483. [4] A. Mekis, S. Fan, J.D. Joannopoulos, Physical Review B 58 (1998) 4809. [5] E. Centeno, D. Felbacq, Optics Communications 160 (1999) 57. [6] R.D. Meade, A. Devenyi, J.D. Joannopoulos, O.L. Alerhand, D.A. Smith, K. Kash, Journal of Applied Physics 75 (1994) 4753. [7] T. Baba, N. Fukaya, J. Yonekura, Electronics Letters 35 (1999) 654. [8] K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, M. Ozaki, Applied Physics Letters 75 (1999) 932. [9] H. Takeda, K. Yoshino, Journal of Applied Physics 92 (2002) 5658. [10] S. Kubo, Z.Z. Gu, K. Takahashi, A. Fujishima, H. Segawa, O. Sato, Journal of the American Chemical Society 126 (2004) 8314. [11] S.W. Leonard, J.P. Mondia, H.M. van Driel, O. Toader, S. John, K. Busch, ¨ A. Birner, U. Gosele, V. Lehmann, Physical Review B 61 (2000) R2389. [12] H. Takeda, K. Yoshino, Physical Review B 67 (2003) 073106. [13] H. Takeda, K. Yoshino, Optics Communications 219 (2003) 177. [14] C.Y. Liu, L.W. Chen, IEEE Photonics Technology Letters 16 (2004) 1849. [15] C.Y. Liu, L.W. Chen, Physical Review B 72 (2005) 045133. [16] C.Y. Liu, L.W. Chen, Optics Communications 256 (2005) 114. [17] A. Sharkawy, S. Shi, D.W. Prather, R. Soref, Optics Express 10 (2002) 1048. [18] K.M. Ho, C.T. Chan, C.M. Soukoulis, Physical Review Letters 65 (1990) 3152. [19] A. Taflove, S.C. Hagness, Computational Electrodynamics: The Finite Difference Time Domain Method, Artech House, Boston, 1998. [20] R.A. Soref, B.E. Little, IEEE Photonics Technology Letters 10 (1998) 1121. [21] C.Y. Liu, Physics Letters A 373 (2009) 3061. [22] R.A. Soref, B.R. Bennett, IEEE Journal of Quantum Electronics QE-23 (1987) 123. [23] S.M. Sze, Physics of Semiconductor Devices, John Wiley & Sons, New Jersey, 1981. [24] S. Fan, P.R. Villeneuve, J.D. Joannopoulos, B.E. Little, H.A. Haus, High Efficiency Channel Drop Filter With Absorption-Induced On/Off Switching and Modulation, US Patent 6101300, August 8, 2000. [25] E. Camargo, H. Chong, R. De La Rue, Optics Express 12 (2004) 588.
Contents lists available at SciVerse ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Electro-optical modulation in two-dimensional photonic crystal linear waveguides Cheng-Yang Liu n Department of Mechanical and Electro-Mechanical Engineering, Tamkang University, New Taipei City, Taiwan
H I G H L I G H T S c
c
c
We report the electro-optical modulation in a 2-D photonic crystal linear waveguide. Line waveguides can be obtained by reducing the radius of a line of rods. The switching mechanism is a change in the conductance in the waveguide region.
G R A P H I C A L
A B S T R A C T
We present the electro-optical modulation in two-dimensional photonic crystal linear waveguides.
a r t i c l e i n f o
abstract
Article history: Received 20 October 2011 Received in revised form 6 August 2012 Accepted 14 September 2012 Available online 24 September 2012
The electro-optical modulation in a two-dimensional photonic crystal linear waveguide is presented. In order to create a linear waveguide, the radius of a line of rods is reduced. The linear waveguide composed of a dielectric cylinder in air is studied by solving Maxwell’s equations using the plane wave expansion method and the finite-difference time–domain method. The switching mechanism is a change in the conductance in the waveguide region and hence modulating the guided modes and eventually switching is achieved. Such a mechanism of modulation should open up a new application for designing tunable components in photonic integrated circuits. & 2012 Elsevier B.V. All rights reserved.
1. Introduction Recently, there has been growing interest in studies of the propagation of electromagnetic waves in periodic dielectric structures [1]. These structures exhibit forbidden frequency regions where electromagnetic waves cannot propagate for both polarizations along any direction. This may bring peculiar physical phenomena to control lightwave and construct integrated optical devices. An important element of optical circuits is a linear waveguide to carry lightwave and photonic crystals (PCs) provide unique advantages for linear waveguides. The electromagnetic
n Correspondence address: No. 151, Ying-chuan Road, Tamsui District, New Taipei City, Taiwan. Tel.: þ886 2 26215656x2061; fax: þ 886 2 26209745. E-mail address: [email protected]
1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.09.018
wave is strongly confined in the defect channel, allowing low loss in linear waveguides and sharp bends [2–5]. Such structures can be used to design highly efficient new optical devices. Optical waveguides in two-dimensional (2-D) PCs produced by insertion of linear defects into PC structures had been proposed [6] and experimentally proved [7]. It is important to obtain tunable PC waveguides for applications in optical devices. Tunable PC structures that utilize synthetic opals and inverse opals infiltrated with functional materials have been proposed [8–10]. The photonic bandgap of a 2-D PC is continuously varied using the temperature dependent refractive index of a liquid crystal (LC) [11]. The propagation of tunable lightwave in Y-shaped waveguides in 2-D PCs by use of LCs and semiconductors as linear defects is discussed [12,13]. A tunable PC waveguide coupler based on nematic LCs is presented by the authors [14], and then we have investigated the tunable bandgap
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C.-Y. Liu / Physica E 46 (2012) 177–181
in a 2-D PC modulated by a nematic LC [15]. These results can be used to obtain a tunable field-sensitive polarizer [16]. Electrooptical switches are the key components of such photonic integrated circuits. A PC channel waveguided 2 2 directional coupler switch that utilizes electrically or optically induced loss in the coupling region between coupled waveguides has been proposed and analyzed [17]. In this paper, we theoretically demonstrate the electro-optical modulation in a 2-D PC linear waveguide. To create a linear waveguide in the PC structure, the radius of a line of rods is reduced. Using a plane wave expansion method (PWE), we have solved Maxwell’s equations for the guided modes of electromagnetic waves in a periodic arrangement of dielectric cylinders. The guided modes can be controlled by changing the conductance in the waveguide region. The propagation properties of electromagnetic waves inside the linear waveguide are calculated by means of finite-difference time–domain simulations (FDTD). The switching mechanism is a change in the conductance in the waveguide region and modulating the guided modes. Such a mechanism of optical modulation should open up a new application for designing components in photonic integrated circuits. Details of the calculations and discussion of the results will be presented in the remainder of the paper.
2. Numerical method The PWE method is illustrated in several papers [2,18]. Here we summarize the theory very briefly. Following the discussion of previous literature, we can express the lightwave equation that is satisfied by the magnetic field in order to determine the photonic bandgaps of periodic structures as 2 1 o r rHðrÞ ¼ HðrÞ ð1Þ eðrÞ c where rHðrÞ ¼ 0. The dielectric tensor eðrÞ ¼ eðr þRÞ is periodic with respect to the lattice vectors R generated by primitive translation. The main numerical problem in obtaining the eigenvalue is the evaluation of the Fourier coefficients of the inverse dielectric tensors. The best method is to calculate the matrix of Fourier coefficients of real space tensors and take its inverse in order to obtain the required Fourier coefficients. This method (HCS) was shown by Ho et al. [18]. The eigenvalues computed with the HCS method for 512 plane waves are estimated to be in error less than 1%. In our calculations, the convergence is quite fast for lowenergy bands. We consider a square lattice of dielectric cylinders in air and examine transverse electric (TE) mode where electric field is parallel to the axis of cylinders. PWE method is used to obtain photonic band structures and guided modes. The electromagnetic wave propagation is simulated by using the FDTD method [19]. FDTD method is a powerful, accurate numerical method that permits computer-aided design and simulation of PC structures. The photonic device is laid out in the x–y plane. The propagation is along the x direction. The computational domain used is rectangular and is bounded by a perfectly matching layer to minimize back reflections. The space steps in the x and y directions are Dx and Dy, respectively. We assumed that Dx and Dy ¼0.05. The sampling in time is selected to ensure numerical stability of the algorithm. A more detailed treatment of the FDTD method is given in Ref. [19]. 3. Electro-optical modulation We consider that the 2-D square lattice PC is composed of dielectric cylinders surrounded by air. The material is
Fig. 1. Schematic view of the proposed 2-D photonic crystal linear waveguide. The linear waveguide is formed by reducing the radius of dielectric rods of a single row whose radius is ri o r.
homogeneous in the z direction, and periodic along x and y with lattice constant. The lattice constant is a and the radius of cylinders is r¼ 0.2a. The refractive index of the dielectric cylinders is n ¼3.4 (Si). This structure has a photonic bandgap for TE modes ranging from 0.2872 to 0.4204 in normalized frequency units (oa/ 2pc), where o is the angular frequency and c is the light velocity in the free space. In this structure, the waveguides made by reducing the radius of a line of rods are inherently single mode because total internal reflection modes are prevented to exist. Fig. 1 depicts the proposed 2-D photonic crystal linear waveguide. The linear waveguide is formed by reducing the radius of dielectric rods of a single row whose radius is ri or. The dispersion relation of the linear waveguide is obtained by using the PWE method. The guided modes in the linear waveguide are computed using a supercell approximation, which consists of placing a large crystal with a defect into a supercell and repeating it periodically in space. In the case below, the supercell contains a 1 7 crystal. In this paper, we propose the loss tangent of dielectric material in the waveguide region can be modified by external commands to spoil the guiding, thereby modulating the lightwave. This is a Da switch in which the change in optical absorption coefficient Da is employed. The change in conductance Ds is proportional to Da. The behavior is analogous to that discussed in Ref. [20] where electric absorption was assumed to reduce the Q factor of microring resonators coupled to strip channel waveguides. Fig. 2 shows the dispersion relation of the guided modes of the linear waveguide at ri ¼0.05a for different conductance s values. The inset of Fig. 2 shows the supercell for this linear waveguide. Shaded regions correspond to the projected band structures of the perfect PCs. It is observed that the guided modes arising from a single waveguide appear inside the photonic bandgap. Inside the photonic bandgap, there exist two kinds of dispersion relations at certain conductance values and there exist cutoff frequencies of guide modes. The cutoff frequencies at s ¼1 10 1 S/cm and 1 106 S/cm are 0.2948 and 0.3864, respectively. For the 1550 nm center wavelength, we assumed a 2-D PC of 217 nm diameter dielectric rods arrayed in a square lattice a¼542.5 nm in the air background. Therefore, the cutoff wavelength lcutoff at s ¼1 10 1 S/cm and 1 106 S/cm are 1840.23 nm and 1403.98 nm, respectively. In order to let the incident lightwave propagate in the linear waveguide, incident wavelength linc%lcutoff must be satisfied. Fig. 3 shows the normalized power response as a function of incident wavelength at ri ¼0.05a for conductivities s ¼1 10 1 S/cm and 1 106 S/cm. The transmissions are in agreement with dispersion relations in Fig. 2. At s ¼1 10 1 S/cm, the incident wavelength linc ¼1550 nm can propagate in the linear waveguide because linc%lcutoff is satisfied. At s ¼ 1 106 S/cm, the incident wavelength linc ¼1550 nm cannot propagate in the linear waveguide because linc%lcutoff is not satisfied. Hence, the cutoff frequencies
C.-Y. Liu / Physica E 46 (2012) 177–181
179
Fig. 4. Normalized power response as a function of conductivity of the linear waveguide with ri ¼0.05a at input wavelength 1550 nm.
Fig. 2. Dispersion relation of the guided modes of the linear waveguide at ri ¼ 0.05a for different conductance s values: s ¼ 1 10 1 S/cm (solid line) and s ¼ 1 106 S/cm (dashed line). Shaded regions correspond to the projected band structures of the perfect photonic crystals. The inset shows the supercell for this linear waveguide.
Fig. 3. Normalized power response as a function of incident wavelength at ri ¼ 0.05a for different conductances: s ¼ 1 10 1 S/cm (solid line) and s ¼ 1 106 S/cm (dashed line).
of PC linear waveguides depend on the conductance in the waveguide region and lightwave propagation in linear waveguides can be controlled by changing the conductance. In this paper, we consider that the 2-D square lattice PC is composed of dielectric cylinders surrounded by air. This perfect PC has complete photonic bandgap for the TE modes, but not for the TM modes. Therefore, we cannot obtain the guided modes in the PC with square lattice for the TM modes. By combining our observations, we can design a PC that has photonic bandgaps for both polarizations. By adjusting the dimensions of the lattice, we can even arrange for the photonic bandgaps to overlap, resulting in an absolute bandgap for all polarizations [21]. If we can obtain the guided modes in an absolute bandgap, the guided modes can be controlled by changing the conductance for all polarizations. Fig. 4 depicts the normalized power response as a function of conductivity of the linear waveguide with ri ¼0.05a at input wavelength 1550 nm. For linc ¼1550 nm, the maximum of output
Fig. 5. Field patterns observed in frequency domain of the electro-optical linear waveguide at input wavelength l ¼ 1550 nm and ri ¼ 0.05a for conductance (a) s ¼1 10 1 S/cm, (b) s ¼5 102 S/cm and (c) s ¼ 1 106 S/cm.
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power is 3.42 dB at s ¼1 10 1 S/cm and the minimum of output power is 188.23 dB at s ¼1 106 S/cm. The results provide a nice illustration of how the linear waveguide can make it possible to control by changing the conductivity of the materials. The power response in Fig. 4 shows that there is a specific value of optical power at output port for a specific value of conductivity (s ¼5 102 S/cm), at this transient value the optical power launched at the input port will be absorbed gradually in the waveguide region and the device suffer high attenuation coefficient a in the waveguide region [22]. An increase or decrease in the conductivity will redirect the optical power to either
transmission or absorption respectively. The FDTD method is used to solve the light propagation in 2-D PC linear waveguide. Fig. 5 shows the field patterns observed in frequency domain of the electro-optical linear waveguide at input wavelength l ¼1550 nm and ri ¼0.05a for different conductance. Fig. 5(a) shows that lightwave can propagate in the linear waveguide at s ¼1 10 1 S/cm. Fig. 5(c) shows that lightwave cannot propagate in the linear waveguide at s ¼1 106 S/cm. These results would provide a basis for the novel application of switching devices in photonic circuits.
Fig. 6. Schematic view of the proposed 2-D photonic crystal waveguide branches. The linear waveguide is formed by reducing the radius of dielectric rods of a single row whose radius is ri o r.
Fig. 7. Normalized power response as a function of conductivity of the waveguide branches with ri ¼0.05a at input wavelength 1550 nm.
Fig. 8. Field patterns observed in frequency domain of the electro-optical waveguide branches at input wavelength l ¼1550 nm and ri ¼ 0.05a for conductance (a) s ¼ 1 10 1 S/cm, (b) s ¼ 5 102 S/cm and (c) s ¼1 106 S/cm.
C.-Y. Liu / Physica E 46 (2012) 177–181
To achieve optical modulation in 2-D PC waveguides for Si/air or Si/SiO2 alternating materials, the free carrier absorption loss of Si can be controlled by three ways [23]. The first way is carrier injection from forward biased PN junctions on the rods. The second way is depletion of doped rods with metal–oxide–semiconductor gates. The third way is generation of electrons and holes by above gap light shining upon the designated rods. If the PC waveguide is implemented in III–V semiconductor hetero-layers, the electric absorption effect could be used. The dependence of the conductance upon doping density is shown in [23]. The prior PC switching device relies upon a point defect resonator or two point defects located between two PC waveguides [24]. They assumed that the Q factor of cavities would be spoiled by loss induced electrically at the defects. The realization of a thermo-optically controlled Mach– Zehnder interferometer based on an AlGaAs/GaAs epitaxial waveguide operating at wavelength of 1550 nm is reported [25]. The fairly low switching power required for the tunable PC device has demonstrated its potential for use in future low cost and compact optical circuits.
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researched for enhancing transmission through waveguide region and reducing reflections.
5. Conclusion We have demonstrated numerically the electro-optical modulation in a 2-D PC linear waveguide with square lattice. In order to create a linear waveguide, the radius of a line of rods is reduced. The operation of the waveguides is controlled by optical loss induced in the dielectric rods in the linear waveguides. Using the FDTD method on a 1550 nm device, we predict low insertion loss in the different switching states although the required change in conductance is large in these non-optimized waveguides. Such an electro-optical mechanism should open up a new application for designing tunable components in photonic integrated circuits. Further theoretical investigations and experimental efforts are needed to bring the tunable linear waveguide into reality.
Acknowledgments 4. Tunable waveguide branches The tunable electro-optical waveguide branches designed by the proposed scheme are depicted in Fig. 6. The dielectric cylinders with the 2-D square lattice and a radius r ¼0.2a are surrounded by air. The waveguides are introduced by reducing the radius of a line of rods in the crystal. The FDTD method is also used to solve the lightwave propagation in 2-D PC waveguide branches. For the T-shaped waveguide shown in Fig. 6, the transmission spectrum thus obtained are shown in Fig. 7. The lightwave of wavelength 1550 nm is launched into input port and the temporal variation of the electromagnetic field in each output port is monitored. The transmission spectra are a function of the conductivity of the waveguide. At linc ¼1550 nm, the optical powers received in the two output ports are identical at s ¼1 10 1 S/cm and the transmittance is 6.96 dB per each output waveguide. When conductance is s ¼ 1 106 S/cm, the output power is almost null and the transmittance is 158.03 dB per each output waveguide. Fig. 8 shows field patterns observed in frequency domain of the electro-optical waveguide branches at input wavelength l ¼1550 nm and ri ¼0.05a for different conductances. Fig. 8(a) shows that the fields are completely confined within the waveguide regions and split equally into the output waveguides at s ¼1 10 1 S/cm. Fig. 8(c) shows that lightwave cannot propagate in the waveguide branches at s ¼1 106 S/cm. We can see that the proposed electro-optical waveguides can not only be a tunable linear waveguide but also be a tunable waveguide branch by controlling the conductivity of waveguides. The waveguides are intended to be interconnected and cascaded in the forward direction into an optical cross-connect network. In this case, further optimization to transmission can be achieved by minimizing the reflections at the waveguide region and waveguide bends. The broadband and narrowband techniques will be
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