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1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal
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1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal Guimin Lin a,∗ , Xiyao Chen a , Dongxia Zhuang b a b
Department of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China School of Physics and Optoelectronics Technology, Fujian Normal University, Fuzhou 350007, China
a r t i c l e
i n f o
Article history: Received 15 August 2013 Accepted 10 March 2014 Available online xxx Keywords: Photonic crystal Self-collimation Mach–Zehnder interferometer Multiplexer
a b s t r a c t In this paper, a novel 1 × 4 optical multiplexer (OMUX) based on the two dimensional photonic crystal embed cascaded Mach–Zehnder interferometer (MZI) employing self-collimation effect was proposed and its performance were numerically demonstrated. The 1 × 4 OMUX consists of four beam splitters and five mirrors. Light propagates in the OMUX employing self-collimation effect. The theoretical transmission spectra at different output ports of OMUX were analyzed with the theory of light interference. Then they were investigated with the finite-difference time-domain (FDTD) simulation technique. The simulation results indicate the cascaded Mach–Zehnder interferometer can work as a 1 × 4 optical multiplexer by selecting path length in the structure properly. For the drop wavelength 1550 nm, the free spectral range of the OMUX is about 24 nm, which almost covers the whole optical communication C-band window. The presented device that has no only a compact size but also a high output efficiency, may have practical applications in photonic integrated circuits. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals (PhCs) [1–3], composed of periodic dielectric materials, have attracted great attention due to their unique ability to manipulate light. PhCs are promising as a platform for compact photonic integrated circuits. In particular, planar PhCs have attracted a great deal of attention for their advantages of small size and easy fabrication using mature microelectronics patterning techniques. The self-collimation (SC) effect in PhC structures allows diffractionless light to propagation in perfect PhCs without “physical” guiding boundaries (e.g. line-defect waveguide) [4]. Additionally, it can also enable two beams intercrossing without cross-talks. One advantage of self-collimation-based devices is that they do not require a physical boundary to achieve narrow lateral confinement. Also, guiding can be achieved over a larger bandwidth when compared to their line-defect counterparts. A variety of photonic devices based on SC effect have been theoretically and experimentally demonstrated [4–14]. Among them, self-collimation Mach–Zehnder interferometers (SMZIs) can work as wavelength divison demultiplexers or power splitters. However, most of these PhCs devices work for monolithic MZI.
Wavelength division multiplexing (WDM) is a technology of transmitting multiple optical signals of different wavelengths on a single mode fiber, which increases the transmission capacity and enhances flexibility of network configurations in optical communication systems. Recently, WDM devices based on silicon have attracted a great attention because of their tremendous potential for low-cost and highly integrated optical components. A multiplexer plays an important role in WDM systems. Several applications for wavelength division multiplexer have been proposed [15–19]. In this paper, we design a 1 × 4 optical multiplexer based on cascade Mach–Zehnder interferometer [5,11,12] (MZI) in a two dimensional silicon photonic crystal employing self-collimation effect. The structure consists of two MZIs with difference path length. The differences in length between the two arms decide the period of the interferometric pattern at the output. The performance of 1 × 4 OMUX was numerically investigated by using finite-difference time-domain (FDTD) technique. 2. 1 × 4 optical multiplexer in rod-type silicon photonic crystal 2.1. Self-collimation frequency range
∗ Corresponding author. E-mail address: [email protected] (G. Lin).
A 2D silicon (Si) PhC we considered consists of a square lattice of dielectric cylindrical rods in air, where the dielectric constant
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Please cite this article in press as: G. Lin, et al., 1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.03.031
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Fig. 1. Dispersion curve of the first band along M direction for TM polarization in the 2D square-lattice rod-type silicon PhC, where the silicon dielectric constant and the ratio of the dielectric cylindrical rods radius r to the lattice constant a are 12.25 and 0.35, respectively.
Fig. 3. Schematic of the proposed 1 × 4 optical multiplexer consisting of two Mach–Zehnder interferometers.
εSi and the ratio of the cylindrical radius r to the lattice constant a are 12.25 and 0.35, respectively, as shown in the inset of Fig. 1. The simulated dispersion curve of the first band along M direction for transverse-magnetic (TM) polarization is shown in Fig. 1. The corresponding equal frequency contours (EFCs) in one fourth of the first Brillouin zone are also shown in Fig. 2. All the calculations are done by the plane-wave expansion (PWE) method [20]. As shown in Fig. 2, the EFCs in the frequency range between 0.190c/a and 0.200c/a are close to straight lines normal to the M direction (where c is the speed of light in vacuum). Hence, at these frequencies and in the direction perpendicular to the flat EFCs, light can propagate within the PhC without diffraction, which is known as self-collimation effect.
2.3. The theory and performance of 1 × 4 OMUX
2.2. Structure of the 1 × 4 optical multiplexer (OMUX) The structure of the 1 × 4 OMUX we proposed is shown in Fig. 3, which consisting two stages of Mach–Zehnder interferometer (MZI). Each MZI is composed of two same mirrors and two same beam splitters. Narrow-width light beams can propagate without diffraction in the structure in dependence on SC effect. Each mirror is formed by inserting a thick air bar, overlapping four-row cylindrical rods along the X direction. Each beam splitter is formed by reducing the radius of one row of several cylindrical rods in the X direction, from normal r = 0.35a, to rs = 0.275a. There are four
0.5
M
0.4 0.3
ky(2π /a)
0.200 0.195 0.190
0.2 0.1 Γ 0.0 0.0
0.10
0.1
0.2
0.13
0.3
kx(2π /a)
0.17
0.4
X 0.5
Fig. 2. Equal-frequency contours (EFCs) of the first band for TM modes in a silicon photonic crystal consisting of a square lattice of cylindrical rods in air.
output ports in the structure, two down ports (Down1, Down2), and two drop ports (Drop1, Drop2).
In the cascaded MZI, the SC light beam incident into first stage is splitted by S1 into two branches, the reflected beam and transmitted beam. The reflected beam will travel along the longer light path while the transmitted beam propagates along the shorter light path. The length of shorter path is equal to l1 and the length of longer path is equal to l2 . It can be seen that l2 = l1 + 2d, where d represents the distance between S1 and M1 , or between the S2 and M2 . The two SC beams form two paths are spliited again by S2 into the right (R) and bottom (B) beam. The right beam and bottom beam which is total reflected by M3 incident into second stage. The similar behavior happens in the second stage. Light interference happens at the four output ports. The first stage has a difference in length between the two light paths of 2d, and the second stage of d. The performance of the beam splitters was first evaluated numerically with the 2D finite-difference time-domain simulation method, with the results shown in Fig. 3. With self-collimation frequency range [0.190c/a, 0.200c/a], the reflectivity and transmissivity of beam splitter are around 50% for TM polarization when rs = 0.275a. The splitter are nearly lossless because RS + TS ≈1. In other word, the splitting ratio of the splitters is 1:1 for TM mode. The theoretical transmission at two output ports in monolithic asymmetric MZI can be obtained based on the theory of light interference in the form [21]: I1 = 1 − 4Rs Ts cos2 (kl + ) I0
(1)
I2 = 1 − 4Rs Ts cos2 (kl + ) I0
(2)
where Rs and Ts are the reflectivity and transmittivity of beam splitter (S1 , S2 , S3 or S4 ). I0 is the intensity of the input SC light; I1 and I2 are the intensities of the SC light at the right and down output ports respectively; k is the Bloch wave vector, based on the dispersion curve (k = k(f)) as shown in Fig. 1, and l represents the half optical path difference; is the total phase shift resulting from the phase jump at the beam splitters and the mirrors. Here we assumed that each of two mirrors is perfect, with reflectivity of 100% and phase shift of for one-time reflection. From Eqs. (1) and (2), the right transmission (I1 /I0 ) and down transmission (I2 /I0 ) are complementary because I1 + I2 = I0 . They have equal amplitude of 4Rs Ts although the down transmission is in the range [0,4Rs Ts ] and the right is in the range [(1 − 4Rs Ts ), 1]. For a lossless splitter, Rs + Ts = 1. When Rs = Ts = 0.5 (i.e., 4Rs Ts = 1), the right and down transmissions have the largest amplitude of 1 and
Please cite this article in press as: G. Lin, et al., 1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.03.031
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1.0
1.0 0.8
0.0 1.0
0.6 0.4 0.2 0.0 0.190
I1/I0
0.5
T R S
Transmission
Reflectivity/Transmisivity
3
I2/I0
0.5 0.0 1.0 I3/I0
0.5 0.0 1.0
I4/I0
0.5 0.192
0.194
0.196
0.198
0.200
0.0 0.190
Frequency(c/a)
0.192
0.194
0.196
0.198
0.200
Frequency(c/a)
Fig. 4. Transmissivity TS and reflectivity RS of a beam splitter vary with frequency when rs = 0.275a.
(a) When l1= 2l2 and l2= 27
2a
1.0
1.0
I2 / I0
Transmission
0.8 0.6 0.4 0.2 0.0 0.190 1.0
0.192
0.194
0.196
Frequency(c/a)
(b)
0.0 1.0
I3/I0
0.5 0.0 1.0
I4/I0
0.0 0.190
0.192
(b) When l1= 54
0.2
Frequency(c/a)
0.198
0.200
2a
and l2= 25
2a
(3)
I2 = 4Rs Ts cos2 (kl1 + 1 )[1 − 4Rs Ts cos2 (kl2 + 2 )] I0
(4)
I3 = [1 − 4Rs Ts cos2 (kl1 + 1 )]4Rs Ts cos2 (kl2 + 2 ) I0
(5)
Theory
1.0 0.5
0.200
Fig. 5. (a) FDTD simulated transmission spectra of the first MZI in OMUX for TM √ modes when l1 = 54 2a. (b) FDTD √ simulated transmission spectra of the second MZI for TM modes when l2 = 27 2a.
FDTD
I1
FSR
(a)
0.0 1.0 0.5
I2
(b)
0.0 1.0 0.5
I3
(c)
0.0 1.0 0.5
0.196
0.198
I1 = [1 − 4Rs Ts cos2 (kl1 + 1 )][1 − 4Rs Ts cos2 (kl2 + 2 )] I0
Transmission
0.4
0.194
0.196
Fig. 6. The theoretical transmission spectra of the 1 × 4 OMUX according √ √ √ to Eqs. (3)–(6). (a) When l = 54 2a and l = 27 2a. (b) When l1 = / 2l2 and l = 27 2a.
I2 / I0
0.192
0.194
Frequency(c/a)
0.200
0.6
0.0 0.190
I2/I0
0.5
0.5
I1 / I0
0.8
Transmission
0.198
I1/I0
0.0 1.0
agrees well with the Eqs. (1) and (2). The peak spacings decrease when the optical path difference increases. From above discussions, the theoretical transmission at four output ports of the OMUX can be obtained based on the theory of light interference in the form [16]:
I1 / I0
(a)
0.5
Transmission
are in the same range [0,1]. When Rs = / 0.5 (i.e., 4Rs Ts < 1), however, their amplitudes diminish to less than 1 and their ranges separate. As a result, the right transmission has higher vertical position than the down transmission. To verify the theoretical analyses aforementioned, the transmission spectra of the monolithic asymmetric MZI are simulated numerically with the FDTD method. A Gaussian optical pulse is launched at the input port. The input power (I0 ), right and down output power (I1 and I2 ) are monitored with three power√monitors respectively. The transmission spectra when l = 54 2a or √ l = 27 2a are shown in Fig. 5(a) and (b), respectively. The right and down transmission curves are plotted as solid lines and dash lines, respectively. For TM modes, both the right and down transmissions are approximately in the range [0,1]. Considering the reflectivity and transmittivity values of the splitters in Fig. 4(b), the simulated results agree very well with the theoretical prediction. In addition, as can be seen in Fig. 5, the√ratio of the √ peak spacings of transmission spectra when l = 54 2a or l = 27 2a is about 1:2, which
0.0 0.190
I4
(d) 0.192
0.194
0.196
0.198
0.200
Frequency(c/a) Fig. 7. Theoretical (dash lines) and FDTD simulated (solid lines) transmission spectra of the 1 × 4 OMUX for TM modes at different output ports.
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I4 = 4Rs Ts cos2 (kl1 + 1 )]4Rs Ts cos2 (kl2 + 2 ) I0
(6)
where I0 is the intensity of the input SC light; I1 , I2 I3 and I4 are the intensities of the SC light at the R-port1, R-port2, D-port1 and D-port2 respectively; k is the Bloch wave vector, based on the dispersion curve (k = k(f)) as shown in Fig. 1; l1 represents the half optical path difference of the first MZI in OMUX, and l2 represents the half optical path difference of the second MZI in OMUX; 1 is the total phase shift resulting from the phase jump at the two beam splitters (S1 ,S2 or S3 ,S4 ) and two the mirrors (M1 ,M2 or M4 ,M5 ) in monolithic asymmetric MZI of OMUX, and 2 is the total phase shift resulting from the phase jump at the two beam splitters (S3 ,S4 ) and three mirrors (M3 ,M4 ,M5 ) in monolithic asymmetric MZI of OMUX. To perform a 1 × 4 beam splitter for the OMUX, we obtained the theoretical transmission spectra according to Eqs. (3)–(6), and con/ 2l2 respectively. The transmission sider the case of l1 = 2l2 and l1 = spectra of four output ports are shown in detail in Fig. 6. We can see from Fig. 6 (a), Ii (i = 1, 2, 3, 4) has the same peak spacing when l1 = 2l2 . But when l1 = / 2l2 we cannot easily measure the peak spacing of each output port, as shown in Fig. 6(b). According the results of theoretical calculations, we note that the half optical path difference of first MZI and second MZI in OMUX l1 , l2 must meet the conditions of l1 = 2l2 to perform beam splitter. To verify the theoretical design of the 1 × 4 OMUXs, the numerical transmissions at four output ports√for TM modes are also simulated with the 2D FDTD √ method. A 5 2a width Gaussian optical pulse which is located 2 2a away from the edge of the photonic crystal structure is launched at the input port. The input power (I0 ) The and each output power (Ii ) are monitored with power monitors.√ half optical√path difference of first MZI and second MZI is l = 54 2a and l = 27 2a, respectively. The output transmission spectra (Ii /I0 ) are plotted in Fig. 7(a)–(d) as solid lines. In Fig. 7, the transmission value at I1 port for the frequency f = 0.19248c/a is equal to 0.940 and the values of other ports are almost equal to zero. When f = 0.19172c/a, the transmission value at I2 port reaches its peak and other ports almost equal to zero. The same case occurs at I3 and I4 port when f = 0.19248c/a and f = 0.19248c/a respectively. It should be pointed out that the simulated transmission values are somewhat lower than the theoretical ones. For Ii ports, the theoretical transmission values at peak frequencies should be 1.0. It indicates that propagation loss exists which may result from the scattering in the line-defect beam splitters. From Fig. 7, we can find that the simulation curves agree well with the theoretical curves in terms of peak positions when 1 = 0.513 and 2 = 0.26. The electric field distributions for the drop frequencies were also simulated and shown in Fig. 8. Notice that when at one port’s peak frequency there is no light exiting from other ports. In addition, for the wavelength 1550 nm, the free spectral range (FSR) of the 1 × 4 OMUX is about 24 nm, which almost covers the whole optical communication C-band window.
3. Conclusion
Fig. 8. Magnetic-field distributions of the steady-state electric field of designed 1 × 4 OMUX at Ii (i = 1, 2, 3, 4) port with different frequencies.
In conclusion, we successfully demonstrated a 1 × 4 optical multiplexer based on cascaded MZI in a rod-type photonic crystal. Selecting properly path length, the light beam with some frequencies can totally exit form one output port. The self-collimation effect was applied in the cascaded MZI. Its performance was evaluated with FDTD simulation technique and well agreements are obtained between the simulation and the theory. It can work at any frequencies by scaling lattice constant and cylindrical rods radius simultaneously for PhC. For the operating wavelength at 1550 nm, the dimension of this structure is about 38.5 m × 35 m. So the
Please cite this article in press as: G. Lin, et al., 1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.03.031
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proposed cascaded MZI have potential applications in high-density integrated photonic circuits. Acknowledgments This work was supported by the Natural Science Foundation of Fujian Province of China (No. 2013J05095), the Research Project of Science & Technology of Fujian Education Office of China (No. JB11149) and the nursery project of Science & Technology of Minjiang University (No. YKY1103). References [1] J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed., Princeton University Press, New Jersey, 2008. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (1987) 4. [3] T.F. Krauss, R.M.D.L. Rue, Photonic crystals in the optical regime-past, present and future, Prog. Quantum Electron. 23 (1999) 46. [4] S.G. Lee, S.S. Oh, J.E. Kim, H.Y. Park, C.S. Kee, Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals, Appl. Phys. Lett. 87 (2005) 3. [5] D.Y. Zhao, J. Zhang, P.J. Yao, X.Y. Jiang, X.Y. Chen, Photonic crystal Mach–Zehnder interferometer based on self-collimation, Appl. Phys. Lett. 90 (2007) 3. [6] H.B. Chen, Z.F. Li, W. Liu, S.L. Feng, H.Z. Zheng, Line defect splitters for selfcollimated beams in photonic crystals, Opt. Commun. 262 (2006) 5. [7] Y.L. Zhang, Y. Zhang, B.J. Li, Optical switches and logic gate based on selfcollimated beams in two dimensional photonic crystals, Opt. Express 15 (2007) 6. [8] V. Zabelin, L.A. Dunbar, N.L. Thomas, R. Houdré, M.V. Kotlyar, L. O’Faolain, T.F. Krauss, Self-collimating photonic crystal polarization beam splitter, Opt. Lett. 32 (2007) 3.
5
[9] X.F. Yu, S.H. Fan, Bends and splitters for self-collimated beams in photonic crystals, Appl. Phys. Lett. 83 (2003) 3. [10] X.Y. Chen, D.Y. Zhao, Z.X. Qiang, G.M. Lin, H. Li, Y.S. Qiu, W.D. Zhou, Polarizationindependent Fabry–Perot interferometer in a hole-type silicon photonic crystal, Appl. Opt. 49 (2010) 4. [11] T.T. Kim, S.G. Lee, H.Y. Park, J.E. Kim, C.S. Kee, Asymmetric Mach–Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals, Opt. Express 18 (2010) 6. [12] X.Y. Chen, Z.X. Qiang, D.Y. Zhao, Y.F. Wang, H. Li, Y.S. Qiu, W.D. Zhou, Polarization beam splitter based on photonic crystal self-collimation Mach–Zehnder interferometer, Opt. Commun. 284 (2011) 4. [13] X.Y. Chen, Z.X. Qiang, D.Y. Zhao, H. Li, Y.S. Qiu, W.Q. Yang, W.D. Zhou, Polarization-independent drop filters based on photonic crystal selfcollimation ring resonators, Opt. Express 17 (2009) 6. [14] P.T. Rakich, M.S. Dahlem, S. Tandon, M. Ibanescu, M.S.c. acute, G.S. Petrich, J.D. Joannopoulos, L.A. Kolodziejski, E.P. Ippen, Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal, Nat. Mater. 5 (2006) 4. [15] E. Centeno, B. Guizal, D. Felbacq, Multiplexing and demultiplexing with photonic crystals, J. Opt. A: Pure Appl. Opt. 1 (1999) 4. [16] S. Kim, I. Park, H. Lim, C.S. Kee, Highly efficient photonic crystal-based multichannel drop filters of three-port system with reflection feedback, Opt. Express 12 (2004) 8. [17] M. Koshiba, Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers, J. Lightwave Technol. 19 (2001) 1. [18] C.J. Jin, S.H. Fan, S.Z. Han, D.Z. Zhang, Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration, IEEE J. Quantum Electron. 39 (2003) 6. [19] A. Sharkawy, S.Y. Shi, D.W. Prather, Multichannel wavelength division multiplexing with photonic crystals, Appl. Opt. 40 (2001) 6. [20] S.G. Johnson, J.D. Joannopoulos, Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis, Opt. Express 8 (2001) 18. [21] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley-interscience, New York, 1991.
Please cite this article in press as: G. Lin, et al., 1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.03.031
ARTICLE IN PRESS Optik xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal Guimin Lin a,∗ , Xiyao Chen a , Dongxia Zhuang b a b
Department of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China School of Physics and Optoelectronics Technology, Fujian Normal University, Fuzhou 350007, China
a r t i c l e
i n f o
Article history: Received 15 August 2013 Accepted 10 March 2014 Available online xxx Keywords: Photonic crystal Self-collimation Mach–Zehnder interferometer Multiplexer
a b s t r a c t In this paper, a novel 1 × 4 optical multiplexer (OMUX) based on the two dimensional photonic crystal embed cascaded Mach–Zehnder interferometer (MZI) employing self-collimation effect was proposed and its performance were numerically demonstrated. The 1 × 4 OMUX consists of four beam splitters and five mirrors. Light propagates in the OMUX employing self-collimation effect. The theoretical transmission spectra at different output ports of OMUX were analyzed with the theory of light interference. Then they were investigated with the finite-difference time-domain (FDTD) simulation technique. The simulation results indicate the cascaded Mach–Zehnder interferometer can work as a 1 × 4 optical multiplexer by selecting path length in the structure properly. For the drop wavelength 1550 nm, the free spectral range of the OMUX is about 24 nm, which almost covers the whole optical communication C-band window. The presented device that has no only a compact size but also a high output efficiency, may have practical applications in photonic integrated circuits. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction Photonic crystals (PhCs) [1–3], composed of periodic dielectric materials, have attracted great attention due to their unique ability to manipulate light. PhCs are promising as a platform for compact photonic integrated circuits. In particular, planar PhCs have attracted a great deal of attention for their advantages of small size and easy fabrication using mature microelectronics patterning techniques. The self-collimation (SC) effect in PhC structures allows diffractionless light to propagation in perfect PhCs without “physical” guiding boundaries (e.g. line-defect waveguide) [4]. Additionally, it can also enable two beams intercrossing without cross-talks. One advantage of self-collimation-based devices is that they do not require a physical boundary to achieve narrow lateral confinement. Also, guiding can be achieved over a larger bandwidth when compared to their line-defect counterparts. A variety of photonic devices based on SC effect have been theoretically and experimentally demonstrated [4–14]. Among them, self-collimation Mach–Zehnder interferometers (SMZIs) can work as wavelength divison demultiplexers or power splitters. However, most of these PhCs devices work for monolithic MZI.
Wavelength division multiplexing (WDM) is a technology of transmitting multiple optical signals of different wavelengths on a single mode fiber, which increases the transmission capacity and enhances flexibility of network configurations in optical communication systems. Recently, WDM devices based on silicon have attracted a great attention because of their tremendous potential for low-cost and highly integrated optical components. A multiplexer plays an important role in WDM systems. Several applications for wavelength division multiplexer have been proposed [15–19]. In this paper, we design a 1 × 4 optical multiplexer based on cascade Mach–Zehnder interferometer [5,11,12] (MZI) in a two dimensional silicon photonic crystal employing self-collimation effect. The structure consists of two MZIs with difference path length. The differences in length between the two arms decide the period of the interferometric pattern at the output. The performance of 1 × 4 OMUX was numerically investigated by using finite-difference time-domain (FDTD) technique. 2. 1 × 4 optical multiplexer in rod-type silicon photonic crystal 2.1. Self-collimation frequency range
∗ Corresponding author. E-mail address: [email protected] (G. Lin).
A 2D silicon (Si) PhC we considered consists of a square lattice of dielectric cylindrical rods in air, where the dielectric constant
http://dx.doi.org/10.1016/j.ijleo.2014.03.031 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
Please cite this article in press as: G. Lin, et al., 1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.03.031
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2
Fig. 1. Dispersion curve of the first band along M direction for TM polarization in the 2D square-lattice rod-type silicon PhC, where the silicon dielectric constant and the ratio of the dielectric cylindrical rods radius r to the lattice constant a are 12.25 and 0.35, respectively.
Fig. 3. Schematic of the proposed 1 × 4 optical multiplexer consisting of two Mach–Zehnder interferometers.
εSi and the ratio of the cylindrical radius r to the lattice constant a are 12.25 and 0.35, respectively, as shown in the inset of Fig. 1. The simulated dispersion curve of the first band along M direction for transverse-magnetic (TM) polarization is shown in Fig. 1. The corresponding equal frequency contours (EFCs) in one fourth of the first Brillouin zone are also shown in Fig. 2. All the calculations are done by the plane-wave expansion (PWE) method [20]. As shown in Fig. 2, the EFCs in the frequency range between 0.190c/a and 0.200c/a are close to straight lines normal to the M direction (where c is the speed of light in vacuum). Hence, at these frequencies and in the direction perpendicular to the flat EFCs, light can propagate within the PhC without diffraction, which is known as self-collimation effect.
2.3. The theory and performance of 1 × 4 OMUX
2.2. Structure of the 1 × 4 optical multiplexer (OMUX) The structure of the 1 × 4 OMUX we proposed is shown in Fig. 3, which consisting two stages of Mach–Zehnder interferometer (MZI). Each MZI is composed of two same mirrors and two same beam splitters. Narrow-width light beams can propagate without diffraction in the structure in dependence on SC effect. Each mirror is formed by inserting a thick air bar, overlapping four-row cylindrical rods along the X direction. Each beam splitter is formed by reducing the radius of one row of several cylindrical rods in the X direction, from normal r = 0.35a, to rs = 0.275a. There are four
0.5
M
0.4 0.3
ky(2π /a)
0.200 0.195 0.190
0.2 0.1 Γ 0.0 0.0
0.10
0.1
0.2
0.13
0.3
kx(2π /a)
0.17
0.4
X 0.5
Fig. 2. Equal-frequency contours (EFCs) of the first band for TM modes in a silicon photonic crystal consisting of a square lattice of cylindrical rods in air.
output ports in the structure, two down ports (Down1, Down2), and two drop ports (Drop1, Drop2).
In the cascaded MZI, the SC light beam incident into first stage is splitted by S1 into two branches, the reflected beam and transmitted beam. The reflected beam will travel along the longer light path while the transmitted beam propagates along the shorter light path. The length of shorter path is equal to l1 and the length of longer path is equal to l2 . It can be seen that l2 = l1 + 2d, where d represents the distance between S1 and M1 , or between the S2 and M2 . The two SC beams form two paths are spliited again by S2 into the right (R) and bottom (B) beam. The right beam and bottom beam which is total reflected by M3 incident into second stage. The similar behavior happens in the second stage. Light interference happens at the four output ports. The first stage has a difference in length between the two light paths of 2d, and the second stage of d. The performance of the beam splitters was first evaluated numerically with the 2D finite-difference time-domain simulation method, with the results shown in Fig. 3. With self-collimation frequency range [0.190c/a, 0.200c/a], the reflectivity and transmissivity of beam splitter are around 50% for TM polarization when rs = 0.275a. The splitter are nearly lossless because RS + TS ≈1. In other word, the splitting ratio of the splitters is 1:1 for TM mode. The theoretical transmission at two output ports in monolithic asymmetric MZI can be obtained based on the theory of light interference in the form [21]: I1 = 1 − 4Rs Ts cos2 (kl + ) I0
(1)
I2 = 1 − 4Rs Ts cos2 (kl + ) I0
(2)
where Rs and Ts are the reflectivity and transmittivity of beam splitter (S1 , S2 , S3 or S4 ). I0 is the intensity of the input SC light; I1 and I2 are the intensities of the SC light at the right and down output ports respectively; k is the Bloch wave vector, based on the dispersion curve (k = k(f)) as shown in Fig. 1, and l represents the half optical path difference; is the total phase shift resulting from the phase jump at the beam splitters and the mirrors. Here we assumed that each of two mirrors is perfect, with reflectivity of 100% and phase shift of for one-time reflection. From Eqs. (1) and (2), the right transmission (I1 /I0 ) and down transmission (I2 /I0 ) are complementary because I1 + I2 = I0 . They have equal amplitude of 4Rs Ts although the down transmission is in the range [0,4Rs Ts ] and the right is in the range [(1 − 4Rs Ts ), 1]. For a lossless splitter, Rs + Ts = 1. When Rs = Ts = 0.5 (i.e., 4Rs Ts = 1), the right and down transmissions have the largest amplitude of 1 and
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1.0
1.0 0.8
0.0 1.0
0.6 0.4 0.2 0.0 0.190
I1/I0
0.5
T R S
Transmission
Reflectivity/Transmisivity
3
I2/I0
0.5 0.0 1.0 I3/I0
0.5 0.0 1.0
I4/I0
0.5 0.192
0.194
0.196
0.198
0.200
0.0 0.190
Frequency(c/a)
0.192
0.194
0.196
0.198
0.200
Frequency(c/a)
Fig. 4. Transmissivity TS and reflectivity RS of a beam splitter vary with frequency when rs = 0.275a.
(a) When l1= 2l2 and l2= 27
2a
1.0
1.0
I2 / I0
Transmission
0.8 0.6 0.4 0.2 0.0 0.190 1.0
0.192
0.194
0.196
Frequency(c/a)
(b)
0.0 1.0
I3/I0
0.5 0.0 1.0
I4/I0
0.0 0.190
0.192
(b) When l1= 54
0.2
Frequency(c/a)
0.198
0.200
2a
and l2= 25
2a
(3)
I2 = 4Rs Ts cos2 (kl1 + 1 )[1 − 4Rs Ts cos2 (kl2 + 2 )] I0
(4)
I3 = [1 − 4Rs Ts cos2 (kl1 + 1 )]4Rs Ts cos2 (kl2 + 2 ) I0
(5)
Theory
1.0 0.5
0.200
Fig. 5. (a) FDTD simulated transmission spectra of the first MZI in OMUX for TM √ modes when l1 = 54 2a. (b) FDTD √ simulated transmission spectra of the second MZI for TM modes when l2 = 27 2a.
FDTD
I1
FSR
(a)
0.0 1.0 0.5
I2
(b)
0.0 1.0 0.5
I3
(c)
0.0 1.0 0.5
0.196
0.198
I1 = [1 − 4Rs Ts cos2 (kl1 + 1 )][1 − 4Rs Ts cos2 (kl2 + 2 )] I0
Transmission
0.4
0.194
0.196
Fig. 6. The theoretical transmission spectra of the 1 × 4 OMUX according √ √ √ to Eqs. (3)–(6). (a) When l = 54 2a and l = 27 2a. (b) When l1 = / 2l2 and l = 27 2a.
I2 / I0
0.192
0.194
Frequency(c/a)
0.200
0.6
0.0 0.190
I2/I0
0.5
0.5
I1 / I0
0.8
Transmission
0.198
I1/I0
0.0 1.0
agrees well with the Eqs. (1) and (2). The peak spacings decrease when the optical path difference increases. From above discussions, the theoretical transmission at four output ports of the OMUX can be obtained based on the theory of light interference in the form [16]:
I1 / I0
(a)
0.5
Transmission
are in the same range [0,1]. When Rs = / 0.5 (i.e., 4Rs Ts < 1), however, their amplitudes diminish to less than 1 and their ranges separate. As a result, the right transmission has higher vertical position than the down transmission. To verify the theoretical analyses aforementioned, the transmission spectra of the monolithic asymmetric MZI are simulated numerically with the FDTD method. A Gaussian optical pulse is launched at the input port. The input power (I0 ), right and down output power (I1 and I2 ) are monitored with three power√monitors respectively. The transmission spectra when l = 54 2a or √ l = 27 2a are shown in Fig. 5(a) and (b), respectively. The right and down transmission curves are plotted as solid lines and dash lines, respectively. For TM modes, both the right and down transmissions are approximately in the range [0,1]. Considering the reflectivity and transmittivity values of the splitters in Fig. 4(b), the simulated results agree very well with the theoretical prediction. In addition, as can be seen in Fig. 5, the√ratio of the √ peak spacings of transmission spectra when l = 54 2a or l = 27 2a is about 1:2, which
0.0 0.190
I4
(d) 0.192
0.194
0.196
0.198
0.200
Frequency(c/a) Fig. 7. Theoretical (dash lines) and FDTD simulated (solid lines) transmission spectra of the 1 × 4 OMUX for TM modes at different output ports.
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I4 = 4Rs Ts cos2 (kl1 + 1 )]4Rs Ts cos2 (kl2 + 2 ) I0
(6)
where I0 is the intensity of the input SC light; I1 , I2 I3 and I4 are the intensities of the SC light at the R-port1, R-port2, D-port1 and D-port2 respectively; k is the Bloch wave vector, based on the dispersion curve (k = k(f)) as shown in Fig. 1; l1 represents the half optical path difference of the first MZI in OMUX, and l2 represents the half optical path difference of the second MZI in OMUX; 1 is the total phase shift resulting from the phase jump at the two beam splitters (S1 ,S2 or S3 ,S4 ) and two the mirrors (M1 ,M2 or M4 ,M5 ) in monolithic asymmetric MZI of OMUX, and 2 is the total phase shift resulting from the phase jump at the two beam splitters (S3 ,S4 ) and three mirrors (M3 ,M4 ,M5 ) in monolithic asymmetric MZI of OMUX. To perform a 1 × 4 beam splitter for the OMUX, we obtained the theoretical transmission spectra according to Eqs. (3)–(6), and con/ 2l2 respectively. The transmission sider the case of l1 = 2l2 and l1 = spectra of four output ports are shown in detail in Fig. 6. We can see from Fig. 6 (a), Ii (i = 1, 2, 3, 4) has the same peak spacing when l1 = 2l2 . But when l1 = / 2l2 we cannot easily measure the peak spacing of each output port, as shown in Fig. 6(b). According the results of theoretical calculations, we note that the half optical path difference of first MZI and second MZI in OMUX l1 , l2 must meet the conditions of l1 = 2l2 to perform beam splitter. To verify the theoretical design of the 1 × 4 OMUXs, the numerical transmissions at four output ports√for TM modes are also simulated with the 2D FDTD √ method. A 5 2a width Gaussian optical pulse which is located 2 2a away from the edge of the photonic crystal structure is launched at the input port. The input power (I0 ) The and each output power (Ii ) are monitored with power monitors.√ half optical√path difference of first MZI and second MZI is l = 54 2a and l = 27 2a, respectively. The output transmission spectra (Ii /I0 ) are plotted in Fig. 7(a)–(d) as solid lines. In Fig. 7, the transmission value at I1 port for the frequency f = 0.19248c/a is equal to 0.940 and the values of other ports are almost equal to zero. When f = 0.19172c/a, the transmission value at I2 port reaches its peak and other ports almost equal to zero. The same case occurs at I3 and I4 port when f = 0.19248c/a and f = 0.19248c/a respectively. It should be pointed out that the simulated transmission values are somewhat lower than the theoretical ones. For Ii ports, the theoretical transmission values at peak frequencies should be 1.0. It indicates that propagation loss exists which may result from the scattering in the line-defect beam splitters. From Fig. 7, we can find that the simulation curves agree well with the theoretical curves in terms of peak positions when 1 = 0.513 and 2 = 0.26. The electric field distributions for the drop frequencies were also simulated and shown in Fig. 8. Notice that when at one port’s peak frequency there is no light exiting from other ports. In addition, for the wavelength 1550 nm, the free spectral range (FSR) of the 1 × 4 OMUX is about 24 nm, which almost covers the whole optical communication C-band window.
3. Conclusion
Fig. 8. Magnetic-field distributions of the steady-state electric field of designed 1 × 4 OMUX at Ii (i = 1, 2, 3, 4) port with different frequencies.
In conclusion, we successfully demonstrated a 1 × 4 optical multiplexer based on cascaded MZI in a rod-type photonic crystal. Selecting properly path length, the light beam with some frequencies can totally exit form one output port. The self-collimation effect was applied in the cascaded MZI. Its performance was evaluated with FDTD simulation technique and well agreements are obtained between the simulation and the theory. It can work at any frequencies by scaling lattice constant and cylindrical rods radius simultaneously for PhC. For the operating wavelength at 1550 nm, the dimension of this structure is about 38.5 m × 35 m. So the
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proposed cascaded MZI have potential applications in high-density integrated photonic circuits. Acknowledgments This work was supported by the Natural Science Foundation of Fujian Province of China (No. 2013J05095), the Research Project of Science & Technology of Fujian Education Office of China (No. JB11149) and the nursery project of Science & Technology of Minjiang University (No. YKY1103). References [1] J.D. Joannopoulos, S.G. Johnson, J.N. Winn, R.D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed., Princeton University Press, New Jersey, 2008. [2] S. John, Strong localization of photons in certain disordered dielectric superlattices, Phys. Rev. Lett. 58 (1987) 4. [3] T.F. Krauss, R.M.D.L. Rue, Photonic crystals in the optical regime-past, present and future, Prog. Quantum Electron. 23 (1999) 46. [4] S.G. Lee, S.S. Oh, J.E. Kim, H.Y. Park, C.S. Kee, Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals, Appl. Phys. Lett. 87 (2005) 3. [5] D.Y. Zhao, J. Zhang, P.J. Yao, X.Y. Jiang, X.Y. Chen, Photonic crystal Mach–Zehnder interferometer based on self-collimation, Appl. Phys. Lett. 90 (2007) 3. [6] H.B. Chen, Z.F. Li, W. Liu, S.L. Feng, H.Z. Zheng, Line defect splitters for selfcollimated beams in photonic crystals, Opt. Commun. 262 (2006) 5. [7] Y.L. Zhang, Y. Zhang, B.J. Li, Optical switches and logic gate based on selfcollimated beams in two dimensional photonic crystals, Opt. Express 15 (2007) 6. [8] V. Zabelin, L.A. Dunbar, N.L. Thomas, R. Houdré, M.V. Kotlyar, L. O’Faolain, T.F. Krauss, Self-collimating photonic crystal polarization beam splitter, Opt. Lett. 32 (2007) 3.
5
[9] X.F. Yu, S.H. Fan, Bends and splitters for self-collimated beams in photonic crystals, Appl. Phys. Lett. 83 (2003) 3. [10] X.Y. Chen, D.Y. Zhao, Z.X. Qiang, G.M. Lin, H. Li, Y.S. Qiu, W.D. Zhou, Polarizationindependent Fabry–Perot interferometer in a hole-type silicon photonic crystal, Appl. Opt. 49 (2010) 4. [11] T.T. Kim, S.G. Lee, H.Y. Park, J.E. Kim, C.S. Kee, Asymmetric Mach–Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals, Opt. Express 18 (2010) 6. [12] X.Y. Chen, Z.X. Qiang, D.Y. Zhao, Y.F. Wang, H. Li, Y.S. Qiu, W.D. Zhou, Polarization beam splitter based on photonic crystal self-collimation Mach–Zehnder interferometer, Opt. Commun. 284 (2011) 4. [13] X.Y. Chen, Z.X. Qiang, D.Y. Zhao, H. Li, Y.S. Qiu, W.Q. Yang, W.D. Zhou, Polarization-independent drop filters based on photonic crystal selfcollimation ring resonators, Opt. Express 17 (2009) 6. [14] P.T. Rakich, M.S. Dahlem, S. Tandon, M. Ibanescu, M.S.c. acute, G.S. Petrich, J.D. Joannopoulos, L.A. Kolodziejski, E.P. Ippen, Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal, Nat. Mater. 5 (2006) 4. [15] E. Centeno, B. Guizal, D. Felbacq, Multiplexing and demultiplexing with photonic crystals, J. Opt. A: Pure Appl. Opt. 1 (1999) 4. [16] S. Kim, I. Park, H. Lim, C.S. Kee, Highly efficient photonic crystal-based multichannel drop filters of three-port system with reflection feedback, Opt. Express 12 (2004) 8. [17] M. Koshiba, Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers, J. Lightwave Technol. 19 (2001) 1. [18] C.J. Jin, S.H. Fan, S.Z. Han, D.Z. Zhang, Reflectionless multichannel wavelength demultiplexer in a transmission resonator configuration, IEEE J. Quantum Electron. 39 (2003) 6. [19] A. Sharkawy, S.Y. Shi, D.W. Prather, Multichannel wavelength division multiplexing with photonic crystals, Appl. Opt. 40 (2001) 6. [20] S.G. Johnson, J.D. Joannopoulos, Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis, Opt. Express 8 (2001) 18. [21] B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, Wiley-interscience, New York, 1991.
Please cite this article in press as: G. Lin, et al., 1 × 4 optical multiplexer based on the self-collimation effect of 2D photonic crystal, Optik - Int. J. Light Electron Opt. (2014), http://dx.doi.org/10.1016/j.ijleo.2014.03.031