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Model of Vernier devices in silicon-on-insulator technology
Infrared Physics & Technology 65 (2014) 83–86
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Infrared Physics & Technology journal homepage: www.elsevier.com/locate/infrared
Model of Vernier devices in silicon-on-insulator technology Guofang Fan a,c,⇑, Yuan Li b, Chunguang Hu c, Lihua Lei c, Dong Zhao d, Hongyu Li c,f, Yunhan Luo e,⇑, Zhen Zhen a a
Key Laboratory of Photochemical Conversion and Optoelectronic Materials, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, 100190 Beijing, China State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Weijin Road, 300072 Tianjin, China c Shanghai Institute of Measurement and Testing Technology, National Center of Measurement and Testing for East China, National Center of Testing Technology, 201203 Shanghai, China d Department of Material Science, Fudan University, Shanghai 200433, China e Institute of Optoelectronic Engineering, Jinan University, Guangzhou 510632, China f College of Mechanical and Electronic Engineering, Shandong University of Science and Technology, Qingdao 266590, China b
h i g h l i g h t s A 3D FVFD model, specifically suited for high index contrast and smaller size waveguides, is presented. The model of coupling length is developed based on 3D FVFD, and verified by the experiments. Detailed and thorough simulations of Vernier devices are discussed.
a r t i c l e
i n f o
Article history: Received 14 January 2014 Available online 13 April 2014 Keywords: FVFD Vernier effect Series-coupled racetrack resonators
a b s t r a c t In order to increase the number of channels that could be multiplexed or demultiplexed in the dense wavelength division multiplexed (DWDM) system based on the resonators on silicon-on-insulator (SOI) technology, the Vernier effect in the series-coupled racetrack resonators is presented to extend the free spectral range (FSR) of the DWDM systems. A method is developed based on a matrix approach to simulate Vernier devices. A three-dimensional full vectorial finite difference (FVFD) model, specifically suited for high index contrast and smaller size waveguides, for example, a waveguide in SOI technology, is developed to obtain the properties of a waveguide. Finally, the Vernier effect in the two series-coupled racetrack resonators is experimentally verified with an improved FSR and interstitial resonance suppression. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction In recent years, ring and racetrack resonators in SOI technology have been under intense investigation to exploit their versatile applications in communications and sensing [1–5]. Among most of the applications, such as DWDM system [2–5], a large FSR of the resonator of a DWDM system is always desired in order to increase the number of channels that could be multiplexed or demultiplexed. However, FSR is inversely proportional to the optical length of the resonator structure and, therefore, to increase the FSR, the total length of the resonator must decrease, ⇑ Corresponding authors. Address: Key Laboratory of Photochemical Conversion and Optoelectronic Materials, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, 100190 Beijing, China. Tel.: +86 13001931753 (G. Fan). E-mail addresses: [email protected] (G. Fan), [email protected] (Y. Luo). http://dx.doi.org/10.1016/j.infrared.2014.04.002 1350-4495/Ó 2014 Elsevier B.V. All rights reserved.
which in turn, increases the bending losses, i.e., decreases the Q factor. By the way, when dimensions of the resonators are enough small as nano-cavities, the resonators can reach no-stop band optical response with extremely high FSR and high Q-factor [6,7]. One solution to this problem is to use the Vernier effect of the series-coupled resonators with different optical length, thus to extend the FSR without decrease of the total length of the resonator structure. Actually, there are some papers reported about Vernier effect [2,4,5,8], however, few detailed and thorough simulations of Vernier devices, for example, in the Ref. [2], the simulation is based on a 2-D finite difference approach. In this paper, we present a matrix model of series-coupled racetrack resonators exhibiting Vernier effect based on a 3D-FVFD, which is specifically suited for high index contrast and smaller size waveguides. Moreover, with increasing integration densities, the resonator in the Vernier devices will have smaller bend radiuses and coupling gaps, hence Vernier devices need more accurate
84
G. Fan et al. / Infrared Physics & Technology 65 (2014) 83–86
description. In addition, in order to obtain the cross- and selfcoupling coefficients in the simulation of Vernier devices, a coupling length according to coupled mode theory (CMT) is presented and calculated, and experiments are performed as a validation of the simulation. Finally, we fabricate a Vernier device based on two series-coupled racetrack resonators in SOI waveguide and take an experiment to demonstrate Vernier effect.
The architecture of the two series-coupled racetracks, commonly used to create Vernier effect, consists of two racetrack resonators in series coupled to one through port bus waveguide and one drop port bus waveguide as shown in Fig. 1.
¼
b2n1
tn
jkn
jkn
tn
a2n a2n1
ðn ¼ 1; 2; 3Þ
ð3Þ
where a1, a2, a3, a4, a5, a6 and b1, b2, b3, b4, b5, b6 are electric field components as shown in Fig. 1, k1, k2, k3 and t1, t2, t3 are crossand self-coupling coefficients, respectively, which describe the interaction intensity in the coupling region. Rearranging Eq. (3), we have
2. Model
b2n
b2n
"
¼
a2n
# jk1n b2n1 ðn ¼ 1; 2; 3Þ a2n1 jktn
tn jkn 1 jkn
ð4Þ
n
We also have
b2nþ1 a2nþ1
2.1. Theory of Vernier devices
"
¼
1
a2n ehn
0 1
#
b2n a2n
an 2 ehn 0
ðhn ¼ jbðLn þ pr n ÞÞðn ¼ 1; 2Þ ð5Þ
In the Vernier devices based on the series-coupled resonators, the FSR of Vernier devices is related to FSR of each racetrack resonator [9]. By considering the Vernier device of two series-coupled resonators as an example, we have
FSRVernier ¼ m1 FSR1 ¼ m2 FSR2
ð1Þ
where FSRVernier is FSR of the Vernier devices, FSR1 and FSR2 are FSR of each racetrack resonator of Vernier devices, m1 and m2 are integers, respectively. m1 and m2 determine the perimeter of each racetrack resonator in Vernier devices by:
m1 P1 ¼ m2 P2
ð2Þ
where P1 and P2 are the perimeter of each racetrack resonator in the Vernier devices, respectively. Hence, one should keep in mind about the design of Vernier devices based on series-coupled resonators: (1) the twin resonance peaks located between the main resonance peaks can be removed by setting that m2 should be equal to m11 (assuming m1 > m2); (2) If we increase m1 (and m2), the interstitial peak suppression decreases. Thus, m1 needs to be small enough to give adequate interstitial peak suppression; (3) the choice of P1 and P2 determines the FSR of Vernier devices; (4) the coupling factors need to be optimized to obtain high main resonance intensity, minimal main resonance splitting, and large interstitial peak suppression. 2.2. Model of Vernier devices We take the architecture as shown in Fig. 1 to explore the model of Vernier devices. Applying the matrix approach to a single coupler [10], we have
Add
a6 a5
b6 k3, t3
where L1, L2 are the length of straight waveguide, r1, r2 are the radiuses of the racetrack resonators; a1, a2 are the intensity loss factors of half of the racetracks, respectively; b is the propagation constant of the waveguides. Combining Eqs. (3)–(5), we have
b6 a6
¼
T1 T2 T3 T4
b1 a1
ð6Þ "
# T1 T2 t 1 a2 eh2 t2 1 0 a1 eh1 0 where ¼ 3 1 T3 T4 1 t3 a12 eh2 0 1 t2 a1 2 eh1 0 2 t 1 1 j is transfer matrix of the two series-coupled 1 t 1 k1 k2 k3 racetracks. If we assume a1 = 1, a6 = 0 (i.e., no added signal), the normalized intensity at the drop port and through port of the device will be, 1 2
#
"
2 2 b6 j T4T1 Pdrop ¼ ¼ ðT2 Þ k1 k2 k3 T3 a1
ð7Þ
2 b1 T42 Pthrough ¼ ¼ T3 a1
ð8Þ
2.3. Coupling length of the racetrack resonator and FVFD mode solver Based on the above discussion, the simulation needs the crossand self-coupling coefficients of the couplers. According to CMT, once the effective indexes of the even and odd modes are known, the cross- and self-coupling coefficients of the racetrack resonators, (those derived considering mode interference, dimensionless
Drop
b5
L2
r2
a4 a3
k2, t2
b4 b3
L1
r1
Input
a2 a1
k1, t1
b2 b1 Through
(a)
1 2
(b)
Fig. 1. The calculation structure (a) and the optical photograph and (b) of the two series-coupled racetrack resonators showing Vernier effect.
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G. Fan et al. / Infrared Physics & Technology 65 (2014) 83–86
Using the finite difference schemes, Eq. (12) can be expressed in an eigenvalue matrix form as:
axxE
axyE
ayxE
ayyE
Ex
Ey
and such that their modulus square represents the fraction of the coupled power), can be approximately given by [11]
k ¼ sin
pL jbL ejbL ; t ¼ cos e 2Lc 2Lc
pL
~ ~ ~ E ¼ jxl0 H r
ð10Þ
~ H ~ ¼ jxe0 er~ r E
ð11Þ
After some algebraic manipulation, the Maxwell’s equation can be expressed as a linear system of two differential equations where the x and y components of the electric and magnetic fields are coupled [13],
2 @2 @ 1 þ @y 2 Ex þ k0 er Ex þ @x er 2 > @ 1 @ @ 1 : @22 Ey þ @y er @y ðer Ey Þ þ k0 er Ey þ @y er @x
e
@ ð r Ex Þ @x
Ex
Ey
ð13Þ
ð9Þ
where Lc ¼ 2ðn kn Þ is the coupling length, b is the propagation effe effo constant of the waveguides, L is the length of straight waveguide of the racetrack resonators, neffe and neffo are the even and odd mode effective indexes, respectively. In our work, neffe and neffo are calculated by a FVFD mode solver, which is developed to simulate a SOI waveguide [12]. The FVFD mode solver is based on Maxwell’s equation for the complex amplitudes and a time dependence as exp(jxt) to solve the optical waveguide. Maxwell’s equations are,
8 @ 1 > < @x er
axxE axyE is the finite difference schemes, ayxE ayyE 2p b ¼ k N, k is free-space optical wavelength, N is effective index of optical waveguide. By solving the eigenvalue matrix of Eq. (13), we can obtain the even and odd effective indexes of the optical waveguide. Then, the coupling length can be calculated using the expression Lc ¼ 2ðn kn Þ for a spacing of 200 nm and 220 nm as shown in effe effo Fig. 2. As a comparison, the experimental results are also presented, which is deduced by fitting the measured throughput signal of the device. One can observe there is very good agreement between the simulation and the experimental results for the spacing of 200 nm and 220 nm. This means our FVFD can work well for the simulation of a SOI waveguide. where the matrix
Fig. 2. Coupling length (Lc) spectral variation of twin waveguide couplers for different spacing.
¼ b2
e
@ ð r Ey Þ @y
e
@ ð r Ex Þ @x
2
@ Ey ¼ b2 Ex @x@y 2
@ Ex ¼ b2 Ey @x@y
ð12Þ
3. Experimental demonstration In order to verify our model, we designed and fabricated the two series-coupled racetrack resonators structure on SOI technology. Here, the perimeter of the first racetrack is half of the perimeter of the second racetrack. As a comparison, we also fabricated a single racetrack resonator with same parameters as the larger racetrack of Vernier device on 200 nm line with microelectronics equipments. In order to increase the measurement accuracy, we designed the measurement setup shown in Fig. 3. In which, the input light is divided into two equal parts using a 1 2 MMI splitter. The first part serves as a reference. In the second part, we put the device into the measurement setup. The outputs from the drop port and the through port of the Vernier device are normalized by the output from the reference port. Also, in order to have efficient coupling, we use a taper waveguide to enlarge the input/output waveguide size to reduce the size of the standard fiber for a good match. In our devices, the size of the optical waveguide is 220 nm 450 nm, the first racetrack has a radius r1 = 7 lm and a coupling section L1 = 10 lm. The second racetrack has a radius r2 = 3.25 lm and a coupling section L2 = 5.8 lm. The longer racetrack has a circumference which is 2 times longer than the one
Fig. 3. The schematic diagram of the measurement setup.
0 -10 -20 -30
Single Vernier
-40 1.565 1.575 1.585 1.595 1.605 1.615
Drop port transmission (dB)
G. Fan et al. / Infrared Physics & Technology 65 (2014) 83–86
Drop port transmission (dB)
86
0 -4 -8 -12 -16 -20 1565
Single Vernier 1575
1585
1595
wavelength (µm)
Wavelength (nm)
(a)
(b)
1605
1615
Fig. 4. TE-polarized spectral response on drop port of a single (red) and Vernier (blue) racetrack configuration of (a) simulation and (b) experiments. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Drop port transmission (dB)
0
k =0.1 2
k2=0.2
-10
k =0.3 2
-20 -30
pling length according to CMT is calculated, and experiments are performed as a validation of the simulation. Compared with other models, our simulation shows more detailed and thorough. In order to validate the simulation, a Vernier device based on two series-coupled racetrack resonators in SOI waveguide has been fabricated, the experimental results shows good agreement with the simulation. Acknowledgments
-40 -50 1.596
1.598
1.6
1.602
1.604
wavelength (µm) Fig. 5. The simulation results of response on drop port of Vernier racetrack configuration for k2 = 0.1, 0.2 and 0.3 with same values of k1 and k3.
of the second racetrack. The gap between the bus waveguide and ring waveguide is chosen to be 0.18 lm. Fig. 4 shows TE-polarized response on the drop port of the Vernier device and a single racetrack resonator as a comparison, respectively. Fig. 4(a) shows the simulation results. In the simulation, we assume that there are no losses in the racetrack resonators (this means that a1 = a2 = 0). Fig. 4(b) shows the measured TE-polarized response on the drop port of the devices. The comparison clearly demonstrates the enlargement of FSR by Vernier effect. The extended FSR of the Vernier device is 2 times larger than the FSR of a single racetrack resonator as expected. The interstitial resonance suppression is greater than 8 dB. We note that in the Vernier device the resonances split due to the strong resonator coupling. The splitting can be avoided by decreasing the coupling between the two resonators as shown in Fig. 5, e.g., by enlarging the gap. The oscillation in Fig. 4(b) is due to Fabry– Perot effects in the periphery of the devices [14]. Although the interstitial peak suppression of our device is poor compared to other devices reported in literature. Anyway, we have developed a simulation method and performed an experiment to validate the model. 4. Conclusion In summary, we have developed a matrix model with a 3DFVFD to describe the Vernier devices based on series-coupled racetrack resonators, which is specifically suited for high index contrast and smaller size waveguides. In order to obtain the cross- and selfcoupling coefficients in the simulation of Vernier devices, a cou-
This work was supported by the Young Scientists Fund of the National Natural Science Foundation of China (Nos. 11104284 and 61107077), the Project PIL1304 and PIL1008 of State Key Laboratory of Precision Measuring Technology and Instruments. References [1] M.N. Vittorio Passaro, Benedetto Troia, Francesco De Leonardis, A generalized approach for design of photonic gas sensors based on Vernier-effect in mid-IR, Sens. Actuators B: Chem. 168 (20) (2012) 402–420. [2] Robi Boeck, Nicolas A. Jaeger, Nicolas Rouger, Lukas Chrostowski, Seriescoupled silicon racetrack resonators and the Vernier effect: theory and measurement, Opt. Express 18 (2010) 25151–25157. [3] S. Mookherjea, Linear and nonlinear light localization in coupled microresonators, in: A. Matsko (Ed.), Practical Applications of Microresonators in Optics and Photonics, Taylor and Francis, 2009. [4] S. Feng, T. Lei, H. Chen, H. Cai, X. Luo, A.W. Poon, Silicon photonics: from a microresonator perspective, Laser Photon. Rev. 6 (2) (2012) 145–177. [5] Michael Gad, Jason Ackert, David Yevick, Lukas Chrostowski, Paul E Jessop, Ring resonator wavelength division multiplexing interleaver, J. Lightwave Technol. 29 (14) (2011) 2102–2108. [6] Y. Chen et al., Ultrasensitive gas-phase chemical sensing based on functionalized photonic crystal nanobeam cavities, ACS Nano 08 (01) (2014) 522–527. [7] W.S. Fegadolli et al., Compact and low power consumption tunable photonic crystal nanobeam cavity, Opt. Express 21 (2013) 3861–3871. [8] William S. Fegadolli, German Vargas, Xuan Wang, Felipe Valini, Luis A.M. Barea, José E.B. Oliveira, Newton Frateschi, Axel Scherer, Vilson R. Almeida, Roberto R. Panepucci, Reconfigurable silicon thermo-optical ring resonator switch based on Vernier effect control, Opt. Express 20 (2012) 14722–14733. [9] O. Schwelb, The nature of spurious mode suppression in extended FSR microring multiplexers, Opt. Commun. 271 (2007) 424–429. [10] A. Yariv, Universal relations for coupling of optical power between microresonators and dielectric waveguides, Electron. Lett. 6 (4) (2000) 321– 322. [11] B.E. Little, W.P. Huang, Coupled-mode theory for optical waveguides, Prog. Electromagn. Res. 10 (1995) 217–270. [12] G. Fan, Y. Li, B. Han, Q. Wang, X. Liu, Z. Zhen, Study of an electro-optic polymer modulator, J. Lightwave Technol. 30 (15) (2012) 2482–2487. [13] J. Niehusmann, A. Vörckel, P. Haring Bolivar, T. Wahlbrink, W. Henschel, H. Kurz, Ultrahigh-quality-factor silicon-on-insulator microring resonator, Opt. Lett. 29 (2004) 2861–2863. [14] K. Okamoto, Fundamentals of Optical Waveguides, Academic Press, San Diego, 2006.
Contents lists available at ScienceDirect
Infrared Physics & Technology journal homepage: www.elsevier.com/locate/infrared
Model of Vernier devices in silicon-on-insulator technology Guofang Fan a,c,⇑, Yuan Li b, Chunguang Hu c, Lihua Lei c, Dong Zhao d, Hongyu Li c,f, Yunhan Luo e,⇑, Zhen Zhen a a
Key Laboratory of Photochemical Conversion and Optoelectronic Materials, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, 100190 Beijing, China State Key Laboratory of Precision Measuring Technology and Instruments, Tianjin University, Weijin Road, 300072 Tianjin, China c Shanghai Institute of Measurement and Testing Technology, National Center of Measurement and Testing for East China, National Center of Testing Technology, 201203 Shanghai, China d Department of Material Science, Fudan University, Shanghai 200433, China e Institute of Optoelectronic Engineering, Jinan University, Guangzhou 510632, China f College of Mechanical and Electronic Engineering, Shandong University of Science and Technology, Qingdao 266590, China b
h i g h l i g h t s A 3D FVFD model, specifically suited for high index contrast and smaller size waveguides, is presented. The model of coupling length is developed based on 3D FVFD, and verified by the experiments. Detailed and thorough simulations of Vernier devices are discussed.
a r t i c l e
i n f o
Article history: Received 14 January 2014 Available online 13 April 2014 Keywords: FVFD Vernier effect Series-coupled racetrack resonators
a b s t r a c t In order to increase the number of channels that could be multiplexed or demultiplexed in the dense wavelength division multiplexed (DWDM) system based on the resonators on silicon-on-insulator (SOI) technology, the Vernier effect in the series-coupled racetrack resonators is presented to extend the free spectral range (FSR) of the DWDM systems. A method is developed based on a matrix approach to simulate Vernier devices. A three-dimensional full vectorial finite difference (FVFD) model, specifically suited for high index contrast and smaller size waveguides, for example, a waveguide in SOI technology, is developed to obtain the properties of a waveguide. Finally, the Vernier effect in the two series-coupled racetrack resonators is experimentally verified with an improved FSR and interstitial resonance suppression. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction In recent years, ring and racetrack resonators in SOI technology have been under intense investigation to exploit their versatile applications in communications and sensing [1–5]. Among most of the applications, such as DWDM system [2–5], a large FSR of the resonator of a DWDM system is always desired in order to increase the number of channels that could be multiplexed or demultiplexed. However, FSR is inversely proportional to the optical length of the resonator structure and, therefore, to increase the FSR, the total length of the resonator must decrease, ⇑ Corresponding authors. Address: Key Laboratory of Photochemical Conversion and Optoelectronic Materials, Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, 100190 Beijing, China. Tel.: +86 13001931753 (G. Fan). E-mail addresses: [email protected] (G. Fan), [email protected] (Y. Luo). http://dx.doi.org/10.1016/j.infrared.2014.04.002 1350-4495/Ó 2014 Elsevier B.V. All rights reserved.
which in turn, increases the bending losses, i.e., decreases the Q factor. By the way, when dimensions of the resonators are enough small as nano-cavities, the resonators can reach no-stop band optical response with extremely high FSR and high Q-factor [6,7]. One solution to this problem is to use the Vernier effect of the series-coupled resonators with different optical length, thus to extend the FSR without decrease of the total length of the resonator structure. Actually, there are some papers reported about Vernier effect [2,4,5,8], however, few detailed and thorough simulations of Vernier devices, for example, in the Ref. [2], the simulation is based on a 2-D finite difference approach. In this paper, we present a matrix model of series-coupled racetrack resonators exhibiting Vernier effect based on a 3D-FVFD, which is specifically suited for high index contrast and smaller size waveguides. Moreover, with increasing integration densities, the resonator in the Vernier devices will have smaller bend radiuses and coupling gaps, hence Vernier devices need more accurate
84
G. Fan et al. / Infrared Physics & Technology 65 (2014) 83–86
description. In addition, in order to obtain the cross- and selfcoupling coefficients in the simulation of Vernier devices, a coupling length according to coupled mode theory (CMT) is presented and calculated, and experiments are performed as a validation of the simulation. Finally, we fabricate a Vernier device based on two series-coupled racetrack resonators in SOI waveguide and take an experiment to demonstrate Vernier effect.
The architecture of the two series-coupled racetracks, commonly used to create Vernier effect, consists of two racetrack resonators in series coupled to one through port bus waveguide and one drop port bus waveguide as shown in Fig. 1.
¼
b2n1
tn
jkn
jkn
tn
a2n a2n1
ðn ¼ 1; 2; 3Þ
ð3Þ
where a1, a2, a3, a4, a5, a6 and b1, b2, b3, b4, b5, b6 are electric field components as shown in Fig. 1, k1, k2, k3 and t1, t2, t3 are crossand self-coupling coefficients, respectively, which describe the interaction intensity in the coupling region. Rearranging Eq. (3), we have
2. Model
b2n
b2n
"
¼
a2n
# jk1n b2n1 ðn ¼ 1; 2; 3Þ a2n1 jktn
tn jkn 1 jkn
ð4Þ
n
We also have
b2nþ1 a2nþ1
2.1. Theory of Vernier devices
"
¼
1
a2n ehn
0 1
#
b2n a2n
an 2 ehn 0
ðhn ¼ jbðLn þ pr n ÞÞðn ¼ 1; 2Þ ð5Þ
In the Vernier devices based on the series-coupled resonators, the FSR of Vernier devices is related to FSR of each racetrack resonator [9]. By considering the Vernier device of two series-coupled resonators as an example, we have
FSRVernier ¼ m1 FSR1 ¼ m2 FSR2
ð1Þ
where FSRVernier is FSR of the Vernier devices, FSR1 and FSR2 are FSR of each racetrack resonator of Vernier devices, m1 and m2 are integers, respectively. m1 and m2 determine the perimeter of each racetrack resonator in Vernier devices by:
m1 P1 ¼ m2 P2
ð2Þ
where P1 and P2 are the perimeter of each racetrack resonator in the Vernier devices, respectively. Hence, one should keep in mind about the design of Vernier devices based on series-coupled resonators: (1) the twin resonance peaks located between the main resonance peaks can be removed by setting that m2 should be equal to m11 (assuming m1 > m2); (2) If we increase m1 (and m2), the interstitial peak suppression decreases. Thus, m1 needs to be small enough to give adequate interstitial peak suppression; (3) the choice of P1 and P2 determines the FSR of Vernier devices; (4) the coupling factors need to be optimized to obtain high main resonance intensity, minimal main resonance splitting, and large interstitial peak suppression. 2.2. Model of Vernier devices We take the architecture as shown in Fig. 1 to explore the model of Vernier devices. Applying the matrix approach to a single coupler [10], we have
Add
a6 a5
b6 k3, t3
where L1, L2 are the length of straight waveguide, r1, r2 are the radiuses of the racetrack resonators; a1, a2 are the intensity loss factors of half of the racetracks, respectively; b is the propagation constant of the waveguides. Combining Eqs. (3)–(5), we have
b6 a6
¼
T1 T2 T3 T4
b1 a1
ð6Þ "
# T1 T2 t 1 a2 eh2 t2 1 0 a1 eh1 0 where ¼ 3 1 T3 T4 1 t3 a12 eh2 0 1 t2 a1 2 eh1 0 2 t 1 1 j is transfer matrix of the two series-coupled 1 t 1 k1 k2 k3 racetracks. If we assume a1 = 1, a6 = 0 (i.e., no added signal), the normalized intensity at the drop port and through port of the device will be, 1 2
#
"
2 2 b6 j T4T1 Pdrop ¼ ¼ ðT2 Þ k1 k2 k3 T3 a1
ð7Þ
2 b1 T42 Pthrough ¼ ¼ T3 a1
ð8Þ
2.3. Coupling length of the racetrack resonator and FVFD mode solver Based on the above discussion, the simulation needs the crossand self-coupling coefficients of the couplers. According to CMT, once the effective indexes of the even and odd modes are known, the cross- and self-coupling coefficients of the racetrack resonators, (those derived considering mode interference, dimensionless
Drop
b5
L2
r2
a4 a3
k2, t2
b4 b3
L1
r1
Input
a2 a1
k1, t1
b2 b1 Through
(a)
1 2
(b)
Fig. 1. The calculation structure (a) and the optical photograph and (b) of the two series-coupled racetrack resonators showing Vernier effect.
85
G. Fan et al. / Infrared Physics & Technology 65 (2014) 83–86
Using the finite difference schemes, Eq. (12) can be expressed in an eigenvalue matrix form as:
axxE
axyE
ayxE
ayyE
Ex
Ey
and such that their modulus square represents the fraction of the coupled power), can be approximately given by [11]
k ¼ sin
pL jbL ejbL ; t ¼ cos e 2Lc 2Lc
pL
~ ~ ~ E ¼ jxl0 H r
ð10Þ
~ H ~ ¼ jxe0 er~ r E
ð11Þ
After some algebraic manipulation, the Maxwell’s equation can be expressed as a linear system of two differential equations where the x and y components of the electric and magnetic fields are coupled [13],
2 @2 @ 1 þ @y 2 Ex þ k0 er Ex þ @x er 2 > @ 1 @ @ 1 : @22 Ey þ @y er @y ðer Ey Þ þ k0 er Ey þ @y er @x
e
@ ð r Ex Þ @x
Ex
Ey
ð13Þ
ð9Þ
where Lc ¼ 2ðn kn Þ is the coupling length, b is the propagation effe effo constant of the waveguides, L is the length of straight waveguide of the racetrack resonators, neffe and neffo are the even and odd mode effective indexes, respectively. In our work, neffe and neffo are calculated by a FVFD mode solver, which is developed to simulate a SOI waveguide [12]. The FVFD mode solver is based on Maxwell’s equation for the complex amplitudes and a time dependence as exp(jxt) to solve the optical waveguide. Maxwell’s equations are,
8 @ 1 > < @x er
axxE axyE is the finite difference schemes, ayxE ayyE 2p b ¼ k N, k is free-space optical wavelength, N is effective index of optical waveguide. By solving the eigenvalue matrix of Eq. (13), we can obtain the even and odd effective indexes of the optical waveguide. Then, the coupling length can be calculated using the expression Lc ¼ 2ðn kn Þ for a spacing of 200 nm and 220 nm as shown in effe effo Fig. 2. As a comparison, the experimental results are also presented, which is deduced by fitting the measured throughput signal of the device. One can observe there is very good agreement between the simulation and the experimental results for the spacing of 200 nm and 220 nm. This means our FVFD can work well for the simulation of a SOI waveguide. where the matrix
Fig. 2. Coupling length (Lc) spectral variation of twin waveguide couplers for different spacing.
¼ b2
e
@ ð r Ey Þ @y
e
@ ð r Ex Þ @x
2
@ Ey ¼ b2 Ex @x@y 2
@ Ex ¼ b2 Ey @x@y
ð12Þ
3. Experimental demonstration In order to verify our model, we designed and fabricated the two series-coupled racetrack resonators structure on SOI technology. Here, the perimeter of the first racetrack is half of the perimeter of the second racetrack. As a comparison, we also fabricated a single racetrack resonator with same parameters as the larger racetrack of Vernier device on 200 nm line with microelectronics equipments. In order to increase the measurement accuracy, we designed the measurement setup shown in Fig. 3. In which, the input light is divided into two equal parts using a 1 2 MMI splitter. The first part serves as a reference. In the second part, we put the device into the measurement setup. The outputs from the drop port and the through port of the Vernier device are normalized by the output from the reference port. Also, in order to have efficient coupling, we use a taper waveguide to enlarge the input/output waveguide size to reduce the size of the standard fiber for a good match. In our devices, the size of the optical waveguide is 220 nm 450 nm, the first racetrack has a radius r1 = 7 lm and a coupling section L1 = 10 lm. The second racetrack has a radius r2 = 3.25 lm and a coupling section L2 = 5.8 lm. The longer racetrack has a circumference which is 2 times longer than the one
Fig. 3. The schematic diagram of the measurement setup.
0 -10 -20 -30
Single Vernier
-40 1.565 1.575 1.585 1.595 1.605 1.615
Drop port transmission (dB)
G. Fan et al. / Infrared Physics & Technology 65 (2014) 83–86
Drop port transmission (dB)
86
0 -4 -8 -12 -16 -20 1565
Single Vernier 1575
1585
1595
wavelength (µm)
Wavelength (nm)
(a)
(b)
1605
1615
Fig. 4. TE-polarized spectral response on drop port of a single (red) and Vernier (blue) racetrack configuration of (a) simulation and (b) experiments. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Drop port transmission (dB)
0
k =0.1 2
k2=0.2
-10
k =0.3 2
-20 -30
pling length according to CMT is calculated, and experiments are performed as a validation of the simulation. Compared with other models, our simulation shows more detailed and thorough. In order to validate the simulation, a Vernier device based on two series-coupled racetrack resonators in SOI waveguide has been fabricated, the experimental results shows good agreement with the simulation. Acknowledgments
-40 -50 1.596
1.598
1.6
1.602
1.604
wavelength (µm) Fig. 5. The simulation results of response on drop port of Vernier racetrack configuration for k2 = 0.1, 0.2 and 0.3 with same values of k1 and k3.
of the second racetrack. The gap between the bus waveguide and ring waveguide is chosen to be 0.18 lm. Fig. 4 shows TE-polarized response on the drop port of the Vernier device and a single racetrack resonator as a comparison, respectively. Fig. 4(a) shows the simulation results. In the simulation, we assume that there are no losses in the racetrack resonators (this means that a1 = a2 = 0). Fig. 4(b) shows the measured TE-polarized response on the drop port of the devices. The comparison clearly demonstrates the enlargement of FSR by Vernier effect. The extended FSR of the Vernier device is 2 times larger than the FSR of a single racetrack resonator as expected. The interstitial resonance suppression is greater than 8 dB. We note that in the Vernier device the resonances split due to the strong resonator coupling. The splitting can be avoided by decreasing the coupling between the two resonators as shown in Fig. 5, e.g., by enlarging the gap. The oscillation in Fig. 4(b) is due to Fabry– Perot effects in the periphery of the devices [14]. Although the interstitial peak suppression of our device is poor compared to other devices reported in literature. Anyway, we have developed a simulation method and performed an experiment to validate the model. 4. Conclusion In summary, we have developed a matrix model with a 3DFVFD to describe the Vernier devices based on series-coupled racetrack resonators, which is specifically suited for high index contrast and smaller size waveguides. In order to obtain the cross- and selfcoupling coefficients in the simulation of Vernier devices, a cou-
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