1. Introduction

2. Principle of Operation

Figure 1. The schematic diagram of a typical Mach-Zehnder interferometer (MZI) sensor based on optical waveguide.

Figure 1. The schematic diagram of a typical Mach-Zehnder interferometer (MZI) sensor based on optical waveguide.

3. Section Dimensions of SOI Rib Waveguide

3.1. Intensity of Evanescent Field

The fundamental principle of optical waveguide MZI sensor is based on the evanescent field sensing. According to the Goos-Hanchen effects, the evanescent field is a fraction of optical field that extends to the cladding layer and the substrate layer of the waveguide. In general, there are two types of evanescent field sensing, homogeneous sensing, and surface sensing [14]. The homogeneous sensing and surface sensing for SOI waveguide with large cross section are shown inFigure 2a,b respectively. For the homogeneous sensing, the effective index variation of the propagating mode is produced by the change of the refractive index in the sensitive region. While for the surface sensing, the effective index variation is produced by the change of the thickness of the ultra-thin sensitive layer which is immobilized on the waveguide surface. The thickness of the surface sensitive layer is denoted ast (shown inFigure 2b), which is changed at a small range, about several nanometers. Homogeneous sensing is generally used to detect the concentration change of gas or liquid in the entire sensitive region, and the surface sensing is often applied to detect protein, DNA, virus, and bacteria with the help of immobilized receptor molecules [15].

Figure 2. The schematic diagram of sensitive region and sensitive layer at silicon-on-insulator (SOI) rib waveguide with large cross section. (a) Homogeneous sensing, the light olive region is the sensitive region; (b) Surface sensing, the blue region represents the sensitive layer, and its thickness is denoted byt.

Figure 2. The schematic diagram of sensitive region and sensitive layer at silicon-on-insulator (SOI) rib waveguide with large cross section. (a) Homogeneous sensing, the light olive region is the sensitive region; (b) Surface sensing, the blue region represents the sensitive layer, and its thickness is denoted byt.

Obviously, the greater the intensity of the evanescent field is, the more sensitive the sensor both in the case of homogeneous sensing and surface sensing will be. The intensity of the evanescent field can be represented by the confinement factorΓ_s, which is the ratio of the electric field intensity in the sensitive region or the sensitive layer to the entire mode distribution of the guiding mode [15], defined as Equation (4).

$${\mathrm{\Gamma }}_s=\frac{{\displaystyle \underset{s}{\iint }{\left|E\left(x,y\right)\right|}^{2}dxdy}}{{\displaystyle \underset{\infty }{\iint }{\left|E\left(x,y\right)\right|}^{2}dxdy}}$$

(4)

Generally, in order to achieve low-loss propagation and avoid negative influences due to multimode transmission or cross-polarization interference, the optical waveguides employed in the senor must possess single mode and single polarization. In this article, the SOI rib waveguides with large cross section will be confined by single mode conditions. Once the mode distributions are solved by mode solver programs, such as finite element method (FEM) and finite difference method (FDM), the intensity of the evanescent field can be calculated through area integration. Therefore, taking the maximization of the intensity of the evanescent field as the target, the optimization of waveguide section dimensions can be realized.

3.2. Optimization of Waveguide Section Dimensions

As an example, suppose the sensor based on SOI rib waveguide with large cross section is used to detect the concentration of glucose solution and the refractive indexn_c of the analyte solution is in the vicinity of 1.33. In order to guarantee high coupling efficiency with the standard single-mode fiber, the total rib height (H) of SOI rib waveguide is set to be 10 μm, and the operating wavelength is selected at 1550 nm so that the refractive index of silicon and SiO₂ are 3.476 and 1.444, respectively. Considering too small a rib width (w) will reduce the restriction of the rib structure on the guiding mode and too big a rib width (w) will be difficult for system integration (such as the waveguide bends and branches analyzed in next section), the rib width (w) of SOI rib waveguide is set to be in the range of 2.5 μm to 10 μm.

The modes of rib waveguide can be denoted as HE_nm or EH_nm [16], wheren = 0, 1, 2…, andm = 0, 1, 2…. HE mode and EH mode are commonly known as quasi-transverse-electric mode and quasi-transverse-magnetic mode respectively. Employing a strict single-mode condition [17], all the SOI rib waveguides with large cross section only support the fundamental guiding modes for each polarization,i.e. HE₀₀ and EH₀₀. Using the full-vector finite difference method (FDM) [18] to solve the fundamental guiding modes with different dimensions (including the outside rib heighth and the rib widthw), the intensity of the evanescent field for homogeneous sensing can be calculated, as shown inFigure 3.

Figure 3. Dependence of the intensity of the evanescent field for homogeneous sensing on rib width (w) with different outside rib height (h). (a) HE polarization; (b) EH polarization.

Figure 3. Dependence of the intensity of the evanescent field for homogeneous sensing on rib width (w) with different outside rib height (h). (a) HE polarization; (b) EH polarization.

It can be seen fromFigure 3, for an SOI rib waveguide with the total rib height of 10 μm, the evanescent field intensity of homogeneous sensing exhibits a maximum of 7.436 × 10⁻³% for HE polarization, corresponding toh = 5 μm andw = 4.5 μm, and 9.407 × 10⁻³% for EH polarization, corresponding toh = 5 μm andw = 2.5 μm.

Using the same method, dependence of the evanescent field intensity for surface sensing on rib width (w) with different outside rib height (h) also can be calculated. Assuming the refractive index of the sensitive layern_m = 1.45 and the thickness of the sensitive layert = 10 nm, the evanescent field intensity for surface sensing of the SOI rib waveguide with the total rib height of 10 μm possesses a maximum of 5.089 × 10⁻³% for HE polarization and 7.924 × 10⁻³% for EH polarization, both corresponding toh = 5 μm andw = 2.5 μm. Both for homogeneous sensing and surface sensing, it is evident that EH polarization of the SOI rib waveguide with large cross section is significantly more sensitive than HE polarization due to the larger evanescent field intensity.

According to the variational theorem for dielectric waveguides, an analytical method can be used to estimate the partial sensitivity of evanescent field sensing based on the calculated evanescent field intensity [17,18]. Therefore, the partial sensitivity of homogeneous sensing (ΔN_eff/Δn) possesses a maximum of 2.86 × 10⁻³ for HE polarization (corresponding toh = 5 μm,w = 4.5 μm) and 3.6 × 10⁻³ for EH polarization (corresponding toh = 5 μm,w = 2.5 μm), and the partial sensitivity of surface sensing (ΔN_eff/Δt) exhibits a maximum of 4.89 × 10⁻⁶ nm⁻¹ for HE polarization and 7.61 × 10⁻⁶ nm⁻¹ for EH polarization (both corresponding toh = 5μm,w = 2.5μm). Therefore, the result of the optimization is that the waveguide mode is the fundamental EH mode, and the cross section dimension of the SOI rib waveguide isH = 10 μm, h = 5 μm, w = 2.5 μm. The simulation of the optimized mode is shown inFigure 4.

Figure 4. Simulation of the fundamental EH mode of the SOI rib waveguide withH = 10 μm,h = 5 μm,w = 2.5 μm at wavelength of 1550 nm, calculating with full-vector finite difference method (FDM).

Figure 4. Simulation of the fundamental EH mode of the SOI rib waveguide withH = 10 μm,h = 5 μm,w = 2.5 μm at wavelength of 1550 nm, calculating with full-vector finite difference method (FDM).

4. MZI Structure Implementation

4.1. Conventional Implementations

In the waveguide MZI sensor, the evanescent field sensing is read out by the MZI configuration. The most critical components of the MZI configuration are the beam splitter and the beam combiner, which are waveguide branches and are often identical. For the conventional MZI configurations, there are three structures to realize waveguide branches, as listed inTable 1. The data in this table show the technical parameters and the minimum required length of a single branch to fulfill the separation distance ofd = 50 μm between the two straight SOI rib waveguides with large cross section. Thus for the SOI rib waveguide with a large cross section, the conventional implementations of waveguide branches lead to an overlong structure which is difficult to be realized.

Table 1. The conventional implementations of waveguide branches for SOI rib waveguide with large cross section. AssumingH = 10 μm,h = 5 μm,w = 5 μm.

Table 1. The conventional implementations of waveguide branches for SOI rib waveguide with large cross section. AssumingH = 10 μm,h = 5 μm,w = 5 μm.

Structure Type	Technology Platform	Technical Parameters & Requirements
Y-junction	Mode-matching	Whend = 50 μm, 2θ = 0.4°, The minimum length of a single branch:L0 = 7.2 mm.
S-bend splitter	Waveguide bending	Whend = 50 μm, R > 0.26 m, The minimum length of a single branch:L0 = 5.1 mm.
Multimode interference	Self-imaging effect [19]	Whend = 50 μm, The minimum length of a single branch:L0 > 12 mm.

4.2. Trench-Based Bend and Branch

Studies have shown that the bends and branches of waveguides can be realized by using the medium trench, slot, or photonic crystal [20,21,22,23,24]. In this paper, media-filled trenches are used to achieve waveguide bends and branches for an SOI rib waveguide with a large cross section, as shown inFigure 5 andFigure 6, respectively. The gray area in these figures represents the filling medium, which can be air, SU8 or other refractive index matching fluid.

Figure 5. 90° trench-based bend geometry of SOI rib waveguide with large cross section.

Figure 5. 90° trench-based bend geometry of SOI rib waveguide with large cross section.

Figure 6. T-shaped branch geometry of SOI rib waveguide with large cross section.

Figure 6. T-shaped branch geometry of SOI rib waveguide with large cross section.

Due to the large dimension of the SOI rib waveguide, the computational memory requirement and time-consumption of a three-dimensional finite difference-time domain (3D-FDTD) simulation are huge. The usual way to overcome this issue is to simplify the 3D structure to 2D by using effective refractive index method (EIM), and then simulate the interested structure using two-dimensional finite difference-time domain (2D-FDTD) [25]. In this paper, a 2D-FDTD method with a perfectly matched layer (PML) boundary condition [26] is employed to numerically simulate the above mentioned waveguide bends and branches.

The medium trench shown inFigure 5 corresponds to a corner mirror, which can change the propagation direction of the waveguide according to total internal reflection. In general, the length and width of the medium trench (L andW as shown inFigure 5) should be large enough to reflect instead of to transmit more mode energy. The parameterD is defined to account for the Goos-Hanchen shift compensation, and it is positive when the trench interface moves away from the bending center (corresponding to the case shown inFigure 5,D > 0). When the trench interface precisely passes through the bending center,D = 0. Setting the width and length of the air trench at 10 μm and 30 μm respectively,i.e. L = 30 μm,W = 10 μm, the electric intensity map of a 90° air-trench bend is shown inFigure 7.

The influence of the parameterD on the bend efficiency of the air-trench bend exhibits as a curve with oscillatory variation in a very small range, as shown inFigure 8. Due to the strong ability of light constraint, there is a high bend efficiency for SOI rib waveguide with large cross section, even whenD = ±500 nm. The reason for this oscillatory variation is that there is an angle between the air-trench and the meshing direction in 2D-FDTD simulation to avoid oblique incident light, and the interface of the air-trench and the waveguide is jagged as shown inFigure 9. It can be verified that this effect will be attenuated if finer mesh sizes are adopted.

Figure 7. The electric intensity map in a plane 4 μm above the SiO₂ layer for a 90° air-trench bend at a wavelength of 1550 nm. The SOI rib waveguide with a large cross section possesses a total rib height of 10 μm, an outside rib height of 5 μm, and a rib width of 2.5 μm. The guiding mode is the fundamental EH mode. The width and length of the air trench is 10 μm and 30 μm respectively, and the Goos-Hanchen shift compensationD = 0. Bend efficiency calculated through the 2D-FDTD simulation with mesh grid of 5 nm is 0.999746.

Figure 7. The electric intensity map in a plane 4 μm above the SiO₂ layer for a 90° air-trench bend at a wavelength of 1550 nm. The SOI rib waveguide with a large cross section possesses a total rib height of 10 μm, an outside rib height of 5 μm, and a rib width of 2.5 μm. The guiding mode is the fundamental EH mode. The width and length of the air trench is 10 μm and 30 μm respectively, and the Goos-Hanchen shift compensationD = 0. Bend efficiency calculated through the 2D-FDTD simulation with mesh grid of 5 nm is 0.999746.

Figure 8. The bend efficiency as a function of the Goos-Hanchen shift compensation (parameterD). There is an oscillatory variation, but the amplitude is very small.

Figure 8. The bend efficiency as a function of the Goos-Hanchen shift compensation (parameterD). There is an oscillatory variation, but the amplitude is very small.

Figure 9. The local meshing in the 2D-FDTD simulation at the interface between air-trench and the bending waveguide. The mesh grid is 5 nm.

Figure 9. The local meshing in the 2D-FDTD simulation at the interface between air-trench and the bending waveguide. The mesh grid is 5 nm.

Similarly, the electric intensity map of the asymmetric T-shaped air-trench branch with the trench width and length of 97 nm and 30 μm respectively is shown inFigure 10. Based on 2D-FDTD simulation with the mesh grid of 5 nm, the reflection efficiency is 0.497357 and the transmission efficiency is 0.502518.

Figure 10. The electric intensity map in a plane 4 μm above the SiO₂ layer for an asymmetric T-shaped air-trench branch at a wavelength of 1550 nm. The SOI rib waveguide with a large cross section possesses total rib height of 10 μm, outside rib height of 5 μm, and rib width of 2.5 μm. The guiding mode is the fundamental EH mode. The width and length of the air trench is 97 nm and 30 μm respectively.

Figure 10. The electric intensity map in a plane 4 μm above the SiO₂ layer for an asymmetric T-shaped air-trench branch at a wavelength of 1550 nm. The SOI rib waveguide with a large cross section possesses total rib height of 10 μm, outside rib height of 5 μm, and rib width of 2.5 μm. The guiding mode is the fundamental EH mode. The width and length of the air trench is 97 nm and 30 μm respectively.

It is found from the simulations that the splitting efficiencies, including the reflection efficiency, the transmission efficiency and the total transmission efficiency, are functions of trench width (w), as shown inFigure 11. Both for air-filled trench and SU8-filled trench, the transmission decreases and the reflection increases as the trench width increases. With the trench width of 96 nm for air-trench branch and 112 nm for SU8-trench branch, a splitting ratio of 50%:50% is achieved.

Figure 11. Splitting efficiency as a function of trench width. A 50%:50% splitting ratio is achieved with 96 nm air-trench width or 112 nm SU8-trench width.

Figure 11. Splitting efficiency as a function of trench width. A 50%:50% splitting ratio is achieved with 96 nm air-trench width or 112 nm SU8-trench width.

Supposing the trench branch is filled with index matching fluid and has a trench width of 120 nm, the influence of the refractive index of the index matching fluid is shown inFigure 12. As expected, the higher refractive the index of the filled material, the smaller the reflection efficiency and the larger the transmission efficiency. Thus, it can be concluded that the higher refractive index of the filled material results in the larger trench width for a 50%:50% splitting ratio. In addition, using index matching fluid as the filled material for the trench branch can prevent pollution of dust or impurities in the air.

Figure 12. Splitter efficiency as a function of refractive index of trench. When the trench width is 0.12 μm, a 50%:50% splitting ratio is achieved with the refractive index of 1.76.

Figure 12. Splitter efficiency as a function of refractive index of trench. When the trench width is 0.12 μm, a 50%:50% splitting ratio is achieved with the refractive index of 1.76.

Overall, the trench-based bends and branches can be used for SOI rib waveguide with large cross section, and their performances are obviously superior to the traditional bending and branching structures, such as the higher transmission efficiency and the shorter length of a single branch. Note that the simulation errors mainly come from two aspects, the coupling between modes and the out-of-plane scattering loss, in the process of simplifying the 3D model to 2D [27]. Because the SOI rib waveguides discussed in this paper are confined by the single-mode condition and have strong capability of light constraint, the influences caused by these two factors are very small, and thus the error of the above results obtained from 2D-FDTD simulations is insignificant.

4.3. Proposed MZI Structure

A new MZI structure based on an SOI rib waveguide with a large cross section that consists of two trench-based waveguide bends, two trench-based waveguide branches, and two straight waveguides is proposed as shown inFigure 13. These two parallel straight waveguides can act as the reference arm and the sensing arm of the MZI sensor, and their spacing is denoted byd. The parameterS represents the horizontal distance of the two waveguide branches. Compared with the traditional MZI configurations, the proposed configuration possesses two out ports, which can be used solely, or synchronously as mutual reference.

Figure 13. Schematic of the MZI using media trenches based on SOI rib waveguide with large cross section.

Figure 13. Schematic of the MZI using media trenches based on SOI rib waveguide with large cross section.

Using a 2D-FDTD simulation, the optical power map of such a trench-based MZI structure withd = 50 μm,S = 80 μm is shown inFigure 14. The guiding mode is the fundamental EH mode of the SOI rib waveguide withH = 10 μm,h = 5 μm,w =2.5 μm at wavelength of 1550 nm. It shows that only one output port has power output due to the interference effect of the two identical guiding modes in the sensing arm and the reference arm. The transmission of the electric field components in the Z directionE_z (out of plane) can be used to clearly explain this interference case, as shown inFigure 15.

Figure 14. The optical power map of the MZI structure with the fundamental EH-polarized mode of an SOI rib waveguide withH = 10 μm,h = 5 μm,w = 2.5 μm at wavelength of 1550 nm. Because there is no phase difference between the sensing arm and the reference arm, only the port “output 1” has an output signal.

Figure 14. The optical power map of the MZI structure with the fundamental EH-polarized mode of an SOI rib waveguide withH = 10 μm,h = 5 μm,w = 2.5 μm at wavelength of 1550 nm. Because there is no phase difference between the sensing arm and the reference arm, only the port “output 1” has an output signal.

Figure 15. The transmission of the electric field components in Z direction Ez (out of plane) of the MZI structure with EH-polarized mode of SOI rib waveguide withH = 10 μm,h = 5 μm,w = 2.5 μm at wavelength of 1550 nm.

Figure 15. The transmission of the electric field components in Z direction Ez (out of plane) of the MZI structure with EH-polarized mode of SOI rib waveguide withH = 10 μm,h = 5 μm,w = 2.5 μm at wavelength of 1550 nm.

Figure 16a,b respectively show the normalized power of the SOI rib waveguide mode that propagates along the sensing arm and the reference arm. This normalized power is the ratio of the power on the transmission cross section to the input power of the light source. It can be seen that there are unstable regions at the waveguide bends and branches, which are caused by the interference of the waveguide modes. In particular, the normalized power of the interference enhancement is more than1. It is found that the length of the unstable transmission segments at the waveguide bends or branches are less than 20 μm, which is a great advantage compared to the conventional implementations of MZI.

Figure 16. Normalized power of the SOI rib waveguide mode propagating along the sensing arm and the reference arm.

Figure 16. Normalized power of the SOI rib waveguide mode propagating along the sensing arm and the reference arm.

In the ideal situation (the splitting ratio of every branch in the MZI is 50%:50%), the normalized output power of each output port is a function of the phase difference (ΔΦ) between the sensing arm and the reference arm, as shown inFigure 17. A complementary relationship between the two output ports can be found. The normalized output powers of the output port “output 1” can be expressed as Equation (5). There,ΔΦ₀ is the initial phase difference caused by machining error or other unbalanced factor.

$$P\text{=}\frac{1}{2}\left[1+\cos (\mathrm{\Delta }\mathrm{\Phi }+\mathrm{\Delta }{\mathrm{\Phi }}_0)\right]$$

(5)

Figure 17. The normalized output power in each output port as a function of the phase difference between the sensing arm and the reference arm.

Figure 17. The normalized output power in each output port as a function of the phase difference between the sensing arm and the reference arm.

5. MZI Sensing Platform

Figure 18. Schematic of the MZI sensing platform based on SOI rib waveguide with large cross section.

Figure 18. Schematic of the MZI sensing platform based on SOI rib waveguide with large cross section.

6. Discussion

7. Conclusions

References

1. Hong, J.; Choi, J.S.; Han, G.; Kang, J.K.; Kim, C.; Kim, T.S.; Yoon, D.S. A Mach-Zehnder interferometer based on silicon oxides for biosensor applications.Anal. Chim. Acta.2006,573–574, 97–103. [Google Scholar] [CrossRef] [PubMed]
2. Yalcin, A.; Popat, K.C.; Aldridge, J.C.; Desai, T.A.; Hryniewicz, J.; Chbouki, N.; Little, B.E.; King, O.; Van, V.; Chu, S.;et al. Optical sensing of biomolecules using microring resonators.IEEE J. Sel. Top. Quant.2006,12, 148–155. [Google Scholar] [CrossRef]
3. De Vos, K.; Bartolozzi, I.; Schacht, E.; Bienstman, P.; Baets, R. Silicon-on-Insulator microring resonator for sensitive and label-free biosensing.Opt. Express2007,15, 7610–7615. [Google Scholar]
4. Claes, T.; Molera, J.G.; de Vos, K.; Schacht, E.; Baets, R.; Bienstman, P. Label-free biosensing with a slot-waveguide-based ring resonator in silicon on insulator.IEEE Photonics J.2009,1, 197–204. [Google Scholar] [CrossRef]
5. Qi, Z.; Matsuda, N.; Itoh, K.; Murabayashi, M.; Lavers, C.R. A design for improving the sensitivity of a Mach-Zehnder interferometer to chemical and biological measurands.Sens. Actuators B Chem.2002,81, 254–258. [Google Scholar] [CrossRef]
6. Irawan, D.; Saktioto, T.; Ali, J.; Yupapin, P. Design of Mach-Zehnder interferometer and ring resonator for biochemical sensing.Photonics Sens.2015,5, 12–18. [Google Scholar] [CrossRef]
7. Fan, X.; White, I.M.; Shopova, S.I.; Zhu, H.; Suter, J.D.; Sun, Y. Sensitive optical biosensors for unlabeled targets: A review.Anal. Chim. Acta.2008,620, 8–26. [Google Scholar] [CrossRef] [PubMed]
8. Liu, Q.; Tu, X.; Kim, K.W.; Kee, J.S.; Shin, Y.; Han, K.; Yoon, Y.; Lo, G.; Park, M.K. Highly sensitive Mach-Zehnder interferometer biosensor based on silicon nitride slot waveguide.Sens. Actuators B Chem.2013,188, 681–688. [Google Scholar] [CrossRef]
9. Densmore, A.; Xu, D.X.; Waldron, P.; Janz, S.; Cheben, P.; Lapointe, J.; Delâge, A.; Lamontagne, B.; Schmid, J.H.; Post, E. A silicon-on-insulator photonic wire based evanescent field sensor.IEEE Photonics Technol. Lett.2006,18, 2520–2522. [Google Scholar] [CrossRef]
10. Densmore, A.; Xu, D.X.; Janz, S.; Waldron, P.; Mischki, T.; Lopinski, G.; Delâge, A.; Lapointe, J.; Cheben, P.; Schmid, J.H.;et al. Spiral-path high-sensitivity silicon photonic wire molecular sensor with temperature-independent response.Opt. Lett.2008,33, 596–598. [Google Scholar] [CrossRef] [PubMed]
11. Barrios, C.A.; Gylfason, K.B.; Sanchez, B.; Griol, A.; Sohlstrom, H.; Holgado, M.; Casquel, R. Slot-waveguide biochemical sensor.Opt. Lett.2007,32, 3080–3082. [Google Scholar] [CrossRef] [PubMed]
12. Barrios, C.A.; Banuls, M.J.; Gonzalez-Pedro, V.; Gylfason, K.B.; Sanchez, B.; Griol, A.; Maquieira, A.; Sohlstrom, H.; Holgado, M.; Casquel, R. Label-free optical biosensing with slot-waveguides.Opt. Lett.2008,33, 708–710. [Google Scholar] [CrossRef] [PubMed]
13. Soref, R.A.; Schmidtchen, J.; Petermann, K. Large single-mode rib waveguides in GeSi-Si and Si-on-SiO₂.IEEE J. Quantum Electron.1991,27, 1971–1974. [Google Scholar] [CrossRef]
14. Chao, C.; Fung, W.; Guo, L.J. Polymer microring resonators for biochemical sensing applications.IEEE J. Select. Top. Quantum Electron.2006,12, 134–142. [Google Scholar] [CrossRef]
15. Dell'Olio, F.; Passaro, V.M. Optical sensing by optimized silicon slot waveguides.Opt. Express2007,15, 4977–4993. [Google Scholar] [CrossRef] [PubMed]
16. Reed, G.T.; Knights, A.P. Silicon-On-Insulator (SOI) Photonics. InSilicon Photonics: An Introduction; John Wiley & Sons, Ltd.: Chichester, UK, 2004. [Google Scholar]
17. Pogossian, S.P.; Vescan, L.; Vonsovici, A. The single-mode condition for semiconductor rib waveguides with large cross section.J. Lightwave Technol.1998,16, 1851–1853. [Google Scholar] [CrossRef]
18. Zhu, Z.; Brown, T. Full-vectorial finite-difference analysis of microstructured optical fibers.Opt. Express2002,10, 853–864. [Google Scholar] [CrossRef] [PubMed]
19. Wei, H.; Yu, J.; Zhang, X.; Liu, Z. Compact 3-dB tapered multimode interference coupler in silicon-on-insulator.Opt. Express2001,26, 878–880. [Google Scholar] [CrossRef]
20. Tang, Y.Z.; Wang, W.H.; Li, T.; Wang, Y.L. Integrated waveguide turning mirror in silicon-on-insulator.IEEE Photonics Technol. Lett.2002,14, 68–70. [Google Scholar] [CrossRef]
21. Lardenois, S.; Pascal, D.; Vivien, L.; Cassan, E.; Laval, S.; Orobtchouk, R.; Heitzmann, M.; Bouzaida, N.; Mollard, L. Low-loss submicrometer silicon-on-insulator rib waveguides and corner mirrors.Opt. Lett.2003,28, 1150–1152. [Google Scholar] [CrossRef] [PubMed]
22. Kiyat, I.; Aydinli, A.; Dagli, N. A compact silicon-on-insulator polarization splitter.IEEE Photonics Technol. Lett.2005,17, 100–102. [Google Scholar] [CrossRef]
23. Qian, Y.; Kim, S.; Song, J.; Nordin, G.P.; Jiang, J. Compact and low loss silicon-on-insulator rib waveguide 90° bend.Opt. Express2006,14, 6020–6028. [Google Scholar] [CrossRef] [PubMed]
24. Qian, Y.; Song, J.; Kim, S.; Nordin, G.P. Compact 90° trench-based splitter for silicon-on-insulator rib waveguides.Opt. Express2007,15, 16712–16718. [Google Scholar] [CrossRef] [PubMed]
25. Lau, S.T.; Ballantyne, J.M. Two-dimensional analysis of a dielectric waveguide mirror.J. Lightwave Technol.1997,15, 551–558. [Google Scholar] [CrossRef]
26. Berenger, J. A perfectly matched layer for the absorption of electromagnetic waves.J. Comput. Phys.1994,114, 185–200. [Google Scholar] [CrossRef]
27. Cai, J.; Nordin, G.P.; Kim, S.; Jiang, J. Three-dimensional analysis of a hybrid photonic crystal-conventional waveguide 90 degree bend.Appl. Opt.2004,43, 4244–4249. [Google Scholar] [CrossRef] [PubMed]
28. Prieto, F.; Sepulveda, B.; Calle, A.; Llobera, A.; Domínguez, C.; Abad, A.; Montoya, A.; Lechuga, L.M. An integrated optical interferometric nanodevice based on silicon technology for biosensor applications.Nanotechnology2003,14. [Google Scholar] [CrossRef]
29. Heideman, R.G.; Lambeck, P.V. Remote opto-chemical sensing with extreme sensitivity: Design, fabrication and performance of a pigtailed integrated optical phase-modulated Mach-Zehnder interferometer system.Sens. Actuators B Chem.1999,61, 100–127. [Google Scholar] [CrossRef]
30. Zinoviev, K.; Carrascosa, L.G.; Sánchez Del Río, J.; Sepúlveda, B.; Domínguez, C.; Lechuga, L.M. Silicon photonic biosensors for lab-on-a-chip applications.Adv. Opt. Technol.2008,2008, 1–6. [Google Scholar] [CrossRef]

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).