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Photonic crystal ring resonator based force sensor: Design and analysis
Optik 155 (2018) 111–120
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Optik journal homepage: www.elsevier.de/ijleo
Full length article
Photonic crystal ring resonator based force sensor: Design and analysis T Sreenivasulu a,∗ , Venkateswara Rao b , Badrinarayana T b , Gopalkrishna Hegde c , T Srinivas b a b c
Department of Electronics and Communication Engineering, SRM University, AP, Amaravati, 522508, India Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru, 560012, India Centre for Nano Science and Engineering, Indian Institute of Science, Bengaluru, 560012, India
a r t i c l e
i n f o
Article history: Received 7 September 2017 Accepted 6 November 2017 Keywords: Photonic bandgap materials Integrate optics Force sensors Photonic crystal ring resonators and cavities
a b s t r a c t In this paper theoretical investigation of photonic crystal based force sensor is presented. Photonic crystal ring resonator design is optimized for the improvement of quality factor considering the fabrication feasibility. For the optimized configuration a high quality factor of 15500 is obtained and it is found that it remains constant over the desired force range. The minimum detectable force is found to be 9 nN for 0.1 nm wavelength resolution. A high sensitivity of 11 nm/N is obtained in the studied force range. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Photonic Crystals (PC) are the periodic optical structures with periodic refractive index variation that affects the motion of light in much the same way that periodic lattices affect electrons in solids. Among these, two dimensional PC slabs with periodic array of air holes are attractive, as they are relatively easy to fabricate compared to the three dimensional PC devices and capable of guiding light effectively in all the three dimensions [1–3]. By fine tuning and controlling the parameters of these PC slabs, many devices are reported such as PC resonator based bio sensors [4,5], channel drop filters [6], refractive index sensor [7], PC slow light devices [8–12]. Apart from these devices, PC devices embedded in microcantilever structures are promising, since they are extremely sensitive to surface deformations and refractive index variations on the surface of cantilever beam. Various PC devices on microcantilever structures for different applications were proposed, as discussed below. An ultra high stress sensitive dual layered InGaAsP PC micro cavity is reported [13]. This structure consists of two PC beams separated with air gap. The minimum detectable stress variation is estimated as small as 0.95 nN. If sensitivity and quality factor of individual PC beams are improved, then the device will be even more sensitive. In the recent past, various cantilever beam sensors integrated with PC devices are proposed. C Lee et al. presented a PC based nano cavity resonator with a minimum detectable force of 62.5 nN [14]. A PC ring resonator integrated on top of Si/SiO2 bilayer cantilever [15] is reported. The reported minimum detectable force is 76 nN for 0.1 nm wavelength resolution with a Q of 3500. On the other hand, PC dual ring resonator integrated on a Si cantilever based sensor with various configurations is proposed [16,17].
∗ Corresponding author. E-mail address: [email protected] (S. T). https://doi.org/10.1016/j.ijleo.2017.11.040 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
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Fig. 1. Photonic Bandstructure of the considered photonic crystal without defects.
Among these configurations the most sensitive configuration gives a minimum detectable force of 7.58 nN in a particular range of applied force, whereas the quality factor was found to be reduced drastically. Apart from these devices with point defects and ring resonator cavities, a shoulder coupled aslant nano cavity based PC stress sensor is proposed, which is capable of sensing in two directions [18]. In horizontal and vertical directions, the sensitivity is found to be 7.5 nm/N and 10 nm/N respectively with a Q factor of 3000. Since quality factor plays crucial role in all the above mentioned sensing applications, Q factor enhancement is of major concern considering the design constraints and fabrication feasibility. In recent years many designs are reported for the improvement of quality factor of a PC based device. High Q resonators are designed by increasing or decreasing the hole sizes, moving the holes around the cavity, removal of holes, addition of extra scatter holes, so that most of the components in the momentum (K-vector) space are pushed away from the light cone region [19–21]. Zhang Y et al. reported a high quality factor photonic crystal ring resonator, with scattering holes in the corners of the hexagonal ring [22]. A high quality factor of 121000 is achieved with a PC ring of large dimension. Even though high quality factors are achieved, when these devices with high Q cavities are surrounded by line defect waveguides, the boundaries of the cavity are altered and hence the K-space distribution, leading to drastic reduction in quality factor. This happens because of the inclusion of more k-space components within the light cone region. This effect can be reduced up to some extent by increasing the coupling distance between the cavity and the bus waveguides, but this leads to poor coupling between them. In order to avoid this problem, one alternate design is proposed in which the quality factor is made independent of the surroundings of the cavity [23]. This cavity design consists of a larger central hole, which is surrounded by hexagons of air holes with decreasing radii along the outward direction. This variation of radii of holes in each hexagon is given by a parabolic relation. The variation in size of the holes fine tunes the cavity, so that two different modes of the cavity become degenerate with equal quality factor. In the similar manner, an optimized PC ring resonator with super defect is proposed to exhibit improved Q factor [24]. The improvement in Q is explained based on forced far field cancellation mechanism. In the present work, a new approach is adopted to get optimal quality factor for a PC ring resonator based force sensor. The main focus is on improving the quality factor of the resonator and keeping it constant for all applied forces of interest. This is achieved by optimizing the air hole radii inside the PC ring resonator. The increase in quality factor is explained by analysis of E-field distribution in momentum space. The design of the sensor consists of two phases. In the first phase 3D Finite-difference time-domain (FDTD) simulations are performed to get the optimized PC resonator design. The second phase consists of Finite Element Method (FEM) simulations to get the deformation data of cantilever surface for various applied forces. For each applied force, FDTD simulations are performed to get the spectral characteristics of the deformed PC device. The second section is dedicated for design and optimization of the PC ring resonator. The force sensing capabilities of the PC ring resonator on cantilever are discussed in the third section. In FDTD simulations we used mesh size of 0.0205 m, 0.0178 m and 0.015 m in X, Y and Z directions respectively. 2. Design, analysis and optimization of photonic crystal ring resonator In our design, a 220 nm thick silicon PC with hexagonal lattice of air holes is used. Hexagonal lattice is chosen since it provides wider photonic band gap compared to square lattice. The lattice constant and air hole radius are chosen to be 410 nm and 120 nm, respectively. The band structure of the considered PC structure is shown in Fig. 1. As shown in the figure, there is a bandgap in the normalised frequency range of 0.26–0.32, correspondingly in the wavelength range of 1250 nm–1600 nm. The proposed PC ring resonator consists of a hexagonal ring formed on the PC slab. The hexagonal ring is formed by removing the air holes on the slab in hexagonal shape and also line defect waveguides are formed on either side of the ring as shown in Fig. 2. The PC resonator consists of four different ports namely input port, transmission port, forward drop and backward drop. The line defects are bent on either side in such a way that there will be sufficient gap between the ports that lie in the same side. This is to avoid cross coupling of light at those ports during characterisation of the PC device.
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Fig. 2. Proposed photonic crystal ring resonator.
Fig. 3. Spectra of signals at various ports of the PC resonator. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
The separation between the waveguides and the ring on either side is chosen as two rows of holes. This coupling distance reduces the number of components in the light cone (leaky) region of the PC and hence reduces radiation loss. In FDTD computations, refractive index of Si is taken as 3.46. A Gaussian light source centred at 1550 nm wavelength is used at the input port for FDTD simulations. The spectra of signals at various ports of the PC resonator are as shown in Fig. 3. As it is discussed in the next section, for the force sensing application we choose the output of backward drop port of the resonator as the output signal. There are two peaks in the spectrum of backward drop at wavelengths of 1541.05 nm and 1549.21 nm. The quality factor of the peak at 1549.21 nm wavelength is found to be 3000, we focus on further improvement of Q factor of this peak. The PC resonator has to be optimized to improve quality factor. The design of the high Q PC ring resonator consists of optimizing the radius of central air hole R0 (shown in red colour) to improve Q factor, then tuning the radii of the holes in the succeeding hexagonal contour of air holes, R1 (shown in green colour) and R2 (shown in yellow colour) respectively, with fixed optimized value of R0 . During this design process the fabrication limitations are taken into account [25], so that there is an upper bound for the spacing between the air holes within the PC ring resonator. Hence it is preferable to have air hole diameters as less as possible during optimization process. To increase the quality factor, R0 is varied from 0.3a to 0.4a, where ‘a’ is the lattice constant. In each case, quality factor of the PC resonator is calculated. In all cases, the Full Width Half Maximum (FWHM) is obtained by using Lorentz fit. The variation of Q factor with R0 is plotted in Fig. 4. The maximum possible Q is found to be 3275 for a corresponding R0 of 0.4a and there is no much difference in the Q values for R0 between 0.35a and 0.4a. Hence as per the fabrication constraints we have optimized R0 to be 0.35a (144 nm) for which the Q is found to be 3200. Keeping R0 as 144 nm, R1 is varied from 0.95R0 to R0 , to get optimal quality factor. The variation of Q factor with the ratio R1 /R0 is as shown in Fig. 5. From the figure it can be noticed that, Q increases with increase in this ratio, touches 4000 at corresponding ratio of 0.975 and there on it remains almost constant. Based on these observations it is clear that, R1 can be chosen in the range of 0.975R0 –R0 . In order to optimize R2 , R1 is varied in the above mentioned range with increment of
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Fig. 4. Variation of Q factor with R0.
Fig. 5. Variation of Q factor as a function of the ratio R1 /R0.
Fig. 6. Variation of Q factor as a function of the ratio R2 /R1.
0.005R0 , in each step R2 is varied relative to R1 , from 0.95R1 to R1 . The Q factor variation as a function of the ratio R2 /R1 for various values of R1 is as shown in Fig. 6. It is clear that as R1 approaches R0 (curve in blue colour) and as the ratio R2 /R1 becomes 1, maximum Q of 18000 is obtained. That is in this case, the ring resonator has air holes of equal dimension. This improvement in quality factor of the PC ring resonator is explained based on multipole cancellation of far field components [24]. As it is shown in Fig. 7, the spectrum of backward drop of PC with this configuration of equal R0 , R1 and R2 , a relatively broader peak, but with high intensity appears closer to the desired high Q resonant peak, which may cause wrong detection in force sensing application. Hence, this configuration is not suitable for force sensing application also. This is because of the multimodal nature of resonator cavity with larger air holes. As the cavity size goes down this nature gets reduced. Instead of choosing equal radii, R1 and R2 can be chosen close to R0 , so that high Q is preserved and the effect of undesired peak is avoided. Accordingly, keeping R0 at 144 nm, R1 is fixed at 0.975R0 , which is equal to 140 nm and correspondingly R2
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Fig. 7. Backward drop spectrum of PC resonator with equal R0 , R1 and R2.
Fig. 8. Backward drop spectrum of optimized PC ring resonator.
is optimized to 0.975R1 , which equals 136.5 nm. Value of Q for this configuration is represented by the intersection of black coloured solid curve and vertical line in Fig. 6, which is given by 15500. Even though this value is less than maximum possible Q, the effect of multimodal nature of PC resonator is avoided. For this combination of R0 , R1 and R2 the spacing between the desired and undesired peak is found to be increased as shown in Fig. 8. In this case the undesired peak is relatively weaker than the desired peak. Hence this configuration is chosen to be the optimized PC ring resonator for force sensing application. Based on these computational results, it is observed that, in PC ring resonators quality factor can be enhanced by fine tuning the air hole sizes within the resonator. Increase in central air hole size within the ring resonator alone increases the Q only by small extent. If this increment of size is applied to the neighbouring holes as well, the Q will be dramatically increased. Because of this optimization of air hole sizes, light is made to confine within the ring for longer duration as radiation losses are minimized. This leads to increase in the total quality factor. The improvement in Q factor is explained based on distribution of electric field in the momentum space of the PC device. The K-space distribution of E-field profile for different configurations of the ring resonator is shown in Fig. 9. It consists of field profiles for the unmodified PC ring resonator, the configuration that gives a Q of 15500 and the configuration with a Q of 18000. The black circle in the plots represents the light cone region in the K-space. Inside the light cone, the in-plane K-vector component is less than the free space wave number. As the radiating modes exist in the region above the light line ( > ck) in the band structure of PC device, the leaky components exist inside the light cone in K-space of the cavity (k < 2/). As the density of the components in the light cone is reduced radiation loss will be reduced, vertical confinement will be improved and hence the quality factor of the PC device will increase. Thus, the interior of light cone can be considered as leaky region. It can be noticed from Fig. 9, the in-plane K-components in the leaky region are reduced in the modified PC ring resonators and hence they have high quality factors. A small difference in E-field distribution can be observed between the optimized PC configuration and the PC ring with equal R0 , R1 and R2 . Even though the last configuration shows less E-field
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Fig. 9. Momentum space distributions of electric field (Ey component) for different ring resonator configurations.
Fig. 10. Schematic of cantilever beam on SOI wafer.
component in the light cone, as it is mentioned previously, the middle configuration is chosen for force sensing application. The next section deals with the force sensing characteristics of the proposed optimized PC based force sensor. 3. Force sensor modeling The proposed force sensor consists of a 220 nm thick silicon cantilever integrated with the optimized PC ring resonator on top. In order to keep the sensor more sensitive, the dimensions of the cantilever are chosen as 50 m, 15 m in length and width, respectively. The length to thickness ratio of the cantilever needs to be very high and the thickness of the Si slab should be between 0.5a–1.2a to have sufficient photonic band gap [26]. Accordingly the slab thickness is chosen as 220 nm, which satisfies both the above mentioned mechanical as well as optical requirements. Modelling of the force sensor consists of FEM based stress distribution analysis of bare cantilever beam and beam integrated with PC ring resonator due to various applied forces. After getting the deformation data of the cantilever beam induced by the applied forces, FDTD computations are performed to get the spectral response of the device to various forces. 3.1. FEM based stress analysis of the cantilever In the design of any cantilever based integrated force sensor, the dimensions of the cantilever and the type of the sensing element play vital role to make it more sensitive device. The principle of operation of these devices is photo-elastic effect, which means that the stress induced refractive index variation changes the optical resonant properties of the sensing element. We choose point load (force) as the load on the tip of the cantilever, in order to generate stress along the cantilever beam. Hence it is expected to have non-uniform distribution of the load and hence non-uniform stress along the cantilever beam. Finite Element Method (FEM) is used to do the stress analysis of more complicated structures. We use commercial FEM software COMSOL Multiphysics [27] for the analysis. It consists of drawing the structure in the cad layout, applying mesh, appropriate boundary conditions and solving the static equation to get displacement and stress parameters. Fig. 10 shows schematic of the cantilever beam with length 50 m, width 15 m and thickness 220 nm released from SOI wafer. As it is seen one end of the cantilever is fixed to the substrate and other end is free. FEM computations are performed to get deformation of the cantilever beam for various applied forces. In these computations Young’s modulus (E) and Poisson’s ratio are taken as 130 GPa and 0.28, respectively. Fig. 11 shows the stress distribution of the cantilever for an applied force of 1 N. As shown in Fig. 12 stress reaches a maximum value 12 × 107 N/m2 at the supporting end of the cantilever and decreases gradually along the cantilever. Since the stress is the maximum at the fixed end, the sensing element is desired to be placed at fixed end to get high sensitivity. Accordingly the optimized PC ring resonator that is presented in the previous section is placed at
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Fig. 11. Stress distribution on the bare cantilever beam surface.
Fig. 12. Stress variation on the surface of cantilever beam along the length.
the fixed end. The following subsection deals with stress distribution analysis of cantilever beam integrated with optimized photonic crystal ring resonator. 3.2. FEM analysis of the cantilever beam integrated with PC ring resonator Integration of photonic crystal device on top of cantilever beam can be done by drilling air holes on silicon cantilever beam with the designed dimensions. The stress induced refractive index changes occur on the surface of the cantilever beam. This change in refractive index induced by the applied force causes shift in the resonant wavelength of the PC ring resonator. The deformation of the PC ring resonator is implemented in the FDTD computations to get resonant characteristics of the deformed resonator. Hence each applied force will have corresponding unique resonant wavelength of the resonator, since the stress or refractive index change varies linearly with respect to the applied force. As shown in Fig. 13, optimized PCRR with high Q factor is integrated on top of cantilever beam in such a way that, it will be elongated in the longitudinal direction and will get affected by maximum force induced stress. When force is applied at the tip of the cantilever beam, the induced stress is distributed on the surface of the beam, as shown in Fig. 14. It is evident that the hexagonal ring gets affected by maximum amount of stress and hence it undergoes maximum deformation. The maximum stress is found to be 1.72GN/m2 . The corresponding maximum refractive index change is calculated to be 0.0285 for an applied force of 1 N. Since the stress and transverse displacement vary linearly with the applied force, the refractive index is also expected to be varied linearly, which is desirable to get linear variation of wavelength shift. For each applied force, the deformation of the ring resonator is computed using FEM, and this deformation is implemented in FDTD computations, to get resonant wavelength of deformed ring resonator. Fig. 15 shows resonant peak shift for various applied forces in the range of 0–1 N. The blue peak at wavelength of 1533 nm represents resonant peak at no-load situation and the peak shifts towards the longer wavelength region (red shift) with increase in the applied force. Also, it is found the resonant wavelength shift and quality factor are uniform throughout the studied force range. A wavelength shift of 11 nm is observed for an applied force of 1 N, from which the sensitivity is derived as 11 nm/N. Variation of resonant wavelength shift versus the applied force is plotted in Fig. 16. Here, wavelength shift for a particular force is defined as the difference between the resonant peaks at that particular force and that when there is no-load applied on the cantilever tip. From these
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Fig. 13. Photonic crystal ring resonator integrated on cantilever beam.
Fig. 14. Stress distribution on the surface of photonic crystal integrated cantilever beam.
Fig. 15. Shift of resonant wavelength with applied force. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
characteristics it is clear that, the shift varies linearly with applied force and the minimum detectable force is derived to be 9 nN for a measurable wavelength shift of 0.1 nm. The variation of Q factor with applied force is plotted in Fig. 17, which conveys that Q is almost constant at 16000 over the studied force range. Hence, the proposed device can be used to sense forces in the range of 0–1 N and forces as small as 10 nN are also can be detected by using any standard spectrum analyzer.
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Fig. 16. Variation of shift in resonant length with applied force.
Fig. 17. Variation of Quality factor (Q) with applied force.
Also it is expected that, if the microcantilever is properly functionalised, the proposed device can be used for bio applications such as protein molecules detection. 4. Conclusion A practically realizable design of optimized hexagonal PC ring resonator for very small (nN) force sensing application is proposed. The PC ring resonator is optimized to get high quality factor of 15500 in the desired force range. This is achieved by optimizing the size of air holes inside the hexagonal ring. The optimization is done by considering the fabrication limitations, the multimodal nature of PC ring resonator. The optimized PC ring resonator on cantilever beam is subjected to simultaneous FEM and FDTD simulations to get the force sensor characteristics. This force sensor with optimized PC ring resonator has a sensitivity of 11 nm/N. The minimum detectable force is found to be 9 nN. The proposed design may also be extended for strain sensing and bio sensing applications such as DNA, protein molecules detection with integrated functionalised cantilever. Acknowledgement This work was funded by Defence Research Development Organization (DRDO), New Delhi, India. References [1] [2] [3] [4]
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Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Full length article
Photonic crystal ring resonator based force sensor: Design and analysis T Sreenivasulu a,∗ , Venkateswara Rao b , Badrinarayana T b , Gopalkrishna Hegde c , T Srinivas b a b c
Department of Electronics and Communication Engineering, SRM University, AP, Amaravati, 522508, India Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru, 560012, India Centre for Nano Science and Engineering, Indian Institute of Science, Bengaluru, 560012, India
a r t i c l e
i n f o
Article history: Received 7 September 2017 Accepted 6 November 2017 Keywords: Photonic bandgap materials Integrate optics Force sensors Photonic crystal ring resonators and cavities
a b s t r a c t In this paper theoretical investigation of photonic crystal based force sensor is presented. Photonic crystal ring resonator design is optimized for the improvement of quality factor considering the fabrication feasibility. For the optimized configuration a high quality factor of 15500 is obtained and it is found that it remains constant over the desired force range. The minimum detectable force is found to be 9 nN for 0.1 nm wavelength resolution. A high sensitivity of 11 nm/N is obtained in the studied force range. © 2017 Elsevier GmbH. All rights reserved.
1. Introduction Photonic Crystals (PC) are the periodic optical structures with periodic refractive index variation that affects the motion of light in much the same way that periodic lattices affect electrons in solids. Among these, two dimensional PC slabs with periodic array of air holes are attractive, as they are relatively easy to fabricate compared to the three dimensional PC devices and capable of guiding light effectively in all the three dimensions [1–3]. By fine tuning and controlling the parameters of these PC slabs, many devices are reported such as PC resonator based bio sensors [4,5], channel drop filters [6], refractive index sensor [7], PC slow light devices [8–12]. Apart from these devices, PC devices embedded in microcantilever structures are promising, since they are extremely sensitive to surface deformations and refractive index variations on the surface of cantilever beam. Various PC devices on microcantilever structures for different applications were proposed, as discussed below. An ultra high stress sensitive dual layered InGaAsP PC micro cavity is reported [13]. This structure consists of two PC beams separated with air gap. The minimum detectable stress variation is estimated as small as 0.95 nN. If sensitivity and quality factor of individual PC beams are improved, then the device will be even more sensitive. In the recent past, various cantilever beam sensors integrated with PC devices are proposed. C Lee et al. presented a PC based nano cavity resonator with a minimum detectable force of 62.5 nN [14]. A PC ring resonator integrated on top of Si/SiO2 bilayer cantilever [15] is reported. The reported minimum detectable force is 76 nN for 0.1 nm wavelength resolution with a Q of 3500. On the other hand, PC dual ring resonator integrated on a Si cantilever based sensor with various configurations is proposed [16,17].
∗ Corresponding author. E-mail address: [email protected] (S. T). https://doi.org/10.1016/j.ijleo.2017.11.040 0030-4026/© 2017 Elsevier GmbH. All rights reserved.
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Fig. 1. Photonic Bandstructure of the considered photonic crystal without defects.
Among these configurations the most sensitive configuration gives a minimum detectable force of 7.58 nN in a particular range of applied force, whereas the quality factor was found to be reduced drastically. Apart from these devices with point defects and ring resonator cavities, a shoulder coupled aslant nano cavity based PC stress sensor is proposed, which is capable of sensing in two directions [18]. In horizontal and vertical directions, the sensitivity is found to be 7.5 nm/N and 10 nm/N respectively with a Q factor of 3000. Since quality factor plays crucial role in all the above mentioned sensing applications, Q factor enhancement is of major concern considering the design constraints and fabrication feasibility. In recent years many designs are reported for the improvement of quality factor of a PC based device. High Q resonators are designed by increasing or decreasing the hole sizes, moving the holes around the cavity, removal of holes, addition of extra scatter holes, so that most of the components in the momentum (K-vector) space are pushed away from the light cone region [19–21]. Zhang Y et al. reported a high quality factor photonic crystal ring resonator, with scattering holes in the corners of the hexagonal ring [22]. A high quality factor of 121000 is achieved with a PC ring of large dimension. Even though high quality factors are achieved, when these devices with high Q cavities are surrounded by line defect waveguides, the boundaries of the cavity are altered and hence the K-space distribution, leading to drastic reduction in quality factor. This happens because of the inclusion of more k-space components within the light cone region. This effect can be reduced up to some extent by increasing the coupling distance between the cavity and the bus waveguides, but this leads to poor coupling between them. In order to avoid this problem, one alternate design is proposed in which the quality factor is made independent of the surroundings of the cavity [23]. This cavity design consists of a larger central hole, which is surrounded by hexagons of air holes with decreasing radii along the outward direction. This variation of radii of holes in each hexagon is given by a parabolic relation. The variation in size of the holes fine tunes the cavity, so that two different modes of the cavity become degenerate with equal quality factor. In the similar manner, an optimized PC ring resonator with super defect is proposed to exhibit improved Q factor [24]. The improvement in Q is explained based on forced far field cancellation mechanism. In the present work, a new approach is adopted to get optimal quality factor for a PC ring resonator based force sensor. The main focus is on improving the quality factor of the resonator and keeping it constant for all applied forces of interest. This is achieved by optimizing the air hole radii inside the PC ring resonator. The increase in quality factor is explained by analysis of E-field distribution in momentum space. The design of the sensor consists of two phases. In the first phase 3D Finite-difference time-domain (FDTD) simulations are performed to get the optimized PC resonator design. The second phase consists of Finite Element Method (FEM) simulations to get the deformation data of cantilever surface for various applied forces. For each applied force, FDTD simulations are performed to get the spectral characteristics of the deformed PC device. The second section is dedicated for design and optimization of the PC ring resonator. The force sensing capabilities of the PC ring resonator on cantilever are discussed in the third section. In FDTD simulations we used mesh size of 0.0205 m, 0.0178 m and 0.015 m in X, Y and Z directions respectively. 2. Design, analysis and optimization of photonic crystal ring resonator In our design, a 220 nm thick silicon PC with hexagonal lattice of air holes is used. Hexagonal lattice is chosen since it provides wider photonic band gap compared to square lattice. The lattice constant and air hole radius are chosen to be 410 nm and 120 nm, respectively. The band structure of the considered PC structure is shown in Fig. 1. As shown in the figure, there is a bandgap in the normalised frequency range of 0.26–0.32, correspondingly in the wavelength range of 1250 nm–1600 nm. The proposed PC ring resonator consists of a hexagonal ring formed on the PC slab. The hexagonal ring is formed by removing the air holes on the slab in hexagonal shape and also line defect waveguides are formed on either side of the ring as shown in Fig. 2. The PC resonator consists of four different ports namely input port, transmission port, forward drop and backward drop. The line defects are bent on either side in such a way that there will be sufficient gap between the ports that lie in the same side. This is to avoid cross coupling of light at those ports during characterisation of the PC device.
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Fig. 2. Proposed photonic crystal ring resonator.
Fig. 3. Spectra of signals at various ports of the PC resonator. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
The separation between the waveguides and the ring on either side is chosen as two rows of holes. This coupling distance reduces the number of components in the light cone (leaky) region of the PC and hence reduces radiation loss. In FDTD computations, refractive index of Si is taken as 3.46. A Gaussian light source centred at 1550 nm wavelength is used at the input port for FDTD simulations. The spectra of signals at various ports of the PC resonator are as shown in Fig. 3. As it is discussed in the next section, for the force sensing application we choose the output of backward drop port of the resonator as the output signal. There are two peaks in the spectrum of backward drop at wavelengths of 1541.05 nm and 1549.21 nm. The quality factor of the peak at 1549.21 nm wavelength is found to be 3000, we focus on further improvement of Q factor of this peak. The PC resonator has to be optimized to improve quality factor. The design of the high Q PC ring resonator consists of optimizing the radius of central air hole R0 (shown in red colour) to improve Q factor, then tuning the radii of the holes in the succeeding hexagonal contour of air holes, R1 (shown in green colour) and R2 (shown in yellow colour) respectively, with fixed optimized value of R0 . During this design process the fabrication limitations are taken into account [25], so that there is an upper bound for the spacing between the air holes within the PC ring resonator. Hence it is preferable to have air hole diameters as less as possible during optimization process. To increase the quality factor, R0 is varied from 0.3a to 0.4a, where ‘a’ is the lattice constant. In each case, quality factor of the PC resonator is calculated. In all cases, the Full Width Half Maximum (FWHM) is obtained by using Lorentz fit. The variation of Q factor with R0 is plotted in Fig. 4. The maximum possible Q is found to be 3275 for a corresponding R0 of 0.4a and there is no much difference in the Q values for R0 between 0.35a and 0.4a. Hence as per the fabrication constraints we have optimized R0 to be 0.35a (144 nm) for which the Q is found to be 3200. Keeping R0 as 144 nm, R1 is varied from 0.95R0 to R0 , to get optimal quality factor. The variation of Q factor with the ratio R1 /R0 is as shown in Fig. 5. From the figure it can be noticed that, Q increases with increase in this ratio, touches 4000 at corresponding ratio of 0.975 and there on it remains almost constant. Based on these observations it is clear that, R1 can be chosen in the range of 0.975R0 –R0 . In order to optimize R2 , R1 is varied in the above mentioned range with increment of
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Fig. 4. Variation of Q factor with R0.
Fig. 5. Variation of Q factor as a function of the ratio R1 /R0.
Fig. 6. Variation of Q factor as a function of the ratio R2 /R1.
0.005R0 , in each step R2 is varied relative to R1 , from 0.95R1 to R1 . The Q factor variation as a function of the ratio R2 /R1 for various values of R1 is as shown in Fig. 6. It is clear that as R1 approaches R0 (curve in blue colour) and as the ratio R2 /R1 becomes 1, maximum Q of 18000 is obtained. That is in this case, the ring resonator has air holes of equal dimension. This improvement in quality factor of the PC ring resonator is explained based on multipole cancellation of far field components [24]. As it is shown in Fig. 7, the spectrum of backward drop of PC with this configuration of equal R0 , R1 and R2 , a relatively broader peak, but with high intensity appears closer to the desired high Q resonant peak, which may cause wrong detection in force sensing application. Hence, this configuration is not suitable for force sensing application also. This is because of the multimodal nature of resonator cavity with larger air holes. As the cavity size goes down this nature gets reduced. Instead of choosing equal radii, R1 and R2 can be chosen close to R0 , so that high Q is preserved and the effect of undesired peak is avoided. Accordingly, keeping R0 at 144 nm, R1 is fixed at 0.975R0 , which is equal to 140 nm and correspondingly R2
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Fig. 7. Backward drop spectrum of PC resonator with equal R0 , R1 and R2.
Fig. 8. Backward drop spectrum of optimized PC ring resonator.
is optimized to 0.975R1 , which equals 136.5 nm. Value of Q for this configuration is represented by the intersection of black coloured solid curve and vertical line in Fig. 6, which is given by 15500. Even though this value is less than maximum possible Q, the effect of multimodal nature of PC resonator is avoided. For this combination of R0 , R1 and R2 the spacing between the desired and undesired peak is found to be increased as shown in Fig. 8. In this case the undesired peak is relatively weaker than the desired peak. Hence this configuration is chosen to be the optimized PC ring resonator for force sensing application. Based on these computational results, it is observed that, in PC ring resonators quality factor can be enhanced by fine tuning the air hole sizes within the resonator. Increase in central air hole size within the ring resonator alone increases the Q only by small extent. If this increment of size is applied to the neighbouring holes as well, the Q will be dramatically increased. Because of this optimization of air hole sizes, light is made to confine within the ring for longer duration as radiation losses are minimized. This leads to increase in the total quality factor. The improvement in Q factor is explained based on distribution of electric field in the momentum space of the PC device. The K-space distribution of E-field profile for different configurations of the ring resonator is shown in Fig. 9. It consists of field profiles for the unmodified PC ring resonator, the configuration that gives a Q of 15500 and the configuration with a Q of 18000. The black circle in the plots represents the light cone region in the K-space. Inside the light cone, the in-plane K-vector component is less than the free space wave number. As the radiating modes exist in the region above the light line ( > ck) in the band structure of PC device, the leaky components exist inside the light cone in K-space of the cavity (k < 2/). As the density of the components in the light cone is reduced radiation loss will be reduced, vertical confinement will be improved and hence the quality factor of the PC device will increase. Thus, the interior of light cone can be considered as leaky region. It can be noticed from Fig. 9, the in-plane K-components in the leaky region are reduced in the modified PC ring resonators and hence they have high quality factors. A small difference in E-field distribution can be observed between the optimized PC configuration and the PC ring with equal R0 , R1 and R2 . Even though the last configuration shows less E-field
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Fig. 9. Momentum space distributions of electric field (Ey component) for different ring resonator configurations.
Fig. 10. Schematic of cantilever beam on SOI wafer.
component in the light cone, as it is mentioned previously, the middle configuration is chosen for force sensing application. The next section deals with the force sensing characteristics of the proposed optimized PC based force sensor. 3. Force sensor modeling The proposed force sensor consists of a 220 nm thick silicon cantilever integrated with the optimized PC ring resonator on top. In order to keep the sensor more sensitive, the dimensions of the cantilever are chosen as 50 m, 15 m in length and width, respectively. The length to thickness ratio of the cantilever needs to be very high and the thickness of the Si slab should be between 0.5a–1.2a to have sufficient photonic band gap [26]. Accordingly the slab thickness is chosen as 220 nm, which satisfies both the above mentioned mechanical as well as optical requirements. Modelling of the force sensor consists of FEM based stress distribution analysis of bare cantilever beam and beam integrated with PC ring resonator due to various applied forces. After getting the deformation data of the cantilever beam induced by the applied forces, FDTD computations are performed to get the spectral response of the device to various forces. 3.1. FEM based stress analysis of the cantilever In the design of any cantilever based integrated force sensor, the dimensions of the cantilever and the type of the sensing element play vital role to make it more sensitive device. The principle of operation of these devices is photo-elastic effect, which means that the stress induced refractive index variation changes the optical resonant properties of the sensing element. We choose point load (force) as the load on the tip of the cantilever, in order to generate stress along the cantilever beam. Hence it is expected to have non-uniform distribution of the load and hence non-uniform stress along the cantilever beam. Finite Element Method (FEM) is used to do the stress analysis of more complicated structures. We use commercial FEM software COMSOL Multiphysics [27] for the analysis. It consists of drawing the structure in the cad layout, applying mesh, appropriate boundary conditions and solving the static equation to get displacement and stress parameters. Fig. 10 shows schematic of the cantilever beam with length 50 m, width 15 m and thickness 220 nm released from SOI wafer. As it is seen one end of the cantilever is fixed to the substrate and other end is free. FEM computations are performed to get deformation of the cantilever beam for various applied forces. In these computations Young’s modulus (E) and Poisson’s ratio are taken as 130 GPa and 0.28, respectively. Fig. 11 shows the stress distribution of the cantilever for an applied force of 1 N. As shown in Fig. 12 stress reaches a maximum value 12 × 107 N/m2 at the supporting end of the cantilever and decreases gradually along the cantilever. Since the stress is the maximum at the fixed end, the sensing element is desired to be placed at fixed end to get high sensitivity. Accordingly the optimized PC ring resonator that is presented in the previous section is placed at
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Fig. 11. Stress distribution on the bare cantilever beam surface.
Fig. 12. Stress variation on the surface of cantilever beam along the length.
the fixed end. The following subsection deals with stress distribution analysis of cantilever beam integrated with optimized photonic crystal ring resonator. 3.2. FEM analysis of the cantilever beam integrated with PC ring resonator Integration of photonic crystal device on top of cantilever beam can be done by drilling air holes on silicon cantilever beam with the designed dimensions. The stress induced refractive index changes occur on the surface of the cantilever beam. This change in refractive index induced by the applied force causes shift in the resonant wavelength of the PC ring resonator. The deformation of the PC ring resonator is implemented in the FDTD computations to get resonant characteristics of the deformed resonator. Hence each applied force will have corresponding unique resonant wavelength of the resonator, since the stress or refractive index change varies linearly with respect to the applied force. As shown in Fig. 13, optimized PCRR with high Q factor is integrated on top of cantilever beam in such a way that, it will be elongated in the longitudinal direction and will get affected by maximum force induced stress. When force is applied at the tip of the cantilever beam, the induced stress is distributed on the surface of the beam, as shown in Fig. 14. It is evident that the hexagonal ring gets affected by maximum amount of stress and hence it undergoes maximum deformation. The maximum stress is found to be 1.72GN/m2 . The corresponding maximum refractive index change is calculated to be 0.0285 for an applied force of 1 N. Since the stress and transverse displacement vary linearly with the applied force, the refractive index is also expected to be varied linearly, which is desirable to get linear variation of wavelength shift. For each applied force, the deformation of the ring resonator is computed using FEM, and this deformation is implemented in FDTD computations, to get resonant wavelength of deformed ring resonator. Fig. 15 shows resonant peak shift for various applied forces in the range of 0–1 N. The blue peak at wavelength of 1533 nm represents resonant peak at no-load situation and the peak shifts towards the longer wavelength region (red shift) with increase in the applied force. Also, it is found the resonant wavelength shift and quality factor are uniform throughout the studied force range. A wavelength shift of 11 nm is observed for an applied force of 1 N, from which the sensitivity is derived as 11 nm/N. Variation of resonant wavelength shift versus the applied force is plotted in Fig. 16. Here, wavelength shift for a particular force is defined as the difference between the resonant peaks at that particular force and that when there is no-load applied on the cantilever tip. From these
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Fig. 13. Photonic crystal ring resonator integrated on cantilever beam.
Fig. 14. Stress distribution on the surface of photonic crystal integrated cantilever beam.
Fig. 15. Shift of resonant wavelength with applied force. (For interpretation of the references to colour in the text, the reader is referred to the web version of this article.)
characteristics it is clear that, the shift varies linearly with applied force and the minimum detectable force is derived to be 9 nN for a measurable wavelength shift of 0.1 nm. The variation of Q factor with applied force is plotted in Fig. 17, which conveys that Q is almost constant at 16000 over the studied force range. Hence, the proposed device can be used to sense forces in the range of 0–1 N and forces as small as 10 nN are also can be detected by using any standard spectrum analyzer.
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Fig. 16. Variation of shift in resonant length with applied force.
Fig. 17. Variation of Quality factor (Q) with applied force.
Also it is expected that, if the microcantilever is properly functionalised, the proposed device can be used for bio applications such as protein molecules detection. 4. Conclusion A practically realizable design of optimized hexagonal PC ring resonator for very small (nN) force sensing application is proposed. The PC ring resonator is optimized to get high quality factor of 15500 in the desired force range. This is achieved by optimizing the size of air holes inside the hexagonal ring. The optimization is done by considering the fabrication limitations, the multimodal nature of PC ring resonator. The optimized PC ring resonator on cantilever beam is subjected to simultaneous FEM and FDTD simulations to get the force sensor characteristics. This force sensor with optimized PC ring resonator has a sensitivity of 11 nm/N. The minimum detectable force is found to be 9 nN. The proposed design may also be extended for strain sensing and bio sensing applications such as DNA, protein molecules detection with integrated functionalised cantilever. Acknowledgement This work was funded by Defence Research Development Organization (DRDO), New Delhi, India. References [1] [2] [3] [4]
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