Open Access
Add to BibSonomyAbstract
In this paper we investigate the optical forces induced by localized optical modes propagating along three parallel waveguides, of which only the central one is free to move. In this configuration, when all three waveguides are identical, the components of the optical-force acting on the free beam are decoupled along the axis of symmetry. As a result, two dimensional optomechanical control of the central waveguide, like single-mode optical trapping, can be achieved. We also study non symmetric configurations, that can be used, for example, to tailor the position of the optical trap. Unlike other techniques that rely on buckling, multi-mode excitation or radiation-pressure, single-mode optomechanical-operation should help the realization of faster and simpler on-chip positioning of a single nanobeam since most of the parameters involved can be controlled with great precision.
© 2013 Optical Society of America
1. Introduction
Optical forces naturally arise as one of the many ways light interacts with matter and, until recently, had remained a topic of theoretical curiosity [1]. In fact, while the relatively weak magnitude and unintuitive formulation [2] (Maxwell stress tensor) of optical forces made integration to older technologies impossible, progressive scaling and new theoretical insights have allowed the realization of devices that make use of such forces in the micro and nano scale, such as interferometers, stable optical memories, accelerometers, pressure sensor, switches and others [3–10]. In all these examples optomechanical interactions are mainly driven by gradient-forces that, unlike radiation pressure, induce forces normal to the direction of propagation of the optical mode that are usually either attractive or repulsive [11,12]. Here we show that gradient forces can also be used to induce useful effects like single-mode optical trapping, levitation (a displacement along the y-axis as shown in Fig. 1) and, in general, to actively control the position of a dielectric waveguide on the whole plane transverse to the direction of propagation of light. As a matter of fact, single-mode optical trapping has been demonstrated to be impossible in systems formed by a waveguide coupled to a resonator [13] and, in our knowledge, has never been observed between parallel dielectric waveguides. To do so we propose a system formed by three interacting modes (waveguides) rather than just two, where the two external waveguides are fixed to a substrate and only the central one is free to move as described in Fig. 1. The force on the central beam (CB) can now be tailored by an appropriate choice of the cross sectional geometry of the waveguides, which can be done via a method we have previously developed [14]. Moreover, in order to guarantee predictability and intense forces among a large number of geometries and modes, we focus on the five lowest-energy, guided eigenmodes. In fact, according to the variational theorem [15] the eigenmode-field distribution should become more complex (more geometry sensitive) and less localized as the mode-frequency increases, inducing smaller forces with less predictable behaviors. In this paper, we start by introducing a commonly used surface-formulation of optical forces [16] and a methodology useful to predict the magnitude and direction of these forces [14]. Such technique is then used to investigate the behavior of the force induced on the CB by the five lowest eigenmodes; although the fifth mode in order of energy is considered separately, due to its distinctive properties. A short comparisons with the well known system formed by two parallel waveguides [11] is also carried out. Finally we quickly consider a simple method useful to tailor the position of the forces along the y and z axis independently via simple modifications of the cross-sectional shape of the waveguides.
Fig. 1 Schematic of the system studied in which only the central waveguide is free to move. The height and thickness of the waveguides is h = a and t = 0.5a. The thickness of the thin SiO2 layers is t0 = 0.5a. The waveguides are made of Silicon and the separation between the external beams is fixed to 2ds + t, where ds = 0.25a.
2. Surface formulation of Gradient-Forces
To understand gradient-forces we make use of a formulation initially developed in the frame work of quantum-optomechanics that can be used to describe classical forces as well [16]:
The gradient-force per unit of electromagnetic energy U/a (for a chosen unit length, a) on the CB is then given by the integral of the norm of E along the surface of the nanobeam, A; it is proportional to the difference between the permittivity ε inside (ε1) and outside (ε2) the waveguides, where Δε = ε1 − ε2; it is directed normally to A, id est along n, which is oriented from the high to the low index media; E|| and D⊥ represent the electric and electric-displacement fields parallel and normal to A. Moreover E is normalized as ∫VεEi · EjdV = δij, V is the volume of the unit-cell and i(j) denotes the order of the eigenmode. Forces are defined positive if oriented toward positive directions of the coordinate system as defined in Fig. 1, and vice versa. Hence, since Eq. (1) depends exclusively on the field distribution, E, if we could predict the electric field distribution, we should be able to make an educated guess of the magnitude and direction of the force as well. In [14] we did so via a set of semi-empirical principles based on the variational theorem and Eq. (1), that can be summarized in a few points. First: the fields tend to be preferably localized in the high refractive index regions, and to have the least amount of spatial variations. Second: those field components that have a node along a symmetry plane will be “pushed away” from it and, vice versa, components that are continuous along such planes will tend to be “localized” along it. Third: the field component with the least number of nodes along the symmetries will dominate the direction and magnitude of the force. Fourth: components of the field that are normal to the interface (E⊥, measured inside the high dielectric index region) have a larger contribution to the force, proportional to ε1/ε2, respect to E||. Based on these principles, although not necessarily in order, we can now discuss and design optical forces in the three-waveguides configuration (3WC).3. Gradient-Forces between three parallel identical waveguides
The system is shown in Fig. 1 is formed by three identical waveguides with a thickness of t = 0.5a and height of h = a; when the CB is at the origin, {dz, dy} = {0, 0}, the beams are evenly spaced along the z-axis with an initial separation of ds = 0.25a and they all lay on the xy-plane. For now we assume that the thin SiO2 (n = 1.5) layer, used to fix the two external waveguides, does not significantly influence the field distribution of the modes we study due to the very large index contrast between such layer and the Silicon waveguides (n = 3.45) so that our system is z-axis symmetric; a more regorous justification will be given when displacements along the y-axis are considered.
The dispersion diagram of the 3WC at {dz, dy} = {0, 0} is shown in Fig. 2(b), where each mode is labeled after its parity [15]. We notice that a different choice of kx will change the order of the modes as a function of frequency. However, this choice should not significantly effect the characteristic field distribution and forces of each mode since these are mainly defined by the symmetry of each mode and by the shape of the waveguides rather than by the mode’s frequency or wavevector [14]. When the forces are measured, all modes are computed at a constant axial wave-vector, kx = π/a. This value is chosen arbitrarily: just large enough to guarantee that all the five modes considered are non radiative. At kx = π/a and when the CB is at the origin of the coordinate system the four lowest-energy modes correspond to the following mirror-symmetries: y-even/z-odd (EO), y-odd/z-odd (OO) and two y-odd/z-even (OE) modes, namely OE(1) and OE(2). We notice that these modes do not match the four distinct sets of mirror-symmetries of the system since the y-even/z-even (EE) mode, whose characteristics will be considered separatelly, lays at higher frequency as shown in Fig. 2(b).
Fig. 2 (a) Force per unit energy as a function of a displacement dz/a for the Yodd-Zeven(1), Yodd-Zodd, Yodd-Zeven(2) and Yeven-Zodd modes at a constant axial wave-vector kx = π/a. In the inset we show a schematic of the geometry in use where a is the unit length, t0 is the thickness of the SiO2 support layer equal to 0.5a and the height and thickness of the waveguides is h = a and t = 0.5a respectively. (b) Dispersion diagram showing the five lowest frequency modes for the same geometry shown in (a) when all waveguides are evenly spaced. The field distribution of these modes at kx = π/a are shown in the first row of Fig. 3 and in Fig. 5(b). (c) Force per unit energy as a function of the displacement dz/a for the Yodd-Zeven, Yeven-Zodd, Yodd-Zodd and Yeven-Zeven fundamental modes at a constant axial wave-vector kx = π/a between two parallel waveguides with identical parameters to those shown in (a).
In the 3WC we identify three remarkable features of the optical forces: trapping, pulling and levitating. Trapping-forces oppose displacements of the CB away from the point of equilibrium, where the forces are 0. These forces can be directed either along one (one dimensional trapping) or both axis (two dimensional trapping). Pulling-forces are directed along the z-axis and towards one of the external waveguides, which is analogue to attractive forces in the two waveguides configuration (2WC). Levitating-forces are oriented along the y-axis such as to pull the beam away from both the point of equilibrium and the external beams.
3.1. z-axis oriented forces
Let’s first consider the case in which the CB is free to move along the z-axis, as shown in Fig. 2(a). In Fig. 3 we also show |E|2 and Eyz for each mode at {dz, dy} = {0, 0} and {dz, dy} = {0.075a, 0} respectively. All pictures share the same normalization of arrows length and color scale: darker colors and longer arrows indicate stronger fields.
Fig. 3 In-plane Eyz vector-field distribution (larger arrows mean more intense) with the total |E|2 in the background (darker is more intense) for the Yodd-Zeven(1), Yodd-Zodd, Yodd-Zeven(2) and Yeven-Zodd modes at kx = π/a. For the first row dz = 0 and for the second row dz = 0.075a. No displacement along the y-axis (dy = 0) is considered. All arrows lengths and color scales share the same normalization.
OE(1) mode
We first consider the OE(1) mode shown in Fig. 3, which induces a pulling force, as shown in Fig. 2(a). We can use such mode to describe the characteristics of the first three principles described in Section 2, which are the most used ones throughout this paper. At the beginning we find the dominant component of the mode via the third principle, which in this case is Ey as a result of the y-odd/z-even symmetry. This means that it is not necessary to consider other components in order to get an idea of the magnitude and direction of the optical force since most of the mode’s energy, U, will be carried by this component. Next we need to roughly guess the spatial distribution of Ey in relation to the waveguides. By using the second principle we expect Ey to be localized long the axis of symmetry since it is continuous along both of them as can be seen from Fig. 3. Finally we need to find the phase of Ey between the CB and the external waveguides. Since OE(1) is the fundamental mode of the y-odd/z-even symmetry-set we can expect Ey to be in phase between all waveguides since this minimizes the spatial variations while localizing most of the fields in the high index media, as dictated by the first principle. Unfortunately, for {dz, dy} ≠ {0, 0}, our principles are not as useful. However we can still extrapolate their distribution from the known field-distribution at {dz, dy} = {0, 0}. We do so, via the first principle, by finding a new distribution that maintains the salient characteristics of each mode while mainly minimizing the spatial variations. In this case, for example, since Ey is in phase between all waveguides, if the CB gets closer to the right most beam, the center of field distribution in the CB will shift to the right too in order to satisfy the first principle. This increases the localization of the fields in the high index media without substantially increasing the spatial field-variations because Ey was originally in phase between all three waveguides. Altogether we observe a pulling force due to the increasing magnitude of the field between the two closer waveguides.
OO mode
Instead the OO mode induces a trapping-force. However this force is tiny: the maximum value of the dimensionless force is Fz = ±0.001 at dz = ∓01a. This is due to the z-odd symmetry which forces Ey, the dominant component, outside the CB. As a result of a negligible magnitude of the fields on the central waveguide, as shown in Fig. 3, a negligible force is formed.
OE(2) mode
The OE(2) mode is the second mode in order of energy associated to the OE symmetry set and, not surprisingly, characterized by Ey out of phase between the CB and the external waveguides as shown in Fig. 3. As a result, for example, as the CB moves closer to the right most waveguide, the fields in the CB will shift to the left to avoid the fast spatial variations at the right wall (first principle), therefore increasing the amplitude of the fields on the left most wall of the CB, inducing a one-dimensional trapping-force along the z-axis. This effect is also shown in Fig. 2(a) where, for positive values of dz the forces are negative and vice versa. The fast change of the force at small separations between the CB and one of the external beams is not treated here but it is clearly explained in [14]. On a section of the waveguide of length a the magnitude of the trapping force is in line with previous reports [17]: for a 0.1 W input power, at dz = 0.15a, the maximum magnitude of the force is ∼ 0.4 nN; measured as in [11]. The group velocity of each mode has also been taken into account when the absolute force per input power is computed, though, for waveguides with constant cross-section, it is fairly constant through all modes.
EO mode
Finally, the EO mode induces large pulling forces, as shown in Fig. 2(a). This mode is dominated by the Ez component (third principle) as can be seen in Fig. 3, which results in the formation of a slot-mode between parallel surfaces (first principle) [14]. However, Ez does not need to quickly decay inside the CB to ensure localization as it happens on the external beams, in accordance to the first principle. Most notably, the large intensity of the Ez component along the walls of the CB then translates into a very large pulling force, accordingly to the fourth principle. In fact, on a section of the waveguide of length a and for an input power of 0.1 W, the attractive force between the two closest waveguides, at a minimum separation of 0.02a (which corresponds to dz = ±0.23a), is ∼ 4 nN. In comparison, the largest attractive force between two parallel waveguides (2WC), shown in Fig. 2(c), with identical cross-sections, separation and wavevector to the ones considered in this work, is ∼ 2.5 nN.
3.2. y-axis oriented forces
In Fig. 4(a) we show the behavior of the force for displacements along the y-axis and a detail of this figure is shown in Fig. 4(c). We notice that for all modes the component of the optical force parallel to the y-axis is not 0 at dy = 0 but at dy ≈ 0.035a. Such a small off-set justifies our initial assumption that, even in the presence of SiO2 layers, we can assume the system to be optically z-axis symmetric. Also the presence of the SiO2 is barelly visible in the first row of Fig. 3. As a result, since one of our objectives remains to illustrate the dynamic of the force via the field distribution of each mode, we conclude that our initial assumption was apropriate and that the SiO2 plates only act as a mechanical support but has no significant impact on the field distribution of the modes considered.
Fig. 4 (a) Normalize force per unit energy as a function of a displacement dy/a for the Yodd-Zeven(1), Yodd-Zodd, Yodd-Zeven(2) and Yeven-Zodd modes at a constant axial wave-vector kx = π/a. (b) Schematic of the geometry in use where a is the height of the blocks and t0 is the thickness of the SiO2 support layer equal to 0.5a and the height and thickness of the waveguides is h = a and t = 0.5a respectively. (c) detail of (a), approximately of the box shown in clear green. (d) In-plane Eyz vector-field distribution (larger arrows mean more intense) with the total |E|2 in the background (darker is more intense) for the same geometry and modes described in (b). All arrows lengths and color scales share the same normalization.
As we have previously mentioned, the components of the force are decoupled along the axis of symmetry, so that induced displacements along the y-axis are independent on the forces along the z-axis, albeit related through their respective field distribution. From Fig. 4(a) we observe that the OE(1), OO and EO induce trapping forces for displacements of the CB along the y-axis while only OE(2) induces levitation. As demonstrated in the previous section this behavior can be described by the field dynamics, shown in Fig. 4(d).
OE(1), OO and EO modes
The trapping force induced by the EO and OE(1) are once more due to the first principle, since, in both cases, the dominant components are in phase through all three waveguides. As a consequence, for a displacement of the CB along the y-axis, the fields will tend to be centered close to {dz, dy} = {0, 0}, inducing a trapping force. A weak but now noticeable trapping force, due to the localization of the fields in the two external beams, is induced by the OO mode since, when CB moves, the fields stay localized along the z-axis.
OE(2) mode
Instead we observe a levitating force for the OE(2) mode. To better understand such behavior let’s consider the case where the CB moves towards the origin from large and positive values of dy (dy ≫ a). In such condition the system can be thought as formed by two waveguides, one formed by the two external beams (that can now be thought as a single waveguide) and the CB, where the dominant field component of OE(2), Ey, is out of phase between the former and the later. As the separation between the waveguides is reduced, in order to satisfy the first-principle, the fields along the CB are expected to move towards its upper surface. The same behavior is observed when the CB penetrates the region between the two parallel waveguides, due to the fact that Ey is still out of phase between the CB and the external waveguides, as shown in Fig. 4(d). As a result OE(2) induces a levitating force along the y-axis as can be seen in Fig. 4(a).
4. 2D trapping, the EE mode
If we were to disregard the magnitude of the force we can notice that the OO mode can be used to trap the nanobeam along both the y and z axis. However, since trapping force along the z direction is very small, two dimensional trapping does not seem feasible. A solution can be found by looking at higher frequency modes. In Fig. 5(a) we plot the magnitude of the force along the y and z axis for the fifth mode in order of energy, which corresponds to the fundamental mode associated with the EE symmetry. From Fig. 5(b) it can be seen that the mode is dominated by Ez, which is out of phase along the z-axis, but in phase along the y-axis. As a result the EE mode exhibits trapping force along both axis, even though the magnitude of the force is nearly two and five fold weaker compared to the trapping force induced by lower energy modes along the y and z axis respectively. However, since no optimization has been done on this system, it is reasonable to expect that larger values of the trapping force could be achieved.
Fig. 5 (a) Normalized force per unit energy as a function of a displacement dz/a and dy/a for the Yeven-Zeven mode at a constant axial wave-vector kx = π/a for the same geometry shown in the inset of Fig. 2(a). (b) In-plane Eyz vector-field distribution (larger arrows mean more intense) with the total |E|2 in the background (darker is more intense) of the Yeven-Zeven mode for three different displacements of the central beam. All arrows lengths and color scales share the same normalization
A summary of the nature of the optical-force along both axis and for all the modes is illustrated in Table 1.
Table 1. Summary of the behavior of the force for each mode.
5. Tailoring forces via geometry
It is important to notice the large scale difference for displacements along the two axis. The CB is free to move between −2a and 2a along the y-axis and only between −0.25a and 0.25a along the z-axis. In fact, in real scale scenarios where the maximum displacement of the beam is at most a fraction of a [7], forces along the y-axis will have little effect on the CB if no further means to enhance or tailor the intensity of the force are used. Here we chose to mold the component of the force along the y-axis and z-axis to have non zero magnitude when the CB is located at {dz, dy} = {0, 0}, thus allowing us to control the position of the trap and the direction of levitation in concordance with the mechanical properties of the CB. For example, the position of the point of equilibrium along the y-axis can be changed by breaking the z-axis symmetry. In fact, the forces at dy = +∞ and −∞ are expected to be of opposite sign, no matter the specific geometry of the system, and, for asymmetric systems, the point of equilibrium (where forces change sign) can happen at any point but the origin. As a case study, in Fig. 6(a), we chose to tilt the walls of the waveguides in order to break the y-axis symmetry as shown in the inset. In this condition the thickness of the top base is tb = 0.25a, the bottom base t = 0.5a and, when the CB lays at the origin, the separation between the walls of the CB and the external waveguides, defined by a line that crosses both walls normally, is 0.25a. In this case we see a significant displacement of the position of the trap, specifically for the EO mode. For the remaining modes the effect is too small to be seen on the chosen scale but are comparable to those observed in Fig. 6(a) for the scale on which dz is represented. Like wise, by breaking the y-axis symmetry, we can expect that the point of equilibrium will shift along the z-axis. A very simple example is shown in Fig. 6(b) where we chose to slightly perturb the thickness or each waveguide with the rule tc < tb < ta. As a result we observe a shift in the position of the trap (OE(2)) toward the left most waveguide (ta), while the effect is opposite for pulling forces (OE(1) and EO modes). We also observe a significant change for the OE mode which now induces a force toward the right most waveguide for all separations.
Fig. 6 (a) Normalized force per unit energy as a function of a displacement dy/a for the Yodd-Zeven(1), Yodd-Zodd, Yodd-Zeven(2) and Yeven-Zodd modes at a constant axial wave-vector kx = π/a. The height of the waveguides is h = a, tb = 0.25a, t = 0.5a, t0 = 0.5a and, when all waveguides lay on the same plane, the walls of the external waveguides are parallel to the walls of the central waveguide. (b) Dimensionless force as a function of a displacement dz/a for the Yodd-Zeven(1), Yodd-Zodd, Yodd-Zeven(2) and Yeven-Zodd modes at a constant axial wave-vector kx = π/a. In the inset we show the schematic of the geometry in use where a is the unit length, h = a, ta = 0.55a, tb = 0.5a, tc = 0.45a and t0 = 0.5a.
6. Conclusion
In conclusion we have studied and explained dynamics of gradient optical forces in the 3WC. This system shows better performance and more complex functionality than the 2WC and can be used to gain optomechanical control along the whole plane transverse to the direction of propagation of the optical mode. Effects like single-mode optical trapping and single-mode optical levitation of the free nanobeam have been discussed and we propose strategies to tailor such forces by breaking the symmetry of the cross-section of the waveguides. More over, all the proposed geometries are amenable to fabrication with standard lithographic techniques. We believe that three waveguides configuration, not only extends the possible application of gradient optical forces, but it can be useful to the development of new reconfigurable optome-chanical devices for optical MEMS applications like switching or multiplexing [18] and might lead the path to develop optomechanical system on stretchable materials [19, 20], where larger optomechanical control should facilitate the manipulation of soft suspended structures.
Acknowledgments
One of the authors (P. A. F.) would like to thank Mian Wang for the useful comments and discussions.
References and links
1. R. N. C. Pfeifer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Colloquium: Momentum of an electromagnetic wave in dielectric media,” Rev. Mod. Phys. 79, 1197–1216 (2007). [CrossRef]
2. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).
3. H. Cai, B. Dong, J. F. Tao, L. Ding, J. M. Tsai, G. Q. Lo, a. Q. Liu, and D. L. Kwong, “A nanoelectromechanical systems optical switch driven by optical gradient force,” Appl. Phys. Lett. 102, 023103 (2013). [CrossRef]
4. X Zhao, J M Tsai, H Cai, X M Ji, J Zhou, M H Bao, Y P Huang, D L Kwong, and Q Liu, “A nano-opto-mechanical pressure sensor via ring resonator,” Opt. Express 20, 8535–8542, (2012). [CrossRef] [PubMed]
5. D. Van Thourhout and J. Roels, “Optomechanical device actuation through the optical gradient force,” Nat. Photonics 4, 211–217 (2010). [CrossRef]
6. J. Ma and M.L. Povinelli, “Applications of optomechanical effects for on-chip manipulation of light signals,” Curr. Op. in. Sol. St. and Mat. Sci 16, 82–90 (2012).
7. V. Intaraprasonk and S. Fan, “Nonvolatile bistable all-optical switch from mechanical buckling,” Appl. Phys. Lett. 98, 241104 (2011). [CrossRef]
8. A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012). [CrossRef]
9. J. Rosenberg, L. Qiang, and P. Oskar, “Static and dynamic wavelength routing via the gradient optical force,” Nat. Photonics 3, 478–483 (2009). [CrossRef]
10. A. Butsch, M. S. Kang, T. G. Euser, J. R. Koehler, S. Rammler, R. Keding, and P. S. J. Russell, “Optomechanical nonlinearity in dual-nanoweb structure suspended inside capillary fiber,” Phys. Rev. Lett. 109, 183904 (2012). [CrossRef] [PubMed]
11. M.L. Povinelli, M. Loncar, M. Ibanescu, E.J. Smythe, S.G. Johnson, F. Capasso, and J.D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005). [CrossRef] [PubMed]
12. G. S. Wiederhecker, L. Chen, A. Gondarenko, and M. Lipson, “Controlling photonic structures using optical forces,” Nature (London) 462, 633–636 (2009). [CrossRef]
13. V. Intaraprasonk and S. Fan, “An optical equilibrium in lateral waveguide-resonator optical force,” http://arxiv.org/abs/1306.2417.
14. P. A. Favuzzi, R. Bardoux, T. Asano, Y. Kawakami, and S. Noda, “Ab-initio design of nanophotonic waveguides for tunable, bidirectional optical forces,” Opt. Express 20, 24488–24495 (2012). [CrossRef] [PubMed]
15. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).
16. M. Eichenfield, J. Chan, R.M. Camacho, K.J. Vahala, and O. Painter, “Optomechanical crystals,” Nature (London) 462, 78–82 (2009). [CrossRef]
17. M. Li, W. H. P. Pernice, and H. X. Tang, “Tunable bipolar optical interactions between guided lightwaves,” Nat. Photonics 3, 464–468 (2009). [CrossRef]
18. M. Li, W. H. P. Pernice, and H. X. Tang, “Broadband all-photonic transduction of nanocantilevers,” Nat. Nanotechnology 4, 377–382 (2009). [CrossRef]
19. C. L. Yu, H. Kim, N. De Leon, I. W. Frank, J. T. Robinson, M. McCutcheon, M. Liu, M. D. Lukin, M. Loncar, and H. Park, “Stretchable photonic crystal cavity with wide frequency tunability,” Nano Lett. 13, 248–252 (2013). [CrossRef]
20. Y. Chen, H. Li, and M. Li, “Flexible and tunable silicon photonic circuits on plastic substrates,” Scientific Reports 2, 622 (2012). [CrossRef] [PubMed]
