The maximum magnetic field sensitivities of −100 pm/mT and −488 pm/mT have been experimentally demonstrated with the PMF lengths of 75 cm and 20 cm respectively in the range of magnetic field strengths from 0 to 200 mT. The dependence of the magnetic field orientation on the performance of the proposed sensor was also examined. The proposed magnetic field sensor is advantageous for applications requiring higher sensitivity over a wide magnetic field range.This paper proposes a novel magnetic field sensor based on a microfiber coupler (MFC) combined with a magnetic fluid (MF) in a Sagnac loop formed from a polarization maintaining fiber (PMF). Thanks to the small (~2.6  Water-based MFs are highly stable colloidal suspensions of magnetic nano-particles distributed evenly throughout the volume, which makes them easy to integrate with optical fibers. Several magnetic field sensors have been developed by combining optical devices with magnetically susceptible fluids utilizing the tunability of the magnetic fluid’s RI to measure the external magnetic field strength. For example, a sensor based on a magnetic fluid clad etched fiber Bragg grating3 and a sensor using tilted fiber grating interacting with magnetic fluid4 have been reported. It should be noted, that since interaction between the fundamental core mode and lower-order cladding modes in such structures is usually quite low, such sensors suffer from relatively low sensitivity to magnetic fields, normally less than 10 pm/mT. Zu et al.5 proposed a magnetic field sensor with a higher sensitivity of ~16.7 pm/Oe but a more complex structure based on a Sagnac interferometer incorporating a free-space MF film with the use of two collimators.Optical fiber magnetic field sensors using magnetic fluids (MFs) as sensing materials have aroused significant research interest due to the magneto-optical properties of MFs, such as field dependent refractive index (RI), optical birefringence and transmission8. Microfiber based magnetic field sensors show higher sensitivity to external RI than traditional optical structures utilizing interference because their much smaller diameter means that a larger proportion of the evanescent field surrounding the microfiber can interact with the local environment. For example, Zheng et al.9 proposed an optical microfiber mode interferometer coated by a MF with a sensitivity of −29.3 pm/Oe and Deng et al.10 reported a magnetic field sensor with a sensitivity of ~162 pm/mT, composed from an asymmetric taper with a waist diameter of 45 μm immersed in a magnetic fluid. However, the fabrication of such an asymmetric taper structure requires accurate manual adjustment of the fibers’ positions within the fusion splicer during the application of multiple discharges leading to poor reproducibility. Luo et al.11 reported a magnetic field sensor with an MFC surrounded with a magnetic fluid, with a sensitivity of 191.8 pm/Oe in a narrow range of weak magnetic fields of up to ~300 Oe. Several researchers13 reported experimental and theoretical studies of fiber sensors where an MFC was combined with a Sagnac loop for sensing of refractive index or magnetic field, showing that a very high sensitivity is achievable with such structures.Among the variety of fiber optic structures suitable for optical sensing, microfiber couplers (MFCs) have become a topic of special interest due to their superior sensing properties compared to other fiber structures and conventional fused fiber couplersIn this paper we propose and examine a novel magnetic field sensing structure formed by connecting two ends of an MFC immersed in a magnetic fluid with a section of polarization maintaining fiber (PMF) to form a Sagnac loop. The introduction of a PMF in an MFC-Sagnac loop sensing structure has not been investigated before and allows for control over both the free spectral range (FSR) of the transmission spectrum and the sensitivity of the sensor by adjusting the length of the PMF section. We investigate for the first time the properties of the proposed magnetic field sensor, its hysteresis behavior and the dependence of the sensor’s performance on the applied magnetic field orientation. The proposed structure offers relative simplicity, good repeatability for fabrication and demonstrates a competitive magnetic field sensitivity of −488 pm/mT in the magnetic field range from 0 to 200 mT. The sensitivity of the proposed sensor could be further improved by optimizing the parameters of the structure using a simple analytical model also presented here.μm. A section of polarization maintaining fiber (PMF) with a length of ~75 cm connects the two ends of the MFC at the same side to form a Sagnac loop interferometer. A broadband source (BBS) with an output spectrum from 1500 nm to 1580 nm serves as the input light source and the output spectrum is observed using an optical spectrum analyzer (OSA) with a resolution of 20 pm. A polarization controller (PC) is inserted into the loop to control the input light polarization state. An optical isolator (ISO) is inserted between the BBS and MFC to prevent backward propagating light from reaching the BBS.A schematic diagram of the proposed magnetic field sensor is shown in Fig. 14. By utilizing the birefringence effect in the PMF section, it is possible to achieve a desired FSR value for the combined MFC-Sagnac structure by variations of the length of the PMF section. When magnetic field acts upon the MFC waist section covered with the MF, the overall interference spectrum changes due to the variation in the magnetic fluid’s RI. As a result, by measuring the parameters of interference spectrum, the strength of the applied magnetic field can be monitored assuming a suitable calibration has taken place.The operating principle of a Sagnac loop is based on the multi-beam interference between the clockwise and counterclockwise beams in the fiber loop. The MFC itself typically has a transmission spectrum containing interference fringes, the visibility and free spectral range (FSR) of which along with the coupling ratio are strongly affected by the coupler’s waist diameter15:l is the length of the PMF section, (no − ne) is the birefringence of the PMF, λ is the operating wavelength and θ1 and θ2 are the angles between light polarizations at both ends of the PMF and r is the coupling ratio of the fiber coupler. Light whose polarization is perpendicular to the optical axis is governed by the refractive index no. Light whose polarization is in the direction of the optic axis sees an optical index ne. Given a fixed length of the PMF, the birefringence (no − ne) is constant. Standard commercial fiber couplers are designed to realize a fixed coupling ratio over a wide wavelength range , in effect r is independent of wavelength over a wide range. Furthermore the r is independent of the RI surrounding the coupler. In our proposed configuration a conventional fiber coupler is replaced by an MFC whose coupling coefficient (rMFC) is both wavelength- and surrounding RI-dependent16:n1 and n2 are the RIs of the silica fiber and surrounding medium, a is the radius of the microfiber waist, d is the center-to-center spacing between fibers forming the MFC, and V is the normalized frequency defined as In a conventional Sagnac loop structure, a standard fiber coupler is used to split the light into two beams which counter-propagate within the PMF loop. Propagating within the PMF loop the two light beams accumulate a phase difference before re-combining at the coupler and they produce interference due to the acquired phase difference. The transmission equation for a conventional coupler-based Sagnac loop is as followsa), the center-to-center spacing between the fibers (d) and length of the waist section (L), is shown in Fig. Equation  allows fEin enters the MFC, the output light in the through port (Eo1) and the coupling port (Eo2) are:Assuming input light with amplitude Eo, Fig. After incorporating the PMF loop, light from both the through and coupling ports of the MFC will counter-propagate within the loop, experiencing interference upon reentering the MFC. The resulting transmission equation for proposed MFC-Sagnac fiber structure ,,5), the Therefore the transmission equation for the proposed structure can be written as:Analysis of Eqs  and 7)  7) showsa of the MFC used in the simulation is 2.6 μm, the coupling length L is 2 mm, the center-to-center spacing d is 1.26 μm, and the length of the PMF l is 0.75 m. As can be seen from the figure increasing the MFC coupling ratio rMFC(λ) which is influenced by the RI surrounding the MFC waist.Figure 4O3 particles within the MF to realign which leads to a change of the magnetic fluid RI and consequently to a change in the resulting interference spectrum of the fiber structure. The changes in the transmission spectrum of the proposed structure then can be related to the value of the magnetic field strength.As can be seen from the above analysis, unlike conventional couplers typically used in a Sagnac loop, the coupling coefficient of the MFC strongly depends on the changes in the surrounding RI due to the large proportion of the evanescent field which extends into the medium in the vicinity of the tapered waist section. This strongly affects the interference conditions for the light entering the two ports of the PMF loop. In the proposed fiber sensor structure, the MFC, whose tapered waist is coated with a layer of a magnetic fluid, acts as a magnetic field sensing element. An external magnetic field applied to the MFC waist region forces the Fe17ϕ is the combined phase difference accumulated due to the birefringence of the PMF and MFC coated with MF layer, Bm and Bp are the birefringence values for the magnetic-fluid-clad-MFC and PMF respectively, ΔBm is the change of the birefringence values for the magnetic-fluid-clad-MFC with the external magnetic field, l is the length of the PMF, D is the length of the coupling region of the magnetic-fluid-clad-MFC of the MF layer, which is ~3.5 mm in our experiments and λ is the wavelength of operation.The sensitivity of the sensor depends on the dip wavelength shift, which can be estimated by the formulaBm ~ 1.2 × 10−3 for magnetic fields above 30 mT)18 is approximately one order of magnitude greater than that of PMF (Bp ~ 10−4). However, since the length of the magnetic-fluid-clad-MFC is D ~ 3.5 mm, which is 100 times shorter than that of PMF (l ~ 700 mm), in our experiment Bpl ≫ BmD, so thatMFS is the difference of the magnetic field strength.Magnetic field-dependent birefringence of MF  and directly proportional to the birefringence of the MFC (ΔBm). Since the birefringence of the MFC is determined by its length and the MF layer thickness both of which are fixed for a packaged MFC, the sensitivity of the structure can be enhanced by shortening the PMF section. On the other hand, since a reduction in the PMF length leads to an increase of the free spectral range of the transmission spectrum, the PMF section should be sufficiently long to ensure the presence of at least one transmission dip within the wavelength range of interest. Thus the length of PMF cannot be shortened without limitations and setting the length of the PMF will involve reaching a compromise between sensitivity and practical need to operate within the wavelength range of an available source. Figure Equation  shows thThe experimental setup for the magnetic field sensor characterization is shown in Fig. Figure 12 in 2015. Similarly for the sample with the PMF length of 20 cm, the maximum sensitivity is estimated as −488 pm/mT in the magnetic field range from 0 to 200 mT. As predicted by the theoretical model, a shorter length of the PMF section leads to a higher sensitivity.Figure The hysteresis behavior of the sensor was also studied by increasing the magnetic field strength from 0 to 200 mT and then decreasing it down to zero with the same step values while recording the wavelength shift for the selected transmission dip. The results are shown in Fig. μm. For this purpose a separate experiment was set up using two permanent magnets to create a parallel and homogeneous magnetic field. A 360° rotating platform (from Thorlabs) was used to hold the two magnets positioned at a fixed distance from each other and in the vicinity of the stationary sensor head. The direction of the magnetic field applied to the sensor was changed by rotating the platform with 5° steps, in the planes X-Z and also Y-Z as shown in Fig. α is defined as the clockwise angle between the normal line of the magnetic field and to the direction the MFC axis. α = 0° when magnetic field axis is normal to the MFC axis, and α = 90° when magnetic field axis is parallel to the MFC axis. For each of the experimental configurations measurements of the wavelength shift of the selected transmission dip were carried out by first setting the magnetic field strength to a certain value  by using different sets of permanent magnets and then changing the rotation angle α from 0° to 360° in 5° steps. The results of the experiment are illustrated in Fig. The dependence of the magnetic field orientation on the performance of the proposed sensor was also examined for the sample with the PMF length of 75 cm and the MFC waist diameter of 2.6 α was set to 0°  and the minimum shift was recorded when α was 90° . This is true for both field rotation experiments in the X-Z and Y-Z planes. The maximum wavelength shift in the X-Z plane is greater than that in the Y-Z plane, which most likely a result of the shape of the MFC placed in the PDMS slot.In order to explain the dependence of the wavelength shift on the magnetic field rotation angle, we need to consider the re-distribution of iron nanoparticles in the MF under the action of the applied field and with respect to the fiber (MFC) axis. When the magnetic field is applied in the vicinity of the PDMS container, the nanoparticles tend to gather around the surface of the PDMS slot, forming multi-particle chains and their density around the MFC surface becomes non-homogeneous and asymmetrical with respect to the fiber axis. This in turn results in a non-homogeneous RI of the MF surrounding the fiber surface. When the magnetic field direction is rotated in the X-Z or Y-Z planes, the chains direction and nanoparticles density around the MFC rotates accordingly, affecting the RI of the MF and ultimately the sensor’s response. The maximum wavelength shift was observed when Another experiment was carried out to study the dependence of the performance of the proposed sensor on the magnetic field direction in the plane perpendicular to the MFC axis. A schematic diagram of the experiment is shown in Fig. μm) and PMF length of 20 cm. We studied the hysteresis behaviour of the sensor and demonstrated that the value of hysteresis is small. We also studied and analysed the influence of the magnetic field direction on the sensor’s response. This proposed sensor offers advantages for applications requiring higher sensitivity and a wide magnetic field range.A MF based sensor of magnetic field using a combination of a MFC and a Sagnac loop has been proposed and studied both theoretically and experimentally. When magnetic field acts upon the MFC waist section covered with the MF, the overall interference spectrum changes due to the variation in the magnetic fluid RI. The small waist diameter of the MFC leads to a large proportion of the evanescent field in contact with the magnetic fluid and this enables a stronger influence of the RI of the medium surrounding the MFC waist on the resulting interference spectrum of the structure, leading to its high magnetic field sensitivity. We have developed a simple theoretical model, allowing for the analysis of the influence of different parameters of the proposed structure on its interference spectrum. The model agrees well with the experimentally measured spectra.The sensitivity of the proposed sensor can be enhanced by shortening the length of the PMF section. The maximum magnetic field sensitivity of −488 pm/mT in the range from 0 to 200 mT has been experimentally demonstrated with a MFC of small waist diameter .19. An MFC sample with a waist diameter ~2.6 μm was used in our experiments. The PDMS container was fabricated to a cuboid with a slot cut through its center along the long side. The MFC was immobilized within the slot by securing the output fiber ends with a UV-curable glue. Two small holes in the top cover of the container are located above the ends of the slot for injecting the MF liquid and exhausting the air.The MFC was fabricated by simultaneously tapering and fusing two standard single-mode fibers (SMF-28) using a method known as the microheater brushing technique4O3 particles at a concentration of 1 mg/ml, which was purchased from Cytodiagnostics Inc.The MF sample (IO-A10-1) was employed with 10 nm Fe


We report a theoretical and experimental study of coupling between a whispering-gallery-mode (WGM) microdisk resonator and a fiber taper which exchange energies at two distinct regions. We observe an oscillatory behavior in the coupling strength as a function of the distance between the two coupling regions when a fiber taper is moved laterally above the resonator at fixed vertical distance. This oscillation is clearly seen in the linewidth of the resonance as well as in the on-resonance transmission. A theoretical model considering for two-point coupling successfully explains the experimental observations as being a result of the interference between the light fields coupled into and out of the resonator at two distinct regions and the light transmitted through the waveguide. Critical coupling in two-region coupling is a collective result of the coupling at two different coupling regions, and does not require critical coupling at both or at any one of the two coupling regions. This relaxes the conditions for achieving critical coupling in waveguide-resonator systems. The discovery of this previously unnoticed oscillatory behavior in two-region coupling between a WGM resonator and a waveguide will benefit both fundamental studies and practical applications based on WGM resonators. Moreover, it is also a basic and highly versatile unit for building more complicated photonic structures and processors, such as add-drop filters11, coupled-resonator optical waveguides12, and parity-time-symmetric structures for unconventional control of light16. Besides high quality Q factor and small mode volume V, maintaining an efficient, stable, and close-to-ideal coupling between the resonator and the waveguide is crucial for the success of the aforementioned studies and applications. Therefore investigating and understanding the behavior of a resonator with a waveguide under various coupling configurations and conditions is of great important from both the fundamental and the practical points of view.A resonator coupled to a waveguide constitutes a very simple yet powerful platform to carry out fundamental research on optomechanics17, the normalized transmission T of a waveguide-coupled resonator is given byEin and Eout are the amplitudes of the light fields in the waveguide right before and after the resonator-waveguide coupling (RWC) region, α0 = eαπr− is the inner circulation factor describing the change of the field amplitude after a round-trip in the resonator with α representing the equivalent absorption coefficient which includes all the intrinsic losses, such as material absorption and the scattering into the surrounding and the waveguide modes other than the resonant mode (RM) of interest18, r is the radius of the path the RM travels, ϕ0 = 2kRπr is the round trip phase with kR = nRk0, nR is the effective refractive index of the RM, k0 = 2π/λ0 indicates the propagation constant in vacuum, λ0 is the wavelength in vacuum, and t is the transmission coefficient through the waveguide and is related to the coupling coefficient κ through |κ|2 + |t|2 = 1. The coupling coefficient κ is mainly determined by the amount of spatial overlap between the RM and the waveguide mode (WG) as well as by how much their propagation constants differ20.According to the coupling theory proposed by A. Yariv18. The linewidth of the resonance at the critical coupling is twice the linewidth at the deep undercoupling regime. Recently, M. Ghulinyan et al.21 showed that the dynamics of the RWC process differs significantly from the conventional picture when the waveguide is moved vertically with respect to the resonator with the lateral distance kept constant. They demonstrated that contrary to the conventional lateral case, the vertical coupling process exhibits oscillatory RWC with alternating under- and over-coupling regions separated by multiple critical points as the vertical distance between the resonator and the waveguide varies. The vertical oscillatory RWC effect was interpreted via the coupling mode theory by considering the RWC as the coupling between two parallel waveguides.Note that coupling between a resonator and a waveguide is through their evanescent fields whose strength decreases exponentially with distance. Thus, the RWC becomes stronger when the waveguide approaches the resonator either horizontally or vertically because spatial overlap between the light leaking out from the resonator and that from the fiber taper increases. As the distance between the resonator and the waveguide decreases laterally, the RWC goes from under coupling, where intrinsic loss is higher than coupling loss, to critical coupling point, where intrinsic loss equals to coupling loss, and then to over-coupling regime, where coupling loss is higher18. The RWC increases monotonically as the waveguide approaches the resonator before they are in contact. In the second case, however, the resonator is placed in a plane above or underneath the rim of the resonator21 and the vertical distance is varied. In both cases, the coupling and the energy exchange between the waveguide and the resonator takes place at only one small region as shown in Fig. Up to this date discussions concerning the RWC have been limited to two cases: In the first case, the waveguide is placed in the same plane as that of the resonator but outside the resonator’s circular rim and the lateral distance between them is varied22 and compound ring resonators with double couplers23, which consist of a resonator coupled to a coplanar curved waveguide via two individual points. The coupling behavior of the resonator influenced by various parameters, such as coupling coefficient at each coupling point, were theoretically investigated. These theoretical investigations have not discussed any effects of shape and position of the waveguide on the coupling, which is essential in practical applications. In this paper, we experimentally studied the two-region coupling behavior between a waveguide and a resonator on different planes and focused more on the untouched aspects, such as the influences of the horizontal position and thickness of the tapered fiber on the coupling, of the previous theoretical studies. We show that one can obtain very efficient RWC as well as critical coupling by placing the waveguide of various thicknesses at different horizontal place even far from the rim of the resonator. The critical coupling in the two-region coupling regime for thicker tapered fibers is due to the collective response of the system at two different coupling regions, and critical coupling at each of the coupling regions is not necessary. As the horizontal position d of the waveguide with respect to the center of the resonator is scanned laterally while the vertical height above the resonator plane is kept constant, RWC experiences periodic oscillation in the d < r region.Here, we investigate a new RWC configuration Fig.  in which2 etchings24. The thickness of the silica microdisk resonator was 2 μm. The fiber taper was prepared by pulling a single mode optical fiber while it was heated by a hydro-oxygen flame. Optical microscope and scanning electron microscope (SEM) were employed to characterize the topographic features of the microdisk resonator and the fiber taper. A tunable laser in the 1450 nm band was used to probe the resonances and the coupling process. The light at the output port of the fiber was detected by a photodiode. An oscilloscope and a computer were employed to monitor and analyze the transmission spectra. The relative position of the microdisk resonator with respect to the fiber taper was precisely controlled by a 3D piezoelectric stage with a resolution of 100 nm.The RWC system in our experiments consists of a microdisk whispering-gallery-mode (WGM) resonator and a fiber taper as the coupling waveguide. The microdisk WGM resonator was fabricated on a silica-on-silicon chip following a procedure consisting of a single UV-lithography step and two rounds of HF and XeF A full-vectorial eigenmode solver based on COMSOL software was employed to simulate the resonance frequencies and the distributions of the energy density of the WGMs supported in the microdisk resonator. The structure parameters were set according to the measurements based on the SEM image and optical microscope image of the resonator.μm. The SEM image depicting the side view of the edge of the microdisk WGM resonator is shown in Fig. 1,453. The 2D distribution of the electric field energy density of this mode is given in Fig. x where we see that in the radial direction the mode distribution of the mode TE1,453 has only one intensity maximum at x = r ~ 74.5 μm.We first characterized the basic properties of the inverted-wedge microdisk resonator used in the experiments. The radius of the top surface of the inverted-wedge microdisk resonator was measured to be 76.5 1,453 mode of the resonator shown in Fig. x direction at fixed heights y above the resonator surface as shown in Fig. x = 0, and the lateral move of the tapers was continued until about x = 80 μm where the coupling between the taper and the resonator was lost completely. Figure μm and 1.2 μm, which clearly show the oscillatory behavior of the coupling effect. More importantly, for both of the fiber tapers critical coupling point at which the on-resonance transmission approaches zero was obtained. We observed that as the fiber taper approached the edge of the resonator the spatial overlap between the WGM and the WM monotonically increased and the RWC became stronger. When the fiber taper moved further towards the resonator center, we observed oscillatory RWC manifested with a series of relatively efficient coupling regions separated by some dramatically weak coupling regions. We also noticed that the position, at which the critical coupling point occurred, varied when the diameter of the fiber taper changed. For the thinner fiber taper the critical coupling appeared for d ~ 74.5 μm whereas for the thicker one it took place at d ~ 72 μm. We attribute this to the fact that the thicker fiber taper has a larger effective refractive index nW than the thinner taper so that, compared to the thin fiber, the thicker taper needs to be closer to the center of the resonator to satisfy the matching condition kR = kW cos θ between the wavevector z direction when θ because the angle θ between d decreases. Note that the overlap between the taper mode and the WGM decreases with decreasing d. This is similar to the case that the coupling efficiency between two parallel identical straight waveguides are stronger than that between the same two waveguides placed such that they are not parallel to each other25. In addition to the difference in the position of the critical coupling, we observe that for d > r the coupling between the resonator and a thicker tapered fiber is always in the undercoupling regime.Using fiber tapers with different diameters we acquired the transmission spectra around the TEμm, we observed efficient energy coupling with on-resonant extinction ratio of −16 dB even when the fiber taper was placed at x = 65.0 μm. The results are shown in Fig. μm placed at x = 74.5 μm is also given in Fig. x of the fiber tapers are shown in the optical micrographs in Fig. 1,453 mode at x = 65 μm is negligibly small, thus no efficient light exchange between the microdisk and the fiber taper can take place at this position for the one-point coupling because the WGM and the waveguide modes do not have overlap.Using a fiber taper of diameter 1.7 26. For simplicity, we assume that the coupling between the resonator and the waveguide takes place at the points 1 and 2 − and α2 = eαrθ− are the inner propagation factor, ϕ1 = 2kRr(π − θ) and ϕ2 = 2kRrθ are the phase changes related to the light along the arcs separated by the two coupling points, and ϕ3 = 2kWr sinθ is the phase shift induced by the light propagation in the waveguide between coupling points 1 and 2. kW = nWk0 denotes the light propagation constant in the waveguide and nW is the effective refractive index of the waveguide mode. For simplicity, we set kW as a constant by neglecting the change of the diameter of the fiber taper with respect to the horizontal position z. The loss in the waveguide is neglected as well.To explain the experimental observations, we developed a theoretical model based on coupled mode theory considering both the one-region and two-region couplingd 2 Fig. . Under tEin. From Eqs , ϕ1 + ϕ2 = 2mπ and To further simplify the model, all the field amplitudes are normalized to From Eqs  and 3),,Ein. Frκi asd < r region), where κ0 represents the strongest coupling one can obtain at either of the two coupling points, κi exponentially decays with the increase of the phase mismatch between the tangential component of kW at the coupling points and kR, γ is a constant determining the decay rate. When the waveguide is placed at somewhere with d > r, we assume that RWC is in the one-point coupling regime where the coupling is achieved via a point on the resonator and a point on the waveguide nearest to it, arriving atnS is the refractive index of the surround materials. Equation  are equal to each other, the resonator-waveguide system is said to be at critical coupling and the transmission becomes zero. It is interesting to note here that in the two-point coupling configuration even though the critical coupling condition is not satisfied at each of the individual coupling points, the system, can still be operated at the critical coupling due to the interference of light fields taking different paths. For example in Fig. d. We attribute these to the fact that the developed model is a very simplified one which does not take into account the change in the spatial overlap of the WGMs and the waveguide modes and other experimental artifacts such as mechanical instabilities and thermal variations which lead to varying coupling efficiencies and resonance wavelength fluctuations as well as the wavelength and intensity fluctuations of the external cavity laser diode used to obtain the transmission spectra. A model which takes those into account will certainly have a better agreement with the experiments.In Fig. In summary, we have introduced a new coupling scheme for the single-waveguide coupled resonator system where a fiber-taper placed at a plane above or blow the resonator can sustain two-point or single-point coupling depending on whether it is placed directly above the resonator or not. The single-point coupling is similar to the traditional approach and exhibits transition from under-coupling to over-coupling regimes through a critical coupling point as the lateral distance between the resonator and the taper is decreased. The two-point coupling, on the other hand, exhibits an oscillatory coupling process and can sustain critical coupling without the requirement that each of the coupling point sustain critical coupling. This is a manifestation of the interference of different paths the light fields takes from the fiber to the detector passing through the resonator. Note that critical coupling is achieved for thicker fiber taper with higher effective refractive index, which gives rise to robust coupling. Our theoretical and experimental results clearly demonstrate the existence of this new coupling behavior which have gone unnoticed up to this date. Because a waveguide-coupled resonator is the basic unit of many more complicated optical processing units and optical sensor networks, we believe that this previously unexplored coupling scheme will help to advance these areas and find usage in applications of microresonators such as tunable feed back system.


The model utilizes graphene as a universal representation of these organic adlayers. It predicts the energy position of the interface state as well as the overlap of its wave function with the bulk metal without free fitting parameters. We show that the energy of the interface state depends systematically on the bond distance between the carbon backbone of the adayers and the metal. The general applicability and robustness of the model is demonstrated by a comparison of the calculated energies with numerous experimental results for a number of flat-lying organic molecules on different closed-packed metal surfaces that cover a large range of bond distances.We present an analytical one-dimensional model potential for the description of electronic interface states that form at the interface between a metal surface and flat-lying adlayers of  It depends crucially on the energy alignment and the wave function overlap of electronic states at such interface, which also governs the binding and even the growth of the molecular layer12DA) on Ag1 for whiIt turned out, however, that a realistic description of organic molecules on metal surfaces by DFT is challenging although it is one of the most widely used approaches for the determination of the geometric and electronic structure at surfaces and interfacesπ-π interaction as in graphene.In order to highlight the main physical mechanism for the formation of the delocalized interface state at organic/metal interfaces without the help of complex DFT calculations, we propose in this letter a one-dimensional description by an analytical model potential. The choice of the potential is inspired by previous work on surface states of clean metals30Our model calculation clearly illustrates how the interface state develops from the former Shockley-type surface state of the bare metal substrate with increasing interaction between the molecular film and the metal. By comparing our model results with available experimental data for different organic molecules, we show that our model is able to describe the systematic dependence of the interface state’s energy on the bond distance between the carbon backbone and the metal with predictive power if this distance and the work function are known. Moreover, the model reveals how the wave function overlap of the interface state with both, the bulk metal and the molecular overlayer depends on the carbon-metal distance.z0 from the position of the topmost atom at z = 032Our one-dimensional model potential is based on the nearly-free-electron model for the bulk. In this approximation, the electronic states within the metal are given by the solution of the Schrödinger equation under the influence of a weak periodic pseudopotential. Already within this simple model, the formation of the Shockley surface state can be described by introducing a potential barrier at the surface. For illustration of the basic properties of this surface state, we first recall the textbook example of a step barrier at a distance g = 2π/a and the distance between lattice planes a. With an appropriate choice of the inner potential V0 and the corrugation Vg, this model provides a good approximation of the energy-momentum dispersion E(k) of electron states for the chosen direction in simple metals such as Al, Ag or Cu which are derived from sp electronsVg in which bulk electronic states are forbidden. The bulk eigenfunctions(Bloch states) at the bottom and at the top of this gap correspond to the k-value at the Brillouin zone boundary Vg > 0), with the reciprocal lattice vector The potential barrier at the surface leads to an additional solution with an energy within the bulk band gap. The corresponding eigenfunction is located at the surface and consists of an exponentially decaying cosine Bloch function in the bulk and a simple decaying exponential on the vacuum side,δ of the bulk cosine function is directly related to the energy E of the surface state. It varies from δ = −π/2 at the bottom to δ = 0 at the top of the bulk band gap. The wavefunctions in the bulk and in the vacuum can only be matched for one specific value of the phase δ or the respective energyV0 and the position z0 of the surface barrier. It is, however, much more sensitive on the latter. V0 and decreasing z0. With parameters for Ag(111) and z0 = a/2, for example, it turns out that changing the surface state’s energy by 1 eV requires a change of the barrier height as large as 4 eV, but only a change of z0 by ∼a/10. The key point for the following discussion is that adlayers of organic molecules modify the distance as well as the height of the barrier, but shift the energy of the surface state predominantly according to their adsorption distance.with et al.For our model potential we use a more realistic description of the potential barrier at metal surfaces as was introduced by Chulkov dC. This potential is composed of four partsRecently, we have proposed such one-dimensional model potential for a description of the image-potential states of graphene(g) on metal substratesVm(z) denotes the metal potential and Vg(z) the potential of the π-conjugated graphene layer. The latter is a parameterized analytic expression of the potential proposed by Silkin et al.dC is the distance of the carbon atoms in the graphene layer with respect to the position of the outermost metal atoms located at z = 0. VΦ(z) and δV(z) are corrections that consider the difference in work function between the bare and the graphene covered metal and the influence of higher-order image-charges, respectively. Beside the distance and change of the workfunction, V(z) is fixed by the separate properties of the metal and the adlayer and does not contain further free fitting parameters. In particular, the metal potential quantitatively reproduce not only the image-potential states, but also the Shockley surface state on the(111) noble metal surfaces.where π-π interaction as in graphene as long as the corresponding carbon-metal distance and work function is considered. For the Ag(111) substrate, we use the parameters given in ref. dC = 3.33 Å as reported in ref. Vg(z), VΦ(z) and δV(z) are determined as described in ref. dC = 5 Å. The wave functions Ψ and energies E of the Shockley surface state(SS) for bare Ag(111) and the interface state(IS) for graphene covered Ag(111) have been calculated at the center of the surface Brillouin zone and then show that the results for this system can be directly related to adlayers of flat-lying organic molecules containing carbon rings that have a similar p of its probability density for z < z0 = a/2. For bare Ag(111) at the p = 76.23% and ESS = −59 meV relative to the Fermi level.We characterize the coupling strength of the interface state to the metal bulk by the the fraction dC. In real systems, dC varies in the range of 2.2–3.7 Å substrate(dotted line) is substantially modified only at distances that are larger than the extension of the Shockley surface state into the vacuum. At this distance, the probability density of the interface state(red solid line) is still basically identical with that of the Shockley surface state of bare Ag(111), but its energy relative to the Fermi level of EIS = −136 meV is slightly reduced. This is caused by the reduction of the barrier in the region between the metal and the carbon layer as can be seen by comparing the dashed and the solid line in dC, there is an interplay between a further reduction of this barrier and the approach of the barrier between the carbon layer and the vacuum closer to the metal. The first leads to a decrease of the interface state’s energy, the latter to an increase. At a distance of dC = 4 ÅEIS = +331 meV. The interface state becomes therefore an unoccupied state. The reallocation of the probability density from the metal surface into the interface and the vacuum region is even stronger as compared to dC = 4.00 Å and the penetration into the metal further decreases to p = 64.70%. The blue solid line shows for comparison the laterally averaged result of a DFT calculation that has been performed for PTCDA on a nine-layer thick Ag(111) slabIn the following we discuss the transition of the Shockley state of the bare metal into the interface state upon approaching the carbon layer to the Ag(111) surface. For this purpose, we have calculated the wavefunction and energy of the former surface state for different carbon-metal distance Åcompare  reflecti dC = 4 Å, which iEIS = EIS−ESS between the interface state and the surface state as well as on the carbon-metal distance dC for a variety of adlayers of organic molecules on metal surfaces. In order to emphasize the prototypic character of a graphene layer on a metal surface for these systems, we added also data of such systems. If available, dC has been taken from x-ray standing wave experiments, otherwise from DFT, low energy electron diffraction or surface x-ray diffraction. EIS and dC for the experimental and the calculated results. The red solid line shows the calculated results for a work function of the combined graphene/Ag(111) system of Φ = 4.24 eV at dC = 3.33 ÅdC, we depict with the gray areas the variation of the calculated results when changing the work function by ±1 eV. As can be clearly seen, the work function, i.e. the height of the potential barrier, has only a minor influence on the energy of the interface state. The latter depends instead much more sensitive on dC and shows a strong increase for distances below ∼3.25 Å. Our model calculation can well reproduce this trend for the majority of the experimental data. We can not confirm the predicted downshift of the interface state at large dC because this regime is not covered by real systems. The agreement between model calculation and experiment described here is rather good, even for systems like PTCDA and NTCDA on Ag(111) that are chemisorbed. A comparison with the results of DFT calculations for these two cases shows that not only the energies but also the laterally averaged wavefunctions are described well by the model38π system which would cause the model to break down.For a comparison with experimental results, we list in dC to the center of the carbon backbone, this might explain the larger deviation between model and experiment for PTCDA/Ag(100). For this system, also DFT calculationsEIS(c.f. 16ZnPc/Cu(111) that are subject to a stronger corrugation or distortion of the molecular layer on the substrateVΦ(z). However, also the metal as well as the graphen part of the potential will deviate slightly from the functional form of the isolated systems Vm(z) and Vg(z − dC) used in our model. One expects Vg(z − dC) to become asymmetric with respect to the carbon plane and, most importantly, the position of the image-plane in the Chulkov potential Vm(z) to move closer the substrate thus leading to an additional upshift of the IS. In order to keep the model potential parameter free we refrained from correcting for this overall small effect. A further possibility for the underestimation of the model to the experimental data is a shift of the last atomic layer of the substrate after molecular adsorption compared to the bulk values. Such a shift is not implemented in the model but it is known that the energetic position of the Shockley state depends on the lattice constant of the substrateSystems with stronger bonds of the functional groups of the organic molecules typically show a pronounced bending of the molecules towards the surface which reduces the carbon-metal distance at the edgessEISc.f. . Even ifk|| = 0. For this purpose, one simply has to solve the Schrödinger equation not only for one potential V(z) containing the metallic part at the Vg(z). It should thus be restricted to the vicinity of the Finally, we note that the present one-dimensional model can, in principle, be used to also predict the effective mass of interface states, i.e. the dispersion How to cite this article: Armbrust, N. et al. Model potential for the description of metal/organic interface states. Sci. Rep.7, 46561; doi: 10.1038/srep46561(2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Establishing the role that different genes play in the development of cancer is a daunting task. A step toward this end is the detection of genes that are important in the illness from high‐throughput biological experiments. Furthermore, it is safe to say that it is highly unlikely that these show expression changes independently, even with a list of potentially important genes. A biological signaling pathway is a more plausible underlying mechanism as favored in the literature. This work attempts to build a mathematical network problem through the analysis of microarray experiments. A preselection of genes is carried out with a multiple criteria optimization framework previously published by our research group  The study of cancer biomarkers is an important issue due to the role they play in the early detection, diagnosis, and prognosis of cancer. A biomarker can be defined as a characteristic that is objectively measured and evaluated as an indicator of normal biologic processes, pathogenic processes or pharmacologic responses to a therapeutic intervention The analysis of high throughput experiments such as microarrays can lead to the identification of potential cancer biomarkers. Microarrays can be used to simultaneously analyze the expression of thousands of genes to select those that are differentially expressed. After selection, differentially expressed genes still go through further experimental validation in order to be confirmed as biomarkers.Unlike other diseases such as cystic fibrosis This work discusses the issue of identifying a potential signaling pathway among a list of potential lung cancer biomarkers through a network representation that leads to optimal configurations. The list of potential biomarkers is determined from microarray databases through the application of Multiple Criteria Optimization There is an inherent difficulty in analyzing microarrays  associated with the large size of these experiments. Microarray experiments simultaneously measure the expression levels of thousands of genes, generating large amounts of data. The analysis of these data present a challenge to biologists; new tools are needed to derive biological insight from these experiments, including signaling pathways Currently several experimental methods for determining signaling pathways exist. Signaling pathways, which according to Baxevanis and Ouellette involve many direct protein‐protein relationships, can be mapped using protein‐protein interaction detection methods such as copurification and nuclear magnetic resonance There are also different computational methods for the identification or analysis of signaling pathways The ArrayXPath is a web‐based tool for profile mapping and visualization of microarray gene expression from biological pathway resources P‐values from three different resources: KEGG, BioCarta, and GenMapp Pathway Miner is another web‐based tool that mines gene associations and networks in biological pathway information. The tool permits the analysis and interpretation of pathway information and networks based on the association with databases, suitable for high‐throughput analysis of gene expression data P‐value is calculated. With the assumption that the number of differentially expressed (NDE) genes in a pathway follows a hyper‐geometric distribution, the probability of the number of differentially expressed (PNDE) genes in the given pathway is calculated. The second probability, probability of pathway perturbation (PPERT), is calculated based on the estimated amount of perturbation in each pathway due to the differential expression of the input gene list Signaling pathway impact analysis (SPIA) is a software tool used to measure the actual perturbation on a given pathway under a particular condition PathRanker is a software tool that can identify genetic pathways that dictate the response of metabolic networks to specific experimental conditions MAPPFinder is another software that can dynamically link gene‐expression data to Gene Ontology (GO) The literature review presented here provides evidence for an opportunity to create a deterministic optimization‐driven approach to the construction of signaling path proxies, even as a way to contrast results from the stochastic/statistic approaches already available. As illustrated in the following sections, the optimization‐driven approach is capable of finding lists of important genes, as well as the most correlated path among them, without the need for the user to adjust parameters that influence the output, thus providing objectivity and repeatability. This characteristic sharply differs from that in artificial intelligence and purely‐statistic procedures, where significance areas, thresholds, assumed underlying distributions, and weights, among other factors, are decisions that importantly affect the results and that are left for the user to manipulate.n cities The Traveling Salesman Problem is one the most famous combinatorial optimization problems As an example, take the network presented in Figure cij represents the cost of traveling from city i to city j and let yij be a binary variable, indicating whether or not the salesperson travels from city i to city j. In addition, let us define flow variables xij on each of the m arcs  and assume that the salesperson has n‐1 units available at node 1 (supply), which is arbitrarily selected as a “source node,” and the salesperson must deliver 1 unit to each of the other nodes or vertices (demand) [1]. The model is as follows [1]: The TSP can be modeled as a network optimization problem. Consider that i,j), then let A’ = {:yij = 0} that is the arcs forming in a solution to the previous model, and let A’’ = {:xij > 0}, that is the arcs with nonnegative flow in a solution to the previous model. The problem tries to minimize the objective function representing the total cost of a solution in Equation A’ leaves and enters any node i; therefore, A’ is the union of node disjoint cycles containing all of the nodes. In general, any integer solution satisfying )Ɲ is a n × m matrix, called the node‐arc incidence matrix of the minimum cost flow problem. Each column Nij in the matrix corresponds to the flow variable xij, in x. The column Nij has a value of +1 in the ith row to indicate incoming flow, and a value of −1 in the jth row to indicate outgoing flow; the rest of its entries are zero. The right hand side vector b with dimensions nx1 contains positive values to indicate supply in particular nodes, negative values to indicate demand in particular nodes, or zeroes in transshipment nodes. Constraint A” is connected as we need to send 1 unit of flow from node 1 to every other node via the arcs in A”. The forcing constraints (5) imply that A” is a subset A.’ These conditions imply that the arc set A’ is connected and thus cannot contain subtours In constraint The minimum spanning tree considers an undirected and connected network, where the given information includes some measure of the positive length  associated to each link nn‐2 where n is the number of nodes in the graph 5‐2 = 125 possible solutionsAn example of the MST is presented in Figure In this work the TSP and the MST are the methods proposed to obtain an optimal proxy for a signaling pathway. In order to begin to apply these methods however, the first step is to identify genes of interest based on the differences in expression when comparing control and cancer tissues. Multiple Criteria Optimization (MCO) based on Pareto conditions, as described in the works of Lorenzo et al. X and Y. Linear correlation values range from −1 to +1. The correlation can be considered more intense the closer the linear correlation values are to either +1 or −1. If we have a series of n measurements of X and Y written as xi and yi where i = 1,2…n, then the sample correlation coefficient can be used to estimate the Pearson correlation ρ between X and Y as follows:x¯ and y¯ are the sample means of X and Y, and sx and sy are the sample standard deviations of X and YStatistical correlation can be defined as a measure of the coordinated behavior between two random variables. It measures the strength or degree of association between two variables, say In this work, as a first approximation, the linear correlations that can be observed among a list of genes considered to be potential biomarkers are used as a base to construct networks such as the one presented in Figure As shown in Figure The TSP can be used to discover a potential signaling pathway from a network of genes, by identifying a sequence that maximizes the linear correlation among the genes For the TSP, a list of potential biomarkers of interest must first be identified. The list was obtained using MCO Once the potential biomarkers were established, the next step was to calculate the differences in genetic expression for each gene. This was done by comparing all of its control samples against all of its cancer samples. Figure d1 represents the difference in genetic expression between a particular pair of control and lung cancer tissue samples for SPP1, and d2 is the difference in genetic expression between the same pair of tissue samples for AGER. Each point can be considered to be .The genetic expression differences in all genes considered to be potential biomarkers were used to calculate the linear correlations among each pair. The values of the differences serve as input for calculating the linear correlations between all genes being considered see eq. . Figure The linear correlations between all genes of interest are calculated using Equation This matrix serves as input for the MCO program in MatLab (developed by our collaborator J. Rodríguez). The software is used to obtain an optimal sequence maximizing linear correlations among the eleven genes considered as potential biomarkers.The TSP has the somewhat restrictive assumption that a signaling pathway behaves as a tour. The MST is an alternative method that results in a noncyclic structure. The MST allows representation of groups of coordinated gene expression changes that may be independent from each other. The MST does not consider feedback because it is an open structure. It was deemed convenient to contrast these models as they have different advantages and could provide complementary information. The MST, as described in Section “According to the National Cancer Institute, lung cancer is the second most common type of cancer and the primary cause of cancer‐related death in both men and women in the United States. Also based on analysis conducted by this institution, an estimated $11.9 billion were spent on lung cancer care in 2014 As detailed previously, the GDS3257 database selected for this work was first reported by TSP and MST were applied to identify potential signaling pathways from the list of eleven potential biomarkers for lung cancer obtained using MCO.By applying the TSP formulation, a sequence that maximizes the linear correlations from a large number of possible solutions was obtained using the methodology described by Lorenzo et al., This sequence was obtained from the (11‐1)! ≈ 3.6 million possible solutions. It represents a solution that maximizes the linear correlations between the expression changes of these 11 genes. This result shows the potential of this work: being able to obtain an optimal cycling path from thousands (or millions) of possible solutions. It must be clarified that because this is an observational analysis of a microarray database, causality or directionality cannot be established at this point.As an alternative method to the TSP, the MST was applied to obtain a tree that maximizes the linear correlations of the eleven genes that were identified previously. The resulting structure is presented in Figure The solutions obtained with the TSP and MST requires biological verification. There are scientific reports for some of the correlation links in the resulting models that will be discussed in Section “Utilizing the combination of methodologies of MCO to determine a list of potential biomarkers from databases, and subsequently employing the TSP and MST to identify possibly signaling pathways from the genes of interest, could serve as a viable alternative to existing methods to provide an original analysis pipeline completely driven by optimization procedures.This section describes the first steps taken to carry out a meta‐analysis in order to identify signaling pathways of interest from the different groups previously described in Section “Initially, MCO was applied to identify genes with high variations in their relative expression levels when utilizing two performance measures, the absolute values of the differences between means and median as described in reference The following step was to apply the TSP and MST optimization methods to identify potential signaling pathways. For each group comparison the TSP and MST were applied and a signaling pathway was obtained. Figure Both methods were applied to identify a signaling pathway when analyzing CNS versus CCS. Once MCO was conducted to determine the genes with the largest changes in relative expression, a total of 20 genes were identified as genes of interest for this particular comparison. The resulting network solution when applying the TSP and MST methods are represented by Figures 20‐2 possible configurations for the TSP and MST respectively.Several similarities exist between both networks’ solutions that are potentially interesting from a biological perspective. These two networks are the best solution from a total of (20‐1)! and 20The next comparison conducted was for HCS versus CCS. The MCO solution produced a list of 16 genes. The same analysis described before was done. Figures The third analysis was the comparison of HCS versus CNS, where a total of 23 genes were considered when carrying out the TSP and MST methods. The resulting signaling pathways are presented in Figures The following comparison was done between HNS versus CCS. When MCO was conducted, a total of 17 genes were identified. The TSP and MST were applied to obtain Figures The HNS versus CNS comparison produced a list of 18 genes as inputs for the TSP and MST Figs. The last analysis carried out for this meta‐analysis was the comparison between HNS versus HCS. When applying the TSP and MST for this comparison, the networks were to be constructed from 30 genes of interest which heightened the number of possible solutions for both methods Figs.  and 21.The similarities that were maintained through the different analyses could prove to be biologically significant once further biological validation can be conducted. Table This comparison of signaling pathways was also done for the resulting pathways when applying the MST, Table There are some arrangements of genes common between the TSP and MST, these are included in the following Table This section describes the search for existing biological evidence in literature to analyze and validate the results described in previous sections.Based on the signaling pathways proposed by the TSP and MST formulations, public databases including KEGG and GeneCards were searched for relevant biological information. Information for the solutions for the TSP see Fig.  and MST Several gene relations were identified to have either a direct or indirect connection. SPP1 and AGER  is one example of a direct relation that has been previously identified and forms part of the IL‐2 pathway. The COL11A1 and SPP1 relation is also documented within several known pathways as described in Table The gene relations included in our results can be categorized into two classifications, the first being gene relationships connected to lung cancer already reported, serving to validate our methodology. The second classification includes gene relationships involved in other types of cancer, which offer the greatest possibility for discovery. In the first type of classification, genes COL11A1, SPP1, AGER, WIF1, SFTPC, FABP4, and CLDN18 are included. Existing literature has previously established the change in expression in lung cancer for several of these genes, as well as being part of the correlated pathways between known driver genes (K‐Ras and EML4‐ALK) and tumor suppressor genes . The correlated pathways between driver genes, tumor suppressor genes, and the genes of the first classification are as follows: COL11A1‐SPP1 through ECM receptor/PI3K‐Akt pathway, AGER through IL‐2 and HGMB1 pathway, WIF1‐SFTPC through WNT and WNT mediated beta catenin signaling pathway, CLDM18 through its function in tight junction on pathways.The second classification of genes includes FCN3, ADH1B, TMEM100, and CYPB1. The expression changes of these genes have been reported for different cancer pathways. These include complement pathway, chemical carcinogenesis, tumor suppression, and cytochrome P450 inflammatory responses respectively. Furthermore, all the genes have been correlated with inflammatory responses. These commonalities infer a possible relation to lung cancer pathways.As described in Section “Sun et al. A relation between the expressions of ABCA8 and ADH1B, was reported by Liu et al. The results for the PANTHER There are several methods or tools for identifying and constructing signaling pathways. GeneMANIA GeneMania uses publicly available databases In GeneMANIA, a researcher must provide a predetermined list of genes of interest. The program then extends the list with genes that are functionally similar or have shared properties with the initial query genes. The output is in the form of a display for the interactive association network, which illustrates the relations among the genes and data sets GeneMANIA is based on a heuristic method that builds a composite functional association network by integrating multiple functional association networks and predicts gene function The list of 11 genes in Table Figure The network that generated the largest number of relationships and the largest network connecting all of the original eleven genes was that of the coexpression factor. In total there are 24 connections that form part of the network that GeneMANIA constructed. It must be noted that GeneMANIA was unsuccessful in including all genes in the list. SPP1 and COL11A1 were not connected to the rest of the genes.The second network constructed by GeneMANIA containing some genes of our query list is the category of colocalization. As can be seen in Figure The third and last network generated by GeneMANIA can be seen in Figure In our case study for lung cancer, GeneMANIA generated three networks based on three separate categories. Only in the first category of coexpression, GeneMANIA was successful in establishing a network linking all eleven of the genes of interest. None of these three networks generated by GeneMANIA was in the category of pathways.GeneMANIA relies on published analysis to generate networks; our methodology does not have this dependence for generating possible signaling pathways. Therefore our method possesses an interesting potential for determining previously undiscovered pathways or gene relations important in a disease, in this case lung cancer. Also, due to its deterministic nature, our methodology is able to obtain a global optimal solution with either the TSP or MST. In addition, both methods were capable of obtaining solutions that include all genes from a list.Identifying cancer biomarkers is an important step in the diagnosis, prognosis, and prevention of this disease; and determining how these biomarkers are related or interact is just as important. A signaling pathway among these biomarkers is, then, a worthy aim. Uncovering signaling pathways for potential biomarkers could further the understanding of the origins and the evolution of cancer.As a first approach to structure a network of potential biomarkers, the linear correlations that exist among these genes are used. Once the network of a preselected list of potential biomarkers is constructed, the TSP can be applied to obtain an optimal sequence that maximizes the linear correlations. This sequence represents the potential signaling pathway. In Section “In addition, the MST formulation is used here as an alternative method to the TSP to discover a signaling pathway from the same list of preselected potential biomarkers. The MST provides an optimal tree, a representation that must be contrasted to that of TSP's tour.In Section “Three important aspects are novel in this work: (1) the joint analyses of different groups of lung cancer states reveal new correlations, biologically evidenced, and previously undocumented; (2) computation of the correlation coefficients from expression differences leads to an effective use of network optimization methods; and (3) the methods yield mathematically optimal correlation structures: no other configuration is better correlated with the available information.None declared.


In this work, we address the ubiquitous phenomenon of Fano resonances in bilayer graphene. We consider that this phenomenon is as exotic as other phenomena in graphene because it can arise without an external extended states source or elaborate nano designs. However, there are not theoretical and/or experimental studies that report the impact of Fano resonances on the transport properties. Here, we carry out a systematic assessment of the contribution of the Fano resonances on the transport properties of bilayer graphene superlattices. Specifically, we find that by changing the number of periods, adjusting the barriers height as well as modifying the barriers and wells width it is possible to identify the contribution of Fano resonances on the conductance. Particularly, the coupling of Fano resonances with the intrinsic minibands of the superlattice gives rise to specific and identifiable changes in the conductance. Moreover, by reducing the angular range for the computation of the transport properties it is possible to obtain conductance curves with line-shapes quite similar to the Fano profile and the coupling profile between Fano resonance and miniband states. In fact, these conductance features could serve as unequivocal characteristic of the existence of Fano resonances in bilayer graphene. The pioneering work of Ugo Fano about auto-ionizing states in atoms came to solve the mystery of the asymmetrical resonance3. Fano suggested a simple model based on the constructive and destructive interference of the propagating and discrete states. That model gave place to what now is known as the Fano profile. This profile is characterized by a rather simple mathematical expression, σ = (ε + q)2/(ε2 + 1). This expression has a phenomenological parameter q that gives the profile shape and a reduced energy ε = (E − ER)/Γ that bears information about the peak position ER and the width of the resonance Γ. Furthermore, according to the value of q we can find different types of resonances: q = 1 is a Fano resonance, q = 0 an antiresonance, and for large values of q (q → ∞) a typical Breit-Wigner resonance. The simplicity and elegance of this formula together with the explosion, continuous refinement and sophistication of spectroscopic techniques make of Fano resonances a preponderant phenomenon in science and engineering4. In fact, Fano resonances can be found in many fields of physics9 and with many potential applications11. In most cases, this special type of resonances arise in systems in which continuum states are provided by a light source and discrete states come from size effects. However, it is also possible to find Fano resonances in quantum transport by special nano designs that provide extended and discrete states simultaneously23. The archetypal nano design is the so called Aharonov-Bohm ring, a nano constriction that can be coupled to quantum barriers, quantum dots, etc., that can give rise to asymmetrical line-shapes in the quantum conductance.Resonances represent an ubiquitous physical phenomenon, since we can find them in very diverse fields, and from the classical world to the quantum world. Resonances are also fundamental characteristics that provide valuable information and help to understand physical systems. The archetypal resonance is the so-called Breit-Wigner resonance, which has as fundamental characteristic a Lorentzian line-shape. However, from the very beginning of atomic physics a resonance with asymmetrical line-shape was challenging26, but also because Fano resonances of different nature can arise in this material31. For instance, in gated bilayer graphene at room temperature light, electrons and phonons can confabulate to give rise to asymmetrical line shapes in the absorption spectra27. Here, it is quite relevant that a band gap be opened by electrical gating25 such that discrete phonons and continuous excitons be coupled. Fano phonon line shapes are also observed in the infrared spectra of bilayer graphene28. However, in this case the fundamental mechanism is the doping induced by gating rather than band gap opening. A radically different Fano resonance is also possible in bilayer graphene31. In fact, this resonance can arise without the need of an external continuum states source or an elaborated nano design31. In this case, it is necessary gapless bilayer graphene as well as low temperatures. The special dispersion relation of bilayer graphene33 makes that electrons have at the same time a propagating and discrete character35. Actually, the degree of coupling between confined and continuum states is ruled by the transversal component of the wave vector. By considering a potential barrier and adjusting the angle of incidence  of the charge carriers on the barrier the propagating and discrete nature of electrons can converge to give rise to a Fano profile in the transmission properties31. This is unprecedented for a material and for graphene constitutes another exotic phenomenon like anti-Klein and Klein tunneling37, atomic collapse38, Hofstadter’s butterfly40 and negative refraction41 to mention a few. However, as far as we know this unusual and unprecedented phenomenon of Fano resonances in bilayer graphene has not been yet confirmed experimentally. This could obey the fact that: the asymmetrical line-shape of Fano resonances is manifested in the transmission probability or transmittance, for which a high degree of control of the angle of incidence of electrons is required31, issue that is not at all possible in current experiments; most of the exotic phenomena in graphene and related 2D materials have been demonstrated through transport measurements41; the lack of theoretical and experimental studies that address in a systematic way the impact of Fano resonances on the transport properties.Within this context, bilayer graphene is a unique material system, not only because of its outstanding properties for technological applicationsIn this work, we address the unique and peculiar phenomenon of Fano resonances in bilayer graphene superlattices (BGSLs). The hybrid matrix method and the Landauer-Büttiker formalism have been used to obtain the transmission and transport properties. The evolution of the asymmetrical line-shape of Fano resonances in the transmission probability or transmittance is studied. In particular, the mentioned evolution is analyzed as a function of the superlattice period, the barriers height and the barriers and wells width. We pay special attention to the impact of Fano resonances on the transport properties. Our findings indicate that the intrinsic energy minibands of the superlattices structure can help to unequivocal identify the contribution of Fano resonances to the transport properties. Specifically, Fano resonances and minibands can couple in such a way that the linear-regime conductance presents signatures of this coupling, and hence of the presence of Fano resonances. Furthermore, by appropriately reducing the angular range for the computation of the transport properties it is possible to obtain conductance curves with line-shapes directly related to the Fano profile and its coupling with the miniband states. We hope that our results can help and encourage experimentalists to test this exotic phenomenon of Fano resonances in bilayer graphene.43 and the Landauer-Büttiker formalism44, respectively. The basic information that we need in order to implement the mentioned methodologies is related to the wave functions and dispersion relations in the two regions that define the unit cell of the superlattice, that is, the region with and without electrostatic potential. So, we will present in first place the wave functions and dispersion relations to proceed with the generals of the hybrid matrix method and the Landauer-Büttiker formalism.The system that we are interested in is a bilayer graphene superlattice. In particular, we will deal with BGSLs conceived by metallic electrodes arranged in periodic fashion, see Fig. 33,qx and qy are the quasiparticle wavevectors along the x and y directions respectively; m is the band effective mass with a value of 0.035 m0, being m0 the bare electron mass45; and V(x) = V0 represents the strength of the electrostatic potential.The wave functions and dispersion relation in the barrier region can be obtained by solving the following Dirac-like equation:The specific dispersion relation for this eigenvalue problem comes as,At this point it is important to mention that, in contrast to monolayer graphene, here in bilayer graphene there are four wave functions, two for propagating states and two more for evanescent-divergent states. The corresponding wave functions for propagating states can be written asqx and βx are:In these expressions A+, A−, B+, B− is a set of expansion coefficients. Furthermore, for regions without electrostatic potential or better known as well regions the wavefunctions as well as the wave vectors can be obtained by simply setting V0 = 0.The general solution for the wave function can be written as a linear combination of the four eigenfunctions:48 this method has numerical instabilities and it is not suitable for the computation of the transmission properties of BGSLs. We have also shown that a better and reliable option is the hybrid matrix method45. The hybrid matrix method relies on writing the Dirac-like equation for bilayer graphene as an ordinary second order differential equation system of the Sturm-Liouville form43. Specifically, for bilayer graphene the matrix Sturm-Liouville equation comes asx) = [B(x) · dF (x)/dx + P(x) · F(x)] is the so called associated-linear-form (ALF) of the operator L, withWith this information we can implement the standard transfer matrix method to compute the transmission properties. However, as we have shownk. In fact, we can find them for the regions with and without electrostatic potential.Within this context, the problem is reduced to find the linearly independent solutionsF(x) and ALFs A(x) we can define the hybrid matrix. In fact, the hybrid matrix is the mathematical entity that relates the solutions and the ALFs between two regions, connected by a certain domain, in mixed fashion. In our case, we have a semi-infinite left region connected to a semi-infinite right region by a domain that is the superlattice structure. In mathematical terms we haveH is determined by the characteristics of the superlattice structure, specifically by the particularities of the solutions and ALFs in the barrier and well regions. Actually, we can define hybrid matrices for the well and barrier regions, and more importantly, irrespective of the particularities of these regions, the hybrid matrix can adopt the following general formd represents the width of the corresponding region. With the hybrid matrices of the well (Hw) and barrier (Hb) we can find the hybrid matrix of the unit cell of the superlattice (Huc) and with it the hybrid matrix of the entire superlattice H, for more details see ref.45.With the solutions a4(L) = 0 to avoid divergence at x → −∞. In similar fashion for the right semi-infinite region,a2(R) = 0 and a3(R) = 0 because there is no reflected wave anymore and to avoid divergence at x → +∞, respectively.In order to obtain the transmission probability or transmittance it is important to remember that the general solution is a linear combination of the four individual solutions. Specifically, the general solution and the general ALF for the left semi-infinite region can be written asWith the help of these expressions it is possible to find relationships between the coefficients of the general solution of the left and right semi-infinite regions, namely:T = T.As the transmittance is given as )|2, eq. , that de44. In concrete, the linear-regime conductance can be obtained by summing over all transmission channels,E0 = V0, and GθΔ), we can reduce the integration limits in eq. , which is the most simple system, then we will study the case of double barriers (BGDBs) and finally a finite superlattice, with nine periods, will be addressed. Here, it is important to mention that even when the transmission properties of BGSBs and BGDBs are already reported2, top and back gates are incorporated to control the barrier characteristics as well as the Fermi energy of electrons, see Fig. As we can corroborate in the methodology there are four solutions or eigenfunctions for gapless bilayer graphene. So, in principle, by appropriately nanostructuring this material we can create the conditions to have in the same energy region propagating and discrete electron states, and hence the conditions to get Fano resonances. In Fig. E ≈ V0/2) of Fano resonances is practically unchanged as the angle of incidence grows. This feature is quite important because the contribution of Fano resonances to the transport properties will be located precisely at that specific energy. Then, in principle, we can track and discriminate the contribution of Fano resonances to the transport properties. In Fig. θ = 3° and V0 = 50 meV, respectively. As we can notice by changing the width of the barrier the Fano resonance is shifting to higher energies, that is, the resonance undergoes a blue shifting. The resonance that was originally located close to 5 meV for 3 nm is gradually shifting to: 15 meV for 6 nm, 25 meV for 10 nm and 40 meV for 20 nm. We can also see that the Fano profile deforms gradually and tends to become a Breit-Wigner resonance profile. Furthermore, a Breit-Wigner resonance arises in the low energy side of the transmission spectrum for the case of 20 nm. A similar shifting for the Fano resonance is obtained by increasing the height of the barrier, see Fig. σx, σxψ± = ±ψ±, as well as by taking advantage that ψ± are uncoupled at normal incidence34. In fact, ψ− is the component that represents the confined states inside the barrier. The details of the method for the calculation of confined states are presented in the supplementary material. Likewise, the specific values of the confined states for the cases of BGSBs treated in Fig. ψ± gives rise to asymmetrical resonances in the transmission spectra. Actually, these resonances are pretty narrow near normal incidence because their width is proportional to 34. Furthermore, for narrow barriers, states with weaker confinement, it seems that the mixing is stronger to such extent that there is not at all correspondence between the energy of confined states and the energy location of Fano resonances.The results of the transmission probability or transmittance for BGSBs are presented in Fig. V0 = 50 meV and (b) V0 = 100 meV. The barrier widths are the same as in the case of the transmittance, that is, the solid-black, dashed-red, dotted-blue and dash-dotted-green curves correspond to 3 nm, 6 nm, 10 nm and 20 nm, respectively. As we can see in both cases V0 = 50 meV and V0 = 100 meV the conductance behaves in the same way in general terms. For instance, in the case of the narrower barrier 3 nm, the conductance is practically zero up to 5 meV (15 meV) for V0 = 50 meV (V0 = 100 meV). At that specific energy the conductance presents a sudden rise that coincides quite well with the energy location of the Fano resonances. In order to highlight the energy location of Fano resonances we have included shaded vertical stripes. Furthermore, the shape of the conductance curves remains the same no matter the width of the barrier. The main change that we can notice is that the sudden increase is taking place at higher energies. Those energies are precisely the energies at which Fano resonances are presented in the transmission spectra. In the case of 20 nm we can also see a peak that is directly related to the Breit-Wigner resonances that arise in the transmittance, see the dash-dotted-green curves in Fig. 52. In particular, by tilting the top-gate electrodes it was possible to determine the mentioned contribution. This is quite appealing for us because we can compute the conductance in the angular range at which Fano resonances are preponderant. In fact, in Fig. π/12, π/12) at which the Fano resonances are relevant. Figure V0 = 50 meV and V0 = 100 meV, respectively. The width of the barrier considered in both cases is dB = 10 nm. By reducing the angular range we can see that an asymmetrical lines-shape also arises for the conductance at precisely the energies at which the Fano resonances are manifested in the transmission spectra. These Fano profiles in the conductance are undoubtedly attributed to Fano resonances in the transmittance curves. Then, it is possible that with angular transport measurements the existence of Fano resonances can be proven.Now, having clear how Fano resonances change by the angle of incidence, the width and height of the barrier, we are in position of evaluate its impact on the transport properties. In Fig. V0 = 50 meV and dB = dW = 10 nm. Here, we are considering dB1 = dB2 = dB as well as VB1 = VB2 = V0, that is, we are dealing with symmetric barriers. As we can notice Fano resonances arise at approximately 20 meV, which represents a small red-shift with respect to BGSBs. We also found that the asymmetrical line-shape of the resonances is preserved only at small angles. Two additional features have arisen as a result of the well region: (1) In the low energy side there is a Breit-Wigner resonance at about 2.5 meV, which remains at the same energy irrespective of the angle of incidence; (2) In the high energy side we can see an extended resonance at 40 meV, which tends to broaden as the angle of incidence increases, see the dash-dotted-green curve. So, in principle, the interplay of these resonances with the asymmetrical line-shape associated to the barrier regions can give rise to new resonance characteristics. These new characteristics can change the conductance landscape and possibly can give place to special features in the transport properties that can serve as hallmarks of the existence of Fano resonances. In fact, in Fig. dB = dW = 3 nm we can see a transmission spectrum typical of BGSBs with a Fano resonance around 10 meV. In this case the well is so narrow that its characteristics are not at all manifested in the transmission spectrum. By increasing the widths to dB = dW = 6 nm resonances in the low and high energy side of the spectrum come into play as in Fig. 53. Similar transmission spectra are obtained for V0 = 100 meV. The main changes with respect to 50 meV are the blue-shifting of the overall spectra as well as that the hybrid Fano resonance arises at narrower widths, see the case of 7 nm. We also notice that the hybrid Fano resonance eventually splits into two weakly coupled Breit-Wigner resonances, see the dash-dotted-green curve that corresponds to 9 nm. Here, it is worth mentioning that hybrid resonances are well defined at small angles of incidence and that they tend to deform and eventually lose their hybrid line-shape as the angle grows, see Fig. Now, it is turn to analyse the case of BGDBs. Here, it is important to stress out that this case is more versatile because the well region can incorporate its own propagating and/or discrete states that will enrich the features of the transmission spectra. Our results for this specific case are shown in Fig. V0 = 50 meV and (b) V0 = 100 meV. The widths of the barriers and the well for Fig. π/12, π/12) we obtain that the conductance also exhibits the hybrid line-shape, see Fig. Now, it is turn to analyse the transport properties of BGDBs. In Fig. dB = dW have been considered: (a) 3 nm, (b) 5 nm and (c) 7 nm. The height of the barriers and the angle of incidence have remained fixed at 50 meV and 3°, respectively. For 3 nm we can see that the Fano resonance and the miniband lie in different energy regions consequently there is no coupling as well as no new features in the transmission spectrum. The Fano resonance is located at 23 meV, while the miniband starts at 28 meV and it has an energy width of 222 meV. By increasing dB = dW to 5 nm the Fano resonance and the miniband coincide in the same energy region, specifically the Fano resonance lies at the onset of the miniband. In this case the resulting line-shape corresponds to the type of hybrid Fano resonances. It is also important to note that the miniband width is reduced around 132 meV. Likewise, the Fano resonance and the miniband blue and red shift with respect to the case of 3 nm. A further increase of the barriers-wells width to 7 nm turns out in an additional reduction of the miniband width of nearly 53 meV as well as extra blue and red shifts of the Fano resonance and the miniband. The net result of these shifts and the miniband narrowing is that the Fano resonance blue shifts with respect to the onset of the miniband, in other words, the Fano resonance tends to be localized at higher energies within the miniband. In fact, by appropriately adjusting the fundamental parameters of the superlattice we can tune the position of the Fano resonance at practically any energy within the miniband. The coupling of the Fano resonance with the states along the miniband gives rise to new and unique features in the transmission spectra. Even more important, we were able to identify and discriminate the contribution of these new transmission line-shapes onto the transport properties. For instance, when the Fano resonance and the miniband are decoupled, Fig. π/18, π/18). As in the case of single and double barriers, the angular reduction modifies greatly the conductance curves to such an extent that the new conductance curves resemble in great extent to the corresponding curves of the transmission spectra.Now it is time to analyse the more general case of bilayer graphene superlattices. Here, it is worth mentioning that one of the most remarkable characteristics of practically any superlattice is the formation of the so-called minibands. In this regard the periodic arrangement of barriers and wells in bilayer graphene is not the exception. The most noteworthy difference of BGSLs with respect to superlattices of conventional materials is that minibands depend strongly on the angle of incidence in the former case. As we have corroborated in the case of single and double barriers Fano resonances represent an intrinsic characteristic of bilayer graphene. Moreover, these special resonances also depend strongly on the angle of incidence. Then, at first instance, these characteristics, minibands and Fano resonances, can be located in different energy regions, however, at second instance, by appropriately adjusting the fundamental parameters of the superlattice such as the angle of incidence, the widths of barriers and wells, the heights of barriers and the superlattice period it is possible to tune the minibands and Fano resonances at the same energy region. In Fig. 23. This formalism solves important problems such as the nature of the parameter q and the width of the resonance. Taking into account that this formalism is based on the Green’s function method for quantum transport and that the Green function and the scattering formalism of quantum transport are equivalent44, in principle the universal formalism can be applied to our problem. In particular, it is quite interesting and relevant how the universal formula for the Fano-resonance profile will depend on the angle of incidence, since as we have corroborated throughout our study, the angle of incidence represents a preponderant parameter that determines in great extent the Fano-resonance profile. Considering the magnitude and relevance of this aspect, a thorough analysis, that go beyond the objectives of the present work, is required.The first aspect that we want to address is the Fano-resonance profile. In fact, there is a formalism that provides a universal formula for the Fano-resonance profile in the context of coherent quantum transport54. On the other hand, if disorder, caused by impurity scattering or temperature, is relevant the transport will be predominantly diffusive rather than ballistic. Therefore, compromising the specific conditions to obtain Fano resonances. Fortunately, high-quality bilayer graphene samples, non-interacting substrates like SiO2 and very low temperatures can guarantee the conditions to obtain a gapless parabolic dispersion relation and coherent quantum transport in bilayer graphene. In fact, unconventional quantum Hall effect and 2π Berry phase associated to the gapless parabolic dispersion relation in bilayer graphene have been demonstrated experimentally24. In addition, it is well known that high quality of graphene samples ensures coherent quantum transport up to 80 K37. So, in principle, the detection of Fano resonances is achievable via low-temperature transport measurements.The second aspect that it is important to discuss is the one related to non-idealities such as substrate effects, impurity scattering and temperature disorder. In fact, these effects can can modify the fundamental characteristic of the band structure of bilayer graphene as well as destroy coherent quantum transport, and consequently jeopardizing the particular conditions that give rise to Fano resonances. For instance, substrates that interact strongly with bilayer graphene open a band gap and modify the dispersion relation55. Regarding Fano resonances our results in BGSBs indicate that the asymmetrical line shape can be deformed and for a certain band gap practically destroyed, see the supplementary material for more details. In addition, band gap opening can activate phonons27 that in principle could be coupled with propagating or discrete electron states and/or could represent an additional scattering mechanism. As far as we know these phonons were observed at room temperature27, but if they are present at temperatures below liquid helium could affect in great extent the asymmetrical line shape associated to confined and continuum electron states. Non parabolicity and warping are effects that can also modify the transmission and transport properties in bilayer graphene structures. In specific, non parabolicity is relevant at energies of 390 meV. Then, if we want to limit this effect it is crucial to work in a reduced energy range, otherwise it will be relevant to know the possible changes that non parabolicity can caused to the transmission and transport properties. Up to this moment, the numerical degradation related to the transfer matrix approach and the Hamiltonian that describe band gap opening and non parabolicity impede us to see to what extent the Fano profile is modified by non parabolicity. The mentioned numerical degradation is presented in the corresponding section of the supplementary material. Lastly, warping in principle it is less relevant than the bandgap opening and non parabolicity32, however, effects such as temperature can induced natural distortions on the graphene sheets56. So, it is also interesting to know the impact of warping on the transmission and transport properties. A full analysis of band gap opening, non parabolicity and warping is needed and its possible consequences will be published elsewhere.The last aspect that we consider it is important to address is the one associated to relevant effects such as the bandgap opening, non parabolicity and warping. For instance, in order to preserve a gapless band structure in bilayer graphene the sheets need to be maintained at the same potential energy. Despite the sophistication of the experimental techniques there will be unavoidable differences between the potential energies of the graphene sheets, causing a bandgap opening in the band structure of bilayer graphene. This bandgap opening can modify significantly the fundamental properties of bilayer graphene. In fact, in the case of anti-Klein tunneling it has been reported that the bandgap opening can destroy itFinally, we want to discuss some important aspects about Fano resonances in bilayer graphene:In summary, we have addressed the exotic phenomenon of Fano resonances in bilayer graphene superlattices. The hybrid matrix method and the Landauer-Büttiker were implemented to unveil the Fano characteristics on the transmission and transport properties. Particularly, we find an asymmetrical line-shape, Fano profile, on the transmittance characteristics. The Fano profile is pretty sensitive to the electron angle of incidence, being well defined for small angles, and deformed and practically lost for large ones. We also find that the Fano resonances can be coupled with the energy minibands of the superlattice, giving rise to special transmission characteristics. Even more important, the Fano resonance by itself as well as its coupling with miniband states have direct and identifiable consequences on the transport properties. Specifically, well-defined conductance characteristics arise at Fermi energies at which the Fano resonance and its coupling are taking place. Furthermore, angular transport calculations provide conductance curves with practically the same line-shape of Fano resonances as well as the same profile of its coupling with miniband states. Then, we have shown that bilayer graphene superlattices are systems that can be helpful to obtain unequivocal characteristics in the transport properties related to existence of Fano resonances. We hope that our findings encourage further studies about this rather exotic phenomenon of Fano resonances in bilayer graphene, and furthermore, that our results help and encourage experimentalists to test this peculiar phenomenon.Supplementary Information