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Abstract
We theoretically investigate the Goos-Hänchen (GH) shifts of optical beam in a defective photonic crystal composed of dielectric multilayers and graphene. The system is non-Hermitian and possesses exceptional points (EPs) as the scattering matrix becomes defective at the zero points of reflection. The reflective wave at EPs experiences an abrupt phase change and there the eigenvalues of scattering matrix coalesce. The GH shifts are extremely large near EPs in parametric space composed of dielectric refractive index and incident angle. The positive and negative maxima of GH shifts could be as high as 103 times of the incident wavelength. The direction of GH shifts switches at EPs and the EPs position can be readily controlled by the chemical potential of graphene. Moreover, the GH shifts should remarkably change as the incident waves impinge on the structure from opposite directions. The study of GH shifts in the graphene incorporated multilayers may find great applications in highly sensitive sensors.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since the Goos-Hänchen (GH) shift was first observed experimentally by Goos and Hänchen in 1947 [1], many efforts have been devoted into the theoretical and experimental studies of the effect [2–10]. The GH shift may manifest a spatial or angular shift [11–14], which could occur individually or simultaneously at the interface of two materials with different permittivities [15]. GH shifts have been reported in various systems, including dielectric slab [16, 17], metamaterials [18], graphene [19, 20], phase-conjugate mirrors [21], and photonic crystals (PCs) [22–24]. In order to enhance the GH effect, one has utilized the Kerr nonlinearity [25, 26] and fabricated defect layers in PCs [22]. Recently, a giant enhancement of GH shift has been achieved in the parity-time (PT)-symmetry cavity [27]. It has also been reported that GH shifts should be large inside the reflection band and became extremely large at exceptional points (EPs) in PT-symmetric photonic crystals [24]. Simultaneously, very large GH shifts may exist in periodic dielectric multilayers with PT-symmetry where the lateral shifts are tightly independent on the incident direction [28].
Although the PT-symmetry stems from the quantum system [29], it could find a counterpart in optics balanced gain and loss, i.e., n(r) = n*(r) [30]. In an open quantum system, non-Hermitian Hamiltonian with PT symmetry can exhibit real eigenvalues as well. But the eigenvalues and eigenfunctions of the Hamiltonian coalesce at EPs, known as PT-symmetry breaking points. The PT-symmetry system has many unique optical features, and found applications in unidirectional invisibility [31], coherent perfect laser absorbers [32], and giant wave amplification [33].
In fact, the two conditions of both non-Hermiticity and PT-symmetry to form EPs are not necessary for the same time. The lossy metamaterials are non-Hermitian and could also be utilized to yield EPs [34]. Since giant GH shifts are present in PT-symmetry and notable at EPs [24, 27, 28], it is necessary to explore the behavior of GH shifts in non-Hermitian systems. Through the unique characteristic of EPs including unidirectional zero reflection and abrupt phase change, we could also in turn find EPs in non-Hermitian systems [35]. Recently, graphene has been widely employed to realize many optoelectronic functionalities for its unique optical, electronic, and mechanical properties [36–39]. Graphene is a lossy material due to the limited relaxation time of electrons. As a result, incorporating the graphene into metamaterials can constitute a non-Hermitian system [40, 41], in which EPs locate at the zero points of reflection [24, 42] and the EPs positions in parametric space can be tuned by the chemical potential of graphene.
In this work, we incorporate graphene in dielectric multilayer to construct a non-Hermitian system and investigate the spatial GH shift of reflected light beam from the multilayer. Although the GH shift in a single layer of two-dimensional material like graphene or boron-nitride has been explored, the shift is demonstrated to be extremely small [20]. Here we show that the GH shift in a bulk multilayer structure and the incorporated graphene plays the role to realize non-Hermiticity of the system. The multilayers are constituted by periodic dielectric layers with a defect in the center of the structure. The reflectance is direction-dependent as light is incident from two sides of opposite directions. The field distribution for defective mode is provided to account for large GH shifts in following study. Then, we find two EPs existing in the parameter space composed of dielectric refractive index and incident angle. The EPs are confirmed through the unique properties, including eigenvalue coalescence and abrupt phase change [40, 41]. We also discuss the GH shifts and their polarity, the singularity at EPs, and the direction-dependence of GH shifts. Especially, the differences in GH shifts with the cases in PT-symmetry multilayers [28] are especially concerned. We further explore the control of EPs in the parameter space by changing the graphene chemical potential. Numerical simulations show that the incident beam width has effect on GH shifts [22]. In the end, we illustrate the application of GH shifts in dielectric refractive index sensing.
2. Non-Hermitian dielectric multilayers
The structure of the dielectric multilayers incorporated with graphene is shown in Fig. 1. The multilayers stack forms a Fabry-Pérot cavity with Bragg gratings. The layers A and B represent the dielectrics with different refractive indices and they are alternatively arranged along the Z-axis, forming the Bragg gratings on two sides of a photonic crystal (PC) defect. The single-layer graphene C is embedded in the middle of the cavity. The right-terminal layer of the PC is replaced by the dielectric D so as to break the geometric symmetry. The dielectric refractive index of layer D is tunable which can be achieved by dynamic modulations [43]. The layers A, B, and D have the same thickness. The optical thicknesses of layers are near quarter-wavelength nλ/4, where λ denotes the free space wavelength of incident waves.
Fig. 1 Schematic of non-Hermitian dielectric multilayers. For the primitive unit-cell layer A, B, graphene C and dielectric D, the thicknesses are d = 0.2 μm and the graphene thickness dc = 1 nm. The refractive indices of dielectrics A and B are na = 2.2, nb = 1.8 and nd = n′ + in″ with n″ = 0.05.
The incident plane wave is obliquely projected at an angle of θ into the dielectric multilayers. The transmission coefficients on both sides (the left and the right) of the reciprocal system are the same and represented by t. But the reflection coefficients at the left and right sides should be different and denoted by r1 and r2, respectively. The reflection coefficients and transmission coefficients could be obtained by the transfer matrix method (TMM).
The scattering matrix for incident and scattered waves in the structure can be expressed as [24, 42]
where r1, r2 represent the reflection coefficients and t is the transmission coefficient. As r1,2 = |r1,2|exp(iφ1,2), the GH shifts of reflected beams can read [28]where φ1,2 are the phases of reflection coefficients r1,2. As the structure is lossy and the geometry is asymmetric, the S-matrix has to be non-Hermitian. The eigenvalues and eigenvectors of S-matrix are β1,2 = t ± (r1r2)1/2 and (r11/2, ± r21/2), respectively. The two physical quantities coalesce at (r1r2)1/2 = 0, that is, the EPs of the eigenvalues. In other words, the EPs appear as either r1 or r2 = 0. Based on the definition of reflectance R1,2 = |r1,2|2, we could derive the condition for realizing EPs, making R1 or R2 = 0. As R1R2 = 0, the related reflection phase φ1 or φ2 has to experience a dislocation, which is a unique property of EPs. The zero reflection and phase dislocation can be utilized to seek out EPs. Since the structure in Fig. 1 is asymmetric, it is hard to make both R1 and R2 be zero simultaneously. Consequently, there are two EPs with new characteristics.3. EPs in non-Hermitian multilayers
For transverse electric (TE) wave, now we study the reflectance and transmittance of the structure when light is injected from the left side. In the simulation, graphene is treated as a thin film with an equivalent thickness dg, then, the relative equivalent permittivity of graphene could be given by εg = 1 + iσgη0/(k0dg) [36, 39], where η0 is the vacuum resistivity and σg is the surface conductivity of graphene. The surface conductivity of graphene can be modeled using Kubo formula [44], which depends on the incidence wavelength λ, chemical potential μc, relaxation time τ, and absolute temperature T. Although the treatment of equivalent thickness for graphene is based on theoretical approximation, the optical response of graphene can be accurately featured as long as the equivalent thickness is small enough, typically smaller than 1 nm [36–38, 45].
Figure 2(a) shows the reflection and transmittance as a function of the incident angle θ. The reflectance spectra reach the minimum values of R1 = 0.034 and R2 = 0.001 at θ = 22.89° and θ = 22.57°, respectively. As a result, the angular GH shift should be very small since the slop of reflectance approaches to zero near the minimum [15]. The difference of the two curves indicates that the reflectance is asymmetric and dependents on the incident direction. Comparably, the transmittance from opposite incident direction is reciprocal. The maximum transmittance reaches T = 0.76 at θ = 22.05°. The peak at the spectrum is aroused by the defect mode in the multilayers. Figure 2(b) illustrates the electric field distribution of the defect mode along the z-axis. It shows that most of the power is tightly confined at the interface of the defect and the field exponentially decays as departing from the center. The incident wavelength is set to λ = 1.55 μm. Other parameters are chosen as dg ≈0.3 nm [38, 39], μc = 0.5 eV, and τ = 0.5 ps.
Fig. 2 (a) The calculated spectra of T (the blue curve), R1 (the red curve) and R2 (the green curve) as a function of incident angle θ for dielectric refractive index n′ = 2.0. R1 and R2 are the reflectance of light incident from the left and right, respectively. (b) Electric fields intensity (|Ez|2) distribution of defective mode.
Now we focuse on the EPs in the system. Figure 3(a) illustrates the reflectance R1 as a function of the incident angle and dielectric refractive index for light incident from the left side of the structure. One sees that there is a nadir locating at θ = 23.978° and n′ = 2.316. The nadir value is as low as R1 = 2.972 × 10−8, which equals nearly to zero considering the limit of calculation accuracy. Figure 3(b) plots the real and imaginary parts of eigenvalues of the scattering matrix vary with the incident angle as n′ = 2.316. One sees that Re(β1,2) and Im(β1,2) almost coalesce as the reflection approaches to zero, validating the appearance of EPs. The EP location is denoted by EP1 in parametric space with θEP1 = 23.978° and nEP1 = 2.316. For light incident from opposite direction of the right side, the nadir value becomes R2 = 1.618 × 10−8 at θEP2 = 22.978° and nEP2 = 2.101, corresponding to EP2 as shown in Fig. 3(c). The eigenvalues also degenerate at EP2 as shown in Fig. 3(d) and the levels of Re(β1,2) and Im(β1,2) experience crossing and anti-crossing in the vicinity of the two EPs [40].
Fig. 3 (a) Reflection spectra R1 in parametric space of incident angle and dielectric refractive index. EP1 locating at θEP1 = 23.978° and nEP1 = 2.316. (b) Real part and imaginary part of eigenvalues for scattering matrix vary with incident angle as n′ = nEP1. (c) Reflection spectra R2 and EP2 locating at θEP2 = 22.978° and nEP2 = 2.101. (d) Real part and imaginary part of eigenvalues for scattering matrix as n′ = nEP2. The black arrows point to the positions of EPs in (a) and (c). Light is incident from the left for (a) and (b), while incident from the right for (c) and (d).
4. GH shifts in the vicinity of EPs
The spatial GH shift relates to the phase of reflection coefficient and is proportional to the slope of reflection phase in accordance with Eq. (2). Figure 4(a) illustrates the reflection phase φ1 in parametric space, where the dielectric refractive index and incident angle can be modified in the vicinity of EP1. There is a phase dislocation around EP1, which is unique for EPs in non-Hermitian systems [38]. The singularity of phase indicates that the phase should abruptly change once the parameters undergo a little variation. Figure 4(b) plots the reflection phase versus the incident angle for several sampling dielectric refractive indices. There is a hop point in the phase curve as n′ = 2.308 and the phase difference is –2π. Neglecting the meaningless phase difference at the hop point, the curve is actually continuous with a positive slope. The phase change can approximately reach to π in a phase variation range around θEP1 as the dielectric refractive index n′ < nEP1. The phase experiences a π abruptly jump at θEP1 as n′ = nEP1. Comparably, the phase change can nearly reach to –π around θEP1 with a negative slope as n′ > nEP1. For light incident from the right, as depicted in Fig. 4(c), there is also a phase dislocation existing in the vicinity of EP2. Contrary to φ1, the hop points lie in the upper region of parametric space, corresponding with higher refractive indices. Figure 4(d) plots the reflection phase varies with the incident angle for several different dielectric refractive indices around EP2. An abrupt phase change also occurs as n′ = nEP2. A π phase roughly climbs in a variation range of dielectric refractive index about nEP2 as n′ < nEP2, but a π phase drops as n′ > nEP2. The slope of phase curve is larger as n′ is closer to nEP1,2 and reaches the positive and negative maxima at EPs. Otherwise, the slope of curve can be negative or positive and the polarity converts as the dielectric refractive index approaches to EP1,2 along the same direction, indicating the direction-dependence of GH shifts, which will be studied shortly.
Fig. 4 (a) Phase of reflection coefficient in the vicinity of EP1. (b) Phase curve versus incident angle for several specific refractive indices around EP1. (c) Phase of reflection coefficient in the vicinity of EP2. (d) Phase curve versus incident angle for several specific refractive indices around EP2. Light is incident from the left for (a) and (b), while incident from the right for (c) and (d).
The GH shift is proportional to the slope of phase curve in accordance with Eq. (2). For light incident from the left, as shown in Fig. 5(a), the GH shifts vary with the incident angle. One sees that the GH shifts are negative and a valley appears in each curve, corresponding to the dielectric refractive indices n′ = 2.190, 2.222 and 2.254 which are below nEP1 = 2.316. It should be mentioned that, for a two-dimensional material, it may find that the negative GH shift is smaller than positive GH shift [20]. However, the situation should be different for bulk material. Here the multilayer structure behaves more like a bulk material and the defect cavity will enhance the confinement of light. The giant negative GH shift results from the permeating of electric field deeply into the multilayer. Actually, the large negative GH shift is common and has been reported in other multilayer structures [28]. For larger dielectric refractive indices, such as n′ = 2.382, 2.414 and 2.446, the peaks emerge as shown in Fig. 5(b). As the dielectric refractive indices are closer to nEP1, the corresponding curves become narrower and sharper (or deeper). Figure 5(c) displays the position of the maximum GH shift can be modified by the dielectric refractive index. The incident angle of the maxima position almost increases linearly with the dielectric refractive index, which may results from the influence of the dielectric refractive index on the reflection. Besides, the maxima of GH shifts can be negative or positive and separated by EP1. Their absolute values sharply climb as the dielectric refractive index approaches to nEP1. In the vicinity of EP1, the negative maximum GH shift can reach as high as –4000 times of the incident wavelength, while the positive maximum can approach to 2000 times. Figure 5(d) depicts the GH shift as a function of the dielectric refractive index and incident angle. The area concerned in the parametric space is divided into two parts labeled by I and II, respectively. The GH shift is negative in part I, while it is positive in part II. The GH shift becomes zero along the dotted curve except for EP1, at which the polarity of GH shifts converts, manifesting the singularity.
Fig. 5 (a) GH shifts versus incident angle for dielectric refractive index n′ < nEP1. (b) GH shifts versus incident angle for dielectric refractive index n′ > nEP1. (c) Maxima of GH shifts varying with dielectric refractive index around nEP1 (in blue dot line). Incident angle of position in parametric space for maxima versus dielectric refractive index (in red dot line). (d) GH shifts in the vicinity of EP1. The GH shifts are negative in the part I, while positive in the part II. It has rescaled the results by taking logarithm log10|Δ1/ λ | for clarity.
For light incident from the right, as shown in Fig. 6(a), the GH shifts vary with the incident angle. The chosen dielectric refractive indices, including n′ = 1.976, 2.008 and 2.040, are lower than nEP2 = 2.101. However, the profile of curves is similar to the case as light impinges from the left for n′ > nEP2. For larger dielectric refractive indices n′ = 2.136, 2.168 and 2.200, as shown in Fig. 6(b), the GH shifts curves change with the incident angle, similar to the profile for light incident from the left as n′ < nEP2. The property manifests that the GH shifts are direction-dependent. In the vicinity of EP2, the maxima of GH shifts are also divided by nEP2 and can reach to ± 103 times of incident wavelength as shown in Fig. 6(c). The polarity converts at EP2, indicating the singularity of GH shifts. Figure 6(d) shows the values are positive in part I and negative in part II. Compared with EP1, the positive and negative positions in parametric space have been turned over. Similarly, the GH shifts are zero along the dotted curve except for EP2.
Fig. 6 (a) GH shifts versus incident angle for dielectric refractive index n′ < nEP2. (b) GH shifts versus incident angle for dielectric refractive index n′ > nEP2. (c) Maxima of GH shifts varying with dielectric refractive index around nEP2 (in blue dot line). Incident angle of position in parametric space for maxima versus dielectric refractive index (in red dot line). (d) GH shifts in the vicinity of EP2. The GH shifts are positive in the part I, while negative in the part II. It has rescaled the results by taking logarithm log10|Δ2/λ| for clarity.
In order to confirm the direction-dependence of GH shifts, we choose θ = 23.5°, 24.0° and 24.5°. The three angles are distributed on both sides of θEP1 = 23.978° but larger than θEP2 = 22.978°. For light incident from the left at a fixed incident angle, as shown in Fig. 7(a), the GH shifts vary with the dielectric refractive index. The curve has a valley as the incident angle θ = 23.5°, but it has a peak as θ = 24.0° and 24.5°. For the incident angle θ = 24.0°, which is closer to θEP1, the peak is sharper than the one for θ = 24.5°. For light incident from the right, as shown in Fig. 7(b), the GH shifts change with the dielectric refractive index. A valley arises in each curve, and the valley becomes deeper as the incident angle more approaches to θEP2. On the whole, the peak and valley positions are identical for θ = 24.5°, attributed to the fields most deeply penetrating into the multilayers at these points. The property shows the reflection is different for light impinging upon the device, so it may find potentials used for optical switches.
Fig. 7 (a) GH shifts versus dielectric refractive index for light incident from the left. (b) GH shifts versus dielectric refractive index for light incident from the right. The curves are in red, green and blue corresponding to incident angles θ = 23.5°, 24.0° and 24.5°, respectively.
The EPs position can be controlled by the graphene chemical potential. In the parametric space composed of incident angle and dielectric refractive index, as shown in Fig. 8(a), the position of the zero-reflectance varies with the graphene chemical potential for light incident from the left. Changing the chemical potential from μc = 0.48 to 0.60 eV, the dielectric refractive index remains constant at n′ = 2.316, while the incident angle decreases with the chemical potential. Accordingly, maintaining the dielectric refractive index n′ = 2.316, one can see lots of continuous zero-reflectance points corresponding to EPs, as shown in Fig. 8(b). Therefore, in the parametric space composed of incident angle and chemical potential, the position of EPs varies with the graphene chemical potential. This is because the surface conductivity of graphene is a function of μc, affecting the reflection of light. Figure 8(c) plots the GH shifts versus the chemical potential as θ = 23.85°, 23.90° and 23.95°. The peaks arise in these curves and the maxima are notable. The peaks have the same values and up to 104 times of incident wavelength around EPs. The EPs are also singular points for GH shifts. Figure 8(d) illustrates the maxima of GH shifts go along the zero-reflective points where EPs locate in parametric space of incident angle and chemical potential.
Fig. 8 (a) Position of zero-reflectance point modifying by graphene chemical potential in parametric space of incident angle and refractive index. Light is incident from the left. Refractive index is shown in blue and incident angle is shown in red. (b) Reflectance varying with incident angle and chemical potential. (c) GH shifts versus graphene chemical potential for three incident angles of θ = 23.85° (in red), 23.90° (in green) and 23.95° (in blue). (d) GH shifts in parametric space of incident angle and chemical potential. The refractive index n′ = 2.316 and it has rescaled the values by taking logarithm for clarity in (b)-(d).
5. GH shifts for sensing
As a Gaussian beam with finite width incident upon the dielectric multilayers at an angle θ from the left side, the distribution of electric field intensity in the calculating region, as shown in Fig. 9(a). Numerical simulations are performed with the commercial software COMSOL Multiphysics based on the finite element method. It shows that the reflective beam has a positive lateral shift from the position predicted by geometrical optics. For the reflected beam much weaker than the incident, we have rescaled the field by taking logarithm. Figure 9(b) gives the profile of the electric field at the incident interface, where the two peaks denote the central points of the incident and reflective beams, respectively. The GH shifts can be derived in accordance with incident angles, propagating distances and central positions of beams. The simulation result Δ = 12λ is lower the theoretical value Δ = 14λ as θ = 24.8° and n′ = 2.46. The difference results from the influence of the Gaussian beams width on simulations. Increasing the width of Gaussian beam, the two results are satisfied well as shown in Fig. 9(c). The width effect is more obvious for larger GH shifts, while the corresponding reflection is much weaker. Here, we choose θ = 22.3° and n′ = 2.29 on account of those factors such as reflection and the value of GH shift comprehensively. Simulations demonstrate that the GH shift can achieve Δ = 51λ in theory as long as the beam width surpasses 70 times of incident wavelength, which agrees with the conclusions mentioned in defective photonic crystals [22].
Fig. 9 (a) Electric field amplitude distribution of Gaussian beam with a finite width as θ = 24.8° and n′ = 2.46. (b) Profile of electric field at the incident interface. (c) Influence of width on GH shifts for Gaussian beam as θ = 22.3° and n′ = 2.29. Simulation of GH shifts for Gaussian beam with different widths (in blue dotted line) and result predicted in theory (in red dotted line).
The GH shift is related largely to the dielectric refractive index and is direction-dependent, so the proposed structure can be utilized for sensors or optical switches. Sensitivity coefficient (SC) is defined as the differential of GH shift changes over dielectric refractive index. For light injected from the left, not only GH shift, but also the SC is sensitive to the dielectric refractive index and incident angle as shown in Fig. 10(a), so the sensitivity of sensors possesses flexible tunability. In the vicinity of EP1, the area is separated into four parts by the dotted lines. The values are positive in parts I and III, while they are negative in parts II and IV. The values are zero along the dotted lines except for EP1, which is a singular point for the SC. The polarity of SC converts at EP1 and the positive and negative maxima of SC can reach 1.3 × 106 and –1.1 × 106, respectively. Figure 10(b) shows that the SC of GH shifts can be modified by the dielectric refractive index. Here we choose three incident angles θ = 23°, 24°and 23.5°, which locate at the two sides of θEP1 = 23.978°, respectively. A peak and a valley exist in each curve at the incident angles θ = 23° and 23.5°, while only a peak appears at θ = 24°. As the incident angle moves closer to θEP1, the peak and valley become sharper and narrower. The SC represents the sensitivity of sensors in detecting physical quantities, so the devices can be used for sensitive detectors. In the study we implement sensing by employing spatial GH shift since the EPs are mainly considered. Moreover, the angular GH shift might also find interesting application in sensors.
Fig. 10 (a) Sensitivity coefficient of GH shifts in parametric space of dielectric refractive index and incident angle around EP1. It has rescaled the values by taking logarithm log10|dΔ/(λdn′) | for clarity. (b) Sensitivity coefficient of GH shifts tuned by dielectric refractive index at incident angles θ = 23°, 24° and 25°, respectively.
6. Conclusion
In conclusion, we construct a non-Hermitian system by dielectric multilayers and graphene, which is lossy and asymmetric. The reflection property in optics is direction-dependent for light incident from the left and right. The EPs locates at zero points of reflectance. The eigenvalues of S-matrix coalescence and phase abruptly changes at EPs. In the vicinity of EPs, the GH shifts can be positive and negative and can reach as high as 103 times of incident wavelength. The polarity of GH shift converts at EPs which has a singularity for GH shift. Moreover, the EPs position can be controlled by the graphene chemical potential. The simulations fairly coincide with the theoretical analysis for Gaussian beam. The GH shifts are direction-dependent and sensitive to a tiny dielectric refractive index variation, so the devices have potential used for optical switches or sensors.
Funding
Program 973 (2014CB921301); National Natural Science Foundation of China (NSFC) (11674117); Natural Science Foundation of Hubei Province (2015CFA040).
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