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Abstract
Local interface response effects are neglected based on the traditional electromagnetic boundary conditions (EMBCs) in an abrupt interface model. In this study, generalized nanoscale EMBCs are derived with interface response functions (IRFs) representing field inhomogeneity across the interface based on integral Maxwell’s equations. They are rewritten in two different forms that correspond to the equivalent abrupt interface models with interface-induced dipoles or charges and currents. Interesting behaviors of Brewster angle shifting, non-extinction at Brewster angle, and unique absorption or gain effects are revealed based on the advanced Fresnel formula. IRFs-controlled GH-shift and angular GH-shift of a Gaussian beam near the Brewster angles are generated by the gradient interface. These unique phenomena provide some guidance for measuring the IRFs and expanding interface photonics at the nanoscale.
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1. Introduction
Electromagnetic boundary conditions (EMBCs) have a wide range of applications in physics. In traditional electromagnetic (EM) theory, the EMBCs are derived from the abrupt interface assumption while neglecting the integral contribution of EM field along the side wall of the integrating box [1,2]. These EMBCs are generally used to describe variations in field intensity on both sides of the medium. However, when it concerns EM phenomena at the nanoscale, EMBCs break down and provide theoretical results that exist in the gap with the experiments, such as Nano-plasmonics [3–6], surface reflectance [7,8], and surface photoexcitation [7]. This gap may be induced by neglecting the electron intrinsic scale, nonlocality property of the electric field response [9,10], and assumption of abrupt interface [7]. To narrow this gap, Feibelman [7] developed the interfacial response functions (IRFs) known as the Feibelman d parameters as an effective approach for explaining nanoscale EM phenomena [4,7,8]. In 2019, scientists further constructed the nanoscale EM framework based on the Feibelman d parameters [6], significantly advancing the development of nanoscale electromagnetism. In 2020, Feibelman d parameters were extended to reveal the plasmon–emitter interactions at the nanoscale [11]. These studies open the era of nanoscale electrodynamics for the metal-dielectric interface. However, nanoscale EMBCs and IRFs are provided in a form with less physical fundamentals. The corresponding IRFs are difficult to calculate and measure directly. Moreover, scientists have focused primarily on IRFs at the jellium–vacuum interface, neglecting that IRFs are universal on all types of interfaces. Therefore, there is an urgent requirement for constructing a smart interface physical model for further development and revealing the reason for neglecting IRFs.
With the development of two-dimensional (2D) optical materials [12,13], their optical parameters are expected to be measured with a distinguishable method from the bulk optical property detection [14]. There are currently two main models in use for the linear optical description of 2D optical materials based on the traditional abrupt interface [15]. The first is a thin film model treating 2D optical materials as an anisotropic or isotropic film with a small but finite thickness and an effective bulk dielectric constant [16,17]. The second model is a 2D sheet with a surface susceptibility or conductivity [18–20]. These two models are used to derive the EMBCs for the interface including 2D material, demonstrating some inconsistencies in the case of oblique incidence of TM (p-polarized) waves. This controversy indicates that the contributions of 2D material on reflection and transmission should be reconsidered from the basic nanoscale EMBCs where the optical response induced by nanoscale material is concerned. Moreover, with the development of nano-photonics, magnetic response becomes more controllable by regulating the optical structures [21,22]. Therefore, the generalized EMBCs including all interfacial electromagnetic responses are highly desired for nano-photonics. In this study, the generalized EMBCs are deduced using the integral Maxwell’s equations and interface model with a transition layer. Two additional IRFs, ${b_ \bot },{b_\parallel }$, are introduced by considering the inhomogeneity of the magnetic fields within the transition layer, reflecting the broken symmetry of the EM fields across the interface, and extending the connotation of the IRFs. The equivalent EMBCs are demonstrated by proposing interface-induced dipoles or charges and currents on the abrupt interface. In addition, the corresponding Fresnel formulas are proposed as a theoretical basis for analyzing the interface reflection and refraction behaviors. Unique non-extinction phenomenon, phase continuous variations, special Goos-Hänchen (GH)-shifts, and Brewster angle shift are induced by the IRFs near Brewster angle. Interestingly, the interface-induced absorption or gain effect is observed at both the Brewster and total internal reflection angles.
2. Nanoscale electromagnetic boundary conditions
2.1 Deduction of nanoscale EMBCs
The interface without free moving charge or current is formed by two isotropic bulk materials with permittivity and permeability of ${\varepsilon _1}$, ${\mu _1}({z > 0} )$ and ${\varepsilon _2}$, ${\mu _2}({z < 0} )$, respectively. The permittivity and permeability change continuously in a transition layer from z20 to z10 in Fig. 1 to demonstrate the inhomogeneity of the EM field across the interface. The EM field in the non-transition region satisfies the traditional EMBCs as the zero-order approximate results. Using Maxwell's equations and considering the contributions of the transition layer as the first-order perturbation of the classical EMBCs from the abrupt interface, the generalized nanoscale EMBCs can be deduced.
Fig. 1. (a) Integrating box across the interface with the transition layer for calculating the discontinuity of tangential electromagnetic (EM) field; (b) Integrating box across the interface with the transition layer is for the discontinuity of normal components of the EM field. (c, d) Magnetic IRFs and corresponding distributions of magnetization charge (c) and magnetization current density (d).
To deduce the the discontinuity of parallel component of electric field across the surface, we assume time-harmonic transverse magnetic (TM) wave with angular frequency ω upon the interface. The electric field $\overrightarrow E$ has x and z components while the magnetic inductive field $\overrightarrow B$ has only y components normally to the interface. A rectangular integrating box including the transition layer is constructed from z2 to z1 and x1 to x2 as shown in Fig. 1(a) for performing the Faraday law [1]
Since the length of the integrating box $\varDelta l = {x_2} - {x_1}$ is much smaller than the wavelength, the variation of ${E_x}$ from ${x_1}$ to ${x_2}$ could be ignored. And 1st-order perturbation approximation is used to describe the ${E_z}$ variation from ${x_1}$ to ${x_2}$ for further considering the transition layer contributions. Equation (1) could be further deduced as
By using partial integral, we can transform Eq. (2) into
Here, the function $\gamma (z) = {E_z}(z)/{E_z}({z_2})$ is introduced to describe the variation of the electric field component ${E_z}$ within the transition layer. The function has $\gamma (z \le {z_{20}}) = 1$ in the bulk medium 2 and $\gamma (z \ge {z_{10}}) = {\varepsilon _B}$ in the non-transition zone of medium 1 based on the traditional EMBC of ${D_z}({z_1}) = {D_z}({z_2})$. The function $\beta = {B_y}(z)/{B_y}({z_1})$ with $\beta (z \le {z_{20}}) = {\mu _B}$ and $\beta (z \ge {z_{10}}) = 1$ is introduced for describing the variation of the magnetic field component ${B_y}$ within the transition region based on ${\mu _B}{B_y}({z_1}) = {B_y}({z_2})$.
Similarly, we use the same integrating box and coordinate system in Fig. 1(a) and consider a time-harmonic transverse electric (TE) wave with the electric field $\overrightarrow E = {E_0}{e^{ - i\omega t}}{\hat{e}_y}$. If there is no free current on the interface, we can obtain the discontinuity of tangential magnetic field
The function $\delta (z) = {H_z}(z)/{H_z}({z_2})$ is introduced with the boundary conditions of $\delta (z \le {z_{20}}) = 1$ and $\delta (z \ge {z_{10}}) = {\mu _B}$ to demonstrate the ${H_z}$ variation across the interface based on the traditional EMBC of ${B_z}({z_1}) = {B_z}({z_2})$. And function $\alpha = {D_y}(z)/{D_y}({z_1})$ with the conditions of $\alpha (z \ge {z_{10}}) = 1$ and $\alpha (z \le {z_{20}}) = {\varepsilon _B}$ is for describing the variation of the displacement component ${D_y}$ within the transition region.
To deduce the discontinuity of normal component of displacement field across the interface, we construct cuboid integrating box as shown in Fig. 1(b). The transition region is from ${z_{20}}$ to ${z_{10}}$. On the interface without free charge, displacement complies with Gauss’s law [1]
Considering that the size of the integrating box in x and y directions are much smaller than the wavelength, the variation of ${D_z}$ can be ignored on the top and the bottom integral surfaces. And Eq. (11) can be further deduced to
According to the definition of the IRF ${d_\parallel }$, this boundary condition is expressed as
Symmetrically, the discontinuity of magnetic induction field can be obtained as
Summarizing the above derivation, we can get the generalized nanoscale EMBCs as follows
2.2 Physical meanings of interfacial response functions
As Liebsch [10] pointed out, Feibelman d parameters can be further expressed as
Among those, ${\rho _{\textrm{ind}}}$ is the surface-induced polarization charge density and ${j_{\textrm{py}}}$ is the tangential polarization current on the surface. These parameters indicate intrisic porperties of the interfaces as ${d_ \bot }$ representing the centroid of interface-induced polarization charge and ${d_\parallel }$ being the centroid of the normal derivative of tangential current. Particularly, on an ideal infinite metal surface, ${d_\parallel }$ is identically zero. But ${d_\parallel }$ should be considered for surface roughness [23], bound screening [9], and even polarization materials [24,25].
In classical electrodynamics [2], the divergence of magnetic field is the equivalent magnetization charge density (not magnetic monopole). Based on the assumption that $\left|{\frac{{\partial {H_\textrm{x}}}}{{\partial x}}} \right|< < \left|{\frac{{\partial {H_\textrm{z}}}}{{\partial z}}} \right|$, we can get
Similarly, the interface-introduced magnetization current density $\overrightarrow {{j_\textrm{m}}} $ can be defined as
According to Eq. (7), ${b_\parallel }$ is the centroid of equivalent magnetization current density (Fig. 1(d)) as
It can be found that the b-parameters can be generated by the magnetic field variation across the interface. When the interface is formed by the materials with magnetic response, some unique optical phenomena will be induced by the b parameters.
2.3 Equivalent abrupt interface models
According to the definition of electric and magnetic dipole moments, the proposed interface-induced dipole moments are introduced as $\overrightarrow {{\pi _ \bot }} \textrm{ = }{\varepsilon _0}{d_ \bot }[\kern-0.15em[{{E_ \bot }} ]\kern-0.15em]\widehat n$ and $\overrightarrow {{\pi _\parallel }} \textrm{ = } - {d_\parallel }[\kern-0.15em[{\overrightarrow {{D_\parallel }} } ]\kern-0.15em]$ for electric dipole, and $\overrightarrow {{m_ \bot }} \textrm{ = }{b_ \bot }[\kern-0.15em[{{H_ \bot }} ]\kern-0.15em]\hat{n}$ and $\overrightarrow {{m_\parallel }} \textrm{ = } - {b_\parallel }[\kern-0.15em[{\overrightarrow {{B_\parallel }} } ]\kern-0.15em]/{\mu _0}$ for magnetic dipole to describe the contributions of the EM field discontinuity across the interface (Fig. 2). Therefore, the nanoscale EMBCs can be expressed as
Thus, the interface with a transition layer can be simply looked as the abrupt interface with interface-induced electric and magnetic dipole moments. The obtained EMBCs are identical in form with those from the proposed sheet model with non-zero surface polarization and magnetization density including both in-plane and out-of-plane components [16,19,20]. But the nanoscale EMBCs are the result of the transition interface model with the inhomogeneous EM field-induced polarization and magnetization, different from the results through assuming polarization and magnetization on the abrupt interface in the basic physical model. Our results might supply a possible physical mechanism for the polarization and magnetization from 2D atom crystal and be used to further specify the contributions from the substrate and the interficial 2D material in the future.
Based on the physical meaning of interface dipole moment, we define the linear density of interface polarization current as $\overrightarrow {{K_1}} \equiv \partial \overrightarrow {{\pi _\parallel }} /\partial t = i\omega {d_\parallel }[\kern-0.15em[{\overrightarrow {{D_\parallel }} } ]\kern-0.15em]$, magnetization current as $\overrightarrow {{K_2}} \equiv \nabla \times \overrightarrow {{m_ \bot }} = {b_ \bot }{\nabla _\parallel }[\kern-0.15em[{{H_ \bot }} ]\kern-0.15em]\times \hat{n}$, the density of interface equivalent magnetic charge currents as $\overrightarrow {{K_{\textrm{m}1}}} ={-} {d_ \bot }{\nabla _\parallel }[\kern-0.15em[{{E_ \bot }} ]\kern-0.15em]\times \hat{n}$ and $\overrightarrow {{K_{\textrm{m}2}}} \equiv {\mu _0}\partial \overrightarrow {{m_\parallel }} /\partial t = i\omega {b_\parallel }[\kern-0.15em[{\overrightarrow {{B_\parallel }} } ]\kern-0.15em]$, equivalent interface polarization charge sheet density as ${\sigma _{\textrm{Ie}}}\textrm{ = }{d_\parallel }{\nabla _\parallel }\cdot [\kern-0.15em[{\overrightarrow {{D_\parallel }} } ]\kern-0.15em]$ and equivalent magnetization charge sheet density as ${\sigma _{\textrm{Im}}} = {b_\parallel }{\nabla _\parallel }\cdot [\kern-0.15em[{\overrightarrow {{B_\parallel }} } ]\kern-0.15em]$ (Fig. 2). The nanoscale EMBCs can be further expressed as
Compared with the traditional EMBCs of the abrupt interface with the free charge sheet density ${\sigma _\textrm{e}}$ and free current line density $\overrightarrow {{\alpha _\textrm{f}}}$ ($[\kern-0.15em[{\overrightarrow {{E_\parallel }} } ]\kern-0.15em]= 0,[\kern-0.15em[{\overrightarrow {{H_\parallel }} } ]\kern-0.15em]= \overrightarrow {{\alpha _\textrm{f}}} \times \hat{n},[\kern-0.15em[{{D_ \bot }} ]\kern-0.15em]= {\sigma _\textrm{e}},[\kern-0.15em[{{B_ \bot }} ]\kern-0.15em]= 0$), the role of interface-induced polarization current $\overrightarrow {{K_1}}$ and magnetization current $\overrightarrow {{K_2}}$ is equivalent to the free current line density $\overrightarrow {{\alpha _\textrm{f}}}$. The role of interface-induced charge sheet density ${\sigma _{\textrm{Ie}}}$ is equivalent to that of ${\sigma _\textrm{e}}$. The nanoscale EMBCs obtained are symmetric and identical to the EM field parameters. The contributions of EM field discontinuity across the interface can be equivalent to the amount of polarization and magnetization charges, current, and magnetic charge currents on the abrupt interface. This is the fundamental for constructing the equivalent model of abrupt interface with the polarization and magnetization charges and currents for calculations.
3. Fresnel formula and unique interface reflection phenomena
3.1 Fresnel formula based on generalized EMBCs
We consider the harmonic plane wave with vacuum wave number ${k_0}$ incident on the interface with the relative permittivity (permeability) discontinuity $\varDelta \varepsilon = {\varepsilon _{\textrm{r}2}} - {\varepsilon _{\textrm{r}1}}$ ($\varDelta \mu = {\mu _{\textrm{r}2}} - {\mu _{\textrm{r}1}}$) at incident angle ${\theta _1}$ and refraction angle ${\theta _2}$. Based on the generalized nanoscale EMBCs, the Fresnel formulas are obtained as
According to the Fresnel formulas, the IRFs contributes a small perturbation to traditional Fresnel transmission coefficients due to the IRFs appearing as a small quantity in the denominator. For the Fresnel reflection coefficients, the contributions of the IRFs can also be neglected for most cases with non-absorption and weak absorption materials due to the extremely small modulation of ${k_0}d$ or ${k_0}b({\sim} 0.001)$. However, the IRFs’ contributions cannot be neglected in some special cases. For the interface with non-absorption and weak absorption materials, the contributions of the IRFs will become significant when the traditional reflection coefficients $|{{r_{ic}}} |\to 0$ ($i = \textrm{p,s}$). This result tells us that the IRFs will greatly affect the reflection behaviors of the waves incident at around Brewster angle (θB). Moreover, the IRFs will obviously affect the reflection behaviors when $\cos {\theta _2}$ is a pure imaginary number or a complex number with large imaginary part. And the important contributions of IRFs can be observed at the conditions of the total internal reflection, and the strong reflection due to the strong absorption of the metal.
3.2 Unique interface reflection phenomena
Considering the similar tuning role of the b parameters (${b_ \bot }$ and ${b_\parallel }$) on rs, and the d parameters (${d_ \bot }$ and ${d_\parallel }$) on rp, we mainly demonstrate the effect of the d parameters on rp. Based on the above analysis, IRF-induced reflection phenomena is demonstrated on the vacuum-SiO2 interfaces with different ${d_ \bot }$ at ${d_\parallel } = 0$.A non-extinction phenomenon at θB resulting from the real part of d-parameter ${d_ \bot }$ ($\textrm{Re(}{d_ \bot })$) is found at this type of interfaces (Fig. 3(a)). This differs from the traditional Fresnel formulas results, but agree with the experimental results [7]. Quantitatively, the coefficient $|{{r_{\textrm{p}, \min }}} |$ at θB is almost proportional to ${k_0}|{A}{\textrm{Re(}{d_ \bot }) - {B}\textrm{Re(}{d_\parallel })} |$ and a large $|{\varDelta \varepsilon } |$ can enhance such phenomenon of non-extinction according to the formula (33).This IRF-induced $|{{r_{\textrm{p}, \min }}} |$ is typically within ${10^{\textrm{ - }3}}$, corresponding to a detectable reflection intensity based on the current technique for a watt or milliwatt laser stimuli. At the same time, the position shifting $\Delta {\theta _{\textrm{B}1}}$ of Brewster angle is caused by the imagery part of ${d_ \bot }$($\textrm{Im(}{d_ \bot })$), following an increment for negative $\textrm{Im(}{d_ \bot })$ and decrease for positive $\textrm{Im(}{d_ \bot })$ shown in Fig. 3(a). In addition, $\Delta {\theta _{\textrm{B}1}}$ is proportional to ${-k_0}({A}{\textrm{Im(}{d_ \bot }) - {B}\textrm{Im(}{d_\parallel })} )$ based on the formula (33). A typical $\Delta {\theta _{\textrm{B}1}}$ can reach 0.1°–1.0° in an observable range.
Fig. 3. Reflection coefficient ${r_\textrm{p}}$ variations with θ1 at the interface with different ${d_ \bot }$. (a, b) $|{{r_\textrm{p}}} |$ (a) and ${\varphi _\textrm{p}}$ (b) vary with θ1 on the vacuum-SiO2 interfaces (${\varepsilon _{\textrm{r}2}} = 2.25$) at λ0 = 589 nm. (c, d) $|{{r_\textrm{p}}} |$ (c) and ${\varphi _\textrm{p}}$ (d) vary with θ1 on the vacuum-Si [27] interfaces (${\varepsilon _{\textrm{r}2}} = 4.30 + 0.073i$) at λ0 = 499 nm.
For the d parameter-tailored interface, the phase variation ${\varphi _\textrm{p}}$ of the reflected light relative to the incident light changes smoothly with θ1 around θB, different from the phase jump at θB predicted by traditional EMBCs at the abrupt interface (Fig. 3(b)). When $\textrm{Re(}{d_ \bot })$>0, the phase ${\varphi _\textrm{p}}$ changes gradually to π . But the phase changes gradually to –π when $\textrm{Re(}{d_ \bot })$<0. The sign of $\textrm{Re(}{d_ \bot })$ determines the phase variation range. And the phase ${\varphi _p}$ equals to π/2 or -π/2 at Brewster angle.The slope of ${\varphi _\textrm{p}} - {\theta _1}$ curve is proportional to $\textrm{1/Re(}{d_ \bot })$ at θB. And the sign of $\textrm{Im(}{d_ \bot })$ affects the shape of phase variations. Therefore, phase variation ${\varphi _\textrm{p}}$ could be modified in $[{0,2\pi } ]$ by intelligently controlling θ1 and tuning the d-parameter ${d_ \bot }$.
These unique phenomena of non-extinction and phase variation at Brewster angle also occur on weak absorption materials. For the weak-absorption vacuum-Si interface with different ${d_ \bot }$, $|{{r_p}} |$ and ${\varphi _p}$ of the p-polarized beams with λ0 = 499 nm vary with the incident angle as shown in Fig. 3(c) and 3(d). The conditions of ${d_ \bot } = 0$ and weak bulk absorption mean $|{{r_{\textrm{p}, \min }}} |\ne 0$ and continuous phase variation predicted by traditional EMBCs (in green). When introducing ${d_ \bot } = - 0.1\; \textrm{nm}$ on the interface, the reflectivity becomes smaller and sharper phase variation appears at Brewster angle (in orange). When introducing ${d_ \bot } ={-} 0.346\; \textrm{nm}$ on the interface as critical IRF point, the reflection vanished and abrupt phase change appears at Brewster point (in black). This amazing extinction phenomenon and phase jump might be the result of the interference coupling between the bulk absorption and interface gain effect from interface-induced dipole.
When the p-waves are incident on the SiO2-vacuum interfaces with different d parameters at total internal reflection angles, module of reflection coefficient $|{{r_\textrm{p}}} |$ varies with the incident angle as shown in Fig. 4(a). The IRF-tailored absorption or gain effect is observed as $|{{r_\textrm{p}}} |$ smaller or larger than unit on the interface formed by non-absorption materials. This unique phenomenon is entirely because of the IRFs properties at the nanoscale.$\textrm{Im(}{d_ \bot }) > 0$ or $\textrm{Im(}{d_\parallel }) < 0$ ($\textrm{Im(}{d_ \bot }) < 0$ or $\textrm{Im(}{d_\parallel }) > 0$) indicates interface-induced absorption effect (gain effect). This tuning role of IRF-induced absorption or gain effect from the interface can also be observed on the noble metal Ag surface [28] with strong absorption (Fig. 4(b)). By controlling the imagery part of IRFs, we may be able to achieve parity–time symmetry [29] photonics at the nanoscale using this interface effect. Notably, the effects of ${d_ \bot }$ are the greatest (in blue), whereas those of ${d_\parallel }$ are the weakest (black) at the critical angle, providing a new approach to distinguish and measure ${d_ \bot }$ and ${d_\parallel }$ on the interface.
Fig. 4. Surface gain/absorption effects induced by the image part of d-parameters. (a) $|{{r_\textrm{p}}} |$ changes with θ1 larger than the critical angle on the SiO2-vacuum interfaces with different ${d_ \bot }\; \textrm{or}\; {d_\parallel }$. (b) $|{{r_\textrm{p}}} |$ varies with θ1 on the vacuum-Ag interfaces (${\varepsilon _{\textrm{r}2}} ={-} 198.189 + 6.76i$ at $\hbar $ω=0.64 eV) with different ${d_ \bot }$.
3.3 IRF-induced GH shift and angular GH shift at Brewster angle
The IRF-tailored phase variation ${\varphi _\textrm{p}}$ at Brewster angle indicates a large GH shift on non-sbsorption/weak-absorption interfaces.When a Gaussian beam is incident on the vacuum-SiO2 interface at $\textrm{z} = 0$ in Fig. 5(a), the Goos-Hänchen (GH) shift around θB is calculated by ${\Delta _{\textrm{GH}}} ={-} \frac{{d{\varphi _p}}}{{kd{\theta _1}}}$ [30]. A large GH shift is obtained on the non-absorption material-formed interface with different ${d_ \bot }$ (Fig. 5(b)). This GH shift is totally induced and tailored by the interface-generated dipole on the interface, different from the bulk weak absorption-induced GH shift [27]. Interestingly, peak position of GH shift is highly dependent on the $\textrm{Im(}{d_ \bot })$ and the sign of ${\varDelta _{\textrm{GH}}}$ can be flexibly controlled by the signs of $\textrm{Re(}{d_ \bot })$. Positive $\textrm{Re(}{d_ \bot })$ means negative ${\varDelta _{\textrm{GH}}}$, indicating the rich GH-shift behaviors tailored by the IRFs. For the vacuum-Si interface with weak-absorption, the bulk absorption induces an obvious negative GH shift as shown the green dashed line in Fig. 5(c). When ${d_ \bot } ={-} 0.7$ nm, the positive GH shift is observed. The result indicates that GH shift can be tailored by the d parameter, distinguishable the bulk absorption induced GH shift.
Fig. 5. IRF-induced GH shift and angular GH shift at Brewster angle. (a) Local Cartesian coordinate systems of xyz for the interface, ${x_i}y{z_i}$ for incident Gaussian beam and ${x_r}y{z_r}$ for the virtual reflected beam. The direction ${\hat{z}_r}$/${\hat{z}_a}$ indicates the ideal/actual ray axis of the reflected beam. (b) GH shift varies with ${\theta _1}$ on the vacuum-SiO2 interface with different ${d_ \bot }$ and at λ0 = 589 nm. (c) GH shift of the p-polarized beams with λ0 = 499 nm varies with incident angle ${\theta _1}$ on the vacuum-Si interface [27] with ${\varepsilon _{r2}} = 4.30 + 0.073i$, and ${\mu _{r2}} = 1$ near θB. (d-f) Angular distribution of the reflected Gaussian beam with λ0 = 449.2 nm at the vacuum-Si interface with ${\varepsilon _{r2}} = 4.68 + 0.15i$ and different ${d_ \bot }$ at ${\theta _1} - {\theta _B} ={-} 0.5^\circ $ (d), ${\theta _1} = {\theta _B}$ (e), and ${\theta _1} - {\theta _B} = 0.5^\circ $ (f).
Angular distribution [31] of the reflected Gaussian beam near θB at the vacuum-Si interface with different ${d_ \bot }$ is demonstrated in Fig. 5(d)-(f). Typically, the reflected beam is divided into two symmetric wave packets at two sides of θB when the incident angle is θ1=θB (Fig. 5(e)) on the weak absorption material with ${d_ \bot } = 0$ (in green). When the beams are incident at ${\theta _1} \ne {\theta _B}$, two asymmetric wave packets are observed due to the bulk weak absorption (in green). Angular distribution of the reflection beam is modified by the IRF ${d_ \bot }$ (Fig. 5(d)-(f)). Interestingly, the amalgamation phenomenon of two wave packets can be observed because of interface-induced dipole radiation (in red). This phenomenon allows us to measure ${d_ \bot }$ by detecting the angular distribution of the reflected light.
4. Conclusions
An interface model with an EM transition layer is constructed through a classical view considering the non-classical interfacial physical processes. The generalized nanoscale EMBCs are developed based on the four IRFs with clear physical meanings beyond the Feibelman parameters. The two equivalent models of interfacial dipole moment and interface-induced charges and currents are introduced for the EMBCs in two more symmetric forms, demonstrating the high duality of electrical and magnetic properties. The generalized EMBCs provide a necessary theoretical basis for studying the EM processes at the nanoscale and further revealing the interface optical properties. The retarded Fresnel formulas are further developed. Unique phenomena induced by the IRFs, that is, non-extinction at θB, Brewster angle shifting, absorption effect and gain effect, GH-shift, and merging behavior of two wave packets, are investigated and predicted. These unique properties demonstrate the rich physical processes generated by IRFs, as well as provide some feasible approaches for measuring and distinguishing the IRFs on the interfaces. These interesting phenomena are attributed to the long-neglected IRFs rather than bulk parameters. Our results provide a basic tool to reveal and regulate the EM phenomena at the nanoscale and may promote interface photonics.
Moreover, the obtained EMBCs can be extended for different complex interface model by giving the interface polarization and magnetization with new physical process. For the configuration of the 2D material on the substrate, the contributions of 2D material on the reflection of oblique incident TM waves can be distinguished from the substrate through the IRF-related reflection coefficient. The in-plane and out-of-plane polarization and/even magnetization can be obtained through looking the 2D material as an anisotropic film. And the difference between the sheet model and the film model might hopefully be eliminated when the appropriate equivalent parameters are chosen for the two models. By extracting the IRFs at the interface with 2D materials or super thin films on the surface of the bulk materials, the photon response parameters of the sandwiched 2D materials or super thin films can be further obtained based on the proposed nanoscale EMBCs.
Funding
National Natural Science Foundation of China (92150109, 61975018, 11574033).
Acknowledgments
Z. Wang would like to thank the National Science Foundation for assisting in identifying collaborators for this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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