Goos–Hänchen effect singularities in transdimensional plasmonic films

1 Introduction

It has been known that the reflection of a linearly polarized optical beam of finite transverse extent incident on a plane surface does not exactly follow the Snell’s law of geometrical optics [1], [2]. Instead, the reflected beam experiences slight lateral in-plane displacement and angular deflection in the plane of incidence – the phenomenon commonly referred to as the Goos–Hänchen (GH) effect. Originating from the spatial dispersion of reflection or transmission coefficients due to the finite transverse size of the beam (and so nonlocal in its nature), the GH effect occurs for both reflected and refracted light in realistic optical systems. It was observed in a variety of systems (see Ref. [2]) including plasmonic metamaterials [3], graphene [4], and even neutron scattering experiments [5]. These days the effect attracts much attention as well [6], [7], [8] due to the new generation of materials being available – quantum nanomaterials of reduced dimensionality, materials that can enhance nonlocal subwavelength light propagation to offer new directions for quantum optics, quantum nanophotonics, and quantum computing application development [9], [10].

Plasmonic transdimensional (TD) quantum materials are atomically thin metal (or semiconductor) films of precisely controlled thickness [11]. Currently available due to great progress in nanofabrication techniques [12], [13], [14], [15], [16], [17], [18], such materials offer high tailorability of their electronic and optical properties not only by altering their chemical and/or electronic composition (stoichiometry, doping) but also by merely varying their thickness (number of monolayers) [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]. They provide a new regime – transdimensional, in between three (3D) and two (2D) dimensions, turning into 2D as the film thickness tends to zero. In this regime, the strong vertical quantum confinement makes the linear electromagnetic (EM) response of the TD film nonlocal, or spatially dispersive, and the degree of nonlocality can be controlled by varying the film thickness [22], [26]. That is what makes plasmonic TD films indispensable for studies of the nonlocal light–matter interactions at the nanoscale [28], [29], [30], [31], [32].

The properties of the TD plasmonic films can be explained by the confinement-induced nonlocal EM response theory [22], [23] built on the Keldysh–Rytova (KR) electron interaction potential [33]. The theory is verified experimentally in a variety of settings [13], [14], [15], [16]. It accounts for vertical electron confinement due to the presence of substrate and superstrate of dielectric permittivities less than that of the film, whereby the thickness of the film becomes a parameter to control its nonlocal EM response. The KR model covers both ultrathin films of thickness much less than the half-wavelength of incoming light radiation and conventional films as thick as a few optical wavelengths [22], [23]. The nonlocal EM response of TD plasmonic systems enables a variety of new effects such as thickness-controlled plasma frequency red shift [15], low-temperature plasma frequency dropoff [16], plasma mode degeneracy lifting [26], a series of quantum-optical [28], [34] and nonlocal magneto-optical effects [23], as well as thermal and vacuum field fluctuation effects responsible for near-field heat transfer [14], [30] and Casimir interaction phenomena [31], [32].

Here, we focus on the confinement-induced nonlocality of the EM response of the TD plasmonic film to study theoretically the GH effect for an incident laser beam. Using the nonlocal KR model, analytical calculations and numerical analysis, we identify and classify topologically protected singularities for the nonlocal reflection coefficient of the system. Such singularities are shown to lead to giant lateral and angular GH shifts in the millimeter and milliradian range, respectively, to greatly exceed those of microscale reported for beams of finite transverse extent with no material-induced nonlocality [2], [3], [4], [5], [6], [7]. They appear in TD materials with broken in-plane reflection symmetry (substrate and superstrate of different dielectric permittivities) where due to the confinement-induced nonlocality the eigenmode degeneracy is lifted to create the points of topological darkness in the visible range not existing in standard local Drude materials.

2 The Goos–Hänchen shift

The theory of the GH shift was originally formulated by Artmann back in 1948 [35]. The geometry of the effect is shown in Figure 1. After reflectance at an interface, the lateral and angular shifts of an incoming p-polarized wave in medium 1 (refractive index n₁) are given by [2], [35], [36], [37]

and

respectively. Here, k = k₀n₁sinθ _i is the wavevector in-plane projection, θ _i is the angle of incidence, k₀ = ω/c, and θ₀ = 2/(w₀k₀n₁) is the angular spread of an incident Gaussian light beam of waist w₀. The p-wave reflection coefficient is written as R p = | R p | e i φ p in the complex exponential form. Both shifts can be seen being spatially dispersive, with Δ_GH being sensitive to reflectivity phase jumps and Θ_GH to zero reflection itself so that large effects are highly likely for all kinds of zero reflection modes in the system. Phase jumps and singularities make the phase ill-defined, in which case the reflection coefficient absolute value must go to zero for causality reasons. In nonlocal materials such as our TD plasmonic films, phase jumps and singularities come from material spatial dispersion in addition to that of the light beam itself. This is precisely what we study here. The respective extra terms are derived for the structure shown in Figure 1 and can be found in Appendix A. Expressions similar to Eqs. (1) and (2) can be written for s-polarized waves as well. However, as they show no peculiarities such as those we are about to discuss, we leave them out (see Appendix A). The derivations of relevance can also be found in Refs. [2], [37].

Figure 1:

GH shifts Δ_GH and Θ_GH with TiN plasmonic slab. A detector placed a distance l from the slab surface measures the shift Δ_total = Δ_GH + l tan(Θ_GH) ≈ Δ_GH + l Θ_GH.

3 Confinement-induced nonlocal electromagnetic response

The electrostatic Coulomb field produced by confined distance-separated charge carriers outside of their confinement region starts playing a perceptible role with confinement size reduction [33], [38]. The Coulomb interaction of charges confined is stronger than that in a homogeneous medium with the same dielectric permittivity constant due to the increased field contribution from outside dielectric environment, given that it has lower dielectric permittivity. That is why to describe the optical properties of TD plasmonic films we use the confinement-induced nonlocal EM response theory built on the Keldysh–Rytova (KR) electron interaction potential [22], [23]. This theory applies nicely to our configuration of an optically dense metallic material slab (plasmonic film) in region 2 of thickness d surrounded by semi-infinite dielectrics of constant permittivities ϵ₁ (top) and ϵ₃ (bottom), to result in the in-plane EM response of medium 2 (confined region) as follows

Here, ϵ _b  (≫ϵ₁, ϵ₃) is the constant background permittivity of the optically dense plasmonic material of the film, Γ _D is its damping constant, and its plasma frequency

is nonlocal (dependent on the in-plane electron momentum k = k₀n₁ sinϑ _i ) due to the vertical electron confinement, turning in the limit of d → ∞ into ω p 3 D = 4 π e 2 N 3 D / ϵ b m * of the standard local (Drude) EM response of 3D metals with electron effective mass m* and volumetric electron density N_3D [22]. More precisely, if ϵ ̃ k d ≫ 1 (relatively thick film), then ω p = ω p 3 D . If ϵ ̃ k d ≪ 1 (thin enough film), then ω p = 4 π e 2 ϵ ̃ N 2 D k / ϵ b m * with N_2D = N_3Dd being the surface electron density, consistent with plasma frequency of 2D electron gas sandwiched between top (ϵ₁) and bottom (ϵ₃) dielectric materials. Films with ϵ ̃ k d = ϵ ̃ d k 0 n 1 sin ϑ i ≲ 1 are referred to as TD films here, so that even relatively thick films can be in the nonlocal TD regime if ϑ _i is small enough.

This theoretical model is verified experimentally [13], [14], [15], [16], which is why we choose to set up

in our numerical studies, with TiN material parameters taken from experimental work [15] and collected in Table 1. The beam waist value is taken from Ref. [37], where it was used to simulate optical reflection processes. Note that Γ _D starts increasing rapidly for decreasing d ≲ 10 nm [15]. However, for the range of d used here, it is equal to the bulk value presented in the table.

With Eqs. (3)–(5), the derivative ∂ϵ₂/∂k in the nonlocal reflection coefficient expressions above reads as follows

It can be seen not only being nonzero at finite d but also being both positive and negative depending on the frequency and direction of the incoming light beam. It disappears for both d → 0 and d → ∞ as it should to indicate the absence of plasmonic material and to make the EM response of thick plasmonic films local in accord with the standard Drude model, respectively.

4 Reflection singularities

For a free standing plasmonic film of thickness d in air (Figure 1), the p-polarized wave reflection coefficient is [39]

with medium 1 (superstrate) and medium 3 (substrate) having the same permittivities ϵ₁ = ϵ₃ = 1. For medium 2 (film), we use ϵ₂ = ϵ_TiN taking a TiN example of TD material that surpasses noble metals such as Au and Ag [40]. The latter have exceptional plasmonic properties but relatively low melting temperatures making them incompatible with semiconductor fabrication technologies. On the contrary, transition metal nitrides have low-loss plasmonic response, high melting point, and structural stability that makes them capable of forming stoichiometrically perfect TD films down to 1 nm in thickness at room temperature [13], [15]. The Fresnel reflection coefficients r p i j (i, j = 1, 2, 3) for interfaces between medium 1 and 2 and between medium 2 and 3 are defined as follows

where γ i = k 0 2 ϵ i − k 2 are the wave vectors components normal to the interface. Here, r p 23 = − r p 12 as ϵ₁ = ϵ₃ = 1, in which case zeroes of R _p are determined by the Brewster mode (BM) condition r p 12 = 0 at the film–air interface, whereby γ₁ϵ₂ = γ₂ϵ₁, leading to the dispersion relation

Also, zeros of R _p can come from the film standing wave (SW) condition, 1 − exp(2iγ₂d) = 0, in which case

Here, n = 1, 2, 3, … and ϵ TiN ′ = Re ϵ TiN . Lastly, zero reflection can also occur at the Christiansen point (CP) where ϵ TiN ′ = 1 . If one uses the local Drude model (a “workhorse” routinely used in plasmonics), then ω_CP = 4 × 10¹⁵ rad/s comes out of it for any angle of incidence, whereas in the nonlocal KR model used here, the confinement-induced nonlocality of the EM response function ϵ_TiN(ω, k) makes the CP depend on k and thus on the incidence angle.

Figure 2 shows the inverse reflectivity function 1/|R _p |² and reflection phase φ _p /π calculated from Eqs. (7) and (8) for the 40 nm thick free standing TiN slab with nonlocal EM response. The 40 nm thickness is chosen here and below as an example to show the GH effect enhancement in the visible range; otherwise, it shifts to either IR or UV as can be seen from Eq. (52) (Appendix B.1). Reflectivity reduction and phase jumps can be seen when the SW, BM, or CP mode is excited in the system. The BM still exists below the bulk TiN plasma frequency ω p 3 D = 3.8 × 1 0 15 r a d / s , making the GH shifts observable for ω < ω p 3 D , including the visible range that is impossible to access with local Drude-like plasmonic materials. Figure 3, calculated from Eqs. (1) and (2) for the same example, shows strong lateral and angular GH shifts when the BM is excited, up to ≈ 80  μm and ≈ 30 m r a d , respectively. TD plasmonic materials thereby open access to the GH effect observation with visible light due to their remarkable property of the confinement-induced nonlocal EM response.

Figure 2:

Inverse reflectivity |R _p |⁻² (a) and reflection phase φ _p /π (b) for the 40 nm thick free-standing nonlocal TiN film. Shown are the (n = 1)-SW (upper dashed line), the nonlocal CP (lower dashed line), and the BM (solid line).

Figure 3:

|Δ_GH| in μm (a) and |Θ_GH| in mrad (b) calculated for p-polarized light wave incident on the 40 nm thick TiN film with inverse reflectivity and reflection phase shown in Figure 2.

For TD plasmonic films sandwiched between superstrates and substrates with ϵ₁ ≠ ϵ₃, the in-plane reflection symmetry is broken and top-bottom interface mode degeneracy is lifted. Figure 4 shows the birth and development of phase singularities in this case as with ϵ₁ = 1 (air) the substrate permittivity rises up to ϵ₃ = 3 (MgO typically used in TiN thin film systems [15]). It can be seen that just a slight substrate–superstrate dielectric permittivity difference gives birth to the two phase singularities close to the BM phase jump. They have opposite winding numbers (topological charges)

where α is a closed integration path around the phase singularity point. There is also another phase singularity on the SW branch, which moves for larger substrate permittivities close to the SW and BM crossing point. Such singularities result in zero reflection previously reported as “points of topological darkness” for specially designed metasurfaces [41], [42] and multilayer nanostructures [43], [44]. Here, we observe these topologically protected singularity points in mere nonlocal TD films. Remarkably, though, due to the plasma frequency decrease with thickness d in our case [15], [22], the gap in Figure 4 between the low-frequency opposite topological charge singularities widens in thinner films (not shown), to red-shift the lower- and blue-shift the higher-frequency singularity points, respectively. By reducing d in a controllable way in our case, it can, therefore, be possible to access the low frequency phase singularity point with He-Ne laser at λ = 632.8 nm (3 × 10¹⁵ rad/s) to observe an enhanced GH effect in the visible range. This remarkable feature is only offered by the TD plasmonic films due to their confinement-induced nonlocality and can never be realized with local Drude-like materials.

Figure 4:

Reflection phase φ _p /π calculated for a TiN film of d = 40 nm with varied ϵ₃ (substrate) and ϵ₁ = 1 (superstrate). Vertical line is the d = 0 BM, dashed lines are the first SWs of Eq. (10) with n = 0.5, 1, and solid line is the gBM of Eq. (14) (see text). Green circles mark the phase singularity points.

5 Discussion

To understand the origin of the topological singularities discussed, it is instructive to represent the condition R _p = 0 in the form

with dissipation temporarily neglected. In this case, the interface reflection coefficients are real numbers for all ω where ϵ₂(ω, k) > 0 (and γ₂ > 0 accordingly) so that our TD film behaves as a dielectric. The LHS of this equation makes a circle in the complex plane of radius r p 23 with phase θ = 2γ₂d, while the RHS is a real number varying between −1 and +1 as ω and θ change. This is why the equality can only be achieved if the LHS is a real number as well, yielding two cases for Eq. (12) to fulfill. They are (1) e 2 i γ 2 d = + 1 , r p 23 = − r p 12 and (2) e 2 i γ 2 d = − 1 , r p 23 = r p 12 . Here, the first equations are the SW condition 1 − e 2 i γ 2 d = 0 and its alternative 1 + e 2 i γ 2 d = 0 . Both of them lead to the same dispersion relation of Eq. (10), but the former for n = 1, 2, 3, … and the latter for n = 0.5, 1.5, 2.5, … describing SWs with multiples of a quarter wavelength and so to be referred to as standing quarter waves (SQW), accordingly. The second equation of case 1 is per Eq. (8) fulfilled if

This is the substrate–superstrate interface BM condition, which can only be realized hypothetically for d = 0, and so to be refereed to as zero-thickness BM (zBM). Similarly, the second equation of case (2) is fulfilled if

For ϵ₃ = ϵ₁, this reduces to the symmetric in-plane interface condition leading to the BM dispersion relation of Eq. (9). However, since it also holds for a broken in-plane reflection symmetry with ϵ₃ ≠ ϵ₁, we refer to the solution of this equation as the generalized BM (gBM).

The two simultaneous equations of cases 1 and 2 are represented by the lines in the 2D configuration space spanned by frequency and angle of incidence. The solutions to Eq. (12) are given by the intersection of these lines. This is where the phase singularity points come from that we obtain at certain frequencies and incident angles. Additionally, there is a singularity coming from the trivial solution of Eq. (12), to give case 3 where r p 23 = r p 12 = 0 , or r p 23 = − r p 12 and r p 23 = r p 12 simultaneously. This leads to the phase singularity at the intersection of the air–TiN and TiN–MgO interface BMs, or more generally, at the intersection of zBM and gBM lines in the configuration space. A complimentary analysis of the Christiansen points can be found in Appendix B.

Figure 5(a) shows the real and imaginary parts of Eq. (12) calculated for a thicker TiN film of d = 150 nm (to include more solution points) as functions of frequency for a few incident angles fixed. In Figure 5(b) that presents a projection of (a), solution cases 1, 2, and 3 discussed above are marked accordingly. Here, case 1 can be seen to generate an infinite number of discrete solution points for a single 60° incident angle as ϵ₁ and ϵ₃ in Eq. (13) are frequency independent constants. Case 2 yields the solution points at different incident angles 68.3°, 74°, 76° (not shown), etc. (also an infinite number, in principle) due to the strong gBM frequency dependence in Eq. (14). The case 3 solution point can be seen at the intersection of the two lines of the 60° incidence angle, to yield the phase singularity point that can be shifted down to the visible ω for thinner films (see below) due to the confinement-induced nonlocal EM response.

Figure 5:

Graphical illustration of the solutions of Eq. (12). (a) Real and imaginary parts of the LHS (solid lines) and RHS (dashed lines) of Eq. (12) for a few incidence angles. (b) Projection of (a) on the plane spanned by ω and real part. Angles are chosen to show the three solution cases to yield phase singularities in an air/TiN/MgO structure with d = 150 nm. Green circles mark the phase singularity points.

The impact of phase singularities on the GH shifts is shown in Figure 6 for the air/TiN/MgO system of the 40 nm thick TiN film with dissipation neglected. Comparing to Figure 4(b), the GH shifts can be seen to be very pronounced at the phase singularity points, which move with dissipation just slightly (not shown), for all three cases discussed. Using thinner TD films with broken in-plane symmetry, it is even possible to make these GH shift singularities observable in the visible range under He-Ne laser excitation (ω = 3 × 10¹⁵ rad/s). Figure 7 shows the GH shifts under such excitation with angle of incidence varied around θ _i = 62.42° for the same air/TiN/MgO system (dissipation included) with TiN film thickness varied around d = 31.7 nm to encounter the case 3 phase singularity point. For d slightly thicker or thinner than that the singularity shows up slightly red- or blue-shifted, depending on its topological charge, making Δ_GH of Eq. (1) change its sign as d decreases. Similarly, the sign of Θ_GH of Eq. (2) changes as the incidence angle varies around θ _i = 62.42° of the case 3 phase singularity point. Most important though is that both lateral and angular GH shifts shown are extremely large, in the millimeter and a few tens of milliradian range, respectively, over an order of magnitude greater than those shown in Figure 3 for a similar in-plane symmetric TD film system.

Figure 6:

Absolute values of lateral (a) and angular (b) GH shifts |Δ_GH| (μm) and |Θ_GH| (mrad) calculated neglecting dissipation for the air/TiN/MgO system with 40 nm thick TiN film. Green (red) dashed line is the SW (SQW) with n = 1 (n = 0.5) of Eq. (10). Vertical orange dashed line is the zBM of Eq. (13). White dashed line is the gBM from Eq. (14). Circles indicate the three cases of phase singularity points.

Figure 7:

Lateral (a) and angular (b) GH shifts Δ_GH (mm) and Θ_GH (rad) calculated for the nonlocal dissipative air/TiN/MgO system with TiN thickness in the vicinity of d = 31.7 nm and θ _i = 62.42° of the He-Ne laser beam (case 3 phase singularity point).

6 Conclusions

To summarize, we show that the TD plasmonic films with broken in-plane reflection symmetry can feature an extraordinarily large GH effect in the visible range due to the topologically protected phase singularities of their reflection coefficient. We classify the phase singularities and provide a detailed analysis for the conditions under which they emerge in the simplest possible system of a single TD plasmonic film (TiN) deposited on a low-permittivity dielectric substrate (MgO) in air. We emphasize that the existence of a phase singularity for a single plasmonic film in the visible stems from the confinement-induced EM response nonlocality (in-plane momentum dependence) of the system, with plasma frequency decreasing as the film gets thinner [15]. We note also that this confinement-induced nonlocality comes as a product of thickness by the in-plane momentum. So, the nonlocal plasma frequency can be red-shifted (and thus brought to the visible) by simultaneous reduction of thickness and in-plane momentum through the incident angle change of the incoming light beam. It is this that provides such a remarkable opportunity to have the strong GH effect in the visible range for not very thin (31–32 nm, Figure 7) TiN films, which can be routinely fabricated in the lab and are considered here theoretically as a demonstrative example. This is not possible for conventional (about an order of magnitude thicker) thin metallic films due to their local EM response with plasma frequency being a constant in the near-UV.

Previous studies reported an overall GH shift of 325 μm by coupling incident light to a surface plasmon polariton of a plasmonic film through the use of a prism [36], as well as lateral and angular GH shifts as large as 70 times 785 nm incident wavelength and 200 μrad, respectively, for artificially designed hybrid multilayer metasurface structures [45]. In contrast, our analysis here reveals even greater lateral and angular GH shifts ∼ 0.4  mm (632 times the wavelength) and ∼ 40  mrad for visible He-Ne laser light with simple TD plasmonic films, where Nature itself does the job to greatly enhance the GH effect, indicating that such systems could provide a new flexible quantum material platform to offer new opportunities for quantum optics, quantum computing, and biosensing application development.