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Abstract
Diffraction of light of a visible spectral range by subwavelength metal gratings has been investigated experimentally and theoretically using rigorous electromagnetic calculations. It has been demonstrated that an effect of surface plasmon resonance (SPR), at which total absorption of light by metal grating can be observed, occurs under certain conditions. Large positive and negative Goos-Hänchen (GH) shifts occur near the SPR. It has been shown that the reflected beam is split into two parts, the relative powers of which depend on the incident beam width and the grating depth. The dependence of the GH shifts on the grating period and grating depth has been investigated for different incident beam widths. The high sensitivity of the GH shift on the incident angle of a light beam near the SPR has been demonstrated.
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1. Introduction
The optical properties of the gratings have been studied extensively by assuming incident fields being in the form of plane waves of infinite extent. However, the fields of light sources are spatially bounded. The excitation of surface plasmons by finite width beams incident on multilayered structure was considered in [1]. The beam reflected off an interface experiences spatial and angular shifts depending on its polarization and profile. The spatial displacements are the well-known Goos-Hänchen (GH) shift in the plane of incidence and the Imbert-Fedorov (IF) shift normal to the plane of incidence [2–4]. The angular shifts that depend on the beam profile were demonstrated in [5].
The lateral displacement of the beams incident upon dielectric and metallic structures was studied in the past. In [6] the first observation of the GH shift of a light beam incident on a bare metal surface was reported. Usually, the GH shift is of the order of the wavelength. Various ways to enhance the shifts were considered. The displacement of the order of the beam width was shown for multilayered structures in [7]. In [8], beam shifts or corrections with respect to geometrical optics caused by the wave effects in a graded-index optical fiber were investigated. The GH effect can be enhanced near the critical angle of total reflection, where the phase of the reflection coefficient changes significantly. The excitation of the surface plasmon-polaritons (SPPs) in the Otto or Kretschmann configuration has a strong effect on the GH shift. When a beam is reflected from a transparent medium the displacement is usually positive and in the forward direction. However there are negative GH shifts in various systems. It was shown in [9] that large lateral optical beam displacements occur at the reflection from a metal (silver) surface due to surface plasmon resonance. A large positive and negative lateral beam displacement is observed on the silver–air interface when the surface plasmon resonance is properly excited [9]. As described in [7], such a negative displacement is due to the backward leaky wave with an opposite sign for the propagating and attenuation constant. In [10] the enhanced spatial and temporal shifts in the reflection of Gaussian wave packets from two-dimensional photonic crystal waveguides are demonstrated theoretically. It is shown that the enhanced GH shift in this type of structures can be related to the imaginary part of the propagation constant of the excited leaky mode.
The enhancement of GH and IF shifts near the SPR angle was also shown for a Laguerre-Gaussian (LG) incident beam with an orbital angular momentum (OAM) [11]. Note that the OAM affects the GH and IF shifts [12–15]. The tunable OAM-dependent spin (polarization) splitting by transmitting higher-order LG beams through graphene metamaterials was demonstrated theoretically in [12].
Large lateral displacements of beams incident upon a diffractive structure under resonance conditions can be also observed [16]. Large GH effects are also expected for dielectric gratings [17]. The plasmon enhanced GH shift of the order of 70 times the wavelength of light incident at a polymethyl methacrylate (PMMA) grating on a gold film was demonstrated experimentally in [18].
The effects of the incident beam width and subwavelength grating height on the plasmon enhanced GH shifts have not been studied before.
In this paper, we present the investigation of the beam displacement at visible wavelengths when the surface plasmon resonance is excited in subwavelength grating. A lateral displacement of the order of the beam width for the reflected beam is demonstrated near the surface plasmon resonance. The reflected beam is split into two beams, the relative powers of which depend on the incident beam width and the grating depth. We also show that the strong negative beam shift can be observed when the grating depth is larger than a critical depth.
2. Plasmon resonance
The momentum conservation for an optical wave exciting a surface plasmon wave in a diffraction grating is expressed as [19]:
2.1 Simulation results
The theory of light diffraction on gratings has now been developed quite fully (see, e.g., [20]). Since the scalar diffraction theory cannot be applied to subwavelength gratings [21], a rigorous electromagnetic theory [22] based on the C- method [23,24] and the rigorous coupled wave analysis (RCWA) method [25] are used for calculations. Since the s- wave does not excite SPWs, the obtained results give the GH shift of the p- polarized incident wave only.
The diffraction efficiency of the grating is affected by a number of factors: duty cycle, microrelief shape and depth, grating material, angle of incidence, wavelength, and polarisation. In [26,27] the diffraction of light of a visible spectral range by subwavelength metal gratings using the RCWA and C-methods was investigated.
In Fig. 1 the diffraction efficiencies in the zero order of silver sinusoidal grating with a period $\Lambda $ = 400 nm and depths h = 50 nm (a) and h = 20 nm (b) depending on the angle of incidence of radiation of TM polarization with a wavelength $\lambda = 632$ nm are presented. It can be seen that almost 100% of the incident energy is absorbed by a silver grating with a depth of relief h = 20 nm. Nonzero reflectivity exists at the grating depths higher and less than h = 20 nm. Similar phenomenon with shallow metallic gratings was demonstrated earlier in [28].
Fig. 1. Diffraction efficiency of a silver sinusoidal gratings with a period $\Lambda $ = 400 nm depending on the incidence angle of radiation.
Figure 1 shows that the efficient excitation of plasmon waves occurs only in a narrow region of the resonance angle of incidence of radiation. Besides, there is a critical grating depth at which effective excitation of surface plasmon occurs. In our case, the resonance angle is ${33.25^ \circ }$ and the critical grating depth is 20 nm. At these parameters, the reflectance disappears completely due to the destructive interference of propagating and surface plasmon waves. Note that the same angle follows from the resonance condition (1) at m = −1. This means that the plasmon wave is excited in −1st diffraction order and runs backwards.
2.2 Experiments
Original gratings were made using electron beam lithography and etching in a polymethylmethacrylate (PMMC) film deposited on a chromatised glass substrate using spin-coating technology [29]. Figure 2 shows an image [top view – (a), grating profile – (b) and 3D image – (c)] of the grating with a period $\Lambda $ = 400 nm, obtained with an atomic-force microscope (AFM). As follows from the AFM measurements, the shape of the grating profiles is well described by the trapezoidal model. Using an optical power meter, the diffraction efficiency was measured as a function of the angle of incidence of the light beams of red, green, and blue colours (R, G, B) emitted by lasers and laser diodes with a power from 5 to 150 mW and diameters of the beams from 1 to 3 mm [22,23].
Fig. 2. Images of gratings obtained with Smena and NT-MDT AFMs: (a) top view, (b) grating profile, and (c) 3D image ($\Lambda $ = 400 nm).
In Fig. 3 the results of calculations and measurements of diffraction efficiency of the zero order of the nickel grating depending on the angle of incidence of the radiation with p- polarization are presented. It is seen that the effect of plasmon resonance at the incidence angle of ∼ 33° occurs [Fig. 3(a)]. At a relief depth of h = 80 nm and angle of incidence ${\theta _i}$ = 31°, almost all incident energy is absorbed by the grating [Fig. 3(b)]. Note that the measured angles of incidence at which the plasmon resonance effect occurs are in good agreement with the calculations.
Fig. 3. Calculated (solid lines) and measured diffraction efficiencies of the zero order depending on the angle of radiation incidence for the nickel grating with a period $\Lambda $ = 400 nm and depth h = 40 nm (a) and h = 80 nm (b) at a wavelength of radiation $\lambda $ = 641 nm with p- polarization.
2.3 Influence of incident beam width
Below the simulation results of diffraction efficiencies of the reflected Gaussian beams are presented. The incident beams are represented by their Fourier spectra so that the reflected field can be evaluated by integrating the corresponding functions in the complex spectral plane. This indicates that the diffraction of plane waves (Fourier components) incident at different angles is calculated at first. Thereafter the reflected beam is constructed by inverse Fourier transform from the calculated reflected/diffracted components. Detailed description of the method of a calculation of the diffraction of a finite Gaussian light beam by a finite grating is presented in [22,30].
Consider a p- polarized Gaussian beam with an electric field in the plane of propagation (Fig. 4). The incident electric field at the interface of the grating can be expressed as
where w0 is the beam waist, $\theta$ is the incident angle of the beam.In Fig. 5 diffraction efficiencies C0 as a function of incidence angle for different values of the Gaussian beam width w0 are presented. It follows from the calculations that the influence of the incident beam width on the diffraction efficiency is significant for the incident angles at which the plasmon resonance occurs. The plasmon resonance effect is greatly reduced for strongly focused beams because the most of the angular spectrum components of the incident beam is not phase matched to surface plasmon modes. It was shown in [1] that the incident beamwidth should be larger than the propagation range of the surface plasmon mode to couple energy effectively into the plasmon field. Indeed, the dip in the reflection spectrum begins to blur when the beam size becomes comparable to this characteristic length. Note that this length depends on the material, the grating period and the grating depth (Table 1 and Table 2). The resonance width at the resonance absorption on the plasmon is determined by the losses in the plasmon. The larger the losses, the wider the resonance, and the weaker the effect of beam width.
Fig. 5. Diffraction efficiencies C0 for Ag gratings depending on the angle of incidence for different values of the Gaussian beam width w0, $\lambda$ = 632.8 nm.
Table 1. Losses and characteristic lengths of plasmons lp in gratings with different periods and a depth h = 30 nm .
Table 2. Losses and characteristic lengths of plasmons lp in Ag gratings with different depths, Λ = 400 nm.
In Fig. 6 the diffraction efficiencies C0 as a function of wavelength for different values of the Gaussian beam width w0 are presented. The incidence angles $\theta$ correspond to the plasmon resonance for the plane wave.
Fig. 6. Diffraction efficiences as function of wavelength for silver grating with a period $\Lambda $ = 400 nm and a depth h = 30 nm. $\theta = {33.25^ \circ }$, $\lambda$ = 632.8 nm.
The characteristic lengths lp are defined by the plasmon losses, i.e ${l_p} \approx {1 \mathord{\left/ {\vphantom {1 \alpha }} \right.} \alpha }$. It is seen (Figs. 5 and 6) that the dip in the reflection spectrum begins to fade when the beam size w0 decreases and becomes comparable to the plasmon characteristic length lp. The characteristic length decreases with increasing grating depth (Table 2).
Note, the losses include both absorption and radiation losses which are related to the intrinsic and radiative damping.
3. Goos-Hänchen shift
The GH shift is affected by a number of parameters of the grating and incident beam: the grating depth and period, grating material, angle of incidence, wavelength, and polarization. The center of the incident Gaussian beam is located at zero (Fig. 4). The incidence angles are chosen from the minimum of the reflectance for the plane wave. At these angles the surface plasmon modes are excited efficiently and large GH shifts can be observed. Spatial shifts $\Delta x$ are determined by the position of the maximum values in the intensity profiles of the reflected beams. Total power of the incident beam is normalized to unity. The reflectivity decreases with the increase of the beam width approaching to zero for plane wave case. The reflectance curve varies very rapidly if the incident angle is phase matched to surface plasmon (Fig. 1), hence the distortion of the reflected beam can then become significant.
3.1 Influence of Gaussian beam width
In Fig. 7 the intensity profiles of reflected beams for the Ag gratings with a period of $\Lambda $ = 400 nm and depths h = 20 nm (a), h = 30 nm (b) and h = 40 nm (c) for incident beams with different waists w0 are presented.
Fig. 7. Intensity profiles of the reflected beams from silver gratings with the depths h = 20 nm (a), h = 30 nm (b) and h = 40 nm (c) for different values of the incident beam width w0. $\lambda = 632.8$ nm, $\theta = {33.25^ \circ }$.
It is seen that the GH shift decreases significantly when the incident beam width decreases. Besides, the reflected beam is split into two parts with positive and negative shifts, accordingly. Such positive and negative spatial shifts indicate the existence of forward and backward surface propagating energy flows along the grating-air interface at resonance. Distortions of the reflected beam become noticeable near the critical depth of the grating ${h_{cr}} \approx 20$ nm even for incident beams with large waists [ Figs. 8(a) and 8(b)]. However, when moving away from the critical depth of the grating, distortions disappear for highly focused beams as well [Figs. 8(c) and 8(d)].
Fig. 8. Intensity profiles of the reflected beams from silver gratings with the depths h = 20 nm (a, b) and h = 40 nm (c, d).
In Fig. 9 the intensity profiles and phase distributions of the reflected beams are presented for different grating depths and incident beam waists. For gratings with a depth noticeably different from the critical one, the reflected beam retains the shape of the incident beam [Fig. 9(a)]. In addition, the phase in the cross section of the reflected beam changes slightly [Fig. 9(b)]. However, close to the critical grating depth the reflected beam is distorted significantly. It is worth noting that splitting a beam into two parts occurs when there is a sharp change in the phase value and its sign of the reflected beam [Fig. 9(d)]. This may indicate the existence of forward and backward surface propagating energy flows along the grating-air interface.
Fig. 9. Intensity profiles (a, c) and phase distributions (b, d) of the reflected beams: h = 40 nm (a, b) and h = 25 nm (c, d).
3.2 Influence of grating height
In Fig. 10 the intensity profiles of the reflected beam for different grating depths at the incidence angle at which surface plasmons are excited are presented. The lateral shift of the reflected beam changes from positive values to negative ones as the grating depth increases through a critical value corresponding to the minimum reflection for the plane wave. For the silver grating with the period $\Lambda $ = 400 nm, the critical depth is equal to hc ${\simeq}$ 20 nm. It is seen from Fig. 10 that near the critical grating depth the reflected beam is split into two separate beams with the relative powers depending on the beam width. The powers of spatially separated beams become equal at some value of the incident beam width. Note that two beams have displacements with different signs.
Fig. 10. Intensity profiles of the beams reflected from the silver gratings with different depths for the incident beam widths w0 = 12 µm (a) and w0 = 40 µm (b). $\lambda = 632.8$ nm; $\theta = {33.25^ \circ }$.
The distance between the peaks of these two separate reflected beams increases with the width of the incident beam.
The similar effects exist for the gratings with other grating periods and materials. We have analyzed the nickel and aluminum gratings with the periods of 450, 500, 550, and 600 nm. We have obtained that the surface plasmon resonance occurs at different incident angles and grating depths in these cases.
In Fig. 11 the spatial shifts as function of the grating depth for different incident beam waists are presented. It can be seen that the GH shift changes its sign as the grating depth increases through the critical value corresponding to the minimum reflection of the plane wave at the resonance incidence angle. For a silver grating, this value is approximately 20 nm [Fig. 1(b)].
Fig. 11. Spatial shifts Δx as function of the grating depth for different incident beam waists: w0 = 40 µm - curve 1, w0 = 60 µm – curve 2. $\lambda = 632.8$ nm; $\theta = {33.25^ \circ }$
3.3 Influence of incidence angle
In Fig. 12 the intensity profiles of the reflected beams for different incident angular ranges: near the SPR, slightly smaller, and slightly larger than the resonance angle are presented.
Fig. 12. Intensity profiles of the beams reflected from the silver grating at different values of the incident angle θ. $\Lambda $ = 400 nm, h = 20 nm, $\lambda = 632.8$ nm, (a) w0 = 12 µm; (b) w0 = 25 µm.
As shown in Figs. 12(a) and 12(b), the beam shift and beam profile are sensitive to the incident angle. The reflected beam retains its Gaussian shape at the angles slightly smaller and larger than the resonance angle. However, close to the surface plasmon resonance, the reflected beam is split into two separate beams with large negative and positive shifts, acoordingly.
In Fig. 13 the intensity profiles and spatial negative GH shifts are presented for different angles of incidence for the beam waist w0 = 40 µm and the grating depth h = 30 nm.
Fig. 13. Intensity profiles (a) and spatial shifts Δx as function of the incident angle (b): w0 = 40 µm; $\lambda = 632.8$ nm; h = 30 nm.
It is seen that negative GH shift of the order of the beam width occurs near the resonance angle of incidence.
In Fig. 14 the intensity profiles and spatial positive GH shifts are presented for different angles of incidence for the beam waist w0 = 60 µm and the grating depth h = 15 nm.
Fig. 14. Intensity profiles (a) and spatial shifts Δx as function of the incident angle (b): w0 = 60 µm; $\lambda = 632.8$ nm; h = 15 nm.
4. Discussion and conclusion
We have examined the influence of the surface plasmon excitation on the GH effect when a Gaussian beam is incident on a metallic subwavelength grating by using rigorous electromagnetic calculation and analysis.
The influence of varying the grating depth and the incident beam width on the GH effect was studied. We identified both the positive and the negative GH shifts around the incidence angle of resonance.
It was shown that, for an incident Gaussian beam, the location of maximum power density on the grating surface shifts with respect to the center of the incident beam by a distance of the order of the beam width. Note that the reflected beam becomes strongly distorted when the incident angle is close to that associated with the plasmon mode. Near the angle of incidence at which surface plasmons are excited, the GH shift of the reflected beam changes from positive to negative values as the grating depth increases through a critical value corresponding to the minimum reflection for a plane wave. The similar effect was observed in [9] at a metal-air interface when the surface plasmon resonance of the metal was excited. The optimal thickness corresponds to the excitation of the surface plasmon, where the resonance absorption is just balanced by radiation damping and internal damping.
We have found that the lateral shift of the reflected beam depends very sensitively on the incidence angle. It was shown that the large positive and negative lateral beam shifts of the order of the beam width occur near the surface plasmon resonance. We have found that as the width of the incident Gaussian beam decreases, the size of the lateral shift decreases sharply. On the other hand, as the beam width increases at the SPR, the reflected beam power decreases rapidly. Therefore, the maximum displacement that can be recorded depends on the sensitivity of the measurement instrument.
Enhancement of the lateral shift with the increase of the beam width can be phenomenologically explained by the increase of coupling efficiency of the incident energy into the plasmon field. Note that the incident beamwidth should be larger than the propagation range of the surface plasmon mode to couple energy effectively into the plasmon field. Indeed, the plasmon resonance effect occurs when the incident beam width becomes larger than the characteristic length lp. We showed (section 2.3) that the plasmon resonance effect is greatly reduced for strongly focused beams because the most of the angular spectrum components of the incident beam is not phase matched to surface plasmon modes. There is a close analogy between GH shifts in grating and prism-waveguide systems. It was found in [31], that the magnitude of the beam shift in prism-waveguide coupling system is closely related to the intrinsic and radiative damping. When the intrinsic damping is larger than the radiative damping, negative lateral beam shift occurs.
It was shown that the reflected beam is split into two spatially separated beams with large negative (backward) and positive (forward) lateral shifts near the surface plasmon resonance. The distance between the intensity peaks increases with the beam width.
Thus, the large positive and negative lateral beam shifts were demonstrated for the subwavelength gratings near the surface plasmon resonance using rigorous electromagnetic calculations. The influence of varying the grating depth and the incident beam width on the GH effect near the SPR, slightly smaller and slightly larger than the resonance angle was investigated. The lateral displacement of the order of the beam width for the reflected beam is demonstrated near the surface plasmon resonance. We also showed that the large negative beam shift can be observed when the grating depth is larger than the critical depth corresponding to the minimum reflection power of an incident plane wave.
Funding
Russian Foundation for Basic Research (19-29-11026); Russian Science Foundation (17-19-01461).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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