1. Introduction

Since the introduction of optical vortex beams in 1989 [1], they have attracted significant attention from many research groups due to their remarkable properties. Laguerre Gaussian (LG) beams as an example of vortex beam possess both orbital angular momentum (OAM) arising from their twisting phase structure and spin angular momentum (SAM) originating from their polarization [2,3]. The angular dependence of LG beams is captured by the term $\exp({il\varphi } )$, where $\varphi $ represents the azimuthal angle and l is an integer known as the topological charge (TC) of the beam. The product $l\hbar $ represents the orbital angular momentum of each photon in the beam, with $\hbar $ denoting the reduced Planck constant. The quantized orbital angular momentum of LG beams provides a complete set of bases analogous to Hilbert space. Similar to quantum mechanics, arbitrary beams can be decomposed to LG beams with different topological charges (TCs). This property makes LG beams highly suitable for communication and quantum information processing applications [4–8]. A prominent characteristic of LG beams is the presence of a phase singularity at the beam's origin, resulting in a central dark region surrounded by a bright ring. This unique intensity profile, resembling a doughnut shape, is well-suited for various applications such as trapping and optical tweezers [9–11], super-resolution imaging and microscopy of delicate microstructures and biological samples [12,13], as well as weather monitoring and atmospheric studies [14,15].

When a beam with orbital angular momentum is reflected from a dielectric or metal interface, due to wave optics effects, the spatial distribution of the reflected beam undergoes some modification. This effect can be illustrated by spatial Fresnel coefficients corresponding to orbital angular momentum or transverse modes similar to polarization Fresnel modes. It was both theoretically and experimentally demonstrated that the state of orbital angular momentum of the incident beam was modified upon reflection and new orbital angular momentum modes were generated in the reflected beam [16]. The strength of the created sidebands was sensitive to the angular spread of the incident wave and reflectivity of the structure illuminated by the beam [16]. Zhu et. al. indicated that when a pure vortex beam with topological charge m transmitted through a thin slab made of epsilon near zero anisotropic metamaterials, new (m + 1) and (m-1) sidebands were generated [17]. In another work, Loffler et. al. demonstrated that the orbital angular momentum sidebands of the reflected beam under total internal reflection strongly depend on the topological charge of the incident beam as well as its waist [18].

Surface plasmon polaritons (SPPs) arising from the strong coupling between photons and collective oscillations of conduction electrons can be propagated at the metal-dielectric interfaces [19]. Their unique properties, such as confinement and guiding the light bellow diffraction limit make them suitable for manipulating the light-matter interactions at small scales. Noble metals like gold and silver are usually used for supporting SPPs in the visible and infrared regions. However, graphene is a promising candidate to observe plasmonic effects in the mid infrared and terahertz region due to its tunability with a voltage gate or applying an external magnetic field [20]. In recent years, the interaction of LG beams with plasmonic structures containing noble metals or graphene layers has been investigated. For example, authors studied Goos-Hänchen (GH) and Imbert- Federov (IF) effects for a LG beam in a Kretschmann configuration composed of a glass prism, an Au thin film, and a liquid analyte [21]. They found that excitation of SPPs through the Au layer led to the enhancement of both spatial shifts. Furthermore, as the beam waist increased, both GH and IF shifts increased and then approached to asymptotic values. In another work, Zhu et. al. reported the enhancement of spin splitting by transmission of higher-order LG beams though a metamaterial consisting of graphene layers [22]. They found that spin splitting could be tuned by varying the OAM of the incident beam as well as its waist. Furthermore, the Fermi energy of graphene provided another tool for tunning of spin splitting. Authors in Ref. [23] examined the reflection of a purely azimuthal LG beam from a three-layered structure consisting of air, hexagonal boron nitride and metal in the presence of a graphene single layer. They found that due to IF effect for the m-mode LG beam, (-m + 1) and (-m-1) sidebands were generated upon reflection. By varying the Fermi energy of graphene, it was possible to change the IF shift from negative to positive values leading to the suppression or enhancement of the sidebands. In another paper, Xiao et. al. found that when a LG beam was incident on an array of graphene ribbons as a hyperbolic meta-surface, the reflected beam acquired orbital angular momentum sidebands whose relative intensity depended on the Fermi energy of graphene, the external magnetic field and the wavelength of the incident wave [24]. Furthermore, they indicated that with increasing the topological charge of the incident wave, the intensity of generated sidebands in the reflected beam was enhanced.

Here, in the present study, we calculate the reflected beam profile of a LG beam employed for excitation of SPPs through a graphene layer embedded in the Otto-configuration for the first time up to our knowledge in the both presence and absence of an external magnetic field. When there is no applied magnetic field, a vertical strip with null intensity emerges in the profile of reflected beam upon optimal excitation of SPPs. Then, we show that the dark strip in the profile of reflected beam translates as the system moves away from the optimal excitation of SPPs by varying the incident angle, the chemical potential of graphene and applying an external magnetic field. Next, OAM spectrum of the reflected beam is calculated for the case of optimal excitation of graphene SPPs and some discussions are made about the features of OAM spectrum. Furthermore, for the cases at which the system is far from the optimal excitation of SPPs the OAM spectrum is calculated and the characteristics of the OAM modes are expressed. Finally, in the presence of external magnetic field, for the optimal excitation of SPPs, the profile of reflected beam and OAM spectrum are calculated and compared with the case at which no external magnetic field exists.

The remaining parts of the present paper are organized as follows. In section 2, the proposed structure is described and the theoretical method is given. In section 3, the results and discussion are presented. Finally, the paper is finished in section 4 with some conclusions.

2. Designed structure and theoretical method

Otto-configuration in our calculations is a three-layer structure composed of a silicon prism, air gap and silica substrate which is deposited by a graphene single layer as shown in Fig. 1. It is assumed that an external magnetic field ${{\boldsymbol B}_0}$ is applied perpendicular to the graphene layer along the z direction. To excite SPPs through the graphene layer, the incident angle of the beam must be greater than the critical angle associated with the beam entering from the silicon prism to the silica substrate. In this situation, evanescent waves penetrating to the air-SiO₂ interface couple to the graphene charge carriers allowing the excitation of SPPs.

 

Fig. 1. A schematic representation of the Otto-configuration consisting of a Si prism, air gap, and a SiO₂ substrate with a graphene single-layer deposited on it for excitation of SPPs. The external magnetic field ${{\boldsymbol B}_0}$ is perpendicularly introduced to the graphene layer in the z direction. ${\theta _i}$ and ${\theta _r}$ denote the incident and reflected angles, respectively. The coordinate system $({{x_i},{y_i},{z_i}} )$ is used to denote the incident LG beam, while the coordinate system $({{x_r},{y_r},{z_r}} )$ corresponds to the reflected beam.

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To obtain the reflection profile of a LG beam incident on the structure depicted in Fig. 1, it is necessary to compute the Fresnel reflection coefficients. In the presence of a static magnetic field ${{\boldsymbol B}_0}$, the conductivity tensor of graphene exhibits non-diagonal elements, resulting in hybrid Fresnel coefficients. It is important to note that due to the small thickness of the graphene layer, we treat it as an infinitely thin surface with a surface current density. By employing the boundary conditions for electromagnetic fields, the Fresnel coefficients can be derived.

The matrix elements of the graphene conductivity tensor in the quantum regime are expressed as follows [25]:

By using the transfer matrix method derived in Ref. [25] based on boundary conditions, we can obtain the hybrid Fresnel coefficients corresponding to the structure shown in Fig. 1 as follows:

The vector potential of the incident LG beam in the coordinate system of the incident beam $({{x_i},\; {y_i},{z_i}} )$ depicted in Fig. 1 can be expressed as follows [26]:

Substituting Eq. (16) in Eq. (14) yields:

Finally, solving integral of Eq. (13) using Eq. (17) gives the vector potential amplitude of the incident LG beam in each arbitrary point of space $({{x_i},{y_i},{z_i}} )$ as follows:

By substituting Eq. (18) into Eq. (12), we obtain the incident vector potential ${{\boldsymbol A}_{\boldsymbol i}}({{x_i},{y_i},{z_i}} )$ at any given point in the space. Then, the electric and magnetic fields of the incident LG beam can be calculated in Lorenz gauge as:

In the similar way, the vector potential of the reflected beam in the coordinate system $({{x_r},\; {y_r},{z_r}} )$ shown in Fig. 1, can be written as:

The amplitudes $({\tilde{u}_r^H,\tilde{u}_r^V} )$ represent angular spectra of the reflected beam which are related to ones of the incident beam by the following matrix relation [27]:

In paraxial approximation, the Fresnel reflection coefficients of each plane waves constituting the beam can be written as:

with

Equations (34) and (35) are used in the next section to obtain the profile or intensity distribution of the reflected beam when the structure in Fig. 1 is illuminated by a LG beam with a topological charge l and p = 0.

Since the LG beams with integer values of l provides a complete set of bases analogous to Hilbert space, one can expand each arbitrary beam in terms of harmonic mode $\exp({il\varphi } )$. Hence, at a given $z = {z_0}$, the reflected beam can be decomposed as [28]:

The energy existing inside each LG mode is given by:

The energy of the electromagnetic field is proportional to square of electric field amplitude. Thus, the total energy of reflected beam is equal to $W = \mathop \sum \nolimits_l {W_l}$. Hence, the mode weight of each OAM state involved in the reflected beam can be expressed as:

Equation (40) is employed to obtain the mode weight in calculation of OAM spectrum of the reflected beam in the next section.

3. Results and discussion

3.1 Reflectance calculation of the structure under illumination with a plane wave

The Otto-configuration, as depicted in Fig. 1, consists of a three-layer structure with ${n_1} > {n_3} > {n_2}$. The layers used in the structure are silicon with refractive index ${n_1} = 3.46$, air with refractive index ${n_2} = 1$, and silica with refractive index ${n_3} = 2$. As shown in Fig. 1, the graphene layer is situated at the interface between air and silica. With respect to the incident angle ${\theta _i}$, we have three different regimes. When ${\theta _i} < {\theta _{c12}}$, the electromagnetic fields in the second and third media can propagate. For ${\theta _{c12}} < {\theta _i} < {\theta _{c13}}$, the fields in the second medium become evanescent while those in the third medium are propagating. Lastly, for ${\theta _i} > {\theta _{c13}}$, the waves in both second and third media are evanescent, that is, the total internal reflection takes place. ${\theta _{cij}}$ denotes the critical angle for the wave incident to the interface between medium i and medium j. Here, for the considered layers mentioned above, the critical angles are ${\theta _{c12}} = {16.8^{\circ}}$ and ${\theta _{c13}} = {35.3^{\circ}}$. To excite SPPs through the graphene layer in the proposed structure using a plane wave, we assume the incident angle to be ${\theta _i} > {\theta _{c13}}$. This is because the evanescent waves in the silica medium can couple with the charge carriers in graphene layer, allowing the excitation of graphene SPPs.

Using Eq. (3–6), one can calculate reflectance ${R_p} = {|{{r_{ps}}} |^2} + {|{{r_{pp}}} |^2}$ and ${R_s} = {|{{r_{sp}}} |^2} + {|{{r_{ss}}} |^2}$ for a plane wave incident onto the structure shown in Fig. 1. The corresponding results are displayed in Fig. 2 in terms of incident angle ${\theta _i}$ and frequency $\nu $ at the fixed chemical potential of graphene $\mu = 0.5\; eV$ and the air gap $d = 5\; \mu m$ for the cases B₀ = 0 and 1 T. ${R_p}$ represents the reflectance of the structure for a p-polarized incident plane wave while ${R_s}$ stands for a s-polarized incident wave. In Fig. 2(a), one can see the corresponding results for R_p in the absence of external magnetic field. As shown in this figure, the blue parabolic region of ${R_p}$ indicates the situations with near zero reflection corresponding to the excitation of SPPs within frequency range from 0.5 to 1.7 THz and incident angle range from ${45^{\circ} }to\; {80^{\circ} }$. In Fig. 2(a), the minimum reflectance occurs at ${\nu _m} = 1.29\; THz$ and ${\theta _i} = {60.4^{\circ} }$ corresponding to the optimal excitation state of graphene SPPs. In this case the coupling of energy from the incident beam to SPPs is maximum, so it is named as the optimal excitation state. Figure 2(b) presents reflectance R_s for the case of s polarized incident plane wave in the absence of external magnetic field. There is no region with near zero reflectance in this figure confirming impossibility of strong excitation of SPPs for incident s polarization in the considered frequency range and incident angles. This means that in the absence of external magnetic field ${{\boldsymbol B}_0}$, the phase matching condition for exciting SPPs through the graphene layer at considered chemical potential and frequency range is only satisfied for the p-polarized incident plane wave. It is well-known that for both S and P polarizations SPPs can be propagated in the graphene layer in the absence of magnetic field. However, in the proposed structure for the given parameters, the excitation of s-polarized SPPs thorough the graphene layer is not possible because the phase matching condition is not satisfied for this polarization.

 

Fig. 2. Reflectance plot corresponding to the structure shown in Fig. 1 under illumination with a plane wave versus incident angle ${\theta _i}$ and frequency $\nu $ at chemical potential $\mu = 0.5\; eV$ and air gap $d = 5\; \mu m$. (a) ${R_p}$ and (b) ${R_s}$ in the absence of magnetic field (c) ${R_p}$ and (d) ${R_s}$ in the presence of the external magnetic field ${B_0} = 1\; T$.

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When an external magnetic field is applied, the graphene layer behaves like anisotropic media resulting in a decrease in R_p, as shown in Fig. 2(c). Additionally, a weak orange parabolic region with ${R_s}$ close to 0.8 emerges in Fig. 2(d), indicating the excitation of SPPs with the s-polarized incident wave.

For more clarification, in Fig. 3, ${R_p}$ and ${R_s}$ are plotted as a function of frequency at different external magnetic fields ${B_0} = 0,\; \; 1,\; 2,\; 3\; and$ $4\; T$. In all curves of Fig. 3, it is assumed the parameters of $\mu = 0.5\; eV$, $d = 5\; \mu m$ and ${\theta _i} = {60.4^{\circ} }$ to be fixed. It is clearly seen in this figure that as the external magnetic field increases, the dip in the reflectance curve is blue-shifted for both p- and s-polarized graphene SPPs. Furthermore, as the external magnetic field increases, the value of reflectance at dip increases for the p-polarized incident wave while it decreases for the s-polarized incident wave. These effects indicate that in the presence of an external magnetic field, one can control the surface plasmon characteristics of the graphene layer without changing the structure geometry.

 

Fig. 3. Reflectance of the Otto-configuration depicted in Fig. 1 under illumination with a plane wave as a function of the incident frequency under different external magnetic fields ${B_0} = 0,1, \ldots ,4T$ for both s and p polarized incident waves at the same parameters of $\mu = 0.5\; eV$, $d = 5\; \mu m$ and ${\theta _i} = {60.4^{\circ}}$.

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3.2 Calculation of the reflected beam profile under illumination with a LG beam

Now we assume that the structure in Fig. 1 is illuminated by a p-polarized LG beam with the topological charge $l = 2$ instead of a plane wave. We first consider the case at which there is no external magnetic field and the LG beam with fixed frequency of ${\nu _m} = 1.29\; THz$ is incident onto the structure at different incident angles in the range from ${\theta _i} = 40^{\circ} $ to ${\theta _i} = 80^{\circ} $. This frequency corresponds to the deepest dip observed in Fig. 3 under illumination with a p-polarized plane wave when no static magnetic field is applied. The distribution of electric field intensity of the incident LG beam is calculated using Eq. (21) and the corresponding result is represented in Fig. 4(a). The central region with zero intensity in which phase singularity occurs is clearly seen in this figure. The intensity profiles of the reflected beam at different incident angles are calculated using Eq. (35) at far field distance ($z = 1000{\lambda _m}$ with ${\lambda _m} = c/{\nu _m}$) and the corresponding results are shown in Fig. 4(b). As shown in this figure, at incident angle ${\theta _{spp}} = 60.4^{\circ} $ in which the optimal excitation of SPPs for frequency of 1.29 THz in the absence of external magnetic field occurs under illumination of the plane wave (according to Fig. 2), the intensity profile of the reflected beam under illumination with the LG beam split into two crescent-shaped lobes separated by a vertical dark strip. For incident angles that are far from ${\theta _{spp}}$ such as ${\theta _i} = 40^{\circ} $ and ${\theta _i} = 80^{\circ} $, the profile of reflected beam is nearly similar to incident LG beam. The emergence of vertical black line in the profile of reflected beam upon excitation of SPPs was previously demonstrated experimentally for Au/air interface in the Kretschmann configuration at visible region [29]. In another work, Gaussian and Bessel–Gauss wave packets have been used to excite SPPs through an Ag/SiO₂/air system in the Kretschmann configuration. For p-polarization incident beam, a vertical dark strip appears in the profile of reflected beam which makes it possible the direct visualization of SPPs excitation [30]. However, up to our knowledge there is no work to theoretically calculate the profile of reflected beam upon excitation of SPPs in the visible and terahertz region. Therefore, the present work can open up an effective and new route for direct observation of SPPs excitation using the visualizing the reflected beam profile.

 

Fig. 4. (a) Electric field intensity profile of the incident LG beam with an orbital angular momentum value of $l = 2$. (b) Spatial distribution of the intensity of refleted beam at $z = 1000{\lambda _m}$ when incident angle varries from ${\theta _i} = {40^{\circ} }$ to ${\theta _i} = {80^{\circ} }$. At ${\theta _i} = {60.4^{\circ} }$ the reflected beam exhibits a distinctive field profile characterized by the formation of two crescent-shaped lobes, which corresponds to the optimal excitation of SPPs in graphene.

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In Fig. 4(b), when incident angle increases from 40° to 57°, the intensity in the left side of profile reduces while in the right side enhances and eventually a crescent-shaped lobe is formed in the right side. The reason for this behavior is that for incident angles smaller than ${\theta _{spp}} = 60.4^{\circ} $, the ray in left hand side of the central wave vector participate in the excitation of SPPs. So, these rays are coupled to the interface and do not appear in the profile of the reflected beam. This effect leads to the darkness of the left part of reflected beam profile when the incident angle is increased from 40° to 57°. When incident angle further increases from 57° to 60.4°, the intensity of left part increases and finally a crescent-shaped lobe appears in the left part of the profile. As shown in Fig. 4(b), for the incident angle 60.4° the left part of the profile is exactly the mirror image of the right part. When incident angle increases from 60.4° to 64°, the intensity of the right part decreases and the right crescent-shaped lobe eventually disappears. This effect results from the fact that for incident angles larger than ${\theta _{spp}} = 60.4^{\circ} $, the rays in the right-hand side of the central wave vector leads to the excitation of SPPs, so they are not seen in the intensity profile of the reflected beam resulting in the darkness of the right-hand side of the intensity profile of the reflected beam. With further increase of incident angle from 64° to 80°, the intensity profile returns to its initial donut shape. Another interesting feature of Fig. 4(b) is that the intensity profile in some incident angles such as 52°, 55° and 59.85° are the mirror image of ones in the incident angles of 68°, 65.5° and 61°, respectively. To understand the reason for this behavior, the reflectance R_p as a function of incident angle at the fixed frequency of 1.29 THz is plotted in Fig. 5.

 

Fig. 5. Reflectance R_p as a function of incident angle at frequency ${\nu _m} = 1.29\; THz$ corresponding to the optimal excitation of SPPs through the graphene layer in the absence of external magnetic field.

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It is clearly seen in Fig. 5 that the reflectance curve is symmetric around the resonance incident angle ${\theta _{spp}} = 60.4^{\circ} $. This effect results in the fact that the profiles of reflected beam at some certain incident angles smaller than ${\theta _{spp}}$ are the mirror image of ones at special incident angles greater than ${\theta _{spp}}$. Consequently, the excitation of SPPs through the graphene layer is possible using the illumination of Otto-configuration with LG beams instead of plane waves and the profile of reflected beam significantly depends on the value of energy coupling between the incident wave and excited SPPs. On the other hand, the investigation of spatial distribution of reflected LG beam provides another tool to confirm the quality of excitation of SPPs instead of measurement of the reflectance using a spectrometer. In addition, with detection of variations in the profile of reflected beam it is possible to realize the presence of biomaterial near the interfaces existing in the graphene based plasmonic structures. To determine the effects of changing the incident angle and excitation of SPPs on the phase of reflected beam, for the structure corresponding to each subfigure of Fig. 4, the phase distribution of x-component of the incident and reflected electric fields at $z = 10{\lambda _m}$ is calculated and the corresponding results are shown in Fig. 6. It is clearly seen in Fig. 6(a) that in the incident LG beam, there are two straight arms which correspond to the topological charge $l = 2$. For the reflected beam, as shown in Fig. 6(b), when the incident angle is far from ${\theta _{spp}} = 60.4^{\circ} $, two spiral arms exist in the phase profile. However, for incident angles near ${\theta _{spp}}$, there are three spiral arms in the phase distribution. This effect confirms the generation of OAM sidebands in the reflected beam upon the optimal SPPs excitation.

 

Fig. 6. Phase distributions of the x-component of electric field of the incident beam (a) and the reflected beam (b) for different configurations corresponding to subfigures in Fig. 4 at $z = 10{\lambda _m}$.

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To investigate the effect of propagation distance of the reflected beam on its electric field and phase distributions, in Fig. 7, we display the phase distribution of the x-component of electric field and the electric field intensity profile at different propagation distances $\textrm{z} = 1{\lambda _m},\; 5\; {\lambda _m},\; 10{\; }{\lambda _m},\; 20{\lambda _m},\; 50{\lambda _m},\; 100{\lambda _m},\; 200{\lambda _m},\; 500{\lambda _m}$ and $1000{\lambda _m}$, for the case corresponding to ${\theta _{spp}} = 60.4^{\circ} $ in Fig. 4. One can see in this figure that both phase and intensity profiles of the reflected beam vary with increase of propagation distance. The variation of phase profile with propagation distance results from the fact that besides vortex phases $\exp({il\varphi } )$ there are other phases in the electric field of the reflected beam such as propagation and Gouy phases which depend on the propagation distance [31]. Therefore, in Fig. 6, we display the phase distributions at the shorter distance $z = 10{\lambda _m}$ for more clarification and better comparison.

 

Fig. 7. Phase distributions of the x-component of electric field of the reflected beam and its electric field intensity profile for the configuration corresponding to the subfigure ${\theta _{spp}} = 60.4^{\circ} $ in Fig. 4 at different propagation distances $\textrm{z} = 1{\lambda _m},\; 5\; {\lambda _m},\; 10{\; }{\lambda _m},\; 20{\lambda _m},\; 50{\lambda _m},\; 100{\lambda _m},\; 200{\lambda _m},\; 500{\lambda _m}$ and $1000{\lambda _m}$

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3.3 Calculation of OAM spectrum of the reflected beam under illumination with a LG beam

As stated in the introduction, when a LG beam is incident into an interface between two different media, new orbital angular momentum sidebands appear in the reflected beam. Now we investigate how the excitation of SPPs through the graphene layer affects the generation of orbital angular momentum sidebands of the LG beam upon reflection in the Otto-configuration represented in Fig. 1. To this end, it is assumed that a LG beam with different OAMs ranging from $l = 0$ to $l = 5$ is incident into the structure. First, we consider the case at which there is no external magnetic field. The values of those parameters used in calculations are $\mu = 0.5\; eV$, ${d_{air}} = 5\; \mu m$, ${\theta _i} = 60.4^{\circ} $ and $\nu = 1.29\; THz$ which correspond to the dip of blue curve in Fig. 3, i.e., required parameters for optimal excitation of SPPs thorough the graphene layer in the absence of the external magnetic field under illumination with the plane wave in section 3.1. The intensities of existing OAM modes in the reflected beam are calculated using Eq. (40) and the corresponding results are shown in Fig. 8(a). As shown in this figure, for each input topological charge ($l$) in the range 0 to 5, there is a superposition of sidebands OAM modes ($-{-}l - 1$) and ($-{-}l + 1$) in the reflected beam while the negative sign of OAM results from the reflection. Therefore, the optimal excitation of SPPs leads to the suppression of the central OAM mode ($- l$) upon reflection. This effect results from the region with null intensity in the middle of profile of the reflected beam due to the excitation of SPPs. In Fig. 8(b), for incident topological charges ${l_i} = 0,\; 1,\; 3,\; 5$, the reflected topological charge mode weights as well as their beam profiles are displayed. As shown in this figure, the weight of central OAM weakly increases with incident topological charge l. In addition, Fig. 8(c, d) represent the OAM spectrum and electric field intensity for the case at which there are no graphene layer and external magnetic field and hence no excitation of SPPs occurs. Comparison of Fig. 8(a) and (c) obviously shows the direct effect of excitation of SPPs on the generation of OAM spectrum and intensity profile of reflected LG beam which is not reported until now. In the absence of graphene layer and excitation of SPPs the dominant phenomenon is total internal reflection from three-layer structure of silicon-air-silica with very weakly OAM sidebands, as depicted in Fig. 8(d).

 

Fig. 8. (a) Orbital angular momentum (OAM) spectrum of the reflected beam for incident topological charges ${l_i} = 0,1,2, \ldots ,5$. (b) Weight of OAM modes and electric field intensity profile of the reflected beam for incident topological charges of ${l_i} = 0,\; 1,\; 3,\; 5$ corresponding to the optimal excitation of graphene SPPs in the absence of external magnetic field with parameters incindent angle ${\theta _i} = 60.4^{\circ} $ and chemical potential $\mu = 0.5\; eV$. (c) OAM spectrum of reflected beam for incident topological charges ${l_i} = 0,1,3, \ldots ,5$ in the absence of graphene layer. (d) Weight of OAM modes and electric field intensity profile of the reflected beam for incident topological charges of ${l_i} = 0,\; 1,\; 3,\; 5$ in the absence of graphene layer. The calculation is carried out at $\textrm{z} = 1000{\lambda _m}$.

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To understand how the deviation from the optimal excitation of graphene SPPs affects the OAM spectrum of the reflected beam, we first examine the effect of increasing the chemical potential from $\mu = 0.5\; eV$ to $\mu = 0.52\; eV$ and calculate the relative weight of OAM modes of the reflected beam for input OAM modes in the range 0 to 5. The other parameters are the same as those in Fig. 8. The relative intensities of generated OAM states in the reflected beam are represented in Fig. 9(a) for different input topological charges. It is clearly seen in Fig. 9(a) that both central and sideband OAM modes exist in the spectrum. However, the intensity of central OAM state undergoes a reduction with increasing the topological charge of the incident beam. For more clarification, the weight of OAM modes and the intensity profile of the reflected beam for input topological charges ${l_i} = 0,\; 1,\; 3,\; 5$ are indicated in Fig. 9(b). As shown in this figure, for the case ${l_i} = 1$, the weight of central OAM mode is 0.2 while for the case ${l_i} = 3$, it is 0.1. As the intensity of central OAM state in the reflected beam decreases, the intensity of sidebands increases due to the energy transfer from central to sideband OAM modes.

 

Fig. 9. (a) OAM spectrum of the reflected beam for incident topological charges ${l_i} = 0,1,2, \ldots ,5$. (b) Weight of OAM modes and electric field intensity profile of the reflected beam for incident topological charges of ${l_i} = 0,\; 1,\; 3,\; 5$ corresponding to the case which becomes far from the optimal excitation of graphene SPPs by a small increase of the chemical potentials from $\mu = 0.5\; eV$ to $\mu = 0.52\; eV$. The other parameters are the same as those in Fig. 8.

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Another way to take the system away from the optimal excitation of SPPs is the increase of incident angle. To understand how the OAM spectrum of the reflected beam changes in this case, we examine the effect of increasing the incident angle from ${60.4^{\circ} }$ to ${63^{\circ} }$. For this case, OAM spectrum of the reflected beam for different incident topological charges is calculated and the corresponding results are shown in Fig. 10(a). The other parameters for calculations of the results in this figure are similar to those in Fig. 8. It can be seen in Fig. 10(a) that both central and sideband OAM modes exist in the spectrum. However, the intensity of central OAM state undergoes a reduction with increasing the topological charge of the incident beam. Figure 10(b) shows the weight of OAM modes and the intensity profile of the reflected beam for input topological charges ${l_i} = 0,\; 1,\; 3,\; 5$.

 

Fig. 10. (a) OAM spectrum of the reflected beam for incident topological charges ${l_i} = 0,1,2, \ldots ,5$. (b) Weight of OAM modes and electric field intensity profile of the reflected beam for incident topological charges of ${l_i} = 0,\; 1,\; 3,\; 5$ corresponding to the case which becomes far from the optimal excitation of graphene SPPs by increasing the incident angle from ${\theta _i} = 60.4^{\circ} $ to ${\theta _i} = 63^{\circ} $. The other parameters are the same as those in Fig. 6.

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Now, to take the system away from the optimal excitation of SPPs, we apply an external magnetic field. So, we increase the external magnetic field to ${B_0} = 1\; T$ for the structure used in Fig. 8 corresponding to the optimal excitation of SPPs and calculate the relative intensity of the generated OAM modes by reflection from the proposed Otto structure. The other parameters are taken to be the same as those in Fig. 8 and the corresponding results for OAM spectrum are presented in Fig. 11(a). One can see in this figure that there are both central and sideband modes in the OAM spectrum of the reflected beam for all incident topological charges in the range 0 to 5. For input topological charges ${l_i} = 0$ and ${l_i} = 1$, the intensity of central OAM mode is stronger than one of sidebands. However, for incident topological charges ${l_i} = 2$ to ${l_i} = 5$, we see the opposite behavior in which the sideband OAM modes are stronger than the central mode. The transfer of energy from the central OAM mode to the sidebands with increase of input topological charge is also clearly observed in Fig. 11(b).

 

Fig. 11. (a) OAM spectrum of the reflected beam for incident topological charges ${l_i} = 0,1,2, \ldots ,5$. (b) Weight of OAM modes and electric field intensity profile of the reflected beam for incident topological charges of ${l_i} = 0,\; 1,\; 3,\; 5$ corresponding to the case far from the optimal excitation of graphene SPPs by applying an external magnetic field ${B_0} = 1\; T$. The other parameters are the same as those in Fig. 8.

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In the following, to further study the effect of applying an external magnetic field on OAM spectrum of the reflected beam, we calculate the OAM sideband spectrum and electric field intensity of the reflected beam for incident topological charges ${l_i} = 0,\; 1,\; 3,\; 5$ at frequencies and magnetic fields corresponding to each dip in Fig. (3) for the p-polarized incident LG beam. The results are shown in Fig. 12(a), (b), (c), (d), (e) and (f) which all belong to (B = 0 T and ν=1.2936 THz), (B = 1 T and ν=1.3639 THz), (B = 2 T and ν=1.5494 THz), (B = 3 T and ν=1.8026 THz) and (B = 4 T and ν=2.0947 THz), respectively. The other parameters are taken to be the same as those in Fig. 8. It is clearly observed that the enhancement of external magnetic field from B = 0 to B = 4 T leads to increasing the weight of central OAM mode and decreasing the weight of the sideband OAM modes. This effect results from the fact that in Fig. 3 the value of reflectance in the dip increases with increasing the external magnetic field for the case of p-polarized incident wave. Therefore, we observe a reduction in the area with null intensity in the reflected beam as the external magnetic field increases leading to the growth of central mode in the OAM spectrum.

 

Fig. 12. The OAM sideband spectrum and electric field intensity of reflected beam for incident topological charges ${l_i} = 0,\; 1,\; 3,\; 5$ at frequencies and magnetic fields (a) $B = 0{\; }T{\; }and{\; }\nu = 1.2936{\; }THz$, (b) $B = 1{\; }T{\; }and{\; }\nu = 1.3639{\; }THz$, (c) $B = 2{\; }T{\; }and{\; }\nu = 1.5494{\; }THz$, (d)${\; }B = 3{\; }T{\; }and{\; }\nu = 1.8026{\; }THz$, (e) $\textrm{B} = 4\mathrm{\;\ T\;\ and\;\ \nu } = 2.0947\; \textrm{THz}$. The other parameters are taken to be the same as those in Fig. 8.

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For all above effects, in the case of optimal SPPs excitation, the sideband mode weight remains independent of the incident beam's OAM. However, after a slight deviation from the optimal state, the sideband mode weight becomes dependent on the value of incident OAM. These characteristics render our system a tunable platform for generating OAM sidebands. By manipulating the properties of graphene SPPs, we can control the mode weight of the OAM sidebands in the reflected LG beam.

For each subfigure in Fig. 12, the phase distributions of x-component of electric field of the reflected beam are displayed in Fig. 13 at $z = 10{\lambda _m}$. For magnetic fields B = 0 to 1 T, for incident topological charges larger than l_i = 0, there exist l_i + 1 spiral arms in the phase profile. However, for magnetic fields B = 2 to 4 T, there are l_i spiral arms in the phase profile of the x-component of electric field of the reflected beam. These results are in agreement with the sidebands spectra represented in Fig. 12.

 

Fig. 13. Phase distributions of the x-component of electric field of the reflected beam corresponding to each subfigure in Fig. 12 at $z = 10{\lambda _m}$.

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The loss effect due to interband transitions is taken into account in the graphene conductivity represented in Eqs. (1) and (2). Therefore, the given results are calculated in the presence of optical loss of graphene. To understand the effect of loss on the results, we neglect the graphene loss and repeat the calculations for the case ${\theta _i} = {60.4^{\circ} }$ in Fig. 4. The corresponding results are shown in Fig. 14 in which the electric field intensity profile and phase distribution of the x-component of electric field for the structure containing a lossy graphene layer are compared with ones for the structure with a lossless graphene layer. As shown in Fig. 14, when the graphene loss is neglected in the structure, the maximum value of electric field intensity of the reflected beam increases by 16 times. But no significant changes occur in the distributions of phase and electric field intensity of the reflected beam.

 

Fig. 14. Phase distributions of the x-component of reflected electric field and its intensity profile for the structure corresponding to the case ${\theta _i} = {60.4^{\circ}}$ in Fig. 4 at $z = 1000{\lambda _m}$ for the cases at which the graphene loss is neglected (a) and included (b).

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So far, different methods have been proposed to determine the OAM spectrum of a complex beam array such as using the refractive elements [32], the vortex spectrometer [33] and measurement of high-order intensity moments and solving a system of linear equations in both nondegenerate and degenerate cases [34,35]. Because these methods were tested experimentally, they can be also employed for measurement of OAM spectrum of LG beams reflected from the plasmonic structure proposed in this paper and other plasmonic structures.

It is well known that OAM multiplexing (MUX) and demultiplexing (DEMUX) lead to the enhancement of capacity and efficiency of systems used for optical and THz communications. It is possible to employ the proposed structure in this paper for developing and designing the OAM based MUX/DMUX systems like previously reported devices based on metasurfaces [36,37]. However, the details for this application are under consideration and will be revealed in the near future.

4. Conclusions

We have theoretically calculated the electromagnetic fields of a LG beam upon reflection from an Otto-configuration including a graphene layer deposited on the SiO₂ substrate. Due to the excitation of SPPs through the graphene layer, a vertical strip with null intensity appears in the electric field intensity profile of the reflected beam for different incident topological charges. In the case of optimal excitation of graphene SPPs in the absence of external magnetic field, the dark strip is located in the middle of reflected beam profile and it is split into two equal crescent-shaped lobes, regardless of the OAM of the incident LG beam. When the system is taken away from the optimal state for excitation of SPPs, the dark strip translates horizontally within the profile. For the cases at which the system is completely far from the optimal state, the dark strip disappears in the profile of reflected beam such that it becomes similar to the profile of incident LG beam. In addition, we have studied OAM spectrum of the reflected beam. In the case of optimal excitation of SPPs through the graphene layer, in the absence of external magnetic field, the central OAM mode does not exist in the spectrum of reflected beam and the weight of sideband modes is independent of input OAM state. The phase distributions of the x-component of electric field of the reflected beam also conform these effects. However, when the system becomes far from the optimal excitation of graphene SPPs by applying the external magnetic field or changing the chemical potentials or variation of incident angle, both central and sideband OAM modes exist in the spectrum. In this case, the weight of central OAM mode decreases with increase of input OAM state and its energy is coupled to the sideband modes leading to the enhancement of their weight. In the presence of external magnetic field, for the cases at which optimal excitation of SPPs occurs, the central OAM mode enhances with external magnetic field while the sidebands experience a reduction. These features are also observed in the phase profile of electric field of the reflected beam. By manipulating the value of deviation from the optimal excitation of graphene SPPs, it is possible to establish a tunable system for OAM sideband generation in the terahertz region based on graphene SPPs. Consequently, our investigation demonstrates the tunable generation of OAM sidebands by using surface plasmon polaritons waves, opening up new avenues for information encoding, manipulation, OAM-based multiplexing and advancement of optical communication systems.