All optical divergence and gradient operators using surface plasmon polaritons

Hadi Mohammadi; Hadi Mohammadi; Mahmood Akbari; Mahmood Akbari; Amin Khavasi; Amin Khavasi

doi:10.1364/OE.456878

1. Introduction

Digital signal processors are nowadays ubiquitous to perform various computational tasks from comparatively simple to highly complicated ones [2–5]. In spite of their versatility, digital processors suffer from a number of drawbacks mostly arisen from analogue-to-digital conversions and vice versa. A/D and D/A convertors consume power and time in an inefficient manner [2,5]. In addition, at frequencies above GHz range, these convertors fail to operate properly due to the rapid variation of signals [3–5]. The aforementioned limitations can be tackled by analogue computation. In most cases, a natural system performs analogue computation thereby eliminating the need for A/D and D/A conversions [3,4]. Opposite to their mechanical and electronic predecessors, optical analogue devices are very fast since the computation process in them takes place at the light speed [6–9]. Due to the aforementioned advantages, the interest toward optical analogue operators have been renewed in the recent years [10,11].

Propagation of light has been of interest for many decades as a means of analogue computations at high speed and low power consumption [12]. Setups based on Fourier optics have tremendously been investigated for their performance in image processing [13–16]. Their bulky structure, however, is a limitation which impede their miniaturization [5,11]. In recent years, on-chip silicon photonic circuits and nano photonic structures has been introduced thereby considerably enhancing the miniaturization of optical analogue operators [17–22].

One important category of optical computing devices is spatial differentiators whereby edge detection in image processing is enabled [23]. Such devices are, in fact, either first-order or second-order derivative operators. In the recent years, a great amount of effort has been dedicated to develop theoretical designs for spatial differentiation either in one [1,3,4,24–34] or two dimensions [7,35–38]. Most of the experimentally fabricated structures, however, operate in one dimension [6,10,39–45] while a less number of them work as two-dimensional differentiators [11,46,47]. Structures introduced in [11] and [36–38,46,47] are first-order and second-order two-dimensional differentiators, respectively, while in [7,35], there are suggestions for both of these cases.

The mentioned two-dimensional differentiators generally fall into three categories: the first and second groups are the differentiators in which either a photonic crystal [36,46] or a metasurface [7,35,37] have been used. In addition in [48], a graphene-on-grating nanostructure has been used as a periodic plasmonic metasurface to perform divergence operation on vectorial paraxial beams. The third group, on the other hand, enjoys a layered structure; in [47] a three layered structure and more recently in [11], a dielectric-air interface have been exploited as two-dimensional differentiators. Due to the simplicity of its fabrication process, the latter is more cost-efficient in comparison with the former two. In addition, the photonic crystal structure in [46], for instance, is not able to perform an isotropic differentiation and in order to do so, a more complex design is required [11]. The first-order, two-dimensional differentiators in [7,11,35] have a low differentiation gain whereas in some applications a high differentiation gain is desirable. Judging by the figure of merit (FOM) in [1], the differentiators in [7,11,35] do not show a good performance in differentiation in x or y directions in comparison with the ideal, yet physical differentiator introduced in [1]. We discuss in this regard later in detail.

Here we propose a plasmonic structure based on Kretschmann configuration and we theoretically show that our structure features the ability to perform high-gain, two-dimensional differentiation. The idea of using Kretschmann prism as a one dimensional differentiator has previously been introduced in [43]. Our tetrahedral structure is, in fact, a combination of two Kretschmann configurations, two of its faces are covered by thin silver coatings. By adequate selection of input and output polarizers, a non-trivial topological charge is created in the transfer function of our structure thereby implementing two dimensional differentiation. In addition, we show that our structure can be utilized as two dimensional gradient and divergence operators, the former computes the gradient of a polarized light beam while the latter calculates the divergence of an unpolarized light beam. It is worth mentioning that divergence operation can reveal polarization singularities in a vector field which is of great importance in comprehension of topological physical phenomena such as ultrahigh-Q resonances in open systems which have been employed in bound states in the continuum (BIC) lasers [48]. In spite of combining two Kretschmann prism configurations, the proposed structure (especially our gradient operator) features almost the same performance in differentiation in any given direction in x-y plane as the one dimensional plasmonic differentiator in [43]. In other words, cascading two Kretschmann configurations, does not have any undesirable impact on the differentiation performance of our proposed structure. Regardless of being one- or two-dimensional, the performance of a given differentiator can quantitatively be evaluated by the FOM in [1]. Finally, in the last section, we compare the FOM values corresponding to our differentiator and a number of its counterparts. The results demonstrate that our structure especially the gradient operator herein has an excellent performance in differentiating in x- and y-directions comparable to the ideal, yet physical differentiator in [1].

2. Plasmonic structure: layout and performance

The structure introduced in [43], is, in fact, a Kretschmann prism configuration which has been exploited as a one-dimensional differentiator. The Kretschmann configuration is simply modelled by a three-layer structure comprised of a dielectric, plasmonic metal and free space in which the dielectric layer is the incident region. The silver coating in the Kretschmann configuration has a thickness of 50 nm and a p-polarized plane wave obliquely illuminates the metal surface from the glass side. At a certain incident angle i.e. 43.72° to which we refer as ${\theta _{SPP}}$, the phase matching condition is met and surface plasmon polaritons (SPP) are stimulated whereby the incident power can transfer to the lossy plasmonic metal and, in turn, dissipated. Therefore, the reflection from glass-silver interface would be approximately zero and consequently, the structure enjoys a transfer function with steep valley around its resonance and can act as a high-gain, one dimensional differentiator. FOM results has demonstrated that this differentiator features an adequate performance, more similar to the ideal, yet physical, differentiator in [1] comparing to its other counterparts [1]. Hence, Kretschmann prism configuration is also an excellent candidate to be exploited as a high-gain, two-dimensional differentiator.

Here, we combine two Kretschmann configurations in one structure to perform high-gain, 2D differentiation. This structure can be implemented by thin silver coatings on two faces of a tetrahedron as schematically shown in Fig. 1. In addition, in Fig. 2, the practical structure of the tetrahedron can be observed. The first and second faces of the tetrahedron are equivalent to three-layered structures which are referred to as “Ag_L₁” and “Ag_L₂” in Fig. 1(a) and Fig. 2(a). Two reflections occur when the structure is illuminated by a light beam. Firstly, the input light beam is reflected by Ag_L₁, then it travels toward the Ag_L₂ where the second reflection happens thereby forming the output beam. Note that if the input bean is p (s)-polarized with respect to Ag_L₁, the reflected beam from Ag_L₁ is s (p)-polarized with respect to Ag_L₂. In addition, the input beam and its reflection from Ag_L₁ both illuminate Ag_L₁ and Ag_L₂ with incident angle ${\theta _{SPP}}$, respectively.

Fig. 1. (a) The schematic structure of the proposed 2D differentiator herein. (b-c) Preview of the two faces of the tetrahedron with silver coatings. Each face is, in fact, a three-layered structure comprised of glass, silver and air, respectively. (b) the unit vectors corresponding to the input and middle coordinate systems at Ag_L₁, (c) the unit vectors corresponding to the middle and output coordinate systems at Ag_L₂.

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Fig. 2. (a) The practical structure of the proposed 2D differentiator herein. The tetrahedron is made of glass and two of its faces i.e. ABC and ACD are covered with silver coatings whose thickness is 50 nm. The black, blue and red lines represent the input light beam, its reflection from Ag_L₁ and the output light beam, respectively. (b-e) The faces of the tetrahedron alongside their corresponding normal vectors. O_1-4 stand for the striking points of the light beam at the corresponding faces of the tetrahedron.

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In Fig. 1(b-c), there are three different Cartesian coordinates as follows:

• Input Coordinate System: this coordinate system is indicated by three orthonormal vectors $({{{{\hat{\mathbf x}}}_{in}},{{{\hat{\mathbf y}}}_{in}},{{{\hat{\mathbf z}}}_{in}}{ }} )$ in Fig. 1(b). ${{\hat{\mathbf z}}_{in}}$ can be chosen every arbitrary unit vector satisfying the equation ${{\hat{\mathbf z}}_{in}} \cdot {{\hat{\mathbf n}}_1} ={-} \cos {\theta _{SPP}}$ where ${{\hat{\mathbf n}}_1}$ is the normal unit vector perpendicular to silver-glass interface as indicated in Fig. 1(b). Then, ${{\hat{\mathbf x}}_{in}}$ and ${{\hat{\mathbf y}}_{in}}$ are determined as follows: $(1)$${{\hat{\mathbf y}}_{in}} = {{\hat{\mathbf z}}_{in}} \times {{\hat{\mathbf n}}_1}\,\,\,\,\,;\,\,\,\,\,\,\,{{\hat{\mathbf x}}_{in}} = {{\hat{\mathbf y}}_{in}} \times {{\hat{\mathbf z}}_{in}}.$$$
• Middle Coordinate System: this coordinate system is indicated by three orthonormal vectors $({{{{\hat{\mathbf x}}}_m},{{{\hat{\mathbf y}}}_m},{{{\hat{\mathbf z}}}_m}{ }} )$ in Fig. 1(b-c) that are defined as follows: $(2)$${{\hat{\mathbf z}}_m} = 2({{{{\hat{\mathbf z}}}_{in}} \cdot {{{\hat{\mathbf n}}}_1}} ){{\hat{\mathbf n}}_1} - {{\hat{\mathbf z}}_{in}}\,\,\,\,\,;\,\,\,\,\,{{\hat{\mathbf y}}_m} = {{\hat{\mathbf n}}_1} \times {{\hat{\mathbf z}}_m}\,\,\,\,\,;\,\,\,\,\,{{\hat{\mathbf x}}_m} = {{\hat{\mathbf y}}_m} \times {{\hat{\mathbf z}}_m}.$$$
• Output Coordinate System: this coordinate system is indicated by three orthonormal vectors $({{{{\hat{\mathbf x}}}_{out}},{{{\hat{\mathbf y}}}_{out}},{{{\hat{\mathbf z}}}_{out}}{ }} )$ in Fig. 1(c) that are defined as follows: $(3)$$ \hat{\mathbf{z}}_{out }=2\left(\hat{\mathbf{z}}_{m} \cdot \hat{\mathbf{n}}_{2}\right) \hat{\mathbf{n}}_{2}-\hat{\mathbf{z}}_{m} \quad ; \quad \hat{\mathbf{x}}_ {out }=\hat{\mathbf{z}}_ {out } \times \hat{\mathbf{n}}_{2} \quad ; \quad \hat{\mathbf{y}}_{ out }=\hat{\mathbf{z}}_ {out } \times \hat{\mathbf{x}}_ {out } . $$$

By this choices, $({{{{\hat{\mathbf x}}}_{in}},{{{\hat{\mathbf y}}}_{in}},{{{\hat{\mathbf z}}}_{in}}{ }} )$, $({{{{\hat{\mathbf x}}}_m},{{{\hat{\mathbf y}}}_m},{{{\hat{\mathbf z}}}_m}{ }} )$ and $({{{{\hat{\mathbf x}}}_{out}},{{{\hat{\mathbf y}}}_{out}},{{{\hat{\mathbf z}}}_{out}}{ }} )$ are right-handed coordinate systems. Note that ${{\hat{\mathbf z}}_{in}}$, ${{\hat{\mathbf z}}_m}$ and ${{\hat{\mathbf z}}_{out}}$ are parallel to the central wave vector of the the incident beam, the reflected beam from Ag_L₁ and the output beam in spatial Fourier space, respectively.

In Sec. 1 in the Supplement 1, we show that the angle between Ag_L₁ and Ag_L₂ i.e. $\theta ^{\prime}$ in Fig. 1(a), is ${\cos ^{ - 1}}({{{\cos }^2}{\theta_{SPP}}} )$. The boundary of the 2D differentiator tetrahedron is, in fact, a closed surface formed by intersection of four planes to which ${{\hat{\mathbf z}}_{in}}$, ${{\hat{\mathbf n}}_1}$, ${{\hat{\mathbf n}}_2}$ and ${{\hat{\mathbf z}}_{out}}$ are normal as it can be seen in Fig. 2(a). The faces of the tetrahedron alongside their corresponding normal vectors have been presented in Fig. 2(b-e). The tetrahedron is made of glass and two of its faces i.e. ABC and ACD are covered with silver coatings whose thickness is 50 nm same as [43]. Note that this is the best silver thickness that can fulfill the critical coupling condition if the accuracy corresponding to the silver thickness is 1 nm. However, our simulation shows that for silver thickness 50.51 nm the critical coupling condition is fulfilled almost perfectly, yet its fabrication requires accuracy level at the order of 0.01 nm which does not seem very practical since it is lower than atomic sizes. Furthermore, our simulations show that silver thickness 50.51 does not enhance the performance of our differentiator notably. The input light beam indicated by a black line in Fig. 2(a) vertically strikes face ABC and enters the tetrahedron. The output light beam which is created by two consecutive reflections from silver coatings i.e. Ag_L₁ and Ag_L₂ in Fig. 1 and Fig. 3(a), respectively, perpendicularly illuminates upon face BCD and exits from the tetrahedron. O_1-4 in Fig. 2(b-e) represent the striking points of the light beam at the corresponding faces of the tetrahedron. The striking points are located far enough from the tetrahedron sides whereby eliminating undesirable edge effects. Table 1 shows the sides’ length of the tetrahedron along with the location of each striking point on the corresponding face (see Sec. 2 in the Supplement 1 for details). All values in Table 1 are stated in term of L which denotes the length of side $\overline {\mathbf{AB}}$.

Fig. 3. (a) the magnitude and (b) the phase of the transfer function corresponding to the proposed two-dimensional differentiator alongside its first order Taylor expansion in Eq. (12) for $|{{k_{x(y )}}} |\,\, \le 0.03{k_0}$ and ${k_{y(x )}} = 0$ in Fourier space domain. The input and output polarizers are selected according to Eq. (17).

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Table 1. The sides’ length of the tetrahedron along with the location of the light beam's striking points on their corresponding faces of the tetrahedron (see Sec. 2 in Supplement 1 for details). All the values are stated in term of the length of side $\overline {{\mathbf AB}}$ which is dented by L.

View Table | View all tables in this article

Now we assume that at Ag_L₁, a plane wave with wave vector ${{\mathbf k}_{in}}$ illuminates the metal surface from the glass side. The reflected plane wave from Ag_L₁ has wave vector ${{\mathbf k}_m}$ which is incident to the metal surface at Ag_L₂, again from the glass side. The reflected plane wave from Ag_L₂ is the output wave and has wave vector ${{\mathbf k}_{out}}$. Let's assume that the electric fields of these three plane waves are as follows:

(4)$$\begin{array}{l} {{{\mathbf E^{\prime}}}_{in}} = {{E^{\prime}}_{in}}{{{\hat{\mathbf a}}}_{in}} = {{E^{\prime}}_{in}}({a_{in}^x{{{\hat{\mathbf x}}}_{in}} + a_{in}^y{{{\hat{\mathbf y}}}_{in}} + a_{in}^z{{{\hat{\mathbf z}}}_{in}}} ),\\ {{{\mathbf E^{\prime}}}_m} = {{E^{\prime}}_m}{{{\hat{\mathbf a}}}_m} = {{E^{\prime}}_m}({a_m^x{{{\hat{\mathbf x}}}_m} + a_m^y{{{\hat{\mathbf y}}}_m} + a_m^z{{{\hat{\mathbf z}}}_m}} ),\\ {{{\mathbf E^{\prime}}}_{out}} = {E_{out}}{{{\hat{\mathbf a}}}_{out}} = {{E^{\prime}}_{out}}({a_{out}^x{{{\hat{\mathbf x}}}_{out}} + a_{out}^y{{{\hat{\mathbf y}}}_{out}} + a_{out}^z{{{\hat{\mathbf z}}}_{out}}} ), \end{array}$$

where ${E^{\prime}_{in}}$ and unit vector ${{\hat{\mathbf a}}_{in}}$ are the electric field phasor and polarization of the incident plane wave, respectively. In the same manner, ${E^{\prime}_{m({out} )}}$ and unit vector ${{\hat{\mathbf a}}_{m({out} )}}$ are the electric field phasor and polarization of the reflected plane wave from Ag_L₁₍₂₎, respectively. ${{\mathbf k}_m}$ and ${{\hat{\mathbf x}}_m}$ are the images of ${{\mathbf k}_{in}}$ and ${{\hat{\mathbf x}}_{in}}$ with respect to the glass-metal interface at Ag_L₁, respectively. ${{\hat{\mathbf y}}_{in}}$ and ${{\hat{\mathbf y}}_m}$, on the other hand, are equal and tangential to the glass-metal interface at Ag_L₁. ${{\mathbf k}_{out}}$ and ${{\hat{\mathbf y}}_{out}}$ are the images of ${{\mathbf k}_m}$ and ${{\hat{\mathbf y}}_m}$ with respect to the glass-metal interface at Ag_L₂, respectively. ${{\hat{\mathbf x}}_{out}}$ and ${{\hat{\mathbf x}}_m}$, on the other hand, are equal and tangential to the glass-metal interface at Ag_L₂. So, one can write:

(5)$$\begin{array}{l} {k_x} \buildrel \Delta \over = {{\mathbf k}_{in}} \cdot {{{\hat{\mathbf x}}}_{in}} = {{\mathbf k}_m} \cdot {{{\hat{\mathbf x}}}_m} = {{\mathbf k}_m} \cdot {{{\hat{\mathbf x}}}_{out}},\\ {k_y} \buildrel \Delta \over = {{\mathbf k}_{in}} \cdot {{{\hat{\mathbf y}}}_{in}} = {{\mathbf k}_m} \cdot {{{\hat{\mathbf y}}}_m} = {{\mathbf k}_m} \cdot {{{\hat{\mathbf y}}}_{out}}. \end{array}$$

According to the above equation, if ${k_x}$ and ${k_y}$ are the x- and y-components of ${{\mathbf k}_{in}}$ in the input coordinate system, ${k_x}$ and ${k_y}$ are also the x- and y-components of ${{\mathbf k}_m}$ and ${{\mathbf k}_{out}}$ in the middle and output coordinate systems, respectively:

(6)$${{\mathbf k}_{in}} = {k_x}{{\hat{\mathbf x}}_{in}} + {k_y}{{\hat{\mathbf y}}_{in}} + {k_z}{{\hat{\mathbf z}}_{in}} \Rightarrow \left\{ \begin{array}{l} {{\mathbf k}_m} = {k_x}{{{\hat{\mathbf x}}}_m} + {k_y}{{{\hat{\mathbf y}}}_m} - {k_z}{{{\hat{\mathbf z}}}_m}\\ {{\mathbf k}_{out}} = {k_x}{{{\hat{\mathbf x}}}_{out}} + {k_y}{{{\hat{\mathbf y}}}_{out}} + {k_z}{{{\hat{\mathbf z}}}_{out}} \end{array} \right.,$$

where ${k_z} = \sqrt {k_0^2 - k_x^2 - k_y^2}$. Thanks to linearity of the 2D differentiator herein, ${{\mathbf E^{\prime}}_{in}}$ and ${{\mathbf E^{\prime}}_{out}}$ in Eq. (4) can be related to each other through transfer matrix ${{\mathbf R}_{3 \times 3}}$ which itself is a function of ${k_x}$ and ${k_y}$:

(7)$${E^{\prime}_{out}}{\left[ {\begin{array}{{ccc}} {a_{out}^x}&{a_{out}^y}&{a_{out}^z} \end{array}} \right]^T} = \overline{\overline {\mathbf R}} ({{k_x},{k_y}} ){\left[ {\begin{array}{{ccc}} {a_{in}^x}&{a_{in}^y}&{a_{in}^z} \end{array}} \right]^T}{E^{\prime}_{in}},$$

where upper index T denotes transpose. One can show that the transfer matrix ${{\mathbf R}_{3 \times 3}}$ can be obtained as follows (see Sec. 3 in Supplement 1 for details):

(8)$$\begin{array}{l} \overline{\overline {\mathbf R}} \left( {{k_x},{k_y}} \right) = \\ \left[ {\begin{array}{cc} {{\mathbf u}_p^{out} \cdot {{{\hat{\mathbf x}}}_{out}}}&{{\mathbf u}_s^{out} \cdot {{{\hat{\mathbf x}}}_{out}}}\\ {{\mathbf u}_p^{out} \cdot {{{\hat{\mathbf y}}}_{out}}}&{{\mathbf u}_s^{out} \cdot {{{\hat{{\mathbf y}}}}_{out}}}\\ {\mathbf {u}_p^{out} \cdot {{\mathbf{\hat{z}}}_{out}}}&{\mathbf{u}_s^{out} \cdot {{\mathbf{\hat{z}}}_{out}}} \end{array}} \right]\left[ {\begin{array}{cc} {{r_p}\left( {{\theta _2}} \right)}&0\\ 0&{{r_s}\left( {{\theta _2}} \right)} \end{array}} \right]\left[ \begin{array}{l} \begin{array}{ccc} {{\mathbf u}_p^{m2} \cdot {{{\hat{\mathbf x}}}_m}}&{{\mathbf u}_p^{m2} \cdot {{{\hat{\mathbf y}}}_m}}&{{\mathbf u}_p^{m2} \cdot {{{\hat{\mathbf z}}}_m}} \end{array}\\ \begin{array}{ccc} {{\mathbf u}_s^{m2} \cdot {{{\hat{\mathbf x}}}_m}}&{{\mathbf u}_s^{m2} \cdot {{{\hat{\mathbf y}}}_m}}&{{\mathbf u}_s^{m2} \cdot {{{\hat{\mathbf z}}}_m}} \end{array} \end{array} \right]\\ \times \left[ {\begin{array}{ccc} {{\mathbf u}_p^{m1} \cdot {{{\hat{\mathbf x}}}_m}}&{{\mathbf u}_s^{m1} \cdot {{{\hat{\mathbf x}}}_m}}\\ {{\mathbf u}_p^{m1} \cdot {{{\hat{\mathbf y}}}_m}}&{{\mathbf u}_s^{m1} \cdot {{{\hat{\mathbf y}}}_m}}\\ {{\mathbf u}_p^{m1} \cdot {{{\hat{\mathbf z}}}_m}}&{{\mathbf u}_s^{m1} \cdot {{{\hat{\mathbf z}}}_m}} \end{array}} \right]\,\left[ {\begin{array}{cc} {{r_p}\left( {{\theta _1}} \right)}&0\\ 0&{{r_s}\left( {{\theta _1}} \right)} \end{array}} \right]\left[ \begin{array}{l} \begin{array}{ccc} {{\mathbf u}_p^{in} \cdot {{{\hat{\mathbf x}}}_{in}}}&{{\mathbf u}_p^{in} \cdot {{{\hat{\mathbf y}}}_{in}}}&{{\mathbf u}_p^{in} \cdot {{{\hat{\mathbf z}}}_{in}}} \end{array}\\ \begin{array}{ccc} {{\mathbf u}_s^{in} \cdot {{{\hat{\mathbf x}}}_{in}}}&{{\mathbf u}_s^{in} \cdot {{{\hat{\mathbf y}}}_{in}}}&{{\mathbf u}_s^{in} \cdot {{{\hat{\mathbf z}}}_{in}}} \end{array} \end{array} \right],\,\, \end{array}$$

where ${\theta _{1(2 )}} \buildrel \Delta \over = {\cos ^{ - 1}}\left( { - \frac{{{{\mathbf k}_{in({m1} )}} \cdot {{{\hat{\mathbf n}}}_{1(2 )}}}}{{{k_0}}}} \right)$, ${\mathbf u}_s^{in({m1} )} \buildrel \Delta \over = \frac{{{\mathbf u}_k^{in({m1} )} \times {{{\hat{\mathbf n}}}_1}}}{{|{{\mathbf u}_k^{in({m1} )} \times {{{\hat{\mathbf n}}}_1}} |}}$ and ${\mathbf u}_s^{out({m2} )} \buildrel \Delta \over = \frac{{{\mathbf u}_k^{out({m2} )} \times {{{\hat{\mathbf n}}}_2}}}{{|{{\mathbf u}_k^{out({m2} )} \times {{{\hat{\mathbf n}}}_2}} |}}.$Additionally, ${\mathbf u}_p^{(\ldots )} \buildrel \Delta \over = {\mathbf u}_s^{(\ldots )} \times {\mathbf u}_k^{(\ldots )}$ and $\mathbf{u}_k^{\left( \ldots \right)} \buildrel \Delta \over = {{{\mathbf{k}_{\left( \ldots \right)}}} / {\left| {{\mathbf{k}_{\left( \ldots \right)}}} \right|}}$ in which $(\ldots )$ can be replaced by one of in, m1, m2 or out.

Similar to [11], we use polarizers through which the input and output beams, each, passes. Let's assume the input (output) polarizer is indicated by unit vector ${{\hat{\mathbf e}}_{in\left( {out} \right)}} = e_{in\left( {out} \right)}^x{{\hat{\mathbf x}}_{in\left( {out} \right)}} + e_{in\left( {out} \right)}^x{{\hat{\mathbf y}}_{in\left( {out} \right)}}$ in the input (output) coordinate system upon which a plane wave with wave vector ${{\mathbf k}_{in({out} )}}$ in Eq. (6) is incident. The output of the polarizer is also a plane wave with the same wave vector i.e. ${{\mathbf k}_{in({out} )}}$. Therefore, its electric field is perpendicular to ${{\mathbf k}_{in({out} )}}$. This fact imposes a z-component to the electric field and consequently, its polarization would be as follows:

(9)$${{\hat{\mathbf e}^{\prime}}_{in({out} )}} = \frac{1}{{{N_{in({out} )}}}}({e_{in({out} )}^x{{{\hat{\mathbf x}}}_{in({out} )}} + e_{in({out} )}^y{{{\hat{\mathbf y}}}_{in({out} )}} + e_{in({out} )}^z{{{\hat{\mathbf z}}}_{in({out} )}}} ),$$

where ${{\hat{\mathbf e}^{\prime}}_{in({out} )}}$ is the electric field polarization of the output plane wave from the polarizer. Also, normalization factor ${N_{in({out} )}}$ and $e_{in({out} )}^z$ are as follows:

(10)$$\begin{array}{l} {{{\hat{\mathbf e}^{\prime}}}_{in({out} )}} \cdot {{\mathbf k}_{in({out} )}} = 0 \Rightarrow e_{in({out} )}^z ={-} \frac{{{k_x}e_{in({out} )}^x + {k_y}e_{in({out} )}^y}}{{{k_z}}} \Rightarrow \\ {N_{in({out} )}} = \sqrt {{{|{e_{in({out} )}^x} |}^2} + {{|{e_{in({out} )}^y} |}^2} + {{|{e_{in({out} )}^z} |}^2}} = \sqrt {1 + {{|{e_{in({out} )}^z} |}^2}} . \end{array}$$

Equation (9) says with the assumption that the incident plane wave to the 2D differentiator has wave vector ${{\mathbf k}_{in}}$ in Eq. (6), the output plane wave from the input (output) polarizer has electric field ${E_{in({out} )}}{{\hat{\mathbf e}^{\prime}}_{in({out} )}}$ where ${E_{in({out} )}}$ is the electric field phasor. Due to the linearity of the structure, the ratio of ${E_{out}}$ to ${E_{in}}$ is independent of the latter and only is a function of ${k_x}$ and ${k_y}$ to which we refer as transfer function $r({{k_x},{k_y}} )$. $r({{k_x},{k_y}} )$ can be computed through the following steps:

• Step 1: applying the effect of the input polarizer. This means the incident plane wave to Ag_L₁ has electric field ${E_{in}}{{\hat{\mathbf e}^{\prime}}_{in}}$ as mentioned earlier.
• Step 2: utilizing transfer matrix $\overline{\overline {\mathbf R}} ({{k_x},{k_y}} )$ in Eq. (8). The transfer matrix can be used to compute the electric field of the reflected plane wave from Ag_L₂.
• Step 3: computing the phasor of the output plane wave from the output polarizer i.e. ${E_{out}}$. ${E_{out}}$ is, in fact, the inner product of ${{\hat{\mathbf e}^{\prime}}_{out}}$ in Eq. (9) and the electric field computed in the previous step.

Mathematically, one can apply steps 1-3 as follows:

(11)$$r({{k_x},{k_y}} )= \frac{1}{{{N_{in}}{N_{out}}}}{\left[ {\begin{array}{{ccc}} {e_x^{out}}&{e_y^{out}}&{e_z^{out}} \end{array}} \right]^ \ast }\overline{\overline {\mathbf R}} ({{k_x},{k_y}} ){\left[ {\begin{array}{{ccc}} {e_x^{in}}&{e_y^{in}}&{e_z^{in}} \end{array}} \right]^T}.$$

Similar to [11], the system introduced herein, is a differentiator for ${k_x},{k_y} \ll {k_0}$. For ${k_x},{k_y} \ll {k_0}$, the transfer matrix and the transfer function in the above equation can be approximated by their first order Taylor expansions in the following equation (see Sec. 4 in the Supplement 1 for details):

(12)$$\begin{array}{l} \overline{\overline {\mathbf R}} ({{k_x},{k_y} \ll {k_0}} )\approx \left[ {\begin{array}{{ccc}} { - {r_s}({{\theta_{ssp}}} ){{\left. {\frac{{\partial {r_p}}}{{\partial \theta }}} \right|}_{\theta = {\theta_{SPP}}}}\frac{{{k_x}}}{{{k_0}}}}&{ - r_s^2({{\theta_{ssp}}} )\left( {\frac{{{k_x}}}{{{k_0}}} + \frac{{{k_y}}}{{{k_0}}}} \right)\cot {\theta_0}}\\ 0&{ - {r_s}({{\theta_{ssp}}} ){{\left. {\frac{{\partial {r_p}}}{{\partial \theta }}} \right|}_{\theta = {\theta_{SPP}}}}\frac{{{k_y}}}{{{k_0}}}} \end{array}} \right] \Rightarrow \\ r({{k_x},{k_y} \ll {k_0}} )= {\left[ {\begin{array}{{ccc}} {e_{out}^x}&{e_{out}^y} \end{array}} \right]^ \ast } \times \overline{\overline {\mathbf R}} ({{k_x},{k_y}} )\times {\left[ {\begin{array}{{ccc}} {e_{in}^x}&{e_{in}^y} \end{array}} \right]^T} \approx \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r({{k_x} = 0,{k_y} = 0} )+ {C_x}\frac{{{k_x}}}{{{k_0}}} + {C_y}\frac{{{k_y}}}{{{k_0}}}, \end{array}$$

where

(13)$$\begin{array}{l} r({{k_x} = 0,{k_y} = 0} )= 0,\\ {C_x} ={-} {r_s}({{\theta_{ssp}}} )\frac{{\partial {r_p}}}{{\partial \theta }}e_{in}^x{({e_{out}^x} )^ \ast } - r_s^2({{\theta_{ssp}}} )\cot {\theta _0}e_{in}^y{({e_{out}^x} )^ \ast },\\ {C_y} ={-} {r_s}({{\theta_{ssp}}} )\frac{{\partial {r_p}}}{{\partial \theta }}e_{in}^y{({e_{out}^y} )^ \ast } - r_s^2({{\theta_{ssp}}} )\cot {\theta _0}e_{in}^y{({e_{out}^x} )^ \ast }. \end{array}$$

3. 2D differentiation condition

In order to make 2D isotropic differentiation possible, the transfer function in Eq. (12) must satisfies the two following conditions:

• Condition I: the value of the transfer function must be zero for the central spatial frequency component of the incident beam, i.e.: $(14)$$r({{k_x} = 0,{k_y} = 0} )= 0.$$$
• Condition II: The second condition states that ${C_x}$ and ${C_y}$ Eq. (12), must have the same magnitude and phase difference 90 degrees i.e.: $(15)$$C \buildrel \Delta \over = {C_x} ={\pm} j{C_y}.$$$

Accordingly, a non-trivial topological Charge is created in the transfer function and consequently, 2D isotropic differentiation can be achieved [11]. As it can be seen in eq. (12-13), condition I in Eq. (14) is instinctively justified. In Eq. (13), ${r_s}({{\theta_{ssp}}} ){\left. {\frac{{\partial {r_p}}}{{\partial \theta }}} \right|_{\theta = {\theta _{SPP}}}} \gg{-} r_s^2({{\theta_{SPP}}} )\cot {\theta _{SPP}}$, therefore the right-handed expression is negligible. On the other hand, the input and output polarizers i.e. ${{\hat{\mathbf e}}_{in}}$ and ${{\hat{\mathbf e}}_{out}}$ can be selected as follows:

(16)$${{\hat{\mathbf e}}_{in}} = \left[ {\begin{array}{{cc}} 1&1 \end{array}} \right]\,\,\,\,\,;\,\,\,\,\,{{\hat{\mathbf e}}_{out}} = \left[ {\begin{array}{{cc}} 1&{ \pm j} \end{array}} \right].$$

By this choice, the approximate values of ${C_x}$ and ${C_y}$ are $\frac{1}{{2}}{r_s}({{\theta_{ssp}}} )\frac{{\partial {r_p}}}{{\partial \theta }}\,$ and ${\pm} \frac{j}{{2}}{r_s}({{\theta_{ssp}}} )\frac{{\partial {r_p}}}{{\partial \theta }}$, respectively. Therefore, condition II in Eq. (15) is also satisfied.

Although ${r_s}({{\theta_{ssp}}} ){\left. {\frac{{\partial {r_p}}}{{\partial \theta }}} \right|_{\theta = {\theta _{SPP}}}} \gg{-} r_s^2({{\theta_{SPP}}} )\cot {\theta _{SPP}}$ in Eq. (13), the right-handed expression can be taken into account, namely one can select the following input and output polarizers to justify the second differentiation condition in Eq. (15) more precisely:

(17)$$e_{in}^y = e_{out}^x = {1 / {\sqrt 2 }}\,\,\,;\,\,\,e_{out}^y = {{\exp ({j\varphi_{out}^y} )} / {\sqrt 2 }}\,\,\,;\,\,\,e_{in}^x = {{\exp ({j\varphi_{in}^x} )} / {\sqrt 2 }},$$

where $\varphi _{out}^y$ and $\varphi _{out}^y$ are defined as follows (see Sec. 5 in Supplement 1 for details):

(18)$$\begin{array}{l} \varphi _{out}^y \buildrel \Delta \over = - {\varphi _0} + \measuredangle\left( {{r_s}\left( {{\theta _{ssp}}} \right)\frac{{\partial {r_p}}}{{\partial \theta }}} \right) - \measuredangle r_s^2\left( {{\theta _{ssp}}} \right) = \\ \varphi _{in}^x = - {\varphi _0} + \measuredangle\left( { - r_s^2\left( {{\theta _{ssp}}} \right)} \right) - \measuredangle\left( {{r_s}\left( {{\theta _{ssp}}} \right)\frac{{\partial {r_p}}}{{\partial \theta }}} \right), \end{array}$$

and ${\varphi _0} = {\cos ^{ - 1}}\frac{{ - \xi \pm \sqrt {2 - {\xi ^2}} }}{2}$. Table 2 shows the exact values of ${C_x}$ and ${C_y}$ calculated by Eq. (13) when the input and output polarizers are selected according to Eq. (16) and Eq. (17). Table 2 demonstrates that the 2D differentiation conditions are satisfied for both cases in Eq. (16) and (17), by the latter, however, the condition in Eq. (15) is met more precisely, thereby achieving a better performance. In Fig. (3), the transfer function has been plotted for the latter case using Eq. (20). Figure 3 demonstrates that the 2D differentiation conditions are adequately justified.

Table 2. the values of ${C_x}$ and ${C_y}$ for the input and output polarizers in Eq. (16) and (Eq. (17))

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4. Gradient operator

There exists an approximation in 2D isotropic differentiation based on justification of conditions I and II in eq. (14-15). Based on these two conditions one can write:

(19)$$ \begin{aligned} &\tilde{S}_{\text {out }}\left(k_{x}, k_{y}\right)=r\left(k_{x}, k_{y}\right) \tilde{S}_{\text {in }}\left(k_{x}, k_{y}\right)=\frac{C}{k_{0}}\left(k_{x} \pm j k_{y}\right) \tilde{S}_{\text {in }}\left(k_{x}, k_{y}\right) \Rightarrow \\ &\tilde{S}_{\text {out }}\left(k_{x}, k_{y}\right) \stackrel{F^{-1}}{\longrightarrow} \frac{C}{k_{0}}\left(-j \frac{\partial}{\partial x} \pm \frac{\partial}{\partial y}\right) S_{\text {in }}(x, y) \Rightarrow \\ &I_{\text {out }}=\left|S_{\text {out }}\left(x,y\right)\right|^{2}=\left|\frac{C}{k_{0}}\right|^{2}\left(\left|\frac{\partial S_{\text {in }}}{\partial x}\right|^{2}+\left|\frac{\partial S_{\text {in }}}{\partial y}\right|^{2} \pm 2 \operatorname{Im}\left\{\frac{\partial S_{\text {in }}}{\partial x} \frac{\partial S_{\text {in }}^{*}}{\partial y}\right\}\right), \end{aligned} $$

where ${\tilde{S}_{in({out} )}}({{k_x},{k_y}} )$ is the spatial Fourier transform of input (output) field distribution on the xy-plane in the input (output) coordinate system. ${I_{out}}$ is the intensity of the output light beam. In [11], it has been assumed that ${\mathop{\rm Im}\nolimits} \left\{ {\frac{{\partial {S_{in}}}}{{\partial x}}\frac{{\partial S_{in}^ \ast }}{{\partial y}}} \right\}$ is zero and the intensity of the output beam is, in turn, proportional to ${|{\nabla {S_{in}}} |^2}$. This assumption, however, is not always true.

In Eq. (13), ${r_s}({{\theta_{ssp}}} ){\left. {\frac{{\partial {r_p}}}{{\partial \theta }}} \right|_{\theta = {\theta _{SPP}}}} \ll{-} r_s^2({{\theta_{SPP}}} )\cot {\theta _{SPP}}$, therefore transfer matrix $\overline{\overline {\mathbf R}} ({{k_x},{k_y}} )$ can be regarded as a diagonal matrix in Eq. (12). Let's delete the output polarizer from the proposed structure i.e. excluding ${\left[ {\begin{array}{{cc}} {e_{out}^x}&{e_{out}^y} \end{array}} \right]^ \ast }$ from Eq. (12). Hence, the field vector of the output light beam can be considered as $\tilde{S}_{out}^{(x )}({{k_x},{k_y}} ){{\hat{\mathbf x}}_{out}} + \tilde{S}_{out}^{(y )}({{k_x},{k_y}} ){{\hat{\mathbf y}}_{out}}$ in the spatial Fourier domain. Let's select $e_{in}^x ={-} e_{in}^y = {1 / {\sqrt 2 }}$ as the input polarizer in Eq. (12), Therefore, one can write:

(20)$$\tilde{S}_{out}^{x(y )}({{k_x},{k_y}} )= {C^{\prime}_{x(y )}}\frac{{{k_{x(y )}}}}{{{k_0}}}{\tilde{S}_{in}}({{k_x},{k_y}} )\Rightarrow s_{out}^{x(y )}({x,y} )={-} j\frac{{C^{\prime}_{x(y )}}}{{k_0}}{\frac{\partial }{{\partial x({\partial y} )}}{s_{in}}({x,y} )} ,$$

where ${C^{\prime}_x} = {C^{\prime}_y} \buildrel \Delta \over = \frac{1}{{\sqrt 2 }}{r_s}({{\theta_{ssp}}} ){\left. {\frac{{\partial {r_p}}}{{\partial \theta }}} \right|_{\theta = {\theta _{SPP}}}}$ and $s_{out}^x({x,y} ){{\hat{\mathbf x}}_{out}} + s_{out}^y({x,y} ){{\hat{\mathbf y}}_{out}}$ is the field vector of the output light beam in the spatial domain. ${s_{in}}({x,y} )$(${\tilde{S}_{in}}({{k_x},{k_y}} )$) is the field distribution of the input light beam in the spatial (Fourier) domain. Therefore, the intensity of the output light beam to which is dented by ${I_{out}}$ would be as follows:

(21)$$ \begin{aligned} &I_{o u t}=\left|s_{o u t}^{x}(x, y)\right|^{2}+\left|s_{o u t}^{y}(x, y)\right|^{2}= \\ &\frac{G^{2}}{k_{0}^{2}}\left|\frac{\partial}{\partial x} s_{i n}(x, y)\right|^{2}+\frac{G^{2}}{k_{0}^{2}}\left|\frac{\partial}{\partial y} s_{i n}(x, y)\right|^{2}=\frac{G^{2}}{k_{0}^{2}}\left|\nabla_{s_{i n}}(x, y)\right|^{2}, \end{aligned} $$

where $G \buildrel \Delta \over = |{{{C^{\prime}}_x}} |= |{{{C^{\prime}}_y}} |$. Eq. (20-21) demonstrate that the proposed structure is capable of computing the gradient of a polarized light beam. In comparison with the differentiator in [11], our structure is able to calculate the exact value of the gradient of the input light. The effect can especially be observed when the differentiator is used for phase mining proposes. Note that in eq.(20), ${{{{C^{\prime}}_{x(y )}}{k_{x(y )}}} / {{k_0}}}$ is the first order Taylor expansion of transfer function ${r_{x(y )}}({{k_x},{k_y}} )$ that can be obtained by exclusion of the out polarizer from Eq. (20):

(22)$${r_{x(y )}}({{k_x},{k_y}} )= \frac{1}{{{N_{in}}}}\left[ {\begin{array}{{ccc}} {1(0 )}&{0(1 )}&0 \end{array}} \right]\overline{\overline {\mathbf R}} ({{k_x},{k_y}} )\left[ {\begin{array}{{c}} {e_x^{in}}\\ {e_y^{in}}\\ {e_z^{in}} \end{array}} \right]\,,$$

where $e_{in}^x = e_{in}^y = {1 / {\sqrt 2 }}$ and also, $e_{in}^z$ and ${N_{in}}$ are determined by Eq. (10). Later in section 6, Eq. (22) is used to calculate the FOM values of the proposed gradient operator thereby appraising its performance in two dimensional differentiation. In Fig. 4(a-b), the magnitude and phase ${r_x}({{k_x},{k_y} = 0} )$ and ${r_y}({{k_x} = 0,{k_y}} )$ in the above equation have been plotted. In addition, Fig. 4(c-d) show the phases and magnitudes of the transfer function and its first order Taylor expansion corresponding to the one-dimensional plasmonic differentiator in [43]. Figure 4 demonstrates that the proposed gradient operator features the same bandwidth in the spatial Fourier domain as the one dimensional plasmonic differentiator in [43]. As we will see later in the next section, $|C |$ and G in Eq. (15) and (21) can be regarded as the differentiation gain corresponding to the proposed two-dimensional differentiator and the gradient operator, respectively. Comparison of Fig. 3(a) and Fig. 4(a), on the other hand, reveals that having the same bandwidth in the spatial Fourier domain, the gradient operator enjoys a greater differentiation gain in contrast with the proposed two-dimensional differentiator.

Fig. 4. (a) the magnitude and (b) the phase of ${r_x}({{k_x},{k_y} = 0} )$ and ${r_y}({{k_x} = 0,{k_y}} )$ in Eq. (22) alongside their first order Taylor expansions in Eq. (20). (c) the magnitudes and (d) the phases of the transfer function and its first order Taylor expansion corresponding to the one-dimensional plasmonic differentiator in [43]. Note that in (a), $G^{\prime} \approx 133$. This Figure demonstrate that the bandwidth of the one-dimensional plasmonic differentiator in [43] has been preserved by the proposed gradient operator in the spatial Fourier domain.

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5. Divergence operator

As mentioned previously, a periodic plasmonic metasurface has been exploited in [48] to perform divergence operation on vectorial paraxial incident beams. The metasurface is comprised of a thick gold-layer substrate and a silicon slot grating onto which a monolayer of graphene is transferred [48]. In order to perform divergence operation, the graphene surface is illuminated by a vectorial paraxial beam and the reflected beam's electric field component parallel to the incident plane is, in fact, the divergence of the electric field vector of the incident beam. Note that the grating period has been selected such that the reflected beam only contains the zeroth diffraction order while higher diffraction orders are evanescent. The differentiation in the direction parallel to the incident plane is carried out by excitation of SPP mode at the air-graphene-silicone interface, therefore it is affected mostly by the slot sizes of the grating [48]. Differentiation along the perpendicular direction to the incident plane, on the other hand, is mostly dependent to the incident angle and additionally, the refractive indices of the materials of the periodic plasmonic metasurface whereas it is insensitive to the grating dimensions. These two derivative results interfere constructively thereby computing the divergence of the incident field. Here in this section, we show that our proposed structure is also capable of performing divergence operation on a unpolarized paraxial incident beam.

Thanks to the diagonality of $\overline{\overline {\mathbf R}} ({{k_x},{k_y}} )$, our structure can calculate the divergence of an unpolarized light beam. For this purpose, the output polarizer is deleted from the proposed structure i.e. excluding ${\left[ {\begin{array}{{cc}} {e_{in}^x}&{e_{in}^y} \end{array}} \right]^t}$ from Eq. (12). Let's assume the field vector of the input light beam and its spatial Fourier transform are $s_{in}^x({x,y} ){{\hat{\mathbf x}}_{in}} + s_{in}^y({x,y} ){{\hat{\mathbf y}}_{in}}$ and $\tilde{S}_{in}^x({{k_x},{k_y}} ){{\hat{\mathbf x}}_{in}} + \tilde{S}_{in}^y({{k_x},{k_y}} ){{\hat{\mathbf y}}_{in}}$, respectively, and additionally $e_{out}^x = e_{out}^y = {1 / {\sqrt 2 }}$ is selected as the output polarizer in Eq. (12). Thus, the field distribution of the output beam i.e. ${s_{out}}({x,y} )$ and its spatial Fourier transform i.e. ${\tilde{S}_{out}}({{k_x},{k_y}} )$ would be as follows:

(23)$$ \begin{aligned} &\tilde{S}_{\text {out }}\left(k_{x}, k_{y}\right)=C_{x}^{\prime} \frac{k_{x}}{k_{0}} \tilde{S}_{i n}^{x}\left(k_{x}, k_{y}\right)+C_{x}^{\prime} \frac{k_{y}}{k_{0}} \tilde{S}_{i n}^{y}\left(k_{x}, k_{y}\right) \Rightarrow \\ &s_{\text {out }}(x, y)=-j \frac{C_{x}^{\prime}}{k_{0}} \frac{\partial}{\partial x} s_{i n}^{x}(x, y)-j \frac{C_{x}^{\prime}}{k_{0}} \frac{\partial}{\partial y} s_{i n}^{y}(x, y) . \end{aligned} $$

The above equation shows the field distribution of the output beam is, in fact, the divergence of the field of the unpolarized input beam.

Note that the gain of our proposed divergence operator (i.e. G in eq. (21, 23) and Fig. 1(a)) is approximately 96 while the gain of the divergence operator in [48] is about 1. On the other hand, the divergence operator in [48] works within the terahertz region (i.e. frequency of 5.368 THz) while our proposed structure operates in the visible light region (i.e. wavelength of 532 nm). Furthermore, our proposed divergence operator is simpler to fabricate in comparison with its counterpart in [48] which require surface patterning.

6. FOM calculation

It is desirable to acquire high gain and low resolution for spatial differentiators. In the real world, however, there is a trade-off between gain and resolution, i.e. An improvement in gain (resolution) entails a deterioration in resolution (gain). The best example to observe this trade-off is the ideal, yet physical differentiators in [1] which have the following transfer function:

(24)$$|{{r_0}({{k_x}} )} |\buildrel \Delta \over = \left\{ \begin{array}{l} G|{{{{k_x}} / {{k_0}}}} |\,\,\,\,\,\,\,\,\,\,|{{{{k_x}} / {{k_0}}}} |\le {1 / G}\\ 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{1 / G} \le |{{{{k_x}} / {{k_0}}}} |\le 1 \end{array} \right.\,\,\,\,;\,\,\,\measuredangle {r_0}({{k_x}} )= \left\{ \begin{array}{l} \frac{\pi }{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,|{{{{k_x}} / {{k_0}}}} |> 0\\ - \frac{\pi }{2}\,\,\,\,\,\,\,\,\,\,\,\,\,\,|{{{{k_x}} / {{k_0}}}} |< 0 \end{array} \right.\,\,\,,$$

where G is the gain of the differentiator. Note that any increase (decrease) in the gain of the differentiator in Eq. (24) leads to a decrease (increase) in the spatial frequency interval in which the system operates as a perfect differentiator. Therefore, an increase (decrease) in the gain is resulted in an increase (decrease) in the resolution of the differentiator. In [1], it has been shown that there is a linear relation between gain and resolution for the differentiator in Eq. (24): $\Delta x - 0.5 = 0.345({G - 1} )$ where $\Delta x$ is resolution of the differentiator that is normalized to the wavelength of the input beam. Note that resolution $\Delta x$ is computed using Rayleigh criterion [1]. Since, most of the differentiators in optics have transfer functions similar to the differentiator in Eq. (24), the linear relation corresponding to the differentiator in Eq. (24) has been used as the figure of merit (FOM) in [1]:

(25)$$FOM = \left|{\frac{{\Delta x - 0.5}}{{G - 1}}} \right|.$$

Hence, FOM is a criterion by which the performances of a given differentiator and the ideal, yet physical differentiator in Eq. (24) are mathematically compared. In other words, it is possible to quantify the similarity between the transfer functions corresponding to a given differentiator and the ideal, yet physical differentiator by means of FOM. The lower FOM means the differentiator has a better performance and the minimum achievable FOM for a given one-dimensional differentiator is 0.345 [1]. Consequently, the closer the FOM value to 0.345, more similar its performance to the ideal, yet physical differentiator in Eq. (24).

In Eq. (25), Rayleigh criterion is used to determine the resolution of a given differentiator. For this purpose, an incident beam with a pulse-shape field distribution is utilized as the input to the differentiator. According to Rayleigh criterion, the differentiator is able to detect the edges of the input pulse beam, if the maximum point of one edge signal and the first zero of another edge signal coincide with one another [1]. In other word, Rayleigh criterion states that the first zero of the step response of the differentiator is, in fact, the resolution of the differentiator. Unlike the even step response of the differentiator in Eq. (24), the step response of a practical differentiator does not often have even or odd symmetry around $x = 0$. In such cases, it is required to determine the first positive and negative zeros of the step response and the zero with higher absolute values are selected as the resolution. Therefore, the worst case scenario is considered as the resolution. On most occasions, however, the step response does not have exact positive or negative zeros. In such cases, the first positive (negative) zero is consider as the maximum (minimum) value of x at which the pulse response magnitude equals 1/20 of its peak value [1].

Although, the FOM in [1] is introduced as a criterion to evaluate the performance of one dimensional differentiators, it can also assess the proposed structures using the following equation:

(26)$$FO{M_x} = \left|{\frac{{\Delta x - 0.5}}{{{G_x} - 1}}} \right|\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,\,FO{M_y} = \left|{\frac{{\Delta y - 0.5}}{{{G_y} - 1}}} \right|,$$

where FOM_x and FOM_y, denote the FOM value for the two dimensional differentiator (or gradient operator) in x- and y-directions. For the two dimensional differentiator, ${G_{x(y )}}$ is $|{{C_{x(y )}}} |$ in Eq. (12) and $\Delta x$ ($\Delta y$) is the resolution of $r({{k_x},{k_y} = 0} )$ ($r({{k_x} = 0,{k_y}} )$) in Eq. (11). For the gradient operator, ${G_{x(y )}}$ is $|{{{C^{\prime}}_{x(y )}}} |$ in Eq. (20) and $\Delta x$ ($\Delta y$) is the resolution of ${r_x}({{k_x},{k_y} = 0} )$ (${r_y}({{k_x} = 0,{k_y}} )$) in Eq. (22). Regarding the differentiators in [11], the gain of the both setups have been calculated in [11] while only the first order expansion of the transfer functions has analytically been obtained which is only valid for ${k_x},{k_y} \ll {k_0}$, yet in order to compute the resolution in x- and y-directions, one must calculate the corresponding transfer function for any given value of ${k_x}$(${k_y}$) when ${k_y} = 0$(${k_x} = 0$). Following a similar procedure to the steps in Sec. 3 in Supplement 1 that has led to Eq. (11), the transfer function of the differentiator in [11] can be obtained as follows:

(27)$$r^{\prime}({{k_x},{k_y}} )= \frac{1}{{{N_{in}}{N_{out}}}}{\left[ {\begin{array}{{ccc}} {e_x^{out}}&{e_y^{out}}&{e_z^{out}} \end{array}} \right]^ \ast }{{\mathbf M^{\prime}}_1}{{\mathbf M^{\prime}}_2}{{\mathbf M^{\prime}}_3}\left[ {\begin{array}{{c}} {e_x^{in}}\\ {e_y^{in}}\\ {e_z^{in}} \end{array}} \right],$$

where

(28)$$\begin{array}{l} {({{{{\mathbf M^{\prime}}}_1}} )^T} \buildrel \Delta \over = \left[ {\begin{array}{{cc}} {\frac{{({k_y^2 + k_z^2} )\sin {\theta_0} + {k_z}{k_x}\cos {\theta_0}}}{{k_0^2\sin \theta }}}&{\frac{{ - {k_y}\cos {\theta_0}}}{{{k_0}\sin \theta }}}\\ {\frac{{{k_z}{k_y}\cos {\theta_0} - {k_y}{k_x}\sin {\theta_0}}}{{k_0^2\sin \theta }}}&{\frac{{{k_z}\sin {\theta_0} + {k_x}\cos {\theta_0}}}{{{k_0}\sin \theta }}}\\ {\frac{{ - ({k_x^2 + k_y^2} )\cos {\theta_0} - {k_z}{k_x}\sin {\theta_0}}}{{k_0^2\sin \theta }}}&{\frac{{ - {k_y}\sin {\theta_0}}}{{{k_0}\sin \theta }}} \end{array}} \right],\\ {{{\mathbf M^{\prime}}}_2} \buildrel \Delta \over = \left[ {\begin{array}{{cc}} {{{r^{\prime}}_p}(\theta )}&0\\ 0&{{{r^{\prime}}_s}(\theta )} \end{array}} \right],\\ {{{\mathbf M^{\prime}}}_3} \buildrel \Delta \over = \left[ {\begin{array}{{cc}} { - \frac{{({k_y^2 + k_z^2} )\sin {\theta_0} + {k_z}{k_x}\cos {\theta_0}}}{{k_0^2\sin \theta }}}&{\frac{{ - {k_y}\cos {\theta_0}}}{{{k_0}\sin \theta }}}\\ {\frac{{ - {k_z}{k_y}\cos {\theta_0} + {k_y}{k_x}\sin {\theta_0}}}{{k_0^2\sin \theta }}}&{\frac{{{k_z}\sin {\theta_0} + {k_x}\cos {\theta_0}}}{{{k_0}\sin \theta }}}\\ {\frac{{ - ({k_x^2 + k_y^2} )\cos {\theta_0} - {k_z}{k_x}\sin {\theta_0}}}{{k_0^2\sin \theta }}}&{\frac{{{k_y}\sin {\theta_0}}}{{{k_0}\sin \theta }}} \end{array}} \right], \end{array}$$

where upper index T denotes transpose and $\theta = {\cos ^{ - 1}}({{{|{{k_x}\sin {\theta_0} - {k_z}\cos {\theta_0}} |} / {{k_0}}}} )$. ${r^{\prime}_{p,s}}(\theta )$ denote Fresnel reflection coefficients from a glass-air interface. In [11], two setups have been introduced as two dimensional differentiator. The first (second) setup acts based on Brewster effect (total internal reflection), therefore the incident medium is air (glass) and ${\theta _0}$ is 57.6° (47.0°.). For the Brewster (total internal reflection) differentiator, $e_x^{in}$, $e_y^{in}$, $e_x^{out}$ and $e_y^{out}$ are -0.38i, 0.93, 1 and 0 (0.39 - 0.59i, -0.19 - 0.68i, -0.71 and 0.71,), respectively. $e_z^{in({out} )}$ and ${N_{in({out} )}}$ are determined by Eq. (10).

To compute FOM_x and FOM_y, there is a consideration one must take into account. In order for a plane wave to be incident onto Ag_L₁, ${{\mathbf k}_{in}} \cdot {{\hat{\mathbf n}}_1} \le 0$ in Fig. 1(b). The reflected plane wave from Ag_L₁ must be incident upon Ag_L₂, therefore, ${{\mathbf k}_m} \cdot {{\hat{\mathbf n}}_2} \le 0$ in Fig. 1(c). Since light is incident from air onto the proposed structure, ${k_x}$ and ${k_y}$ cannot be greater than ${k_0}$. Consequently, the value of ${k_x}$ (${k_y}$) in $r({{k_x},{k_y} = 0} )$ and ${r_x}({{k_x},{k_y} = 0} )$($r({{k_x} = 0,{k_y}} )$ and ${r_y}({{k_x} = 0,{k_y}} )$) in Eq. (11) and Eq. (22) lies in interval $({ - {k_0},\min ({{k_0}{n_g}\cos {\theta_{SPP}},{k_0}} )} )$ where ${n_g}$ is the refractive index of glass. Similarly, for the differentiators in [11], the intervals in which ${k_x}$ and ${k_y}$ vary in Eq. (28), are $({ - {k_0},\min ({{k_0}{n_{inc}}\cos {\theta_0},{k_0}} )} )$ and $({ - {k_0},{k_0}} )$, respectively, where ${n_{inc}}$ is the refractive index of the incident medium.

Using Eq. (11) and Eq. (22), the magnitudes and phases of transfer functions $r({{k_x} = 0,{k_y}} )$, $r({{k_x},{k_y} = 0} )$, ${r_x}({{k_x},{k_y} = 0} )$ and ${r_y}({{k_x} = 0,{k_y}} )$ corresponding to the proposed two-dimensional differentiator and gradient operator have been plotted in Fig. 5(a-d). Figure 6 shows the output beam field distribution of the proposed two-dimensional differentiator and gradient operator when the input beam has a pulse-form distribution with a pulse width equal to the resolution in the input coordinate system i.e. $\Pi ({{x / {\Delta x}}} )$ ($\Pi ({{y / {\Delta y}}} )$). Figure 6 demonstrates that the Rayleigh criterion has been met. Table 3 shows the features of the proposed structure alongside the differentiators in [7,11,35]. In Table 3, the gain and resolution of the first-order differentiators in [7,35] are estimated from the corresponding reported data. Furthermore, FOM results corresponding to a number of first-order, one-dimensional differentiators have been calculated in [1] and are summarized in Table 4 for comparison purposes. FOM values in Table 3 demonstrate that our proposed structures enjoys a much better performance in edge detection and two dimensional differentiation comparing to its other counterparts introduced in [7,11,35]. In addition, the proposed structure enjoy a much greater differentiation gain in contrast with its other competitors in [7,11,35]. On the other hand, Table 4 shows the FOM values corresponding to the proposed structure are comparable to one dimensional differentiators with lower values of FOM and consequently, better performances. Therefore, the proposed structure can be an excellent option for two-dimensional edge detection purposes.

Fig. 5. (a) The magnitudes and (b) the phases of transfer functions $r({{k_x},{k_y} = 0} )$ and $r({{k_x} = 0,{k_y}} )$ corresponding to the plasmonic two-dimensional differentiator in Eq. (11).(c) The magnitudes and (d) the phases of transfer functions ${r_x}({{k_x},{k_y} = 0} )$ and ${r_y}({{k_x} = 0,{k_y}} )$ corresponding to the plasmonic gradient operator in Eq. (22).

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Fig. 6. The output beam filed distribution of (a) the plasmonic two-dimensional differentiator and (b) the gradient operator corresponding to an input beam with a pulse-form field distribution whose pulse width is $\Delta x$($\Delta y$) i.e. $\Pi ({{x / {\Delta x}}} )$ ($\Pi ({{y / {\Delta y}}} )$) in the input coordinate system. Note that λ denotes the wavelength of the incident beam.

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Table 3. features of the proposed 2D differentiator alongside its counterparts

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Table 4. FOM values corresponding to different types of one-dimensional differentiators calculated in [1].

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7. Conclusion

Herein, we have theoretically demonstrated high-gain two dimensional isotropic differentiation based on surface plasmon polariton excitation on two plasmonic metal-air interfaces using Kretschmann configuration. We have shown that two dimensional differentiation can be performed either by generating a non-trivial topological charge in the transfer function or by diagonality of the transfer matrix of the proposed structure. The latter, however, is in fact a gradient operator whereby a more precise two dimensional edge detection can be obtained. In addition, our gradient operator has especially provided an adequate platform for phase mining applications. Furthermore, it enjoys a simpler structure since its input light beam is unpolarized thereby eleminating the need for an input polarizer. The proposed structure can be exploited as a divergence operator thanks to diagonality of its transfer function. The performance of the proposed structure has been assessed and compared with its other counterparts by the figure of merit developed in [1]. The FOM results have proven the dominance of the proposed structure over its other components.

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.

Supplemental document

See Supplement 1 for supporting content.

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Line Segment	Length
$\bar{A B}$	L
$\bar{A C}$	1.45L
$\bar{A D}$	1.78L
$\bar{B C}$	L
$\bar{B D}$	1.45L
$\bar{C D}$	L
$\bar{A M_{1}}$	0.62L
$\bar{O_{1} M_{1}}$	0.25L
$\bar{A M_{2}}$	0.62L
$\bar{O_{2} M_{2}}$	0.35L
$\bar{A M_{3}}$	0.87L
$\bar{O_{3} M_{3}}$	0.22L
$\bar{B M_{4}}$	0.59L
$\bar{O_{4} M_{4}}$	0.22L

${\hat{e}}_{i n}$ and ${\hat{e}}_{o u t}$	$\| C_{x} \|$	$\| C_{y} \|$	$∡ C_{x}$	$∡ C_{y}$
(Eq. (16)) (+ sign)	67.4511	67.8959	94.7763^°	184.9161^°
Equation (16) (- sign)	68.0618	67.8959	274.5415^°	184.9161^°
Eq. (17)	68.2098	68.2098	205.3204^°	115.0874^°

Type	Resolution in x-direction	Resolution in y-direction	$F O M_{x}$	$F O M_{y}$	Gain
Proposed two-dimensional differentiator	$Δ x =$ 62.30	$Δ y =$ 64.1	0.92	0.95	68.21
Proposed divergence Operator	$Δ x =$ 62.60	$Δ y =$ 64.06	0.65	0.67	96.46
Brewster differentiator in [11]	$Δ x =$ 4.14	$Δ y =$ 1.01	4.85	0.68	0.25
Total internal reflection differentiator in [11]	$Δ x =$ 11.12	$Δ y =$ 2.66	13.40	2.72	0.21
1^st order, 2D differentiator in [7]	$Δ x >$ 32.3	$Δ y >$ 32.3	>61	>61	0.48
1^st order, 2D differentiator in [35]	$Δ x \approx 20$	$Δ y \approx 20$	≈442	≈442	1.043

Type	FOM
Ideal, physical differentiator in [1]	0.345
Brewster differentiator in [3]	41.35
Dielectric slab differentiator in [1]	0.385
Half-wavelength slab differentiator in [4]	19.9
Plasmonic differentiator in [43]	0.47
Spin hall effect differentiator in [34]	0.52

Line Segment	Length
$\bar{A B}$	L
$\bar{A C}$	1.45L
$\bar{A D}$	1.78L
$\bar{B C}$	L
$\bar{B D}$	1.45L
$\bar{C D}$	L
$\bar{A M_{1}}$	0.62L
$\bar{O_{1} M_{1}}$	0.25L
$\bar{A M_{2}}$	0.62L
$\bar{O_{2} M_{2}}$	0.35L
$\bar{A M_{3}}$	0.87L
$\bar{O_{3} M_{3}}$	0.22L
$\bar{B M_{4}}$	0.59L
$\bar{O_{4} M_{4}}$	0.22L

All optical divergence and gradient operators using surface plasmon polaritons

Abstract

1. Introduction

2. Plasmonic structure: layout and performance

3. 2D differentiation condition

4. Gradient operator

5. Divergence operator

6. FOM calculation

7. Conclusion

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (6)

Tables (4)

Equations (28)

Optics Express