Flat metal-dielectric grating with 100% retro-diffraction efficiency: rigorous theory

Let us start with the analysis of the TM polarization. In this case, the magnetic field intensity has the only component ${H_z}$ while the electric field intensity has two components, ${E_x}{\text{ and }}{E_y}.$ The electromagnetic field vectors of the incident plane wave are

$$\begin{align} {{\mathbf{H}}_i}(x,y) & = {H_i}{{\mathbf{z}}^0}{e^{i{k_0}({p_0}x - {q_0}y)}},\nonumber \\ {{\mathbf{E}}_i}(x,y) & = \frac{{{Z_0}}}{{{\varepsilon _a}}}({q_0}{{\mathbf{x}}^0} + {p_0}{{\mathbf{y}}^0}){H_i}{e^{i{k_0}({p_0}x - {q_0}y)}}, \end{align} \tag{ 1 }$$

where ${Z_0} = \sqrt {{\mu _0}/{\varepsilon _0}}$ is the free-space impedance. The complex representation of the time harmonic dependence of the form $\exp ( - i\omega {} t)$ is supposed and suppressed. The layered structure of the grating allows us to calculate the distributions of tangential field components ${H_z}{\text{ and }}{E_x}$ only; the vertical field component ${E_y}$ can then be easily found from Maxwell equations. Due to the periodicity of the grating structure, the field distribution of the reflected wave in the region $y > 0$ above the grating is given by the superposition of plane waves—diffraction orders of the grating:

$$\begin{equation}\begin{array}{*{20}{l}} {{H_r}(x,y)}&{ = \sum\limits_{m = - \infty }^\infty {{H_{rm}}{e^{i{k_0}({p_m}x + {q_{am}}y)}}} ,{\textrm{ }}{p_m} = {p_0} + m\lambda {\textrm{ /}}\Lambda ,}\\ {{E_{xr}}(x,y)}&{ = \sum\limits_{m = - \infty }^\infty {{E_{xrm}}{e^{i{k_0}({p_m}x + {q_{am}}y)}}} ,{\textrm{ }}{q_{am}} = \sqrt {{\varepsilon _a} - p_m^2} ,}\\ {{E_{xrm}}}&{ = - \frac{{{Z_0}}}{{{\varepsilon _a}}}{q_{am}}{H_{rm}}.} \end{array}\end{equation} \tag{ 2 }$$

The field within the dielectric layer between the metal layers is composed of plane waves propagating upwards and downwards with respect to the coordinate axis y:

$$\begin{align} {H_d} & = \sum\limits_{m = - \infty }^\infty {(H_{dm}^ + {e^{i{k_0}{q_{dm}}y}} + H_{dm}^ - {e^{ - i{k_0}{q_{dm}}y}}){} {} {e^{i{k_0}{p_m}x}}} , \nonumber\\ {E_{xd}} & = \sum\limits_{m = - \infty }^\infty {(E_{dm}^ + {e^{i{k_0}{q_{dm}}y}} + E_{dm}^ - {e^{ - i{k_0}{q_{dm}}y}}){} {} {e^{i{k_0}{p_m}x}}} , \\ E_{dm}^ \pm & = \mp \frac{{{Z_0}}}{{{\varepsilon _d}}}{q_{dm}}H_{dm}^ \pm ,{\text{ }}{q_{dm}} = \sqrt {{\varepsilon _d} - p_m^2} . \nonumber \end{align} \tag{ 3 }$$

In the last equation in (3), all upper or all lower signs are to be taken simultaneously. At the bottom interface with the metal, $y = - d,$ the tangential component of the electric field intensity, ${E_{xd}},$ must be equal to zero for any x. From (3) it immediately follows that

$$\begin{equation}H_{dm}^ + = H_{dm}^ - {e^{2i{k_0}{q_{dm}}d}}.\end{equation} \tag{ 4 }$$

The tangential field components just below the metal stripes at $y = 0$ can then be expressed by using of only one set of amplitudes:

$$\begin{align} {H_d}(x,0) & = \sum\limits_{m = - \infty }^\infty {\left( {1 + {e^{2i{k_0}{q_{dm}}d}}} \right)H_{dm}^ - {e^{i{k_0}{p_m}x}}} , \nonumber\\ {E_{xd}}(x,0) & = \frac{{{Z_0}}}{{{\varepsilon _d}}}\sum\limits_m {{q_{dm}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right)H_{dm}^ - } {e^{i{k_0}{p_m}x}}. \end{align} \tag{ 5 }$$

At this interface, the boundary conditions are more complicated. At the metalized sections, the tangential components of the electric field intensity must vanish both above and below the interface:

$$\begin{equation}\begin{array}{l} {E_{xi}}(x,0) + {E_{xr}}(x,0) = 0,{\textrm{ }}{E_{xd}}(x,0) = 0,\\ x \in \left\langle { - a/2 + m\Lambda ,{\textrm{ }}a/2 + m\Lambda } \right\rangle . \end{array}\end{equation} \tag{ 6 }$$

Within the openings between the metal stripes, the tangential components of both electric and magnetic fields must be continuous:

$$\begin{equation}\begin{array}{l} {E_{xi}}(x,0) + {E_{xr}}(x,0) - {E_{xd}}(x,0) = 0,\\ {H_i}(x,0) + {H_r}(x,0) - {H_d}(x,0) = 0,\\ x \notin \left\langle { - a/2 + m\Lambda ,{\textrm{ }}a/2 + m\Lambda } \right\rangle . \end{array}\end{equation} \tag{ 7 }$$

In order to cope with these complicated boundary conditions, we introduce a periodic rectangular function $f(x)$

$$\begin{equation}f(x) = \left\{ {\begin{array}{*{20}{c}} {1,{\textrm{ }}x \in \left\langle { - a/2 + m\Lambda ,{\textrm{ }}a{\textrm{/}}2 + m\Lambda } \right\rangle ,}\\ {0,{\textrm{ }}x \notin \left\langle { - a/2 + m\Lambda ,{\textrm{ }}a/2 + m\Lambda } \right\rangle ,} \end{array}} \right.\end{equation} \tag{ 8 }$$

the graph of which is shown in figure 2.

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Then, the boundary conditions at the metalized parts can be rewritten in the following way:

$$\begin{equation}\begin{gathered} f(x)\left[ {{E_{xi}}(x,0) + {E_{xr}}(x,0)} \right] = 0, \\ f(x){E_{xd}}(x,0) = 0. \\ \end{gathered} \end{equation} \tag{ 9 }$$

As long as the conditions (9) are satisfied, we can avoid the specification of the region of validity of the continuity condition (7) for the electric field since outside the openings, the tangential components of the electric field vanish. Thus, we can write the continuity conditions (7) in the form

$$\begin{equation}\begin{gathered} {E_{xi}}(x,0) + {E_{xr}}(x,0) - {E_{xd}}(x,0) = 0, \\ \left[ {{H_i}(x,0) + {H_r}(x,0) - {H_d}(x,0)} \right]\left[ {1 - f(x)} \right] = 0, \\ \end{gathered} \end{equation} \tag{ 10 }$$

which is now valid for any x. In view of the first equation in (10), it is sufficient to satisfy just one of the equation (9). However, it means that we still have three equations for two unknown fields ${H_r}{\text{ and }}{H_d}$. Inspired with [17], we reduce the number of equations by introducing a dimensionless parameter g and joining two of the three equations into one. We obtain

$$\begin{equation}\begin{gathered} {E_{xi}}(x,0) + {E_{xr}}(x,0) - {E_{xd}}(x,0) = 0, \\ g{Y_0}f(x){E_{xd}}(x,0) + {\text{ }} \\ + \left[ {1 - f(x)} \right]\left[ {{H_i}(x,0) + {H_r}(x,0) - {H_d}(x,0)} \right] = 0, \\ \end{gathered} \end{equation} \tag{ 11 }$$

where ${Y_0} = 1/{Z_0}$ is the vacuum admittance. The value of g is rather arbitrary; the number close to one is a good choice.

Equations (11) can be expressed with the help of the complex amplitudes ${H_i},{\text{ }}{H_{rm}}{\text{, and }}H_{dm}^ -$ only as follows:

$$\begin{equation}\begin{gathered} \frac{{{q_0}}}{{{\varepsilon _a}}}{H_i} - \frac{1}{{{\varepsilon _a}}}\sum\limits_m {{q_{am}}{H_{rm}}{e^{imKx}}} - \hfill \\ - \frac{1}{{{\varepsilon _d}}}\sum\limits_m {{q_{dm}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right)H_{dm}^ - } {e^{imKx}} = 0, \hfill \\ f(x)\frac{g}{{{\varepsilon _d}}}\sum\limits_m {{q_{dm}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right)H_{dm}^ - } {e^{imKx}} + \hfill \\ + \left[ {1 - f(x)} \right]\left[ {{H_i} + \sum\limits_m {{H_{rm}}{e^{imKx}}} } \right. - \hfill \\ - \left. {\sum\limits_m {\left( {1 + {e^{2i{k_0}{q_{dm}}d}}} \right)H_{dm}^ - {e^{imKx}}} } \right] = 0. \hfill \\ \end{gathered} \end{equation} \tag{ 12 }$$

Here, we used the substitution ${k_0}{p_m} = {k_0}{p_0} + mK,$ where $K = 2\pi /\Lambda$ is the grating constant, and the exponential ${e^{i{k_0}{p_0}x}}$ (occurring in all terms) was omitted. Now we can use the first equation in (12) to exclude the terms ${H_{rm}}$ from the second one. By comparing of the exponential terms ${e^{imKx}}$, we obtain

$$\begin{equation}{H_{rm}} = {H_i}{\delta _{m0}} - \frac{{{\varepsilon _a}{q_{dm}}}}{{{\varepsilon _d}{q_{am}}}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right)H_{dm}^ - .\end{equation} \tag{ 13 }$$

The substitution for ${H_{rm}}$ from (13) into the second equation in (12) results after some further manipulations in the following equation:

$$\begin{equation}\begin{gathered} g\,f(x)\sum\limits_m {\frac{{{q_{dm}}}}{{{\varepsilon _d}}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right)H_{dm}^ - } {e^{imKx}} + \hfill \\ \quad + \left[ {1 - f(x)} \right]\sum\limits_m {{e^{imKx}}\left\{ {2{\delta _{m0}}{H_i} - \begin{array}{*{20}{c}} {} \\ {} \end{array}} \right.} \hfill \\ \quad \left. { - \left[ {\frac{{{\varepsilon _a}{q_{dm}}}}{{{q_{am}}{\varepsilon _d}}}\left( {1 - {e^{2i{k_0}q{\eta _{dm}}d}}} \right) + \left( {1 + {e^{2i{k_0}{q_{dm}}d}}} \right)} \right]H_{dm}^ - } \right\} = 0. \hfill \\ \end{gathered} \end{equation} \tag{ 14 }$$

In order to solve this equation for $H_{dm}^ -$, the periodic functions $f(x)$ and $1 - f(x)$ have to be expanded into Fourier series:

$$\begin{equation}\begin{gathered} f(x) = \sum\limits_{l = - \infty }^\infty {{f_l}{e^{ilKx}}} ,{\text{ }}{f_l} = \frac{a}{\Lambda }\operatorname{sinc} \left( {\frac{{l\pi a}}{\Lambda }} \right), \\ 1 - f(x) = \sum\limits_{l = - \infty }^\infty {\left( {{\delta _{0l}} - {f_l}} \right){e^{ilKx}}} . \\ \end{gathered} \end{equation} \tag{ 15 }$$

After introducing these expansions into (14) and subsequent comparing the complex harmonic terms we obtain the following infinite set of linear equations for $H_{dm}^ -$:

$$\begin{equation}\begin{gathered} \sum\limits_{m = - \infty }^\infty {\left\{ {g\,{f_{n - m}}\frac{{{q_{dm}}}}{{{\varepsilon _d}}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right) + \left( {{f_{n - m}} - {\delta _{nm}}} \right) \times } \right.} \hfill \\ \quad\times \left. {\left[ {\frac{{{q_{dm}}{\varepsilon _a}}}{{{q_{am}}{\varepsilon _d}}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right) + \left( {1 + {e^{2i{k_0}{q_{dm}}d}}} \right)} \right]} \right\}H_{dm}^ - = \hfill \\ = 2\left( {{f_n} - {\delta _{0n}}} \right){H_i},{\text{ }}n = 0,{\text{ }} \pm 1,{\text{ }} \pm 2, \ldots \hfill \\ \end{gathered} \end{equation} \tag{ 16 }$$

For numerical solution the number of harmonics and correspondingly also the number of equations has to be limited to a finite number: $\left| m \right| \leq M,{\text{ }}\left| n \right| \leq M.$ Note that this is the only approximation to the solution of the problem. Having solved equation (16), the amplitudes $H_{dm}^ +$ are easily determined from (4) and the amplitudes of the reflected waves ${H_{rm}}$ from (13). The electric field amplitudes can be evaluated from (2) and (3). The most important parameters of the grating are the efficiencies of reflected diffraction orders. Since we consider the grating unlimited in the x-direction, the diffraction efficiency of the mth order is determined as a ratio of the power flow density of the diffracted and incident plane waves in the direction perpendicular to the grating plane, respectively, i.e. as a ratio of y-components of the respective time-averaged Poynting vectors. Taking into account equations (1) and (2), we easily obtain the following expression:

$$\begin{equation}\eta _m^{\text{TM}} = \frac{{\tfrac{1}{2}\operatorname{Re} \{ {E_{xrm}}H_{rm}^*\} }}{{\tfrac{1}{2}\operatorname{Re} \{ {E_{xi}}H_i^*\} }} = \frac{{\operatorname{Re} \{ {q_{am}}\} {{\left| {{H_{rm}}} \right|}^2}}}{{{q_0}{{\left| {{H_i}} \right|}^2}}}.\end{equation} \tag{ 17 }$$

As expected, the diffraction efficiency is nonzero only for propagating diffraction orders for which ${q_{am}}$ are real.

The analysis for the case of TE polarization is very similar, we thus present here only the important relations. The electric and magnetic fields of the incident wave are

$$\begin{align} {{\mathbf{E}}_i}(x,y) & = {E_i}{{\mathbf{y}}^0}{e^{i{k_0}({p_0}x - {q_0}y)}}, \nonumber\\ {{\mathbf{H}}_i}(x,y) & = - {Y_0}({q_0}{{\mathbf{x}}^0} + {p_0}{{\mathbf{y}}^0}){E_i}{e^{i{k_0}({p_0}x - {q_0}y)}}. \end{align} \tag{ 18 }$$

In analogy with (2), the tangential field components of the reflected diffraction orders are expressed as

$$\begin{align} {E_r} & = \sum\limits_{m = - \infty }^\infty {{E_{rm}}{} {e^{i{k_0}({p_m}x + {q_{am}}y)}}} , \nonumber\\ {H_{xr}} & = {Y_0}\sum\limits_{m = - \infty }^\infty {{q_{am}}{E_{rm}}{e^{i{k_0}({p_m}x + {q_{am}}y)}}} , \end{align} \tag{ 19 }$$

and the tangential components of the fields in the dielectric layer just below the metallic stripes are

$$\begin{align} {E_d}(x,0)& = \sum\limits_{m = - \infty }^\infty {\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right)E_{dm}^ - {e^{i{k_0}{p_m}x}}} , \nonumber\\ {H_{xd}}(x,0)& = - {Y_0}\sum\limits_m {{\eta _{dm}}\left( {1 + {e^{2i{k_0}{q_{dm}}d}}} \right)E_{dm}^ - } {e^{i{k_0}{p_m}x}}. \end{align} \tag{ 20 }$$

Here, the boundary condition at the bottom metal interface, ${E_d}(x, - d) = 0,$ giving $E_{dm}^ + = - {e^{2i{k_0}{q_{dm}}d}}E_{dm}^ -$, has already been implemented. The boundary conditions at the upper interface with the periodic array of metal stripes result now in the following equations:

$$\begin{equation}\begin{gathered} {E_i}(x,0) + {E_r}(x,0) - {E_d}(x,0) = 0, \hfill \\ g{Y_0}f(x){E_d}(x,0) + \hfill \\ + \left[ {1 - f(x)} \right]\left[ {{H_{xi}}(x,0) + {H_{xr}}(x,0) - {H_{xd}}(x,0)} \right] = 0, \hfill \\ \end{gathered} \end{equation} \tag{ 21 }$$

where g is an arbitrary number. By inserting of the expansions (19) and (20) into (21) we obtain after rearrangements of the exponential terms the equation

$$\begin{equation}{E_{rm}} = - {E_i}{\delta _{m0}} + \left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right)E_{dm}^ - \end{equation} \tag{ 22 }$$

for ${E_{rm}}$ and the set of linear equations for the complex amplitudes $E_{dm}^ -$

$$\begin{equation}\begin{gathered} \sum\limits_{m = - \infty }^\infty {\left\{ { - g{f_{n - m}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right) + \left( {{f_{n - m}} - {\delta _{nm}}} \right) \times } \right.} \hfill \\ \quad\times \left. {\left[ {{q_{am}}\left( {1 - {e^{2i{k_0}{q_{dm}}d}}} \right) + {q_{dm}}\left( {1 + {e^{2i{k_0}{q_{dm}}d}}} \right)} \right]} \right\}E_{dm}^ - = \hfill \\ = 2{q_0}\left( {{f_n} - {\delta _{0n}}} \right){E_i},{\text{ }}n = 0,{\text{ }} \pm 1,{\text{ }} \pm 2, \ldots \hfill \\ \end{gathered} \end{equation} \tag{ 23 }$$

Having solved this set of equations (properly limited to $\left| m \right| \leq M,$ $\left| n \right| \leq M,$ where M is of the order of a few hundreds), we obtain the formula

$$\begin{equation}\eta _m^{\text{TE}} = \frac{{\operatorname{Re} \{ {q_{am}}\} {{\left| {{E_{rm}}} \right|}^2}}}{{{q_0}{{\left| {{E_i}} \right|}^2}}}\end{equation} \tag{ 24 }$$

for the efficiency of the mth diffraction order.

We are interested in the grating behaviour from the point of view of its intended application as a tuning retroreflecting grating mirror for (fibre) lasers operating at the near infrared wavelength range. The wavelength of the incident wave is thus chosen as 1550 nm. In order to avoid the existence of more than a single (−1st) propagating diffraction order (except the 0th order reflected wave), the grating period is fixed at $\Lambda = 960{\text{ nm}}{\text{.}}$ In this case, the incidence angle for the Littrow (retroreflection) configuration in air $({\varepsilon _a} = 1)$ is ${\vartheta _i} = \arcsin [\lambda /(2\Lambda )] = {53.83^\circ }.$ Silica with the relative permittivity ${\varepsilon _d} = 2.08541$ is chosen as a dielectric layer supporting the metalized surfaces. The diffraction efficiency of the grating depends then on the two remaining parameters—the metal filling factor of the grating, $f = a/\Lambda,$ and the thickness of the dielectric layer d. The dependence of diffraction efficiencies on these parameters are shown in the plots in figure 3 for both TM and TE polarizations.

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It is obvious that the diffraction efficiency can reach values very close to one for both TE and TM polarizations, although for different parameters. This situation is apparently more tolerant to variations of the filling factor than to variations of the thickness of the dielectric layer. Figure 4 shows the graphs of the retro-diffraction efficiencies for TM and TE polarizations for the values of the filling factors possessing somewhat relaxed tolerances of the dielectric layer thickness. For TM polarization, the optimum filling factor $f = 0.45$ corresponds to the width a of the metal stripes (see figure 1) of 432 nm, while their optimum width for TE polarization, corresponding to $f = 0.045,$ is only about 43 nm.

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The periodically repeating patterns in dependence of the dielectric layer thickness in the diffraction efficiency plots in figures 3 and 4 are primarily due to the Fabry–Perot type resonances between the metal sheets supported by the −1st and 0th propagating diffraction orders. Except for the first pattern for the TM polarization, which will be discussed separately, the period can be unanimously determined as $\Delta d = 647{\text{ nm}}{\text{.}}$ It exactly corresponds to the phase increment $2{k_y}\Delta d = 2\pi$ of the −1st and 0th diffraction orders propagating in the dielectric layer with the tangential wave vector components ${k_x} = \mp {k_0}\sin {\vartheta _i} = \mp \pi /\Lambda$ in the Littrow configuration. Namely, ${k_y} = {k_0}\sqrt {{\varepsilon _d} - {{\left[ {\lambda /(2\Lambda )} \right]}^2}} ,$ and $\Delta d = \pi /{k_y} = 647{\textrm{ nm,}}$ as expected.

According to figure 3(a) for the TM polarization, the 100% retro-diffraction efficiency takes place for a particular value of the filling factor $f$ even for an extremely small thickness $d$ of the order of a few tens of nanometre. It is thus not surprising that in this case, higher diffraction orders, evanescent in the y-direction (perpendicular to the grating), play an important role in the field distribution, affect the resulting phase of the −1st and the 0th diffraction orders, and thus also the position and shape of the first maximum in figure 4(a).

The field distributions in the vicinity of the grating in the Littrow configuration for the optimum parameters for TM and TE polarizations are shown in figure 5. In figure 5(a) the metal grating is represented by black lines while in figure 5(b) the narrow metal stripes are shown by small white rectangles, to improve their visibility. The interface between the dielectric SiO₂ layer and air is marked by a magenta line. In figure 5(a) it is shown the distribution of the absolute value of the magnetic field intensity $\left| {{H_z}(x,y)} \right|$ for the TM polarization at $d = 150{\text{ nm}}$ and $f = 0.45,$ while figure 5(b) displays the distribution of the absolute value of the electric field intensity $\left| {{E_z}(x,y)} \right|$ for the TE polarization at $d = 375{\text{ nm}}$ and $f = 0.045.$ Note that in the case of a 100% retro-diffraction efficiency, the optical field has the character of a pure standing wave; the incident wave is totally diffracted backwards, and no power is propagating along the grating plane.

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The maximum field intensity of the incident wave is normalized to unity, and the total field maxima above the grating are thus twice larger (corresponding to the light orange colour in figure 5). The far-fields for TM and TE polarizations are thus identical, and in both figures, the influence of evanescent waves near the dielectric interface $(y = 0)$ is apparent. However, there is a marked difference in the field distributions within the dielectric layer. In the case of the TM polarization, the metal stripes and slots have similar widths (which makes its fabrication easier), but there is quite strong accumulation of optical energy within the dielectric layer between the metal layers; the field intensity there is several times stronger than the intensity of the incident wave. On the other hand, it is surprising that in the case of TE polarization, very tiny metal stripes of the width of about ${1/20}$ of the grating period (∼40 nm) lead to 100% retro-diffraction efficiency. There is only very weak field concentration between the metal stripes; however, the fabrication of such a grating is not easy. Let us note for completeness that—in accordance with figure 3(a)—the retro-diffraction efficiency close to 100% can be obtained also for TM polarization with the dielectric layer only 50 nm thick with a filling factor as high as 0.97, i.e. with slits only about 30 nm wide.

Although the geometry of the grating is very simple, the wave processes leading to such a high diffraction efficiency deserve deeper discussion. Obviously, in order to obtain a 100% retro-diffraction efficiency, the reflected wave (the 0th reflected diffraction order) has to be cancelled out by the destructive interference of all waves propagating in the same direction after (multiple) reflections and diffractions at the metal sheets. (Note that in the far field zone of the grating, $y \gg \lambda ,$ which is utilized for technical applications, only the propagating −1st and 0th reflected diffraction orders are important.)

In order to investigate the role of the permittivity of the dielectric layer ${\varepsilon _d}$ between the metal sheets, we calculated the retro-diffraction efficiency of the grating for the two more cases, ${\varepsilon _d} = 1$ (air) and ${\varepsilon _d} = 12.0854$ (silicon). The results are presented in figure 6 for the TM polarization; the results for the TE polarization are fully analogical and for brevity they are not presented.

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The hypothetic case of ${\varepsilon _d} = 1$ was chosen in order to demonstrate that the existence of the dielectric interface in the slits between metal stripes does not play any important role; in this case it is namely missing. The main effect of the lower permittivity is that, in order to preserve the necessary optical path between the metal layers, the optimum thickness of the layer is increased. On the other hand, the dominant effect of the high permittivity of silicon is due to the existence of more than the −1st and 0th propagating diffraction orders in this layer; for the particular case considered here, the 1st and −2nd orders are also propagating, thus making the interference effects within the dielectric layer much more complicated. Nevertheless, the retro-diffraction efficiency may reach 100% for particular combinations of the filling factor and the dielectric layer thickness d too.

In reality, the incident optical wave is not a pure plane wave but a beam with a finite width. Such a beam can be represented by a superposition of plane waves. To keep the analysis simple, only waves propagating in the plane of incidence will be considered. Thus, the beam diffracted into the $- {1{{\text{st}}}}$ diffraction order can be obtained as a superposition of plane waves diffracted by the grating into this diffraction order. For each plane wave we introduce a complex diffraction coefficient ${R_m}$ into the mth (reflected) diffraction order analogously with the definition of the standard Fresnel reflection coefficient from a dielectric interface. For the (more practical) case of the TM polarization, this diffraction coefficient can be defined in accordance with (1) and (2) as

$$\begin{equation}R_m^{\text{TM}} = \frac{{{H_{r,m}}}}{{{H_i}}}.\end{equation} \tag{ 25 }$$

Obviously, the diffraction coefficient depends on the incidence angle ${\vartheta _i}$ of the incident plane wave. The beam diffracted into the −1st reflected order can then be constructed as a superposition of all incident plane waves, each of them is multiplied by the diffraction coefficient (25).

The absolute value and the (negatively taken) phase of the reflection coefficient $R_{ - 1}^{{\text{TM}}}$ are plotted in figure 7 in dependence of the incidence angle ${\vartheta _i}$. The diffraction efficiency is also shown there (black line, right vertical axis). For incidence angles smaller than 39°, there is no propagating reflected diffraction order. For the Littrow configuration, ${\eta _{{\text{Litt}}}} = 53.83^\circ$, both the absolute value of the diffraction coefficient and the diffraction efficiency are equal to one. Both the diffraction efficiency and the phase of the diffraction coefficient have very flat maxima in the vicinity of the Littrow angle. The change of the absolute value of the diffraction coefficient in the vicinity of ${\vartheta _{{\text{Litt}}}}$ is due to the difference between the angles of incidence and diffraction according to the grating equation. As a consequence of this effect, the cross-section profile of the diffracted beam incident under the Littrow angle is very slightly modified in comparison with that of the incident beam, but the diffraction efficiency is essentially unaffected. And since the phase of the diffraction coefficient in the vicinity of ${\vartheta _{{\text{Litt}}}}$ does not depend in the first approximation on the incidence angle, the Goos–Hänchen shift [18] of the reflected beam with respect to the incident one is negligible. It is obvious from figure 7 that these conclusions are valid even for incident beams with the divergence angles as large as a few degrees, which, for the wavelength of 1.55 µm, corresponds to beam diameters of the order of a few tens of micrometres only.

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In practical applications, the metal grating has to be protected by a thin but hard enough dielectric layer, such as SiO₂, as shown in figure 8.

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Since we do not want to overload this article with detailed numerical analysis of this structure, we present here only selected results of practical importance.

In figure 9 we show the diffraction efficiency of the grating in dependence of the bottom dielectric layer thickness (as in figure 4) for several values of the thickness of the SiO₂ protective layer $t$ for both TM and TE polarizations.

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Obviously, although the protective layer slightly modifies the shape of the curves, the maximum diffraction efficiency of the grating close to 100% is retained.