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Abstract
Two-dimensional multi-element phase gratings can be engineered to show an even symmetry along one direction while an odd symmetry along the other direction in terms of offset refractive indices in each unit cell. The interplay of such even and odd symmetries has been explored to tailor diffraction columns and rows on demand by making offset refractive indices to satisfy specific requirements and hence attain different types of destructive interference. The resultant tailoring effects include the directional column elimination, the grouped column elimination, and the directional column selection as well as the natural row absence, the grouped row elimination, and the central row selection.
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1. Introduction
Diffraction gratings, owing to engineered beam splitting functions, exhibit important applications in many frontier science and technology fields including physics, astronomy, chemistry, and biology [1–4]. The underlying physics of far-field light diffraction is known in particular as Fourier transform optics [5], which has been employed to accurately predict and control diverse diffraction behaviors indispensable in realistic applications. In other words, it provides a theoretical basis for designing etched phase gratings with neither loss nor gain, like holographic gratings, blazed gratings, and Dammann gratings [6–10], to achieve specific diffraction effects characterized by different intensity distributions along different directions. This is relevant to binary optics [11–13] allowing for making micron-level relief phase elements based on computer-aided designs and has been extended to metasurfaces composed of subwavelength building blocks to gain a more flexible and complete control over the flow of light [14–16].
Diffraction patterns of traditional phase gratings are generally symmetric, but asymmetric diffractions are becoming more and more essential in light-wave communications, micro-nano lithography, and other application scenarios though basically difficult to attain. This prompts us to recall the Friedel’s law [17] arguing that light diffraction from a real potential is expected to be symmetric when (a) scattering is weak and (b) there is no absorption in the corresponding optical crystals or structures [18–20]. That indicates, we have two possible methods to realize asymmetric diffractions by breaking different necessary conditions of the Friedel’s law. The familiar one is to utilize parity-time (PT) symmetric or more general non-Hermitian gratings [21–23] where absorption and/or gain is introduced to display an out-of-phase spatial modulation with respect to refractive index, hence yielding a complex optical potential with misaligned real and imaginary parts. The method of non-Hermitian modulation, when incorporated into electromagnetically induced gratings (EIGs) [24–26], owns the advantage of coherent generation and dynamic manipulation, including both one-dimensional (1D) [27,24,28–30] and two-dimensional (2D) [31–33] examples. A main limitation of this method lies in that it is not viable to tailor specific diffraction orders on demand though diffraction asymmetry may be controlled on the whole, e.g. to quench diffracted beams in one side (1D case) or some quadrants (2D case).
To date, few works have been done to attain asymmetric diffractions, especially in the optical regime, by modulating the spatial distributions of only refractive indices in the presence of strong scattering [34–36]. In a recent work, we have shown that 1D multi-element phase gratings could exhibit asymmetric diffractions as offset refractive indices of different elements in a unit cell are not in the spatially even symmetry and largest asymmetries occur always in the case of spatially odd-symmetric offset refractive indices [37]. It is more important that appropriate engineerings of odd-symmetric offset refractive indices allow us to attain a few interesting effects including the directional eliminations of a single diffraction order, all even or odd diffraction orders, and all but one or two diffraction orders. Then a question arises, what will happen and what are new if we consider 2D multi-element phase gratings?
In the spirit of extended binary optics, here we examine appropriate ways for engineering 2D multi-element phase gratings with offset refractive indices in each unit cell being odd symmetric along both diagonal directions $y=\pm x$, i.e. odd and even symmetric along the $y$ and $x$ directions, respectively. First, we show that it is viable to eliminate a few equidistant diffraction columns via paired destructive interference, eliminate all even or odd diffraction columns via dual paired destructive interference, and select a few equidistant diffraction columns via successive destructive interference as offset refractive indices satisfy specific requirements along the $y$ direction. Second, we show that these diffraction effects on tailored columns are accompanied by the natural absence of paired symmetric diffraction rows and it is viable to further attain the elimination of all even or odd diffraction rows and the definite selection of the central diffraction row by making offset refractive indices to satisfy some extra requirements along the $x$ direction. The joint tailorings of diffraction columns and rows allow for a flexible and versatile control over all diffracted beams and could even leave only two diffracted beams at desired output angles for instance.
2. Model and method
We start by considering a 2D phase grating of same period $a$ along both $x$ and $y$ directions, which will result in the far-field diffraction of a light beam incident along the $z$ direction into a few discrete orders marked by $i\in [-I,I]$ along the $x$ direction and $j\in [-J,J]$ along the $y$ direction as shown in Fig. 1(a). Each unit cell of this grating is designed to have $2m_{g}\times 2n_{g}$ elements with identical areas $\delta x\times \delta y=\frac {a}{2m_{g}}\times \frac {a}{2n_{g}}$ but different refractive indices $n_{m,n}=n_{0}+\delta n_{m,n}$ for $m\in \pm [1,m_{g}]$ and $n\in \pm [1,n_{g}]$, with $n_{0}$ being the mean value while $\delta n_{m,n}$ the offset values, as shown in Fig. 1(b). Taking $\mathcal {L}$ as the common thickness of all elements and considering a probe field with wavelength $\lambda _{p}$ and wavenumber $k_{p}=2\pi /\lambda _p$, for the $(m,n)$ element, we can write down the individual transmission function
where the common and hence unimportant phase factor $e^{\imath n_{0}k_{p}\mathcal {L}}$ has been removed and $\imath$ stands for the imaginary unit. Then, each unit cell of the 2D multi-element phase grating can be described by the total transmission function
Fig. 1. $(a)$ Schematic of a phase grating of period $a$ along both $x$ and $y$ directions, which can diffract a light beam of amplitude $E_{in}$ incident along the $z$ direction into a few light beams of amplitudes $E_{i,j}$ deviating with angles $(\theta _{i},\theta _{j})$ from the $z$ direction. $(b)$ Fine structure in a period consisting of $2m_{g}\times 2n_{g}$ elements with a common thickness $\mathcal {L}$, identical widths $\delta x=a/2m_{g}$ and $\delta y=a/2n_{g}$, while different refractive indices $n_{m,n}=n_{0}+\delta n_{m,n}$ restricted by $\delta n_{m,n}=\delta n_{-m,n}$ and $\delta n_{m,n}=-\delta n_{m,-n}$.
Further considering the translational invariance, we now make a Fourier transform to expand the total transmission function into $T(x,y)=\int _{-k_{p}}^{k_{p}}\int _{-k_{p}}^{k_{p}}E(\theta _{x},\theta _{y}){e^{\imath k_{x}x}}{e^{\imath k_{y}y}}dk_{x}dk_{y}$. Here $\theta _{x}$ ($\theta _{y}$) corresponds to an angle of diffracted beams deviating from the $z$ direction in the $xz$ ($yz$) plane, while $k_{x}=k_{p}\sin \theta _{x}$ ($k_{y}=k_{p}\sin \theta _{y}$) denotes a projection of probe wavenumber $k_{p}$ from the direction determined by angle $\theta _{x}$ ($\theta _{y}$) to the $x$ ($y$) direction. Then, an inverse procedure of this Fourier transform yields the following dimensionless field amplitude
In what follows, we restrict our discussions to the specific case where $\delta n_{\pm m,n}=-\delta n_{\pm m,-n}$ and $\delta n_{-m,\pm n}=\delta n_{m,\pm n}$ are simultaneously satisfied. That means, our 2D phase grating exhibits an even symmetry along the $x$ direction and meanwhile an odd symmetry along the $y$ direction, which then yields an odd symmetry along both diagonal directions defined by $y=\pm x$. In this case, Eq. (3) can be recast into a more compact form
3. Tailored asymmetric diffraction columns
In this section, we consider the case where the odd symmetry along the $y$ direction, namely $\delta n_{\pm m,n}=-\delta n_{\pm m,-n}$, is modulated while the even symmetry along the $x$ direction, namely $\delta n_{m,\pm n}=\delta n_{-m,\pm n}$, remains fixed.
Directional column elimination.-First, we note from Eq. (5) that the diffraction amplitude $E_{i,j}$ will become vanishing for a certain column order $j\in [-J,J]$, independent of the row order $i\in [-I,I]$, if we require $\delta n_{m,n}k_{p}\mathcal {L}=\delta n_{m,n}^{j}k_{p}\mathcal {L}\equiv k_{j}y_{n}+(l_{m,n}^j-1/2)\pi$, with $l_{m,n}^j$ being an integer. A simple rearrangement of this requirement yields
where $l_{m,n}^j$ can be chosen at will for $m\in [1,m_{g}]$ and $n\in [1,n_{g}]$ to realize the directional elimination of a diffraction column of order $j\in [-J,J]$. Sufficient conditions for realizing a directional column elimination are summarized in the left column of Table 1, according to which two examples on the integer matrices $\hat {l}^{1}_{m,n}$ and $\hat {l}^{-2}_{m,n}$ have been chosen to show that the $1$st and $-2$nd diffraction columns can be eliminated on demand in Fig. 2($a_1$) and Fig. 2($a_2$), respectively. Note, however, that the $-5$th and $4$th diffraction columns also disappear in Fig. 2($a_1$) and Fig. 2($a_2$), respectively, which can be easily understood by considering $k_{j\pm 2n_{g}}y_{n}-k_{j}y_{n}=\pm (2n-1)\pi$ in Eq. (5) so that $\cos (\delta n_{m,n}k_{p}\mathcal {L}-k_{j}y_{n})=0$ must result in $\cos (\delta n_{m,n}k_{p}\mathcal {L}-k_{j\pm 2n_{g}}y_{n})=0$. Moreover, we note from Fig. 2($a_1$) and Fig. 2($a_2$) that other surviving diffraction columns are clearly asymmetric as manifested by $E_{i,j'}\ne E_{i,-j'}$ with $j'\ne j$.Fig. 2. Intensity $I$ against $\sin \theta _x$ and $\sin \theta _y$ with $E_{i,-5}=E_{i,1}=0$ $(a_1)$ while $E_{i,-2}=E_{i,4}=0$ ($a_2$) as we set integer matrices $\hat {l}^{1}_{m,n}=\hat {l}^{-2}_{m,n}=\{-1,0,2;-2,0,1;0,1,2\}$ to attain offset refractive indices $\delta n_{m,n}^{1}$ in ($b_1$) while $\delta n_{m,n}^{-2}$ in ($b_2$). Other parameters used in calculations are $\lambda _{p}=0.8$ $\mu$m, $m_{g}=3, n_{g}=3$, $R=30$, $M_x=M_y=10$, $\mathcal {L}=10\lambda _{p}$, and $n_{0}=2.0$.
Table 1. Conditions for three types of tailored asymmetric diffraction columns.
As in the 1D case [37], the directional column elimination arises from the destructive interference between paired elements in each unit cell while the asymmetric diffraction columns are resulted from a violation of the Friedel’s law. The paired destructive interference can be understood as follows: (i) different parts of a light beam scattered by the two $(m,\pm n)$ elements, respectively, suffer the opposite deflected phases $\pm \beta _{n}^{j}=\pm k_{j}y_{n}$ and the opposite forward phases $\pm \alpha _{m, n}^{j}=\pm \delta n_{m,n}^{j}k_{p}\mathcal {L}$ but share a common deflected phase $\gamma _{m}^{i}=k_{i}x_{m}$, which then results in a combined amplitude proportional to $e^{\imath \gamma _{m}^{i}}\cos {(\alpha _{m,n}^{j}-\beta _{n}^{j})}$; (ii) similarly, the two $(-m,\pm n)$ elements will contribute a combined amplitude proportional to $e^{-\imath \gamma _{m}^{i}}\cos {(\alpha _{m,n}^{j}-\beta _{n}^{j})}$ so that we have $\mathcal {E}^{i,j}_{m,n}\propto \cos {(\gamma _{m}^{i})}\cos {(\alpha _{m,n}^{j}-\beta _{n}^{j})}$ for the four $(\pm m,\pm n)$ elements as shown in Eq. (5); (iii) hence, $E_{i,j}$ for a certain column order $j\in [-J,J]$ will be vanishing when perfect destructive interference occurs between the $(m,n)$ and $(m,-n)$ elements as well as the $(-m,n)$ and $(-m,-n)$ elements in the case of $\alpha _{m,n}^{j}-\beta _{n}^{j}=(l_{m,n}^j-1/2)\pi$ as indicated by Eq. (6). On the other hand, we know that a violation of the Friedel’s law requires non-negligible higher-order scattering processes [37], which can be seen by making a power series expansion upon $\delta n_{m,n}k_{p}\mathcal {L}$ in Eq. (5) and explains why asymmetric diffraction columns are observed in Fig. 2.
Grouped column elimination.-Second, on the basis of $E_{i,j}=0$ for a certain value of integer $j$, we find from Eq. (5) and Eq. (6) that the diffraction column of order $j\pm 2k$ with $k\in \{1,2,3,\ldots \}$ exhibits the amplitude
Fig. 3. Intensity $I$ against $\sin \theta _x$ and $\sin \theta _y$ with all odd columns eliminated on the basis of $E_{i,1}=0$ $(a_1)$ while all even columns eliminated on the basis of $E_{i,-2}=0$ $(a_2)$. Parameters are the same as in Fig. 2 except $\hat {l}^{1}_{m,n}=\hat {l}^{-2}_{m,n}=\{-1,0,1;-2,0,0;0,1,2\}$ restricted by $l^{1,-2}_{m,n}+l^{1,-2}_{m,n^\ast }\in \{0,\pm 2,\pm 4,\ldots \}$ are reset to attain $\delta n_{m,n}^{1}$ in ($b_1$) while $\delta n_{m,n}^{-2}$ in ($b_2$).
The grouped column elimination arises instead from the destructive interference between dual paired elements and can be understood as follows: (i) the $j$th diffraction column has been eliminated with $E_{i,j}=\sum _{m=1}^{m_{g}}\sum _{n=1}^{n_{g}}\mathcal {E}^{i,j}_{m,n}=0$ by taking $\alpha ^{j}_{m,n}-\beta ^{j}_{n}=(l_{m,n}^{j}-1/2)\pi$ in $\mathcal {E}^{i,j}_{m,n}\propto \cos {(\alpha _{m,n}^{j}-\beta _{n}^{j})}$ for the four $(\pm m,\pm n)$ elements; (ii) as to the diffraction column of order $j\pm 2k$, we have $\mathcal {E}^{i,j\pm 2k}_{m,n}\propto \sin (l_{m,n}^{j}\pi \mp k\delta _{n})$ for the $(\pm m,\pm n)$ elements and $\mathcal {E}^{i,j\pm 2k}_{m,n^{\ast }}\propto \sin (l_{m,n^{\ast }}^{j}\pi \mp k\delta _{n^{\ast }})$ for the $(\pm m,\pm n^{\ast })$ elements by considering $\alpha ^{j\pm 2k}_{m,n}=\alpha ^{j}_{m,n}$ and $\beta ^{j\pm 2k}_{m,n}=\beta ^{j}_{n}\pm k\delta _{n}$; (iii) taking $\delta _{n}+\delta _{n^{\ast }}=2\pi$ into account, it is not difficult to find that the $(\pm m,\pm n)$ and $(\pm m,\pm n^{\ast })$ elements jointly contribute a beam superposition of amplitude $\mathcal {E}^{i,j\pm 2k}_{m,n}+\mathcal {E}^{i,j\pm 2k}_{m,n^{\ast }}\propto \sin [(l_{m,n}^{j}+l_{m,n^{\ast }}^{j})\pi /2]$; (iv) hence, the diffraction column of order $j\pm 2k$ will disappear with $E_{i,j\pm 2k}=0$ due to perfect destructive interference between the ‘dual’ paired $(m,\pm n)$ and $(m,\pm n^{\ast })$ elements as well as $(-m,\pm n)$ and $(-m,\pm n^{\ast })$ elements provided $l^j_{m,n}+l^j_{m,n^{\ast }}$ is an even integer.
Directional column selection.-Finally, on the basis of $E_{i,j}=0$ for a certain value of integer $j$, we find from Eq. (5) and Eq. (6) that the diffraction column of order $j\pm k$ with $k\in \{1,2,3,\ldots \}$ exhibits the amplitude
This diffraction amplitude, upon the multiplication of a non-zero term $\cos (k\delta _{1}/2)$ in the case of $k\ne \widetilde {n}_{g}\in \{1,3,5,\ldots \}n_{g}$, will become vanishing if we further require
By defining $l_{m,n_{g}+1}^{j}\equiv l_{m,1}^{j}\pm k$, we can translate this requirement into
Fig. 4. Intensity $I$ against $\sin \theta _x$ and $\sin \theta _y$ with two even columns selected on the basis of $E_{i,1}=0$ $(a_1)$ while two odd columns selected on the basis of $E_{i,-2}=0$ $(a_2)$. Parameters are the same as in Fig. 2 except $\hat {l}^{1}_{m,n}=\hat {l}^{-2}_{m,n}=\{-1,0,1;-2,-1,0;0,1,2\}$ restricted by $l^{1,-2}_{m,n+1}-l^{1,-2}_{m,n}\in \{\pm 1,\pm 3,\ldots \}$ are reset to attain $\delta n_{m,n}^{1}$ in ($b_1$) while $\delta n_{m,n}^{-2}$ in ($b_2$).
The directional column selection arises however from the destructive interference between successive paired elements and can be understood as follows: (i) the $j$th diffraction column has been eliminated with $E_{i,j}=\sum _{m=1}^{m_{g}}\sum _{n=1}^{n_{g}}\mathcal {E}^{i,j}_{m,n}=0$ by taking $\alpha ^{j}_{m,n}-\beta ^{j}_{n}=(l_{m,n}^{j}-1/2)\pi$ in $\mathcal {E}^{i,j}_{m,n}\propto \cos {(\alpha _{m,n}^{j}-\beta _{n}^{j})}$ for the four $(\pm m,\pm n)$ elements; (ii) as to the diffraction column of order $j\pm k$, we have $\mathcal {E}^{i,j\pm k}_{m,n}\propto \sin (l_{m,n}^{j}\pi \mp k\delta _{n}/2)$ for the $(\pm m,\pm n)$ elements, which can be rewritten as a sum of inward $\mathcal {E}^{i,j\pm k,in}_{m,n}\propto \sin [l_{m,n}^{j}\pi \mp k(n-1)\pi /n_{g}]$ and outward $\mathcal {E}^{i,j\pm k,out}_{m,n}\propto \sin (l_{m,n}^{j}\pi \mp kn\pi /n_{g})$ amplitudes after multiplied by a non-zero $\cos (k\delta _{1}/2)$ for $k\ne \widetilde {n}_{g}$; (iii) it is not difficult to find that the sum of $\mathcal {E}^{i,j\pm k,out}_{m,n}$ contributed by the $(\pm m,\pm n)$ elements and $\mathcal {E}^{i,j\pm k,in}_{m,n+1}$ contributed by the $(\pm m,\pm n\pm 1)$ elements is proportional to $\cos [(l_{m,n+1}^{j}-l_{m,n}^{j})\pi /2]$; (iv) hence, all diffraction columns except those of orders $j\pm \widetilde {n}_{g}$ will disappear due to destructive interference between the successive $(\pm m,\pm n)$ and $(\pm m,\pm n\pm 1)$ elements provided $l^j_{m,n+1}-l^j_{m,n}$ is an odd integer.
4. Tailored symmetric diffraction rows
In this section, we examine whether it is viable to further tailor the diffraction rows while reserving the phenomena of directional column elimination, grouped column elimination, and directional column selection. This will be done by exerting extra requirements on integers $l_{m,n}^{j}$ with respect to subscript $m$ in the case of $\delta n_{\pm m,n}=-\delta n_{\pm m,-n}$ (odd symmetry along the $y$ direction) and $\delta n_{m,\pm n}=\delta n_{-m,\pm n}$ (even symmetry along the $x$ direction).
Natural row absence.-First, we note that Eq. (9) turns into
Grouped row elimination.-Second, on the basis of $E_{i,j}=0$ for a certain value of integer $j$, we rewrite Eq. (9) into $E_{i,j\pm k}=\sum _{m=1}^{m_{g}}\sum _{n=1}^{n_{g}}\mathcal {E}^{i,j\pm k}_{m,n}$ where $\mathcal {E}^{i,j\pm k}_{m,n}\propto \cos (k_i x_m)\sin (l_{m,n}^{j}\pi \mp k\delta _{n}/2)$ is a contribution from the four $(\pm m,\pm n)$ elements. Similarly, we have $\mathcal {E}^{i,j\pm k}_{m^{\ast },n}\propto \cos (k_i x_{m^{\ast }})\sin (l_{m^{\ast },n}^{j}\pi \mp k\delta _{n}/2)$ contributed by other four $(\pm m^{\ast },\pm n)$ conjugate elements with $m^{\ast }=m_{g}+1-m$. Then, it is easy to prove $\cos (k_i x_{m^{\ast }})=\cos (i\pi )\cos (k_i x_{m})$ and attain
Fig. 5. Intensity $I$ against $\sin \theta _x$ and $\sin \theta _y$ with all odd rows eliminated on the basis of $E_{i,2}=0$ in the presence of directional column elimination $(a_1)$; grouped column elimination $(a_2)$; directional column selection $(a_3)$. Parameters are the same as in Fig. 2 except corresponding integer matrices restricted by $l^{2}_{m,n}-l^{2}_{m^\ast,n}\in \{0,\pm 2,\pm 4,\ldots \}$ are shown in $(b_1)$, $(b_2)$, and $(b_3)$ while corresponding offset refractive indices are shown in $(c_1)$, $(c_2)$, and $(c_3)$.
Fig. 6. Intensity $I$ against $\sin \theta _x$ and $\sin \theta _y$ with all even rows eliminated on the basis of $E_{i,1}=0$ in the presence of directional column elimination $(a_1)$; grouped column elimination $(a_2)$; directional column selection $(a_3)$. Parameters are the same as in Fig. 2 except $m_g=n_g=4$ and corresponding integer matrices restricted by $l^{1}_{m,n}-l^{1}_{m^\ast,n}\in \{\pm 1,\pm 3,\ldots \}$ are shown in $(b_1)$, $(b_2)$, and $(b_3)$ while corresponding offset refractive indices are shown in $(c_1)$, $(c_2)$, and $(c_3)$.
Table 2. Conditions for three types of tailored symmetric diffraction rows.
The grouped row elimination arises from the destructive interference between dual paired elements in each column of a unit cell and can be understood as follows: (i) in $E_{i,j\pm k}=\sum _{m=1}^{m_{g}}\sum _{n=1}^{n_{g}}\mathcal {E}^{i,j\pm k}_{m,n}$ with $j\pm k$ covering all diffraction columns, we have $\mathcal {E}^{i,j\pm k}_{m,n}\propto \cos (k_{i}x_{m})\sin (l_{m,n}^{j}\pi \mp k\delta _{n}/2)$ and $\mathcal {E}^{i,j\pm k}_{m^\ast,n}\propto \cos (k_{i}x_{m^\ast })\sin (l_{m,n}^{j}\pi \mp k\delta _{n}/2)$ contributed by the $(\pm m,\pm n)$ and $(\pm m^\ast,\pm n)$ elements, respectively; (ii) taking $\cos (k_i x_{m^{\ast }})=\cos (i\pi )\cos (k_i x_{m})$ into account, it is not difficult to find that the $(\pm m,\pm n)$ and $(\pm m^\ast,\pm n)$ elements jointly contribute a beam superposition of amplitude $\mathcal {E}^{i,j\pm k}_{m,n}+\mathcal {E}^{i,j\pm k}_{m^{\ast },n}\propto \cos [(l_{m,n}^{j}-l_{m^\ast,n}^{j})\pi /2]$ for $i\in \{0,\pm 2,\pm 4,\ldots \}$ while $\mathcal {E}^{i,j\pm k}_{m,n}+\mathcal {E}^{i,j\pm k}_{m^{\ast },n}\propto \sin [(l_{m,n}^{j}-l_{m^\ast,n}^{j})\pi /2]$ for $i\in \{\pm 1,\pm 3,\ldots \}$; (iii) hence, all odd (even) diffraction rows will disappear due to perfect destructive interference between the dual paired conjugate $(\pm m, n)$ and $(\pm m^\ast,n)$ elements as well as the dual paired conjugate $(\pm m, -n)$ and $(\pm m^\ast, -n)$ elements provided $l^j_{m,n}-l^j_{m^{\ast },n}$ is an even (odd) integer, independent of $j\pm k$.
Central row selection.-Finally, we note that Eq. (9) will turn into
Fig. 7. Intensity $I$ against $\sin \theta _x$ and $\sin \theta _y$ with the central row selected on the basis of $E_{i,1}=0$ in the presence of directional column elimination $(a_1)$; grouped column elimination $(a_2)$; directional column selection $(a_3)$. Parameters are the same as in Fig. 2 except $m_g=n_g=4$ and corresponding integer matrices restricted by $l^{1}_{m+1,n}-l^{1}_{m,n}\in \{0,\pm 2,\pm 4,\ldots \}$ are shown in $(b_1)$, $(b_2)$, and $(b_3)$ while corresponding offset refractive indices are shown in $(c_1)$, $(c_2)$, and $(c_3)$.
The central row selection arises from the collective destructive interference between all $m_{g}$ elements in each column as the total contribution of all $n_{g}$ elements is identical for each row. This can be understood as follows: (i) we note from Eq. (15) that the summation $\sum _{m=1}^{m_g}\cos (k_i x_m)$ is independent of $n$ and the summation $\sum _{n=1}^{n_g}\sin {(l_{m_{g},n}^{j}\pi \mp k\delta _{n}/2)}$ is independent of $m$ in the case of $l^{j}_{m+1,n}-l^{j}_{m,n}\in \{0,\pm 2,\pm 4,\ldots \}$; (ii) the separable summation $\sum _{m=1}^{m_g}\cos (k_i x_m)$ is definitely zero for $i\ne 0$ because the angles $\{k_{i}x_{1},k_{i}x_{2},\ldots,k_{i}x_{m_{g}}\}$ are symmetrically distributed on a circle so that their cosine functions must cancel each other out; (iii) hence, all diffraction rows except the central one will disappear due to collective destructive interference as a joint contribution of all $m_g$ elements in each column.
5. Conclusions
In summary, a 2D multi-element phase grating has been designed to exhibit diverse odd-symmetric structures along two diagonal $y=\pm x$ directions in terms of offset refractive indices in each unit cell. It is of interest that this grating with neither loss nor gain is characterized by asymmetric diffraction columns along the $y$ direction while symmetric diffraction rows along the $x$ direction. By engineering offset refractive indices along the $y$ direction to satisfy specific requirements $\delta _{m,n}k_p\mathcal {L}-k_jy_n= (l_{m,n}^{j} -1/2)\pi$, we can realize on demand the elimination of a few equidistant diffraction columns, the elimination of all even or odd diffraction columns, or the selection of a few equidistant diffraction columns, all accompanied by the natural absence of symmetric row pairs. Similar engineerings of offset refractive indices along the $x$ direction allow us to further eliminate all even or odd diffraction rows, or definitely select the central diffraction row, on the basis of tailored diffraction columns mentioned above. These diffraction effects of tailored columns and rows can be explained in terms of paired, dual paired, successive, or collective destructive interference between different elements in each column or row of a unit cell, and may find applications in devising photonic devices requiring peculiar light filtering functions like eliminating or selecting one or more specific diffracted beams, especially when extended to some materials with tunable refractive indices [38,39]. Our results can also be generalized to realize diagonal row selection with other 2D multi-element phase gratings exhibiting, e.g., odd-symmetric offset refractive indices along a certain diagonal direction. A combination of our analytical method and numerical approaches based on iterative Fourier-transform algorithms (IFTA) [40] will be more efficient in realizing various beam filtering and shaping functions.
Funding
National Natural Science Foundation of China (12074061); National Key Research and Development Program of China (2021YFE0193500).
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.
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