Engineered asymmetric diffractions of diagonal-line odd-symmetric phase gratings

A two-dimensional multi-element phase grating has been designed in terms of the offset refractive index to exhibit the spatially odd symmetry (antisymmetry) along one transparent diagonal line with an even number of rectangular elements while leaving other elements in a unit cell opaque. This grating can be engineered to attain a few intriguing phenomena of asymmetric diffraction, including the elimination of equally spaced oblique diffraction lines, the elimination of alternately crossed oblique diffraction lines, and the selection of equally spaced oblique diffraction lines. These phenomena of engineered asymmetric diffraction are well explained via destructive interference between transmitted field amplitudes from paired, dual-paired, and successive elements along the transparent diagonal line.

As in the scattering case, asymmetric diffraction gratings provide effective modification of the propagation dynamics following Friedel's law [20][21][22][23].It states that essential conditions for achieving asymmetric diffraction are the existence of either (i) absorption/gain or (ii) the significant contributions from higher-order scatterings.
The former one can be realized in one-dimensional [24][25][26][27][28] or two-dimensional [29][30][31][32][33] non-Hermitian optical systems with parity-time symmetry.For example, using electromagnetically induced gratings which are famous for the dynamic modulation [34][35][36], one can achieve optical parity-time symmetry modulation via periodicity of standing waves and that of atomic density, in virtue of which absorption/gain is introduced simultaneously.Since the refractive index depends on the intensity of the applied field, this challenges the accuracy of controlling or constructing the refractive index.
On the other hand, a more precisely implemented grating with asymmetric diffraction patterns can be achieved in a discrete Hermitian optical system using binary optics and metasurface optics techniques.By introducing certain symmetry into the scattering potential, significant higher-order scattering takes place and leads to asymmetric diffraction according to the second condition of Friedel's law.In our previous work [37], we employed a pure phase grating with an odd symmetric (antisymmetric) distribution of refractive index in space (n = n + δn, δn(−x) = −δn(x)) to achieve higher order scattering, resulting in one-dimensional asymmetric diffraction and enabling specific functions, including directional elimination, group elimination, and directional diffraction, whose considerable robustness against disorders in offset refractive indices is nearly up to 30%.Moreover, many researchers have successfully synthesized media with tunable refractive indices through various materials and methods [38,39], thereby allowing for the feasible fabrication of such gratings in experiments.
It is natural to recognize the potential of a two-dimensional pure phase grating with an odd-symmetric distribution to facilitate two-dimensional asymmetric diffraction and unlock a broader range of functionalities.In this paper, we present a design proposal for such a grating within the framework of extended binary optics, which enables the generation of asymmetric diffraction modes under multiple scattering conditions.First, we describe a theoretical framework to resolve the diffraction asymmetry by incorporating a multi-element periodic structure into our phase grating.Second, we provide an elaborate procedure to derive specific conditions under which diffraction patterns can be tailored.We discuss three examples of diffraction asymmetries, which are validated via numerical calculations, including the elimination of equally spaced oblique diffraction lines, the elimination of alternately crossed oblique diffraction lines, and the selection of equally spaced oblique diffraction lines.

Model and equations
We start by considering a 2D phase grating composed of M x × M y unit cells (figure 1(a)).Each one of them has identical fine structures, as shown in figure 1(b).With different optical properties, 2m g × 2m g rectangular elements join together to form a unit (a x × a y ) with the spatial period a x and a y along the x and y direction, respectively.It results in the diffraction of an incident light traveling along the z direction into a few symmetric orders i ∈ [−I, I] and j ∈ [−J, J] as illustrated in figure 1(a).Therefore, each rectangular elements has width δx = a x /2m g and height δy = a y /2m g in one unit.Figure 1(b) exhibits the fine structure of each unit where the 2m g elements on the diagonal (numbered in sequence of coordinates) with various refractive indexes are transparent but the others are opaque.The refractive index of any mth subunit is defined as n m = n 0 + δn m for |m| ∈ [1, m g ], with n 0 being the mean value while δn m the offset values.To have an asymmetric diffraction pattern, we need to take m g > 1.Otherwise, in the case of m g = 1 (binary), the diffraction is always symmetric since the overall pattern (over several units on the diagonal) essentially belongs to an ordinary grating.
Taking L as the common thickness for all elements and considering a probe field with wavelength λ p (wave-vector k p = 2π/λ p ), we can write down the individual transmission function of the mth element, where i 0 stands for the imaginary unit and the immaterial common phase factor e i0n0kpL has been removed for simplicity.Each unit of this multi-element phase grating can be described by the total transmission function with the center of the mth element located at (x m , y m ) and width (height) δx = a x /2m g (δy = a y /2m g ), which is on the diagonal of each unit cell.Here, rect[τ ] is defined as a rectangular function, having the expression where τ = (x − x m )/δx or (y − y m )/δy.When the elements of the grating are distributed on the diagonal line in the first or third quadrants, it is referred to as type A grating as shown in figure 1(b), with the coordinate of the mth element recorded as (x m , y m ) = (x ′ m , y ′ m ).When the element of the grating is distributed on the diagonal in the second or fourth quadrants, we call it the type B grating, and the corresponding coordinate is recorded as for a normally incident light beam diffracted into angle (θ x , θ y ).It is straightforward to write down the diffraction intensity (I d ) contributed by (M x × M y ) irradiated unit cells where the beam width W ζ has been taken into account in terms of the ratio Diffraction peaks are known to occur at discrete angles θ i (θ j ) determined by with R x = a x /λ p , R y = a y /λ p .For convenience, we adopt the coordinate on the diffraction plane (i, j) to represent the diffraction order corresponding to the diffraction angle (θ i , θ j ) in the following discussion.Based on this consideration, we would focus on the (i, j)th order diffraction by examining In what follows, we restrict our discussions to diffraction peaks with k x = k i (k y = k j ) only in the case of δn −m = −δn m (spatially odd symmetry along the diagonal line) such that equation ( 4) can be translated into for the (i, j)th diffraction order, with setting , where '±' corresponds to type A/B grating respectively.
From equation (7), we can easily attain E i,j = 0 by requiring m − 1/2)π to eliminate the (i, j)th diffraction order (zero diffraction intensity), with l i,j m being any integer.On the basis of E i,j = 0, we can further obtain the formula for any other orders (i + ξ, j + η) for example, with ∆ m = (2m − 1)π/m g and ξ, η ∈ ∀Z, where '±' corresponds to type A/B grating.Equations ( 7) and ( 8) allow us to analyze and control the diffraction behavior in the diffraction plane.The validity of this method has been verified through full-wave simulations using COMSOL in the previous work [37] conducted in a 1D scenario.

Results and discussion
Numerical calculations and qualitative discussions can provide further insights into the analytical results derived in the previous section.Our calculations and discussions will be restricted to the weak modulation case of |δn m,n | ≪ n 0 to avoid multiple reflections between the phase grating and the surrounding medium of refractive index n = n 0 .In this section, we will discuss in detail three special diffraction patterns and the derivation of their formation conditions as well as the underlying physical mechanisms.

Elimination of equally spaced oblique direction lines
We try to explore how to eliminate the oblique diffraction line with an arbitrary slope k l in the diffraction plane.Firstly, we restrict the following discussions to diffraction peaks under the condition E i,j = 0 by requiring m ∈ Z) from equation (7).After a simple rearrangement, the refractive index distribution conditions of the elements in the first/second quadrant (m ∈ [1, m g ]) are obtained as to realize elimination of the (i, j)th diffraction order (i It is easy to achieve those in the third/fourth quadrant according to spatial odd symmetry along the diagonal line The sign '±' in equation ( 8) also corresponds to type A/B grating here.For the convenience of explanation, we take type A grating as an example, that is, the positive sign is taken before i in equation (9).It is evident that the diffraction intensity of orders (i ± ∆, j ∓ ∆) will also vanish concurrently due to the fact that the summation of the two indices (S = i + j) remains unchanged and satisfies equation ( 9) as well, with ∆ ∈ ∀Z.Consequently, we can assert that all absent orders (i ± ∆, j ∓ ∆) are located on a contiguous oblique line that goes through the (i, j)th order and has a slope of y) is defined as the interval in the horizontal or vertical direction between the adjacent diffraction orders according to the equation (6).Additionally, the negative sign of k l is determined by the inverse increase of the (i ± ∆, j ∓ ∆)th order in relation to the (i, j)th order, both horizontally and vertically, which restricts the scope of the oblique angle's range to (π/2, π).
Similarly, for type B grating, the minus sign is taken in front of i in equation ( 9), then the range of the oblique angle is (0, π/2).In the diffraction plane of the grating, all (i + ξ, j + η)th orders with ±ξ + η = m g vanish collectively consisting of a series of oblique line arrays paralleled to the vanishing line going through the (i, j)th order, where m g = (0, ±2, ±4, . ..)m g .
We verify via numerical calculations that it is viable to eliminate an oblique diffraction line 'going through' (i, j) with appropriate values of δn i,j m according to equation (9).It tells the requirement δn i,j m = −δn i,j −m can be designed for various combinations of R x , R y , i, j and l i,j m while m g and L/λ p are fixed parameters.For instance, in figure 2(a 1 ), we can choose R x = 30 and in figure 2(a 2 ), we can select R x = 50, while keeping R y = 30, i = −1, j = 1, and l −1,1 m ≡ (1, 2, 4) constant for type A grating.This configuration effectively eliminates the oblique diffraction line that passes through the (−1, 1)th order, with a slope of Similarly, we can also achieve the elimination of the oblique diffraction line at the (1, 1)th order, with a slope of k l = R x /R y = 1, in figures 2(a 3 ) and (a 4 ) by setting i = 1 while keeping other parameters unchanged.This is accomplished using type B grating.Surviving diffraction orders are highly asymmetric and other choices of l i,j m will also eliminate equally spaced oblique direction lines, but with various diffraction asymmetries or slopes.
The underlying physics for the elimination of oblique diffraction line may be attributed to the destructive interference between each pair of light beams scattered by the (±m)th elements in the same unit as illustrated in figure 2(b).This paired destructive interference can be understood as described below.(i) We note from equation (6) that the (i, j)th light beam scattered by the (±m)th element acquires a (exact) forward phase α  i,j m − 1/2)π.(iv) In the same way, the intensity of the other (i ± ∆, j ∓ ∆)th (or (−i ± ∆, j ± ∆)th) diffraction orders will vanish simultaneously, having the identical sum of diffraction order coordinates S = i + j (or S = −i + j).Consequently, we would observe the elimination of equally spaced oblique direction lines of grating.

Elimination of alternately crossed oblique direction lines
Secondly, we further explore the feasibility of realizing alternately crossed oblique direction lines with E i,j = 0 as the initial condition, further requiring E i+ξ,j+η = 0 with ±ξ + η = 2k, k ∈ {±1, ±2, . ..},where '±' corresponds to the A/B type grating, respectively.To achieve this goal, we note with ∆ m = (2m − 1)π/m g being the mean phase shift of the 2kth diffraction orders relative to the (i, j)th diffraction orders contributed by the ±mth elements of the type of grating shown in figure 1. Accordingly, E i+ξ,j+η (±ξ + η = 2k) will become vanishing if we require If m g is an even integer, then the conjugate terms match each other perfectly, where m * = m g − m + 1 denotes the conjugate element of the mth element shown in figure 3(b).On the contrary, if m g is an odd integer, the elimination of alternately crossed oblique direction lines will also need sin with m c = (m g + 1)/2 for the unpaired term.It is thus clear that the (i + ξ, j + η)th diffraction orders will disappear (E i+ξ,j+η = 0) at intervals when (l i,j m + l i,j m * ) is chosen as an even integer, whether m g is even or odd, with ±ξ + η = 2k.
The elimination of alternately crossed oblique direction lines should be attributed to a dual-paired destructive interference as illustrated in figure 3(b), which can also be understood in three steps.

Selection of equally spaced oblique diffraction lines
Finally, we consider whether it is possible to realize a more interesting case of E i+ξ,j+η = 0 with (±ξ + η) ∈ {±1, ±2 . ..}, based on E i,j = 0, realizing thus selection of equally spaced oblique diffraction lines with arbitrary slopes.According to equation (8), the intensity of the (i + ξ, j + η)th order will become vanish (E i+ξ,j+η = 0) with requiring in the case of (±ξ + η) ̸ = m g ∈ {±1, ±3, ±5, . ..}m g so as to guarantee cos[(±ξ after a sequential summation of the 2m g sine functions.It is thus clear that we can attain E i+ξ,j+η = 0 being ±ξ + η ̸ = m g as long as (l  In the case of ±ξ + η = m g , however, a straightforward calculation from equation ( 14) yields E i+ξ,j+η = 8 sin(k i+ξ δx/2) sin(k j+η δy/2) a x a y k i k j mg m=1 which cannot be zero because all terms in the summation are equal when (l i,j m+1 − l i,j m ) is an odd integer.As a result, we can attain one or equally spaced oblique diffraction lines while eliminating all others only if just a specific value of (i + ξ) falls within {−I, I} and (j + η) falls within {−J, J}, being ±ξ + η = m g , depending critically on R x = a x /λ p , R y = a y /λ p .
Combining equation (9) with the above conclusion, it is easily found that all [i + ξ, j + η]th orders with (±ξ + η) = m g stand in a series of parallel oblique lines.Here m g represents a set of numerical values, each corresponding to one of the paralleled oblique diffraction lines.In the diffraction plane, the interval of adjacent lines can be modulated with R x and R y values, with the slope of oblique lines k l = ∓ Rx Ry which can be attributed to the opposite or identical order interval increment in x and y direction in diffraction plane.The entire set of parallel oblique lines can be shifted simultaneously in position, by modulating (i, j) value with identical interval 2m g .Additionally, the rotation for the angle of equally spaced oblique lines can also be achieved by modulating R x and R y values.
We verify via numerical calculations that it is viable to select one or more desired oblique diffraction lines with appropriate values of δn i,j m according to equation (9).It tells that the requirement δn  1,2,3 = (1, 2, 3) for instance, we can see from figure 4(a 1 ) that the survive orders, whose coordinates in the diffraction plane satisfy ξ + η = m g , form an oblique diffraction line with the slope k l = −1 going through the origin point (0, 0) in the diffraction plane.Figure 4(a 2 ) shows the case taking i = 1, j = 2, l 1,2 1,2,3 = (1, 2, 3) that three equally spaced oblique diffraction lines will be obtained with the slope k l = − 8  3 due to ξ + η = m g .Another case without going through the origin is shown in figure 4(a 4 ), depicting two oblique diffraction lines with the slope k l = − 8  3 surviving with i = −1, j = 1, l −1,1 1,2,3 = (1, 2, 3).Furthermore, figure 4(a 3 ) displays the surwive orders, whose coordinates satisfy −ξ + η = m g , form a single oblique diffraction line with the slope of k l = 1 (arctan ϕ = π/4) using type B grating with R x = R y = 30, l 0,3 1,2,3 = (1, 2, 3).The (i + ξ, j + η)th orders (±ξ + η = m g ) of type A/B grating will form equally spaced oblique diffraction lines with a selected slope while the elimination of other orders arises from a sequential destructive interference as illustrated in figure 4(b), which will be explained once again in three steps.(i) We

Conclusions
In summary, a 2D multi-element phase grating has been designed to exhibit spatially odd-symmetric offset refractive indices for 2m g squared elements along one transparent diagonal line in each unit cell.It is of interest that this unique grating with neither loss nor gain can be well engineered on demand, e.g. to eliminate equally spaced oblique diffraction lines, eliminate alternately crossed oblique diffraction lines, and select equally spaced oblique diffraction lines.Analytical discussions of these intriguing asymmetric-diffraction phenomena are verified via numerical calculations and explained in terms of paired, dual paired, and sequential destructive interference between different elements along each transparent diagonal line.In terms of practical fabrication, the proposed approach can be experimentally realized through silicon-based material doping [40,41] or by employing high-transmittance thin-film perovskite materials [42][43][44][45].A main advantage of this grating design is the possibility of achieving multiple diffraction functions by flexible spatial engineering of just offset refractive indices.These sub-wavelength refractive index-modulating gratings have the potential to establish a new theoretical basis for attaining accurate control over imaging, optical routing, optical quick response codes, and similar aspects in scattering-type optical devices [46], thus catalyzing progress in interconnected disciplines.
2m+1 4mg a y ) for m < 0. Note that m ̸ = 0 by definition.Further considering the translational invariance, we now make a Fourier transform of the total transmission function to obtain E(θ x , θ y ) = ´ax/2 −ax/2 dx ´ay/2 −ay/2 dy T(x, y)e −i0(kxx+kyy) where θ ζ (ζ = x, y) refers to an angle deviating from the z direction in the ζ − z plane while k ζ = k p sin θ ζ denotes a projection of k p

Figure 1 .
Figure 1.(a) Schematic of a 2D phase grating with period ax and ay along the x direction and y direction, which can diffract a light beam with amplitude E in incident along the z direction into several beams with amplitudes E i,j deviating with angles (θ i , θ j ) relative to the z direction.(b) Fine structure in a unit consisting of 2mg elements on the diagonal with a common width/height δx = ax/2mg, δy = ay/2mg while the other part of the unit is opaque.The thickness of each element along the z direction is L, while the refractive indices of them nm = n0 + δnm, is restricted by δnm = −δn −m .
p L and a (mean) deflected phase −β i,j m = ∓k i x ′ m − k j y ′ m in addition to a common amplitude independent of m. (ii) We write down E i,j m ∝ cos (α m − β i,j m ) for the (±m)th elements by further considering δn i,j m = −δn i,j −m .(iii) We find that the (±m)th elements contribute a beam superposition, to the (i, j)th-order diffraction, of amplitude (E i,j m + E i,j −m ) ∝ cos(α i,j m − β i,j m ), which will become vanishing in the case of α i,j m = β i,j m + (l

Figure 3 .
Figure 3. Diffraction intensity I d against sin θx and sin θy with the elimination of alternately crossed oblique diffraction lines using type A grating based on a missing (1, 1)th diffraction order in panel (a1) or (1, 2)th in panel (a2).(b) Schematic of a dual-paired destructive interference between diffracted amplitudes contributed by the ±mth and (±m * )th elements.The parameters are the same as in figure 2 except l 1,1 m = l 1,2 m = (1, 2, 2, 1), mg = 4.

Table 1 .m
Sufficient conditions for realizing three types of asymmetric diffraction patterns.Equally spaced elimination lines Alternately crossed elimination lines Equally spaced lines selection l i,j m = 0, ±1, ±2, ±3, . . .2)π has been taken to eliminate the (i, j)th-order diffraction, yielding δn i,j m in equation(10).(ii)On this basis, we then write down E i+ξ,j+η m ∝ sin[l i,j m π − (±ξ + η)∆ m /2] for the (±m)th elements, which upon the multiplication of a non-vanishing cos[(±ξ + η)∆ 1 /2] being (±ξ + η) ̸ = m g can be rewritten as a sum of inward E i+ξ,j+η,in m ∝ sin[l i,j m π − (±ξ + η)(m − 1)π/m g ] and outward E i+ξ,j+η,out m ∝ sin[l i,j m π − (±ξ + η)mπ/m g ] components.(iii) We find that the sum of E i+ξ,j+η,out m + E i+ξ,j+η,in m−1 contributed by the (±m)th and (±(m − 1))th elements is proportional to cos[(l i,j m − l i,j m−1 )π/2].Consequently, only the (i + ξ, j + η)th order with (±ξ + η) = m g can survive due to a sequential destructive interference of the adjacent ±(m − 1)th and ±mth elements when l are of the opposite parity).So, all (i + ξ, j + η)th orders standing in lines with (±ξ + η) = m g can survive when l i,j m − l i,j m−1 is an odd integer.Sufficient conditions derived above for realizing the elimination of equally spaced oblique direction lines, the elimination of alternately crossed oblique direction lines, and the selection of equally spaced oblique diffraction lines are summarized in table 1 in terms of δn i,j m and l i,j m .