Goal-driven method for decoding the configuration of coherent wave groups required for the generation of arbitrary-order vortex lattices

Together, the number of waves, wave vectors, amplitudes, and additional phases constitute the coherent wave group configuration and determine the pattern of the interference field. Identifying an appropriate wave group configuration is key to generating vortex lattices via interferometry. Previous studies have approached this task by first assigning the four elements, then calibrating the vortex state of the interference field. However, this method has failed to progress beyond generating third-order vortex lattices, which are insufficient for some practical applications. Therefore, this study proposes a method for determining the proper wave group configurations corresponding to arbitrary-order vortex lattices. We adopt a goal-driven approach: First, we set a vortex lattice as the target field and model it, before decomposing the target field into a sum of multiple harmonics using Fourier transforms. These harmonics constitute the wave group required to generate the target vortex lattice. As vortex lattices of any order can be set as the target field, the proposed method is compatible with any mode order. Simulations and experiments were conducted for fourth- and fifth-order vortex lattices, thus demonstrating the effectiveness of the proposed method.


On Proof of Eq. (31) in the manuscript and the principle of choosing parameters
Proof: Here we are dealing with the open statement The proof process requires the use of trigonometric formulas as follow: 2sin sin cos( ) cos( ) 2 cos sin sin( ) sin( ) .2sin cos sin( ) sin( ) 2 cos cos cos( ) cos( ) When the field contains an odd number group of edge dislocation lines in the form of parallel line family, m is an odd number.a r =-2 m sinξ r , b r =2 m cosξ r .
When the field contains an even number group of edge dislocation lines in the form of parallel line family, m is an even number.a r =2 m cosξ r , b r =-2 m sinξ r .
1.1.We proof the situation when m is an odd number.
Inductive step: Now we assume the truth of (2 1) St k + , for some (particular) k ∈N , that ( Using the induction hypothesis (2 3) St k + , we find the real part ' . The last term is similar.Thus For the imaginary part and its first term is ' ' .The last term is similar.Thus The assumption leads us to the following, when m is an odd number: 1.2.We proof the situation when m is an even number.
Basis Step: we star with the statement (2)

St
when m=2 and find that

St
is true.
Inductive step: Now we assume the truth of (2 ) St k , for some (particular) k ∈N , that is and its first term is '' For the imaginary part and its first term is '' The assumption leads us to the following, when m is an even number: We give a 1 =-b 2 =a, b 1 =a 2 =b and ξ 2 =ξ 1 +0.5π for two reasons: First, it is obtained by analyzing the first-and third-order vortex lattices.Secondly, bring the parameters into the mth-order case and solve Eq. (34) = 0 and Eq.(35) = 0. We can get m zero lines of the imaginary and real parts intersecting at the preset vortex points, i.e., U mth = E mth g1 +E mth g2 is the mth-order vortex lattice.
We first look at the case of the first-order vortex array presented in the manuscript.For E 1st g1 , Eqs. ( 11) and ( 12) can be rewritten as For E 1st g2 , Eqs. ( 13) and ( 14) can be rewritten as For the third-order vortex lattice in the manuscript.Eqs. ( 20) and ( 21) related E 3th g1 , can be rewritten as 3th 1 2 1 1 For E 3th g2 , Eqs. ( 22) and ( 23) can be rewritten as In the manuscript, we superimpose two edge dislocation field E mth g1 [Eq.( 29)] and E mth g2 [Eq.( 30)] to obtain the vortex lattice U mth .We use a qualitative analysis in third-order vortex lattice.We obtain three zero lines of imaginary part and the real part intersect at the vortex point using Equivalent Infinitesimal Replacement and the Existence Theorem of Root.
It is also similar to the case of mth-order.We still divide the field taking the edge dislocation line as the boundary, determine the positivity and negativity of the real and imaginary parts on the boundary, and then use the Existence Theorem of Root to determine the number and distribution of zero lines, i.e., solve the solution of Eq. (29)=0 and Eq.(30)=0.We take the point (0, 0) as an example for analysis as usual.
Based on Equivalent Infinitesimal Replacement, sinε may replace ε when ε is getting close to zero.Also, there are sin(A j x+B j y) ~ A j x+B j y and sin(A j 'x+B j 'y) ~ A j 'x+B j 'y when (x,y) approach (0,0).At the same time, making a 1 =-b 2 =a, b 1 =a 2 =b, ξ 2 =ξ 1 +0.5π, we can get the real part and the imaginary part of U mth , which is expressed as mth 1 1 In order to get the number and distribution of the zero lines of the real and imaginary parts, solving Eq. (29) = 0 and Eq.(30) = 0 becomes solving Eq. (S29) = 0 and Eq.(S30) = 0.
We first analyze the simplest case: a or b is equal to 0. Since the case sinξ = cosξ = 0 does not happen, a and b cannot be zero at the same time.We choose to discuss the case of a= 0 and we can directly get the solution of Eq.(S29) = 0 is ' ' =0, and the solution of Eq.(S30) = 0 is =0, where j=1,2,3….m.In this way, we can easily get that m zero lines of the real and imaginary parts to intersect at point (0,0), i.e., the (0,0) is the mth-order vortex.It is similar for b = 0. Now we analyze the case when a, b is not zero.The coordinate relations on the boundaries satisfy A j x+B j y=0 or A j 'x+B j 'y=0.We bring the coordinate relations to equations (S29) and (S30) in turn and obtain the algebraic expressions for the real and imaginary parts on each boundary.
Re [ ( : We set t = 1,2,3.... 2m.When t is odd number, the results of numerical multiplication of the real or imaginary part on the two adjacent boundaries are When t is even number, the results of numerical multiplication on the two adjacent boundaries are In addition, due to the periodicity of rotation around (0, 0), B 2m+1 =B 1 .In either case, the positive and negative of Re(B t )Re (B t+1 ) and Im(B t )m (B t+1 ) are always opposite.According to the Existence Theorem of Root, if the product of two boundaries of an interval is negative, then the inte distribution of
Step 2: According to the distribution of edge dislocation lines [Fig.S3(b3) and (c3)] and the parameter selection rules, we can obtain the real and imaginary parts of the two fields E where κ=2π/d.Figure S3 shows the simulation results.We can see the vortex order and position in the field U 6th =E 6th g1 +E 6th g2 are consistent with the target field of the initial set.
Step 3: Decomposition process.By using Eqs.( 29)-(32), we can obtain each wave expression for the 6th-order vortex lattice generation  5)], exp[ ( 5)], exp[ ( 5)], exp[ ( 5 ) where κ = 2π/d.It can be seen that the amplitude of the wave is not equal.The threedimensional coordinate components of the wave vector satisfy k 0 = (k 2 x +k 2 y +k 2 z ) 0.5 .Then, by combining Eq. (33) with the preset vortex spacing, d, and the wavelength, λ, we can obtain the number of waves, the amplitude, the wave vectors, and the additional phases of the wave group.
Step 2: According to the distribution of edge dislocation lines [Fig.S4(b3) and (c3)] and the parameter selection rules, we can obtain the real and imaginary parts of the two fields E 7th g1 and E 7th g2 7 1 where κ=2π/d.Figure S4 shows the simulation results.We can see the order and position in the field U 7th =E 7th g1 +E 7th g2 are consistent with the target field of the initial set.The construction process for the vortex lattice of any order is the same as the example of the fourth-order vortex lattice.
Step 3: Decomposition process.By calculation, we can obtain each wave expression for the 7th-order vortex lattice generation  (S44-2) where κ = 2π/d.It can be seen that the amplitude of the wave is not equal.The threedimensional coordinate components of the wave vector satisfy k 0 = (k 2 x +k 2 y +k 2 z ) 0.5 .Then, by combining Eq. (33) with the preset vortex spacing, d, and the wavelength, λ, we can obtain the number of waves, the amplitude, the wave vectors, and the additional phases of the wave group.
After obtaining the appropriate configuration in practical applications, we can use optics to modulate the interference waves used to generate the OVL.However, the modulation of multiple engineered waves using conventional refractive/diffractive optical elements remains a challenge.At present, relevant studies have explored the potential of micro-and nanostructured devices for realizing multi-beam modulation in the generation of OVLs, but the interference fields are still limited to low-order modes [1,2].In short, determining the appropriate configuration is a fundamental and crucial step for realizing the generation of arbitrary-order OVLs via engineered beam interference.Our next study will focus on designing a suitable metasurface device to modulate multiple beams in order to maximize the number of attainable OVL patterns.

Fig. S1 .
Fig. S1.Schematic diagram of the vortex lattices generated by four wave interference with different additional phases.
from the principle of Mathematical Induction that ( ) St m is true for all positive integers m. 3. On Proof of "a 1 =-b 2 =a, b 1 =a 2 =b and ξ 2 =ξ 1 +0.5π" and "U mth = Fig. S differe imagi Fig. S2 (a parts on the co

Fig. S3 .
Fig. S3.Construction of sixth-order vortex lattice U 6th .(a1) The target field is set to be a sixthorder OVL.The vortex order is six, and the vortex spacing is d.(a2) Zero-crossing topology at each vortex point.(a3) Extension of the local zero-lines in (a2) to generate the zero-line topology for the whole field, where the circles indicate the positions of vortices.Then, (a3) is split into two sets of array edge dislocation lines, as shown in (b3) and (c3).The phase map (b1) and intensity distribution (b2) of E 6thg1 were modeled using (b3).Similarly, the phase map (c1) and intensity distribution (c2) of E 6th g2 were modeled using (c3).Thus, U 6th = E 6th g1 +E 6th g2 .(d1) Phase map of U 6th .(d2) Intensity distribution of U 6th .(d3) Zero-line topology of U 6th (red lines: Re(U 6th ) = 0; green lines: Im(U 6th ) = 0).The insets in (d1) and (d3) show magnifications of the phase and topology at the vortex points, respectively.

Fig. S4 .
Fig. S4.Construction of seventh-order vortex lattice U 7th .(a1) The target field is set to be a seventh-order OVL.The vortex order is seven, and the vortex spacing is d.(a2) Zero-crossing topology at each vortex point.(a3) Extension of the local zero-lines in (a2) to generate the zero-line topology for the whole field, where the circles indicate the positions of vortices.Then, (a3) is split into two sets of array edge dislocation lines, as shown in (b3) and (c3).The phase map (b1) and intensity distribution (b2) of E 7thg1 were modeled using (b3).Similarly, the phase map (c1) and intensity distribution (c2) of E 7th g2 were modeled using (c3).Thus, U 7th = E 7th g1 +E 7th g2 .(d1) Phase map of U 7th .(d2) Intensity distribution of U 7th .(d3) Zero-line topology of U 7th (red lines: Re(U 7th ) = 0; green lines: Im(U 7th ) = 0).The insets in (d1) and (d3) show magnifications of the phase and topology at the vortex points, respectively.