Generation of cylindrical optical lattices: from shape-invariant to propagation-invariant

Huiqing Li; Yile Shi; Yongsheng Dong; Yefeng Liu; Bo Zhang; Zhijun Ren

doi:10.3788/COL202523.082601

1. Introduction

A plane wave passing through a periodic grating is diffracted to induce the Talbot effect in the near-field region. The Talbot effect is also known as self-imaging. Multiple and regular repetition of self-images results in the formation of two-dimensional (2D) periodic intensity distributions in the near field behind the grating, also called Talbot carpets^[1], or three-dimensional (3D) optical lattices in the case of 2D gratings^[2,3]. Many efforts have been made to further develop the potential of this effect, such as generating carpet beams with radial gratings^[4].

As a class of structured optical field, from their introduction in 2018^[4,5], carpet beams have grown significantly and received considerable attention because of their carpet-shaping transverse intensity distribution^[6,7]. Researchers have then generated various carpet beams with special intensity distributions by modulating beams with different gratings^[4–10]. Carpet beams are not a kind of beam with a determined beam shape. Although they have various beam shapes, all carpet beams have a common feature that they have a radiating shape with central symmetry, similar to actual carpets.

In generating carpet beams, gratings with periodic structures are the key elements. As is known, the “diffraction grating” is an optical element that imposes a “periodic” variation in the amplitude or phase of an incident electromagnetic wave. In the past, people mainly used transmissive-type amplitude gratings (TAGs) to produce carpet beams^[4,8–10]. However, a significant amount of optical energy is lost when a laser passes through an amplitude grating. It is worth mentioning that compared with TAGs, phase gratings (PGs) have higher diffraction efficiency because there is no opaque slit scattering of light; hence, PGs can manipulate the phase of the incident light and ideally maintain a constant intensity. PGs have been widely used in photonic integrated circuits^[11,12], beam manipulation^[13], and augmented reality (AR) displays^[14]. In this study, we will construct a new PG, instead of an amplitude grating (AG), to produce a novel carpet beam to ensure higher light energy utilization efficiency.

We first introduce a kind of even-symmetric radial phase (ESRP) gratings. Then, on the basis of the pure-phase modulation of ESRP gratings, we present the mathematical foundation of the work by considering the Fresnel diffraction integral formula in polar coordinates, and the complex amplitude expression of shape-invariant (not non-diffracting) beams is obtained. To construct non-diffracting carpet beams, we use the stationary phase principle. Through the analytical formula obtained on the basis of the stationary phase principle, a family of carpet beams with different parameters is numerically simulated. The beam distribution of this new non-diffracting carpet beam has the optical structure of radial lattices with polar symmetry; thus, they are named radial lattices.

It is well-known that the optical lattice is an artificial light structure with periodic structures. Since their emergence, optical lattices have captured considerable attention and exhibited great application value in many areas, such as quantum computation^[15], trapping ultracold atomic gases^[16,17], quantum phase transition^[18], spin–exchange interaction^[19], super-resolution microscopy^[20], and the optical induction technique^[21]. In the past, the most common method used by researchers to generate desired optical lattices was multi-beam interference^[22,23]. Other possible approaches to producing optical lattices were Fourier transform of the amplitude mask^[24,25] and holographic technology^[26,27]. Here, a stationary phase technique is introduced to generate optical lattices, which provides a new way of obtaining propagation-invariant optical lattices. To verify the effectiveness of this approach, we generate radial lattices using the introduced phase plate (PP) and axicon. The experimental results verify the theoretical predictions, and a family of diffraction-free radial lattices with a high light energy utilization rate is generated. Moreover, their optical structures, propagation characteristics, and distinctive phase characteristics are studied.

2. Theory

In this study, we focus on generating a novel type of optical lattice and obtaining the corresponding non-diffracting radial lattices on the basis of the stationary phase principle. We assume that the complex amplitude distribution of the initial light field ( $z = 0$ ) is $U (r^{'}, φ)$ , and $U (r, θ)$ is the complex amplitude distribution at transmission distance $z$ , where $(r^{'}, φ)$ and $(r, θ)$ are the polar coordinates at the initial and observed planes, respectively. The Fresnel–Kirchhoff diffraction integral in polar coordinates can be expressed as $U (r, θ) = h e^{i α r^{2}} \int_{0}^{+ \infty} \int_{0}^{2 π} U (r^{'}, φ) e^{i α r'^{2}} e^{- 2 i α r r' \cos (φ - θ)} r^{'} d r^{'} d φ,$ (1)where $h = \exp (i k z) / i λ z$ , $k = 2 π / λ$ is the wavenumber, $λ$ is the wavelength of the light beam, and $α = k / 2 z = π / λ z$ . When light incident on a 2D radial structure is collimated, the initial light field can be expressed as $U (r^{'}, φ) = A (φ) T (r^{'}) .$ (2)In Eq. (2), $A (φ)$ is a periodic function with a period of $2 π$ as the amplitude transmittance function of the grating, expanded as a Fourier series, $A (φ) = \sum_{t = - \infty}^{+ \infty} c_{t} e^{i t φ} .$ (3)Through substituting Eq. (3) into Eq. (2) and then substituting the results into Eq. (1), the following equation can be obtained: $U (r, θ) = h e^{i α r^{2}} \int_{0}^{+ \infty} T (r^{'}) e^{i α r'^{2}} r^{'} d r^{'} \int_{0}^{2 π} \sum_{t = - \infty}^{+ \infty} c_{t} e^{i t φ} e^{- 2 i α r r' \cos (φ - θ)} d φ .$ (4)In the above equation, $c_{t}$ is the $t$ th Fourier series coefficient. According to the Jacobi–Anger expansion^[28], $e^{- 2 i α r r^{'} \cos (φ - θ)} = \sum_{n = - \infty}^{+ \infty} {(- i)}^{n} e^{- i n (φ - θ)} J_{n} (2 π ρ r^{'}),$ (5)where $ρ = r / λ z$ and $J_{n} (*)$ is the $n$ th Bessel function of the first kind. On the basis of Eq. (5), the following expression can be acquired: $U (r, θ) = h e^{i α r^{2}} \sum_{n = - \infty}^{+ \infty} {(- i)}^{n} c_{n} e^{i n θ} 2 π \int_{0}^{+ \infty} f (r^{'}) J_{n} (2 π ρ r^{'}) r^{'} d r^{'},$ (6)where $f (r^{'}) = T (r^{'}) \exp (i α r^{' 2})$ . If we use the Hankel transform of the $n$ th order of the function $f (r^{'})$ ^[29], $H_{n} {f (r^{'})} = 2 π \int_{0}^{+ \infty} f (r^{'}) J_{n} (2 π ρ r^{'}) r^{'} d r^{'},$ (7)Eq. (6), which is about an integral of $r^{'}$ , can be written in the form of a Hankel transformation as follows: $U (r, θ) = h e^{i α r^{2}} \sum_{n = - \infty}^{+ \infty} {(- i)}^{n} c_{n} e^{i n θ} H_{n} {f (r^{'})} .$ (8)Equation (8) represents that the diffraction of 2D structures is separable in the polar coordinates.

In this paper, to construct a new optical lattice, we introduce a type of radial PG with the absolute value of the cosine function. The modulation function of the introduced PG can be written as $γ | \cos (q φ) | = γ \cos (q φ) sgn [\cos (q φ)],$ (9)where “sgn” indicates the sign function, $γ$ is the grating phase modulation amplitude, and $q$ determines the number of grating spokes. Figure 1 gives the diagrams of $γ | \cos (q φ) |$ when $q$ takes different values with $γ = π / 2$ . The grating spoke value in the diagram is twice the given $q$ value; hence, we call this ESRP grating. The main feature of the ESRP grating is the continuous change of the spatial period along the radial direction.

Figure 1.ESRP gratings with different q values with γ = π/2. (a) q = 3; (b) q = 4; (c) q = 5; (d) q = 6; (e) q = 7; (f) q = 8.

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When parallel light is incident on the ESRP gratings, the transformation function $A (φ)$ of the modulated beams by the introduced ESRP grating can be written as $A (φ) = e^{i γ | \cos (q φ) |} = \sum_{m = - \infty}^{+ \infty} i^{m} e^{i m q φ} J_{m} (β),$ (10)where $β = γ \cdot sgn [\cos (q φ)]$ . By utilizing the mathematical properties of Bessel functions^[29], as shown below: $J_{- a} (*) = {(- 1)}^{a} J_{a} (*),$ (11)we can rewrite Eq. (10) as $A (φ) = \sum_{m = - \infty}^{+ \infty} i^{m} e^{i m q φ} J_{m} (β) = J_{0} (β) + \sum_{m = 1}^{+ \infty} i^{m} (e^{- i m q φ} + e^{i m q φ}) J_{m} (β) .$ (12)

From the comparison of Eqs. (3) and (12), $c_{0} = J_{0} (β)$ , $c_{t} = c_{- t} = i^{m} J_{m} (β)$ , and other coefficients are 0. By plugging these coefficients into Eq. (8), using Eq. (11), we can obtain $U (r, θ) = h e^{i α r^{2}} {J_{0} (β) H_{0} [f (r^{'})] + \sum_{m = 1}^{+ \infty} {(- i)}^{m q - m} J_{m} (β) (e^{- i m q θ} + e^{i m q θ}) H_{m q} [f (r^{'})]} .$ (13)

Now, we suppose $T (r^{'}) = 1$ , which is $f (r^{'}) = e^{i α r^{' 2}}$ , with only the radial grating at the transmission distance $z = 0$ , to obtain the diffraction pattern of the radial structure. On the basis of the integral tables for $\int_{0}^{+ \infty} x \cos (α x^{2}) J_{v} (b x) d x$ and $\int_{0}^{+ \infty} x \sin (α x^{2}) J_{v} (b x) d x$ ^[30], we obtain $H_{0} {e^{i α r^{' 2}}} = \frac{i π}{α} e^{- i \frac{{(π ρ)}^{2}}{α}},$ (14) $H_{m q} {e^{i α r^{' 2}}} = \frac{π ρ}{2} {(\sqrt{\frac{π}{α}})}^{3} e^{i [\frac{m q π}{4} - \frac{{(π ρ)}^{2}}{2 α}]} [J_{\frac{m q + 1}{2}} (\frac{π^{2} ρ^{2}}{2 α}) + i J_{\frac{m q - 1}{2}} (\frac{π^{2} ρ^{2}}{2 α})] .$ (15)Then, we plug the above two equations into Eq. (13) to generate $U (r, θ) = e^{i k z} {J_{0} (β) + \frac{R}{2} e^{i R^{2}} \sum_{m = 1}^{+ \infty} ϕ_{m} (e^{- i q m θ} + e^{i q m θ}) [J_{\frac{m q + 1}{2}} (R^{2}) + i J_{\frac{m q - 1}{2}} (R^{2})]},$ (16)where $ϕ_{m} = \sqrt{2} π (- i)^{m (q / 2 - 1) + 1} J_{m} (β)$ , and $R = r \sqrt{π} / 2 λ z$ . Finally, through using Euler’s formula $e^{\pm i θ'} = \cos θ^{'} \pm i \sin θ^{'}$ , Eq. (16) can be transformed into the following form: $U (r, θ) = e^{i k z} {J_{0} (β) + R e^{i R^{2}} \sum_{m = 1}^{+ \infty} ϕ_{m} \cos (m q θ) [J_{\frac{m q + 1}{2}} (R^{2}) + i J_{\frac{m q - 1}{2}} (R^{2})]} .$ (17)Equation (17) is the Fresnel diffraction formula of a radial PG with the cosine function in a polar coordinate system. The diffraction complex amplitude is an explicit function of $R$ . Equation (17) shows that the form of the generated optical pattern remains unchanged under propagation. However, the diffraction pattern expands with a factor $\sqrt{z}$ in the propagation. Thus, the radial lattices are not strictly non-diffracting beams, although they are shape-invariant beams propagating in free space.

Given that the non-diffracting characteristics are very important when people use laser beams in some scientific experiments, the transformation from an optical lattice to a non-diffracting radial lattice is a key leap. We take the stationary phase principle to construct non-diffracting radial lattices using the introduced gratings. Here, the axicon phase is necessary in our scheme. The phase transform function of the axicon is $T (r^{'}) = {\begin{matrix} e^{[- i k (n_{0} - 1) θ_{0} r^{'}]}, & r^{'} \leq r_{1} \\ 0, & r^{'} > r_{1} \end{matrix},$ (18)where $n_{0}$ is the refractive index of the axicon, $θ_{0}$ is the base angle of the axicon, which is usually a small angle, and $r_{1}$ is an aperture radius of the entrance pupil of the axicon. Substituting Eq. (18) into $f (r^{'}) = T (r^{'}) e^{i α r^{' 2}}$ , we can easily obtain $f (r^{'}) = e^{- i k (n_{0} - 1) θ_{0} r^{'}} e^{i α r^{' 2}} = e^{i k [\frac{r^{' 2}}{2 z} - (n_{0} - 1) θ_{0} r^{'}]} .$ (19)The $m q$ -order Hankel transform of $f (r^{'})$ can be expressed as $H_{m q} {e^{i k [\frac{r^{' 2}}{2 z} - (n_{0} - 1) θ_{0} r^{'}]}} = 2 π \int_{0}^{+ \infty} e^{i k [\frac{r^{' 2}}{2 z} - (n_{0} - 1) θ_{0} r^{'}]} J_{m q} (2 π ρ r^{'}) r^{'} d r^{'} .$ (20)

In mathematics, Eq. (2) is a complex integral expression, and its analytic solution is difficult to obtain directly. The stationary phase principle can be used to simplify oscillatory integrals that are difficult to solve directly, i.e., it can approximately solve integral equations with the form $\int g (r) \exp [i k j (r)] d r$ when $k \to \infty$ . The mathematical expression of the stationary phase technique is given as $\int g (r) \exp [i k j (r)] d r \approx \sqrt{\frac{2 π}{k j^{''} (r_{0})}} \exp (\pm i \frac{π}{4}) g (r_{0}) \exp [i k j (r_{0})] .$ (21)

For Eq. (2), we set $g (r^{'}) = 2 π J_{m q} (2 π ρ r^{'})$ and $j (r^{'}) = {r^{'}}^{2} / 2 z - (n_{0} - 1) θ_{0} r^{'}$ . Through taking the derivative $j (r^{'})$ , the stationary phase point can be obtained when $r_{0} = (n_{0} - 1) θ_{0} z$ . When $r^{'} = r_{0} \in (0, r_{1})$ , a maximum non-diffracting distance $z_{\max} = r_{1} / (n_{0} - 1)$ $θ_{0}$ is generated. Substituting $r^{'} = (n_{0} - 1) θ_{0} z$ , $g (r^{'})$ , and $j (r^{'})$ into Eq. (21), we can obtain $H_{m q} {e^{i k [\frac{r'^{2}}{2 z} - (n_{0} - 1) θ_{0} r^{'}]}} \approx k_{r} {(λ z)}^{\frac{3}{2}} e^{i {\frac{π}{4} - k \frac{{[(n_{0} - 1) θ_{0}]}^{2} z}{2}}} J_{m q} (k_{r} r),$ (22)where $k_{r} = k (n_{0} - 1) θ_{0}$ . The distribution of optical fields at $0 < z < z_{\max}$ is shown below: $U (r, θ) \approx - i k_{r} \sqrt{λ z} e^{i (k z + \frac{π r^{2}}{λ z})} e^{i {\frac{π}{4} - k \frac{{[(n_{0} - 1) θ_{0}]}^{2} z}{2}}} \times {2 \sum_{m = 1}^{+ \infty} {(- i)}^{m q - m} J_{m} (β) \cos (m q θ) J_{m q} (k_{r} r) + J_{0} (β) J_{0} (k_{r} r)} .$ (23)From Eq. (23), the main feature of these diffraction patterns is that the radial wavenumber of all Bessel functions is independent of the propagation distance $z$ . Apparently, these kinds of radial lattices are non-diffracting beams. Different from previous studies^[31–33], the constructed radial lattices do not have the orbital angular momentum (OAM) characteristics since they do not contain a helical phase term in Eq. (23).

The intensity distribution of the different kinds of constructed radial lattices using ESRP gratings is $I (r, θ) = U (r, θ) U^{*} (r, θ) .$ (24)where $*$ denotes a complex conjugate. On the basis of Eq. (17), the diffraction patterns of radial lattices at a given $z$ can be calculated using Eq. (24).

In Fig. 2, shape-invariant [Eq. (17)] and propagation-invariant cylindrical lattice [see Eq. (23)] patterns for an ESRP grating with $q = 7$ spokes and $γ = π / 2$ at different distances are shown. Without loss of generality, from the first row of Fig. 2, although the shape of the cylindrical lattices remains unchanged during propagation, the radius of the “patternless area” near the $z$ axis increases with the increase in the propagation distance^[6]. By contrast, the second row of Fig. 2 illustrates that the cylindrical lattices keep their non-diffraction propagation characteristics. From shape-invariant to propagation-invariant lattices, the axicon is a key element.

$Simulated patterns of diffraction intensity at different propagation distances. (a) z = 60 cm; (b) z = 80 cm; (c) z = 100 cm. First row: shape-invariant lattice. Second row: propagation-invariant lattice.$

Figure 2.Simulated patterns of diffraction intensity at different propagation distances. (a) z = 60 cm; (b) z = 80 cm; (c) z = 100 cm. First row: shape-invariant lattice. Second row: propagation-invariant lattice.

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For the evaluation of the propagation characteristics, Fig. 3(a) presents a sectional diagram at any location of the propagation-invariant lattice. Every part of the radial lattice is non-diffracting during propagation, but the phase of the lattice will jump [Fig. 3(b)], which is called phase anomaly. In general, the Gouy phase shift is the phenomenon in which a converging wave undergoes a drastic phase change of an integer multiple of $π / 2$ as it passes through the focal region^[34].

Figure 3.Phase and intensity diagrams of the x–z plane for q = 7 and γ = π/2. (a) Sectional intensity pattern; (b) sectional phase pattern.

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Since the discovery of the anomalous axial phase behavior of optical beams by Gouy more than a hundred years ago^[35], research on the Gouy phase has been continuously carried out, including the Gouy phase research of Airy^[36] and Laguerre–Gaussian beams^[37]. In addition, Martelli et al. found the Gouy phase shift in Bessel beams up to the third order using an interferometric technique in 2010^[38]. As seen from Fig. 3(b), the phase jumps along the $z$ axis are close to $π / 2$ for radial lattices^[34]. The existence of this abrupt phase transition causes a redistribution of the optical field energy, resulting in a slight change in the radial lattice’s light intensity over the diffract-free propagation distance.

Moreover, the effects of $γ$ and $q$ values on the radial lattice form are also studied. We know that hollow beams with zero central intensity have received extensive attention owing to their wide applications in atomic optics^[39,40]. Various efficient methods for generating hollow beams, such as off-axis coupling of multimode fibers^[41], superposition of multi-off-axis vortex beams^[42], and cross-phase modulation in atoms^[43], have been investigated. Through adjusting the phase amplitude $γ$ of the grating, a new idea for the formation of hollow beams is provided in our research.

In Fig. 4, simulated patterns of the radial lattice generated by ESRP gratings with $q = 7$ spokes at $z = 80 cm$ for different $γ$ are given. For a non-diffracting radial lattice, the hollow region (i.e., the patternless area) can be constructed given $γ = 2.42, 5.52, 8.65, \dots$ for the grating phase amplitude. The reason is simply that the value of the zero-order Bessel function $J_{0} (β)$ in Eq. (23) is equal to 0 in this situation.

$Simulated patterns of diffraction intensity from radial PGs at different values of γ. (a) γ = π/2; (b) γ = 2.40; (c) γ = π; (d) γ = 3π/2; (e) γ = 5.52; (f): γ = 2π. First row: shape-invariant lattice. Second row: propagation-invariant lattice.$

Figure 4.Simulated patterns of diffraction intensity from radial PGs at different values of γ. (a) γ = π/2; (b) γ = 2.40; (c) γ = π; (d) γ = 3π/2; (e) γ = 5.52; (f): γ = 2π. First row: shape-invariant lattice. Second row: propagation-invariant lattice.

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Besides, we also discovered that various radial lattices with different optical patterns can be generated by tuning $γ$ . Figure 5 demonstrates several examples of radial lattices with different $q$ values based on $γ = π / 2$ and $z = 80 cm$ . By tuning the $q$ value of the PG, we can construct a polar symmetric diffraction pattern with different 2D optical lattices around the central region. With the increase in $q$ , the number of spokes of non-diffracting radial beams increases accordingly. Compared with the radius of shape-invariant lattices, the radius of the central region of the non-diffracting radial lattices increases slowly with the increase in $q$ .

$Simulated patterns of diffraction intensity from radial PGs at different numbers of spokes q. (a) q = 3; (b) q = 4; (c) q = 5; (d) q = 6; (e) q = 7; (f) q = 8. First row: intensity patterns. Second row: phase patterns.$

Figure 5.Simulated patterns of diffraction intensity from radial PGs at different numbers of spokes q. (a) q = 3; (b) q = 4; (c) q = 5; (d) q = 6; (e) q = 7; (f) q = 8. First row: intensity patterns. Second row: phase patterns.

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3. Experiments

We experimentally explore the evolution of generating cylindrical lattices based on the stationary phase technique. Figure 6 shows a schematic of the experimental setup. A low-power continuous laser with a wavelength of 632 nm passes through a polarizer to make the beam become a linearly polarized monochromatic light. Then, a beam expander is employed to collimate the laser beam with a divergence angle $< 0.1 mrad$ . The collimated beam illuminates a pure PP. The axicon with a base angle of 1° is closely installed behind the PP. Moreover, the entrance pupil diameter of the axicon is 2.50 cm, and the refractive index is 1.457. The axicon will be installed only in the generation of propagation-invariant radial lattices. The CCD (Microvision 130FC, pixel unit size of 5.4 µm) is placed at varying distances to record the intensity distribution of cylindrical optical lattices. Two polarizers in the entire experimental system can improve the quality of the generated lattices.

Figure 6.Experimental setup for generating the radial lattice.

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In the experimental system, the phase plates are key elements. We machined PPs by applying a homemade digital holographic direct printing system^[44]. The working principle of the holographic micro output system is as follows. The hologram is divided into several pieces according to the number of spatial light modulation (SLM) pixels, and the SLM is controlled by a computer to display frame by frame. Then the holograms on SLM are microfilmed by the optical system and imaged on the photosensitive material in sequence. The recording medium is installed on the computer numerical control stepping platform. With the input of holograms one by one, the stepping platform moves on the plane, and the whole hologram can be spliced. By this splicing method, a hologram with a high resolution and a large size can be obtained. The pixel size of the hologram output by the system is 318 nm.

According to the optical parameters of the experimental system, we can readily obtain the maximum non-diffracting propagation distance of the generated propagation-invariant lattice as $z_{\max} \approx 1539 mm$ . In order to verify whether the radial optical lattice generated in this work has non-diffracting characteristics, we record the intensity distribution of the radial lattice with spoke value $q = 7$ at different distances ( $z < z_{\max}$ ) using a CCD, as shown in the first row of Fig. 7. The optical lattice produced by Fresnel diffraction methods is diffused during propagation. The second row of Fig. 7 indicates that the pattern of the radial lattice generated using the stationary phase method can keep invariant during propagation. Moreover, the radial optical lattice can recover the original intensity distribution when it encounters partial obstacles in the propagation process, i.e., the lattice has a self-healing property.

$Experimental diffraction intensity patterns at different propagation distances. (a) z = 60 cm; (b) z = 80 cm; (c) z = 100 cm. First row: shape-invariant lattice. Second row: propagation-invariant lattice.$

Figure 7.Experimental diffraction intensity patterns at different propagation distances. (a) z = 60 cm; (b) z = 80 cm; (c) z = 100 cm. First row: shape-invariant lattice. Second row: propagation-invariant lattice.

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Grating PPs with different spokes $q$ are placed in front of the axicon, and different kinds of diffraction-free radial optical lattices with various structures are generated, as shown in Fig. 8. Figure 8 is consistent with the intensity diagram of the theoretical simulation results given in Fig. 5.

$Experimental patterns of diffraction intensity from radial PGs with γ = π/2 at different numbers of spokes q. (a) q = 3; (b) q = 4; (c) q = 5; (d) q = 6; (e) q = 7; (f) q = 8.$

Figure 8.Experimental patterns of diffraction intensity from radial PGs with γ = π/2 at different numbers of spokes q. (a) q = 3; (b) q = 4; (c) q = 5; (d) q = 6; (e) q = 7; (f) q = 8.

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4. Conclusion

Using the Fresnel diffraction method and the stationary phase principle, we derive two complex analytical formulas for accurately representing the propagation evolution from ESRP gratings. These two formulas correspond to a shape-invariant lattice and a propagation-invariant lattice, respectively. With the analytical formula derived from the stationary phase method, the formation of non-diffracting quasi-crystalline structures in the focal area of an axicon can be explicitly verified. Propagation-invariant optical lattices with a cylindrical structure can be generated. In addition to the theoretical analysis work, we employ the derived optical beam function to numerically simulate optical patterns. Furthermore, we set up an experiment to generate lattices with various wave patterns and observe the non-diffracting characteristic of the radial lattice based on the stationary phase method. The resulting diffraction patterns and experimental work verify each other. The practically generated radial lattice demonstrates that it favors the experimental study in comparison with the classical carpet beam. Finally, several interesting characteristics exhibited by the radial lattices are revealed through this study. Lattices with these features have the potential to contribute to optical tweezers and optical coherence tomography applications. Thus, the proposed practical generation procedure for radial lattices will serve as an important stepping stone for optically induced photonic structures.

Category: Physical Optics

Received: Feb. 20, 2025

Accepted: Apr. 14, 2025

Published Online: Aug. 1, 2025

The Author Email: Yile Shi (syl@zjnu.cn), Zhijun Ren (renzhijun@zjnu.cn)

DOI:10.3788/COL202523.082601

CSTR:32184.14.COL202523.082601