Non-diffracting integer-order and half-integer-order carpet beams obtained by even-type sinusoidal amplitude radial gratings

Yefeng Liu; Huiqing Li; Rijian Chen; Changjiang Fan; Yile Shi; Zhijun Ren

doi:10.3788/COL202422.072601

1. Introduction

As one of the most important optical elements, gratings with periodic structures have been widely used in many application fields, such as spectral analysis, laser tuning, optical displays, optical sensing, and optical storage. In nature, a grating is an optical diffractive element. When beams pass through different diffraction gratings, remarkable and intriguing diffractive behavior can be observed. Talbot self-imaging effects can be regarded as diffraction phenomena at the near-field of a periodic grating^[1-3]. Referring to the principle of the Talbot effect, Rasouli et al. studied the near-field and far-field propagation behavior of radial periodic gratings with two-dimensional periodic structures and constructed and generated several kinds of carpet beams^[1,2]. Subsequently, various carpet beams with special intensity distributions, even carpet beams with tunable two-dimensional optical lattice structures, have been generated by modulating beams with different phase and amplitude gratings^[4-9]. Since the inception of carpet beams, they have inspired numerous applications and intriguing concepts owing to their intricate optical structures and distinctive characteristics. For example, carpet beams have been used in optical tweezers for multiple optical trapping (i.e., multiple trapping of particles over an annular array)^[9,10], optical communication^[11], indoor optical communication systems^[7], optical communication in an underwater medium^[12], and production of multiple filament plasma channels^[4]. Indeed, carpet beam generation and application have generated considerable experimental and theoretical interest.

Despite that classical carpet beams with different optical structures have been used in some research fields, the above-mentioned carpet beams are not non-diffracting beams. The term “non-diffracting beam” was first introduced to designate zero-order Bessel beams^[13]. In the past few decades, rapid progress in the generation and use of various types of non-diffracting beams has been achieved. In some fields, non-diffracting beams play essential roles, even fundamentally changing the working paradigms of some scientific instruments. For example, non-diffracting Bessel beams have been used in optical tweezers^[14], super–resolution imaging^[15], light sheet microscopy^[16], and space-division-multiplexing optical communication systems^[17]. Non-diffracting airy beams are used in the micrometer-sized “snowblower,”^[18] optical coherence tomography^[19], and curved plasma channel production^[20]. Non-diffracting Lommel beams are used in optical wireless communication^[21], and non–diffracting parabolic beams are used as optical tweezers for manipulating the dielectric particles^[22]. Many non–diffracting array beams are used in the generation of optical lattices^[23-26], ultracold atom trapping^[27], and three–dimensional microstructure design^[28].

It is readily known that special beams with non-diffracting propagation characteristics can play a more unique role in optical application field compared with other ordinary beams. However, classical carpet beams are not entirely non-diffracting beams. The optical field distribution of classical carpet beams slowly expands during propagation, although the optical structures of classical carpet beams are not altered during propagation. In 2022, Gong et al. explored non-diffracting petal-like beams, which is one type of carpet beam^[29,30]. The transition of carpet beams from structure-invariant beams (not non-diffracting beams) to non-diffracting beams (of course, being structure-invariant beams) is an especially critical progress development^[12].

Given that gratings with different structures can generate varying diffraction patterns, we designed a novel type of grating named the even-type amplitude sinusoidal amplitude radial (ETASR) grating, which is a kind of amplitude radial grating with an absolute sinusoidal profile. Using the designed ETASR gratings, we theoretically constructed new carpet beams with non-diffracting characteristics based on the principle of stationary phase. To substantiate our findings, we generated non-diffracting carpet beams. This research offers a novel type of carpet beam, especially for the non-diffracting carpet beam family, while providing theoretical insights into its potential applications to some research fields.

2. Theory

In this study, we focused on the creation and exploration of novel and special carpet beams by designing radial gratings. The diffraction theory of designed grating can be researched from Fresnel–Kirchhoff’s diffraction integral in cylindrical coordinates: $U_{0} (ρ, θ, z) = \frac{(- i)}{λ z} \exp (i k z) \times \int_{0}^{\infty} \int_{0}^{2 π} U_{0} (r, φ) \times \exp [\frac{i k}{2 z} (r^{2} + ρ^{2})] \times \exp [- \frac{i k}{z} ρ r \cos (φ - θ)] r d r d φ,$ (1)where $λ$ is the wavelength of the beam and $k = 2 π / λ$ is the wavenumber. $U_{0} (r, φ)$ is the complex amplitude distribution of the initial light field ( $z = 0$ ) modulated by the mask, $r$ and $φ$ are the radial and azimuth coordinates of the initial field source, respectively, $U_{0} (ρ, θ, z)$ is the complex amplitude distribution at the transmission distance $z$ from the mask, and $ρ$ and $θ$ are the radial and azimuth coordinates of the observed plane, respectively.

To construct non-diffracting carpet beams with the newly designed gratings, we used the principle of stationary phase. Hence, the axicon phase was necessary in our scheme. The collimated parallel light enters the amplitude mask and axicon successively, and hence the initial light field can be written as $U_{0} (r, φ) = A (φ) \exp [- i k T (r)],$ (2)where $T (r)$ is the phase transform function of the axicon. The radial phase distribution of the axicon is^[31] $T (r) = {\begin{cases} (n_{0} - 1) θ_{0} r, & r \leq R, \\ 0, & r > R, \end{cases}$ (3)where $n_{0}$ is the refractive index of the axicon, $θ_{0}$ is the base angle of the axion, which is usually a small angle, and $R$ is an aperture radius of the entrance pupil of the axicon. In Eq. (2), $A (φ)$ is the amplitude transmittance function of the grating. In our scheme, to construct new carpet beams, we introduced ETASR gratings. Its transformation function $A (φ)$ can be written as $A (φ) = | \sin (q φ) |,$ (4)where $q$ determines the number of gratings’ spokes.

Based on mathematical expansion, Eq. (4) can be rewritten as $| \sin (q φ) | = \frac{2}{π} - \frac{4}{π} \sum_{n = - \infty}^{+ \infty} \frac{\cos (2 n q φ)}{(2 n - 1) (2 n + 1)} .$ (5)

Using the Euler formula $\cos (x) = \frac{e^{i x} + e^{- i x}}{2}$ , Eq. (5) can be transformed into the following form: $| \sin (q φ) | = \frac{2}{π} - \frac{2}{π} \sum_{n = - \infty}^{+ \infty} \frac{\exp (- 2 i n q φ) + \exp (2 i n q φ)}{(2 n - 1) (2 n + 1)} .$ (6)

By substituting Eqs. (6) and (4) into Eq. (2) and then substituting the results into Eq. (1), the following expression is obtained: $U_{0} (ρ, θ, z) = - \frac{i}{λ z} \exp (i k z) \frac{2}{π} \int_{0}^{\infty} \int_{0}^{2 π} \sum_{n = - \infty}^{+ \infty} [1 - \frac{\exp (- 2 i n q φ) + \exp (2 i n q φ)}{(2 n - 1) (2 n + 1)}] \times \exp [\frac{i k}{2 z} (r^{2} + ρ^{2})] \exp [- i k (n_{0} - 1) θ_{0} r] \times \exp [- \frac{i k}{z} ρ r \cos (φ - θ)] r d r d φ .$ (7)

For the convenience of mathematical writing, unnecessary terms before integral symbol, which do not affect the distribution of light intensity, were omitted. Moreover, complex amplitude distribution $U_{0}$ given in Eq. (7) can be split into three separate equations: $U_{1}$ , $U_{2}$ , and $U_{3}$ : $U_{0} (ρ, θ, z) = U_{1} (ρ, θ, z) + U_{2} (ρ, θ, z) + U_{3} (ρ, θ, z),$ (8)where $U_{1} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{2 π} \sum_{n = - \infty}^{+ \infty} \frac{\exp (- 2 i n q φ)}{(2 n - 1) (2 n + 1)} \exp [\frac{i k}{2 z} (r^{2} + ρ^{2})] \times \exp [- i k (n_{0} - 1) θ_{0} r] \times \exp [- \frac{i k}{z} ρ r \cos (φ - θ)] r d r d φ,$ (9.1) $U_{2} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{2 π} \sum_{n = - \infty}^{+ \infty} \frac{\exp (2 i n q φ)}{(2 n - 1) (2 n + 1)} \exp [\frac{i k}{2 z} (r^{2} + ρ^{2})] \times \exp [- i k (n_{0} - 1) θ_{0} r] \times \exp [- \frac{i k}{z} ρ r \cos (φ - θ)] r d r d φ,$ (9.2) $U_{3} (ρ, θ, z) = \int_{0}^{\infty} \int_{0}^{2 π} \exp [\frac{i k}{2 z} (r^{2} + ρ^{2})] \times \exp [- i k (n_{0} - 1) θ_{0} r] \times \exp [- \frac{i k}{z} ρ r \cos (φ - θ)] r d r d φ .$ (9.3)

Here, we mainly derive Eq. (9.1). Given that the analytic solution of Eq. (9.1) can be extended to Eq. (9.2) given that the two equations have similar mathematical forms, we derive Eq. (9.3) using the Bessel function.

To derive Eq. (9.1), we use the following Jacobi–Anger expansion^[32]: $\exp (i z \cos θ) = \sum_{m = - \infty}^{+ \infty} {(i)}^{m} J_{m} (z) \exp (i m θ),$ (10)where $J_{m}$ is the $m$ th order Bessel function of the first kind. From Eq. (10), the following expression is obtained: $\int_{0}^{2 π} \exp [- \frac{i k}{z} ρ r \cos (φ - θ)] d φ = \sum_{m = - \infty}^{+ \infty} {(- i)}^{m} J_{m} (\frac{k}{z} ρ r) \int_{0}^{2 π} \exp [- i m (φ - θ)] d φ .$ (11)

By substituting Eq. (11) into Eq. (9.1), there is $U_{1} (ρ, θ, z) = \sum_{n = - \infty}^{+ \infty} \sum_{m = - \infty}^{+ \infty} \frac{2 i}{λ z π} \exp (i k z) \times \exp (\frac{i k}{2 z} ρ^{2}) \int_{0}^{\infty} {(- i)}^{m} J_{m} (\frac{k}{z} ρ r) \times \exp {i k [\frac{r^{2}}{2 z} - (n_{0} - 1) θ_{0} r]} r d r \times \exp (i m θ) \int_{0}^{2 π} \exp [i (2 n q - m) φ] d φ .$ (12)

To solve Eq. (12), the following equation is used: $\int_{0}^{2 π} \exp [i (m - q) φ] d φ = 2 π δ_{m, q},$ (13)where $δ_{m, q}$ is the Kroneck function. In the Kroneck function, the function has a value of $2 π$ when $m = q$ , and the function has a value of 0 when $m \neq q$ . Thus, Eq. (12) is written as $U_{1} (ρ, θ, z) = \sum_{n = - \infty}^{+ \infty} \frac{2 i k}{z π} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \exp (i 2 n q θ) \times \int_{0}^{\infty} {(- i)}^{2 n q} J_{2 n q} (\frac{k}{z} ρ r) \times \exp {i k [\frac{r^{2}}{2 z} - (n_{0} - 1) θ_{0} r]} r d r .$ (14)

In mathematics, Eq. (14) is a complex integral expression. Hence, obtaining its analytic solution by directly solving the integral equation is difficult. In near-field diffraction, the principle of stationary phase approximation is an important method for simplifying the oscillatory integral with the from $\int g (r) \exp [i k f (r)] d r$ when $k \to \infty$ ^[33]. To solve the complex Fresnel integral equation using the stationary phase method, one can set $f (r) = r^{2} / (2 z) - (n_{0} - 1) θ_{0} r$ and $g (r) = J_{2 n q} (k ρ r / z) r$ for Eq. (14). By taking the derivative $f (r)$ , the stationary phase point can be obtained when $r_{0} = (n_{0} - 1) θ_{0} z$ . Given the size of the entrance pupil of axicon, $r = r_{0} \in (0, R)$ , we obtain the maximum non-diffraction distance $z_{\max} = R / [(n_{0} - 1) θ_{0}]$ of non-diffracting carpet beams. Hence, the analysis result of Eq. (14) at $0 < z < z_{\max}$ is $U_{1} (ρ, θ, z) = \frac{2 i k}{π} \sqrt{λ z} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times \exp {i {- \frac{k {[(n_{0} - 1) θ_{0}]}^{2} z}{2} + 0.25 π}} \times \sum_{n = - \infty}^{+ \infty} {(- i)}^{2 n q} \exp (2 i n q θ) J_{2 n q} (k_{r} ρ) .$ (15)

Using the similar method, the analytical result of Eqs. (9.2) and (9.3) can be obtained: $U_{2} (ρ, θ, z) = \frac{2 i k}{π} \sqrt{λ z} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times \exp {i {- \frac{k {[(n_{0} - 1) θ_{0}]}^{2} z}{2} + 0.25 π}} \times \sum_{n = - \infty}^{+ \infty} {(- i)}^{- 2 n q} \exp (- 2 i n q θ) J_{- 2 n q} (k_{r} ρ),$ (16) $U_{3} (ρ, θ, z) = \frac{2 i k}{π} \sqrt{λ z} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times J_{0} (k_{r} ρ) \exp {i {- \frac{k {[(n_{0} - 1) θ_{0}]}^{2} z}{2} + 0.25 π}} .$ (17)

By bringing Eqs. (15 )–(17) into Eq. (8), we obtain the final expression of a kind of new carpet beam using the amplitude modulation element of ETASR gratings: $U_{0} (ρ, θ, z) = \frac{2 i k}{π} \sqrt{λ z} \exp (i k z) \exp (\frac{i k}{2 z} ρ^{2}) \times \exp {i {- \frac{k {[(n_{0} - 1) θ_{0}]}^{2} z}{2} + 0.25 π}} \times \sum_{n = - \infty}^{+ \infty} [{(- i)}^{2 n q} \exp (- 2 i n q θ) J_{- 2 n q} (k_{r} ρ) + {(- i)}^{2 n q} \exp (2 i n q θ) J_{2 n q} (k_{r} ρ) + J_{0} (k_{r} ρ)] .$ (18)

Equation (18) is the main analytical result that describes the diffracted amplitude of the ETSAR gratings. One can obtain the main physical insight from this above mathematical expression. From the sum expression, we know that all Bessel functions have the same transverse wavenumber. Moreover, the radial wavenumber of each order Bessel function is independent of the propagation distance $z$ . It is readily known that the superposition of non-diffracting beams with the same transverse wavenumber is still a non-diffracting beam. The intensity distribution of the kind of newly constructed carpet beams using ETASR gratings is $I (ρ, θ, z) = U_{0} (ρ, θ, z) U_{0}^{*} (ρ, θ, z),$ (19)where $*$ denotes a complex conjugate. From the sum expression given in Eq. (18), the main feature of these diffraction patterns is that the radial wavenumber of all Bessel functions is independent of the propagating distance $z$ . In addition, all Bessel functions have the same transverse wavenumber, indicating that these kinds of carpet beams are novel non-diffracting beams. In addition, Eq. (18) shows that $U_{0} (ρ, θ, z)$ can be calculated when $2 n q$ is an integer, that is, $2 q$ must be an integer. When $2 q$ is an even number, $q$ is an integer number. When $2 q$ is an odd number, $q$ is a half-integer. Hence, these carpet beams can be divided into two categories according to whether $q$ is an integer or a half-integer: integer-order and half-integer-order carpet beams.

3. Simulations and Experiments

To validate the correctness of the mathematical derivation, we numerically simulated and experimentally investigated the carpet beams. We selected multiple samples to showcase this type of carpet, starting from different ETASR gratings. Referring to Eq. (4), we typically drew ETASR gratings with different $q$ values (Fig. 1). In Fig. 1, $2 q$ determines the number of spokes of a radial grating. According to the number of grating spokes, they were named radial even or odd-type gratings.

Figure 1.ETASR gratings with different q values. (a) q = 6; (b) q = 8; (c) q = 10; (d) q = 6.5; (e) q = 8.5; (f) q = 10.5.

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From Eq. (19), the corresponding intensity and phase distribution of carpet beams were simulated with the same parameters as Fig. 1, as shown in Fig. 2. The carpet beams given in Fig. 2 can be generated using ETASR gratings given in Fig. 1. Apparently, each ETASR grating is accompanied by its own characteristic diffraction pattern. The main characteristic of all diffraction carpet patterns was that they are shape-invariant during propagation. The structure of a carpet beam strongly depends on the number of grating spokes ( $2 q$ ). The main characteristic of the integer-order and half-integer-order carpet beams is that the central part of the beam is a main lobe, in which the intensity becomes predominant. The main lobe is axial caustic owing to the interference of the intersecting portions of the optical wave. Moreover, the horizontal and vertical symmetry of both types of carpet beams is observed.

Figure 2.Simulation graphs of carpet beams. First row: the intensity patterns. Second row: the phase patterns. The parameters correspond to Fig. 1.

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The beam structures of the integer-order or half-integer-order carpet beams were explored in detail. For integer-order carpet beams, the transverse plane of the diffraction patterns of the gratings can be divided into three distinct areas. The central area is a concentric ring surrounding the main lobe. The area of the concentric ring increases with parameter $q$ . On the periphery of the concentric ring (middle area of beams), a radial divergent toroidal lattice structure forms. In the outward area, another divergent toroidal lattice is present. The radial row of lattice in the outward area is situated at the intermediate area between two adjacent radial lattices of the middle area. As for half-integer order carpet beams, the optical structure of the central area is similar to the central area of the integer-order carpet beams, but they differ in toroidal lattice structure. On the periphery of a concentric ring, half-integer-order carpet beams only exist in an area with a radial divergent ring lattice structure. In the central areas of both types of carpet beams, the phase of each concentric ring along the angular direction is invariant constant.

The anomalous axial phase behavior of optical beams has been drawing attention since Gouy’s discovery in 1890^[34] and is often referred to as the Gouy phase or phase anomaly. This peculiar phase behavior is worth studying because it plays an essential role in some physical problems and application areas^[8,35]. A similar phase anomalous behavior can be observed in integer-order and half-integer-order carpet beams (Fig. 3). Except the central area, the general trend of the phase change along the $r$ -axis and $ρ$ -axis is the same for integer-order and half-integer-order carpet beams, and the phase jumps are close to $2 π$ , which are also integer multiples of $π / 2$ ^[8,35]. The existence of abrupt phase changes provides an amazing behavior for the corresponding intensity distribution of the light field during propagation.

Figure 3.Phase anomalies of the carpet beams. The phase profiles along the x axis with (a) q = 8; (b) q = 8.5.

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Moreover, a detailed study of the phase anomaly of the light field in the radial direction has been conducted, discussing the anomaly in integer-order and half-integer-order carpet beams. For integer-order carpet beams, all phases in the area in the concentric ring and divergent toroidal lattice exhibit anomalous behavior, as shown in Fig. 3(a). In the central and peripheral areas, the phase of the concentric ring periodically changes, and the value of this phase change is close to $π$ . However, a phase jump at the junction of the two adjacent areas is close to $2 π$ . Similar to the phase of the integer order carpet beams, the phase of the concentric ring in the radial direction in the half-integer-order carpet beams’ central areas periodically changes, as shown in Fig. 3(b). In the whole peripheral area, the phase change of the half-integer-order carpet beams is greater than that of the integer-order carpet beams in the radial direction. Moreover, the phase shift of the half-integer-order carpet beams is clearer than that of the integer-order carpet beams, which may originate from the asymmetry of the half-integer-order carpet beams. These phase anomalies may guarantee an invariant intensity distribution during propagation. The phase is still symmetrical for two types of carpet beams.

We also investigate the propagation characteristics of the two types of carpet beams. Without loss of generality, we arbitrarily selected a cross-section diagram at the position of a beam. The theoretically simulated cross-sectional propagation graph is shown in Fig. 4. Each lattice of the carpet beams is non-diffracting during propagation. However, in the non-diffractive propagating distance, the optical intensity slightly increases for each lattice.

Figure 4.x–z cross-section through the intensity volume carpet beam propagation. The location is taken from the white line in Fig. 2. (a) q = 8; (b) q = 8.5.

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The experimental setup of generating carpet beams Is illustrated in Fig. 5. After He–Ne laser expansion and collimation (the wavelength is 632.8 nm, and the radius of the expanded beam is approximately 4 cm), the plane wave illuminated the ETASR grating, which was tightly attached to the axicon. The base angle of the axicon is 1°, the entrance pupil radius of the axicon is 1.25 cm, and the refractive index is 1.457. The centers of the two elements were strictly aligned. The masks of the ETASR gratings were imposed on an amplitude-type spatial light modulator (SLM). These masks were loaded on the SLM one by one, and hence different carpet beams were generated. To improve the quality of the generated carpet beams, we added two polarizers before the grating and after the axicon. An optimal carpet beam was obtained after the polarizer was rotated. After the axicon, the modulated beam was Fresnel diffracted. Thus, the obtained beams were captured using a high-resolution CCD camera (Fig. 6). According to the parameters of the axicon, the maximum non-diffraction distance of carpet beams was 1083 mm. According to Eq. (18), one can know that the shape of the beam is independent of the wavelength. It means that lasers in other wavelengths that can be modulated by the SLM can also be used to generate the kind of non-diffracting carpet beams.

Figure 5.Experimental system of generating carpet beams.

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Figure 6.(a)–(f) Experimentally recorded graphs of the integer-order and half-integer-order carpet beams with different spokes and the same parameters corresponding to Fig. 2.

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The carpet beams were recorded at 80 cm after the axicon (Fig. 6). The recorded experimental diffraction patterns of the ETASR grating with different parameters were in agreement with the theoretical simulated carpet beams. The introduced ETASR gratings were used to generate the novel carpet beams.

To verify the non-diffracting transmission characteristics of the generated carpet beams in the range of $z < z_{\max}$ , we recorded the intensity distribution of 8-order carpet beams in the different propagation distances after the axicon (Fig. 7). The shapes of the optical structure of the carpet beams were almost preserved during propagation, although the light intensity slightly increased. Hence, the generated caustic beams on the basis of the stationary phase principle can be regarded as non-diffracting beams.

Figure 7.Experimentally recorded graphs of 8-order carpet beams at different propagation distances after axicon. (a) z = 70 cm; (b) z = 80 cm; (c) z = 90 cm.

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4. Conclusion

We constructed novel non-diffracting carpet beams by introducing ETASR gratings. As a proof of concept, a theoretical derivation of this kind of carpet beam was first presented according to the principle of stationary phase. Then, using an axicon and the ETASR gratings, non-diffracting integer-order and half-integer-order carpet beams were generated. The distinctive optical characteristics and phase anomalies of integer-order and half-integer-order carpet beams were investigated for different cases. The generated non-diffracting carpet beams with the propagation-invariant divergent ring lattice structures and an extended depth of multiple focuses may open numerous potential applications for optical manipulation, optical processing, lattice light-sheet microscopy, and even other related applications.

Category: Physical Optics

Received: Jan. 29, 2024

Accepted: Mar. 12, 2024

Published Online: Jul. 17, 2024

The Author Email: Zhijun Ren (renzhijun@zjnu.cn)

DOI:10.3788/COL202422.072601

CSTR:32184.14.COL202422.072601