Since the generation of Bessel beams in 1987,^[1] diffraction-free beams, which can propagate over an extended distance with an invariant transverse profile, have attracted much attention because of their especially useful value in some application fields, such as laser surgery, optical coherence tomography, and optical trapping.^[2,3] Each diffraction-free beam is a fundamental solution of the wave equation. The most familiar example of diffraction-free beams is the plane wave, which is the solution of the wave equation in Cartesian coordinates. Bessel beams are the fundamental solutions of the wave equation in circular cylindrical coordinates. In elliptic cylindrical coordinates, Gutiérrez–Vega found another special solution of wave equation, which is the Mathieu function. Accordingly, diffraction-free Mathieu beams are also introduced and researched in theory.^[4,5]

The propagation-invariant nature of the transverse field profile of diffraction-free beams is a result of their cone-like angular spectrum: all plane wave components have equal longitudinal wavenumber, which can be described by the Dirac delta function δ(|k| − k_t). Here, we will extend the original experiment of Durnin to create an annular pupil aperture in the Fourier plane with arbitrary azimuthal amplitude components. We know that by using an annular pupil in focal plane of Fourier transform lens illuminated by a collimated laser beam, diffraction-free Bessel beams can be generated after Fourier-transform lens.^[1] Compared with Bessel beams generation, Mathieu beams generation can be especially difficult because the annular slit required in modulation profiles is the spectra distribution with angular Mathieu functions. To generate Mathieu beams using the experimental setup of Durnin, researchers had to introduce several approximate methods. For example, Gutiérrez–Vega make a one-dimensional (1D) Gaussian aperture and optimized its parameters to match the angular spectra of Mathieu functions to approximately generate zero-order Mathieu beams.^[6] Compared with classical two-dimensional (2D) radial Gaussian beams, 1D Gaussian beams in Ref. [6] are also difficult to obtain. In 2008, Anguiano-Morales generated a field distribution that is close to the diffraction-free Mathieu beam by using a combination of an axion and amplitude mask with two circular sectors of 70°.^[7] In Ref. [8], Florian even generated sectioned Bessel beams by blocking sections of a beam’s angular spectrum, which are a kind of quasi diffraction-free beams similar to optical distribution of Mathieu beams.

A spatial light modulator (SLM) is an important element to precisely modulate the phase or amplitude of the beam wavefront. To generate Mathieu beams, we extend the original experimental setup of Durnin^[1] to generate ring slit apertures with amplitude distribution of angular Mathieu function using amplitude-type SLM placed in the front focal plane of a Fourier-transform lens. Hence, we obtain a family of ideal Mathieu beams after lens by accurately generating angular Mathieu function. The optical characteristics of Mathieu beams are subsequently analyzed.

We begin our analysis on Mathieu beams generation with the scalar wave equation (∇² + k²)E(r) = 0, where r is the position vector and k is the wavenumber. We assume a solution of the form E(r) = U(x,y)Z(z), thus separating the wave equation into two equations, which is

Equation (6), derived from Eq. (1), represents a transverse Mathieu field distribution propagating in the z direction with a phase velocity v_p = w/k_z. Since the amplitude distribution is not related to z, equation (6) represents an ideal propagation-invariant beam that can be described by the known Mathieu functions. Similar with the plane wave solution of wave equation in Cartesian coordinate and the Bessel solution in circular cylindrical coordinates of wave equation, equation (6) shows another fundamental solution of wave equation in elliptic cylindrical coordinates.

On the basis of Eq. (6), we plot the transverse intensity distributions of the even-parity Mathieu beams of the first kind with parameters m = 0, 1, 2, 3 and q = 0, 15, 30, 45 (as show in Fig. 1). Each stimulating and recording photo (including that in Figs. 1, 4, and 5) shows an area size of 1 mm × 1 mm in this study. When m = 0 in Eq. (6), their fundamental solutions are the angular Mathieu function ce₀(ν,q) and radial Mathieu function Je₀(u,q). When constant coefficient is ignored, equation (6) can be referred to as the zero-order Mathieu beams

Fig. 1.

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	Fig. 1. (color online) Theoretical simulation of even mode Mathieu beams of the first kind in elliptic coordinate. The first to fourth rows represent Mathieu beams for m = 0, 1, 2, 3. The first to fourth columns represent the beams for q = 0, 15, 30, 45. The red regions represent the crests (maxima intensity of beams), whereas the blue regions represent the valleys (minima intensity of beams).

Fig. 2.

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	Fig. 2. (color online) The extended experimental setup of Durnin for generating Mathieu beams.

Fig. 3.

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	Fig. 3. Narrow annular pupils with angular spectrum that generates Mathieu beams, their parameters being same with those in Fig. 1.

Fig. 4.

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	Fig. 4. (color online) Experimentally recording cross-sections of Mathieu beams for the same parameters with those in Fig. 1.

Fig. 5.

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	Fig. 5. (color online) The transverse intensity distribution of Mathieu beams for the parameters (m = 0, q = 15) at the distance after Fourier transform lens: (a) z = 35 mm, (b) z = 39 mm, (c) z = 43 mm, (d) z = 47 mm.

As shown in Figs. 1(a)–1(d), the zero-order Mathieu beams’ distribution is characterized by the ellipticity parameter q. With increasing q, the elliptical main lobes of zero-order Mathieu beams become increasingly flat. Radial unsymmetrical zero-order Mathieu beams transform into radial symmetry zero-order Bessel beams when q = 0, as shown in Figs. 1(a)–1(d). The central main lobes of the zero-order Bessel beams are circular spots. Similar to the Airy beams being a special type of the Weber beams,^[12] the circularly symmetric zero-order Bessel beams are a special case of zero-order Mathieu beams when q = 0.

As shown in Figs. 1(e)–1(p), the transverse intensity pattern distribution of high-order Mathieu beams is fairly complicated. The morphology of high-order Mathieu beams is dependent on the even or odd modes, order number m, and the parameter q of the Mathieu function. Considering the even-parity Mathieu beams for q = 0. Figures 1(a), 1(e), 1(i), and 1(m) show that an m-order Mathieu beam has m angular nodal lines over the beam center (m angular valleys over the center of beam); thus 2m crests are generated. All Mathieu beams for q = 0, including zero-order and high-order Mathieu beams, are axis-symmetry and centrosymmetry beams. Moreover, q is an important factor to decide the optical distribution of high-order Mathieu beams. When q = 0, the radical nodal lines of Mathieu beams are circular, and the crests of high-order Mathieu beam arrays are on a circle. All Mathieu beams for q ≠ 0 are axis-symmetric beams and not centrosymmetric beams because the radical nodal lines of high-order Mathieu beams are characterized by a set of confocal elliptical and hyperbolic lines and the crests of high-order Mathieu beams array on elliptical and hyperbolic lines. Mathieu functions are the fundamental solutions of wave equation in cylindrical coordinates. With increasing q, the eccentricity of the elliptical and hyperbolic nodal lines of Mathieu beams becomes large. Thus, q is an important parameter that controls the ellipticity of radical nodal lines of Mathieu beams.

In short, Mathieu beams are important diffraction-free beams that have more flexibly and better controllable beam morphology than diffraction-free Bessel and Airy beams, and also for form-invariant Gaussian and Pearcey beams. Hence, precisely generating each types of Mathieu beams is especially valuable and will have wide uses in scientific researches. For example, the optical lattices are theoretically proposed to generate by applying diffraction-free Mathieu beams.^[13,14]

The history of generating diffraction-free beams started with the experimental setup of Durnin,^[1] which demonstrated that the principal properties are derived from Whittaker’s solution of Helmholtz equation. The function of exp(·) is a fundamental solution of Helmholtz equation (∂_xx + ∂_yy + k²)U = 0 and describes the transverse field of an ideal plane wave. Any linear superposition of a finite or infinite number of plane waves with the equal wave vector k is a solution of Helmholtz wave equation, which may generate propagation-invariant patterns (i.e., diffraction-free beams).^[15] Fundamental diffraction-free optical fields of Helmholtz equation in free space can be written as a superposition of plane waves with a single transverse wave number k_t as follows^[4,16]

Equation (8) shows that the several known fundamental solutions of the wave equation that can describe as diffraction-free beams are fully determined by their angular spectrum A(ν).^[18] The function A(ν) describes the field distribution on the circular ring. From the equation θ₀ = tan⁻¹(R/f_L), the spectrum parameter is the single radial frequency that is related to both the radius of that circle ring and the focal length of Fourier transform lens. In our case, the focal length f_L of the Fourier transform lens is 40 mm. The diffraction-free beams can be understudied as an interference field generated by the coherent superposition of ideal plane waves for a single radial spatial frequency whose relative phase differences remain unchanged upon propagation in the free space.^[19] In 1987, diffraction-free Bessel beams were generated by using the setup of annular slit and adding a Fourier transform lens for the case of A(ν) = const. to obtain the constant angular spectrum experimentally. Similarly, for the case of A(ν) = ce_m(ν,q) or se_m(ν,q), the integral resulting field amplitude is proportional to Je_m(u;q)ce_m(v;q) or Jo_m(u,q)se_m(ν,q).^[4,20] This result implies that the diffraction-free even and odd-parity Mathieu beams can be generated when the amplitudes in the circular ring with radius R = k_t · f_L are modulated by the angular spectrum ce_m(ν,q)δ(|k| − k_t) or se_m (ν,q)δ (|k| − k_t). This distribution of the angular Mathieu spectrum may be difficult to obtain. Therefore, some researchers only approximately generate zero-order Mathieu beams by approximately modulating the annular slit by the angular Mathieu functions.^[6,7] In this study, we will precisely modulate the annular distribution of slit by accurately calculating the angular Mathieu functions. We typically calculate any angular transmission distribution of the even type angular Mathieu function of the first kind (i.e., ce_m (ν,q)) according to the following equation

In Durnin’s extended experimental setup (see Fig. 2), the angular spectrum is an azimuthally modulated narrow annular pupil with radius R = 3 mm and thickness ΔR = 80 μm, as shown in Fig. 3. Mathieu beams can be generated by the complex Fourier-transform of the aperture function. We load the amplitude distribution with angular Mathieu functions in an amplitude-type SLM (GC-SLM-T-XGA, with a 22 μm × 22 μm pitch for each pixel) to generate the transmittance function of a narrow annular pupil ce_m (ν,q)δ [(x² + y²)) − R]. In the focal plane of the Fourier transform lens, the transverse intensity distributions of several Mathieu beams are recorded using scientific CCD (Microvision 130FC, with a 5.4 μm × 5.4 μm pitch for each pixel), see Fig. 4.

As shown in Fig. 4, we can find that the experimental results agree well with the corresponding numerical results in Fig. 1 if we can eliminate the measurement error induced by the limited resolution of CCD. Moreover, figure 4(a) shows that Bessel beams are the special case of Matheiu beams, which are generated by using a narrow annular pupil modulated by A(ν) = ce₀(ν,0) = 1 as shown in Fig. 3(a). Apparently, it is especially simple case for uniform amplitude over the circular ring. Bessel beams can be regarded as zero-order Mathieu beams for q = 0.

To show the propagation character of Mathieu beams, we typically record the beams with the parameters m = 0 and q = 15 using CCD after the Fourier transform lens. Four output intensity distributions on different transverse planes measured by the CCD are given in Fig. 5. These results clearly demonstrate that the fabricated Mathieu beams can satisfactorily produce the desired diffraction-free beams within a certain range of propagation.

Using the extended Durnin’s experimental setup, we have generated a family of Mathieu beams by spectra amplitude modulation of the incident field using an amplitude-type SLM. As a special case of the Mathieu beams, the Bessel beam is also generated with the same experimental setup. The optical morphology and propagation characteristics of Mathieu beams are also studied. As important diffraction-free beams, Mathieu beams have more flexibility and controllable beam morphology than other known diffraction-free beams. The precise generation of various types of Mathieu beams is expected to lay the foundation for their various applications in both science and technology.