Spiral phase plates for the generation of Laguerre–Gaussian (LG) beam with non-null radial index were designed and fabricated by electron beam lithography on polymethylmethacrylate over glass substrates. The optical response of these phase optical elements was theoretically considered and experimentally measured, and the purity of the experimental beams was investigated in terms of LG modes contributions. The far-field intensity pattern was compared with theoretical models and numerical simulations, whereas interferometric analyses confirmed the expected phase features of the generated beams. The high quality of the output beams confirms the applicability of these phase plates for the generation of high-order LG beams.
© 2014 Optical Society of America
In the last decades a considerable interest has been shown in the generation of coherent beams carrying phase singularities [1,2]. In particular Laguerre–Gaussian (LG) beams have gained increasing attention, especially after Allen et al.  demonstrated that these modes carry orbital angular momentum (OAM) of light. LG modes form a complete set of orthonormal solutions of the paraxial wave equation whose complex amplitude, in cylindrical coordinates, is given by 5]. The azimuthal index , corresponding to the topological charge of the embedded singularity, indicates the number of twists of the helical wavefront within a wavelength and represents the amount of OAM, in units of , carried by each single photon. The index represents the number of radial nodes on a plane perpendicular to the direction of propagation and is thus related to the distribution of the intensity pattern in concentric rings around the central dark zone of the phase singularity (when ).
The first method for LG beam generation was demonstrated by Beijersbergen et al.  with the conversion of Hermite–Gaussian beams by means of a couple of cylindrical lenses. Intensive interest in LG beams has required the development of simple and reliable methods for their generation from conventional laser beams, e.g., Gaussian beams. One of the most widespread approaches is based on the use of the so-called fork-holograms, holographic structures encoding the interference pattern of the considered LG beam with the generating reference wave, e.g., a plane wave or a Gaussian beam. The interference pattern presents the structure of a grating with one or several central dislocations corresponding to the amount of topological charge [7,8]. Other groups exploit refractive optical elements, the so-called spiral phase plates (SPP), for the shaping of a Gaussian beam into an OAM-carrying optical vortex beam. These SPPs are helicoidal transmission optical elements that look like a spiral staircase with a certain number of steps that progressively build the total phase gap in air ,
Since SPPs are transmission phase elements, they do not alter the direction of propagation and moreover show a higher efficiency; therefore they are much more preferred especially in applications such as optical coronagraphy , spiral-phase contrast microscopy , and stimulated emission depletion microscopy  to improve both contrast and resolution of images.
So far, however, azimuthally higher and radially lowest-order LG beams have been extensively studied, while radially higher-order LG beams have not attracted much attention. Specifically, a few groups described the generation of high-order LG beams with the use of spatial light modulators  or fabricated fork-holograms [14,15]. Multi-ringed LG beams can be holographically generated with phase patterns involving radial phase discontinuities; however, this is technically difficult, especially when the azimuthal and radial mode indices become larger. In this case, the holographic phase patterns involve more phase discontinuities to cause nonideal diffraction of light.
In this Letter, we extend the use of SPPs for the generation of high-order LG beams with nonzero radial index and we present the work of design, fabrication, and optical characterization of such phase optical elements.
To design SPPs for the control of the radial index , we started considering some basic properties of LG beams and the relations between far-field and near-field of a given phase element.
In the paraxial and Fraunhofer approximations, the far-field of an illuminated optical element is described as the Fourier transform (FT) of its transmission function multiplied by the impinging illumination .
It is worth noting that an LG beam with radial and azimuthal indices and , described by the complex function , is eigenfunction of the FT with eigenvalue .
Therefore, since the inverse FT of an LG beam of radial index and topological charge , is an LG beam again with the same indices, the transmission function of the generating optical element should be exactly the complex function of the generated beam in the far-field. On the other hand, since we are looking for a phase-type element, by considering only the phase term of the complex function , we get the following expression for the transmission function of such a SSP:
In the case of , we get the usual form . For instead, it presents a number of radial phase discontinuities of , in correspondence to the change of sign of the associated Laguerre polynomial.
By illuminating the SPP with a Gaussian beam , the far-field is given by the FT of the product between the incident beam and the transmission function :18] we get 19]:
Electron beam lithography (EBL) represents a powerful tool to generate this kind of structures, because of the possibility to realize continuous surface profiles, high flexibility in the element’s design, and good optical quality of the fabricated reliefs [20,21]. 3D profiles can be generated modulating the local dose distribution, inducing different dissolution rates in the polymer exposed, giving rise to different resist thicknesses at the end of the development process. A dose correction to compensate for proximity effect is required to obtain a good shape definition, especially at the phase steps and at the central anomaly of the spiral phase mask. A JEOL JBX-6300FS EBL machine was used to fabricate the SPPs . The 3D-spiral phase mask profiles were written on a 2.0 μm thick poly (methyl methacrylate) (PMMA) resist layer. As a substrate, we chose a 1.1 mm thick ITO coated soda lime float glass slide. For the considered wavelength of the laser beam (), PMMA refractive index results from spectroscopic ellipsometry analysis; therefore according to Eq. (2) the total height of the spiral as a function of the topological charge is .
Electron-beam exposures were carried out at 100 keV, in high resolution mode, with a beam probe current of 100 pA and a beam diameter of 2 nm. Single- and multi-step SPPs have been realized, with different and , with 256 phase steps in a period (Fig. 1). Careful inspection of exposure parameters, along with accurate proximity effect correction and optimization of developing conditions, has allowed obtaining SPPs with superior optical quality.
The fabricated samples were characterized with an optical setup assembled with commercial components mounted on an optical table. The Gaussian beam (, beam waist , power 0.8 mW) emitted by a HeNe laser source (HNR008R, Thorlabs) was resized by a lens of focal length placed between the laser and the SPP at such a distance that the beam illuminates the SPP with the prescribed waist size and flat wavefront. The far-field intensity distribution was recorded with a CCD camera (AxioCam MRc 5, Zeiss, pixels, -bit RGB color depth) placed at the back-focal plane of a second lens of focal length .
Figure 2 exhibits some results of the characterization process for the SPP sample of radial index and topological charge . For the given phase pattern of the optical element [Fig. 2(a)], the far-field was obtained by implementing in MATLAB environment a custom code exploiting commercial algorithm for fast Fourier transform (FFT) calculation. The result is reported in Fig. 2(b) in a false-color scale where the brightness indicates the field amplitude and the color exhibits its phase. The intensity pattern measured on the camera [Fig. 2(d)] well reproduces the simulated intensity distribution; however, a more significant comparison is given by Fig. 2(c), where an experimental cross section on a plane perpendicular to the beam axis and crossing the central singularity reproduces the numerical trend well. While the far-field trend perfectly reproduces the theoretical LG mode close to the central singularity, this overlap is less accurate in correspondence to the outer ring. A stronger overlap could be achieved by illuminating the SSP with a top-hat input beam .
We estimated the purity of the generated modes by using an amplitude overlap integral along a selected radial direction on the plane perpendicular to the beam axis in the far-field, calculated as the scalar product between the measured and the normalized theoretical distributions (Eq. 1):
Since the topological charge is conserved, for a given we limited the decomposition into the set of functions of different and constant . In Fig. 3 the efficiency values are reported for SPPs of and radial index in the range for a decomposition into LG functions with the same topological charge and radial index up to 9. The plot exhibits a high generation efficiency of the dominant mode for the fabricated SPPs, with out-of-diagonal values below 5% and greater than 0.5% for no more than a few -values closed to the highest-efficiency one. While it is well known that, for , the radius of maximum field is proportional to , no experimental data have been presented in the literature concerning the behavior of the inner-ring radius as a function of . In Fig. 4 we reported the experimental data of the inner-radius of maximum field , normalized to the beam waist radius, for increasing values of the radial index and fixed . The experimental data well reproduce the theoretical trend of analytical LG modes ; moreover the linear dependence of on the radial index suggests, for a given and in case of , the behavior of as .
We confirmed the phase structures of the generated beams by observing the interference patterns of the output beams and a plane reference wave with a Mach–Zehnder interferometer system. Figure 5 displays typical interference patterns of the output beams generated by SSPs with phase patterns of [Figs. 5(a) and 5(b)] and [Fig. 5(c)] modes, respectively. In Fig. 5, we can observe fork-like patterns of fringes corresponding to azimuthal mode index of the observed beams as well as radial phase discontinuities of half a period corresponding to the radial mode index , which reflect the phase properties of LG beams.
The usual SPPs are designed for a certain wavelength; hence they induce topological-charge dispersion for short pulses or broadband sources, leading to a superposition of contributions with different values of . Solutions presented in the literature, e.g., , could be considered in order to extend the bandwidth and assure integer phase steps of the singularity and phase shift between adjacent annular rings of the SPP.
In conclusion, the possibility of fabricating SSPs for the generation of high-order Laguerre–Gauss beams has been considered and investigated. The far-field intensity and phase patterns are in good accordance with the expected distributions and present a remarkable quality with a high-purity level of the dominant LG term. These optical elements could find interest and applications not only in astronomy and microscopy, as above mentioned, but also in other fields where optical beams carrying OAM are exploited, such as optical trapping and tweezing , encoding of optical quantum information , and telecommunications .
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