## Abstract

A multi-element design scheme is proposed to produce optical vortices of large spectrum width. The key component within the approach is a radially modulated spiral phase plate. Apart from a conventional spiral phase plate having an azimuthal phase function, the proposed element possesses an additional change of phase in the radial direction.

©2008 Optical Society of America

Optical vortices or vortex beams have been receiving increasing interest partially because of the orbital angular momentum directly related to the helical wavefront of a vortex beam. It has been pointed out that the amount of orbital angular momentum carried by each photon in a vortex beam is controlled by the topological charge of the optical vortex (in particular, ℓ*ħ* per photon when the topological charge is equal to *ℓ*) [1]. While most of current research is focusing on optical vortices generated by monochromatic light, the ability to produce an optical vortex with polychromatic light or broadband sources is essential for some studies [2, 3]. An achromatic design comprising a hologram, a 4-f system and a prism was implemented to generate white-light optical vortices [4]. In this method, the angular dispersion of the diffractive grating in the hologram is compensated by the prism. However, the uneven diffractive efficiency experienced by different spectral components could influence the composition of spectrum. Recently, an achromatic design combining two refractive spiral phase plates (SPPs) [5] that have different dispersion properties is proposed [6]. The approach is power efficient and straightforward in the generation of optical vortices for a spectrum width up to 140 nm which is limited by the materials used to construct SPPs. In this paper, we propose a new design of spiral phase plate that could work under broadband illumination with the help of axicons.

Various unconventional spiral phase plates exhibiting radial phase variations have been used to reduce the sidelobes of an optical vortex in the far-field [7, 8] and create higher order Bessel beams [9, 10]. Most of their phase functions take the summation of a radial function *f*(*r*)and the linear azimuthal phase term ℓ*θ* characterizing the nature of an optical vortex. In this paper, we study another possible form, i.e. *f*(*r*)×*θ*, that will be referred to as radial spiral phase plate (RSPP). If an infinitely narrow annular aperture with the radius equal to *r*
_{ο} is applied to such a phase element, the transmittance within the aperture can be approximated as exp[*i*
*f*(*r*
_{ο})×*θ*], which is very similar to that of a conventional spiral phase plate while the constant *f*(*r*
_{ο}) takes the position of topological charge *ℓ*. In other words, various topological charges can be achieved by the same phase element when the radius of the annular aperture is carefully selected to a series of discrete values. Meanwhile, it is reported that a pair of identical axicons transforms an incident Gaussian beam into an annular beam [11, 12]. Therefore, a combination of axicon pairs and an RSPP would enable the achromatic generation of optical vortices as proposed in Fig. 1. Compared to light with long wavelength (depicted in red), spectral component having shorter wavelength (depicted in blue) results in a larger annulus after passing through the first axicon pair. The phase delay conveyed by a thin phase element e.g. RSPP in this paper can be given by

where Δ*n* is the difference in refractive index between the phase element and its surrounding medium, *h*(*r*,*θ*) is the thickness of the phase element and λ is the wavelength of light in vacuum. Normally, such an element will present different phase delay function for various wavelengths. However, Fig. 1 shows that the blue ray representing light with shorter wavelength will pass through part of the phase element thinner than that experienced by the red ray, which makes it possible to have an equivalent phase delay for these two wavelengths. It is noted that this cannot be achieved by the conventional SPP as its thickness does not change in the radial direction. The second axicon pair combines the annular beams back to one laser beam.

Although the RSPP is designed for more than one spectral component, for simplicity the element can be expressed by a phase-only transmittance function at a fixed wavelength. In this paper, we will focus on a straightforward example when the transmittance of the RSPP at the wavelength λ_{ο} is

with the constant *k* chosen to be ℓ_{ο}/*r*
_{ο}. Here ℓ_{ο} is the topological charge of the optical vortex generated from the infinitely narrow annular zone at *r*=*r*
_{0}. The phase profiles of an SPP and the RSPP with the parameter *k*>0 are illustrated in Fig. 2 while the case of *k*<0 has been demonstrated in Fig. 1. It is noted that a similar phase function was first proposed by multiplying the phase function of an SPP with that of an axicon and such an element results in a spiral intensity pattern in the far-field when working with a full circular aperture [13]. For the optical field produced by the RSPP illuminated by a uniform annular beam with mean radius *r*
_{ο} and width *w*, the weight of the optical vortex with topological charge *n* can be calculated by the mode analysis method [14–16] with the thin-element approximation [17]:

which can be given in an explicit form:

$$+\mathrm{In}\left(2{\ell}_{0}-2n+\frac{{\ell}_{0}w}{{r}_{0}}\right)-\mathrm{In}\left(2{\ell}_{0}-2n-\frac{{\ell}_{0}w}{{r}_{0}}\right)+2n\pi \mathrm{Si}\left(2{\ell}_{0}\pi -2n\pi +\frac{{\ell}_{0}\pi w}{{r}_{0}}\right)$$

$$-2n\pi \mathrm{Si}\left(2{\ell}_{0}\pi -2n\pi -\frac{{\ell}_{0}\pi w}{{r}_{0}}\right)+\frac{8{\ell}_{0}n{r}_{0}w}{4{r}_{0}^{2}{\left(n-{\ell}_{0}\right)}^{2}-{\ell}_{0}^{2}{w}^{2}}{\mathrm{sin}}^{2}\left(\frac{{\ell}_{0}\pi w}{2{r}_{0}}\right)],$$

where *Ci*(*x*) and *Si*(*x*) are cosine integral and sine integral, respectively. Therefore, the weight of the major topological charge can be calculated by setting *n*=ℓ_{ο}:

with the dimensionless width *y*=πℓ_{ο}
*w*/*r*
_{ο} and *Sinc*(*x*)=sin*x*/*x*.

Figure 3(a) is the plot of Eq. 5 that shows the weight of the specific optical vortex as a function of the dimensionless width. As expected, the weight reaches 100% when the width of the annular aperture is infinitely narrow and decreases with the increasing width. Since Eq. 5 is irrelevant to the value of the topological charge (as the variable *y*=πℓ_{ο}
*w*/*r*
_{ο}=π*kw*), this analysis applies to other topological charges generated by the same RSPP, e.g. the topological charge 2ℓ_{ο} excited from the vicinity of *r*=2*r*
_{ο}. The high concentration of power on the target topological charge is illustrated in Fig. 3(b) when a narrow annular aperture is imposed. Figure 4 shows the Fraunhofer diffraction patterns of optical vortices produced by the RSPP with different annular apertures. The topological charge of the generated optical vortex increases with the mean radius of the annular aperture when *k*>0. The slightly asymmetry of intensity distribution is due to the mixture of modes other than the desired one. As a result of the annular aperture, the optical vortices in the far-field are embedded in the profile of Bessel beams that exhibits concentric intensity circles [18].

An RSPP is fabricated by the technique of direct electron beam writing in negative resist with the phase profile wrapped within the range of 0~2π [19]. The surface relief phase element with the size of 7.68*mm*×7.68*mm* is designed to support the topological charge up to 5 at the wavelength of 633nm. The maximum topological charge is limited by the size of the element and the value of *k*, in our case, equal to 1.3mm^{-1}. The confinement of RSPP in the radial direction imposed by an annular aperture is replaced by an incident annular beam which is converted from a Gaussian beam by two identical axicons as shown in Fig. 5. A beam of wavelength λ=633*nm* from a HeNe laser is collimated and enlarged by a telescope. After that, the laser beam is truncated by a circular diaphragm that controls the width of the annular beam. One of the axicons is set to be movable so that the distance between axicons can be adjusted according to the desired size of the annular beam [11]. A central stop is placed after the axicon pair in order to remove the residual fractional intensity at the beam axis. The transverse intensity profile of the annular beam right before the RSPP is shown in Fig. 6. The optical field created after the annular beam passing through the RSPP sample is then focused into a CCD beam profiler by a lens. Optical vortices produced by the same RSPP while changing the mean radius of the annular beam are shown in Fig. 7. The corresponding topological charge of each optical vortex is measured by interfering with an oblique reference beam (See Fig. 8).

In summary, we have demonstrated that an RSPP is capable of switching the topological charge of the resulting optical vortex at a fixed wavelength of light when the radius of the incident annular beam is changed. This result suggests that the RSPP can be used to generate optical vortices of an identical topological charge for various wavelengths. Based on the RSPP discussed in this paper, we propose a design scheme for the generation of an optical vortex with a wideband light source. The dispersion of the material used to construct the axicons makes the mean radius of the annular beam differ from one wavelength to another. The size change of the annular beam will be compensated by the radial variation of an RSPP in a way that the resultant optical vortex has the same topological charge within a broad range of wavelength. Although it may not be suitable for the imaging application such as the optical vortex coronagraph [20] because of the distortion of images introduced by apertures, stops and axicons, the proposed method (Fig. 1) of compensating chromatic dispersion is still able to find applications in reducing the topological dispersion of optical vortices generated by pulse laser [21].

## Acknowledgments

This work is partially supported by National Natural Science Foundation of China for grant 60778045 and National Research Foundation of Singapore under Grant No. NRF-G-CRP 2007-01 and Ministry of Education under ARC 3/06, RGM6/05 and RGM37/06. XCY acknowledges the support given by Nankai University (China) and Nanyang Technological University (Singapore). The authors would like to thank Yuyang Sun for data conversion for electron beam lithography.

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