1. Introduction

Optical vortices, i.e. laser beams having a helical wavefront with a phase term $\exp (iL\theta )$ , have been extensively studied owing to their applications in such areas as optical trapping [1], image processing [2], spiral phase contrast microscopy [3], and free-space communication [4]. The most convenient way to generate an optical vortex with topological charge L is by passing a fundamental mode (TEM₀₀) laser beam through an external spiral phase element with transmittance $\exp (iL\theta )$ , such as a computer generated hologram (CGH) [5] or a spiral phase plate (SPP) [6,7].

The ideal SPP has a continuous surface thickness transition topology that imposes the desired azimuthal phase. However, due to the limitations of fabrication techniques, usually the SPP takes a multilevel quantized form where the spiral is divided into a number M of segments, each having a relative quantized phase level of $2\pi L/M$ , to approximate the continuous transition of phase values. This kind of multilevel SPP can be fabricated using various methods, including direct electron-beam writing on negative photoresists [3] and multi-stage vapor deposition process [8].

However the quantized phase levels degrade the mode purity of the optical vortices. Guo et al. [9] examined the vortex properties of an optical beam transmitted by a multilevel SPP and obtained expressions for the intensities of the vortex components as a function of the size of the phase step between adjacent spiral segments. Kotlyar et al. [10] approximated the diffraction efficiency of the quantized SPP using quantized polygon regions and derived somewhat similar results using Fraunhofer diffraction theory. However, neither group provided experimental results showing the validity of the theory nor the characteristics of the output topological charge composition of the quantized SPP.

In this work, we have three objectives. First in Section (2), we analyze the topological charge composition and the charge components’ efficiency of the output from a quantized SPP using the Fourier series approach of Ref [9]. Second in Section (3), to demonstrate the accuracy of this model, we study the vortex content of the beam produced by these SPP designs using a Dammann vortex grating spectrum analyzer approach that is well documented [11–13]. Third, in Section (4), we report the experimental detection of a quantized SPP’s charge composition using a fabricated Dammann vortex grating.

2. Fourier analysis of the beam transmitted by a multilevel SPP

The simplified transmittance function of a conventional SPP can be represented in polar coordinates as $u\text{(}r,\theta )=\exp (iL\theta )$ , where L stands for the topological charge which can take any integer value. For a multilevel SPP with a total number of M phase steps, each step has a phase height of $\Delta \Phi =2\pi L/M$ . Consequently each region covering a phase of $2\pi$ has a total of $M/L$ steps. This stepped or quantized SPP function can be described mathematically as the function $u_s(r,\theta )$ as:

As this transmittance function can be considered as a periodic function of variable θ with period $2\pi$ , it is possible to expand the function into a Fourier series as [14]:

Equation (2) shows that the beam generated with the stepped SPP mask can be regarded as the superposition of perfect vortex beams $\exp (im\theta )$ with integer charges m, each one with a weight $C_m$ . Following the treatment in Ref [9], the relative intensity of each component with topological charge of m is given by

We examine the case where $L=1$ and the SPP is quantized into M steps as shown in Fig. 1 . Figures 2(a) –2(d) show the harmonic content intensity distribution for various values of M and several characteristics are easily discovered. First, we see that the intensity of the desired $m=L=1$ harmonic increases as ${\text{sinc}}^2(1/M)$ as the number of steps increases. In addition, there are a number of orders where the intensity is zero. For example when the SPP is quantized into 2 steps, i.e. $M=2$ , the intensity has non-zero values at $m=+1,-1,+3,-3,\mathrm{...}$ and the intensity of the desired component $m=1$ is 40%. When $M=3$ , the intensity has non-zero values at $m=+1,-2,+4,-5,\mathrm{...}$ and the intensity of charge 1 component is now 68%. Finally, Fig. 2(e) shows how the efficiency of the $m=1$ component increases as the number of steps increases, reaching 95% when $M=8$ and 99% when $M=16$ .

 

Fig. 1 Sketches of quantized SPPs (first row) and the respective optical vortices generated experimentally (second row), with topological charge L = 1 and step number from M = 2 to M = 5, the grayscale represents 0 (black) to 2π (white) phase values.

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Fig. 2 (a) - (d): The intensity spectrum of topological charge components of an L = 1 quantized SPP with step number M = 2 to M = 5. (e): The intensity of topological charge 1 component I_{m = 1,L = 1} versus number of steps M of a multilevel SPP.

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In fact, these intensity distributions are exactly the same as we see in the diffraction efficiency of a one-dimensional linear binary step grating [15,16]. We also see that the important quantity regarding the efficiency of any multilevel SPP is the number of phase steps per 2π section. This is depicted in Fig. 3 , where the Fourier spectra are shown for SPPs with $L=1$ , $L=2$ , and $L=3$ at a constant $M/L$ ratio of 3. The designed vortex components with topological charges 1, 2 and 3 respectively all have the same efficiency of 68%. When this number becomes $M/L=8$ , then that optical vortex will be generated with 95% efficiency in agreement with the predictions of both references [9,10]. In fact, Fig. 1 of Ref [9] agrees exactly with Fig. 5 of Ref [16] as well as our Fig. 2. Next we discuss an experimental technique for detecting this harmonic vortex intensity spectrum.

 

Fig. 3 The Fourier spectra of SPPs with (a) M = 3, L = 1; (b) M = 6, L = 2; and (c) M = 9, L = 3. The diffraction efficiencies are the same when the M/L value is constant (M/L = 3).

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Fig. 5 Experimental detection results using the Dammann vortex sensing grating. The input vortex beam has topological charge L = 1. The number of steps of the quantized SPP are listed on the right, and the number of diffracted order are shown at the bottom. The arrows indicate the detected vortex components.

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3. Dammann vortex grating spectrum analyzer

The Vortex Grating Spectrum Analyzer [11–13] is an excellent probe of the values and signs of the topological charge of an arbitrary input optical vortex beam. Consequently it is an excellent probe of the harmonic content of the beam transmitted by the segmented multilevel SPP. In this approach, the incident vortex beam produced by the SPP is transmitted through a specially designed binary phase vortex sensing diffraction grating. This binary phase grating combines a spiral phase pattern for a given vortex charge with a linear grating as $g\left(x\right)=\exp \left[i\left(q\theta +\gamma x\right)\right]$ . Here q is the topological charge of the spiral phase and $\gamma =2\pi /d$ , where d is the period of the grating. The binarized version of this grating can be written as

Now the diffraction pattern from the product of the incident beam (from Eq. (2)) and the grating (from Eq. (5)) forms a delta function when the input beam has the vortex sign opposite to the vortex sign of the diffracted order (i.e. when $nq=-mL$ ).

In order to increase the diffraction efficiency of the vortex grating, we encode a Dammann grating structure [17,18] to produce equally distributed energy across the useful diffraction orders, allowing easy comparison of the efficiency values. These structures are formed by encoding phase values of 0 or π radians at selected transition points within each period. These transition points are chosen to maximize the total diffracted efficiency for a given number of orders and to provide equal energy in each diffracted order.

We choose to generate 9 diffracted orders using transition points listed in Ref [19]. over a half period of x₁ = 0.099750, x₂ = 0.159152, x₃ = 0.369009, and x₄ = 0.491743 for displaying the 9 diffraction orders from −4 to + 4, with a total efficiency η = 66.32%. Figure 4(a) shows the standard Dammann grating. Using the approach of Ref [20], we encode the spiral phase grating combining the diffraction grating and the spiral phase. The resulting Dammann vortex grating is shown in Fig. 4(b). The grating can also be designed in a 2-D format similar as Fig. 1(a) in Ref [14], which requires lower spatial resolution. However, for simplicity we only consider the 1-D grating here.

 

Fig. 4 (a) 1-D Dammann grating and (b) Dammann vortex grating with embedded topological charge 1, with phase values shown as 0 (black) and π (gray). (c) and (d) are their diffraction patterns respectively.

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4. Experimental results

In the experiment, we use a Quantum Ventus 532 laser system to generate an expanded and collimated laser beam having wavelength of 532nm. The desired incident vortex beam is formed by passing the laser beam through a Hamamatsu X8267-11 reflection type parallel aligned liquid crystal spatial light modulator (PAL-LCSLM) which operates in a pure phase modulation mode with a dimension of 20mm by 20mm and pixel size 26μm. Therefore, we can encode the desired multilevel SPP phase functions onto the SLM, as shown in Fig. 1.

The generated optical vortex beam is then passed through the fabricated Dammann vortex sensing grating as discussed earlier. We use UV lithography technique to fabricate SU-8 photoresist medium coated on glass slide, which gives us a transmissive grating. The respective height for π phase difference is calculated to be h = 0.5λ/Δn = 450.85nm, with the refractive index of the SU-8 photoresist is 1.59 at 532nm.

The transmitted beam is then focused by a 30mm focal length converging lens to a CCD camera. Figures 4(a) and 4(c) show the standard Dammann grating and its diffraction patterns. We see the 9 equally intense diffracted orders. Figures 4(b) and 4(d) show the Dammann vortex grating and its output. Each order forms a vortex beam having a charge of $nq$ where n is the diffracted order and q is the embedded charge of the grating ( $q=1$ in our case). All these vortex beams show a null intensity at the center of the corresponding diffraction order.

When the input vortex from a quantized SPP is incident on the grating, the detection pattern will occur at the diffraction order having opposite charge number, as explained in Ref [16]. For example when $M/L=2$ , the components of the input vortex beam are at $m=+1,-1,+3,-3,\mathrm{...}$ from Fig. 2(a). So the product of the incident beam from the multilevel SPP and the Dammann vortex grating will yield delta function outputs at the diffracted orders corresponding to $n=-1,+1,-3,+3,\mathrm{...}$ This is shown in Fig. 5, first row, where now these orders show a spot on the center of the diffraction order, whose intensity is proportional to ${\left|C_n\right|}^2$ (in this case it is noticeable the two bright spots located at the $n=\pm 1$ orders). Those harmonic components that are absent are noticed as diffraction orders that keep the null intensity at the center ( $n=\pm 2,\pm 4,\mathrm{...}$ in this case). Similar results are shown for cases where $M/L$ = 3, 4, 5 and 100. In each case, the strength of the delta function increases as $M/L$ increases while the intensities of the other orders decrease and agree with all theoretical predictions.

After obtaining the experimental results, the intensity values of the resultant delta functions are measured for comparison with theory. When a continuous SPP is used, the maximum intensity of the $n=-1$ diffraction order recorded at the CCD camera is 198, according to the 8-bit grayscale value, and this gives us a benchmark. When the number of steps increases from 2 to 5, the recorded intensity values are 84, 135, 162, and 168, which translate to intensity values of 42.4%, 68.2%, 81.8% and 84.8%. Comparing these values with the calculated value using Eq. (5), the errors are below 5%, which concludes the validity of the detection technique.

5. Conclusions

In this work, we have examined the capability of a quantized SPP to generate an optical vortex beam. The SPP is marked by an azimuthal phase shift of $2\pi L$ where L is the desired vortex beam charge. We first expanded the transmission function of a quantized SPP using a Fourier series approach, and then calculated the Fourier spectra and diffraction efficiencies of the generated vortex beams. We find that the important quantity is the number of segments per 2π phase level with the requirement that 16 steps are required for 99% generation of the desired vortex beam.

To experimentally identify the vortex components, we utilize a vortex sensing Dammann grating approach to clearly reveal the vortex pattern of the incident beam. The Dammann vortex grating distributes energy equally across a number of designated orders and clearly verifies the theoretical predictions for a vortex beam having a topological charge of $L=1$ . Experimental results agree with the theoretical model.