1. Introduction

The orbital angular momentum (OAM) carried by a twisted light beam is associated with the helical phase structure of$\exp (i\ell \phi )$, where$\ell$ is an integer and$\phi$is the azimuthal angle. In such a beam, each photon carries a well-defined OAM of $\ell \hslash$, and in principle, the OAM can exist within an inherently infinite dimensional Hilbert space [1]. Over the past two decades, optical OAM has attracted a lot of research interest due to its promising applications ranging from biophysics [2], micromechanics [3], microscopy imaging [4] and quantum communications [5]. The OAM has been readily generated and extensively investigated with the monochromatic laser source [6–10], however, there were few studies reporting the connection of OAM with the white-light sources. Based on the combination of a single fork grating with a dispersion compensator, Leach et al. observed the chromatic effects near a white-light vortex with co-axial spectral components [11]. Besides, Wright et al. illustrated the mechanical effect of a white-light vortex beam, as a result of OAM transfer from light to microscopic particles held in optical tweezers [12]. Spangenberg et al. used a spatial light modulator (SLM) to shape any wavelength of a super-continuum fiber laser, and to produce a rotating white-light Bessel beams carrying orbital angular momentum [13]. Very recently, Kobashi et al. described the polychromatic generation of optical vortices in cholesteric liquid crystals, owing to the Bragg-reflection-based conversion mechanism [14]. Here we use a white-light LED source to produce chromatic OAM beams as well as their superpositions. In contrast with previous schemes, our approach is based on the combination of fork holographic gratings with the technique of theta-modulation in a 4f optical system. This method enables the OAM coloration with various topological charges, appearing as a tunable mixture of red, green and blue (RGB) in different proportions.

The theta-modulation was actually not a new optical technique, which was first proposed early in 1965 by Armitage and Lohmann [15]. This technique originated from the famous Abbe-Porter experiment, and were explored for production of a color image from a black and white film and for multiplex storage. Based on theta-modulation, the techniques of color mixture [16], optical logical processing [17], and multiplexing in multiple image [18] were subsequently demonstrated. Recently, theta-modulation was also extended to realize a dynamical optical encrypting technique to color time evolving phenomena [19]. Here we incorporate the theta-modulation into the programmable holographic gratings addressed by a spatial light modulator (SLM), and produce chromatic OAM beams of various topological charges from a white-light source. We anticipate that these chromatic OAM beams could find direct application in wavelength multiplexing for OAM-based optical communication.

2. Method and setup

There have been several methods established to generate OAM beams, such as spiral phase plate [9, 10], cylindrical lenses mode converter [20], computer-generated holograms [6] and q-plates [21]. Among these techniques, the spatial light modulator (SLM) has been a convenient and reliable way to shape the phase and amplitude of a light beam [22]. Here we employ a single SLM to perform the theta-modulation with OAM generation. Figure 1 is our experimental setup used for producing the chromatic OAM beams from a white-light source, which is basically a 4f optical system consisting of two lenses with focal length f₁ = 500 mm and f₂ = 300 mm. The white light derived from a 1 W LED (Daheng, GCI-060411) has a bandwidth ranging from 440 nm to 670 nm. A telescope is used to collimate the white-light beam, which then illuminates the SLM uniformly. The SLM is put at the object plane of the 4f system, while the demodulation of spatial filtering is performed at the Fourier plane. Then we record the generated OAM beams by placing a color CCD at the image plane.

 

Fig. 1 Optical setup of 4f system used for theta-modulated chromatic OAM generation. Inset (a) and (b) Typical examples of holograms displayed by SLM; (c) Frequency spectrum in the Fourier plane, where the pinholes are exaggerated by the color circles.

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The SLM (Hamamatsu, X10486-1) we use here is a phase-only modulator, which has an array of pixels (800 × 600) with an effective area of 16 mm × 12 mm and a pixel pitch of 20 μm. Each pixel imprints individually the incoming light with a phase modulation (0 ∼2π) according to the 8-bit grayscale (0 ∼255). And the whole SLM acts as a reconfigurable diffractive element, allowing an interactive manipulation with a response time comparable to the video displays. For producing helically phased OAM beams, the frequently used design is to add a blazed grating modulo 2π to a spiral phase of$\exp (i\ell \phi )$, then we obtain a forked hologram with a $\ell -pronged$dislocation on the beam axis. When illuminated with a monochromatic Gaussian beam, the first-order diffracted beam acquires the desired $\exp (i\ell \phi )$ phase structure and therefore carries $\ell \hslash$ OAM per photon. It is noted that the SLM is not achromatic diffractive device such that the diffraction can create “rainbow” colors with a broadband white-light illumination. And the mth-order diffraction angle ${\theta }_m$ for wavelength $\lambda$ is determined by the well-known grating equation, $d(\sin {\theta }_m+\sin {\theta }_i)=m\lambda$, where d denotes the grating period, ${\theta }_i$ and ${\theta }_m$ are the incident and diffracted angles, respectively, with respect to the normal to the grating surface [23]. However, it is just this spectral dispersion that enables the spatial filtering to generate the chromatic OAM with various RGB mixtures. To perform the theta-modulation, we divide the pixel array into four rectangle sub-domains, each of which has an area of 8 mm × 6 mm, consisting of 400 × 300 pixels, as was shown by inset (a) of Fig. 1. In each fork holographic grating, the $\ell -pronged$dislocation is responsible for the desired transverse phase structure of the diffracted light, while the blazed grating only serves as the carrier and its orientation determines the diffraction direction. Generally, we have the transmission function,$T(x,y)=\exp [iH(x,y)]$ for a pure phase grating, and specifically for our case,$H(x,y)$ can be written as,

For theta-modulation, the blazed grating in each domain has been specially designed to have a different orientation, i.e.,$\theta =0°$,$\text{45}°$,$\text{90}°$, and$\text{135}°$, respectively, see inset (a) of Fig. 1. All zero-order diffractions are trivially blocked out. In such a configuration, their diffraction patterns will be angularly separated, namely, situating on the lines along the angle of$\text{90}°$,$\text{135}°$,$\text{0}°$, and $\text{45}°$, respectively. Each holographic grating can be programmed according to Eq. (1) to generate the desired OAM individually in the 1st-order diffraction. We estimate a conversion efficiency of about 20%. With a white-light illumination, these theta-modulating diffractions are also spectrally dispersed in the Fourier plane, see inset (c) of Fig. 1. In the demodulation process, we insert a tailed-made opaque mask with a set of pinholes to select a certain portion of each frequency spectrum. The size and position of the pinholes is closely related to the spectrum distribution we considered. As was shown by inset (c) of Fig. 1, we have marked these pinholes by color circles such that light beams with varied frequencies and different OAM can be generated effectively. In the image plane of the 4f system, we use a color CCD to record the intensity patterns of the colored OAM beams.

3. Experimental observations and RGB separation

Based on the optical setup of Fig. 1, we produce the chromatic beams with single and multiple OAM, respectively. For single OAM, the holographic gratings are simply the fork ones, which can be prepared according to Eq. (1) with $\Phi (x,y)=\ell \phi$. The prong number in the center of the fork hologram determines the generated OAM number and the orientation of the grating is responsible for the theta-modulation. One typical example was shown by the inset (a) of Fig. 1, which are used for generation of single OAM $\ell =2,$3, 4 and 5, respectively, see Fig. 2(b). We put an opaque disk in the Fourier plane and filter out suitable frequency components with four punctured pinholes in different locations of the disk. We use a color CCD camera to record the generated OAM beams in the image plane, and present our observations in Fig. 2. We successfully generated single OAM with different OAM numbers ranging from $\ell =1$ to$\ell =5$, and their colors are approximately blue, red, light green and light yellow. Besides, the reliability of our results can also be verified by these observed bright intensity rings, from which one can see that the ring radius scales with $\sqrt{\ell }$ generally [26].

 

Fig. 2 Generation of single chromatic OAM, with the numbers being the topological charges.

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We perform a further experiment to generate the chromatic OAM mixtures. Without loss of generality, we have considered the coherent superposition of two different OAM, i.e., ${\text{LG}}_{p=0}^{{\ell }_1}$ and${\text{LG}}_{p=0}^{{\ell }_2}$, where ${\text{LG}}_p^{\ell }$denotes the LG beam with radial and azimuthal indices $p$ and$\ell$, respectively. The holograms loaded on SLM are made according to Eq. (2), with$\Phi (x,y)=\arg \left({\text{LG}}_{p=0}^{{\ell }_1}+{\text{LG}}_{p=0}^{{\ell }_2}\right)$ and$I(x,y)={\left|{\text{LG}}_{p=0}^{{\ell }_1}+{\text{LG}}_{p=0}^{{\ell }_2}\right|}^2$. For example, the hologram shown by inset (b) of Fig. 1 was used to generate the beams in Fig. 3(b). The recoded patterns result essentially from the interference of an$\exp (i{\ell }_1\phi )$ beam with the other $\exp (i{\ell }_2\phi )$ beam, where the complete constructive and destructive interference occurs at the angle $\phi$ determined by$\cos ({\ell }_1-{\ell }_2)\phi =\pm 1$, respectively. As a consequence, a pattern like a $({\ell }_1-{\ell }_2)$-petal flower is formed [27, 28]. The observations of Fig. 3 confirm again the good feasibility of our scheme. Besides, it is noted that $\pm \ell$ OAM superstitions, as shown by Figs. 3(a) and 3(b), have been extensively employed to the measurement of rotational Doppler shifts [29].

 

Fig. 3 Interference petal-like patterns of two OAM, ${\text{LG}}_{p=0}^{{\ell }_1}$ and${\text{LG}}_{p=0}^{{\ell }_2}$, where the OAM combinations are indicated above each figure. The pixels indicated by white squares in (a) are used for RGB analysis in Fig. 4.

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In the frame of RGB color model, we analyze the color composition of the generated chromatic OAM beams. A color in the RGB model is described by indicating how much of each of the red (R), green (G), and blue (B) is included [30], which is expressed as an RGB triplet$(r,g,b)$, and each component vary in the range 0 to 255. If all the components are at zero the result is black; if all are at maximum, the result is the brightest representable white. Without loss of generality, we pick up several pixels in Fig. 3(a) to specifically illustrate the RGB components. We present the RGB histograms in Fig. 4, and it is just these various RGB combinations that indicates the effective generation of chromatic OAM beams by our scheme. We also recorded the full spectral data as was shown in Fig. 4(e). Based on the grating function, we have estimated and marked the wavelengths of Figs. 4(a)-4(c) in the spectrum. Besides, it is noted that the generated chromatic OAM beams are displaced in space. Thus a possible extension of our scheme is to use additionally a serious of beam splitters to combine these beams directly for OAM multiplexing. Besides, we think that a similar corporation of theta-modulation with two-dimensional complex gratings may offer a simpler scheme for the same purpose, and we would like to leave this in our further studies.

 

Fig. 4 The RGB analysis. (a)-(d) are the top-left, top-right, bottom-left and bottom-right pixels indicated by white squares in Fig. 3(a), respectively; (e) Measured rainbow spectrum.

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4. Conclusion

In summary, we have incorporated the theta-modulation technique into the fork holograms to generate the chromatic OAM beams from a white light source based on a basic 4f optical system. The principle of preparing the desired holograms has been discussed. We have observed the generation of both single chromatic OAM and multiple chromatic OAM superpositions, and the RGB color model is utilized to show the feasibility of our scheme. The agreement between theoretical and experimental results can be clearly seen. Our work may find direct application in the OAM-based optical communication, where the multi-state and multi-color OAM holds promise for increasing the data capacity significantly.