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Abstract
In this manuscript, the generation of an optical vortex beam with high order and reconfigurable orbital angular momentum (OAM) is studied. Multi-waveguide holographic gratings (MWHG) are deployed to generate OAM beams with high order. The generation of the OAM beam with an order l from +4 to +8 is demonstrated by numerical simulations, and the generated OAM order is manipulable and configurable by incident phase. The working bandwidths of the MWHG for different OAM orders are at the level of 40 nm. This work could provide valuable references for practical implementation of OAM in integrated optics.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the optical communication, optical information is usually encoded by the property of photon, such as frequency, energy, spin angular momentum and so on [1]. The orbital angular momentum (OAM) is a special intrinsic property of photon, which can expands infinite dimensions for information encoding by the eigen state. The optical vortex beam containing OAM has a factor of $exp(il\phi )$ with $l$ as the order of OAM, which can be $\pm 1,\pm 2,\pm 3,\ldots$ in theory [2]. It is possible to encode information in a single photon with OAM in a high dimensional space, so more information can be carried by the photon with more OAM components. So a great efforts have been proposed to prepare and detect the OAM beam with high order for its potential applications in the high dimensional optical information encoding [3,4].
With the development of integrated information, OAM is introduced to be applied in integrated optics. There have been great endeavors devoted to study the application of OAM in integrated optics and various device based on the OAM arises, such as integrated OAM multiplexer [5–7], OAM modes emitter [8,9], information encoder [10], and so on. X. Cai et al. propose a microring with gratings to generate vortex beam on a chip in 2012 [11]. P. Miao et al. report that the microring laser can lase the single-mode OAM beam [12]. The holographic grating is introduced to generate the optical vortex beam with broad bandwidth [13,14]. The broadband polarization diversity OAM generator is compacted to a footprint with several-micrometer by N. Zhou et al. [15]. By specially designing the on-chip devices, the OAM of the generated vortex beam can be switched freely and detected in real time [16–19]. In the integrated optics, it is still a challenge to manipulate the optical vortex beam with high order OAM by the waveguide holographic grating. Because the generation of OAM beam with high order requires a wide waveguide, which will introduce high guided modes to make the generated OAM beam complex and increase the background noise [20].
In this paper, an on-chip multi-waveguide holographic grating (MWHG) is proposed to generate OAM beam with high order and reconfigurable OAM. The amplitude and phase of the obtained OAM beam are given by numerical results. By the designed MWHG, the obtained OAM beams with orders $l$ from $+4$ to $+8$ are illustrated with the fidelity above 0.8 and the modes crosstalk below 0.2. The generated OAM order can be controlled by manipulating the incident phase on the waveguides of MWHG. The working bandwidths of the designed MWHGs for different OAM orders are given, and they contain a similar bandwidth of $40$nm with the fideltiy above 0.5.
2. Principle of theory
The proposed optical vortex beam generator of high order OAM is a multi-waveguide surface holographic gratings (MWHG), which is composed of several waveguides ranged in a square and the holographic gratings are etched on the surface of the waveguides in the center square area. Figure 1(a) gives the illustration of a MWHG with four-waveguide. The guided waves in the waveguides (in plane) will be scattered to free space in vertical direction by the holographic gratings and form an OAM beam.
Fig. 1. (a)The schematic illustration of the proposed optical vortex beam generator on the integrated multi-waveguide. (b1)-(b4) The principle of generating OAM beam by MWHG.
The basic principle of the holographic grating is explained in Figs. 1(b1)-(b4). At first step, the time-reversal of the target OAM beam
is incident at the center of the square and vertically to the waveguide, where $A(x,y,z)$ is the amplitude of the OAM beam. At the same time, there are four guided waves propagating in the four waveguides respectively from inside to outside of the squares (Fig. 1(b2)). The electric field in the waveguides can be given as where $B_{j}(x, y, z)$ is the amplitude of the electric field in the cross section of the $j_{th}$ waveguide and $k_{j}$ is the wave vector along the $j_{th}$ waveguide, N is the number of the waveguides. The azimuthal angle $\varphi$ in polar coordinates is related to the Cartesian coordinates by $tan\phi = y/x$. The interference of the guided wave and the target OAM beam interfere to form the holographic grating on top of each waveguide ($z=0$) as shown in Fig. 1(b3). The distribution of the holographic grating can be given as where $\theta = k_{j}r_{j}-l\varphi$. The MWHG is designed by dividing the interference region on the square into many pixels (60 nm $\times$ 60 nm) [21]. Since the size of pixel is much smaller than the wavelength, these pixels can be seen as new "point" sources, which can scatter light to form the generated light field. A gray value of G(x, y) is set for each pixel. According to the phase difference between the target OAM beam and the guided wave, the simple binary function is applied, i.e., $(G(x, y)=0$ if $\delta \theta <0.5\pi$, or else $G(x, y) = 1$. The inset of Fig. 1(a) gives the binary gray image of the MWHG for OAM beam with $l=+4$. The MWHG with designed scale is practical with current nanotechnology [14,22]. In the generation of the OAM beam, the conjugate guided waves $E_{w}^{*}$ propagate along the waveguides (the directions are shown by the red arrows in Fig. 1(b4)) and are scattered by the MWHG to produce the OAM beam, Due to the multi-incident waveguides of the MWHG, the phases incident to the waveguides can be manipulated separately and freely by the phase modulator. When the phase $\varphi (n)=\varphi _{0}+2n\Delta \pi /N$ is incident to the n-th (n=0, 1, 2,…, N-1) waveguide, the generated OAM beam can be given asThe MWHG has the advantage in generating OAM beam with a high order due to the special arrangement of the holographic gratings on multi-waveguide. The intensity distribution of the OAM beam is a donut, and the radio of the hollow increases as the order of the OAM order $l$. With waveguide holographic grating, the generation of OAM beam with high order needs a wide waveguide to cover the electromagnetic field. In the wide waveguide, high order guided modes may arise to introduce unexpected scattering light to the OAM beam obtained from holographic grating. The proposed MWHG can cover a maximum electromagnetic field of the OAM beam with high order by designing proper waveguides combination according to the intensity distribution of OAM beam.
3. Numerical results
In the following, we verify this idea by numerical simulations based on the scalar diffraction theory on the platform of the Wolfram Mathematica 8.0. The MWHG is composed of four $Si3N4$ waveguides on a silica substrate as shown in Fig. 1(a). The refractive index of $Si3N4$ is taken as 2.0351 at the wavelength of $\lambda _{0}=670$ nm [23]. All of the waveguides have a width of $d=1.5$ $\mu$m [13], and the width of the inner square formed by the waveguides is $q=1.8$ $\mu$m. The gap between the end of the one waveguide and the side of the adjacent waveguides is set to be $g=200$ nm, where optical opacity should be filled to prevent the coupling. The holographic gratings are fabricated on the surfaces of the waveguides within the square. In the current study, we focus on the fundamental TE mode and the field distribution on the top surface of each waveguide is approximated by a Gaussian function with $d >\lambda _{0}$. The OAM beam considered in the simulation is a Laguerre-Gaussian (LG) mode with radial index $p=0$ and the waist diameter of the OAM beam is $2.0$ $\mu$m and positions on the up surface of the waveguides [24]. The insets of Fig. 1(b) give the phase and intensity distributions of the target OAM beam with $l=+4$, which contains a donut amplitude and a helical phase front with a singularity in the center.
Figure 2 gives the phase and intensity distributions of the OAM beams, where the OAM orders are $+4, +5, +6, +7, +8$ from the first to the fifth rows, respectively. In each row of Fig. 2, the first and second pictures are the phase and intensity distributions of the target OAM beam, and the third and fourth pictures correspond to the OAM beam generated from the MWHG. The image plane is $10\lambda _{0}$ far away from the up surface of the waveguide in the vertical direction. For the OAM beam with low order, the deficiency of holographic grating in the inner vacant square and in the gap have obvious destruction on the generation of the OAM beam, so the results for the OAM beam with lower order are not given. The phase and intensity distributions of the generated OAM beams agree with those of the target OAM beams at their intensity cross-sections as shown in Fig. 2. There is a peak intensity in the center of the obtained OAM beam, which is expected to be hollow. These unexpected OAM modes is due to the geometric imperfections of the MWHG.
Fig. 2. The phase (1st and 3rd columns) and intensity (2nd and 4th columns) distributions of the target (1st and 2nd columns) and generated (3rd and 4th columns) OAM beams with the orders from $l=+4$ to $l=+8$.
In order to qualify the quality of the generated OAM beam, the fidelity of OAM is introduced as
Fig. 3. The histograms of the $5\times 8$ crosstalk matrix. $l_{in}$ denotes the same OAM order as the input mode.
Table 1. The fidelity of generated OAM.
Because the MWHG is composed of multi-incident waveguides, the generated OAM order can be controlled by manipulating the phase of the incident light. For the MWHG with four waveguides, the generated OAM beam with $l+\Delta$ can be obtained with the phase $\varphi (n)=\varphi _{0}+n\Delta \pi /2$ incident to the n-th waveguide of the MWHG designed with l. The manipulted order $\Delta$ of the OAM beam can be $0, \pm 1, \pm 2, \pm 3\cdots$ For example, with the same incident phase $\varphi _{0}$ on all waveguides of the MWHG designed with $l=+4$, an OAM beam with the same order $l=+4$ is obtained as shown in Fig. 4(b). When the phase $\varphi (n)=\varphi _{0}-n\pi /2$ incident to the n-th waveguide of the MWHG designed with $l=+4$, an OAM beam with $l=+3$ is obtained as shown in Fig. 4(a). When the incident phase is set to be $\varphi (n)=\varphi _{0}+n\pi /2$, an OAM beam with $l=+5$ is obtained as shown in Fig. 4(c). The order of OAM beam generated from MWHG designed with $l=+4$ can be manipulated by the incident phase, so is the MWHG designed with other $l$. The reconfigurable generation of OAM beam makes the MWHG have great potential application in the OAM based integrated optics.
Fig. 4. The intensity (up row) and phase distributions (down row) of the OAM beam obtained from the MWHG for $l=+4$ with manipulated incident phase. (a) $l=+3$, (b) $l=+4$, (c) $l=+5$.
For different OAM order, the optimal MWHG is different. In order to make the MWHG more suitable for genenrating OAM beam, it is necessary to optimize the structure of the MWHG. The missing holographic gratings in the center vacant square reduce the quality of the OAM beam, which will decrease the fidelity of the obtained OAM beam directly. Figure 5(a) gives the phase and intensity distributions of the OAM beam with $l=+4$ obtained from the MWHGs with different center square width $q$. For simplicity, the same phase is incident to the waveguides of MWHG, so is in the following. The intensity distribution seems to be a square for $q=1.0$ $\mu$m as shown in Fig. 5(a1). When the center square width $q$ increases, the intensity distribution become a circular just as that of the target OAM beam, as shown in Figs. 5(a3) and 5(a4) for $q=1.8$ $\mu$m and $q=2.2$ $\mu$m, respectively. The intensity distribution returns to be a square when $q$ increases further as shown in Fig. 5(a5) for $q=2.6$ $\mu$m. Figure 5(b) gives the fidelity of the OAM beam as a function of the square width $q$, where the fidelities of OAM beam with different orders are given by different lines. As the square width $q$ increases, the fidelity increases at first, then fluctuates around the maxmun and then decreases at last. That is, the designed MWHG is robust to the square width $q$ and practical in the experiment.
Fig. 5. The relation of the obtained OAM beam and the square width of MWHG. (a1)-(a5)The phase (up row) and intensity (down row) distributions of the OAM beams with $l=+4$ obtained from MWHG with the square width $q=1.0$ $\mu$m,1.4 $\mu$m, 1.8 $\mu$m, 2.2 $\mu$m, 2.6 $\mu$m, respectively. (b) The fidelity of the generated OAM as a function of the square width $q$.
Further, we analyze the property of the MWHG working in different wavelength. Considering the preparation practicality and the effect of missing holographic gratings in the center vacant square, the vacant square width of the MWHG is set to be $1800$ nm. Figure 6 gives the fidelity of OAM beam as a function of the incident wavelength. For different OAM orders, the maximal fidelities are obtained at the wavelength of $\lambda =670$ nm since the grating structure of MWHG is designed by the interference of light with $\lambda _{0}=670$ nm. The fidelity of OAM beam is above 0.5 with the incident wavelength among $650$ nm and $690$ nm for the OAM order from $l=+4$ to $l=+8$. The MWHG has a working bandwidth of $40$ nm, which indicates the MWHG is feasible for preparation.
Fig. 6. The fidelity of generated optical OAM beam as a function of the incident wavelength for different OAM orders.
4. Conclusion
A composite holographic gratings has been proposed to generate reconfigurable OAM with high orders. Numerical simulation results show that the fidelities of the generated OAM beams are all above 0.8 for $l=+4, +5, +6, +7$ and $+8$. With the MWHG, the generated OAM order was illustrated to be manipulated freely by the incident phase. The center vacant square of the MWHG was anylized in effecting the quality of the generated OAM beam. The MWHG has a working bandwidth of $40$ nm at the working wavelength of $\lambda _{0}=670$ nm. The designed MWHG will play a key role in the application of OAM with high order and reconfigurable OAM in integrated optics.
Funding
Leading-edge technology Program of Natural Science Foundation of Jiangsu Province (BK20192001); China Postdoctoral Science Foundation (2019M651911); National Natural Science Foundation of China (11774180, 61590932).
Acknowledgments
We thank Dr. Chang-Ling Zou for enlightening discussions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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