1. Introduction

The full degrees of freedoms of light include frequency, intensity, polarization and orbital angular momentum (OAM). It has been shown that a beam with azimuthal phase of $e^{il\phi }$ carries $l\hslash$ OAM [1]. Singularities in intensity and phase distributions of OAM-carried beams have stimulated many exciting applications such as optical manipulation and trapping [2,3], high precision optical metrology [4–6] and quantum information processing [7–14]. Mutual-orthogonal OAM modes offer the possibility of spatial mode multiplexing for high capacity classical optical communications [15–17]. The unlimited dimensions of OAM modes also hold promising for dense coding [18,19] and coordinate independent quantum key distribution [20].

For various applications based on multi-OAM modes, efficiently discriminating and separating different OAM modes are of fundamental importance. In history, there are many methods for this target. For example, a hologram grating and a single mode fiber can be used as a mode detector for a specific OAM mode [15,h21]. In this case, the mode detector is a projector, we have to use N projection measurements to measure N OAM modes. One disadvantage of this way is that the original state is destroyed completely after measurement, besides, this way is not efficient. A more efficient method, which uses the wave front transformation from Cartesian to log-polar coordinate, is present in [22]. By this way, the azimuthal phase profile of an OAM mode is mapped to a tilted plane wavefront. As a result, different OAM modes are mapped to different transverse tilts being proportional to their topological charges in the image plane. One disadvantage is that the separation efficiency is limited to the theoretical limit of 77%. In [23], an improved scheme is proposed, which uses the multi-copy of the unitary transformed refractive beams. By which the separation efficiency is increased to be 97%. In addition to these methods, another method for separating OAM modes using the spin-orbital coupling effect is proposed, which is based on a Mach-Zehdner interferometer with Dove prisms in each arm [24–28]. This method can separate different OAM modes according to their parity, the separation efficiency can reach 100% in principle. N OAM modes can also be separated by cascading N-1 interferometers. However, the difficulty in designing and aligning such a system limits its applications. Later on, in [29], M. Lavery and colleagues reported an improved version of the interference method as reported in [24], which has advantages of robust and adjustment-free.

In this letter, we propose and experimentally demonstrate a different OAM modes splitting scheme by introducing Dove prisms inside a ring cavity. The rotational symmetry for two OAM modes with opposite topological charges is broken by relatively rotating the prisms’ axes, leading to non-degenerate of the cavity’s transmission spectrum. One mode being resonant in cavity transmits, while the other mode is reflected. This scheme has several advantages: 1. Two modes are split non-destructively, therefore the splitter acts like a polarized beam splitter. 2. Our method is able to split both symmetry and asymmetry superposition modes. 3. The transmitting mode can be selected at your will. 4. Beam carries multi-OAMs can also be separated by cascading cavities. In the following, we firstly give a numerical simulation of the performance of this device with cavity loss and impedance mismatch, then we show proof of principle experiments to give a glance at the performance of our device.

2. Theoretical description and simulations

For a typical ring cavity, OAM modes are also eigenmodes of the cavity, the frequency of a certain OAM mode being resonant with cavity which contains two Dove prisms can be expressed as [30]

For the transmittance and reflectance parameters shown in Fig. 1(a), the transmission and reflection coefficients of the ring cavity can be expressed as following:

 

Fig. 1 Non-degenerate cavity configuration and transmission spectra. (a) Ring cavity with two Dove prisms; (b) transmission spectra for two OAM modes with opposite topological charges, here we assume perfect mode and impedance matching of the cavity. M1-M4: mirrors; DP1-DP2: Dove prisms; R1,R4: reflectance of mirrors M1 and M4; T: internal cavity transmittance excludes losses of mirrors M1 and M4.

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Where$R_1$,$R_4$ and $T_1$,$T_4$are the transmittance and reflectance of the cavity mirrors M1 and M4 respectively; $T$represents the internal cavity losses including the loss of the Dove prisms and loss from other cavity mirrors;$\delta \text{=}{\delta }_{\text{1}}\text{+}\Delta \Phi$is the phase shift in one round trip, ${\delta }_{\text{1}}\text{=2}\pi \text{c/L}$is the normal phase shift depends on the round trip optical path,$\Delta \Phi$ is arising from the relative rotation of the two Dove prisms, this phase leads to splitting of the transmission spectra of different OAM modes (See Fig. 1(b)). For an impedance matched cavity, the reflection coefficient CR is 0, which means$R_1=R_{4}T$in Eq. (2). For$T_i+R_i=1$, the impedance matching condition can also be expressed as$T_1=T_4+R$.

We first numerically simulate the transmission and reflection coefficients (CT, CR) of the cavity for various cavity loss and impedance mismatch. The simulation results are shown in Figs. 2(a)-2(c). In Fig. 2(a), the internal cavity loss and the transmittance of the output mirror are fixed and the input mirror transmittance is an adjustable parameter. While in Fig. 2(b), the internal loss is an adjustable parameter, the transmittances of the input and output mirrors are fixed. We can see that CR reaches minimum but CT reaches the maximum for impedance matching case in Figs. 2(a) and 2(b), CT decreases while CR increases with the increase of impedance mismatching. In Fig. 2(c), we show the transmission coefficient for various cavity losses at the condition of impedance matching$T_1=T_4+R$. In this situation, CR keeps zero and CT decreases with the increase of internal loss. In Fig. 2(d), we simulate the transmission and reflection mode shapes for various phase shifts $\Delta \Phi$for input mode$1/\sqrt{2}(|1〉+|-1〉$ when the cavity is on resonance with mode$|1〉$. where we define$|l〉=L{G}_0^l$, here we use quantum mechanical language to represent different OAM modes for simplicity. When $\Delta \Phi$ increases, the transmission spectra of modes$|1〉$ and$|-1〉$ become splitting from the degenerate states to non-degenerate states. When the transmission spectra of two modes are totally separated, the transmitting mode will only contain the mode which is resonant with cavity, the other mode is totally reflected. The reflection coefficient of mode $|-1〉$ increases from 0 to 1. In Fig. 2(e), we simulate the reflection and transmission spatial shapes for various matching coefficient M ranging from 0 to 0.995 for input mode$1/\sqrt{2}(|1〉+|-1〉$. For M = 0, both modes are totally reflected, there is no transmission of modes from the cavity; while for nearly perfect mode matching M = 0.995, one of the mode is totally reflected and the other mode transmits completely.

 

Fig. 2 Simulation results for various cavity parameters. (a)CR and CT of the cavity as function of reflectance R₁, R₄ = 0.95, T = 0.98; (b) CR and CT of the cavity as function of internal transmittance T, R₁ = R₄ = 0.95; (c) CT as function of transmittance T at the condition of impedance matching, R₄ = 0.95 in the calculation; (d) transmission and reflection spatial shapes of the cavity for various phase shift between modes $|1〉$ and$|-1〉$for input mode$1/\sqrt{2}(|1〉+|-1〉$, R₁ = R₄ = 0.90, T = 1 in the calculation;(e) transmission and reflection spatial shapes of the cavity for different mode matching efficiency M ranges from 0.995 to 0, R₁ = R₄ = 0.90, T = 1 in the simulations.

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3. Experimental results

Next, we perform proof of principle experiments for a few superposition modes. The experimental setup is depicted in Fig. 3. The 795 nm laser is from a continuous wave Ti: Sapphire laser (Coherent, MBR110), the polarization of the beam is controlled with quarter and half wave plates (QWP, HWP) before it is transformed to an OAM-carried beam by a spatial light modulator (SLM); the OAM-carried beam is mode-matched to the ring cavity using lenses L1 and L2. The ring cavity consists of mirrors M1-M4. Two Dove prisms (DP1, DP2) are placed between planar mirrors M1 and M2, a piezo-transducer (PZT) is attached to mirror M2 for scanning and locking the cavity. Two concave mirrors M3 and M4 have radius of 80 mm. The input coupling mirror M1 has 85% reflectance at 795 nm, the output coupler mirror M4 has 95% reflectance at 795 nm, mirrors M2 and M3 are high reflection (R>99.9%) coated at 795 nm. The total length of the cavity is 600 mm. The transmitting beam from the cavity is imaged using CCD camera.

 

Fig. 3 Experimental setup of our experiments. QWP(HWP): quarter (half) wave plate; M1-M7:mirrors; L1-L2:lenses; SLM: spatial light modulator; DP1, DP2: Dove prisms; PZT: piezo-transducer; CCD; PD: photodiode; charge coupler device camera.

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The experimental and simulation results for different input modes are shown in Fig. 4. Figure 4(a)-4(d) are the transmission spectra for input modes of $|0〉+|1〉$, $|1〉+|2〉$, $|-1〉+|1〉$and$|-2〉+|2〉$, respectively. A triangle signal is applied to the PZT for scanning the cavity. The transmission spectra are detected using fast photodiode (PD) and acquired by an oscilloscope. For high topological charge OAM modes, there will be more irrelevant higher cavity modes in the transmission spectra arising from the input mode impurity and resonance of other set of modes. The internal cavity loss for higher OAM mode is also higher as a result of larger diffraction and scattering losses, which leads to lower transmission efficiency. In our experiments, the internal cavity transmittance T = 0.90, the total mode and impedance matching coefficient is about 0.80 to 0.90, which means there are about 0.10 to 0.20 ratio of the resonance mode is reflected. In our experiment, the cavity is locked to one of input mode using Hasch-Coillaud (HC) method [31], the reflected and transmitted spatial shapes acquired using CCD camera are shown in left group of images in Fig. 4(e). We find that the transmitting beam has a well distinguished donut shape and high mode purity, while the reflected beam is distorted because of partially reflected resonant modes arising from mode mismatching (the mode matching coefficient is the overlap of the input mode with the cavity’s eigen-mode). To increasing the mode-impedance matching coefficient, the mode purity of the reflected beam will be improved. Another reason for poor reflected beam shape is the impurity of the input mode, as our SLM act as a reflecting mirror, there is the un-modulated part in the input beam, which is reflected from the cavity, contaminating the reflected modes. The purity of the input beam can be improved by adding a tilted phase front of the SLM [32], we will using this technique in our future researches. Right group of images are the corresponding theoretical simulation results based on experimental parameters.

 

Fig. 4 Experimental results for different modes input. (a)-(d) are transmission spectral for different input modes; (e) left group of images shows the intensity distribution for input mode, reflected mode and transmitted mode, right group of images are the corresponding numerical simulation images, the cavity is resonance for modes $|1〉$,$|2〉$,$|1〉$ and $|2〉$ for the four input superposition mode respectively.

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4. Discussions and conclusion

In the present experiments, the internal cavity loss is about 10%, which is mainly from surface reflections of two Dove prisms. The transmission efficiency of the resonance mode is not very high. We are also prevented from testing higher topological charges because of higher modes losses and poor surface quality of the cavity mirrors, but these problems can be solved with the state-of –art mirror fabrication and coating technology. Another optimizing method is to minimize the size of the device and encapsulate it in a confined environment, then the cavity does not need active locking. Our method will be of great promising in chip scale realization by using integrated waveguide technology [33], then the present method will be compatible with telecommunication networks. If 95% transmission efficiency of the resonance mode is reached in the future, such device acts like a polarized beam splitter, which will be of great importance in many OAM-carried light based applications especially in optical communications [15–20], quantum simulations [34–36].

By comparison with the work of J. Leach et al in [24], the key point of the two works is to induce a phase shift for different OAM beam by using Dove prisms, the differences are arising from the operation principle of a MZ interferometer and a cavity. Both J. Leach’s and our device have advantage that OAM modes do not been destroyed after they are separated, another common feature between J. Leach’s and our device is that by cascading the basic device, one can achieve an OAM modes sorter for multi-OAM carried beams. The distinguished property of our device is that the transmitting mode can be selected at your will by locking the cavity to the mode on your demand, we can filter out a specific mode for a beam carrying multi-OAM modes by using a single cavity, but Leach’s method need multi-interferometers. Our device is very sensitive to wavelength changes, but Leach’s method is not sensitive to wavelength. Both disadvantage of the present work and J. Leach’s work is the difficulty in align multi-cavities or MZ interferometers, but this problem can be well solved with integrated fabrication technology. The improved version of J. Leach’s method is demonstrated in [29], which has advantages of robust and adjustment-free.

In conclusion, we propose and experimentally verify a promising splitter for OAM-carried light based on spatial symmetry broken in a ring cavity. The symmetry broken is caused by two Dove prisms, which leads to non-degenerate cavity modes for two OAM-carried beams with opposite topological charges. Both theoretical simulations and experimental verification clearly demonstrate its workability. We believe that our device will be feasible to broadly used in both classical and quantum regime for OAM-carried beam based optical communications and quantum information processing.