Two-dimension and high-resolution demultiplexing of coaxial multiple orbital angular momentum beams

Jishun Yang; Zhibing Liu; Shecheng Gao; Xincheng Huang; Yuanhua Feng; Weiping Liu; Zhaohui Li

doi:10.1364/OE.27.004338

1. Introduction

Over the past few decades, the rapid growth of applications such as social networking, online video, cloud computing, and the Internet of things has led to a dramatic increasing demand for the capacity in communication networks. The communication capacity of the modern optical fiber-based communication network has increased by almost ten times every four years [1], which is quickly going to approach its capacity limit [2]. It means that the available channel sources supplied by the degrees of freedom [3] of amplitude, time, frequency [4–6] and polarization [7,8] grow increasingly scarce. Therefore, to break through the forthcoming capacity limit in the traditional communication system, developing new channel sources becomes more and more urgent. Recently, a spatial division multiplexing (SDM) technology that takes advantage of the spatial degrees of the photons has been developed [9]. Among them, OAM multiplexing gets more and more attention [10]. In order to effectively multiplex and demultiplex the channels labeled with different OAM states, the combination and separation of OAM beams with high efficiency are of great importance [11]. In other words, it should be able to redirect the propagation direction of the incident beams carrying different OAM to one direction to couple into the transmission system at the input end and separate each of them to different directions at output end.

Various OAM demultiplexing methods are put forward. A method based on integrated photonics devices suffers serious energy loss and complex fabrication of the integrated photonics devices [12]. Other schemes including interferometric methods [13,14] and multi-plane light conversion [15] technique which requires more elements for more OAM states are hard to deal with a great quantity of OAM states. Another more convenient method is by using the Daman optical vortex grating (DOVG) [16], which makes the Gaussian beams incident from certain directions into the same direction or diffracts coaxial multiple OAM beams into different directions. With using the DOVG, the capacity of the free space optical communication system has been upgraded to 160T bit/s [17]. However, the DOVG based demultiplexing technique suffers large energy loss. The energy efficiency of DOVG is $1 / N$ , where $N$ is the number of channels, meaning that the more channels the DOVG used, the lower the energy efficiency will be.

Considering the energy efficiency in the process of OAM demultiplexing, another method based on the coordinate transform/optical geometric transformation (OGT) technique attracts a lot of attention [18,19]. The OGT performs the transformation from Cartesian coordinates to log-polar coordinates. After the OGT, an OAM beam with a helical wavefront $e x p (i l θ)$ becomes a tilted plane wave, with its donut intensity distribution shaped to a rectangle. The normal direction of the tilted plane wave is dependent on the topological charge of OAM, leading to an OAM topological charge dependent displacement on the focal plane after being focused by a convex lens. However, the OGT only works perfectly for the plane wave. For an OAM beam with a helical wavefront, the upper limit for an acceptable performance of OGT should satisfy the condition: $l ≪ 2 π r^{2} / L λ$ [20] ( $l$ is the absolute value of topological charge of the OAM beam, r is its radius, L is the distance between the two devices which complete the OGT, and $λ$ is the wavelength of the light beam). In order to break though the limit and increase spatial channels available, besides the method that only consider the azimuthal structure of OAM beam in the OGT, a radial phase structure for the OAM beam can also be taken into account as another dimension [21]. Based on the OGT method, the OAM beam with radial phase structure gets transformed to the plane wave titled both along horizontal and vertical directions, and accordingly can be discriminated by the position of its focal spot in the focal plane of a lens. Consequently, the space for OAM demultiplexing is enlarged from one dimension to two dimensions. The other important issue is the crosstalk which comes from the inter-channel overlap when the OGT is used as the demultiplexing method. Due to the overlap, the interval of the topological charge of OAM channels has to keep large enough. In other words, many OAM channel sources cannot be exploited when considering the inter-channel overlap that may be expected to be effectively minimized, for example, with the coherent copy technology as in the work [22].

In this paper, firstly, we design a CGH to generate coaxial multiple OAM beams with logarithmic spiral phase structures, which extends the separation of coaxial OAM beams from one dimension to two dimensions. Then, a high-resolution demultiplexing method which is based on the coherently copy and the coordinate transform is put forward to separate OAM beams with lower overlap between adjacent channel. Therefore, the number of available spatial channels free from crosstalk gets increased.

2. Principle

The OAM beam with the logarithmic spiral phase structure can be generated by a CGH shown in Fig. 1 which is encoded with a logarithmic spiral phase [21]:

ϕ (r, φ) = l φ + m 2 π \frac{\ln r}{C},

where

l

characterizes the azimuthal part of the logarithmic spiral phase structure, the integer

m

characterizes the radial part of the logarithmic spiral phase structure, and the constant C is related to the width of the OAM ring. As the state of the generated beam is associated with two parameters

l

and

m

of the logarithmic spiral phase, let |l, m> represent a certain generated beam, and |l₁, m₁; l₂, m₂> represent generated coaxial two optical beams and so on. Taking advantage of the large size screen of the used spatial light modulator (SLM), coaxial multiple OAM beams with different spiral phase structures can be generated by use of the ‘complementary method’ [23]. For example, to produce two beams of |-1, −1; 0, 0>, the modulation region of the SLM is divided into two parts, region 1 and region 2 with complementary transmittance, as shown in Fig. 1. Both the white region (transmittance is 1) and the black region (transmittance is 0) are equally spaced in region 1 and region 2. Subsequently, the phase structure, shown in Eq. (1), consisting of the azimuthal phase and the radial phase fills the region 1 or region 2. In Fig. 1, the region 1 is filled with the phase hologram corresponding to the state |-1, −1> while the region 2 corresponds to the state |0, 0>. Finally, the two regions filled with phase holograms are combined to obtain the final hologram.

Fig. 1 Scheme of the hologram composition to generate coaxial two OAM beams |-1, −1; 0, 0>. (a) The hologram generating the OAM beam with $l$ = −1. (b) The hologram imprinting radial phase ( $m$ = −1) into the OAM beam. (c), (d) The blank holograms for state |0, 0>. (e) The final hologram for generating the OAM beams |-1, −1; 0, 0>.

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In order to efficiently separate coaxial multiple OAM beams with logarithmic spiral phases, for instance, the state |l1, m1> and |l2, m2>, an improved optical transformation (IOT) based on the OGT and a coherent copy technology is proposed. The IOT can be divided into two steps. The first step implemented by a SLM and a lens is to meanwhile perform unwrapping and copying functions. Based on [18] and [24], the required phase loaded on the SLM is expressed as

\begin{matrix} ϕ_{1} (x, y) = \frac{2 π a}{λ f} [y arc \tan (\frac{y}{x}) - x \ln (\frac{\sqrt{x^{2} + y^{2}}}{b}) + x] \\ + arc \tan [\frac{\sum_{m = - M}^{m = + M} γ_{m} \sin (\frac{2 π θ}{λ} m y + α_{m})}{\sum_{m = - M}^{m = + M} γ_{m} \cos (\frac{2 π θ}{λ} m y + α_{m})}] + arc \tan [\frac{\sum_{n = - N}^{n = + N} η_{n} \sin (\frac{2 π Θ}{λ} n x + β_{n})}{\sum_{n = - N}^{n = + N} η_{n} \cos (\frac{2 π Θ}{λ} n x + β_{n})}] . \end{matrix}

where

θ

and

Θ

are the angle of separation between neighboring copies along the vertical and horizontal direction separately,

α_{m} / β_{n}

and

γ_{m} / η_{n}

are relative phases and amplitudes which make sure that every copy has the same amplitude and phase,

a

and

b

are the parameters to determine the size and position of single unwrapped beam,

λ

is the wavelength in vacuum of the laser source, and

f

is the focal length of the lens (L3 in Fig. 3) which implements the first step with a SLM (SLM2 in Fig. 3). The first term on the right side of the above equation represents the ordinary OGT [18] which unwraps the donut-shaped OAM beam into a rectangle shaped beam with an inclined plane phase, and the last two terms stand for the coherent copy [24] which makes

2 M + 1

and

2 N + 1

coherent copies of the coaxial rectangle shaped beams along both vertical and horizontal directions respectively. The coherent copy technology converts an input beam into multiple output beams with the same phase and nearly uniform intensity. Therefore, it can narrow light spots in the focal plane of the Fourier lens (L4 in Fig. 3) according to the Fourier transform theory [25], without changing the spacing between spots, leading to an evident enhancement of the resolution of light spots and therefore reduce the overlap between the adjacent channel.

The second step is to perform the phase correction. To correct the phase of the light field in the back focal plane of the lens (L3 in Fig. 3), a corresponding correction phase which is similar with [26] is added here, which written as:

ϕ_{2} (u, v) = \sum_{m = - M}^{m = + M} \sum_{n = - N}^{n = + N} [\frac{- 2 π a b}{λ f} \exp (- \frac{u - m w}{a}) \cos (\frac{v - 2 π a n}{a}) - α_{m} - β_{n}] r e c t [\frac{u - m w}{w}, \frac{v - 2 π a}{2 π a}],

with

u

and

v

be the Cartesian coordinate of the back focal plane of the lens (L3 in Fig. 3),

w

and

2 π a

be the respective width and length of the rectangle beam. (The parameter values: a = 0.6515mm, b = 0.5mm, w = 1.7mm,f = 150mm, M = N = 1. The value of the

θ

,

Θ

,

α_{m}

,

β_{n}

,

γ_{m}

,

η_{n}

can be found in [24]).

As shown in Fig. 2, coaxial two OAM beams, |l1, m1> and |l2, m2> become coaxial rectangle beams with different inclined plane wave fronts which get copied along both horizontal and vertical directions after the IOT. Then, a Fourier lens focus the inclined plane waves into its focus plane.

Fig. 2 The concept of the IOT. Image c is the resultant light fields of coaxial two OAM beams, |l₁, m₁> and |l₂, m₂>. Image c can be decomposed into image a and b which are the resultant light fields of |l₁, m₁> and |l₂, m₂> after the IOT, respectively.

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3. Experiments and results

The experimental setup is illustrated in Fig. 3. A laser beam operated at the wavelength of 1550nm is directed into a reflective spatial light modulator (SLM1) where the logarithmic phase mask is loaded on to generate the designed coaxial multiple OAM beams (The parameter C = 1.98). The generated coaxial multiple OAM beams, after passing through a 4f system to maintain the phase structure of the beams and an iris placed in between to filter out unwanted light beams, are reflected by a beam splitter(BS1) to another spatial light modulator (SLM2). The incident coaxial beams on SLM2 are modulated with the loaded transformation phase formulated in Eq. (2) and get unwrapped and coherently copied in the back focal plane of the third lens (L3) where the third spatial light modulator (SLM3) is placed. The correction phase is loaded to SLM3 for the correction of the phase distortion introduced by SLM2. After the transformation and phase correction, the OAM beams with different logarithmic spiral wave fronts are mapped into the corresponding tilted plane waves with the tilt angle depending on the topological charge of OAM. Therefore, with a lens (L4), the tilted plane waves are focused to corresponding light spots in the focal plane of L4 which will be captured by a charge coupled device (CCD).

Fig. 3 Experimental setup. HWP: Half-wave plate. SLM: spatial light modulator. L:Lens. BS: beam splitter. CCD: charge coupled device.

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Figure 4 shows the simulation and experimental results of the separation of coaxial two OAM beams. Experimental results are shown in the upper row and simulation results are displayed in the bottom row. The coaxial two OAM beams shown in the first, second, third and fourth column are: |0, 0; 0, 1>; |0, 0; 1, 0>; |0, 0; −1, −1>; |0, 0; −1, 1>, respectively. The position of the focused optical spot is up to the parameters of logarithmic spiral phase, that is, the value of parameter $l$ and $m$ determine the horizontal and vertical position of the spots, respectively. The relative position of the two spots is displayed in Fig. 4.

Fig. 4 Separation results of two coaxial OAM beams. (a) and (e) the state |0, 0; 0, 1>;(b) and (f) the state |0, 0; 1, 0>;(c) and (g) the state |0, 0; −1, −1>;(d) and (h) the state |0, 0; −1, 1>.

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To demonstrate the reliability of the IOT, coaxial three OAM beams’ separation experiments were performed. Simulation and experimental results are shown in Fig. 5. Just as the way above, the form |l₁, m₁; l₂, m₂; l₃, m₃> is used to represent the state of the coaxial three beams. The first column in Fig. 5 shows the result of separation of |0, 0; −1, 1; −1, −1>, and the second column corresponds to |0, 0; 2, 0; 0, −1>. The experimental results are highly consistent with the simulation results. As can be seen from the Fig. 5, the spots are indeed distributed along two directions, indicating that our method can separate the coaxial multiple OAM beams two dimensionally.

Fig. 5 Separation results of coaxial three OAM beams. Experimental results are shown in the upper row and simulation results are displayed in the bottom row. (a) and (c) the state |0, 0; −1, 1; −1, −1>; (b) and (d) the state |0, 0; 2, 0; 0, −1>.

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A comparison between the IOT and the OGT is made. Separation results of the coaxial two OAM beams |0, 0; −1, −1> are shown in Fig. 6. The phase holograms used by the two methods are also shown in Fig. 6. Pictures in the left column are related to the OGT, and those in the right column correspond to the IOT. Figure 6 (a)/(b) shows the unwrapping phase without/with a coherent copy. Correction phases are exhibited in Figs. 6(c) and 6(d). Figure 6(e) exhibits the result using the OGT method and Fig. 6(f) corresponds the IOT. It can be seen from Figs. 6(e) and 6(f) that, with the IOT, the overlap of two separated spots is greatly reduced compared with the separation result using the OGT. The IOT method leads to a less space occupation for every single channel, and accordingly a less overlap between adjacent spot while the spacing between spots remains unchanged.

Fig. 6 Experimental results of the separation with two methods. (a) and (c) the unwrapping phase and correction phase of the OGT, respectively. (b) and (d) the unwrapping phase and correction phase of the IOT, respectively. (e) the separating result with the OGT (f) the separating result with IOT. The axes of the azimuthal and radial parameters are marked in (e) and (f).

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4. Conclusions

In this work, we design a CGH to generate coaxial multiple OAM beams with different logarithmic spiral phases and propose an improved optical transformation (IOT) which combines a coherent copy technique and an optical geometrical transform for the separation of coaxial multiple OAM beams. The coaxial multiple OAM beams possess different logarithm spiral phase structures which have both the azimuthal and radial phase gradient that enable two-dimensional separation of the coaxial multiple OAM beams. We carry out simulations and experiments to confirm the reliability of our method. The simulation and experimental results show that the coaxial multiple OAM beams can be effectively separated in the specific plane, manifesting as optical spots, and their positions are determined by the parameters of the logarithm spiral phase. Besides, the overlap between the adjacent channel is also effectively reduced by the coherent copy technique which is part of the IOT. The proposed method has great potential in improving the capacity of future optical communication and networks.

Funding

National Natural Science Foundation of China (NSFC) (61525502, 61435006, U1701661, 61490710, 61775085, 61875076, 61490715, 61705088); Local Innovation and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X121); the Science and Technology Planning Project of Guangdong Province (2017B010123005, 2018BT010114002).

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Two-dimension and high-resolution demultiplexing of coaxial multiple orbital angular momentum beams

Abstract

1. Introduction

2. Principle

3. Experiments and results

4. Conclusions

Funding

References

Cited By

Figures (6)

Equations (3)

Optics Express