- Email: [email protected]
Design of the multiplexing communication system with non-coherent vortex beams
Optics Communications 378 (2016) 5–9
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design of the multiplexing communication system with non-coherent vortex beams Hongdong Zhao n, Xiaocan Peng, Li Ma, Mei Sun School of Electronic and Information Engineering, Hebei University of Technology, Tianjin 300401, China
art ic l e i nf o
a b s t r a c t
Article history: Received 2 March 2016 Received in revised form 4 May 2016 Accepted 6 May 2016 Available online 25 May 2016
In order to enlarge the communication capability, a model of the multiplexing communication system with non-coherent vortex beams is established. One detector for measurement the signal of the vortex beam with topological charge of 0, which is a Gaussian beam, is located in the center of the cross sectional plane of vortex beam. The other three detectors are set around the first detector in the same plane to receive the power of the vortex beam with topological charge of 1. The principle of determining the emitting power of vortex beams, the radii and the positions of the detectors are suggested to increase the signals and reduce the interchannel crosstalk noise at the detectors. The signal powers as well as the interchannel crosstalk noise in a receiver channel are identical to that in another channel, respectively. This research may have applications in free space optical communications. & 2016 Elsevier B.V. All rights reserved.
Keywords: Vortex beam Communication system Multiplexing Interchannel crosstalk
1. Introduction Vortex beams have attracted considerable attention because of their potential applications and the generation of vortex beams have been reported. The vortex beams may be generated by the coherent-superposition of multi beams, a spatial filtering method and a self-Raman laser with an off axis pumping scheme [1–6]. On the other hand, free space optical communication systems with using light propagating in free space to transmit data between two points are rapidly gaining popularity for its advantages such as large capacity, high security, low cost and without frequency applications [7–10], there are growing interests in the propagation and detection of vortex beam. The intensity distributions of off axial partially coherent vortex beam were numerically studied [11]. The properties of a vortex beam through the diffraction gratings and a high numerical-aperture lens can be changed [12– 14]. The intensity distribution of a vortex beam passing through a paraxial ABCD optical system was calculated [15]. The analytical formulae for the average intensity of the propagation vortex beams in turbulent atmosphere were also derived [16–19]. The changes in the spectral degree of polarization as a vortex beam propagating through a misaligned optical system with aperture were given [20] and the intensity evolution of the optical vortex beam on a hard-edge screen was diffracted [21].
Intensity and phase distributions of vortex beam were studied in its generation and the propagation, which does not meet the requirements the communication system with vortex beam. The bit error will be introduced at the receiver, when the vortex beam is distorted in the propagation medium, however, it can be shown to be insignificant over shorter ranges, considered in this paper. Most of bit errors come from the principles of communication system when the vortex beam can keep its shape in the propagation distance of free space optical communication. The detectors receive not only the signal power in its channels, but also the crosstalk noise from the other channels. The incident signal power at the detector is only a part of vortex beam energy as the detectors of one channel can not cover the whole cross sectional plane. Increasing the ratio of signal power to crosstalk noise at the detector can enlarge the communication capability. In this paper, a novel multiplexing communication system with non-coherent vortex beams is proposed, which can fit point-to-point multiplexing optical communications. The rest of the paper is organized as follows: Section 2 gives a theoretical model of a multiplexing communication system with non-coherent vortex beams. Section 3 provides the numerical simulations and analyses. Section 4 draws conclusions of this paper.
2. Theoretical model n
Corresponding author. E-mail address: [email protected] (H. Zhao).
http://dx.doi.org/10.1016/j.optcom.2016.05.012 0030-4018/& 2016 Elsevier B.V. All rights reserved.
The fields of the complete non-coherent vortex beam with the
6
H. Zhao et al. / Optics Communications 378 (2016) 5–9
0.7
0.3
I
2 0 (a.u)
0.4
I
0.5
2 0 /a.u.
………… m=0 m=1 _______ Superposition
0.6
0.2
x/
0.1 0
0
1
2
3
4
r/
5
6
7
0
Fig. 1. Intensity distributions of vortex beam m¼ 0,1 with the same power and their superposition.
topological charge m can be expressed as [13]
Em + 1 (r , φ) =
⎛ r2 ⎞ ⎛ 2 ⎞1/2 1 ⎛ r 2 ⎞ m Pm ⎜ ⎜ ⎟ exp ⎜− 2 ⎟ exp (imφ) ⎟ ⎝ π m ! ⎠ σm ⎝ σm ⎠ ⎝ σm ⎠
(1)
where Pm is the total power with the topological charge m, respectively, φ is the azimuth angle and i is the imaginary unit, sm is the beam width of vortex beam in the center of the cross section plane of vortex beam propagation, (r, φ) is the polar coordinate at the cross sectional plane of vortex beams. Here we take the coaxial linearly superposition vortex beam as the double vortex beam model. The fields of vortex beams with the topological charge m ¼0,1, generated from the two lasers, are given as
P0
⎛ r2 ⎞ 1 ⎛⎜ 2 ⎞⎟1/2 exp ⎜− 2 ⎟ exp (imφ) σ0 ⎝ π ⎠ ⎝ σ0 ⎠
(2)
E2 (r , φ) =
P1
⎛ r2 ⎞ 2 ⎛ 1 ⎞1/2 ⎛ r ⎞ ⎜ ⎟ ⎜ ⎟ exp ⎜− 2 ⎟ exp (imφ) ⎝ ⎠ ⎝ σ1 ⎠ σ1 π ⎝ σ1 ⎠
(3)
where P0 and P1 are the total powers of vortex beam with the topological charge m ¼0,1, s0 and s1 are the beam widths. In the case of m ¼0, the vortex beam reduces to a Gaussian beam with some amplitude. The emitting power distributions of vortex beam with the topological charge m ¼0,1 are given as
I1 = E1* (r , φ) E1 (r , φ)
(4)
I2 = E2* (r , φ) E2 (r , φ)
(5)
The total powers of coaxial non-coherent beam, mixed by two vortex beams with the topological charge m ¼ 0,1, can be written as
It = I1 + I2
(6)
The non-coherent communication is the preferred methodology of commercial implementation in practice, as more detectors are provided to measure the orbital angular momentum of vortex beam based on holographic separation of vortex beams. In our proposed design, the communication system consists of only four detectors, which are separated into two groups to measure the signal in the channels. It is obvious that high intensity region of I1 and the dark core of I2 are on axial propagation. One detector is located in the center of the cross sectional plane as the first group detector to measure the Gaussian beam, which is the vortex beam with the topological charge m ¼ 0. The other three detectors as the second group, which are put in the apexes of an equilateral triangle, are set around the first the detector with the distance of r0
y/
I
x/
0
y/
0
0
Fig. 3. Intensity distributions of vortex beam with topological charge of 0 at the detectors.
0
2 0 /a.u.
E1 (r , φ) =
y/
0
x/
0
Fig. 2. Intensity distributions of vortex beam with topological charge of 0 from (a) three dimensions and (b) top view.
7
I
2 0
y/
0
H. Zhao et al. / Optics Communications 378 (2016) 5–9
x/
0
y/
0
x/
0
I
I
2 0 /a.u.
2 0 /a.u.
Fig. 4. Intensity distributions of vortex beam with topological charge of 1 from (a) three dimensions and (b) top view.
x/
y/
0
x/
0
y/
0
0
Fig. 5. Intensity distributions of vortex beam with topological charge of 1 at the detectors.
Fig. 7. Intensity distributions of vortex beam with topological charge of 0 and 1 respectively at the detectors.
to receive the vortex beam with topological charge m ¼ 1. The signal power Si in the i-th channel, together with the noise power Ni from the other channel, is also received at the detectors (i¼ 1,2). The output power PDi at the i-th group detector can be written as
detector in the cross section plane, the signal power Si and the noise power Ni (i¼ 1,2) can be expressed as:
(7)
PDi = Si + Ni
Si = ki
Ri
∫0 ∫0
2π
Ii ρ sin θdρdθ
(8)
I
y/
0
2 0 /a.u.
As the received power is integrated the intensity over the
x/
0
y/
0
x/
0
Fig. 6. Total power intensity distributions of coaxial non-coherent beam mixed by two vortex beams with the topological charge of 0 and 1 from (a) three dimensions and (b) top view.
8
Ni = ki
H. Zhao et al. / Optics Communications 378 (2016) 5–9
Ri
∫0 ∫0
2π
I j ρ sin θdρdθ
(9)
where i, j = 1, 2, i ≠ j , k i are the number of detectors in the group, here k1 = 1 and k2 = 3, (ρ,θ) are the polar coordinate at the detector in the cross section plane, Ri is radii of the i-th detector. In a multiplexing communication, the signal is improved and at the same time interchannel crosstalk noise is suppressed. The signal power and the interchannel crosstalk in the one channel should be the same values as that in the other channel, respectively. The detector can not be overlapped in the cross section plane and the following problem should be optimized
⎧max (Si ) ⎪ min (Ni ) ⎪ ⎪ ⎨ S1 = S2 , ⎪ ⎪ N1 = N2 ⎪ ⎩ Si ≫ Ni
(10)
where i ¼1,2. As the key feature on the free space optical communication system is that the incident optical power is converted into a maximum photocurrent, r0 is selected as the position of the maximum value of I2. The main flow can be described as follows: (1) Get an initial R1, R2 and P1 (2) Calculate Si and Ni from Eqs. (2)–(9) (3) Determine whether R1, R2 and P1 are accepted a) If satisfy Eq. (10) the state corresponding to the global maximum b) Si < Nj , then reduce R1, R2 or P1 go to step (2) c) Si > Ni , S1 > S2, N1 > N2, then reduce R1 or enlarge R2 go to step (2) d) Si > Ni , S1 < S2, N1 < N2, then enlarge R1 and P1 or reduce R2 go to step (2) e) Si > Ni , S1 > S2, N1 < N2, then enlarge P1 go to step (2) f) Si > Ni , S1 < S2, N1 > N2, then reduce P1 go to step (2) The iteration will be ended if reaches the ending standard, otherwise, update R1, R2 and P1 then go back to step (3). In the following Section, we will give some numerical calculation results to optimize.
described in Section 2. The intensity distributions at detectors is shown in Fig. 3 for the emitting vortex beam with topological charge m ¼ 0 in Fig. 2. The maximum intensity of the vortex beam with m ¼0 is on the beam axis and the intensity will decrease with increasing the distances off the beam axis. As the part energy of vortex beam can be received only within the detector along the ways of vortex beam transmitting, the detected energy is limited by the size of the detector. The vortex beam with m ¼0 also induces the interchannel crosstalk N2 in the second group detector. Our simulation results show that the signal power S1 in the first group detector is 0.51P0 and the interchannel crosstalk N2 in the second group detector is 0.06P0, as the result of the emitting vortex beam with topological charge m ¼ 0 in the communication system, in which the distance between the first group detector and the second detector r0 is 1.73s0, the radii of the two group detector R1 and R2 are 0.60s0 and 0.71s0, respectively. As the maximum intensity with m ¼1 is off the beam axis, the intensity distributions of vortex beam with m ¼1 are in a larger space than the Gaussian beam, which is a vortex beam with m ¼0 (see Fig. 2). When the emitting power of the vortex beam with m ¼1 is improved to 2.25P0, S2 ¼S1 and N1 ¼ N2 can be realized in the communication system. The emitting intensity distributions of vortex beam with m ¼1 are shown in Fig. 4 and its distributions at the detectors are illustrated in Fig. 5. After studying only one vortex beam to be received, we also display the intensity distribution at detector by two vortex beams with m ¼0,1 (see Figs. 6 and 7). The emitting intensity is improved as the mixtures of two non-coherent vortex beams are turned on. As the intensity distributions of vortex beam shown in Fig. 5 is the sum of the beam in Figs. 2 and 4, the received power in each group detector is 0.57P0 including the signal and the interchannel crosstalk noise. The data bits {0, 1} is emitted in the multiplexing communication system with the vortex beam m ¼0 turned on and vortex beam of m ¼1 turned off in Fig. 2. The detected signal data bits {0, 1} is showed in the Fig. 3. The signal data bits {1, 0} and {1, 1} in the multiplexing communication system are also described from the Figs. 4 to 7. From the above mentioned results, it can be seen that data bits {0, 1}, {1, 0} and {1, 1} can be decoded by the generally the optimum threshold level of 0.285P0 in Figs. 3, 5 and 7, respectively.
3. Numerical results and analysis 4. Conclusion Generally, the distributions of coaxial non-coherent vortex beam with m ¼0,1 overlapped in the cross section plane. It is beneficial to improve the ratio of the signal to noise with increasing the difference between the beam width s0 and s1, however the vortex beam with m ¼1 will extend the large range in the cross sectional plane with increasing the beam width s1. A large number of detectors are required to receive the most power of vortex beam and the communication capability in the cross sectional plane is also limited. In numerical simulations, we select the beam widths of vortex beam s1 ¼ 2s0. The intensity distributions of vortex beam with m¼ 0,1, which are used to design the multiplexing communication system, are shown in Fig. 1. Although there is the same total emitting power P0 for two vortex beams, the maximum intensity of vortex beams with m ¼ 1 at r 40 is weaker than that of vortex beams with m ¼0 at r ¼ 0 (see Fig. 1). To find the optimal parameters of the multiplexing communication system with non-coherent vortex beams in practice, the S1 and N2 from the vortex beam with m ¼ 0 (the Gaussian beam) and the S2 and N1 from the vortex beam with m ¼ 1 are calculated, respectively. The R1, R2 and P1 are obtained as the processing
In conclusion, we have designed a binary multiplexing communication system with non-coherent vortex beam. The numerical results show the emitting power is P0 for the Gaussian beam with waist radius of s0, which is the vortex beam with topological charge of 0. The emitting power P1 is improved to 2.25P0 for the vortex beam with topological charge of 1 and the beam waist radius of 2s0. The radius of the center detector is 0.60s0, the radii of the other three detectors around the center at a distance of 1.73s0 are 0.71s0, respectively. The signal power of 0.51P0 is received and the interchannel crosstalk noise is 0.06P0. The balances between signal powers as well as the interchannel crosstalks in the two receiver channels are achieved. This research examination of a binary is beneficial to the free space density communication of vortex beam.
Acknowledgments The research was supported by the Natural Science Foundation of Hebei Province, China (Grant no. F2013202256).
H. Zhao et al. / Optics Communications 378 (2016) 5–9
References [1] L.G. Wang, L.Q. Wang, S.Y. Zhu, Formation of optical vortices using coherent laser beam arrays, Opt. Commun. 282 (2009) 1088–1094. [2] S.J. Huang, Z. Miao, C. He, F.F. Pang, Y.H. Li, T.Y. Wang, Composite vortex beams by coaxial superposition of Laguerre–Gaussian beams, Opt. Laser Technol. 78 (2016) 132–139. [3] T. Ohtomo, S.C. Chu, K. Otsuka, Generation of vortex beams from lasers with controlled Hermite- and Ince-Gaussian modes, Opt. Express 16 (2008) 5082–5094. [4] C.S. Guo, Y. Zhang, Y.J. Han, J.P. Ding, H.T. Wang, Generation of optical vortices with arbitrary shape and array via helical phase spatial filtering, Opt. Commun. 259 (2006) 449–454. [5] C.Y. Lee, C.C. Chang, C.Y. Cho, C.Y. Cho, P.H. Tuan, Y.F. Chen, Generation of higher order vortex beams from a YVO4/Nd:YVO4 self-raman laser via off-axis pumping with mode converter, IEEE J. Sel. Top. Quant. 21 (2015) 1600305. [6] J.L. Andrew, O. Takashige, M.P. Helen, Direct generation of a first-Stokes vortex laser beam from a self-Raman laser, Opt. Express 21 (1) (2013) 12401–12408. [7] S. Kazemlou, S. Hranilovic, S. Kumar, All-optical multihop free-space optical communication systems, J. Lightwave Technol. 29 (2011) 2663–2669. [8] T. Shang, J. Jia, X. Wang., Analysis and design of a multi-transceiver optical cylinder antenna for mobile free space optical communication, Opt. Laser Technol. 44 (2012) 2384–2392. [9] W. Liu, W. Shi, B. Wang, K. Yao, Y. Lv, J. Wang, Free space optical communication performance analysis with focal plane based wavefront measurement, Opt. Commun. 309 (2013) 212–220. [10] J. Singh, N. Kumar, Performance analysis of different modulation format on free space optical communication system, Optik 124 (2013) 4651–4654. [11] P.F. Ding, Z.Y. Chen, J.X. Pu, Propagation properties of off axial partially
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coherent vortex beam, Opt. Commun. 357 (2015) 172–176. [12] M. Liu, Measuring the fractional and integral topological charges of the optical vortex beams using a diffraction grating, Optik 126 (2015) 5263–5268. [13] S. Lyubomir, T. Suzana, S. Ivan, J. Ljiljana, D. Alexander, Far field diffraction of an optical vortex beam by a fork-shaped grating, Opt. Commun. 350 (2015) 301–308. [14] L.Z. Rao, J.X. Pu, Z.Y. Chen, P. Yei, Focus is helping of cylindrically polarized vortex beams by a high numerical-aperture lens, Opt. Laser Technol. 41 (2009) 241–246. [15] L.N. Guo, Z.L. Tang, W. Wan, Propagation of a four-petal Gaussian vortex beam through a paraxial ABCD optical system, Optik 125 (2014) 5542–5545. [16] Z.Y. Qin, R.M. Tao, P. Zhou, X.J. Xu, Z.J. Liu, Propagation of partially coherent Bessel–Gaussian beams carrying optical vortices in non-Kolmogorov turbulence, Opt. Laser Technol. 56 (2014) 182–188. [17] D.J. Liu, Y.C. Wang, H.M. Yin, Propagation properties of partially coherent fourpetal Gaussian vortex beams in turbulent atmosphere, Opt. Laser Technol. 78 (2016) 95–100. [18] C. Wen, J.W. Haus, Q.W. Zhan, Propagation of vector vortex beams through a turbulent atmosphere, Opt. Express 17 (2009) 17829–17836. [19] G.J. Ghur, R.K. Tyson, Vortex beam propagation through atmospheric turbulence and topological charge conservation, J. Opt. Soc. Am. A 25 (2008) 225–230. [20] M.L. Luo, D.M. Zhao, Changes in the polarization and coherence of a stochastic electromagnetic vortex beam propagating through a misaligned optical system with aperture, Opt. Commun. 328 (2014) 57–61. [21] A. Ya.Bekshaev, K.A. Mohammed, Spatial profile and singularities of the edgediffracted beam with a multicharged optical vortex, Opt. Commun. 341 (2015) 284–294.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Design of the multiplexing communication system with non-coherent vortex beams Hongdong Zhao n, Xiaocan Peng, Li Ma, Mei Sun School of Electronic and Information Engineering, Hebei University of Technology, Tianjin 300401, China
art ic l e i nf o
a b s t r a c t
Article history: Received 2 March 2016 Received in revised form 4 May 2016 Accepted 6 May 2016 Available online 25 May 2016
In order to enlarge the communication capability, a model of the multiplexing communication system with non-coherent vortex beams is established. One detector for measurement the signal of the vortex beam with topological charge of 0, which is a Gaussian beam, is located in the center of the cross sectional plane of vortex beam. The other three detectors are set around the first detector in the same plane to receive the power of the vortex beam with topological charge of 1. The principle of determining the emitting power of vortex beams, the radii and the positions of the detectors are suggested to increase the signals and reduce the interchannel crosstalk noise at the detectors. The signal powers as well as the interchannel crosstalk noise in a receiver channel are identical to that in another channel, respectively. This research may have applications in free space optical communications. & 2016 Elsevier B.V. All rights reserved.
Keywords: Vortex beam Communication system Multiplexing Interchannel crosstalk
1. Introduction Vortex beams have attracted considerable attention because of their potential applications and the generation of vortex beams have been reported. The vortex beams may be generated by the coherent-superposition of multi beams, a spatial filtering method and a self-Raman laser with an off axis pumping scheme [1–6]. On the other hand, free space optical communication systems with using light propagating in free space to transmit data between two points are rapidly gaining popularity for its advantages such as large capacity, high security, low cost and without frequency applications [7–10], there are growing interests in the propagation and detection of vortex beam. The intensity distributions of off axial partially coherent vortex beam were numerically studied [11]. The properties of a vortex beam through the diffraction gratings and a high numerical-aperture lens can be changed [12– 14]. The intensity distribution of a vortex beam passing through a paraxial ABCD optical system was calculated [15]. The analytical formulae for the average intensity of the propagation vortex beams in turbulent atmosphere were also derived [16–19]. The changes in the spectral degree of polarization as a vortex beam propagating through a misaligned optical system with aperture were given [20] and the intensity evolution of the optical vortex beam on a hard-edge screen was diffracted [21].
Intensity and phase distributions of vortex beam were studied in its generation and the propagation, which does not meet the requirements the communication system with vortex beam. The bit error will be introduced at the receiver, when the vortex beam is distorted in the propagation medium, however, it can be shown to be insignificant over shorter ranges, considered in this paper. Most of bit errors come from the principles of communication system when the vortex beam can keep its shape in the propagation distance of free space optical communication. The detectors receive not only the signal power in its channels, but also the crosstalk noise from the other channels. The incident signal power at the detector is only a part of vortex beam energy as the detectors of one channel can not cover the whole cross sectional plane. Increasing the ratio of signal power to crosstalk noise at the detector can enlarge the communication capability. In this paper, a novel multiplexing communication system with non-coherent vortex beams is proposed, which can fit point-to-point multiplexing optical communications. The rest of the paper is organized as follows: Section 2 gives a theoretical model of a multiplexing communication system with non-coherent vortex beams. Section 3 provides the numerical simulations and analyses. Section 4 draws conclusions of this paper.
2. Theoretical model n
Corresponding author. E-mail address: [email protected] (H. Zhao).
http://dx.doi.org/10.1016/j.optcom.2016.05.012 0030-4018/& 2016 Elsevier B.V. All rights reserved.
The fields of the complete non-coherent vortex beam with the
6
H. Zhao et al. / Optics Communications 378 (2016) 5–9
0.7
0.3
I
2 0 (a.u)
0.4
I
0.5
2 0 /a.u.
………… m=0 m=1 _______ Superposition
0.6
0.2
x/
0.1 0
0
1
2
3
4
r/
5
6
7
0
Fig. 1. Intensity distributions of vortex beam m¼ 0,1 with the same power and their superposition.
topological charge m can be expressed as [13]
Em + 1 (r , φ) =
⎛ r2 ⎞ ⎛ 2 ⎞1/2 1 ⎛ r 2 ⎞ m Pm ⎜ ⎜ ⎟ exp ⎜− 2 ⎟ exp (imφ) ⎟ ⎝ π m ! ⎠ σm ⎝ σm ⎠ ⎝ σm ⎠
(1)
where Pm is the total power with the topological charge m, respectively, φ is the azimuth angle and i is the imaginary unit, sm is the beam width of vortex beam in the center of the cross section plane of vortex beam propagation, (r, φ) is the polar coordinate at the cross sectional plane of vortex beams. Here we take the coaxial linearly superposition vortex beam as the double vortex beam model. The fields of vortex beams with the topological charge m ¼0,1, generated from the two lasers, are given as
P0
⎛ r2 ⎞ 1 ⎛⎜ 2 ⎞⎟1/2 exp ⎜− 2 ⎟ exp (imφ) σ0 ⎝ π ⎠ ⎝ σ0 ⎠
(2)
E2 (r , φ) =
P1
⎛ r2 ⎞ 2 ⎛ 1 ⎞1/2 ⎛ r ⎞ ⎜ ⎟ ⎜ ⎟ exp ⎜− 2 ⎟ exp (imφ) ⎝ ⎠ ⎝ σ1 ⎠ σ1 π ⎝ σ1 ⎠
(3)
where P0 and P1 are the total powers of vortex beam with the topological charge m ¼0,1, s0 and s1 are the beam widths. In the case of m ¼0, the vortex beam reduces to a Gaussian beam with some amplitude. The emitting power distributions of vortex beam with the topological charge m ¼0,1 are given as
I1 = E1* (r , φ) E1 (r , φ)
(4)
I2 = E2* (r , φ) E2 (r , φ)
(5)
The total powers of coaxial non-coherent beam, mixed by two vortex beams with the topological charge m ¼ 0,1, can be written as
It = I1 + I2
(6)
The non-coherent communication is the preferred methodology of commercial implementation in practice, as more detectors are provided to measure the orbital angular momentum of vortex beam based on holographic separation of vortex beams. In our proposed design, the communication system consists of only four detectors, which are separated into two groups to measure the signal in the channels. It is obvious that high intensity region of I1 and the dark core of I2 are on axial propagation. One detector is located in the center of the cross sectional plane as the first group detector to measure the Gaussian beam, which is the vortex beam with the topological charge m ¼ 0. The other three detectors as the second group, which are put in the apexes of an equilateral triangle, are set around the first the detector with the distance of r0
y/
I
x/
0
y/
0
0
Fig. 3. Intensity distributions of vortex beam with topological charge of 0 at the detectors.
0
2 0 /a.u.
E1 (r , φ) =
y/
0
x/
0
Fig. 2. Intensity distributions of vortex beam with topological charge of 0 from (a) three dimensions and (b) top view.
7
I
2 0
y/
0
H. Zhao et al. / Optics Communications 378 (2016) 5–9
x/
0
y/
0
x/
0
I
I
2 0 /a.u.
2 0 /a.u.
Fig. 4. Intensity distributions of vortex beam with topological charge of 1 from (a) three dimensions and (b) top view.
x/
y/
0
x/
0
y/
0
0
Fig. 5. Intensity distributions of vortex beam with topological charge of 1 at the detectors.
Fig. 7. Intensity distributions of vortex beam with topological charge of 0 and 1 respectively at the detectors.
to receive the vortex beam with topological charge m ¼ 1. The signal power Si in the i-th channel, together with the noise power Ni from the other channel, is also received at the detectors (i¼ 1,2). The output power PDi at the i-th group detector can be written as
detector in the cross section plane, the signal power Si and the noise power Ni (i¼ 1,2) can be expressed as:
(7)
PDi = Si + Ni
Si = ki
Ri
∫0 ∫0
2π
Ii ρ sin θdρdθ
(8)
I
y/
0
2 0 /a.u.
As the received power is integrated the intensity over the
x/
0
y/
0
x/
0
Fig. 6. Total power intensity distributions of coaxial non-coherent beam mixed by two vortex beams with the topological charge of 0 and 1 from (a) three dimensions and (b) top view.
8
Ni = ki
H. Zhao et al. / Optics Communications 378 (2016) 5–9
Ri
∫0 ∫0
2π
I j ρ sin θdρdθ
(9)
where i, j = 1, 2, i ≠ j , k i are the number of detectors in the group, here k1 = 1 and k2 = 3, (ρ,θ) are the polar coordinate at the detector in the cross section plane, Ri is radii of the i-th detector. In a multiplexing communication, the signal is improved and at the same time interchannel crosstalk noise is suppressed. The signal power and the interchannel crosstalk in the one channel should be the same values as that in the other channel, respectively. The detector can not be overlapped in the cross section plane and the following problem should be optimized
⎧max (Si ) ⎪ min (Ni ) ⎪ ⎪ ⎨ S1 = S2 , ⎪ ⎪ N1 = N2 ⎪ ⎩ Si ≫ Ni
(10)
where i ¼1,2. As the key feature on the free space optical communication system is that the incident optical power is converted into a maximum photocurrent, r0 is selected as the position of the maximum value of I2. The main flow can be described as follows: (1) Get an initial R1, R2 and P1 (2) Calculate Si and Ni from Eqs. (2)–(9) (3) Determine whether R1, R2 and P1 are accepted a) If satisfy Eq. (10) the state corresponding to the global maximum b) Si < Nj , then reduce R1, R2 or P1 go to step (2) c) Si > Ni , S1 > S2, N1 > N2, then reduce R1 or enlarge R2 go to step (2) d) Si > Ni , S1 < S2, N1 < N2, then enlarge R1 and P1 or reduce R2 go to step (2) e) Si > Ni , S1 > S2, N1 < N2, then enlarge P1 go to step (2) f) Si > Ni , S1 < S2, N1 > N2, then reduce P1 go to step (2) The iteration will be ended if reaches the ending standard, otherwise, update R1, R2 and P1 then go back to step (3). In the following Section, we will give some numerical calculation results to optimize.
described in Section 2. The intensity distributions at detectors is shown in Fig. 3 for the emitting vortex beam with topological charge m ¼ 0 in Fig. 2. The maximum intensity of the vortex beam with m ¼0 is on the beam axis and the intensity will decrease with increasing the distances off the beam axis. As the part energy of vortex beam can be received only within the detector along the ways of vortex beam transmitting, the detected energy is limited by the size of the detector. The vortex beam with m ¼0 also induces the interchannel crosstalk N2 in the second group detector. Our simulation results show that the signal power S1 in the first group detector is 0.51P0 and the interchannel crosstalk N2 in the second group detector is 0.06P0, as the result of the emitting vortex beam with topological charge m ¼ 0 in the communication system, in which the distance between the first group detector and the second detector r0 is 1.73s0, the radii of the two group detector R1 and R2 are 0.60s0 and 0.71s0, respectively. As the maximum intensity with m ¼1 is off the beam axis, the intensity distributions of vortex beam with m ¼1 are in a larger space than the Gaussian beam, which is a vortex beam with m ¼0 (see Fig. 2). When the emitting power of the vortex beam with m ¼1 is improved to 2.25P0, S2 ¼S1 and N1 ¼ N2 can be realized in the communication system. The emitting intensity distributions of vortex beam with m ¼1 are shown in Fig. 4 and its distributions at the detectors are illustrated in Fig. 5. After studying only one vortex beam to be received, we also display the intensity distribution at detector by two vortex beams with m ¼0,1 (see Figs. 6 and 7). The emitting intensity is improved as the mixtures of two non-coherent vortex beams are turned on. As the intensity distributions of vortex beam shown in Fig. 5 is the sum of the beam in Figs. 2 and 4, the received power in each group detector is 0.57P0 including the signal and the interchannel crosstalk noise. The data bits {0, 1} is emitted in the multiplexing communication system with the vortex beam m ¼0 turned on and vortex beam of m ¼1 turned off in Fig. 2. The detected signal data bits {0, 1} is showed in the Fig. 3. The signal data bits {1, 0} and {1, 1} in the multiplexing communication system are also described from the Figs. 4 to 7. From the above mentioned results, it can be seen that data bits {0, 1}, {1, 0} and {1, 1} can be decoded by the generally the optimum threshold level of 0.285P0 in Figs. 3, 5 and 7, respectively.
3. Numerical results and analysis 4. Conclusion Generally, the distributions of coaxial non-coherent vortex beam with m ¼0,1 overlapped in the cross section plane. It is beneficial to improve the ratio of the signal to noise with increasing the difference between the beam width s0 and s1, however the vortex beam with m ¼1 will extend the large range in the cross sectional plane with increasing the beam width s1. A large number of detectors are required to receive the most power of vortex beam and the communication capability in the cross sectional plane is also limited. In numerical simulations, we select the beam widths of vortex beam s1 ¼ 2s0. The intensity distributions of vortex beam with m¼ 0,1, which are used to design the multiplexing communication system, are shown in Fig. 1. Although there is the same total emitting power P0 for two vortex beams, the maximum intensity of vortex beams with m ¼ 1 at r 40 is weaker than that of vortex beams with m ¼0 at r ¼ 0 (see Fig. 1). To find the optimal parameters of the multiplexing communication system with non-coherent vortex beams in practice, the S1 and N2 from the vortex beam with m ¼ 0 (the Gaussian beam) and the S2 and N1 from the vortex beam with m ¼ 1 are calculated, respectively. The R1, R2 and P1 are obtained as the processing
In conclusion, we have designed a binary multiplexing communication system with non-coherent vortex beam. The numerical results show the emitting power is P0 for the Gaussian beam with waist radius of s0, which is the vortex beam with topological charge of 0. The emitting power P1 is improved to 2.25P0 for the vortex beam with topological charge of 1 and the beam waist radius of 2s0. The radius of the center detector is 0.60s0, the radii of the other three detectors around the center at a distance of 1.73s0 are 0.71s0, respectively. The signal power of 0.51P0 is received and the interchannel crosstalk noise is 0.06P0. The balances between signal powers as well as the interchannel crosstalks in the two receiver channels are achieved. This research examination of a binary is beneficial to the free space density communication of vortex beam.
Acknowledgments The research was supported by the Natural Science Foundation of Hebei Province, China (Grant no. F2013202256).
H. Zhao et al. / Optics Communications 378 (2016) 5–9
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