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Abstract
In the paper, we experimentally demonstrate the propagation property of Laguerre-Gaussian (LG) beams carrying fractional orbital angular momentum (FOAM) in an underwater environment. The effects of topological charge (TC), temperature gradient, and salinity on the transmission of the LG-FOAM beam in underwater turbulence are analyzed, and the optimum interval of TC for FOAM communication and their help for the improvement of system capacity are discussed. The results show that both the salinity and the temperature gradient have serious impacts on the beam, and meanwhile, the temperature gradient plays a heavier influence. Although the detection probability of one FOAM mode at the receiver side is lower than that nearest integer OAM mode, in the case of that topological charges used are limited, the interval-based mode-multiplexed communication using FOAM can increase the channel capacity.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
It is now well appreciated that light beams with helical phase fronts would carry an orbital angular momentum (OAM). Light beams with an azimuthal phase structure $\exp (i\ell \phi )$, where $\ell$ is an integer number, carry OAM of $\ell \hbar$ per photon [1], which arises directly from the azimuthal component of their Poynting vector. A common example for such light beams is Laguerre-Gaussian (LG) mode, which can be produced in the laboratory using spiral phase plates or computer generated holograms [2,3].
With a huge increase of underwater applications, such as divers, unmanned underwater vehicles, submarines, and sensors in the oceanic environment [4,5], underwater wireless optical communication (UWOC) has currently received more and more attention. Recently published results concerning UWOC have demonstrated the UWOC system employing OAM has greatly improved the system’s capacity [6,7]. For example, J. Baghdady et al. reported a 3-Gbit/s UWOC system by employing 2 OAM modes multiplexing [6]. Y. Ren et al. further increased the UWOC transmission capacity to 4 Gbit/s (directly modulated laser diode) and 40 Gbit/s (external modulation & frequency doubling) by multiplexing 4 OAM modes [7]. In addition, the propagation properties of the beams carrying OAM in the underwater environment are also studied to understand the quality of the UWOC system using OAM mode effected by oceanic turbulence [8–11]. For instance, Zhao et al. studied the propagation and self-healing properties of Bessel-Gaussian beam carrying orbital angular momentum in an underwater environment [8]. Huang et al. investigated the propagation property of Gaussian Schell-model vortex beams through oceanic turbulence [9]. The authors in [10] investigated the propagation of Airy vortex beam carrying orbital angular momentum passed through the oceanic turbulence, and the authors in [11] discussed the effect of ocean turbulence on the partially coherent LG beam carrying OAM mode.
In 2004, Berry [12] first theoretically analysed that the beams had intricate-phase structures comprising chain of alternating charge vortices along the direction of the initial radial discontinuity after propagation. This chain of vortices affects the intensity and phase profiles of light emerging from a fractional phase step and makes the light unstable on propagation, which was we called fractional orbital angular momentum (FOAM). Leach et al. [13] later observed such non-integer vortex structure in experiment. After that, Oemrawsingh et al. [14] presented the experimental demonstration of FOAM entanglement of two photons. Götte et al. [15] presented the quantum formulation of FOAM. And Zhou et al. reported a concise high-efficient experiment converter to realize the detection of both high-order and FOAM [16]. Additionally, the free-space optical communications carrying FOAM multiplexing was reported [17]. However, the effects of topological charge, temperature gradient, and salinity on the propagation property of FOAM in the underwater turbulence has not been discussed yet.
In this paper, we experimentally demonstrate the propagation property of LG-FOAM beam in the underwater environment. We use a one-meter water pipe with the distilled water to simulate the underwater environment. The temperature fluctuations is controlled by a heater over the water pipe, and salinity fluctuations is obtained by dissolving different weights of salt in the water. We discuss the effects of temperature gradients, salinity and topological charge (TC) on the propagation property of LG-FOAM. We also compare those results with integer OAM (IOAM) beam at the same underwater condition. With the cross probability of the LG-FOAM beam through the underwater environment, we present the impact of the temperature gradient and salinity on the channel’s capacity.
2. Experimental setup
In this section, we do the experiment to study the propagation property of FOAM in the underwater channel.
Fig. 1 is the experimental setup for the underwater channel with LG-FOAM mode. A Gaussian beam with wavelength $532nm$ is emitted from a green laser (Thorlabs, CPS532), and then is attenuated by a neutral density filter. After passing through the polarizer and halfwave plate, the Gaussian beam matches its polarization to the optimized working polarization of the selected spatial light modulator (SLM). Rather than using several optical elements to generate each LG modes separately, we employ a single SLM programmed with a hologram that sets the phase and intensity structure for the superposition, the FOAM mode. When the Gaussian beam is illuminated on a SLM (Holoeye, PLUTO-VIS-006-A), the desired FOAM mode is generated, where the special phase pattern of the desired FOAM is displayed on the liquid crystal of the SLM. A spatial filter consisting of two lenses and a pinhole is used to produce a clean LG-FOAM. A water pipe of length 1m is used to simulate the underwater channel. When the LG-FOAM beam passes through the water pipe, another SLM (SLM2, Holoeye, PLUTO-VIS-006-A) with the same spiral phase pattern is used to demodulate the LG-FOAM beam at the receiver. The resultant beam has a bright center since the output beam is turned back to Gaussian mode, while other centers are still dark due to the phase singularity. A power detector (Thorlabs, BC106N-VIS/M) with a pinhole is used to detect the power of the output beam. In addition, a heating device is placed over the water pipe, a thermometer is used to detect the temperature, and some salts (NaCl) are dissolved to produce different salinity. Utilizing the orthogonality of OAM modes, the FOAM amplitude field function can be expressed as a superposition mode of different OAM modes, that is,
where the fractional topological charge $M=m_0+\mu$ is a fractional number, $m_0$ is its integer part, and $\mu$ lies between 0 and 1, $\alpha$ is the orientation of the edge dislocation, $u_{m}(r,\varphi ,z)$ is an IOAM mode amplitude field function with $TC=m$, $C_{m}[M(\alpha )]$ is the coefficient on $m$ IOAM mode, which is described as
Fig. 1. The experimental setup for the propagation property of LG-FOAM in the an underwater environment. NDF, neutral density filter; Pol., polarizer; HWP, half-wave plate; SLM, spatial light modulator.
For an underwater communication channel, the interactions caused by underwater turbulence are mainly dependant on the temperature gradient and the salinity [19]. Nikishov et al presented the spectrum of turbulent fluctuation of the refractive index of seawater [20], in which the high-power polynomial expression for the refractive index $n$ as a function of temperature fluctuation $T'$ and salinity fluctuation $S'$ was limited to a linear approximation. It takes the form $\delta n = -AT'+BS'$ where $A=2.6 \times 10^{-4} l/deg, B=1.7 \times 10^{-4} l/g$ [19]. Hence, we mainly discuss the propagation property of FOAM with temperature ingredient and salinity in the paper.
With interferences, the crosstalk among FOAM modes would occur when the beam carrying FOAM passing through the underwater environment. Because the FOAM modes with integer TC differences are orthogonal, like FOAM with $TC=-0.5$ is orthogonal to FOAM with $TC=0.5$, and is also orthogonal to FOAM with $TC=1.5$, thereafter, at the receiver side, the FOAM mode components could be obtained by orthogonal decomposition, that is,
where $C_{M^{'}}$ is the expansion coefficient of FOAM mode $M'$, and $M'$ is a fractions who has an integer difference to $M$, which could be given by the inner products3. Results and discussions
Fig. 2 shows the FOAM power spectrum of a FOAM beam with $\ell =+2.5$ under various underwater conditions. The environment temperature was 20.1$^{\circ }$C. The energy of FOAM mode $\ell = +2.5$ at the receiver side decrease by 0.24 dB and 3 dB when the salinity was $2.8\%$ (the temperature gradient was 0) and the temperature gradient was 0.16$^{\circ }$C/m (the salinity was 0), respectively, in comparison with those both the temperature gradient and the salinity were 0. Additionally, the crosstalk caused by the temperature gradient was much greater than that caused by the salinity, and even a little increase of the temperature gradient would have a huge effect on the crosstalk.
We then change the temperature gradient from 0 to 0.16$^{\circ }$C/m, the salinity from 0 to 2.8$\%$, and TC from 2.5 to 5.0. We measure and compute the detection probability of FOAM at the receiver side. Fig. 3 shows the propagation property of LG-FOAM beams against salinity and temperature gradient in the underwater channel, where TC are $\ell$ = 2.5, 2.9, 3.0, 4.5, 4.9, 5.0. Fig. 3.(a) shows the propagation property against salinity and Fig. 3.(b) shows the propagation property against temperature gradient. The temperature of water was 19.9$^{\circ }$C. The experimental results showed that there was the feasibility of FOAM mode transmitted through the underwater environment. The detection probability was better when the topological charge of LG-FOAM beam was closer to an integer. The detection probability performance of $\ell =3$ was better than those of $\ell =2.5$ and $\ell =2.9$, meanwhile, the FOAM with $\ell =2.9$ had better detection probability than that of $\ell =2.5$. With the same conditions, the detection probability of $\ell =3$ was better than that of $\ell =5$.
Fig. 3. The propagation property of LG-FOAM beams against salinity and temperature gradient in the underwater environment.
The salinity had a little effect on the propagation property of LG-FOAM beam whereas the temperature gradient had a heavier effect. When the salinity varied from $0\%$ to $2.8\%$, the detection probability gradually decreased from 0.768 to 0.746 for $\ell =2.5$, from 0.768 to 0.746 for $\ell =2.9$, which were smaller than that for $\ell =3$ (from 0.8 to 0.75). At the same time, when the temperature gradient varied from $0.0$ to $0.16 K/m$, the detection probability decreased quickly from from $0.729$ to $0.437$ for $\ell =2.5$, from $0.768$ to $0.513$ for $\ell =2.9$, and they were 0.8, 0.75 for $\ell =3$, respectively.
Furthermore, we experimentally demonstrate the feasibility of the usage of FOAM in multiplexing communications system. We set a FOAM mode at the transmitter, say $\ell =2$, and detect the probability of different FOAM modes at the receiver due to the crosstalk caused by the underwater channel. In order to introduce the fractions, we detected the FOAM modes with 2+${\Delta }$, where ${\Delta }$ is a small quantity.
Figure 4 shows the detection probability against different $\Delta$ values for different salinity and temperature gradient. The results showed that the detection probability was still very large, and close to the IOAM mode, when the interval $\Delta$ was 0.05 or 0.10. But when the interval $\Delta$ was larger, for example, $\Delta$ was greater than 0.15, the detection probability was significantly reduced. It also could be seen that the interference caused by salinity was smaller than that by temperature gradient.
Fig. 4. The detection probability against different $\Delta$ values for different salinity and temperature gradient. The topological charge of FOAM at the transmitter is 2, and the interval value at the receiver is ${\Delta }$ =0,0.05,0.1,0.15,0.20
According to the possibility to receive a FOAM mode with $\Delta$ to the transmitter, we explore the system capacity of the underwater channel with LG-FOAM modes, together with the comparison with the channel capacity of the original IOAM system, when the topological charges used for multiplexing system are limited. Fig. 5 shows the channel capacity of a multiplexed communication system with a limited topological charges, say, ${0<\ell <5}$, versus salinity and temperature gradient. For the multiplexing communication system using IOAM modes, the allowed input modes were $\ell =1,2,3,4$, therefore, we set $10$ corresponding detection IOAM modes (from $\ell =-2$ to $\ell =7$) at the receiver to obtain the energies of the transmitting modes. On the other hand, if the FOAM mode was allowed, we found that there were more FOAM modes ($0<\ell <5$) could be used. For instance, we might used $\ell =0.05,1.00,1.95, 2.9,3.85,4.8$ as transmitting FOAM modes(the input FOAM mode break was $0.95$), and we used $\ell =-1.9,-0.95,0.05,1.00,1.95,2.9,3.85,4.8,5.75,6.7$ these $10$ corresponding FOAM modes to detect. Similarly, we could used $\ell =0.1, 1.00, 1.9, 2.8, 3.7, 4.6$ (the input FOAM mode break was $0.9$) as the sending FOAM modes and $\ell = -1.8, -0.9$, 0.1, 1.9, 2.8, 3.7, 4.6, 5.7, 6.5 as the corresponding detection FOAM modes. The results showed that the capacity of the underwater channel with LG-FOAM modes beam was greatly improved when the input mode break was $0.95$, in comparison with those with only integer OAM modes. The FOAM system capacity was also better than the IOAM system when the input mode interval was $0.9$. However, as the mode break decreased (less than $0.85$), the capacity of the underwater system became worse. Additionally, the results also showed that the capacity of the underwater system with FOAM modes beam greatly decreased with temperature gradient, and gradually decreased with salinity. The underwater channel with FOAM modes beam had the advantage of increasing the channel capacity in case of the available topological charge was limited.
Fig. 5. The channel capacity of a multiplexed communication system with a limited topological charges, say, ${0<\ell <5}$, versus salinity and temperature gradient.
4. Conclusion
In the paper, we have experimentally demonstrated the propagation property of LG-FOAM beam in the underwater environment. The effects of TC, temperature gradient and salinity on the propagation of FOAM mode in the underwater turbulence have been analyzed. The experimental results have showed that temperature gradient has more serious impacts on LG-FOAM beam than that of salinity. Moreover, the detection probability of LG-FOAM mode beam has reduced in comparison to the transmission the nearest IOAM mode. The closer the topological charge to an integer, the bigger the detection probability the LG-FOAM beam has. Additionally, The capacity using FOAM modes has been greatly improved when the input mode break is smaller, since the detection probability has reduced a little when the interval $\Delta$ is smaller. The capacity of the underwater system with FOAM modes beam has greatly decreased with the increase of temperature gradient, and has gradually decreased with the increase of salinity.
Funding
National Natural Science Foundation of China (61871234, 61475075); State Key Laboratory of Low-Dimensional Quantum Physics (KF201909); Natural Science Foundation of Jiangsu Province (BK20180755).
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