## I Introduction

Future wireless communication systems require high data rate, ultra-reliability, and low latency to satisfy the needs of advanced applications while available spectrum is limited. For short range communications, one of the candidates to satisfy these demands is a visible light communication (VLC) system that usually exploits light emitting diodes (LEDs) as transmitters and photodiodes (PDs) as receivers [13, 7]. In the VLC systems, the shortage of spectrum is not an issue due to the extremely high frequency, i.e., about 400 to 790 THz, of visible light that enables the high speed data transmission. Moreover, the inherent directivity and impermeability of VLC signals prevent information leakage, which enhances the secrecy rate compared to the wireless communication systems using radio frequency (RF) bands [18]. There are additional benefits for the VLC systems such as low implementation cost by using existing infrastructure and dual function of illumination/communications with low power consumption using LEDs [8].

There have been many works to improve the secrecy rate of VLC systems. To achieve the maximum average secrecy rate in a large indoor room, simultaneous beamforming and jamming were considered in [1]. In [2]

, a zero-forcing beamformer was designed to minimize the secrecy outage probability with the signal-to-noise ratio (SNR) constraint for the legitimate receiver (Bob) by sampling the SNR space. A robust beamforming for the VLC systems was proposed using imperfect location information of eavesdropper (Eve) in

[12]. These works did not put much attention to the polarization of visible light though. Only a few works exploited perpendicular polarizations to eliminate mutual interference [17, 16, 15, 3]. It would be possible, however, to improve the performance of VLC systems by judiciously exploiting the polarization of visible light.In this paper, we propose a gold nanoparticle (GNP)-based secure VLC system with linear polarizers for indoor environments. The GNPs have chiroptical properties that can yield different circular dichroism (CD) and optical rotatory dispersion (ORD) for the incident light, which denote the effects of differential absorption and refraction for left circularly polarized (LCP) and right circularly polarized (RCP) light, by controlling the type and size of GNPs [9]. We first derive the effects of GNP plate, which is a panel composed of GNPs, and linear polarizers in the circular polarization (CP) domain. By integrating these effects, we model an overall VLC channel when each transmitter is equipped with a GNP plate and a linear polarizer and Bob is equipped with a linear polarizer. We then optimize the angles of linear polarizers to improve the secrecy rate when the transmitters simultaneously transmit intended symbols (to Bob) and artificial noise (to confuse Eve). Simulation results show that the proposed VLC system using the GNP plates can significantly improve the secrecy rate of all positions in a room, even near Bob.

This paper is organized as follows. In Section II, we first present the chiroptical properties of GNPs and then derive the transmit signal, VLC channel, and received signal models. Taking these models into account, we optimize the linear polarizer angles to improve the secrecy rate in Section III. In Section IV, we show the secrecy rate improvement of GNP-based VLC system in an indoor scenario via simulation results. Finally, we conclude the paper in Section V.

Notation: , , and

denote a scalar, vector, and matrix.

and denote the magnitude of and the Euclidean norm of . , , , , andare the transpose, complex conjugate, conjugate transpose, inverse, and pseudo inverse. A normal distribution with mean

and variance

is denoted as . represents the set of positive real numbers. , , and denote theall-ones vector, all-zeros matrix with the size

, and identity matrix with the size

. and are the operations extracting the real and imaginary parts of a given complex number. The Hadamard product of and is represented as . is the max function that returns the largest value between and . denotes the diagonal matrix.## Ii System Model

We focus on an indoor GNP-based VLC system with transmitters located separately, the legitimate receiver Bob, and a randomly located eavesdropper Eve. The transmitters are uniformly located on the ceiling and equipped with GNP plates and linear polarizers, while Bob and Eve are both equipped with only a linear polarizer. The block diagram of overall system is described in Fig. 1. We first explain some important GNP properties in Section II-A and elaborate the signal and channel models in Sections II-B, II-C, and II-D.

### Ii-a GNP properties

The chiroptical properties of GNPs are the CD and ORD, which are the differential absorption and refraction of LCP and RCP light [11]. The differential refraction caused by the GNPs results in differential phase retardation between the incident LCP and RCP light. A GNP plate is made by judiciously stacking GNPs with different types and sizes. This elaborate arrangement of GNPs with different types and sizes makes it possible to precisely control the amplitudes and phases of LCP and RCP light. In the CP domain, the effect of chiroptical properties of GNP plate for the polarized incident light can be represented as

(1) |

where and are the absorption factors for the LCP and RCP light with the range of , and is the difference of phase retardation between the LCP and RCP light. Figs. 2 and 3 show the CD and ORD for GNP plates with distinct sizes of GNPs and patterns of GNP plate. It is clear from the figures that, by using different sizes of GNPs and patterns of GNP plate, the CD and ORD significantly change with the wavelength of light. We assume the transmit signals toward Bob are less absorbed than the signals toward other directions by the GNP plates due to well-structured GNP plates.

We transform the GNP properties specified in the Stokes parameters to the Jones vector to handle the VLC signals in the CP domain. The Stokes parameters, which are associated with the intensity and polarization ellipse parameters of light, indicate the polarization state of electromagnetic field. The Jones vector represents the polarization of light as transverse waves. The Stokes parameters have the relation with the Jones vector as [4]

(2) |

where is the total intensity of light, is the degree of polarization for the state of polarization, e.g., completely polarized, partially polarized, and unpolarized, and denote the ORD and CD, and and are the complex amplitudes of Jones vector in the Cartesian coordinates. By using and in the Jones vector, the LCP and RCP complex amplitudes and can be obtained using the transformation matrix as

(3) |

Note that the parameters related to the GNP properties in (1) can be extracted from and for the polarized incident light.

### Ii-B Transmit signal model

In the scenario of interest, LEDs that work as the transmitters transmit direct current (DC)-biased signals, which are composed with intended symbols for Bob and artificial noise to disguise Eve. The DC bias is necessary to obtain positive real-valued transmit signals after applying real-valued precoding techniques [14]. The real-valued symbol modulated transmit signals with the DC bias are represented as

(4) |

where is the current-to-light conversion efficiency, is the optical power at the LED, is the DC bias, and and are the precoding matrices for the intended symbol and the artificial noise , respectively.

### Ii-C VLC channel model

To increase the secrecy rate by the phase retardation effect of GNP plates, we employ the linear polarizer that can control the phase retardation of light in the CP domain. The effect of linear polarizer with the angle in the Cartesian coordinates is well represented as the Jones matrix

(5) |

We need the linear polarizer representation in the CP domain to derive the overall channel model with the GNP plates. From the fact that the linear polarization in the CP domain after the transformation by equals to the transformation by after linear polarization in the Cartesian coordinates, the linear polarizer in the CP domain is derived as

(6) |

which retards the RCP light by with respect to the LCP light.

Due to strong directivity of signals in the visible light spectrum, we assume only the line-of-sight (LOS) path exists between an LED and a PD [6]. Since most of commercial LEDs are assumed to have the Lambertian emission beam pattern, the path-loss of VLC channel between the -th LED and a PD, which corresponds to the VLC geometric channel in Fig. 1, can be denoted as [10]

(7) |

where is the Lambertian emission order, is the half power angle of LED, represents the area of PD, and are the distance and the radiation and incidence angle pair between the -th LED and a PD, is the field of view of PD, and denotes the rectangular function being for .

The wireless channel between LEDs and a PD can be represented in the CP domain by taking the propagation delay into account as

(8) |

with

(9) |

where , and denote Bob and Eve, with where is the path-loss for , where denotes the phase due to the propagation delay for , and , , and are the absorption factors and the phase retardation differences for the LCP and RCP light toward .

### Ii-D Received signal model

The received signals before arriving at the PD are expressed as

(11) |

Since the PD can measure only the intensity of signals, the received signal at the PD is represented as[5]

(12) |

where is the PD’s responsivity, and is the thermal noise at the PD [10]. The received signal model in (12) clearly shows that the phase due to the propagation delay in (8) has no impact on the received signal at the PD. Using (11) for (12), the received signal at the PD with the first-order approximation is shown in (II-D) at the top of next page, where is the -th component of , and is the effective channel including the effects of path-loss, linear polarizers, and GNP plates.

(13) |

The final received signal model after eliminating the DC bias is represented as

(14) |

## Iii Precoder Design and Linear Polarizer Angle Optimization

We design the precoding matrices and with respect to the effective channel of Bob since the information of location and channel of Eve cannot be known at the transmitters. To maximize the signal-to-interference-plus-noise ratio (SINR) of Bob, the precoders and for the intended symbol and the artificial noise are chosen as

with the singular value decomposition (SVD) of

and that is the orthogonal projection matrix with respect to . With the above precoders, the achievable rates of Bob and Eve are given as(15) |

where .

The SINRs of Bob and Eve are highly affected from the angles of linear polarizers. In the angle optimization, we simply consider the effective channels to verify the feasibility of GNP plates in the wiretapping scenario. We first optimize the linear polarizer angles at the LEDs to minimize the effective channel of Eve. The components of are independent of each other due to the receiving mechanism of PD that measures each intensity of received signal as in (12). Assuming the angle of linear polarizer at Eve (because this information cannot be known at the transmitters), the optimization problem to minimize the effective channel of Eve is represented as

(16) |

Since (16) has the trivial solutions of , which severely restrict the receive power at Bob, we consider a closely related problem given as

(17) |

In (17), in the effective channel is discarded, and the constraints do not include . Then, the optimal solutions of (17) are the angles that satisfy , i.e.,

(18) |

where is an integer to satisfy the constraints in (17).

With obtained in (18), the linear polarizer angle at Bob can be optimized as

(19) |

with

(20) |

The difference of phase retardation between LCP and RCP light has the range with the lower bound and the upper bound . Because of the narrow range of as in [11], it is reasonable to assume that , which approximates in (20) as

(21) |

Then, the linear polarizer angle at Bob can be set as

(22) |

where is an integer to satisfy the constraint in (19).

## Iv Simulation Results

We present simulation results considering an indoor eavesdropping scenario to evaluate the proposed GNP-based VLC system in this section. The performance metric is the secrecy rate, i.e., the difference between the achievable rates of Bob and Eve defined as

(23) |

The size of indoor room is m m, the number and locations of LEDs are and (, , ) m, and Bob and Eve are located in (0, 0, 1) m and (, , 1) m for and with the range of . The rest of simulation parameters are set as follows: , the half power angle of LED , the area of PD , the current-to-light conversion efficiency W/A, the PD’s responsivity A/W, and the thermal noise variance dBm. The lower and upper bounds of phase retardation difference are and in radian for a multiple-layered GNP plate [11]. We set the optical power dBm for both VLC systems with/without the GNP plates.

Fig. 4 shows the secrecy rates depending on the location and linear polarizer angle at Eve without the GNP plates at the transmitters. Both Figs. 3(a) and 3(b) have extremely small secrecy rates, i.e., the order of , because the optimization in (17) forces close to when (i.e., the case of no GNP plates), resulting in low received signal power even at Bob. If Eve can set the linear polarizer angle same as that of Bob, which is unrealistic in practice, Fig. 3(b) shows that the secrecy rate further decreases near Bob. These results clearly show that it is not possible to implement secure VLC systems by only using linear polarizers.

Fig. 5 verifies the secure VLC system can be developed using the GNP plates. It is shown in Fig. 4(a) that even if Eve is located at the same position with Bob, the secrecy rate is quite high as long as . As shown in Fig. 4(b), it is possible to obtain a high secrecy rate for most of locations even with the pessimistic case of . This is because the various arrangement of GNPs with different types and sizes inside the GNP plates results in significantly different channels for all radiation directions, making the artificial noise designed in Section III highly effective to Eve. These results clearly show that it is possible to implement extremely secure VLC systems by exploiting the GNP plates.

## V Conclusion

GNPs are recently synthesized metamaterials that can differentially absorb and retard left and right circularly polarized light. In this paper, we proposed a secure GNP-based VLC system with linear polarizers. The transmitters are equipped with well-structured GNP plates, which consist of various types of GNPs, to prevent wiretapping by exploiting the chiroptical properties of GNP plate. A novel VLC channel model considering the GNP plates and linear polarizers was derived in the CP domain to obtain the received signal model. Then, the precoder design and linear polarizer angle optimization were performed to achieve the high secrecy rate. Numerical results showed that even with the pessimistic case that Eve sets the linear polarizer angle same as Bob, the proposed GNP-based VLC system can achieve high secrecy rates for most of locations, even near Bob. The proposed GNP-based VLC system can be adapted to many wireless communication scenarios that require high level security, e.g., military and vehicular communications.

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