## Abstract

A parallel bit-interleaved filter-bank multicarrier/offset quadrature amplitude modulation (FBMC/OQAM) security strategy based on four-dimensional chaos is proposed in this paper. After the QAM constellation point distribution is disturbed, the modulated FBMC bits and symbols are interleaved and encrypted to realize the improvement of the FBMC/OQAM system physical layer security performance. The chaotic sequence generated by the four-dimensional hyperchaotic system is optimized and calculated to control the disturbance process, which enhances the performance of the system against illegal malicious attacks. The parallel encryption scheme proposed in this scheme increases the encryption efficiency by 1.43 times; can provide a keyspace of 10^{90} size, which effectively resists brute force attacks; and improves the physical layer security of the system. The proposed FBMC/OQAM parallel bit interleaved encryption scheme using a 5 km weakly coupled four-mode fiber achieves a 3×10 Gb/s multiple-input multiple-output-free transmission. The experimental results show that this scheme can effectively improve the security performance of the system, and combines the few-mode multiplexing technology with advanced modulation. It is a candidate for the future large-capacity and high-security optical transmission system.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Multi-carrier modulation (MCM) has become a key physical layer transmission technology in communication systems because of its ability to transmit data at high bit rates on multiple parallel sub-channels. Among them, the filter-bank multicarrier/offset quadrature amplitude modulation (FBMC/OQAM) modulation technology based on the filter bank is a promising MCM method in the current optical fiber communication system [1–3]. Compared with the commonly used orthogonal frequency division multiplexing (OFDM) modulation technology, FBMC/OQAM does not need to add cyclic prefix (CP) and guard interval in the system, which improves the throughput of the system, and also has good time-frequency limitation and better sidelobe suppression, which can provide higher useful data rate and spectral efficiency [4–6]. And in the FBMC system, the real part and the imaginary part of the QAM symbol are continuously transmitted in the form of real numbers. Therefore, the transmission symbol rate of FBMC is twice that of OFDM [7].

In addition to the consideration of communication system performance improvement, information security has also been the focus of research in recent years. In the current encryption schemes, most of them encrypt the upper layer of the optical network in the communication system [8–10], but the upper layer encryption scheme cannot directly protect the control data, and there are potential security threats [11–13]. A physical layer security solution that can transparently encrypt all types of data is a very promising method. Among them, chaos technology is widely used in various encryption schemes due to its high sensitivity to initial values, pseudo-randomness, and ergodicity [14]. In the FBMC/OQAM system, offline data signal processing (DSP) can be used to flexibly process digital signals. While signal generation and modulation are completed, chaotic disturbances can be easily added to improve data security. Compared with encryption schemes in the optical domain that require the use of specific devices to complete encryption, such as laser chaos [15,16], the electrical encryption scheme using DSP omits additional optical components and significantly reduces system costs. Therefore, chaotic encryption schemes based on electrical domain DSP has received extensive attention, such as bit encryption based on deoxyribonucleic acid algorithm [17], multi-scroll chaotic based high-speed encryption system [18], chaotic shifting of QAM constellation [19], and I/Q encryption scheme based on hyperchaos [20], etc. But These studies are basically based on the security of the OFDM system, tend to focus on the encryption level, and use the original data generated by the chaotic system for disturbance. Therefore, in addition to the encryption level, we also need to pay attention to the calculation time and cost of the encryption system. Based on the research of existing encryption schemes, we propose a parallel bit-interleaved FBMC/OQAM security strategy based on four-dimensional chaos, which improves encryption efficiency while ensuring encryption performance and reduces the time cost of system encryption.

In recent years, to solve the problem of the capacity limit of the transmission system based on single-mode fiber, the research community has turned its attention to the space division multiplexing technology on the new type of fiber. The commonly used method is to use multi-core fiber to multiplex and transmit data in multiple uncoupled cores in the optical fiber. Each core can be considered as a separate channel so that the communication capacity of the system can be doubled [21]. The other is the mode division multiplexing technology, which uses the different spatial modes in the few-mode fiber as independent channels to transmit data [22], or use the high-order angular momentum mode in the vortex fiber for multiplexing transmission, and the transmission capacity of the few-mode channel is proportional to the number of stable transmission modes in the fiber [23]. Ideally, as long as there are enough modes supported by the fiber, the spatial spectrum efficiency can be greatly improved to meet future capacity requirements. To achieve high-efficiency and low-loss conversion and channel switching between different modes, a new spatial mode conversion multiplexing system based on unitary conversion is proposed: a multi-plane light conversion (MPLC) system. Any mode in the system can achieve specific mode conversion through continuous multiple unitary matrix transformations. Since the entire system has no intrinsic loss, the energy loss of the system is low. And the MPLC system can realize the fusion operation of multiple modes simultaneously in the same system, has greatly improved the mode conversion efficiency compared with the traditional multiplexing technology [24,25].

In this paper, a parallel bit-interleaved encryption scheme based on four-dimensional chaos is proposed for the first time, which can obtain a keyspace of 10$^{90}$, and effectively improve the security of the system. After perturbing the probability distribution of the QAM modulated constellation points, perform FBMC/OQAM modulation, and then perform double encryption through parallel bit interleaving. After that, the MPLC module was used to multiplex the signal with four modes, and the 5 km weakly coupled four-mode fiber was used to realize the 3$\times$10 Gb/s encrypted signal transmission for the FBMC/OQAM security system proposed in this paper. Experimental results show that the proposed encryption scheme can effectively improve the transmission system physical layer security, increase the encryption efficiency by 1.43 times, and effectively resist illegal reception while ensuring the security of the system.

## 2. Principle

The principle of the parallel bit-interleaved FBMC/OQAM security strategy based on four-dimensional chaos is shown in Fig. 1. At the transmitter of the FBMC/OQAM system, the DSP is used to generate pseudo-random binary sequence (PRBS) data, then perform 16QAM mapping on it, and the optimized hyperchaotic sequence is used to perturb the distribution probability of the constellation points, and then the FBMC/OQAM modulation is performed. Simultaneously perturb the modulated FBMC bit signal and FBMC symbol. The perturbation sequence is generated from the original hyperchaotic sequence after optimization and block dislocation calculation. At this time, the encryption of the signal is completed. Pass the encrypted FBMC/OQAM signal matrix through the parallel-to-serial (P/S) conversion, and then use the 1:4 optical power splitter to realize the four-mode multiplexing coupling through the MPLC module and then enter the 5 km weakly coupled four-mode fiber to realize the signal transmission. The receiver with the correct key can decrypt the information from the ciphertext and obtain the original data using a calculation method that is inverse to that of the encryption end.

#### 2.1 Hyperchaotic sequence generation

Use the following new hyperchaotic system to generate the original hyperchaotic sequences:

The original hyperchaotic sequences generated by the hyperchaotic system does not match the value type of the FBMC signal, and the distribution and pseudo-random characteristics of the digitized original hyperchaotic sequences are not ideal, so it is not suitable for direct data encryption, need to calculate and optimize the original hyperchaotic sequences first. The original hyperchaotic sequences are optimized, and the steps to generate the intermediate chaotic key sequences are as follows.

In order to eliminate the harmful effects of the sensitivity of the initial conditions of the chaotic sequences transient process, the first $L_0$ values of the original hyperchaotic sequences are removed, and 4 original chaotic sequences of length $L/4$ are obtained, expressed as $S_j(i)=\{x(i), y(i), z(i), w(i): i=1, 2,\ldots , L/4; j=1, 2, 3, 4\}$, Where $L$ is the required sequences length. Perform the following operations on the original hyperchaotic sequence to obtain an improved sequence:

Among them, $max\_S_j$ and $min\_S_j$ are the maximum and minimum values in the $j^{th}$ sequence, respectively. Re-optimize the obtained improved sequence to obtain 4 chaotic key subsequences:

#### 2.2 QAM constellation point distribution disturbance

The constellation symbol after 16QAM modulation is expressed as $X = \{x_1, x_2,\ldots , x_{16}\}$, and the probability mass function (PMF) of the signal point satisfying the Maxwell-Boltzmann distribution is:

Among them $\mu$ is the adjustment coefficient of the probability distribution of the constellation symbols, and the non-uniform constellation point distribution can be obtained after the 16QAM symbol distribution is chaotically disturbed. Take the sequence $K_1$ from the hyperchaotic key sequence $K$, and perform the following calculations to obtain the probability perturbation parameter $\varphi (i)$:

According to the value of the disturbance parameter $\varphi (i)$, the disturbance mode of the corresponding QAM symbol is determined (1 means unchanged, -1 means negative), and the signal PMF after the probability distribution disturbance is:

#### 2.3 FBMC bit and symbol interleaved encryption

The bit data after the FBMC modulation is divided into two sub-blocks, and the two sub-blocks are simultaneously subjected to ciphertext interleaving diffusion perturbation to realize data-parallel encryption. Assuming that the size of the FBMC data matrix is $R=M\times N$, the sequence obtained by scanning the data matrix row by row is marked as $\{Q(i), i=1, 2,\ldots , R\}$, then the first half of the data sequence to be encrypted is $\{Q(1), Q(2),\ldots , Q(R/2)\}$, denoted as block A, The second half block is composed of data sequence $\{Q(R/2+1), Q(R/2+2),\ldots , Q(R)\}$ in turn, denoted as block B. Take the sequence of length $R$ from the hyperchaotic sequence $K$ and record it as $K_2$.

The first data of block A and block B are respectively used to generate the final encryption key calculate using Eqs. (8a) and (9a), and the encryption operation is performed through Eqs. (8b) and (9b) respectively:

Among them, $C_0\in [1,255]$ is a preset positive integer, and $Q(i)$ and $C(i)$ are the $i^{th}$ original data and encrypted data, respectively.

Next, Eq. (10) and Eq. (8b) are used to generate the final encryption key for the $i^{th}$ data of block A and the encryption operation is performed, and the corresponding data points of block B are generated and encrypted by Eqs. (9a) and (9b) respectively.

Repeat the above steps until $i=L/2$, at which time the first round of encryption operation is completed. To improve the encryption performance and enhance the robustness of the system against proportional cracking, the data is double-encrypted. For the first data of block A and block B, Eqs. (11a) and (12a) are used to generate the final encryption keys, and the encryption operations are performed through Eqs. (11b) and (12b) respectively:

After that, perform Eqs. (10) and (11b) on the data of block A, Eqs. (12a) and (12b) on the data of block B respectively to obtain the final encryption key and perform encryption operation until $i=L/2$, at which time the double encryption is completed, and the final key and ciphertext $C$ are obtained. Among them, the final encryption key named $key(i)$ is not only related to the current chaotic key $K_2(i)$ but also related to the $(i-1)^{th}$ encrypted ciphertext data of another sub-block, that is, the ciphertext interleaving diffusion mechanism is introduced. Therefore, after double encryption, any change in data value will affect the ciphertext value of all other data.

Fig. 2 is a schematic diagram of bit and symbol interleaving perturbation, where the Rubik’s Cube structure is used to represent the FBMC data frame, each small square in the Rubik’s Cube represents the FBMC bit data after the modulation is completed, and each layer in the Rubik’s Cube represents the FBMC symbol. After dividing the FBMC bit data into two sub-blocks, bit interleaving and scrambling are performed on the two sub-blocks respectively, and then FBMC symbol interleaving is performed to complete encryption. At the receiver, the decryption process is the reverse operation of encryption. Using the correct key, the original data can be recovered from the ciphertext, and the decryption can be completed. The transmitter and the legal receiver share the security key.

## 3. Experiment setup

In this article, a new MPLC-based spatial mode multiplexer with low inherent loss and high mode selectivity is used, Fig. 3 is the MPLC module used in the experiment. In Fig. 3(d), the devices 1-4 are collimating couplers, fiber arrays, mirrors, and phase plates. Fundamentally speaking, spatial multiplexing is an inter-mode conversion that converts N individual input beams into N orthogonal modes. Since spatial multiplexing converts one mode into another, it can be considered as space unitary transformation. It has been theoretically proven that the optical Fourier separation of the transverse phase profile can be used to achieve any desired spatial unitary transformation between the input plane and the output plane. In this paper, four-mode multiplexing coupled MPLC module is used to achieve mode conversion and transmission.

Fig. 4 is the experimental setting of the parallel bit-interleaved FBMC/OQAM security strategy based on four-dimensional chaos. The illustrations (a) and (f) are the light field diagrams of mode LP01 before and after transmission, the illustrations (b) - (e) are the light field diagrams of mode LP01, LP11a, LP11b and LP21 in transmission, respectively, the mode field edges of the four modes can be clearly seen. In the experiment, we set up a normal receiver with a secure key, and an illegal receiver without a secure key can only obtain information through brute force to verify the feasibility of the security strategy proposed in this article. At the transmitter, the encrypted FBMC/OQAM signal is generated by offline DSP using MATLAB program. Other FBMC/OQAM system data settings are as follows: the subcarriers number is 150, the symbols number is 100, the IFFT/FFT points number is 512, 16QAM modulation, the prototype filter based on the PHYDYAS filter [26] with an overlap factor of $O = 4$. The encrypted data is converted to analog radio frequency (RF) signals using an AWG with a sampling rate of 10 GSa/s. The RF signal is then loaded via MZM onto a continuous optical carrier generated by a tunable light source with a line width of less than 100 kHz, an output power of 14.5 dBm, and a wavelength of 1550 nm. The electro-optic conversion is realized while the optical carrier is modulated by the optical intensity modulator. After that, a 1:4 optical power splitter is used to divide the optical signal into 4 beams, and then pass it to the MPLC module for four-mode multiplexing. Then the 5 km weakly coupled four-mode fiber is coupled through the collimator coupler for transmission, and the MPLC module is used at the receiver to demultiplex the signal. Afterward, a 5.5 dB noise EDFA is used to adjust the power of the modulated optical signal, and then the PD is used for the photoelectric conversion of the received signal. Then the analog-to-digital conversion is realized by the MSO with the sampling rate of 50 GSa/s. The original transmission signal can be obtained by offline inverse operation of the electrical signal in Matlab.

## 4. Results and discussion

Take the initial state variable $S_0 = (x_0, y_0, z_0, w_0) = (2.5, 5.2, 3, 7.3)$, Fig. 5 shows the phase diagram of the 4-D hyperchaotic map used in this paper. It can be seen that after the initial state variables are driven by the hyperchaotic system shown in Eq. (1), the generated values show complex bifurcation and chaotic dynamics.

Figure 6(a) is a graph of the numerical value distribution curve in the hyperchaotic sequence $K$ calculated by Eqs. (2)–(4), compared with the original hyperchaotic sequences $S$ calculated by using Eq. (1) alone as shown in Fig. 6(b), the optimized key sequence has a more uniform distribution. Figure 6(c) and (d) are the histograms of the normalized distribution of the data values of the original FBMC signal and the encrypted signal, respectively. It can be seen that the data value distribution of the original signal is not uniform, while the data value of the ciphertext presents a flat and uniform distribution. That is, the probability of the ciphertext data value is almost equal, which proves that the encryption model proposed in this paper can effectively resist statistical analysis attacks.

In addition, the encryption model should be strongly sensitive to the initial state, that is, a small change in the initial state will cause the system to generate a completely different final key. The initial state variable $S_0$ to encrypt the image used in this article, and then uses the slightly modified key $S^{'}_{0} = (2.5+10e{-15}, 5.2, 3, 7.3)$ to decrypt the encrypted image. Figure 7 is the image encrypted with the correct key $S_0$ and the wrong key $S^{'}_{0}$, respectively. Using the correct key can achieve complete and accurate decryption, while the image decrypted with the wrong key is very different from the original image. Even with a small change of the key, no information can be obtained from the ciphertext, proving that the chaotic encryption scheme has strong initial state sensitivity, which is sufficient to ensure the security of the system.

In the encryption model, the 4 state variables $x_0, y_0, z_0, w_0$, the number of pre-iterations $N_0$, and the positive integer $C_0$ of the hyperchaotic system are the original keys, which are defined as double-precision real numbers with 15 decimals in DSP calculations. Therefore, the system can provide a keyspace of $(10^{15})^6 = 10^{90}\approx 2^{299}$, which is equivalent to a key length of 299 bits, and sufficient to resist brute force attacks by illegal attackers. And because the traditional perturbation of the data position is changed to the bit interleaved, with the sub-block parallel encryption method is adopted to improve the encryption efficiency, it takes about 0.92 s to encrypt a 512$\times$512 grayscale image using this scheme, while it takes approximately 1.32 s to encrypt the same image using the serial bit shift scheme. This scheme increases the encryption efficiency by 1.43 times. In the process of bit-interleaved parallel encryption, the key required for each encryption is updated in real time by the DSP according to the result of the previous encryption, and there is no need to store a large number of keys. At the same time, it will not cause the number of keys to be greater than the amount of data that needs to be encrypted and cause data overflow, which can save the calculation space of the DSP, and reduce the complexity. Experiments show that the chaotic system proposed in this paper has the advantages of large keyspace, high encryption efficiency, strong resistance to brute force cracking and statistical analysis attacks, and can be applied to information encryption transmission.

Fig. 8 shows the inter-mode isolation of the MPLC module in the article measured experimentally. When the LP01, LP11a, LP11b and LP21 modes are loaded with light with a power of 14.5 dBm, respectively, the optical power of the other three modes after transmission by the MPLC module is measured. It can be seen from the figure that when only the LP01 mode is loaded with light, the optical power of the LP21 is the smallest after being transmitted by the MPLC module, indicating that the isolation between the LP01 and LP21 modes is the largest and the inter-mode crosstalk (IMC) is the smallest. On the contrary, only when the light is loaded on the LP21 mode, the optical power of the LP01 mode is the smallest after being transmitted by the MPLC module.

When the light is loaded on the LP11a mode only, the LP11b mode transmitted by the MPLC module receives the maximum optical power. When the LP11b mode is loaded with light, it is found that the received optical power of the LP11a mode is the largest after transmission, indicating that the isolation between the LP11a and LP11b modes is very small and is greatly affected by the IMC. In addition, the isolation between LP11b and LP21 has a smaller increase compared with the isolation between LP11b and LP11a, that is, the LP11b mode will be affected by the IMC between the LP11a and LP21 modes during transmission.

Fig. 9 is the bit error rate (BER) curve and constellation diagram after the signal is transmitted back-to-back (B2B) and the normal receiver decodes the signal. Figure 9(a) is the result of separate transmission of the four modes after the conversion using the MPLC module. It can be clearly seen that the BER trends of the four modes LP01, LP11a, LP11b, and LP21 are the same. When the optical power is -20 dBm, the BER of the LP01 mode is about $10^{-4}$, and the BER performance is the best, and the higher-order mode LP21 has a BER of $10^{-3}$ and the BER performance is slightly worse. In the case of the same BER, compared to the mode LP01, the mode LP21 will have a power loss of about 0.8 dB.

Considering the IMC between the two modes LP11a and LP11b shown in Fig. 8, use the MPLC module to multiplex and transmit the three modes. The first experiment is to multiplex and transmit the three modes of LP01, LP11a and LP21 at the same time, the second is to multiplex and transmit the three modes of LP01, LP11b and LP21. Since the BER curves of the LP01 and LP21 modes after transmission through these two multiplexing schemes basically overlap, only the BER curves of the LP11a and LP11b modes have significant changes. Therefore, for the LP01 and LP21 modes, after taking the average of the corresponding data points obtained from the two experiments, then plot the BER curves of the four modes LP01, LP11a, LP11b and LP21 into the same graph for easy reading. The experimental results of three-mode transmission when B2B are shown in Fig. 9(b). For the mode LP01, when the BER is $10^{-4}$, the optical power required for mode separate transmission is -20 dBm, while the three-mode multiplexing transmission requires -17 dBm. Compared with the mode separate transmission, the three-mode multiplex transmission will bring a power penalty of 3 dB. And because the mode isolation between the mode LP01 and the mode LP21 is relatively large, their BER curves are roughly the same as the trends when the modes are transmitted alone. The LP11 mode-group (LP11a and LP11b) is greatly affected by the IMC, which shows that the BER curve becomes steeper, and the constellation diagram under the same BER has different degrees of deformation. Specifically, when the BER is around $10^{-4}$, compared with the constellation diagram of the mode LP01 in Fig. 9(a), the constellation diagram of LP11a spreads outward, and the interval between constellation points becomes smaller. When the BER is around 0.45, the constellation diagram of LP11b is compressed and narrowed in the Y-axis direction compared with the constellation diagram of the mode LP21 in the single transmission in Fig. 9(a).

Figure 10 is the BER curve and constellation diagram after the signal is transmitted through a 5 km weakly coupled four-mode fiber, and the signal is received and decoded by the normal receiver. Figure 10(a) is the result of the four modes being transmitted separately. In the case of the same BER, compared to the mode LP01, the mode LP11b has a power loss of 1 dB. This shows that the BER performance of the LP11b mode is greatly affected by the increase in the transmission distance and the nonlinear effect of the fiber.

Fig. 10(b) is the result of using the MPLC module to multiplex and transmit the three modes. The first experiment was to multiplex and transmit the three modes LP01, LP11a and LP21 simultaneously, and the second was to multiplex and transmit the three modes LP01, LP11b and LP21. After taking the average of the corresponding data of LP01 and LP21 after two experiments, plot them together with the data of LP11a and LP11b. When the BER is $10^{-4}$, the three-mode multiplexing transmission has a power loss of 3 dBm compared to the mode single transmission. When the optical power is -16 dBm, the BER of the LP01 and LP21 modes are about $10^{-4}$, while the BER of the LP11 mode-group is around $10^{-2}$, and the constellation diagram becomes blurred. The reason for the different BER performance of each mode is that the mode isolation between the mode LP01 and the mode LP21 is relatively large, which is less affected by the IMC and channel, while the LP11 mode-group is more affected by the IMC and channel. When the BER is the same, compared with the LP01 mode, the LP11 mode-group will have a power loss of about 1.8 dB.

Figure 11 shows the BER curves of the LP01 and LP11b modes when the modes are transmitted separately, and the channel is B2B and 5 km weakly coupled four-mode fiber. Comparing Fig. 9(a) and Fig. 10(a), it can be seen that the BER curve trends of LP01, LP11a and LP21 modes under different channels are the same, and less affected by the channel distance. However, after 5km weakly coupled four-mode fiber channel transmission, the BER curve trend of the LP11b mode changes greatly, and the BER performance deteriorates. In order to facilitate analysis, the BER curves of the LP01 and LP11b modes are plotted in Fig. 11. When the BER is the same, the power loss of the LP11b mode after transmission via the optical fiber channel is as high as 1.7 dB. Fig. 12 shows the BER curves of the LP01 and LP11a modes after modes transmitted through 5 km weakly coupled four-mode fiber. The hollow circles and solid circles respectively represent the results of separate transmission and three-mode multiplexing transmission for each mode. Comparing the BER curves of each mode in Fig. 10(a) and Fig. 10(b) under different transmission conditions, it can be seen that for the LP01 and LP21 modes, the trend of the BER curve after three-mode multiplexing transmission is basically the same as the trend of separate transmission of the modes. Because the isolation between LP01 and LP21 is relatively large, and it is less affected by the increase in channel distance and IMC. However, the BER curve of the LP11 mode-group becomes steep after three-mode multiplexing transmission. Because the isolation between the LP11 mode-group is small, and it is also affected by the IMC between the LP01 and LP21 modes, it is greatly affected by the cumulative effect of channel interference and IMC. For a clearer comparison, the BER curves of the LP01 and LP11a modes after three-mode multiplexing transmission are plotted in Fig. 12. Under the same BER, the power loss of the LP11a mode is about 2 dB higher than that of the LP01 mode.

Fig. 13(a) and (b) are the BER curve and constellation diagram after receiving and decoding the fundamental mode LP01 signal by the normal and the illegal receiver, respectively. Figure 13(a) shows that at the normal receiver, the ciphertext can be correctly decrypted using the security key to obtain the original information. The BER of the illegal receiver without the security key is around 0.5, and the data cannot be recovered and decoded through forward error correction. This shows that the parallel bit-interleaved encryption scheme based on four-dimensional chaos proposed in this paper can resist the eavesdropping of illegal receivers and effectively protect information security.

## 5. Conclusion

A parallel bit-interleaved FBMC/OQAM security strategy based on four-dimensional chaos was introduced in this paper to enhance the FBMC/OQAM system physical layer security. In this paper, the four-dimensional chaotic sequence optimized by the secondary calculation is used to perturb the QAM constellation points, then FBMC/OQAM modulation is performed, and then the modulated FBMC bits and symbols are parallel interleaved and encrypted. Subsequently, the proposed security strategy is verified by experimental transmission, and $3\times 10$ Gb/s MIMO-free transmission was achieved through 5 km weakly coupled four-mode fiber with a loss of 1.7 dB. The results show that the parallel encryption method can increase the encryption efficiency by 1.43 times, and has a keyspace of $10^{90}$ size, effectively resistance to illegal acceptance, and has the high sensitivity, guarantees the security. At the same time, after using the MPLC module for three-mode multiplexing coupling, compared to the mode-separate transmission, the three-mode simultaneous transmission will bring a power penalty of 3 dB. The LP11 mode-group is greatly affected by the crosstalk between modes and the channel distance. Under the same BER, compared to the mode LP01, there will be a power loss of 2 dB. This scheme combines few-mode multiplexing technology with advanced modulation and is a candidate for future large-capacity and high-security optical transmission systems.

## Funding

National Key Research and Development Program of China (2018YFB1800905); National Natural Science Foundation of China (U2001601, 61822507, 61835005, 61875248, 61775098, 61727817, 62035018, 61975084, 61720106015, 61935011, 61935005); Open Fund of IPOC (BUPT); Jiangsu talent of innovation and entrepreneurship; Jiangsu team of innovation and entrepreneurship.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

## References

**1. **T.-H. Nguyen, L. Bramerie, M. Gay, M. Kazdoghli-Lagha, C. Peucheret, R. Gerzaguet, S.-P. Gorza, J. Louveaux, and F. Horlin, “Experimental demonstration of the tradeoff between chromatic dispersion and phase noise compensation in optical fbmc/oqam communication systems,” J. Lightwave Technol. **37**(17), 4340–4348 (2019). [CrossRef]

**2. **H. Saeedi-Sourck, “Multiple carrier frequency offsets and multipath channels estimation for coordinated multi-point fbmc systems,” Phys. Commun. **25**, 511–518 (2017). [CrossRef]

**3. **L. Gong, X. Zhou, X. Liu, W. Zhao, and Z. Zhu, “Efficient resource allocation for all-optical multicasting over spectrum-sliced elastic optical networks,” J. Opt. Commun. Netw. **5**(8), 836–847 (2013). [CrossRef]

**4. **H. Mohammadi, H. Saeedi-Sourck, and H. Shayanfar, “Frequency synchronization for fbmc-based massive mimo system uplink,” Phys. Commun. **41**, 101096 (2020). [CrossRef]

**5. **K. A. Alaghbari, H.-S. Lim, and T. Eltaif, “Compensation of chromatic dispersion and nonlinear phase noise using iterative soft decision feedback equalizer for coherent optical fbmc/oqam systems,” J. Lightwave Technol. **38**(15), 3839–3849 (2020). [CrossRef]

**6. **M. K. Al-Haddad and H. T. Ziboon, “Joint carrier frequency and symbol timing synchronization sequence for fbmc based system,” Phys. Commun. **42**, 101165 (2020). [CrossRef]

**7. **T.-H. Nguyen, F. Rottenberg, S.-P. Gorza, J. Louveaux, and F. Horlin, “Efficient chromatic dispersion compensation and carrier phase tracking for optical fiber fbmc/oqam systems,” J. Lightwave Technol. **35**(14), 2909–2916 (2017). [CrossRef]

**8. **Z. Zhu, W. Lu, L. Zhang, and N. Ansari, “Dynamic service provisioning in elastic optical networks with hybrid single-/multi-path routing,” J. Lightwave Technol. **31**(1), 15–22 (2013). [CrossRef]

**9. **L. Gong and Z. Zhu, “Virtual optical network embedding (vone) over elastic optical networks,” J. Lightwave Technol. **32**(3), 450–460 (2014). [CrossRef]

**10. **Y. Yin, H. Zhang, M. Zhang, and M. Xia, “Spectral and spatial 2d fragmentation-aware routing and spectrum assignment algorithms in elastic optical networks [invited],” J. Opt. Commun. Netw. **5**(10), A100–A106 (2013). [CrossRef]

**11. **M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in ofdm-pon,” IEEE Photonics J. **9**(1), 1–10 (2017). [CrossRef]

**12. **K. Zhang, J. Zhang, G. Gao, and A. Fei, “Physical layer security based on chaotic spatial symbol transforming in fiber-optic systems,” IEEE Photonics J. **10**(3), 1–10 (2018). [CrossRef]

**13. **P. Lu, L. Zhang, X. Liu, J. Yao, and Z. Zhu, “Highly efficient data migration and backup for big data applications in elastic optical inter-data-center networks,” IEEE Network **29**(5), 36–42 (2015). [CrossRef]

**14. **J. Ren, B. Liu, D. Zhao, S. Han, S. Chen, Y. Mao, Y. Wu, X. Song, J. Zhao, X. Liu, and X. Xin, “Chaotic constant composition distribution matching for physical layer security in a ps-ofdm-pon,” Opt. Express **28**(26), 39266–39276 (2020). [CrossRef]

**15. **Z. Zhao, M. Cheng, C. Luo, L. Deng, M. Zhang, S. Fu, M. Tang, P. Shum, and D. Liu, “Semiconductor-laser-based hybrid chaos source and its application in secure key distribution,” Opt. Lett. **44**(10), 2605–2608 (2019). [CrossRef]

**16. **X. Jiang, D. Liu, M. Cheng, L. Deng, S. Fu, M. Zhang, M. Tang, and P. Shum, “High-frequency reverse-time chaos generation using an optical matched filter,” Opt. Lett. **41**(6), 1157–1160 (2016). [CrossRef]

**17. **Y. Xiao, Y. Chen, C. Long, J. Shi, J. Ma, and J. He, “A novel hybrid secure method based on dna encoding encryption and spiral scrambling in chaotic ofdm-pon,” IEEE Photonics J. **12**(3), 1–15 (2020). [CrossRef]

**18. **F. Wang, B. Zhu, K. Wang, M. Zhao, L. Zhao, and J. Yu, “Physical layer encryption in dmt based on digital multi-scroll chaotic system,” IEEE Photonics Technol. Lett. **32**(20), 1303–1306 (2020). [CrossRef]

**19. **A. Sultan, X. Yang, A. A. E. Hajomer, and W. Hu, “Chaotic constellation mapping for physical-layer data encryption in ofdm-pon,” IEEE Photonics Technol. Lett. **30**(4), 339–342 (2018). [CrossRef]

**20. **Z. Hu and C.-K. Chan, “A 7-d hyperchaotic system-based encryption scheme for secure fast-ofdm-pon,” J. Lightwave Technol. **36**(16), 3373–3381 (2018). [CrossRef]

**21. **J. Li, C. Cai, J. Du, S. Jiang, L. Ma, L. Wang, L. Zhu, A. Wang, M.-J. Li, H. Chen, J. Wang, and Z. He, “Ultra-low-noise mode-division multiplexed wdm transmission over 100-km fmf based on a second-order few-mode raman amplifier,” J. Lightwave Technol. **36**(16), 3254–3260 (2018). [CrossRef]

**22. **G. Rademacher, B. J. Puttnam, R. S. Luis, J. Sakaguchi, W. Klaus, T. A. Eriksson, Y. Awaji, T. Hayashi, T. Nagashima, T. Nakanishi, T. Taru, T. Takahata, T. Kobayashi, H. Furukawa, and N. Wada, “Highly spectral efficient c + l-band transmission over a 38-core-3-mode fiber,” J. Lightwave Technol. **39**(4), 1048–1055 (2021). [CrossRef]

**23. **X. Wu, S. Gao, J. Tu, L. Shen, C. Hao, B. Zhang, Y. Feng, J. Zhou, S. Chen, W. Liu, and Z. Li, “Multiple orbital angular momentum mode switching at multi-wavelength in few-mode fibers,” Opt. Express **28**(24), 36084–36094 (2020). [CrossRef]

**24. **R. Tanomura, R. Tang, T. Suganuma, K. Okawa, E. Kato, T. Tanemura, and Y. Nakano, “Monolithic inp optical unitary converter based on multi-plane light conversion,” Opt. Express **28**(17), 25392–25399 (2020). [CrossRef]

**25. **G. Labroille, B. Denolle, P. Jian, P. Genevaux, N. Treps, and J. F. Morizur, “Efficient and mode selective spatial mode multiplexer based on multi-plane light conversion,” Opt. Express **22**(13), 15599–607 (2014). [CrossRef]

**26. **M. Bellanger, “FBMC physical layer: a primer,” PHYDYAS, 2010.