2.1.

OAM Configured Chaotic Laser

In order to tackle the shortcomings of the traditional chaotic laser, we propose a conceptual paradigm of OAM-CCL, as shown in Fig. 1(a). After OAM configured, the wavefront of the chaotic laser becomes a helical structure, and the temporal properties of the chaotic signal are preserved. Thus the OAM-CCL can be used as a broadband chaotic laser source driving a pair of lasers to generate synchronous masking signals for communication in free space. The OAM-CCL is schematically shown in Fig. S1 in the Supplementary Material. The OAM-CCL possesses the following characteristics: (i) it is difficult to eavesdrop in free space, providing additional security. (ii) It can be spatial quadrature multiplexed with other OAM beams, bringing additional freedom to the optical communication system, similar to Fig. 1(b).

Fig. 1

OAM-CCL. (a) Concept of OAM-CCL and (b) intensity profiles and phase fronts for different OAM beams.

2.2.

OAM-CCL-Based Communication System

The schematic diagram of the communication system based on OAM-CCL is shown in Fig. 2. An all-optical feedback chaotic laser ${\mathrm{SL}}_{\mathrm{D}}$ with a wavelength of 1550 nm generates a high-bandwidth chaotic laser as the driving chaotic source.³⁴ The driving chaotic signal is split into two parts: one part is injected into the laser ${\mathrm{SL}}_{\mathrm{T}}$ to generate an encrypted chaotic signal $E_{\mathrm{ch}}(t)$, whereas the other part is sent to the receiver and injected into ${\mathrm{SL}}_{\mathrm{R}}$ to generate a synchronous chaotic signal $E_{\mathrm{ch}}^{\prime }(t)$.³⁵ Continuous-wave lasers ${\mathrm{SL}}_1-{\mathrm{SL}}_{10}$ are modulated to produce $10\mathrm{Gb}/\mathrm{s}$ nonzeroing on–off keying signals $E_n(t)$, respectively. After that, the information signals are masked as encrypted signals $E_n^{\mathrm{enc}}(t)=E_n(t)+a{E}_{\mathrm{ch}}(t)$. Note that due to the presence of the attenuator the masking coefficients differ from channel to channel, avoiding the possibility of the eavesdropper removing the chaotic signal by simple differencing.

Fig. 2

Schematic diagram of the communication system based on OAM-CCL. SL, laser; VOA, variable optical attenuator; FM, fiber optic mirror; PC, polarization controller; DL, delay line; Col., collimator; Pol., linear polarizer; SLM, spatial light modulator; EDFA, erbium-doped fiber amplifier; AM, amplitude modulator; BS, free-space beam splitter; M, mirror; PD, photodetector; DIF, current differential; and OSC, oscilloscope.

Then the encrypted signals are coupled from single-mode fibers to free space via collimators that can be described as Eq. (S1) in the Supplementary Material. After the construction of spatial light modulators, the beams carrying the encrypted signals is, respectively, converted into vortex beams of OAM $\pm 3$, OAM $\pm 5$, OAM $\pm 7$, OAM $\pm 9$, and OAM $\pm 11$, as shown in Eq. (S2) in the Supplementary Material. Similarly, the driving chaotic signal is also converted from a Gaussian beam to an OAM $+1$ beam, multiplexed with other encrypted signals through beam splitters and transmitted to the receiver via free space, which can be described as Eq. (S3) in the Supplementary Material. At the receiver, the multiplexed OAM beams are subjected to an opposite topological charge and filtered to demultiplexing, the optical field can be expressed as Eq. (S4) in the Supplementary Material. The demultiplexed driving chaotic signal will be injected into the laser ${\mathrm{SL}}_{\mathrm{R}}$ to generate synchronous chaotic signal $E_{\mathrm{ch}}^{\prime }(t)$. The demultiplexed encrypted signals are differenced from the synchronous chaotic signal to recover information signals $E_n^{\prime }(t)=E_n^{\mathrm{en}{\mathrm{c}}^{\prime }}(t)-a_n^{\prime }{E}_{\mathrm{ch}}^{\prime }(t)$.

The commonly driven chaotic synchronization mechanism is used in the OAM-CCL-based communication system.^36,37 This scheme has the advantages of simplicity, robustness, fast synchronization, and suitability for multichannel encryption. When the driving chaotic signal is injected into a pair of parameter matching slave lasers with a certain intensity, the two slave lasers will produce highly synchronous chaotic signals for encryption and decryption. We use Eqs. (S5)–(S7) in the Supplementary Material to describe the complex chaotic synchronization dynamics and stable spatial modes in the OAM-CCL optical communication system.

3.1.

Multidimensional High Security

The OAM-CCL-based communication system inherits the security of traditional chaotic secure optical communication. Figure 3 shows the simulation results of the OAM-CCL-based communication system in the time domain. Figure 3(a) shows the time waveform of the chaotic lasers ${\mathrm{SL}}_{\mathrm{D}}$, ${\mathrm{SL}}_{\mathrm{T}}$, and ${\mathrm{SL}}_{\mathrm{R}}$. The synchronization coefficient of the two slave lasers ${\mathrm{SL}}_{\mathrm{R}}$ and ${\mathrm{SL}}_{\mathrm{T}}$ is 0.976, and the synchronization coefficient between the driving laser ${\mathrm{SL}}_{\mathrm{D}}$ and the slave laser ${\mathrm{SL}}_{\mathrm{T}}$ is 0.763, which means that attackers cannot use the driving chaotic signal for decryption, ensuring the security of the system. The effective bandwidths (80% energy) for driving and masking chaotic signal are 11.73 and 13.35 GHz, respectively. Figure 3(b) shows the variation of the power spectrum of the information signal before and after encryption with $\rho =0.187$. The masking coefficient $\rho$ is defined as the ratio of the amplitude of the information signal to the amplitude of the chaotic signal.³⁸ The encrypted signal exhibits a noticeably flat power spectrum, indicating effective masking of the information signal by the chaotic signal. In Fig. 3(c), the impact of different masking coefficients on the BER is shown for both legitimate and illegitimate decryption at an OSNR of 20 dB. It can be seen that even if the signal is completely eavesdropped, the system still achieves secure communication within a masking coefficient range of 0.31 to 0.56. In Fig. 3(d), the BER for legitimate decryption of the system is presented. It can be observed that increasing the masking coefficient enhances the system’s tolerance to noise, but it may reduce the system’s security level.

Fig. 3

System performance. (a) Timing waveforms of ${\mathrm{SL}}_{\mathrm{D}}$, ${\mathrm{SL}}_{\mathrm{T}}$, and ${\mathrm{SL}}_{\mathrm{R}}$; (b) power spectrum before and after encryption; (c) BER versus masking coefficient at OSNR = 20 dB; and (d) BER versus masking coefficient at different OSNRs.

In addition to the conventional temporal domain security discussed above, the OAM-CCL communication system also demonstrates remarkable resistance against eavesdropping in the spatial domain. These outcomes are elucidated further in Sec. S5 of the Supplementary Material. We substantiate this eavesdropping resilience by constructing an eavesdropping channel. Our results indicate that compared to the free-space optical communication system, Eve’s eavesdropping on the OAM-CCL-based communication system for high-quality interception requires a larger size and positioning closer to the optical axis. In this case, Eve’s interception of the beam at such a close position will cause Bob’s alarm.

3.2.

Massive Capacity with Compatibility

For the OAM-CCL-based communication system, the limit value of communication rate $C_{\lim }$ can be described as Eq. (S9) in the Supplementary Material. The parameters $C_{\max }$ and DoF determine the value of $C_{\lim }$, which represents the system single-mode maximum rate and the system degree of freedom, respectively. $C_{\max }$ is related to the bandwidth of the chaotic carrier and masking coefficient.³⁹ Figure S6 in the Supplementary Material shows the relationship between a single-mode information rate and BER for the system with different masking coefficients at OSNR = 20 dB. The relationship between the mode number and the system capacity of the system with different masking coefficients is shown in Fig. 4(a). It is clear that the communication rate increases in proportion to the mode number.

Fig. 4

System capacity and degrees of freedom. (a) System capacity versus mode number for different masking coefficients at OSNR = 20 dB. (b) The relationship between receiver size and transmission distance for different beam waists, $R_R=10\mathrm{cm}$ and $R_T=10\mathrm{cm}$.

Theoretically, OAM modes have infinite degrees of freedom, but in practice, because of the divergence nature of OAM beams, the degrees of freedom of the system are limited under finite physical conditions. Here, we modify the study result of Sawan et al.⁴⁰ and considering the beam limitation by the transmitter, define the DoF for the OAM-CCL-based communication system:

Figure 4(b) shows the relationship between transmission distance and DoF for different initial beam waists, as the transmission distance increases, the DoF tends to decrease. At close range ($<345\mathrm{m}$), the smaller the initial beam waist is, the higher the DoF in the system is, and the opposite is true at long range. And the receiver size plays a role in constraining the degrees of freedom, as depicted in Fig. S7 in the Supplementary Material. Therefore, in order to ensure the freedom of the system, the appropriate transmitter size, receiver size, and initial beam waist radius should be selected depending on the communication distance.

3.3.

Optimizable Misalignment Robustness

In practical applications, mechanical vibrations and wind variations can cause minor displacements or jitters, which may cause misalignment between the transmitter and receiver. Such a misalignment leads to a decrease in orthogonality between OAM modes,^43–47 thus giving rise to a deterioration of chaotic synchronization and communication quality in the OAM-CCL-based communication system. The basic forms of misalignment include angular error and displacement error between the transmitter and the receiver, shown as Fig. S10 in the Supplementary Material.

By adjusting the basic parameters, we find that the OAM-CCL-based communication system exhibits robustness to misalignment. Figure 5(a) shows the variation of the synchronization coefficient when displacement error and angle error occur simultaneously. Figure 5(b) shows the BER variation with a mode spacing of 2 under the presence of displacement error and angular error, when $\rho =0.187$, ${\omega }_0=2\mathrm{cm}$. Beyond that, we have also investigated the change in system performance when displacement and angular errors occur independently, and the specific results are shown in Figs. S11–S14 in the Supplementary Material. We find that the larger the beam waist is, the more robust the system is to displacement error, and the less robust it is to angular error. Larger mode space can reduce the crosstalk between modes, but it may increase the overall power loss of the system. Therefore, the choice of initial beam waist and mode space should be carefully considered in the system design.

Fig. 5

System performance with misalignment. (a) Synchronization coefficient under simultaneous displacement and angular errors. (b) BER of OAM +3 mode under simultaneous displacement and angular errors, with $\rho =0.187$, ${\omega }_0=2\mathrm{cm}$, and mode space = 2.

3.4.

Discussions and Prospects

As a conceptual illustration, we built a prototype system of OAM-CCL-based communication system, as shown in Fig. 6. The prototype system is used to simulate image transfer experiment, and the simulations together confirmed the above conclusions, as detailed in Sec. S11 in the Supplementary Material. More interestingly, the OAM-CCL, as a new type of hybrid laser, is in a position to realize the coupling of temporal chaotic laser and spatial OAM in an all-optical way, which means that more dimensions can be fused together. For example, it is theoretically possible to incorporate up to 256 polarization dimensions to the OAM-CCL using commercially available spatial light modulators, and we predict that the security of the OAM-CCL-based communication system will be further improved⁴⁸ (legal $\mathrm{BER}<10^{-4}$ and illegal $\mathrm{BER}\gg \mathrm{FEC}$), and the system capacity can theoretically reach the order of $66.56\mathrm{Tb}/\mathrm{s}$³³ (26 OAM × 256 Pol.), while the requirements of the system on the environment to be more loosened. In addition, we simulate the communication performance of the system under turbulent conditions, details of which are given in Fig. S16 in the Supplementary Material. The results show that the system still performs well under the moderately turbulent condition, meaning that the system has the potential to be used in a variety of environments, such as underwater and atmospheric.

Fig. 6

Illustration of the indoor communication simulation for the OAM-CCL-based communication system.

In order to improve the robustness of chaotic synchronization, we use the commonly driven chaotic synchronization mechanism as the core of the encryption and decryption mechanism in the OAM-CCL-based communication system. Interestingly, the OAM modes allow the OAM-CCL-based communication system to have multiple channels, so we can use different channels to transmit the driving chaotic signal and the encrypted signal, respectively, reducing the interference between each other, thus improving the quality of chaotic synchronization.

So far, the OAM-CCL-based communication system is in its infancy. Its practical implementation still faces several significant challenges, including beam divergence, alignment issues, atmospheric turbulence, and the lack of standardized protocol and interface. Therefore, in future endeavors, it is crucial to integrate the OAM-CCL-based communication with beam shaping technology,⁴⁹ adaptive optics technology,⁵⁰ and a tracking and capture system⁵¹ to overcome the challenges posed by divergence, misalignment, and turbulence. Furthermore, there is a pressing necessity to formulate communication protocol standards aimed at augmenting interoperability and compatibility between the OAM-CCL-based communication system and the established optical communication infrastructure.

Our team has done extensive preliminary research on chaotic secure communication and OAM light beams, respectively. The former mainly includes long-range chaotic synchronization,⁵² chaotic lasers for high-speed classical key distribution,⁵³ and coherent chaotic optical communication.³⁷ The latter mainly consists of OAM generation,⁵⁴ manipulation,²⁹ and application.^30,31,33 These above achievements have laid a solid foundation for the experimental realization of OAM-CCL and OAM-CCL-based communication systems. As a result, we believe it is reasonable to use the conceptual paradigm of OAM-CCL to establish a superior-security, massive-capacity, high-fidelity, and robust optical communication.