1. Introduction

Optical chaos attracts great attention due to its unique characteristics of the wide spectrum and quasi-noise, and it has been widely applied in chaotic secure communication [1,2], pseudo-random number generation [3,4], lidar [5], reservoir computing [6], etc. Compared with chaotic laser communication systems composed of discrete devices, photonic integrated circuits have their unique advantages, such as smaller size, lower cost, better stability, and suitability for mass production, so the miniaturization and integration of chaotic laser sources is a research hotspot. Commonly, chaotic laser generators can be broadly subdivided into two categories: all-optical feedback and optoelectronic ones. The all-optical feedback system primarily relies on rate equations, as described in Refs. [2,7]. This system directly employs optical signals as feedback, routing them back to the laser through optical paths to influence its dynamic behavior. In contrast, the optoelectronic feedback system is primarily based on electronic filter equations, as detailed in Refs. [1,8]. In this system, the optical output signal is first converted into an electrical signal. The electrical signal processed by an electronic filter is subsequently fed back to the modulator to modulate the laser output. For available integration technology, photonic integrated chaotic semiconductor lasers with all-optical feedback come into being. A photonic monolithic integrated device consisting of a distributed feedback laser (DFB), a gain-absorption section, a phase section, and a long passive waveguide has been studied as a compact potential chaotic optical communication emitter [9], which firstly confirms high-dimensional broadband chaos can be generated by chaotic laser photonic integrated circuits (PIC). Common-signal-induced synchronization in two photonic integrated circuits comprising a DFB and a short external cavity is experimentally achieved [10]. The aforementioned PIC can be classified as a 4-section chaotic laser generator structure, and it’s reported that a chaotic source operated in the long-cavity regime reveals better performance than in the short-cavity regime [11]. Besides the 4-section structure, Wu, etc. proposed a 3-section solitary monolithic integrated amplified feedback laser (AFL) chip consisting of a DFB section, a phase section, and an amplification section. The 25 GHz broadband chaos generated by the AFL possesses significant dimension and complexity, and further research on dynamic transition routes under different amplifier section currents is presented [12]. In addition, a monolithically integrated laser with a distributed Bragg reflector (DBR) is designed [13]. DBR grating can provide wavelength detuning to generate mode beating, which enhances chaotic bandwidth and distributed feedback to suppress time delay signature (TDS). Apart from monolithic integration, the hybrid integrated chaotic semiconductor laser is designed and fabricated as well [14]. By simply replacing the inbuilt isolator with a translative mirror in a conventional DFB laser module configuration, this laser has advantages such as easy production and low cost. Compared with the single-cavity feedback mentioned above, the multi-cavity optical feedback structure [15] is more suitable for eliminating or suppressing the TDS of the chaotic laser. For instance, an introduced air gap and a highly reflective film constitute the three-cavity feedback which can generate a high-dimensional chaotic laser with lower feedback intensity [15]. Besides straight-cavity structure, it’s reported that a novel chaos semiconductor laser chip in which a DFB laser, two semiconductor optical amplifiers (SOAs) and a photodiode (PD) are monolithically integrated with a passive ring waveguide [16]. The passive ring works as a delayed feedback loop in replace of cleaved facet mirror and high-reflection coating. To get more compact device, chaotic laser chip with a two-dimensional external cavity can be achieved sufficient delay time for multiple reflections in a small area [17]. Furthermore, there are some reports about secure data transmission by integrated chaotic laser chip based on all-optical feedback. 2.5 Gbps data sequences with small amplitudes are completely encrypted within these chaotic carriers [18], and the efficiency of chaos encryption utilizing all-optical feedback technique was investigated as well [19]. Recent research has been intensive in pursuing high-capacity chaotic secure optical communications. In 2022, a wavelength-tunable chaotic semiconductor laser chip is designed and fabricated to enable chaos wavelength division multiplexing [20,21]. In 2023, 400 Gbps physical random number generation based on deformed square self-chaotic lasers is proposed [22]. Furthermore, a 15-channel random number generation system with 20 Gbps bit channel rate is achieved using silicon photonic WDM and receivers [23]. These advancements hold immense potential in elevating the aggregate rate of on-chip random number generators. In addition, recent years have seen some studies on chaotic semiconductor quantum dot light-emitting diodes (QD-LEDs). In 2011, it’s experimentally demonstrated that chaos can be observed for quantum-dot microlasers operating close to the quantum limit at nW output powers [24]. Chaos synchronization of delayed quantum dot light-emitting diodes has been studied theoretically in 2020 [25]. In 2021, the nonlinear dynamics of the sole ground-state emitting QDL (GS-QDL) under external optical injection is numerically studied, which can be used in isolator-free photonic integrated circuits [26]. In 2023, it’s proposed to use four parallel optical reservoirs to predict the complex chaotic dynamics of the four polarization components output by a driving QD spin-VCSEL, respectively [27].

All-optical feedback system takes advantage of the very large bandwidth offered by optical fiber but suffers from limitations in reliability, stability, and mechanical robustness [28]. The all-optical feedback system is particularly sensitive to optical polarization and optical phase. However, the chaotic laser source based on optoelectronic feedback can suppress sensitivity to optical phase and polarization through the photoelectric conversion of the photodetector and exhibits chaotic behavior within a certain parameter range. In addition, another advantage of the chaos source system based on optoelectronic feedback is its flexibility. All-optical feedback systems mainly rely on the physical properties of the laser. These properties are fixed and determined during the laser manufacturing process, but their control is often imprecise. Nevertheless, chaos sources based on optoelectronic feedback can flexibly control the output state by adjusting the laser power or the bias voltage of the modulator. Therefore, the integrated chaos sources based on optoelectronic feedback are more suitable for practical long-distance communication. Based on the integration technology nowadays [29], the hybrid photonic integration design method can realize the integration of a laser chaotic communication system based on optoelectronic feedback. Si-based passive devices are integrated with InP-based active devices, which can make better use of the currently mature Silicon-on-Insulator (SOI) process on Si or insulating substrates. However, there is a lattice mismatch between InP-based materials and silicon-based materials, thus, efficient coupling of III-V light sources to silicon photonic circuits is one of the key challenges of integrated optics. Moreover, the luminous surface and volume of the integrated device are small, so the alignment is difficult and the lack of alignment accuracy cause large optical loss. Photonic wire bonding (PWB) [30], analogous to electrical wire bonding, is an optical bonding technology that enables the interconnection of different optical chips, optical chips and optical fibers. PWB technique is based on three-dimensional (3D) in-situ structuring of negative-tone resist materials with two-photon polymerization (TPP). By combining accurate positioning and alignment techniques, it achieves three-dimensional bonding between polymer and photonic chip end faces. In this way, this scheme avoids the time-consuming alignment adjustment in the traditional scheme, saves the lenses required for beam shaping, and is conducive to large-scale production for a simple and fast preparation process. Efficient optocouplers can be realized automatically [31]. PWB enables insertion losses below 0.4 dB between InP-based horizontal-cavity surface-emitting lasers and passive silicon photonic circuits [32], making it suitable for coupling laser arrays and modulator arrays.

In this paper, a hybrid integrated chaotic laser chip based on optoelectronic feedback is proposed. Through the hybrid integration of a 2-channel DFB laser array, a 2-channel single-drive intensity modulator array, an optical coupler, and a photodetector based on photonic wire bonding technology, the proposed chip consists of the chaos generation module and signal modulation module of the intensity-chaotic laser communication system. To keep a small size, the chip is integrated without external components, such as optical fiber or external cavity. However, chip-scale chaotic systems based on optoelectronic feedback are subject to additional size and feedback gain limitations compared with discrete systems, so it is important to study the relevant design of this chip. The overall length of our proposed chip is only 5 mm. Details about chaos performance both in the time domain and frequency domain are reported, and the revolution of the output chaotic waveform from determination to period to quasi-period to chaotic state can be observed. Under the short-delay condition of the photonic integrated circuit scale, this chip can generate a complex chaotic signal, which is similar to Gaussian white noise. Moreover, the output chaotic signal has the better masking capacity of TDS. Furthermore, the performance of 1 Gbps back-to-back communication transmission using the proposed photonic integrated chips is demonstrated, and it can be seen that the authorizer can achieve qualified synchronization and the eavesdropper can achieve BER higher than 10⁻², which is generally considered as a condition where communication cannot be achieved [33]. Finally, 100 MHz chaos generation experiment is completed with a short delay, which verifies the integration feasibility preliminarily.

2. Experimental setup and principle

2.1 Experimental setup

The architecture of the hybrid integrated chaotic laser chip is illustrated in Fig. 1(a), incorporating an InP waveguide-integrated DFB laser array (consisting of two DFB lasers) and a monolithic silicon chip onto the sub-mount. The silicon chip consists of three key devices, a single-drive Mach-Zehnder modulator (SDMZM) array (consisting of two SDMZMs), a 3 dB Optical coupler (OC), and a PD. Figure 1(b) presents the operation principles of the integrated chip. Continuous light with a center wavelength of 1550 nm generated by two DFB lasers is routed to the SDMZMs, separately. The input radio-frequency (RF) signal is applied to the upper SDMZM to implement the modulation of the signal light. The drive voltage of the lower SDMZM is generated by the optoelectronic oscillators (OEO) with time-delayed feedback. An OEO is composed of the lower SDMZM, the OC, and the PD. Electrical-to-optical conversion performs in the MZM and optical-to-electrical conversion occurs in the PD. Both the optical path and electrical path induce the time delay. The pseudorandom voltage from the PD is generated by this closed-loop system which is always nonlinear, dissipative, and infinite dimensional because of the time delay. The offset phase Φ_DC of the SDMZMs is adjusted via thermal phase shifters. Coupling between the modulated signal light from the upper SDMZM and the modulated chaotic light from the lower SDMZM occurs in the OC. Subsequently, the transmitted light with the chaotic carrier is split into two paths, and 50% of optical power is emitted to the channel while the other 50% returns to the feedback loop.

 

Fig. 1. (a) Schematic diagram of the hybrid integrated chaotic laser chip. (b) The operation principle of the integrated chaotic laser chip.

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In the conceived hybrid integrated chaotic laser chip, the integrated InP DFB laser array (size: 500 µm × 250 µm) is based on a buried-heterostructure waveguide structure, whose wavelength is around 1550 nm. The monolithic silicon photonics chip is fabricated on a SOI platform leveraging CMOS-compatible processes and the total size is 2.5 mm × 1.5 mm. The PWB length between InP chip and SOI chip is around 300 µm, and the diameter of the spot is better between 3 µm and 9 µm. The optical waveguides of SDMZM array are around 300 µm long. The whole length of the OC waveguide is 300 µm. The footprint of PD is about 1 mm × 1 mm. The InP laser chip and the monolithic silicon chip were hybrid integrated with the help of PWB and assembled into a compact package and the optical waveguide is 300 µm. The optical waveguide between SDMZ array and OC and between OC and PD are 300 µm long. The RF and DC pads on the chip are wire-bonded to a custom-designed print circuit board (PCB) for electrical connections.

2.2 Principle

Whether it is a discrete chaotic communication system or a photonic chip-scale chaotic communication system, the satisfied chaotic dynamics equations are consistent, as shown in [34]:

Since the form of the chaotic dynamic Eq. (1) is an integrodifferential equation, according to the relevant mathematical properties of the delay differential equation, the solution of the equation is affected by the parameters of the equation. When the parameters meet certain conditions, the system can output chaotic signals. To simplify the form, β=πR_dGLP/2V_π is noted as the feedback gain, and x(t)=πV(t)/2V_π. As shown in Eq. (3) [35], it is the characteristic equation corresponding to Eq. (1) at the single stationary point (x_st, y_st) = (0, βcos²Φ_DC/ɛ),

3. Simulation analyses and experimental results

3.1 Time-domain and frequency-domain analysis

Here, the time constants τ and θ are set to 5 µs and 25 ps, respectively. Assuming that the other initial conditions of the system are unchanged, the system output status can be observed intuitively in Fig. 2 as the delay time T changes. The time series evolutions of the system are presented in Fig. 2(a)-(c), whereas the corresponding phase space trajectories are depicted in Fig. 2(d)-(f). Given that the dynamic equation is expressed as an integrodifferential function, the three axes of the three-dimensional phase space correspond to the amplitude of the output signal, the derivative of x with respect to time t, and the integral of x with respect to t, respectively. To gain a clearer distinction among various output states, the associated power spectrum density (PSD) is exhibited in Fig. 2(g)-(i). As shown in Fig. 2(a), when the delay time T is set to 8 ps, the system output exhibits a periodic state. The corresponding phase diagram, depicted in Fig. 2(d), reveals a limit cycle. Based on Ref. [37], from a fixed point to a limit cycle, the first Hopf bifurcation occurs. Consequently, distinct characteristic peaks are observed in the PSD shown in Fig. 2(g). As T increases to 11 ps, the system output transitions to a quasi-periodic state as shown in Fig. 2(b), undergoing the second Hopf bifurcation. The corresponding phase space shown in Fig. 2(e) manifests as a periodic attractor [38]. Additionally, due to the superposition of multiple waveforms with distinct periods, the power spectrum shown in Fig. 2(h) becomes more intricate than that of a periodic signal, although some characteristic frequencies can still be discerned. When the delay is further increased to 40 ps in Fig. 2(c)(f)(i), the output series enters a chaotic state. The corresponding phase diagram evolves into a singular attractor, and no distinct frequency peaks are observed in the PSD depicted in Fig. 2(i) [39,40].

 

Fig. 2. (a)-(c) Time series of the output signals when T = 8 ps/11 ps/40 ps. (d)-(f) Phase space trajectory when T = 8 ps/11 ps/40 ps. (g)-(i) Power spectrum when T = 8 ps/11 ps/40 ps.

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Permutation entropy (PE) [41] can be used to quantitatively assess the complexity of the signal. A larger permutation entropy indicates a higher level of complexity. PE is calculated with an ordinal pattern length of L = 6, and the total data length is 10⁶. When T is 11 ps, PE reaches 0.6612. As T increases to 40 ps, the output signal PE reaches 0.9136. Meanwhile, the Lyapunov exponent [42] is a unique quantitative indicator for measuring the complexity of output signals in chaotic systems. The largest Lyapunov exponents at T = 8 ps and T = 11 ps are negative (−0.2254 and −0.0001 respectively), while the largest Lyapunov exponent at T = 40 ps is positive (0.4796), indicating that the output signal on this condition is chaos. In summary, the system can generate chaotic output when the delay time is comparable to the inverse bandwidth τ. To study the effect of feedback gain β and time delay T on the output state, the time-domain and frequency-domain properties of the system's output signals under various initial conditions are analyzed.

The probability density functions (PDF) of the system output signal are shown in Fig. 3, and the orange curves shown in the diagrams are the PDF of the system output signal when T = 40 ps. According to [43], among distributions with given expected value and variance, the one with the maximum entropy is necessarily the Gaussian distribution. Therefore, if the PDF of an output signal fits well with a Gaussian distribution function, it indicates that the signal exhibits a relatively high complexity. When the feedback gain β is set to 4 as shown in Fig. 3(a), the amplitude range of the output signal gradually expands as the delay time T varies from 0 ps to 100 ps. Specifically, at T = 40 ps, the orange curve exhibits irregularity and fails to align with a Gaussian distribution. Even if T further extends to 100 ps, the PDF does not fit Gaussian distribution. This indicates that with a reduced feedback gain, the generation of high-complexity chaotic signals under conditions of short delay becomes impractical. When β is 8 as shown in Fig. 3(b), with the increase of time delay T, the system output signal is closer to the unimodal Gaussian distribution. There is a delay time, beyond which the state of the delayed system output state is almost unchanged. In other words, after the delay reaches a certain threshold, the system output state is mainly affected by the feedback gain β. When β reaches 10. The PDF curve at T = 40 ps is closer to the Gaussian distribution as shown in Fig. 3(c). As β is further increased to 12, the distribution curve shown in Fig. 3(d) corresponding to T = 40 ps experiences minimal changes. This indicates the existence of a feedback gain threshold of around 10. Once β reaches this threshold, the system can produce a satisfactory chaotic waveform.

 

Fig. 3. The probability density functions (PDF) of the output signal at (a) β = 4 (b) β = 8 (c) β = 10 (d) β = 12.

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Figure 4 presents the power spectrum density of the output signal under different time delays with feedback gains of 10 and 12. Both in Fig. 4(a) and Fig. 4(b), prominent characteristic peaks can be observed within the range of T < 40 ps. These characteristic peak frequencies correspond to the inherent time parameters of the system and are crucial keys that need to be masked in a confidential communication system. As the delay time T increases, the characteristic frequency peaks gradually disappear, evolving into a relatively flat power spectrum. Additionally, as the frequency increases, the amplitude of the spectrum gradually decreases due to the bandwidth limitation. This spectral distribution is similar to the spectrum of white noise. Therefore, for systems with a delay time of tens of picoseconds, chaotic signals resembling Gaussian white noise can be generated.

 

Fig. 4. The power spectrum density of the output signal. (a) β = 10 (b) β = 12.

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3.2 Time delay signature concealment analysis

In addition to using the PSD to observe the eigenfrequency, the autocorrelation function (ACF) [44] is a more effective mathematical-statistical method commonly used to find TDSs. ACF is used to calculate the correlation of signal sequences with different delay intervals, and the delay point with a large absolute value of the correlation coefficient is related to the TDS. The ACF is defined as follows [44],

 

Fig. 5. Autocorrelation function (ACF) T = (a) 50 ps (b) 100 ps (c) 500 ps (d) 1000 ps.

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3.3 Confidentiality and synchronization analysis

To evaluate the confidentiality and synchronization of the chip, the bit error rate (BER) of 50 ps (corresponding to 1 cm) of the authorizer and the eavesdropper is discussed under the short-term delay condition as shown in Fig. 6. Mismatch noise [46] occurs even in a back-to-back transmission due to the unmatched device parameters at the transmitter and receiver, the shot noise, thermal noise, and dark current of PD [47]. The mismatch ratio is defined as the ratio of the device parameters at the transmitter and receiver ends. When the mismatch ratio is 0.985 as shown in Fig. 6(a), the BER of the authorizer is lower than 10⁻⁶, while the BER of the eavesdropper can be maintained higher than 10⁻⁴. When the message masking ratio α is 1.025 shown in Fig. 6(b), the eavesdropper's BER is still higher than 10⁻². Stable BER performance is better maintained at low feedback gain due to lower mismatch noise and detector noise. Under the better mismatch ratio (0.990), the BER trend is the same, but the communication performance of the system is less affected by the gain condition. The above results show that the system can ensure good confidentiality and synchronization under the condition of short delay time.

 

Fig. 6. BER of 50 ps (corresponding to 1 cm) of the authorizer and the eavesdropper at (a) mismatch radio: 0.985, (b) mismatch radio: 0.990.

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While the research on discrete chaos generation systems based on the classical optoelectronic feedback structure is well-developed, there is a scarcity of studies on integrated chaos chips using this classical structure. The results demonstrate the feasibility of generating chaos signal output through the hybrid integration of lasers, MZ modulators, couplers, and photodetectors. The proposed chip avoids bulky delay components such as long optical fibers, free-space propagation, and external cavities. Instead, it utilizes internal devices and waveguides to provide the optical path for feedback signal delay, simplifying integration and maintaining a smaller chip size.

3.4 Equivalent experiment results

Finally, to further verify the feasibility of our proposed chaotic integrated chip conception, we use an intensity-chaotic laser generation system composed of discrete components to verify that optical chaos can be generated under short-delay conditions. The time delay T is expected to be close to τ. Due to the existence of the pigtail of the optoelectronic devices, the total loop length of the optical path and electric path, including the intensity MZM, 50:50 OC, PD and low-pass electric filter, is about 2.25 m, and the corresponding delay time T is about 11.25 ns. To keep the inverse bandwidth and loop delay time in the same order of magnitude, a 100 MHz low-pass filter is chosen to limit the bandwidth of the entire system, with a corresponding high-frequency cutoff time τ of 1.6 ns. The T/τ was within the range of 1∼10, which was consistent with our previous numerical simulation. As shown in Fig. 7, under this work condition, the generator achieved successful excitation. The signal waveform captured by the oscilloscope, depicted in Fig. 7(a), exhibits irregular and random fluctuations. Additionally, Fig. 7(b) illustrates the frequency spectrum of the signal, revealing the absence of discernible peak frequencies. Furthermore, Fig. 7(c) presents the ACF of the collected data, indicating the absence of distinct TDS. Similarly, the positive value of the largest Lyapunov Exponent (0.3035), as shown in Fig. 7(d), confirms the chaotic nature of the output signal. Lastly, the PE exceeding 0.9 demonstrates a certain degree of complexity.

 

Fig. 7. (a) Signal waveform captured from the oscilloscope. (b) Frequency spectrum of the captured signal. (c) Autocorrelation function (ACF) of the captured signal. (d) The largest Lyapunov Exponent of the captured signal.

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4. Conclusions

In summary, we propose a hybrid integrated chaotic laser chip using optoelectronic feedback. The chip size design and power consumption are depended on feedback gain and delay time, respectively. We study the system output under the two parameters mentioned above by numerical simulation. Simulations show that the chip's output signal has high permutation entropy, revealing a singular attractor at T = 40 ps. The signal distribution is Gaussian-like and the power spectrum is flat, indicating signal’s complexity. TDS concealment is improved with similar time delay and inverse bandwidth settings, sacrificing some signal complexity. The chip meets the need for security and synchronization performance in secure communications. Furthermore, the equivalent experiments confirm its feasibility. Leveraging the advantages of low loss and automation offered by PWB, the realization of a hybrid integrated photonic chip based on optoelectronic feedback is expected. Our work is significant for practical chip-based chaotic laser systems in communications.