Ultrafast Physical Random Bit Generation Based on an Integrated Mutual Injection DFB Laser

Abstract

Ultrafast physical random bit generators (PRBGs) are essential components for modern applications in secure communication, quantum cryptography, encrypted optical fiber sensing and artificial intelligence. While optical chaos-based PRBGs offer high-speed capabilities, conventional systems often rely on discrete components that suffer from system complexity and environmental instability. This paper proposes and experimentally demonstrates a robust, integrated solution using a two-section mutual injection DFB laser. The device was fabricated using the reconstruction equivalent chirp (REC) technique, which provides precise control over grating phase variation while utilizing low-cost, high-volume fabrication methods. The laser sections, each measuring 450 μm in length, were designed with a free-running wavelength difference of 0.3 nm to ensure a flat optical spectrum and enhanced chaotic dynamics. By optimizing the bias currents, we achieved a chaos RF bandwidth of 20.1 GHz. Notably, the resulting chaotic signal lacks time-delayed signatures, which simplifies the randomness extraction process. To generate random bits, the chaotic waveform was sampled by an 8-bit analog-to-digital converter at 100 GSa/s. Following post-processing through delay-subtracting and the extraction of the four least significant bits (4-LSBs), we realized a total physical random bit rate of 400 Gb/s. The randomness of the generated sequence was successfully verified using the NIST SP 800-22 statistical test suite. This approach offers a compact, energy-efficient, and high-performance integrated chaotic source suitable for secure communication and high-performance computation.

1. Introduction

Ultrafast physical random bit generators (PRBGs) are key components for the Monte Carlo method, making them suitable for applications in secure communication, quantum cryptography, encrypted optical fiber sensing, and artificial intelligence [1,2,3,4]. PRBGs can be realized through two main mechanisms, thermal noise and optical chaos, which rely on the intricate dynamics of chaotic physical systems [5,6]. The latter has been studied for decades to overcome the limitations of electronic approaches by utilizing the ultra-fast nature of optical processes. The chaotic lasers have attracted considerable attention for generating high-speed PRBGs owing to their large bandwidth and intensive randomness. In 2008, Uchida et al. first proposed the high-rate generation of random bit sequences of 1.7 Gb/s using two independent chaotic semiconductor lasers, which demonstrated the simplicity and high-speed capabilities inherent in PRBGs based on optical chaos [7]. Then, the total PRBG rate was pushed up to several hundreds of gigabits per second and even terabits per second by using chaotic lasers with various external perturbations, such as optical feedback, optical injection, and optoelectronic feedback [8,9,10,11,12]. However, in most of the above-mentioned situations, the discrete devices were used for PRBGs, which will cause a complex system and vulnerability to environmental influence. Recently, based on integrated chaotic lasers, such as microcomb [13], amplified spontaneous emission (ASE) [14], and microlaser [15,16], led to the realization of ultrafast PRBGs characterized by simplified systems and enhanced stability. Moreover, the integrated mutually coupled distributed feedback (DFB) laser was used in the chaos synchronization due to its ultra-short coupling delay, which shows the enormous potential for applying it to the ultrafast PRBGs [17].

In this paper, we propose and experimentally demonstrate a 400 Gb/s PRBG based on an integrated mutual injection DFB laser. The chaotic laser consists of the two-section mutual DFB laser with a free-running wavelength difference of 0.3 nm, which has a chaos radio-frequency (RF) bandwidth of 20.1 GHz. We successfully realize 400 Gbps random number sequences with verified randomness by adopting a multi-bit extraction method. The demonstrated performance suggests the proposed scheme is a strong candidate for integrated PRBGs.

2. Design and Fabrication

The reconstruction equivalent chirp (REC) technique has recently been introduced as a method to effectively achieve grating chirp and phase shifts, offering precise control over grating phase variation [18,19]. This approach utilizes low-cost, high-volume fabrication methods involving holographic exposure and micrometer-level contact exposure, steering clear of the more expensive and intricate electron beam lithography. It employs the high-order frequency response that the sampling pattern generates on a homogeneous grating to achieve laser resonance. The index modulation change n(z) of the REC-based Bragg grating is given by [20]:

$$\Delta n(z)={\displaystyle \frac{\Delta n_0}{2}}{\displaystyle \sum _m{F}_m}\exp \left[j2\pi \left({\displaystyle \frac{z}{{\Lambda }_0}}+{\displaystyle \frac{mz}{P}}-{\displaystyle \frac{m\Delta P}{P}}\right)\right]+c.c.$$

(1)

where F_m represents the mth order Fourier coefficient, Λ₀ is the period of the seed grating, n₀ is the index modulation of the seed grating, P and ΔP are the sampling period and the phase shift in the sampling structure, respectively. According to Equation (1), an equivalent phase shift m = 2πmΔP/P can be generated in the m^th order sub-grating with the variable length of the phase shift ΔP. The Bragg wavelength of +1^st order is usually used and its order period can be expressed as,

$${\lambda }_{+1}=2n{\Lambda }_{+1}=2n{\displaystyle \frac{{\Lambda }_{0}P}{{\Lambda }_0+P}}$$

(2)

where λ₊₁, n, Λ₁, Λ₀, and P are the Bragg wavelength of +1^st order sub-grating, the effective index, the period of the +1^st order grating, the period of the uniform seed grating, and the sampling period, respectively. When the period of a uniform grating is determined, the lasing wavelength is solely dependent on the sampling period; thus, the grating fabrication tolerance is relaxed by the factor (P/Λ₀ + 1)², with the value of several hundred. In our device, the Bragg wavelength of the 0^th order sub-grating is designed at 1635 nm and the center of the gain spectrum is at 1550 nm. Figure 1a shows the schematic of the proposed integrated two-section mutual injection DFB laser, which consists of the LD₁ and LD₂, respectively. The free-running wavelength difference in the LD₁ and LD₂ is set to 0.3 nm. The lengths of the LD₁ and LD₂ are both 450 μm. There is a 5-μm-long electrical isolation region between the two lasers. The anti-reflection films (AR/AR) are coated on both facets with their reflectivities of ~1% to avoid the influence of the random facet phase. Figure 1b depicts the microscope image of the fabricated two-section mutual injection DFB laser chip. The device was fabricated in the EPIHOUSE (Xiamen, China), a commercial InP foundry in China. Two-step metal–organic chemical vapor deposition (MOCVD) was employed to grow a standard 1.55 μm InGaAsP/InP laser on a 2-inch n-InP substrate. During the first epitaxy growth process, an n-InP buffer layer, an n-InAlGaAs lower optical confinement layer, a compressive InAlGaAs multiple-quantum-well active structure, a p-InGaAsP upper optical confinement layer, and a p-InGaAsP grating layer were successively grown. Subsequently, the pre-designed sample grating was defined by the standard photolithography and inductive coupling plasma. A p-InP cladding layer, a p-InGaAsP transition layer, and a p-InGaAs contact layer were successively grown. The wafer was then further processed into the ridge waveguide lasers with a ridge width of 2 μm. After that, the Ti/Pt/Au electrodes were deposited through the standard photolithography and lift-off processes, in which the two adjacent electrodes were isolated by the electrical isolation layer. Finally, the lasers were obtained by wafer cleaving. The chaotic laser chip was further Butterfly packaged, which can protect the delicate chip from environmental factors and ensure stable operation.

3. Experiment Setup and Laser Test

The integrated two-section mutual injection DFB laser is tested under 25 °C controlled by the thermoelectric cooler using the experimental setup shown in Figure 2. DC bias is applied to both LD₁ and LD₂. Then, the optical signal is coupled into a tapered single-mode fiber, with the incorporation of an isolator to prevent optical feedback from other components within the system. The AQ6370 optical spectrum analyzer (OSA, Yokogawa, Tokyo, Japan) records the laser output with a resolution of 0.02 nm. Finally, an electrical spectrum analyzer (ESA, Rohde & Schwarz, Munich, Germany) and an oscilloscope (OSC, Tektronix, Beaverton, OR, USA) are adopted to record the RF spectrum and the time series of the electrical signal from a high-speed photodetector (PD, Finisar, Sunnyvale, CA, USA). The free-running wavelengths of the LD₁ and LD₂ are tested.

As shown in Figure 3a, when the LD₁ is biased at 49 mA and LD₂ is biased at a small current (~16 mA) to compensate for the material absorption, the free-running wavelength of LD₁ is 1539.85 nm. Under the same state, when the LD₂ is biased at 46 mA and LD₁ is biased at 16 mA, the free-running wavelength of LD₂ is 1539.54 nm. When they are free running, the −20 dB linewidths of LD₁ and LD₂ are 0.078 nm and 0.074 nm, respectively. Under the biased currents of LD₁ and LD₂ at 49 mA and 46 mA, the spectrum of the two-section DFB laser at a chaotic state subjected to mutual injection is shown in Figure 3a by the black line. The −20 dB linewidth is 0.568 nm, which is widely broadened compared to that of free-running of LD₁ and LD₂. The corresponding RF spectrum of the chaotic laser is shown in Figure 3b. The broad peak on the spectrum at approximately 8.5 GHz is related to the relaxation oscillation frequency of the chaotic laser, which is larger than that of free-running lasers due to the optical injection. The calculation of the 80% bandwidth considers the span between zero and the frequency where 80% of the total energy, excluding the noise floor, is encompassed [21]. In the optical feedback chaos depicted in Figure 3b, the 80% bandwidth measures 20.1 GHz. The AC waveform of the chaotic laser system is illustrated in Figure 3c, demonstrating a random distribution of large values characteristic of a typical chaotic state. Meanwhile, an auto-correlation function (ACF) curve is depicted in Figure 3d. Without the need for a feedback loop to estimate a chaotic state, such as the integrated chaotic laser, no time-delayed signature (TDS) is observed in the ACF of the two-section mutual DFB laser [22]. The inset in Figure 3d represents the entire ACF curve for 500 ns. The chaotic behavior of this integrated mutual-injection architecture has been theoretically and numerically verified in previous studies, showing high-dimensional complexity and positive Lyapunov exponents [17]. Our experimental results, characterized by significant linewidth broadening to 0.568 nm and a broad RF spectrum of 20.1 GHz, are in excellent agreement with the established chaotic regimes for this device structure.

We also investigate the influence of injection current on the behavior of the bandwidth of the chaotic laser. Here, the bias current of LD₁ is fixed at 49 mA and the bias current of LD₂ varies from 46 to 52 mA with the step of 2 mA. As shown in Figure 4a, the distinct multi-modal peaks of the lasing spectra can be observed with the increase in bias current. The corresponding radio frequency (RF) spectra are depicted in Figure 4b. The calculated 80% bandwidths of the chaotic laser are 20.1, 19.38, 18.14, and 16.3 GHz at 46, 48, 50, and 52 mA respectively. Owing to the modal beating of the multi-modal peaks, it causes the reduction in the flatness of the spectrum and the bandwidth. In the case of the two-section mutual injection laser, if the wavelengths of LD₁ and LD₂ are too close, it will lead to intensified competition between the two wavelengths and generate multi-modal peaks. Thus, in our design, the free-running wavelength difference in the LD₁ and LD₂ is set to 0.3 nm, which can provide a flat optical spectrum and a large bandwidth in the chaotic state.

4. Random Bit Generation

The two-section chaotic laser is utilized to generate physical random numbers. The bias currents of LD₁ and LD₂ are set to 49 mA and 46 mA, respectively. The chaotic laser waveform is converted into a binary stream through an 8-bit ADC operating at a sample rate of 100 GSa/s and the intensity histogram distribution of the 100 μs long raw data stream is illustrated in Figure 5a. The observed distribution displays asymmetry, a characteristic often associated with chaotic systems, particularly when subjected to a chaotic signal featuring random spikes. The inherent asymmetry in the distribution poses the risk of introducing bias into the generated random sequence. To mitigate this, additional post-processing techniques, including delay subtracting and extraction of the least significant bits (LSBs), were implemented for the PRBG [23]. To rectify the asymmetry and achieve a more balanced distribution, a delay difference is introduced to the raw data. It is crucial to note that the selection of the delay time is pivotal in eliminating the periodicity of the original chaotic signals. Theoretically and experimentally demonstrated criteria for delay time selection suggest that weak periodicity can be eliminated if the cross-correlation coefficient between the chaotic signal and its delayed counterpart is less than 0.007 [24]. Considering the very low correlation coefficient at 1 ns in Figure 3d, we select a delay time of 1 ns and obtain a symmetric distribution as shown in Figure 5b. The symmetric distribution is favorable for unbiased bit extraction. By retaining four LSBs, we can obtain a nearly uniform probability distribution, as shown in Figure 5c. Figure 5d shows the ACF of the generated bit sequence and the correlation is around the lower limit 1/$\sqrt{n}$, where n = 10⁶ is the length of the bit sequence.

Figure 6 shows the number of statistical tests passed by the National Institute of Standards and Technology Special Publication (NIST SP) 800-22 of random bits generated for different cases of M-LSBs. To pass the tests, the composite p-value (the uniformity of p-values) should be larger than 0.0001 and the corresponding proportion should be limited to the range of 0.99 ± 0.0094392 under the significance level α = 0.01 [25]. In this experiment, we use the 1000 sequences of 1-Mbit data to execute the standard statistical test suite with a significance level α = 0.01. It can be seen that, when M ≤ 4, the random bits generated can pass the 15 NIST tests, indicating that the random bits with verified randomness are achieved with minimal post-processing.

As shown in Figure 7a, the p-values of the 15 NIST tests are larger than 0.0001 under the case of extracting 4-LSBs. Meanwhile, the proportions of the tested random bits are within 0.99 ± 0.0094392. These results indicate the true randomness of the proposed PRBG.

In

Table 1. Comparison of different schemes of random bit generation based on an integrated source.

Scheme	Post-Processing	Sampling Rate (GSa/s)	Bandwidth (GHz)	Number of Channels	Bit Rate (Tb/s)	TDS	System Complexity
Quarter-wavelength-shifted DFB laser [11]	16-bit ADC + 13-LSB + XOR	80	11.2	1	1.04	High	Medium
Microlaser [26]	8-bit ADC + 2-LSB	10	11.6	1	0.01	Low	Low
Chaotic microcomb [13]	8-bit ADC + 3-LSB + XOR	40	9.6	32	3.84	No	High
Chaotic microcomb [27]	16-bit ADC + 8-LSB + XOR	40	~1	7	2.24	No	High
This work	8-bit ADC + 4-LSB	100	20.1	1	0.5	No	Low

, we present a comparison of different random bit generation schemes using an integrated source. Due to the lack of data in published articles, the power consumption is not listed in Table 1. Compared to single-section DFB laser and microlaser, our device utilizes mutual-injection dynamics to achieve a 20.1 GHz bandwidth, directly enabling a state-of-the-art 400 Gb/s bit rate [11,26]. While microcomb-based generators achieve higher bit rates through parallelism, they involve significantly higher system complexity [13,27]. Furthermore, by employing the low-cost REC technique for grating definition, our scheme offers a more practical and scalable alternative for mass-produced high-speed PRBGs compared to systems requiring expensive fabrication or complex external control.

5. Conclusions

In summary, we propose and experimentally demonstrate an integrated two-section mutual injection DFB laser. The −20 dB linewidth of 0.568 nm and the 80% bandwidth of 20.1 GHz can be achieved under the chaotic state. By retaining 4-LSBs post-processing, we have realized a 400 Gb/s physical random bit sequence, which passed the NIST SP 800-22 statistical test of randomness. This work provides a compact and energy-efficient integrated chaotic source for ultrafast PRBGs.