Chaos has attracted considerable research interest because of its many potential applications, such as secure communication,^[1,2] high-speed random number generation,^[3–5] chaotic radar,^[6–8] optical fiber detection,^[9–11] weak signal detection^[12,13] and distributed optical sensors.^[14–18] Semiconductor lasers can easily generate chaos^[19] with additional degrees of freedom, such as optical feedback, optical injection, electro-optical feedback and mutual coupling.^[20–24] Nowadays, most chaotic lasers are fabricated by using discrete optical components in a laboratory. However, such chaotic lasers are bulky and easily affected by the external environment, which is not conducive to their practical application and large-scale production.^[25–27] To solve these problems, many research groups have put forward the integrated chaotic semiconductor laser.

Recently, some monolithic integrated chaotic semiconductor laser chips have been reported. A laser chip with optical feedback was developed by Satoshi Sunada et al.,^[28] which consists of a distributed feedback laser (DFB), two semiconductor optical amplifiers, a photodiode and a passive ring waveguide cavity. The chaos with a flatness of ±6.5 dB and a bandwidth of 10 GHz was obtained. Wu et al. also proposed a laser chip,^[27] which includes a DFB section, a phase section, an amplification section and a high-reflection coating to the face of the amplifier section. The chaotic bandwidth increased to more than 26.5 GHz when using this monolithic integrated structure. Yin et al. researched the generation of wideband chaotic signal and generated a chaotic signal of 36 GHz by using a three-section monolithic integrated semiconductor laser under external optical injection.^[29] The development of monolithic integrated chaotic semiconductor laser chips has reduced the size of the chaotic lasers and improved the quality of the chaotic signal. However, the production of monolithic integrated chaotic semiconductor laser chips requires sophisticated and expensive instruments, and every step of the growth process must be strictly controlled, which restricts the mass production of monolithic integrated chaotic semiconductor lasers.

Therefore, it is necessary to design a new simple integrated chaotic laser. The feasibility verification of the structure through numerical simulation is an indispensable step in the design process. In the structure that produces chaos, the short cavity optical feedback structure is easiest to integrate.^[30] However, it is very sensitive to the external cavity length and the feedback intensity required for the generation of chaos.^[31] A small change of the external conditions may have a significant effect on the dynamic characteristics of this structure. Due to the different internal parameters of different laser chips, the length of the external cavity and the feedback intensity are different. This makes previous simulation methods no longer applicable. And we should utilize real internal parameters in the numerical simulation to ensure that the designed integrated chaotic laser can generate chaos.

In this paper, we propose an easy-to-implement hybrid integrated chaotic semiconductor laser with a short external feedback cavity. The frequency test method of the network analyzer is used for acquiring the internal parameters of the chaotic semiconductor laser chip 1 (CSLC1) and CSLC2, and the extracted parameters are used for the following simulation. We obtain the optimal external cavity length and the optimal feedback intensity for the generation of chaos, corresponding to the different internal parameters of each chip. In addition, the results provide a theoretical foundation for the production of the hybrid integrated chaotic semiconductor lasers.

By measuring the P–I curve and the small signal frequency response curve of the CSLC, we can obtain the threshold current, relaxation oscillation frequency and damping factor. These three parameters will be used in the subsequent internal parameter calculation.

The steady-state laser output power versus bias current relationship has been established as

The experimental setup of the small signal intensity modulated frequency response curve is shown in Fig. 2, the low-cost laser diode driver (ILX Lightwave, LDX3412) and the vector network analyzer (ROHDE & SCHWARZ, ZVA24, 10 MHz–24 GHz bandwidth) provide direct current and radio frequency signals, respectively. The two signals mixed by a bias-T which is fixed on the vector network analyzer. The two signals are supplied to the CSLC by a high-frequency probe (Cascade Microtech, ACP40-GS-200, DC to 40 GHz).

Fig. 2.

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	Fig. 2. (color online) Experimental device diagram for measuring frequency responses. The optical fiber and EDFA are used only to measure the fiber frequency response.

CSLC1 emits continuous light which is converted into an electrical signal by a photodetector (Finisar, XPDV2120RA, 50 GHz bandwidth). Then, the electrical signal is measured by the vector network analyzer. The measured result is the laser frequency response curve, as shown in Fig. 3(a).

Fig. 3.

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	Fig. 3. (color online) (a) The laser frequency responses of CSLC1 for different bias currents. (b) Fitted results when the laser frequency response of 13 mA is subtracted from the laser frequency response for other higher bias currents. (c) Damping factor versus the relaxation oscillation frequency squared (the slope gives a value of 0.3 ns for K-factor).

The relative frequency response curves can be obtained by subtracting the frequency response at a lower biased current from the ones biased at higher biased current, as shown in Fig. 3(b). The method of the frequency response subtraction can eliminate all the adverse effects that are caused by the equipment and do not vary with the bias current during the experiment. The relative frequency response curves satisfy

Figure 3(a) shows the results of the laser frequency response curves when the bias current is 13 mA, 16 mA, 22 mA and 28 mA. Figure 3(b) depicts the fitted results of CSLC1 relative to the response of 13 mA biased current. The blue curves represent the measured results and the red curves are the fitted results.

The relaxation oscillation frequency squared versus the damping factor is linear, as shown in Fig. 3(c). The slope of the line is the K-factor^[34] of 0.3 ns. The K-factor is equal to .

The threshold current I_th of CSLC1 is 11.3 mA. The active volume of the laser chip is provided. The two values, together with the measured values ( , v and K-factor), can be used to calculate the carrier lifetime τ_c and the threshold carrier density of CSLC1 according to the reported calculation methods.^[35–37]

The light-wave signal generated by CSLC is coupled into the dispersive optical fiber and amplified by an erbium doped optical fiber amplifier (EDFA, KEOPSYS, CEFA-C-HG). The amplified signal is converted into an electrical signal by a photodetector. Finally, the electrical signal is tested by the vector network analyzer. The electrical signal is the fiber frequency response.

The fiber frequency responses use the absolute value on the logarithmic domain with subtraction of the laser frequency response at the same bias current. The obtained curves fit to

The fitted results of the fiber frequency responses of CSLC1 are shown in Fig. 4(a). The blue and red curves represent the measured and the fitted results at 16 mA, respectively. The values of α and f_c together with the measured values above can be used to calculate the internal parameters (N₀, τ_p, g₀, Γ and ε) of CSLC1. Then all the internal parameters of CSLC1 can be extracted by the experiment, which are listed in Table 1.

Fig. 4.

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	Fig. 4. (color online) (a) Fiber frequency response subtracts the laser frequency response at 16 mA. (b) Result of simulated laser frequency response (red star) at 13 mA and the measured laser frequency response (blue ball) at the same bias current.

Table 1.

Table 1.

Table 1. Internal parameters of CSLC1. . Parameter Value Threshold current Ith/mA 11.3 Wavelength λ/nm 1550 Linewidth enhancement factor α 3.04 Threshold carrier density 2.943×1024 Carrier density at transparency N0/m−3 0.905×1024 Photon lifetime τp/s 5.2×10−12 Carrier lifetime τc/s 0.2×10−9 Field confinement factor Γ 0.01 Differential gain g0/ 9.434×10−12 Gain saturation coefficient ε/m3 2.271×10−23 Round-trip time in internal cavity τin/s 7.1×10−12 Active volume /m3 4.8×10−18

Table 1. Internal parameters of CSLC1. .

To verify the accuracy of the parameters, we compare the laser frequency response of calculation and experiment at the same bias current, as shown in Fig. 4(b). At the bias current of 13 mA, we use the extracted parameters to simulate the laser frequency response which is represented by the red star. The experimental results deviate slightly from the simulation results due to the systematic errors in the experimental system. The systematic errors are mainly caused by the frequency response nonuniformity of the electric adapter and cable. However, the internal parameters used in the simulation are obtained after eliminating the systematic errors, so the simulation result is accurate. Then, we utilize the extracted internal parameters for the following design.

Furthermore, we extract the internal parameters of CSLC2 by experiment, which are listed in Table 2. We also utilize the internal parameters of CSLC2 for the rate equations, and investigate its dynamic characteristics.

Table 2.

Table 2.

Table 2. Internal parameters of CSLC2. . Parameter Value Threshold current Ith/mA 11.4 Wavelength λ/nm 1550 Linewidth enhancement factor α 3.85 Threshold carrier density 3.8×1024 Carrier density at transparency N0/m−3 0.16×1024 Photon lifetime τp/s 5.2×10−12 Carrier lifetime τc/s 0.256×10−9 Field confinement factor Γ 0.01 Differential gain g0/ 5.28×10−12 Gain saturation coefficient ε/m3 3.938×10−23 Round-trip time in internal cavity τin/s 7.1×10−12 Active volume /m3 4.8×10−18

Table 2. Internal parameters of CSLC2. .

To obtain an insight into the route to chaos of the system, we present the bifurcation diagram of CSLC1 for and L = 4 mm in Fig. 5. The feedback strength K_ap is taken as the variable. It is shown that the system experiences steady state (A), one-period oscillation (B), two-period oscillation (C), four-period oscillation (D), eight-period oscillation (E), and finally into the chaotic state.

Fig. 5.

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	Fig. 5. (color online) Bifurcation diagrams of CSLC1: steady state (A), one-period oscillation (B), two-period oscillation (C), four-period oscillation (D), eight-period oscillation (E).

Figure 6 describes the dynamic characteristics of CSLC1 in the time domain and frequency domain, the columns from (I) to (III) plot the time series, the optical spectra and the frequency spectra operated at different state, respectively. The bias current and the external cavity are set at and L = 4 mm for all cases. The values of I and L are optimized by the simulation. The external cavity frequency is , and the relaxation oscillation frequency is , corresponding to a short cavity with .

Fig. 6.

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	Fig. 6. (color online) Calculated time series (I), optical spectra (II), and frequency spectra (III) of the chaotic semiconductor laser chip 1: (a) the steady state, (b) the one-period oscillation, (c) two-period oscillation, (d) chaotic state, . The bias current is and the short external cavity length is L = 4 mm.

Figure 6(a) (the first row) is the case with feedback intensity . It can be seen from Fig. 6(a-I) that the output power is constant. The optical spectrum (Fig. 6(a-II)) shows a typical single mode shape without deviation. The power of the frequency spectrum reflected in Fig. 6(a-III) is very low. These indicate that the laser oscillation is damped and the dynamics of the system is steady state.

As K_ap increases from 0.015 to 0.07, the laser appears non-damped oscillations and the system enters the one-period oscillation. As shown in Fig. 6(b-I), the laser shows an uniform pulsation with period of , it has only one peak. The optical spectrum (Fig. 6(b-II)) shows peaks at and . The frequency spectrum (Fig. 6(b-III)) shows peaks at , , and , where , 3 and are higher harmonics of the laser. The sharp peak is not necessarily equal to the relaxation oscillation frequency , only close to it.

Figure 6(c) (the third row) shows different dynamics when K_ap increases to 0.11. As can be seen from Fig. 6(c-I), the laser output shows two peaks of different heights with period of . The optical spectrum (Fig. 6(c-II)) and frequency spectrum (Fig. 6(c-III)) are characterized by two sharp peaks at and ( is its half-harmonic ).^[38,39] These correspond to the double-period state.

The dynamics of CSLC1 at are shown in Fig. 6(d) (the fourth row), which is more complicated and can be called the chaotic state. The chaotic state is characterized by irregular time variations and larger amplitudes, as shown in Fig. 6(d-I). The optical spectrum (Fig. 6(d-II)) and frequency spectrum (Fig. 6(d-III)) are continuous. It is visible that the frequency spectrum covers a broad frequency range with a sharp peak at 6.97 GHz.

Figure 6 plots the typical examples of the period-doubling route to chaos. Moreover, the peak frequency decreases with the increase of .^[39]

The numerical result of the largest Lyapunov exponent of CSLC1 is shown in Fig. 7, the length of the feedback cavity L is 4 mm and the current I is . If the largest Lyapunov exponent is positive,^[40] it indicates that the system is in the chaotic state. The largest Lyapunov exponent is negative or close to zero for the regions of the steady-state and periodic oscillations. It can be seen from Fig. 7, when the feedback strength is less than 0.04, the largest Lyapunov exponent is less than 0, the system is in the steady state; , the largest Lyapunov exponent is around 0 and the system is in the periodic state; when the feedback strength is greater than 0.125, the largest Lyapunov exponent is larger than 0, and thus the system is in the chaotic state.

Fig. 7.

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	Fig. 7. (color online) Largest Lyapunov exponent as a function of the feedback strength Kap of CSLC1. Feedback cavity L is 4 mm and bias current I is .

To investigate the comprehensive effects of bias current (15–41 mA) and K_ap (0–0.2) on CSLC1, we apply the maps of the largest Lyapunov exponent to estimate the dynamics of this system under different external cavities,^[40] as shown in Fig. 8. Different colors in the dynamic maps represent different values of the largest Lyapunov exponent. When the Lyapunov exponent is far less than zero, it is expressed in blue; near zero in green; and greater than zero with red and yellow. As shown in Figs. 8(a) and 8(b), when the external cavity L = 2 mm and L = 3 mm, the largest Lyapunov exponent is far less than 0, which corresponds to the steady state of CSLC1.

Fig. 8.

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	Fig. 8. (color online) Dynamic maps of the chaotic semiconductor laser chip 1 calculated by the largest Lyapunov exponent: (a) L = 2 mm, (b) L = 3 mm, (c) L = 4 mm, (d) L = 5 mm, (e) L = 6 mm and (f) L = 4 cm. The blue represents that the largest Lyapunov exponent is far less than zero; green is near zero; red and yellow are greater than zero.

When the external cavity length is equal to 4 mm, the result is shown in Fig. 8(c). Figure 8(c) indicates that the largest Lyapunov exponent remains negative when optical feedback . Among them, K_ap is and the bias current ranges from 15 mA to 41 mA, the system is in a steady state, which is represented in blue; and the bias current increases from 15 mA to 41 mA, the system is in the period-doubling state, which is marked by green. Only when and the bias current increases from 22 mA to 41 mA, CSLC1 can generate chaos which is marked by yellow and red.

When the external-cavity length increases to 5 mm, the chaotic state disappears and the region of steady state increases, which is reflected in Fig. 8(d). When K_ap is and the bias current ranges from 15 mA to 41 mA, the system is in a steady state, which is represented in blue; and the bias current increases from 25 mA to 41 mA, the system is in the period-doubling state, which is marked by green. When the length of the external cavity increases to 6 mm, the region of steady state increases and the period-doubling state reduces, which are reflected in Fig. 8(e).

When L = 4 cm, the external cavity frequency is about 3.7 GHz, corresponding to of CSLC1 at 12 mA. As the current increases, is , which is in the long-cavity regime. It can be seen from the output of CSLC1 in Fig. 8(f), the long cavity is more likely to produce chaos. However, the 4-cm cavity is larger than the butterfly shell and unable to be integrated in a commercial butterfly shell.

Figure 9 shows the numerical result of the largest Lyapunov exponent of CSLC2, where the length of the feedback cavity L is 5 mm and the current . As shown in Fig. 9, when K_ap is less than 0.03, the largest Lyapunov exponent is less than 0, and thus the region is in the steady state; when and , the largest Lyapunov exponent is around 0, which corresponds to the periodic state; , the largest Lyapunov exponent is greater than 0, and the system is in the chaotic state.

Fig. 9.

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	Fig. 9. (color online) Largest Lyapunov exponent as a function of the feedback strength of CSLC2. Feedback cavity L = 5 mm and the bias current I is .

Figure 10 plots the dynamic characteristics of CSLC2. As shown in Figs. 10(a) and 10(b), when the external cavity L = 2 mm and 3 mm, the largest Lyapunov exponent remains negative, which is the same phenomenon as CSLC1. However, when the external cavity length increases to 4 mm, the system has no chaotic state, but steady state and periodic state. This is completely different from the dynamic characteristics of the CSLC1, as shown in Fig. 10(c).

Fig. 10.

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	Fig. 10. (color online) Dynamic maps of the chaotic semiconductor laser chip 2 calculated by the largest Lyapunov exponent: (a) L = 2 mm, (b) L = 3 mm, (c) L = 4 mm, (d) L = 5 mm, (e) L = 6 mm, and (f) L = 4 cm. The blue represents that the largest Lyapunov exponent is far less than zero; green is near zero; red and yellow are greater than zero.

When the external cavity length is equal to 5 mm (Fig. 10(d)), the dynamic characteristics of CSLC2 are quite different from those of CSLC1. When and the bias current is greater than 25 mA, CSLC2 is prone to generate chaos.

When the length of the external cavity increases to 6 mm, the region of steady state increases and the period-doubling state reduces, which are reflected in Fig. 10(e). As the external cavity increases to 4 cm, we also investigate the dynamic characteristics of CSLC2, and the wide regions of chaos can be observed in Fig. 10(f).

The results of this section are obtained by theoretical simulation. We prove the feasibility of this structure by a great mass of theoretical simulations. For a long cavity, we can find that the external-cavity length and the feedback strength have little effect on different chips to generate chaos by comparing the dynamic maps in Figs. 8(f) and 10(f). However, for the regime of short cavity, the optimal external cavity length and feedback intensity are completely different for each chip. We find that the optimal external cavity length and optimal feedback intensity of CSLC1 are L = 4 mm and , respectively. The optimal external cavity length and feedback intensity of CSLC2 are L = 5 mm and , respectively. Based on this, we must extract the internal parameters of the chips and use them for simulation to obtain the precise production parameters.

The route to chaos of this structure is systematically investigated. The results show that when is , the hybrid integrated chaotic semiconductor laser has a period-doubling route to chaos. We use the maps of the largest Lyapunov exponent to estimate the dynamics of two CSLCs of different external-cavity lengths. The results indicate that the optimal external cavity length and the feedback intensity of CSLC1 are 4 mm and for CSLC2, the optimal external cavity length and feedback intensity are 5 mm and , respectively. These results indicate that for the short-cavity structure, the optimal external cavity length and feedback intensity required for each chip to produce chaos are completely different. Therefore, we should extract the internal parameters of the chips and use them for simulation to ensure that the fabricated chaotic laser can generate chaos.

We propose a hybrid integrated chaotic semiconductor laser with short-cavity optical feedback. This is a new structure to achieve integration through a simple, low-cost manufacturing process, and it is more conducive to mass production of integrated chaotic semiconductor lasers. To make the simulation results more accurate, we extract the internal parameters of the chaotic semiconductor laser chips (CSLCs) and utilize them for numerical simulation. Based on this method, we can provide accurate external cavity length and feedback intensity for the fabrication of the hybrid integrated chaotic laser and ensure that it can generate chaos. Practically, we have fabricated a hybrid integrated chaotic laser in collaboration with the Institute of Semiconductors of the Chinese Academy of Science.^[41] And the experimental phenomena agree well with the theoretical prediction.

We propose the method of combining extraction parameters with numerical simulation for the first time. This study lays a theoretical foundation for the fabrication of hybrid integrated chaotic semiconductor lasers with short-cavity optical feedback.