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Dynamics and stability conditions of a spectrally combined beam based on a diode laser array

Peng Feng, Danni Li, Qinqi Peng, Chengshuang Yang, Yinghao Zhuang, Qingsong Zhang, Zhen Wu, and Zheqiang Zhong

Open Access Open Access

Abstract

In the spectral beam combining (SBC) system based on a diode laser array (DLA), optical feedback is required to lock the wavelengths of the diode lasers, but excessive feedback may lead to instability. To analyze the dynamics and stability conditions of the SBC system based on DLA, a model for the round-trip propagation of the feedback beam in the external cavity is established, and the optical feedback is calculated using the diffraction integral method. On this basis, a stability condition (the feedback-factor is smaller than 0.07) is introduced to assess the potential stability of the SBC system. If the feedback-factor exceeds 0.07, the combined laser can be in chaos. Based on the stability condition, we analyzed the influence of external-cavity parameters on the external-cavity feedback. The results indicate that the optical feedback becomes increasingly strong with the increasing focal length of the transform lens and the reflectivity of the output coupler, while it decreases with the increasing reflectivity of the front facet of the diode emitter. This work paves the way toward the stabilized spectral beam combining system based on a diode laser array.

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1. INTRODUCTION

Diode lasers are extensively used in diverse fields such as communication, medicine, and industry due to their compact size, low energy consumption, high efficiency, and long lifespan [1]. The continuous development in diode laser materials processing, laser radar, and optical imaging has led to high requirements for the output power and beam quality of diode lasers [2,3]. However, the beam quality of a diode laser typically deteriorates as its power grows due to thermal and nonlinear effects [4]. In order to achieve high brightness and good beam quality, various methods have emerged. The spectral beam combining technology is widely studied because it does not require precise control of phase and polarization [5]. This technique employs an external-cavity structure and utilizes a grating to achieve incoherent beam combining of arrayed beams [6]. Unfortunately, for diode lasers with optical feedback, Lang found that external optical feedback has the potential to trigger instability and chaotic behavior in these lasers [7,8]. The output of chaotic lasers exhibits highly irregular and unpredictable characteristics, limiting their applications [9,10]. Prof. Nianqiang Li studied the nonlinear dynamics of dual-element laser arrays and found that compared to individual diode lasers, diode laser arrays with optical injection are more prone to exhibit nonlinear dynamic characteristics [11,12]. These nonlinear phenomena can lead to problems such as coherence collapse, spectral broadening, and efficiency reduction in SBC systems [13,14]. In the SBC system, beam output from a diode emitter is reflected by the external cavity and can be injected into another emitter, called the crosstalk effect [15], which can make combined power unstable, output spectra complex, and phase disarray.

In order to analyze beam properties, we use the fourth-order Runge-Kutta method to numerically solve the Lang-Kobayashi rate equations for the diode laser with optical feedback. It employs a method to record local extreme points to numerically simulate and analyze the relationship between the output optical intensity ($I$) of the diode laser and the feedback-factor ($\kappa$). By establishing a computational model for the round-trip propagation of the external-cavity beam, the study analyzes the relationships between various parameters of the SBC system and the feedback-factor. Based on this analysis, the impact of external-cavity parameters on the stability of the SBC system is assessed.

2. BEAM PROPAGATION MODEL IN THE EXTERNAL CAVITY

Figure 1(a) illustrates a schematic of a SBC system, which mainly consists of a diode laser array, a fast-axis collimator (FAC), a slow-axis collimator (SAC), a transform lens, a grating, a filter, and an output coupler [16]. The beams from the laser array are collimated and then overlap onto the grating at different incident angles. The beams reach the coupler at the same angle due to the dispersion of the grating. Then the beams are partly reflected backward to the laser array, which makes the arrayed emitters maintain stable spectral characteristics. The SBC system can be represented as a three-mirror cavity model as shown in Fig. 1(b). The space between the front and rear facets of the diode laser is regarded as an internal cavity, and that between the front facet and the coupling mirror essentially acts as an external cavity. ${I_1}$, ${I_2}$, and ${I_3}$ represent the optical intensities emitted from the front facet of the laser emitter, those output from the coupling mirror, and those coupled back into the internal cavity, respectively. ${R_1}$ and ${R_2}$ are the reflectivities of the rear facet and front facet, and ${R_3}$ is the equivalent reflectivity of the external cavity, with ${R_3} = {I_3}/{I_1}$. ${L_{\rm{int}}}$ and ${L_{\rm{ext}}}$ represent the lengths of the internal and external cavities, respectively. $z$ denotes the distance between the coupler and the grating. $f$ stands for the focal length of the transform lens.

 figure: Fig. 1.

Fig. 1. (a) Schematic of a SBC system and (b) its equivalent cavity.

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 figure: Fig. 2.

Fig. 2. Beams propagating in the external cavity.

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A. Calculating Feedback of the External Cavity

A coordinate is established as shown in Fig. 2 to analyze beams passing in the external cavity. The ${x_0}$ axis represents the front focal plane of the transform lens, with its main optical axis as the $z^{\prime \prime}$ axis, and axis as its back focal plane. Emitters distributed along the ${x_0}$ axis are numbered as ${-}M, \ldots, - {1}, {0}, {1}, \ldots, M$, respectively. The emitters have wavelengths of $\lambda _{-M}, \ldots, {\lambda _0}, \ldots, {\lambda _M}$. The spacing between two adjacent emitters is $\Delta p$. $NN^\prime$ denotes the normal, $d$ is the grating period, ${\theta _0}$ is the incident angle at the central emitter, ${\theta _m}$ is the incident angle of the $m$’th emitter on the grating, and $\beta$ is the diffraction angle. $R_{c}$, ${T_4}$, and ${T_5}$ denote the coupling mirror’s reflectivity, the transmittance of the lens, and the transmittance of the filter.

The field distribution of the beam is assumed to have a Gaussian shape with a waist width of ${w_0}$ and an amplitude of ${E_0}$ [17] and is expressed as

$${E_m}({x_0},0) = {E_0}\exp \!\left[{- \frac{{{{({x_0} - {p_m})}^2}}}{{w_0^2}}} \right].$$

According to the geometric relationship, the incident angle of the $m$’th emitter on the grating is

$${\theta _m} = {\theta _0} + \gamma = {\theta _0} - {\rm atan}\frac{{m\Delta p}}{f}.$$

By utilizing the grating equation, the center wavelength of the $m$’th emitter is

$${\lambda _m} = {\lambda _0} + d\left[{\sin \!\left({{\theta _0} - {\rm atan}\frac{{m\Delta p}}{f}} \right) - \sin {\theta _0}} \right].$$

The $\textit{ABCD}$ matrix between the diode laser array and grating can be written as

$$\left({\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right) = \left({\begin{array}{*{20}{c}}1&{z^{\prime \prime} - f}\\0&1\end{array}} \right)\left({\begin{array}{*{20}{c}}1&0\\{- \frac{1}{f}}&1\end{array}} \right)\left({\begin{array}{*{20}{c}}1&f\\0&1\end{array}} \right) = \left({\begin{array}{*{20}{c}}{\frac{{2f - z^{\prime \prime}}}{f}}&f\\{- \frac{1}{f}}&0\end{array}} \right).$$

Using the Collins diffraction integral formula, the light field distribution on the grating can be obtained as

$$\begin{split}{E_m}(\mu) &= \sqrt {\frac{{{T_4}}}{{i{\lambda _m}B}}} \exp (i{k_m}z^{\prime \prime})\int_{- \infty}^{+ \infty} {E({x_0}} ,0)\\&\quad\times\exp \!\left({\frac{{i{k_m}}}{{2B}}(Ax_0^2 - 2{x_0}x^{\prime \prime} + D{{x^{\prime \prime}}^2})} \right){\rm d}{x_0},\end{split}$$
where $\mu$ represents the numerical order of the relief, $x^{\prime \prime } = \mu d \cos \theta_0$, and ${k_m} = {2}\pi /{\lambda _m}$. Assuming that the complex amplitudes of the beams are equal in each relief, the field distribution of the $m$’th beam on the coupler can be expressed as
$$\begin{split} {E_m}(x,z) & = \sum\limits_{\mu = - N}^N {\sqrt {- \frac{{i{k_m}{T_5}}}{{2\pi z}}}} \exp (i{k_m}z) \int_{\mu d\cos \beta}^{(\mu + 1)d\cos \beta} {{E_m}(\mu})\\& \quad\times\exp (ik{\Delta _{\rm{total}}})\exp \!\left[{i{k_m}\frac{{{{(x - x^\prime)}^2}}}{{2z}}} \right]{\rm d}x^\prime ,\end{split}$$
where ${\Delta _{\rm{total}}}$ represents the optical path difference introduced by the grating structure [6]. Then, the beam is partly reflected by the coupler and injected into the internal cavity to assist the laser emitter in locking its wavelength.

Similarly, the reflected beams propagating from the output coupler to the grating, field distribution can be written as

 figure: Fig. 3.

Fig. 3. (a) Feedback-factor of each emitter influenced by the focal length of the transform lens and (b) distance from the grating to the output coupler.

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$$\begin{split}{E^\prime _m}(x^\prime ,z) &= \sum\limits_{\mu = - N}^N {\sqrt {- \frac{{i{k_m}{R_c}{T_5}}}{{2\pi z}}}} \exp (i{k_m}z) \int_{- \infty}^{+ \infty} {{E_m}(x,z})\\&\quad\times\exp\! \left[{i{k_m}\frac{{{{(x - x^\prime)}^2}}}{{2z}}} \right]{\rm d}x.\end{split}$$

The $A^\prime B^\prime C^\prime D^\prime$ matrix from the grating to the laser array is represented as

$$\left({\begin{array}{*{20}{c}}{A^\prime}&{B^\prime}\\{C^\prime}&{D^\prime}\end{array}} \right) = \left({\begin{array}{*{20}{c}}1&f\\0&1\end{array}} \right)\left({\begin{array}{*{20}{c}}1&0\\{- \frac{1}{f}}&1\end{array}} \right)\left({\begin{array}{*{20}{c}}1&{z^{\prime \prime} - f}\\0&1\end{array}} \right).$$

The field distribution of the beam after diffraction through the grating and feedback into the internal cavity is as follows:

$$\begin{split} {{E_m^{\prime \prime}}}(x^{\prime \prime} ,z) &= \sum\limits_{\mu = - N}^N {\sqrt {\frac{{{T_4}}}{{i{\lambda _m}B^\prime}}}} \exp (i{k_m}z^{\prime \prime})\\&\quad \times \int_{\mu d\cos {\theta _0}}^{(\mu + 1)d\cos {\theta _0}} {{{E_m^\prime}}} (x^\prime ,z)\exp (i{k_m}{{\Delta ^\prime_m}})\\&\quad\times\exp \!\left[{\frac{{i{k_m}}}{{2B^\prime}}\!\left({A^\prime {{x^\prime}^2} - 2x^\prime x^{\prime \prime} + D^\prime {{x^{\prime \prime}}^2}} \right)} \right]{\rm d}x^\prime .\end{split}$$

Combining Eqs. (1) and (9), the equivalent reflectivity of the external cavity grating is calculated as follows:

$${R_{3m}} = \frac{{\int_{- \infty}^{+ \infty} {|{{E_m^{\prime \prime}}}(x^{\prime \prime} ,z){|^2}{\rm d}x^{\prime \prime}}}}{{\int_{- \infty}^{+ \infty} {|{E_m}({x_0},0){|^2}{\rm d}{x_0}}}}.$$

A feedback-factor ${\kappa _m}$ is defined as

$${\kappa _m} = (1 - {R_2})\sqrt {\frac{{{R_{3m}}}}{{{R_2}}}} .$$

The feedback-factor is a measure of the coupling strength between the external cavity and the internal cavity [17].

B. Calculation and Analysis of the Feedback

A laser array with seven emitters from the Hamamatsu company is taken for calculation and its gap between two adjacent emitters is 0.5 mm. Simulations are performed based on parameters [18]: ${\lambda _0} = {808}\;{\rm nm}$, ${\theta _0} = {50}\;\deg$, ${T_4} = {0.99}$, ${T_5} = {0.9}$, $f = {0.15}\;{\rm m}$, ${d^{- 1}} = {1600}\;{\rm line/mm}$, $z = {0.3}\;{\rm m}$, and the waist width ${w_0} = {150}\;\unicode{x00B5}{\rm m}$. Figure 3 depicts the change of the feedback-factor with the focal length of the transform lens and the distance from the grating to the coupler when ${R_c} = {10}\%$ and ${R_2} = {3.5}\%$. In Fig. 3(a), the increasing focal length strengthens the feedback-factor of each emitter. However, in Fig. 3(b), as the coupler gets further from the grating, the feedback-factor of each emitter gradually weakens with a periodic fluctuation. Combining Figs. 3(a) and 3(b), it can also be seen that the feedback-factor of the central emitter is the largest. This is because an increase in the focal length of the transform lens increases the equivalent reflectivity of the external cavity [17], ultimately leading to an increase in the feedback-factor. The grating has an optimal angle of incidence (the angle at which diffraction efficiency is maximized). Compared with the incident angles of other emitters, the incident angle of the central emitter is closer to the optimal incident angle of the grating, so the central emitter has greater diffraction efficiency and feedback-factor. On the other hand, when the diffraction distance $z$ increases, the beam radius increases, resulting in a weakened light intensity fed back to the emitters. Moreover, both the internal and external cavities resemble Fabry–Perot cavities, exhibiting periodic fluctuation in the feedback-factor as the cavity length changes.

In order to further analyze the impact of system parameters on the feedback-factor, the central emitter with the largest feedback-factor is taken as an example. Figure 4 gives the variation of feedback-factor ${\kappa _0}$ of the central emitter with the reflectivities of the laser cavity’s front facet and the output coupler. The feedback-factor of the central emitter diminishes as the reflectivity of the front facet increases, but it grows linearly with that of the output coupler, which can be attributed to that the impact of the grating-external feedback weakens with the increasing reflectivity of the front facet of the laser cavity, resulting in a reduction in the feedback-factor [19]. Additionally, a coupler with an increasing reflectivity gradually strengthens the coupling between the internal cavity and the external cavity. An excessive feedback-factor can cause complex changes in the phase and amplitude of the beam, resulting in an unstable laser output [20]. Therefore, in practical design, by slightly raising the reflectivity of the front facet of the diode emitter and reducing that of the output coupler, its output stability can be adjusted.

 figure: Fig. 4.

Fig. 4. Feedback-factor of the central emitter influenced by the reflectivity ${R_2}$ of the front facet of the diode emitter and the reflectivity ${R_c}$ of the coupler.

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 figure: Fig. 5.

Fig. 5. Time-series intensity of the laser array for different the feedback-factors.

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3. DYNAMIC BEHAVIOR IN THE EXTERNAL-CAVITY SBC SYSTEM

We employ the Lang-Kobayashi rate equations to describe the dynamic behavior of diode lasers with optical feedback [21]:

$$\begin{split}\frac{{{\rm d}\tilde E(t)}}{{{\rm d}t}} &= \frac{1}{2}(1 + i\alpha)\left[{\frac{{{G_N}({N(t) - {N_0}} )}}{{1 + \varepsilon |\tilde E(t){|^2}}} - \frac{1}{{{\tau _p}}}} \right]\tilde E(t) \\&\quad+ \frac{\kappa}{{{\tau _{{\mathop{\rm int}}}}}}\tilde E(t - {\tau _{\rm{ext}}})\exp (- i{\omega _0}{\tau _{\rm{ext}}}) + \frac{{{\kappa _{\textit{ij}}}}}{{{\tau _{{\mathop{\rm int}}}}}}\tilde E(t - {\tau _{\textit{ij}}})\\&\quad\times\exp (- i{\omega _{\textit{ij}}}{\tau _{\textit{ij}}}),\end{split}$$
where $\varepsilon$ denotes the gain compression factor, ${\kappa _{\textit{ij}}}$ is the crosstalk optical feedback-factor, ${\tau _{\textit{ij}}}$ is the round-trip time of the crosstalk beam in the external cavity, $E(t)$ denotes the complex electric field inside the laser cavity, $\alpha$ is the linewidth enhancement factor, ${G_N}$ is the differential gain coefficient, and $N$ and ${N_0}$ denote the carrier densities in the internal cavity with and without optical feedback, respectively. ${\tau _p}$, ${\tau _{\rm{in}}}$, and ${\tau _{\rm{ex}}}$ represent the photon lifetime, the round-trip time in the internal cavity, and that in the external cavity (${\tau _{\rm{in}}} = {2}{n_{\rm{eff}}}\;{L_{\rm{int}}}/c$; ${\tau _{\rm{ex}}} = {2}{L_{\rm{ex}}}/c$). Here, ${n_{\rm{eff}}}$ represents the refractive index of the medium. ${\omega _0}$ is the center angular frequency while there is no feedback. Specially, for a SBC system in which the crosstalk has been well suppressed, Eq. (12) can be simplified as
$$\begin{split}\frac{{{\rm d}\tilde E(t)}}{{{\rm d}t}} &= \frac{1}{2}(1 + i\alpha)\!\left[{\frac{{{G_N}\!\left({N(t) - {N_0}} \right)}}{{1 + \varepsilon |\tilde E(t){|^2}}} - \frac{1}{{{\tau _p}}}} \right]\tilde E(t) \\&\quad+ \frac{\kappa}{{{\tau _{{\mathop{\rm int}}}}}}\tilde E(t - {\tau _{\rm{ext}}})\exp (- i{\omega _0}{\tau _{\rm{ext}}}).\end{split}$$

Substituting $E({t}) = E(t)\exp[i\varphi (t)]$ into Eq. (13), the real and imaginary parts can be given as

$$\begin{split}\frac{{{\rm dE}(t)}}{{{\rm d}t}} &= \frac{1}{2}\left[{\frac{{{G_N}(N(t) - {N_0})}}{{1 + \varepsilon |E(t){|^2}}} - \frac{1}{{{\tau _p}}}} \right]E(t) \\&\quad+ \frac{\kappa}{{{\tau _{{\mathop{\rm int}}}}}}E(t - {\tau _{\rm{ext}}})\cos (\psi (t)),\end{split}$$
$$\begin{split}\frac{{{\rm d}\varphi (t)}}{{{\rm d}t}}& = \frac{1}{2}\alpha \left[{\frac{{{G_N}(N(t) - {N_0})}}{{1 + \varepsilon |E(t){|^2}}} - \frac{1}{{{\tau _p}}}} \right] \\&\quad- \frac{\kappa}{{{\tau _{{\mathop{\rm int}}}}}}\frac{{E(t - {\tau _{\rm{ext}}})}}{{E(t)}}\sin (\psi (t)),\end{split}$$
 figure: Fig. 6.

Fig. 6. Bifurcation diagram of extremal values of diode laser output intensity with changes in feedback-factor.

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 figure: Fig. 7.

Fig. 7. Bifurcation diagrams for different reflectivities of the front facet.

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where $\psi (t) = {\omega _0}{\tau _{\rm{ext}}} + \varphi (t) - \varphi (t - {\tau _{\rm{ext}}})$. A usual form of a rate equation has also been adopted for carrier density $N$ and expressed as
$$\begin{split}\frac{{{\rm d}N{\rm (}t{\rm)}}}{{{\rm d}t}} &= P{J_{\rm{th}}} - \frac{{N(t)}}{{{\tau _c}}} \\&\quad- \frac{{{G_N}(N(t) - {N_0})}}{{1 + \varepsilon |E(t){|^2}}}|E(t){|^2},\end{split}$$
where $P$ is the pump factor, ${\tau _c}$ is the carrier lifetime, ${J_{\rm{th}}}$ is the threshold current density, and ${J_{\rm{th}}} = {N_{\rm{th}}}/{\tau _c}$, where ${N_{\rm{th}}} = {N_0} + {1/}({G_N}{\tau _p})$.

Diode laser emitters with different materials and structures always have different parameters. The following values are taken for the simulation [22]: $\alpha = {5.0}$, ${G_N} = {8.4} \times {{10}^{- 13}}\;{{\rm m}^3}/{\rm s}$, ${N_0} = {1.4} \times {{10}^{24}}\;{{\rm m}^{- 3}}$, ${\tau _p} = {1.927} \times {{10}^{- 12}}\;{\rm s}$, ${\tau _{\rm{ext}}} = {2.1}\;{\rm ns}$, ${\tau _{\rm{in}}} = {8.0} \times {{10}^{- 12}}\;{\rm s}$, ${\tau _c} = {2.04} \times {{10}^{- 9}}\;{\rm s}$, $\varepsilon = {2.5} \times {{10}^{- 23}}\;{{\rm m}^3}$, $P = {1.6}$. By using the fourth-order Runge-Kutta method to numerically solve Eqs. (14)–(16), the time series of the output laser for different $\kappa$ values are shown in Fig. 5. When the feedback-factor is 0.01, the intensity of the output laser stays constant in a stable state. Increasing the feedback-factor to 0.025 results in the laser output showing a single-period state with oscillations in a time waveform. As the feedback-factor is further raised to 0.06, the laser output exhibits quasi-periodic behavior, and the amplitude starts to fluctuate. As the feedback-factor further rises to 0.12 in Fig. 5(d), the laser output undergoes a transition into a chaotic oscillation state, with an irregular and erratic time-series waveform. Obviously, with an increasing feedback-factor, the laser goes through various outputs, including steady state, single-periodic state, and quasi-periodic state, and ultimately enters a state of chaotic oscillation, which can be attributed to the diode laser’s high intrinsic gain, relaxation oscillation characteristics, and exceptionally high chirping effect (i.e., intensity changes causing frequency variations), making it highly sensitive to optical feedback [23,24]. Therefore, for a SBC system that uses optical feedback to lock spectra, it is necessary to analyze the coupling strength between the internal and external cavities.

To visualize the laser output under different feedback-factors, we used a method of recording local extreme points for numerical simulations and provided a bifurcation diagram to exhibit the variation of the output laser with the feedback factor, as shown in Fig. 6. The red and blue colors represent the maximum and minimum values in the time-series waveforms of the output intensity, respectively. As the feedback-factor increases from zero, the diode laser initially exhibits a stable output. Its time-series waveform shows a straight line in the bifurcation diagram. As the feedback-factor increases from 0.02, there is one maximum and one minimum value in the time-series laser, resulting in two curves in the bifurcation diagram. With the feedback-factor exceeding 0.04, the laser output displays a discrete distribution, which means its time-series laser has multiple maxima and minima. The diode laser changes from a single periodic state to a quasi-periodic state. As the feedback-factor continues to increase, the number of extreme points in the bifurcation graph continues to increase. When the feedback-factor falls within the range of 0.07–0.15, the autocorrelation function (ACF) curve and time series of the diode laser output begin to be chaotic [25–27]. Therefore, the black line, which corresponds to $\kappa = {0.07}$, serves as a stability condition in this paper. It is crucial to maintain the external-cavity feedback below 0.07 in order to avoid causing chaos and instability in the SBC system.

Since the feedback-factor depends largely on the parameters of the optical element and the feedback-factor of the central emitter is the largest, the central emitter is taken as an example. Figure 7 depicts a bifurcation diagram illustrating the variation of the output intensity of the central emitter with changes in the transform lens focal length. As the focal length of the transform lens increases, the output beam undergoes a transition from a steady state to a single-periodic state, then to a quasiperiodic state, and finally enters a chaotic oscillation state. This phenomenon occurs because as the focal length of the transform lens increases, the feedback-factor gradually increases, ultimately causing the diode laser output beam to exhibit rich nonlinear dynamic characteristics. When the front facet of the diode laser has different reflectivities, the state of the output laser switches at different focal length values. Therefore, in Fig. 8 we further give the bifurcation points for different reflectivities of the front facet of the diode laser, including the bifurcation point from steady state to single-periodic state, that from single-periodic state to quasi-periodic state, and that from quasi-periodic state to chaotic state. In Fig. 8,

 figure: Fig. 8.

Fig. 8. Bifurcation points vary with the reflectivity of the front facet and the focal length of the transform lens.

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 figure: Fig. 9.

Fig. 9. Bifurcation diagrams for different reflectivities of the coupler.

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as the reflectivity increases, the focal length of the transform lens can change over a larger range without causing system chaos. Therefore, to evade system chaos, it is advisable to appropriately select transform lenses with smaller focal lengths and laser emitters with relatively higher reflectivity of its front facet.

Furtherly, Fig. 9 depicts bifurcation diagrams under different reflectivities of the coupler and the front facet of the central emitter. It is evident that with the rise in the reflectivity of the coupler, the output beam goes through a progression from a state of stability to that of a single-period state, followed by a quasi-periodic state, and ultimately reaches chaotic oscillations. Figure 10 demonstrates the changes in bifurcation points due to the front facet with different reflectivities. From Fig. 10, it is evident that with the increase in the reflectivity of the front facet, the values of the respective bifurcation points also exhibit an increment. Based on the above analysis, a coupler with low reflectivity is beneficial to suppressing system chaos. However, in practical applications, optical feedback is necessary to lock wavelengths of arrayed emitters, so the coupler generally has a minimum reflectivity of about 10% [28]. In order to suppress a chaotic laser and ensure spectral locking it is suggested to adopt a transform lens with a focal length ranging from 5 to 15 cm, a coupler with its reflectivity between 10% and 15%, and the front facet of the laser array with an effective reflectivity from 3.5% to 5%.

From Figs. 7–10 we observe that for various DLAs with different reflectivities ${R_2}$, the SBC can utilize a transform lens with a focal length shorter than that indicated by the black line in Fig. 8. Additionally, a coupler with a reflectivity below the line shown in Fig. 10 can be employed. This allows the external-cavity feedback of all seven emitters to be controlled below 0.07, thus preventing the combined output from being unstable.

4. CONCLUSION

In a SBC system based on a diode laser array, the optical feedback is essential for wavelength locking, but excessive feedback may lead to chaotic behavior. To analyze the dynamics and stability conditions of SBC systems based on DLA, a model for SBC systems based on the grating external cavity is established using the diffraction integral method and the incoherent superposition principle. The influence of the feedback-factor on the output intensity of the diode laser was analyzed in detail. The results indicate that as the focal length of the transform lens and the reflectivity of the output coupler increase, the feedback-factor also increases. However, with an increase in the reflectivity of the front facet of the diode emitter, the feedback-factor decreases. The variation of the distance between the grating and the coupler has less significant impact on the output intensity of the diode laser. As the feedback-factor increases, the output of the diode laser transitions from a stable state to period-doubling bifurcations. When the feedback-factor exceeds 0.07, it may evolve into chaotic output. To maintain the feedback-factor below 0.07, it is recommended to design the transform lens with a focal length of less than 15 cm, utilize an output coupler with a reflectivity below 15%, and ensure the front facet of the diode emitter has a reflectivity above 3.5%, in order to avoid beam instability.

Funding

National Undergraduate Innovation and Entrepreneurship Training Program of Sichuan (S202310623102); National Natural Science Foundation of China (61905203); Department of Science and Technology of Sichuan Province (2021ZYD0036).

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.

 figure: Fig. 10.

Fig. 10. Bifurcation points vary with the reflectivities of the front facet and the coupler.

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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.

Bifurcation points vary with the reflectivities of the front facet and the coupler.

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Figures (10)
Fig. 1.
Fig. 1. (a) Schematic of a SBC system and (b) its equivalent cavity.

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Fig. 2.
Fig. 2. Beams propagating in the external cavity.

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Fig. 3.
Fig. 3. (a) Feedback-factor of each emitter influenced by the focal length of the transform lens and (b) distance from the grating to the output coupler.

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Fig. 4.
Fig. 4. Feedback-factor of the central emitter influenced by the reflectivity ${R_2}$ of the front facet of the diode emitter and the reflectivity ${R_c}$ of the coupler.

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Fig. 5.
Fig. 5. Time-series intensity of the laser array for different the feedback-factors.

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Fig. 6.
Fig. 6. Bifurcation diagram of extremal values of diode laser output intensity with changes in feedback-factor.

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Fig. 7.
Fig. 7. Bifurcation diagrams for different reflectivities of the front facet.

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Fig. 8.
Fig. 8. Bifurcation points vary with the reflectivity of the front facet and the focal length of the transform lens.

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Fig. 9.
Fig. 9. Bifurcation diagrams for different reflectivities of the coupler.

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Fig. 10.
Fig. 10. Bifurcation points vary with the reflectivities of the front facet and the coupler.

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Equations (16)

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E m ( x 0 , 0 ) = E 0 exp [ − ( x 0 − p m ) 2 w 0 2 ] .
θ m = θ 0 + γ = θ 0 − a t a n m Δ p f .
λ m = λ 0 + d [ sin ( θ 0 − a t a n m Δ p f ) − sin ⁡ θ 0 ] .
( A B C D ) = ( 1 z ′ ′ − f 0 1 ) ( 1 0 − 1 f 1 ) ( 1 f 0 1 ) = ( 2 f − z ′ ′ f f − 1 f 0 ) .
E m ( μ ) = T 4 i λ m B exp ⁡ ( i k m z ′ ′ ) ∫ − ∞ + ∞ E ( x 0 , 0 ) × exp ( i k m 2 B ( A x 0 2 − 2 x 0 x ′ ′ + D x ′ ′ 2 ) ) d x 0 ,
E m ( x , z ) = ∑ μ = − N N − i k m T 5 2 π z exp ⁡ ( i k m z ) ∫ μ d cos ⁡ β ( μ + 1 ) d cos ⁡ β E m ( μ ) × exp ⁡ ( i k Δ t o t a l ) exp [ i k m ( x − x ′ ) 2 2 z ] d x ′ ,
E m ′ ( x ′ , z ) = ∑ μ = − N N − i k m R c T 5 2 π z exp ⁡ ( i k m z ) ∫ − ∞ + ∞ E m ( x , z ) × exp [ i k m ( x − x ′ ) 2 2 z ] d x .
( A ′ B ′ C ′ D ′ ) = ( 1 f 0 1 ) ( 1 0 − 1 f 1 ) ( 1 z ′ ′ − f 0 1 ) .
E m ′ ′ ( x ′ ′ , z ) = ∑ μ = − N N T 4 i λ m B ′ exp ⁡ ( i k m z ′ ′ ) × ∫ μ d cos ⁡ θ 0 ( μ + 1 ) d cos ⁡ θ 0 E m ′ ( x ′ , z ) exp ⁡ ( i k m Δ m ′ ) × exp [ i k m 2 B ′ ( A ′ x ′ 2 − 2 x ′ x ′ ′ + D ′ x ′ ′ 2 ) ] d x ′ .
R 3 m = ∫ − ∞ + ∞ | E m ′ ′ ( x ′ ′ , z ) | 2 d x ′ ′ ∫ − ∞ + ∞ | E m ( x 0 , 0 ) | 2 d x 0 .
κ m = ( 1 − R 2 ) R 3 m R 2 .
d E ~ ( t ) d t = 1 2 ( 1 + i α ) [ G N ( N ( t ) − N 0 ) 1 + ε | E ~ ( t ) | 2 − 1 τ p ] E ~ ( t ) + κ τ int E ~ ( t − τ e x t ) exp ⁡ ( − i ω 0 τ e x t ) + κ ij τ int E ~ ( t − τ ij ) × exp ⁡ ( − i ω ij τ ij ) ,
d E ~ ( t ) d t = 1 2 ( 1 + i α ) [ G N ( N ( t ) − N 0 ) 1 + ε | E ~ ( t ) | 2 − 1 τ p ] E ~ ( t ) + κ τ int E ~ ( t − τ e x t ) exp ⁡ ( − i ω 0 τ e x t ) .
d E ( t ) d t = 1 2 [ G N ( N ( t ) − N 0 ) 1 + ε | E ( t ) | 2 − 1 τ p ] E ( t ) + κ τ int E ( t − τ e x t ) cos ⁡ ( ψ ( t ) ) ,
d φ ( t ) d t = 1 2 α [ G N ( N ( t ) − N 0 ) 1 + ε | E ( t ) | 2 − 1 τ p ] − κ τ int E ( t − τ e x t ) E ( t ) sin ⁡ ( ψ ( t ) ) ,
d N ( t ) d t = P J t h − N ( t ) τ c − G N ( N ( t ) − N 0 ) 1 + ε | E ( t ) | 2 | E ( t ) | 2 ,
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