Demonstration of regularity of bifurcation in chaotic laser diode

Demonstration of regularity of bifurcation in chaotic laser diode | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Demonstration of regularity of bifurcation in chaotic laser diode Chol-Hyon Kim, Myong-Il Kim, Kwang-Myong Ho This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4081906/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The behavior of bifurcation and chaos in a laser diode subject with delayed optical feedback is very chaotic and unpredictable. We have theoretically and experimentally demonstrated an extremely significant and useful rule of the growth of bifurcation and chaos displayed in the bifurcation diagram with the injection current. Such a significant rule is attributed to the resonance between the relaxation oscillation frequency and the external cavity frequency. This regularity can provide a principle for the suppression and control of bifurcation and chaos in semiconductor lasers with delayed optical feedback. Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1. INTRODUCTION Since a laser diode has a large linewidth enhancement factor (the value of which ranges typically between 2 and 7 [1]), it exhibits an interesting dynamics unlike other lasers. In the presence of optical feedback, particularly, the dynamics of a laser diode is ready to be accompanied by bifurcations or chaos [2]. The study on such laser systems is very useful for communication encryption systems [3]-[5], random number generation [6],[7], remote sensing [8][9], and photonic information processing [10]. There are many parameters that have an effect on the dynamic of a laser diode; the external parameters of which are the injection current, the optical feedback strength, and the delayed time (or external cavity length), and the internal parameters of which are the linewidth enhancement factor, the laser diode cavity length (or diode cavity round trip time), the carrier lifetime, the photon lifetime, and the carrier density at threshold. The dynamics of a laser diode is very sensitive to the above-mentioned parameters and therefore a small change of any parameter of them may cause the laser system to stay irregularly on a stable state, bifurcation or chaos. To find a regularity in such an irregularity is of great significance and it can provide an available tool for many applications (e.g. neuro-inspired information processing techniques [11]). In the paper we report a regularity of the bifurcation diagram with the injection current. First, we examine the characteristics of the development of bifurcations and chaos in the diagram for different linewidth enhancement factors, feedback strengths, and external cavity lengths. Then we explain the regularity reported based on the resonance relation between the oscillation frequency and the external cavity frequency. 2. REGULARITY OF BIFURCATION DIAGRAM IN NUMERICAL SIMULATION: PRINCIPLE OF EXISTENCE OF STABLE CURRENT POINT The dynamics of a laser diode with delayed optical feedback is depicted by [1],[2] where E 0 ( t ) is the amplitude of the laser electric field which is normalized so that equals the photon density in the laser diode cavity, φ ( t ) the phase of the laser electric field, N ( t ) the carrier density, τ the external cavity round trip time, κ the feedback strength, α the linewidth enhancement factor, and J the injection current. We set in the numerical calculation the angular frequency of the laser light, ω 0 =2.89×10 24 rad s -1 (corresponding to a wavelength of 632 nm), the modal gain coefficient, G N =8×10 -13 m 3 s -1 , the carrier density at transparency, N 0 =1.4×10 24 m -3 , the carrier density at threshold for the unperturbed laser, N T =2.3×10 24 m -3 , the photon lifetime, τ P =1/[ G N ( N T - N 0 )], the laser cavity round trip time, τ L =8.3 ps, the carrier lifetime, τ S =2 ns. Also the electric field and the carrier density are normalized by the steady-state electric field and the carrier density at the threshold for the unperturbed laser, respectively. We have carried out numerical simulations of Eq. (1) to (3) under the above conditions using SIMULINK supported by MATLAB to examine bifurcation diagrams with the injection current. We have observed the bifurcation diagram with the injection current while changing the linewidth enhancement factor α from 2 to 7 and the feedback strength κ from 0 to 10×10 -3 for different external cavity length L s. First, we have examined bifurcation diagrams with changing only κ when L =50 cm, and α =3. Fig. 1 illustrates the development of bifurcations and chaos in some of the bifurcation diagrams. The current injection is normalized by the threshold current injection J th in Fig. 1, where the range without bifurcation is called an inter-bifurcation range . As seen in Fig. 1 (a) to (c), there alternately are appearance and disappearance of bifurcations while the bifurcations ranges expand and the inter-bifurcation ranges reduce as κ increases. If the bifurcation ranges expand to merge into each other, chaos can evolve in such bifurcation ranges alone. The chaotic regions marked blue arrows in Fig. 1 (b) to (e) grow greater in given bifurcation ranges as κ increases. In addition, the chaos regions develop to some extent to divide into two parts (see the red dashed rectangle in Fig. 1 (b) and (c)). Fig. 1 indicates some important rules in the birth and evolution of bifurcations and chaos: i) there exist stable current points (marked with red arrows in Fig. 1 where the birth of bifurcations is suppressed while there are unstable current points (not indicated in Fig. 1, but found in Fig. 2 and 3) ready for bifurcation birth; ii) the chaos states develop around the middle of the merged bifurcation ranges (to be more specific, they lean to the right as seen in Fig. 1 (d) and (e)); and iii) the intervals between stable current points become wider as the relative injection current increases. The most important thing in these rules is that the positions of the stable and unstable current points are independent of the value of κ . The stable current points marked with red arrows situate within the inter-bifurcation ranges and their positions are constant regardless of κ from Fig. 1 (a), (b) and (c), and so are the positions of the unstable current points. In addition, the total number of possible bifurcations is constant in a given range of the injection current, which is the result of the mechanism of the bifurcation evolution above-explained. Second, we have observed bifurcation diagrams varying only α when L =30 cm, and κ =3×10 -3 . Fig. 2 shows some of them. The positions of stable and unstable current points are constant independent of α as in the case of Fig. 1. As a result, the analysis of Fig. 1 and 2 draws a conclusion that the positions of stable and unstable current points are independent of both κ and α . Fig. 3 shows the bifurcation diagrams plotted for different sets of κ and α , which will convince the above conclusion. As evident in Fig. 3, although the values of α and κ differs greatly, respectively, and, what is more, mature chaos exist as in Fig. 3 (b), the positions of stable and unstable current positions can be easily identified and are invariable. Third, we have investigated bifurcation diagrams for different external cavity length, some of which is shown in Fig. 4. It can be seen from Fig. 4 that the bifurcation number per unit relative injection current increases as the external cavity length grows. We claim through the above consideration that the positions of stable and unstable current points and the bifurcation number density depend on only the external cavity length irrespective of κ or α . 3. THEORETICAL ANALYSIS OF STABLE CURRENT POINTS Now we explain why the laser diode with delayed optical feedback follows the bifurcation scenario as reported above. When a bifurcation or a chaos occurs its fundamental frequency is almost equal to the relaxation oscillation frequency [12] where J 0 is the injection current at transparency, which is proportional to N 0 . It should be noted in Eq. (4) that the relaxation oscillation frequency f r is proportional to the square root of the injection current J . Whenever the relaxation oscillation frequency f r coincides with integer multiples of the external frequency, the semiconductor laser system at such injection current points exhibits very strong tendency to maintain a stable state which is the most bifurcation-proof. Let us explain it in more detail. Fig. 5 illustrates the plot of the relative injection current J / J th versus the relaxation oscillation frequency f r for L =50cm. In this case the frequency space Δν between the external cavity modes is The red asterisks in Fig. 5 represent the positions of the relaxation oscillation frequencies being integer multiples of the external cavity frequency space Δν , which are none other than the stable injection current points. To manifest this, Fig. 6 shows stable injection current points obtained in the way of plotting Fig. 5 for different sets of κ , α , and L . It is clear from Fig. 6 that bifurcations are suppressed in stable injection current points. It should be noted that a stable injection current point cannot be identified in the regions where bifurcations are mature and merged. Meanwhile, the stable current points in Fig. 5 get sparse as the injection current increases, for the curve follows the form of a function This accounts for the above-mentioned reason why an interval between stable injection currents points increases with the injection current. Moreover the total number, n bif , of bifurcations in the variation range of the normalized injection current is defined by where f r,max is the maximum of the relaxation oscillation frequency in the injection current variation range, which is calculated as 16 for Fig. 6. Fig. 7 shows the positions of unstable injection current points in the bifurcation diagrams plotted for different sets of κ , α , and L . It should be noted that the latest and smallest bifurcations in Fig. 7 (a) and (b) demonstrate well a birth of bifurcation. Here we emphasize that unstable injection current points, where bifurcations begin to grow, lie halfway between stable injection current points because there is greatest differences between the relaxation oscillation frequencies and the external cavity frequency in the positions in contrast to the cases of stable injection points. 4. EXPERIMENTAL DEMONSTRATION We demonstrated experimentally the new regularity of bifurcation in a chaotic laser diode. In the experiment, a Hitachi HL6312G, whose typical threshold current was 45 mA, was served as a laser diode and the optical output power of the laser diode was monitored by an u2t photodiode of 9 ps response time. The temperature of the laser diode was stabilized at 25℃ by a precision temperature controller (TED 200, Thorlabs). The electrical signal from the photodiode was measured by a digital 50 GHz sampling oscilloscope (HP 54120B). Note that the oscilloscope ability was insufficient to obtaining the exact waveform of the optical power of the laser diode but was enough to distinguish whether there was a bifurcation or not. The external cavity length was set 50 cm corresponding the external cavity frequency 0.3 GHz. The injection current of the laser diode was adjusted by a laser diode controller (LDC210) from 45 mA to 80 mA. In this case, the relaxation oscillation frequency ranges between 0 and about 4.1 GHz. We observed the bifurcation with the injection current through the oscilloscope while changing the feedback strength by using a neutral density filter placed between the laser diode and the external mirror. Table 1 shows the stable current points observed in the experiment for different feedback strengths. Table 1. Stable current points observed in the experiment for different feedback strengths κ 1 , κ 2 and κ 3 . κ Stable current points observed (mA) κ 1 56.20 59.02 62.42 66.12 70.06 74.27 κ 2 56.14 59.12 62.43 66.06 70.02 74.31 κ 3 56.16 59.15 62.38 66.03 70.10 74.28 The experimental results manifested clearly that there were stable current points when increasing the injection current. The observation of the bifurcation diagram was made from 55 mA to 75 mA since a chaos accompanied at small current around the threshold current disturbed exact measurement. 5. CONCLUSION We reported a principle of stable injection current where bifurcations are suppressed based on the investigation of the bifurcation scenario with the injection current. The existence of stable and unstable injection current points was clearly explained by the resonance between the relaxation oscillation frequency and the external cavity frequency. The positions of stable injection current points are defined by only the external cavity length regardless of both the linewidth enhancement factor and the feedback strength. Also, we suggested the formula of computing the total number of possible bifurcations in the variation interval of the injection current. The facts reported will be very significant and useful for applications of bifurcation and chaos in a laser diode. Declarations C.-H. Kim is funded by Natural Science Grant. C.-H. Kim acknowledges experimental support by S.-C. Ri. References S. Donati, G. Giuliani, and S. Merlo, IEEE J. Quantum Electron. 31, 113(1995). M. Virte, A. K. D. Bosco, D. Wolfersberger, and M. Sciamanna, Phys. Rev. A 84, 043836(2011). S. Sunada, T. Fukushima, S. Shinohara, T. Harayama, K. Arai, and M. Adachi, Appl. Phys. Lett. 104, 241105 (2014). Y. Liu, Y. Takiguchi, P. Davis, T. Aida, and S. Saito, Appl. Phys. Lett. 80, 4306 (2002). A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature (London)438, 343 (2005). A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, Nat. Photonics 2, 728 (2008). I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, Nat. Photonics 4, 58 (2010). F.-Y. Lin and J.-M. Liu, IEEE J. Sel. Top. Quantum Electron.10, 991 (2004). Y. Wang, B. Wang, and A. Wang, IEEE Photonics Technol. Lett.20, 1636 (2008). D. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, Nat. Commun. 4, 1364 (2013). N. Oliver, T. Jüngling, and I. Fischer, Phys. Rev. Lett. 114, 123902(2015). J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, 2 nd ed. Springer, Berlin, 2007. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4081906","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":278733815,"identity":"37c8043a-53a1-4460-8aa0-4a772f338ede","order_by":0,"name":"Chol-Hyon Kim","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA80lEQVRIiWNgGAWjYHACxsc/KmzkUIUa8GthNmY4k2bMwIYQIKiFTZix7XBiA9FazNtPpzEXnDmcPn9+87MPDDU2DObs/QcYZ+7ArUXmTO62xzMq0nM3HGMznsFwLI3BsucwA+PGM7i1SEjwbjfgOWOdu4GNwZj5b8NhBoMbyQyMD9vwatkmwdvGnC7fxv4Z6AeglvuPCWuR5m1zTmA4xmMM0XID6P2N+LTw5G42nHEmzXDDsZxiBqBfeAzOJBscnInPL+xnNz74UGEjL998fDMDMMTkDI4ffPiwF0+IYQAeEHGAUFxiAWRoGQWjYBSMguELADX9Tx1Eaz0FAAAAAElFTkSuQmCC","orcid":"","institution":"Kim Chaek University of Technology","correspondingAuthor":true,"prefix":"","firstName":"Chol-Hyon","middleName":"","lastName":"Kim","suffix":""},{"id":278733816,"identity":"467a1f20-3b19-41be-a5f8-6409e0c67e9a","order_by":1,"name":"Myong-Il Kim","email":"","orcid":"","institution":"","correspondingAuthor":false,"prefix":"","firstName":"Myong-Il","middleName":"","lastName":"Kim","suffix":""},{"id":278733818,"identity":"bb5570d0-b0a0-475c-941f-b65d3eba654b","order_by":2,"name":"Kwang-Myong Ho","email":"","orcid":"","institution":"Kim Chaek University of Technology","correspondingAuthor":false,"prefix":"","firstName":"Kwang-Myong","middleName":"","lastName":"Ho","suffix":""}],"badges":[],"createdAt":"2024-03-12 08:24:53","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4081906/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4081906/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":52691460,"identity":"ffa32603-ad09-4678-be53-bd7a02b68056","added_by":"auto","created_at":"2024-03-14 15:06:23","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":437517,"visible":true,"origin":"","legend":"\u003cp\u003eDevelopment of bifurcation and chaos in bifurcation diagrams with the normalized injection current when \u003cem\u003eL\u003c/em\u003e=50 cm, and \u003cem\u003eα\u003c/em\u003e=3 for (a) \u003cem\u003eκ\u003c/em\u003e=2×10\u003csup\u003e-3\u003c/sup\u003e, (b)\u003cem\u003e κ\u003c/em\u003e=2.5×10\u003csup\u003e-3\u003c/sup\u003e, (c)\u003cem\u003e κ\u003c/em\u003e=3×10\u003csup\u003e-3\u003c/sup\u003e, (d)\u003cem\u003e κ\u003c/em\u003e=3.35×10\u003csup\u003e-3\u003c/sup\u003e, and (e) \u003cem\u003eκ\u003c/em\u003e=3.4×10\u003csup\u003e-3\u003c/sup\u003e. (d) and (e) display detailed processes of chaos development. The red arrows represent the positions where the development of bifurcations is suppressed at most and the blue arrows the positions where the evolution of chaos occurs and grows. The red dashed rectangle in (b) and (c) shows the division of a mature chaos region.\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4081906/v1/05524cb10cf1569846c07193.jpg"},{"id":52691459,"identity":"8d57d091-10d2-4920-89ca-aea83c3354aa","added_by":"auto","created_at":"2024-03-14 15:06:23","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":189501,"visible":true,"origin":"","legend":"\u003cp\u003eBifurcation diagrams with the normalized injection current for \u0026nbsp;(a) \u003cem\u003eα\u003c/em\u003e=3.5, (b)\u003cem\u003e α\u003c/em\u003e=5, and (c) \u003cem\u003eα\u003c/em\u003e=7 when \u003cem\u003eL\u003c/em\u003e=30 cm and \u003cem\u003eκ=\u003c/em\u003e3×10\u003csup\u003e-3\u003c/sup\u003e. The red dashed lines indicate stable current points and the black ones unstable current points. The positions of unstable current points are in the middle of bifurcation ranges.\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4081906/v1/8cac2c41da104fb4083dd43f.jpg"},{"id":52691463,"identity":"50d37247-9e9b-4d03-9244-fab25daeb2f3","added_by":"auto","created_at":"2024-03-14 15:06:24","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":141569,"visible":true,"origin":"","legend":"\u003cp\u003eBifurcation diagrams with the normalized injection current for different sets of \u003cem\u003eκ\u003c/em\u003e and \u003cem\u003eα\u003c/em\u003e. (a) \u003cem\u003eα=\u003c/em\u003e3 and \u003cem\u003eκ=\u003c/em\u003e4×10\u003csup\u003e-3\u003c/sup\u003e, and (b) \u003cem\u003eα=\u003c/em\u003e4 and \u003cem\u003eκ=\u003c/em\u003e7×10\u003csup\u003e-3\u003c/sup\u003e. The red and black dashed lines indicate stable and unstable current points, respectively.\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4081906/v1/d1e4ffac0202d7f7ba00ad27.jpg"},{"id":52691461,"identity":"17e03fde-ccf3-4ed9-ae95-fb02995c6674","added_by":"auto","created_at":"2024-03-14 15:06:24","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":177438,"visible":true,"origin":"","legend":"\u003cp\u003eBifurcation diagrams with the normalized injection current for (a) \u003cem\u003eL=\u003c/em\u003e40cm, (b)\u003cem\u003e L=\u003c/em\u003e100cm, and (c) \u003cem\u003eL=\u003c/em\u003e250cm when \u003cem\u003eα\u003c/em\u003e=2.3, and \u003cem\u003eκ\u003c/em\u003e=3.5×10\u003csup\u003e-3\u003c/sup\u003e.\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4081906/v1/82c17d997effadf0d112d91b.jpg"},{"id":52691462,"identity":"57a20c08-e929-4cbd-b08a-f8695e652b24","added_by":"auto","created_at":"2024-03-14 15:06:24","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":170147,"visible":true,"origin":"","legend":"\u003cp\u003eRelation curve between the normalized injection current \u003cem\u003eJ\u003c/em\u003e/\u003cem\u003eJ\u003c/em\u003e\u003csub\u003e\u003cem\u003eth\u003c/em\u003e\u003c/sub\u003e and the relaxation oscillation frequency \u003cem\u003ef\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e for \u003cem\u003eL\u003c/em\u003e=50 cm. The frequency axis is divided equally at intervals of 0.3 GHz. The red asterisks, which corresponds to stable injection current points, indicates the positions where the relaxation oscillation frequencies are integer multiples of the external cavity frequency space.\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4081906/v1/94c1e282843387789d148d9a.jpg"},{"id":52691465,"identity":"0c78e438-d60f-42e5-905b-0e14c64d4bf4","added_by":"auto","created_at":"2024-03-14 15:06:24","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":180918,"visible":true,"origin":"","legend":"\u003cp\u003eBifurcation diagram with normalized injection current for (a) \u003cem\u003eα\u003c/em\u003e=2, \u003cem\u003eκ\u003c/em\u003e=3.5×10\u003csup\u003e-3\u003c/sup\u003e, and \u003cem\u003eL\u003c/em\u003e=50 cm, and (b)\u003cem\u003e α\u003c/em\u003e=5, \u003cem\u003eκ\u003c/em\u003e=2×10\u003csup\u003e-3\u003c/sup\u003e, and \u003cem\u003eL\u003c/em\u003e=90 cm. The red asterisks are stable injection current points found using Eq. (4) and Fig. 5. Bifurcations are clearly suppressed in the positions where intersect the red dashed lines.\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4081906/v1/4959344d88b3fc0794b9a419.jpg"},{"id":52691464,"identity":"85ea4fc9-a374-430d-be10-42bb5639e44d","added_by":"auto","created_at":"2024-03-14 15:06:24","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":161570,"visible":true,"origin":"","legend":"\u003cp\u003ePositions of unstable injection current points in bifurcation diagram with normalized injection current when \u003cem\u003eα\u003c/em\u003e=2.3, \u003cem\u003eκ\u003c/em\u003e=3.5×10\u003csup\u003e-3\u003c/sup\u003e, for (a) \u003cem\u003eL\u003c/em\u003e=40 cm, and (b) \u003cem\u003eL\u003c/em\u003e=130 cm. The red asterisks are stable injection current. The black dashed lines represent of the positions of unstable injection current points which lie halfway between stable injection current points.\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4081906/v1/cc8c6909a38420cf0389f1a1.jpg"},{"id":54792667,"identity":"6bd7419e-89c2-4d68-b861-fa19638d77ec","added_by":"auto","created_at":"2024-04-16 22:07:38","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":798102,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4081906/v1/893bcc1d-16dc-480e-a5ff-045a0e3bb5f1.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Demonstration of regularity of bifurcation in chaotic laser diode","fulltext":[{"header":"1. INTRODUCTION","content":"\u003cp\u003eSince a laser diode has a large linewidth enhancement factor (the value of which ranges typically between 2 and 7 [1]), it exhibits an interesting dynamics unlike other lasers. In the presence of optical feedback, particularly, the dynamics of a laser diode is ready to be accompanied by bifurcations or chaos [2]. The study on such laser systems is very useful for communication encryption systems [3]-[5], random number generation [6],[7], remote sensing [8][9], and photonic information processing [10].\u003c/p\u003e\n\u003cp\u003eThere are many parameters that have an effect on the dynamic of a laser diode; the external parameters of which are the injection current, the optical feedback strength, and the delayed time (or external cavity length), and the internal parameters of which are the linewidth enhancement factor, the laser diode cavity length (or diode cavity round trip time), the carrier lifetime, the photon lifetime, and the carrier density at threshold. The dynamics of a laser diode is very sensitive to the above-mentioned parameters and therefore a small change of any parameter of them may cause the laser system to stay irregularly on a stable state, bifurcation or chaos. To find a regularity in such an irregularity is of great significance and it can provide an available tool for many applications (e.g. neuro-inspired information processing techniques [11]).\u003c/p\u003e\n\u003cp\u003eIn the paper we report a regularity of the bifurcation diagram with the injection current. First, we examine the characteristics of the development of bifurcations and chaos in the diagram for different linewidth enhancement factors, feedback strengths, and external cavity lengths. Then we explain the regularity reported based on the resonance relation between the oscillation frequency and the external cavity frequency.\u003c/p\u003e"},{"header":"2. REGULARITY OF BIFURCATION DIAGRAM IN NUMERICAL SIMULATION: PRINCIPLE OF EXISTENCE OF STABLE CURRENT POINT","content":"\u003cp\u003eThe dynamics of a laser diode with delayed optical feedback is depicted by [1],[2]\u003c/p\u003e\n\u003cp\u003e\u003cimg 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\" width=\"394\" height=\"156\"\u003e\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003eE\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e(\u003cem\u003et\u003c/em\u003e) is the amplitude of the laser electric field which is normalized so that\u0026nbsp;\u003cimg src=\"data:image/png;base64,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\"\u003e equals the photon density in the laser diode cavity, \u003cem\u003eφ\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) the phase of the laser electric field, \u003cem\u003eN\u003c/em\u003e(\u003cem\u003et\u003c/em\u003e) the carrier density, \u003cem\u003eτ\u003c/em\u003e the external cavity round trip time, \u003cem\u003eκ\u003c/em\u003e the feedback strength, \u003cem\u003eα\u003c/em\u003e the linewidth enhancement factor, and \u003cem\u003eJ\u003c/em\u003e the injection current. We set in the numerical calculation the angular frequency of the laser light, \u003cem\u003eω\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e=2.89×10\u003csup\u003e24\u003c/sup\u003e rad s\u003csup\u003e-1\u003c/sup\u003e (corresponding to a wavelength of 632 nm), the modal gain coefficient, \u003cem\u003eG\u003csub\u003eN\u003c/sub\u003e\u003c/em\u003e=8×10\u003csup\u003e-13\u003c/sup\u003e m\u003csup\u003e3\u003c/sup\u003e s\u003csup\u003e-1\u003c/sup\u003e, the carrier density at transparency, \u003cem\u003eN\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e=1.4×10\u003csup\u003e24\u003c/sup\u003e m\u003csup\u003e-3\u003c/sup\u003e, the carrier density at threshold for the unperturbed laser, \u003cem\u003eN\u003csub\u003eT\u003c/sub\u003e\u003c/em\u003e=2.3×10\u003csup\u003e24\u003c/sup\u003e m\u003csup\u003e-3\u003c/sup\u003e, the photon lifetime, \u003cem\u003eτ\u003csub\u003eP\u003c/sub\u003e\u003c/em\u003e=1/[\u003cem\u003eG\u003csub\u003eN\u003c/sub\u003e\u003c/em\u003e(\u003cem\u003eN\u003csub\u003eT\u003c/sub\u003e\u003c/em\u003e-\u003cem\u003eN\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e)], the laser cavity round trip time, \u003cem\u003eτ\u003csub\u003eL\u003c/sub\u003e\u003c/em\u003e=8.3 ps, the carrier lifetime, \u003cem\u003eτ\u003csub\u003eS\u003c/sub\u003e\u003c/em\u003e=2 ns. Also the electric field and the carrier density are normalized by the steady-state electric field and the carrier density at the threshold for the unperturbed laser, respectively. We have carried out numerical simulations of Eq. (1) to (3) under the above conditions using SIMULINK supported by MATLAB to examine bifurcation diagrams with the injection current.\u003c/p\u003e\n\u003cp\u003eWe have observed the bifurcation diagram with the injection current while changing the linewidth enhancement factor \u003cem\u003eα\u003c/em\u003e from 2 to 7 and the feedback strength \u003cem\u003eκ\u003c/em\u003e from 0 to 10×10\u003csup\u003e-3\u003c/sup\u003e for different external cavity length \u003cem\u003eL\u003c/em\u003es.\u003c/p\u003e\n\u003cp\u003eFirst, we have examined bifurcation diagrams with changing only \u003cem\u003eκ\u003c/em\u003e when \u003cem\u003eL\u003c/em\u003e=50 cm, and \u003cem\u003eα\u003c/em\u003e=3. Fig. 1 illustrates the development of bifurcations and chaos in some of the bifurcation diagrams. The current injection is normalized by the threshold current injection \u003cem\u003eJ\u003csub\u003eth\u003c/sub\u003e\u0026nbsp;\u003c/em\u003ein Fig. 1, where the range without bifurcation is called an \u003cem\u003einter-bifurcation range\u003c/em\u003e. As seen in Fig. 1 (a) to (c), there alternately are appearance and disappearance of bifurcations while the bifurcations ranges expand and the inter-bifurcation ranges reduce as \u003cem\u003eκ\u003c/em\u003e increases. If the bifurcation ranges expand to merge into each other, chaos can evolve in such bifurcation ranges alone. The chaotic regions marked blue arrows in Fig. 1 (b) to (e) grow greater in given bifurcation ranges as \u003cem\u003eκ\u0026nbsp;\u003c/em\u003eincreases. In addition, the chaos regions develop to some extent to divide into two parts (see the red dashed rectangle in Fig. 1 (b) and (c)). Fig. 1 indicates some important rules in the birth and evolution of bifurcations and chaos: i) there exist \u003cem\u003estable current points\u003c/em\u003e (marked with red arrows in Fig. 1 where the birth of bifurcations is suppressed while there are \u003cem\u003eunstable current points\u003c/em\u003e (not indicated in Fig. 1, but found in Fig. 2 and 3) ready for bifurcation birth; ii) the chaos states develop around the middle of the merged bifurcation ranges (to be more specific, they lean to the right as seen in Fig. 1 (d) and (e)); and iii) the intervals between stable current points become wider as the relative injection current increases. The most important thing in these rules is that the positions of the stable and unstable current points are independent of the value of \u003cem\u003eκ\u003c/em\u003e. The stable current points marked with red arrows situate within the inter-bifurcation ranges and their positions are constant regardless of \u003cem\u003eκ\u003c/em\u003e from Fig. 1 (a), (b) and (c), and so are the positions of the unstable current points. In addition, the total number of possible bifurcations is constant in a given range of the injection current, which is the result of the mechanism of the bifurcation evolution above-explained.\u003c/p\u003e\n\u003cp\u003eSecond, we have observed bifurcation diagrams varying only \u003cem\u003eα\u003c/em\u003e when \u003cem\u003eL\u003c/em\u003e=30 cm, and \u003cem\u003eκ\u003c/em\u003e=3×10\u003csup\u003e-3\u003c/sup\u003e. Fig. 2 shows some of them. The positions of stable and unstable current points are constant independent of \u003cem\u003eα\u003c/em\u003e as in the case of Fig. 1. As a result, the analysis of Fig. 1 and 2 draws a conclusion that the positions of stable and unstable current points are independent of both \u003cem\u003eκ\u003c/em\u003e and \u003cem\u003eα\u003c/em\u003e. Fig. 3 shows the bifurcation diagrams plotted for different sets of \u003cem\u003eκ\u003c/em\u003e and \u003cem\u003eα\u003c/em\u003e, which will convince the above conclusion. As evident in Fig. 3, although the values of \u003cem\u003eα\u003c/em\u003e and \u003cem\u003eκ\u003c/em\u003e differs greatly, respectively, and, what is more, mature chaos exist as in Fig. 3 (b), the positions of stable and unstable current positions can be easily identified and are invariable.\u003c/p\u003e\n\u003cp\u003eThird, we have investigated bifurcation diagrams for different external cavity length, some of which is shown in Fig. 4. It can be seen from Fig. 4 that the bifurcation number per unit relative injection current increases as the external cavity length grows.\u003c/p\u003e\n\u003cp\u003eWe claim through the above consideration that the positions of stable and unstable current points and the bifurcation number density depend on only the external cavity length irrespective of \u003cem\u003eκ\u003c/em\u003e or \u003cem\u003eα\u003c/em\u003e.\u003c/p\u003e"},{"header":"3. THEORETICAL ANALYSIS OF STABLE CURRENT POINTS","content":"\u003cp\u003eNow we explain why the laser diode with delayed optical feedback follows the bifurcation scenario as reported above. When a bifurcation or a chaos occurs its fundamental frequency is almost equal to the relaxation oscillation frequency [12]\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" height=\"66\" width=\"273\"\u003e\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003eJ\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is the injection current at transparency, which is proportional to \u003cem\u003eN\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e. It should be noted in Eq. (4) that the relaxation oscillation frequency \u003cem\u003ef\u003csub\u003er\u003c/sub\u003e\u003c/em\u003e is proportional to the square root of the injection current \u003cem\u003eJ\u003c/em\u003e. Whenever the relaxation oscillation frequency \u003cem\u003ef\u003csub\u003er\u003c/sub\u003e\u003c/em\u003e coincides with integer multiples of the external frequency, the semiconductor laser system at such injection current points exhibits very strong tendency to maintain a stable state which is the most bifurcation-proof.\u003c/p\u003e\n\u003cp\u003eLet us explain it in more detail. Fig. 5 illustrates the plot of the relative injection current \u003cem\u003eJ\u003c/em\u003e/\u003cem\u003eJ\u003csub\u003eth\u003c/sub\u003e\u003c/em\u003e versus the relaxation oscillation frequency \u003cem\u003ef\u003csub\u003er\u003c/sub\u003e\u003c/em\u003e for \u003cem\u003eL\u003c/em\u003e=50cm. In this case the frequency space \u003cem\u003eΔν\u003c/em\u003e between the external cavity modes is\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,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\" height=\"47\" width=\"132\"\u003e\u003c/p\u003e\n\u003cp\u003eThe red asterisks in Fig. 5 represent the positions of the relaxation oscillation frequencies being integer multiples of the external cavity frequency space \u003cem\u003eΔν\u003c/em\u003e, which are none other than the stable injection current points. To manifest this, Fig. 6 shows stable injection current points obtained in the way of plotting Fig. 5 for different sets of \u003cem\u003eκ\u003c/em\u003e, \u003cem\u003eα\u003c/em\u003e, and \u003cem\u003eL\u003c/em\u003e. It is clear from Fig. 6 that bifurcations are suppressed in stable injection current points. It should be noted that a stable injection current point cannot be identified in the regions where bifurcations are mature and merged. Meanwhile, the stable current points in Fig. 5 get sparse as the injection current increases, for the curve follows the form of a function \u003cimg src=\"data:image/png;base64,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\" width=\"64\" height=\"24\"\u003e\u0026nbsp;This accounts for the above-mentioned reason why an interval between stable injection currents points increases with the injection current.\u003c/p\u003e\n\u003cp\u003eMoreover the total number, \u003cem\u003en\u003csub\u003ebif\u003c/sub\u003e\u003c/em\u003e, of bifurcations in the variation range of the normalized injection current is defined by\u003c/p\u003e\n\u003cp\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARYAAAA2CAYAAAAClrbNAAAHdElEQVR4Ae2cDVEsOxCFsYAGLOABCWjAAg5wgAMUoAADGMABHubVt/edqqbJ/G6WZGZPV01NJukknZPuk85y694MFiNgBIxAZQRuKo/n4YyAETACg4nFTmAEjEB1BEws1SH1gEbACJhY7ANGwAhUR8DEUh1SD2gEjICJxT5gBIxAdQRMLNUh9YBGwAiYWOwDRsAIVEfAxFIdUg9oBIyAicU+YASMQHUETCzVIfWARsAImFjsA0bACFRHwMRSHVIPaASMgInFPmAEjEB1BEws1SH1gEbACJhYOvOB19fX4fPzsxur3t/fBx6LEViDgIllDVoX1n16ehpeXl5+zALJPDw8DDc3N7/afihe8AO7eCxGYCkCJpalSF1Y7/n5uRi8kApZTGvpxY7WOHj+ZQiYWJbhdFGtj4+P4fb2dvj+/v4xz/39/SlTaZmtyKCvr6+TLbwtRmAOARPLHEJ/0E42kK9ATMs16O7u7g8sWDaFr0TLcLLW4P+asrUTkK2QkeRsBbu4AvX02wZEh63OWlp7Tf/zO2NpvEcQBxlLSah/e3s7NSl7eXx8PNVxdYpZDr/REPT8BYcrFO0qU09/iXTRQReBLPimDZKjnnIWMqgefvPJdvm7LwSujlgIGoKTYNMzFth/sVWZIDQndmKfsgMyG4gCfciGhzoJbayDh765LF3IiPWjQx/KCGWRC2QHEUXi0jxTRCgdv/eBAPudr9p8Ky54I+y5DrilK7s6YiHgCBgCCxA5lalrIWwsm6egjzaQFeRNZ3PZ5DFBX0REWVkKZdaLQEzSYe05+yBTKdmjOdGXw6nO7/0hgC/lfcyHVVwVMTLle1GX8lURC8GVAxmSaUUsZAnYo0DX5rCBkRhifT45ZDtj0AehDIEgkAs6vKlnPnCAPNDBBjCAZHjIYPimviT0YwyRVknHdX0jID/IVs4RB36U/S+Poe+rIhYFRTyRWxILcxOkSwXiwCmiiFjYcDkGZV1xIAAIRG3UMydzU2ZMshB00OWhHZIpidojhiU91/WLAD7D/kfBZ9h3+UZsU3mMkNQe34u9mlMOg5iY0wwHwymnDIkT1Shr4VPvDJjmjfaqP23oKzilS53WRsDlIGMsBShjcX2gTqK+BB990clzaG7atgpzLj1Bts5R6ofN+QpV0nNdfwiIHPIBpW+1j/klvr3E5xZ7NQGCIxMgnH58Qza8S0HTH6TDKf0HsHjaZmIBNMhEOrz5joQFuGDA+nkoo0MZoQ/zgAskTHsJI8Yc28Ae8ZNN2BzxUL3f/SOgzGTOUvl41sOXeeZkMbFoICZUmk3d0onUv+VbAS/SwJZMLKU/s0KerBuBKAgskQh1lKkTk2uesd8pTgP9P3eJWKjr5ZGt8Y1tJpaIyH7K7Jt8ecpqfLikN3ZI5rFWEYvSpBiYTK6AYnAmxvHi1UCT0p+TPQuLpZ5+U7Ik2KYcXgEf7Uc/ZhOloEFHtsVytDX2K80TdVUeG0vtvb7jWnu10XaVEcDnSoRR0i7pXYRYdE2QESKaeHrTViIP9clvgpAF5DGyXo3vUsADdCQWbCFDicI3mQyiVDLaq4xFGUppnjieynsmFv/Gol3c11v+O2c1PgyJZKGuVJ/1plOEpM2A8Rqkf9NAYCkYMYhARZeTLdbzHR1SREU9z6WlFPDYF5mZYIcY0UV48y27WSvfrI8yD+VIjqV5SmvbI7EsXVtpva5rj4CSAd5TMhaP+DlxOyeroplBFWAMrKCLfxUhWPTNCR6zFwgnX5EgqiWGzi1kSTv2Axg26S8qIjXIQQLZoEMbfVhTFPqyRvVlDXFdeZ7YN5YVpLFvbO+xLJvnHLNH223TPwSIw+zT8ln5dAkrkVKpLdetIpbcufSN0TgfArHwLcH4LATwtTqpNkpXqIxNj984ZDwserTRNk0jIL+b1vrdyuGbCem31r+a6sQC43E9wHhOdWUjnMoiHREJBFQimzFjj1jP+pdu1tT65SzCdkr3nDays3gdPmcs922HwNrMk9iNWf2c5VWJBWN1ReBUiwEDsUA6+s0FwyivMXZuMXtsBwMwO1fAms2PV9Vzxyz1Z191WJTaXbcvBJYc7MTo2j2vSixrICWrYVFrDV4zxx50a2Ua+p0nkhSkBZnragr5nJNt6Dcz9s5iBKYQaEIsCqaYvUwZefQ2Mo2Y3a1dL+QsgoaswZfgJ0sEY41NWXpr50AfUvKebUHu+vo0IZbrg3l6xRDAOT9ixyyEwI/XIUiFB6KBwLYKWQ82OlvZiuB19TOxdLLfkMGWwCfglZGwFLKVeB0iQ4FsGHsrKdCPMff016tOtvVqzTCxdLT1EAA/lK0hAPT1bw/iG4JBlGmQFW0Rkco5V6gt87rPvhEwsXS2fxCBSKEH0yCkraTUg/22oQ0CJpY2uHtWI3BoBEwsh95eL84ItEHAxNIGd89qBA6NgInl0NvrxRmBNgiYWNrg7lmNwKERMLEcenu9OCPQBgETSxvcPasRODQCJpZDb68XZwTaIGBiaYO7ZzUCh0bAxHLo7fXijEAbBEwsbXD3rEbg0Aj8BxqrrW79REkrAAAAAElFTkSuQmCC\" width=\"278\" height=\"54\"\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003ef\u003csub\u003er,max\u003c/sub\u003e\u003c/em\u003e is the maximum of the relaxation oscillation frequency in the injection current variation range, which is calculated as 16 for Fig. 6.\u003c/p\u003e\n\u003cp\u003eFig. 7 shows the positions of unstable injection current points in the bifurcation diagrams plotted for different sets of \u003cem\u003eκ\u003c/em\u003e, \u003cem\u003eα\u003c/em\u003e, and \u003cem\u003eL\u003c/em\u003e. It should be noted that the latest and smallest bifurcations in Fig. 7 (a) and (b) demonstrate well a birth of bifurcation. Here we emphasize that unstable injection current points, where bifurcations begin to grow, lie halfway between stable injection current points because there is greatest differences between the relaxation oscillation frequencies and the external cavity frequency in the positions in contrast to the cases of stable injection points.\u003c/p\u003e"},{"header":"4. EXPERIMENTAL DEMONSTRATION","content":"\u003cp\u003eWe demonstrated experimentally the new regularity of bifurcation in a chaotic laser diode. In the experiment, a Hitachi HL6312G, whose typical threshold current was 45 mA, was served as a laser diode and the optical output power of the laser diode was monitored by an u2t photodiode of 9 ps response time. The temperature of the laser diode was stabilized at 25℃ by a precision temperature controller (TED 200, Thorlabs). The electrical signal from the photodiode was measured by a digital 50 GHz sampling oscilloscope (HP 54120B). Note that the oscilloscope ability was insufficient to obtaining the exact waveform of the optical power of the laser diode but was enough to distinguish whether there was a bifurcation or not. The external cavity length was set 50 cm corresponding the external cavity frequency 0.3 GHz. The injection current of the laser diode was adjusted by a laser diode controller (LDC210) from 45 mA to 80 mA. In this case, the relaxation oscillation frequency ranges between 0 and about 4.1 GHz. We observed the bifurcation with the injection current through the oscilloscope while changing the feedback strength by using a neutral density filter placed between the laser diode and the external mirror.\u003c/p\u003e\n\u003cp\u003eTable 1 shows the stable current points observed in the experiment for different feedback strengths.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eTable 1. \u003c/strong\u003eStable current points observed in the experiment for different feedback strengths \u003cem\u003e\u0026kappa;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003e\u0026kappa;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e and \u003cem\u003e\u0026kappa;\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e.\u003c/p\u003e\n\u003ctable border=\"0\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e\u0026kappa;\u003c/em\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"85.71428571428571%\" colspan=\"6\" valign=\"top\"\u003e\n \u003cp\u003eStable current points observed (mA)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e\u0026kappa;\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e56.20\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e59.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e62.42\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e66.12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e70.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e74.27\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e\u0026kappa;\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e56.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e59.12\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e62.43\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e66.06\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e70.02\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e74.31\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e\u003cem\u003e\u0026kappa;\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e56.16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e59.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e62.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e66.03\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e70.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"14.285714285714286%\" valign=\"top\"\u003e\n \u003cp\u003e74.28\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eThe experimental results manifested clearly that there were stable current points when increasing the injection current. The observation of the bifurcation diagram was made from 55 mA to 75 mA since a chaos accompanied at small current around the threshold current disturbed exact measurement.\u0026nbsp;\u003c/p\u003e"},{"header":"5. CONCLUSION","content":"\u003cp\u003eWe reported a principle of stable injection current where bifurcations are suppressed based on the investigation of the bifurcation scenario with the injection current. The existence of stable and unstable injection current points was clearly explained by the resonance between the relaxation oscillation frequency and the external cavity frequency. The positions of stable injection current points are defined by only the external cavity length regardless of both the linewidth enhancement factor and the feedback strength. Also, we suggested the formula of computing the total number of possible bifurcations in the variation interval of the injection current. The facts reported will be very significant and useful for applications of bifurcation and chaos in a laser diode.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eC.-H. Kim is funded by Natural Science Grant. C.-H. Kim acknowledges experimental support by S.-C. Ri.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eS. Donati, G. Giuliani, and S. Merlo, IEEE J. Quantum Electron. 31, 113(1995).\u003c/li\u003e\n \u003cli\u003eM. Virte, A. K. D. Bosco, D. Wolfersberger, and M. Sciamanna, Phys. Rev. A 84, 043836(2011).\u003c/li\u003e\n \u003cli\u003eS. Sunada, T. Fukushima, S. Shinohara, T. Harayama, K. Arai, and M. Adachi, Appl. Phys. Lett. 104, 241105 (2014).\u003c/li\u003e\n \u003cli\u003eY. Liu, Y. Takiguchi, P. Davis, T. Aida, and S. Saito, Appl. Phys. Lett. 80, 4306 (2002).\u003c/li\u003e\n \u003cli\u003eA. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Nature (London)438, 343 (2005).\u003c/li\u003e\n \u003cli\u003eA. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, Nat. Photonics 2, 728 (2008).\u003c/li\u003e\n \u003cli\u003eI. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, Nat. Photonics 4, 58 (2010).\u003c/li\u003e\n \u003cli\u003eF.-Y. Lin and J.-M. Liu, IEEE J. Sel. Top. Quantum Electron.10, 991 (2004).\u003c/li\u003e\n \u003cli\u003eY. Wang, B. Wang, and A. Wang, IEEE Photonics Technol. Lett.20, 1636 (2008).\u003c/li\u003e\n \u003cli\u003eD. Brunner, M. C. Soriano, C. R. Mirasso, and I. Fischer, Nat. Commun. 4, 1364 (2013).\u003c/li\u003e\n \u003cli\u003eN. Oliver, T. J\u0026uuml;ngling, and I. Fischer, Phys. Rev. Lett. 114, 123902(2015).\u003c/li\u003e\n \u003cli\u003eJ. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, 2\u003csup\u003end\u003c/sup\u003e ed. Springer, Berlin, 2007.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-4081906/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4081906/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The behavior of bifurcation and chaos in a laser diode subject with delayed optical feedback is very chaotic and unpredictable. We have theoretically and experimentally demonstrated an extremely significant and useful rule of the growth of bifurcation and chaos displayed in the bifurcation diagram with the injection current. Such a significant rule is attributed to the resonance between the relaxation oscillation frequency and the external cavity frequency. This regularity can provide a principle for the suppression and control of bifurcation and chaos in semiconductor lasers with delayed optical feedback.","manuscriptTitle":"Demonstration of regularity of bifurcation in chaotic laser diode","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-14 15:06:19","doi":"10.21203/rs.3.rs-4081906/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"63731c03-8848-418b-98ff-dfa3b713a1af","owner":[],"postedDate":"March 14th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-04-16T21:59:29+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-14 15:06:19","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4081906","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4081906","identity":"rs-4081906","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}