{"id":261,"date":"2021-02-01T11:26:57","date_gmt":"2021-02-01T10:26:57","guid":{"rendered":"https:\/\/www.multimodeoptics.com\/erc-stems\/?page_id=261"},"modified":"2021-03-05T12:57:49","modified_gmt":"2021-03-05T11:57:49","slug":"spatiotemporal","status":"publish","type":"page","link":"https:\/\/www.multimodeoptics.com\/erc-stems\/spatiotemporal\/","title":{"rendered":"SPATIOTEMPORAL"},"content":{"rendered":"\n
\n

Spatiotemporal Mode-Locking in a Fiber Laser<\/p>\n<\/div><\/div>\n\n\n\n

Kerr-lens mode-locking (KLM) [1] provided unprecedented progress in the generation of ultrashort pulses in solid-state oscillators, with remarkable fundamental and practical consequences [2]. KLM utilizes self-focusing inside a nonlinear medium, which causes an effective loss decrease with power growth. This involves using the inherent nonlinearity of a dissipative system for producing highly coherent localized structures (i.e., dissipative solitons, or DSs). Simultaneously, a remarkable breakthrough has been achieved by introducing ultrafast fiber lasers, which permit to avoid thermal effects and environmental sensitivity issues, while providing high gains and covering a broad spectral range [2]. However, optical nonlinearities limit energy harvesting in ultrafast fiber lasers. The typical method to suppress nonlinearity by means of increasing the mode-size may results in multimode instabilities [3]. Nevertheless, current studies have demonstrated that the introduction of distributed nonlinearity, in combination with a careful control of dissipative effects, could allow for stable energy harvesting (by means of the so-called distributed KLM, DKLM mode of operation) [4]. As a result, spatiotemporal mode-locking (ML) in both solid-state and fiber lasers could be realized [3, 5\u20136].<\/p>\n\n\n\n

In this work, we demonstrate, for the first time to our knowledge, that the transversely-graded dissipation provided by doping of a graded-index (GRIN) fiber allows for DKLM in a fiber laser, operating in either the anoma- lous or the normal group-delay dispersion (GDD) regime (ANDR and NDR), respectively. We investigate, both analytically and numerically, the parameters and stability of spatiotemporal DSs. The problem of ML self-starting, and the outlook for the experimental demonstration of fiber DKLM are discussed.<\/p>\n\n\n\n

\"\"
Fig. 1<\/strong>. Possible realizations of DKLM in MMF laser: a) disjointed structure with a free beam evolving in a highly nonlinear crystal, leading to highly reduced beam overlap with the subsequent fiber for a low-intensity field (LI mode); b) tapered fiber with reduced leaking mode losses for a high-intensity field (HI mode); c) MMF or photonic-crystal-fiber (PCF) structure with graded losses, providing a loss decrease due to switching to a \u201cself-focused\u201d HI mode. The common principle is a self-amplitude modulation by loss decrease for the HI field, in analogy with the KLM principle.<\/small><\/figcaption><\/figure><\/div>\n\n\n\n

<\/p>\n\n\n\n

As was discussed broadly, the trapping parabolic-like profile of the refractive index in a GRIN fiber assists spa- tiotemporal soliton stabilization [3]. Here we propose using a gradient-gain-profile in a doped fiber amplifier. This produces a distributed amplification, akin to the action of an aperture in a KLM oscillator (see Fig.1). As a result, we pre- dict the formation of a stable spatiotemporal DS in an all-fiber laser, without any additional ML mechanism, thus implementing fiber-based DKLM. We consider a GRIN fiber with axially symmetrical parabolic profile of the refractive index (s<\/em>). In combination with GDD (\u03b4) and Kerr-nonlinearity (\u03bd), propagation of the laser field a <\/em>can be described by the Gross-Pitaevskii equation [3], generalized with taking into account of the net-loss, including saturated gain on the fiber axis (\u039b), spectral dissipation (\u03c4), and parabolic profiled loss (\u03ba).<\/p>\n\n\n\n

\"\"
Fig.2.<\/strong> Dependence of the DS temporal width an intensity (inset) oN the grading dissipationp arameter \u03ba, for two values of the saturated net-loss \u039b, and \u03b4 = \u22121.<\/small><\/figcaption><\/figure><\/div>\n\n\n\n

<\/p>\n\n\n\n

Our model admits an analysis based on the variational approximation in combination with the Kantarovich\u2019s method, taking into account dissipative factors. The obtained results are briefly summarized in Figs.1-2. Fig. 2 shows the dependencies of the DS temporal width and peak power on the graded dissipation parameter \u03ba for the case of normal-GVD are shown in Fig. 2. A DS is positively chirped for both anomalous- and normal-GVD regimes (\u03c8 > 0), and it has a negative wave-front curvature (\u03b8 < 0).<\/p>\n\n\n\n

On the other hand, Fig.3 shows the dependence of DS stability boundaries on the \u201crelative aperture size\u201d \u03c7 = \udbff\udc04<\/sup>|\u039b|\/\u03ba. This defines the size of the dissipative guiding window and the graded-loss coefficient \u03ba, for both ANDR and NDR regimes.<\/p>\n\n\n\n

\"\"
Fig. 3.<\/strong> Dependence of the upper boundaries of DS stability in either anomalous- (\u03b4 = 1, a<\/em>) and normal-GVD (\u03b4 = \u22121, b<\/em>) dispersion regime, respectively, upon the dimensionless \u201caperture size\u201d \u03c7 for different spectral dissipation parameters \u03c4 . The curve and scatter point plots trace stability boundaries for different initial field amplitude (\u03b10<\/sub>) values. Insets show the DS intensity evolution in the vicinity of the upper stability boundary obtained on the VA based simulations, and contour plots illustrate the DS intensity on the t <\/em>versus r plane, obtained by full-dimensional Gross-Pitaevskii equation simulations for the shown parameters.<\/small><\/figcaption><\/figure><\/div>\n\n\n\n

<\/p>\n\n\n\n

Our principal results can be summarized in the following way. i) Spectral dissipation enhances DS stability, and broadens the \u201cattraction basin,\u201d i.e., the ML self-starting ability, which is not possible for \u03c4 = 0. ii) The growth of the graded-dissipation index \u03ba reduces the stability region, and increases the DS duration. However, it also reduces the sensitivity to the initial condition, which could mean an enhancement of the ML self-starting ability. iii) The DS stability region is broader for the case of NDR. Direct numerical simulations demonstrate the self-starting of DS generation for both regimes, as well as the contribution of higher-order modes, causing DS oscillations (insets in Fig. 3, b<\/em>). The main destabilizing scenarios are provided by i) multipulsing in the ANDR (inset in Fig. 3, a<\/em>), ii) DS collapse in the NDR, owing to strong Q-switching on the initial stage of ML, and iii) unlimited temporal spreading of the DS.<\/p>\n\n\n\n

We anticipate that the nonlinear coupling of spatial modes in both GRIN and photonic-crystal fibers could lead implementing DKLM in a fiber laser in the regime of multimode self-cleaning [3] supported by the graded dissipation, thus providing highly-efficient and stable energy harvesting in an all-fiber laser, without using any additional ML mechanism (i.e., a self-ML is realized).<\/p>\n\n\n\n

  1. D. E. Spence, et al., \u201c60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,\u201d Opt. Lett. 16<\/strong>, 42\u201344 (1991).<\/li>
  2. see chapters in Ultrafast Lasers: Technology and Applications<\/em>, M. E. Fermann, A. Galvanauskas, G. Sucha, eds. (Marcel Dekker, 2003).<\/li>
  3. K. Krupa, et al., \u201cMultimode nonlinear fiber optics, a spatiotemporal avenue,\u201d APL Photonics 4<\/strong>, 110901 (2019).<\/li>
  4. J. Zhang, et al., \u201cGeneration of 49-fs pulses directly from distributed Kerr-lens mode-locked Yb:YAG thin- disk oscillator,\u201d in Advanced Solid State Lasers Conference<\/em>, OSA Technical Digest (online) (Optical Society of America, 2015), paper ATh4A.7.<\/li>
  5. W. H. Renninger, F. W. Wise, \u201cSpatiotemporal soliton laser,\u201d Optica 1<\/strong>, 101\u2013104 (2014).<\/li>
  6. L. G. Wright, et al., \u201cSpatiotemporal mode-locking in multimode fiber lasers,\u201d Science 358<\/strong>, 94\u201397 (2017).<\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

    Kerr-lens mode-locking (KLM) [1] provided unprecedented progress in the generation of ultrashort pulses in solid-state oscillators, with remarkable fundamental and practical consequences [2]. KLM utilizes self-focusing inside a nonlinear medium, which causes an effective loss decrease with power growth. This involves using the inherent nonlinearity of a dissipative system for producing highly coherent localized structures … Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_mi_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_links_to":"","_links_to_target":""},"yoast_head":"\nSPATIOTEMPORAL - ERC-STEMS<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.multimodeoptics.com\/erc-stems\/spatiotemporal\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"SPATIOTEMPORAL - ERC-STEMS\" \/>\n<meta property=\"og:description\" content=\"Kerr-lens mode-locking (KLM) [1] provided unprecedented progress in the generation of ultrashort pulses in solid-state oscillators, with remarkable fundamental and practical consequences [2]. 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