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Coherent Gas-Laser Interactions via Stimulated Raman Scattering in Hollow-Core Photonic Crystal Fibers Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Marta Ziemieńczuk aus Bielawa, Polen Als Dissertation genehmigt von der naturwissenschaftlichen Fakultät der Universität Erlangen-Nürnberg Tag der mündlichen Prüfung: 23. Mai 2012 Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink Erstberichterstatter: Prof. Dr. Philip St.J. Russell Zweitberichterstatter: Prof. Dr. Clemens F. Kaminski Abstract Invention of hollow-core photonic crystal fibers (HC-PCF) has been a milestone for studies in light-matter interactions and in particular stimulated Raman scattering (SRS). Tight confinement of both intense laser light and Raman-active medium in the micronsize fiber core has led to extremely high frequency conversion efficiencies as well as lowering the threshold for the SRS process by several orders of magnitude. The objective of this thesis is to investigate novel aspects of stimulated Raman scattering in gas-filled hollow-core photonic crystal fibers. In the theoretical part of this thesis Maxwell-Bloch equations are solved for rotational SRS and the mathematical foundation of this phenomenon is formulated both in classical and quantum picture. I provide a full description of space-time evolution of laser light (pump and Stokes) and material response (population inversion and coherence). The formalism described in this chapter serves as a basis for treating more complex experimental and theoretical situations, such as intermodal SRS. HC-PCF comes in two main varieties: hollow-core photonic bandgap (HC-PBG) fiber and kagome lattice fiber. The former is characterized by a narrowband transmission window and low loss within it. The latter has higher confinement loss but broadband transmission. For both types of these fibers the guiding mechanism, loss and dispersion properties are explained. The experimental part of this work starts with a description of the advantages of using HC-PCF fibers for SRS experiments. Diffractionless propagation over a long distance, low confinement loss, high intensity along the entire length of the fiber and good quality transverse beam profiles make HC-PCF an excellent candidate for examining gaslaser interactions. Moreover, HC-PCF significantly reduces the complications which arise when using ultrashort laser pulses with high peak power, i.e. the onset of SRS generation is well below the threshold for other detrimental nonlinear effects such as self-focusing, self-phase modulation, or spectrum broadening. Next, a case of intermodal Stimulated Raman scattering (IM-SRS) is investigated in a multi-mode gas-filled HC-PGB fiber. IM-SRS accounts for an energy dependent mode pattern at Stokes frequency. In particular we observe that by increasing the pump energy, the second-order Stokes light switches from the fundamental mode to a higherorder two-lobed mode. Conversion to the higher-order mode is explained by the presence of a two-lobed spatial pattern of coherence wave in the gas. This provides both phase-matching and an intermodal pump-Stokes overlap. Quantitative agreement between numerical simulations and experimental results supports this interpretation. The results suggest new opportunities for all-fiber-based nonlinear processes requiring phasematching, such as coherent anti-Stokes Raman scattering (CARS), as well as providing a means of efficiently converting light from a higher order pump mode to a fundamental Stokes mode. In the next chapter frequency comb generation in kagome HC-PCF is demonstrated. Optical frequency comb applications require accurate control of light across a broad spectrum of frequencies. We assure this by seeding the comb generation with a weak signal at Stokes frequency which provides phase-locking. Starting from a microchip laser and a Stokes seed pulse (generated separately from noise in a hydrogen-filled HCPBG fiber) a phase-locked, octave-spanning, purely rotational frequency comb is generated. Using a spectral interferometric technique based on second-harmonic generation, it is shown that the relative phases between the comb components are fixed and locked to a well-defined value. This is irrespective of the large fluctuations in the relative phase of the pump and the Stokes seed pulse. In the final chapter summarize the conclusions drawn from this work are summarized, together with possible directions for future work, such as intermodal CARS and frequency comb compression for ultrashort pulse generation. Zusammenfassung Die Erfindung von photonischen Kristallfasern mit hohlem Kern (engl. HC-PCF) ist ein Meilenstein für die Erforschung von Licht-Materie-Wechselwirkungen und im Besonderen von stimulierter Raman-Streuung (SRS) in Gasen. Indem Laserlicht und Ramanaktives Medium auf sehr engem Raum im mikrometergroßen Kern der Faser überlagert werden, können sowohl sehr hohe Effizienzen der Frequenzumwandlung erreicht als auch die Schwellwerte für stimulierte Ramanstreuung um Größenordnungen gesenkt werden. Ziel dieser Arbeit ist die Untersuchung von neuen Aspekten von stimulierter Raman-Streuung in gasgefüllten Hohlkern-PCF. Im theoretischen Teil dieser Dissertation werden die Maxwell-Bloch Gleichungen für rotierende SRS gelöst und die mathematische Grundlage dieses Phänomens sowohl in klassischer als auch in quantenmechanischer Betrachtung gelegt. Ich beschreibe die Entwicklung des Laserlichts (Pumpe und Stokes) in Raum und Zeit sowie die Auswirkung auf die Materie (Besetzungsinversion und Kohärenz). Der hier beschriebene Formalismus bildet die Basis für die spätere Behandlung komplexerer experimenteller und theoretischer Situationen, wie z. B. intermodale SRS. HC-PCF gibt es in zwei Varianten: Hohlkern-Bandlückenfasern (HC-PBG) und Fasern mit Kagome-Gitter. HC-PBG-Fasern sind charakterisiert durch ein schmales Transmissionsfenster mit niedrigen Verlusten, während Kagome-Fasern höhere Verluste aber eine breitbandigere Transmission aufweisen. Für beide Fasertypen werden der Lichtleitungsmechanismus, Verlust und Dispersionseigenschaften beschrieben. Der experimentelle Teil dieser Arbeit beginnt mit einer Beschreibung der Vorteile von HC-PCF für SRS-Experimente. Beugungsfreie Ausbreitung über lange Strecken, niedrige Confinement-Verluste, hohe Intensität über die gesamte Länge und saubere transversale Strahlprofile machen HC-PCF zu exzellenten Kandidaten, um Gas-Laser-Wechselwirkungen zu untersuchen. Außerdem erleichtern HC-PCF signifikant die Verwendung von ultrakurzen Laserpulsen mit hoher Spitzenleistung, da hier der Schwellwert für SRS-Erzeugung weit unter der Schwelle anderer nachteiliger nichtlinearer Effekte liegt, wie z.B. Selbstfokussierung, Selbstphasenmodulation und spektrale Verbreiterung. Als Nächstes wird ein Fall von intermodaler stimulierter Ramanstreuung (IM-SRS) in einer multimodigen gasgefüllten HC-PBG-Faser untersucht. IM-SRS erzeugt eine energieabhängige Modenform in der Stokes-Frequenz. Insbesondere beobachten wir, dass mit ansteigender Pumpenergie das Licht der zweiten Stokes-Ordnung von der fundamentalen in eine Mode höherer Ordnung mit LP11 -Feldverteilung wechselt. Diese Umwandlung in eine Mode höherer Ordnung wird erklärt durch das Vorhandensein einer ähnlichen räumlichen Feldverteilung der Kohärenzwelle im Gas. Dadurch wird sowohl die Phasenanpassung als auch die intermodale Überlappung von Pumpe und Stokes-Licht gewährleistet. Quantitative Übereinstimmung von numerischer Simulation und experimentelle Ergebnissen untermauert diese Interpretation. Die Resultate weisen auf neue Möglichkeiten für faserbasierende nichtlineare Prozesse mit Phasenanpassung hin, wie z.B. kohärente Anti-Stokes Ramanstreuung (CARS), und erlauben eine effiziente Umwandlung von Licht aus einer Pumpmode höherer Ordnung in eine fundamentale Stokes-Mode. Im nächsten Kapitel wird die Erzeugung eines Frequenzkamms in Kagome-Hohlkernfasern demonstriert. Zur Anwendung optischer Frequenzkämme bedarf es einer exakten Kontrolle des Lichts über ein breites Frequenzspektrum. Dies kann sichergestellt werden, indem wir die Kammerzeugung über einen schwachen Signalpuls mit StokesFrequenz zur Phasenregelung anregen. Ausgehend von einem Mikrochip-Laser und einem Stokes-Anregungspuls (welcher separat aus dem Rauschen in einer wasserstoffgefüllten HC-PBG-Faser erzeugt wird) wird ein phasengeregelter, oktavenbreiter, nur auf Rotation beruhender Frequenzkamm erzeugt. Mit einer spektralen Interferometriemethode basierend auf Frequenzverdopplung zeige ich, dass die relativen Phasen zwischen den Kammkomponenten starr auf einen definierten Wert geregelt sind, unab- hängig von den großen Schwankungen in der relativen Phase zwischen Pump- und Stokes-Anregungspuls. Im letzten Kapitel dieser Dissertation werden die Ergebnisse und Schlussfolgerungen der Arbeit zusammengefasst und Möglichkeiten für weiterführende Forschung aufgezeigt, z.B. intermodales CARS und Frequenzkammkompression zur Erzeugung von ultrakurzen Pulsen. Acknowledgements First and foremost I would like to thank my supervisor Prof. Philip St.J. Russell whose advice and guidance were invaluable throughout the course of my research. Philip, thank you for giving me the opportunity to work in such a great place and fantastic scientific environment, where new ideas pop out all the time. Andy and Amir, thank you for your time and patience to introduce me to Raman scattering, for answering all my questions, even the very silly ones. And then aswering them again. For countless hours we spent in the lab together. I couldn’t have done it without you. Andy, thank you for your calculations and your help with my code. Also, this thesis has benefited greatly from your critical eye and patient reading of early drafts. Thank you for all those delicious Swiss chocolates you brought. They sweetened many of my days. Amir, for all our religion/life philosophy discussions, for making me try lamb, for listening to all my stories about Poland and Polish people (who are the best at everything, never forget that!) and for all the wine we drank together. Thanks, Prince of Persia ;-) John, for putting up with me and my craziness. For our discussions about English language and priorities in scientific presentations. And for all your good advice that I hardly ever took. Thanks! Nick, thanks for those few friendly pushes when I was getting off the track, for the Jolytaxi and for not leaving me in the middle of the corn field, despite all your threats. Marcinek, thank you for being a good friend I could always count on, for helping me on numerous occasions, for your hospitality and for Gimlets. Markus, thank you for all our fashion and shoes discussions. Whenever I see socks and sandals worn together, I think of you (not that I ever suspect you of wearing them together). Josip, for all the design advice (especially for the non-casual Friday) and all the new things. And for the glasses. Thank you very much. Many thanks to you, Benito. For all the great fun we had, for all your good advice about my future prospects. And for all those meters we drank in Kanapee. Thanks! Olga, thank you for making me do things I didn’t want to do and make me enjoy them a lot. Michael, thank you for believing in me. Finally, I would like to thank my parents. Mom and Dad, you always wanted me to be a doctor, just the one that treats people. Nevertheless, you supported me in following my dream, not yours. Thank you for that. I would like to dedicate this thesis to you. Moim Rodzicom “Physics would be dull and life most unfulfilling if all physical phenomena around us were linear. Fortunately, we are living in a nonlinear world. While linearization beautifies physics, nonlinearity provides excitement in physics.” Y. R. Shen in The Principles of Nonlinear Optics xii Contents 1 Introduction 1 2 Theory of Raman scattering 7 2.1 The origin of Raman spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.1 Rigid rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.2 Nonrigid rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.3 Rotational Raman scattering . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Raman scattering - classical approach . . . . . . . . . . . . . . . . . . . . . 15 2.4 2.5 2.6 2.3.1 Raman gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.2 Raman cross-section . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Molecular excitations via Raman scattering . . . . . . . . . . . . . . . . . . 21 2.4.1 Quantum picture of Raman scattering . . . . . . . . . . . . . . . . . 21 2.4.2 Distribution of rotational levels . . . . . . . . . . . . . . . . . . . . . 22 2.4.3 Optical phonons and coherence wave . . . . . . . . . . . . . . . . . 23 2.4.4 Raman scattering vs. fluorescence . . . . . . . . . . . . . . . . . . . 25 Maxwell-Bloch equations for Raman scattering . . . . . . . . . . . . . . . . 26 2.5.1 Density matrix formalism . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5.2 Two-level system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Coherent anti-Stokes Raman scattering (CARS) . . . . . . . . . . . . . . . . 34 xiv 3 Contents Properties of hollow-core photonic crystal fibers 3.1 Photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Photonic crystal fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Fiber fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4 Guidance mechanisms in photonic crystal fibers . . . . . . . . . . . . . . . 42 3.5 4 5 37 3.4.1 Modified total internal reflection vs photonic bandgap guidance . . 42 3.4.2 Kagomé lattice PCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Characteristics of hollow-core photonic bandgap fibers . . . . . . . . . . . 46 3.5.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5.2 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5.3 Modal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.5.4 Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.6 Optical properties of kagomé lattice fibers . . . . . . . . . . . . . . . . . . . 51 3.7 PCF applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 SRS in gas-filled hollow-core photonic crystal fibers 55 4.1 Gas-laser interactions in HC-PBG fibers . . . . . . . . . . . . . . . . . . . . 55 4.2 Regimes of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Forward stimulated Raman scattering . . . . . . . . . . . . . . . . . . . . . 60 4.4 Numerical modeling of forward SRS . . . . . . . . . . . . . . . . . . . . . . 63 4.5 Backward stimulated Raman scattering . . . . . . . . . . . . . . . . . . . . 65 Intermodal stimulated Raman scattering 67 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2.1 Fiber characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.2.2 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.2.3 Regime of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.2.4 SRS measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.3 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.4 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Contents xv 6 83 Phase-locked frequency comb generation via SRS 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.2 Frequency comb initiated from quantum fluctuations . . . . . . . . . . . . 84 6.3 Seeded comb generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.4 7 6.3.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.3 Second harmonic generation . . . . . . . . . . . . . . . . . . . . . . 89 Analysis of comb coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.4.1 Sum-frequency components . . . . . . . . . . . . . . . . . . . . . . . 91 6.4.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Conclusions and outlook 95 7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2.1 Intermodal SRS switching . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2.2 Intermodal CARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.2.3 Two-color CARS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.2.4 Short pulse generation from frequency comb . . . . . . . . . . . . . 99 A MATLAB code for forward SRS 101 List of Abbreviations Term Definition CARS coherent anti-Stokes Raman scattering CEO CSRS carrier envelope offset coherent Stokes Raman scattering DOS density of states GVD group velocity dispersion HC-PBG hollow-core photonic bandgap HC-PCF hollow-core photonic crystal fiber IM-SRS intermodal stimulated Raman scattering IR infrared PBG photonic bandgap PCF photonic crystal fiber SC-PCF solid-core photonic crystal fiber SLM spatial light modulator SRS stimulated Raman scattering UV ZDW ultraviolet zero (group velocity) dispersion wavelength Chapter 1 Introduction When propagating through the medium, light can behave in various ways depending on the properties of the light itself and the medium. Light can be absorbed, transmitted, reflected, refracted, diffracted, scattered, etc. Scattering can be elastic (without energy exchange between light and matter) or inelastic (involving energy exchange due to collisions between photons and molecules). The former, called Rayleigh scattering gives rise to the blue appearance of the sky; it scales as the fourth power of the light frequency and is more effective at short wavelengths. The latter is called Raman scattering where energy is either gained or lost so that the scattered photons are shifted in frequency.1 C. V. Raman discovered the inelastic scattering phenomenon which bears his name in 1928 [1]. In the paper Raman called it a new type of secondary radiation and made an observation that the signal is “feeble”. Indeed, only 1 out of 106 photons of incident light get spontaneously scattered in Raman fashion. Maximization of that number had been a long-standing challenge of nonlinear optics and was finally successful with intense laser light, which could access the stimulated regime of Raman scattering. Here conversions to another frequency is much higher due to gain and the emission is in a narrow cone in forward and backward directions [2]. This is contrary to spontaneous Raman scattering where light is emitted in 4π space in a pattern characteristic for dipole radiation. 1 Here, we do not consider Brilloiun scattering. 2 1. Introduction Stimulated Raman scattering (SRS) was first observed in nitrobenzene in the ruby laser [3]. Shortly after that discovery it was realized that the SRS process is accompanied by optical phonons [4]. These are spatially and temporally coherent non-acoustic excitation of the material internal degrees of freedom such as molecular vibrations or rotations in the case of gases and liquids [5].2 Those excitations serve as a source term for nonlinear polarization in Maxwell equations that describe SRS. Moreover, the dephasing time of those oscillations, T2 has a crucial impact on the dynamics of the process, as will be shown in Section 4.2. High intensities which are required to initiate the SRS process can be obtained by tighly focusing the laser beam to a small spot in the medium. Therefore, initial SRS experiments in gases were done in free space or gas cells to maintain control over gas pressure. Raman active gas, such as hydrogen, was trapped in a long gas cell and laser light (few tens of mJ) was focused inside. The amplified Stokes energy did not exceed few % of the initial pump energy [6]. This low efficiency can be easily explained. When increasing the light intensity by focusing the beam, one has to pay the price of shorter interaction length, which can be approximated with the Rayleigh length, see Fig. 1.1. For tightly focused beams the Rayleigh length is usually not longer than few mm. Figure 1.1: Focused Gaussian beam in free-space. Light intensity is high enough for gas-laser interactions only in the focus of the beam, which is limited by the Rayleigh length. 2 Optical phonons will be described in more detail in Section 2.4.3. 3 Another option is to perform SRS experiments in a capillary to confine gas and provide guidance of light [7]. However, the attenuation constant for capillaries scales with λ2 /a3 , where a is the capillary inner radius [8], making losses of the capillary very high for small inner radius.3 One can say that light “leaks out” of the capillary, see Fig. 1.2. Figure 1.2: Gaussian beam focused in a capillary of diameter 2a. The attenuation constant for capillary scales with λ2 /a3 and is very high for small cores. In 1991, the idea emerged to trap light in a hollow-core by means of a two-dimentional photonic crystal of microscopic air capillaries running along the entire length of a glass fiber. Appropriately designed, this array would support a photonic bandgap for incidence from air, preventing the escape of light from a hollow-core into the photoniccrystal cladding and avoiding the need for total internal reflection. The first convincing hollow-core photonic crystal fiber (HC-PCF) structure emerged from the fiber drawing tower in November 1995 and provided access to many previously unreachable areas of gas-laser interactions [9, 10]. Figure 1.3: Gaussian beam focused in a hollow-core photonic crystal fiber. Rayleigh length is practically infinite and low losses assure high light transmission. 3 For a wavelength λ = 1 µm capillary of a radius 75 µm would lose 40% of light in the fundamental mode after 1 m propagation. To lower the losses, radius of the capillary can be increased, but then the light intensity would not be high enough for SRS. 4 1. Introduction First succesful HC-PCF [11] opened new posilibities for SRS. Due to their unique properties, HC-PCF turned out to be perfect candidates for performing high efficiency SRS experiments in gases [12]. They not only provide diffraction-free propagation of light (Rayleigh length is basically infinite) but also low power loss, high intensity along the entire length of the fiber, good quality transverse beam profiles and high overlap between transmitted light and gas, see Fig. 1.3. Moreover, using HC-PCF significantly lowers the threshold of SRS, which avoids the need of using high peak power lasers. The latter means that one does not reach the threshold for higher-order SRS, backward SRS, self-focusing and self-phase modulation that would make the physical picuture of SRS complex [5, 12]. Furtermore, HC-PCF are quite often multimode which opens up new posibilities for in-fiber phase matching. This idea will be developed in Chapter 5. Fig. 1.4 shows the simple picture of rotational SRS in gas-filled HC-PCF. Gas molecules are trapped in the core of the fiber where they are excited with the pump light. Molecules start to coherently rotate (or vibrate in case of vibrational SRS) and scatter light to lower frequency output signals. The length of the pump pulse is selected in a way to access the so called transient regime of SRS - that is the pump pulse duration is long enough to create a stable phase correlation between pump and signal field while being short enough to ensure that molecular dephasing cannot destroy the coherence during signal generation.4 Figure 1.4: Scheme of rotational SRS in gas-filled HC-PCF. Pump excites the molecules trapped in the fiber core which start coherent oscillations scattering the input light to lower frequency output signal. 4 Regimes of SRS are described in Section 4.2.
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