Nonlinear optics in KTiOPO4 for spectral management of ultra-short pulses in the near- and mid-IR Anne-Lise Viotti
Doctoral Thesis in Physics
Laser Physics Department of Applied Physics School of Engineering Science KTH
Stockholm, Sweden 2019
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Nonlinear optics in KTiOPO4 for spectral management of ultra-short pulses in the near- and mid-IR © Anne-Lise Viotti, 2019
Laser Physics Department of Applied Physics KTH – Royal Institute of Technology 106 91 Stockholm Sweden
ISBN: 978-91-7873-183-1 TRITA-SCI-FOU 2019:15
Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan framlägges till offentlig granskning för avläggande av teknologie doktorsexamen i fysik, fredagen 14 juni 2019 kl. 10:00 i sal FA31, Albanova, Roslagstullsbacken 21, KTH, Stockholm. Avhandlingen kommer att försvaras på engelska.
Cover pictures: KTiOPO4 samples; optical parametric generation and second harmonic generation phase-matched at specific angles in single-domain KTiOPO4.
Printed by Universitetsservice US AB, Stockholm 2019.
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Anne-Lise Viotti Nonlinear optics in KTiOPO4 for spectral management of ultra-short pulses in the near- and mid- IR Department of Applied Physics, KTH – Royal Institute of Technology 106 91 Stockholm, Sweden ISBN: 978-91-7873-183-1, TRITA-SCI-FOU 2019:15 Abstract This thesis explores the possibilities of controlling nonlinear optical interactions in ferroelectric materials for bandwidth tailoring of ultrashort pulses in the pico- and femto-second range. The control is achieved through quasi-phase matching, which is based on alternating the material’s spontaneous polarization into different domains. In the presented work, KTiOPO4 (KTP) is the material of choice as it provides high optical nonlinearity, a wide transparency window in the near- and mid-infrared (IR), as well as a high damage threshold. Furthermore, KTP also enables fabrication of uniform high aspect ratio and fine-pitch domain structures of high quality. These qualities make KTP highly attractive for a vast range of applications and enabled much of the work presented in this thesis. The propagation of ultrashort pulses in domain-structured ferroelectrics was studied numerically with a model based on a single nonlinear envelope equation. This model accounts for the absorption and the dispersion of the material, as well as the second- and the third-order nonlinearities. Supercontinuum generation in the near- and mid-IR was studied in periodically structured KTP for femtosecond pulses at 1.5 µm. The numerical results showed the potential for pulse self-compression with octave-spanning spectral broadening. This process is enabled by cascaded second-order nonlinearities and can be tailored by the phase-matching parameters, which are set by the structure’s periodicity. The proposed design presented in this work resulted in a negative effective Kerr nonlinear coefficient with a magnitude of 1.65 1014 cm2/W in the positive dispersion regime, which is one order of magnitude higher than the natural Kerr coefficient in KTP. Experimental characterization of single pass propagation of 128 fs-long pulses at 1.52 µm through a periodically structured KTP sample, with a periodicity of 36 µm based on the proposed design, are also presented. The results show a spectral broadening from 1.1 µm to 2.7 µm and a simultaneous compression down to 18.6 fs, thus confirming the numerical findings. Bandwidth tailoring of ultrashort IR pulses in the picosecond range was also studied through devices known as backward-wave optical parametric oscillators (BWOPOs). These devices rely on sub-micrometer domain periods to generate counter-propagating signal and idler waves. In a BWOPO, the forward-generated wave inherits the phase modulation of the pump wave, while the backward-generated wave is inherently narrowband and basically insensitive to pump wavelength tuning. In this work, BWOPOs operated in a cascaded manner, with the forward-generated wave being employed as a pump in a single pass configuration, were studied. The tunability issue of the narrowband backward wave was solved by employing a broadband optical parametric amplifier seeded by the BWOPO forward wave. A tunability of 2.7 THz for a wave at 1.87 µm with a
iii bandwidth of 28 GHz was demonstrated. The coherent phase transfer from the pump to the BWOPO forward wave was investigated in the context of pulse compression. In this experiment, a 220 GHz bandwidth was transferred from 800 nm to 150 ps-long pulses at 1.4 µm, which could be compressed down to 1.3 ps with µJ energy, in a single-grating compressor.
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Sammanfattning I den här avhandlingen undersöks möjligheterna att kontrollera ickelinjära optiska processer i ferroelektriska material. Detta för att kunna anpassa bandbredden hos ultrakorta pulser i piko- och femtosekundsområdena. Processerna kontrolleras genom kvasifasmatchning, vilket innebär att materialets spontana polarisation alterneras på ett sätt som ger upphov till olika domäner. Arbetet som presenteras här avser främst användandet av kristallen KTiOPO4 (KTP), då den har en hög optisk ickelinjäritet, en bred transmission i de nära- och mellan-infraröda områdena samt en hög skadetröskel. Dessutom möjliggör KTP tillverkning av uniforma samt finstrukturerade domäner med hög kontrast och kvalitet. Dessa egenskaper gör KTP till ett attraktivt material för en stor mängd tillämpningar och har möjliggjort mycket av arbetet i denna avhandling. Utbredning av ultrakorta pulser i domänstrukturerade ferroelektriska material studerades numeriskt med en modell som baserades på en ickelinjär amplitudekvation. Modellen tar hänsyn till materialets absorption och dispersion så väl som andra- och tredje ordningens ickelinjäriteter. Superkontinuumgenerering i det nära- och mellan-infraröda området undersöktes för periodiskt strukturerat KTP i fallet med femtosekundspulser vid 1.5 µm. Beräkningarna visade att pulserna kan genomgå självkomprimering i samband med en oktavsträckande spektral breddning. Dessa processer möjligörs av kaskaderade andra ordningens ickelinjäriteter och kan anpassas med kvasifasmatchningsparametrarna, vilka bestäms av strukturens periodicitet. Designen som presenteras i detta arbete gav upphov till en negativ effektiv ickelinjär Kerr-koefficient med en storlek på 1.65 1014 cm2/W i det positiva dispersionsområdet, vilket är en storleksordning högre än den naturliga Kerr-koefficienten i KTP. Karaktäriseringen av ett enkelpasseringsexperiment med en 128 fs-puls vid 1.52 µm propagerandes genom en periodiskt strukturerad KTP-kristall, med en periodicitet på 36 µm, presenteras också. Resultaten visar på en spektral breddning från 1.1 µm till 2.7 µm samt en komprimering ner till 18.6 fs, vilket således verifierar de numeriska resultaten. Möjligheten att anpassa bandbredden för ultrakorta infraröda pulser studerades även i pikosekundsområdet med s.k. baklängesvågs-optiska parametriska oscillatorer (BWOPO). Den här sortens system bygger på domänstrukturer med sub-mikrometerperioder för att generera motpropagerande signal- och komplementärsignalvågor. I dessa system får den framåtgenererade vågen samma fasmodulation som pumpvågen, medan den bakåtgenererade vågen är intrinsiskt smalbanding och i stort sett okänslig för ändringar i pumpvåglängden. I detta arbete studeras kaskaderade BWOPO-system, där den framåtgenererade vågen används som pump i en enkelpasseringskonfiguration. Problemet med att ändra våglängden på den smalbandiga bakåtpropagerande signalen löstes genom att passera den genom en bredbanding optisk parametrisk förstärkare, vilket möjliggjorde ett frekvensskift på 2.7 THz för en våglängd kring 1.87 µm med en bandbredd på 28 GHz. Den koherenta fasöverföringen från pumpen till den framåtpropagerande BWOPO-vågen undersöktes även genom pulskomprimering. I detta experiment så överfördes 220 GHz bandbredd från 800 nm till en 150 ps lång puls vid 1.4 µm, vilken kunde komprimeras till 1.3 ps i en enkel-gitterkompressor med µJ energi.
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Preface
The research presented in this thesis has been performed in the Laser Physics division, at the Applied Physics department of the Royal Institute of Technology (KTH), in Stockholm from 2015 to 2019.
This work was funded by the Swedish Research Council (VR) through its Linnaeus Center of Excellence ADOPT.
Some of the work presented is the result of international and national collaborations: The experimental investigation of supercontinuum generation in the mid-IR with periodically poled KTP was performed together with Light Conversion and the University of Vilnius, in Lithuania. The study of 1µm pumped supercontinuum in single-domain KTP was the result of collaboration with the Atomic Physics division of the faculty of engineering at Lund Technical University, in Sweden.
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Pour Mamie et Emma, qui me manquent chaque jour.
Welcome to your life There’s no turning back Even while we sleep We will find you acting on your best behaviour Turn your back on mother nature Everybody wants to rule the world
It’s my own desire It’s my own remorse Help me to decide Help me make the most of freedom and of pleasure Nothing ever lasts forever Everybody wants to rule the world
(Chris Hughes, Ian Stanley, Roland Orzabal, 1985)
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List of Publications This thesis is based on the following journal articles:
I. A. Zukauskas, A.-L. Viotti, C. Liljestrand, V. Pasiskevicius, and C. Canalias, "Cascaded counter-propagating nonlinear interactions in highly-efficient sub-µm periodically poled crystals," Sci. Rep. 7, 8037 (2017).
II. A.-L. Viotti, R. Lindberg, A. Zukauskas, R. Budriunas, D. Kucinskas, T. Stanislauskas, F. Laurell, and V. Pasiskevicius, “Supercontinuum generation and soliton self- compression in -structured KTiOPO4,” Optica 5(6), 711-717 (2018).
III. A.-L. Viotti, A. Zukauskas, C. Canalias, F. Laurell, and V. Pasiskevicius, “Narrowband, tunable infrared radiation by parametric amplification of a chirped backward-wave OPO signal,” Opt. Express 27, 10602-10610 (2019).
IV. A.-L. Viotti, F. Laurell, A. Zukauskas, C. Canalias, and V. Pasiskevicius, "Coherent phase transfer and pulse compression at 1.4 µm in a backward-wave OPO," (submitted).
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Description of Author Contribution My contributions to the original papers were the following:
Paper I I performed the measurements and wrote the manuscript with the assistance of the first author.
Paper II I wrote the numerical model, performed the theoretical analysis and took part in the experimental investigations. I wrote the manuscript with the assistance of the coauthors.
Paper III I performed the measurements and wrote the manuscript with the assistance of the coauthors.
Paper IV I performed the measurements and wrote the manuscript with the assistance of the coauthors.
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Publications not Included in this Thesis
A. A. Barbet, F. Balembois, A. Paul, J.-P. Blanchot, A.-L. Viotti, J. Sabater, F. Druon, and P. Georges, “Revisiting of LED pumped bulk laser: first demonstration of Nd:YVO4 LED pumped laser,” Optics Letters 39, 6731-6734 (2014).
B. A. Barbet, F. Balembois, A. Paul, J.-P. Blanchot, A.-L. Viotti, J. Sabater, F. Druon, 3+ th and P. Georges, “LED pumped QCW Nd :YVO4 laser,” 6 EPS-QEOD EUROPHOTON Conference, Neuchâtel, 2014.
C. A.-L. Viotti, V. Pasiskevicius, “Cascaded supercontinuum in mid-IR with periodically th structured KTiOAsO4 and KTiOPO4,” 7 EPS-QEOD EUROPHOTON Conference, Vienna, 2016.
D. A.-L. Viotti, V. Pasiskevicius, “Cascaded supercontinuum in mid-IR with structured KTP and KTA,” Optics and Photonics Sweden, Linköping, 2016.
E. H. Jang, A.-L. Viotti, G. Strömqvist, A. Zukauskas, C. Canalias, and V. Pasiskevicius, "Counter-propagating parametric interaction with phonon-polaritons in periodically poled KTiOPO4", Optics Express 25, 2677-2686 (2017).
F. A.-L. Viotti, R. Lindberg, F. Laurell, and V. Pasiskevicius, “Cascaded self- compression and mid-infrared supercontinuum generation in (2) -structured KTP and KTA,” CLEO San Jose, 2017.
G. A.-L. Viotti, A. Zukauskas, C. Canalias, and V. Pasiskevicius, “Generation of widely- tunable narrowband infrared radiation by PPRKTP mirrorless OPO and broadband chirped-pulse OPA,” CLEO/Europe-EQEC, Munich, 2017.
H. A. Zukauskas, C. Liljestrand, A.-L. Viotti, V. Pasiskevicius, and C. Canalias, “Highly- efficient cascaded mirrorless OPO in sub-µm periodically poled RKTP crystals,” CLEO/Europe-EQEC, Munich, 2017.
I. A.-L. Viotti, R. Lindberg, F. Laurell, and V. Pasiskevicius, “Mid-infrared supercontinuum generation in (2) -structured KTP and KTA using cascaded soliton compression,” ICO-24 Japan, Tokyo, 2017.
J. A.-L. Viotti, R. Lindberg, F. Laurell, and V. Pasiskevicius, “Cascaded soliton self- compression and mid-infrared supercontinuum generation in (2) -structured KTP and KTA,” Optics and Photonics Sweden, Kista, 2017.
K. R. Budriunas, D. Kucinskas, T. Stanislauskas, D. Gadonas, A.-L. Viotti, A. Zukauskas, F. Laurell, V. Pasiskevicius, and A. Varanavicius, “Broadband noncollinear optical
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parametric amplification in GaSe pumped at 1.5 µm,” OSA High-brightness sources and light-driven interactions congress, Strasbourg, 2018.
L. C. Canalias, A. Zukauskas, S. Tjörnhammar, A.-L. Viotti, V. Pasiskevicius, and F. Laurell, “Highly-efficient periodically poled KTP-isomorphs with large apertures and extreme domain aspect-ratios,” SPIE Photonics West, San Francisco, 2018.
M. A.-L. Viotti, R. Lindberg, A. Zukauskas, R. Budriunas, D. Kucinskas, T. Stanislauskas, C. Canalias, F. Laurell, and V. Pasiskevicius, “Octave-spanning mid-infrared supercontinuum generation and self-compression in (2) -structured Rb-doped KTP,” CLEO, San Jose, 2018.
N. C. Canalias, A. Zukauskas, A.-L. Viotti, R. Coetzee, C. Liljestrand, and V. Pasiskevicius, “Periodically poled KTP with sub-wavelength periodicity: nonlinear optical interactions with counter-propagating waves,” Advanced Photonic Congress, Zurich, 2018.
O. A.-L. Viotti, F. Laurell, and V. Pasiskevicius, “Supercontinuum generation and pulse self-compression of femtosecond pulses at 1030 nm in KTP crystals,” 8th EPS-QEOD EUROPHOTON Conference, Barcelona, 2018.
P. A.-L. Viotti, R. Lindberg, A. Zukauskas, R. Budriunas, D. Kucinskas, T. Stanislauskas, C. Canalias, F. Laurell, and V. Pasiskevicius, “Mid-IR supercontinuum generation and (2) cascaded soliton self-compression in -modulated KTiOPO4,” Northern Optics, Lund, 2018.
Q. V. Pasiskevicius, A. Zukauskas, A.-L. Viotti, R. S. Coetzee, and C. Canalias, “Advances in backward-wave optical parametric oscillators,” Advanced Laser Technologies conference, Tarragona, 2018.
R. A. Zukauskas, A.-L. Viotti, R. S. Coetzee, C. Liljestrand, V. Pasiskevicius, and C. Canalias, “recent advances in sub-µm PPKTP for non-linear interactions with counter- propagating photons,” SPIE Photonics West, San Francisco, 2019.
S. A.-L. Viotti, F. Laurell, A. Zukauskas, C. Canalias, and V. Pasiskevicius, “Coherent temporal phase transfer in backward-wave parametric oscillator at 1.4 µm,” CLEO, San Jose, 2019.
T. A.-L. Viotti, P. Mutter, A. Zukauskas, V. Pasiskevicius, and C. Canalias, “Degenerate mirrorless optical parametric oscillator,” CLEO/Europe-EQEC, Munich, 2019.
U. A.-L. Viotti, B. Hessmo, S. Mikaelsson, C. Guo, C. Arnold, A. L’huillier, B. Momgaudis, A. Melnikaitis, F. Laurell, and V. Pasiskevicius, “Soliton self- compression and spectral broadening of 1 µm femtosecond pulses in single-domain KTiOPO4,” CLEO/Europe-EQEC, Munich, 2019.
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List of Abbreviations
BBO: BaB2O4 (Beta Barium Borate) BWOPO: Backward Wave Optical Parametric Oscillator CARS: Coherent Anti-Stokes Raman Scattering CEP: Carrier Phase Envelope CEO: Carrier Phase Offset CPA: Chirped Pulse Amplification DCA: Duty Cycle Apodization DFG: Difference Frequency Generation FROG: Frequency Resolved Optical Gating fs: femtosecond (10-15 second) FWHM: Full Width at Half Maximum FWM: Four Wave Mixing GDD: Group Delay Dispersion GVD: Group Velocity Dispersion IR: Infrared KTA: KTiOAsO4 (Potassium Titanyl Arsenate) KTP: KTiOPO4 (Potassium Titanyl Phosphate) LIDAR: LIght Detection And Ranging LN: LiNbO3 (Lithium Niobate) LT: LiTaO3 (Lithium Tantalate) MOPO: Mirrorless Optical Parametric Oscillator NEE: Nonlinear Envelope Equation OPA: Optical Parametric Amplification OPCPA: Optical Parametric Chirped Pulse Amplification OPG: Optical Parametric Generation OPO: Optical Parametric Oscillator OR: Optical Rectification OSA: Optical Spectrum Analyzer PM: Phase-Matching PPA: Poling Period Apodization PPKTP/PPRKTP: Periodically Poled KTiOPO4/Rb-doped KTiOP4 ps: picosecond (10-12 second) QPM: Quasi-Phase Matching RKTP: Rubidium-doped KTiOPO4 SFG: Sum Frequency Generation SFG-XFROG: Sum Frequency Generation Cross-Correlation Frequency Resolved Optical Gating
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SHG: Second Harmonic Generation SPM: Self-Phase Modulation SROPO: Singly-Resonant Optical Parametric Oscillator SRS: Stimulated Raman Scattering SSFM: Split-Step Fourier Method SVEA: Slowly Varying Envelope Approximation THG: Third Harmonic Generation TOD: Third-Order Dispersion UV: Ultra-Violet XFROG: Cross-Correlation Frequency Resolved Optical Gating YAG: Yttrium Aluminium Garnet Yb:KGW: Ytterbium-doped Potassium Gadolinium Tungstate
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Acknowledgements I have received a lot of help and support during these last years for which I am deeply grateful. First of all, I would like to thank my two supervisors Prof. Valdas Pasiskevicius and Prof. Fredrik Laurell for accepting me as a PhD student in the Laser Physics division. I learnt a lot from them and they helped me develop precious skills for a researcher. They also gave me a lot of opportunities and responsibilities to become a bit more than a standard PhD student and I really appreciate their trust. Lastly, they also let me travel the world without complaining too much. I would also like to thank Dr. Björn Hessmo for his help during this past year. Not only did you save a worried student, both feet in the water in a flooded lab on a very (very!) humid summer day, but you also provided advice and fun experiment ideas. It is always good to have someone to put things in perspective and who constantly laugh at French weirdness (and politics). Another important person of my PhD life was Prof. Carlota Canalias, who provided support and advice for about anything, ranging from “How should I measure that?” to “Which movie should I watch during this year’s film festival?”. I made a rule of always disturbing you at the busiest times, and I am very grateful that your door was always open for me. Carlota also ran the female network of our school and created a mentorship program. Without her, I would never have met Lucie Delemotte, another member of the French mafia in Stockholm, who became my mentor and my official “complaints bureau”. Merci Lucie, pour ton aide et tous tes précieux conseils! I would like to thank Robert Lindberg, not only for sharing the office during almost 5 years, but for all the laughs and the good times we had in the lab or at conferences. I sincerely hope that our business plan to “solve problems that you don’t have” will work out, otherwise we can always go bar hoping in Vienna… I would like to also thank the biggest and most generous heart I have ever met, Hoda Kianirad. Thank you for being here in the hard times and in the good times too, you are gold! I would then like to thank all my colleagues, present or former, for making the work place even more enjoyable. Thank you, Andrius, for the time in the lab, for putting up with countless cross- correlation measurements and for telling me where not to walk alone in Vilnius. Thank you, Michael Fokine, for being the first one to discover my sweet tooth and bribe me with muffins, doughnuts and Thorlabs candies. Thank you also for reading and commenting this beast! Thank you, Cristine, for being so easy going and sharing so much fun at conferences. Thank you, Max, for reminding me that I have been here for quite some time and for making the best chocolate fudge brownies. Thank you, Kjell, for being an always curious student and asking questions often overlooked. Thank you, Patrick, for the fun in the lab. I guess I will see you at the gym or at our next common work place! Thank you, Walter, for speaking French to me and for always catching me while eating. Thank you Korbi for joining my club of Germans! Thanks also to all the office mates, past and present: Riaan, sweet Jingyi and Cherrie. Thank you, Staffan, for trying to convince me to move to Rimbo and raise horses. Thank you, Xiong, for always joining our adventures outside of work and for being the only one sharing my basketball craziness. Deep thanks to Jens, for his sweetness and his unlimited patience with my broken Swedish. Thank you also for being my number one fan on TV. Thank you, Lars-Gunnar, for keeping me updated on French news
xiv when I barely had time to read the newspaper headlines. I would also like to thank former colleagues: Sebastian (the most relax PhD student I have ever met), Gustav (always in for a beer), Micke and Peter! Special thanks also to Eleonora, for sharing the same passion for travelling. I am ready for a second round in sunny San Diego whenever you want! I would also like to thanks collaborators in Lund, who always welcomed me with open arms during measurement campaigns: Sara, Chen, Neven, Emma and of course Cord and Anne! However, I learnt that earning a PhD also requires a lot of support from outside the working world. So, I want to take time to also thank people who made this adventure quite unique for me. Being far away from home, I am glad to have found a second family! Thank you, Vanesa, for being the best neighbor and the best kind of friend one could wish for. From cook to private counsellor, from tanning buddy on hot summer days to provider of spicy chilies and personal accountant, you have done it all and I am so grateful for your friendship. Thanks also to Madeleine, who is not only the best/funniest admin, but also the best at French impressions and jokes planning! Thanks to all my Stockholm friends, starting with my German crew: Sonja for the brunch therapies, Jan, David and Jonas for the unforgettable parties, hangovers and following tough football sessions, Melanie for pushing me at the gym and the lengthy swimming sessions. Thank you Zofia for endless basketball fun all year round, for all the Rio souvenirs and for all the diverse activities you dragged me to around Stockholm (some of which I cannot erase from my memory). Thanks Christie for being one of the first friend I made in Stockholm and for helping me progress with my game. Plus, you have the loveliest family! I am sorry that I was never able to understand your French Canadian accent… Thanks Marianna and Oliver for joining in countless concerts! Thanks James for all the fun, and sorry for the fake birthday present! Thanks to all the basketball players I played with across Stockholm. But special thanks to Fryshuset division 3 women team for those past two years and this year’s title (tack Maja!), SSIF basketball, KTH-hallen basketball sessions (Tack Isaac!) with a dislocated finger and KTH University official team. It was always such great fun to play with all of you! I would like to thanks friends back home in France, or in other parts of the world, who made the PhD journey also enjoyable. My childhood friends Ingrid and Inès, always here for me and who came to visit me up in the cold north. Jessica who opened her home to me in Ottawa, on a very hot summer. Sarah, who has the winning combo of being German and playing basketball. Sanne, the Belgian architect always gone discovering new places. Last but not least, I would like to thanks my family for being here for me and supporting me. Merci en particulier à papy pour être toujours curieux et intéressé par mes projets. Je continuerai d’envoyer des cartes postales du monde entier. Merci maman et papa, pour absolument tout. Je ne suis pas quelqu’un au caractère facile, un vrai boulot à temps plein ! Merci pour les colis de France qui ont toujours fait un bien fou, les vacances au soleil et les heures passées au téléphone.
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Table of Contents
Table of Contents ...... 1 1. Introduction ...... 4 1.1 Background ...... 4 1.2 Cascaded Second-Order Nonlinearities ...... 5 1.3 Backward-Wave Optical Parametric Oscillators ...... 6 1.4 Main Directions of the Thesis Work ...... 7 1.5 Outline of the Thesis ...... 8 References to Chapter 1 ...... 9 2. Nonlinear Optics ...... 12 2.1 Nonlinear Polarization ...... 12 2.2 Second-Order Nonlinear Optical Processes ...... 13 2.2.1 Second Harmonic Generation ...... 14 2.2.2 Sum Frequency Generation...... 14 2.2.3 Difference Frequency Generation ...... 15 2.3 Third-Order Nonlinear Processes ...... 15 2.3.1 Optical Kerr Effect ...... 16 2.3.2 Self-Phase Modulation ...... 17 2.3.3 Cascaded Second-Order Nonlinearities ...... 19 2.3.4 Four-Wave Mixing...... 19 2.4 Coupled Wave Equations ...... 20 2.5 Phase-Matching ...... 21 2.5.1 Phase-Matching Considerations ...... 21 2.5.2 Birefringent Phase-Matching ...... 22 2.5.3 Quasi-Phase Matching ...... 24 2.6 Optical Parametric Oscillators and Backward-Wave Optical Parametric Oscillators ... 25 2.6.1 Optical Parametric Oscillators ...... 25 2.6.2 Concept of Backward-Wave Optical Parametric Oscillators ...... 26 2.6.3 BWOPO Wavelengths ...... 27 2.6.4 Spectral Features of BWOPOs...... 28
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References to Chapter 2 ...... 30
3. KTiOPO4 ...... 32 3.1 Ferroelectric Materials ...... 32
3.2 Properties of KTiOPO4 ...... 32 3.2.1 Crystallographic Properties ...... 32 3.2.2 Optical Properties ...... 33
3.2.3 Periodic Poling of KTiOPO4 ...... 35 References to Chapter 3 ...... 37 4. Characterization of Ultrashort Pulses ...... 38 4.1 Definitions for Ultrashort Optical Pulses ...... 38 4.2 Characterizing Ultrashort Pulses ...... 39 4.2.1 Phase and Intensity ...... 40 4.2.2 Pulse Intensity Autocorrelation ...... 41 4.2.3 Pulse Cross-Correlation ...... 42 4.2.4 SFG Cross-Correlation Frequency Resolved Optical Gating ...... 42 References to Chapter 4 ...... 45 5. Supercontinuum Generation and Pulse Self-Compression in the IR ...... 46 5.1 Supercontinuum Generation ...... 46 5.2 Numerical Model for Ultra-Short Pulse Propagation in QPM Media ...... 48 5.2.1 Outline of the Numerical Model ...... 49 5.2.2 Extended Numerical Model with Cubic Nonlinearities ...... 50 5.2.3 Raman Parameters for KTP ...... 51 5.3 Application to Mid-IR Supercontinuum Generation in Periodically Poled KTP...... 52 5.3.1 Strategy for Supercontinuum Generation and Pulse Self-Compression ...... 52 5.3.2 Simulation Results ...... 53 5.3.3 Experimental Setup...... 56 5.3.4 Experimental Results and Comparison with the Numerical Model ...... 57 5.3.5 Conclusions ...... 61 5.4 Aperiodic Structures ...... 62 5.4.1 Poling Period Apodization...... 63 5.4.2 Numerical Results for Poling Period Apodization ...... 64
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5.4.3 Duty Cycle Apodization ...... 65 5.5 Mid-IR Supercontinuum Generation with Other Ferroelectrics ...... 67 5.5.1 Materials ...... 67 5.5.2 Numerical Results...... 69 References to Chapter 5 ...... 71 6. Studies on Backward Wave Optical Parametric Oscillators ...... 75 6.1 Cascaded BWOPO ...... 75 6.1.1 Cascaded BWOPO Scheme ...... 75 6.1.2 Experimental Setup ...... 77 6.1.3 Conversion Efficiencies ...... 78 6.1.4 Spectral Characterization ...... 80 6.2 Backward Wave Tunability ...... 83 6.2.1 BWOPO Tuning Issues ...... 84 6.2.2 Experimental Setup...... 85 6.2.3 BWOPO Characterization ...... 85 6.2.4 OPA Characterization ...... 87 6.2.5 Chirp Measurements ...... 88 6.2.6 Tunable BWOPO Idler ...... 90 6.2.7 Conclusions ...... 91 6.3 Coherent Phase Transfer for Femtosecond Near-IR Pulse Compression ...... 92 6.3.1 Experimental Setup...... 93 6.3.2 Results ...... 94 6.3.3 Conclusions ...... 98 6.4 Homogeneity Check of 509 nm QPM Samples ...... 98 6.5 Conclusions ...... 100 References to Chapter 6 ...... 101 7. Conclusions and Outlook...... 104
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1. Introduction 1.1 Background Interest in ultra-short optical pulses, with durations in the picosecond and femtosecond range, has been continuously growing since the invention of the laser [1]. Ultra-short pulses allow high temporal resolution as they enable “pictures” to be taken of fast-moving event or objects. They can be used to measure, e.g., the relaxation process of carriers in semiconductors [2], and to study chemical reaction dynamics [3,4] at the femtosecond and attosecond time scales. In 1999, the Nobel Prize in chemistry was awarded to A. H. Zewail for his work where ultra-short laser pulses were used to investigate the dissociation dynamics of molecules. Another particularity of ultra- short pulses is their broad optical spectrum. This property provides high spatial resolution, e.g., in optical coherence tomography for non-invasive imaging of biological tissues [5]. Ultrafast laser systems generate trains of pulses that result in a comb-shaped spectrum where each individual longitudinal mode separation matches exactly the repetition rate of the laser. The optical comb acts as a multi-wavelength light source which can be used, e.g., in telecommunications [6]. Such broadband frequency combs are also employed for high precision optical frequency metrology as a “ruler” in the frequency domain, helping to determine unknown optical frequencies [7]. In 2005, part of the Nobel Prize in physics was awarded to both J. L. Hall and T. W. Hänsch for their contribution to the development of the optical frequency comb technique. Broadband frequency combs are nowadays used to stabilize the electric field under the pulse envelope [8]. Photo- ionization and high harmonic generation are nonlinear processes, which require such stabilization in order to, e.g., generate single attosecond pulses in the x-ray region [9]. Moreover, being able to generate ultra-short pulses also means that potentially very high peak intensities can be reached. High peak intensity, ultra-short pulses allow for non-thermal ablation and fabrication of microstructures of superior quality compared to using nanosecond pulses [10]. Intense, ultra-short optical pulses are now commonly available thanks to three technical breakthroughs. In the end of the 80s, (I) Ti:Sapphire started to be employed as a solid-state laser medium due to its broad gain bandwidth that could support femtosecond pulses [11]. Kerr lens mode-locking (II) is a fast, saturable absorber mechanism that enables generation of ultra-short pulses from lasers [12]. Direct amplification of ultra-short pulses is limited due to saturation and self-focusing effects in the optical amplifier, which is detrimental when high powers are required for high-field science applications [13]. However, if the pulses are temporally stretched before amplification to lower the circulating peak power (while maintaining the same total energy) inside the amplifier, then the amplification process is distributed over a longer period of time and the limitations are overcome. After the amplifier, the pulses are compressed back to their original time duration, with higher peak powers. This technique is called (III) chirped pulse amplification [14], or CPA, and in 2018 the Nobel Prize in physics was partly awarded to D. Strickland and G. Mourou for developing this method. The combination of Kerr lens mode-locking and CPA, applied to Ti:Sapphire-based oscillators, opened the route to many new exciting applications. In particular, ultra-short pulses with high peak
4 intensities are commonly used in the field of nonlinear optics to investigate new nonlinear interactions and new frequency conversion schemes [15]. Laser-based applications often require a specific wavelength range and particular spectral properties in order to perform optimally. Most wavelength regions are not directly accessible using available laser materials. However, nonlinear optics provides the way towards tailoring the optical pulses in terms of color, tunability or spectral management. In ultrafast nonlinear optics the pulsed source is essential, but choosing the appropriate nonlinear material is also vital depending on the targeted applications. Efficient frequency conversion is achieved when the relative phase between the fundamental wave and the generated wave is kept constant. This is automatically the case when both waves experience the same refractive indices, i.e. the waves are phase-matched. Such a configuration can be achieved in birefringent media when appropriate beam polarizations are selected. When this is not possible, efficient frequency conversion can also be achieved using quasi-phase matching (QPM), where the sign of the material nonlinear susceptibility is periodically reversed in order to reset the phase error. QPM is commonly performed with ferroelectric crystals artificially structured using the electric field poling method [16]. The crystal family of KTiOPO4 (KTP) is one of the most promising for nonlinear applications employing the QPM approach [17]. KTP has several advantages that make it a material of choice for many QPM applications. It has a wide transparency range, from 360 nm to about 4 µm [18], it has a high resistance to optical damage, and a large nonlinearity [19]. Moreover, the KTP crystal structure is suitable for periodic poling of fine-pitch gratings. Indeed, it was recently demonstrated that gratings with sub-µm periods could be fabricated in KTP, allowing to compensate large phase-mismatches between interacting waves, leading to the first experimental realization of counter-propagating parametric devices with first- order QPM in bulk crystals [20]. QPM materials are employed in numerous efficient and compact parametric devices, and integrated in systems, with time scales ranging from continuous wave down to femtosecond operation. They are employed for a wide range of practical applications, such as atmospheric gas sensing [21], time-resolved spectroscopy [22] or microscopy. 1.2 Cascaded Second-Order Nonlinearities One advantage of the QPM method is that it allows tailoring of the nonlinear interactions for specific applications by designing the phase-matching conditions in given dispersion regimes. Nonlinear optics is usually discussed in terms of second-order nonlinearities, for frequency conversion for instance [23], or in terms of third-order nonlinearities, for which the refractive index becomes intensity-dependent [24]. Nevertheless, cascaded second-order processes can effectively act as third-order nonlinearities and lead to the generation of soliton-like waves [25,26]. In practice, this phenomenon happens via nonlinear phase-shifting [27,28] and amplitude modulation of the fundamental wave during two successive second-order processes. Typically, this occurs via second harmonic generation (SHG), i.e. up-conversion, followed by down-conversion back to the fundamental wave. This is experimentally achieved by self-phase modulation arising from phase- mismatch, and here QPM can play an important role in the realization of cascaded nonlinear 5 devices. The cascaded nonlinear phase-shift, tuned away from the phase-matching conditions, can be used for mode-locking [29]. Moreover, the controllable effective third-order nonlinearity due to cascaded second-order interactions provides the possibility to tailor the spatial and temporal conditions for generating nonlinear waves, including spatial and temporal solitons [30]. Cascaded solitons are created when the fundamental and the SHG waves are propagating simultaneously in the second-order nonlinear material, while being trapped, balancing dispersion and nonlinearities [31]. Soliton pulse compression has been numerically studied by Bache et al. [32] for highly phase- mismatched SHG in bulk media (in particular in -barium borate crystal) in which the quadratic nonlinearities dominate over the cubic Kerr nonlinearity. Mid-IR bulk nonlinear crystals such as LiInS2 were also numerically investigated for pulse compression and supercontinuum generation [33]. Octave-spanning spectra have been studied and experimentally demonstrated by the Fejer group at Stanford University using QPM lithium niobate waveguides in the 1 µm wavelength range [34,35]. One aim of this thesis has been to explore more versatile approaches to the use of cascaded second- order nonlinearities for supercontinuum generation and pulse self-compression by employing QPM techniques in bulk ferroelectrics. 1.3 Backward-Wave Optical Parametric Oscillators Backward-wave optical parametric oscillators (BWOPOs) were first used in the field of electronics [36,37] where they allowed frequency tuning over a wide band even reaching the THz range [38], and for low-background radio astronomy observations and remote sensing [39]. In the field of optics, BWOPOs were first theoretically proposed in 1966 [40] and provide a self-established positive feedback mechanism, which results in a single-pass device with no required external cavity [41]. The counter-propagating interaction in a BWOPO gives a signal and idler wave, potentially spanning the entire transmission window of the material, and propagating in opposite directions. However, in order to practically realize a BWOPO, the inherently large momentum mismatch characteristic of counter-propagating interactions needs to be compensated for. The electric field poling technique employed to achieve first-order QPM was essential to the first experimental realization of these optical counter-propagating devices, which required sub-µm poling periods [20]. In this Letter, Canalias and Pasiskevicius demonstrated a conversion efficiency of 16% when the BWOPO was pumped by 47-ps pump pulses using a KTP sample poled with a periodicity of 800 nm. The threshold intensity to start the BWOPO process was 1.6 GW/cm2. Counter-propagating interactions have also been studied in waveguide geometries [42]. The first sub-µm periods were achieved in KTP waveguides, with periodicity of 0.7 µm for 6th and 7th order QPM for backward SHG [43]. Very recently, QPM structures in lithium niobate were also used to demonstrate backward narrowband THz parametric oscillation [44]. Besides generating a backward wave, BWOPOs show two main spectral features; a coherent phase transfer from the pump to the forward-generated wave, and an intrinsic narrowband backward- generated wave which is very insensitive to pump wavelength tuning [45-47].
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In this thesis, various properties of BWOPOs were studied. In particular, cascaded BWOPO processes have been observed. Moreover, a simple solution to drastically increase the backward- wave tuning range was developed. Finally, the unique property of frequency modulation transfer to the BWOPO forward wave was investigated to include the group velocity dispersion effects and was further used for pulse compression in the near-IR. 1.4 Main Directions of the Thesis Work The work presented here addresses research challenges of domain-engineered ferroelectrics for the spectral management of ultrashort pulses in the picosecond and femtosecond ranges. This was implemented along two directions. First, ultra-broadband femtosecond pulses are investigated in the near- and mid-IR through supercontinuum generation and pulse self-compression. Second, BWOPOs with different poling periods are used to generate either narrowband mid-IR pulses or to study coherent phase transfer and broadband near-IR ultra-short pulse generation in single-pass devices. I investigated a route to directly achieve self-compressed supercontinuum pulses with an octave- exceeding spectrum extending into the mid-IR by employing self-defocusing nonlinearities obtained through cascaded (2) interactions in domain-structured ferroelectrics. Domain- engineering of KTP provides the interesting property of being able to strongly phase-mismatch the fundamental wave in the nonlinear process. This implies that cascaded second-order nonlinearities can be tailored, allowing a change in the sign and magnitude of the effective Kerr nonlinearity of the second-order nonlinear crystal. The combination of the intrinsic material dispersion properties and tailored nonlinearities gives rise to the phenomenon of pulse self-compression, or soliton compression. A numerical model was first developed based on a single-wave nonlinear envelope equation that accounts for cascaded (2) nonlinearities, the native Kerr response from the third-order nonlinear polarization, and the delayed Raman response. Experimental validation of the model was carried out by using an in-house fabricated, periodically poled, Rb-doped KTP crystal with a period of 36 μm. A supercontinuum spectrum spanning from 1.1 µm to 2.7 μm was achieved, as well as self- compression down to 18.6 fs, from a 128-fs pump pulse at 1.52 μm. Using the actual pump pulse and sample parameters, excellent agreement was reached between the model and the experimental results, proving the validity of the model. Spectral management of picosecond pulses in the IR from BWOPOs was then studied and focus was put on the two main spectral features of BWOPOs. The forward-generated wave is a frequency-translated copy of the pump wave, i.e. there is a coherent phase transfer from the pump to the forward wave. On the other hand, the backward-generated wave is inherently narrowband and very insensitive to pump frequency tuning. Being able to generate tunable narrowband picosecond pulses in the near- and mid-IR is interesting for a number of practical applications. I showed for the first time a simple method to circumvent the tunability limitation of the backward- wave in a BWOPO. I demonstrated that a broadband optical parametric amplifier seeded by the BWOPO forward wave can help tuning the backward wave, having a wavelength of 1.87 µm, by
7
2.7 THz, while still maintaining the attractive BWOPO backward wave bandwidth and µJ-level output energies. Additionally, the property of coherent phase transfer in BWOPOs was extensively investigated in the context of near-IR pulse compression. The maximum transferrable bandwidth from the pump to the forward wave in a BWOPO was determined by the group dispersion mismatch. A 1.88 THz bandwidth transfer from 800 nm to 1.4 µm was experimentally demonstrated. Subsequently, a single-grating compressor was used to compress the BWOPO signal 115 times, leading to near-IR pulses with µJ energy as short as 1.3 ps. 1.5 Outline of the Thesis The thesis is written as a summary of the investigations described in the original publications labelled I to IV. The primary focus is put on the use of domain-structured KTiOPO4 crystals as nonlinear devices for the manipulation of short optical pulses with tailored spectral bandwidths in the infrared. Properties such as cascading second-order nonlinearities, supercontinuum generation, pulse self-compression and counter-propagating nonlinear interactions, enabled by versatile quasi- phase matching designs in KTP, are studied and explained. The thesis is organized in the following manner. Chapter 1 provides an introduction to the thesis. Chapter 2 gives an overview of the basic concepts of nonlinear optics necessary for the understanding of the thesis work. Chapter 3 introduces ferroelectric materials and specifically reviews the properties of KTP. Chapter 4 describes essential features of ultrashort optical pulses and how they are commonly characterized in the lab environment. Chapter 5 describes the work on supercontinuum generation and self-compression in domain-engineered and single-domain KTP, in connection with paper II. In particular, my numerical model employed for femtosecond pulse propagation in structured ferroelectrics is detailed. Chapter 6 presents the investigations on backward-wave optical parametric oscillators, together with the findings of papers I, III and IV. Tunability properties, as well as the very interesting coherent phase transfer feature of BWOPOs are extensively studied. Finally, chapter 7 is a concluding chapter summarizing the findings of the thesis work and outlining some remaining challenges and possibilities.
8
References to Chapter 1
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[16] M. Yamada, N. Nada, M. Saitoh, and K. Watanabe, “First-order quasi-phase matched LiNbO3 waveguide periodically poled by applying an external field for efficient blue second-harmonic generation,” Appl. Phys. Lett. 62, 435-436 (1993).
[17] F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1-xTiOPO4: A new nonlinear optical material,” J. Appl. Phys. 47, 4980-4985 (1976).
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[20] C. Canalias and V. Pasiskevicius, “Mirrorless optical parametric oscillator,” Nat. Phot. 1, 459-462 (2007).
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[21] D. Richter, D. G. Lancaster, and F. K. Tittel, “Development of an automated diode-laser-based multicomponent gas sensor,” Appl. Opt. 39, 4444-4450 (2000).
[22] D. Brida, S. Bonora, C. Manzoni, M. Marangoni, P. Villoresi, S. De Silvestri, and G. Cerullo, “Generation of 8.5-fs pulses at 1.3 µm for ultrabroadband pump-probe spectroscopy,” Opt. Express 17, 12510-12515 (2009).
[23] J. A. Armstrong, N. Bloembergen, J. Duduing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918-1939 (1962).
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[29] G. Cerullo, S. De Silvestri, A. Monguzzi, D. Segala, and V. Magni, “Self-starting mode locking of a cw Nd:YAG laser using cascaded second-order nonlinearities,” Opt. Lett. 20, 746-748 (1995).
[30] L. Torner, C. R. Menyuk, and G. I Stegeman, “Bright solitons with second-order nonlinearities,” J. Opt. Soc. Am. B 12, 889-897 (1995).
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[33] M. Bache, H. Guo, and B. Zhou, “Generating mid-IR octave-spanning supercontinua and few-cycle pulses with solitons in phase- mismatched quadratic nonlinear crystals,” Opt. Mater. Express 3, 1647-1657 (2013).
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2. Nonlinear Optics In common linear optics, an electromagnetic wave (or light wave) interacts with a molecule or atom which then vibrates and emits its own light that interferes with the incoming wave. If the light intensity is high enough so that many molecules or atoms are in a high-energy excited state, this excited state can become a lower level for a new excitation process. Thus, there are vibrations at frequencies associated to the various energy differences between all the populated states. In other words, in the presence of intense enough laser light, the optical properties of a dielectric material can become intensity-dependent. These phenomena relate to the field of nonlinear optics. In nonlinear optics, there is an energy exchange happening between the different optical inputs of the system. This phenomenon is qualified as nonlinear because the response of the material to the optical field strength occurs in a nonlinear fashion. Optical nonlinearity of a material is best described by its nonlinear polarization, a quantity which is essential in order to understand the nonlinear interactions. The Maxwell’s equations for the displacement field D , where D 0 EP, form the basis of all the following derivations.
2.1 Nonlinear Polarization The polarization of a dielectric material, also known as the dipole moment per unit volume, can be induced by an electromagnetic wave and depends on the strength of the applied electric field. It can be expressed as a power series in the electric field strength Et():
LNL (1) (2) 2 (3) 3 (1) (2) (3) Pt() P P 0 [ Et () E () t E ()...] t P () t P () t P ()..., t (2.1)
L (1) where P 0 Et() is the induced polarization which depends linearly on the electric field, is the vacuum permittivity and (1) is the linear susceptibility. At low light intensities, the higher order terms can be neglected. On the other hand, in the presence of intense laser radiation, the nonlinear terms P NL must be taken into account [1]. (2) and (3) are the second- and third-order nonlinear susceptibilities. We assumed here that the medium is dispersion-less and lossless given the instantaneous character of the medium susceptibilities. Here the fields are treated as scalars. When considering vectors, the susceptibilities become third- and fourth-rank tensors for (2) and (3) , respectively. Second-order nonlinear interactions, i.e. non-zero P (2) , occur in materials which do not have inversion symmetry, also called non-centrosymmetric media. However, third- order nonlinear interactions can occur in both centrosymmetric and non-centrosymmetric materials. The magnetic field contribution is omitted in Eq. (2.1), as it has a much weaker effect.
As we will see in Chapter 5, the usual way to describe nonlinear processes is to study the evolution of the electric field of the electromagnetic wave in the considered nonlinear optical medium. Indeed, the nonlinear response of the material and the polarization will drive the electric field and
12 add new components. That is why we commonly investigate the wave equation for the electric field in a nonlinear medium: 22 2NL 2 nE1 P E 22 2 2, (2.2) ct0 c t where n is the refractive index of the medium, c is the speed of light in vacuum and the field are vector-objects here.
In the following, we assume a lossless nonlinear optical medium and describe simply the main nonlinear processes that can be encountered in the lab. 2.2 Second-Order Nonlinear Optical Processes Let us consider a combined optical field which consists of two electric fields at two different optical frequencies 1 and 2 . We assume monochromatic plane waves and write the total field as:
it12 i t Et() Ee12 Ee cc .., (2.3) where cc.. denotes the complex conjugate. The second-order nonlinear polarization is expressed (2) (2) 2 as P ()tEt 0 (), which then gives: P(2)() t (2) [ Ee 222it12 Ee 2 i t 2 EEe i () 12 t 01 2 12 (2.4) *(2)**it()12 2..]2[].E12Ee cc 0 EE 11 EE 22
One can conveniently separate Eq. (2.4) into different groups of frequency components to highlight the physical processes:
Process Nonlinear polarization component (2) 2 Second Harmonic Generation (SHG) PE(210 ) 1 (2) 2 Second Harmonic Generation (SHG) PE(220 ) 2 (2) Sum Frequency Generation (SFG) PEE()212 0 12 (2) * Difference Frequency Generation (DFG) PEE()212 0 12 (2) * * Optical Rectification (OR) PEEEE(0) 201122 ( ) Table 2.1. Different second-order processes [2].
Nonlinearity thus leads to the generation of new frequency components. Realistically, only one of these processes will preferably occur and generate a measurable intensity for one (or two) of the above frequency components through the nonlinear optical interaction. Efficient output signal is produced by the nonlinear polarization for a given phase-matching condition so that there is a preferential second-order process for a given configuration. Phase-matching is further discussed in section 2.5. All the above-mentioned nonlinear processes are parametric. This means that the
13 quantum state of the material remains unchanged, the total photon-energy is conserved, and no net energy is transferred to the medium.
The frequency independent term, corresponding to the process of optical rectification, translates to the creation of a quasi-static electric field across the nonlinear material, also called the electro-optic effect that is used, e.g., to generate THz radiation [3,4].
2.2.1 Second Harmonic Generation
The case of second harmonic generation (SHG) occurs when 12 in Eq. (2.4). As illustrated in Fig. 2.1 (b), two photons at the fundamental frequency are annihilated when a photon at twice the frequency is generated. Thus, a typical application for SHG is the conversion of a fixed frequency laser output to a different spectral region, for instance from infrared (IR) to visible as for the Nd:YAG laser (1064 nm converted to 532nm). Such a laser source can then be used as a pump for Ti:Sapphire oscillators, among other applications [5].
(a) (b)
Material
Fig.2.1 Geometric configuration for the SHG process (a) and quantum-mechanical depiction of the SHG process (b). The dashed lines represent virtual energy levels and the solid line represents the atomic ground state.
2.2.2 Sum Frequency Generation Sum frequency generation (SFG) is very similar to SHG and is illustrated in Fig.2.2. This time, the input waves have different optical frequencies (1 and 2 ), while the new generated wave has a frequency which is the sum of the input waves: 312.
(a) (b)
Material
Fig.2.2 Geometric configuration for the SFG process (a) and quantum-mechanical depiction of the SFG process (b). The dashed lines represent virtual energy levels and the solid line represents the atomic ground state.
14
A typical application of SFG is ultraviolet (UV) generation. One can choose two laser sources in the visible to generate the UV light. Additionally, if one of the visible laser sources is tunable, the resulting SFG signal can also be tunable in the UV region [6,7].
2.2.3 Difference Frequency Generation Difference frequency generation (DFG) corresponds to the case where the frequency of the generated wave is the difference of the frequencies of the input waves: 312. Here, a photon with a higher frequency 1 is annihilated while a photon with a lower frequency2 is generated, in order to fulfil energy conservation, as shown in Fig.2.3.
(a) (b)
Material
Fig.2.3 Geometric configuration for the DFG process (a) and quantum-mechanical depiction of the DFG process (b). The dashed lines represent virtual energy levels and the solid line represents the atomic ground state.
As opposed to SFG, DFG can be used to generate radiation in the IR by mixing two visible light sources. The output can also be made tunable by having one of the visible sources tunable [8,9].
In the DFG process, the input wave with lower frequency (2 ) is amplified. Fig.2.3 (b) shows that the absorbed photon at 1 excites the atom to the highest virtual level. It then decays by emitting a photon at 2 . This process is stimulated by the presence of an input wave at frequency 2 . This is what we call optical parametric amplification (OPA). In the case where there is no preceding seeding wave at 2 , the weaker generated fields are created through spontaneous two-photon emission. This is what we call optical parametric generation (OPG), parametric down-conversion or parametric fluorescence [10,11]. 2.3 Third-Order Nonlinear Processes As mentioned in section 2.1, the third-order nonlinear polarization is expressed as (3) (3) 3 P ()tEt 0 (). If we now consider an input electric field with three different frequency components, the resulting expression for the nonlinear polarization gets more complicated. Let us assume the following applied electric field:
it12 i t it3 Et() Ee12 Ee Ee 3 cc .. (2.5)
15
Combining Eq. (2.1) and Eq. (2.5) results in 44 derived frequency components for the electric field, which are listed in ref [2]. The simplest term to describe is third harmonic generation (THG), where
123. Essentially, three identical photons of frequency are annihilated to create a photon at three times the fundamental frequency, as seen in Fig.2.4. This nonlinear interaction can be used to generate, e.g., light in the blue-UV wavelength region [12]. (a) (b)
Material 3
Fig.2.4 Geometric configuration for the THG process (a) and quantum-mechanical depiction of the THG process (b). The dashed lines represent virtual energy levels and the solid line represents the atomic ground state.
2.3.1 Optical Kerr Effect
For the case where 12 , the third-order nonlinear polarization can be written: (3) (3) (3) (3) Pt() P (213 ) P (2 13 ) P ( 3 ), (2.6) where the first two terms represent two components at shifted frequencies and the last term is a component at an unshifted frequency. In the case that all three frequencies are the same, Eq. (2.6) reduces to two terms: one component at the third harmonic (3 ) and one component at the fundamental frequency ( ). Considering the latter and a non-centrosymmetric medium, the total polarization can be expressed as: 2 PEEE() (1) () 3 (3) () (). (2.7) 0
An effective susceptibility eff can then be defined so that P eff E :
2 (1)3(). (3) E (2.8) eff 0
2 The resulting effective refractive index of the medium is thus: n 1 eff :
(3) (1) (3)22 (1) 3 nE1 3 () 1 1 (1) E () 1 (2.9) 3 (3) nI0 2 , 2nc00
2 (1) where I 1/2ncE00 is the intensity of the input wave and n0 1 is the normal refractive index. This shows that there is a nonlinear contribution to the refractive index, with an intensity-
16 dependent term: nn02 nI. The nonlinear refractive index n2 has units of inverse intensity (cm2/W) and it is an optical constant for the considered material. The index of refraction of the medium thus changes when the input radiation is very intense. This is known as the optical Kerr effect. Intensity-dependent refractive index can lead to effects such as self-focusing or self-lensing [13], which can be detrimental and trigger optical damage of the nonlinear material. In particular, if the material response is very fast, dramatic effects can be observed when using short light pulses [14], such as severe beam distortion or destruction of the nonlinear material itself..
2.3.2 Self-Phase Modulation Self-phase modulation (SPM) describes the change in phase of the optical pulse as a consequence of the nonlinearity of the refractive index in the medium. SPM is often used to broaden the spectrum of a pulse, i.e. to introduce new frequency components. To understand where this comes from, let us consider a light wave which propagates a distance z in the medium. It acquires a phase that can be described as: n tkzt z. (2.10) c
With sufficiently intense light wave (typically short optical pulses), the optical Kerr effect comes into play and the refractive index n is now intensity-dependent. Thus, the acquired phase becomes: tnnIz(). (2.11) 02c
The corresponding instantaneous frequency, which is the derivative of the phase in time is then: ddIt nz () ()tt2 (). (2.12) inst dt c dt 0
This means that when the intensity is time-dependent, i.e. a short pulse, then the center frequency changes with time by ()t , effect which is known as SPM. Let us assume a pulsed light source with a temporal intensity profile as seen in Fig.2.5 (a). The instantaneous frequency is initially smaller than the central frequency 0 , and then becomes larger than 0 .
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Fig.2.5 Temporal intensity profile (a) and corresponding temporal change (b) in instantaneous frequency.
We will see in Chapter 5 that if the time-dependent intensity changes rapidly as it is the case with femtosecond pulses, then the derivative of the intensity with respect to time is large, meaning that the overshot of the frequency can be even larger than the initial bandwidth of the input optical pulse. This translates to the fact that the spectrum of the light pulse is broadened and a supercontinuum can eventually be generated [15,16], as illustrated in Fig.2.6.
Fig.2.6 Illustration of the self-phase modulation effect for spectral broadening.
Additionally, let us assume that this broadening effect happens in a regime of anomalous dispersion where dn/0 d . In Fig.2.6, the upper part of the curve represents the new frequency components
18 generated by the falling edge of the pulse (at 0 ), while the lower part of the curve shows the new frequency components generated by the rising edge of the pulse (at 0 ). In the anomalous dispersion regime, the frequencies on the leading edge travel slower: they are caught up by the pulse. On the other hand, the frequencies on the trailing edge of the pulse travel faster: they catch up with the pulse. This means that the resulting pulse shape is stable at all times, this is called a soliton wave [17], and it is made possible by the combination of anomalous dispersion and the optical Kerr effect.
2.3.3 Cascaded Second-Order Nonlinearities In non-centrosymmetric materials where the second-order nonlinearity is non-zero, a third-order- like nonlinear phase shift can be observed. Second-order cascaded processes, due to SHG of the input wave and DFG of the second harmonic with the input wave, trigger this nonlinear phase shift [18]. This process occurs when a large degree of phase-mismatch is present. There is then an alternate exchange of power between the input wave and the newly generated waves (SHG, etc.). The generated waves propagate periodically in and out of phase with respect to the nonlinear polarization that drives the nonlinear process. This creates an intensity-dependent phase retardation on the input wave, i.e. a quasi-third order process (nonlinear refractive index) due to a cascaded second-order effect. The phase-shift is proportional to the intensity of the input wave and creates an additional contribution to the nonlinear refractive index n2 [19]: 2 SHG 41 L deff n2 . (2.13) ckL 22 00nn
In Eq. (2.13), 0 is the wavelength of the input fundamental wave, L is the length of the nonlinear 2 medium, deff is the effective nonlinear coefficient, kk k is the phase-mismatch (see section 2.5) and k is the wave-vector of the corresponding wave in the medium. This cascaded second-order nonlinear effect will be used, and additionally commented in Chapter 5, for supercontinuum generation and pulse self-compression.
2.3.4 Four-Wave Mixing Four-wave mixing (FWM) is also a third-order nonlinear process which can occur when at least two different frequency components travel together in a nonlinear material. If we assume two input frequency components 1 and 2 with 21 , then a refractive index modulation occurs due to
DFG, generating two additional frequency terms: 31()2 21 12 and
4 2()2 21 21 as shown in Fig.2.7.
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Fig.2.7 Illustration of four-wave mixing process.
FWM is related to SPM, notably in optical fibers [20], and can also be involved in spectral broadening for supercontinuum generation [16], as we will see in Chapter 5. 2.4 Coupled Wave Equations We already established that the propagation of an electromagnetic wave through a dielectric medium without free charges and currents can be described by the wave equation (Eq. (2.2)). This equation must be true for each frequency component involved in the considered nonlinear process. Let us assume a second-order nonlinear process, i.e. SFG as in subsection 2.2.2, and an electric field expressed as a plane wave propagating in the z-direction:
ikz()33 t E(,)zt Ae3 cc .. (2.14)
Here, kn333 / c is the wavenumber at frequency 3 . On the other hand, and in accordance with Fig.2.2, the applied input electric fields are written as: Ezt(,) Aeikz()11 t cc .. 11 (2.15) ikz()22 t Ezt22(,) Ae cc ..
The nonlinear driving term in Eq. (2.2) is expressed as (see Eq. (2.4)):
(2) ik()12 k z it33 it P(,) z t 4012 deff AAe e cc .. Pe 3 cc .. (2.16)
1 d (2) is the effective nonlinear coefficient. It depends on the polarization of the interacting eff 2 electric fields and phase-matching conditions. By using the Slowly-Varying-Envelope- Approximation (SVEA), we can neglect the second derivative terms: dA2 dA 11 k dz2 1 dz dA2 dA 22 k (2.17) dz2 2 dz dA2 dA 33 k dz2 3 dz
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In the wave equation (Eq. (2.2)), we substitute the electric field and nonlinear polarization terms (Eq. (2.15), (2.16)), within the SVEA approximation [1], to obtain the coupled wave equations for the three wave amplitudes: 2d 2 dA1 eff 1 * ikz iAAe2 32 dz k1 c 2 dA2 2deff 2 * ikz iAAe2 31 (2.18) dz k2 c 2 dA3 2deff 3 ikz iAAe2 12 , dz k3 c
where kkk123 k is the phase-mismatch between the three interacting waves. Those equations are called coupled-wave equations and they describe the evolution of the different waves interacting in the nonlinear medium. 2.5 Phase-Matching In nonlinear optics, photons must fulfil conservation laws. The photon energy must be conserved, as well as the photon momentum. In the previous example of the SFG process, these laws can be written as:
312 nnn (2.19) kkk k 11 2 2 33 0 123 ccc
Satisfying this last relation is called phase-matching (PM). Maximum efficiency for the considered nonlinear interaction is achieved when k 0 . This happens when the spatial wavelength of the polarization and the spatial wavelength of the field driven by this polarization are the same. If this is not the case, then energy is repeatedly flowing from the polarization to the field along the nonlinear crystal over a distance of one coherence length and then back to the polarization, etc. In practice, the coherence length of a crystal is so short that the newly generated field cannot be built up in a useful way. To obtain longer coherence lengths, phase-matching schemes are usually implemented, as detailed in the following sub-sections.
2.5.1 Phase-Matching Considerations In section 2.4, the coupled wave equation for the SFG wave was presented. If we assume a weak conversion of the input waves to the generated wave, one can take the amplitudes A1 and A2 as constant. Let us now consider two cases: perfect phase-matching and non-zero phase-mismatch.
For perfect phase-matching, k 0 and A3 grows linearly with z as shown in Eq. (2.18). This means that the intensity of the generated wave grows quadratically with the travelled distance z (see Fig.2.10 (b)). In this particular case, the generated wave and the nonlinear polarization
21 conserve a fixed phase relation and that is when the generated wave can extract energy from the applied wave in the most efficient way. On the other hand, when k 0 , the intensity of the generated wave is lower. If one integrates 2 Eq. (2.18) over a nonlinear medium of length L and uses the relation IncE1/2 00 , the intensity of the generated wave can then be expressed as: 22 8dIIeff 312 22kL IL3 2 sinc . (2.20) nnn1230 c 2
The intensity is represented in Fig.2.8. It decreases strongly when we go away from the perfect phase-matching position, i.e. when the wave gets out of phase with respect to the nonlinear polarization. Intensity then starts to flow back to the input waves after a given coherence length L for the medium. coh k
Fig.2.8 Intensity of the generated wave depending on the phase-mismatch.
2.5.2 Birefringent Phase-Matching Let us consider the second-order nonlinear process for SHG, where the energy and momentum conservation laws require: 2 (2.21) kkk(2 ) ( ) ( )
That is to say: 2(ncnc )*/ (2)*2/ . In other words, one must fulfil the following refractive index condition: nn(2 ) ( ) (2.22)
This collinear phase-matching condition is impossible to achieve in lossless isotropic materials whose refractive index shows normal dispersion, i.e. the refractive index n is increasing as a function of the frequency (see Fig.2.9 (a)).
22
Thus, the most common way to reach phase-matching is to use the birefringence property of some crystals. Birefringence means that there are different refractive indices for different polarizations, called the ordinary and extraordinary refractive indices. If one wants to phase-match SHG, for instance, one needs to have the wave with the lower frequency, i.e. the fundamental wave at , polarized along the direction which gives the highest of the two refractive indices, that is the extraordinary polarization. And vice versa for the second harmonic. For the case of a positive birefringent crystal and assuming a normal dispersion regime, the new phase-matching relation to fulfil becomes:
nn0 (2 ) e ( ), (2.23)
where n0 and ne refer to the ordinary and extraordinary refractive indices, respectively. The corresponding refractive index configuration is shown in Fig.2.9 (b).
Fig.2.9 In an isotropic material, phase-matching is not possible in the normal dispersion regime (a). Phase-matching in a birefringent material (b).
The extraordinary refractive index depends on the propagation angle of the corresponding polarized wave in the material [2]:
nn0 e ne () , (2.24) 2222 nnn00e cos ( )
where ne is the principal value of the refractive index for an extraordinary polarized wave. If the birefringent crystal is rotated, the red curve in Fig.2.9 (b) will move up and down with respect to the blue curve, making it possible to adjust the phase-matching condition. Although the birefringent phase-matching technique seems simple, it suffers from the walk-off effect [21]. The direction of the Poynting vector of the extraordinary wave diverges from the direction of that of the ordinary wave, as well as from the directions of the wave vectors (also known as pulse-front tilt). In most cases, this results in undesirable effects such as beam distortion and reduced conversion efficiency for the considered nonlinear process due to decreased overlap
23 of the waves during propagation. The noncritical phase-matching scheme can be employed to avoid spatial walk-off. Also called temperature phase-matching, this method relies on adjusting the temperature of the birefringent nonlinear crystal so that the phase velocities of the interacting waves are the same, thus reducing the phase-mismatch. However, some materials of interest may not display any birefringence, or insufficient birefringence in the considered wavelength range. Furthermore, some nonlinear optical applications may require the use of the highest nonlinear coefficient in a material, d33 , which can only be accessed by using waves with the same polarization, thus the birefringence method cannot be used in this case.
2.5.3 Quasi-Phase Matching Another solution to fulfil phase-matching requirements, when birefringence phase-matching fails, is to artificially structure the nonlinear medium. This technique is called quasi-phase matching (QPM) and was first proposed in 1962 [22]. Several approaches have been studied to obtain QPM structures. The nonlinear medium can be cut into thin slices which are then rotated by 180 degrees and stacked. However, this method has practical limitations as the slices have to be very thin and very many, which makes the realization quite difficult. One approach is to use special crystal growth techniques to obtain periodic, alternate orientations of the crystal axis [23]. Though, the most common method to achieve QPM today is to use periodic poling, usually performed by applying an external electric field over a periodic electrode on a ferroelectric crystal [24]. In the periodically poled crystal, the orientation of one of the crystal axes is reversed periodically, as shown in Fig.2.10 (a). Thus, the sign of the nonlinear coefficient, deff , is inverted accordingly, allowing to compensate for the phase-mismatch.
(a)
Fig.2.10 Unpoled and periodically poled crystal (a), with periodicity . Different phase- matching conditions for a nonlinear process (b). The blue line represents perfect phase- matching, the red line represents phase-mismatch and the orange line represents first- order QPM (m=1).
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Figure 2.10 (b) shows the comparison between the different phase-matching cases. In the case of perfect phase-matching, the amplitude of the generated wave grows linearly with the propagation distance z in the crystal (blue line). On the other hand, if there is phase-mismatch, the amplitude of the generated wave oscillates with z (red line), with a period of two coherence lengths. Finally, for a QPM configuration where the periodicity corresponds to twice the coherence length Lcoh , the amplitude is allowed to grow thanks to the artificial reversal of the sign of deff (orange line). A simple analytical description of the spatial dependence of the nonlinear coefficient d in a first- order QPM periodically poled crystal will prove useful in Chapter 5:
2 z dz() deff sign cos . (2.25)
Thus, the second-order nonlinearity can be spatially modulated along the nonlinear medium and custom-made designs can be implemented depending on the applications. An example is phase- matching for more than one wavelength with chirped grating periods [25]. 2.6 Optical Parametric Oscillators and Backward-Wave Optical Parametric Oscillators This section introduces the basic features of the optical parametric oscillators (OPOs) and the backward-wave optical parametric oscillators (BWOPOs). In the BWOPOs, the phase-matching is performed in a periodically poled crystal where the grating wave-vector will force one of the generated waves to propagate backwards.
2.6.1 Optical Parametric Oscillators Let us consider an OPG process (see subsection 2.2.3) where two new frequencies are generated from an input pump frequency p . The newly generated down-converted photons are referred to as signal and idler with frequencies s and i , respectively. The signal photon has higher energy and all the frequencies must fulfil the energy conservation: p si (2.26) si
Through second-order nonlinear interaction, the pump interacts with vacuum states of both signal and idler waves, in order to start the down-conversion process. From the coupled amplitude equations of A1 (signal) and A2 (idler) in Eq. (2.18), and for the phase-matching case
kkpsiQPM k kk 0 under the assumption of no initial idler ( A2 (0) 0 ), the solution for the evolution of the signal and idler waves becomes:
25
1 Az() A (0) egz 112 (2.27) nA11 3 * gz A21()zi A (0) e , nA22 3
22*2 22idA13eff idA 22 eff k where g is the gain coefficient, expressed as g . Thus, the 22 kc12 kc 2 signal and idler grow exponentially with the interaction length. This is exploited in an OPO to increase the conversion efficiency and reduce the bandwidths of the generated waves. In practice, the nonlinear medium is placed into a cavity which is resonant at least for one of the generated waves, as shown in Fig.2.11. The optical cavity thus provides feedback for this wave at the frequency defined by the cavity design.
Fig.2.11 Schematic of a conventional OPO with a periodically structured nonlinear medium and a cavity. The input coupler (on the left) is a highly reflective mirror and the output coupler (on the right) is a partially reflective mirror for the signal and/or the idler wave. The corresponding momentum conservation is also shown.
2.6.2 Concept of Backward-Wave Optical Parametric Oscillators In this thesis, the focus is put on a special type of OPO, called the backward-wave OPO (BWOPO) or the mirrorless OPO (MOPO). Such a device relies on a counter-propagating three-wave nonlinear interaction where no resonant cavity is needed in order to provide positive feedback mechanism to the process. The BWOPO concept is schematically presented in Fig.2.12. BWOPOs do not require alignment or additional optical component apart from the second-order nonlinear medium itself. This device allows to generate signal and idler waves in the near- to mid-IR. Backward-wave oscillators in the field of electronics have been studied since the 50’s [26].They are still used as tunable oscillators to cover the mm and sub-mm ranges, up to the THz regime [27,28]. The optical counterpart (BWOPO) was first proposed in 1966 [29], but only recently experimentally demonstrated [30].
26
Fig.2.12 Illustration of the BWOPO in a structured nonlinear medium with sub-µm periodicity, and the k-vector configuration shown below. Here f and b refer to forward and backward waves, respectively.
The parametric waves propagate in opposite directions, which automatically establish the distributed feedback mechanism, necessary to start the oscillation above a certain pump threshold. As seen in Fig.2.12, the phase-matching condition imposes an important constraint on the grating k-vector. The later must be large in order for counter-propagating interactions to happen, leading to very short grating periodicities, on the order of magnitude of the pump wavelength, i.e. sub-µm periodicities, if we assume first-order QPM. Very short pump pulses are not suitable for BWOPO pumping. Indeed, to establish the feedback mechanism, the backward generated wave must interact with the pump wave along the full length of the crystal. Thus, the pump pulse duration, p , must be longer than the time it takes for the pump pulse to travel the full length of the sample plus the time it takes for the backward wave to travel through the crystal in the opposite direction: 11 L . (2.28) p vvgp gb
In Eq. (2.29), L denotes the crystal length, vgp and vgb are the group velocities for the pump and the backward wave, respectively. For the wavelengths and samples used in this thesis (see Chapter 6), the minimum pump pulse duration is approximately 85 ps. Using shorter pulses is possible, but leads to increased oscillation thresholds. In the opposite case, using a continuous wave light source could be interesting, provided that the BWOPO threshold can be reached, i.e. if the continuous wave laser source has enough power.
2.6.3 BWOPO Wavelengths The BWOPOs studied in this thesis are based on QPM interaction. The wavelengths at stake must fulfil the QPM relation and the energy conservation: kkkk p fbQPM (2.29) pfb.
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With knjjj 2/ and jj 2/c , where jpfb ,,, we get the following equation:
1 nnpf nn bf . (2.30) pb
We note that the pump wavelength and the QPM grating periodicity determine which generated wave is the idler and which wave is the signal (most energetic wave). The different cases are summarized in Table 2.2.
pp/ n pp/ n pp/ n Degeneracy Configuration Backward idler Backward signal fbp/2 Table 2.2 Different BWOPO configurations depending on the QPM periodicity.
2.6.4 Spectral Features of BWOPOs The BWOPO has some unique spectral properties: The forward generated wave is a frequency translated copy of the pump wave, i.e. there is coherent phase transfer from the pump to the forward wave. This means that the spectral chirp of the forward generated wave is similar to the pump wave. The backward generated wave is thus inherently narrowband and it is very insensitive to pump frequency tuning, as shown in Fig.2.13.
Fig.2.13 Wavelength tuning of backward and forward waves (idler and signal, respectively) as a function of the pump wavelength in a KTP sample.
The phase of the backward wave in BWOPO is essentially constant, regardless of the QPM period. We can show this by comparing the derivatives of the parametric frequencies versus the pump frequency:
28
f vvgf gb v gp 1 p vvgp gb v gf (2.31) vvgb gp v gf b , p vvgp gb v gf
where vgi (ipfb ,,) are the waves group velocities, and is a dimensionless parameter which characterizes the difference in group velocity between pump and forward wave. In this thesis, we use a low dispersion nonlinear medium for the wavelengths studied, i.e. 1. The phases of the interacting waves in a down-conversion process are linked through: . (2.32) pfb 2
If we assume 0 then the phase derivatives for the generated waves become: tf tp (2.33) tb 0.
This means that the phase of the backward wave does not change, and ultimately the backward generated pulses are transform-limited. The forward wave is, however, a frequency-shifted replica of the pump wave [31,32].
Given the particular spectral features of BWOPOs and the potential of the counter-propagating scheme, such devices could be used for a number of applications, e.g., intrinsic waves with narrow bandwidths can be employed for the generation of ultrabright biphotons [33] for quantum applications or spectroscopy [34]. BWOPOs can also be used as perfect frequency translators [35].
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[35] H. J. McGuinness, M. G. Raymer, and C. J. McKinstrie, “Theory of quantum frequency translation of light in optical fiber: application to interference of two photons of different color,” Opt. Express 19, 17876-17907 (2011).
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3. KTiOPO4 The second-order nonlinear material used in this thesis is Potassium Titanyl Phosphate (KTiOPO4), also called KTP. KTP belongs to the family of ferroelectric materials and is extensively used for nonlinear optical conversion with birefringence phase-matching or QPM. This chapter gives an overview of the properties of ferroelectric materials and present KTP in more details. 3.1 Ferroelectric Materials Ferroelectrics are part of a wider class of materials known as ferroics which includes ferromagnets, ferroelectrics, ferroelastics and multiferroics. All these materials have one physical quantity which is stable in the absence of applied fields: magnetization for ferromagnets, polarization for ferroelectrics and strain for ferroelastics. However, the direction of such physical quantity can be reversed if an external field is applied along certain axes of the material. In multiferroics, then more than one of these physical quantities can be found in the same material. Ferroics are used as memory devices, phase shifters, sensors, etc [1]. In ferroelectric materials, the spontaneous polarization can be switched by the application of an external electric field [2]. Ferroelectrics are a subgroup of the class of pyroelectrics, thus they exhibit both pyroelectric and piezoelectric features.
3.2 Properties of KTiOPO4 KTP is a ferroelectric [3], positive biaxial crystal that was first synthetized in 1890 [4], and started being used in the late 70’s as a nonlinear medium for frequency conversion [5].
3.2.1 Crystallographic Properties KTP has an orthorhombic crystal structure [6] and is a non-centrosymmetric material. Each unit cell contains eight units of KTiOPO4 and the structure is characterized by chains of TiO6 octahedra bonded directly to each other or via bridges of PO4 tetrahedra. The alternation of long and short Ti-O bonds gives rise to the ferroelectric character of the crystal and the spontaneous electric polarization along the c-axis. In the crystal, regions where the spontaneous polarization has the same direction constitute the ferroelectric domains. The polarity of the domain determines the sign of the nonlinear coefficients. The TiO6-PO4 chains form a three-dimensional network (Fig.3.1), in which channels are created where the potassium ions (K+) can move, meaning that the crystal has a high ionic conductivity in the presence of potassium vacancies.
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Fig.3.1 Crystal structure of KTiOPO4. PS shows the orientation of the spontaneous electric polarization. The lengths of the Ti-O bonds can alternatively be long or short,
which gives rise to the ferroelectricity and a spontaneous electric polarization, PS , along
the c-axis. Regions where PS is in the same direction form ferroelectric domains of the
crystal. The TiO6-PO4 chains create a 3D network, in which channels are formed where the potassium ions are located. The potassium ions can move in these channels, leading to a rather high ionic conductivity. This motion of the potassium ions can be used for ion exchange [7].
In the rest of the thesis, we will consider the principal dielectric axes x , y and z parallel to the crystallographic axes a , b and c , respectively.
3.2.2 Optical Properties KTP has a wide transparency window [8], which ranges from about 365 nm to 4.3 µm, with transmission dips at 2.8 µm and 3.4 µm as can be seen in Fig.3.2. The strong absorption feature at 2.8 µm is due to OH- impurities which are incorporated in the crystal during the growth process.
33
Fig.3.2 Transmission window of KTP in the z-direction.
KTP shows high damage threshold and resistance to photorefractive effects [9]. Damage thresholds depend on the material quality, wavelength and pulse durations. For nanosecond pulses at 1 µm, the damage threshold for KTP is 10 J/cm2 [10]. For approximately 100 fs pulses at 1 µm, the damage threshold is 600-800 GW/cm2 [11]. The nonlinear coefficient matrix for KTP is given by:
0000d15 0 dd 000 00. (3.1) 24 ddd31 32 33 000
The corresponding values of the non-zero coefficients are given in Table 3.1.
Material d15 (pm/V) d24 (pm/V) d31 (pm/V) d32 (pm/V) d33 (pm/V) KTP (1064 nm) 1.91 3.2 2.8 4.35 16.9 Table 3.1 Nonlinear coefficients in KTP [12].
For nonlinear applications, it is advantageous to use z-polarized waves as they allow access to the largest nonlinear coefficient, d33 , in KTP and Rb-doped KTP. The refractive index corresponding to this polarization can be calculated from the Sellmeier equations [13]:
22BD nAz 2 F , (3.2) 1 C 2 1 E where the coefficients A, B, C, D, E and F are specific to the material and can be found in [13,14], and is the wavelength in vacuum, with units in microns. The refractive index is also temperature-
34 dependent and thus, the Sellmeier equation needs to be adjusted accordingly. For instance, for wavelengths above 1 µm, the temperature correction is [15]: 2 nnT12(25)(25), C nT C (3.3)
3 am where n1,2 m and the coefficients am are material specific. Establishing the correct Sellmeier m0 equations is very important in order to know the exact dispersion of the nonlinear material. In Chapter 5, a numerical model investigating ultra-short pulse propagation in ferroelectrics, and more particularly in KTP, will be presented. The results from such calculations, depend strongly on the dispersion data, therefore accurate refractive indices measurements are of paramount importance here.
3.2.3 Periodic Poling of KTiOPO4 In KTP, the spontaneous polarization direction can be inverted by applying an external electric field with opposite polarity to the polarization direction, and with a magnitude larger than the + + coercive field, Ec 2/kV mm [16]. Rubidium ions (Rb ) can replace K in the crystal structure and with low Rb+ concentrations (<0.3%) the linear and nonlinear properties of KTP are unchanged, while the ionic conductivity is substantially lower (by two orders of magnitude) due to the larger ionic radius of Rb+, which is blocking the K+ ions from moving in the crystal. This is beneficial for lowering the domain broadening during electric field poling [17] and improving the growth of domains in the z-direction. Such ion-exchanged samples are also less susceptible to grey-tracking [18]. The Rb doping in KTP leads to a substantial coercive field increase in the ion-exchanged regions (3.8 kV/mm). This can be exploited for a selectively induced domain-switching in the low- coercive field regions [7]. Rb-doped KTP is a promising material for the fabrication of sub-µm periodically poled samples. Photolithographic patterning technique for periodic poling technique becomes challenging due to the fine-pitch domain structures that need to be realized in the sub-µm range. Thus, a method based on coercive-field engineering was developed instead [7]. Details regarding the fabrication process can be found in the method section of paper I. Example of such a domain structure can be seen in Fig.3.3.
35
Fig.3.3 Atomic force microscope images of a chemically etched (a) patterned face and (b) un-patterned face of a PPRKTP crystal with sub-µm domain structure.
In this thesis, Rb-doped KTP samples with the following periods were used: 509 nm, 755nm, 25.5 µm, 26.3 µm, 36 µm, 38.52 µm. They were fabricated by the method of room temperature electric field poling.
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References to Chapter 3
[1] J. F. Scott, “Prospects for ferroelectrics: 2012-2022,” Mater. Sci., 187313 (2013).
[2] ANSI/IEEE std. 180-1986, IEEE Standard Definitions of Primary Ferroelectric Terms, The Institute of Electrical and Electronics Engineers, Inc., (New York, 1986).
[3] V. K. Yanovskii and V. I. Voronkova, “Ferroelectric phase transitions and properties of crystals of the KTiOPO4 family,” Phys. Status Solidi 93, 665-668 (1986).
[4] I. J. Nordborg, “Non-linear optical titanyl arsenates,” Doctoral thesis, Department of Inorganic Chemistry, Chalmers University of Technology (Göteborg, Sweden, 2000).
[5] F. C. Zumsteg, J. D. Bierlein, and T. E. Gier, “KxRb1-xTiOPO4: a new nonlinear optical material,” J. Appl. Phys. 47, 4980-4985 (1976).
[6] P. I. Tordjman, R. Masse, and J. C. Guitel, “Structure cristalline du monosphate KTiPO5,” Zeitschrift für Kristallographie 139, 103-115 (1974).
[7] C. Liljestrand, F. Laurell, and C. Canalias, “Periodic poling of Rb-doped KTiOPO4 by coercive field engineering,” Opt. Express 24, 4664- 4670 (2016).
[8] G. Hansson, H. Karlsson, S. Wang, and F. Laurell, “Transmission measurements in KTP and isomorphic compounds,” Appl. Opt. 39, 5058- 5069 (2000).
[9] X. B. Hu, J. Y. Wang, H. J. Zhang, H. D. Jiang, H. Liu, X. D. Mu, and Y. J. Ding, “Dependence of photochromic damage on polarization in KTiOPO4 crystals,” J. Cryst. Growth 247, 137-140 (2003).
[10] A. Hildenbrand, F. R. Wagner, H. Akhouayri, J.-Y. Natoli, M. Commandre, F. Theodore, and H. Albrecht, “Laser-induced damage investigation at 1064 nm in KTiOPO4 crystals and its analogy with RbTiOPO4,” Appl. Opt. 48, 4263-4269 (2009).
[11] F. Bach, M. Mero, V. Pasiskevicius, A. Zukauskas, and V. Petrov, “High repetition rate, femtosecond and picosecond laser induced damage thresholds of Rb:KTiOPO4 at 1.03 µm,” Opt. Mater. Express 7, 744-750 (2017).
[12] H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear- optical coefficients of KTiOPO4,” Opt. Lett. 17, 982-984 (1992).
[13] T. Y. Fan, C. E. Huang, B. Q. Hu, R. C. Eckardt, Y. X. Fan, R. L. Byer, and R. S. Feigelson, “Second harmonic generation and accurate index of refraction measurements in flux-grown KTiOPO4,” Appl. Opt. 26, 2390-2394 (1987).
[14] K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74, 914-916 (1999).
[15] S. Emanueli and A. Arie, “Temperature-dependent dispersion equations for KTiOPO4 and KTiOAsO4,” Appl. Opt. 42, 6661-6665 (2003).
[16] H. Karlsson and F. Laurell, “Electric field poling of flux grown KTiOPO4,” Appl. Phys. Lett. 71, 3474-3476 (1997).
[17] Q. Jiang, P. A. Thomas, K. B. Hutton, and R. C. C. Ward, “Rb-doped potassium titanyl phosphate for periodic ferroelectric domain inversion,” J. Appl. Phys. 92, 2717-2723 (2002).
[18] A. Zukauskas, V. Pasiskevicius, and C. Canalias, “Second-harmonic generation in periodically poled bulk Rb-doped KTiOPO4 below 400 nm at high peak-intensities,” Opt. Express 21, 1395-1403 (2013).
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4. Characterization of Ultrashort Pulses The importance of ultrashort pulses was mentioned in the introduction and, to use them properly for experimental investigations, it is necessary to be able to carefully characterize them. Indeed, simply knowing the pulse duration is most of the time not sufficient, as fundamentally different short pulses can present the exact same temporal trace. Full characterization of the pulse temporal shape, and phase is thus required. This chapter aims at reviewing the most essential properties of short pulses, and it describes some characterization setups that were used during this thesis work. 4.1 Definitions for Ultrashort Optical Pulses Ultrashort pulses typically have pulse durations below a few picoseconds and are generated with mode-locked lasers [1]. Such pulses have a broad spectrum and quite high intensity. Spectral width and pulse duration are related by Fourier Transform: 1 E()te ( ) it 2 (4.1) () E ()teit dt .
The time-bandwidth product is the product of the pulse duration by the spectral bandwidth of the pulse: 1 (4.2) 2
In order to generate a given short pulse, a certain spectral bandwidth is thus necessary. When Eq. (4.2) is an equality, the pulse is a transform-limited pulse, i.e. the phase variation of such a pulse is linear in time. Another important definition is the pulse duration. The most common, and the one used in this thesis is the full width at half maximum (FWHM), as described in Fig.4.1. Equation (4.2) can then be rewritten tK , where t and are the duration and frequency spans at FWHM, and K is a constant which depends on the shape of the pulse. For a Gaussian pulse K 0.441 and for a hyperbolic secant pulse K 0.315 .
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Fig.4.1 Definition of the pulse FWHM.
As ultrashort pulses have broad spectral bandwidth, they are often subjected to dispersion phenomena. As a consequence, they will get longer by passing through a medium such as glass, where the refractive index, and thus the velocity of light in the material, depends on the wavelength. This is called dispersion. For the materials considered in this thesis work, the refractive index n() , calculated from the Sellmeier equations, increases monotonically as the frequency increases (normal dispersion, see Fig.2.9 (a)). We define the group velocity as the slope of this curve: dc v , (4.3) g n dk n and the group velocity dispersion (GVD) as the second derivative: dk2 GVD . (4.4) d 2
The group velocity describes the velocity of the wave packet (distribution of wavelengths on each side of a central wavelength with a certain spectral width), while the GVD describes the varying rate of change of the phases of the different wavelength components. In practice, while crossing a normal dispersion medium, the redder parts of the pulse (longer wavelengths) will have a higher velocity than the bluer parts (shorter wavelengths), which broadens temporally the optical pulse. 4.2 Characterizing Ultrashort Pulses Before the ultrashort pulses can be used for any experiments, they need to be fully characterized. This will determine the temporal resolution of the experiment. Being able to actually “measure” an
39 ultrashort pulse helps to determine if it can be made even shorter (as seen in Chapter 6), allows to verify numerical models for ultrashort pulse propagation (as in Chapter 5), and to better understand the nonlinear media we study. Indeed, if we are able to measure the intensity and the spectral phase of a given pulse, we can sufficiently reconstruct the pulse. For instance, characterizing the pulse before and after a given medium, allows one to gain information on the linear and nonlinear effects of the medium. For a linear medium, one will gain knowledge about the absorption coefficient and refractive index while in a nonlinear material, such pulse characterization will help determine the SPM. Measuring such short pulses is not straightforward because in order to measure a given event in time (the ultrashort pulse), we will need an even shorter event to resolve it. Additionally, the existing photo-detectors are commonly used to resolve longer optical pulses and are not fast enough for ultrashort pulses. One approach is to use the ultrashort pulse to measure itself, through different techniques. In this section, the techniques used in the thesis work are described.
4.2.1 Phase and Intensity Let us assume a time-domain representation of the optical pulse electric field: Et() Re Ite ()(())itit0 , (4.5) where It() is the intensity and ()t is the phase. In the frequency domain, this becomes:
i() 0 () ReSe ( 0 ) , (4.6)
where S() 0 is the spectrum of the pulse and () 0 is the spectral phase. The spectral phase determines the time-dependence of the pulse frequency. We already mentioned the instantaneous phase in Chapter 2: ()tdtdt0 ()/ . When we have an ultrashort pulse, we basically need to know how its frequency changes over time, i.e. what the chirp is. It is important to be able to measure linearly chirped pulses (see Fig.4.2), but also pulses with more complex phases.
40
Fig.4.2 Time-dependent phase (a) and corresponding instantaneous frequency (b) of the electric field (c) of a linearly chirped pulse.
4.2.2 Pulse Intensity Autocorrelation As mentioned at the beginning of this chapter, in order to measure ultrashort pulses, the pulse itself can be employed for the measurement, which results in an autocorrelation [2,3]. The pulse to be measured is split into two identical copies (by a beam splitter for instance). One of the copies is delayed with respect to the other. Both pulses are then recombined to spatially overlap into a second-order nonlinear medium (traditionally BBO but also KTP in this thesis) to generate a SHG signal. A typical setup for an intensity autocorrelator is presented in Fig.4.3. The electric field envelope of the outcoming SHG beam can be expressed as: Et(, ) EtEt () ( ), (4.7) where the second term on the right-hand side corresponds to the delayed pulse, and is the inserted delay. The total field intensity then becomes: It(, ) ItIt () ( ). (4.8)
As regular detectors are too slow to resolve this intensity, the autocorrelation is measured instead, as follows: AItItdt() ()( ) (4.9)
The autocorrelation depends only on the delay and its trace is in practice obtained by plotting the SHG energy measured against the delay. Unknown Pulse
H L A BS
P T M PPKTP
Fig.4.3 Illustration of the intensity autocorrelation experimental setup used in paper IV. BS: beam splitter; H: hollow-roof mirror; T: translation stage; M: mirror; L: lens; A: aperture; P: photodiode.
In paper IV, an intensity autocorrelator with a large aperture 5 mm-thick PPKTP crystal [4] was built to characterize the output pulses of a single-grating compressor, at a central wavelength of 1.4 µm.
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However, different intensity distributions can give the same autocorrelation results, so this technique does not help gaining information on the pulse shape. This is intensity ambiguity, i.e. very different intensities yield the same autocorrelations [5]. Furthermore, it does not give any spectral resolution. Depending on the assumed pulse shape, the measured autocorrelation FWHM
A can relate to the pulse FWHM p according to Table 4.1.
Hyperbolic Pulse shape Square Gaussian Lorentzian secant
A / p 1 1.41 1.54 2 Table 4.1 Relationship between pulse FWHM and autocorrelation FWHM for different pulse shapes.
4.2.3 Pulse Cross-Correlation Another technique, used in this thesis work (papers III and IV), that can be applied when a shorter reference pulse is available is the intensity cross-correlation [2]. In the case where the reference pulse is much shorter than the unknown pulse to be measured, the intensity cross-correlation method allows to fully determine the pulse intensity. The corresponding experimental setup is very similar to the intensity autocorrelation setup and is shown in Fig.4.4. This time, the second-order nonlinear process used is SFG. L A Unknown Pulse P Reference M Pulse PPKTP
H T
Fig.4.4 Illustration of the intensity cross-correlation experimental setup used in papers III and IV. H: hollow-roof mirror; T: translation stage; M: mirrors; L: lens; A: aperture; P: photodiode.
In addition, this experimental setup was used to measure chirp rates of unknown pulses by using a spectrometer, instead of the photodiode, in order to record the SFG spectrum for different delays. This gives a measure of the pulse “color” versus the time, but no rigorous quantitative information on the pulse.
4.2.4 SFG Cross-Correlation Frequency Resolved Optical Gating Intensity autocorrelation and cross-correlation methods give information on the pulse length and shape, while interferometric autocorrelation gives some information on temporal phase distortion.
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However, in all cases, these are insufficient to be able to recover the complex electric field envelope of the pulse. One convenient solution is to use Frequency Resolved Optical Gating (FROG), which operates in a time-frequency domain in order to achieve both temporal and spectral resolution [2]. A basic FROG setup is very similar to an intensity autocorrelation setup. The main difference is that the SHG (or SFG signal in the case of cross-correlation FROG, also called SFG-XFROG) is the detection part. Instead of measuring energy as a function of delay (with a power meter or a photodiode), FROG measures the spectrum versus the delay to obtain a FROG trace measured in the plane. The intensity autocorrelation previously described can be retrieved by projecting the FROG trace onto the frequency axis. The FROG technique must rely on an iterative algorithm to be able to reconstruct the phase and amplitude of the unknown pulse [6,7]. In paper II, a custom-built SFG-XFROG setup, as shown in Fig.4.5, was used in order to characterize complex supercontinuum pulses having a large bandwidth. A 10µm-thick nonlinear BBO crystal was used to resolve the entire frequency content of the short pulses.
Unknown PM Pulse Reference M Pulse
H T BBO
L S A
Fig.4.5 Illustration of the SFG-XFROG experimental setup used in paper II. H: hollow- roof mirror; T: translation stage; M: mirrors; L: lens; A: aperture; PM: parabolic mirror; S: spectrometer.
The SFG signal is imaged onto the spectrometer which disperses the light to produce a 2D image whose axes are the time delay ( ) and the frequency ( ). The obtained spectrogram of the pulse of envelope Et() is:
2 SEt(,) ()(g tdt )eit , (4.10) XFROG where gt() is a gate function, which varies with the delay between the pulse replicas. When the unknown pulse is very complex, gating with a smooth reference pulse is bound to generate a trace that can be easily interpreted, as opposed to SHG-FROG which has a time direction ambiguity
43 which requires an independent measurement to resolve it. Other advantages are that the unknown pulse can be quite weak and have a complicated structure. Indeed, in the case of SFG-XFROG, the gate function is simply the reference pulse and thus the XFROG signal is proportional to it:
2 SEtEtedt(,) () ( )it . (4.11) XFROG ref
In standard FROG general projection algorithms, there is a trade-off with the choice of the reference: if the reference pulse is shorter than the unknown pulse, then there is a good resolution in the delay axis of the spectrogram, but details might be hidden in the frequency direction. On the other hand, if the reference pulse is narrowband, the spectral resolution is good but not the temporal delay resolution. However, this reference pulse dependency is removed when using ptychographic reconstruction [8]. Such algorithm allows XFROG setups to reconstruct attosecond pulses using femtosecond gate pulses or even to characterize ultra-broadband and complex supercontinuum pulses. As XFROG traces are not necessarily symmetric, the direction-of-time ambiguity is not a problem.
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References to Chapter 4
[1] A. J. D. Maria, D. A. Stetser, and H. Heynau, “Self mode-locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174-176 (1966).
[2] R. Trebino, Frequency-resolved optical gating: the measurement of ultrashort laser pulses, (Springer, 2000).
[3] K. L. Sala, G. A. Kenney-Wallace, and G. E. Hall, “CW autocorrelation measurements of picosecond laser pulses,” IEEE J. Quantum Electron. 16, 990-996 (1980).
[4] A. Zukauskas, N. Thilmann, V. Pasiskevicius, F. Laurell, and C. Canalias, “5 mm thick periodically poled Rb-doped KTP for high energy optical parametric frequency conversion,” Opt. Mater. Express 1, 201-206 (2011).
[5] J.-H. Chung and A. M. Weiner, “Ambiguity of ultrashort pulse shapes retrieved from the intensity autocorrelation and power spectrum,” IEEE J. Quantum. Electron. 7, 656-666 (2001).
[6] R. Trebino and D. J. Kane, “Using phase retrieval to measure the intensity and phase of ultrashort pulses: frequency- resolved optical gating,” J. Opt. Soc. Am. A 10, 1101-1111 (1993).
[7] D. J. Kane and R. Trebino, “Single-shot measurement of the intensity and phase of an arbitrary ultrashort pulse by using frequency-resolved optical gating,” Opt. Lett. 18, 823-825 (1993).
[8] A. M. Heidt, D.-M. Spangenberg, M. Brügmann, E. G. Rohwer, and T. Feurer, “Improved retrieval of complex supercontinuum pulses from XFROG traces using a ptychographic algorithm,” Opt. Lett. 41, 4903-4906 (2016).
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5. Supercontinuum Generation and Pulse Self-Compression in the IR In this chapter, we focus on ultra-short pulsed sources with the two following dominant features: Ultra-short time duration Ultra-broad spectral bandwidth Ultra-short pulse durations, typically few-cycle durations, allow outstanding temporal resolution for optical coherence tomography [1] for instance, while extremely broad spectra give access to a large panel of applications such as molecular spectroscopy [2], or seeding of OPAs in order to generate broadband carrier-envelope phase-stable signals [3]. These ultrafast broadband sources, especially in the IR, are indispensable tools for spectroscopic or medical applications. They can also be employed for attosecond pulse generation and detection [4, 5, 6], particle acceleration [7], streaking methods [8] and dual-comb spectroscopy [9], provided that sufficiently high repetition rates can be achieved. This chapter will first review how supercontinuum spectra are conventionally generated. Then, it will focus on supercontinuum generation and simultaneous femtosecond pulse self-compression in the mid-IR, through second-order nonlinear structured KTP. Specifically, we investigate femtosecond supercontinuum generation in (2) -structured ferroelectrics, both theoretically and experimentally by employing cascaded second-order nonlinear interactions in order to obtain a negative effective Kerr nonlinearity and pulse self-compression in the normal GVD regime. Extension of the supercontinuum generation in the anomalous GVD region is achieved by a judicious choice of spatial structuring of the second-order nonlinearity, i.e. by having an appropriate QPM periodicity. 5.1 Supercontinuum Generation Basically, supercontinuum generation is the creation of very wide optical spectra through nonlinear effects. It was first realized 50 years ago in bulk media (borosilicate glass), where a 300 nm-wide spectrum was generated over the entire visible region [10] through non-collinear FWM and Raman effects, reinforced and cascaded by Kerr-related diffraction. However, very high energies were needed to achieve such broadening, leading to optical damages in the material. Such supercontinuum also suffered from diffraction effects which limited its use. Thus, optical fibers were being developed and later used for supercontinuum generation [11]. Pulse propagation in optical fibers undergo no diffraction and benefit from long interaction lengths. The dispersion and the nonlinearity can be controlled by engineering the fiber: core size, doping, use of tapered fibers or photonic crystal fibers, etc. In the end, the propagation dynamics into the optical fiber depend on the pump wavelength with respect to the zero-dispersion wavelength of the fiber, as shown in Fig.5.1.
46
Fig.5.1 Dispersion regimes depending on the pulse wavelength.
The spectral broadening mechanisms rely on third-order nonlinear effects in Kerr media: SPM, FWM and Raman scattering. SPM refers to the situation in which the light pulse modulates its own phase and the pulse frequency is time-dependent. Typically, SPM will broaden the spectrum of the pulse. FWM will create additional frequencies through the nonlinear mixing of two input optical frequencies. This effect needs to fulfil two conditions to efficiently generate new spectral content: energy conservation and phase-matching condition. Finally, the Raman effect or Stimulated Raman Scattering (SRS) arises from the interaction of the light pulse with the vibrational modes of molecules in the considered nonlinear medium. The Raman gain depends on the material and can be as narrow as a few GHz in crystals with strong bonds and little disorder (such as defects, vacancies, etc.). If the input wave has a high enough intensity, the Raman gain will broaden the pulse towards longer wavelengths. Broad supercontinua are of paramount importance for seeding ultrafast OPAs. Such radiation, combining broad spectral widths and compressibility led to the invention of ultra-broadband non- collinear OPAs, which can nowadays produce few cycle optical pulses, in the visible or in the mid- IR region [12,13]. In optical fibers, the femtosecond pulse propagation dynamics are essentially one-dimensional and stem from soliton generation which is a result of SPM and material dispersion. In bulk media, the physics is more complex and combines spatial and temporal effects. In solids, liquids and gases, the nonlinear propagation and spectral broadening of intense femtosecond laser light can be understood through the concept of femtosecond filamentation [14- 17]. In this case, different processes interplay: self-focusing, SPM, plasma generation, multiphoton absorption and ionization, etc. This leads to the generation of intricate structures that can propagate long distances while keeping a narrow beam size [15]. The first stage of filamentation is established by self-focusing (intensity-dependent refractive index, see Chapter 2). Typically, radiation needs to reach a critical power for self-focusing, in which case the self-focusing effect perfectly balances the diffraction. For a symmetric Gaussian beam, the self-focusing critical power is given by: 3.72 2 Pcr , (5.1) 8 nn02
47 where is the laser wavelength, n0 is the linear refractive index and n2 is the nonlinear refractive index, as defined in sub-section 2.3.1 [18]. In this work, we studied supercontinuum generation in the infrared using domain-engineered KTP. The following sections aim at understanding the complex pulse propagation dynamics in such media and the phenomenon of pulse self-compression. 5.2 Numerical Model for Ultra-Short Pulse Propagation in QPM Media Exploiting SPM in large-bandgap dielectric materials is difficult in the mid-IR spectral region as the nonlinear phase is inversely proportional to the input pump wavelength. Starting the broadening mechanism through SPM would require operation well above the critical power for self-focusing, which is detrimental to obtain a stable and reliable broadband light source. Broad supercontinuum generation has been reported in Yttrium Aluminium Garnet (YAG) through a filamentation process [19] by using a pump power exceeding three times the critical power. Materials with a large Kerr coefficient can be employed, but they are associated with lower bandgap energies and thus suffer from lower optical damage threshold [20]. On the contrary, cascaded (2) interactions in structured ferroelectric materials such as lithium niobate (LiNbO3, LN) and KTP can be used to enhance the third-order nonlinear response. The use of domain-engineered ferroelectrics allows to tailor the effective Kerr nonlinearity so that it can be substantially larger than the natural Kerr coefficient present in the bulk crystals. In addition, the eff sign of the effective Kerr nonlinearity, n2 , resulting from cascaded second-order nonlinear interactions, can also be tailored. Common numerical models for optical pulse propagation usually involve nonlinear coupled wave equations [21] which are to be solved for the different frequency domains relevant to the nonlinear interactions at stake. However, such methods are not suitable for ultra-broadband optical pulses due to the overlap of the various frequency bands. This is why a single-wave nonlinear envelope equation (NEE) is the most appropriate numerical tool for studying ultra-broadband cascaded processes [22]. The NEE was employed in [23,24] to investigate solitons generated through (2) - cascaded processes, giving rise to Cherenkov radiation when subjected to nonlinear pulse compression. In Ref. [25] a NEE was also used to study ultrafast pulse propagation in lithium tantalate (LiTaO3, LT) and LN. This model was further developed to account for both second- and third-order nonlinearities in BBO [26]. A similar numerical model was developed for waveguides, which included the Raman response for the material as well [27]. The same model was applied to periodically poled LN waveguides to verify the octave-spanning spectra experimentally obtained in [28]. Finally, broadband OPG was reported in periodically poled MgO-doped stochiometric LT, with a broad spectrum extending from 1.1 µm to 3.5 µm [29]. Once again, a NEE-based numerical model was used to show good agreement with the experimental results, both in MgO-doped LT [30] and in bulk LN [31].
48
5.2.1 Outline of the Numerical Model In order to study ultra-broadband pulses propagation in domain-engineered materials, a numerical model involving a single-wave NEE was developed based on a total-field version of the nonlinear Schrödinger equation [25]: (2) 2 Ai 2 iD A i0 12. A2 eiti0010'' Z A e iti 0010 Z (5.2) 2 Zct4'00
Here 0 and 1 refer, respectively, to the wave-vector and its first derivative evaluated at the (2) reference frequency 0 , and Z is the propagation direction, while describes the magnitude and spatial modulation of the second-order nonlinearity. Equation (5.2) describes the propagation of the complex envelope A in [V/m] derived through an analytic continuation of the real electric field of the optical pulse. It is defined in a co-propagating coordinate frame ( Z,'t ), which corresponds to the reference frequency 0 . The time t ' in this frame is given by tt' 1 z. If one considers the example of SHG process, the reference frequency can be set at twice the pump frequency, meaning that the generated SH wave will be stationary in the new temporal reference frame. In the context of this study, we defined conveniently 0 to coincide with the central frequency of the pump wave. The complex envelope AZt(,') is defined by applying a standard procedure used for analytical signals [25]. The real time- dependent electric field is Fourier transformed and all the negative frequency components are suppressed. Then, the imaginary part of the envelope is obtained using a Hilbert transform [32]. A final Fourier transform of the latter gives the time-dependent analytic signal, which fully describes the real electric field. Moreover, only positive frequencies will appear in the simulation results and causality is automatically taken care of by the Hilbert transform. The left-hand side of Eq. (5.2) takes into account the dispersion of the envelope that the electric field undergoes, corrected by the absorption of the considered material. In the equation, we use the time-domain linear response function related to the dispersion ( D ) and absorption ( ), but those quantities are more conventionally expressed as functions of frequency, and thus evaluated in the frequency domain. In the Fourier space, these translate to:
FT iD i k() 01 k ( 0 ) (), (5.3) where FT stands for Fourier Transform and kn() () / c is the wave-vector, with n being the refractive index. All dispersion orders are automatically accounted for by the numerical model because the explicit dispersion relation is implemented through the temperature-dependent Sellmeier equations [33] over the wavelength range of 0.4-5 µm. The zero- and first-order dispersion terms are subtracted from the dispersion relation in Eq. (5.3) in order to be consistent with the definition of Eq. (5.2). The absorption spectrum of the material is used to define the absorption term () . It delimits the transparency window of the considered material and the term
49
() impacts the amplitude of the electric field envelope, whereas the dispersion term D influences the phase of the wave. For the KTP absorption spectrum, see reference [34] and Fig.3.2. The right-hand side of Eq. (5.2) represents the nonlinear polarization and describes the field envelope distribution due to (2) nonlinearity. In the case of periodically poled materials, (2) is implemented as a square-wave function with alternating sign:
(2) 2Z dsign33 cos , (5.4)
where d33 is the highest nonlinear coefficient [pm/V] in KTP, corresponding to an input wave polarized along the c-axis of the crystal, and is the poling period for first-order QPM. In the visible, as well as in the near- and mid-IR part of the spectrum up to 3.5 µm, (2) is essentially dispersion-free. This is why an instantaneous (2) response was assumed in this thesis work. The interaction picture of the 4th-order Runge-Kutta algorithm [35,36] was used to solve Eq. (5.2). This scheme is a more accurate version of the Split-Step Fourier Method (SSFM). It allows separating the dispersion term (evaluated in the frequency domain) from the nonlinear terms (evaluated in the time domain) to solve them in an interaction picture. The sample is basically numerically sliced in steps of length h and the calculation of the complex envelope is performed for each predefined step. A strength of the 4th-order Runge-Kutta algorithm lies in the fact that the number of required Fourier transforms is divided in two by choosing midpoint steps to evaluate the Runge-Kutta coefficients.
5.2.2 Extended Numerical Model with Cubic Nonlinearities Besides the second-order nonlinear response, we also decided to include cubic nonlinearities and delayed Raman response into the numerical model due to high peak intensities and broad spectra involved. The studied process of supercontinuum generation involved a FWM conversion channel, which is a third order nonlinear process issued form a cascaded quadratic process in our case. Other third order nonlinearities should not be omitted as they might play a non-negligible part and we should also add the electric Kerr effect. In the following we account for the instantaneous electric (3) nonlinearity as well as the delayed (3) Raman response. The usual redshift witnessed with Raman effect may lead to an extended supercontinuum spectrum towards mid-IR. In particular, for ferroelectric materials, the relative fractional response of the Raman effect with respect to the Kerr effect is quite large [37]. This addition to the initial Eq. (5.2) was necessary to achieve good agreement between the numerical calculations and the experimental results. Using Agrawal derivations for third-order nonlinear effects [38,39] we derive the single-wave NEE:
50
(2) 2 Ai 2 iD A i0 12 A2 eitiZitiZ0'' 0 10 A e 0 0 10 2 Zct4'00 (5.5) i 2 i()1 A Rt ()(,') AZt t dt . 0 0 t '
Now, the second term on the right-hand side describes the third-order nonlinear polarization, where
()0 is the third-order nonlinear parameter and is given by: 1 () nn () nat , (5.6) 000022
nat 2 -1 where n2 in [m W ] is the native Kerr nonlinearity for a given material. The nonlinear response function Rt() is associated with the delayed third-order nonlinearities and the instantaneous Kerr, and Rt() can be expressed as a sum of Raman resonance modes: t N 22 t Rt() 1 f () t f12jj Hte ()2 j sin , (5.7) RT Rj 2 j1 12jj 1 j where is the Dirac function and H is the Heaviside step function, N is the total number of
Raman modes taken into account. The factors fRj are defined by the oscillator strengths for the N considered resonances [40,41] and verify f f , where f represents the total fractional RTRj j1 RT contribution of the Raman response with respect to the nonlinear polarization. The quantities 2 j are the decay parameters used to fit the Raman gain spectrum with a Lorentzian profile, and the quantities 1 j are related to the period of the phonon vibrations.
5.2.3 Raman Parameters for KTP In the following section, we investigate supercontinuum generation in domain-engineered KTP. Here one needs to define the material parameters for the simulation of the Raman delayed response. In our case, the pump pulses will propagate along the a-axis of the crystal and be polarized along its c-axis. This means that we are interested in the A1 vibration symmetry of the crystal, also referred to as the X(ZZ)X Raman scattering configuration. Indeed, KTP presents two strong Raman modes, with resonances at 8 THz and 21 THz, which are the ones considered for the calculations in sections 5.3 and 5.4. The different parameters used for the simulations are listed in Table 5.1 [40-43].
Material 11 (fs) 12 (fs) fR1 21 (ps) 22 (ps) fR2 fRT KTP 20 7.6 0.13 7.4 1.2 0.37 0.5 Table 5.1 Parameters used for the calculations of the Raman delayed response including two resonances for KTP.
51
5.3 Application to Mid-IR Supercontinuum Generation in Periodically Poled KTP
5.3.1 Strategy for Supercontinuum Generation and Pulse Self-Compression In order to obtain the broadest supercontinua in the mid-IR range, one needs to start with a positive GVD and a large enough negative Kerr effect (see Chapter 2) so that the combination of both eventually leads to soliton self-compression and spectral broadening. Additionally, it is possible to reverse the sign of the Kerr nonlinearity for longer wavelengths, when the dispersion becomes negative for KTP. Therefore, we chose an input pump wavelength close to, but shorter than, the zero GVD wavelength and used domain-engineering of a KTP crystal in order to tailor the sign and magnitude of the effective Kerr nonlinearity, through cascaded second-order nonlinear interactions. Indeed, in the following sub-section, we will present simulation results for a 100 fs- long (FWHM) Gaussian transform-limited pump pulse centered at a wavelength of 1.58 µm. casc 2 -1 By periodic poling, the effective Kerr nonlinearity, or cascaded n2 nonlinearity in [m .W ] can be designed according to the phase-mismatch k [44,45]: 2 casc 2pd33 sin kL n2 1, (5.8) nn22 c k kL pp20 where n and n are the refractive indices of the material at the pump frequency (fundamental) p 2p
p and its second harmonic 2p , respectively, and L is the propagation length. The total effective eff eff nat casc Kerr nonlinearity n2 can be expressed as nnn222. Furthermore, the phase-mismatch k is given by:
kk22ppQPM k k , (5.9)
where kQPM is the grating wave-vector, defined as kQPM 2/ . In our configuration, the grating period was chosen to phase-match SHG close to the zero GVD wavelength, to ensure a large phase-mismatch at the pump wavelength in order to trigger cascaded second-order nonlinearities (see chapter 2, sub-section 2.3.3). The effective Kerr coefficient was calculated from Eq. (5.8) and was found to be negative at the casc 14 2 input pump wavelength (1.58 µm) in KTP: n2 1.65 10 cm /W, almost one order of magnitude larger than the natural Kerr coefficient in the corresponding bulk crystal. The native nat 15 2 Kerr nonlinearity in bulk KTP is [46]: n2 2.4 10 cm /W. Figure 5.2 (a) shows the group index and the GVD curves for KTP. The zero GVD wavelength for KTP is located at 1.8 µm. As mentioned previously, a pump wavelength of 1.58 µm is initially chosen for supercontinuum generation. At this wavelength, the GVD is positive. The QPM period was chosen to be 36 µm, which provides phase-matching of SHG for 2 µm radiation. At this casc casc wavelength, it provides a pronounced n2 resonance (Fig.5.2 (c)), and changes the sign of n2
52 as well. In this wavelength range, the sign of the GVD is also reversed accordingly. It should be noted that different grating periods can be used as long as they correspond to a phase-matched process close to the zero GVD point. Namely, poling periods between 32 µm and 40 µm will lead to similar supercontinuum generation and pulse compression (see following sub-section 5.3.2 and Fig. 5.4).
Fig.5.2 (a) Group index for c-polarized waves (dashed line) and GVD (solid line) as a function of the wavelength for KTP. The horizontal black line represents the zero GVD. (b) KTP third-order dispersion (TOD). The vertical dotted red line in (a) and (b) casc highlights the zero GVD wavelength. (c) Phase-mismatch (dashed line) and n2 casc spectrum for KTP (solid line). (d) Zoomed-in plot of the n2 spectrum for the region of interest.
5.3.2 Simulation Results In this section, we present the simulation results for a 100 fs-long Gaussian transform-limited pump pulse centered at 1.58 µm. The peak intensity of the pulse at the entrance of the crystal was set to 53
15 GW/cm2. This is much lower than the optical damage threshold for femtosecond pulses in KTP (600 to 800 GW/cm2 [47,48]). The input pump pulses are polarized along the crystal c-axis, in order to take advantage of the material highest nonlinearity d33 16.9 pm/V, and propagate along the crystal a-axis. The crystal grating is 10 mm-long, while the QPM period is 36 µm. The spectral and temporal distributions are presented in Fig.5.3. The Raman parameters used are presented in Table 5.1.
Fig.5.3 Calculated spectral evolution (a) along the 10 mm-long PPKTP sample. The pump spectrum broadens solely through SPM up to around 6 mm propagation within the sample. Once the broadened spectrum of the pulse reaches and passes the zero GVD wavelength, where SHG phase-matching is fulfilled, the corresponding IR component at 1 µm (SHG of 2 µm radiation) is generated and appears around 7 mm propagation into the crystal. The leaking spectral components at longer wavelengths, past the zero GVD point, take part in a phase-matched cascaded FWM process, transferring power from the spectral region of the pump towards the mid-IR around 2.5 µm to 3 µm. Calculated temporal evolution (b) of the femtosecond pump pulse, output supercontinuum spectrum (c) after the 10 mm propagation and (d) temporal comparison of input pump pulse and
output supercontinuum pulse for fRT 0.5 .
54
The pump pulse starts by propagating in the positive GVD and negative Kerr environment. Thus, it experiences spectral broadening until it extends through the zero GVD point. Due to resonances casc in the n2 spectrum (see Fig.5.2 (d)), the cascaded FWM process will become efficient at that point and transfer energy from the short-wavelength supercontinuum part to longer wavelengths, while reversing the remnant chirp in the process. The Kerr nonlinearity is then positive while the GVD is negative. These conditions are suitable for solitary pulse compression and further spectral broadening. For the pump conditions used in these calculations, the optimum compression was obtained for a 10 mm-long sample. The fractional response of the Raman effect relative to the Kerr response was set to fRT 0.5 because it was found to produce the best match with the experimental data. This is a free parameter in the calculations, but such high fRT value is in accordance with previous observations for ferroelectric materials, where the relative fractional response is indeed quite large [37]. Fig.5.3 (a) depicts the spectral pump evolution. As soon as the pump is launched into the crystal, the non-phase-matched SHG and SFG are generated at 790 nm and 526 nm, respectively, due to high peak intensity and the large nonlinearity of KTP. In Fig.5.3 (a), the SH amplitude at 790 nm is periodically modulated, which is the result of the periodic cycling of the SHG and SFG processes, i.e. cascading processes which give rise to the effective Kerr nonlinearity. This is associated to the pump pulse self-compression in the temporal domain, as shown in Fig.5.3 (b). The pump spectrum broadens solely through SPM up to around 6 mm propagation within the sample. Once the broadened spectrum of the pulse reaches and passes the zero GVD wavelength, where SHG phase-matching is fulfilled, the corresponding IR component at 1 µm (SHG of 2 µm radiation) is generated and appears in Fig.5.3 (a) around 7 mm propagation into the crystal. The leaking spectral components at longer wavelengths, past the zero GVD point, take part in a phase- matched cascaded FWM process, transferring power from the spectral region of the pump towards the mid-IR around 2.5 µm to 3 µm. The increasing third-order dispersion in the mid-IR wavelength range, as seen in Fig.5.2 (b), shapes this wave into an Airy-type wave-packet which propagates with compression of the central peak, deceleration in the reference frame and formation of tailing sub-pulses (Fig.5.3 (b)). The parabolic trajectory is characteristic of Airy-type solitons [49]. On the other hand, the weak remnants of the non-phase-matched visible radiation at 790 nm and 526 nm, as well as the phase-matched emission at 1 µm, are rapidly walking away from the forming soliton pulse due to the large GV difference from the soliton pulse. The temporal periodic modulation along the crystal length in Fig.5.3 (b) originates from the SHG and back-conversion phenomenon, i.e. from the cascading processes. Fig.5.3 (c) shows that the generated supercontinuum spectrum at the output of the sample spans from 1.2 µm to 3 µm. The temporal width of the output main peak of the pulse is 8 fs FWHM, as shown in Fig.5.3 (d), thus compressing the pump pulse more than 10 times. It was previously mentioned that quite a wide range of poling periods can be used to generate such broad supercontinuum, while still reaching a good compression of the pump pulse. Fig.5.4 presents simulation results for QPM periods between 32 µm and 40 µm. The zero-dispersion wavelength
55 for KTP is 1.8 µm. Phase-matching for SHG from 1.8 µm gives a poling period of 32 µm. In the same fashion, phase-matching for SHG from 1.9 µm, 2 µm, 2.1 µm and 2.2 µm lead to poling periods of 35 µm, 36 µm, 38.5 µm and 40 µm, respectively. While investigated, those different poling periods gave quite similar output spectra and self-compressed pulses as seen in Fig.5.4 (b). A 40 µm QPM period gives the broadest spectrum, but with additional modulations at longer wavelengths.
(a) (b)
Fig.5.4 Supercontinuum spectra (a) and self-compressed output pulses (b) for different poling periods. The pulses are slightly offset in time to be able to visualize them correctly. In (a), the noise at short wavelengths is due to the cascaded effect for the non-phase- matched SHG and SFG processes.
5.3.3 Experimental Setup Given the interesting numerical results for ultra-broadband spectrum and pulse self-compression, experiments were performed to validate the simulations. The experimental setup actually uses PPRKTP due to all the favorable properties presented in Chapter 3. The engineered samples were 1 mm-thick, with QPM periods of 36 µm and grating lengths of 11 mm. The setup is presented in Fig.5.5. It consisted of an OPA (Orpheus F, Light Conversion) pumped by an Ytterbium-doped potassium gadolinium tungstate (Yb:KGW) femtosecond laser system operating at 50 kHz (Pharos, Light Conversion). The OPA generated 128 fs-long pulses (FWHM) centered at 1.52 µm. The OPA beam was split into two parts using a wave-plate and a polarizer. The largest part of the energy (4 µJ) was used as a pump for supercontinuum generation in the PPRKTP crystal, while the remaining part (1 µJ) served as a probe pulse for SFG-XFROG measurements.
56
10 µm BBO Yb:KGW 50 kHz SHG FROG/ fs laser SFG XFROG
1 µJ 760 nm OPA BBO Short pass f=200 mm filter R=‐500 mm
1520 nm Supercontinuum 128 fs 4 µJ λ/2 λ/2 PPRKTP Long pass filter
Fig.5.5 Schematic of the experimental setup with the SHG-FROG/SFG-XFROG device.
The pump beam, of radius of approximately 1.5 mm, was loosely focused into the PPRKTP sample by a lens with a 200 mm focal length. The crystal was located at a distance of about 25mm in front of the focal point so that the self-defocusing in PPRKTP was, to some extent, counteracted by the convergence of the beam. A wave-plate was used to adjust properly the polarization of the pump pulses before the focusing lens so that the pump input polarization matches the c-axis of the crystal. The pump input intensity at the entrance of the crystal was estimated to be 13 GW/cm2. The supercontinuum beam exiting the sample was collimated using a metallic spherical mirror with a radius of curvature of - 500 mm to avoid adding chromatic aberration. With these experimental parameters, one can use Agrawal’s derivations for soliton dynamics [38] 2 and estimate the soliton order. It is defined as NLL D / NL where the dispersion length is given 2 by LTD 02/(0) and the nonlinear length is defined by LPNL 1/( 0 ). We used the following 25 2 experimental parameters: the pulse length is T0 128 fs, the dispersion is 2 3.2 10 s /m, the intensity is 13 GW/cm2, the central wavelength is 1.52 µm and the nonlinearity is 1.65 1014 cm2/W. This gives a soliton order of 6.6 (i.e. 7). The nonlinearity dominates with respect to dispersion and leads to substantial spectral broadening. The SFG-XFROG method was employed to characterize the output supercontinuum pulses [50,51]. The 1 µJ deflected pump was frequency doubled to 760 nm in a 1.5 mm BBO crystal and filtered to remove any remaining fundamental. It was then used as the probe to get the SFG-XFROG in the visible range. The supercontinuum pulse and the probe pulse were overlapped in a 10 µm-thin BBO crystal to generate the sum frequency signal. The XFROG device was designed in such a way that it could also be used to fully characterize the pump pulses via SHG-FROG, which ensured compatibility of the characterized probe pulses and the XFROG measurements.
5.3.4 Experimental Results and Comparison with the Numerical Model The pump characteristics were retrieved through SHG-FROG measurements and are presented in Fig.5.6. In the next sub-section, such information will be used as input for the numerical model to
57 compare simulations to experimental results. The FROG error was 0.48% for the retrieved FROG trace in Fig.5.7.
Fig.5.6 SHG-FROG retrieved spectrum and spectral phase (a) of the pump beam centered at 1.52 µm. One notices that the pump spectrum contains additional spectral features at longer wavelengths, which come from the OPA. Temporal profile (b) of the input pump pulse. The FWHM is 128 fs.
Fig.5.7 Measured (a) and retrieved (b) SHG-FROG traces for the pump pulse.
The measured and retrieved SFG-XFROG traces, with a retrieval error of 3%, are presented in Fig.5.8 (a) and (b), respectively. It should be noted that the time axes are reversed there, due to how the measurement was performed.
58
Fig.5.8 Measured (a) and retrieved (b) SFG-XFROG traces of the supercontinuum pulse. The time axes are reversed and the color bars represent the normalized spectrogram amplitude in log scale. Retrieved spectrum and spectral phase (c) of the supercontinuum pulse. A long-pass filter cuts away the wavelengths shorter than 1.1 µm. The supercontinuum spectrum spans from 1.1 µm to about 2.7 µm and the spectral phase is smooth, with quadratic contribution for wavelengths above 2 µm.
The SFG-XFROG traces reveal that at a zero delay, the generated spectrum is indeed very broad. A SFG wavelength of 600 nm in the traces would here corresponds to an actual signal wavelength at 2.85 µm. The weak components at longer wavelengths are trailing the main peak, being split into a dispersive wave by third-order dispersion. We do not observe intensity modulation on the trailing side of the pulse, characteristic of an Airy soliton, which was obtained under ideal Gaussian beam excitation conditions in the simulations, shown in Fig.5.3. The corresponding supercontinuum spectrum extends from 1.1 µm to 2.7 µm as seen in Fig.5.8 (c). It also shows that the spectral phase is relatively smooth, with a predominant contribution of the quadratic phase for wavelengths beyond 2 µm. This implies that the supercontinuum pulse can be further compressed in the mid-IR range, using suitable dispersion compensation. Taking into account the spectrum and the phase of the pump pulse retrieved from the SHG-FROG measurements, we performed numerical simulations based on the real experimental conditions. The results are shown in Fig.5.9. It should be noted that the input pump pulses are chirped and that,
59 apart from the peak at the central wavelength of 1.52 µm, the spectrum also contains additional features coming from the OPA.
Fig.5.9 Calculated spectral distribution (a) and temporal evolution (b) of the pump pulse
along the 11 mm-long PPRKTP sample with a Raman fractional response of fRT 0.5 for the “real pump”. Calculated spectral distribution (c) and temporal evolution (d) of the pump pulse along the 10 mm-long PPRKTP sample with a Raman fractional response
of fRT 0.5 for an ideal Gaussian beam excitation.
Even if the pump spectrum is not an ideal Gaussian, as assumed in the previous numerical simulations, the results of Fig. 5.9 (a) and (b) can be compared to the theoretical results of sub- section 5.3.2 (Fig. 5.9 (c) and (d)) to confirm the broadening mechanisms. The pump spectrum first broadens, while the non-phase-matched SHG and SFG are generated in the visible wavelength range. After propagation of about 9 mm into the crystal, the pulse spectrum reaches 2 µm and the quasi-phase-matched 1 µm wavelength is then generated while the supercontinuum keeps expanding towards mid-IR wavelengths. The additional spectral components initially present in the pump spectrum are gradually amplified during the propagation. The nonlinear compression process is highlighted in Fig.5.9 (b) and matches well with the experimental observations. Indeed, under the same experimental conditions, the calculations show that the Airy soliton is not formed, in agreement with the SFG-XFROG traces of Fig.5.8. Finally, the experimental results and the simulation findings, adapted to the real pump pulse, match very well and confirm the broadening and compression mechanisms. A comparison between the
60 experimental and calculated temporal shapes and spectra of the supercontinuum pulse is presented in Fig.5.10.
Fig.5.10 Comparison of (a) the output temporal profiles and (b) the output supercontinuum spectra for both the numerical model (blue solid curves) and the experimental measurements (orange solid curves). The experimental spectrum does not show any content below 1 µm because a long-pass filter was used in the SFG-XFROG setup, cutting away shorter wavelengths.
The self-compressed experimental supercontinuum pulse was 18.6 fs (FWHM), while the transform limit would correspond to a 13 fs long pulse. The numerical calculations show resulting pulses with FWHM of 11.4 fs. The spectral content of the simulated supercontinuum was studied for different values of the fractional contribution of the Raman response, fRT . The best agreement in terms of output spectral shape, extent and temporal width was achieved for fRT 0.5 and is shown in Fig.5.10.
5.3.5 Conclusions A single-wave nonlinear envelope equation was developed to investigate the propagation of ultra- broadband optical pulses in (2) -modulated non-centrosymmetric media. It was applied to PPRKTP in order to study its performance in supercontinuum generation and pulse self- compression in the mid-IR range. To get a good fit with experiments, the two strongest Raman modes were used in the calculations with the Raman contribution fRT 0.5 . Proper design of the domain grating was implemented to obtain a strong negative effective Kerr nonlinearity, which led to drastic spectral broadening in the normal dispersion regime, where the pump was initially launched. The model shows a potential for obtaining an octave-exceeding spectrum extending to 3 µm while providing a reasonably clean output temporal profile. At the same time, this led to self- 61 compression of 100 fs Gaussian pump pulses down to 8 fs (FWHM) in PPKTP. Previous experimental results with periodically poled LN waveguides showed a supercontinuum extending up to 3 µm, but with a large dip in magnitude around 2.5 µm [25]. Moreover, utilizing structured bulk samples instead of waveguides would be beneficial, as one can utilize large beam sizes and thereby obtain higher energy without risking catastrophic material damage. Experimental validation of the model was performed using an OPA generating 128 fs pulses centered at 1.52 µm and an 11 mm-long PPRKTP crystal. A broad supercontinuum spectrum extending from 1.1 µm to 2.7 µm was generated, and an associated nonlinear self-compression was observed, as predicted from the numerical model. Pulses as short as 18.6 fs were measured using a SFG-XFROG setup. The retrieved spectral phase proved to be quite smooth in the mid-IR range. Consequently, this supercontinuum pulse could be considered for further experimental work, for instance, as a seed for optical parametric chirped pulse amplifiers. Moreover, this approach is energy scalable for periodically poled ferroelectrics thanks to large-aperture samples [52-54]. In addition, our numerical model and experimental setup could be applied to other periodically poled ferroelectrics. Particularly promising is the KTP isomorph KTiOAsO4, which presents a flatter GVD profile and a broader transmission window in the mid-IR, features that are of interest for octave-spanning supercontinuum generation. Finally, the model is designed such that different poling structures can be investigated. In particular, one can focus on apodized, aperiodically poled structures [55-58], as described in the following section. 5.4 Aperiodic Structures Another part of the theoretical work regarding mid-IR supercontinuum generation and pulse self- compression was dedicated to the theoretical design of aperiodic QPM structures. In particular, we focused on apodized chirped gratings in order to get cleaner and smoother output supercontinuum spectra. Moreover, chirped gratings should offer the potential for even larger phase-matched bandwidths, without having to shorten the crystal length and reduce the conversion efficiency of the nonlinear process. In the spatial Fourier domain, the dispersion of the material gives a mapping of the different phase-matched frequencies to the grating k-vector. By using chirped (aperiodic) grating structures, the k-vector is smoothly swept over the required range and thus broadens the Fourier spectrum of the grating, which means that the supported phase-matching bandwidth is also widened. Apodization techniques are used to avoid accumulation of spectral phase ripples by smoothly bringing the effective nonlinearity to zero in the material [59-61]. Apodization allows an adiabatic change of the nonlinear coupling, providing smooth spectrum and spectral phase. In order to get higher efficiencies for supercontinuum generation as well as the broadest achievable spectra, several design solutions found in literature were investigated. Indeed, it has been reported that adiabatic nonlinear conversions, for instance in sum frequency generation, allow robust transfer of energy between the two input wavelengths and the output wave [55]. In practice, the rapid adiabatic passage is performed through adiabatic variation of the phase-mismatch magnitude along the crystal in the direction of propagation. This requires specific poling structures. In particular, we will focus on aperiodic (or chirped) designs characterized by a given sweep rate of the phase-mismatch, which allow efficient broadening of the spectrum. However, such structures 62 present abrupt variation of the nonlinear polarization at the edges of the crystal, leading to ripples in the output spectra. Apodization techniques are then commonly used to reduce those unwanted spectral features and smoothly tune the nonlinearity down to zero on the crystal edges [58]. Apodization, coupled with chirped poling, allows one to conserve smooth spectra and cleaner temporal profiles throughout the amplification process. We decided to compare poling period apodization (PPA) involving a constant chirp rate along the grating with an apodization function on the edges, and duty cycle apodization (DCA) where the duty cycle ratio is adiabatically detuned away from 50%. Those techniques are used to turn the nonlinear coefficient on and off, in an adiabatic way, as the optical pulse is entering and leaving the medium. Numerical calculations are implemented with KTP for comparison with the periodic poling case and follow the methods presented in [57].
5.4.1 Poling Period Apodization The poling period apodization method consists in a slight detuning of the phase-matching condition so that different frequencies are phase-matched at different locations inside the crystal, as the optical pulse propagates. The grating periodicity is defined along the crystal of length L as
()zk 2 / QPM , where kQPM is given by the following expressions depending on the position in the grating: L 1kkkkapo0 apo center k edge z 2 kQPM (5.10) L 1kkkkapo0 apo center k edge z 2
In Eq. (5.10), kcenter is the wave-vector which corresponds to a phase-matching for which the chosen QPM period lies in the center of the grating. In other words, if one decides to have a QPM period of 36 µm in the center of the grating, then the QPM period is chirped around this central value, as shown in Fig.5.11. k0 is the linear chirp of the period and is defined as: dk L kk0 center z , (5.11) dz 2
-2 where dk/ dz is the chirp rate in [m ], and kedge describes the amount of phase-mismatch desired on the edges of the grating. Apodization is also performed on these edges through the term kapo : zl Lzl 1 tanhrr tanh ww k rr (5.12) apo l 1tanh r wr
The apodization profile is shaped so that it smoothly detunes the QPM period on a width wr located at a distance lr from the edges of the crystal.
63
Fig.5.11 Profile of the QPM period for poling period apodization at a central period of 36 µm in KTP.
5.4.2 Numerical Results for Poling Period Apodization Modelling of the supercontinuum was performed using the type of variation of the QPM period, shown in Fig. 5.11, which allows tuning of the phase-mismatch from k 0 to k 0 through k 0 (for SHG close to the zero GVD point of the given material) along the whole crystal. Aperiodic poling was investigated for 10 mm-long crystals for the same pump pulse used in the previous section: a Gaussian transformed limited 100 fs pulse at a wavelength of 1.58 µm, carrying a peak intensity of 15 GW/cm2. The different parameters for the apodized structure are the -2 following: chirp rate -100 cm , center period 36 µm, lr 1 mm, wr 0.5 mm, and the amount of phase-mismatch was 10 cm-1 as shown in Fig.5.11. The resulting supercontinuum spectrum in KTP is slightly flatter at the output of the crystal as the peak intensity is more evenly distributed, as shown in Fig.5.12 (a), compared to the case of the periodically poled grating. In the calculations, two Raman resonances are taken into account as mentioned previously. However, the poling period apodization technique gives a slightly shorter supercontinuum spectrum on the long wavelength side, but with a flatter profile. The output pulse temporal FWHM is 9 fs. The compression mechanism is the same as described in section 5.3 and the results of the calculations are presented in Fig.5.12.
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Fig.5.12 Profiles of the supercontinuum spectrum (a) at the output of the PPKTP crystal for poling period apodization and regular periodic poling for a QPM period of 36 µm. Temporal distribution of the pump pulse (b) and spectral evolution (c) along the 10 mm- long KTP crystal. Comparison of input pump pulse and output supercontinuum pulse in time (d).
5.4.3 Duty Cycle Apodization Another apodization technique is called duty cycle apodization and is also presented in [57]. In this case, the duty cycle is adiabatically tuned away from the 50% usual ratio on the edges of the crystal as seen in Fig.5.13, while the phase-mismatch magnitude is linearly chirped. The duty cycle variation used in [57] is similar to the one used for period poling apodization, with hyperbolic tangent behavior on the edges of the crystal: zl 1tanh r 1 w Dz() r . (5.13) apo 3 L zl 1tanh r wr
Because a duty cycle of 50% is wanted, the final variation of the duty cycle is defined as:
DzDCA() 0.5 Dz apo ().
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Fig.5.13 Duty cycle (red solid line) across the crystal and QPM period (blue solid line) for KTP.
Such a material design was implemented for a KTP sample with the aforementioned QPM central periods of 36 µm. The results are shown in Fig.5.14.
Fig.5.14 Profiles of the supercontinuum spectrum (a) at the output of the PPKTP crystal for duty cycle apodization and regular periodic poling for a QPM period of 36 µm. Temporal distribution of the pump pulse (b) and spectral evolution (c) along the 10 mm- long KTP crystal. Comparison of input pump pulse and output supercontinuum pulse in time (d).
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The supercontinuum spectrum at the output of the crystal is also flatter than in the periodic poling case. The duration of the output pulse is 10.5 fs FWHM and the temporal profile shows less ripples. In conclusion, the supercontinuum spectra obtained through poling period apodization and duty cycle apodization techniques are comparable to the ones reached with conventional periodic poling; except that the output pulse might look cleaner (smoothed ripples) and the output spectra flatter. This is due to the apodization. The trailing oscillations observed in KTP are reduced and result in time profiles with lower pedestal and fewer oscillations. In general, the supercontinuum spectra are slightly narrower, but they still lead to comparable FWHM in the range 9 to 11 fs, that is to say a compression factor close to 10. One limitation of the duty cycle apodization technique lies in the very small ferroelectric domains (< 5 µm) which potentially need to be manufactured towards the edges of the grating. 5.5 Mid-IR Supercontinuum Generation with Other Ferroelectrics Given the interesting results observed in domain-engineered KTP for mid-IR supercontinuum generation and pulse self-compression, numerical analyses were also performed to compare the performances of different ferroelectrics for the periodic poling case: KTP, KTiOAsO4 (KTA, a KTP isomorph), lithium niobate, LiNbO3 (LN), and lithium tantalate, LiTaO3 (LT).
5.5.1 Materials KTA is the arsenate isomorph of KTP, for which the PO4 tetrahedra are replaced by AsO4 tetrahedra. In this sense, it behaves very much like KTP and presents similar parameters, listed in Table 5.2, which we used in the numerical model. In the same fashion, material data is available for LN and we consider the same values for LT, as default, considering the strong analogy between both materials. The last column shows the QPM periods for the supercontinuum generation in the four cases, chosen with the same criterion as in section 5.3.
nnat (10- 0 2 d 11 12 21 22 14 33 QPM Material fR1 fR2 fRT GVD (fs) (fs) (ps) (ps) (pm/V) (µm) cm2/W) (µm) KTP 20 7.6 0.13 7.4 1.2 0.37 0.5 2.4 16.9 1.8 36 KTA 23 7.7 - 7.4 1.2 - - 1.7 16.2 2 38 LN 22 8.4 - 0.74 0.72 - - 3.46 19.5 1.9 29 LT 22 8.4 - 0.74 0.72 - - 3.46 16 1.8 31 Table 5.2 Different parameters used in the simulations for KTP, KTA, LN and LT. Two Raman resonances are taken into account for KTP from [41] and KTA [62]. Analogy is made between KTA and KTP for the damping and oscillator strength terms due to the same type of TiO bounds involved in both material [63-65]. Similar analogy is made
between LN and LT. One can notice that the decay parameters 2 j are much longer for KTP and KTA than for LN and LT.
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In Fig.5.15, the group indices, GVD, phase-mismatches and cascaded second-order nonlinearities spectra are plotted for the QPM periods chosen in Table 5.2 and an input wavelength of 1.58 µm.
casc Fig.5.15 Group indices (a), GVD (b), phase-mismatches (c) and n2 spectra (d) for KTP, KTA, LN and LT. All the curves show the same tendency, which means that we expect to see a similar behavior from the four different materials for mid-IR supercontinuum generation.
As the strength of nonlinear interactions depends not only on the effective nonlinear coefficient but also on the refractive indices of the waves involved in the process, one can use the nonlinear figure of merit to compare the different nonlinear materials in the given spectral range: d 2 FOM eff . (5.14) n3
Thus, the highest nonlinearity observed in LN is compensated by its higher nonlinear refractive index values, while LT will be the least effective crystal with the lowest effective nonlinear coefficient and high values of the refractive indices. On the other hand KTP and KTA have refractive indices of similar magnitude and similar effective nonlinear coefficients.
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5.5.2 Numerical Results Supercontinuum generation was numerically studied in the four different materials for a grating length of 10 mm and the same pump parameters (1.58 µm, 100 fs) as stated in section 5.3. The input pump intensities for KTP, KTA, LN and LT were 16 GW/cm2, 14 GW/cm2, 10 GW/cm2 and 16 GW/cm2, respectively. The resulting output supercontinuum spectra are shown in Fig.5.16.
Fig.5.16 Comparison of the output supercontinuum spectra in KTP, KTA, LN and LT.
The KTP isomorphs give the broadest spectra, the widest span being achieved with KTA, up to 3.5 µm. All the spectra look very similar below 2 µm, with the 1µm phase-matched radiation and the depleted main pump peak due to the FWM process. However, in the longer wavelength range, spectral modulations appear differently for the various materials. The corresponding output temporal profiles are presented in Fig.5.17, where the different pulses are intentionally delayed in order to display the results in a readable fashion.
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Fig.5.17 Output temporal profiles for the supercontinuum pulses in KTP, KTA, LN and LT.
The corresponding output pulse durations for the main peak are 13 fs, 14.3 fs, 18 fs and 13 fs for KTP, KTA, LN and LT, respectively. One can notice that the trailing oscillations are somewhat smaller in LN and LT as compared to those in KTP isomorphs. This can be explained by the faster decay parameters for LN and LT (see Table 5.2).
Figure 5.15 (d) shows that KTP and KTA also have the highest n2 resonances, or at least comparable to LN. Finally, KTP isomorphs present higher damage threshold. Considering the above comparisons, one can conclude that PPKTP isomorphs, and particularly PPKTA, could be the most suitable material for applications similar to supercontinuum generation and simultaneous pulse self-compression.
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6. Studies on Backward Wave Optical Parametric Oscillators As introduced in Chapter 2, this thesis work also focused on studies regarding a special type of oscillators: BWOPOs, also known as mirrorless optical parametric oscillators (MOPOs) [1,2]. Those counter-propagating parametric devices rely on a self-established feedback mechanism and do not require any external cavity. The unique spectral and coherence features of these devices are attractive for applications in spectroscopy, and even for quantum communications. Such counter- propagating devices rely on strong constraint regarding the phase-matching scheme and thus the momentum conservation. As discussed in Chapter 2, material birefringence cannot be used in this case and the large momentum mismatch requires employing QPM structures with sub-µm periodicities. Fortunately, a reliable and efficient fabrication technique for sub-µm periodically poled Rb-doped KTiOPO4 (RKTP) was recently demonstrated [3] and led to the fabrication of high-quality and homogeneous structures enabling low-threshold BWOPO devices with centimeter-scale interaction length. This chapter addresses the different investigations and findings with three different samples having periodicities of 755 nm and 509 nm. 6.1 Cascaded BWOPO If a homogeneous sub-µm grating can be efficiently poled, one can demonstrate cascading counter- propagating interactions. This means that the initially generated forward wave is further used as a pump for a second BWOPO process, generating new forward- and backward-propagating waves. In conventional OPOs, where the interaction is co-propagating, cascaded phase-matching processes require complex cavity arrangements and multiple QPM grating structures. This section explains the findings of paper I, regarding cascaded BWOPO processes where primary-pump conversion efficiencies reach above 60%. This cascading situation happens when the BWOPO signal is generated in the forward direction.
6.1.1 Cascaded BWOPO Scheme The cascading scheme in our BWOPO process can be explained as follows: the signal wave (generated in the forward direction) from the primary BWOPO process is pumping a secondary BWOPO process. In both stages, the QPM momentum conservation must be fulfilled, which, in turn, determines the generated frequencies: kkkk ps11 i QPM (6.1) kkkks122siQPM.
In Eq. (6.1), the subscripts p , s and i refer to pump, signal and idler wave, respectively, and indices 1 and 2 refer to primary and secondary BWOPO processes. The corresponding energy conservation is: p si11 (6.2) s122si.
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A sketch of the cascaded scheme is shown in Fig.6.1, together with the wave-vector diagrams for both BWOPO processes.
Primary BWOPO Secondary BWOPO z x ‐y
Fig.6.1 Illustration of the cascaded BWOPO scheme and wave-vector diagrams. Here, two cascaded processes are shown: the primary BWOPO where the pump (p) generates a forward-propagating signal ( s1) and a backward-propagating idler (i1). In the secondary BWOPO process, the signal ( s1) is then used as the new “pump” to generate a forward-propagating signal ( s2 ) and a backward-propagating idler (i2 ).
Photon energy of the forward- and backward-generated waves are calculated depending on the pump photon energy for the case of a QPM period of 755 nm and presented in Fig.6.2 (a). As mentioned in Chapter 2, the forward wave carries the same phase information as the pump wave, so its frequency modulation will vary according to the one of the pump. On the other hand, the backward-generated wave is insensitive to pump tuning. Therefore, for each cascading BWOPO process, the forward wave will lose energy approximately equal to the photon energy of the backward-generated wave, as shown in Fig.6.2 (b). In these graphs, we considered a QPM period of 755 nm and an initial pump photon energy of 1.55 eV.
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Fig.6.2 Cascaded BWOPO processes from a primary pump at 1.55 eV in a QPM structure with a periodicity of 755 nm. Cascaded steps in photon energy (a) for the forward and backward waves. Energy levels (b) of the generated photons in the BWOPO cascaded processes.
As the forward-propagating wave frequency depends linearly on the pump frequency, any frequency modulation will be efficiently transmitted through the cascading effect to lower wave frequencies.
6.1.2 Experimental Setup The sample used in the experiments was a 1 mm-thick PPRKTP with a QPM periodicity of 755 nm. The length ( L ) of the grating was 6.4 mm. The crystal was pumped by the uncompressed output of a Ti:Sapphire regenerative amplifier, which delivered 240 ps-long FWHM stretched pulses at a repetition rate of 1 kHz. The pump pulses had a positive linear frequency chirp of 123 mrad/ps2 and a spectral bandwidth of 10 nm FWHM around a central wavelength of 800 nm. The pulses were polarized along the crystal z-axis, in order to take advantage of the highest nonlinear coefficient in KTP ( d33 ) and propagated along the x-axis. The pump polarization and energy were controlled thanks to a half-wave plate and polarizer arrangement. Part of the pump input light was reflected off a partially reflecting optics onto a reference power-meter in order to monitor the incident pump energy impinging on the sample. The pulses were focused into the 2 PPRKTP crystal using a CaF2 lens (f = 200 mm) down to a focal spot radius of 105 µm (1/e ). The forward-generated waves (residual pump and signals) were collimated by a BK7 lens (f = 200 mm) and passed through a BK7 prism to achieve spatial separation of the different wavelengths. The backward-generated waves (idlers) were separated from the pump beam using a dichroic mirror. The setup is illustrated in Fig.6.3.
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Fig.6.3 Sketch of the experimental setup. PM: power-meter, L1 & L2: lenses f= 200 mm, PR1: partial reflector, OSA: optical spectrum analyzer, iHR550: free-space spectrometer.
In the primary BWOPO process, the pump wave at 800 nm generated a forward signal s1 1125 nm and a backward idler, i12763 nm. In the secondary BWOPO, s1 generated a forward signal, s2 1898 nm and a backward idler, i22739 nm.
6.1.3 Conversion Efficiencies The pump depletion was estimated by measuring the pump energy before the sample and after the prism, with two calibrated power-meters (Melles Griot 13PEM001). The calibration for the transmission losses for the signal energies was obtained by measuring the pump energy loss when propagated through the unpoled region of the crystal. The same method was used to measure the energies of the signals waves to estimate the conversion efficiency. The energies of the idlers waves were calculated using the Manley-Rowe relations. The threshold of the first BWOPO process was reached for a pump energy of 39.8 µJ, i.e. a peak intensity of 0.64 GW/cm2. The measured BWOPO efficiency and pump depletion versus the pump energy, or pump peak intensity, is presented in Fig.6.4. The threshold for the secondary BWOPO process is reached for a s1 energy of 22 µJ, which corresponds to a pump energy of 108 µJ.
Peak intensity (GW/cm2) 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5
60
50
40
30
20 Pump depletion Efficiency s1 10 Efficiency s1 + i1 Efficiency s1 + i1 + s2 + i2 0
Pump depletion, efficiency (%) efficiency Pump depletion, 20 40 60 80 100 120 140 160 180 200 Pump energy (J) Fig.6.4 Conversion efficiencies and pump depletion for the cascaded BWOPO processes.
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At the maximum pump energy of 203 µJ the pump depletion reached 61%, s1 then had an energy of 62 µJ and the total si11 conversion efficiency was 43%. The energy of the cascaded signal s2 was 9 µJ, and the si22 conversion efficiency reached 8%. The maximum total conversion efficiency for the primary BWOPO process was achieved at a pump energy of 130 µJ, as can be seen in Fig.6.4. While increasing the pump energy, this conversion efficiency rolled off due to the depletion of s1 for the cascaded secondary BWOPO process. When the pump energy reached 150 µJ, a discrepancy appeared between the pump depletion curve and the total efficiency of both processes. This is attributed to a third cascaded BWOPO process where s2 now acts as a pump. In this case, the third process leads to the generation of a backward-propagating signal s3 at 2762 nm and a forward-propagating idler i3 at 5.9 µm. The latter is mostly absorbed by the material as the 5.9 µm radiation is outside the IR transmission window of RKTP. The effective second-order nonlinearity can be estimated from the threshold of the primary BWOPO process. In the plane wave approximation, the threshold intensity can be expressed as [4]:
0cnpisis n n IPth, 22 , (6.3) 32Ldeff
where L is the interaction length (or grating length in our case), deff is the effective second-order nonlinearity, n j is the refractive index corresponding to the different wavelengths j ( j psi,, for pump, signal and idler). The estimated effective second-order nonlinearity is 9.77 pm/V in the PPRKTP sample, which is close to the expected value (10.7 pm/V) for a perfect QPM grating in KTP [5]. A plane-wave model for a non-degenerate BWOPO [6] was used to simulate the intensity distributions of the different waves inside the crystal, and is shown in Fig.6.5.
79
pth pth (x)/I (x)/I j p I I
Fig.6.5 Normalized intensity distributions of the different waves for a pump input intensity of 2.1 GW/cm2 (120 µJ), right above the threshold for the secondary BWOPO process.
Operation close to the threshold of the second process is assumed, as well as similar beam sizes for the different waves inside the crystal. Due to the second cascaded process, the calculations of the intensity distributions for the secondary BWOPO are not entirely decoupled from the primary process. Indeed, one has to account for the depletion of s1. The normalized intensities are presented in Fig.6.5. The calculations were performed for an ideal case and lead to a reduced effective grating length for the secondary BWOPO process of 5.4 mm. Using our experimentally measured oscillation threshold and the previously estimated effective nonlinear coefficient (Eq. (6.3)), one finds a reduced grating length of 5.1 mm, which compares well to the simulations.
6.1.4 Spectral Characterization The spectral measurements of the pump and signals were performed using a fiber-coupled optical spectrum analyzer (OSA, ANDO AQ-6315A) with 0.05 nm resolution. The spectrum of the second signal was measured with a free-space imaging spectrometer (Jobin-Yvon iHR550), with a 0.14 nm resolution. The spectra of the idlers were obtained using a fiber-coupled OSA (Yokogawa AQ6376), with 0.1 nm resolution. As previously discussed, one of the unique features of BWOPO is the coherent transfer of the phase modulation of the pump to the forward wave, while the backward wave has an inherently narrow bandwidth [7,8]. As the input pump pulses are chirped, this frequency chirp is transferred to s1 and ultimately to s2. This effect can be seen in the spectral measurements performed in this experiment. Fig. 6.6 compares the spectra of the pump, s1 and s2 for different pump energies. Due to the pump pulses being linearly chirped, the pump spectrum is not uniformly depleted. It
80 starts to deplete on the short wavelength side, which corresponds to the trailing edge of the pulse as a given time delay is required to establish the distributed feedback mechanism. As explained in subsection 2.2.6, the cumulative time needed for an efficient BWOPO process is 80 ps.
Fig.6.6 Spectra of (a) the pump, (b) s1 and (c) s2 for different pump energies. The inset in (c) presents the normalized s1 spectra which show depletion of s1.
As the pump energy increases, the build-up time reduces and longer wavelengths from the pump spectrum are converted more efficiently. At the maximum pump energy of 203 µJ, the undepleted part of the pump spectrum has a FWHM length of 93 ps, which compares well to the above- mentioned time delay of 80 ps. At the pump energy of 76 µJ (primary BWOPO process only), the s1 pulse length is estimated from the depleted part of the pump spectrum, which is 141 ps-long (FWHM). Correspondingly, the FWHM bandwidth of s1 at the same pump energy is 2.8 THz, which gives a chirp rate of 125 mrad/ps2. It is very close to the calculated chirp rate of 124.5 mrad/ps2 from Eq. (2.31). At a pump energy of 131 µJ, the spectrum of s1 broadens towards the longer wavelengths, reflecting the increasing pump depletion on the longer wavelength side of the spectrum. At this pump energy, the secondary BWOPO process already comes into play and starts depleting the s1 spectrum on the lower wavelength side, transferring the phase modulation to s2. This effect is
81 even more visible at the maximum pump energy, where the obtained s1 spectral bandwidth (FWHM 2.55 THz) is narrower than the one at 76 µJ. The measured bandwidth of s2 is 0.863 THz (FWHM), which gives a FWHM pulse length of about 43 ps, according to the chirp rate transfer from s1 to s2 of 1.005 (Eq. (2.31)). At the maximum pump energy of 203 µJ, the spectra of the backward-propagating waves were measured and are presented in Fig.6.7. As expected from Fig.6.2, the wavelengths of the backward wave generated in cascaded BWOPO processes changes very little compared to the drastic wavelength variations in pump wavelengths (from 800nm, to 1125 nm, to 1898 nm).
4 i1 i2 3
2
1
Intensity (a.u.) s3
0
2720 2730 2740 2750 2760 2770 2780 Wavelength (nm)
Fig.6.7 Spectra of the backward-generated waves at a pump energy of 203 µJ.
The measured spectral bandwidths of i1 and i2 were 17 GHz and 10 GHz, respectively. From the BWOPO forward and backward chirp ratios (Eq. (2.31)), one can estimate the expected bandwidths for i1 and i2 . The theoretical chirp ratio is given by: () fgfgbgp , (6.4) bgbgpgf()
where gj represent the group velocities for the different waves. For the primary process (1,1)s i the chirp ratio is 84, while for the secondary process (2,2)s i it becomes 201. This means that the bandwidth of i2 is narrower than that of i1, which is what the measurement also tells us. By looking at the spectral widths of the signals s1 and s2 in Fig.6.6, one estimates the bandwidths of i1 and i2 to be 33 GHz and 4.3 GHz, respectively. However, it was noted that the pulse length for the generated waves of the secondary BWOPO process is expected to be around 43 ps, which gives a corresponding transform-limited bandwidth of 10 GHz, which was measured and is displayed in Fig.6.7. 82
In Fig.6.7, the spectrum of i1 consists of two separated peaks. The one with the highest amplitude is i1 from the primary BWOPO process, while the one at slightly shorter wavelength, and with a spectral bandwidth of 28 GHz , can be attributed to a third cascaded process where s2 acts as a pump. In such a QPM structure, with a 755 nm grating periodicity, a third cascaded process will lead to a forward i3 idler wave at 5.9 µm and a backward s3 signal at 2762 nm. Due to the large absorption of KTP at wavelengths longer than 4.5 µm (see Fig.3.2), it was not possible to observe i3 . However, two hints can make us think that the third process is actually happening. The expected s3 central wavelength of 2762 nm ends up within the spectral bandwidth of the first idler i1, thus the third BWOPO process is helped by cross-seeding from i1 of the primary BWOPO process. The second hint is the above-mentioned discrepancy between the total conversion efficiency sis11 2 i 2 with respect to the pump depletion above a pump energy of 140 µJ in Fig.6.4. Below this energy, the idler spectrum did not contain the s3 peak. 6.2 Backward Wave Tunability The strict momentum conservation constraints for BWOPOs gives an inherently narrowband backward-generated wave, generally in the mid-IR, even with broadband pumping, as seen in the previous section. Such a feature would be of interest for stimulated Raman scattering spectroscopy [9,10], studies of phonon-polariton modes in ferroelectrics [11,12] and other linear and nonlinear materials. Narrowband tunable sub-nanosecond pulses in the near- to mid-IR can be employed in imaging techniques such as coherent anti-Stokes Raman scattering (CARS) where longer excitation wavelengths would help to reduce the Rayleigh scattering efficiency, avoiding damage to biological samples and potentially provide higher measurement sensitivity [13,15]. The BWOPO narrowband backward wave could also be utilized for seeding mid-IR optical parametric amplifiers used in MIR LIDARs [16], or directly in MIR imaging LIDARs [17]. However, as mentioned in chapter 2, and demonstrated in the previous section, the wavelength of this narrowband wave is very insensitive to the wavelength of the pump. This makes tuning difficult. Narrowband signal and idler waves can be generated in over-constrained doubly-resonant OPO cavities where the signal and idler phases are fixed by the boundary conditions of the cavity [18]. However, stable steady-state operation in the sub-nanosecond regime with such a device would be difficult to obtain. A method for bandwidth tailoring using a fiber-based four-wave mixing (FWM) OPO cavity has been suggested in [19], where the FWM has to operate at degeneracy and at the zero GVD point of the photonic crystal fiber, and harnessing the large dispersive propagation length in the rest of the cavity in order to achieve substantial stretching. However, such a device is inherently sensitive to pump frequency fluctuations and environmental factors. Additionally, it would be very challenging to generate picosecond, narrowband and tunable pulses with energies in the µJ range with this technique. Synchronously-pumped OPOs with intra-cavity spectral filtering have been used to generate dual narrowband, tunable picosecond pulses [20,21], but with low output powers. Reaching µJ energies would require employing a separate synchronized laser to pump an additional OPA and a pulse-picker to reduce the repetition rate appropriate for the pump laser.
83
Bandwidth tailoring of infrared idler waves in OPA can be done by adjusting the chirps of the pump and the seed through the concept of double-chirp optical parametric amplification [22,23]. BWOPO is the only device which can generate a properly chirped OPA seed directly from the pump, i.e. without including any additional OPA stage that in turn has to be seeded with an external, narrowband and synchronized laser source [24]. Other techniques, including OPA seeded with supercontinuum [25], or exploiting OPG [26,27], would require additional spectral phase tailoring steps to properly adjust the chirp of the seed. Moreover, in the OPG case, the temporal coherence is limited by the amplified quantum noise which, by definition, has a random phase with respect to the pump. The BWOPO-OPA configuration as described in this section does not have any of these limitations. The narrowband and tunable pulses are automatically produced due to the physical properties of the BWOPO and dual-chirp OPA. Such pulses can be generated in the near-infrared as well as mid- infrared by choosing different QPM periodicities of the samples. Moreover, the arrangement is incomparably simpler and more robust that any of the other methods which could produce a comparable result.
6.2.1 BWOPO Tuning Issues Different tuning approaches were investigated to find a way around the inherent wavelength stability of the backward wave. A study of the chirp rates of the generated waves and experimental validation show that the backward wave tunes at -1% of the pump tuning rate [2,7]. This means that in order to be able to tune the backward wave by 1 THz, one would need to tune the pump frequency by 100 THz, which is comparable to the full Ti:Sapphire gain bandwidth. Output wavelengths from a BWOPO can be tuned by changing the sample temperature. Indeed, the refractive index is temperature-dependent and thermal expansion in the material will lead to change in the QPM periodicities. However, for BWOPOs which are environmentally-stable devices, temperature tuning only leads to fine tuning of the backward wave by 100 GHz, or so [28]. Angular tuning by implementing non-collinear interaction schemes in BWOPOs was also investigated in [28]. A tuning range of 63 GHz was demonstrated. In paper III, a simple method is presented to circumvent the tuning limitations. It heavily relies on the coherent phase transfer property of BWOPOs from the pump to the forward-generated wave. While pumping the BWOPO with broadband, strongly chirped pulses, the frequency modulation of the pump pulses (i.e. their chirp) is transferred to the forward wave (the signal in our case), but at a different central wavelength. The signal is then used to seed a conventional broadband co- propagating OPA which is pumped by the same chirped pulses as the BWOPO and thus becomes a dual-chirp OPA. The co-propagating idler generated in the OPA keeps the narrowband property of the BWOPO backward wave but becomes widely tunable by simply adjusting the time delay between the OPA pump and the seed pulses.
84
6.2.2 Experimental Setup The experimental setup is illustrated in Fig.6.8. It consists of a BWOPO followed by an OPA, both realized in PPRKTP crystals. The BWOPO crystal had a 7.3 mm-long QPM grating with a periodicity of 509 nm, while the OPA crystal was 8 mm-long with a grating period of 26.3 µm. Both uncoated samples were pumped by 240 ps-long (FWHM) stretched pulse generated by a Ti:Sapphire regenerative amplifier operating at 1 kHz. The pulses were centered at a wavelength of 800 nm, with a spectral bandwidth of 5.5 nm (FWHM). The uncompressed output of the Ti:Sapphire amplifier provided the chirped pump pulses. A small fraction of the pump power was used for BWOPO pumping, while the rest was employed for pumping of the OPA after being appropriately delayed. Delay line Pump 800 nm Signal 1.4 µm Idler 1.87 µm f=500 mm PPRKTP 26.3 µm
BWOPO Filter f=200 mm OPA
f=200 mm PPRKTP 509 nm f=200 mm
Fig.6.8 Illustration of the experimental setup including the BWOPO and the OPA stages.
The pump pulses were loosely focused with an f = 200 mm lens into the BWOPO crystal to a beam waist radius of 105 µm. The pump polarization and power were controlled by a half-wave plate and polarizer arrangement. The pump pulses were polarized along the crystal z-axis and propagated along its x-axis. The pump pulses at 800 nm generated a forward-propagating signal at 1.4 µm and a backward- propagating idler at 1.867 µm. The threshold of the BWOPO process was reached for a pump peak intensity of 0.72 GW/cm2.
6.2.3 BWOPO Characterization The BWOPO total conversion efficiency (signal + idler) reached 42% at a maximum pump energy of 196 µJ, as can be seen in Fig.6.9 (a). At this pump energy the measured BWOPO signal and idler energies were 47 µJ and 35 µJ, respectively.
85
Idler wavelength [nm]
50 1360 1380 1400 1850 1860 1870 1880 (a) Efficiency, signal (b) Efficiency, signal + idler 40 100.4 µJ 1,0 1,0 195.6 µJ
30
20 0,5 0,5
Efficiencies [%] Efficiencies 10 Normalized signal intensity [a.u.] intensity signal Normalized Normalized idler intensity [a.u.] intensity idler Normalized 0 0,0 0,0
0 50 100 150 200 1360 1380 1400 1850 1860 1870 1880 Pump energy [µJ] Signal wavelength [nm]
(c) Undepleted pump 1,0 100.4 µJ 195.6 µJ
0,5
Normalized Intensity [a.u.] 0,0
785 790 795 800 805 810 Wavelength [nm]
Fig.6.9 Signal efficiency and total conversion efficiency (a) for the BWOPO process. Normalized spectra of the BWOPO signal (b) for different pump energies and idler spectrum. Normalized BWOPO pump spectra (c) showing the pump depletion.
Figures 6.9 (b) and (c) show the spectra for the different waves. The signal bandwidth was measured to be 2.15 THz at a pump energy of 120 µJ, while the measured FWHM bandwidth of the corresponding idler was 0.34 nm (Fig. 6.9 (b)). Correcting for the spectrometer resolution of 0.1 nm (Yokogawa AQ6376), gives a spectral bandwidth of 0.325 nm, or approximately 28 GHz. As shown in Fig. 6.9 (b), at larger pump energies the BWOPO-generated signal broadens. This is a direct consequence of increasing depletion of the chirped pump pulses and the frequency transfer to the BWOPO signal. As can be seen from the pump spectra in Fig. 6.9 (c), the depletion starts on the higher frequency side which corresponds to the trailing part of the pulse. This is due to the time required to establish the distributed feedback and oscillation in the BWOPO. As the pump energy increases, this delay is reduced, and the lower frequency components of the pump spectrum are also depleted and converted to BWOPO signal and idler. Thus, the signal spectrum gradually broadens on the longer wavelength side in conjunction with the pump depletion (Fig. 6.9 (c)). On the other hand, the spectrum of the BWOPO idler remains stable and fixed regardless of the pump depletion.
86
The expected bandwidth of the backward-generated BWOPO idler, i , can be estimated from the bandwidth of the depleted pump, p , and the momentum conservation condition. Expanding i 2 in power series with respect to p to the second-order gives iip '0.5'', i p where the first- and second-order derivatives i ' and i '' of the idler frequency with respect to the pump frequency are given by:
1,sp 1, i ', 1,si 1, 2 (6.5) 2,spiis 2,'2' 2, 2, is 2, i '' . 1,si 1,
22 In Eq. (6.5), ij, k j/ and 2, jjk / are the wave-vector derivatives over the frequency for the pump, the signal and the idler, with the corresponding indices jpsi ,,, respectively. From the energy conservation, the derivative for the BWOPO forward signal wave is then si'1 '. According to the wavelengths used in this work and employing the Sellmeier 3 expansion from [29], the derivatives of the idler frequencies are: i '1.5310 and 5 i '' 1.5 10 . The second-order corrections are three orders of magnitude smaller and can be safely neglected. From the measured FWHM spectral width of the depleted pump of 1.88 THz, at the incident energy of 100.4 µJ, the expected bandwidth of the backward idler wave is 28.8 GHz, in good agreement with the experimentally measured data of 28 GHz.
6.2.4 OPA Characterization As illustrated in Fig.6.8, the BWOPO signal was used as a seed in the co-propagating OPA after filtering out the remaining pump. The OPA sample was 8 mm-long, with a QPM period of 26.3 µm, well adapted to phase-match amplification of the seed at 1.4 µm (see Fig.6.10).
Fig.6.10 Calculated phase-matching gain in an 8 mm-long PPRKTP crystal with poling period 26.3 µm. The Sellmeier equations of [29] where used in the calculations. 87
Both pump and seed pulses were focused to beam waist radii of 130 µm and collinearly superimposed in the OPA in order to take advantage of the largest OPG gain in the PPRKTP sample. The parametric gain bandwidth is about 50 THz in the 8 mm-long PPRKTP crystal for the spectral range investigated. The parametric gain scales with the inverse square root of the crystal length [30]. In order to reduce this gain bandwidth to 28 GHz (which corresponds to the BWOPO idler bandwidth) and directly generate a narrowband OPA idler, one would need to increase the sample length 310 6 times, which cannot be done in practice. The OPG without the seed reached a threshold of 120 µJ. The OPG spectrum is shown in Fig. 6.11 (a).
-30 (a) -10 (b) OPA signal -35 -15 OPA seed -40 -20 -45 -25 -50 -30 -55 -35
-60 -40
OPG spectral gain [dB] gain OPG spectral -65 -45
-70
Relative spectral amplitude [dB] -50
-75 -55 1200 1300 1400 1500 1600 1700 1360 1380 1400 1420 1440 Wavelength [nm] Wavelength [nm] Fig.6.11 Parametric super-fluorescence spectrum (a). Spectrum of the OPA signal (purple curve) compared to the OPA seed (black curve), showing the amplification process (b).
At a pump energy of 85 µJ and seed energy of 1 µJ, the OPA pump depletion reached 23.5% and the total OPA gain was 25 dB for the peak of the signal spectrum. A 25 µJ output amplified signal energy was measured, corresponding to an OPA idler energy of 19 µJ. The signal spectrum after the OPA is shown by the purple curve in Fig. 6.11 (b). The apparent wavelength shift of the OPA signal with respect to the BWOPO signal (i.e. the OPA seed) is due to the larger gain available at longer wavelengths, as seen in Fig. 6.11 (a).
6.2.5 Chirp Measurements Chirp measurements for the pump, BWOPO signal and OPA signal were performed using a home- built cross-correlation setup with a large-aperture KTP crystal as the nonlinear medium. The setup is shown in Fig.4.4, in Chapter 4. A small fraction of the Ti:Sapphire regenerative amplifier was compressed and served as the reference pulse with pulse duration of approximately 70 fs FWHM. Cross-correlation measurements allowed us to extract the chirps of the pump, the BWOPO signal and the OPA signal (Fig.6.12).
88
3695
BWOPO signal Pump 3690 OPA signal
3685 [rad/ps]
3680 Rescaled Rescaled 3675
0 100 200 300 Delay [ps] Fig.6.12 Cross-correlation measurements for the pump pulses, the BWOPO signal and the amplified signal after the OPA stage.
Both the pump and the BWOPO signal have positive linear chirp of similar magnitude, approximately 59 mrad/ps2. This was expected as the phase information is coherently transferred from the pump to the forward wave in the BWOPO process. We also note that the BWOPO signal is twice as short as the pump pulse with a FWHM of 120 ps at maximum pump energy. This is even more visible in Fig.6.13, where the corresponding spectrograms or XFROG traces of the BWOPO pump and signal are plotted. These chirp measurements indicate that adjusting the time delay ( t ) between the pump and the seed (i.e. the BWOPO signal) before the broadband amplifier will allow tuning of the OPA idler central wavelength, while preserving the narrow bandwidth of the idler. The spectral width would still be defined by the difference in chirp rates between the pump and the seed [22]: (,tt ) t ( t t ) ipsps (6.6) psst,
where p and s are the chirp rates of the pump and the seed, respectively. The BWOPO frequency translation properties ensure that the linear chirp rates of the OPA seed and pump are essentially equal (see Fig. 6.12), which leads to the approximation made in Eq. (6.6). To verify that the OPA pumped with the chirped pump pulses does not impart additional chirp, we also measured the chirp rate of the amplified signal. As presented in Fig. 6.12, the measured chirp rate of the OPA signal is found to be 59 mrad/ps2, i.e. the same as that of the seed before amplification. The broadband OPA hence conserves the chirp of the seed and therefore allows tuning of the idler according to Eq. (6.6) while preserving the narrow bandwidth. In fact, the chirp rates of the pump and the seed, generated by the BWOPO would be slightly different. Specifically, a chirp rate ratio sp/1.0153 is estimated to the first order from
89 equations in [2,7]. This difference in the chirp rates determines the OPA idler bandwidth. First, the
OPA idler bandwidth will have a constant component isps1 , given the variation in the frequency difference between the pump and the seed. Here s is the seed pulse length. Second, this idler frequency variation will be modified during the OPA process due to the group velocity mismatch between the pump and the seed. As both the pump and the seed are located in the normal dispersion region of PPRKTP, the group velocity mismatch will result in an OPA idler bandwidth increase. The total idler bandwidth can then be estimated by: 1 iOPA,1,p1,s s s p s L , (6.7) 2 where L is the OPA crystal length. For the parameters of our experiment, this bandwidth will be about 32 GHz, i.e. only somewhat larger than the bandwidth of the backward wave in the BWOPO.
Fig.6.13 XFROG traces of (a) the pump and (b) the signal of the BWOPO stage. The color scale is linear and corresponds to 60% of the highest values in the normalized scale of the measured spectra.
6.2.6 Tunable BWOPO Idler As shown in the experimental setup in Fig. 6.8, the OPA pump pulses go through a variable delay line before reaching the OPA stage. Adjusting the temporal delay between the pump and the seed pulses with the delay line allows tuning of the central wavelength of the OPA idler. This method provides a much larger tuning range than the one achievable by tuning the central wavelength of
90 the BWOPO pump. For instance, in Fig. 6.14 we show a tuning range of 2.7 THz (about 28 nm) obtained by changing the delay by 137 ps. In the BWOPO-OPA configuration, the limit of the tuning range comes from the bandwidth of the pump itself. In order to achieve the same tuning range in BWOPO, one would need to tune the BWOPO pump by approximately 177 THz, as given by the relation:
gi gs pigp , (6.8) gi gs gp
where gj are the group velocities for j psi,,, respectively.
163 (a) 1022 (b) MOPOBWOPO idler idler 1020 MOPO idler 1,0 OPA idler 1018 162 1016
1014
1012 161
0,5 [rad/ps]
1010
1008 [THz] Frequency Normalized intensity [a.u.] 1006 160
1004 0,0 1840 1845 1850 1855 1860 1865 1870 1875 1880 -80 -60 -40 -20 0 20 40 60 80 OPA Idler [nm] Relative delay pump/signal [ps]
Fig.6.14 Tuning of the idler central wavelength (a) for different time delays. The BWOPO idler spectrum is shown in dashed red line. Angular frequency of the tunable OPA idler and the BWOPO idler (b) for different pump-seed delays, where 0 delay corresponds to the middle of the total achieved tuning range.
The OPA idler spectra in Fig. 6.14 were measured with a Jobin-Yvon iHR550 spectrometer with a resolution of 0.25 nm, except for the peak at largest delay of 117 ps (leftmost peak in Fig. 6.14 (a)) where the resolution was decreased to 0.5 nm owing to the lower average power. This was caused by the decreasing temporal overlap between the pump and the seed. The median value of the OPA idler bandwidth throughout the tuning range was 38 GHz, in line with the expectations of Eq. (6.7). The measured tunability of the idler peak position is shown in Fig. 6.14 (b) with red data points. A small fraction of the BWOPO backward-propagating idler was sent to the OPA crystal and amplified so that the BWOPO idler frequency could be measured simultaneously after the OPA sample (black squares in Fig. 6.14 (b)). In this way, two narrowband near-IR pulses are generated, with a frequency separation that can be tuned over 2.7 THz.
6.2.7 Conclusions Exploiting counter- and co-propagating parametric processes as well as a double-chirped OPA configuration allowed to generate a narrowband infrared radiation tunable over 2.7 THz with 19 µJ
91 of energy. This method is easily scalable with the OPA stage and the tuning range is now limited by the bandwidth of the pump itself. The bandwidth of the narrowband idler can be varied down to the transform limit by adjusting the chirp of the OPA seed so that it matches exactly that of the OPA pump. This can be done by inserting a dispersive medium on the seed beam path if required. As described in this section, an interesting feature of the setup is the efficient phase transfer from the pump to the OPA signal, via the BWOPO counter-propagating interaction. As the BWOPO, or even the OPA signal, show similar chirp compared to the pump, one should be able to compress the signal in order to get an easily power-scalable femtosecond broadband source at 1.4 µm. This is the topic of the next section. 6.3 Coherent Phase Transfer for Femtosecond Near-IR Pulse Compression The unique counter-propagating interaction in a BWOPO provides a frequency modulation transfer feature from the pump to the forward-generated wave. As mentioned in the previous sections, the forward-generated wave has a frequency response which closely follows that of the pump wave. This can be of great interest for applications in telecommunication and microwave technology involving frequency translators [31,32], for which the signal encoded in the pump can be faithfully translated to another spectral range simply by launching it in a single pass through a BWOPO. This is in large contrast to the spectral features of standard OPOs where the momentum conservation condition is much less restrictive and typically results in output signal and idler spectra that are much broader than that of the pump, in addition to a poor temporal coherence. Moreover, optical frequency translators are mostly realized in waveguide structures with lower achievable powers. Single-pass BWOPOs can easily provide µJ energies for the near-IR frequency translated bandwidths (see papers I and III) and the accessible wavelength range is much broader thanks to the tuning capabilities of the forward wave with respect to the pump. Additionally, BWOPOs can be used to generate short and tunable broadband pulses in the IR, useful for time-resolved spectroscopy, stimulated Raman spectroscopy, remote sensing and bio- photonics, to mention a few applications [33-35]. Typical biological applications require short (< 10 ps), stable high-power pulses with reasonable repetition rates (kHz range) and high pulse-to- pulse stability [36,37]. Such light sources are primarily based on OPOs to provide the necessary tunability where the spectral coverage is ultimately set by the phase-matching bandwidth of the nonlinear crystal [38]. An efficient single-pass nonlinear device, such as a BWOPO, would be a simpler solution and provide means for new applications. In paper IV, we investigate this interesting frequency transfer feature of a BWOPO and prove that a coherent phase transfer is occurring from the pump to the forward-generated signal. This is demonstrated by using strongly chirped pump pulses at 800 nm and studying the resulting chirp for the BWOPO signal. Additionally, we built a single-grating compressor in which we compressed the signal from 150 ps (FWHM) down to 1.3 ps. Current setup limitations for sub-ps pulse compression are also investigated. Indeed, a nonlinear temporal phase, due to group dispersion
92 mismatch, emerged in the counter-propagating interaction, leading to a limited linear frequency modulation transfer bandwidth of 220 GHz when going from 800 nm to 1.4 µm.
6.3.1 Experimental Setup In order to study the phase-transfer property and its limitations in a BWOPO process, the sample with a QPM periodicity of 509 nm, from section 6.2.2, was pumped by strongly chirped pulses and generated a forward signal which then was passed through a single-grating compressor. A home- built autocorrelation setup identical to the one presented in Chapter 4 (see Fig.4.3) was used to characterize the resulting compressed signal pulses. The full experimental setup is shown in Fig.6.15.
Fig.6.15 Illustration of the experimental setup. The BWOPO crystal, pumped by 800 nm pulses, generates a forward signal at 1.4 µm and a backward idler at 1.87 µm. The remaining pump is filtered out after the crystal by a RG1000 long-pass filter. The signal is then sent to a single-grating compressor which includes a corner mirror and a periscope to provide four passes onto the grating. The compressed output pulse is further characterized with the home-made auto-correlator.
Once again, the uncompressed output of the Ti:Sapphire regenerative amplifier operating at 1 kHz is used as a pump. The pulses are centered at a wavelength of 800 nm, with a spectral bandwidth of 5.5 nm (FWHM) and a duration of 240 ps (FWHM). The BWOPO crystal is the same as in section 6.2 (PPRKTP with 509 nm QPM period and a 7.3 mm-long grating). In the BWOPO process, a forward signal at 1.4 µm and a backward idler at 1.87 µm are generated. Focusing geometry, pump pulses polarization, conversion efficiency and estimated intensity threshold are the same as described in section 6.2. Given the chirp measurements presented in Fig.6.12, and given the accuracy of the measurement, the pump and the BWOPO signal have the same positive linear chirp of 59 mrad/ps2. The estimated magnitude of the GDD was 16.9 106 fs2. However, this measurement does not discriminate the possible presence of a third-order phase in the BWOPO signal. In order to further
93 investigate the contribution of the GDD mismatch, a single-grating Treacy compressor [39] was built, with four passes on the reflection grating, as shown in Fig.6.15. The compressor consisted of a corner mirror to retro-reflect the angularly-dispersed beam onto a gold-coated reflection grating (Spectrogon AB, 900 lines/mm, 30 110 16 mm) and a periscope which reversed the beam path but at a different height on the grating. The corner mirror was placed on a translation stage in order to carefully adjust the amount of negative GDD introduced in the compressor. This scheme is free of spatial chirp and pulse front tilt, and the length of the compressor was reduced by half compared to a conventional design based on a grating pair [40].
6.3.2 Results The spectrum of the BWOPO signal was measured before and after the compressor and is presented in Fig.6.16. The entire spectral content is preserved through the compressor. The measured 2.8 THz bandwidth of the pulse corresponds to a Fourier transform limit of 157 fs (FWHM), thus it should be possible to obtain sub-ps pulses after the compressor if the dispersion can be fully compensated for. The input power to the compressor was 30 mW and the output power was approximately 6 mW, which corresponds to a pulse energy of 6 µJ at 1 kHz. The power loss is primarily ascribed to an unoptimized grating reflectivity. The grating had a maximum efficiency of 93% at a narrow reflection band around 1350 nm, slightly off our signal wavelength.
1,0 Before After
0,5 Normalized spectrum (linear scale) spectrum Normalized
0,0 1360 1380 1400 1420 1440 Wavelength (nm) Fig.6.16 Spectrum of the BWOPO signal before (black) and after (red) the compressor.
In order to optimize the GDD compensation for a double-pass configuration and diffraction order m the compressor length must be correctly chosen according to [41,42]: 3 m lcomp GDD 22 3 , (6.9) cdcos d
94 where d is the diffraction angle measured from the normal to the grating surface, lcomp is the compressor length, is the wavelength, d is the groove spacing on the grating, and c is the speed of light in vacuum. In our case, the angle of incidence on the grating was 34.5 degrees, giving a total compressor length of about 800 mm, which we folded in two for the single-grating design. The compressor adds third-order dispersion (TOD) according to the following formula [42]: 3 //sindd TOD GDD 1, i (6.10) 2c 2 1/sind i which gives a magnitude of the TOD of 0.1 ps3. According to [43], high-quality pulse compression is obtained in a grating compressor when the following criterion is fulfilled:
1 3 TOD , (6.11) 6 where is the pulse bandwidth. Given the results of Eq. (6.10), the above expression has a value of 155 rad, which refutes the criterion and most likely limits the compression in our setup. The output of the compressor was passed through the home-made auto-correlator based on a large aperture (5 mm) PPRKTP and presented in Chapter 4, Fig.4.3. Figure 6.17 shows the autocorrelation traces of (a) the FWHM of the BWOPO signal as measured before the compressor, and (b) the FWHM of the compressed BWOPO signal.
80 (b)(a) (b) Data points 70 1,0 Lorentzian fit
60
50
0,5 40
30 Voltage (mV) 20
Normalized voltage (mV) voltage Normalized 0,0 10 Data points Gaussian fitting 0 0 50 100 150 0246 Delay (ps) Delay (ps) Fig.6.17 Autocorrelation traces of (a) the BWOPO signal with a Gaussian amplitude fit of FWHM 150 ps and (b) the shortest compressed pulse with a Lorentzian fit of FWHM 1.3ps.
Before compression, the signal was 150 ps-long (Gaussian fit FWHM). This corresponds to about half the duration of the BWOPO pump pulses, matching the wide spectrum of Fig.6.16. In contrast, the shortest compressed BWOPO signal had an autocorrelation trace FWHM of 1.3 ps, with the best fit obtained for a Lorentzian shape. This would correspond to a pulse length of 650 fs (see Chapter 4, Table 4.1) and a compressed bandwidth of about 218 GHz. This means that the added
95 cubic phase in the grating compressor forms an Airy beam and limits the maximum compression achievable. This auto-correlation trace was obtained for a compressor length of 39.7 cm, corresponding to a compression of almost 115 times. The compressed output pulse (1.3 ps, 6 mW) would translate to a peak power of 4.6 MW. By varying the compressor length by 0.5 cm, the autocorrelation trace FWHM could be tuned from 1.3 ps to 1.7 ps, as can be seen in Fig.6.18.
Fig.6.18 Evolution of the FWHM of the autocorrelation trace of the compressed BWOPO signal for different compressor lengths.
When a BWOPO is pumped by broadband chirped pump pulses, the group velocity mismatch (GVM) of the interacting waves will result in slightly different chirp rate for the forward-generated (signal) wave, i.e. only the second-order temporal and spectral phases will be affected without compromising the compressibility of the wave. On the other hand, the group dispersion mismatch (GDM) will induce higher order phase distortions. For a purely linearly chirped pump, the GDM contributes with a third-order temporal (and spectral) phase. The temporal phases of the BWOPO signal and idler are estimated analytically to the third-order using a power series expansion: t ' 2 (')tt 0.5222 tdt , (6.12) fb,,0, fb p fb t p p fb , t p p fb , t p 0
where p and t denote derivatives over the pump frequency and time, respectively. The calculations assumed that the pump only contains linear frequency chirp, meaning that the last term in the kernel is reduced to zero. The appropriate derivatives for Eq. (6.12) can be obtained from the momentum conservation. In Fig.6.19, the calculated phases for the pump and the BWOPO signal, as well as their phase difference are shown. The phase difference has primarily a quadratic time dependence, which indicates a slightly larger chirp rate of 59.7 mrad/ps2 for the BWOPO signal. The third-order temporal phase is shown in Fig. 6.19 (b) and can be readily visualized by subtracting the second- order term in Eq. (6.12), related to the GVM, from the full signal phase.
96
3000 (a) BWOPO pump BWOPO forward wave Phase difference 0
2000
-20 1000 Phases (rad) Phases
0 -40 Pump-forward phase difference (rad) difference phase Pump-forward
0 100 200 300 Time (ps)
0,04 (b) Full phase-GVM phase Cubic polynomial fit Quadratic polynomial fit 0,03
0,02
0,01 Full phase-GVM (rad) phase Full phase-GVM 0,00
0 50 100 150 200 250 300 350 Time (ps) Fig.6.19 Time-dependence of the pump (red) and the BWOPO signal (black) phases (a), and phase difference between the pump and the signal (blue) as a function of time. (b) Calculated third-order phase (full phase minus GVM phase) of the signal (black), cubic polynomial fit (red circles) and quadratic polynomial fit (blue). The signal acquires an excess nonlinear temporal phase which fits very well with a cubic polynomial.
The value of the third-order derivative of the temporal phase is 8.24 106 mrad/ps3, i.e. five orders of magnitude smaller than the second-order derivative. The calculations show that the total accumulated third-order phase would be below 40 mrad. For the chirp rates obtained in the experiment, the fidelity of the chirp transfer from the pump to the forward wave will eventually be limited by the accumulated third-order phase. In the case of a pump chirp rate of 59 mrad/ps2, a third-order phase of will be accumulated over a pump length of 1.32 ns, corresponding to a pump bandwidth of 77.7 THz. The BWOPO can only accommodate a limited pump chirp rate. If
97 too large, the backward idler wave generated at the end of the crystal will encounter the pump frequencies at the beginning of the sample, which would then require generation of the signal at frequencies detuned from the phase-matching. From this picture, one can estimate the bandwidth of the idler as: bb5.57 / (2L 1, ). This can be considered as a stationary bandwidth of the BWOPO idler and, for the specific BWOPO employed in this work, the bandwidth would be 18 GHz. This bandwidth, associated with the backward wave tunability [2], provides the limiting value for the pump chirp rate, where the conversion efficiency would decrease by half, given by:
pfb5.57 1, 1, 22 . (6.13) tLmax 1,bp 1, 1, f
For the BWOPO used here, the maximum chirp rate can be 149 mrad/ps2, i.e. 2.5 times larger than our actual pump chirp rate.
6.3.3 Conclusions In conclusion, a linearly chirped, broadband BWOPO signal at 1.4 µm was compressed 115 times down to 1.3 ps in a simple single-grating compressor. This demonstrates the excellent coherent phase transfer of approximately 220 GHz bandwidth for the BWOPO process, converting the 800 nm pump to the 1.4 μm (forward) signal wave. The BWOPO contributes to much less TOD on the forward-generated pulse than the compressor itself. Furthermore, the grating compressor adds a non-negligible third-order phase which limits the achievable compression. To overcome this a tailored compressor design would be required. A chirped volume Bragg grating could be employed to reach full dispersion compensation at 1.4 µm [44] or the TOD could be tailored beforehand through phase-modulation of the pump pulses in an acousto-optic programmable dispersive filter [45]. The BWOPO bandwidth transfer was limited by the effects of GDM, contributing to additional nonlinear phase for the BWOPO forward signal. This limitation can be overcome by designing an aperiodic QPM structure, or by optimizing the length of the PPRKTP crystal. Additionally, the setup could be energy-scalable by seeding a broadband optical parametric amplifier with the BWOPO signal, ensuring chirp rate conservation and compressibility of the near- IR wave. 6.4 Homogeneity Check of 509 nm QPM Samples In order to demonstrate the homogeneity of the grating structures in the periodically poled samples, uniformity scans were performed on a batch of crystals with a QPM period of 509 nm. The 800 nm pump beam, issued from the same laser system as the two previous sections, was focused to a beam waist radius of 125 µm into each sample with an input power of 121 mW. The power of the generated forward wave, i.e. signal at 1.4 µm was monitored at the output of the crystals. Then the end facets of the crystals were sampled every 300 µm, using a crystal holder mounted on a translation stage. The method of scanning is schematically shown in Fig.6.20.
98
c
a b
c‐
x
y
Fig.6.20 Scanning setup for characterizing the homogeneity of the 509 nm samples. The red circles represent the scanning input beam at 800 nm, (a,b,c) are the crystallographic axis, and (x,y) are the translation stage axis.
The output signal powers are normalized to the maximum measured power and the resulting scans in the (x,y) plane for the six investigated PPRKTP crystals are shown in Fig.6.21.
Fig.6.21 Homogeneity scans for six PPRKTP crystal with a QPM periodicity of 509 nm.
One notices that most of the samples present uniform areas of nonlinear generation for the BWOPO process (crystals II, III, V and VI), meaning that there is no “lucky spot” but rather wider regions where the pump can be efficiently converted to signal and idler waves.
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6.5 Conclusions This chapter summarized the work on the properties of backward-wave parametric interaction, which is the basis for BWOPO operation. In this device, a narrowband idler and a signal whose chirp is precisely controlled by managing that of the pump are generated. In a conventional co- propagating parametric geometry, both signal and idler experience symmetric wide-band gain. Thus, the gain bandwidth and the temporal coherence of the signal and idler generated in unseeded co-propagating parametric down-conversion schemes are lower than those of the pump. On the contrary, an oscillator based on counter-propagating interaction generates a backward wave with temporal coherence two orders of magnitude higher than that of the pump. Indeed, in an OPO with co-propagating geometry, the phase difference between the pump and the signal, or the pump and the idler can drift without bounds [46]. This means that the signal and idler generated in background-seeded singly resonant OPOs (SROPOs) are in general incoherent and with spectral widths larger than that of the pump wave. Chirp manipulation is then only possible in a seeded co- propagating process where phases of the pump and seed are carefully controlled. In this work, we took advantage of these particular spectral properties to perform diverse studies on BWOPOs. The high quality and homogeneity of the QPM structures with sub-µm periodicities was demonstrated by the observation of at least three cascaded BWOPO processes with high efficiencies and low thresholds. In co-propagating geometries, such cascaded interaction would instead require quite complex cavity arrangements. Moreover, when coupled to the concept of double-chirp amplification in an OPA [20,21], a BWOPO can provide a tunable idler, potentially with adjustable bandwidth as well as a fixed- wavelength idler directly from the BWOPO. Therefore, a BWOPO-OPA system gives access to two narrowband infrared pulses with tunable frequency separation. The BWOPO coherent phase transfer was also studied by investigating the limitations of the bandwidth transfer due to GDM. The effects of GDM contribute to additional nonlinear phase for the BWOPO signal. The forward-generated signal was compressed using a single-grating compressor from 150 ps (FWHM) down to 1.3 ps (FHWM), demonstrating good frequency modulation transfer from the pump to the signal. The compression was limited by the inherent TOD introduced by the grating compressor. The realized near-IR source at 1.4 µm, with 1.3 ps pulse duration and µJ pulse energy, can be used for advanced microscopy and spectroscopy where it can provide high spectral resolution. Such reliable frequency transfer to longer wavelengths is also highly interesting for telecommunications devices, such as frequency translators.
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7. Conclusions and Outlook
In this thesis, nonlinear processes with ultra-short pulses, in the picosecond to femtosecond range, were studied in domain-engineered KTiOPO4 (KTP) in the infrared part of the spectrum. As ultra- short pulses with longer wavelengths are increasingly important tools for a broad range of applications, accent was put on spectral management, i.e. bandwidth tailoring, of such optical pulses. The choice of KTP as a nonlinear medium was evident due to its favorable properties for mid-IR pulse generation and its ability to be periodically poled with fine-pitch and high aspect ratio structures. Domain-engineered KTP was used to tailor the effective Kerr nonlinear response in different parts of the spectrum. Indeed, choosing an appropriate poling period allowed us to design the sign and magnitude of the cascaded effective Kerr nonlinearity in the two dispersion regimes. We succeeded in designing a crystal structure for which positive dispersion would be associated with negative effective Kerr and negative dispersion with positive effective Kerr response. This configuration is the key to achieve simultaneous spectral broadening and pulse self-compression in both dispersion regimes, leading to soliton-like pulse propagation. A numerical model based on a single nonlinear envelope equation was developed to account for all dispersion orders of the material, all second-order nonlinearities as well as cubic nonlinearities due to the large spectra considered. Domain-engineering of the nonlinear material is conveniently implemented in the model through the second-order susceptibility term. In this way, different spatial modulations of the second-order nonlinearity can be tested. Supercontinuum generation and pulse self-compression was subsequently investigated for a pump wavelength in the near-IR (approximately 1.5 µm) with 100 fs pulse duration (FWHM). Theoretical calculations in periodically poled Rb-doped KTP (RKTP) with periodicity 36 µm showed a potential for obtaining an octave-exceeding spectrum extending to 3 µm while providing a reasonably clean output temporal profile. The 100 fs Gaussian pump pulses were compressed down to 8 fs (FWHM) in the crystal through single-pass interaction. The expected spectral broadening and pulse compression mechanisms were confirmed by experimentally demonstrated supercontinuum generation in periodically poled RKTP (PPRKTP). Pump pulses at a central wavelength of 1.52 µm with 128 fs duration were used in an 11 mm-long PPRKTP sample. A wide spectrum spanning from 1.1 µm to 2.7 µm was measured and output pulses as short as 18.6 fs were characterized employing a sum-frequency generation cross-correlation frequency-resolved optical gating setup. Those experimental results were compared to numerical simulations which used the “real” experimental pump characteristics as inputs and showed a very good agreement in terms of generated supercontinuum span and compressed output pulse durations. Consequently, these supercontinuum pulses could be considered for further experimental work, e.g., as a seed for optical parametric chirped pulse amplifiers. Moreover, the approach presented in the thesis is energy scalable for periodically poled ferroelectrics. Using large-aperture samples, which are readily available, and increasing the pump beam size would allow to scale up the generated supercontinuum energy even further.
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Such motivating preliminary results also open the path to interesting studies for mid-IR pulse generation in single pass domain-engineered nonlinear devices. In particular, the developed numerical model is versatile and allows to test different sample designs and nonlinear materials. A promising material is the KTP isomorph KTiOAsO4 (KTA), which presents a flatter group velocity dispersion profile and a broader transmission window in the mid-IR, features that are of interest for octave-spanning supercontinuum generation. Some numerical results were presented for different ferroelectric materials commonly used in nonlinear optics (KTP, KTA, lithium niobate and lithium tantalate) which showed comparable behaviors. A future work could be to compare these materials performances experimentally. Another direction of research would be the design of the spatial modulation of the second-order nonlinearity. In this work, focus was mainly put on periodic poling. However, chirped, apodized grating structures could be of great interest as they can allow larger phase-matched bandwidths. Apodization techniques are known to remove spectral ripples and obtain cleaner output pulses by bringing the variations of the effective nonlinearity smoothly to zero at the sample edges. Numerical calculations showed promising results with KTP and experimental validation would be a next step towards achieving broader, smoother spectra in the mid-IR, with almost oscillation-free output temporal profiles. Targeting even longer wavelength ranges in the IR can also be considered. KTP is limited to about 4.5 - 5 µm, due to multi-phonon absorption. One way to overcome this limitation is to employ a semiconductor nonlinear crystal, GaP, which has a large nonlinearity in the IR (approximately 40 pm/V) and a very wide transparency window in the IR, up to 20 µm. In addition, the GaP nonlinearity can be engineered through the technique of orientation-patterning. Thus, cascaded second-order interactions can be implemented with appropriate “periods” to generate supercontinuum with associated pulse compression in the range 2-6 µm with a pump wavelength around 2.4 µm in the positive dispersion regime (the zero-dispersion wavelength for GaP is approximately 4.7 µm). Cascaded second-order interactions through intra-pulse DFG could also be considered to target longer spectral ranges around 8 µm, with sub-20 fs generated output pulses. A second part of the thesis was dedicated to experiments with special types of oscillators called backward-wave optical parametric oscillators (BWOPOs), which have the particularity of generating counter-propagating signal and idler waves, in our case in the near- and mid-IR, within a single-pass. The different studies performed dealt with short pulses in the picosecond range generated by BWOPOs and the opportunities that such device offers regarding bandwidth tailoring. BWOPOs were realized thanks to state-of-the-art domain-engineered Rb-doped KTP crystals, with sub-µm periodicities. The thesis work highlighted the unique feature of coherent phase transfer from the pump wave to the forward-generated wave in BWOPOs. This property allowed to demonstrate a 2.7 THz tuning range for a 28 GHz narrowband wave at a wavelength of 1.87 µm via a double-chirp BWOPO-OPA configuration. Moreover, coherent phase transfer was investigated to generate ultra-short pulses at 1.4 µm with 1.3 ps duration and µJ energy. Additional investigations with BWOPOs also showed how efficient those devices are. Indeed, cascaded BWOPO processes were demonstrated in a single pass thanks to the low threshold of the BWOPO interaction in high-quality poled devices. Such cascaded operation would require complicated experimental setups if realized in common optical parametric oscillators. 105
It has been suggested that degenerate BWOPO designs could be used to generate coherent biphoton sources for quantum applications. In order to investigate this path, PPRKTP crystals with QPM period of 420 nm can be realized. While pumped with 800 nm pulses, those BWOPO crystals would generate a broad forward wave at 1.6 µm and a corresponding narrowband backward wave also at 1.6 µm. Future work will focus on preliminary findings and characterization for such a degenerate BWOPO.
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