Ultrafast near- and mid-infrared laser
sources for linear and nonlinear
spectroscopy
Von der Fakult¨at Mathematik und Physik der Universit¨at Stuttgart zur Erlangung der W ¨urde eines Doktors der
Naturwissenschaften (Dr. rer. nat.) genehmigte Abhandlung
vorgelegt von Tobias Steinle aus Bad Saulgau
Hauptberichter: Prof. Dr. Harald Giessen Mitberichter: Prof. Dr. Peter Michler Mitberichter: Prof. Dr. Karsten Buse Tag der m ¨undlichen Pr ¨ufung: 17.11.2016
4. Physikalisches Institut der Universit¨at Stuttgart November 2016
A B S T R A C T
This work aims for the development of novel infrared laser systems tar-geting applications in laser spectroscopy. The method of optical para-metric frequency conversion of a near-infrared femtosecond laser is em-ployed to generate broadly tunable radiation in nonlinear crystals. The parametric light sources that are investigated in this thesis are distin-guished by their extremely high single-pass gain, which enables frequency conversion with very low or even without feedback.
This work demonstrates, to the best of our knowledge, the most broad-band oscillator-pumped mid-infrared laser system in terms of tuning range (1.33 − 20 µm) that is currently available. It further features record-level output power of several tens of mW in the deep mid-infrared as well as excellent stability. The latter is a result of the fiber-feedback optical parametric oscillator that represents the core unit of the frequency con-version system. We show that this laser system can be combined with a FOURIER-transform infrared spectrometer to allow very sensitive spectro-scopic investigation of tiny effects. In direct comparison it outperforms synchrotron radiation sources under certain conditions, which shows the great potential not only regarding the increased measurement sensitivity, reduced measurement time, and permanent availability, but also regard-ing operatregard-ing and energy cost.
In another approach we investigate “cw-seeding”, which is a very sim-ple way of frequency conversion. While most systems require optical feedback, the conversion process can also be initiated and stabilized by an external continuous-wave laser. This reduces complexity of the sys-tem, improves robustness, and minimizes cost. It is demonstrated at the example of stimulated Raman scattering microscopy that this source shows equal performance compared to conventional methods when ap-plied for biological and biomedical imaging.
Under certain conditions the above mentioned fiber-feedback optical parametric oscillators can be manipulated to show dynamics that are known to only show up in nonlinear systems with self-feedback. Com-plex phenomena such as period doubling, limit cycles, and chaos are ob-served. These phenomena are investigated in this work for the first time in an optical parametric oscillator. Moreover, it is demonstrated that such systems can be employed for all-optical modulation to perform ultra-low
noise spectroscopy with minimum use of electronics. Finally, the system is used for intrinsically unbiased hardware (“true”) random number gen-eration.
Z U S A M M E N FA S S U N G
Ziel dieser Arbeit ist die Entwicklung von neuartigen Infrarotlasersyste-men f ¨ur die Anwendung in der Laserspektroskopie. Dazu wird optisch-parametrische Frequenzkonversion eingesetzt, um nahinfrarote Laser-strahlung aus einem gepulsten Femtosekundenlaser mittels nichtlinearer Kristalle in breitbandig durchstimmbare Strahlung zu konvertieren. Ein besonderer Fokus dieser Arbeit liegt auf der Untersuchung von para-metrischen Lichtquellen mit sehr hoher Einzeldurchlaufverst¨arkung, wo-durch Frequenzkonversion mit sehr geringer oder gar ganz ohne R ¨uck-kopplung m ¨oglich wird.
In dieser Arbeit wird das, nach unserem besten Wissen, derzeit bre-itbandigste (1.33 − 20 µm) oszillatorgepumpte Mittelinfrarotlasersystem demonstriert, welches zudem Rekordleistungswerte von mehreren zehn Milliwatt im tiefen mittleren Infrarot erreicht. Ein auszeichnendes Merk-mal dieser Quelle ist ihre exzellente Stabilit¨at, welche auf den Einsatz eines sogenannten optisch-parametrischen Oszillators mit Faserr ¨uckkop-plung als Kerneinheit f ¨ur die Frequenzkonversion zur ¨uckzuf ¨uhren ist. Der optisch-parametrische Oszillator mit Faserr ¨uckkopplung wird zu-n¨achst mit verschiedenen Pumplasersystemen kombiniert, wobei Pump-pulsdauern von 100 fs bis zu 1 ps erfolgreich getestet werden. In einem weiteren Experiment wird Leistungsausbeute der Frequenzkonversion durch zweistufige Frequenzkonversion wesentlich gesteigert, bei gleich-zeitiger Verbesserung der Rausch- und Drifteigenschaften. Es wird ge-zeigt, dass dieses System unter bestimmten Voraussetzungen in Kom-bination mit einem FOURIER-Transform Infrarotspektrometer pr¨azisere Messungen von sehr kleinen Effekten erm ¨oglicht, als Synchrotronstrah-lungsquellen. Neben dem wissenschaftlichen Gewinn durch verk ¨urzte Messzeiten, erh ¨ohte Messgenauigkeit und permanente Laborverf ¨ugbar-keit, k ¨onnen so massiv Betriebs- und Energiekosten eingespart werden.
Ein weiteres Konzept, das sogenannte
”cw-seeding“, stellt eine sehr einfache Alternative der Frequenzkonversion dar. W¨ahrend die meisten Systeme auf optische R ¨uckkopplung angewiesen sind, wird hier der Kon-versionsprozess durch einen Dauerstrichlaser initiiert. Insgesamt sind nur sehr wenige Komponenten notwendig, wodurch das System robust, leicht zu bedienen und g ¨unstig wird. Dieser einfache Ansatz wird hin-sichtlich seiner Effizienz und Rauscheigenschaften mit
trischen Oszillatoren verglichen, Hier ergibt sich eine ¨uberlegene Lang-zeitstabilit¨at, w¨ahrend die Rauscheigenschaften auf der MHz-Zeitskala deutlich schlechter ausfallen. Es wird nachfolgend am Beispiel der stim-ulierten RAMAN-Streuung gezeigt, dass die Verwendung dieser Quelle f ¨ur die biologische und biomedizinische Bildgebung im Vergleich zu her-k ¨ommlichen Ans¨atzen her-keine Nachteile mit sich bringt. Dazu werden sowohl Standardcharakterisierungsproben, im Speziellen wenige Mikrom-eter messende Polymerkugeln, untersucht, sowie in einem weiteren Ex-periment auch stark streuende biologische Proben.
Unter bestimmten Bedingungen ist es m ¨oglich, die oben erw¨ahnten optisch-parametrischen Oszillatoren mit Faserr ¨uckkopplung so zu ma-nipulieren, dass sie Dynamiken zeigen, die nur in selbstr ¨uckgekoppelten nichtlinearen Systemen auftreten. Dabei ergeben sich komplexe Ph¨ano-mene wie Periodenverdopplung, Grenzzyklen und Chaos. Diese werden in der vorliegenden Arbeit erstmals in einem optisch-parametrischen Os-zillator gezeigt und untersucht. Dabei zeigt sich, dass diese Erschein-ungsformen nichtlinearen Verhaltens kartiert werden k ¨onnen, um bes-timmte Zust¨ande, so genannte ,,Attraktoren”, gezielt pr¨aparieren zu k ¨on-nen. F ¨ur eine Reihe dieser Attraktoren wird das Einschwingverhalten untersucht, wobei sehr unterschiedliche Zeitspannen zwischen weniger als 20 bis hin zu ¨uber 100 Uml¨aufen festgestellt werden. Des Weiteren wird demonstriert, dass ein solches System als rein optischer Modulator eingesetzt werden kann, um sehr rauscharme Spektroskopie mit mini-malem Einsatz von Elektronik zu betreiben. Zun¨achst wird gezeigt, dass sich die erreichbare Modulationstiefe mithilfe von Frequenzverdopplung auf das Niveau konventioneller Modulatoren bringen l¨asst.
Anschließend wird mittels stimulierter RAMAN-Streuung ein Vergle-ich zwischen dem rein optischen und einem konventionellen akustooptis-chen Modulator gezogen, wobei sich ein gleichwertiges Signal-zu-Rausch-Verh¨altnis ergibt. Der rein optische Ansatz bringt großes Skalierungspo-tential hin zu Modulationsfrequenzen in den GHz-Bereich, ben ¨otigt keine spezielle Abschirmung der Detektionselektronik und muss nicht aktiv mit einer Referenz synchronisiert werden.
Als weitere Anwendung wird gezeigt, dass der optisch-parametrische Oszillator unter Ausnutzung der Phase der Periodenverdopplung als hardwarebasierter Zufallszahlengenerator einsetzbar ist. In diesem “proof-of-principle”-Experiment wird gezeigt, dass mit diesem bistabilen System Zufallszahlen erzeugt werden k ¨onnen, die keiner algorithmischen Nachbearbeitung bed ¨urfen.
C O N T E N T S Abstract iii Zusammenfassung v 1 I N T R O D U C T I O N 1 2 O U T L I N E 3 2.1 Dissertation Outline . . . 3 3 B A S I C S 5 3.1 Light and Pulses . . . 5
3.1.1 Ultrashort Pulses . . . 7
3.2 Nonlinear Optics . . . 12
3.2.1 Second-Order Nonlinear Effects . . . 12
3.2.2 Optical Parametric Amplification . . . 14
3.3 Optical Parametric Oscillators . . . 23
3.3.1 Singly-Resonant Optical Parametric Oscillators . . 23
3.3.2 Fiber-Feedback Optical Parametric Oscillators . . . 23
3.4 Nonlinear Dynamics in Optical Systems . . . 30
3.4.1 Modeling the ffOPO with Nonlinear Feedback . . . 33
3.5 Laser Spectroscopy . . . 40
3.5.1 Fourier-Transform Infrared Spectroscopy . . . 40
3.5.2 Stimulated Raman Scattering Spectroscopy . . . 41
3.6 Signal-to-Noise Ratio in Laser-Based Measurements . . . . 43
3.6.1 Noise . . . 43
3.6.2 Noise in FTIR spectroscopy . . . 48
3.6.3 Noise in SRS spectroscopy . . . 53
4 S O U R C E S F O R F O U R I E R-T R A N S F O R M I N F R A R E D S P E C T R O S C O P Y 55 4.1 Chances and Challenges for Laser-Based FTIR Spectroscopy 55 4.2 Fiber-Feedback Optical Parametric Oscillator . . . 56
4.2.1 Design Parameters . . . 58
4.2.2 Pump Laser Systems . . . 58
4.2.3 Experimental Results . . . 60
4.3 Optical Parametric Power Amplifier . . . 71
4.3.1 Seeding Unit . . . 71
4.3.2 Amplifier . . . 74
4.4 Mid-Infrared Generation by Difference Frequency Mixing 78 4.4.1 Experimental Results . . . 79
4.4.2 Summary and Comparison to State-of-the-Art . . . 82
4.5 FTIR Spectroscopy in the 2-4 µm Range . . . 84
4.6 FTIR Spectroscopy in the 5-10 µm Range . . . 89
4.6.1 Comparison to the 2-4 µm Range . . . 89
4.6.2 Stitching . . . 90
4.6.3 FTIR Spectroscopy of the Amide Bands . . . 91
4.7 Summary and Outlook . . . 93
5 S O U R C E S F O R S T I M U L AT E D R A M A N S C AT T E R I N G M I C R O S C O P Y 95 5.1 State-of-the-Art . . . 95
5.2 Requirements for a Single-Frequency SRS Source . . . 96
5.3 Switchable OPA/ffOPO Source . . . 97
5.3.1 Comparison of OPG and cw-Seeded OPA . . . 98
5.3.2 Experimental Results in cw-Seeded Mode . . . 101
5.4 Stimulated Raman Scattering Experiments . . . 103
5.4.1 SRS Microscope . . . 104
5.4.2 Detection Noise Figures . . . 105
5.4.3 SRS Spectroscopy of Acetone . . . 105
5.4.4 SRS Microscopy: Polymer Beads . . . 108
5.4.5 SRS Microscopy: Plant Cells . . . 110
5.5 Summary and Outlook . . . 111
6 N O N L I N E A R D Y N A M I C S A N D A L L-O P T I C A L M O D U L AT I O N B A S E D S P E C T R O S C O P Y 115 6.1 State of the Art . . . 115
6.2 All-Optical Modulator Based on Period Multiplication . . 116
6.2.1 Optical Setup . . . 116
6.2.2 Nonlinear Dynamics in a ffOPO . . . 117
6.3 Modulation Spectroscopy . . . 126
6.3.1 Enhancing the Modulation Depth . . . 126
6.3.2 Experimental Results in SRS Microscopy . . . 128
6.4 True Random Number Generation . . . 130
6.4.1 TRNG concept . . . 130
6.4.2 TRNG Test . . . 132
6.5 Summary and Outlook . . . 135
Contents ix
A A P P E N D I X 143
A.1 Pulse Propagation in Optical Fibers . . . 143
A.1.1 Self-Phase Modulation . . . 144
A.2 Instruments and Characterization . . . 147
A.3 Tunable Cw-Seed Laser . . . 149
A.4 SRS Microscopy: Plasmonic Signal Enhancement . . . 151
A.4.1 General Remarks and Expected Signal Enhancement 151 A.4.2 Experimental Results . . . 152
A.4.3 Outlook . . . 155
A.5 Electronics . . . 156
A.5.1 Automation: pMOPA . . . 156
A.5.2 Noise Amplitude Sweep Measurements . . . 157
A.5.3 SRS Detection Chain . . . 158
A.5.4 Dispersive Fourier Oscilloscope . . . 159
Publications 165
B I B L I O G R A P H Y 169
1
I N T R O D U C T I O NNumerous advances in science over the past decades were enabled, real-ized, and enforced by laser technology. Not surprisingly, the list of NO -BELprizes includes a constantly growing share of discoveries and inven-tions directly or indirectly enabled by the laser.
There is, however, a growing number of applications, particularly in science and environmental and medical diagnostics, that demand the property of simultaneous multi-color emission, a property, naturally met by optical parametric frequency conversion.
Generally, parametric amplification and oscillation are not only found in optics, but also in electronic and mechanical systems. The amplitude of a periodic oscillation, in the following referred to as “signal”, is modified indirectly by a second periodically oscillating quantity, called “pump”, that changes one or more system parameters. This time-dependent type of coupling between pump and signal via a system parameter established the term “parametric” amplification. There is no restriction to the fre-quencies of the pump and the signal oscillator, although in some cases the pump frequency may be required to be half or twice the signal frequency to achieve amplification. More generally, additional signal or pump os-cillators may be present leading to a more complex situation, where ad-ditional frequencies such as the sum- or difference frequency can also be amplified.
A characteristic property of parametric processes is the need for an ini-tial nonzero signal amplitude. Further, parametric amplification is highly sensitive to the relative phase of the oscillators. Parametric amplification is suitable to achieve extremely efficient energy transfer and thus high gain coefficients.
While its practical applications in mechanics are rather limited, para-metric amplification was employed in electronics to realize low-noise amplifiers in the 1950s. Intensive theoretical work, such as the discov-ery of the MANLEY-ROWErelations, was done in these years. In 1958 a traveling-wave parametric amplifier for microwaves was described [1]. Although parametric amplifiers were outperformed by the silicon
sistor, they should soon become relevant in optics with the realization of the first lasers in the early 1960s.
About one year after the ruby laser was presented by Theodore Maiman [2], Peter Franken followed up with the demonstration of second har-monic generation, describing the first observation of an optical paramet-ric process based on the χ(2)-nonlinearity of quartz [3]. This was accom-panied by comprehensive theoretical work on nonlinear optical effects in dielectrics [4]. In 1965 optical parametric amplification based on differ-ence frequency generation was demonstrated and nearly simultaneously the first optical parametric oscillator was reported [5, 6]. Two years later, Harris et. al. observed the quantum-noise-seeded optical parametric su-perfluorescence [7] in the visible, by directly watching at the radiation through a monochromator.
These experiments were performed with pulsed lasers with nano- to microsecond pulse duration. Today, femtosecond laser systems are avail-able, which has changed the opportunities of parametric frequency con-version drastically. On the one hand, synchronously-pumped optical para-metric oscillators [8, 9] transform lasers into broadly tunable light sources at tens of MHz repetition rate, on the other hand optical parametric am-plifiers are employed at kHz repetition rate to generate ultra-short highly energetic pulses.
The applications of these sources range from time-resolved spectro-scopy in material sciences and solid-state physics, multi-photon micro-scopy for biochemical studies and medical diagnostics, remote sensing in the atmospheric windows to high-field and attosecond physics, to name a few.
Despite their great opportunities and a wide field of applications, para-metric sources, however, are until today rather complex, require pro-found knowledge of the operator, and substantial maintenance for re-liability. While in industry highly developed automation is employed to make the optical parametric oscillator more user-friendly, more fun-damental approaches are pursued in this thesis to simplify parametric sources and improve their reliability in very close coordination with the potential applications.
2
O U T L I N E2.1 D I S S E R TAT I O N O U T L I N E
The core of this PhD project is the investigation of novel optical paramet-ric frequency converters and their application to spectroscopy and imag-ing. Figure 2.1 shows the project tree in approximate chronological order. Two main approaches, one based on the cw-seeding concept and a sec-ond based on the fiber-feedback optical parametric oscillator technology, are pursued. FOURIER-transform infrared spectroscopic experiments are carried out with the aim to compare laser-based sources to state-of-the-art thermal and synchrotron sources. A stimulated RAMAN scattering microscope is realized to evaluate and demonstrate the source’s perfor-mance for multi-photon microscopy.
The thesis is divided into the following chapters:
Chapter 3introduces the relevant basics of optical parametric frequency conversion and its dynamics with ultrashort pulses. It further discusses the applied spectroscopic techniques from a fundamental point of view with a specific focus on measurement uncertainties and noise.
Chapter 4introduces the fiber-feedback optical parametric oscillator as well as methods on power scaling and deep mid-infrared generation. Moreover, the source’s applicability to FOURIER-transform infrared spec-troscopy is investigated.
Chapter 5investigates cw-seeding of optical parametric amplifiers as a simple alternative to optical parametric oscillators. Both types of fre-quency converters are compared with respect to their performance in stimulated RAMANscattering microscopy.
Chapter 6reports on the investigation of nonlinear dynamics that can be found in the fiber-feedback optical parametric oscillator. Its appli-cability to pump-probe spectroscopy is demonstrated with stimulated RAMANscattering microscopy experiments. Lastly, direct unambiguous
hardware (“true”) random number generation using nonlinear dynamics is shown.
Chapter 7summarizes the results of this work and provides an outlook for future experiments.
FTIR spectroscopy SRS spectroscopy & imaging nonlinear optics high brilliance infrared laser spectroscopy cw-seeding power amplifier ECDL driven tunable OPA near field enhanced SRS mid-infrared DFG stage extension to the mid-infrared nonlinear all-optical modulator fiber-feedback OPO SRS without electronic modulation true random number generation compact high-speed SRS microscope
Figure 2.1.:Projects in this thesis. DFG: difference frequency generation; ECDL: external
cav-ity diode laser; FTIR: FOURIER-transform infrared; OPA: optical parametric amplifier; OPO: optical parametric oscillator; SRS: stimulated RAMANscattering.
3
B A S I C SIn this chapter, the fundamentals of ultrafast pulses (3.1) and nonlinear frequency conversion (3.2), particularly emphasizing optical parametric amplification and oscillation (3.2.2, 3.3), will be recapitulated. In this con-text, the basics of laser-based linear FOURIER-transform infrared spec-troscopy (3.5.1) as well as stimulated RAMAN scattering spectroscopy (3.5.2) are addressed. Finally, laser noise and its influence on spectro-scopic measurements are discussed (3.6).
3.1 L I G H T A N D P U L S E S
At the most fundamental level, the interaction of light and matter is given by MAXWELL’Sequations [10] ∇ · ~D = ρ (3.1a) ∇ · ~B = 0 (3.1b) ∇ × ~E = −∂ ∂t~B (3.1c) ∇ × ~H = ~j + ∂ ∂tD.~ (3.1d)
MAXWELL’Sequations relate the electric field ~E and magnetic field ~Hto the respective electric and magnetic flux density ~Dand ~B by the vacuum permittivity ε0and vacuum permeability µ0
~ D = ε0~E + ~P (3.2a) ~ H = 1 µ0 ~B − ~M (3.2b)
and the electric and magnetic polarization (density) ~P and ~M, as well as the electric current density ~j and charge density ρ. In this work, the electric polarization ~P = ε0χ(1)~E + ε0χ(2)~E2+ ε0χ(3)~E3+ ... | {z } ~ PNL (3.3) 5
is of major importance, since it contains, besides the linear share, the non-linear polarization ~PNL, which is the origin of nonlinear optical phenom-ena. Details on nonlinear optics and the following derivations can be found in [11]. Note that the notation in Eq. (3.3) is symbolic in the sense that for higher orders the tensor nature of the susceptibility χ needs to be taken into account.
To model the propagation of light in a non-magnetic, uncharged di-electric, Eq. (3.3) is substituted into ~Din Eq. (3.1d). Using a rotation with (3.1c) along with the identity ∇ × ∇ × ~E = ∇∇ · ~E− ∆ ~E and substitut-ing Eq. (3.1a), we obtain
∆ ~E = ε0µ0 χ(1)+ 1 ∂ 2 ∂t2~E + µ0 ∂2 ∂t2~P NL. (3.4)
Substituting the speed of light in vacuum c = (ε0µ0)−1/2and the refrac-tive index in a non-magnetic dielectric n = (χ(1)+ 1)1/2into Eq. (3.4) results in ∆ ~E = n 2 c2 ∂2 ∂t2~E + 1 ε0c2 ∂2 ∂t2~P NL, (3.5)
which is often referred to as the nonlinear wave equation. In general, the susceptibility χ(ω) is frequency-dependent. The resulting frequency de-pendence of the refractive index n(ω) is known as dispersion.
Plane Waves
In the case of vanishing nonlinear polarization ~PNL= 0, the solution of the wave equation (3.4) are plane waves
~E(~x, t) = ~E0cos(ωt − ~k · ~x + φ) = ~E0Re exp(i(ωt − ~k · ~x + φ)) = ~E0 1 2 exp(i(ωt − ~k · ~x + φ)) + c.c (3.6)
with the real amplitude ~E0and the wave vector ~k giving the direction of propagation and the wavenumber k = |~k| = nωc . In spectroscopy, the spectroscopic wavenumber ˜ν = 2πk is a frequently used quantity. Further very important optical quantities are the wavelength λnin matter
λn=2π k = 2πc nω = c nf, (3.7)
and the optical frequency f = 2πω. It is common, however, to use the vacuum wavelength λ = cf.
3.1 L I G H T A N D P U L S E S 7
For simplicity, we will use kx= ky= 0 and kz= k, assuming propa-gation in z-direction in the following. Further, we assume linear polariza-tion in x-direcpolariza-tion ~E0= E0ˆex, which turns Eq. (3.6) into a scalar quantity
E(z, t) = E0
2 (exp(i(ωt − kz + φ)) + c.c.) . (3.8) The intensity I of an electric field E is
I = 1 2nε0c|E|
2. (3.9)
3.1.1 Ultrashort Pulses
In this thesis, ultrashort pulses in the femto- and picosecond range are employed, since they provide giant electric fields on the order of 108 V/m when focused into a dielectric, while maintaining moderate average pow-ers at the Watt level. To obtain such short pulses, the concept of a mode-locked frequency comb is employed. As shown in Fig. 3.1 the ultrashort
electric field amplitude (arb. units)
time (arb. units)
frep
amplitude (arb. units)
T = 1/fR rep
-1.5 -1.0 -0.5 0 0.5 1.0 1.5
frequency (arb. units)
0 1 -1 1 0 fc a b c
Figure 3.1.:Representation of a frequency comb in time and frequency domain. The
coher-ent combination of phase-locked plane waves (a) results in a periodic pattern of short pulses (b). The blue shaded line indicates constructive interference of all components, while the red shaded line indicates π phase shift between adjacent components. The frequency comb (c) represents a sech-pulse centered at frequency fcwith a distance of two comb lines of the frequency frep, which determines the temporal pulse-to-pulse distance TR.
pulse is synthesized from a linear superposition of electric fields with equally spaced frequencies ωn, also called modes, weighted with their in-dividual field amplitude ˜En(ωn). The temporal electric field amplitude is thus given by
E(z, t) =X n
1
2( ˜En(ωn) exp(i(ωnt − knz + φn)) + c.c.), (3.10) with kn=nωcn. The phases φnof the individual modes are in a fixed re-lation (“mode-locked”, formerly “cw mode-locked”), such that at a given point in time, where all components interfere constructively, a short and intense laser pulse is synthesized. The spectral modes ωn = 2π(fCEP+ nfrep) are determined by the repetition rate frepof the laser resonator and the (random, if not actively stabilized) carrier-envelope phase (CEP) fCEP. In practice, mode-locking is achieved in lasers by using intensity-dependent feedback, for instance a semiconductor saturable absorber mir-ror (SESAM) [12], nonlinear polarization rotation [13], or KERR-lens mode-locking (KLM) [14, 15].
A mode-locked laser emitting such pulses is usually characterized by the quantities shown in table 1. Typical repetition rates frep of mode-locked lasers are in the 10 MHz–10 GHz region, which leads to a much denser frequency comb than shown exemplary in Fig. 3.1. For instance there are approximately 15, 700 modes within the intensity FWHM as-suming 1 µm center wavelength, 40 MHz repetition rate and 500 fs pulse duration. Generally speaking, the shorter the temporal pulse duration and the lower the repetition rate, the more modes contribute with signif-icant amplitude.
repetition rate frep
round-trip time TR =1/frep
pulse energy Ep
pulse duration τ
average power Pavg = Ep· frep center wavelength λc =c/fc spectral bandwidth ∆λ =c/f2
c· ∆f
3.1 L I G H T A N D P U L S E S 9
Phase and Group Velocity
For dense frequency combs or single ultrashort pulses continuous distri-bution in frequency space can be assumed, so the electric field in time domain E(z, t) is given by
E(z, t) = 1 2π
Z
˜E(z, ω − ω0) exp (−i(ω − ω0)t) dω, (3.11) with carrier frequency ω0and spectral amplitude
˜E(z, ω − ω0) = ˜A(z, ω − ω0) exp(ik0z) (3.12) that contains the slowly varying amplitude
˜
A(z, ω − ω0) = ˜A(z)exp(i(k(ω − ω0) − k0)z). (3.13) The slowly varying amplitude approximation (∂∂z2A2 ≈ 0) is intensively used in the study of pulse propagation dynamics in the sections 3.2.2 and A.1. For these processes, dispersive effects are easier to handle in the frequency domain, which is regularly done by a power series expansion of the wavenumber at the carrier frequency
k(ω − ω0) = k0+ k1(ω − ω0) + 1 2k2(ω − ω0) 2+ ..., (3.14) where k0= k(ω0) and km= d mk dωm ω=ω0 (m = 1, 2, ...). (3.15) The individual terms in Eq. (3.14) are physically meaningful and shall be addressed in the following. The phase velocity that describes the speed of the wavefronts, the planes of equal phase, is given by
vp(ω0) = ω0
k0 = c
n. (3.16)
The speed of the pulse envelope, known as group velocity vg(ω0) = dω dk ω=ω0 = 1 k1 = c n |{z} vp −ck n2 dn dk = c n + ωdωdn (3.17) will usually differ from the phase velocity. Originating from the similarity of the right hand side of Eq. (3.17) and Eq. (3.16), the group index is defined as [11]
ng= n + ωdn
dω = n − λ dn
The group velocity dispersion (GVD) k2(ω0) = d2k dω2 ω=ω0 = − 1 v2 g dvg dω ! ω=ω0 (3.19)
is a measure of the strength of pulse stretching in a dispersive medium. The GVD is usually given in mmfs2, or, more suitable when working with wavelengths, as the dispersion parameter
D(λ) = dk1 dλ = − 2πc λ2k2= − λ c d2n dλ2. (3.20)
in nm·kmps . The group delay dispersion (GDD) is the accumulated GVD after a propagation distance z
GDD(z, ω0) = k2(ω0)z,
GDD(z, λ) = D(λ)z, (3.21)
in the appropriate unit fs2 orps/nm. Note that GDD and GVD are fre-quently confused or even treated as synonyms, so that the unit needs to be known to clearly identify the supposed quantity.
a c
b d
Figure 3.2.:If an unchirped pulse in temporal (a) and spatial (b) domain experiences positive
chirp, its temporal profile changes (c). The red line represents the electric field, while the black line illustrates its envelope. Subfigure (d) shows the increase in pulse duration as well as the spatial separation of different frequencies. The black arrow indicates the direction of motion.
3.1 L I G H T A N D P U L S E S 11
Chirp
When propagating through a medium of thickness L, a pulse will accu-mulate the spectral phase
φ(ω) = k(ω)L. (3.22)
A formerly unchirped (FOURIER-limited) pulse ˜E(0, ω) will thus read ˜E(L, ω) = ˜E(0, ω) exp(iφ(ω)). (3.23) It will be called chirped, if its spectral phase φ(ω) is neither constant nor linear. “Positive” chirp dωd22φ(ω) > 0refers to the case where “blue” components are delayed with respective to the center frequency, as shown in Fig. 3.2.
In case of constant or linear spectral phase, the pulse will be FOURIER -limited, which ensures the shortest possible temporal pulse duration τFL for a given spectral bandwidth ∆f. Hence, so called time-bandwidth prod-uct (TBP) τ ∆f with the actual pulse duration τ is a measure for the flat-ness of the spectral phase. Any ultrashort pulse could be shaped to fulfill the FOURIERlimit, if its (arbitrarily complex) spectral phase φ(ω) was known and compensated accordingly. Table 2 gives an overview of com-mon pulse shapes for ultrashort pulses and their most important proper-ties.
Gaussian sech2
temporal shape P(t) P0exp
− Tt2
P0sech2 Tt
spectral shape P(f) ˜P0exp −δff2 ˜P0sech2 f δf FWHM (τ, ∆f) (2√ln 2 T , 2√ln 2 δf) (1.76 T , 1.76 δf) FOURIERlimit τFL∆f 2ln(2)π ≈ 0.44 ≈ 1.76 2 π2 ≈ 0.315 peak power P0 ≈ 0.94Eτp ≈ 0.88 Ep τ
3.2 N O N L I N E A R O P T I C S
In nonlinear optics, the higher order terms of Eq. (3.3) are utilized to ma-nipulate not only the amplitude, but also the frequency of light. We still assume linearly polarized light propagating along the z-axis to facilitate the discussion without loss of insight. Also, we will first focus on second order nonlinear effects, which reduces the nonlinear polarization to
PNL= ε0χ(2)E2= 2ε0dE2. (3.24) Here, the nonlinear coefficient d = χ(2)/
2is introduced that describes the strength of nonlinear coupling between the incident electric field and the polarization of the dielectric. Note that a large class of media, includ-ing all liquids, gases, glasses and many crystals exhibit d = 0 resultinclud-ing from centrosymmetric structure [11]. In practice, the effective nonlinear co-efficient deffis of major importance. It is a scalar quantity that is derived from the d-tensor by taking into account the direction of propagation and the polarization of the interacting waves with respect to the crystal axes. 3.2.1 Second-Order Nonlinear Effects
The nonlinear electric polarization PNL, when excited by an electric field with two frequencies
E(t) = 1
2(E1exp(iω1t) + E2exp(iω2t)) + c.c. (3.25) acts as a driving force in the wave equation at any sum or difference of these frequencies, as shown in Fig. 3.3. The nonlinear polarization in-cludes the terms for second harmonic generation (SHG), sum frequency gen-eration (SFG), difference frequency gengen-eration (DFG), and optical rectifi-cation (OR) as shown in Eq. (3.26).
PNL(t) = ε0( 12d(2ω1)E21exp(i2ω1t) (SHG) +12d(2ω2)E22exp(i2ω2t) (SHG) +d(ω1+ ω2)(E1E2+ E1∗E∗2) exp(i(ω1+ ω2)t) (SFG) +d(ω1− ω2)(E1E∗2+ E∗1E2) exp(i(ω1− ω2)t) (DFG) +12d(0)(E1E∗1+ E2E∗2) ) (OR) + c.c. (3.26)
3.2 N O N L I N E A R O P T I C S 13
While the occurrence of these terms results in generation of coherent frequency-shifted radiation, it does not yet imply information about the overall efficiency of the individual processes during propagation. There-fore, the conservation of momentum, in this context also known as phase matching, needs to be considered, which will be reviewed in the following section exemplary for DFG.
nonlinear
medium
ω
1ω
22
ω
1ω + ω
1 22
ω
2ω
1-
ω
2(2)
χ
3.2.2 Optical Parametric Amplification
Although the term “difference frequency generation” implies that a wave at the difference frequency ω3 = ω1− ω2of the incident frequencies ω1 > ω2 is generated, it is not evident that during this process the incident lower frequency component ω2is also amplified. This led to the occurrence of the term optical parametric amplification (OPA) whenever the amplification of ω2was of prior importance to the generation of ω3. While strictly classified, also SFG and SHG could be included into the realm of OPA, it is usually applied to the aforementioned case of DFG.
In this scope, OPA is an interaction of three waves called pump, signal, and idler with frequencies, by definition, ωp > ωs > ωi.
Figure 3.4.:Energy scheme
of an OPA process. While propagating through a nonlinear medium
(d 6= 0) energy is transferred from the pump into both signal and idler wave. Energy conservation demands
hωp= hωs+ hωi, (3.27) with the reduced PLANCK constant h that re-lates photon frequency and energy E{p,s,i} = hω{p,s,i}. The process is illustrated in Fig. 3.4. Note that from a mathematical point of view sig-nal and idler are interchangeable.
Coupled Amplitude Equations
The evolution of the spatially varying amplitudes of pump, signal, and idler Ap(z, t), As(z, t), Ai(z, t) during the OPA process can be modeled by the coupled amplitude equations [16]
∂Ap ∂z + 1 vg,p ∂Ap ∂t + αpAp= ideffωp npc AsAiexp(−i∆kz) (3.28a) ∂As ∂z + 1 vg,s ∂As ∂t + αsAs= ideffωs nsc A∗ iApexp(i∆kz) (3.28b) ∂Ai ∂z + 1 vg,i ∂Ai ∂t + αiAi= ideffωi nic ApA∗sexp(i∆kz). (3.28c) Here, the effective nonlinear coefficient deffand the wavevector mismatch ∆k = kp− ks− kiare the basis for efficiency considerations of the para-metric conversion process. These equations further account for group velocity effects and linear absorption due to the absorption coefficients
3.2 N O N L I N E A R O P T I C S 15
α{p,s,i}The Eqs. (3.28) are derived by transforming both (3.5) and the electric field envelope A(z, t) to the FOURIERdomain, taking into account the frequency dependence of the refractive index. This is followed by a power series expansion of dispersive effects in frequency that can be trun-cated to neglect second and higher order dispersion (cf. Eq. (3.14)). Af-ter inverse transformation and slowly varying amplitude approximation Eqs. (3.28) are obtained.
Parametric Gain in Neglected Pump Depletion Approximation
As mentioned in the introduction of this thesis, parametric amplification provides enormous single-pass gain under certain conditions. This be-comes evident, if the small-signal gain is evaluated by solving the Eq. (3.28) assuming no pump depletion (Ap = const) and neglecting both the dispersive term and the absorption. Hence, the following equations are valid for the loss-free cw regime. The evolution of the signal and idler intensities propagating in z direction is then given by
Is(z) = Is(0) 1 +Γ 2 g2sinh 2(gz) (3.29a) Ii(z) = Is(0) λs λi Γ2 g2sinh 2(gz) (3.29b)
assuming an incident signal intensity Is(0) 6= 0 while neglecting incident idler [16]. The (small-signal) gain coefficient g reads
g = s Γ2− ∆k 2 2 (3.30a) Γ2= 4π 2d2 eff nsniλsλi |Ap|2= 8π2d2eff nsninpλsλiε0c Ip. (3.30b) Note that for gz ≫ 1 and vanishing phase mismatch ∆k = 0 both sig-nal and idler grow exponentially as sinh2(Γ z) ≈ 14exp(2Γ z). The small-signal single-pass gain
G = Is(z)
Is(0)= 1 + sinh
2(Γ z) ≈ 1
4exp(2Γ z) (3.31)
can easily exceed 106, which allows to choose I
s(0) on the order of the vacuum fluctuations. In this case, both Is(0) and Ii(0) will contribute equally as seed, which is the basis for the optical parametric generator (OPG). This will be further discussed in section 3.2.2.
Manley-Rowe Relations
The change of intensity is given by dI{p,s,i} dz = n{p,s,i}ε0c 2 (A ∗ {p,s,i} dA{p,s,i} dz + A{p,s,i} dA∗ {p,s,i} dz ). (3.32) Substituting Eq. (3.28), while neglecting absorption and dispersion, into Eq. (3.32) and comparing the resulting set of equations results in a repre-sentation of the MANLEY-ROWErelations
− 1 ωp dIp dz = 1 ωs dIs dz = 1 ωi dIi dz. (3.33)
These relations demand that the loss in pump intensity must exactly be balanced by the gain of signal and idler intensity in the ratio ∆Ii
∆Is= ωi ωs = λs
λi. This is also very intuitive from the photon picture as illustrated in Fig. 3.4 due to energy conservation.
Phase Matching
The phase matching condition [11, 17]
kp= ks+ ki (3.34a)
ωpnp= ωsns+ ωini (3.34b)
sets the boundary conditions for the refractive index requirements of the nonlinear medium. Evidently, it is not possible to fulfill this condition in bulk materials in the normal dispersion regime. However, birefringent crystals can satisfy Eq. (3.34) for some spectral ranges, by choosing the orientation of the crystal properly with respect to the propagation and the polarizations for the three waves. It is distinguished between type I (os+ oi→ ep) and type II (es+ oi→ ep); (os+ ei→ ep) phase match-ing, with o and e denoting the ordinary and extraordinary crystal axes. Another type of phase matching is based on periodic modulation of the crystal susceptibility, as shown in Fig. 3.5. With this technique, called quasi phase matching (QPM), a nonzero wavevector mismatch is compen-sated by the contribution of the grating vector km= 2πmΛ , with Λ being the periodicity of the grating.
∆k = kp− ks− ki+ km (m=1)
= kp− ks− ki− 2π
Λ (3.35)
In case that first order phase matching is used (m = 1), the effective non-linear coefficient is reduced by a factor of2/π
dQPM= 2
3.2 N O N L I N E A R O P T I C S 17
0 0.5 1
0 1 2 3 4 5 6 7
signal amplitude (arb. units) crystal length z/L coh
phase matchedquasi phase matched
not phase matched
Λ = 2Lcoh kp a b ks ki kp ks ki km c
Figure 3.5.:Comparison of ideal phase matching (a) and quasi phase matching (b). The signal
amplitude gain is exemplary shown (c) for the ideally phase matched case (black), quasi phase matched (orange) and not phase matched (red) case.
Note that if “deff” is given, it usually already contains the factor for QPM. These equations are easily derived in a FOURIERdomain picture [11, 18], which is very recommendable as it provides deep insight also to the op-portunities based on aperiodic poling [19, 20]. An intuitive explanation in the spatial domain is that the phases of pump, signal, and idler, af-ter matching initially, walk off on the length scale of the coherence length Lcoh. This leads to destructive interference of newly generated signal and idler radiation with co-propagating previously generated radiation and hence to a decrease of the overall amplitudes, as illustrated in Fig. 3.5. Of course, the phase matching condition may reversely been
em-22 24 26 28 30 32 1 2 3 4 5 6 PPLN PPLT signal wavelength ( µm ) periodicity (µm) 1.4 1.6 1.8 2.0 0 20 40 60 80 100 120 140 b PPLN PPLT signal bandwidth (nm) signal wavelength (µm) a
Figure 3.6.:Comparison of signal and idler wavelength versus periodicity for a 1.03 µm
pumped PPLN and PPLT crystal (a) and phase matching bandwidth for both crystals (b), as-suming an intensity of 1GW/cm2, 10 mm crystal length, and 300 K crystal temperature. The data was calculated using [21–23].
ployed to tune the wavelength of a parametric device. The relation of pol-ing periodicity and wavelength is exemplary shown for periodically poled lithium niobate (PPLN) and periodically poled lithium niobate (PPLT) in Fig. 3.6. The upper branch represents the idler, while the lower branch rep-resents the signal. Both branches meet at the point of degeneracy, where λs= λi= 2λpand signal and idler become indistinguishable.
Group Velocity Mismatch
For ultrafast pulses, the group velocity mismatch (GVM) between the pump and signal or idler
GVMp,{s,i}= 1 vg,p− 1 vg,{s,i} = ng,p− ng,{s,i} c , (3.37)
should be considered, as it determines the pulse splitting length lp,{s,i}= τp
GVMp,{s,i}, (3.38)
based on the pump pulse duration τp. Crystals that are longer than the pulse splitting length will provide less gain than estimated by Eq. (3.31) due to the limited pulse interaction length. The direction of the so called temporal walk-off leads to different gain dynamics due to pulse stretching effects that are further discussed in [16]. For PPLN and PPLT the GVM is shown in Fig. 3.7, based on the SELLMEIERdata from [22, 23]. Both crys-tals show rather moderate GVM between 1400 nm and 2000 nm. Strong
1200 1300 1400 1500 1600 1700 1800 1900 2000 -1000 -800 -600 -400 -200 0 200 0.45 0.56 0.75 1.13 2.25 2.25 p u lse sp lit ti n g l e n g th ( m m ) G V M ( fs/ m m ) signal wavelength (nm) PPLN PPLT p-sig p-sig p-idl p-idl
Figure 3.7.:Comparison of the GVM in PPLN and PPLT for signal (solid) and idler (dashed)
versus signal wavelength. A pump wavelength of 1.03 µm is assumed. The pulse splitting length (right scale) is given for a 450 fs pulse.
3.2 N O N L I N E A R O P T I C S 19
negative GVM, resulting in slow idler pulses, is expected for idler wave-lengths above 4 µm or signal wavewave-lengths lower than 1400 nm, if 1030 nm pump wavelength are assumed.
Phase Matching Bandwidth
It can be shown that the GVM influences the phase matching bandwidth, in first order approximation, by
∆f =2 √ ln2 π s Γ Lcry 1 1 vg,s− 1 vg,i , (3.39)
for a given crystal length Lcry[16]. Equation (3.39) shows that small GVM between signal and idler leads to a high phase matching band-width. Naturally, the phase matching bandwidth increases towards de-generacy, when the phase velocities of signal and idler become equal. This effect is shown in Fig. 3.6. For small GVM, Eq. (3.39) is not ap-propriate anymore and the second order
∆f =(ln2) 1 4 π Γ Lcry 14 1 ∂2k s ∂ω2 s + ∂2k i ∂ω2 i 2, (3.40)
should be considered. Equation (3.40) will only provide the correct result for zero GVM of signal and idler.
Exact Solution with Pump Depletion and Backconversion
The general solution for the (cw) three-wave interaction in a nonlinear di-electric was published 1962 by Armstrong and coworkers [4]. It includes non-collinear geometry, phase mismatch, and is, naturally, not limited to describe DFG, so that also backconversion due to SFG is included. The complex equation (3.28) is split into real and imaginary parts, neglecting dispersion and absorption
∂ ∂z|Ap|= − deffωp npc |As||Ai| sin(θ) ∂ ∂z|As|= deffωs nsc |Ai||Ap| sin(θ) ∂ ∂z|Ai|= deffωi nic |Ap||As| sin(θ), (3.41)
with the phase θ fulfilling θ = ∆k + φp− φs− φi, ∂ ∂zθ = −∆k −cos(θ) deff c ωp|As||Ai| np|Ap| −ωs|Ap||Ai| ns|As| − ωi|Ap||As| ni|Ai| . (3.42)
These equations provide decent insight to understand “parasitic” back-conversion, since the sign of θ will either result in DFG (θ = π2) or SFG (θ = −π2). For the OPA process with negligible incident idler, the system will automatically adjust the phase of the generated idler such that DFG is favored. In the cw case or for short crystals with negligible temporal walk-off and pulses of sufficiently narrow spectral bandwidth, the phases will stay constant during propagation. In a second case, where there is no incident pump wave, but signal and idler, SFG will generate pump pho-tons, since the phase will automatically adjust to this situation. Thus, for short pulses in long crystals, phase mismatch and GVM may cause back-conversion generating a leading or trailing pump pulse via SFG if signal and idler exhibit the same temporal walk-off direction with respect to the pump.
Equations (3.41) can be solved analytically, by integrating Eq. (3.42) [4] and using the transformation for the intensities
ˆIp,s,i= 8π 2d2 eff npnsniλsλiε0c λp,s,i λp Ip,s,i. (3.43) Note that this convention comes with [ˆIp,s,i] =1/m2. Thus, we get
∂
∂zˆIp= − ˆIpˆIsˆIi 1/2 ∂
∂zˆIs= ˆIpˆIsˆIi 1/2 ∂
∂zˆIi= ˆIpˆIsˆIi 1/2
(3.44)
assuming perfect phase matching. The solution for (3.44) for ˆIi(0) = 0, as given by [24], reads
ˆIp(z) = ˆIp(0)sn2(α(z); β)
ˆIs(z) = ˆIs(0) + ˆIp(0) − ˆIp(0)sn2(α(z); β) ˆIi(z) = ˆIp(0) − ˆIp(0)sn2(α(z); β),
3.2 N O N L I N E A R O P T I C S 21 0 2 4 6 8 10 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 pump signal
signal (neg. pump dep.) idler
intensity (GW/cm²)
crystal length (mm)
Figure 3.8.:Comparison of OPA with and without pump depletion, assuming 1.03 µm pump
and 1.5 µm signal wavelength, perfect phase matching, plane waves, and deff= 14pm/V.
with the JACOBIelliptic function sinus amplitudinis sn(α; β) and α(z) = q ˆIp(0) + ˆIs(0)(z − z0) β = ˆIp(0) ˆIp(0) + ˆIs(0) . (3.46)
The constant z0ensures that sn2(0, β) = 1 to achieve consistence for the initial values at z = 0. Figure 3.8 shows a comparison of the exact solu-tion according to Eq. (3.45), showing good agreement for pump deple-tion below 50% and sufficiently short crystals. Further, it also shows the periodicity of the solution due to backconversion. Note that
sn(α, 0) = sin(α) and
sn(α, 1) = tanh(α).
These relations are particularly important as they simplify Eq. (3.45) dras-tically for ˆIs(0) ≫ ˆIp(0) or β ≈ 0, which is reasonable for most optical parametric oscillators (OPOs), and ˆIs(0) ≪ ˆIp(0) or β ≈ 1 resulting in an approximate solution including pump depletion.
Optical Parametric Generation and Weak Seeding
So far, it has been assumed that the seeding intensity at either the signal or idler wavelength was comparable to the pump intensity at least within
a few orders of magnitude. It is, however, also possible to observe para-metric amplification starting from quantum noise [7, 16, 25–27]. This is re-ferred to “parametric (super)fluorescence” or “optical parametric genera-tion” (OPG). It can be described in the fully quantum-mechanical frame-work [28, 29], or, with equal results, in a semi-classical picture [29, 30], where broadband noise fields are assumed for incident signal and idler. This process of OPG, however, intrinsically carries strong stochastic el-ements such as high uncertainty of the shot-to-shot spectra, extremely high relative intensity noise that is only constrained by the single-pass gain saturation, and missing spectral and temporal control for the pulse parameters. For sufficiently strong saturation the output pulse energies stabilize, but a timing jitter is found [26, 31]. This can be understood by assuming different “build-up” times for the signal pulse during propa-gation through the nonlinear crystal and thus different effective propaga-tion lengths leading to different timing due to GVM. The CEP is random for every single pulse [31], due to the stochastic nature of the seeding fields. Nevertheless, OPG is a very appealing and simple way to realize a tunable frequency converter.
The stochastic behavior is changed for minute amounts of seed, which do not necessarily have to be phase-locked to the pump beam or even be coherent [32]. It is for instance possible to reduce the noise level of an OPG by seeding with a cw laser diode [25, 33], which is also known as OPG with injection-seeding, derived from the similar technique of “in-jecting” a cw laser diode into a laser or OPO resonator. We will show later in this thesis (cf. section 5.3.1) that cw-seeded OPAs exhibit better noise properties (10 − 20 dB) and reduced spectral shot-to-shot jittering compared to OPGs.
3.3 O P T I C A L PA R A M E T R I C O S C I L L AT O R S 23
3.3 O P T I C A L PA R A M E T R I C O S C I L L AT O R S
Optical parametric amplification can not only be realized in single-pass systems, but also within a cavity with optical feedback. The generated signal will thus act as the seed during the next round-trip, removing the need for the generation of a separate seed. The initial field builds up from parametric fluorescence [7].
In femto- and picosecond OPOs synchronous pumping is required, which means that the OPO resonator length needs to match the resonator length of the pump laser. Such a device was demonstrated with 40 ps pump pulses in 1988 by Piskarskas et al. [8]. The first femtosec-ond OPO followed one year later by Edelstein et al. using the intracavity beam of a colliding-pulse mode-locked dye laser as pump [9].
There are also OPOs that are pumped with nanosecond pulses. In this case the optical parametric oscillation is realized within a single pulse [6, 34]. As this type of OPO is not within the scope of this thesis, the term “OPO” will always refer to the synchronously-pumped implementation.
3.3.1 Singly-Resonant Optical Parametric Oscillators
An OPO is called singly-resonant if only one field, signal or idler, is reso-nantly oscillating in the cavity. In practice, singly-resonant OPOs are of-ten preferred to dualy-resonant systems, which employ both signal and idler feedback. In particular, they are less susceptible to stability prob-lems and provide broad tunability, while dualy-resonant OPOs feature lower oscillation threshold, but suffer from stability issues. The following discussion is thus limited to singly-resonant OPOs. Further, the discus-sion of gain and threshold apply to cw OPOs, but they are well applicable to synchronously-pumped OPOs, as long as the regime of significant par-asitic backconversion and excessive GVM is avoided.
3.3.2 Fiber-Feedback Optical Parametric Oscillators
In this thesis the actual implementation of an OPO is the fiber-feedback optical parametric oscillator (ffOPO). As indicated in the name, the ffOPO employs an optical fiber within the resonator that provides a significant share of the feedback path length. The investigation of this concept is
favorable, since it combines broad tunability, excellent stability, and com-pact footprint without the use of GDD-optimized optics. However, it re-quires careful handling of nonlinearities. The first realization of a ffOPO was given by S ¨udmeyer et al. in 2001 using a thin-disk Yb pump laser and periodically poled lithium tantalate (PPLT) as nonlinear medium [35, 36]. While per definition it is classified as OPO, as it uses self-feedback to seed the parametric conversion, it differs significantly from the OPO devices based on free-space resonators, as illustrated in Fig. 3.9. The main reason for this difference is that the intracavity fiber induces very high round-trip GDD and quite high losses, which are compensated by high single-pass gain. In contrast to other systems in literature [37, 38] we employ a variable output coupler. Compared to classical OPOs different types of
QWP NL crystal PBS pump output fiber feedback b a NL crystal pump output
Figure 3.9.:Schematic comparison of a conventional bulk OPO setup (a) and ffOPO (b) in
linear cavity configuration. The variable output coupling unit consisting of a quarter-wave plate (QWP) and a polarizing beam splitter (PBS) is a substantial part of the ffOPO, since a tunable feedback ratio is required, while bulk OPOs use broadband dielectric optics for output coupling.
nonlinear crystals, mirrors, and output couplers are required. The delay mirror that ensures synchronization between the pump and OPO pulses needs significantly more travel in ffOPOs due to the strong dispersion (cf. section “tuning and wavelength stability”). As a result, an ffOPO rather shares its most characteristic properties with OPAs than OPOs, as shown in table 3.
Basic equations
Since ffOPOs models are not treated in literature, some important quan-tities are defined in the following.
The feedback ratio (FBR) is defined as the power Pfbthat is fed back into the resonator and the amplified power after the crystal Pcry. It is
3.3 O P T I C A L PA R A M E T R I C O S C I L L AT O R S 25
OPO ffOPO OPA
rep. rate 10MHz 1MHz 1Hz −1 GHz −100 MHz −100 MHz single-pass gain 1.1 − 2 10 − 1000 1 − 10 6 intensities Is(0) ≫ Ip(0) Is(0) ≪ Ip(0) Is(0) ≪ Ip(0) |GDDrt| < 104fs2 < 108fs2 N/A
Table 3.:Comparsion of ffOPOs to both OPOs and OPAs, taking into account the repetition
rate, single-pass gain, the relation of seed Isand pump Ipintensities, as well as the round-trip dispersion |GDDrt|. The given numbers are very generally chosen values that may nevertheless be exceeded by specialized individual implementations.
determined by the angle θOCof the quarter-wave plate (QWP) that is used for output coupling
FBR = Pfb
Pcry = cos 2(θOC
− θOC,0)2
. (3.47)
The angle θOC,0is the angle of maximum feedback.
The round-trip transmission T is given by the transmission of the in-tracavity elements of one round-trip
T = FBR · Y i RMi ! Y j ηFj Y k ηARk ! (3.48)
with the (intensity) reflectivity of the ith mirror RM
i , the fiber coupling efficiencies ηF
j of all fiber-to-free-space transitions of one round-trip, and the coating transmission ηARk of the crystal. Typically, T in a ffOPO will be on the order of 0.01 − 0.1, which, obviously, is a very low value for a resonator.
The delay mismatch between pump oscillator and ffOPO can be used to change the overlapping section of the feedback pulse with the pump pulse. Usually, this will lead to a shift in center wavelength due to disper-sion. Generally, positive delay mismatch corresponds to lOPOres > lpumpres .
The minimum threshold wavelength, which is the wavelength at the parametric gain maximum, is important, since it is the ”native“ operating wavelength at low feedback. It depends on the poling periodicity of the
PPLN crystal and the crystal temperature. It is measured by optimiza-tion of delay and feedback ratio and subsequently lowering the pump power, while optical parametric oscillation is still maintained. For strong feedback it is used as reference wavelength, since the actual wavelength can depend strongly on the feedback ratio, pump power, and delay mis-match.
The oscillation threshold condition as usually derived for synchro-nously-pumped OPOs [24] loses validity for the ffOPO, as the common assumptions Is(0) ≫ Ip(0) and G ≈ 1 are not fulfilled. However, we may use the neglected pump depletion for the single-pass gain (Eq. (3.31)). The threshold condition is thus given by
1 +sinh2(Γ Lcry) = 1
T, (3.49)
neglecting also effects of focusing and GVM and assuming perfect phase matching.
Oscillating Conditions
The neglected pump depletion is a good approximation for the oscillation threshold, but it can not be applied to describe the steady-state properly, as it is usually undesirable to operate OPOs close to the oscillation thresh-old. There is no straightforward analytical model to describe the ffOPO dynamics with ultrashort pulses, since the following effects can not be neglected:
• pump depletion,
• group velocity mismatch,
• feedback dispersion, and
• feedback nonlinearity.
A numerical model will be discussed in section 3.4.1. Nevertheless, a qualitative discussion of the effects of the above properties is feasible. De-spite the more complex dynamics, the control of a ffOPO is simpler com-pared to its classical counterpart. Figure 3.10 shows the relation of control parameters for both systems. The ideal system would posses one control parameter per output parameter corresponding to exactly one blue ar-row per parameter in Fig. 3.10. Clearly, the ffOPO concept is closer to
3.3 O P T I C A L PA R A M E T R I C O S C I L L AT O R S 27 output power pulse duration center wavelength pump power pump pulse duration resonator losses resonator length phase matching round-trip GDD crystal length pump power pump pulse duration resonator losses crystal length resonator length phase matching round-trip GDD sync sync adjus t compensa te
ffOPO
OPO
Figure 3.10.:Comparison of the influence of control parameters (blue) and fixed parameters
(black) on the OPO signal properties (green).
this case, which is by far more than just a technical advantage, since it also enables extremely stable operation.
Stability
The output power is stabilized by strong gain saturation and a high de-gree of output coupling, which minimizes the effects of resonator losses. The center wavelength is fixed due to steep slope of dldλres due to high round-trip GDD. Both effects are briefly discussed in the following sec-tions. In practice, these two simple but important effects lead to supe-rior free-running performance of ffOPOs compared to their classical bulk counterparts.
Conversion Efficiency and Gain Saturation
The solution of the coupled amplitude equations with pump depletion, as discussed in section 3.2.2 shows that oscillatory behavior is expected for the pump, signal, and idler intensities for during propagation in the nonlinear crystal. One effect of the variable output coupler in ffOPOs is that it can be tuned such that the parametric gain process always stops at the maximum signal and idler power. This is feasible, since the
pump seed: 25 MW/cm² 6.6 MW/cm² 2.3 MW/cm² 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 intensity (GW/cm²) distance (mm) 0 1 2 3 4 5 a c distance (mm) b 0 1 2 3 4 5 distance (mm)
Figure 3.11.:Analytical solution of Eq. 3.45 for various input pump intensities (black) of
0.51GW/cm2(a), 0.77GW/cm2(b), and 1.03GW/cm2(c). Variation of the seed power (red) shows that there is an specific optimum value (orange) for each pump intensity level and crystal length. The pump and signal wavelength are 1.03 µm and 1.5 µm, respectively, and deffwas set to 14pm/V.
odicity of the energy transfer between pump and signal depends on the initial seed power that is indirectly controlled by the FBR. As shown in Fig. 3.11, there is always an optimum seed power that grants the maxi-mum output power. Thus, the differences in single-pass gain, either due to different pump intensity, different operating wavelength, or both, can easily be compensated by adjusting the FBR, which is a unique feature of the ffOPO.
In case that GVM is present, i.e. if ultrashort pulses are involved, the pulses may walk off temporally before backconversion sets on. This lim-its the quantitative applicability of the analytical model used for Fig. 3.11. For PPLN and PPLT there are different regimes of temporal walk-off (cf. Fig. 3.7), where signal and idler walk off into either the same direction with respect to the pump (λs& 1.5 µm) or oppositely (λs. 1.5 µm). In the latter case temporal walk-off is no issue, since backconversion will not oc-cur after one pulse splitting length. In the first case, however, signal and idler will walk off from the pump and substantial backconversion will occur, if the initial seed conditions are not optimal. In any case, the vari-able output coupler helps to suppress the undesired effects, by setting the initial conditions and ensuring operation at the maximum saturation of conversion efficiency. This will greatly enhance the output power sta-bility, since at this point of operation fluctuations of both seed and pump power show minimum impact on the signal output power.
3.3 O P T I C A L PA R A M E T R I C O S C I L L AT O R S 29
Tuning and Wavelength Stability
In contrast to OPAs, the resonantly oscillating signal in synchronously-pumped OPOs is subject to the intracavity GDD that is induced by the nonlinear crystal, the mirrors, and, generally, by any other dispersive ele-ment in the resonator. As the OPO resonator length lresis thus a function of the wavelength, but needs to be synchronized to the fixed resonator length of the pump laser, detuning this length leads to a shift in the oscil-lating signal wavelength λs
dlres= −c GDDrt(λs) dλs = −c D(λs) lfiberdλs = 2πc 2 λ2 s k2(λs) lfiberdλs, (3.50)
with the overall round-trip group delay dispersion GDDrtinps/nm(cf. Eq. 3.19 – 3.21). Integrating Eq. (3.50)
lres(λ) = l(λref) − Zλ
λref
GDDrt(λ′)dλ′, (3.51) provides the resonator length lresat a wavelength λ, relative to a refer-ence wavelength λref. As GDDrt(λ) is not necessarily a strictly mono-tonic function, the integration does not always result in an injective func-tion. This may be used to engineer a dual- or multi-signal-wavelength OPO system [39]. Note that in addition, the phase matching condition in the nonlinear crystal needs to be fulfilled for the oscillating signal wave-length. For ffOPOs, typical values are dldλres ≈ 50pm/µm, which is about 100 times lower compared to the classical OPO. Thus, the effects of vi-brations, thermal drifts or other distortions of the resonator length on the OPO center wavelength will be reduced by this factor.
3.4 N O N L I N E A R D Y N A M I C S I N O P T I C A L S Y S T E M S
Nonlinearities are substantial part of everyday life and are further found in all conceivable fields of science. Turbulences, traffic, coupled pendu-lums, but also the famous FEIGENBAUMdiagrams are examples of such systems [40]. Nonlinear systems in general may exhibit features such as bifurcations, hysteresis, bistability, limit cycles, period multiplication, and chaos. For this reason nonlinearities are usually avoided or mini-mized in optical systems, particularly in lasers, with the obvious excep-tion of spectral broadening and frequency conversion. In this secexcep-tion, we will focus on the special case of nonlinear feedback effects in optical res-onators that mainly arise from self-phase modulation (SPM, cf. A.1.1).
Generally, SPM modifies a pulse while propagating through the nonlin-ear feedback medium. Self-phase modulation affects the spectral power distribution, peak power, pulse duration, chirp, and the CEP. If the pulse is circulating in a resonator, it needs to be amplified periodically, e.g., ev-ery round-trip. Thus, when a pulse is amplified by a gain medium that is sensitive to changes in any of these parameters, the feedback and the gain dynamics couple to one another. Note that the ”amplification“ can also result from linear combination with a pump pulse, where a beam splitter would be the ”gain medium“. Due to the feedback-dependent gain, such a system can exhibit oscillations depending on its initial parameters.
In the simplest case, where the pulse is not modified at all (i.e. negligi-ble nonlinearity) and is reproduced every round-trip the system operates in ”steady-state“ (or formally also ”P1 cycle“, for ”period-1-cycle“). It is the situation that is found in mode-locked oscillators. If an optical system is prepared in a P2-cycle, it will emit a pulse train where every second pulse is identical. In contrast to the former case, this requires the nonlin-earity of the feedback to be nonzero. These two situations are illustrated in Fig. 3.12. The oscillation period is exactly twice the period of a single round-trip, which has led to the term ”period doubling“.
In a very simple, yet intuitive, but partially incomplete picture, the P2-cycle can be understood by thinking of a spectrally narrow gain window that interacts with the SPM broadened seed spectrum. Assume that the resonator contains a high energy pulse A. Driven by the feedback non-linearity, this pulse changes its spectrum such that at the former peak wavelength, the characteristic SPM dip occurs. Thus, this pulse will be an effectively weak seed pulse for the next round-trip, since the spectrally narrow gain window will only accept the weak center wavelengths as seed. Consequently, a weak pulse B is generated, which thus undergoes
3.4 N O N L I N E A R D Y N A M I C S I N O P T I C A L S Y S T E M S 31 gain
steady-state
OC 1/frep gainP2-cycle
OC 1/frep nonlinear feedback a bFigure 3.12.:Schematic illustration of the steady-state (a) and a P2-cycle (b). In the latter case
the distance between two adjacent pulses is still identical, i.e. the resonator round-trip time τrt=1/frep. However, it takes two round-trips and thus the effective period of the pulse train is doubled.
less nonlinear broadening due to SPM when passing the feedback path. Accordingly, it will represent a stronger effective seed compared to pulse A, although starting out with a lower pulse energy. As a result the seed obtained from pulse B will be amplified to a strong pulse A, which closes the cycle. Again, be aware that this simple picture neglects temporal and spatial effects, chirp, gain saturation, and any nonlinear feedback effects beyond SPM.
Attractors
For a proper choice of conditions a multitude of attractors can be ob-served, which are listed in table 4 and illustrated in Fig. 3.13. An attractor refers to a subspace of the phase space of a system, which ”traps“ the sys-tem on a certain trajectory in phase space, once it is reached. Such an attractor may consist of a single point (steady-state), a finite set of points (PN cycle), or a closed trajectory (limit cycle). Chaotic attractors include an entire volume of the phase space. The phase space diagrams shown in Fig. 3.13 relate the amplitude and phase of a single comb line (cf. 3.1.1) at a defined position in the resonator. Since all comb lines are in fixed phase relation the entire pulse is fully defined, if the optical spectrum and chirp are known. A thorough analysis of these attractors is provided
for instance in [41]. The RF spectrum as shown in 3.13(b) is an impor-tant measure to quickly identify the attractor that currently dominates the dynamics of the system.
φ A0 π π/2 3π/2 steady-state φ A0 π π/2 3π/2 P2 / P4 cylce φ A0 π π/2 3π/2 limit cycle φ A0 π π/2 3π/2 chaos phase space RF spectrum
frep f frep f frep f frep f
a
b
f /2rep
f /4rep fLC,1
fLC,2
Figure 3.13.:Qualitative illustration of possible attractors in phase space (a) with
correspond-ing RF spectrum. The phase can be interpreted as the phase of scorrespond-ingle frequency comb line at a defined position in the resonator. Illustration adopted from [41].
type of attractor
phase space
characteristics RF spectrum
steady-state initial state is reproduced every round-trip
single peak at frep
PNcycle
initial state is reproduced every (integer) N round-trips
peaks at m ·frep N (m integer)
limit cycle defined (periodic) trajectory in phase space
one or more peaks at arbitrary f <frep/2
chaos aperiodic distribution of
phase and amplitude flat plateau
Table 4.:Overview of different attractors with their typical characteristics in the phase space
and their appearance in the RF spectrum. Spectral components with f > frepare not consid-ered. The RF spectrum is symmetric with respect tofrep/2.
3.4 N O N L I N E A R D Y N A M I C S I N O P T I C A L S Y S T E M S 33
3.4.1 Modeling the ffOPO with Nonlinear Feedback
The standard approach in modeling a nonlinear optical resonator is to start with the nonlinear SCHRODINGER¨ equation [42] or the GINZBURG -LANDAUequation [43] and model the building blocks of the system. In case of the ffOPO these blocks are:
• parametric gain (crystal),
• linear loss (coatings, output coupler, mirrors, fiber coupling),
• third-order nonlinearity (feedback fiber), and
• linear loss (coatings, mirrors).
Hence, the nonlinear wave equation (Eq. 3.5) needs to take into account discrete linear losses, second- and third-order nonlinearity, and disper-sion. It is usually sufficient to model the evolution of the pulse enve-lope. As shown in Fig. 3.14, the feedback leads to a relation for the inci-dent signal amplitude before amplification En
s(0) and after amplification En−1
s (Lcry) in the nth round-trip
Ens(0) = tEn−1s (Lcry) Ins(0) = T In−1s (Lcry),
(3.52) with the round-trip amplitude transmission of the resonator t, which is related to the intensity transmission T via
|t|2= T . (3.53)
GD(λ)
n-1T
I (L )
s cry n= I (0)
sloss
OC
SPM(λ, )
I
s nI (L ) = G
s cry nI (0)
sT
I (0)
pI (L )
p cry n,outI
s= OC
nI (L )
s crynonlinear crystal
feedback
resonator
G
(
n,
I (0)
s nI (0)
p, L )
cryFigure 3.14.:Schematic elements of an OPO with symbolic building blocks for a round-trip
model based on the intensities In(z). For phase-sensitive processes, the electric field ampli-tudes En(z) need to be considered instead.
Generally, t will be a function of (wavelength-dependent) amplitude losses due to
• output coupling,
• resonator losses, and a phase factor considering
• linear phase shift,
• group delay (GD),
• nonlinear phase shift (SPM).
The parametric single-pass gain G = |g|2= Ins(Lcry) In
s(0) is required to com-pensate the effects of the round-trip transmission function
Ens(0) = tEn−1s (Lcry) = tgEn−1s (0) Ins(0) = T In−1s (Lcry) = T GIn−1s (0).
(3.54) In steady-state, the amplitudes will not change with the round-trip num-ber n except for a constant phase φn, giving
Ens(z) = En−1s (z) · exp(iφn) Ins(z) = In−1s (z).
(3.55) In this case, we obtain T G = 1. Note that for any other attractor En
s 6= En−1
s . For PN-cycles every Nthpulse is equal Ens(z) = En−Ns exp iφn,
Ins(z) = In−Ns .
(3.56) In this case, also the transmission t and the single-pass gain g will vary periodically.
Implementation of the Model
The above system is modeled using the commercially available software RP Pro Pulse V2, which uses a 1+1 dimensional numeric solver for the nonlinear SCHRODINGER¨ equation.
Table 5 gives an overview of the parameters that are used for the fol-lowing numeric investigations. The crystal length is chosen quite short
