New rapidly wavelength-swept light sources for optical coherence tomography and picosecond pulse generation

This thesis deals with research on novel, semiconductor-based, ultrafast and widely tunable wavelength-swept light sources with respect to different applications. The main focus was on the young technology of Fourier domain mode locked (FDML) lasers, where the insertion of a kilometer-long fiber delay line enables to tune a narrowband spectral filter synchronously to the roundtrip time of light in the resonator. In this way, very high sweep speeds become feasible. A very successful application in the field of biomedical imaging is optical coherence tomography (OCT), where FDML lasers allow for very large image acquisition rates. One important part of the research work was the development and characterization of novel concepts of wavelength-swept light sources improving performance and applicability in OCT. In this context, two novel modes of operation of FDML lasers have been demonstrated. On the one hand, an FDML laser with a highly linear time-frequency sweep characteristic was realized for the first time and allowed for OCT imaging at 1300 nm based on simplified numerical image processing. On the other hand, the first subharmonic FDML laser was implemented and successfully used for OCT imaging at 1300 nm. Here, light passes the same fiber delay line several times during each laser cavity roundtrip. In case of reduced sweep range, subharmonic FDML operation enabled an inherent multiplication of the effective sweep rate by a factor of ten, reaching 570 kHz. Another important achievement was the demonstration of a new type of ultrafast wavelength-swept light sources, where superluminescent light alternately passes a cascade of different gain elements and spectral filters which have to be tuned out of phase in order to compensate for the transit time of light. Different implementations operated at 1300 nm and at 1060 nm enabled effective sweep rates of up to 340 kHz. Ultrafast OCT imaging of the human retina was shown. The second part of the research work focused on the demonstration and investigation of a novel approach of short pulse generation, where light within the wavelength sweeps of an FDML laser is temporally compressed by a subsequent pass through 15 km of highly dispersive fiber. The achievable temporal pulse width was an indicator for the coherence properties and the quality of mode-locking of the FDML laser. This became evident in the very critical dependence on the FDML sweep frequency as well as due to the results of comparable pulse generation experiments based on using an incoherent wavelength-swept light source. With a dispersion compensated FDML laser, operated at 1560 nm, pulse durations of 60-70 ps at a repetition rate of 390 kHz were achieved. Although the generation of bandwidth-limited pulses was not feasible, it was shown that the electric field within the wavelength sweeps of the FDML laser must at least be partially coherent. Due to remaining uncompensated higher order chirp, the optical bandwidth was limited to 6 nm and the pulse energy was restricted. Pulse energies of 5.6 nJ have been achieved using erbium-doped fiber amplification prior to temporal compression

Snap Rounding of Bézier Curves

We present an extension of snap roundingfrom straight-line segments (see Guibas and Marimont, 1998)to Bézier curves of arbitrary degree, and thus the first method for geometric roundingof curvilinear arrangements.Our algorithm takes a set of intersecting Bézier curvesand directly computes a geometric rounding of their true arrangement, without the need of representing the true arrangement exactly.The algorithm's output is a deformation of the true arrangementthat has all Bézier control points at integer pointsand comes with the same geometric guarantees as instraight-line snap rounding: during rounding, objects do not movefurther than the radius of a pixel, and features of thearrangement may collapse but do not invert

Real root isolation for exact and approximate polynomials using descartes' rule of signs

Collins und Akritas (1976) have described the Descartes method for isolating the real roots of an integer polynomial in one variable. This method recursively subdivides an initial interval until Descartes' Rule of Signs indicates that all roots have been isolated. The partial converse of Descartes' Rule by Obreshkoff (1952) in conjunction with the bound of Mahler (1964) and Davenport (1985) leads us to an asymptotically almost tight bound for the resulting subdivision tree. It implies directly the best known complexity bounds for the equivalent forms of the Descartes method in the power basis (Collins/Akritas, 1976), the Bernstein basis (Lane/Riesenfeld, 1981) and the scaled Bernstein basis (Johnson, 1991), which are presented here in a unified fashion. Without losing correctness of the output, we modify the Descartes method such that it can handle bitstream coefficients, which can be approximated arbitrarily well but cannot be determined exactly. We analyze the computing time and precision requirements. The method described elsewhere by the author together with Kerber/Wolpert (2007) and Kerber (2008) to determine the arrangement of plane algebraic curves rests in an essential way on variants of the bitstream Descartes algorithm; we analyze a central part of it.Collins und Akritas (1976) haben das Descartes-Verfahren zur Einschließung der reellen Nullstellen eines ganzzahligen Polynoms in einer Veränderlichen angegeben. Das Verfahren unterteilt rekursiv ein Ausgangsintervall, bis die Descartes'sche Vorzeichenregel anzeigt, dass alle Nullstellen getrennt worden sind. Die partielle Umkehrung der Descartes'schen Regel nach Obreschkoff (1952) in Verbindung mit der Schranke von Mahler (1964) und Davenport (1985) führt uns auf eine asymptotisch fast scharfe Schranke für den sich ergebenden Unterteilungsbaum. Daraus folgen direkt die besten bekannten Komplexitätsschranken für die äquivalenten Formen des Descartes-Verfahrens in der Monom-Basis (Collins/Akritas, 1976), der Bernstein-Basis (Lane/Riesenfeld, 1981) und der skalierten Bernstein-Basis (Johnson, 1991), die hier vereinheitlicht dargestellt werden. Ohne dass die Korrektheit der Ausgabe verloren geht, modifizieren wir das Descartes-Verfahren so, dass es mit "Bitstream"-Koeffizienten umgehen kann, die beliebig genau angenähert, aber nicht exakt bestimmt werden können. Wir analysieren die erforderliche Rechenzeit und Präzision. Das vom Verfasser mit Kerber/Wolpert (2007) und Kerber (2008) an anderer Stelle beschriebene Verfahren zur Bestimmung des Arrangements (der Schnittfigur) ebener algebraischer Kurven fußt wesentlich auf Varianten des Bitstream-Descartes-Verfahrens; wir analysieren einen zentralen Teil davon

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based method, and \sOB(N^{12}) for two subresultant-based methods: this notation ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure

True logarithmic amplification of frequency clock in SS-OCT for calibration

With swept source optical coherence tomography (SS-OCT), imprecise signal calibration prevents optimal imaging of biological tissues such as coronary artery. This work demonstrates an approach using a true logarithmic amplifier to precondition the clock signal, with the effort to minimize the noises and phase errors for optimal calibration. This method was validated and tested with a high-speed SS-OCT. The experimental results manifest its superior ability on optimization of the calibration and improvement of the imaging performance. Particularly, this hardware-based approach is suitable for real-time calibration in a high-speed system where computation time is constrained

Frequency comb swept lasers

We demonstrate a frequency comb (FC) swept laser and a frequency comb Fourier domain mode locked (FC-FDML) laser for applications in optical coherence tomography (OCT). The fiber-based FC swept lasers operate at a sweep rate of 1kHz and 120kHz, respectively over a 135nm tuning range centered at 1310nm with average output powers of 50mW. A 25GHz free spectral range frequency comb filter in the swept lasers causes the lasers to generate a series of well defined frequency steps. The narrow bandwidth (0.015nm) of the frequency comb filter enables a ~-1.2dB sensitivity roll off over ~3mm range, compared to conventional swept source and FDML lasers which have −10dB and −5dB roll offs, respectively. Measurements at very long ranges are possible with minimal sensitivity loss, however reflections from outside the principal measurement range of 0-3mm appear aliased back into the principal range. In addition, the frequency comb output from the lasers are equally spaced in frequency (linear in k-space). The filtered laser output can be used to self-clock the OCT interference signal sampling, enabling direct fast Fourier transformation of the fringe signals, without the need for fringe recalibration procedures. The design and operation principles of FC swept lasers are discussed and designs for short cavity lasers for OCT and interferometric measurement applications are proposed.National Institutes of Health (U.S.) (R01-CA75289-12)National Institutes of Health (U.S.) (R01-EY011289-24)United States. Air Force Office of Scientific Research (FA9550-07-1-0014)United States. Dept. of Defense. Medical Free Electron Laser Program (FA9550-07-1-0101)National Science council of Taiwan. Taiwan Merit ScholarshipCenter for Integration of Medicine and Innovative Technolog

Optimization of excitation of fiber Fabry–Perot tunable filters used in swept lasers using a phase-correction method

In this paper, we investigate a phase correction method for compensation of the nonlinearity of conventional wavelength swept laser sources based on a fiber Fabry-Perot tunable filter as a wavelength selective element. A triangular waveform signal is commonly used to drive the filter. We however extract the zero crossings from the interferograms and modify the shape of the triangular signal accordingly. This algorithm was tested for different values of the optical path length difference (OPD) in the interferometer set-up. Significant compensation for the nonlinearity of the filter was obtained

Corneal topography with high-speed swept source OCT in clinical examination

We present the applicability of high-speed swept source (SS) optical coherence tomography (OCT) for quantitative evaluation of the corneal topography. A high-speed OCT device of 108,000 lines/s permits dense 3D imaging of the anterior segment within a time period of less than one fourth of second, minimizing the influence of motion artifacts on final images and topographic analysis. The swept laser performance was specially adapted to meet imaging depth requirements. For the first time to our knowledge the results of a quantitative corneal analysis based on SS OCT for clinical pathologies such as keratoconus, a cornea with superficial postinfectious scar, and a cornea 5 months after penetrating keratoplasty are presented. Additionally, a comparison with widely used commercial systems, a Placido-based topographer and a Scheimpflug imaging-based topographer, is demonstrated

Ultrahigh speed 1050nm swept source / Fourier domain OCT retinal and anterior segment imaging at 100,000 to 400,000 axial scans per second

We demonstrate ultrahigh speed swept source/Fourier domain ophthalmic OCT imaging using a short cavity swept laser at 100,000 – 400,000 axial scan rates. Several design configurations illustrate tradeoffs in imaging speed, sensitivity, axial resolution, and imaging depth. Variable rate A/D optical clocking is used to acquire linear-in-k OCT fringe data at 100kHz axial scan rate with 5.3um axial resolution in tissue. Fixed rate sampling at 1 GSPS achieves a 7.5mm imaging range in tissue with 6.0um axial resolution at 100kHz axial scan rate. A 200kHz axial scan rate with 5.3um axial resolution over 4mm imaging range is achieved by buffering the laser sweep. Dual spot OCT using two parallel interferometers achieves 400kHz axial scan rate, almost 2X faster than previous 1050nm ophthalmic results and 20X faster than current commercial instruments. Superior sensitivity roll-off performance is shown. Imaging is demonstrated in the human retina and anterior segment. Wide field 12×12mm data sets include the macula and optic nerve head. Small area, high density imaging shows individual cone photoreceptors. The 7.5mm imaging range configuration can show the cornea, iris, and anterior lens in a single image. These improvements in imaging speed and depth range provide important advantages for ophthalmic imaging. The ability to rapidly acquire 3D-OCT data over a wide field of view promises to simplify examination protocols. The ability to image fine structures can provide detailed information on focal pathologies. The large imaging range and improved image penetration at 1050nm wavelengths promises to improve performance for instrumentation which images both the retina and anterior eye. These advantages suggest that swept source OCT at 1050nm wavelengths will play an important role in future ophthalmic instrumentation.National Institutes of Health (U.S.) (5R01-EY011289-23)National Institutes of Health (U.S.) (5R01-EY013178-10)National Institutes of Health (U.S.) (2R01-EY013516-07)National Institutes of Health (U.S.) (1R01-EY019029-02)United States. Air Force Office of Scientific Research (Contract Number FA9550-07-1-0014)United States. Dept. of Defense. Medical Free Electron Laser Program (Contract Number FA9550-07-1-0101