Interferometric applications with a femtosecond frequency comb laser in complementary spaces

In the recent years, precision measurement techniques in the field of optical science have improved with the advances in the phase stabilization of femtosecond lasers. Due to their unique properties, phase stabilized femtosecond lasers, also known as frequency comb lasers, are used as a versatile tools, not only for time and frequency metrology, but also in fundamental physics, high-precision spectroscopy, distance metrology and laser noise characterization. In this thesis we mainly focus on two fundamental applications that are encountered in optical metrology: In the first part, we present a study of the formation of correlation patterns in a dispersive unbalanced Michelson interferometer using a frequency comb laser as a source. This was intended for accurate distance measurement in dispersive media. In the second part, we focus on the precise determination of laser spectra where we demonstrate measurement of single frequency comb modes using a conventional Fourier transform spectrometer. This can be used for broadband molecular spectroscopy with high resolution. The study of correlation patterns is specifically intended for distance measurement in dispersive media. There are several ways to determine distances using optical means. With pulsed lasers, the measurement of the time of flight-of-pulses provides a good estimation on the pulse propagation distance within a non-ambiguity range that corresponds to the pulse-to-pulse distance or equivalently the laser cavity length. For precision measurements, interferometric techniques are used to provide accurate information on the distance between the source and the target. These interferometric techniques can be implemented in two different domains: (1) in the time domain where the interference signal or the correlation pattern (in case of pulsed lasers) is measured using a single detector. (2) In the frequency domain where a spectrometer is placed at the output of the interferometer to provide spectral interference patterns. Regardless of the measurement method, the acquired data require further analysis in order to get the final results. Moreover, placing the measurement system in a dispersive medium such as air adds more complexity to the analysis. In the first part of this thesis, we study the case of distance measurement using correlation patterns where we focus on the effects of dispersion on the measured data. We will mainly try to understand (1) how correlations are formed using the frequency comb laser as a source, (2) whether the definition of the group refractive index is sufficient to correct for the dispersion effects and (3) to understand how the shape of the correlation patterns vary with the propagation distance in the dispersive medium. Therefore, we study the formation of correlation patterns using a rigorous mathematical model where results will be compared to experimental measurements. We first show that the discrete spectrum of the frequency comb laser yield correlation patterns that are mathematically described by a discrete series. This series shows that the knowledge of the electric field of the optical pulses is unnecessary and only an accurate knowledge of the power spectral density of the laser source together with the refractive index of the dispersive medium are sufficient to provide an accurate model of the correlation patterns. Using the discrete model of correlation patterns we have carried out simulation work by considering various spectral shapes, environmental conditions and distances. As a result, we show that for accurate distance determination, precise knowledge of the position of the brightest fringe that is the fringe having maximum contrast, of the correlation pattern is required. In vacuum, the distance can be accurately determined using the repetition rate of the laser and the position of the brightest fringe. In the case of pulse propagation in a dispersive medium such as air, the position of the brightest fringe is influenced by the shape of the power spectral density and the environmental parameters. Only in the case of perfectly symmetric power spectral densities, one can use the group refractive index definition to correct for the dispersion effects. With perfectly symmetric spectra, correlations appear to broaden linearly where no distortion effects can be seen even if the phase refractive index variation is nonlinear. Distortion effects are only seen when the power spectral density of the laser source has an asymmetric distribution. Moreover, we have experimentally and numerically observed that after a certain delay distance in air, the envelope of the correlation pattern takes a particular structure which is the shape of the power spectral density of the laser source. The discrete model of correlation patterns has been shown to be accurate enough to reproduce the correlation patterns as compared to experimental data. From this model we were able to extract the absolute measured distance in air. However, the discrete model has failed to explain some physical effects such as the shift of the central fringe and the envelope shape convergence after a certain delay distance in the dispersive medium. To explain these results, we extend the discrete model to a continuous model of cross-correlation functions and use the Poisson summation formula. This allows us to show that even for a homogenous dispersive medium the position of the brightest fringe varies non-linearly for small delay distances and stabilizes for longer ones. The distance where non-linear effects are important is shown to be dependent on the properties of the dispersive media and the initial spectrum of the transmitted pulse. In case of very large delay distances the particular values of the frequencies that are present in the spectrum play an important role since only specific frequencies contribute to specific fringes in the cross-correlation. These specific frequencies will be addressed as stationary frequencies where the method of stationary phase is applied to the continuous model aiming to describe the correlation patterns in the asymptotic regime of large delay distances. In the second part of this work, a new experimental scheme has been built allowing the measurement of single frequency comb modes with sub-MHz resolution. This study is specifically intended for molecular spectroscopy with high resolution using a broadband light. Spectroscopy involves the determination of accurate spectral lines on a frequency scale. The experimental setup consists of a conventional Fourier transform spectrometer where the measuring arm of the Michelson interferometer has a scanning length of 10 m. Depending on the repetition rate of the frequency comb laser (typically 100 MHz to 1GHz ), many correlation patterns (4 to10) are captured within this large scanning length. By applying a Fourier transform on the measured train of correlations (or series of correlations) one can obtain the frequency comb mode resolved spectrum. We also provide a complete study on the analysis steps that are required to get the final spectrum such as, accurate sampling, apodization and phase correction. Results have been validated by measuring the absorption lines of molecular transitions using a sample cell. As time and frequency domains are complementary, we therefore entitle this thesis "Interferometric applications with a femtosecond frequency comb laser in complementary spaces".