A phase-modulated interferometer for high-precision spectroscopy

We present a novel spectroscopic method based on a phase-modulated interferometer which is suitable for the high-precision measurement of absorption and index of refraction profiles. A comparison with competing methods, that is with interferometry and FM spectroscopy, is given. The combination of these two methods, the phase-modulated interferometer, is shown to be best suited for experiments aiming at the realization of novel optical media, for example media exhibiting an ultra-large index of refraction or strong dispersion without absorption. The theory of operation and a theoretical and experimental signal-to-noise analysis are presented. Current detection limits for optical phase shift and relative absorption are 1.1×10−5 rad/√Hz and 2×10−5 /√Hz, respectively. We demonstrate the effectiveness of this novel technique by investigating the absorption and index of refraction profiles of the 4s2 1S0→ 4s4p P1 resonance transition at 423 nm in calcium. PACS: 07.65; 07.57.Pt; 32.70.Jz Over the last years it has been shown both theoretically and experimentally that the optical properties of atoms and molecules can be “designed” simply by introducing atomic coherence or utilizing quantum interference. Examples are cancellation of absorption [1], enhancing the index of refraction and the dispersion [2–4] and the realization of ultralarge non-linearities without absorption [5]. These media provide a variety of new applications, such as an ultra-sensitive magnetometer [6] or high-finesse broadband optical cavities (white-light cavities [7]). For their realization, precise knowledge and control of the absorption coefficient and of the index of refraction of these media are necessary. Since it overcomes some of the problems related to competing spectroscopic methods while offering the potential for shot-noiselimited measurement of optical spectra, the novel spectroscopic method that we present in this paper is best suited for the measurement of these properties. Among the large number of existing spectroscopic methods there are only two providing highly sensitive and simultaneous detection of the absorption and phase shift (index of refraction) which a “probe field” experiences as it passes through the medium under investigation. The first method is given by the category of phase and/or amplitude modulation techniques whereas the second is an interferometric method. Based on the pioneering work of Bjorklund [9] phase modulation techniques in the optical domain have been used and optimized for about two decades [8]. By choosing the phase modulation index M and the modulation frequency ωm one can distinguish between two distinct regimes of phase modulation spectroscopy. The wavelength modulation spectroscopy utilizes a large phase modulation index (M 1) and a modulation frequency small compared to the width Γ of spectral feature under investigation: M×ωm Γ . This method reveals only the derivative of the absorption with respect to the optical frequency and does practically not provide information about the index of refraction [10]. The second limiting case where M 1 and a ωm Γ , is referred to as frequency modulation spectroscopy (FMS). For this case, only the carrier and the first-order modulation sidebands of the phase-modulated optical field have to be considered [10]. When passing through the sample, these three spectral components will experience a certain phase shift and damping, which can be described by a complex amplitude transmission coefficient Tn = e−δn × eiφn , where n =−1, 0,+1 designates the low-frequency, the carrier, and the high-frequency component, respectively. Assuming small variation of the absorption and the phase shift with frequency (|δn− δn+1|, |φn−φn+1| 1), the beat signal at the modulation frequency detected with a fast photodiode takes the form [10]: Iωm(t)∝ M (δ−1− δ+1) cosωm t +M (φ−1+φ+1−2φ0) sinωm t . (1) From (1) it follows that a simultaneous measurement of the phase shift φ(ω) (index of refraction) and the absorption δ(ω) can only be achieved when one of the sidebands is swept across the spectral feature of interest, with the carrier and the other sideband not experiencing any phase shift or damping at all. One of the most important features of FMS is the