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Realisation of the Metre by Using a Femtosecond Laser Frequency Comb: Applications in Optical Frequency Metrology

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Realisation of the Metre by Using a Femtosecond Laser Frequency Comb: Applications in Optical Frequency Metrology Int. J. Metrol. Qual. Eng. 8, 16 (2017) International Journal of © B. Samoudi, published by EDP Sciences, 2017 Metrology and Quality Engineering DOI: 10.1051/ijmqe/2017008 Available online at: www.metrology-journal.org RESEARCH ARTICLE Realisation of the metre by using a femtosecond laser frequency comb: applications in optical frequency metrology Bousselham Samoudi* National School of Applied Sciences of Safi, Cadi Ayyad University, Sidi Bouzid’s Road, BP 63, Safi 46000, Morocco Received: 18 September 2016 / Accepted: 2 May 2017 Abstract. The appearance of the frequency comb technology, awarded the Nobel Prize in Physics 2005, has enormously revolutionized the metrology of optical frequencies, eliminating the need for complicated frequency chains. By direct linking to the unit of time, the second, through frequency standards (Cs, Rb), by using femtosecond mode-locked lasers and frequency comb technology, the Spanish Centre of Metrology (CEM) has established a new way of practical realisation of the National Standard of Length, the metre. By stabilising and characterising two free parameters – the repetition frequency fr and the offset frequency f0, the frequency comb generator thereby was successfully put into operation. After such realization, the accuracy of the length unit will be increased in two orders of magnitude, that is 2 Â 10À13 instead of 2.1 Â 10À11. In this paper we present the results of applying comb generator to the absolute measurement of the three Zeeman stabilized He–Ne lasers operating at 633 nm with a nominal frequency of 473.612 THz. A comparison of these results with those obtained by the current system based on standard iodine stabilized lasers is in good compatibility. A treatment for the evaluation of measurement uncertainty of laser frequency in calibration using a comb in accordance with Guide of Uncertainty Measurement ISO/BIPM is also presented. Keywords: frequency combs / length standard / uncertainty / optical frequency metrology / mode-locked laser 1 Introduction (c) by means of one of the radiations from a given list, whose stated wavelength in vacuum or whose stated fi frequency can be used with the uncertainty shown, The de nition of the unit of length and its practical provided that the given specifications and accepted realisation are based on the adopted value of the speed of good practice are followed. light, c0 = 299 792 458 m/s, and the frequency of an optical transition. Thus, length measurements are intrinsically Various sources of radiation have been recommended as related to the unit of time. It was adopted by the 17th standards of wavelength and have been updated by the Conference Générale des Poids et Mesures in 1983 [1]. At the CIPM over time [2,3]. One of the most important same time, the Comité International des Poids et Mesures recommended radiations in the field of length metrology (CIPM) made recommendations for the practical realization and worldwide precision measurements is that at 474 THz of the metre, referred to as the mise en pratique (in later text (633 nm) from a He–Ne laser, stabilized on an absorbing fi 127 referenced as MeP) of the definition. The metre should be hyper ne component in I2. This laser is used in many realized by one of the following methods [2]: laboratories around the world as a practical means of (a) by means of the length l of the path travelled in vacuum realizing the SI metre and is commonly used for calibrating by a plane electromagnetic wave in a time t; this length the frequency of lasers employed in length measurement is obtained from the measured time t, using the relation and precision measurement in atomic physics. The lasers l = c0 · t and the value of the speed of light in vacuum standards work according to method (c) of the Comité c0 = 299 792 458 m/s; consultatif de Longueurs (CCL)/CIPM recommendation (b) by means of the wavelength in vacuum l of a plane for the realization of SI of the metre definition. electromagnetic wave of frequency f; this wavelength is To check the values of the standards and the possible obtained from the measured frequency f using the improvement of uncertainty, one needs to absolutely relation l = c /f and the value of the speed of light in measure laser frequency (hundreds of terahertz) relative to 0 fi vacuum c0 = 299 792 458 m/s; caesium clocks (realizing SI de nition of the second). This measurement (method b in MeP) was made easier by the invention of optical frequency comb technique [4–6] * Corresponding author: [email protected] awarded the Nobel Prize in Physics 2005. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 B. Samoudi: Int. J. Metrol. Qual. Eng. 8, 16 (2017) With this system (frequency comb), the Spanish Centre Because the envelope is periodic of Metrology (CEM) is establishing a new practical hiX realization of the metre improved accuracy in two orders EðtÞ¼Re An expði2pðf þ nf Þt ; ð3Þ of magnitude (i.e. 2 Â 10À13 instead of 2.1 Â 10À11) with n c r respect to the current system based on iodine stabilized where A are Fourier components of A(t). This equation lasers [7,8]. n shows that, under the assumption of a periodic pulse In this paper, the state of art of femtosecond optical envelope, the resulting spectrum consists of a comb of laser frequency comb, including a theoretical background study, modes that are separated by the pulse repetition frequency. fl is brie y described. The absolute frequency measurement Since fc is not necessarily an integer multiple of fr, the of Zeeman-stabilized He–Ne lasers operating at 633 nm modes are shifted with respect to the exact harmonics by the frequency comb and the comparison of the results of the repetition frequency by an offset which can be obtained by the current method of beating frequency, using selected in such a way that f0 < fr simply by renumbering a standard iodine stabilized lasers (CEM2) have been the modes: reported. In both cases, a stability study of the signal beat ¼ þ ; ð4Þ frequency is presented. Finally, we estimate the measure- fn f0 nfr ment uncertainty for comb and laser frequency under 6 calibration using the Guide of Uncertainty Measurement where n is a large integer of order 10 that indexes the (GUM) ISO/BIPM. comb line. This equation maps two radio frequencies (RF) fr and f0 onto the optical frequencies fn. Although the detection of fr is rather effortless and usually lies between 2 Theoretical background a few ten MHz and a few GHz depending on the length of the laser resonator, the determination of f0 is far more challenging to access unless the frequency comb contains Optical frequency combs are routinely used to generate more than an optical octave. f then is detected by the self- optical radiation with a frequency uncertainty limited only 0 referencing technique (Fig. 2) with an f–2f interferometer by the uncertainty of the master clock that sets the setup [9,10]. Therefore, a mode, with mode number n at repetition frequency of the combs. These frequency combs, the red wing of the comb, whose frequency is given working in mode-lock regime, have to be designed so that according to equation (4), is frequency doubled in a non- the round trip time for all longitudinal modes is the same. linear crystal. If the frequency comb covers a full optical To understand the mode structure of a femtosecond octave, a mode with the number 2n should oscillate frequency comb and the techniques applied for its simultaneously at stabilization one can look at the idealized case of a pulse circulating in a laser cavity with length L and carrier ¼ þ 2 : ð5Þ f2n f0 nfr frequency fc as shown in Figure 1. The output of this laser is a sequence of pulses that are essentially copies of the same The beat note between the frequency doubled mode and pulse separated by the round trip time given by: the mode at 2n yields the offset frequency T ¼ 2L=vg; ð1Þ 2ð þ Þð þ 2 Þ¼ : ð6Þ f0 nfr f0 nfr f0 fi where vg is the cavity mean group velocity de ned by the The frequency of a laser under calibration (continuous round trip time and the cavity length. wave), fcw, is determined by observing a beat frequency, The pulses however are not quite identical. This is fbeat, with the nearest mode of the frequency comb system. because the pulse envelope A(t) propagates with v while g Therefore, the fcw is expressed as the carrier wave travels with its phase velocity. As a result, the carrier shifts with respect to the pulse envelope after ¼ ± ± : ð7Þ fcw nfr f0 fbeat each round trip by a phase angle Df as shown in Figure 1. The shape of the pulse train in time domain is related to the The signs of the equation (5) depend on the measurement amplitude and phase spectra by complex Fourier trans- conditions and can be determined by changing f0 and fr form. while monitoring the resulting change in the beat signal. Let us describe the time evolution of the electric field at If the visible cw lasers are measured after second one point in space by carrier frequency fc and periodic harmonic generation (SHG), that is our case, equation (5) amplitude/envelope A. Unlike the envelope function, has to be modified to take the SHG process into fi which provides us with a more rigorous de nition of the consideration.
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