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We proposed a new primary calibration method of a laser Doppler vibrometer (LDV) using the optical modulated excitation of an electro-optical modulator and validated this approach by comparing mechanical modulated excitation with homodyne laser interferometer at the National Metrology Institute of Japan (NMIJ). The velocity sensitivity of the LDV was evaluated with the uncertainty budget at a high-frequency range up to 1 MHz, which is indicated as the upper limit of 50 kHz in ISO (International Organization for Standardization) 16063–41. As the results, we confirmed that velocity sensitivities between optical and mechanical modulated excitations are flat within their uncertainty from 100 Hz up to 1 MHz. Moreover, in order to evaluate the accurate phase shift of the accelerometer up to 20 kHz using optical modulated excitation, we also revealed the reliability of two time shifts between homodyne and heterodyne laser interferometers.

The new definition of the kilogram was implemented on May 20, 2019. The kilogram is presently defined by a fixed value of the Planck constant. On the basis of the new definition, the kilogram will be realized at the National Metrology Institute of Japan by the X-ray crystal density method using 28Si-enriched spheres. For the realization, the volume of 28Si-enriched spheres is measured by optical interferometry. The sphere surface characterization by X-ray photoelectron spectroscopy and ellipsometry is also performed. The relative standard uncertainty of the realization is estimated to be 2.4 × 10−8. Details of the realization and future dissemination of mass standards in Japan based on the 28Si-enriched spheres are described.

The new definition of the kilogram was implemented on May 20, 2019. The kilogram is presently defined by the Planck constant. On the basis of the new definition, the kilogram will be realized at the National Metrology Institute of Japan by the X-ray crystal density method using 28Si-enriched spheres. For the realization, the volume of 28Si-enriched spheres is measured by optical interferometry. The sphere surface is also characterized by X-ray photoelectron spectroscopy and ellipsometry. The relative standard uncertainty of the realization is estimated to be 2.1 × 10−8. This corresponds to the realization of the new kilogram with a standard uncertainty of 21 μg for 1 kg. Details of the realization and future dissemination of mass standards in Japan based on the 28Si-enriched spheres are described.

We carry out infrared-infrared double-resonance spectroscopy of the $2{\ensuremath{\nu}}_{3}{A}_{1}\text{\ensuremath{-}}{\ensuremath{\nu}}_{3}$ band of methane. Ten transitions of Q(1) to Q(4) are observed with an 88.5-THz difference-frequency-generation (DFG) source, while the lower levels are pumped through the ${\ensuremath{\nu}}_{3}$ band transition by another 90.5-THz DFG source phase-locked to an optical frequency comb (OFC). The transition frequencies are determined using the OFC with an uncertainty of 10 kHz, which is an improvement of 4 orders of magnitude compared with the previous studies.

We report improved frequency ratio measurement with $^{87}$Sr and $^{171}$Yb optical lattice clocks at the National Metrology Institute of Japan (NMIJ). The $^{87}$Sr optical lattice clock is enhanced with several major modifications and is re-evaluated with a reduced uncertainty of $1.1\times10^{-16}$. We employed a $^{171}$Yb optical lattice clock with an uncertainty of $4\times10^{-16}$ that was developed for contributing to International Atomic Time (TAI). The measurement result is $\nu_{\mathrm{Yb}}/\nu_{\mathrm{Sr}}$ = 1.207 507 039 343 338 58(49)$_{\mathrm{sys}}$(6)$_{\mathrm{stat}}$ with a fractional uncertainty of $4.1\times10^{-16}$, which is 3.4 times smaller than our previous measurement result., Comment: 8 pages, 6 figures, 1 table

We report the status on an Yb optical lattice clock that was designed to run for a long-term operation towards contribution to the International Atomic Time (TAI) regularly. Nearly continuous operation of the Yb clock was realized with 90% uptime over several weeks. This enables to reduce the uncertainty in the link between the Yb clock and TAI to the low 10−16 level.

The new definition of the kilogram based on the Planck constant was implemented in May 2019. The National Metrology Institute of Japan (NMIJ) will realize the new kilogram by the X-ray crystal density method using a 28Si-enriched sphere. An international comparison of the realizations of the new kilogram by national metrology institutes was organized. As a participant in the comparison, the NMIJ realized the kilogram by the volume measurement and surface characterization of a 28Si-enriched sphere. Details of the measurement results and the uncertainty evaluation of the realization are described. The comparison of the realization results with previous realization results in 2016 is also provided.

The kilogram based on the Plank constant will be realized at the National Metrology Institute of Japan (NMIJ) by using the X-ray crystal density method. In this study, for the realization, the volume of a 28Si-enriched sphere, named AVO28-S5c, was measured. The volume was determined from diameter measurements taken in many directions using an optical interferometer. The apparent volume of the sphere, which does not consider the influence of the surface layers on the volume measurement, was determined with a relative standard uncertainty of $1.8\times 10^{-8}$ . The reproducibility of the kilogram realization is important for the reliable dissemination of the kilogram at the NMIJ. Therefore, the volume determined in this study was compared with that determined in 2016. These volumes were shown to agree with the uncertainty. This article provides details on the measurements, improvements of the interferometer, data analysis, and uncertainty evaluation.

We evaluate the uncertainty of our Sr optical lattice clock at the 10 -17 level, through robust clock operation with assistance of optical frequency combs. This clock will be operated for contribution to the International Atomic Time.

For the determination of the Avogadro constant by the x-ray crystal density method, an accurate volume measurement of 1 kg Si spheres is of primary importance. For this purpose, an optical interferometer was improved and used to measure the volumes of two 1 kg silicon spheres which were manufactured from a silicon crystal highly enriched in 28Si. The apparent volumes of the spheres, which do not take into account the influence of the surface layers on the volume measurement by interferometry, were determined with a relative standard uncertainty of as small as 2.0 × 10−8. The surface of the spheres was characterized by using an improved spectroscopic ellipsometer. By considering the influence of the surface layers, the core volumes of the spheres, which exclude the surface layers, were determined. These results were used for the determination of the Avogadro constant in 2015 as a framework organized by the International Avogadro Coordination project. This paper provides details on the measurements, the improvements made to the apparatus, the data analysis and the uncertainty evaluation.

We report the status of our 171Yb optical lattice clock. The frequency stability of the Yb lattice clock is $1\times10^{-14}(\tau$/s)-1/2 where $\tau$ is an averaging time. The systematic uncertainty of the Yb lattice clock is evaluated to be $3.6\times10^{-16}$. Towards contribution to the International Atomic Time, some contrivances are incorporated in order for the Yb lattice clock to run for a long term. For example, automatic relocking schemes are recently prepared for phase locking and injection locking.

We report an uncertainty evaluation of an $^{171}$Yb optical lattice clock with a total fractional uncertainty of $3.6\times10^{-16}$, which is mainly limited by the lattice-induced light shift and the blackbody radiation shift. Our evaluation of the lattice-induced light shift, the density shift, and the second-order Zeeman shift is based on an interleaved measurement where we measure the frequency shift using the alternating stabilization of a clock laser to the $\mathrm{6s^{2}\,^{1}S_{0}-6s6p\,^{3}P_{0}}$ clock transition with two different experimental parameters. In the present evaluation, the uncertainties of two sensitivity coefficients for the lattice-induced hyperpolarizability shift $d$ incorporated in a widely-used light shift model by RIKEN and the second-order Zeeman shift $a_{\mathrm{Z}}$ are improved compared with the uncertainties of previous coefficients. The hyperpolarizability coefficient $d$ is determined by investigating the trap potential depth and the light shifts at the lattice frequencies near the two-photon transitions $\mathrm{6s6p^{3}P_{0}-6s8p^{3}P_{0}}$, $\mathrm{6s8p^{3}P_{2}}$, and $\mathrm{6s5f^{3}F_{2}}$. The obtained values are $d=-1.1(4)$ $\mathrm{\mu}$Hz and $a_{\mathrm{Z}}=-6.6(3)$ Hz/mT$^{2}$. These improved coefficients should reduce the total systematic uncertainties of Yb lattice clocks at other institutes.

We describe frequency noise measurements for an optical frequency comb by using a delayed self-heterodyne method with a Michelson-type fiber-optic delay line interferometer without a low-noise reference laser. We measured the frequency noise power spectral densities (PSDs) for free-running and frequency-stabilized comb modes and estimated the uncertainties of the measurement results. For example, for the frequency-stabilized comb, the measured frequency noise PSD and its uncertainty were 23 dBHz2 Hz−2 and 0.66 dB, respectively, at a Fourier frequency of 10 kHz. We also measured the frequency noise PSDs of the comb modes by the conventional method using an ultrastable reference laser and compared the results with those that we measured with the delayed self-heterodyne method. The measurement results that we obtained with the two methods were consistent within their uncertainties, which shows that the delayed self-heterodyne method provides high reliability. This combined with its simplicity and user-friendliness suggest that the method has the potential to become a standard frequency noise measurement approach for frequency combs.

This paper investigates the history of statistical mechanics from the viewpoint of the development of the ensemble theory from 1871 to 1902. In 1871, Ludwig Boltzmann introduced a prototype model of an ensemble that represents a polyatomic gas. In 1879, James Clerk Maxwell defined an ensemble as copies of systems of the same energy. Inspired by H.W. Watson, he called his approach “statistical”. Boltzmann and Maxwell regarded the ensemble theory as a much more general approach than the kinetic theory. In the 1880s, influenced by Hermann von Helmholtz, Boltzmann made use of ensembles to establish thermodynamic relations. In Elementary Principles in Statistical Mechanics of 1902, Josiah Willard Gibbs tried to get his ensemble theory to mirror thermodynamics, including thermodynamic operations in its scope. Thermodynamics played the role of a “blind guide”. His theory of ensembles can be characterized as more mathematically oriented than Einstein’s theory proposed in the same year. Mechanical, empirical, and statistical approaches to foundations of statistical mechanics are presented. Although it was formulated in classical terms, the ensemble theory provided an infrastructure still valuable in quantum statistics because of its generality.

We report wavelength-dependent pump-induced responses of repetition and carrier-envelope-offset frequencies of a soliton fiber laser comb. Lasing wavelength adjustment can tune a fixed point and design controlling orthogonality between a pump-laser current and another actuator.

We demonstrate an 8-branch Er:fiber frequency comb with seven application ports, which can be individually optimized for applications with different wavelengths. The beat between the comb and a cw laser has a signal-to-noise ratio exceeding 30 dB at a resolution bandwidth of 300 kHz. The 8-branch frequency comb is used to perform frequency locking for four repumping and lattice lasers, and the frequency measurement of two clock lasers of strontium and ytterbium optical lattice clocks. We have achieved reliable optical lattice clock operation, thanks to the stable frequency locking and measurement obtained by using the 8-branch frequency comb. The developed frequency comb is a powerful experimental tool for various applications, including not only optical lattice clocks, but also research on quantum optics that use many frequency-stabilized lasers.

We have developed a simple method for measuring relatively large phase noise, which is usually observed in a RF signal down-converted from optical frequency. The noise floor is sufficiently low for observing phase noise of optical frequencies (−120 dBc/Hz) and its dynamic range is practically unlimited. A time-domain phase noise down-converted from optical frequencies is measured by using a high-speed digitizer with sampling rate of 220 MHz. The observed phase noise has difference only within 1 dB compared with that using a conventional method.

In multi-branch combs, the comb outputs from the branches suffer from different fiber noises, which often limit the uncertainty of the combs referring a highly-stable optical frequency. To overcome this limitation, we introduced fiber noise difference cancellation to multi-branch fiber combs. We detected and phase-locked the beat notes between the branch outputs and a common 1542 nm continuous wave laser. A piezo-electric transducer-based fiber stretcher was installed in each branch except for the branch used as the cancellation reference. We fabricated two quasi-identical combs with this mechanism and confirmed the relative frequency uncertainty by comparing them. The cancellation improved the frequency uncertainty to a low level of 10-20 at a 100000-s averaging time.

We demonstrate a self-referencing locking method for an optical frequency comb that leads to an offset-free frequency comb. It works with a standard f−2f interferometer and phase-locks the carrier–envelope offset frequency to the repetition frequency. This technique uses neither an acousto-optic modulator nor difference frequency generation. No synthesizer signals are needed aside from a reference signal to stabilize the repetition rate. We test the performance by comparing it with an independent comb system; it does not result in any performance degradation when compared with the conventional phase-locking method. The offset-free comb realized by this technique may become a powerful and universal tool for use in frequency-comb-related research.

We had reported in a previous paper the precision measurement of line profiles of a transition of 13 C acetylene in the near-infrared region, employing a tunable diode-laser spectrometer, the frequency of which is locked to an optical-comb signal [K.M.T. Yamada et al., J. Mol. Spectrosc. 249 (2008) 95–99]. In the present article we review the methodology for the experiment and the analysis in determining the Boltzmann constant from the Doppler width of a rovibrational transition on the basis of our experiences. Problems to be surmounted for further improvement in accuracy are discussed. To cite this article: K.M.T. Yamada et al., C. R. Physique 10 (2009).

We apply a modified adaptive sampling method for dual optical comb spectroscopy in order to make signal integration of interferogram in time domain for improvement of signal-to-noise ratio.

We performed an absolute frequency measurement of the $^1S_{0}$-$^3P_{0}$ transition in $^{87}$Sr with a fractional uncertainty of $1.2 \times 10^{-15}$, which is less than one third that of our previous measurement. A caesium fountain atomic clock was used as a transfer oscillator to reduce the uncertainty of the link between a strontium optical lattice clock and the SI second. The absolute value of the transition frequency is 429 228 004 229 873.56(49) Hz., Comment: accepted for publication in Journal of the Physical Society of Japan, 7 pages, 2 figures

Real-time measurement of the absolute frequency of continuous-wave terahertz (CW-THz) waves is required for characterization and frequency calibration of practical CW-THz sources. We proposed a method for real-time monitoring of the absolute frequency of CW-THz waves involving temporally parallel, i.e., simultaneous, measurement of two pairs of beat frequencies and laser repetition frequencies based on dual THz combs of photocarriers (PC-THz combs) with different frequency spacings. To demonstrate the method, THz-comb-referenced spectrum analyzers were constructed with a dual configuration based on dual femtosecond lasers. Regardless of the presence or absence of frequency control in the PC-THz combs, a frequency precision of 10-11 was achieved at a measurement rate of 100 Hz. Furthermore, large fluctuation of the CW-THz frequencies, crossing several modes of the PC-THz combs, was correctly monitored in real time. The proposed method will be a powerful tool for the research and development of practical CW-THz sources, and other applications., Comment: 27 pages

A continuous-wave optical parametric oscillator (OPO) was phase locked to an optical frequency comb in the 830-nm region. The optical frequency of the OPO was controlled by changing the cavity length of the pump laser. The residual phase noise under phase locking was 220 mradrms and the energy concentration to the carrier was 95%. Furthermore, the optical frequency fluctuations of a free-running OPO were measured by using an optical frequency comb that was phase locked to an atomic clock. The measured fluctuations were around 10 MHz in an hour.

We demonstrate an erbium-based offset-free frequency comb using a fiber-coupled acousto-optic modulator. The comb has two f–2f interferometers; one is for carrier-envelope offset beat detection, and the other is for frequency-shifted offset beat detection. The frequency was shifted by placing the acousto-optic modulator in front of an amplifier and a highly nonlinear fiber for spectral broadening. We confirmed that the offset frequency was stabilized at zero and measured it by shifting it from zero. The Allan deviations of the measured offset frequency were 0.10 and 0.18 Hz with a 1 s averaging time and using feedback and feed-forward stabilization, respectively.

The frequency ratio of the 1S0(F=1/2)-3P0(F=1/2) clock transition in 171Yb and the 1S0(F=9/2)-3P0(F=9/2) clock transition in 87Sr is measured by an optical-optical direct frequency link between two optical lattice clocks. We determined the ratio (\nu_{Yb}/\nu_{Sr}) to be 1.207 507 039 343 340 4(18) with a fractional uncertainty of 1.5x10^{-15}. The measurement uncertainty of the frequency ratio is smaller than that obtained from absolute frequency measurements using the International Atomic Time (TAI) link. The measured ratio agrees well with that derived from the absolute frequency measurement results obtained at NIST and JILA, Boulder, CO using their Cs-fountain clock. Our measurement enables the first international comparison of the frequency ratios of optical clocks, and we obtained a good agreement between the two measured ratios with an uncertainty smaller than the TAI link. The measured frequency ratio will be reported to the International Committee for Weights and Measures for a discussion related to the redefinition of the second., Comment: accepted for publication in Opt. Express. 8 pages, 3 figures, 1 table

We demonstrated a frequency measurement of CW-THz wave referring to THz frequency comb. Effectiveness of the proposed method is demonstrated by measurement of sub-THz test sources. The achieved precision of frequency measurement was 2.0*10-11.

We demonstrated a frequency measurement of CW-THz wave referring to THz frequency comb. Effectiveness of the proposed method is demonstrated by measurement of sub-THz test sources. The achieved precision of frequency measurement was 2.0*10-11.

Heterodyne interferometry of 61-m path-length based on two-color optical frequency combs is developed for air-refractive-index correction. Measured two-color optical-path-differences agreed with calculations with 10-11 for 10 hour. Corrected distance variation agreed with thermal expansion of base-plate.

We demonstrate an improved absolute frequency measurement of the 1S 0 -3P 0 clock transition at 578 nm in 171Yb atoms in a one-dimensional optical lattice. The absolute frequency is determined as 518 295 836 590 863.1(2.0) Hz relative to the SI second.

We introduce a method of polarization-sensitive dual-comb spectroscopy. This technique is a combination of dual-comb spectroscopy and rotating compensator polarimetry, in which the rotation frequency of the compensator is optimized to establish the frequency sidebands of each comb tooth. Thus, we can determine the polarization state of each comb tooth by analyzing the amplitudes and phases of the sidebands. For a frequency resolution of 48 MHz and a measurement time of 726.5 s, the precisions of the orientation and the ellipticity angles of the individual comb teeth are ∼0.01 rad.

We report absolute frequency measurements of 81 hyperfine components of the rovibrational transitions of molecular iodine at 578 nm using the second harmonic generation of an 1156-nm external-cavity diode laser and a fiber-based optical frequency comb. The relative uncertainties of the measured absolute frequencies are typically $1.4\times10^{-11}$. Accurate hyperfine constants of four rovibrational transitions are obtained by fitting the measured hyperfine splittings to a four-term effective Hamiltonian including the electric quadrupole, spin-rotation, tensor spin-spin, and scalar spin-spin interactions. The observed transitions can be good frequency references at 578 nm, and are especially useful for research using atomic ytterbium since the transitions are close to the $^{1}S_{0}-^{3}P_{0}$ clock transition of ytterbium.

Laser frequency stabilization and atom manipulation are important techniques for optical frequency metrology. We report on the development of an ytterbium (171Yb) and a strontium (87Sr) optical lattice clock at the National Metrology Institute of Japan (NMIJ). The result of absolute frequency measurement of the 171Yb lattice clock is shown. We also report on narrow-linewidth fiber-based frequency combs that are used not only to measure the absolute frequency of the optical lattice clocks but also to servo control the frequency of the lasers for cooling and trapping Yb and Sr atoms.

In the determination of the Avogadro constant using the x-ray crystal density method, volume measurements of Si spheres by optical interferometry play a crucial role. For the accurate volume determination, geometrical shape of the optical components of the interferometer has been evaluated by using ray-tracing method. To reduce the noise in the interference fringes, a new etalon has been developed. An optical frequency comb has been also introduced in the interferometer as the optical frequency standard for more reliable and precise volume measurement. Details of the improvements are described.

Long-path pulse-to-pulse interferometers of two-color frequency combs are developed using fundamental and second harmonics of a mode-locked fiber laser. Interferometric phase difference between two-color frequency combs was precisely measured by stabilizing the fundamental fringe phase by controlling the repetition frequency of the comb, and a stability of 10(-10) for 1000 s was achieved in the measurement of an optical path length difference between two wavelengths. In long-term measurements performed for 10 h, results of phase variation of interferometric measurements were highly consistent with the fluctuations in the calculated difference of refractive indices of air at two wavelengths with an accuracy of 10(-10). The difference between the measured optical distances corresponding to two wavelengths and the optical distance corresponding to the fundamental wavelength were used in the two-color method; high-accuracy self-correction of the fluctuation of refractive index of air was performed with an uncertainty of 5 × 10(-8) for 10-h measurements when the maximum refractive index change was on the order of 10(-6).

Experimental efforts to reduce the uncertainty of the ytterbium (171Yb) optical lattice clock at NMIJ, AIST are shown. The signal-to-noise ratio of the spectrum was increased by 10 times by an atom number normalization scheme.

We report as regards an optical-optical frequency transfer with relative frequency stability at 10−21-level by canceling fiber noise between plural optical amplifier branches.

We report the current status of our ytterbium optical lattice clock at the NMIJ, AIST. After the first measurement of the clock transition frequency and the estimation of the uncertainty, we have been improving our clock. For an increased signal to noise ratio of the observed spectrum, we employed an atom number normalization scheme. We stabilized the frequency of the lattice laser using a fiber-based optical frequency comb. We also stabilized the intensity of the lattice laser.

We have being developing optical frequency combs, especially fiber-based frequency combs, for laser frequency measurement and laser frequency control. We are also working on Yb and Sr optical lattice clocks for precision frequency metrology.

We demonstrate the spectroscopy of 171Yb atoms confined in a one-dimensional (1D) optical lattice and the absolute frequency measurement of the clock transition. The absolute frequency of the 1S 0 (F = 1/2) − 3P 0 (F = 1/2) clock transition in 171Yb is determined to be 518 295 836 590 864(28) Hz with respect to the SI second.

We have been developing a new clock laser system for Yb and Sr optical lattice clocks. The system is based on an infrared laser with a narrow linewidth, and a fiber-based frequency comb transferring the linewidth to required plural wavelengths. We demonstrate that fiber-based frequency combs with a multi-branch configuration can transfer laser linewidth to another wavelength at the millihertz level. An intra-cavity electro-optic modulator is employed to obtain a broad servo bandwidth for repetition rate control.

Experimental efforts to reduce the uncertainty of the ytterbium (171Yb) optical lattice clock at NMIJ, AIST are shown. Especially, atom number normalization scheme is presented.

We have developed a one-dimensional optical lattice clock using a fermionic isotope of ytterbium (171Yb). The absolute frequency of the 1S0 – 3P0 clock transition in 171Yb is determined with respect to the SI second.

We measure the absolute frequency of 12 sub-Doppler lines of the $P(6)$ and $P(7)$ transitions in the ${\ensuremath{\nu}}_{3}$ band of ${^{12}\text{C}\text{H}}_{4}$ using difference-frequency-generation (DFG) of $1.064\text{\ensuremath{-}}\ensuremath{\mu}\text{m}$ Nd:YAG and $1.55\text{\ensuremath{-}}\ensuremath{\mu}\text{m}$ DFB lasers and a fiber-based frequency comb referenced to International Atomic Time. The relative uncertainties are typically $9.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$, and the determined value for the $P(7)$ ${F}_{2}^{(2)}$ line agrees with the International Committee for Weights and Measures recommendation with a discrepancy of $(1.2\ifmmode\pm\else\textpm\fi{}8.4)\text{ }\text{kHz}$. The combination of the DFG spectrometer and the frequency comb enables determining energy levels as accurately as microwave spectroscopy.

We have developed a one-dimensional optical lattice clock with ultracold 171 Yb atoms. The absolute frequency of the 1 S 0 ( F = 1/2) - 3 P 0 ( F = 1/2) clock transition in 171 Yb is determined to be 518 295 836 590 864(28) Hz with respect to the SI second. Details of the experimental setups and atom trapping results are also described.

A stable light source obtained using sum-frequency generation (SFG) is developed for high-resolution spectroscopy at 578 nm. Hyperfine transitions of molecular iodine are observed by using the SFG light source with saturation spectroscopy. The light source is frequency stabilized to the observed hyperfine transition and achieves a stability of 2*10-12 for a 1-s averaging time. The absolute frequency of the light source stabilized on the a1 component of the R(37)16-1 transition is determined as 518304551833 (2) kHz. This transition serves as a frequency reference for the 1S0 - 3P0 optical clock transition in neutral ytterbium (Yb)., Comment: 8 pages

We demonstrate a precision frequency measurement using a phase-stabilized 120-km optical fiber link over a physical distance of 50 km. The transition frequency of the 87Sr optical lattice clock at the University of Tokyo is measured to be 429228004229874.1(2.4) Hz referenced to international atomic time (TAI). The measured frequency agrees with results obtained in Boulder and Paris at a 6*10^-16 fractional level, which matches the current best evaluations of Cs primary frequency standards. The results demonstrate the excellent functions of the intercity optical fibre link, and the great potential of optical lattice clocks for use in the redefinition of the second., 14 pages, 3 figures