1. Introduction

Absolute length calibration of gauge blocks traceable to the definition of meter is an important task of the national metrology institutes responsible for providing reliable length artifacts for industrial use. The calibration task has been accomplished by way of optical interferometry with various working techniques that have been progressively evolving in response to the ever-increasing demand on calibration accuracy. In early days, multi-wavelength interferometry was popularly used with a sequence of wavelengths selected from a spectral lamp by use of a diffraction grating [1]. Traditional homodyne interferometry adopting a single wavelength-stabilized laser was also exploited in combination with a white light source providing short-coherence start/stop trigger signals needed for absolute determination of the length of a gauge block [2]. There were several other techniques, found in literature but of less wide use, such as multi-frequency heterodyne interferometry [3], two-side ring interferometry with a modified Michelson-or Fizeau-type interferometer [4], and differential wavelengthscanning heterodyne interferometry [5]. Nowadays the state-of-the-art practice being most widely used worldwide is based on the long-used multi-wavelength interferometry, in which the conventional spectral lamp is being replaced by multiple lasers stabilized to the welldefined absorption lines of selected atoms or molecules [6].

In this paper we propose a new way of gauge block calibration, exploiting the concept of optical frequency synthesizer based on the optical comb of a femtosecond pulse laser [7–9]. As the wavelength ruler, the optical comb is stabilized by locking the pulse repetition rate to the Rb clock of frequency standard. At the same time, the carrier offset frequency is also secured to the same Rb clock through a self-referencing f-2f interferometer fed by a photonic crystal fiber broadening the input spectrum over an octave span. Then a single-frequency laser generated from a tunable external-cavity laser diode is employed as the working source light, whose frequency is tuned consecutively to a sequence of selected modes of the stabilized optical comb. Then the principle of multi-wavelength interferometry is implemented so as to determine the absolute length of gauge blocks with calibration uncertainty readily traced to the Rb clock of frequency standard.

2. Optical frequency synthesizer

Figure 1 shows the schematic of the hardware system configured in this investigation for the task of absolute length calibration of gauge blocks. The hardware system consists of three main units; a femtosecond pulse laser source, a tunable external-cavity laser diode (ECLD), and a gauge block interferometer. The femtosecond laser is to provide an optical comb, all the modes of which are collectively locked to the Rb clock of frequency standard. The ECLD unit is to produce an independent single-frequency laser beam, of which frequency is precisely tuned to a series of selected modes of the optical comb in sequence. The ECLD laser is used as the working source for the gauge block interferometer configured to perform the absolute length calibration on the basis of multi-wavelength interferometry.

The femtosecond pulse laser source in use (Del Mar Photonics, Trestles-50) contains a Ti:sapphire (Ti:Al₂O₃) crystal, emitting a train of pulses of 35 fs duration with a central wavelength of 780 nm at a repetition rate of 81 MHz. The ultrashort pulse train yields an optical comb of 24 THz spectral width being centered at 384 THz with a uniform mode spacing of 81 MHz. For stabilization of the optical comb, the pulse repetition rate f _r is locked to a Rb clock signal at 81 MHz by translating the output coupler of the oscillator cavity using the PLL (Phase Locked Loop) control technique. At the same time, the carrier offset frequency f _o caused by the pulse-to-pulse phase shift is measured through a self-referencing f-2f interferometer and secured to the same Rb clock signal by adjusting the tilt angle of the cavity end mirror. A photonic crystal fiber (Crystal Fibre, NL-PM-750) is used to broaden the spectrum of the optical comb over an octave span for adequate operation of the f-2f interferometer. This scheme of frequency stabilization, first proposed in 2000 [8], allowed all the comb modes in our hardware setup to be collectively stabilized to a stability of 1.3×10^–12 at 10 s. This level of frequency stabilization is found enough to be used as the wavelength ruler for the absolute length calibration of gauge blocks as will be discussed later on.

 

Fig. 1. Overall hardware configuration for absolute length calibration of gauge blocks using the optical comb of a femtosecond laser. Blue dotted lines represent electric signals and orange curved lines are optical fibers. PLL: phase locked loop, APD: avalanche photo detector, ECLD: external cavity laser diode, CCD: charge coupled detector, CL: collimation lens, BS: beam splitter, M: mirror, IL: imaging lens, PH: pinhole, SMF: single mode fiber, PCF: photonic crystal fiber.

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The average power of the femtosecond laser in use is ~200 mW, which offers a mere share of ~10 nW to each single mode that is too weak to be used as the working source of the gauge block interferometer. Thus an ECLD (New Focus, TLB-6312) is adopted as the working source, which provides 12 mW average power with 300 kHz line width. The ECLD working laser is tunable within the range of 765 nm to 781 nm in wavelength continuously by varying the external cavity length using a dc motor in tandem with a piezoelectric micro-actuator. Tuning the working laser to a selected mode of the optical comb is performed following the two-step procedure; firstly coarse tuning by use of a wavelength meter and secondly fine tuning by locking the beat note with the chosen comb mode to the Rb clock using the PLL technique. The wavelength meter (Angstrom, WS Ultimate 30 Lt) in use allows the working laser to be tuned to a resolution of 8 MHz (0.016 pm in wavelength), which is rather coarse but accurate enough to obtain an unambiguous access to the selected optical comb mode. Then, for the purpose of fine tuning, the beat note of the working laser with the selected comb mode is observed with a typical signal-to-noise (S/N) ratio of ~40 dB in a 300 kHz detection bandwidth as illustrated in Fig. 2. Using the PLL technique, the beat note is fixed to a predetermined value, for example 5 MHz, by feedback-controlling the current input to the ELCD. This fine tuning of the beat note offers a frequency fluctuation of ~70 kHz at 10 s, equivalent to an instability of 1.9×10^-10, which is in fact two orders of magnitude larger than that of the optical comb. The main cause of the relatively low stability of the fine wavelength tuning in comparison to the wavelength ruler provided by the optical comb is attributable to the limited controllability of the ECLD.

 

Fig. 2. Observation of a 5 MHz beat note using a spectrum analyzer (LG Precision Co., SA-7270). Also seen are other peaks corresponding to the acoustic length variation of the ECLD at 10 MHz, (f _r-f _b) at 76 MHz, the repetition rate at 81 MHz, and (f _r+f _b) at 86 MHz.

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3. Gauge block interferometer

The gauge block interferometer constructed is basically of Twyman-Green type (Fig. 1). The working laser is collimated into a parallel beam of 50 mm diameter. The horizontal arm holds a flat mirror, reflecting a plane wave that is used as the reference wave. In the vertical arm, the gauge block under calibration is wrung on to a flat base plate so that the measurement wave is reflected from the top surface of the gauge block but also from the base plate as well. The 2-D interference between the reference and measurement waves is observed using a CCD camera of 640×480 pixels. For a given wavelength λ of the working laser, the absolute height of the gauge block is given as L=(λ/2)(m+f), where m and f denote an integer (m=0,1,2,…) and an excess fraction (1>f≥0), respectively. The excess fraction of a gauge block is obtained directly by analyzing the resulting interferogram, but the integer m is not the case due to the 2π-ambiguity of single-frequency interferometry. For the reason, multiple wavelengths are provided from the optical synthesizer so that the absolute height L is now given in the form of simultaneous equations as

where the subscript N indicate the total number of individual wavelengths in use. As all m _i (i=1,2, …,N) should be positive integer numbers, a unique solution of L can be determined by solving Eq.(1) numerically in association with a proper estimation for the feasible range of L [10]. In doing that, the required minimum number of wavelengths increases as the extent of the unknown range of L increases. As a general rule, four equations are found enough when a good assumption of L is available within an error of less than ±1.0 mm.

 

Fig. 3. Fringe analysis procedure to determine the excess fraction between the gauge block and base plate. Three representative lines are selected; line G along the center line of the gauge block, and line Ba and line Bb from the base plate at a same distance from line G. Fourier transform is adopted to obtain the phase value of each line, which finally leads to determination of the exact value of the excess fraction.

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Figure 3 depicts the procedure of obtaining the excess fraction of a gauge block from its sampled interferogram. For convenience of fringe analysis, a small amount of tilt is purposely given to the reference mirror so as to generate sinusoidal carrier fringes of a dominating frequency. Three representative lines crossing the carrier fringes are then selected; the top and bottom lines along the base plate and the middle line along the top surface of the gauge block. The sampled data of interference intensity along each selected line is then Fourier transformed and its phase value φ is subsequently picked up at the amplitude peak corresponding to the spatial carrier frequency. This Fourier-transform method of fringe analysis allow the excess fraction f to be finally determined as

where the subscripts, G, Ba, and Bb indicate the lines as depicted in Fig. 3. Note that line Ba and line Bb are separated at a same distance from line G, so that the phase value for the base plate is decided by averaging φ _Ba and φ _Bb without being affected by the non-orthogonal misalignment between the selected three lines and carrier fringes.

Table 1 summarizes a calibration result in which four wavelengths were used for a gauge block of 25 mm nominal length. The excess fraction part for each wavelength was taken 20 times repeatedly with each measurement taking 1 s and averaged to minimize the effect of vibration and thermal fluctuation. Solving Eq. (1) numerically using the measured excess fractions led to the mean nominal length of the gauge block worked out to be 24.999890. Note that the individual length for each wavelength yields a maximum discrepancy of 1.5 nm from each other, which is converted to a dispersion of 0.0035 in terms of excess fraction. The refractive index of air was compensated using the updated Edlen’s equation in association with actual monitoring of temperature, atmosphere pressure, humidity, and CO2 composition in air [11].

Table 1. Measurement result of a gauge block of 25 mm length.

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4. Uncertainty evaluation

In accordance with the conventional and ISO-recommended guidelines [12, 13], the overall uncertainty evaluation was made for the gauge block calibration performed in this investigation. The result is summarized in Table 2, which includes individual contributions from various sources of errors that should be taken into consideration. First of all, the wavelength uncertainty is worked out to be 1.9×10^-10·L ₀ where L ₀ is the nominal length of the gauge block given in meter. It is important to note that the wavelength uncertainty yields an insignificant, practically negligible effect on the overall calibration uncertainty, justifying the use of the optical comb of a femtosecond laser as the wavelength ruler as attempted in this work.

A major contribution comes from the fringe-analyzing error induced in determining the excess fraction using Eq. (1), which is estimated to be 0.005 that is converted to as large as 2.0 nm in length. The uncertainty for the refractive index of air turns out to be 1.4×10^-8·L ₀, which is mainly caused by the inherent imprecision of the updated Edlen’s equation together with measurement errors pertaining to the temperature, pressure, and composition of air. The uncertainty for estimating the thermal expansion of gauge blocks is another major error source, which is usually one order of magnitude larger than that of the refractive index of air. The gauge block interferometer itself is subject to a total wavefront error of ~12 nm, which is regarded as the most dominating error source being attributable to the imperfect fabrication and assembly of the optical components involved. Other practical error sources are caused by the wringing process of a gauge block to the base plate and also by the geometrical error of the gauge block itself, of which contributions amount to as much as 8.5 nm. To conclude, the overall calibration uncertainty is worked out to be 15 nm (k=1).

Table 2. Uncertainty evaluation of absolute calibration of gauge blocks. L₀ denotes the nominal length of a gauge block whose length is given in meter.

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5. Conclusions

The concept of optical frequency synthesizer based on the optical comb of a femtosecond pulse laser was exploited for the task of absolute length calibration of gauge blocks. The optical comb was stabilized to the Rb clock of frequency standard and used as the wavelength ruler, to which a single-frequency cw laser was tuned consecutively to provide a sequence of selected wavelengths needed to implement the calibration practice on the basis of multiwavelength interferometry. The wavelength uncertainty finally achieved was 1.9×10^-10, which permits the gauge block calibration to be implemented with no significant error contribution from the source. This result demonstrates a successful industrial application of the optical frequency synthesis employing a femtosecond laser, which offers many possibilities of carrying out precision length metrology with traceability to the well-defined international definition of time.