1. Introduction

The meter is defined by the distance over which light propagates in vacuum. The wavelength of an iodine-stabilized He–Ne laser has been used as an achievement means of the meter. The frequency of a He–Ne laser is stable. Because the wavelength is inversely proportional to the frequency, the wavelength is also stable; therefore, the wavelength can be used to measure the length.

The national standard of length in Japan changed from the iodine-stabilized He–Ne laser to a femtosecond optical frequency comb (FOFC) in 2009. The characteristics of an FOFC and He–Ne laser are different. A He–Ne laser is a single-frequency source; however, an FOFC is a pulsed laser source with a coherent combination of several hundreds of thousands of ultra-stable wavelengths. Because of this difference, it is possible to measure length with an FOFC in two ways. One is by using the wavelength, and another is by using the adjacent pulse repetition interval length (APRIL), which is a coherent representation of individual wavelengths.

Several methods for generating a single frequency from an FOFC or multiple FOFCs have been proposed. For example, an individual wavelength can be extracted from an FOFC to measure length [1, 2]. In addition, two FOFCs can be used together to generate a beat signal [3, 4]. Further, the possibility of measuring length with an APRIL has also been explored. Yamaoka et al. [5] first tested the possibility of measuring length by an APRIL. Later, Ye [6] independently proposed using the integer time of an APRIL for the absolute length measurement. In addition, various experiments [7–12] were performed for distance detection. We examined the stability of an APRIL in air [13] and the group refractive-index characteristic of an APRIL in air [14].

In this work, we compare the wavelength and APRIL of an FOFC to test their characteristics for length measurement. This paper is organized as follows. First, a theoretical estimation of the frequency parameter required to stabilize the wavelength and APRIL is presented in Section 2. Next, the stability of the APRIL in air is described in Section 3 when only the repetition frequency is stabilized. In Section 4, the ambiguity in the length measurement that uses an APRIL is examined. Finally, the main conclusions are summarized in Section 5.

2. Frequency stability for stabilizing the wavelength and APRIL

First, we examine the frequency stability required to obtain a stabilized wavelength and an APRIL. For convenience, we summarize the features of an FOFC, the details of which can be found elsewhere [15]. In the frequency domain, a mode-locked laser generates equidistant frequency comb lines with the pulse repetition frequency $f_{\text{rep}}$, and the entire equidistant frequency comb is shifted by the offset frequency $f_{\text{CEO}}$ from the zero frequency. In the time domain, when the electric-field packet repeats at the pulse repetition period $T_R=1/f_{\text{rep}}$, the carrier phase changes according to $\Delta {\phi }_{\text{ce}}=2\pi f_{\text{CEO}}/f_{\text{rep}}$ to the carrier-envelope phase.

In vacuum, the relation ${\lambda }_{\text{vac}}=c_{\text{vac}}/f$holds between the wavelength ${\lambda }_{\text{vac}}$ and the frequency $f$. Here, $c_{\text{vac}}$ is the speed of light in vacuum. The uncertainty in the wavelength is given by $u({\lambda }_{\text{vac}})/{\lambda }_{\text{vac}}=u(f)/f$, where $u(x)$ is the uncertainty of the variable $x$.

One of the frequencies of an FOFC $f_P$ is expressed as

In vacuum, the following relation holds between an APRIL ${\delta }_{\text{vac}}$ and the pulse repetition frequency $f_{\text{rep}}$:

Without the stability requirement for $f_{\text{CEO}}$, the frequencies (wavelengths) of an FOFC change in time. In the following, we examine how this change influences an APRIL in air.

3. Error in the APRIL in air caused by a change in the central frequency of an FOFC

The wavelength in air ${\lambda }_{\text{air}}$ is a function of ${\lambda }_{\text{vac}}$ and the phase refractive index of air $n_p({\lambda }_{\text{vac}})$ according to ${\lambda }_{\text{air}}={\lambda }_{\text{vac}}/n_p({\lambda }_{\text{vac}})$. The phase refractive index of air can be derived from the temperature, atmospheric pressure, etc., by using empirical equations [16–19].

The APRIL in air ${\delta }_{\text{air}}$ is a function of ${\delta }_{\text{vac}}$ and the group refractive index of air $n_{\text{g}}({\lambda }_{\text{cen\_vac}})$ according to

In the above, we showed that the stability of $f_{\text{CEO}}$ is not required to stabilize an APRIL. Without the stability of $f_{\text{CEO}}$, we know from Eq. (1) that the central wavelength of the FOFC will change. This change introduces an error into the group refractive index of air according to Eq. (10), and subsequently, ${\delta }_{\text{air}}$ in Eq. (9) will change. In the following, we consider this error caused by the change in $f_{\text{CEO}}$.

We assume that $f_{\text{rep}}$ = 100 MHz; then, we obtain $f_{\text{CEO}}\in [0,100)$ MHz. When ${\lambda }_{\text{cen\_vac\_1}}$ = 1560 nm, $f_{\text{1}}=c_{\text{vac}}/{\lambda }_{\text{cen\_vac\_1}}$ = 192,174.6526 GHz. In order to simplify the calculations and obtain the maximum error caused by the change in $f_{\text{CEO}}$, we assume $f_{\text{CEO}}\approx$100 MHz and obtain $f_2=f_1+f_{\text{CEO}}$ = 192,274.6525 GHz. ${\lambda }_{\text{cen\_vac\_2}}=c/f_2$ = 1559.18866 nm. Using Eq. (10), we obtain $n_{\text{g}}({\lambda }_{\text{cen\_vac\_1}})$ = 1.00026689 ± 0.00000003 and $n_{\text{g}}({\lambda }_{\text{cen\_vac\_2}})$ = 1.00026689 ± 0.00000003. The uncertainty of $\pm 30\times {10}^{-9}$ is the empirical value.

This result shows that the change of $f_{\text{CEO}}\approx 100$ MHz introduces no error into the value of the group refractive index of air (and the value of the APRIL in air). Generally, $f_{\text{CEO}}$ and $f_{\text{rep}}$ are controlled separately. (The change in $f_{\text{CEO}}$ will not affect the value of $f_{\text{rep}}$ and ${\delta }_{\text{vac}}=c/f_{\text{rep}}$.) Fig. 1 shows the calculated values of the group index for different changes in $f_{\text{CEO}}$. We find that the influence of the APRIL due to the deviation in the offset frequency can be ignored up to $f_{\text{CEO}}=498$ MHz.

 

Fig. 1 Change in the group refractive index according to the shift in the offset frequency.

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4. Ambiguity problem by using an ARPIL

A wavelength-based interferometer suffers from the $2\pi$ ambiguity problem. We consider the ambiguity problem in the length measurement by using an APRIL. We rewrite Eq. (9) as

$\Delta L_{\text{air}}(t_1,t_2)$ is the length difference caused by the changes in the environmental parameters $T$, $P$, and $H$. $c_{\text{vac}}/f_{\text{rep}}(t_1)={\delta }_{\text{vac}}(t_1)$ is the length of the APRIL. When $\Delta L_{\text{air}}(t_1,t_2)<{\delta }_{\text{vac}}(t_1)$, there is no ambiguity in the APRIL because $L_{\text{air}}(t_2)$ and $L_{\text{air}}(t_1)$ have the same integral part.

For example, in a measurement in a laboratory, we can assume that $T\in \left[10,30\right]$ °C, $P\in \left[80,120\right]$ kPa, and $H\in \left[10,80\right]$%. We find the maximum value $\max (n_{\text{g}}({\lambda }_{\text{cen\_vac}}(t_1),T(t_1),P(t_1),H(t_1)))=\max (n_{\text{g}}(t_1))$ and the minimum value $\min (n_{\text{g}}({\lambda }_{\text{cen\_vac}}(t_2),T(t_2),P(t_2),H(t_2)))=\min (n_{\text{g}}(t_2))$ of the group refractive index in air for changes in $T$, $P$, and $H$. Then, we calculate $\Delta n_g^{-1}(t_1,t_2)=1/\min (n_{\text{g}}(t_2))-1/\max (n_{\text{g}}(t_1))$, which is approximately $1.3\times {10}^{-4}$. For $M<5\times {10}^3$, Eq. (16) is satisfied, and there is no ambiguity in the APRIL for the length measurement. If the APRIL is one meter [$c_{\text{vac}}/f_{\text{rep}}(t_1)={\delta }_{\text{vac}}(t_1)=1$ m], and the range without ambiguity is less than 5 km ($M<5\times {10}^3$), $\Delta L_{\text{air}}(t_1,t_2)/\left[c_{\text{vac}}/f_{\text{rep}}(t_1)\right]=(M+N)\times \Delta n_g^{-1}(t_1,t_2)<1$, and the condition in Eq. (15) meets the requirement. In other words, for a length measurement greater than hundreds of meters ($\ll 5\times {10}^3$) using an APRIL (namely, one meter) in the laboratory, the measured length changes caused by the changes in the environmental parameters only become changes in the fractional parts.

For the wavelength, we assume ${\lambda }_{\text{vac}}=1560$ nm and $\Delta n_{\text{p}}^{-1}(t_1,t_2)=1.3\times {10}^{-4}$. When $M<5\times {10}^3$, $\Delta L_{\text{air}}(t_1,t_2)/{\lambda }_{\text{vac}}<1$ is satisfied, and there is no ambiguity in the wavelength for length measurement. The range without ambiguity is less than 7.8 mm. In other words, for a length measurement greater than 8 mm using the wavelength (namely, 1560 nm) in the laboratory, the measured length changes caused by the changes in the environmental parameters may become the changes in the integral parts.

5. Conclusion

We examined the wavelength and APRIL of an FOFC used for length measurements. We conclude that an APRIL can be used as a low-cost method to measure the length compared to the wavelength because only one frequency parameter is necessary for stabilization. If the offset frequency is small enough (<498 MHz), we can disregard the influence of the group refractive index according to the changes in the offset frequency. In addition, we do not need to consider the ambiguity caused by the changes in the environmental parameters in the APRIL for a displacement measurement greater than hundreds of meters in laboratory conditions. This allows for the realization of absolute length measurements with an APRIL.