1. Introduction

Fabry-Perot (FP) based refractometry is a technique that can be used for assessment of gas refractivity, molar density, and pressure. With the latest revision of the SI-system, it also provides an attractive path to realize the Pascal [1,2]. By measuring the refractivity and the temperature of a gas it is possible to calculate its pressure by the use of the Lorentz-Lorenz equation and an equation of state. Such a realization of the Pascal does not comprise any mechanical actuator but depends instead on gas parameters, which potentially can decrease uncertainties and shorten calibration chains [3–12].

FP-based refractometry is often performed in cavities using high reflectivity mirrors that are produced by coating the mirrors with a distributed Bragg grating (DFB), often comprising a quarter wave stack (QWS). In addition to provide a high reflectivity, the DFB will also affect the light by giving rise to phase shifts upon reflection. Such phase shift can be interpreted in terms of a group delay (GD), which represents the time delay a narrow-band light pulse experiences upon reflection, or a frequency penetration depth, which represents the elongation of the cavity length the light experiences when it is scanned across a pair of cavity modes [13]. The latter is therefore the entity that most naturally is used when the effect of mirror phase shifts is implemented in refractometry. This adversely affects the ability to accurately assess refractivity. To properly account for this, the properties of the coating, and their influence on the assessment of refractivity, need to be known with high accuracy; in particular if the systems are to be used as primary standards.

However, although some mirror suppliers provide information about their coatings, this is in general calculated from the the coating materials and the design parameters with no given uncertainty. In the case such information cannot be obtained, it is alternatively possible to perform an element-resolving material analysis of the coating. However, such an analysis does not directly provide information about the optical properties of the coating, it can only provide information about the elemental constituents of the coating. In either case, uncertainties and imperfections in the thicknesses and refractivity of the coating will affect the assessed (or estimated) entities and their uncertainties. This implies that the true properties of the mirrors can differ from the specified ones. To correctly take the optical properties of a specific set of mirrors into account when refractivity or pressure is to be assessed, they should therefore preferably be experimentally assessed.

Although it has previously been shown that dielectric mirrors can be characterized with respect to their GD and group delay dispersion (GDD) by white light interferometry [14,15], e.g. by changing the length of micro FP cavities [13], or by measuring resonance frequencies of a micro cavity over a large frequency range [16], these methods require removing the mirrors from the refractometer. Furthermore, local variations in the mirror coating and the spatial distribution of the light can affect the GD and GDD. The mirrors, with their coatings, should therefore preferably be characterized under the same conditions as when they are used.

In this work we present an experimental procedure for in situ measurement of the relevant optical properties of distributed Bragg reflector (DBR) equipped mirrors in a FP cavity used for assessment of refractivity and pressure comprising a QWS of type H (for which the outermost layer of the stack, $n_H$, has a higher index of refraction than the subsequent one, $n_L$) that affect the assessment of refractivity; one denoted $\gamma _s'$ and another the frequency penetration depth, which both are related to the GD of the coating and, in particular for the case when the working range of the laser is not centered on the mirror center frequency, also its dispersion. It is specifically shown that the presented procedure provides a means to in situ characterize a given set of mirrors with respect to their optical properties with such an accuracy that the uncertainty in its assessment solely contributes to the uncertainty in the assessed refractivity with a pressure independent value of $8 \times 10^{-13}$, which corresponds to an uncertainty in pressure of 0.3 mPa (when nitrogen is addressed). Since FP-based refractometers have been demonstrated with uncertainties down to [(2.0 mPa)$^2$ + (8.8 $\times 10^{-6}P)^2$]$^{1/2}$ up to 100 kPa [5], this means that the procedure presented provides a means to characterize QWS equipped mirrors to such an extent that they no longer contribute to the uncertainty in the assessed pressure to any significant extent.

In addition, there has been ambiguities about how the properties of mirror coatings affect assessments of refractivity in FP-based refractometry and how to assess refractivity from measurements when the light used is not centered on the mirror center frequency of the coatings. To both clarify the latter, and to explicate the basis of the mirror characterizing procedure, this paper also provides a mathematical description of how the GD (and thereby the frequency penetration depth) from QWS mirrors of type H, together with the Gouy phase, affect the assessment of refractivity in FP-based refractometry, both when the working range of the laser is centered on and off the mirror center frequency [17].

2. Theory

2.1 Penetration depth of a quarter wave stack equipped mirror

The complex reflection coefficient from a QWS-equipped mirror can be expressed in terms of the phase shift the light experiences when it reflects at its front facet, $\phi$, as [18]

The GD and the frequency penetration depth of a QWS, $\tau$ and $L_{\tau }$ respectively, are defined as $\frac {\partial \phi }{\partial \omega }$ and $\frac {1}{2}\frac {\partial \phi }{\partial k}$, where $\omega$ is the angular frequency of the light. Since $k$ is related to $\omega$ by $k=\frac {n \omega }{c}$, where $n$ is the index of refraction of the medium in front of the mirror (for the case with a FP cavity, of the gas in the cavity) [13], $L_{\tau }(n)$ and $\tau (n)$ are, in general, related to each other through

2.1.1 On the mirror center frequency

It has previously been shown that, for an ideal QWS of type H, which is schematically illustrated in Fig. 1, and on the mirror center frequency, the group delay, here referred to as the ideal GD, can be written as [13]

 

Fig. 1. Schematic illustration of a QWS of type H, consisting of alternating layers of material with higher and lower index of refraction, $n_H$ and $n_L$, respectively. The red dashed line represents the frequency penetration depth at which the light seems to be reflected during scans.

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As was alluded to above, in reality, because of uncertainties and imperfections in the thicknesses and the refractivity of the coating, this expression does not always provide the actual value of the GD of a specific set of QWS-equipped mirrors. To account for the possibility that the actual GD, henceforth denoted $\tau _c (n)$, can differ from the ideal one, it is suitable to define a fully material-dependent entity, $\gamma _c$, as

As can be concluded by comparison with Eq. (3), under ideal cases and on the mirror center frequency, $\gamma _c$ takes a value of $(n_H - n_L)^{-1}$. However, under various pertinent conditions, and as was indicated by Hood et al. [16], it might in practice take a value that differs from this.

The introduction of $\gamma _c$ implies that both the actual GD and the actual frequency penetration depth can be expressed in terms of $\gamma _c$, the former defined by Eq. (4), and the latter, which is independent of $n$ and schematically illustrated in Fig. 1, denoted $L_{\tau,c}$, as

2.1.2 Off the mirror center frequency

For all frequencies sufficiently close to the mirror center frequency, the frequency penetration depth has the value given by Eq. (5). For the cases when the measurements are not performed at (or in the closest proximity of) the mirror center frequency, but instead around an off-center frequency (although still within the passband), here denoted $\nu _s$, the frequency penetration will, primarily because of dispersion in the coating materials, differ from this value. For frequencies close to (or around) any off-center frequency at which the light experiences a GD of $\tau _s(n)$, it has been found suitable to define a similar material-dependent entity, $\gamma _s$, that is given by

It this case, the frequency penetration depth, then denoted $L_{\tau,s}$, becomes

By this, both $L_{\tau,c}$ and $L_{\tau,s}$ are related to the true GD (given by $\tau _c$ or $\tau _s$, respectively), according to Eq. (2). Moreover, since the case with "on the mirror center frequency" can be seen as a special case of "off the mirror center frequency", it has, in this work, been found convenient to express the influence of a QWS on the assessment of the refractivity by $\gamma _s$.

2.2 Gouy phase

When the influence of mirror coatings on the assessment of refractivity is considered, also the Gouy phase, which is the phase advance, compered to a plane wave, that light gradually acquires when passing the focal region, needs to be taken into account. For a FP cavity comprising two identical concave mirrors, the single pass Gouy phase, $\Theta _{G}$, can be written as [19]

2.3 Refractivity assessed by a Fabry-Perot cavity comprising mirrors with a high-reflectivity quarter wave stack coating in the presence of the Gouy phase and cavity deformation

2.3.1 Frequencies of the cavity modes and the free-spectral-range

The frequency of a given mode in an FP cavity when pressure induced cavity deformation, the phase shift of light from the mirrors, and the Gouy phase, are taken into account, can be obtained by the use of a round-trip resonance condition for the phase of the light. As is shown in part 1 of the Supplement 1, such a condition can, for the $m^{th}$ $TEM_{00}$ mode of an FP cavity with DBR mirrors, be written as [13,20]

As is shown by the Eqs. (S14) and (S15) in the Supplement 1, for the case with two identical mirrors with DBR coatings comprising a QWS of type H, and for the case when the frequency working range of the laser is not necessarily centered on the mirror center frequency (referred to as an "off-center frequency"), the frequencies of the modes of the cavity the laser addresses in the absence and in the presence of gas (i.e. the $m_0^{th}$ and the $m^{th}$ modes), $\nu _0$ and $\nu$, respectively, can be written as

Equation (10) shows that if the number of the cavity mode the laser addresses in an evacuated cavity is changed by one unit, the laser will shift a frequency $\nu _0(m_0)-\nu _0(m_0-1)$, referred to as the free spectral range (FSR), here denoted $\nu _{FSR}$, that is given by

2.3.2 Assessment of refractivity

By defining the shift in the frequency of the laser that takes place when the gas is let into the cavity, $\Delta \nu$, as $\nu _0-\nu$, which is suitable when repetitive assessments are made, and the number of modes the laser has jumped during the filling, $\Delta m$, as $m-m_0$, it is possible, as is shown by the Eq. (S17) in the Supplement 1, to express, with a minimum of approximations (which are on the relative $10^{-9}$ to low $10^{-8}$ level), the refractivity in terms of measurable quantities and material parameters as

It can be noticed that the deformation dependence of this expression agrees with that of Eq. (2) in Egan and Stone [4,21]; series expanding Eq. (13) in terms of the distortion ($n\varepsilon '$) and making use of the definition of $\varepsilon '$ gives

This indicates that the $\varepsilon '$-concept is a fully analogous alternative to the $\frac {\delta L}{L_0'}$-concept to describe the influence of cavity distortion in refractometry.

As is shown by Eq. (SM-15) in the Supplement 1 to Zakrisson et al. [22], by using an equation of state and the Lorentz-Lorenz expression, it is possible to conclude that $\varepsilon '$ is an entity that has a weak dependence on refractivity (for low pressures it acts as a constant and for higher it is weakly dependent on the refractivity) that, under the condition that the relative physical distortion $\delta L/L_0^\prime$ can be written as $\kappa P$, where $\kappa$ is the deformation coefficient for the cavity, can be written as $\varepsilon _0' \left [1+\xi _{2}(T) (n-1) \right ]$, where $\varepsilon _0'$ is given by $\kappa RT \frac {2}{3A_{R}}$ and $\xi _{2}(T)$ is given by a combination of density and refractivity virial coefficients. This implies that $n\varepsilon '$ can be written as $[1+(n-1)] \varepsilon _0' [1+\xi _{2}(T) (n-1)]$. Since, for nitrogen, $\xi _{2}(T)$ takes a value of −1.00(4) at a temperature of 296.15 K, it can be concluded that $n \varepsilon ' = \varepsilon _0'$. This implies that, for this species, Eq. (13) can alternatively be rewritten as

Since the deformation coefficient of the cavity, $\varepsilon _0'$, is a constant (i.e. an index of refraction independent) entity, this shows that, by using the concept of $\varepsilon '$, it is possible to derive an expression for refractivity that does not contain any recursivity.

It is finally of interest to note that, by defining an "effective" empty cavity frequency, $\nu _0'$, given by $\nu _{0}/(1+\frac {\Theta _{G}}{\pi m_0}+\frac {\gamma _s'}{m_0})$, where $(1+\frac {\Theta _{G}}{\pi m_0}+\frac {\gamma _s'}{m_0})$ henceforth will be referred to as the phase compensation factor, it is alternatively possible to write Eq. (15) in a more succinct forms, e.g. as

All this implies that, to assess refractivity from a measurement that provides values to the $\Delta \nu$ and the $\Delta m$ entities, one needs, in addition to the laser frequency and the mode number addressed for an empty cavity, i.e. $\nu _0$ and $m_0$, respectively, knowledge about the deformation parameter, $\varepsilon _0'$, the Gouy phase, $\Theta _{G}$, and, the $\gamma _s'$ entity, where the latter, in turn, through Eq. (7), is a measure of the frequency penetration depth. While $\Delta \nu$ and $\Delta m$ are repeatedly measured during a measurement campaign, $\nu _0$ and $m_0$ are typically assessed once per campaign, $\Theta _{G}$ is given by geometrical properties, and $\varepsilon _0'$ is assessed by the use of a characterization routine, e.g. that given by Zakrisson et al. [22], $\gamma _s'$ needs to be known (or assessed) in order to provide an accurate assessment of $n-1$. An experimental procedure for assessment of $\gamma _s'$ is presented in Section 4 below.

3. Experimental setup

A schematic illustration of the experimental setup is shown in Fig. 2. The setup used is utilizing the same Invar-based dual FP cavity setup as in [23–25], to where the reader is referred for a more in-depth technical description of the system. In short, the lasers used were Er-doped fiber lasers emitting light within the C34 communication channel, i.e., around 1.55 $\mathrm {\mu }$m. The mirrors were produced by a major producer of mirrors, with a reflectivity of 99.97% at the mirror center wavelength, which is 1.525 $\mathrm {\mu }$m. The lasers are locked to the cavity modes by a Pound-Drever-Hall (PDH) locking scheme. Furthermore, the system comprises a relocking routine that automatically performs a controlled jump of the laser to a neighbouring cavity mode when the scanning is outside a given preset range. It can also induce mode jumps to measure the FSR of the cavities. This means that the utilized scanning ranges of the lasers are in the order of the FSR of the cavities, typically 1 GHz.

 

Fig. 2. Schematic illustration of the experimental setup. EDFL: Er-doped fiber laser; AOM: acousto-optic modulator; 90/10: 90/10 fiber splitter; EOM: electro-optic modulator; RD: fast photodetector for the reflected light; TD: large area photodetector for the transmitted light; FPGA: field programmable gate array, VCO: voltage controlled oscillator; 50/50: 50/50 fiber coupler; BD: fast fiber-coupled photodetector for the beat signal; TD; transmission detector; Freq.Counter: frequency counter; and Er:fiber comb: Er:fiber frequency comb: 25/75: 25/75 fiber coupler;. Black arrows represent electrical signals, blue and green curves represent optical fibers, and red solid lines free-space beam paths. The dashed blue curves represent two different modes of operation; 1) when the FSR was assessed, the two fibers from the two lasers were sent to the 50/50 beam coupler, or 2) when the laser frequency was assessed, the laser, here exemplified by the upper EDFL, was sent to a 25/75 beam coupler where it was combined with the frequency comb.

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In addition, to mitigate the influence of disturbances, the gas modulation refractometry (GAMOR) methodology was used [11,26,27]

To measure the frequencies of the locked lasers, they were beat with the light from an Er:fiber frequency comb (Menlo Systems, FC1500-250-WG), referenced to a GPS-disciplined Rb-clock with a relative accuracy of $5 \times 10^{-12}$ over 1 s. This was done by sequentially merging the light from one of the locked lasers (the blue dashed curves) with the light from the frequency comb (green curve) by use of a 25/75 beam combiner that combined the two laser fields and sent them onto a beat detector (BD) whose beat signal was detected by a separate frequency counter. Both frequency counters were referenced to the Rb clock.

4. Procedure for assessment of $\gamma _s'$

4.1 Basic principles of the procedure

The procedure for in situ determination of $\gamma _s'$ of the mirrors is based on a monitoring of the assessed refractivity during forced mode jumps for the case with an empty measurement cavity. For such a case, a mode jump should not give rise to any change in the refractivity, which additionally and unceasingly should be zero. As can be seen from the Eq. (16), this takes place when $\overline {\Delta \nu }+\overline {\Delta m} = 0$.

Although all four mirrors used in the dual FP cavity came from the same coating batch, it is possible that they show dissimilar properties. To investigate the possibility of any such differences, the combined penetration depths of the mirrors in each cavity, $\gamma _{s,i}'$ (where $i$ represents the cavity investigated, $m$ or $r$, referred to as the measurement and the reference cavity under ordinary working conditions, respectively), was assessed, one cavity at the time.

For the case when there is a forced mode jump (of one mode) in cavity $i$ (i.e. $\Delta m = \mp 1$), for which the shift in the frequency of the laser is equal to the FSR of cavity $i$, denoted $\nu _{FSR,i}$, (i.e. $\Delta \nu = \pm \nu _{FSR,i}$) the condition given above can, by use of Eq. (16), be written as

This shows that $\gamma _{s,i}'$ can be retrieved once the other quantities, i.e. the $\nu _{0i}$, $\nu _{FSR,i}$, $m_{0i}$, and $\Theta _{G}$, are known. However, a difficulty is that solely three of these, viz. $\nu _{0i}$, $\nu _{FSR,i}$, and $\Theta _{G}$, are quantities that can either by measured or directly estimated; $m_{0i}$ is not necessarily fully known. This precludes a direct assessment of $\gamma _{s,i}'$ from Eq. (18).

4.2 Assessment of the number of the mode addressed

A means to assess $m_{0i}$ is to utilize the fact that it must be an integer. This implies that it has to fulfill the condition

Hence, if $\frac {\nu _{0i}}{\nu _{FSR,i}}$ and $\frac {\Theta _{G}}{\pi }$ can be experimentally assessed and the value for $\gamma _{s,i}'$ can be estimated (then denoted $\gamma _{est}$) with such accuracies that their combined expanded standard uncertainty, $U\left (\frac {\nu _{0i}}{\nu _{FSR,i}} - \frac {\Theta _{G}}{\pi } - \gamma _{est}\right )$, where $U$ represents two standard deviations, is well below 0.5, $m_{0i}$ can be uniquely assessed, by the use of Eq. (19), in which $\gamma _{s,i}'$ is replaced by $\gamma _{est}$.

4.3 Assessment of $\gamma _{s,i}'$

When the mode numbers have been uniquely assessed, it is possible to retrieve the actual values for the $\gamma _{s,i}'$ from the expression

Since $\frac {\Theta _{G}}{\pi }$ can be assessed with a negligible uncertainty (see below) and since $m_{0i}$ is an integer without uncertainty, the accuracy of $\gamma _{s,i}'$ will be given by the accuracy by which the $\nu _{0i}$ and the $\nu _{FSR,i}$ can be assessed.

4.4 Assessment of the group delay and the (frequency) penetration depth

As is illustrated below, if the $\gamma _{s,i}$ can be retrieved from $\gamma _{s,i}'$ with sufficiently good accuracy, the group delays, $\tau _{s,i}$, and the (frequency) penetration depths, $L_{\tau,s,i}$, for the two cavities can be assessed, by the use of the Eqs. (6) and (7), with the same relative uncertainty as for the $\gamma _{s,i}'$.

It is worth to note that the proposed procedure, in which the $\gamma _{s,i}$ entity is assessed from experimental data, provides a value of $\gamma _{s,i}$, which, in turn, according to Eq. (13), gives the required information about the influence of the penetration depth on the assessment of refractivity, without any influence of (and thereby without any prior knowledge of) any higher order dispersion terms (GDD etc.).

5. Results

The procedure above thus indicates that to assess the influence of the penetration depth of the mirrors on the assessment of refractivity by refractometry, which is encapsulated in the $\gamma _{s,i}'$ entities, it is necessary to assess $\nu _{0i}$ and $\nu _{FSR,i}$ by experimental means with good accuracy, to estimate $\frac {\Theta _{G}}{\pi }$ from geometrical considerations, and to have a reasonable initial estimate of $\gamma _{s,i}'$ (i.e. a sufficiently appropriate value of $\gamma _{est}$).

5.1 Assessment and estimates or system parameters

5.1.1 Assessment of the cavity mode frequencies

The cavity mode frequencies were measured by beating each laser, one at a time, locked to a cavity mode in the pertinent cavity, with an optical frequency comb, according to a standard procedure, carried out as follows.

By measuring the beat between the laser addressed and the frequency comb, denoted $f_{b}$, it is possible to conclude that the frequency of the laser, denoted $f_{cw}$, is given by

When this is done, the comb tooth number addressed is estimated by use of the expression

To verify that the estimated comb tooth number is the correct one, measurements of the frequency of the laser, $f_{cw}$, were made, based on Eq. (21), for five different repetition rates, $f_{rep}$. Each measurement was evaluated for nine different (candidate) cavity tooth numbers, denoted $\Delta k$, centered on the assumed one, given by Eq. (22) (i.e. for $\Delta k$ ranging from −4 to 4). Since the five measurements provide the same laser frequency only for the correct tooth number, this can provide a verification of the assumed comb tooth number.

To visualize this, it was found convenient to plot the difference in assessed frequency between two measurements performed with dissimilar repetition rates, denoted $\Delta f_{cw}$, as a function of $\Delta k$. Figure (3) shows this entity for four pairs of measurements addressing the measurement cavity, where the various $\Delta f_{cw}$ represent the differences $f_{cw,2}-f_{cw,1}$ to $f_{cw,5}-f_{cw,1}$, respectively, as a function for $\Delta k$. As can be seen in the figure, they produce the same laser frequency (i.e., a $\Delta f_{cw}=0$) for a $\Delta k$ value of zero. This confirms that the initial estimate of the tooth number was correct.

 

Fig. 3. The difference between the predicted frequency of the laser locked to the measurement cavity for two different repetition rates, $\Delta f_{cw}$, [the four set of curves thus represent $f_{cw,2}-f_{cw,1}$, $f_{cw,3}-f_{cw,1}$, $f_{cw,4}-f_{cw,1}$, and $f_{cw,5}-f_{cw,1}$, respectively] as a function of comb tooth number, $\Delta k$, referred to the estimated comb mode number. The correct comb tooth number is given by the one for which all four predicted laser frequency differences, i.e. $\Delta k$, is zero. The frequency of the laser, and thereby the empty cavity mode of the measurement cavity, was assessed to 193401.674224(42) GHz.

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This type of analysis indicated that the laser frequency addressing the measurement cavity on its $m_{0m}$ mode was 193401.67422(4) GHz, where the error represents two standard deviations of the five measurements. The corresponding value for the reference cavity, when addressing its $m_{0r}$ mode, was found to be 193398.03063(4) GHz. It can be noticed that the procedure used provides the laser frequencies with a relative uncertainty of $2 \times 10^{-10}$.

Even if the PDH technique is a powerful technique to lock lasers to cavity modes, it is possible that, for the case with unbalanced sidebands, generated either by the EOM or by etalons in the optical system, the locking does not take place at the center of the cavity mode. However, since the sidebands are generated by an EOM, which only has a small tendency of generating non-symmetrical sidebands, the locking is expected to take place to within a fraction of the width of a cavity mode. Since the estimated width of the cavity modes in this system is 100 kHz, this deviation is expected to be significantly smaller than this, and thereby smaller than the uncertainty in the assessment of the absolute frequency of the laser, which is the requirement that it should not affect the assessment. Hence, we do not consider that this process influences the presented procedure for assessment of penetration depth significantly.

5.1.2 Assessment of the free spectral ranges

The FSR:s of the two cavities, $v_{FSR,i}$, were measured, as described in section 4.1 above, by detecting the change in the beat frequency, $\Delta f$, while repeatedly changing the mode number of the cavity addressed by one (performed by repeatedly unlocking and relocking the laser to an adjacent cavity mode). Figure (4(a)) displays, by the colored curves, the assessed shift in beat frequency from ten consecutive measurements in which the number of the cavity mode addressed in the measurement cavity is changed by one unit (back and forth displayed at 15 and 30 s, respectively). To improve on the accuracy of the assessment, i.e. to compensate for drifts in the cavity length, interpolation was used in a similar manner as for the GAMOR methodology [11,27], although here it was the cavity mode number that was modulated instead of the amount of gas in the cavity.

 

Fig. 4. Measurement of the FSR from the measurement cavity. Panel (a) displays, by the colored curves, the change in beat frequency, $\Delta f_b$, when the number of the cavity mode addressed in the cavity was changed by one unit (back and forth, in the figure displayed at 15 and 30 s, respectively), thus the FSR, for ten consecutive measurements. The solid curve shows the average of the individual assessments. Panel (b) shows the measured FSR over 18 hours of measurements of the type of data displayed in panel (a). The solid line represents the mean while the dashed lines indicate $\pm 2\sigma$. Panel (c) illustrates the Allan deviation of the data presented in panel (b). The dashed line represents a deviation of 75 Hz.

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To estimate the stability of the FSR measurements this measurement was repeated 2000 times (thus being performed over a period of 18 h). Panel (b) displays the assessed shift in frequency, thus the FSR, over this period of time. The standard deviation of this series of data was found to be 350 Hz. To estimate the precision of the averaged value, and to assess optimal conditions, an Allan deviation of the measurements was created, shown in panel (c). It was found that, after 10 min of averaging, the system experienced flicker noise of 75 Hz, which thus represents the limit of the accuracy of the FSR measurement under the pertinent conditions. A comparable set of data was obtained for the reference cavity (not shown).

Based on the data displayed in Fig. (4), it was found that the FSR for the measurement cavity was 1 012 595 490 $(150)$ Hz. The corresponding value for the reference cavity was assessed to 1 012 587 160 $(150)$ Hz. This shows that the relative uncertainty of the measurement of the FSR was 1.5 $\times$ $10^{-7}$.

5.1.3 Estimate of the Gouy phase

As was indicated by Eq. (8) above, the Gouy phase, $\Theta _{G}$, is given in terms of the geometry of the cavity, its length, $L$, and the radius of curvature of the mirrors, $R$. For our cavities, $L$ and $R$ are 0.148 and 0.500 m, respectively. This implies that $\Theta _{G}$ takes, for both cavities, a value of 0.7954, whereby the $\frac {\Theta _{G}}{\pi }$ term, to be used in the Eqs. (19) and (20), takes a value of 0.253. The $\frac {\Theta _{G}}{\pi m_{0i}}$ term, to be used in the phase compensation factor, becomes $1.33 \times 10^{-6}$.

Since the cavity length can be assessed with significantly better uncertainty than the radius of curvature of the mirrors, the uncertainty in the Gouy phase is dominated by the uncertainty in the latter, which is estimated to $\pm$0.5%. This implies that the uncertainty of the $\frac {\Theta _{G}}{\pi }$ and the $\frac {\Theta _{G}}{\pi m_{0i}}$ terms, i.e. $U\left (\frac {\Theta _{G}}{\pi }\right )$ and $U\left (\frac {\Theta _{G}}{\pi m_{0i}}\right )$, can be assessed to 0.002 and $1 \times 10^{-8}$, respectively.

5.1.4 Estimate of $\gamma _{est}$

To get a sufficiently accurate estimate of $\gamma _{est}$, before mounting the mirrors on the cavity spacer, the coating of one mirror from the batch was analyzed with energy-dispersive X-ray spectroscopy (EDXS). This indicated that the DBR layers were made of Ta$_2$O$_2$ and SiO$_2$. The indices of refraction for these materials at the pertinent wavelength for this study (i.e. at 1.55 $\mathrm {\mu }$m) are 2.0573 and 1.4431 respectively [28]. This implies that, under the condition that the QWS is made with "perfect" layer thicknesses, the estimated value for $\gamma _s'$, i.e. $\gamma _{est}$, evaluated by use of the ideal GD, i.e $\tau _c^{id} (n)$, which, according to Eq. (3), is given by $\frac {1}{n_H - n_L}$, takes a value of $1.628$.

The main contribution to the uncertainty in this value comes from variations or deviations in the indices of refraction and the thicknesses of the individual layers in the QWS as well as the fact that the experimental assessments performed in this work are performed at 1550 nm, i.e. 25 nm from the mirror center frequency (1525 nm), at which $\Delta \nu _{cs}$ = $- 3.2 \times 10^{12}$ Hz. Based on this, a conservative estimate is that the estimated value is within 10% of the actual value. This implies that the uncertainty of $\gamma _{est}$, i.e. $U\left (\gamma _{est}\right )$, can be estimated to 0.16.

5.2 Assessment of mode number and mirror entities by use of the presentedmethodology

5.2.1 Assessment of the number of the mode addressed

According to the analysis given above, the uncertainty of the measurement of the $\frac {\nu _{0i}}{\nu _{FSR,i}}$ term is dominated by the uncertainty of the FSR measurement, which has a relative uncertainty of 1.5 $\times$ $10^{-7}$. Since $\frac {\nu _{0i}}{\nu _{FSR,i}}$ takes a value close to $m_{0i}$, which is around 2 $\times$ $10^{5}$, this implies that $U\left (\frac {\nu _{0i}}{\nu _{FSR,i}}\right )$ can be estimated to 0.03.

Regarding the uncertainty from the Gouy phase and the $\gamma _{est}$ terms, i.e. the $U\left (\frac {\Theta _{G}}{\pi }\right )$ and $U\left (\gamma _{est}\right )$, it was concluded above that they could be estimated to 0.002 and 0.16, respectively.

This implies that the $U(\frac {\nu _{0i}}{\nu _{FSR}} - \frac {\Theta _{G}}{\pi } - \gamma _{est})$ entity is dominated by $U\left (\gamma _{est}\right )$, which implies that it takes a value of 0.16. Since this is significantly below the requirement of 0.5 for utilization of Eq. (19), the $m_{0i}$ can be assessed by use of the same expression with no uncertainties.

It was found that the empty cavity mode numbers for the measurement and reference cavities, i.e. $m_{0m}$ and $m_{0r}$, could be assessed, by using Eq. (19), to 190 994 and 190 992, respectively.

5.2.2 Determination of the $\gamma _s'$ entity for the mirrors

Once the numbers of the modes addressed, i.e. the $m_{0i}$, had been uniquely assessed, it was possible, by use of Eq. (20), to obtain a more accurate (i.e. the actual) value of the $\gamma _{s,i}'$ of each cavity. These entities were assessed, for each cavity separately, by the use of four consecutive assessments of the FSR. The result of these measurements are shown in Fig. 5.

 

Fig. 5. Four separate measurement of the $\gamma _{s,i}'$ from the reference and the measurement cavities (red and blue markers, respectively). The error bars represent the $2\sigma$ uncertainty of the individual measurements. The solid line indicate the average value of all measurements, while the dashed lines show $\pm$ two standard deviations of the measurement values.

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This assessment provided mean values for the $\gamma _{s,i}'$ for the measurement and reference cavity, i.e. for $\gamma _{s,m}'$ and $\gamma _{s,r}'$, of 1.737(27) and 1.720(30), respectively, where the uncertainties correspond to two standard deviations. Since the uncertainties of the various assessments overlap, it was found appropriate to assess a mean value of the two $\gamma _{s,i}'$, representing $\gamma _{s}'$, which was found to be 1.728(32). This shows that the $\gamma _{s}'$ entity, which is the one that is needed for accurate assessment of the refractivity by use of either of the Eq. (13) or (16), could be assessed with a relative uncertainty of $2\%$. This is an almost one order of magnitude more accurate value than what previous characterizations of FP-cavities for refractometry have had to use [4].

This also implies that the contribution to the uncertainty of the phase compensation factor from the penetration depth, given by the $U\left (\frac {\gamma _{s}'}{m_{0i}}\right )$ term, can be estimated to be $1.6 \times 10^{-7}$.

5.2.3 Determination of other mirror entities — the group delay and the frequency penetration depth

Although it is sufficient to assess the value for $\gamma _{s}'$ to provide the required input to the expressions for the refractivity, based on the assessed value of this entity, it is additionally possible to determine the GD of the mirrors, i.e. $\tau _s$, and the frequency penetration depth, $L_{\tau,s}$, at the pertinent wavelength (i.e. at 1550 nm). However, as can be seen from the Eqs. (4) and (5), these two entities encompass the $\gamma _{s}$ entity (and not the assessed $\gamma _{s}'$). Since it is possible to estimate the former one from the experimentally assessed latter one by use of the $\left (1+\frac {1+\chi _0}{1+\chi _1}\frac {\Delta \nu _{cs}}{\nu _s}\right )$ factor, an accurate assessment of $\tau _s$ and $L_{\tau,s}$ requires, to a certain extent, knowledge about the both $\chi _0$ and $\chi _1$, which represent the effect of dispersion of the DFB coatings.

Since, in our case, it was possible to retrieve calculated information from the mirror supplier about the GD of the mirrors, it could be concluded that $\tau _s$ (at 1550 nm) was 3$\%$ larger than $\tau _c$. Based on data in Table 1 in the Supplement 1, this implied that $\chi _1$ could be assessed to 0.03. Moreover, since the non-linearity of the GD supplied by the mirror manufacturer has the same frequency dependence as the $D_3$-dispersion term, it can be surmised that $\chi _1$ is mainly dependent on this term. This implies that $\chi _0$ is the same, whereby it can be concluded that it takes a value of 0.01. This implies that the $\left (1+\frac {1+\chi _0}{1+\chi _1}\frac {\Delta \nu _{cs}}{\nu _s}\right )$ term takes a value of 0.9839, whose uncertainty is below that for the $\gamma _s'$ ($2\%$). This implies that $\gamma _{s}$ is 1.756(33), which, in turn, implies that the mean GD of the mirrors in the two cavities at the pertinent wavelength (i.e. at 1550 nm), $\tau _s$, could be assessed to 4.54(8) fs while the frequency penetration depth, $L_{\tau,s}$, could be assessed to 0.681(13) $\mathrm {\mu }$m. The value of $\tau _s$ can alternatively be expressed as $\frac {1.756}{2 \nu _s}$, which is in good agreement with assessments of the same entity for a similar type of mirrors made with a mirror center wavelength of 852 nm [16].

It is worth to note that even without the information from the mirror supplier about the values of $\chi _0$ and $\chi _1$, these estimates do not change markedly; for the case when the influence of dispersion is neglected, hence neglecting the influence of $\chi _0$ and $\chi _1$, i.e. assuming the aforementioned factor to be given by $\left (1+\frac {\Delta \nu _{cs}}{\nu _s}\right )$, it becomes 0.9836. This implies that $\gamma _{s}$ changes marginally to 1.757(33), while $\tau _s$ and $L_{\tau,s}$ remain the same.

This shows that, irrespective of whether the dispersion was accounted for or not, the value for $\tau _s$ [4.54(8) fs] is in good agreement with the one given in the specifications of the mirrors (provided by the manufacturer) at the wavelength addressed, which was given to 4.5 fs. Most importantly, it is assessed with a low uncertainty ($2\%$), which is of highest relevance for refractometry. It can moreover be noticed that it is markedly larger than the theoretically calculated ideal group delay for this type of mirror, $\tau _c^{id}(n)$, which, when calculated by use of Eq. (4), becomes 4.21 fs. Reasons for this discrepancy comprise that the latter is calculated at the mirror center wavelength and that it is assumed that the mirror is a ’perfect’ QWS.

The fact that the properties of the mirrors could be assessed by the presented methodology to values that are in good agreement with that provided by the manufacturer shows that the methodology presented is adequate and accurately carried out. It also verifies the assumption that the coating is made by the assumed types of materials, i.e. Ta$_2$O$_2$ and SiO$_2$. This implies that it is possible to use the presented methodology for both in situ experimental verification of the assumed layer compositions and assessments of not only the $\gamma _s'$ entity that is used in the expressions for the refractivity, but also the GD and penetration depth of the coating, with high accuracy.

5.3 Uncertainty in $(n-1)$

An important consequence of the presented procedure is that it is capable of providing a value of $\gamma _s'$ with such a small uncertainty that the effect of penetration of light into the mirror coating can be taken into account to such an extent that it currently does not contribute to the uncertainty in the assessment of refractivity or pressure when the refractometry system is used for assessments of these entities.

An estimate of to which extent the uncertainty in the penetration of light will affect the uncertainty in the assessment of refractivity can most conveniently be assessed based on Eq. (16). This equation shows that the leading part of the expression for refractivity comprises the numerator, given by $\overline {\Delta \nu }+\overline {\Delta m}$. This implies that the uncertainty in the assessment of refractivity due to an uncertainty in $\gamma _s'$, henceforth denoted $U(n-1)_{\gamma }$, is given by the corresponding uncertainty in the aforementioned expression, i.e. by $U\left (\overline {\Delta \nu }+\overline {\Delta m}\right )_{\gamma }$.

Since the $\overline {\Delta m}$ term solely comprises entities that are integers, with no uncertainty, it can be concluded that the uncertainty in the assessment of refractivity from that of the penetration depth is solely given by $U\left (\overline {\Delta \nu }\right )_{\gamma }$. Hence.

The relative uncertainty in $\nu _{0m}'$, i.e. $\frac {U(v_{0m}')_{\gamma }}{v_{0m}'}$, where $\nu _{0m}'$ is defined by $\nu _{0m} / (1+\frac {\Theta _{G}}{\pi m_{0m}}+\frac {\gamma _s'}{m_{0m}})$, is then given by the relative uncertainty of the cavity mode frequency $U(\nu _{0m})/\nu _{0m}$, which was found to be $2 \times 10^{-10}$, and that of the phase compensation factor, which in turn depends on the uncertainties in the Gouy phase, $U\left (\frac {\Theta _G}{\pi m_{0m}}\right )$, and the penetration depth, $U\left (\frac {\gamma _s'}{m_{0m}}\right )$, which were found to be $1 \times 10^{-8}$ and $1.6 \times 10^{-7}$, respectively. This implies that the relative uncertainty in $\nu _{0m}'$ due to the uncertainty in $\gamma _s'$, i.e. $\frac {U(\nu _{0m}')_{\gamma }}{\nu _{0m}'}$, is $1.6 \times 10^{-7}$.

Since the shift in the beat frequency, $\Delta f$, is normally not larger than one FSR, which in this case is $10^9$ Hz, this implies, according to Eq. (23), that the uncertainty in the refractivity attributed to the uncertainty in $\gamma _s'$, i.e. $U(n-1)_{\gamma }$, stays below $8 \times 10^{-13}$, which for nitrogen corresponds to 0.3 mPa.

Finally, regarding the effect of dispersion in the coatings, it can be added that since the $\gamma _s'$ entity is experimentally assessed by the same instrumentation that later is used for assessment of refractivity and pressure, dispersion is automatically taken into account in subsequent assessments of refractivity and pressure; there is no need to specifically assess its influence.

6. Summary and conclusions

The mode frequency of a FP cavity depends on a number of entities, e.g. some optical properties of the mirror coating, which, in turn, comprise the penetration depth or the GD, and the Gouy phase. Expressions for refractivity assessed in a FP cavity taking such entities into account, both on- and off- the mirror center frequency, are presented. These show that, to use such a device for accurate assessments of refractivity and pressure, the properties need to be accurately assessed at the wavelength addressed.

An experimental methodology for assessment of the joint influence of the Gouy and the penetration depth of mirror coatings on the assessment of refractivity through a single material-dependent entity, here denoted $\gamma _s'$, is presented. The procedure encompasses accurate assessments of the FSR and the frequency of the empty cavity mode together with the use of the mode number, $m_0$, which, since it is an integer, can be assessed without uncertainty. The first entity was measured by the use of induced mode jumps while the second one was assessed by referencing the locked laser to an optical frequency comb.

Using the presented methodology, the $\gamma _s'$ entity for the mirrors addressed in this work could be assessed to 1.728(32), thus with a relative uncertainty of $2\%$, under the same conditions as when refractivity measurements are performed and without modifying the set-up. This implies that the optical properties of the mirror coatings do not significantly influence the uncertainty of assessments of refractivity and pressure; they contribute to the expanded uncertainties of these entities with contributions that solely are $< 8 \times 10^{-13}$ and (for nitrogen) $< 0.3$ mPa, respectively. This implies that the methodology presented is suitable for FP-based refractometry; as long as FP-based refractometers that claim uncertainties of a few mPa (or more) are used, as for example is the case for the instrumentation used in this work [for which the uncertainty primarily has been assessed to [(10 mPa)$^2$ + (10 $\times 10^{-6}P)^2$]$^{1/2}$] [11], the presented procedure can be applied to eliminate the influence of penetration depth on the uncertainty of the system [29].