Correction for stress-induced optical path length changes in a refractometer cell at variable external pressure

In the field of interferometry for high-precision length measurements the influence of the refractive index of air can either be eliminated by completely measuring in a vacuum or by the application of a refractometer cell which enables the precise in situ determination of the refractivity. In the latter case, systematic effects can occur in connection with the operation of the refractometer cell due to mechanical stress induced at different pressures. Operation at a pressure range beyond the usual atmospheric pressure variation is needed when, for instance, the compressibility of a material measure is investigated or the pressure dependence of the refractive index of gases is aimed at. In this work an approach is shown for the correction of pressure-induced effects for the type of interference refractometer being used at PTB when starting from a reference state in vacuum. The required correction of the refractivity’s value was found to be of the order of for a refractometer cell of 420 mm length which corresponds to a relative contribution of when determining the air refractivity at standard laboratory conditions. For instance, in the application of high-precision length measurements (up to 1000 mm) the effect amounts to a length correction of several nanometers.

In the derivation of equation (10) a term is missing, which takes into account the replacement of glass material by the gas medium when the refractometer cell is compressed by the external pressure. As a consequence, instead of the change of the optical path length, i.e. the distance between two fixed points independent of the material boundaries, unintentionally only the change of the optical thickness is represented by equation (10). This can be resolved by setting up equation (4) as follows: [A V] = [n air · (l w,out −l w,out ) +ñ f.s.,out ·l w,out + n air ·l c,out + n air · (l c,out −l c,out )] − [n air · (l w,in −l w,in ) +ñ f.s. ·l w,in + 1 ·l c,in + n air · (l c,in −l c,in )]. (1) Then, proceeding with the subsequently described conversion steps and additionally applying the relation δl c,in = δl c,out + δl c,bend yields the expression for the refractivity n air − 1 (previously equation (12) − 2 · L f.s. · (l w,out − l w,in ) · κ 3 δp out (2) which is approximated by in analogy to the previous equation (13). Note that the factor

Corrigendum
Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. demonstrated that particularly for the interior beam path of our refractometer cell the calculation of the window's refractive index change described above may lack accuracy due to the non-isotropic distribution of strain inside the window. Instead of our previously used average representation of that effect, the FEM-based numerical integration appears to evaluate it more accurately. Applying their approach to our particular window geometry at 1000 hPa external pressure yields δn f.s.,in = (7.6 ± 0.2) · 10 −7 and δn f.s.,out = (8.7 ± 0.2) · 10 −7 with the uncertainties taking into account the mesh-dependent variability of the FEM results. The air refractivity can then be calculated via the adapted equation The resulting implications are pointed out below.

Corrigendum: Appendix. Refractive index changes
Following [2] and [3] we applied the Lorentz-Lorenz relation to calculate the change of the refractive index of fused silica induced by external pressure. Unfortunately, this approach is in conflict with experimental data of Vedam et al [4], Ritland [5], Spinner et al [6] and Waxler et al [7]. These publications provide experimental evidence that the relation between the refractive index and the density of solid materials, in par ticular of fused silica, is not compatible with the Lorentz-Lorenz relation (i.e. with the assumption of a constant value of the polarizability). Therefore, instead of the previous equation (A.2) the relation between the relative change of the refractive index and the relative density change should be expressed by (similar to [8]) with a = 0.226 ± 0.010 being a fit parameter which is determined by linear regression of the exper imental results from [4] for fused silica. Consequently, the experimentally-based factor L must read

Conclusion
Considering the corrections described above, the partially FEM-based approach from [9] yields the updated figure 6 which shows the dependence between the resulting corrections and the external gas pressure affecting the cell windows of our particular cell geometry. However, as mentioned above, the all-FEM-based approach from Egan et al [1] provides more reliable results and yields the pressure-dependent correction shown in figure 7.
Compared to the previously published correction in [9] its magnitude is decreased so that, for instance, at 1000 hPa  the contribution to the air refractivity is of the order of 2.4 nm/420 mm ≈ 6 × 10 −9 which corresponds to a relative effect of approximately 2 × 10 −5 at standard conditions.

Acknowledgments
We appreciate a thorough and critical discussion of our previously published results from [9] with J Stone and P Egan from NIST, USA. This interaction has lead to the revealing of the corrections shown in the present corrigendum and the FEM data from P Egan enabled us to improve the accuracy of the correction.

Introduction
The determination of the refractive index of air by interferometric techniques began in the 19th century as stated in the introductory review in [1]. Still nowadays a common technique for the determination of the refractive index of air is the application of interference refractometers which allow the precise measurement of the refractivity of a gas by the comparison of the optical path lengths through the gaseous medium and a (vacuum) reference path. More than 30 years ago widespread investigations came up to determine the accuracy of such refractometers in the field of interferometric dimensional measurements [2] and to compare the results with values from an empirical formula [3] (which has been followed by modifications since then [4][5][6]). Depending on the design of the refractometer cell these paths may appear very different and there is a variety of realisations (e.g. [6][7][8][9][10]). Some consist of a single evacuated light path which is compared to an outer path through a free gas, while others combine a path through a vacuum and a gas filled cavity side by side. Depending on the design principle the operation mode differs from type to type and, hence, the occurring systematic effects also do.
For instance pressure changes of the gas under examination can either mechanically affect the own gas cavity (internal change) or the whole refractometer cell (external change). While differences caused by geometric and stress-induced In the field of interferometry for high-precision length measurements the influence of the refractive index of air can either be eliminated by completely measuring in a vacuum or by the application of a refractometer cell which enables the precise in situ determination of the refractivity. In the latter case, systematic effects can occur in connection with the operation of the refractometer cell due to mechanical stress induced at different pressures. Operation at a pressure range beyond the usual atmospheric pressure variation is needed when, for instance, the compressibility of a material measure is investigated or the pressure dependence of the refractive index of gases is aimed at. In this work an approach is shown for the correction of pressure-induced effects for the type of interference refractometer being used at PTB when starting from a reference state in vacuum. The required correction of the refractivity's value was found to be of the order of 2 × 10 −8 for a refractometer cell of 420 mm length which corresponds to a relative contribution of 7 × 10 −5 when determining the air refractivity n − 1 at standard laboratory conditions. For instance, in the application of high-precision length measurements (up to 1000 mm) the effect amounts to a length correction of several nanometers.
Keywords: refractive index, refractivity, interference refractometer, refractometry, length measurement (Some figures may appear in colour only in the online journal) Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. optical path length changes have been taken into account in former measurements at PTB for pressure changes inside the cavity [6], the stress-induced path length changes resulting from variations in the environment of the refractometer cell must be handled differently. On the one hand, the influence of the geometric difference can be determined by a simple in-place measurement with the refractometer cell being in a reference state with vacuum inside and around the cell, so that a detected path difference can be used as a correction for this influence [6,11,12]. On the other hand, stress-induced influences require detailed consideration. Hence, this work deals with the necessary corrections for systematic effects on the measurement result, which occur for a refractometer cell exposed to external pressure changes. The focus is on the derivation of a theoretical model equation enabling the identification of these effects. The application of the model requires knowledge of a quantity which is determined for practical purposes by the finite-element method (FEM). Eventually the validity of the FEM approach is supported by means of experimental data.

Optical paths in the refractometer cell
The refractometer cells used at PTB consist of a fused silica tube with polished end faces ( figure 1). The open ends of the tube are sealed by fused silica plates which are wrung onto the end faces so that the molecular forces keep the bodies in tight contact (similar to the wringing of gauge blocks). In the middle of the tube there is a junction to a vacuum pump for the high vacuum evacuation of the inner cavity. For the application during measurements the complete device is placed in a pressure tight chamber which can be operated between atmospheric and high vacuum conditions [12].
The cell depicted in figure 1 is 420 mm long, has an inner diameter of 10 mm and an outer diameter of 19 mm. The sealing windows have lateral dimensions of 20 mm by 50 mm and a thickness of 5 mm. In order to deflect light reflexions from the windows away from the optical axis, they are slightly inclined to the tube's axis and not perfectly parallel to each other. Moreover, the windows are wedge-shaped along the long edge in reverse direction to each other. As a consequence, the optical paths through the centre of and along the tube differ by the various geometric conditions.
For the following theoretical model the geometric paths are defined by l out = l c,out + l w1,out + l w2,out = l c,out + l w,out (1) and as sketched in figure 2, where l c,in is the reference path through vacuum and l c,out is the path through a gas at a specific pressure. As described in [12] the use as an interference refractometer yields the measured optical path length difference between the path through air and the reference path through vacuum, which is defined by the symbol [A V] in the expression in terms of the measured integral and fractional orders of interference N and the vacuum wavelength λ of the used light. Apart from that, the difference can also be written in terms of the respective optical path lengths n x l y with x and y being placeholders for the different indices of the refractive index n and the geometric length l: = (ñ f.s.,out ·l w,out + n air ·l c,out ) − (ñ f.s. ·l w,in + 1 ·l c,in ).
(4) The quantities marked with a tilde specify a state under the influence of external pressure and 'f.s.' refers to fused silica. Consequently, a similar expression can be derived for the refractometer cell being in equilibrium, i.e. inside and outside under vacuum conditions  [V V] = (n f.s.,out · l w,out + 1 · l c,out ) − (n f.s. · l w,in + 1 · l c,in ) (5) which can serve as a reference state to determine geometric differences along the inner and outer path.

Stress-induced optical path length changes
When the refractometer cell is exposed to external pressure, the optical paths undergo mechanical stress in the glass material (fused silica). Therefore, using integration over the total differential of the optical path length d(n y l x ) = l x dn y + n y dl x (again with x and y being placeholders for the different indices) the expression δ(n y l x ) = d(n y l x ) = l x δn y + n y δl x (6) can be derived assuming that the small δ-changes follow a linear relation, which is then equivalent to δ(n y l x ) =ñ ylx − n y l x .
The combination of the two previous equations yields n ylx = n y l x + l x δn y + n y δl x (8) or, when δn y = 0, i.e. n y is not affected by pressure changes, With the equations (8) and (9), the former equation (4) can be converted to [A V] = n f.s.,out · l w,out + n f.s.,out · δl w,out + δn f.s.,out · l w,out + n air · (l c,out + δl c,out ) − (n f.s. · l w,in + n f.s. · δl w,in + l w,in · δn f.s. ) and by replacing n f.s.,out · l w,out − n f.s. · l w,in with help of equation (5) and approximating n f.s.,out · δl w,out = (n f.s. + ∆n)· δl w,out ≈ n f.s. · δl w,out (justified by the small inherent material inhomogeneity ∆n n f.s. ) we eventually have Aimed at an expression to determine the gas refractive index n air based on the measured optical path differences [A V] and [V V] the additional quantities in the previous equation have to be collected in correction terms. δl w,out is simply the geometric length change of the window thickness due to external pressure in the outer region of the window. As the mechanical stress is quasi-isotropic in this region, one can define δl w,out = −l w,out takes account of the geometric length change of the tube due to external pressure and for the path replacement which comes from the external pressure bending the window into the tube by the amount δl c,bend (δp out ). As illustrated in figure 3, a small fraction of vacuum path inside the tube is replaced by air beyond the window. In the outer region, the path along the tube is shortened by δl c,out = −l c,out κ 3 δp out due to the compression of the tube's length. The bending of the windows does not have a significant effect on the external path length, as the occurring tilting angles are too small. δl w,in = δl w,in (δp out ) is the geometric length change of the window thickness on the central axis of the tube, but cannot be defined analogously to the previous terms, because the influence of the external pressure is not isotropic in this region. Therefore, it has to be determined by different means as described in section 4.2. The definition of the two remaining quantities specifying the changes of the refractive index of the glass material, is derived in the appendix. With these definitions inserted into equation (11) it can be converted to an expression to determine the gas refractive index n air or, here, the refractivity n air − 1: computing time and to set suitable boundary conditions, it is common to divide the model at the symmetry planes, which is why only an eighth of the original geometry was needed as shown in figure 4. Subsequently, the acting pressures were applied to the outer or rather the inner and outer surfaces. Then a linear static analysis was carried out and the displacements of the relevant geometry points along the components' axes were extracted. With these data the deformation of the structure (figure 5), such as the window curvature (δl c,bend ) and the change of its thickness (δl w,in ), for instance, was calculated. The calculations are based on material parameters for amorphous fused silica (Young's modulus: 73 GPa, Poisson's ratio: 0.16) with conservatively estimated uncertainties of 10% to cover the variation in the data from literature. To estimate the uncertainty of the FEM results the simulations are performed repeatedly with the extreme values of the input data.

Validation
In order to validate the FEM approach one can refer to the measured data reported in [6] combined with the previous FEM model, but matching the dimensions of the respective  refractometer cell (10 mm window thickness, 18 mm inner diameter). Moreover, it has to be taken into account that the reference state of the refractometer cell was atmospheric pressure inside and outside the cell. Nevertheless, the described measurements yielded the optical path length change through one of the cell windows while it was exposed to pressure changes inside the tube which was evacuated, so that the results are appropriate for the validation purpose. By combining equations (6) and (A.4) and considering that the pressure change outside the refractometer cell is zero in this case, one gets the relation between the measured optical path length change δOPL and the geometric length change δl w,in of the window thickness. On the one hand, the data of figure 3 in [6] shows that the resulting path length change is approximately δOPL = 8.2 nm ± 0.9 nm. The uncertainty of about 10% considers the variation in the repeated evacuation cycles, the approximated linear approach of the refractive index change and the estimate of the actual atmospheric pressure during the measurement. On the other hand, the FEM simulations yield δl w,in = 8.7 nm ± 0.5 nm at a pressure difference of 1000 hPa; i.e. the window becomes thicker when the cavity is evacuated. Then, with n f.s. = 1.458 and by applying equation (14), one gets δOPL FEM = 7.3 nm ± 0.5 nm. As the values of δOPL FEM and δOPL are in mutual agreement within their uncertainties, the validity of the approach from section 3 proves to be trustworthy.

Results for the correction
Consequently, in the next step the required correction terms (iii) and (iv) from equation (12) can be determined for the refractometer cell described in section 2. As there are two windows which are passed by the light on the way through the cell, the values of l w,out and δl w,in are considered as the sum of both windows, respectively. From the FEM simulations one obtains δl w,in = 2 · (−0.7 nm ± 0.2 nm) at 1000 hPa external pressure which means that the thickness of the windows becomes smaller on the refractometer cell's central axis compared to the vacuum reference state. Because the pressure range between vacuum and atmospheric conditions only provokes linear deformations in the material-in contrary to [14] where pressures of several atmospheres are investigated-, the result for 1000 hPa can be scaled accordingly to smaller pressures. Then, with κ = 2.78 × 10 −11 Pa −1 and l w,out = 2 · (5 mm ± 0.2 mm), the corrections (iii) and (iv) from equation (12) can be calculated depending on the actual pressure difference. Figure 6 shows the dependence between the respective contributions and the external gas pressure affecting the cell windows.
As it turns out, the total correction '(iii) + (iv)' is negligibly small at low pressures, i.e. somewhat below 50 hPa. Therefore, in [12]-where pressures below 13 hPa were dealt with-the consideration of this contribution could be omitted and only the correction (ii) was required. While the contribution of (iv) is comparatively small, the dominating part of (iii) shows the relevance of the compression of the window material. Therefore, towards higher pressures the correction becomes significant and needs to be taken into account in determining the gas refractivity. For instance, at 1000 hPa the contribution to the refractivity's value is of the order of 7.3 nm/420 mm ≈ 2 × 10 −8 which corresponds to a relative contribution of 7 × 10 −5 to the refractivity of air at standard conditions.

Conclusion
In the previous method of operation of the refractometer cells at PTB, the reference state used for the measurement of geometric corrections was atmospheric air pressure. This is convenient for typical measurements in the field of calibration of length standards (e.g. gauge blocks) under atmospheric conditions. But for highly precise measurements of the compressibility of a material the pressure in the environment of the sample under test and, hence, the refractometer cell is varied beyond the range of usual atmospheric pressure variations. Moreover, starting from a reference state in vacuum is reasonable to obtain small measurement uncertainties. Due to the design of the refractometer cell the derived correction is then to be used for avoiding systematic deviations of the measurement results. The derived theoretical model is kept simple, could be validated by experimental data and the applicability only requires the following constraints: high homogeneity of the cell's material and a pressure range in accordance with pure-linear material deformations. Then, for a refractometer cell with a geometry as shown in section 2, the calculation of the correction yields pressure-induced optical path length changes in the size of up to a few nanometers which affect the measurement of the air refractivity by a relative contribution on the order of up to 7 × 10 −5 .

Appendix. Refractive index changes
The mechanical stress on the refractometer cell due to the external pressure induces changes of the material's density ρ and, therefore, of its refractive index n as can be deduced from the Lorentz-Lorenz relation [15] written in the form with R being the specific refraction of the material. Following the approach from [13] via the total differential dρ = ∂ρ ∂n dn one can derive the expression While in [13] the pressure was changed inside the refractometer cell, in the present case the derivation has to deal with external pressure. Therefore, the relative differential volume change of a cylindrical volume segment V = (π · (D tube /2) 2 ) · l w,in of the window in front of the tube with an inner diameter D tube is Then, because of − dV V = dρ ρ = 1 L · dn and dD tube = − κ 3 · D tube · dp out one gets after carrying out the integration δn = L · − 1 l w,in δl w,in + 2 · κ 3 · δp out (A. 4) and, in the special case of dl w,in = − κ 3 · l w,in · dp out , δn = L · κ · δp out . (A.5)