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The Edlén empirical equations and the two-color method are the commonly used approaches to converting a length measured in air to the corresponding length in vacuum to eliminate the influence of the refractive index of air. However, it is not well known whether the two-color method is superior to empirical equations in refractive index compensation. We investigated the uncertainties of these approaches via numerical calculations of their sensitivity coefficients of environmental parameters. On the basis of a comparison of their uncertainties, we found that in a 0% humidity environment, the two-color method had potential to provide greater measurement accuracy than the empirical equations.

Meter, the unit of length, is defined in vacuum. However, measurements of length are often carried out in air, which presents some problems. Let us assume that we want to compare two geometric distances

One approach to obtaining the value of RIA is to use empirical equations [

Another approach to suppressing the influence of RIA is to apply the two-color method, which was first proposed by Bender and Owens [

Recently, high-precision length measurements based on fem to second optical frequency comb (FOFC) have been carried out (e.g., [^{−11} throughout hours [

One question arises naturally: theoretically, is the two-color method superior to the empirical equations in RIA compensation? We employed a numerical approach to investigate this possibility.

The distance between two points measured in air is an optical distance

where n represents the RIA. By applying the law of propagation of uncertainty [

where

The uncertainty of refractive index can be evaluated by the following equation [

where

where

The distances between two points measured in air by using different wavelengths are optical distances

where A is the so-called A-factor defined as

Equation (7) can be rewritten as follows.

By applying the law of propagation of uncertainty to Equation (9), we have

The uncertainties of the first and second terms of the right-hand side of Equation (10) are, respectively,

Because we have

By substituting Equations (11) and (13) into Equation (10), we obtain

The first and third terms of the right-hand side of Equation (14) are the uncertainty due to the A-factor, and the second and fourth terms are the uncertainty due to the length measurement. These two are defined as follows, respectively.

The uncertainty of A-factor is as follows.

where

where

In Equation (4), the uncertainty due to the length measurement is multiplied by the factor

is satisfied, the two-color method can be shown to obtain measurements with a smaller error than that of the empirical equations. We performed numerical calculations to check whether Equation (19) is feasible.

We used the following parameters for simulation. By referring to Ref. [

On the basis of Equations (6) and (18), we calculated the change in the sensitivity coefficients when environmental parameters change in a realistic range (T ∊ [10, 30] ˚C, P ∊ [90,115] kPa, H = 0%). The calculations of the derivative of each refractive index have been validated in Ref. [

As shown in

On the basis of Equations (3) and (15), we calculated the uncertainties due to the A-factor and refractive indices, respectively. The geometric distance G was set to 1 m. We assumed that

We analyzed the uncertainties of length conversion based on the Edlén empirical equations and the two-color

method, in which the uncertainties due to length measurement and refractive index compensation were decomposed. Using numerical calculations of sensitivity coefficients of the A-factor and refractive indices of the environmental parameters, we found for the first time that in a realistic environmental parameter range (T ∊ [10, 30] ˚C, P ∊ [90, 115] kPa, H = 0%), the uncertainty of the two-color method due to the A-factor was smaller than that of the empirical equations due to refractive indices. This result suggests that in a 0% humidity environment, the two-color method has potential to provide greater measurement accuracy than the empirical equations, with the cooperation of suppressing the uncertainties of length measurements (compared with uncertainties of refractive index compensation) to a negligible level. The findings of this study provide a better insight into the two- color method, and will create opportunities for further development of application of this method.

This research work was partially financially supported by a grant (R1501) from the Mitutoyo Association for Science and Technology.

Dong Wei,Kiyoshi Takamasu,Hirokazu Matsumoto, (2016) Is the Two-Color Method Superior to Empirical Equations in Refractive Index Compensation?. Optics and Photonics Journal,06,8-13. doi: 10.4236/opj.2016.68B002