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Determination of calculation parameters for difference approximation of group refractive index of air based on twopoint central method
Journal of the European Optical SocietyRapid Publications volume 14, Article number: 14 (2018)
Abstract
Background
The accurate knowledge of the refractive index of air is very important for environmental compensation in length measurements.
Methods
In this study, we apply difference approximation methods to facilitate the calculation of the group refractive index of air (GRA) over all calculable wavelengths. Our approach involves the determination of a suitable combination of the step size Δλ and numbers of significant digits of calculations of the phase refractive index of air (PRA) by balancing the two main errors (roundoff and truncation errors) over the entire calculable wavelength range.
Results
Based on our calculations, we find that the GRA computation over the range of all calculable wavelengths (301.51698.5 nm) can be easily approximated by the twopoint central difference method with Δλ = 1.5 nm and 12digit PRA accuracy.
Conclusion
The approximation accuracy is less than 5 × 10^{9}. Our approach can be used by nonexpert users to obtain the GRA with sufficient accuracy.
Background
The accurate knowledge of the refractive index of air is very important in length measurements. The wavelength of a laser (which is commonly used for length measurements) propagating in air changes depending on the air refractive index. A lower refractive index of air corresponds to a higher speed of light, and as the speed of light changes, its wavelength also changes. When measuring linear displacements in air with a laser, it is particularly necessary to perform wavelength compensation. For most applications, the refractive index of air is directly calculated by the Edlén equation (for e.g., [1,2,3,4]).
Here, we consider the case of a length measurement based on the use of the wavelength of a singlemode laser. Firstly, we determine values for air pressure, temperature, and humidity using sensors. We next calculate the phase refractive index of air (PRA) as n_{p}. Secondly, we know the laser vacuum wavelength to be λ_{vac}, and therefore, using n_{p} and λ_{vac}, we calculate the laser wavelength in air, λ_{air}. For a linear displacement L, we measure L as L_{air} = λ_{air} × (M + N)/2. Here, M and N denote the integer and fraction parts, respectively, of the length measurement. Next, it is required to compensate for the influence of n_{p} on L_{air}. By calculating L_{vac} = L_{air} × n_{p}, we can convert the length measured in air (L_{air}) to the length in vacuum, L_{vac}. Without the influence of n_{p}, the values of lengths in vacuum are comparable with each other.
Probably the most exciting recent advancement in the field of length measurement is the development of the distance ranging technique based on the femtosecond optical frequency comb (FOFC) [5]. This technique enables us to measure length based on not only wavelength (e.g., [6,7,8,9]) but also the adjacent pulse repetition interval length (APRIL) (e.g., [10,11,12,13,14,15,16,17,18,19,20]), where APRIL denotes the physical length between two adjacent pulses.
In the study, we consider the case of a length measurement based on the use of the APRIL of an FOFC laser. We acquire values for air pressure, temperature, and humidity, and we calculate the group refractive index of air (GRA), n_{g}, using the following equation [21].
where λ_{cen_vac} represents the central wavelength of the FOFC. Further, \( {\left({\mathrm{dn}}_{\mathrm{p}}\left({\lambda}_{\mathrm{vac}}\right)/\mathrm{d}{\lambda}_{\mathrm{vac}}\right)}_{\lambda_{\mathrm{cen}\_\mathrm{vac}}} \) represents the derivative of the function y = n_{p}(λ_{vac}) at λ_{vac} = λ_{cen_vac}.
We know that the vacuum value of APRIL is Λ_{vac}. Using n_{p} andΛ_{vac}, we calculate the FOFC’s APRIL in air, Λ_{air}. For a linear displacement L, we represent L as L_{Λair} = Λ_{air} × (M_{Λ} + N_{Λ})/2. Here, M_{Λ} and N_{Λ} denote the integer and fraction parts, respectively, of the length measurement, and they are required when L_{Λair} is measured in units ofΛ_{air}. Next, we compensate for the influence of n_{g} on L_{Λair}. By calculating L_{Λvac} = L_{Λair} × n_{g}, we can convert the length measured in air L_{Λair} to the length in vacuum L_{Λvac}. Without the influence of n_{g}, the values of the lengths in vacuum are comparable with each other.
In essence, for APRILbased distance evaluation, the GRA is required, which can be calculated via the derivative of the PRA with respect to the wavelength.
The expression of the PRA given by the empirical formula is a complicated function, and mistakes in differentiation can lead to transformation and calculation mistakes. One way to avoid this problem is to use the difference approximation. The differential approximation is a method of approximating derivatives with algebraic operation of discrete values. In a previous research [22], we determined that by means of the twopoint central difference with a step size of 10 nm, the GRA could approximated by an order of 60 × 10^{− 9} for both the visible light range (380770 nm) and the optical fiber light transmittance range (12601625 nm). In addition [23], we found that the GRA computation can be easily approximated by the fourpoint central difference approximation method and that approximation accuracies of less than 30 × 10^{− 9} can be achieved over the entire wavelength range (3201680 nm) with a step size of 10 nm.
The question remains as to whether the applications of the twopoint difference approximation could be sufficiently accurate to cater to nonexpert users over the entire abovementioned wavelength range. When compared with the fourpoint central difference approximation method, the calculation of twopoint difference approximation method is very simple. In previous researches [22, 23], we calculated the PRA with ninedigit accuracy. In this study, we verify the achievable accuracy of the twopoint difference approximation method for different step sizes and number of significant digits (NSD) of the PRA.
The rest of the paper is organized as follows. Section 2 presents an overview of the definition of twopoint difference approximation. Section 3 presents our numerical calculations. Section 4 presents the results of our numerical calculations and discussions. Finally, Section 5 presents our conclusions.
Methods
Having understood the importance of environmental compensation in length measurement, we next focus our attention on differential calculations by difference approximations (e.g., [24, 25]). The complexity of the calculation of the twopoint difference methods (namely, twopoint forward, twopoint backward, and twopoint central difference methods) is the same. As can been inferred from previous researches [22, 23], when compared with the twopoint central difference method, no significant improvement in the calculation accuracy can be expected with the twopoint forward and twopoint backward difference methods. Based on these two facts, we only consider the twopoint central difference method in the following discussion. We first present the difference approximation equations for the convenience of users. The expression for the GRA calculated by means of the twopoint central n_{p_2cent}(λ_{vac}) difference method is presented as Eqs. (2).
With this understanding of the calculation of n_{g} for environmental compensation and the concept of the difference approximation, we turn to the numerical calculations.
The goal of this study is to determine a specific step size and NSD of n_{p} that can be applied to most wavelengths, thereby resulting in an approximated calculation of n_{g} with a negligible difference from the actual value. The Edlén formula affords an accuracy of approximately 3~ 5 × 10^{− 8} [26]. We set a difference of < 5 × 10^{− 9} between differential calculations and the differential approximation as an acceptable level of difference.
Due to length constraints of this paper, we only discuss the Edlén empirical equations [2,3,4] in which the PRA can be derived as a function of the wavelength λ_{vac}, temperature T, barometric pressure P, and humidity H. For the phase refractive index, we use the equations given in Ref. [4].
Numerical experiments
The absolute value of (the) difference (between the theoretical and approximated values) is a function of the approximation method, step size, differential wavelength point, and NSD of n_{p}. The calculable range of the empirical formula [4] is 300 to 1700 nm, which means that λ_{vac} + Δλ and/or λ_{vac} − Δλ must lie within this range. In other words, for the twopoint central difference method, the calculable ranges of the GRA is [λ_{min}, λ_{max}] (λ_{max} + Δλ ≤ 1700, λ_{min} − Δλ ≥ 300). Since our objective is the determination of a specific step size that can be applied to almost all wavelengths, as a test, we apply step sizes that are multiples of 10 and less than 10 nm.
In the study, the first numerical calculations were performed under the following environmental conditions: T = 20 °C, P = 101.325 kPa, and H = 50%. The theoretical value of n_{ g }(λ_{vac}) (λ_{vac} : 300 ≤ λ_{vac} ≤ 1700 nm) was calculated with the use of Eq. (1). The approximate values of n_{g_2cent}(λ_{vac}) were calculated based on Eqs. (2), for different λ_{vac} values. Subsequently, by using these values, we calculated the absolute value of the difference between the theoretical and difference approximation values as n_{ g }(λ_{vac}) − n_{g_2cent}(λ_{vac}) for the twopoint central difference method. The approximate values change when the step sizes and NSD of the PRA change.
Results and discussion
Figure 1 shows the variation in the absolute value of difference with the calculable wavelength with Δλ = 1 nm upon calculating the PRA with 12digit accuracy. In Fig. 1, we note that the maximum absolute value of the twopoint central difference method is obtained at around 1700 nm. As mentioned in a previous study [23], the “zigzag” nature of the curves is due to the roundoff error in calculating the PRA. The roundoff error originates from numerical calculations that use a limited number of digits in a computer. In the case of this combination (namely, Δλ = 1 nm and n_{p} with 12 digits), the absolute value of the difference of the twopoint central difference method is mainly affected by the roundoff error. The reason underlying the high roundoff error occurring at larger wavelengths is that the curve of PRA at larger wavelengths is “gentler” than that at smaller wavelengths. A gentle curve means the difference between these two values used for calculation is too small. If we do not consider more digits, we cannot accurately evaluate their difference.
We next change the step size Δλ and NSD of the PRA to observe the change in the maximum absolute value of difference. For example, Fig. 2 shows the variation in the absolute value of difference with Δλ = 0.1 nm upon calculating the PRA with 16digit accuracy. From the figure, we note that the maximum absolute value of difference obtained with the twopoint central difference method exhibits a corresponding change, with the corresponding position shifting to around 300 nm. The absolute values of difference obtained at smaller wavelengths are mainly affected by the truncation error. Truncation error exists because the difference approximation is calculated based on the linear Taylor expansion. This shift of the maximum value indicates that the main error contributor is now the truncation error and not the roundoff error. In the case of this combination (namely, Δλ = 0.1 nm and n_{p} with 16 digits), the maximum absolute value of difference for the twopoint central difference method is mainly affected by the truncation error. The reason underlying high truncation error at smaller wavelengths is that the curve of the PRA at smaller wavelengths is “steeper” than that at larger wavelengths. Since the difference approximation is a linear approximation, its precision is suitable for a gentle curve, and its accuracy is poor for a steep curve.
When the combination of Δλ and the NSD of PRA is changed, the main contributor of the error in the maximum absolute value of difference also changes. It is difficult to estimate the absolute value of difference based on the law of propagation of uncertainty. For this reason, we perform numerical simulations to evaluate the maximum absolute value of difference without considering the main cause of error. Here, we remark that the PRA is a continuous function, and therefore, we only test the range around the upper and lower limits of the variation range of the environmental parameters. Since the maximum absolute value of the difference is obtained around 300 nm and 1700 nm, we address only these two cases in the following discussion.
Figures 1 and 2 indicate that there is an optimal combination of Δλ and NSD of PRA that can minimize the total error. We can determine a suitable estimate of this combination by balancing the truncation error and the roundoff error of the PRA; this forms the basis of our strategy. We test different pairs of Δλ and PRA digits. Figure 3 shows a candidate result that can be optimized to almost all wavelengths with Δλ = 1.5 nm and 12digit PRA accuracy. We note that the maximum absolute value of difference is 2.7 × 10^{− 9}. Since the purpose of this research is to determine a combination of the step size Δλ and NSD of the PRA satisfying the error conditions, we only consider this combination in the following discussion.
Figures 4 and 5 show the variation in the absolute value of difference as a function of temperature. The maximum absolute values of difference in Figs. 4 and 5 are 3.5 × 10^{− 9} and 2.8 × 10^{− 9}, respectively. The maximum absolute value of difference with λ_{vac} = 301.5 nm is larger than that of λ_{vac} = 1698.5 nm. In the following discussion, we only present the results for λ_{vac} = 301.5 nm.
Figures 6 and 7 depict the variation in the absolute value of difference with pressure. The maximum absolute values of difference in Figs. 6 and 7 are 3.8 × 10^{− 9} and 3.7 × 10^{− 9}, respectively. The maximum absolute value of difference at T = 10 °C is larger than that at T = 30 °C, and therefore, in the following discussion, we subsequently address only the case of T = 10 °C.
Figure 8 shows the variation in the absolute value of difference as a function of humidity. The maximum absolute value of difference in Fig. 8 is 4.1 × 10^{− 9}.
Here, we again recall that the PRA is a continuous function. The maximum or minimum value of the function appears at the upper and lower bounds of the variable range. The roundoff error leads to the zigzagging of the curve. As can be observed in the above figures, there is no significant deviation in the curve. Based on these facts, we can conclude that under environmental conditions of T : 10 ≤ T ≤ 30 °C, P : 60 ≤ P ≤ 120 kPa, and H : 5 ≤ H ≤ 85 %, the maximum absolute value of difference is less than 5 × 10^{− 9} over the entire wavelength range (301.51698.5 nm). This result proves the feasibility of our approach.
Conclusion
Under environmental conditions of T : 10 ≤ T ≤ 30 °C, P : 60 ≤ P ≤ 120 kPa, and H : 5 ≤ H ≤ 85 %, we confirmed the accuracy of the twopoint central difference approximation method via numerical calculations. Our strategy was to determine a suitable combination of the step size Δλ and NSD of PRA by balancing the truncation error and the roundoff error. We determined that the GRA computation can be easily approximated by the twopoint central difference approximation method and that approximation accuracies of less than 5 × 10^{− 9} can be achieved over the entire wavelength range (301.51698.5 nm) with Δλ = 1.5 nm and 12digit PRA accuracy. Our method can be used by nonexpert users to obtain the GRA with sufficient accuracy for environmental compensation in length measurements.
Abbreviations
 APRIL:

Adjacent pulse repetition interval length
 FOFC:

Femtosecond optical frequency comb
 GRA:

Group refractive index of air
 NSD:

Number of significant digits
 PRA:

Phase refractive index of air
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Acknowledgments
This research work was partially financially supported by Japan Society for the Promotion of Science (JSPS) KAKENHI GrantinAid for Young Scientists (B) (Grant Number 17 K17743).
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WD proposed the idea, carried out the experiments and wrote the manuscript. All authors participated in the discussion of the experiments. All authors read and approved the final manuscript.
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Correspondence to Dong Wei.
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Dr. Dong WEI received his BSc and MSc in Engineering from the University of ElectroCommunications of Japan, in 2006 and 2008, respectively, and he received his DSc from the university of Tokyo, Japan, in 2011. He is now an assistant professor in the Department of Mechanical Engineering at Nagaoka University of Technology in Japan. His interest includes applied optics and intelligent signal processing.
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The authors declare that they have no competing interests.
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Keywords
 Group refractive index
 Difference approximation
 Frequency comb
 Length measurement
 Metrology