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The absolute group refractive index of air at 1563 nm is measured by dispersive interferometry, and a combined uncertainty of 1.2 × 10−8 is achieved. The group refractive index of air is calculated from the dispersive interferograms of the two beams passing through the inner and outer regions of a vacuum cell by fast-Fourier-transform. Experimental results show that the discrepancies between our method and modified Edlén equation are less than 3.43 × 10−8 and 4.4 × 10−8 for short-term and long-term experiments, respectively. The interferogram update rate is 15 ms, which makes it suitable for application of real-time monitoring. Furthermore, it is promising to improve the measurement uncertainty to 3.0 × 10−9 by changing the material of the vacuum cell and measuring its length more accurately through optical interferometry.
© 2015 Optical Society of America
1. Introduction
There has been increasing concern about the accurate knowledge of air refractive index, which is of great importance to optical interferometry, precision instruments manufacturing and industrial applications like the control of step-and-repeat mask positioning for integrated circuit production. For example, the uncertainty of ultra-precise interferometric absolute distance measurements in air are determined mainly by the accuracy of the air refractive index [1–4].
Conventionally, the refractive index of air has been determined according to Edlén equation, which can reach an uncertainty of a few parts in 10−8 by means of relating the refractive index of air to pressure, temperature, humidity and carbon dioxide concentration [5–8]. For example, Hieta et al. utilizes direct absorption spectroscopy of oxygen to measure the average temperature of air and of water vapor to measure relative humidity, thus compensating the air refractive index by Edlén equation [9]. However, the modified Edlén’s equation is only applicable to standard air with definite gas compositions and its uncertainty is limited by the measurement uncertainties of environmental parameters. Therefore, the direct measurements are usually preferred in place where high accuracy or in situ monitoring is required. The direct methods for measuring air refractive index can be classified into two general methodologies. The first approach is derived from the incremental interferometry, where the phase change of interference fringes along with the process of air pumping or filling should be recorded continuously [10,11]. For example, J. Zhang et al. measured the air refractive index with high precision by simultaneous measuring the pulse delay and the spectral phase in a highly unbalanced Mach-Zehnder interferometer with an optical frequency comb [12,13]. Nevertheless, this approach is time-consuming and complex because the pressure and temperature fluctuation cannot be avoided in the process of air filling or pumping. The second approach is measuring air refractive index absolutely, where the continuous phase-recording in the process of air filling or pumping is not necessary [14,15]. In 2013, Jitao Zhang et al. developed a gas refractometer based on a synthetic pseudo-wavelength method [16,17], but the lengths of the vacuum cells should be designed properly so that the highest order synthetic pseudo-wavelength is larger than the refractive index of the measured gas. More recently, a group from Institute of Scientific Instruments of the Czech Academy of Sciences presented a new method to measure the air refractive index by low-coherence interferometry combined with laser interferometry [18], but the measuring mirror should be scanned by a piezoelectric movable stage.
In recent years, femtosecond optical frequency combs have revolutionized the field of optical precision measurement [19,20]. In addition to its many applications in absolute distance measurement [21–23], it is possible to extend the use of dispersive interferometry with optical frequency combs to measure the refractive index of air precisely. In this paper, we demonstrate an air refractometer based on the Michelson interferometer combined with an optical frequency comb and a double-spaced glass cell whose inner area is evacuated to vacuum in advance and outer area is connected to the ambient air. Utilizing dispersive interferometry, the group refractive index of air can be absolutely obtained by measuring the geometrical length of the double-spaced glass cell and the optical path difference between the inner and outer regions of the double-spaced glass cell. In this method, as neither gas-filling nor pumping process is required during the measurement, a complete measurement could be finished within 15 ms. The simple configuration makes it compact and easy to handle. In addition, taking advantage of the optical frequency comb with a broad spectrum bandwidth and a stable repetition rate, the refractometer can perform absolute measurement of the air refractive index in a large measurement range with a high accuracy. Moreover, the air refractometer reported here has a potential to be applied to in situ measurement where the environmental parameters vary within a large scale.
2. Methods
2.1 Experimental setup
Figure 1 illustrates the experimental setup of the air refractometer. The vacuum cell is made of BK7 glass, and its diameter is 18 mm with a wall thickness of 5 mm. The window has a diameter of 42 mm and thickness of 5mm, and its transmittance is better than 99.8% after anti-reflecting coating. The windows are attached to the end faces of the vacuum cell in parallel by epoxy resin, and the inner space of the tube is pumped down to 10−2 Pa before sealed. The actual length is measured from the inner surfaces of the windows as 185.140 mm by a Coordinate Measuring Machine (CMM) (HEXAGON Metrology Optiv Performance 222). In the experiment, a ~86 fs femtosecond optical frequency comb (MenloSystems, M-Comb) is used as the light source, whose repetition frequency is locked at 250MHz referenced to an Rb atomic clock (Symmetricom 8040C). The center wavelength and the FWHM (Full Width Half Maximum) are about 1563 nm and 50.3 nm, respectively. After collimation, the output of the light source is sent to the BSP (Beam Splitter Plate) which is coated with 50% reflective film on the upper surface and >99% high-reflecting film on the bottom surface. When the incident angle is 45 degree, the light beam can be divided by the BSP into two parallel beams with equal intensity, which then pass through the inner and outer regions of the vacuum cell. After reflected by the corner cube with gold coating, these two beams recombine on the upper surface of the BSP and interfere with each other. The spectral interferograms are coupled into a single mode fiber and recorded by the OSA (Agilent 86140B), which has a wavelength resolution of 0.2 nm.
Fig. 1 Optical setup of the air refractometer. Iso, isolator; M1, M2, mirror; BSP, beam splitter plate; OSA, optical spectrum analyzer. The red line corresponds to the light that propagates in the free space (the dashed line denotes the light going through the inner region of the vacuum cell) and the yellow line corresponds to the light that propagates in the single mode fiber. The photograph of the vacuum cell is also shown.
2.2 Measurement principle
The dispersive interference information of the two beams passing through the inner and outer regions of the vacuum cell provided by the OSA is represented in the form of the spectral power density, which is a function of the optical frequency expressed by the equation as follows [21]:
where is the mean intensity of the two beams, is the modulation amplitude, and is the phase difference between the measurement beam and the reference beam, which is generally expressed as: where is the optical path delay between the measurement and reference beams. is the geometrical length of the vacuum cell. n(v) and are refractive index of air and the speed of light in vacuum respectively. Considering Eq. (2), Eq. (1) is described in the complex form as:where j = (−1)1/2. Then Eq. (4) is Fourier-transformed and the result is as follows:where is Dirac delta function and is the variable representing the optical path delay. is the Fourier transform of and is the Fourier transform of . From Eq. (5), we know has three peaks at , and . To determine the exact value of from the t-domain, only the peak at is isolated by use of a band-pass filter of finite width and then inverse Fourier-transformed into the -domain, whose result is derived as:Subsequently, the phase difference can readily be obtained through the arctangent operation in the following [24]:However, the arctangent operation gives the phase difference in its wrapped value confined within the range of –π to π. Consequently, the unwrapped value of bears an offset from the true absolute phase value of . Thus the first-order slope of the unwrapped phase is taken, which can be related to L with no effect of the offset as:where . Note N represents the group index of refraction of air, which is a function of the center frequency of the light source. As we can see from Eq. (8), if the geometrical length of the vacuum cell is measured with a high accuracy, then (N-1) can be obtained precisely from the relation of:In order to minimize the uncertainties introduced by Fast Fourier Transform, a filtering window has to be used to get the best result [23]. In addition, the described Fourier-transform method of determining (N-1) demands that the peak at in the t-domain be securely separable from its neighboring peak appearing at the origin. This is, however, not always the case when α is too small. Equation (5) indicates that the peak at α is convoluted by B(t), so the peak at α and the peak at the origin go overlapped when α reduces below a certain threshold (w) that is related to the temporal width of B(t) [21]. This situation limits the minimum measurable (N-1) following the relation of (N-1)min = c/(2Lw), which is found ~8 × 10−6 in our configuration.
The maximum extent of α is also restricted by the Nyquist sampling limit that is described as α≤1/(2P). The frequency resolution P represents the spacing of the two neighboring modes actually sampled by the OSA. This upper limit of α is converted to (N-1) as (N-1)max = c/(4PL), which is calculated as ~0.022. Therefore, the flexible measurement range of N is from 1.000008 to 1.022, which can cover almost all of the routine gases at 1563 nm including the ambient air fluctuating within a large range.
3. Results
3.1 Experimental results
Two measurement experiments were carried out to evaluate the performance of the proposed method in short-term and long-term measurements. The measurement results were compared with reference data calculated by the modified Edlén equation derived by Ciddor [8]. In our experiment, the environmental parameters (temperature, pressure, humidity and CO2 concentration) were monitored by several sensors: a thermometer with an uncertainty of 0.01 ◦C (Fluke 1523), pressure gauge with an uncertainty of 14 Pa (Setra Model 470), hygrometer with an uncertainty of less than 2% RH (Rotronic HP23), and a CO2 sensor with an accuracy of less than 50 ppm (TEL 7001), respectively, all of which were carefully calibrated by the National Institute of Metrology of China and the calibrated ranges are 14-26 ◦C, 80-110 KPa, 20-60% RH and 0-10000 ppm. However, since the fluctuation of the CO2 concentration is 27 ppm (593 ppm~620 ppm) and is very small, we assume the CO2 concentration is homogenous and equals to the mean value of the measured CO2 concentration. According to the modified Edlén equation, what the above assumption results in the air refractive index residuals is less than 2 × 10−9. Therefore, for convenience and better visualization, we didn’t give the CO2 concentration data in the paper.
Figure 2(a) shows the spectral interferogram monitored by the OSA at some time. The corresponding Fourier transform with three peaks are clearly shown in Fig. 2(b). The peak at is picked by band-pass filtering and then perform inverse Fourier transform. Figure 2(c) shows the wrapped phase of the inverse Fourier transform and Fig. 2(d) shows the unwrapped phase. The first-order slope of the unwrapped phase is 2.036088699 × 10−12, so that the group refractive index of air is calculated as 1.000262547 using Eq. (9). Compared with the result from the modified Edlén equation of 1.000262545(temperature: 25.547 ◦C, pressure: 101.132 KPa, humidity: 43.71%), the difference is 2 × 10−9.
Fig. 2 Data processing procedure for measurement of group refractive index of air. (a) Dispersed interference intensity monitored by the OSA. (b) Fourier transform of the spectral interferogram. (c) Wrapped phase of the filtered peak. (d) The unwrapped phase (red line denotes the fitted straight line).
In Fig. 3, continuous measurement was performed for approximately 70 minutes. During the experiment, the fluctuations of the temperature, pressure and humidity were 0.333 ◦C, 54 Pa and 0.92%, respectively. According to Fig. 3(a), the results of the proposed method are consistent with those obtained by the modified Edlén equation within 3.43 × 10−8, and the standard deviation of the difference is 1.4 × 10−8.
Fig. 3 Experimental results of the refractive index of ambient air in short term. (a) Comparison of experimental and reference data of ambient air. Dashed circle with arrow indicates the corresponding y axis of the data curve.(b) Pressure and temperature data in experiment. (c) Humidity data in experiment.
For the long-term experiment, the fluctuations of the temperature, pressure and humidity were 0.375 ◦C, 97 Pa and 4.87%, respectively. It took about 10 h for this continuous measurement and the result is shown in Fig. 4. Note that an offset of 0.2 × 10−6 between our measured data and reference data is given for better visibility so that the two curves do not overlap. As is shown in Fig. 4(b), the measurement difference between the two methods is less than 4.4 × 10−8 and long-term measurement of the group refractive index of ambient air indicates a good stability of the performance (better than1.66 × 10−8). It should be stated that, besides the measurement data, the reference data partially accounts for the above discrepancy. Since the reference data are calculated by the modified Edlén’s equation with environmental parameters, the calibrated uncertainties of these sensors will contribute an uncertainty of about 4.5 × 10−8 to the reference data. In addition, the response times of sensors are also responsible for the difference, since the air refractive index calculated by the modified Edlén’s equation lags behind the real-time value. In other words, the refractive index is really fluctuating on the same level and at the same rate as the environmental parameters (there is no control of environmental parameters in the lab and the air follow caused by vibration and people walking by can also contribute to the fluctuations of the environmental parameters) and the slow environmental sensors used in the Edlén method can’t follow these changes.
Fig. 4 Long-term measurement of the refractive index of ambient gas at 1563 nm for about 10 h. (a) Experimental and reference data of ambient air. (b) Difference between experimental and reference data. (c) Pressure and temperature data in experiment. (d) Humidity data in experiment.
3.2 Uncertainty evaluation
The uncertainty was evaluated on the basis of the Guide to the Expression of Uncertainty in Measurement (GUM), as recommended by the International Organization for Standardization [25]. According to Eq. (9) and with an assumption that all of the input quantities were independent, the combined uncertainty of our air refractometer can be described as:
where ,,and are the measurement uncertainty of group air refractive index, calculation accuracy of phase difference by FFT, measurement uncertainty of length of vacuum cell and frequency, respectively. In our experiment, four main error sources are considered: the frequency error, the length error of vacuum cell, the calculation error of phase difference by FFT, and the residual gas in the vacuum cell. Their contributions to the relative uncertainty of (N-1) can be evaluated as follows.- • Frequency error: this error source incorporates the sampling frequency linearity of the spectrometer and the frequency stability of the optical comb. The frequency sampling linearity of the spectrometer results in wrong readings of the dispersed frequencies [21], which causes the same level of error as that of the frequency instability of the frequency comb. In our experiment, the linearity error contributes 1 × 10−5 to the measurement uncertainty of (N-1). However, the repetition rate of the optical comb is locked to an Rb atomic clock (Symmetricom 8040C), whose frequency stability is 2.4 × 10−12 (1s). Since the carrier offset frequency within a range of 10 MHz is not stabilized, it contributes 5 × 10−8 to the relative measurement uncertainty of (N-1) and can be ignored, considering the frequency is around 2 × 1014 Hz. In summary, the frequency error contributes 1 × 10−5 to the measurement uncertainty.
- • Calculation error of phase difference by FFT: the phase difference is calculated from the Fourier Transform method and the result is mainly affected by two factors: the hardware resolution of analog-to-digital conversion of the OSA and the computational accuracy of the adopted Fourier transform algorithm. Before the experiment, we have simulated the calculation process on the computer and found that the calculation accuracy of the phase difference is ~1 × 10−9 and can be ignored. Note that the appropriate use of Hanning window can reduce the effect of noise, thus improving the calculation accuracy.
- • Length error of the vacuum cell: the length of the vacuum cell is measured by a Coordinate Measuring Machine (CMM) (HEXAGON Metrology Optiv Performance 222). The CMM has a measurement range of 250 mm × 200 mm × 200 mm in XYZ directions and an uncertainty of less than 2 μm. During the measurement, four points located symmetrically around the circle of each end of the cell are measured repeatedly, and the length is finally determined by the mean of four repeated results. Besides, if the two ends of the vacuum cell are inclined in the direction parallel to the incident plane (Fig. 5(a)), the average path is still equal to the center length of the vacuum cell when the light beams pass through the vacuum cell twice, that is: l1 + l2 = 2l. Similarly, if the two transmission surfaces of vacuum cell’s one end are not parallel (Fig. 5(b)), the whole path of the two beams passing through the inner and outer space of the vacuum cell is twice the center length of the vacuum cell. Therefore, the parallelism of vacuum cell’s ends has no influence on the measurement accuracy. In addition, if the vacuum cell is not in parallel with the optical path, the real path passed by the beam will be larger than the geometric length of the cell and the path difference is (1/cosγ-1)L (γ: tilt angle). In our experiment, the vacuum cell is mounted on an adjustable V-clamp mount (Throslabs, KM200V). By adjusting the tip and tilt of the vacuum cell, we first use an autocollimator (ELCOMAT 3000) to guarantee that the vacuum cell is parallel with the optical path. After careful adjustment, the tilt angle γ could be confined within 0.1 degree, resulting in a path difference of less than 1 μm. Furthermore, the length of the vacuum cell will also vary with temperature fluctuation. The cell is made of BK7 glass with a thermal expansion coefficient of 7.1 × 10−6/K. The vacuum cell is measured by the CMM at the temperature of 298.15 K, but the environmental temperature is around 298.64 K-301.38 K, thus introducing an error of 4.25 μm. Moverover, because there is an atmospheric pressure difference between inside and outside of the vacuum cell, it will cause window deformation [26]. Therefore, the windows can’t stay flat with the cell evacuated and will cause corresponding optical path length changes. According to Mechanics of materials, the window deformation dl = PL/E(P is the pressure difference, L is the length of the vacuum cell and E is the Young’s modulus of the material). Since the windows are made from fused silica, the Young’s modulus is 73 GPa, then dl = 1.01 × 105/(73 × 109) × 185.140 mm = 0.26 μm. Considering what has been mentioned above, the total length error will contribute∼4.2 × 10−5 to the relative measurement uncertainty of (N-1), which is the most dominating error source.
- • Residual gas in the vacuum cell: During the fabrication of the vacuum cell, the ambient air inside was first evacuated below 0.02 Pa using a turbomolecular vacuum pump, then the glass pipe connected to the vacuum cell was melted and sealed. According to the modified Edlén’s equation, the relative uncertainty of (N-1) caused by residual gas with 0.02 Pa would be no larger than 2 × 10−7. Nevertheless, note that the error caused by residual gas is systematic and can be modified.
To sum up, the combined relative uncertainty of (N-1) induced by the four errors should be 4.3 × 10−5. According to Eq. (10), the direct result of this method is (N-1) but not N, thus bringing a significant advantage. For ambient air, the refractive index usually approximates 1.00027 and (N-1) will be 2.7 × 10−4. Therefore, the relative uncertainties of the four error sources above contribute to the relative uncertainty of (N-1) at a ratio of 1:1, but contribute to the relative uncertainty of N at a ratio of 1: 2.7 × 10−4, thus improving the measurement accuracy of N under same circumstances. In conclusion, the measurement uncertainty of our air refractermeter is around 1.2 × 10−8. Since the length error of the vacuum cell is the most dominating error source, the measurement uncertainty of the air refractive index will become less than 3.0 × 10−9 if the vacuum cell is made of Zerodur glass (thermal expansion coefficient at 5 ◦C to 35 ◦C: 1 × 10−8/K) [27] or fused Silica glass (thermal expansion coefficient at 5 ◦C to 35 ◦C: 5.4 × 10−7/K), and the length of the vacuum cell is measured by optical interferometry with a measurement uncertainty of 10−6 [28].
Fig. 5 Parallelism error caused by two ends of the vacuum cell. (a) The two ends of the vacuum cell are inclined in the direction parallel to the incident plane. l1, geometrical path of the upper light; l2, geometrical path of the lower light;l, center length of the vacuum cell. (b) The two transmission surfaces of vacuum cell’s one end are not parallel.
4. Conclusions
We demonstrate an absolute air refractometer capable of measuring the air refractive index from 1.000008 to 1.022 with an uncertainty of ~1.2 × 10−8 at 1563 nm, using dispersive interferometry with an optical frequency comb. Since there is no gas-filling, gas-pumping process and mechanical movement during the measurement, a complete measurement could be carried out within 15 ms, which is applicable in conditions where air refractive index fluctuates very fast. The simple configuration makes it easy to handle. Besides, it is not necessary to control the environmental parameters precisely, so it can be used out of the well-controlled optical laboratory. If the vacuum cell is mounted in a gas chamber, the refractermeter can measure refractive indices of other gases by filling the measured gas into the chamber. Besides, we can use a homemade portable femtosecond fiber laser [29] as the light source of the refractometer, thus the refractometer can be applied to in situ measurement for industrial applications. In addition, the measurement uncertainty can be improved up to 3.0 × 10−9 or even less if the vacuum cell is made of fused Silica glass or Zerodur glass and the length of the vacuum cell is measured by optical interferometry with a measurement uncertainty of 10−6. All of these possible improvements are under consideration for future work.
Acknowledgments
The authors are grateful for the support of National Natural Science Foundation of China (NSFC) (Grant No. 51575311). Author Lijun Yang would also like to thank Dr. Jitao Zhang from the University of Maryland for his helpful comments and suggestions.
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