Refractive Index of Air and Precision Length Measurements

Jennifer E. Decker

Science, Technology & Innovation Division Foreign Affairs & International Trade Canada 13 May 2011 Vancouver

• Historical overview of International System of Units (SI) definition of the metre

• Interferometric length measurement / calibration

• Impact of refractive index of air on accurate length measurement – What is the magnitude of the correction? – How is the correction applied? – Recent Developments

1872 prototype Kilogramme des Archives and Mètre des Archives (c.1799)

1875 Convention of the Metre signed

1887 Michelson proposed using optical interferometers for length measurement; received 1907 Nobel Prize for physics

1892 Michelson interferometer at BIPM (Michelson & Benoît) measured the metre in terms of red line of cadmium; confirmed in 1906 by Benoît, Fabry & Perot

1960 Definition of the metre in terms of wavelength in vacuum of specific radiation from krypton 86

1975 CGPM recommended value for speed of light in vacuum based on wavelength and frequency of laser radiation

1983 Definition of the metre as length of path travelled by light in vacuum during a specific fraction of a second

The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

• Consultative Committee for Length (CCL) formed in 1952 – Provides recommendations for practical realization of the metre (Mise en Pratique) – Provides wavelengths, frequencies and associated uncertainties of recommended laser radiations, spectral lamps

http://www.bipm.org/en/si/base_units/ © Government of Canada, 2011 Definition Gauge Block Length

• “. . . the perpendicular distance between point of the measuring face and the plane surface of an auxiliary plate of the same material and surface texture upon which the other measuring face has been wrung …includes the effect of one face wringing.” ISO 3650(E)

• lengths of gauge blocks directly compared with internationally- recommended wavelength standards

• traceability established via comparison measurements

10-14 Time Frequency standards c 10-12 Laser frequency stabilized   on molecular transition  (Mise en Pratique) 10-9 Polarization-stabilized lasers c  299792458 m/s (distance measuring) 10-8 Equation for Air Refractive Index of Air 10-6 Gauge Block length   nair • 25 mm ± 20 nm • 1 m ± 70 nm

10-6 Frequency of a free-running laser

• Main influences: – air temperature, pressure and humidity on the wavelength of light refractive index, n – gauge block temperature deviation from 20ºC – optical phase change on reflection of light from surface – wringing Gauge length – design parameters of the optical Block instrument (obliquity correction)

Optical Flat

Schödel PTB-Mitteilungen 120 (2010), Heft 1 © Government of Canada, 2011 Method of Exact Fractions

• ‘Fringe fraction’ fi observed with wavelength i for several laser sources, each in turn

• Length evaluation by finding the best match for integer interference orders  to satisfy:  lf() i ii2

Wavelength Measured Example of a 10 mm gauge block: /nm fraction 1st Step: Measure fractions from the 543 0.2 interference patterns 612 0.7 633 0.5 Interference Orders Length Red Orange Green /mm 31592.5 32676.6 36828.8 9.999 03 31593.5 32677.6 36830.0 9.999 35 2nd Step: Evaluate 31594.5 32678.7 36831.2 9.999 67 gauge length based on 31595.6 32679.7 36832.4 10.000 00 Fractions and “known” 31596.6 32680.8 36833.6 10.000 31 Nominal length. 31597.6 32681.8 36834.8 10.000 64 31598.6 32682.9 36835.9 10.000 96

Decker et al., Applied Optics 42 (2003) 5670-5678

• 1966: Edlén published empirical equation for n of standard dry air and corrections for water vapour, based on experimental data (Barrell and Sears 1939, Hilsenrath 1955)

• 1988: Birch and Downs revise water vapour constants - higher accuracy

• 1994: Updated by Birch and Downs (1993, 1994) – accommodates the SI units (Pascal vs. Torr) – replaced IPTS-1948 temperature scale with ITS-90

– corrects for increased levels of CO2 in laboratory air (62 ppm) – includes improved experimental data on density of air and the refractivity of water vapour

1 ppm  1 part per million = 1x10-6 © Government of Canada, 2011 An Empirical Equation

 

8 1 2  1 2 n 1 N 10  8091.37  2333983/130   / m 15518/38.9   / m 

n 1 x  n 1 N 1 0.5327x  0.0004

n 1 p / Pa 1108 0.5953 0.009876t /C p / Pa n 1  x  tp 93214.60 1 0.0036610 t/C

 1 2 10 ntpf  ntp  f / Pa 3.8020  0.0384 / m 10

Bönsch & Potulski, Metrologia 35 (1998) 133-139

Bönsch & Potulski, 1998, “Measurement of the refractive index of air and comparison with modified Edlén’s formulae,” Metrologia, 35, 133-139

Ciddor P. E., 1996, “Refractive index of air: new equations for the visible and near infrared,” Appl. Opt., 35, 1566-1573.

Birch K. P. and Downs M. J., 1994, “Correction to the Updated Edlén Equation for the Refractive Index of Air,” Metrologia, 31, 315-316.

Decker et al., NRC Document No. 42753 (2000) Edlén B., 1966, “The Refractive Index of Air,” Metrologia, 2, 72-80.

• Vacuum wavelengths are adjusted for refractive index of air, n

 vac = n air • Correction for refractive index of air = (n-1)L n  1.000 27 ~300L nm for L [mm] (about 300 nm on 1 mm)

Change for which Nominal Value n=+1x10-6 Temperature 20.0°C -1.0°C Pressure 101.3 kPa +0.4 kPa (760 Torr) (+3 Torr) Relative Humidity 40% -100%

Estler Applied Optics 24 (1985) 808

NRC Gauge Block Interferometer Pressure Measurement 101800 Rate of change = 101700 200 Pa in 30 minutes 101600 50 nm for L=100 mm 101500 101400 101300 Pressure /Pa Pressure 101200 101100 101000 8:09 9:38 11:08 12:37 14:06 15:36 Time of Day

Molar Concentrations

N2 O2 Ar CO2 0.781 0.209 0.009 0.0004

Other gases such as He, CH4, etc. make up the remaining air composition.

Pendrill Metrologia 25 (1988) 87-93

•CO2 largest single source of variation & significant contaminant • Operators are not at rest up to 5 ~5 nm on times higher respiration rate! 100 mm • Operator near the instrument

• CO2 paired with  O2 results in additional change in composition

Birch and Downs “The Precise Determination of the Refractive Index of Air,” National Physical Laboratory (NPL) Report MOM90 (1988) Teddington, UK © Government of Canada, 2011 PTB Kösters Interferometer

• Kösters design allows real time evaluation of the refractive index of air in the proximity of the gauge block by direct comparison with vacuum. • Frequency-stabilized laser sources. • High-accuracy thermometry equipment: thermocouples paired with Pt-25 and precision bridge. • Cooled CCD camera.

• Refractive index of air is the largest single correction to 1 n length measurements by m 2L interferometry (3x10-4 L) v   Cell mair   2LCell v v • Measure fringe fractions – determine the difference in m  m  m number of interference orders n air v

mn between air and vacuum  1 • More accurate than estimation  v  n 1  mn  by empirical formulas (Edlén LCell  2  equations).  1  v   n  fn  C  LCell  2 

• Popular technique for interference fringe analysis – flatness-measuring interferometers – optical component evaluation • Displacement of fringes when the optical path length is changed – Sample interferogram at each of 5 phase steps – Change in grey-scale is related to the change in optical path length and the wavelength of light • Fringe fraction measurand for: – Gauge block length – Refractive index of air

Influence  Type dy/dxi ui Refractive index of air -9 Window optics correction 3 nm B L/ 3x10L -9 Fringe fraction 0.007 fringe B L/2 2x10 L -9 -11 3.7x10 L Vacuum cell length,  250 nm B (n-1)L/ 7x10 L -10 Vacuum wavelength 10  B (n-1)L/ negl.

Influence  Type dy/dxi ui Refractive index of air -8 -8 Edlen Equation 1x10 BL1x10 L -9 Air Temperature 4 mK B -9.5x10 -7L3x10L -9 -8 -8 Air Pressure 6 Pa B 2.7x10 L1.6x10L 2.4x10 L -8 Relative Humidity 2% B -8.5x10 -9L1.3x10L -8 Vacuum Wavelength 10  B -1.2x10-5Lnegl.

Decker et al., Metrologia 41 (2004) L11-L17

120 Ciddor Equation Refractometer 100

/nm 60

40 84 20

0 Expanded (k=2) Uncertainty 0 200 400 600 800 1000 Nominal Gauge Block Length /mm

• Gauge block length is defined at the standard atmospheric P pressure of P0=101325 Pa LP   L ISO 3650 3K P1 2 • Artifact length can change LP as   L a result of a change in pressure E • Gauge block length increases with Example: 900 mm gauge block decreasing pressure P (higher measured in Boulder, CO (altitude altitude = lower pressure) 1500 m; 83 kPa) is 34 nm longer than when measured in Paris (at sea level, 101 kPa)

Bayer-Helms Über den Einfluss von Luftdruck und Gewichtskraft auf Endmasse PTB-Mitt. 2/73 (1973) 97-8 Darnedde Metrologia 29 (1992) 349, Decker Metrologia 40 (2003) 1 Pressure calculator: http://www.dangermouse.net/gurps/science/pressure.html © Government of Canada, 2011 GB Length Change Post Travel

• Length change for 900 mm gauge block following travel from Rio de Janeiro

• Total length change during 5 days of settling = 94 nm  78 nm

• Temperature measurements included as demonstration that they are not correlated

Decker Metrologia 38 (2001) 269

• Improve dispersion components (wavelength dependence) via frequency comb measurements – Refractivity of constituent gases – Validation of Lorentz-Lorentz equations for a dilute gas

• Improved correction for humidity

Historical International Prototype Metre bar, made of an alloy of platinum and iridium, that was the standard from 1889 to 1960. Zhang et al., Appl. Opt. 47 (2008) 3143. Schödel et al., Opt. Lett. 31 (2006) 1979.

• Canada-Germany S&T Cooperation Agreement signed in 1971 • Governed by the German- Canadian Commission for Scientific & Technological Cooperation – Meets annually to – Reviews progress – Ensures coordination and identifies priorities for cooperative action

http://www.bmbf.de/pubRD/30_jahre_deutsch-kanadische_wtz_1971-2001_en-fr.pdf

• Over 500 joint research projects in priority areas: environment, energy, nanotechnology, health, genomics, photonics • Future: innovation-based business activities and commercialisation through: tech transfer, partnerships, bilateral R&D cooperation

• Visit of German Federal Minister of Education and Research Prof. Annette Schavan (tentative: Oct 2011) – Public lecture at NRC, 100 Sussex Dr. Auditorium “Reaching the ultra-small and the ultra-fast using intense light” Paul Corkum, Joint Attosecond Science Laboratory, University of Ottawa & National Research Council of Canada – Announcement & registration: http://www.science.gc.ca/

• Closing Ceremony: Berlin Feb 2012 • Website: www.de-can-fti.com

Hou W., Thalmann R., 1994, “Accurate measurement of the refractive index of air,” Measurement, 13, pp. 307-314. Hirukawa H., Ogita E., “Measurement of the Absolute Air Refractivity,” Proceedings of the 33rd SICE Annual Conference, International Session, Tokyo Metropolitan Institute of Technology, 26-28 July 1994, pp. 931-936. Leibengardt G. I., Naidenov A. S., Fedorin V. L., Shur V. L., 1993, “Investigations of a Two-wave Laser Refractometer,” Translated from Izmeritel’naya Tekhnika, 7, pp. 29- 31. Schellekens P., Wilkening G., Reinboth F., Downs M. J., Birch K. P., Spronck, J., 1986, “Measurements of the Refractive Index of Air Using Interference Refractometers,” Metrologia, 22, pp. 279-287; and Rischel C., Ramanujam P. S., 1989, “Refractive Index of Air - Errata,” Metrologia, 26, p. 263. Hilsenrath J., 1955, US Natl. Bur. Stand. Circ., 564. Barrell H., Sears J. E., 1939, Philos. Trans. R. Soc. London, A238, pp. 1-64.

Other Evaluations : Birch K. P., Downs M. J., Ferriss, D. H., 1988, “Optical path length changes induced in cell windows and solid etalons by evacuation,” J. Phys. E: Sci. Instrum., 21, pp. 690-692.

* Birch K. P. Reinboth F., Ward R. E., Wilkening G., 1993, “The Effect of Variations in the Refractive Index of Industrial Air upon the Uncertainty of Precision Length Measurement,” Metrologia, 30, pp. 7-14.

Estler W. T., 1985, “High-accuracy displacement interferometry in air,” Applied Optics, 24, pp. 808-815.