Volume 101, Number 5, September–October 1996 Journal of Research of the National Institute of Standards and Technology
[J. Res. Natl. Inst. Stand. Technol. 101, 671 (1996)] Corrections for Wavelength Variations in Precision Interferometric Displacement Measurements
Volume 101 Number 5 September–October 1996
Jack Stone and Steven D. Phillips Precision interferometric displacement mea- correction necessary to fully account for surements require deadpath corrections to variations in wavelength. National Institute of Standards and account for variations in wavelength during Technology, the course of the measurement. This pa- Key words: deadpath; interferometry. Gaithersburg, MD 20899-0001 per discusses common errors in applying deadpath corrections and describes the Accepted: May 28, 1996 and
Gary A. Mandolfo Hewlett Packard Co., Santa Clara, CA 95052-9952
1. Introduction
Commercially available interferometer systems are recting for traditional deadpath errors. This paper at- used to measure the displacement of a moving reflector tempts to clarify this issue by explicitly describing the in terms of the wavelength (usually in air) of a laser or deadpath correction procedure. other light source. Corrections are required to account for wavelength variations during the measurement. (Wavelength variations most commonly arise from 2. Analysis changes in the index of refraction of air due to changing atmospheric conditions—particularly pressure.) Wave- Any interferometer for displacement measurements length variations are usually discussed in the context of senses the phase difference between light reflected from ‘‘deadpath’’ corrections, which account for shifts in the a moving reflector and from a fixed reference reflector. interferometer zero position caused by wavelength A typical geometry for a single-pass interferometer is changes. As discussed below, the standard deadpath cor- shown in Fig. 1. The phase difference changes by 2 for rection is sensible only if displacement is computed us- each half-wavelength displacement of the moving re- ing an appropriate value for the time-varying wave- flector; the net displacement can be determined by mea- length. Operating procedures for commercial suring the accumulated phase change as the mirror interferometers may tend to result in an incorrect choice moves from its initial position to its final position. for the wavelength, so that error remain even after cor-
671 Volume 101, Number 5, September–October 1996 Journal of Research of the National Institute of Standards and Technology
Equation (3) can be solved for the displacement
⌬L = Lf Ϫ Li in terms of the phase change ⌬ measured by the interferometer and the change in wavelength
⌬ = f Ϫ i:
⌬ ⌬L = ⌬ f +(L ϪL ) . (4) ͩ ͪ i R 4 i
This result may be interpreted as follows. The first term on the right converts a measured phase change to a Fig. 1. A typical Michelson interferometer with a cube beamsplitter displacement by multiplying the phase change ⌬ by and a moving corner cube to measure displacement. The length L of f/4, the displacement for each radian of phase. This the measurement arm is L = Li at the beginning of the displacement. term would clearly give a correct value for displacement The dashed triangle represents the position of the moving reflector at if the wavelength had a constant value f throughout the the end of the measurement (L = L ). f measurement. The second term can be thought of as a correction for motion of interference fringes past the When the distances from the beamsplitter to the mov- initial position of the moving reflector, due to the ing reflector and reference reflector are L and L respec- R change in wavelength. This term is the ‘‘deadpath’’ cor- tively, the phase difference between the light beams rection for the situation described here and depicted in traversing the two arms of the interferometer is Fig. 1; the quantity (Li Ϫ LR) is the deadpath, and the deadpath correction accounts for an apparent shift in the 4 = (L Ϫ L )+( Ϫ ). (1) position L caused by wavelength variations. (This will ͩ ͪ R M R i be referred to as a ‘‘zero-shift’’ error—an error arising from the apparent shift of the zero position L —in the Here is the wavelength of light in air, the quantity i M following discussion.) ‘‘Deadpath’’ is formally defined represents a phase shift of light in the measurement arm as the difference in optical path length between the due to reflection and transmission through glass optics, measurement and reference arms at the beginning of the and represents a similar quantity for the reference R measurement, when the interferometer is set to zero [1]. arm. We assume that the environment is sufficiently Thus far we have explicitly considered only displace- homogeneous that the wavelength is the same in any ments of the moving reflector away from the beamsplit- part of the air path. ter, but Eq. (4) is also valid for a displacement toward Displacement is determined by measuring the change the beamsplitter. In this case the second term on the in the phase . Ignoring possible thermal drifts of the right in Eq. (4) is again the deadpath correction with optics or thermal expansion of the reference arm, we (L Ϫ L ) equal to the deadpath, but now the initial posi- may assume that , ,andL are constant. The phase i R M R R tion L is the farthest distance of the moving reflector will then vary only in response to changes in L or : i from the beamsplitter. Although this is in accord with the formal definition of deadpath, in practice ‘‘dead- 4 L Ϫ L d = dL Ϫ 4 R d. (2) path’’ is sometimes used to refer to the distance of ͩ ͪ ͩ 2 ͪ closest approach of the moving reflector to the beam- splitter or, somewhat more precisely, to the distance of A displacement interferometer effectively integrates closest approach minus L , where L is often negligibly the differential d and thus measures the total change in R R small. The distinction between the formal definition of as the reflector moves from initial position L to a final i deadpath and this second usage of the term is not impor- position L (while the wavelength may also be changing f tant for measurements with low accuracy (standard un- from an initial value to a final value ). The resulting i f certainties, i.e., estimated standard deviations, larger phase change is simply the difference between the initial than1or2ϫ10Ϫ6), but it is significant in high-accuracy and final values of [which could be obtained directly applications. from Eq. (1)], independent of the path of integration; With either meaning of deadpath an additional cor- that is, the change in depends only on the initial and rection may be required to fully account for wavelength final values of L and and is independent of how these variations. The formal definition allows the two correc- quantities vary at intermediate times. tions to be clearly separated into two categories; the formal deadpath error can be interpreted unambiguously L ϪL L ϪL ⌬ =4 f R Ϫ i R . (3) as a ‘‘zero-shift’’ error, independent of displacement, in ͭ ͮ f i contrast to errors that are proportional to displacement.
672 Volume 101, Number 5, September–October 1996 Journal of Research of the National Institute of Standards and Technology
In this paper we will adopt the formal definition of ing phase change to distance [/4 in Eq. (5)] is not deadpath. In order to avoid possible confusion, we will calculated with = f. refer to the deadpath error as a ‘‘zero-shift’’ error. In using either Eq. (4) or Eq. (6) to calculate displace- Equation (4) can be used to calculate displacement if ment, several common errors must be avoided: (1) When the phase change ⌬ is known. If the interferometer calculating the zero-shift correction, the initial position readout gives displacement in units of /4 or some other Li should not be confused with the distance of closest fraction of a wavelength, the reading may easily be con- approach; (2) In order to be consistent with the conven- verted to a corresponding phase change ⌬. Often the tions adopted in this paper, the interferometer should interferometer readout is given as a distance, in which read positive for displacements of the moving reflector case it is not quite so straightforward to determine the away from the beamsplitter and negative for displace- phase change. The distance displayed by commercial ments toward the beamsplitter; (3) Contrary to popular interferometers is calculated from ⌬ using belief, using an average wavelength rather than the final
wavelength f in Eq. (4) does not improve accuracy; using an average wavelength will not give the correct L = ⌬ . (5) 0 ͩ4ͪ displacement. Only the wavelengths at the beginning and end of the measurement are important. Values of
Here L0 is the displacement shown by the interferometer wavelength at intermediate times have no effect on the display and is usually either i or f. The value of final result. In fact, if Li Ϫ LR is zero, then displacement depends on the nature of the interferometer and how it can be calculated from Eq. (4) knowing only a single is used; this situation potentially creates confusion in wavelength f,orifLfϪLR is zero only i need be calculating deadpath corrections. For example, some known. (The latter statement is not immediately obvious interferometers are equipped with external environmen- from Eq. (4).) tal sensors for determining the index of refraction; the If the positive direction of the interferometer has been value for the wavelength in air is continually updated as reversed, so that displacements of the moving reflector conditions change, and displacement is thus calculated toward the beamsplitter are positive, then either the sign with = f. Other interferometers compute from in- of the zero-shift correction must be changed or else the dex of refraction information (velocity-of-light-compen- interferometer reading [⌬ in Eq. (4) or L0 in the first sation value) entered by the user before the displacement two terms on the right in Eq. (6)] must be multiplied by begins. If this information accurately reflects current Ϫ1. atmospheric conditions, then the displacement is calcu- lated with = i in Eq. (5). Some users prefer to use an artificial value for the velocity-of-light compensation so 3. Discussion and Conclusions that has some constant value independent of actual conditions. (For example, may be the vacuum wave- In summary, some care is required in making correc- length.) tions for wavelength variations. The scale factor /4 Substituting Eq. (5) into Eq. (4) yields a formula that required for calculating displacement from a phase corrects the readout L0 to give ⌬L, the true displace- change must be computed with = f, assuming that ment: one calculates the zero-shift correction using the proper (formal) definition of deadpath. (In passing it may be Ϫ Ϫ noted that, if the zero-shift correction were to be com- ⌬L = L + f L +(L ϪL ) f i . (6) 0 ͩ ͪ 0 i R ͩ ͪ i puted using the distance of closest approach in place of the formally defined deadpath, then the appropriate Here is the wavelength value used by the interferome- scale factor would be given by the value of wavelength ter to calculate L0. As explained above, it is usually when the moving reflector is positioned farthest from either the initial or final value of wavelength depending the beamsplitter, independently of whether this is the on the method of operation of the interferometer. If initial or final position.) It is essential to use the appro-
= f, only the rightmost term in Eq. (6) is required to priate scale factor or errors proportional to displacement correct the interferometer reading L0 for wavelength will result. For example, if atmospheric pressure variations; this term is the zero-shift correction. How- changes by 133 Pa (1 mm Hg) during a measurement, ever, it is clear from Eq. (6) that the zero-shift correction the computed displacement will be in error by alone will not fully correct the interferometer reading 0.4 ϫ 10Ϫ6⌬L. The resulting error is thus small but not when Þ f. The second term on the right in Eq. (6) negligible for the most demanding commercial and sci- represents a correction proportional to the measured entific applications of interferometry. displacement that is required when the scale factor relat-
673 Volume 101, Number 5, September–October 1996 Journal of Research of the National Institute of Standards and Technology
Two cases may be distinguished where corrections for About the Authors: Jack Stone is a physicist in the wavelength variations are required: Precision Engineering Division of the Manufacturing (1) Corrections are clearly needed for applications Engineering Laboratory at NIST. Steven Phillips is the requiring relative errors smaller than 1 ϫ 10Ϫ6. Al- leader of the Large Scale Coordinate Metrology Group though wavelength variations can often be eliminated by in the Precision Engineering Division of the Manufac- working in vacuum (so that the wavelength only varies turing Engineering Laboratory at NIST. Gary Mandolfo due to imperfect laser stabilization), it is not always is a laser application engineer with the Hewlett Packard possible to do so. Special applications of interferometry Company’s Precision Motion Control Group in Santa require working under atmospheric conditions while Clara, California. The National Institute of Standards keeping relative errors below a few parts in 108 and it and Technology is an agency of the Technology Adminis- would be impossible to maintain this high level of accu- tration, U.S. Department of Commerce. racy without a proper treatment of wavelength varia- tions. (2) Lower accuracy measurements may still require zero-shift corrections if the deadpath is very large rela- tive to the displacement. For example, ifa1mmdis- placement must be measured witha1mdeadpath, a change in wavelength of 1 ϫ 10Ϫ6 will produce a zero- shift error that is 1 ϫ 10Ϫ3 of the displacement. In this type of situation, with Li approximately equal to Lf, the common errors mentioned previously (confusion regard- ing the definition of deadpath or neglect of scale errors) are usually of negligible importance, but complete ne- glect of the zero-shift correction could be catastrophic. Finally, it may be noted that the analysis given here is largely independent of interferometer details. Equation (4) or Eq. (6) can be used to correct the reading of any single-pass interferometer.
Acknowledgments
This research was funded in part by NIST’s computa- tional metrology project. Partial funding was also pro- vided by the U.S. Air Force’s Calibration Coordination Group. The research was originally motivated by the development of interferometer intercomparison tech- niques by John Beers and Ted Doiron of the NIST Di- mensional Metrology Group (along with Jack Stone, one of the authors); we are indebted to them for several useful discussions and comments.
4. References
[1] C. R. Steinmetz, Sub-Micron Position Measurement and Control on Precision Machine Tools With Laser Interferometry, Precision Eng. 12(1), 12–24 (1990).
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