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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 2p 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 DL = Lf 2 Li in terms of the phase change Df measured by the interferometer and the change in wavelength Dl = lf 2 li: l Dl DL = Df f +(L 2L ) . (4) S pD i R l 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 Df by and a moving corner cube to measure displacement. The length L of lf/4p, 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 lf 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 2 LR) is the deadpath, and the deadpath correction accounts for an apparent shift in the 4p f = (L 2 L )+(f 2f ). (1) position L caused by wavelength variations. (This will S l D 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 l is the wavelength of light in air, the quantity f 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 f 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 f. 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 2 L ) equal to the deadpath, but now the initial posi- may assume that f , f ,andL are constant. The phase i R M R R tion L is the farthest distance of the moving reflector f will then vary only in response to changes in L or l: i from the beamsplitter. Although this is in accord with the formal definition of deadpath, in practice ‘‘dead- 4p L 2 L df = dL 2 4p R dl. (2) path’’ is sometimes used to refer to the distance of S l D S l 2 D 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 df and thus measures the total change in R R small. The distinction between the formal definition of f 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 l to a final value l ). The resulting i f certainties, i.e., estimated standard deviations, larger phase change is simply the difference between the initial than1or231026), but it is significant in high-accuracy and final values of f [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 f depends only on the initial and rection may be required to fully account for wavelength final values of L and l 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 2L L 2L Df =4p f R 2 i R . (3) as a ‘‘zero-shift’’ error, independent of displacement, in H l l J 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 [l/4p in Eq. (5)] is not deadpath. In order to avoid possible confusion, we will calculated with l = lf. 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 Df is known. If the interferometer calculating the zero-shift correction, the initial position readout gives displacement in units of l/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 Df. 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 Df using belief, using an average wavelength rather than the final wavelength lf in Eq. (4) does not improve accuracy; l using an average wavelength will not give the correct L = Df . (5) 0 S4pD 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 l is usually either li or lf. The value of l final result. In fact, if Li 2 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 lf,orifLf2LR is zero only li 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 l = lf.

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