Precision Interferometer

Demands on dimensional stability and on the detailed knowledge of thermal expansion of "high tech" materials are growing constantly.
The method of absolute length measurements by optical interferometry is the appropriate tool for extracting these properties in a direct way. This is done by observing the absolute length of macroscopic samples as a function of the temperature and time. Our ambition is to continuously increase the level of certainty in the range of sub-nanometers. From the length measurements the coefficient of thermal expansion (CTE) and its uncertainty can be extracted as a function of the temperature. In addition, conclusions regarding homogeneity of thermal expansion, compressibility, length relaxation and long term stability of samples can be extracted.

Fig. 1 shows the configuration of PTB´s Precision Interferometer. In this versatile Twyman-Green Interferometer the length of gauge block shaped samples is determined as a multiple of the wavelength of light. It is equipped with three lasers whose frequency is stabilized to hyperfine energy levels of Iodine- and Rubidium-molecules, respectively. This stability and the accurate knowledge of the vacuum wavelength is one prerequisite for high accuracy (< 1 nm) length measurements by interferometry.
At measurements under air conditions the wavelength of the light is inversely proportional to the refractive index of air which has to be determined accurately. For this purpose a vacuum cell is positioned within the field of view along the measurement path. This allows for the simultaneous measurement of air refractivity from the same interference map. Another important prerequisite for the highest level definition of a sample length is the nearly perfectly perpendicular incidence of the light. This is done by a special autocollimation technique developed recently at PTB.

Fig. 1

The interference within the field of view of 65 mm is projected to a high dynamic
(16 bit) CCD camera array. Phase stepping interferometry using 10 camera data frames is applied to accurately evaluate the interference phase assigned to each camera pixel. Thereby the reference path of the interferometer is shifted by a constant amount from frame to frame. For this purpose the compensation plate is tilted by a certain amount. The tilt angle is servo controlled by an additional angle sensitive interferometer. In this way errors of phase stepping are reduced to a negligible level.

Fig. 2, left, shows a typical example of a gauge block shaped sample onto the back side of which a platen is attached by wringing. The right figure shows the result of a phase evaluation. Data within the highlighted, symmetrically arranged, regions of interest are averaged. From the difference of these means obtained at the front phase and the end plate, respectively, the interference fraction is obtained.

Fig. 2

The interferometer is situated in a vacuum tight enclosure which is temperature controlled by two different water circulations. This ensures a homogenous temperature equilibration in the temperature range from 5 °C to 45 °C. Using an AC-bridge with pt25 standard thermometers (fix point calibrated according to ITS 90) and a special designed thermo couple measurement system the sample temperature is measured with an uncertainty of less than 1 mK.

Besides measurements under vacuum conditions (p < 10^-3 hPa), measurements under well defined pressures up to 110 kPa are possible. In this case a pressure balance is used to measure/control the pressure with an uncertainty of 2 Pa.

Fig. 3

Fig. 3 shows the measuring instruments for temperature, pressure, humidity and CO₂ content connected to the interferometer.

  High accuracy determination of thermal expansion coefficients and
     compressibility

From measurements of the absolute length of gauge block shaped samples as a function of temperature using the Precision Interferometer the coefficient
of thermal expansion (CTE) can be obtained with exceptional small uncertainty (typically < 10^-9 K^-1). These measurements are mostly performed under vacuum conditions.

Fig. 4

Fig 4 shows an example for the length change of a 200 mm long sample as a function of temperature (data points). It is pointed out that the length at 20 °C is subtracted in this plot while from the measurements a total (absolute) sample length is returned at each temperature.
The solid lines in Fig. 4 depict fit polynomials of 3rd (pink) and 4th (blue) degree of order. The deviation of the data points from the fits is always within 1 nm. Based on the assumption that the mentioned fit polynomials are valid for the description of the length as a function of temperature, the CTE can be extracted according to its definition, i.e. from the derivation of the length over the temperature and normalizing to the length itself.


Fig. 5 shows the CTE-curves resulting from the polynomials displayed in the bottom part of Fig. 4 (change of sign!). The middle part shows the difference between the two CTE-curves and in the top part of Fig. 4 the CTE-uncertainty for case of the 3rd polynomial degree is displayed.

Fig. 5

The accurate determination of the refractive index of air enables precise length measurements under different air pressures. From the differences of lengths derived under vacuum and under air pressure, respectively, the material compressibility can be obtained directly and accurately.

  Investigation of the homogeneity of thermal expansion

Regions of interest (ROIs) within the interference map are defined to evaluate the length of a sample (see Fig. 2). From the availability of the phase topography it is obvious to investigate the phase topography of the sample's front face as a function of the temperature. For this purpose an extended ROI, covering a relatively large area at the front face, is used. A length topography can be defined at each particular wavelengths, where the orientation of the end plate in the phase map is used as reference in each case. As for the length evaluation, the exact position of the sample with respect to pixel coordinates must be taken into account carefully (sub-pixeling). This is very important, otherwise even very small lateral shifts of the sample would pretend a change of the topography.

Fig. 6 shows a somewhat extraordinary example for a sample body that has a cylindrical shape. Top, left shows the phase map measured at a wavelength of 532 nm. Top, right, shows the corresponding "length topography" (use of the extended ROI at the sample's front face, mean length is subtracted). Bottom, left shows the difference with respect to the length topographies resulting from measurements using 633 nm. This illustrates that errors of measured interference phases can be neglected. Bottom, right shows the difference length topography resulting from measurements at different temperatures. The latter allows for conclusions regarding homogeneity of thermal expansion.

Fig. 6

  Investigation of length relaxation and long term stability

Solid state bodies (besides single crystals) and amorphous bodies are a subject to a permanent change of the internal structure. This involves changes of their macroscopic dimensions. Measurements of the long term stability of the length give information about the status of internal processes. As an advantage of the principle of absolute length measurements by interferometry, the sample body can be removed from the Interferometer and later reinserted, the reference (absolute length) always remains. This enables investigations of long term stability of materials using an arbitrary large interval between the measurements.
Temperature steps can also initiate/influence internal changes of the structure. This manifests itself in a length change which is delayed with respect to the temperature change which may be referred to as length relaxation.

Fig. 7 illustrates both phenomena together. Initially, the long term stability at 20 °C is observed which is quasi linearly at the time scale of days/weeks. Next, a temperature step from 20 °C to 30 °C is applied, resulting in a length change which clearly exhibits relaxation. A similar behavior is observed for the temperature step back to 20 °C.

Fig. 7

  Selected publications regarding these topics:

• Schödel, R.:
  "Accurate extraction of thermal expansion coefficients and their uncertainties from high precision interferometric length measurements"
  Proc. of SPIE 5879, 1-11 (2005)



• Schödel, R.:
  "Investigation of thermal expansion homogeneity by optical interferometry"
  Proc. of SPIE 5858, 0Q-1 - 0Q-8 (2005)



• Schödel, R.; Abou-Zeid, A.:
  "PTB’s precision interferometer for high accuracy characterization of thermal expansion properties of low expansion materials", In: Nanoscale calibration standards and methods: dimensional and related measurements in the micro- and nanometer range, Wilkening, G. and Koenders, L., eds.; ISBN 3-527-40502-X; ISBN 978-3-527-40502-2, pp. 500 - 514 (2005)



• Schödel, R.; Decker, J. E.:
  "Methods to recognize the sample position for most precise interferometric length measurements", Proc. of SPIE, 5532, 237 - 247 (2004)



• Schödel, R.; Bönsch, G.:
  "Highest accuracy interferometer alignment by retroreflection scanning", Applied Optics, 43, 5738 - 5743 (2004)



• Schödel, R.; Nicolaus, A.; Bönsch, G.:
  "Phase stepping interferometry: Methods to reduce errors caused by camera nonlinearities", Applied Optics, 41, 55-63 (2002)



• Schödel, R.; Bönsch, G.:
  "Precise interferometric measurements at single crystal silicon yielding thermal expansion coefficients from 12 °C to 28 °C and compressibility", Proc. SPIE 4401, 54 - 62 (2001) 

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© Physikalisch-Technische Bundesanstalt Page created: 26-Sep-2006, last update: 03-Apr-2008 11:28 AM, K. Eggert