Fundamentals of Optical Interferometry for Thermal Expansion Measurements

Fundamentals of Optical Interferometry for Thermal Expansion Measurements

Ernest G. Wolff

For presentation at the 27th International Thermal Conductivity Conference and 15th International Thermal Expansion Symposium , October 26-29, 2003 Knoxville, Tennessee

ABSTRACT

Measurement techniques based on optical interferometry are widely used for the determination of thermal expansion coefficients smaller than about 1 ppm/oC. The principle methods are Fabry-Perot, Fizeau (including Abbe-Pulfrich and Priest), holographic, Michelson, Moiré, and speckle interferometers, and diffraction techniques. Each has advantages and disadvantages depending on the sample size and shape, temperature range, resolution, and whether linear, surface or volumetric expansion is required. Auxiliary analysis is needed to convert a fringe pattern into an accurate strain versus temperature curve. This paper reviews the basic theory, the major experimental approaches, possible errors and typical results. Since the relevant ASTM standard (E 289) has recently (1995) been extended to cover Michelson interferometry, this technique is emphasized for its high resolution and test sample versatility.

INTRODUCTION

Albert A. Michelson, in 1881, described an optical arrangement that used a partial mirror to create interfering beams – an interferometer which could detect possible changes in the speed of light [1]. Since then the development of lasers, solid state electronics, high speed photodetectors, CCD cameras and computerized data acquisition systems have considerably improved the versatility, range and resolution of interferometers, allowing a wide variety of applications (Table I). A major advantage of laser interferometry for length measurements is the accuracy - traceability to a reference standard which is the wavelength of the laser used . The meter was defined in 1983 at the Conference Generale des Poids et Mesures as the distance traveled by light in free space during 1/c of a second, where c is the defined speed of light (299, 792, 458 ms-1 ). Lasers can also be made very stable with time. For example, a frequency stabilized ( Lamb dip) device using isotope 20Ne had a wavelength of 632.991410 nm which drifted to 632.991430 nm in 3 years [2].

Since in principle any method which can measure strain can measure thermal expansion, [3-7], interferometers compete with other methods (Table II) Besides accuracy, interferometers reduce sample size/shape restrictions and contact problems. Probe contacts, for example, may deform the sample through pressure and also alter both its and the sensor’s temperature distribution.

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Today, near zero CTE values are found in many materials such as silica, glass ceramics and many fiber reinforced composites. All materials approach a zero CTE as T → 0 K. CTE uniformity is important for many structures such as mirror substrates, and this also implies changes of ≤ ppb/K. Nanotechnology and micromechanical systems development suggest the need to thermally characterize ever smaller sample shapes, such as fibers and whiskers, nanotubes and thin films. This means the total strains to be measured are constantly decreasing. The potential of interferometry to measure the smallest strains or displacements is witnessed by current studies to measure gravity waves. Certainly detection of 10-15 m is state- of- the- art [3]. It is therefore worthwhile to review the fundamentals of interferometry and assess its potential to measure strain of a material, component or structure over all possible temperatures with the highest possible resolution and accuracy.

Table I – APPLICATIONS OF INTERFEROMETRY

Thermal Expansion Measurement

Temporal Stability

Material Behavior (e.g., radiation shrinkage, sintering, moisture desorption)

Gravity Wave Research

Thermal Analysis of Microelectronic Devices

Velocimetry (Laser Doppler Interferometry)

Vibration Analysis

Microlithography

Non Destructive Testing and Evaluation

Wavelength Calibration (and spectral analysis)

Surface Profiling (Flatness/Figure, Kosters, Twyman –Green)

Wavefront Quality (Mach-Zehnder)

Refractive Index Measurement

Machine Tool Calibration (including PZT calibration)

General Metrology

Table II – CTE METHODS COMPETING WITH INTERFEROMETERS

Dilatometers (e,g. thermomechanical analysis (TMA, LVDT based)

Scanning Laser Dilatometers (single, multiple beams)

Optical levers/ Comparators

Capacitance

Strain gages

Thermoelastic Methods

Telemicroscope (long focal length microscopes)

Microwave resonance

Figure 1. Optical configuration of a (Michelson ) interferometer

GENERAL THEORY

Interferometry may be defined as interference between electromagnetic waves. It may be regarded as a superposition of coplanar electric fields or a product of complex amplitudes. Suppose we have two coherent waves of amplitudes A1 and A2 with identical polarization and equal wave velocity (ν) traveling in a z-axis direction:

E1(z,t) = A1 cos [ (2π/λ) ( z – ν t) ] (1)

E2 (z,t) = A2 cos [ (2π / λ) ( z - ν t - δ ) ] (2)

Where “δ” is the distance the second wave lags behind the other. The phase difference φ between the waves may be described as ;

φ = 2 π δ / λ = 2 π (n ΔL) / λ ≡ ω t (3)

where n is the index of refraction of thel path length difference (ΔL = L1 – L2) and ω the angular frequency. All interferometers compare the phases of two light beams that travel different optical paths; thus (NnΔL) is also called the optical path length difference (OPLD). Constructive interference occurs when the path difference ΔL is a multiple (N) of λ. Destructive interference occurs when ΔL = N λ/2, leading to a minima in intensity or a fringe. In practice, phase differences of optical fields are transformed into detectable intensity variations. The irradiance distribution I = |E1 + E2 |2 uses time averages to remove undetectable optical frequency oscillations. Local irradiance or light intensity is the dot product of the two electric field intensities of the two beams at that point, thus:

I = I1 + I2 + 2 √(I1 I2 ) cos φ (4)

If the amplitudes A1 = A2 (hence I1 = I2) we can set both to Io, and

I = 2 Io (1 + cos φ) = 4 Io cos2 (φ / 2) (5)

Equation 4 also shows that the interference pattern will have a DC component (I1 + I2) and an AC component . Equation (5) indicates the interference pattern caused by the maxima and minima of this irradiance varies as the cosine squared. Different types of interferometers give variations in the fringe patterns. Their visibility , defined as {(Imax – Imin) / (I max + I min)} is also variable.

Linear thermal expansion may be considered as the relative displacement of two points on a material. Let us suppose attached points are reflective, such as the mirrors in Figure 1. If the phase of one (reflected) wave is shifted by one wavelength, we have passed through one minima and one maxima; thus a wave displacement of one wavelength corresponds to one fringe. Conversely, if we see the fringe pattern moving by one fringe, and this is caused by a reflective mirror displacing the (collinear) optical path length of one beam by distance ΔX, we can conclude that the mirror has moved by λ/2 (since the beam has traveled over this distance twice). Interferometers thus measure the (OPLD) changes between two beams, one of which could be considered a reference beam. Hence

Δ OPLD = 2 (Δ X1 cos θ1 + ΔX2 cos θ2 ) = (λv / n) ( ΔN + (Δφ/2π)) – ( Δn/n) (6)

Thus ΔN is the number of fringes passing a point in space. The subscript “v” refers to vacuum (where n = 1). Here θ represents the angles the beams make from their normals and these are usefully made as close to zero or identical as possible. In a Michelson interferometer (Figure 1) the object is to equate ΔL of a sample to ΔX1 + ΔX2 when both beams are oriented collinearly. General principles of optical interferometry are covered in texts such as [1, 8, 9].

TYPES OF INTERFEROMETERS

Table III lists the major types of interferometers suitable for thermal expansion measurement. A brief description of each follows, along with advantages (PRO) and disadvantages (CON) for CTE testing [10,11]. Conventional techniques involve a single laser beam and one or two photodetectors to record the phase changes caused by the movement of a single spot. Full field imaging techniques use a laser beam to illuminate a larger area and CCD cameras to record the information. Each pixel of the camera becomes an interferometer by combining the sample image and a reference image When the illumination and observation directions coincide (as in a Michelson arrangement) , out-of-plane motion is recorded; when the two directions differ one can measure in-plane displacements. Thus imaging interferometry provides the three displacement components (u,v,w) of any point (x,y, or z) on a sample surface.

Table III – INTERFEROMETERS FOR THERMAL EXPANSION MEASUREMENTS

A) Linear Expansion: Fabry-Perot [44]

Fizeau [13 - 15,31], including Abbe-Pulfrich [13], and Priest[13,17]

Michelson (single/double) [5, 13, 19, 10, 12, 20, 21]

B) Full Field Imaging Methods: Moiré [22, 55 - 59]

Holographic Interferometry [16, 67,68, 72]

Speckle (e.g., ESPI) [1, 61-66]

Shearography [70]

C) Diffraction Methods [18]

Fabry-Perot: Charles Fabry and Alfred Perot invented, in 1897, an interferometer where interference of multiply reflected beams occurs within a cavity or etalon with partially reflecting coated flat or spherical end plates. Constructive interference occurs whenever the difference in optical path length between the rays transmitted at successive reflections is such that the emerging waves are in the same phase, producing a transmission maximum. If the wavelength shifts, or if the end plates move relatively to each other, the multiple reflections mean multiple interference and either the same wavelength is no longer transmitted or it needs to be changed to follow the relative motion. In the first instance we have an interference or band-pass filter, in the second a means to measure the CTE of the wall materials.

Fringe patterns may be viewed either in transmission or reflection. Temporal stability is measured by tracking the transmission maximum of a spherical confocal Fabry-Perot interferometer using the sample as a spacer between two mirrors. For CTE, the cavity is heated [36] over a small temperature interval and stabilized. Figure 2 shows an experimental arrangement used for low expansion materials such as fused silica. Here mirrors are attached to the ends of the sample to form the Fabry-Perot etalon and the frequency of a slave laser is locked to a transmission peak, so that the wavelength of the slave laser is an integral submultiple of the optical path difference in the interferometer. Sample expansion changes the wavelength and hence the frequency of the slave laser. This is measured by mixing the beam from the slave laser with the beam from the frequency stabilized reference laser at a fast photodiode and measuring the beat frequency [77].

PRO: Very high precision - < 10 parts in 109 [39,40] and partial immunity to thermal perturbations on external optics. Useful for measuring CTE variations in sample.

CON: extensive sample preparation needed, sample size and shape restrictions , can not use polarization techniques for fringe counting, frequency modulation does not allow bi-directional counting, suitable optical reflective coatings for > 600K difficult to find. Range limited when the transmission peak of the interferometer moves outside the gain profile of the laser.

Figure 2. Fabry-Perot Interferometer for Measurement of Thermal Expansion [9,77].

Fizeau: Fused silica optical flats (uncoated) are used as the reflection surfaces in a multiple beam system. Expansion can be measured absolutely or relative to standard reference materials (SRM) [73]. Figure 3 shows differential approach when an optical flat is supported by the sample and one or two reference samples [14]. The pedestal supports another optical flat and the slight difference in sample and reference length produces a small angle θ (< 1 degree) between the flats. Reflections from the bottom of the top flat and the pedestal form an interference pattern consisting of parallel fringes. (The samples could also be supported by the reference flat). Interference fringes can be detected by a TV camera [11].

PRO: CTE to ± 3 x 10-8/K. Sharpness of fringes allows accurate fringe motion measurement (to < λ /40). Any sample height if pedestal height and reference samples are adjusted. Partial immunity to thermal perturbations on external optics.

Commercially available through Gaertner or Theta Industries.

CON: Needs isothermal holds or a maximum temperature ramp of 1-3 K/min, hence time consuming. Sample size and shape restrictions. Can not use polarization techniques for fringe counting. Difficult for anisotropic materials. Significant sample preparation.

Figure 3. Fizeau type interferometer for thermal expansion [14].

Michelson This is a double beam technique which can be used to measure the relative displacements of sample ends or mirrors placed on the sample away from the ends. [1, 5, 10, 12, 13, 19, 20, 23, 45]. (See Figures 1, 4, 5 and 10). Methods to keep a sample stationary during heating/cooling are described in [42]. The PZT here is used to scan the fringe pattern by changing the ΔOPL (phase shifting) by λ/2. The purpose is to monitor beam alignment and help in fringe interpolation. Thermal drift of the PZT and other external optics may be a problem. Alternatives to the PZT for phase shifting included electro-optic modulators, frequency shifting, rotation of half wave plates (between two quarter wave plates) or polarizers [9, 75], but each has its problems. Calibration is best done with NIST- SRM-739 (fused silica) [24]. Handling of warping samples is outlined in [30].

Vertical Michelson or double pass arrangements are described in [10, 11, 24, 32, 34, 52]. Figure 5 illustrates the use of an external mechanically mounted corner cube prism.

The prism partially compensates for movements in sample support which may redirect the return beams. A temperature controller for the optical window of the thermal bath helps to reduce uncertainties in the optical path length when the external optics are employed [52].