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
1
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.
Precision Measurements & Instruments Corporation, 3665 SW Deschutes Street, Corvallis OR
97333
2
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
3
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
4
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.
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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 bidirectional 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.
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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.
7
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].
PRO: Polarization techniques can be used to count fringes. Arbitrary size and shape
sample. Wide temperature range, to >1300K [22]. Absolute or differential method.
High resolution. Minimal sample preparation. Commercially available from HewlettPackard, PMIC. Suitable for simultaneous opto-acoustic emission. [3] .
CON: Sensitivity is limited by the accuracy of fringe interpolation. Long term stability
of all optical components in optical path required. Mirror surface stability needed at
high temperatures. Reflections back into the laser destabilize fringe pattern. Method is
not completely non-contacting since a specular reflecting surface is required. Vacuum is
required for accuracy due to sensitivity to index of refraction changes in optical paths.
Vertical systems are sensitive to transverse temperature gradients, which cause sample
bending and loss of the sample beam relative to the reference beam.
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Figure 4. Michelson interferometer used for CTE measurements
Figure 5. Schematic of a double path interferometer [52].
Moiré: Moiré fringes appear as black bands whenever two sets of closely spaced lines
(or gratings) are brought together. Their movement relates to the relative motion of these
sets of lines, one of which can be placed on a sample then exposed to changes in
temperature. Figure 6 shows the reference grating is placed in close proximity to the
specimen [4]. Daniel Post at VPI.[56] used a cast silicone rubber with 600 lines/mm on
the sample and a reference grating of fused quartz with 1200 lines/mm, placed 1-2 mm
away in the oven (good to 422K). Silicone rubber is also useful for flexible substrates
[57]. Electron beam lithography offers advantages in control of gratings, such as
generation of high frequency line patters (to >10,000 lines/mm, [60]) and control of
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pitch. The sample grating must remain photographable throughout the test cycle. A
ceramic refractory paint covering for a nickel mesh (up to 40 lines per mm) was found to
be suitable to 1643K with medium sensitivity ( e.g., for turbine alloys) [55].) There are
many Moiré techniques, including hybrid approaches combining other optical techniques
such as diffraction and Fourier optics.
PRO: Three displacement components with wide field of measurement. No stabilized
laser or vacuum oven is required. Wide range of deformations. Edge/end effects can be
studied separately,
CON: Formation of gratings and their placement on sample. Reinforcing effect of
specimen grating. High temperature limitations. Reduction of fringe patterns to strains
complex. Sensitivity to air turbulence. Magnification drift with time may produce errors
in CTE measurements [58].
Figure 6. Schematic diagram of a Moiré interferometric dilatometer [4].
Holographic Interferometry Holography is well known as a means to display a 3D
image of a sample of arbitrary shape or size. If the object is distorted in some way during
the hologram exposure, or from one exposure to the next, fringes appear which provide
“maps’ to the distortions. Real time holographic interferometry using CW lasers
compares the OPLD of points on a diffusely reflecting surface to the sample before
thermal or mechanical deformation. As shown in Figure 7, the data are recorded on film
or videotape, etc., to create a (transmission) hologram which must be positioned with
respect to the reference beam exactly as it was before the deformation took place.
Holographic interferometry has often been used to measure thermally induced
warpage [25, 67, 69]. Goggin [69] and Hsu and Lewak [67] used this technique to
monitor thermal distortion of graphite-epoxy/honeycomb paraboloid aerospace reflectors
and graphite-epoxy laminated plates. Hsu and Moyer [25] used double exposure
10
holographic interferometry to measure the deflection of graphite to 800oC . The sample
incandescence did not affect the quality of the holograms because the Kodak 649F film is
more sensitive to laser light than the light from the glowing sample. A filter plus a more
powerful laser was recommended for higher temperatures. Sensitivity in general has been
on the order of λ/4. The use of holographic interferometry to measure CTEs has been
limited [16, 68]. Grunewald et al [68] report an accuracy of ± 0.05 μm and studied the
CTE of glass reinforced Kapton by floating it in a heated fluid (-10 to 30oC). Heflinger
and Wuerker [16] studied metals and claim an accuracy of better than 10% in CTE of
samples of arbitrary shape.
PRO: Can be noncontacting if surface sufficiently reflective. Minimal sample
preparation. Real time deformations. Arbitrary size or shape of sample,
CON: Double exposure holographic interferometry is difficult to use due to thermal
warpage of photographic plate between exposures [26] . Sensitive to vibrations, rotations
and solid body motions of sample support systems. Needs independent indication of sign
of CTE. Complex data analysis.
Figure 7. Holographic interferometry for CTE Measurements S = sample , L = lens
Speckle Interferometry : Laser speckle is formed by multiple interference of wavelets
arising from small surface elements when illuminated by spatially coherent radiation.
Surface movement causes changes in phase relationships in the individual speckles. This
in turn leads to changes in speckle irradiances which can be related to the displacement
through cross correlation of the patterns before and after the displacement, e.g., by
imaging with a large aperture lens and photographing. The combined speckle pattern
and the original pattern will correlate to produce fringes when;
OPLD = 2 u sin θ = n λ (in-plane displacement)
(7)
where u is the in-plane component of the displacement and 2θ is the angle between the
illumination beams. For example, when θ = 18.3o. ESPI sensitivity (u/n) of 1.00
μm/fringe was achieved [27]. The intensity of the light in digital speckle interferometry is
11
given by equation (3) [1]. The lens in Figure 8 forms an image of the surface of interest
on the detector plane (photodetector). The speckle there is produced by interference
between the two coherent and symmetrically situated laser beams.
When a granular surface has features on the order of magnitude of the optical
wavelength, one may also regard each speckle or grain as an interferometer in its own
right, and indeed changes in intensity from a single speckle can be used as a measure of
surface displacement. Digital speckle interferometry (DSI), (also called Electronic
Speckle Pattern Interferometry (ESPI)), digital speckle pattern interferometry (DSPI) ,
and digital speckle shearography (DDS) are electronic analogs of film based speckle
interferometry and shearography. ESPI can be defined as the generation of Moiré fringes
resulting from electronic subtraction between two speckle patterns obtained before and
after deformation.
Speckle interferometry has been used for CTE measurements[27, 50, 61-64]. ESPI
was used to measure CTEs from 10-7 to 10-4 /K between 100 and 300K [27]. It was used
to 900K [63], 1300K [66], 1473K [61], and as high as 2750K on tungsten filaments[65].
The incoherent background light at high temperatures reduces the quality of speckle
patterns but a color camera with an interference filter at the relevant laser frequency helps
[61]. Slight changes in optical configuration provide either in-plane or out-of-plane
deformations. A completely noncontacting approach was used to follow the CTE
through the glass transition of polymers [62]. High sensitivity of a laser speckle system
for CTE measurement was also achieved by rotation of a probe due to differential thermal
expansion of sample and reference rods [50]. Speckle size, laser beam width and spectral
width of the laser beam affect the sensitivity. High (diffuse) reflectivity and surface
roughness on the order of the wavelength are needed to obtain a fine speckle.
PRO: Can use rough and/or flat samples. Non-contacting. No calibration needed. Large
dimension samples or even components possible. Minimal sample preparation allows
fragile or chemically reactive materials to be tested. Relative ease of optical alignment.
CON: Complex fringe analysis. Sensitive to vibrations and air turbulence. Noise levels
interfere with higher displacement levels or require small thermal loading steps. Sensitive
to rigid body motions (unless dual beams are used). Precision somewhat less than
multiple beam interferometers. Double exposure analysis is discontinuous in time,
although continuous video recording is possible.
12
Figure 8. Double illumination speckle interferometer for measuring in-plane
displacement (after Leendertz [76])
Shearography: Shearography permits measurement of the spatial derivatives (i.e., ∂u/∂x,
∂u/∂y, ∂v/∂x, etc) of any surface point displacement . This is done via interference
between two spatially displaced (∂x or ∂y) images of the sample as recorded on an
image-shearing CCD camera.[70]. One approach is to align the two mirrors of a
Michelson interferometer so that the camera sees a double image of the object. While this
technique has been chiefly used for NDT at ambient temperature, there is no inherent
reason why it could not be adopted to thermal expansion studies.
PRO: Measure strains on an arbitrarily curved surface.
CON: In-plane displacements difficult to measure independently.
Diffraction: Diffraction is an interference phenomenon where we deviate from
rectilinear propagation. The beam “bends” around an obstacle and superposition of
several wavelets occurs. X-ray and neutron diffraction techniques have been commonly
used for CTE measurement through lattice parameter changes [29]. The CTE is derived
by the subtraction of two nearly equal numbers (the lattice parameters) and this limits
precision). These techniques are sensitive to local composition, lattice defects such as
vacancies and crystallite orientation variations. They are difficult to use for highly
anisotropic materials.
Laser diffraction has been sparsely used for CTE, but has potential for high
temperatures, variable environments and thin film applications [18]. Diffraction from
two circular apertures or an internal slit aperture is a also potentially useful to measure
the CTE of thin films, provided the disadvantage of small values of Lo can be overcome.
PRO: Noncontacting. Linear displacement method. High temperatures (to >1300K).
Minimal sample preparation. Thin film potential, variable environment.
CON: Free edge required, hence possible material edge effects. Accurate fringe spacing
measurement needed. Precision a function of aperture spacing
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AUXILIARY TECHNIQUES
Table IV lists techniques utilized in order to convert fringe formation into useful
displacement information. Since the CTE is by definition a thermodynamic or
equilibrium property, the temperature is assumed to be uniform in the sample when the
strain is measured. For noncontacting methods, the temperature distribution can be
estimated by standard heat flow calculations base on heat/cooling rate, geometry, thermal
properties, etc. Minimal heat losses due to sample support are achievable with
Michelson and speckle methods. Fabry-Perot and Fizeau techniques involve sample
contacts and their thermal stabilization is normally required, preventing acquisition of
time dependent data such as microcracking, phase transformations, etc. When the latter
phenomena are important, data acquisition must be sufficiently rapid (e.g., a high CCD
camera rate).
Table IV – TECHNIQUES USED WITH INTERFEROMETRY FOR CTE
Thermal cycling apparatus
Retroreflectors (open, corner cube, cat’s eye [28], )
Simultaneous measurement of other properties (e.g., index of refraction)
Differential techniques (e.g., extra reference beams, double pass [10, 34, 48, 52, 54])
Fiber optics (single-mode fibers as one leg of interferometer)
Laser frequency stabilization (e.g., Lamb dip, CH4 doping, two-mode)
Antireflection coatings (e. prevent light to laser cavity)
Remote optics positioning systems
Photodetectors ( e.g., shot noise suppression)
Phase Modulation/shifting [9,75]
Laser Pulsing [51]
Homodyne or single frequency detection (polarization [33])
Two-frequency (heterodyne) [21,34,35]
Simultaneous pulsing [28]
Virtual reference gratings [59]
Wavelength scanning [9]
Recording media (films, computer data, charts, digital CCD)
Fringe counting/interpolation [37, 38] (See Fig. 9)
Thermal strain data must be derived from the fringe motion. Figure 9, for example,
shows the fringe movement typical of a temperature reversal in a thermal expansion
measurement. Tracking of sample length must account for similar fringe movement
changes which may be caused by phase transformations or microcracking. Methods
include homodyne, heterodyne and frequency modulation. The homodyne approach
involves a single frequency and conversion of linear laser polarization to circular. It
permits easy automatic and bi-directional fringe counting (e.g., through phase
quadrature counting methods. However, it is sensitive to DC shifts (which may affect
the trigger levels of automatic counters. Two photodetectors per channel are needed.
14
Figure 9. Fringe motion change corresponding to temperature reversal. Quadrature
signals from photodetectors change relative phase.
Heterodyne implies mixing two different ac signals for purposes for generating two
new frequencies, a difference and a sum frequency. In two beam heterodyne
interferometry, the two light fields may be expressed as variations of equations 1-3 [72];
E1 = A1 cos (ω1t + φ1 )
E2 = A2 cos [ ω1 t + Δω t + φ2 + φ(t) ]
(8)
(9)
Where Δω is the angular frequency shift. Detection by a square law detector gives
I = 2 A1 A2 cos [Δωt + φ(t) + (φ2 - φ1) ]
(10)
A heterodyne detection system is one where a carrier beat frequency carries the phase
information φ(t). For example, one may use two laser beams of slightly different
frequencies f1 and f2 (with a typical difference of 2 MHz) with opposite linear
perpendicular polarizations so that, with a polarizing beam splitter, it is possible to
separate the reference beam f2 from the measuring beam f1. If the two beams are directed
along different optical paths and then made to interfere, the two frequencies are phase
shifted by Δf1 and Δf2 due to the motion of respective mirrors in the furnace, The
change in the difference frequencies gives a signal proportional to the relative
displacement [34, 41]. An example is the Hewlett-Packard interferometer designed to
operate in a nonvacuum environment. An axial magnetic field (Zeeman effect) splits the
laser frequency into two frequencies of opposite circular polarizations. One is the
measurement frequency which is Doppler shifted by the moving reflector and then
optically mixed and compared to the reference frequency. The required electronics detect
and amplify the signals and need only be sensitive to frequency changes. Direction
sensing and distance information is carried on high gain, stable AC electronics rather
15
than DC signals since two slightly different frequencies (near 5 x 1014 Hz) are mixed.
This means that a high level of loss can be tolerated in the optical system (such as
caused by air turbulence, optical misalignment, sample warping, etc.). Generally the
frequency difference must be small enough to be resolved by a photodetector to at least
λ/1000. Other methods of digital fringe counting/analysis include phase modulation,
where the phase is modulated at a frequency ωm << ωo [78]. Such methods are applicable
to several interferometers, e.g., Michelson and Fabry-Perot.
The work of Miiler and Cezairliyan [28] at NIST has led to a use for interferometry at
very high temperatures (>2000K). A double Michelson-type interferometer (Figure 10)
together with a pulse resistive self-heating system, polarized optics, interference filters
for 632.8 nm, and high speed recording of signals by a digital oscilloscope was used to
measure thermal expansion of metallic samples.
Figure 10 . Michelson based interferometer with pulse heated sample. Polarizing
beamsplitters PB1 and PB2, quarter wave plates QP1 and QP2, pentaprisms PP1 and
PP2. Polarization states of the beams are indicated by double-headed arrows (ppolarization), heavy dots (s-polarization) and curved arrows (circular polarization) [22].
Single mode fiber optics can be used as a leg of an interferometer, but the optical path
length and polarization in the fiber is sensitive to fiber strains, pressure and temperature.
Imbedded fibers used as Fabry-Perot etalons can measure localized thermal expansion,
and once the proper optical coupling is made to the ends, a wide variety of thermal
distortions can be measured.
Mirror alignment is critical in most interferometers; hence corner cube retroreflectors
are often used to replace mirrors. This is satisfactory for room temperature metrology
but generally unacceptable for thermal expansion, especially if the measuring beams pass
through glass whose index of refraction changes with temperature. Open retroreflectors
16
are a slight improvement and cat’s eye reflectors additionally allow sample rotations, but
the lenses used must be compensated.
DISCUSSION
ASTM Standard guide E2208-02 [43] provides a common framework for quantitative
comparison of optical measurement systems. It lists the error sources that are involved in
making non-contact optical strain measurements (using a Moiré interferometer as an
example). Errors are commonly introduced by the recording process, extraneous
vibrations, lighting variations, rigid body motions, extraction of derived data
(displacement field) from basic data (e.g., image intensity data) , and in (strain) data
processing. Errors in the use of interferometers for CTE measurements are also outlined
in [2, 45, 47]. The total CTE measurement system accuracy may be determined by
summing the errors due to the sources listed in Table V. The theoretical noise-limited
sensitivity does not greatly depend on the type of interferometer used and is typically
10-15 m Hz-1/2 [53].
Table V– CONSIDERATIONS IN THE USE OF INTERFEROMETRY FOR
THERMAL EXPANSION
Alignment (Abbé error, cosine error, deadpath errors [2]).
Mirror or surface quality (figure, fog-over, recrystallization, oxidation [61])
Frequency Stability (warm-up, coherence length, Lamb dip)
Unwanted Reflections (laser source instability)
Optical path (pressure, temperature, humidity, index of refraction)
Temperature gradients in sample (e.g.,asymmetrical heating [49])
Temperature measurement
Thermal drift of interferometer components (beam splitters, modulators [54], windows)
Nonlinear sample motions/distortions (sample support stability, curvature [30].)
Electronic/mechanical noise
Edge effects [78]
Sensitivity to amplitude changes
Sample support [42]
Sample thermal distribution stabilization [62]
Temperature measurement requirements for accurate CTE measurement suggest
that contacting methods such as Fizeau interferometry will have thermal interference
from sample supports and optical flats. At high temperatures, the sample incandescence
creates optical noise through increasing non-coherent radiation. At very low
temperatures, heat transfer is poor and cooling rates are restricted or temperature
gradients may occur. How does the proposed technique handle other sample distortions,
such as bowing or warping due to asymmetrical midplane (composite) layups or
17
transverse expansion due to Poisson’s effects? Microcracking may occur during cooling
- can the technique be modified to record the corresponding discontinuous dimensional
changes ? Other considerations include edge/end effects – these may distort CTE
measurement of composites panels or sandwich structures if length changes are based on
sample ends. Another consideration for high accuracy CTE is whether the interferometer
is a “common-path” device (such as basic shearography) or “separate path” such as
holography or ordinary speckle interferometry. Since the path length differences are
affected unequally, the latter are more susceptible to vibration [1].
Real time strain measurement has advantages over isothermal holds at various
temperatures. In the latter approach, important materials’ behavior can be missed, such
as dimensional changes due to phase transformations or microcracking. An example of
the former is the discontinuity in CTE at the Tg of adhesives used in honeycomb
sandwich structures. On the other hand a potential drawback of continuous strain
recording may be thermal equilibration, since the CTE is by definition an equilibrium
property.
CONCLUSIONS
Interferometry has major advantages in the field of thermal expansion measurement. It
is capable of extremely high accuracy and can be designed to measure the thermal
dimensional response of any solid material of any size or shape. With auxiliary signal
processing, many related materials behavior can be simultaneously monitored, including
microcracking or phase changes. Since length determination is an absolute measurement,
being based on the laser wavelength, routine calibration is unnecessary (although
occasional use of a fused silica standard (NBS-SRM-739) to 1300K is recommended
[46]) While there are a number of interferometer techniques available, Michelson laser
interferometry is recommended for linear CTE determination as it is not restricted by the
sample preparation and size/shape restrictions common to Fabry-Perot or Fizeau
methods. The complexity of full field image techniques must be justified by the added
displacement information obtained. A full understanding of these techniques leads to a
broad capability for accurate measurements of linear, surface, in-plane or out-of-plane
expansions, temperature ranges, ease of setup, costs, thermal cycling data and related
requirements.
The ITES symposia have provided forums for the application of interferometry to the
CTE measurement of high temperature refractories [28], asymmetrical laminates [30],
thin films, [74], sandwich structures [23], and edge and end effects [71]. Future
developments in interferometric measurements for thermal expansion include adoption of
these techniques to new materials, conditions, and sample forms and shapes.
Interferometric methods to measure expansions of nanomaterials, aerogels, smart
composites, functionally graded composites, simultaneous multidirectional expansions,
simultaneous moisture/stress impositions, fibers/whiskers/nanotubes, and cryogenic
applications are being developed.
18
ACKNOWLEDGMENTS
The author would like to thank the staff of PMIC for review of this paper and valuable
suggestions.
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