Laser interferometric dilatometry

We develop a measurement system to investigate the coefficient of thermal expansion (CTE) of ultra stable samples. Low CTE materials are important for stable structures in space applications to enable precise measurements.

Various material samples were determined with our heterodyne interferometer which offers a noise level below 2pm Hz-1/2 at frequencies above 0.1Hz. Our setup is designed such that we are able to measure e.g. CFRP (Carbon Fiber Reinforced Plastic) samples with CTEs of 108K1<math xmlns=""><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>8</mn></mrow></msup><mspace width="thickmathspace"></mspace><msup><mi>K</mi><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup></math>.



Photograph of our interferometer and dilatometer setup: IFO: Interferometer with all the optical components and the photodetectors mounted on an aluminum breadboard with M3 threads (grid 10mm x 10mm), 1: device under test mounted vertically in the thermally highly stable support (2) made of Zerodur.


Structural materials change their geometric dimensions due to a temperature change which can be described via the linear coefficient of thermal expansion (CTE):


where ΔL<math xmlns=""><mi mathvariant="normal">Δ</mi><mi>L</mi></math> is the dimensional change depending on a thermal change ΔT<math xmlns=""><mi mathvariant="normal">Δ</mi><mi>T</mi></math> over the length L of the sample. High dimensional stability is needed for several terrestrial and space-based applications. In space missions such as NGO/LISA or GRACE-FO up to pm stability of their optical instruments is necessary to meet the mission requirements. Materials with such ultra low CTEs are rare, glass, ceramics such as Zerodur or Clearceram enable high thermal stability but on the other hand they are very heavy. Light weight materials like CFRP can be tuned to have CTE values below 10-7 K-1 in at least on dimension.

To measure CTEs with an accuracy down to 10 ppb/K, not only a high sensitive measurement system is needed, also a thermally and mechanically stable support of the device under test (DUT) and corresponding mirror mounts are necessary. In our setup we use a heterodyne interferometer to measure the displacement of two mirrors as a result of an expansion of the device under test. The sensitivity of our interferometer is demonstrated with a noise level below 2 pm Hz-1/2 at frequencies above 0.1 Hz. The additional dilatometer setup, the support of the device under test decreases the accuracy of the whole setup. The support of the device under test is made of Zerodur to achieve a high mechanical stability and minimize thermal dependency. The measurement mirrors are fixed using clamps made of Invar36 in a tube shaped sample with a maximum length of 120 mm.

Measurement Principle

The schematic of our CTE measurement setup is shown in figure 2. The heterodyne interferometer, the heating system and the device under test (DUT) are placed inside a vacuum chamber. The heating system is placed around the DUT and shielded with MLI (multi layer insulation) to minimize a radiative heat transfer to the interferometer setup.

The expansion of the DUT is measured with the two beams of the interferometer. One beam is reflected by a mirror on the lower end and the other beam is reflected by a mirror at the upper end of the DUT. The expansion is calculated in a LabView program with the phases of the laser beams which are detected by the quadrant photo detectors (QPD). The QPDs also enable a tip and tilt measurement of the two mirrors using differential wavefront sensing (DWS) method.

The heating system applies a temperature variation around room temperature only by radiation to the sample. The temperature at the DUT is measured by Pt100 sensors.

Schematic of our measurement setup, heterodyne interferometer and thermal DUT setup. f1, f2: beams with two different frequencies, BS: beam splitter, PBS: polarizing beam splitter, QPD1, QPD2: quadrant photo detectors, H: heater, MM: mirror of the measurement arm, RM: mirror of the reference arm, DUT: device under test.

Mechanical Setup

The mechanical setup of our dilatometer (cf. figure above) is designed to enable a high resolution. Therefore the whole setup is made of Zerodur or Invar. The Zerodur support is adjustable in height with three feet such that the Invar parts have no influence to the thermal stability and only the behavior of a Zerodur transfer function is performed from breadboard to the DUT.

The Invar mirrors mounts are clamped into the sample. The design of our clamps enables measurements where the mount itself has no influence to the CTE of the sample. Also, all kind of materials, especially CFRPs where the end parts of the sample have different characteristics than the middle part can be determined.


With our dilatometer we measured several low-CTE materials. In our focus we determined CFRP materials and validated our measurement setup with ultra low CTE glass ceramics.

In the table below glass ceramics and composite materials are shown with measured and expected CTE values. Systematic errors are currently further investigated.

MaterialMeasured CTE [1/K]Measured CTE [1/K]
CFRP(3.1±0.1)106<math xmlns=""><mrow><mo>(</mo><mrow><mn>3.1</mn><mo>±</mo><mn>0.1</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>6</mn></mrow></msup></mrow></math>(2.5)106<math xmlns=""><mrow><mo>(</mo><mn>2.5</mn><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>6</mn></mrow></msup></mrow></math>
CFRP (Meteosat)(0.519±0.024)106<math xmlns=""><mrow><mo>(</mo><mrow><mo>−</mo><mn>0.519</mn><mo>±</mo><mn>0.024</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>6</mn></mrow></msup></mrow></math>(0.647)106<math xmlns=""><mrow><mo>(</mo><mrow><mo>−</mo><mn>0.647</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>6</mn></mrow></msup></mrow></math>
Zero-CTE CFRP(0.335±0.004)106<math xmlns=""><mrow><mo>(</mo><mrow><mo>−</mo><mn>0.335</mn><mo>±</mo><mn>0.004</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>6</mn></mrow></msup></mrow></math>108<math xmlns=""><mo>≈</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>8</mn></mrow></msup></math>
C-SiC0.05106<math xmlns=""><mo>−</mo><mn>0.05</mn><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>6</mn></mrow></msup></mrow></math>-
CFRP II(0.2±0.02)106<math xmlns=""><mrow><mo>(</mo><mrow><mo>−</mo><mn>0.2</mn><mo>±</mo><mn>0.02</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>6</mn></mrow></msup></mrow></math>-
Zerodur1.75108<math xmlns=""><mo>−</mo><mn>1.75</mn><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>8</mn></mrow></msup></mrow></math>(0±3)108<math xmlns=""><mrow><mo>(</mo><mrow><mn>0</mn><mo>±</mo><mn>3</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>8</mn></mrow></msup></mrow></math>
Clearcedam(3.205±0.039)108<math xmlns=""><mrow><mo>(</mo><mrow><mn>3.205</mn><mo>±</mo><mn>0.039</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>8</mn></mrow></msup></mrow></math>(0±2)108<math xmlns=""><mrow><mo>(</mo><mrow><mn>0</mn><mo>±</mo><mn>2</mn></mrow><mo>)</mo></mrow><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>8</mn></mrow></msup></mrow></math>

Measured CTE of several materials