Optical dynamic deformation measurements at translucent materials

Katrin Philipp; Nektarios Koukourakis; Robert Kuschmierz; Christoph Leithold; Andreas Fischer; Jürgen Czarske

doi:10.1364/OL.40.000514

Contactless deformation measurements of translucent materials are required in a variety of fields, from biology [1] to material science [2]. One particular research topic is the dynamic deformation measurement at rotors of glass fiber-reinforced polymers (GFRP). These are an attractive alternative to conventional materials, since they offer high stiffness-to-weight ratios [3]. The understanding of their dynamic material behavior requires deformation measurements at high rotation speeds. The translucence of the material makes optical measurements difficult, since the main part of incoming light propagates into the material and might be backscattered by inhomogenities in the interior. Common optical measurement techniques such as triangulation [4], chromatic confocal sensing [5], and optical coherence tomography [6] fail at high-surface velocities due to increasing measurement uncertainties. The usability of interferometers is also limited due to speckle. Since the rotor deformation is small compared to its diameter, the application of shadow projection techniques is restricted. Nonoptical techniques like strain gauges are well established but invasive and may change the dynamic behavior of the rotor [7]. Electric sensors like capacitive or inductive ones are well suited for metallic surfaces, but not adequate for composites due to their low permeability [8].

In contrast to other optical measurement systems, the laser Doppler distance (LDD) sensor [9,10] is an adequate system for measurements at fast moving surfaces, because the measurement uncertainty is independent from the surface velocity. Consequently, LDD sensors were already successfully applied for dynamic deformation measurements of metallic rotors with measurement rates of several 10 kHz [11] and surface velocities of up to 600 m/s [12]. The aim of this report is to demonstrate the applicability of the LDD sensor at translucent materials, which has not yet been investigated. The influence of surface properties on the behavior of the sensor is evaluated by experiments and simulations. To underline the potential of the sensor, dynamic deformation measurements of a fast-rotating GFRP rotor with surface speeds of more than 300 m/s are demonstrated.

The LDD sensor, a two-wavelength Mach–Zehnder interferometer (see Fig. 1) is based on the well-established laser Doppler velocimetry technique. The LDD sensor consists of two interference systems with converging and diverging fringes to additionally obtain the position inside the measurement volume. When a moving object passes the measurement volume, the light scattered at the surface is modulated with a Doppler frequency. Distinguishing of the two interference systems is achieved by wavelength multiplexing. This enables simultaneous determination of surface position $z$ and velocity $v$ via

z = q^{- 1} (\frac{f_{D -}}{f_{D +}}), v \approx \frac{1}{2} [f_{D +} d_{+} (z) + f_{D -} d_{-} (z)]

with the inverted calibration function

q^{- 1}

, the fringe spacing functions

d_{\pm} (z)

, and the measured Doppler frequencies

f_{D \pm}

. The unambiguous calibration function

q (z) = \frac{d_{+} (z)}{d_{-} (z)} = \frac{f_{D -}}{f_{D +}}

as well as the fringe spacing functions

d_{\pm} (z)

are a priori determined by measuring the Doppler frequencies

f_{D \pm}

at a rotating calibration disc with known velocity

v

.

Fig. 1. Measurement setup of the laser Doppler distance sensor.

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In order to investigate the influence of the material properties on the measurement volume and the occurrence of secondary signals due to backscattering inside the volume, a simplified model setup is introduced. At first, the surface is considered perfectly flat and later extended to different surface roughnesses. Without loss of generality, the surface of the sample is assumed to be located at the crossing plane of the two laser beams, i.e., at $z = 0$ . As illustrated in Fig. 2 for a perfectly flat surface, the refractive index step between air and sample leads to a deformation of the measurement volume due to refraction of the beams. Starting from simulated fringe spacings $d_{\pm}$ for Gaussian beams according to Eq. (18) in [13] and applying Snell’s law lead in second order Taylor approximation to fringe spacings

Fig. 2. Scheme (not to scale) of undisturbed diverging interference fringe system (left) and deformation due to refraction at the material (right).

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d_{\pm} (z) = \frac{λ}{2 θ} [1 + \frac{(n^{2} - θ^{2}) [z^{2} (n^{2} - θ^{2}) \mp n^{4} z z_{R}]}{n^{8} z_{R}^{2} + (n^{8} - n^{6} θ^{2}) (n^{2} z_{R}^{2} \mp z_{R} z)}]

for

z \geq 0

with the half-crossing angle

θ

, the vacuum laser wavelength

λ

, and the Rayleigh length

z_{R} = \frac{π w_{0}^{2}}{λ}

. Note that the displacement between beam waist and crossing plane

z_{w}

was chosen to be the Rayleigh length

z_{R}

, i.e.,

z_{w} = \pm z_{R}

, in order to provide maximum spatial resolution. The measured fringe spacings

d_{\pm}

of the applied LDD sensor in air and calculated fringe spacings inside a medium with

n = 1.5

using Eq. (3) are shown in Fig. 3. The refraction of the laser beams leads to a flattening of the fringe spacing curve and thus to a slope change of the calibration function inside the medium compared to air. A scatterer located inside the sample volume therefore leads to a secondary signal with a shifted Doppler frequency compared to that of an object in air. To validate this model, a 1-mm-thick acrylic glass sample with surface roughness

R_{A} \approx 100 nm

is used. The refractive index equals

n \approx 1.5

. A layer of paper is glued on the backside of the sample in order to mimic a volume-scattering layer (e.g., fibers). The front surface of the acrylic glass is placed at position

z = - 469 μm

of the measurement volume and moves with

1.47 m / s

. Taking into account uncertainties of velocity and layer thickness, the calculated Doppler frequencies

f_{1} = (137 \pm 0.2) kHz

for scattering at the front surface and

f_{2} = (148.6 \pm 0.3) kHz

for scattering at the paper on the back side are in good agreement to the measured ones, as can be seen in Fig. 4 (green curve) for the converging interference fringe system (+).

Fig. 3. Measured fringe spacings $d_{\pm} (z)$ in air and calculated fringe spacings according to Eq. (3) with medium placed at $z = 0$ .

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Contrary to a simple refractive index step, the surface of real materials is rough and leads to scattering of light. In order to understand the influence of surface roughness on the measurement volume, a set of acrylic glass samples is used that have different surface roughnesses on their front-side. The samples are referred to as #1, #2, and #3, sorted with ascending surface roughness. It can be seen in Fig. 4 that with increasing roughness, the intensity of the second Doppler peak decreases (blue curve) until the signal completely disappears (red curve). These observations can be explained by the interferometric nature of the LDD sensor. After scattering at the surface, the light penetrates into the sample in an angular distribution with the scattering angle $α$ that strongly depends on surface properties. Increasing the scattering angle affects the interference pattern and eventually leads to its destruction above the threshold angle $α_{th}$ . This occurs as soon as the resulting broadening $Δ d$ of the interference fringe spacings $d_{\pm}$ meets the condition

Δ d_{\pm} \leq \frac{d_{\pm}}{2} with Δ d_{\pm} = | \frac{\partial d_{\pm}}{\partial θ} | α_{th} .

These observations are confirmed by simulations implemented in Matlab. Here the interference pattern of two beams is calculated and the waves are propagated in axial direction

z

for different surface characteristics. Static surfaces are generated based on measured surface parameters (see Table 1). In Fig. 5, slices of the measurement volume inside the medium for different surface roughness of acrylic glass are depicted. The results show that with increasing surface roughness, the measurement volume inside the translucent medium strongly degrades. In Fig. 6 (top left) line-scans across one specific slice of the measurement volume at distance

z = 20 μm

from the sample surface for increasing roughness at acrylic glass are shown. As can be seen in the Fourier transformation of the slices depicted in Fig. 6 (bottom left), the interference pattern strongly degrades at roughness above the threshold roughness

R_{acrylic, th} \approx 1 μm

. This is in agreement with the measurements at the acrylic glass samples (cf. Fig. 4).

Fig. 4. Spectrum of interference fringe system with diverging fringe spacing $d_{+}$ for varying surface roughness. Calculated Doppler frequencies and corresponding error margins are represented by orange bars. The variation of the Doppler peak values is caused by slightly different layer thicknesses (left). Measurement setup (right).

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Fig. 5. Simulation of measurement volume inside acrylic glass samples for surfaces located at $z = 0$ . The interference fringe system deteriorates with increasing surface roughness.

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Fig. 6. Line profiles simulated at $z = 20 μm$ for increasing surface roughness (top) and corresponding Fourier transform (bottom) for acrylic glass (left) and GFRP (right). The roughness $R_{A}$ of the acrylic glass samples #1−#3 are marked by orange lines, the roughness of the investigated rotor is marked green, and corresponding thresholds are indicated by white dashed lines.

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Table 1. Surface Parameters of Acrylic Glass and GFRP Rotors^a

View Table

In order to apply the experimentally verified simulation to the investigated rotor, its material properties are introduced. The rotor consist of glass fiber-reinforced epoxy composite material with a total thickness of 4 mm and a diameter of 50 mm. As specified by the manufacturer, the infusion Resin RIM 135 that constitutes the main part of the matrix material (77% of total mass) has a refractive index of $n \approx 1.55$ . Surface properties are listed in Table 1. As can be seen in Fig. 6 (right), the measured roughness of the investigated rotor is greater than the threshold $R_{A, GFRP, th} = 200 nm$ , i.e., the simulation predicts that the measurement volume diminishes inside the volume of the investigated GFRP rotor. Applying Eqs. 3 and 4 leads to the threshold angle $α_{th, GFRP} = 2.7 °$ for GFRP, i.e., the scattering angle $α_{GFRP} = 4 °$ of the investigated rotor is above this threshold. As a result, both calculations and simulations predict the absence of secondary signals from the rotor volume above a certain threshold roughness and scattering angle, respectively. Consequently, the sensor signal contains information only from the surface of the rotor.

In order to quantify the influence of translucency on the measurement precision, uncertainties due to random and systematic errors for measurements with a single LDD sensor at aluminum and GFRP rotors are shown in Fig. 7, whereby the measured position was averaged over 60 rounds. The measurement uncertainties at GFRP are about twice as big as those of the reference measurements at aluminum. This is mainly caused by the lower back-scattered light power of the translucent material despite using the maximum provided laser power of the laser diodes of about 5 mW (in contrast to measurements at aluminium). Backscattering at glass fibers adds noise to the obtained signal, resulting in a decreased signal-to-noise ratio and thus increased uncertainties.

Fig. 7. Uncertainties due to systematic (left) and random (right) errors for measurements on translucent rotors. The apparent decrease of random errors with velocity is an effect of systematic rotor instabilities at low velocities.

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Due to this low, velocity-independent measurement uncertainties, the LDD sensor provides a powerful and reliable method for dynamic deformation measurements of fast rotating, translucent GFRP rotors. Hence, an in situ measurement system [14] consisting of three evenly distributed LDD sensors was applied for dynamic deformation measurements of rotors with surface velocities of up to 314 m/s. 48.7% of the fibers are orientated in 90° direction, 23% in each $- 45 °$ and $+ 45 °$ direction, and only 4.8% in 0° direction (mass percent). Consequently, the centrifugal force of the rotor is expected to result in greater expansion in 0° than in 90° direction, leading to an elliptic deformation. As can be seen in Fig. 8, the measurement system is capable of detecting this expansion. This is particularly remarkable, since the rotor expansion of only a few hundred microns is small compared to the initial diameter of 50 cm.

Fig. 8. Schematic depiction of measurement setup consisting of three evenly distributed LDD sensors (left). Dynamic expansion of rotor at different rotation frequencies (right). The initial radius is $r_{0} = 25 cm$ , and $Δ r$ denotes radius expansion.

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In conclusion, the laser Doppler distance sensor provides a powerful and reliable method to monitor radial expansion of translucent materials rotating at high speeds. For surface roughnesses above a certain threshold (which is the case at the investigated rotor), the interference fringe system inside the material diminishes, and consequently, backscattering on glass fibers does not lead to secondary signals in the obtained spectra. Due to resulting low measurement uncertainties, it was possible to apply an in situ measurement system for dynamic deformation monitoring.

The authors thank Mr. Filippatos, Dr. Langkamp and Prof. Hufenbach of the Institute of Lightweight Engineering and Polymer Technology (ILK) for providing a sample rotor and enabling measurements at the high-speed rotor test rig. They are thankful of Dr.-Ing. Davids and Mrs. Vogt of the laboratory of Production Metrology and Quality Management at Technische Universität Dresden for performing surface measurements. The financial support from the Deutsche Forschungsgemeinschaft (funding code CZ 55/23-1) is gratefully acknowledged.

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Optical dynamic deformation measurements at translucent materials

Abstract

References

Cited By

Figures (8)

Tables (1)

Equations (4)

Optics Letters

Sample	$R_{A} [nm]$	${RP}_{c} [{cm}^{- 1}]$	$α$ [°]
Acrylic glass 1	100	35	–
Acrylic glass 2	900	33	–
Acrylic glass 3	3000	42	–
GFRP	300	8	4