2.1

Sensor Concept

The pure bending model⁵ describes the behavior of bent beams. One requirement is that the material must be homogenous or isotropic in elasticity, so the model is not applicable to compound materials which possess different mechanical properties. In the case of ultra-thin glass (AS87eco by Schott), the sensor elements are integrated using femtosecond laser pulses which changes the mechanical properties only negligibly. The refractive index of waveguide and Bragg grating are marginally larger than the substrate or cladding material and therefore the ultra-thin glass curvature sensor can be treated as homogenous and the pure bending model can be applied.

When such beam is under load, the plane inside the beam which experiences no load is also the plane of symmetry for homogenous materials. Compression and stretching are present only above and below the plane of symmetry, further called “neutral bending line” for two-dimensional representations.

A Bragg grating reflects wavelengths λ_B according to the Bragg formula

where n_eff, m, and Λ are the effective refractive index, the reflection order, and the grating distance, respectively.

Bending the beam leads to a compression of the Bragg grating for a positive curvature and to stretching for a negative curvature. Upon compression, the grating distance is reduced and the effective refractive index increases whereas for stretching, the grating points lie farther apart from one another and the effective refractive index shrinks. As the grating distance is affected much more than the effective refractive index, the Bragg wavelength change can be estimated from the bending of the material. A change of temperature also affects both the grating distance and the effective refractive index but is not in the scope of this work as a constant temperature was maintained during the experiments.

In Figure 1, the concept of bending is shown as a cross section for the ultra-thin glass with waveguide and Bragg grating. The waveguide and the Bragg grating are positioned parallel to the neutral bending line in Figure 1a. The side length of the ultra-thin glass is 50 mm with a thickness of 100 μm and the length L_FBG of the Bragg grating is 5 mm. The distance y between the Bragg grating and the neutral bending line is 23 μm. When negative bending of radius R from the perspective of the Bragg grating is performed, the situation shown in Figure 1b is obtained. In this case, the Bragg grating is stretched by

Figure 1.

Pure bending model. a) Both waveguide and Bragg grating are positioned parallel to the neutral bending line. b) Under (positive) bending, the Bragg grating is compressed.

When R >> y, the radius or its inverted, curvature C, can be approximated to be the same for both the neutral bending line and the Bragg grating, yielding

When the Bragg grating is compressed, the distance between the grating points Λ shrinks. At the same time, the refractive index is increased but this effect is small enough to be neglected for estimating wavelength shifts under bending. As the wavelength of reflected light is proportional to the grating distance, the wavelength shift can be estimated through

The curvature C can be calculated from the wavelength shift either through a calibration for different curvatures or through the determination of the distance y. The size of y determines the sensitivity of the sensor.

2.2

Fabrication

For a thin substrate, we chose 100 μm-thick aluminosilicate glass (Schott AG, AS87eco, refractive index n = 1.504) with side lengths of 50 mm, which stands out due to its high flexibility and low thickness. We used a regeneratively amplified Ti:Sa femtosecond laser (Spectra Physics, Tsunami/Spitfire, wavelength: 800 nm, repetition rate: 5 kHz, pulse duration: 100 fs) to integrate both a waveguide and a Bragg grating into the thin glass substrate. After transferring the laser beam into circular polarization and attenuating the energy down to 7.04 μJ, the laser beam is partly cut off by a slit and is subsequently transmitted through a microscope objective (Zeiss, LD Plan-NEOFLUAR 20x/0.4) which focuses the beam 27 μm or 73 μm (23 μm off the midpoint) deep into the glass substrate. The glass substrate is mounted on an air-bearing axis (PI, PIglide A-110). Because of the cutoff at the slit, the beam focus and therefore interaction volume between beam and glass are circular⁶. The beam-glass interaction leads to a positive refractive index change of the glass inside the interaction volume. To create light waveguides, the sample is moved perpendicularly to the laser beam at a speed of 2 mm/s. Eight waveguides were integrated, four above and four below the middle. The waveguides were positioned equidistantly along the sides of the glass. Using a laser scanning microscope, the waveguide depth was determined to be 23 μm off the neutral bending line. For curvature sensing elements, four 5 mm-long plane-by-plane⁷ Bragg gratings are inscribed along each waveguide equidistantly with the same setup and the same slit opening at a pulse energy of 7.41 μJ. In Figure 2, the ultra-thin glass is shown with glass fibers and the positions of the Bragg gratings are indicated.

Figure 2.

Ultra-thin glass with bi-directional sensor array. Four waveguides in both direction with four Bragg gratings each are indicated (black: x-direction, red: y-direction). A glass fiber is connected to each waveguide for sensor readout.

To interrogate the Bragg gratings, fiber pigtails (Fibercore, 800 nm, 5.6 μm) were put in proximity to the waveguide facets. At the other end, the fiber pig tails were connected to an interrogation system consisting of a broadband light source (SUPERLUM, FSLD-840-20-FC), a spectrometer (Ocean Insight, OceanFX), and a polarization randomizer. The polarization randomizer enables a polarization-independent monitoring of the Bragg wavelength shifts and consists of a half-waveplate and a quarter-waveplate rotating in opposite directions. A transparent UV-curing adhesive (Dymax, OP-4-20632-GEL, glass transition temperature at 79 °C) was added between fiber and waveguide, the fiber position was optimized for best Bragg reflection signals and the adhesive was cured by UV light.

2.3

Measurement Principle

For Bragg wavelength probing, the interrogation system from section 2.2 is used. A Gaussian fit is applied to the acquired Bragg wavelengths for peak detection.

To measure the sensitivity of the curvature sensor, the sensor was mounted on a 3D-printed cylinders with curvatures of 0 m^-1, 5 m^-1, 10 m^-1, 15 m^-1, and 20 m^-1 (Figure 3a). 20 m^-1 corresponds to a radius of 50 mm.

Figure 3.

Measurement setups. a) 3D-printed cylinders with curvatures of 0 m^-1, 5 m^-1, 10 m^-1, 15 m^-1, 20 m^-1, and 17.1 m^-1. b) Three-point bending setup.

The wavelength shifts are then compared to the curvatures, which give the sensitivities of the Bragg gratings. From detaching and remounting the sensor onto the cylinder, the remounting repeatability can be obtained.

To determine the linearity of the sensor and the precision or repeatability of measurements when the sensor stays mounted, the curvature sensor is put in a three-point bending setup (see Figure 3b). The bending setup consists of three posts (6 mm diameter) of which the outer two are fixed at a distance of 42 mm and the middle one can be moved with a linear stage (OWIS, LIMES 120-200-MSM), which allows a precise and reproducible change of bending.

2.4

3D-Shape Calculation

The curvatures of all four Bragg gratings along one waveguide were used to create a curve through extra- and interpolation². In this approach, the curvature of each Bragg grating is assumed for the length in which it resides up to the midpoint to the adjacent Bragg grating or the end of the sample. At their intersection, the curves of two Bragg gratings are on the same tangent line.

To move from a two-dimensional representation of a waveguide to a three-dimensional shape consisting of all Bragg gratings from all waveguides, the requirements⁸ were assumed that one side of the sample is fixed in a straight line and that the middle line along the bent direction experiences no effect of torsion.