II. Working Principle Figure 1 illustrates the principle of strain measurement using a diffraction grating and a position sensitive detector. The grating is attached to the surface of the sample. When a monochromatic collimated beam is incident perpendicularly onto the linear grating (>40 lines/mm), a set of diffracted spots appears on a screen parallel to the grating plane. In the setup shown, a laser beam is directed perpendicularly onto a reflective diffraction grating. For high-frequency gratings, only the ±1 order diffracted beams are used for strain measurement. These beams are captured by a high-resolution detector. As the grating deforms along with the sample, the movement of the diffracted beam occurs due to in-plane deformation or out-of-plane displacement. The displacement of the ±1 order beam along the sensor length is given by equation (1): (1) where p is the grating’s spatial frequency, B is the diffraction angle, l is the laser wavelength. A small deformation causes a change in the grating line spacing (Dp), leading to a change in the diffraction angle (Db). From equation (1), we get: (2) which implies: (3) where ex is the strain along the x-direction. Assuming the diffracted beam is perpendicular to the position sensor plane, the displacement along sensor 1 is: (4) For sensor 2, simply replace b with -b to obtain: (5) Combining equations (4) and (5) gives the basic strain measurement equation:

III. Sensor System and Measurement Method

1. Sensor System Hardware Figure 2 shows the configuration of the sensor system, suitable for both laboratory and industrial applications. It includes a laser source, two position sensitive detectors, two 633 nm bandpass filters, a converging lens, and a grating. The grating has a spatial frequency of 1200 lines/mm and is bonded to the sample surface. A He-Ne laser beam (632.8 nm) with a diameter of about 1 mm is incident on any point of the grating plane. The position sensitive detector is an optoelectronic device based on a single photodiode. Key features of the system include: high spatial resolution compared to devices like CCD, use of two voltage signals for rapid signal processing, compact size, high relative position resolution (1/5000), immunity to light intensity changes, wide spectral sensitivity (300–1100 nm), and fast response time (<20 ms) for dynamic measurements. The output voltages from the two position sensitive detectors are sent to a computer via an A/D converter, with a maximum sampling rate of 105 samples/s. Two 633 nm filters help reduce background noise.

2. Adjustment Method If the laser beam is not perpendicular to the sample surface, it can cause significant measurement errors. To correct this, the zero-order reflected beam from the grating must align with the incident beam. This alignment must occur vertically to ensure symmetric distribution of the ±1 order diffracted beams. The key to system tuning is ensuring the incident laser beam is perpendicular to the specimen surface. Care must be taken to confirm the grating is firmly adhered and the sample is properly positioned. Additionally, the position sensitive sensor can be adjusted so that the ±1 order diffracted beam is centered on the sensor planes.

3. Measurement Methods The main steps in the measurement process are as follows: 1) Prepare the sample and diffraction grating, similar to a Moiré interferometer; 2) Determine the distance L between the position sensor and the grating (100–500 mm), and input it into the software; note that L=250 mm cannot be selected; 3) Perform an initial test to measure the average of x10 and x20; 4) Apply pressure to the sample and measure the average of x1 and x2; 5) Calculate the strain using equation (6). All calculations are automatically performed by the software.

4. Interface Software The interface software is developed using LabVIEW, handling tasks such as data sampling, filtering, calculation, memory reading/writing, and display. The data processing speed is very high, with a full cycle taking approximately 0.1 seconds. All signal processing and data acquisition are automated, and the strain results are continuously displayed on the PC screen in numerical and graphical form.

IV. System Characteristics Several factors influence the performance of the sensor system, including random noise from the position sensitive detector, A/D converter noise, and systematic errors caused by misalignment of the incident laser beam.

1. Random Noise The random noise in the system limits its sensitivity and spatial resolution. Four main sources of noise in the position sensitive detector include: 1) intensity noise from the light source; 2) amplifier voltage noise; 3) thermal noise from the feedback resistor; and 4) shot noise caused by DC current. The noise level varies depending on the spot's position on the detector, with the lowest noise at the center and the highest at the edges. The A/D converter noise variance is D²/12, where D is the digitized value, and 12-bit conversion is used.

2. Position Resolution With a recorder, the position detector has a relative resolution of 1/5000. The double-ended output voltage signal ranges from -5V to +5V, corresponding to a spot position from -5mm to +5mm. A 12-bit A/D converter can resolve 2.4mm. Considering the impact of position sensor noise, the overall system resolution is approximately 0.3mm.

3. Strain Sensitivity The average residual noise is independent of the spot position on the detector. Let x represent the noise, x* be the noisy position signal, then x* = x + x. Equation (6) becomes: (7) where x₀ is the initial center position of the diffracted beam (treated as a constant), and x is the final averaged position after loading. The strain error due to random noise is: (8) The standard deviation of the strain error is: (9) where sx is the standard noise deviation (approximately 0.3 mm), and r is the correlation coefficient of noise from channels 1 and 2, measured as r=0.4. Using actual parameters—grating frequency of 1200 lines/mm, laser wavelength of 632.8 nm, b=49.4°, tan(b)=0.9492, L=150 mm—the maximum noise error is ss=0.9 me. This value represents the strain sensitivity, which varies with L, as shown in Table 1. Table 1: Variation of Strain Sensitivity (ss) with Distance (L) L (mm) | 150 | 200 | 250 | 300 | 350 | 400 | 450 | 500 ss (me) | 0.9 | 0.7 | 0.6 | 0.5 | 0.4 | 0.4 | 0.3 | 0.3

4. System Error When the incident laser beam deviates from the normal direction of the sample, a systematic error arises. If the angle of deviation is q, the resulting strain error is derived from equation (3): (10) where Db1 and Db2 are changes in the diffraction angle due to sample deformation and angular deviation. Equation (6) becomes: (11) Assuming no other errors, and considering only the effect of q, Db1 can be calculated as: (12) Ignoring higher-order terms, we get: (13) Similarly: (14) Substituting equations (13) and (15) into (11) gives the strain error: (16)

5. Spatial Resolution Measurement The spatial resolution is determined by the diameter of the incident laser beam. A typical laser beam has an original diameter of 1–2 mm. To improve resolution, the beam is focused using a lens before being incident on the sample. A low-loss plastic lens with a focal length of 10 cm can reduce the beam diameter from 1.5 mm to 0.1 mm, significantly enhancing spatial resolution.

V. Technical Parameters and Features The sensor system has the following specifications: 1) sensitivity of 1 me; 2) variable spatial resolution (0.1–2 mm); 3) maximum strain capacity of 15%; 4) flexible measurement positions, allowing any point on the grating plane to be measured; 5) capability for dynamic and continuous strain measurement; 6) automated data acquisition and processing; 7) user-friendly interface; and 8) compact and small-sized structure.

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