Optical Model to Correct the Blue Shift Effect of Interference Bandpass Filters for Full-Field, Ratio Method Phosphor Thermography

The ratio method is a sector of phosphor thermography in which the phosphor emission intensities in two or more spectral bands are ratioed, such that the intensity ratio is proportional to temperature. For full-field measurements, interference bandpass filters mounted to two cameras are used to filter the emission. However, interference filters exhibit a shift in their transmission windows toward shorter (bluer) wavelengths when light enters the filter at non-perpendicular angles. This so-called "blue shift" results in spatial heterogeneity across intensity ratio maps, which is influenced by the geometry of the imaging configuration. To address this issue, we created an optical model with six key components: 1) The geometry of the imaging system provides the entrance angles of chief rays passing through the filter and impinging on each pixel of each camera. 2) The model intensity ratio is computed from a set of spectrally- and thermographically-resolved phosphor emission signatures by integrating over the shifted transmission windows of the filters. 3) Digital image correlation (DIC) registers the experimental image pairs to match each material point on the test piece to the corresponding subpixel location in each image. 4) The experimental intensity ratio for the test piece is computed from the matched points. 5) Eight free model parameters are tuned to the specific geometry of the imaging system. 6) The calibrated model is used to infer the temperature of the test piece. The optical model was first verified with a phosphor calibration coupon of manganese-doped magnesium fluorogermanate (MFG) heated uniformly to six different temperatures from 25-150°C. The uncorrected intensity ratio led to a temperature bias up to 27°C from the center to the edge of the image. When the optical model was calibrated and applied to the data, the temperature error was reduced to less than 3-5°C across four different imaging systems using different focal length lenses and standoff distances. The model was also validated with a tensile dog bone of stainless steel, with an applied MFG DIC pattern, subjected to mechanical deformation resulting in a temperature rise due to the conversion of plastic work into heat. The temperature inferred with the proposed model agreed with previous measurements using a pixel-wise calibration approach to within ±5°C and with thermocouples to within 3-4°C. Building on prior methods to address the blue shift effect of interference filters, our optical model provides precise temperature measurements across the entire field of view of the images. This result is achieved for a phosphor that demonstrates a temperature-dependent spectral shift, utilizing commercially available bandpass filters. Additionally, our calibration process is three times more efficient than the alternative pixel-wise approach. These advancements facilitate wider adoption of the ratio method for phosphor thermography across academia and industry.