Wavelength sensitivity of the speckle patterns produced by an integrating sphere

Abstract

Speckle metrology is a powerful tool in the measurement of wavelength and spectra. Recently, speckle produced by multiple reflections inside an integrating sphere has been proposed and showed high performance. However, to our knowledge, a complete characterisation of speckle sensitivity to wavelength in that geometry has not been performed to date. In this work, we derive a general model predicting the variation in a speckle pattern as a result of a generic transformation. Applying this to a shift in the incident wavelength, we show that the speckle sensitivity is mainly governed by the radius and surface reflectivity of the sphere. We show that integrating spheres offer sensitivity four orders of magnitude above that of multimode fibres of a similar size, and discuss analogies with the transmission profile of a Fabry–Pérot interferometer.

Measuring picometre-level displacements using speckle patterns produced by an integrating sphere

“…We note that the HWHM is not the resolution limit of the approach -the ultimate resolution of the method depends on the smallest detectable variation in similarity (see Section IV). Note that the sensitivity of the speckle pattern to displacement is independent of the size of the sphere, contrary to what is found for wavelength and refractive index variation [26].…”

“…In the case of speckle patterns produced by multiple reflections of light in an integrating sphere, it was shown in [26] that for two speckle patterns taken before and after some generic transformation, the similarity is given by…”

“…Here we show the effect of a deviation from either the Lambertian reflectance or the uniform reflectivity on the similarity profile. Following the derivation given in [26], the general expression of the similarity (2) comes from the integral…”

“…Although speckle can be generated in many ways, we have recently analytically shown that an integrating sphere offers an optimal scattering medium, due to the large spread of optical path lengths inherent in this geometry [26]. In the context of measuring the wavelength of monochromatic light, the speckle from a typical integrating sphere is four orders of magnitude more sensitive to wavelength changes than the speckle produced in more-ubiquitous multi-mode fibre [26]. The integrating sphere has also been used recently to advance the speckle-based measurement of changes in refractive index of gases at the level of one part in 10 9 [19].…”

Measurement of variations in gas refractive index with $10^{-9}$ resolution using laser speckle

“…In this work, to further increase the spread in path lengths and enable more finely-resolved measurements of refractive index, we take inspiration from recent progress in speckle-based measurements of the wavelength of monochromatic light [22][23][24][25][26][27][28]. In that context, we have recently analytically shown the advantages of using an integrating sphere to generate the speckle [29]. The integrating sphere creates a particularly broad path length distribution which offers orders of magnitude improvement in sensitivity compared with other speckle-based techniques.…”