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How To Tolerance for Material Inhomogeneity
- By Dan Hill
- Published 16 December 2005
- Surface Tolerances
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Modeling Inhomogeneity via Surface Irregularity
Given a single number or value representing the inhomogeneity of a material, it is impossible to exactly predict the index profile of the glass.
Therefore, the most accurate and superior approach to modeling the inhomogeneity of a material can be performed via the statistical results of Monte Carlo Tolerance Analysis using tolerances on surface irregularity.
For example, the TEZI tolerance operand in ZEMAX is used to analyze random irregular deviations of small amplitude on a surface that is either type Standard or Even Aspheric. Within the Monte Carlo Analysis, the specified surface is converted to a Zernike Standard Sag surface, and each polynomial term is assigned a coefficient randomly chosen between zero and one. The resulting coefficients are normalized to yield the exact specified RMS tolerance.
So how are these related? Basically, the surface irregularity model will “warp” the wavefront much like the non-homogeneity of a specific material. Index variation within a piece of glass will lead to a deformation of the wavefront passing through it [1]. From SCHOTT’s technical information article on homogeneity (also reference [1]), the deformation to the wavefront as a function of inhomogeneity is given by:
Δs = d • Δn
where Δs is the wavefront deviation, d is the thickness of the glass, and Δn is the peak to valley refractive index variation in the glass. In ZEMAX terms, we might write this as:
ΔOPL = t • Δn
where ΔOPL is the change in optical path length, t is the thickness of the glass, and Δn is the peak to valley refractive index variation in the glass. For example, if we considered a plane wave incident on a 25mm thick plane parallel plate of glass of +/- 5*10-6 index varation, the maximum ΔOPL would be equal to:
ΔOPL = 25mm • 10 • 10-6 = 250nm