A STUDENT EXERCISE IN CONSTRUCTING A MICHEL-LÉVY CHART FROM FIRST PRINCIPLES

Students may construct their own Michel-Lévy (ML) chart from first principles through a simple computer exercise as part of an exploration in optical mineralogy. In addition to examining the details of this useful interference scale, students learn about the polarizing light microscope, optical interference, vector math, and color perception. The exercise also fosters spreadsheet skills and exposes students to digital image construction and manipulation. This exercise is based on Sorensen (2012, Eur. J. Min.)’s elegant technique constructed in MATLAB, but utilizes more widely-accessible spreadsheet (e.g. Excel, OpenOffice Calc) and freeware image processing (ImageJ) programs.

This method constructs interference diagrams of any size, but the following instructions produce a small, novice-friendly 590 x 400 pixel chart. For a given visible wavelength (λ) and retardation (Δ) in nm, the transmission (L) by the analyzer for light from a crystal between crossed polarizers is L = sin² [180° Δ/λ]. This formula produces a spectral color matrix spreadsheet with a λ column from 390 to 700 nm in 1 nm increments and a Δ row from 0 to 2945 nm in 5nm increments. This spectral color matrix is mapped to vision color using the CIEXYZ color-matching function using tables from UCL’s Color and Vision Research Lab (cvrl.org). The exercise then constructs an LX matrix from the product of L(λ, Δ) and CIEX(λ), repeating for CIEY(λ) and CIEZ(λ) to form LY and LZ matrices. These are each saved as a text file and imported into ImageJ as the first three channels of a color merged image. This produces an intensity image as function of in λ and Δ CIEXYZ. The color is then mapped from XYZ to RGB (plugin from Lindbloom, 2016, brucelindbloom.com) to produce an RGB transmission diagram (LRGB).

A Michel Levy (ML) diagram is the sum all of the transmission values for each retardation increment. The exercise splits the transmission diagram into LR, LG, and LB, saving each as a text image file. The sum of LR for each Δ is duplicated downward for 400 rows to produce a 590 x 400 pixel MLR, and is repeated for MLB and MLG. These are merged in ImageJ to produce a ML diagram. Students then import this into a spreadsheet or other program to be annotated with specific Δ, thickness, and birefringence values and lines.