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We present a brief discussion of the transmission ellipsometric function of an unsupported film/pellicle optical
structure. We also briefly discuss different ellipsometric techniques that could be used to characterize an unsupported
film/pellicle. The current state of data reduction either uses forward curve-fitting techniques or other numerical methods
to obtain the refractive index of the optical slab and its thickness. Both methods are dependent on a good starting point
and use an iterative approach to minimize a merit function that consumes much valuable time and memory resources.
We present closed-form formulas to obtain both the refractive index and thickness. We spare the reader successive and
involved transformations and algebraic manipulations to arrive at the closed forms. We provide the reader with an easy-to-
follow step-by-step algorithm to obtain the system parameters. Also, we present a closed-form formula for the
refractive index using two, and more, sets of measurements. In addition, we discuss the effect of film-thickness
multiplicity and its separation. Other technique-specific closed-form formulas are given for different ellipsometric
techniques. We also present numerical simulation results that prove the accuracy of the closed-form formulas, and that
revealed an interesting and useful characteristic that we utilize. We close by introducing a closed-form formula to
calculate the ratio of the unsupported film/pellicle to that of the ambient, which could be used to determine either
experimentally. The advantages of closed-form inversion over forward curve fitting and numerical methods are
numerous, including: 1) a much higher speed of obtaining the problem solution that allows for real-time applications, 2)
it does not require human judgments or intervention, 3) absolute stability, 4) much higher accuracy, 5) no need for
close-to-solution starting values of the unknown parameter(s), 6) no errors introduced by the formulas themselves, 7)
smart, simple, and concise software programs, 8) use in new material characterization where starting-point-dependent
numerical methods fail or require much trial and error.
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