Mach-Zehnder data reduction method for refractively inhomogeneous test objects: author's reply to comments Allen M. Hunter II Air Force Weapons Laboratory (AFSC), Kirtland Air Force Base, New Mexico 87117. Received 25 June 1975. Kahl 1 correctly identified an error in the paper by Hun- ter and Schreiber. 2 The error leads to incorrect conclu- sions even though the basic approach is correct. The article mentioned above advocates using an iterative data reduction scheme to remove the systematic optical system errors from the measured fringe shifts. For an iter- ative analysis to converge, however, the most significant fringe shift contributions must be included in the forma- tion for the first data inversion. As Kahl points out, at 2336 APPLIED OPTICS / Vol. 14, No. 10 / October 1975
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Mach-Zehnder data reduction method for refractively inhomogeneous test objects: author's reply to comments
Allen M. Hunter II Air Force Weapons Laboratory (AFSC), Kirtland Air Force Base, New Mexico 87117. Received 25 June 1975.
Kahl1 correctly identified an error in the paper by Hunter and Schreiber.2 The error leads to incorrect conclusions even though the basic approach is correct.
The article mentioned above advocates using an iterative data reduction scheme to remove the systematic optical system errors from the measured fringe shifts. For an iterative analysis to converge, however, the most significant fringe shift contributions must be included in the formation for the first data inversion. As Kahl points out, at
least the term ynR (tan α)/λ0 should be added to the right-hand side of Eq. (5) in the article by Hunter and Schreiber. Other terms also become important if the interferometer is misfocused or lensless, contains thick optical plates, or has a reference index of refraction different from the index of refraction at the periphery of the test medium.1 The image position yi of a ray can significantly differ from its impact parameter y if the interferometer is misfocused or contains no focusing lens.1
A suitable integral for the first data inversion in an iterative analysis is obtained by inverting the fringe shift equation, which includes zero and first-order terms in the angle of refraction for a well-focused interferometer. Using Eqs. (5) and (9) in the article by Hunter and Schreiber, the fringe shift becomes
where n is the index of refraction, r is the radius, R is the outer radius of the test medium, nR = n(r = R), u ≡ rn(r), UR = RNR, and υ ≡ ynR. This approximate equation has the formal inverse
The equation for n(u) is similar to the conventional inverted Abel equation for very small values of n - 1, and therefore the inverted Abel equation should yield reasonable results for well-focused single pass two-beam interferometers.
An iterative data analysis scheme that uses the n(u) equation above as well as Eq. (8) from the Hunter and Schreiber article and the correct Kahl and Mylin fringe shift contributions from Ref. 3 has been developed. The results obtained from using the iterative method show that for \η - 1| < 10 - 2 an inverted Abel equation provides acceptably accurate estimates of n(r) for well-focused interferometers. The image position errors, however, can be significant for misfocused or lensless interferometers, and the inverted Abel equation alone is generally insufficient to analyze accurately these data.
References 1. G. D. Kahl, Appl. Opt. 14, 2336 (1975). 2. A. M. Hunter II and P. W. Schreiber, Appl. Opt. 14, 634 (1975). 3. G. D. Kahl and D. C. Mylin, J. Opt. Soc. Am. 55, 364 (1965).
