Optics Communications 270 (2007) 16 www.elsevier.com/locate/optcom
Discussion
Comments on ‘‘A new method for phase extraction from a single fringe pattern’’ Jose´ A. Ferrari
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Instituto de Fı´sica, Facultad de Ingenierı´a, J. Herrera y Reissig 565, Montevideo, Uruguay Received 2 May 2006; received in revised form 22 July 2006; accepted 22 August 2006
Abstract We comment on the recent paper by Tay et al. [Opt. Commun. 239 (2004) 251], in which a phase extraction method from a single fringe pattern is proposed. Their claimed ‘‘new method’’ is based on a couple of assumptions on the behavior of the fringe pattern, which are closely related to the well-known Fourier-transform technique. Ó 2006 Elsevier B.V. All rights reserved.
I have some critical remarks regarding to a recent paper by Tay et al. [1], in which a method for phase extraction from a single fringe pattern is presented. The intensity distribution of a fringe pattern can be written as: Iðx; yÞ ¼ aðx; yÞ þ bðx; yÞ cos½/ðx; yÞ;
ð1Þ
where a(x, y), b(x, y) and /(x, y) are three unknown functions of the spatial coordinates (x, y). Hence, it is clear that at least three phase-shifted interferograms (or three assumptions on the behavior of the unknown functions, or a mix of both things) are required for a complete wavefront reconstruction. Firstly, we have to say that there are not ‘‘miracles’’ in interferometry; if only one interferogram is used, as claimed in the title of the paper, then one requires two additional assumptions for phase retrieval. In the paper by Tay et al., the first assumption is that a(x, y) is a low-frequency function. The second assumption is that b(x, y) ‘‘can be regarded as a constant in a fringe period’’ (see paragraph after Eq. (2) in [1]), which is equivalent to say that b(x, y) is also a low-frequency function. In the mentioned paper, the function a(x, y) is directly filtered out, and the function b(x, y) is (implicitly) filtered by *
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setting b(x, y) = 1 in a fringe period. With these two assumptions the retrieval of /(x,y) from Eq. (1) is not a difficult task. Although no use of a spatial carrier frequency is explicitly mentioned in the paper by Tay et al., they require a (local) spatial carrier frequency (i.e., high fringe density) so that a(x, y) and b(x, y) can be regarded as constants over one fringe period, which resembles the old method proposed by Takeda et al. [2]. Obviously, the applicability of such simplifications is restricted to conveniently chosen situations. The point to be stressed is that the paper of Tay et al. gives an impression as if their techniques were almost universal and applicable to a wide class of single fringe patterns without carrier frequency. However, this is not the case, and this sense we say that there are not ‘‘miracles’’ in interferometry. Acknowledgement The author thanks the reviewer for the constructive remarks, which helped us to clarify and improve our comments. References [1] C.J. Tay, C. Quan, F.J. Yang, X.Y. He, Opt. Commun. 239 (2004) 251. [2] M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 (1982) 156.