The physical theory of the Foucault test has been investigated to represent the complex amplitude and irradiance of the shadowgram in terms of the wavefront error; however, most of the studies have limited the treatment for the particular case of nearly diffraction-limited optical devices (i.e., aberrations smaller than the wavelength). In this paper we discard this restriction, and in order to show a more precise interpretation from the physical theory we derive expressions for the complex amplitude and the irradiance over an optical device with larger aberrations. To the best of our knowledge, it is the first time an expression is obtained in closed form. As will be seen, the result of this derivation is obtained using some properties of the Hilbert transform that permit representing the irradiance in a simple form in terms of the partial derivatives of the wavefront error. Additionally, we briefly describe from this point of view a methodology for the quantitative analysis of the test.
© 2014 Optical Society of America
As a Schlieren technique the Foucault test is very simple, highly sensitive, and very useful for testing optical devices, such as lenses and mirrors. The Foucault test has been widely applied over several decades to assess the quality of telescope mirrors by means of detecting transverse aberrations using a simple knife edge that blocks out part of the rays to form a shadow pattern, usually called shadowgram. Despite its advantages, the Foucault test has been mostly used as a first qualitative test before the use of more sophisticated interferometric techniques.
One of the main advantages of the Foucault test is its easy interpretation from the geometrical point of view [1–3], which states that the transverse ray aberrations are functions of the partial derivatives of the wavefront. The geometrical theory has been widely used to evaluate large aberrations.
The first advances in the analysis from the point of view of the physical theory were reported by Gascoigne  and Linfoot [5–8]. Later, some other advances were reported by Barakat , Welford , Katzoff , and Wilson ; in particular, Katzoff  proposed an inversion method of Linfoot’s formula using a linear approximation that permits one to obtain an expression of the wavefront error in terms of the irradiance distribution. Unfortunately, the physical theory has been mostly used to analyze the complex amplitude and the irradiance in the image plane considering the particular case of wavefront errors smaller than the wavelength.
Opticians have long used the Foucault test in the geometrical sense; however, the interpretation of the test from the physical theory has been almost discarded for most practical purposes.
Despite its great potential, the Foucault test has been little explored for use as a sophisticated technique for quantitative evaluations. Although the geometrical sense has been widely used to analyze the Foucault test, it has some limitations for this purpose. For this reason it is necessary to explore in depth the physical theory to overcome these limitations and to develop more sophisticated quantitative evaluation techniques using the Foucault test, which is part of our the current research.
In order to show a more precise interpretation from the physical theory and its application to quantitative evaluations, we now derive irradiance distribution models that are obtained in closed form. As will be shown, the models are deduced in this way using some properties of the Hilbert transform.
2. PHYSICAL THEORY OF THE FOUCAULT TEST
Consider the basic Foucault arrangement to test spherical mirrors, as depicted in Fig. 1.
For simplicity, if we assume uniform illumination the complex amplitude of the aberrated wavefront coming from the mirror can be defined as3] 6) represents the radius of the reference sphere.
The knife edge is a filter in the Fourier plane blocking out part of the converging light. If it is placed vertically coinciding with the axis it can be represented by the Heaviside unit step function:7) in Eq. (5) we obtain 8) results in the well-known form
3. SHADOWGRAM MODELING FROM THE PHYSICAL THEORY
As can be seen, the analysis of the second term in Eq. (8) is the key to deriving an expression for , so we shall first establish a result concerning such term.
The Hilbert transform is a useful mathematical tool to construct an analytic function from a real function , the two being associated in the following way:
For simplicity in the following derivation we assume that is large enough that the size of the Airy disc is much smaller than the spectral lobe due to . Then, taking into account that represents an analytic signal, we now use some properties of the Hilbert transform  to obtain an expression for , in particular for the case10) is not satisfied (i.e., is not a monotonically increasing function for ). In a similar way, it can also be verified that for the case . As a consequence of the Hilbert transform properties (i.e., it represents the quadrature of the function ) the local sign of determines the sign of Eq. (13) (see Fig. 2).
Therefore, in general8) as 14) is not well defined for , Eq. (16) is an approximation to , so that we can see in real experiments that the shadowing is gradual instead of occurring a sharp edge. This can be explained from the fact that
4. QUANTITATIVE ANALYSIS FROM THE PHYSICAL THEORY
A quantitative method to detect at a given point is to record the intensity as a function of the knife-edge position as indicated in Fig. 3; then the edge in the plot of intensity versus the knife-edge position is used to determine [1,2].
Using the dummy variable , where , the complex amplitude as a function of in the image plane is4 and 5).
We have derived expressions to model the irradiance of shadowgrams using the physical theory, which allows a more precise interpretation from this point of view. The signum function that represents the knife edge in the Fourier plane permits expression of the complex amplitude in the image plane in terms of the Hilbert transform. As shown, the use of some basic properties of this transform has been the key to deriving such image models. From this point of view we have also described a methodology for the quantitative analysis of the test.
The authors wish to acknowledge the partial support for the realization of this work by the Consejo Zacatecano de Ciencia y Tecnologia (COZCYT), through the project ZAC-2013-C02-203103, and the Consejo Nacional de Ciencia y Tecnología (CONACYT) of México for the scholarship of Gustavo Rodríguez.
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