Foucault test: shadowgram modeling from the physical theory for quantitative evaluations

Jesús Villa; Gustavo Rodríguez; Ismael de la Rosa; Rumen Ivanov; Tonatiuh Saucedo; Efrén González

doi:10.1364/JOSAA.31.002719

Journal of the Optical Society of America A
Vol. 31,
Issue 12,
pp. 2719-2722
(2014)
•https://doi.org/10.1364/JOSAA.31.002719

Foucault test: shadowgram modeling from the physical theory for quantitative evaluations

Jesús Villa, Gustavo Rodríguez, Ismael de la Rosa, Rumen Ivanov, Tonatiuh Saucedo, and Efrén González

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Jesús Villa, Gustavo Rodríguez, Ismael de la Rosa, Rumen Ivanov, Tonatiuh Saucedo, and Efrén González, "Foucault test: shadowgram modeling from the physical theory for quantitative evaluations," J. Opt. Soc. Am. A 31, 2719-2722 (2014)

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Abstract

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.

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Fig. 1. The Foucault test for a spherical mirror.

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Fig. 2. The horizontal axis represents the

x

variable in meters for

y = 0

. (a) Profile of a simulated wavefront. (b) The normalized complex amplitude

ψ_{o}

. (c) The normalized complex amplitude

i H {ψ_{o}}

. Note the sign inversion with respect to

ψ_{o}

for

W_{x} < 0

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Fig. 3. Displacement

x_{s}

of the knife edge for quantitative evaluations.

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Fig. 4. Simulation of a sequence of shadowgrams (from left-top to right-bottom) for several positions (

x_{s}

) of the knife edge, using a mirror with spherical aberration.

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Fig. 5. Simulation of a typical normalized

I - x_{s}

graph of a given site

(x, y)

in the sequence of shadowgrams.

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Equations (20)

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ψ_{o} (x, y) = a (x, y) e^{i \frac{2 π}{λ} W (x, y)},

a (x, y) = a_{o} \times circ (\frac{r}{r_{p}}),

circ (\frac{r}{r_{p}}) = {\begin{cases} 1 & if r \leq r_{p} \\ 0 & otherwise \end{cases},

ψ_{i} (x, y) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} ψ_{o} (x^{'}, y^{'}) h (x - x^{'}, y - y^{'}) d x^{'} d y^{'} = ψ_{o} (x, y) * h (x, y),

ψ_{i} (x, y) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} H (u, v) Ψ_{o} (u, v) e^{i 2 π (x u + y v)} d u d v = F^{- 1} {H (u, v) Ψ_{o} (u, v)},

u = \frac{x}{λ R} and v = \frac{y}{λ R} .

H (u, v) = {\begin{matrix} 0 & if u < 0 \\ \frac{1}{2} & if u = 0 \\ 1 & if u > 0 \end{matrix} = H (u) = \frac{1}{2} (1 + sgn (u)) .

ψ_{i} (x, y) = \frac{1}{2} (ψ_{o} (x, y) + F^{- 1} {sgn (u) Ψ_{o} (u, v)}) = \frac{1}{2} (ψ_{o} (x, y) + i H {ψ_{o} (x, y)}) = \frac{1}{2} (ψ_{o} (x, y) + \frac{i}{π} \int_{- \infty}^{\infty} \frac{ψ_{o} (x^{'}, y)}{x^{'} - x} d x^{'}),

H {ψ_{o} (x, y)} = \frac{1}{π} \ln | \frac{x + \sqrt{r_{p}^{2} - y^{2}}}{x - \sqrt{r_{p}^{2} - y^{2}}} |,

\hat{g} (x) = g (x) - i H {g (x)},

W_{x} (x, y) = \frac{\partial W}{\partial x} > 0, \forall x .

ψ_{o} (x, y) = α (x, y) + i β (x, y),

H {ψ_{o} (x, y)} = H {α (x, y)} + i H {β (x, y)} = β (x, y) + i H {H {α (x, y)}} = β (x, y) - i α (x, y) = - i ψ_{o} (x, y) .

H {ψ_{o} (x, y)} \approx - i sgn (W_{x}) ψ_{o} (x, y) .

ψ_{i} (x, y) \approx \frac{1}{2} [ψ_{o} (x, y) + sgn (W_{x}) ψ_{o} (x, y)] \approx H (W_{x}) ψ_{o} (x, y) .

I (x, y) \approx H (W_{x}) a^{2} (x, y) .

Ψ_{o} (u, v) = a_{0} F {circ (r / r_{p})} * F {e^{i \frac{2 π}{λ} W (x, y)}} = a_{0} r_{p} \frac{J_{1} (2 π r_{p} ρ)}{ρ} * F {e^{i \frac{2 π}{λ} W (x, y)}},

Ψ_{o} (u, v) \approx a_{0} F {e^{i \frac{2 π}{λ} W (x, y)}} .

ψ_{i} (x, y; x_{s}) \approx \frac{1}{2} F^{- 1} {[1 + sgn (\hat{u})] Ψ_{o} (\hat{u} + u_{s}, v)} \approx \frac{1}{2} (ψ_{o} (x, y) e^{i 2 π x u_{s}} + i H {ψ_{o} (x, y) e^{i 2 π x u_{s}}}) \approx H (W_{x} + λ u_{s}) ψ_{o} (x, y) e^{i 2 π x u_{s}},

I (x, y; x_{s}) \approx H (W_{x} + x_{s} / R) a^{2} (x, y) .

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Equations (20)

Journal of the Optical Society of America A