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.

© 2014 Optical Society of America

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References

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  1. E. M. Granger, “Wave-front measurements from a knife-edge test,” Proc. SPIE 429, 174–177 (1983).
  2. D. E. Vandenberg, W. D. Humbel, and A. Wertheimer, “Quantitative evaluation of optical surfaces by means of an improved Foucault test approach,” Opt. Eng. 32, 1951–1954 (1993).
    [Crossref]
  3. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley-Interscience, 2007), Chap. 8.
  4. S. C. B. Gascoigne, “The theory of the Foucault test,” Mon. Not. R. Astron. Soc. 104, 326–334 (1944).
    [Crossref]
  5. E. H. Linfoot, “Astigmatism under the Foucault test,” Mon. Not. R. Astron. Soc. 105, 193–199 (1945).
    [Crossref]
  6. E. H. Linfoot, “A contribution to the theory of the Foucault test,” Proc. R. Soc. A 186, 72–99 (1946).
  7. E. H. Linfoot, “On the interpretation of the Foucault test,” Proc. R. Soc. A 193, 248–259 (1948).
  8. E. H. Linfoot, “The Foucault test,” in Recent Advances in Optics (Oxford, 1958), Chap. II, pp. 128–174.
  9. R. Barabak, “General diffraction theory of optical aberration tests, from the point of view of spatial filtering,” J. Opt. Soc. Am. 59, 1432–1439 (1969).
    [Crossref]
  10. W. T. Welford, “A note on the theory of the Foucault knife-edge test,” Opt. Commun. 1, 443–445 (1970).
    [Crossref]
  11. S. Katzoff, “Quantitative determination of optical imperfections by mathematical analysis of the Foucault knife-edge test pattern,” (NASA, August 1971).
  12. R. G. Wilson, “Wavefront-error evaluation by mathematical analysis of experimental Foucault-test data,” Appl. Opt. 14, 2286–2297 (1975).
    [Crossref]
  13. F. W. King, Hilbert Transforms, Vol. 1–2 of Encyclopedia of Mathematics and Applications (Cambridge University, 2009), pp. 145–251.

1993 (1)

D. E. Vandenberg, W. D. Humbel, and A. Wertheimer, “Quantitative evaluation of optical surfaces by means of an improved Foucault test approach,” Opt. Eng. 32, 1951–1954 (1993).
[Crossref]

1983 (1)

E. M. Granger, “Wave-front measurements from a knife-edge test,” Proc. SPIE 429, 174–177 (1983).

1975 (1)

1970 (1)

W. T. Welford, “A note on the theory of the Foucault knife-edge test,” Opt. Commun. 1, 443–445 (1970).
[Crossref]

1969 (1)

1948 (1)

E. H. Linfoot, “On the interpretation of the Foucault test,” Proc. R. Soc. A 193, 248–259 (1948).

1946 (1)

E. H. Linfoot, “A contribution to the theory of the Foucault test,” Proc. R. Soc. A 186, 72–99 (1946).

1945 (1)

E. H. Linfoot, “Astigmatism under the Foucault test,” Mon. Not. R. Astron. Soc. 105, 193–199 (1945).
[Crossref]

1944 (1)

S. C. B. Gascoigne, “The theory of the Foucault test,” Mon. Not. R. Astron. Soc. 104, 326–334 (1944).
[Crossref]

Barabak, R.

Gascoigne, S. C. B.

S. C. B. Gascoigne, “The theory of the Foucault test,” Mon. Not. R. Astron. Soc. 104, 326–334 (1944).
[Crossref]

Granger, E. M.

E. M. Granger, “Wave-front measurements from a knife-edge test,” Proc. SPIE 429, 174–177 (1983).

Humbel, W. D.

D. E. Vandenberg, W. D. Humbel, and A. Wertheimer, “Quantitative evaluation of optical surfaces by means of an improved Foucault test approach,” Opt. Eng. 32, 1951–1954 (1993).
[Crossref]

Katzoff, S.

S. Katzoff, “Quantitative determination of optical imperfections by mathematical analysis of the Foucault knife-edge test pattern,” (NASA, August 1971).

King, F. W.

F. W. King, Hilbert Transforms, Vol. 1–2 of Encyclopedia of Mathematics and Applications (Cambridge University, 2009), pp. 145–251.

Linfoot, E. H.

E. H. Linfoot, “On the interpretation of the Foucault test,” Proc. R. Soc. A 193, 248–259 (1948).

E. H. Linfoot, “A contribution to the theory of the Foucault test,” Proc. R. Soc. A 186, 72–99 (1946).

E. H. Linfoot, “Astigmatism under the Foucault test,” Mon. Not. R. Astron. Soc. 105, 193–199 (1945).
[Crossref]

E. H. Linfoot, “The Foucault test,” in Recent Advances in Optics (Oxford, 1958), Chap. II, pp. 128–174.

Malacara, D.

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley-Interscience, 2007), Chap. 8.

Vandenberg, D. E.

D. E. Vandenberg, W. D. Humbel, and A. Wertheimer, “Quantitative evaluation of optical surfaces by means of an improved Foucault test approach,” Opt. Eng. 32, 1951–1954 (1993).
[Crossref]

Welford, W. T.

W. T. Welford, “A note on the theory of the Foucault knife-edge test,” Opt. Commun. 1, 443–445 (1970).
[Crossref]

Wertheimer, A.

D. E. Vandenberg, W. D. Humbel, and A. Wertheimer, “Quantitative evaluation of optical surfaces by means of an improved Foucault test approach,” Opt. Eng. 32, 1951–1954 (1993).
[Crossref]

Wilson, R. G.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Mon. Not. R. Astron. Soc. (2)

S. C. B. Gascoigne, “The theory of the Foucault test,” Mon. Not. R. Astron. Soc. 104, 326–334 (1944).
[Crossref]

E. H. Linfoot, “Astigmatism under the Foucault test,” Mon. Not. R. Astron. Soc. 105, 193–199 (1945).
[Crossref]

Opt. Commun. (1)

W. T. Welford, “A note on the theory of the Foucault knife-edge test,” Opt. Commun. 1, 443–445 (1970).
[Crossref]

Opt. Eng. (1)

D. E. Vandenberg, W. D. Humbel, and A. Wertheimer, “Quantitative evaluation of optical surfaces by means of an improved Foucault test approach,” Opt. Eng. 32, 1951–1954 (1993).
[Crossref]

Proc. R. Soc. A (2)

E. H. Linfoot, “A contribution to the theory of the Foucault test,” Proc. R. Soc. A 186, 72–99 (1946).

E. H. Linfoot, “On the interpretation of the Foucault test,” Proc. R. Soc. A 193, 248–259 (1948).

Proc. SPIE (1)

E. M. Granger, “Wave-front measurements from a knife-edge test,” Proc. SPIE 429, 174–177 (1983).

Other (4)

E. H. Linfoot, “The Foucault test,” in Recent Advances in Optics (Oxford, 1958), Chap. II, pp. 128–174.

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley-Interscience, 2007), Chap. 8.

S. Katzoff, “Quantitative determination of optical imperfections by mathematical analysis of the Foucault knife-edge test pattern,” (NASA, August 1971).

F. W. King, Hilbert Transforms, Vol. 1–2 of Encyclopedia of Mathematics and Applications (Cambridge University, 2009), pp. 145–251.

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Figures (5)

Fig. 1.
Fig. 1.

The Foucault test for a spherical mirror.

Fig. 2.
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 ψo. (c) The normalized complex amplitude iH{ψo}. Note the sign inversion with respect to ψo for Wx<0.

Fig. 3.
Fig. 3.

Displacement xs of the knife edge for quantitative evaluations.

Fig. 4.
Fig. 4.

Simulation of a sequence of shadowgrams (from left-top to right-bottom) for several positions (xs) of the knife edge, using a mirror with spherical aberration.

Fig. 5.
Fig. 5.

Simulation of a typical normalized Ixs graph of a given site (x,y) in the sequence of shadowgrams.

Equations (20)

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ψo(x,y)=a(x,y)ei2πλW(x,y),
a(x,y)=ao×circ(rrp),
circ(rrp)={1ifrrp0otherwise,
ψi(x,y)=ψo(x,y)h(xx,yy)dxdy=ψo(x,y)*h(x,y),
ψi(x,y)=H(u,v)Ψo(u,v)ei2π(xu+yv)dudv=F1{H(u,v)Ψo(u,v)},
u=xλRandv=yλR.
H(u,v)={0ifu<012ifu=01ifu>0=H(u)=12(1+sgn(u)).
ψi(x,y)=12(ψo(x,y)+F1{sgn(u)Ψo(u,v)})=12(ψo(x,y)+iH{ψo(x,y)})=12(ψo(x,y)+iπψo(x,y)xxdx),
H{ψo(x,y)}=1πln|x+rp2y2xrp2y2|,
g^(x)=g(x)iH{g(x)},
Wx(x,y)=Wx>0,x.
ψo(x,y)=α(x,y)+iβ(x,y),
H{ψo(x,y)}=H{α(x,y)}+iH{β(x,y)}=β(x,y)+iH{H{α(x,y)}}=β(x,y)iα(x,y)=iψo(x,y).
H{ψo(x,y)}isgn(Wx)ψo(x,y).
ψi(x,y)12[ψo(x,y)+sgn(Wx)ψo(x,y)]H(Wx)ψo(x,y).
I(x,y)H(Wx)a2(x,y).
Ψo(u,v)=a0F{circ(r/rp)}*F{ei2πλW(x,y)}=a0rpJ1(2πrpρ)ρ*F{ei2πλW(x,y)},
Ψo(u,v)a0F{ei2πλW(x,y)}.
ψi(x,y;xs)12F1{[1+sgn(u^)]Ψo(u^+us,v)}12(ψo(x,y)ei2πxus+iH{ψo(x,y)ei2πxus})H(Wx+λus)ψo(x,y)ei2πxus,
I(x,y;xs)H(Wx+xs/R)a2(x,y).

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