Abstract

We describe the interference between two caustics of diffraction fields that are generated in a double-curve-shaped-slit interferometer. One of the slits is obtained from the other through a projective mapping in the normal direction. In this way the diffraction fields present the same caustic region because the slit functions have a common evolute. Catastrophe functions are used for describing the optical path length of the rays involved. We show that the phase difference between the diffracted rays can also be considered a catastrophe function, which permits us to describe the geometry of the interference pattern because its singular points correspond to organized regions of the interference pattern. Experimental results are shown.

© 1999 Optical Society of America

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References

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  1. A. V. Gaponov-Grekhov, M. I. Rabinovich, Nonlinearities in Action (Springer-Verlag, Berlin, 1991), p. 25.
  2. B. J. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 16 (1962).
    [CrossRef]
  3. Y. A. Kratsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), pp. 29–40.
  4. G. Martı́nez-Niconoff, “Interference between caustics of diffraction fields,” Opt. Lett. 23, 750–752 (1998).
    [CrossRef]
  5. G. Martı́nez-Niconoff, J. Carranza-Gallardo, A. Cornejo-Rodriguez, “Caustics of diffraction fields,” Opt. Commun. 114, 194–198 (1995).
    [CrossRef]
  6. V. I. Arnold, Singularities of Caustics and Wave Fronts (Kluwer Academic, Dordrecht, The Netherlands, 1990) p. 17.
  7. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259.
  8. N. Piskunov, Calculo Diferencial e Integral (Mir, Moscow, 1977), p. 216.
  9. R. Gilmore, Catastrophe Theory for Scientists and Engineers (Wiley, New York, 1981), p. 319.

1998 (1)

1995 (1)

G. Martı́nez-Niconoff, J. Carranza-Gallardo, A. Cornejo-Rodriguez, “Caustics of diffraction fields,” Opt. Commun. 114, 194–198 (1995).
[CrossRef]

1962 (1)

B. J. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 16 (1962).
[CrossRef]

Arnold, V. I.

V. I. Arnold, Singularities of Caustics and Wave Fronts (Kluwer Academic, Dordrecht, The Netherlands, 1990) p. 17.

Berry, M. V.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

Carranza-Gallardo, J.

G. Martı́nez-Niconoff, J. Carranza-Gallardo, A. Cornejo-Rodriguez, “Caustics of diffraction fields,” Opt. Commun. 114, 194–198 (1995).
[CrossRef]

Cornejo-Rodriguez, A.

G. Martı́nez-Niconoff, J. Carranza-Gallardo, A. Cornejo-Rodriguez, “Caustics of diffraction fields,” Opt. Commun. 114, 194–198 (1995).
[CrossRef]

Gaponov-Grekhov, A. V.

A. V. Gaponov-Grekhov, M. I. Rabinovich, Nonlinearities in Action (Springer-Verlag, Berlin, 1991), p. 25.

Gilmore, R.

R. Gilmore, Catastrophe Theory for Scientists and Engineers (Wiley, New York, 1981), p. 319.

Keller, B. J.

B. J. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 16 (1962).
[CrossRef]

Kratsov, Y. A.

Y. A. Kratsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), pp. 29–40.

Marti´nez-Niconoff, G.

G. Martı́nez-Niconoff, “Interference between caustics of diffraction fields,” Opt. Lett. 23, 750–752 (1998).
[CrossRef]

G. Martı́nez-Niconoff, J. Carranza-Gallardo, A. Cornejo-Rodriguez, “Caustics of diffraction fields,” Opt. Commun. 114, 194–198 (1995).
[CrossRef]

Orlov, Y. I.

Y. A. Kratsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), pp. 29–40.

Piskunov, N.

N. Piskunov, Calculo Diferencial e Integral (Mir, Moscow, 1977), p. 216.

Rabinovich, M. I.

A. V. Gaponov-Grekhov, M. I. Rabinovich, Nonlinearities in Action (Springer-Verlag, Berlin, 1991), p. 25.

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

J. Opt. Soc. Am. (1)

B. J. Keller, “Geometrical theory of diffraction,” J. Opt. Soc. Am. 52, 16 (1962).
[CrossRef]

Opt. Commun. (1)

G. Martı́nez-Niconoff, J. Carranza-Gallardo, A. Cornejo-Rodriguez, “Caustics of diffraction fields,” Opt. Commun. 114, 194–198 (1995).
[CrossRef]

Opt. Lett. (1)

Other (6)

A. V. Gaponov-Grekhov, M. I. Rabinovich, Nonlinearities in Action (Springer-Verlag, Berlin, 1991), p. 25.

Y. A. Kratsov, Y. I. Orlov, Geometrical Optics of Inhomogeneous Media (Springer-Verlag, Berlin, 1990), pp. 29–40.

V. I. Arnold, Singularities of Caustics and Wave Fronts (Kluwer Academic, Dordrecht, The Netherlands, 1990) p. 17.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1980), Vol. XVIII, p. 259.

N. Piskunov, Calculo Diferencial e Integral (Mir, Moscow, 1977), p. 216.

R. Gilmore, Catastrophe Theory for Scientists and Engineers (Wiley, New York, 1981), p. 319.

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Figures (3)

Fig. 1
Fig. 1

Geometric description of the projective curves with a common evolute.

Fig. 2
Fig. 2

Schematic spatial evolution of the interference fringes’ contour and description of the caustic region as an envelope of inflection points of the interference fringes.

Fig. 3
Fig. 3

(a) Diffraction field of a single curve-shaped slit. (b) Interference pattern between two diffraction fields with a common caustic region.

Equations (22)

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t(x, y)=δ[y-f(x)].
ϕ(x, y, z)= 1r δ[y-f(x)]exp(ikr)dydx,
rz+x2+y22z+a2+b22z-2xa+2ybz.
ϕ(x, y, z)=-δ[y-f(x)]×exp (ikz)expikx2+y22z+a2+b22z+xa+ybzdydx=- expikx2+f(x)22z+a2+b22z+xa+f(x)bzdx.
Lx=0,
2Lx2=0,
(x-a)+f(x)[f(x)+b]=0,
1+f(x)[f(x)+b]+f(x)2=0.
a=x-f(x)[1+f(x)2]f(x),
b=f(x)+1+f(x)2f(x).
t(x, y)=δ(y-x2/2).
φ(a, b, z)=- expikx2+x4/42z+a2+b22z+xa+bx2/2zdx;
φ(a, b, z)=- expiπλz x44+x21+b2+x2a+a2+b2dx,
Lx=iπλz x3+2x1+b2+2a=0,
2Lx2=iπλz 3x2+21+b2=0.
b=(3/2)a2/3+1.
ϕi,p(x, a)=Ai,p exp ikf(Ri,p, z)×[catastrophefunction(x, a)]=Ai,p exp ikf(Ri,p, z)×[catastrophe germ(x)+pert(x, a)],
I=2I0{1+Re exp ik[f(Ri, z)-f(Rp, z)]×[catastrophegerm(x)+pert(x, a)]}.
Δδ=k[f(Ri, z)-f(Rp, z)]×[catastrophe germ(x)+pert(x, a)].
R=αR0.
φ(x, a, z)=A exp ikf(Ri, z)(x4/4+ax2+bx)+A exp ikf(Rp, z)(x4/4+ax2+bx),
I(x, z, a, b)=2I{1+Re exp ik[f(Ri, z)-f(Rp, z)]×(x4/4+ax2+bx)}.

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