Abstract

Scattering from an initially plane surface of a thin Plexiglas plate damaged by an impinging coherent light beam of a laser is considered. The morphology and the characteristic dimensions of the damaged speckles was studied during the evolution of the phenomenon by interferometry and the creation of diffraction patterns. The cusped interferogram created by superposition of evoluting two initial diffraction patterns, created during successive damage steps, forms caustics corresponding to hyperbolic umbilic catastrophes. An experimental study of the evolution of the damage phenomenon clarifies the mechanisms of elastic and plastic deformations of the affected zone.

© 1997 Optical Society of America

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References

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  1. J. P. Toennis, “Scattering of molecular beams from surfaces,” Appl. Phys. 3, 91–114 (1974).
    [CrossRef]
  2. U Garifalde, A. G. Levi, R Spadacini, G. E. Tommei, “Correlation functions in atom scattering from surfaces,” Surf. Sci. 38, 269–274 (1973); “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).
  3. J. N. Smith, D. R. O’Keefe, H. Salgburg, R. L. Palimer, “Preferential scattering of Ar from LiF: correlation with lattice properties,” J. Chem. Phys. 50, 4667–4671 (1969).
    [CrossRef]
  4. M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics5th ed. (Pergamon, London, 1975).
  6. J. C. Maxwell, “On hills and dales,” Philos. Mag. 40, 421–427 (1870).
  7. R. Thom, Stabilité structurelle et Morphogénèse (Benjamin, New York, 1972).
  8. H. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. GPO, Wash., D.C., 1964).
  9. G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Proc. Cambridge Philos. Soc. 6, 379–402 (1838).
  10. K. W. Ford, J. A. Wheeler, “Application of semiclassical scattering analysis,” Ann. Phys. N.Y. 7, 287–322 (1959).
    [CrossRef]
  11. T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).
  12. P. S. Theocaris, J. Michopoulos, “Generalization of the theory of far-field caustics by the catastrophy theory,” Appl. Opt. 21, 1080–1092 (1982).
    [CrossRef] [PubMed]
  13. P. S. Theocaris, “Francisco Maurolyco, a precursor of Newton and Kepler. Four hundred years from the date of his death,” Proc. Natl. Acad. Athens 53, 110–127 (1978).

1982 (1)

1978 (1)

P. S. Theocaris, “Francisco Maurolyco, a precursor of Newton and Kepler. Four hundred years from the date of his death,” Proc. Natl. Acad. Athens 53, 110–127 (1978).

1975 (1)

M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[CrossRef]

1974 (1)

J. P. Toennis, “Scattering of molecular beams from surfaces,” Appl. Phys. 3, 91–114 (1974).
[CrossRef]

1973 (1)

U Garifalde, A. G. Levi, R Spadacini, G. E. Tommei, “Correlation functions in atom scattering from surfaces,” Surf. Sci. 38, 269–274 (1973); “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).

1969 (1)

J. N. Smith, D. R. O’Keefe, H. Salgburg, R. L. Palimer, “Preferential scattering of Ar from LiF: correlation with lattice properties,” J. Chem. Phys. 50, 4667–4671 (1969).
[CrossRef]

1959 (1)

K. W. Ford, J. A. Wheeler, “Application of semiclassical scattering analysis,” Ann. Phys. N.Y. 7, 287–322 (1959).
[CrossRef]

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

1870 (1)

J. C. Maxwell, “On hills and dales,” Philos. Mag. 40, 421–427 (1870).

1838 (1)

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Proc. Cambridge Philos. Soc. 6, 379–402 (1838).

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Proc. Cambridge Philos. Soc. 6, 379–402 (1838).

Berry, M. V.

M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics5th ed. (Pergamon, London, 1975).

Ford, K. W.

K. W. Ford, J. A. Wheeler, “Application of semiclassical scattering analysis,” Ann. Phys. N.Y. 7, 287–322 (1959).
[CrossRef]

Garifalde, U

U Garifalde, A. G. Levi, R Spadacini, G. E. Tommei, “Correlation functions in atom scattering from surfaces,” Surf. Sci. 38, 269–274 (1973); “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).

Levi, A. G.

U Garifalde, A. G. Levi, R Spadacini, G. E. Tommei, “Correlation functions in atom scattering from surfaces,” Surf. Sci. 38, 269–274 (1973); “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).

Maxwell, J. C.

J. C. Maxwell, “On hills and dales,” Philos. Mag. 40, 421–427 (1870).

Michopoulos, J.

O’Keefe, D. R.

J. N. Smith, D. R. O’Keefe, H. Salgburg, R. L. Palimer, “Preferential scattering of Ar from LiF: correlation with lattice properties,” J. Chem. Phys. 50, 4667–4671 (1969).
[CrossRef]

Palimer, R. L.

J. N. Smith, D. R. O’Keefe, H. Salgburg, R. L. Palimer, “Preferential scattering of Ar from LiF: correlation with lattice properties,” J. Chem. Phys. 50, 4667–4671 (1969).
[CrossRef]

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

Salgburg, H.

J. N. Smith, D. R. O’Keefe, H. Salgburg, R. L. Palimer, “Preferential scattering of Ar from LiF: correlation with lattice properties,” J. Chem. Phys. 50, 4667–4671 (1969).
[CrossRef]

Smith, J. N.

J. N. Smith, D. R. O’Keefe, H. Salgburg, R. L. Palimer, “Preferential scattering of Ar from LiF: correlation with lattice properties,” J. Chem. Phys. 50, 4667–4671 (1969).
[CrossRef]

Spadacini, R

U Garifalde, A. G. Levi, R Spadacini, G. E. Tommei, “Correlation functions in atom scattering from surfaces,” Surf. Sci. 38, 269–274 (1973); “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).

Theocaris, P. S.

P. S. Theocaris, J. Michopoulos, “Generalization of the theory of far-field caustics by the catastrophy theory,” Appl. Opt. 21, 1080–1092 (1982).
[CrossRef] [PubMed]

P. S. Theocaris, “Francisco Maurolyco, a precursor of Newton and Kepler. Four hundred years from the date of his death,” Proc. Natl. Acad. Athens 53, 110–127 (1978).

Thom, R.

R. Thom, Stabilité structurelle et Morphogénèse (Benjamin, New York, 1972).

Toennis, J. P.

J. P. Toennis, “Scattering of molecular beams from surfaces,” Appl. Phys. 3, 91–114 (1974).
[CrossRef]

Tommei, G. E.

U Garifalde, A. G. Levi, R Spadacini, G. E. Tommei, “Correlation functions in atom scattering from surfaces,” Surf. Sci. 38, 269–274 (1973); “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).

Wheeler, J. A.

K. W. Ford, J. A. Wheeler, “Application of semiclassical scattering analysis,” Ann. Phys. N.Y. 7, 287–322 (1959).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics5th ed. (Pergamon, London, 1975).

Ann. Phys. N.Y. (1)

K. W. Ford, J. A. Wheeler, “Application of semiclassical scattering analysis,” Ann. Phys. N.Y. 7, 287–322 (1959).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

J. P. Toennis, “Scattering of molecular beams from surfaces,” Appl. Phys. 3, 91–114 (1974).
[CrossRef]

J. Chem. Phys. (1)

J. N. Smith, D. R. O’Keefe, H. Salgburg, R. L. Palimer, “Preferential scattering of Ar from LiF: correlation with lattice properties,” J. Chem. Phys. 50, 4667–4671 (1969).
[CrossRef]

J. Phys. A (1)

M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[CrossRef]

Philos. Mag. (2)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Philos. Mag. 37, 311–317 (1946).

J. C. Maxwell, “On hills and dales,” Philos. Mag. 40, 421–427 (1870).

Proc. Cambridge Philos. Soc. (1)

G. B. Airy, “On the intensity of light in the neighbourhood of a caustic,” Proc. Cambridge Philos. Soc. 6, 379–402 (1838).

Proc. Natl. Acad. Athens (1)

P. S. Theocaris, “Francisco Maurolyco, a precursor of Newton and Kepler. Four hundred years from the date of his death,” Proc. Natl. Acad. Athens 53, 110–127 (1978).

Surf. Sci. (1)

U Garifalde, A. G. Levi, R Spadacini, G. E. Tommei, “Correlation functions in atom scattering from surfaces,” Surf. Sci. 38, 269–274 (1973); “Quantum theory of atom-surface scattering: diffraction and rainbow,” Surf. Sci. 48, 649–675 (1975).

Other (3)

M. Born, E. Wolf, Principles of Optics5th ed. (Pergamon, London, 1975).

R. Thom, Stabilité structurelle et Morphogénèse (Benjamin, New York, 1972).

H. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (U.S. GPO, Wash., D.C., 1964).

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Figures (9)

Fig. 1
Fig. 1

Schematic of the hyperbolic umbilic caustic surfaces.

Fig. 2
Fig. 2

Caustic curves derived by the intersection of the caustic surfaces by a plane normal to the axis of a light bundle.

Fig. 3
Fig. 3

Diffraction pattern at the vicinity of a cusp of the internal caustic C2 where the classical rainbow line is developed.

Fig. 4
Fig. 4

(a) Interferogram of the reflected rays from the speckle created by a laser beam and the pair of caustics formed by the diffraction patterns of the interferograms. (b) Detail of the inner cuspoid caustic C2.

Fig. 5
Fig. 5

Same patterns as in Fig. 4 but under different angles of incidence of the light bundles on the reference screen.

Fig. 6
Fig. 6

Impinging light flux from a laser bundle and the damage morphology of the surface at (a) the initial step of formation of the speckle and (b) after development of the crater.

Fig. 7
Fig. 7

Interferograms at (a) the initial phase of creation of the protrusion and their respective shapes and (b) a subsequent phase of the phenomenon of creation of the speckle when a crater starts to develop.

Fig. 8
Fig. 8

Talysurf tracings of the thickness variation of the speckle hill by successive traverses: (a) initial step and (b) step with a crater.

Fig. 9
Fig. 9

Radial profile of the speckle.

Equations (20)

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λ 0 = 2 π k 0 = h ( 2 m E 0 ) 1 / 2 ,
I ( k 0 , ω 0 ; k , ω ) = δ ( ω - ω 0 ) G S G 2 δ ( k - k 0 - G ) .
S G = 1 A cell d R exp { - i [ G · R + ( k 0 n + k n ) h ( R ) ] } .
N = m A E 0 2 π h 2
G = - ( k 0 + k z ) h ( R i ) .
S G = 2 π A ( k 0 z + k z ) i γ i × exp { - i [ G · R ] + ( k 0 z + k z ) h ( R i ) } { K ( R i ) } 1 / 2 ,
S G clas 2 = 4 π 2 A 2 ( k 0 z + k z ) 2 i 1 K ( R i ) ,
K ( R ) = 2 h x 2 2 h y 2 - ( 2 h x y ) 2 = 0
G x = ± 2 π h p a ( k 0 + k z ) ,             G y = ± 2 π h p b ( k 0 + k z ) ,
h p = ɛ h p ( R ) = h 0 cos ( 2 π x a ) cos ( 2 π y b ) ,
Δ c 16 π 2 ɛ 2 h 0 2 a λ 0 ,             Δ x 16 π 2 ɛ h 0 a λ 0 ,             Δ y 16 π 2 ɛ h 0 b λ 0 .
S G R = 2 π π exp [ 1 2 i ( Φ 1 + Φ 2 - 3 2 π ) ] A ( k 0 z + k z ) × ( [ 1 K 1 1 / 2 + 1 ( - K 2 ) 1 / 2 ] [ 3 ( Φ 2 - Φ 1 ) 4 ] 1 / 6 × A i [ - 3 4 ( Φ 2 - Φ 1 ) 2 / 3 ] - i [ 1 K 1 1 / 2 - 1 ( - K 2 ) 1 / 2 ] × [ 4 3 ( Φ 2 - Φ 1 ) ] 1 / 6 A i { - [ 3 4 ( Φ 2 - Φ 1 ) ] 2 / 3 } ) ,
Φ - [ G · R + ( k 0 z + k z ) h ( R ) ] .
C ( x , y ) - exp [ i ( t 4 8 - x t 2 2 + y t ) ] d t ,
I 2 = I 0 λ 2 | - A ( r ) exp ( - i k r sin θ ) d r | 2 ,
A ( r ) = exp ( [ - ( r 2 / 2 a 2 ) ]             for r > r 0 , A ( r ) = exp { [ - ( r 2 / 2 a 2 ) ] - 2 i k r 0 }             for r < r 0 ,
- r 0 r 0 exp { [ - ( r 2 / 2 a 2 ) ] - i k r sin θ } d r
I ( θ ) = E 0 sin 2 k d [ sin ( k r 0 sin θ ) k r 0 sin θ ] 2
I ( r ) = I a ( r ) + I d ( r ) ,
I d ( r ) = I 0 [ 2 J 1 ( 2 π r 0 r λ f ) ( 2 π r 0 r λ f ) ] 2 ,

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