Abstract

The appearance of interference blots and of the dislocation of interference fringes is studied by means of modeling an arbitrary inhomogeneous birefringent medium with rotating principal optical axes by a system of two birefringent plates. A theoretical explanation of these irregularities is given. Conditions for the appearance of interference blots and fringe dislocations are derived. A method of classification of singular points in the fringe patterns is presented.

© 1998 Optical Society of America

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  1. J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.
  2. H. Aben, J. Josepson, “Nonlinear optical phenomena in integrated photoelasticity,” in Proceedings of the International Symposium on Advanced Technology in Experimental Mechanics (The Japanese Society of Mechanical Engineers, Tokyo, 1995), pp. 49–54.
  3. H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
    [CrossRef]
  4. H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.
  5. H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).
  6. H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).
  7. H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414–1421 (1986).
    [CrossRef]
  8. J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
    [CrossRef]
  9. N. N. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, B. Ya. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. 73, 525–528 (1983).
    [CrossRef]
  10. P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
    [CrossRef]
  11. V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).
  12. F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
    [CrossRef] [PubMed]
  13. I. Freund, “Optical vortices in Gaussian random wave fields: statistical probability densities,” J. Opt. Soc. Am. A 11, 1644–1652 (1994).
    [CrossRef]
  14. M. Tlidi, P. Mandel, “Spatial patterns in nascent optical bistability,” in Chaos, Solitons & Fractals (Pergamon, London, 1994), Vol. 4, pp. 1475–1486.
  15. A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3:Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
    [CrossRef]
  16. A. Poincaré, Théorie mathématique de la lumière, II (Georges Carré, Paris, 1892).
  17. D. Hull, D. J. Bacon, Introduction to Dislocations (Butterworth-Heinemann, Oxford, UK, 1984).
  18. D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, New York, 1987).

1997 (1)

1996 (1)

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3:Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

1994 (1)

1991 (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

1990 (1)

V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

1989 (1)

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

1986 (1)

1983 (1)

1974 (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Aben, H.

H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
[CrossRef]

H. Aben, “Characteristic directions in optics of twisted birefringent media,” J. Opt. Soc. Am. A 3, 1414–1421 (1986).
[CrossRef]

H. Aben, J. Josepson, “Nonlinear optical phenomena in integrated photoelasticity,” in Proceedings of the International Symposium on Advanced Technology in Experimental Mechanics (The Japanese Society of Mechanical Engineers, Tokyo, 1995), pp. 49–54.

H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).

Anton, J.

H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.

Arecchi, F. T.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

Bacon, D. J.

D. Hull, D. J. Bacon, Introduction to Dislocations (Butterworth-Heinemann, Oxford, UK, 1984).

Baranova, N. N.

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Berry, M. V.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Coullet, P.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Freund, I.

Giacomelli, G.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

Gil, L.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Guillemet, C.

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).

Hull, D.

D. Hull, D. J. Bacon, Introduction to Dislocations (Butterworth-Heinemann, Oxford, UK, 1984).

Ilyenkov, A. V.

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3:Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

Jordan, D. W.

D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, New York, 1987).

Josepson, J.

H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous birefringent objects,” Appl. Opt. 36, 7172–7179 (1997).
[CrossRef]

H. Aben, J. Josepson, “Nonlinear optical phenomena in integrated photoelasticity,” in Proceedings of the International Symposium on Advanced Technology in Experimental Mechanics (The Japanese Society of Mechanical Engineers, Tokyo, 1995), pp. 49–54.

H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.

J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.

Khiznyak, A. I.

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3:Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

Kreminskaya, L. V.

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3:Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

Mamaev, A. V.

Mandel, P.

M. Tlidi, P. Mandel, “Spatial patterns in nascent optical bistability,” in Chaos, Solitons & Fractals (Pergamon, London, 1994), Vol. 4, pp. 1475–1486.

Nye, J. F.

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Pilipetsky, N. F.

Poincaré, A.

A. Poincaré, Théorie mathématique de la lumière, II (Georges Carré, Paris, 1892).

Ramazza, P. L.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

Residori, S.

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

Rocca, F.

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Shkunov, V. V.

Smith, P.

D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, New York, 1987).

Soskin, M. S.

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3:Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Tlidi, M.

M. Tlidi, P. Mandel, “Spatial patterns in nascent optical bistability,” in Chaos, Solitons & Fractals (Pergamon, London, 1994), Vol. 4, pp. 1475–1486.

Vasnetsov, M. V.

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3:Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Zel’dovich, B. Ya.

Appl. Opt. (1)

Appl. Phys. B: Lasers Opt. (1)

A. V. Ilyenkov, A. I. Khiznyak, L. V. Kreminskaya, M. S. Soskin, M. V. Vasnetsov, “Birth and evolution of wave-front dislocations in a laser beam passed through a photorefractive LiNbO3:Fe crystal,” Appl. Phys. B: Lasers Opt. 62, 465–471 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

JETP Lett. (1)

V. Yu. Bazhenov, M. V. Vasnetsov, M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Opt. Commun. (1)

P. Coullet, L. Gil, F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Phys. Rev. Lett. (1)

F. T. Arecchi, G. Giacomelli, P. L. Ramazza, S. Residori, “Vortices and defect statistics in two-dimensional optical chaos,” Phys. Rev. Lett. 67, 3749–3752 (1991).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

J. F. Nye, M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. London, Ser. A 336, 165–190 (1974).
[CrossRef]

Other (9)

J. Josepson, “Curious optical phenomena in integrated photoelasticity,” in Recent Advances in Experimental Mechanics, J. F. Silva Gomes, F. B. Branco, F. Martins de Brito, J. Gil Saraiva, M. Luerdes Eusébio, J. Sousa Cirne, A. Correia de Cruz, eds. (A. A. Balkema, Rotterdam, The Netherlands, 1994), Vol. 1, pp. 91–94.

H. Aben, J. Josepson, “Nonlinear optical phenomena in integrated photoelasticity,” in Proceedings of the International Symposium on Advanced Technology in Experimental Mechanics (The Japanese Society of Mechanical Engineers, Tokyo, 1995), pp. 49–54.

H. Aben, J. Anton, J. Josepson, “Ambiguity of the fringe order in integrated photoelasticity,” in Proceedings of the Third International Workshop on Automatic Processing of Fringe Patterns (Bremen, Germany 1997), W. Jüptner, W. Osten, eds. (Akademie Verlag, Berlin, 1997), pp. 309–317.

H. Aben, Integrated Photoelasticity (McGraw-Hill, New York, 1979).

H. Aben, C. Guillemet, Photoelasticity of Glass (Springer-Verlag, Berlin, 1993).

A. Poincaré, Théorie mathématique de la lumière, II (Georges Carré, Paris, 1892).

D. Hull, D. J. Bacon, Introduction to Dislocations (Butterworth-Heinemann, Oxford, UK, 1984).

D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations (Clarendon, New York, 1987).

M. Tlidi, P. Mandel, “Spatial patterns in nascent optical bistability,” in Chaos, Solitons & Fractals (Pergamon, London, 1994), Vol. 4, pp. 1475–1486.

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Figures (20)

Fig. 1
Fig. 1

Computer-generated fringe pattern for an optical system containing two birefringent plates: β=x+y, Δ1=x-2y, and Δ2=2x+y.

Fig. 2
Fig. 2

Computer-generated fringe pattern: β=x+y, Δ1=x-2y, and Δ2=30x+y.

Fig. 3
Fig. 3

Interference blots in the computer-generated fringe pattern: β=30x+y, Δ1=x-2y, and Δ2=2x+y.

Fig. 4
Fig. 4

Interference blots in the computer-generated fringe pattern: β=y, Δ1=14x, and Δ2=15x.

Fig. 5
Fig. 5

Interference blots in the form of stripes in the computer-generated fringe pattern: β=y, Δ1=16x, and Δ2=16x+π.

Fig. 6
Fig. 6

(a) Burgers circuit ABCDE around a fringe dislocation, (b) Burgers circuit in a perfect fringe pattern.

Fig. 7
Fig. 7

Fringe beat in an optical system containing two birefringent plates: β=π/4, Δ1=x, and Δ2=30x.

Fig. 8
Fig. 8

Envelope for light intensity: β=y, Δ1=x, Δ2=30x, and I0=1.

Fig. 9
Fig. 9

Fringe dislocation at a saddle point: β=1.5x+y-0.75π, Δ1=2x+2y-π, Δ2=30x, and I=-1.

Fig. 10
Fig. 10

Fringe dislocation at a center: β=y, Δ1=x, Δ2=30x, and I=+1.

Fig. 11
Fig. 11

Fringe dislocation at a focus: β=0.5x+0.5y-π/8, Δ1=x-y+π/4, Δ2=30x, and I=+1.

Fig. 12
Fig. 12

Fringe dislocation at a node: β=0.5x-y+0.25π, Δ1=-3x+4y+π, Δ2=30x, and I=+1.

Fig. 13
Fig. 13

Fringe dislocation for a degenerate case q=0: β=0.5x-0.5y+π/8, Δ1=x-y+π/4, and Δ2=30x.

Fig. 14
Fig. 14

Argand diagram representing the function 21(0.1 cos θ, 0.1 sin θ)-iγ1(0.1 cos θ, 0.1 sin θ) around the saddle point in Fig. 9.

Fig. 15
Fig. 15

Argand diagram representing the function 21(0.1 cos θ, 0.1 sin θ)-iγ1(0.1 cos θ, 0.1 sin θ) around the center in Fig. 10.

Fig. 16
Fig. 16

Argand diagram representing the function 21(0.1 cos θ, 0.1 sin θ)-iγ1(0.1 cos θ, 0.1 sin θ) around the focus in Fig. 11.

Fig. 17
Fig. 17

Argand diagram representing the function 21(0.1 cos θ, 0.1 sin θ)-iγ1(0.1 cos θ, 0.1 sin θ) around the node in Fig. 12.

Fig. 18
Fig. 18

Fringe dislocation at the singular point with index I=+2: β=π/4+0.5(x-π/2)(y-π/4), Δ1=π/2+0.25(x-π/2)2-(y-π/4)2, and Δ2=30x.

Fig. 19
Fig. 19

Argand diagram representing the function 21(0.1+0.002θ, θ)-iγ1(0.1+0.02θ, θ) around the singular point with index I=+2 in Fig. 18.

Fig. 20
Fig. 20

Interference blot in the computer-generated integrated fringe pattern for the case of the Boussinesq problem, I=+2.

Equations (60)

Equations on this page are rendered with MathJax. Learn more.

E*=UE0.
U=exp(iξ)cos θexp(iζ)sin θ-exp(-iζ)sin θexp(-iξ)cos θ,
U=S(α*)G(γ)S(-α0),
S(α)=cos α-sin αsin αcos α,
G(γ)=exp(iγ)00exp(-iγ).
Ex=G(γ)E0,
E*=S(-α*)E*,E0=S(-α0)E0.
cos Δ*=cos 2ξ cos2 θ+cos 2ζ sin2 θ,
0Δ*π.
U=G(Δ2)S(β)G(Δ1),
Δ1=ξ-ζ,Δ2=ξ+ζ,β=-θ.
cos Δ*=cos(Δ1+Δ2)cos2 β+cos(Δ2-Δ1)sin2 β.
I1=12I0(1-cos Δ*)
I2=12I0(1+cos Δ*).
β(x, y)=π/2+η(x, y),
cos β=-sin η-η,
sin β=cos η1-η2/2.
cos Δ*cos(Δ2-Δ1)-2η2 cos Δ1 cos Δ2.
Δ2-Δ1=±2nπ+μ(x, y),
cos Δ*1-μ2/2-2η2 cos2 Δ1.
I1I0(14μ2+η2 cos2 Δ1).
Δ2-Δ1=(1±2n)π+μ,
I1I0(1-14μ2-η2 cos Δ1).
β(x, y)=π/2,Δ2(x, y)=Δ1(x, y)±2nπ
β(x, y)=π/2,Δ2(x, y)=Δ1(x, y)+(1±2n)π,
Δ2Δ1±2nπ
Δ2Δ1+(1±2nπ),
cos Δ*=cos Δ1 cos Δ2-cos 2β sin Δ1 sin Δ2.
Δ1(x, y)=±2nπ+(x, y),
Δ2(x, y)=(1±2m)π+ν(x, y),
cos Δ*-1+22+ν22+ν cos 2β.
I1=I01-24-ν24-12ν cos 2β.
Δ1(x, y)=±n2π,Δ2(x, y)=1±2mπ,
Δ1(x, y)=±2nπ,Δ2(x, y)=±2mπ,
cos Δ*=A sin(Δ2+ϕ).
A=(cos2 2β sin2 Δ1+cos2 Δ1)1/2,
tan ϕ=-cos Δ1cos 2β sin Δ1.
I1c=12I0(1-A).
cos 2β(x, y)=0,cos Δ1(x, y)=0,
tan ϕ=0/0.
2β(x, y)=π/2±nπ,Δ1(x, y)=π/2±mπ.
β(x0, y0)=π/4,Δ2(x0, y0)=π/2.
β(x, y)=π/4+(x, y),Δ1(x, y)=π/2+γ(x, y),
cos 2β=-sin 2-2,
sin Δ1=cos γ1-γ22,
cos Δ1=-sin γ-γ.
A(x, y)={4[(x, y)]2+[γ(x, y)]2}1/2,
tan ϕ(x, y)=-γ(x, y)2(x, y).
x˙=2(x, y),y˙=-γ(x, y).
dydx=-γ(x, y)2(x, y).
γ(x, y)=0,(x, y)=0
[ϕ]=2πI,
(x, y)=x0(x-x0)+y0(y-y0),
γ(x, y)=γx0(x-x0)+γy0(y-y0),
x˙=2x0(x-x0)+2y0(y-y0),
y˙=-γx0(x-x0)-γy0(y-y0).
p=2x0-γy0,q=-2x0γy0+2y0γx0,
d=p2-4q.
x=x0+ρ cos ψ,y=y0+ρ sin ψ.
(x, y)=1(ρ, ψ),γ(x, y)=γ1(ρ, ψ).

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