Abstract

The feasibility of noninterferometric methods to measure phase distribution in a laser beam cross section for visualization of the vortex dislocations of an optical speckle-field wave front is analyzed. Peculiarities of the phase retrieved from the measured intensity distribution (the phase problem in optics) and from the wave-front slopes measured by a Hartmann sensor are discussed. A concept of the vortex and the potential parts of the phase is introduced. An analytic formula to retrieve the potential phase from the measured intensity has been obtained. We show that the considered means of measurements allow the positions of the dislocation centers to be sensed and the spatial configuration of the intensity zero lines to be reconstructed.

© 1998 Optical Society of America

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References

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  1. M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.
  2. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  3. B. D. Bobrov, “Screw dislocations of laser speckle fields on interferograms with circular line structure,” Quantum Electron. 18, 886–890 (1991).
  4. I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  5. M. A. Vorontsov, A. V. Koryabin, V. I. Shmal’gauzen, Controlled Optical System (Nauka, Moscow, 1988), Chaps. 2 and 3.
  6. V. V. Voitsekhovich, “Hartmann test in atmospheric research,” J. Opt. Soc. Am. A 13, 1749–1757 (1996).
    [CrossRef]
  7. M. R. Teaque, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  8. G. A. Korn, T. M. Korn, Mathematical Handbook (McGraw-Hill, New York, 1961), Chap. 5.
  9. E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
    [CrossRef]
  10. V. P. Aksenov and Yu. N. Isaev, “Analytical representation of the phase and its mode components reconstructed according to the wave front slopes,” Opt. Lett. 17, 1180–1182 (1992).
    [CrossRef]
  11. A. N. Bogaturov, “Solution of the system of equations of the wave front zone reconstruction in the adaptive optics,” Izv. Vysh. Uchebn. Zaved. Fiz. 28, 86–95 (1985).
  12. V. P. Lukin, B. V. Fortes, “Influence of wave front dislocations on phase conjugation unstability at thermal compensation,” Atmos. Ocean Opt. 8, 435–447 (1995).
  13. M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
    [CrossRef] [PubMed]
  14. Yu. A. Anan’ev, Optical Resonators and Problem of Laser Radiation Divergence (Nauka, Moscow, 1979), Chap. 2.
  15. N. N. Bautin, E. A. Leontovich, Methods and Procedures of Qualitative Study of Dynamical System at Plane (Nauka, Moscow, 1990), Chap. 1.
  16. V. A. Zhuravlev, I. K. Kobozev, Yu. A. Kravtsov, “Statistical characteristics for dislocations of the wave field phase front,” Zh. Eksp. Teor. Fiz. 102, 483–494 (1992).

1996 (1)

1995 (1)

V. P. Lukin, B. V. Fortes, “Influence of wave front dislocations on phase conjugation unstability at thermal compensation,” Atmos. Ocean Opt. 8, 435–447 (1995).

1993 (1)

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

1992 (3)

1991 (2)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

B. D. Bobrov, “Screw dislocations of laser speckle fields on interferograms with circular line structure,” Quantum Electron. 18, 886–890 (1991).

1989 (1)

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

1985 (1)

A. N. Bogaturov, “Solution of the system of equations of the wave front zone reconstruction in the adaptive optics,” Izv. Vysh. Uchebn. Zaved. Fiz. 28, 86–95 (1985).

1983 (1)

Abramochkin, E. G.

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Aksenov and Yu. N. Isaev, V. P.

Anan’ev, Yu. A.

Yu. A. Anan’ev, Optical Resonators and Problem of Laser Radiation Divergence (Nauka, Moscow, 1979), Chap. 2.

Basistiy, I. V.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Battipede, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Bautin, N. N.

N. N. Bautin, E. A. Leontovich, Methods and Procedures of Qualitative Study of Dynamical System at Plane (Nauka, Moscow, 1990), Chap. 1.

Bazhenov, V. Yu.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Berry, M.

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.

Bobrov, B. D.

B. D. Bobrov, “Screw dislocations of laser speckle fields on interferograms with circular line structure,” Quantum Electron. 18, 886–890 (1991).

Bogaturov, A. N.

A. N. Bogaturov, “Solution of the system of equations of the wave front zone reconstruction in the adaptive optics,” Izv. Vysh. Uchebn. Zaved. Fiz. 28, 86–95 (1985).

Brambilla, M.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Fortes, B. V.

V. P. Lukin, B. V. Fortes, “Influence of wave front dislocations on phase conjugation unstability at thermal compensation,” Atmos. Ocean Opt. 8, 435–447 (1995).

Fried, D. L.

Kobozev, I. K.

V. A. Zhuravlev, I. K. Kobozev, Yu. A. Kravtsov, “Statistical characteristics for dislocations of the wave field phase front,” Zh. Eksp. Teor. Fiz. 102, 483–494 (1992).

Korn, G. A.

G. A. Korn, T. M. Korn, Mathematical Handbook (McGraw-Hill, New York, 1961), Chap. 5.

Korn, T. M.

G. A. Korn, T. M. Korn, Mathematical Handbook (McGraw-Hill, New York, 1961), Chap. 5.

Koryabin, A. V.

M. A. Vorontsov, A. V. Koryabin, V. I. Shmal’gauzen, Controlled Optical System (Nauka, Moscow, 1988), Chaps. 2 and 3.

Kravtsov, Yu. A.

V. A. Zhuravlev, I. K. Kobozev, Yu. A. Kravtsov, “Statistical characteristics for dislocations of the wave field phase front,” Zh. Eksp. Teor. Fiz. 102, 483–494 (1992).

Leontovich, E. A.

N. N. Bautin, E. A. Leontovich, Methods and Procedures of Qualitative Study of Dynamical System at Plane (Nauka, Moscow, 1990), Chap. 1.

Lugiato, L. A.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Lukin, V. P.

V. P. Lukin, B. V. Fortes, “Influence of wave front dislocations on phase conjugation unstability at thermal compensation,” Atmos. Ocean Opt. 8, 435–447 (1995).

Penna, V.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Prati, F.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Shmal’gauzen, V. I.

M. A. Vorontsov, A. V. Koryabin, V. I. Shmal’gauzen, Controlled Optical System (Nauka, Moscow, 1988), Chaps. 2 and 3.

Soskin, M. S.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Tamm, C.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Teaque, M. R.

Vasnetsov, M. V.

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Vaughn, J. L.

Voitsekhovich, V. V.

Volostnikov, V. G.

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Vorontsov, M. A.

M. A. Vorontsov, A. V. Koryabin, V. I. Shmal’gauzen, Controlled Optical System (Nauka, Moscow, 1988), Chaps. 2 and 3.

Weiss, C. O.

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Zhuravlev, V. A.

V. A. Zhuravlev, I. K. Kobozev, Yu. A. Kravtsov, “Statistical characteristics for dislocations of the wave field phase front,” Zh. Eksp. Teor. Fiz. 102, 483–494 (1992).

Appl. Opt. (1)

Atmos. Ocean Opt. (1)

V. P. Lukin, B. V. Fortes, “Influence of wave front dislocations on phase conjugation unstability at thermal compensation,” Atmos. Ocean Opt. 8, 435–447 (1995).

Izv. Vysh. Uchebn. Zaved. Fiz. (1)

A. N. Bogaturov, “Solution of the system of equations of the wave front zone reconstruction in the adaptive optics,” Izv. Vysh. Uchebn. Zaved. Fiz. 28, 86–95 (1985).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

I. V. Basistiy, V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Optics of light beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

M. Brambilla, F. Battipede, L. A. Lugiato, V. Penna, F. Prati, C. Tamm, C. O. Weiss, “Transverse laser patterns. I. Phase singularity crystal,” Phys. Rev. A 43, 5090–5113 (1991).
[CrossRef] [PubMed]

Quantum Electron. (1)

B. D. Bobrov, “Screw dislocations of laser speckle fields on interferograms with circular line structure,” Quantum Electron. 18, 886–890 (1991).

Zh. Eksp. Teor. Fiz. (1)

V. A. Zhuravlev, I. K. Kobozev, Yu. A. Kravtsov, “Statistical characteristics for dislocations of the wave field phase front,” Zh. Eksp. Teor. Fiz. 102, 483–494 (1992).

Other (5)

Yu. A. Anan’ev, Optical Resonators and Problem of Laser Radiation Divergence (Nauka, Moscow, 1979), Chap. 2.

N. N. Bautin, E. A. Leontovich, Methods and Procedures of Qualitative Study of Dynamical System at Plane (Nauka, Moscow, 1990), Chap. 1.

M. A. Vorontsov, A. V. Koryabin, V. I. Shmal’gauzen, Controlled Optical System (Nauka, Moscow, 1988), Chaps. 2 and 3.

M. Berry, “Singularities in waves and rays,” in Physics of Defects, R. Balian, M. Kleman, J.-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.

G. A. Korn, T. M. Korn, Mathematical Handbook (McGraw-Hill, New York, 1961), Chap. 5.

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Figures (3)

Fig. 1
Fig. 1

(a) Intensity distribution and (b) phase distribution at the beam cross section.

Fig. 2
Fig. 2

Energy streamlines for the (a) potential, (b) vortex, (c) whole fields of the Umov–Poynting vector.

Fig. 3
Fig. 3

(a) Potential and (b) divergent phases.

Equations (19)

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2 ik   U z + Δ U + k 2 ˜ ρ ,   z U ρ ,   z = 0
2 kI 2 S z + I 2 S 2 = k 2 I 2 ˜ ρ ,   z + 1 2   I Δ I ρ ,   z - 1 4 I ρ ,   z 2 ,
I ρ ,   z S = - k   L z ,
I ρ ,   z S = I ρ ,   z S p + I ρ ,   z S v ,
p = grad   φ ,
Δ φ = - k   I ρ ,   z z ;
p ρ ,   z = - k 2 π - d ξ d η   z   I ξ ,   η ,   z × l x - ξ + m y - η x - ξ 2 + y - η 2 .
d y d z = - 1 2 π z   d ξ d η   I ξ ,   η ,   z y - η x - ξ 2 + y - η 2 ,
d x d z = - 1 2 π z   d ξ d η I ξ ,   η ,   z x - ξ x - ξ 2 + y - η 2 ,
d y d x = z   d ξ d η I ξ ,   η ,   z y - η x - ξ 2 + y - η 2 z   d ξ d η I ξ ,   η ,   z x - ξ x - ξ 2 + y - η 2 .
S p ρ ,   z = k 4 π 2 D     d ρ 0 I ρ 0 ,   z z × - d ρ 0 I ρ 0 ,   z ρ 0 - ρ 0 ρ - ρ 0 ρ 0 - ρ 0 2 ρ - ρ 0 2 .
1 2 π Γ S Γ ξ ,   η ,   z ξ - x 2 + η - y 2 ξ - x d η - η - y d ξ .
Δ S x ,   y = x   μ x ,   y + y   v x ,   y ,
S x ,   y = S x ,   y + S s x ,   y .
S ρ ,   z = D x   μ r ,   z + y   v r ,   z G r ,   ρ d r , G r ,   ρ = 1 4 π ln R 2 + r 2 ρ 2 / R 2 - 2 r ρ   cos φ - φ r 2 + ρ 2 - 2 r ρ   cos φ - φ , r r   cos   φ ,   r   sin   φ ,     ρ   ρ   cos   φ ,   ρ   sin   φ .
U r ,   φ = m ,   n   g mn ± A mn ± r ,   φ , A mn ± r ,   φ = 4 r 2 m ! π m + 1 ! 1 / 2 L m n 2 r 2 exp - r 2 ± in φ ,
A 10 = 2 π 1 - 2 r 2 exp - r 2 , A 021 = 1 π   2 r 2 exp - r 2 + 2 i φ , A 022 = 1 π   2 r 2 exp - r 2 - 2 i φ , U r ,   φ ,   0 = A 10 r ,   φ g 1 + A 021 r ,   φ g 2 + A 022 r ,   φ g 3 .
U x ,   y ,   z = Ω 1 + Ω 2 - 3 / 2 × exp 3 i   arctan   Ω + Ω 2 x 2 + y 2 1 + Ω 2 i - Ω g , g = 3 i + 2 Ω + i Ω 2 1 - 2 x 2 + y 2 g 1 + i Ω 2 2 g 2 x - iy 2 + g 3 x + iy 2 ,
x ,   y = I x ,   y S x ,   y = I x ,   y S x ,   y + I x ,   y S s x ,   y ,

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