Abstract

Nonlinear-optical methods for input field phase-distortion suppression and corresponding improvement of spatial coherence properties are presented. Analytical results obtained for different nonlinear two-dimensional feedback systems are verified by direct numerical simulation. Our experiments have demonstrated evidence of phase-distortion suppression.

© 1995 Optical Society of America

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References

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  1. N. B. Abraham and W. J. Firth, "Overview of transverse effects in nonlinear optical systems," J. Opt. Soc. Am. B 7, 951–962 (1990).
    [CrossRef]
  2. J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1991).
  3. L. A. Lugiato, "Spatio-temporal structures. Part 1," Phys. Rep. 219, 293–310 (1992).
    [CrossRef]
  4. G. Grynberg, A. Maître, and A. Petrossian, "Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror," Phys. Rev. Lett. 72, 2379–2382 (1994).
    [CrossRef] [PubMed]
  5. W. J. Firth and M. A. Vorontsov, "Adaptive phase distortion suppression in nonlinear system with feedback mirror," J. Mod. Opt. 40, 1841–1846 (1993).
    [CrossRef]
  6. M. A. Vorontsov and I. P. Nikolaev, "Nonlinear 2-D feedback optical systems: new approaches for adaptive wavefront correction," in Atmospheric Propagation and Remote Sensing III, W. A. Flood and W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2222, 413–422 (1994).
    [CrossRef]
  7. A. D. Fisher and C. Warde, "Technique for real-time highresolution adaptive phase compensation," Opt. Lett. 8, 353–355 (1983).
    [CrossRef] [PubMed]
  8. M. A. Vorontsov and K. V. Shishakov, "Phase-distortion suppression in nonlinear cavities with gain," J. Opt. Soc. Am. B 9, 71–77 (1992).
    [CrossRef]
  9. G. Giusfredi, J. F. Valley, R. Pon, G. Khitrova, and H. M. Gibbs, "Optical instabilities in sodium vapor," J. Opt. Soc. Am. B 5, 1181–1191 (1988).
    [CrossRef]
  10. W. J. Firth, "Spatial instabilities in a Kerr medium with single feedback mirror," J. Mod. Opt. 37, 151–153 (1990).
    [CrossRef]
  11. G. D'Alessandro and W. J. Firth, "Hexagonal spatial patterns for a Kerr slice with a feedback mirror," Phys. Rev. A 46, 537–548 (1992).
    [CrossRef]
  12. M. A. Vorontsov and W. J. Firth, "Pattern formation and competition in nonlinear optical systems with twodimensional feedback," Phys. Rev. A 49, 2891–2903 (1994).
    [CrossRef] [PubMed]
  13. M. Tamburrini, M. Bonavita, S. Wabnitz, and E. Santamato, "Hexagonally patterned beam filamentation in a thin liquidcrystal film with a single feedback mirror," Opt. Lett. 18, 855–857 (1993).
    [CrossRef] [PubMed]
  14. For an input field with homogeneous intensity distribution the term R|A(r,0,t)|2 gives rise to an insignificant spatially uniform phase shift ū = R|A0|2, which can be removed by redefinition of u.
  15. M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhausen, "Rotatory instability of the spatial structure of light fields in nonlinear media with two-dimensional feedback," in Laser Optics of Condensed Matter: Proceedings of the 3rd Binational USA–USSR Symposium (Plenum, New York, 1988), pp. 507–517.
    [CrossRef]
  16. A. A. Vasiliev, D. Casasent, I. N. Kompanets, and A. V. Parfionov, Spatial Light Modulators (Radio i Sviaz, Moscow, 1987).
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  18. E. V. Degtiarev and M. A. Vorontsov, "Optical design kit of nonlinear spatial dynamics," in Self-Organization in Optical Systems and Applications in Information Technology, M. A. Vorontsov and W. B. Miller, eds. (Springer-Verlag, Berlin, 1995).
    [CrossRef]
  19. G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
    [CrossRef]
  20. R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).
  21. A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Mass., 1975), Vol. 2.
  22. W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, San Diego, Calif., 1992).
  23. M. A. Vorontsov, "Akhseals' as a new class of spatialtemporal instabilities of optical fields," Sov. J. Quantum Electron. 23, 269–271 (1993).
    [CrossRef]

1994 (2)

G. Grynberg, A. Maître, and A. Petrossian, "Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror," Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

M. A. Vorontsov and W. J. Firth, "Pattern formation and competition in nonlinear optical systems with twodimensional feedback," Phys. Rev. A 49, 2891–2903 (1994).
[CrossRef] [PubMed]

1993 (3)

M. A. Vorontsov, "Akhseals' as a new class of spatialtemporal instabilities of optical fields," Sov. J. Quantum Electron. 23, 269–271 (1993).
[CrossRef]

W. J. Firth and M. A. Vorontsov, "Adaptive phase distortion suppression in nonlinear system with feedback mirror," J. Mod. Opt. 40, 1841–1846 (1993).
[CrossRef]

M. Tamburrini, M. Bonavita, S. Wabnitz, and E. Santamato, "Hexagonally patterned beam filamentation in a thin liquidcrystal film with a single feedback mirror," Opt. Lett. 18, 855–857 (1993).
[CrossRef] [PubMed]

1992 (3)

L. A. Lugiato, "Spatio-temporal structures. Part 1," Phys. Rep. 219, 293–310 (1992).
[CrossRef]

G. D'Alessandro and W. J. Firth, "Hexagonal spatial patterns for a Kerr slice with a feedback mirror," Phys. Rev. A 46, 537–548 (1992).
[CrossRef]

M. A. Vorontsov and K. V. Shishakov, "Phase-distortion suppression in nonlinear cavities with gain," J. Opt. Soc. Am. B 9, 71–77 (1992).
[CrossRef]

1990 (2)

W. J. Firth, "Spatial instabilities in a Kerr medium with single feedback mirror," J. Mod. Opt. 37, 151–153 (1990).
[CrossRef]

N. B. Abraham and W. J. Firth, "Overview of transverse effects in nonlinear optical systems," J. Opt. Soc. Am. B 7, 951–962 (1990).
[CrossRef]

1988 (1)

1983 (1)

Abraham, N. B.

Ames, W. F.

W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, San Diego, Calif., 1992).

Bonavita, M.

Casasent, D.

A. A. Vasiliev, D. Casasent, I. N. Kompanets, and A. V. Parfionov, Spatial Light Modulators (Radio i Sviaz, Moscow, 1987).

D'Alessandro, G.

G. D'Alessandro and W. J. Firth, "Hexagonal spatial patterns for a Kerr slice with a feedback mirror," Phys. Rev. A 46, 537–548 (1992).
[CrossRef]

Degtiarev, E. V.

E. V. Degtiarev and M. A. Vorontsov, "Optical design kit of nonlinear spatial dynamics," in Self-Organization in Optical Systems and Applications in Information Technology, M. A. Vorontsov and W. B. Miller, eds. (Springer-Verlag, Berlin, 1995).
[CrossRef]

DeVelis, J. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

Firth, W. J.

M. A. Vorontsov and W. J. Firth, "Pattern formation and competition in nonlinear optical systems with twodimensional feedback," Phys. Rev. A 49, 2891–2903 (1994).
[CrossRef] [PubMed]

W. J. Firth and M. A. Vorontsov, "Adaptive phase distortion suppression in nonlinear system with feedback mirror," J. Mod. Opt. 40, 1841–1846 (1993).
[CrossRef]

G. D'Alessandro and W. J. Firth, "Hexagonal spatial patterns for a Kerr slice with a feedback mirror," Phys. Rev. A 46, 537–548 (1992).
[CrossRef]

N. B. Abraham and W. J. Firth, "Overview of transverse effects in nonlinear optical systems," J. Opt. Soc. Am. B 7, 951–962 (1990).
[CrossRef]

W. J. Firth, "Spatial instabilities in a Kerr medium with single feedback mirror," J. Mod. Opt. 37, 151–153 (1990).
[CrossRef]

Fisher, A. D.

Gibbs, H. M.

Giusfredi, G.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grynberg, G.

G. Grynberg, A. Maître, and A. Petrossian, "Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror," Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

Ivanov, V. Yu.

M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhausen, "Rotatory instability of the spatial structure of light fields in nonlinear media with two-dimensional feedback," in Laser Optics of Condensed Matter: Proceedings of the 3rd Binational USA–USSR Symposium (Plenum, New York, 1988), pp. 507–517.
[CrossRef]

Khitrova, G.

Kompanets, I. N.

A. A. Vasiliev, D. Casasent, I. N. Kompanets, and A. V. Parfionov, Spatial Light Modulators (Radio i Sviaz, Moscow, 1987).

Lugiato, L. A.

L. A. Lugiato, "Spatio-temporal structures. Part 1," Phys. Rep. 219, 293–310 (1992).
[CrossRef]

Maître, A.

G. Grynberg, A. Maître, and A. Petrossian, "Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror," Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

Moloney, J. V.

J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1991).

Monin, A. S.

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Mass., 1975), Vol. 2.

Newell, A. C.

J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1991).

Nikolaev, I. P.

M. A. Vorontsov and I. P. Nikolaev, "Nonlinear 2-D feedback optical systems: new approaches for adaptive wavefront correction," in Atmospheric Propagation and Remote Sensing III, W. A. Flood and W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2222, 413–422 (1994).
[CrossRef]

Parfionov, A. V.

A. A. Vasiliev, D. Casasent, I. N. Kompanets, and A. V. Parfionov, Spatial Light Modulators (Radio i Sviaz, Moscow, 1987).

Parrent, G. B.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

Petrossian, A.

G. Grynberg, A. Maître, and A. Petrossian, "Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror," Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

Pon, R.

Reynolds, G. O.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

Santamato, E.

Shishakov, K. V.

Shmalhausen, V. I.

M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhausen, "Rotatory instability of the spatial structure of light fields in nonlinear media with two-dimensional feedback," in Laser Optics of Condensed Matter: Proceedings of the 3rd Binational USA–USSR Symposium (Plenum, New York, 1988), pp. 507–517.
[CrossRef]

Tamburrini, M.

Thompson, B. J.

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).

Valley, J. F.

Vasiliev, A. A.

A. A. Vasiliev, D. Casasent, I. N. Kompanets, and A. V. Parfionov, Spatial Light Modulators (Radio i Sviaz, Moscow, 1987).

Vorontsov, M. A.

M. A. Vorontsov and W. J. Firth, "Pattern formation and competition in nonlinear optical systems with twodimensional feedback," Phys. Rev. A 49, 2891–2903 (1994).
[CrossRef] [PubMed]

W. J. Firth and M. A. Vorontsov, "Adaptive phase distortion suppression in nonlinear system with feedback mirror," J. Mod. Opt. 40, 1841–1846 (1993).
[CrossRef]

M. A. Vorontsov, "Akhseals' as a new class of spatialtemporal instabilities of optical fields," Sov. J. Quantum Electron. 23, 269–271 (1993).
[CrossRef]

M. A. Vorontsov and K. V. Shishakov, "Phase-distortion suppression in nonlinear cavities with gain," J. Opt. Soc. Am. B 9, 71–77 (1992).
[CrossRef]

M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhausen, "Rotatory instability of the spatial structure of light fields in nonlinear media with two-dimensional feedback," in Laser Optics of Condensed Matter: Proceedings of the 3rd Binational USA–USSR Symposium (Plenum, New York, 1988), pp. 507–517.
[CrossRef]

E. V. Degtiarev and M. A. Vorontsov, "Optical design kit of nonlinear spatial dynamics," in Self-Organization in Optical Systems and Applications in Information Technology, M. A. Vorontsov and W. B. Miller, eds. (Springer-Verlag, Berlin, 1995).
[CrossRef]

M. A. Vorontsov and I. P. Nikolaev, "Nonlinear 2-D feedback optical systems: new approaches for adaptive wavefront correction," in Atmospheric Propagation and Remote Sensing III, W. A. Flood and W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2222, 413–422 (1994).
[CrossRef]

Wabnitz, S.

Warde, C.

Yaglom, A. M.

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Mass., 1975), Vol. 2.

J. Mod. Opt. (2)

W. J. Firth, "Spatial instabilities in a Kerr medium with single feedback mirror," J. Mod. Opt. 37, 151–153 (1990).
[CrossRef]

W. J. Firth and M. A. Vorontsov, "Adaptive phase distortion suppression in nonlinear system with feedback mirror," J. Mod. Opt. 40, 1841–1846 (1993).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (2)

Phys. Rep. (1)

L. A. Lugiato, "Spatio-temporal structures. Part 1," Phys. Rep. 219, 293–310 (1992).
[CrossRef]

Phys. Rev. A (2)

G. D'Alessandro and W. J. Firth, "Hexagonal spatial patterns for a Kerr slice with a feedback mirror," Phys. Rev. A 46, 537–548 (1992).
[CrossRef]

M. A. Vorontsov and W. J. Firth, "Pattern formation and competition in nonlinear optical systems with twodimensional feedback," Phys. Rev. A 49, 2891–2903 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

G. Grynberg, A. Maître, and A. Petrossian, "Flowerlike patterns generated by a laser beam transmitted through a rubidium cell with single feedback mirror," Phys. Rev. Lett. 72, 2379–2382 (1994).
[CrossRef] [PubMed]

Sov. J. Quantum Electron. (1)

M. A. Vorontsov, "Akhseals' as a new class of spatialtemporal instabilities of optical fields," Sov. J. Quantum Electron. 23, 269–271 (1993).
[CrossRef]

Other (11)

J. V. Moloney and A. C. Newell, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1991).

M. A. Vorontsov and I. P. Nikolaev, "Nonlinear 2-D feedback optical systems: new approaches for adaptive wavefront correction," in Atmospheric Propagation and Remote Sensing III, W. A. Flood and W. B. Miller, eds., Proc. Soc. Photo-Opt. Instrum. Eng. 2222, 413–422 (1994).
[CrossRef]

For an input field with homogeneous intensity distribution the term R|A(r,0,t)|2 gives rise to an insignificant spatially uniform phase shift ū = R|A0|2, which can be removed by redefinition of u.

M. A. Vorontsov, V. Yu. Ivanov, and V. I. Shmalhausen, "Rotatory instability of the spatial structure of light fields in nonlinear media with two-dimensional feedback," in Laser Optics of Condensed Matter: Proceedings of the 3rd Binational USA–USSR Symposium (Plenum, New York, 1988), pp. 507–517.
[CrossRef]

A. A. Vasiliev, D. Casasent, I. N. Kompanets, and A. V. Parfionov, Spatial Light Modulators (Radio i Sviaz, Moscow, 1987).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

E. V. Degtiarev and M. A. Vorontsov, "Optical design kit of nonlinear spatial dynamics," in Self-Organization in Optical Systems and Applications in Information Technology, M. A. Vorontsov and W. B. Miller, eds. (Springer-Verlag, Berlin, 1995).
[CrossRef]

G. O. Reynolds, J. B. DeVelis, G. B. Parrent, Jr., and B. J. Thompson, The New Physical Optics Notebook: Tutorials in Fourier Optics (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1989).
[CrossRef]

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).

A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, Mass., 1975), Vol. 2.

W. F. Ames, Numerical Methods for Partial Differential Equations (Academic, San Diego, Calif., 1992).

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Figures (13)

Fig. 1
Fig. 1

Nonlinear-optical systems with 2-D feedback. (a) Kerr slice/feedback-mirror model. (b) Kerr slice-type system having a spatial Fourier filter in the feedback loop. (c) LCLV-based system.

Fig. 2
Fig. 2

Spectral coefficient of phase transformation for a system with (a), (b) a defocusing medium and (c) a self-focusing nonlinear medium. (a), (b) With no spatial filter, (c) system with a Fresnel filter (|K| = 1.2, d = D/σ = 0.2). The low-pass amplitude filter transfer function with cutoff frequency q0 (θ0 = σq02) is shown by the solid curves in (a).

Fig. 3
Fig. 3

LCLV with two feed back circuits.

Fig. 4
Fig. 4

Spectral coefficient of phase transformation for systems with one (dashed curve) and two (solid curve) feedback circuits (|K| = 2 and d = 0.2).

Fig. 5
Fig. 5

Dependence of the suppression coefficient S on the parameter |K| from direct numerical simulation (shown by dots) and from calculation with Eq. (38) for 2σφ2 = 0.13 and d = 0.2.

Fig. 6
Fig. 6

Normalized spatial correlation functions versus dimensionless coordinate r ˜ = rq0. (a) Incident field (dashed curve) and corrected field for different |K|. lin and lc are correlation lengths for input and corrected fields, respectively. (b) Correlation functions for the corrected field with one (dashed curve) and two (solid curve) feedback circuits (|K| = 2, d = 0.2).

Fig. 7
Fig. 7

Ratio of correlation lengths η as a function of |K| for a system with one (dashed curve) and two (solid curve) feedback circuits (d = 0.2).

Fig. 8
Fig. 8

Averaged spectral component amplitudes of the input 〈|Φ|〉 and residual 〈| Ψ ¯ |〉 phases versus the diffraction parameter θ =σq2 for (a), (b) uniform and (c) random modulated input wave intensity distribution. The dashed curves correspond to the linear approach [Eq. (20)], and the solid curves correspond to numerical simulation (|K| = 0.8, d = 0.2). (a) σφ2 = 0.13, (b) σφ2 = 0.53, (c) σφ2 = 0.13; intensity fluctuation variance σν2 = 0.04.

Fig. 9
Fig. 9

Spectral energy distribution for incident (dashed curve) and corrected (solid curve) waves (|K| = 1.2, σφ2 = 0.53).

Fig. 10
Fig. 10

(a) Incident φ(r) and (b) corrected ψ ¯(r) phases for a Gaussian beam (|K| = 4, d = 0.2, ka2/L = 0.1).

Fig. 11
Fig. 11

Log spectra of (1) incident and (2) corrected waves for an input beam with a Gaussian intensity distribution (parameters are the same as in Fig. 10). The dashed curves represent results pertaining to the undistorted phase φ(r) = 0

Fig. 12
Fig. 12

Experimental setup for observation of phase-distortion suppression: OFB, optical feedback.

Fig. 13
Fig. 13

Intensity distribution in the output plane for (a) closed and (b) open feedback.

Equations (50)

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A ( r , 0 , t ) = A 0 exp [ i u ( r , t ) + i φ ( r ) ] ,
τ u t + u = D 2 u + R [ A ( r , 0 , t ) 2 + A FB ( r , t ) 2 ] ,
- 2 i k A z = 2 A ,
A FB ( r , t ) = A ( r , L , t ) h ( r - r ) d 2 r ,
A FB ( q , t ) = F ( q ) A ( q , L , t ) ,
F ( q ) = P ( q ) exp [ i Q ( q ) ]
v ( r , t ) u ( r , t ) + φ ( r ) .
v ( r , t ) = v ¯ ( t ) + ψ ( r , t ) .
A ( r , 0 , t ) = A 0 e i v ¯ [ 1 + i ψ ( r , t ) ] .
- 2 i k A z = - q 2 A
A ( q , z , t ) = A ( q , 0 , t ) exp ( - i q 2 z / 2 k ) ,
A ( q , 0 , t ) = A 0 e i v ¯ [ δ ( q ) / ( 2 π ) 2 + i Ψ ( q , t ) ] ,
A FB ( q , t ) = F ( q ) A ( q , L , t ) = A 0 e i v ¯ [ δ ( q ) / ( 2 π ) 2 + i Ψ ( q , t ) ] P ( q ) × exp [ - i σ q 2 + i Q ( q ) ] ,
A FB ( r , t ) = A 0 e i v ¯ { F ( 0 ) + i e i qr P ( q ) × exp [ - i σ q 2 + i Q ( q ) ] Ψ ( q , t ) d 2 q } .
A FB ( r , t ) 2 = A 0 2 P ( 0 ) { P ( 0 ) + 2 e i qr P ( q ) × sin [ σ q 2 - Q ( q ) + Q ( 0 ) ] Ψ ( q , t ) d 2 q } .
u ( r , t ) = v ¯ ( t ) + ψ ( r , t ) - φ ( r ) .
v ¯ = R A 0 2 P 2 ( 0 ) ,
τ ψ ( r , t ) t + ( 1 - D 2 ) [ ψ ( r , t ) - φ ( r ) ] = 2 R A 0 2 P ( 0 ) e i qr P ( q ) sin [ σ q 2 - Q ( q ) + Q ( 0 ) ] × Ψ ( q , t ) d 2 q .
τ Ψ ˙ ( q , t ) + ( 1 + D q 2 ) [ Ψ ( q , t ) - Φ ( q ) ] = 2 K P 0 ( q ) sin [ σ q 2 - Q ( q ) + Q ( 0 ) ] Ψ ( q , t ) .
Ψ ( q , t ) = exp [ - λ ( q ) τ t ] Ψ ( q , 0 ) + ( 1 + D q 2 ) λ ( q ) Φ ( q ) ,
λ ( q ) = 1 + D q 2 - 2 K P 0 ( q ) sin [ σ q 2 - Q ( q ) + Q ( 0 ) ]
Ψ ¯ ( q ) = 1 + D q 2 λ ( q ) Φ ( q ) .
T ( q ) = Ψ ¯ ( q ) Φ ( q ) = 1 + D q 2 1 + D q 2 - 2 K P 0 ( q ) sin [ σ q 2 - Q ( q ) + Q ( 0 ) ] .
K sin [ σ q 2 - Q ( q ) + Q ( 0 ) ] < 0.
sin [ σ q 2 - Q ( q ) + Q ( 0 ) ] > 0
2 m π < σ q 2 - Q ( q ) + Q ( 0 ) < ( 2 m + 1 ) π ,             m = 0 , 1 , 2 .
F ( q ) = { 1 q q 0 0 q > q 0 .
Q ( q ) = { δ q = 0 0 q 0 ,
T ( q ) = 1 + D q 2 1 + D q 2 - 2 K P 0 ( q ) sin ( σ q 2 + δ ) .
T ( q ) = 1 + D q 2 1 + D q 2 - K P 0 ( q ) [ sin ( σ q 2 ) + sin ( α σ q 2 ) ] .
A in ( r ) = A 0 exp [ i φ ( r ) ] ,
Φ ( q ) Φ ( q ) = δ ( q - q ) G φ ( q ) ,
B φ ( r ) = 2 π 0 G φ ( q ) J 0 ( r q ) q d q ,
σ φ 2 = B φ ( 0 ) = 2 π 0 G φ ( q ) q d q .
Ψ ¯ ( q ) Ψ ¯ ( q ) = δ ( q - q ) G ψ ( q ) = T 2 ( q ) Φ ( q ) Φ ( q ) ,
G ψ ( q ) = T 2 ( q ) G φ ( q ) .
σ ψ 2 = 2 π 0 G ψ ( q ) q d q = 2 π 0 T 2 ( q ) G φ ( q ) q d q .
S = σ ψ 2 σ ψ 2 = 0 T 2 ( q ) G φ ( q ) q d q 0 G φ ( q ) q d q .
G φ ( q ) = { G 0 q q 0 0 q > q 0 .
S = q 0 2 2 0 q 0 ( 1 + D q 2 ) 2 [ 1 + D q 2 - 2 K sin ( σ q 2 ) ] 2 q d q .
A in ( r ) = A 0 [ 1 + i φ ( r ) ] .
B in ( r ) = A ˜ in ( r 1 ) A ˜ in * ( r 2 ) ,
B in ( r ) = A 0 2 B φ ( r ) = 2 π A 0 2 0 G φ ( q ) J 0 ( r q ) q d q .
B c ( r ) = A 0 2 B ψ ( r ) = 2 π A 0 2 0 G ψ ( q ) J 0 ( r q ) q d q ,
l = 1 B ( 0 ) 0 B ( r ) d r .
η = 1 S 0 T 2 ( q ) G φ ( q ) d q / 0 G φ ( q ) d q ,
I 0 ( r ) = I 0 + ν ( r ) ,
E ( q ) = 0 q 0 2 π A ( q ) 2 q d q d ϕ ,
τ u t + u = D 2 u + R A FB ( r , t ) 2 .
A in ( r ) = A 0 exp [ - x 2 + y 2 2 a 2 + i φ ( x , y ) ] ,

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