Abstract

Concerning the problem of transmitting a laser beam from one telescope to another telescope through a turbulent medium, it is established that using an adaptive optical system on both telescopes to precompensate an outgoing laser beam based on the aberrations measured on the received laser beam leads to an iteration that maximizes the transmission (neglecting attenuation losses) of laser power between the telescopes. Simulation results are presented demonstrating the effectiveness of this technique when the telescopes are equipped with either phase-only or full-wave compensation systems. Simulation results are shown that indicate that for a uniform distribution of the strength of turbulence, 95% transmission of laser power is attained when both telescopes can achieve full-wave compensation provided that the aperture diameter D of the two telescopes is greater than twice the Fresnel length λL, where λ is the wavelength of propagation and L is the distance between the two telescopes.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. M. Levine, E. A. Martinsen, A. Wirth, A. Jankevics, M. Toledo-Quinones, F. Landers, “Horizontal line-of-sight turbulence over near-ground paths and implications for adaptive optics corrections in laser communications,” Appl. Opt. 37, 4553–4560 (1998).
    [CrossRef]
  2. M. C. Roggemann, D. J. Lee, “A two deformable mirror concept for correcting scintillation effects in laser beam pro-jection through the turbulent atmosphere,” Appl. Opt. 37, 4577–4585 (1998).
    [CrossRef]
  3. J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
    [CrossRef]
  4. J. D. Barchers, “Evaluation of the impact of finite resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A 18, 3098–3109 (2001).
    [CrossRef]
  5. J. D. Barchers, “Application of the parallel generalized projection algorithm to the control of two finite resolution deformable mirrors for scintillation compensation,” J. Opt. Soc. Am. A 19, 54–63 (2002).
    [CrossRef]
  6. J. D. Barchers, B. L. Ellerbroek, “Increase in the compensated field of view in strong scintillation by use of two deformable mirrors,” in Beyond Conventional Adaptive Optics, R. Ragazonni, ed. (Astronomical Observatory and Department of Astronomy, Padova, Italy, 2001). Available online at http://www.adaopt.it/venice2001/proceedings/pdf/barchers_pap.pdf , 2001.
  7. J. D. Barchers, “Closed loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations,” J. Opt. Soc. Am. A 19, 926–945 (2002).
    [CrossRef]
  8. R. H. Dicke, “Phase-contrast detection of telescope seeing error and their correction,” Astrophys. J. 198, 605–615 (1975).
    [CrossRef]
  9. J. M. Beckers, “Detailed compensation of atmospheric seeing using multiconjugate adaptive optics,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
    [CrossRef]
  10. B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [CrossRef]
  11. D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
    [CrossRef]
  12. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).
  13. H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).
  14. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  15. A convex set C is one in which, given any two points within said convex set, every point on the line connecting the two points is also contained in the set, i.e., ∀x1,x2∈C,αx1+(1-α)x2∈C,∀α∈[0,1].As an example, the set describing the unit ball in a Hilbert space, C={x∈H| |x|⩽1}, is a convex set, whereas the set describing the boundary of the unit ball in a Hilbert space, C={x∈H| |x|=1}, is nonconvex.
  16. J. Von Neumann, Functional Operators, Vol. II of Annals of Mathematics Studies (Princeton U. Press, Princeton, N.J., 1950.
  17. L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
    [CrossRef]
  18. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [CrossRef]
  19. T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. 34, 3883–3895 (1995).
    [CrossRef] [PubMed]
  20. D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).
  21. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, Berlin, 1994).
  22. T. R. O’Meara, “The multi-dither principle in adaptive optics,” J. Opt. Soc. Am. 67, 306–315 (1977).
    [CrossRef]
  23. M. A. Vorontsov, G. Carhart, J. W. Gowens, J. C. Ricklin, “Adaptive correction of wavefront phase distortions for beam position stabilization and improved focusing in a duplex laser communication link,” U.S. patent pending.
  24. M. A. Vorontsov, V. P. Sivokon, “Stochastic parallel gradient descent technique for high-resolution wavefront phase distortion correction,” J. Opt. Soc. Am. A 15, 2745–2758 (1998).
    [CrossRef]
  25. J. D. Barchers, “Convergence rates for iterative vector space projection methods for control of two deformable mirrors for compensation of both amplitude and phase fluctuations,” Appl. Opt. 41, 2213–2218 (2002).
    [CrossRef] [PubMed]

2002 (3)

2001 (2)

1998 (4)

1995 (1)

1994 (2)

1984 (1)

1982 (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

1977 (1)

1975 (1)

R. H. Dicke, “Phase-contrast detection of telescope seeing error and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).

1967 (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Barchers, J. D.

Beckers, J. M.

J. M. Beckers, “Detailed compensation of atmospheric seeing using multiconjugate adaptive optics,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
[CrossRef]

Carhart, G.

M. A. Vorontsov, G. Carhart, J. W. Gowens, J. C. Ricklin, “Adaptive correction of wavefront phase distortions for beam position stabilization and improved focusing in a duplex laser communication link,” U.S. patent pending.

Dicke, R. H.

R. H. Dicke, “Phase-contrast detection of telescope seeing error and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

Ellerbroek, B. L.

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
[CrossRef]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

J. D. Barchers, B. L. Ellerbroek, “Increase in the compensated field of view in strong scintillation by use of two deformable mirrors,” in Beyond Conventional Adaptive Optics, R. Ragazonni, ed. (Astronomical Observatory and Department of Astronomy, Padova, Italy, 2001). Available online at http://www.adaopt.it/venice2001/proceedings/pdf/barchers_pap.pdf , 2001.

Fried, D. L.

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).

Gowens, J. W.

M. A. Vorontsov, G. Carhart, J. W. Gowens, J. C. Ricklin, “Adaptive correction of wavefront phase distortions for beam position stabilization and improved focusing in a duplex laser communication link,” U.S. patent pending.

Gubin, L. G.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Jankevics, A.

Johnston, D. C.

Kotzer, T.

Landers, F.

Lee, D. J.

Levi, A.

Levine, B. M.

Martinsen, E. A.

O’Meara, T. R.

Polyak, B. T.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Raik, E. V.

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Ricklin, J. C.

M. A. Vorontsov, G. Carhart, J. W. Gowens, J. C. Ricklin, “Adaptive correction of wavefront phase distortions for beam position stabilization and improved focusing in a duplex laser communication link,” U.S. patent pending.

Roggemann, M. C.

Rosen, J.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, Berlin, 1994).

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).

Shamir, J.

Sivokon, V. P.

Stark, H.

Toledo-Quinones, M.

Von Neumann, J.

J. Von Neumann, Functional Operators, Vol. II of Annals of Mathematics Studies (Princeton U. Press, Princeton, N.J., 1950.

Vorontsov, M. A.

M. A. Vorontsov, V. P. Sivokon, “Stochastic parallel gradient descent technique for high-resolution wavefront phase distortion correction,” J. Opt. Soc. Am. A 15, 2745–2758 (1998).
[CrossRef]

M. A. Vorontsov, G. Carhart, J. W. Gowens, J. C. Ricklin, “Adaptive correction of wavefront phase distortions for beam position stabilization and improved focusing in a duplex laser communication link,” U.S. patent pending.

Webb, H.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Welsh, B. M.

Wirth, A.

Yang, Y.

H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).

Youla, D. C.

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Appl. Opt. (4)

Astrophys. J. (1)

R. H. Dicke, “Phase-contrast detection of telescope seeing error and their correction,” Astrophys. J. 198, 605–615 (1975).
[CrossRef]

Atmos. Oceanic Opt. (1)

D. L. Fried, “Scaling laws for propagation through turbulence,” Atmos. Oceanic Opt. 11, 982–990 (1998).

IEEE Trans. Med. Imaging (1)

D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,” IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

M. A. Vorontsov, V. P. Sivokon, “Stochastic parallel gradient descent technique for high-resolution wavefront phase distortion correction,” J. Opt. Soc. Am. A 15, 2745–2758 (1998).
[CrossRef]

J. D. Barchers, “Closed loop stable control of two deformable mirrors for compensation of amplitude and phase fluctuations,” J. Opt. Soc. Am. A 19, 926–945 (2002).
[CrossRef]

A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
[CrossRef]

J. D. Barchers, B. L. Ellerbroek, “Improved compensation of turbulence induced amplitude and phase distortions by means of multiple near field phase adjustments,” J. Opt. Soc. Am. A 18, 399–411 (2001).
[CrossRef]

J. D. Barchers, “Evaluation of the impact of finite resolution effects on scintillation compensation using two deformable mirrors,” J. Opt. Soc. Am. A 18, 3098–3109 (2001).
[CrossRef]

J. D. Barchers, “Application of the parallel generalized projection algorithm to the control of two finite resolution deformable mirrors for scintillation compensation,” J. Opt. Soc. Am. A 19, 54–63 (2002).
[CrossRef]

B. L. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

D. C. Johnston, B. M. Welsh, “Analysis of multiconjugate adaptive optics,” J. Opt. Soc. Am. A 11, 394–408 (1994).
[CrossRef]

Optik (Stuttgart) (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane images,” Optik (Stuttgart) 35, 225–246 (1972).

USSR Comput. Math. Math. Phys. (1)

L. G. Gubin, B. T. Polyak, E. V. Raik, “The method of projections for finding the common point in convex sets,” USSR Comput. Math. Math. Phys. 7, 1–24 (1967).
[CrossRef]

Other (7)

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence (Springer-Verlag, Berlin, 1994).

A convex set C is one in which, given any two points within said convex set, every point on the line connecting the two points is also contained in the set, i.e., ∀x1,x2∈C,αx1+(1-α)x2∈C,∀α∈[0,1].As an example, the set describing the unit ball in a Hilbert space, C={x∈H| |x|⩽1}, is a convex set, whereas the set describing the boundary of the unit ball in a Hilbert space, C={x∈H| |x|=1}, is nonconvex.

J. Von Neumann, Functional Operators, Vol. II of Annals of Mathematics Studies (Princeton U. Press, Princeton, N.J., 1950.

J. M. Beckers, “Detailed compensation of atmospheric seeing using multiconjugate adaptive optics,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 215–217 (1989).
[CrossRef]

H. Stark, Y. Yang, Vector Space Projections (Wiley, New York, 1998).

J. D. Barchers, B. L. Ellerbroek, “Increase in the compensated field of view in strong scintillation by use of two deformable mirrors,” in Beyond Conventional Adaptive Optics, R. Ragazonni, ed. (Astronomical Observatory and Department of Astronomy, Padova, Italy, 2001). Available online at http://www.adaopt.it/venice2001/proceedings/pdf/barchers_pap.pdf , 2001.

M. A. Vorontsov, G. Carhart, J. W. Gowens, J. C. Ricklin, “Adaptive correction of wavefront phase distortions for beam position stabilization and improved focusing in a duplex laser communication link,” U.S. patent pending.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Configuration of the optimal beam propagation system (OBPS). An adaptive optical (AO) system on each telescope is used to compensate the received beam from the other telescope, simultaneously precompensating an outgoing beam.

Fig. 2
Fig. 2

Example of results for (a) the Strehl ratio and (b) the normalized power in the bucket as a function of D/λL for a Rytov number of 0.7. The solid curves are, from top to bottom, the OBPS with the adaptive optical system on the [transmit, receive] telescopes equal to [phase and amplitude, phase and amplitude], [phase and amplitude, phase only], and [phase only, phase only], respectively. The dashed curves utilized a collimated source propagated from the receive telescope for wave-front sensing and do not have an adaptive optical system on the receive telescope. The transmit telescope had an adaptive optical system capable of amplitude and phase compensation (top curve) and phase-only compensation (bottom curve). The dotted–dashed curves utilized a Gaussian point source for wave-front sensing, and again the receive telescope is not equipped with an adaptive optical system. The transmit telescope had an adaptive optical system capable of amplitude and phase compensation (top curve) and phase-only compensation (bottom curve).

Fig. 3
Fig. 3

Strehl ratio shown for the OBPS as a function of D/λL for values of the Rytov parameter of 0.1 (top curve) and 0.7 (bottom curve). Intermediate values of the Rytov parameter were evaluated and found to fall between these curves. The adaptive optical system in the [transmit, receive] telescopes is (a) [amplitude and phase, amplitude and phase], (b) [amplitude and phase, phase only], and (c) [phase only, phase only]. MCAO refers to amplitude and phase compensation, and PO refers to phase-only compensation.

Fig. 4
Fig. 4

Strehl ratio (solid curves) and normalized power in the aperture (dashed curves) shown as a function of D/λL for values of the Rytov parameter of 0.1 (upper curves) and 0.7 (lower curves). Intermediate values of the Rytov parameter were evaluated and found to fall between these lines. A collimated source is propagated from the receive telescope. The transmit telescope is equipped with an adaptive optical system capable of (a) amplitude and phase and (b) phase-only compensation.

Fig. 5
Fig. 5

The advantage, in terms of minimum required aperture diameter, of using an adaptive optical system on both the transmit and receive telescopes is emphasized by plotting the ratio D/λL as a function of the Strehl ratio for (right to left) systems with [transmit, receive] telescopes equipped with [amplitude and phase, amplitude and phase], [amplitude and phase, phase only], [phase only, phase only], [amplitude and phase, none], and [phase only, none]. For the case when the receive telescope does not have an adaptive optical system, a collimated source was propagated from the receive telescope for wave-front sensing.

Equations (53)

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

PCjx=arg minPCjxCjPCjx-x.
J(xk)=PC1xk-xk+PC2xk-xk.
U(r¯2)=dr¯1GA(r¯1, r¯2)U(r¯1).
U(r¯2)=TA[U(r¯1), r¯2].
V(r¯1)=dr¯2GA*(r¯1, r¯2)V(r¯2).
dr¯2GA*(r¯1, r¯2)GA(r¯3, r¯2)=δ(r¯1-r¯3),
U(r¯)-V(r¯)=dr¯|U(r¯)-V(r¯)|21/2.
C1U(r¯1)L2|U(r¯1)=0|r¯1|>R1, and dr¯1|U(r¯1)|2=I1,
M1(r¯1)=1|r¯1|R1 and0otherwise.
U1(r¯1)=M1(r¯1)V(r¯1)I1dr¯1|M1(r¯1)V(r¯1)|21/2.
U(r¯1)-V(r¯1)2=dr¯1|U(r¯1)-V(r¯1)|2
=dr¯1M1(r¯1)|U(r¯1)-V(r¯1)|2+dr¯1[1-M1(r¯1)]|V(r¯1)|2
=dr¯1M1(r¯1)|U(r¯1)|2+dr¯1M1(r¯1)|V(r¯1)|2+dr¯1[1-M1(r¯1)]|V(r¯1)|2-2 Re dr¯1M1(r¯1)U(r¯1)V*(r¯1).
dr¯1M1(r¯1)U(r¯1)V*(r¯1)=η11/2I11/2IV1/2,
η1=dr¯1M1(r¯1)U(r¯1)V*(r¯1)2I1IV,
IV=dr¯1|V(r¯1)|2.
dr¯1M1(r¯1)U(r¯1)V*(r¯1)dr¯1|U(r¯1)|21/2×dr¯1M1(r¯1)|V(r¯1)|21/2.
PC1[V(r¯1)]=M1(r¯1)V(r¯1)I1dr¯1|M1(r¯1)V(r¯1)|21/2.
PC1[V(r¯1)]-V(r¯1)2=dr¯1[1-M1(r¯1)]|V(r¯1)|2+dr¯1M1(r¯1)|V(r¯1)|2×1-I1dr¯1M1(r¯1)|V(r¯1)|21/2.
PC1[V(r¯1)]-V(r¯1)2=IV(1-KV)+KVIV1-I1KVIV1/2,
C˜2V˜(r¯2)L2|V˜(r¯2)=0|r¯2|>R2, and dr¯2|V˜(r¯2)|2=I2.
M2(r¯2)=1|r¯2|R2 and0otherwise.
V˜2(r¯2)=M2(r¯2)U˜(r¯2)I2dr¯2|M2(r¯2)U˜(r¯2)|21/2.
PC˜2[U˜(r¯2)]=M2(r¯2)U˜(r¯2)I2dr¯2|M2(r¯2)U˜(r¯2)|21/2.
V(r¯1)=dr¯2GA*(r¯1, r¯2)V˜(r¯2).
C2V(r¯1)L2|TA[V(r¯1),r¯2]=0|r¯2|>R2,dr¯2|TA[V(r¯1),r¯2]|2=I2.
U(r¯1)-V(r¯1)2=dr¯1|U(r¯1)-V(r¯1)|2.
U˜(r¯2)-V˜(r¯2)2=dr¯2dr¯1GA(r¯1, r¯2)[U(r¯1)-V(r¯1)]2
=dr¯2dr¯1dr¯1GA(r¯1, r¯2)GA*(r¯1, r¯2)×[U(r¯1)-V(r¯1)][U*(r¯1)-V*(r¯1)].
U˜(r¯2)-V˜(r¯2)2=dr¯1dr¯1δ(r¯1-r¯1)[U(r¯1)-V(r¯1)]×[U*(r¯1)-V*(r¯1)]
=dr¯1|U(r¯1)-V(r¯1)|2.
PC2[U(r¯1)]=TA*(PC˜2{TA[U(r¯1),r¯2]},r¯1).
Uk+1(r¯1)=PC1{PC2[Uk(r¯1)]}.
PC2[Uk+1(r¯1)]-Uk+1(r¯1)
PC1{PC2[Uk(r¯1)]}-PC2[Uk(r¯1)]
PC2[Uk(r¯1)]-Uk(r¯1).
C1,POU(r¯1)L2|U(r¯1)|=M1(r¯1)×I1dr¯1M1(r¯1)1/2.
U1(r¯1)=exp{i arg[V(r¯1)]}M1(r¯1)I1dr¯1M1(r¯1)1/2.
PC1,PO[V(r¯1)]=exp{i arg[V(r¯1)]}M1(r¯1)×I1dr¯1M1(r¯1)1/2.
PC1,PO[V(r¯1)]-V(r¯1)2=dr¯1|V(r¯1)|2+I1M1(r¯1)2 dr¯1M1(r¯1)-2 I11/2M1(r¯1) dr¯1|V(r¯1)|M1(r¯1).
η1=dr¯1PC1,PO[V(r¯1)]V*(r¯1)2PC1,PO[V(r¯1)]V(r¯1)
=dr¯1|V(r¯1)|M1(r¯1)2V(r¯1)M1(r¯1).
PC1,PO[V(r¯1)]-V(r¯1)2=dr¯1|V(r¯1)|2+I1M1(r¯1)2 dr¯1M1(r¯1)-2I11/2V(r¯1)η11/2.
C2,POV(r¯1)L2|TA[V(r¯1),r¯2]|=M2(r¯2)I2dr¯2M2(r¯2)1/2.
PC2,PO[U(r¯1)]=TA*exp(i arg{TA[U(r¯1),r¯2]})×M2(r¯2)I2dr¯2M2(r¯2)1/2,r¯1.
PC2,PO[Uk+1(r¯1)]-Uk+1(r¯1)
PC1,PO{PC2,PO[Uk(r¯1)]}-PC2,PO[Uk(r¯1)]
PC2,PO[Uk(r¯1)]-Uk(r¯1).
Snone=dr¯2M2(r¯2)TA[U(r¯1),r¯2]2dr¯1U*(r¯1)U(r¯1)dr¯2M2(r¯2).
SPO=dr¯2M2(r¯2)|TA[U(r¯1),r¯2]|2dr¯1U*(r¯1)U(r¯1)dr¯2M2(r¯2).
SAP=dr¯2M2(r¯2)|TA[U(r¯1),r¯2]|22dr¯1U*(r¯1)U(r¯1)dr¯2M2(r¯2)|TA[U(r¯1),r¯2]|2.
PIB=dr¯2M2(r¯2)|TA[U(r¯1),r¯2]|2dr¯1U*(r¯1)U(r¯1).
x1,x2C,αx1+(1-α)x2C,α[0,1].

Metrics