Abstract

Scattering prevents light from being focused in turbid media. The effect of scattering can be negated through wavefront shaping techniques when a localized form of feedback is available. Even in the absence of such a localized reporter, wavefront shaping can blindly form a single diffraction-limited focus when the feedback response is nonlinear. We developed and experimentally validated a model that accurately describes the statistics of this blind focusing process. We show that maximizing the nonlinear feedback signal only results in the formation of a focus when a limited number of reporters are contributing to the signal. Using our model, we can calculate the minimal requirements for the number of controlled spatial light modulator segments and the order of nonlinearity to blindly focus light through strongly scattering media.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express 23, 12189–12206 (2015).
    [Crossref] [PubMed]
  2. R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
    [Crossref] [PubMed]
  3. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
    [Crossref]
  4. H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
    [Crossref] [PubMed]
  5. J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112, 9236–9241 (2015).
    [Crossref] [PubMed]
  6. Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21, 085001 (2016).
    [Crossref]
  7. J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. U. S. A. 109, 8434–8439 (2012).
    [Crossref] [PubMed]
  8. D. Sinefeld, H. P. Paudel, D. G. Ouzounov, T. G. Bifano, and C. Xu, “Adaptive optics in multiphoton microscopy: comparison of two, three and four photon fluorescence,” Opt. Express 23, 31472–31483 (2015).
    [Crossref] [PubMed]
  9. D. E. Milkie, E. Betzig, and N. Ji, “Pupil-segmentation-based adaptive optical microscopy with full-pupil illumination,” Opt. Lett. 36, 4206–4208 (2011).
    [Crossref] [PubMed]
  10. O. Katz, E. Small, Y. Guan, and Y. Silberberg, “Noninvasive nonlinear focusing and imaging through strongly scattering turbid layers,” Optica 1, 170–174 (2014).
    [Crossref]
  11. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
    [Crossref]
  12. J. Aulbach, B. Gjonaj, P. Johnson, and A. Lagendijk, “Spatiotemporal focusing in opaque scattering media by wave front shaping with nonlinear feedback,” Opt. Express 20, 29237–29251 (2012).
    [Crossref]
  13. D. B. Conkey, A. N. Brown, A. M. Caravaca-Aguirre, and R. Piestun, “Genetic algorithm optimization for focusing through turbid media in noisy environments,” Opt. Express 20, 4840–4849 (2012).
    [Crossref] [PubMed]
  14. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32, 2309–2311 (2007).
    [Crossref] [PubMed]
  15. J. Bosch, S. A. Goorden, and A. P. Mosk, “Frequency width of open channels in multiple scattering media,” Opt. Express 24, 26472–26478 (2016).
    [Crossref] [PubMed]
  16. I. N. Papadopoulos, J.-S. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
    [Crossref]
  17. J. W. Goodman, Speckle phenomena in optics (Roberts & Company, 2007), chap. 3.
  18. E. G. van Putten, “Disorder-enhanced imaging with spatially controlled light,” Ph.D. thesis, University of Twente (2011).
  19. S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
    [Crossref]
  20. R. F. H. Fisher, Precoding and signal shaping for digital transmission (John Wiley & Sons, Inc., 2002), chap. Appendix A.
    [Crossref]
  21. I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
    [Crossref] [PubMed]

2017 (3)

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

I. N. Papadopoulos, J.-S. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

2016 (2)

J. Bosch, S. A. Goorden, and A. P. Mosk, “Frequency width of open channels in multiple scattering media,” Opt. Express 24, 26472–26478 (2016).
[Crossref] [PubMed]

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21, 085001 (2016).
[Crossref]

2015 (4)

D. Sinefeld, H. P. Paudel, D. G. Ouzounov, T. G. Bifano, and C. Xu, “Adaptive optics in multiphoton microscopy: comparison of two, three and four photon fluorescence,” Opt. Express 23, 31472–31483 (2015).
[Crossref] [PubMed]

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112, 9236–9241 (2015).
[Crossref] [PubMed]

I. M. Vellekoop, “Feedback-based wavefront shaping,” Opt. Express 23, 12189–12206 (2015).
[Crossref] [PubMed]

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[Crossref] [PubMed]

2014 (1)

2012 (3)

2011 (3)

D. E. Milkie, E. Betzig, and N. Ji, “Pupil-segmentation-based adaptive optical microscopy with full-pupil illumination,” Opt. Lett. 36, 4206–4208 (2011).
[Crossref] [PubMed]

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[Crossref]

2008 (1)

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[Crossref] [PubMed]

2007 (1)

Aulbach, J.

Betzig, E.

Bifano, T. G.

Bosch, J.

Brake, J.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

Bromberg, Y.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[Crossref]

Brown, A. N.

Caravaca-Aguirre, A. M.

Cižmár, T.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

Conkey, D. B.

Cui, M.

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112, 9236–9241 (2015).
[Crossref] [PubMed]

J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. U. S. A. 109, 8434–8439 (2012).
[Crossref] [PubMed]

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

Fisher, R. F. H.

R. F. H. Fisher, Precoding and signal shaping for digital transmission (John Wiley & Sons, Inc., 2002), chap. Appendix A.
[Crossref]

Germain, R. N.

J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. U. S. A. 109, 8434–8439 (2012).
[Crossref] [PubMed]

Gigan, S.

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Gjonaj, B.

Goodman, J. W.

J. W. Goodman, Speckle phenomena in optics (Roberts & Company, 2007), chap. 3.

Goorden, S. A.

Gradinaru, V.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

Guan, Y.

Horstmeyer, R.

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[Crossref] [PubMed]

Jang, M.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

Ji, N.

Johnson, P.

Jouhanneau, J.-S.

I. N. Papadopoulos, J.-S. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Judkewitz, B.

I. N. Papadopoulos, J.-S. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Katz, O.

O. Katz, E. Small, Y. Guan, and Y. Silberberg, “Noninvasive nonlinear focusing and imaging through strongly scattering turbid layers,” Optica 1, 170–174 (2014).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[Crossref]

Lagendijk, A.

Liu, Y.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21, 085001 (2016).
[Crossref]

Ma, C.

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21, 085001 (2016).
[Crossref]

Milkie, D. E.

Mosk, A. P.

Ouzounov, D. G.

Papadopoulos, I. N.

I. N. Papadopoulos, J.-S. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Park, J.-H.

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112, 9236–9241 (2015).
[Crossref] [PubMed]

Paudel, H. P.

Piestun, R.

Poulet, J. F. A.

I. N. Papadopoulos, J.-S. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Robinson, J. E.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

Rotter, S.

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Ruan, H.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[Crossref] [PubMed]

Shen, Y.

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21, 085001 (2016).
[Crossref]

Silberberg, Y.

O. Katz, E. Small, Y. Guan, and Y. Silberberg, “Noninvasive nonlinear focusing and imaging through strongly scattering turbid layers,” Optica 1, 170–174 (2014).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[Crossref]

Sinefeld, D.

Small, E.

O. Katz, E. Small, Y. Guan, and Y. Silberberg, “Noninvasive nonlinear focusing and imaging through strongly scattering turbid layers,” Optica 1, 170–174 (2014).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[Crossref]

Sun, W.

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112, 9236–9241 (2015).
[Crossref] [PubMed]

Tang, J.

J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. U. S. A. 109, 8434–8439 (2012).
[Crossref] [PubMed]

van Putten, E. G.

E. G. van Putten, “Disorder-enhanced imaging with spatially controlled light,” Ph.D. thesis, University of Twente (2011).

Vellekoop, I. M.

Wang, L. V.

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21, 085001 (2016).
[Crossref]

Xiao, C.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

Xu, C.

Yang, C.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[Crossref] [PubMed]

Zhou, C.

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

J. Biomed. Opt. (1)

Y. Shen, Y. Liu, C. Ma, and L. V. Wang, “Focusing light through biological tissue and tissue-mimicking phantoms up to 9.6 cm in thickness with digital optical phase conjugation,” J. Biomed. Opt. 21, 085001 (2016).
[Crossref]

Nat. Photonics (4)

R. Horstmeyer, H. Ruan, and C. Yang, “Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue,” Nat. Photonics 9, 563–571 (2015).
[Crossref] [PubMed]

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics 5, 335–342 (2011).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, “Focusing and compression of ultrashort pulses through scattering media,” Nat. Photonics 5, 372–377 (2011).
[Crossref]

I. N. Papadopoulos, J.-S. Jouhanneau, J. F. A. Poulet, and B. Judkewitz, “Scattering compensation by focus scanning holographic aberration probing (F-SHARP),” Nat. Photonics 11, 116–123 (2017).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Optica (1)

Phys. Rev. Lett. (1)

I. M. Vellekoop and A. P. Mosk, “Universal optimal transmission of light through disordered materials,” Phys. Rev. Lett. 101, 120601 (2008).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. U. S. A. (2)

J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. U. S. A. 109, 8434–8439 (2012).
[Crossref] [PubMed]

J.-H. Park, W. Sun, and M. Cui, “High-resolution in vivo imaging of mouse brain through the intact skull,” Proc. Natl. Acad. Sci. U. S. A. 112, 9236–9241 (2015).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Sci. Adv. (1)

H. Ruan, J. Brake, J. E. Robinson, Y. Liu, M. Jang, C. Xiao, C. Zhou, V. Gradinaru, and C. Yang, “Deep tissue optical focusing and optogenetic modulation with time-reversed ultrasonically encoded light,” Sci. Adv. 3, eaao5520 (2017).
[Crossref] [PubMed]

Other (3)

R. F. H. Fisher, Precoding and signal shaping for digital transmission (John Wiley & Sons, Inc., 2002), chap. Appendix A.
[Crossref]

J. W. Goodman, Speckle phenomena in optics (Roberts & Company, 2007), chap. 3.

E. G. van Putten, “Disorder-enhanced imaging with spatially controlled light,” Ph.D. thesis, University of Twente (2011).

Supplementary Material (2)

NameDescription
» Visualization 1       Video displaying the speckle images and corresponding feedback signals of the iterative blind focusing experiments using feedback from a small region of interest
» Visualization 2       Video displaying the speckle images and corresponding feedback signals of the iterative blind focusing experiments using feedback from a small region of interest

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Figures (4)

Fig. 1
Fig. 1 Illustration of the blind focusing experiment. SLM: Spatial light modulator.
Fig. 2
Fig. 2 Schematic of the experimental setup. λ/2: half wave plate, Pol: Polarizer, SLM: Spatial light modulator, CMOS: Complementary metal oxide semiconductor camera.
Fig. 3
Fig. 3 Results of the blind focusing experiment using two feedback targets. (a) Example image of the initial intensity distribution ( I ( 0 )), and (b) the corresponding intensity distribution after wavefront shaping ( I ( 1 )). The red (right) and blue (left) circles mark the locations of targets 1 and 2. (c)-(e) The optimized target intensities plotted as function of the ratio of target intensities before the optimization. The optimized intensities at target 1 and 2 are represented by the red circles and blue squares, respectively. The experiments were performed using (c) first-order, (d) second-order, and (e) third-order feedback. All intensities are normalized to I m a x = N | γ | 2 I 0. The predicted mean value for the optimized intensity and the corresponding standard deviation, as given by Eq. (9) and Eq. (10), arerepresented by the colored solid lines and the shaded areas, respectively.
Fig. 4
Fig. 4 Results of the blind focusing experiments with a pre-optimized focus using the total second-order feedback signal from a region of interest containing M targets. (a) Example speckle pattern with a pre-optimized focus, where the red and purplecircles represent the small (M = 96) and large (M = 2400) ROIs, respectively. (b)-(c) The enhancement η in the pre-optimized focus before and after blind focusing with M = 96 (red circles) and M = 2400 (purple squares), with (b) N = 80, and (c) N = 208. The predicted mean value for η(1) and the corresponding standard deviation are represented by the colored solid lines and the shaded areas, respectively. The black solid line indicates the identity line.

Equations (49)

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

S b M | E b | 2 n ,
E b ( k ) = a N t b a E a ( k ) .
S ˜ ( k + 1 ) ( ϕ ) = b M ( | E b ( k ) | 2 + | t b a E a ( k ) | 2 | e i ϕ 1 | 2 + ( E b ( k ) t b a * E a ( k ) * [ e i ϕ 1 ] + c . c . ) ) n ,
S ˜ ( k + 1 ) ( ϕ ) S ( k ) + b M ( W b ( k ) t b a * E a ( k ) * [ e i ϕ 1 ] + c . c . ) ,
E a ( k + 1 ) = c ( k + 1 ) E a ( k ) * P p P S ˜ ( k + 1 ) ( ϕ p ) e i ϕ p = c ( k + 1 ) b M W b ( k ) t b a * ,
E b ( k + 1 ) = c ( k + 1 ) a N b M t b a t b a * W b ( k ) .
P ( E b ( k + 1 ) ) = 1 π I 0 exp  ( | E b ( k + 1 ) μ b ( k + 1 ) | 2 I 0 ) .
μ b ( k + 1 ) W b ( k ) b M | W b ( k ) | 2 γ N I 0 and I 0 | t b a | 2 ,
I b ( k + 1 ) = | μ b ( k + 1 ) | 2 + I 0 = I 0 ( N | γ | 2 | W b ( k ) | 2 b M | W b ( k ) | 2 + 1 ) ,
σ I = I 0 | μ b ( k + 1 ) | 2 + I 0 2 = I 0 N | γ | 2 | W b ( k ) | 2 b M | W b ( k ) | 2 + 1 ,
η ( 1 ) = N | γ | 2 ( η ( 0 ) ) 2 n 1 ( η ( 0 ) ) 2 n 1 + ( 2 n 1 ) ! ( M 1 ) + 1.
M m a x = ( 2 n 2 ) 2 n 2 ( 2 n 1 ) ! ( 2 n 1 ) 2 n 1 ( | γ | 2 N ) 2 n 1 + 1.
b = Ta ,
S ( a , a ¯ ) = b M [ ( T * a ¯ ) b ( Ta ) b ] n ,
S ( a + Δ , a ¯ + Δ ¯ ) = S ( a , a ¯ ) + S a Δ + S a ¯ Δ ¯ + O ( Δ 2 ) .
S a i = n b M [ ( T * a ¯ ) b ( Ta ) b ] n 1 t b i = b M w b * t b i ,
w b ( a , a ¯ ) = n ( Ta ) b n ( T * a ¯ ) b ( n 1 ) and w ¯ b ( a , a ¯ ) = n ( T * a ¯ ) b n ( Ta ) b ( n 1 ) .
S a = w ¯ T T and S a ¯ = w T T * .
Δ i = { a i ( e i ϕ 1 ) for i = a 0 otherwise .
f ( a , a ¯ ) = T w ( a , a ¯ ) T w ( a , a ¯ ) ,
f ( a * + Δ , a ¯ * + Δ ¯ ) f ( a * , a ¯ * ) + [ f a f a ¯ f ¯ a f ¯ a ¯ ] a = a * a ¯ = a ¯ * [ Δ Δ ¯ ] ,
Δ ¯ T f a | a = a * a ¯ = a ¯ * = Δ ¯ T [ T T w w a + T w a 1 T w ] a = a * a ¯ = a ¯ * = Δ ¯ T T T w w a | a = a * a ¯ = a ¯ * ,
S L ( a , a ¯ ) = S ( a , a ¯ ) λ ( a ¯ T a 1 ) ,
w ¯ * T T λ a ¯ * T = 0
w * T T * λ a * T = 0
a ¯ * T a * 1 = 0 ,
H [ 2 S L a a ¯ 2 S L a ¯ 2 2 S L a 2 2 S L a ¯ a ] a = a * a ¯ = a ¯ * = [ T w a λ T w a ¯ T T w ¯ a T T w ¯ a ¯ λ ] a = a * a ¯ = a ¯ * .
[ Δ ¯ Δ ] T H [ Δ Δ ¯ ] = [ Δ ¯ Δ ] T λ ( J f I ) [ Δ Δ ¯ ] < 0 ,
E ^ b ( k + 1 ) = γ c ( k + 1 ) a N b M t b a t b a * W b ( k ) + 1 | γ | 2 ζ b .
E ^ b ( k + 1 ) = γ c ( k + 1 ) a N χ a ( k ) + 1 | γ | 2 ζ b with χ a ( k ) b M t b a t b a * W b ( k ) .
χ a ( k ) = b M W b ( k ) t b a t b a * = W b ( k ) | t b a | 2 ,
| χ a ( k ) | 2 = b M b M W b ( k ) W b ( k ) * | t b a | 2 t b a * t b a
= b M | W b ( k ) | 2 | t b a | 2 | t b a | 2
= b b M | W b ( k ) | 2 | t b a | 2 | t b a | 2 + | W b ( k ) | 2 | t b a | 4 .
var ( χ a ( k ) ) = | χ a ( k ) | 2 | χ a ( k ) | 2 = | t b a | 2 2 b M | W b ( k ) | 2 .
c ( k + 1 ) = 1 a N | b M t b a * W b ( k ) | 2 1 N | t b a | 2 b M | W b ( k ) | 2 .
E ^ b ( k + 1 ) = γ c ( k + 1 ) a N χ a ( k ) + 1 | γ | 2 ζ b = W b ( k ) b M | W b ( k ) | 2 γ N | t b a | 2 .
var ( E ^ b ( k + 1 ) )   = | γ | 2 ( c ( k + 1 ) ) 2 a N var ( χ a ( k ) ) + ( 1 | γ | 2 ) var ( ζ b )
= | γ | 2 N | t b a | 2 2 b M | W b ( k ) | 2 N | t b a | 2 b M | W b ( k ) | 2 + ( 1 | γ | 2 ) | t b a | 2
= | t b a | 2 .
N | γ | 2 ( η ( 0 ) ) 2 n 1 ( η ( 0 ) ) 2 n 1 + ( 2 n 1 ) ! ( M 1 ) + 1 η ( 0 ) + 1
( η ( 0 ) ) 2 n 1 + N | γ | 2 ( η ( 0 ) ) 2 n 2 ( 2 n 1 ) ! ( M 1 ) .
max η ( 0 ) ( ( η ( 0 ) ) 2 n 1 + N | γ | 2 ( η ( 0 ) ) 2 n 2 ) ,
( 2 n 1 ) ( η ( 0 ) ) 2 n 2 = ( 2 n 2 ) N | γ | 2 ( η ( 0 ) ) 2 n 3
η ( 0 ) = 2 n 2 2 n 1 N | γ | 2 .
( 2 n 2 2 n 1 N | γ | 2 ) 2 n 1 + N | γ | 2 ( 2 n 2 2 n 1 N | γ | 2 ) 2 n 2 ( 2 n 1 ) ! ( M 1 ) .
( 2 n 1 ) ! ( M 1 ) < ( 2 n 2 2 n 1 ) 2 n 2 ( 1 2 n 2 2 n 1 ) ( N | γ | 2 ) 2 n 1
( 2 n 1 ) ! ( M 1 ) < ( 2 n 2 ) 2 n 2 ( 2 n 1 ) 2 n 1 ( N | γ | 2 ) 2 n 1
M < ( 2 n 2 ) 2 n 2 ( 2 n 1 ) ! ( 2 n 1 ) 2 n 1 ( | γ | 2 N ) 2 n 1 + 1.

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