Abstract

The transmission matrix is a unique tool to control light through a scattering medium. A monochromatic transmission matrix does not allow temporal control of broadband light. Conversely, measuring multiple transmission matrices with spectral resolution allows fine temporal control when a pulse is temporally broadened upon multiple scattering, but requires very long measurement time. Here, we show that a single linear operator, measured for a broadband pulse with a co-propagating reference, naturally allows for spatial focusing and interestingly generates a two-fold temporal recompression at the focus, compared with the natural temporal broadening. This is particularly relevant for non-linear imaging techniques in biological tissues.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
  2. I. M. Vellekoop and A. P. Mosk, Opt. Lett. 32, 2309 (2007).
    [Crossref]
  3. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
    [Crossref]
  4. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
    [Crossref]
  5. D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
    [Crossref]
  6. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
    [Crossref]
  7. J. Aulbach, B. Gjonaj, P. Johnson, and A. Lagendijk, Opt. Express 20, 29237 (2012).
    [Crossref]
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    [Crossref]
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2017 (2)

S. Rotter and S. Gigan, Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, Sci. Adv. 3, e1600743 (2017).
[Crossref]

2016 (5)

H. B. de Aguiar, S. Gigan, and S. Brasselet, Phys. Rev. A 94, 043830 (2016).
[Crossref]

M. Mounaix, H. Defienne, and S. Gigan, Phys. Rev. A 94, 041802 (2016).
[Crossref]

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

J. Carpenter, B. J. Eggleton, and J. Schröder, Opt. Lett. 41, 5580 (2016).
[Crossref]

J. Bosch, S. A. Goorden, and A. P. Mosk, Opt. Express 24, 26472 (2016).
[Crossref]

2015 (4)

Z. Shi and A. Z. Genack, Phys. Rev. B 92, 184202 (2015).
[Crossref]

X. Tao, D. Bodington, M. Reinig, and J. Kubby, Opt. Express 23, 14168 (2015).
[Crossref]

C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, Phys. Rev. Lett. 115, 223901 (2015).
[Crossref]

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
[Crossref]

2013 (3)

2012 (2)

J. Aulbach, B. Gjonaj, P. Johnson, and A. Lagendijk, Opt. Express 20, 29237 (2012).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[Crossref]

2011 (5)

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[Crossref]

N. Curry, P. Bondareff, M. Leclercq, N. F. van Hulst, R. Sapienza, S. Gigan, and S. Grésillon, Opt. Lett. 36, 3332 (2011).
[Crossref]

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

F. Van Beijnum, E. G. Van Putten, A. Lagendijk, and A. P. Mosk, Opt. Lett. 36, 373 (2011).
[Crossref]

2010 (1)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

2009 (1)

F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, Phys. Rev. Lett. 103, 173902 (2009).
[Crossref]

2007 (1)

Andreoli, D.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
[Crossref]

Aulbach, J.

J. Aulbach, B. Gjonaj, P. Johnson, and A. Lagendijk, Opt. Express 20, 29237 (2012).
[Crossref]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[Crossref]

Austin, D. R.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

Bifano, T.

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Bodington, D.

Bondareff, P.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

N. Curry, P. Bondareff, M. Leclercq, N. F. van Hulst, R. Sapienza, S. Gigan, and S. Grésillon, Opt. Lett. 36, 3332 (2011).
[Crossref]

Bosch, J.

Brasselet, S.

H. B. de Aguiar, S. Gigan, and S. Brasselet, Sci. Adv. 3, e1600743 (2017).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, Phys. Rev. A 94, 043830 (2016).
[Crossref]

Bromberg, Y.

C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, Phys. Rev. Lett. 115, 223901 (2015).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[Crossref]

Cao, H.

C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, Phys. Rev. Lett. 115, 223901 (2015).
[Crossref]

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Carpenter, J.

Chatel, B.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

Curry, N.

Davy, M.

de Aguiar, H. B.

H. B. de Aguiar, S. Gigan, and S. Brasselet, Sci. Adv. 3, e1600743 (2017).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, Phys. Rev. A 94, 043830 (2016).
[Crossref]

de Rosny, J.

F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, Phys. Rev. Lett. 103, 173902 (2009).
[Crossref]

Defienne, H.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

M. Mounaix, H. Defienne, and S. Gigan, Phys. Rev. A 94, 041802 (2016).
[Crossref]

Eggleton, B. J.

Fink, M.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, Phys. Rev. Lett. 103, 173902 (2009).
[Crossref]

Genack, A. Z.

Gigan, S.

S. Rotter and S. Gigan, Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, Sci. Adv. 3, e1600743 (2017).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, Phys. Rev. A 94, 043830 (2016).
[Crossref]

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

M. Mounaix, H. Defienne, and S. Gigan, Phys. Rev. A 94, 041802 (2016).
[Crossref]

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
[Crossref]

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

N. Curry, P. Bondareff, M. Leclercq, N. F. van Hulst, R. Sapienza, S. Gigan, and S. Grésillon, Opt. Lett. 36, 3332 (2011).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Gjonaj, B.

J. Aulbach, B. Gjonaj, P. Johnson, and A. Lagendijk, Opt. Express 20, 29237 (2012).
[Crossref]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[Crossref]

Goetschy, A.

C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, Phys. Rev. Lett. 115, 223901 (2015).
[Crossref]

A. Goetschy and A. Stone, Phys. Rev. Lett. 111, 063901 (2013).
[Crossref]

Goorden, S. A.

Grésillon, S.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
[Crossref]

N. Curry, P. Bondareff, M. Leclercq, N. F. van Hulst, R. Sapienza, S. Gigan, and S. Grésillon, Opt. Lett. 36, 3332 (2011).
[Crossref]

Hsu, C. W.

C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, Phys. Rev. Lett. 115, 223901 (2015).
[Crossref]

Johnson, P.

Johnson, P. M.

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[Crossref]

Katz, O.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[Crossref]

Kubby, J.

Lagendijk, A.

J. Aulbach, B. Gjonaj, P. Johnson, and A. Lagendijk, Opt. Express 20, 29237 (2012).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[Crossref]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[Crossref]

F. Van Beijnum, E. G. Van Putten, A. Lagendijk, and A. P. Mosk, Opt. Lett. 36, 373 (2011).
[Crossref]

Leclercq, M.

Lemoult, F.

F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, Phys. Rev. Lett. 103, 173902 (2009).
[Crossref]

Lerosey, G.

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, Phys. Rev. Lett. 103, 173902 (2009).
[Crossref]

McCabe, D. J.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

Mertz, J.

Mosk, A. P.

J. Bosch, S. A. Goorden, and A. P. Mosk, Opt. Express 24, 26472 (2016).
[Crossref]

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[Crossref]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[Crossref]

F. Van Beijnum, E. G. Van Putten, A. Lagendijk, and A. P. Mosk, Opt. Lett. 36, 373 (2011).
[Crossref]

I. M. Vellekoop and A. P. Mosk, Opt. Lett. 32, 2309 (2007).
[Crossref]

Mounaix, M.

M. Mounaix, H. Defienne, and S. Gigan, Phys. Rev. A 94, 041802 (2016).
[Crossref]

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

Paudel, H. P.

Popoff, S.

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
[Crossref]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Reinig, M.

Rotter, S.

S. Rotter and S. Gigan, Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Sapienza, R.

Schröder, J.

Shi, Z.

Silberberg, Y.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[Crossref]

Small, E.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[Crossref]

Stockbridge, C.

Stone, A.

A. Goetschy and A. Stone, Phys. Rev. Lett. 111, 063901 (2013).
[Crossref]

Stone, A. D.

C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, Phys. Rev. Lett. 115, 223901 (2015).
[Crossref]

Tajalli, A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

Tao, X.

Van Beijnum, F.

van Hulst, N. F.

Van Putten, E. G.

Vellekoop, I. M.

Volpe, G.

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
[Crossref]

Walmsley, I. A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

Wang, J.

Nat. Commun. (1)

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[Crossref]

Nat. Photonics (2)

A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, Nat. Photonics 6, 283 (2012).
[Crossref]

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. A (2)

M. Mounaix, H. Defienne, and S. Gigan, Phys. Rev. A 94, 041802 (2016).
[Crossref]

H. B. de Aguiar, S. Gigan, and S. Brasselet, Phys. Rev. A 94, 043830 (2016).
[Crossref]

Phys. Rev. B (1)

Z. Shi and A. Z. Genack, Phys. Rev. B 92, 184202 (2015).
[Crossref]

Phys. Rev. Lett. (6)

A. Goetschy and A. Stone, Phys. Rev. Lett. 111, 063901 (2013).
[Crossref]

F. Lemoult, G. Lerosey, J. de Rosny, and M. Fink, Phys. Rev. Lett. 103, 173902 (2009).
[Crossref]

C. W. Hsu, A. Goetschy, Y. Bromberg, A. D. Stone, and H. Cao, Phys. Rev. Lett. 115, 223901 (2015).
[Crossref]

M. Mounaix, D. Andreoli, H. Defienne, G. Volpe, O. Katz, S. Grésillon, and S. Gigan, Phys. Rev. Lett. 116, 253901 (2016).
[Crossref]

J. Aulbach, B. Gjonaj, P. M. Johnson, A. P. Mosk, and A. Lagendijk, Phys. Rev. Lett. 106, 103901 (2011).
[Crossref]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Rev. Mod. Phys. (1)

S. Rotter and S. Gigan, Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

Sci. Adv. (1)

H. B. de Aguiar, S. Gigan, and S. Brasselet, Sci. Adv. 3, e1600743 (2017).
[Crossref]

Sci. Rep. (1)

D. Andreoli, G. Volpe, S. Popoff, O. Katz, S. Grésillon, and S. Gigan, Sci. Rep. 5, 10347 (2015).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1.

Principle of the measurement of the broadband transmission matrix (BBTM). (a) An input pulse of duration δt (spectral width Δλ) illuminates a spatial light modulator (SLM) before propagation through a thick scattering medium via a microscope objective (MO). The active part of the SLM will modulate a subpart of the input beam iM with respect to a static reference part iR. The transmitted light results in an intensity speckle with low contrast C0 on a CCD. Every single grain is the resulting broadband interference of the transmitted speckle from the input reference part oR and from the input modulated part oM. The BBTM is the linear relation connecting iM to oM. In the temporal domain, the speckle grain is characterized by its temporal feature δt and by the averaged traversal time of photons through the medium τm. In the spectral domain, the speckle has a characteristic width δλm, which is the spectral speckle correlation bandwidth of the medium. (b) Phase-conjugation of the BBTM enables spatial focusing in a given speckle grain, whose efficiency is inversely related to Nλ. The images show the results for four different samples whose spectral degrees of freedom Nλ1/C02 are indicated with corresponding markers in (c). Scale bar: 2 μm. (c) Log of signal-to-noise ratio (SNR) is plotted as a function of log(Nλ). Error bars are standard deviation for SNR over 50 different foci. Dashed line: linear fit, with a 1 slope.

Fig. 2.
Fig. 2.

Broadband focusing leads to a shortened averaged confinement time of focused photons. (a) Apparatus for measuring the BBTM and temporal/spectral characterization of the achieved focus. An ultra-short pulse generated by a Ti:sapphire laser (λ0=800  nm, τ0100  fs FWHM, MaiTai, Spectra-Physics) is reflected by a phase-only SLM (X10468-2, Hamamatsu) and injected into a thick scattering sample of randomly distributed ZnO nanobeads using a MO (Olympus, UMPlan FI, 10×, NA 0.3). Transmitted scattered light is collected with another MO (Olympus, LMPlan FI, 100×, NA 0.85), and imaged on a CCD after a polarizer (P). A reference arm is added by selecting part of the laser light before the SLM on a polarizing beam splitter and recombining with the output scattered light on a beam splitter (BS) before the CCD camera. A delay line (DL) controls the relative optical path delay between the reference and the scattered pulse. Importantly, this arm is blocked by a shutter (S) during both BBTM measurement and the focusing experiment and is only used for interferometric cross correlation (ICC) of the output speckle. L: lens; Ph: pinhole. (b) Time-of-flight distributions retrieved with ICC, in a single speckle grain in the case of (red) focusing with the BBTM and (black) without shaping. The two plots are shifted for better visibility. (c) Corresponding averaged time-of-flight distributions over 25 different foci plotted in semilog. τm (without shaping) and τf (broadband focusing) are measured with a linear fit. (d) Comparison between τm and τf for various samples. While the dashed black line represents the expected τm=τf, experimental data (red markers) evidence that τf is twice as small as τm. The red dashed line represents a linear fit, and error bars indicate the standard deviation for estimation of τf.

Fig. 3.
Fig. 3.

Broadband focusing changes the envelope spectral response. (a) Temporal field autocorrelation of both (red) foci averaged over 25 different foci, and (black) without shaping averaged over 200 speckle grains. The temporal width of the autocorrelation peak for the focus is broader than without shaping. (b) Spectral intensity measurement. Monochromatic intensity in a single grain (with or without shaping) is recorded as a function of λ. Plots are spectral intensities (black) without wavefront shaping, (red) at the focus, and (blue) intensity spectrum of the input pulse. (c) (red) Spectral intensity at the focus, averaged over 100 different focus positions, compared to (blue) the spectrum of the input pulse. On average, the output pulse has a narrower spectrum than the input pulse.

Equations (1)

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I=λi=1NλI(λi)=λi=1Nλ|Eout,R(λi)+Eout,M(λi)|2=I0+λi=1Nλ[(Eout,R(λi))*Eout,M(λi)+c.c.]=I0+λi=1Nλ[(Eout,R(λi))*T(λi)T(λi)Ein(λi)].