Abstract

We develop and experimentally verify a theory of evolution of polarization in artificially-disordered multi-mode optical fibers. Starting with a microscopic model of photo-induced index change, we obtain the first and second order statistics of the dielectric tensor in a Ge-doped fiber, where a volume disorder is intentionally inscribed via UV radiation transmitted through a diffuser. A hybrid coupled-power & coupled-mode theory is developed to describe the transient process of de-polarization of light launched into such a fiber. After certain characteristic distance, the power is predicted to be equally distributed over all co-propagating modes of the fiber regardless of their polarization. Polarization-resolved experiments, confirm the predicted evolution of the state of polarization. Complete mode mixing in a segment of fiber as short as ∼ 10cm after 3.6dB insertion loss is experimentally observed. Equal excitation of all modes in such a multi-mode fiber creates the conditions to maximize the information capacity of the system under e.g. multiple-input-multiple-output (MIMO) transmission setup.

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).
  2. A. F. Garito, J. Wang, and R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
    [CrossRef] [PubMed]
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    [CrossRef]
  4. S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. M. Limonov and R. D. L. Rue, eds., Optical Properties of Photonic Structures: Interplay of Order and Disorder (Francis & Taylor, 2012).
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    [CrossRef]
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  10. G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun. 6, 311–335 (1998).
    [CrossRef]
  11. E. Telatar, “Capacity of multi-antenna gaussian channels,” Euro. Trans. Telecommun. 10, 585–595 (1999).
    [CrossRef]
  12. S. H. Simon, A. L. Moustakas, M. Stoychev, and H. Safar, “Communication in a disordered world,” Phys. Today 54, 38–43 (2001).
    [CrossRef]
  13. H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289, 281–283 (2000).
    [CrossRef] [PubMed]
  14. M. Nazarathy and A. Agmon, “Coherent transmission direct detection MIMO over short-range optical interconnects and passive optical networks,” J. Lightwave Technol. 26, 2037–2045 (2008).
    [CrossRef]
  15. M. Greenberg, M. Nazarathy, and M. Orenstein, “Multimode fiber as random code generator–application to massively parallel MIMO transmission,” J. Lightwave Technol. 26, 882–890 (2008).
    [CrossRef]
  16. R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006).
    [CrossRef]
  17. N. P. Puente, E. I. Chaikina, S. Herath, and A. Yamilov, “Fabrication, characterization, and theoretical analysis of controlled disorder in the core of optical fibers,” Appl. Opt. 50, 802–810 (2011).
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    [CrossRef]

2011 (2)

2010 (1)

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

2008 (3)

2006 (2)

O. Korotkova, “Changes in statistics of the instantaneous stokes parameters of a quasi-monochromatic electromagnetic beam on propagation,” Opt. Commun. 261, 218–224 (2006).
[CrossRef]

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006).
[CrossRef]

2005 (1)

2004 (1)

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

2001 (1)

S. H. Simon, A. L. Moustakas, M. Stoychev, and H. Safar, “Communication in a disordered world,” Phys. Today 54, 38–43 (2001).
[CrossRef]

2000 (2)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289, 281–283 (2000).
[CrossRef] [PubMed]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[CrossRef] [PubMed]

1999 (1)

E. Telatar, “Capacity of multi-antenna gaussian channels,” Euro. Trans. Telecommun. 10, 585–595 (1999).
[CrossRef]

1998 (2)

G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun. 6, 311–335 (1998).
[CrossRef]

A. F. Garito, J. Wang, and R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[CrossRef] [PubMed]

1997 (3)

M. Fink, “Time reversed acoustics,” Phys. Today 50, 34–40 (1997).
[CrossRef]

K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277 –1294 (1997).
[CrossRef]

1994 (3)

1991 (1)

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
[CrossRef]

1978 (1)

1975 (1)

1972 (1)

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).

Agmon, A.

Amphawan, A.

Azana, J.

Belhadj, N.

Cao, H.

H. Cao, “Lasing in disordered media,” in Progress in Optics, E. Wolf, ed. (North Holland, 2003), Vol. 45.
[CrossRef]

Chaikina, E. I.

Dossou, K.

Erdogan, T.

Fink, M.

M. Fink, “Time reversed acoustics,” Phys. Today 50, 34–40 (1997).
[CrossRef]

Foschini, G. J.

G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun. 6, 311–335 (1998).
[CrossRef]

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
[CrossRef]

Gans, M. J.

G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun. 6, 311–335 (1998).
[CrossRef]

Gao, R.

A. F. Garito, J. Wang, and R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[CrossRef] [PubMed]

Garito, A. F.

A. F. Garito, J. Wang, and R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[CrossRef] [PubMed]

Gloge, D.

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Coberts & Co, Englewood, 2007).

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[CrossRef] [PubMed]

Greenberg, M.

Herath, S.

Hill, K. O.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Horowitz, M. A.

Hsu, R. C. J.

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006).
[CrossRef]

Inniss, D.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).

Jalali, B.

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006).
[CrossRef]

Kahn, J. M.

Kamal, A.

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[CrossRef] [PubMed]

Korotkova, O.

O. Korotkova, “Changes in statistics of the instantaneous stokes parameters of a quasi-monochromatic electromagnetic beam on propagation,” Opt. Commun. 261, 218–224 (2006).
[CrossRef]

Kosinski, S. G.

Lagendijk, A.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

LaRochelle, S.

Lemaire, P. J.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (Univ. California Press, 1964).

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).

Marcuse, D.

Meltz, G.

K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Mizrahi, V.

Mosk, A. P.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Moustakas, A. L.

S. H. Simon, A. L. Moustakas, M. Stoychev, and H. Safar, “Communication in a disordered world,” Phys. Today 54, 38–43 (2001).
[CrossRef]

Nazarathy, M.

Olshansky, R.

Orenstein, M.

Park, Y.

Poole, C. D.

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
[CrossRef]

Puente, N. P.

Reed, W. A.

Russell, P. S. J.

Safar, H.

S. H. Simon, A. L. Moustakas, M. Stoychev, and H. Safar, “Communication in a disordered world,” Phys. Today 54, 38–43 (2001).
[CrossRef]

Sayed, A. H.

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006).
[CrossRef]

Shah, A.

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006).
[CrossRef]

Shen, X.

Simon, S. H.

S. H. Simon, A. L. Moustakas, M. Stoychev, and H. Safar, “Communication in a disordered world,” Phys. Today 54, 38–43 (2001).
[CrossRef]

Skipetrov, S. E.

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

Stoychev, M.

S. H. Simon, A. L. Moustakas, M. Stoychev, and H. Safar, “Communication in a disordered world,” Phys. Today 54, 38–43 (2001).
[CrossRef]

Stuart, H. R.

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289, 281–283 (2000).
[CrossRef] [PubMed]

Tarighat, A.

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006).
[CrossRef]

Telatar, E.

E. Telatar, “Capacity of multi-antenna gaussian channels,” Euro. Trans. Telecommun. 10, 585–595 (1999).
[CrossRef]

Vellekoop, I. M.

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Vengsarkar, A. M.

Wang, J.

A. F. Garito, J. Wang, and R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[CrossRef] [PubMed]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).

Yamilov, A.

Zhong, Q.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).

Euro. Trans. Telecommun. (1)

E. Telatar, “Capacity of multi-antenna gaussian channels,” Euro. Trans. Telecommun. 10, 585–595 (1999).
[CrossRef]

IEEE Commun. Lett. (1)

R. C. J. Hsu, A. Tarighat, A. Shah, A. H. Sayed, and B. Jalali, “Capacity enhancement in coherent optical MIMO (COMIMO) multimode fiber links,” IEEE Commun. Lett. 10, 195–197 (2006).
[CrossRef]

J. Lightwave Technol. (5)

K. O. Hill and G. Meltz, “Fiber Bragg grating technology: Fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277 –1294 (1997).
[CrossRef]

G. J. Foschini and C. D. Poole, “Statistical theory of polarization dispersion in single mode fibers,” J. Lightwave Technol. 9, 1439–1456 (1991).
[CrossRef]

M. Nazarathy and A. Agmon, “Coherent transmission direct detection MIMO over short-range optical interconnects and passive optical networks,” J. Lightwave Technol. 26, 2037–2045 (2008).
[CrossRef]

M. Greenberg, M. Nazarathy, and M. Orenstein, “Multimode fiber as random code generator–application to massively parallel MIMO transmission,” J. Lightwave Technol. 26, 882–890 (2008).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

I. M. Vellekoop, A. Lagendijk, and A. P. Mosk, “Exploiting disorder for perfect focusing,” Nat. Photonics 4, 320–322 (2010).
[CrossRef]

Nature (1)

S. E. Skipetrov, “Disorder is the new order,” Nature 432, 285–286 (2004).
[CrossRef] [PubMed]

Opt. Commun. (1)

O. Korotkova, “Changes in statistics of the instantaneous stokes parameters of a quasi-monochromatic electromagnetic beam on propagation,” Opt. Commun. 261, 218–224 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Today (2)

M. Fink, “Time reversed acoustics,” Phys. Today 50, 34–40 (1997).
[CrossRef]

S. H. Simon, A. L. Moustakas, M. Stoychev, and H. Safar, “Communication in a disordered world,” Phys. Today 54, 38–43 (2001).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[CrossRef] [PubMed]

Science (2)

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289, 281–283 (2000).
[CrossRef] [PubMed]

A. F. Garito, J. Wang, and R. Gao, “Effects of random perturbations in plastic optical fibers,” Science 281, 962–967 (1998).
[CrossRef] [PubMed]

Wireless Personal Commun. (1)

G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun. 6, 311–335 (1998).
[CrossRef]

Other (7)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, 1974).

H. Cao, “Lasing in disordered media,” in Progress in Optics, E. Wolf, ed. (North Holland, 2003), Vol. 45.
[CrossRef]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, 1978).

M. Limonov and R. D. L. Rue, eds., Optical Properties of Photonic Structures: Interplay of Order and Disorder (Francis & Taylor, 2012).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York, 1995).

R. K. Luneburg, Mathematical Theory of Optics (Univ. California Press, 1964).

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Coberts & Co, Englewood, 2007).

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Figures (5)

Fig. 1
Fig. 1

UV irradiation geometry is shown. Expanded unpolarized λUV = 244nm UV light from an Ar laser illuminates an elongated area on the surface of a diffuser. A complex interference pattern is incident onto the core of the Ge-doped photosensitive fiber.

Fig. 2
Fig. 2

Correlations between field components of the UV light used to fabricate disorder in the core of the photo-sensitive optical fiber. Equation (9) and similar expression for the other field components, originating from Rayleigh-Sommerfeld integrals in Eq. (5) and Eq. (6), are evaluated numerically under the experimentally relevant conditions – Lx = 3mm, Lz = 5mm, D = 5mm. Not shown are E x ( U V ) ( r ) E z ( U V ) * ( r ) which vanishes completely and E z ( U V ) ( r ) E z ( U V ) * ( r ) which is identical to E x ( U V ) ( r ) E x ( U V ) * ( r ) .

Fig. 3
Fig. 3

Statistical properties of the spatially fluctuating dielectric tensor are described by correlators 〈δεij(r⃗)δεij(r⃗)〉 in Eqs. (12). To evaluate these expression we used the same set of parameters as in Fig. 2. Only the xy part of the entire tensor relevant to inter-mode coupling are shown.

Fig. 4
Fig. 4

(a) Runge-Kutta numerical solution of Eqs. (20) with α = 0. Only x-polarized LP modes are exited: cν,x(0) = (2/N)1/2. The power becomes equally distributed among the modes of both polarizations (x – blue, y – gold) before equilibrating at 1/N level shown as a dashed line. ν P ν , i ( δ ) ( z )converging at 1/2 are shown in the inset. The ballistic component ν P ν , x ( b ) ( z ) and the total power in the x polarization ν P ν , x ( b ) ( z ) + ν P ν , x ( δ ) ( z ) are depicted with dashed and dotted lines respectively. (b) Evolution along the fiber length of elements of the Stokes vector s⃗(z). The Poincare sphere plot describes the transition from the linearly ′ polarized light with the degree of polarization �� = |s⃗| = 1 at z = 0 to unpolarized light with �� = 0 in the limit z → ∞. Blue, green and red lines correspond to the light linearly polarized at φ = 0, π/8, and π/4 with respect to the x primary axis of the average dielectric tensor respectively.

Fig. 5
Fig. 5

(a) Experimental data in L = 2,4 and 12cm samples for the ensemble-averaged transmission for xx′ (circles) and xy′ (squares) polarization channels as the function of the angle φ between x′ and the principal axis x. In all cases the incident light is polarized along x′ axis. For clarity, the data is normalized as Pxx (z) → Pxx (z)/Pxx (z) + Pxy (z) and similarly for Pxy (z) to eliminate the effect of attenuation ×exp(−αz). Solid lines show the theoretical fit with Eq. (23) by combining the ∑ν term into a single fitting parameter. Two converging surface envelopes 1/2 ± exp(−σ2z)/2 are shown a guide for an eye de-mostrating almost complete mode mixing at L = 12cm. (b) Normalized by Px(z) + Py(z) to eliminate the exp(−αz) factor, Px(z), Py(z) show convergence toward 1/2. To acheive alighnment with the principal axes of fiber, the experimental data (symbols) was obtained in-situ during fabrication of the additional segments of disordered fiber. Solid lines obtained for φ = 0 from the fit in (a) show somewhat slower decay, that is attributed to unintentional twisting in process of generating ensemble realizations by bending of the fiber.

Equations (23)

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Δ χ i j ( r ) = α b Δ ρ ( r , θ , ϕ ) u i u j d Ω ,
Δ χ x x ( r ) = C 0 [ 3 | E x ( U V ) ( r ) | 2 + | E y ( U V ) ( r ) | 2 + | E z ( U V ) ( r ) | 2 ] ,
Δ χ x y ( r ) = C 0 [ E x ( U V ) ( r ) E y ( U V ) * ( r ) + E x ( U V ) * ( r ) E y ( U V ) ( r ) ] ,
δ ε i j ( r ) δ ε i j ( r ) ( Δ χ i j ( r ) Δ χ i j ( r ) ) ( Δ χ i j ( r ) Δ χ i j ( r ) ) .
E x ( U V ) ( r ) = D + y 2 π i k exp [ i k R ] R 2 E x ˜ ( U V ) ( x ˜ , z ˜ ) d x ˜ d z ˜ ,
E y ( U V ) ( r ) = 1 2 π i k exp  [ i k R ] R 2 ( E x ˜ ( U V ) ( x ˜ , z ˜ ) ( x x ˜ ) + E z ˜ ( U V ) ( x ˜ , z ˜ ) ( z z ˜ ) ) d x ˜ d z ˜ ,
E x ˜ ( U V ) ( x ˜ , z ˜ ) E x ˜ ( U V ) * ( x ˜ , z ˜ ) = E z ˜ ( U V ) ( x ˜ , z ˜ ) E z ˜ ( U V ) * ( x ˜ , z ˜ ) = ( 1 / 2 ) I 0 ( U V ) ( x ˜ , z ˜ ) κ δ ( x ˜ x ˜ ) δ ( z ˜ z ˜ ) ,
E x ˜ ( U V ) ( x ˜ , z ˜ ) E x ˜ ( U V ) ( x ˜ , z ˜ ) = E z ˜ ( U V ) ( x ˜ , z ˜ ) E z ˜ ( U V ) ( x ˜ , z ˜ ) = E x ˜ ( U V ) ( x ˜ , z ˜ ) E z ˜ ( U V ) ( x ˜ , z ˜ ) = 0 ,
E x ( U V ) ( r ) E x ( U V ) * ( r ) = κ ( D + y ) ( D + y ) λ U V 2 exp  [ i k ( R R ) ] R 2 R 2 I 0 ( U V ) ( x ˜ , z ˜ ) d x ˜ d z ˜ .
Δ ε x x = ε x x n core 2 = C 0 ( 4 | E x ( U V ) | 2 + 2 | E y ( U V ) | 2 )
Δ ε y y = ε y y n core 2 = C 0 ( 2 | E x ( U V ) | 2 + 2 | E y ( U V ) | 2 )
x x : 10 | E x ( U V ) E x ( U V ) * | 2 + | E y ( U V ) E y ( U V ) * | 2 + 6 | E x ( U V ) E y ( U V ) * | 2 + 2 | E y ( U V ) E z ( U V ) * | 2 y y : 2 | E x ( U V ) E x ( U V ) * | 2 + 9 | E y ( U V ) E y ( U V ) * | 2 + 6 | E x ( U V ) E y ( U V ) * | 2 + 6 | E y ( U V ) E z ( U V ) * | 2 x y : R e [ E x ( U V ) E x ( U V ) * E y ( U V ) E x ( U V ) * + E x ( U V ) E x ( U V ) * 2 c . c . ] ,
E i ( x , y , z ) ν c ν , i ( z ) e i ( ω t β ν ) z ν , i ( x , y ) .
d c ν , i ( z ) / d z = ( ± i Δ β ν / 2 α / 2 ) c ν , i ( z ) + ν , i 𝒦 ν ν , i i ( z ) c ν , i ( z ) e i ( β ν β ν ) z ,
c ν , i ( z ) = c ν , i ( z ) + δ c ν , i ( z ) ,
d c ν , i ( z ) / d z = ( ± i Δ β ν / 2 α / 2 h ν , i / 2 ) c ν , i ( z ) ,
h ν , i = ν , i h ν ν , i i
h ν ν , i i = ( ω 4 / 4 c 4 ) δ ε i i δ ε i i V i i i i ν , i 2 ( x , y ) v , i 2 ( x , y ) d x d y ,
c ν , i ( z ) c ν , i ( 0 ) exp  [ ( ± i Δ β ν / 2 α / 2 h ν , i / 2 ) z ]
d P ν , i ( δ ) / d z = h ν , i P ν , i ( b ) α P ν , i ( δ ) + ν , i h ν ν , i i ( P ν , i ( δ ) P ν , i ( δ ) ) .
S 0 = ν = 1 N / 2 ( P ν , x ( b ) + P ν , x ( δ ) P ν , y ( b ) P ν , y ( δ ) ) ; S 1 = ν = 1 N / 2 ( P ν , x ( b ) + P ν , x ( δ ) P ν , y ( b ) P ν , y ( δ ) ) ; S 2 = ν = 1 N / 2 ( c ν , x c ν , y * + c ν , x * c ν , y ) ; S 3 = i ν = 1 N / 2 ( c ν , x c ν , y * c ν , x * c ν , y ) .
P [ x y ] = P x [ cos ( ϕ ) 2 sin ( ϕ ) 2 ] + P y [ sin ( ϕ ) 2 cos ( ϕ ) 2 ] ± 2 sin ( ϕ ) cos ( ϕ ) [ ν c ν , x c ν , y * ] ,
P [ x y ] ( z ) = e α z { ( P x ( 0 ) [ cos ( ϕ ) 2 sin ( ϕ ) 2 ] + P y ( 0 ) [ sin ( ϕ ) 2 cos ( ϕ ) 2 ] 1 2 ) e σ 2 z + 1 2 ± 2 sin ( ϕ ) cos ( ϕ ) 2 N ν P ν , y 1 / 2 ( 0 ) P ν , y 1 / 2 ( 0 ) exp ( h ν , x + h ν , y 2 z ) cos ( Δ β ν z ) } ,

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