Abstract

We present an approach for numerically solving the multimode generalized nonlinear Schrödinger equation (MM-GNLSE). We propose to transform the MM-GNLSE to a system of first-order ordinary differential equations (ODEs) that can then be solved using readily available ODE solvers, thus making modeling of pulse propagation in multimode fibers easier. The solver is verified for the simplest multimode case in which only the two orthogonal polarization states in a non-birefringent microstructured optical fiber are involved. Also, the nonlinear dynamics of the degree and state of spectral polarization are presented for this case.

© 2013 OSA

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2013 (3)

2012 (4)

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

X.-h. Fang, M.-l. Hu, L.-l. Huang, L. Chai, N.-l. Dai, J.-y. Li, A. Y. Tashchilina, A. M. Zheltikov, and C.-y. Wang, “Multiwatt octave-spanning supercontinuum generation in multicore photonic-crystal fiber,” Opt. Lett.37, 2292–2294 (2012).
[CrossRef] [PubMed]

C. Agger, C. Petersen, S. Dupont, H. Steffensen, J. K. Lyngsø, C. L. Thomsen, J. Thøgersen, S. R. Keiding, and O. Bang, “Supercontinuum generation in ZBLAN fibers—detailed comparison between measurement and simulation,” J. Opt. Soc. Am. B29, 635–645 (2012).
[CrossRef]

2010 (2)

2009 (2)

2008 (2)

2007 (4)

2006 (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

2005 (1)

M. Tianprateep, J. Tada, and F. Kannari, “Influence of polarization and pulse shape of femtosecond initial laser pulses on spectral broadening in microstructure fibers,” Opt. Rev.12, 179–189 (2005).
[CrossRef]

2004 (3)

2003 (5)

2002 (3)

2001 (1)

2000 (1)

Agger, C.

Agrawal, G. P.

Akimov, D. A.

Apolonski, A.

Arriaga, J.

Bang, O.

C. Agger, C. Petersen, S. Dupont, H. Steffensen, J. K. Lyngsø, C. L. Thomsen, J. Thøgersen, S. R. Keiding, and O. Bang, “Supercontinuum generation in ZBLAN fibers—detailed comparison between measurement and simulation,” J. Opt. Soc. Am. B29, 635–645 (2012).
[CrossRef]

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

Barthélémy, A.

Blandin, P.

Brambilla, G.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Brocklesby, W. S.

Brown, T. G.

Chai, L.

Chaipiboonwong, T.

Chau, A. H. L.

Cherif, R.

Coen, S.

Couderc, V.

Dai, N.-l.

Degiorgio, V.

Drexler, W.

Druon, F.

Dudley, J. M.

Dukel’skii, K. V.

Dupont, S.

Efimov, A.

Eggleton, B. J.

Essiambre, R.-J.

Fang, X.-h.

Feng, X.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Friberg, A. T.

Frosz, M. H.

Genty, G.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
[CrossRef]

M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, “Supercontinuum generation in a highly birefringent microstructured fiber,” Appl. Phys. Lett.82, 2197–2199 (2003).
[CrossRef]

Georges, P.

Grossard, N.

Guobin, R.

Hanna, M.

Harvey, J. D.

Heidt, A.

Heidt, A. M.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Holdsworth, J.

Holzlohner, R.

Horak, P.

Hu, M.-l.

Huang, L.-l.

Hult, J.

Ibsen, M.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Jakobsen, C.

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

Johansen, J.

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

Kaivola, M.

M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, “Supercontinuum generation in a highly birefringent microstructured fiber,” Appl. Phys. Lett.82, 2197–2199 (2003).
[CrossRef]

Kannari, F.

M. Tianprateep, J. Tada, and F. Kannari, “Influence of polarization and pulse shape of femtosecond initial laser pulses on spectral broadening in microstructure fibers,” Opt. Rev.12, 179–189 (2005).
[CrossRef]

Keiding, S. R.

Kiefer, W.

King, B.

Knight, J. C.

Kondrat’ev, Y. N.

Konorov, S. O.

Lacroix, S.

Laegsgaard, J.

Larsen, C.

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

Legge, S.

Lehtonen, M.

M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, “Supercontinuum generation in a highly birefringent microstructured fiber,” Appl. Phys. Lett.82, 2197–2199 (2003).
[CrossRef]

Leonhardt, R.

Leproux, P.

Lesvigne, C.

Li, J.-y.

Loh, W. H.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Ludvigsen, H.

M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, “Supercontinuum generation in a highly birefringent microstructured fiber,” Appl. Phys. Lett.82, 2197–2199 (2003).
[CrossRef]

Lyngsø, J. K.

Maillotte, H.

Maksimenka, R.

Menyuk, C. R.

Mills, J. D.

Møller, U.

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

Moores, M. D.

Moselund, P. M.

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

Mumtaz, S.

Omenetto, F. G.

Petersen, C.

Petropoulos, P.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Petrovich, M.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Poletti, F.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

F. Poletti and P. Horak, “Dynamics of femtosecond supercontinuum generation in multimode fibers,” Opt. Express17, 6134–6147 (2009).
[CrossRef] [PubMed]

F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B25, 1645–1654 (2008).
[CrossRef]

Ponzo, G.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Povazay, B.

Price, J. H. V.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Provino, L.

Ranka, J. K.

Richardson, D. J.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Russell, P. St. J.

Rutt, H. N.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Schmitt, M.

Serebryannikov, E. E.

Setälä, T.

Shevandin, V. S.

Shi, J.

J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
[CrossRef]

Shuisheng, J.

Shuqin, L.

Sinkin, O. V.

Sørensen, S. T.

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

Steffensen, H.

Stentz, A. J.

Tada, J.

M. Tianprateep, J. Tada, and F. Kannari, “Influence of polarization and pulse shape of femtosecond initial laser pulses on spectral broadening in microstructure fibers,” Opt. Rev.12, 179–189 (2005).
[CrossRef]

Tarasevitch, A. P.

Tartara, L.

Tashchilina, A. Y.

Taylor, A. J.

Thøgersen, J.

Thomsen, C. L.

C. Agger, C. Petersen, S. Dupont, H. Steffensen, J. K. Lyngsø, C. L. Thomsen, J. Thøgersen, S. R. Keiding, and O. Bang, “Supercontinuum generation in ZBLAN fibers—detailed comparison between measurement and simulation,” J. Opt. Soc. Am. B29, 635–645 (2012).
[CrossRef]

U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
[CrossRef]

Tianprateep, M.

M. Tianprateep, J. Tada, and F. Kannari, “Influence of polarization and pulse shape of femtosecond initial laser pulses on spectral broadening in microstructure fibers,” Opt. Rev.12, 179–189 (2005).
[CrossRef]

Tonello, A.

Travers, J. C.

J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt.12, 113001 (2010).
[CrossRef]

Unterhuber, A.

Voipio, T.

von der Linde, D.

Wadsworth, W. J.

Wang, C.-y.

Windeler, R. S.

Zghal, M.

Zheltikov, A. M.

Zhi, W.

Zhou, P.

Zhu, Z.

Zwan, B.

Zweck, J.

Appl. Phys. Lett. (1)

M. Lehtonen, G. Genty, H. Ludvigsen, and M. Kaivola, “Supercontinuum generation in a highly birefringent microstructured fiber,” Appl. Phys. Lett.82, 2197–2199 (2003).
[CrossRef]

J. Lightwave Technol. (4)

J. Opt. (1)

J. C. Travers, “Blue extension of optical fibre supercontinuum generation,” J. Opt.12, 113001 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (6)

Opt. Express (10)

F. Poletti and P. Horak, “Dynamics of femtosecond supercontinuum generation in multimode fibers,” Opt. Express17, 6134–6147 (2009).
[CrossRef] [PubMed]

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express11, 1310–1321 (2003).
[CrossRef] [PubMed]

A. Efimov, A. J. Taylor, F. G. Omenetto, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Nonlinear generation of very high-order UV modes in microstructured fibers,” Opt. Express11, 910–918 (2003).
[CrossRef] [PubMed]

S. O. Konorov, E. E. Serebryannikov, A. M. Zheltikov, P. Zhou, A. P. Tarasevitch, and D. von der Linde, “Mode-controlled colors from microstructure fibers,” Opt. Express12, 730–735 (2004).
[CrossRef] [PubMed]

R. Cherif, M. Zghal, L. Tartara, and V. Degiorgio, “Supercontinuum generation by higher-order mode excitation in a photonic crystal fiber,” Opt. Express16, 2147–2152 (2008).
[CrossRef] [PubMed]

B. Zwan, S. Legge, J. Holdsworth, and B. King, “Spatio-spectral analysis of supercontinuum generation in higher order electromagnetic modes of photonic crystal fiber,” Opt. Express21, 834–839 (2013).
[CrossRef] [PubMed]

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J. H. V. Price, X. Feng, A. M. Heidt, G. Brambilla, P. Horak, F. Poletti, G. Ponzo, P. Petropoulos, M. Petrovich, J. Shi, M. Ibsen, W. H. Loh, H. N. Rutt, and D. J. Richardson, “Supercontinuum generation in non-silica fibers,” Opt. Fiber Technol.18, 327–344 (2012).
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U. Møller, S. T. Sørensen, C. Larsen, P. M. Moselund, C. Jakobsen, J. Johansen, C. L. Thomsen, and O. Bang, “Optimum PCF tapers for blue-enhanced supercontinuum sources,” Opt. Fiber Technology18, 304–314 (2012). Fiber Supercontinuum sources and their applications.
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M. Tianprateep, J. Tada, and F. Kannari, “Influence of polarization and pulse shape of femtosecond initial laser pulses on spectral broadening in microstructure fibers,” Opt. Rev.12, 179–189 (2005).
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Rev. Mod. Phys. (1)

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys.78, 1135–1184 (2006).
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J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibers (Cambridge University Press, New York, USA, 2010).
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Supplementary Material (3)

» Media 1: MP4 (613 KB)     
» Media 2: MP4 (688 KB)     
» Media 3: MP4 (612 KB)     

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Figures (7)

Figure 1
Figure 1

(a) Calculated GVD parameter for the six lowest-order modes of the MOF. Inset shows the fiber geometry used for the eigenmode analysis. (b) Mode fields (upper) and polarization (lower) of the two lowest-order modes M1 and M2.

Figure 2
Figure 2

Simulated spectral contents of modes M1 to M6 after 15 cm of propagation with the initial pump pulse fully coupled to mode M1.

Figure 3
Figure 3

Ensemble average of supercontinuum spectra obtained from 100 simulations with pulse propagation either restricted to mode M1 only (red line) or including coupling between modes M1 and M2 (blue line) for input pulses with 10 kW peak power fully coupled to mode M1.

Figure 4
Figure 4

Evolution of energy distribution among modes M1 and M2 for 1 kW (solid line), 10 kW (dotted line) and 50 kW (dash-dotted line) peak pump powers and up to a propagation length of 30 cm. The respective gray lines represent the total energy ∑ of the two modes for each launched pulse. Input power is originally coupled to mode M1 only.

Figure 5
Figure 5

Spectral and temporal evolution of the supercontinuum in modes M1 and M2 generated from 10 kW pulses originally launched into mode M1.

Figure 6
Figure 6

(a) Evolution of the spectral degree of polarization Ps (Eq. (11)) along the fiber. (b) Spectral degree of polarization as a function of wavelength at the end of the fiber. (c) ( Media 1) Graphical illustration of the evolution of the polarization state at 0.8 μm along the fiber. The tip of the Poincaré vector is traced by the purple line. The green line on the Poincaré sphere indicates the corresponding polarization state of the polarized part of the light. (d) Ensemble average of the spectral contents in mode M1 (red line) and M2 (blue line).

Figure 7
Figure 7

Graphical illustration of the evolution of the polarization state at wavelengths (a) ( Media 2) 0.7 μm and (b) ( Media 3) 0.9 μm along the fiber. For further explanations see Fig. 6(c).

Equations (12)

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A p ( z , T ) z = [ i ( β 0 ( p ) β 0 ) A p ( z , T ) ( β 1 ( p ) β 1 ) A p ( z , T ) T + i n 2 β n ( p ) n ! ( i T ) n A p ( z , T ) ] + [ i n 2 ω 0 c l , m , n { ( 1 + i τ p l m n ( 1 ) T ) Q p l m n ( 1 ) 2 A l ( z , T ) R ( T ) A m ( z , T T ) A n * ( z , T T ) d T + ( 1 + i τ p l m n ( 2 ) T ) Q p l m n ( 2 ) A l * ( z , T ) R ( T ) A m ( z , T T ) A n ( z , T T ) e 2 i ω 0 T d T } ] = [ 𝒟 ( p ) ( z , T ) ] + [ 𝒩 ( p ) ( z , T ) ] .
Q p l m n ( 1 ) ( ω ) = ε 0 2 n 0 2 c 2 12 [ E p * ( ω ) E l ( ω ) ] [ E m ( ω ) E n * ( ω ) ] d S N p ( ω ) N l ( ω ) N m ( ω ) N n ( ω ) Q p l m n ( 2 ) ( ω ) = ε 0 2 n 0 2 c 2 12 [ E p * ( ω ) E l * ( ω ) ] [ E m ( ω ) E n ( ω ) ] d S N p ( ω ) N l ( ω ) N m ( ω ) N n ( ω )
N p 2 ( ω ) = 1 4 [ E p * ( ω ) × H p ( ω ) + E p ( ω ) × H p * ( ω ) ] e z d S .
d y ( z ) d z = f ( y , z ) ,
A p ( z , T ) = 1 { A ˜ p ( z , ω ω 0 ) } = 1 2 π A ˜ p ( z , ω ω 0 ) exp [ i ( ω ω 0 ) T ] d ω .
A ˜ p ( z , ω ω 0 ) z i [ β ( p ) ( ω ) β 0 ( ω ω 0 ) β 1 ] A ˜ p ( z , ω ω 0 ) = 𝒩 ˜ ( p ) ( z , ω ω 0 ) ,
A ˜ p = A ˜ p exp [ L ( p ) ( ω ) z ] , L ( p ) ( ω ) = i [ β ( p ) ( ω ) β 0 ( ω ω 0 ) β 1 ] .
A ˜ p ( z , ω ω 0 ) z = exp [ L ( p ) ( ω ) z ] i n 2 ω c × × l , m , n M { Q p l m n ( 1 ) 2 A l ( z , T ) [ R * ( A m A n * ) ] + Q p l m n ( 2 ) A l * ( z , T ) [ ( R e 2 i ω 0 T ) * ( A m A n ) ] } .
y ( z ) = ( A ˜ 1 A ˜ M ) , A ˜ p = ( A ˜ p ( z , ω 1 ω 0 ) A ˜ p ( z , ω N ω 0 ) )
y ( 0 ) = ( A ˜ 1 ( 0 , ω 1 ω 0 ) A ˜ M ( 0 , ω N ω 0 ) ) .
P s ( z , ω ) = 2 tr Φ 2 ( z , ω ) tr 2 Φ ( z , ω ) 1 ,
Φ i j ( z , ω ) = A ˜ i * ( z , ω ) A ˜ j ( z , ω ) ,

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