Abstract

Design of a mid-wave IR (MWIR) broad-band fiber-based light source exploiting degenerate four-wave mixing (D-FWM) in a meter long suitably designed highly nonlinear (NL) chalcogenide microstructured optical fiber (MOF) is reported. This superior FWM bandwidth (BW) was obtained through precise tailoring of the fiber’s dispersion profile so as to realize positive quartic dispersion at the pump wavelength. We consider an Erbium (Er3+) - doped continuous wave (CW) ZBLAN fiber laser emitting at 2.8 μm as the pump source with an average power of 5 W. Amplification factor as high as 25 dB is achievable in the 3 – 3.9 μm spectral range with average power conversion efficiency > 32%.

© 2013 OSA

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  8. J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Computational study of 3-5 microm source created by using supercontinuum generation in As2S3 chalcogenide fibers with a pump at 2 microm,” Opt. Lett.35(17), 2907–2909 (2010).
    [CrossRef] [PubMed]
  9. C. M. B. Cordeiro, W. J. Wadsworth, T. A. Birks, and P. S. J. Russell, “Engineering the dispersion of tapered fibers for supercontinuum generation with a 1064 nm pump laser,” Opt. Lett.30(15), 1980–1982 (2005).
    [CrossRef] [PubMed]
  10. D. I. Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. B. Fu, and B. J. Eggleton, “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires,” Opt. Lett.33(7), 660–662 (2008).
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    [CrossRef] [PubMed]
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    [CrossRef]
  22. G. Tao, S. Shabahang, E. H. Banaei, J. J. Kaufman, and A. F. Abouraddy, “Multimaterial preform coextrusion for robust chalcogenide optical fibers and tapers,” Opt. Lett.37(13), 2751–2753 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
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2013 (1)

A. Barh, S. Ghosh, G. P. Agrawal, R. K. Varshney, I. D. Aggarwal, and B. P. Pal, “Design of an efficient mid-IR light source using chalcogenide holey fibers: a numerical study,” J. Opt.15(3), 035205 (2013).
[CrossRef]

2012 (2)

2011 (3)

B. J. Eggleton, B. L. Davies, and K. Richardson, “Chalcogenide photonics,” Review Articles - Nat. Photonics5, 141–148 (2011).

C. S. Brès, S. Zlatanovic, A. O. J. Wiberg, and S. Radic, “Continuous-wave four-wave mixing in cm-long chalcogenide microstructured fiber,” Opt. Express19(26), B621–B627 (2011).
[CrossRef] [PubMed]

D. W. Hewak, “The promise of chalcogenides,” Nat. Photonics5(8), 474 (2011).
[CrossRef]

2010 (4)

2009 (1)

2008 (2)

T. Yamashita and Y. Ohishi, “Cooperative energy transfer between Tb3+ and Yb3+ ions co-doped in borosilicate glass,” J. Non-Cryst. Solids354(17), 1883–1890 (2008).
[CrossRef]

D. I. Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. B. Fu, and B. J. Eggleton, “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires,” Opt. Lett.33(7), 660–662 (2008).
[CrossRef] [PubMed]

2007 (1)

2005 (2)

2004 (1)

G. Boudebs, S. Cherukulappurath, M. Guignard, J. Troles, F. Smektala, and F. Sanchez, “Linear optical characterization of chalcogenide glasses,” Opt. Commun.230(4-6), 331–336 (2004).
[CrossRef]

2003 (2)

2002 (1)

2001 (1)

J. S. Sanghera, I. D. Aggarwal, L. B. Shaw, L. E. Busse, P. Thielen, V. Nguyen, P. Pureza, S. Bayya, and F. Kung, “Applications of chalcogenide glass optical fibers at NRL,” J. Optoelectron. Adv. Mater.3, 627–640 (2001).

1991 (1)

1981 (1)

Abouraddy, A. F.

Achille, M.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Aggarwal, I. D.

Agrawal, G. P.

A. Barh, S. Ghosh, G. P. Agrawal, R. K. Varshney, I. D. Aggarwal, and B. P. Pal, “Design of an efficient mid-IR light source using chalcogenide holey fibers: a numerical study,” J. Opt.15(3), 035205 (2013).
[CrossRef]

Banaei, E. H.

Barh, A.

A. Barh, S. Ghosh, G. P. Agrawal, R. K. Varshney, I. D. Aggarwal, and B. P. Pal, “Design of an efficient mid-IR light source using chalcogenide holey fibers: a numerical study,” J. Opt.15(3), 035205 (2013).
[CrossRef]

Bayya, S.

J. S. Sanghera, I. D. Aggarwal, L. B. Shaw, L. E. Busse, P. Thielen, V. Nguyen, P. Pureza, S. Bayya, and F. Kung, “Applications of chalcogenide glass optical fibers at NRL,” J. Optoelectron. Adv. Mater.3, 627–640 (2001).

Birks, T. A.

Boudebs, G.

G. Boudebs, S. Cherukulappurath, M. Guignard, J. Troles, F. Smektala, and F. Sanchez, “Linear optical characterization of chalcogenide glasses,” Opt. Commun.230(4-6), 331–336 (2004).
[CrossRef]

Brès, C. S.

Brilland, L.

Broderick, N. G. R.

Busse, L.

Busse, L. E.

J. S. Sanghera, I. D. Aggarwal, L. B. Shaw, L. E. Busse, P. Thielen, V. Nguyen, P. Pureza, S. Bayya, and F. Kung, “Applications of chalcogenide glass optical fibers at NRL,” J. Optoelectron. Adv. Mater.3, 627–640 (2001).

Cappellini, G.

Chaudhari, C.

Cherukulappurath, S.

G. Boudebs, S. Cherukulappurath, M. Guignard, J. Troles, F. Smektala, and F. Sanchez, “Linear optical characterization of chalcogenide glasses,” Opt. Commun.230(4-6), 331–336 (2004).
[CrossRef]

Coen, S.

Cordeiro, C. M. B.

Davies, B. L.

B. J. Eggleton, B. L. Davies, and K. Richardson, “Chalcogenide photonics,” Review Articles - Nat. Photonics5, 141–148 (2011).

de Sterke, C. M.

Désévédavy, F.

Eggleton, B. J.

El-Amraoui, M.

Elliott, S. R.

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids330(1-3), 1–12 (2003).
[CrossRef]

Fatome, J.

Finazzi, V.

Florea, C.

Fortier, C.

Frédéric, S.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Fu, L. B.

Gadret, G.

Gao, W.

Ghosh, S.

A. Barh, S. Ghosh, G. P. Agrawal, R. K. Varshney, I. D. Aggarwal, and B. P. Pal, “Design of an efficient mid-IR light source using chalcogenide holey fibers: a numerical study,” J. Opt.15(3), 035205 (2013).
[CrossRef]

Gilles, R.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Guignard, M.

G. Boudebs, S. Cherukulappurath, M. Guignard, J. Troles, F. Smektala, and F. Sanchez, “Linear optical characterization of chalcogenide glasses,” Opt. Commun.230(4-6), 331–336 (2004).
[CrossRef]

Harbold, J. M.

Harvey, J. D.

Hervé, O.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Hewak, D. W.

D. W. Hewak, “The promise of chalcogenides,” Nat. Photonics5(8), 474 (2011).
[CrossRef]

Hu, J.

Ilday, F. Ö.

Jackson, S. D.

S. D. Jackson, “Towards high-power mid-infrared emission from a fibre laser,” Review Articles - Nat. Photonics6(7), 423–431 (2012).
[CrossRef]

Jean-Christophe, S.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Johann, T.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Jules, J. C.

Julien, F.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Kaufman, J. J.

Knight, J. C.

Kung, F.

J. S. Sanghera, I. D. Aggarwal, L. B. Shaw, L. E. Busse, P. Thielen, V. Nguyen, P. Pureza, S. Bayya, and F. Kung, “Applications of chalcogenide glass optical fibers at NRL,” J. Optoelectron. Adv. Mater.3, 627–640 (2001).

Lamont, M. R. E.

Laurent, B.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Leonhardt, R.

Lin, C.

Mägi, E. C.

Menyuk, C. R.

Messaddeq, Y.

Miklos, F.

Monro, T. M.

Nam, N. T.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Nguyen, V.

J. S. Sanghera, I. D. Aggarwal, L. B. Shaw, L. E. Busse, P. Thielen, V. Nguyen, P. Pureza, S. Bayya, and F. Kung, “Applications of chalcogenide glass optical fibers at NRL,” J. Optoelectron. Adv. Mater.3, 627–640 (2001).

Nguyen, V. Q.

Ohishi, Y.

Pal, B. P.

A. Barh, S. Ghosh, G. P. Agrawal, R. K. Varshney, I. D. Aggarwal, and B. P. Pal, “Design of an efficient mid-IR light source using chalcogenide holey fibers: a numerical study,” J. Opt.15(3), 035205 (2013).
[CrossRef]

Patrick, H.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Pearson, A. D.

Poletti, F.

Pureza, P.

J. S. Sanghera, I. D. Aggarwal, L. B. Shaw, L. E. Busse, P. Thielen, V. Nguyen, P. Pureza, S. Bayya, and F. Kung, “Applications of chalcogenide glass optical fibers at NRL,” J. Optoelectron. Adv. Mater.3, 627–640 (2001).

Quentin, C.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Radic, S.

Reed, W. A.

Richardson, D. J.

Richardson, K.

B. J. Eggleton, B. L. Davies, and K. Richardson, “Chalcogenide photonics,” Review Articles - Nat. Photonics5, 141–148 (2011).

Roelens, M. A. F.

Russell, P. S. J.

Sanchez, F.

G. Boudebs, S. Cherukulappurath, M. Guignard, J. Troles, F. Smektala, and F. Sanchez, “Linear optical characterization of chalcogenide glasses,” Opt. Commun.230(4-6), 331–336 (2004).
[CrossRef]

Sanghera, J. S.

Sen, R.

R. Sen and Central Glass and Ceramic Research Institute, Kolkata, India, Personal Communication, (2013).

Shabahang, S.

Shang, H. T.

Shaw, B.

Shaw, L. B.

Skripatchev, I.

Smektala, F.

St .J. Russell, P.

Suzuki, T.

Tao, G.

Thielen, P.

J. S. Sanghera, I. D. Aggarwal, L. B. Shaw, L. E. Busse, P. Thielen, V. Nguyen, P. Pureza, S. Bayya, and F. Kung, “Applications of chalcogenide glass optical fibers at NRL,” J. Optoelectron. Adv. Mater.3, 627–640 (2001).

Thierry, C.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Thierry, P.

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Trillo, S.

Troles, J.

Tse, V.

Varshney, R. K.

A. Barh, S. Ghosh, G. P. Agrawal, R. K. Varshney, I. D. Aggarwal, and B. P. Pal, “Design of an efficient mid-IR light source using chalcogenide holey fibers: a numerical study,” J. Opt.15(3), 035205 (2013).
[CrossRef]

Wadsworth, W. J.

Wiberg, A. O. J.

Wise, F. W.

Wong, G. K. L.

Yamashita, T.

T. Yamashita and Y. Ohishi, “Cooperative energy transfer between Tb3+ and Yb3+ ions co-doped in borosilicate glass,” J. Non-Cryst. Solids354(17), 1883–1890 (2008).
[CrossRef]

Yeom, D. I.

Zakery, A.

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids330(1-3), 1–12 (2003).
[CrossRef]

Zlatanovic, S.

J. Lightwave Technol. (1)

J. Non-Cryst. Solids (2)

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids330(1-3), 1–12 (2003).
[CrossRef]

T. Yamashita and Y. Ohishi, “Cooperative energy transfer between Tb3+ and Yb3+ ions co-doped in borosilicate glass,” J. Non-Cryst. Solids354(17), 1883–1890 (2008).
[CrossRef]

J. Opt. (1)

A. Barh, S. Ghosh, G. P. Agrawal, R. K. Varshney, I. D. Aggarwal, and B. P. Pal, “Design of an efficient mid-IR light source using chalcogenide holey fibers: a numerical study,” J. Opt.15(3), 035205 (2013).
[CrossRef]

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

J. Optoelectron. Adv. Mater. (1)

J. S. Sanghera, I. D. Aggarwal, L. B. Shaw, L. E. Busse, P. Thielen, V. Nguyen, P. Pureza, S. Bayya, and F. Kung, “Applications of chalcogenide glass optical fibers at NRL,” J. Optoelectron. Adv. Mater.3, 627–640 (2001).

Nat. Photonics (1)

D. W. Hewak, “The promise of chalcogenides,” Nat. Photonics5(8), 474 (2011).
[CrossRef]

Opt. Commun. (1)

G. Boudebs, S. Cherukulappurath, M. Guignard, J. Troles, F. Smektala, and F. Sanchez, “Linear optical characterization of chalcogenide glasses,” Opt. Commun.230(4-6), 331–336 (2004).
[CrossRef]

Opt. Express (5)

Opt. Lett. (7)

G. Tao, S. Shabahang, E. H. Banaei, J. J. Kaufman, and A. F. Abouraddy, “Multimaterial preform coextrusion for robust chalcogenide optical fibers and tapers,” Opt. Lett.37(13), 2751–2753 (2012).
[CrossRef] [PubMed]

J. D. Harvey, R. Leonhardt, S. Coen, G. K. L. Wong, J. C. Knight, W. J. Wadsworth, and P. St .J. Russell, “Scalar modulation instability in the normal dispersion regime by use of a photonic crystal fiber,” Opt. Lett.28(22), 2225 (2003).
[CrossRef] [PubMed]

C. Lin, W. A. Reed, A. D. Pearson, and H. T. Shang, “Phase matching in the minimum-chromatic-dispersion region of single-mode fibers for stimulated four-photon mixing,” Opt. Lett.6(10), 493–495 (1981).
[CrossRef] [PubMed]

J. M. Harbold, F. Ö. Ilday, F. W. Wise, J. S. Sanghera, V. Q. Nguyen, L. B. Shaw, and I. D. Aggarwal, “Highly nonlinear As-S-Se glasses for all-optical switching,” Opt. Lett.27(2), 119–121 (2002).
[CrossRef] [PubMed]

J. Hu, C. R. Menyuk, L. B. Shaw, J. S. Sanghera, and I. D. Aggarwal, “Computational study of 3-5 microm source created by using supercontinuum generation in As2S3 chalcogenide fibers with a pump at 2 microm,” Opt. Lett.35(17), 2907–2909 (2010).
[CrossRef] [PubMed]

C. M. B. Cordeiro, W. J. Wadsworth, T. A. Birks, and P. S. J. Russell, “Engineering the dispersion of tapered fibers for supercontinuum generation with a 1064 nm pump laser,” Opt. Lett.30(15), 1980–1982 (2005).
[CrossRef] [PubMed]

D. I. Yeom, E. C. Mägi, M. R. E. Lamont, M. A. F. Roelens, L. B. Fu, and B. J. Eggleton, “Low-threshold supercontinuum generation in highly nonlinear chalcogenide nanowires,” Opt. Lett.33(7), 660–662 (2008).
[CrossRef] [PubMed]

Proc. SPIE (1)

C. Quentin, B. Laurent, H. Patrick, N. T. Nam, C. Thierry, R. Gilles, M. Achille, F. Julien, S. Frédéric, P. Thierry, O. Hervé, S. Jean-Christophe, and T. Johann, “Fabrication of low losses chalcogenide photonic crystal fibers by molding process,” Proc. SPIE7598, 75980O, 75980O-9 (2010).
[CrossRef]

Review Articles - Nat. Photonics (2)

B. J. Eggleton, B. L. Davies, and K. Richardson, “Chalcogenide photonics,” Review Articles - Nat. Photonics5, 141–148 (2011).

S. D. Jackson, “Towards high-power mid-infrared emission from a fibre laser,” Review Articles - Nat. Photonics6(7), 423–431 (2012).
[CrossRef]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics, Academic, San Diego, Calif., (2007).

R. Sen and Central Glass and Ceramic Research Institute, Kolkata, India, Personal Communication, (2013).

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Figures (5)

Fig. 1
Fig. 1

Cross sectional view of the designed MOF. Cladding consists of 4 rings of borosilicate rods (white circles) embedded in the As2S3 matrix (black background). The diameter of the rod is d, and the centre to centre separation is denoted as pitch (Λ).

Fig. 2
Fig. 2

Dispersion characteristics of As2S3 and Borosilicate based solid core MOF for d/Λ = 0.5 and Λ = 2.5 μm. (a) D (blue solid curve) and β2 (pink dashed curve) variation with operating wavelength (λ); λZD = 2.792 μm. (b) Variation of β4 withλ.

Fig. 3
Fig. 3

(a) Variation of signal amplification factor (AFs) for different λp is shown. With λp coinciding at λZD ( = 2.792 μm), output signal spectrum is almost uniform around λp. With increase in λp from λZD, the BW as well as fluctuation increases. (b) Variation of AFs for a pump power of 5 W at 2.797 μm for different L (0.6 – 1.0 m).

Fig. 4
Fig. 4

Variations of (a) amplitude, (b) output power (Pout) and (c) amplification factor (AF) of pump, signal and idler with the fiber length (L) are shown. A weak idler of 10 mW at 2.19 μm is assumed along with 5 W of pump to initiate this D-FWM.

Fig. 5
Fig. 5

Variation of amplification factor (AF) for different PI,in with pumping at 2.792 μm, L = 1 m and P0 = 5 W. (a) The idler side of spectrum. (b) The signal side of spectrum. AFs as high as 25 dB is achievable, where overlap between two signal spectrum (one around 3.12 μm and another around 3.85 μm) makes the entire spectrum (3 – 3.9 μm) almost uniform.

Tables (1)

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Table 1 Variation of output signal power with input idler power

Equations (9)

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Δκ=γ P 0 +Δ k L
Δ k L = m=2,4,6... 2 β m ( ω P ) Ω S m m! +Δ k W
Ω s = 6| β 2 | β 4 ( 1± 1 β 4 γ P 0 3 β 2 2 )
A F S = P S,out / P I,in = ( γ P 0 /g ) 2 sin h 2 ( gL )
g= ( γ P 0 ) 2 ( Δκ /2 ) 2
d A p dz = α p A p 2 + i n 2 ω p c [ ( f pp | A p | 2 +2 k = i, s f pk | A k | 2 ) A p +2 f ppis A p * A i A s e jΔ k L z ]
d A i dz = α i A i 2 + i n 2 ω i c [ ( f ii | A i | 2 +2 k = p, s f ik | A k | 2 ) A i + f ispp A s * A p 2 e jΔ k L z ]
d A s dz = α s A s 2 + i n 2 ω s c [ ( f ss | A s | 2 +2 k = p, i f sk | A k | 2 ) A s + f sipp A i * A p 2 e jΔ k L z ]
f jk = | F j | 2 | F k | 2 | F j | 2 | F k | 2 f ijkl = F i * F j * F k F l [ | F i | 2 | F j | 2 | F k | 2 | F l | 2 ] 1/2

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