Abstract

The Gaussian approximation of waveguide modes is used to analyze a Kerr-type nonlinearity of an optical fiber. This analysis gives physical insights into the behavior of the optimal transverse size of a waveguide providing the maximum enhancement of nonlinear-optical processes as a function of the core-cladding refractive-index step and the radiation wavelength. We show that, for the maximum enhancement of nonlinear-optical processes in a fiber where only the core is nonlinear, the effective mode area needs to be slightly larger than its minimum value. In the case of fused-silica fibers, the maximum enhancement of nonlinear-optical processes is achieved with fiber core diameters that are less than the radiation wavelength.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2004 (6)

2003 (12)

L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2003), http://www.opticsexpress.org.
[CrossRef]

S. O. Konorov and A. M. Zheltikov, "Frequency conversion of subnanojoule femtosecond laser pulses in a microstructure fiber for photochromism initiation," Opt. Express 11, 2440-2445 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

D. A. Akimov, E. E. Serebryannikov, A. M. Zheltikov, M. Schmitt, R. Maksimenka, W. Kiefer, K. V. Dukel'skii, V. S. Shevandin, Yu. N. Kondrat'ev, "Efficient anti-Stokes generation through phase-matched four wave mixing in higher-order modes of a microstructure fiber," Opt. Lett. 28, 1948-1950 (2003).
[CrossRef] [PubMed]

M. D. Nielsen and N. A. Mortensen, "Photonic crystal fiber design based on the V-parameter," Opt. Express 11, 2762-2768 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

M. D. Nielsen, N. A. Mortensen, J. R. Folkenberg, and A. Bjarklev, "Mode-field radius of photonic crystal fibers expressed by the V-parameter," Opt. Lett. 28, 2309-2311 (2003).
[CrossRef] [PubMed]

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

V. C. Sundar, A. D. Yablon, J. L. Grazul, M. Ilan, and J. Aizenberg, "Fibre-optical features of a glass sponge," Nature (London) 424, 899-900 (2003).
[CrossRef]

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

A. M. Zheltikov, "The physical limit for the waveguide enhancement of nonlinear-optical processes," Opt. Spectrosc. 95, 410-415 (2003).
[CrossRef]

A. M. Zheltikov, "Limiting efficiencies of nonlinear-optical processes in microstructure fibers," JETP 97, 505-521 (2003).
[CrossRef]

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photonics Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

2002 (1)

J. C. Johnson, H.-J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, "Single gallium nitride nanowire lasers," Nat. Mater. 1, 106-110 (2002).
[CrossRef]

2001 (1)

X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, "Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices," Nature (London) 409, 66-69 (2001).
[CrossRef]

1993 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

Aizenberg, J.

V. C. Sundar, A. D. Yablon, J. L. Grazul, M. Ilan, and J. Aizenberg, "Fibre-optical features of a glass sponge," Nature (London) 424, 899-900 (2003).
[CrossRef]

Akimov, D.

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

Akimov, D. A.

Ashcom, J. B.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

Birks, T. A.

Bjarklev, A.

Bolger, J. A.

Bures, J.

Choi, H.-J.

J. C. Johnson, H.-J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, "Single gallium nitride nanowire lasers," Nat. Mater. 1, 106-110 (2002).
[CrossRef]

Cui, Y.

X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, "Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices," Nature (London) 409, 66-69 (2001).
[CrossRef]

Duan, X.

X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, "Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices," Nature (London) 409, 66-69 (2001).
[CrossRef]

Dukel'skii, K.

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

Dukel'skii, K. V.

Dumais, P.

Eggleton, B. J.

Finazzi, V.

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photonics Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

Folkenberg, J. R.

Foster, M. A.

Gaeta, A. L.

Gattass, R. R.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

Gonthier, F.

Grazul, J. L.

V. C. Sundar, A. D. Yablon, J. L. Grazul, M. Ilan, and J. Aizenberg, "Fibre-optical features of a glass sponge," Nature (London) 424, 899-900 (2003).
[CrossRef]

He, S.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

Huang, Y.

X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, "Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices," Nature (London) 409, 66-69 (2001).
[CrossRef]

Ilan, M.

V. C. Sundar, A. D. Yablon, J. L. Grazul, M. Ilan, and J. Aizenberg, "Fibre-optical features of a glass sponge," Nature (London) 424, 899-900 (2003).
[CrossRef]

Johnson, J. C.

J. C. Johnson, H.-J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, "Single gallium nitride nanowire lasers," Nat. Mater. 1, 106-110 (2002).
[CrossRef]

Kiefer, W.

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

D. A. Akimov, E. E. Serebryannikov, A. M. Zheltikov, M. Schmitt, R. Maksimenka, W. Kiefer, K. V. Dukel'skii, V. S. Shevandin, Yu. N. Kondrat'ev, "Efficient anti-Stokes generation through phase-matched four wave mixing in higher-order modes of a microstructure fiber," Opt. Lett. 28, 1948-1950 (2003).
[CrossRef] [PubMed]

Knutsen, K. P.

J. C. Johnson, H.-J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, "Single gallium nitride nanowire lasers," Nat. Mater. 1, 106-110 (2002).
[CrossRef]

Kohkhlov, A.

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

Kondrat'ev, Y.

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

Kondrat'ev, Yu. N.

Konorov, S. O.

Koshiba, M.

Lacroix, S.

Leon-Saval, S. G.

Lieber, C. M.

X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, "Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices," Nature (London) 409, 66-69 (2001).
[CrossRef]

Lizé, Y. K.

Lou, J.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2003), http://www.opticsexpress.org.
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, Dordrecht, The Netherlands, 1983).

Mägi, E. C.

Maksimenka, R.

D. A. Akimov, E. E. Serebryannikov, A. M. Zheltikov, M. Schmitt, R. Maksimenka, W. Kiefer, K. V. Dukel'skii, V. S. Shevandin, Yu. N. Kondrat'ev, "Efficient anti-Stokes generation through phase-matched four wave mixing in higher-order modes of a microstructure fiber," Opt. Lett. 28, 1948-1950 (2003).
[CrossRef] [PubMed]

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

Mason, M. W.

Maxwell, I.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

Mazur, E.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2003), http://www.opticsexpress.org.
[CrossRef]

Moll, K. D.

Monro, T. M.

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photonics Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

Mortensen, N. A.

Nielsen, M. D.

Richardson, D. J.

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photonics Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

Russell, P. St. J.

Saitoh, K.

Saykally, R. J.

J. C. Johnson, H.-J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, "Single gallium nitride nanowire lasers," Nat. Mater. 1, 106-110 (2002).
[CrossRef]

Schaller, R. D.

J. C. Johnson, H.-J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, "Single gallium nitride nanowire lasers," Nat. Mater. 1, 106-110 (2002).
[CrossRef]

Schmitt, M.

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

D. A. Akimov, E. E. Serebryannikov, A. M. Zheltikov, M. Schmitt, R. Maksimenka, W. Kiefer, K. V. Dukel'skii, V. S. Shevandin, Yu. N. Kondrat'ev, "Efficient anti-Stokes generation through phase-matched four wave mixing in higher-order modes of a microstructure fiber," Opt. Lett. 28, 1948-1950 (2003).
[CrossRef] [PubMed]

Serebryannikov, E. E.

Shen, M.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

Shevandin, V.

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

Shevandin, V. S.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, Dordrecht, The Netherlands, 1983).

Stegeman, G. I.

Steinvurzel, P.

Sundar, V. C.

V. C. Sundar, A. D. Yablon, J. L. Grazul, M. Ilan, and J. Aizenberg, "Fibre-optical features of a glass sponge," Nature (London) 424, 899-900 (2003).
[CrossRef]

Ta'eed, V. G.

Tong, L.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2003), http://www.opticsexpress.org.
[CrossRef]

Villeneuve, A.

Wadsworth, W. J.

Wang, J.

X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, "Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices," Nature (London) 409, 66-69 (2001).
[CrossRef]

Wigley, P. G. J.

Yablon, A. D.

V. C. Sundar, A. D. Yablon, J. L. Grazul, M. Ilan, and J. Aizenberg, "Fibre-optical features of a glass sponge," Nature (London) 424, 899-900 (2003).
[CrossRef]

Yang, P.

J. C. Johnson, H.-J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, "Single gallium nitride nanowire lasers," Nat. Mater. 1, 106-110 (2002).
[CrossRef]

Zheltikov, A. M.

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

D. A. Akimov, E. E. Serebryannikov, A. M. Zheltikov, M. Schmitt, R. Maksimenka, W. Kiefer, K. V. Dukel'skii, V. S. Shevandin, Yu. N. Kondrat'ev, "Efficient anti-Stokes generation through phase-matched four wave mixing in higher-order modes of a microstructure fiber," Opt. Lett. 28, 1948-1950 (2003).
[CrossRef] [PubMed]

S. O. Konorov and A. M. Zheltikov, "Frequency conversion of subnanojoule femtosecond laser pulses in a microstructure fiber for photochromism initiation," Opt. Express 11, 2440-2445 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

A. M. Zheltikov, "The physical limit for the waveguide enhancement of nonlinear-optical processes," Opt. Spectrosc. 95, 410-415 (2003).
[CrossRef]

A. M. Zheltikov, "Limiting efficiencies of nonlinear-optical processes in microstructure fibers," JETP 97, 505-521 (2003).
[CrossRef]

Appl. Phys. B: Lasers Opt. (1)

D. Akimov, M. Schmitt, R. Maksimenka, K. Dukel'skii, Y. Kondrat'ev, A. Kohkhlov, V. Shevandin, W. Kiefer, and A. M. Zheltikov, "Supercontinuum generation in a multiple-submicron-core microstructure fiber: toward limiting waveguide enhancement of nonlinear-optical processes," Appl. Phys. B: Lasers Opt. 77, 299-305 (2003).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

V. Finazzi, T. M. Monro, and D. J. Richardson, "The role of confinement loss in highly nonlinear silica holey fibers," IEEE Photonics Technol. Lett. 15, 1246-1248 (2003).
[CrossRef]

JETP (1)

A. M. Zheltikov, "Limiting efficiencies of nonlinear-optical processes in microstructure fibers," JETP 97, 505-521 (2003).
[CrossRef]

Nat. Mater. (1)

J. C. Johnson, H.-J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, and R. J. Saykally, "Single gallium nitride nanowire lasers," Nat. Mater. 1, 106-110 (2002).
[CrossRef]

Nature (London) (3)

X. Duan, Y. Huang, Y. Cui, J. Wang, and C. M. Lieber, "Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices," Nature (London) 409, 66-69 (2001).
[CrossRef]

V. C. Sundar, A. D. Yablon, J. L. Grazul, M. Ilan, and J. Aizenberg, "Fibre-optical features of a glass sponge," Nature (London) 424, 899-900 (2003).
[CrossRef]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

Opt. Express (8)

S. O. Konorov and A. M. Zheltikov, "Frequency conversion of subnanojoule femtosecond laser pulses in a microstructure fiber for photochromism initiation," Opt. Express 11, 2440-2445 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

E. C. Mägi, P. Steinvurzel, and B. J. Eggleton, "Tapered photonic crystal fibers," Opt. Express 12, 776-784 (2004), http://www.opticsexpress.org.
[CrossRef] [PubMed]

L. Tong, J. Lou, and E. Mazur, "Single-mode guiding properties of subwavelength-diameter silica and silicon wire waveguides," Opt. Express 12, 1025-1035 (2003), http://www.opticsexpress.org.
[CrossRef]

S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, "Supercontinuum generation in submicron fibre waveguides," Opt. Express 12, 2864-2869 (2004), http://www.opticsexpress.org.
[CrossRef] [PubMed]

M. A. Foster, K. D. Moll, and A. L. Gaeta, "Optimal waveguide dimensions for nonlinear interactions," Opt. Express 12, 2880-2887 (2004), http://www.opticsexpress.org.
[CrossRef] [PubMed]

M. A. Foster and A. L. Gaeta, "Ultra-low threshold supercontinuum generation in sub-wavelength waveguides," Opt. Express 12, 3137-3143 (2004), http://www.opticsexpress.org.
[CrossRef] [PubMed]

Y. K. Lizé, E. C. Mägi, V. G. Ta'eed, J. A. Bolger, P. Steinvurzel, and B. J. Eggleton, "Microstructured optical fiber photonic wires with subwavelength core diameter," Opt. Express 12, 3209-3217 (2004), http://www.opticsexpress.org.
[CrossRef] [PubMed]

M. D. Nielsen and N. A. Mortensen, "Photonic crystal fiber design based on the V-parameter," Opt. Express 11, 2762-2768 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

Opt. Lett. (4)

Opt. Spectrosc. (1)

A. M. Zheltikov, "The physical limit for the waveguide enhancement of nonlinear-optical processes," Opt. Spectrosc. 95, 410-415 (2003).
[CrossRef]

Science (1)

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

Other (2)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, Dordrecht, The Netherlands, 1983).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

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Figures (3)

Fig. 1
Fig. 1

Normalized nonlinearity γ λ 2 ( 8 π n co Δ n 2 co ) (curve 1), and the normalized effective mode area n co Δ A λ 2 (curve 2) as functions of the V parameter for Gaussian modes.

Fig. 2
Fig. 2

Optimal diameter of a fused-silica waveguide ( n co 1.45 ) providing the maximum nonlinearity normalized to the radiation wavelength λ as a function of the ratio n cl n co of the refractive indices of the cladding and the core calculated with the use of the Gaussian-mode expression [Eq. (17)] (curve 1) and according to the FMG formula [Eq. (18)] (curve 2).

Fig. 3
Fig. 3

Normalized nonlinearities γ 1 λ 2 ( 8 π n co Δ n 2 co ) (curve 1) and γ 2 λ 2 ( 8 π n co Δ n 2 co ) (curve 2), and the total nonlinearity γ λ 2 ( 8 π n co Δ n 2 co ) (curve 3) as functions of the V parameter for Gaussian modes with n 2 cl n 2 co = 10 .

Equations (21)

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γ = 2 π λ n 2 ( r ) S z 2 d 2 r S z d 2 r 2 ,
S z = ( E H * ) z
E ( x , y , z ) = E 0 F ( r ) e ( x , y ) exp ( i β z ) ,
H ( x , y , z ) = H 0 F ( r ) h ( x , y ) exp ( i β z ) ,
F ( r ) = exp [ 1 2 ( r r 0 ) 2 ] ,
S z a 2 exp ( r 2 r 0 2 ) ,
n 2 ( r ) = { n 2 co , r a n 2 cl , r > a } ,
γ = γ 1 + γ 2 ,
γ 1 n 2 co λ r 0 2 [ 1 exp ( 2 a 2 r 0 2 ) ] ,
γ 2 n 2 cl λ r 0 2 exp ( 2 a 2 r 0 2 ) .
n 2 ( r ) = n co 2 [ 1 2 Δ f ( r ) ] ,
f ( r ) = { 0 , r a 1 , r > a } ,
r 0 = a ( 2 ln V ) 1 2 ,
V = k a n co ( 2 Δ ) 1 2 ,
γ 1 2 n 2 co ln V λ a 2 ( 1 1 V 4 ) ,
γ 2 2 n 2 cl ln V λ a 2 V 4 .
d opt = 2 a opt = 0.59 λ ( n co n cl ) 1 2 ( n co + n cl ) 1 2 .
d opt FMG = 0.854 λ ( n co n cl ) 0.4 ( n co + n cl ) 0.6 .
A = ( F ( r ) d 2 r ) 2 F ( r ) 4 d 2 r
d A = λ exp ( 1 2 ) π ( n co n cl ) 1 2 ( n co + n cl ) 1 2 .
d ¯ opt = 0.32 λ ( n co n cl ) 1 2 ( n co + n cl ) 1 2 .

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