Abstract

A detailed investigation of the nonlinear multimodal interference in a short graded-index multimode optical fiber is presented. The analysis is performed for a specific device geometry, where the light is coupled in and out of the multimode fiber via single-mode fibers. The same device geometry was recently used to obtain ultra-low-loss coupling between two single-mode optical fibers with very different mode-field diameters. Our results indicate the potential application of this simple geometry for nonlinear devices, such as in nonlinear switching, optical signal processing, or as saturable absorbers in mode-locked fiber lasers. Saturable absorption in this all-fiber configuration is discussed and it is shown that it provides attractive properties that can potentially be used in high pulse energy mode-locked fiber lasers.

© 2013 Optical Society of America

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References

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2012

2011

2010

2009

X. Zhu, A. Schülzgen, L. Li, and N. Peyghambarian, “Generation of controllable nondiffracting beams using multimode optical fibers,” Appl. Phys. Lett. 94, 201102 (2009).
[CrossRef]

2008

2007

2006

2005

J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933–8950 (2005).
[CrossRef]

G. J. Liu, B. M. Liang, Q. Li, and G. L. Jin, “Beam propagation in nonlinear multimode interference waveguide,” J. Opt. A 7, 457–462 (2005).
[CrossRef]

2004

2003

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef]

A. Mehta, W. S. Mohammed, and E. G. Johnson, “Multimode interference based fiber optic displacement sensor,” IEEE Photon. Technol. Lett. 15, 1129–1131 (2003).
[CrossRef]

1998

F. Kartner, J. aus der Au, and U. Keller, “Slow and fast saturable absorbers for modelocking of solid state lasers—what’s the difference?,” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

1992

1990

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

1984

1983

M. I. Dzhibladze, Z. G. Esiashvili, E. S. Teplitskii, S. K. Isaev, and V. R. Sagaradze, “Mode-locking in a fiber laser,” Kvantovaya Elektron. 10, 432–434 (1983).
[CrossRef]

1982

1975

1973

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2011).

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

Ashkin, A.

aus der Au, J.

F. Kartner, J. aus der Au, and U. Keller, “Slow and fast saturable absorbers for modelocking of solid state lasers—what’s the difference?,” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

Botineau, J.

Buckley, J.

Chong, A.

Christodoulides, D. N.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef]

Dzhibladze, M. I.

M. I. Dzhibladze, Z. G. Esiashvili, E. S. Teplitskii, S. K. Isaev, and V. R. Sagaradze, “Mode-locking in a fiber laser,” Kvantovaya Elektron. 10, 432–434 (1983).
[CrossRef]

Esiashvili, Z. G.

M. I. Dzhibladze, Z. G. Esiashvili, E. S. Teplitskii, S. K. Isaev, and V. R. Sagaradze, “Mode-locking in a fiber laser,” Kvantovaya Elektron. 10, 432–434 (1983).
[CrossRef]

Farahi, F.

M. Mayeh and F. Farahi, “Laser beam shaping and mode conversion in optical fibers,” Photonic Sens. 1, 187–198 (2011).
[CrossRef]

Farrell, G.

Gloge, D.

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Gu, X.

Han, L.

Hofmann, P.

Isaev, S. K.

M. I. Dzhibladze, Z. G. Esiashvili, E. S. Teplitskii, S. K. Isaev, and V. R. Sagaradze, “Mode-locking in a fiber laser,” Kvantovaya Elektron. 10, 432–434 (1983).
[CrossRef]

Jin, G. L.

G. J. Liu, B. M. Liang, Q. Li, and G. L. Jin, “Beam propagation in nonlinear multimode interference waveguide,” J. Opt. A 7, 457–462 (2005).
[CrossRef]

Johnson, E. G.

Kartner, F.

F. Kartner, J. aus der Au, and U. Keller, “Slow and fast saturable absorbers for modelocking of solid state lasers—what’s the difference?,” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

Keller, U.

F. Kartner, J. aus der Au, and U. Keller, “Slow and fast saturable absorbers for modelocking of solid state lasers—what’s the difference?,” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

Kutz, J. N.

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef]

Li, H.

Li, L.

Li, Q.

G. J. Liu, B. M. Liang, Q. Li, and G. L. Jin, “Beam propagation in nonlinear multimode interference waveguide,” J. Opt. A 7, 457–462 (2005).
[CrossRef]

Liang, B. M.

G. J. Liu, B. M. Liang, Q. Li, and G. L. Jin, “Beam propagation in nonlinear multimode interference waveguide,” J. Opt. A 7, 457–462 (2005).
[CrossRef]

Liu, G. J.

G. J. Liu, B. M. Liang, Q. Li, and G. L. Jin, “Beam propagation in nonlinear multimode interference waveguide,” J. Opt. A 7, 457–462 (2005).
[CrossRef]

Mafi, A.

Marcatili, E. A. J.

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

Mayeh, M.

M. Mayeh and F. Farahi, “Laser beam shaping and mode conversion in optical fibers,” Photonic Sens. 1, 187–198 (2011).
[CrossRef]

Mehta, A.

Mohammed, W. S.

Mollenauer, L. F.

Moloney, J. V.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic, 2006).

Olshansky, R.

Peyghambarian, N.

Proctor, J.

Ramachandran, S.

Renninger, W.

Sagaradze, V. R.

M. I. Dzhibladze, Z. G. Esiashvili, E. S. Teplitskii, S. K. Isaev, and V. R. Sagaradze, “Mode-locking in a fiber laser,” Kvantovaya Elektron. 10, 432–434 (1983).
[CrossRef]

Salvin, C.

Salvin, C. J.

Schülzgen, A.

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef]

Smith, P. W. E.

Stegeman, G. I.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

Stolen, R. H.

Sumetsky, M.

Teplitskii, E. S.

M. I. Dzhibladze, Z. G. Esiashvili, E. S. Teplitskii, S. K. Isaev, and V. R. Sagaradze, “Mode-locking in a fiber laser,” Kvantovaya Elektron. 10, 432–434 (1983).
[CrossRef]

Tiess, T.

Walton, D. T.

Wang, Q.

Wei, H.

Winful, H. G.

Wise, F.

Wright, E. M.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

Yilmaz, Y. O.

Zhu, X.

Appl. Opt.

Appl. Phys. Lett.

X. Zhu, A. Schülzgen, L. Li, and N. Peyghambarian, “Generation of controllable nondiffracting beams using multimode optical fibers,” Appl. Phys. Lett. 94, 201102 (2009).
[CrossRef]

Bell Syst. Tech. J.

D. Gloge and E. A. J. Marcatili, “Multimode theory of graded-core fibers,” Bell Syst. Tech. J. 52, 1563–1578 (1973).

IEEE J. Sel. Top. Quantum Electron.

F. Kartner, J. aus der Au, and U. Keller, “Slow and fast saturable absorbers for modelocking of solid state lasers—what’s the difference?,” IEEE J. Sel. Top. Quantum Electron. 4, 159–168 (1998).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Mehta, W. S. Mohammed, and E. G. Johnson, “Multimode interference based fiber optic displacement sensor,” IEEE Photon. Technol. Lett. 15, 1129–1131 (2003).
[CrossRef]

J. Lightwave Technol.

J. Opt. A

G. J. Liu, B. M. Liang, Q. Li, and G. L. Jin, “Beam propagation in nonlinear multimode interference waveguide,” J. Opt. A 7, 457–462 (2005).
[CrossRef]

Kvantovaya Elektron.

M. I. Dzhibladze, Z. G. Esiashvili, E. S. Teplitskii, S. K. Isaev, and V. R. Sagaradze, “Mode-locking in a fiber laser,” Kvantovaya Elektron. 10, 432–434 (1983).
[CrossRef]

Nature

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

G. I. Stegeman and E. M. Wright, “All-optical waveguide switching,” Opt. Quantum Electron. 22, 95–122 (1990).
[CrossRef]

Photonic Sens.

M. Mayeh and F. Farahi, “Laser beam shaping and mode conversion in optical fibers,” Photonic Sens. 1, 187–198 (2011).
[CrossRef]

Other

K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic, 2006).

G. P. Agrawal, Fiber-Optic Communication Systems, 4th ed. (Wiley, 2011).

G. P. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2012).

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Figures (18)

Fig. 1.
Fig. 1.

GIMF of length L is used as an intermediate coupler between two SMF fibers. In [9,10], this geometry was used to create very low-loss couplers between two SMFs with very different mode-field diameters. In this paper, nonlinear MMI effects for identical input and output SMFs are explored.

Fig. 2.
Fig. 2.

Excitation amplitudes of the LGp0 modes from Eq. (17) are plotted as a function of the radial number p, for three different values of the η parameter.

Fig. 3.
Fig. 3.

Relative power transmission is plotted as a function of the normalized GIMF length L˜ for the case of LG00 and LG10 modes when p0=p1=0.5 at ζ=0, for γ˜=0 (solid), γ˜=0.7 (dashed), and γ˜=3 (dotted).

Fig. 4.
Fig. 4.

Same as Fig. 3, except for p0=0.75 and p1=0.25 at ζ=0.

Fig. 5.
Fig. 5.

Same as Fig. 3, except for p0=0.25 and p1=0.75 at ζ=0.

Fig. 6.
Fig. 6.

This figure shows the exchange of power between the LG00 and LG10 modes, where p0 at the output of the GIMF is plotted as a function of p0 at the input for L˜=4.5π.

Fig. 7.
Fig. 7.

Same as Fig. 6, except for L˜=50π, and different choices of γ˜.

Fig. 8.
Fig. 8.

Relative power transmission is plotted as a function of γ˜ (thus the total power) for L˜=4.5π for the case of LG00 and LG10 modes, when p0=0.5, 0.25, 0.75 at ζ=0, in solid, dashed, and dotted lines, respectively.

Fig. 9.
Fig. 9.

Same as Fig. 8, except for L˜=100.5π.

Fig. 10.
Fig. 10.

Relative power transmission is plotted as a function of the normalized GIMF length L˜ for the case of LG00 through LG40 modes (five zero angular modes) when p0=0.5 at ζ=0 (η=3+8), for γ˜=0 (solid), γ˜=0.7 (dashed), and γ˜=3 (dotted). The results should be compared with Fig. 3 where only two modes, LG00 and LG10, were considered.

Fig. 11.
Fig. 11.

Same as Fig. 10, except for p0=0.75 at ζ=0 (η=3). The results should be compared with Fig. 4 where only two modes, LG00 and LG10, were considered.

Fig. 12.
Fig. 12.

Same as Fig. 10, except for p0=0.25 at ζ=0 (η=7+48). The results should be compared with Fig. 5 where only two modes, LG00 and LG10, were considered.

Fig. 13.
Fig. 13.

Relative power transmission is plotted as a function of γ˜ (thus the total power) for L˜=4.5π for the case of five modes, when p0=0.5, 0.25, 0.75 at ζ=0, in solid, dashed, and dotted lines, respectively.

Fig. 14.
Fig. 14.

Same as Fig. 13, except for L˜=100.5π.

Fig. 15.
Fig. 15.

Relative change in the normalized power transmission in the SMF-GIMF-SMF geometry defined in Eq. (22) plotted as a function of δL˜, where the length of the GIMF section is given by L˜=300π+δL˜. A large positive value of (δτ/τ) is desirable for saturable absorption. The plots are presented for η=2 (solid) and η=3 (dashed).

Fig. 16.
Fig. 16.

Relative power transmission is shown as function of δL˜, where the length of the GIMF section is given by L˜=300π+δL˜. τ presented in this figure is the average value between τ|γ˜=0.001 and τ|γ˜=0. This figure is intended to show that the value of τ is not too low near the peak value of (δτ/τ) in Fig. 15. The plots are presented for η=2 (solid) and η=3 (dashed).

Fig. 17.
Fig. 17.

Same as Fig. 15, except for η=4.

Fig. 18.
Fig. 18.

Same as Fig. 16, except for η=4.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

n2(ρ)=n02[12Δ(ρR)α],
Ep,m(ρ,ϕ)=Npmρ|m|ρ0|m|+1exp(ρ22ρ02)Lp|m|(ρ2ρ02)eimϕ,
ρ0=R1/2(k0n0)1/2(2Δ)1/4,Npm=p!π(p+|m|)!,
β(g)=n0k(12ΔXg)1/2,
Xg=(gNα)2α/(α+2),
Nα=αα+2n02k02R2Δ.
Aμz=iδβμAμ+iγν,κ,ξη˜μνκξAνAκAξ,
γμνκξ=(n2ω0c)d2xEμEνEκEξ.
Bμ(z)=1P˜Aμ(z)eiγP˜z,
ζ=z×(β(1)β(2)).
ζBμ=i(rμ+γ˜)Bμ+iγ˜ν,κ,ξη˜μνκξBνBκBξ,
rμ=βμβ(1)β(2)β(1),γ˜=γP˜β(1)β(2).
τ=1P˜|out|E(ρ,ϕ,L)|2,
E(ρ,ϕ,L)=eiβ(1)LμAμ(L)Eμ(ρ,ϕ).
τ=1P˜2|μAμ(0)Aμ(z)|2=|μBμ(0)Bμ(z)|2.
|in=2πw2exp(ρ2w2).
Bp(0)=2ηη+1Ψp,η=ρc2w2,Ψ=η1η+1.
ζB0=iγ˜(|B0|2+14|B1|2)B1+iγ˜2(B02B1+B12B0),ζB1=ir1B1+iγ˜2B0+iγ˜2(|B1|2B1+B02B1+12B12B0).
B0(ζ)=B0(0),B1(ζ)=eir1ζB1(0).
τ=14p0p1sin2(r1L˜2).
τmax=(1Ψ2P)21,
(δτ/τ)=1τ(τ|γ˜=0.001τ|γ˜=0),

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