Abstract

We address the theory of temporal soliton switching in a planar geometry directional coupler constructed from silica and doped silica glass and operating at the central wavelength of 1.55 µm, significant for erbium-doped amplification. We formulate the field in the coupler in terms of the supermodes of the total structure and take account of the two transverse dimensions of the rectangular channels. In the case of the weak coupling between channels consistent with elimination of pulse breakup, the effect of the fields in the outer corner regions of the channels results in a switching intensity that differs significantly from that derived from coupled-mode theory on the basis of a slab model of the coupler.

© 1999 Optical Society of America

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  1. A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Vol. 7 of Oxford Series in Optical and Imaging Sciences (Clarendon, Oxford, 1995).
  2. S. Trillo, S. Wabnitz, E. M. Wright, G. I. Stegeman, “Soliton switching in fiber nonlinear directional couplers,” Opt. Lett. 13, 672–674 (1988).
    [CrossRef] [PubMed]
  3. E. Caglioti, S. Trillo, S. Wabnitz, B. Crosignani, P. Di Porto, “Finite-dimensional description of nonlinear pulse propagation in optical-fiber couplers with applications to soliton switching,” J. Opt. Soc. Am. B 7, 374–385 (1990).
    [CrossRef]
  4. F. S. Locati, M. Romagnoli, A. Tajani, S. Wabnitz, “Adiabatic femtosecond soliton active nonlinear directional coupler,” Opt. Lett. 17, 1213–1215 (1992).
    [CrossRef] [PubMed]
  5. J. S. Aitchison, A. Villeneuve, G. I. Stegeman, “All-optical switching in two cascaded nonlinear directional couplers,” Opt. Lett. 20, 698–700 (1995).
    [CrossRef] [PubMed]
  6. P. L. Chu, Y. S. Kivshar, B. A. Malomed, G.-D. Peng, M. L. Quiroga-Teixeiro, “Soliton controlling, switching, and splitting in nonlinear fused-fiber couplers,” J. Opt. Soc. Am. B 12, 898–903 (1995).
    [CrossRef]
  7. R. Schiek, “Time-resolved switching characteristic of the nonlinear directional coupler under consideration of susceptibility dispersion,” IEEE J. Quantum Electron. 27, 2150–2158 (1991).
    [CrossRef]
  8. A. L. Sala, M. G. Mirkov, B. G. Bagley, R. T. Deck, “Derivation of dimensional and material requirements for propagation and processing of temporal optical solitons in planar geometry channel waveguides,” Appl. Opt. 36, 7846–7852 (1997); erratum, 37, 1626 (1998).
    [CrossRef]
  9. Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
    [CrossRef]
  10. A. Kumar, A. N. Kaul, A. K. Ghatak, “Prediction of coupling length in a rectangular-core directional coupler: an accurate analysis,” Opt. Lett. 10, 86–88 (1985).
    [CrossRef] [PubMed]
  11. A. L. Sala, “Propagation and switching of light in rectangular waveguiding structures,” Ph.D. dissertation (University of Toledo, Toledo, Ohio, 1998).
  12. Higher-order dispersion terms, connected to additional terms in Eq. (23), are significant only in the case of pulse lengths of the order of 10 fs or less. See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989); M. Piche, J. Cormier, X. Zhu, “Bright optical soliton in the presence of fourth-order dispersion,” Opt. Lett. 21, 845–847 (1996).
    [CrossRef] [PubMed]
  13. J. Yang, “Improved theory of rectangular directional coupler,” M. S. thesis (University of Toledo, Toledo, Ohio, 1994).
  14. Under this condition the nonlinear response of the coupler becomes independent of the sign of n2 as demonstrated in F. J. Fraile-Pelaez, G. Assanto, D. R. Heatley, “Sign-dependent response of nonlinear directional couplers,” Opt. Commun. 77, 402–406 (1990). More generally, in the present formalism, a dependence on the sign of the nonlinearity is produced by the terms in Eqs. (14) proportional to Qll′ and Rll′, whereas in Eqs. (37) any arbitariness in the sign of n2 is eliminated by the requirement that n2 and β″ have opposite signs.
    [CrossRef]
  15. The switching power can be lowered somewhat, and the power remaining in channel 1 at powers above the switching threshold can be increased, at the expense of a further increase in the coupling length of the coupler. For example, in Ref. 2 the critical power for switching equals the peak power of the fundamental soliton, but the coupler has a coupling length Lc ≅ 6 LD. In our case, Lc ≅ 2LD and the critical switching power is approximately 2.7P0. The coupling length can be decreased significantly only for pulse lengths less than 100 fs, for which the present analysis cannot be assumed to apply.
  16. K. Yasumoto, N. Mitsunaga, H. Maeda, “Coupled-mode analysis of power-transfer characteristics in a three-waveguide nonlinear directional coupler,” J. Opt. Soc. Am. B 13, 621–627 (1996).
    [CrossRef]

1997

1996

1995

1992

1991

R. Schiek, “Time-resolved switching characteristic of the nonlinear directional coupler under consideration of susceptibility dispersion,” IEEE J. Quantum Electron. 27, 2150–2158 (1991).
[CrossRef]

1990

Under this condition the nonlinear response of the coupler becomes independent of the sign of n2 as demonstrated in F. J. Fraile-Pelaez, G. Assanto, D. R. Heatley, “Sign-dependent response of nonlinear directional couplers,” Opt. Commun. 77, 402–406 (1990). More generally, in the present formalism, a dependence on the sign of the nonlinearity is produced by the terms in Eqs. (14) proportional to Qll′ and Rll′, whereas in Eqs. (37) any arbitariness in the sign of n2 is eliminated by the requirement that n2 and β″ have opposite signs.
[CrossRef]

E. Caglioti, S. Trillo, S. Wabnitz, B. Crosignani, P. Di Porto, “Finite-dimensional description of nonlinear pulse propagation in optical-fiber couplers with applications to soliton switching,” J. Opt. Soc. Am. B 7, 374–385 (1990).
[CrossRef]

1988

1987

Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

1985

Agrawal, G. P.

Higher-order dispersion terms, connected to additional terms in Eq. (23), are significant only in the case of pulse lengths of the order of 10 fs or less. See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989); M. Piche, J. Cormier, X. Zhu, “Bright optical soliton in the presence of fourth-order dispersion,” Opt. Lett. 21, 845–847 (1996).
[CrossRef] [PubMed]

Aitchison, J. S.

Assanto, G.

Under this condition the nonlinear response of the coupler becomes independent of the sign of n2 as demonstrated in F. J. Fraile-Pelaez, G. Assanto, D. R. Heatley, “Sign-dependent response of nonlinear directional couplers,” Opt. Commun. 77, 402–406 (1990). More generally, in the present formalism, a dependence on the sign of the nonlinearity is produced by the terms in Eqs. (14) proportional to Qll′ and Rll′, whereas in Eqs. (37) any arbitariness in the sign of n2 is eliminated by the requirement that n2 and β″ have opposite signs.
[CrossRef]

Bagley, B. G.

Caglioti, E.

Chu, P. L.

Crosignani, B.

Deck, R. T.

Di Porto, P.

Fraile-Pelaez, F. J.

Under this condition the nonlinear response of the coupler becomes independent of the sign of n2 as demonstrated in F. J. Fraile-Pelaez, G. Assanto, D. R. Heatley, “Sign-dependent response of nonlinear directional couplers,” Opt. Commun. 77, 402–406 (1990). More generally, in the present formalism, a dependence on the sign of the nonlinearity is produced by the terms in Eqs. (14) proportional to Qll′ and Rll′, whereas in Eqs. (37) any arbitariness in the sign of n2 is eliminated by the requirement that n2 and β″ have opposite signs.
[CrossRef]

Ghatak, A. K.

Hasegawa, A.

A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Vol. 7 of Oxford Series in Optical and Imaging Sciences (Clarendon, Oxford, 1995).

Heatley, D. R.

Under this condition the nonlinear response of the coupler becomes independent of the sign of n2 as demonstrated in F. J. Fraile-Pelaez, G. Assanto, D. R. Heatley, “Sign-dependent response of nonlinear directional couplers,” Opt. Commun. 77, 402–406 (1990). More generally, in the present formalism, a dependence on the sign of the nonlinearity is produced by the terms in Eqs. (14) proportional to Qll′ and Rll′, whereas in Eqs. (37) any arbitariness in the sign of n2 is eliminated by the requirement that n2 and β″ have opposite signs.
[CrossRef]

Kaul, A. N.

Kivshar, Y. S.

Kodama, Y.

A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Vol. 7 of Oxford Series in Optical and Imaging Sciences (Clarendon, Oxford, 1995).

Kumar, A.

Locati, F. S.

Maeda, H.

Malomed, B. A.

Mirkov, M. G.

Mitsunaga, N.

Peng, G.-D.

Quiroga-Teixeiro, M. L.

Romagnoli, M.

Sala, A. L.

Schiek, R.

R. Schiek, “Time-resolved switching characteristic of the nonlinear directional coupler under consideration of susceptibility dispersion,” IEEE J. Quantum Electron. 27, 2150–2158 (1991).
[CrossRef]

Silberberg, Y.

Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

Stegeman, G. I.

Tajani, A.

Trillo, S.

Villeneuve, A.

Wabnitz, S.

Wright, E. M.

Yang, J.

J. Yang, “Improved theory of rectangular directional coupler,” M. S. thesis (University of Toledo, Toledo, Ohio, 1994).

Yasumoto, K.

Appl. Opt.

Appl. Phys. Lett.

Y. Silberberg, G. I. Stegeman, “Nonlinear coupling of waveguide modes,” Appl. Phys. Lett. 50, 801–803 (1987).
[CrossRef]

IEEE J. Quantum Electron.

R. Schiek, “Time-resolved switching characteristic of the nonlinear directional coupler under consideration of susceptibility dispersion,” IEEE J. Quantum Electron. 27, 2150–2158 (1991).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

Under this condition the nonlinear response of the coupler becomes independent of the sign of n2 as demonstrated in F. J. Fraile-Pelaez, G. Assanto, D. R. Heatley, “Sign-dependent response of nonlinear directional couplers,” Opt. Commun. 77, 402–406 (1990). More generally, in the present formalism, a dependence on the sign of the nonlinearity is produced by the terms in Eqs. (14) proportional to Qll′ and Rll′, whereas in Eqs. (37) any arbitariness in the sign of n2 is eliminated by the requirement that n2 and β″ have opposite signs.
[CrossRef]

Opt. Lett.

Other

The switching power can be lowered somewhat, and the power remaining in channel 1 at powers above the switching threshold can be increased, at the expense of a further increase in the coupling length of the coupler. For example, in Ref. 2 the critical power for switching equals the peak power of the fundamental soliton, but the coupler has a coupling length Lc ≅ 6 LD. In our case, Lc ≅ 2LD and the critical switching power is approximately 2.7P0. The coupling length can be decreased significantly only for pulse lengths less than 100 fs, for which the present analysis cannot be assumed to apply.

A. Hasegawa, Y. Kodama, Solitons in Optical Communications, Vol. 7 of Oxford Series in Optical and Imaging Sciences (Clarendon, Oxford, 1995).

A. L. Sala, “Propagation and switching of light in rectangular waveguiding structures,” Ph.D. dissertation (University of Toledo, Toledo, Ohio, 1998).

Higher-order dispersion terms, connected to additional terms in Eq. (23), are significant only in the case of pulse lengths of the order of 10 fs or less. See, for example, G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989); M. Piche, J. Cormier, X. Zhu, “Bright optical soliton in the presence of fourth-order dispersion,” Opt. Lett. 21, 845–847 (1996).
[CrossRef] [PubMed]

J. Yang, “Improved theory of rectangular directional coupler,” M. S. thesis (University of Toledo, Toledo, Ohio, 1994).

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Figures (5)

Fig. 1
Fig. 1

(a) Geometry of a rectangular NLDC. (b) Dielectric function profile transverse to the direction of propagation.

Fig. 2
Fig. 2

Graph of the linear coupling length of the coupler L c as a function of the separation distance d between the channels, with (solid curve) and without (dashed curve) the effects of the corner fields included. Parameters of the waveguide are given in the text.

Fig. 3
Fig. 3

Percentage of the incident power that exits from channel 1 versus the input power into channel 1 (in units of the fundamental soliton power P 0) for distinct cases. Curves 1 and 2 are soliton input signals with (solid curve) and without (dashed curve) corner field corrections. Curves 3 and 4 are continuous-wave input signals with (dot–dash curve) and without (dotted curve) corner field corrections. Curve 5 is the soliton input signal with corner field corrections in the case that dispersion is negligible.

Fig. 4
Fig. 4

Longitudinal profiles of the fields that exit from channels 1 (solid curve) and 2 (dotted curve) compared with the profile of the input field in channel 1 (dashed curve) under the condition that the dispersion length L D is equal to the nonlinear length L NL.

Fig. 5
Fig. 5

Longitudinal profiles of the fields that exit from channels 1 (solid curve) and 2 (dotted curve) compared with the profile of the input field in channel 1 (dashed curve) under the condition that the dispersion length L D is far greater than the nonlinear length L NL.

Equations (54)

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2Er, t-1c22DLr, tt2=4πc22PNLr, tt2,
DLr, tEr, t+4πPLr, t.
Er, t=ω0r, texp-iω0t+ω0*r, texpiω0t,
PNLr, t=3χ3|ω0r, t|2ω0r, texp-iω0t+c.c.
D˜Lr, ω=εLr, ωE˜r, ω=εLx, y, ωE˜r, ω
2x2+2y2+2z2+ω2c2 εLx, y, ωE˜r, ω=-4π ω2c2 P˜NLr, ω.
lx, yexpiβlz-iωt, lx, yexpiβlz-iωt
2x2+2y2+ω2c2 εLx, y, ω-βl2lx, y=0,
ω0r, t=l=12 aω0,lz, tlx, yexpiklz,
ω0r, t=n=1N aω0,nz, tnx, yexpik0z,
2x2+2y2+ω2c2 εnx, y-βn2nx, y=0,
-dx -dylx, ylx, y=δll.
PNLr, t=3χ3l=12 aω0,lz, tlx, yexpiklz2×l=12 aω0,lz, tlx, yexpiklz-iω0t+c.c.
ãlz, ω-ω012π-dt aω0,lz, texpiω-ω0t.
dã1dz+β12-k122ik1 ã1=i2π-dt expiω-ω0t 1,
dã2dz+β22-k222ik2 ã2=i2π-dt expiω-ω0t 2,
 l=Ql|aω0,lz, t|2+Qll|aω0,lz, t|2+2Rll Reaω0,lz, taω0,l*z, t×expikl-klzaω0,lz, t+Rll|aω0,lz, t|2+klkl Rll|aω0,lz, t|2+2QllReaω0,lz, taω0,l*z, t×expikl-klzaω0,lz, t×expikl-klz,
Ql6π ω02/c2kl-dx -dy χ3l4x, y,
Qll6π ω02/c2kl-dx -dy χ3l2x, yl2x, y,
Rll6π ω02/c2kl-dx -dy χ3l3x, ylx, y.
βlβlωkl+βω-ω0+12 βω-ω02,
βdβdωω=ω0, βd2βdω2ω=ω0,
βl2-kl22klβl-klβω-ω0+12 βω-ω02, l=1, 2.
aω0,1z+i 12 β 2aω0,1t2=i 1,
aω0,2z+i 12 β 2aω0,2t2=i 2,
zz, tt=t-βz, aω0,lz, taω0,lz, t.
 1=Q1|aω0,1z, t|2+Q12|aω0,2z, t|2×aω0,1z, t+2Q12 Reaω0,1z, taω0,2*z, t×expik1-k2zaω0,2z, texpik2-k1z,
 2=Q2|aω0,2z, t|2+Q21|aω0,1z, t|2×aω0,2z, t+2Q21 Reaω0,1z, taω0,2*z, t×expik1-k2zaω0,1z, texpik1-k2z.
aω0,1z+i 12 β 2aω0,1t2-iQ1|aω0,1|2aω0,1=0,
Qn6π ω02/c2k0×-dx -dy χ3n4x, y, n=1, 2.
Aeff1/-dx -dy n4x, y.
aω0,1z, t=A0 secht-βzt0expiκz,
P0=cn02π |A0|2,
κ=-β2t02,
|A0|2=2κQl=-βt02Ql
1x, y=A11x, y+2x, y, 2x, y=A21x, y-2x, y,
-dx -dy nx, ynx, y=1, n=nΔ, nn=|Al|21+1±2Δ=1,
A1=121+Δ, A2=121-Δ.
ω0r, t=aω0,1z, t1x, yexpik1z+aω0,2z, t2x, yexpik2z=A1z, t1x, y+A2z, t2x, y,
A1z, t=121+Δ aω0,1z, texpik1z+121-Δ aω0,2z, texpik2z, A2z, t=121+Δ aω0,1z, texpik1z-121-Δ aω0,2z, texpik2z.
Lc=πk1-k2.
δkl=ω02c2δε2kl  dx  dy lx, y2,
z¯z/LD, t¯t/t0, a¯ω0,laω0,l/A0,
a¯ω0,1z¯=i22a¯ω0,1t¯2+iμ1|a¯ω0,1|2+ν12|a¯ω0,2|2a¯ω0,1+2ν12 Rea¯ω0,1a¯ω0,2* expik1-k2LDz¯×a¯ω0,2 expik2-k1LDz¯,
a¯ω0,2z¯=i22a¯ω0,2t¯2+iμ2|a¯ω0,2|2+ν21|a¯ω0,1|2a¯ω0,2+2ν21 Rea¯ω0,1a¯ω0,2* expik1-k2LDz¯×a¯ω0,1 expik1-k2LDz¯,
μl=LD|A0|2Ql=Ql/Ql, νll=LD|A0|2Qll=Qll/Ql,
aω0,l0, t=A secht/t0.
Ix, y, z=-c28πωReiEy* Eyz.
aω0,1z+i 12 β 2aω0,1t2-ic1aω0,1-ic2aω0,2=id1|aω0,1|2aω0,1+d2|aω0,2|2aω0,2+e1|aω0,2|aω0,1+e2|aω0,1|2aω0,2+2e2 Reaω0,1aω0,2*aω0,1+2e1 Reaω0,1aω0,2*aω0,2,
aω0,2z+i 12 β 2aω0,2t2-ic1aω0,2-ic2aω0,1=id1|aω0,2|2aω0,2+d2|aω0,1|2aω0,1+e1|aω0,1|2aω0,2+e2|aω0,2|2aω0,1+2e1 Reaω0,1aω0,2*aω0,1+2e2 Reaω0,1aω0,2*aω0,2,
c1κ1-Δκ121-Δ2=κ2-Δκ211-Δ2, c2κ21-Δκ21-Δ2=κ12-Δκ11-Δ2, d1Q1-ΔR121-Δ2=Q2-ΔR211-Δ2, d2R21-ΔQ21-Δ2=R12-ΔQ11-Δ2, e1Q12-ΔR211-Δ2=Q12-ΔR121-Δ2, e2R12-ΔQ121-Δ2=R21-ΔQ121-Δ2,
κnω02/c22k0-dx -dy Δεnx, yn2x, y, κnn=κnnω02/c22k0×-dx -dy Δεnx, ynx, ynx, y,
Qn6π ω02/c2k0-dx -dy χ3n4x, y, Qnn=Qnn6π ω02/c2k0×-dx -dy χ3n2x, yn2x, y,
Rnn=Rnn6π ω02/c2k0×-dx -dy χ3n3x, ynx, y,

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