Abstract

A recently developed, simple and easily programmable method based on the resonance technique for the analysis of nonlinear guided modes for fiber geometry is extended to include more complicated configurations. As an example of the proposed methodology, the following configuration is treated: a circular optical fiber consisting of a nonlinear core bounded by a nonlinear cladding medium. The calculation of the nonlinear modes in this case has some peculiarities and is presented in detail. The nonlinearity considered is of the more general type, including the saturable mode, which is of current significant importance. The numerical results obtained by this method correspond to the weak-guidance approximation and include both field distributions and power calculations. Some useful conclusions are deduced.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric structure,” Sov. Phys. JETP 56, 299–303 (1982).
  2. F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab,” Appl. Phys. B 31, 69–73 (1983).
    [CrossRef]
  3. A. D. Boardman, G. S. Cooper, and D. J. Robbins, “Novel nonlinear waves in optical fibers,” Opt. Lett. 11, 112–114 (1986).
    [CrossRef] [PubMed]
  4. A. W. Snyder, D. J. Mitchell, and L. Poladian, “Linear approach for approximating spatial solutions and nonlinear guided modes,” J. Opt. Soc. Am. B 8, 1618–1620 (1991).
    [CrossRef]
  5. A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 26, 643–644 (1990).
    [CrossRef]
  6. J. D. Kanellopoulos and N. A. Stathopoulos, “Calculation of nonlinear waves guided by optical fibers using the resonance technique,” J. Mod. Opt. 42, 141–155 (1995).
    [CrossRef]
  7. J. D. Kanellopoulos and N. A. Stathopoulos, “Application of the resonance technique for the evaluation of the TE and TM modes guided by successive nonKerr nonlinear dielectric planar layer structures,” J. Mod. Opt. 40, 743–760 (1993).
    [CrossRef]
  8. J. D. Kanellopoulos and N. A. Stathopoulos, “Calculation of nonlinear modes guided by multilayer dielectric structures using the resonance technique,” J. Opt. Quantum Electron. 24, 755–773 (1992).
  9. G. I. Stegeman, C. T. Seaton, J. Ariyasu, T. P. Shen, and J. V. Moloney, “Saturation and power law dependence of nonlinear waves guided by a single interface,” Opt. Commun. 56, 365–368 (1986).
    [CrossRef]
  10. Th. Peschel, P. Dannberg, U. Langbein, and F. Lederer, “Investigation of optical tunneling through nonlinear films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
    [CrossRef]
  11. C. D. Papageorgiou, J. D. Kanellopoulos, and J. G. Fikioris, “Equivalent circuits in Fourier space for the study of inhomogeneous cylindrical layered structures,” J. Opt. Soc. Am. 73, 1291–1295 (1983).
    [CrossRef]

1995 (1)

J. D. Kanellopoulos and N. A. Stathopoulos, “Calculation of nonlinear waves guided by optical fibers using the resonance technique,” J. Mod. Opt. 42, 141–155 (1995).
[CrossRef]

1993 (1)

J. D. Kanellopoulos and N. A. Stathopoulos, “Application of the resonance technique for the evaluation of the TE and TM modes guided by successive nonKerr nonlinear dielectric planar layer structures,” J. Mod. Opt. 40, 743–760 (1993).
[CrossRef]

1991 (1)

1990 (1)

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 26, 643–644 (1990).
[CrossRef]

1988 (1)

1986 (2)

G. I. Stegeman, C. T. Seaton, J. Ariyasu, T. P. Shen, and J. V. Moloney, “Saturation and power law dependence of nonlinear waves guided by a single interface,” Opt. Commun. 56, 365–368 (1986).
[CrossRef]

A. D. Boardman, G. S. Cooper, and D. J. Robbins, “Novel nonlinear waves in optical fibers,” Opt. Lett. 11, 112–114 (1986).
[CrossRef] [PubMed]

1983 (2)

1982 (1)

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric structure,” Sov. Phys. JETP 56, 299–303 (1982).

Akhmediev, N. N.

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric structure,” Sov. Phys. JETP 56, 299–303 (1982).

Ariyasu, J.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, T. P. Shen, and J. V. Moloney, “Saturation and power law dependence of nonlinear waves guided by a single interface,” Opt. Commun. 56, 365–368 (1986).
[CrossRef]

Boardman, A. D.

Chen, Y.

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Cooper, G. S.

Dannberg, P.

Fikioris, J. G.

Kanellopoulos, J. D.

J. D. Kanellopoulos and N. A. Stathopoulos, “Calculation of nonlinear waves guided by optical fibers using the resonance technique,” J. Mod. Opt. 42, 141–155 (1995).
[CrossRef]

J. D. Kanellopoulos and N. A. Stathopoulos, “Application of the resonance technique for the evaluation of the TE and TM modes guided by successive nonKerr nonlinear dielectric planar layer structures,” J. Mod. Opt. 40, 743–760 (1993).
[CrossRef]

C. D. Papageorgiou, J. D. Kanellopoulos, and J. G. Fikioris, “Equivalent circuits in Fourier space for the study of inhomogeneous cylindrical layered structures,” J. Opt. Soc. Am. 73, 1291–1295 (1983).
[CrossRef]

Langbein, U.

Th. Peschel, P. Dannberg, U. Langbein, and F. Lederer, “Investigation of optical tunneling through nonlinear films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
[CrossRef]

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

Lederer, F.

Th. Peschel, P. Dannberg, U. Langbein, and F. Lederer, “Investigation of optical tunneling through nonlinear films,” J. Opt. Soc. Am. B 5, 29–36 (1988).
[CrossRef]

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

Mitchell, D. J.

A. W. Snyder, D. J. Mitchell, and L. Poladian, “Linear approach for approximating spatial solutions and nonlinear guided modes,” J. Opt. Soc. Am. B 8, 1618–1620 (1991).
[CrossRef]

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Moloney, J. V.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, T. P. Shen, and J. V. Moloney, “Saturation and power law dependence of nonlinear waves guided by a single interface,” Opt. Commun. 56, 365–368 (1986).
[CrossRef]

Papageorgiou, C. D.

Peschel, Th.

Poladian, L.

A. W. Snyder, D. J. Mitchell, and L. Poladian, “Linear approach for approximating spatial solutions and nonlinear guided modes,” J. Opt. Soc. Am. B 8, 1618–1620 (1991).
[CrossRef]

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Ponath, H. E.

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

Robbins, D. J.

Seaton, C. T.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, T. P. Shen, and J. V. Moloney, “Saturation and power law dependence of nonlinear waves guided by a single interface,” Opt. Commun. 56, 365–368 (1986).
[CrossRef]

Shen, T. P.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, T. P. Shen, and J. V. Moloney, “Saturation and power law dependence of nonlinear waves guided by a single interface,” Opt. Commun. 56, 365–368 (1986).
[CrossRef]

Snyder, A. W.

A. W. Snyder, D. J. Mitchell, and L. Poladian, “Linear approach for approximating spatial solutions and nonlinear guided modes,” J. Opt. Soc. Am. B 8, 1618–1620 (1991).
[CrossRef]

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 26, 643–644 (1990).
[CrossRef]

Stathopoulos, N. A.

J. D. Kanellopoulos and N. A. Stathopoulos, “Calculation of nonlinear waves guided by optical fibers using the resonance technique,” J. Mod. Opt. 42, 141–155 (1995).
[CrossRef]

J. D. Kanellopoulos and N. A. Stathopoulos, “Application of the resonance technique for the evaluation of the TE and TM modes guided by successive nonKerr nonlinear dielectric planar layer structures,” J. Mod. Opt. 40, 743–760 (1993).
[CrossRef]

Stegeman, G. I.

G. I. Stegeman, C. T. Seaton, J. Ariyasu, T. P. Shen, and J. V. Moloney, “Saturation and power law dependence of nonlinear waves guided by a single interface,” Opt. Commun. 56, 365–368 (1986).
[CrossRef]

Appl. Phys. B (1)

F. Lederer, U. Langbein, and H. E. Ponath, “Nonlinear waves guided by a dielectric slab,” Appl. Phys. B 31, 69–73 (1983).
[CrossRef]

Electron. Lett. (1)

A. W. Snyder, Y. Chen, L. Poladian, and D. J. Mitchell, “Fundamental mode of highly nonlinear fibers,” Electron. Lett. 26, 643–644 (1990).
[CrossRef]

J. Mod. Opt. (2)

J. D. Kanellopoulos and N. A. Stathopoulos, “Calculation of nonlinear waves guided by optical fibers using the resonance technique,” J. Mod. Opt. 42, 141–155 (1995).
[CrossRef]

J. D. Kanellopoulos and N. A. Stathopoulos, “Application of the resonance technique for the evaluation of the TE and TM modes guided by successive nonKerr nonlinear dielectric planar layer structures,” J. Mod. Opt. 40, 743–760 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

G. I. Stegeman, C. T. Seaton, J. Ariyasu, T. P. Shen, and J. V. Moloney, “Saturation and power law dependence of nonlinear waves guided by a single interface,” Opt. Commun. 56, 365–368 (1986).
[CrossRef]

Opt. Lett. (1)

Sov. Phys. JETP (1)

N. N. Akhmediev, “Novel class of nonlinear surface waves: asymmetric modes in a symmetric structure,” Sov. Phys. JETP 56, 299–303 (1982).

Other (1)

J. D. Kanellopoulos and N. A. Stathopoulos, “Calculation of nonlinear modes guided by multilayer dielectric structures using the resonance technique,” J. Opt. Quantum Electron. 24, 755–773 (1992).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

(a) Thin homogeneous cylindrical dielectric layer; (b) equivalent T circuit of a transmission line (magnetic and electric circuit); (c) a nonlinear core bounded by a nonlinear cladding.

Fig. 2
Fig. 2

Equivalent T circuits connected in tandem for the nonlinear problem (HE and EH case).

Fig. 3
Fig. 3

(a) Total guided power for NLP01 mode versus β/k0 and for various values of Δnsat(2). In this configuration ncr=1.51, ncd=1.50, n¯cr=10-9m2/W, and core radius a=0.5 µm. (b) Field distributions along the core and the cladding for the configuration described in Fig. 3(a) (Kerr-type core). For mode NLP01 (A=1.51, B=1.515, C=1.52, and D=1.53).

Fig. 4
Fig. 4

(a) Field distributions along the core and the cladding for the same configuration described in Fig. 3. For mode NLP02 and A=1.506, B=1.51, C=1.52, and D=1.53. (b) Total guided power for NLP02 mode versus β/k0 and for various values of Δnsat(2).

Fig. 5
Fig. 5

(a) Total guided for the NLP01 mode versus β/k0 for various values of Δnsat(2). In this configuration ncr=1.51, ncd=1.50, n¯cr=n¯cd=10-9m2 W-1, and Δnsat(1)=0.03. (b) Field distributions along the core and the cladding for the same configuration described in Fig. 5(a). For mode NLP01 and A=1.515, B=1.517, C=1.518, D=1.519, E=1.52, and F=1.521, Δnsat(2)=0.01.

Equations (86)

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

Ei(r, θ, z, t)=Re{E˚i(r)exp[j(-ωt+βz+lθ)]},
Hi(r, θ, z, t)=Re{H˚i(r)exp[j(-ωt+βz+lθ)]}.
V˚MlH˚θ+βrH˚zjFZ0,
I˚Mωμ0rH˚rjZ0,
V˚ElE˚θ+βrE˚zF,
I˚En2(r)0ωrE˚r,
Z0=μ001/2=120πΩ,F=l2+(βr)2r.
dV˚Mdr=-γ2jωμ0(F/Z02)I˚M-j 2βlZ0[(βr)2+l2]I˚E,
dI˚Mdr=-jωμ0FZ02V˚M,
dV˚Edr=-γ2jωn2(r)0FI˚E-j 2βlZ0[(βr)2+l2]I˚M,
dI˚Edr=-jωn2(r)0FV˚E.
γ2β2+(l/r)2-ω2n2(r)μ00,
ZMγjωμ0FZ02,ZEγjωn2(r)0F,
M=2βlZ0rF2.
V˚HEV˚M[n(r)]1/2+V˚E[n(r)]1/2,
V˚EHV˚M[n(r)]1/2-V˚E[n(r)]1/2,
I˚HEI˚M[n(r)]1/2+I˚E[n(r)]1/2,
I˚EHI˚M[n(r)]1/2-I˚E[n(r)]1/2.
γHE2γ2-0ωn(r)FM,
γEH2γ2+0ωn(r)FM,
ZHE=γHEjω0n(r)F,ZEH=γEHjω0n(r)F.
ZS,HEZHE tanhγHEΔr2,
ZP,HEZHEsinhγHEΔr,
ZS,EHZEH tanhγEHΔr2,
ZP,EHZEHsinh(γEHΔr).
ncdn=ncd[1+a1f(|E˚|)]1/2,
ncrn=ncr[1+a2f(|E˚|)]1/2,
a1=c0n¯cd,
a2=c0n¯cr,
|E˚|=(|E˚r|2+|E˚θ|2+|E˚z|2)1/2.
f(|E˚|)=|E˚|p,
f(|E˚|)=|E˚|2,
ncdn=ncd2+(nsat(1))21-exp-a1ncdnsat(1)|E˚|21/2,
ncrn=ncr2+(nsat(2))21-exp-a2ncrnsat(2)|E˚|21/2,
(nsat(1))2=2ncdΔnsat(1)+(Δnsat(1))2,
(nsat(2))2=2ncrΔnsat(2)+(Δnsat(2))2,
ncdn[ncd2+(nsat(1))2]1/2,
ncrn[ncr2+(nsat(1))2]1/2,
ncdnncd(1+a1|E˚|2)1/2,
ncrnncr(1+a2|E˚|2)1/2
ZHE,EH(r)=0.
(a)β2+lrk2>ncdn2(k)k02-2ncdn(k)βk0lβ2rk2+l2.
ZS,k(1)=γEH(1)(rk)rkjω0ncdn(k)[(βrk)2+l2]tanhγEH(1)(rk)Δrk2,
ZP,k(1)=γEH(1)(rk)rkjω0ncdn(k)[(βrk)2+l2]sinh[γEH(1)(rk)Δrk],
γEH(1)(rk)=β2+lrk2-ncdn2(k)k02+2ncdn(k)βk0ll2+β2rk21/2.
(b)β2+lrk2<ncdn2(k)k02-2ncdn(k)βk0lβ2rk2+l2.
ZS,k(1)=jβEH(1)(rk)rkω0ncdn(k)(β2rk2+l2)tanβEH(1)(rk)Δrk2,
ZP,k(1)=βEH(1)(rk)rkjω0ncdn(k)(β2rk2+l2)sin[βEH(1)(rk)Δrk],
βEH(1)(rk)=ncdn2(k)k02-2ncdn(k)βk0lβ2rk2+l2-β2-lrk21/2.
(a)β2+lrk2>ncdn2(k)k02+2ncdn(k)βk0lβ2rk2+l2.
ZS,k(1)=γHE(1)(rk)rkjω0ncdn(k)[(βrk)2+l2]tanhγHE(1)(rk)Δrk2,
ZP,k(1)=γHE(1)(rk)rkjω0ncdn(k)[(βrk)2+l2]sinh[γHE(1)(rk)Δrk],
γHE(1)(rk)=β2+lrk2-ncdn2(k)k02-2ncdn(k)βk0ll2+β2rk21/2.
(b)β2+lrk2<ncdn2(k)k02+2ncdn(k)βk0lβ2rk2+l2.
ZS,k(1)=jβHE(1)(rk)rkω0ncdn(k)(β2rk2+l2)tanβHE(1)(rk)Δrk2,
ZP,k(1)=βHE(1)(rk)rkjω0ncdn(k)(β2rk2+l2)sin[βHE(1)(rk)Δrk],
βHE(1)(rk)=ncdn2(k)k02+2ncdn(k)βk0lβ2rk2+l2-β2-lrk21/2.
ncdn(k)=ncd{1+a1[|E˚r(1)(k-1)|2+|E˚θ(1)(k-1)|2+|E˚z(1)(k-1)|2]1/2}1/2,
ncrn(k)=ncr{1+a2[|E˚r(2)(k-1)|2+|E˚θ(2)(k-1)|2+|E˚z(2)(k-1)|2]1/2}1/2,
ncdn(k)=ncd2+(nsat(1))21-exp-a1 ncd2(nsat(1))2×[|E˚r(1)(k-1)|2+|E˚θ(1)(k-1)|2+|E˚z(1)(k-1)|2]1/2,
ncrn(k)=ncr2+(nsat(2))21-exp-a2 ncr2(nsat(2))2×[|E˚r(2)(k-1)|2+|E˚θ(2)(k-1)|2+|E˚z(2)(k-1)|2]1/2,
|E˚r(1)(k-1)|=|I˚HE/EH(1)(k-1)|2[ncdn3(k-1)]1/20ωrk-1,
|E˚θ(1)(k-1)|=l2β2rk-12+l2rk-12|V˚HE/EH(1)(k-1)|2+Z02β2rk-12|I˚HE/EH(1)(k-1)|24ncdn(k-1)(β2rk-12+l2)21/2,
|E˚z(1)(k-1)|=β2rk-12+l2rk-12|V˚E/M(1)(k-1)|2+l2|E˚θ(1)(k-1)|2β2rk-121/2,
|V˚E/M(1)(k-1)|=|V˚HE/EH(1)(k-1)|2[ncdn(k-1)]1/2
Ze=Z0jncnr(re)l,
Z0=120πΩ,
Zk(m)=(Zk-1(m)+ZS,k(m))ZP,k(m)Zk-1(m)+ZS,k(m)+ZP,k(m)+ZS,k(m)
Z0(1)=0,
ZN1(1)=Z0(2).
I˚HE/EH(m)(k)=I˚HE/EH(m)(k-1)1+Zk-1(m)+ZS,k(m)ZP,k(m),
V˚HE/EH(m)(k)=V˚HE/EH(m)(k-1)-I˚HE/EH(m)(k-1)ZS,k(m)-I˚HE/EH(m)(k)ZS,k(m),
I˚HE/EH(2)(0)I˚HE/EH(1)(N1),
V˚HE/EH(2)(0)V˚HE/EH(1)(N1).
Zin=ZHE/EH*(r0)=[Z0jncrn(re)l]*,
S=12Re S PzdS=12Re 002π(ErHθ*-EθHr*)rdrdϕ=Re 0π(ErHθ*-EθHr*)rdr,
Er=E˚r(r)exp(jβz+jlθ),
Eθ=E˚θ(r)exp(jβz+jlθ),
Hr*=H˚r*(r)exp(-jβz-jlθ),
Hθ*=H˚θ*(r)exp(-jβz-jlθ).
S=Re 0π(E˚rH˚θ*-E˚θH˚r*)rdr.
SHE/EH=Re 0π[-β(β2r2+l2)4n(r)0ω|I˚HE/EH(r)|2-Z02β4n(r)(β2r2+l2)ωμ0|I˚HE/EH(r)|2j l4n(r)0ωr2Z0I˚HE/EH(r)V˚HE/EH*(r)±j Z0l4n(r)ωμ0r2I˚HE/EH*(r)V˚HE/EH(r)]rdr,
SHE/EH=π Rem=12k=1Nm×-β(β2rk,m2+l2)4n(rk,m)0ω|I˚HE/EH(m)(k)|2-Z02β4n(rk,m)(β2rk,m2+l2)ωμ0|I˚HE/EH(m)(k)|2jl4n(rk,m)ω0rk,m2Z0I˚HE/EH(m)(k)V˚HE/EH(m) *(k)±j Z0l4n(rk,m)ωμ0rk,m2I˚HE/EH(m) *(k)V˚HE/EH(m)(k)×rk,mΔrk,m,
rk,1rkandrk,2rk,
n(rk,1)=ncdn(k),
n(rk,2)=ncrn(k).

Metrics