Abstract

The scalar Maxwell's equation, including the often-neglected second-derivative paraxial term, is solved. This term is shown to increase the loss of the amplitude of the light beam inside a single-mode optical fiber. The reflected (backward-traveling) wave that is due to the second-derivative term is obtained from the calculation of the scattering matrix as formulated in quantum-mechanical dynamical problems.

© 1997 Optical Society of America

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References

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  1. A. W. Snyder and S. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  2. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  3. R. E. Prange and S. Fishman, “Experimental realization of kicked quantum chaotic systems,” Phys. Rev. Lett. 63, 704 (1989).
    [CrossRef] [PubMed]
  4. M. D. Feit, J. A. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Compu. Phys. 47, 412 (1982); M. D. Feit and J. A. Fleck, Jr., “Simple spectral method for solving propagation problems in cylindrical geometry with fast Fourier transforms,” Opt. Lett. 14, 662 (1989).
    [CrossRef] [PubMed]
  5. R. Kosloff, in Time-Dependent Quantum Molecular Dynamics, NATO ASI Series B299, J. Broeckhave and L. Lathouwers, eds. (Plenum, New York, 1992), p. 92.
  6. I. Vorobeichik, U. Peskin, and N. Moiseyev, “Modal losses and design of modal irradiance patterns in an optical fiber by the complex scaled (t, t) method,” J. Opt. Soc. Am. B 12, 1133 (1995); “Propagation of light beam in optical fiber by the (t, t) method,” Non-Linear Opt. 11, 79 (1995);
    [CrossRef]
  7. See, for example, recent simulations of large-scale strongly coupled scattering problems (as occur in four-center reactions) involving millions of basis functions: U. Manthe, T. Seideman, and W. H. Miller, “Full-dimensional quantum mechanical calculation of the rate constant for the H2+OH→H2O+H reaction,” J. Chem. Phys. 99, 10078 (1193); D. H. Zhang and J. Z. H. Zhang, “Full-dimensional time-dependent treatment for diatom–diatom reaction—the H2+OH reaction,” J. Chem. Phys. 101, 1146 (1994); D. Neuhauser, “Fully quantal initial-state-selected reaction probabilities (j=0) for a four-atom system—H2(v= 0, 1, j=0)+OH(v=0, 1, j=0)→H + H2O,” J. Chem. Phys. JCPSA6 100, 9272 (1994).
    [CrossRef]
  8. U. Peskin and N. Moiseyev, “The solution of the time-dependent Schrödinger equation by the (t, t) method—theory, computational algorithm and applications,” J. Chem. Phys. 99, 4590 (1993).
    [CrossRef]
  9. R. Ratowsky and J. A. Fleck, “Accurate numerical solution of the Helmholtz equation by iterative Lanczos reduction,” Opt. Lett. 16, 787 (1991).
    [CrossRef] [PubMed]
  10. Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335 (1990).
    [CrossRef]
  11. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett. 17, 1743 (1992).
    [CrossRef] [PubMed]
  12. U. Peskin, W. H. Miller, and Å Edlund, “Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner,” J. Chem. Phys. 103, 10030 (1995).
    [CrossRef]

1995 (1)

U. Peskin, W. H. Miller, and Å Edlund, “Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner,” J. Chem. Phys. 103, 10030 (1995).
[CrossRef]

1993 (1)

U. Peskin and N. Moiseyev, “The solution of the time-dependent Schrödinger equation by the (t, t) method—theory, computational algorithm and applications,” J. Chem. Phys. 99, 4590 (1993).
[CrossRef]

1992 (1)

1991 (1)

1990 (1)

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335 (1990).
[CrossRef]

1989 (1)

R. E. Prange and S. Fishman, “Experimental realization of kicked quantum chaotic systems,” Phys. Rev. Lett. 63, 704 (1989).
[CrossRef] [PubMed]

Chung, Y.

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335 (1990).
[CrossRef]

Dagli, N.

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335 (1990).
[CrossRef]

Edlund, Å

U. Peskin, W. H. Miller, and Å Edlund, “Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner,” J. Chem. Phys. 103, 10030 (1995).
[CrossRef]

Fishman, S.

R. E. Prange and S. Fishman, “Experimental realization of kicked quantum chaotic systems,” Phys. Rev. Lett. 63, 704 (1989).
[CrossRef] [PubMed]

Fleck, J. A.

Hadley, G. R.

Miller, W. H.

U. Peskin, W. H. Miller, and Å Edlund, “Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner,” J. Chem. Phys. 103, 10030 (1995).
[CrossRef]

Moiseyev, N.

U. Peskin and N. Moiseyev, “The solution of the time-dependent Schrödinger equation by the (t, t) method—theory, computational algorithm and applications,” J. Chem. Phys. 99, 4590 (1993).
[CrossRef]

Peskin, U.

U. Peskin, W. H. Miller, and Å Edlund, “Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner,” J. Chem. Phys. 103, 10030 (1995).
[CrossRef]

U. Peskin and N. Moiseyev, “The solution of the time-dependent Schrödinger equation by the (t, t) method—theory, computational algorithm and applications,” J. Chem. Phys. 99, 4590 (1993).
[CrossRef]

Prange, R. E.

R. E. Prange and S. Fishman, “Experimental realization of kicked quantum chaotic systems,” Phys. Rev. Lett. 63, 704 (1989).
[CrossRef] [PubMed]

Ratowsky, R.

IEEE J. Quantum Electron. (1)

Y. Chung and N. Dagli, “An assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 1335 (1990).
[CrossRef]

J. Chem. Phys. (2)

U. Peskin, W. H. Miller, and Å Edlund, “Quantum time evolution in time-dependent fields and time-independent reactive-scattering calculations via an efficient Fourier grid preconditioner,” J. Chem. Phys. 103, 10030 (1995).
[CrossRef]

U. Peskin and N. Moiseyev, “The solution of the time-dependent Schrödinger equation by the (t, t) method—theory, computational algorithm and applications,” J. Chem. Phys. 99, 4590 (1993).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

R. E. Prange and S. Fishman, “Experimental realization of kicked quantum chaotic systems,” Phys. Rev. Lett. 63, 704 (1989).
[CrossRef] [PubMed]

Other (6)

M. D. Feit, J. A. Fleck, and A. Steiger, “Solution of the Schrödinger equation by a spectral method,” J. Compu. Phys. 47, 412 (1982); M. D. Feit and J. A. Fleck, Jr., “Simple spectral method for solving propagation problems in cylindrical geometry with fast Fourier transforms,” Opt. Lett. 14, 662 (1989).
[CrossRef] [PubMed]

R. Kosloff, in Time-Dependent Quantum Molecular Dynamics, NATO ASI Series B299, J. Broeckhave and L. Lathouwers, eds. (Plenum, New York, 1992), p. 92.

I. Vorobeichik, U. Peskin, and N. Moiseyev, “Modal losses and design of modal irradiance patterns in an optical fiber by the complex scaled (t, t) method,” J. Opt. Soc. Am. B 12, 1133 (1995); “Propagation of light beam in optical fiber by the (t, t) method,” Non-Linear Opt. 11, 79 (1995);
[CrossRef]

See, for example, recent simulations of large-scale strongly coupled scattering problems (as occur in four-center reactions) involving millions of basis functions: U. Manthe, T. Seideman, and W. H. Miller, “Full-dimensional quantum mechanical calculation of the rate constant for the H2+OH→H2O+H reaction,” J. Chem. Phys. 99, 10078 (1193); D. H. Zhang and J. Z. H. Zhang, “Full-dimensional time-dependent treatment for diatom–diatom reaction—the H2+OH reaction,” J. Chem. Phys. 101, 1146 (1994); D. Neuhauser, “Fully quantal initial-state-selected reaction probabilities (j=0) for a four-atom system—H2(v= 0, 1, j=0)+OH(v=0, 1, j=0)→H + H2O,” J. Chem. Phys. JCPSA6 100, 9272 (1994).
[CrossRef]

A. W. Snyder and S. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

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Figures (6)

Fig. 1
Fig. 1

Amplitude of the field |ψ(r)|2 as a function of r after 1000 µm when the Maxwell equation is solved within the framework of the paraxial approximation and beyond it. The radius of the fiber is r0=3 µm when the index-of-refraction modulation parameter f0 [see Eq. (9)] is equal to 0.5. The box size in the r coordinate is 100 µm, and optical potential is located at 60 µm <r<100µm. Sixty eigenfunctions of the z-independent operator [see Eq. (14)] were used to describe the transverse-direction dependence of the fiber modes. The propagation step size was 0.1 µm, which corresponds to approximately six points per wavelength of the trapped mode. This size was found to be satisfactory for convergence even for the largest studied variations of the refractive index.

Fig. 2
Fig. 2

Effect of the second-derivative term in z in the Maxwell equation on the modal losses. The relative difference in loss is a function of the index-of-refraction modulation parameter f0. % stands for [(I1-I2)/I2]100%, where I1 denotes 0r0|ψ(r, z)|2dr, which was calculated when the paraxial approximation was used, and I2 stands for results obtained when the z-second-derivative term was included. The circles (connected by a dashed curve) stand for the results obtained after propagation to 300 µm, whereas the squares (connected by a solid curve) stand for the same results after 1000 µm.

Fig. 3
Fig. 3

Schematic representation of the potential barrier obtained when the index of refraction varies along the fiber axis. → stands for the incoming light beam, ← (R) stands for the reflected beam, and → (T) denotes the transmitted light beam.

Fig. 4
Fig. 4

Reflection effect |R1|2 as a function of the barrier steepness parameter, η, which is defined in Eq. (30). |R1|2l=1|R1,l|2 when {Rli=1,l} are the elements of the reflection matrix associated with the l=1 trapped incoming optical mode. The inner plot represents the change in the potential barrier like term as η varied (η=0, solid curve; η=1, dotted curve; η=5, dashed curve).

Fig. 5
Fig. 5

Probability of the l=1 trapped incoming optical mode being reflected into the 1th mode. l=1 stands for a trapped mode; whereas l>1 stand for the radiative modes. The inset emphasizes that the entire reflection is into the trapped mode and not the radiative ones.

Fig. 6
Fig. 6

Same as in Fig. 5 for the transmitted light beam. Most of the transmitted light is into the l=1 trapped mode. The loss in transmission is associated with the probability of populating the radiative l>1 modes.

Equations (33)

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2E+k2n2E=0,
[x, y,z2+k2(nco2-ncl2)]ψ=-2ik0 ψz,
φ(x, y, z)=ψz,
zφ+Hˆψ=-2ik0φ,
Hˆ=x, y2+k2[nco2(x, y, z)-ncl2].
Hχ=i zχ,
H=0i-iHˆ2k0.
χ(0)=ψ(x, y, z)=0φ(x, y, z)=0.
k2[nco2(r, z)-ncl2]=k022Δ01-2Δ0×[1-g(r)]1+f0 cos 2πaz,
Vopt(r)=A1+exp[(rabs-r)/η]
H(x, y, z=0)=H(x, y, z=L),
H=2x2+2y2+k02n(x, y, z)n02.
2z2+H(x, y, z=0)Ψi(0)(x, y, z)=0,
Ψi(0)(x, y, z)=12βi exp(±iβiz)Φi(x, y),
H(x, y)Φi(x, y)=[x, y2+k2n2(x, y, z=0)-βi2]×Φi(x, y)=0.
exp(iβiz)2βiΦi(x, y)-lRi,l exp(-iβlz)2βlΦl(x, y),
mTi,m exp(iβmz)2βmΦm(x, y),
2z2I+H(z)C(z)=0,
Hij(z)=Φi|H(x, y, z)|Φjx, y,
Ψj(x, y, z)=iCi,j(z)Φi(x, y).
H=[β(d)]2+k2V,
Vi,j=k2Φi|n2(x, y, z)-n2(x, y, z=0)|Φjx, y,
C(z=L)=TE+(z=L),
C(z=0)=E+(z=0)-RE-(z=0),
2z2I+H(z)D(z)=0,
D(z=L)=I.
D=T-1C(z=0)=T-1E+(z=0)-T-1RE-(z=0).
D(z=0)=T-1[E+(z=0)]-T-1R[E-(z=0)],
n(x, y, z)n02=1-2Δg(r)1-2Δ×1-f01+exp[-(z-L/2)/η],
g(r)=rr02ifr<r01ifrr0,
2Δ=n2(0, 0, 0)-n02n02.
f(z)n(r=rm, z=0)n02,
f(z)=f01+exp[-(z-L/2)/η].

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