Abstract

We develop a novel perturbation theory formulation to evaluate polarization-mode dispersion (PMD) for a general class of scaling perturbations of a waveguide profile based on generalized Hermitian Hamiltonian formulation of Maxwell’s equations. Such perturbations include elipticity and uniform scaling of a fiber cross section, as well as changes in the horizontal or vertical sizes of a planar waveguide. Our theory is valid even for discontinuous high-index contrast variations of the refractive index across a waveguide cross section. We establish that, if at some frequencies a particular mode behaves like pure TE or TM polarized mode (polarization is judged by the relative amounts of the electric and magnetic longitudinal energies in the waveguide cross section), then at such frequencies for fibers under elliptical deformation its PMD as defined by an intermode dispersion parameter τ becomes proportional to group-velocity dispersion D such that τ=λδ|D|, where δ is a measure of the fiber elipticity and λ is a wavelength of operation. As an example, we investigate a relation between PMD and group-velocity dispersion of a multiple-core step-index fiber as a function of the core–clad index contrast. We establish that in this case the positions of the maximum PMD and maximum absolute value of group-velocity dispersion are strongly correlated, with the ratio of PMD to group-velocity dispersion being proportional to the core–clad dielectric contrast.

© 2002 Optical Society of America

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References

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  1. S. J. Savori, “Pulse propagation in fibers with polarization-mode dispersion,” J. Lightwave Technol. 19, 350–357 (2001).
    [CrossRef]
  2. A. W. Synder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  3. C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).
  4. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998), p. 357.
  5. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  6. M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
    [CrossRef]
  7. D. Chowdhury, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Top. Quantum Electron. 6, 227–232 (2000).
    [CrossRef]
  8. V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Electron. 13, 109–111 (1983).
    [CrossRef]
  9. V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Electron. 14, 427–430 (1984).
    [CrossRef]
  10. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  11. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (June 2002).
    [CrossRef]
  12. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001), http://www.opticsexpress.org.
    [CrossRef] [PubMed]
  13. Note, multiplying from the left by B⁁−1 on both sides of Eq. (8) and, thus, trying to make it look as a standard eigenvalue problem is erroneous as, in general, a resulting matrix on the right would not be Hermitian.
  14. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory) (Butterworth-Heinemann, Stoneham, Mass., 2000), p. 140.
  15. F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution if the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
    [CrossRef]
  16. D. Q. Chowdhury and D. A. Nolan, “Perturbation model for computing optical fiber birefringence from a two-dimensional refractive-index profile,” Opt. Lett. 20, 1973–1975 (1995).
    [CrossRef] [PubMed]
  17. G. B. Arfken and H. J. Webber, Mathematical Methods for Physicists (Academic, New York, 1995).

2002 (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (June 2002).
[CrossRef]

2001 (3)

2000 (1)

D. Chowdhury, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Top. Quantum Electron. 6, 227–232 (2000).
[CrossRef]

1999 (1)

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

1995 (1)

1990 (1)

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution if the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

1984 (1)

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Electron. 14, 427–430 (1984).
[CrossRef]

1983 (1)

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Electron. 13, 109–111 (1983).
[CrossRef]

Bahlmann, N.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

Chowdhury, D.

D. Chowdhury, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Top. Quantum Electron. 6, 227–232 (2000).
[CrossRef]

Chowdhury, D. Q.

Curti, F.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution if the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

Daino, B.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution if the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

De Marchis, G.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution if the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

Engeness, T. D.

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (June 2002).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001), http://www.opticsexpress.org.
[CrossRef] [PubMed]

Hertel, P.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (June 2002).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001), http://www.opticsexpress.org.
[CrossRef] [PubMed]

Jacobs, S. A.

Joannopoulos, J. D.

Johnson, S. G.

Kalosha, V. P.

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Electron. 14, 427–430 (1984).
[CrossRef]

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Electron. 13, 109–111 (1983).
[CrossRef]

Khapalyuk, A. P.

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Electron. 14, 427–430 (1984).
[CrossRef]

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Electron. 13, 109–111 (1983).
[CrossRef]

Lohmeyer, M.

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

Matera, F.

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution if the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

Nolan, D. A.

Savori, S. J.

Skorobogatiy, M.

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (June 2002).
[CrossRef]

Soljacic, M.

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (June 2002).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001), http://www.opticsexpress.org.
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

D. Chowdhury, “Comparison between optical fiber birefringence induced by stress anisotropy and geometric deformation,” IEEE J. Sel. Top. Quantum Electron. 6, 227–232 (2000).
[CrossRef]

J. Lightwave Technol. (2)

F. Curti, B. Daino, G. De Marchis, and F. Matera, “Statistical treatment of the evolution if the principal states of polarization in single-mode fibers,” J. Lightwave Technol. 8, 1162–1166 (1990).
[CrossRef]

S. J. Savori, “Pulse propagation in fibers with polarization-mode dispersion,” J. Lightwave Technol. 19, 350–357 (2001).
[CrossRef]

Opt. Commun. (1)

M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. 163, 86–94 (1999).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65, 066611 (June 2002).
[CrossRef]

Sov. J. Quantum Electron. (2)

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence in a single-mode elliptic optical fiber,” Sov. J. Quantum Electron. 13, 109–111 (1983).
[CrossRef]

V. P. Kalosha and A. P. Khapalyuk, “Mode birefringence of a three-layer elliptic single-mode fiber waveguide,” Sov. J. Quantum Electron. 14, 427–430 (1984).
[CrossRef]

Other (7)

Note, multiplying from the left by B⁁−1 on both sides of Eq. (8) and, thus, trying to make it look as a standard eigenvalue problem is erroneous as, in general, a resulting matrix on the right would not be Hermitian.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory) (Butterworth-Heinemann, Stoneham, Mass., 2000), p. 140.

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

C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998), p. 357.

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

G. B. Arfken and H. J. Webber, Mathematical Methods for Physicists (Academic, New York, 1995).

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Figures (5)

Fig. 1
Fig. 1

General scaling perturbation defined by a scaling of coordinates xscaled=x(1+δx) and yscaled=y(1+δy): (a) the particular case of δx=-δy corresponds to uniform elliptical perturbation of a fiber, and (b) scaling perturbations can also be used to analyze size variations of planar waveguides.

Fig. 2
Fig. 2

Dual-core dielectric waveguide. The inner core is a dielectric cylinder of radius R1 and index n1, whereas the outer core is a ring of index n2 with inner and outer radii R2 and R3, respectively. Two cores are separated by cladding with an index ncl.

Fig. 3
Fig. 3

Value of a split |Δβe|=|βm=1-βm=-1| in a propagation constant of a doubly degenerate m=1 fundamental mode of a dual core (n1=2.5) fiber (see Fig. 2) that is due to a uniform elliptical perturbation of δ=2%. |Δβe| is measured in units of 2π/a, where a=0.2046 µm. The solid curve corresponds to |Δβe| as calculated by the first-order perturbation theory in Eq. (35). The crosses correspond to |Δβe| as calculated by the frequency-domain plane-wave expansion code MPB. Excellent agreement was observed between the first-order perturbation theory and the frequency-domain code across a wide frequency range.

Fig. 4
Fig. 4

PMD parameter De (dotted curves) and group-velocity dispersion D (solid curves) plotted for the m=1 fundamental mode as a function of frequency for different core–clad index contrasts. The value of scaling factor a was chosen to maximize the absolute value of the negative dispersion of the waveguide for the m=1 fundamental mode at λ=1.55 µm. For a fixed nclad=1.5, varying the dielectric constant of the core leads to the De curves that approach the D curves for the high values of index contrast.

Fig. 5
Fig. 5

Ratio of the PMD parameter and group-velocity dispersion De/D plotted as a function of the core–clad dielectric contrast relative to the dielectric constant of the core. The solid curve corresponds to the results of a strong perturbation theory (circles). The crosses correspond to a numerical simulation that was performed with the frequency-domain plane-wave expansion code MPB, and they closely match a curve that is due to a strong perturbation theory. Weak perturbation theory (squares) predicts correct results only for the relative dielectric contrasts that are less than ∼10%.

Equations (43)

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E(x, y, z, t)H(x, y, z, t)=E(x, y)H(x, y)exp(iβz-iωt)
E=Et+Ez,Ez=zˆEz,Et=(zˆ×E)×zˆ,
×E=i ωcμH,×H=-i ωcE,
·μH=0,·E=0,
Etz+i ωcμzˆ×Ht=tEz,Htz-i ωczˆ×Et=tHz,
zˆ(t×Et)=i ωcμHz,zˆ(t×Ht)=-i ωcEz,
t·Et=-Ezz,t·μHt=-μHzz.
-i z EtHt=0-ωcμ(zˆ×)+cωt1zˆ·(t×)ωc(zˆ×)-cωt1μzˆ·(t×)0EtHt.
β0-zˆ×zˆ×0EtHt=ωc-cωt×zˆ1μzˆ·(t×)00ωcμ-cωt×zˆ1zˆ·(t×)EtHt.
Bˆ=0-zˆ×zˆ×0,
Aˆ=ωc-cωt×zˆ1μzˆ·(t×)00ωcμ-cωt×zˆ1zˆ·(t×),
|ψ=EtHt,
Aˆ|ψβ=βBˆ|ψβ,
ψβ|Bˆ|ψβ=β|β|δβ,β.
β˜=β+ψβ|ΔAˆ|ψβψβ|Bˆ|ψβ
|ψβ±=12(|ψβ,m±|ψβ,-m),
β±=β+ψβ,m|ΔAˆ|ψβ,mψβ,m|Bˆ|ψβ,m±ψβ,m|ΔAˆ|ψβ,-m|ψβ,m|Bˆ|ψβ,m|.
τ=1vg+-1vg-=(β+-β-)ω=Δβeω,
ωc 0-zˆ×zˆ×0DtBt
=β-1βt 1t·00βμ-1βt 1μt·DtBt,
(Aˆo+ΔAˆ)|ψβ˜=β˜Bˆ|ψβ˜,
ΔAˆ=ωcΔ00cωt×zˆΔ+Δzˆ·(t×).
ψβ|Bˆ|ψβ=SdsEtHtβ 0-zˆ×zˆ×0EtHtβ=Sdszˆ·(Et,β*×Ht,β+Et,β×Ht,β*),
a(×b)=(b×a)+b(×a),
ψβ|Aˆo|ψβ=SdsEtHtβ ωc-cωt×{zˆ[zˆ·(t×)]}00ωc-cωt×zˆ1zˆ·(t×)EtHtβ=ωc SdsEzEtHzHtβ -00000000-100001EzEtHzHtβ,
β=ψβ|Aˆo|ψβψβ|Bˆ|ψβ,
ψβ|ΔAˆ|ψβ=SdsEtHtβ ωcΔ00cωt×zˆΔ+Δzˆ·(t×)EtHtβ=ωc SdsEzEtHzHtβ Δ+Δ0000Δ0000000000EzEtHzHtβ.
xscaled=x(1+δx),yscaled=y(1+δy).
t,scaled×F=detxˆyˆzˆxscaledyscaled0Fx,scaledFy,scaledFz=detxˆyˆzˆ(1+δx)x(1+δy)y0FxFyFz=t×F-Oˆ,
Oˆ=detxˆyˆzˆηx xηy y0FxFyFz.
Aˆscaled=Aˆo+ΔAˆ,
ΔAˆ=cω t×[zˆ(zˆ·Oˆ)]+Oˆ[zˆ(zˆ·t×)]00t×zˆ1zˆ·Oˆ+Oˆzˆ1zˆ·t×.
ψβ,m|ΔAˆ|ψβ,m=SdsExEyHxHyβ 2 ωcηy00-(βηx+βηy)02 ωcηxβηy+βηx00βηx+βηy2 ωcηy0-(βηy+βηx)002 ωcηxExEyHxHyβ,
E(r, t)H(r, t)β,m=E(ρ)H(ρ)β,m exp(iβz-iωt+imθ).
ψβ,m|ΔAˆ|ψβ,m=drErEθHrHθβ,m (ηx+ηy)2δm,m2 ωc00-(β+β)02 ωc(β+β)00(β+β)2 ωc0-(β+β)002 ωc+(ηx-ηy)2δm,m2-2 ωc±i ωci (β-β)2-(β-β)2±i ωc2 ωc-(β-β)2±i (β-β)2±i (β-β)2(β-β)2-2 ωc±i ωc(β-β)2i (β-β)2±i ωc2 ωc×ErEθHrHθβ,m.
Δβs=ψβ,m|ΔAˆ|ψβ,m=2ηdrErEθHrHθβ,m×ωc00-β0ωcβ00βωc0-β00ωcErEθHrHθβ,m=2η ωc dr(|Ez|2+|Hz|2),
Δβs=δω βω-β.
Δβsω=δω 2βω2=-λδD(ω),
Δβe=2|ψβ,1|ΔAˆ|ψβ,-1|=2η ωc drErEθHrHθβ,1×--i00-i0000-1-i00-i1ErEθHrHθβ,-1=2η ωc dr[(-|Ez|2+|Hz|2)+2Im(Er *Eθ-Hr *Hθ)],
Ez-m=-Ezm,Er-m=-Erm,Eθ-m=Eθm,
Hz-m=Hzm,Hr-m=Hrm,Hθ-m=-Hθm,
dr(|Ez|2+|Et|2)=dr(|Hz|2+|Ht|2).
τ=ΔβeωΔβsω=λδ|D|.

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