Abstract

Axisymmetric periodic media such as annular layers have axi-symmetric birefringence. I report the theoretical analysis of mode coupling in axisymmetric birefringent waveguides. Any of the LP11 modes in a perturbed system can couple with a corresponding LP11 mode that is orthogonal to the original mode with respect to both the polarization direction and the node line of an electric field. The coupling beat length depends on the amount of birefringence, which can be as short as 5 mm for a practical structure.

© 1997 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14.
  2. M. Tateda, T. Kimura, “Optical wave propagation in form-birefringent media and waveguides,” J. Lightwave Technol. LT-1, 402–407 (1983).
    [CrossRef]
  3. M. Tateda, “Birefringence in a dielectric with periodic structure,” J. Lightwave Technol. LT-2, 522–527 (1984).
    [CrossRef]
  4. J. B. MacChesney, P. B. O’Connor, F. V. DiMarcello, J. R. Simpson, P. D. Lazay, “Preparation of low loss optical fibers using simultaneous vapor phase deposition and fusion,” in Proceedings of the Tenth International Congress on Glass (Ceramic Society of Japan, 1974), Vol. 6, pp. 40–45.
  5. E. Snitzer, “Cylindrical dielectric waveguide modes,” J. Opt. Soc. Am. 51, 491–498 (1961).
    [CrossRef]
  6. A. W. Snyder, “Asymptotic expression for eigenfunctions and eigenvalues of a dielectric on optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
    [CrossRef]
  7. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef] [PubMed]
  8. J. Sakai, T. Kimura, “Birefringence and polarization characteristics of single-mode optical fibers under elastic deformations,” IEEE J. Quantum Electron. QE-17, 1041–1051 (1981).
    [CrossRef]

1984 (1)

M. Tateda, “Birefringence in a dielectric with periodic structure,” J. Lightwave Technol. LT-2, 522–527 (1984).
[CrossRef]

1983 (1)

M. Tateda, T. Kimura, “Optical wave propagation in form-birefringent media and waveguides,” J. Lightwave Technol. LT-1, 402–407 (1983).
[CrossRef]

1981 (1)

J. Sakai, T. Kimura, “Birefringence and polarization characteristics of single-mode optical fibers under elastic deformations,” IEEE J. Quantum Electron. QE-17, 1041–1051 (1981).
[CrossRef]

1971 (1)

1969 (1)

A. W. Snyder, “Asymptotic expression for eigenfunctions and eigenvalues of a dielectric on optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[CrossRef]

1961 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14.

DiMarcello, F. V.

J. B. MacChesney, P. B. O’Connor, F. V. DiMarcello, J. R. Simpson, P. D. Lazay, “Preparation of low loss optical fibers using simultaneous vapor phase deposition and fusion,” in Proceedings of the Tenth International Congress on Glass (Ceramic Society of Japan, 1974), Vol. 6, pp. 40–45.

Gloge, D.

Kimura, T.

M. Tateda, T. Kimura, “Optical wave propagation in form-birefringent media and waveguides,” J. Lightwave Technol. LT-1, 402–407 (1983).
[CrossRef]

J. Sakai, T. Kimura, “Birefringence and polarization characteristics of single-mode optical fibers under elastic deformations,” IEEE J. Quantum Electron. QE-17, 1041–1051 (1981).
[CrossRef]

Lazay, P. D.

J. B. MacChesney, P. B. O’Connor, F. V. DiMarcello, J. R. Simpson, P. D. Lazay, “Preparation of low loss optical fibers using simultaneous vapor phase deposition and fusion,” in Proceedings of the Tenth International Congress on Glass (Ceramic Society of Japan, 1974), Vol. 6, pp. 40–45.

MacChesney, J. B.

J. B. MacChesney, P. B. O’Connor, F. V. DiMarcello, J. R. Simpson, P. D. Lazay, “Preparation of low loss optical fibers using simultaneous vapor phase deposition and fusion,” in Proceedings of the Tenth International Congress on Glass (Ceramic Society of Japan, 1974), Vol. 6, pp. 40–45.

O’Connor, P. B.

J. B. MacChesney, P. B. O’Connor, F. V. DiMarcello, J. R. Simpson, P. D. Lazay, “Preparation of low loss optical fibers using simultaneous vapor phase deposition and fusion,” in Proceedings of the Tenth International Congress on Glass (Ceramic Society of Japan, 1974), Vol. 6, pp. 40–45.

Sakai, J.

J. Sakai, T. Kimura, “Birefringence and polarization characteristics of single-mode optical fibers under elastic deformations,” IEEE J. Quantum Electron. QE-17, 1041–1051 (1981).
[CrossRef]

Simpson, J. R.

J. B. MacChesney, P. B. O’Connor, F. V. DiMarcello, J. R. Simpson, P. D. Lazay, “Preparation of low loss optical fibers using simultaneous vapor phase deposition and fusion,” in Proceedings of the Tenth International Congress on Glass (Ceramic Society of Japan, 1974), Vol. 6, pp. 40–45.

Snitzer, E.

Snyder, A. W.

A. W. Snyder, “Asymptotic expression for eigenfunctions and eigenvalues of a dielectric on optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[CrossRef]

Tateda, M.

M. Tateda, “Birefringence in a dielectric with periodic structure,” J. Lightwave Technol. LT-2, 522–527 (1984).
[CrossRef]

M. Tateda, T. Kimura, “Optical wave propagation in form-birefringent media and waveguides,” J. Lightwave Technol. LT-1, 402–407 (1983).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

J. Sakai, T. Kimura, “Birefringence and polarization characteristics of single-mode optical fibers under elastic deformations,” IEEE J. Quantum Electron. QE-17, 1041–1051 (1981).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, “Asymptotic expression for eigenfunctions and eigenvalues of a dielectric on optical waveguide,” IEEE Trans. Microwave Theory Tech. MTT-17, 1130–1138 (1969).
[CrossRef]

J. Lightwave Technol. (2)

M. Tateda, T. Kimura, “Optical wave propagation in form-birefringent media and waveguides,” J. Lightwave Technol. LT-1, 402–407 (1983).
[CrossRef]

M. Tateda, “Birefringence in a dielectric with periodic structure,” J. Lightwave Technol. LT-2, 522–527 (1984).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14.

J. B. MacChesney, P. B. O’Connor, F. V. DiMarcello, J. R. Simpson, P. D. Lazay, “Preparation of low loss optical fibers using simultaneous vapor phase deposition and fusion,” in Proceedings of the Tenth International Congress on Glass (Ceramic Society of Japan, 1974), Vol. 6, pp. 40–45.

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Figures (2)

Fig. 1
Fig. 1

Structure of an axisymmetric birefringent waveguide: (a) cross section, (b) dielectric constant distribution.

Fig. 2
Fig. 2

Electric field vector (→) and mode-coupling pair (ao-36-15-3452-i001) for (a) the LP mode, (b) TE, TM, and HE modes.

Equations (44)

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ε=ε0εrεrθ0εrθεθ000εθ  ra,
εrθ=0.
εr=f1/ε1+f2/ε2-1,  εz=εθ=f1ε1+f2ε2,
f1=t1/t1+t2,  f2=t2/t1+t2.
εcl=ε0εcl100010001  r>a.
εco=ε0εθ100010001.
εˆ=ε-εco=ε0Δ100000000ra0r>a,
Δ=εr-εθ.
EH=AzE1H1+BzE2H2,
ddzAB=-iPTT*QAB,
P=β1+ω/4E1*εˆE1ds,  Q=β2+ω/4E2*εˆE2ds,  T=ω/4E1*εˆE2ds,
E1=Er1Eθ1Ez1,  E2=Er2Eθ2Ez2,  H1=Hr1Hθ1Hz1,  H2=Hr2Hθ2Hz2,
P=β1+Δωε0/4Er12ds,  Q=β2+Δωε0/4Er22ds,  T=Δωε0/4Er1*Er2ds.
ErpEθpEzpexpiωt-βz  p=x, y,
Erx=Eθy=A0Rrcos θ,  Eθx=-Ery=-A0Rrsin θ.
Rr=J0ur/a/J0uraK0wr/a/K0wr>a.
T=Δωε0/4Erx*Eryds=-Δωε0/4A02Rr2rdr02πsin θ cos θdθ=0,
ErpEθpEzpexpiωt-βz  p=R, L,
ErR=1/2Erx-iEry=1/2A0Rrcos θ-i sin θ,  ErL=1/2Erx+iEry=1/2A0Rrcos θ+i sin θ.
T=Δωε0/4ErR*ErLds=1/2ωε0/4A02Rr2rdr02π exp2iθdθ=0.
EjA1Rrexpiωt-βz  j=1, 2, 3, 4,
E1=sc-s2*,  E2=scc2*,  E3=s2sc*,  E4=c2-sc*,
s=sin θ,  c=cos θ,  Rr=J1ur/a/J1uraK1wr/a/K1wr>a.
A12=2β/ωεclε0π/0Rr2rdr
T12=Δωε0/4A120aRr2rdr02π sin2 θ cos2 θdθ=ΔI0π/2,
I0=ωε0/4A120aRr2rdr.
I0  β/2πεcl.
T12=T21=T34=T43=ΔI0π/2βΔ/4εcl,  Tij=0  otherwise.
δβ=Δωε0/4Er2ds,  δβ1=δβ2=Δωε0/4A12Rr2rdrsin2 θ cos2 θdθ=ΔI0π/2βΔ/4εcl,  δβ3=δβ4=Δωε0/4A12Rr2rdr sin4 θdθ=ΔI03π/43βΔ/8εcl.
Az=exp-iβ0z×A0cos δz+i sin δz/1+2T/P-Q21/2-iB0 sin δz/1+P-Q/2T21/2,  Bz=exp-iβ0z×A0i sin δz/1+P-Q/2T21/2+B0cos δz+i sin δz/1+2T/P-Q21/2,
β0=P+Q/2,  δ=T2+P-Q/221/2.
δ=T,  Az=exp-iβ0zA0 cosTz-iB0 sinTz,
LB=2π/T=8πεcl/Δβ.
ε1/εSiO2-1=5×10-2,  ε2/εSiO2-1=-1×10-2.
εθ=1.02×εSiO2,  εr=1.0191×εSiO2,  Δ/εSiO2=εr-εθ/εSiO2=-9×10-4.
Sz=1/2 ReE×H*ezds=1/2 ReErHθ*-EθHr*ds=1.
Er=A0Rrcos θ expiωt-βz,  Eθ=A0Rr-sin θexpiωt-βz,  Hr=-ωεrε0/βEθ,  Hθ=ωεrε0/βEr,
Sz=ωεclε0/2βA0202π×cos2 θ+sin2 θdθ0Rr2rdr=ωεclε0π/βA020Rr2rdr=1.
A02=β/ωεclε0π/0Rr2rdr.
Er=A1Rrsin θ cos θ expiωt-βz,  Eθ=A1Rr-sin2 θexpiωt-βz,  Hr=-ωεrε0/βEθ,  Hθ=ωεrε0/βEr,
Sz=ωεclε0/2βA1202πsin2 θ cos2θ+sin4 θdθ0Rr2rdr=ωεclε0/2βA12π0Rr2rdr=1.
EjA2Rrexpiωt-βz  j=1, 2, 3, 4,
E1=01*,  E2=10*,  E3=sin 2θcos 2θ*,  E4=cos 2θ-sin 2θ*.
Tij=0  i, j=1, 2, 3, 4.

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