Abstract

The infinite set of linear algebraic equations for solving the boundary-value problem for electromagnetic radiation transmitted in a system of two identical parallel dielectric cylindrical waveguides, which replaces the usual coupled-mode perturbative theory, is explicitly written and truncated to a finite size. For the first time, to my knowledge, its exact numerical solutions are obtained, discussed, and illustrated. Two symmetries of the problem are employed to simplify its numerical analysis and conveniently classify the existing global modes. In the two-waveguide system the degeneracy of modes of a single cylindrical waveguide is removed. The dependence of modes on the waveguides' radius and their separation is illustrated. The solutions include, as a special case, the solution of the problem of a single cylindrical waveguide placed parallel to the planar surface of a perfect conductor.

© 1997 Optical Society of America

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References

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  1. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 10, pp. 421–426.
  2. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 568–575.
  3. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
    [Crossref]
  4. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [Crossref]
  5. K. Yasumoto, “Coupled-mode analysis of two-parallel circular dielectric waveguides using a singular perturbation technique,” J. Lightwave Technol. 12, 74–81 (1994).
    [Crossref]
  6. A. G. Bulushev, E. M. Dianov, O. G. Okhotnikov, “Propagation of the radiation in two identical coupled waveguides,” Quantum Electron. 15, 1433–1441 (1988); Sov. J. Quantum Electron. 18, 900 (1988).
    [Crossref]
  7. G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1962), Chap. 11, p. 361.

1994 (2)

K. Yasumoto, “Coupled-mode analysis of two-parallel circular dielectric waveguides using a singular perturbation technique,” J. Lightwave Technol. 12, 74–81 (1994).
[Crossref]

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[Crossref]

1988 (1)

A. G. Bulushev, E. M. Dianov, O. G. Okhotnikov, “Propagation of the radiation in two identical coupled waveguides,” Quantum Electron. 15, 1433–1441 (1988); Sov. J. Quantum Electron. 18, 900 (1988).
[Crossref]

1972 (1)

Bulushev, A. G.

A. G. Bulushev, E. M. Dianov, O. G. Okhotnikov, “Propagation of the radiation in two identical coupled waveguides,” Quantum Electron. 15, 1433–1441 (1988); Sov. J. Quantum Electron. 18, 900 (1988).
[Crossref]

Dianov, E. M.

A. G. Bulushev, E. M. Dianov, O. G. Okhotnikov, “Propagation of the radiation in two identical coupled waveguides,” Quantum Electron. 15, 1433–1441 (1988); Sov. J. Quantum Electron. 18, 900 (1988).
[Crossref]

Huang, W.-P.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 568–575.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 10, pp. 421–426.

Okhotnikov, O. G.

A. G. Bulushev, E. M. Dianov, O. G. Okhotnikov, “Propagation of the radiation in two identical coupled waveguides,” Quantum Electron. 15, 1433–1441 (1988); Sov. J. Quantum Electron. 18, 900 (1988).
[Crossref]

Snyder, A. W.

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
[Crossref]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 568–575.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1962), Chap. 11, p. 361.

Yasumoto, K.

K. Yasumoto, “Coupled-mode analysis of two-parallel circular dielectric waveguides using a singular perturbation technique,” J. Lightwave Technol. 12, 74–81 (1994).
[Crossref]

J. Lightwave Technol. (1)

K. Yasumoto, “Coupled-mode analysis of two-parallel circular dielectric waveguides using a singular perturbation technique,” J. Lightwave Technol. 12, 74–81 (1994).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Quantum Electron. (1)

A. G. Bulushev, E. M. Dianov, O. G. Okhotnikov, “Propagation of the radiation in two identical coupled waveguides,” Quantum Electron. 15, 1433–1441 (1988); Sov. J. Quantum Electron. 18, 900 (1988).
[Crossref]

Other (3)

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1962), Chap. 11, p. 361.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972), Chap. 10, pp. 421–426.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983), pp. 568–575.

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Figures (6)

Fig. 1
Fig. 1

Geometry of a two-waveguide system.

Fig. 2
Fig. 2

Example of Ez and Bz fields in a two-waveguide system: koa=1.2, koD=2.0. β1++=1.2810, β1--=1.2189, β1+-=1.2203, and β1-+=1.2775.

Fig. 3
Fig. 3

Transverse fields E and B for three {++} modes of a two-waveguide system: koa=2 and koD=5. β1++ =1.6578, β2++=1.2011, and β3++=1.0945. The field directions are represented by tails, and magnitudes by tails’ length (after logarithmic transformation elog(1+10e) magnifying smaller fields outside the waveguide).

Fig. 4
Fig. 4

Propagation constant β as a function of the waveguide radius a for fixed separation, koh=2ko(d-a)=1.

Fig. 5
Fig. 5

β as a function of waveguide separation for a fixed radius, single-symmetry-mode case.

Fig. 6
Fig. 6

β as a function of waveguide separation for a fixed radius, multimode case. Different families of curves are associated with values of βn of a single waveguide corresponding to (a) n=1, (b) n=0(TE), (c) n=0(TM), and (d) n=2.

Tables (2)

Tables Icon

Table 1 Symmetry of Field Components with Respect to Px and Py Reflections

Tables Icon

Table 2 Accuracy of the Method

Equations (53)

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2x2+2y2+ko2-β2EzBz=0.
2r2+1rr+ko2(-β2)+1r22ϕ2EzBz=0.
Er=1koWβiβ Ezr+irBzϕ,
Eϕ=1koWβiβrEzϕ-i Bzr,
Br=1koWβ-irEzϕ+iβ Bzr,
Bϕ=1koWβi Ezr+iβrBzϕ,
E(r,ϕ)B(r,ϕ)=exp(inϕ)En(r)Bn(r).
EznBzn=AniBnWβJn(qdr)exp(inϕ)
EznBzn=CniDnW1βKn(qvr)exp(inϕ)
qd=kod-β2,qv=koβ2-v.
1:r1=|R+d|<a,
2:r2=|R-d|<a,
3:r1>a,r2>a,
Ez(R, ϕ, z)
=exp(ikoβz)×n=-1:Wβan1Jn(qdr1)exp(inϕ1)2:Wβan2Jn(qdr2)exp(inϕ2)3:W1β[cn1Kn(qvr1)exp(inϕ1) +cn2Kn(qvr2)exp(inϕ2)],
Bz(R, ϕ, z)
=i exp(ik0βz)×n=-1:Wβbn1Jn(qdr1)exp(inϕ1)2:Wβbn2Jn(qdr2)exp(inϕ2)3:W1β[dn1Kn(qvr1)exp(inϕ1) +dn2Kn(qvr2)exp(inϕ2)].
Ez(R, ϕ, z)=exp(ikoβz)n=-[αn1Jn(qdr1)exp(inϕ1)+αn2Jn(qdr2)exp(inϕ2)],
Kn(qr2)exp(inϕ2)=(-1)nm=- Im(qr1)×Kn-m(qD)exp(imϕ1),
Kn(qr1)exp(inϕ1)=m=-(-1)mIm(qr2)×Kn-m(qD)exp(imϕ2),
Ez(r1, ϕ1)=exp(ikoβz)n=- exp(inϕ1)cn1Kn(qvr1)+In(qvr1)m=-(-1)mcm2Kn-m(qvD),
Ez(r2, ϕ2)=exp(ikoβz)n=- exp(inϕ2)cn2Kn(qvr2)+(-1)nIn(qvr2)m=- cm1Kn-m(qvD),
WdβJnaan1-WvβKnacn1-WvβIna m(-)mKm-ndcm2=0,
WdβJnabn1-WvβKnadn1-WvβIna m(-)mKm-nddm2=0,
-nβaJnaan1+qdJnabn1+nβaKnacn1
+nβaIna m(-)mKm-ndcm2-qvKnadn1
-qvIna m(-1)mKm-nddm2=0,
qddJnaan1-nβaJnabn1-qvvKnacn1
-qvvIna m(-)mKm-ndcm2+nβaKnadn1
+nβaIna m(-1)mKm-nddm2=0,
Jna=Jn(qda), Jna=Jn(qda), Knd=Kn(qvD),
Ina=In(qva), Kna=Kn(qva).
Mˆ·x=0,
det{Mˆ}=0.
an2bn2cn2dn2=-τC(-1)nan1bn1cn1dn1,
a-n1b-n1c-n1d-n1=τS(-)nan1-(-)nbn1cn1-dn1.
MˆτCτS·y=0,
{a0, , aN, b0, , bN, , dN}.
det{MˆτCτS(β)}=0.
1<βτCτS<d1/2,
r1=r2r1r2+ϕ2r1ϕ2.
WdJnaan-WvKnacn+τCWvInaKndc0
+τCWvIna m=1(K|n-m|p+τSKn+mp)cm=0,
WdJnabn-WvKnadn+τCWvInaKndd0
+τSWvIna m=1(K|n-m|p-τSKn+mp)dm=0,
-nβaJnaan+qdJnabn+nβaKnacn-qvKnadn
-τC nβaInaKndc0-τC nβaIna m=1(K|n-m|p
+τSKn+mp)cm+τCqvInaKndd0
+τCqvIna m=1(K|n-m|p-τSKn+mp)dm=0,
dqdJnaan-nβaJnabn-vqvKnacn+nβaKnadn
+τCvqvInaKndc0+τCvqvInam=1(K|n-m|p
+τSKn+mp)cm-τC nβaInaKndd0
-τC nβaInam=1(K|n-m|p-τSKn+mp)dm=0.

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