Abstract

We report what is to our knowledge a novel dielectric waveguiding structure consisting of periodic arrays of parallel planar fins placed on both sides of and perpendicular to a central planar core. This structure is analyzed, and the behavior of its bound modes is investigated. This finned guide exhibits many of the same features as a recently reported all-silica photonic crystal fiber. For example, it can be designed to support only the fundamental transverse bound mode over the entire electromagnetic frequency spectrum. This new class of endlessly single-mode waveguide may prove to have important applications in many areas of optoelectronics.

© 1998 Optical Society of America

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  1. J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996);errata, Opt. Lett. 22, 484–485 (1997).
    [CrossRef] [PubMed]
  2. T. A. Birks, J. C. Knight, P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
    [CrossRef] [PubMed]
  3. J. C. Knight, T. A. Birks, P. St. J. Russell, J. G. Rarity, “Bragg scattering from an obliquely illuminated photonic crystal fiber,” Appl. Opt. 37, 449–452 (1998).
    [CrossRef]
  4. Although the photonic crystal fiber can support modes trapped by Bragg scattering, this requires a full two-dimensional photonic bandgap, which is possible only at air-filling fractions >40%. [See T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1942 (1995).] Thus Bragg modes in photonic crystal fibers with smaller air-filling fractions will always be leaky.
    [CrossRef]
  5. D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
    [CrossRef]
  6. P. St, J. Russell, D. M. Atkin, T. A. Birks, “Bound modes of photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  7. P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Applications, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

1998 (1)

1997 (1)

1996 (2)

J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996);errata, Opt. Lett. 22, 484–485 (1997).
[CrossRef] [PubMed]

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

1995 (1)

Although the photonic crystal fiber can support modes trapped by Bragg scattering, this requires a full two-dimensional photonic bandgap, which is possible only at air-filling fractions >40%. [See T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1942 (1995).] Thus Bragg modes in photonic crystal fibers with smaller air-filling fractions will always be leaky.
[CrossRef]

Atkin, D. M.

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996);errata, Opt. Lett. 22, 484–485 (1997).
[CrossRef] [PubMed]

Although the photonic crystal fiber can support modes trapped by Bragg scattering, this requires a full two-dimensional photonic bandgap, which is possible only at air-filling fractions >40%. [See T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1942 (1995).] Thus Bragg modes in photonic crystal fibers with smaller air-filling fractions will always be leaky.
[CrossRef]

P. St, J. Russell, D. M. Atkin, T. A. Birks, “Bound modes of photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Birks, T. A.

J. C. Knight, T. A. Birks, P. St. J. Russell, J. G. Rarity, “Bragg scattering from an obliquely illuminated photonic crystal fiber,” Appl. Opt. 37, 449–452 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996);errata, Opt. Lett. 22, 484–485 (1997).
[CrossRef] [PubMed]

Although the photonic crystal fiber can support modes trapped by Bragg scattering, this requires a full two-dimensional photonic bandgap, which is possible only at air-filling fractions >40%. [See T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1942 (1995).] Thus Bragg modes in photonic crystal fibers with smaller air-filling fractions will always be leaky.
[CrossRef]

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Applications, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

P. St, J. Russell, D. M. Atkin, T. A. Birks, “Bound modes of photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

Knight, J. C.

Lloyd-Lucas, F. D.

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Applications, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

Rarity, J. G.

Roberts, P. J.

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

Although the photonic crystal fiber can support modes trapped by Bragg scattering, this requires a full two-dimensional photonic bandgap, which is possible only at air-filling fractions >40%. [See T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1942 (1995).] Thus Bragg modes in photonic crystal fibers with smaller air-filling fractions will always be leaky.
[CrossRef]

Russell, J.

P. St, J. Russell, D. M. Atkin, T. A. Birks, “Bound modes of photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Applications, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

Russell, P. St. J.

J. C. Knight, T. A. Birks, P. St. J. Russell, J. G. Rarity, “Bragg scattering from an obliquely illuminated photonic crystal fiber,” Appl. Opt. 37, 449–452 (1998).
[CrossRef]

T. A. Birks, J. C. Knight, P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996);errata, Opt. Lett. 22, 484–485 (1997).
[CrossRef] [PubMed]

Although the photonic crystal fiber can support modes trapped by Bragg scattering, this requires a full two-dimensional photonic bandgap, which is possible only at air-filling fractions >40%. [See T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1942 (1995).] Thus Bragg modes in photonic crystal fibers with smaller air-filling fractions will always be leaky.
[CrossRef]

Shepherd, T. J.

Although the photonic crystal fiber can support modes trapped by Bragg scattering, this requires a full two-dimensional photonic bandgap, which is possible only at air-filling fractions >40%. [See T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1942 (1995).] Thus Bragg modes in photonic crystal fibers with smaller air-filling fractions will always be leaky.
[CrossRef]

St, P.

P. St, J. Russell, D. M. Atkin, T. A. Birks, “Bound modes of photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Applications, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

Appl. Opt. (1)

Electron. Lett. (1)

Although the photonic crystal fiber can support modes trapped by Bragg scattering, this requires a full two-dimensional photonic bandgap, which is possible only at air-filling fractions >40%. [See T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, T. J. Shepherd, “Full 2-D photonic band gaps in silica/air structures,” Electron. Lett. 31, 1941–1942 (1995).] Thus Bragg modes in photonic crystal fibers with smaller air-filling fractions will always be leaky.
[CrossRef]

J. Mod. Opt. (1)

D. M. Atkin, P. St. J. Russell, T. A. Birks, P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

Opt. Lett. (2)

Other (2)

P. St, J. Russell, D. M. Atkin, T. A. Birks, “Bound modes of photonic crystal waveguides,” in Quantum Optics in Wavelength Scale Structures, J. G. Rarity, C. Weisbuch, eds. (Kluwer Academic, Dordrecht, The Netherlands, 1996).

P. St, J. Russell, T. A. Birks, F. D. Lloyd-Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons: New Physics and Applications, E. Burstein, C. Weisbuch, eds. (Plenum, New York, 1995), pp. 585–633.

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Figures (9)

Fig. 1
Fig. 1

Finned structure. The thickness of the central waveguiding layer (the core) is hco. The fins have width h1 and extend to x=±, and the pitch in the multilayer cladding regions is Λ. The y axis points normal to the fins, the x axis points perpendicular to the core layer, and propagation along the z axis is considered. To mimic the photonic crystal fiber, nco=n1 throughout this paper.

Fig. 2
Fig. 2

Wave-vector diagrams for a finned structure with h1/Λ=0.8, hco/Λ=0.8, and Λ/λ=0.25. The diagrams have been calculated for a value of n1 equal to twice the refractive index of silica. n2 remains equal to 1. The left-hand plots show all the allowed real wave vectors in the (ky, β0) plane for (a) TE Bloch modes, electric field in the x direction; (b) TM Bloch modes, magnetic field in the x direction. The right-hand plots (systems of concentric circles) show the stop-band edges for the periodic cladding regions for (c) TE and (d) TM. In all the figures the outermost dashed circle represents the locus of maximum wave vectors in the pure silica core. The two arrows in (c) and (d) represent the real-valued components of the wave vectors of the two fundamental zigzag rays (i.e., those with a zero ky component) of the guided mode.

Fig. 3
Fig. 3

Power-density plots for quasi-TM guided modes in two cases: (a) Λ/λ=2 and (b) Λ/λ=6. The silica filling fraction is h1/Λ=0.8; the relative core width is hco/Λ=0.3. The white dashed lines represent the outline of the finned structure. Note that, despite a factor of 3× increase in frequency, the mode patterns are very similar.

Fig. 4
Fig. 4

Plot of effective refractive index versus Λ/λ for a situation in which the finned waveguide supports a pair of fundamental modes with quasi-TE (dashed curve) and quasi-TM (solid curve) polarization states. The silica filling fraction is h1/Λ =0.8, and the relative core width is hco/Λ=0.8. The upper and lower (dotted) curves represent the refractive index of the core and of the cladding, respectively; note the strong dispersion of the effective index of the periodic cladding. Inset, power-density plot of the quasi-TM mode for Λ/λ=2 (encircled point gives mode index). Note that the fields are much more tightly confined to the vicinity of the core than in Fig. 3.

Fig. 5
Fig. 5

Plots of effective refractive index versus Λ/λ for a step-profile planar waveguide with the same core index and thickness as the finned structure in Fig. 4, but with cladding indices given by the appropriate low-frequency average values (see text) for quasi-TM and quasi-TE modes. These values represent the mean cladding indices in the long-wavelength limit; unlike for the finned structure, they are independent of frequency. As expected, more and more guided modes are supported as the wavelength falls. The dashed curves represent the quasi-TE modes; the solid curves, the quasi-TM modes. Except for the different cutoff line (lower dotted line), the lowest-order pair of modes has characteristics quite similar to those for the higher-order pair in the equivalent finned guide (Fig. 4).

Fig. 6
Fig. 6

Plot of effective refractive index versus Λ/λ for a situation in which the finned waveguide supports two pairs of spatial modes with quasi-TE (dashed curve) and quasi-TM (solid curve) polarization states. The silica filling fraction is h1/Λ=0.8, and the relative core width is hco/Λ=1.8. Once again, the upper and the lower (dotted) curves represent the refractive index of the core and of the cladding. In the limit of infinite frequency, the structure supports only two pairs of spatial modes. Insets, power-density plots at Λ/λ=2 of the fundamental and first-order spatial quasi-TM modes (encircled points give mode indices). Once again, the white dashed lines represent the structure.

Fig. 7
Fig. 7

Plots of the V parameter against Λ/λ for hco/Λ=0.8 and discrete values of h1/Λ between 0.1 and 0.9 (solid curves). The dashed horizontal lines represent the short-wavelength asymptotic limits of the V parameter predicted by Eq. (15) for each value of h1/Λ.

Fig. 8
Fig. 8

Plot of the V parameter against Λ/λ for fixed fin width h1/Λ=0.8 and discrete values of core width hco/Λ between 0.1 and 2.3 (dashed curves). The solid curves represent the values of V at which each successive guided mode cuts off.

Fig. 9
Fig. 9

Plot of the V parameter against Λ/λ for fixed core width hco/Λ=0.8 and discrete values of fin width h1/Λ between 0.1 and 0.9 (dashed curves, as in Fig. 7). The solid curves passing through the filled points represent the parameter values at which each guided mode cuts off. We can see how pairs of quasi-TE and quasi-TM modes appear at the same frequency and at the same V parameter in the limit h1/Λ0, as expected. The intersection of the thin solid lines [defined, respectively, by V=k0(hco/2)nco2-12 (the long, slanting line) and V =mπ/2, m=1 (the short, horizontal line)] marks the cutoff of the m=1 mode pair in a step-profile waveguide with nonperiodic air cladding (i.e., zero silica fin width h1=0).

Equations (39)

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TE:E=(1, 0, 0)fe,H=1ik0z0(0,-iβ0,-y)fe,
TM:E=-z0ik0n2(y)(0,-iβ0,-y)fm,
H=(1, 0, 0)fm,
ft(β0, ky; x, y, z)=exp(-iβ0z)exp(-ikyy)Bt[β02,sign(ky); y],
A(t)(β02)=cos(kyΛ),
ft(β0; x, y, z)=exp(-iβ0z)Bt(β02; y),
A(t)(β02)=1.
β02=kx2+β2.
TE:E=(β, 0,-kx)fe,
H=1ik0z0(-kxy,-iβ02,-βy)fe,
TM:E=-z0ik0n2(y)(-kxy,-iβ02,-βy)fm,
H=(β, 0,-kx)fm,
ft(kx, β;x, y, z)=exp[-i(kxx+βz)]Bt(β02; y).
kx(t, i)=±[β02(t, i)-β2]1/2,
E=exp(-iβz)i=1Ve,i exp[-ikx(e, i)x]×[β, 0,-kx(e, i)]Be(i; y)+Vm,i exp[-ikx(m, i)x]z0ik0n2(y)×[kx(m, i)y, iβ02(m, i), βy]Bm(i; y),
H=exp(-iβz)i=1Vm,i exp[-ikx(m, i)x]×[β, 0,-kx(m, i)]Bm(i; y)+Ve,i exp[-ikx(e, i)x]×-1ik0z0[kx(e, i)y, iβ02(e, i), βy]Be(i; y),
k˜=[±κ(n), ky(n), β],
κ(n)=+[k02nco2-ky2(n)-β2]1/2.
E=exp(-iβz)n=- exp[-iky(n)y]g+p,n(x)×Ue,n[0, β,-ky(n)]+Um,nz0κ(n)k0nco2×[0, ky(n), β]+g-p,n(x)Um,n-z0k0nco2×[β2+ky2(n), 0, 0],
H=exp(-iβz)n=- exp[-iky(n)y]g-p,n(x)×Um,n[0, β,-ky(n)]+Ue,n-κ(n)k0z0×[0, ky(n), β]+g+p,n(x)Ue,n1k0z0×[β2+ky2(n), 0, 0],
V=k0hco2 nco2-ncl2,
k0ncl<βk0nco.
ncl2n12-πh1k02
V=π2 hcoh1.
nclqTM=[(n12h1+n22h2)/Λ]1/2,
nclqTE=[Λ/(h1n1-2+h2n2-2)]1/2
gt(y)=ft(β0, ky; x, y, z)exp(iβ0z)=ajN cos[pj(y-yjN)]+bjN sin[pj(y-yjN)]ξjpjΛ,
a2Nb2N=M21a1Nb1N=A21B21C21D21a1Nb1N,
A21=c1c2-(ξ1 p1Λ/ξ2 p2Λ)s1s2,
B21=s1c2/(ξ1 p1Λ)+c1s2/(ξ2 p2Λ),
C21=-ξ1 p1Λs1c2-ξ2 p2Λc1s2,
D21=c1c2-(ξ2 p2Λ/ξ1 p1Λ)s1s2,
det(M21)=1,
cj=cos(pjhj/2),sj=sin(pjhj/2).
a1N+1b1N+1=M12a2Nb2N=D21B21C21A21a2Nb2N.
a1N+1b1N+1=Ma1Nb1N=ABCDa1Nb1N,
A=D=A21D21+B21C21,B=2D21B21, C=2A21C21.
A=cos(p1h1)cos(p2h2)-12 p1ξ1p2ξ2+p2ξ2p1ξ1×sin(p1h1)sin(p2h2),
A=D=A,B=2A21B21,C=2D21C21.

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