Abstract

Single-mode optical wave guiding properties of silica and silicon subwavelength-diameter wires are studied with exact solutions of Maxwell’s equations. Single mode conditions, modal fields, power distribution, group velocities and waveguide dispersions are studied. It shows that air-clad subwavelength-diameter wires have interesting properties such as tight-confinement ability, enhanced evanescent fields and large waveguide dispersions that are very promising for developing future microphotonic devices with subwavelength-width structures.

© 2004 Optical Society of America

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References

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Anal. Chem. (1)

A. P. Abel, M. G. Weller, G. L. Duveneck, M. Ehrat, and H. M. Widmer, �??Fiber-optic evanescent wave biosensor for the detection of oligonucleotides,�?? Anal. Chem. 68, 2905-2912 (1996).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

K. K. Lee, D. R. Lim, H. C. Luan, A. Agarwal, J. Foresi, and L. C. Kimerling, �??Effect of size and roughness on light transmission in a Si/SiO2 waveguide: experiments and model, �?? Appl. Phys. Lett. 77, 1617-1619 (2000). Erratum: Appl. Phys. Lett. 77, 2258 (2000).
[CrossRef]

Bell Syst. Tech. J. (2)

D. Marcuse, �??Mode conversion caused by surface imperfections of a dielectric slab waveguide, �?? Bell Syst. Tech. J. 48, 3187-3215 (1969).

D. Marcuse, and R. M. Derosier, �??Mode conversion caused by diameter changes of a round dielectric waveguide, �?? Bell Syst. Tech. J. 48, 3217-3232 (1969).

J. Am. Chem. Soc. (1)

Z. W. Pan, Z. R. Dai, C. Ma, and Z. L. Wang, �??Molten gallium as a catalyst for the large-scale growth of highly aligned silica nanowires,�?? J. Am. Chem. Soc. 124, 1817-1822 (2002).
[CrossRef] [PubMed]

J. Lightwave Technol. (2)

Nature (2)

X. F. Duan, Y. Huang, R. Agarwal, and C. M. Lieber, �??Single-nanowire electrically driven lasers," Nature 421, 241-245 (2003).
[CrossRef] [PubMed]

L. M. Tong, R. R. Gattass, J. B. Ashcom, S. L. He, J. Y. Lou, M. Y. Shen, I. Maxwell, and E. Mazur, �??Subwavelength-diameter silica wires for low-loss optical wave guiding,�?? Nature 426, 816-819 (2003).
[CrossRef] [PubMed]

Opt. Lett. (2)

Science (3)

L. F. Mollenauer, �??Nonlinear optics in fibers,�?? Science 302, 996-997 (2003).
[CrossRef] [PubMed]

M. J. Levene, J. Korlach, S. W. Turner, M. Foquet, H. G. Craighead, and W. W. Webb, �??Zero-mode waveguides for single-molecule analysis at high concentrations, �?? Science 299, 682-686 (2003).
[CrossRef] [PubMed]

A. M. Morales, and C. M. Lieber, �??A laser ablation method for the synthesis of crystalline semiconductor nanowires,�?? Science 279, 208-211 (1998).
[CrossRef] [PubMed]

Sensors Actuat. (1)

Z. M. Qi, N. Matsuda, K. Itoh, M. Murabayashi, and C. R. Lavers, �??A design for improving the sensitivity of a Mach-Zehnder interferometer to chemical and biological measurands, �?? Sensors Actuat. B81, 254-258 (2002).
[CrossRef]

Other (8)

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, New York, NY 1991).
[CrossRef]

A. Ghatak, and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University Press, Cambridge, 1998).

P. P. Bishnu, Fundamentals of Fibre Optics in Telecommunication and Sensor Systems (John Wiley & Sons, New York, NY 1993).

R. G. Hunsperger, Photonic Devices and Systems (Marcel Dekker, New York, NY 1994).

J. S. Sanghera, and I. D. Aggarwal, Infrared Fiber Optics (CRC Press, New York, NY 1998).

A. W. Snyder, and J. D. Love, Optical waveguide theory (Chapman and Hall, New York, NY 1983).

P. Klocek, Handbook of infrared optical materials (Marcel Dekker, New York, NY 1991).

E. D. Palik, Handbook of optical constants of solids (Academic Press, New York, NY 1998).

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Figures (12)

Fig. 1.
Fig. 1.

Mathematic model of an air-clad cylindrical wire waveguide.

Fig. 2.
Fig. 2.

Numerical solutions of propagation constant (β) of air-clad silica wire at 633-nm wavelength. Solid line, fundamental mode. Dotted lines, high-order modes. Dashed line, critical diameter for single-mode operation (DSM ).

Fig. 3.
Fig. 3.

Numerical solutions of propagation constant (β) of air-clad silicon wire at 1.5-µm wavelength. Solid line, fundamental mode. Dotted lines, high-order modes. Dashed line, critical diameter for single-mode operation (DSM ).

Fig. 4.
Fig. 4.

Single mode condition of an air-clad silica and silicon wires. Solid line, critical diameter for single-mode operation. Dotted line, wavelength in media.

Fig. 5.
Fig. 5.

Electric components of HE11 modes of silica wires at 633-nm wavelength with different diameters in cylindrical coordination. Normalizations are applied as: εer(r=0)=1 and eΦ(r=0)=1. Wire diameters are arrowed to each curve in unit of nm.

Fig. 6.
Fig. 6.

Z-direction Poynting vectors of silica wires at 633-nm wavelength with diameters of (A) 400 nm and (B) 200 nm. Mesh, field inside the core. Gradient, field outside the core.

Fig. 7.
Fig. 7.

Fractional power of the fundamental modes inside the core of (A) silica wire at 633-nm wavelength, (B) silica wire at 1.5-µm wavelength and (C) silicon wire at 1.5-µm wavelength. Dashed line, critical diameter for single mode operation.

Fig. 8.
Fig. 8.

Effective diameters of the light fields of the fundamental modes. Solid line, Deff. Dotted line, real diameter. Dashed line, critical diameter for single mode operation. (A) silica wire at 633-nm wavelength, (B) silica wire at 1.5-µm wavelength and (C) silicon wire at 1.5-µm wavelength.

Fig. 9.
Fig. 9.

Diameter-dependent group velocities of the fundamental modes of air-clad (A) silica wire at 633- nm and 1.5-µm wavelengths and (B) silicon wire at 1.5-µm wavelength.

Fig. 10.
Fig. 10.

Wavelength-dependent group velocities of the fundamental modes of air-clad (A) lica wire and (B) silicon wire with different diameters (wire diameters are labeled on each rve in unit of nm)

Fig. 11.
Fig. 11.

Diameter-dependent waveguide dispersion of fundamental modes of air-clad (A) silica wire at 633-nm and 1.5-µm wavelengths and (B) silicon wire at 1.5-µm wavelengths.

Fig. 12.
Fig. 12.

Wavelength-dependent waveguide dispersion of fundamental modes of air-clad (A) silica wire and (B) silicon wire with different wire diameters (the wire diameter is labeled on each curve in unit of nm). Material dispersion is plotted in dotted line.

Equations (23)

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n ( r ) = { n 1 , 0 < r < a , n 2 , a r <
( 2 + n 2 k 2 β 2 ) e = 0 ,
( 2 + n 2 k 2 β 2 ) h = 0
{ J v ( U ) UJ v ( U ) + K v ( W ) WK v ( W ) } { J v ( U ) UJ v ( U ) + n 2 2 K v ( W ) n 1 2 WK v ( W ) } = ( v β kn 1 ) 2 ( V UW ) 4
J 1 ( U ) UJ 0 ( U ) + K 1 ( W ) WK 0 ( W ) = 0
n 1 2 J 1 ( U ) UJ 0 ( U ) + n 2 2 K 1 ( W ) WK 0 ( W ) = 0
n 2 1 = 0.6961663 λ 2 λ 2 ( 0.0684043 ) 2 + 0.4079426 λ 2 λ 2 ( 0.1162414 ) 2 + 0.8974794 λ 2 λ 2 ( 9.896161 ) 2
n 2 = 11.6858 + 0.939816 λ 2 + 0.000993358 λ 2 1.22567
V = 2 π · a λ 0 · ( n 1 2 n 2 2 ) 1 2 2.405 .
{ J 1 ( U ) UJ 1 ( U ) + K 1 ( W ) WK 1 ( W ) } { J 1 ( U ) UJ 1 ( U ) + n 2 2 K 1 ( W ) n 1 2 WK 1 ( W ) } = ( β kn 1 ) 2 ( V UW ) 4
{ E ( r , ϕ , z ) = ( e r r ̂ + e ϕ ϕ ̂ + e z z ̂ ) e i β z e i ω t , H ( r , ϕ , z ) = ( h r r ̂ + h ϕ ϕ ̂ + h z z ̂ ) e i β z e i ω t
e r = a 1 J 0 ( UR ) + a 2 J 2 ( UR ) J 1 ( U ) · f 1 ( ϕ ) ,
e ϕ = a 1 J 0 ( UR ) + a 2 J 2 ( UR ) J 1 ( U ) · g 1 ( ϕ ) ,
e z = iU a β J 1 ( UR ) J 1 ( U ) · f 1 ( ϕ )
e r = U W a 1 K 0 ( WR ) a 2 K 2 ( WR ) K 1 ( W ) · f 1 ( ϕ ) ,
e ϕ = U W a 1 K 0 ( WR ) a 2 K 2 ( WR ) K 1 ( W ) · g 1 ( ϕ ) ,
e z = iU α β K 1 ( WR ) K 1 ( W ) · f 1 ( ϕ )
S z 1 = 1 2 ( ε 0 μ 0 ) 1 2 kn 1 2 β J 1 2 ( U ) [ a 1 a 3 J 0 2 ( UR ) + a 2 a 4 J 2 2 ( UR ) + 1 F 1 F 2 2 J 0 ( UR ) J 2 ( UR ) cos ( 2 ϕ ) ]
S z 2 = 1 2 ( ε 0 μ 0 ) 1 2 kn 1 2 β K 1 2 ( W ) U 2 W 2 [ a 1 a 5 K 0 2 ( WR ) + a 2 a 6 K 2 2 ( WR ) 1 2 Δ F 1 F 2 2 K 0 ( WR ) K 2 ( WR ) cos ( 2 ϕ ) ]
η = 0 a S z 1 dA 0 a S z 1 dA + a S z 2 dA
{ 0 D eff S z 1 dA 0 a S z 1 dA + a S z 2 dA = 86.5 % , ( if D eff a ) , 0 a S z 1 dA + a D eff S z 1 dA 0 a S z 1 dA + a S z 2 dA = 86.5 % , ( if D eff > a ) .
v g = c n 1 2 · β k · 1 1 2 Δ ( 1 η ) .
D w = d ( v g 1 ) d λ .

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