Abstract

Closed silicon V grooves are proposed as new hollow waveguides suitable for optical microelectromechanical systems applications. These easily fabricated guides with large index contrast could be designed to work with very low loss for the fundamental mode. A ray optics model is developed for the loss analysis of such guides. The model is tested using the beam propagation method. The model allows one to obtain approximate design equations for the fundamental mode losses in equilateral triangles as well as the practical waveguide and thus greatly simplifies the design effort.

© 2006 Optical Society of America

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References

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  1. H. Schmidt, D. Yin, J. P. Barber, and A. Hawkins, "Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids," IEEE J. Sel. Top. Quantum Electron. 11, 519-527 (2005).
    [CrossRef]
  2. R. Bernini, S. Campopiano, and L. Zeni, "Silicon micromachined hollow optical waveguides for sensing applications," IEEE J. Sel. Top. Quantum Electron. 8, 106-110 (2002).
    [CrossRef]
  3. K. Madkour, H. Maaty, and D. Khalil, "Silicon hollow waveguide for MEMS applications," in Proceedings of the European Conference on Optical Communication-International Conference on Integrated Optics and Optical Fibre Communication (ECOC-IOOC) (2003).
    [PubMed]
  4. J. N. Mcmullin, R. Narendra, and C. R. James, "Hollow metallic waveguides in silicon V grooves," IEEE Photon. Technol. Lett. 5, 1080-1082 (1993).
    [CrossRef]
  5. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).
  6. M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).
  7. K. D. Laakmann and W. Steier, "Waveguides: characteristic modes of hollow rectangular dielectric waveguides," Appl. Opt. 15, 1334-1340 (1976).
    [CrossRef] [PubMed]
  8. A. W. Snyder and X.-H. Zheng, "Optical fibers of arbitrary cross sections," J. Opt. Soc. Am. A 3, 600-609 (1986).
    [CrossRef]
  9. P. L. Overfelt and D. J. White, "TE and TM modes of some triangular cross-section waveguides using superposition of plane waves," IEEE Trans. Microwave Theory Tech. MTT-34, 161-167 (1986).
    [CrossRef]
  10. P. Benech, D. A. M. Khalil, and F. Saint Andre, "An exact simplified method for the normalization of radiation modes in planar multilayer structures," Opt. Commun. 88, 96-100 (1992).
    [CrossRef]
  11. H. Haferkorn, Optik: Physikalisch-Technische Grundlagen und Anwedungen (Barth, 1994).
  12. L. Thylen, "The beam propagation method: an analysis of its applicability," Opt. Quantum Electron. 15, 433-439 (1983).
    [CrossRef]
  13. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  14. P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, and A. Poudoulec, "Collective micro-optics on fibre ribbon for optical interconnecting devices," J. Lightwave Technol. 17, 924-928 (1999).
    [CrossRef]
  15. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).
  16. R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1990).
    [CrossRef]
  17. M. H. Ahmed, Ain Shams University, Cairo, Egypt (personal communication, 2006).

2005 (1)

H. Schmidt, D. Yin, J. P. Barber, and A. Hawkins, "Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids," IEEE J. Sel. Top. Quantum Electron. 11, 519-527 (2005).
[CrossRef]

2002 (1)

R. Bernini, S. Campopiano, and L. Zeni, "Silicon micromachined hollow optical waveguides for sensing applications," IEEE J. Sel. Top. Quantum Electron. 8, 106-110 (2002).
[CrossRef]

1999 (1)

1993 (1)

J. N. Mcmullin, R. Narendra, and C. R. James, "Hollow metallic waveguides in silicon V grooves," IEEE Photon. Technol. Lett. 5, 1080-1082 (1993).
[CrossRef]

1992 (1)

P. Benech, D. A. M. Khalil, and F. Saint Andre, "An exact simplified method for the normalization of radiation modes in planar multilayer structures," Opt. Commun. 88, 96-100 (1992).
[CrossRef]

1986 (2)

A. W. Snyder and X.-H. Zheng, "Optical fibers of arbitrary cross sections," J. Opt. Soc. Am. A 3, 600-609 (1986).
[CrossRef]

P. L. Overfelt and D. J. White, "TE and TM modes of some triangular cross-section waveguides using superposition of plane waves," IEEE Trans. Microwave Theory Tech. MTT-34, 161-167 (1986).
[CrossRef]

1983 (1)

L. Thylen, "The beam propagation method: an analysis of its applicability," Opt. Quantum Electron. 15, 433-439 (1983).
[CrossRef]

1976 (1)

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

Ahmed, M. H.

M. H. Ahmed, Ain Shams University, Cairo, Egypt (personal communication, 2006).

Andre, F. Saint

P. Benech, D. A. M. Khalil, and F. Saint Andre, "An exact simplified method for the normalization of radiation modes in planar multilayer structures," Opt. Commun. 88, 96-100 (1992).
[CrossRef]

Barber, J. P.

H. Schmidt, D. Yin, J. P. Barber, and A. Hawkins, "Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids," IEEE J. Sel. Top. Quantum Electron. 11, 519-527 (2005).
[CrossRef]

Benech, P.

P. Benech, D. A. M. Khalil, and F. Saint Andre, "An exact simplified method for the normalization of radiation modes in planar multilayer structures," Opt. Commun. 88, 96-100 (1992).
[CrossRef]

Bernini, R.

R. Bernini, S. Campopiano, and L. Zeni, "Silicon micromachined hollow optical waveguides for sensing applications," IEEE J. Sel. Top. Quantum Electron. 8, 106-110 (2002).
[CrossRef]

Campopiano, S.

R. Bernini, S. Campopiano, and L. Zeni, "Silicon micromachined hollow optical waveguides for sensing applications," IEEE J. Sel. Top. Quantum Electron. 8, 106-110 (2002).
[CrossRef]

Chanclou, P.

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1990).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Gadonna, M.

Haferkorn, H.

H. Haferkorn, Optik: Physikalisch-Technische Grundlagen und Anwedungen (Barth, 1994).

Hawkins, A.

H. Schmidt, D. Yin, J. P. Barber, and A. Hawkins, "Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids," IEEE J. Sel. Top. Quantum Electron. 11, 519-527 (2005).
[CrossRef]

James, C. R.

J. N. Mcmullin, R. Narendra, and C. R. James, "Hollow metallic waveguides in silicon V grooves," IEEE Photon. Technol. Lett. 5, 1080-1082 (1993).
[CrossRef]

Khalil, D.

K. Madkour, H. Maaty, and D. Khalil, "Silicon hollow waveguide for MEMS applications," in Proceedings of the European Conference on Optical Communication-International Conference on Integrated Optics and Optical Fibre Communication (ECOC-IOOC) (2003).
[PubMed]

Khalil, D. A. M.

P. Benech, D. A. M. Khalil, and F. Saint Andre, "An exact simplified method for the normalization of radiation modes in planar multilayer structures," Opt. Commun. 88, 96-100 (1992).
[CrossRef]

Laakmann, K. D.

Lostec, J.

Love, J. D.

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

Maaty, H.

K. Madkour, H. Maaty, and D. Khalil, "Silicon hollow waveguide for MEMS applications," in Proceedings of the European Conference on Optical Communication-International Conference on Integrated Optics and Optical Fibre Communication (ECOC-IOOC) (2003).
[PubMed]

Madkour, K.

K. Madkour, H. Maaty, and D. Khalil, "Silicon hollow waveguide for MEMS applications," in Proceedings of the European Conference on Optical Communication-International Conference on Integrated Optics and Optical Fibre Communication (ECOC-IOOC) (2003).
[PubMed]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

Mcmullin, J. N.

J. N. Mcmullin, R. Narendra, and C. R. James, "Hollow metallic waveguides in silicon V grooves," IEEE Photon. Technol. Lett. 5, 1080-1082 (1993).
[CrossRef]

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Narendra, R.

J. N. Mcmullin, R. Narendra, and C. R. James, "Hollow metallic waveguides in silicon V grooves," IEEE Photon. Technol. Lett. 5, 1080-1082 (1993).
[CrossRef]

Overfelt, P. L.

P. L. Overfelt and D. J. White, "TE and TM modes of some triangular cross-section waveguides using superposition of plane waves," IEEE Trans. Microwave Theory Tech. MTT-34, 161-167 (1986).
[CrossRef]

Pavy, D.

Poudoulec, A.

Schmidt, H.

H. Schmidt, D. Yin, J. P. Barber, and A. Hawkins, "Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids," IEEE J. Sel. Top. Quantum Electron. 11, 519-527 (2005).
[CrossRef]

Snyder, A. W.

Steier, W.

Thual, M.

Thylen, L.

L. Thylen, "The beam propagation method: an analysis of its applicability," Opt. Quantum Electron. 15, 433-439 (1983).
[CrossRef]

White, D. J.

P. L. Overfelt and D. J. White, "TE and TM modes of some triangular cross-section waveguides using superposition of plane waves," IEEE Trans. Microwave Theory Tech. MTT-34, 161-167 (1986).
[CrossRef]

Yin, D.

H. Schmidt, D. Yin, J. P. Barber, and A. Hawkins, "Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids," IEEE J. Sel. Top. Quantum Electron. 11, 519-527 (2005).
[CrossRef]

Zeni, L.

R. Bernini, S. Campopiano, and L. Zeni, "Silicon micromachined hollow optical waveguides for sensing applications," IEEE J. Sel. Top. Quantum Electron. 8, 106-110 (2002).
[CrossRef]

Zheng, X.-H.

Appl. Opt. (1)

IEEE J. Sel. Top. Quantum Electron. (2)

H. Schmidt, D. Yin, J. P. Barber, and A. Hawkins, "Hollow-core waveguides and 2-D waveguide arrays for integrated optics of gases and liquids," IEEE J. Sel. Top. Quantum Electron. 11, 519-527 (2005).
[CrossRef]

R. Bernini, S. Campopiano, and L. Zeni, "Silicon micromachined hollow optical waveguides for sensing applications," IEEE J. Sel. Top. Quantum Electron. 8, 106-110 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. N. Mcmullin, R. Narendra, and C. R. James, "Hollow metallic waveguides in silicon V grooves," IEEE Photon. Technol. Lett. 5, 1080-1082 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. L. Overfelt and D. J. White, "TE and TM modes of some triangular cross-section waveguides using superposition of plane waves," IEEE Trans. Microwave Theory Tech. MTT-34, 161-167 (1986).
[CrossRef]

J. Lightwave Technol. (1)

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

Opt. Commun. (1)

P. Benech, D. A. M. Khalil, and F. Saint Andre, "An exact simplified method for the normalization of radiation modes in planar multilayer structures," Opt. Commun. 88, 96-100 (1992).
[CrossRef]

Opt. Quantum Electron. (1)

L. Thylen, "The beam propagation method: an analysis of its applicability," Opt. Quantum Electron. 15, 433-439 (1983).
[CrossRef]

Other (8)

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

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1990).
[CrossRef]

M. H. Ahmed, Ain Shams University, Cairo, Egypt (personal communication, 2006).

H. Haferkorn, Optik: Physikalisch-Technische Grundlagen und Anwedungen (Barth, 1994).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, 1991).

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

K. Madkour, H. Maaty, and D. Khalil, "Silicon hollow waveguide for MEMS applications," in Proceedings of the European Conference on Optical Communication-International Conference on Integrated Optics and Optical Fibre Communication (ECOC-IOOC) (2003).
[PubMed]

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Figures (14)

Fig. 1
Fig. 1

(a) Three-dimensional view of the ray trajectory of the fundamental mode inside the hollow waveguide. (b) Path projection of the rays of the fundamental mode on the transverse plane. To trace the ray, follow the arrows in any direction. This path is peculiar to this particular geometry, and no other choice of angles would lead to such a path. The boundaries of the triangle are labeled 1, 2, and 3.

Fig. 2
Fig. 2

Incident wave vector k ¯ and normal to the boundary n ^ . β ¯ eq is a vector along the interface.

Fig. 3
Fig. 3

Rays of the fundamental mode incident on the upper and lower boundaries of the waveguide. n ^ is a unit vector normal to the boundary.

Fig. 4
Fig. 4

Attenuation coefficient in dB / mm of the fundamental mode of the hollow triangular waveguide against the side length for Δ n = 0.01 .

Fig. 5
Fig. 5

Attenuation coefficient in dB∕mm against the side length for Δ n = 0.05 .

Fig. 6
Fig. 6

(Color online) Plane of incidence containing the incident and the reflected wave vectors.

Fig. 7
Fig. 7

Losses of the hollow equilateral triangle and the losses of the hollow square with the same side length for an air core and a silicon substrate.

Fig. 8
Fig. 8

Side length required for maximum coupling to fiber spot size versus spot size diameter.

Fig. 9
Fig. 9

Losses of the hollow equilateral triangle mode and the losses of the hollow square waveguide mode versus fiber spot size diameter in micrometers.

Fig. 10
Fig. 10

Cross section of the original (bold) and perturbed waveguide (dashed).

Fig. 11
Fig. 11

Contour integration counterclockwise along C and clockwise along C o .

Fig. 12
Fig. 12

Original equilateral waveguide and its perturbed version with a head angle of 70.52° (shaded).

Fig. 13
Fig. 13

Contour of integration corresponding to the shaded area in Fig. 12. The integral vanishes on lines 1 and 2, and only line 3 survives the integration. The result of integration is identical for both subcontours.

Fig. 14
Fig. 14

Loss coefficient against waveguide width.

Tables (5)

Tables Icon

Table 1 Loss Comparison Using Different Techniques for Δ n = 0.01

Tables Icon

Table 2 Loss Comparison Using Different Techniques for Δ n = 0.05

Tables Icon

Table 3 Loss Coefficient for Δ n = 0.5

Tables Icon

Table 4 Comparison of Transverse Resonance Constant as Calculated from Eq. (58) and FEM

Tables Icon

Table 5 Comparison of Losses as Calculated from Eq. (63) to Simulation

Equations (73)

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E = E o [ exp ( j 4 π y s 3 ) exp ( j 4 π y s 3 ) + exp [ j 2 π s ( y 3 + x ) ] exp [ j 2 π s ( y 3 + x ) ] exp [ j 2 π s ( y 3 x ) ] + exp [ j 2 π s ( y 3 x ) ] ,
α = Δ P P 1 Δ z .
R eq = ( h eq ρ eq h eq + ρ eq ) 2 ,
R eq = ( n s 2 h eq n g 2 ρ eq n s 2 h eq + n g 2 ρ eq ) 2 ,
R = ( h ρ h + ρ ) 2 ,
R = ( n s 2 h n g 2 ρ n s 2 h + n g 2 ρ ) 2 ,
k ¯ = h y ^ + β z ^ .
n ^ = 3 2 x ^ + 1 2 y ^ .
h eq = k ¯ · n ^ = h / 2 ,
β ¯ eq = k ¯ - ( k ¯ · n ^ ) n ^ = 3 4 h x ^ + 3 h 4 y ^ + β z ^ ,
β eq = | β ¯ eq | = 3 h 2 / 4 + β 2 ,
ρ eq = k o 2 n s 2 β eq 2 = k o 2 n s 2 ( 3 h 2 / 4 + β 2 ) ,
Δ P P = 1 R 2 R eq 2 ,
α = h ( 1 R 2 R eq 2 ) β s 3 ,
1 R 2 R eq 2 = ( 1 R R eq ) ( 1 + R R eq ) 2 ( 1 R R eq )
R = 1 δ , R eq = 1 σ ,
1 R 2 R eq         2 = 2 ( 1 R R eq ) = 2 [ 1 ( 1 δ ) ( 1 σ ) ] 2 δ + 2 σ = 2 ( 1 R ) + 2 ( 1 R eq ) .
α 2 h ( 1 R ) β s 3 + 2 h ( 1 R eq ) β s 3 .
| C A | 2 = h ρ ( 1 R ) .
k ¯ = h y ^ + β z ^ .
P ¯ = | A | 2 2 Z g ( h y ^ + β z ^ ) k o n g ,
P core = P z d x d y = P ¯ z ^ d x d y = | A | 2 2 ω μ o β s 2 3 4 ,
ρ y ^ + β z ^ k o n s .
P ¯ = | C | 2 2 Z s ( ρ y ^ + β z ^ k o n s ) ,
d P d z = 0 s P ¯ · ( y ^ ) d x = ρ 2 ω μ o | C | 2 s .
α = 4 ρ β s 3 | C A | 2 .
α = 4 h ( 1 R ) β s 3 .
P ¯ = | A | 2 2 Z g ( h y ^ + β z ^ ) k o n g ,
s ^ = s ^ + Γ n ^ ,
Γ = [ ( n g n s ) 2 ( s ^ · n ^ ) 2 ( n g n s ) 2 + 1 ] 1 / 2 ( s ^ · n ^ ) .
P ¯ = | C | 2 2 Z s s ^ .
d P d z = 2 0 s P ¯ · n ^ d l = 2 0 s | C | 2 2 Z s s ^ · n ^ d l
= 2 0 s | C | 2 2 Z s ( s ^ · n ^ + Γ ) d l ,
Γ = ρ eq k o n s h eq k o n g ,
s ^ · n ^ = h eq k o n g ,
d P d z = ρ eq 2 ω μ o | C | 2 ( 2 s ) .
α = 8 ρ eq β s 3 | C A | 2 .
α = 8 h eq β s 3 ( 1 R eq ) .
α = 4 h β s 3 ( 1 R eq ) .
h c s 3 + 2 φ + 2 φ eq = 4 n π .
h c s 3 = 4 tan 1 ( γ h c ) + 4 tan 1 ( γ eq h c eq ) .
E reflected = Γ E incident ,
E reflected = Γ E incident ,
| E ¯ reflected | 2 = R eq | E incident | 2 + R eq | E incident | 2 .
r ^ = k ¯ × n ^ | k ¯ × n ^ | = β 2 β eq x ^ 3 β 2 β eq y ^ + h 3 2 β eq z ^ .
| E incident | 2 = | E ¯ incident · r ^ | 2 = | E o | 2 β 2 4 β eq 2 ,
| E incident | 2 = | E o | 2 | E incident | 2 = | E o | 2 ( 1 β 2 4 β eq 2 ) .
| E ¯ reflected | 2 | E o | 2 = 1 4 R eq + 3 4 R eq .
α x = h β s 3 [ 1 R ( 1 4 R eq + 3 4 R eq ) 2 ( 1 4 R + 3 4 R ) ] .
α y = h β s 3 [ 1 R ( 1 4 R eq + 3 4 R eq ) 2 ( 1 4 R + 3 4 R ) ] .
α x = h β s 3 [ 5 4 ( 1 R ) + 3 4 ( 1 R ) + 1 2 ( 1 R eq ) + 3 2 ( 1 R eq ) ] .
α y = h β s 3 [ 5 4 ( 1 R ) + 3 4 ( 1 R ) + 1 2 ( 1 R eq ) + 3 2 ( 1 R eq ) ]
1 R = 4 h / ρ ,
1 R = 4 n s 2 h / n g 2 ρ ,
1 R eq = 2 h / ρ ,
1 R eq = 2 n s 2 h / n g 2 ρ ;
α x = α y = 6 h 2 β ρ s 3 [ 1 + ( n s n g ) 2 ] .
α = 8 3 ( λ 2 s 3 ) ( n s 2 + n g 2 ) n g 3 n s 2 n g 2 .
α = λ 2 w 3 ( 1 + n s 2 n s 2 1 ) ,
( h 2 h o 2 ) Ω o ψ φ o d S + C o ( φ o tr ψ ψ tr φ o ) · n ^ d l = 0 ,
( h 2 h o 2 ) Ω o ψ φ o d S C o ψ tr φ o · n ^ d l = 0 .
C ψ tr φ o · n ^ d l = C o ψ tr φ o · n ^ d l + δΩ tr · ( ψ tr φ o ) d S ,
( h 2 h o 2 ) Ω o ψ φ o d S = δΩ tr · ( ψ tr φ o ) d S = δ C ( ψ tr φ o ) · n ^ d l ,
[ ( h 1 ) 2 h o     2 ] = δ C ( φ o tr φ o ) · n ^ d l Ω o φ o 2 d S ,
Ω o φ o 2 d S = 3 s 2 3 8 ,
( 9 a 4 30 a 2 + 9 ) δ C ( φ o tr φ o ) · n ^ d l = 2 a [ ( 9 a 4 + 6 a 2 3 ) cos 2 ( π 3 a ) 24 ( 1 + a 2 ) × cos ( π 3 a ) 9 a 4 30 a 2 21 ] .
h = 6.49 / s .
h = 6.883 L .
β r = [ k o 2 n g 2 ( 6.883 / L ) 2 ] 1 / 2 .
α = h β ( 1 s 3 ) ( 6 h ρ ) [ 1 + ( n s n g ) 2 ] .
h ( 1.8257 L ) = 4 π .
α = 3.9438 L 3 λ 2 ( n s 2 + n g 2 n g 3 n s 2 n g 2 ) .
α = 6.07 w 3 λ 2 ( n s 2 + n g 2 n g 3 n s 2 n g 2 ) .

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