Abstract

We present a detailed study on the behavior of coupling-induced resonance frequency shift (CIFS) in dielectric microring resonators. CIFS is related to the phase responses of the coupling region of the resonator coupling structure, which are examined for various geometries through rigorous numerical simulations. Based on the simulation results, a model for the phase responses of the coupling structure is presented and verified to agree with the simulation results well, in which the first-order coupled mode theory (CMT) is extended to second order, and the important contributions from the inevitable bent part of practical resonators are included. This model helps increase the understanding of the CIFS behavior and makes the calculation of CIFS for practical applications without full numerical simulations possible.

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References

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  1. G. T. Reed, and A. P. Knights, Silicon Photonics: An Introduction, (John Wiley, West Sussex, 2004).
  2. L. Pavesi, and D. J. Lockwood, Silicon Photonics, (Springer-Verlag, New York, 2004).
  3. K. Vahala, Optical Microcavities, (World Scientific, Singapore, 2004).
  4. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
    [CrossRef]
  5. F. Xia, M. Rooks, L. Sekaric, and Y. Vlasov, “Ultra-compact high order ring resonator filters using submicron silicon photonic wires for on-chip optical interconnects,” Opt. Express 15(19), 11934–11941 (2007).
    [CrossRef] [PubMed]
  6. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
    [CrossRef]
  7. B. Liu, A. Shakouri, and J. E. Bowers, “Passive microring-resonator-coupled lasers,” Appl. Phys. Lett. 79(22), 3561–3563 (2001).
    [CrossRef]
  8. K. De Vos, I. Bartolozzi, E. Schacht, P. Bienstman, and R. Baets, “Silicon-on-Insulator microring resonator for sensitive and label-free biosensing,” Opt. Express 15(12), 7610–7615 (2007).
    [CrossRef] [PubMed]
  9. M. A. Popovic, C. Manolatou, and M. R. Watts, “Coupling-induced resonance frequency shifts in coupled dielectric multi-cavity filters,” Opt. Express 14(3), 1208–1222 (2006).
    [CrossRef] [PubMed]
  10. Q. Li, S. Yegnanarayanan, A. Atabaki, and A. Adibi, “Calculation and correction of coupling-induced resonance frequency shifts in traveling-wave dielectric resonators,” in Integrated Photonics and Nanophotonics Research and Applications, (Optical Society of America, 2008), paper IWH3. http://www.opticsinfobase.org/abstract.cfm?URI=IPNRA-2008-IWH3
  11. Q. Li, M. Soltani, S. Yegnanarayanan, and A. Adibi, “Design and demonstration of compact, wide bandwidth coupled-resonator filters on a siliconon- insulator platform,” Opt. Express 17(4), 2247–2254 (2009).
    [CrossRef] [PubMed]
  12. Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20(3), 463–468 (2002).
    [CrossRef]
  13. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).
  14. A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
    [CrossRef]
  15. S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5(1), 5–15 (1987).
    [CrossRef]
  16. D. Marcuse, Theory of Dielectric Optical Waveguides, second edition (Academic, Boston, 1991), Chap. 6.
  17. D. Khalil, “On the radiation mode effects in integrated optical directional couplers,” Opt. Quantum Electron. 31(2), 151–159 (1999).
    [CrossRef]
  18. A. Melloni, F. Carniel, R. Costa, and M. Martinelli, “Determination of bend mode characteristics in dielectric waveguides,” J. Lightwave Technol. 19(4), 571–577 (2001).
    [CrossRef]
  19. For example, for the material system used in Ref. [9], Δβ is numerically found to be negative and small (~-0.015 μm−1 at 1578nm for TE polarization, w1/w2 = 400/400 nm, and gap = 100 nm). The corresponding simulation results are shown in Fig. 6 in Ref. [9]. We already know Δψ is negative for this coupling geometry (“CB” coupler), and therefore both terms in Eqs. (23) are negative and the second term is dominant. The two features that ϕ11 (ring-ring phase) is negative (corresponding CIFS positive) and ϕ11 and ϕ22 are of opposite signs are readily understood from Eqs. (23) and (24).
  20. M. Soltani, Novel integrated silicon nanophotonic structures using ultra-high Q resonator, Ph.D. dissertation, Georgia Institute of Technology (2009)
  21. M. Popovic, “Complex-frequency leaky mode computations using PML boundary layers for dielectric resonant structures,” in Proceedings of Integrated Photonics Research (Washington, DC, June 17, 2003).

2009 (1)

2007 (3)

2006 (1)

2002 (1)

2001 (2)

B. Liu, A. Shakouri, and J. E. Bowers, “Passive microring-resonator-coupled lasers,” Appl. Phys. Lett. 79(22), 3561–3563 (2001).
[CrossRef]

A. Melloni, F. Carniel, R. Costa, and M. Martinelli, “Determination of bend mode characteristics in dielectric waveguides,” J. Lightwave Technol. 19(4), 571–577 (2001).
[CrossRef]

1999 (1)

D. Khalil, “On the radiation mode effects in integrated optical directional couplers,” Opt. Quantum Electron. 31(2), 151–159 (1999).
[CrossRef]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

1987 (1)

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5(1), 5–15 (1987).
[CrossRef]

1985 (1)

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Adibi, A.

Baets, R.

Bartolozzi, I.

Bienstman, P.

Bowers, J. E.

B. Liu, A. Shakouri, and J. E. Bowers, “Passive microring-resonator-coupled lasers,” Appl. Phys. Lett. 79(22), 3561–3563 (2001).
[CrossRef]

Carniel, F.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Chuang, S. L.

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5(1), 5–15 (1987).
[CrossRef]

Costa, R.

De Vos, K.

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Hardy, A.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Khalil, D.

D. Khalil, “On the radiation mode effects in integrated optical directional couplers,” Opt. Quantum Electron. 31(2), 151–159 (1999).
[CrossRef]

Koshiba, M.

Laine, J.-P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Li, Q.

Little, B. E.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Liu, B.

B. Liu, A. Shakouri, and J. E. Bowers, “Passive microring-resonator-coupled lasers,” Appl. Phys. Lett. 79(22), 3561–3563 (2001).
[CrossRef]

Manolatou, C.

Martinelli, M.

Melloni, A.

Popovic, M. A.

Rooks, M.

Schacht, E.

Sekaric, L.

Shakouri, A.

B. Liu, A. Shakouri, and J. E. Bowers, “Passive microring-resonator-coupled lasers,” Appl. Phys. Lett. 79(22), 3561–3563 (2001).
[CrossRef]

Soltani, M.

Streifer, W.

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

Tsuji, Y.

Vlasov, Y.

Watts, M. R.

Xia, F.

Yegnanarayanan, S.

Appl. Phys. Lett. (1)

B. Liu, A. Shakouri, and J. E. Bowers, “Passive microring-resonator-coupled lasers,” Appl. Phys. Lett. 79(22), 3561–3563 (2001).
[CrossRef]

J. Lightwave Technol. (5)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Y. Tsuji and M. Koshiba, “Finite element method using port truncation by perfectly matched layer boundary conditions for optical waveguide discontinuity problems,” J. Lightwave Technol. 20(3), 463–468 (2002).
[CrossRef]

A. Melloni, F. Carniel, R. Costa, and M. Martinelli, “Determination of bend mode characteristics in dielectric waveguides,” J. Lightwave Technol. 19(4), 571–577 (2001).
[CrossRef]

A. Hardy and W. Streifer, “Coupled mode theory of parallel waveguides,” J. Lightwave Technol. 3(5), 1135–1146 (1985).
[CrossRef]

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5(1), 5–15 (1987).
[CrossRef]

Nat. Photonics (1)

F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007).
[CrossRef]

Opt. Express (4)

Opt. Quantum Electron. (1)

D. Khalil, “On the radiation mode effects in integrated optical directional couplers,” Opt. Quantum Electron. 31(2), 151–159 (1999).
[CrossRef]

Other (9)

D. Marcuse, Theory of Dielectric Optical Waveguides, second edition (Academic, Boston, 1991), Chap. 6.

Q. Li, S. Yegnanarayanan, A. Atabaki, and A. Adibi, “Calculation and correction of coupling-induced resonance frequency shifts in traveling-wave dielectric resonators,” in Integrated Photonics and Nanophotonics Research and Applications, (Optical Society of America, 2008), paper IWH3. http://www.opticsinfobase.org/abstract.cfm?URI=IPNRA-2008-IWH3

G. T. Reed, and A. P. Knights, Silicon Photonics: An Introduction, (John Wiley, West Sussex, 2004).

L. Pavesi, and D. J. Lockwood, Silicon Photonics, (Springer-Verlag, New York, 2004).

K. Vahala, Optical Microcavities, (World Scientific, Singapore, 2004).

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, Englewood Cliffs, NJ, 1984).

For example, for the material system used in Ref. [9], Δβ is numerically found to be negative and small (~-0.015 μm−1 at 1578nm for TE polarization, w1/w2 = 400/400 nm, and gap = 100 nm). The corresponding simulation results are shown in Fig. 6 in Ref. [9]. We already know Δψ is negative for this coupling geometry (“CB” coupler), and therefore both terms in Eqs. (23) are negative and the second term is dominant. The two features that ϕ11 (ring-ring phase) is negative (corresponding CIFS positive) and ϕ11 and ϕ22 are of opposite signs are readily understood from Eqs. (23) and (24).

M. Soltani, Novel integrated silicon nanophotonic structures using ultra-high Q resonator, Ph.D. dissertation, Georgia Institute of Technology (2009)

M. Popovic, “Complex-frequency leaky mode computations using PML boundary layers for dielectric resonant structures,” in Proceedings of Integrated Photonics Research (Washington, DC, June 17, 2003).

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Figures (8)

Fig. 1
Fig. 1

Two typical coupler architectures with different coupling geometries. a 1, a 2 are the input signals and b 1, b 2 are the corresponding output signals; L is the length of the middle straight part; R is the outer bend radius; gap is the spacing between edges of the two waveguides, and w 1, w 2 are the waveguide widths as shown. The brown region is the waveguide core at the resonator side, and the yellow region, which shows the access waveguide or part of the adjacent resonator, can be changed to be either waveguide core or cladding to obtain the net phase shift from the coupling effect. (a) Coupler CA with vertical symmetry axis V and horizontal symmetry axis H. (b) Coupler CB with vertical symmetry axis V.

Fig. 2
Fig. 2

(a) Simulation parameters for the coupler CA shown in Fig. 1(a). The bend radius R is 6 μm; the waveguide width w 1 is 400 nm, and the gap is 100 nm. n WG and n Cladding are the refractive indices of the waveguide core and cladding, respectively, as obtained from the effective index method. Simulation (marker) and modeling (dotted line) results of the coupler CA as a function of the straight part length L are shown for (b) phase responses (for the sake of clarity, ϕ 12 (ϕ 21) is plotted relative to its value at L = 0), and (c) power coupling coefficient κ2 .

Fig. 3
Fig. 3

(a) Simulation parameters for the coupler CB shown in Fig. 1(b). The bend radius R is 6 μm; the waveguide widths w 1 and w 2 are both 400 nm, and the gap is 100 nm. Simulation (marker) and modeling (dotted line) results of the coupler CB as a function of the straight part length L are shown for (b) phase responses ϕ 11, ϕ 22, (c) phase responses ϕ 12 ,ϕ 21, and (ϕ 11 + ϕ 22)/2 (for the sake of clarity, ϕ 12 (ϕ 21) is plotted relative to its value at L = 0), and (d) power coupling coefficient κ2 .

Fig. 4
Fig. 4

Three components comprising the coupler CA (lower part of input and output bends by dashed line) and CB (lower part of input and output bends by solid line) originally defined in Fig. 1: input bend, middle straight part, and output bend, which are modeled by transfer matrices B, T and BT , respectively. The two fundamental system modes of the middle parallel waveguide coupler are schematically shown on the middle part of the coupler.

Fig. 5
Fig. 5

(a) Input bend of the coupler CB shown in Fig. 1(b), which consists of the upper bent waveguide with width w 1 and the lower straight waveguide with width w 2. β 1, β 2 are the propagation constants of individual straight waveguides with widths w 1 and w 2, respectively. The reference planes of [a 1, a 2] T and [c 1, c 2] T are shown by the dashed line. From the simulation and modeling results of Fig. 3 we conclude that the length of propagation path from a 1 to c 1 is smaller than that from a 2 to c 2, resulting a negative Δψ when β 1 = β 2. (b) A schematic drawing Δψ of as a function of (β 1-β 2). The numbers are just shown for illustrative purpose and not exact.

Fig. 6
Fig. 6

(a) Simulation parameters for the coupler CB shown in Fig. 1(b). The bend radius R is 6 μm; the gap is 100 nm; the upper waveguide width w 1 is fixed at 400 nm, and the simulations are performed for three different values of the lower waveguide width w 2: 390 nm, 400 nm, and 410 nm. Simulation (marker) and modeling (dashed line) results of the coupler CB as a function of the straight part length L are plotted in (b), (c) and (d): (b) phase response ϕ 11, (c) phase response ϕ 22, and (d) power coupling coefficient κ2 .

Fig. 7
Fig. 7

(a) Simulation parameters for the coupler CB shown in Fig. 1(b) with L = 0. The bend radius R is 6 μm; the gap is 100 nm; the upper waveguide width w 1 is fixed to be 400 nm, and the lower waveguide width w 2 varies. Simulation results as a function of w 2 are plotted in (b), (c): (b) phase responses ϕ 11 and ϕ 22 (the red dotted line denotes the waveguide width that Δψ = 0), and (c) power coupling coefficient κ2 (the two red dotted lines denote the waveguide widths corresponding to maximum power transfer and Δψ = 0, respectively).

Fig. 8
Fig. 8

(a) Structure used for CIFS simulation using a complex-frequency eigenmode solver: a microring with an outer radius R coupled to a straight waveguide. The brown region is the waveguide core with refractive index n WG, and the yellow region can be either waveguide core or cladding to obtain the coupling effect on the resonance frequency shift of the resonator. (b) Simulation parameters for the structure shown in (a). (c) CIFS obtained via the eigenvalue solution (dashed line) and the phase response method (cross), respectively (note that CIFS is shown in shift in wavelength, and a positive shift in wavelength means a negative CIFS).

Tables (1)

Tables Icon

Table 1 Modeling parameters and fitting results for the coupled structures shown in Figs. 2, 3 and 6

Equations (29)

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[ b 1 b 2 ] = U [ a 1 a 2 ] = [ u 11 e i ϕ 11 u 21 e i ϕ 21 u 12 e i ϕ 12 u 22 e i ϕ 22 ] [ a 1 a 2 ]
C I F S ϕ 11 2 π Δ f F S R = ϕ 11 2 π c n g S
U = [ u 11 e i ϕ 11 u 21 e i ϕ 21 u 12 e i ϕ 12 u 22 e i ϕ 22 ] = e i θ 0 [ 1 κ 2 e i θ 1 i κ e i θ 2 i κ e i θ 2 1 κ 2 e i θ 1 ]
u 12 = u 21 = κ
ϕ 11 + ϕ 22 = ϕ 12 + ϕ 21 π
ϕ 11 + ϕ 22 2 = ϕ 12 π 2 = ϕ 21 π 2
ϕ 11 = ϕ 22 = ϕ 12 π 2 = ϕ 21 π 2
U = N B T T B
T = e i ( β s + β a s 2 ) L [ cos ( β 0 L ) + i δ β 0 sin ( β 0 L ) i κ ¯ β 0 sin ( β 0 L ) i κ ¯ β 0 sin ( β 0 L ) cos ( β 0 L ) i δ β 0 sin ( β 0 L ) ]
β 0 β s β a s 2 = κ ¯ 2 + δ 2
T = e i β 1 L e i Δ β L [ cos ( β 0 L ) i sin ( β 0 L ) i sin ( β 0 L ) cos ( β 0 L ) ]
β s = β 1 + β 0 + Δ β
β a s = β 1 β 0 + Δ β
B = e i α 0 e i Δ β z p [ cos ( β 0 z a ) e i Δ ψ i sin ( β 0 z a ) i sin ( β 0 z a ) cos ( β 0 z a ) e i Δ ψ ]
N = [ e i [ 2 ( α 0 + Δ ψ ) + β 1 L ] 0 0 e i [ 2 ( α 0 Δ ψ ) + β 2 L ] ]
ϕ 11 = ϕ 22 = ϕ 12 π 2 = ϕ 21 π 2 = Δ β ( L + 2 z p )
κ 2 = sin 2 ( β 0 ( L + 2 z a ) )
ϕ 12 π 2 = ϕ 21 π 2 = Δ β ( L + 2 z p )
ϕ 11 Δ β ( L + 2 z p ) + 2 Δ ψ sin ( β 0 z a ) sin ( β 0 ( L + z a ) ) cos ( β 0 ( L + 2 z a ) )
ϕ 22 Δ β ( L + 2 z p ) 2 Δ ψ sin ( β 0 z a ) sin ( β 0 ( L + z a ) ) cos ( β 0 ( L + 2 z a ) )
ϕ 11 Δ β ( L + 2 z p ) + 2 Δ ψ sin ( β 0 z a ) sin ( β 0 ( L + z a ) ) cos ( β 0 ( L + 2 z a ) ) + δ L ( sin ( β 0 L ) β 0 L cos ( β 0 ( L + 2 z a ) ) 1 )
ϕ 22 Δ β ( L + 2 z p ) 2 Δ ψ sin ( β 0 z a ) sin ( β 0 ( L + z a ) ) cos ( β 0 ( L + 2 z a ) ) δ L ( sin ( β 0 L ) β 0 L cos ( β 0 ( L + 2 z a ) ) 1 )
ϕ 11 2 Δ β z p + 2 Δ ψ sin 2 ( β 0 z a ) cos ( 2 β 0 z a )
ϕ 22 2 Δ β z p 2 Δ ψ sin 2 ( β 0 z a ) cos ( 2 β 0 z a )
κ 2 = sin 2 ( 2 β 0 z a ) cos 2 ( Δ ψ )
ϕ 11 Δ β ( L + 2 z p ) + 2 Δ ψ sin ( β 0 z a ) sin ( β 0 ( L + z a ) ) cos ( β 0 ( L + 2 z a ) ) + 2 ( Δ ψ Δ ψ ' )
ϕ 22 Δ β ( L + 2 z p ) 2 Δ ψ sin ( β 0 z a ) sin ( β 0 ( L + z a ) ) cos ( β 0 ( L + 2 z a ) ) 2 ( Δ ψ Δ ψ ' )
( Δ ψ Δ ψ ' ) δ L ( tan ( β 0 L ) β 0 L 1 )
( Δ ψ Δ ψ ' ) δ z a ( β 0 z a ) 2 / 3

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