Abstract

A vertically-stacked multi-ring resonator (VMR), which is a sequence of ring resonators stacked on top of each other, is investigated. The light in the VMR propagates horizontally in the plane of rings and at the same time propagates vertically between the adjacent rings due to evanescent coupling. If fabricated, the VMR may be advantageous compared to the conventional planar arrangement of coupled rings due to its dramatic compactness and more flexible transmission characteristics. In this paper, the uniform VMR, which consists of N rings coupled to the input and output waveguides, is studied. The uniform VMR is a 3D version of a coupled resonator optical waveguide (CROW). Closed analytical expressions for the transmission amplitudes and eigenvalues are obtained by solving coupled wave equations. In the approximation considered, it is shown that, in contrast to the conventional planar ring CROW, a VMR can possess eigenmodes even when interring coupling as well as coupling between rings and waveguides is strong. For the isolated VMR, the eigenvalues of the propagation constant are shown to change linearly with the interring coupling coefficient. The resonance transmission near the VMR eigenvalues is investigated. The dispersion relation of a VMR with an infinite number of rings is found. For weak coupling, the VMR dispersion relation is similar to that of a planar ring CROW (leading, however, to a much smaller group velocity), while for stronger coupling, a VMR does not possess bandgaps.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. C. K. Madsen and G. Lenz, �??Optical All-Pass Filters for Phase Response Design with Applications for Dispersion Compensation,�?? IEEE Photon. Technol. Lett. 10, 994-996 (1998).
    [CrossRef]
  2. C. K. Madsen, S. Chandrasekhar, E. J. Laskowski, M. A. Cappuzzo, J. Bailey, E. Chen, L. T. Gomez, A. Griffin, R. Long, M. Rasras, A. Wong-Foy, L. W. Stulz, J. Weld, and Y. Low, �??An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ,�?? Optical Fiber Communication Conference, Postdeadline papers, Paper FD9, Anaheim (2002).
  3. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, �??Very High-Order Microring Resonator Filters for WDM Applications,�?? IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
    [CrossRef]
  4. J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, �??Ultrahigh-quality-factor silicon-on-insulator microring resonator,�?? Opt. Lett. 29, 2861-2863 (2004).
    [CrossRef]
  5. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, �??Coupled-resonator optical waveguide: a proposal and analysis,�?? Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  6. J. E. Heebner, R. W. Boyd, and Q-H. Park, �??SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,�?? J. Opt. Soc. Am. B 19, 722-731 (2002).
    [CrossRef]
  7. S. Mookherjeaa, �??Semiconductor coupled-resonator optical waveguide laser,�?? Appl. Phys. Lett. 84, 3265-3267 (2004).
    [CrossRef]
  8. P. C. Sercel, K. J. Vahala, D. W. Vernooy, G. Hunziker, and R. B. Lee, �??Fiber ring optical resonators,�?? United States Patent Application Publication, US 2002/0041730 A1 (2002).
  9. M. Sumetsky, �??Optical fiber microcoil resonator,�?? Opt. Express 12, 2303-2316 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303</a>.
    [CrossRef] [PubMed]
  10. M. Sumetsky, �??Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,�?? Opt. Express 13, 4331-4340 (2005), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331</a>.
    [CrossRef] [PubMed]
  11. M. Raburn, B. Liu, K. Rauscher, Y. Okuno, N. Dagli, and J. E. Bowers, �??3-D Photonic Circuit Technology,�?? IEEE J. Sel. Top. Quant. Electron. 8, 935-942 (2001).
    [CrossRef]
  12. P. Koonath, K. Kishima, T. Indukuri, and B. Jalali, �??Sculpting of three-dimensional nano-optical structures in silicon,�?? Appl. Phys. Lett. 83, 4909-4911 (2003).
    [CrossRef]
  13. R.Kashyap, Fiber Bragg Gratings, Academic Press, 1999.
  14. M. Sumetsky, B. J. Eggleton, and P. S. Westbrook, �??Holographic methods for phase mask and fiber grating fabrication and characterization,�?? Proc. SPIE 4941, 1 (2003).
    [CrossRef]
  15. S. Shoji and S. Kawata, �??Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,�?? Appl. Phys. Lett. 76, 2668-2670 (2000).
    [CrossRef]
  16. G.Kakarantzas, T.E.Dimmick, T.A.Birks, R.LeRoux, and P.St.J.Russell "Miniature all-fiber devices based on CO2 laser microstructuring of tapered fibers," Opt. Lett. 26, 1137-1139 (2001).
    [CrossRef]
  17. M. Sumetsky, �??Whispering-gallery-bottle microcavities: the three-dimensional etalon,�?? Opt. Lett. 29, 8-10 (2004).
    [CrossRef] [PubMed]
  18. J. K. S. Poon, J. Scheuer, Y. Xu, and A. Yariv, �??Designing coupled-resonator optical waveguide delay lines,�?? J. Opt. Soc. Am. B 21, 1665-1672 (2004).
    [CrossRef]
  19. K. J. Vahala, �??Optical microcavities,�?? Nature 424, 839-846 (2003).
    [CrossRef] [PubMed]
  20. A. W. Snyder and J. D. Love, Optical waveguide theory, Chapman & Hall, London, 1983.
  21. J. M. Bendickson, J. P. Dowling, and M. Scalora, �??Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,�?? Phys. Rev. E 53, 4107-4121 (1996).
    [CrossRef]
  22. Y. Chen and S. Blair, �??Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,�?? Opt. Express 12, 3353-3366 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3353">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3353</a>
    [CrossRef] [PubMed]
  23. L. D. Landau and E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1958.
  24. M. Sumetskii, �??Modeling of complicated nanometer resonant tunneling devices with quantum dots�??, J. Phys. Condens. Matter 3, 2651�??2664 (1991).
    [CrossRef]
  25. M. Sumetsky and B. J. Eggleton, �??Modeling and optimization of complex photonic resonant cavity circuits,�?? Opt. Express 11, 381-391 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-381">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-381</a>.
    [CrossRef] [PubMed]
  26. F. Seitz, The modern theory of solids, McGraw-Hill Book Company, New York, 1940.

Appl. Phys. Lett. (3)

S. Mookherjeaa, �??Semiconductor coupled-resonator optical waveguide laser,�?? Appl. Phys. Lett. 84, 3265-3267 (2004).
[CrossRef]

P. Koonath, K. Kishima, T. Indukuri, and B. Jalali, �??Sculpting of three-dimensional nano-optical structures in silicon,�?? Appl. Phys. Lett. 83, 4909-4911 (2003).
[CrossRef]

S. Shoji and S. Kawata, �??Photofabrication of three-dimensional photonic crystals by multibeam laser interference into a photopolymerizable resin,�?? Appl. Phys. Lett. 76, 2668-2670 (2000).
[CrossRef]

IEEE J. Sel. Top. Quant. Electron. (1)

M. Raburn, B. Liu, K. Rauscher, Y. Okuno, N. Dagli, and J. E. Bowers, �??3-D Photonic Circuit Technology,�?? IEEE J. Sel. Top. Quant. Electron. 8, 935-942 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

C. K. Madsen and G. Lenz, �??Optical All-Pass Filters for Phase Response Design with Applications for Dispersion Compensation,�?? IEEE Photon. Technol. Lett. 10, 994-996 (1998).
[CrossRef]

B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, �??Very High-Order Microring Resonator Filters for WDM Applications,�?? IEEE Photon. Technol. Lett. 16, 2263-2265 (2004).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. Condens. Matter (1)

M. Sumetskii, �??Modeling of complicated nanometer resonant tunneling devices with quantum dots�??, J. Phys. Condens. Matter 3, 2651�??2664 (1991).
[CrossRef]

Nature (1)

K. J. Vahala, �??Optical microcavities,�?? Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

OFC 2002 (1)

C. K. Madsen, S. Chandrasekhar, E. J. Laskowski, M. A. Cappuzzo, J. Bailey, E. Chen, L. T. Gomez, A. Griffin, R. Long, M. Rasras, A. Wong-Foy, L. W. Stulz, J. Weld, and Y. Low, �??An integrated tunable chromatic dispersion compensator for 40 Gb/s NRZ and CSRZ,�?? Optical Fiber Communication Conference, Postdeadline papers, Paper FD9, Anaheim (2002).

Opt. Express (4)

M. Sumetsky, �??Optical fiber microcoil resonator,�?? Opt. Express 12, 2303-2316 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-10-2303</a>.
[CrossRef] [PubMed]

M. Sumetsky, �??Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,�?? Opt. Express 13, 4331-4340 (2005), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4331</a>.
[CrossRef] [PubMed]

Y. Chen and S. Blair, �??Nonlinearity enhancement in finite coupled-resonator slow-light waveguides,�?? Opt. Express 12, 3353-3366 (2004), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3353">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3353</a>
[CrossRef] [PubMed]

M. Sumetsky and B. J. Eggleton, �??Modeling and optimization of complex photonic resonant cavity circuits,�?? Opt. Express 11, 381-391 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-381">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-4-381</a>.
[CrossRef] [PubMed]

Opt. Lett (1)

J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, �??Ultrahigh-quality-factor silicon-on-insulator microring resonator,�?? Opt. Lett. 29, 2861-2863 (2004).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (1)

J. M. Bendickson, J. P. Dowling, and M. Scalora, �??Analytical expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures,�?? Phys. Rev. E 53, 4107-4121 (1996).
[CrossRef]

Proc. SPIE (1)

M. Sumetsky, B. J. Eggleton, and P. S. Westbrook, �??Holographic methods for phase mask and fiber grating fabrication and characterization,�?? Proc. SPIE 4941, 1 (2003).
[CrossRef]

Other (5)

R.Kashyap, Fiber Bragg Gratings, Academic Press, 1999.

P. C. Sercel, K. J. Vahala, D. W. Vernooy, G. Hunziker, and R. B. Lee, �??Fiber ring optical resonators,�?? United States Patent Application Publication, US 2002/0041730 A1 (2002).

L. D. Landau and E. M. Lifshitz, Quantum mechanics, Pergamon Press, 1958.

A. W. Snyder and J. D. Love, Optical waveguide theory, Chapman & Hall, London, 1983.

F. Seitz, The modern theory of solids, McGraw-Hill Book Company, New York, 1940.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1.

Illustration of (a) VMR and (b) PMR

Fig. 2.
Fig. 2.

Surface plots of the PMR through transmission power, |T (thru) |2, in the (B,K) plane for N = 2,3,5, and 10; (a) - transmission power plots for the ring-waveguide coupling parameter K 0 having sin(K 0)=0.2; (b) - transmission power plots for the ring-waveguide coupling parameter K 0 with sin(K 0)=0.8.

Fig. 3.
Fig. 3.

Eigenvalues of the isolated VMR consisting of 5 rings, N=5, in the (B,K) plane. Bold lines correspond to n 1=2 and n 2=5. These lines are tilted with respect to the B-axis by the angles α 1=π/4 and α 2=5π/6, respectively. Line crossing correspond to the eigenvalues which are degenerated in the considered approximation.

Fig. 4.
Fig. 4.

Surface plots of the VMR through transmission power, |T (thru)|2, in the (B,K) plane for N = 2,3,4, and 5; (a) - transmission power plots for the ring-waveguide coupling parameter K 0 having sin(K 0)=0.2; (b) - transmission power plots for the ring-waveguide coupling parameter K 0 with sin(K 0)=0.8.

Fig. 5.
Fig. 5.

Enlarged surface plots of behavior of transmission power, |T (thru)|2, and average electromagnetic filed intensity, P, near resonances. (a), (a1), and (a2) - N = 4; (b), (b1), and (b2) - N = 5. (a1) and (b1) show the transmission power, while (a2) and (b2) show the average field intensity. The eigenvalues of VMR correspond to the infinite values of P.

Fig. 6.
Fig. 6.

Characteristic singular behavior near resonances in the (B,K) plane plotted using (a) Eq. (19) for oe-13-17-6354-i017.jpg singularity, (b) Eq.(20) for oe-13-17-6354-i018.jpg singularity, and (c) Eq.(23) for oe-13-17-6354-i019.jpg singularity (arbitrary units).

Fig. 7.
Fig. 7.

Dispersion relation for (a) VMR and (b) PMR for different interring coupling parameter, K.

Equations (62)

Equations on this page are rendered with MathJax. Learn more.

d A 1 ds = κ A 2
d A n ds = κ ( A n 1 + A n + 1 ) , n = 2,3 N 1 ,
d A N ds = κ A N 1
A n ( 0 ) = A n ( S ) exp ( i β S ) , n = 2,3 , , N 1 .
A 0 ( out ) = cos ( K 0 ) A 0 ( in ) + i sin ( K 0 ) exp ( iβS ) A 1 ( S )
A 1 ( 0 ) = i sin ( K 0 ) A 0 ( in ) + cos ( K 0 ) exp ( iβS ) A 1 ( S )
A N + 1 ( out ) = i sin ( K 0 ) exp ( iβS ) A N ( S )
A N ( 0 ) = cos ( K 0 ) exp ( iβS ) A 1 ( S )
K 0 = S c ( 0 ) κ 0 ( s ) ds ,
T ( thru ) = A 0 ( out ) / A 0 ( in )
T ( drop ) = A N + 1 ( out ) / A 0 ( in ) .
T ( thru ) = ξ + ξ + i μ + , T ( drop ) = ξ + ξ + i μ + ,
F m ( s ) = A 0 ( in ) sin ( K 0 ) n = 1 N Q n sin πmn N + 1 exp [ i s S ( B + 2 K cos πn N + 1 ) ] .
ξ ± = ( N + 1 ) 2 ( g + ) 2 ± 16 ( g ) 2 [ ( σ + ) 2 ( σ ) 2 ) , μ ± = 8 ( N + 1 ) g + g σ ± ,
g ± = 1 ± cos ( K 0 ) ,
σ ± = n = 1 N ( ± 1 ) n cot ( 1 2 B + K c n ) s n 2 , Q n = 2 [ cot ( 1 2 B + K c n ) + i ] ( N + 1 ) g + + 4 i g ( σ + + ( 1 ) n σ ) ,
c n = cos ( πn N + 1 ) , s n = sin ( πn N + 1 ) .
B = βS , K = κS .
T ( thru ) 2 + T ( drop ) 2 = 1 .
P = 1 NS m = 1 0 N 0 S ds F m ( s ) 2 = N + 1 2 N ( A 0 ( in ) sin ( K 0 ) ) 2 n = 1 N Q n 2
B ln + 2 K c n = 2 πl , n = 1,2 , , N ,
B l 1 n 1 , l 2 n 2 = 2 π l 2 c n 1 l 1 c n 2 c n 1 c n 2 , K l 1 n 1 , l 2 n 2 = π l 1 l 2 c n 1 c n 2 ,
F m ( ln ) ( s ) = C sin πmn N + 1 exp ( ils S ) ,
T ( thru ) Δ B Δ B + i 2 Γ , T ( drop ) ( 1 ) n + 1 i 2 Γ ΔB + i 2 Γ , P = 2 A 0 ( in ) 2 K 0 2 N ( N + 1 ) [ ΔB 2 + 1 4 Γ 2 ] ,
ΔB = B B ln ( K ) , Γ = 4 K 0 2 N + 1 s n 2 .
T ( thru ) = i 2 Γ ΔB + i 2 Γ ( if l 1 l 2 odd ) .
T ( thru ) = ΔB ΔB + i 2 Γ ( if l 1 l 2 is even and n 1 + N + 1 2 is odd ) .
Γ = ( N + 1 ) g + 4 g ( c n 1 s n 1 ) 2 ( ΔK ) 2 .
ΔB = θ ± ΔK , θ ± = ± 2 c n 1 1 + 2 s n 1 2 .
T ( thru ) = i 2 Γ ΔB + θ ± ΔK + i 2 Γ ( if l 1 l 2 is even and n 1 + N + 1 2 is even ) ,
Γ = ( N + 1 ) g + 2 g c n 1 s n 1 2 ( ΔK ) 2 ( 1 + 2 s n 1 2 ) 2 .
B = 2 π l 2 K cos ( ξd ) ,
B = πl arcsin [ sin ( K ) cos ( ξ d 0 ) ] ,
A s ( s ) = m = 1 N a m exp ( 2 i κ s cos πm N + 1 ) sin πmn N + 1 ,
m = 1 N a m ( E m 1 ) sin πmn N + 1 = 0 , n = 2,3 , N 1 ,
x 1 = m = 1 N a m ( E m 1 ) sin πm N + 1
x 2 = m = 1 N a m ( E m 1 ) sin πNm N + 1 .
a m = 2 ( N + 1 ) ( E m 1 ) ( x 1 sin πm N + 1 + x 2 sin πNm N + 1 ) .
a m = i A 0 ( in ) sin ( K 0 ) ( E m 1 ) [ 1 4 ( N + 1 ) cos ( K 0 ) + i 2 ( 1 cos ( K 0 ) ) ( σ + + ( 1 ) m σ ] sin πm N + 1 ,
( A n ( + ) A n ( ) ) = M ( A n + 1 ( + ) A n + 1 ( ) ) , M = i sin ( K ) ( exp ( i B ) cos ( K ) cos ( K ) exp ( i B ) ) .
( A 0 ( in ) A 0 ( out ) ) = M 0 ( A 1 ( + ) A 1 ( ) ) , ( A N ( + ) A N ( ) ) = M 0 ( A N + 1 ( out ) 0 ) , M 0 = i sin ( K 0 ) ( exp ( iB ) cos ( K 0 ) cos ( K 0 ) exp ( iB ) ) .
( A 0 ( in ) A 0 ( out ) ) = P ( A N + 1 ( out ) 0 ) , P = M 0 M N 1 M 0 ,
T ( thru ) = P 21 P 11 , T ( drop ) = 1 P 11 .
σ ± ΔB , ΔK 0 ν n 1 n 2 ± + ( ± 1 ) n 1 2 s n 1 2 ΔB + 2 ΔK c n 1 + ( ± 1 ) n 2 2 ss n 2 2 ΔB + 2 ΔK c n 2 ,
ΔB = B B l 1 n 1 l 2 n 2 , ΔK = K K l 1 n 1 l 2 n 2 ,
ν n 1 n 2 ± = n = 1 N n n 1 , n 2 ( ± 1 ) n cot ( 1 2 B l 1 n 1 l 2 n 2 + K l 1 n 1 l 2 n 2 cos ( πn N + 1 ) ) s n 2 .
( σ + ) 2 ( σ ) 2 4 [ 1 ( 1 ) n 1 + n 2 ] s n 1 2 s n 2 2 ( ΔB + 2 ΔK c n 1 ) ( ΔB + 2 ΔK c n 2 )
+ 4 [ ν n 1 n 2 + ( 1 ) n 1 ] s n 1 2 ( ΔB + 2 ΔK c n 1 ) + 4 [ ν n 1 n 2 + ( 1 ) n 2 ] s n 2 2 ( ΔB + 2 ΔK c n 2 ) .
n 2 = N + 1 n 1 .
B l 1 n 1 , l 2 n 2 = π ( l 1 l 2 ) , K l 1 n 1 , l 2 n 2 = π l 1 l 2 2 c n 1 .
σ ± ( ± 1 ) n 1 cot ( π l 1 + 1 2 ΔB + ΔK c n 1 ) s n 1 2
+ ( ± 1 ) n 1 cot ( π l 2 + 1 2 ΔB - ΔK c n 1 ) s n 1 2 ,
+ ( ± 1 ) N + 1 2 cot ( π 2 ( l 1 + l 2 ) + 1 2 ΔB )
σ ± ( ± 1 ) n 1 ΔB s n 1 2 1 4 ( ΔB ) 2 ( ΔK ) 2 c n 1 2 .
σ ± ( ± 1 ) N + 1 2 1 4 ( ΔB ) 2 ( 1 + 2 ( ± 1 ) n 1 + N + 1 2 s n 1 2 ) ( ΔK ) 2 c n 1 2 1 2 ΔB [ 1 4 ( ΔB ) 2 ( ΔK ) 2 c n 1 2 ] .
( σ + ) 2 ( σ ) 2 8 s n 1 2 [ 1 2 ( ΔB ) 2 ( ΔK ) 2 c n 1 2 ] .
T ( thru ) = ( N + 1 ) g + Δ Λ 1 ( 2 ) ( N + 1 ) g + Δ Λ 1 ( 2 ) + 8 i g Δ Λ ( 1 ) ( if l 1 l 2 is odd ) ,
T ( thru ) = 8 i g - Δ Λ ( 1 ) ( N + 1 ) g + Δ Λ 2 ( 2 ) + 8 i g Δ Λ ( 1 ) ( if l 1 l 2 is even and n 1 + N + 1 2 is odd ) ,
T ( thru ) = ( N + 1 ) g + ΔBΔ Λ 1 ( 2 ) ( N + 1 ) g + ΔBΔ Λ 1 ( 2 ) + 16 i g Δ Λ 2 ( 2 ) ( if l 1 l 2 is even and n 1 + N + 1 2 is even ) ,
Δ Λ ( 1 ) = ΔB s n 1 2 ,
Δ Λ 1 ( 2 ) = 1 4 ( ΔB ) 2 ( ΔB ) 2 c n 1 2 ,
Δ Λ 2 ( 2 ) = 1 4 ( ΔB ) 2 ( 1 + 2 s n 1 2 ) ( ΔK ) 2 c n 1 2 ,

Metrics