Abstract

We develop a Hamiltonian optics formalism to quantitatively analyze a recently proposed scheme for increasing the delay-time-bandwidth product for microring resonator structures with varying ring resonances [Yang and Sipe, Opt. Lett. 32, 918 (2007)]. This theory is formally compact, simple and physically intuitive. We compare this formalism with the more rigorous transfer matrix method, and conclude that the Hamiltonian optics formalism correctly gives the average dispersion, which essentially determines the group delay as well as the dispersive distortion for pulses in the ps regime or longer.

© 2011 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  9. M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  13. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  15. N. H. Kwong, J. E. Sipe, R. Binder, Z. S. Yang, and A. L. Smirl, ”Stopping and Storing Light in Semiconductor Quantum Wells and Optical Resonators” in Slow Light: Science and Applications (Optical Science and Engineering), J. B. Khurgin and R. S. Tucker, eds. (CRC Press, 2009).
  16. J. A. Arnaud, Beam and Fiber Optics, (Academic, 1976).
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    [Crossref]
  18. C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier–Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998).
    [Crossref]
  19. G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo “Theory of photon Bloch oscillations in photonic crystals,” Phys. Rev. B 63, 035108 (2001).
    [Crossref]
  20. R. Sapienzav, P. Costantino, and D. Wiersma, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
    [Crossref]
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2009 (3)

2007 (3)

2006 (3)

2005 (2)

2004 (3)

2003 (1)

R. Sapienzav, P. Costantino, and D. Wiersma, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[Crossref]

2001 (2)

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo “Theory of photon Bloch oscillations in photonic crystals,” Phys. Rev. B 63, 035108 (2001).
[Crossref]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[Crossref]

1999 (1)

1998 (1)

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier–Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998).
[Crossref]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics, (Academic, 1976).

Baba, T.

Binder, R.

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum well Bragg structures,” Opt. Lett. 30, 2790–2792 (2005).
[Crossref] [PubMed]

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Stopping, storing, and releasing light in quantum-well Bragg structures,” J. Opt. Soc. Am. B 22, 2144–2156 (2005).
[Crossref]

N. H. Kwong, J. E. Sipe, R. Binder, Z. S. Yang, and A. L. Smirl, ”Stopping and Storing Light in Semiconductor Quantum Wells and Optical Resonators” in Slow Light: Science and Applications (Optical Science and Engineering), J. B. Khurgin and R. S. Tucker, eds. (CRC Press, 2009).

Birks, T. A.

Borel, P. I.

Boyd, R. W.

Bright, J. N.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier–Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998).
[Crossref]

Chak, P.

Chamorro-Posada, P.

Costantino, P.

R. Sapienzav, P. Costantino, and D. Wiersma, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[Crossref]

de Sterke, C. M.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier–Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998).
[Crossref]

Di Carlo, A.

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo “Theory of photon Bloch oscillations in photonic crystals,” Phys. Rev. B 63, 035108 (2001).
[Crossref]

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[Crossref]

Fage-Pedersen, J.

Fan, S. H.

Fietz, C.

Fraile-Pelaez, F. J.

Frandsen, L. H.

Gao, D.

J. Hou, D. Gao, H. Wu, R. Hao, and Z. Zhou, “Flat band slow light in symmetric line defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 21, 1571–1573 (2009).
[Crossref]

Hamachi, Y.

Hammon, T. E.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier–Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998).
[Crossref]

Hao, R.

J. Hou, D. Gao, H. Wu, R. Hao, and Z. Zhou, “Flat band slow light in symmetric line defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 21, 1571–1573 (2009).
[Crossref]

Heebner, J. E.

Hou, J.

J. Hou, D. Gao, H. Wu, R. Hao, and Z. Zhou, “Flat band slow light in symmetric line defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 21, 1571–1573 (2009).
[Crossref]

Huang, Y.

Kavokin, A.

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo “Theory of photon Bloch oscillations in photonic crystals,” Phys. Rev. B 63, 035108 (2001).
[Crossref]

Krug, P. A.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier–Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998).
[Crossref]

Kubo, S.

Kwong, N. H.

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum well Bragg structures,” Opt. Lett. 30, 2790–2792 (2005).
[Crossref] [PubMed]

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Stopping, storing, and releasing light in quantum-well Bragg structures,” J. Opt. Soc. Am. B 22, 2144–2156 (2005).
[Crossref]

N. H. Kwong, J. E. Sipe, R. Binder, Z. S. Yang, and A. L. Smirl, ”Stopping and Storing Light in Semiconductor Quantum Wells and Optical Resonators” in Slow Light: Science and Applications (Optical Science and Engineering), J. B. Khurgin and R. S. Tucker, eds. (CRC Press, 2009).

Lavrinenko, A. V.

Lenz, G.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[Crossref]

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[Crossref]

Malpuech, G.

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo “Theory of photon Bloch oscillations in photonic crystals,” Phys. Rev. B 63, 035108 (2001).
[Crossref]

Mookherjea, S.

Paloczi, G. T.

Panzarini, G.

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo “Theory of photon Bloch oscillations in photonic crystals,” Phys. Rev. B 63, 035108 (2001).
[Crossref]

Pereira, S.

Poon, J. K. S.

Povinelli, M. L.

Russell, P. St. J.

Sandhu, S.

Sapienzav, R.

R. Sapienzav, P. Costantino, and D. Wiersma, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[Crossref]

Scheuer, J.

Sekaric, L.

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

Shvets, G.

Sipe, J. E.

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[Crossref]

Smirl, A. L.

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Stopping, storing, and releasing light in quantum-well Bragg structures,” J. Opt. Soc. Am. B 22, 2144–2156 (2005).
[Crossref]

Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum well Bragg structures,” Opt. Lett. 30, 2790–2792 (2005).
[Crossref] [PubMed]

N. H. Kwong, J. E. Sipe, R. Binder, Z. S. Yang, and A. L. Smirl, ”Stopping and Storing Light in Semiconductor Quantum Wells and Optical Resonators” in Slow Light: Science and Applications (Optical Science and Engineering), J. B. Khurgin and R. S. Tucker, eds. (CRC Press, 2009).

Vlasov, Y.

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

Wiersma, D.

R. Sapienzav, P. Costantino, and D. Wiersma, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[Crossref]

Wu, H.

J. Hou, D. Gao, H. Wu, R. Hao, and Z. Zhou, “Flat band slow light in symmetric line defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 21, 1571–1573 (2009).
[Crossref]

Xia, F.

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

Yang, Z. S.

Yanik, M. F.

Yariv, A.

Zhou, Z.

J. Hou, D. Gao, H. Wu, R. Hao, and Z. Zhou, “Flat band slow light in symmetric line defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 21, 1571–1573 (2009).
[Crossref]

IEEE J. Quantum. Electron. (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. Hou, D. Gao, H. Wu, R. Hao, and Z. Zhou, “Flat band slow light in symmetric line defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. 21, 1571–1573 (2009).
[Crossref]

J. Lightwave Technol. (1)

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

Nat. Photonics (1)

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

Opt. Express (2)

Opt. Lett. (7)

Phys. Rev. B (1)

G. Malpuech, A. Kavokin, G. Panzarini, and A. Di Carlo “Theory of photon Bloch oscillations in photonic crystals,” Phys. Rev. B 63, 035108 (2001).
[Crossref]

Phys. Rev. E (1)

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier–Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998).
[Crossref]

Phys. Rev. Lett. (2)

R. Sapienzav, P. Costantino, and D. Wiersma, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. 91, 263902 (2003).
[Crossref]

M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
[Crossref] [PubMed]

Other (2)

N. H. Kwong, J. E. Sipe, R. Binder, Z. S. Yang, and A. L. Smirl, ”Stopping and Storing Light in Semiconductor Quantum Wells and Optical Resonators” in Slow Light: Science and Applications (Optical Science and Engineering), J. B. Khurgin and R. S. Tucker, eds. (CRC Press, 2009).

J. A. Arnaud, Beam and Fiber Optics, (Academic, 1976).

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Figures (8)

Fig. 1
Fig. 1 Schematic of a SCISSOR unit.

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Fig. 2
Fig. 2 Photonic band structure of the periodic SCISSOR system.

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Fig. 3
Fig. 3 Schematic of: (a) an inclined SCISSOR structure with varying ring resonances; (b) an AR block; (c) a finite SCISSOR sequence with AR blocks on both ends.

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Fig. 4
Fig. 4 The simulation results from the Hamiltonian optics formalism: (a) K as a function of t; (b) K as a function of z; (c) z as a function of t.

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Fig. 5
Fig. 5 Delay time as a function of frequency: the Hamiltonian delay τHam for the inclined structure (dashed), the actual delay τ for the inclined structure (solid), and the actual delay for the uniformly periodic structure (dotted).

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Fig. 8
Fig. 8 Time dependence of pulse intensities: the input (dotted), transmitted in the 10-unit-cell uniformly periodic system (dashed), reflected in the 10-unit-cell inclined system (solid), transmitted in the 100-unit-cell uniformly periodic system (dash-dotted), and Eout,Ham(t) in Eq. (34) (filled triangles, almost indistinguishable from the solid curve).

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Fig. 6
Fig. 6 Reflection of the inclined structure (solid) and transmission of the uniformly periodic structure (dashed).

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Fig. 7
Fig. 7 Phase of the reflectivity in the inclined structure: the Hamiltonian phase ϕHam (solid) and the actual phase ϕ (dashed).

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Equations (34)

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( E 3 up / low E 2 up / low )   =   ( σ i κ i κ σ )   ( E 4 up / low E 1 up / low ) ,
( E R + E R − )   =   M ( ω )   ( E L + E L − ) ,   M ( ω )   =   ( α β β ★ α ★ ) ,
α   =   e i π ω ω B 2 i σ   sin ( π ω ω r )   [ e i π   ω ω r   −   σ 2 e − i π ω ω r ] ,   β   =   κ 2 2 i σ   sin ( π ω ω r ) .
cos ( K L )   =   1 2 T r { M ( ω ) }   =   ℜ { α } ,
ω B   =   2 π   ×   1.973 THz ,   ω r   =   2 π   ×   3.942 THz ,   σ   =   0 . 97 ;   m r   =   50 ,   m B   =   100 .
α   =   e i π ω ω B   { 1   −   i Γ m r m r ω r ω   −   m r ω r } ,
ω ( K )   =   cos ( K L )   −   ( − 1 ) m B cos ( K L )   −   ( − 1 ) m B   ( 1   +   π m B m r Γ )   ( m r ω r   −   m B ω B )   +   m B ω B .
ω   =   g ( K ; z ) .
d z d t   =   ∂ g   ( K ; z ) ∂ K ,
d K d t   =   − ∂ g   ( K ; z ) ∂ z .
g   ( K ; z )   =   cos   ( K L )   −   ( − 1 ) m B cos   ( K L )   −   ( − 1 ) m B   ( 1   +   π m B m r Γ )   [ m r ω r   ( z )   −   m B ω B ]   +   m B ω B ,
m r ω r , j   =   m r ω r   +   ( j − 1 )   ×   m B ω B   −   m r ω r N ,
m r ω r   ( z )   =   m r ω r   +   ( z   −   L )   ×   m B ω B   −   m r ω r N L ,
d z d t   =   Δ Ω N   ( − 1 ) m B + 1   π m B m r Γ   sin ( K L ) [ cos   ( K L )   −   ( − 1 ) m B   ( 1   +   π m B m r Γ ) ] 2   [ z   −   ( N + 1 ) L ] ,
d K d t   =   Δ Ω N L   cos   ( K L )   −   ( − 1 ) m B cos   ( K L )   −   ( − 1 ) m B   ( 1   +   π m B m r Γ ) ,
Δ Ω N t = [ K L   −   ( − 1 ) m B   π m B m r Γ   ( tan   K L 2 ) ( − 1 ) m B + 1 ] − [ K 0 L   −   ( − 1 ) m B   π m B m r Γ   ( tan K 0 L 2 ) ( − 1 ) m B + 1 ] ,
d K d z   =   ( − 1 ) m B + 1   [ cos   ( K L )   −   ( − 1 ) m B   ( 1   +   π m B m r Γ ) ]   [ cos ( K L )   −   ( − 1 ) m B ] π m B m r Γ L sin ( K L ) 1 z   −   ( N + 1 ) L ,
z L   =   1   +   N   { 1   −   cos   ( K 0 L )   −   ( − 1 ) m B cos   ( K 0 L )   −   ( − 1 ) m B   ( 1   +   π m B m r Γ )   cos   ( K 0 L )   −   ( − 1 ) m B   ( 1   +   π m B m r Γ ) cos   ( K L )   −   ( − 1 ) m B } .
50 ω r , j   =   50 ω r   +   ( j − 1 )   ( 100 ω B   −   50 ω r ) / 12 ,
f ( K )   =   Δ Ω N L cos ( K L )   −   ( − 1 ) m B cos   ( K L )   −   ( − 1 ) m B ( 1   +   π m B m r Γ ) .
τ Ham = N Δ Ω   [ ( − K 0 L   −   2 π )   −   ( − 1 ) m B π m B m R Γ   ( tan   − K 0 L   −   2 π 2 ) ( − 1 ) m B + 1 ] − [ K 0 L   −   ( − 1 ) m B π m B m R Γ   ( tan   K 0 L 2 ) ( − 1 ) m B + 1 ] = 2 N ( − Δ Ω )   [ K 0 L   −   ( − 1 ) m B π m B m R Γ   ( tan   K 0 L 2 ) ( − 1 ) m B + 1   +   π ] .
M   ( ω )   =   ( M 11 M 12 M 21 M 22 ) ,
r   =   − M 21 M 22 ,   t   =   det M M 22 ,
( E R + E R − )   =   M a r   ( E L + E L − )
M a r   =   1 i κ a r   [ − exp   { i 1 2 π   ( ω ω r , R a r   +   ω ω r , L a r ) } , σ a r   exp   { i 1 2 π ( ω ω r , R a r   −   ω ω r , L a r )   } − σ a r   exp   { − i 1 2 π   ( ω ω r , R a r   −   ω ω r , L a r ) } , exp   { − i 1 2 π   ( ω ω r , R a r   +   ω ω r , L a r ) } ] .
ω r , L a r   =   c n r R a r , L ,   ω r , R a r   =   c n r R a r , R ,
M uni ,   tot   ( ω )   =   M a r , B   ( ω ) M uni N s   ( ω )   M a r , A   ( ω ) .
M inc , tot   ( ω )   =   M a r , B   ( ω ) M N s   ( ω ) M N s − 1   ( ω )   … M 1   ( ω ) M a r , A   ( ω ) ,
ω r , L a r   =   2 π   ×   3.942 THz ,   ω r , R a r   =   2 π   ×   3.866 THz ,   σ a r   =   0.42 ,
d ϕ Ham d ω   =   τ Ham   =   2 N ( − Δ Ω )   [ K 0 L   −   ( − 1 ) m B   π   m B m R Γ   ( tan   K 0 L 2 ) ( − 1 ) m B + 1   +   π ] .
E in   ( ω )   =   1 π δ ω   exp   ( − ( ω   −   ω 0 ) 2 δ ω 2 )   exp   [ i   ( ω   −   ω 0 ) t 0 ] ,
E in   ( t )   =   ∫ d ω   E in   exp ( − i ω t ) .
E out , T   ( t )   =   ∫ d ω   E in   ( ω )   exp   [ i ϕ   ( ω )   ]   exp   ( − i ω t ) .
E out , Ham   ( t )   =   ∫ d ω   E in   ( ω )   exp   [ i ϕ Ham   ( ω ) ]   exp   ( − i ω t ) .

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