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 OSA

Full Article  |  PDF Article
OSA Recommended Articles
Increasing the delay-time-bandwidth product for microring resonator structures by varying the optical ring resonances

Zhenshan Yang and J. E. Sipe
Opt. Lett. 32(8) 918-920 (2007)

Analysis of optical ARMA architectures in the slow-light regime

Vishnupriya Govindan and Steve Blair
J. Opt. Soc. Am. B 25(12) C116-C126 (2008)

Side coupled adjacent resonators CROW - formation of mid-band zero group velocity

Ori Weiss and Jacob Scheuer
Opt. Express 17(17) 14817-14824 (2009)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. P. Chamorro-Posada and F. J. Fraile-Pelaez, “Fast and slow light in zigzag microring resonator chains,” Opt. Lett. 34, 626–628 (2009).
    [Crossref] [PubMed]
  2. C. Fietz and G. Shvets, “Simultaneous fast and slow light in microring resonators,” Opt. Lett. 32, 3480–3482 (2007).
    [Crossref] [PubMed]
  3. F. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2007).
    [Crossref]
  4. J. K. S. Poon, J. Scheuer, S. Mookherjea, G. T. Paloczi, Y. Huang, and A. Yariv “Matrix analysis of microring coupled-resonator optical waveguides,” Opt. Express 12, 90–103 (2004).
    [Crossref] [PubMed]
  5. J. E. Heebner, P. Chak, S. Pereira, J. E. Sipe, and R. W. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optic,” J. Opt. Soc. Am. B 21, 1818–1832 (2004).
    [Crossref]
  6. 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]
  7. S. Sandhu, M. L. Povinelli, M. F. Yanik, and S. H. Fan, “Dynamically tuned coupled-resonator delay lines can be nearly dispersion free,” Opt. Lett. 31, 1985–1987 (2006).
    [Crossref] [PubMed]
  8. 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]
  9. M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92, 083901 (2004).
    [Crossref] [PubMed]
  10. 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]
  11. Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. 34, 1072–1074 (2009).
    [Crossref] [PubMed]
  12. 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]
  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]
  14. Z. S. Yang and J. E. Sipe, “Increasing the delay-time-bandwidth product for micro-ring resonator structures by varying the optical ring resonances,” Opt. Lett. 32, 918–920 (2007).
    [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).
  17. P. St. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol. 17, 1982–1988 (1999).
    [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]
  21. P. Chak and J. E. Sipe, “Minimizing finite-size effects in artificial resonance tunneling structures,” Opt. Lett. 31, 2568–2570 (2006).
    [Crossref] [PubMed]

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, “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).

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, “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).

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, “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).

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)

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

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

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).

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 (8)

Fig. 1
Fig. 1

Schematic of a SCISSOR unit.

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2

Photonic band structure of the periodic SCISSOR system.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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).

Download Full Size | PPT Slide | PDF

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).

Download Full Size | PPT Slide | PDF

Fig. 6
Fig. 6

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

Download Full Size | PPT Slide | PDF

Fig. 7
Fig. 7

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

Download Full Size | PPT Slide | PDF

Equations (34)

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

( 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 ) .

Metrics