Abstract

The optical buffer is a key component in all-optical information processing systems. Recently a number of schemes that use slow light propagation in various media and structures have been proposed as means toward implementation of all-optical buffers. We rigorously analyze the similarities and differences in approaches that use electromagnetically induced transparency (EIT) and coupled resonant structures (CRS). We introduce the figure of merit, finesse, that is common to both approaches and obtain fundamental limitations on bit rates and storage capacities of optical buffers. We show that at very low bit rates and storage capacities EIT outperforms CRS, but at rates of 10Mbitss and above the EIT medium becomes quite inefficient, and the situation is reversed. Two types of CRS based on high-index-contrast fiber gratings and high-index semiconductor–air photonic crystals and (or) microring resonators are found to hold promise for applications in the 11000-Gbitss range, but only if the losses can be drastically reduced.

© 2005 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2005

2004

M. F. Yanik and S. Fan, "Stopping light all-optically," Phys. Rev. Lett. 92, 083901-083904 (2004).
[CrossRef]

2003

S. J. McNab, N. Moll, and Yu. A. Vlasov, "Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides," Opt. Express 11, 2927-2939 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

Yu. A. Vlasov and S. J. McNab, "Losses in single-mode silicon-on-insulator strip waveguides and bends," Opt. Express 11, 2927-2939 (2003), http://www.opticsexpress.org.

T. Li, H. Wang, N. H. Kwong, and R. Binder, "EIT via electron spin coherence in QW waveguide," Opt. Express 11, 3298-3303 (2003), http://www.opticsexpress.org.
[CrossRef] [PubMed]

L. Tong, R. Gattass, J. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

A. Melloni, F. Morichetti, and M. Martnelli, "Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365-378 (2003).
[CrossRef]

Z. Wang and S. Fan, "Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines," Phys. Rev. E 68, 066616-066623 (2003).
[CrossRef]

Y. Tao, Y. Sugimoto, S. Lan, N. Ikeda, Y. Tanaka, and Y. K. Asakawa, "Transmission properties of coupled-cavity waveguides based on two-dimensional photonic crystals with a triangular lattice of air holes," J. Opt. Soc. Am. B 20, 1992-1998 (2003).

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature (London) 426, 638-641 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of ultraslow light propagation in a ruby crystal at room temperature," Phys. Rev. Lett. 90, 113903-113904 (2003).
[CrossRef] [PubMed]

M. R. Poulsen and P. I. Borel, "Advances in silica based integrated optics," Opt. Eng. (Bellingham) 42, 2821-2856 (2003).
[CrossRef]

2002

P. C. Ku, C. J. Chang-Hasnain, and S. L. Chuang, "Variable semiconductor all-optical buffers," Electron. Lett. 38, 1581-1583 (2002).
[CrossRef]

A. V. Turukhin, V. S. Sudarshanam, and P. R. Hemmer, "Observation of ultraslow and stored light in a solid," Phys. Rev. Lett. 88, 023602-1-4 (2002).
[CrossRef]

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1948-1950 (2002).
[CrossRef]

M. Soljacic, S. G. Johnson, S. Fan, M. Inanescu, E. Ippen, and J. D. Joannopulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002).
[CrossRef]

S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, "Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects," Opt. Lett. 27, 2079-2081 (2002).
[CrossRef]

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, "Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures," Opt. Quantum Electron. 34, 123-130 (2002).
[CrossRef]

2001

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]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, "Storage of light in atomic vapor," Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

M. Notomi, K. Yamada, A. Shinya, J. Takajhashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 235902-1-4 (2001).
[CrossRef]

2000

J. B. Khurgin, "Light slowing down in Moire fiber gratings and its implications for nonlinear optics," Phys. Rev. A 62, 3821-3824 (2000).
[CrossRef]

1999

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]

N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, "Dispersion of cascaded fiber gratings in WDM lightwave systems," J. Lightwave Technol. 16, 1523-1529 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-596 (1999).
[CrossRef]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, "Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas," Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

1998

C. K. Madsen and G. Lenz, "Optical all-pass filters for phase response design with applications for dispersion compensation," IEEE Photonics Technol. Lett. 10, 994-996 (1998).
[CrossRef]

1997

M. D. Lukin, M. Fleichhauer, A. S. Zibrov, and M. O. Scully, "Spectroscopy in dense coherent media: line narrowing and interference effects," Phys. Rev. Lett. 79, 2959-2962 (1997).
[CrossRef]

1996

R. Langenhorst, M. Eiselt, W. Pieper, G. Grosskopf, R. Ludwig, L. Kuller, E. Dietrich, and H. G. Weber, "Fiber loop optical buffer," J. Lightwave Technol. 14, 324-335 (1996).
[CrossRef]

1995

M. Scalora, R. J. Flynn, S. B. Reinhard, and R. L. Fork, "Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss," Phys. Rev. E 54, R1078-R1081 (1995).
[CrossRef]

R. W. Equal, R. L. Cone, and R. M. MacFarlane, "Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5," Phys. Rev. B 52, 3963-3970 (1995).
[CrossRef]

Y.-Q. Li and M. Xiao, "EIT in 3-level Lambda-type system in Rb atoms," Phys. Rev. A 51, R2703-R2706 (1995).
[CrossRef]

A. Kasapi, M. Jain, G. Y. Jin, and S. E. Harris, "EIT: Propagation dynamics," Phys. Rev. Lett. 74, 2447-2450 (1995).
[CrossRef] [PubMed]

1992

S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of EIT," Phys. Rev. A 46, R29-R32 (1992).
[CrossRef]

1991

K.-J. Boller, A. Imamoglu, and S. E. Harris, "Observation of EIT," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

P. J. Lemaire, "High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers," Electron. Lett. 29, 1191-1193 (1991).
[CrossRef]

Agrawal, G. P.

Asakawa, K.

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1948-1950 (2002).
[CrossRef]

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, "Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures," Opt. Quantum Electron. 34, 123-130 (2002).
[CrossRef]

S. Nishikawa, S. Lan, N. Ikeda, Y. Sugimoto, H. Ishikawa, and K. Asakawa, "Optical characterization of photonic crystal delay lines based on one-dimensional coupled defects," Opt. Lett. 27, 2079-2081 (2002).
[CrossRef]

Asakawa, Y. K.

Y. Tao, Y. Sugimoto, S. Lan, N. Ikeda, Y. Tanaka, and Y. K. Asakawa, "Transmission properties of coupled-cavity waveguides based on two-dimensional photonic crystals with a triangular lattice of air holes," J. Opt. Soc. Am. B 20, 1992-1998 (2003).

Ashcom, J.

L. Tong, R. Gattass, J. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics (Saunders, New York, 1996), p. 551.

Bajcsy, M.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature (London) 426, 638-641 (2003).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-596 (1999).
[CrossRef]

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of ultraslow light propagation in a ruby crystal at room temperature," Phys. Rev. Lett. 90, 113903-113904 (2003).
[CrossRef] [PubMed]

Binder, R.

Boller, K.-J.

K.-J. Boller, A. Imamoglu, and S. E. Harris, "Observation of EIT," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

Borel, P. I.

M. R. Poulsen and P. I. Borel, "Advances in silica based integrated optics," Opt. Eng. (Bellingham) 42, 2821-2856 (2003).
[CrossRef]

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, "Observation of ultraslow light propagation in a ruby crystal at room temperature," Phys. Rev. Lett. 90, 113903-113904 (2003).
[CrossRef] [PubMed]

Carlsson, N.

N. Carlsson, N. Ikeda, Y. Sugimoto, K. Asakawa, T. Takemori, Y. Katayama, N. Kawai, and K. Inoue, "Design, nano-fabrication and analysis of near-infrared 2D photonic crystal air-bridge structures," Opt. Quantum Electron. 34, 123-130 (2002).
[CrossRef]

Chang-Hasnain, C. J.

P. C. Ku, C. J. Chang-Hasnain, and S. L. Chuang, "Variable semiconductor all-optical buffers," Electron. Lett. 38, 1581-1583 (2002).
[CrossRef]

Chuang, S. L.

P. C. Ku, C. J. Chang-Hasnain, and S. L. Chuang, "Variable semiconductor all-optical buffers," Electron. Lett. 38, 1581-1583 (2002).
[CrossRef]

Cone, R. L.

R. W. Equal, R. L. Cone, and R. M. MacFarlane, "Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5," Phys. Rev. B 52, 3963-3970 (1995).
[CrossRef]

Dietrich, E.

R. Langenhorst, M. Eiselt, W. Pieper, G. Grosskopf, R. Ludwig, L. Kuller, E. Dietrich, and H. G. Weber, "Fiber loop optical buffer," J. Lightwave Technol. 14, 324-335 (1996).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-596 (1999).
[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]

N. M. Litchinitser, B. J. Eggleton, and G. P. Agrawal, "Dispersion of cascaded fiber gratings in WDM lightwave systems," J. Lightwave Technol. 16, 1523-1529 (1999).
[CrossRef]

Eiselt, M.

R. Langenhorst, M. Eiselt, W. Pieper, G. Grosskopf, R. Ludwig, L. Kuller, E. Dietrich, and H. G. Weber, "Fiber loop optical buffer," J. Lightwave Technol. 14, 324-335 (1996).
[CrossRef]

Equal, R. W.

R. W. Equal, R. L. Cone, and R. M. MacFarlane, "Homogeneous broadening and hyperfine structure of optical transitions in Pr3+:Y2SiO5," Phys. Rev. B 52, 3963-3970 (1995).
[CrossRef]

Fan, S.

M. F. Yanik and S. Fan, "Stopping light all-optically," Phys. Rev. Lett. 92, 083901-083904 (2004).
[CrossRef]

Z. Wang and S. Fan, "Compact all-pass filters in photonic crystals as the building block for high-capacity optical delay lines," Phys. Rev. E 68, 066616-066623 (2003).
[CrossRef]

M. Soljacic, S. G. Johnson, S. Fan, M. Inanescu, E. Ippen, and J. D. Joannopulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002).
[CrossRef]

Feinberg, J.

E. Salik, D. S. Starodubov, V. Grubsky, and J. Feinberg, "Increase of photosensitivity in Ge-doped fibers under strain," in Optical Fiber Communications Conference, Postconference Digest, Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 124-125.

Field, J. E.

S. E. Harris, J. E. Field, and A. Kasapi, "Dispersive properties of EIT," Phys. Rev. A 46, R29-R32 (1992).
[CrossRef]

Fleichhauer, M.

M. D. Lukin, M. Fleichhauer, A. S. Zibrov, and M. O. Scully, "Spectroscopy in dense coherent media: line narrowing and interference effects," Phys. Rev. Lett. 79, 2959-2962 (1997).
[CrossRef]

Fleischhauer, A.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, "Storage of light in atomic vapor," Phys. Rev. Lett. 86, 783-786 (2001).
[CrossRef] [PubMed]

Flynn, R. J.

M. Scalora, R. J. Flynn, S. B. Reinhard, and R. L. Fork, "Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss," Phys. Rev. E 54, R1078-R1081 (1995).
[CrossRef]

Fork, R. L.

M. Scalora, R. J. Flynn, S. B. Reinhard, and R. L. Fork, "Ultrashort pulse propagation at the photonic band edge: Large tunable group delay with minimal distortion and loss," Phys. Rev. E 54, R1078-R1081 (1995).
[CrossRef]

Fry, E. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, "Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas," Phys. Rev. Lett. 82, 5229-5232 (1999).
[CrossRef]

Gattass, R.

L. Tong, R. Gattass, J. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
[CrossRef]

Grosskopf, G.

R. Langenhorst, M. Eiselt, W. Pieper, G. Grosskopf, R. Ludwig, L. Kuller, E. Dietrich, and H. G. Weber, "Fiber loop optical buffer," J. Lightwave Technol. 14, 324-335 (1996).
[CrossRef]

Grubsky, V.

E. Salik, D. S. Starodubov, V. Grubsky, and J. Feinberg, "Increase of photosensitivity in Ge-doped fibers under strain," in Optical Fiber Communications Conference, Postconference Digest, Vol. 37 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), pp. 124-125.

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metres per second in an ultracold atomic gas," Nature 397, 594-596 (1999).
[CrossRef]

A. Kasapi, M. Jain, G. Y. Jin, and S. E. Harris, "EIT: Propagation dynamics," Phys. Rev. Lett. 74, 2447-2450 (1995).
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L. Tong, R. Gattass, J. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, "Subwavelength-diameter silica wires for low-loss optical wave guiding," Nature (London) 426, 816-819 (2003).
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S. J. McNab, N. Moll, and Yu. A. Vlasov, "Ultra-low loss photonic integrated circuit with membrane-type photonic crystal waveguides," Opt. Express 11, 2927-2939 (2003), http://www.opticsexpress.org.
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D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, "Storage of light in atomic vapor," Phys. Rev. Lett. 86, 783-786 (2001).
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M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, "Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas," Phys. Rev. Lett. 82, 5229-5232 (1999).
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Y.-Q. Li and M. Xiao, "EIT in 3-level Lambda-type system in Rb atoms," Phys. Rev. A 51, R2703-R2706 (1995).
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M. Notomi, K. Yamada, A. Shinya, J. Takajhashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 235902-1-4 (2001).
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M. Bajcsy, A. S. Zibrov, and M. D. Lukin, "Stationary pulses of light in an atomic medium," Nature (London) 426, 638-641 (2003).
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M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, "Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas," Phys. Rev. Lett. 82, 5229-5232 (1999).
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Appl. Phys. Lett.

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, "Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines," Appl. Phys. Lett. 81, 1948-1950 (2002).
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Electron. Lett.

P. C. Ku, C. J. Chang-Hasnain, and S. L. Chuang, "Variable semiconductor all-optical buffers," Electron. Lett. 38, 1581-1583 (2002).
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IEEE Photonics Technol. Lett.

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J. Lightwave Technol.

R. Langenhorst, M. Eiselt, W. Pieper, G. Grosskopf, R. Ludwig, L. Kuller, E. Dietrich, and H. G. Weber, "Fiber loop optical buffer," J. Lightwave Technol. 14, 324-335 (1996).
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Figures (7)

Fig. 1
Fig. 1

(a) Two-level resonant atomic system, (b) the dispersion of the slow wave (polariton).

Fig. 2
Fig. 2

EIT in the Λ scheme: (a) atomic-level diagram, (b) dispersion of the slow wave.

Fig. 3
Fig. 3

(a) Bragg reflector, (b) its dispersion.

Fig. 4
Fig. 4

Various coupled-resonator-based slow-light structures: (a) coupled Fabry–Perot cavities (Moiré pattern), (b) microring lines, (c) defect-mode waveguide in photonic crystal, (d) characteristic dispersion of CRS.

Fig. 5
Fig. 5

Required lengths of various optical buffers with storage capacity of 10   bits as functions of bit rate. Bps bits per second .

Fig. 6
Fig. 6

Same as Fig. 5 but for 50   bits . Bps bits per second .

Fig. 7
Fig. 7

Same as Fig. 5 but for 200   bits . Bps bits per second .

Tables (3)

Tables Icon

Table 1 Performance Characteristics of Various EIT Slow-Light Optical Buffers

Tables Icon

Table 2 Performance Characteristics of Various CRS Slow-Light Optical Buffers

Tables Icon

Table 3 Comparison of EIT and CRS Slow-Light Optical Buffers

Equations (72)

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k = ω n ¯ c ,
v g = ω k ,
γ 12 1 T 2 ( 12 ) ,
β 2 = v g 1 ω = 2 k ω 2 ,
β 3 = 3 k ω 3 > 0 ,
ϵ ( δ ω ) = n ¯ 2 + N μ A ̇ G 2 ϵ 0 δ ω + i γ B G Ω 2 ( δ ω + i γ B G ) ( δ ω + i γ A G ) ,
n ̃ ( δ ω ) = [ ϵ ( δ ω ) ] 1 2 = n ¯ + N e 2 f A G 4 n ¯ m 0 ω 0 ϵ 0 δ ω + i γ B G Ω 2 ( δ ω + i γ B G ) ( δ ω + i γ A G ) ,
Ω p 2 = N e 2 f A G 4 n ¯ 2 m 0 ϵ 0 .
n g ( 0 ) = c k ω = n ¯ + ω Re n ̃ ω δ ω = 0 = n ¯ + n ¯ Ω p 2 Ω 2 γ B G 2 ( Ω 2 + γ B G γ A G ) 2 ,
α ( 0 ) = 2 ω c Im ( n ̃ ) δ ω = 0 = 2 n ¯ Ω p 2 c γ B G Ω 2 + γ B G γ A G .
t abs = n g ln 2 ( α c ) ln 2 2 γ G B 1 Ω 2 γ B G 2 Ω 2 + γ B G γ A G ,
t abs , max = ln 2 2 γ B G 1 ,
n g n ¯ ( t abs ) = 1 + 2 ln 2 Ω p 2 t d ( γ B G + γ A G ) 1 ( 1 t abs t abs , max ) = 1 + Ω p 2 t abs t A G ( 1 t abs t abs , max ) ,
t abs = 1 2 t abs , max ,
Ω max = [ γ B G ( γ A G + 2 γ B G ) ] 1 2 ( γ B G γ A G ) 1 2
n g , max n ¯ = 1 + 1 4 Ω p 2 t abs , max t A G .
n g ( EIT ) ( 0 ) = n ¯ + n ¯ Ω p 2 Ω 2 .
n g ( EIT ) ( 0 ) = n ¯ + n ¯ Ω p 2 a 2 B 2 ,
β 2 ( EIT ) ( δ ω ) = c 1 d d ( δ ω ) [ ( ω 0 + δ ω ) n ( δ ω ) ] = 2 Ω p 2 n ¯ c 1 Ω 2 δ ω 2 [ 1 ω 0 + δ ω 3 Ω 2 + δ ω 2 ( Ω 2 δ ω 2 ) 2 ] .
β 3 ( EIT ) ( 0 ) = c 1 d β 2 ( EIT ) ( δ ω ) d ( δ ω ) δ ω = 0 = 6 Ω p 2 n ¯ c 1 Ω 4 .
B [ β 3 L max ] 1 3 < γ = 0.324 .
t d , max = L max c n g = γ 3 B 3 c β 3 n g = γ 3 Ω 2 6 B 3 ( 1 + Ω 2 Ω p 2 ) ,
N st = B t d , max = γ 3 Ω 2 6 B 2 ( 1 + Ω 2 Ω p 2 ) = γ 3 Ω 2 6 B 2 n g n g n ¯ .
Ω = 6 1 2 γ 3 2 N st 1 2 n g n ¯ n g B 13.2 N st 1 2 n g n ¯ n g B .
n g ( EIT ) n ¯ = 1 + B EIT , max 2 B 2 n g ( EIT ) n g ( EIT ) n ¯ ,
B EIT , max ( N st ) = γ 3 2 6 1 2 Ω p N st 1 2 = 0.075 Ω p N st 1 2 ,
n g ( EIT ) n ¯ N st = 1 2 B EIT , max 2 B 2 + 1 + 1 2 B EIT , max 2 B 2 ( 1 + 4 B 4 B EIT , max 4 ) 1 2 { B EIT , max 2 B 2 + 2 B B EIT , max B EIT , max B + 1 B B EIT , max } .
L EIT ( N st , B ) = c N st B 1 n g ( EIT ) 6 c γ 3 n ¯ N st 2 B Ω p 2 ,
L 0 ( N st , B ) = c N st B 1 n ¯ .
B EIT ( N st n g ) γ 3 2 6 1 2 ( N st n g ) 1 2 Ω p 0.075 ( N st n g ) 1 2 Ω p .
B EIT , min ( N st ) = N st t abs , max .
γ A G 200 N st 3 γ B G .
n g , max ( N st ) = γ 3 6 F EIT 2 N st 3 ,
F EIT = Ω p t d , max .
N st , max = γ 6 1 3 F EIT = 0.18 F EIT 2 3 ,
B max , cap = N st , max t d , max = γ 6 1 3 Ω p 2 3 t d , max 1 3
δ ω B = ω 0 2 π sin 1 ( n H n L n H + n L ) .
n g n ¯ ω ω 0 ( ω ω 0 ) 2 δ ω B 2 .
β 2 = sign ( ω 0 ω ) n ¯ c δ ω B 2 [ ( ω ω 0 ) 2 δ ω B 2 ] 3 2 ,
ω FSR = π c d n ¯ .
sin π ω ω FSR = t cos β d ,
m = ω 0 ω FSR = d n ¯ λ 0 2 ,
β = 2 n ¯ λ 0 m cos 1 ( t 1 sin m π ω ω 0 ) .
Ω t = ω 0 m π sin 1 t ,
Ω t Ω t , 0 = ω 0 t m π = ω FSR t π .
n g = c β ω = n ¯ t 1 = n ¯ ω 0 m π Ω t , 0 .
β 3 = 3 β ω 3 = n ¯ m 2 π 2 c ω 0 2 1 t 2 t 3 n ¯ ω 0 m π c Ω t , 0 3 .
r = ( n H n L ) 2 ( m 1 ) 1 ( n H n L ) 2 ( m 1 ) + 1 ,
t = ( 1 r 2 ) 1 2 = 2 ( n H n L ) m 1 ( n H n L ) 2 ( m 1 ) + 1 .
m = 1 + ln 1 + ( 1 t 2 ) 1 2 t ln ( n H n L ) .
m ln 2 n g n ¯ ln n H n L .
n g n ¯ Ω CRS π Ω t , 0 ,
Ω CRS = ω 0 ln ( n H n L ) π ln 2 n g n ¯
β 3 Ω CRS c Ω g , 0 3 .
Ω CRS = ω 0 ln ( n H n L ) π ln 2 n g ln n g n av n av ,
n av
N st = B t d , max = γ 3 B 2 c β 3 n g = γ 3 B 2 ω 0 2 m 2 π 2 t 2 1 t 2 = γ 3 Ω t 0 2 B 2 n g 2 n g 2 n ¯ 2 .
Ω t 0 = γ 3 2 N st 1 2 ( 1 n ¯ 2 n g 2 ) 1 2 B 5.4 N st 1 2 ( 1 n ¯ 2 n g 2 ) 1 2 B .
n g = n ¯ B CRS , max B ( 1 n ¯ 2 n g 2 ) 1 2 × 1 ln n H n L + ln [ n g n ¯ ( 1 + 1 n ¯ 2 n g 2 ) ] ,
B CRS , max ( N st ) = γ 3 2 ω 0 ln ( n H n L ) π N st 1 2
N st = n ¯ B L c t
N st 3 γ 3 = ( 2 n ¯ L λ 0 m ) 2 ( 1 t 2 ) 1 = ( L d ) 2 ( 1 t 2 ) 1 = N res 2 ( 1 n ¯ 2 n g 2 ) 1 ,
N res 5 N st 3 2 ,
L = L D ( N st ) { ln n H n L + ln [ n g n ¯ ( 1 + 1 n ¯ 2 n g 2 ) ] } ( 1 n ¯ 2 n g 2 ) 1 2 ,
L D ( N st ) = c N st n ¯ B CRS , max = λ 0 2 n ¯ ln n H n L ( N st γ ) 3 2 .
L L D B B CRS , max = c N st n g c N st n ¯ = n ¯ n g .
L L D = [ 1 ( L L D B B CRS , max ) 2 ] 1 2 { ln n H n L + ln [ 1 + 1 ( L L D B B CRS , max ) 2 ] ln L L D B B CRS , max } .
t scat , max CRS = ln 2 ( α c n ¯ ) 1 .
B CRS , min ( N st ) = N st t scat , max CRS .
n g , max ( N st ) = γ 3 2 F CRS N st 3 2 ,
F CRS = Ω CRS t scat , max CRS .
N st , max CRS γ F CRS 2 3 .

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