Abstract

We show that there is a general limit to the performance of linear optical components, based only on their size, shape, and dielectric constants. The limit is otherwise independent of the design. The mathematics involved applies generally to linear systems with arbitrarily strong multiple scattering. Relevant optical structures include dielectric stacks, photonic crystals, nanometallics, metamaterials, and slow-light structures. The limit also covers acoustic and quantum-mechanical waves, and electromagnetic waves of any frequency. In an example, a one-dimensional glass/air structure, a thickness of at least 41.7μm is required for the separation of pulses of 32 different frequencies near 1.55μm center wavelength. Larger available dielectric constants would lead to correspondingly shorter limits.

© 2007 Optical Society of America

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  5. T. Baba and T. Matsumoto, "Resolution of photonic crystal superprism," Appl. Phys. Lett. 81, 2325-2327 (2002).
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  8. L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, "Dual lattice photonic-crystal beam splitters," Appl. Phys. Lett. 86, 211106 (2005).
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    [CrossRef]
  16. Y. Jiao, S. H. Fan, and D. A. B. Miller, "Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator," Opt. Lett. 30, 141-143 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  33. A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, "Delay limit of slow light in semiconductor optical amplifiers," IEEE Photon. Technol. Lett. 18, 731-733 (2006).
    [CrossRef]
  34. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
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    [CrossRef] [PubMed]
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    [CrossRef]
  37. D. A. B. Miller, "Communicating with waves between volumes - evaluating orthogonal spatial channels and limits on coupling strengths," Appl. Opt. 39, 1681-1699 (2000).
    [CrossRef]
  38. R. Piestun and D. A. B. Miller, "Electromagnetic degrees of freedom of an optical system," J. Opt. Soc. Am. A 17, 892-902 (2000).
    [CrossRef]
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2006 (5)

R. E. Klinger, C. A. Hulse, C. K. Carniglia, and R. B. Sargent, "Beam displacement and distortion effects in narrowband optical thin-film filters," Appl. Opt. 45, 3237-3242 (2006).
[CrossRef] [PubMed]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Systematic photonic crystal device design: global and local optimization and sensitivity analysis," IEEE J. Quantum Electron. 42, 266-279 (2006).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, "Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities," J. Phys.: Condens. Matter 18, 3117-3126 (2006).
[CrossRef]

A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, "Delay limit of slow light in semiconductor optical amplifiers," IEEE Photon. Technol. Lett. 18, 731-733 (2006).
[CrossRef]

J. B. Khurgin, "Performance limits of delay lines based on optical amplifiers," Opt. Lett. 31, 948-950 (2006).
[CrossRef] [PubMed]

2005 (13)

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

J. Shamir, "Analysis of volume holographic storage allowing large-angle illumination," J. Opt. Soc. Am. B 22, 975-986 (2005).
[CrossRef]

R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers: capabilities and fundamental limitations," J. Lightwave Technol. 23, 4046-4066 (2005).
[CrossRef]

M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, "Distortion management in slow-light pulse delay," Opt. Express 13, 9995-10002 (2005).
[CrossRef] [PubMed]

M. Povinelli, S. Johnson, and J. Joannopoulos, "Slow-light, band-edge waveguides for tunable time delays," Opt. Express 13, 7145-7159 (2005).
[CrossRef] [PubMed]

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]

M. R. Fisher and S.-L. Chuang, "Variable group delay and pulse reshaping of high bandwidth optical signals," IEEE J. Quantum Electron. 41, 885-891 (2005).
[CrossRef]

J. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. Willner, and A. Gaeta, "All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion," Opt. Express 13, 7872-7877 (2005).
[CrossRef] [PubMed]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator," Opt. Lett. 30, 141-143 (2005).
[CrossRef] [PubMed]

M. Gerken and D. A. B. Miller, "Limits on the performance of dispersive thin-film stacks," Appl. Opt. 44, 3349-3357 (2005).
[CrossRef] [PubMed]

M. Gerken and D. A. B. Miller, "The relationship between the superprism effect in one-dimensional photonic crystals and spatial dispersion in nonperiodic thin-film stacks," Opt. Lett. 30, 2475-2477 (2005).
[CrossRef] [PubMed]

L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, "Dual lattice photonic-crystal beam splitters," Appl. Phys. Lett. 86, 211106 (2005).
[CrossRef]

B. Momeni and A. Abidi, "Systematic design of superprism-based photonic crystal demultiplexers," IEEE J. Sel. Areas Commun. 23, 1355-1364 (2005).
[CrossRef]

2004 (6)

C. Y. Luo, M. Soljacic, and J. D. Joannopoulos, "Superprism effect based on phase velocities," Opt. Lett. 29, 745-747 (2004).
[CrossRef] [PubMed]

O. Schwelb, "Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters-a tutorial overview," J. Lightwave Technol. 22, 1380-1394 (2004).
[CrossRef]

K. Yu and O. Solgaard, "Tunable optical transversal filters based on a Gires-Tournois interferometer with MEMS phase shifters," IEEE J. Sel. Top. Quantum Electron. 10, 588-597 (2004).
[CrossRef]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method," Phys. Rev. E 70, 036612-1-036612-9 (2004).
[CrossRef]

M. Gerken and D. A. B. Miller, "Photonic nanostructures for wavelength division multiplexing," Proc. SPIE 5597, 82-96 (2004).
[CrossRef]

K. Tian and G. Barbastathis, "Cross talk in resonant holographic memories," J. Opt. Soc. Am. A 21, 751-756 (2004).
[CrossRef]

2003 (3)

2002 (1)

T. Baba and T. Matsumoto, "Resolution of photonic crystal superprism," Appl. Phys. Lett. 81, 2325-2327 (2002).
[CrossRef]

2001 (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical fibers," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

2000 (2)

1999 (3)

1998 (1)

1989 (1)

H. Lee, X. G. Gu, and D. Psaltis, "Volume holographic interconnections with maximal capacity and minimal cross talk," J. Appl. Phys. 65, 2191-2194 (1989).
[CrossRef]

1969 (1)

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

1963 (1)

Abidi, A.

B. Momeni and A. Abidi, "Systematic design of superprism-based photonic crystal demultiplexers," IEEE J. Sel. Areas Commun. 23, 1355-1364 (2005).
[CrossRef]

B. Momeni and A. Abidi, "Optimization of photonic crystal demultiplexers based on the superprism effect," Appl. Phys. B 77, 555-560 (2003).
[CrossRef]

Baba, T.

T. Baba and T. Matsumoto, "Resolution of photonic crystal superprism," Appl. Phys. Lett. 81, 2325-2327 (2002).
[CrossRef]

Barbastathis, G.

Bigelow, M. S.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, "Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities," J. Phys.: Condens. Matter 18, 3117-3126 (2006).
[CrossRef]

Binder, R.

Boyd, R. W.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, "Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities," J. Phys.: Condens. Matter 18, 3117-3126 (2006).
[CrossRef]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Campbell, S.

X. M. Yi, P. Yeh, C. Gu, and S. Campbell, "Crosstalk in volume holographic memory," Proc. IEEE 87, 1912-1930 (1999).
[CrossRef]

Carniglia, C. K.

Chang-Hasnain, C. J.

A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, "Delay limit of slow light in semiconductor optical amplifiers," IEEE Photon. Technol. Lett. 18, 731-733 (2006).
[CrossRef]

R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, "Slow-light optical buffers: capabilities and fundamental limitations," J. Lightwave Technol. 23, 4046-4066 (2005).
[CrossRef]

Chuang, S.-L.

M. R. Fisher and S.-L. Chuang, "Variable group delay and pulse reshaping of high bandwidth optical signals," IEEE J. Quantum Electron. 41, 885-891 (2005).
[CrossRef]

Dawes, A. M. C.

Eggleton, B.

Eggleton, B. J.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical fibers," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Fan, S. H.

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Systematic photonic crystal device design: global and local optimization and sensitivity analysis," IEEE J. Quantum Electron. 42, 266-279 (2006).
[CrossRef]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator," Opt. Lett. 30, 141-143 (2005).
[CrossRef] [PubMed]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method," Phys. Rev. E 70, 036612-1-036612-9 (2004).
[CrossRef]

Fisher, M. R.

M. R. Fisher and S.-L. Chuang, "Variable group delay and pulse reshaping of high bandwidth optical signals," IEEE J. Quantum Electron. 41, 885-891 (2005).
[CrossRef]

Gaeta, A.

Gaeta, A. L.

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Gallet, J.-F.

L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, "Dual lattice photonic-crystal beam splitters," Appl. Phys. Lett. 86, 211106 (2005).
[CrossRef]

Gauthier, D. J.

M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, "Distortion management in slow-light pulse delay," Opt. Express 13, 9995-10002 (2005).
[CrossRef] [PubMed]

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Gerken, M.

Gu, C.

X. M. Yi, P. Yeh, C. Gu, and S. Campbell, "Crosstalk in volume holographic memory," Proc. IEEE 87, 1912-1930 (1999).
[CrossRef]

Gu, X. G.

H. Lee, X. G. Gu, and D. Psaltis, "Volume holographic interconnections with maximal capacity and minimal cross talk," J. Appl. Phys. 65, 2191-2194 (1989).
[CrossRef]

Hanson, G. W.

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002), p. 172.

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002), p. 259.

Hulse, C. A.

Jiao, Y.

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Systematic photonic crystal device design: global and local optimization and sensitivity analysis," IEEE J. Quantum Electron. 42, 266-279 (2006).
[CrossRef]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator," Opt. Lett. 30, 141-143 (2005).
[CrossRef] [PubMed]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method," Phys. Rev. E 70, 036612-1-036612-9 (2004).
[CrossRef]

Joannopoulos, J.

Joannopoulos, J. D.

Johnson, S.

Khurgin, J. B.

Klinger, R. E.

Kogelnik, H.

H. Kogelnik, "Coupled wave theory for thick hologram gratings," Bell Syst. Tech. J. 48, 2909-2947 (1969).

Krauss, T. F.

L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, "Dual lattice photonic-crystal beam splitters," Appl. Phys. Lett. 86, 211106 (2005).
[CrossRef]

Ku, P.-C.

Kwong, N. H.

Lee, H.

H. Lee, X. G. Gu, and D. Psaltis, "Volume holographic interconnections with maximal capacity and minimal cross talk," J. Appl. Phys. 65, 2191-2194 (1989).
[CrossRef]

Lee, R. K.

Lenz, G.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical fibers," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

G. Lenz and C. K. Madsen, "General optical all-pass filter structures for dispersion control in WDM systems," J. Lightwave Technol. 17, 1248-1254 (1999).
[CrossRef]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, "Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities," J. Phys.: Condens. Matter 18, 3117-3126 (2006).
[CrossRef]

Luo, C. Y.

Madsen, C. K.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical fibers," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

G. Lenz and C. K. Madsen, "General optical all-pass filter structures for dispersion control in WDM systems," J. Lightwave Technol. 17, 1248-1254 (1999).
[CrossRef]

Matsumoto, T.

T. Baba and T. Matsumoto, "Resolution of photonic crystal superprism," Appl. Phys. Lett. 81, 2325-2327 (2002).
[CrossRef]

Mazilu, M.

L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, "Dual lattice photonic-crystal beam splitters," Appl. Phys. Lett. 86, 211106 (2005).
[CrossRef]

Miller, D. A. B.

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Systematic photonic crystal device design: global and local optimization and sensitivity analysis," IEEE J. Quantum Electron. 42, 266-279 (2006).
[CrossRef]

M. Gerken and D. A. B. Miller, "The relationship between the superprism effect in one-dimensional photonic crystals and spatial dispersion in nonperiodic thin-film stacks," Opt. Lett. 30, 2475-2477 (2005).
[CrossRef] [PubMed]

M. Gerken and D. A. B. Miller, "Limits on the performance of dispersive thin-film stacks," Appl. Opt. 44, 3349-3357 (2005).
[CrossRef] [PubMed]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator," Opt. Lett. 30, 141-143 (2005).
[CrossRef] [PubMed]

Y. Jiao, S. H. Fan, and D. A. B. Miller, "Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method," Phys. Rev. E 70, 036612-1-036612-9 (2004).
[CrossRef]

M. Gerken and D. A. B. Miller, "Photonic nanostructures for wavelength division multiplexing," Proc. SPIE 5597, 82-96 (2004).
[CrossRef]

M. Gerken and D. A. B. Miller, "Multilayer thin-film structures with high spatial dispersion," Appl. Opt. 42, 1330-1345 (2003).
[CrossRef] [PubMed]

D. A. B. Miller, "Communicating with waves between volumes - evaluating orthogonal spatial channels and limits on coupling strengths," Appl. Opt. 39, 1681-1699 (2000).
[CrossRef]

R. Piestun and D. A. B. Miller, "Electromagnetic degrees of freedom of an optical system," J. Opt. Soc. Am. A 17, 892-902 (2000).
[CrossRef]

D. A. B. Miller, "Spatial channels for communicating with waves between volumes," Opt. Lett. 23, 1645-1647 (1998).
[CrossRef]

Momeni, B.

B. Momeni and A. Abidi, "Systematic design of superprism-based photonic crystal demultiplexers," IEEE J. Sel. Areas Commun. 23, 1355-1364 (2005).
[CrossRef]

B. Momeni and A. Abidi, "Optimization of photonic crystal demultiplexers based on the superprism effect," Appl. Phys. B 77, 555-560 (2003).
[CrossRef]

Neifeld, M. A.

Okawachi, Y.

Piestun, R.

Povinelli, M.

Psaltis, D.

H. Lee, X. G. Gu, and D. Psaltis, "Volume holographic interconnections with maximal capacity and minimal cross talk," J. Appl. Phys. 65, 2191-2194 (1989).
[CrossRef]

Sargent, R. B.

Scherer, A.

Schwelb, O.

Sedgwick, F. G.

A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, "Delay limit of slow light in semiconductor optical amplifiers," IEEE Photon. Technol. Lett. 18, 731-733 (2006).
[CrossRef]

Shamir, J.

Sharping, J.

Shin, H.

M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, "Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities," J. Phys.: Condens. Matter 18, 3117-3126 (2006).
[CrossRef]

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, "Optical delay lines based on optical fibers," IEEE J. Quantum Electron. 37, 525-532 (2001).
[CrossRef]

Smirl, A. L.

Solgaard, O.

K. Yu and O. Solgaard, "Tunable optical transversal filters based on a Gires-Tournois interferometer with MEMS phase shifters," IEEE J. Sel. Top. Quantum Electron. 10, 588-597 (2004).
[CrossRef]

Soljacic, M.

Stenner, M. D.

Sumetsky, M.

Tian, K.

Tucker, R. S.

Uskov, A. V.

A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, "Delay limit of slow light in semiconductor optical amplifiers," IEEE Photon. Technol. Lett. 18, 731-733 (2006).
[CrossRef]

van Heerden, P. J.

van Howe, J.

Wang, Y.

Willner, A.

Willner, A. E.

R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, "Maximum time delay achievable on propagation through a slow-light medium," Phys. Rev. A 71, 023801 (2005).
[CrossRef]

Wu, L.

L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, "Dual lattice photonic-crystal beam splitters," Appl. Phys. Lett. 86, 211106 (2005).
[CrossRef]

Xu, C.

Xu, Y.

Yakovlev, A. B.

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Other (2)

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002), p. 172.

G. W. Hanson and A. B. Yakovlev, Operator Theory for Electromagnetics (Springer-Verlag, 2002), p. 259.

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Figures (5)

Fig. 1
Fig. 1

Illustration of scattered (a) pulses and (b) beams for temporal and spatial dispersers, respectively. The straight-through and single-scattered pulses or beams may not actually be present physically, but the theory first considers outputs that are orthogonal to what both of these would be mathematically. Here we show the case where the straight-through and single-scattered pulses or beams miss the receiving volume, though in general they may not.

Fig. 2
Fig. 2

Illustration of the physical volumes used for the one-dimensional example.

Fig. 3
Fig. 3

Sketch of four input pulses of identical envelope shapes in time but different center frequencies, with their corresponding spectral ranges.

Fig. 4
Fig. 4

Sketch of dispersion of two different pulses. An input pulse (solid black curve) consisting of a superposition of four pulses of the same shape but different center wavelengths is dispersed by the scatterer to give the resulting set of pulses (solid colored curves) in the receiving volume. A similar but delayed input pulse (dashed black curve) would, however, similarly be dispersed to give a delayed set of dispersed pulses (dashed colored curves). The simple limit formulas [Eqs. (41, 42)] count both of these as distinct results, an overcounting that must be corrected for a useful limit.

Fig. 5
Fig. 5

Sketch of number of orthogonal basis functions (or degrees of freedom) required for separating pulses of different center frequencies in the receiving volume, shown for separation of N b = 4 pulses of different center frequencies. Pulse basis functions sketched in solid parts of curves are set to finite amplitudes, while those in dashed parts are set to zero amplitude, hence requiring N b 2 = 16 basis functions altogether (neglecting carrier phase).

Equations (117)

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I H = j ψ j ψ j .
ϕ S C m = G S ψ S m .
ϕ S m = ϕ I m + ϕ S C m .
ψ = C ϕ .
ψ S m = C ϕ S m = C ϕ I m + C ϕ S C m = C ϕ I m + CG S ψ S m = C ϕ I m + A S ψ S m ,
A S = CG S .
ϕ R m = G S R ψ S m .
ϕ R m = G S R ψ S m = G S R C ϕ I m + G S R A S ψ S m .
ϕ R m G S R C ϕ I m = 0 .
ϕ R m ϕ R m = ψ S m G S R G S R ψ S m = 0 + ψ S m G S R G S R A S ψ S m .
ψ S m G S R G S R ψ S m = j ψ S m G S R G S R ψ S j ψ S j A S ψ S m = ψ S m G S R G S R ψ S m ψ S m A S ψ S m ,
ψ S m G S R G S R ψ S j = ϕ R m ϕ R j = 0 , unless m = j ,
ψ S m G S R G S R ψ S m 0 ,
ψ S m A S ψ S m = 1 ,
ψ S m A S ψ S m 2 = 1 .
i ψ S i A S ψ S i 2 S A
M S A ,
ψ S m A S ψ S m = ψ S m CG S ψ S m .
C ψ S m = c S m * ϕ C m
ψ S m C = c S m ϕ C m ,
G S ψ S m = g S m ϕ G m
ψ S m A S ψ S m = c S m g S m ϕ C m ϕ G m ,
ψ S m A S ψ S m 2 = c S m 2 g S m 2 ϕ C m ϕ G m 2 .
c S m 2 g S m 2 ψ S m A S ψ S m 2 = 1 .
c S m 2 = ψ S m CC ψ S m = ψ S m C C ψ S m .
g S m 2 = ψ S m G S G S ψ S m .
m c S m 2 = N c Tr ( C C )
m g S m 2 = N G S Tr ( G S G S ) ,
m ψ S m A S ψ S m 2 c S max 2 m g S m 2 = c S max 2 N G S .
P = c S m 2 g S m 2 ¯ = N C N G S M 2 .
M N C N G S .
d 2 ϕ d z 2 + k o 2 ϕ = k o 2 η ( z , ω o ) ϕ ,
η ( z , ω o ) Δ ε ( z , ω o ) ε r o ,
N G S = Δ k Δ z R π k c 2 ( Δ z S ) 2 12
N C Δ k Δ z R π η r m s 2 Δ k Δ z R π η max 2 ,
η r m s 2 = 1 Δ z S Δ ω ω = ν o ( k c Δ k 2 ) ν o ( k c + Δ k 2 ) z = Δ z S 2 Δ z S 2 η ( z , ω ) 2 d z d ω
M N C N G S n t o t N o ,
N o = π 3 N S λ η r m s π 3 N S λ η max ,
n t o t = Δ k Δ z R π ,
N S λ = k c Δ z S 2 π .
M r e f t o t n t o t ( 1 + N o ) ,
M t r a n s t o t n t o t ( 2 + N o ) .
N b = Δ ω δ ω = Δ k δ k
n b = δ k Δ z R π
M r e f t o t N b ( 1 + N o )
M t r a n s t o t N b ( 2 + N o )
N t o t = N b ( 2 N b 1 )
N b 3 2 + N o 2 = 3 2 + π 2 3 N S λ η max ,
N b 1 + N o 2 = 1 + π 2 3 N S λ η max .
ψ S m G S R G S R ψ S m = f m + ψ S m G S R G S R ψ S m ψ S m A S ψ S m ;
1 = h m + ψ S m A S ψ S m ,
ψ S m A S ψ S m 1 + h m = 1 + h m ,
1 h m ψ S m A S ψ S m 1 + h m .
h m = ϕ R m G S R ψ I m ψ S m G S R G S R ψ S m u
( 1 u ) 2 ψ S m A S ψ S m 2 ( 1 + u ) 2 ,
M S A ( 1 u ) 2 ,
t max ( z S ) = 1 ν o ( Δ z R 2 + z S ) ,
t min ( z S ) = 1 ν o ( Δ z R 2 + z S ) ,
1 k o 2 d 2 ϕ d z 2 + ϕ = η ( z , ω o ) ϕ ,
G ω o ( z , z o ) = ( i k o 2 ) exp ( i k o z z o ) .
G δ ( z , t ; z o , t o ) i ν o 4 π k o exp { i [ ν o ( t t o ) z z o ] } d k o .
G ( z , t ; z o , t o ) ν o k c 2 π k c Δ k 2 k c + Δ k z sin { k o [ ν o ( t t o ) z z o ] } d k o .
G S R G S R = ν o 2 k c 2 Δ t R 4 π × k = Δ k 2 Δ k 2 cos { ( k c + k ) [ ( ν o t 1 z S 1 ) ( ν o t 2 z S 2 ) ] } d k .
z S = Δ z S 2 Δ z S 2 t S = 1 ν o ( Δ z R 2 + z S ) 1 ν o ( Δ z R 2 + z S ) d t S d z S .
z S = Δ z S 2 Δ z S 2 τ S = Δ z R 2 ν o Δ z R 2 ν o d τ S d z S ,
G S R G S R = ν o 2 k c 2 Δ t R 4 π × k = Δ k 2 Δ k 2 cos [ ( k c + k ) ν o ( τ S 1 τ S 2 ) ] d k .
S S R = m ψ S m G S R G S R ψ S m .
s m 2 ψ S m G S R G S R ψ S m > 0 .
n ψ A n G S R G S R ψ A n = S S R .
ψ p G S R G S R ψ p = G S R G S R ( z p , τ p ; z p , τ p ) .
S S R = p ψ p G S R G S R ψ p = ν 0 2 k c 2 Δ t R 4 π z S = Δ z S 2 Δ z S 2 τ S = Δ z R 2 ν o Δ z R 2 ν o k = Δ k 2 Δ k 2 d k d τ S d z S cos [ ( k c + k ) ν o ( τ s τ s ) ] = ν o k c 2 Δ t R Δ z S Δ z R Δ k 4 π .
ψ A ( z S , τ S ) = { ψ A s n ( z S , τ S ) = 2 ν o Δ z S Δ z R sin [ ( k c + 2 n π Δ z R ) ν o τ S ] ψ A c n ( z S , τ S ) = 2 ν o Δ z S Δ z R cos [ ( k c + 2 n π Δ z R ) ν o τ S ] ) ,
Δ k Δ z R 4 π n Δ k Δ z R 4 π .
μ c n = ψ A c n G S R G S R ψ A c n = ν o 2 k c 2 Δ t R 4 π 2 ν o Δ z S Δ z R z S 2 = Δ z S 2 Δ z S 2 τ S 2 = Δ z R 2 ν o Δ z R 2 ν o z S 1 = Δ z S 2 Δ z S 2 τ S 1 = Δ z R 2 ν o Δ z R 2 ν o k = Δ k 2 Δ k 2 d k d τ S 1 d z S 1 d τ S 2 d z S 2 cos [ ( k c + 2 m π Δ z R ) ν o τ S 2 ] cos [ ( k c + k ) ν o ( τ S 1 τ S 2 ) ] cos [ ( k c + 2 m π Δ z R ) ν o τ S 1 ] .
μ c n = ν o k c 2 Δ t R Δ z S 4
n t o t ψ A G S R G S R ψ A = Δ k Δ z R π ν o k c 2 Δ t R Δ z S 4 = S S R ,
N G S Tr ( G S G S ) = m γ m ,
G S G S ( z , t ; z o , t o ) ν o k c 2 π Δ k 2 Δ k 2 sin { ( k c + k ) [ ν o ( t t o ) z z o ] } d k .
I S = sin { k c [ ν o ( t t o ) z z o ] } × cos { k [ ν o ( t t o ) z z o ] } + cos { k c [ ν o ( t t o ) z z o ] } × sin { k [ ν o ( t t o ) z z o ] } .
I S = Θ ( z z o ) sin { k c [ ν o ( t t o ) ( z z o ) ] } cos { k [ ν o ( t t o ) ( z z o ) ] } + Θ ( z o z ) sin { k c [ ν o ( t t o ) ( z o z ) ] } cos { k [ ν o ( t t o ) ( z o z ) ] } ,
I S = Θ ( z z o ) sin [ k c ν o ( τ τ o ) ] cos [ k ν o ( τ τ o ) ] + Θ ( z o z ) sin [ k c ν o ( τ τ o ) + 2 k c ( z z o ) ] cos [ k ν o ( τ τ o ) + 2 k ( z z o ) ] .
G S G S ( z , τ ; z o , τ o ) k c ν o 2 π Θ ( z z o ) sin [ k c ν o ( τ τ o ) ] Δ k 2 Δ k 2 cos [ k ν o ( τ τ o ) ] d k .
G S G S ( z 2 , τ 2 ; z 1 , τ 1 ) = k c 2 ν o 2 ( 2 π ) 2 1 4 cos [ k c ν o ( τ 1 τ 2 ) ] Δ z S 2 Δ z S 2 Θ ( z z 2 ) Θ ( z z 1 ) d z I D ,
I D = τ = Δ z R 2 ν o Δ z R 2 ν o k 2 = Δ k 2 Δ k 2 k 1 = Δ k 2 Δ k 2 d k 1 d k 2 d τ { cos [ ( k 2 k 1 ) ν o τ ] cos [ k 1 ν o τ 1 k 2 ν o τ 2 ] + cos [ ( k 2 + k 1 ) ν o τ ] cos [ k 1 ν o τ 1 k 2 ν o τ 2 ] } .
τ = Δ z R 2 ν o Δ z R 2 ν o cos [ ( k 2 k 1 ) ν o τ ] d τ = 2 sin [ Δ z R 2 ( k 2 k 1 ) ] ( k 1 k 1 ) ν o 2 π ν o δ ( k 2 k 1 ) ,
I D 2 π ν o k = Δ k 2 Δ k 2 2 cos [ k ν o ( τ 1 τ 2 ) ] d k ,
G S G S ( z 2 , τ 2 ; z 1 , τ 1 ) = k c 2 ν o 4 π cos [ k c ν o ( τ 1 τ 2 ) ] Δ z S 2 Δ z S 2 Θ ( z z 2 ) Θ ( z z 1 ) d z k = Δ k 2 Δ k 2 cos [ k ν o ( τ 1 τ 2 ) ] d k .
γ c n = ψ A c n G S G S ψ A c n = k c 2 ν o 4 π 2 ν o Δ z S Δ z R I z I C ,
I z = z = Δ z S 2 Δ z S 2 z 1 = Δ z S 2 Δ z S 2 z 1 = Δ z S 2 Δ z S 2 d z 1 d z 2 d z Θ ( z z 2 ) Θ ( z z 1 ) ,
I C = τ 2 = Δ z R 2 ν o Δ z R 2 ν o τ 1 = Δ z R 2 ν o Δ z R 2 ν o k = Δ k 2 Δ k 2 d k d τ 1 d τ 2 cos [ ( k c + 2 π n ν o ) ν o τ 1 ] cos [ k c ν o ( τ 1 τ 2 ) ] cos [ k ν o ( τ 1 τ 2 ) ] cos [ ( k c + 2 π n ν o ) ν o τ 2 ] .
z 1 = Δ z S 2 Δ z S 2 Θ ( z z 1 ) d z 1 = Δ z S 2 z d z 1 = z + Δ z S 2 ,
I z = z = Δ z S 2 Δ z S 2 ( z + Δ z s 2 ) 2 d z = ( Δ z S ) 2 3 .
I C = π Δ z R 2 ν o 2 .
γ c n = ψ A c n G S G S ψ A c n = k c 2 ν o 4 π 2 ν o Δ z S Δ z R ( Δ z S ) 3 3 π Δ z R 2 ν o 2 = k c 2 ( Δ z S ) 2 12 .
N c Tr ( C C ) = m ψ S m C C ψ S m .
N C = p ψ A p C C ψ A p .
n ψ A n C C ψ A n = p ψ A p C C ψ A p + q ψ A q C C ψ A q = N C + q ψ A q C C ψ A q .
ψ C C ψ = j ψ C α j α j C ψ = j α j C ψ 2 ,
N c n ψ A n C C ψ A n .
ψ A c n C C ψ A c n = 2 ν o Δ z S Δ z R z = Δ z S 2 Δ z S 2 τ = Δ z R 2 ν o Δ z R 2 ν o d τ d z cos 2 [ ( k c + 2 n π Δ z R ) ν o τ ] η ( z , [ k c + 2 n π Δ z R ] ν o ) 2 1 Δ z S z = Δ z S 2 Δ z S 2 η [ z , ( k c + 2 n π Δ z R ) ν o ] 2 d z .
N C m ψ A m C C ψ A m = 2 n ψ A c n C C ψ A c n .
ψ A c n C C ψ A c n η max 2 ,
n = n min n max d n d ω ω min ω max d ω .
d ω d n = 2 π ν o Δ z R ,
ψ S q = ( n a n ψ A n ) + ψ e x t r a ,
ψ A n ψ e x t r a = 0 , for all n ,
S A = n ψ A n G S R G S R ψ A n = n , m , p ψ A n ψ S m ψ S m G S R G S R ψ S p ψ S p ψ A n ,
ψ S m G S R G S R ψ S p = s m * s p ϕ R m ϕ R p = s m * s p δ m p = s m 2 δ m p ,
S A = m s m 2 ψ S m n ψ A n ψ A n ψ S m .
n ψ A n ψ A n ψ S q = n a n ψ A n = ψ S q ψ e x t r a ,
ψ S q ψ e x t r a = [ ( n a n * ψ A n ) + ψ e x t r a ] ψ e x t r a = ψ e x t r a ψ e x t r a > 0 .
ψ S q n ψ A n ψ A n ψ S q = ψ S q ψ S q ψ e x t r a ψ e x t r a = 1 ψ e x t r a ψ e x t r a < 1 .
ψ S j n ψ A n ψ A n ψ S j = n b n 2 = 1 .
ψ A n G S G S ψ A n = k c 2 ( Δ z S ) 2 = β .
ψ A m G S G S ψ A n = 0 m n .
ψ P q ψ P q = n , p ψ A p a q p * a q n ψ A n = n a q n 2 = 1 .
ψ P q G S G S ψ P q = n , p a q p * a q n ψ A p G S G S ψ A n = n a q n 2 ψ A n G S G S ψ A n = β n a q n 2 = β .

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