Abstract

We demonstrate a new technique of coherent pulse stacking (CPS) amplification to overcome limits on achievable pulse energies from optical amplifiers. CPS uses reflecting resonators without active cavity-dumpers to transform a sequence of phase- and amplitude-modulated optical pulses into a single output pulse. Experimental validation with a single reflecting resonator demonstrates a near-theoretical stacked peak-power enhancement factor of ~2.5 with 92% and 97.4% efficiency for amplified nanosecond and femtosecond pulses. We also show theoretically that large numbers of equal-amplitude pulses can be stacked using sequences of multiple reflecting resonators, thus providing a new path for generating very high-energy pulses from ultrashort pulse fiber amplifier systems.

© 2015 Optical Society of America

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References

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  1. D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
    [Crossref]
  2. F. Stutzki, F. Jansen, A. Liem, C. Jauregui, J. Limpert, and A. Tünnermann, “26 mJ, 130 W Q-switched fiber-laser system with near-diffraction-limited beam quality,” Opt. Lett. 37(6), 1073–1075 (2012).
    [Crossref] [PubMed]
  3. F. Röser, T. Eidam, J. Rothhardt, O. Schmidt, D. N. Schimpf, J. Limpert, and A. Tünnermann, “Millijoule pulse energy high repetition rate femtosecond fiber chirped-pulse amplification system,” Opt. Lett. 32(24), 3495–3497 (2007).
    [Crossref] [PubMed]
  4. M. Kienel, A. Klenke, T. Eidam, S. Hädrich, J. Limpert, and A. Tünnermann, “Energy scaling of femtosecond amplifiers using actively controlled divided-pulse amplification,” Opt. Lett. 39(4), 1049–1052 (2014).
    [Crossref] [PubMed]
  5. Y. Zaouter, F. Guichard, L. Daniault, M. Hanna, F. Morin, C. Hönninger, E. Mottay, F. Druon, and P. Georges, “Femtosecond fiber chirped- and divided-pulse amplification system,” Opt. Lett. 38(2), 106–108 (2013).
    [Crossref] [PubMed]
  6. R. J. Jones and J. Ye, “Femtosecond pulse amplification by coherent addition in a passive optical cavity,” Opt. Lett. 27(20), 1848–1850 (2002).
    [Crossref] [PubMed]
  7. S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
    [Crossref]
  8. A. E. Siegman, Lasers (University Science Books, 1986), Chap. 11.
  9. X. Ma, C. Zhu, I. N. Hu, A. Kaplan, and A. Galvanauskas, “Single-mode chirally-coupled-core fibers with larger than 50 µm diameter cores,” Opt. Express 22(8), 9206–9219 (2014).
    [Crossref] [PubMed]
  10. J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005), Chap. 9.
  11. L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
    [Crossref]
  12. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
    [Crossref]
  13. S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D Appl. Phys. 35(8), R43–R59 (2002).
    [Crossref]
  14. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 9.
  15. F. Kuyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6(1), 87–93 (1988).
    [Crossref]

2014 (3)

M. Kienel, A. Klenke, T. Eidam, S. Hädrich, J. Limpert, and A. Tünnermann, “Energy scaling of femtosecond amplifiers using actively controlled divided-pulse amplification,” Opt. Lett. 39(4), 1049–1052 (2014).
[Crossref] [PubMed]

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

X. Ma, C. Zhu, I. N. Hu, A. Kaplan, and A. Galvanauskas, “Single-mode chirally-coupled-core fibers with larger than 50 µm diameter cores,” Opt. Express 22(8), 9206–9219 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2007 (1)

2002 (2)

R. J. Jones and J. Ye, “Femtosecond pulse amplification by coherent addition in a passive optical cavity,” Opt. Lett. 27(20), 1848–1850 (2002).
[Crossref] [PubMed]

S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D Appl. Phys. 35(8), R43–R59 (2002).
[Crossref]

2001 (1)

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

1988 (1)

F. Kuyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6(1), 87–93 (1988).
[Crossref]

1985 (1)

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[Crossref]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

Bergquist, J. C.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Breitkopf, S.

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Carstens, H.

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Cundiff, S. T.

S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D Appl. Phys. 35(8), R43–R59 (2002).
[Crossref]

Curtis, E. A.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Daniault, L.

Diddams, S. A.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

Drullinger, R. E.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Druon, F.

Eidam, T.

Fill, E.

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

Galvanauskas, A.

Georges, P.

Guichard, F.

Hädrich, S.

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

Hanna, M.

Hollberg, L.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Holzberger, S.

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Hönninger, C.

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

Hu, I. N.

Itano, W. M.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Ivanov, E. N.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Jansen, F.

Jauregui, C.

Jones, R. J.

Kaplan, A.

Kienel, M.

Klenke, A.

M. Kienel, A. Klenke, T. Eidam, S. Hädrich, J. Limpert, and A. Tünnermann, “Energy scaling of femtosecond amplifiers using actively controlled divided-pulse amplification,” Opt. Lett. 39(4), 1049–1052 (2014).
[Crossref] [PubMed]

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

Krausz, F.

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Kuyama, F.

F. Kuyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6(1), 87–93 (1988).
[Crossref]

Liem, A.

Limpert, J.

Ma, X.

Morin, F.

Mottay, E.

Mourou, G.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[Crossref]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

Oates, C. W.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Oga, K.

F. Kuyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6(1), 87–93 (1988).
[Crossref]

Pupeza, I.

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Rafac, R. J.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Robinson, H. G.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Röser, F.

Rothhardt, J.

Schimpf, D. N.

Schmidt, O.

Schreiber, T.

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Strickland, D.

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[Crossref]

Stutzki, F.

Tünnermann, A.

Udem, T.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

von Grafenstein, L.

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

Wineland, D. J.

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

Ye, J.

Zaouter, Y.

Zhu, C.

Appl. Phys. B (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31(2), 97–105 (1983).
[Crossref]

IEEE J. Quantum Electron. (1)

L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37(12), 1502–1513 (2001).
[Crossref]

J. Lightwave Technol. (1)

F. Kuyama and K. Oga, “Frequency chirping in external modulators,” J. Lightwave Technol. 6(1), 87–93 (1988).
[Crossref]

J. Phys. D Appl. Phys. (1)

S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D Appl. Phys. 35(8), R43–R59 (2002).
[Crossref]

Light Sci. Appl. (1)

S. Breitkopf, T. Eidam, A. Klenke, L. von Grafenstein, H. Carstens, S. Holzberger, E. Fill, T. Schreiber, F. Krausz, A. Tünnermann, I. Pupeza, and J. Limpert, “A concept for multiterawatt fibre lasers based on coherent pulse stacking in passive cavities,” Light Sci. Appl. 3(10), e211 (2014).
[Crossref]

Opt. Commun. (1)

D. Strickland and G. Mourou, “Compression of amplified chirped optical pulses,” Opt. Commun. 56(3), 219–221 (1985).
[Crossref]

Opt. Express (1)

Opt. Lett. (5)

Other (3)

J. Ye and S. T. Cundiff, Femtosecond Optical Frequency Comb: Principle, Operation, and Applications (Springer, 2005), Chap. 9.

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 11.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1984), Chap. 9.

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Figures (10)

Fig. 1
Fig. 1 Coherent pulse stacking in a traveling-wave Gires-Tournois interferometer, showing the stacking-signal pulse sequence at its input and the stacked solitary pulse at its output.
Fig. 2
Fig. 2 Amplitudes of all incident, in-cavity circulating, and output pulses in a conceptualized 2-mirror GTI cavity.
Fig. 3
Fig. 3 Experimental coherent pulse stacking amplification system. fs EXP: femtosecond experiment; ns EXP: nanosecond experiment; BPF: band-pass filter; HWP: half-wave plate; SMF Amp.: single-mode fiber amplifier; LMA Amp.: large-mode-area fiber amplifier; CCC Amp.: chirally-coupled-core fiber amplifier.
Fig. 4
Fig. 4 Peak-power enhancement η and satellite-pulse contrast γ vs the phase error of the input pulse burst.
Fig. 5
Fig. 5 Nanosecond experiment: the input pulse burst and output stacked pulse.
Fig. 6
Fig. 6 Femtosecond experiment: the input pulse burst and output stacked pulse.
Fig. 7
Fig. 7 Femtosecond experiment: normalized autocorrelation signals of the output stacked pulse, and the input stacking pulse of the GTI stacker.
Fig. 8
Fig. 8 (a) m cascaded equal-roundtrip GTI cavities; (b) m x m multiplexed different-roundtrip GTI cavities.
Fig. 9
Fig. 9 The calculated intensities and phases of the input pulse burst and the output pulse intensities of 4 cascaded cavities (cavity parameters given in the text).
Fig. 10
Fig. 10 The calculated intensities and phases of the input pulse burst and the output pulse intensities of 4 x 4 multiplexed cavities (cavity parameters given in the text).

Equations (30)

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[ T ]=[ 1 0 0 α e iδ ].
[ S ]=[ r it it r ].
p ˜ in (t)= j p ˜ in j (t) p ˜ out (t)= p ˜ s (t) e i( ω 0 t+ ϕ 0 ) .
p ˜ in (t)= n=0 p ˜ in n (t) = n=0 A ˜ in n p ˜ s (t+nΔT) e i ω 0 (t+nΔT) ,
[ A ˜ out n A ˜ cav. n1 ]=[ 1 0 0 α e iδ ][ r it it r ][ A ˜ in n A ˜ cav. n ],
[ 1 0 ]=[ 1 0 0 α e iδ ][ r it it r ][ A ˜ in 0 A ˜ cav. 0 ], for n= 0,
[ 0 A ˜ cav. n1 ]=[ 1 0 0 α e iδ ][ r it it r ][ A ˜ in n A ˜ cav. n ], for n= 1, 2, .
{ A ˜ in 0 =r A ˜ in n =(1 r 2 ) r n1 e inδ α n ,forn=1,2,3,... A ˜ cav. n =it ( r α e iδ ) n ,forn=0,1,2,...
{ B in 0 =R B in n = (1R) 2 R n1 ( α 2 ) n ,forn=1,2,3,...
η= B out 0 / { B in n } max =1/ { B in n } max ,
B in 0 = B in 1 R opt = (1 R opt ) 2 α 2 R opt 2 (2+ α 2 ) R opt +1=0.
R opt = (2+ α 2 ) (2+ α 2 ) 2 4 2 .
η max = 1 R opt = α 2 (1 R opt ) 2 .
χ N 1 = 1R(2 α 2 ) α 2 R + ( R α 2 ) N1 α 2 (2R)1 α 2 R .
χ exp = χ N ·[1+ ( R α 2 ) N1 ].
[ A ˜ k out n A ˜ k cav n1 ]=[ 1 0 0 α k e i δ k ][ r k i t k i t k r k ][ A ˜ k1 out n A ˜ k cav n ].
[ 1 0 ]=[ 1 0 0 α m e i δ m ][ r m i t m i t m r m ][ A ˜ m1 out 0 A ˜ m cav 0 ],forn=0andk=m
[ A ˜ k out 0 0 ]=[ 1 0 0 α k e i δ k ][ r k i t k i t k r k ][ A ˜ k1 out 0 A ˜ k cav 0 ],forn=0andk=2,3,...m-1
[ A ˜ 1 out n 0 ]=[ 1 0 0 α 1 e iδ1 ][ r 1 i t 1 i t 1 r 1 ][ A ˜ 1 in 0 A ˜ 1 cav 0 ],forn=0andk=1
[ 0 A ˜ m cav n1 ]=[ 1 0 0 α m e i δ m ][ r m i t m i t m r m ][ A ˜ m1 out n A ˜ m cav n ],forn=1,2,...andk=m
[ A ˜ k out n A ˜ k cav n1 ]=[ 1 0 0 α k e i δ k ][ r k i t k i t k r k ][ A ˜ k1 out n A ˜ k cav n ],forn=1,2,...andk=2,3,...m-1
[ A ˜ 1 out n A ˜ 1 cav n1 ]=[ 1 0 0 α 1 e i δ 1 ][ r 1 i t 1 i t 1 r 1 ][ A ˜ 1 in n A ˜ 1 cav n ],forn=1,2,...andk=1
[ a ˜ out N1 A ˜ cav. N2 ]=[ 1 0 0 α e iδ ][ r it it r ][ a ˜ in N1 0 ].
{ a ˜ in N1 = A ˜ in N1 1R a ˜ out N1 = ( r α e iδ ) N1
ε stck. = B out 0 + p s 2 (t)dt = B out 0 ε S = ε S .
ε in = + | p ˜ in (t) | 2 dt = + n=0 | A ˜ in n p ˜ S (t+nΔT) | 2 dt= n=0 | A ˜ in n | 2 ε S = n=0 B in n ε S .
χ = ε stck. ε in = ( n=0 B in n ) 1 .
χ N = ε stck. ε in = ( n=0 N2 B in n + b N1 ) 1 = ( n=0 N2 B in n + B N1 (1R) 2 ) 1 .
χ = ( B 0 + n=1 B in n ) 1 = ( R+ (1R) 2 α 2 R ) 1 = α 2 R 1R(2 α 2 ) ,
χ N 1 = χ 1 + ( R α 2 ) N1 α 2 (2R)1 α 2 R .

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