Abstract

Different approaches to qualify coherent beam combining performance in tiled and filled aperture combining experiments are discussed. The dependence of the combining efficiency on different misalignments and the number of combining pulses has been investigated and analytical equations for its evaluation have been obtained. The results provide design guidelines for laser systems based on coherent beam combining and allow comparison of experiments performed in different combining approaches. The analysis shows that there are good prospects to scale achieved peak intensity.

© 2015 Optical Society of America

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2015 (1)

V. E. Leshchenko, V. A. Vasiliev, N. L. Kvashnin, and E. V. Pestryakov, “Coherent combining of relativistic-intensity femtosecond laser pulses,” Appl. Phys. B 118(4), 511–516 (2015).
[Crossref]

2014 (2)

V. E. Leshchenko, V. I. Trunov, S. A. Frolov, E. V. Pestryakov, V. A. Vasiliev, N. L. Kvashnin, and S. N. Bagayev, “Coherent combining of multimillijoule parametric amplified femtosecond pulses,” Laser Phys. Lett. 11(9), 095301 (2014).
[Crossref]

S. N. Bagayev, V. E. Leshchenko, V. I. Trunov, E. V. Pestryakov, and S. A. Frolov, “Coherent combining of femtosecond pulses parametrically amplified in BBO crystals,” Opt. Lett. 39(6), 1517–1520 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (2)

2011 (7)

Y. Ma, X. Wang, J. Leng, H. Xiao, X. Dong, J. Zhu, W. Du, P. Zhou, X. Xu, L. Si, Z. Liu, and Y. Zhao, “Coherent beam combination of 1.08 kW fiber amplifier array using single frequency dithering technique,” Opt. Lett. 36(6), 951–953 (2011).
[Crossref] [PubMed]

S. M. Redmond, K. J. Creedon, J. E. Kansky, S. J. Augst, L. J. Missaggia, M. K. Connors, R. K. Huang, B. Chann, T. Y. Fan, G. W. Turner, and A. Sanchez-Rubio, “Active coherent beam combining of diode lasers,” Opt. Lett. 36(6), 999–1001 (2011).
[PubMed]

C. X. Yu, S. J. Augst, S. M. Redmond, K. C. Goldizen, D. V. Murphy, A. Sanchez, and T. Y. Fan, “Coherent combining of a 4 kW, eight-element fiber amplifier array,” Opt. Lett. 36(14), 2686–2688 (2011).
[Crossref] [PubMed]

J. Bourderionnet, C. Bellanger, J. Primot, and A. Brignon, “Collective coherent phase combining of 64 fibers,” Opt. Express 19(18), 17053–17058 (2011).
[PubMed]

A. Klenke, E. Seise, J. Limpert, and A. Tünnermann, “Basic considerations on coherent combining of ultrashort laser pulses,” Opt. Express 19(25), 25379–25387 (2011).
[Crossref] [PubMed]

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

G. Genoud, F. Wojda, M. Burza, A. Persson, and C.-G. Wahlström, “Active control of the pointing of a multi-terawatt laser,” Rev. Sci. Instrum. 82(3), 033102 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
[Crossref]

2007 (2)

2006 (2)

Y. Akahane, Ma. Jinglong, Y. Fukuda, M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, “Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2,” Rev. Sci. Instrum. 77, 023102 (2006).

T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D. Pilkington, J. Spring, D. J. Nelson, and C. A. Lu, “First experimental demonstration of self-synchronous phase locking of an optical array,” Opt. Express 14(25), 12015–12021 (2006).
[Crossref] [PubMed]

2005 (1)

2000 (1)

G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70(1), 1–9 (2000).
[Crossref]

Aflatouni, F.

Akahane, Y.

Y. Akahane, Ma. Jinglong, Y. Fukuda, M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, “Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2,” Rev. Sci. Instrum. 77, 023102 (2006).

Akturk, S.

Aoyoma, M.

Y. Akahane, Ma. Jinglong, Y. Fukuda, M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, “Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2,” Rev. Sci. Instrum. 77, 023102 (2006).

Augst, S. J.

Bagayev, S. N.

S. N. Bagayev, V. E. Leshchenko, V. I. Trunov, E. V. Pestryakov, and S. A. Frolov, “Coherent combining of femtosecond pulses parametrically amplified in BBO crystals,” Opt. Lett. 39(6), 1517–1520 (2014).
[Crossref] [PubMed]

V. E. Leshchenko, V. I. Trunov, S. A. Frolov, E. V. Pestryakov, V. A. Vasiliev, N. L. Kvashnin, and S. N. Bagayev, “Coherent combining of multimillijoule parametric amplified femtosecond pulses,” Laser Phys. Lett. 11(9), 095301 (2014).
[Crossref]

Baker, J. T.

Bellanger, C.

Benham, V.

Bhardwaj, S.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Birge, J. R.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Bourderionnet, J.

Breitkopf, S.

Brignon, A.

Brocklesby, B.

G. Mourou, B. Brocklesby, T. Tajima, and J. Limpert, “The future is fiber accelerators,” Nat. Photonics 7(4), 258–261 (2013).
[Crossref]

Burza, M.

G. Genoud, F. Wojda, M. Burza, A. Persson, and C.-G. Wahlström, “Active control of the pointing of a multi-terawatt laser,” Rev. Sci. Instrum. 82(3), 033102 (2011).
[Crossref] [PubMed]

Cerullo, G.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Chang, W. Z.

Chann, B.

Chen, L.-J.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Cirmi, G.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Connors, M. K.

Creedon, K. J.

Culpepper, M. A.

Dong, X.

Du, W.

Eggleton, B. J.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Eidam, T.

Eikema, K. S. E.

Fan, T. Y.

Frolov, S. A.

S. N. Bagayev, V. E. Leshchenko, V. I. Trunov, E. V. Pestryakov, and S. A. Frolov, “Coherent combining of femtosecond pulses parametrically amplified in BBO crystals,” Opt. Lett. 39(6), 1517–1520 (2014).
[Crossref] [PubMed]

V. E. Leshchenko, V. I. Trunov, S. A. Frolov, E. V. Pestryakov, V. A. Vasiliev, N. L. Kvashnin, and S. N. Bagayev, “Coherent combining of multimillijoule parametric amplified femtosecond pulses,” Laser Phys. Lett. 11(9), 095301 (2014).
[Crossref]

Fukuda, Y.

Y. Akahane, Ma. Jinglong, Y. Fukuda, M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, “Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2,” Rev. Sci. Instrum. 77, 023102 (2006).

Gabolde, P.

Galvanauskas, A.

Genoud, G.

G. Genoud, F. Wojda, M. Burza, A. Persson, and C.-G. Wahlström, “Active control of the pointing of a multi-terawatt laser,” Rev. Sci. Instrum. 82(3), 033102 (2011).
[Crossref] [PubMed]

Goldizen, K. C.

Goodno, G. D.

Gottschall, T.

Gu, X.

Hädrich, S.

Hashemi, H.

Hogervorst, W.

Hong, K.-H.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Huang, R. K.

Huang, S.-W.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Ivanov, M.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
[Crossref]

Jinglong, Ma.

Y. Akahane, Ma. Jinglong, Y. Fukuda, M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, “Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2,” Rev. Sci. Instrum. 77, 023102 (2006).

Kandula, D. Z.

Kansky, J. E.

Kärtner, F. X.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Kasper, A.

G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70(1), 1–9 (2000).
[Crossref]

Kewitsch, A.

Kienel, M.

Kiriyama, H.

Y. Akahane, Ma. Jinglong, Y. Fukuda, M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, “Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2,” Rev. Sci. Instrum. 77, 023102 (2006).

Klenke, A.

Krausz, F.

F. Krausz and M. Ivanov, “Attosecond physics,” Rev. Mod. Phys. 81(1), 163–234 (2009).
[Crossref]

Kudryashov, A. V.

Y. Akahane, Ma. Jinglong, Y. Fukuda, M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, “Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2,” Rev. Sci. Instrum. 77, 023102 (2006).

Kvashnin, N. L.

V. E. Leshchenko, V. A. Vasiliev, N. L. Kvashnin, and E. V. Pestryakov, “Coherent combining of relativistic-intensity femtosecond laser pulses,” Appl. Phys. B 118(4), 511–516 (2015).
[Crossref]

V. E. Leshchenko, V. I. Trunov, S. A. Frolov, E. V. Pestryakov, V. A. Vasiliev, N. L. Kvashnin, and S. N. Bagayev, “Coherent combining of multimillijoule parametric amplified femtosecond pulses,” Laser Phys. Lett. 11(9), 095301 (2014).
[Crossref]

Leng, J.

Leshchenko, V. E.

V. E. Leshchenko, V. A. Vasiliev, N. L. Kvashnin, and E. V. Pestryakov, “Coherent combining of relativistic-intensity femtosecond laser pulses,” Appl. Phys. B 118(4), 511–516 (2015).
[Crossref]

V. E. Leshchenko, V. I. Trunov, S. A. Frolov, E. V. Pestryakov, V. A. Vasiliev, N. L. Kvashnin, and S. N. Bagayev, “Coherent combining of multimillijoule parametric amplified femtosecond pulses,” Laser Phys. Lett. 11(9), 095301 (2014).
[Crossref]

S. N. Bagayev, V. E. Leshchenko, V. I. Trunov, E. V. Pestryakov, and S. A. Frolov, “Coherent combining of femtosecond pulses parametrically amplified in BBO crystals,” Opt. Lett. 39(6), 1517–1520 (2014).
[Crossref] [PubMed]

Li, E.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Liang, W.

Limpert, J.

Liu, Z.

Lu, C. A.

Ma, P.

Ma, Y.

Missaggia, L. J.

Moses, J.

S.-W. Huang, G. Cirmi, J. Moses, K.-H. Hong, S. Bhardwaj, J. R. Birge, L.-J. Chen, E. Li, B. J. Eggleton, G. Cerullo, and F. X. Kärtner, “High-energy pulse synthesis with sub-cycle waveform control for strong-field physics,” Nat. Photonics 5(8), 475–479 (2011).
[Crossref]

Mourou, G.

G. Mourou, B. Brocklesby, T. Tajima, and J. Limpert, “The future is fiber accelerators,” Nat. Photonics 7(4), 258–261 (2013).
[Crossref]

Murphy, D. V.

Nelson, D. J.

Persson, A.

G. Genoud, F. Wojda, M. Burza, A. Persson, and C.-G. Wahlström, “Active control of the pointing of a multi-terawatt laser,” Rev. Sci. Instrum. 82(3), 033102 (2011).
[Crossref] [PubMed]

Pestryakov, E. V.

V. E. Leshchenko, V. A. Vasiliev, N. L. Kvashnin, and E. V. Pestryakov, “Coherent combining of relativistic-intensity femtosecond laser pulses,” Appl. Phys. B 118(4), 511–516 (2015).
[Crossref]

V. E. Leshchenko, V. I. Trunov, S. A. Frolov, E. V. Pestryakov, V. A. Vasiliev, N. L. Kvashnin, and S. N. Bagayev, “Coherent combining of multimillijoule parametric amplified femtosecond pulses,” Laser Phys. Lett. 11(9), 095301 (2014).
[Crossref]

S. N. Bagayev, V. E. Leshchenko, V. I. Trunov, E. V. Pestryakov, and S. A. Frolov, “Coherent combining of femtosecond pulses parametrically amplified in BBO crystals,” Opt. Lett. 39(6), 1517–1520 (2014).
[Crossref] [PubMed]

Pilkington, D.

Pretzler, G.

G. Pretzler, A. Kasper, and K. J. Witte, “Angular chirp and tilted light pulses in CPA lasers,” Appl. Phys. B 70(1), 1–9 (2000).
[Crossref]

Primot, J.

Rakuljic, G.

Redmond, S. M.

Renault, A.

Rothenberg, J. E.

Rothhardt, J.

Sanchez, A.

Sanchez, A. D.

Sanchez-Rubio, A.

Satyan, N.

Seise, E.

Shay, T. M.

Sheldakova, J. V.

Y. Akahane, Ma. Jinglong, Y. Fukuda, M. Aoyoma, H. Kiriyama, J. V. Sheldakova, A. V. Kudryashov, and K. Yamakawa, “Characterization of wave-front corrected 100 TW, 10 Hz laser pulses with peak intensities greater than 1020 W/cm2,” Rev. Sci. Instrum. 77, 023102 (2006).

Shih, C.-C.

Si, L.

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Figures (19)

Fig. 1
Fig. 1 General schemes of filled (a) and tiled (b) aperture combining.

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Fig. 2
Fig. 2 Analytically calculated coherent combining efficiency in terms of peak intensity (ηI): (a) as a function of only phase instability (σφ); (b) of only optical path lengths mismatch (σL); and of both parameters for 2 (c), 8 (d) and 100 (e) channels. The white and black lines define η = 0.95 and η = 0.9, respectively.

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Fig. 3
Fig. 3 The profile of a combined beam in tiled aperture combining of two beams with phase difference equal to 0 (a) and π (b).

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Fig. 4
Fig. 4 Exact (numerically calculated) coherent combining efficiency in terms of peak intensity (ηI): (a) as a function of only phase instability (σφ); (b) of only optical path lengths mismatch (σL); and of both parameters for 2 (c), 8 (d) and 100 (e) channels. The white and black lines define η = 0.95 and η = 0.9, respectively.

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Fig. 5
Fig. 5 Coherent combining efficiency in terms of peak fluency (ηF) and energy (ηW): (a) as a function of only phase instability (σφ), (b) of only optical path lengths mismatch (σL), and of both parameters for 2 (c), 8 (d) and 100 (e) channels. The white and black lines define η = 0.95 and η = 0.9, respectively.

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Fig. 6
Fig. 6 Principal scheme of the stabilization system to achieve zero ceo-phase in the combined beam.

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Fig. 7
Fig. 7 The dependence of the coherent combining efficiency in terms of peak intensity (ηI, blue lines), and in terms of peak fluency and energy (ηF and ηW, green lines) on dispersion compensation accuracy for second (a), third (b), fourth (c) and fifth (d) dispersion orders. (blue lines in (a) are analytical results (Eq. (26), the other results are obtained numerically)

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Fig. 8
Fig. 8 The dependence of the coherent combining efficiency in terms of peak intensity (ηI, blue lines), and in terms of peak fluency and energy (ηF and ηW, green lines) on dispersion compensation accuracy for second (a), third (b) and fourth (c) dispersion orders.

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Fig. 9
Fig. 9 The dependence of the coherent combining efficiency on beam pointing instability. (a) In terms of peak intensity (ηI) and peak fluency (ηF). (b) In terms of energy (ηW).

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Fig. 10
Fig. 10 The dependence of the coherent combining efficiency on pointing instability for different beam profiles with the aperture diameter of 10 mm, N = 100. (a) In terms of peak intensity (ηI) and peak fluency (ηF). (b) In terms of energy (ηW).

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Fig. 11
Fig. 11 The dependence of the combining efficiency in terms of peak intensity and fluency (a), and energy (b) on beam aperture diameter for different beam profiles.

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Fig. 12
Fig. 12 The dependence of the coherent combining efficiency on angular chirp compensation accuracy (σθ). (a) In terms of peak intensity (ηI,). (b) In terms of peak fluency (ηF). (c) In terms of energy (ηW). (Ψ is the FWHM beam divergence which for a Gaussian beam equal to Ψ = 4ln(2)/(k0D0)).

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Fig. 13
Fig. 13 The dependence of the coherent combining efficiency in terms of peak intensity (ηI) and peak fluency (ηF) on rms value of wave front distortions for Gaussian profile for spherical aberration (a) and astigmatism (b).

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Fig. 14
Fig. 14 The dependence of the coherent combining efficiency in terms of peak intensity (ηI) and peak fluency (ηF) on rms aberrations in beam aperture for Gaussian (a), super-Gaussian (b) and top-hat (c) profiles.

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Fig. 15
Fig. 15 The dependence of the coherent combining efficiency in terms of energy (ηW) on rms aberration in beam aperture for Gaussian (a), super-Gaussian (b) and top-hat (c) profiles.

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Fig. 16
Fig. 16 Coherent combining efficiency dependence on: (a) polarization error under fixed number of channels; (b) the number of channels under fixed polarization error.

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Fig. 17
Fig. 17 The dependence of the coherent combining efficiency in filled aperture combining (ηfilled) on spatial chirp.

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Fig. 18
Fig. 18 The relation between the efficiency of the coherent combining of two (N = 2) and large (N>>1) number of beams for tiled (a) and filled (b) aperture combining. Gray filled aria shows the range of possible values.

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Fig. 19
Fig. 19 (a) The relation between the efficiency of the coherent beam combining in terms of peak fluency (ηF) and energy(ηW). (b) The relation between the efficiency of the coherent beam combining in terms of peak fluency and peak intensity (ηI). Gray filled aria shows the range of possible values.

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Tables (1)

Tables Icon

Table 1 Coherent combining loss for each misalignments discussed in the paper and tolerances for each effect individually under 10 fs pulses combining (τ is the FWHM pulse duration; Ψ is the FWHM beam divergence).

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Equations (73)

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η filled W = W Σ W Σ max = W Σ ∑ n=1 N W n ,
FOM= W ∑ − ∑ n=1 N W secondaryn W ∑ + ∑ n=1 N W secondaryn ,
FOM=2 η filled W −1.
∑ n=1 N W n = W ∑ + ∑ n=1 N W secondaryn .
η I = I Σ I Σ max .
η F = F Σ F Σ max .
IFC= F max − F min F max + F min ,
IFC=2 η F −1.
E(t,δ T n ,δ φ n )= I 0 exp[ −2ln(2) (t−δ T n ) 2 τ 2 −i ω 0 t+iδ φ n ],
E(ω,δ T n ,δ φ n )= I 0 τ 4ln(2) exp[ − (ω− ω 0 ) 2 τ 2 8ln(2) +iδ T n (ω− ω 0 )+iδ φ n ],
I ¯ Σ peak ( N, σ T , σ φ )= ∫ | ∑ n=1 N E(0,δ T n ,δ φ n ) | 2 e − ∑ n=1 N δ T n 2 2 σ T 2 ( 2π σ T ) N dδ T 1 ×...×dδ T N e − ∑ n=1 N δ φ n 2 2 σ φ 2 ( 2π σ φ ) N dδ φ 1 ×...×dδ φ N = = I 0 ( N 1+ 8ln(2) σ T 2 τ 2 + N(N−1) 1+ 4ln(2) σ T 2 τ 2 exp[ − σ φ 2 ] ).
I ¯ Σ peak ( N, σ L , σ φ )= I 0 ( N 1+ 8ln(2) σ L 2 c 2 τ 2 + N(N−1) 1+ 4ln(2) σ L 2 c 2 τ 2 exp[ − σ φ 2 ] ).
I Σ_max peak = N 2 I 0 .
η I ( N, σ L , σ φ )= 1 N 2 ( N 1+ 8ln(2) σ L 2 c 2 τ 2 + N(N−1) 1+ 4ln(2) σ L 2 c 2 τ 2 exp[ − σ φ 2 ] ).
η N>>1, σ L cτ <<1, σ φ <<1 I ( σ L , σ φ )=1− σ φ 2 − 4ln(2) σ L 2 c 2 τ 2 .
I ¯ Σ peak = 1 M ∑ m=1 M Max [ | ∑ n=1 N E(x,t,δ T n m ,δ φ n m ) | 2 ],
W Σ (N,δ T 1 ,...,δ T N ,δ φ 1 ,...,δ φ N )∝ F Σ (x=0)= ∫ −∞ ∞ ∑ n=1 N ∑ m=1 N E(t,δ T n ,δ φ n ) E * (t,δ T m ,δ φ m )dt= = I 0 π 4ln(2) τ ∑ n=1 N ∑ m=1 N exp[ i( δ φ n −δ φ m ) ]exp[ −4ln(2) (δ T n −δ T m ) 2 4 τ 2 ].
W Σ max = N 2 I 0 π 4ln(2) τ.
W ¯ Σ ( N, σ T , σ φ )= ∫ W Σ (δ T 1 ,...,δ T N ,δ φ 1 ,...,δ φ N ) e − ∑ n=1 N δ T n 2 2 σ t 2 ( 2π σ T ) N dδ T 1 ×...×dδ T N × × e − ∑ n=1 N δ φ n 2 2 σ φ 2 ( 2π σ φ ) N dδ φ 1 ×...×dδ φ N = I 0 π 4ln(2) τ ( N+ N( N−1 ) 1+4ln(2) σ T 2 τ 2 exp[ − σ φ 2 ] )
η filled W ( N, σ L , σ φ )= η F ( N, σ L , σ φ )= 1 N 2 ( N+ N( N−1 ) 1+4ln(2) σ L 2 c 2 τ 0 2 exp[ − σ φ 2 ] ).
η filled W N>>1, σ L c τ 0 <<1, σ φ <<1 ( σ L , σ φ )= η N>>1, σ L c τ 0 <<1, σ φ <<1 F ( σ L , σ φ )=1− σ φ 2 − 2ln(2) σ L 2 c 2 τ 0 2 .
σ L 2 → σ L 2 + σ L_ceo 2 .
I peak (δ k 2_n )= I 0 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 ,
I ¯ Σ peak (N, σ k 2 )= ∫ | ∑ n=1 N I peak (δ k 2_n ) | 2 e − ∑ n=1 N δ k 2_n 2 2 σ k 2 2 ( 2π σ k 2 ) N dδ k 2_1 ×...×dδ k 2_N = = I 0 ( N τ 0 2 2π 4ln(2) σ k 2 exp[ τ 0 4 64 ln 2 (2) σ k 2 2 ] K 0 [ τ 0 4 64 ln 2 (2) σ k 2 2 ]+ +N(N−1) τ 0 4 32 ln 2 (2) σ k 2 2 U 2 [ 1 2 , 5 4 , τ 0 4 32 ln 2 (2) σ k 2 2 ] ),
U[a,b,z]= 1 Γ(a) ∫ 0 ∞ e −zt t a−1 (1+t) b−a−1 dt .
η I (N, σ k 2 )= 1 N 2 ( N τ 0 2 2π 4ln(2) σ k 2 exp[ τ 0 4 64 ln 2 (2) σ k 2 2 ] K 0 [ τ 0 4 64 ln 2 (2) σ k 2 2 ]+ +N(N−1) τ 0 4 32 ln 2 (2) σ k 2 2 U 2 [ 1 2 , 5 4 , τ 0 4 32 ln 2 (2) σ k 2 2 ] ).
η N>>1, σ k 2 τ 0 2 <<1 I ( σ k 2 )=1− 1 2 ( 4ln(2) σ k 2 τ 0 2 ) 2 .
W Σ ∝ F Σ (x=0)= ∫ −∞ ∞ I Σ (t)dt= ∫ −∞ ∞ I Σ (ω)dω= = I 0 ( τ 0 4ln(2) ) 2 ∑ n=1 N ∑ m=1 N ∫ −∞ ∞ exp[ − (ω− ω 0 ) 2 τ 0 2 4ln(2) +i( δ Φ n [ ω− ω 0 ]−δ Φ m [ ω− ω 0 ] ) ]dω,
W ¯ Σ = ∫ W Σ e − ∑ n=1 N δ Φ n 2 2 σ Φ 2 ( 2π σ Φ ) N dδ Φ 1 ×...×dδ Φ N ,
W ¯ Σ = I 0 ( τ 0 4ln(2) ) 2 ∑ n=1 N ∑ m=1 N ∫ ∫ ∫ −∞ ∞ exp[ − (ω− ω 0 ) 2 τ 0 2 4ln(2) +i( δ k 2_n −δ k 2_m ) (ω− ω 0 ) 2 2 ] × × e − δ k 2_n 2 +δ k 2_m 2 2 σ k2 2 ( 2π σ k2 ) 2 dω×dδ k 2_n ×dδ k 2_m .
η filled W = η F = W ¯ Σ N 2 I 0 ( π τ 0 4ln(2) ) .
E(ψ,δ ψ n )= I 0 exp[ −2ln(2) (ψ−δ ψ n ) 2 Ψ 2 +i α n ψ ],
I Σ (N,ψ,δ ψ 1 ,...,δ ψ N )∝ | ∑ n=1 N E(ψ,δ ψ n ) | 2 .
I ¯ Σ peak (N, σ ψ )= ∫ I Σ (N,0,δ ψ 1 ,...,δ ψ N ) e − ∑ n=1 N δ ψ n 2 2 σ ψ 2 ( 2π σ ψ ) N dδ ψ 1 ×...×dδ ψ N ,
F ¯ Σ peak (N, σ ψ )= π 4ln(2) τ 2 I ¯ Σ peak (N, σ ψ ).
η I (N, σ ψ )= η F (N, σ ψ )= 1 N 2 ( N ( 1+ ( k 0 D 0 σ ψ ) 2 2ln(2) ) 1/2 + N(N−1) ( 1+ ( k 0 D 0 σ ψ ) 2 4ln(2) ) ).
η I N>>1, k 0 D 0 σ ψ <<1 ( σ ψ )= η F N>>1, k 0 D 0 σ ψ <<1 ( σ ψ )=1− ( k 0 D 0 σ ψ ) 2 4ln(2) .
E(x,δ ψ n )= I 0 exp[ − 2ln(2) x 2 D 0 2 +i k 0 xδ ψ n ],
W Σ (N,δ ψ 1 ,...,δ ψ N )∝ ∫ −∞ ∞ | ∑ n=1 N E(x,δ ψ n ) | 2 dx= I 0 π 4ln(2) D 0 ∑ n=1 N ∑ m=1 N exp[ − D 0 2 k 0 2 4ln(2) ( δ ψ n −δ ψ m ) 2 4 ].
W ¯ Σ (N, σ ψ )= ∫ W Σ (δ ψ 1 ,...,δ ψ N ) e − ∑ n=1 N δ ψ n 2 2 σ ψ 2 ( 2π σ ψ ) N dδ ψ 1 ×...×dδ ψ N = = I 0 π 4ln(2) D 0 ( N+N(N−1) 1 1+ ( k 0 D 0 σ ψ ) 2 4ln(2) ).
η filled W (N, σ ψ )= 1 N 2 ( N+N(N−1) 1 1+ ( k 0 D 0 σ ψ ) 2 4ln(2) ).
η filled W N>>1, k 0 D 0 σ ψ <<1 ( σ ψ )=1− ( k 0 D 0 σ ψ ) 2 8ln(2) .
I n peak (Са)= I 0 1+ ( Са× λ FWHM π D 0 2ln(2) λ 0 ) 2 = I 0 1+ ( δ θ n k 0 D 0 4ln(2) ) 2 = I 0 1+ ( δ θ n Ψ ) 2 ,
I Σ = ( ∑ n=1 N I n ) 2 .
η I (N, σ θ )= 1 N 2 ( N 4πln(2) k 0 D 0 2π σ θ exp[ 1 2 ( 4ln(2) k 0 D 0 σ θ ) 2 ]( 1−Erf[ 4ln(2) 2 k 0 D 0 σ θ ] )+ +N(N−1) ( 4ln(2) k 0 D 0 2π σ θ ) 2 exp[ 2 ( 2ln(2) k 0 D 0 σ θ ) 2 ] K 0 2 [ ( 2ln(2) k 0 D 0 σ θ ) 2 ] ),
η I N>>1, k 0 D 0 σ θ <<1 ( σ θ )=1− ( k 0 D 0 σ θ 4ln(2) ) 2 .
E(x=0,t,δ θ n )= I 0 1+ ( δ θ n k 0 D 0 4ln(2) ) 2 exp[ − 2ln(2) t 2 τ 0 ( 1+ ( δ θ n k 0 D 0 4ln(2) ) 2 ) −i ω 0 t ],
F Σ peak (N,δ θ 1 ,...,δ θ N )∝ ∫ −∞ ∞ | ∑ n=1 N E(0,t,δ θ n ) | 2 dt= ∫ −∞ ∞ ∑ n=1 N ∑ m=1 N E(0,t,δ θ n ) E * (0,t,δ θ m )dt.
F ¯ Σ peak (N, σ θ )= ∑ n=1 N ∑ m=1 N ∭ E(0,t,δ θ n ) E * (0,t,δ θ m ) e − δ θ n 2 +δ θ m 2 2 σ θ 2 ( 2π σ θ ) 2 dtdδ θ n dδ θ m .
E(x,t,δ θ n )= I 0 exp[ −2ln(2) x 2 D 0 2 −2ln(2) ( t− xδ θ n k 0 τ 4ln(2) ) 2 τ 2 +i k 0 x−i ω 0 t ].
W Σ (N,δ θ 1 ,...,δ θ N )∝ ∫ −∞ ∞ ∫ −∞ ∞ | ∑ n=1 N E(x,t,δ θ n ) | 2 dxdt.
η filled W (N, σ θ )= 1 N 2 ( N+N(N−1) 4ln (2) π k 0 D 0 σ θ e 1 2 ( 4ln (2) k 0 D 0 σ θ ) 2 K 0 ( 1 2 ( 4ln (2) k 0 D 0 σ θ ) 2 ) ).
η filled W N>>1, k 0 D 0 σ Δθ <<1 ( σ θ )=1− ( k 0 D 0 σ θ 8ln(2) ) 2 .
Φ n (r,φ)=δ A n Z m k ( r,φ ).
Z coma ( r,φ )= 1 8 ( 3 ( r a ) 3 −2( r a ) )cos(φ),
η F ( σ Φ )= η I ( σ Φ )= I ¯ Σ peak ( σ Φ ) I ¯ Σ peak ( σ Φ =0 ) .
I ¯ Σ peak = 1 M ∑ m=1 M I Σ peak_m .
η rel I = I ¯ Σ peak / ( ∑ n=1 N I n ) 2 .
W max = ∑ n=1 N W n .
W Σ (N,δ A 1 ,...,δ A N )∝ ∫ | ∑ n=1 N E n (x,y) | 2 dxdy= I 0 ∫ 0 ∞ | Pr(x,y) | 2 ∑ n=1 N ∑ m=1 N exp[ i( δ A n −δ A m )Z(x,y) ]dxdy.
η filled W ( N, σ A )= 1 I 0 N 2 ∫ W Σ (δ A 1 ,...,δ A N ) e − ∑ n=1 N δ A n 2 2 σ A 2 ( 2π σ A ) N dδ A 1 ×...×dδ A N = = 1 N 2 ( N+N(N−1) ∫ | Pr(x,y) | 2 exp[ − Z 2 (x,y) σ A 2 ]dxdy ).
η filled_N>>1 W ( σ A )= ∫ | Pr(x,y) | 2 exp[ − Z 2 (x,y) σ A 2 ]dxdy.
E → (δ χ n )= I 0 ( cos( δ χ n ) e → + x sin( δ χ n ) e → y ).
η I (N, σ χ )= η F (N, σ χ )= η W (N, σ χ )= 1 N 2 ( N+N(N−1)exp[ − σ χ 2 ] ).
E(x,t,δ x n )= I 0 1+ ( δ x n D 0 ) 2 exp[ − 2ln(2) ( x D 0 − it τ 0 δ x n D 0 ) 2 ( 1+ ( δ x n D 0 ) 2 ) − 2ln(2) t 2 τ 0 2 ].
η filled (N, σ x )= 1 N 2 ( N+N(N−1) D 0 π σ x e 1 2 ( D 0 σ x ) 2 K 0 ( 1 2 ( D 0 σ x ) 2 ) ).
η filled N>>1, σ x / D 0 <<1 ( σ x )=1− ( σ x 2 D 0 ) 2 .
E → (t,ψ,δ T n ,δ φ n ,...)= I 0 ( cos( δ χ n ) e → + x sin( δ χ n ) e → y )exp[ −2ln(2) (t−δ T n ) 2 τ 0 2 −i ω 0 t ]exp[ iδ φ n ]× × exp[ −2ln(2) (ψ−δ ψ n ) 2 Ψ 2 +i α n ψ ] ( 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 ) 1/4 ( 1+ ( δ x n D 0 ) 2 ) 1/4 1+ ( δ θ n k 0 D 0 4ln(2) ) 2 = I 0 ∏ i E i ( δ X n_i ).
η total = ∏ i η( σ i ) .
I ¯ Σ peak (N, σ k 2 )= ∫ | ∑ n=1 N I peak (δ k 2_n ) | 2 e − ∑ n=1 N δ k 2_n 2 2 σ k 2 2 ( 2π σ k 2 ) N dδ k 2_1 ×...×dδ k 2_N .
| ∑ n=1 N I peak (δ k 2_n ) | 2 = | ∑ n=1 N 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 4 | 2 = ∑ n=1 N ∑ m=1 N 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 4 1+ ( 4ln(2) δ k 2_m τ 0 2 ) 2 4 = = ∑ n=1 N 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 + ∑ n=1 N ∑ m≠n N 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 4 1+ ( 4ln(2) δ k 2_m τ 0 2 ) 2 4 .
I ¯ Σ peak (N, σ k 2 )= I 0 ( ∑ n=1 N [ ∫ −∞ ∞ 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 e − δ k 2_n 2 2 σ k 2 2 ( 2π σ k 2 ) dδ k 2_n ] + + ∑ n=1 N ∑ m≠n N [ ∫ −∞ ∞ 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 4 e − δ k 2_n 2 2 σ k 2 2 ( 2π σ k 2 ) dδ k 2_n × ∫ −∞ ∞ 1 1+ ( 4ln(2) δ k 2_m τ 0 2 ) 2 4 e − δ k 2_m 2 2 σ k 2 2 ( 2π σ k 2 ) dδ k 2_m ] )= = I 0 ( ∑ n=1 N [ ∫ −∞ ∞ 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 e − δ k 2_n 2 2 σ k 2 2 ( 2π σ k 2 ) dδ k 2_n ]+ ∑ n=1 N ∑ m≠n N [ ∫ −∞ ∞ 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 4 e − δ k 2_n 2 2 σ k 2 2 ( 2π σ k 2 ) dδ k 2_n ] 2 )= = I 0 ( 2N ∫ 0 ∞ 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 e − δ k 2_n 2 2 σ k 2 2 ( 2π σ k 2 ) dδ k 2_n +N(N−1) [ 2 ∫ 0 ∞ 1 1+ ( 4ln(2) δ k 2_n τ 0 2 ) 2 4 e − δ k 2_n 2 2 σ k 2 2 ( 2π σ k 2 ) dδ k 2_n ] 2 ).
I ¯ Σ peak (N, σ k 2 )= I 0 ( N τ 0 2 4ln(2) 2π σ k 2 e τ 0 4 64 ln 2 (2) σ k 2 2 ∫ 0 ∞ e − τ 0 4 cosh(q) 64 ln 2 (2) σ k 2 2 dq+ +N(N−1) [ τ 0 2 32 ln(2) σ k 2 π ∫ 0 ∞ e − τ 0 4 q ′ 32 ln 2 (2) σ k 2 2 1 q ′ 1 1+ q ′ 4 d q ′ ] 2 ).

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