Abstract

Coherent combining is a novel approach to scale the performance of laser amplifiers. The use of ultrashort pulses in a coherent combining setup results in new challenges compared to continuous wave operation or to pulses on the nanosecond timescale, because temporal and spectral effects such as self-phase modulation, dispersion and the optical path length difference between the pulses have to be considered. In this paper the impact of these effects on the combining process has been investigated and simple analytical equations for the evaluation of this impact have been obtained. These formulas provide design guidelines for laser systems using coherent combining. The results show that, in spite of the temporal and spectral effects mentioned above, for a carefully adjusted and stabilized system an excellent efficiency of the combining process can still be achieved.

© 2011 OSA

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  1. C. R. E. Baer, Ch. Kränkel, C. J. Saraceno, O. H. Heckl, M. Golling, R. Peters, K. Petermann, Th. Südmeyer, G. Huber, and U. Keller, “Femtosecond thin-disk laser with 141 W of average power,” Opt. Lett. 35(13), 2302–2304 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-13-2302 .
    [CrossRef] [PubMed]
  2. P. Russbueldt, T. Mans, G. Rotarius, J. Weitenberg, H. D. Hoffmann, and R. Poprawe, “400W Yb:YAG Innoslab fs-Amplifier,” Opt. Express 17(15), 12230–12245 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12230 .
    [CrossRef] [PubMed]
  3. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, Th. Gabler, Ch. Wirth, Th. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94 .
    [CrossRef] [PubMed]
  4. T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255 .
    [CrossRef] [PubMed]
  5. R. Xiao, J. Hou, M. Liu, and Z. F. Jiang, “Coherent combining technology of master oscillator power amplifier fiber arrays,” Opt. Express 16(3), 2015–2022 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-2015 .
    [CrossRef] [PubMed]
  6. R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18(13), 13547–13553 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13547 .
    [CrossRef] [PubMed]
  7. E. Seise, A. Klenke, J. Limpert, and A. Tünnermann, “Coherent addition of fiber-amplified ultrashort laser pulses,” Opt. Express 18(26), 27827–27835 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-27827 .
    [CrossRef] [PubMed]
  8. E. Seise, A. Klenke, S. Breitkopf, M. Plötner, J. Limpert, and A. Tünnermann, “Coherently combined fiber laser system delivering 120 μJ femtosecond pulses,” Opt. Lett. 36(4), 439–441 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-439 .
    [CrossRef] [PubMed]
  9. I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. 35(12), 2052–2054 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?&uri=ol-35-12-2052 .
    [CrossRef] [PubMed]
  10. G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25403 .
    [CrossRef] [PubMed]
  11. L. Daniault, M. Hanna, L. Lombard, Y. Zaouter, E. Mottay, D. Goular, P. Bourdon, F. Druon, and P. Georges, “Coherent beam combining of two femtosecond fiber chirped-pulse amplifiers,” Opt. Lett. 36(5), 621–623 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-621 .
    [CrossRef] [PubMed]
  12. T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
    [CrossRef]
  13. G. P. Agrawal, Nonlinear Fiber Optics 2nd Edition (Academic Press, 1995)
  14. D. N. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, “Self-phase modulation compensated by positive dispersion in chirped-pulse systems,” Opt. Express 17(7), 4997–5007 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4997 .
    [CrossRef] [PubMed]
  15. M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped-pulse amplification,” Opt. Lett. 19(24), 2149–2151 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=ol-19-24-2149 .

2011 (3)

2010 (6)

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, Th. Gabler, Ch. Wirth, Th. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94 .
[CrossRef] [PubMed]

I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. 35(12), 2052–2054 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?&uri=ol-35-12-2052 .
[CrossRef] [PubMed]

G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25403 .
[CrossRef] [PubMed]

R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18(13), 13547–13553 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13547 .
[CrossRef] [PubMed]

E. Seise, A. Klenke, J. Limpert, and A. Tünnermann, “Coherent addition of fiber-amplified ultrashort laser pulses,” Opt. Express 18(26), 27827–27835 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-27827 .
[CrossRef] [PubMed]

C. R. E. Baer, Ch. Kränkel, C. J. Saraceno, O. H. Heckl, M. Golling, R. Peters, K. Petermann, Th. Südmeyer, G. Huber, and U. Keller, “Femtosecond thin-disk laser with 141 W of average power,” Opt. Lett. 35(13), 2302–2304 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-13-2302 .
[CrossRef] [PubMed]

2009 (2)

2008 (1)

1994 (1)

1980 (1)

T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[CrossRef]

Alahmed, Z. A.

Alley, T. G.

Andersen, T. V.

Apolonski, A.

Azzeer, A. M.

Baer, C. R. E.

Bernhardt, B.

Bourdon, P.

Bratcher, A.

Breitkopf, S.

Carstens, H.

Couillaud, B.

T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[CrossRef]

Daniault, L.

Ditmire, T.

Druon, F.

Eidam, T.

Fill, E.

Flores, A. S.

Gabler, Th.

Georges, P.

Golling, M.

Goodno, G. D.

Goular, D.

Hädrich, S.

Hanf, S.

Hanna, M.

Hänsch, T. W.

Heckl, O. H.

Hoffmann, H. D.

Hou, J.

Huber, G.

Jansen, F.

Jauregui, C.

Jiang, Z. F.

Keller, U.

Klenke, A.

Kränkel, Ch.

Krausz, F.

Limpert, J.

E. Seise, A. Klenke, S. Breitkopf, M. Plötner, J. Limpert, and A. Tünnermann, “Coherently combined fiber laser system delivering 120 μJ femtosecond pulses,” Opt. Lett. 36(4), 439–441 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-439 .
[CrossRef] [PubMed]

T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255 .
[CrossRef] [PubMed]

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, Th. Gabler, Ch. Wirth, Th. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94 .
[CrossRef] [PubMed]

E. Seise, A. Klenke, J. Limpert, and A. Tünnermann, “Coherent addition of fiber-amplified ultrashort laser pulses,” Opt. Express 18(26), 27827–27835 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-27827 .
[CrossRef] [PubMed]

I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. 35(12), 2052–2054 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?&uri=ol-35-12-2052 .
[CrossRef] [PubMed]

D. N. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, “Self-phase modulation compensated by positive dispersion in chirped-pulse systems,” Opt. Express 17(7), 4997–5007 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4997 .
[CrossRef] [PubMed]

Liu, M.

Lombard, L.

Mans, T.

Mottay, E.

Ozawa, A.

Perry, M. D.

Petermann, K.

Peters, R.

Plötner, M.

Poprawe, R.

Pulford, B.

Pupeza, I.

Rauschenberger, J.

Rotarius, G.

Rothenberg, J. E.

Rothhardt, J.

Russbueldt, P.

Sanchez, A. D.

Saraceno, C. J.

Schimpf, D. N.

Schreiber, Th.

Seise, E.

Shih, C. C.

Stuart, B. C.

Stutzki, F.

Südmeyer, Th.

Tünnermann, A.

T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255 .
[CrossRef] [PubMed]

E. Seise, A. Klenke, S. Breitkopf, M. Plötner, J. Limpert, and A. Tünnermann, “Coherently combined fiber laser system delivering 120 μJ femtosecond pulses,” Opt. Lett. 36(4), 439–441 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-439 .
[CrossRef] [PubMed]

E. Seise, A. Klenke, J. Limpert, and A. Tünnermann, “Coherent addition of fiber-amplified ultrashort laser pulses,” Opt. Express 18(26), 27827–27835 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-27827 .
[CrossRef] [PubMed]

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, Th. Gabler, Ch. Wirth, Th. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94 .
[CrossRef] [PubMed]

I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. 35(12), 2052–2054 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?&uri=ol-35-12-2052 .
[CrossRef] [PubMed]

D. N. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, “Self-phase modulation compensated by positive dispersion in chirped-pulse systems,” Opt. Express 17(7), 4997–5007 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4997 .
[CrossRef] [PubMed]

Uberna, R.

Udem, T.

Weitenberg, J.

Wirth, Ch.

Xiao, R.

Zaouter, Y.

Opt. Commun. (1)

T. W. Hänsch and B. Couillaud, “Laser frequency stabilization by polarization spectroscopy of a reflecting reference cavity,” Opt. Commun. 35(3), 441–444 (1980).
[CrossRef]

Opt. Express (7)

D. N. Schimpf, E. Seise, J. Limpert, and A. Tünnermann, “Self-phase modulation compensated by positive dispersion in chirped-pulse systems,” Opt. Express 17(7), 4997–5007 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-4997 .
[CrossRef] [PubMed]

T. Eidam, J. Rothhardt, F. Stutzki, F. Jansen, S. Hädrich, H. Carstens, C. Jauregui, J. Limpert, and A. Tünnermann, “Fiber chirped-pulse amplification system emitting 3.8 GW peak power,” Opt. Express 19(1), 255–260 (2011), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-1-255 .
[CrossRef] [PubMed]

R. Xiao, J. Hou, M. Liu, and Z. F. Jiang, “Coherent combining technology of master oscillator power amplifier fiber arrays,” Opt. Express 16(3), 2015–2022 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-3-2015 .
[CrossRef] [PubMed]

R. Uberna, A. Bratcher, T. G. Alley, A. D. Sanchez, A. S. Flores, and B. Pulford, “Coherent combination of high power fiber amplifiers in a two-dimensional re-imaging waveguide,” Opt. Express 18(13), 13547–13553 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13547 .
[CrossRef] [PubMed]

E. Seise, A. Klenke, J. Limpert, and A. Tünnermann, “Coherent addition of fiber-amplified ultrashort laser pulses,” Opt. Express 18(26), 27827–27835 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-26-27827 .
[CrossRef] [PubMed]

P. Russbueldt, T. Mans, G. Rotarius, J. Weitenberg, H. D. Hoffmann, and R. Poprawe, “400W Yb:YAG Innoslab fs-Amplifier,” Opt. Express 17(15), 12230–12245 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-15-12230 .
[CrossRef] [PubMed]

G. D. Goodno, C. C. Shih, and J. E. Rothenberg, “Perturbative analysis of coherent combining efficiency with mismatched lasers,” Opt. Express 18(24), 25403–25414 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-25403 .
[CrossRef] [PubMed]

Opt. Lett. (6)

L. Daniault, M. Hanna, L. Lombard, Y. Zaouter, E. Mottay, D. Goular, P. Bourdon, F. Druon, and P. Georges, “Coherent beam combining of two femtosecond fiber chirped-pulse amplifiers,” Opt. Lett. 36(5), 621–623 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-5-621 .
[CrossRef] [PubMed]

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, Th. Gabler, Ch. Wirth, Th. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-2-94 .
[CrossRef] [PubMed]

E. Seise, A. Klenke, S. Breitkopf, M. Plötner, J. Limpert, and A. Tünnermann, “Coherently combined fiber laser system delivering 120 μJ femtosecond pulses,” Opt. Lett. 36(4), 439–441 (2011), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-36-4-439 .
[CrossRef] [PubMed]

I. Pupeza, T. Eidam, J. Rauschenberger, B. Bernhardt, A. Ozawa, E. Fill, A. Apolonski, T. Udem, J. Limpert, Z. A. Alahmed, A. M. Azzeer, A. Tünnermann, T. W. Hänsch, and F. Krausz, “Power scaling of a high-repetition-rate enhancement cavity,” Opt. Lett. 35(12), 2052–2054 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?&uri=ol-35-12-2052 .
[CrossRef] [PubMed]

M. D. Perry, T. Ditmire, and B. C. Stuart, “Self-phase modulation in chirped-pulse amplification,” Opt. Lett. 19(24), 2149–2151 (1994), http://www.opticsinfobase.org/abstract.cfm?URI=ol-19-24-2149 .

C. R. E. Baer, Ch. Kränkel, C. J. Saraceno, O. H. Heckl, M. Golling, R. Peters, K. Petermann, Th. Südmeyer, G. Huber, and U. Keller, “Femtosecond thin-disk laser with 141 W of average power,” Opt. Lett. 35(13), 2302–2304 (2010), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-35-13-2302 .
[CrossRef] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics 2nd Edition (Academic Press, 1995)

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Figures (5)

Fig. 1
Fig. 1

Schematic setup of coherent addition of ultrashort laser pulses; Δφ: element for path length matching

Fig. 2
Fig. 2

Combining process using (a) a partially reflective surface and (b) a polarization dependent cube.

Fig. 3
Fig. 3

Dependency of the local maxima of the FOM on a OPD for Gaussian pulses with a bandwidth up to 10nm at a typical center wavelength of 1030nm. The white line defines the boundary where the FOM falls below 95%.

Fig. 4
Fig. 4

Analytically calculated FOM for Gaussian pulses with a bandwidth of (a) 5nm, (b) 10nm, (c) 15nm propagating through fused silica, when a B-Integral or LDE difference is introduced.

Fig. 5
Fig. 5

Dependency of the absolute change of the B-Integral and of the FOM on the value of the B-Integral, if a fluctuation of the input power and amplification coefficient of 5% is introduced.

Equations (22)

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

P(ω)= P 0 (ω)( 1+cos( ΔΦ( ω ) ) )
FOM= P comb P secondary P comb + P secondary
FOM=DOLP= P max P min P max + P min
FOM( ω )= P max (ω) P min (ω) P max (ω)+ P min (ω) =± 2 P 0 (ω)cos( ΔΦ( ω ) ) 2 P 0 (ω) =±cos( ΔΦ( ω ) )
FOM=C s(ω)FOM( ω) dω=±C s(ω) cos( ΔΦ( ω ) ) dω
ΔΦ( ω )= Δl c 0 ω
E 1 ( t )= 1 2π E 0 ( ω ) e i Φ chirp ( ω ) e iωt dω E ( t ) 2 = 1 2π E 0 ( ω ) e i Φ chirp ( ω ) e iΔ Φ delay ( ω ) e iωt dω E( t )= 1 2 ( E 1 ( t )+E ( t ) 2 )
F~ | E( t ) | 2 dt= | E( ω ) | 2 dω= 1 2 | E 0 ( ω ) | 2 | e i Φ chirp ( ω ) + e i( Φ chirp ( ω )+Δ Φ delay ( ω ) ) | 2 dω = | E 0 ( ω ) | 2 ( 1+cos( Δ Φ delay ( ω ) ) ) dω
s( ω )= e 4ln2 ( ω ω 0 ω FWHM ) 2
FOM=C s(ω) cos( ΔΦ( ω ) ) dω= e ( 2πk ω FWHM ω 0 ) 2 16ln2
Φ( ω )= 1 2 β 2 L ( ω ω 0 ) 2 +Bs( ω ω 0 )
ΔΦ( ω )= 1 2 β 2 ΔL ( ω ω 0 ) 2 +ΔBs( ω ω 0 )
Δ Φ c ( ω 0 )= Δl c 0 ω 0 =ΔB Δ Φ c ( ω )= Δl c 0 ω=ΔB ω ω 0 =ΔB( 1+ ω ω 0 ω 0 )ΔB for | ω ω 0 |<< ω 0
FOM=C s(ω) cos( ΔΦ( ω )+Δ Φ c ( ω ) ) dω 1+( 1 2 1 2 1 2 3 )Δ B 2 3 512ln (2) 2 ω FWHM 4 β 2 2 Δ L 2 + 1 16ln (2) ( 1 1 2 2 ) ω FWHM 2 β 2 ΔLΔB
ΔB< π 4 ( 1 e 1 ) 1 1.2 rad
ΔL< 2πln2 β 2 ω FWHM 2 1.0 m
B=C P 0 ( g1 )ln ( g ) 1 with C= 8 n 2 L λ 0 MF D 2 f rep τ
B P 0 Δ P 0 =B Δ P 0 P 0 and B g ΔgB Δg g for g>>1
E( ω )= 1 N n=1 N E 0 ( ω ) e i Φ n ( ω ) = E 0 ( ω 0 ) s( ω ) N n=1 N e i Φ n ( ω )
I( ω )~ E * ( ω )E( ω )= | E 0 ( ω 0 ) | 2 s( ω ) N ( N+ i=1,ij N j=1 N cos( Δ Φ ij ( ω ) ) )
FOM= I(ω)( Ns( ω )I(ω) ) dω Ns( ω ) dω = C N s( ω )( ( 2-N )+ 2 N i=1,ij N j=1 N cos( Δ Φ ij ( ω ) ) ) dω =( 2 N 1 )+ 2 N 2 i=1,ij N j=1 N FO M ij
FOM( 2 N 1 )+2( 1 1 N )FO M 12

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