Abstract

Using second-order coherence theory of nonstationary light we examine in detail the coherence properties of supercontinuum radiation generated in nonlinear fibers. We show that the supercontinuum can be divided into quasi-coherent and quasi-stationary parts and that the relative contributions depend on the dynamics involved in the spectral broadening process. We establish the correspondence between the quasi-coherent part of the two-frequency correlation function of the second-order theory and the usual Dudley–Coen degree of coherence used to characterize the shot-to-shot stability of supercontinuum sources. Experimental implementation for measuring separately the quasi-coherent and quasi-stationary contributions is further addressed. Our results open the route for complete characterization of supercontinuum coherence.

© 2011 Optical Society of America

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References

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  1. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184(2006).
    [CrossRef]
  2. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered optical fibers,” Opt. Lett. 27, 1180–1182 (2002).
    [CrossRef]
  3. J. M. Dudley and S. Coen, “Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers,” IEEE J. Sel. Top. Quantum Electron. 8, 651–659 (2002).
    [CrossRef]
  4. Z. Zhu and T. G. Brown, “Experimental studies of polarization properties of supercontinua generated in a birefringent photonic crystal fiber,” Opt. Express 12, 791–796 (2004).
    [CrossRef] [PubMed]
  5. S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005).
    [CrossRef]
  6. S. M. Kobtsev and S. V. Smirnov, “Coherent properties of super-continuum containing clearly defined solitons,” Opt. Express 14, 3968–3980 (2006).
    [CrossRef] [PubMed]
  7. X. Gu, M. Kimmel, E. Zeek, P. O’shea, A. P. Shreenath, R. Trebino, and R. S. Windeler, “Frequency-resolved optical gating and single-shot spectral measurements reveal fine structure in microstructure-fiber supercontinuum,” Opt. Lett. 27, 1174–1176(2002).
    [CrossRef]
  8. X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. S. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express 11, 2697–2703 (2003).
    [CrossRef] [PubMed]
  9. Z. Zhu and T. G. Brown, “Polarization properties of supercontinuum spectra generated in birefringent photonic crystal fibers,” J. Opt. Soc. Am. B 21, 249–257 (2004).
    [CrossRef]
  10. F. Lu and W. H. Knox, “Generation of a broadband continuum with high spectral coherence in tapered single-mode optical fibers,” Opt. Express 12, 347–353 (2004).
    [CrossRef] [PubMed]
  11. I. Zeylikovich, V. Kartazaev, and R. R. Alfano, “Spectral, temporal, and coherence properties of supercontinuum generation in microstructure fiber,” J. Opt. Soc. Am. B 22, 1453–1460(2005).
    [CrossRef]
  12. F. Lu and W. H. Knox, “Generation, characterization, and application of broadband coherent femtosecond visible pulses in dispersion micromanaged holey fibers,” J. Opt. Soc. Am. B 23, 1221–1227 (2006).
    [CrossRef]
  13. D. Türke, S. Pricking, A. Husakou, J. Teipel, J. Herrmann, and H. Giessen, “Coherence of subsequent supercontinuum pulses generated in tapered fibers in the femtosecond regime,” Opt. Express 15, 2732–2741 (2007).
    [CrossRef] [PubMed]
  14. M. Bellini and T. W. Hänsch, “Phase-locked white-light continuum pulses: toward a universal optical frequency-comb synthesizer,” Opt. Lett. 25, 1049–1051 (2000).
    [CrossRef]
  15. M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
    [CrossRef]
  16. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andr`es, “Pulseby-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express 18, 14979–14991 (2010).
    [CrossRef] [PubMed]
  17. G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Second-order coherence of supercontinuum light,” Opt. Lett. 35, 3057–3059 (2010).
    [CrossRef] [PubMed]
  18. C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
    [CrossRef]
  19. M. Erkintalo, B. Wetzel, G. Genty, and J. M. Dudley, “Limitations of the linear Raman gain approximation in modeling broadband nonlinear propagation in optical fibers,” Opt. Express 18, 25449–25460 (2010).
    [CrossRef] [PubMed]
  20. G. Genty, S. Coen, and J. M. Dudley, “Fiber supercontinuum sources,” J. Opt. Soc. Am. B 24, 1771–1785 (2007).
    [CrossRef]
  21. G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub 30 fs pulses,” Opt. Express 12, 4614–4624(2004).
    [CrossRef] [PubMed]
  22. A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photon. 1, 653–657 (2007).
    [CrossRef]
  23. R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
    [CrossRef]
  24. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437(2009).
    [CrossRef]

2010

2009

2007

2006

2005

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005).
[CrossRef]

I. Zeylikovich, V. Kartazaev, and R. R. Alfano, “Spectral, temporal, and coherence properties of supercontinuum generation in microstructure fiber,” J. Opt. Soc. Am. B 22, 1453–1460(2005).
[CrossRef]

2004

2003

2002

2000

1998

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[CrossRef]

1997

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Alfano, R. R.

Andr`es, P.

Bellini, M.

Bertolotti, M.

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

Brown, T. G.

Coen, S.

Delong, K. W.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Dorrer, C.

Dudley, J. M.

Erkintalo, M.

Fateev, N. V.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005).
[CrossRef]

Ferrari, A.

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

Fittinghoff, D. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Friberg, A. T.

Genty, G.

Giessen, H.

Gorbach, A. V.

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photon. 1, 653–657 (2007).
[CrossRef]

Gu, X.

Hänsch, T. W.

Herrmann, J.

Husakou, A.

Iaconis, C.

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[CrossRef]

Kane, D. J.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Kartazaev, V.

Kimmel, M.

Knox, W. H.

Kobtsev, S. M.

S. M. Kobtsev and S. V. Smirnov, “Coherent properties of super-continuum containing clearly defined solitons,” Opt. Express 14, 3968–3980 (2006).
[CrossRef] [PubMed]

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005).
[CrossRef]

Krumbügel, M. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Kukarin, S. V.

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005).
[CrossRef]

Lajunen, H.

Lancis, J.

Lehtonen, M.

Lu, F.

Ludvigsen, H.

O’shea, P.

Pricking, S.

Richman, B. A.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Sereda, L.

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

Shreenath, A. P.

Silvestre, E.

Skryabin, D. V.

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photon. 1, 653–657 (2007).
[CrossRef]

Smirnov, S. V.

S. M. Kobtsev and S. V. Smirnov, “Coherent properties of super-continuum containing clearly defined solitons,” Opt. Express 14, 3968–3980 (2006).
[CrossRef] [PubMed]

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005).
[CrossRef]

Surakka, M.

Sweetser, J. N.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

Teipel, J.

Torres-Company, V.

Trebino, R.

Türke, D.

Turunen, J.

Walmsley, I. A.

I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437(2009).
[CrossRef]

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[CrossRef]

Wetzel, B.

Windeler, R. S.

Wong, V.

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[CrossRef]

Zeek, E.

Zeylikovich, I.

Zhu, Z.

Adv. Opt. Photon.

Appl. Phys. B

S. M. Kobtsev, S. V. Kukarin, N. V. Fateev, and S. V. Smirnov, “Coherent, polarization and temporal properties of self-frequency shifted solitons generated in polarization-maintaining microstructured fibre,” Appl. Phys. B 81, 265–269 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. M. Dudley and S. Coen, “Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers,” IEEE J. Sel. Top. Quantum Electron. 8, 651–659 (2002).
[CrossRef]

C. Iaconis, V. Wong, and I. A. Walmsley, “Direct interferometric techniques for characterizing ultrashort optical pulses,” IEEE J. Sel. Top. Quantum Electron. 4, 285–294 (1998).
[CrossRef]

J. Opt. Soc. Am. B

Nat. Photon.

A. V. Gorbach and D. V. Skryabin, “Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres,” Nat. Photon. 1, 653–657 (2007).
[CrossRef]

Opt. Express

G. Genty, M. Lehtonen, and H. Ludvigsen, “Effect of cross-phase modulation on supercontinuum generated in microstructured fibers with sub 30 fs pulses,” Opt. Express 12, 4614–4624(2004).
[CrossRef] [PubMed]

X. Gu, M. Kimmel, A. P. Shreenath, R. Trebino, J. M. Dudley, S. Coen, and R. S. Windeler, “Experimental studies of the coherence of microstructure-fiber supercontinuum,” Opt. Express 11, 2697–2703 (2003).
[CrossRef] [PubMed]

F. Lu and W. H. Knox, “Generation of a broadband continuum with high spectral coherence in tapered single-mode optical fibers,” Opt. Express 12, 347–353 (2004).
[CrossRef] [PubMed]

S. M. Kobtsev and S. V. Smirnov, “Coherent properties of super-continuum containing clearly defined solitons,” Opt. Express 14, 3968–3980 (2006).
[CrossRef] [PubMed]

Z. Zhu and T. G. Brown, “Experimental studies of polarization properties of supercontinua generated in a birefringent photonic crystal fiber,” Opt. Express 12, 791–796 (2004).
[CrossRef] [PubMed]

D. Türke, S. Pricking, A. Husakou, J. Teipel, J. Herrmann, and H. Giessen, “Coherence of subsequent supercontinuum pulses generated in tapered fibers in the femtosecond regime,” Opt. Express 15, 2732–2741 (2007).
[CrossRef] [PubMed]

M. Erkintalo, B. Wetzel, G. Genty, and J. M. Dudley, “Limitations of the linear Raman gain approximation in modeling broadband nonlinear propagation in optical fibers,” Opt. Express 18, 25449–25460 (2010).
[CrossRef] [PubMed]

H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andr`es, “Pulseby-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express 18, 14979–14991 (2010).
[CrossRef] [PubMed]

Opt. Lett.

Pure Appl. Opt.

M. Bertolotti, L. Sereda, and A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

Rev. Mod. Phys.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. 78, 1135–1184(2006).
[CrossRef]

Rev. Sci. Instrum.

R. Trebino, K. W. Delong, D. N. Fittinghoff, J. N. Sweetser, M. A. Krumbügel, B. A. Richman, and D. J. Kane, “Measuring ultrashort laser pulses in the time frequency domain using frequency-resolved optical gating,” Rev. Sci. Instrum. 68, 3277–3295 (1997).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Overall degrees of spectral coherence μ ¯ (red diamonds) and | g 12 ( 1 ) | (black circles) versus input pulse peak power. A, B, and C mark the three cases of quasi-coherent, partially coherent, and quasi-incoherent SC light investigated in detail.

Fig. 2
Fig. 2

Left: normalized cross-spectral density | μ ¯ ( ω , Δ ω ) | for (a) case A, (c) case B, and (e) case C. Right: normalized mutual coherence function | γ ( t ¯ , Δ t ) | for (b) case A, (d) case B, and (f) case C. For clarity, the absolute frequency and times axes are indicated. Note the different color scale in (a) and (b) compared to (c)–(f).

Fig. 3
Fig. 3

Left: normalized cross-spectral density | μ ¯ ( ω , Δ ω ) | and right: normalized mutual coherence function | γ ( t ¯ , Δ t ) | for 960 nm input pulses with (a),(b) P P = 22 kW and (c),(d) P P = 88 kW .

Fig. 4
Fig. 4

Left: normalized cross-spectral density | μ ¯ ( ω , Δ ω ) | and right: normalized mutual coherence function | γ ( t ¯ , Δ t ) | for (a),(b) 50 fs input pulses, (c),(d) 1 ps input pulses.

Fig. 5
Fig. 5

Left: mean spectra (black lines), coherent (red lines) and quasi-stationary (blue lines) spectral contributions for (a) case A, (c) case B, and (e) case C. Right: mean temporal intensity (black lines), coherent (red lines) and quasi-stationary (blue lines) intensity contributions for (b) case A, (d) case B, and (f) case C.

Fig. 6
Fig. 6

False color representation of the evolution of (a) mean spectrum, (b) quasi-coherent contribution, and (c) quasi- stationary contribution for case C ( P P = 22 kW ).

Fig. 7
Fig. 7

Left: calculated cross-spectral density | W ( ω ¯ , Δ ω ) | for (a) case A, (c) case B, and (e) case C. Right: coherent part of the cross-spectral density | W ( ω ¯ , Δ ω ) | retrieved from g 12 ( 1 ) ( ω ) for (b) case A, (d) case B, and (f) case C.

Fig. 8
Fig. 8

Schematic for complete experimental characterization of the SC coherence properties. OSA, optical spectrum analyzer; FROG, frequency-resolved optical gating; MZI, Mach– Zehnder interferometer.

Fig. 9
Fig. 9

Reconstructed normalized CSD when measuring separately the quasi-coherent and quasi-stationary contributions for (b) case B and (d) case C. For comparison, the original CSD as directly numerically computed from the ensemble of simulation realizations is also shown for (a) case B and (c) case C.

Tables (1)

Tables Icon

Table 1 Taylor-series Expansion Coefficients of the Fiber Dispersion

Equations (29)

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

Γ ( t 1 , t 2 ) = E * ( t 1 ) E ( t 2 ) ,
Γ ( t ¯ , Δ t ) = E * ( t ¯ Δ t / 2 ) E ( t ¯ + Δ t / 2 ) .
γ ( t ¯ , Δ t ) = Γ ( t ¯ , Δ t ) I ( t ¯ Δ t / 2 ) I ( t ¯ + Δ t / 2 ) ,
γ ¯ 2 = | Γ ( t ¯ , Δ t ) | 2 d t ¯ d Δ t [ I ( t ) d t ] 2 .
W ( ω 1 , ω 2 ) = E ˜ * ( ω 1 ) E ˜ ( ω 2 ) ,
W ( ω ¯ , Δ ω ) = E ˜ * ( ω ¯ Δ ω / 2 ) E ˜ ( ω ¯ + Δ ω / 2 ) .
μ ( ω ¯ , Δ ω ) = W ( ω ¯ , Δ ω ) S ( ω ¯ Δ ω / 2 ) S ( ω ¯ + Δ ω / 2 ) .
μ ¯ 2 = 0 | W ( ω ¯ , Δ ω ) | 2 d ω ¯ d Δ ω [ 0 S ( ω ) d ω ] 2 ,
Γ ( t ¯ , Δ t ) = 0 W ( ω ¯ , Δ ω ) × exp [ i ( ω ¯ Δ t + Δ ω t ¯ ) ] d ω ¯ d Δ ω .
g 12 ( 1 ) ( ω ) = E ˜ i * ( ω ) E ˜ j ( ω ) i j | E ˜ ( ω ) | 2 ,
| g 12 ( 1 ) | = 0 | g 12 ( 1 ) ( ω ) | S ( ω ) d ω 0 S ( ω ) d ω .
A z k 2 i k + 1 k ! β k k A T k = i γ ( 1 + i τ shock T ) ( A R ( T ) × | A ( z , T T ) | 2 d T ) .
μ ( ω ¯ , Δ ω ) μ c ( ω 1 , ω 2 ) + μ q ( Δ ω ) ,
γ ( t ¯ , Δ t ) γ c ( t 1 , t 2 ) + γ q ( Δ t ) .
g m ( ω ) = 1 N 2 N N 2 | E ˜ ( ω ) | 2 N | E ˜ ( ω ) | 2 | E ˜ ( ω ) | 2 ,
g m ( ω ) = | E ˜ ( ω ) | 2 | E ˜ ( ω ) | 2 = | E ˜ ( ω ) | 2 S ( ω ) .
| μ c ( ω 1 , ω 2 ) | g m ( ω 1 ) g m ( ω 2 ) .
μ q ( Δ ω ) = 1 2 π E 0 I q ( t ¯ ) exp ( i Δ ω t ¯ ) d t ¯ ,
γ q ( Δ t ) = 1 E 0 0 S q ( ω ¯ ) exp ( i ω ¯ Δ t ) d ω ¯ ,
g m ( ω ) = | g 12 ( ω ) | .
g m ( ω ) = [ g 12 ( ω ) ] = E ˜ i * ( ω ) E ˜ j ( ω ) i j | E ˜ ( ω ) | 2 + E ˜ j * ( ω ) E ˜ i ( ω ) i j | E ˜ ( ω ) | 2 .
g m ( ω ) = 1 n pairs i j N [ E ˜ i * ( ω ) E j ( ω ) + E ˜ i ( ω ) E ˜ j * ( ω ) ] | E ˜ ( ω ) | 2 .
g m ( ω ) = 1 n pairs | i = 1 N E ˜ i ( ω ) | 2 i = 1 N | E ˜ i ( ω ) | 2 | E ˜ ( ω ) | 2 ,
g m ( ω ) = 1 n pairs | N E ˜ ( ω ) | 2 N | E ˜ ( ω ) | 2 | E ˜ ( ω ) | 2 .
g m ( ω ) = 1 N 2 N N 2 | E ˜ ( ω ) | 2 N | E ˜ ( ω ) | 2 | E ˜ ( ω ) | 2 .
| W c ( ω 1 , ω 2 ) | S ( ω 1 ) g m ( ω 1 ) S ( ω 2 ) g m ( ω 2 ) .
μ ¯ 2 = 0 0 | W ( ω 1 , ω 2 ) | 2 d ω 1 d ω 2 [ 0 S ( ω ) d ω ] 2 = 0 0 | W c ( ω 1 , ω 2 ) + W q ( ω 1 , ω 2 ) | 2 d ω 1 d ω 2 [ 0 S ( ω ) d ω ] 2 ,
μ ¯ 2 0 | W c ( ω ¯ , Δ ω ) | 2 d ω ¯ d Δ ω [ 0 S ( ω ) d ω ] 2 = 0 0 g m ( ω 1 ) S ( ω 1 ) g m ( ω 2 ) S ( ω 2 ) d ω 1 d ω 2 [ 0 S ( ω ) d ω ] 2 ,
μ ¯ 2 [ 0 g m ( ω ) S ( ω ) d ω 0 S ( ω ) d ω ] 2 = | g 12 ( 1 ) | 2 .

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