Abstract

We describe the second-order coherence functions of supercontinuum (SC) in terms of elementary fields that can be obtained from measurable average quantities. The representation is based on the partition of the second-order correlation functions of SC into quasi-coherent and quasi-stationary contributions. Numerical simulations of statistical ensembles of SC pulses with different coherence properties are used to illustrate the elementary field model. Comparison with the SC coherent-mode expansion is presented, and we also simulate the propagation of the elementary fields in a dispersive fiber to demonstrate the benefits of the model.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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2012 (1)

2011 (2)

2010 (4)

2009 (1)

2008 (1)

2007 (2)

2006 (3)

2005 (1)

2002 (1)

2000 (1)

1998 (1)

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 (1)

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]

1994 (1)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

1989 (1)

Aalto, A.

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]

Coen, S.

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

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]

Douay, M.

Dudley, J. M.

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Special Topics 185, 135–144 (2010).
[CrossRef]

J. M. Dudley, G. Genty, and B. J. Eggleton, “Harnessing and control of optical rogue waves in supercontinuum generation,” Opt. Express 16, 3644–3651 (2008).
[CrossRef]

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

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]

Eggleton, B. J.

Erkintalo, M.

M. Erkintalo, M. Surakka, J. Turunen, A. T. Friberg, and G. Genty, “Coherent-mode representation of supercontinuum,” Opt. Lett. 37, 169–171 (2012).
[CrossRef]

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Special Topics 185, 135–144 (2010).
[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]

Friberg, A. T.

Frosz, M.

Genty, G.

Hänsch, T. W.

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974), Chap. 23, p. 161.

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]

Jalali, B.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Kolner, B. H.

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
[CrossRef]

Kolobov, M.

Koonath, P.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Kudlinski, A.

Lajunen, H.

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974), Chap. 23, p. 161.

Louvergneaux, E.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mussot, A.

Nazarathy, M.

Ropers, C.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[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]

Solli, D. R.

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Surakka, M.

Taki, M.

Tervo, J.

Toivonen, J.

Torres-Company, V.

Trebino, R.

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer, 2002).

Turunen, J.

Vahimaa, P.

Walmsley, I. A.

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]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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]

Eur. Phys. J. Special Topics (1)

M. Erkintalo, G. Genty, and J. M. Dudley, “On the statistical interpretation of optical rogue waves,” Eur. Phys. J. Special Topics 185, 135–144 (2010).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

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. Mod. Opt. (1)

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Nature (1)

D. R. Solli, C. Ropers, P. Koonath, and B. Jalali, “Optical rogue waves,” Nature 450, 1054–1057 (2007).
[CrossRef]

Opt. Express (7)

Opt. Lett. (5)

Pure Appl. Opt. (1)

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. (1)

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

Other (5)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

R. R. Alfano, ed., The Supercontinuum Laser Source, 2nd ed. (Springer, 2006).

J. M. Dudley and J. R. Taylor, eds., Supercontinuum Generation in Optical Fibers (Cambridge University, 2010).

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer, 2002).

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974), Chap. 23, p. 161.

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Figures (3)

Fig. 1.
Fig. 1.

Intensities of the quasi-coherent elementary field (first column), the quasi-stationary elementary field (second column), and the quasi-stationary weight function (third column). The average SC temporal intensity computed from the ensembles (gray) and with the elementary field method (black) is also shown (last column). The peak power of the input pulses used to generate the SC is 1 kW (top row), 500 W (middle row), and 250 W (bottom row).

Fig. 2.
Fig. 2.

Absolute values of the normalized MCFs of the three simulated ensembles of SC pulses (1 kW, 500 W, and 250 W) constructed from the ensemble data (top), the elementary field model (middle), and the coherent-mode expansion (bottom).

Fig. 3.
Fig. 3.

Intensities of pulses after propagating a distance of 1 m in a linear fiber, computed from the simulated ensembles (top row), with elementary field reconstruction (middle row) and with coherent-mode expansion (bottom row).

Tables (2)

Tables Icon

Table 1. Taylor-Series Expansion Coefficients of the Dispersion of the Fiber Used in the Numerical Simulations

Tables Icon

Table 2. Comparison between the Degree of Coherence Calculated Directly from the SC Ensembles, from the Elementary Representation, and from the Coherent-Mode Expansiona

Equations (24)

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W(ω1,ω2)=E˜*(ω1)E˜(ω2),
Γ(t1,t2)=E*(t1)E(t2),
μ(ω1,ω2)=W(ω1,ω2)/S(ω1)S(ω2),
γ(t1,t2)=Γ(t1,t2)/I(t1)I(t2),
μ¯2=1E02|Γ(t1,t2)|2dt1dt2,
W(ω1,ω2)=Wqc(ω1,ω2)+Wqs(ω1,ω2),
Γ(t1,t2)=Γqc(t1,t2)+Γqs(t1,t2),
S(ω)=Sqc(ω)+Sqs(ω),
I(t)=Iqc(t)+Iqs(t).
Γ(t1,t2)=pqc(t)fqc*(t1t)fqc(t2t)dt+pqs(t)fqs*(t1t)fqs(t2t)dt,
Γ(t1,t2)=fqc*(t1)fqc(t2)+pqs(t)fqs*(t1t)fqs(t2t)dt.
Iqc(t)=|fqc(t)|2,
Iqs(t)=pqs(t)|fqs(tt)|2dt,
fqc(t)=Iqc(t)exp[iϕqc(t)],
fqs(t)=Sqs(ω)exp(iωt)dω.
Δμ¯=|μ¯μ¯e,c|,
e2(z)=|I(t;z)Ie,c(t;z)|2dt|I(t;z)|2dt,
Γ(t1,t2)=n=1λnψn*(t1)ψn(t2),
Γ(t1,t2)ψn(t1)dt1=λnψn(t2).
f(t;z)=0F(ω;0)exp[iD(ω)z]exp(iωt)dω,
Ie(t;z)=|fqc(t;z)|2+pqs(t)|fqs(tt;z)|2dt.
Ic(t;z)=n=1Nλn|ψn(t;z)|2,
ψ(t;z)=Ψ(ω;0)exp[iD(ω)z]exp(iωt)dω,
Ψ(ω;0)=ψ(t;0)exp(iωt)dt.

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