Abstract

We have developed a numerical framework that allows estimation of coherence in spatiotemporal and spatiospectral domains. Correlation properties of supercontinuum (SC) pulses generated in a bulk medium are investigated by means of second-order coherence theory of non-stationary fields. The analysis is based on simulations of individual space–time and space–frequency realizations of pulses emerging from a 5 mm thick sapphire plate, in the regimes of normal, zero, and anomalous group velocity dispersion. The temporal and spectral coherence properties are analyzed in the near field (as a function of spatial position at the exit plane of the nonlinear medium) and as a function of propagation direction (spatial frequency) in the far field. Unlike in fiber-generated SC, the bulk case features spectacularly high degrees of temporal and spectral coherence in both the spatial and spatial-frequency domains, with increasing degrees of coherence at higher pump energies. When operating near the SC generation threshold, the overall degrees of temporal and spectral coherence exhibit an axial dip in the spatial domain, whereas in the far field, the degree of coherence is highest around the optical axis.

© 2019 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
  3. 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]
  4. 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]
  5. I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009).
    [Crossref]
  6. G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Second-order coherence of supercontinuum light,” Opt. Lett. 35, 3057–3059 (2010).
    [Crossref]
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    [Crossref]
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    [Crossref]
  9. M. Korhonen, A. T. Friberg, J. Turunen, and G. Genty, “Elementary field representation of supercontinuum,” J. Opt. Soc. Am. B 30, 21–26 (2013).
    [Crossref]
  10. M. Närhi, J. Turunen, A. T. Friberg, and G. Genty, “Experimental measurement of the second-order coherence of supercontinuum,” Phys. Rev. Lett. 116, 243901 (2016).
    [Crossref]
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    [Crossref]
  12. G. Genty, A. T. Friberg, and J. Turunen, “Coherence of supercontinuum light,” Progr. Opt. 61, 71–112 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  22. R. Dutta, J. Turunen, and A. T. Friberg, “Two-time coherence of pulse trains and the integrated degree of temporal coherence,” J. Opt. Soc. Am. A 32, 1631–1637 (2015).
    [Crossref]
  23. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
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    [Crossref]
  25. M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host material with sub-μJ pump pulses,” Appl. Phys. B 97, 561–574 (2009).
    [Crossref]
  26. A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  33. I. B. Gonzalo, R. D. Engelsholm, M. P. Sørensen, and O. Bang, “Polarization noise places severe constraints on coherence of all-normal dispersion femtosecond supercontinuum generation,” Sci. Rep. 8, 6579 (2018).
    [Crossref]

2019 (1)

M. Koivurova, C. Ding, J. Turunen, and A. T. Friberg, “Cross-spectral purity of nonstationary light,” Phys. Rev. A 99, 043842 (2019).
[Crossref]

2018 (1)

I. B. Gonzalo, R. D. Engelsholm, M. P. Sørensen, and O. Bang, “Polarization noise places severe constraints on coherence of all-normal dispersion femtosecond supercontinuum generation,” Sci. Rep. 8, 6579 (2018).
[Crossref]

2017 (2)

A. M. Heidt, J. S. Feehan, J. H. V. Price, and T. Feurer, “Limits of coherent supercontinuum generation in normal dispersion fibers,” J. Opt. Soc. Am. B 34, 764–775 (2017).
[Crossref]

A. Dubietis, G. Tamoauskas, R. Suminas, V. Jukna, and A. Couairon, “Ultrafast supercontinuum generation in bulk condensed media,” Lith. J. Phys. 57, 113–157 (2017).
[Crossref]

2016 (2)

M. Närhi, J. Turunen, A. T. Friberg, and G. Genty, “Experimental measurement of the second-order coherence of supercontinuum,” Phys. Rev. Lett. 116, 243901 (2016).
[Crossref]

G. Genty, A. T. Friberg, and J. Turunen, “Coherence of supercontinuum light,” Progr. Opt. 61, 71–112 (2016).
[Crossref]

2015 (3)

2014 (1)

2013 (2)

2012 (1)

2011 (3)

2010 (1)

2009 (2)

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

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host material with sub-μJ pump pulses,” Appl. Phys. B 97, 561–574 (2009).
[Crossref]

2007 (2)

K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: general theory and application to electromagnetic fields,” J. Opt. A 9, 907–919 (2007).
[Crossref]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–190 (2007).
[Crossref]

2006 (1)

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

2004 (3)

2003 (2)

2002 (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]

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]

1995 (1)

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

1972 (1)

Aitchison, J. S.

Alfano, R. R.

I. S. Zeylikovich and R. R. Alfano, “Coherence properties of the supercontinuum source,” Appl. Phys. B 77, 265–268 (2003).
[Crossref]

Antonnetti, A.

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

Audebert, P.

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

Bang, O.

I. B. Gonzalo, R. D. Engelsholm, M. P. Sørensen, and O. Bang, “Polarization noise places severe constraints on coherence of all-normal dispersion femtosecond supercontinuum generation,” Sci. Rep. 8, 6579 (2018).
[Crossref]

Baum, P.

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host material with sub-μJ pump pulses,” Appl. Phys. B 97, 561–574 (2009).
[Crossref]

Blomstedt, K.

K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: general theory and application to electromagnetic fields,” J. Opt. A 9, 907–919 (2007).
[Crossref]

Bradler, M.

M. Bradler and E. Riedle, “Temporal and spectral correlations in bulk continua and improved use in transient spectroscopy,” J. Opt. Soc. Am. B 31, 1465–1475 (2014).
[Crossref]

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host material with sub-μJ pump pulses,” Appl. Phys. B 97, 561–574 (2009).
[Crossref]

Brambilla, E.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Coen, S.

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

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]

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]

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]

Corti, T.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Couairon, A.

A. Dubietis, G. Tamoauskas, R. Suminas, V. Jukna, and A. Couairon, “Ultrafast supercontinuum generation in bulk condensed media,” Lith. J. Phys. 57, 113–157 (2017).
[Crossref]

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–190 (2007).
[Crossref]

Daguzan, P.

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

Ding, C.

M. Koivurova, C. Ding, J. Turunen, and A. T. Friberg, “Cross-spectral purity of nonstationary light,” Phys. Rev. A 99, 043842 (2019).
[Crossref]

Dorrer, C.

Dos Santos, A.

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

Dubietis, A.

Dudley, J. M.

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

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]

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]

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]

Dutta, R.

Engelsholm, R. D.

I. B. Gonzalo, R. D. Engelsholm, M. P. Sørensen, and O. Bang, “Polarization noise places severe constraints on coherence of all-normal dispersion femtosecond supercontinuum generation,” Sci. Rep. 8, 6579 (2018).
[Crossref]

Erkintalo, M.

Feehan, J. S.

Feurer, T.

Forget, N.

Friberg, A. T.

M. Koivurova, C. Ding, J. Turunen, and A. T. Friberg, “Cross-spectral purity of nonstationary light,” Phys. Rev. A 99, 043842 (2019).
[Crossref]

M. Närhi, J. Turunen, A. T. Friberg, and G. Genty, “Experimental measurement of the second-order coherence of supercontinuum,” Phys. Rev. Lett. 116, 243901 (2016).
[Crossref]

G. Genty, A. T. Friberg, and J. Turunen, “Coherence of supercontinuum light,” Progr. Opt. 61, 71–112 (2016).
[Crossref]

R. Dutta, J. Turunen, and A. T. Friberg, “Two-time coherence of pulse trains and the integrated degree of temporal coherence,” J. Opt. Soc. Am. A 32, 1631–1637 (2015).
[Crossref]

R. Dutta, J. Turunen, and A. T. Friberg, “Michelson’s interferometer and temporal coherence of pulse trains,” Opt. Lett. 40, 166–169 (2015).
[Crossref]

M. Korhonen, A. T. Friberg, J. Turunen, and G. Genty, “Elementary field representation of supercontinuum,” J. Opt. Soc. Am. B 30, 21–26 (2013).
[Crossref]

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

G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Complete characterization of supercontinuum coherence,” J. Opt. Soc. Am. B 28, 2301–2309 (2011).
[Crossref]

G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Second-order coherence of supercontinuum light,” Opt. Lett. 35, 3057–3059 (2010).
[Crossref]

K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: general theory and application to electromagnetic fields,” J. Opt. A 9, 907–919 (2007).
[Crossref]

Geindre, J. P.

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

Genty, G.

Georges, P.

Gonzalo, I. B.

I. B. Gonzalo, R. D. Engelsholm, M. P. Sørensen, and O. Bang, “Polarization noise places severe constraints on coherence of all-normal dispersion femtosecond supercontinuum generation,” Sci. Rep. 8, 6579 (2018).
[Crossref]

Gu, X.

Guichard, F.

Guizard, S.

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

Hanna, M.

Heidt, A. M.

Jukna, V.

A. Dubietis, G. Tamoauskas, R. Suminas, V. Jukna, and A. Couairon, “Ultrafast supercontinuum generation in bulk condensed media,” Lith. J. Phys. 57, 113–157 (2017).
[Crossref]

D. Majus, V. Jukna, E. Pileckis, G. Valiulis, and A. Dubietis, “Rogue-wave-like statistics in ultrafast white-light continuum generation in sapphire,” Opt. Express 19, 16317–16323 (2011).
[Crossref]

Kimmel, M.

Koivurova, M.

M. Koivurova, C. Ding, J. Turunen, and A. T. Friberg, “Cross-spectral purity of nonstationary light,” Phys. Rev. A 99, 043842 (2019).
[Crossref]

Kolesik, M.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Korhonen, M.

Lajunen, H.

Major, A.

Majus, D.

Mandel, L.

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

Martin, P.

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

Mysyrowicz, A.

A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–190 (2007).
[Crossref]

Närhi, M.

M. Närhi, J. Turunen, A. T. Friberg, and G. Genty, “Experimental measurement of the second-order coherence of supercontinuum,” Phys. Rev. Lett. 116, 243901 (2016).
[Crossref]

Nikolakakos, I.

Petite, G.

S. Guizard, P. Martin, P. Daguzan, G. Petite, P. Audebert, J. P. Geindre, A. Dos Santos, and A. Antonnetti, “Contrasted behaviour of an electron gas in MgO, Al2O3 and SiO2,” Europhys. Lett. 29, 401 (1995).
[Crossref]

Pileckis, E.

Price, J. H. V.

Ramírez-Góngora, O. de J.

A. Couairon, E. Brambilla, T. Corti, D. Majus, O. de J. Ramírez-Góngora, and M. Kolesik, “Practitioner’s guide to laser pulse propagation models and simulation,” Eur. Phys. J. Spec. Top. 199, 5–76 (2011).
[Crossref]

Riedle, E.

M. Bradler and E. Riedle, “Temporal and spectral correlations in bulk continua and improved use in transient spectroscopy,” J. Opt. Soc. Am. B 31, 1465–1475 (2014).
[Crossref]

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host material with sub-μJ pump pulses,” Appl. Phys. B 97, 561–574 (2009).
[Crossref]

Setälä, T.

K. Blomstedt, T. Setälä, and A. T. Friberg, “Effective degree of coherence: general theory and application to electromagnetic fields,” J. Opt. A 9, 907–919 (2007).
[Crossref]

Shreenath, A. P.

Smith, P. W. E.

Smith, R. G.

Sørensen, M. P.

I. B. Gonzalo, R. D. Engelsholm, M. P. Sørensen, and O. Bang, “Polarization noise places severe constraints on coherence of all-normal dispersion femtosecond supercontinuum generation,” Sci. Rep. 8, 6579 (2018).
[Crossref]

Suminas, R.

A. Dubietis, G. Tamoauskas, R. Suminas, V. Jukna, and A. Couairon, “Ultrafast supercontinuum generation in bulk condensed media,” Lith. J. Phys. 57, 113–157 (2017).
[Crossref]

Surakka, M.

Tamoauskas, G.

A. Dubietis, G. Tamoauskas, R. Suminas, V. Jukna, and A. Couairon, “Ultrafast supercontinuum generation in bulk condensed media,” Lith. J. Phys. 57, 113–157 (2017).
[Crossref]

Tervo, J.

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
[Crossref]

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[Crossref]

Thai, A.

Trebino, R.

Turunen, J.

Vahimaa, P.

H. Lajunen, J. Tervo, and P. Vahimaa, “Overall coherence and coherent-mode expansion of spectrally partially coherent plane-wave pulses,” J. Opt. Soc. Am. A 21, 2117–2123 (2004).
[Crossref]

P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
[Crossref]

Valiulis, G.

van de Walle, A.

Walmsley, I. A.

Weber, M. J.

M. J. Weber, Handbook of Optical Materials (CRC Press, 2002).

Windeler, R. S.

Wolf, E.

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

Yoshino, F.

Zaouter, Y.

Zeylikovich, I. S.

I. S. Zeylikovich and R. R. Alfano, “Coherence properties of the supercontinuum source,” Appl. Phys. B 77, 265–268 (2003).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Opt. (1)

Appl. Phys. B (2)

M. Bradler, P. Baum, and E. Riedle, “Femtosecond continuum generation in bulk laser host material with sub-μJ pump pulses,” Appl. Phys. B 97, 561–574 (2009).
[Crossref]

I. S. Zeylikovich and R. R. Alfano, “Coherence properties of the supercontinuum source,” Appl. Phys. B 77, 265–268 (2003).
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Figures (8)

Fig. 1.
Fig. 1. Evolution of the output pulse intensity and spectral density at the center of the beam as a function of pump pulse energy. (a), (c), (e) Temporal intensity and (b), (d), (f) spectral density. Here (a), (b) correspond to 800 nm, (c), (d) to 1300 nm, and (e), (f) to 2000 nm pump pulses.
Fig. 2.
Fig. 2. Evolutions of beam radius over propagation distance for pulses with central wavelengths of 800 nm (blue), 1300 nm (green), and 2000 nm (red), having input energies of 0.282 μJ, 0.569 μJ, and 1.175 μJ, respectively.
Fig. 3.
Fig. 3. (a), (c), (e) Spatiotemporal intensity profiles of the pulses at the exit plane of a crystal and (b), (d), (f) corresponding spatial frequency-resolved spectra. Subplots (a), (b) correspond to 800 nm, (c), (d) to 1300 nm, and (e), (f) to 2000 nm pump wavelengths, having input energies of 0.282 μJ, 0.569 μJ, and 1.175 μJ, respectively.
Fig. 4.
Fig. 4. Absolute values of normalized degrees of coherence. Spatial degrees of temporal coherence at (a) ρ=0 and (b) ρ=0.018  mm. Angular degrees of spectral coherence at (c) κ=0 and (d) κ=300  mm1. Red curves show temporal pulse amplitude profiles in (a) and (b), and spectral amplitude profiles in (c) and (d).
Fig. 5.
Fig. 5. Overall degrees of coherence at the exit plane, plotted as functions of spatial position ρ (left, red) and spatial frequency κ (right, red) together with the corresponding normalized field intensity distribution (blue) integrated over time (Ft) and integrated over wavelengths (Fs). The pump wavelength is 800 nm for the upper row, where we have considered pump energies just below the threshold at 0.277 μJ (solid), at threshold 0.280 μJ (dashed), and just above threshold 0.282 μJ (dotted). Similarly, for the 1300 nm pump wavelength in the middle row, the energies are 0.565 μJ (solid), 0.567 μJ (dashed), and 0.569 μJ (dotted), and at 2000 nm in the bottom row, they are 1.173 μJ (solid), 1.175 μJ (dashed), and 1.18 μJ (dotted). Note that the horizontal axes in the left column have different scales.
Fig. 6.
Fig. 6. Comparison between the spectral density distributions taken at particular spatial frequencies κρ (blue) and at corresponding real diffraction angles θ (red). Pump wavelength is 1300 nm. (a) θ=0.32°, κρ=26  mm1. (b) θ=1.78°, κρ=150  mm1. (c) θ=3.60°, κρ=300  mm1.
Fig. 7.
Fig. 7. Comparison between the overall degree of coherence as a function of spatial frequency (νκ, blue line), and as a function of real diffraction angle (νθ, red crosses) calculated from the spatial frequency component at 1300 nm.
Fig. 8.
Fig. 8. Overall degrees of temporal coherence calculated for pump energies well above the SC generation threshold: 0.31 μJ at 800 nm, 0.65 μJ at 1300 nm, and 1.25 μJ at 2000 nm. The notations are the same as in Fig. 5.

Tables (1)

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Table 1. Relevant Linear and Nonlinear Parameters of Sapphire Crystal at the Wavelengths of Interesta

Equations (36)

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E(ρ;t)=0E˜(ρ;ω)exp(iωt)dω.
A(κ;t)=1(2π)2E(ρ;t)exp(iκ·ρ)d2ρ
A˜(κ;ω)=1(2π)2E˜(ρ;ω)exp(iκ·ρ)d2ρ
Γ(ρ1,ρ2;t1,t2)=E*(ρ1;t1)E(ρ2;t2).
W(ρ1,ρ2;ω1,ω2)=E˜*(ρ1;ω1)E˜(ρ2;ω2).
G(κ1,κ2;t1,t2)=A*(κ1;t1)A(κ2;t2)
T(κ1,κ2;ω1,ω2)=A˜*(κ1;ω1)A˜(κ2;ω2).
f(x)=limN1Nn=1Nfn(x),
γ(ρ1,ρ2;t1,t2)=Γ(ρ1,ρ2;t1,t2)I(ρ1;t1)I(ρ2;t2),
μ(ρ1,ρ2;ω1,ω2)=W(ρ1,ρ2;ω1,ω2)S(ρ1;ω1)S(ρ2;ω2),
g(κ1,κ2;t1,t2)=G(κ1,κ2;t1,t2)I(κ1;t1)I(κ2;t2),
ν(κ1,κ2;ω1,ω2)=T(κ1,κ2;ω1,ω2)S(κ1;ω1)S(κ2;ω2).
G(κ1,κ2;t1,t2)=1(2π)4Γ(ρ1,ρ2;t1,t2)×exp[i(κ1·ρ1κ2·ρ2)]dρ1dρ2
T(κ1,κ2;ω1,ω2)=1(2π)4W(ρ1,ρ2;ω1,ω2)×exp[i(κ1·ρ1κ2·ρ2)]dρ1dρ2.
γ¯2(ρ)=|Γ(ρ,ρ;t1,t2)|2dt1dt2I(ρ;t1)I(ρ;t2)dt1dt2
μ¯2(ρ)=0|W(ρ,ρ;ω1,ω2)|2dω1dω20S(ρ;ω1)S(ρ;ω2)dω1dω2.
g¯2(κ)=|G(κ,κ;t1,t2)|2dt1dt2I(κ;t1)I(κ;t2)dt1dt2
ν¯2(κ)=0|T(κ,κ;ω1,ω2)|2dω1dω20S(κ;ω1)S(κ;ω2)dω1dω2.
s^=(sx,sy,sz)=(σ^,sz)=(sinθcosϕ,sinθsinϕ,cosθ)
k=(kx,ky,kz)=(κ,kz)=ks^,
r=(x,y,z)=(ρ,z)=rs^,
E˜()(r;ω)=i2πkszA˜(kσ^;ω)exp(ikr)r,
W()(r,r;ω1,ω2)=E˜()*(r;ω1)E˜()(r;ω2).
W()(r,r;ω1,ω2)=(2πsz)2ω1ω2c2T(k1σ^,k2σ^;ω1,ω2)×exp[i(k2k1)r]r2,
W()(r,r;ω,ω)=S()(r;ω)=(2πkszr)2S(kσ^;ω),
μ()(r,r;ω1,ω2)=T(k1σ^,k2σ^;ω1,ω2)S(k1σ^;ω1)S(k2σ^;ω2).
E()(r;t)=0E˜()(r;ω)exp(iωt)dω,
E()(r;t)=i2πszrc0ωA˜(kσ;ω)×exp[iω(tr/c)]dω,
Γ()(r,r;t1,t2)=E()*(r;t1)E()(r;t2),
Γ()(r,r;t1,t2)=(2πszrc)20T(k1σ^,k2σ^;ω1,ω2)×ω1ω2exp[ir(k2k1)]×exp[i(ω1t1ω2t2)]dω1dω2.
Γ()(r,r;t1,t2)=(2πszk0r)2G(k0σ^,k0σ^;t1,t2).
T(k1σ^1,k2σ^2;ω1,ω2)=Tσ(k0σ^1,k0σ^2)Tω(ω1,ω2).
A˜(κ,ω;z)z=i[k2(ω)κ2k(ω0)kg(ω0)]×A˜(κ,ω;z)+iω2n(ω)cϵ01[P˜(κ,ω;z)+iJ˜(κ,ω;z)ω].
ϵ01P(ρ,t;z)=2n0n2|E(ρ,t;z)|2E(ρ,t;z),
ϵ01J(ρ,t;z)=n0c[σB(1+iω0τc)ρe+UgW(ρ,t;z)|E(ρ,t;z)|2(1ρeρnt)]E(ρ,t;z),
ρet=(1ρeρnt)×[W(ρ,z;t)+σBUg|E(ρ,z;t)|2ρe]ρeτrec,