Coherence of bulk-generated supercontinuum

Atri Halder; Vytautas Jukna; Matias Koivurova; Audrius Dubietis; Jari Turunen

doi:10.1364/PRJ.7.001345

Coherence of bulk-generated supercontinuum

Atri Halder, Vytautas Jukna, Matias Koivurova, Audrius Dubietis, and Jari Turunen

Photonics Research
Vol. 7,
Issue 11,
pp. 1345-1353
(2019)
•https://doi.org/10.1364/PRJ.7.001345

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Atri Halder, Vytautas Jukna, Matias Koivurova, Audrius Dubietis, and Jari Turunen, "Coherence of bulk-generated supercontinuum," Photon. Res. 7, 1345-1353 (2019)

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Related Topics
Table of Contents Category
- Nonlinear Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: June 5, 2019
- Revised Manuscript: September 2, 2019
- Manuscript Accepted: September 8, 2019
- Published: November 1, 2019

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Open Access

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.

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References

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|
Author
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Publication

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fibers,” Rev. Mod. Phys. 78, 1135–1184 (2006).
[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]
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]
I. A. Walmsley and C. Dorrer, “Characterization of ultrashort electromagnetic pulses,” Adv. Opt. Photon. 1, 308–437 (2009).
[Crossref]
G. Genty, M. Surakka, J. Turunen, and A. T. Friberg, “Second-order coherence of supercontinuum light,” Opt. Lett. 35, 3057–3059 (2010).
[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]
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. 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. 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]
I. S. Zeylikovich and R. R. Alfano, “Coherence properties of the supercontinuum source,” Appl. Phys. B 77, 265–268 (2003).
[Crossref]
G. Genty, A. T. Friberg, and J. Turunen, “Coherence of supercontinuum light,” Progr. Opt. 61, 71–112 (2016).
[Crossref]
A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441, 47–190 (2007).
[Crossref]
D. Majus and A. Dubietis, “Statistical properties of ultrafast supercontinuum generated by femtosecond Gaussian and Bessel beams: a comparative study,” J. Opt. Soc. Am. B 30, 994–999 (2013).
[Crossref]
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]
A. van de Walle, M. Hanna, F. Guichard, Y. Zaouter, A. Thai, N. Forget, and P. Georges, “Spectral and spatial full-bandwidth correlation analysis of bulk-generated supercontinuum in the mid-infrared,” Opt. Lett. 40, 673–675 (2015).
[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]
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]
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]
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]
R. Dutta, J. Turunen, and A. T. Friberg, “Michelson’s interferometer and temporal coherence of pulse trains,” Opt. Lett. 40, 166–169 (2015).
[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]
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
M. Koivurova, C. Ding, J. Turunen, and A. T. Friberg, “Cross-spectral purity of nonstationary light,” Phys. Rev. A 99, 043842 (2019).
[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]
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]
M. J. Weber, Handbook of Optical Materials (CRC Press, 2002).
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]
A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29, 602–604 (2004).
[Crossref]
R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
[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]
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]
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]

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]

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]

2010 (1)

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

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)

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]

A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29, 602–604 (2004).
[Crossref]

2003 (2)

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]

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

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)

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
[Crossref]

Aitchison, J. S.

A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29, 602–604 (2004).
[Crossref]

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.

Audebert, P.

Bang, O.

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.

Coen, S.

J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fibers,” 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]

Corti, T.

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

Daguzan, P.

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.

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

Dos Santos, A.

Dubietis, 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]

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]

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]

Dutta, R.

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

Engelsholm, R. D.

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]

Feehan, J. S.

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]

Feurer, T.

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]

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.

Genty, G.

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

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]

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

Georges, P.

Gonzalo, I. B.

Gu, X.

Guichard, F.

Guizard, S.

Hanna, M.

Heidt, A. M.

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]

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]

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.

Korhonen, M.

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]

Lajunen, H.

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]

Major, A.

A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29, 602–604 (2004).
[Crossref]

Majus, D.

Mandel, L.

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

Martin, P.

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.

A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29, 602–604 (2004).
[Crossref]

Petite, G.

Pileckis, E.

Price, J. H. V.

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]

Ramírez-Góngora, O. de J.

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.

A. Major, F. Yoshino, I. Nikolakakos, J. S. Aitchison, and P. W. E. Smith, “Dispersion of the nonlinear refractive index in sapphire,” Opt. Lett. 29, 602–604 (2004).
[Crossref]

Smith, R. G.

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
[Crossref]

Sørensen, M. P.

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.

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]

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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.

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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.

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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.

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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 {mm}^{- 1}

. Red curves show temporal pulse amplitude profiles in (a) and (b), and spectral amplitude profiles in (c) and (d).

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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 (

F_{t}

) and integrated over wavelengths (

F_{s}

). 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.

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Fig. 6. Comparison between the spectral density distributions taken at particular spatial frequencies

κ_{ρ}

(blue) and at corresponding real diffraction angles

θ

(red). Pump wavelength is 1300 nm. (a)

θ = 0.32 °

κ_{ρ} = 26 {mm}^{- 1}

. (b)

θ = 1.78 °

κ_{ρ} = 150 {mm}^{- 1}

. (c)

θ = 3.60 °

κ_{ρ} = 300 {mm}^{- 1}

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Fig. 7. Comparison between the overall degree of coherence as a function of spatial frequency (

ν_{κ}

, blue line), and as a function of real diffraction angle (

ν_{θ}

, red crosses) calculated from the spatial frequency component at 1300 nm.

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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.

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Tables (1)

Table 1. Relevant Linear and Nonlinear Parameters of Sapphire Crystal at the Wavelengths of Interest^a

View Table

Equations (36)

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E (ρ; t) = \int_{0}^{\infty} \tilde{E} (ρ; ω) \exp (- i ω t) d ω .

A (κ; t) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} E (ρ; t) \exp (- i κ \cdot ρ) d^{2} ρ

\tilde{A} (κ; ω) = \frac{1}{{(2 π)}^{2}} \int_{- \infty}^{\infty} \tilde{E} (ρ; ω) \exp (- i κ \cdot ρ) d^{2} ρ

Γ (ρ_{1}, ρ_{2}; t_{1}, t_{2}) = ⟨ E^{*} (ρ_{1}; t_{1}) E (ρ_{2}; t_{2}) ⟩ .

W (ρ_{1}, ρ_{2}; ω_{1}, ω_{2}) = ⟨ {\tilde{E}}^{*} (ρ_{1}; ω_{1}) \tilde{E} (ρ_{2}; ω_{2}) ⟩ .

G (κ_{1}, κ_{2}; t_{1}, t_{2}) = ⟨ A^{*} (κ_{1}; t_{1}) A (κ_{2}; t_{2}) ⟩

T (κ_{1}, κ_{2}; ω_{1}, ω_{2}) = ⟨ {\tilde{A}}^{*} (κ_{1}; ω_{1}) \tilde{A} (κ_{2}; ω_{2}) ⟩ .

⟨ f (x) ⟩ = \lim_{N \to \infty} \frac{1}{N} \sum_{n = 1}^{N} f_{n} (x),

γ (ρ_{1}, ρ_{2}; t_{1}, t_{2}) = \frac{Γ (ρ_{1}, ρ_{2}; t_{1}, t_{2})}{\sqrt{I (ρ_{1}; t_{1}) I (ρ_{2}; t_{2})}},

μ (ρ_{1}, ρ_{2}; ω_{1}, ω_{2}) = \frac{W (ρ_{1}, ρ_{2}; ω_{1}, ω_{2})}{\sqrt{S (ρ_{1}; ω_{1}) S (ρ_{2}; ω_{2})}},

g (κ_{1}, κ_{2}; t_{1}, t_{2}) = \frac{G (κ_{1}, κ_{2}; t_{1}, t_{2})}{\sqrt{I (κ_{1}; t_{1}) I (κ_{2}; t_{2})}},

ν (κ_{1}, κ_{2}; ω_{1}, ω_{2}) = \frac{T (κ_{1}, κ_{2}; ω_{1}, ω_{2})}{\sqrt{S (κ_{1}; ω_{1}) S (κ_{2}; ω_{2})}} .

G (κ_{1}, κ_{2}; t_{1}, t_{2}) = \frac{1}{{(2 π)}^{4}} \iint_{- \infty}^{\infty} Γ (ρ_{1}, ρ_{2}; t_{1}, t_{2}) \times \exp [i (κ_{1} \cdot ρ_{1} - κ_{2} \cdot ρ_{2})] d ρ_{1} d ρ_{2}

T (κ_{1}, κ_{2}; ω_{1}, ω_{2}) = \frac{1}{{(2 π)}^{4}} \iint_{- \infty}^{\infty} W (ρ_{1}, ρ_{2}; ω_{1}, ω_{2}) \times \exp [i (κ_{1} \cdot ρ_{1} - κ_{2} \cdot ρ_{2})] d ρ_{1} d ρ_{2} .

{\bar{γ}}^{2} (ρ) = \frac{\iint_{- \infty}^{\infty} {| Γ (ρ, ρ; t_{1}, t_{2}) |}^{2} d t_{1} d t_{2}}{\iint_{- \infty}^{\infty} I (ρ; t_{1}) I (ρ; t_{2}) d t_{1} d t_{2}}

{\bar{μ}}^{2} (ρ) = \frac{\iint_{0}^{\infty} {| W (ρ, ρ; ω_{1}, ω_{2}) |}^{2} d ω_{1} d ω_{2}}{\iint_{0}^{\infty} S (ρ; ω_{1}) S (ρ; ω_{2}) d ω_{1} d ω_{2}} .

{\bar{g}}^{2} (κ) = \frac{\iint_{- \infty}^{\infty} {| G (κ, κ; t_{1}, t_{2}) |}^{2} d t_{1} d t_{2}}{\iint_{- \infty}^{\infty} I (κ; t_{1}) I (κ; t_{2}) d t_{1} d t_{2}}

{\bar{ν}}^{2} (κ) = \frac{\iint_{0}^{\infty} {| T (κ, κ; ω_{1}, ω_{2}) |}^{2} d ω_{1} d ω_{2}}{\iint_{0}^{\infty} S (κ; ω_{1}) S (κ; ω_{2}) d ω_{1} d ω_{2}} .

\hat{s} = (s_{x}, s_{y}, s_{z}) = (\hat{σ}, s_{z}) = (\sin θ \cos ϕ, \sin θ \sin ϕ, \cos θ)

k = (k_{x}, k_{y}, k_{z}) = (κ, k_{z}) = k \hat{s},

r = (x, y, z) = (ρ, z) = r \hat{s},

{\tilde{E}}^{(\infty)} (r; ω) = - i 2 π k s_{z} \tilde{A} (k \hat{σ}; ω) \frac{\exp (i k r)}{r},

W^{(\infty)} (r, r; ω_{1}, ω_{2}) = ⟨ {\tilde{E}}^{(\infty) *} (r; ω_{1}) {\tilde{E}}^{(\infty)} (r; ω_{2}) ⟩ .

W^{(\infty)} (r, r; ω_{1}, ω_{2}) = {(2 π s_{z})}^{2} \frac{ω_{1} ω_{2}}{c^{2}} T (k_{1} \hat{σ}, k_{2} \hat{σ}; ω_{1}, ω_{2}) \times \frac{\exp [i (k_{2} - k_{1}) r]}{r^{2}},

W^{(\infty)} (r, r; ω, ω) = S^{(\infty)} (r; ω) = {(\frac{2 π k s_{z}}{r})}^{2} S (k \hat{σ}; ω),

μ^{(\infty)} (r, r; ω_{1}, ω_{2}) = \frac{T (k_{1} \hat{σ}, k_{2} \hat{σ}; ω_{1}, ω_{2})}{\sqrt{S (k_{1} \hat{σ}; ω_{1}) S (k_{2} \hat{σ}; ω_{2})}} .

E^{(\infty)} (r; t) = \int_{0}^{\infty} {\tilde{E}}^{(\infty)} (r; ω) \exp (- i ω t) d ω,

E^{(\infty)} (r; t) = - \frac{i 2 π s_{z}}{r c} \int_{0}^{\infty} ω \tilde{A} (k σ; ω) \times \exp [- i ω (t - r / c)] d ω,

Γ^{(\infty)} (r, r; t_{1}, t_{2}) = ⟨ E^{(\infty) *} (r; t_{1}) E^{(\infty)} (r; t_{2}) ⟩,

Γ^{(\infty)} (r, r; t_{1}, t_{2}) = {(\frac{2 π s_{z}}{r c})}^{2} \iint_{0}^{\infty} T (k_{1} \hat{σ}, k_{2} \hat{σ}; ω_{1}, ω_{2}) \times ω_{1} ω_{2} \exp [i r (k_{2} - k_{1})] \times \exp [i (ω_{1} t_{1} - ω_{2} t_{2})] d ω_{1} d ω_{2} .

Γ^{(\infty)} (r, r; t_{1}, t_{2}) = {(\frac{2 π s_{z} k_{0}}{r})}^{2} G (k_{0} \hat{σ}, k_{0} \hat{σ}; t_{1}, t_{2}) .

T (k_{1} {\hat{σ}}_{1}, k_{2} {\hat{σ}}_{2}; ω_{1}, ω_{2}) = T_{σ} (k_{0} {\hat{σ}}_{1}, k_{0} {\hat{σ}}_{2}) T_{ω} (ω_{1}, ω_{2}) .

\frac{\partial \tilde{A} (κ, ω; z)}{\partial z} = i [\sqrt{k^{2} (ω) - κ^{2}} - k (ω_{0}) - k_{g} (ω_{0})] \times \tilde{A} (κ, ω; z) + i \frac{ω}{2 n (ω) c} ϵ_{0}^{- 1} [\tilde{P} (κ, ω; z) + i \frac{\tilde{J} (κ, ω; z)}{ω}] .

ϵ_{0}^{- 1} P (ρ, t; z) = 2 n_{0} n_{2} {| E (ρ, t; z) |}^{2} E (ρ, t; z),

ϵ_{0}^{- 1} J (ρ, t; z) = n_{0} c [σ_{B} (1 + i ω_{0} τ_{c}) ρ_{e} + U_{g} \frac{W (ρ, t; z)}{{| E (ρ, t; z) |}^{2}} (1 - \frac{ρ_{e}}{ρ_{n t}})] E (ρ, t; z),

\frac{\partial ρ_{e}}{\partial t} = (1 - \frac{ρ_{e}}{ρ_{n t}}) \times [W (ρ, z; t) + \frac{σ_{B}}{U_{g}} {| E (ρ, z; t) |}^{2} ρ_{e}] - \frac{ρ_{e}}{τ_{rec}},

$λ$ (μm)	800	1300	2000
$n_{0}$	1.76	1.75	1.73
$g$ ( ${fs}^{2} / mm$ )	$+ 58.1$	$+ 1.8$	−121.8
$K$	7	11	16
$n_{2}$ ( $\times 10^{- 16} {cm}^{2} / W$ )	3.11	2.89	2.7
$σ_{B}$ ( $\times 10^{- 22} m^{2}$ )	9.2	19.6	32.4

Abstract

References

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