Abstract

Diffraction and interferometry with fast pulses are analyzed for the case that the fields are partially correlated in time and in space. This generalizes a previous work [ Schoonover et al., J. Mod. Opt. 55, 1541 (2008) ], where only the temporal correlations of pulsed fields were considered in a Young’s interferometer. The meaning of the interferograms is addressed for measurements taken in the near, Fresnel, and far zones of the source. It is shown that single-shot measurements cannot generally be used to infer statistical properties of the source, rather, data averaged over many pulses must be used.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  2. E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge Univ. Press, 2007).
  3. N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117-258 (1930).
    [CrossRef]
  4. A. Khintchine, “Korrelationstheorie der stationären stochastischen Prozesse,” Math. Ann. 109, 604-615 (1934).
    [CrossRef]
  5. A. Einstein, “Méthode pour la détermination de valeurs statistiques d'observation concernant des grandeurs soumises à des fluctuations irrégulières,” Arch. Sci. Phys. Nat. 37, 254-256 (1914).
  6. H. Lajunen, J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Spectral coherence properties of temporally modulated stationary light sources,” Opt. Express 11, 1894-1899 (2003).
    [CrossRef] [PubMed]
  7. R. Schoonover, B. Davis, R. Bartels, and P. Carney, “Optical interferometry with pulsed fields,” J. Mod. Opt. 55, 1541-1556 (2008).
    [CrossRef]
  8. I. G. Fuss, “An interpretation of the spectral measurement of optical pulse-train noise,” IEEE J. Quantum Electron. 30, 2707-2710 (1994).
    [CrossRef]
  9. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
    [CrossRef] [PubMed]
  10. T. Udem, R. Holzwarth, and T. Haensch, “Optical frequency metrology,” Nature 416, 233-237 (2002).
    [CrossRef] [PubMed]
  11. V. Devrelis, M. O'Connor, and J. Munch, “Coherence length of single laser pulses as measured by CCD interferometry,” Appl. Opt. 34, 5386-5389 (1995).
    [CrossRef] [PubMed]
  12. V. Papadakis, A. Stassinopoulos, D. Anglos, S. H. Anastasiadis, E. P. Giannelis, and D. G. Papazoglou, “Single-shot temporal coherence measurements of random lasing media,” J. Opt. Soc. Am. B 24, 31-36 (2007).
    [CrossRef]
  13. R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
    [PubMed]
  14. X. Zhang, A. Libertun, A. Paul, E. Gagnon, S. Backus, I. Christov, M. Murnane, H. Kapteyn, R. Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matched high-harmonic generation,” Opt. Lett. 29, 1357-1359 (2004).
    [CrossRef] [PubMed]
  15. L. Le Déroff, P. Salières, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61, 35 (2000).
    [CrossRef]
  16. T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
    [CrossRef] [PubMed]
  17. T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
    [CrossRef]
  18. Y. Nagata, K. Furusawa, Y. Nabekawa, and K. Midorikawa, “Single-shot spatial-coherence measurement of 13 nm high-order harmonic beam by a Young's double-slit measurement,” Opt. Lett. 32, 722-724 (2007).
    [CrossRef] [PubMed]
  19. D. Eliyahu, R. A. Salvatore, and A. Yariv, “Noise characterization of a pulse train generated by actively mode-locked lasers,” J. Opt. Soc. Am. B 13, 1619-1626 (1996).
    [CrossRef]
  20. 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]
  21. R. Schoonover, B. Davis, and P. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705-4711 (2009).
    [CrossRef] [PubMed]
  22. W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
    [CrossRef]
  23. A. Yaglom, Correlation Theory of Stationary and Related Random Functions (Springer, 1987).
  24. B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
    [CrossRef]
  25. J. Eberly and K. Wodkiewicz, “The time-dependent physical spectrum of light,” J. Opt. Soc. Am. 67, 1252-1261 (1977).
    [CrossRef]
  26. M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117-2132 (1994).
    [CrossRef] [PubMed]
  27. J. Degnan, N. Center, and M. Greenbelt, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
    [CrossRef]
  28. V. Torres-Company, H. Lajunen, and A. Friberg, “Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B 24, 1441-1450 (2007).
    [CrossRef]
  29. V. Torres-Company, H. Lajunen, and A. Friberg, “Effects of partial coherence on frequency combs,” J. Eur. Opt. Soc. Rapid Pub. 2 (2007).
  30. T. Young, “The Bakerian lecture. Experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. London 94 (1804).
  31. T.Jannson, ed., Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics (SPIE Press, 2004).
  32. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge Univ. Press, 1999).
    [PubMed]
  33. B. Davis, “Simulation of vector fields with arbitrary second-order correlations,” Opt. Express 15, 2837-2846 (2007).
    [CrossRef] [PubMed]

2009 (1)

2008 (1)

R. Schoonover, B. Davis, R. Bartels, and P. Carney, “Optical interferometry with pulsed fields,” J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

2007 (6)

2006 (1)

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

2004 (2)

2003 (1)

2002 (2)

T. Udem, R. Holzwarth, and T. Haensch, “Optical frequency metrology,” Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

2000 (2)

L. Le Déroff, P. Salières, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61, 35 (2000).
[CrossRef]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

1997 (1)

T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
[CrossRef]

1996 (2)

D. Eliyahu, R. A. Salvatore, and A. Yariv, “Noise characterization of a pulse train generated by actively mode-locked lasers,” J. Opt. Soc. Am. B 13, 1619-1626 (1996).
[CrossRef]

T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (2)

I. G. Fuss, “An interpretation of the spectral measurement of optical pulse-train noise,” IEEE J. Quantum Electron. 30, 2707-2710 (1994).
[CrossRef]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

1989 (1)

J. Degnan, N. Center, and M. Greenbelt, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
[CrossRef]

1977 (1)

1934 (1)

A. Khintchine, “Korrelationstheorie der stationären stochastischen Prozesse,” Math. Ann. 109, 604-615 (1934).
[CrossRef]

1930 (1)

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117-258 (1930).
[CrossRef]

1914 (1)

A. Einstein, “Méthode pour la détermination de valeurs statistiques d'observation concernant des grandeurs soumises à des fluctuations irrégulières,” Arch. Sci. Phys. Nat. 37, 254-256 (1914).

1804 (1)

T. Young, “The Bakerian lecture. Experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. London 94 (1804).

Anastasiadis, S. H.

Anglos, D.

Attwood, D.

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Attwood, D. T.

Backus, S.

X. Zhang, A. Libertun, A. Paul, E. Gagnon, S. Backus, I. Christov, M. Murnane, H. Kapteyn, R. Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matched high-harmonic generation,” Opt. Lett. 29, 1357-1359 (2004).
[CrossRef] [PubMed]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Balcou, P.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

Bartels, R.

R. Schoonover, B. Davis, R. Bartels, and P. Carney, “Optical interferometry with pulsed fields,” J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

X. Zhang, A. Libertun, A. Paul, E. Gagnon, S. Backus, I. Christov, M. Murnane, H. Kapteyn, R. Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matched high-harmonic generation,” Opt. Lett. 29, 1357-1359 (2004).
[CrossRef] [PubMed]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge Univ. Press, 1999).
[PubMed]

Carney, P.

R. Schoonover, B. Davis, and P. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705-4711 (2009).
[CrossRef] [PubMed]

R. Schoonover, B. Davis, R. Bartels, and P. Carney, “Optical interferometry with pulsed fields,” J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

Carré, B.

L. Le Déroff, P. Salières, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61, 35 (2000).
[CrossRef]

Center, N.

J. Degnan, N. Center, and M. Greenbelt, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
[CrossRef]

Christov, I.

X. Zhang, A. Libertun, A. Paul, E. Gagnon, S. Backus, I. Christov, M. Murnane, H. Kapteyn, R. Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matched high-harmonic generation,” Opt. Lett. 29, 1357-1359 (2004).
[CrossRef] [PubMed]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Corkum, P. B.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

Cundiff, S. T.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Davis, B.

R. Schoonover, B. Davis, and P. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705-4711 (2009).
[CrossRef] [PubMed]

R. Schoonover, B. Davis, R. Bartels, and P. Carney, “Optical interferometry with pulsed fields,” J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

B. Davis, “Simulation of vector fields with arbitrary second-order correlations,” Opt. Express 15, 2837-2846 (2007).
[CrossRef] [PubMed]

Degnan, J.

J. Degnan, N. Center, and M. Greenbelt, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
[CrossRef]

Devrelis, V.

Diddams, S. A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Ditmire, T.

T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
[CrossRef]

T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
[CrossRef] [PubMed]

Eberly, J.

Einstein, A.

A. Einstein, “Méthode pour la détermination de valeurs statistiques d'observation concernant des grandeurs soumises à des fluctuations irrégulières,” Arch. Sci. Phys. Nat. 37, 254-256 (1914).

Eliyahu, D.

Friberg, A.

V. Torres-Company, H. Lajunen, and A. Friberg, “Effects of partial coherence on frequency combs,” J. Eur. Opt. Soc. Rapid Pub. 2 (2007).

V. Torres-Company, H. Lajunen, and A. Friberg, “Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B 24, 1441-1450 (2007).
[CrossRef]

Furusawa, K.

Fuss, I. G.

I. G. Fuss, “An interpretation of the spectral measurement of optical pulse-train noise,” IEEE J. Quantum Electron. 30, 2707-2710 (1994).
[CrossRef]

Gagnon, E.

Gardner, W.

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

Giannelis, E. P.

Green, H.

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Greenbelt, M.

J. Degnan, N. Center, and M. Greenbelt, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
[CrossRef]

Gumbrell, E.

T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
[CrossRef]

T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
[CrossRef] [PubMed]

Haensch, T.

T. Udem, R. Holzwarth, and T. Haensch, “Optical frequency metrology,” Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

Hall, J. L.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Holzwarth, R.

T. Udem, R. Holzwarth, and T. Haensch, “Optical frequency metrology,” Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

Hutchinson, M.

T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
[CrossRef]

T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
[CrossRef] [PubMed]

Ivanov, M. Y.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

Jacobsen, C.

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Jones, D. J.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Joyeux, D.

L. Le Déroff, P. Salières, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61, 35 (2000).
[CrossRef]

Kapteyn, H.

X. Zhang, A. Libertun, A. Paul, E. Gagnon, S. Backus, I. Christov, M. Murnane, H. Kapteyn, R. Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matched high-harmonic generation,” Opt. Lett. 29, 1357-1359 (2004).
[CrossRef] [PubMed]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Khintchine, A.

A. Khintchine, “Korrelationstheorie der stationären stochastischen Prozesse,” Math. Ann. 109, 604-615 (1934).
[CrossRef]

Lajunen, H.

Le Déroff, L.

L. Le Déroff, P. Salières, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61, 35 (2000).
[CrossRef]

Lewenstein, M.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

L'Huillier, A.

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

Libertun, A.

Liu, Y.

X. Zhang, A. Libertun, A. Paul, E. Gagnon, S. Backus, I. Christov, M. Murnane, H. Kapteyn, R. Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matched high-harmonic generation,” Opt. Lett. 29, 1357-1359 (2004).
[CrossRef] [PubMed]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Mandel, L.

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

Meyerhofer, D.

T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
[CrossRef]

T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
[CrossRef] [PubMed]

Midorikawa, K.

Munch, J.

Murnane, M.

X. Zhang, A. Libertun, A. Paul, E. Gagnon, S. Backus, I. Christov, M. Murnane, H. Kapteyn, R. Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matched high-harmonic generation,” Opt. Lett. 29, 1357-1359 (2004).
[CrossRef] [PubMed]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Nabekawa, Y.

Nagata, Y.

Napolitano, A.

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

O'Connor, M.

Papadakis, V.

Papazoglou, D. G.

Paul, A.

X. Zhang, A. Libertun, A. Paul, E. Gagnon, S. Backus, I. Christov, M. Murnane, H. Kapteyn, R. Bartels, Y. Liu, and D. T. Attwood, “Highly coherent light at 13 nm generated by use of quasi-phase-matched high-harmonic generation,” Opt. Lett. 29, 1357-1359 (2004).
[CrossRef] [PubMed]

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

Paura, L.

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

Phalippou, D.

L. Le Déroff, P. Salières, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61, 35 (2000).
[CrossRef]

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Salières, P.

L. Le Déroff, P. Salières, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61, 35 (2000).
[CrossRef]

Salvatore, R. A.

Schoonover, R.

R. Schoonover, B. Davis, and P. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705-4711 (2009).
[CrossRef] [PubMed]

R. Schoonover, B. Davis, R. Bartels, and P. Carney, “Optical interferometry with pulsed fields,” J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

Smith, R.

T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
[CrossRef]

T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
[CrossRef] [PubMed]

Stassinopoulos, A.

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Tervo, J.

Tisch, J.

T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
[CrossRef]

T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
[CrossRef] [PubMed]

Torres-Company, V.

V. Torres-Company, H. Lajunen, and A. Friberg, “Effects of partial coherence on frequency combs,” J. Eur. Opt. Soc. Rapid Pub. 2 (2007).

V. Torres-Company, H. Lajunen, and A. Friberg, “Coherence theory of noise in ultrashort-pulse trains,” J. Opt. Soc. Am. B 24, 1441-1450 (2007).
[CrossRef]

Turunen, J.

Udem, T.

T. Udem, R. Holzwarth, and T. Haensch, “Optical frequency metrology,” Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

Vahimaa, P.

Wiener, N.

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117-258 (1930).
[CrossRef]

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Wodkiewicz, K.

Wolf, E.

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

E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge Univ. Press, 2007).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge Univ. Press, 1999).
[PubMed]

Wyrowski, F.

Yaglom, A.

A. Yaglom, Correlation Theory of Stationary and Related Random Functions (Springer, 1987).

Yariv, A.

Young, T.

T. Young, “The Bakerian lecture. Experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. London 94 (1804).

Zhang, X.

Acta Math. (1)

N. Wiener, “Generalized harmonic analysis,” Acta Math. 55, 117-258 (1930).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B: Photophys. Laser Chem. (1)

T. Ditmire, J. Tisch, E. Gumbrell, R. Smith, D. Meyerhofer, and M. Hutchinson, “Spatial coherence of short wavelength high-order harmonics,” Appl. Phys. B: Photophys. Laser Chem. 65, 313-328 (1997).
[CrossRef]

Arch. Sci. Phys. Nat. (1)

A. Einstein, “Méthode pour la détermination de valeurs statistiques d'observation concernant des grandeurs soumises à des fluctuations irrégulières,” Arch. Sci. Phys. Nat. 37, 254-256 (1914).

IEEE J. Quantum Electron. (2)

I. G. Fuss, “An interpretation of the spectral measurement of optical pulse-train noise,” IEEE J. Quantum Electron. 30, 2707-2710 (1994).
[CrossRef]

J. Degnan, N. Center, and M. Greenbelt, “Theory of the optimally coupled Q-switched laser,” IEEE J. Quantum Electron. 25, 214-220 (1989).
[CrossRef]

J. Eur. Opt. Soc. Rapid Pub. (1)

V. Torres-Company, H. Lajunen, and A. Friberg, “Effects of partial coherence on frequency combs,” J. Eur. Opt. Soc. Rapid Pub. 2 (2007).

J. Mod. Opt. (1)

R. Schoonover, B. Davis, R. Bartels, and P. Carney, “Optical interferometry with pulsed fields,” J. Mod. Opt. 55, 1541-1556 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. B (3)

Math. Ann. (1)

A. Khintchine, “Korrelationstheorie der stationären stochastischen Prozesse,” Math. Ann. 109, 604-615 (1934).
[CrossRef]

Nature (1)

T. Udem, R. Holzwarth, and T. Haensch, “Optical frequency metrology,” Nature 416, 233-237 (2002).
[CrossRef] [PubMed]

Opt. Express (3)

Opt. Lett. (2)

Philos. Trans. R. Soc. London (1)

T. Young, “The Bakerian lecture. Experiments and calculations relative to physical optics,” Philos. Trans. R. Soc. London 94 (1804).

Phys. Rev. A (3)

B. Davis, “Measurable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[CrossRef]

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L'Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117-2132 (1994).
[CrossRef] [PubMed]

L. Le Déroff, P. Salières, B. Carré, D. Joyeux, and D. Phalippou, “Measurement of the degree of spatial coherence of high-order harmonics using a Fresnel-mirror interferometer,” Phys. Rev. A 61, 35 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

T. Ditmire, E. Gumbrell, R. Smith, J. Tisch, D. Meyerhofer, and M. Hutchinson, “Spatial coherence measurement of soft x-ray radiation produced by high order harmonic generation,” Phys. Rev. Lett. 77, 4756-4759 (1996).
[CrossRef] [PubMed]

Science (2)

R. Bartels, A. Paul, H. Green, H. Kapteyn, M. Murnane, S. Backus, I. Christov, Y. Liu, D. Attwood, and C. Jacobsen, “Fully spatially coherent EUV beams generated using a small-scale laser,” Science 297, 376-378 (2002).
[PubMed]

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635-639 (2000).
[CrossRef] [PubMed]

Signal Process. (1)

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: half a century of research,” Signal Process. 86, 639-697 (2006).
[CrossRef]

Other (5)

A. Yaglom, Correlation Theory of Stationary and Related Random Functions (Springer, 1987).

T.Jannson, ed., Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics (SPIE Press, 2004).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th (expanded) ed. (Cambridge Univ. Press, 1999).
[PubMed]

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

E. Wolf, Introduction to the Theory of Coherence and Polarization (Cambridge Univ. Press, 2007).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Diagrammatic sketch of the source and interferometer.

Fig. 2
Fig. 2

Simulated interferograms for a source with T 0 = 200 fs , T = 50 fs , τ c = 600 fs , and σ g = σ s 2 = 1 mm . The top panel contains the interferogram as would be measured in the far zone ( 100 m away from the source), the middle panel contains the interferogram as would be measured in the Fresnel zone ( 500 mm away from the source), and the bottom panel contains the interferogram as would be measured in the near zone ( 2 mm away from the source). This source is in Regime I.

Fig. 3
Fig. 3

Simulated interferograms for a source with T 0 = 200 fs , T = 50 fs , τ c = 600 fs , and σ g = 2 σ s = 4 mm . Panel descriptions as for Fig. 2. This source is in Regime I.

Fig. 4
Fig. 4

Simulated interferograms for a source with T 0 = 200 fs , T = 50 fs , τ c = 100 fs , and σ g = σ s 2 = 1 mm . Panel descriptions as for Fig. 2. This source is in Regime II.

Fig. 5
Fig. 5

Simulated interferograms for a source with T 0 = 200 fs , T = 50 fs , τ c = 100 fs , and σ g = 2 σ s = 4 mm . Panel descriptions as for Fig. 2. This source is consistent with being in Regime II.

Fig. 6
Fig. 6

Simulated interferograms for a source with T 0 = 500 fs , T = 200 fs , τ c = 50 fs , and σ g = σ s 2 = 1 mm . Panel descriptions as for Fig. 2. This source is in Regime III.

Fig. 7
Fig. 7

Simulated interferograms for a source with T 0 = 500 fs , T = 200 fs , τ c = 50 fs , and σ g = 2 σ s = 4 mm . Panel descriptions as for Fig. 2. This source is in Regime III.

Fig. 8
Fig. 8

Simulated interferograms for three sources with T 0 = 500 fs , T = 200 fs , τ c = 50 fs and σ s = 2 mm . In the top panel, the simulated source has coherence length σ g = 1 mm , in the middle panel, the simulated source has coherence length σ g = 1.5 mm , and in the bottom panel, the simulated source has coherence length σ g = 2 mm .

Fig. 9
Fig. 9

Plot of the visibility versus the magnitude of the complex degree of coherence for a source characterized by T 0 = 200 fs , T = 50 fs , τ c = 100 fs , σ s = 2 mm and σ g = 1 mm . Each ‘×’ denotes a simulation run with 8000 pulses. The interferometer is simulated to be 2 mm away from the secondary source (one beam width). The straight line represents a perfect match between the simulated visibility and the calculated magnitude of the complex degree of coherence.

Fig. 10
Fig. 10

Results of three simulations of a single-shot measurement for a field parameterized by T 0 = 500 fs , T = 200 fs , τ c = 50 fs , σ s = 2 mm and σ g = 1 mm as would be measured in the Fresnel zone.

Fig. 11
Fig. 11

Results of a simulation of the time-averaged intensity (taken over 800 pulses) for a field parameterized by T 0 = 500 fs , T = 200 fs , τ c = 50 fs , σ s = 2 mm and σ g = 1 mm as would be measured in the Fresnel zone.

Equations (20)

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

Γ ¯ P ( ρ 1 , ρ 2 , τ ) = U ¯ * ( ρ 1 , t τ ) U ¯ ( ρ 2 , t ) ,
Γ P ( ρ 1 , ρ 2 , t τ , t ) = U * ( ρ 1 , t τ ) U ( ρ 2 , t ) = Γ ¯ P ( ρ 1 , ρ 2 , τ ) h * ( t τ ) h ( t ) ,
h ( t ) = n h n e i ω 0 n t , ω 0 = 2 π T 0 ,
W ¯ P ( ρ 1 , ρ 2 , ω ) = d τ Γ ¯ P ( ρ 1 , ρ 2 , τ ) e i ω τ .
W P ( ρ 1 , ρ 2 , ω , ω + Ω ) = d t d τ Γ P ( ρ 1 , ρ 2 , t τ , t ) e i ( ω τ + Ω t ) = m , n h n * h m + n W ¯ ( ρ 1 , ρ 2 , ω ω 0 n ) δ ( Ω m ω 0 ) ,
U S ( r , t ) = d t d 3 r G ( r , r , t , t ) U P ( r , t ) + b.c. ,
G ( r , r , t , t ) = δ ( t t | r r | c ) | r r | ,
U ̃ S ( r , ω ) = d 3 r G ̃ ( r , r ; k ) U ̃ P ( r , ω ) + b.c. ,
G ̃ ( r , r ; k ) = e i k | r r | | r r | ,
K ( r 1 , r 1 , r 2 , r 2 , t 1 , t 1 , t 2 , t 2 ) = G * ( r 1 , r 1 , t 1 , t 1 ) G ( r 2 , r 2 , t 2 , t 2 ) ,
K ̃ ( r 1 , r 1 , r 2 , r 2 ; k 1 , k 2 ) = G ̃ * ( r 1 , r 1 ; k 1 ) G ̃ ( r 2 , r 2 ; k 2 ) .
Γ S ( r 1 , r 2 , t 1 , t 2 ) = d t 1 d t 2 d 3 r 1 d 3 r 2 K ( r 1 , r 1 , r 2 , r 2 , t 1 , t 1 , t 2 , t 2 ) Γ P ( r 1 , r 2 , t 1 , t 2 e )
W S ( r 1 , r 2 , ω 1 , ω 2 ) = d 3 r 1 d 3 r 2 K ̃ ( r 1 , r 1 , r 2 , r 2 , k 1 , k 2 ) W P ( r 1 , r 2 , ω 1 , ω 2 ) .
Γ D ( P , P , t , t ) = | κ 1 | 2 Γ A ( Q 1 , Q 1 , t R 1 c , t R 1 c ) + | κ 2 | 2 Γ A ( Q 2 , Q 2 , t R 2 c , t R 2 c ) + κ 1 * κ 2 Γ A ( Q 1 , Q 2 , t R 1 c , t R 2 c ) + κ 1 κ 2 * Γ A ( Q 2 , Q 1 , t R 2 c , t R 1 c ) ,
V = I max I min I max + I min .
Γ ¯ ( r 1 , r 2 , τ ) = Γ ( τ ) F ( r 1 , r 2 ) ,
Γ ( τ ) = exp ( τ 2 2 τ c 2 ) exp ( i ω c τ ) ,
F ( r 1 , r 2 ) = exp ( r 1 2 4 σ s 2 ) exp ( r 2 2 4 σ s 2 ) exp ( | r 1 r 2 | 2 2 σ g 2 ) ,
z 0 a a 4 λ ,
W ( ) ( ω , ω + Ω ) = F ̃ ( 0 , 0 ) A ( ω , ω + Ω ) e i Ω r r 2 ,

Metrics