Abstract

Even so-called “complete” ultrashort laser pulse-measurement techniques actually have ambiguities and so are not truly complete. In particular, the spectral-interferometry technique called scanning SEA TADPOLE measures the “complete” spatiotemporal intensity and phase of arbitrary ultrashort pulses (using a previously characterized spatially uniform reference pulse), but the difficulty of maintaining the stability of the required interferometer to submicron resolution while scanning in space usually blurs the frequency-independent spatial component of the pulse phase. We show here, however, that this information is actually still contained in the measured SEA TADPOLE data, and using a simple Gerchberg–Saxton-like phase-diversity algorithm, it can be recovered from measurements in only two planes, yielding a truly complete spatiotemporal measurement of the pulse field, limited only by any possible ambiguities present in the reference pulse.

© 2012 Optical Society of America

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References

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  1. R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, 2002).
  2. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
    [CrossRef]
  3. T. Feurer, J. C. Vaughan, R. M. Koehl, and K. A. Nelson, “Multidimensional control of femtosecond pulses by use of a programmable liquid-crystal matrix,” Opt. Lett. 27, 652–654 (2002).
    [CrossRef]
  4. J. C. Vaughan, T. Feurer, and K. A. Nelson, “Automated two-dimensional femtosecond pulse shaping,” J. Opt. Soc. Am. B 19, 2489–2495 (2002).
    [CrossRef]
  5. J. C. Vaughan, T. Feurer, and K. A. Nelson, “Automated spatiotemporal diffraction of ultrashort laser pulses,” Opt. Lett. 28, 2408–2410 (2003).
    [CrossRef]
  6. P. Bowlan, P. Gabolde, M. A. Coughlan, R. Trebino, and R. J. Levis, “Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution,” J. Opt. Soc. Am. B 25, A81–A92 (2008).
    [CrossRef]
  7. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009).
    [CrossRef]
  8. P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16, 13663–13675 (2008).
    [CrossRef]
  9. P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express 15, 10219–10230 (2007).
    [CrossRef]
  10. P. Bowlan and R. Trebino, “Extreme pulse-front tilt from an etalon,” J. Opt. Soc. Am. B 27, 2322–2327 (2010).
    [CrossRef]
  11. P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measurement of the spatio-temporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278(2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. P. Gabolde and R. Trebino, “Single-frame measurement of the complete spatio-temporal intensity and phase of ultrashort laser pulse(s) using wavelength-multiplexed digital holography,” J. Opt. Soc. Am. B 25, A25–A33 (2008).
    [CrossRef]
  18. W. Amir, T. A. Planchon, C. G. Durfee, J. A. Squier, P. Gabolde, R. Trebino, and M. Müller, “Simultaneous visualization of spatial and chromatic aberrations by 2D Fourier transform spectral interferometry,” Opt. Lett. 31, 2927–2929 (2006).
    [CrossRef]
  19. C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Direct space time-characterization of the electric fields of ultrashort optical pulses,” Opt. Lett. 27, 548–550 (2002).
    [CrossRef]
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    [CrossRef]
  21. F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. D. Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. 33, 2952–2954 (2008).
    [CrossRef]
  22. D. Faccio, A. Lotti, A. Matijosius, F. Bragheri, V. Degiorgio, A. Couairon, and P. Di Trapani, “Experimental energy-density flux characterization of ultrashort laser pulse filaments,” Opt. Express 17, 8193–8200 (2009).
    [CrossRef]
  23. R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).
  24. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  25. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef]
  26. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58(2004).
    [CrossRef]

2010

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

P. Bowlan and R. Trebino, “Extreme pulse-front tilt from an etalon,” J. Opt. Soc. Am. B 27, 2322–2327 (2010).
[CrossRef]

2009

2008

2007

2006

2004

2003

2002

1996

1994

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1971

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Aeschlimann, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Amir, W.

Antonetti, A.

Audebert, P.

Bauer, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Bayer, D.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Bonacina, L.

Bonaretti, F.

Bowlan, P.

Bragheri, F.

Brixner, T.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Clerici, M.

Couairon, A.

Coughlan, M. A.

Courvoisier, F.

Cunovic, S.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Degiorgio, V.

Di Trapani, P.

Dimler, F.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Dorrer, C.

Dorsch, R. G.

Dos Santos, A.

Durfee, C. G.

Extermann, J.

Faccio, D.

Falliés, F.

Feurer, T.

Fischer, A.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Fuchs, U.

Gabolde, P.

Galez, C.

Gauthier, J. C.

Geindre, J. P.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Hamoniaux, G.

Henin, S.

Jedrkiewicz, O.

Kiselev, D.

Koehl, R. M.

Kosik, E. M.

Le Dantec, R.

Levis, R. J.

Liberale, C.

Lõhmus, M.

Lotti, A.

Matijosius, A.

Mendlovic, D.

Mugnier, Y.

Müller, M.

Mysyrowicz, A.

Nelson, K. A.

Pfeiffer, W.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Piksarv, P.

Planchon, T. A.

Rohmer, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Rousse, A.

Saari, P.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Schneider, C.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Squier, J. A.

Steeb, F.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Strüber, C.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Tartara, L.

Trapani, P. D.

Trebino, R.

P. Bowlan and R. Trebino, “Extreme pulse-front tilt from an etalon,” J. Opt. Soc. Am. B 27, 2322–2327 (2010).
[CrossRef]

P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009).
[CrossRef]

P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measurement of the spatio-temporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278(2009).
[CrossRef]

P. Bowlan, U. Fuchs, R. Trebino, and U. D. Zeitner, “Measuring the spatiotemporal electric field of tightly focused ultrashort pulses with sub-micron spatial resolution,” Opt. Express 16, 13663–13675 (2008).
[CrossRef]

P. Gabolde and R. Trebino, “Single-frame measurement of the complete spatio-temporal intensity and phase of ultrashort laser pulse(s) using wavelength-multiplexed digital holography,” J. Opt. Soc. Am. B 25, A25–A33 (2008).
[CrossRef]

P. Bowlan, P. Gabolde, M. A. Coughlan, R. Trebino, and R. J. Levis, “Measuring the spatiotemporal electric field of ultrashort pulses with high spatial and spectral resolution,” J. Opt. Soc. Am. B 25, A81–A92 (2008).
[CrossRef]

P. Bowlan, P. Gabolde, and R. Trebino, “Directly measuring the spatio-temporal electric field of focusing ultrashort pulses,” Opt. Express 15, 10219–10230 (2007).
[CrossRef]

W. Amir, T. A. Planchon, C. G. Durfee, J. A. Squier, P. Gabolde, R. Trebino, and M. Müller, “Simultaneous visualization of spatial and chromatic aberrations by 2D Fourier transform spectral interferometry,” Opt. Lett. 31, 2927–2929 (2006).
[CrossRef]

P. Gabolde and R. Trebino, “Single-shot measurement of the full spatiotemporal field of ultrashort pulses with multispectral digital holography,” Opt. Express 14, 11460–11467 (2006).
[CrossRef]

P. Gabolde and R. Trebino, “Self-referenced measurement of the complete electric field of ultrashort pulses,” Opt. Express 12, 4423–4429 (2004).
[CrossRef]

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

Valtna-Lukner, H.

Vaughan, J. C.

Voronine, D. V.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Walmsley, I. A.

Wolf, J.-P.

Zalevsky, Z.

Zeitner, U. D.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

W. Amir, T. A. Planchon, C. G. Durfee, J. A. Squier, P. Gabolde, R. Trebino, and M. Müller, “Simultaneous visualization of spatial and chromatic aberrations by 2D Fourier transform spectral interferometry,” Opt. Lett. 31, 2927–2929 (2006).
[CrossRef]

J. C. Vaughan, T. Feurer, and K. A. Nelson, “Automated spatiotemporal diffraction of ultrashort laser pulses,” Opt. Lett. 28, 2408–2410 (2003).
[CrossRef]

F. Bragheri, D. Faccio, F. Bonaretti, A. Lotti, M. Clerici, O. Jedrkiewicz, C. Liberale, S. Henin, L. Tartara, V. Degiorgio, and P. D. Trapani, “Complete retrieval of the field of ultrashort optical pulses using the angle-frequency spectrum,” Opt. Lett. 33, 2952–2954 (2008).
[CrossRef]

J. P. Geindre, P. Audebert, A. Rousse, F. Falliés, J. C. Gauthier, A. Mysyrowicz, A. Dos Santos, G. Hamoniaux, and A. Antonetti, “Frequency-domain interferometer for measuring the phase of a femtosecond pulse probing a laser-produced plasma,” Opt. Lett. 19, 1997–1999 (1994).
[CrossRef]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef]

C. Dorrer, E. M. Kosik, and I. A. Walmsley, “Direct space time-characterization of the electric fields of ultrashort optical pulses,” Opt. Lett. 27, 548–550 (2002).
[CrossRef]

T. Feurer, J. C. Vaughan, R. M. Koehl, and K. A. Nelson, “Multidimensional control of femtosecond pulses by use of a programmable liquid-crystal matrix,” Opt. Lett. 27, 652–654 (2002).
[CrossRef]

P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measurement of the spatio-temporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278(2009).
[CrossRef]

P. Bowlan, H. Valtna-Lukner, M. Lõhmus, P. Piksarv, P. Saari, and R. Trebino, “Measuring the spatiotemporal field of ultrashort Bessel-X pulses,” Opt. Lett. 34, 2276–2278 (2009).
[CrossRef]

Optik

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Proc. Natl. Acad. Sci. USA

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, S. Cunovic, F. Dimler, A. Fischer, W. Pfeiffer, M. Rohmer, C. Schneider, F. Steeb, C. Strüber, and D. V. Voronine, “Spatiotemporal control of nanooptical excitations,” Proc. Natl. Acad. Sci. USA 107, 5329–5333 (2010).
[CrossRef]

Other

H. Stark, ed., Image Recovery: Theory and Application(Academic, 1987).

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

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

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Figures (4)

Fig. 1.
Fig. 1.

Measurement of the drift in the (a) absolute phase, φ0, (b) delay, φ1, and (c) group delay dispersion, φ2, in a typical SEA TADPOLE measurement. The rms variation of each of the curves, σ, is given with respect to the pulse duration τ. While the phase drift in our interferometer randomizes the absolute phase, φ0, all higher-order, frequency-dependent terms are unaffected and accurately measured.

Fig. 2.
Fig. 2.

SEA TADPOLE spatial-phase-retrieval algorithm. We use the measured spatiospectral intensity and phase, E(x,λ,z1), at one plane, z1, and the spatiospectral amplitude at another, z2, to recover the spatial phase φ(x,y,ω0). On the first iteration, we use the measured spatiotemporal amplitude and phase for the field at z1, whose spatial phase is random. This field is propagated to the plane z2, where we replace the spatiospectral amplitude with the measured amplitude and use the spatiospectral phase of the propagated field. Then we back-propagate this to plane z1, replacing the spatiospectral amplitude with the measured amplitude at z1. This is repeated until the rms difference between the amplitudes of the propagated and measured fields is minimal.

Fig. 3.
Fig. 3.

Spatial phase retrieval results. First column: measured spatiospectral amplitudes at z1 (top) and z2 (bottom). Second column: spatiospectral intensities obtained by propagating the field at z2 to z1 (top) and vice versa (bottom) using the measured spatial phase after only one iteration. Third column: same as the previous, but using the retrieved spatial phase after 29 iterations. Note the excellent agreement between the first and last columns.

Fig. 4.
Fig. 4.

Measured (top) and numerically propagated spatiospectral intensities using the field from z=1.1mm, including the retrieved spatial phase.

Equations (1)

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φ(x,y,ω)=φ0(x,y,ω0)+(ωω0)φ1(x,y,ω0)+12(ωω0)2φ2(x,y,ω0)+

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