Abstract

Small-angle, noncollinear, first- and second-order interferometric autocorrelation experiments with Ti:sapphire laser pulses of 9–80-fs duration have been performed with microaxicon arrays. Predictions of short-pulse spatial frequency effects were verified by comparison of interference patterns of single elements and matrices. An angular spectrum of Gaussian-shaped axicons was analyzed on the basis of linear refraction. Experimental data indicate contributions to autocorrelation by nonlinear refraction and travel-time differences. The influence of the spectral bandwidth was separated from the pulse-duration-dependent effects. Spatially resolved information about the coherence time was delivered by the multichannel structure.

© 2001 Optical Society of America

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2000 (5)

1999 (4)

Z. Jiang, X.-C. Zhang, “2D measurement and spatio-temporal coupling of few-cycle THz pulses,” Opt. Express 5, 243–248 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

K. Oba, P.-C. Sun, Y. T. Mazurenko, Y. Fainman, “Femtosecond single-shot correlation system: a time-domain approach,” Appl. Opt. 38, 3810–3817 (1999).
[CrossRef]

C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space–time couplings in Gaussian pulse beams,” J. Mod. Opt. 56, 1881–1891 (1999).
[CrossRef]

A. Baltuska, M. S. Pshenichnikov, D. A. Wiersma, “Second harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459–478 (1999).
[CrossRef]

1998 (2)

1997 (2)

E. T. J. Nibbering, O. Dühr, G. Korn, “Generation of intense tunable 20-fs pulses near 400 nm using a gas-filled hollow waveguide,” Opt. Lett. 22, 1335–1337 (1997).
[CrossRef]

P. Saari, K. Reivelt, “Evidence of x-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

1993 (1)

T. K. Song, J.-Y. Lu, J. F. Greenleaf, “Modified x-waves with improved field properties,” Ultrason. Imaging 15, 36–47 (1993).
[CrossRef] [PubMed]

1991 (1)

1987 (1)

1954 (1)

J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. A 44, 592–597 (1954).
[CrossRef]

Baltuska, A.

A. Baltuska, M. S. Pshenichnikov, D. A. Wiersma, “Second harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459–478 (1999).
[CrossRef]

Caron, C. F. R.

C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space–time couplings in Gaussian pulse beams,” J. Mod. Opt. 56, 1881–1891 (1999).
[CrossRef]

Dühr, O.

Durnin, J. E.

Ehlert, R.

R. Grunwald, S. Woggon, R. Ehlert, “Fabrication of thin-film microlens arrays by mask-shaded vacuum deposition,” in Diffractive Optics and Optical Microsystems, S. Martellucci, A. N. Chester, eds. (Plenum, New York, 1997), pp. 169–177.

Eichler, H. J.

H. J. Eichler, P. Günter, D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 15–17.

Elsaesser, T.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays,” Opt. Lett. 25, 981–983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, P. Hamm, T. Elsaesser, V. Kebbel, “Thin-film microoptics for the generation of femtosecond Bessel beam arrays,” in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 208–210.

Fainman, Y.

Fan, D. Y.

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, L17–L21 (1998).
[CrossRef]

Fischer, D. J.

Friesem, A. A.

Gallmann, L.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, I. A. Walmsley, “Spatially resolved amplitude and phase characterization of ultrashort optical pulses using SPIDER,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 583–584.

Greenleaf, J. F.

T. K. Song, J.-Y. Lu, J. F. Greenleaf, “Modified x-waves with improved field properties,” Ultrason. Imaging 15, 36–47 (1993).
[CrossRef] [PubMed]

Griebner, U.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays,” Opt. Lett. 25, 981–983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, P. Hamm, T. Elsaesser, V. Kebbel, “Thin-film microoptics for the generation of femtosecond Bessel beam arrays,” in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 208–210.

Grunwald, R.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays,” Opt. Lett. 25, 981–983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, P. Hamm, T. Elsaesser, V. Kebbel, “Thin-film microoptics for the generation of femtosecond Bessel beam arrays,” in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 208–210.

R. Grunwald, S. Woggon, R. Ehlert, “Fabrication of thin-film microlens arrays by mask-shaded vacuum deposition,” in Diffractive Optics and Optical Microsystems, S. Martellucci, A. N. Chester, eds. (Plenum, New York, 1997), pp. 169–177.

Günter, P.

H. J. Eichler, P. Günter, D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 15–17.

Hamm, P.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, P. Hamm, T. Elsaesser, V. Kebbel, “Thin-film microoptics for the generation of femtosecond Bessel beam arrays,” in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 208–210.

Harkrider, C. J.

Hartmann, H.-J.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays,” Opt. Lett. 25, 981–983 (2000).
[CrossRef]

V. Kebbel, H.-J. Hartmann, W. Jüptner, “Characterization of micro-optics using digital holography,” in Laser Interferometry X: Applications, G. M. Brown, W. Jüptner, R. J. Pryputniewicz, Proc. SPIE4101, 477–487 (2000).

Herman, R. M.

Iaconis, C.

C. Iaconis, I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, I. A. Walmsley, “Spatially resolved amplitude and phase characterization of ultrashort optical pulses using SPIDER,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 583–584.

Jiang, Z.

Jüptner, W.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays,” Opt. Lett. 25, 981–983 (2000).
[CrossRef]

V. Kebbel, H.-J. Hartmann, W. Jüptner, “Characterization of micro-optics using digital holography,” in Laser Interferometry X: Applications, G. M. Brown, W. Jüptner, R. J. Pryputniewicz, Proc. SPIE4101, 477–487 (2000).

Kebbel, V.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays,” Opt. Lett. 25, 981–983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, P. Hamm, T. Elsaesser, V. Kebbel, “Thin-film microoptics for the generation of femtosecond Bessel beam arrays,” in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 208–210.

V. Kebbel, H.-J. Hartmann, W. Jüptner, “Characterization of micro-optics using digital holography,” in Laser Interferometry X: Applications, G. M. Brown, W. Jüptner, R. J. Pryputniewicz, Proc. SPIE4101, 477–487 (2000).

Keller, U.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, I. A. Walmsley, “Spatially resolved amplitude and phase characterization of ultrashort optical pulses using SPIDER,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 583–584.

Korn, G.

Liu, Z. Y.

Z. Y. Liu, D. Y. Fan, “Propagation of pulsed zeroth-order Bessel beams,” J. Mod. Opt. 45, L17–L21 (1998).
[CrossRef]

Lohmann, A. W.

Lu, J.-Y.

T. K. Song, J.-Y. Lu, J. F. Greenleaf, “Modified x-waves with improved field properties,” Ultrason. Imaging 15, 36–47 (1993).
[CrossRef] [PubMed]

Matuschek, N.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, I. A. Walmsley, “Spatially resolved amplitude and phase characterization of ultrashort optical pulses using SPIDER,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 583–584.

Mazurenko, Y. T.

McCarthy, N.

M. Piché, G. Rousseau, Ch. Varin, N. McCarthy, “Conical wave packets: their propagation speed and their longitudinal fields,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 332–343 (1999).
[CrossRef]

McLeod, J. H.

J. H. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. A 44, 592–597 (1954).
[CrossRef]

Moore, D. T.

Nibbering, E. T. J.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays,” Opt. Lett. 25, 981–983 (2000).
[CrossRef]

E. T. J. Nibbering, O. Dühr, G. Korn, “Generation of intense tunable 20-fs pulses near 400 nm using a gas-filled hollow waveguide,” Opt. Lett. 22, 1335–1337 (1997).
[CrossRef]

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, P. Hamm, T. Elsaesser, V. Kebbel, “Thin-film microoptics for the generation of femtosecond Bessel beam arrays,” in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 208–210.

O’Shea, P.

P. O’Shea, R. Trebino, “Extremely simple intensity-and-phase ultrashort-pulse measurement device with no spectrometer, thin crystal, or delay line,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 587–588.

Oba, K.

Oberlé, J.

L. Sarger, J. Oberlé, “How to measure the characteristics of laser pulses,” in Femtosecond Laser Pulses, C. Rullière, ed. (Springer-Verlag, Berlin, 1998), pp. 177–201.

Pe’er, A.

Piché, M.

M. Piché, G. Rousseau, Ch. Varin, N. McCarthy, “Conical wave packets: their propagation speed and their longitudinal fields,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 332–343 (1999).
[CrossRef]

Pohl, D. W.

H. J. Eichler, P. Günter, D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 15–17.

Potvliege, R. M.

C. F. R. Caron, R. M. Potvliege, “Free-space propagation of ultrashort pulses: space–time couplings in Gaussian pulse beams,” J. Mod. Opt. 56, 1881–1891 (1999).
[CrossRef]

Pshenichnikov, M. S.

A. Baltuska, M. S. Pshenichnikov, D. A. Wiersma, “Second harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459–478 (1999).
[CrossRef]

Reivelt, K.

P. Saari, K. Reivelt, “Evidence of x-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

Rousseau, G.

M. Piché, G. Rousseau, Ch. Varin, N. McCarthy, “Conical wave packets: their propagation speed and their longitudinal fields,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 332–343 (1999).
[CrossRef]

G. Rousseau, “Impulsions coniques: considérations théoretiques et expérimentales,” M. Sc. Thesis (Laval University, Quebec, 2000), p. 54.

Saari, P.

P. Saari, K. Reivelt, “Evidence of x-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79, 4135–4138 (1997).
[CrossRef]

P. Saari, “Spatially and temporally nondiffracting ultrashort pulses,” in Ultrafast Processes in Spectroscopy, O. Svelto, S. De Silvestri, G. Denardo, eds. (Plenum, New York, 1996), pp. 151–155.

Sarger, L.

L. Sarger, J. Oberlé, “How to measure the characteristics of laser pulses,” in Femtosecond Laser Pulses, C. Rullière, ed. (Springer-Verlag, Berlin, 1998), pp. 177–201.

Singer, W.

W. Singer, “Entwicklung und experimentelle Überprüfung nicht-paraxialer Wellenausbreitungsmethoden zur Analyse mikrooptischer Komponenten,” doctoral dissertation (Friedrich-Alexander-University, Erlangen–Nuremberg, Germany, 1995), pp. 13–18.

Song, T. K.

T. K. Song, J.-Y. Lu, J. F. Greenleaf, “Modified x-waves with improved field properties,” Ultrason. Imaging 15, 36–47 (1993).
[CrossRef] [PubMed]

Steinmeyer, G.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, I. A. Walmsley, “Spatially resolved amplitude and phase characterization of ultrashort optical pulses using SPIDER,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 583–584.

Sun, P.-C.

Sutter, D. H.

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, I. A. Walmsley, “Spatially resolved amplitude and phase characterization of ultrashort optical pulses using SPIDER,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 583–584.

Tanaka, T.

T. Tanaka, S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
[CrossRef]

Trebino, R.

P. O’Shea, R. Trebino, “Extremely simple intensity-and-phase ultrashort-pulse measurement device with no spectrometer, thin crystal, or delay line,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 587–588.

Tschirschwitz, F.

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, T. Elsaesser, V. Kebbel, H.-J. Hartmann, W. Jüptner, “Generation of femtosecond Bessel beams with microaxicon arrays,” Opt. Lett. 25, 981–983 (2000).
[CrossRef]

R. Grunwald, U. Griebner, F. Tschirschwitz, E. T. J. Nibbering, P. Hamm, T. Elsaesser, V. Kebbel, “Thin-film microoptics for the generation of femtosecond Bessel beam arrays,” in Diffractive Optics and Micro-Optics, Postconference Digest, Vol. 41 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 208–210.

Varin, Ch.

M. Piché, G. Rousseau, Ch. Varin, N. McCarthy, “Conical wave packets: their propagation speed and their longitudinal fields,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 332–343 (1999).
[CrossRef]

Walmsley, I. A.

C. Iaconis, I. A. Walmsley, “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses,” Opt. Lett. 23, 792–794 (1998).
[CrossRef]

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, C. Iaconis, I. A. Walmsley, “Spatially resolved amplitude and phase characterization of ultrashort optical pulses using SPIDER,” in Conference on Lasers and Electro-Optics (CLEO), Postconference Digest, Vol. 39 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2000), pp. 583–584.

Wang, D.

Wiersma, D. A.

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T. Tanaka, S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
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Appl. Phys. B (1)

L. Gallmann, D. H. Sutter, N. Matuschek, G. Steinmeyer, U. Keller, “Techniques for the characterization of sub-10-fs optical pulses: a comparison,” Appl. Phys. B 70, S67–S75 (2000).
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IEEE J. Quantum Electron. (1)

A. Baltuska, M. S. Pshenichnikov, D. A. Wiersma, “Second harmonic generation frequency-resolved optical gating in the single-cycle regime,” IEEE J. Quantum Electron. 35, 459–478 (1999).
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Figures (10)

Fig. 1
Fig. 1

Schematic of a spatially resolving microaxicon matrix autocorrelator with a BBO crystal for multichannel second-order small-angle noncollinear SHG autocorrelation.

Fig. 2
Fig. 2

Angular transformation of a plane wave generated by a Gaussian-shaped thin-film microaxicon in a geometric-optics approximation: (a) Thickness profile and angular distributions of a single refractive microaxicon (vertex directed in the direction opposite the incoming laser beam). Layer, SiO2 on quartz; h, thickness measured with a Zygo interferometer (solid curve); αG, calculated internal angle with the optical axis (dashed curve); βG, calculated refracted angle with the optical axis (dotted curve). (b) Transverse deviation Δx of partial beams starting at three radial coordinates x in the axicon plane for three axial distances z.

Fig. 3
Fig. 3

Geometrical conditions for interference at an axial distance of z=9 mm [schematically shown in ray tracing for selected rays of the bundle after the rays have passed through a single refractive microaxicon of Gaussian thickness distribution according to Fig. 2(a)]. Solid lines, partial beams with small angles (interfering near the optical axis); dashed lines, partial beams with large angles (interfering far from the optical axis).

Fig. 4
Fig. 4

Difference in propagation time for rays passing through the microaxicon at different transverse initial positions in the plane of the axicon (dotted curve, caused by path differences within the uniform SiO2 layer; dashed curve, caused by the free-space path differences that resulted from the angular distribution; solid curve, total time difference).

Fig. 5
Fig. 5

(a) Interferometric autocorrelation trace and (b) averaged autocorrelation (circles) with an approximate Gaussian fit function (solid curve), and (c) time-integrated spectrum for a femtosecond Ti:sapphire laser with gas-filled hollow-fiber and prism compression (FWHM, 15.8 fs; τ=11.2 fs).

Fig. 6
Fig. 6

Time-integrated patterns of (a), (c) first-order interference for the cw reference laser and (b), (d) second order interference for a 17-fs Ti:sapphire laser pulse generated with a solitary microaxicon (row 1) and a hexagonal array of microaxicons of 405-µm pitch (row n), measured in the central region of the beam.

Fig. 7
Fig. 7

Array-specific effects indicated by the spatial frequency shift: comparison of radial cuts of first-order cw interference [dashed curves, according to Figs. 6(a) and 6(c)] and SHG ultrashort-pulse pattern [17-fs pulse, solid curves, according to Figs. 6(b), and 6(d)], for (a) an array and (b) a single element. x, radial coordinate; p, pitch.

Fig. 8
Fig. 8

Comparison of calculated and measured time-integrated first-order interference patterns perpendicular to the propagation direction with a hexagonal array of microaxicons (z=9 mm, p=405 μm pitch, overexposed). (a), (b) Simulation without and with six neighboring elements and data from a synthetic spectrum; image size, 1024×1024 pixels. (c), (d) Measured fringes for a single element and an array for a pulse duration of 17 fs (∼50 elements illuminated; 7 elements imaged; radius of field of view, 0.81 mm).

Fig. 9
Fig. 9

Averaged spatial frequency from a Fourier analysis for an ultrashort-pulse Bessel beam array. Dashed curve, theory (seven elements, measured time-integrated spectral data, without travel-time effects). Solid curves, experiment with ∼50 illuminated elements. Circles, first-order interference; triangles, second-order interference (SHG; 100-µm BBO crystal).

Fig. 10
Fig. 10

Experimentally measured spatial frequencies of first-and second-order interference patterns for individual fringe spacings generated by an array of microaxicons at a pulse duration of 11 fs (fringe spacing numbered starting from the central peak, simulation with time-integrated spectra without travel-time effects; SHG crystal, 100-µm BBO; microscope objective, 100×).

Equations (9)

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β(x)=arcsin(n sin α)-α(x),
α(x)=arctan ddxf(x)
αG(x)=arctan-4w02-xhmax exp-2x2w02.
ν0=2 sin βλ0=1Λ0
ΔL=λ0/2
Δt=λ0/2c.
f(x, y)=hmax exp[-2(x2+y2)/w02],
p(x, y, λ)=f(x, y) 2π(n-n0)/λ,
E(x, y, λ)=g(λ)A exp[ip(x, y, λ)]=g(λ)A{cos[p(x, y)]+i sin[p(x, y)]},

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