April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
1877
Realization of laterally nondispersing ultrabroadband Airy pulses Andreas Valdmann,1,* Peeter Piksarv,1 Heli Valtna-Lukner,1,2 and Peeter Saari1,3 1
2
Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia Max Planck Institute for the Science of Light, Günther-Scharowsky Str. 1, 91058 Erlangen, Germany 3
Estonian Academy of Sciences, Kohtu 6, Tallinn 10130, Estonia *Corresponding author: [email protected]
Received December 16, 2013; revised February 21, 2014; accepted February 25, 2014; posted February 26, 2014 (Doc. ID 203028); published March 21, 2014 We present the measurements of the spatiotemporal impulse response of a system creating nondispersing Airy pulses, i.e., ultrabroadband Airy beams whose main lobe size remains constant over propagation. A custom refractive element with a continuous surface profile was used to impose the cubic phase on the input beam. The impulse response of the Airy pulse generator was spatiotemporally characterized by applying a white-light spatial-spectral interferometry setup based on the SEA TADPOLE technique. The results were compared with the theoretical model and previously spatiotemporally characterized Airy pulses generated by a spatial light modulator. © 2014 Optical Society of America OCIS codes: (320.5550) Pulses; (320.7100) Ultrafast measurements; (350.5500) Propagation; (260.0260) Physical optics. http://dx.doi.org/10.1364/OL.39.001877
Over the last few years, several wave fields have been discovered whose intensity maximum is transversely confined and moves along a curved trajectory during propagation. The first and most well-known—the Airy beam—has attained enduring attention since its introduction in optics by Siviloglou et al. [1]. Pulsed versions of the Airy beam have also been discussed and experimentally realized [2–4]. Although direct light-field modulation can be used to create Airy beams [5], a more common experimental setup involves imposing a cubic phase profile on the input Gaussian beam and performing an optical Fourier transform with a lens. The desired spatial phase modulation can be acquired, e.g., with spatial light modulators (SLM) [1], lithographically etched micro-optical phase plates [6], combinations of cylindrical lenses [7], and even by a purposeful use of lens aberrations [3,8]. Airy beams have been put to use for generation of curved plasma channels [9] and optical micromachining [10]. For many possible applications, it is important that the main lobe of the beam remains laterally confined. In the monochromatic case, even the truncated version of an Airy beam resists diffraction over propagation distances much larger than the Rayleigh length and its main intensity lobe retains its lateral confinement [1]. However, the spectrum of an ultrashort optical pulse, only a few femtosecond long, spans over more than an octave and dispersion becomes evident. Therefore, wavelength dependence of the system for creating ultrashort Airy pulses must be taken into consideration to ensure spatial and temporal confinement of the main intensity lobe. For example, we recently showed that the Airy pulses shaped with an SLM are affected by strong lateral dispersion and the main lobe will quickly spread during propagation [11]. The most straightforward way of creating broadband Airy beams would involve an optical element that has a continuous surface profile following a cubic function. In this Letter, we study the impulse response of such an 0146-9592/14/071877-04$15.00/0
element and show that ultrashort Airy pulses shaped with this element retain a highly localized main intensity lobe. A spatial-spectral interferometry technique based on the SEA TADPOLE method [12] was used to fully characterize the created Airy wave field with high spatial and temporal resolution. A broadband 1D Airy beam in free space can be expressed as a superposition of its monochromatic constituents of wavenumber k over spectrum Sk [2]: Z Ψx; z; t
∞ 0
dkSkΦx; z; keikz−ct ;
(1)
where 2 x z z − ia Φx; z; k Ai x0 2z0 z0 2 x az × exp a − 2 iφx; z; k ; x0 2 z0 x z 1 z 3 a2 z φx; z; k − : 2 z0 2x0 z0 12 z0
(2)
(3)
Here Ai… is the Airy function and a is small parameter describing the lateral exponential decay of the beam. Scale of the beam is determined by lateral and longitudinal characteristic lengths x0 and z0 . The parabolic trajectory of the Airy beam main lobe is characterized by the parameter b0 x0 ∕4z20 . The wave function of the 2D Airy beams is obtained as the product of two 1D wave functions Ψx; y; z; t Ψx x; z; tΨy y; z; t. The Airy beam is typically created with a lens performing an optical Fourier transform of the initial field [1]: 2 2
1
3 3
2
3
Φ0 ξ; k e−ac0 ξ ei3c0 ξ −3a c0 ξ−ia ; © 2014 Optical Society of America
(4)
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OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
where ξ is the coordinate in the input plane of the lens. The lateral scale of the cubic phase function in the input plane is described by parameter c0 . The focal length f of the lens imposes a relation c0 x0 k∕f on the input and output fields. Four types of Airy pulses have been defined according to the dependence of c0 on the wavenumber [2,4]. Wavelength constituents of an Airy pulse can spatially overlap only in certain regions, which is determined by c0 k as shown in Fig. 1. Type I Airy pulses have c0 ∝ k, for type II c0 is constant, type III pulses have c0 ∝ k1∕2 , and for type IV c0 ∝ k1∕3 . The type of an Airy pulse is determined by properties of the optical system creating it. It was previously shown that an SLM with a cubic phase mask containing discontinuities will produce type II Airy pulses that exhibit strong lateral dispersion [11]. This behavior is not unique to SLM and lateral dispersion of the main lobe is present in Airy pulses that are shaped with any cubic phase element that has phase discontinuities, i.e., phase wrapping is used. For applications that require a spatially and temporally localized ultrashort pulse, the type IV Airy beam would be most suitable, as all its spectral components propagate along the same trajectory, and its main intensity lobe is not subjected to lateral dispersion. The cubic phase term in Eq. (4) is proportional to c30 ; therefore, for the nondispersing type IV Airy pulses with c0 ∝ k1∕3 , the phase modulation amplitude for different wavelengths in the Fourier plane is proportional to k. This particular dependence on the wavenumber k could be achieved by using a suitably curved mirror as the cubic phase element. A refractive phase plate with a continuous cubic surface profile can also be used, but dispersion
Fig. 1. Four types of broadband Airy beams in the visible spectral range (simulations). Colors are shown as perceived by a human eye. In certain regions, interference maxima of all wavelength constituents overlap, i.e., the beam is white. For the four types of Airy beams this occurs in: I: beam waist at z 0, II: far field, III: axis x 0, and IV: parabolic trajectory of the main lobe. Plots are compressed 350 times along z axis.
From Laser
BS1
To SEA TADPOLE BS2
P
L1
PCF1
D PCF2 z
y x
L2
A
Fig. 2. Experimental setup. The expanded beam from the laser is split into two using a UV fused-silica window BS1. On the measurement arm, a custom-made continuous-phase element A was used to introduce cubic spatial phase. The Fourier transforming lens L2 on the measurement arm and the focusing lens L1 on the reference arm were chosen to be from the same type and with the same focal length. A fused-silica wedge prism pair P was used to fine-tune the dispersion between two arms of the interferometer.
in the substrate material must be taken into account. We chose to use a custom-made refractive phase element, as it can be directly placed into the beam path. The phase plate was fabricated using the PowerPhotonic LightForge service. The bulk dispersion of the fused-silica substrate was compensated in the setup. The deviation from the pure type IV Airy pulse caused by the dispersion of the phase element was less than 2.0% in the value of the parameter b0 over the full spectral range. The experimental setup for generating and measuring laterally nondispersing Airy pulses is show in Fig. 2. The measurement setup follows the SEA TADPOLE arrangement for spatiotemporal ultrashort pulse characterization using spatial spectral interferometry [12]. The current measurement setup is a refined version of our previous setup used to measure ultrabroadband Airy pulses generated using an SLM [11]. In order to improve the transverse beam quality, the laser beam was spatially filtered through a single-mode photonic crystal fiber and expanded using off-axis parabolic mirrors. In addition, a pair of fused-silica wedge prisms were added into the reference arm to precisely control the dispersion in the reference arm and compensate for the residual dispersion arising from the possible inequalities in the length of fiber and from the substrate of the phase element. Therefore, any additional calibration of the device, i.e., subtracting a spectral phase of a blank substrate from the measurement results, was not necessary. A more detailed description of the SEA TADPOLE method using supercontinuum laser source (Fianium SC400-2-PP) can be found in [13] and references therein. In short, the device involves sampling a small spatial region of the field under study with a single-mode photonic crystal fiber and recording its interference pattern with a reference pulse in a spectrometer from which the complex spectral response and hence also the impulse response is reconstructed at every point in space. The fiber is scanned transversely along the x and y axis
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
and longitudinally along z. For longitudinal scanning, the delay on the reference arm was adjusted correspondingly to keep the relative path lengths of the interferometer arms constant. Our measurable spectral range was from 428…1088 nm, yielding a near-cycle temporal resolution of 2.5 fs. The spatial resolution is determined by the mode size of the fibers and is approximately 3 μm. Two sets of achromatic lenses (f 500 mm and f 300 mm) were used in the setup, effectively allowing us to obtain desired parameters of the generated Airy pulses even with the given single-phase element. As the linear dimensions of the generated Airy beam scale inversely proportionally to the focal length of the Fourier transforming lens, it is a convenient way to generate a beam with a desired size of the focal spot. The simulations were carried out by evaluating Eq. (1) using the measured spectrum. The parameters x0 and y0 were calculated from the phase profile of the cubic phase element by taking into account the dispersion relation of the fused-silica substrate of the element. It was assumed that the Fourier transform was performed by an ideal achromatic thin lens. The parameter a was determined for each wavelength by measuring the beam profile with a separate CCD camera and a set of interference filters. These data were thereafter inter- and extrapolated to cover the full spectral range of the SEA TADPOLE. The measured spatiotemporal response of the system for generating type IV Airy pulses is shown and compared with simulations in Fig. 3. The propagation characteristics depend on the focal length of the Fourier lens. With a lens of f 500 mm, the pulse cross section retained a distinct pattern typical to Airy beams throughout the longitudinal measurement range of 200 mm. With f 300 mm, the pulse exhibited stronger lateral
1879
acceleration, but the pattern on the cross section began to deteriorate when moving farther away from the apex of the parabolic trajectory along the z axis. The main lobe was distinguishable throughout a propagation range of 100 mm. Spatial and temporal localization of the main intensity maximum during propagation was examined to demonstrate the nondispersive characteristics of the created pulse. The pulse intensity profile was averaged over time for each longitudinal measurement point, and the transverse size of the main lobe was calculated as an area surrounded by the half-maximum intensity contour. The measured and simulated cross-section areas are shown in Fig. 4. For comparison, the main lobe cross-section area of an Airy wave field created with an SLM was calculated. The main lobe size of the newly created pulse remains constant throughout the whole propagation range. This was not the case for type II pulses shaped with an SLM, where the main lobe quickly spreads out while moving away from the focus of the Fourier lens. This was caused by strong lateral dispersion as the wavelength constituents of the pulse showed a varying parabolic deflection rate during propagation. The temporal duration of the pulse intensity maximum remained almost constant at 3.3 fs (FWHM). This, along with the aforementioned lateral confinement of the main lobe, suggests that creation of curvilinearly propagating localized ultrashort pulses, i.e., “light bullets,” is feasible with the refractive phase element having a continuous cubic surface profile. When using a femtosecond laser to create the pulses, bulk dispersion of the cubic phase element can be compensated by adjusting the pulse compressor, which is typically present in the setup. It should be pointed out that the damage threshold of the used
Fig. 3. Measured and simulated spatiotemporal impulse response of Airy pulse generators at two distances. (a) and (d) Measured nondispersing ultrashort Airy pulse generated by the special phase element (snapshots of Media 1). (b) and (e) Simulation of the previous (Media 2). (c) and (f) Measured ultrashort Airy pulse generated by an SLM [11]. Color represents the amplitude of the electric field. The images show projections of the pulses to the corresponding planes.
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OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
curved trajectory, which can be used for high-precision micromachining, particle manipulation, and exploring nonlinear optical processes. This work has been supported by the Estonian Science Foundation and by the European Regional Development Fund. A. V. has been partially supported by the European Social Fund.
Fig. 4. Measured (markers) and simulated (lines) main lobe cross-section area of the ultrashort Airy pulse. Rhombuses and dotted line: Airy pulse generated by an SLM and Fourier lens with f 500 mm [11]; nondispersing Airy pulse generated by the special phase element and Fourier lens with f 500 mm (circles and dashed line); and with f 300 mm (squares and solid line).
fused-silica phase plate is much higher than of an SLM, which is promising for applications where high-intensity pulses are needed. In conclusion, we have measured the spatiotemporal impulse response of an optical system for shaping ultrabroadband nondispersing Airy pulses. It was shown that a continuous cubic phase plate creates type IV Airy pulses. The temporal duration and cross-section area of the main lobe of the pulse remained constant during propagation. Properties of the created pulse can be adjusted by selecting a suitable Fourier lens. Choosing a lens with a shorter focal length will result in Airy pulses with stronger lateral acceleration and smaller main lobe, but aberrations are more noticeable at shorter propagation distances. The presented results are important for creating nondispersing ultrashort pulses that follow a
References 1. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007). 2. P. Saari, Opt. Express 16, 10303 (2008). 3. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010). 4. Y. Kaganovsky and E. Heyman, J. Opt. Soc. Am. A 28, 1243 (2011). 5. D. M. Cottrell, J. A. Davis, and T. M. Hazard, Opt. Lett. 34, 2634 (2009). 6. J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, Appl. Opt. 50, 6627 (2011). 7. B. Yalizay, B. Soylu, and S. Akturk, J. Opt. Soc. Am. A 27, 2344 (2010). 8. D. G. Papazoglou, S. Suntsov, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 81, 061807 (2010). 9. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009). 10. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, Appl. Phys. Lett. 101, 071110 (2012). 11. P. Piksarv, A. Valdmann, H. Valtna-Lukner, R. Matt, and P. Saari, Opt. Lett. 38, 1143 (2013). 12. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, Opt. Express 14, 11892 (2006). 13. P. Piksarv, H. Valtna-Lukner, A. Valdmann, M. Lõhmus, R. Matt, and P. Saari, Opt. Express 20, 17220 (2012).
1877
Realization of laterally nondispersing ultrabroadband Airy pulses Andreas Valdmann,1,* Peeter Piksarv,1 Heli Valtna-Lukner,1,2 and Peeter Saari1,3 1
2
Institute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia Max Planck Institute for the Science of Light, Günther-Scharowsky Str. 1, 91058 Erlangen, Germany 3
Estonian Academy of Sciences, Kohtu 6, Tallinn 10130, Estonia *Corresponding author: [email protected]
Received December 16, 2013; revised February 21, 2014; accepted February 25, 2014; posted February 26, 2014 (Doc. ID 203028); published March 21, 2014 We present the measurements of the spatiotemporal impulse response of a system creating nondispersing Airy pulses, i.e., ultrabroadband Airy beams whose main lobe size remains constant over propagation. A custom refractive element with a continuous surface profile was used to impose the cubic phase on the input beam. The impulse response of the Airy pulse generator was spatiotemporally characterized by applying a white-light spatial-spectral interferometry setup based on the SEA TADPOLE technique. The results were compared with the theoretical model and previously spatiotemporally characterized Airy pulses generated by a spatial light modulator. © 2014 Optical Society of America OCIS codes: (320.5550) Pulses; (320.7100) Ultrafast measurements; (350.5500) Propagation; (260.0260) Physical optics. http://dx.doi.org/10.1364/OL.39.001877
Over the last few years, several wave fields have been discovered whose intensity maximum is transversely confined and moves along a curved trajectory during propagation. The first and most well-known—the Airy beam—has attained enduring attention since its introduction in optics by Siviloglou et al. [1]. Pulsed versions of the Airy beam have also been discussed and experimentally realized [2–4]. Although direct light-field modulation can be used to create Airy beams [5], a more common experimental setup involves imposing a cubic phase profile on the input Gaussian beam and performing an optical Fourier transform with a lens. The desired spatial phase modulation can be acquired, e.g., with spatial light modulators (SLM) [1], lithographically etched micro-optical phase plates [6], combinations of cylindrical lenses [7], and even by a purposeful use of lens aberrations [3,8]. Airy beams have been put to use for generation of curved plasma channels [9] and optical micromachining [10]. For many possible applications, it is important that the main lobe of the beam remains laterally confined. In the monochromatic case, even the truncated version of an Airy beam resists diffraction over propagation distances much larger than the Rayleigh length and its main intensity lobe retains its lateral confinement [1]. However, the spectrum of an ultrashort optical pulse, only a few femtosecond long, spans over more than an octave and dispersion becomes evident. Therefore, wavelength dependence of the system for creating ultrashort Airy pulses must be taken into consideration to ensure spatial and temporal confinement of the main intensity lobe. For example, we recently showed that the Airy pulses shaped with an SLM are affected by strong lateral dispersion and the main lobe will quickly spread during propagation [11]. The most straightforward way of creating broadband Airy beams would involve an optical element that has a continuous surface profile following a cubic function. In this Letter, we study the impulse response of such an 0146-9592/14/071877-04$15.00/0
element and show that ultrashort Airy pulses shaped with this element retain a highly localized main intensity lobe. A spatial-spectral interferometry technique based on the SEA TADPOLE method [12] was used to fully characterize the created Airy wave field with high spatial and temporal resolution. A broadband 1D Airy beam in free space can be expressed as a superposition of its monochromatic constituents of wavenumber k over spectrum Sk [2]: Z Ψx; z; t
∞ 0
dkSkΦx; z; keikz−ct ;
(1)
where 2 x z z − ia Φx; z; k Ai x0 2z0 z0 2 x az × exp a − 2 iφx; z; k ; x0 2 z0 x z 1 z 3 a2 z φx; z; k − : 2 z0 2x0 z0 12 z0
(2)
(3)
Here Ai… is the Airy function and a is small parameter describing the lateral exponential decay of the beam. Scale of the beam is determined by lateral and longitudinal characteristic lengths x0 and z0 . The parabolic trajectory of the Airy beam main lobe is characterized by the parameter b0 x0 ∕4z20 . The wave function of the 2D Airy beams is obtained as the product of two 1D wave functions Ψx; y; z; t Ψx x; z; tΨy y; z; t. The Airy beam is typically created with a lens performing an optical Fourier transform of the initial field [1]: 2 2
1
3 3
2
3
Φ0 ξ; k e−ac0 ξ ei3c0 ξ −3a c0 ξ−ia ; © 2014 Optical Society of America
(4)
1878
OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
where ξ is the coordinate in the input plane of the lens. The lateral scale of the cubic phase function in the input plane is described by parameter c0 . The focal length f of the lens imposes a relation c0 x0 k∕f on the input and output fields. Four types of Airy pulses have been defined according to the dependence of c0 on the wavenumber [2,4]. Wavelength constituents of an Airy pulse can spatially overlap only in certain regions, which is determined by c0 k as shown in Fig. 1. Type I Airy pulses have c0 ∝ k, for type II c0 is constant, type III pulses have c0 ∝ k1∕2 , and for type IV c0 ∝ k1∕3 . The type of an Airy pulse is determined by properties of the optical system creating it. It was previously shown that an SLM with a cubic phase mask containing discontinuities will produce type II Airy pulses that exhibit strong lateral dispersion [11]. This behavior is not unique to SLM and lateral dispersion of the main lobe is present in Airy pulses that are shaped with any cubic phase element that has phase discontinuities, i.e., phase wrapping is used. For applications that require a spatially and temporally localized ultrashort pulse, the type IV Airy beam would be most suitable, as all its spectral components propagate along the same trajectory, and its main intensity lobe is not subjected to lateral dispersion. The cubic phase term in Eq. (4) is proportional to c30 ; therefore, for the nondispersing type IV Airy pulses with c0 ∝ k1∕3 , the phase modulation amplitude for different wavelengths in the Fourier plane is proportional to k. This particular dependence on the wavenumber k could be achieved by using a suitably curved mirror as the cubic phase element. A refractive phase plate with a continuous cubic surface profile can also be used, but dispersion
Fig. 1. Four types of broadband Airy beams in the visible spectral range (simulations). Colors are shown as perceived by a human eye. In certain regions, interference maxima of all wavelength constituents overlap, i.e., the beam is white. For the four types of Airy beams this occurs in: I: beam waist at z 0, II: far field, III: axis x 0, and IV: parabolic trajectory of the main lobe. Plots are compressed 350 times along z axis.
From Laser
BS1
To SEA TADPOLE BS2
P
L1
PCF1
D PCF2 z
y x
L2
A
Fig. 2. Experimental setup. The expanded beam from the laser is split into two using a UV fused-silica window BS1. On the measurement arm, a custom-made continuous-phase element A was used to introduce cubic spatial phase. The Fourier transforming lens L2 on the measurement arm and the focusing lens L1 on the reference arm were chosen to be from the same type and with the same focal length. A fused-silica wedge prism pair P was used to fine-tune the dispersion between two arms of the interferometer.
in the substrate material must be taken into account. We chose to use a custom-made refractive phase element, as it can be directly placed into the beam path. The phase plate was fabricated using the PowerPhotonic LightForge service. The bulk dispersion of the fused-silica substrate was compensated in the setup. The deviation from the pure type IV Airy pulse caused by the dispersion of the phase element was less than 2.0% in the value of the parameter b0 over the full spectral range. The experimental setup for generating and measuring laterally nondispersing Airy pulses is show in Fig. 2. The measurement setup follows the SEA TADPOLE arrangement for spatiotemporal ultrashort pulse characterization using spatial spectral interferometry [12]. The current measurement setup is a refined version of our previous setup used to measure ultrabroadband Airy pulses generated using an SLM [11]. In order to improve the transverse beam quality, the laser beam was spatially filtered through a single-mode photonic crystal fiber and expanded using off-axis parabolic mirrors. In addition, a pair of fused-silica wedge prisms were added into the reference arm to precisely control the dispersion in the reference arm and compensate for the residual dispersion arising from the possible inequalities in the length of fiber and from the substrate of the phase element. Therefore, any additional calibration of the device, i.e., subtracting a spectral phase of a blank substrate from the measurement results, was not necessary. A more detailed description of the SEA TADPOLE method using supercontinuum laser source (Fianium SC400-2-PP) can be found in [13] and references therein. In short, the device involves sampling a small spatial region of the field under study with a single-mode photonic crystal fiber and recording its interference pattern with a reference pulse in a spectrometer from which the complex spectral response and hence also the impulse response is reconstructed at every point in space. The fiber is scanned transversely along the x and y axis
April 1, 2014 / Vol. 39, No. 7 / OPTICS LETTERS
and longitudinally along z. For longitudinal scanning, the delay on the reference arm was adjusted correspondingly to keep the relative path lengths of the interferometer arms constant. Our measurable spectral range was from 428…1088 nm, yielding a near-cycle temporal resolution of 2.5 fs. The spatial resolution is determined by the mode size of the fibers and is approximately 3 μm. Two sets of achromatic lenses (f 500 mm and f 300 mm) were used in the setup, effectively allowing us to obtain desired parameters of the generated Airy pulses even with the given single-phase element. As the linear dimensions of the generated Airy beam scale inversely proportionally to the focal length of the Fourier transforming lens, it is a convenient way to generate a beam with a desired size of the focal spot. The simulations were carried out by evaluating Eq. (1) using the measured spectrum. The parameters x0 and y0 were calculated from the phase profile of the cubic phase element by taking into account the dispersion relation of the fused-silica substrate of the element. It was assumed that the Fourier transform was performed by an ideal achromatic thin lens. The parameter a was determined for each wavelength by measuring the beam profile with a separate CCD camera and a set of interference filters. These data were thereafter inter- and extrapolated to cover the full spectral range of the SEA TADPOLE. The measured spatiotemporal response of the system for generating type IV Airy pulses is shown and compared with simulations in Fig. 3. The propagation characteristics depend on the focal length of the Fourier lens. With a lens of f 500 mm, the pulse cross section retained a distinct pattern typical to Airy beams throughout the longitudinal measurement range of 200 mm. With f 300 mm, the pulse exhibited stronger lateral
1879
acceleration, but the pattern on the cross section began to deteriorate when moving farther away from the apex of the parabolic trajectory along the z axis. The main lobe was distinguishable throughout a propagation range of 100 mm. Spatial and temporal localization of the main intensity maximum during propagation was examined to demonstrate the nondispersive characteristics of the created pulse. The pulse intensity profile was averaged over time for each longitudinal measurement point, and the transverse size of the main lobe was calculated as an area surrounded by the half-maximum intensity contour. The measured and simulated cross-section areas are shown in Fig. 4. For comparison, the main lobe cross-section area of an Airy wave field created with an SLM was calculated. The main lobe size of the newly created pulse remains constant throughout the whole propagation range. This was not the case for type II pulses shaped with an SLM, where the main lobe quickly spreads out while moving away from the focus of the Fourier lens. This was caused by strong lateral dispersion as the wavelength constituents of the pulse showed a varying parabolic deflection rate during propagation. The temporal duration of the pulse intensity maximum remained almost constant at 3.3 fs (FWHM). This, along with the aforementioned lateral confinement of the main lobe, suggests that creation of curvilinearly propagating localized ultrashort pulses, i.e., “light bullets,” is feasible with the refractive phase element having a continuous cubic surface profile. When using a femtosecond laser to create the pulses, bulk dispersion of the cubic phase element can be compensated by adjusting the pulse compressor, which is typically present in the setup. It should be pointed out that the damage threshold of the used
Fig. 3. Measured and simulated spatiotemporal impulse response of Airy pulse generators at two distances. (a) and (d) Measured nondispersing ultrashort Airy pulse generated by the special phase element (snapshots of Media 1). (b) and (e) Simulation of the previous (Media 2). (c) and (f) Measured ultrashort Airy pulse generated by an SLM [11]. Color represents the amplitude of the electric field. The images show projections of the pulses to the corresponding planes.
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OPTICS LETTERS / Vol. 39, No. 7 / April 1, 2014
curved trajectory, which can be used for high-precision micromachining, particle manipulation, and exploring nonlinear optical processes. This work has been supported by the Estonian Science Foundation and by the European Regional Development Fund. A. V. has been partially supported by the European Social Fund.
Fig. 4. Measured (markers) and simulated (lines) main lobe cross-section area of the ultrashort Airy pulse. Rhombuses and dotted line: Airy pulse generated by an SLM and Fourier lens with f 500 mm [11]; nondispersing Airy pulse generated by the special phase element and Fourier lens with f 500 mm (circles and dashed line); and with f 300 mm (squares and solid line).
fused-silica phase plate is much higher than of an SLM, which is promising for applications where high-intensity pulses are needed. In conclusion, we have measured the spatiotemporal impulse response of an optical system for shaping ultrabroadband nondispersing Airy pulses. It was shown that a continuous cubic phase plate creates type IV Airy pulses. The temporal duration and cross-section area of the main lobe of the pulse remained constant during propagation. Properties of the created pulse can be adjusted by selecting a suitable Fourier lens. Choosing a lens with a shorter focal length will result in Airy pulses with stronger lateral acceleration and smaller main lobe, but aberrations are more noticeable at shorter propagation distances. The presented results are important for creating nondispersing ultrashort pulses that follow a
References 1. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007). 2. P. Saari, Opt. Express 16, 10303 (2008). 3. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010). 4. Y. Kaganovsky and E. Heyman, J. Opt. Soc. Am. A 28, 1243 (2011). 5. D. M. Cottrell, J. A. Davis, and T. M. Hazard, Opt. Lett. 34, 2634 (2009). 6. J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, Appl. Opt. 50, 6627 (2011). 7. B. Yalizay, B. Soylu, and S. Akturk, J. Opt. Soc. Am. A 27, 2344 (2010). 8. D. G. Papazoglou, S. Suntsov, D. Abdollahpour, and S. Tzortzakis, Phys. Rev. A 81, 061807 (2010). 9. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009). 10. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, Appl. Phys. Lett. 101, 071110 (2012). 11. P. Piksarv, A. Valdmann, H. Valtna-Lukner, R. Matt, and P. Saari, Opt. Lett. 38, 1143 (2013). 12. P. Bowlan, P. Gabolde, A. Shreenath, K. McGresham, R. Trebino, and S. Akturk, Opt. Express 14, 11892 (2006). 13. P. Piksarv, H. Valtna-Lukner, A. Valdmann, M. Lõhmus, R. Matt, and P. Saari, Opt. Express 20, 17220 (2012).
