We present measurements of the impulse response of a circular phase diffraction grating in dependence of the field point location behind it. These measurements were carried out using a white-light spectral interferometry set-up, which employs photonic crystal fibers in both the signal and reference arms, and achieves a few micron spatial and almost one-wave-cycle temporal resolution. Our study shows that the grating as a simple and robust single-element optical device (i) suppresses the material-induced spread of ultrashort pulses, (ii) thereby generates the Airy–Bessel light bullets, and (iii) enables temporal focusing of the pulses at the prescribed propagation depth.
© 2012 OSA
For numerous applications in non-linear optics, imaging, lithography, communications, ultrafast photonics, etc. high localization of pulses—both in space and time—is desired. Laser pulses lasting only about one optical cycle have been accomplished [1–4]. In order to achieve such ultrashort pulses close to the fundamental wave-optical limits, whose spectrum spans over the whole visible region, every optical element is a challenge of its own and requires careful pulse dispersion compensation and compression as well as appropriate measurement methods (see, e.g., reviews [5, 6]). Here, we show how temporal focusing of a broadened pulse can be accomplished for certain propagation depths by a single circular phase diffraction grating. We propose a version of spatio-spectral interferometry technique employing supercontinuum laser source and photonic crystal fibers for full spatio-temporal measurement of impulse responses of optical systems with almost one-wave-cycle temporal resolution. Our results open up new possibilities in the characterization of ultrafast optical processes, devices, and in their applications.
Optical pulse compression with a pair of diffraction gratings is well established ever since the pioneering work by Treacy . However, when an ultrashort broadband pulse diffracts from a grating, it broadens due to the acquired chirp, which is rather substantial in the reality [8, 9]. The properties of circular diffraction gratings have been first considered by Dyson . The suppression of temporal spread for 30...200fs duration pulses has been accomplished with Bessel pulses generated by circular phase elements , and in a set-up of two refractive axicons .
Under broadband illumination using a circular grating as a diffractive axicon is advantageous for pulsed Bessel beam generation as the resulting radial profile remains constant over entire spectrum and propagation length . However, the temporal profile of the pulsed Bessel beam is not propagation invariant contrary to the Bessel-X pulses [14, 15]. Previous experiments with circular amplitude and phase gratings [16, 17], have been performed with limited spectral range and lacked the temporal resolution to clearly distinguish diffraction orders or to observe temporal broadening and focusing.
High-resolution spatio-temporal measurement of light pulses is essential for understanding complex time and space couplings they exhibit over propagation . The recent advancements in the ultrashort pulse characterization techniques which enable the measurement of the complex light-field simultaneously in both spatial and temporal domain include SEA TADPOLE [19, 20], STARFISH [16, 21], SEA-SPIDER [6, 22], and others. For example, superluminal propagation of the Bessel-X pulse and an ultrashort counterpart of Arago spot have been exquisitely demonstrated [15, 23–25].
2. Pulsed Bessel beam
The Bessel beam can be formed from plane waves in various ways—either by a planar circular diffraction grating, or by an annular slit in conjunction with a collimating lens, or by a conical lens (refractive axicon) or mirror. While in the monochromatic case it is all the same, under ultrashort pulsed illumination the grating generates what is named “Bessel pulse” or “pulsed Bessel beam” [13, 26–28]. Figuratively speaking, it is like a more or less thin slice cut from a Bessel beam. In distinction from the superluminal Bessel-X pulse [14,15], the spatial profile of the Bessel pulse has no X-like section, its group velocity is subluminal but not the same over its spectrum—the effect of group velocity dispersion (GVD) takes place—and therefore the pulse is not propagation-invariant but broadens along the axis z in the course of propagation.
Let us take a closer look at a pulsed Bessel beam generator comprising a circular phase grating under broadband illumination. If the grating constant is K = 2π/d, d being the groove spacing, then for a given diffraction order m the normally incident light acquires a radial wave vector component k⊥ = m × K. The resulting pulsed Bessel beam can be expressed in the form
The complex spectral amplitude of the pulse is29], 30],
The grating induced negative chirp may partially compensate for the positive chirp present due to the material dispersion at certain propagation distances. Figure 1 shows the calculated evolution of the on-axis amplitude of the electric field of the pulsed Bessel beam with various initial spectral phases corresponding to a dispersion induced by 0, 0.92 and 1.57mm thick fused silica samples. The simulations agree with the initial assumptions for temporal focusing. In the propagation range near the temporal focus the pulses can be described as the Airy–Bessel wave packets [31, 32]. It is evident that shorter temporal focus region can be obtained by using higher diffraction order or higher density grating. In addition, the more dispersive the grating, the smaller the lateral size of the maximum is obtained.
3. Methods and experimental set-up
The underlying principle of our technique for measuring spatiotemporally resolved impulse responses is the well known spectrally resolved white-light interferometry (see, e.g., [33–36]). However, since the output field of an optical element generally depends not only on time and/or the longitudinal coordinate, but also on the transversal coordinates, the interferometry has to be generalized into three dimensions as follows.
Let E0 (t) be the electric field of the homogeneous, transversally coherent light beam in the input plane z = 0 of the optical system. The field E (r,t) in the output half-space z > 0, which generally has acquired a 3D dependence on the field point coordinates r = (x,y,z) and a specific temporal dependence in every point, is given as temporal convolution
Let us suppose that somehow we have measured the cross-correlation function K (r,τ) between the input and output fields K (r,τ) ≡ 〈E (r,t)E0 (t − τ)〉. Thereby we have also found out the impulse response function sought, since
Our set-up follows the scheme of SEA TADPOLE, where the cross-correlation K (r,τ) has been obtained through three procedures: (i) quadratic detection of the sum E (r,t) + E0 (t − τ), where the τ-dependence appears in the overlap region of the two beams tilted with respect to each other; (ii) temporal averaging due to the “slowness” of the detector (a CCD camera); (iii) spectral-domain separation of the cross-correlation term from the two non-oscillating autocorrelation terms. The spatial dependence of h(r,t) ≡ h(r,τ) is obtained by simply scanning the light-collecting tip of the signal-carrying fiber in the output half-space.
It should be pointed out that Eq. (6) in the case of time averaging is actually insensitive to the nature of the temporal dependence of the input light—be it deterministic ultrashort pulses or stationary noise or periodical bursts of temporally incoherent noise (which is the case of our set-up with a supercontinuum fiber laser). This favorable independence from the spectral phase of the light source follows from the basic principle of the spectrally resolved white-light interferometry, or—more generally—from the properties of time-averaged responses of linear systems. The resulting equivalence between the short pulse duration and the short coherence length of the illuminating light was used already in the 1970-ies for the holographic imaging of impulse responses , and is widely exploited nowadays in the optical coherence tomography (see, e.g., [38–40]).
Therefore the measurement part of our experimental set-up (Fig. 2) is nothing but a version of that of SEA TADPOLE technique [19, 20]. However, for accommodating the challenges the ultra-broadband spectrum of the supercontinuum laser poses, all optical components were carefully chosen. For example, the single mode operation of the light collecting fibers turned out to be achievable in the whole usable spectral range only by introducing photonic crystal fibers (PCF) .
The spectral range of the spectrometer determines the available temporal resolution of the method. In the current configuration the spectral amplitude and phase response is measurable from 450nm to 1020nm, which means that a resolution of 2.7fs, i.e. 1.4 optical cycles, can be achieved. The typical spectral and impulse response function of the measurement system is shown in Fig. 3. As being a linear method, this serves as a calibration and can be subtracted from subsequent measurements . The spectral range is limited by the diminishing quantum efficiency of Si-based CCD camera in the near-infrared, and by the increasing dispersion of the glass prism in the blue. Spectral resolution, which determines the maximum pulse length, ranges between , resulting that pulses up to 4ps duration and with a time-bandwidth product of 1500 could be measured. A method similar to MUD TADPOLE , could be applied to enhance the technique for even longer temporal range. The spatial resolution—currently 3μm—is determined by the mode size of the fibers.
In our set-up we have used Fianium ultra-broadband supercontinuum fiber laser SC-400-2-PP. For single mode pulse delivery NKT Photonics polarization-maintaining endlessly single-mode photonic crystal fibers LMA-PM-5 were used, which have an almost constant mode field diameter ∼ 4.1μm from 400nm to 1200nm. Although polarization-maintaining properties were not used in the current experiments, they may prove useful for measuring the polarization response of optical systems. In the measurement set-up shown on Fig. 2, an additional broadband polarizer with a negligible wave-front distortion was placed in the spectrometer right before the CCD camera (Allied Vision Technologies Stingray F-504B). As the sensitivity of the camera abruptly decays at longer wavelengths than 1000nm, the whole spectrum of the laser source was not used, resulting in some loss of temporal resolution. The non-uniform coupling of all angular-spectral components into fiber has the same result. However, as these circumstances affect the two arms equally, they do not cause measurement errors or artifacts. For more detailed consideration of the effects of numerical aperture of fibers in SEA TADPOLE set-up see Ref. .
In contrast to previous SEA TADPOLE configurations, a prism was chosen as a dispersing element in the spectrometer in order to avoid overlapping diffraction orders. The spectrometer calibration was carried out by using a red He-Ne laser to determine the crossing angle on the CCD camera between the two light beams, and by the wavelength dependent interference fringe spacing on the recorded trace. The result was verified with interference filters and by measuring the dispersion of known glass samples. Due to the difficulty of maintaining the submicron stability while scanning in space and thermal fluctuations of fibers the measurements exhibit a slow drift in time in the measured absolute phase of a pulse similarly to SEA TADPOLE method . However, this issue can be overcome by the use of iterative algorithm for complete phase retrieval from SEA TADPOLE measurement .
Circularly symmetric binary phase gratings have the groove spacing of d = 20μm and were manufactured in Fraunhofer Heinrich Hertz Institute. The groove depth varies in the range δ = 940...945nm in a 1.57mm thick fused silica substrate was initially optimized for illumination around 800nm. Under electron microscope the duty cycle of the grating was measured to be κ = 47.5%, therefore yielding weak even diffraction orders.
High-precision UV fused silica windows were used as beam splitters. They have been put into the beam path at an almost normal incidence in order to minimize unnecessary wavelength dependent polarization effects and spatial chirp.
For measurement series “a”, the dispersion of grating substrate was compensated by another part cut out from the same substrate. In order to obtain partial dispersion compensation in “b” series, a 6mm thick fused silica substrate was placed in the reference arm of the interferometer and two fused silica substrates about 2mm and 3mm thick were placed in the measurement arm which resulted an effective thickness of 0.92mm dispersion uncompensated grating substrate. For series “c”, the dispersion from grating substrate was entirely uncompensated.
In order to make measurements at different propagation distances z after the diffraction grating, the grating was moved away from the fiber tip, while keeping the lengths of the interferometer arms constant. The lateral scans were performed with 1μm step. The FWHM temporal duration of pulses was measured from cubic spline interpolation of data points. For data smoothing purposes zero filling was used by adding 2870 zeros to both ends of retrieved spectral amplitude and phase.
The simulations were carried out by evaluating the spectral phase and amplitude of pulsed Bessel beam in Eq. (1), which was Fourier transformed into time-domain. Additional spectral term that accounts for the numerical aperture of fibers was also introduced in the simulations. In the calculations only the zeroth to fifth diffraction orders were taken into account. Experimentally measured spectral phase for given configurations was used in the simulations.
The measurement and simulation results on the impulse response of a circular phase grating with a period of 20μm and an effective fused silica substrate thickness of 0.92mm on several distances behind the grating are compared in Fig. 4. It shows that initially long pulse focuses in time for some propagation distance z and thereafter spreads out again. The theoretical predictions show good agreement with the experimental results.
The small discrepancy in the pulse intensities of different diffraction orders and in the z dependence of intensity arises from the non-Gaussian pulse profile in the experiments as the Gaussian radial profile was assumed in the simulations. Also neither the finite radius of the grating nor the small radial spectral variation of the laser pulse were accounted for. The small tilt in the t–x plane for the pulses registered at higher z values might be due to the asymmetric intensity profile of the laser beam.
To further study the temporal focusing of diffraction gratings the FWHM duration of the pulse intensities were calculated from each measurement. The results are shown in Fig. 5. The prevailing 1st diffraction order with a transform limited duration of 3.7fs, which initially after propagation through 0.92mm substrate had a duration of 27.8fs achieves average duration of 5.9fs for propagation distances z = 20...65mm. For the fully uncompensated grating substrate with a thickness of 1.57mm we can observe temporal focusing from 44.8fs down to an average of 7.0fs within the propagation range z = 40...110mm. In conjunction with the non-diffracting properties of the Airy–Bessel wave packets from the theoretical study , the temporal focusing effect was achieved in a quite long propagation extent.
We have studied the temporal broadening and focusing effect of ultrashort pulsed Bessel beams generated by a circularly symmetric binary phase grating. We have shown that a 7-fold decrease in the temporal spread of ultrashort pulse can be achieved and the Airy–Bessel wave packet with a prescribed propagation depth formed by an extremely simple device consisting only of one optical element—a diffractive axicon. Our white-light spectral interferometry set-up for characterization of ultrashort impulse responses of optical devices was built following the SEA TADPOLE scheme adopted for ultra-wide (> 1 octave) passband and with its present realization we achieved sub-3fs temporal resolution. Using incoherent supercontinuum laser source exhibits advantages over conventional sources of few-femtosecond pulses in terms of cost, simplicity of the set-up and ease of operation. Generally, the set-up should be especially useful for examining the underlying physics of light field transformation by optical systems or in designing and selecting suitable optical elements for obtaining prescribed ultrashort light-fields.
The authors acknowledge Pamela Bowlan and Rick Trebino for valuable insights into the SEA TADPOLE pulse measurement method. This work has been supported by the Estonian Science Foundation grant 7870 and by the European Regional Development Fund under project 3.2.0101.11-0037. P. P. has been partially supported by the European Social Fund under project 1.2.0401.09-0079. This work was carried out in part in the High Performance Computing Center of the University of Tartu.
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