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Abstract
We demonstrate a new method for a systematic, dynamic, high-speed, spatio-temporal control of femtosecond light filamentation in BK7 as a particular example of nonlinear medium. This method is based on using coherent conjugate asymmetric Bessel-Gaussian beams to control the far-field intensity distribution and in turn control the filamentation location. Such spatio-temporal control allows every femtosecond pulse to have a unique intensity distribution that results in the generation of structured filamentation patterns on demand. The switching speed of this technique is dependent on the rise time of the acousto-optic deflector, which can operate in the MHz range while having the ability to handle high peak power pulses that are needed for nonlinear interactions. The proposed and demonstrated spatio-temporal control of structured filaments can enable generation of large filament arrays, opto-mechanical manipulations of water droplets for fog clearing, as well as engineered radiofrequency plasma antennas.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser induced filamentation is a nonlinear process that was first observed by Pilipetskii and Rustamov in 1965 [1]. Filamentation occurs when an intense laser pulse propagating in a transparent media exceeds the critical power and undergoes self-focusing due to the Kerr effect and plasma generation resulting in beam defocusing. The critical power is defined as Pcrit = α(λ2/4πn0n2) where λ is the wavelength of light, n0 and n2 are the linear and nonlinear refractive index, respectively, and α is a constant that depends on the spatial profile of the beam and is independent of the material properties. By numerically integrating the nonlinear Schrödinger equation, it has been determined that α ≈ 1.8962 for a Gaussian distribution [2]. For more complicated distributions such as Laguerre-Gauss (LG) beams, Sa and Vieira defined the critical power as a function of the radial and azimuthal indices [3]. The balance between self-focusing and plasma defocusing results in a filament with propagation distances much longer than the diffraction limit. It has been shown that when laser pulses have peak power several time larger than Pcrit, multiple filaments will form, and their transverse locations will vary from pulse to pulse [4]. Because of this, there has been a great deal of research conducted on the control of single and multiple filamentation and the distribution of filaments [5–7]. It has been shown that a simulated array of filaments could be used to form virtual hyperbolic metamaterial (VHMMs) that could manipulate a continuous microwave beam [8] and a pulsed microwave system for radar [9]. Experimentally, filamentation has been used for fog clearing [10] and structured filamentation was used to demonstrate high-power optical waveguides in air [11], along with an increase of the plasma density and plasma survival time using two co-propagating femtosecond beams [12]. With the proper control of the number of filaments and their spatial location, applications such as these could be exploited.
Some of the earliest work trying to control filamentation used a vortex phase plate as an effective method to improve the stability and repeatability of filamentation of high peak power femtosecond laser pulses [13]. This was followed by the use of a concentric vortex phase element to manipulate the intensity distribution to control the filamentation in air [14]. Others have manipulated the shape of laser filaments with a combination of a concentric vortex phase plate in series with a coaxially aligned Fresnel axicon [15]. This resulted in the superposition of two Bessel-Gauss beams that caused a pair of helical filaments which rotate about the optical axis. Helical filaments in CS2 were generated using a phase-only spatial light modulator (P-SLM), along with a common path interferometric configuration with a 4f optical system that resulted in a collinear coherent superposition of two Bessel-Gaussian beams with opposite topological charges and different radial indices [16]. Ruiz-Jimenez et. al. reported on theoretical work that showed the existence of what they called rotating dissipative azimuthons [17]. In this work, they showed these modes propagate steadily, with a constant rotation velocity, in nonlinearly absorbing Kerr media, being excited by the coherent superposition of two Bessel vortex beams with opposite topological charges and slightly different cone angles. The difference in conical angles leads to a rotation as the beam propagates. Filament formation was also controlled by applying a π phase step across a SLM [18]. This phase step generates a Fresnel diffraction pattern that redistributed the laser intensity causing the critical threshold to be exceeded and filament formation. High-order Bessel beams with weak azimuthal modulation of the transverse intensity patterns were generated using a SLM and resulted in the formation of regular filament patterns [19]. These patterns formed on the dominant intensity rings of the beams. An arbitrary array of filaments was created in fused silica by using a SLM to generate an array of diffractive phase elements which focused parts of the original beam at different spatial locations [20]. Polarization has been shown to be an important parameter in the formation of filaments [21]. Polarization structured light fields have been used to manipulate the optical field collapse and produce more robust and controllable filaments [22–24]. Beams with nonuniform transverse polarization distribution have also been shown to suppress small-scale filamentation and beam breakup, increasing the stability of the filaments [25,26]. These techniques listed above to generate and control structured filamentation either lack true dynamic control or have limited control in regard to speed and power handling capability. Therefore, dynamically engineered light filamentation with simultaneously controlled beam parameters and shape remains largely unaddressed.
In order to address this challenge, in this paper, we introduce a new method for a systematic, dynamic, high-speed, spatio-temporal control of femtosecond light filamentation in BK7. The generation of asymmetric BG beams was described in our former work [27]. We are able to generate asymmetric BG beams with the Higher Order Bessel-Gaussian Beams Integrated in Time (HOBBIT) setup using log-polar elements. The method described in this paper integrates a femtosecond laser source at 517 nm with the HOBBIT system. A diagram of the concept is shown in Fig. 1, with the acousto-optic deflector (AOD), the beam shaping optics, and the log-polar optics comprising the HOBBIT system. By applying a time-varying RF signal to the AOD, single or multiple asymmetric BG beams are generated that result in controlled filamentation in the far-field. This can be seen in the inset for Fig. 1. By changing the RF signal the distribution of filaments is changed on demand at a time scale that corresponds to the pulse repetition frequency of the laser source. In Fig. 1, the near-field intensity distribution is shown just after the log-polar optics. The HOBBIT system allows spatio-temporal control of the nonlinear interaction resulting in structured filamentation that can be controlled on a pulse-by-pulse basis. This is shown in BK7, but this method can be applied to other nonlinear media such as air and water.
Fig. 1. Diagram of the controllable structured filamentation concept made possible by using a time-varying RF signal to generate multiple OAM modes that result in filamentation. The inset shows structured filamentation at three different instances in time.
2. Theory and simulations
The HOBBIT system is comprised of a set of optics used to precondition the input light incident on a pair of log-polar transformation optics. The HOBBIT system takes an array of Gaussian beams and converts each one to an asymmetric higher-order Bessel-Gaussian (BG) beam through a series of optical transformations, resulting in a superposition of coaxially propagating higher-order BG beams that can have integer or fractional OAM states. To perform a numerical simulation of the HOBBIT pulse with nonlinear propagation in BK7 glass, let a pulse with OAM charge m at the surface $z = 0$ be expressed by
It can be shown that using Maxwell’s equations and constitutive relations for the transparent nonlinear medium the behavior of the electric field E can be described by the following [28]:
3. Experiment and results
3.1 Experimental setup
In our experiments, the ultra-short pulse source was integrated with the HOBBIT architecture to generate spatio-temporal controlled femtosecond pulses at 517 nm. The experimental setup is shown in Fig. 2(a). The output beam from the laser source (Monaco 517) was measured to be 2.1 mm in diameter. The beam was expanded to approximately 3.9 mm with a telescope. The height of the beam is adjusted using a periscope before passing through a crystal quartz acousto-optic deflector (AOD) (Brimrose CQD-80-30-520-WC). The AOD has a clear aperture of 6 mm x 6 mm, a center frequency of 80 MHz, bandwidth of 30 MHz and a measured diffraction efficiency of 66%. In general, the time varying signal driving the AOD can be expressed as $S(t) = \sum\limits_n {{c_n}} \sin ({2\pi {f_n}(t)t + {\phi_n}} ),$where ${c_n}$ is the weighting factor used to control the amplitude of each sinusoidal component, ${f_n}(t)$ is the time varying frequency function for each sine wave and ${\phi _n}$ is the initial phase of each sine wave. In this paper, the values of n will be limited to n = 1 for generating a single OAM mode or n = 2 for generating a coherent superposition of two OAM modes. After the linear phase tilt was added to the beam with the AOD, the beam was shaped into an elliptical Gaussian (3.9 mm x 416 µm) using a cylindrical telescope. This elliptical Gaussian was incident on the log-polar optics, which preformed the geometrical transformation that wraps the beam into a ring with azimuthally varying phase. A Fourier transform is performed with a f = 150 mm lens to generate the asymmetric BG beams that are used for the nonlinear interaction. Because of the size of the beam through the optical system, a 1 mm aperture is inserted just before the BK7 sample to block the 0th order diffraction from the AOD. The imaging setup shown in Fig. 2(b) is used to image the BK7 during the nonlinear interaction. This includes a high-speed camera (Phantom C110) that has a maximum frame rate of 1 kHz using the full frame resolution. A notch filter (ThorLabs NF514-17) that had a center wavelength of 514 nm and a full width at half maximum of the blocking region of 17 nm was placed just before the camera to block any the 517 nm light from the laser so that the camera captured the spectral broadening. Using a scanning autocorrelator (GECO from Light Conversion), the pulse width before and after the HOBBIT system was measured to see if there was any pulse broadening due to the HOBBIT system. The pulse width directly from the laser was measured to be 239 fs. After the HOBBIT setup, the pulse width was measured to be 265 fs. This slight increase in pulse width was attributed to the chromatic dispersion from the AOD in the HOBBIT system.
Fig. 2. a) Experimental setup for generation of dynamic asymmetric Bessel Gaussian Beams and b) imaging setup for filamentation. c) Pulse duration measurement before (orange) and after (blue) the HOBBIT optical setup using a Gaussian fit.
3.2 Static filamentation
Beam profiles in the far-field generated from the HOBBIT system are shown in Fig. 3(a) for (left to right) m = 0, m = ±1, and m = ±2. At λ = 517 nm, n2 = (2.8 ± 0.3) × 10−20 m2/W for BK7 [30]. The critical power is calculated to be Pcrit ≈ 0.95 MW. For a pulse width of 265 fs, the pulse energy would need to exceed 0.25 µJ for filamentation to occur. When these modes are incident on the 9.5 mm thick BK7 glass, Fig. 3(b) shows the resulting structured filamentation that occurred because of the energy in each lobe exceeding Pcrit and each lobe formed individual filaments. These images were recorded just before the exit face of the BK7 sample and show that the beams collapse into approximately 6 µm diameter filaments. Figure 3(c) presents the numerically calculated beam profiles and Fig. 3(d) shows the corresponding structured filamentation for each mode or mode combination as described in Section 2. The experimental results show good agreement with numerical predictions. Single or multiple filaments are generated and contained to a volume that corresponds to the peak optical intensity.
Fig. 3. a) Experimental static beam profiles and b) the associated filaments generated from the intensity distributions. c) Numerical simulation results of the static beam profiles and d) the associated filaments.
During the filamentation process, the optical pulse undergoes spectral broadening resulting in supercontinuum (SC) generation. The SC generation can be used to determine the onset of filamentation. The SC was collected using an Ocean Optics USP4000 fiber spectrometer with a 400 µm core fiber. Between the multimode collection fiber and the fact that the fiber spectrometer did not have a slit installed, the spectral width of the SC generation measurements appears broader than in reality. However, since the SC generation was just used to ensure the onset of filamentation, the exact amount of spectral broadening was less important than the fact that there was spectral broadening. Figure 4 shows the laser spectrum and a representative set of the SC generation for m = 0, m = ±1, and m = ±2 using a pulse energy per lobe of 1.5 µJ. The dip in the middle of the spectra when using the HOBBIT modes was due to the notch filter that blocked the 517 nm laser light and allowed the camera to capture the SC generation.
Fig. 4. Laser spectrum (blue) and spectral broadening for different modes with 1.5 µJ pulse energy per lobe.
3.3 Dynamic filamentation
The timing of this experimental setup was critical to ensure the proper control of the generated modes on a pulse-to-pulse basis using the femtosecond source. To ensure the proper timing, a synchronization (sync) signal from the laser was sent to an arbitrary waveform generator (AWG). This sync signal corresponds to the optical output pulse, but with a constant temporal delay. The sync signal triggered the AWG to send the RF signal to the AOD to give the deflection to generate the proper mode(s) and the signal to the high-speed camera to capture the filament images. With this specific hardware, the maximum pulse repetition frequency (PRF) of the laser source was limited to 500 kHz due to the relatively large constant delay between the optical pulse and the electrical sync pulse. However, this technique for the generation of asymmetric Bessel-Gaussian modes is not limited to this frequency. The switching limit is related to the rise time of the AOD. For the high power AOD used in these experiments, the rise time was 0.68 µs for a beam diameter of 3.9 mm which corresponds to switching speeds in the MHz range. This rise time can be further reduced with smaller beam diameters. With the proper control to be able to change the mode or combination of modes for each pulse, dynamic structured filamentation is now possible. The modes were generated by applying a single RF signal to the AOD. To generate a coherent combination of conjugate modes, this RF signal was composed of two frequencies. By changing the relative phase between these two frequencies, an apparent rotation of the interference pattern was observed. Just like in the static case, Fig. 3, the filaments were formed at the maximum intensity locations, but in this dynamic case, the filaments changed their spatial location depending on the time varying RF signal. Still images captured from the dynamic filamentation can be seen in Fig. 5(a) for m = ±1 (Visualization 1) and Fig. 5(b) for m = ±2 (Visualization 2). In Fig. 5(c), not only was the relative phase changed for each pulse, but also the mode (Visualization 3). For reference, the relative phase for each pulse is shown in the lower right corner of each frame. All the Visualization videos captured consecutive pulses at the pulse repetition rate (PRF) of the laser. In this case, the PRF is reduced to 1 kHz to match the frame rate of the high-speed camera.
Fig. 5. Sill images captured from dynamic filamentation showing pulse-to-pulse control enabled by changing the relative phase between two frequencies for a) m = ±1 (see Visualization 1), b) m = ±2 (see Visualization 2) and c) with additional switching between modes (m = ±2, ±1, 0) (see Visualization 3) for 5 instances in time that corresponds to the laser pulse repetition frequency.
The images of the filamentation events shown so far were captured with the imaging camera focused at a fixed location just inside the exit face of the BK7. By translating the camera with a linear stage, a series of images were captured and stacked to show the filaments in the 3D volume using the maximum intensity projection. An example of this is shown in Fig. 6 for m = ±2 with a pulse energy of approximately 6 µJ. The direction of the pulse propagation through the 9.5 mm thick BK7 glass is indicated. It is important to note that the apparent transverse undulations of the filaments in Fig. 6 are artifacts caused by the motion of the linear stage as the camera was scanned through the volume.
With the current technique for visualizing the volume, the filaments can be examined under different experimental conditions. Figure 7(a) shows two representative plots of experimental data for the filament formation using m = ±2 at different pulse energies, Epulse. Assuming the energy was equally divided into each lobe, the pulse energy per lobe is also listed for each condition. As Epulse increases, the filaments tended to form sooner and closer to the entrance face. In all cases, the filaments were restricted to specific regions that correspond to the high intensity lobes of the beam profiles. For a given pulse energy one can see that the filaments formed using m = ±2 do not start at the same location along the propagation direction. This can be explained by looking at the experimental profile of this mode in Fig. 3(a). There is a slight asymmetry in the lobe sizes. Since some of the lobes are larger than others, the energy density for those larger lobes is less, causing the filament to form at a larger distance from the entrance face of the BK7. It is important to note that the energy distribution is reasonably equal for each lobe, with the values differing by less than l0%. The numerical simulations under similar conditions are shown in Fig. 7(b), confirming the same general trend that as the peak power is increased, the distance need for the filaments to form is decreased.
Fig. 7. a) Experimental and b) simulation results of the filamentation volume using m = ±2 with different Epulse. The optical beam propagates left to right and the length scale for all plots (experimental and simulation) is 9.5 mm.
Up to this point, only integer mode combinations have been used in this paper to create structured filaments. The HOBBIT system is also capable of generating fractional OAM modes which can be considered as a superposition of beams carrying integer OAM. This capability can be exploited for additional spatio-temporal control of the structured filaments that would not be possible with integer modes alone. By varying the coherent superposition of modes from m = 0 to m = ±1 in steps of 0.1, the intensity profile changed from a single maximum (m = 0) to a pattern with two distinct lobes (m = ±1), Fig. 8(a) (top). Because of the modification of the intensity profile, distinct filaments were seen with increased separation distance as shown in Fig. 8(a) (bottom). In Fig. 8(b), the distance between filaments was plotted against the difference in the charge numbers for the given combination. The separation distance was measured to be 16 µm using m = ±0.6 up to almost 70 µm using m = ±1. The intensity profiles and the pairs of filaments shown in Fig. 8(a) should ideally be in the vertical position. However, due to a maximum timing jitter of 5 ns between the optical pulse and the RF signal sent to the AOD, there was a slight rotation in the intensity profile and therefore the filaments’ locations for some pulses. The pulse energy was held constant at 1.5 µJ for each mode combination. Another method to adjust the spacing between filaments was to use a fractional mode combination and adjust the phase between the modes. Figure 8(c) (top) shows the beam intensity profiles for different relative phase, θ. When θ = 0, there were two maxima with a slight depression in the intensity between them. As the relative phase was increased, the two maxima separated as they rotated. This resulted in the generated filaments having a larger separation as they rotated about the propagation axis as seen in Fig. 8(c) (bottom) for m = ±0.7. Using this method, the distance between filaments varied from 40 µm to 70 µm as the phase was adjusted from 0 to π radians as shown in Fig. 8(d). Being able to generate fractional OAM modes with the HOBBIT system makes it possible to adjust the filament spacing beyond the limit when using integer modes.
Fig. 8. (a) Beam profiles (top) and generated filaments (bottom) for fractional mode combinations, m = 0 to m = ±1 in steps of 0.1 with (b) the distance between filaments for the different fractional mode combinations. (c) Beam profiles (top) and generated filaments (bottom) for m = ±0.7 with relative phase between modes from 0 to π radians with (d) the distance between filaments for the different relative phase values.
4. Discussion and conclusion
This paper introduced a unique approach for the spatio-temporal control of laser filaments. Specifically, we combine a source at 517 nm with the HOBBIT system and were able to manipulate the intensity distribution by using either a single OAM mode or a coherent superposition of OAM modes. When the peak power exceeded Pcrit of BK7, nonlinear self-focusing occurred that resulted in filamentation. By having spatio-temporal control of the intensity distribution of the femtosecond laser pulses using fractional and/or integer modes, we were able to achieve structured filamentation on demand. One of the critical components of the HOBBIT system is the AOD, which adds a linear phase tilt to the femtosecond pulse before the geometric transform. The AOD is driven by a single time varying RF signal that can be changed for each individual pulse. Given the rise time of the AOD, this switching time is in the MHz range. Therefore, by changing the driving signal to the AOD, the OAM mode(s) can be changed for each individual femtosecond pulse, resulting in structured filamentation that can be controlled on a pulse-by-pulse basis. By having this spatio-temporal control of structured filaments, applications such as high-power optical waveguides in air, forming virtual hyperbolic metamaterials to be able to remotely focus and steer radar signals, and opto-mechanically move droplets for fog clearing can be exploited to the fullest potential. Further research is currently underway to exploit the spatial filament control in liquids and other dynamic environments that are typical in maritime regimes.
Funding
Multidisciplinary University Research Initiative (N00014-20-1-2558); Office of Naval Research (N00014-18-1-2225, N00014-20-1-2037).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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