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Abstract
While conventional complex phase masks are chromatic, we present an achromatic holographic phase mask capable of performing optical beam transformations in a spectral range exceeding 1000 nm. The system consists of a holographic phase mask fabricated by encoding the desired phase profiles into volume Bragg gratings, inserted in between two surface gratings. This device automatically adjusts each spectral component diffracted by the surface grating to the Bragg angle of the volume Bragg grating and equalizes phase incursion for all diffracted wavelengths. Transverse mode conversion is demonstrated and compared with theory for multiple narrow line laser sources operating from 488 to 1550 nm and for a broadband femtosecond source.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Corrections
25 June 2019: A typographical correction was made to the author listing.
1. Introduction
Optical phase masks or phase shifting masks have been used for many decades and have numerous applications in the fields of imaging [1–5], encryption [6–9], beam shaping [10–14], and mode conversion [15–18]. In order to create these optical elements, there are two typically used methods: controlling the local geometrical path length by generating a contoured surface [5,10–15], or introducing variations in the local refractive index in a photosensitive material such as lithium niobate or photosensitive glass [19,20]. A phase mask of almost any profile can be made with these two methods and phase differences can be achieved with extreme accuracy. However, these elements are inherently monochromatic due to their reliance on changes in optical path length in order to create phase differences. This makes optical phase masks ineffective for broadband sources, limiting the range of potential applications. Achromatic phase plates (trivial phase masks with a flat spatial profile) have been previously produced with birefringent materials or birefringence in diffraction gratings with periods below the working wavelength [21,22]. Another technique is multiplexing many computer-generated holograms in one medium [23–25]. However, for a fully achromatic device several holograms have to be recorded, resulting in high losses in this device.
In this manuscript, we demonstrate a novel achromatic phase mask system that is capable of performing wavefront conversions over 1000 nm. The central element of the achromatic system is a holographic phase mask (HPM) that is produced by encoding binary phase profiles into a transmitting volume Bragg grating (TBG) [26,27]. HPMs can modify the wavefront of the diffracted beam over a wide range of wavelengths, as long as the Bragg condition of the volume grating is satisfied. This is in contrast to complex holograms that cannot generally reconstruct the same phase profile at wavelengths different than the recording wavelength. The HPM utilizes the diffraction characteristics of TBG, which can diffract up to 100% of a beam into a single order, and can diffract over a broad range of wavelengths by changing the angle of incidence (with the diffraction efficiency depending on the wavelength and strength of the grating) [28]. We build upon further and demonstrate in this paper that an HPM can become fully achromatic if it is placed between a pair of surface gratings which provide the necessary dispersion for a collimated broadband laser beam [29,30]. This combination results in a new optical element that is a complex achromatic holographic phase mask with properties that cannot be achieved by conventional phase masks technologies.
Moreover, the narrow angular selectivity of a TBG also allows for several HPMs to be multiplexed into the same element with little to no cross-talk between elements; each grating is accessed by altering the beam’s angle of incidence onto the element. By recording such elements in a suitably thick photosensitive medium, it is also possible to ensure that the diffraction efficiency of each element is not affected.
To explain the unique properties of the HPMs, we note that a Bragg grating is the simplest volume hologram, which, unlike more complex holograms, can diffract multiple wavelengths without distorting the beam profile by changing the incident angle. By encoding phase levels which cover a macroscopic area, the HPM acts locally as a standard TBG with a given phase shift. Thus, the HPM will diffract in the same manner as a standard TBG except at the relatively small number of phase discontinuities, and the diffracted beam’s phase profile will match the encoded phase level profile regardless of incident wavelength. The HPM therefore acts as a spectrally addressable phase mask, and by applying surface gratings with double the period of the HPM, a truly achromatic phase mask with complex spatial phase profile is created.
In this paper, we focus first on the properties of the HPM itself, and demonstrate HPMs that, though recorded at 325 nm, preserve a binary phase profile in the diffracted beam at different wavelengths when the angle of incidence corresponds to the Bragg angle and can operate with laser sources having wavelengths spaced more than 500 nm apart. Secondly, we show that pairing the HPM with transmitting surface diffraction gratings allows the Bragg condition to be met for all the spectral components of a broadband source, eliminating the need for angular tuning. TEM mode conversion on spectrally narrow and broad sources with the HPM itself and with its surface grating pair is demonstrated and discussed as a potential application of the technology.
2. Holographically encoded phase masks
To encode the phase profile into a TBG, consider the holographic recording setup in Fig. 1. Here, a multi-level phase mask has been placed into one arm of a two-beam interference system (the object beam), where the two beams interfere at an angle θ relative to the normal of the holographic sample. The two-beam interference equation describing the fringe pattern in the sample is then
where is the intensity, is the wavevector for each beam, and is the phase variation introduced by the phase mask after the object beam has propagated to the sample. Since the phase mask’s profile is located in the phase mask’s x0-y plane, which is rotated with respect to the sample plane, the recorded phase profile will generally be different than the phase mask. However, if the thickness of the sample, the axial distance between the phase mask and the sample, and are all small then , so the phase profile recorded in the hologram will be approximately the same as that of the original phase mask. The recorded hologram will have a refractive index profile ofwhere is the background refractive index, is the refractive index modulation, and is the grating vector.
Fig. 1 Recording setup for encoding a conventional phase mask into a volume Bragg grating. BS – beam splitter, M – mirror.
Once the hologram is recorded, it is placed in a system with a probe beam to be diffracted, which may or may not have the same wavelength as the recording beams. If the recorded phase profile contains macroscopic regions of constant phase, and relatively few sharp phase transitions, then the hologram recorded will act locally as a Bragg grating, except at the phase transition regions. If the probe beam is incident at the local Bragg condition (see Kogelnik [28]), the total electric field will satisfy the scalar Helmholtz wave equation,
Here kp is the wavenumber of the probe beam. The Helmholtz equation has a general solution of the form [28]
where A and B are the complex amplitudes of the transmitted and diffracted waves, respectively, and is the wavevector of the diffracted beam. Inserting Eq. (4) into Eq. (3) results in a set of coupled wave equations between the amplitudes A and B. Kogelnik has solved these equations when A and B depend solely on the axial distance z (homogenous gratings). However, when a phase mask is placed in the recording system, the phase term is not a constant across the entire hologram aperture and consequently it cannot be assumed that this one-dimensional dependence will still hold. In this particular study we are only interested in probe beams that exactly satisfy the Bragg condition. In this case, the coupled wave equations becomeHere κ = πn1/λ0 is the coupling coefficient of the grating. Note that we have assumed that the second derivatives are negligibly small in the same manner as Kogelnik, as we still expect the transfer of energy between the transmitted and diffracted waves to be slow.
We solve the coupled equations numerically by converting Eq. (5) into Fourier space along the transverse dimensions x and y, giving
where and are the Fourier transforms of A and B, respectively, and fx and fy are the spatial frequencies along the x and y axes, respectively. To solve these equations numerically, we split the propagation and energy transfer between the waves into two discrete steps and successively propagate and transfer energy between waves over several small propagation steps. To calculate the propagation of the beam, the right side of Eq. (6) is assumed to be zero. In this case the Fourier amplitudes have a solution of the formNote that this is only exact in the case where the right side of Eq. (6) truly equals zero, but for small (~100 nm) propagation steps, this is a reasonable approximation. To account for energy transfer, we piecewise integrate the right side of Eq. (5) with the Euler method and add it to the inverse Fourier transform of Eq. (7):
Calculations indicate that for a propagation step size of 100 nm, our numerical method conserves energy to within 0.01% after propagating the coupled waves through the entire system, which is sufficient for the phase profiles discussed here.Numerical simulations were performed to determine the diffracted beam phase profile and diffraction efficiency of an HPM in the case where a binary phase profile is encoded. The numerical method described previously was first applied to simulate a standard TBG with an 8 µm period, a refractive index modulation of 250 ppm, and a thickness of 2 mm. The probe beam with a wavelength of 1064 nm is incident at the Bragg angle and propagates in the x-z plane, see Fig. 2(a). Using a propagation step size of 100 nm, the simulated diffraction efficiency of this TBG is 99.13%, which is consistent with the 99.14% peak diffraction efficiency of a homogenous TBG described by Kogelnik. A π phase step was then introduced to the grating at a recording wavelength of 325 nm. This binary step was first introduced along the x-axis and then along the y-axis to determine if there would be any orientation-dependent variations in diffraction efficiency or phase profile. The calculations were then repeated for two probe beams with wavelengths of 632 nm and 975 nm. In all cases the grating parameters are the same but the incident angle was changed so that all beams were incident at their respective Bragg angles.
Fig. 2 (a) A probe beam incident at the Bragg angle is diffracted by a holographic phase mask (HPM) with a single phase dislocation along one axis. Numerical simulations results demonstrating: (b) the diffracted beam phase profile and (c) the local diffracted intensity of a plane wave for beams of different wavelength. (d) The diffraction efficiency of an HPM at 1064 nm relative to a standard transmitting volume Bragg grating as a function of beam diameter when a binary phase dislocation is encoded along the x-axis. Here, the coordinate origin is the center of the front surface of the HPM. (e) The diffracted beam phase profile and (f) the local diffracted intensity when a binary phase dislocation is encoded along the y-axis for beams of different wavelength [27].
As shown in Fig. 2(b), each diffracted beam contains a π phase shift when the phase step is introduced along the x-axis. The location of the phase step is slightly offset from the origin due to the propagation of the beam through the sample, and changes for each beam due to their different Bragg angles. For the local intensities of the diffracted waves, shown in Fig. 2(c) for a plane wave, (note that the relative strengths are different due to the wavelength-dependent diffraction efficiency) a general decrease in diffracted energy is observed at the phase discontinuity. This is unsurprising, as any region of the probe beam crossing the phase discontinuity will not be able to satisfy the Bragg condition at that point. This inability to satisfy the Bragg condition locally will result in a decrease in diffraction efficiency (which is the total energy in the diffracted beam divided by the total energy in the incident beam) that is dependent on the fraction of the beam energy which crosses the phase discontinuity. This decrease in diffraction efficiency will also mask the slight differences in phase for each wavelength in the phase transition region. However, as shown in Fig. 2(d), this loss in efficiency becomes very small with increasing beam size and the diffraction efficiency of an HPM will asymptotically approach the diffraction efficiency of a standard TBG as the beam diameter approaches infinity.
When the phase discontinuity is oriented along the y-axis, the phase profile of the diffracted beam will also have a π phase shift, as shown in Fig. 2(e). Here the phase discontinuity is located at the origin because the probe beam has no component propagating along the y-axis. This zero y-axis component results in the diffracted beam having a near zero-width transition region that is identical for every wavelength. It also results in only a single infinitesimal fraction of the beam ever encountering the phase discontinuity, giving constant local intensities for the diffracted waves, as shown in Fig. 2(f). The simulated diffraction efficiency of the HPM in this case is constantly within 0.01% of the predicted efficiency for a TBG at each wavelength as given by Kogelnik regardless of beam diameter, and given that there is a 0.01% uncertainty in energy conservation in the numerical method we conclude that the diffraction efficiency is identical to a standard TBG. These two cases indicate that the diffracted beam from an HPM will always inherit the phase profile of the original phase mask, and will have some orientation-specific decrease in diffraction efficiency that is dependent on beam size and largely negligible for most typical beam diameters. Repeating these cases for other wavelengths at their respective Bragg angles show similar results, with the diffraction efficiency of both the TBG and HPM changing with respect to wavelength as described by Kogelnik. Thus the HPM will maintain the diffraction characteristics of a TBG, including the wavelength and angular spectrum, while preserving the desired phase profile over the whole bandwidth of possible Bragg wavelengths.
For experimental confirmation, an HPM was recorded in a 1.97 mm thick photo-thermo-refractive (PTR) glass sample as discussed above. PTR glass is a multicomponent photosensitive glass with a transparency window from the near UV to the near IR which has the ability to sustain high power beams [31–33]. To simplify fabrication and provide a clear demonstration of the phenomenon we chose to use binary phase profiles here, but the approach is fully applicable for multilevel phase masks as well. To fabricate the HPM, a four-sector binary phase mask20 designed for the recording wavelength of 325 nm was placed in one arm of the recording setup and the half angle of interference was set to 0.786°, giving a grating period of 8 µm. The phase mask was placed approximately 150 mm from the sample.
Three beams (3 mm in diameter at 1/e2) at wavelengths in the visible and the infrared regions were applied to study the wavelength dependence of diffraction and mode conversion using the HPM. As shown in Figs. 3(b)-3(d), for the three extremely different Bragg wavelengths (632.8 nm, 975 nm, and 1064 nm), the diffracted beam profiles exhibited the predicted four-lobed pattern [Fig. 3(a)]. This clearly confirms our initial thesis that the binary phase profile is being preserved in the diffracted order for an extremely broad range of wavelengths.
Fig. 3 (a) Simulated far field profile of a beam after passing through an ideal four-sector binary mask and the diffracted beam from a four-sector HPM at (b) 632.8 nm, (c) 975 nm, and (d) 1064 nm (Sizes are not to scale) [27].
In order to directly and accurately compare the diffraction efficiency of the HPM with that of a standard TBG, we fabricated a homogeneous TBG in the same volume of PTR glass during the recording of the HPM. This was done by recording an HPM, and then removing the phase mask from the object beam and rotating the PTR glass sample without lateral shifting to record a tilted TBG with the same recording dosage. Recording both elements in the same volume ensured that the local refractive index change and any sample inhomogeneities would be shared between the elements, and further demonstrated a new opportunity for holographic phase mask multiplexing. The diffraction efficiencies of each element were approximately 78% at 1064 nm, showing good agreement with theoretical predictions, and indicating that the HPM does not suffer from reduced diffraction efficiency relative to standard TBGs. There is also no cross-talk between the two holograms, demonstrating that the HPM preserves the narrow angular acceptance of standard TBGs.
3. Achromatization with surface gratings
True achromatization of HPMs can be achieved by pairing the Bragg grating with two surface gratings [29,34]. The use of thin gratings with a thick volume transmission grating was demonstrated for broadband spatial filtering, and is the same approach in this paper [35]. According to coupled wave theory [28], incident light onto a TBG undergoes diffraction under the Bragg condition,
Where is the TBG’s grating period, is the Bragg angle, and λ is the incident wavelength. A TBG’s Bragg condition is similar to the dispersion equation for a surface grating:Here is the period of the surface grating, is the diffracted angle, and is the order of diffraction. When the two gratings periods satisfy the condition, the first order diffracted angle from the surface grating will match the Bragg condition for the TBG for all wavelengths. This circumvents the broad but still finite spectral selectivity of the TBG and results in increased diffracted spectral bandwidth and overall diffraction efficiency for sources with a bandwidth larger than the TBG’s spectral selectivity. Although using surface gratings with exactly twice the period of the TBG’s period is ideal, gratings with periods in close proximity to this ratio have also been shown to be effective [29,30].
The proposed system of the two surface gratings and the TBG is laid out in Fig. 4. Here the broadband input is diffracted through the first surface grating, where the diffracted angle for each spectral component is equal to the Bragg angle of the TBG with an incorporated phase mask. Figure 5 shows the theoretical spectra of diffraction efficiency for a surface grating compared to that for a TBG. The gratings simulated had similar properties as the ones used for experimental proof of concept described in detail in Section 4. The simulated TBG had a spectral bandwidth of approximately 25 nm (FWHM). After pairing with a single surface grating, the total spectral bandwidth of the two elements equals the bandwidth of the surface grating (~750 nm FWHM), corresponding to a 30X improvement in bandwidth. However, pairing the TBG with just a single surface grating does not eliminate the dispersion from the system; though the surface grating/TBG pair has a bandwidth of ~750 nm, this bandwidth is spread over a large angular range. This limits the effective bandwidth of the system to a couple of nanometers for applications needing minimal chromatic aberrations, such as mode conversion. To eliminate this dispersion, we add a second surface grating identical to the first after the TBG. This counters the dispersion from the first element, providing a collinear output while still preserving the full bandwidth of the system. When replacing the TBG with an HPM, which has the same diffraction properties as a simple TBG, the resulting system becomes an achromatic HPM (AHPM).
Fig. 4 Concept of using surface gratings pairs to meet the Bragg condition for various wavelengths at the same incident angle, allowing applications such as achromatic mode conversion.
Fig. 5 Normalized spectra of diffraction efficiency for a surface grating and a transmitting volume Bragg grating.
4. Broadband mode conversion
To experimentally demonstrate the potential applications of the AHPM, we constructed a mode converter using a 4-sector binary HPM. While a purely binary phase profile cannot completely convert from one Gaussian mode to another, it can convert it such that the dominant mode is the desired transformed mode; the TEM00 mode is nearly 70% converted to the TEM11 mode with a 4-sector binary profile [20]. The HPM used for the following mode conversions had a grating period of 3.4 µm and an 87% diffraction efficiency at 765 nm. Two commercial transmitting surface gratings with a groove spacing of 150 lines/mm, corresponding to a period of 6.66 µm, were chosen as a close match to the periodicity condition (6.8 μm). The surface gratings were blazed for a wavelength of 725 nm, giving diffraction efficiencies between 20% and 80% in the wavelength range of 425 nm to 1550 nm, where our experiments were performed. The AHPM was used to convert both multiple laser sources and a femtosecond laser source.
4.1 Multiple laser sources
The goal was to achieve successful mode conversion from a fundamental TEM00 to a TEM11 mode profile for laser sources of different wavelengths without needing to angularly tune the HPM. In our experiment, five different narrow linewidth laser sources were used to cover a wavelength range of over 1000 nm (488-1550 nm). Each laser’s output was expanded to a beam diameter of approximately 6 mm to ensure a high HPM efficiency while remaining within the clear aperture of the surface gratings (10 × 10 mm). The first surface grating was aligned so that the HPM’s Bragg condition would be met for all wavelengths; the second grating successfully nullified the beam deviation and dispersion. To achieve far field imaging, the beam was focused with an achromatic doublet (f = 500 mm) onto a CCD camera.
A commercial Argon-ion (488 nm), He-Ne (543 nm and 632 nm), and diode (765 nm, 1064 nm and 1550 nm) laser sources were used to show the achromatic properties of the AHPM. Successful TEM00 to TEM11 mode conversion was achieved for each source, as shown in Fig. 6. A total of three diffraction efficiency measurements were performed for each of the six sources - the surface diffraction grating, the HPM itself, and lastly the full AHPM combination. These measurements are shown together with the corresponding theoretical calculations in Fig. 7. The theoretical predictions and measurements for the surface gratings matched the ones provided by the manufacturer, measuring 80% efficiency at 765 nm. When measuring the HPM’s efficiency, the HPM had to be angularly tuned to the Bragg condition of each individual source’s wavelength. Without angular tuning, the HPM would only be able to diffract one of these sources, as each source is spectrally separated by far more than the bandwidth of the HPM. Due to the lack of anti-reflection coatings on the HPM, the efficiency measurements of the HPM took into account any reflection losses. The HPM showed a maximum diffraction efficiency of 87% at 765 nm. Coupled wave theory was used to model the expected diffraction efficiency, and results showed a similar trend of a higher efficiency near 600-800 nm region. Due to the HPM’s diffraction efficiency properties being the same as of a TBG, optimization of the grating’s strength and length can be made to reach efficiencies near 100%. The measured absolute diffraction efficiencies of the AHPM (that, by implementing the surface gratings on both sides of the HPM, eliminated the need to angularly tune the HPM) showed a maximum of 47% at 765 nm and followed the behavior of the surface gratings efficiency profile. The theoretical efficiency was calculated by multiplying the efficiency of the HPM to that of the surface gratings, and the results were consistent with measured values. For many applications, a high diffraction efficiency is desired, and becomes even more critical in systems with multiple diffractive elements. Even though our measured diffraction efficiency was only 47%, optimization of the system can be done to reach higher efficiencies. Due to the HPM’s diffraction efficiency properties being the same as of a TBG, optimization of the grating’s strength and length can be made in order to reach diffraction efficiencies near 100%. Surface gratings have been shown capable of reaching higher diffraction efficiencies near 95-99% theoretically [36] and experimentally [37]. This was done by controlling the gratings periodic structure and groove depth in order to suppress the diffraction in non-desired orders [36]. As well, multilayer surface dielectric transmission gratings, which is the combination of a thin film dielectric coatings and an etched periodic grating structure, has achieved diffraction efficiencies of >95% [38,39]. While our experiment was to show proof of concept of the AHPM, a fully optimized system can reach estimated diffraction efficiency between 90% and 98%.
Fig. 6 Far field images of the Gaussian beams converted to TEM11 mode for an (a) 488 nm (b) 543 nm (c) 632 nm, and (d) 765 nm diode, (e) 1064 nm, and (f) 1550 nm laser (Sizes are not to scale).
Fig. 7 Comparison of measured and simulated diffraction efficiencies for the (a) surface grating at normal incidence, (b) the HPM at each source’s wavelength Bragg condition, and (c) the AHPM system at normal incidence for all sources.
4.2 Femtosecond laser source
Ultrafast femtosecond laser systems have enabled many breakthroughs in the fields of science and technology. Their continued development and growth has led to industrial applications such as high-precision micro-machining [40], industrial processing [41], ultra-fast detection [42], biology [43], and material processing [44]. Currently, further research is being done in manipulating the transverse mode structure of femtosecond systems in order to further their applicability [45]. The zero-intensity center of higher order transverse modes provides an optical trapping effect for a wide range of potential applications in subwavelength nonlinear microscopy [46], optical tweezers for micro- and nano-manipulation [47], and for creating filamentation of radiation [48,49]. In order to show that the AHPM could be applicable in creating such non-Gaussian beam profiles where broad-band laser sources are used, we performed tests with a commercial femtosecond laser system (FemtoFErb 780 from Toptica Photonics, Inc), with specifications listed in Table 1. Figure 8(a) shows the image of the fundamental mode output after imaging with an achromatic doublet. Initially, when using only the HPM (no surface gratings), the chromatic dispersion causes the beam to smear in the diffraction plane, resulting in an unsuccessful mode-converted beam, as shown in Fig. 8(b). This dispersion was eliminated, showing proper mode conversion, when using the full AHPM system, as shown in Fig. 9. The four sector HPM allows three different mode conversions to be achieved depending on beam location relative to the phase shift boundaries; Fig. 9 shows the mode transformations of the femtosecond beam (TEM00 to TEM01, TEM10, and TEM11) using the AHPM system.
Table 1. FemtoFErb 780 Femtosecond Laser Specifications
Fig. 8 Far field images of (a) the sources Gaussian output and (b) the resulting beam after attempting TEM00 to TEM11 mode conversion using only the HPM. Angular dispersion results in beam smearing in plane of diffraction.
Fig. 9 Far field images of femtosecond beam mode conversion from TEM00 to (a) TEM10, (b) TEM11, and (c) TEM01 after passing the single mode beam through the AHPM (Sizes are not to scale).
We also investigated the spectral width of the transformed femtosecond beam, with the purpose of verifying that no spectral bandwidth was lost. Figure 10 shows spectra of the original laser pulse, the pulse transformed with HPM, and the pulse transformed with AHPM. The spectral measurements of the beams were done using a fiber coupled optical spectrum analyzer (OSA). The original laser pulse had a bandwidth of 12.8 nm. If converted by the HPM only, the output bandwidth measured decreased to 1.8 nm over the angular range captured by the OSA. This is due to the dispersion of the HPM; when shifting the OSA collection optics to measure the diffraction at different angles, the center frequency of the measured spectrum would shift to other portions of the source’s original spectrum. However, the spectrum of the beam after the AHPM showed negligible alteration to the spectral structure at a fixed diffraction angle, confirming that achromatic mode conversion for a broadband femtosecond pulse was achieved. The overall diffraction efficiency of the AHPM when converting the femtosecond pulses was 41%, which was due to the gratings’ efficiencies (particularly the surface gratings) not being optimized for use around 780 nm. An optimized AHPM (where both the volume hologram and surface gratings are appropriately designed) is expected to exhibit over 99% diffraction efficiency at the desired peak wavelength.
Fig. 10 The original femtosecond pulse spectrum and output spectra after conversion with HPM and AHPM.
5. Conclusions
We have successfully demonstrated that complex (e.g., binary) phase profiles may be encoded into transmitting volume Bragg gratings to form holographic phase masks. These masks provide the same phase profile to the diffracted beam regardless of wavelength as long as the Bragg condition is satisfied, resulting in a tunable holographic phase mask (HPM). By pairing the HPM with surface gratings having twice the period of the HPM, a new element functioning as an achromatic holographic phase mask (AHPM) for broadband sources was demonstrated. These AHPMs may be used to simultaneously diffract and convert transverse laser modes. This new optical element dramatically increases the potential applications of optical phase masks in broadband sources including ultrashort pulse lasers.
Funding
DEPS Graduate Fellowship and HEL JTO grant’s W911NF-10-1-0441 and W911NF-12-1-0450.
Acknowledgments
We would like to thank Dr. Boris Zeldovich for his fruitful discussions in developing the theoretical model for this work, the Laser Plasma Laboratory for use of the femtosecond laser, and OptiGrate Corp. for supplying the PTR glass used in our experiments. The portion of this work regarding HPM theory and numerical simulation was previously published in SeGall [27].
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