System and method of characterizing a light pulse

Abstract

A method of characterizing an ultrafast laser pulse includes emitting, via a laser, laser pulses that are carrier-envelope phase-stabilized; directing, via a mirror, a portion of the laser pulses into a non-linear gas comprising carbon dioxide to generate plasma; generating, via a detector, an electrical signal representative of a phenomena produced at the plasma; and determining, via a processor, a characteristic of the laser pulse based at least in part on the electrical signal.

Description

SYSTEM AND METHOD OF CHARACTERIZING A LIGHT PULSECROSS-REFERENCE TO RELATED APPLICATIONS

[0001] The current patent application is a non-provisional utility patent application which claims priority benefit of U.S. Provisional Patent Application Serial No. 63 / 683,015 entitled “PLUG AND PLAY DEVICE FOR LIGHT PULSE CHARACTERIZATION BASED ON AIR FLUORESCENCE,” filed August 14, 2024, U.S. Provisional Patent Application Serial No. 63 / 713,138 entitled “MEASUREMENT OF CARRIER-ENVELOPE-OFFSET FREQUENCY AND PHASE BY MICROPHONE,” filed October 29, 2024, and U.S. Provisional Patent Application Serial No. 63 / 723,682 entitled “SINGLE-SHOT CEP TAGGING APPARATUS AND SUPERCONTINUUM ULTRAVIOLET LIGHT SOURCES BASED ON AIR LASING,” filed November 22, 2024, the entire disclosures of which are incorporated herein by reference.BACKGROUND OF THE INVENTION

[0002] Despite rapid advances in laser science, no new methods have been developed for characterizing ultrafast laser pulses (having a duration less than one picosecond), such as measuring a pulse duration, determining chirp, measuring carrier-envelope frequency, etc. Existing techniques generally require utilizing the principle of f-to-2f interferometry (i.e., frequency beating). Traditional f-to-2f interferometry techniques require the spectrum to be broadened over one octave using photonic crystal fibers or nonlinear crystals before measurement, making it highly unstable. Existing techniques for carrier-envelope phase measurement, such as use of a stereo-above-threshold ionization phase meter, require a complex apparatus and high- vacuum conditions, making it challenging to extend carrier-envelope phase measurements to the frequency domain for determining carrier-envelope frequency.

[0003] Thus, there is a need for improved systems and methods for characterizing ultrafast laser pulses. This background discussion is intended to provide information related to the present invention which is not necessarily prior art.SUMMARY OF THE INVENTION

[0004] Embodiments of the current invention address one or more of the above-mentioned problems and provide a distinct advance in the art of ultrafast laser science and spectroscopy.

[0005] One embodiment of the invention is a method of characterizing an ultrafast laser pulse. The method includes emitting, via a laser, laser pulses that are carrier-envelope phase- stabilized; directing, via a mirror, the laser pulses into a non-linear gas comprising carbon dioxide to generate plasma; generating, via a transducer, an electrical signal representative of acoustic waves produced at the plasma; and determining, via a processor, an estimated duration of the laser pulses based on the electrical signal.

[0006] Another embodiment of the invention is a method of characterizing an ultrafast laser pulse. The method includes emitting, via a laser, laser pulses that are carrier-envelope phase- stabilized; directing, via a mirror, a portion of the laser pulses into a non-linear gas comprising carbon dioxide to generate plasma; generating, via a spectrometer, an electrical signal representative of fluorescence of the plasma; and determining, via a processor, an estimated duration of one of the laser pulses based on the electrical signal.

[0007] Another embodiment of the invention is a system for characterizing an ultrafast laser pulse. The system includes a laser, a non-linear medium, a transducer, and a processor. The laser is configured to emit carrier-envelope phase-stabilized laser pulses. The non-linear medium includes gaseous carbon dioxide operable to absorb the laser pulses to generate plasma. The transducer is configured to generate an electrical signal representative of acoustic waves produced at the plasma. The processor is configured to determine an estimated duration of one of the laser pulses based on the electrical signal.

[0008] Another embodiment of the invention is a system for characterizing an ultrafast laser pulse. The system includes a laser, a non-linear medium, a spectrometer, and a processor. The laser is configured to emit carrier-envelope phase-stabilized laser pulses. The non-linear medium includes gaseous carbon dioxide operable to absorb the laser pulses to generate plasma. The spectrometer is configured to generate an electrical signal representative of fluorescence of the plasma. The processor is configured to determine an estimated duration of one of the laser pulses based on the electrical signal.

[0009] Another embodiment of the invention is a method of determining a carrier envelope phase of a single-shot, ultrafast laser pulse. The method includes splitting, via a mirror, the laser pulse into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity; directing, via a mirror, the ionizing beam into a non-linear gas comprising carbon dioxide to generate plasma; directing, via a mirror, the probingbeam at the plasma; generating, via a spectrometer, an electrical signal representative of fluorescence of the plasma; and determining, via a processor, the carrier envelope phase of the laser pulse based on the electrical signal.

[0010] Another embodiment of the invention is a method of generating an ultraviolet supercontinuum. The method includes emitting, via a laser, a laser pulse that is carrier-envelope phase-stabilized. The laser pulse has a duration that is less than 4 femtoseconds. The method further includes directing, via a mirror, a portion of the laser pulse into a non-linear gas comprising carbon dioxide to generate the ultraviolet supercontinuum having wavelengths from 250 nanometers (nm) to 500 nm.

[0011] This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the detailed description. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter. Other aspects and advantages of the current invention will be apparent from the following detailed description of the embodiments and the accompanying drawing figures.BRIEF DESCRIPTION OF DRAWINGS

[0012] Embodiments of the current invention are described in detail below with reference to the attached drawing figures, wherein:

[0013] FIG. 1 is a schematic diagram depicting selected components of a system constructed in accordance with embodiments of the present invention;

[0014] FIG. 2 is a perspective view of a system constructed according to another embodiment of the invention and having one or more filters;

[0015] FIG. 3 is a flowchart depicting exemplary steps of a method of characterizing a laser pulse according to an embodiment of the present invention;

[0016] FIG. 4 is a perspective view of a system according to another embodiment of the invention and having wedge mirrors;

[0017] FIG. 5 is a flowchart depicting exemplary steps of a method of carrier-envelope phase tagging according to an embodiment of the present invention; and

[0018] FIG. 6 is a flowchart depicting exemplary steps of a method of generating an ultraviolet supercontinuum according to an embodiment of the present invention.

[0019] The drawing figures do not limit the current invention to the specific embodiments disclosed and described herein. The drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the invention.DETAILED DESCRIPTION OF THE INVENTION

[0020] The following detailed description of the technology references the accompanying drawings that illustrate specific embodiments in which the technology can be practiced. The embodiments are intended to describe aspects of the technology in sufficient detail to enable those skilled in the art to practice the technology. Other embodiments can be utilized and changes can be made without departing from the scope of the current invention. The following detailed description is, therefore, not to be taken in a limiting sense. The scope of the current invention is defined only by the appended claims, along with the full scope of equivalents to which such claims are entitled.

[0021] Turning to FIG. 1, a system 10 constructed in accordance with an embodiment of the invention is schematically depicted. The system 10 is operable to characterize an ultrafast laser pulse. The system 10 broadly comprises a laser source 14, a mask plate 16, one or more mirrors 18, 20, 22, a non-linear medium 24, one or more detectors 26, 28, and one or more processing element 30.

[0022] The system 10 is operable to receive one or more laser beams 12 from an internal and / or external source. For example, the laser beam 12 may be produced by a laser 14 as described in the article “Waveform-dependent air fluorescence from neutral and ionic nitrogen molecules” by Hao Liang, Ming-Shian Tsai, Chun-Chia Tseng, Ming-Chang Chen, Uwe Thumm, and Meng Han, and published on the Science website (https: / / www.science.org) on July 7, 2025, which is hereby incorporated by reference herein. The laser beam 12 may be an ultrafast laser pulse having a duration less than 1 picosecond and in some embodiments less than 4 femtoseconds. The laser pulse may have a frequency from about 0.25 picohertz (PHz) to about 0.5 PHz with a wavelength of about 600 nanometers (nm) up to about 1,200 nm. In one or more embodiments, the laser 14 is configured to emit carrier-envelope phase-stabilized laser pulses. The pulses may have a repetition rate of around 2 kilohertz (kHz). The intensity may be at least aboutl x 1014Watts per squared centimeter (W / cm2).

[0023] The mask plate 16 is configured to split the laser beams into two or more beams. The mask plate 16 may be a plate with physical apertures and / or one or more patterned optical elements for beam shaping.

[0024] The mirrors 18, 20, 22 are configured to direct, focus, and / or split portions of the laser beam 12. In one or more embodiments, the mirror 18 is configured to direct the split beams of the laser pulses into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity. The mirror 18 is also configured to focus the beams to a focal point in the non-linear medium 24. In one or more embodiments, the mirror 18 is mounted on shiftable and / or motorized delay stage 19 operable to shift the mirror 18 relative to the laser 14 and / or the mask plate 16. This introduces a delay between the ionizing beam and the probing beam. In one or more embodiments, the mirror 18 includes two or more D-shaped mirrors. The wedge mirrors may be fused-silica mirrors. However, the system 10 may include other types of mirrors without departing from the scope of the present invention. For example, one or more of the mirrors may be wedge mirrors, as discussed elsewhere herein.

[0025] The other mirrors 20, 22 may be turning mirrors configured to redirect the beams of the pulses that pass through the non-linear medium 24 to one of the detectors 28. The system 10 may have different mirror arrangements without departing from the scope of the present invention. For example, one or more or the mirrors may be parabolic mirrors, as discussed elsewhere herein.

[0026] The non-linear medium 24 is operable to absorb at least a portion of the laser beam 12 to form plasma. The non-linear medium 24 may be contained within a transparent housing or be located within the device. However, the non-linear medium 24 may be introduced into the system 10 and / or contained within the system 10 any number of ways without departing from the scope of the present invention. In one or more embodiments, the non-linear medium 24 comprises gaseous carbon dioxide , such as air, operable to absorb one or more of the laser pulses to generate plasma. For example, the ionizing beam may ionize the non-linear medium 24, and the probing beam may affect aspects of the plasma as it passes therethrough, which can be detected, as discussed in further detail below.

[0027] The detectors 26, 28 are configured to capture data related to the plasma induced in the non-linear medium 24 and / or characteristics of the beam itself. The detector 26 may comprise a transducer, such as a microphone, for capturing sound waves emanating from the plasma and / ora spectrometer for capturing fluorescence of the plasma. For example, the detector 26 may be configured to generate one or more electrical signals representative of the acoustic waves and / or the fluorescence for analysis by the processing element 30, as discussed below. In one or more embodiments, the detector 28 is a spectrometer for measuring the spectral phase using frequency- resolved optical gating (FROG). The detector 28 may likewise generate electrical signals based on the detected light pulses reflected off the mirrors 20, 22. The FROG spectrometer 28 is optional and may be used for verification of the data.

[0028] The processing element 30 is configured to determine one or more characteristics of the laser beam 12 based on data captured by the detectors 26, 28. The processing element 30 may also be configured to control operations of the system 10. The processing element 30 may comprise one or more communication elements, one or more memory elements, and a user interface. The processing element 30 may be in communication with the delay stage 19, the detectors 26, 28, and / or the laser 14. In one or more embodiments, the processing element 30 is configured to determine an estimated duration of one or more of the laser pulses based at least in part on the data captured by the detectors 26, 28. The processing element 30 may additionally or alternatively be configured to determine a carrier envelope offset frequency and / or the carrier envelope phases of the laser pulses.

[0029] It is foreseen that the system 10 may have any number of setups without departing from the scope of the present invention. For example, the delay stage of the system 10 may be implemented using any type of delay stage known in the art.

[0030] A system 10A constructed in accordance with another embodiment of the invention is shown in FIG. 2. The system 10A may comprise substantially similar components as system 10; thus, the components of system 10A that correspond to similar components in system 10 have an ‘A’ appended to their reference numerals.

[0031] The system 10A includes all the features of system 10 except that it further includes an alignment plate 32 and one or more filters 34, 36. Additionally, as shown in FIG. 2, some of the components of the system 10A may be secured to a base plate 38. The alignment plate 32 is positioned to help align the mirrors 18B, 20B, 22B. One of the filters 34 is a selection wheel with one or more filters positioned therein for selecting one or more wavelengths of the laser pulses to be removed from beams going to spectrometer 28B. The other filter 36 may comprise a neutral density filter.

[0032] The flow chart of FIG. 3 depicts the steps of an exemplary method 300 of characterizing one or more ultrafast laser pulse. In some alternative implementations, the functions noted in the various blocks may occur out of the order depicted in FIG. 3. For example, two blocks shown in succession in FIG. 3 may in fact be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order depending upon the functionality involved. In addition, some steps may be optional.

[0033] The method 300 is described below, for ease of reference, as being executed by exemplary devices and components introduced with the embodiments illustrated in FIGS. 1 and 2. Some of the steps of the method 300 may be performed by the processing element through the utilization of processors, transceivers, hardware, software, firmware, or combinations thereof. However, some of such actions may be distributed differently among such devices or other devices without departing from the spirit of the present invention. Control of the system may also be partially implemented with computer programs stored on one or more non-transient computer- readable medium(s). The computer-readable medium(s) may include one or more executable programs stored thereon, wherein the program(s) instruct one or more processing elements to perform all or certain of the steps outlined herein. The program(s) stored on the computer-readable medium(s) may instruct processing element(s) to perform additional, fewer, or alternative actions, including those discussed elsewhere herein.

[0034] Referring to step 302, laser pulses are emitted from the laser. In one or more embodiments, the laser pulses may be carrier-envelope phase-stabilized, have durations that are less than 200 femtoseconds, have pulse energies of less than 2.5 millijoule, and spectral ranges from 400 nm to 1,500 nm. This step may include adjusting the delay stage to determine the carrierenvelope phase and adjusting an intensity of the laser to obtain carrier-envelope phase stabilization.

[0035] Referring to step 304, at least a portion of the laser pulses are directed, via one or more of the mirrors, into the non-linear medium. As discussed above, the non-linear medium is preferably a gas comprising carbon dioxide. The non-linear medium absorbs the laser pulses to generate plasma. In one or more embodiments, the fluorescence of the plasma has a wavelength of around 337 nm. In one or more embodiments, this step may include first splitting, via one or more mirror and / or a mask plate, the laser pulses into ionizing beams having ionizing intensities and probing beams having probing intensities that are less than the ionizing intensities. Theionizing beams may be used to generate the plasma while the probe beams are used to capture data about the laser pulses as the probe beams pass through the plasma. For example, an estimated duration of one or more of the laser pulses may be based at least in part on a delay between the ionizing beam and the probing beam. In one or more embodiments, this step further includes adjusting a distance between the one or more wedge mirror and the laser and / or the mask plate.

[0036] Referring to step 306, one or more electrical signal is generated by one or more of the detectors representative of some characteristic of the plasma. For example, the transducer or microphone may detect and generate electrical signals based at least in part on acoustic waves produced at the plasma, such as sound waves produced by the ionizing beam(s) and / or sound waves generated due to the probing beam(s). The transducer may sample the sound waves at a sample rate of less than 100 kilohertz. Additionally or alternatively, this step may include generating, via one or more spectrometer, one or more electrical signals representative of fluorescence of the plasma, such as the fluorescence of the plasma generated by the ionizing beam(s) and / or fluorescence affected by the probing beam(s).

[0037] Referring to step 308, a characteristic of one or more of the laser pulses is determined, via the processing element, based at least in part on the one or more electrical signal generated by one or more of the detectors. For example, in one or more embodiments, the characteristic is an estimated duration of the laser pulse(s). This step may also include determining, via the processing element, an estimated chirp or an estimated intensity of one or more laser pulses based at least in part on the one or more electrical signal. This step may also include determining, via the processing element, a carrier envelope offset frequency of the laser pulses based at least in part on the electrical signals. This step may also include determining, via the processing element, the carrier envelope phase(s) of the laser pulse(s) based at least in part on the electrical signal(s). In one or more embodiments, some of the characteristics gleaned from the data collected by the detectors may be compared, via the processing element, to improve accuracy. For example, this step may include comparing, via the processing element, a first estimated duration of one or more of the laser pulses determined using data from the transducer with a second estimated duration of the laser pulse using data from the spectrometer.

[0038] The method 300 may include additional, less, or alternate steps and / or device(s), including those discussed elsewhere herein.EXAMPLE 1

[0039] In experiments performed according to one or more embodiments of the present invention, carrier-envelope phase (CEP)-stabilized few-cycle pulses were generated at a repetition rate frep of 2.0008 kHz in air, corresponding to a frequency comb with the space of frep, ranging from 600 nm (0.5 PHz) to 1200 nm (0.25 PHz) were focused on. The peak intensity at the laser focus exceeds 1 x 1014W / cm2, resulting in a plasma filament.

[0040] A typical result of new frequency generation is the high-order harmonic generation (HHG), originating from the carrier-wave-driven electronic sub-cycle dynamics. In contrast, the acoustic wave is considered to originate from the gas pressure oscillation due to the heat dissipation after strong-field ionization. Additionally, fluorescence is another mechanism for releasing the injected energy in the laser-induced plasma.

[0041] Acoustic harmonics and side-emission fluorescence were measured simultaneously as a function of laser CEP or time delay between two pulses.

[0042] The measured spectrum intensity of the acoustic harmonics were measured using a state-of-the-art microphone with a dynamic range of up 170 dB. The frequency spacing of the acoustic harmonics corresponded to the laser repetition rate and the harmonic order can exceed one hundred, forming an acoustic frequency comb. Notably, these acoustic harmonics are audible to the human ear.

[0043] The measured spectrum intensity was also measured by a common off-the-shelf microphone, where the cutoff frequency is around 22 kHz, aligning with the frequency range of human hearing.

[0044] The static fluorescence spectrum was measured in a background-free ultraviolet (UV) regime, where the photon lines can be roughly assigned into two categories. The dominant category is the transition from the C state to the B state in excited neutral N2 molecules, such as 337 nm (v = 0 — > v' = 0), 358 nm (v = 0 — > v' = 1), 317 nm (v = 1 v' = 0) and 297 nm (v = 2 v' = 0). The second category is the transition from the B state to the X state in N2+ ions, such as 391 nm (v = 0 — ► v' = 0). v and v' donate the vibrational quantum numbers of the upper and lower states, respectively.

[0045] The CEP -resolved differential distribution of the fluorescence spectrum was defined as [Y (CEP, X)- Y (X)] / Y ( ), where Y (CEP, ) and Y (X) are the CEP-resolved and - averaged fluorescence spectrum intensities at wavelength X. The differential distribution revealsthat there is a notable phase shift between the photon lines from N2 and N2+, indicating their generation mechanisms are different. The fluorescence of neutral N2 relies on collision excitation (20, 21), while the fluorescence of N2+ is proportional to the total ionization probability.

[0046] The acoustic harmonic intensities as a function of laser relative CEP was determined. The first order harmonics at frepwere partly contaminated by the background noise in the lab, therefore the CEP dependence curve was not perfectly smooth. Starting from the second- order harmonic (4 kHz), the CEP dependence is very close to a standard sine or cosine oscillation and significant chirps were not observed among these harmonics.

[0047] A comparison was made of the CEP dependence between the integrated sound signal and the 391-nm and 337-nm fluorescence signals. The experimental results reveal that the sound signal almost synchronized with the ionic signal, against with the signal from excited neutral molecules.

[0048] The generation of acoustic harmonics can be understood through a two-step process. The first step involves energy injection via strong-field ionization, which produces a large number of hot electrons and ions within the ionization volume. This process is common to both acoustic waves and fluorescence. The second step encompasses energy relaxation and temperature cooling. These relaxation processes occur on different time scales. Fluorescence typically involves energy relaxation on picosecond to nanosecond scales, as excited ions return to lower energy states by emitting photons. In contrast, acoustic waves are related to the macroscopic motion of the gas medium, such as expansion and compression, which even occurs on a slower time scale. Despite the wide range of time scales — from attoseconds to milliseconds-the experimental results showed that the acoustic harmonic waves retain their distinctive characteristics from the moment of the energy injection.

[0049] Since the intensity of acoustic harmonics depends on the electric field of a laser pulse not only its intensity profile, this property can be used to characterize laser pulses. A straightforward waveform-characterization technique called TIPTOE (tunneling ionization with a perturbation for the time-domain observation of an electric field) can be used for measuring fewcycle pulses by collecting the ion current. Measuring the plasma fluorescence can also be used. Existing setups are very complex, use pure N2 gas, have unclear mechanisms, and the pulse chirp effect is not clear.

[0050] In the TIPTOE measurements, two identical pulses (except for the intensity) split from the same laser beam were used. The strong pulse (>1 x 1014W / cm2) ionizes the air and thus gives rise to a laser plasma filament. A very weak pulse (<1 x 1012W / cm2) perturbs the strongfield ionization, affecting the productions of the ionization, such as electron and ion yields, fluorescences and also the acoustic harmonic waves.

[0051] The second-order acoustic harmonic spectrum as a function of time delay between the two pulses was determined. Dominant oscillations around time zero correspond to the electric field of the laser pulse, while oscillations beyond time zero are relate to the pedestal of the fewcycle pulse. The 391-nm fluorescence spectrum obtained from the TIPTOE measurement was also determined. These TIPTOE results consist of a cross-correlation DC background and a fast oscillation component, which reflects the temporal coherence of the laser beam. A 10-point smooth curve was used to represent the DC background and then subtracted to extract the residuals, which correspond to the fast oscillation component. The residuals from the laser-acoustic wave and the 391-nm fluorescence were compared. The close agreement between the two curves indicates that the laser-acoustic wave is a reliable observable for probing strong interactions in the laser-induced air plasma.

[0052] The laser system used in these experiments is the ATTO-PHAROS at Kansas State University, which is based on an industrial-level CEP-stabilized Yb laser system (PHAROS, Light Conversion). This system provides single-pulse energies of 2 mJ with a pulse duration of 170 fs. The laser pulse duration was post-compressed to 4 fs (full-width at half-maximum of the intensity envelope), as evaluated from TG-FROG and TIPTOE measurements, using cascaded 4-pass cells. After post-compression, the pulse energy was 1.4 mJ, and the spectral range extended from 450 nm to 1400 nm. The CEP stabilization achieved was better than 200 mrad for the single-shot standard deviation over 24 hours. To verify the observed CEP dependence, two independent mechanisms were used for varying the laser CEP. For the experiments, a pulse energy of 100 pj was used, split from the reflection by a fused-silica wedge mirror. The laser pulse was focused using a silver concave mirror with a focal length of 10 cm. A silicon-based spectrometer (BROADCOM, Qmini) was employed to collect side-emission fluorescence. Acoustic waves were detected using a common off-the-shelf microphone (FIFINE) and a state-of-the-art microphone (GRAS, 46DD 1 / 8”) positioned 10 mm from the laser focus. The GRAS microphone operated with a sample rate of 1 MHz, using a DAQ card (NI-9223), while the FIFINE microphone operated at40 kHz with the built-in sound card of a laptop. For TIPTOE measurements, the strong pulse energy was maintained at 100 pJ, and the weak pulse energy was adjusted to the perturbation level, approximately 100 nJ. A piezoelectric stage (PI-625) with nanometer precision was used to control the time delay between the two pulses.

[0053] The CEP of the laser pulses was actively stabilized using both fast and slow feedback loops, providing two independent controls for adjusting the CEP. The first method involved varying the CEP at the laser oscillator, while the second method involved altering the CEP at the experimental end station by adjusting the thickness of a fused-silica wedge (3.5 degrees) inserted into the beam path. The group and phase velocities differed slightly in fused silica, allowing for CEP modulation by changing the wedge thickness. The wedge was mounted on a piezoelectric stage (Newport CONEX-SAG-LS32P). To verify the CEP dependence of the laseracoustic signal, a two-dimensional CEP scan was conducted by varying both the pulse CEP from the oscillator and the wedge position. The results of the total sound intensity measurement was determined. The diagonal fringes observed indicate a periodic dependence on both the laser CEP and the wedge position, confirming the reported CEP dependence of the laser-acoustic wave.

[0054] The residuals between the laser-acoustic signal and the 391-nm fluorescence were compared. The raw data was determined as a 10-point smoothed curve as the cross-correlation background. The residuals, obtained by subtracting this background from the raw data, were also determined. Various methods for extracting the cross-correlation background were explored and consistently found that the residual results for the two signals were identical.

[0055] To conclude, the experimental observation of carrier-envelope phase dependence in laser-acoustic waves using intense sub-4 femtosecond pulses was demonstrated. The measurements revealed acoustic harmonics up to the 100th order, spanning from kilohertz to megahertz. By altering the electric-field waveform with attosecond precision, its signatures were observed as persisting for up to milliseconds, which are typically associated with the macroscopic properties of the target. Based on the acoustic harmonics, a novel pulse characterization method was introduced that leverages the acoustic frequency comb in the near-zero frequency regime. The findings offer significant implications for frequency metrology and ultrafast science, providing a new approach to pulse characterization and deepening the understanding of high-intensity laser interactions with matter.EXAMPLE 2

[0056] A laser-acoustic experiment in accordance with embodiments of the inventionusing CEP-stabilized sub-4-femtosecond pulses was conducted, allowing direct observation of the carrier-envelope offset frequency in the acoustic spectrum and demonstrating unexpected long coherence preservation.

[0057] The harmonic oscillator is one of the most fundamental physical models, extensively used to describe laser-driven bound and free electronic systems. When the light intensity is sufficiently strong to introduce some distortions on the simple harmonic motion, nonlinear interactions can generate new frequencies beyond the spectrum of the driving field, significantly making our world “colorful” with nonlinear phenomena such as sum frequency generation. Laser-induced plasma is a typical extremely nonlinear system, which can generate high-order harmonics in the extreme-ultraviolet and soft-x-ray regimes, as well as terahertz radiation through four-wave optical rectification or transient photocurrent. These new frequencies not only provide spectroscopic tools for studying the mechanisms underlying their generation but also serve as novel light sources for probing various dynamic processes in other systems.

[0058] Laser-induced acoustic wave generation has been studied since the 1960s, shortly after the invention of the laser. Previous studies typically utilized pulse durations ranging from several tens of femtoseconds to picoseconds and nanoseconds. In these cases, the CEP of light pulses was not stabilized, preventing the formation of a frequency comb. For such few-cycle pulses, the CEP significantly influences the electric-field waveform under the intensity envelope on the attosecond precision. The experiments revealed that the intensities of acoustic harmonics are dependent on the CEP of the driving light pulses. Moreover, the carrier-envelope offset frequency (fceo) of optical frequency combs within the acoustic spectrum was determined, bypassing the need for interferometric or frequency -beating techniques. This observation indicates an ultralong coherence preservation from attoseconds to milliseconds in the light-induced plasma. Utilizing the acoustic harmonic waves, a pulse characterization method by effectively ’’hearing” the light in ambient air was demonstrated.

[0059] In the experiments, focused CEP-stabilized few-cycle pulses were generated at a repetition rate frepof 2.0008 kHz in air, corresponding to an optical frequency comb with thespacing of frep, ranging from 600 nm (0.5 PHz) to 1300 nm (0.23 PHz). The peak intensity at the laser focus exceeds 1 x 1014 W / cm2, resulting in a plasma filament. A typical result of new frequency generation is the high-order harmonic generation (HHG), originating from the carrier- wave-driven electronic sub-cycle dynamics. In contrast, the acoustic wave is considered to originate from the gas pressure oscillation due to the heat dissipation after strong-field ionization. Additionally, fluorescence was an another mechanism for releasing the inj ected energy in the laser- induced plasma. Acoustic harmonics and side-emission fluorescence were measured simultaneously as a function of laser CEP or time delay between two pulses. The measured spectrum intensity of the acoustic harmonics were determined using a state-of-the-art microphone with a dynamic range of up 170 dB. The frequency spacing of the acoustic harmonics corresponds to the laser repetition rate and the harmonic order can exceed one hundred, forming an acoustic frequency comb. Notably, these acoustic harmonics are audible to the human ear. The measured spectrum intensity was determined by a common off-the-shelf microphone, where the cutoff frequency was around 22 kHz, aligning with the frequency range of human hearing. The phenomena were very robust and can be observed with a common off-the-shelf microphone.

[0060] The acoustic harmonics and side-emission fluorescence were measured simultaneously as a function of CEP or time delay between two pulses. Spectrum intensity of acoustic harmonics were measured by the state-of-the-art microphone. Spectrum intensity of acoustic harmonics were also measured by a common off-the-shelf microphone.

[0061] The static fluorescence spectrum in the background-free ultraviolet regime was measured, where the photon lines can be roughly assigned into two categories. The dominant category is the transition from the C3nustate to the B3ngstate in excited neutral N2 molecules, such as 337 nm (v = 0 — > v' = 0), 358 nm (v = 0 — > v' = 1), 317 nm (v = 1 — > v' = 0), 297 nm (v = 2 — >• v' = 0) and 380 nm (v = 0 — >• v' = 2). The second category is the transition from the B state to the X state in N2+ ions, such as 391 nm (v = 0 — > v' = 0). v and v' donate the vibrational quantum numbers of the upper and lower states, respectively. The CEP-resolved differential distribution of the fluorescence spectrum was determined, which is defined as [Y (CEP, ) - Y ( )] / Y (X), where Y (CEP, X) and Y (X) are the CEP -resolved and -averaged fluorescence spectrum intensities at wavelength X. The differential distribution reveals that there is a notable phase shift between the photon lines from N2 and N2+, indicating their generation mechanisms are different. Thefluorescence of neutral N2 relies on collision excitation, while the fluorescence of N +is proportional to the total ionization probability.

[0062] CEP dependence of fluorescence and acoustic waves. The CEP-averaged fluorescence spectrum was determined. The CEP-resolved differential distribution of the fluorescence spectrum was also determined as well as the requency-resolved acoustic harmonic intensity as a function of laser CEP. Each harmonic intensity was normalized to its maximum value. The CEP dependence of the total acoustic intensity was compared with the fluorescences from ionized and excited nitrogen molecules. The two-dimensional CEP scan of the total sound intensity was determined by varying both the pulse CEP from the oscillator and the inserted thickness of a fused silica wedge.

[0063] The acoustic harmonic intensities as a function of the laser relative CEP was determined. The first order harmonic at frepwas partly contaminated by the background noise in the lab, therefore the CEP dependence was not perfectly smooth. Starting from the second-order harmonic 4 kHz), the CEP dependence closely resembled a standard sine or cosine function and significant chirps among these harmonics was not observed, which demonstrates the teeth in the acoustic frequency comb are phase locked. The CEP dependence between the frequency -integrated sound signal and the 391-nm and 337-nm fluorescence signals were compared. The experimental result revealed that the sound signal almost synchronized with the ionic signal, against with the signal from excited neutral molecules. The two-dimensional scan of the total sound intensity as a function of the laser CEP and the thickness of a fused-silica wedge pair inserted into the beam path was determined. The slight difference between the group and phase velocities in fused silica allowed for CEP modulation by adjusting the wedge thickness. The diagonal fringes observed indicated an equal and periodic dependence on both the laser CEP and the wedge thickness, confirming the reported CEP dependence of the laser-acoustic wave. Additionally, by varying the wedge thickness, dispersion and correspondingly stretched the pulse duration were introduced. The results show that the CEP dependence remains visible when the pulse is stretched to durations longer than 5 fs.

[0064] In the attosecond science community, the carrier-envelope offset phase is a crucial parameter for generating isolated attosecond light pulses. Conversely, in the frequency metrology community, the focus is more on the carrier-envelope offset frequency (fceo), which directly determines the frequency precision of the comb teeth. The experiments demonstrated that one candirectly hear fceo within the acoustic spectrum. The experiments illustrated the CEP variation of the CEP-stabilized light pulses, corresponding to fceo= 0. The experiments showed the CEP variations when a modulation at frequencies of approximately 3 Hz and 5 Hz were introduced, respectively. The experiments determined the sound intensity spectra around a specific acoustic harmonic order for fceo= 0, 3, and 5 Hz. The experiments observed that acoustic sidebands appeared at frequencies of nfeeo(n = ±1, 2, . . . ), with the n = ±1 sidebands being the most prominent. The appearance of higher-order sidebands is attributed to the imperfect harmonic nature of the CEP modulation. By monitoring the sideband frequencies, one can directly measure fceo in the sound spectrum using a low-cost microphone, eliminating the need for interferometric techniques.

[0065] The generation of the acoustic harmonics indeed originated from the non-linear interaction, which was supported by the energy scaling rule of the acoustic intensity as a function of laser intensity presented in SM. A very similar trend with the ADK rate of tunneling ionization was found. Therefore, the experiments suggest the laser-acoustic waves can be understood through a twostep process. The first step involves energy injection via strong-field ionization, which produces a large number of hot electrons and ions within the ionization volume. This process is common to both acoustic waves and fluorescence. The second step encompasses energy relaxation and temperature cooling. These relaxation processes occur on different time scales. Fluorescence typically involves energy relaxation on picosecond to nanosecond scales, as excited ions return to lower energy states by emitting photons. In contrast, acoustic waves are related to the macroscopic motion of the gas medium, such as expansion and compression, which occurs on a slower time scale. Despite the wide range of time scales — from attoseconds to milliseconds — the experimental results showed that the acoustic harmonic waves retained their distinctive characteristics from the moment of the energy injection.

[0066] Since the intensity of acoustic harmonics is sensitive to the electric field of a laser pulse — not only its intensity profile — this property can be used to characterize laser pulses. A straightforward waveform-characterization technique called TIPTOE (tunneling ionization with a perturbation for the time-domain observation of an electric field) can be used for measuring fewcycle pulses by collecting the ion current. In the TIPTOE measurements, the experiments used two identical pulses (except for the intensity) split from the same laser beam. The strong pulse (~ 1 x 1014W / cm2) ionized the air and thus gave rise to a laser plasma filament. A very weak pulse (~ 1x IO10W / cm2) perturbed the strong-field ionization, affecting the productions of the ionization, such as electron and ion yields, fluorescences and also the acoustic harmonic waves. The experiments showed the second-order acoustic harmonic spectrum as a function of time delay between the two pulses. Dominant oscillations around time zero correspond to the electric field of the main pulse, while oscillations beyond time zero are relate to the pedestal of the few-cycle pulse. The experiments determined the 391-nm fluorescence spectrum obtained from the TIPTOE measurement. These TIPTOE results consisted of a cross-correlation DC background and a fast oscillation component. The latter reflects the temporal coherence of the laser beam. The experiments used a period-averaged curve to represent the DC background and subtract it to extract the residuals, which correspond to the fast oscillation components. The experiments compared the residuals from the laser-acoustic wave and the 391-nm fluorescence. The close agreement between the two curves indicates that the laser-acoustic wave is a reliable observable for probing strong interactions in the laser-induced air plasma.

[0067] To conclude, the experiments demonstrated the experimental observation of carrier-envelope offset frequency and phase in laser-acoustic waves using intense sub-4 femtosecond pulses. The measurements revealed a phase-locked acoustic frequency comb, spanning from kilohertz to megahertz. By altering the electric-field waveform with attosecond precision, the experiments observed its signatures persisting for up to milliseconds, which are typically associated with the macroscopic properties of the target. The experiments offer a novel way to directly ’hear’ the offset frequency of the comb lasers without relying on the traditional f- to-2f interferometry method. Based on the acoustic harmonics, the experiments introduced a novel pulse characterization method that leverages the acoustic frequency comb in the near-zero frequency regime. The findings offer significant implications for frequency metrology and ultrafast science, deepening the understanding of laser-induced shock waves, which has broad applications in laser wakefield acceleration.

[0068] The laser system used in these experiments was the ATTO-PHAROS at KSU, which is based on an industrial-level CEP-stabilized Yb laser system (PHAROS, Light Conversion). This system provided single-pulse energies of 2 mJ with a pulse duration of 170 fs. The experiments post-compressed the laser pulse duration to 3.7 fs (full-width at half-maximum of the intensity envelope), as evaluated from transient-grating FROG and TIPTOE measurements, using cascaded 4-pass cells. After post-compression, the pulse energy is 1.4 mJ, and the spectralrange extends from 600 nm to 1300 nm. The CEP stabilization achieved is better than 200 mrad for the single-shot standard deviation over 24 hours. The CEP of the laser pulses was actively stabilized using both fast and slow feedback loops, providing two independent controls for adjusting the CEP. The first method involved varying the CEP at the laser oscillator, while the second method involved altering the CEP at the experimental end station by adjusting the thickness of a fused-silica wedge (3.5 degrees) inserted into the beam path. The group and phase velocities differ slightly in fused silica, allowing for CEP modulation by changing the wedge thickness. One wedge was mounted on a piezoelectric stage (Newport CONEX-SAG-LS32P), and the other paired wedge was fixed into the beam path. For the CEP modulation results of the main text, the experiments flipped the CEP value between — TT / 2 and +K / 2 with the specific frequencies in the slow feedback loop. The experiments utilized a pulse energy of 100 pj, split from the reflection by a fused-silica wedge prism (Eksma). The laser pulse was focused using a silver concave mirror with a focal length of 10 cm. The experiments employed a silicon-based spectrometer (BROADCOM, Qmini) to collect side-emission fluorescence. Acoustic waves were detected using a common off-the-shelf microphone (FIFINE) and a state-of-the-art microphone (GRAS, 46DD 1 / 8”) positioned 10 mm from the laser focus. The GRAS microphone operated with a sample rate of 1 MHz, using a DAQ card (NI-9223), while the FIFINE microphone operated at 40 kHz with the built-in sound card of a laptop. For TIPTOE measurements, the strong pulse energy was maintained at 100 pj, and the weak pulse energy was adjusted to the perturbation level, approximately 10 nJ. A piezoelectric stage (PI-625) with nanometer precision was used to control the time delay between the two pulses.

[0069] The experiments showed the power spectrum of acoustic harmonics as a function of the light intensity I. The experiments gave the double-logarithmic plot of the total acoustic intensity W. Its shape demonstrates that the laser acoustic harmonic generation is an extremely nonlinear and non-perturbative process. The experiments fit it according to the ADK formulaInW = A + ln - - ] lnI - 2 / (3n3V7)

[0070] where n = 1 / ^21^ is the effective principle quantum number and Ip = 15.58 eV is the ionization potential for N2. The molecular structure only gives a correction to the prefactor, which can be absorbed into A. Note that only the pulse energy E can be measured directly in experiment, which is proportional to light intensity I = B xE. By fitting A and B from experimentdata with the above equation, the laser intensity can be calibrated. The good agreement between the experimental data and the theoretical fit indicates that the nonlinearity begins with strong-field ionization and preserves its characteristics up to the millisecond timescale.

[0071] A system 10B constructed in accordance with another embodiment of the invention is shown in FIG. 4. The system 10B may comprise substantially similar components as system 10; thus, the components of system 10B that correspond to similar components in system 10 have a ‘B’ appended to their reference numerals.

[0072] The system 10B includes all the features of system 10 except that the mirrors 18B, 20B comprise a pair of wedge mirrors 18B, 20B that form the delay and a perforated parabolic mirror 22B. The laser 14B may be configured to emit a single-shot laser that is carrier-envelope phase-stabilized. As used herein, “single-shot” means that only a single laser pulse is emitted and not multiple laser pulses. The laser pulse may have a duration of less than 4 femtoseconds. The wedge mirrors 18B, 20B are configured to split the laser pulse into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity. The parabolic mirror 22B is configured to direct the ionizing beam and the probing beam into a non-linear gas comprising carbon dioxide to generate plasma.

[0073] The detector 26B comprises a spectrometer for detecting side emissions and is configured to generate one or more electrical signal representative of fluorescence of the plasma. The processing element is configured to determine the carrier envelope phase of the laser pulse based at least in part on the one or more electrical signal from the spectrometer.

[0074] In one or more embodiments, the system 10B is additionally or alternatively configured to generate a supercontinuum. The laser 14B may be configured to emit a laser pulse that is carrier-envelope phase-stabilized and that has a duration under 4 femtoseconds. The laser pulse may have a wavelength of 300 nm to 500 nm, 375 nm to 430 nm, and in preferred embodiments, a wavelength of 391 nm to 428 nm.

[0075] However, in one or more embodiments, the laser pulse may not be initially carrierenvelope phase-stabilized, and the system 10B may be operable to stabilize the carrier-envelope phase. The mirrors 18B, 20B, 22B may split the laser pulse into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity and direct the beams into the non-linear gas to generate plasma. The processing element 30B may be configured to use the data collected by spectrometer to determine an estimated carrier-envelope phase of the laser pulse based at least in part on the one or more electrical signal and adjust an intensity of the laser so that subsequent laser pulses from the laser are carrier-envelope phase-stabilized. Once the intensity of the laser 14B is adjusted to enable carrier-envelope phasestabilization, the carrier-envelope phase-stabilized laser pulse may be generated.

[0076] The mirrors 18B, 20B, 22B are configured to direct at least a portion of the carrierenvelope phase-stabilized laser pulses into a non-linear gas comprising carbon dioxide to generate the supercontinuum. The supercontinuum may be an ultraviolet supercontinuum having wavelengths from 250 nanometers (nm) to 500 nm. In one or more embodiments, the supercontinuum is a forward-emission of the plasma generated by the laser pulses.

[0077] The flow chart of FIG. 5 depicts the steps of an exemplary method 500 of determining a carrier envelope phase of a single-shot, ultrafast laser pulse. In some alternative implementations, the functions noted in the various blocks may occur out of the order depicted in FIG. 5. For example, two blocks shown in succession in FIG. 5 may in fact be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order depending upon the functionality involved. In addition, some steps may be optional.

[0078] The method 500 is described below, for ease of reference, as being executed by exemplary devices and components introduced with the embodiments illustrated in FIG. 4. Some of the steps of the method 500 may be performed by the control system through the utilization of processors, transceivers, hardware, software, firmware, or combinations thereof. However, some of such actions may be distributed differently among such devices or other devices without departing from the spirit of the present invention. Control of the system may also be partially implemented with computer programs stored on one or more non-transient computer-readable medium(s). The computer-readable medium(s) may include one or more executable programs stored thereon, wherein the program(s) instruct one or more processing elements to perform all or certain of the steps outlined herein. The program(s) stored on the computer-readable medium(s) may instruct processing element(s) to perform additional, fewer, or alternative actions, including those discussed elsewhere herein.

[0079] Referring to step 502, the laser pulse is split into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity. In one or more embodiments, the laser pulse has a duration of less than 4 femtoseconds. It is foreseen that embodiments of the invention may include systems or devices for measuringexternal lasers; thus, the laser may be generated by an external laser under test. Alternatively, this step may include emitting the laser pulse via the laser of the system.

[0080] Referring to step 504, the ionizing beam is directed, via one or more of the mirrors, into a non-linear medium. In one or more embodiments, the non-linear medium comprises gas including carbon dioxide to generate plasma. Referring to step 506, the probing beam is directed, via one or more of the mirrors, at the plasma. The splitting of the of laser pulses may introduce a delay between the ionizing beam and the probing beam, which is used for determining characteristics of the laser pulse, as discussed below and elsewhere herein.

[0081] Referring to step 508, one or more electrical signal representative of fluorescence of the plasma is generated, via one or more spectrometer. The spectrometer may be positioned adjacent to a focal point of the beams and capture data related to the fluorescence of the plasma. Additionally or alternatively, a spectrometer may be used to capture data related to the fluorescence directly in the path of the beam. This step may include filtering one or more of the beams.

[0082] Referring to step 510, the carrier-envelope phase of the laser pulse is determined based at least in part on the one or more electrical signal. The carrier-envelope phase may be determined via one or more processing element in communication with spectrometer and may process the data captured by the spectrometer to determine or tag the carrier-envelope phase of the laser pulse.

[0083] The method 500 may include additional, less, or alternate steps and / or device(s), including those discussed elsewhere herein.

[0084] The flow chart of FIG. 6 depicts the steps of an exemplary method 600 of generating an ultraviolet supercontinuum. In some alternative implementations, the functions noted in the various blocks may occur out of the order depicted in FIG. 6. For example, two blocks shown in succession in FIG. 6 may in fact be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order depending upon the functionality involved. In addition, some steps may be optional.

[0085] The method 600 is described below, for ease of reference, as being executed by exemplary devices and components introduced with the embodiments illustrated in FIG. 4. Some of the steps of the method 600 may be performed by the control system through the utilization of processors, transceivers, hardware, software, firmware, or combinations thereof. However, someof such actions may be distributed differently among such devices or other devices without departing from the spirit of the present invention. Control of the system may also be partially implemented with computer programs stored on one or more non-transient computer-readable medium(s). The computer-readable medium(s) may include one or more executable programs stored thereon, wherein the program(s) instruct one or more processing elements to perform all or certain of the steps outlined herein. The program(s) stored on the computer-readable medium(s) may instruct processing element(s) to perform additional, fewer, or alternative actions, including those discussed elsewhere herein.

[0086] Referring to step 602, one or more laser pulses that are carrier-envelope phase- stabilized are emitted. In one or more embodiments, the laser pulses have durations under 4 femtoseconds. The laser pulse may have a wavelength of 300 nm to 500 nm, 375 nm to 430 nm, and in preferred embodiments, a wavelength of 391 nm to 428 nm. In one or more embodiments, this step includes adjusting an intensity of the laser in order to stabilize the carrier-envelope phase. This step may include emitting, via the laser, a laser pulse that is not carrier-envelope phase- stabilized; determining the carrier-envelope phase based on spectrometer data; and adjusting the intensity of the laser accordingly.

[0087] In one or more embodiments, the carrier-envelope phase is determined by splitting, via the mirrors, the laser pulse into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity; directing, via the mirrors, the ionizing beam into the non-linear medium to generate plasma; directing, via the one or more mirror, the probing beam at the plasma; generating, via the spectrometer, one or more electrical signal representative of fluorescence of the plasma; determining, via one or more processor, an estimated carrier envelope phase of the laser pulse based at least in part on the one or more electrical signal; and adjusting an intensity of the laser so that a subsequent laser pulse from the laser is carrier-envelope phase-stabilized.

[0088] Referring to step 604, at least a portion of the laser pulse is directed, via one or more of the mirrors, into a non-linear medium to generate the supercontinuum. The non-linear medium may comprise gas having carbon dioxide. In one or more embodiments, the supercontinuum is an ultraviolet supercontinuum having wavelengths from 250 nanometers (nm) to 500 nm. In one or more embodiments, the supercontinuum is a forward-emission of the plasma.

[0089] The method 600 may include additional, less, or alternate steps and / or device(s), including those discussed elsewhere herein.EXAMPLE 3

[0090] Experiments conducted in accordance with one or more embodiments of the invention observated backward-directed, lasing-like radiation at 391 nm and 428 nm from nitrogen molecular ions in ambient air, driven by near-single-cycle pulses. Unlike typical fluorescence, these backward emissions show partial polarization and a strong dependence on the carrierenvelope phase (CEP) of the driving pulses. This distinctive emission behavior reveals a new pathway for remotely generating and detecting polarized, phase-sensitive emissions in ambient air, opening a novel approach to atmospheric diagnostics. For forward-emission air lasing, the experiments produced a bright UV supercontinuum spanning from 500 nm down to 200 nm, corresponding to a Fourier-transform-limited pulse duration of 2 femtoseconds. The strong CEP dependence of the driving pulses enables single-shot CEP tagging.

[0091] A 0.5-mJ, sub-4-femtosecond light pulse spanning wavelengths from 600 nm to 1300 nm was focused in ambient air using a perforated parabolic mirror with a 4-inch focal length. The estimated peak intensity in the focal volume was approximately 0.5 PW / cm2The carrierenvelope phase (CEP) of the laser pulses was actively stabilized with two feedback loops, achieving a standard deviation below 100 mrad over 14 hours. This stabilization was used for observing CEP dependence at a contrast level of about 1%. To vary the pulse CEP, the experiments scanned the thickness of a fused-silica wedge pair (wedge angle: 3.5 degrees) inserted into the beam path, using a piezoelectric stage. The slight difference between group and phase velocities in fused silica allows for CEP modulation by adjusting the wedge thickness. Two silicon-based spectrometers were used to simultaneously record side-emission fluorescence and backward emission as functions of the wedge thickness. To analyze the polarization state, a broad-spectrum wire-grid polarizer was mounted on a motorized rotation stage.

[0092] When the intense light pulses were focused in air, the strong electric field ionized most of the air constituents, generating a bright plasma filament. If the light pulse was a very short, near-single-cycle pulse, it produced white light in air. The incident beam appeared slightly orange due to the central wavelength, but after focusing, the laser beam transitions to white.

[0093] The experiments presented typical radiation spectra measured in both the side and backward directions, which can be categorized into two distinct regions. The most dominant peaks were the transitions from C state to B state in the neutral N2 molecule, such as 337 nm, 358 nm, 316 nm. The second category is the transition from the B state to the X state in N2+ ions, such as 391 nm and 428 nm. The backward-emission spectrum closely resembles the side-emission fluorescence.

[0094] In conventional fluorescence, excited molecular states relax isotropically, emitting light in all directions with minimal dependence on the properties of the driving laser pulses. To distinguish between fluorescence and lasing-like superfluorescence, one method is to examine the polarization using a polarizer. The experiments showed the spectral intensities of side-emission fluorescence and backward radiation as a function of the polarizer angle. The side-emission fluorescence was independent of the polarizer angle and was therefore unpolarized for both ionic and neutral signals. In contrast, the backward emission exhibited different polarization states. The neutral signals remained close to unpolarized, whereas the ionic signals were partially polarized. For the 391 nm transition, polarization exceeded 85%, and for the 428 nm transition, it approached 65%.

[0095] In addition to the evidence of polarization, the CEP dependence provided further insight into the distinct mechanisms underlying side-emission fluorescence and backward radiation. The ionic fluorescence was closely tied to the total ionization yield, which, as observed in previous studies, is maximized when the driving pulses exhibit a cosine-like intensity profile. In contrast, side-emission fluorescence was more uniformly distributed and was less sensitive to the pulse shape or phase. The experiments observed that the backward radiation exhibited a clear phase offset relative to the side-emission fluorescence, indicating that the underlying processes governing the two emissions are different. This phase shift further supports the idea that backward radiation is influenced by a different ionization and recombination mechanism, possibly linked to the dynamics of the plasma or the nature of the backward-propagating waves. These observations suggest that the CEP plays a crucial role in modulating the emission characteristics and reveals the complex interplay between the laser pulse and the ionized medium.

[0096] In the forward emission direction, the experiments observed a broad UV supercontinuum when the driving laser intensity exceeded 0.5 PW / cm2This UV supercontinuum spanned from 500 nm to 200 nm, corresponding to an ultrashort 2-fs UV pulse in the Fourier-transform limit. The measured UV supercontinuum exhibited a strong dependence on the carrierenvelope phase (CEP), enabling single-shot CEP tagging. The experiments showed the UV spectra at a lower intensity (0.5 PW / cm2), revealing a rich set of discrete photon lines, which demonstrated that embodiments of the invention can operate effectively across a wide range of parameters.

[0097] In conclusion, the experiments demonstrated an ultraviolet (UV) supercontinuum light source based on air lasing, driven by intense near-single-cycle light pulses. The UV supercontinuum spanned from 500 nm to 200 nm, corresponding to a 2-femtosecond light pulse in the Fourier-transform limit. The strong dependence on the carrier-envelope phase (CEP) of the driving pulses enabled single-shot CEP tagging. Furthermore, the experiments observed backward-emission lasing-like radiation at 391 nm and 428 nm from nitrogen molecular ions in ambient air, driven by intense sub-4-femtosecond pulses. This backward radiation exhibited partial polarization, distinguishing it from typical isotropic fluorescence. Another key indicator of its lasing-like nature is its CEP dependence; specifically, the backward radiation oscillates nearly out of phase with the side-emission fluorescence, a characteristic signature of stimulated emission processes. The discovery of backward-emission lasing-like radiation opens new possibilities for remote molecular sensing in ambient air. This phenomenon could enable more sensitive detection of atmospheric nitrogen compounds and enhance our ability to study and monitor environmental conditions.

[0098] Throughout this specification, references to “one embodiment”, “an embodiment”, or “embodiments” mean that the feature or features being referred to are included in at least one embodiment of the technology. Separate references to “one embodiment”, “an embodiment”, or “embodiments” in this description do not necessarily refer to the same embodiment and are also not mutually exclusive unless so stated and / or except as will be readily apparent to those skilled in the art from the description. For example, a feature, structure, act, etc. described in one embodiment may also be included in other embodiments, but is not necessarily included. Thus, the current invention can include a variety of combinations and / or integrations of the embodiments described herein.

[0099] Although the present application sets forth a detailed description of numerous different embodiments, it should be understood that the legal scope of the description is defined by the words of the claims set forth at the end of this patent and equivalents. The detailed description is to be construed as exemplary only and does not describe every possible embodimentsince describing every possible embodiment would be impractical. Numerous alternative embodiments may be implemented, using either current technology or technology developed after the fding date of this patent, which would still fall within the scope of the claims.

[0100] As used herein, the phrase “and / or,” when used in a list of two or more items, means that any one of the listed items can be employed by itself or any combination of two or more of the listed items can be employed. For example, if a composition is described as containing or excluding components A, B, and / or C, the composition can contain or exclude A alone; B alone; C alone; A and B in combination; A and C in combination; B and C in combination; or A, B, and C in combination.

[0101] The present description also uses numerical ranges to quantify certain parameters relating to various embodiments of the invention. It should be understood that when numerical ranges are provided, such ranges are to be construed as providing literal support for claim limitations that only recite the lower value of the range as well as claim limitations that only recite the upper value of the range. For example, a disclosed numerical range of about 10 to about 100 provides literal support for a claim reciting “greater than or equal to about 10” (with no upper bounds) and a claim reciting “less than or equal to about 100” (with no lower bounds).

[0102] Furthermore, unless otherwise specified, any directional references (e.g., upper, lower, above, below, etc.) are used herein solely for the sake of convenience and should be understood only in relation to each other. For instance, a component might in practice be oriented such that faces referred to as “upper” and “lower” are sideways, angled, inverted, etc. relative to the chosen frame of reference.

[0103] Throughout this specification, plural instances may implement components, operations, or structures described as a single instance. Although individual operations of one or more methods are illustrated and described as separate operations, one or more of the individual operations may be performed concurrently, and nothing requires that the operations be performed in the order illustrated. Structures and functionality presented as separate components in example configurations may be implemented as a combined structure or component. Similarly, structures and functionality presented as a single component may be implemented as separate components. These and other variations, modifications, additions, and improvements fall within the scope of the subject matter herein.

[0104] Certain embodiments are described herein as including logic or a number of routines, subroutines, applications, or instructions. These may constitute either software (e.g., code embodied on a machine-readable medium or in a transmission signal) or hardware. In hardware, the routines, etc., are tangible units capable of performing certain operations and may be configured or arranged in a certain manner. In example embodiments, one or more computer systems (e.g., a standalone, client or server computer system) or one or more hardware modules of a computer system (e.g., a processor or a group of processors) may be configured by software (e g., an application or application portion) as computer hardware that operates to perform certain operations as described herein.

[0105] In various embodiments, computer hardware, such as a processing element, may be implemented as special purpose or as general purpose. For example, the processing element may comprise dedicated circuitry or logic that is permanently configured, such as an applicationspecific integrated circuit (ASIC), or indefinitely configured, such as an FPGA, to perform certain operations. The processing element may also comprise programmable logic or circuitry (e.g., as encompassed within a general -purpose processor or other programmable processor) that is temporarily configured by software to perform certain operations. It will be appreciated that the decision to implement the processing element as special purpose, in dedicated and permanently configured circuitry, or as general purpose (e.g., configured by software) may be driven by cost and time considerations.

[0106] Accordingly, the term “processing element” or equivalents should be understood to encompass a tangible entity, be that an entity that is physically constructed, permanently configured (e.g., hardwired), or temporarily configured (e.g., programmed) to operate in a certain manner or to perform certain operations described herein. Considering embodiments in which the processing element is temporarily configured (e.g., programmed), each of the processing elements need not be configured or instantiated at any one instance in time. For example, where the processing element comprises a general -purpose processor configured using software, the general- purpose processor may be configured as respective different processing elements at different times. Software may accordingly configure the processing element to constitute a particular hardware configuration at one instance of time and to constitute a different hardware configuration at a different instance of time.

[0107] The processing element may include processors, microprocessors (single-core and multi-core), microcontrollers, DSPs, field-programmable gate arrays (FPGAs), analog and / or digital application-specific integrated circuits (ASICs), or the like, or combinations thereof. The processing element may generally execute, process, or run instructions, code, code segments, software, firmware, programs, applications, apps, processes, services, daemons, or the like. The processing element may also include hardware components such as finite-state machines, sequential and combinational logic, and other electronic circuits that can perform the functions necessary for the operation of the current invention. The processing element may be in communication with the other electronic components through serial or parallel links that include address busses, data busses, control lines, and the like.

[0108] Computer hardware components, such as communication elements, memory elements, processing elements, and the like, may provide information to, and receive information from, other computer hardware components. Accordingly, the described computer hardware components may be regarded as being communicatively coupled. Where multiple of such computer hardware components exist contemporaneously, communications may be achieved through signal transmission (e.g., over appropriate circuits and buses) that connect the computer hardware components. In embodiments in which multiple computer hardware components are configured or instantiated at different times, communications between such computer hardware components may be achieved, for example, through the storage and retrieval of information in memory structures to which the multiple computer hardware components have access. For example, one computer hardware component may perform an operation and store the output of that operation in a memory device to which it is communicatively coupled. A further computer hardware component may then, at a later time, access the memory device to retrieve and process the stored output. Computer hardware components may also initiate communications with input or output devices, and may operate on a resource (e.g., a collection of information).

[0109] The memory device or element may include data storage components, such as readonly memory (ROM), programmable ROM, erasable programmable ROM, random-access memory (RAM) such as static RAM (SRAM) or dynamic RAM (DRAM), cache memory, hard disks, floppy disks, optical disks, flash memory, thumb drives, universal serial bus (USB) drives, or the like, or combinations thereof. In some embodiments, the memory element may be embedded in, or packaged in the same package as, the processing element. The memory element may include,or may constitute, a “computer-readable medium”. The memory element may store the instructions, code, code segments, software, firmware, programs, applications, apps, services, daemons, or the like that are executed by the processing element.

[0110] The communication element may generally allow communication with systems and / or external devices. The communication element may include signal or data transmitting and receiving circuits, such as antennas, amplifiers, filters, mixers, oscillators, digital signal processors (DSPs), and the like. The communication element may establish communication wirelessly by utilizing RF signals and / or data that comply with communication standards such as cellular 2G, 3G, 4G, 5G, or LTE, WiFi, WiMAX, Bluetooth®, BLE, or combinations thereof. The communication element may be in communication with the processing element and the memory element.[OHl] The user interface generally allows the user to utilize inputs and outputs to interact with the device and is in communication with the one or more processing element. Inputs may include buttons, pushbuttons, knobs, jog dials, shuttle dials, directional pads, multidirectional buttons, switches, keypads, keyboards, mice, joysticks, microphones, or the like, or combinations thereof. The outputs of the present invention may include a display and / or any number of additional outputs, such as audio speakers, lights, dials, meters, printers, or the like, or combinations thereof, without departing from the scope of the present invention.

[0112] The various operations of example methods described herein may be performed, at least partially, by one or more processing elements that are temporarily configured (e.g., by software) or permanently configured to perform the relevant operations. Whether temporarily or permanently configured, such processing elements may constitute processing element- implemented modules that operate to perform one or more operations or functions. The modules referred to herein may, in some example embodiments, comprise processing element-implemented modules.

[0113] Similarly, the methods or routines described herein may be at least partially processing element-implemented. For example, at least some of the operations of a method may be performed by one or more processing elements or processing element-implemented hardware modules. The performance of certain of the operations may be distributed among the one or more processing elements, not only residing within a single machine, but deployed across a number of machines. In some example embodiments, the processing elements may be located in a singlelocation (e.g., within a home environment, an office environment or as a server farm), while in other embodiments the processing elements may be distributed across a number of locations.

[0114] Unless specifically stated otherwise, discussions herein using words such as “processing,” “computing,” “calculating,” “determining,” “presenting,” “displaying,” or the like may refer to actions or processes of a machine (e.g., a computer with a processing element and other computer hardware components) that manipulates or transforms data represented as physical (e.g., electronic, magnetic, or optical) quantities within one or more memories (e.g., volatile memory, non-volatile memory, or a combination thereof), registers, or other machine components that receive, store, transmit, or display information.

[0115] As used herein, the terms “comprises,” “comprising,” “includes,” “including,” “has,” “having” or any other variation thereof, are intended to cover a non-exclusive inclusion. For example, a process, method, article, or apparatus that comprises a list of elements is not necessarily limited to only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.

[0116] The patent claims at the end of this patent application are not intended to be construed under 35 U.S.C. § 112(f) unless traditional means-plus-function language is expressly recited, such as “means for” or “step for” language being explicitly recited in the claim(s).

[0117] Although the technology has been described with reference to the embodiments illustrated in the attached drawing figures, it is noted that equivalents may be employed and substitutions made herein without departing from the scope of the technology as recited in the claims.

[0118] Having thus described various embodiments of the technology, what is claimed as new and desired to be protected by Letters Patent includes the following:

Claims

CLAIMS1. A method of characterizing one or more ultrafast laser pulse, the method comprising: emitting, via a laser, laser pulses that are carrier-envelope phase-stabilized; directing, via one or more mirror, at least a portion of the laser pulses into a non-linear gas comprising carbon dioxide to generate plasma; generating, via one or more transducer, one or more electrical signal representative of acoustic waves produced at the plasma; and determining, via one or more processor, an estimated duration of one or more of the laser pulses based at least in part on the one or more electrical signal.

2. The method of claim 1, further comprising determining, via the one or more processor, an estimated chirp of the one or more laser pulse based at least in part on the one or more electrical signal.

3. The method of claim 1, wherein the estimated duration of the one or more laser pulse is less than 200 femtoseconds.

4. The method of claim 1, wherein the one or more transducer has a sample rate of less than 100 kilohertz.

5. The method of claim 1, further comprising determining, via the one or more processor, an estimated intensity of the one or more laser pulse based at least in part on the one or more electrical signal.

6. The method of claim 1, wherein the one or more laser pulse has a pulse energy of less than 2.5 millijoule.

7. The method of claim 1, wherein the one or more laser pulse has a spectral range from 400 nanometers to 1,500 nanometers.

8. The method of claim 1 , wherein the one or more mirror comprises a fused-silica wedge mirror.

9. The method of claim 1, further comprising: splitting, via one or more wedge mirror, one or more laser pulse of the laser pulses into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity, wherein the step of directing, the via one or more mirror, the at least a portion of the laser pulses into the non-linear gas includes directing, via the one or more mirror, the ionizing beam into the nonlinear gas; and directing, via one or more mirror, the probing beam at the plasma, wherein the estimated duration of the one or more laser pulse is based at least in part on a delay between the ionizing beam and the probing beam.

10. The method of claim 9, further comprising adjusting a distance between the one or more wedge mirror and the laser.

11. The method of claim 1, further comprising determining, via the one or more processor, a carrier envelope offset frequency of the laser pulses based at least in part on the one or more electrical signal.

12. The method of claim 1, further comprising determining, via the one or more processor, carrier envelope phases of the laser pulses based at least in part on the one or more electrical signal.

13. A method of characterizing one or more ultrafast laser pulse, the method comprising: emitting, via a laser, laser pulses that are carrier-envelope phase-stabilized; directing, via one or more mirror, at least a portion of the laser pulses into a non-linear gas comprising carbon dioxide to generate plasma; generating, via one or more spectrometer, one or more electrical signal representative of fluorescence of the plasma; and determining, via one or more processor, an estimated duration of one or more of the laser pulses based at least in part on the one or more electrical signal.

14. The method of claim 13, wherein the one or more laser pulse has a spectral range from 400 nanometers to 1,500 nanometers.

15. The method of claim 14, wherein the one or more electrical signal is representative of fluorescence of the plasma having a wavelength of 337 nanometers.

16. The method of claim 13, wherein the estimated duration is a first estimated duration, further comprising: generating, via one or more transducer, one or more electrical signal representative of acoustic waves produced at the plasma; and determining, via the one or more processor, a second estimated duration of the one or more laser pulse based at least in part on the one or more electrical signal representative of the acoustic waves.

17. The method of claim 16, further comprising comparing, via the one or more processor, the first estimated duration with the second estimated duration.

18. The method of claim 13, further comprising: splitting, via one or more wedge mirror, one or more laser pulse of the laser pulses into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity, wherein the step of directing, the via one or more mirror, the at least a portion of the laser pulses into the non-linear gas includes directing, via the one or more mirror, the ionizing beam into the nonlinear gas; and directing, via one or more mirror, the probing beam at the plasma, wherein the estimated duration of the one or more laser pulse is based at least in part on a delay between the ionizing beam and the probing beam.

19. The method of claim 13, further comprising determining, via the one or more processor, a carrier envelope offset frequency of the laser pulses based at least in part on the one or more electrical signal.

20. The method of claim 13, further comprising determining, via the one or more processor, carrier envelope phases of the laser pulses based at least in part on the one or more electrical signal.

21. A system for characterizing one or more ultrafast laser pulse, the system comprising: a laser configured to emit carrier-envelope phase-stabilized laser pulses; a non-linear medium comprising gaseous carbon dioxide operable to absorb one or more of the laser pulses to generate plasma; one or more transducer configured to generate one or more electrical signal representative of acoustic waves produced at the plasma; and one or more processor configured to determine an estimated duration of the one or more laser pulse based at least in part on the one or more electrical signal.

22. The system of claim 21 , wherein the estimated duration is a first estimated duration, further comprising one or more spectrometer configured to generate one or more electrical signal representative of fluorescence of the plasma, wherein the one or more processor is configured to determine a second estimated duration of the one or more laser pulse based at least in part on the one or more electrical signal representative of the fluorescence of the plasma.

23. The system of claim 21, further comprising: one or more wedge mirror configured to split one or more laser pulse of the laser pulses into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity; one or more first mirror configured to direct the ionizing beam at the non-linear medium; and one or more second mirror configured to direct the probing beam at the plasma, wherein the one or more processor is configured to determine an estimated duration of the laser pulse based at least in part on a delay between the ionizing beam and the probing beam.

24. The system of claim 23, wherein the one or more wedge mirror is configured to shift to adjust a distance between the one or more wedge mirror and the laser.

25. The system of claim 23, wherein the one or more wedge mirror comprises a fused- silica one or more wedge mirror.

26. The system of claim 23, wherein the one or more wedge mirror comprises two D- shaped wedge mirrors.

27. The system of claim 21, wherein the one or more processor is configured to determine a carrier envelope offset frequency of the laser pulses based at least in part on the one or more electrical signal.

28. The system of claim 21, wherein the one or more processor is configured to determine carrier envelope phases of the laser pulses based at least in part on the one or more electrical signal.

29. A system for characterizing one or more ultrafast laser pulse, the system comprising: a laser configured to emit carrier-envelope phase-stabilized laser pulses; a non-linear medium comprising gaseous carbon dioxide operable to absorb one or more of the laser pulses to generate plasma; one or more spectrometer configured to generate one or more electrical signal representative of fluorescence of the plasma; and one or more processor configured to determine an estimated duration of the one or more laser pulse based at least in part on the one or more electrical signal.

30. The system of claim 29, further comprising: one or more wedge mirror configured to split one or more laser pulse of the laser pulses into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity; one or more first mirror configured to direct the ionizing beam at the non-linear medium; and one or more second mirror configured to direct the probing beam at the plasma, wherein the one or more processor is configured to determine an estimated duration of the laser pulse based at least in part on a delay between the ionizing beam and the probing beam.

31. The system of claim 30, wherein the one or more wedge mirror is configured to shift to adjust a distance between the one or more wedge mirror and the laser.

32. The system of claim 29, wherein the one or more processor is configured to determine a carrier envelope offset frequency of the laser pulses based at least in part on the one or more electrical signal.

33. The system of claim 29, wherein the one or more processor is configured to determine carrier envelope phases of the laser pulses based at least in part on the one or more electrical signal.

34. A method of determining a carrier envelope phase of a single-shot, ultrafast laser pulse, the method comprising: splitting, via one or more mirror, the laser pulse into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity; directing, via the one or more mirror, the ionizing beam into a non-linear gas comprising carbon dioxide to generate plasma; directing, via the one or more mirror, the probing beam at the plasma; generating, via one or more spectrometer, one or more electrical signal representative of fluorescence of the plasma; and determining, via one or more processor, the carrier envelope phase of the laser pulse based at least in part on the one or more electrical signal.

35. The method of claim 34, wherein the laser pulse has a duration of less than 4 femtoseconds.

36. The method of claim 34, further comprising emitting, via a laser, the laser pulse.

37. A method of generating an ultraviolet supercontinuum, the method comprising: emitting, via a laser, a laser pulse that is carrier-envelope phase-stabilized and having a duration under 4 femtoseconds; and directing, via one or more mirror, at least a portion of the laser pulse into a non-linear gas comprising carbon dioxide to generate the ultraviolet supercontinuum having wavelengths from 250 nanometers (nm) to 500 nm.

38. The method of claim 37, further comprising: emitting, via the laser, a laser pulse that is not carrier-envelope phase-stabilized; splitting, via one or more mirror, the laser pulse into an ionizing beam having an ionizing intensity and a probing beam having a probing intensity that is less than the ionizing intensity; directing, via the one or more mirror, the ionizing beam into the non-linear gas to generate plasma; directing, via the one or more mirror, the probing beam at the plasma; generating, via one or more spectrometer, one or more electrical signal representative of fluorescence of the plasma; determining, via one or more processor, an estimated carrier envelope phase of the laser pulse based at least in part on the one or more electrical signal; and adjusting an intensity of the laser so that a subsequent laser pulse from the laser is carrierenvelope phase-stabilized.

39. The method of claim 37, wherein the laser pulse has a wavelength of 391 nm to 428 nm.

40. The method of claim 37, wherein the supercontinuum is a forward-emission of plasma generated by the laser pulse.

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