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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
NASA NTRS1992

Pulsed laser spectral measurement using a Fabry-Perot interferometer: Limits to resolution

We are developing a Doppler lidar system using the edge technique to measure atmospheric wind profiles. The edge technique requires a laser with a narrow spectral bandwidth and a high resolution optical filter. The lidar system will use a Nd:YAG laser operating at 1.064 microns and a high resolution Fabry-Perot interferometer for the edge filter. The Doppler shift measurement is made by locating the laser on the edge of the filter's spectral response function. Due to the steep slope on the edge, large changes in the filter transmission will be observed for small changes in frequency. The Doppler shift can be determined from a measurement of this change in filter transmission if the filter spectral response function in the region of the measurement is well known. Recently developed injection seeded solid state lasers have made near transform limited laser output readily available for lidar work. Injection-seeded Nd:YAG laser exhibit single mode output with smooth Gaussian temporal pulse shapes. Results of an experiment we conducted to evaluate the effects of a short Gaussian temporal input pulse on the spectral response of a high resolution Fabry-Perot interferometer are presented.

Notari, Anthony↗
NASA NTRS1994

An EXOSAT detection of the transient X-ray pulsar GRO J1008-57

The hard X-ray transient, GRO J1008-57, was discovered during a bright outburst in 1993 July by the Burst and Transient Source Experiment (BATSE) instrument on board the Compton Gamme-Ray Observatory (CGRO). There are no published reports of previous emission from this 93.6 sec X-ray pulsar. Recent optical results have suggested a Be star as the companion. A search of the EXOSAT archives shows that an ME observation centered on the star HD 88661 includes GRO J1008-57 within the field of view. The characteristics of the medium-energy detection including a hard spectrum and pulsed emission at 91.36 sec (with chance probability of 3 x 10(exp -4)) indicates that EXOSAT detected GRO J1008-57 rather than the field star. The estimated flux is 2.4 x 10(exp -11) ergs/cm sq/sec for a luminosity of 1.1 x 10(exp 34) (D/2 kpc)(exp 2) ergs/sec. The X-ray spectrum is hard (photon index approximately equals 1.2) and highly absorbed (N(sub H) approximately equals 0.7 x 10(exp 22)). The detection of this transient suggests that there is a pulse shape dependence on luminosity, a spectral shape independent of luminosity, a large period derivative of P-dot/P = 2.3 x 10(exp -3)/yr, and a dynamic range of at least 100 in L(sub chi). We discuss GRO J1008-57 data in the context of wind-accretion models for this new member of the Be class of X-ray binaries.

Macomb, D. J.↗
DOE OSTI2020

Ablating nanoscale pores in crystalline quartz using laser-induced micro-plasmas in tri-layer structures

We demonstrate a novel approach for achieving rapid, consistent, and controllable micro-pore fabrication in single-crystalline quartz. These micro-pores are essential for applications in biology, i.e., studying ion channels in general and mechano-sensitive channels (MSC) in particular. The fabrication process consists of direct material ablation using pulsed UV light from a 193 nm excimer laser. These pulses ablate single-crystalline quartz chips by burning a laser-induced plasma in a tri-layer structure. Controllable plasma confinement and thus pore size is achieved by sandwiching a thin layer of a selected organic solution between the quartz chip and different substrates. This solution causes the confined micro-plasma to generate special ablation conditions, to create uniformly sized and shaped nanopores.

36 MATERIALS SCIENCE↗
NASA NTRS1982

Plasma ignition for laser propulsion

For a specific optical system a pulsed carbon dioxide laser having an energy output of up to 15 joules was used to initiate a plasma in air at one atmosphere pressure. The spatial and temporal development of the plasma were measured using a multiframe image converter camera. In addition the time dependent velocity of the laser supported plasma front which moves opposite to the direction of the laser pulse was measured in order to characterize the type of wavefront developed. Reliable and reproducible spark initiation was achieved. The lifetime of the highly dense plasma at the initial focal spot was determined to be less than 100 nanoseconds. The plasma front propagates toward the laser at a variable speed ranging from zero to 1.6 x 1,000,000 m/sec. The plasma front propagates for a total distance of approximately five centimeters for the energy and laser pulse shape employed.

Askew, R. F.↗
NASA NTRS2001

Measuring Rocket Engine Temperatures with Hydrogen Raman Spectroscopy

Optically accessible, high pressure, hot fire test articles are available at NASA Marshall for use in development of advanced rocket engine propellant injectors. Single laser-pulse ultraviolet (UV) Raman spectroscopy has been used in the past in these devices for analysis of high pressure H2- and CH4-fueled combustion, but relies on an independent pressure measurement in order to provide temperature information. A variation of UV Raman (High Resolution Hydrogen Raman Spectroscopy) is under development and will allow temperature measurement without the need for an independent pressure measurement, useful for flows where local pressure may not be accurately known. The technique involves the use of a spectrometer with good spectral resolution, requiring a small entrance slit for the spectrometer. The H2 Raman spectrum, when created by a narrow linewidth laser source and obtained from a good spectral resolution spectrograph, has a spectral shape related to temperature. By best-fit matching an experimental spectrum to theoretical spectra at various temperatures, a temperature measurement is obtained. The spectral model accounts for collisional narrowing, collisional broadening, Doppler broadening, and collisional line shifting of each Raman line making up the H2 Stokes vibrational Q-branch spectrum. At pressures from atmospheric up to those associated with advanced preburner components (5500 psia), collisional broadening though present does not cause significant overlap of the Raman lines, allowing high resolution H2 Raman to be used for temperature measurements in plumes and in high pressure test articles. Experimental demonstrations of the technique are performed for rich H2-air flames at atmospheric pressure and for high pressure, 300 K H2-He mixtures. Spectrometer imaging quality is identified as being critical for successful implementation of technique.

Wehrmeyer, Joseph A.↗
NASA NTRS1991

Precluding nonlinear ISI in direct detection long-haul fiber optic systems

Long-distance, high-rate fiber optic systems employing directly modulated 1.55-micron single-mode lasers and conventional single-mode fiber suffer severe intersymbol interference (ISI) with a large nonlinear component. A method of reducing the nonlinearity of the ISI, thereby making linear equalization more viable, is investigated. It is shown that the degree of nonlinearity is highly dependent on the choice of laser bias current, and that in some cases the ISI nonlinearity can be significantly reduced by biasing the laser substantially above threshold. Simulation results predict that an increase in signal-to-nonlinear-distortion ratio as high as 25 dB can be achieved for synchronously spaced samples at an optimal sampling phase by increasing the bias current from 1.2 times threshold to 3.5 times threshold. The high SDR indicates that a linear tapped delay line equalizer could be used to mitigate ISI. Furthermore, the shape of the pulse response suggests that partial response precoding and digital feedback equalization would be particularly effective for this channel.

Swenson, Norman L.↗
NASA NTRSDate not supplied

Progress of 2.05 uM Fiber Laser Development for A Martian Co2 Dial

We have proposed a new concept for differential absorption lidar (DIAL) operating in the 2050 nm CO 2 absorption band for atmospheric CO 2 and pressure observations on Mars. This concept has earned us funding from NASA's PICASSO Program to advance 2050 nm fiber laser technology for future space applications. The laser design is an all-fiber master oscillator and power amplifier (MOPA) system. The master oscillator with pulse shaping and several stages of fiber preamplifiers has been built. Preliminary tests show the output pulse energy can reach 1 mJ, meeting our baseline goal. However, the current power amplifier design has an issue with undesired parasitic lasing at a different frequency, which exhausts pump energy and limits the laser energy. We have modified the power amplifier design to suppress parasitic lasing, and higher laser power energy is expected. To achieve high measurement accuracy, the laser frequency must be stabilized. A fraction of the master oscillator is split to lock the laser frequency to the center of the selected absorption line. The laser can then be shifted to a frequency far from the absorption line center, where extinction due to CO 2 and other trace gases is minimal, serving as the offline reference. The online wavelength is optimized at 2050.44156 nm, ensuring a CO 2 absorption optical depth (AOD) of approximately 1.1 at 3 km, which maximizes the signal-to-noise ratio (SNR) for measurements in the lower Martian atmosphere. At the conference, we will provide more details about this project and report on the progress of instrument development.

DIAL↗
NASA NTRS1997

Discovery of the 198 s X-Ray Pulsar GRO J2058+42

GRO J2058+42, a transient 198 second x-ray pulsar, was discovered by the Burst and Transient Source Experiment (BATSE) on the Compton Gamma-Ray Observatory (CGRO), during a "giant" outburst in 1995 September-October. The total flux peaked at about 300 mCrab (20-50 keV) as measured by Earth occultation. The pulse period decreased from 198 s to 196 s during the 46-day outburst. The pulse shape evolved over the course of the outburst and exhibited energy dependent variations. BATSE observed five additional weak outbursts from GRO J2058+427 each with two week duration and peak pulsed flux of about 15 mcrab (20-50 keV), that were spaced by about 110 days. An observation of the 1996 November outburst by the Rossi X-ray Timing Explorer (RXTE) Proportional Counter Array (PCA) localized the source to within a 4' radius error circle (90% confidence) centered on R.A. = 20 h 59 m.0, Decl. = 41 deg 43 min (J2000). Additional shorter outbursts with peak pulsed fluxes of about 8 mCrab were detected by BATSE halfway between the first four 15 mCrab outbursts. The RXTE All-Sky Monitor detected 8 weak outbursts with approximately equal durations and intensities. GRO J2058+42 is most likely a Be/X-ray binary that appears to outburst at periastron and apastron. No optical counterpart has been identified to date and no x-ray source was present in the error circle in archival ROSAT observations.

Wilson, Colleen A.↗
NASA NTRS2020

Fast Neutron Spectroscopy with Organic Scintillation Detectors in a High-Radiation Environment

Organic scintillators with pulse shape discrimination capability are widely used in both research and practical applications of neutron detection. The neutron and gamma-ray identification performance of the detector depends on the classification algorithms, noise filters and pileup rejection criteria in a high-flux bremsstrahlung radiation environment. In this paper, a technique has been developed and implemented for the neutron detection with multiple filter and discrimination steps, which to a high confidence level eliminates counting of gamma-ray pulses. Such a technique is merited when making measurements in a high flux bremsstrahlung and secondary fluorescence environment. The EJ-309 and stilbene detectors coupled to the digital data acquisition system were used for the calibration assessments with standard gamma-ray and neutron sources such as 137Cs, 60Co, 252Cf, and Am-Be. The MCNPX-PoliMi and GEANT4 toolkits were used to simulate the light output and the optical photon transport in the scintillators and create detector response functions for each type of detector. The neutron spectrum unfolding algorithm, GRAVELW, was used to recreate and calibrate with the Am-Be as final step before applying the neutron detection system to extract fusion neutron spectra generated in an intense bremsstrahlung radiation environment. This new technique described offers the user the ability to measure neutron spectra in a high-flux gamma-ray field and tune the parameters to meet required filtering needs.

Neutron Detector, Organic Scintillator, PulseShape↗
DOE OSTI2023

The Third Fermi Large Area Telescope Catalog of Gamma-Ray Pulsars

We present 294 pulsars found in GeV data from the Large Area Telescope (LAT) on the Fermi Gamma-ray Space Telescope. Another 33 millisecond pulsars (MSPs) discovered in deep radio searches of LAT sources will likely reveal pulsations once phase-connected rotation ephemerides are achieved. A further dozen optical and/or X-ray binary systems colocated with LAT sources also likely harbor gamma-ray MSPs. This catalog thus reports roughly 340 gamma-ray pulsars and candidates, 10% of all known pulsars, compared to ≤11 known before Fermi. Half of the gamma-ray pulsars are young. Of these, the half that are undetected in radio have a broader Galactic latitude distribution than the young radio-loud pulsars. The others are MSPs, with six undetected in radio. Overall, ≥236 are bright enough above 50 MeV to fit the pulse profile, the energy spectrum, or both. For the common two-peaked profiles, the gamma-ray peak closest to the magnetic pole crossing generally has a softer spectrum. The spectral energy distributions tend to narrow as the spindown power $\dot{E}$ decreases to its observed minimum near 10 33 erg s -1 , approaching the shape for synchrotron radiation from monoenergetic electrons. We calculate gamma-ray luminosities when distances are available. Our all-sky gamma-ray sensitivity map is useful for population syntheses. The electronic catalog version provides gamma-ray pulsar ephemerides, properties, and fit results to guide and be compared with modeling results.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1998

Discovery of the 198 Second X-Ray Pulsar GRO J2058+42

GRO J2058+42, a transient 198 s X-ray pulsar, was discovered by the Burst and Transient Source Experiment (BATSE) on the Compton Gamma Ray Observatory (CGRO) during a "giant" outburst in 1995 September-October. The total flux peaked at about 300 mcrab (20-50 keV) as measured by Earth occultation. The pulse period decreased from 198 to 196 s during the 46 day outburst. The pulse shape evolved over the course of the outburst and exhibited energy-dependent variations. BATSE observed five additional weak outbursts from GRO J2058 + 42, each with a 2 week duration and a peak-pulsed flux of about 15 mcrab (20-50 keV), that were spaced by about 110 days. An observation of the 1996 November outburst by the Rossi X-Ray Timing Explorer (RXTE) proportional counter array (PCA) localized the source to within a 4 s radius error circle (90% confidence) centered on R.A. = 20h 59m.0, decl. = 41 deg 43 s (J2000). Additional shorter outbursts with peak-pulsed fluxes of about 8 mcrab were detected by BATSE halfway between the first four 15 mcrab outbursts. The RXTE All-Sky Monitor detected all eight weak outbursts with approximately equal durations and intensities. GRO J2058 + 42 is most likely a Be/X-ray binary that appears to outburst at periastron and apastron, No optical counterpart has been identified to date, and no X-ray source was present in the error circle in archival ROSAT observations.

Wilson, Colleen A.↗
NASA NTRS1992

Experimental investigation of a simulated LOX injector flow field and other nonintrusive measurement efforts

Efforts to improve the characteristics of fuel-oxidizer mixing in liquid rocket combustors have lead to a swirl element design for a liquid oxygen injector which is being considered for use on the STME. For the design which is the subject of this investigation, the oxygen enters the injector element perpendicular to the injector axis and nearly tangent to the circular injector wall. This swirl element is at one end of a tube and the injector exit is at the other. This geometric configuration creates a plume in the shape of a conical sheet. This sheet is either primarily contiguous liquid or droplets depending on the pressure drop in the injector and the distance from the injector exit. Probe-based devices such as two-dimensional grid patternators have been used to investigate simulated LOX injector flow fields (Hulka). The primary work described herein is an effort to use optical techniques to investigate the plume of a swirl injector element. For this investigation, a high pressure (500 psig) cold flow test facility was constructed. Water was used as the LOX simulate and air pressure was used to drive the injector flow field. Laser-induced fluorescence (LIF) from dye seeded into the water was used to obtain quantitative measurements of the time-averaged water concentration distribution in the plume. Scattered laser light and LIF were used for time averaged plume visualization and scattered light from a strobe with a 1 microsecond pulse was used for time-resolved plume visualization. During the Summer Faculty Fellowship for which this report was developed, an additional effort, unrelated to the swirl injector investigation, was made to resolve fluctuations in the combustion product composition in the exhaust of a hybrid rocket motor. A brief description of this effort is included herein.

Hartfield, Roy J., Jr.↗
DOE OSTI2023

Four-dimensional dynamics of multirotational transition stimulated rotational Raman scattering in air

Stimulated rotational Raman scattering in air is a powerful parasitic process that degrades high intensity laser beams and pulses propagated over significant distances. Conversely, it is used beneficially in the context of Raman lasers. Through this inelastic scattering process, laser photons are converted to higher (anti-Stokes) or lower (Stokes) energies, according to rotational mode transitions in nitrogen and oxygen diatomic molecules. The full wave-mixing problem involves numerous frequencies, and it is consistently assumed that only one rotational mode contributes to the conversion process. We instead present a dynamic 4D multirotational model that is implemented in a parallelized manner within the Virtual Beamline++ optical modeling package allowing high-resolution 4D studies. We highlight the effect that spontaneous emission plays in large and small beam-width setups, even in the highly saturating regime. The weaker transition modes play a large role in the persistent dynamics and can lead to complex spatiotemporal coupling through nonlinear competition of the modes. We highlight how and why these weaker modes persist, how the size and shape of speckle patterns depends highly on the initial beam profile, and how weaker modes can transiently become stronger as a result of such competition.

47 OTHER INSTRUMENTATION↗
DOE OSTI2020

The effect of annealing on optical transmittance and structure of ZLANI fluorozirconate glass thin films

In this work, we report the effect of thermal annealing on structure, composition, optical transmittance and thickness of a novel fluorozirconate glass (ZLANI) containing Zr, La, Al, Na and In fluorides. In this work, pulsed laser depo- sition was used to grow thin films of ZLANI, and thermal annealing at different temperatures was performed on the films. Annealing did not change the composition, but a clear structural evolution of the ZLANI glass was observed by transmission electron microscopy (TEM), showing that we can control microstructure independent of composition. An increase in transmittance after the film was subject to a 100 °C thermal anneal was ascribed to the removal of defects and structural relaxation in the amorphous state. Following an anneal of 200 °C, the transmittance decreases due to heterogeneous formation of crystalline nuclei and changes in the local bonding. After the final annealing at 300 °C, a wider-scale crystallization occurred, with some major crystal phases formed as Zr 2 F 8 (H 2 O) 6 and ZrO 2 , which alters the shape of the transmittance curve. The crystalline content of the crystal phases that form in the annealed films was quantified using hollow cone dark field TEM imaging. The 100 °C or 200 °C annealing decreases the film thickness by inducing structural relaxation and densification of the amorphous films, while the thickness increase of the 300 °C annealed film resulted from the formed large crystals. These results provide insights for the design of multilayer nanocomposites with a ZLANI glass matrix, which have potential applications as up-/down-conversion luminescent materials and X-ray storage phosphors.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Pulse shaping in strong-field ionization: Theory and experiments

Intense ultrafast pulses cause dissociative ionization and shaping the pulses may allow control of both electronic and nuclear dynamics that determine ion yields. We report on a combined experimental and theoretical effort to determine how shaped laser pulses affect tunnel ionization, the process that precedes many strong-field phenomena. We carried out experiments on Ar, N 2 , H 2 O, and O 2 using a phase-step function of amplitude 3/4π that is scanned across the spectrum of the pulse. In addition, we changed the amount of chirp in the pulses. Semiclassical as well as fully quantum mechanical time-dependent Schrödinger equation calculations are found to be in excellent agreement with experimental results. We find that precise knowledge of the field parameters in the time and frequency domains is essential to afford reproducible results and quantitative theory and experiment comparisons.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS1993

Plasma-puff initiation of high Coulomb transfer switches

The plasma-puff triggering mechanism based on a hypocycloidal pinch geometry was investigated to determine the optimal operating conditions for an azimuthally uniform surface flashover which initiates plasma-puff under wide ranges of fill gas pressures of Ar, He and N2. The optimal fill gas pressures for the azimuthally uniform plasma-puff were about 120 mTorr less than P(sub opt) less than 450 Torr for He and N2. For Argon 20 mTorr is less than P(sub opt) is less than 5 Torr. The inverse pinch switch was triggered with the plasma-puff and the switching capability under various electrical parameters and working gas pressures of Ar, He and N2 was determined. It was also shown that the azimuthally uniform switching discharges were dependent on the type of fill gas and its fill pressure. A new concept of plasma-focus driven plasma-puff was also discussed in comparison with hypocycloidal pinch plasma-puff triggering. The main discharge of the inverse pinch switch with the plasma-focus driven plasma-puff trigger is found to be more azimuthally uniform than that with the hypocycloidal pinch plasma-puff trigger in a gas pressure region between 80 mTorr and 1 Torr. In order to assess the effects of plasma current density on material erosion of electrodes, emissions from both an inverse-pinch plasma switch (INPIStron) and from a spark gap switch under test were studied with an optical multichannel analyzer (OMA). The color temperature of the argon plasma was approximately 4,000 K which corresponded with the peak continuum emission near 750 nm. There are the strong line emissions of argon in the 650 - 800 nm range and a lack of line emissions of copper and other solid material used in the switch. This indicates that the plasma current density during closing is low and the hot spot or hot filament in the switch is negligible. This result also indicates considerable reduction of line emission with the INPIStron switch over that of a spark-gap switch. However, a strong carbon line emission exists due to vaporization of the plastic insulator used. In order to reduce the vaporization of the insulator, the plexiglass insulating material of INPIStron was replaced with Z-9 material. A comparative study of the INPIStron and a spark gap also reveals that the INPIStron, with a low impedance of Z = 9 ohms, can transfer a high voltage pulse with a superior pulse-shape fidelity over that of a spark gap with Z = 100 ohms.

Han, Kwang S.↗
NASA NTRS1993

Plasma-puff initiation of high Coulomb transfer switches

The plasma-puff triggering mechanism based on a hypocycloidal pinch geometry was investigated to determine the optimal operating conditions for an azimuthally uniform surface flashover which initiates plasma-puff under wide ranges of fill gas pressures of Ar, He and N2. The optimal fill gas pressures for the azimuthally uniform plasma-puff were about 120 mTorr less than P(opt) less than 450 Torr for He and N2. For Argon 120 mTorr less than P(opt) less than 5 Torr for argon. The inverse pinch switch was triggered with the plasma-puff and the switching capability under various electrical parameters and working gas pressures of Ar, He and N2 was determined. It was also shown that the azimuthally uniform switching discharges were dependent on the type of fill gas and its fill pressure. A new concept of plasma-focus driven plasma-puff was also discussed in comparison with hypocycloidal pinch plasma-puff triggering. The main discharge of the inverse pinch switch with the plasma-focus driven plasma-puff trigger is found to be more azimuthally uniform than that with the hypocycloidal pinch plasma-puff trigger in a gas pressure region between 80 mTorr and 1 Torr. In order to assess the effects of plasma current density on material erosion of electrodes, emissions from both an inverse-pinch plasma switch (INPIStron) and from a spark gap switch under test were studied with an optical multichannel analyzer (OMA). The color temperature of the argon plasma was approximately 4,000 K which corresponded with the peak continuum emission near 750 nm. There are the strong line emissions of argon in the 650 - 800 nm range and a lack of line emissions of copper and other solid material used in the switch. This indicates that the plasma current density during closing is low and the hot spot or hot filament in the switch is negligible. This result also indicates considerable reduction of line emission with the INPIStron switch over that of a spark-gap switch. However, a strong carbon line emission exists due to vaporization of the plastic insulator used. In order to reduce the vaporization of the insulator, the plexiglass insulating material of INPIStron was replaced with Z-9 material. A comparative study of the INPIStron and a spark gap also reveals that the INPIStron, with a low impedance of Z equals 9 ohms, can transfer a high voltage pulse with a superior pulse-shape fidelity over that of a spark gap with Z equals 100 ohms.

Venable, D. D.↗
DOE OSTI2024

Photonic curing for innovative fabrication of flexible metal oxide optoelectronics

Abstract Flexible optoelectronics, based on non-planar substrates, hold promise for diverse applications such as wearables, health monitors, and displays due to their cost-effective manufacturing methods. Despite the superior properties of metal oxides, the challenge of processing them at high temperatures incompatible with plastic substrates necessitates innovative annealing approaches. Photonic curing, which delivers microsecond to millisecond broadband (200–1500 nm) light pulses on a sample, emerges as a viable solution. Depending on the optical properties, the targeted film absorbs the radiant energy resulting in rapid heating while the transparent substrate absorbs a minimal amount of light and remains at ambient temperature. The light intensity can be high, but since the light pulse is short, the total energy absorbed by the sample remains low and will not damage the plastic substrate. This perspective explores the innovative application of photonic curing to fabricate flexible metal oxide optoelectronics, including thin-film transistors, metal–insulator–metal devices, solar cells, transparent conductors, and Li batteries, emphasizing the conversion of sol–gel precursors to metal oxides. However, this technique was initially developed for sintering metal nanoparticles to conductive patterns and poses intriguing challenges in explaining its mechanism for metal oxide conversion, especially considering the limited absorption of visible light by most sol–gel precursors. The review delves into UV-induced photochemistry, common flexible metal-oxide optoelectronic components, and non-intuitive distinctions between photonic curing and thermal annealing. By elucidating the distinctive role of photonic curing in overcoming temperature-related challenges and advancing the fabrication of flexible metal oxide optoelectronics, this perspective offers valuable insights that could shape the future of flexible optoelectronics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗