Phosphate glass is an attractive material for rare-earth-doped fiber manufacturing because high doping levels are possible without introducing negative effects such as up-conversion or increased non-radiative recombination. In this paper we present a novel PM heavily Yb-doped polarization maintaining large mode area phosphate fiber and a < 100 W power level amplifier based on this fiber. The fiber was fabricated by a rod-in-tube technique. An 18 cm long piece of the fiber was used to build a high-power all-fiber amplifier. 106 W of output power at 1030 nm was achieved with 55 % slope efficiency with respect to the launched pump power. To the best of our knowledge, this is the highest average power ever demonstrated for short phosphate fiber lasers.
High power laser systems operating at mid IR wavelengths are required for medical applications, environmental monitoring, and military applications. All of these systems require optical isolators to avoid feedback into the pump laser cavity. We present measurements of the Verdet coefficient of germanate glass with Dy concentrations varying from 20-50% at wavelengths between .4 and 1.5 microns. The results indicate a linear increase of the Verdet coefficient with impurity concentration and a Sellmeier like dependence on wavelength.
A high-energy, single-frequency fiber laser with long pulse duration of 100 ns has been experimentally investigated in an all-fiber architecture. Only 34-cm long heavily Yb-doped phosphate fiber was employed in power scaling stage to efficiently suppress the Stimulated Brillouin effect (SBS). In the experiment, 0.47 mJ single pulse energy was achieved in power scaling stage at the pump power of 16 W. The pre-shaped pulse was gradually broadened from 103 to 140 ns during the amplification without shape distortion.
We demonstrate an all fiber picosecond laser with megawatt-level peak power based on highly ytterbium-doped phosphate fiber. The phosphate fiber used in the power amplifier stage has the cladding absorption 50 dB/m at 976 nm. The length of the gain fiber in the main amplifier is only 34 cm, which will effectively reduce the nonlinear effect during the process of picosecond pulse amplification. The core diameter of the gain fiber is 25 μm with numerical aperture of 0.04, which is helpful to obtain high beam quality laser. The pulse energy of the seed is 0.2 μJ at repetition rate of 25 kHz, which is amplified to 21 μJ with the pulse width of 20 ps and the peak power is 1.05 MW. High beam quality is also demonstrated with M2 factor measured at highest pulse energy of 1.4 in the both X and Y directions. This kind of laser source with high peak power and high beam quality has a wide range of applications in the field of material processing.
Heavy doping of common silica gain fibers is not practical; therefore long fibers are required for efficient amplification (usually 5-10m). This is undesirable due to nonlinearities that grow with fiber length. In contrast, NP Photonics phosphate-glass based fibers can be heavily doped without any side-effects, and hence can provide very high gain in short lengths (less than 0.5m). This enables an ideal pulsed fiber amplifier for a MOPA system that maximizes the energy extraction and minimizes the nonlinearities. We demonstrate 1W average power, 200μJ energy, and >10kW peak power from a SBS-limited all-fiber MOPA system at 1550nm, and 32W average power, 90μJ energy, and 45kW peak power from a SRS and SPM limited all-fiber MOPA system at 1064 nm. These results were limited by the seed and pump sources.
Mid-infrared sources are a key enabling technology for various applications such as remote chemical sensing, defense communications and countermeasures, and bio-photonic diagnostics and therapeutics. Conventional mid-IR sources include optical parametric amplifiers, quantum cascade lasers, synchrotron and free electron lasers. An all-fiber approach to generate a high power, single mode beam with extremely wide (1μm-5μm) and simultaneous wavelength coverage has significant advantages in terms of reliability (no moving parts or alignment), room temperature operation, size, weight, and power efficiency. Here, we report single mode, high power extended wavelength coverage (1μm to 5μm) supercontinuum generation using a tellurite-based dispersion managed nonlinear fiber and an all-fiber based short pulse (20 ps), single mode pump source. We have developed this mid IR supercontinuum source based on highly purified solid-core tellurite glass fibers that are waveguide engineered for dispersion-zero matching with Tm-doped pulsed fiber laser pumps. The conversion efficiency from 1922nm pump to mid IR (2μm-5μm) supercontinuum is greater than 30%, and approaching 60% for the full spectrum. We have achieved > 1.2W covering from 1μm to 5μm with 2W of pump. In particular, the wavelength region above 4μm has been difficult to cover with supercontinuum sources based on ZBLAN or chalcogenide fibers. In contrast to that, our nonlinear tellurite fibers have a wider transparency window free of unwanted absorption, and are highly suited for extending the long wavelength emission above 4μm. We achieve spectral power density at 4.1μm already exceeding 0.2mW/nm and with potential for higher by scaling of pump power.
Mid Infrared (MIR) fiber optics has gained a great deal of interest over the past several decades. Applications range from passive transport to fiber lasers and nonlinear applications. These fibers have found use in a wide array of fields such as sensing, military countermeasures, scientific instrumentation, medical instrumentation, and in research laboratories. As with all fiber development there is a continual urge to seek better performance characteristics including transparency over a wide wavelength range, corrosion resistance, high power handling and low loss. We report on development of tellurite glass fibers displaying exceptionally high performance for various applications including wide band, low loss passive transport for mid IR, high efficiency, wide wavelength range and high power supercontinuum generation from visible to MIR wavelengths >4.5um, and active doping in fibers for use in laser cooling. High performance in each of these areas of interest has been brought about by development of a stable glass formulation and advanced processing techniques to remove impurities ions, entrapped hydroxyl, and scatter centers which allow fibers to be made with exceptionally low losses ~0.2dB/m.
We present our experimental and theoretical results of optical cooling in Tm-doped glass fibers as optical cooler pumped by single-mode, high efficiency and high power Tm-doped glass fiber lasers. The effects of impurities including OHabsorption and transition metals have been investigated systematically using different purified glasses for fiber fabrication. Our experimental results of spectroscopic measurements show temperature drops of more than 30 degrees from room temperature with pump powers of less than 3W. The results are in good agreement with theoretical simulation.
We present the performance of a single frequency, single-polarization holmium (Ho3+)-doped ZBLAN (ZrF4-BaF2-LaF3- AlF3-NaF) fiber laser at 1200 nm. This distributed Bragg reflector (DBR) fiber laser was developed by splicing a 22 mm long highly Ho3+-doped ZBLAN fiber to a pair of silica fiber Bragg gratings (FBG). The successful fusion splicing of silica fiber to ZBLAN fiber, with their very different melting temperatures, was accomplished by using NP Photonics proprietary splicing technique. The 3 mol% Ho3+-doped ZBLAN fiber had a core diameter of 6.5 μm and a cladding diameter of 125 μm. The threshold of this laser was seen to be about 260 mW, and when the pump power was 520 mW, the output power was about 10 mW. The efficiency of the 1200 nm single-frequency fiber laser, i.e. the ratio of the output power to the launched pump power, was about 3.8%. The linewidth of the 1200 nm single-frequency fiber laser was estimated to be about 100 kHz by comparing the measured frequency noise of the 1200 nm single-frequency fiber laser with that of 1 μm NP Photonics single-frequency fiber lasers whose linewidths have been measured to be in the 1- 10 kHz range. The relative intensity noise of this DBR all-fiber laser was measured to be < 110 dB/Hz at the relaxation oscillation peak and the polarization extinction ratio was measured to be > 19 dB. Due to its low phonon energy and long radiative lifetimes, rare-earth-doped ZBLAN allows various transitions that are typically terminated in silica glass, resulting in ultraviolet, visible, and infrared rare-earth doped ZBLAN lasers. Therefore, our results highlight the exciting prospect that the accessible wavelength range of single-frequency DBR fiber lasers can be expanded significantly by using rare-earth-doped ZBLAN fibers.
Linearly polarized wavelength stable single frequency ytterbium (Yb3+) doped fiber lasers below 1 μm, namely threelevel Yb3+ fiber lasers, are highly demanded for nonlinear wavelength conversion to generate coherent blue light or even deep ultraviolet coherent sources. We present performance of a 976 nm single-frequency core-pumped distributed Bragg reflector (DBR) fiber laser consisting of a 2-cm long highly ytterbium-doped phosphate fiber and a pair of silica fiber Bragg gratings (FBGs) and their use for frequency doubling experiment. The high reflection (HR > 99%) and partial reflection (PR = 60%) FBGs were cleaved very close to the index modulation region and directly spliced to a 2-cm-long highly Yb3+-doped phosphate fiber. Over 100 mW of linearly polarized output with a linewidth less than 2 kHz can be obtained when the launched pump power is about 450 mW. The efficiency of the 976 nm single-frequency fiber laser (the output power vs the launched pump power) is about 25%. The relative intensity noise was measured to be -110 dB/Hz at 1 MHz and the variation of the center wavelength is less than 0.0005 nm during a measurement period of 2.5 hours. This single-frequency fiber laser has an SNR of over 50 dB and there is no strong ASE or spurious lasing at long wavelengths even at the maximum pump power. This all-fiber single-frequency DBR laser with attractive features can be used for efficient blue and UV generation through nonlinear frequency conversion. Moreover, this high-performance 976 nm single-frequency fiber laser can be used as a single-frequency, low RIN pump laser for long wavelength Yb3+-, Er3+-, or Yb3+/Er3+-doped fiber lasers and amplifiers.
Holmium (Ho3+)-doped ZBLAN glasses have been investigated for the purpose of achieving efficient fiber lasers at
1.2 μm. Because of the long lifetime of the upper laser level and the small phonon energy in Ho3+-doped ZBLAN
glasses, strong fluorescence at 1.2 μm that usually cannot be observed in Ho-doped silica glass has been measured.
Fluorescence of 1 mol%, 3 mol%, and 6 mol% Ho3+-doped ZBLAN glasses are reported. The effect of cerium and
terbium ions on the emission of Ho3+-doped ZBLAN glass has also been studied. Obstacles to achieving an efficient
Ho3+-doped ZBLAN laser are analyzed and discussed. In studies of a commercial Ho3+-doped ZBLAN fiber laser, it was
found that the 3 μm four-energy-level laser can easily overwhelm the 1.2 μm laser, which is a three-energy-level system
having the same upper laser level with the 3 μm laser. In order to effectively suppress the competiting 3 μm transition,
advanced Ho3+-doped ZBLAN fiber has been designed and fabricated for 1.2 μm fiber lasers. Fiber lasers at 1.2 μm
using the new Ho3+-doped ZBLAN fiber have been developed. Our experiments demonstrate that the new Ho3+-doped
ZBLAN fiber is an efficient gain medium for lasers at 1.2 μm.
We present a new approach in which the nonlinear Schrodinger equation (NLSE) coupled with beam
propagation method (BPM) is used to quantitatively simulate pulse propagation in multimode cladding
pumped fiber amplifiers. The complexity of fiber structures including fiber shapes and multiple cores
which strongly affects the pump propagation and pump absorption and therefore also affects the fiber gain
can be described by BPM modeling. The signal gain obtained from BPM is then used for modeling pulse
amplification using the standard Step Split Fourier Method (SSFM). The simulation results are in
quantitative agreement with our experiments.
An all fiber approach to optical cooling is being investigated experimentally and theoretically using Tm-doped fiber
laser and Tm-doped fiber cooler. A single mode, high efficiency and high power Tm-doped fiber laser is used to pump
at the absorption edge of Tm-doped fiber coolers, one made by germanate and the other by tellurite glasses. The glass
characterization shows that the quenching effect, which is negative for cooling processes in the fiber, in germanate
glass is much stronger than that in tellurite glass. The preliminary results of experiments indicate cooling effects could
occur in the fiber, but net cooling in the system has not been achieved. A theoretical framework aimed at
understanding the nature of cooling in this laser cooling system has been developed which shows that the temperature
in the sample could increase even if the fiber core is indeed cooling. The details of the temperature dynamics depend
on many factors such as background loss and absorption of scattered light by the heat spreader.
We report a monolithic specialized high stimulated Brillouin scattering (SBS) threshold fiber laser/amplifier in the C and
L band based on highly co-doped phosphate glass fibers. This represents an important new development for coherent
LIDAR and remote sensing applications. By using single mode polarization-maintaining large core highly Er/Yb codoped
phosphate fibers in the power amplifier stages, we have achieved the highest peak power of 2.02 kW at 1530 nm
for 105 ns pulses with transform-limited linewidth, and with a corresponding pulse energy of about 0.212 mJ. The
achieved high-energy pulses were frequency doubled by using a commercial periodically poled lithium niobate (PPLN)
crystal, and the highest SHG peak power of 271 W has been achieved for the SHG pulses at 765 nm that can be used for
oxygen coherent remote sensing. In the L band, more than 80 μJ fiber laser pulses at 1572 nm with 1-2 μs pulse width
and transform-limited linewidth have been achieved by using a monolithic fiber laser system in MOPA configuration,
which can be used for CO2 coherent remote sensing.
We report an all fiber-based single-frequency Q-switched 2 μm pulsed laser based on highly Tm-doped germanate fiber
by using a piezo to induce stress in fiber laser cavity. The pulse width of this Q-switched fiber laser can be tuned from 10's
ns to sub-μs. The repetition rate can be tuned from 100 Hz to 100's kHz. The average power is ~ mW-level, peak power wattlevel,
and pulse energy 30-75 nJ without any amplifier. Moreover, this transform-limited fiber laser pulses has been scaled up
to 220 μJ by using a newly developed SM PM highly Tm-doped germanate fiber 25/250μm for transform-limited 80 ns pulses
at repetition rate 20 kHz. This narrow linewidth high energy MOPA-based pulsed fiber laser can be used for LIDAR and laser
remote sensing.
The noise power spectrum of solid-state lasers - including fiber lasers - exhibits a characteristic peak at the relaxation
oscillation frequency. The tails associated with this peak extend to neighboring spectral ranges and may increase the
noise level above acceptable limits in applications using weak signals. One of the key factors to reduce the relative
intensity noise (RIN) amplitude is a low loss laser resonator. We describe a method to ultimately reduce the intensity
noise in single frequency phosphate fiber lasers by minimizing intra-cavity losses caused by fusion splices between
fibers made of different materials. Conventional fiber Bragg gratings written in silica fibers have been replaced with
gratings written in phosphate glass fibers. The quality of the intra-cavity fusion splice has been improved due to material
similarity. All-phosphate fiber laser devices have been built and tested utilizing the new gratings. The results show
relative intensity noise amplitudes that are very similar to those of conventionally fabricated devices. Challenges in the
grating writing process are currently preventing the new devices from surpassing their commercial counterparts in terms
of performance. However, this type of all phosphate glass fiber lasers may ultimately lead to a new generation of
commercial single frequency fiber lasers with improved intensity noise performance.
We report coherent monolithic THz generation in GaP QPM bonded structures based on difference-frequency generation
(DFG) using two pulsed fiber lasers in the C-band. We observed that the QPM-GaP crystals effectively increase the THz
generation power and efficiency with increasing periods of QPM structures. The azimuthal dependence of the THz
generation for the GaP QPM bonded structure has been measured when the polarization directions of the two pump
beams are orthogonal and parallel, respectively. Moreover, we observed the external cavity enhanced THz DFG when
we put the QPM-GaP crystal in an external ring cavity, for the first time. The THz cavity enhancement factor of ~250
has been achieved compared with the single-pass THz DFG. The maximum THz average power can reach 339 μW,
corresponding to a power conversion efficiency of 2.43×10-4 and quantum efficiency of 3.16%.
Optical cooling in an all fiber system using fiber laser pumps and cooling fibers doped with rare earth ions has been
investigated both theoretically and experimentally. A 2% Tm doped germanate glass was selected from glasses with
different Tm concentrations 0.5, 1, 2, 3, 4, 5, 6, 8 and 10% wt for fabrication of the cooling fiber. A high efficiency,
single mode Tm-doped fiber laser has been built to pump a Tm-doped fiber cooler. The cooling experiments done in a
vacuum chamber show indications that cooling has occurred in the fiber. A theoretical framework to understand the
nature of cooling in this laser cooling system has been developed which highlights the cooling power available as a
function of various material and fiber parameters including background loss and absorption saturation effects in the
cooling fiber. Cooling characteristics, with special emphasis on the fiber's saturation behavior, have been studied using
theoretical models of Tm3+-doped glass (4-level models) and Tm3+ doped KLa(WO4)2 crystals (20-level model).
Rare-earth doped fiber lasers and fiber amplifiers are highly attractive due to their efficiency,
compactness, and, particularly, for their potential to various applications including communication
systems, biomedical equipment, materials processing, LIDAR, and fiber-optic sensing. At the heart
of these devices is the active fiber - most commonly based on silica host glass. However, the ability
to dope silica glass fibers with high concentrations of erbium is limited due to clustering and
nonlinear up-conversion - both of which degrade the efficiency of the gain fiber. Over many years,
we have focused on developing highly doped phosphate glass fibers. The erbium concentrations can
reach 4-5% weight erbium concentration without any negative effect to the optical gain. As a result,
highly erbium doped phosphate glass fibers can produce large gain per unit length (typically 5
dB/cm) [1, 2]. This characteristic is a key enabler for a variety of optical devices that can make use
of high optical gain in a short length - most notably high power single frequency fiber lasers and
short length fiber amplifiers. In this presentation, we focus on two applications of the highly doped
phosphate fiber. One is a high power, narrow linewidth single frequency fiber laser. The second is a
fiber amplifier for coherent LIDAR applications capable of power scaling transform limited pulses
without deleterious nonlinear effects. Both are examples of how this type of active fiber can lead to
unique and superior performance.
We report a monolithic high SBS-threshold pulsed fiber laser in MOPA for longer nanosecond pulses with transformlimited
linewidth. By using a single mode polarization-maintaining large core 25 μm highly Er/Yb co-doped phosphate
fiber in the power amplifier stage, we have achieved the highest peak power of 1.2 kW at 1530 nm for 105 ns pulses
with transform-limited linewidth, and with a corresponding pulse energy of about 0.126 mJ. The achieved high-energy
pulses were frequency doubled by using a commercial periodically poled lithium niobate (PPLN) crystal, and the highest
SHG peak power of 271 W has been achieved for the SHG pulses at 765 nm.
We have implemented monolithic narrow linewidth single-mode single-frequency pulsed fiber lasers in master oscillator
and power amplifier (MOPA) configuration based on highly Er/Yb co-doped phosphate fiber with core size of 25 μm.
The narrow linewidth pulsed fiber laser seed has been achieved by directly modulating single-frequency CW fiber laser.
An arbitrary waveform generator (AWG) was used to pre-shape the pulse shapes in order to avoid the pulse distortion
and the dynamic gain saturation in the cascade fiber amplifiers. Based on the newly developed large core single-mode
highly Er/Yb co-doped phosphate fiber in the power amplifier stage, the peak power of single-mode pulses can be scaled
to more than 100 kW with transform-limited linewidth and diffraction-limited beam quality. These high power narrow
linewidth single-mode fiber laser pulses have been successfully used to generate coherent THz waves based on
difference-frequency generation (DFG) in GaSe crystal. The single-pass generated THz peak power can reach 0.12 W.
Moreover, we have observed the external cavity enhancement of DFG THz generation by using ZnGeP2 for the first
time, and implemented a high spectral resolution THz spectrometer based on the developed fiber-based tunable narrow
linewidth THz source.
Compact, efficient, narrow linewidth, fiber based THz sources have been achieved by using the monolithic high power single-frequency pulsed fiber lasers in MOPA based on difference-frequency generation (DFG) in nonlinear optical crystals. We have observed the external cavity enhancement of DFG THz generation by using ZnGeP2 for the first time, and implemented a high spectral resolution THz spectrometer based on the developed fiber-based tunable narrow linewidth THz source.
A single frequency fiber laser operating near 2 micron with over 50 mW output power has been demonstrated by using a short piece of newly developed single mode holmium-doped germanate glass fiber. Laser from 2004 nm to 2083 nm was demonstrated from a short Ho-doped fiber laser cavity. A heavily thulium-doped germanate fiber was used as an in-band pump source for the holmium-doped fiber laser. The single frequency fiber laser can be thermally tuned.
We report a new pulsed, narrow linewidth, single-mode, polarization maintaining (PM) all-fiber laser source in master
oscillator and power amplifier (MOPA) configuration that can operate over the C-band. The single-frequency pulsed
fiber laser seed was achieved by actively Q-switching a fiber laser using a piezo, with a wide pulse duration tuning range
of 7.5 ns - 1.24 μs. We use single-mode PM large core highly Er/Yb
co-doped phosphate glass fiber (LC-EYPhF) in the
power amplifier stage of MOPA to achieve 54 μJ/pulse for 153-ns pulses at 1538 nm with repetition rate of 20 kHz and
an estimated linewidth of ~ 5 MHz.
A novel fiber optic sensor has been developed to be used in superconducting magnets for fusion reactors and other large
cable-in-conduit superconductor (CICC) magnet applications. These large superconducting magnets need a diagnostic
that can measure the temperature and strain throughout the magnet in real-time, which was not possible until now.
Simultaneous temperature and strain measurements at cryogenic temperatures have been demonstrated, using
spontaneous Brillouin scattering in an optical fiber. Using an extremely narrow (100 Hz) linewidth Brillouin laser with
very low noise as a frequency shifted local oscillator, the frequency shift of spontaneous Brillouin scattered light was
measured using heterodyne detection. A pulsed laser was used to probe the fiber using Optical Time Domain
Reflectometry (OTDR) to determine spatial resolution. The spontaneous Brillouin frequency shift and linewidth as a
function of temperature agree with previous literature on stimulated Brillouin scattering data from room temperature
down to 4 K. For the first time, the spontaneous Brillouin frequency shift, linewidth, and intensity as a function of strain
have been measured down to 4 K. Analyzing the frequency spectrum of the scattered light after an FFT gives the
Brillouin frequency shift, linewidth, and intensity of the scattered light. 65,000 pulses, with 53 ns pulse widths, were
averaged in under one second, providing a 5 meter spatial resolution along a fiber that was about 100 m long. Measuring
these three parameters allow the simultaneous determination of temperature and strain in real-time throughout a fiber
with a spatial resolution on the order of several meters.
This paper describes a new approach to amplify optical images by using optically pumped doped cores in a multi-core
optical fiber structure. This approach combines the high gain and high efficiency properties of cladding pumped optical
amplifiers with the imaging properties of coherent fiber bundles. The individual cores correspond to the pixels in the image amplifier. We have demonstrated 3x3 arrays in an ytterbium-doped phosphate fiber energized by one multimode
semiconductor diode. Each pixel is capable of high gain (> 20 dB), low noise, and large acceptance angle (>12 degrees).
We expect our glass and preform fabrication method to scale to over 100 pixels. The amplified image can preserve
coherence (phase and wavelength) - or scramble the coherence depending on the design of the cores. This image
amplifier is an enabling technology for any type of imaging system that is photon-starved and requires a compact and
low noise image amplifier.
In vivo retinal imaging with ~ 8 μm axial resolution at 1030 nm is demonstrated for the first time, enabling
enhanced penetration into the choroid. A new high power, broad bandwidth light source based on amplified spontaneous
emission (NP Photonics, λc = 1030 nm, Δλ= 50 nm, Pout = 25 mW) has been interfaced to a time domain ophthalmic
OCT system. In vivo retinal OCT tomograms performed at 800 nm are compared to those achieved at 1030 nm. Retinal
OCT at longer wavelengths, e.g. 1030 nm significantly improves the visualization of the retinal pigment
epithelium/choriocapillaris/choroid interface and might therefore provide new insight into choroidal/choriocapillary
changes in age-related macular degeneration and other diseases of the retinal pigment epithelium (RPE)-choroid
complex. 1030 nm OCT could also become a valuable tool in monitoring treatment effects on the choroids as in
Verteporfin therapy.
Typical fiber Bragg sensing systems, in the 1500nm to 1600nm wavelength range, use gratings spaced approximately 5nm apart. This large spacing is preferred since optical power illuminating the gratings is limited and monitoring systems often have low performance and hence have difficulty in differentiating between closely spaced
gratings.
In this paper, we present a high power ASE source in combination with an extremely high performance Optical Spectrum Analyzer (OSA), that allows sensing gratings to be closely spaced so many sensing elements can be monitored while maintaining very good wavelength and power detection. The integrated unit can be provided on a PCI card, in a box or as an OEM module to enable a robust, handheld unit.
The OSA is built around proprietary Compliant Micro Electro Mechanical (CMEMs) tunable filters, primarily developed for very densely populated Telecommunications applications. The OSA is wavelength calibrated and can be matched to the ASE source to compensate for any spectral power characteristic of the source. This integrated OSA and ASE source will allow the sensing industry to increase grating array sizes while maintaining low cost.
Nanometer-sized features as small as 400Ahave been fabricated in single-quantum-well GaAs/A1GaAs
heterostructures for studies of quantum confinement effects in quantum dots. The features have been
fabricated by dry-etching techniques using nanometer-sized etch masks by a novel surface deposition of
colloidally-suspended spherical particles. SEM was used to examine the feature size.
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