Structural,optical properties and optical modelling of hydrothermal chemical growth derived Zn O nanowires

Zn O nanowire films were produced at 90°C using a hydrothermal chemical deposition method,and were characterised with scanning electron microscopy,optical transmission spectrometry and X-ray diffraction.The results showed that the optical band gap is 3.274-3.347 e V.Film porosity and microstructure can be controlled by adjusting the p H of the growth solution.Zn O nanowire films comprise a 2-layer structure as demonstrated by SEM analysis,showing different porosities for each layer.XRD analysis shows preferential growth in the(002)orientation.A comprehensive optical modelling method for nanostructured Zn O thin films was proposed,consisting of Bruggeman effective medium approximations,rough surface light scattering and O’Leary-Johnson-Lim models.Fitted optical transmission of nanostructured Zn O films agreed well with experimental data.

Theoretical uncertainties of(d,^(3)He)and(^(3)He,d)reactions owing to the uncertainties of optical model potentials

The theoretical uncertainties of single proton transfer cross sections of the(^(3)He,d)and(d,^(3)He)reactions,owing to the uncertainties of the entrance-and exit-channel optical model potentials,are examined with the^(30)Si(^(3)He,d)^(31)P,^(13)B(d,^(3)He)^(12)Be,and^(34)S(^(3)He,d)^(35)Cl reactions at incident energies of 25,46,and 25 MeV,respectively,within the framework of the distorted wave Born approximation.The differential cross sections at the first peaks in the angular distributions of these reactions are found to have uncertainties of approximately 5%,owing to the uncertainties in the optical model potentials from 20,000 calculations of randomly sampled parameters.This amount of uncertainty is found to be nearly independent of the angular momentum transfer and the target masses within the studied range of incident energies.Uncertainties in the single proton spectroscopic factors obtained by matching the theoretical and experimental cross sections at different scattering angles are also discussed.

A novel method of rapidly modeling optical properties of actual photonic crystal fibres

The flexible structure of photonic crystal fibre not only offers novel optical properties but also brings some difficulties in keeping the fibre structure in the fabrication process which inevitably cause the optical properties of the resulting fibre to deviate from the designed properties. Therefore, a method of evaluating the optical properties of the actual fibre is necessary for the purpose of application. Up to now, the methods employed to measure the properties of the actual photonic crystal fibre often require long fibre samples or complex expensive equipments. To our knowledge, there are few studies of modeling an actual photonic crystal fibre and evaluating its properties rapidly. In this paper, a novel method, based on the combination model of digital image processing and the finite element method, is proposed to rapidly model the optical properties of the actual photonic crystal fibre. Two kinds of photonic crystal fibres made by Crystal Fiber A/S are modeled. It is confirmed from numerical results that the proposed method is simple, rapid and accurate for evaluating the optical properties of the actual photonic crystal fibre without requiring complex equipment.

Diagnostic of capacitively coupled radio frequency plasma from electrical discharge characteristics：comparison with optical emission spectroscopy and fluid model simulation

The capacitively coupled radio frequency（CCRF）plasma has been widely used in various fields.In some cases,it requires us to estimate the range of key plasma parameters simpler and quicker in order to understand the behavior in plasma.In this paper,a glass vacuum chamber and a pair of plate electrodes were designed and fabricated,using 13.56 MHz radio frequency（RF）discharge technology to ionize the working gas of Ar.This discharge was mathematically described with equivalent circuit model.The discharge voltage and current of the plasma were measured atdifferent pressures and different powers.Based on the capacitively coupled homogeneous discharge model,the equivalent circuit and the analytical formula were established.The plasma density and temperature were calculated by using the equivalent impedance principle and energy balance equation.The experimental results show that when RF discharge power is 50–300 W and pressure is 25–250 Pa,the average electron temperature is about 1.7–2.1 e V and the average electron density is about 0.5？×10^17–3.6？×10^17m^-3.Agreement was found when the results were compared to those given by optical emission spectroscopy and COMSOL simulation.

Response of multi-step compound pre-equilibrium reaction cross sections for the(p,n)reactions to forms of optical model parameters

In furtherance to improving agreement between calculated and experimental nuclear data, the nuclear reaction code GAMME was used to calculate the multistep compound(MSC) nucleus double differential cross sections(DDCs) for proton-induced neutron emission reactions using the Feshbach-Kerman-Koonin(FKK) formalism. The cross sections were obtained for reactor structural materials involving ^(52)Cr(p, n)^(52)Mn,^(56)Fe(p,n)^(56)Co, and ^(60)Ni(p, n)^(60)Cu reactions at 22.2 MeV incident energy using the zero-range reaction mechanism. Effective residual interaction strength was 28 MeV, and different optical potential parameters were used for the entrance and exit channels of the proton-neutron interactions. The calculated DDCs were fitted to experimental data at the same backward angle of 150°, where the MSC processes dominate. The calculated and experimental data agree well in the region of pre-equilibrium(MSC) reaction dominance against a weaker fit at the lower emission energies. We attribute underestimations to contributions from the other reaction channels and disagreement at higher outgoing energies to reactions to collectively excited states. Contrary to the FKK multi-step direct calculations, contributions from the higher stages to the DDCs are significant. Different sets of parameters resulted in varying levels of agreement of calculated and experimental data for the considered nuclei.

A model for estimating the rate constant between CO_2-CO gas and molten slag containing iron oxides using optical basicity

A simple model for estimating the rate constant between CO2-CO gas and molten slag containing iron oxides was developed using optical basicity only. In this model, the temperature dependence of the rate constant can be described by the Arrhenius law, and the activation energy can be expressed with a linear function of the slag＇s optical basicity. The model was applied to some molten slag systems, such as FeO, FeO-CaO, FeO-SiO2, FeO-Na2O, FeO-CaO-SiO2, FeO-SiO2-P2O5, FeO-SiOE-Na2O, and FeO-CaO-SiOE-P2O5. A comparison between the predicted results and measured data showed that the model worked well.

Accommodation in human eye models:a comparison between the optical designs of Navarro,Arizona and Liou-Brennan

AIM： To simulate and compare accommodation in accommodative and non-accommodative human eye models.METHODS： Ray tracing and optical design program was used. Three eye models were designed and studied： the Navarro, the Arizona and the Liou-Brennan. In order to make the Navarro and Liou-Brennan models to accommodate, specific geometric parameters of the models were altered with values that were chosen from the literature. For the Arizona model, its’ mathematical functions for accommodation were used for the same accommodative demands. The simulation included four distances of accommodation for each model： at infinity, 3, 1 and 0.5 m.The results were diffraction images of a “letter F” for graphical comparison, spot diagrams on the retinal field and Modulation Transfer Function （MTF） graphs.RESULTS：Zernike coefficients for the aberrations, Airy disk diameter, root mean square （RMS） error diameter and total axial length of the model were provided from the program. These were compared between them in all distances. The Navarro model had the smallest axial length change as a simple model. The Arizona did not change its axial length because it is designed to be accommodative. The Liou-Brennan model had different results concerning the aberrations because of the decentration of the pupil. The MTF graphs showed small differences between the models because of the differences in their designs.CONCLUSION： All the three models are able to simulate accommodation with the expected results. There is no model that can be assumed as the best choice. Accommodation can be simulated in non-accommodativemodels and in customized ones.

Quantum-dot Semiconductor Optical Amplifiers in State Space Model

A state space model(SSM) is derived for quantum-dot semiconductor optical amplifiers(QD-SOAs).Rate equations of QD-SOA are formulated in the form of state update equations,where average occupation probabilities along QD-SOA cavity are considered as state variables of the system.Simulations show that SSM calculates QD-SOA′s static and dynamic characteristics with high accuracy.

Establishing an experimental model of photodynamic induced anterior ischemic optic neuropathy

BACKGROUND： Scholars have supposed to establish animal models of optic neuropathy by pressing and partially amputating optic nerve, increasing intraocular pressure and injecting vasoconstrictor, etc., but the models are greatly different from anterior ischemia optic neuropathy. Therefore, a more ideal method is needed to establish animal model of anterior ischemic optic neuropathy （AION）. OBJECTIVE ： To establish AION models in rats, observe the functional changes of fundus, fundus fluorescein angiography （FFA）, optical coherence tomography （OCT）, flash visual evoked potential （F-VEP）, and histopathologically confirm its reliability. DESIGN： A randomized control tria SETTINGS ： Department of Ophthalmology, Xi＇an Fourth Hospital; Xi＇an Institute of Ocular Fundus Diseases MATERIALS ： The experiments were carried out in the research room of Xi＇an Institute of Ocular Fundus Diseases from February 2005 to May 2006. Thirty healthy male SD rats of 4-5 weeks old, weighing 140-160 g, were provided by the animal experimental center of the Fourth Military Medical University of Chinese PLA [SCXK （Military）2002-005], and those without eye disease examined by slit lamp and direct ophthalmoscope after mydriasis were enrolled. The conditions for feeding mice without special pathogen were strictly followed. The rats were randomly divided into blank control group （n =5）, laser group （n =5）, hematoporphyrin derivative （HPD） group and AION group （n =15）, each group was numbered randomly. For each rat, the right eye was taken as the experimental eye, and the left one as the control one. METHODS： In the AION group, the rats were injected with HPD （10 mg/kg） via caudal vein, and then the optic discs were exposed to krypton red （647 nm, 80 mV） for 120 s, and the rats were in avoidance of light for 2 weeks postoperatively. Rats in the laser group were only exposed to krypton red （647 rim, 80 mV） for 120 s, and in avoidance of light for 2 weeks postoperatively; Those in the HPD group were only injected with HPD （10 mg/kg） via caudal vein; Those in the blank control group were untouched. （1） Visual electrophysiological test： The F-VEP was used to evaluate the function of visual nerve. （2） FFA： After mydriasis and anesthesia as describe above, the fluorescein sodium parenteral solution （1 mL/kg） was injected v/a caudal vein and finished within about 3 s, the time of FFA was recorded from the beginning of injection, the video sight aimed at the optic disc and the surrounding area. （3） After mydriasis and anesthesia as describe above, the rats were examined with OCT. （4） Histological observation： After hematoxylin and eosin （HE） staining, the optic disc and surrounding blood vessels of retina were observed under light microscope at high power field. MAIN OUTCOME MEASURES： The results of fundus, FFA, visual electrophysiological test and OCT detection within 90 days after model establishment were observed. RESULTS： Of the 30 rats, 1 died after anesthesia in the laser group and 2 died in the AION group respectively, and finally 27 rats were involved in the analysis of results. （1） Changes in fundus： In the AION group, there was edema in upper optic disc and unclear boundary at 1 day after establishment, edema still could be observed at 6 days, and upper optic disc atrophied and appeared as pale at 90 days. （2） FFA results： In the AION group, early ＂low fluorescence＂, middle and late ＂high fluorescence＂ were observed in upper optic disc 1 day after model establishment, and there was ＂low fluorescence＂ at 6 days, and the low fluorescence could be observed all the time at 23 days. （3） Visual electrophysiological changes： In the AION group as compared with the control eyes, the experimental ones had prolonged F-VEP P100 latency [（71.65±8.81）, （57.58±8.38） ms, t =3.148, P =0.012], and decreased wave amplitude [（4.77±1.90）, （10.06±3.66） μV, t =4.082, P =0.003], and these changes lasted to 35 days after model establishment. （4） OCT results： In the AION group, the reflection surface of part nerve fiber layer was higher than the retina plane, the surface was rough and the thickness was increased at 6 days after model establishment. （5） Histopathological results： At 1 day after model establishment, part optic discs had highly edema, edema of nerve fibers, and loose tissue, also accompanied by the displacement of surrounding retina; At 23 days, the optic disc and surrounding nerve fiber layers became thinner, and the numbers of ganglion nuclei in the retina tissue sections were obviously decreased. These changes were not observed in the laser group, HPD group and blank control group. CONCLUSION ： The changes of fundus, FFA, OCT, visual electrophysiology and histopathology confirmed that the krypton red laser irradiation （647 nm） at 2 hours after HPD was injected via caudal vein can establish more ideal animal models of AION.

Establishing a cat model of acute optic nerve injury

BACKGROUND： In order to investigate the progress in optic nerve injury and the following regeneration and repair, many kinds of animal models of optic nerve injury have been established, such as models of acute and chronic ocular hypertension, compression, amputating wound, ischemia reperfusion or hypoxia, intravitreal injection of excitatory amino acids, etc. However, most of these models are established by squeezing intraorbital optic nerve, and suitable for ophthalmology, and there are fewer models suitable for the acute cranial contusion in neurosurgery. OBJECTIVE： To observe the changes of optic nerve after acute injury, and the characteristics of methods for establishing model of acute optic nerve injury in cats. DESIGN： A complete randomized grouping and controlled animal trial. SETTING： Department of Neurosurgery, General Hospital of Ji＇nan Military Area Command of Chinese PLA. MATERIALS： Twenty-eight healthy adult cats, common degree, either sex, weighing 2.0-3.5 kg, were provided by the animal experimental center of Fudan University. The cats were randomly divided into control group （n =3） and model group （n =25）, and 5 cats in the model group were observed at 6 hours and 1, 3, 7 and 14 days after injury respectively. JX-2000 biological signal processing system （Department of Physiology, Second Military Medical University of Chinese PLA, Shanghai）; Inverted phase contrast microscope （Olympus）; Axioplan 2 imaging microgram analytical system （Labsystems）. METHODS： The experiments were carried out in the Department of Neurosurgery, General Hospital of Jinan Military Area Command of Chinese PLA from June 2004 to June 2005. The cats in the model groups were made into models of acute optic nerve injury： The cats were anesthetized, then the limbs were fixed in a lateral recumbent position. Pterion approach in human was imitated, the operative incision was made along the line between lateral canthus and tragus, and it could be seen deep along the skull base that white optic nerve （about 3 mm） went through optic foramen and entered into brain tissue. It was squeezed with noninvasive vascular clip for 20 seconds, then the clip was removed, and then the skull was closed after it was examined to be no bleeding. The size of bilateral pupils, direct and indirect light reflexes were observed postoperatively. Successfully established models were judged by larger operated pupil than controlateral one, disappearance of direct light reflex and the existence of indirect light reflex. No model establishment was performed in the control group. Each cat was tested with flash visual evoked potential （F-VEP） to observe the electrophysiological changes before and after experiment. All the cats in the control group and model groups were killed under anesthesia before model establishment and at 6 hours, 1, 3, 7 and 14 days after model establishment respectively, and the pathological changes of the optic nerve after injury were observed under electron microscope and light microscope. MAIN OUTCOME MEASURES： VEP and the ultrastructural changes of optic nerve after acute optic nerve injury in both groups. RESULTS： All the 28 cats were involved in the analysis of results. ① VEP results： The VEP latencies were obviously different between the control group and model group at each time point （P 〈 0.05）, whereas there were no obvious differences among different time points in the model group （P 〉 0.05）. The VEP amplitudes were obviously different between the control group and model group at each time point （P 〈 0.05）, whereas there were no obvious differences among different time points in the model group （P 〉 0.05）. ② Ultrastructural changes of the optic nerve： Under electron microscope, normal optic nerve myelin sheath had complete structure, tramal plates were clear and arranged tightly, axolemma was complete, whereas the structures of endoneurium, myelin sheath, tramal plates, axolemma and axon were in disorders after optic nerve injury. CONCLUSION： Models of acute optic nerve injury established by squeezing intracranial optic nerve with noninvasive vascular clip can be applied in studying intracranial acute optic nerve injury.

The Nucleon-Actinide Global Optical Model Potential Below 300 MeV

A set of new global phenomenological optical model potential parameters has been obtained in the mass range of target nuclei 220≤A≤260 with incident energies below 300 MeV, by simultaneously fitting the experimental data of 232Th and 23Su, and these potential parameters are analyzed and used to calculate the reaction cross sections, energy spectra and double differ- ential cross sections for p＋232Th reaction. Comparison of calculated results using these potential parameters with available experimental data shows that the present form of global optical model potential could reproduce experimental data for both the neutron and the proton.

Optical Modeling and Analysis of Peripheral Optics of Contact Lenses

Background: Degraded peripheral vision has been hypothesized to be a stimulus for the development of foveal refractive error. Contact lenses have been widely used to correct central vision, but their impacts on peripheral vision are still unknown. The purpose of this study was to use optical model software to evaluate the peripheral optics of rigid gas permeable (RGP) and soft contact lenses (SCLs) in isolation. This will better assist us in understanding their peripheral optical performances on human eyes. Methods: An optical design software package (Zemax EE) was used to model peripheral optics of Menicon RGP lens and Acuvue 2 SCLs. Profiles of sphero-cylindrical power and major higher-order aberrations were computed in 10osteps out to 40o off-axis eccentricity for –3.0 D central focal power contact lens. The results of optical modeling were analyzed and compared with previously published experimental data. Results: –3.0 D RGP lenses and SCLs had –1.4 D and –2.0 D dioptric power at 40o eccentricity, respectively. The reduced dioptric power in the periphery of the analyzed contact lenses quantitatively matched with the reduced amount of hyperopic field curvature found from experimental data when these contact lenses fitted on human eyes. Cylindrical power increased to 0.3 D ~ 0.4 D at 40o eccentricity for both lens types. In addition, both contact lens types produced higher order aberrations, namely 1.2 μm coma and 0.15 μm spherical aberration at 40o eccentricity. Conclusions: Compared to SCLs, RGP lenses with equal focal power had less dioptric power in the periphery. Both RGP lenses and SCLs produced the same amount of major higher-order aberrations with increasing of the field angle. Some of these results can be used to predict and understand the peripheral optical performance of contact lenses on human eyes.

An Optical Model for the Remote-Sensing of Absorption Coefficients of Phytoplankton in Oceanic/Coastal Waters

A new model for the remote sensing of absorption coefficients of phytoplankton aph (λ) in oceanic and coastal waters is developed and tested with SeaWiFS and MODIS-Aqua data. The model is derived from a rela-tionship of the remote sensing reflectance ratio Rrs (670)/Rrs (490) and aph (490) and aph (670) (from large in-situ data sets). When compared with over 470 independent in-situ data sets, the model provides accurate retrievals of the aph (λ) across the visible spectrum, with mean relative error less than 8%, slope close to unity and R2 greater than 0.8. Further comparison of the SeaWiFS-derived aph (λ) with in-situ aph (λ) values gives similar and consistent results. The model when used for analysis of MODIS-Aqua imagery, provides more realistic values of the phytoplankton absorption coefficients capturing spatial structures of the massive algal blooms in surface waters of the Arabian Sea. These results demonstrate that the new algorithm works well for both the coastal and open ocean waters observed and suggest a potential of using remote sensing to provide knowledge on the shape of phytoplankton absorption spectra that are a requirement in many inverse models to estimate phytoplankton pigment concentrations and for input into bio-optical models that predict carbon fixation rates for the global ocean.

NUMERICAL MODEL OF THERMAL TRANSIENT EFFECT IN FIBER OPTIC GYRO

The sensitivity of the interferometric fiber optic gyro in the presence of time varying thermal gradients plays a key role in its performance. It is well known that this sensitivity is due to the difference of index changes between the points symmetrical with respect to the middle of the coil. In order to reduce this sensitivity, different winding patterns, such as quadrupolar winding, were introduced to keep the thermal environment of the symmetrical points. In this paper, a numerical model of the transient temperature distribution in the gyro was established. The temperature gradient of the coil was solved in conjugation with the nature convection heat transfer in the aperture between the coil and the case. Effects of the winding pattern and the design of its case were investigated to optimize the design of the interferometric fiber optic gyro.

Application of a microscopic optical potential of chiral effective field theory in (p, d) transfer reactions

The microscopic global nucleon–nucleus optical model potential(OMP)proposed by Whitehead,Lim,and Holt,the WLH potential(Whitehead et al.,Phys Rev Lett 127:182502,2021),which was constructed in the framework of many-body per-turbation theory with state-of-the-art nuclear interactions from chiral effective field theory(EFT),was tested with(p,d)transfer reactions calculated using adiabatic wave approximation.The target nuclei included both stable and unstable nuclei,and the incident energies reached 200 MeV.The results were compared with experimental data and predictions using the phenomenological global optical potential of Koning and Delaroche,the KD02 potential.Overall,we found that the micro-scopic WLH potential described the(p,d)reaction angular distributions similarly to the phenomenological KD02 potential;however,the former was slightly better than the latter for radioactive targets.On average,the obtained spectroscopic factors(SFs)using both microscopic and phenomenological potentials were similar when the incident energies were below approxi-mately 120 MeV.However,their difference tended to increase at higher incident energies,which was particularly apparent for the doubly magic target nucleus 40Ca.

Ultrasonic sensitivity-improved fiber-optic Fabry-Perot interferometer using a beam collimator and its application for ultrasonic imaging of seismic physical models

An ultrasonic sensitivity-improved fiber-optic Fabry-Perot interferometer （FPI） is proposed and employed for ultra- sonic imaging of seismic physical models （SPMs）. The FPI comprises a flexible ultra-thin gold film and the end face of a graded-index multimode fiber （MMF）, both of which are enclosed in a ceramic tube. The MMF in a specified length can collimate the diverged light beam and compensate for the light loss inside the air cavity, leading to an increased spectral fringe visibility and thus a steeper spectral slope. By using the spectral sideband filtering technique, the collimated FP1 shows an improved ultrasonic response. Moreover, two-dimensional images of two SPMs are achieved in air by recon- structing the pulse-echo signals through using the time-of-flight approach. The proposed sensor with easy fabrication and compact size can be a good candidate for high-sensitivity and high-precision nondestructive testing of SPMs.

Properties of Optical and X-ray Selected AGN-Probing the Unified Model of AGN

We have compiled a sample of two subsets of AGN selected from their optical and X ray data. The first subset was selected for very broad and/or peculiar optical emission line profiles, the second for a high X ray flux. Here we will discuss properties of these galaxies and show that both subsets are very similar in the multi wavelength view. Furthermore, we will discuss differences between the two subsets and their implications for a Unified Model of AGN.

A Study on Sulfate Optical Properties and Direct Radiative Forcing Using LASG-IAP General Circulation Model

The direct radiative forcing （DRF） of sulfate aerosols depends highly on the atmospheric sulfate loading and the meteorology, both of which undergo strong regional and seasonal variations. Because the optical properties of sulfate aerosols are also sensitive to atmospheric relative humidity, in this study we first examine the scheme for optical properties that considers hydroscopic growth. Next, we investigate the seasonal and regional distributions of sulfate DRF using the sulfate loading simulated from NCAR CAM-Chem together with the meteorology modeled from a spectral atmospheric general circulation model （AGCM） developed by LASG-IAP. The global annual-mean sulfate loading of 3.44 mg m-2 is calculated to yield the DRF of -1.03 and -0.57 W m-2 for clear-sky and all-sky conditions, respectively. However, much larger values occur on regional bases. For example, the maximum all-sky sulfate DRF over Europe, East Asia, and North America can be up to -4.0 W m-2. The strongest all-sky sulfate DRF occurs in the Northern Hemispheric July, with a hemispheric average of -1.26 W m-2. The study results also indicate that the regional DRF are strongly affected by cloud and relative humidity, which vary considerably among the regions during different seasons. This certainly raises the issue that the biases in model-sinmlated regional meteorology can introduce biases into the sulfate DRF. Hence, the model processes associated with atmospheric humidity and cloud physics should be modified in great depth to improve the simulations of the LASG-IAP AGCM and to reduce the uncertainty of sulfate direct effects on global and regional climate in these simulations.

DISSYMMETRY MODEL OF MOLECULAR POLARIZABILITY AND OPTICAL ACTIVITY

Dissymmetry model of molecular polarizability divided into some layers within a sphere,some rules and sequence according to the magnitude of polarizability replaced by bond refraction for many groups have been suggested.The relationship between the dissymmetry of molecular polarizability arrounding the dissymmetric carbon atom and the direction of optical activity has been discussed .The accuracy is above 95 persent to use our model and rules to determine over 6000 compounds of optical activity.

Modeling, Simulation and Experimental Studies of Refractometric Fiber Optic Sensor

Refractometric fiber optic sensors have a number of applications in industry due to advantages like remote sensing ability, compact size, easy to fit, etc. A refractometric sensor contains a pair of parallel fibers and a gap between the sensor probe and reflector, wherein the liquid whose refractive index is to be measured is filled. This paper describes the importance of mathematical modeling of this sensor. Ray tracing approach is used to model the sensor mathematically. This mathematical model is generalized for any scenario which is useful to avoid tedious trial and error techniques to design the sensor prototype. Mathematical modelling is a useful tool to optimize the gap distance for a detection of refractive index of liquid. The model is developed and analyzed rigorously considering adulteration of diesel by kerosene where refractive index varies from 1.44 to 1.46. Simulation experiments are carried out to optimize the gap distance which is found to be 6.8 mm using both models. Experiments are carried out where sensor probe is fabricated and results are analyzed. It is observed that for suggested gap distance sensor output varies almost linear over the entire range.