The high intensity and ultrashort laser pulse duration (fs) make direct observation of plasma dynamics a challenging task. In the field of laser-plasma physics and especially for the acceleration of ions, the spatio-temporal intensity distribution is one of the most critical aspects. We describe a novel method based on a single-shot (i.e. single laser pulse) chirped probing scheme, taking nine sequential frames at frame rates up to THz. This technique, to which we refer as temporally resolved intensity contouring (TRIC) enables single-shot measurement of laser-plasma dynamics. Using TRIC, we demonstrate the reconstruction of the complete spatio-temporal intensity distribution of a high-power laser pulse in the focal plane at full pulse energy with sub-picosecond resolution.Today’s high-power laser systems are capable of reaching photon intensities up to 10 Multi-GeV electrons3, x-rays4, neutrons5 and energetic ions7 emerge from intense laser-plasma interactions with targets. A milestone for acceleration of multi MeV ions has been demonstrated 20008. Short acceleration lengths (MeV/µm), small source sizes (µm), high particle flux10, poly-energetic energy distributions and broad angular divergence11 mark the key differences compared to the output from conventional accelerators. Substantial improvements of laser and target technology have enabled higher repetition rates also for solid targets16. With the new generation of petawatt class laser systems18 proton energies up to 100 MeV19 in a single shot and repetition rates up to 1 Hz20 at lower kinetic energies are now available, fueling enthusiasm on the way to developing laser-driven systems11.Since the advent of chirped pulse amplification21 and the transverse intensity distribution in the focal plane. Due to sub-picosecond duration and the high peak intensity a direct measurement of such remains challenging. In fact the spatial and temporal intensity distributions are often determined independently, for separate shots and with attenuated beams. The temporal intensity contrast is typically measured relatively (in relation to the peak intensity) with autocorrelation methods23 which integrate over the spatial distribution. Newest approaches aiming for single-shot measurements25 seem very promising for monitoring the temporal shape of the intensity during the interaction. The spatial intensity distribution is usually measured with time-integrating beam profilers using an attenuated laser pulse and the anticipated high intensities require mapping with high dynamic range (at least five orders of magnitude) in order to avoid an overestimation of the peak intensity at focus26. Also several methods have been developed to measure the peak intensity of high-power laser pulses29. A combined spatio-temporal measurement of an attenuated high-power laser focus has recently been demonstrated for the first time30. Although several techniques allow the analysis of the spatial and temporal intensity distribution of a high-power laser focus32, it has not been directly accessible during a laser-plasma interaction. Here we describe a technique that measures the spatio-temporal evolution of a laser-induced plasma on target for a single-shot. This not only yields information about the evolution of plasma formation, but can be further interpreted to retrieve the absolute spatio-temporal intensity distribution (STID) of a high-power laser in the focal plane at full intensity. Within a single-shot we measure temporally resolved intensity contours with sub-picosecond temporal resolution and about 25 µm spatial resolution. With each additional measurement with modified laser intensity (in our example five pulses) we add about one order of magnitude to the covered dynamic range of our measurement. Because there are no direct comparisons with our novel technique, we extract the transverse spatial and the temporal profiles of the three-dimensional intensity distribution function, and compare them to the standard techniques, i.e. a spatial high dynamic range (HDR) focus image (integrated over the full laser pulse duration) and a temporal contrast measurement taken with a third-order autocorrelator.Two of the most influential aspects relevant to laser-ion acceleration are the temporally resolved intensity contrast15 with a diameter of 2 mm which is positioned for irradiation at 45 degree incidence. A small fraction of the 30 fs laser beam is coupled out in advance to provide the probe beam. After traversing a glass rod, the resulting chirped probe pulse of 2 ps duration is overlapped perpendicularly with the main laser pulse at the target. The transmitted probe light is collected with a lens, guided from the vacuum chamber and imaged with another lens onto a camera  with a magnification of ≈6.5 and a spatial resolution of roughly 1.5 µm. A 2D grating  is positioned between the lens and the CCD chip, such that nine spatially separated replicated images are accommodated on the camera chip. Each image still contains the complete spectrum. By adding nine different narrow band-pass filters (bandwidth of 10 nm) in front of the camera chip at the positions of the replicas they will be spectrally filtered and thus correlated temporal information is imprinted34 onto each image. The time delay between subsequent images is defined by the set spectral chirp and in our case set to 222 fs (see methods). We thus measure nine sequenced images with a frame rate of 5 THz, revealing the plasma evolution in response to a high-power laser pulse interacting with a solid density target. It is worth mentioning, that this implementation can considerably simplify single-shot probe schemes that have been realized in the past38. An exemplary raw image of such measurement is displayed in Fig. ne > nc, the plasma turns reflective for the incoming probe pulse. The horizontal dimension recorded in the image contains convoluted information since the laser and the orthogonal probe pulse both hit the target under an angle of 45 degree. The probe image is thus a projection. At the same time, the orientation of the target with respect to the drive beam implies that non-central parts of the target interact with the laser in out-of-focus planes causing a complex two-fold convolution of time and space. A spatial difference in horizontal dimension of 60 µm in the image, corresponds to a time difference of 200 fs of both the laser and probe pulse hitting the target. For simplicity we therefore only consider the vertical dimension, where those effects do not play a role.The experimental setup resembles a typical pump-probe configuration and is shown in Fig. The measured average growth of the overdense plasma area (dark region) of the single-shot image sequence in Fig. I5 is shown in Fig. t = 0 (largest contour) with an intensity of I4 of the main pulse. The hole in the target foil after the shot is shown in Fig. ne ≥ nc can be related to the contour of the hole in the target after the shot. As a conclusion of this empirical observation we state, that the intensity at the measured contour of the plasma is the isosurface of the intensity distribution of the laser focus reaching a certain threshold intensity. This threshold −2 was determined by reducing the laser energy stepwise and thus the intensity until no hole was observed at the target after the shot (see methods). In the previous part we have simplified a few physical aspects. This is mostly valid since the threshold intensity (−2) is significantly lower than relativistic intensities ≈1018 · W cm−2, where other effects such as relativistic transparency39, relativistic self-focusing40 and self-phase-modulation41 can play a role. We further assume, that the enlargement of the plasma is caused, when the intensity distribution of the laser pulse reaches the threshold intensity and the target is ionized to reach free electron density beyond the critical density ne ≥ nc. Another cause of an advancing plasma diameter could be the transverse expansion of the plasma itself by hot electrons43. In the center of the laser focus at t = 0 the laser eventually reaches relativistic intensities for the shots with highest laser energy and thus in the inner part relativistic transparency, self-focusing and other effects can play a significant role. Also the electron temperature at the central part of the laser focus can be beyond the MeV level, resulting in expansion velocities approaching a large fraction of c. However, since the interesting region for TRIC is the outer contour of the plasma  = Ith) those effects do not play a role and the electron temperature is in the low keV region or smaller. In principle, fast electrons that are excited in the center, where the intensity is higher, could contribute to the transversely expanding front. Because in our case the transverse expansion primarily in vertical dimension stagnates after t = 0 when the peak of the pulse has hit the target, we ignore this effect but note that it could become relevant for transversely steep intensity distributions. We observed a growth in horizontal dimension after t = 0 s for the highest intensities  is used. The intensity map features measuring the spatial intensity profiles at select time steps and reveals the temporal dynamics at a given position. The most interesting cases are the spatial distribution of the peak intensity (t = 0) and the temporal contrast at the central point (x = 0 and y = 0). We therefore evaluated I as displayed in Fig. t = 0) and I as displayed in Fig. −2, which is orders of magnitude below the classically estimated peak intensity of −2.Repeating this for several g(t) and a HDR image of the laser in the focal plane (integrated over time) f26. Both of those standard measurements yield only relative information, which we scale to absolute values by calculatingELaser = 374 mJ is the full laser pulse energy measured for each shot (see methods). The spatial shape function of the laser focus f and the temporal laser contrast-curve g(t) are both normalized, 44 by definition.To assess our method we compared the extracted temporal and spatial intensity distribution to data obtained by conventional methods, a contrast profile of a third-order autocorrelator  f. Assuming a Gaussian profile for the temporal distribution of the laser pulse, with a full-width at half-maximum t = 0 isThe HDR focus image is the integrated spatial distribution over the full duration of the laser pulse and can thus be normalized to t = 189 fs. The focus, measured offline with the camera, appears continuous since it has more data points, while TRIC contributes a small number of points (dictated by the number of shots with different attenuation coefficients). In the overlap region, both methods yield similar comparable results. With TRIC we can measure down to Ith, which exceeds the capability of HDR imaging as this is limited by damage of the camera. With an increased spatial resolution of TRIC, we expect to resolve also the peak of the pulse such that a 7 orders of magnitude dynamic range seems feasible. The horizontal error bar originates from the readout of the diameter of the contour. The rather large error bar for the evaluated intensities is due to the error of the determination in the threshold intensity.Figure t = 0) is equal to one. The extent of temporal error bars of TRIC is attributed to the time interval correlated to the width of the bandpass filters for each time step in addition to an increment from time delay uncertainty. The intensity error bars originate from the uncertainty of the determined damage threshold. The TRIC values follow the shape of the autocorrelation trace but indicate reduced temporal contrast. The reason for this disagreement between the autocorrelation trace and the trace extracted from TRIC is not clear but one of the interesting features that can be studied with TRIC. Spatio-temporal coupling44, which can have diverse origins, manifests in focus (the simplest example is pulse lengthening by a pulse front tilt). This is where TRIC can yield complementary information. It highlights the necessity of monitoring and controlling the laser contrast on target at full energy, especially in the focal plane. More detailed measurements with an improved spatial resolution will help to shed light on this interesting aspect.The Sequoia is a scanning third order autocorrelator and measures the temporal intensity contrast relative to the laser peak. Comparison to the absolute temporal intensity evolution at a fixed spatial position, as accessible with TRIC, requires more assumptions. We thus solely compare the autocorrelator contrast curve to a normalized curve measured with TRIC as indicated in Fig. 16 or other nonlinear techniques45 that aim to manipulate the laser pulse close to target. TRIC gives access to the spatial intensity distribution at energies slightly above Ith. We note that it requires still analytical effort to account for the horizontal dimension. Because the ignition of the plasma and thus the change of opacity is irreversible on ultra-fast timescales, TRIC only yields information of the rising slope of the laser pulse and therefore requires the assumption of an intensity distribution that rises monotonically over time. Two short prepulses with intensities around 1014 · W cm−2 and thus just above the threshold intensity occurring ≈600 ps prior to the main pulse have been identified in the laser-system using third-order autocorrelation. If they were focused to a similar spot as the main pulse, they would have reached 1014 · W cm−2 for the shots with highest energy of 374 mJ. Although exceeding the intensity threshold for TRIC, we have not observed a respective contour when probing at times as early as 100 ps before the main pulse. We note though, that the contour with a size of a few µm could have not been detected with TRIC in its current spatial resolution limit. The absence of any sign of target damage/modification due to these prepulses on the other hand reconfirms that the measurement is not disturbed by transverse plasma-expansion initiated by very early prepulses at sufficiently low intensity, since no plasma shape is recorded in the probe beam until 1.3 ps prior to the peak of the pulse as evidenced in the last line of Fig. Ith. A more detailed determination of the threshold intensity would further allow a reduction of this systematic error and thus an even more accurate determination of the absolute intensity distribution on target.By using a simple method of beam multiplexing we enable recording of nine images of the plasma evolution during the interaction of a high-power laser pulse with a thin solid density target in a single-shot. Our interpretation of the such obtained images in the framework of TRIC yields information about the absolute STID in the focus of a high-power laser pulse. We note though, that if the spatial intensity distribution on target was significantly steeper, for example in the case of a perfect Gaussian pulse, this interpretation can be misleading and the transversely streaming electrons could contribute or even dominate the observes expansion front. Determining the STID in focus at high peak intensity is specially interesting in combination with plasma mirrors47 instead of binary shadowgraphy. It can therefore add a temporal component to diverse probing experiments. Thus, we see TRIC as a very first step that is accessible with this technique. In the future we foresee a potential use of TRIC for different pump-probe modalities. Especially the intrinsic synchronization in laser-driven particle acceleration processes enables temporal resolved investigations of the interaction of ions, electrons and x-rays with matter. The demonstrated sub-ps temporal resolution could for example also be exploited further for development of a time-of-flight measurement49 of ion bunches with combined temporal and 2D spatial resolution.The technique of acquiring multiple images by exploiting the natural features of a broadband highly intense laser-pulse, can be adapted to other, more complicated probing techniques, e.g. using holography31 and the temporal contrast was measured with a third-order autocorrelator (Sequoia or Tundra). Several Pockels cells are implemented in the system to clean the ns contrast, the fastest one has a rise time of about 500 ps. The temporal laser contrast has been found to be in the order or 10−9 from a few ns up to shortly before the main pulse ≈50 ps measured over 2 ns prior to the main pulse. Two short prepulses with ≈10−6 of peak power have been identified about 600 ps prior to the main pulse, originating from reflections in the regenerative amplifier. Further measurements with saturated photodiodes have not shown any significant prepulses in the ns range prior to the pulse. Therefore the majority of the energy is in the main pulse and the measured energy (determined with an energy meter) is used for calculations of the focus intensity. To monitor the energy of each shot a second energy meter is positioned behind a leakage mirror close to the experiment. At lower energies the sensitivity of the energy meter was not sufficient. In this case, the energy was estimated by integrating over the spectrum measured with an Ocean Optics spectrometer for each shot. The neutral density filters that are used to attenuate the beam during the experiment are positioned in the stretched beam before the compressor (≈400 ps). The probe beam is coupled out by a pickoff mirror about 2 meters before the laser-target interaction at the edge of the beam. The main laser is focused with a 90°, f = 2 silver parabola with 20 cm focal length impinging onto target with p-polarization. The resulting HDR focus picture is depicted in Fig. 26.The ATLAS-300 is a single CPA laser system based on a Titanium Sapphire system and can be operated at a 5 Hz repetition rate. The pulse length of 30 fs is measured with FROGThe probe pulse was coupled out with a pick off mirror at the edge of the main laser pulse in the target chamber . The timing of the probe beam with respect to the main laser pulse was controlled with a delay stage. The probe beam was then guided through an aperture of 7.5 mm diameter. This aperture was imaged onto the interaction point (IP) with a demagnification of 9, resulting in an illuminated area of ≈830 µm. A glass rod of 3 cm length was introduced in the beam path prior to target in order to chirp the pulse up to 2 ps. The energy of the probe beam was ≈0.5% of the energy of the main pulse, resulting in a maximum of 20 mJ and an intensity of 1 · 1012 · W cm−2 on target. The intensity of the probe was thus 5 orders of magnitude lower than the intensity of the main pulse and further distinctly smaller than the threshold intensity. Due to the modified energy of the laser and thus also the probe beam, it was further filtered in front of the camera (for high energies). An intended good contrast between the image created by the probe and self-emission from the target favored filtering in front of the camera instead of filtering prior to the target. The imaging system of the probe beam was based on a two-lens-system resulting in a magnification of the probe image  of ≈6.5. The first lens (2 inch) with a focal length of 10 cm is used to collect the light so it can thus further be guided out of the vacuum chamber. A second lens is used to image the IP onto the camera. A transmission grating  is placed in the beam path of the second lens to the camera, creating 9 nearly identical replicas on the camera chip. The narrow bandpass (10 nm FWHM) filters with dimensions 5 × 5 · mm−2  are placed directly in front of the camera chip. The spatial resolution of the complete setup was determined experimentally to about 25 µm. This is significantly lower than the theoretical resolution of the imaging system, which would support ≈2 µm (Abbe limit). The main limiting factor is the low quality of the beam profile of the probe beam, which shows strong intensity fluctuations on the order of 10 µm spatial scale.The zero timing describes the coincidence of the incident of probe and main pulse on the target. It was measured with the use of an air plasma ignited by the attenuated pump pulse in air. A high-power laser pulse can generate an air plasma when a certain intensity threshold50 is reached. Therefore the laser intensity was diminished until the air plasma was solely visible in one or two of the nine frames and therefore marking the peak of the laser pulse with an accuracy of ±222 fs. It should be noted that the probe beam is coupled out in the vacuum chamber and has thus also been in air for the determination of the zero timing.The frame rate of the camera is set by the temporal spacing of the spectral images and the chirp of the probe pulse. Therefore the group velocity dispersion has to be calculated including complete knowledge of the stretching material in the probe beam and respective material characteristics. Since multiple error sources are introduced, we chose a different approach. Two nine-frame images are created, differing in delay by 1000 steps of the motor of the delay stage. If a specific plasma size is observed in two different frames of both nine-frame images, one can check for the change in the frame number and unambiguously correlate frame-number and temporal delay. Knowing that 1000 steps equal a delay of 666 fs the frame rate can be simply calculated. In the setup described above, a 3 cm glass rod resulted in a delay of 222 fs between each frame and a total observation time of 2 ps within a single-shot.The threshold intensity was determined by focusing the main laser pulse onto a 200 nm thick plastic foil with a peak intensity that was calculated via equation −2, whereas a clear hole was visible after a shot with I6. We thus, supported by the extensions of the hole in the target after one shot with I6, determined the threshold intensity of the 200 nm plastic target to be −2. We further note, that we empirically found that the intensity, needed to create an overdense and thus opaque plasma, also caused a hole in the target. The laser-induced damage threshold of different materials has been discussed widely over the last decades. The correlation between the conduction band (free) electron density reaching the critical density and damage due to ablation, is broadly accepted52. However, newer research indicates that higher electron densities are required until damage occures54. Because we only use the transition from transparent to opaque for determining the contours, which happens for any free electron density ne ≥ nc, our method is not sensitive to the exact damage mechanism. The threshold intensity −2 should therefore not be associated with a damage threshold. Note, that the large error bar of Ith results from the course intensity filtering steps.


We theoretically investigate magneto-optical rotation (MOR) of a linearly polarized probe field in the four-level N-type cold atoms. By applying a static magnetic field and a weak coupling field, it is shown that the birefringence enhancement is induced in the system. Moreover, we show that the static magnetic field has a major role in switching the dichroism to enhanced birefringence in the system. We also obtain a large intensity for the output field with nearly perpendicular MOR angle by 88 degrees with subnatural width. It is demonstrated that Doppler broadening has a destructive effect on the MOR of the polarization direction of the probe field. The results of our study can be used for selecting narrow band of wavelengths and polarization converter for efficient switching of TM/TE polarization modes in optical communication, the depolarization backscattering lidar, polarization spectroscopy and precision measurements. Faraday and Voigt effects depend respectively on the longitudinal and transverse orientation of the external magnetic field with respect to the propagation direction of the light. In MOR, the optical fields interact with atoms in the presence of an externally static magnetic field. Agarwal et al. have reported the possibility of the control of MOR in atomic system using a coherent coupling field and the static magnetic3. The effect of temperature of the atomic system on MOR has been studied in multiple three-level electromagnetically induced transparency subsystems and the maximum rotation angle have been reported 45 degrees at T = 65 °C4. The coherent control of optical polarization has been studied in GaAs quantum well nanostructures5, metamaterials6, single- and multilayer graphene7, mesogenic organic molecules8, diamond nitrogen vacancy centers9 and cavity QED10. Moreover, several experimental studies have been reported in generation and control of the MOR15.It is well known that the polarization plane of a linearly polarized field is rotated passing through an anisotropic medium. In fact, asymmetry between refractive indices of the circular components of the linear field is the cornerstone of the rotation of the polarization plane. When a medium is subjected to a magnetic field, Zeeman splitting of the magnetic degenerate sub-levels is the responsible for the asymmetry leading to the Faraday rotation16 in magnetometry18 atomic clocks19, optical limitation21, optical atomic filters based on birefringence23 and dichroism24. One of the interesting applications of the MOR is magnetometry. The output of nonlinear MOR is a signal that depends on both light and static magnetic field. It allows us to accurate measurements of non-zero magnetic fields25. Petrosyan et al. proposed an experimentally feasible setup that can induce large MOR and therefore be used for ultrasensitive optical magnetometry26. The effect of the intensity and atomic density on the MOR have been reported in a three-level lambda-type system27. Another mechanism in removing degeneracy and making asymmetry is based on the AC-stark shift28. It was shown that a circularly polarized control light plays the role of a static magnetic field via a Zeeman-like ac Stark interaction and induces the optical Faraday rotation of the probe light29. Recently, the polarization rotation with the subnatural width, in the absence of the static magnetic field, has been reported at room temperature and the several 10−3 rad rotation has been measured in the hyperfine transition of the Rb vapor cell30. More recently, the effect of microwave field on the nonlinear optically induced Faraday rotation was presented and it was demonstrated that the optical activity due to the enhanced birefringence can be generated in a wide range of frequency31.Numerous studies have reported that manipulating of the polarization of a field has a wide variety of applicationIn this paper, MOR of a linearly polarized field is theoretically investigated passing through the four-level N-type cold atoms in the presence of a static magnetic field. It is shown that using a weak coupling field and a static magnetic field, a birefringence enhancement is induced in the system. We show that the linearly polarized probe field is transmitted through the medium by 0.94 of the intensity of the input field while its polarization direction is rotated up to 88 degrees due to the large birefringence induced in a narrow subnatural linewidth. We also find that the generation and control of the dichroism or birefringence in the system depend on the applied static magnetic field. Finally, the effect of the Doppler broadening is studied on the MOR and it is observed that increase of the Doppler width has a destructive role in MOR and reduces dramatically the intensity of the output field. Despite the decrease in MOR by increasing the temperature, it is reported a MOR angle by 17 degrees at room temperature. Our obtained results can be used for designing a monochromator as well as in the TM/TE polarization converters in optical communication, the depolarization backscattering lidar, polarization spectroscopy and precision measurements.D2 lines, including the transitions 87Rb atoms in a vapor medium. Two states F and F′ are the quantum numbers of the total angular momentum and mF denotes magnetic quantum number of the corresponding states. An externally static magnetic field B. On the Faraday geometry, a strong linearly polarized probe field μB is Bohr magneton, gs is Lande’ factor and 26. Simultaneously, a static magnetic field is applied to the medium parallel to the probe field. The transmitted probe field is then passed through a polarization analyzer cube (P2). Finally, the transmitted probe fields with different polarizations are detected by PhotoDiodes PD1 and PD2 from both channels of P2. It allows simultaneous measurements of the polarization rotation angle and the transmitted fields intensities.The considered system is a four-level N-type system shown schematically in Fig. N-type system can be written asUnder the dipole and the rotating wave approximation, the Hamiltonian describing system in the interaction picture is:S± are the normalized susceptibilities defined as:αl = 4πkplμ2N/ℏγ is the field absorption at resonance in which l and kp are the length of the atomic medium and probe field wave number respectively. Moreover, N indicates electron number density in the medium and S± represent the dispersion and absorption of the right- and left- circular component of the probe field. As mentioned above, it is assumed that the polarization direction of the input probe field is in Ty (Tx) is given by3:Here, entclass1pt{minimaS± leads to difference between the dispersions (absorptions) of the right- and left- circular components of the linearly polarized probe field making the medium to show birefringence (dichroism) behavior. It is noteworthy that to obtain the lossless MOR, the birefringence is preferred with respect to the dichroism. In current study, we try to enhance the birefringence and reduce the dichroism in the medium. Here, we are going to determine the suitable conditions for birefringence enhancement in the system. An interesting case takes place when β is positive and β in which the propagating probe field experiences a gain, leading to amplification of the rotated probe field.Rotation of the polarization plane of the probe field passing through a medium can be resulted in either birefringence or dichroism induced in the system. Difference between the real (imaginary) part of the normalized susceptibilities quations  and 6 arγ which is taken as D2 transition of 87Rb. Furthermore, it is supposed that S+ (dotted), S− (dashed) and their difference (solid) are plotted versus detuning of the probe field Δp. The other parameters are S+ and S− are maximally different in magnitude and different in sign. Difference in dispersions of the right- and left- circular components of the probe field, accompanied by nearly zero absorption, implies a birefringence enhancement in the system at p. It should be noted that Ty is the intensity of the output field with rotated polarization direction, while Tx is the intensity of the output field in input polarization direction. Figure Ty enhances around Tx attenuates extremely at In this section, we study MOR of the polarization direction of a linearly polarized probe field crossing through a medium by numerically solving the equation  in the sentclass1pt{minimaS+ (dotted), S− (dashed) and their difference (solid) are plotted versus ΔB. The other parameters are considered as S+ and S− is different. But the situation is different in the presence of the static magnetic field and their difference gradually decreases by increasing the static magnetic field. For the region of higher Zeeman splitting i.e., In Fig. B. It is seen from Fig. Ty enhances while Tx attenuates, leading to an enhancement in MOR angle as shown in Fig. Ty , at resonance and weak coupling approximation, are given byHere, we are going to introduce an analytical expression to the physical interpretations of our numerical results. The normalized susceptibilities v/c denotes propagation vector of the probe and weak coupling field with the same (opposite) direction of the atom motion. Throughout the results, we suppose that v, v and v in equation (S+ and S− are averaged over the Maxwell-Boltzmann distribution for velocities and are given bym is mass of moving atom, T is temperature of the cell and kB is Boltzmann constant.Finally, we are interested in taking into account the effect of Doppler broadening on the MOR in the system. Through the numerical calculations, the effect of the Doppler broadening was ignored. In a sample, atoms may move with velocity entclass1pt{minima87Rb vapor at room temperature. The other parameters are the same taken in Fig. Ty is extremely suppressed while Tx is enhanced at In Fig. entclass1pt{minimaThe lower row of the Fig. N-type cold atoms. It was shown that in the presence of a static magnetic field and using a weak coupling field, a birefringence enhancement is induced in the system. We have illustrated that by applying the static magnetic field the dichroism switches to the enhanced birefringence in the system. Then, a higher intensity of the output field by 0.94 with roughly perpendicular rotated polarization direction by 88 degrees has been obtained with subnatural width. It was shown that the Doppler broadening has a destructive effect on the MOR so that MOR angle is reduced to 17 degrees at room temperature. The results can be used for designing a monochromator to select a wavelength with subnatural band width. Moreover, it can be used as a polarization converter for efficient switching of TM/TE polarization modes in optical communications.In summary, we have theoretically investigated the MOR of a linearly polarized probe field passing through the four-level


This allowed us to avoid confounds and biases involved in relying on observers’ response , which contaminated previous measurements of this window. We presented letters to the right and left of fixation, each surrounded by task-irrelevant disks with varying distances. The disks were bright on one side and dark on the other. A central cue indicated which letter to attend. Luminance levels were identical across trials. We found that pupil size was modulated by the disks’ luminance when they were 1° away from the attended letter, but not when this distance was larger. This suggests that the diameter of the attentional window is at least 2°, which is twice as large as that established with behavioral measurements.This study measured the size of the attentional window when attention is narrowly focused, using attentional modulation of the pupillary light response – pupillary constriction when  Others showed that the size of the attentional window can be voluntarily modulated3. Additionally, it was suggested that the size of the attentional window plays a critical role in attentional capture, as attention can only be captured by stimuli appearing within the attentional window5. Eriksen and Hoffman6 further suggested that the size of the attentional window can be narrowed down to a diameter of about 1° of visual angle . Critically, most of the previous studies explored the characteristics of the attentional window using behavioral measurements, particularly reaction time (RT). For example, Eriksen and Hoffman6 presented a target letter flanked on each of its sides by two irrelevant letters (distractors). Target and distractors were chosen from the same set of four letters , and target identification RT was measured under three target-distractor distances: 0.53°, 1.0°, and 1.4°. They found no significant difference between the second and third distance conditions, but RT was significantly slower with the smallest distance. This led Eriksen and Hoffman to conclude that only with the distance of 0.53° the distractors were included within the attentional window. However, behavioral measurements could be affected by various factors that are not directly related to the size of the attentional window, such as higher-level strategies, decisional criteria, experience, learning, speed accuracy tradeoffs, response biases, etc. This is particularly so with RT, as it also involves factors related to motor preparation. The goal of the current study, therefore, was to estimate the minimal size of the attentional window , with a measurement that is independent of performance – the pupillary light response (PLR).Spatial covert attention refers to processes that prioritize information gathered from a specific location, in the absence of eye movements. The attentional window is defined as a restricted region in our visual field to which spatial attention is deployed. Many studies examined the characteristics of the attentional window. For instance, several studies suggested that the size of the attentional window varies with task demands7. For many years the PLR was considered a reflex that is mediated by low level, subcortical, mechanisms9. Interestingly, recent studies demonstrated that spatial covert attention can also affect the PLR. Under identical overall luminance conditions, covertly shifting spatial attention to a brighter area produces pupillary constriction relative to shifting covert attention to a darker area14. For instance, Binda et al.10 presented two disks, each with a dot at its center, against a uniform gray background. One disk was presented to the right and the other to the left of a central fixation mark. Additionally, one disk was bright and the other one was dark. A central cue instructed observers to covertly attend one of the disks while fixating the central fixation mark. The task was to count the number of times the dot on the attended disk changed its color. The results showed that pupil size depended on where spatial covert attention was focused: when the bright disk was attended pupil size was smaller relative to when the dark disk was attended. Similar results were found by Mathôt et al.11 and further extended to involuntary allocation of covert spatial attention12. We used these attentional modulations of the PLR to assess the spatial extent of attention without having to rely on performance. Specifically, we employed a task requiring the participants to focus their attention narrowly on a target, and added to the display task-irrelevant stimuli of different levels of luminance. The irrelevant stimuli were presented at varying distances from the target. As long as these distances were small enough so that the irrelevant stimuli fell within the attentional window, we expected to find a brightness effect: smaller pupil size when covert attention is directed to a target surrounded by bright than dark irrelevant stimuli. Once the irrelevant stimuli appear at a distance larger than the attentional window, their luminance should not affect pupil size. This allowed us to estimate the size of the attentional window.It is well known that the pupils change their size in response to light – they constrict when light levels are high and dilate when light levels are lowWe conducted two such experiments, in which we presented two squares to the right and left of the fixation Fig. . Each sqTwelve observers participated in this experiment. All observers were students from the University of Haifa, with normal vision and without diagnosis of attention deficit disorder. All observers signed an informed consent form. Observers were naive to the purpose of the study and were either paid or received course credit for participating. This study adhered to the Declaration of Helsinki, and was approved by the ethics committee of the University of Haifa.15 in MATLAB on a 19-in. monitor of an IBM-compatible PC . Pupil diameter and eye movements were recorded from the right eye with an EyeLink 1000 eye tracker . Two squares  were presented at an eccentricity of 6.75° to the right and left of a central square fixation mark (1° × 1°). An arrow (0.7° × 0.7°) pointing to the right or the left appeared inside the central square, and a rotating letter T (1° × 1°) was displayed inside each of the other squares against a static random dot noise that filled each square . The Ts could be oriented upright, inverted, 90° to the right or to the left, and they changed their orientation randomly and independently. The luminance of the Ts was set individually for each observer to avoid performance that is too high or too low . This was established based on the practice session. If performance was either very high or very low in the practice session the experimenter changed the luminance of the T in the experimental session accordingly to make the task harder or easier . The contours of all squares and the arrow were colored in isoluminant red (24.5 cd/m2), and the background was a uniform gray (24.5 cd/m2). A group of four disks surrounded each T with varying inter-stimuli distances. The distances from the center of each T to the inner edge of the disks that appeared above and below the T were 1°, 4°, 7° and 11° of visual angle. The distances of the disks that appeared diagonally above and below each T were 1.5°, 4.5°, 7.5° and 11.5°. Each disk had a diameter of 1°. The disks were bright (90 cd/m2) on one side of the screen and dark (0.04 cd/m2) on the other side.Stimuli were presented using the Psychophysics Toolbox extensionsAt the beginning of the session, observers were instructed to look only at the center and avoid blinking throughout each trial, but were encouraged to take short breaks between trials and long breaks between blocks. Then, a 9 points calibration was conducted. Each trial was preceded by a drift check to ensure the accuracy of the calibration is maintained and that the participant looks at the center. A trial started with the central fixation mark and the squares. After 1 sec, the arrow appeared inside the central square indicating the side to which the observers have to attend, and it was presented till the end of the trial. After 2 sec, the disks and rotating Ts appeared simultaneously. The Ts changed their orientation every 500 ms, and were visible for 6 sec. The target was the letter T on the attended side, and the task was to count the number of times the attended target assumed an upright orientation, while looking at the center of the fixation mark. Following a response, an auditory feedback indicated whether or not the response was correct. The distance between the target and disks was constant within a block and varied between blocks. The experiment included 448 experimental trials. For 10 out of the 12 participants these 448 experimental trials were divided into 4 sessions of 112 trials. These sessions were conducted on different days to minimize fatigue or at the same day if the participant could rest for at least two hours between sessions. In each session there were two blocks, each of a different target-disks distance. The order of distances was random, but to avoid order effects each distance appeared once as the first block of a session and once as the second block of a session. For 2 observers the 448 experimental trials were divided into 2 sessions of 224 trials, which included 4 blocks each. For all 12 participants, each experimental session was preceded by a practice session that included 20 practice trials. These practice trials did not include the disks in order to motivate the observers to concentrate only on the task. In addition, before the beginning of each experimental block, there were 5 practice trials that were identical to those of the upcoming experimental block. The side at which the bright (dark) disks were displayed, and the side to which the arrow pointed were chosen randomly for each trial and were counterbalanced across the entire experiment.10 . At first, blinks events were linearly interpolated17. Then, the sampling points of pupil size were averaged within 50 ms time bins. The baseline was the average pupil size of the 100 ms preceding the onset of the disks and Ts. For each trial, the average pupil size of each bin was divided by the baseline value. We performed a linear mixed effects analyses (LME) for each bin, using R’s lme4 package19. The variable of pupil size was entered into the model as the dependent measure and the variable of subjects served as a random effect. To test for a difference between the disk brightness conditions , as well as the effect of target-disks distance, the variables of brightness and target-disks distance as well as their interaction were entered to the model as fixed effects. Additionally, to examine in more details the relationship among these two variables, we conducted a separate analysis for each target-disks distance with disks brightness at attended side as a fixed effect. Similar to previous studies, we defined the significance criterion as t > 2 for at least 200 ms time interval21.We included in the analyses only trials in which fixation was not broken more than 1° and pupil size ranged between 2 to 8 mmhttps://osf.io/tg5n9/?view_only=3039239e39104ee292700b5cdf022f8c.The data that support the findings of this study are available in the Open Science Framework repository, 23. In addition, with all four distances there was a gradual increase in pupil size as the trial advanced, regardless of disks brightness. Such gradual increase was also demonstrated before, and it is probably due to cognitive effort associated with the task25. More important for the goal of this study, we found a significant interaction between brightness and distance. Further analyses performed for each distance separately revealed a significant brightness effect with the smallest distance of 1°: Pupil size was smaller when the attended target was surrounded by bright than dark disks. Critically, this difference was not significant at any time along the trial with the other larger distances. Hence, the results suggest that disks that were presented at a distance of 1° from the target fell within the attentional window, whereas disks that were presented at a distance of 4° or larger fell outside the attentional window.The results are presented in Fig. To establish a more accurate estimation of the attentional window, we performed Experiment 2, which was similar to Experiment 1, but examined the attentional effect on the PLR with a finer scale of distances .Twelve observers participated in this experiment, of which 6 also participated in Experiment 1. All observers were students from the University of Haifa, with normal vision and without diagnosis of attention deficit disorder. All observers signed an informed consent form. Observers were naive to the purpose of the study and were either paid or received course credit for participating. This study adhered to the Declaration of Helsinki, and was approved by the ethics committee of the University of Haifa.2 in Experiment 2 with a mode of 19 cd/m2. The distances from the center of each T to the inner edge of the disks that appeared above and below the T were 1°, 2°, 3° and 4°, and from the diagonally upper and lower disks were 1.5°, 2.5°, 3.5° and 4.5°. The presentation duration of the Ts was shortened to 4 sec. This was motivated by the fact that no effects of interest emerged at the last 2 sec of the trials in Experiment 1. With all the participants, the 448 experimental trials were divided into 4 sessions of 112 trials, as described for Experiment 1.The stimuli, apparatus and procedure were similar to Experiment 1 except for the following. The luminance of the Ts ranged between 15 to 21 cd/mStatistical analyses were identical to those of Experiment 1 . A similar significant brightness × distance interaction emerged Fig. : With th6. Experiment 2 further shows that disks that where presented at a distance of 2° or more, fell outside the attentional window, suggesting that the diameter of the attentional window is smaller than 4°. It is important to note that the distribution of attention is likely not uniform but rather falls gradually from the center of the attentional focus27, and that there is evidence for individual differences in the spreading of attention29. Such individual differences and general noise in the data may deem impractical an attempt to establish a finer estimation of the size of the attentional window, at least when looking for the minimal size.The results of both experiments demonstrate that disks that where presented at a distance of 1° from the target fell within the attentional window, even though they were task-irrelevant. If the attentional window is viewed as a round window centered at the target, then our findings propose that the diameter of this window is at least 2°. This suggests that the minimal size of the attentional window is twice as large as that found with behavioral measurements33. The typical way by which these studies measured the size of the attentional window was to measure distractor interference as a function of target-distractor distance; when distracting stimuli influenced performance it was concluded that they fell inside the attentional window33. However, this conclusion was debatable, because a large body of evidence suggests that unattended visual stimuli are also processed and can affect performance37. Hence, the mere fact that performance was affected by distracting stimuli does not necessarily mean that these stimuli fell within the attentional window. Likewise, the lack of response interference may not necessarily mean that the distracting information fell outside the attentional window. For example, Merikle and Gorewich38 suggested an alternative critical factor –the size of the distractors, or more specifically whether or not they meet the limitations of visual acuity. They presented a color name above and below a central color patch with varying distances: 0.5° or 2.5°. The color names were either incompatible with respect to the color patch  or neutral (a string of the letter ‘l’: “1111”), and they could be small (0.24°) or large (0.57°). Participants had to rapidly identify the color patch while ignoring the color names. The results revealed that when the target-distractors distance was 0.5°, distractor interference emerged for both large and small distractors: RT was significantly slower in the incompatible than neutral condition. However, when the target-distractors distance was 2.5° an interference effect was found only with the large distractors. These findings suggest that the lack of interference effect with the small distractors at the large distance was not necessarily due to the fact that they were presented outside the attentional window. More likely, the lack of effect was due to the fact that at this distance from the center the color names were too small to be resolved, and therefore too small to generate response interference. Likewise, Gatti and Egeth39 employed a similar paradigm and found that the interference generated by incompatible color names decreased as the distance between the color patch and color names increased. This led them to conclude that visual acuity plays an important role in distractor interference, and may compromise measurement of the size of the attentional window. Visual acuity limitations are not a problem in our study because we employed relatively large distractors  and there were four such distractors around the target. Moreover, because our method for measuring the size of the attentional window does not rely on response interference generated by distractors with conflicting semantic meaning, our measurements are not affected by the ability/inability to resolve the distractors’ fine details.A substantial part of our knowledge about spatial covert attention was established in studies that inferred attention allocation based on differences in performance. This is especially so regarding experiments that attempted to measure the size of the attentional window35 concluded that: “…complete elimination of unattended stimuli from semantic processing is a rare occurrence, probably requiring a very special combination of circumstances”. (p. 286). Moreover, Driver and Tipper36 showed that both distractors that generated response interference and those that did not, produced equivalent negative priming, indicating that both interfering and noninterfering distractors were processed and represented. This finding suggests that the presence or absence of distractor interference cannot serve as an indication of attention allocation or processing level (see Fox37 for similar conclusion). The current study overcomes this problem by not relying on differences in performance but on modulations of pupil size. In our study, the disks were neither compatible nor incompatible with the required response, and they were always bright on one side and dark on the other side, so the overall luminance level was always the same and it always affected the PLR. Critically, we tested effects on PLR beyond the overall luminance level – effects that are specific to covert attentional modulations13. If the disks were displayed close enough to the focus of attention so that they fell inside the attentional window  then pupil size was modulated by their brightness. Therefore, the measurement of the PLR seems more informative for our current goal than differences in performance. Additionally, contrary to performance, pupil size does not appear to be under a direct volitional control7 and therefore could serve as an uncontaminated measurement of attentional effects.As mentioned above, even when distractor interference is found, this by itself does not necessarily mean that the distractors were attended, because unattended stimuli are also processed. Indeed, after conducting an impressive serious of experiments that look at different factors that may affect the processing of the distractors, including their distance from the target, Miller26 demonstrated that the characteristics of the attentional window vary as a function of perceptual load; LaBerge1 demonstrated that the size of the attentional window varies as a function of task demands; and a study by Sagi and Julesz31 showed that the size of the window changes as a function of eccentricity. Previous studies who examined the effects of such factors on the size of the attentional window employed only behavioral measurements. Given the limitations of the behavioral measurements detailed above, our understanding of the characteristics of the attentional window would greatly benefit from future studies that will employ the PLR to carefully examine the various factors that mediate the size of the attentional window.Prior research of the attentional window suggests that its size, at a given moment, depends on several different factors. For instance, Caparos and Linnell40 and the attentional attraction field (AAF) model41, a parameter that represents the attentional spread plays an important role. Such models have to ensure that the value they assign to this parameter, when attempting to simulate performance, comply with the minimal size found here. Finally, our findings are important for cross-validation of neurophysiological studies that attempt to examine the characteristics of the attentional window using functional neuroimaging data44.Taken together, this study has important methodological, theoretical and applied implications. In terms of research methods, this study demonstrates the effectiveness of PLR examination as a method to study the nature of spatial attention – a method that is not contaminated by all the above-mentioned factors that are not directly related to attention. Moreover, the results of this study are important for cases in which it is essential to know the exact size of the attentional window in order to ensure that stimuli are falling/not falling within the attentional window. This is critical for the design of experiments that wish to control which information is attended/unattended at any given moment. Additionally, from a more applied point of view, it is also important for the design of optimal visual displays of any form , who wish to minimize clutter. Furthermore, our study has important theoretical implications for the development of physiological and computational models of spatial attention, because the minimal size of the attentional window poses a fundamental constrain on such models. For instance, neurophysiological models of attention have to ensure that the neural substrate suggested to underlie spatial attention can support the minimal size of the attentional window. Similarly, in several computational models, such as the normalization model of attentionTo sum, using the attentional modulation of the PLR we found that the minimal size of the attentional window when attention is narrowly focused has a diameter of about 2°, which is twice the size found when measured based on performance differences.


The process of rumor spreading among people can be represented as information diffusion in social network. The scale of rumor spread changes greatly depending on starting nodes. If we can select nodes that contribute to large-scale diffusion, the nodes are expected to be important for viral marketing. Given a network and the size of the starting nodes, the problem of selecting nodes for maximizing information diffusion is called influence maximization problem.We propose three new approximation methods  for influence maximization problem in dynamic networks. These methods are the extensions of previous methods for static networks to dynamic networks.When compared with the previous methods, MC Greedy and Osawa, our proposed methods were found better than the previous methods: Although the performance of MC greedy was better than the three methods, it was computationally expensive and intractable for large-scale networks. The computational time of our proposed methods was more than 10 times faster than MC greedy, so they can be computed in realistic time even for large-scale dynamic networks. When compared with Osawa, the performances of these three methods were almost the same as Osawa, but they were approximately 7.8 times faster than Osawa.Based on these facts, the proposed methods are suitable for influence maximization in dynamic networks. Finding the strategies of choosing a suitable method for a given dynamic network is practically important. It is a challenging open question and is left for our future work. The problem of adjusting the parameters for Dynamic CI and Dynamic RIS is also left for our future work. Diffusion of rumors (or information) can be represented as information propagation in a social network where its nodes are people and its edges are contacts among the people. The scale of information propagation depends on where to start the propagation. In order to propagate as much as possible, starting nodes should be carefully selected. Selecting starting nodes for large-scale information propagation is important as one of the methods for viral marketing.From given network, selecting such starting nodes for large-scale information propagation was formalized as “influence maximization problem” by Kempe et al. . The oriIn order to find better solutions for the information maximization problem, we propose three new methods for dynamic networks as the extension of the methods for static networks. Dynamic Degree Discount is a heuristic method based on node degree. Dynamic CI is a method based on a node’s degree and the degrees of reachable nodes from the node within specific time. Dynamic RIS uses many similar networks generated by random edge removal. We compare the proposed methods with previous methods. The number of propagated nodes based on our method is about 1.5 times of that of previous methods. And computational time of our method is about 7.8 times faster than previous methods.The authors discuss the extended methods for influence maximization in dynamic networks . In addiThe structure of this paper is as follows. “t, u, v) at time u is in state I and node v in state S, information is propagated from u to v with probability v is changed from S to I at time We use SI model as the model of information propagation on networks. In SI model, each node in networks is either in state S (susceptible) or in state I (infected). Nodes in state S do not know the information and those in state I know the information. At the beginning of information propagation (at time S of state I based on SI model. (Please keep in mind that S in S in SI model is susceptible state.) Influence maximization problem in a dynamic network is to search for a set of seed nodes S of size k that maximizes G, duration of the network T, susceptibility of SI model k are given.Based on the above notations, we can formulate influence maximization problem as follows. We define There are some problems related to influence maximization in dynamic networks. Instead of giving item (or information) to seed nodes for free, revenue maximization  is the pOpinion formation –7 is anoIt is often pointed out that the properties of dynamic networks are quite different from those in static networks. Braha and Bar-Yam , 10 poinJalili presents a survey on spreading dynamics of rumor and disease based on centrality . There aS is given as a set of seed nodes, simulations of information propagation are repeated R times and the average number of infected nodes is defined as v which maximizes the difference Monte-Carlo simulation method is proposed by Kempe et al. . \documeSince mentclass2pt{minimv is selected as one of seed nodes and u is its neighbor, it is highly likely that v propagates information to u, so selecting nodes other than u as seed nodes is better for information diffusion.Heuristic methods are proposed in order to search for seed nodes at high speed. Chen et al.  proposesn et al.  proposedi. i after giving penalty. Algorithm of Degree Discount is shown as follows. v is l. The example of v and the degrees of both nodes are 8. Therefore, Morone et al.  proposedmentclass2pt{minimv itself is low in the network in Fig. v is effective for information propagation because it is connected with some high degree nodes with distance two. This method thus selects seed nodes with wider propagation compared with the cases when seed nodes are selected based on the degree of node v only.The degree of node These heuristic methods compute seed nodes faster than the methods based on Monte-Carlo simulation. However, it is experimentally confirmed that the scale of propagation of the methods depends on network structures and parameters.S in static network G is set as S on network where edges are removed at constant ratio from the network G is set as RR numbers of networks with edges removed at constant ratio. SCC is a subgraph where each node in the subgraph can be reachable to and from any other nodes.Ohsaka et al.  proposedv are computed, and then seed nodes are selected based on the nodes. More specifically, the algorithm is as follows. Borgs et al.  and TangThere are other approaches for influence maximization problem in different problem settings. Chen et al.  proposedt, u, v) represents an edge from node u to v at time t. A path from node Notations of edges and paths in dynamic networks are the same as the ones in ref. .  is a collection of nodes adjacent to node v at time t. Figures A do not change during the period. The difference of adjacent nodes A in Fig. A change over time. So the degree of node A is bigger than that in Fig. A is the same every time, so the average degree of node A is the same in Figs. Habita et al.  define dentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaentclass1pt{minimaS is shown as follows. There are two approaches for influence maximization problem in dynamic networks: methods based on Monte-Carlo simulation and heuristic-based methods. The former method is proposed by Habiba and Berger-Wolf . The metd Murata  proposedAfter G: dynamic network, T: duration of the dynamic network, k: the size of seed nodes, S: seed nodes.We propose new methods for influence maximization problem in dynamic networks in this section. We propose three new methods  which are the extensions of static network methods to dynamic network methods. We use the following notations: Dynamic Degree Discount is the extension of Degree Discount by Chen et al.  to dynamv and the degrees of nodes with distance l from v. Dynamic CI defines an index DBall. DBall represents nodes where their shortest duration of time  in our algorithm. RR is a set of all nodes that are reachable to v within the shortest duration of time d in all durations of dynamic networks, which is defined as follows:v at time t” within the shortest period of d.The difference between Borgs’s and Tang’s algorithm and Dynamic RIS is where The computational complexities of these methods are as follows.k is the number of seed nodes, n is the number of nodes, and m is the number of edges, respectively. Dynamic Degree Discount is an extension of Degree Discount. Static degree is replaced with dynamic one , respectively. Therefore, the total computational complexity of Dynamic RIS is According to the paper of Tang et al. , the comhttp://www.sociopatterns.org).We perform experiments for comparing the proposed methods with previous ones in order to confirm their effectiveness. Dynamic networks used for the experiments are shown in Table Hospital  is a netHospital , 29 is aHospital  is a netS. The simulation of influence maximization based on SI model is repeated R times with the obtained seed nodes and set the average of the number of nodes in state I as Methods used in the experiments are previous two methods (Monte-Carlo simulation (MC Greedy) and Osawa) for dynamic network explained in “k changes.Comparison of k changes.Comparison of computational time when the size of seed nodes Parameters in the experiments are set as follows. The number of repetition of the simulations for information propagation is set as k is set from 0 to 20%. Susceptibility l in Dynamic CI which takes the value of d in Dynamic RIS, d, values l of Dynamic CI.Experiments are performed for the following purposes:CELF  is used k with fixed susceptible x axis of the Figure shows the percentage of seed nodes, and the y axis shows the number of infected nodes. Values of the x axis is y axis is l in Dynamic CI and d in Dynamic RIS are used in our experiments. As shown in Fig. The results of information propagation for each size of seed nodes There is not much difference in the scale of diffusion among each of the three proposed methods. Dynamic RIS achieves the highest in High School 2013, for example, but the difference among proposed methods is small compared with the difference between proposed methods and previous methods (MC Greedy and Osawa).x axis shows the value of y axis shows the percentage of diffusion. Parameters l and d are the same as the ones used in the previous experiments. As shown in Fig. Figure As the result of comparison with proposed methods and Osawa, our proposed methods achieve higher scale of diffusion than Osawa in Hospital and High School 2013 when X axis shows the percentage of seed nodes, and y axis shows the computational time .Figure Figure Regarding the comparison among three proposed algorithm, computational time of Dynamic Degree Discount and Dynamic CI is almost the same in all datasets. Dynamic RIS is about the same computational time as the other two proposed methods in Hospital, and is faster in Primary School and High School 2013. Regarding the comparison with proposed methods and Osawa, Dynamic RIS is approximately 7.8 times faster than Osawa except very small network .l of Dynamic CI, and d in Dynamic RIS.Diffusions of proposed methods with different parameters are shown in this section. We change parameters l in Dynamic CI changes to 1, 5, 10, 20 are shown in Fig. l is changed in each network. Right bar graphs show computational time. Left line graphs show that diffusion depends on the value of l. Therefore, it is important to find appropriate l in Dynamic CI. Since there is no simple correlation between the scale of diffusion and the value of l (such as diffusion becomes larger as l becomes large), diffusions for various values of l should be investigated and compared. Right bar graphs show that there are no big differences of execution time when the value of l changes.Diffusion and computational time when RR. Diffusion and computational time when parameter d in Dynamic RIS is a parameter for the number of time steps for looking back. Figure d changes to 0, 5, 10, 20. Left line graphs show that there is almost no difference in diffusion when d changes, while right bar graphs show that computational time increases as the value of d becomes bigger. The scale of diffusion does not change even if the value of d becomes bigger in our experiments.In the experiments when susceptibility changes in “v when Monte-Carlo simulation is used. X axis shows the percentage of diffusion from node v to the whole network .It requires two parameters to be adjusted (Finding the strategies of choosing suitable method for given dynamic network is practically important. It is a challenging open question and is left for our future work.The problem of adjusting the parameters for Dynamic CI and Dynamic RIS is also left for our future work.


Single-molecule stretching experiments on DNA, RNA, and other biological macromolecules opened up the possibility of an impressive progress in many fields of life and medical sciences. The reliability of such experiments may be crucially limited by the possibility of determining the influence of the apparatus on the experimental outputs. Here we deduce a model that let us analytically evaluate such influence, fundamental for the interpretation of Single Molecule Force Spectroscopy experiments and intermolecular interactions phenomena. As we show, our model is coherent with previous numerical results and quantitively reproduce AFM experimental tests on titin macromolecules and P-selectin with variable probe stiffnesses. Fundamental phenomena such as DNA and RNA hairpins unfolding and refolding in enzymatic activity, sensing of metabolites, transcription termination and attenuation, morphogenic phenomena, cell motility, focal adhesion and so on cannot be described without a comprehension of mechanical fields effect and macromolecular and cells force-responseIn recent years the possibility of accurate experimental tests detecting protein and cells mechanical responses through Single Molecule Force Spectroscopy (SMFS) techniques has given a great impulse to understand the effects of mechanical forces in the natural environment. This scientific breakthrough has been achieved thanks to an increasing evolution of high-precision instruments such as Atomic Force Microscopes (AFM), optical tweezers, magnetic tweezers and microneedles.bumpy energy manifolds, with the possibility of analyzing the relative stability of locally stable configurations and topological properties of the entire energy landscape4. An important example of this possibility is observed in the folding and refolding phenomena of the RNA or DNA secondary structure underlying gene transcription. Indeed, in this case SMFS experiments show that unzipping propagates through a single stranded/unstranded front (di-block behavior)5 whereas this is not observed for temperature of chemically induced denaturation. Moreover SMFS experiments are fundamental to analyze the mechanical response of the many proteins that are designated to withstand or transmit forces such as cytoskeletal or muscle protein macromolecules.Differently from temperature or chemical denaturant based classical manipulation techniques, SMFS allows for the choice of specific trajectories in complex 7, these effects are often neglected in many theoretical approaches and underestimated in the experimental field. In particular, the two main aspects to be analyzed in this perspective are rate of loading and probe stiffness effects. Regarding the former, we point out that, schematically, two different behaviors can be observed9. At high rate of loading, a rate dependent regime is observed and Kramer’s type relations are adopted with unfolding forces growing  with the loading rate11. More recently, in13 the authors compare analytical results with numerical simulations based on Brownian dynamics and show that it is possible to recover the unfolding rates taking into account corrections from the measuring apparatus.One of the main drawbacks of SMFS is that the observed force-end to end distance diagrams are strongly influenced by unavoidable effects due to the pulling device. Although a significant effort, both from a theoretical and experimental perspective, has been pursued onkdv < 10 pN, where kd is the device stiffness and v the pulling rate8), when the relaxation rate to the energy minimum of the molecular chain is much higher than the external loading rate, the force is rate-independent and it is influenced only by the stiffness kd. This rate-independent regime characterizes many SFMS experiments as well as many unfolding14 and detaching ligand-receptor phenomena in physiological regimes8 and is the focus of this paper.At low rate of loading . To the knowledge of the authors, the first theoretical analysis of the device stiffness effect on SMFS was delivered in16. There, based on a Statistical Mechanics approach, the behavior of a chain of elements with convex potential energy in series with an harmonic springs with variable stiffness reproducing the probe effect is studied. Numerical examples to describe the experimental behavior of poly (ethylene glycol) molecules have also been performed.It is important to remark that, as reported 16. In the case of stiff cantilevers (hard device) the system can be described using the Helmholtz ensemble: the molecule is held at a fixed extension with the corresponding force representing a ‘fluctuating’ dual variable. In the opposite case of soft cantilever (soft device), the Gibbs ensemble is adopted with an assigned force acting on the molecule and a fluctuating end-to-end length. The two ensembles can be shown to be equivalent in the thermodynamical limit with the Helmholtz and Gibbs free energies correlated by a Legendre transform17. As we show in this paper, completely different experimental responses can be observed for the same macromolecule under such different boundary conditions. For example in the fundamental case of protein macromolecules constituted by two-states (folded/unfolded) modules, the modification of the probe stiffness can induce a very different change in the response of the chain: the behavior can range from force-plateaux to sawtooth force-elongation diagrams corresponding to a cooperative and a non cooperative transition, respectively19. This results also in important variations of the force inducing the dissociation of a molecular complex20, the unfolding of a protein21, or of higher-order protein structure22.As we show in this paper, even more important is the effect of probe stiffness when one considers molecules constituted by elements exhibiting two (or more) ‘stable’ configurations. Two ideal extreme cases can be considered, depending on the ratio between the cantilever and the molecule stiffness16 by considering a chain of elements with non convex, two wells potential energies in series with an harmonic spring representing the device stiffness. To fix the ideas, we associate a different shape chain configuration to each energy well such as in the case of protein macromolecules21. In this case the chain is constituted by (α-helices or β-sheets) folded elements undergoing a conformational (unfolding) transition due to the elongation imposed by the pulling device. On the other hand, our approach can be extended to other fields where non convex (free) energies have been successfully adopted such as force induced martensitic phase transformation in shape-memory alloys23.In this perspective, in this paper we extend the analysis in16 in the deduction of a model predicting the behavior of single polyethylene glycol chains AFM experiment. The analysis is more subtle when one considers local energy minimizers (metastable configurations) that can survive also in the thermodynamical limit24. The case of a chain with two-wells energy, in the limit of hard device, has been studied in25. In particular, in order to obtain an analytical expression of the partition function and analyze the thermodynamical limit, the authors use the approximation that we adopt here to extend the quadratic energy wells beyond the spinodal point. The transition to the soft regime has been studied using Monte Carlo numerical simulations in26. The importance of intermediate stiffness regimes is obvious: in real experiment one cannot expect a regime completely described by the hard or soft device limits. On the contrary, one would like to have a theory describing and predicting the behavior of the unfolding phenomenon in all intermediate regimes. In this paper we obtain explicit expressions for the partition function and force-strain relations that are valid in the whole range of values of the measuring device stiffness.We recall that the possibility of deducing the hard and soft cases as limit models for a system with variable probe stiffness, was numerically shown inInterestingly, despite the simplicity of the proposed model, the comparison with AFM pulling behavior on titin macromolecule and P-selectin single module unfolding at different probe stiffnesses demonstrate its effectiveness in describing the influence of device stiffness in the experimental response of real biological molecules with particularly accurate predictivity.It is important to remark the importance of analytical results regarding the mechanical interaction between an elastic link and a multidomain unfolding macromolecule. Indeed this is fundamental both in the perspective of bioinspired material design and in the important topic of evaluating the mechanical interaction between protein macromolecules and external elastic domains.N masses connected by bistable springs and a probe acting on the N-th oscillator through a spring with stiffness kd.Consider a SMFS experiment on a molecule constituted by bi-stable elements see Fig. . We modeL is the reference length of the molecule, εi is the strain of the i-th spring and φ its potential energy. Since the folded/unfolded transition of protein domains is typically an all or none transition, we assume for each domain a two wells energy χi is a ‘spin variable’ such that χi = 0 if the i-th spring is in the first (folded) state and χi = 1 if it is in the second (unfolded) state, with εu the reference strain of the second well. Observe that, to get analytical results, we approximate the energy wells by parabolic laws and that for simplicity of notation we assume that the spring stiffness kp is the same in the two states. We also assume that the transition (Maxwell) force of the springs is zero. The following analysis is easily extended to the case when different stiffnesses in the two wells and non zero Maxwell force18 are considered.The potential energy of the chain is The crucial aspect of our model is to consider the macromolecule and the measurement device as a single system  with a total potential energyVd is the energy of the measurement device, εd is the probe deformation and αL is its undeformed length. For small displacements (~150 nm) we can approximate the probe as a linear spring27. The potential energy clearly depends on the configuration assumed by the domains (folded or unfolded) described by the  variables {χ1, .. χN}. In the following, we sum over all configurations in order to obtain the canonical partition function of the system.Here d that is related to the strains by the following relation:We model the SMFS experiment by assigning the total displacement kd in the framework of equilibrium Statistical Mechanics28. In the following we first consider the zero-temperature case, especially interesting when rate-independent hysteresis is observed18, then we consider the important temperature effects.As already mentioned, we analyze the fundamental effect of different values of 19, we consider a simplified analysis, neglecting entropic effects and considering two classical approaches based on the ability of the system to cross energy barriers. In the first case, the so-called Maxwell convention, the system is able to overcome all the energy barriers so that the configurations of the system always correspond to the global energy minima. On the other hand, the presence of local minima is fundamental to analyze the emergence of hysteresis cycles. In order to describe this possibility, we here consider the opposite hypothesis, maximum delay convention, where the system is not able to overcome any energy barrier, so that it stays in a local energy minimum until it becomes unstable and switches to the nearest energy well19.The response of the considered system is regulated by its possibility to ‘explore’ the bumpy energy landscape. This, in turn, depends on three different time scales: the time scale of the external loading, the time scale of relaxation to the local minimizer and the time scale of exploring the whole energy landscape to relax to the global energy minimum. While, as we will show in the following section, temperature effects are crucial for the description of the system response (characterized by folded configurations where entropic effects play the major role), in this paragraph, followingF withAssume first that the configurations of the system are the global minima of the total potential energy. The equilibrium configurations are characterized by a constant force We then obtain the force-strain relationδ = d/L andmain non dimensional parameter of the paper measuring the relative stiffness of the probe vs the entire (chain plus probe) system. The ideal limits of ‘hard’ and ‘soft’ devices correspond to γ → 1 and γ → 0, respectively.By using –5 we theUsing  and 5)  5) it isObserve that all two-phase solutions with The fully folded state is defined for One can deduce that all these solutions are metastable  due to the local convexity of the energy of both springs and probe. The global minima of the energy can be evaluated using . The twoδ from the fully folded state then the chain starts to unfolds at the thresholdThe fully folded state represents the global minimum for δ decreases, the chain start to refold to go back to the primary folded state at the thresholdOn the other hand, if we let the system refold by relaxing the probe and let Observe that these values should be considered as incremental force from the transition force of the bistable spring, here assumed null for simplicity of analytical expressions.γ. Dashed lines represent the ‘jumps’ between different equilibrium branches under the Maxwell convention hypothesis that the system configurations correspond always to the global energy minima. Interestingly, as the device stiffness is decreased, not only the overall stiffness decreases (slope of the branches), but the force transition thresholds decrease in amplitude with an unavoidable influence of the pulling device stiffness.In Fig. 29.We remark that in the considered model we neglect non local interactions. As a result, the configurations with the same fraction of unfolded domains are energetically independent on the number of folded/unfolded interfaces. The introduction of non local energy terms allows to determine the role of the loading device in determining also the number of interfaces of the energy minimum configurationsγ defined in . On the other hand, in the following, we focus on the features of hysteresis in the rate-independent behavior which is dominant in the case of low temperature regimes24. Specifically, in the limit of zero temperature we assume, according with the so called maximum delay convention, that the system is not able to overcome any energy barrier.Let us show how the hysteresis phenomenon emerges in our model and its dependence on the relative stiffness of the probe with respect to the chain measured by the parameter fined in . As a geN = 5 in the cases of γ = 0.8 and γ = 0.3 . If the system is loaded starting from the fully folded state, it follows the corresponding equilibrium path (AB in the figure) until the configuration becomes unstable and the system starts unfolding along a ‘sawtooth’ transition path (BC in the figure). If the displacement is further increased, the system follows elastically the fully folded branch (CD in the figure). If then the system is unloaded, it remains in this configuration until it becomes unstable (DE in the figure) and the reverse (unfolded/folded) transition is attained along another sawtooth plateau (EF in the figure). In this case, each jump from one branch to the other (dashed lines in the figure) occurs when the value of the total rescaled displacement δ reaches the bound of the existence domain defined in  corresponds to a reduction of the size of the jumps and to a decrease of the slope  the Gaussian terms are ‘squeezed’ and contributions to ZN are relevant only for values of δ very close to p = 0, …, N.Detailed calculations (see SM) deliver the canonical partition function F applied to the system, conjugate to the displacement δ, is The free energy is defined as From Eq.  we thus n in Eq.  is16\doccentral result of the paper giving the force-strain relation at assigned temperature and relative stiffness γδ. This equation fully characterizes the material response of the macromolecule for the whole range of stiffness ratio γ.Finally, from Eqs  and 16)16) we obγ corresponding to different stiffnesses of the probe kd. As described previously, different values of the device stiffness alter quantitatively and qualitatively the macromolecule response. Thus the unfolding force decreases as kd decreases with the transition becoming cooperative. Observe that, as shown in the Fig. kd (i.e. softer devices) each unfolding transition corresponds to localized strain jumps. In the opposite case of hard devices the strain evolution is smoother.In Fig. N. As expected, the amplitude of the oscillations decreases as N grows. We notice that in one case (left figure) we have increased N keeping l fixed. On the other hand, in Fig. N keeping L = Nl fixed. As the figure shows, the effect of stiffness variation is particularly important for small systems whereas it becomes negligible in the thermodynamical system analyzed in the following paragraph.In Fig. 21, the values of the local maxima of F (corresponding in the experiment to the forces for the successive unfolding of β-sheets domains) are not constant, but increase with hardening effect, see Fig. p of unfolded domains is increased. Correspondingly the number of available configurations with p + 1 unfolded domains decreases. Interestingly, our tests show that the hardening effect is more evident for higher temperatures  is the minimum of the functionWe begin by searching for the asymptotic expression of the function 〉 in Eq. . Using tN = 100 and the thermodynamical limit case in Eqs . In the figure we have indicated 〈ΔF〉 because in our model the Maxwell stress of the two-wells potential energy is set to zero, but the average unfolding force of the proposed model can be easily calibrated on the experimental results by simply considering a non-zero Maxwell stress.In order to show the predictive properties of the proposed model, in Fig. F*〉 for dissociating P-selectin-PSGL-1 bond on the stiffness of the transducer, experimentally determined in34. To this scope, following35, we describe the protein rupture as a transformation of a system with a two-wells potential energy and second minimum ‘far’ from the first one, i.e. F*〉 with the maximum attained value of 〈F〉 as the rescaled total displacement δ is increased. The values of the parameters are taken from34 except εu (fixing the energy barrier) whose value is chosen in order to fit the experimental results. As the figure shows we obtain a linear trend quantitatively reproducing the experimental behavior.To further test our results against the experimental behavior, in Fig. 36. In this case, following the approach proposed in31, we neglect the entropic energy terms as in the zero-temperature approximation considered in Fig. 31 for analytical details. Notwithstanding the fact that one observes a deviation from the linear behavior of the force obtained from our model, the predictions about the values of the measured force and the location of the jumps are consistent. On the other hand, we remark that, differently from the theoretical two wells model and the experimental behavior, in this case the unfolding force decreases with the unfolded fraction. This is due to the neglect of the temperature effects. Moreover, the adoption of the WLC energy does not let us obtain analytical formulations regarding the partition function that is the main aim of this paper.We remark that in order to keep analytical simplicity we have assumed a parabolic behavior of the energy for folded and unfolded configurations of the domains. While the experimental comparison in previous subsection confirmed the ability of determining the influence of the external loading device on the unfolding thresholds, we may observe that our energetic assumption leads to a linear global force displacement diagrams. In Fig. β-sheets and α-helices unfolding). SFMS techniques constituted in the last decades the main experimental tools to determine the protein macromolecules folding and refolding behavior and the mechanical behavior of biomolecules and cells. As we have shown in this paper, the only way to interpret such experiments is to model the macromolecule and the measuring device as a unique system . Here, for the first time, we deliver explicit analytic solutions describing the force vs end-to-end distance diagram as a function of a main non-dimensional parameter measuring the relative device versus macromolecule’s stiffness. As this parameter is varied, the model describes the experimentally observed regimes going from sawtooth transitions in the case of isometric (hard device ≡ kd → ∞) conditions to the cooperative type (force plateau) transition observed for the assigned force experiments (soft device ≡ kd → 0). Such crucial effect is often ignored in the interpretation of the experimental results26. We believe that the approach here proposed and its possible extensions can represent an important step forward in the field. In particular, the presence of mechanical interactions for proteins, cells and biological tissues play a crucial role in many fundamental diseases and biological functions37. In this perspective, we observe that a direct extension of our approach to such problems can give a new theoretical framework correctly considering the boundary conditions, e.g. for biomaterial growth and medical pathologies.The comprehension of the response of macromolecules is fundamental in many fields of biology, medicine and bio-inspired materials engineering. Indeed, the incredible elasticity and recovery properties of bio-materials result from the features of the constituent macromolecules. In this paper, we deduced a simple, analytical and effective description of the macromolecule behavior modeled as a chain of elements undergoing conformational transitions (such as in Supplementary Material