# Cherenkov Emission Generated by a Modulated Source in 2D Dispersive Photonic Crystal Slabs

Authors: Erika Martínez-Sánchez; Gennadiy Burlak; Carlos Rodríguez-Garcia; Marisol Gallardo-Heredia; Ulises Avila-López
Journal: Advances in Materials Science and Engineering
 (2020)

---

## Abstract

This work presents a systematic numerical study of Cherenkov optical radiation generated by a modulated source that moves with uniform velocity on a two-dimensional (2D) photonic crystal (PCr) slab surface. We apply the FDTD technique with emphasis on the dispersion properties of the periodic medium to perform our numerical analysis. The field oscillations generated at the passage of a modulated source in the PCr produce a series of spectral resonances corresponding to the eigenmodes in the spatial frequency domain for the photonic slab. The amplitudes of the field oscillations have maximal values in the group cone closely to the path of the moving charge.


---

## Body

## 1. Introduction

Dielectric two-dimensional photonic crystals, composed of either dielectric rods or air columns, are widely used to obtain a high control over the propagation of light without loss of energy. It has been shown theoretically and experimentally that photonic crystal slab structural defects behave as waveguides [1], resonant devices [2], nanocavity laser [3], photonic circuits, and various functional devices [4–6]. In a conventional material, the Cherenkov radiation is coherent, and the charged particle movement is associated with a velocity threshold, a forward-pointing radiation cone, and a forward direction of emission [7, 8]. Cherenkov radiation generated by a charged particle in dielectric homogeneous medium has been extensively studied. In a more complex periodic medium, e.g., a meta-material, we can also get a variety of anomalous effects [9], such as backward direction Cherenkov radiation [10–15]. However, the Cherenkov radiation (CR) has a different behavior in a photonic crystal. Photonic crystals, where very complex Bragg scattering is possible, exhibit a new medium for unusual radiation pattern under different particle-velocity regimes. [16–19].There are various approaches to study the influence of a periodic medium on the Cherenkov effect. For example, in [20], there is a complete description of stimulated Cherenkov radiation and laser oscillation in a photonic crystal in the range 10 GHz, getting significant power and high spectral and spatial coherence [20]. The fiber-optical Cherenkov radiation also was of great interest in recent years [17, 21].Other studies are devoted to the analysis of the Cherenkov emission spectrum through the photonic band structures calculation in photonic crystals. In such simulations, the method of the finite differences in time domain (FDTD) was applied, and it was shown that the spectral and spatial distributions can be varied over different ranges of velocities [16, 22].However, in the aforementioned studies, the case of a modulated source moving through a slab of photonic crystal has not been studied. We propose a simpler approach based on the geometrical design of the photonic crystal and characteristics of the source, allowing obtaining a coherent radiation in the specific frequency ranges. Unlike previous works, our approach does not require an analysis of the crystal band structure. Getting the output of desired wavelengths is of high interest for creation of novel light sources, e.g., in the extreme ultraviolet (EUV) radiation range [23]. Among other applications, such a study includes the velocity-sensitive particle detection and radiation generation at selected range of frequencies.Askar’yan demonstrates theoretically that Cherenkov radiation generated from a modulated source moving with velocityv0 occurs when the medium has an inhomogeneous structure. For example, when the wave packet is transmitted through the boundary between two media, then the transition radiation should be produced [24]. It is interesting to investigate the optical modes in the photonic crystal associated with the Cherenkov radiation generated by a modulated wave train, since this kind of wave packet is produced in most cases of practical interest [24].In this paper, we study numerically the Cherenkov optical radiation in a two-dimension (2D) photonic crystal slab (PCr). The modulated source is nonrelativistic, having the eigenfrequencyω0. We performed the finite differences time domain (FDTD) simulations [25], where the size of the photonic crystal structure is fixed, and the values of dielectric permittivity inside and outside the holes of the structure are varied. With the use of spectral Fast Fourier Transform (FFT) analysis we have found that the spatial frequencies of the oscillations produced by Cherenkov emission also oscillates correspondingly to the eigenmodes located in the gap zone of the photonic crystal slab.


## 2. Basic Equations

In this paper, we exploit the FDTD 3D algorithm that became a widely used algorithm for modelling in complex materials. The Maxwell equations are(1)∇×Er,t=−μ0∂∂tHr,t,∇×Hr,t=ε0ε∂∂tEr,t+qv0fr,t,where ε=εr is a dielectric permittivity of the PCr structure [26]. We consider the charge q that moves with uniform velocity v0 parallel to y direction: v0e^y, closely to the x,y surface. In the numerical grid, the point particle (bunch) is simulated by the Gaussian impulse fr,t=cosω0tW−3exp−x2+y−v0t2+z2/W2, where ω0 is the modulating frequency and W is the width.To perform the numerical simulations, we used the dimensionless variables for normalization, the vacuum light velocityc=ε0μ0−0.5, and the typical spatial scale for nano-optic objects l0=500nm were used. The electrical and magnetic fields are renormalized with the electrical scale E0=ql0ε0, and magnetic scale H0=ε0/μ00.5E0, respectively.


## 3. Numerical Results

The considered photonic crystal slab structure perforated by periodical air rods in a dielectric medium is shown in Figure1.Figure 1
No magnetic photonic crystal slab structure with average dielectric permittivity, where dielectric slabεs=3 is perforated by holes εh=3 ordered periodically.2D photonic crystals have been extensively studied because they are relatively easier to fabricate using existing techniques and to operate in optical frequencies [27–30]. We focus our attention on the 2D lattice of air holes that is a widely used structure, since the triangular lattice of air holes has a large band gap [31], see also experiments [32, 33]. The slab thickness, radius, and refractive index contrast determine the properties of photonic crystal slabs [34]. In our approach, we use a high index contrast, that is required between the dielectric material, and the holes to open a bandgap in the xy-plane [33]. Since the lattice constant of the photonic crystal must be comparable to the wavelength of the light [34], we use, a = 500 nm. The radius and height of the holes we have chosen are in the experimentally accessible range for reasonable values [31, 35, 36], such that, r=0.3 and h=1a, which normally is used within the resolution range of electron beam lithography. Such parameter values also allow improving the accuracy of numerical simulation, [31]. We should note that, since the spatial period of PCr is comparable with the field wavelength l0/λ≤1, such a 2D system acts as a distributed 2D plane resonator having a special eigenfrequency spectrum.Since the refractive index of a photonic system varies in space and it is well-defined only locally (or in average), we only know numerical methods to calculate the Cherenkov emission in such 3D inhomogeneous photonic systems. To realize the FDTD simulation, the scheme from [2] was followed, which we have elaborated for our purposes. The renormalized variables were used dx/dt⟶3dx/dt=3dl/dt=cn≡1.73 that is the vacuum light velocity in the numerical approach, such that v0<cn, where v0 is the velocity of charge.


### 3.1. Cherenkov Radiation in a Photonic Crystal Slab

The practically interesting case of the Cherenkov radiation (CR) in the case of dielectric permittivity of the slab and holes with distinct valuesεs≠εh was considered. Besides, the critical velocity of the Cherenkov radiation in a homogeneous medium was used as a reference value.Figure2 shows the field oscillations within the Cherenkov cone v0=1.5 generated at the time when the particle touches the output boundary of the computational system. Figure 2(a) shows the 3D field component Ex, with the uses radius r=0.25 and εs=3, εh=1 as permittivity of slab and holes, respectively. Figure 2(b) displays the field Ex when r=0.35 and εs=15. It is worth noting that the increase in hole radius does not significantly affect the field of Cherenkov radiation. However, when the difference in dielectric constants is increased, the angle of the cone is reduced, but the amplitude of oscillations becomes larger. We observe that such a geometry on the field within the Cherenkov cone acquires a highly inhomogeneous structure due to the discontinuity of dielectric permittivity in the boundary of holes.Figure 2
(Color on line) snapshots the fieldExx,y in plane x,y for a particle moving with v0=1.5>vc and frequency source ω0=0. (a) r=0.25, εs=3. (b) r=0.35, εs=15. The Cherenkov group cone is well defined with oscillations in the path of the charge.
(a)
(b)Cherenkov radiation (CR) in a photonic crystal arises from a coherent excitation of its eigenmodes by the moving charge. Its origin lies in both the transition radiation, which occurs when the charge crosses a dielectric boundary or experiences at an inhomogeneous dielectric environment, and the conventional CR. In these cases, the coherence is preserved throughout the medium as we see in [37]. Unlike the case of Cherenkov radiation in a homogeneous medium, there exist radiating modes in this inhomogeneous dielectric case. It is interesting to study the behavior of these oscillations within the Cherenkov group cone [38].A simple physical interpretation of the group cone can be put forward in terms of group velocity considering that for each direction around the charge velocity, the burst of Cherenkov light is emitted into a group of modes, while the peak of the pulse moves in space with a velocity equal to the group velocityvg experiencing an almost negligible absorption [38].In the following, the spectra of electromagnetic oscillations associated with the Cherenkov emission for different perforated slab systems are calculated.In Figure3 the values of optical eigenmodes of PCr excited by the moving charge calculated for different PCr configurations given by variation in the dielectric permittivity of slab εs and cases of radiuses r=0.25;0.3;0.35 are displayed. A decreasing value of the spectral peak’s values for greater contrast of the permittivities is observed. The inset panel of Figure 3 shows the curve of the set of spatial frequencies obtained as a result of our numerical simulations (red line) and the values by applying the formula f=c/λεp (blue line), which is a simple dimension relationship, where εp is a weighted dielectric permittivity of PCr εp=εsVs−Vc−εhVc/Vs+Vc [37]. Here, Vs and Vc are the volume of the slab and holes, respectively.Figure 3
Frequencyks calculated numerically for different values of εs in a slab of holes εh=1 for radius r=0.25,0.3 and 0.35. It is seen that for a greater contrast of the dielectric constant, the value of the ks decreases, until reaching the minimum value in ks∼0.43 for r=0.25 and the maximum ks∼0.87 for r=0.35. The graph inside the panels show the case of ks for r=0.3 where the red curve is the experimental data, while the blue one shows the results following from the equation εp=εsVs−Vc−εhVc/Vs+Vc. See more details in text.


### 3.2. Modulated Source for a Hole Slab of Photonic Crystal

Uniform motion of a modulated source in the medium can produce radiation when the medium is inhomogeneous, for example, when the wave packet is transmitted through the boundary between two media, then transition radiation should be created [24]. In the following, we analyze in more detail the Cherenkov field for a modulated source with ω0=0 and ω0≠0. A geometry with r=0.3 and a permittivity of the slab εs=3 were studied.Figure4 shows the time dependence of field at a long-time simulation when a particle with velocity v=1.5 and modulation ω0=0 passes over a slab. The Figures 4(a) and 4(b) display the components Ex and Ey, respectively. We observe that Ex,y amplitudes have an oscillating shape shift with a pedestal defined by a point position. Panel (c), (d) shows that the field Hy,z (x, y) has smoother shape due to the continuity of magnetic field in the system.Figure 4
The time dependence for the components (a)Ex, (b) Ey, (c) Hx, and (d) Hz, in various points along the charge path with velocity v0=1.5 and frequency ω0=0.
(a)
(b)
(c)
(d)Figure5 displays the frequency spectrum of electromagnetic oscillations shown in Figure 4 for various points along the charged path, having the highest spectral peaks at ks∼0.82. Such oscillations correspond to optical eigenmodes of PCr excited by the moving charged particle for components (a) Ex, (b) Ey, (c) Hx, and (d) Hz for values v0=1.5 and ω0=0.Figure 5
(Color on line) the spatial frequency spectrum of electromagnetic oscillations shown in Figure4 for various points along the charge path, having highest spectral peaks at ks∼0.82. Such oscillations correspond to optical eigenmodes of PCr excited by the moving charged particle for components (a) Ex, (b) Ey, (c) Hx, and (d) Hz, for values v0=1.5 and ω0=0.
(a)
(b)
(c)
(d)Figure6 shows the 3D spatial structure of the field Ex at long-time simulation L2=10l0×10l0, where l0=500nm. The cases (a) v0=1.5, ω0=0; (b) v0=1.5, ω0=0.82; (c) v0=1, ω0=0; (d) v0=1, ω0=0.82 are displayed. Figure 6 shows that the distribution of the field Ex in PCr is structurally stable, and this is due to discontinuity in the boundary of holes, for all cases shown.Figure 6
(Color on line) the spatialx,y shape of field component Ex for particle that moves at different speeds and modulated frequencies. (a) v0=1.5, ω0=0. (b) v0=1.5, ω0=0.82. (c) v0=1, ω0=0. (d) v0=1, ω0=0.82. (a), (c) Oscillations are very similar when ω0=0, and we observe the same amplitude. (b), (d) The oscillations do not correspond to the positions of the holes in the structure, and this is due to the modulation of the source. The number of oscillations and amplitude of Ex has dependence of ω0≠0.
(a)
(b)
(c)
(d)Figure7, shows the frequency spectrum of electromagnetic oscillations corresponding to Figures 6(c) and 6(d), for various points along the charge path at x±1/4,xc,x3/4,xs,xf at x=±L/4,0. In Figure 7(a), we observe the existence of near-static, no radiating field artificial peak around ks=0 [38], in panel (b), ks≃0.82, and such oscillations correspond to optical eigenmodes in PCr excited by moving charged particle; this evince the existence of the Cherenkov radiant field for a modulated source in an inhomogeneous medium.Figure 7
Fourier spectrum (wavenumbers domain) for (a)v0=1, ω0=0. (b) v0=1, ω0=0.82. In panel (a) we observe the existence of near-static no radiating field artificial peak around ks=0 while that of (b) well-defined peaks ks≃0.82 corresponds to optical eigenmodes of PCr.
(a)
(b)Figure8(a) shows the characteristic Cherenkov cone when ω0=0, and v0=1.5. In Figure 8(b), the group cone is well defined but with fewer oscillations in the charged path, this is due to the modulation of the particle bunch. The situation changes when the velocity of the source is decreased. In this case, when v0=1, we do not observe the Cherenkov group cone (see Figure 8(c)). In periodic dispersive materials, the critical velocity in the medium is not well defined. We observe that it is possible to extend the velocity range v0 and observe the group cone for a modulated source, and this situation is shown in Figure 8(d).Figure 8
(Color on line) snapshots the fieldExx,y in plane x,y for particle moving with (a)-(b) v0=1.5 and (c)-(d) v0=1. (a) Field Ex for ω0=0, the Cherenkov cone is well defined with oscillations in the path of the charge. (b) Field Ex for ω0=0.82, the Cherenkov emission is observed, with a lower number of oscillations than subsection (a) due to the modulation of the source. (c) ω0=0, oscillations in the path of the charge; however, in this case, we did not observe the group cone. (d) Cherenkov group cone for ω0=0.82 and v0=1.
(a)
(b)
(c)
(d)Figure9 shows the group cone for different PCr systems, when ω0∼ks. Cases are shown: (a) εs=12.25,ω0=0.52; (b) εs=9,ω0=0.57; (c) εs=6,ω0=0.67; (d) εs=3,ω0=0.82. Cherenkov group cone is well defined for all cases.Figure 9
(Color on line) snapshots the fieldExx,y in plane x,y for particle bunch moving with parameters (a) εs=12.25,v0=0.6,ω0=0.52; (b) εs=9,v0=0.62,ω0=0.57; (c) εs=6,v0=0.8,ω0=0.67; (d) εs=3,v0=0.1,ω0=0.82. The Cherenkov wave cone is well defined for all cases shown.
(a)
(b)
(c)
(d)


## 3.1. Cherenkov Radiation in a Photonic Crystal Slab

The practically interesting case of the Cherenkov radiation (CR) in the case of dielectric permittivity of the slab and holes with distinct valuesεs≠εh was considered. Besides, the critical velocity of the Cherenkov radiation in a homogeneous medium was used as a reference value.Figure2 shows the field oscillations within the Cherenkov cone v0=1.5 generated at the time when the particle touches the output boundary of the computational system. Figure 2(a) shows the 3D field component Ex, with the uses radius r=0.25 and εs=3, εh=1 as permittivity of slab and holes, respectively. Figure 2(b) displays the field Ex when r=0.35 and εs=15. It is worth noting that the increase in hole radius does not significantly affect the field of Cherenkov radiation. However, when the difference in dielectric constants is increased, the angle of the cone is reduced, but the amplitude of oscillations becomes larger. We observe that such a geometry on the field within the Cherenkov cone acquires a highly inhomogeneous structure due to the discontinuity of dielectric permittivity in the boundary of holes.Figure 2
(Color on line) snapshots the fieldExx,y in plane x,y for a particle moving with v0=1.5>vc and frequency source ω0=0. (a) r=0.25, εs=3. (b) r=0.35, εs=15. The Cherenkov group cone is well defined with oscillations in the path of the charge.
(a)
(b)Cherenkov radiation (CR) in a photonic crystal arises from a coherent excitation of its eigenmodes by the moving charge. Its origin lies in both the transition radiation, which occurs when the charge crosses a dielectric boundary or experiences at an inhomogeneous dielectric environment, and the conventional CR. In these cases, the coherence is preserved throughout the medium as we see in [37]. Unlike the case of Cherenkov radiation in a homogeneous medium, there exist radiating modes in this inhomogeneous dielectric case. It is interesting to study the behavior of these oscillations within the Cherenkov group cone [38].A simple physical interpretation of the group cone can be put forward in terms of group velocity considering that for each direction around the charge velocity, the burst of Cherenkov light is emitted into a group of modes, while the peak of the pulse moves in space with a velocity equal to the group velocityvg experiencing an almost negligible absorption [38].In the following, the spectra of electromagnetic oscillations associated with the Cherenkov emission for different perforated slab systems are calculated.In Figure3 the values of optical eigenmodes of PCr excited by the moving charge calculated for different PCr configurations given by variation in the dielectric permittivity of slab εs and cases of radiuses r=0.25;0.3;0.35 are displayed. A decreasing value of the spectral peak’s values for greater contrast of the permittivities is observed. The inset panel of Figure 3 shows the curve of the set of spatial frequencies obtained as a result of our numerical simulations (red line) and the values by applying the formula f=c/λεp (blue line), which is a simple dimension relationship, where εp is a weighted dielectric permittivity of PCr εp=εsVs−Vc−εhVc/Vs+Vc [37]. Here, Vs and Vc are the volume of the slab and holes, respectively.Figure 3
Frequencyks calculated numerically for different values of εs in a slab of holes εh=1 for radius r=0.25,0.3 and 0.35. It is seen that for a greater contrast of the dielectric constant, the value of the ks decreases, until reaching the minimum value in ks∼0.43 for r=0.25 and the maximum ks∼0.87 for r=0.35. The graph inside the panels show the case of ks for r=0.3 where the red curve is the experimental data, while the blue one shows the results following from the equation εp=εsVs−Vc−εhVc/Vs+Vc. See more details in text.


## 3.2. Modulated Source for a Hole Slab of Photonic Crystal

Uniform motion of a modulated source in the medium can produce radiation when the medium is inhomogeneous, for example, when the wave packet is transmitted through the boundary between two media, then transition radiation should be created [24]. In the following, we analyze in more detail the Cherenkov field for a modulated source with ω0=0 and ω0≠0. A geometry with r=0.3 and a permittivity of the slab εs=3 were studied.Figure4 shows the time dependence of field at a long-time simulation when a particle with velocity v=1.5 and modulation ω0=0 passes over a slab. The Figures 4(a) and 4(b) display the components Ex and Ey, respectively. We observe that Ex,y amplitudes have an oscillating shape shift with a pedestal defined by a point position. Panel (c), (d) shows that the field Hy,z (x, y) has smoother shape due to the continuity of magnetic field in the system.Figure 4
The time dependence for the components (a)Ex, (b) Ey, (c) Hx, and (d) Hz, in various points along the charge path with velocity v0=1.5 and frequency ω0=0.
(a)
(b)
(c)
(d)Figure5 displays the frequency spectrum of electromagnetic oscillations shown in Figure 4 for various points along the charged path, having the highest spectral peaks at ks∼0.82. Such oscillations correspond to optical eigenmodes of PCr excited by the moving charged particle for components (a) Ex, (b) Ey, (c) Hx, and (d) Hz for values v0=1.5 and ω0=0.Figure 5
(Color on line) the spatial frequency spectrum of electromagnetic oscillations shown in Figure4 for various points along the charge path, having highest spectral peaks at ks∼0.82. Such oscillations correspond to optical eigenmodes of PCr excited by the moving charged particle for components (a) Ex, (b) Ey, (c) Hx, and (d) Hz, for values v0=1.5 and ω0=0.
(a)
(b)
(c)
(d)Figure6 shows the 3D spatial structure of the field Ex at long-time simulation L2=10l0×10l0, where l0=500nm. The cases (a) v0=1.5, ω0=0; (b) v0=1.5, ω0=0.82; (c) v0=1, ω0=0; (d) v0=1, ω0=0.82 are displayed. Figure 6 shows that the distribution of the field Ex in PCr is structurally stable, and this is due to discontinuity in the boundary of holes, for all cases shown.Figure 6
(Color on line) the spatialx,y shape of field component Ex for particle that moves at different speeds and modulated frequencies. (a) v0=1.5, ω0=0. (b) v0=1.5, ω0=0.82. (c) v0=1, ω0=0. (d) v0=1, ω0=0.82. (a), (c) Oscillations are very similar when ω0=0, and we observe the same amplitude. (b), (d) The oscillations do not correspond to the positions of the holes in the structure, and this is due to the modulation of the source. The number of oscillations and amplitude of Ex has dependence of ω0≠0.
(a)
(b)
(c)
(d)Figure7, shows the frequency spectrum of electromagnetic oscillations corresponding to Figures 6(c) and 6(d), for various points along the charge path at x±1/4,xc,x3/4,xs,xf at x=±L/4,0. In Figure 7(a), we observe the existence of near-static, no radiating field artificial peak around ks=0 [38], in panel (b), ks≃0.82, and such oscillations correspond to optical eigenmodes in PCr excited by moving charged particle; this evince the existence of the Cherenkov radiant field for a modulated source in an inhomogeneous medium.Figure 7
Fourier spectrum (wavenumbers domain) for (a)v0=1, ω0=0. (b) v0=1, ω0=0.82. In panel (a) we observe the existence of near-static no radiating field artificial peak around ks=0 while that of (b) well-defined peaks ks≃0.82 corresponds to optical eigenmodes of PCr.
(a)
(b)Figure8(a) shows the characteristic Cherenkov cone when ω0=0, and v0=1.5. In Figure 8(b), the group cone is well defined but with fewer oscillations in the charged path, this is due to the modulation of the particle bunch. The situation changes when the velocity of the source is decreased. In this case, when v0=1, we do not observe the Cherenkov group cone (see Figure 8(c)). In periodic dispersive materials, the critical velocity in the medium is not well defined. We observe that it is possible to extend the velocity range v0 and observe the group cone for a modulated source, and this situation is shown in Figure 8(d).Figure 8
(Color on line) snapshots the fieldExx,y in plane x,y for particle moving with (a)-(b) v0=1.5 and (c)-(d) v0=1. (a) Field Ex for ω0=0, the Cherenkov cone is well defined with oscillations in the path of the charge. (b) Field Ex for ω0=0.82, the Cherenkov emission is observed, with a lower number of oscillations than subsection (a) due to the modulation of the source. (c) ω0=0, oscillations in the path of the charge; however, in this case, we did not observe the group cone. (d) Cherenkov group cone for ω0=0.82 and v0=1.
(a)
(b)
(c)
(d)Figure9 shows the group cone for different PCr systems, when ω0∼ks. Cases are shown: (a) εs=12.25,ω0=0.52; (b) εs=9,ω0=0.57; (c) εs=6,ω0=0.67; (d) εs=3,ω0=0.82. Cherenkov group cone is well defined for all cases.Figure 9
(Color on line) snapshots the fieldExx,y in plane x,y for particle bunch moving with parameters (a) εs=12.25,v0=0.6,ω0=0.52; (b) εs=9,v0=0.62,ω0=0.57; (c) εs=6,v0=0.8,ω0=0.67; (d) εs=3,v0=0.1,ω0=0.82. The Cherenkov wave cone is well defined for all cases shown.
(a)
(b)
(c)
(d)


## 4. Conclusions

The optical field of Cherenkov by a modulated source in a perforated slab of photonic crystal is studied with details. Similar field structures can be produced in various cases of practical interest; therefore, we can conclude that considered periodic medium with studied optical properties may by adequate to this scope. We have shown that the Cherenkov group cone exists when the modulation frequencyω0≠0 in a wider range of charge velocities. In our approach, the slab parameters are fixed, while the spatial frequency spectrum is a result of numerical simulations. Such oscillations of the Cherenkov field correspond to optical eigenmodes of the photonic crystal excited by the moving modulated source. Applications may include velocity-sensitive particle detectors, laser oscillation, fiber-optic Cherenkov, detection, and field radiation generation at selectable frequency ranges.



---

*Source: 7431915-2020-02-26.xml*