k A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. = A transmission mask containing the FT of the second function, g(x,y), is placed in this same plane, one focal length behind the first lens, causing the transmission through the mask to be equal to the product, F(kx,ky) G(kx,ky). k ) As we know, the electric force per unit charge describes the electric field. An electric field is defined as the electric force per unit charge. {\displaystyle ~(k_{x},k_{y})} As a side note, electromagnetics scientists have devised an alternative means to calculate an electric field in a far zone which does not involve stationary phase integration. Five Flagler field school students assisted, where they immersed themselves in what the lives of the multicultural community who lived and worked at Fort Mose from 1752-1763 were like. e All FT components are computed simultaneously - in parallel - at the speed of light. Volt per meter (V/m) is the SI unit of the electric field. Mantenha-se ao corrente das ltimas notcias da poltica europeia, da economia e do desporto na euronews ( Outside the charged sphere, the electric field is given by whereas the field within the sphere is zero. x ( The first is ordinary focused optical imaging systems (e.g., cameras), wherein the input plane is called the object plane and the output plane is called the image plane. . Shop by department, purchase cars, fashion apparel, collectibles, sporting goods, cameras, baby items, and everything else on eBay, the world's online marketplace To show this more explicitly, note that a test charge q t q t at the point P in space has distances of r 1, r 2, , r N r 1, r 2, , r N from the N charges fixed in space above, as shown in Figure 7.19. , + Consider a charged spherical shell with a surface charge density and radius R. Consider a spherical Gaussian surface with any arbitrary radius r, centered with the spherical shell. ) or a higher NA imaging system is required to image finer features of integrated circuits on a photoresist on a wafer. However, their speed is obtained by combining numerous computers which, individually, are still slower than optics. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity.The SI unit for electric dipole moment is the coulomb-meter (Cm). It means that, for a given light frequency, only a part of the full feature of the pattern can be imaged because of the above-mentioned constraints on for most combinations of frequency and wavenumber, but will also be singular (I.e., it does not have the inverse matrix.) k News from San Diego's North County, covering Oceanside, Escondido, Encinitas, Vista, San Marcos, Solana Beach, Del Mar and Fallbrook. {\displaystyle k_{z}} {\displaystyle k={\omega \over c}={2\pi \over \lambda }} ( z r having both magnitude and direction), it follows that an electric field is a vector field. The finer the features in the transparency, the broader the angular bandwidth of the plane wave spectrum. Home of Entrepreneur magazine. The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity.The SI unit for electric dipole moment is the coulomb-meter (Cm). k In this regard, the far-field criterion is loosely defined as: Range = 2D2/ where D is the maximum linear extent of the optical sources and is the wavelength (Scott [1998]). then we can rewrite intensity as follows: Note that for the point at the origin of the film plane ( : 46970 As the electric field is defined in terms of force, and force is a vector (i.e. , The alert reader will note that the integral above tacitly assumes that the impulse response is NOT a function of the position (x',y') of the impulse of light in the input plane (if this were not the case, this type of convolution would not be possible). (2.1). As a result, the elementary product solution It also measures how far from the optic axis the corresponding plane waves are tilted, and so this type of bandwidth is often referred to also as angular bandwidth. By finding which combinations of frequency and wavenumber drive the determinant of the matrix to zero, the propagation characteristics of the medium may be determined. 0 The electric field inside the conductor is equal to zero. {\displaystyle k_{y}} d ) 2 k In turn, that electric field creates a changing magnetic field through Maxwell's addition to Ampre's law. Working 40 hours per week in the field for six weeks will help each student advance in their career in archeology through the Fort Mose Field School. The electric field between parallel plates depends on the charged density of plates. Note: this logic is valid only for small sources, such that the lens is in the far field region of the source, according to the 2D2/ criterion mentioned previously. All spatial dependence of each plane wave component is described explicitly by an exponential function. {\displaystyle h} . y Light can be described as a waveform propagating through a free space (vacuum) or a material medium (such as air or glass). However, the FTs of most wavelets are well known and could possibly be shown to be equivalent to some useful type of propagating field. from its spectrum): The normalizing factor of: Choisissez votre sige sur tous les vols is indeed due solely to the plane wave component with the wave vector ) {\displaystyle k={\omega \over c}={2\pi \over \lambda }} k In Ragnarsson' s work, this method is based on the following postulates: By these postulates, we have the following relationship: Finally, we get a amplitude transmittance with the form of a Wiener filter: Electrical fields can be represented mathematically in many different ways. Rsidence officielle des rois de France, le chteau de Versailles et ses jardins comptent parmi les plus illustres monuments du patrimoine mondial et constituent la plus complte ralisation de lart franais du XVIIe sicle. In the HuygensFresnel or Stratton-Chu viewpoints, the electric field is represented as a superposition of point sources, each one of which gives rise to a Green's function field. / For a charged particle with charge q, the electric field formula is given by. Working 40 hours per week in the field for six weeks will help each student advance in their career in archeology through the Fort Mose Field School. x T Loss of the high (spatial) frequency content causes blurring and loss of sharpness (see discussion related to point spread function). is determined in terms of , e On the other hand, since the wavelength of visible light is so minute in relation to even the smallest visible feature dimensions in the image i.e.. Optical processing is especially useful in real time applications where rapid processing of massive amounts of 2D data is required, particularly in relation to pattern recognition. k Here we describe the reversible and non-volatile electric-field control of dual-ion (oxygen and hydrogen) phase transformations, with associated electrochromic2 and magnetoelectric11 effects. Rsidence officielle des rois de France, le chteau de Versailles et ses jardins comptent parmi les plus illustres monuments du patrimoine mondial et constituent la plus complte ralisation de lart franais du XVIIe sicle. In the matrix equation case in which A is a square matrix, eigenvalues ) ) The boundary of no escape is called the event horizon.Although it has a great effect on the fate and finding where the matrix has no inverse. The force experienced by a unit test charge placed at that point, without altering the original positions of charges q 1, q 2,, q n, is described as the electric field at a point in space owing to a system of charges, similar to the electric field at a point in space due In the near field, no single well-defined spherical wave phase center exists, so the wavefront isn't locally tangent to a spherical ball. 2 = i In this far-field case, truncation of the radiated spherical wave is equivalent to truncation of the plane wave spectrum of the small source. unc ( A curved phasefront may be synthesized from an infinite number of these "natural modes" i.e., from plane wave phasefronts oriented in different directions in space. , z The twin subjects of eigenfunction expansions and functional decomposition, both briefly alluded to here, are not completely independent. Definition of Electric Field. That spectrum is then formed as an "image" one focal length behind the first lens, as shown. {\displaystyle f=1/\tau } The lists do not show all contributions to every state ballot measure, or each independent expenditure committee formed to support or time dependence in wave solutions at the angular frequency with the constraints of They have devised a concept known as "fictitious magnetic currents" usually denoted by M, and defined as. A plane wave spectrum does not necessarily mean that the field as the superposition of the plane wave components in that spectrum behaves something like a plane wave at far distances. , 2 = which basically translates the impulse response function, hM(), from x' to x = Mx. The coefficient of the exponential is a function of only two components of the wave vector for each plane wave (since other remained component can be determined via the above mentioned constraints), for example 1 / but close to (2.1) - the full plane wave spectrum - accurately represents the field incident on the lens from that larger, extended source. Need to confirm if this is the right understanding. The third-order (and lower) Zernike polynomials correspond to the normal lens aberrations. In the case of a short dipole, that is essentially a linear drop from at the feedpoint to zero at the end. {\displaystyle u(x,y)} 2 By the convolution theorem, the FT of an arbitrary transparency function - multiplied (or truncated) by an aperture function - is equal to the FT of the non-truncated transparency function convolved against the FT of the aperture function, which in this case becomes a type of "Greens function" or "impulse response function" in the spectral domain. is. Live news, investigations, opinion, photos and video by the journalists of The New York Times from more than 150 countries around the world. {\displaystyle \lambda } E = F Q E = F Q. An optical field in the image plane (the output plane of the imaging system) is desired to be a high-quality reproduction of an optical field in the object plane (the input plane of the imaging system). ) Solutions to the Helmholtz equation in the Cartesian coordinate system may readily be found via the principle of separation of variables for partial differential equations. {\textstyle \psi _{0}(x,y)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~dk_{x}dk_{y}} 2 x In the case of a short dipole, that is essentially a linear drop from at the feedpoint to zero at the end. The electric field inside the conductor is equal to zero. y Search our huge selection of new and used video games at fantastic prices at GameStop. = The actual impulse response function of an imaging system typically resembles an Airy function, whose radius is on the order of the wavelength of the light used. It is at this stage of understanding that the previous background on the plane wave spectrum becomes invaluable to the conceptualization of Fourier optical systems. k is the intensity distribution of its image which is blurred by a space-invariant point-spread function k . {\displaystyle o(x,y)} 2 x k particularly since both the scalar Laplacian Again, this is true only in the far field, roughly defined as the range beyond {\displaystyle {\frac {1}{(2\pi )^{2}}}} This is because any source bandwidth which lies outside the bandwidth of the optical system under consideration won't matter anyway (since it cannot even be captured by the optical system), so therefore it's not necessary in determining the impulse response. {\displaystyle (x,y,z)} ( The optical system output image g is related to the input image f by convolving the input image with the optical impulse response function of the optical system, h (known as the point-spread function, for focused optical systems). and a noise The electric field is the region where a force acts on a particle placed in the field. To justify this, let's say that the first quotient is not a constant, and is a function of x. , and whose plane is tangent to the phasefront at y The 4F correlator is based on the convolution theorem from Fourier transform theory, which states that convolution in the spatial (x,y) domain is equivalent to direct multiplication in the spatial frequency (kx, ky) domain (aka: spectral domain). i {\displaystyle k} where , This principle says that in separable orthogonal coordinates, an elementary product solution to this wave equation may be constructed of the following form: It may now be argued that each quotient in the equation above must, of necessity, be constant. z Image blurring by a point spread function is studied extensively in optical information processing, one way to alleviate the blurring is to adopt Wiener Filter. The D of the transparency is on the order of cm (102 m) and the wavelength of light is on the order of 106 m, therefore D/ for the whole transparency is on the order of 104. ) 2 The electric field is defined at each point in space as the force per unit charge that would be experienced by a vanishingly small positive test charge if held stationary at that point. Consider a charged spherical shell with a surface charge density and radius R. Consider a spherical Gaussian surface with any arbitrary radius r, centered with the spherical shell. There is a striking similarity between the Helmholtz equation (2.3) above, which may be written. i y such as {\displaystyle k_{i}} In this case, the dispersion relation is linear, as in section 1.3. {\displaystyle H} k In certain physics applications such as in the computation of bands in a periodic volume, it is often a case that the elements of a matrix will be very complicated functions of frequency and wavenumber, and the matrix will be non-singular (I.e., it has the inverse matrix.) y A large number of human relationships in any social group are still upon the machine-like plane. 5. The equation (2.2) above is critical to making the connection between spatial bandwidth (on the one hand) and angular bandwidth (on the other), in the far field. where Whenever a function is discontinuously truncated in one FT domain, broadening and rippling are introduced in the other FT domain. ('4.2), hM() will be a magnified version of the impulse response function h() of a similar, unmagnified system, so that hM(x,y) = h(x/M,y/M). The interested reader may investigate other functional linear operators (so for different equations than the Helmholtz equation) which give rise to different kinds of orthogonal eigenfunctions such as Legendre polynomials, Chebyshev polynomials and Hermite polynomials. Then if one wants the electric field, it is obtained from the derivatives of $\phi$. y , k {\displaystyle H(\omega )} is present whenever angular frequency (radians) is used, but not when ordinary frequency (cycles) is used. Ray optics is the very first type of optics that most of us encounter in our lives; it's simple to conceptualize and understand, and works very well in gaining a baseline understanding of common optical devices. As a result, the two images and the impulse response function are all functions of the transverse coordinates, x and y. = Depending on the operator and the dimensionality (and shape, and boundary conditions) of its domain, many different types of functional decompositions are, in principle, possible. k The changing magnetic field creates a changing electric field through Faraday's law. Reasoning in a similar way for the y and z quotients, three ordinary differential equations are obtained for the fx, fy and fz, along with one separation condition: Each of these 3 differential equations has the same solution form: sines, cosines or complex exponentials. , In this figure, the light blue plane represents the charged plane, and only electric field vectors are represented above the plane. D i k As an example, light travels at a speed of roughly 1ft (0.30m). 1 The same logic is used in connection with the HuygensFresnel principle, or Stratton-Chu formulation, wherein the "impulse response" is referred to as the Green's function of the system. . Whenever bandwidth is expanded or contracted, image size is typically contracted or expanded accordingly, in such a way that the space-bandwidth product remains constant, by Heisenberg's principle (Scott [1998] and Abbe sine condition). Likely to electrical signals, bandwidth in optics is a measure of how finely detailed an image is; the finer the detail, the greater the bandwidth required to represent it. (2.2), Then, the lens passes - from the object plane over onto the image plane - only that portion of the radiated spherical wave which lies inside the edge angle of the lens. Below are lists of the top 10 contributors to committees that have raised at least $1,000,000 and are primarily formed to support or oppose a state ballot measure or a candidate for state office in the November 2022 general election. And, as mentioned above, the impulse response of the correlator is just a picture of the feature we're trying to find in the input image. In the case of a short dipole, that is essentially a linear drop from at the feedpoint to zero at the end. 2 This is somewhat like the point spread function, except now we're really looking at it as a kind of input-to-output plane transfer function (like MTF), and not so much in absolute terms, relative to a perfect point. Consider a system of charges q 1, q 2,, qn with position vectors r 1, r 2,, r n with respect to some origin O. where The {\displaystyle k_{i}} Causality means that the impulse response h(t - t') of an electrical system, due to an impulse applied at time t', must of necessity be zero for all times t such that t - t' < 0. takes an infinite range of real numbers. y The imaging is the reconstruction of a wave on the object plane (having information about a pattern on the object plane to be imaged) on the image plane via the proper wave propagation from the object to the image planes, (E.g., think about the imaging of an image in an aerial space.) {\displaystyle \tau } This is a concept that spans a wide range of physical disciplines. , {\displaystyle k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}} Substituting this expression into the Helmholtz equation, the paraxial wave equation is derived: The equation (2.1) above may be evaluated asymptotically in the far field (using the stationary phase method) to show that the field at a distant point + For optical systems, bandwidth also relates to spatial frequency content (spatial bandwidth), but it also has a secondary meaning. The transmittance function in the front focal plane (i.e., Plane 1) spatially modulates the incident plane wave in magnitude and phase, like on the left-hand side of eqn. : 46970 As the electric field is defined in terms of force, and force is a vector (i.e. radial dependence is a spherical wave - both in magnitude and phase - whose local amplitude is the FT of the source plane distribution at that far field angle. = Note that this is NOT a plane wave. Even though the input transparency only occupies a finite portion of the x-y plane (Plane 1), the uniform plane waves comprising the plane wave spectrum occupy the entire x-y plane, which is why (for this purpose) only the longitudinal plane wave phase (in the z-direction, from Plane 1 to Plane 2) must be considered, and not the phase transverse to the z-direction. In the frequency domain, with an assumed time convention of f Rsidence officielle des rois de France, le chteau de Versailles et ses jardins comptent parmi les plus illustres monuments du patrimoine mondial et constituent la plus complte ralisation de lart franais du XVIIe sicle. ) Rservez des vols pas chers sur le site officiel easyJet.com vers plus de 130 destinations en Europe. y The propagating plane waves that we'll study in this article are perhaps the simplest type of propagating waves found in any type of media. , However, the plus sign in the Helmholtz equation is significant.) Propagation of light in homogeneous, source-free media, The complete solution: the superposition integral, Connection between Fourier optics and imaging resolution, Paraxial wave propagation (optic axis assumed as z axis), The plane wave spectrum: the foundation of Fourier optics, Eigenfunction (natural mode) solutions: background and overview, Optical systems: general overview and analogy with electrical signal processing systems, The 2D convolution of input function against the impulse response function, Applications of Fourier optics principles, Fourier analysis and functional decomposition, Hardware implementation of the system transfer function: The 4F correlator, Afterword: Plane wave spectrum within the broader context of functional decomposition, Functional decomposition and eigenfunctions. 2 Optical systems typically fall into one of two different categories. So the spatial domain operation of a linear optical system is analogous in this way to the HuygensFresnel principle. and the matrix A are linear operators on their respective functions / vector spaces. A low-pass filter is the complement of a high . k Literally, the point source has been "spread out" (with ripples added), to form the Airy point spread function (as the result of truncation of the plane wave spectrum by the finite aperture of the lens). are transverse wave numbers satisfying k To put it in a slightly more complex way, similar to the concept of frequency and time used in traditional Fourier transform theory, Fourier optics makes use of the spatial frequency domain (kx, ky) as the conjugate of the spatial (x, y) domain. News from San Diego's North County, covering Oceanside, Escondido, Encinitas, Vista, San Marcos, Solana Beach, Del Mar and Fallbrook. k k Similarly, Gaussian wavelets, which would correspond to the waist of a propagating Gaussian beam, could also potentially be used in still another functional decomposition of the object plane field. , 2 do not exist for the given light of The lists do not show all contributions to every state ballot measure, or each independent expenditure committee formed to support or f , View all results for thinkgeek. + 5. . is the angle (in radian) between the wave vector k and the z-axis as the optical axis of an optical system under discussion. {\displaystyle k_{z}} , Rservez des vols pas chers sur le site officiel easyJet.com vers plus de 130 destinations en Europe. {\displaystyle \lambda } In optical imaging this function is better known as the optical transfer function (Goodman). Outside the charged sphere, the electric field is given by whereas the field within the sphere is zero. If there would be an electric field inside the conductor, the free charges would move and produce an electric field of their own opposite to the initial electric field. . and {\displaystyle \psi (\mathbf {r} )e^{i\omega t}} . , = x , An electric field is defined as the electric force per unit charge. ) A sinusoidal plane wave is one special solution of these equations. . y As a result, machines realizing such an optical lithography have become more and more complex and expensive, significantly increasing the cost of the electronic component production. , can not be fully imaged since waves with such Maxwell's equations explain how these waves can physically propagate through space. Note that electric potential follows the same principle of superposition as electric field and electric potential energy. z The dipole moment, , is a vector which points perpendicular to the plane of. 2 {\displaystyle k} y We'll consider one such plane wave component, propagating at angle with respect to the optic axis. ): A synthesis equation (reconstructing the function Electrodynamics is the physics of electromagnetic radiation, and electromagnetism is the physical phenomenon associated with the theory of electrodynamics. Fourier optics forms much of the theory behind image processing techniques, as well as finding applications where information needs to be extracted from optical sources such as in quantum optics. Fourier optics is somewhat different from ordinary ray optics typically used in the analysis and design of focused imaging systems such as cameras, telescopes and microscopes. G ) 2 is the maximum linear extent of the optical sources and It is assumed that the source is small enough that, by the far-field criterion, the lens is in the far field of the "small" source. The output of the system, for a single delta function input is defined as the impulse response of the system, h(t - t'). For example, assume that A low-pass filter is the complement of a high = D ), the recorded wave from the lower portion should be much stronger than that from the upper portion because the wave passing through the lower path is focused, which leads to the relationship In this figure, the light blue plane represents the charged plane, and only electric field vectors are represented above the plane. {\displaystyle f} Electric and magnetic fields obey the properties of superposition.Thus, a field due to any particular particle or time-varying electric or magnetic field contributes to the fields present in the same space due to other It is assumed that is small (paraxial approximation), so that, In the figure, the plane wave phase, moving horizontally from the front focal plane to the lens plane, is. k r 2 k From this equation, we'll show how infinite uniform plane waves comprise one field solution (out of many possible) in free space. having both magnitude and direction), it follows that an electric field is a vector field. = The debye (D) is another unit of measurement used in atomic physics and chemistry.. Theoretically, an electric dipole is defined by the first-order This step truncation can introduce inaccuracies in both theoretical calculations and measured values of the plane wave coefficients on the RHS of eqn. Electrodynamics is the physics of electromagnetic radiation, and electromagnetism is the physical phenomenon associated with the theory of electrodynamics. Electric and magnetic fields obey the properties of superposition.Thus, a field due to any particular particle or time-varying electric or magnetic field contributes to the fields present in the same space due to other z where Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the waveform being considered is regarded as made up of a combination, or superposition, of plane waves. 0 Live news, investigations, opinion, photos and video by the journalists of The New York Times from more than 150 countries around the world. Perhaps a lens figure-of-merit in this "point spread function" viewpoint would be to ask how well a lens transforms an Airy function in the object plane into an Airy function in the image plane, as a function of radial distance from the optic axis, or as a function of the size of the object plane Airy function. Notice that the electric field is uniform and independent of distance from the infinite charged plane. {\displaystyle u(\mathbf {r} ,t)} All of these functional decompositions have utility in different circumstances. Notice that the electric field is uniform and independent of distance from the infinite charged plane. Electrodynamics is the physics of electromagnetic radiation, and electromagnetism is the physical phenomenon associated with the theory of electrodynamics. As notcias de ltima hora disponveis em acesso livre em video on demande. This source of error is known as Gibbs phenomenon and it may be mitigated by simply ensuring that all significant content lies near the center of the transparency, or through the use of window functions which smoothly taper the field to zero at the frame boundaries. (The minus sign in this matrix equation is, for all intents and purposes, immaterial. ), and (2) spatial frequencies with If there would be an electric field inside the conductor, the free charges would move and produce an electric field of their own opposite to the initial electric field. A solution to the Helmholtz equation as the spatial part of a complex-valued Cartesian component of a single frequency wave is assumed to take the form: where u in the Cartesian coordinate system may be formed as a weighted superposition of all possible elementary plane wave solutions as. In practice, it is not necessary to have an ideal point source in order to determine an exact impulse response. This equation takes on its real meaning when the Fourier transform, Individuals use one another so as to get desired results, without reference to the emotional and intellectual disposition and consent of those used. z ( k is the weight factor or the amplitude of the plane wave component with the wave vector for certain specific combinations. The electric field is the region where a force acts on a particle placed in the field. Consider a charged spherical shell with a surface charge density and radius R. Consider a spherical Gaussian surface with any arbitrary radius r, centered with the spherical shell. is then split into two parts: Therefore, the total intensity distribution is, Assume f The current along dipole arms are approximately described as proportional to ( ) where z is the distance to the end of the arm. {\displaystyle \omega =2\pi f} ( Definition of Electric Field. {\displaystyle e^{i\omega t}} k The plane wave spectrum arises naturally as the eigenfunction or "natural mode" solution to the homogeneous electromagnetic wave equation in rectangular coordinates (see also Electromagnetic radiation, which derives the wave equation from Maxwell's equations in source-free media, or Scott [1998]). ) On the object plane, the spatial part of a complex-valued Cartesian component of a wave is, as shown above, Bandwidth truncation causes a (fictitious, mathematical, ideal) point source in the object plane to be blurred (or, spread out) in the image plane, giving rise to the term, "point spread function." Once the concept of angular bandwidth is understood, the optical scientist can "jump back and forth" between the spatial and spectral domains to quickly gain insights which would ordinarily not be so readily available just through spatial domain or ray optics considerations alone. ) The dipole moment, , is a vector which points perpendicular to the plane of. y , r (4.1) may be Fourier transformed to yield: The system transfer function, u Consider a system of charges q 1, q 2,, qn with position vectors r 1, r 2,, r n with respect to some origin O. 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