The Experts below are selected from a list of 33 Experts worldwide ranked by ideXlab platform
Amnon Yariv - One of the best experts on this subject based on the ideXlab platform.
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Synthesis of an Arbitrary Axial Field Profile By Computer-Generated Holograms
1994Co-Authors: Joseph Rosen, Amnon YarivAbstract:arency g(x,, y,) is placed at the first focal plane of a lens of infinite diameter (see Fig. 1). Using the Fresnel approximation, we obtain the three-dimensional complex ampli- tude distribution past the lens7: exp(jkz) u(x,y,z)- jAz f f g'(x,,y2) jk (1) where (x2, y, are the coordinates of the lens's plane and g'(x2, Y2) is the field amplitude just in front of the lens, which is given by (2) Substituting Eq. (2) into Eq. (1), changing the order of integration, and performing the integration over x2 and y2, we obtain u(x,y,z)= exp[jk(z + f)] /_ j Af f g(x, y) - . k(z - k yyl dxidy x exPb-j 2f2f)(x + y, 2) - j-(xx + . (3) We note that, at the second focal plane z = f, Eq. (3) yields the familiar Fourier integral relation between u(x,y,z = f) and g(x,y). Here we are interested in the field distribution along the z axis (x, y = 0). Changing to polar coordinates, (x, Yl) (r, 0), we see that Eq. (3) becomes u(z) = exp[jk(z + f] exp[ k(f).]d. 2jAf fo t(p) -j - = exp[
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Synthesis of an arbitrary axial field profile by computergenerated holograms
1994Co-Authors: Joseph Rosen, Amnon YarivAbstract:Computer-generated holograms are employed to design any desired intensity distribution along the propagation axis for a finite specified distance. A computer-generated hologram is most commonly a transmission mask employed to construct a de-sired transverse image. ' In this Letter we consider the problem of constructing a longitudinal image from a computer-generated hologram. That is, we design a mask that results in a desired arbitrary complex amplitude distribution along the propaga-tion (z) direction at a given single transverse (x,y) point. Holographic methods for creating a longitu-dinal profile appear in Refs. 2-5. However, these holograms were synthesized to construct the spe-cific case of Bessel beams only. Here we extend this concept to the general case of constructing any desired longitudinal profile by computer-generated holograms. Rosen6 proposed another technique to control the axial distribution, using synthetic apertures. This method was based on the Rayleigh-Sommerfeld diffraction theory and thus was applicable to the entire axial region (from the near to the far field). However, two problems arise with this method. First, the iterative algorithm for cal-culating the hologram sometimes diverges, and secondly, during the synthesis procedure the variable z must be changed to z2, which causes differences between the simulated axial profile and the actual profile and leads to a degradation in the image quality. In this study these problems are eliminated. Assume that an arbitrary transparency g(x,, yl) is placed at the first focal plane of a lens of infinite diameter (see Fig. 1). Using the Fresnel approxima-tion, we obtain the three-dimensional complex ampli-tude distribution past the lens7: g'(X 2, Y 2) =exp(jkf) f 7 g(xi,,y,)x j[f-. x) X expi j kf [ (X2- X1) + (y2- yl)2] dxldyl- (2) Substituting Eq. (2) into Eq. (1), changing the order of integration, and performing the integration over x2 and Y2, we obtain, exp[jk(z + f)] A Ai) u~x~yz) = jAf J JL.,,
Joseph Rosen - One of the best experts on this subject based on the ideXlab platform.
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Synthesis of an Arbitrary Axial Field Profile By Computer-Generated Holograms
1994Co-Authors: Joseph Rosen, Amnon YarivAbstract:arency g(x,, y,) is placed at the first focal plane of a lens of infinite diameter (see Fig. 1). Using the Fresnel approximation, we obtain the three-dimensional complex ampli- tude distribution past the lens7: exp(jkz) u(x,y,z)- jAz f f g'(x,,y2) jk (1) where (x2, y, are the coordinates of the lens's plane and g'(x2, Y2) is the field amplitude just in front of the lens, which is given by (2) Substituting Eq. (2) into Eq. (1), changing the order of integration, and performing the integration over x2 and y2, we obtain u(x,y,z)= exp[jk(z + f)] /_ j Af f g(x, y) - . k(z - k yyl dxidy x exPb-j 2f2f)(x + y, 2) - j-(xx + . (3) We note that, at the second focal plane z = f, Eq. (3) yields the familiar Fourier integral relation between u(x,y,z = f) and g(x,y). Here we are interested in the field distribution along the z axis (x, y = 0). Changing to polar coordinates, (x, Yl) (r, 0), we see that Eq. (3) becomes u(z) = exp[jk(z + f] exp[ k(f).]d. 2jAf fo t(p) -j - = exp[
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Synthesis of an arbitrary axial field profile by computergenerated holograms
1994Co-Authors: Joseph Rosen, Amnon YarivAbstract:Computer-generated holograms are employed to design any desired intensity distribution along the propagation axis for a finite specified distance. A computer-generated hologram is most commonly a transmission mask employed to construct a de-sired transverse image. ' In this Letter we consider the problem of constructing a longitudinal image from a computer-generated hologram. That is, we design a mask that results in a desired arbitrary complex amplitude distribution along the propaga-tion (z) direction at a given single transverse (x,y) point. Holographic methods for creating a longitu-dinal profile appear in Refs. 2-5. However, these holograms were synthesized to construct the spe-cific case of Bessel beams only. Here we extend this concept to the general case of constructing any desired longitudinal profile by computer-generated holograms. Rosen6 proposed another technique to control the axial distribution, using synthetic apertures. This method was based on the Rayleigh-Sommerfeld diffraction theory and thus was applicable to the entire axial region (from the near to the far field). However, two problems arise with this method. First, the iterative algorithm for cal-culating the hologram sometimes diverges, and secondly, during the synthesis procedure the variable z must be changed to z2, which causes differences between the simulated axial profile and the actual profile and leads to a degradation in the image quality. In this study these problems are eliminated. Assume that an arbitrary transparency g(x,, yl) is placed at the first focal plane of a lens of infinite diameter (see Fig. 1). Using the Fresnel approxima-tion, we obtain the three-dimensional complex ampli-tude distribution past the lens7: g'(X 2, Y 2) =exp(jkf) f 7 g(xi,,y,)x j[f-. x) X expi j kf [ (X2- X1) + (y2- yl)2] dxldyl- (2) Substituting Eq. (2) into Eq. (1), changing the order of integration, and performing the integration over x2 and Y2, we obtain, exp[jk(z + f)] A Ai) u~x~yz) = jAf J JL.,,
Canute Vaz - One of the best experts on this subject based on the ideXlab platform.
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Correction to Quantifying Short-Lived Events in Multistate Ionic Channel Measurements.
ACS nano, 2015Co-Authors: Arvind Balijepalli, Jessica Ettedgui, Andrew T. Cornio, Joseph W. F. Robertson, Kin P. Cheung, John J. Kasianowicz, Canute VazAbstract:An inconsistent boundary condition in Eq 2 was corrected. This change does not in any way alter the results or the conclusions of the paper. Page 1549, column 1, line 11. The text reads as follows: “In Laplace space, each transition is modeled with a Heaviside step function, Rp(s) = ΔRp/s, where ΔRp is the instantaneous change in pore resistance, per unit time. We can obtain an expression for the nanopore current response of a single transition by Substituting Eq 1 into I(s) = Va/Z(s) and simplifying 2 where α = (1/ΔRp + Cm)Va and τ = (Rcis + Rtrans)(1/ΔRp + Cm). The inverse Laplace transform of Eq 2, yields an exponentially decaying time-domain current response 3 where i0 is the open channel current offset.” It should be replaced with the following: “The circuit response for each transition (with characteristic pore resistance Rp) is modeled with a voltage step, Va(s) = Va/s. From Eq 1 and Ohm’s law, the ionic current is then I(s) = Va(s)/Z(s). Upon simplification 2 where a = Va/(Rcis + Rtrans), b = 1/(RpCm), and τ = RpCm(Rcis + Rtrans)/(Rp + Rcis + Rtrans). The inverse Laplace transform of Eq 2 yields an exponentially decaying time-domain current response 3 where α = a(1 – bτ) and i0 = abτ is the open channel current offset.”
Arvind Balijepalli - One of the best experts on this subject based on the ideXlab platform.
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Correction to Quantifying Short-Lived Events in Multistate Ionic Channel Measurements.
ACS nano, 2015Co-Authors: Arvind Balijepalli, Jessica Ettedgui, Andrew T. Cornio, Joseph W. F. Robertson, Kin P. Cheung, John J. Kasianowicz, Canute VazAbstract:An inconsistent boundary condition in Eq 2 was corrected. This change does not in any way alter the results or the conclusions of the paper. Page 1549, column 1, line 11. The text reads as follows: “In Laplace space, each transition is modeled with a Heaviside step function, Rp(s) = ΔRp/s, where ΔRp is the instantaneous change in pore resistance, per unit time. We can obtain an expression for the nanopore current response of a single transition by Substituting Eq 1 into I(s) = Va/Z(s) and simplifying 2 where α = (1/ΔRp + Cm)Va and τ = (Rcis + Rtrans)(1/ΔRp + Cm). The inverse Laplace transform of Eq 2, yields an exponentially decaying time-domain current response 3 where i0 is the open channel current offset.” It should be replaced with the following: “The circuit response for each transition (with characteristic pore resistance Rp) is modeled with a voltage step, Va(s) = Va/s. From Eq 1 and Ohm’s law, the ionic current is then I(s) = Va(s)/Z(s). Upon simplification 2 where a = Va/(Rcis + Rtrans), b = 1/(RpCm), and τ = RpCm(Rcis + Rtrans)/(Rp + Rcis + Rtrans). The inverse Laplace transform of Eq 2 yields an exponentially decaying time-domain current response 3 where α = a(1 – bτ) and i0 = abτ is the open channel current offset.”
Aoxue Peng - One of the best experts on this subject based on the ideXlab platform.
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Fan Sub-Equation method with Improved Algorithms for Travelling Wave Solutions of Jimbo-Miwa Equation
Proceedings of the 2015 3rd International Conference on Machinery Materials and Information Technology Applications, 2015Co-Authors: Sheng Zhang, Aoxue PengAbstract:In this paper, the (3+1)-dimensional Jimbo-Miwa Equation is solved by Fan sub-Equation method with improved algorithms. As a result, many new and more general travelling wave solutions are obtained including kink-shaped soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions. At a certain limit condition, the obtained Jacobi elliptic periodic wave solutions can degenerate into soliton solutions. It is shown that the improved algorithms of Fan sub-Equation method can lead to such solutions with external linear functions possessing two remarkable evolutionary properties: (i) the wave propagation is skew; (ii) the amplitude enlarges along with the increasing time. Introduction With the development of computer science, some symbolic computation systems like Mathematica or Maple have been used to perform the complex and tedious computation on computers for constructing exact solutions of nonlinear evolution Equations (NLEEs), such as those in [1-10]. Searching for exact travelling wave solutions of NLEEs plays an important role in the study of nonlinear physical phenomena. In 2003, the so-called Fan sub-Equation method [11] was proposed for solving NLEEs and received many applications [12-14]. Recently, Zhang and Peng [15] improved Fan sub-Equation by modifying its algorithms. One of the advantages of this improved algorithms can lead to such solutions with external linear functions of some given NLEEs. This present paper is motivated by the desire to show the effectiveness and advantage of the improved algorithms [15] through the (3+1)-dimensional Jimbo-Miwa Equation [9]: 3 3 2 3 = 0. xxxy y xx x xy yt xz u u u u u u u + + + − (1) Exact solutions In this section, we consider the (3+1)-dimensional Jimbo-Miwa Equation (1). We take the following travelling wave transformation: = ( , , , ) = ( ), = , u u x y z t u ax by cz t x x ω + + − (2) where , a , b c and ω are constants, then Eq. (1) is reduced into an ODE 3 (4) 2 6 (3 2 ) = 0. a bu a bu u ac b u ω ′ ′′ ′′ + − + (3) We then integrate Eq. (3) once with respect to x and set the integration constant to zero, Eq. (3) becomes 3 (3) 2 2 3 ( ) (3 2 ) = 0. a bu a b u ac b u ω ′ ′ + − + (4) Setting = u v ′ , we have 3 2 2 3 (3 2 ) = 0. a bv a bv ac b v ω ′′ + − + (5) 3rd International Conference on Machinery, Materials and Information Technology Applications (ICMMITA 2015) © 2015. The authors Published by Atlantis Press 674 According to the improved Fan sub-Equation method [15] we suppose that Eq. (5) has the following formal solution 2 2 1 0 = ( ) ( ) . v α φ x α φ x α + + (6) where ( ) φ x satisfis a second-order linear ordinary differential Equation (ODE): 2 2 3 4 0 1 2 3 4 ( ) = ( ) ( ) ( ) ( ). h h h h h φ x φ x φ x φ x φ x ′ + + + + (7) Substituting Eq. (6) along with Eq. (7) into Eq. (5) and collecting all terms with the same order of ( ) φ x together, then setting each coefficient of the polynomial to zero, we derive a set of algebraic Equations for , a , b , c , ω 0 , α 1 α and 2 α as follows: 2 2 3 4 0 1 2 3 4 ( ) = ( ) ( ) ( ) ( ). h h h h h φ x φ x φ x φ x φ x ′ + + + + (8) 0 3 3 2 2 1 1 2 0 0 0 0 ( ) : 4 6 4 6 = 0, a b h a b h a b b ac φ x α α α ωα α + + − − (9) 1 3 3 2 1 2 2 1 1 2 1 1 ( ) : 3 6 2 3 = 0, a b h a b h a b b ac φ x α α α α ωα α + + − − (10) 2 3 3 2 2 2 1 3 2 2 1 0 2 2 2 ( ) : 3 8 6 12 4 6 = 0, a b h a b h a b a b b ac φ x α α α α α ωα α + + + − − (11) 3 3 3 2 1 4 2 3 1 2 ( ) : 2 5 6 = 0, a b h a b h a b φ x α α α α + + (12) 4 3 2 2 2 4 2 ( ) : 2 = 0. a b h a b φ x α α + (13) With the help of Mathematica, from this set of algebraic Equations we obtain five cases as follows: Case 3.1, when 3 1 0 = = = 0 h h h : 3 2 2 4 1 0 3 4 = 2 , = 0, = 0, = , 2 ac a bh ah b α α α ω − + − (14) 3 2 2 4 1 0 2 3 4 4 = 2 , = 0, = , = . 3 2 ac a bh ah ah b α α α ω − − − − (15) We then obtain 3 2 2 4 3 4 = 2 ( ), = , 2 ac a bh v ah b φ x ω − + − (16) 3 2 2 4 2 3 4 4 = 2 ( ) , = . 3 2 ac a bh v ah ah b φ x ω − − − − (17) We substitute the general solutions [11] of Eq. (7) into Eqs. (16) and (17), respectively, and use Eq. (6), then three types of travelling wave solutions of Jimbo-Miwa Equation (1) are obtained. (i) If 2 > 0, h 4 0, h we obtain two triangular periodic solutions 2 2 1 = 2 tan( ) , u a h h d x − − − + 3 2 3 4 = , 2 ac a bh ax by cz t b x − + + + (20)
