Abstract:Based on the Snyder and Mitchell model, a closed-form propagation expression of astigmatic sin-Gaussian beams through strongly nonlocal nonlinear media (SNNM) is derived. The evolutions of the intensity distributions and the corresponding wave front dislocations are discussed analytically and numerically. It is generally proved that the light field distribution varies periodically with the propagation distance. Furthermore, it is demonstrated that the astigmatism and edge dislocation nested in the initial sin-… Show more
“…The extra dislocation closest to the center on the right is found on the horizontal axis and is 'being prepared' to become the sixth dislocation as soon as the fractional part gets close to 1. Figure 2(c) depicts the field amplitude in (7), with its phase presented in figure 2(a). Here, just three central intensity nulls can be seen on the horizontal axis and although the rest nulls are not seen but their position can be extracted from the phase distribution in figure 2 wavelength of light was λ = 532 nm, and the Gaussian beam radius was w = 500 µm.…”
Section: Results Of the Numerical Simulationmentioning
confidence: 99%
“…A significant conclusion that can be made from the numerical simulation results is as follows: considering that the fractional number ν enters equation ( 7) only as a term of the constant before the exponential function and the first parameter of the Tricomi function, it is the latter which is responsible for the birth of extra dislocations and their evolution with varying fractional part of ν. From equation (7), the OVs are seen to be born at Tricomi function zeros lying on the horizontal axis because the Tricomi function argument becomes real at y = 0. For the case of an integer-order edge dislocation, ν = n, the Tricomi function zeros coincide with real zeros of the Hermite polynomial H n (x), lying on the horizontal axis at the crosssection of beam (8).…”
Section: Discussion Of Resultsmentioning
confidence: 99%
“…For instance, the propagation of a HG beam in a 4 × 4 optical system, including an astigmatic one, was investigated in [2,3], whereas [4] was concerned with a Hermite-LG beam traveling in an astigmatic mode converter. Also, focusing of a high-order optical vortex (OV) with an astigmatic lens [5,6], conversion of an astigmatic sin-Gaussian beam in a nonlinear medium [7], and an astigmatic mode conversion in a laser cavity [8,9] have been reported. Elliptic optical Gaussian beams with astigmatic phase were investigated in [10,11], at first with the conversion of an HG beam (0, n) by means of a tilted cylindrical lens and later by studying a mode beam in which a conventional OV with the charge n. In the last case, the charge embedded into an elliptic astigmatic Gaussian beam remains unchanged upon propagation, not splitting into a bunch of simple OVs.…”
In this work, it is theoretically and numerically demonstrated that an astigmatic transformation of a νth-order edge dislocation (shaped as a zero-intensity straight line) of a coherent light field – where ν = n+α is a real positive number, n is integer, and 0 < α < 1 is fractional – produces n optical elliptic vortices (screw dislocations) with topological charge –1, which are arranged on a straight line perpendicular to the edge dislocation and found at Tricomi function zeros. We also reveal that at a distance from the said optical vortices (OV), an extra OV with charge –1 is born on the same straight line, which departs to the periphery with α tending to zero, or gets closer to the n OVs with α tending to 1. Additionally, we find that a countable number of OVs (intensity nulls) with charge –1 are produced at the field periphery and arranged on diverging hyperbolic curves equidistant from the straight line of the n main intensity nulls. These additional OVs, which we term as “escort”, either approach the beam center, accompanying the extra “companion” OV if 0 < α < 0.5, or depart to the periphery, whereas the “companion” keeps close to the main OVs if 0.5 < α < 1. At α = 0 or α = 1, the “escort” OVs are shown to be at infinity. At fractional ν, the topological charge (TC) of the whole optical beam is theoretically shown to be infinite. Numerical simulation results are in agreement with the theoretical findings.
“…The extra dislocation closest to the center on the right is found on the horizontal axis and is 'being prepared' to become the sixth dislocation as soon as the fractional part gets close to 1. Figure 2(c) depicts the field amplitude in (7), with its phase presented in figure 2(a). Here, just three central intensity nulls can be seen on the horizontal axis and although the rest nulls are not seen but their position can be extracted from the phase distribution in figure 2 wavelength of light was λ = 532 nm, and the Gaussian beam radius was w = 500 µm.…”
Section: Results Of the Numerical Simulationmentioning
confidence: 99%
“…A significant conclusion that can be made from the numerical simulation results is as follows: considering that the fractional number ν enters equation ( 7) only as a term of the constant before the exponential function and the first parameter of the Tricomi function, it is the latter which is responsible for the birth of extra dislocations and their evolution with varying fractional part of ν. From equation (7), the OVs are seen to be born at Tricomi function zeros lying on the horizontal axis because the Tricomi function argument becomes real at y = 0. For the case of an integer-order edge dislocation, ν = n, the Tricomi function zeros coincide with real zeros of the Hermite polynomial H n (x), lying on the horizontal axis at the crosssection of beam (8).…”
Section: Discussion Of Resultsmentioning
confidence: 99%
“…For instance, the propagation of a HG beam in a 4 × 4 optical system, including an astigmatic one, was investigated in [2,3], whereas [4] was concerned with a Hermite-LG beam traveling in an astigmatic mode converter. Also, focusing of a high-order optical vortex (OV) with an astigmatic lens [5,6], conversion of an astigmatic sin-Gaussian beam in a nonlinear medium [7], and an astigmatic mode conversion in a laser cavity [8,9] have been reported. Elliptic optical Gaussian beams with astigmatic phase were investigated in [10,11], at first with the conversion of an HG beam (0, n) by means of a tilted cylindrical lens and later by studying a mode beam in which a conventional OV with the charge n. In the last case, the charge embedded into an elliptic astigmatic Gaussian beam remains unchanged upon propagation, not splitting into a bunch of simple OVs.…”
In this work, it is theoretically and numerically demonstrated that an astigmatic transformation of a νth-order edge dislocation (shaped as a zero-intensity straight line) of a coherent light field – where ν = n+α is a real positive number, n is integer, and 0 < α < 1 is fractional – produces n optical elliptic vortices (screw dislocations) with topological charge –1, which are arranged on a straight line perpendicular to the edge dislocation and found at Tricomi function zeros. We also reveal that at a distance from the said optical vortices (OV), an extra OV with charge –1 is born on the same straight line, which departs to the periphery with α tending to zero, or gets closer to the n OVs with α tending to 1. Additionally, we find that a countable number of OVs (intensity nulls) with charge –1 are produced at the field periphery and arranged on diverging hyperbolic curves equidistant from the straight line of the n main intensity nulls. These additional OVs, which we term as “escort”, either approach the beam center, accompanying the extra “companion” OV if 0 < α < 0.5, or depart to the periphery, whereas the “companion” keeps close to the main OVs if 0.5 < α < 1. At α = 0 or α = 1, the “escort” OVs are shown to be at infinity. At fractional ν, the topological charge (TC) of the whole optical beam is theoretically shown to be infinite. Numerical simulation results are in agreement with the theoretical findings.
“…Асимптотика функции Трикоми (5), которая входит как сомножитель в амплитуду поля (7), имеет вид [26]:…”
Section: нули функций куммера и трикомиunclassified
“…В [5,6] исследовалась фокусировка астигматической линзой оптического вихря высокого порядка. Преобразования астигматического sin-Гауссова пучка в нелинейной среде рассматривались в [7]. В [8,9] исследовалось астигматическое модовое преобразование внутри лазерного резонатора.…”
It is shown theoretically that an astigmatic transformation of an edge dislocation (straight line of zero intensity) of the ν-th order (ν=n+α is a real positive number, n is integer, 0<α<1 is the fractional part of the number) forms at twice the focal length from a cylindrical lens n optical elliptical vortices (screw dislocations) with a topological charge of –1, located on a straight line perpendicular to the edge dislocation. Coordinates of these points are zeros of the Tricomi function. At some distance from these vortices and on the same straight line, another additional vortex with a topological charge of –1 is also generated, which moves to the periphery if α decreases to zero, or approaches n vortices if α tends to 1. In addition, at the periphery in the beam cross-section, a countable number of optical vortices (intensity zeros) are formed, all with a topological charge of –1, which are located on diverging curved lines (such as hyperbolas) equidistant from a straight line on which the main n intensity zeros are located. These "accompanying" vortices approach the center of the beam, following the additional "passenger" vortex, if 0<α<0.5, or move to the periphery, leaving the "passenger" next to the main vortices, if 0.5<α<1. At α=0 and α=1, the "accompanying" vortices are situated at infinity. The topological charge of the entire beam at fractional ν is infinite. The numerical simulation confirms theoretical predictions.
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