Richards-Wolf Formulae Research Articles

Vector Beams with Only Transverse Intensity at Focus

In this work, the tight focusing of vector beams with azimuthal polarization and beams with a V-line of polarization singularity (sector azimuthal polarization) was simulated numerically using the Richards–Wolf formulas. It was demonstrated that in a tight focus for these beams, there is no longitudinal component of the electric field. Previously, a similar effect was demonstrated for azimuthally polarized light only. The longitudinal component of the spin angular momentum for these beams was calculated, and the possibility of creating sector azimuthally polarized beams (beams with V-line singularities) using vector waveplates was demonstrated.

Vector Light Field Immediately behind an Ideal Spherical Lens: Spin–Orbital Conversion, Additional Optical Vortices, Spin Hall Effect, Magnetization

The Richards–Wolf formulas not only adequately describe a light field at a tight focus, but also make it possible to describe a light field immediately behind an ideal spherical lens, that is, on a converging spherical wave front. Knowing all projections of light field strength vectors behind the lens, the longitudinal components of spin and orbital angular momenta (SAM and OAM) can be found. In this case, the longitudinal projection of the SAM immediately behind the lens either remains zero or decreases. This means that the Spin–Orbital Conversion (SOC) effect where part of the “spin goes into orbit” takes place immediately behind the lens. And the sum of longitudinal projections of SAM and OAM is preserved. As for the spin Hall effect, it does not form right behind the lens, but appears as focusing occurs. That is, there is no Hall effect immediately behind the lens, but it is maximum at the focus. This happens because two optical vortices with topological charges (TCs) 2 and −2 and with spins of different signs (with left and right circular polarization) are formed right behind the lens. However, the total spin is zero since amplitudes of these vortices are the same. The amplitude of optical vortices becomes different while focusing and at the focus itself, and therefore regions with spins of different signs (Hall effect) appear. A general form of initial light fields which longitudinal field component is zero at the focus was found. In this case, the SAM vector can only have a longitudinal component that is nonzero. The SAM vector elongated only along the optical axis at the focus is used in magnetization task.

Tight focusing of circularly polarized light limited by semicircular aperture

In this work, the focusing of a circularly polarized plane wave (wavelength 532 nm) was simulated by a lens with a numerical aperture NA = 0.95. The wave front was considered flat. When integrating according to the Richards-Wolf formulas, the semicircular aperture was set by limiting the azimuthal angle from 0 to π. It was shown that when focusing light with right and left circular polarization, the focal spot turns out to be elliptical - elongated along the y axis, and, depending on the direction of polarization, its center shifts by about 0.05 μm in different directions along the x axis. It was also shown that the reverse flow region is located near the focal spot (at a distance of 0.25 μm from the center). Depending on the direction of polarization, it is located either to the right or to the left of the focal spot. Thus, the polarization state of the incident radiation can be determined from the displacement of the spot in focus.

An orbital energy flow and a spin flow at the tight focus

We have shown that a reverse energy flow (negative projection of the Poynting vector onto the optical axis) at the sharp focus of an optical vortex with topological charge 2 and left-hand circular polarization arises because the axial spin flow has a negative projection onto the optical axis and is greater in magnitude than positive projection onto the optical axis of the orbital energy flow (canonical energy flow). Also, using the Richards-Wolf formulas, it is shown that when focusing a left-handed circularly polarized light, in the region of the on-axis reverse energy flow, the light is right-handed circularly polarized.

Transverse intensity at the tight focus of a second-order cylindrical vector beam

In this paper, an effect of a reverse energy flow at the focus of a second-order cylindrical vector beam which passed through amplitude zone plate was investigated with a scanning near-field optical microscope. A comparison of the intensity distribution detected with a pyramidal metallized cantilever with a hole and the characteristics of the light field calculated using a FDTD method and the Richards-Wolf formulas suggests that the cantilever is sensitive to the transverse intensity component rather than the total intensity or the components of the Poynting vector in the backflow region.

Toroidal polarization vortices in tightly focused beams with singularity

In this paper, we numerically investigated tight focusing of cylindrical vector beams of the sec-ond order using Richards-Wolf formulae. It was shown that intensity rings where the Poynting vector was equal to zero appeared not only in the focal plane but also in nearby planes. For example, a lens with numerical aperture NA=0.95 was shown to generate periodical toroidal vortices with a 0.8-mkm period along the z-axis at a distance of about 0.45 mkm from the axis. The vortices were generated pairwise, with the closest-to-focus vortex having clockwise helicity and the subsequent being anti-clockwise. The vortices were also characterized by saddle points. When focusing an optical beam passed through a narrow annular aperture, no toroidal vortices were observed.

Strong negative longitudinal component of the Poynting vector in a tightly focused cylindrical vector beam

It was shown that tightly focused cylindrical vector beam of m-th order produces reverse energy flow in the focal plane (m=2). If m=3 the reverse energy flow on the axis is equal to zero and near the axis it increases in proportion to a square of the distance to the axis. The azimuthally polarized beam of the m-th order is an example of a polarization vortex. Previously, the reverse energy flow in the focus was obtained only for vortex beams with a topological charge m and circular polarization. Numerically, using the FDTD method and the Richards-Wolf formulae, it was shown that in the focus of the zone plate such laser beams produce regions the Poynting vector opposite to the direction of propagation of the beam.

Vortex energy flow in the tight focus of a non-vortex field with circular polarization

Using Richards-Wolf formulas, we show that an axisymmetric circularly polarized vortex-free field can be focused into a sharp subwavelength focal spot, around which there is a region where the light energy flow propagates along a spiral. This effect can be explained by the conversion of the spin angular momentum of the circularly polarized field into the orbital angular momentum near the focus, although the on-axis orbital angular momentum remains zero. It is also shown that a linearly polarized optical vortex with topological charge 2 forms near the focal plane an on-axis reverse energy flow (defined by the negative longitudinal component of the Poynting vector) whose amplitude is comparable with the direct energy flow.

Exploiting the circular polarization of light to obtain a spiral energy flow at the subwavelength focus

Using Richards–Wolf formulas, we show analytically that when strongly focusing a circularly polarized axisymmetric non-vortex field (e.g., a Gaussian beam or zero-order Bessel–Gaussian beam), a spiral energy flow is produced around the sub-wavelength focal spot. This phenomenon can be explained by the conversion of spin angular momentum associated with the circular polarization state into the near-focus orbital angular momentum, with the on-axis angular momentum being strictly zero. It is also shown that a linearly polarized optical vortex with topological charge 2 produces an on-axis near-focus energy backflow whose amplitude (defined by the negative longitudinal component of the Poynting vector) is comparable with that of the incident beam.

Formation of an elongated region of energy backflow using ring apertures

In this paper, we have investigated the focusing of a second-order cylindrical vector beam by using a high numerical aperture (NA) lens limited by a ring aperture using the Richards-Wolf formulae. It was shown that the range of negative on-axis projections of the Poynting vector could be increased by increasing the depth of focus through the use of a ring aperture. It was shown that when focusing light with a lens with NA = 0.95, the use of a ring aperture limiting the entrance pupil angle to 0.9 of maximum, allows the depth of the region of negative on-axis Poynting vector projections to be four times increased, with the region width remaining almost unchanged and varying from 0.357 to 0.352 of the incident wavelength. Notably, the magnitude of the reverse energy flow was found to be larger than the direct one by a factor of 2.5.

Longitudinal component of the Poynting vector of tightly focused cylindrical vector beam

We numerically investigated the tight focusing of cylindrical vector beam with wavelength of 532 nm by a diffractive lens with a numerical aperture 0.95. The simulation was carried out using the Richards-Wolf formulae. It was shown that the focusing of cylindrical vector beam by a wide-aperture lens can produce intensity distribution with a negative component of the longitudinal component of the Poynting vector in the center of the focal spot.

Inverse energy flux of focused radially polarized optical beams

We report the formation of a negative energy flux in the case of tight focusing of an arbitrary-order radially polarized annular beam. We consider not only the longitudinal component of the Poynting vector (energy flux density) and the region of its negative values, but also the local inverse energy flux as an integral characteristic over the region of negative values. Theoretically, the Richards-Wolf formulas in the Debye approximation are used to maximize the negative value of the energy flux density on the optical axis in the focal region for the second-order radial polarization ($m=2$). In this case, the local inverse energy flux, an integral characteristic in the bulk region of negative values, increases with increasing radial polarization order, i.e., for $mg2$. Jones matrices are employed to show that the result obtained will be valid for azimuthal polarization, as well as for a circularly polarized vortex beam. The results of numerical simulation are in good agreement with theoretical calculations. In addition, taking into account the approximate nature of the Richards-Wolf formulas, we also additionally model the tight focusing of a circular beam based on the solution of Maxwell's equation using the finite element method. It is shown that the local inverse energy flux in the case of the second-order radial polarization will be 1.5 times less than that in the case of the third-order radial polarization. In turn, the use of the fourth-order radial polarization makes it possible to increase the local inverse energy flux by a factor of 2.

Energy backflow in the focus of an optical vortex

By employing Richards-Wolf formulae for an aplanatic system, we show that when tightly focusing a left-hand circularly polarized optical vortex with the topological charge 2 and an arbitrary apodization function, an on-axis energy backflow occurs in the focus plane, which is verified by a negative optical-axis projection of the Poynting vector. This result is numerically reconfirmed using the rigorous FDTD-based analysis of diffraction of a left-hand circularly polarized plane wave by a spiral zone plate with the topological charge 2 and a numerical aperture of ~1. Significantly, the back and direct flows of energy are comparable in magnitude. While the on-axis energy backflow is shown to take place on the entire axis, its maximum is attained in the focal plane, quickly decaying with distance from the focus. The on-axis depth of backflow on which it drops by a factor of two is nearly the same as the depth of focus, while the transverse spot in which the energy backflow occurs is approximately equal to the Airy disk.

Manipulation of the transverse energy flow of azimuthally polarized beam in tight focusing system

Abstract A simple method is proposed to manipulate the transverse energy flow in the focal plane of a tightly focused azimuthally polarized beam. Using the Richards–Wolf formulas, we obtain explicit expressions for all components of the electromagnetic field and the Poynting vector near focus of the tightly focused input field with azimuthal polarization and optical vortex. Further, the redistributions of the transverse energy flow by designing phase masks, including traditional and annular vortex phase masks are explored. Transverse energy flow in the shape of a regular polygon is achieved. As a result, transport absorptive particles can be transported along more flexible paths, which might be help in optical tweezers manipulations.

Energy density and energy flux in the focus of an optical vortex: reverse flux of light energy.

Using the Richards-Wolf formulas for an arbitrary circularly polarized optical vortex with an integer topological charge m, we obtain explicit expressions for all components of the electric and magnetic field strength vectors near the focus, as well as expressions for the intensity (energy density) and for the energy flux (components of the Poynting vector) in the focal plane of an aplanatic optical system. For m=2, from the obtained expressions it follows that the energy flux near the optical axis propagates in the reversed direction, rotating along a spiral around the optical axis. On the optical axis itself, the reversed flux is maximal and decays rapidly with the distance from the axis. For m=3, in contrast, the reversed energy flux in the focal plane is minimal (zero) on the optical axis and increases (until the first ring of the light intensity) as a squared distance from the axis.

Energy backflow in a focal spot of the cylindrical vector beam

Using Richards-Wolf formulae it is shown that a tightly focused azimuthally-radially polarized m-th order laser beam with an arbitrary apodization function produces a reverse energy flow in the focal plane (m=2). If m=3, the reverse energy flow on the axis is equal to zero, increasing in the axis vicinity as the square of the distance to the axis. The azimuthally-radially polarized beam of the m-th order is an example of a polarization vortex. Previously, the reverse energy flow in the focus was obtained only for circularly polarized vortex beams with the topological charge m. Using the FDTD method and the Richards-Wolf formulae, we show numerically that in the focus of a zone plate such laser beams produce regions where the Poynting vector is opposite to the direction of the beam propagation.

A spirally rotating backward flow of light

Using Richards-Wolf formulas, we show that when a left-hand circularly polarized optical vortex with the topological charge 3 is sharply focused in an aplanatic system, a backward near-axis energy flow is observed in the focal plane. While being zero on the axis, the backward flow is only 2-3 times smaller in magnitude than the incident energy flow coming to the focus. It is also shown that near the optical axis the reverse flow propagates spiraling counter-clockwise about the optical axis. The presence of the near-axis backward flow of energy is also shown by the FDTD-aided numerical simulation of diffraction of a circularly polarized plane wave by a third-order spiral zone plate with the NA ≈ 1. A Rayleigh microparticle captured in the focus vicinity is expected to move in the opposite direction to the beam propagation.

Longitudinal component of the Poynting vector of a tightly focused optical vortex with circular polarization

We numerically investigated the focusing of optical vortices with the wavelength λ = 532 nm and right-hand circular polarization by using a diffractive lens with the numerical aperture NA = 0.95. The simulation was carried out using the Richards-Wolf formulae and the FDTD-method. The focusing of optical vortices with topological charges equal to ±1 and ±2 was investigated. It was shown that the focusing of optical vortices with circular polarization by a wide-aperture diffractive lens can produce an intensity distribution with the negative value of the longitudinal component of the Poynting vector.

The near-axis backflow of energy in a tightly focused optical vortex with circular polarization

Using the Richards-Wolf formulae for a diffractive lens, we show that in the focal plane of a sharply focused left-hand circularly polarized optical vortex with the topological charge 2 there is an on-axis backflow of energy (as testified by the negative axial projection of the Poynting vector). The result is corroborated by the FDTD-aided rigorous calculation of the diffraction of a left-hand circularly polarized plane wave by a vortex zone plate with the topological charge 2 and the NA≈1. Moreover, the back- and direct flows of energy are comparable in magnitude. We have also shown that while the backflow of energy takes place on the entire optical axis, it has a maximum value in the focal plane, rapidly decreasing with distance from the focus. The length of a segment along the optical axis at which the modulus of the backflow drops by half (the depth of backflow) almost coincides with the depth of focus, and the transverse circle in which the energy flow is reversed roughly coincides with the Airy disk.

Tightly focused laser light with azimuthal polarization and singular phase

Using simplified Richards-Wolf formulas we show that laser light with azimuthal polarization and singular phase can produce a smaller focal spot than that from a laser beam with radial polarization, other conditions remaining the same. It is numerically shown that when focusing an azimuthally polarized laser beam with phase singularity using a zone plate a 1.3 times smaller focal spot can be attained than when an aplanatic lens is used. A spiral phase plate can be replaced with a phase step with a π-phase shift. In this case the subwavelength focal spot from a laser beam with azimuthal polarization, which is formed near the zone plate surface, loses circular symmetry, while becoming smaller and acquiring an elliptical form with radiuses of 0.273λ and 0.314λ (NA = 1).