Aplanatic System Research Articles

Equations to design an aplanatic catadioptric freeform optical system.

The present paper introduces a set of equations to design an aplanatic catadioptric freeform optical system. These equations form a partial differential equation system, in which a numerical solution defines the first and last surfaces of the catadioptric freeform optical system, composed of an arbitrary number of reflective/refractive surfaces with arbitrary shapes and orientations. The solution of the equation can serve as an initial setup of a more complex design that can be optimized. An illustrative example is presented to show the methodology introduced in this paper.

Inverse Design of Focused Vector Beams for Mode Excitation in Optical Nanoantennas

We propose a free-space, inverse design of nanostructure's effective mode-matching fields via a backward propagation of tightly focused vector beams to the pupil plane of an aplanatic system of high numerical aperture. First, we study the nanostructure's eigenmodes without considering any excitation fields and then extract the modal near fields in the focal plane. Each modal field is then taken as the desired focal field, the band-limited waves of which are backward propagated to the pupil plane via a reversal of the Richards--Wolf vector diffraction formula. The pupil fields can be designed to be genuinely paraxial by associating the longitudinal electric/magnetic field component with the radial one on the reference sphere. The inversely designed pupil field in turn is propagated forwardly into the focal region to generate the designed focal field, whose distribution over the nanostructure's surface is used to evaluate the overlap between the designed focal field and the modal fields, i.e., the modal expansion coefficients. Studies for a silicon nanodisk monomer, dimer, and tetramer demonstrate the ability of our inverse approach to design the necessary tightly focused vector field that can effectively and exclusively match a certain eigenmode of interest. Compared with the forward beam-shaping method, the inverse design approach tends to yield quantitatively more precise mode-matching field profiles. This work can have a significant impact on optical applications that rely on controllable and tunable mode excitation and light scattering.

Design of a pair of aplanatic mirrors.

In this paper, a new, to the best of our knowledge, differential equation for designing a pair of aplanatic mirrors is introduced. The differential equation is a direct consequence of the Fermat principle and Abbe sine condition. If it is solved, the solution expresses the shape of a pair of mirrors such that they form an aplanatic system. The differential equation has been solved numerically. We have also tested the performance of the pair of mirrors, which is as predicted by the theory.

Exact equations to design aplanatic sequential optical systems

We present the exact differential equations to design an aplanatic sequential optical system, a system that is free of spherical aberration and linear coma. We get the exact set of equations from the Fermat principle and the Abbe sine condition. We solve the mentioned set of equations by implementing the Runge-Kutta algorithm. We test the solutions using commercial ray-tracing software and confirm the expected behavior of the optical system.

Three-dimensional transfer function of optical microscopes in reflection mode.

Three-dimensional (3D) transfer functions build the basis for a comprehensive characterization of optical imaging systems in the spatial frequency domain. Utilizing the projection-slice theorem, the 2D modulation transfer function of an incoherent imaging system can be derived from a 3D transfer function by integration with respect to the axial spatial frequency. For a diffraction limited microscope with homogeneous incoherent pupil illumination, the modulation transfer function equals the 2D autocorrelation function of a circular disc. However, until now to the best of our knowledge no 3D transfer function has been published, which exactly leads to the 2D modulation transfer function of a diffraction limited microscope in reflection mode. In this article, we derive a formula, which after integration with respect to the axial spatial frequency coordinate perfectly fits to the diffraction limited 2D modulation transfer function. The inverse three-dimensional Fourier transform of the 3D transfer function results in a complex-valued 3D point spread function, from which the depth of field, the lateral resolution and, in addition, the corresponding 3D point spread function of both, a conventional and an interference microscope, can beobtained.

Extremely fast aplanatic space telescope system with a 10-m-diameter spherical primary mirror

This paper proposes an optical system of the telescope for space research. The system comprises a segmented spherical mirror with a diameter of 10 m and a secondary aspherical mirror. The focal length of the telescope is equal to the radius of a spherical primary mirror. The paraxial focal point of the spherical mirror is aligned with the focal point of the whole system. The closest analog of the proposed optical system is the optical system of the Millimetron space observatory (Spektr-M). The physical aperture ratio of the proposed system is ten times that of the Millimetron telescope with the same overall dimensions. The proposed system rigorously satisfies the Abbe sine condition. The focal length of the system is 7070 mm, and the numerical aperture is 0.7071. The Airy disk diameter of the proposed system is more than ten times smaller than the diffraction circle of the Millimetron telescope. The angular field of the system is determined by the selected image quality criteria and specific wavelength.

Generalized Lorenz–Mie theory of photonic wheels

Plane light waves can carry angular momentum which is usually longitudinal (i.e. oriented along the axis of propagation). However, in confined (focused) light waves it can be oriented transversely (or perpendicular to the direction of propagation). Such electromagnetic fields therefore are called ‘photonic wheels’. We create photonic wheels with an aplanatic system and expand them into vector spherical harmonics. This decomposition is applied to create a generalized Lorenz–Mie theory of photonic wheels. We study the interaction of these beams with gold and silicon spheres and reveal an appearance of localized longitudinal angular momentum in the scattered field. Lastly, we employ the multiple scattering method and simulate a chiral cluster of nanoparticles in the focal plane of a photonic wheel.

Hybrid polarization induced transverse energy flow

Nonzero transverse energy flow, which describes phenomenon in which the energy flux of localized light propagates in a plane perpendicular to the optical axis, has attracted enormous interest recently due to its useful application in micromanipulation. We show that the appearance of transverse energy flow in the focal plane of an aplanatic high numerical aperture focusing system is possible. We demonstrate our approach by specially tailoring the input state of polarization. Calculations reveal that number of transverse energy flow rings is controllable and depend on azimuthal index of the input field, thereby giving rise to tunable manipulating locations in optical trapping.

Analytic aplanatic singlet lens: setting and design for three-point objects and images in the meridional plane

We present the setting and design of an aplanatic optical system for three-point objects and images. The system is composed of a singlet aspheric lens and its entrance pupil. The aspheric lens and the entrance pupil are such that the three-point images and objects lay on the same plane. The image at the center is free of spherical aberration, and the other off-axis images are free of coma.

Spin angular momentum density in the tight focus of a light field with phase and polarization singularities

For a light field with both phase and polarization singularities at its center, expressions are obtained that describe the distribution of the spin angular momentum (SAM) density in the sharp focal spot of an aplanatic system. These expressions include the radial, azimuthal, and longitudinal SAM components. As special cases, focusing of optical vortices with radial, azimuthal, and saddle polarizations is studied. Using the Bessel beam as an example, it is shown that in some areas in the focal plane the longitudinal SAM component is zero (resulting in a photonic wheel), while in others it is an order of magnitude less than the transverse component.

Analytical inversion of the focusing of high-numerical-aperture aplanatic systems.

We propose a method for an analytical inversion of the electric and magnetic fields at the focus of a high-NA aplanatic system to obtain incident light beam distribution. Our approach is based on an inverse Fourier transform of the Richards-Wolf formalism for targeted longitudinal fields along the radial or axial directions at the non-paraxial focus. Analytical solutions are discussed for both axial and radial focal fields for a radially polarized incident light beam, and a criterion is defined to access a physically valid solution. We also validate the method according to results found in the literature. Finally, we show how the method can be generalized to other incident field distributions.

The Non-Vortex Inverse Propagation of Energy in a Tightly Focused High-Order Cylindrical Vector Beam

In this paper, the tight focusing of high-order cylindrical vector beams (beams with polarization singularity) was investigated. Using the Richards–Wolf formalism, there were obtained expressions for all projections of the electric and magnetic light fields with m -order polarization singularity in the focus of the aplanatic system. Also expressions for the longitudinal component of the Poynting vector were obtained. It was shown that these beams produce in the focal areas with the direction of the Poynting vector opposite to the direction of propagation of the beam. Moreover, the negative values could be comparable in absolute value with positive values; however, this strong inverse energy flow is obtained only while laser light is focused by a lens with high numerical aperture. Of particular interest is the case when the beam order is two ( m = 2). In this case, the region where the Poynting vector longitudinal projection is negative is located on the optical axis. If the order of the beam is more than two ( m > 2), than the reverse flow occurs near the optical axis and has a shape of a “tube.” Moreover, the width of the negative values region (the diameter of the “tube”) increases with increasing order of the beam, however, the absolute value of energy backflow decreases. Earlier, we reported on the observation of a spiral negative energy flow from the center of the focal plane of a focused vortex beam. In this paper, the negative propagation of a laminar near-axis energy flow is reported.

Manipulating the magnetic energy density and energy flux by cylindrically symmetric state of polarization

Using the Richards and Wolf formulas for an arbitrary cylindrical vector (CV) beam, we obtain explicit expressions for all components of the electric and magnetic field strength vectors near the focus, as well as expression for the energy flux in an aplanatic optical system. Based on such analytical models, it reveals that the intensity pattern of the magnetic field at the focus can be tailored by appropriately adjusting the initial phase, peak-centered, doughnut, and flat-topped magnetic fields can be achieved using this method. For the energy flows, in contrast, they are almost the same for arbitrary CV beams, which exhibit hollow shaped patterns for both the transverse and longitudinal components. Unlike the longitudinal component, however, the hollow shaped pattern of the transverse component is separated into two regions, arriving from the null transverse energy flow in the focal plane. Besides, the directions of the transverse energy flow are reversed between these two regions, which are directed inwardly and outwardly along the radial direction, respectively.

Pancharatnam-Berry phase shaping for control of the transverse enhancement of focusing.

We show that elongating a tightly focused field in the direction perpendicular to the optical axis is possible. We demonstrate our approach by specially shaping the Pancharatnam-Berry (PB) phase. Moreover, the analytical formulae required to calculate the strength vectors and energy flux of the three-dimensional electromagnetic fields near the focus of an aplanatic optical system are derived using the Richards and Wolf vectorial diffraction methods. Calculations reveal that the transverse enhancement is controllable and depend on the phase index in the PB phase, thereby giving rise to a focus with tunable length and subwavelength width in the focal plane.

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.

Helical reverse flux of light of a focused optical vortex

Using the Richards–Wolf formalism, we show that if an optical vortex with a topological charge of 3 and with left (opposite) circular polarization is sharply focused by an aplanatic system, then in the focal plane there is a reversed flux of light energy near the optical axis (on the axis itself it is zero), which is only 2–3 times less than the forward energy flux in the focus. It is also shown that near the optical axis the reverse flux propagates along a counterclockwise helix (spiral) around the optical axis. The reverse flux near the optical axis is also demonstrated by the FDTD-simulation of diffraction of a circularly polarized plane wave by a third-order spiral zone plate with a numerical aperture of about 1. A Rayleigh microparticle trapped in the area of such a focus should move in the opposite direction with respect to the direction of beam propagation.

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.

Синтез и исследование сверхширокополосной планарной трехзеркальной системы.

Cylindrical mirror and mirror - lens systems are used as radiating systems for multibeam antenna arrays. In this case, the planar waveguide, in which the feed element is located, makes a 180-degree turn and forms a two-layer structure, the second floor of which is occupied by an array. Such a design is more compact than a one-story one, in which the Rotman lens is usually used as a focusing element. A two-mirror aplanatic system allows reducing aberrations and expanding the scanning sector in comparison with a single-mirror one, however, for the arrangement of the grating above the planar waveguide, one more rotation of the planar waveguide is required, i.e. a transition to a three-mirror system is necessary. This construction is considered in this paper. It is shown that the third mirror allows reducing aberrations in comparison with the two-mirror system. The work consists of two parts. In the first part, problems of synthesis and minimization of aberrations of a cylindrical three-mirror aplanatic system are considered. As a result of optimization of the parameters, the angle of view of 100 degrees is realized. In the second part, the scanning properties of a four-layer antenna based on a synthesized three-mirror system were studied. The results of the experiment confirmed the size of the scanning sector and the broadband properties of the multi-beam antenna based on a synthesized three-mirror system.

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.

用双液芯柱透镜测量蔗糖水溶液的液相扩散系数

Based on the consideration of the high resolution of the spatial resolution of the refractive index of the double liquid-core cylindrical lens(DLCL), the liquid-phase diffusion coefficients of different concentrations of aqueous sucrose solution are measured at room temperature(25℃) using two methods. Method 1:equivalent RI(refractive index) method is used to calculate the liquid phase diffusion coefficient by recording the time-dependent change of a specific refractive index layer during diffusion. Method 2:instantaneous diffusion analytical method is used to determine the liquid diffusion coefficient by reading the relationship between image width and diffusion position in an instantaneous diffusion image. The front liquid core of the DLCL serves as a diffusion cell and a main imaging element, and the rear liquid core serves as an aplanatic auxiliary system. The spherical aberration at a particular thin liquid layer can be reduced as needed with a DLCL, and the spherical aberration advantage within a certain range of refractive index can also be reduced. Both methods have the characteristics of high measurement accuracy. The relative errors between the measured results and the literature values of the two methods are less than 1.3% and 3.9%, respectively, indicating that the measurement system is stable and reliable and the measurement results are accurate when the liquid-phase diffusion coefficient is measured with a DLCL.