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Lenses
Published in Roshan L. Aggarwal, Kambiz Alavi, Introduction to Optical Components, 2018
Roshan L. Aggarwal, Kambiz Alavi
Focal length of a lens is a function of the distance h of the object rays from the optical axis. Normally, the focal length of a lens is specified for values of h, which are much smaller than the focal length, that is, in the paraxial approximation. For large values of h, the paraxial approximation is not valid. Consequently, the image of an on-axis object is degraded if the diameter of the lens is not much smaller than its focal length. This image defect is known as spherical aberration. The magnitude of the spherical aberration depends on the shape of the lens. The shape factor is given by (Jenkins and White 1976) () q=R2+R1R2−R1
Geometrical optics
Published in Timothy R. Groves, Charged Particle Optics Theory, 2017
Assuming r′2≪1,C=0in (2.136), and retaining only terms through order r, one obtains (2.123, 2.130, 2.136, 2.137) the approximation 2+Φ1+ΦΦr′′+Φ′r′+12Φ′′r+B24(1+Φ)r=0. This is a linear second order equation for the radial position r(z) of a meridional ray. It is only accurate for rays close to the optic axis, and this approximation is therefore known as the paraxial approximation. A purely electrostatic field has Β = 0, and a purely magnetic field has Φ = const.
Rays and Wavefronts
Published in Julio Chaves, Introduction to Nonimaging Optics, 2017
In the paraxial approximation, it is assumed that light rays have paths through the optical system such that the trajectories keep almost parallel to the optical axis, and that the changes in direction are small. This does not mean that light rays cannot spatially propagate far from the optical axis; only the angles with them are small. Light rays can become distant from the optical axis by traveling along a long path. Since x1′ and x2′ describe the slope of the light ray relative to the optical axis x3, these quantities should be small. We then have x1′≪1andx2′≪1. From Equation 15.30, we get the following approximation:
Propagation dynamics of vortex electromagnetic waves in dispersive left-handed materials
Published in Waves in Random and Complex Media, 2023
Yuanfei Hui, Zhiwei Cui, Shenyan Guo, Yiping Han
On the other hand, much attention has also been paid to the metamaterials [33–36]. As a subset of metamaterials, left-handed materials (LHMs) usually refer to those materials whose permittivity and permeability are negative simultaneously [37]. In such materials, the electric field, the magnetic field, and the wave vector form a left hand triad. Compared with ordinary right-handed materials (RHMs), LHMs exhibit a number of unusual electromagnetic (EM) effects, including negative refraction [38–40], inverse Doppler effect [41,42], inverse Cherenkov radiation [43], and reversed Goos- Hänchen shift [44]. Predictably, these anomalous features of LHMs allow considerable control over the EM waves propagation and change their dynamical properties, such as the energy, momentum, and angular momentum. Now a question arises: how the propagation and dynamical properties of vortex EM waves are influenced by the negative refraction index as well as the dispersive property of LHMs? Luo et al. have examined the propagation and dynamical properties of Laguerre-Gaussian (LG) vortex beams in LHMs [45] within the framework of the paraxial approximation [46]. Note that LG beams with paraxial condition, which are the most common form of vortex waves in optics, are not always applicable in radio frequency domain [25]. As well known, the paraxial approximation is invalid for beams with a large divergent angle that is comparable with the wavelength of beams. However, the radio EM waves carrying OAM have a larger divergence angle compared with optical vortex beams. Moreover, within the paraxial approximation, the inherent vectorial property of EM beams is lost because their longitudinal component vanishes, and there is no restriction within the two transverse components. Therefore it is necessity to consider the vortex EM waves without the limitation of the paraxial condition for a better description of their interactions with LHMs. However, up to now, to our knowledge, the propagation dynamics of radio vortex waves without the limitation of the paraxial condition have not been studied yet. It is the purpose of this article to focus on this topic.