Polarized waves linearly

Figure 7 shows an aberration-free intensity distribution at the focus of a typical objective lens similar to that used for DLW lithography. Calculations were carried out using a vectorial Debye theory, which accounts for the polarization effects. For the linearly polarized wave it can be seen that the spot is elongated along the polarization vector. To reduce this asymmetry, a X/4-plate can be used to convert the polarization of the incident beam to circular, which can be interpreted as a combination of two mutually perpendicular linearly polarized components. Thus, width of the photomodified line becomes independent of the beam scanning direction in the sample. 

These two cases imply that two linearly polarized waves at right angles to each other, differing in amplitude and differing in phase by 0, ir or a multiple of vt, compound to give a resultant wave which is also linearly polarized but the plane of polarization lies at an angle tan-1 (b/a) or tan-1 ( b/a) to one of them, depending whether the phase difference is an even or an odd multiple of n. 

The decomposition of linearly polarized wave is the reverse of compounding of two plane polarized waves of the same phase angle (8 = 0). Depending on the slope tan-1 (b/a), the amplitudes a and b of the two waves, will differ and can be computed. For fluorescence depolarization studies, these amplitudes will correspond to Ij. and Ig components of the emitted radiation. 

This is a monochromatic linearly polarized wave with electric field vibrating in the x—z plane, and the magnetic field vibrating in the y—z plane. We have adopted the practice of explicitly identifying the plane of vibration [62, p. 29]. Current density associated with E and B is given by Eq. (16) as J = 0. It is stressed that J = 0 is obtained here from the fields, whereas the conventional approach is to assume the current to be zero on the grounds that pe = 0. 

This example is a nonplanar linearly polarized wave. The direction of vibration of the electric field is still along the x axis, while the magnetic field and the Poynting vector are both contained on the y-z plane. The instantaneous direction of magnetic field is along angle 0 given by... 

When R is rational, there is a surprising find for a linearly polarized wave specifically, the average energy along the direction of propagation (the z axis) is the same for the unperturbed [Eq. (41)] and the perturbed cases [Eq. (48)]. This is important because a direct measurement of intensity cannot distinguish between the two physically different situations. 

In Equations 6.1 and 6.2, sp is the electric permittivity and /10 is the magnetic permeability. The solutions of Equations 6.1 and 6.2 are vectors vibrating in coordinate planes perpendicular to each other (Figure 6.1). E oscillates in the xz plane and H in the yz plane when a linearly polarized wave is propagating in the positive direction of the z-axis. 

If the polarization directions of two lineariy polarized light waves, 1 and 2, with identical amplitudes , " = , frequencies v, = vj, and directions of propagation x, are mutually orthogonal, and if the phases of the two waves are identical, B, = Bj = their superposition will produce a new linearly polarized wave... 

According to Equation (I.S) circularly polarized light may be described by the superposition of two linearly polarized waves that are mutually orthogonal and show a phase shift of tt/2 (Figure 3.1). Conversely, linearly polarized light can be viewed as a superposition of right-handed and left-handed circularly polarized light of identical frequency and identical amplitude. 

Figure 8.2 (a) The helix described by the tip of the real electric vector of a plane electromagnetic wave with right-handed polarization in 0, (p, t) coordinates at a fixed point in space, (b) As in (a), but in 6, (p, s) coordinates at a fixed moment in time, (c) As in (b), but for a linearly polarized wave. 

The PEM is made of a piezoelectric transducer that is glued to a ZnSe crystal. The piezoelement converts a periodic voltage to a periodic mechanical (acoustic) wave, which compresses or expands the crystal. This movement changes the refractive index in the x direction and imposes a periodic retardation (or acceleration) of the fix component of the incident linearly polarized wave. The fiy component remains unchanged. The PEM is operated at its resonant frequency (50 kHz). If the optical element is at rest, the polarization of the radiation remains unchanged. If the optical element undergoes compression or expansion, the component fix has a positive (retardation) or negative (acceleration) phase shift relative to the phase component of the component fiy. 

Fig. 2. An Illustration of Electromagnetic Radiation can be imagined as a self-propaj ting transverse oscillating wave of electric and magnetic fields. This diagram shows a plane linearly polarized wave propagating from left to light. The electric field is in a vertical plane (E) blue and the magnetic field in a horizontal plane (M) red...

Fig. 9.28 Schematic diagram of polarization gradient (Sisyphus) cooling (a) two counter-propagating linearly polarized waves with orthogonal polarization create a standing wave with z-dependent polarization, (b) Atomic level scheme and Clebsch-Gordan coefficients for a Jg = 1/2 Je = 3/2 transition, (c) Atomic Sisyphus effect in the lin J lin configuration [1169]...

A polarizer is a device that transforms a linear polarized wave into a circular polarized wave, or vice versa. The common principle is simply to decompose the incident field into two components where the phase of one is advanced and the other is delayed such that their difference is 90° while their amplitudes are the same. It appears that Pakan [128] was the first to utilize this principle. Later improvements were introduced by Lemer [129]. These devices were not of the meander-line type, as will be discussed here. These seem to appear first in a paper by Young et al. [130] and were subsequently unproved by Epis [131]. Later, a paper by Terret et al. [132] discussed how to calculate the susceptance of a meander line. All of these contributions were primarily focused on normal angle of incidence while Chu and Lee [133] extended the calculation to include oblique angle of incidence. A recent contribution was supplied by Marino [134], It was apparent that meander-line polarizers gradually deteriorate for higher angles of incidence. The present appendix will demonstrate that introduction of a dielectric profile can greatly improve this calamity. 

Figure 2.8 Schematics ofpolarized lightwaves, with view in direction toward the source on the right. From top to bottom horizontal linear-polarized wave (oscillation in Y-Z plane) vertical linear polarized wave (oscillation in X-Z plane) left-hand circular-polarized wave and right-hand circular-polarized wave...

Figure 10.14 Principle of interference of two Linear polarized waves, Ei(t) and E2(t) with frequency w, for observation at a crossing point P. Cases for the resulting interference intensity are summarized in Box 10.2...

The follow ing step consists in a second change of matrix representation, by introducing a more convenient basis whose eigenvectors are linearly polarized waves propagating within the medium in the forward and backward directions. 

Eq and Bq are constant complex amplitudes, and e- and 2 shall be constant unit vectors. As a consequence the electric field vector always points in the direction of Cy and such an electromagnetic wave is said to be linearly polarized. By superpositions of two such linearly polarized waves with different phases and amplitudes so-called elliptically polarized waves may be constructed. However, we do not need to further discuss this possibility here. 

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