Field modes

Weinacht T C, White J L and Bucksbaum P H 1999 Toward strong field mode-selective chemistry J. Phys. Chem. A 103 10 166-8... 

The result of this approximation is that each mode is subject to an effective average potential created by all the expectation values of the other modes. Usually the modes are propagated self-consistently. The effective potentials governing die evolution of the mean-field modes will change in time as the system evolves. The advantage of this method is that a multi-dimensional problem is reduced to several one-dimensional problems. 

Fig.44 TEM image for binary blend of H-shaped (PS-PI)3-PS-(PI-PS)3 with 35 wt% PS stained with 0s04 in bright field mode. From [122], Copyright 2002 Elsevier... 

Figure 7.5 shows images of a Pd/AI2CF catalyst in bright and dark field mode. In the latter, the geometry was chosen such to selectively detect the (111) and (200) diffraction rings of Pd. Features indicated by arrows clearly demonstrate the better contrast and resolution of the dark field image [14],... 

Transmission electron microscopy (TEM) resembles optical microscopy, except that electromagnetic instead of optical lenses are used to focus an electron beam on the sample. Two modes are available in TEM, a bright-freld mode where the intensity of the transmitted beam provides a two-dimensional image of the density or thickness of the sample, and a dark-field mode where the electron diffraction pattern is recorded. A combination of topographic and crystallographic information, including particle size distributions, can be obtained in this way [32],... 

Let us consider the large volume limit L —> oo. In this limit we have a continuous spectrum for the field modes k, and we replace summations over modes by integrations... 

One can see the physical meaning of the operator Y for the case where the field is in a thermal equilibrium state. Indeed, by taking the ensemble average with a Bose-Einstein distribution of field modes at temperature T,... 

Single-mode operation of the waveguides is identified from near-field mode patterns (NFPs). The light intensity of an NFP has a Gaussian distribution with the strongest intensity located at the core center. This waveguide shows single-mode behavior at 1.3 pm. 

In the following, (13.15) is specialized for the case of a homogeneous medium. Field modes are plane waves labeled by transverse wavenumbers kx, ky, a polarization index s = 1,2, and a sign depending on the direction of propagation along z ... 

As particular solutions of Eq. (1.19) we can choose plane waves (also called field modes )... 

That is, each field mode is represented by a classical harmonic oscillator of mode frequency cok = ck. 

Each field mode must also satisfy the Coulomb gauge condition, V Ak(r, t) = 0, which, when substituted into Eq. (1.20), implies that... 

Since, according to Eq. (1.25), the polarization vector is perpendicular to the direction of propagation, each field mode can have only two independent polarization directions. 

Because Eq. (1.19) is linear, its general solution can be expressed as a sum over pll the field modes and their complex conjugates when the radiation field is in a pavity of a finite volume. That is, since A must be real, it satisfies... 

Here, in accord with Eq. (2.9), we have added an extra phase, (j>k, to each plane wave field mode in Eq. (12.15), representing the phase shifts accumulated by the light in the k mode as it travels from the source to the sample. f... 

Figure 3.4. Schematic presentation of the TEM a) bright field and b) dark field modes. The difference lies in the position of the aperture in the bright field mode, the central spot of the diffraction pattern is selected whereas the dark field mode is based on the selection of a diffraction spot.

Alternatively, images can be formed with electrons scattered at high angles (HAADF mode, for High Angle Annular Dark Field mode). An annular detector is placed below the sample and the focused electron beam is scanned over the sample. Interestingly, the contrast is not altered by diffraction phenomena and depends both the thickness and the composition (25). 

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