Non-absorbing Case Research Articles

Mitigation of hysteresis due to a pseudo-photochromic effect in thermochromic smart window coatings

The aim of thermochromic window coatings is to reduce the energy consumption in the built environment by passively switching between a high solar transmitting state at low temperatures and low solar transmitting state at high temperatures. Previous studies have highlighted the negative impact of phase transition hysteresis on the performance of reflection based thermochromic films. However in the literature, the best reported results have depended on vanadium dioxide nanoparticle composites and anti-reflective structures that modulate light via changes in absorption rather than reflection. In light of these factors, this work aims to demonstrate theoretically, how the effects of phase transition hysteresis and gradient differ between absorbing and non-absorbing thermochromic films. To quantify and compare the performance of films with different transition characteristics, we define a metric based on the varying net energy flux through the window over the course of a year, including thermal energy re-radiated into the building from the film. Specifically, and importantly for the field, we demonstrate that a pseudo-photochromic effect in absorbing thermochromic films mitigates the detrimental effects of phase transition hysteresis and gradient that have been reported for reflection based thermochromic films. We find that for moderate hysteresis widths of 15 °C, the performance of the non-absorbing case drops to ~60% of its initial value whilst the performance of the absorbing film only drops to ~95%. As a result we find that the absorbing case outperforms the non-absorbing case when hysteresis widths are greater than 8 °C.

On an equivalent representation of the Green’s function for the Helmholtz problem in a non-absorbing impedance half-plane

The Green’s function associated with the Helmholtz problem in a non-absorbing impedance half-plane can be expressed as an integral form. In the non-absorbing case, the presence of surface waves presents a challenge in order to obtain accurate approximations. In this work, we present an equivalent representation for this Green’s function, expressed as a sum of analytical terms and bounded integrals. The resulting representation is numerically stable and it can be estimated by any well known robust integration rule for bounded intervals. We provide a detailed description of the equivalent representation and we validate it with numerical experimentation.

Reflection by absorbing periodically stratified media

Existing theory gives the optical properties of a periodically stratified medium in terms of a two by two matrix. This theory is valid also for absorbing media, because the matrix remains unimodular. The main effect of absorption is that the reflection (of either polarization) becomes independent of the number of periods N, and of the substrate properties, provided N exceeds a certain value which depends on the absorption. The s and p reflections are then given by simple formulae. The stop-band structure, which gives total reflection in bands of frequency and angle of incidence in the non-absorbing case, remains influential in weakly absorbing media, causing strong variations in reflectivity. The theory is applied to the frequency dependence of the normal-incidence reflectivity of a quarter-wave stack in which the high-index and low-index layers both absorb weakly. Analytical expressions are obtained for the frequency at which the reflectivity is maximum, the maximum reflectivity, and also for the reflectivity at the band edges of the stop band of the non-absorbing stack.

Negative acoustic radiation forces produced by Bessel beams: acoustic tractor beams and scattering

The theory for negative acoustic radiation forces on objects in various Bessel beams is reviewed [P. L. Marston, J. Acoust. Soc. Am. 120, 3518–3524 (2006); L. K. Zhang and P. L. Marston, Phys. Rev. E 84, 035601 (2011)] together with geometric insight provided by researchers in optics. When examining conditions favorable for negative radiation forces it can be helpful to consider the far field scattering for axisymmetric objects of interest because the axial radiation force can be directly related to the asymmetry in the scattering and the conic angle of the beam. The analysis also serves to clarify the way in which absorption by the object contributes to positive forces and to clarify the influence of helicity. Considering the symmetry properties of the scattering provides additional insight as does the evaluation of radiation forces and wave fields using the computational method of finite elements. In the non-absorbing case it can also be helpful to express the partial-wave amplitudes in terms of phase shifts since in acoustics there is only one phase shift for each partial-wave. [Marston and Thiessen were supported by ONR and Zhang was supported by NASA.]

Scattering of Electromagnetic Waves from Two-Dimensional Randomly Rough Penetrable Surfaces

An accurate and efficient numerical simulation approach to electromagnetic wave scattering from two-dimensional, randomly rough, penetrable surfaces is presented. The use of the Müller equations and an impedance boundary condition for a two-dimensional rough surface yields a pair of coupled two-dimensional integral equations for the sources on the surface in terms of which the scattered field is expressed through the Franz formulas. By this approach, we calculate the full angular intensity distribution of the scattered field that is due to a finite incident beam of p-polarized light. We specifically check the energy conservation (unitarity) of our simulations. Only after a detailed numerical treatment of both diagonal and close-to-diagonal matrix elements is the unitarity condition found to be well satisfied for the nonabsorbing case (U>0.995), a result that testifies to the accuracy of our approach.

Ultrafast propagation of Schrödinger waves in absorbing media

We identify the characteristic times of the evolution of a quantum wave generated by a point source with a sharp onset in an absorbing medium. The "traversal'' or "B\"uttiker-Landauer'' time (which grows linearly with the distance to the source) for the Hermitian, non-absorbing case is substituted by three different characteristic quantities. One of them describes the arrival of a maximum of the density calculated with respect to position, but the maximum with respect to time for a given position becomes independent of the distance to the source and is given by the particle's ``survival time'' in the medium. This later effect, unlike the Hartman effect, occurs for injection frequencies under or above the cut-off, and for arbitrarily large distances. A possible physical realization is proposed by illuminating a two-level atom with a detuned laser.

Horizontal radiative fluxes in clouds and accuracy of the independent pixel approximation at absorbing wavelengths

For absorbing wavelengths, we discuss the effect of horizontal solar radiative fluxes in clouds on the accuracy of a conventional plane‐parallel radiative transfer calculation for a single pixel, known as the Independent Pixel Approximation (IPA). Vertically integrated horizontal fluxes can be represented as a sum of three components: the IPA accuracies for reflectance, transmittance and absorptance. We show that IPA accuracy for reflectance always improves with more absorption, while the IPA accuracy for transmittance is less sensitive to the changes in absorption: with respect to the non‐absorbing case, it may first deteriorate for weak absorption and then improve again for strongly absorbing wavelengths. IPA accuracy for absorptance always deteriorates with more absorption.

Neutron diffraction in flat and bent mosaic crystals for asymmetric geometry

Exact solutions of power transfer equations for plane-parallel and bent crystals in symmetric and asymmetric Bragg and Laue geometries are obtained, for the nonabsorbing case. Conceptual and mathematical errors in the solutions for asymmetric diffraction given by Bacon & Lowde [Acta Cryst. (1948). 1, 303–314] are pointed out. A very significant result is that, for asymmetrical Laue geometry, over 50% reflecting power is proved to be possible when b < 1. The integrated reflecting power and expressions for the optimum value of the curvature radius for bent mosaic crystals in the Bragg and Laue cases are also given.

Backscattering of electromagnetic waves by a random dielectric medium.

The averaged two-photon Green's function is calculated for a random medium. An analytic expression for the backscattering intensity is found. The backscattering intensity is determined by the ladder and maximal crossed diagrams. Our calculation applies to the scalar equation as well as to the full Maxwell's equation which has been investigated in the lowest order of the 1/N expansion. We show that for the absorbing case approaching the mobility edge (the critical frequency ${\ensuremath{\omega}}^{\mathrm{*}}$) the backscattering intensity decreases. This is opposite to the nonabsorbing case, where the backscattering intensity increases as one approaches the mobility edge ${\ensuremath{\omega}}^{\mathrm{*}}$. The line shape for the scattering intensity is obtained for different cases: We find that the line shape for a finite slab (finite in the Z direction and infinite in the X,Y directions) becomes narrower as the size of the slab increases.

Angular dependence of the absorption-induced nodal plane shifts of X-ray stationary waves

The classical equations of dynamical theory in the Bragg case are rederived in a way which is valid even in the presence of absorption where every quantity used is complex. The properties of the phase and the amplitude of the diffracted wave are discussed in detail. It is shown that for non-centrosymmetric crystals the position of the nodes at the centre of the reflection domain is strongly absorption dependent. The limit of the phase of the ratio of the diffracted to the incident amplitude far from the reflection domain is calculated to be equal to ϕh + π and ϕh, below and above the reflection domain, respectively, where ϕh is the phase of the structure factor calculated taking into account anomalous dispersion and the imaginary part of the form factor. It is therefore absorption dependent and so is the position of the nodes of stationary waves which is never invariant with the angular position of the crystal, even outside the total reflection domain. The example of a non-centrosymmetric crystal, GaAs, opposite reflections, 111 and {\bar 1}{\bar 1}{\bar 1}, and of wavelengths close to the absorption edges of gallium and arsenic is used to illustrate the results. It shows that the absorption-induced shifts of the nodal planes are in general directed towards the surface of the crystal, independently of the sense of the diffraction vector. The variation of the penetration depth within the total reflection is interpreted in the non-absorbing case by means of a new surface which completes the dispersion surface within the Bragg gap.

Intensity of X Rays Diffracted from an Elastically Vibrating Single-Crystal Plate

The intensity of the x-ray beam diffracted from a single crystal changes when the crystal lattice is distorted. An elastically vibrating quartz plate driven piezoelectrically was used to study relationships between diffracted intensity and lattice distortions for the case of Laue geometry with μt∼1. The main advantages in studying the distortions of a plate vibrating at resonance are that the shape of the displacement of the lattice is amenable to mathematical analysis and that the amplitude of vibration can be varied continuously. Integrated intensity of the diffracted x-ray beam was measured as functions of the position in the crystal plate, the amplitude of vibration, and the phase of a vibration cycle. The following conclusions were drawn, which demonstrate and verify some predictions made by the dynamical theory of x-ray diffraction for a distorted single crystal for the nonabsorbing case: (1) For an intensity change to occur, the distortion in the net plane must have a component along the diffraction vector and have a nonzero second derivative in the plane of diffraction. (2) The measured integrated intensity is a monotonically increasing function of lattice distortions and closely follows the theoretical curve. (3) Increases in the diffracted intensity are independent of the sign of the second derivative of the displacement.