Optical superposition, principle

This expression is the main tool used in describing diffraction effects associated with Fourier optics. Holographic techniques and effects can, likewise, be approached similarly by describing first the plane wave case which can then be generalized to address more complex distribution problems by using the same superposition principle. 

The first of these was van t Hoff s principle of optical superposition, namely, that, in a compound having two or more asymmetric carbon atoms, the optical activities of the individual atoms can be added algebraically. This principle was applied with considerable success to carbohydrates by Hudson, in the form of his well known isorotation, lactone, and amide rules. These rules have been reviewed elsewhere, " and will not be discussed here. 

Then they applied van t Holf s principle of optical superposition [43-45]. The sum of the molecular rotation values of the three fragments 93,94 [46] and 95 was calculated (Fig. 4). The results were applicable because the value of synthetic BST-C (58) (+211) was in good agreement with the calculated value... 

The hypothesis of the additive nature of the rotatory contributions of the individual asymmetric centers of steroisomers in making up the total rotation of each isomer was formulated by van t Hoff and has been known as the principle of optical superposition. In its full generalization as applied to all substances, the hypothesis of optical superposition is definitely unsound and thus it is not a principle nevertheless, it has been shown by Hudson (116) that the hypothesis holds in first approximation for a large number of carbohydrates, and the approximation is sufficiently close to permit valuable inferences concerning structure and configuration to be drawn from comparisons of the rotations of carbohydrates through the application of his Isorotation Rules. 

In the past SO years wood chemists have learned much about the composition and physical properties of hardwood xylans. Partially acetylated glucuronoxylans are model native hardwood xylans. They exhibit thermoplasticity, film forming properties, crystallization potential and are oriented in the secondary cell wall. Their crystal structure has been determined and the hydration of this crystalline polysaccharide has been defined. The structural regularity of these abundant polysaccharides can be interpreted using the principle of optical superposition. 

The configuration of amino acids relative to that of D-glyceraldehyde has also been established. The work of Freudenberg and Kuhn which is based on a careful and circumscribed application.of the principle of optical superposition is reviewed and it is su ested that it is not by itself conclusive. A more decisive proof for the n configuration of amino acids is obtained from the X-ray structure of glucosamine and enzymic specificity. The configuration of secondary asymmetric centers in certain amino-acids is considered and it is concluded that the carboxyl and hydroxyl groups in hydroxy-proline are in trans position to each other. 

The essential principle of coherent control in the continuum is to create a linear superposition of degenerate continuum eigenstates out of which the desired process (e.g., dissociation) occurs. If one can alter the coefficients a of the superposition at will, then the probabilities of processes, which derive from squares of amplitudes, will display an interference term whose magnitude depends upon the a,. Thus, varying the coefficients a, allows control over the product properties via quantum interference. This strategy forms the basis for coherent control scenarios in which multiple optical excitation routes are used to dissociate a molecule. It is important to emphasize that interference effects relevant for control over product distributions arise only from energetically degenerate states [7], a feature that is central to the discussion below. 

The principle of superposition, a fundamental of classical optics holds. 

Once this discussion of the space-inversion operator in the context of optically active isomers is accepted, it follows that a molecular interpretation of the optical activity equation will not be a trivial matter. This is because a molecule is conventionally defined as a dynamical system composed of a particular, finite number of electrons and nuclei it can therefore be associated with a Hamiltonian operator containing a finite number (3 M) of degrees of freedom (variables) (Sect. 2), and for such operators one has a theorem that says the Hamiltonian acts on a single, coherent Hilbert space > = 3 (9t3X)51). In more physical terms this means that all the possible excitations of the molecule can be described in . In principle therefore any superposition of states in the molecular Hilbert space is physically realizable in particular it would be legitimate to write the eigenfunctions of the usual molecular Hamiltonian, Eq. (2.14)1 3 in the form of Eq. (4.14) with suitable coefficients (C , = 0. Moreover any unitary transformation of the eigen-... 

Multisite excitation can be easily achieved by mixing together several frequency shifted pulses according to the principle of superposition, known from optics [6,7] ... 

The next field of applications of elementary catastrophe theory are optical and quantum diffraction phenomena. In the description of short wave phenomena, such as propagation of electromagnetic waves, water waves, collisions of atoms and molecules or molecular photodissociation, a number of physical quantities occurring in a theoretical formulation of the phenomenon may be represented, using the principle of superposition, by the integral... 

Figure 5. Light micrograph of a few facets of a fly s compound eye. Dark spots that represent the rhabdomeres of the photoreceptors have been superimposed onto each corneal lens to demonstrate the principle of neuro-superposition. The central corneal facet has been removed from the photograph to depict the underlying lamina cartridge. Anatomically, six peripheral photoreceptor axon terminals (R1-R6) synapse with two second-order cells (LI and L2) in the underlying neuropil that is called the lamina. Each of these six photoreceptors is illuminated by a different lens, but optically they share the same visual axis that is, they look at the same point in space. This lamina subunit is known as neuroommatidium. Axons of the central receptor cells (R7 and R8) from the overlying ommatidium pass close to this cartridge, but simply bypass the lamina and do not contribute synapses at this neural level.

In interferometric autocorrelation the coherent superposition of the two collinear partial beams is realized. The basic principle is shown in Fig. 6.65. The incoming laser pulse is split by the beamsplitter BSl into two parts, which travel through two different pathlengths and are then collinearly superimposed at BS2. When they are focused by the lens L into a nonlinear optical crystal, the output signal (6.39) is generated at 2co. Instead of the delay line arrangement in Fig. 6.65 a Michelson interferometer in Fig. 6.70 can also be used. The second harmonics are detected by a photomultiplier, while the fundamental wavelength is rejected by a filter. 

It is impossible by means of any instrument to distinguish between various incoherent superpositions of wave fields, having the same frequency, that may together form a beam with the same Stokes parameters. This is known as the principle of optical equivalence. 

The basic principle of all interferometers may be summarized as follows (Fig. 4.24). The indicent lightwave with intensity Iq is divided into two or more partial beams with amplitudes Ak, which pass different optical path lengths Sic = nxk (where n is the refractive index) before they are again superimposed at the exit of the interferometer. Since all partial beams come from the same source, they are coherent as long as the maximum path difference does not exceed the coherence length (Sect. 2.8). The total amplitude of the transmitted wave, which is the superposition of all partial waves, depends on the amplitudes Ak and on the phases 0 = 0o + 27r A / of the partial waves. It is therefore sensitively dependent on the wavelength X. 

For measurements of optical pulse widths below 1 ps the best choice is a correlation technique that is based on the following principle the optical pulse with the intensity profile I t) =ceo E t) and the halfwidth AT is split into two pulses I t) and hit), which travel different path lengths S and S2 before they are again superimposed (Fig. 11.41). For a path difference As = S —S2 the pulses are separated by the time interval r = As/c and their coherent superposition yields the total intensity... 

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