Stable and Unstable Resonators

In a stable resonator the field amplitude A(x, y) reproduces itself after each round-trip apart from a constant factor C, which represents the total diffraction losses but does not depend on x or y, see (5.27). 

This reveals that for gi = 0 or g2 = 0 and for gigi = E the spot sizes become infinite at one or at both mirror surfaces, which implies that the Gaussian beam diverges the resonator becomes unstable. An exception is the confocal resonator with = g2= 0, which is metastable , because it is only stable if both parameters gi are exactly zero. For gig2 1 or g g2 0, the right-hand sides of (5.42) become imaginary, which means that the resonator is unstable. The condition for a stable resonator is therefore 

With the general stability parameter G = 2gig2 — we can distinguish stable resonators 0 G 1, unstable resonators G 1, metastable res- 

Type of resonator Mirror radii Stability parameter  

Let us consider the simple example of a symmetric unstable resonator depicted in Fig. 5.14 formed by two mirrors with radii Rj separated by the distance d. Assume that a spherical wave with its center at F is emerging from mirror M. The spherical wave geometrically reflected by M2 has its center in F2. If this wave after ideal reflection at M is again a spherical wave with its center at F, the field configuration is stationary and the mirrors image the local point F into F2, and vice versa. 

The beam waist wl of a confocal nonsymmetric resonator with 1 R2 is no longer at the center of the resonator (as for symmetric resonators). Its distance from Ml is 

If in a symmetric confocal resonator a plane mirror is placed at the beam waist (where the phase front is a plane), a semiconfocal resonator results (Fig.5.14), with = oo, d= Ri/2, g = 1, g2 = 1/2, w = Xdln, u 2 = 2XdlTi. 

For the magnification of the beam diameter on the way from mirror Mj to M2 we obtain from Fig.5.14 the relation 

We define the magnification factor M = P found trip as the ratio 

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It seems essential to include unstable particles in any coherent schema. Even the simplest isotopic doublet, neutron-proton, contains one stable and one unstable particle, and the more refined group theoretical classifications introduce both stable and unstable particles on the same footing.17 This is especially true since the discovery of resonances which are not only produced by strong interactions but also decay through strong interactions. 

The technique of electron-spin resonance (e.s.r.), well known as an important tool for studying the structure of paramagnetic transition-metal complexes, has recently been used in the detection and study of a wide range of organic radicals, both stable and unstable, in fluids and solids. 

We can obtain additional insight into the nature of the solutions of Mathieu s equation by extending the asymptotic solution described previously to small, but nonzero, values of e. Of particular interest is the behavior of solutions near the first resonant instability point ak = 1/4. Referring to Fig. 4-17, we see that there is a finite region around ak = 1/4 where solutions for nonzero values of ebk are predicted to be unstable. In this section, we seek an asymptotic expression in terms of e for the critical values of ak 1 /4 that separate the regions of stable and unstable solutions. Thus we suppose that... 

Figure 3.25(c) exhibits a case where no dynamical correlation exists. Here, the stable and unstable manifolds fall into the potential well, and encounter densely populated resonances. Then, the movement within the web takes so much time that dynamical correlation would be completely lost there, which is the case where the statistical reaction theory is applicable. 

Controlled-potential electrolysis yields a product which may be identified in situ, e.g., spectrometrically, or after isolation from the solution. In the former case, both stable and unstable products may be studied, whereas isolation is usually limited to stable compounds. The methods used for identification of the product will also depend upon the stability of that product. Electron spin resonance, ultraviolet spectrophotometry, and cyclic voltammetry have proved useful techniques for the identification of unstable (radical) species. The presence of water in the electrolyzed solution usually prevents the use of in situ infrared (but not Raman) spectrophotometric analysis, and the use of such powerful techniques as nuclear magnetic resonance and mass spectrometry is also excluded unless the product can be isolated in a reasonably pure state. 

A stability diagram in the 9 92 plane, as shown in Fig.6.9, allows the stable and unstable regions to be identified. In Fig.6.10, the ratio y q/yqq of the diffraction losses for the TEM q and the TEMqq modes is plotted for different values of g as a function of the Fresnel number N. From this diagram one can obtain for any given resonator the diameter 2a of an aperture which suppresses the TEM q mode but still has sufficiently small losses for the fundamental TEMqq mode. In gas lasers, the diameter 2a of the discharge tube generally forms the limiting aperture. One has to choose the resonator parameters in such a way that a 3w/2 (see Sect.5.11) because this assures that the fundamental mode nearly fills the whole active medium but still suffers less than 1% diffraction losses. 

Fig. 10.5.4. Topology of the stable and unstable manifold of a resonant tt/3 fixed point in...

The boundary of the resonant zone corresponds to a coalescence of the stable and unstable periodic orbits on the invariant circle, i.e. to the saddle-node bifurcation of the same type we consider here. Besides, if there were more than two periodic orbits, saddle-node bifurcations may happen at the values of parameters inside the resonant zone. By the structure of the Poincare map (12.2.26) on the invariant curve,... 

The various resonator geometries can then be represented diagrammatically as in Fig.12.6 where the boundaries between the stable and unstable (shaded regions) are determined by equation (12.16). We find that the symmetric confocal, (Ri=R2=L), the symmetric concentric, (Rj=R2=L/2), and the plane-parallel resonators are all on the verge of instability and may, by accidental misalignment, become extremely lossy. Thus it is virtually impossible to obtain laser oscillation... 

Further insights into reaction dynamics can be obtained by analyzing the stability of classical trajectories. Presumably, stable periodic orbits will be restricted to KAM tori and therefore be nonreactive and unstable periodic orbits will provide information about the location of resonances and therefore some quahtative features of the intramolecular energy flow. 

The intermediates, formed by electrophilic reaction at the ortho or para positions, are not as stable as the intermediate formed by reaction at the meta position because, for ortho and para attack, one of the three resonance forms has positive charge on the carbon bearing the positively charged sulfur of the sulfonic acid group. For example, for the intermediate resulting from reaction at the para position we can draw three resonance forms. Of these 4-69-2 is particularly unstable because of the destabilizing effect of two centers of positive charge in close proximity. Because the intermediate involved in meta substitution lacks this unfavorable interaction, it is more stable and meta substitution is favored. 

Triafulvene (methylenecyclopropene, 1) is the smallest member of the fulvenoid series and is, therefore, generally expected to gain some 7t-delocalization energy by resonance contribution of the dipolar structure IB. The first known system of this type was quinocyclopropene 3 reported in 1963. The parent compound 1 was synthesized in 1984 as a highly reactive and unstable substance, whose spectra could only be recorded at low temperature (< — 75 °C in solution). - Compound 1 is much less stable than cyclopropenone (2), which is fairly stable in solution at room temperature and can be stored in its crystalline state for a long period at below... 

Fig. S.4. Schematic view of a quadrupole system with stable (resonant ion) and unstable (non-resonant ion) ion trajectories. 

Aminoindole exists mainly as the 3H-tautomer, presumably because of the energy advantage conveyed by amidine-type resonance. 3-Aminoindole is very unstable, and easily autoxidised. Both 2- and 3-acylamino-indoles are stable and can be obtained by catalytic reduction of nitro-precursors in the presence of anhydrides. 1-Amino-indoles can be prepared by direct amination of the indolyl anion. ... 

On a resonant invariant curve, out of the infinite set of r-multiple fixed points, only a finite (even) number survive, half stable and half unstable, as a consequence of the Poincare-Birkhoff fixed point theorem (Arnold and Avez, 1968 Lichtenberg and Lieberman, 1983), as shown schematically in Figure 14b. 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

In contrast to the cyanate ion (NCO ), which is stable and found in many compounds, the fulminate ion (CNO ), with its different atom sequence, is unstable and forms compounds with heavy metal ions, such as Ag and Hg " ", that are explosive. Like the cyanate ion, the fulminate ion has three resonance structures. Which is the most important contributor to the resonance hybrid Suggest a reason for the instability of fulminate. 

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