Open optical resonator

These open resonators are, in principle, the same as the Fabry-Perot interferometers discussed in the previous chapter, and we shall see that several relations derived in Sect.4.2 apply here. However, there is an essential difference with regard to the geometrical dimensions. While in a common FPI the distance between both mirrors is small compared with their diameter, the relation is generally reversed for laser resonators. The mirror diameter 2a is small compared with the mirror separation d. This implies that diffraction losses of the wave, which is reflected back and forth between the mirrrors, will play a major role in laser resonators while they can be completely neglected in the conventional FPI. 

In order to estimate the magnitude of diffraction losses let us make use of a simple example. A plane wave incident onto a mirror with diameter 2a exhibits, after being reflected, a spatial intensity distribution which is determined by diffraction and which is completely equivalent to the intensity distribution of a plane wave passing through an aperture with diameter 2a (Fig.5.5). The central diffraction maximum at = 0 lies between the two 

N gives the number of Fresnel zones [5.15,16] across a resonator mirror, as seen from the center A of the opposite mirror. For the mirror separation d these zones have radii s = VmAd and the distances p = i(m+q)A (m = 0,1,2... q) from A (Fig.5.6). 

If the number of transits, which a photon makes through the resonator, is n, the maximum diffraction angle 20 should be smaller than a/(nd). With 20 = A/a we obtain the condition 

For our first example the diffraction losses of the plane FPI are about 5 10 and therefore completely negligible, whereas for the second example they reach 25% and may already exceed the gain for many laser transitions. This means that a plane wave would not reach threshold in such a resonator. However, these high diffraction losses cause non-negligible distortions of a plane wave, and the amplitude A(x, y) is no longer constant across the mirror surface (Sect.5.2.2), but decreases towards the mirror edges. This decreases the diffraction losses, which become, for example, 7oiffr for N 20. 

Besides these walk-ojf losses, reflection losses also cause a decrease of the energy stored in the resonator modes. With the reflectivities R and R2 of the resonator mirrors Mi and M2, the intensity of a wave in the passive resonator has decreased after a single ronnd-trip to 

In order to estimate the magnitude of diffraction losses let us make use of a simple example. A plane wave incident onto a mirror with diameter la exhibits, after being reflected, a spatial intensity distribution that is determined by diffraction and that is completely equivalent to the intensity distribution of a plane wave passing through an aperture with diameter la (Fig. 5.5). The central diffraction maximum at 0 = 0 lies between the two first minima at d = Xlla (for circular apertures a factor 1.2 has to be included, see, e.g., [306]). About 16 % of the total intensity transmitted through the aperture is diffracted into higher orders with 0 X/la. Because of diffraction the outer part of the reflected wave misses the second mirror M2 and is therefore lost. This example demonstrates that the diffraction losses depend on the values of a, d, X, and on the amplitude distribution A x,y) of the incident wave across the mirror surface. The influence of diffraction losses can be characterized by the dimensionless Fresnel number 

The meaning of this is as follows (Fig. 5.6a). If cones around the resonator axis are constructed with the side length = ( + m)Xfl and the apex point A on a resonator mirror they intersect the other resonator mirror at a distance d = qX11 in circles with radii = q- -m)-X. The annular zone on mirror Mi between two circles is called Fresnel zone. The quantity Ap gives the number of Fresnel zones [306, 307] across a resonator mirror with diameter 2a, as seen from the center A of the opposite mirror. For the mirror separation d these zones have radii pm = JmXd and the distances = 2 (m - -q)X (m = 0,1,2. ) from A (Fig. 5.6). 

If a photon makes n transits through the resonator, the maximum diffraction angle 19 should be smaller than ajind). With 19 = Xja we obtain the condition Xja a I in d) which gives with (5.23) 

The fractional energy loss per transit due to diffraction of a plane wave reflected back and forth between the two plane mirrors is approximately given by 

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Figure 2.3 Stability criterion for open optical resonators. The curves plot the ratio of mirror spacing L to radius of curvature R of the two mirrors making up the cavity in this case Rj = In the hatched sectors and inside the curve boundaries the stability criterion is not met (see text) resulting in increased loss from the cavity and reduced Q (Adapted from Yariv and Pozar )...

The function of the optical resonator is the selective feedback of radiation emitted from the excited molecules of the active medium. Above a certain pump threshold this feedback converts the laser amplifier into a laser oscillator. When the resonator is able to store the EM energy of induced emission within a few resonator modes, the spectral energy density p(v) may become very large. This enhances the induced emission into these modes since, according to (2.22), the induced emission rate already exceeds the spontaneous rate for p(v) > hv. In Sect. 5.1.3 we shall see that this concentration of induced emission into a small number of modes can be achieved with open resonators, which act as spatially selective and frequency-selective optical filters. 

The fine-tuning of the effective refractive index with the template, temperature, and rate of thermal treatment or growing mixed MTFs opens new opportunities for the design of low- materials, that is, eff < 1.2 valuable for many apphcations, such as waveguide claddings or optical resonators [36]. 

A chemical sensor is a device that transforms chemical information into an analytically useful signal. Chemical sensors contain two basic functional units a receptor part and a transducer part. The receptor part is usually a sensitive layer, therefore a well founded knowledge about the mechanism of interaction of the analytes of interest and the selected sensitive layer has to be achieved. Various optical methods have been exploited in chemical sensors to transform the spectral information into useful signals which can be interpreted as chemical information about the analytes [1]. These are either reflectometric or refractometric methods. Optical sensors based on reflectometry are reflectometric interference spectroscopy (RIfS) [2] and ellipsometry [3,4], Evanescent field techniques, which are sensitive to changes in the refractive index, open a wide variety of optical detection principles [5] such as surface plasmon resonance spectroscopy (SPR) [6—8], Mach-Zehnder interferometer [9], Young interferometer [10], grating coupler [11] or resonant mirror [12] devices. All these optical... 

These both key intermediates were opened with H2S in the presence of diisopro-pylamine. This reaction is known to proceed with full retention of configuration. Therefore we assume, that the obtained thiols 210 and 211 are of the assigned absolute stereochemistry. The optical purity of each enantiomer was directly determined from the relative peak areas and senses of nonequivalence of the resonances of enantiotopic nuclei in chiral solvent, e. g. Eu(TBC)3. We observe optical purities for 210p = 85% and for 211 p = 75%. The addition of 210 to the optically active 212 gave after column chromatography the desired 8 R, 11 R, 12 R, 15 S-13-thiaprostanoid E 213. 

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