Spectral mode density

It is often interesting to know the number n(a))dco of modes per unit volume within a certain frequency interval dco, for instance, within the width of a spectral line. The spectral mode density n(co) can be obtained directly from (2.6) by differentiating N(co)/L with respect to co. N(co) is assumed to be a continuous function of co, which is, strictly speaking, only the case for L 00. We get... 

In Fig. 2.3 the spectral mode density (v) is plotted against the frequency v on a double-logarithmic scale. This illustrates, that the spectral mode density (number of modes within the frequency interval dv = 1 s ) in the visible range is about n(y) = l(Fm , which gives inside the Doppler widths of a spectral line dv =... 

Figure 2.3 Spectral mode density n(v) = N(v)/L as a function of the frequency v...

In Fig. 2.3 the spectral mode density is plotted against the fiequen( v on a double-logarithmic scale. 

According to Eqnation (2.1), the mode density at that frequency in a spectral interval of 10 s Ms... 

Moreover, in recent years broad band lasers have appeared which lack any frequency modal structure, at the same time retaining such common properties of lasers as directivity and spatial coherence of the light beam at sufficiently high spectral power density. The advantages of such a laser consist of fairly well defined statistical properties and a low noise level. In particular, the authors of [245] report on a tunable modeless direct current laser with a generation contour width of 12 GHz, and with a spectral power density of 50 /xW/MHz. The constructive interference which produces mode structure in a Fabry-Perot-type resonator is eliminated by phase shift, introduced by an acoustic modulator inserted into the resonator. 

Local non-uniformities present the danger of hot spot formation where the current forms filaments. Two adverse scenarios can develop (1) local thermal run away that engenders regions of high defect density which gradually grow in size, and (2) spectral hole burning in which certain spectral modes are locally depleted. For all of these reasons, uniform material properties have always been a mandatory virtue. 

As already pointed out, the exponent v, calculated for three proteins above, is related to the fractal dimension of the object, D, and the spectral dimension, d, by v = d/2D [35,140]. The latter is defined from the form in which the density of states varies with frequency as p(co) o> / 1. We thus seek a value for the spectral dimension, d, which for many fractal objects is d k, 4/3 [35]. In Fig. 12, we plot log p(co) versus log co for the normal modes of cytochrome c, myoglobin, and GFP the normal mode density is computed in 2-cm 1 intervals... 

The impact of lasers on spectroscopy can hardly be overestimated. Lasers represent intense light sources with spectral energy densities which may exceed those of incoherent sources by several orders of magnitude. Furthermore, because of their extremely small bandwidth, single-mode lasers allow a spectral resolution which far exceeds that of conventional spectrometers. Many experiments which could not be done before the application of lasers, because of lack of intensity or insufficient resolution, are readily performed with lasers. 

This chapter deals with basic considerations about absorption and emission of electromagnetic waves interacting with matter. Especially emphasized are those aspects that are important for the spectroscopy of gaseous media. The discussion starts with thermal radiation fields and the concept of cavity modes in order to elucidate differences and connections between spontaneous and induced emission and absorption. This leads to the definition of the Einstein coefficients and their mutual relations. The next section explains some definitions used in photometry such as radiation power, intensity, and spectral power density. 

Assume the isotropic emission of a pulsed flashlamp with spectral bandwidth Ak = 100 nm around k = 400 nm amounts to 100-W peak power out of a volume of 1 cm. Calculate the spectral power density p(v) and the spectral intensity I(v) through a spherical surface 2 cm away from the center of the emitting sphere. How many photons per mode are contained in the radiation field ... 

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. 

In Sect. 2.1 it was shown that in a closed cavity a radiation field exists with a spectral energy density p(v) that is determined by the temperature T of the cavity walls and by the eigenfrequencies of the cavity modes. In the optical... 

The problem of control of Mossbauer nuclei gamma decay for the first time was considered about 25 years ago in the work [6]. In this work, we have made (without detailed analysis) investigated the possibility of controlling the probability of decay of excited Mossbauer nuclei by controlled mode restriction of electromagnetic vacuum (controlled change of spectral-volume density of quantized field modes p co)) by action of additional resonant or nonresonant screen. 

Qi) Radiance of a HeNe laser. We assume that the output power of 1 mW is emitted from 1 mm of the mirror surface into an angle of 4 minutes of arc, which is equivalent to a solid angle of 1 xl0 sr. The maximum radiance in the direetion of the laser beam is then L = 10 /(10 -10 ) = 10 W/(m sr). This is about 50 times larger than the radiance of the sun. For the speetral density of the radiance the comparison is even more dramatie. Sinee the emission of a stabilized single-mode laser is restricted to a speetral range of about 1 MHz, the laser has a spectral radiance density Zv = lxlO W- s/(m sr ), whereas the sun, which emits within a mean spectral range of 10 Hz, only reaches Ly = l xlO W- s/(m sr ). 

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