Cavity mode

Consider a cubic cavity with the sides L at the temperature T. The walls of the cavity absorb and emit electromagnetic radiation. At thermal equilibrium the absorbed power Pq, oo) has to be equal to the emitted power P ico) for all frequencies co. Inside the cavity there is a stationary radiation field E, which can be described at the point r by a superposition of plane waves with the amplitudes Ap, the wave vectors A p, and the angular frequencies o) as 

The waves are reflected at the walls of the cavity. For each wave vector k = (kxjky.kz), this leads to eight possible combinations kt = - kx, - ky, kz) that interfere with each other. A stationary-field configuration only occurs if these superpositions result in standing waves (Fig. 2.1a,b). This imposes boundary conditions for the wave vector, namely 

The magnitudes of the wave vectors allowed by the boundary conditions are 

These standing waves are called cavity modes (Fig. 2.1b). 

Since the amplitude vector of a transverse wave E is always perpendicular to the wave vector k, it can be composed of two components a and d2 with the unit vectors e and 2 

The waves are reflected at the walls of the cavity. For each wave vector Ki leads to 8 possible combinations k = ( k, k, k ) 

This chapter deals with basic considerations about absorption and emission of electromagnetic waves interacting with matter. Especidly 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 coeflhoients and their mutual relations. The next section explains some definitions used in photometry such as radiation power, intensity, and spectral power density. 

It is possible to understand many phenomena in optics and spectroscopy in terms of classical models based on concepts of classical electrodynamics. For example, the absorption and emission of electromagnetic waves in matter can be described using the model of damped oscillators for the atomic electrons. In most cases, it is not too difficult to give a quantum-mechanical formulation of the classical results. The semiclassical approach will be outlined briefly in Sect. 2.7. 

Many experiments in laser spectroscopy depend on the coherence properties of the radiation and on the coherent excitation of atomic or molecular levels. Some basic ideas about temporal and spatial coherence of optical fields and the density-matrix formalism for the description of coherence in atoms are therefore discussed at the end of this chapter. 

Throughout this text the term light is frequently used for electromagnetic radiation in all spectral regions. Likewise, the term molecule in general statements includes atoms as well. We shall, however, restrict the discussion and most of the examples to gaseous media, which means essentially fi ee atoms or molecules. 

For more detailed or more advanced presentations of the subjects summarized in this chapter, the reader is referred to the extensive literature on spectroscopy [2.1-2.11]. Those interested in light seattering from solids are directed to the sequence of Topics volumes edited by Cardona and cowoikers [2.12]. 

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The cavity of a laser may resonate in various ways during the process of generation of radiation. The cavity, which we can regard as a rectangular box with a square cross-section, has modes of oscillation, referred to as cavity modes, which are of two types, transverse and axial (or longitudinal). These are, respectively, normal to and along the direction of propagation of the laser radiation. 

Although 0-switching produces shortened pulses, typically 10-200 ns long, if we require pulses in the picosecond (10 s) or femtosecond (10 s) range the technique of mode locking may be used. This technique is applicable only to multimode operation of a laser and involves exciting many axial cavity modes but with the correct amplitude and phase relationship. The amplitudes and phases of the various modes are normally quite random. 

Figure 9.5 Suppression of five out of seven axial cavity modes by mode locking...

An optical microcavity produced by the latter process has been applied to tune the emission from erbium-doped PS [Zh6], Erbium compounds like Er203 are known to exhibit a narrow emission band at 1.54 pm, which is useful for optical telecommunications. Several methods have been used to incorporate erbium in PS. A simple and economical way is cathodic electrochemical doping. External quantum efficiencies of up to 0.01% have been shown from erbium-doped PS films under electrical excitation [Lo2]. The emission band, however, is much broader than observed for Er203. This drawback can be circumvented by the use of an optical cavity formed by PS multilayers. In this case the band is narrowed and the intensity is increased because emission is only allowed into optical cavity modes [Lo3]. 

Let us consider thermal radiation in a certain cavity at a temperature T. By the term thermal radiation we mean that the radiation field is in thermal equilibrium with its surroundings, the power absorbed by the cavity walls, Fa (v), being equal to the emitted power, Pe v), for all the frequencies v. Under this condition, the superposition of the different electromagnetic waves in the cavity results in standing waves, as required by the stationary radiation field configuration. These standing waves are called cavity modes. 

The laser interferometer consists of two coupled resonators, one containing the laser, the other the plasma under investigation (Fig. 10). The laser radiation, reflected back from mirror A/s, which contains phase information about the refractive index of the plasma, interferes with the laser wave in cavity A, resulting in an amplitude modulation of the laser output 267). This modulation can be related to the refractive index and therefore to the plasma frequency and electron density. With a curved rather than a planar mirror, the sensitivity can be increased by utilizing transverse cavity modes 268). 

In the case of a common lower level, the second absorption transition would show this narrowing effect when probed with a tunable monochromatic laser line. This example can be realized if atoms or molecules in a magnetic field are pumped by a laser, oscillating simultaneously on two cavity modes 324). if the Zeeman splitting of the probe equals the mode spacing of the laser, both transitions are pumped simultaneously and each laser mode selectively eats... 

The frequency of a single-mode laser inside the spectral gain profile of its active medium is mainly determined by the eigenfrequency of the active laser cavity mode. Therefore any instability of resonator parameters, such as variation of cavity length, mirror vibrations or thermal drifts of the refractive index will show up as frequency fluctuations and drifts of the laser line. 

These structures were recorded by a vectorial focal spot scanning in a spiral-by-spiral method rather in a raster layer-by-layer mode using a PZT stage. Such spiral structures fabricated in SU-8 have optical spot bands in near-lR [24], telecommunication [25], and 2-5 pm-IR region [26] or can be used as templates for Si infiltration [11]. It is obvious, that direct laser scanning is well suited for defect introduction into 3D PhC, as demonstrated in resin where a missing rod of a logpUe structure resulted in the appearance of a cavity mode in an optical transmission spectriun [27]. 

Fig. 2. Schematic representations of artificiai receptors with various cavity modes (Types A-C)...

Since the discovery of lasers it has been known that a derivation of time-dependent equations governing interaction of molecules with electromagnetic cavity modes leads to the so-called spontaneous instabilities. These laser instabilities were also observed experimentally—even for the first laser built by Maiman in 1960. A random, periodic, or quasiperiodic train of spikes in a laser generation is a fundamental instability due to nonlinearity of laser equations. A comprehensive review of this specific laser-related topics was published in 1983 [14]. 

The case of a frequency mismatch between laser pumps and cavity modes was investigated [83], and for the first time, chaos in SHG was found. When the pump intensity is increased, we observe a period doubling route to chaos for Ai = 2 = 1. Now, for/i = 5.5, Eq. (3) give aperiodic solutions and we have a chaotic evolution in intensities (Fig. 5a) and a chaotic attractor in phase plane (Imaj, Reai) (Fig. 5b). 

We have described the effects of black body radiation in free space. In a closed cavity the radiation is confined to the allowed modes of the cavity. In essence all the thermal radiation is forced into the cavity modes, raising the intensity at the... 

This equation fixes the optical frequency u)n = 2Tmvp(u)n)/2L and the wave number k(u)n) = u>n/vp(u)n) of the nth cavity mode, where vp(u>n) is the phase velocity for a monochromatic wave at u)n. The following expansion about some mean frequency u>m is generally used to take dispersion into account ... 

The achievable pulse length is determined by the total number of modes that can contribute to the pulse. The broader the frequency comb the shorter the possible pulse length, ideally reaching the so-called Fourier limit. In fact, the spectral width is usually limited by the width over which the GVD and higher order terms can be compensated for by mode pulling [5,6]. Cavity modes that are outside this bandwidth are suppressed without the help of the Kerr-lens effect and do not oscillate. 

We have developed FM lasers based on a commercial ring laser (Coherent 699-21). In this case all the intracavity etalons are removed and replaced by a lithium niobate phase modulator. This modulator can be resonantly driven at a frequency close to the cavity mode spacing. A simple theory of FM operation of a laser suggests that the modulation index is given by [12]... 

Schematic plot of laser intensity as a function of frequency and of five equally spaced longitudinal cavity modes.

Figure 141 shows the EL spectra from a microcavity (a) and conventional LED (b) based on the emission from an NSD dye forming a thin emitting layer of a three-organic layer device. It is apparent that the half-width of emission spectra from the diode with microcavity is much narrower than those from the diode without cavity. With 0 = 0°, for example, the half-width of the spectrum of the diode with cavity is 24 nm whereas that of the sample without cavity increases to 65 nm. According to Eq. (275), the resonance wavelength, A, decreases with an increase of 0 in agreement with the experimental data of Fig. 141. We note that no unique resonance condition in the planar microcavity is given due to broad-band emission spectrum of the NSD emission layer. Multiple matching of cavity modes with emission wavelengths occurs. Thus, a band emission is observed instead a sharp emission pattern from the microcavity structure as would appear when observed with a monochromator the total polychromic emission pattern is a superposition of a range of monochromatic emission patterns. The EL spectra...

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