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Polarization

Andreas F. Molisch, Bernhard P Oehry

Published Nov 26, 1998• 0 citations
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Abstract

Abstract In theory, all radiation trapping problems must be treated under consideration of the polarization state of the radiation. Even with an unpolarized excitation source, the geometry of the problem can introduce some degree of polarization to the fluorescence radiation. Of course, when the excitation source itself is polarized, as is usually the case for a laser, polarization can become even more important. One should not forget, however, that incorporation of polarization makes for a very complicated problem. The Holstein equation will become a Fredholm integral equation for a four-component vector instead of the usual scalar function (the excited-state density). Having seen the difficulties in solving the comparatively simple ‘usual’ trapping problem, we see that polarization should only be included when it is not to be avoided.

DOI
10.1093/oso/9780198538660.003.0012

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Abstract In theory, all radiation trapping problems must be treated under consideration of the polarization state of the radiation. Even with an unpolarized excitation source, the geometry of the problem can introduce some degree of polarization to the fluorescence radiation. Of course, when the excitation source itself is polarized, as is usually the case for a laser, polarization can become even more important. One should not forget, however, that incorporation of polarization makes for a very complicated problem. The Holstein equation will become a Fredholm integral equation for a four-component vector instead of the usual scalar function (the excited-state density). Having seen the difficulties in solving the comparatively simple ‘usual’ trapping problem, we see that polarization should only be included when it is not to be avoided.
From prehistoric times has come recognition that sunlight and firelight provide warmth, and that such illumination casts shadows. Expressed in more contemporary terms one would say that light travels in rays, and that this radiation has the potential to provide heat energy to absorbing material. From the time of Newton it has been known that light sources emit radiation comprising a distribution of colors. During the nineteenth century it was recognized that radiation had characteristics of transverse waves (with wavelength associated to color) but until the late twentieth century, when lasers became laboratory tools, it was hardly necessary to delve into the equations of electromagnetic theory to treat such experiments as were then possible; interest lay primarily with thermodynamic considerations of energy flow or with measurements of the dark or bright lines seen in the distribution of light that, after passing through a slit, was dispersed by a prism or grating into constituent colors. Although laser light sources are essential for the types of atomic excitation considered in this monograph, the legacy from thermal radiation still influences many interpretations of the interaction between radiation and matter, and it is therefore useful to summarize some of those concepts. The mathematics needed for describing laser radiation, or polarized light in general, has much in common with the mathematics of quantum theory discussed starting in Sec. 3.5 and specialized to two-level atoms in Sec. 5.6. In both cases one deals with two complex-valued functions – independent electric field amplitudes or probability amplitudes – whose absolute squares are measurable.
Many ion species commonly used for laser-cooled ion-trapping studies have a low-lying metastable 2D3/2{}^{2}{D}_{3/2}2D3/2​ state that can become populated due to spontaneous emission from the 2P1/2{}^{2}{P}_{1/2}2P1/2​ excited state. This requires a repumper laser to maintain the ion in the Doppler cooling cycle. Typically, the 2D3/2{}^{2}{D}_{3/2}2D3/2​ state, or some of its hyperfine components if the ion has nuclear spin, has a higher multiplicity than the upper state of the repumping transition. This can lead to dark states, which have to be destabilized by an external magnetic field or by modulating the polarization of the repumper laser. We propose using unpolarized, incoherent amplified spontaneous emission (ASE) to drive the repumping transition. An ASE source offers several advantages compared to a laser. It prevents the buildup of dark states without external polarization modulation even in zero magnetic field, it can drive multiple hyperfine transitions simultaneously, and it requires no frequency stabilization. These features make it very compact and robust, which is essential for the development of practical, transportable optical ion clocks. We construct a theoretical model for the ASE radiation, including the possibility of the source being partially polarized. Using 88Sr+{}^{88}{\mathrm{Sr}}^{+}88Sr+ as an example, the performance of the ASE source compared to a single-mode laser is analyzed by numerically solving the eight-level density-matrix equations for the involved energy levels. Finally, a reduced three-level system is derived, yielding a simple formula for the excited-state population and scattering rate, which can be used to optimize the experimental parameters. The required ASE power spectral density can be obtained with current technology.
We compute radiation trapping in a two-level atomic vapour excited by a laser beam, taking into account saturation effects and using a realistic vapour-cell geometry, the finite cylinder. By using an approximation for the angular distribution of the fluorescence radiation, CPU-time requirements are kept low. We then analyse various practical cases and present useful practical rules of thumb when certain simplifications are valid. When we consider the decay of an initial excitation, the decay at late times can be computed by using an equivalent one-dimensional geometry. For early times, it is usually important to include the three dimensionality of the problem. Finally, we present calculations how the `burn-through' effect of a strong laser affects the approximation of the geometry.
In the absence of incoherent relaxation processes, the emission, absorption, and transport of resonant radiation by an extended medium are all cooperative processes in which the medium participates as a whole. Such radiative processes depend sensitively on the geometry of the source, on its size and density, and on the interplay of electric polarization resonances with the ambient electromagnetic field. By making a succession of simple approximations, we reduce the radiative interaction problem to one of solving a linear integral equation for the time-dependent polarization of the radiating medium. For the example of one-dimensional geometry, we show that the polarization possesses a sequence of exponentially decaying eigenmodes that oscillate at frequencies slightly shifted from that of the fundamental resonance. Only a few of these modes, for any radiating medium, are strongly coupled to the electromagnetic field, and thereby radiate efficiently. All of the other modes, infinitely many of them, are found to radiate quite slowly, or not at all. We consider in detail the radiation emitted by an arbitrary initial polarization distribution when the electromagnetic field is initially empty. Analytical and numerical results are presented for the time-dependent fields that follow from a wide variety of initial excitations. A locally oscillatory exchange of energy is found to take place between the electric and polarization fields, which gives rise to a ringing of the radiated intensity. The spectra of the emitted light are shown accordingly to be complex in structure, and to have a frequency gap which becomes independent of the size of the medium and its initial polarization distribution for media that are sufficiently large. The gap is caused by the interference and interaction of oppositely propagating excitations.