Spontaneous emission
The rate of spontaneous emission of an atom inside a dielectric is a crucial property for optical/electrical applications and requires fundamental understanding of its quantum aspects. Experimental and theoretical efforts have focused up to now on the modification of the emission in (micro) cavities filled with a homogeneous dielectric. Recently, three-dimensional inhomogeneous dielectric structures with characteristic length scales matching the wavelength of the luminescence have been put forward as structures in which spontaneous emission can be modified. In such 'photonic materials' one identifies, for instance, Bragg reflection, band gaps in the density of states, and many other phenomena in close analogy with the well-known wavelike propagation of electrons in a crystalline structure. It has been shown that the emission of an atom with an emission frequency inside the photonic band gap of such photonic crystal would be fully suppressed and may serve as the basis for designing a laser without threshold. It can be demonstrated that suppression of spontaneous emission can be obtained under a much weaker condition than a full bandgap.
Lorentz cavity.
The investigation of the fundamental relation between microscopic properties of individual polarizable building blocks (such as atoms, dipoles, and dielectric spheres) and the macroscopic electromagnetic response of matter has been considered a long-standing problem. It is particularly important in the interpretation of the dielectric constant (from zero frequency up to the X-ray regime), the Einstein coefficient for spontaneous emission (1), the absorption of an impurity in a host material, and all non-linear optical susceptibility.
where Nnm are the number of molecular transitions, Anm the Einstein coefficient of spontaneous emission and Nm the number of molecules in the lower energy level (diagram 1).
Diagram 1.
The two energy levels have the ground state (Em) occupied (a). When an electron is excited by stimulated absorption (b), the electron moves to the excited state (En) with the change in energy being
.
The number of electrons in En when a large population of identical systems are present is given by the Boltzmann distribution.
Lorentz made a significant contribution by introducing a local field. The introduction of the local field was an attempt to explain the conductivity by the classical approach of using the equations of motion for a particle in an applied electric field, E. Up to today the validity of the concept of a local field, and the particular value suggested by Lorentz, is still a matter of debate. The reason for the debate follows these lines :
- The velocity ud of the electrons in the average velocity attained between collision times, with the relaxation time, t, giving a measure of time between the collisions. When the field is constant, the accelation of an electron is given by (2.1) which therefore will give (2.2)
2.1
2.2
- Since the electrical density, j, is the charge on N electrons per unit volume, -Ne, multiplied by the velocity nd, (2.3) can be written. The equations are then combined to give (2.4). s (in 2.4) is the electrical conductivity,
2.3
2.4
- (2.4) is consistent with Ohm's law. The drift velocity per unit electrical field is known as the mobility, m (2.5) and is related to the electrical conductivity s by (2.6).
2.5
2.6
The debate arises as while the theory explains certain aspects of the conductivity, it fails to take into account heat capacity calculations. When these are applied, the theory fails. It does also not explain the difference between metals, insulators and semi-conductors. Neither does it explain why the conductivity decreases with temperature in metals, but rises with insulators and semi-conductors.
Materials that have a dielectric constant, that is strongly varying periodically in space (photonic materials), are known to reduce the Einstein coefficient for spontaneous emission. However the appropriate local field correction are not known.
The ideal experiment would be a measurement of the quantum efficiency and radiative lifetime as a function of dielectric constant keeping other contributions constant. The heavier noble gases in the liquid or solid phase fulfil the requirements. Complexes of rare earth ions can be dissolved in supercritical xenon.
The treatment in literature of the transfer functions for interferometers does not allow for a proper use of the Kramers-Kronig (KK) relations between amplitude and phase. It can be shown that if the full frequency dependence of all the optical parameters, including the complex index of refraction, are properly taken into account, KK relations do exist.
The Kramers-Kronig equation and relationships.
The two components of the anomalous scattering factors f" and f' are related by the KK relationship (3). Once the f" spectrum (diagram 2) is obtained experimentally from a single crystal via fluorescence measurements, the corresponding f' spectrum can be calculated using an integrated form of (3).
3
Diagram 2. Graph of the KK relationship between f' and f"
The KK Transform.[1]
The KK Transform is useful for obtaining both absorbance and refractive index (RI) information from reflectance data. [2]
When the reflectance spectrum of an optically thick sample is measured, the data will consist of two components : the absorbance spectrum and refractive index spectrum. In reflectance spectroscopy, the RI spectrum dominates and makes the qualitative interpretation of data near impossible.
The RI of a typical mid-IR absorber tends to change rapidly in regions of strong absorbance which causes the major absorbance peaks to appear as strong as the first derivative shaped features in the measured data.
The KK Transform can decompose the reflectance spectrum into the separate extinction co-efficient and RI spectra (called the K and N spectra respectively). These can be then used for the qualitative evaluation of a sample. The extinction coefficient spectrum is then used to produce the absorbance spectrum.
The KK Transform assumes that the relectance angles near zero and is based on a double FFT approach with the main advantage being speed of calculation.
The real (n = RI) and imaginary (k = extinction) parts of the index of refraction are calculated from the reflectance spectrum via (4a) and (4b).
4a
4b
where R = the reflectance spectrum, u = wave number, and q the phase shift angle of the sample.
For a given wavenumber, the phase shift is calculated using (4c)
4c
The evaluation of this presents two problems.
- In practice, the spectra is over a finite range, so approximations are required at either end.
- There is a pole a n = nm which also requires an approximation.
By use of a computer program[3], these can be calculated. The program takes the Fourier Transform, FT, of the argument in the phase integral, sets the points for t < 0 to 0 and takes the inverse FT.