Since the pump power is absorbed in a small region in the fiber core, a high concentration of the Rare Earth Ions is required. As a result boding to Er+3 ions is difficult and an inhomogeneity in ion concentration results which causes a clustering of these ions in the host lattice. Clustering depopulates the upper laser level. A solution to this problem is to confine erbium ions in the fiber core center. The Vapor Axial Deposition technique provides this requirement and as a result we have an increase in the amplifier gain. Our aim in this work is to provide a model that considers a symmetrical distribution for erbium ions which obeys a gaussian profile. We then solve the rate and propagation equations analytically. Our results show that the overlap integral between the dopant and optical mode ((Gamma) k) is independent of the pump power over the (b/w < 1) range, where b is the equivalent radius of the doped region and w is the minimum spot size of the incident radiation. We use this result to find an analytic gain equation which agrees well with the result of other models obtained by standard numerical integration techniques.
The superiority of gain or loss-coupled DFB lasers over the index coupled ones are evident in many respects. However the fabrication of first order gain or loss coupled gratings involves many difficulties. Therefore we are practically restricted to employing higher order gain or loss-coupled gratings. One of the main disadvantages of higher order gratings is radiation losses. To take into account both the effects of higher order modes on coupling and losses in linear regime, we apply a well-known coupled wave analysis due to Striefer et al. In 1977 to loss coupled DFB lasers. This theory using the first order perturbation analysis is suitable for analyzing shallow gratings. It leads to a pair of coupled wave equations, where coefficients indicate all the possible coupling losses.
The gain coupled gratings are analyzed by the extension of a well-known coupled mode theory, first developed by Streifer et al. in 1977. We present a new matrix method that takes into account the polarization nature of the non-dominant modes, that has been ignored in the previous theories. Our method can analyze any grating irrespective of its shape, its order and its type. It is based on slicing the grating layer such that in each layer, the equivalent waveguide refractive index and excitation can be considered as constants. We obtain different coupled wave equations, as we applied this method to pure gain coupled DFB lasers. It is worth noting that our theory is presented in the under threshold regime.
Analysis of reflection gratings in As-S thin films photodissolved with silver for a rectangular profile is presented. This analysis is performed by solving the scalar wave equation and by matching the amplitude of the electric and magnetic fields at the boundary of the grating with the external medium. In order to find the amplitudes of the reflected waves, an infinite system of coupled wave equations should be solved. This system of the differential equations can be put in a matrix form and the matrix of coefficients can be obtained using a Gauss elimination method. Our numerical results show that for a pure reflection grating, only the zero transmitted and reflected orders have appreciable efficiency, when the diffraction efficiency is plotted versus the grating strength parameter. Angular response of the diffraction efficiency shows that at each angle of incidence, the diffraction efficiencies of the transmitted and the reflected waves add to unity for each diffraction order. Our results show that the diffraction efficiency increases as the concentration of silver increases in the arsenic sulfide layer.
We have used a coupled wave analysis and solved numerically the coupled wave equations to model the performance of the diffraction gratings produced by the metal photo-dissolution effect in As - S thin films. This analysis is based on the material properties of arsenic sulphide silver doped films. We have analyzed both a sinusoidal and a square wave profile to consider both the holographic gratings and gratings produced by a mask exposure technique. Our model computes the diffraction efficiency versus two parameters (Omega) and (xi) where (Omega) is a thickness parameter and (xi) is a modulation parameter which is related to the change in the refractive index of the films. For the case of the sinusoidal profile the result of our model for very small values of (Omega) fits very well with the Bessel functions of the first kind which is the expected analytic results. As we change (Omega) from a small value of on the order of 0.01 to a value of on the order of 10, fewer diffraction orders become important in the replay of the grating with a red wavelength. For (Omega) on the order of 10 only one significant order is seen in the replay. The same results are generally obtained for the square grating. The angular response of the efficiency for a typical grating shows that the efficiency is a maximum near the Bragg angle. This result is in good agreement with the experimental results of the diffraction efficiency measurements obtained on a grating with the same parameters.
The authors report on techniques used to manufacture IR diffractive elements in chalcogenide glasses and on measurements of the material properties relevant to the performance of these elements. The characteristics of the elements produced are also presented and compared with theoretical predictions. The fabrication process used is based on the photodissolution of Ag into amorphous As-S films. Both surface relief and volume phase modulated transmission elements have been made. The transmission of Ag photodoped and undoped As-S films was found to be >80% over the range 2-12 micrometers for films up to 2 micrometers thick, the main loss mechanism being reflection. The difference in refractive index between Ag photodoped and undoped As-S over the range 0.5-12 micrometers was 0.5 for the most heavily doped material, so that high modulations are achievable for phase gratings. Theory suggests that for these As-S materials, green illumination (e.g., 514.5 nm) is the most efficient for producing the deep structures required for many of these IR elements. Surface relief structures can be produced by removing undoped material with an alkali etchant (e.g., NaOH). For transmission gratings, any remaining metallic Ag must be removed, to avoid high losses: the most successful Ag etchant was found to be Fe(NO3)3 in water. For the bulk holographic transmission gratings produced, efficiencies of >33% were observed for first diffraction orders measured in air at 632.8 nm, the main loss mechanisms being absorption and reflection, with some scatter. Measurements at 1.5 micrometers have given efficiencies of >30%, stability requirements during holographic recording currently being the main limitation to higher efficiencies at these and longer wavelengths. The results of a theoretical analysis based on numerical solution of the appropriate coupled-wave equations and taking into account bulk losses with phase and absorption modulation are in good agreement with the observed diffraction efficiency data. Given the low material absorption in the IR, theoretical studies show that, with suitable coatings, >95% efficiency should be possible for properly optimized bulk gratings and blazed zone plates.