@article{saghir_shannon_2012, title={The impact of Langmuir probe geometries on electron current collection and the integral relation for obtaining electron energy distribution functions}, volume={21}, ISSN={0963-0252 1361-6595}, url={http://dx.doi.org/10.1088/0963-0252/21/2/025003}, DOI={10.1088/0963-0252/21/2/025003}, abstractNote={The Druyvesteyn relation for obtaining electron energy distribution functions (EEDFs) from Langmuir probes is derived based on a model that assumes spherical probe geometry and extends this formulation to arbitrary geometries including the more commonly used planar and cylindrical probes. In this paper, we revisit the formulation of the relationship between electron current, probe potential and EEDF for a cylindrical geometry that considers geometric differences between the spherical and cylindrical case and provides an identical integral relationship to that posed by Mott-Smith and Langmuir in 1926. Comparing the spherical and cylindrical integral relationships and EEDFs reconstructed from them, noticeable differences in EEDF shape are seen using the Druyvesteyn relation for cylindrical probes that becomes more pronounced for highly non-Maxwellian distributions. In order to minimize this geometry-induced distortion, a solution of the integral relation between EEDF and probe current may be needed in place of the more commonly used derivative formulation of Druyvesteyn.}, number={2}, journal={Plasma Sources Science and Technology}, publisher={IOP Publishing}, author={Saghir, Ahmed El and Shannon, Steven}, year={2012}, month={Mar}, pages={025003} }
 @article{el saghir_shannon_2011, title={Limitations of Regularization Methods for the Reconstruction of Electron Velocity Distribution Function}, volume={39}, ISSN={0093-3813 1939-9375}, url={http://dx.doi.org/10.1109/TPS.2011.2125970}, DOI={10.1109/tps.2011.2125970}, abstractNote={The extraction of electron energy distribution functions (EEDFs) from Langmuir probe data is a discrete ill-posed problem. This problem rises due to the integral relationship between electron current and the probe voltage known as the Druyvesteyn relation. There have been a number of methods for the solution of this ill-posed problem ranging from data smoothing to a priori solution conditioning. Such methods include truncated singular value decomposition, truncated generalized singular value decomposition, and various regularization techniques. When these methods are extended to solve for similar integral relationships between electron current and electron distributions, complications arise due to their slightly different integral characteristics. For example, the electron velocity distribution function (EVDF) presents a similar ill-posed integral relationship. However, the EVDF integral presents an additional complication of rank deficiency that can` make accurate solutions of the inverse problem extremely challenging. In this paper, the ill-posed and rank deficiency problems of EEDF and EVDF reconstructions, respectively, are compared to highlight these challenges.}, number={4}, journal={IEEE Transactions on Plasma Science}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={El Saghir, A and Shannon, S}, year={2011}, month={Apr}, pages={1034–1037} }
 @article{el saghir_shannon_2011, title={Reduction of EEDF Measurement Distortion in Regularized Solutions of the Druyvesteyn Relation}, volume={39}, ISSN={0093-3813 1939-9375}, url={http://dx.doi.org/10.1109/TPS.2010.2090906}, DOI={10.1109/tps.2010.2090906}, abstractNote={Electron energy distribution function (EEDF) extraction from Langmuir probe data is an ill-posed problem due to the integral relationship between electron current and probe voltage. Both curve fitting of experimental data and reconstruction of the integral problem through methods, such as Tikhonov regularization, address this to some measure, with regularized solutions offering an advantage in overall EEDF accuracy over curve fitting. Although Tikhonov regularization provides a more accurate estimation of EEDF overall energy space, it typically also can distort the overall shape of the reconstructed distribution, particularly at high energies and energies below the distribution peak. This, combined with the relative ease of use that simple data smoothing algorithms provide, has limited the use of the more advanced reconstruction algorithms in EEDF analysis. In this paper, we will shed some light on these limitations and offer an alternative method to overcome these limitations.}, number={1}, journal={IEEE Transactions on Plasma Science}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={El Saghir, Ahmed and Shannon, Steven}, year={2011}, month={Jan}, pages={596–602} }
 @article{el saghir_kennedy_shannon_2010, title={Electron Energy Distribution Function Extraction Using Integrated Step Function Response and Regularization Methods}, volume={38}, ISSN={0093-3813 1939-9375}, url={http://dx.doi.org/10.1109/TPS.2009.2036013}, DOI={10.1109/tps.2009.2036013}, abstractNote={Recently, electron energy distribution function (EEDF) extraction techniques have been evaluated using regularized solutions to the integral problem. These techniques do not assume any mathematical representation of the EEDF and solve the integral problem for any function that best represents the EEDF. Also, unlike the more widely used point-by-point extraction of the second-derivative relationship, the integrated relationship between electron current and the EEDF is used, instead of a relatively small fraction of the integrated data in the point-by-point method. In this paper, the electron current for an arbitrary distribution function is derived, assuming that the distribution is a sum of step functions representing such a function. This technique for EEDF extraction is validated by adding noise to numerically generated data and using a regularized least squares (RLS) method to calculate the original function by solving for the individual step function contribution to the total electron current. Comparisons are then made between the expected and the reconstructed solution to evaluate its accuracy with respect to EEDF reconstruction and integrated normalization of the electron density.}, number={2}, journal={IEEE Transactions on Plasma Science}, publisher={Institute of Electrical and Electronics Engineers (IEEE)}, author={El Saghir, A. and Kennedy, C. and Shannon, S.}, year={2010}, month={Feb}, pages={156–162} }