An apparatus, comprising: a light source arranged to emit light in a direction parallel to an axis x; a chamber to define a first part of an interaction volume that attenuates light emitted by the light source as a function of path length of the light through a material sample in the first part of the interaction volume, to provide attenuated light energy after the light has traveled through the interaction volume along a plurality of paths; and an interacted light energy transformation element (240, 340) to define a second part of the interaction volume, to intercept the attenuated light energy along a plurality of path lengths, wherein the interacted light energy transformation element has a logarithmic surface (120, 320) that forms an entrance surface of the second part of the interaction volume, the entrance surface having a shape such that the plurality of path lengths vary approximately logarithmically in a radial direction that is perpendicular to the axis x, wherein the entrance surface is axisymmetric, and the shape of the entrance surface being such that a transmittance of the transformation element is substantially linear with respect to absorption of light by the material sample such that the transformation element simultaneously transforms the attenuated light energy by simultaneous detection from the plurality of paths, which attenuated light energy is characterized by a substantially exponential intensity function of the absorption coefficient of the material sample, into resultant light energy that is characterized by a substantially polynomial intensity function of said absorption coefficient. a light source arranged to emit light in a direction parallel to an axis x; a chamber to define a first part of an interaction volume that attenuates light emitted by the light source as a function of path length of the light through a material sample in the first part of the interaction volume, to provide attenuated light energy after the light has traveled through the interaction volume along a plurality of paths; and an interacted light energy transformation element (240, 340) to define a second part of the interaction volume, to intercept the attenuated light energy along a plurality of path lengths, wherein the interacted light energy transformation element has a logarithmic surface (120, 320) that forms an entrance surface of the second part of the interaction volume, the entrance surface having a shape such that the plurality of path lengths vary approximately logarithmically in a radial direction that is perpendicular to the axis x, wherein the entrance surface is axisymmetric, and the shape of the entrance surface being such that a transmittance of the transformation element is substantially linear with respect to absorption of light by the material sample such that the transformation element simultaneously transforms the attenuated light energy by simultaneous detection from the plurality of paths, which attenuated light energy is characterized by a substantially exponential intensity function of the absorption coefficient of the material sample, into resultant light energy that is characterized by a substantially polynomial intensity function of said absorption coefficient.