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tokens: several tens of percent at high optical depths. An accurate calculation of the dichroic optical depth requires the full solution of the intensity profile along the path. In the general case of a dichroic medium, the radiative transfer equation becomes a set of four coupled first-order differential equations with varying coefficients, and a closed expression for the dichroic optical depth cannot be derived. However, the exact solution corresponding to a medium with uniform grain alignment can be used to find the full solution in an elegant way without any further numerical integration. There is no benefit in using τ ext , τ tot , or any other approximation instead of the dichroic optical depth to calculate the next interaction location in a Monte Carlo radiative transfer simulation. Our results have implications for Monte Carlo radiative transfer coders that wish to incorporate the attenuation by elongated dust grains. If scattering polarisation by spherical grains already adds some complexity to Monte Carlo radiative transfer codes, dealing with non-spherical grains increases this complexity to a new level. Compared to spherical grains, the scattering process is significantly more complex. The scattering properties of spherical grains are fully described by just the albedo and the scattering phase function; for elongated grains, a full 4 × 4 scattering matrix comes into play ( Mishchenko et al. 2000 ), and the random determination of a new propagation direction after a scattering event is not trivial ( Wolf et al. 2002 ; Whitney Wolff 2002 ; Lucas 2003 ). A related complexity concerns the amount of data that needs to be stored and accessed. Each of the elements of the extinction matrix and the scattering matrix is not only dependent on grain material, size, and wavelength, but also on shape and incidence angle. Moreover, each element of the scattering matrix needs to be discretised on the unit sphere. Finally, the process of dichroic extinction adds yet another level of complexity, and the results in this paper show that this also affects the random generation of the next interaction location. So far, only a limited number of Monte Carlo radiative transfer codes have attempted to actually calculate dichroic attenuation by non-spherical aligned grains 4 . Wolf et al. (2002) were the first to include non-spherical aligned grains in their Monte Carlo radiative transfer calculations. They presented multiple light scattering calculations and demonstrated that the incorporation of elongated grains is important to explain the circular polarisation of light. They discussed the concepts of dichroism and birefringence, but they did not include these effects in the simulations they presented. Whitney Wolff (2002) presented a Monte Carlo code that models the effects of scattering and dichroic absorption by aligned grains in circumstellar environments. While their code is presented for general geometries, they only discussed models with a uniform grain