An important point to note here is that an FDTD simulation of in Ref. 6, where a classical simulation of Maxwell equations using Finite Difference Time Domain (FDTD) methods was done. Moreover, it compares very well with Ref. We have showed that the analytic formula gives us an excellent match with both photon and electron parameters used in Ref. The analytic formula now makes the understanding of the deviation from the naive application of the superposition principle more tractable, shedding more light on the “black-box” like understanding that numerical simulations could afford. This would enable experimentalists to have a feel for how errors in the precise knowledge of various parameters can affect the distribution on the detector plane thus making it easier to compare theory with experiments. Thus having an analytic handle is very important as this makes it a much more accessible quantity to experimentalists. We find that the quantity is very sensitive to certain length parameters. In the current work, we have derived an analytic formula for as a function of detector position in the Fraunhofer regime. Only the first order correction term was considered in which paths of the kind shown in the inset of fig. The analysis using path integrals was restricted to the far field regime i.e., the Fraunhofer regime in optics and considered cases in which the thickness of the slits is negligible.
However 7, was restricted to semi-analytic and numerical methods. 7 to detect the presence of the non-classical paths uses a triple slit configuration as shown in fig. This would be zero if only the classical paths contribute and would be non-zero when the non-classical paths are taken into account. 7, 9, 10, the normalized version of the Sorkin parameter (defined later) was estimated. This discussion makes it clear that our approach is equally applicable to physics described by Maxwell theory and Schrödinger equation-see supplementary material of Ref. This way the naive application of the superposition principle is violated. Adding to Fresnel theory, we take into account a higher order effect and also account for influence by waves arriving through neighboring slits. In the usual Fresnel theory of diffraction, the assumption is that the wave amplitude at a particular slit would be the same as it would be away from the slits. In the nomenclature used, paths which extremize the classical action are called “classical” paths whereas paths which do not extremize the action are called “non-classical” paths. According to the path integral formalism, the probability amplitude to travel from point A to B should take into account all possible paths with proper weightage given to the different paths. More recently 7, dealt with the quantification of this correction in the quantum mechanical domain where the Feynman path integral formalism 8 was used to solve the problem of scattering due to the presence of slits.
This incorrect application was pointed out in a physically inaccessible domain by 5 and in a classical simulation of Maxwell equations by 6. However, the conditions described here correspond to different boundary conditions (or different Hamiltonians) and as such the superposition principle should not be directly applicable in this case.
For example, in a double slit experiment, the amplitude at the screen is usually obtained by adding the amplitudes corresponding to the slits open one at a time. It is not widely appreciated that the superposition principle is incorrectly applied in most textbook expositions of interference experiments both in optics and quantum mechanics 1, 2, 3, 4.
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