Refinement of a double-slit test proposal for bicomplex phase coupling

The bicomplex extension of quantum mechanics introduces an ideal sector J1 whose phases are algebraically coupled to the complex sector C1 via the algebraic identity e^(iα)*k*e^(jβ) = k*e^j(α+β). In an earlier work, a double-slit test proposal for a possible physical realization of this phase coupling was formulated, which contained a calculation error. The present work refines this proposal and presents the corrected theoretical interpretation. The starting point is a bicomplex state Y = S + R, where S represents the complex component and R the ideal component, which is phenomenologically associated with a monochromatic laser source. In a nonlinear, frequency-doubling crystal, the state is phenomenologically mapped to Y^2 = S^2 + 2SR + R^2. The component S^2 lies in the complex sector, while the mixing term SR according to the above equation lies in the ideal sector and carries phase information transferred by the algebraic phase coupling. For a measurement at a detector or screen, this ideal mixing term must be transferred into a complex measurable channel with an effective coupling strength ηeff . This detector coupling is a phenomenological assumption and does not follow from the algebra alone. The mathematical evaluation of the double-slit interference, taking into account the given field structure (superposition of two slit contributions, cosine profile), shows that within the used Fraunhofer effective model the principal maxima follow the classical geometric shift to first approximation. An effectively coupled SR term at the detector primarily generates contrast and brightness modulations of the interference pattern. The double slit is interesting as a spatial structure test because a brightness modulation of the interference fringes depending on the position of the source is at least surprising.