What if the beam splitter has no losses

A lossless beam splitter is an optical device that splits an incident light beam into transmitted and reflected components without absorbing or dissipating any energy.Definition and PrincipleA lossless beam splitter is designed so that the total optical power of the output beams equals the power of the input beam. This means that no energy is lost to absorption or scattering within the device. Mathematically, the beam-splitter transfer matrix is unitary, ensuring energy conservation, and the reflection and transmission coefficients satisfy the relation |r|² + |t|² = 1 for each input port . In quantum optics, this corresponds to a transformation that preserves the commutation relations of the photonic operators .Classical DescriptionIn classical optics, a lossless beam splitter can be modeled as a dielectric plate or a cube with partially reflective coatings. When a light beam hits the splitter, part of it is transmitted and part is reflected. The phase shifts introduced upon reflection and transmission are accounted for in the complex coefficients r and t, which can include amplitude and phase factors. The total output intensity is equal to the input intensity, confirming the absence of loss .Quantum Optical PerspectiveIn quantum optics, a lossless beam splitter transforms the annihilation operators of the input modes (â₁, â₂) into output modes (b̂₁, b̂₂) via a unitary transformation. This ensures that the quantum state of light is preserved in terms of total photon number, although the photons are redistributed between the output ports. The determinant of the transfer matrix is 1, and the transformation belongs to the SU(2) group, reflecting the conservation of probability amplitudes .Practical ConsiderationsWhile ideal lossless beam splitters are a theoretical model, in practice, high-quality dielectric coatings can achieve negligible losses, making them effectively lossless for most applications. Lossless beam splitters are crucial in interferometry, quantum information processing, and optical experiments, where precise control of amplitude and phase is required . In summary, a lossless beam splitter ensures energy conservation, maintains the unitarity of optical transformations, and is essential for both classical and quantum optical systems where minimizing loss is critical.