Abstract
We develop geometric optics expansion up to the subleading order for circularly polarized electromagnetic waves on curved spacetime. This subleading order geometric optics expansion, in which the conventional eikonal function is modified by inserting a carefully chosen helicity-dependent correction, is called spin optics. We derive the propagation and polarization equations in the spin optics approximation as electromagnetic waves travel in curved spacetime. Polarization-dependent deviation of the light ray trajectory from the geodesic, describing the gravitational spin Hall effect, is observed. We also establish an analogy with the related phenomena (optical Magnus effect) of condensed matter physics.
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PKD is supported by an International Macquarie University Research Excellence Scholarship.
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Appendices
Geometric optics limit
All the results of geometrical optics could be retrieved by taking Eqs. (12)–(14) and substituting \(m_1^\beta =0=l_1^\beta\). The Lorenz condition Eq. (12) and the wave equation Eq. (13) in the leading order approximation in \(\omega\) reduces to
Next, we calculate \(\tilde{m}_{0\alpha }j^\alpha\) from Eq. (13) by taking \(m_1^{\alpha }=0=l_{1\mu }\) as they are subleading order terms in \(\omega\) and thus irrelevant in geometric optics approximation, to obtain
Since the term \(\tilde{m}_{0 \alpha } m^\alpha _{0;\beta }l_0^\beta\) is purely imaginary and the remaining terms
are purely real, they should be separately zero, thereby giving
In the geometric optics approximation, these are the entire set of equations for electromagnetic waves in curved spacetime.
Checking self-duality
To verify that the tetrad \(\left( \dot{x}^\alpha , n^\alpha , m^\alpha , \tilde{m}^\alpha \right)\), satisfying Eqs. (62), (63) and (65) in the subleading order, is not a self-dual solution, let us first calculate Eq. (34):
where Eq. (A3) is used to obtain this identity. Similarly, Eq. (33) gives
where Eq. (66) is used to arrive at this identity. Finally, Eq. (32) gives
where \(\lambda\) denotes the Newman–Penrose scalar. Hence, Eq. (32) is not satisfied unless \(\tilde{\lambda }=0\).
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Dahal, P.K. Covariant formulation of spin optics for electromagnetic waves. Appl. Phys. B 129, 11 (2023). https://doi.org/10.1007/s00340-022-07952-2
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DOI: https://doi.org/10.1007/s00340-022-07952-2