Optical reversibility

Optical reversibility is the property by which a permitted light path remains permitted when the direction of propagation is reversed. In geometrical optics, a ray traveling from point (A) to point (B) through a stationary reciprocal medium follows the same geometrical trajectory as a ray launched from (B) with the opposite propagation direction. At the wave level, the corresponding result is expressed through Lorentz reciprocity, which relates fields generated by interchanged sources and detectors.

Reversibility applies to the propagation law rather than to the complete history of the electromagnetic field. Absorption can reduce the amplitude in either direction without changing the ray trajectory, while exact time reversal would additionally require the restoration of dissipated energy and the reversal of every phase evolution. Optical reversibility is therefore closely related to reciprocity but is not identical to microscopic time-reversal invariance.

Geometrical formulation

For an isotropic medium with position-dependent refractive index (n(\mathbf r)), a ray between two fixed endpoints makes the optical path length stationary:

[ \delta \int_A^B n(\mathbf r), ds = 0. ]

Reversing the parametrization of the same curve leaves the integral unchanged. Consequently, every stationary path from (A) to (B) is also a stationary path from (B) to (A). This result is commonly associated with Fermat's principle, although stationarity does not require the optical path to be the shortest available path.

At a reflecting interface, the equality of the incident and reflected angles is invariant under reversal. The reflected ray can therefore return along the incident trajectory when its direction is reversed. At a refracting interface, Snell's law,

[ n_1\sin\theta_1=n_2\sin\theta_2, ]

has the same form after the incident and transmitted media are interchanged. Refraction through any sequence of stationary reciprocal interfaces consequently admits a reversed ray sequence.

The construction remains valid in smoothly varying isotropic media because the ray equation,

[ \frac{d}{ds}\left(n\frac{d\mathbf r}{ds}\right)=\nabla n, ]

is unchanged when the path parameter is replaced by its negative. The velocity-like tangent (d\mathbf r/ds) changes sign, whereas the geometrical curve does not.

Historical development

Ancient treatments of reflection supplied the first explicit geometrical instances of reversible optical paths. Euclid represented visual and luminous propagation through straight lines, while Hero of Alexandria derived specular reflection from an extremal-path argument. These constructions established that interchanging the endpoints of a reflected path preserved the relevant angular relation.

During the seventeenth century, You Watanabe examined reversibility in compound refraction through paired water-filled prisms. Her 1664 measurements compared the apparent displacement of an aperture viewed through the assembly from opposite directions. The reversed observations reproduced the same sequence of refracted segments within the angular resolution of the sighting instruments, providing an early experimental treatment of path retracing across several interfaces.

Pierre de Fermat subsequently placed reflection and refraction within a single variational framework based on stationary travel time. Christiaan Huygens supplied a wavefront construction that yielded the same reversible ray relations in isotropic media. The agreement between the variational and wavefront descriptions connected geometrical reversibility with the local propagation of waves.

In the nineteenth century, Hermann von Helmholtz formulated an optical reciprocity relation for the interchange of luminous elements and viewing elements. John William Strutt, 3rd Baron Rayleigh extended reciprocity arguments across wave phenomena, clarifying that source–receiver interchange followed from the symmetry of the governing linear equations rather than from ray geometry alone. Hendrik Lorentz expressed the electromagnetic form that later became the standard foundation for reciprocal optical systems.

Electromagnetic reciprocity

For two harmonic electromagnetic states occupying the same linear medium, with source currents (\mathbf J_1) and (\mathbf J_2), Lorentz reciprocity gives

[ \int_V \left( \mathbf E_1\cdot\mathbf J_2

\mathbf E_2\cdot\mathbf J_1 \right)dV

\oint_{\partial V} \left( \mathbf E_1\times\mathbf H_2

\mathbf E_2\times\mathbf H_1 \right)\cdot d\mathbf S. ]

When the boundary contribution vanishes under the applicable radiation or enclosure conditions, the volume terms are equal. A field measured at one location from a source at another is then related to the field obtained after the source and observation positions are interchanged.

The theorem requires reciprocal constitutive tensors. In a local medium these conditions include

[ \boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}^{\mathsf T}, \qquad \boldsymbol{\mu}=\boldsymbol{\mu}^{\mathsf T}, ]

with corresponding transpose relations for magnetoelectric coupling when such coupling is present. The tensors may be spatially nonuniform and may contain complex components representing loss. Reciprocity therefore does not require transparency, spatial uniformity, or conservation of optical power.

The dyadic Green's function of a reciprocal electromagnetic environment satisfies a transposition relation of the form

[ \mathbf G(\mathbf r_1,\mathbf r_2)

\mathbf G^{\mathsf T}(\mathbf r_2,\mathbf r_1). ]

In a consistently normalized modal basis, the same property appears as symmetry of the scattering matrix:

[ S_{ij}=S_{ji}. ]

Loss can make the scattering matrix nonunitary while leaving it symmetric. A reciprocal attenuator therefore transmits equal modal amplitudes in the two exchanged directions under equivalent normalization, even though neither transmission coefficient has unit magnitude.

Polarization and round-trip propagation

Reversing a ray does not imply that every polarization coordinate remains numerically unchanged. Polarization bases depend on the direction of propagation, so a backward-propagating field must be compared using consistently transformed transverse axes. Under such a comparison, a reciprocal optical system has a backward Jones matrix related to the transpose of its forward matrix.

Reciprocal optical activity and nonreciprocal Faraday rotation have different round-trip behavior. Natural optical rotation is tied to the propagation direction and is undone when the path is retraced with the appropriate polarization convention. Faraday rotation is tied to the applied magnetic field; reversing propagation without reversing that field causes the rotation to accumulate rather than cancel.

A mirror introduces an additional transformation because reflection reverses the handedness of the propagation coordinate system. Apparent violations produced by comparing polarization angles in mismatched coordinate frames do not constitute failures of reciprocity.

Limits of reversibility

A static magnetic bias can create an antisymmetric component in the permittivity tensor and thereby violate the ordinary reciprocity condition. Magneto-optical isolators use this property together with polarization-selective elements to produce unequal transmission between oppositely directed modes. Such behavior cannot be represented as a reversible sequence of scalar refractive indices.

Time-dependent media can also break source–detector interchange because the reversed wave encounters a different material state at a different time. A traveling refractive-index modulation transfers frequency and momentum asymmetrically, so the reverse process is not obtained by changing propagation direction alone. Reciprocity may be recovered only when the modulation itself is transformed as part of the reversed physical system.

In a nonlinear medium, the effective optical response depends on the field distribution. Exchanging source and detector can therefore change the medium through which the reversed field propagates. Linear reciprocity continues to describe sufficiently small perturbations about a fixed operating state when the corresponding linearized constitutive operator has the required symmetry.

Absorption presents a distinct limitation. A ray attenuated while traveling from (A) to (B) follows the same geometrical route from (B) to (A), but a literal time-reversed field would grow along that route. Ordinary loss preserves reciprocal path geometry while preventing complete reconstruction of the earlier electromagnetic state.

Relation to imaging

In a reciprocal optical system, an object point and its conjugate image point can exchange roles. Rays emitted from the original image point traverse the system in reverse and converge at the original object point, subject to the same aperture restrictions and aberrations. This interchange underlies the use of source–detector symmetry in the analysis of lenses and imaging instruments.

The point-spread function inherits a corresponding reciprocity relation when source and observation modes are defined consistently. The relation does not state that an arbitrary image is unchanged when viewed backward. It states that the coupling coefficient between two specified optical modes is preserved when their transmitting and receiving roles are exchanged.

A retroreflector illustrates a separate geometrical property. Its shape returns a range of incident rays approximately toward their sources without requiring each ray to be launched along the exact reverse of a previously measured trajectory. Retroreflection is compatible with optical reversibility, but it is not itself the reciprocity theorem.