Geometrical optics is an extensively used concept of optics where the light propagation is described with geometric light rays. This article discusses the importance of geometrical optics, recent developments in this field, and the application of geometrical optics for light propagation analysis in microscale structures, specifically multi-mode fibers.
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Importance of Geometrical Optics
Geometrical optics/ray optics is primarily an optics model that describes light propagation in terms of rays without considering quantum effects, polarization, diffraction, and interference. In geometrical optics, the ray is an abstraction crucial to approximate the paths along which the light propagates under specific circumstances.
The assumptions that are made in geometrical optics include that light rays have no transverse extension/zero thickness, propagate in straight lines in homogeneous optical materials, can be stopped upon heating optical apertures, can cross without influencing each other, and can be refracted or reflected at smooth optical interfaces, with specific laws determining the outgoing ray directions.
Geometrical optics is based on the short-wavelength approximation of electromagnetic theory and can be defined in terms of a package of rules/geometrical optics rules that can be derived from the Maxwell equations in a consistent approximation scheme, which is known as the eikonal approximation. Geometrical optics is a convenient and effective scheme for describing the electromagnetic field in several practical applications.
In an accurate description of the electromagnetic field for a specific setup, the Maxwell equations must be solved considering the boundary and initial conditions that are appropriate to the setup, which can be challenging in real-world scenarios, necessitating the use of proper approximation schemes, such as geometrical optics.
The approximation scheme/eikonal approximation that relates ray optics to the general electromagnetic theory is not specific either to electromagnetic theory or to optics as it is applicable to approximating solutions to a class of differential equations. Specifically, the approximation is applicable to vector and scalar wave equations, with the vector wave equations including wave equations satisfied by the electromagnetic field variables.
The approximation scheme is based on the characterization of visible light by a range of extremely short wavelengths/wavelengths shorter compared to the dimensions of devices used commonly in optical instruments or objects typically viewed with the help of light.
For instance, if the dimension of these objects is on the order of one cm, then the wavelength of the light is typically on the order of 10−4 times of this dimension. This smallness of wavelength makes the eikonal approximation relevant in the formulation of a number of working rules related to the light propagation/ray optics rules.
Thus, the eikonal approximation leads to wave equation solutions in the limit of zero wavelength. Although the approximation has limitations as it is only applicable to a limited class of optical phenomena, most of the regular and common experience falls within this class.
Geometrical Optics in Microscale Structures
In a study published in the journal Optical Fiber Technology, researchers constructed encircled-flux (EF) ray-based compliant sources for multi-mode optical fiber launches using classic geometrical optics. Specifically, they developed a linearly polarized (LP)-mode conforming launch that excited only guided rays by distributing the β¯ ray invariant based on a scaled counterpart of the LP-mode modal power distribution.
The geometrical-optics-based source models developed by researchers generated spatially stable ray launches in graded-index regular multimode optical fibers. Additionally, the LP-mode conforming model generated rays that satisfied the interpolated near-field pattern at every arbitrary cross-section along the fiber length due to fewer requirements on the near-field pattern.
The far-field ray-angle distribution was connected to the near-field ray-position distribution through a scaled counterpart of the LP-mode modal power distribution, and similar near-field and far-field patterns were produced as the LP-mode conforming source.
In the circular-skew ray model, 20% of the optical power was carried by core-confined tunneling rays, while the LP-mode conforming source excited only guided rays. Moreover, the LP-mode conforming source generated a spatially stable ray density distribution and enabled the prediction of far-field measurements only by applying Snell’s law at the fiber-air interface.
Although the classic geometrical optics do not allow describing rays beyond the core-cladding interface, the ray simulation of the one mm far-field distribution tail matched effectively. The LP-mode conforming source also allowed the prediction of physical-contact connection attenuation measurements.
Researchers demonstrated that both the attenuation and modal noise predictions compare effectively to a vectorial full-wave approach through the proper configuration of a geometrical-optics-based LP-mode conforming source even when the launch consisted of coherent spatio-temporal speckle patterns.
Specifically, the configuration of the LP-mode conforming source to reproduce the measured/actual EF of the launch fiber significantly impacted the attenuation simulation outcome.
The connection attenuation curve shape better matched the measurements after considering a marginal deviation of the EF away from the ideal target EF. Subsequently, a 50.4 ± 0.2 μm core diameter setting allowed researchers to match the lower- and upper-bound of the connection attenuation measurements.
Thus, the model-based comparison approach can be feasibly used as a more accurate and new alternative to existing core-diameter measurement methods, such as the transmitted near-field method, which has a measurement uncertainty of approximately one μm.
Recent Developments in Geometrical Optics
The continuum-mechanical formulation of wave mechanics indicates the presence of an intermediate stage of theoretical generality between point mechanics and wave mechanics, which is referred to as continuum mechanics. If this argument can be applied to the transition from wave optics to geometrical optics, the corresponding intermediate stage would be the geometrical theory of diffraction/the theory of diffracted geodesics.
In a paper published in the Journal of Physics: Conference Series, the author discussed this optical-mechanical analogy for wave mechanics, which could lead to new insights and applications. In geometrical optics approximation, the presence of the quantum potential in the absence of approximation can potentially indicate a deeper difference between quantum and classical mechanics.
The primary difference between geometrical optics and wave optics can be characterized as the difference between a system of ordinary differential equations whose solutions represent the light rays normal to the wave surfaces and the amplitude propagation along those rays and a system of partial differential equations whose solutions define the amplitude and phase of waves in space.
The basic form of the ordinary differential equations of light rays is similar to geodesic equations for a spatial metric conformal to the geometric background metric through a conformal factor that equals the square of the refraction index for the medium. Thus, a good intermediate stage between geometrical optics and wave optics can be a theory of diffracted geodesics.
Specifically, the geometrical optics approximation would correspond to considering a single geodesic/light ray, while considering a congruence of geodesics with a finite cross section would represent an intermediate step between wave optics and geometrical optics, which would require the study of diffracted geodesics considering the impact of diffraction on the solutions to the Helmholtz equation.
The paper concluded that the difference between the geometrical optics approximation and wave optics can possibly be attributed to the inclusion of quantum potential, which can be theoretically expanded into an asymptotic series.
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References and Further Readings
Lahiri, A. (2015). Foundations of Ray Optics. Basic Optics, 141-202. https://doi.org/10.1016/B978-0-12-805357-7.00002-2
Floris, S. J., De Hon, B. P., Bolhaar, T., Smink, R. W. (2019). A geometrical optics approach to encircled-flux compliant source modeling for multi-mode fiber illumination and connection attenuation. Optical Fiber Technology, 54, 102116. https://doi.org/10.1016/j.yofte.2019.102116
Delphenich, D. H. (2022). The optical-mechanical analogy for wave mechanics: a new hope. Journal of Physics: Conference Series, 2197, 012005. https://doi.org/10.1088/1742-6596/2197/1/012005
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