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In optics, a ray is an idealized geometrical model of light or other electromagnetic radiation, obtained by choosing a curve that is perpendicular to the wavefronts of the actual light, and that points in the direction of energy flow.[1][2] Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very complex optical systems to be analyzed mathematically or simulated by computer. Ray tracing uses approximate solutions to Maxwell's equations that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light's wavelength. Ray optics or geometrical optics does not describe phenomena such as diffraction, which require wave optics theory. Some wave phenomena such as interference can be modeled in limited circumstances by adding phase to the ray model.
A light ray is a line (straight or curved) that is perpendicular to the light's wavefronts; its tangent is collinear with the wave vector. Light rays in homogeneous media are straight. They bend at the interface between two dissimilar media and may be curved in a medium in which the refractive index changes. Geometric optics describes how rays propagate through an optical system. Objects to be imaged are treated as collections of independent point sources, each producing spherical wavefronts and corresponding outward rays. Rays from each object point can be mathematically propagated to locate the corresponding point on the image.
A slightly more rigorous definition of a light ray follows from Fermat's principle, which states that the path taken between two points by a ray of light is the path that can be traversed in the least time.[3]
There are many special rays that are used in optical modelling to analyze an optical system. These are defined and described below, grouped by the type of system they are used to model.
An incident ray is a ray of light that strikes a surface. The angle between this ray and the perpendicular or normal to the surface is the angle of incidence.
The reflected ray corresponding to a given incident ray, is the ray that represents the light reflected by the surface. The angle between the surface normal and the reflected ray is known as the angle of reflection. The Law of Reflection says that for a specular (non-scattering) surface, the angle of reflection is always equal to the angle of incidence.
The refracted ray or transmitted ray corresponding to a given incident ray represents the light that is transmitted through the surface. The angle between this ray and the normal is known as the angle of refraction, and it is given by Snell's law. Conservation of energy requires that the power in the incident ray must equal the sum of the power in the refracted ray, the power in the reflected ray, and any power absorbed at the surface.
If the material is birefringent, the refracted ray may split into ordinary and extraordinary rays, which experience different indexes of refraction when passing through the birefringent material.
Single lens imaging with the aperture stop. The entrance pupil is an image of the aperture stop formed by the optics in the front of it, and the location and size of the pupil are determined by chief rays and marginal rays, respectively.
A meridional ray or tangential ray is a ray that is confined to the plane containing the system's optical axis and the object point from which the ray originated.[4] This plane is called meridional plane or tangential plane.
A skew ray is a ray that does not propagate in a plane that contains both the object point and the optical axis (meridional or tangential plane). Such rays do not cross the optical axis anywhere and are not parallel to it.[4]
The exit pupil is the image of the aperture stop formed by the optics behind it, and the location and size of the pupil are determined by chief rays and marginal rays.The marginal ray (sometimes known as an a ray or a marginal axial ray) in an optical system is the meridional ray that starts from an on-axis object point (the point where an object to be imaged crosses the optical axis) and touches an edge of the aperture stop of the system.[5][6][7] This ray is useful, because it crosses the optical axis again at the location where a real image will be formed, or the backward extension of the ray path crosses the axis where a virtual image will be formed. Since the entrance pupil and exit pupil are images of the aperture stop, for a real image pupil, the lateral distance of the marginal ray from the optical axis at the pupil location defines the pupil size. For a virtual image pupil, an extended line, forward along the marginal ray before the first optical element or backward along the marginal ray after the last optical element, determines the size of the entrance or exit pupil, respectively.
The principal ray or chief ray (sometimes known as the b ray) in an optical system is the meridional ray that starts at an edge of an object and passes through the center of the aperture stop.[5][8][7] The distance between the chief ray (or an extension of it for a virtual image) and the optical axis at an image location defines the size of the image. This ray (or forward and backward extensions of it for virtual image pupils) crosses the optical axis at the locations of the entrance and exit pupils. The marginal and chief rays together define the Lagrange invariant, which characterizes the throughput or etendue of the optical system.[9] Some authors define a "principal ray" for each object point, and in this case, the principal ray starting at an edge point of the object may then be called the marginal principal ray.[6]
A sagittal ray or transverse ray from an off-axis object point is a ray propagating in the plane that is perpendicular to the meridional plane for this object point and contains the principal ray (for the object point) before refraction (so along the original principal ray direction).[4] This plane is called sagittal plane. Sagittal rays intersect the pupil along a line that is perpendicular to the meridional plane for the ray's object point and passes through the optical axis. If the axis direction is defined to be the z axis, and the meridional plane is the y-z plane, sagittal rays intersect the pupil at yp= 0. The principal ray is both sagittal and meridional.[4] All other sagittal rays are skew rays.
A paraxial ray is a ray that makes a small angle to the optical axis of the system and lies close to the axis throughout the system.[10] Such rays can be modeled reasonably well by using the paraxial approximation. When discussing ray tracing this definition is often reversed: a "paraxial ray" is then a ray that is modeled using the paraxial approximation, not necessarily a ray that remains close to the axis.[11][12]
A finite ray or real ray is a ray that is traced without making the paraxial approximation.[12][13]
A parabasal ray is a ray that propagates close to some defined "base ray" rather than the optical axis.[14] This is more appropriate than the paraxial model in systems that lack symmetry about the optical axis. In computer modeling, parabasal rays are "real rays", that is rays that are treated without making the paraxial approximation. Parabasal rays about the optical axis are sometimes used to calculate first-order properties of optical systems.[15]
A leaky ray or tunneling ray is a ray in an optical fiber that geometric optics predicts would totally reflect at the boundary between the core and the cladding, but which suffers loss due to the curved core boundary.
Geometrical optics, or ray optics, is a model of optics that describes lightpropagation in terms of rays. The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances.
The simplifying assumptions of geometrical optics include that light rays:
propagate in straight-line paths as they travel in a homogeneous medium
bend, and in particular circumstances may split in two, at the interface between two dissimilar media
follow curved paths in a medium in which the refractive index changes
may be absorbed or reflected.
Geometrical optics does not account for certain optical effects such as diffraction and interference, which are considered in physical optics. This simplification is useful in practice; it is an excellent approximation when the wavelength is small compared to the size of structures with which the light interacts. The techniques are particularly useful in describing geometrical aspects of imaging, including optical aberrations.
In physics, ray tracing is a method for calculating the path of waves or particles through a system with regions of varying propagation velocity, absorption characteristics, and reflecting surfaces. Under these circumstances, wavefronts may bend, change direction, or reflect off surfaces, complicating analysis.
Historically, ray tracing involved analytic solutions to the ray's trajectories. In modern applied physics and engineering physics, the term also encompasses numerical solutions to the Eikonal equation. For example, ray-marching involves repeatedly advancing idealized narrow beams called rays through the medium by discrete amounts. Simple problems can be analyzed by propagating a few rays using simple mathematics. More detailed analysis can be performed by using a computer to propagate many rays.
^Moore, Ken (25 July 2005). "What is a ray?". ZEMAX Users' Knowledge Base. Retrieved 30 May 2008.
^Greivenkamp, John E. (2004). Field Guide to Geometric Optics. SPIE Field Guides. p. 2. ISBN0819452947.
^Arthur Schuster, An Introduction to the Theory of Optics, London: Edward Arnold, 1904 online.
^ abcdStewart, James E. (1996). Optical Principles and Technology for Engineers. CRC. p. 57. ISBN978-0-8247-9705-8.
^ abGreivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN0-8194-5294-7., p. 25 [1].
^ abRiedl, Max J. (2001). Optical Design Fundamentals for Infrared Systems. Tutorial texts in optical engineering. Vol. 48. SPIE. p. 1. ISBN978-0-8194-4051-8.
^ abHecht, Eugene (2017). "5.3.2 Entrance and Exit Pupils". Optics (5th ed.). Pearson. p. 184. ISBN978-1-292-09693-3.
^ abAtchison, David A.; Smith, George (2000). "A1: Paraxial optics". Optics of the Human Eye. Elsevier Health Sciences. p. 237. ISBN978-0-7506-3775-6.
^Welford, W. T. (1986). "4: Finite Raytracing". Aberrations of Optical Systems. Adam Hilger series on optics and optoelectronics. CRC Press. p. 50. ISBN978-0-85274-564-9.
^Buchdahl, H. A. (1993). An Introduction to Hamiltonian Optics. Dover. p. 26. ISBN978-0-486-67597-8.
In optics, a ray is defined as a line representing the direction along which light energy flows, modeling light propagation as straight-line paths in homogeneous media.[1] Ray optics, also known as geometrical optics, is the branch of optics that uses this ray approximation to study light behavior, valid when the wavelength of light is much smaller than the dimensions of the optical elements involved.[2] This model treats light rays as non-interacting lines that originate from sources, travel at speed c/n (where c is the speed of light in vacuum and n is the refractive index), and interact with matter through processes like reflection, refraction, absorption, or scattering.[3]The foundational principles of ray optics are governed by two primary laws: the law of reflection and the law of refraction. The law of reflection states that the angle of incidence equals the angle of reflection for a light ray striking a surface, with both angles measured relative to the normal, enabling the formation of images in mirrors.[4] The law of refraction, known as Snell's law, describes how a ray bends at the interface between two media: n1sinθ1=n2sinθ2, where n1 and n2 are the refractive indices and θ1 and θ2 are the angles of incidence and refraction, respectively; this bending occurs toward the normal when entering a denser medium and away otherwise.[1] A related phenomenon, total internal reflection, arises when light attempts to refract from a denser to a rarer medium at an angle greater than the critical angle (sinθc=n2/n1), fully reflecting the ray back into the original medium.[3]Ray optics finds extensive applications in imaging systems and optical technologies, such as lenses and mirrors in cameras, microscopes, and telescopes, where rays are traced to determine image location, size, and orientation using paraxial approximations for small angles.[2] It underpins fiber optic communications, relying on total internal reflection to guide light signals through thin glass cores with cladding layers.[3] While powerful for macroscopic phenomena, ray optics is an approximation of the more complete wave theory of light, breaking down for diffraction or interference effects where wavelengths are comparable to obstacle sizes.[4]
Basic Concepts
Definition
In geometrical optics, a ray represents an idealized model of light propagation as a narrow beam of energy traveling along a straight line path that is perpendicular to the wavefronts, effectively approximating the direction of light's advancement without considering its wave nature.[5][6] This abstraction treats light as having negligible thickness and propagating in straight lines within uniform conditions, serving as a foundational tool for analyzing image formation and optical systems.[5][7]The concept of rays traces its origins to ancient Greek optics, where Euclid, in his work Optica around 300 BCE, first posited that light travels in straight lines and described the law of reflection based on visual rays emanating from the eye.[8] Ptolemy, in the 2nd century CE, expanded on these ideas in his treatise Optics, incorporating early notions of refraction and maintaining an extramission theory of vision where rays originate from the observer.[9] The framework was formalized in the 17th century by René Descartes, who integrated rays into a particle-like model of light in La Dioptrique (1637), while Pierre de Fermat, in 1657, derived the laws of reflection and refraction from his principle of least time, stating that light follows the path minimizing travel duration between points.[10][11]Ray optics distinguishes itself from wave optics and the full electromagnetic theory by ignoring phenomena such as diffraction and interference, which arise from light's wave properties; this approximation holds when the wavelength of light is much smaller than the scale of obstacles or apertures, allowing straight-line propagation to dominate./25:_Geometric_Optics)[12] In contrast, wave models account for these effects explicitly, providing a more complete description but at greater computational complexity./13:_Optics/13.05:_Wave_Optics)The validity of ray optics relies on specific prerequisites, including the use of monochromatic light to eliminate dispersion effects that vary with wavelength, and propagation through isotropic media where the refractive index remains uniform in all directions, ensuring consistent ray paths.[5][13] These assumptions simplify the analysis while maintaining accuracy for macroscopic optical phenomena.[5]
Mathematical Representation
In geometrical optics, a light ray is mathematically modeled as a directed line segment representing the path of light propagation. The position along the ray can be described by the parametric vector equation r(t)=r0+td, where r0 is the position vector of the ray's origin, d is the unit direction vector pointing along the ray, and t≥0 is a scalar parameter proportional to the distance traveled.[14] This formulation allows for precise computation of ray positions in three-dimensional space, with the unit vector d ensuring normalization to maintain directional consistency.[15]The direction vector d of the ray is aligned with the unit wave vectork^, which is perpendicular to the wavefronts and points in the direction of phase propagation.[16] Specifically, d=k^=k/∣k∣, where k is the wave vector with magnitude k=2π/λ related to the wavelengthλ. This connection bridges the ray model to the underlying wavenature of light, as the ray direction corresponds to the local propagation direction of the phase velocity. In homogeneous media, rays propagate in straight lines along this direction, facilitating simple trajectory calculations.[14]Along a ray in lossless, isotropic media, the specific intensity (or radiance) I satisfies the transport equation, leading to the invariance of I/n2, where n is the refractive index. This quantity remains constant, reflecting conservation of the etendue (phase space volume) for a ray bundle, even across refractive index variations.[17] The invariance arises from the geometric optics approximation, where energy flux through infinitesimal areas and solid angles is preserved, adjusted by the medium's optical density.[18]For computational purposes, rays are often parameterized in specific coordinate systems depending on the optical setup. In general cases, Cartesian coordinates (x,y,z) are used to describe arbitrary ray paths via the parametric equation. For rotationally symmetric systems, such as those involving spherical wavefronts from point sources, spherical coordinates (r,θ,ϕ) provide a natural framework, with the ray direction expressed in terms of radial and angular components.[14]
Ray Propagation
In Homogeneous Media
In homogeneous, isotropic media, light rays propagate in straight lines without deviation, serving as the foundational assumption of geometrical optics. This rectilinear propagation occurs because the medium's uniform refractive indexn ensures that the ray's direction remains constant along its path. The speed of the ray in such a medium is given by v=c/n, where c is the speed of light in vacuum, reflecting the medium's influence on light's velocity without altering its trajectory.[5][19][20]The path taken by a ray in a homogeneous medium adheres to Fermat's principle, which states that rays follow trajectories of stationary optical path length, defined as ∫n[ds](/page/Infinitesimal), where ds is an infinitesimal element along the path. In a uniform medium, this principle simplifies to the shortest geometric distance, as n is constant, reinforcing the straight-line behavior./03%3A_Geometrical_Optics/3.05%3A_Fermats_Principle)[21]Ray optics breaks down near caustics, where rays converge and the approximation fails due to singularities in the ray field, requiring wave-based corrections. Additionally, the model neglects wavelength-dependent effects like diffraction, which become prominent when the wavelength is comparable to structural scales in the medium.[22][23]Examples of ray propagation in homogeneous media include light traveling through air, where n≈1 allows near-vacuum speeds and straight paths over long distances, or through a glass block with n≈1.5, where rays maintain their direction but slow to v≈(2/3)c. Parallel rays, such as those from a distant point source, remain parallel throughout, preserving beam collimation in uniform glass or air.[24][25]
At Interfaces
When a ray encounters an interface between two media, it can undergo reflection, where the ray bounces off the surface, or refraction, where the ray passes into the second medium while changing direction. In homogeneous media, rays travel in straight lines, but at boundaries, these deviations occur due to the change in the medium's refractive index.[26]The law of reflection states that the incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane, and the angle of incidence equals the angle of reflection, given by θi=θr, where angles are measured from the normal vector nperpendicular to the interface.[27] This law applies to smooth surfaces like mirrors and can be derived from Fermat's principle of least time, which posits that light takes the path minimizing travel time.[27]For refraction, Snell's law describes the bending of the ray: n1sinθi=n2sinθr, where n1 and n2 are the refractive indices of the first and second media, respectively, and θi and θr are the angles of incidence and refraction relative to the normal.[27] This relation emerges from Fermat's principle by considering paths across the interface that minimize the total time, accounting for the speed of lightc/n in each medium, leading to the equality of nsinθ on both sides.[28]If the ray travels from a denser medium (n1>n2) and the angle of incidence exceeds the critical angle θc=sin−1(n2/n1), total internal reflection occurs, with the entire ray reflecting back into the first medium and no refracted ray transmitting.[29] The critical angle is derived by setting θr=90∘ in Snell's law, yielding sinθc=n2/n1.[30]The amplitudes of the reflected and transmitted rays are governed by the Fresnel equations, which provide the reflection coefficientr as the ratio of reflected to incident electric field amplitude, depending on the angle and refractive indices, while emphasizing that ray paths follow the reflection and refraction laws without delving into polarization effects.[31]A common example is a ray from air (n1≈1) incident on glass (n2≈1.5) at 30° to the normal; the refracted ray bends toward the normal at θr=sin−1(sin30∘/1.5)≈19.5∘, illustrating refraction's role in image formation.[32] In a prism, such as an equilateral glass prism with apex angle 60°, an incident ray undergoes refraction at the first face and refraction at the second, resulting in an overall deviation angle that disperses white light into colors due to wavelength-dependent n.[33]
Special Rays
Paraxial Rays
Paraxial rays in optics are defined as light rays that propagate close to the optical axis and make small angles θ with it, typically satisfying θ≪1 radian.[34] This condition enables the paraxial approximation, which simplifies ray tracing by replacing trigonometric functions with their small-angle limits: sinθ≈θ and tanθ≈θ.[35] These rays form the basis for first-order optical analysis, where higher-order terms in the ray paths are neglected to focus on primary imaging properties.[15]In paraxial ray tracing, the position and direction of a ray are tracked using a two-element vector (y,α), where y is the transverse height from the optical axis and α is the paraxial angle (optical direction cosine) with the axis.[36] Propagation through a homogeneous medium of refractive indexn over distance d is described by the ray transfer matrix:(y′α′)=(10d/n1)(yα)This matrix preserves the ray's angle while linearly updating its height based on the propagation distance scaled by the refractive index.[37] Similar matrices apply to refraction and reflection at interfaces under the paraxial approximation, allowing the entire optical system to be represented as a product of such 2×2 matrices for efficient computation of ray paths.[3]Paraxial rays are particularly suited for first-order optics in lens systems, where they enable straightforward predictions of image location, magnification, and focal length without considering wavefront distortions.[38] This approach ignores aberrations, providing an ideal model for thin lenses and centered optical elements aligned along the axis, as commonly used in basic telescope and microscope designs.[39]However, the paraxial approximation becomes inaccurate for rays at wider field angles or larger apertures, where the neglected higher-order terms in the sine expansion—such as sinθ≈θ−θ3/6—introduce significant deviations.[40] Exceeding the small-angle regime leads to third-order errors manifested as Seidel aberrations, including spherical aberration and coma, which blur the image and limit the approximation's validity to narrow fields of view.[41]
Chief and Marginal Rays
In optical systems, the chief ray is defined as the ray originating from the edge of the object field that passes through the center of the aperture stop, as well as the centers of the entrance and exit pupils.[42] This ray determines the field angle and image height, providing a reference for the off-axis position in the image plane.[43] In the paraxial approximation, the chief ray passes undeviated through the optical center of symmetric thin lenses due to the negligible deviation for rays near the axis.[44]Marginal rays, in contrast, are those that originate from the on-axis point of the object and pass through the edges of the aperture stop, entrance pupil, and exit pupil.[42] These rays delineate the beam width and the extent of the ray bundle, which are essential for calculating the f-number of the system—defined as the ratio of the focal length to the diameter of the entrance pupil—and assessing resolution limits.[42] The f-number, often denoted as f/#=f/DEP, where f is the focal length and DEP is the entrance pupildiameter, quantifies the light-gathering capability and depth of field.[42]Pupil rays refer to the rays that bound the entrance and exit pupils, forming the conical sections of the ray bundle between the chief ray (central axis of the off-axis bundle) and the marginal rays (defining the radial extent).[45] Rays intermediate between the chief and marginal rays are used in vignetting analysis to evaluate partial obscuration of the bundle by system apertures, where unvignetted bundles satisfy aperture radius a≥y+y′ (with y and y′ as chief ray heights in object and image space), half-vignetted at a=y, and fully vignetted at a≤∣y−y′∣.[45]In a camera lens, for instance, the chief ray from an off-axis object point traces the corresponding image height, establishing the field of view, while marginal rays from the on-axis point define the light collection efficiency and limit the bundle entering the system.[42] Marginal rays are particularly relevant to aberrations, as their greater deviation compared to paraxial rays in spherical surfaces leads to spherical aberration, where they focus at a different point than central rays, degrading image sharpness.[46]
Applications
Geometrical Optics
Geometrical optics, also known as ray optics, models light propagation through optical systems by treating light as rays that follow straight-line paths in homogeneous media and obey the laws of reflection and refraction at interfaces. This approximation is valid when the wavelength of light is much smaller than the dimensions of the optical elements involved, allowing the prediction of image formation without considering wave interference effects.[47]In geometrical optics, rays from an object point converge or appear to diverge to form an image, governed by object-image relations such as the thin lens equation for a lens with focal lengthf, object distance o, and image distance i:o1+i1=f1.This equation derives from Snell's law applied to paraxial rays and determines whether the image is real (rays actually converge) or virtual (rays appear to diverge). For converging systems like convex lenses, positive f yields real images when o>f, while diverging systems like concave lenses produce virtual images with negative f. Magnificationm=−oi quantifies the image size and orientation.[47]Ray paths in mirror and lens systems illustrate image formation. For a concave mirror (converging, focal length f=R/2>0 where R is the radius of curvature), a ray parallel to the optical axis reflects through the focal point, a ray through the focal point reflects parallel to the axis, and a ray through the center of curvature reflects back along itself; objects beyond f form real, inverted images on the same side, while closer objects yield virtual, upright images behind the mirror. Convex mirrors (diverging, f<0) always produce virtual, upright, diminished images behind the mirror, with parallel incident rays diverging as if from the focal point. In convex lenses, parallel rays converge to the focal point on the opposite side, forming real images for distant objects; concave lenses cause parallel rays to diverge, creating virtual images on the same side. These paths enable applications in telescopes and microscopes by controlling image location and size.[48][47]The foundations of modern geometrical optics were laid by Carl Friedrich Gauss in his 1841 treatise Dioptrische Untersuchungen, which introduced the paraxial approximation—assuming small angles relative to the optical axis—and derived invariants like the Helmholtz-Lagrange invariant, relating object and image positions through conserved quantities such as nyuˉ, where n is the refractive index, y the ray height, and uˉ the angle with the axis. These invariants simplify the analysis of ray bundles across optical systems. Gauss also defined key elements like focal lengths in terms of ray intersections, establishing a systematic framework for lens and mirror design.[49]Cardinal points further characterize optical systems in Gaussian optics, consisting of two focal points (where parallel input rays converge or appear to diverge), two principal points (defining effective optical planes where ray heights are scaled by magnification), and two nodal points (where rays pass undeviated, with input and output directions parallel). These points are derived from ray transfer matrix analysis (ABCD matrices), where the front focal point is found by tracing parallel rays from infinity and locating their intersection with the axis after refraction; principal points emerge from extrapolating undeviated rays, and nodal points from rays entering and exiting parallel to the axis. For a thin lens, principal and nodal points coincide at the lens center, but in thick systems or compound lenses, they separate, aiding in the prediction of image shifts without tracing every ray.[50]Despite ideal assumptions, real systems exhibit aberrations—deviations of ray bundles from perfect focus—limiting image quality. Spherical aberration occurs when marginal rays focus closer to the lens than paraxial rays due to the spherical surface's varying curvature, causing a blurred disk instead of a point; ray diagrams show peripheral rays intersecting the axis ahead of central ones. Coma, an off-axis aberration, produces asymmetric teardrop-shaped images as rays from an off-axis point fan out with varying magnifications in tangential and sagittal planes, resembling a comet tail in ray fans. Astigmatism arises off-axis from elliptical apertures, focusing rays in meridional and sagittal planes at different distances, resulting in line-like images rather than points; ray bundles split into two foci, with astigmatic difference increasing with field angle. These monochromatic aberrations are analyzed via Seidel coefficients in third-order theory, guiding aspheric designs to minimize deviations.[51]
Ray Tracing
Ray tracing serves as a fundamental computational method in optics for simulating the propagation of light rays through complex systems, enabling the prediction of image formation, aberrations, and overall performance by iteratively applying geometric principles. Developed as an extension of manual tracing techniques, it has evolved with computing power to handle intricate designs in lenses, mirrors, and assemblies.[52]Sequential ray tracing follows a predefined sequence of optical surfaces, where each ray is transferred from one surface to the next using local coordinate systems defined relative to the previous surface. At each intersection—calculated via exact equations for conic or aspheric profiles—the ray direction is updated through refraction or reflection, with optical path lengths accumulated for wavefront analysis. This approach excels in ordered systems like telescopes or microscopes, producing outputs such as spot diagrams to assess focusing quality. Chief rays are often traced first to establish field positions, followed by marginal rays for aberration evaluation.[52]Non-sequential ray tracing accommodates disordered light paths, allowing rays to scatter, reflect multiply, or absorb without a fixed order, which is essential for modeling stray light, diffusers, or illumination optics. Monte Carlo techniques enhance efficiency by statistically sampling a large bundle of rays from sources, approximating irradiance distributions and tolerancing effects through repeated simulations of parameter variations. This method's computational intensity is mitigated by parallel processing on clusters, achieving near-interactive speeds for design iterations.[53]Commercial software implements these algorithms via real ray tracing, eschewing paraxial approximations to compute exact intersections and path deviations for high-fidelity simulations. Tools like Zemax and Code V employ direction cosines and iterative solvers to trace rays through user-defined geometries, supporting analyses from basic propagation to advanced tolerancing.[54]Aberration correction leverages traced ray data to minimize deviations from ideal performance, using merit functions that quantify errors such as spot size or wavefront error across field points. Optimization algorithms adjust surface parameters—curvatures, separations, or tilts—to reduce the merit function, often the least-squares sum of ray deviations, balancing aberrations like spherical and coma for improved image quality.[55]Polarization ray tracing extends geometric models by tracking the electric field state along each ray path, employing Jones vectors to represent linear, circular, or elliptical polarization in local coordinates. At interfaces, 3x3 transformation matrices account for diattenuation, retardance, and rotation due to reflection or refraction, enabling analysis of birefringent or coated systems. This calculus generalizes traditional Jones methods for curved ray bundles in three dimensions.[56]
Fiber Optics
In optical fibers, the ray model describes lightpropagation through a core of refractive indexn1 surrounded by a cladding of lower refractive indexn2<n1, where rays are confined by repeated total internal reflection at the core-cladding interface. Meridional rays lie in a plane containing the fiber axis and undergo planar zig-zag paths, crossing the axis with each reflection, while skew rays follow helical, non-planar trajectories that do not intersect the axis, resulting in more complex bouncing patterns.[57][58] This model applies to multimode fibers, where multiple ray paths coexist, enabling the guidance of light over long distances without significant divergence.[59]The acceptance angle θa defines the maximum angle at which an incident ray in the surrounding medium (refractive index n0) can enter the fiber core and still undergo total internal reflection, given by θa=sin−1n12−n22 for n0=1 (air).[60] This angle delineates the entry cone for bound rays, ensuring propagation without leakage. The numerical apertureNA=n0sinθa quantifies the fiber's light-gathering capacity and links directly to the cone's half-angle, with typical values around 0.2 for telecommunications fibers.[61][62]In ray optics terms, modes correspond to distinct ray paths classified as bound if their invariants—such as the skew invariant βˉ (axial component of the wave vector)—satisfy conditions for confinement within the core. The V-number, or normalized frequency, V=λ2πaNA (where a is the core radius and λ the wavelength), determines the number of supported modes: V<2.405 for single-mode operation (few or no bound rays) and higher V for multimode fibers supporting numerous meridional and skew rays.[63][64]Attenuation in fibers arises from material absorption and ray-dependent losses at reflections, while dispersion primarily manifests as modal dispersion in multimode fibers, where rays follow paths of varying lengths—axial rays travel straight, while bouncing rays cover longer distances—leading to pulse broadening over distance.[65][66] For example, in a step-index multimode fiber, the differential path length can limit bandwidth to hundreds of MHz·km.[65]Ray optics in fibers underpins applications in telecommunications, where multimode fibers transmit data over short links (up to ~500 m) via multiple ray paths, and single-mode fibers (approximated by paraxial rays) enable long-haul, high-bit-rate signals exceeding 100 Gbps.[67] In medicine, fiber bundles using meridional rays for illumination and imaging form endoscopes, allowing minimally invasive visualization of internal structures with diameters as small as 1 mm.[68] However, modal dispersion confines reliable ray-based guidance to multimode fibers over limited distances, beyond which wave optics is required for precision.[66]
