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Ray (optics)

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</style><div role="note" class="hatnote navigation-not-searchable">"Ray of light" redirects here. For other uses, see <a href="Ray_of_light_(disambiguation)" class="mw-disambig" title="Ray of light (disambiguation)">Ray of light (disambiguation)</a>.</div><div role="note" class="hatnote navigation-not-searchable">"Lightray" redirects here. For the superhero, see <a href="Lightray_(character)" title="Lightray (character)">Lightray (character)</a>.</div><div role="note" class="hatnote navigation-not-searchable">"Incident light" redirects here. For the film, see <a href="Incident_Light_(film)" class="mw-redirect" title="Incident Light (film)">Incident Light (film)</a>.</div>

In <a href="Optics" title="Optics">optics</a>, a ray is an idealized geometrical model of <a href="Light" title="Light">light</a> or other <a href="Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>, obtained by choosing a <a href="Curve" title="Curve">curve</a> that is perpendicular to the <a href="Wavefront" title="Wavefront">wavefronts</a> of the actual light, and that points in the direction of <a href="Energy_transfer" class="mw-redirect" title="Energy transfer">energy flow</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-Greivenkamp2-2">[2]</a> Rays are used to model the <a href="Propagation_of_light" class="mw-redirect" title="Propagation of light">propagation of light</a> through an optical system, by dividing the real <a href="Light_field" title="Light field">light field</a> up into discrete rays that can be computationally propagated through the system by the techniques of <a href="Ray_tracing_(physics)" title="Ray tracing (physics)">ray tracing</a>. This allows even very complex optical systems to be analyzed mathematically or simulated by computer. Ray tracing uses approximate solutions to <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a> that are valid as long as the <a href="Light_wave" class="mw-redirect" title="Light wave">light waves</a> propagate through and around objects whose dimensions are much greater than the light's <a href="Wavelength" title="Wavelength">wavelength</a>. <a href="Ray_optics" class="mw-redirect" title="Ray optics">Ray optics</a> or <a href="Geometrical_optics" title="Geometrical optics">geometrical optics</a> does not describe phenomena such as <a href="Diffraction" title="Diffraction">diffraction</a>, which require <a href="Wave_optics" class="mw-redirect" title="Wave optics">wave optics</a> theory. Some wave phenomena such as <a href="Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a> can be modeled in limited circumstances by adding <a href="Phase_(waves)" title="Phase (waves)">phase</a> to the ray model.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

A light ray is a line (<a href="Straight_line" class="mw-redirect" title="Straight line">straight</a> or <a href="Curved_line" class="mw-redirect" title="Curved line">curved</a>) that is <a href="Perpendicular" title="Perpendicular">perpendicular</a> to the light's <a href="Wavefront" title="Wavefront">wavefronts</a>; its <a href="Tangent_(geometry)" class="mw-redirect" title="Tangent (geometry)">tangent</a> is <a href="https://en.wiktionary.org/wiki/collinear" class="extiw external" title="wiktionary:collinear">collinear</a> with the <a href="Wave_vector" title="Wave vector">wave vector</a>. Light rays in <a href="Homogeneous_medium" class="mw-redirect" title="Homogeneous medium">homogeneous media</a> are straight. They bend at the <a href="https://en.wiktionary.org/wiki/interface" class="extiw external" title="wiktionary:interface">interface</a> between two dissimilar <a href="Optical_medium" title="Optical medium">media</a> and may be curved in a medium in which the <a href="Refractive_index" title="Refractive index">refractive index</a> changes. <a href="Geometric_optics" class="mw-redirect" title="Geometric optics">Geometric optics</a> 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 <a href="Fermat's_principle" title="Fermat's principle">Fermat's principle</a>, 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.<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Special_rays">Special rays</h2></div>

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.

<div class="mw-heading mw-heading3"><h3 id="Interaction_with_surfaces">Interaction with surfaces</h3></div>

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</style>incident ray is a ray of light that strikes a <a href="Surface" title="Surface">surface</a>. The angle between this ray and the perpendicular or <a href="Surface_normal" class="mw-redirect" title="Surface normal">normal</a> to the surface is the <a href="Angle_of_incidence_(optics)" title="Angle of incidence (optics)">angle of incidence</a>.</li>

<li>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 <a href="Angle_of_reflection" class="mw-redirect" title="Angle of reflection">angle of reflection</a>. The Law of Reflection says that for a <a href="Specular_reflection" title="Specular reflection">specular</a> (non-scattering) surface, the angle of reflection is always equal to the angle of incidence.</li>

<li>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 <a href="Angle_of_refraction" class="mw-redirect" title="Angle of refraction">angle of refraction</a>, and it is given by <a href="Snell's_law" title="Snell's law">Snell's law</a>. <a href="Conservation_of_energy" title="Conservation of energy">Conservation of energy</a> 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.</li>

<li>If the material is <a href="Birefringence" title="Birefringence">birefringent</a>, the refracted ray may split into ordinary and extraordinary rays, which experience different <a href="Index_of_refraction" class="mw-redirect" title="Index of refraction">indexes of refraction</a> when passing through the birefringent material.</li></ul>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Reflection_(physics)" title="Reflection (physics)">Reflection (physics)</a>, <a href="Refraction" title="Refraction">Refraction</a>, <a href="Absorption_(optics)" class="mw-redirect" title="Absorption (optics)">Absorption (optics)</a>, <a href="Birefringence" title="Birefringence">Birefringence</a>, <a href="Specular_reflection" title="Specular reflection">Specular reflection</a>, and <a href="Plane_of_incidence" title="Plane of incidence">Plane of incidence</a></div>

<div class="mw-heading mw-heading3"><h3 id="Optical_systems">Optical systems</h3></div>

<ul><li>A meridional ray or tangential ray is a ray that is confined to the plane containing the system's <a href="Optical_axis" title="Optical axis">optical axis</a> and the object point from which the ray originated.<a href="#cite_note-Stewart-4">[4]</a> This plane is called meridional plane or tangential plane.</li>

<li>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.<a href="#cite_note-Stewart-4">[4]</a></li>

<li>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 <a href="Aperture_stop" class="mw-redirect" title="Aperture stop">aperture stop</a> of the system.<a href="#cite_note-Greivenkamp25-5">[5]</a><a href="#cite_note-Riedl-6">[6]</a><a href="#cite_note-:0-7">[7]</a> This ray is useful, because it crosses the optical axis again at the location where a <a href="Real_image" title="Real image">real image</a> will be formed, or the backward extension of the ray path crosses the axis where a <a href="Virtual_image" title="Virtual image">virtual image</a> will be formed. Since the <a href="Entrance_pupil" title="Entrance pupil">entrance pupil</a> and <a href="Exit_pupil" title="Exit pupil">exit pupil</a> are <a href="Image" title="Image">images</a> 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.</li>

<li>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.<a href="#cite_note-Greivenkamp25-5">[5]</a><a href="#cite_note-8">[8]</a><a href="#cite_note-:0-7">[7]</a> 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 <a href="Lagrange_invariant" title="Lagrange invariant">Lagrange invariant</a>, which characterizes the throughput or <a href="Etendue" title="Etendue">etendue</a> of the optical system.<a href="#cite_note-Greivenkamp28-9">[9]</a> 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.<a href="#cite_note-Riedl-6">[6]</a></li>

<li>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).<a href="#cite_note-Stewart-4">[4]</a> 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.<a href="#cite_note-Stewart-4">[4]</a> All other sagittal rays are skew rays.</li>

<li>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.<a href="#cite_note-10">[10]</a> Such rays can be modeled reasonably well by using the <a href="Paraxial_approximation" title="Paraxial approximation">paraxial approximation</a>. 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.<a href="#cite_note-11">[11]</a><a href="#cite_note-Atchison-12">[12]</a></li>

<li>A finite ray or real ray is a ray that is traced without making the paraxial approximation.<a href="#cite_note-Atchison-12">[12]</a><a href="#cite_note-13">[13]</a></li>

<li>A parabasal ray is a ray that propagates close to some defined "base ray" rather than the optical axis.<a href="#cite_note-14">[14]</a> 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.<a href="#cite_note-15">[15]</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Fiber_optics">Fiber optics</h3></div>

<ul><li>A meridional ray is a ray that passes through the <a href="Optical_axis" title="Optical axis">axis</a> of an <a href="Optical_fiber" title="Optical fiber">optical fiber</a>.</li>

<li>A skew ray is a ray that travels in a non-planar zig-zag path and never crosses the <a href="Optical_axis" title="Optical axis">axis</a> of an <a href="Optical_fiber" title="Optical fiber">optical fiber</a>.</li>

<li>A <a href="Guided_ray" title="Guided ray">guided ray</a>, bound ray, or trapped ray is a ray in a <a href="Multi-mode_optical_fiber" title="Multi-mode optical fiber">multi-mode optical fiber</a>, which is confined by the <a href="Fiber_optics" class="mw-redirect" title="Fiber optics">core</a>. For <a href="Step-index_profile" title="Step-index profile">step index fiber</a>, light entering the fiber will be guided if it makes an angle with the fiber axis that is less than the fiber's <a href="Guided_ray" title="Guided ray">acceptance angle</a>.</li>

<li>A leaky ray or tunneling ray is a ray in an optical fiber that geometric optics predicts would <a href="Total_internal_reflection" title="Total internal reflection">totally reflect</a> at the boundary between the <a href="Fiber_optics" class="mw-redirect" title="Fiber optics">core</a> and the <a href="Cladding_(fiber_optics)" title="Cladding (fiber optics)">cladding</a>, but which suffers loss due to the curved core boundary.</li></ul>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Numerical_aperture" title="Numerical aperture">Numerical aperture</a></div>

<div class="mw-heading mw-heading2"><h2 id="Geometrical_optics">Geometrical optics</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="Geometrical_optics" title="Geometrical optics">Geometrical optics</a>.[<a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Geometrical_optics&action=edit">edit</a>]</div><div class="excerpt">

<a href="Geometrical_optics" title="Geometrical optics">Geometrical optics</a>, or ray optics, is a model of <a href="Optics" title="Optics">optics</a> that describes <a href="Light" title="Light">light</a> <a href="Wave_propagation" class="mw-redirect" title="Wave propagation">propagation</a> in terms of rays. The ray in geometrical optics is an <a href="Abstract_object" class="mw-redirect" title="Abstract object">abstraction</a> useful for approximating the paths along which light propagates under certain circumstances.

The simplifying assumptions of geometrical optics include that light rays:

<ul><li>propagate in straight-line paths as they travel in a <a href="Homogeneity_(physics)" title="Homogeneity (physics)">homogeneous</a> medium</li>

<li>bend, and in particular circumstances may split in two, at the <a href="Interface_(matter)" title="Interface (matter)">interface</a> between two dissimilar <a href="Optical_medium" title="Optical medium">media</a></li>

<li>follow curved paths in a medium in which the <a href="Refractive_index" title="Refractive index">refractive index</a> changes</li>

<li>may be absorbed or reflected.</li></ul>

Geometrical optics does not account for certain optical effects such as <a href="Diffraction" title="Diffraction">diffraction</a> and <a href="Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a>, which are considered in <a href="Physical_optics" title="Physical optics">physical optics</a>. 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 <a href="Image" title="Image">imaging</a>, including <a href="Optical_aberration" title="Optical aberration">optical aberrations</a>.</div></div>

<div class="mw-heading mw-heading2"><h2 id="Ray_tracing">Ray tracing</h2></div>

<div class="excerpt-block"><div role="note" class="hatnote navigation-not-searchable dablink excerpt-hat selfref">This section is an excerpt from <a href="Ray_tracing_(physics)" title="Ray tracing (physics)">Ray tracing (physics)</a>.[<a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Ray_tracing_(physics)&action=edit">edit</a>]</div><div class="excerpt">

In physics, <a href="Ray_tracing_(physics)" title="Ray tracing (physics)">ray tracing</a> is a method for calculating the path of <a href="Wave" title="Wave">waves</a> or <a href="Subatomic_particle" title="Subatomic particle">particles</a> through a system with regions of varying propagation <a href="Velocity" title="Velocity">velocity</a>, absorption characteristics, and reflecting surfaces. Under these circumstances, <a href="Wavefront" title="Wavefront">wavefronts</a> may bend, change direction, or <a href="Reflection_(physics)" title="Reflection (physics)">reflect</a> off surfaces, complicating analysis.

Historically, ray tracing involved analytic solutions to the ray's trajectories. In modern <a href="Applied_physics" title="Applied physics">applied physics</a> and <a href="Engineering_physics" title="Engineering physics">engineering physics</a>, the term also encompasses numerical solutions to the <a href="Eikonal_equation" title="Eikonal equation">Eikonal equation</a>. For example, <a href="Ray_marching" title="Ray marching">ray-marching</a> involves repeatedly advancing idealized narrow beams called rays through the <a href="Medium_(optics)" class="mw-redirect" title="Medium (optics)">medium</a> 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.

When applied to problems of <a href="Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>, ray tracing often relies on approximate solutions to <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a> such as <a href="Geometric_optics" class="mw-redirect" title="Geometric optics">geometric optics</a>, that are valid as long as the light waves propagate through and around objects whose dimensions are much greater than the light's <a href="Wavelength" title="Wavelength">wavelength</a>. Ray theory can describe <a href="Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a> by accumulating the <a href="Phase_(waves)" title="Phase (waves)">phase</a> during ray tracing (e.g., complex-valued <a href="Fresnel_coefficients" class="mw-redirect" title="Fresnel coefficients">Fresnel coefficients</a> and <a href="Jones_calculus" title="Jones calculus">Jones calculus</a>). It can also be extended to describe edge <a href="Diffraction" title="Diffraction">diffraction</a>, with modifications such as the <a href="Geometric_theory_of_diffraction" class="mw-redirect" title="Geometric theory of diffraction">geometric theory of diffraction</a>, which enables tracing diffracted rays.

More complicated phenomena require methods such as <a href="Physical_optics" title="Physical optics">physical optics</a> or <a href="Wave_optics" class="mw-redirect" title="Wave optics">wave theory</a>.</div></div>

<ul><li><a href="Collimated_beam" title="Collimated beam">Collimated beam</a></li>

<li><a href="Optical_path" title="Optical path">Optical path</a></li>

<li><a href="Optical_path_length" title="Optical path length">Optical path length</a></li>

<li><a href="Paraxial_approximation" title="Paraxial approximation">Paraxial approximation</a></li>

<li><a href="Pencil_beam" class="mw-redirect" title="Pencil beam">Pencil beam</a></li>

<li><a href="Ray_transfer_matrix_analysis" title="Ray transfer matrix analysis">Ray transfer matrix analysis</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFMoore2005" class="citation web cs1">Moore, Ken (25 July 2005). <a rel="nofollow" class="external text" href="http://www.zemax.com/os/resources/learn/knowledgebase/what-is-a-ray">"What is a ray?"</a>. ZEMAX Users' Knowledge Base. Retrieved 30 May 2008.</cite>

</li>

<li id="cite_note-Greivenkamp2-2"><a href="#cite_ref-Greivenkamp2_2-0">^</a> <cite id="CITEREFGreivenkamp2004" class="citation book cs1">Greivenkamp, John E. (2004). Field Guide to Geometric Optics. SPIE Field Guides. p. 2. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0819452947</bdi>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="Arthur_Schuster" title="Arthur Schuster">Arthur Schuster</a>, An Introduction to the Theory of Optics, London: Edward Arnold, 1904 <a rel="nofollow" class="external text" href="https://archive.org/details/anintroductiont02schugoog/page/n62">online</a>.

</li>

<li id="cite_note-Stewart-4">^ <a href="#cite_ref-Stewart_4-0">a</a> <a href="#cite_ref-Stewart_4-1">b</a> <a href="#cite_ref-Stewart_4-2">c</a> <a href="#cite_ref-Stewart_4-3">d</a> <cite id="CITEREFStewart1996" class="citation book cs1">Stewart, James E. (1996). Optical Principles and Technology for Engineers. CRC. p. 57. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-9705-8</bdi>.</cite>

</li>

<li id="cite_note-Greivenkamp25-5">^ <a href="#cite_ref-Greivenkamp25_5-0">a</a> <a href="#cite_ref-Greivenkamp25_5-1">b</a> <cite id="CITEREFGreivenkamp2004" class="citation book cs1">Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8194-5294-7</bdi>.</cite>, p. 25 <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=1YfZNWZAwCAC&dq=Greivenkamp%20optics&pg=PA25">[1]</a>.

</li>

<li id="cite_note-Riedl-6">^ <a href="#cite_ref-Riedl_6-0">a</a> <a href="#cite_ref-Riedl_6-1">b</a> <cite id="CITEREFRiedl2001" class="citation book cs1">Riedl, Max J. (2001). Optical Design Fundamentals for Infrared Systems. Tutorial texts in optical engineering. Vol. 48. SPIE. p. 1. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8194-4051-8</bdi>.</cite>

</li>

<li id="cite_note-:0-7">^ <a href="#cite_ref-:0_7-0">a</a> <a href="#cite_ref-:0_7-1">b</a> <cite id="CITEREFHecht2017" class="citation book cs1">Hecht, Eugene (2017). "5.3.2 Entrance and Exit Pupils". Optics (5th ed.). Pearson. p. 184. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-292-09693-3</bdi>.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFMalacara2003" class="citation book cs1">Malacara, Daniel and Zacarias (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7aa2nDZoAHEC&q=%22Handbook%20of%20Optical%20Design%22&pg=PA25">Handbook of Optical Design</a> (2nd ed.). CRC. p. 25. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-4613-1</bdi>.</cite>

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<li id="cite_note-Greivenkamp28-9"><a href="#cite_ref-Greivenkamp28_9-0">^</a> Greivenkamp (2004), p. 28 <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=1YfZNWZAwCAC&dq=Greivenkamp%20optics&pg=PA28">[2]</a>.

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> Greivenkamp (2004), pp. 19–20 <a rel="nofollow" class="external autonumber" href="https://books.google.com/books?id=1YfZNWZAwCAC&dq=Greivenkamp%20optics&pg=PA19">[3]</a>.

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<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFNicholson2005" class="citation web cs1">Nicholson, Mark (21 July 2005). <a rel="nofollow" class="external text" href="http://www.zemax.com/kb/articles/18/1/Understanding-Paraxial-Ray-Tracing/Page1.html">"Understanding Paraxial Ray-Tracing"</a>. ZEMAX Users' Knowledge Base. Retrieved 17 August 2009.</cite>

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<li id="cite_note-Atchison-12">^ <a href="#cite_ref-Atchison_12-0">a</a> <a href="#cite_ref-Atchison_12-1">b</a> <cite id="CITEREFAtchisonSmith2000" class="citation book cs1">Atchison, David A.; Smith, George (2000). "A1: Paraxial optics". Optics of the Human Eye. Elsevier Health Sciences. p. 237. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-3775-6</bdi>.</cite>

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<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFWelford1986" class="citation book cs1">Welford, W. T. (1986). "4: Finite Raytracing". Aberrations of Optical Systems. Adam Hilger series on optics and optoelectronics. CRC Press. p. 50. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-85274-564-9</bdi>.</cite>

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<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFBuchdahl1993" class="citation book cs1">Buchdahl, H. A. (1993). An Introduction to Hamiltonian Optics. Dover. p. 26. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-67597-8</bdi>.</cite>

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<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFNicholson2005" class="citation web cs1">Nicholson, Mark (21 July 2005). <a rel="nofollow" class="external text" href="http://www.zemax.com/kb/articles/18/2/Understanding-Paraxial-Ray-Tracing/Page2.html">"Understanding Paraxial Ray-Tracing"</a>. ZEMAX Users' Knowledge Base. p. 2. Retrieved 17 August 2009.</cite>

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