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Ray (optics)
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<h2 id="definition">Definition</h2> <p>A light ray is a line (<a href="/facts/Straight_line/gJqsr4Gj">straight</a> or <a href="/facts/Curved_line/xBQ0pTBW">curved</a>) that is <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a> to the light's <a href="/facts/Wavefront/5XVoVtDQ">wavefronts</a>; its <a href="/facts/Tangent_(geometry)/sF0jH3EI">tangent</a> is collinear with the <a href="/facts/Wave_vector/Jk8wUEft">wave vector</a>. Light rays in <a href="/facts/Homogeneous_medium/J4v4DwfF">homogeneous media</a> are straight. They bend at the interface between two dissimilar <a href="/facts/Optical_medium/gm6LqshC">media</a> and may be curved in a medium in which the <a href="/facts/Refractive_index/yjkzvgeE">refractive index</a> changes. <a href="/facts/Geometric_optics/gIFu84Tm">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. </p><p>A slightly more rigorous definition of a light ray follows from <a href="/facts/Fermat%2527s_principle/UL9XumdW">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 class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="special-rays">Special rays</h2> <p>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. </p> <h3>Interaction with surfaces</h3> <ul><li>An incident ray is a ray of light that strikes a <a href="/facts/Surface/0XA8ce89">surface</a>. The angle between this ray and the perpendicular or <a href="/facts/Surface_normal/jhtLIhcu">normal</a> to the surface is the <a href="/facts/Angle_of_incidence_(optics)/DTaCjhTU">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="/facts/Angle_of_reflection/BhcjDs79">angle of reflection</a>. The Law of Reflection says that for a <a href="/facts/Specular_reflection/dwVIqLVb">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="/facts/Angle_of_refraction/cE0rhDXd">angle of refraction</a>, and it is given by <a href="/facts/Snell%2527s_law/cE0rhDXd">Snell's law</a>. <a href="/facts/Conservation_of_energy/12CdN4Ik">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="/facts/Birefringence/J4lLuR3a">birefringent</a>, the refracted ray may split into ordinary and extraordinary rays, which experience different <a href="/facts/Index_of_refraction/yjkzvgeE">indexes of refraction</a> when passing through the birefringent material.</li></ul> <p class="note">See also: <a href="/facts/Reflection_(physics)/BhcjDs79">Reflection (physics)</a>, <a href="/facts/Refraction/S8kvSKzM">Refraction</a>, <a href="/facts/Absorption_(optics)/xBYKt7yS">Absorption (optics)</a>, <a href="/facts/Birefringence/J4lLuR3a">Birefringence</a>, <a href="/facts/Specular_reflection/dwVIqLVb">Specular reflection</a>, and <a href="/facts/Plane_of_incidence/d2tJAscI">Plane of incidence</a></p> <h3>Optical systems</h3> <ul><li>A meridional ray or tangential ray is a ray that is confined to the plane containing the system's <a href="/facts/Optical_axis/fKFqnagT">optical axis</a> and the object point from which the ray originated.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></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 class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li>The marginal ray (sometimes known as an <i>a ray</i> or a <i>marginal axial ray</i>) 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="/facts/Aperture_stop/8PV11GQm">aperture stop</a> of the system.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> This ray is useful, because it crosses the optical axis again at the location where a <a href="/facts/Real_image/G8KApAFz">real image</a> will be formed, or the backward extension of the ray path crosses the axis where a <a href="/facts/Virtual_image/hldBr4zj">virtual image</a> will be formed. Since the <a href="/facts/Entrance_pupil/8KSLrjSJ">entrance pupil</a> and <a href="/facts/Exit_pupil/gSQbtt60">exit pupil</a> are <a href="/facts/Image/hbjEdTXK">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 <i>b ray</i>) 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 class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></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="/facts/Lagrange_invariant/11ny9bP2">Lagrange invariant</a>, which characterizes the throughput or <a href="/facts/Etendue/nApbAC9m">etendue</a> of the optical system.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> Some authors define a "principal ray" for <i>each</i> object point, and in this case, the principal ray starting at an edge point of the object may then be called the <i>marginal principal ray</i>.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></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 class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></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 <i>z</i> axis, and the meridional plane is the <i>y</i>-<i>z</i> plane, sagittal rays intersect the pupil at <i>yp</i>= 0. The principal ray is both sagittal and meridional.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></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 class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> Such rays can be modeled reasonably well by using the <a href="/facts/Paraxial_approximation/Mj2XrtWm">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 class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a></li> <li>A finite ray or real ray is a ray that is traced without making the paraxial approximation.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a></li> <li>A parabasal ray is a ray that propagates close to some defined "base ray" rather than the optical axis.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></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 class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a></li></ul> <h3>Fiber optics</h3> <ul><li>A meridional ray is a ray that passes through the <a href="/facts/Optical_axis/fKFqnagT">axis</a> of an <a href="/facts/Optical_fiber/5ruuIFbj">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="/facts/Optical_axis/fKFqnagT">axis</a> of an <a href="/facts/Optical_fiber/5ruuIFbj">optical fiber</a>.</li> <li>A <a href="/facts/Guided_ray/4NlctfrB">guided ray</a>, bound ray, or trapped ray is a ray in a <a href="/facts/Multi-mode_optical_fiber/tgSR57m6">multi-mode optical fiber</a>, which is confined by the <a href="/facts/Fiber_optics/5ruuIFbj">core</a>. For <a href="/facts/Step-index_profile/tJ0f5JnD">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="/facts/Guided_ray/4NlctfrB">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="/facts/Total_internal_reflection/Tm0M62NQ">totally reflect</a> at the boundary between the <a href="/facts/Fiber_optics/5ruuIFbj">core</a> and the <a href="/facts/Cladding_(fiber_optics)/URSEucdL">cladding</a>, but which suffers loss due to the curved core boundary.</li></ul> <p class="note">See also: <a href="/facts/Numerical_aperture/EvW1oVBC">Numerical aperture</a></p> <h2 id="geometrical-optics">Geometrical optics</h2> <p class="note">This section is an excerpt from <a href="/facts/Geometrical_optics/gIFu84Tm">Geometrical optics</a>.[<a href="https://en.wikipedia.org/w/index.php?title=Geometrical_optics&action=edit">edit</a>]</p> <p><a href="/facts/Geometrical_optics/gIFu84Tm">Geometrical optics</a>, or ray optics, is a model of <a href="/facts/Optics/p0vq4S1T">optics</a> that describes <a href="/facts/Light/jeofpMn4">light</a> <a href="/facts/Wave_propagation/PXhWIt7k">propagation</a> in terms of <i>rays</i>. The ray in geometrical optics is an <a href="/facts/Abstract_object/uNshak6S">abstraction</a> useful for approximating the paths along which light propagates under certain circumstances. </p><p>The simplifying assumptions of geometrical optics include that light rays: </p> <ul><li>propagate in straight-line paths as they travel in a <a href="/facts/Homogeneity_(physics)/J4v4DwfF">homogeneous</a> medium</li> <li>bend, and in particular circumstances may split in two, at the <a href="/facts/Interface_(matter)/HnIAndVA">interface</a> between two dissimilar <a href="/facts/Optical_medium/gm6LqshC">media</a></li> <li>follow curved paths in a medium in which the <a href="/facts/Refractive_index/yjkzvgeE">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="/facts/Diffraction/Mk80xXHT">diffraction</a> and <a href="/facts/Interference_(wave_propagation)/lI5f78I4">interference</a>, which are considered in <a href="/facts/Physical_optics/Al5EFMyy">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="/facts/Image/hbjEdTXK">imaging</a>, including <a href="/facts/Optical_aberration/YIqd9cA8">optical aberrations</a>. <h2 id="ray-tracing">Ray tracing</h2> <p class="note">This section is an excerpt from <a href="/facts/Ray_tracing_(physics)/VKdjk6Sc">Ray tracing (physics)</a>.[<a href="https://en.wikipedia.org/w/index.php?title=Ray_tracing_(physics)&action=edit">edit</a>]</p> <p>In physics, <a href="/facts/Ray_tracing_(physics)/VKdjk6Sc">ray tracing</a> is a method for calculating the path of <a href="/facts/Wave/fU0HH6St">waves</a> or <a href="/facts/Subatomic_particle/tMEzcTWQ">particles</a> through a system with regions of varying propagation <a href="/facts/Velocity/Q3o1LVDe">velocity</a>, absorption characteristics, and reflecting surfaces. Under these circumstances, <a href="/facts/Wavefront/5XVoVtDQ">wavefronts</a> may bend, change direction, or <a href="/facts/Reflection_(physics)/BhcjDs79">reflect</a> off surfaces, complicating analysis. </p><p>Historically, ray tracing involved analytic solutions to the ray's trajectories. In modern <a href="/facts/Applied_physics/47NRAMzF">applied physics</a> and <a href="/facts/Engineering_physics/eoQEWCwq">engineering physics</a>, the term also encompasses numerical solutions to the <a href="/facts/Eikonal_equation/ITjReqSB">Eikonal equation</a>. For example, <a href="/facts/Ray_marching/bnCjkUUR">ray-marching</a> involves repeatedly advancing idealized narrow beams called <i>rays</i> through the <a href="/facts/Medium_(optics)/gm6LqshC">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. </p><p>When applied to problems of <a href="/facts/Electromagnetic_radiation/Ynx7ujCW">electromagnetic radiation</a>, ray tracing often relies on approximate solutions to <a href="/facts/Maxwell%2527s_equations/LfJ1drHS">Maxwell's equations</a> such as <a href="/facts/Geometric_optics/gIFu84Tm">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="/facts/Wavelength/kcJU0ymz">wavelength</a>. Ray theory can describe <a href="/facts/Interference_(wave_propagation)/lI5f78I4">interference</a> by accumulating the <a href="/facts/Phase_(waves)/cpGtuVll">phase</a> during ray tracing (e.g., complex-valued <a href="/facts/Fresnel_coefficients/6SEao7ua">Fresnel coefficients</a> and <a href="/facts/Jones_calculus/zUBSR95j">Jones calculus</a>). It can also be extended to describe edge <a href="/facts/Diffraction/Mk80xXHT">diffraction</a>, with modifications such as the <a href="/facts/Geometric_theory_of_diffraction/HGogdyX6">geometric theory of diffraction</a>, which enables tracing <i>diffracted rays</i>. </p> More complicated phenomena require methods such as <a href="/facts/Physical_optics/Al5EFMyy">physical optics</a> or <a href="/facts/Wave_optics/Al5EFMyy">wave theory</a>. <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Collimated_beam/lhd2hhPt">Collimated beam</a></li> <li><a href="/facts/Optical_path/tXu2qmnN">Optical path</a></li> <li><a href="/facts/Optical_path_length/qpXIMzqz">Optical path length</a></li> <li><a href="/facts/Paraxial_approximation/Mj2XrtWm">Paraxial approximation</a></li> <li><a href="/facts/Pencil_beam/wQtPHxsa">Pencil beam</a></li> <li><a href="/facts/Ray_transfer_matrix_analysis/k8MtAnYt">Ray transfer matrix analysis</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Moore, Ken (25 July 2005). "What is a ray?". ZEMAX Users' Knowledge Base. Retrieved 30 May 2008. <a href="http://www.zemax.com/os/resources/learn/knowledgebase/what-is-a-ray" target="_blank">http://www.zemax.com/os/resources/learn/knowledgebase/what-is-a-ray</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Greivenkamp, John E. (2004). Field Guide to Geometric Optics. SPIE Field Guides. p. 2. ISBN 0819452947. <a href="0819452947" target="_blank">0819452947</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Arthur Schuster, An Introduction to the Theory of Optics, London: Edward Arnold, 1904 online. <a href="/wiki/Arthur_Schuster" target="_blank">/wiki/Arthur_Schuster</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Stewart, James E. (1996). Optical Principles and Technology for Engineers. CRC. p. 57. ISBN 978-0-8247-9705-8. <a href="978-0-8247-9705-8" target="_blank">978-0-8247-9705-8</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Stewart, James E. (1996). Optical Principles and Technology for Engineers. CRC. p. 57. ISBN 978-0-8247-9705-8. <a href="978-0-8247-9705-8" target="_blank">978-0-8247-9705-8</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN 0-8194-5294-7., p. 25 [1]. <a href="0-8194-5294-7" target="_blank">0-8194-5294-7</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Riedl, Max J. (2001). Optical Design Fundamentals for Infrared Systems. Tutorial texts in optical engineering. Vol. 48. SPIE. p. 1. ISBN 978-0-8194-4051-8. <a href="978-0-8194-4051-8" target="_blank">978-0-8194-4051-8</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hecht, Eugene (2017). "5.3.2 Entrance and Exit Pupils". Optics (5th ed.). Pearson. p. 184. ISBN 978-1-292-09693-3. <a href="978-1-292-09693-3" target="_blank">978-1-292-09693-3</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. ISBN 0-8194-5294-7., p. 25 [1]. <a href="0-8194-5294-7" target="_blank">0-8194-5294-7</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Malacara, Daniel and Zacarias (2003). Handbook of Optical Design (2nd ed.). CRC. p. 25. ISBN 978-0-8247-4613-1. <a href="978-0-8247-4613-1" target="_blank">978-0-8247-4613-1</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Hecht, Eugene (2017). "5.3.2 Entrance and Exit Pupils". Optics (5th ed.). Pearson. p. 184. ISBN 978-1-292-09693-3. <a href="978-1-292-09693-3" target="_blank">978-1-292-09693-3</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Greivenkamp (2004), p. 28 [2]. <a href="https://books.google.com/books?id=1YfZNWZAwCAC&dq=Greivenkamp%20optics&pg=PA28" target="_blank">https://books.google.com/books?id=1YfZNWZAwCAC&dq=Greivenkamp%20optics&pg=PA28</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Riedl, Max J. (2001). Optical Design Fundamentals for Infrared Systems. Tutorial texts in optical engineering. Vol. 48. SPIE. p. 1. ISBN 978-0-8194-4051-8. <a href="978-0-8194-4051-8" target="_blank">978-0-8194-4051-8</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Stewart, James E. (1996). Optical Principles and Technology for Engineers. CRC. p. 57. ISBN 978-0-8247-9705-8. <a href="978-0-8247-9705-8" target="_blank">978-0-8247-9705-8</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Stewart, James E. (1996). Optical Principles and Technology for Engineers. CRC. p. 57. ISBN 978-0-8247-9705-8. <a href="978-0-8247-9705-8" target="_blank">978-0-8247-9705-8</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Greivenkamp (2004), pp. 19–20 [3]. <a href="https://books.google.com/books?id=1YfZNWZAwCAC&dq=Greivenkamp%20optics&pg=PA19" target="_blank">https://books.google.com/books?id=1YfZNWZAwCAC&dq=Greivenkamp%20optics&pg=PA19</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Nicholson, Mark (21 July 2005). "Understanding Paraxial Ray-Tracing". ZEMAX Users' Knowledge Base. Retrieved 17 August 2009. <a href="http://www.zemax.com/kb/articles/18/1/Understanding-Paraxial-Ray-Tracing/Page1.html" target="_blank">http://www.zemax.com/kb/articles/18/1/Understanding-Paraxial-Ray-Tracing/Page1.html</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Atchison, David A.; Smith, George (2000). "A1: Paraxial optics". Optics of the Human Eye. Elsevier Health Sciences. p. 237. ISBN 978-0-7506-3775-6. <a href="978-0-7506-3775-6" target="_blank">978-0-7506-3775-6</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Atchison, David A.; Smith, George (2000). "A1: Paraxial optics". Optics of the Human Eye. Elsevier Health Sciences. p. 237. ISBN 978-0-7506-3775-6. <a href="978-0-7506-3775-6" target="_blank">978-0-7506-3775-6</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Welford, W. T. (1986). "4: Finite Raytracing". Aberrations of Optical Systems. Adam Hilger series on optics and optoelectronics. CRC Press. p. 50. ISBN 978-0-85274-564-9. <a href="978-0-85274-564-9" target="_blank">978-0-85274-564-9</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Buchdahl, H. A. (1993). An Introduction to Hamiltonian Optics. Dover. p. 26. ISBN 978-0-486-67597-8. <a href="978-0-486-67597-8" target="_blank">978-0-486-67597-8</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Nicholson, Mark (21 July 2005). "Understanding Paraxial Ray-Tracing". ZEMAX Users' Knowledge Base. p. 2. Retrieved 17 August 2009. <a href="http://www.zemax.com/kb/articles/18/2/Understanding-Paraxial-Ray-Tracing/Page2.html" target="_blank">http://www.zemax.com/kb/articles/18/2/Understanding-Paraxial-Ray-Tracing/Page2.html</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> </ol>