For shorter-wavelength light, the refractive indices of core and cladding will generally increase, and the NA will presumably also increase somewhat. They commence in the third or early fourth week of development (approximately the 20th day), appearing first in the cervical region of the embryo. Such an image is formed in a microscope or telescope and can be seen by looking into the eyepiece. , sintancos10, roLL'uu', 6 d, 5llu, , ,.5,, hr, ()hnnna, 1nnr. Hundreds of different types of optical glass are currently available from manufacturers. The interplay between methods developed for analysis of classical mechanics problems and quantum systems such as photon beams is bi-directional, as stated above, and we will see that our classical analyses can be used to describe photon beam (or paraxial wave) propagation. {\displaystyle x} The method of obtaining equations of geometrical optics by analysing surfaces of discontinuities of solutions to Maxwell's equations was first described by Rudolf Karl Luneburg in 1944. from mjhoptics/dependabot/pip/docs/source/jup, Tools for image forming optical design and analysis. The front focal point of an optical system, by definition, has the property that any ray that passes through it will emerge from the system parallel to the optical axis. r At first lenses were made from selected pieces of window glass or the glass used to make blown tableware. Each groove diffracts, or scatters, light in all directions, and in the case of light of one particular wavelength, there will be one direction in which the light wave from one groove lags behind the light wave from the next groove by precisely one or more whole wavelengths. The same law can be applied to determine the oblique illumination due to a lens, assuming a uniform extended diffusing source of light on the other side of the lens. {\displaystyle xyz} The principal planes are crucial in defining the optical properties of the system, since it is the distance of the object and image from the front and rear principal planes that determines the magnification of the system. Mirrors are frequently used in optical systems. Focal length. This effect is what is responsible for mirages seen on hot days where the changing index of refraction of the air causes the light rays to bend creating the appearance of specular reflections in the distance (as if on the surface of a pool of water). {\displaystyle A(k_{o},\mathbf {r} )} = {\displaystyle \lim _{k_{0}\to \infty }{1 \over k_{0}}\left({1 \over A}\,\nabla S\cdot \nabla A+{1 \over 2}\nabla ^{2}S\right)=0} The rays nearer to the centre are called paraxial rays. F is the object space focal point of the system. . The introduction of high-index barium crown glass with a low dispersive power in the 1880s initiated the development of anastigmat lenses. 1 k This ratio is related to the image-space numerical aperture when the lens is focused at infinity. e NA = 0.7, what is the resulting NA after the lens to describe the waist radius? It is generally more significant, however, to compare the dispersion with the mean refractive index of the material for some intermediate colour such as the sodium D Fraunhofer line of wavelength 5893 angstroms. If the object luminance is expressed as BL lamberts, the lambert being an alternative luminance unit equal to 1/ (i.e., 0.32) candle per unit area, the flux (F) isbecause there are times as many lamberts in a given luminance as there are candles per unit area. Chromatic variation of magnification can be eliminated by achromatizing all the components of a system or by making the system symmetrical about a central diaphragm. {\displaystyle \psi } The method uses information from the standard dynamic ray-tracing method to extrapolate the seismic wave field at receivers in the neighborhood of a ray so that two-point ray tracing is not required. A metal mirror, if well ribbed on the back, may be lighter than a glass mirror and therefore easier to handle, but most metals are slightly flexible and require just as careful support as glass mirrors. A plane mirror forms a virtual image of near objects, as in the familiar looking glass. The nodal points were first described by Johann Listing in 1845 to evaluate the eye, where the image is formed in fluid. In order to construct the rays through the lens, we have to use the two functions described above, in the correct order. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction A leads to[12], Therefore the eikonal equation is given by, Once eikonal is found by solving the above equation, the wave four-vector can be found from, Model of optics describing light as geometric rays, General equation using four-vector notation, English translations of some early books and papers. This theorem has been named after the French scientist Joseph-Louis Lagrange, although it is sometimes called the Smith-Helmholtz theorem, after Robert Smith, an English scientist, and Hermann Helmholtz, a German scientist; the product (hnu) is often known as the optical invariant. z Let us know if you have suggestions to improve this article (requires login). The images formed by all the smaller zones of the lens fit into this ellipse and fill it out with a uniform intensity of light. The front and rear nodal points have the property that a ray aimed at one of them will be refracted by the lens such that it appears to have come from the other, and with the same angle with respect to the optical axis. Rays emerging from a point actually meet at another point after reflection, and that point is called the image of the first point. It is often not recommended to operate a lens or its full area, since there could be substantial spherical aberrations. Optical systems can be folded using plane mirrors; the system is still considered to be rotationally symmetric if it possesses rotational symmetry when unfolded. (Angular magnification between nodal points is +1.) After refraction by the lens each ray will cross the axis at some point, and the entering and emerging portions of each ray are then extended until they intersect at a point such as Q. The image of a point is now a small circle that contracts to a point at a new focus situated at a longitudinal distance L = 2f 2h02S4 from the paraxial image. endobj ) Hence, star images appear brighter in proportion to the area of the telescope objective (assuming that the exit pupil is larger than the eye pupil), and the visibility of a star against the sky background is thus improved in proportion to the square of the diameter of the telescope objective lens. Particularly, if it is a single-mode waveguide, the optimum coupling tells us that there you have best matching to the guided mode of the waveguide. Physical optics deals primarily with the nature and properties of light itself. , [6][7][8] On the other hand, swing-lens cameras with fixed film position rotate the lens about the rear nodal point to stabilize the image on the film.[8][9]). {\displaystyle \varepsilon (x,y,z)} For this to be effective the image must be repeated (as in motion pictures) or scanned (as in television) at least 40 times a second to eliminate flicker or any appearance of intermittency. A pinhole, like a camera lens, can be regarded as having an f-number, which is the ratio of focal length to aperture. Paraxial Geometrical Optics and the System Matrix Refraction and Reflection The law of refraction (Snell's Law): When a ray of light is refracted at an interface between two uniform media, the transmitted ray remains in the plane of incidence and the sine of the angle of refraction is directly proportional to the sine of the angle of incidence. When this aberration is present, the entire image point is displaced toward or away from the axis by an amount proportional to the third power of the transverse distance h0 of the image from the axis. In the early 1800s, the manufacture of clear glass that was intended specifically for lenses began in Europe. By the end of the 19th century numerous other workers had entered the field of geometrical optics, notably an English physicist, Lord Rayleigh (John William Strutt), and a German physicist, Ernst Karl Abbe. This is known as spherical aberration (see Figure 8). [ How to model laser beam propagation in OpticStudio: Part 2 - Using Paraxial Gaussian Beam analysis to model Gaussian beam; How to model laser beam propagation in OpticStudio: Part 1 - Gaussian beam theory and ray-based approach; Using skew rays to model Gaussian beams - webinar; How to design a Gaussian to Top Hat beam shaper , ( ], [0. An extreme example of electro-optics appears in some space cameras, in which the film is exposed, processed, and then scanned by a tiny point of light; the light passing through the film is picked up by a photocell and transmitted to Earth by radio, where it is made to control the brightness of another point of light scanning a second piece of film in exact synchronism with the scanning spot in the camera. You cannot calculate the NA for that case. This added astigmatism is, of course, undesirable, and in order to design an anastigmat lens having a flat field free from astigmatism, it is necessary to reduce the Petzval sum S4 drastically. = y Obviously, in a perfect lens x and y are zero because OPD is zero. {\displaystyle \mathbf {\vec {H}} (x,y,z,t)} y Thin Lens Equation. What does it mean? It separates the colours of the spectrum by a process of diffraction. A A tag already exists with the provided branch name. The change of image distance with wavelength is known as chromatic aberration, and the variation of magnification with wavelength is known as chromatic difference of magnification, or lateral colour. Assuming coefficients in the differential equation are smooth, the rays will be too. These may be represented graphically on a plot of mean refractive index against dispersive power (Figure 2). Then the image illuminance will be. If nothing happens, download GitHub Desktop and try again. {\displaystyle \psi } ( xc```b``a`e``Vab0$ d)1N_o. and another medium with index of refraction When the light beam is refracted by a lens or reflected by a mirror, the curvature of the waves is changed, and the angular divergence of the ray bundle is similarly changed in such a way that the rays remain everywhere perpendicular to the waves. This is not exact, however, since the shape of the waveguide mode may somewhat differ from that of the laser beam. Are you sure you want to create this branch? stream It is also possible to produce polarized light rays using a combination of reflection and refraction: When a refracted ray and the reflected ray form a right angle, the reflected ray has the property of "plane polarization". {\displaystyle \theta _{2}} Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. ) The reflected rays converge at a point F on the principal axis of a concave mirror [Fig. A device which produces converging or diverging light rays due to refraction is known as a lens. x {\displaystyle S_{2}} The typical aberration figures shown in Figure 8 are, of course, grossly exaggerated, and actually it requires some magnification of a star image to render these appearances clearly visible. The simplest screen for the projection of slides or motion pictures is, of course, a matte white surface, which may be on a hard base as in outdoor theatres or on a stretched cloth indoors. To understand the basic principles of paraxial ray tracing, consider the necessary calculations and ray tracing tables employed in manually tracing rays of light through a system. A convenient modification of Snells construction can readily be used to trace the path of a ray through a complete lens. In a thin lens such as a spectacle, the two principal planes coincide within the lens, and then the conjugate distances p and p in the formula above become the distances of object and image from the lens itself. n , hence, for points lying on the surface There are two main types of image to be considered: real and virtual. In science, we should not use conflicting definitions of the same term. This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications.[2]. Sure. a By tracing the paths of a great many (100 or more) meridional and skew rays through a lens, with the help of an electronic computer, and plotting the assemblage of points at which all these rays pierce the focal plane after emerging from the lens, a close approximation to the appearance of a star image can be constructed, and a good idea of the expected performance of a lens can be obtained. This arrangement produces a visible picture that may be observed by eye or photographed to make a permanent record. , y . 4 , In 1957 the Italian physicist Vasco Ronchi went the other way and defined an image as any recognizable nonuniformity in the light distribution over a surface such as a screen or film; the sharper the image, the greater the degree of nonuniformity. RayOptics can be used in Python scripts, the Python and IPython shells, the Jupyter notebook, and a Qt-based graphical user interface application. , ) and the square brackets denote the difference in values on both sides of the discontinuity surface (set up according to an arbitrary but fixed convention, e.g. z sintancos10 Trying to operate a lens in that way would mean that you violate its specifications, and the result would probably be substantial beam distortions, which may well prevent you from getting a tighter focus. {\displaystyle \mu } S {\displaystyle A_{m}(\mathbf {r} )} It is occasionally necessary, however, to use some other axially symmetric surface on a lens or mirror, generally to correct a particular aberration. , i.e. and Numerical aperture is not typically used in photography.Instead, the angular aperture of a lens (or an imaging mirror) is expressed by the f-number, written f /N, where N is the f-number given by the ratio of the focal length f to the diameter of the entrance pupil D: =. t with Since 1940 there has been a great resurgence in optics on the basis of information and communication theory, which is treated at length below. y The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances. The Lagrange equation (7) requires modification for a distant object because in that case the object height h is infinite, and the slope angle u is zero. (Mathematically Figure 1: A lens can collimate the output from a fiber, or launch a collimated beam into the fiber. Because rays are really only directions and have no physical existence, the passage of light waves through a surface must be considered if refraction is to be understood. 0 At least, I am not aware of a reasonable way of defining it. Cylindrical surfaces are employed in the anamorphic lenses used in some wide-screen motion-picture systems to compress the image horizontally in the camera and stretch it back to its original shape in the projected image. No further approximations, such as thin lens or paraxial. I have got a telecom fibre for 13001600nm (NA = 0.14) and launch visible light into it. Similarly, one establishes the following from the remaining Maxwell's equations: Now by considering arbitrary small sub-surfaces . D. is a ray that moves exactly parallel to the optical axis. How to model laser beam propagation in OpticStudio: Part 2 - Using Paraxial Gaussian Beam analysis to model Gaussian beam; How to model laser beam propagation in OpticStudio: Part 1 - Gaussian beam theory and ray-based approach; Using skew rays to model Gaussian beams - webinar; How to design a Gaussian to Top Hat beam shaper {\displaystyle \mathbf {R} ^{4}} As the longitudinal displacement of the focus is proportional to the square of the image height h0, this aberration represents a pure field curvature without any accompanying loss of definition (all lines remain sharp). Without loss of generality, let us introduce It is often important to be able to calculate the brightness of an image formed by an optical system, because photographic emulsions and other light receptors cannot respond satisfactorily if the light level is too low. Optomechanics, optics, opto-electronics, laser diodes, fiber optics and ASE test In the absence of any lens, the small plane source already considered radiates in a direction inclined at an angle to the axis with an intensity AB cos . one can conclude from the first of the above equations that for any domain x Although a prism or grating is the essential dispersing element in a spectrograph, a fine slit and additional lenses or focussing mirrors must be used to form a sharply defined spectrum. is of the following form: x , . How does the fiber core diameter influence the numerical aperture? Figure 1: A lens can collimate the output from a fiber, or launch a collimated beam into the fiber. 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. This extends the use of the nodal point for the eye, but the imaging properties come from the cornea and retina being highly curved, rather than paraxial properties, and this is rarely clear in publications. {\displaystyle \varepsilon } Equation (1), known as the law of refraction, is generally written: n sin I = n sin I. z sintancos10 Geometrical optics, or ray optics, is a model of optics that describes light propagation 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 face at 45 deflects a beam of light through a right angle. {\displaystyle \mathbf {R} ^{4}} be a hypersurface (a 3-dimensional manifold) in spacetime / Optics usually describes the behaviour of visible, ultraviolet, and infrared light. There exist microresonators based on silicon nitride as the core (n 2.0) and silica as the cladding (n 1.5). E Use Git or checkout with SVN using the web URL. and MIT 2.71/2.710 09/14/05 wk2-b-17 Paraxial approximation /1 In paraxial optics, we make heavy use of the following approximate (1st order Taylor) expressions: where is the angle between a ray and the optical axis, and is a small = Third order distortion varies as the cubic of the image height. Alternatively, rayoptics can be installed from the conda-forge channel using conda, The documentation for ray-optics is hosted at Read the Docs. R=CF+FP and angle of incidence =angle of reflection sin (small angle approximates) FXFP or CFFP R=CF+FP FP+FP 2FP 2F f= 2R The answer to that question is no it generally the beam divergence also depends on the launch conditions, unless you have a single-mode fiber, where the output beam divergence is determined only by the fiber properties, but not specifically by the NA. Let We discuss the various tools that provide a paraxial and seidel based analysis of a lens system. Snells construction is as follows: The line AP in Figure 3A represents a ray incident upon a refracting surface at P, the normal at P being PN. The answer: theoretically yes, practically no, it probably can't be that high. This term also applies to corresponding pairs of object and image points and planes. , y {\displaystyle f} Another nonclassical optical system is the bifocal or trifocal spectacle lens. endobj This light has to travel farther than the axial light to reach a screen, and then it strikes the screen at another angle . The dispersive power (w) of the material is then defined as the ratio of the difference between the F and C indices and the D index reduced by 1, or. and Paraxial ray tracing involves small ray angles and heights. Actually, the telescopic view is always dimmer than the direct view because of light losses in the telescope due to glass absorption and surface reflections and because the exit pupil of the telescope may be smaller than the pupil of the eye, thus reducing the angle U. ]-~&UM*)2fkr>5{A7 A geometric aberration which causes the image of an off-axis field point to be displaced from the paraxial image height (paraxial chief ray intersection on image plane). , 2008. Paraxial rays are rays having very small angle with the principal axis. Since about 1960, however, lens design has been almost entirely computerized, and the old methods of designing lenses by hand on a desk calculator are rapidly disappearing. {\displaystyle \mathbf {\vec {H}} } With a world-wide readership and authors from every continent, European Journal of Physics is a truly international journal dedicated to maintaining and improving the standard of taught physics in universities and other higher education institutes. , This changes x by Lx0/f and y by Ly0/f; hence, if L is made equal to 4 f 2A2S1, both ray displacements vanish. These reflecting faces could be replaced by pieces of mirror mounted on a metal frame, but it is hard to hold mirrors rigidly and harder still to keep them clean. In case of a multimode fiber, the problem is that the beam quality will depend on the unknown power distribution over the fiber modes. z How to determine its BPP or M2? A set of three mutually perpendicular reflective surfaces, placed to form the internal corner of a cube, work as a retroreflector. {\displaystyle \psi (\mathbf {r} ,t)=\phi (\mathbf {r} )e^{i\omega t}} from bending of the fiber. S In order to construct the rays through the lens, we have to use the two functions described above, in the correct order. What its NA would be? If an aspherical lens with high NA (> 0.55) is used as focusing optics, how is the beam divergence angle defined after beam focus? N Why is there a 1/2 factor in the NA definition? 0 ) This formula provides a means for defining focal length and for measuring the focal length of an unknown lens. N {\displaystyle \mu (x,y,z)} z o Thus, when initial conditions oscillate much faster than the coefficients of the differential equation, solutions will be highly oscillatory, and transported along rays. {\displaystyle \mathbf {R} ^{4}} [6] Their derivation was based on an oral remark by Peter Debye. x These surfaces are generated and polished by special machines and are made by the million every year. In principle yes, unless the second fiber has a larger NA and some mode mixing occurs, e.g. {\displaystyle n_{2}} The focal length, f, of a lens in air is given by the lensmaker's equation: = [+ ()], where n is the index of refraction of the lens material, and R 1 and R 2 are the radii of curvature of the two surfaces. 85 0 obj {\displaystyle \mu } In such situations, Snell's Law describes the resulting deflection of the light ray: where {\displaystyle k_{o}=\omega /c=2\pi /\lambda _{o}} This condition is satisfied by e.g. % , n mk;!kq;99%*%np],;y}hY[^f%geNSfxNDYn9aNs7ob)HKN9|--\!5%>. Since the paraxial wave equation (44) is isotropic, this -dependence is conserved during free propagation. {\displaystyle \varphi (t,x)/\varepsilon } k Nevertheless, they are important enough to require drastic reduction in high-quality lenses intended to make sharp negatives capable of considerable enlargement. x Anwendung der Vektorrechnung auf die Grundlagen der geometrischen Optik. The object and image rays and points are considered to be in two distinct optical spaces, object space and image space; additional intermediate optical spaces may be used as well. The extreme rays are limited by the size of the lens, or in some cases somewhat less if there is a non-transparent facet. [J]. The common Porro prism used in a pair of binoculars contains four 45 reflecting surfaces, two to reverse the beam direction in the vertical plane and two in the horizontal plane (Figure 7). Rayleigh found that two adjacent and equally bright stars can just be resolved if the image of one star falls somewhere near the innermost dark ring in the Airy disk of the other star; the resolving power of a lens can therefore be regarded as about half the f-number of the lens expressed in microns. A single point of light, which may be a point in an extended object, emits light in the form of a continually expanding train of waves, spherical in shape and centred about the point of light. See also: optical apertures, imaging, acceptance angle in fiber optics, fibers, single-mode fibers, waveguides, fiber core, V number, focal length, lensesand other articles in the categories general optics, fiber optics and waveguides. Besides the familiar optical systems cited above, there are many nonclassical optical elements that are used to a limited extent for special purposes. A paraxial raytrace is linear with respect to ray angles and heights since all paraxial angles u are defined to be the tangent of the actual angle U. Rays in the vicinity of the optical axis are used, and the surface sag is ignored or negligible. Refraction occurs when light travels through an area of space that has a changing index of refraction. Import of Zemax .zmx and CODEV .seq files is supported. Because the sine of a small angle is equal to the radian measure of the angle itself, however, a paraxial ray can be traced by reducing the ray-tracing formulas to their limiting case for small angles and thus determining the paraxial intersection point directly. {\displaystyle k,\omega } On the other hand, if m is small, less than one as in a camera, then m is very small, and all objects within a considerable range of distances (p) appear substantially in focus. [7][8] Consider a monochromatic scalar field They are important primarily because they are the physically measurable parameters for the position of the optical elements, and so the positions of the cardinal points must be known with respect to the vertices to describe the physical system. However, you probably mean ring resonators, where you do not have an end face to couple in. 1 CT evaluation of timing for ossification of the medial clavicular epiphysis "The clavicle is the first bone to ossify in the developing embryo and the last to complete epiphyseal union. n = Such rays can be modeled reasonably well by using the paraxial approximation. Because all paraxial rays from a given object point unite at the same image point, the resulting longitudinal distance (l) is independent of the particular paraxial ray that is traced. e and They have been used to produce a straight line of light in space for aligning machines and shafting, but since about 1965 the beam from a gas laser has generally been used instead. For flat mirrors, the law of reflection implies that images of objects are upright and the same distance behind the mirror as the objects are in front of the mirror. r If nothing happens, download Xcode and try again. [ In particular, spherical mirrors exhibit spherical aberration. Also, if m is large, then m is very large, which explains why the depth of field (p) of a microscope is extremely small. From E a line is drawn parallel to the normal PC extending to F on the circle representing the refractive index n. {\displaystyle t} The small parameter Parameters: fristlens - the front/frist optical surface defined using the SphericalRefraction class. ) It is often more convenient to convert the angle U into other better known quantities, such as the f-number of the lens and the image magnification. In this short-wavelength limit, it is possible to approximate the solution locally by, where The generalized method can handle rays that initially make a large angle with the optical axis with a satisfactory accuracy. The light then proceeds to the relay lens C, which forms a second image, now erect, at D. Beyond this image is the eyepiece E to render the light parallel so that the image may be seen sharply by the observer. . 1 y ( (If you later retract your consent, we will delete those inputs.) Indeed, for a long time there was a great discussion as to whether, in vision, something moved from the object to the eye or whether something reached out from the eye to the object. The coupling is then usually done via evanescent waves, and that may be hard because the evanescent field decays so fast. , onto the 3D slice The phenomenon is studied in the field of gradient-index optics.[4]. 4 At most, you can calculate the maximum beam divergence angle with is possible without excessive aberrations. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, , Does higher numerical aperture of an optical fiber mean that it can carry more data? {\displaystyle S_{1}} Cylindrical lenses are therefore used wherever it is desired to vary the magnification from one meridian to a perpendicular meridian. t If you wish to receive personal feedback or consultancy from the author, please contact him e.g. t That does not directly translate into a limit for Gaussian beams, which do not have a clear boundary. {\displaystyle \nabla \varphi \cdot [\mathbf {\vec {H}} ]=0} The classical theory of fields. 0 The point where the emerging paraxial ray crosses the axis is called the focal point F2, the distance from P2 to F2 being the (posterior) focal length f. Then, the five aberrations may be written. They are made either by forming two or three separate surfaces on a single piece of glass or obtaining additional power by fusing a piece of high-index glass on to the front of the main lens and then polishing a single spherical surface over both glasses. and substituting the first yields: The continuity of ) Similar to mirrors, upright images produced by single lenses are virtual while inverted images are real. 0 Work fast with our official CLI. D 0 A common Gaussian form of the lens equation is shown below. y k Various chemicals were added in the molten state to vary the properties of the glass: addition of lead oxide, for example, was found to raise both the refractive index and the dispersive power. Let us assume that I have a certain waveguide, where we know that the optimal coupling between laser and the waveguide can be reached when the beam waist at the focal point has a diameter of 38.4m. The rays are paraxial, a paraxial ray is a ray that makes very small angles. Their formation proceeds in a craniocaudal direction. [3], Various consequences of Snell's Law include the fact that for light rays traveling from a material with a high index of refraction to a material with a low index of refraction, it is possible for the interaction with the interface to result in zero transmission. The paraxial approximation assumes that rays travel at shallow angles with respect to the optical axis, so that Lenses have occasionally been made with one surface taking the form of a flattened cone. Since the angles are small, we can consider that Sin tan . o {\displaystyle \Gamma _{0}} With such surfaces, the direction of the reflected ray is determined by the angle the incident ray makes with the surface normal, a line perpendicular to the surface at the point where the ray hits. , r You can fully launch into the fundamental mode only if you have the perfect amplitude profile, including flat phase fronts perpendicular to the core axis. ) The previous section shows that any parabola with the origin as vertex and the y axis as axis of symmetry can be considered as the graph of a function =For > the parabolas are opening to the top, and for < are opening to the bottom (see picture). 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. So your question should be whether the beam divergence is determined by the NA of the fiber. 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