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Topic: Paraxial approximation

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	Paraxial approximation - Wikipedia, the free encyclopedia
		In geometric optics, the paraxial approximation is an approximation used in ray tracing of light through an optical system (such as a lens).
		A paraxial ray is a ray which makes a small angle (θ) to the optical axis of the system, and lies close to the axis throughout the system.
		The paraxial approximation is used in first-order raytracing and Gaussian optics.
	en.wikipedia.org /wiki/Paraxial_approximation (209 words)

	Small-angle approximation - Wikipedia, the free encyclopedia
		Small-angle approximation is a useful simplification of the laws of trigonometry which is only approximately true for finite angles, but correct in the limit as the angle approaches zero.
		Small angle approximation is useful in many areas of physical science, including optics (where it forms the basis of the paraxial approximation), cartography, and astronomy.
		The short leg is approximately equal to the arc from the long leg to the hypotenuse, so the sine and tangent are both approximated by the value of the angle in radians.
	en.wikipedia.org /wiki/Small_angle_approximation (313 words)

	Topic 28. Aberrations
		The aberration is caused by the fact that the magnification of a lens (that is the size of the image for a given object) is different for rays that strikes the lens in the paraxial region and for rays that strike the lens near its edge.
		The result is a bright image formed by the paraxial rays and a series of smaller (or larger) images formed by the rays that hit the lens far away from the axis.
		In the paraxial approximation, the rays passing through the lens strike this image surface near its center where the curvature of this surface is not great, and the approximation that it is a flat plane is not too bad.
	www.colorado.edu /physics/phys1230/phys1230_fa01/topic28.html (1194 words)

	Topic 20. Spherical Mirrors, Part 1.
		The scale factor is the ratio of the distance of the ray from the axis divided by the focal length, and the paraxial ray approximation requires that this ratio be small compared to 1.
		This ratio often appears as its reciprocal: the ratio of the focal length to the size of the object (or to the diameter of the lens or the mirror), and the paraxial ray approximation implies that this reciprocal ratio be much larger than 1.
		Using this notation, the paraxial ray approximation is equivalent to the statement that this simple model of constructing the image produced by a lens or mirror is valid only when the f-number is large, and the approximation becomes increasingly poor as the f-number decreases.
	www.colorado.edu /physics/phys1230/phys1230_fa01/topic20.html (987 words)

	Paraxial is not Perfect (Site not responding. Last check: 2007-10-15)
		Although paraxial rays can be used to predict the location and magnification of images, paraxial ray trajectories do not obey Fermat's principle, except in a region near the optical axis.
		The paraxial approximation applies strictly to rays that are infinitesimally displaced from the optical axis of a system.
		Sometimes the discrepancy between the real ray trajectories and the paraxial trajectories is minimal; sometimes (as in the drawing in the note) it is quite large.
	www.sinopt.com /learning1/desnotes/pnotperf.htm (580 words)

	aberrations
		In the paraxial approximation all rays behave like meridional rays, since the equations for the projections of the rays on the xz-plane and on the yz-plane decouple and the projections can be treated independently.
		Spherical aberrations are the consequence of the failure of meridional rays to obey the paraxial approximation.
		Spherical aberration is the failure of meridional rays to obey the paraxial approximation, and coma is the failure of skew rays to behave like meridional rays.
	electron9.phys.utk.edu /optics421/modules/m2/aberration.htm (1585 words)

	Optics - The Physics Community (Site not responding. Last check: 2007-10-15)
		Geometric optics is often simplified even further by making the paraxial approximation, or "small angle approximation." The mathematical behavior then becomes linear, allowing optical components and systems to be described by simple matrices.
		While still using the paraxial approximation, this technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused.
		Approximations are still generally used, however, so this is not a full electromagnetic wave theory model of the propagation of light.
	physicscommunity.org /index.php?title=Optics (779 words)

	Weidlinger Associates Inc., EMFlex Theory - Section 8 (Site not responding. Last check: 2007-10-15)
		The 4th order paraxial and Sandler conditions are demonstrated to be roughly equivalent in accuracy, although certain implementation issues appear to favor the latter.
		This condition is superior to the paraxial approach described above because it eliminates wavespeed from the formulation (which in nonlinear calculations is problematic), and it provides better directionality than the 2nd order paraxial or normal incidence condition.
		By comparing the snapshots, it is evident that the 4th order paraxial and Sandler absorbers are roughly equivalent and that both are superior to the normal incidence condition.
	www.wai.com /appliedscience/Software/Emflex/Reports/emflex-sec8.html (1298 words)

	CameraLecture/005 - cs348b-06 (Site not responding. Last check: 2007-10-15)
		Essentially under paraxial approximation, we know that tan(u) = sin(u) = u, so we can simply compute tan(u) and plug in that value into snell's law.
		However, this is only true for paraxial rays (i.e rays near the axis) and for rays further away (i.e peripheral rays), 'a' is better approximated by a - a
		This better approximation is then used when modeling the wavelet abberations caused by rays arriving at the lens periphery.
	graphics.stanford.edu /cs348b-06/CameraLecture/005 (180 words)

	REVIEW OF SYMMETRIC PROPERTIES OF ELLIPSES
		The horizontal velocity of the paraxial elliptic approximation (not the same thing as the true horizontal velocity) is what is required for NMO.
		On the left is the standard vertical paraxial elliptic approximation.
		In the center is the horizontal paraxial elliptic approximation.
	sepwww.stanford.edu /public/docs/sep70/joe1/paper_html/node2.html (317 words)

	Jens Nöckel's Microdome page
		In the world of optics, one of the limitations to diversity is that designers like to stay within the confines of the paraxial approximation.
		Paraxial optics is a way of solving the propagation of light with the help of an approximate solution to the Maxwell equations of electrodynamics.
		The paraxial approximation is just one possible way to exploit the ray limit of Maxwell's equations.
	darkwing.uoregon.edu /~noeckel/microdome.php (3057 words)

	LASER COMPONENTS: Linsentheorie
		In the paraxial approximation, all ray angles are small.
		This approximation permits the modelling of the principal surfaces as principal planes.
		The beam width in the paraxial focal plane PP is the transverse spherical aberration TSAm,determined by the heights of the marginal rays.
	www.laser-components.com /faq/linsth.htm (1657 words)

	Computer Simulations for Atoms Inside a Laser Light Potential
		Close enough [8] to the anti-node of the standing wave light field the potential has an approximately parabolic shape along the x–direction and acquires the Gaussian profile of the laser beam light field along the propagation direction, which is the z–direction.
		Assuming both that the kinetic energy of the incoming atoms in the z–direction is much larger than the interaction energy and that the focusing region is uniform in the y-direction, the focusing is then described by an effectively one dimensional model of the quantum mechanical motion along the x–direction.
		Although the [1,1] Pade approximant produces unwanted finite oscillations near points of discontinuity it is adequate for this simulation since the wave function is continuous and smooth over the grid and zero at the edges.
	phycomp.technion.ac.il /~phr76ja/KAR/Article4.html (2418 words)

	paraxial vergence equations (English translation glossary) IMPLANTABLE CONTACT LENS,Medical: Instruments,Medical
		To achieve this, an assumption known as the paraxial approximation is made.
		paraxial: Characteristic of optical analyses that are limited to infinitesimally small apertures.
		Paraxial, I think, refers to the original axis of the beam of light, from which the light rays deflect upon passing through the lens.
	www.proz.com /kudoz/1073119 (682 words)

	OPTIMIZATION OF ION SOURCE EXTRACTION AND TRANSPORT WITH SYMBOLIC MANIPULATION PROGRAMS: ELECTROSTATIC LENSES FORMULAS
		This paper describes the nontrivial approximations and equations that we used in simulating round electrostatic elements in some detail.
		Paraxial analysis of einzel lens was indeed possible, by decomposing the lens in seven regions (or elements), where the axial field
		Non paraxial analysis is more easily applied to the remarkable case of the anode lens [4], a hole of radius R in a conducting metal sheet (at
	www94.web.cern.ch /Linac96/Proceedings/Monday/MOP29/Paper.html (1370 words)

	Understanding Paraxial Ray-Tracing (Site not responding. Last check: 2007-10-15)
		These approximations are all made to make the numerical computation easier, but at the cost of generality.
		In summary, paraxial ray data is computed using first order approximations to the surface power for tracing rays, while parabasal rays are real, exact ray traces close to a chief or reference ray.
		Most paraxial data, such as EFL, F/#, and magnification, use paraxial rays and the data is invalid if the optical system is not well described by the vertex power of every surface.
	www.zemax.com /kb/articles/18/4/Understanding-Paraxial-Ray-Tracing/Page4.html (330 words)

	OPTI 546
		Topics range from paraxial ray optics and diffraction theory to laser beam propagation with application to resonators and interferometers, and culminate in a treatment of nonlinear wave interactions in media.
		Paraxial wave optics: The paraxial wave equation, Gaussian beam solution, Guoy phase.
		Relation of paraxial ray and wave optics, path integrals.
	www.optics.arizona.edu /Classes/Grad/Opti_546.htm (220 words)

	The Ray and Wave Theory of Lenses - Cambridge University Press (Site not responding. Last check: 2007-10-15)
		Calculations on lens systems are often marred by the unjustifiable use of the small-angle approximation.
		This book describes in detail how the ray and wave pictures of lens behaviour can be combined and developed into a theory capable of dealing with the large angles encountered in real optical systems.
		The propagation of waves through lenses is then treated by exploiting the close connection between eikonal function theory and the stationary phase approximation.
	www.cambridge.org /aus/catalogue/catalogue.asp?isbn=0521028299 (253 words)

	PARAXIAL RAY APPROXIMATION
		To compute the traveltimes and amplitudes off but near the ray, we use the paraxial ray approximation.
		The direction of a ray follows the direction of the traveltime gradient that is determined by
		The next section explains how these equations are used in the local paraxial ray method.
	sepwww.stanford.edu /public/docs/sep72/lin3/paper_html/node2.html (225 words)

	The Paraxial Approximation for Radiation of a Planar Ultrasonic Transducer at Oblique Incidence Through an Interface, ... (Site not responding. Last check: 2007-10-15)
		The Paraxial Approximation for Radiation of a Planar Ultrasonic Transducer at Oblique Incidence Through an Interface, by Terrence P
		In the paraxial approximation, the calculation of the radiation fields of transducers with a boundary diffraction wave model is particularly simple and efficient.
		Here, we will compare the "exact" boundary diffraction wave model predictions for transmission at oblique incidence through a planar interface with those based on the paraxial assumption and with Gauss-Hermite theory, which is also based on the paraxial approximation.
	www.cnde.iastate.edu /qndedatabase/Vol.14A/p1067-1074.html (286 words)

	RR-5547 : Modeling and Numerical Simulation of Space Charge Dominated Beams in the Paraxial Approximation
		This work is devoted to the modeling of space charge dominated particle beams in the paraxial approximation with several types of external focusing fields (uniform, periodic and alternating gradient).
		A solid mathematical background for numerical beam simulation is established.
		Then the matched KV beam is used to approximately match arbitrary beams.
	www.inria.fr /rrrt/rr-5547.html (299 words)

	[No title]
		Fowles, Introduction to Modern Optics, Dover, 1989, (Holt, Rinehart, Winston, 1968, 1975) We have been studying the propagation of paraxial waves.
		In the paraxial approximation it can be shown that the propagation of the field amplitude and vector can be handled as separate entities.
		A 11, 1365-1370 (1975) Davis, "Theory of electromagnetic beams," Phy.s Rev. A 19, 1177-1170 (March, 1979) Agrawal and Patanayak, "Gaussian beam propagation beyond the paraxial approximation," J. Opt.
	uhaweb.hartford.edu /ltownsend/Polarization.doc (594 words)

	I P M - Papers (Site not responding. Last check: 2007-10-15)
		Beyond The Paraxial Approximation in Laser Plasma Interaction
		A generalization of the paraxial approximation based on stationary transverse distribution of the laser pulse in laser-plasma interaction is addressed.
		The laser beam is assumed to have a non-uniform intensity distribution (along its wave front) in a plane transverse to the direction of propagation.
	www.ipm.ac.ir /ipm/publications/ViewPaperInfo.jsp?PTID=7668&school=Physics (168 words)

	Gaussian Optics / Paraxial Approximation Text - Physics Forums Library
		Gaussian Optics / Paraxial Approximation Text - Physics Forums Library
		View Full Version : Gaussian Optics / Paraxial Approximation
		Because of the paraxial approximation you have that si >>> R
	www.physicsforums.com /archive/index.php/t-54470.html (332 words)

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