Convergence tests - Revision history

193.205.210.44: /* Limit comparison test */

2014-12-05T18:38:00Z

Limit comparison test

← Older revision		Revision as of 20:38, 5 December 2014
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147.161.1.49: /* Alternating series test */ This test was phrased as "if and only if", which is incorrect-- the test only proves convergence, not divergence.

2014-02-26T09:49:57Z

Alternating series test: This test was phrased as "if and only if", which is incorrect-- the test only proves convergence, not divergence.

← Older revision		Revision as of 11:49, 26 February 2014
Line 1:		Line 1:
	~~{{Optical aberration}}~~		Hi there. Allow me begin by introducing the author, her title is Sophia Boon but she by no means truly favored that title. For years she's been working as a travel agent. To play lacross is the thing I love most of all. Her family members life in Ohio but her husband wants them to move.<br><br>Feel free to visit my webpage; [http://clothingcarearchworth.com/index.php?document_srl=441551&mid=customer_review free psychic reading]
	~~In [[optics]]~~, '~~''tilt'''~~ is ~~a deviation in~~ the ~~direction a beam~~ of ~~light propagates~~. ~~Tilt quantifies the [[average]] [[slope]]~~ in ~~both the X and Y directions of a [[wavefront]] or [[Phase (waves)\|phase]] profile across the pupil of an optical system~~. ~~In conjunction with [[Piston (optics)\|piston]] (the first [[Zernike polynomials\|Zernike polynomial]] term), X and Y tilt can be modeled using the second and third Zernike polynomials:~~

	~~:X-Tilt:~~ <~~math~~>~~a_1 \rho \cos(\theta)~~<~~/math~~>

	~~:Y-Tilt~~: ~~<math>a_2 \rho \sin(\theta)<~~/~~math>~~

	~~where <math>\rho<~~/~~math> is the normalized radius with <math>0 \le \rho \le 1</math> and <math>\theta</math> is the azimuthal angle with <math>0 \le \theta \le 2\pi</math>~~.

	~~The <math>a_1</math> and <math>a_2<~~/~~math> coefficients are typically expressed as a fraction of a chosen [[wavelength]] of light.~~

	Piston and tilt are not actually true optical [[Aberration in optical systems\|aberrations]], as they do not represent or model [[curvature]] in the wavefront. [[Defocus]] is the lowest order true optical aberration. If piston and tilt are subtracted from an otherwise perfect wavefront, a perfect, aberration-free image is formed.

	Rapid optical tilts in both X and Y directions are termed '''jitter'''. Jitter can arise from three-dimensional mechanical vibration, and from rapidly varying 3D [[refraction]] in aerodynamic flowfields. Jitter may be compensated in an [[adaptive optics]] system by using a flat mirror mounted on a dynamic two-axis mount that allows small, rapid, computer-controlled changes in the mirror X and Y angles. This is often termed a "fast steering mirror", or FSM. A [[gimbal]]ed optical pointing system cannot mechanically track an object or stabilize a projected laser beam to much better than several hundred [[microradian]]s. Buffeting due to aerodynamic turbulence further degrades the pointing stability.

	Light, however, has no appreciable momentum, and by reflecting from a computer-driven FSM, an image or laser beam can be stabilized to single microradians, or even a few hundred [[nanoradian]]s. This almost totally eliminates image blurring due to motion, and [[far-field]] laser beam jitter. Limitations on the degree of line-of-sight stabilization arise from the limited dynamic range of the FSM tilt, and the highest frequency the mirror tilt angle can be changed. ~~Most FSM's can be driven to several wavelengths of tilt, and at frequencies exceeding one [[kilohertz]].~~

	As the FSM mirror is optically flat, FSM's need not be located at [[pupil images]]. Two FSM's can be combined to create an '''anti-beamwalk''' pair, which stabilizes not only the beam pointing angle but the location of the beam center. Anti-beamwalk FSM's are positioned prior to a [[deformable mirror]] (which must be located at a pupil image) to stabilize the position of the pupil image on the deformable mirror and minimize correction errors resulting from wavefront movement, or [[Shearing interferometer\|shearing]], on the deformable mirror faceplate.

	==~~References==~~
	*Malacara, D., ''Optical Shop Testing - Second Edition'', John Wiley and Sons, 1992, ISBN 0-471-52232-5.

	~~[[Category:Optics]~~]

en>Cheerioswithmilk: /* List of tests */

2014-01-26T20:24:16Z

List of tests

← Older revision		Revision as of 22:24, 26 January 2014
Line 1:		Line 1:
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			In [[optics]], '''tilt''' is a deviation in the direction a beam of light propagates. Tilt quantifies the [[average]] [[slope]] in both the X and Y directions of a [[wavefront]] or [[Phase (waves)\|phase]] profile across the pupil of an optical system. In conjunction with [[Piston (optics)\|piston]] (the first [[Zernike polynomials\|Zernike polynomial]] term), X and Y tilt can be modeled using the second and third Zernike polynomials:

			:X-Tilt: <math>a_1 \rho \cos(\theta)</math>

			:Y-Tilt: <math>a_2 \rho \sin(\theta)</math>

			where <math>\rho</math> is the normalized radius with <math>0 \le \rho \le 1</math> and <math>\theta</math> is the azimuthal angle with <math>0 \le \theta \le 2\pi</math>.

			The <math>a_1</math> and <math>a_2</math> coefficients are typically expressed as a fraction of a chosen [[wavelength]] of light.

			Piston and tilt are not actually true optical [[Aberration in optical systems\|aberrations]], as they do not represent or model [[curvature]] in the wavefront. [[Defocus]] is the lowest order true optical aberration. If piston and tilt are subtracted from an otherwise perfect wavefront, a perfect, aberration-free image is formed.

			Rapid optical tilts in both X and Y directions are termed '''jitter'''. Jitter can arise from three-dimensional mechanical vibration, and from rapidly varying 3D [[refraction]] in aerodynamic flowfields. Jitter may be compensated in an [[adaptive optics]] system by using a flat mirror mounted on a dynamic two-axis mount that allows small, rapid, computer-controlled changes in the mirror X and Y angles. This is often termed a "fast steering mirror", or FSM. A [[gimbal]]ed optical pointing system cannot mechanically track an object or stabilize a projected laser beam to much better than several hundred [[microradian]]s. Buffeting due to aerodynamic turbulence further degrades the pointing stability.

			Light, however, has no appreciable momentum, and by reflecting from a computer-driven FSM, an image or laser beam can be stabilized to single microradians, or even a few hundred [[nanoradian]]s. This almost totally eliminates image blurring due to motion, and [[far-field]] laser beam jitter. Limitations on the degree of line-of-sight stabilization arise from the limited dynamic range of the FSM tilt, and the highest frequency the mirror tilt angle can be changed. Most FSM's can be driven to several wavelengths of tilt, and at frequencies exceeding one [[kilohertz]].

			As the FSM mirror is optically flat, FSM's need not be located at [[pupil images]]. Two FSM's can be combined to create an '''anti-beamwalk''' pair, which stabilizes not only the beam pointing angle but the location of the beam center. Anti-beamwalk FSM's are positioned prior to a [[deformable mirror]] (which must be located at a pupil image) to stabilize the position of the pupil image on the deformable mirror and minimize correction errors resulting from wavefront movement, or [[Shearing interferometer\|shearing]], on the deformable mirror faceplate.

			==References==
			*Malacara, D., ''Optical Shop Testing - Second Edition'', John Wiley and Sons, 1992, ISBN 0-471-52232-5.

			[[Category:Optics]]

en>LutzL: Undid revision 495854265 by 68.45.215.249 (talk) This is not about the divergence operator of vector calculus.

2012-06-04T07:16:57Z

Undid revision 495854265 by 68.45.215.249 (talk) This is not about the divergence operator of vector calculus.

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