Optics - Revision history

Gll059 at 19:51, 18 August 2009

2009-08-18T19:51:04Z

← Older revision		Revision as of 14:51, 18 August 2009
Line 104:		Line 104:


			In the above computation, r is actually the radius of the sensing area, so if the cylinder is actually larger than the sensing area,
	~~In the above computation, r is actually the radius of the sensing area, so if the cylinder is actually larger than the sensing area,~~ only the range of angles where the light converges on the sensor will be included. This is because light reflected from a point in space toward the lens will have a narrow range of angles, which will converge on a single point in the image plane, resulting in clear images at that distance. The angular distance from the origin of the light to the axis of the lens will determine where in the image plane the light converges, so imagining that for a sensor at the image plane, light simply passes through an infinitesimally small hole at the lens position and emerges (inverted) to be projected on the sensor is an appropriate approximation.		only the range of angles where the light converges on the sensor will be included. This is because light reflected from a point in space toward the lens will have a narrow range of angles, which will converge on a single point in the image plane, resulting in clear images at that distance. The angular distance from the origin of the light to the axis of the lens will determine where in the image plane the light converges, so imagining that for a sensor at the image plane, light simply passes through an infinitesimally small hole at the lens position and emerges (inverted) to be projected on the sensor is an appropriate approximation.

	<br clear=all>		<br clear=all>

EricN: /* Field of View */

2008-12-12T22:09:26Z

Field of View

← Older revision		Revision as of 17:09, 12 December 2008
Line 99:		Line 99:

	The field of view (FoV) of a lens is a function of the focal length and the diameter of the imaging surface. Imagine that the lens is a small aperture in one end of an opaque cylinder of large diameter with the other end of the cylinder at the image plane. Assuming that the object being imaged is approximately at a distance of infinity, this would make the focal plane and the image plane the same. The radius of the cylinder is r, the focal length is f, and the field of view is 2*<math>\theta</math>.		The field of view (FoV) of a lens is a function of the focal length and the diameter of the imaging surface. Imagine that the lens is a small aperture in one end of an opaque cylinder of large diameter with the other end of the cylinder at the image plane. Assuming that the object being imaged is approximately at a distance of infinity, this would make the focal plane and the image plane the same. The radius of the cylinder is r, the focal length is f, and the field of view is 2*<math>\theta</math>.


	<math> \theta=arctan \frac{r}{f}</math>		<math> \theta=arctan \frac{r}{f}</math>


			In the above computation, r is actually the radius of the sensing area, so if the cylinder is actually larger than the sensing area, only the range of angles where the light converges on the sensor will be included. This is because light reflected from a point in space toward the lens will have a narrow range of angles, which will converge on a single point in the image plane, resulting in clear images at that distance. The angular distance from the origin of the light to the axis of the lens will determine where in the image plane the light converges, so imagining that for a sensor at the image plane, light simply passes through an infinitesimally small hole at the lens position and emerges (inverted) to be projected on the sensor is an appropriate approximation.

			<br clear=all>

EricN: /* Field of View */

2008-12-12T22:02:01Z

Field of View

← Older revision		Revision as of 17:02, 12 December 2008
Line 96:		Line 96:

	==Field of View==		==Field of View==
			[[Image:field_of_view.gif\|right\|Determining the field of view of an optical sensor with a converging lens]\|thumb\|350px]]

	The field of view (FoV) of a lens is a function of the focal length and the diameter of the imaging surface. Imagine that the lens is a small aperture in one end of an opaque cylinder of large diameter with the other end of the cylinder at the image plane. Assuming that the object being imaged is approximately at a distance of infinity, this would make the focal plane and the image plane the same. The radius of the cylinder is r, the focal length is f, and the field of view is 2*<math>\theta</math>.		The field of view (FoV) of a lens is a function of the focal length and the diameter of the imaging surface. Imagine that the lens is a small aperture in one end of an opaque cylinder of large diameter with the other end of the cylinder at the image plane. Assuming that the object being imaged is approximately at a distance of infinity, this would make the focal plane and the image plane the same. The radius of the cylinder is r, the focal length is f, and the field of view is 2*<math>\theta</math>.

EricN: /* Focal Length */

2008-12-12T21:56:13Z

Focal Length

← Older revision		Revision as of 16:56, 12 December 2008
Line 80:		Line 80:

	When the path of the light beam is not parallel to the axis of the lens, the beam still converges, however the convergence point is also off-axis. This results in a focal plane which is at the focal length. This plane is where parallel beams of light converge. Real images are all located beyond the focal plane, provided that the object emitting or reflecting the light is more than the focal distance in front of the lens. It is important to note that the location of the image plane is a function of the focal length of the lens and the distance from the lens to the object being imaged.		When the path of the light beam is not parallel to the axis of the lens, the beam still converges, however the convergence point is also off-axis. This results in a focal plane which is at the focal length. This plane is where parallel beams of light converge. Real images are all located beyond the focal plane, provided that the object emitting or reflecting the light is more than the focal distance in front of the lens. It is important to note that the location of the image plane is a function of the focal length of the lens and the distance from the lens to the object being imaged.
		⚫	<br clear=all>
		⚫	[[Image:image_plane.gif\|right\|Relationship between object distance, focal plane, and image plane]\|thumb\|350px]]

	Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':		Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':


⚫	<br clear=all>
	<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>		<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>
⚫	[[Image:image_plane.gif\|right\|Relationship between object distance, focal plane, and image plane]\|thumb\|350px]]

EricN: /* Focal Length */

2008-12-12T21:55:12Z

Focal Length

← Older revision		Revision as of 16:55, 12 December 2008
Line 83:		Line 83:
	Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':		Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':

			<br clear=all>

	<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>		<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>
	[[Image:image_plane.gif\|right\|Relationship between object distance, focal plane, and image plane]\|thumb\|350px]]		[[Image:image_plane.gif\|right\|Relationship between object distance, focal plane, and image plane]\|thumb\|350px]]

EricN: /* Focal Length */

2008-12-12T21:46:20Z

Focal Length

← Older revision		Revision as of 16:46, 12 December 2008
Line 83:		Line 83:
	Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':		Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':

⚫	[[Image:image_plane.gif\|right\|Relationship between object distance, focal plane, and image plane]\|thumb\|350px]]

	<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>		<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>
		⚫	[[Image:image_plane.gif\|right\|Relationship between object distance, focal plane, and image plane]\|thumb\|350px]]


Line 91:		Line 91:

	For more information, see http://en.wikipedia.org/wiki/Lens_(optics) where you can find information about some of the common sources of distortions in lenses and the properties of diverging lenses.		For more information, see http://en.wikipedia.org/wiki/Lens_(optics) where you can find information about some of the common sources of distortions in lenses and the properties of diverging lenses.

			<br clear=all>

	==Field of View==		==Field of View==

EricN: /* Focal Length */

2008-12-12T21:36:40Z

Focal Length

← Older revision		Revision as of 16:36, 12 December 2008
Line 83:		Line 83:
	Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':		Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':

			[[Image:image_plane.gif\|right\|Relationship between object distance, focal plane, and image plane]\|thumb\|350px]]

	<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>		<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>

EricN: /* Focal Length */

2008-12-12T21:22:24Z

Focal Length

← Older revision		Revision as of 16:22, 12 December 2008
Line 82:		Line 82:

	Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':		Assuming a negligible lens thickness, the relationship between the object distance, image distance, and focal length is given by the '''thin lens equation''':


	<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>		<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}</math>


	Additionally, there is a resolution to any optical sensors, which permits a certain amount of blurring in the image, permitting objects at distances similar to that of the imaged object to be "acceptably sharp" as well. The hyperfocal distance is the distance at which a lens is to be focused to make all objects from half the [http://en.wikipedia.org/wiki/Hyperfocal_distance hyperfocal distance] to infinity acceptably sharp. This is heavily dependent on the definition of "acceptably sharp" which is the size which an infinitesimally small point spreads out to on the imaging plane. So for crude applications, the hyperfocal distance will be relatively near, while for high precision optics the hyperfocal distance will be relatively far.		Additionally, there is a resolution to any optical sensors, which permits a certain amount of blurring in the image, permitting objects at distances similar to that of the imaged object to be "acceptably sharp" as well. The hyperfocal distance is the distance at which a lens is to be focused to make all objects from half the [http://en.wikipedia.org/wiki/Hyperfocal_distance hyperfocal distance] to infinity acceptably sharp. This is heavily dependent on the definition of "acceptably sharp" which is the size which an infinitesimally small point spreads out to on the imaging plane. So for crude applications, the hyperfocal distance will be relatively near, while for high precision optics the hyperfocal distance will be relatively far.

EricN: /* Focal Length */

2008-12-12T21:22:09Z

Focal Length

← Older revision		Revision as of 16:22, 12 December 2008
Line 73:		Line 73:

	Most applications will employ converging (or positive) lenses, which is what will be described here. A converging lens causes a convergence of collimated (parallel) light, for example light emitted from a source an infinite distance from the lens. Such a beam of light will be refracted as it enters and again as it leaves the lens, resulting in paths which converge on a single point. Assuming that the path of the light beam is parallel to the axis of the lens, the point at which the beam converges is known as the focal point, and the distance from that point to the center of the lens is known as the focal length. The focal length is a function of the radii of curvature of the two faces of the lens, which is known as the [lens maker's equation, and holds for lenses in air. Under the conditions that the lens thickness is negligible with respect to the focal length, <math>r_1</math> is the radius of curvature of the entry side of the lens, <math>r_2</math> is the radius of curvature of the exit side of the lens, and positive radius of curvature represents convex curvature, the thin lens form of the lens maker's equation results in:		Most applications will employ converging (or positive) lenses, which is what will be described here. A converging lens causes a convergence of collimated (parallel) light, for example light emitted from a source an infinite distance from the lens. Such a beam of light will be refracted as it enters and again as it leaves the lens, resulting in paths which converge on a single point. Assuming that the path of the light beam is parallel to the axis of the lens, the point at which the beam converges is known as the focal point, and the distance from that point to the center of the lens is known as the focal length. The focal length is a function of the radii of curvature of the two faces of the lens, which is known as the [lens maker's equation, and holds for lenses in air. Under the conditions that the lens thickness is negligible with respect to the focal length, <math>r_1</math> is the radius of curvature of the entry side of the lens, <math>r_2</math> is the radius of curvature of the exit side of the lens, and positive radius of curvature represents convex curvature, the thin lens form of the lens maker's equation results in:


	<math>\frac{1}{f} = (n-1)\left [\frac{1}{r_1}-\frac{1}{r_2}\right ]</math>		<math>\frac{1}{f} = (n-1)\left [\frac{1}{r_1}-\frac{1}{r_2}\right ]</math>

EricN: /* Focal Length */

2008-12-12T21:20:25Z

Focal Length

← Older revision		Revision as of 16:20, 12 December 2008
Line 70:		Line 70:

	==Focal Length==		==Focal Length==
	[[Image:focal_length.gif\|right\|~~Light~~ beam converging to the focal point, ~~from~~ ~~[http://en.wikipedia.org/wiki/Lens_(optics)~~ ~~Wikipedia~~]\|thumb\|350px]]		[[Image:focal_length.gif\|right\|Collimated light beam parallel to lens axis converging to the focal point of a biconvex lens]\|thumb\|350px]]

	Most applications will employ converging (or positive) lenses, which is what will be described here. A converging lens causes a convergence of collimated (parallel) light, for example light emitted from a source an infinite distance from the lens. Such a beam of light will be refracted as it enters and again as it leaves the lens, resulting in paths which converge on a single point. Assuming that the path of the light beam is parallel to the axis of the lens, the point at which the beam converges is known as the focal point, and the distance from that point to the center of the lens is known as the focal length. The focal length is a function of the radii of curvature of the two faces of the lens, which is known as the [lens maker's equation, and holds for lenses in air. Under the conditions that the lens thickness is negligible with respect to the focal length, <math>r_1</math> is the radius of curvature of the entry side of the lens, <math>r_2</math> is the radius of curvature of the exit side of the lens, and positive radius of curvature represents convex curvature, the thin lens form of the lens maker's equation results in:		Most applications will employ converging (or positive) lenses, which is what will be described here. A converging lens causes a convergence of collimated (parallel) light, for example light emitted from a source an infinite distance from the lens. Such a beam of light will be refracted as it enters and again as it leaves the lens, resulting in paths which converge on a single point. Assuming that the path of the light beam is parallel to the axis of the lens, the point at which the beam converges is known as the focal point, and the distance from that point to the center of the lens is known as the focal length. The focal length is a function of the radii of curvature of the two faces of the lens, which is known as the [lens maker's equation, and holds for lenses in air. Under the conditions that the lens thickness is negligible with respect to the focal length, <math>r_1</math> is the radius of curvature of the entry side of the lens, <math>r_2</math> is the radius of curvature of the exit side of the lens, and positive radius of curvature represents convex curvature, the thin lens form of the lens maker's equation results in:

	<math>\frac{1}{f} = (n-1)\left [\frac{1}{r_1}-\frac{1}{r_2}\right ]</math>		<math>\frac{1}{f} = (n-1)\left [\frac{1}{r_1}-\frac{1}{r_2}\right ]</math>

			[[Image:focal_plane.gif\|right\|Collimated light beam not parallel to lens axis converging to the focal plane of a biconvex lens]\|thumb\|350px]]

	When the path of the light beam is not parallel to the axis of the lens, the beam still converges, however the convergence point is also off-axis. This results in a focal plane which is at the focal length. This plane is where parallel beams of light converge. Real images are all located beyond the focal plane, provided that the object emitting or reflecting the light is more than the focal distance in front of the lens. It is important to note that the location of the image plane is a function of the focal length of the lens and the distance from the lens to the object being imaged.		When the path of the light beam is not parallel to the axis of the lens, the beam still converges, however the convergence point is also off-axis. This results in a focal plane which is at the focal length. This plane is where parallel beams of light converge. Real images are all located beyond the focal plane, provided that the object emitting or reflecting the light is more than the focal distance in front of the lens. It is important to note that the location of the image plane is a function of the focal length of the lens and the distance from the lens to the object being imaged.