| Line 12: | Line 12: | ||
[[Image:Lens1.jpeg|400px|thumb|left|A Biconvex Lens]] | [[Image:Lens1.jpeg|400px|thumb|left|A Biconvex Lens]] | ||
| + | |||
When parallel light rays are incident upon a biconvex lens, they converge at one point after passing through. This point is called the '''focal point''' of the lens. The distance from the optical center of the lens to the focal point is called the '''focal distance'''. | When parallel light rays are incident upon a biconvex lens, they converge at one point after passing through. This point is called the '''focal point''' of the lens. The distance from the optical center of the lens to the focal point is called the '''focal distance'''. | ||
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== Image Formation == | == Image Formation == | ||
| − | Most of your photography subjects however, will not be an infinite distance away. In that case your object's distance from the lens is some finite quantity. | + | Most of your photography subjects however, will not be an infinite distance away. In that case your object's distance from the lens is some finite quantity and the image plane has to be adjusted so that the image is in focus. |
[[Image:Lens2.jpeg|400px|thumb|left|Image Formation via a Biconvex Lens]] | [[Image:Lens2.jpeg|400px|thumb|left|Image Formation via a Biconvex Lens]] | ||
| − | Let <math>d_f</math> be the focal length of the lens. | + | |
| + | Let <math>d_f</math> be the focal length of the lens. This quantity is intrinsic to the lens. | ||
Let <math>d_o</math> be the distance of the object plane from the optical center of the lens. | Let <math>d_o</math> be the distance of the object plane from the optical center of the lens. | ||
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Let <math>d_i</math> be the distance of the image plane from the optical center of the lens. | Let <math>d_i</math> be the distance of the image plane from the optical center of the lens. | ||
| − | + | So for your image to be in focus, the following equation must be satisfied. | |
<math>\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{d_f}</math> | <math>\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{d_f}</math> | ||
| + | |||
| + | Notice that | ||
| + | |||
| + | <math> | ||
| + | \begin{align} | ||
| + | d_o &= \infty \\ | ||
| + | \Rightarrow \frac{1}{d_i} &= \frac{1}{d_f} \\ | ||
| + | \Rightarrow d_i &= d_f | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | which is what we stated earlier when defining focal length. | ||
| + | |||
| + | Also note that | ||
| + | |||
| + | <math>d_i = (\frac{1}{d_f} - \frac{1}{d_o})^{-1}</math> | ||
| + | |||
| + | So the smaller the <math>d_o</math>, the smaller the reciprocal of <math>d_i</math> and hence the larger <math>d_i</math> is. In other words, as the object plane moves in, the image plane moves out and vice versa. | ||
Revision as of 20:26, 28 March 2013
Lenses in Optical Imaging Systems
keyword: ECE 637, digital image processing
Some Basic Properties and Assumptions
First let's review some basics about lenses. For these notes, we will discuss lenses assuming the thin lens model (i.e. assume that the thickness of the lens is negligible compared to its focal length).
When parallel light rays are incident upon a biconvex lens, they converge at one point after passing through. This point is called the focal point of the lens. The distance from the optical center of the lens to the focal point is called the focal distance.
Note that the focal distance can be different for the two sides of the lens if the lens is not perfectly symmetric.
We can approximate light rays from a star that is many light years away to be parallel but ideally, for a point source to produce perfectly parallel light rays, it would have to be an infinite distance away from your lens.
Image Formation
Most of your photography subjects however, will not be an infinite distance away. In that case your object's distance from the lens is some finite quantity and the image plane has to be adjusted so that the image is in focus.
Let $ d_f $ be the focal length of the lens. This quantity is intrinsic to the lens.
Let $ d_o $ be the distance of the object plane from the optical center of the lens.
Let $ d_i $ be the distance of the image plane from the optical center of the lens.
So for your image to be in focus, the following equation must be satisfied.
$ \frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{d_f} $
Notice that
$ \begin{align} d_o &= \infty \\ \Rightarrow \frac{1}{d_i} &= \frac{1}{d_f} \\ \Rightarrow d_i &= d_f \end{align} $
which is what we stated earlier when defining focal length.
Also note that
$ d_i = (\frac{1}{d_f} - \frac{1}{d_o})^{-1} $
So the smaller the $ d_o $, the smaller the reciprocal of $ d_i $ and hence the larger $ d_i $ is. In other words, as the object plane moves in, the image plane moves out and vice versa.
