We recently examined the way a spherical glass surface focusses light; however, this focussing property is not unique for light that travels from air into glass. Instead, this phenomenon occurs anytime light travels between two surroundings separated by a spherical boundary.
For light that originates from a point source, the general equation that specifies the relationship between the location of the source and the location of the image is:
which, in turn, depends on the curvature radius and refractive indices that are shown in the following figure.
After a few algebraic manipulations we can use this equation to predict the distance to the image for a fixed curvature radius and source distance:
This equation works for all source locations and curvature radii; however, we need to use care when assigning the sign — positive or negative — to the distances. If the source is to the left of the spherical surface, for example, then its distance should be a positive number. If the source is to the right of the spherical surface, then its distance should be a negative number. If, for instance, the source is two meters to the right of the surface, then we should specify
and be careful to define the surface radius with units of meters also.
Suppose, for example, that the region to the left is air (refractive index = 1.0), the region to the right is glass (refractive index = 1.5), the spherical surface has a radius equal to 0.2 meters, and the source is 10 meters to the left. According to the imaging equation, the distance to the image will satisfy the following relationship:
We use ray diagrams like the one above to show how rays travel through optical elements from left to right. If we specify a negative source distance, then the light rays encounter the boundary in a way that converges toward the source location on the right side of the surface.
This important concept — a virtual source to the right of the boundary — is crucial for understanding the way glass lenses work.
If the spherical surface curves to the right as it does in the figure above, then the curvature radius should be a positive number. If, however, the spherical surface curves to the left as it does in the figure below, then the curvature radius should be a negative number.
For the example in this figure, the source distance and surface curvature cause the imaging equation to specify a negative image distance, which, in turn, corresponds to rays on the right side of the surface that diverge from a virtual source on the left.
Here, then, is a summary of the conventions we need to follow when using the imaging equation for a spherical boundary between two different media:
- All of the rays in our diagrams flow from left to right.
- A positive source distance corresponds to rays that diverge from a source on the left side of the boundary. A negative source distance corresponds to rays that converge toward a virtual source on the right side of the boundary.
- A positive curvature radius corresponds to a boundary that curves to the right. A negative curvature radius corresponds to a boundary that curves to the left.
- A positive image distance corresponds to rays that converge to an image on the right side of the boundary. A negative source distance corresponds to rays that diverge from a virtual image on the left side of the boundary.
© 2011 Timothy Schulz
The Digital Camera
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