The Digital Camera: The Spherical Lens (Part I)

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:

\frac{n_1}{d_s} + \frac{n_2}{d_i} = \frac{n_2 - n_1}{R},

which, in turn, depends on the curvature radius and refractive indices that are shown in the following figure.

The general situation for refraction by a spherical boundary.

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:

d_i = \frac{n_2 d_s R}{(n_2-n_1)d_s - n_1R}.

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

d_s = -2 ,

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:

d_i = \frac{(1.5)(10)(0.2)}{(1.5-1.0)(10) - (1.0)(0.2)} = 0.625meters.

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.

A negative source distance corresponds to a group of rays that are converging toward the source.

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.

A spherical surface with a negative curvature.

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.
The imaging equation for a spherical boundary and the necessary conventions can be tedious. But these important concepts are all we need to determine how a spherical lens — or a collection of several spherical lenses — will refract rays and form images.

© 2011 Timothy Schulz

The Digital Camera
Previous: Refraction by a Spherical Surface
Next: The Spherical Lens (Part II)

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