The Digital Camera: Refraction by a Spherical Surface

Having learned about refraction in our recent posts, we can now begin to appreciate the imaging capability of a lens by applying the principle of Snell’s law to understand the way in which a glass surface can bend light rays to form an image. To see how this works, we first think about a cylindrical piece of glass with one end polished into a spherical shape, and then envision the consequence of pointing the spherical end of the cylinder toward a small source of light. Each ray from the source will bend when it encounters the spherical surface, and Snell’s law can provide a mathematical description for the precise amount of bending. In a qualitative sense, though, something very special happens when the cylinder is long enough and the surface of the glass has a spherical shape: the rays of light rejoin (or nearly so) inside the glass to form an image of the source.

Image formation in glass with a spherical lens

If the refractive index for air is 1 and the refractive index for the glass is n, then Snell’s law predicts the following relationship between the distance to the source and the distance to the image:

\frac{1}{d_s} + \frac{n}{d_i} = \frac{n-1}{R},

where the distance to the source ds is measured to the left of the glass surface, the distance to the image di is measured to the right of the glass surface, and R is the radius of curvature for the spherical surface. If R is equal to 1 meter, for instance, then the surface of the lens will have the same curvature as the surface of a ball with a 1 meter radius or a 2 meter diameter. The curvature is more pronounced for small positive values of R and less pronounced for large values. A negative value for R corresponds to a surface that curves outward, away from the glass cylinder. This mathematical relationship between the source and image distances is actually an approximation, but the approximation is good enough to guide our understanding of the basic principles of image formation with lenses. (Leave a comment if you’d like to see a subsequent post that shows how to derive this relationship from Snell’s law.)

An important thing happens when the source is sufficiently far from the surface of the glass. When this happens ds is very large and 1/ds is very small; consequently, the distance to the image is

d_i = \frac{n}{n-1}R.

This distance behind the glass surface where an image is formed for a very distance object is called the image focal length:

f_i = \frac{n}{n-1}R.

The image focal length. When the source is very far from the lens the image is formed at the focal length.

A small image focal length requires the surface curvature to be more pronounced; a large image focal length requires the curvature to be less pronounced.

Another interesting thing happens when the distance to the source is exactly R/(n-1). To satisfy the relationship between the source and image distance for this situation the term n/dneeds to be zero. For this to happen the distance to the image has to be infinite, which, in turn, causes the rays to be parallel within the glass.

The source focal length. When the source distance is equal to the source focal length the image is an infinite distance behind the lens and the refracted rays are parallel.

If we move the source closer than the source focal length for a spherical surface with a positive radius of curvature, the image will not be inside the glass; instead, the image distance in the equation

\frac{1}{d_s} + \frac{n}{d_i} = \frac{n-1}{R},

will be negative, and the rays will diverge within the glass as though they originated from a virtual image on the source side of the glass.

The virtual image that is formed when the source is closer than the source focal length.

The graphical depiction of the virtual image in the preceding figure exhibits an important characteristic that often appears in images that are formed by lenses with spherical surfaces. Because of a phenomenon called spherical aberration, all of the rays — especially those far from the center of the lens — do not come together in the same location. Lenses with spherical surfaces are especially susceptible to this when the image focal length is small compared to the diameter of the lens.

Spherical aberration.

The ratio of image focal length to lens diameter is referred to as the f-number for the lens, and is usually denoted with the notation f/#. Small f-numbers cause a problem because it is difficult to shape the surface of a lens so that it redirects all of the rays to the same image point when the lens diameter is large compared with the image focal length. The effects of spherical aberration can be reduced by decreasing the size of the lens aperture, and one way that cameras do this is by using an aperture stop to prevent the rays from the outer part of the lens from contributing to the image.

Spherical aberration is reduced by using an aperture stop.

The hypothetical glass cylinder we’ve used in these examples is useful for understanding the process by which a spherical surface bends light to form images, but it is not practical for making a camera. We can not, after all, place a photographic detector inside the glass to capture the image. But if we use a narrow piece of glass with an appropriate curvature on both surfaces, the rays will form an image after exiting from glass to air and we can use a detector behind the lens to capture the image. We’ll explore this important idea in our next post on The Digital Camera.

© 2011 Timothy Schulz

The Digital Camera
Previous: The Pinhole Camera
Next:  The Spherical Lens (Part I)

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