Lens (optics) - Imaging Properties

Imaging Properties

As mentioned above, a positive or converging lens in air will focus a collimated beam travelling along the lens axis to a spot (known as the focal point) at a distance f from the lens. Conversely, a point source of light placed at the focal point will be converted into a collimated beam by the lens. These two cases are examples of image formation in lenses. In the former case, an object at an infinite distance (as represented by a collimated beam of waves) is focused to an image at the focal point of the lens. In the latter, an object at the focal length distance from the lens is imaged at infinity. The plane perpendicular to the lens axis situated at a distance f from the lens is called the focal plane.

If the distances from the object to the lens and from the lens to the image are S₁ and S₂ respectively, for a lens of negligible thickness, in air, the distances are related by the thin lens formula

.

This can also be put into the "Newtonian" form:

where and .

What this means is that, if an object is placed at a distance S₁ along the axis in front of a positive lens of focal length f, a screen placed at a distance S₂ behind the lens will have a sharp image of the object projected onto it, as long as S₁ > f (if the lens-to-screen distance S₂ is varied slightly, the image will become less sharp). This is the principle behind photography and the human eye. The image in this case is known as a real image.

Note that if S₁ < f, S₂ becomes negative, the image is apparently positioned on the same side of the lens as the object. Although this kind of image, known as a virtual image, cannot be projected on a screen, an observer looking through the lens will see the image in its apparent calculated position. A magnifying glass creates this kind of image.

The magnification of the lens is given by:

,

where M is the magnification factor; if |M|>1, the image is larger than the object. Notice the sign convention here shows that, if M is negative, as it is for real images, the image is upside-down with respect to the object. For virtual images, M is positive and the image is upright.

In the special case that S₁ = ∞, then S₂ = f and M = −f / ∞ = 0. This corresponds to a collimated beam being focused to a single spot at the focal point. The size of the image in this case is not actually zero, since diffraction effects place a lower limit on the size of the image (see Rayleigh criterion).

The formulas above may also be used for negative (diverging) lens by using a negative focal length (f), but for these lenses only virtual images can be formed.

For the case of lenses that are not thin, or for more complicated multi-lens optical systems, the same formulas can be used, but S₁ and S₂ are interpreted differently. If the system is in air or vacuum, S₁ and S₂ are measured from the front and rear principal planes of the system, respectively. Imaging in media with an index of refraction greater than 1 is more complicated, and is beyond the scope of this article.