structural:reformatting, structural:scholar_edition, quality:correction, structural:summarization, information:classification
Imaging properties

As mentioned above, a positive or converging lens in air focuses 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 is 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 S and S respectively, for a lens of negligible thickness (thin lens), in air, the distances are related by the thin lens formula:

1 f = 1 S 1 + 1 S 2. {\displaystyle {1 \over f}={1 \over S_{1}}+{1 \over S_{2}}\,.}

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

f 2 = x 1 x 2, {\displaystyle f^{2}=x_{1}x_{2}\,,}

where x 1 = S 1 − f {\displaystyle x_{1}=S_{1}-f} and x 2 = S 2 − f. {\displaystyle x_{2}=S_{2}-f\,.} It can be derived by using a similarity between triangles P P F and L L F and another similarity between triangles L L F and P P F in the right figure. The similarities give the following equations and combining these results gives the Newtonian form of the lens equation.

y 1 x 1 = | y 2 | f y 1 f = | y 2 | x 2 {\displaystyle {\begin{array}{lcr}{\frac {y_{1}}{x_{1}}}={\frac {\left\vert y_{2}\right\vert }{f}}\\{\frac {y_{1}}{f}}={\frac {\left\vert y_{2}\right\vert }{x_{2}}}\end{array}}}

The above equations also hold for a thick lens if S 1 {\textstyle S_{1}}, S 2 {\textstyle S_{2}}, and f {\textstyle f} are with respect to the principal planes of the lens (f {\textstyle f} is the effective focal length in this case). If an object is placed at a distance S > f from a positive lens of focal length f, we will find an image at a distance S according to this formula. If a screen is placed at a distance S on the opposite side of the lens, an image is formed on it. This sort of image, which can be projected onto a screen or image sensor, is known as a real image. This is the principle of the camera, and also of the human eye, in which the retina serves as the image sensor.

The focusing adjustment of a camera adjusts S, as using an image distance different from that required by this formula produces a defocused (fuzzy) image for an object at a distance of S from the camera. Put another way, modifying S causes objects at a different S to come into perfect focus.

In some cases, S is negative, indicating that the image is formed on the opposite side of the lens from where those rays are being considered. Since the diverging light rays emanating from the lens never come into focus, and those rays are not physically present at the point where they appear to form an image, this is called a virtual image. Unlike real images, a virtual image cannot be projected on a screen, but appears to an observer looking through the lens as if it were a real object at the location of that virtual image. Likewise, it appears to a subsequent lens as if it were an object at that location, so that second lens could again focus that light into a real image, S then being measured from the virtual image location behind the first lens to the second lens. This is exactly what the eye does when looking through a magnifying glass. The magnifying glass creates a (magnified) virtual image behind the magnifying glass, but those rays are then re-imaged by the lens of the eye to create a real image on the retina.

Using a positive lens of focal length f, a virtual image results when S < f, the lens thus being used as a magnifying glass (rather than if S ≫ f as for a camera). Using a negative lens (f < 0) with a real object (S > 0) can only produce a virtual image (S < 0), according to the above formula. It is also possible for the object distance S to be negative, in which case the lens sees a so-called virtual object. This happens when the lens is inserted into a converging beam (being focused by a previous lens) before the location of its real image. In that case even a negative lens can project a real image, as is done by a Barlow lens.

For a thin lens, the distances S and S are measured from the object and image to the position of the lens, as described above. When the thickness of the lens is not much smaller than S and S or there are multiple lens elements (a compound lens), one must instead measure from the object and image to the principal planes of the lens. If distances S or S pass through a medium other than air or vacuum a more complicated analysis is required.