Sign Conventions Used for Mirrors,
Refracting Surfaces, and Lenses

We study optical surfaces, i.e., mirrors, curved refracting surfaces, and lenses to understand how and why they produce images of objects; therefore, we must agree on how to describe the locations, and sizes, of the objects and the locations and sizes of the images that they create. In addition, we need parameters to describe the size and/or shape of the optical surface, and an agreement as to which kind it is. To simplify this part we will only consider surfaces which are parts of spheres, which can then be described by the radius of that sphere. Lastly, we will need the index of refraction of the material if it allows light to pass through it. We will ask, and answer, these questions for each optical surface separately. Therefore a system containing more than one such surface will be studied one surface at a time, sequentially, following the actual path of a light ray through the system. These notes will try to make the sign conventions used by your textbook as clear as possible, and to also, perhaps, try to explain why the authors have chosen these particular conventions. Remember, of course, that what really happens, i.e., the physics of the situation, is entirely independent of what sign conventions you choose to use; however, the form of the formulas, and the ease of explaining the physics to you or to your friend does indeed depend on the sign conventions. Because of this there are many, many different sets of conventions about positive and negative signs in the field of geometric optics, i.e., when dealing with rays propagating near mirrors and lenses. Each one of these conventions was created by some author/physicist who believed that his/her choices made your understanding the easiest. Considering a single optical surface, we first use it to divide all the space around it into two parts:

• the space where the light ray travels as it is incident on the surface;
  refer to this as Incoming Space;
• the space where the light ray travels as it leaves the optical surface;
  refer to this as Outgoing Space.

Notice that for a mirror these two spaces are identical, while for a transparent material they are on opposite sides of that surface. We will measure distances along a symmetric normal, i.e., a normal line that divides the surface in half, and use the following symbols:

• p to denote the distance of the object from the optical surface, and
• i for the distance of the image from that surface, and
• R for the radius of curvature of the surface, and
• f₁ and f₂ for the distances to the focal points of the surface. i.e., the point at which light rays from infinity will be focused.
• The focal point, F₁, at a distance f₁ from the surface, is the point from which light would exit the surface parallel, commonly described as saying that it will focus at infinity;
• The focal point, F₂, at a distance f₂ from the surface, is the point to which light will be focused if it is incident on the surface parallel, again commonly described as saying that it is coming in from infinity.
• Also notice that the two focal points are

  1.

  at the same physical point, for a mirror;

  2.

  on opposite sides of the surface, and at different distances, for a single refracting surface; and,

  3.

  on opposite sides of a thin lens, but at the same distance.

We then use the following sign conventions for all these sorts of surfaces:

• An object distance, p, is positive if the object lies in the Incoming Space for the surface being studied;
• an image distance, i, is positive if the image, either real or virtual, lies in the Outgoing Space for the surface;
• and the radius, R, is positive if the center of curvature lies in the Outgoing Space for the surface;
• the first focal distance, f₁, is an object point, for light going to infinity, so it satisfies the object sign conventions.
• the second focal point, f₂, is an image point, for light coming from infinity, so it satisfies the image sign conventions.

The three major formulas that use these sign conventions, and help us understand image formation by single surfaces are the following. [Do remember that these are approximations for paraxial rays, i.e., for surfaces that are only small portions of a sphere.]

1.

A spherical mirror [which has $f_1 = f_2\equiv f$]

$$\frac{1}{p} + \frac{1}{i} = \frac{1}{f} = \frac{2}{R}$$

2.

A single refracting surface (between medium 1 and medium 2):

$$\frac{n_1}{p} + \frac{n_2}{i} = \frac{n_2-n_1}{R} = \frac{n_2}{f_2} = \frac{n_1}{f_1}$$

3.

The thin-lens formula (in air) [which has $f_1 = f_2\equiv f$]:

$$\frac{1}{p} + \frac{1}{i} = \frac{1}{f} = (n-1)\left( \frac{1}{R_1} - \frac{1}{R_2}\right)$$

A useful quick Summary for Single Surfaces is the following, where we assume that all object distances are positive and that light rays from that object are incident from air.

• If the focal length is positive, then it creates images at a positive distance if the object is farther than the focal point, and at a negative distance if the object is closer than the focal point.
• If the focal length is negative, then all images created are at a negative distance.