9.2.1. Sign convention

To derive the relevant formulae for reflection by spherical mirrors and refraction by spherical lenses, we must first adopt a sign convention for measuring distances. In this book, we shall follow the Cartesian sign convention.

The sign convention is a standard way of measuring distances in optical systems. It helps in consistently interpreting and calculating various physical quantities related to lenses and mirrors. For this context, we will specifically use the Cartesian sign convention.

Think of measuring distances like navigating a city grid. If you start at a central point and move north, those measurements are positive. If you move south, they become negative. Similarly, the sign convention helps in determining where measurements start and how they progress in optical systems.

According to this convention, all distances are measured from the pole of the mirror or the optical centre of the lens. The distances measured in the same direction as the incident light are taken as positive and those measured in the direction opposite to the direction of incident light are taken as negative.

In optics, the pole of the mirror or the lens's optical center acts like our origin point. Measurements taken in the direction that light is coming from are positive. Conversely, measurements taken away from the light's direction are considered negative. For example, if light travels to the right towards a mirror, distances to the right are positive; those to the left are negative.

Imagine you are on a skateboard going downhill. As you roll down in the direction you're facing, it's positive progress. If you were to roll back uphill against the direction you're facing, it would be like reversing your progress, hence a negative measure.

The heights measured upwards with respect to x-axis and normal to the principal axis (x-axis) of the mirror/lens are taken as positive. The heights measured downwards are taken as negative.

In this sign convention, we also have a way of measuring heights. If you imagine the optical axis (the line that goes through the lens/mirror), any height measured above this line is considered positive, while any height measured below this line is negative. This is essential as it helps to define the size and position of the images created by the optical systems.

Think of a graph where everything above the line is considered good news (positive) and everything below is bad news (negative). In optics, heights above the axis are where the image forms positively, while images that appear below are negative images, just like a poor report card.

With a common accepted convention, it turns out that a single formula for spherical mirrors and a single formula for spherical lenses can handle all different cases.

Having a standardized sign convention allows us to derive universal formulas that apply to both mirrors and lenses. This means we can use the same equations regardless of whether we're working with a concave mirror or a convex lens, simplifying our calculations and understanding.

Think of cooking recipes that have common ingredients. Whether you're making pasta or pizza, many recipes use the same base. Similarly, with a unified formula, you can handle various optical problems without needing separate equations for each scenario.

--