Sign Conventions For Lenses

With thin lenses, we generally come across three distances parallel to and two distances perpendicular to the principal axis. Each of these distances we define a particular quantity and a symbol is assigned to it. These quantities are given in table.

Distance	Quantity	Symbol
From the optical centre to the object	object-distance	$u$
From the optical centre to the image	image-distance	$v$
From the optical centre to second principal focus	focal length	$f$
Height of the object	object-height	$h_{o}$
Height of the image	image-height	$h_{i}$

The above definitions of u and v are for a point object placed on the principal axis. If an object is placed away from the principal axis, u corresponds to the distance from the optical centre to the foot of the perpendicular drawn from the object to the principal axis. Similarly, v corresponds to the distance from the optical centre to the foot of the perpendicular drawn from the image to the principal axis. The quantities $h_{o}$ and $h_{i}$ are defined for a linear, extended object placed perpendicular to the principal axis.

For finding a particular distance when some other are given, we need to assign these quantities $(u,v,f,h_{o},h_{i})$ positive or negative signs according to a fixed set of rules, or a convention. Several conventions for the signs of these quantities are in use. We shall follow the Cartesian coordinates convention.

1 Transverse distances are measured from the optical centre.If the object or image is present above the principal axis than it is taken as positive that is positive y axis while those are below taken as negative that is negative y axis.

2 Longitudinal distances are also measured from the optical centre and taken as positive if they are in the direction of propagation of incident light that is positive x axis and taken as negative if they are in opposite direction that is negative x-axis..

Coordinate convention

For u, v, f

u,v and f are measured from the optical centre.

If a distance is measured along the incident rays, the corresponding quantity is positive. If a distance is measured opposite to the incident rays, the corresponding quantity is negative.

Coordinate Convention

for $h_{o},h_{i}$

The object-height $h_{o}$ is always taken as positive. If the image is formed on the same side of the principal axis (erect image), the image-height $h_{i}$ is also positive. If the image is formed on the opposite side (inverted image), $h_{i}$ is negative.

Note that in this convention, f is negative for a concave lens and positive for a convex lens. Also, heights above the principal axis are positive and those below it are negative.