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The longitudinal lens formula and sign conventions

The simple lens formula for thin lenses is included in some advanced level physics courses, though it is rarely used by contemporary optical designers. It provides a source of examination questions and a wrangle about sign conventions. Conventionally, u is the distance from lens to object, v is the distance from lens to image, and f is the focal length of the lens. 
 
If treated lightly, the 'formula' can be put to good use as an encouragement to students to practise placing virtual images. Nearly everyone uses an optical instrument sometimes. In most optical instruments (telescope, microscope, magnifying glass, spectrometer) the observer looks at a virtual image. 
 
The important thing in the argument about conventions is to choose one convention and stick to it. Advantages and disadvantages of the two common conventions are discussed below. 
 
The Cartesian convention 
 
The Cartesian convention emphasizes the point of view which looks at a lens as changing the curvature of wave fronts going through it. One reason to do this is that it makes good sense of the reciprocal quantities in: 
 
1/v=1/u+1/f 
 
Thinking of a lens as adding curvature, the natural formulation is: 
 
curvature after = curvature before + curvature added 
 
This convention also expresses the fact that the effective power of two thin lenses in contact is found by adding their powers. 
 
In both conventions, diverging lenses have negative powers and converging lenses have positive powers. In the Cartesian system, it is advisable not to restrict the unit dioptre to lens powers alone, but extend it to 1/v and 1/u. Indeed, this must be correct if the equation is to have consistent units. Opticians always measure lens powers in dioptres, and so the unit itself is more than respectable. 
 
It is better to express the lens equation in the form above rather than as: 
 
1/v - 1/u = 1/f 
 
where it may be less easy to recall which term is subtracted (though reading this as 'change in curvature = curvature provided by the lens' is quite natural). 
 
In the Cartesian convention, distances to the right are positive and distances to the left are negative, just the same as for cartesian graphs. For a converging lens forming a real image, u is negative and v is positive. 
 
The 'real is positive' convention 
 
1/u + 1/v = 1/f 
 
In the 'real is positive', the symbols are taken at face value and the fact that these reciprocals are related receives no attention. The sign is taken as positive for a real object or image distance, and negative for a virtual object or image distance. 
 
Advantages: The merit of this convention is that it makes the lens equation simple and easy to remember. Double negatives, which can confuse students, do not arise in as many cases as with the Cartesian system. 
 
Disadvantages: The 'real is positive' sign convention is not used at all in professional work in optics, nor in ray tracing software which is readily available, and it obscures what is going on underneath. Some students get confused between the sign of (-1/u) and the sign of u itself. 

Thanks to Dave Martindale for pointing out an error on this page, now corrected. Editor