# image formation by lenses

Why is the image formed by an object as described in the picture can be derived only from these three ray of lights?Every point on the object has multiple ray of lights going in every direction, how do we know that the other ray of lights(other then these three) don't converge at some other points, or alternatively create a big splotch of light the would hide the image?

• That is not a given. With an irregular piece of glass exactly that happens. This is only true for perfect lenses. ('Perfect' being defined by exactly that metric - every ray from one object point converging on one image point, moslty with the additional requirement that all image points for a flat vertical object are also flat and vertical – bukwyrm Nov 20 '18 at 10:58
• @bukwyrm That should be an answer – Aaron Stevens Nov 20 '18 at 11:00
• @bukwyrm So how can it be proven that all the other light rays(Other than the three shown in the picture) converge to the same point on the image? – Venuce Nov 20 '18 at 11:02
• @Venuce You are somewhat mistaken. Not all light rays emitted from the object meet at a single point. Rather, all light rays emitted from a single point on the object will meet at the same single point in the image. – Aaron Stevens Nov 20 '18 at 11:05
• With only the information from the image above, it cannot be 'proven'. If this is a homework, can you include all pertinent information, and the exact question? – bukwyrm Nov 20 '18 at 11:05

With an irregular piece of glass it happens exactly as you describe. Three rays may converge onto one point, but others may go anywhere else. The convergence onto one point for all rays is only true for perfect lenses. ('Perfect' being defined by exactly that metric - every ray from one object point converging on one image point, mostly with the additional requirement that all image points for a flat vertical object are also flat and vertical.

What makes these three rays of light unique is that they are very easy to draw.

Why is the image formed by an object as described in the picture can be derived only from these three ray of lights?

Actually two rays are enough. But no real lens (or system of lenses) will behave that way. I wonder if your teacher or textbook didn't tell you that this is an approximation, called after Gauss, good under special conditions. There is a prerequisite: that the optical system has an axis of symmetry (the dashed line in your figure). This is known as a centred optical system. In order Gauss' approximation holds good it is required

1. That source and all rays are near optical axis.
2. That all rays are at small angles wrt optical axis.

Under these hypotheses it can be shown that all rays starting from a point (the source) converge to another point (the image). It may also happen that rays exit the system diverging; in this case they will all meet if prolonged backwards (virtual image).

The construction you are showing uses some other properties of Gauss' approximation:

• All rays entering the lens parallel to optical axis converge to a point, named rear focus.
• There is a point (front focus) such that all rays passing through it before entering the lens exit parallel to optical axis.
• For a thin lens rays passing through its center are not deflected. For a general Gaussian system this property is expressed in a more complicated way, involving the nodal points.

All this given, for any point source you can find its image (for a thin lens) using two rays among the following three

• a ray starting parallel to optical axis
• a ray passing through front focus
• a ray passing through lens' centre.