## Monday, February 24, 2020

### The Virtual Image: Keeping It Real

When recently asked to explain the operation of a magnifying glass, one of us (MHB) revisited an old conundrum from secondary school: the virtual image. When a viewed object is within the focal length of the lens, an image is formed that is not inverted relative to the viewed object. Far more mystifying, it is on the same side of the lens as the light source, not on the side to which actual light is conveyed. Yet we see virtual images all the time.

You can see the traditional picture of a virtual-image situation in the part of Fig. 1 that includes the “eyepiece” lens and everything to the left of it.  The “object” is a short, downward-pointing black arrow. The “virtual image” is the longer, dashed black arrow at the far left.  The base of the object and of the image lie on the axis of the lens. The point of the arrow in object and image are connected by two rays (dashed black lines delimiting a pink area). One of these rays passes through the center of the lens, and in the ideal (thin-lens) case this ray will be undeflected by the lens.  The other ray starts out parallel to the lens axis, and is bent by the lens so it passes through the focal point (not shown) that is to the right of the lens.  (A collection of such rays from the Sun could concentrate enough to cause a fire.)  If the object had been to the left of the left-hand focal point of the lens, this bent axial ray would have intersected the undeflected ray so as to form a real image: inverted relative to the object, on the same side of the lens as the light propagates, and generally understandable. But when, as in Fig. 1, the object lies to the right of the left-hand focal point, the two rays diverge to the right of the lens, and instead, we are told, we have to backtrack both rays until they intersect on the left-hand side, and this  intersection is part of the virtual image.

How can an image form that is coincident neither with the eye nor with the light that should be generating that image? Virtual or not, this deserves an explanation.

As optical engineers will tell you (if pressed), the virtual image is a way of expressing a geometric relationship without involving the actual eye.  As soon as you include the eye in the explanation (see Fig. 1) the paradox is resolved.  The rays that diverge as they pass to the right of the eyepiece are brought together in focus by the lens of the eye.  Where these rays meet, one finds a real image on the retina. This image is shown in Fig. 1 as a white arrow. It is a real image, being where the light ends up, where vision takes over, and in an inverted configuration that the visual system interprets correctly. (That is a subject for another essay.)

The important thing to remember about the virtual image is that it is a shorthand to replace a fairly intricate relationship of the rays that form the real image if the explanation includes the eye. That relationship is only partially apparent in Fig. 1.  One gets the false impression that the bent axial ray from the eyepiece lens becomes the undeflected ray of the eye’s lens, and vice versa.  Actually, the bent axial ray from the eyepiece lens passes through the right-hand focal point of the eyepiece, which only accidentally happens to be near the center of the eye lens.

A clearer picture of the eye’s focusing of the diverging rays is shown in Fig. 2, a ray-tracing simulation.  Here the optical elements are more stylized than in Fig. 1: the two lenses are red-tipped line segments, the object consists of a pair of radiating green dots, and rays from the dots proceed through both lenses and meet at the right-hand convergence points that depict the real image on the retina.  The virtual image is too far to the left to be captured by the figure.  We think that both Figs. 1 and 2 are needed to show how the eye acts in a virtual-image situation.

In short, experts know the virtual image and how to “keep it real.” Now you know too.

Michael H. Brill and Nilesh Dhote
Datacolor

Fig. 1. Drastically abbreviated depiction of a virtual image and the real counterpart that emerges when the eye is included. Please ignore the yellow region. The figure in context can be found at https://micro.magnet.fsu.edu/primer/anatomy/components.html.

Fig. 2.  Simulation of formation of a real image using two lenses. The divergence of the rays through the left-hand (eyepiece) lens is evident from the rays that proceed outside the lens aperture of the eye.  (See https://ricktu288.github.io/ray-optics/simulator/.)