Do convex lenses make light from different points converge to different points?

Question

I'm starting to study camera and lenses. Most of the sources online introduce you to the concept of convex lenses, with pictures similiar to this:

This picture just show you that parallel rays (of same wavelength) converge to a single point. That doesn't seem to help anyone to understand how an image is formed.
I'm just trying to understand if convex lenses, apart form making parallel rays converge to a point of focus, make light rays departing from a single point converge to its corresponding image.

Answer 1

That fact that lenses focus images of objects follows from the basic diagram you already show:

Think of this as the definition of what a lens does. It also happens to show it focusing the image of something at infinity to a single point.

Now imagine you have a object at some finite distance and position relative to the lens. You can use the few special cases of rays you already know about to find where the lens will focus the image of that object:

The square at left is the object. We already know from the previous diagram what will happen to two of the rays from that object in this cross-sectional view. The ray going parallel to the center line will be bent to pass thru the focal point at right. The ray thru the left focal point will be bent to go parallel to the center line. Where those two rays meet at right is where the image of the object will be focused.

The math for calculating where the other rays from the object will go after being bent by the lens gets more complicated, but they converge at the same point for a ideal infinitely thin lens.

Of course real lenses aren't infinitely thin, so there are various approximations and tradeoffs in real lens design. The further the focused image gets from the center line, the less this ideal approximation holds. This is why real lenses have a maximum sensor size they are specified to work with. It is also why characteristics generally get worse at the edges of pictures.

Note that the rays getting bent in the center of the lens is a convenient simplification. They actually get bent at the two interfaces. When taken from far enough away and the lens isn't too fat, the bent at the center simplification applies well enough.

Rays are actually bent at the air/glass interface due to the different index of refraction of the two materials. Since the index of refraction changes as a function of wavelength, the focal points of a lens are effectively in different places for different colors. Of course we choose materials for lenses that have relatively flat index of refraction across the visible light spectrum, but there is no perfect material. One of the reasons for multi-element lenses is to use different materials with different refraction indexes to cancel out the variations.

The simplified view of a lens is easy to understand and requires only high school geometry. Real lenses get very complicated.

Answer 2

The first diagram is an example of parallel light rays all emminnating from the same point source of light at a distance of infinity. That is, the light rays are collimated. Additionally, the point source of light is directly on the line of the lens' optical axis.

This type of diagram does not show that a convex lens focuses all light to a single point, it shows that a convex lens focuses all light coming from the same direction to a single point. Light coming from a different direction will also be focused to a single point, but the point at which it is focused will be a different point from the light coming from other directions. If all of the light is collimated (parallel to other rays coming from the same point), it will all be focused at the same distance behind the lens, but the light from each different point source will be focused on different points that are all the same distance behind the lens.

If light from another, discrete point source at infinity not on the center of the lens' optical axis strikes the lens it would also converge at the same distance behind the lens, but it would not converge at the same point. In the following diagram the red rays are light from a star on the lens' optical axis. The green rays are light from a different star. The green rays are also parallel to each other when they strike the lens, but they are at an angle to the lens' optical axis. Thus they converge behind the lens at a point not on the lens' optical axis.

Collimated light is defined as light rays from a point source that all strike the front of a lens parallel to other light rays from the same point source. When collimated light strikes a lens, the parallel rays from that collimated light will converge to a point at a distance behind the lens equal to the focal length of the lens.

Light rays from a point source that are not collimated strike the lens at varying angles. This causes them to produce a blur circle, rather than a point, at a distance behind the lens equal to its focal length. If the differences between the angles of light rays from a point source are small enough then the blur circle will be so small that the imaging system can not distinguish the blur circle from a singular point. If light from a point source strikes the lens at varying angles too small to be discernable from truly collimated light, then the distance of that point source of light can be said to be at infinity.

Your second diagram is different. It shows multiple non-collimated light rays coming from each of various point sources and striking the lens at various angles.

The tree is closer to the lens than infinity. These light rays, even the ones that are all coming from the same point source, are not collimated and strike the lens at varying angles. Since they are not collimated, they will not all converge to a single point at a distance of the lens' focal length behind the lens. Rather, they will be projected by the lens to various different points behind the image plane when the image plane is at a distance behind the lens equal to the lens' focal length (or would if the image plane does not block them from passing through it).

By changing the distance between the lens and the image plane at varying distances greater than the lens' focal length, we can control the specific distance at which a closer-than-infinity point source's rays converge on the image plane. The distance at which rays coming from the same point source converge at a point on the image plane is said to be our focus distance or, in the nomenclature of creative photography, our subject distance.

Rays coming from point sources closer than our focus distance converge behind the image plane. Rays coming from further than our focus distance converge in front of the image plane. At the image plane these front and back focused rays create a blur circle. The size of the blur circle is determined by how far in front of or behind the focus distance the point source is and how wide the aperture of our lens is.

• The further the point source is from the focus distance, the larger (and blurrier) the circle will be. Think of the light rays as a cone. The closer the point of the cone is to the image plane, the smaller the diameter of the cone will be where it intersects the image plane. The further from the image plane the point of the cone is, the larger the cone will be where it intersects the image plane.
• The wider the lens' aperture, the larger the blur circle will be. This is because as the aperture is made smaller, the rays striking the front of the lens at the greatest angles are prevented from passing through the lens. This reduces the angle of the point of our cone of light. Compared to the cone allowed to enter by the larger aperture, we need to move further from the point of the cone for the cone to be the same diameter with the smaller aperture.

Answer 3

A useful mathematical model is the thin lens equation, which states that the for a lens focal length \$f\$, lens-to-subject distance \$s\$, and lens-to-image distance (distance to film/sensor plane) \$i\$, they are related by

$$ \frac{1}{f} = \frac{1}{s} + \frac{1}{i} $$

In a strictly mathematical sense^*, the focal length of a (simple) lens is actually defined by this equation. If you think of parallel rays as coming from a light source an infinite distance away, you can choose to set the object distance to infinity (\$s = \infty\$). Then the equation simplifies to \$1/f = 1/i\$, or even simpler, \$f = i\$.

As you decrease the subject distance \$s\$ from infinity to more terrestrial distances, the lens must move slightly further away from the film/sensor — that is, \$i\$ increases. When \$s\$ becomes fairly close to the lens, within a few multiples of \$f\$ (but never equal to or less than \$f\$), then the image distance \$i\$ increases quickly.

Thus, with the thin lens equation, you can see that rays from any source (\$f < s < \infty\$) will converge to a specific image distance, the focus distance, the distance the lens must be placed in front of the film/sensor plane to bring those specific light rays into focus in the image.

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* The mathematical model doesn't actually define the focal length of a particular lens. A physical lens's focal length is determined by its shape/geometry (i.e., how much curvature both lens faces have), and by the optical properties of its material (i.e., the refractive index of the lens material). For more information, see the section regarding the lensmaker's equation at the Lens (optics) article from Wikipedia.

Answer 4

Envision that the vista you are about to image has been dotted everyplace using a wax pencil. We are taking about countless image points. These either reflect light or they are themselves illuminates. Now draw imaginary lines from each to the center of the camera lens. Don’t stop there, continue your ray trace, rays from each point will hit the top, bottom, left, right, all over the lens. Now trace each ray as they pass through the glass. You will discover that each subject point traces out a cone of light. I said each. Thus your ray trace drawing is a hodgepodge of ray traces so dense that we can’t make much sense of it. To simplify, most ray trace teaching aids just show one image point and one cone of light. The key point is, every countless image point transverses the lens and we can trace out a cone of image forming rays from each.

In the old days, opticians picked a few such points and made ray traces so they could fine tune the lens they were designing. With side rule and trig tables it took months of hard work. Today, computer based optical design software does the job in a split second.

Look at this ray trace, it shows only two cones of light from two separated image points.