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IMAGE FORMATION Part 1

 

In the previous tutorial, I was explaining, that light is usually changing its direction of propagation if and when it comes to the borderline of a medium with a different refractive index to the one it is traveling in. Snellius’ law was explained and this helps us now to understand how image formation works.

We know that in photographic devices, images are usually formed by means of lenses, lens systems or mirrors and mirror/lens combinations. And just to make it clear, in photography, it is common, that the imaging rays come from a medium of lower refractive index (air, water) into a medium of higher refractive index (glass).

 

The classic definition of “lens” in the discipline of optics is ….it is an optical system, consisting of at least two refracting (changing the direction of light) interfaces where at least one of them is curved (curved can be spherical but also aspherical or even any other non-linear curvature, like parabolic, elliptic etc.

 

A lens is an optical system, consisting of at least two refracting interfaces where at least one of them is curved

 

As you can see below and should remember, light is refracted according to the laws we have discussed in the previous tutorial but as the surface elements approach smaller and smaller dimensions, the quasi-plano surface elements form a spherical surface – the surface of, what we can call - a lens.

 

A lens is therefore an optical element, which has the ability to alter the previously straight path of a ray of light depending on its position and angle of incidence to its surface. With other words – if light is hitting the lens surface exactly at the rotational center of the lens, on its optical axis, and perpendicular to its surface, nothing is going to happen – the ray of light is passing the lens without any change of direction. But if the same ray of light is either hitting the surface of the lens under any other angle than 90degrees (perpendicular) or at any other position outside of the optical axis, then it will be refracted according to the previously discussed laws.

 

This very brief explanation does certainly not cover image formation deeply enough. It also leaves the question open how an image is formed by the mirrors in super-tele lenses or if and how images are coming into existence when light is passing other types of lenses. It also does not reflect the fact that lenses are never ideally thin but always have a certain thickness – and this thickness in the middle (or at the outer border) must be taken into consideration.

 

To enable people in optics meaning the same when talking to each other, a certain terminology has been established over the past several hundred of years.

 

Lenses can have various shapes and are classed usually into categories according to their ability to diffract light – “convex” and “concave” More detailed, we talk about plano-convex, plano-concave, bi-convex, bi-concave, convex-concave and concave-convex

 

 

A plano plate, a glass plate with two plano parallel surfaces (like a window) can be seen as a lens with infinite curvature (interesting, right?).

 

Picture 1: Canon FD 1:1,4/50mm

 

The picture on the left, a highly corrected camera lens is a good example to explain the various definitions:

 

A convex surface is a spherical surface with an outside oriented bending of its curvature (seen from the lower refractive index towards the higher one) – left surface 1 on the left picture

 

A concave surface is a spherical surface with an inside oriented bending of its curvature (seen from the lower refractive index towards the higher one) - right surface of lens 3 and left surface of lens 4 on the picture

A lens, which consists only of one single element, which has two refracting surfaces, is called a simple lens – lenses 1, 2, 3, 6 and 7

A lens (system) which consists of two single elements, cemented together or at least mounted in such way that they represent one element, is called a doublet, if three lenses are cemented together, they are called a triplet – lenses 4 and 5 in the picture represent a doublet.

Without seeing the lens design, it could possible. That lenses 2 and 3 are also a doublet, but as they are not cemented together, they would be called air-spaced doublet.

A convex lens (also called converging or positive) is a lens, which is thicker at its center and thinner at its outside – look at the lenses 1, 2, 5, 6 and 7!

There are two forms of convex lenses, the plano-convex (lens 5) and the bi-convex lens (lens 7), the first one has got one plane surface on one side and a convex surface on the other one, the bi-convex lens has got two convex surfaces. For both forms, the center is thicker than the edges. The lens 5 is plano-convex and the lens 7 is bi-convex.

 

A concave lens (also called diverging or negative) is a lens, which is thinner at its center than at its outside – see lenses 3 and 4.

There are two forms of concave lenses, the plano-concave (lens 4) and the bi-concave lens (see the two arrows in the image below), the first one has got one plane surface on one side and a concave surface on the other one, the bi-concave lens has got two concave surfaces. For both forms, the center is thinner than the edges.

 

Picture 2:

 

If the lens’ surfaces are a combination of concave and convex, there are two possible combinations - Concave-convex (lens 2, 6 and 1 in the picture 1 before) and convex-concave (lens 3 in the same picture 1) lenses are both having one concave and one convex surface. The concave-convex lens is also called positive meniscus, the convex-concave lens negative meniscus.

Concave-convex lenses are thicker in the middle than on their outside whereas convex-concave lenses are thinner in the middle than on their outside.

 

Concave-convex lens is also called positive meniscus Convex-concave lens negative meniscus Concave-convex lenses are thicker in the middle than on their outside Convex-concave lenses are thinner in the middle than on their outside

 

As this chapter is called image formation, we need now to talk about the abilities of lenses to form an image - please read the additional explanation as well.

 

As mentioned before, all lenses until now have not been considered “thick”, and for the following basic explanations about image formation, we stick to this simplification.

 

When light enters a lens, what happens, depends on the shape of the lens – this is a short summary. Now the details – look at the following pictures. Bundles of light pass through a positive, bi-convex and a negative, bi-concave lens. The positive lens produces a location of least diameter, called focal point (very precisely called principal point of focus). A negative lens does not produce such a “real” point.

 

Principal Point of Focus = Focal point It is the point to which incident parallel light rays converge or from which they diverge after being acted upon by a lens (or a mirror). A lens has always one focal point on each side of the lens (whereas a mirror has but one)

 

 

Picture 3: Bundles of light pass through a positive and a negative lens:

 

 

As one can see, the focal point is a location, where light rays come together after passing a lens. There are two kinds of focal points – real ones, like the one on the left side of the picture, and those, which cannot be seen directly, but calculated and drawn – the virtual focal point. One can find the later ones with negative lenses and under certain circumstances also with positive lenses.

 

The focal point(s) are one of the important points and locations in optics – but there are others. See the following picture:

 

Picture 4: Image construction with spherical surfaces

This picture shows the construction of the focal points F and F’ when light is passing spherical surfaces.

Picture part a – light is coming from an object y1 and the resulting image is in y2. The focal points F and F’. What makes them special – if light passes through the focal point, it will be parallel to the optical axis after it passed the spherical surface – see drawing a and d!

This means that if an object is placed in the focal point (plane), its image will be in the infinity. Vice-versa of course, if an object is in infinity, then its image will be in the focal point (plane).

 

Why is it written focal point (plane) – well, a single lens does have a focal point (more or less, we will talk about it later) but the plane in the focus is not a plane but a spherical surface as well. A reason for this is that the lengths of all light passes must be the same from object to image – and as the amount of glass passed (and the distances in air) for oblique rays are different and depending on the angle of incidence, the resulting sum of focal points is not a plane but a spherical surface.

 

Looking at picture 4, one can easily see that this very oblique ray does have a longer path in glass than one, which enters parallel to the optical axis.

 

 

 

Picture 5: Oblique ray passing a thick lens

 

A few more important points can be explained on this picture as well: M1 and M2 are the centers of curvature of the two surfaces, which have the radii r1, and r2, O is the optical center of the lens. This optical center is not the geometrical center of the lens unless the lens is bi-convex/concave with identical spherical curvatures on both surfaces. The optical center O divides the thickness of the lens according to the radii of curvature of the two surfaces. Therefore an oblique ray does not pass the geometrical center of the lens based on its thickness but it passed the optical center of the lens based on the two radii of the surfaces.

Two more points are of interest – they are marked K1 and K2 and are called Nodal points (the K is from the German word, Knoten, which is node in English). The nodal points are determined by letting the incoming and exiting ray cross the optical axis – as one can see, the incoming and exiting ray are parallel to each other. The points, where the optical axis intersects with the surface of the lens on each side, are called Vertices (singular=vertex), S1 and S2 and the reader can see that the ray crosses the optical axis at the optical center O.  Looking more precisely at this drawing, it can be seen that the relation OS1:OS2 = r1:r2, which means that the position of the optical center is, as already mentioned, determined by the radii of the two surfaces of the lens.

 

It is now important to understand how an image is formed – not so much experimentally, but to understand the laws of image formation. For this purpose, it is necessary to go back to the drawings in picture 4 – showing the image formation of a simple arrow depending on its position and distance from the lens surface. y1 is the object and y2 its image, M is the center of curvature of the lens’ surface and S is again the vertex.

 

Picture 6 (as pict.4)

 

As the picture demonstrates, there is a close relationship between the distance of the object y1 and its image y2. Experiments several hundred years ago have shown that if the object is in “infinity” – very, very far away, then the image of this object is for a given lens always in the same place (“plane”). That point on the optical axis is called the focal point or focus F. And of course, as all optical paths are reversible, all images of an object placed in the focal point F / focal plane “F” are in infinity. Geometrically, this means that they rays from infinity are shown as parallel lines in reference to the optical axis. Scientists in the past centuries have experimented a lot with placing objects and searching for their images – several interesting special positions have been found – if the object is placed in a distance which is exactly 2xF, the image is as well in 2xF, just on the other side and – as you can see, inverted. If the object is placed inside the focal distance, what happens then? Well, then the lens acts as magnifying glass, delivering a virtual image, which is larger than the object.

All these considerations are valid for positive lens surfaces, if the lens surface has got a negative power, is a negative lens, then the laws are similar but the images are not real but virtual. A real image is an image, which can be seen on a ground-glass, on a piece of paper but a virtual image can be calculated and not been projected on a ground-glass or piece of paper.

Drawings b and c show surfaces which produce virtual images – b is the opposite situation to c as one can determine by checking the refractive indices – light comes from the left, from an object and in b, n1 is smaller than n2, which means that the surface of the optical element has got a negative diffracting power whereas in c the refractive index on the right is smaller, which means that the object is within an environment of higher refraction and therefore the surface, despite the fact that it is of opposite shape to b, has got again a negative diffraction power for the rays leaving the medium of higher refractive index.

 

In reality, all lenses have some thickness and therefore their actual dimension has to be taken into account as well. What happens, once the rays of light are propagating inside the glass – assuming that the optical glass is homogeneous (has no internal differences regarding its refractive index) and completely clear (does not absorb light – neither selectively nor any specific wavelength)? Well, not much happens, light is traveling at a reduced speed – compared to air – and travels without further changes in its direction. That means, that once light has changed its direction when it entered the lens, it does not change it any further until it exits the lens.

 

 

Pict.7: Image formation – Thick negative lens

 

 

The geometrical construction of an image formed by a thick lens shows that there are two special planes with corresponding points at the optical axis, which reflect what was said before. These planes H and H’ are called Principal Planes and the two points therefore are called Principal Points.

 

 

 

 

 

Pict. 8: Image formation – Thick positive lens

 

The location of these two planes H and H’ – sometimes also written as H1 and H2 – is determined by the location of the intersection of those rays which come from infinity with those ones coming from the other side’s focal point – as there are two focal points (object side and image side focus) and since optical paths are reversible, there must be two sets of intersections – image side rays coming from infinity and intersecting with the object side rays from the object side focus and object side infinity rays intersecting with the image side rays from the image side focus. This is very well shown in picture 9 below:

 

 

Pict 9: A think lens with both principal planes

 

In the first picture, the Primary principal plane is shown and in the second one, the Secondary principal plane. For completion ‘ the two intersection points of the optical axis with the surface of the lens are V1 and V2, with V standing for vertex and the two focal lengths are called f.f.l. and b.f.l. - front focal length and back focal length.

As we have found out before, rays, which come from a focal point, are leaving the lens parallel on the other side – these parallel rays indicate that the image (or object) is in infinity. As one can see, the extensions of these rays create an intersection plane – which is in reality not a plane surface – and where this plane intersects with the optical axis, the Principal points H1 and H2 are located and (simplified) the planes perpendicular through these points are the before mentioned Principal planes. It should be noted as well, that taking the change of size of an object image into consideration, one can conclude from the before said facts that the space between H1 and H2 as well as H1 and H2 themselves do not cause any change of the size of an image – with other words, the magnification factor between H1 and H2 is 1. This leads to the definition of H1 and H2:

 

Principal Plane A plane of unit magnification. Any ray directed at the first principal plane appears to emerge from the second principal plane at the same height Principal Point The intersection of the principal plane with the optical axis

 

 

The fact that these Principal planes are not really planes, is of importance and it is necessary to remember this fact later for understanding image aberrations and related topics.

In picture 5 it was shown already that those rays, which enter oblique and travel through the optical center, if extended from the entrance point, cross the optical axis at two special points – the Nodal points. This is shown again very nicely in picture 10 – please note that the incoming and the exiting rays are parallel to each other.

 

Pict 10: Nodal points

 

O is again the optical center – in a symmetrical lens, it is identical with the geometrical center of the lens and N1 and N2 are the nodal points. The denomination N1 and N2 is the English one, in the classical German literature, one can find K1 and K2 for the same points.

 

Nodal Points Off all the rays passing through a lens from an off-axis object point there is always one ray whose direction in the object space is equal to that in the image space. The two points at which these two rays appear to intersect the optical axis are called the Nodal Points

 

If the lens is in air – on both sides – or inside the same medium on both sides, the nodal points and the principal points are coincident. If you like, you can think about the reason for this circumstance.

 

These special six points – the two focus points, the two nodal points and the two principal points are called Cardinal points of a lens system.

 

Do the principal points always lay inside of the lens – not at all, if the lens bending is different from a symmetrical one, these points can be even outside of the lens as shown in picture 11 – if the lens is a meniscus, then even both points are located on one side of the lens.

 

Pict.11: Location of the principal points in lenses of various shapes

 

 

The dotted lines show the location of the Principal planes depending on the geometry of the lens –both inside, one inside or both outside of the lens.

 

How far are these planes apart from each other – there exists a thumb rule which says that the Principal planes are approximately 1/3 of the thickness of the lens apart, assuming a more or less standard optical glass and air surrounding the lens.

 

Having explained all this, it is now easy for the reader to understand that if more than one lens is used, the image is formed as a result of the combination of two, three or even more lenses. The rules are always the same; it is just getting more complex. 

 

Pict.12: Image formation – two lenses

Both drawings on the left show two lenses – just in different distance from each other.

In the upper drawing, the lenses are closer than in the lower drawing but the principle of image formation are of course identical. I leave it up to the reader to look at these two sets of lenses and understand why the image in the second case is larger than in the first one. It is necessary to pay attention to the fact that in the first example, the right lens is position that its focus is within the focal length of the left lens, whereas in the second example, the right lens is position further away with its focus well outside of the left lens’ focus.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Pict.13: Image formation by two thick lenses

 

In this example, two thick lenses have been placed apart from each other.

In example (a) the two focal points of the lenses are separated from each other, in example (b), the rear focal point of the left lens is on the same side as the rear focal point of the right lens – you can see that the paths of the rays are totally different. Do you see that the actual geometrical distance of the two lenses is identical? It would be beneficial for your understanding of image formation to at least mentally construct the image in both cases to see why there is such a difference in the formation of an image.

As you also can see, the incoming light is in both cases parallel, meaning the object is in infinity.

 

Towards the end of this very brief introduction into image formation (by lenses), we need to touch also the topic of the orientation of an image formed by a lens. Well, you might have guessed it already – the image is certainly not oriented the same way as the object is. Remember the old time photographers with their large wooden cameras and their heads under a large black cover to look at the groundglass – they looked at an inverted image – upside is down, left is right. Here the explanation for this inversion of the orientation of an image compared to the object

 

Pict 14: Orientation of image and object

 

As all light rays pass through the lens, some of them pass through a center point – we have talked about which ones they are. The rays through the center of the lens are now showing clearly that the image is fully inverted compared to the object.

Later, in one of the following tutorials, it will be shown how to invert back the image to have an upright and correctly oriented image available for focusing through a camera.

 

Another topic, which we have until now quietly accepted and not explained is the image magnification. If the photographer wants to image a person to sell a portrait photograph, it is of course necessary to get an image which is much smaller than the object. On the other side, if you want to take a shot of a tiny insect, you want to see it properly on the image – the term magnification needs to explained.

 

Magnification The ratio of the size of an object to that of an image. This ratio is a linear relation

 

Pict 15: Image magnification

The height of an object be h, then image that object – the resulting height of its image be h’ – the magnification is the ratio of h to h’. As you can conclude, the magnification can be positive as well as negative. Therefore one can get an enlarged image of an object or an image, which is smaller than the object. The sizes h and h’ are lateral dimensions, which means that the resulting magnification is called Lateral Magnification.

 

Lateral Magnification The ratio of the linear size of an image to the linear size of its object

 

Linked to this term is the Magnification power. It is the ability of an optical system to make an object appear larger than without that optical system. As example, if an image of an object makes is appear to be 2x larger and 2x wider than in reality – without that optical system, one can say, the optical system has a magnification of 2 (-power). Some cheap, incorrectly labeled systems (mostly toys etc) use instead of this correct magnification power its square – to appear better and stronger – and would call this before described magnification power 4x, the result of multiplying the two single values. One can find such an incorrect labeling often with cheap microscopes – displaying enormous magnification values which are simply wrong and misleading.

 

Magnification Power The ability of an optical system to make an object appear larger

 

Another term needs to be mentioned – Angular Magnification. Think about a binocular with which you look at a remote bear (hopefully remote enough). Without these binoculars, you could (and should) see that bear under a certain viewing angle – this viewing angle is grossly enlarged by means of the binoculars. The ratio of these two viewing angles is called Angular Magnification.

 

Angular Magnification Angular magnification is the ratio of the apparent angular size of an image observed through an optical system to that of the object viewed without that optical system

 

Since optical systems with enlargement power enlarge all dimensions, there is yet another term which required identification and explanation – it is the term Longitudinal Magnification. As you might have guessed again, it is the ratio of the longitudinal dimension of the object versus the one of the image.

 

Longitudinal Magnification The ratio of the axial dimension of an image to the corresponding dimension of the object

 

 

What was described and explained in this short tutorial – here the list of all of them: Lens, shapes of lenses, concave, convex, plano, meniscus, principal point, focus, focal point, vertex, optical center, nodal points, principal points, principal planes, thin lens, thick lens, optical axis, image orientation, magnification, lateral magnification, magnification power, angular magnification, longitudinal magnification.

 

A note from the author:

I have in mind to deepen these tutorials with hotlinks to more and detailed explanations including the most important formulas. These links will be made active during the next few months as I progress with the work. All what I am presenting here, was lectured by me for about 7 years at the Technical University Vienna in Austria and was/is therefore in German. Translation only would not the right approach – the entire course has to be rewritten in English – and that is quite some work….