Optical Basics: From the Converging Lens to the Compound Microscope

April 12, 2017

Knowing the optical basics can be a great help for working at the microscope. This article describes the main components of optical microscopes, such as lenses, objectives and eyepieces, and explains elementary functions such as refraction, diffraction and reflection. You will read about the basic functioning principles of magnifiers and the “simple microscopes” derived from them, and finally also the principles of the “compound microscopes”.

Light as a wave movement

A study of the light microscope is naturally incomplete without considering the element that is used to convey the information – light itself. For the time being, we will concentrate here on the wave model and the ray model of geometric optics.

Figure 1: The light that is visible to the human eye is only a small part of the electromagnetic spectrum.
Figure 2a: The wavelength (λ) represents the distance after which the same vibrational states are repeated on the light wave.
Figure 2b: Points with the same vibrational state (marked A and B) are on the same phase of the light wave. By contrast, between the two points marked A and C there is a phase difference of half a wavelength.
Figure 2c: The maximum upward and downward variation of the light wave from the baseline is the wave’s “amplitude“.
Figure 2d: The vibration plane of the light wave is perpendicular to the propagation direction.

From light wave to light ray

If you look at drawings of light paths in optical instruments, you do not find light waves, but light rays that describe the propagation of the light. This is the manner of representation of "Geometric optics" ("ray optics"). At first glance, these light rays seem quite familiar from everyday experience. Think of the “light ray” from a torch, for example. However, the following diagram shows what is meant by light rays for the purpose of geometric optics.

Figure 3: Light rays show the propagation of light and are therefore always perpendicular to the wavefronts they ultimately describe. The rays are thus a simplification of the wave model to illustrate the propagation of light in an easily understood way.

Light refraction and light reflection

The job of the light microscope is to produce a magnified image of an observed object. To do this, the optical components of the microscope capture part of the light emanating from the object and reunite the rays at the image plane. The light is “guided” by light-refracting and also partly by light-reflecting components.

Let‘s see what happens when light from a transparent medium passes into a second medium that is also transparent. As an example, we will look at light entering glass from air.

Figure 4a: The speed of light in glass is less than in air. We therefore say that glass is an “optically denser” medium than air. As the light frequency remains unchanged, the reduction of the light speed is a result of a shortening of the wavelength. The light waves are “crushed”, so to speak.

In the last view, the direction of light incidence was perpendicular to the air/glass interface.

It is also interesting to observe what happens when the light enters the glass at an oblique angle. The glass block is tilted for this purpose. As the propagation of light in glass is slower than in air, the light waves are not only “crushed” but also “tilted” when they enter the glass at an oblique angle.

Figure 4b: The ray-optic view shows rays of light that are perpendicular to the waves and show the direction of the light propagation. In this example, the light rays change direction at the air/glass interface. This is called refraction of light.
Figure 4c: If you draw the normal through the point of incidence at the interface of the two media, the angles between the normal and the refracted ray will give you the angle of incidence and the angle of refraction. When light passes from an optically thinner medium such as air into an optically denser medium (e.g. glass) it is refracted toward the normal and the angle of refraction is smaller than the angle of incidence. Conversely, when light passes from glass to air, the angle of refraction is greater than the angle of incidence.
Figure 4d: According to the law of refraction: n1 * sinα = n2 * sinα’

The light refraction index “n“ is a dimensionless value. The light refraction index of a material is obtained from the equation: n = c0 / c

n: Light refraction index of the examined material
c0: Speed of light in a vacuum
c: Speed of light in the examined material

Lenses: Guiding light in a specific direction

Refraction can be used to guide light rays in a specific direction. Two basic types of lenses are used for this: converging lenses and diverging lenses.

Figure 5: Lens shapes – the basic types. Most lenses are spherical, i.e. their concave or convex surfaces are segments of a sphere. It is relatively easy to produce this type of lens, as, for example, a convex converging lens fits perfectly into a concave grinding wheel with the same curvature and the two elements can be moved against each other for the grinding process without any problems.

The key dimensions and reference points of a spherical lens:

Figure 6a: The optical axis connects the centers of curvature C1 and C2 of the lens surfaces.
Figure 6b: Rather than the actually occuring double refraction at the lens interfaces, the light refraction can be reduced to a single refraction at the principal plane for the sake of simplicity.
Figure 6c: Rays running parallel to the optical axis behind the lens intersect this optical axis at the focal point of the object F.
Figure 6d: Rays running parallel to the optical axis when they strike the lens intersect the optical axis at the focal point of the image F'.
Figure 6e: The focal planes are perpendicular to the optical axis at the focal points.
Figure 6f: The object-side focal length f connects the principal point H with the focal point of the object F. The image-side focal length f' connects the principal point H with the focal point of the image F'.

The above diagrams show the extremely simple case of a “thin” converging lens. In practice, the situation is often much more complex. For instance, from a certain lens thickness onwards it is not sufficient to have just one principal plane. These lenses then have two principal planes – one for the incident rays and one for the emerging rays. For our purposes of a model-based description, however, the simpler case with only one principal plane is adequate. 

The formation of an image through a lens is easy to draw by using the so-called “design rays” (parallel, central and focusing ray). The path of these rays follows simple rules that are easy to remember. Basically, one only ever needs two of these design rays to determine the image position as their point of intersection always lies at the image position.

Image formation in a converging lens:

Figure 7a: Parallel ray. Light rays running parallel to the optical axis are refracted in such a way that they pass through the focal point in the image space F'.
Figure 7b: Central ray. Light rays passing through the principal point H of the converging lens are not refracted and their direction is not changed.
Figure 7c: Focusing ray. Rays passing through the object-side focal point F are refracted in such a way that they run parallel to the optical axis in the image space.

The illustration of image formation with a converging lens is simple and readily understood. The image on the right of the lens is simply formed at the point where the design rays meet and can be captured there on a piece of paper, for example. We call this a “real image”. The imaging process with a diverging lens is much less simple, as the rays coming from a point on the object do not intersect on the right of the lens. On the contrary: because of the diverging effect of the lens, they move more and more apart. Nevertheless, a point of intersection of the design rays as image position can be determined here, too by projecting the refracted rays backwards. From the perspective on the right of the lens, the rays then seem to emanate from this (image) point. Unlike the converging lens of course, the diverging lens does not produce a real image that can be captured on a projection surface, but a “virtual image” that is reduced in scale in the case of the diverging lens.

Image formation in a diverging lens:

Figure 8a: Parallel ray. Light rays running parallel to the optical axis when they strike the lens are refracted in such a way that they pass through F'.
Figure 8b: Central ray. Light rays running through the principal point H of the diverging lens are not refracted and their direction is not changed.
Figure 8c: Focusing ray. Rays running parallel to the optical axis after leaving the lens are refracted in such a way that they run through the focal point F.

Finally, it should be said again that light rays are only a way of illustrating the path of light waves in a simple and manageable form.

Figure 9: Light waves and light rays passing through a converging lens. The light rays are perpendicular to the wavefronts, therefore having the effect of guide rails along which the waves run.

The magnifier and the “simple microscope”

The relative size that we see things as having depends on the size of their image on the retina of our eye. To enlarge this retina image, we can bring the object we are looking at as close as possible to our eye. By the process of accommodation, the focal length of the eye changes and we see a sharp image of the object at any distance. Of course, accommodation is not infinitely possible and only up to a certain point of proximity (near point).

Accommodation is normally controlled without the conscious control of the observer via the state of contraction of the ciliary muscle, which encompasses the lens like a ring with a small space in between. The eye lens itself is elastic and has a natural “spherical” shape. However, the normal inner pressure of the eye presses the suspensory frame of the lens (olive and red in the illustration below) apart, thereby also ‘squashing’ the lens. In this state it is relatively flat and focused on the distance, while the ciliary muscle (red) is relaxed. When we look at a near object, the ring-shaped ciliary muscle contracts, counteracting the expansion direction of the inner pressure of the eye. The tension on the lens diminishes and it takes on a more “spherical” shape. For long periods however, looking into the distance is much less tiring than looking at an object at the near point of the eye, because the ciliary muscle is less contracted in this state and therefore does not have to work as hard.


Figure 10: The visual angle. If an object moves closer to the eye, the focal length of the eye lens is involuntarily altered to ensure that the image of the object on the retina is sharply focused. This process is called “accommodation”. However, the accommodation capacity of the eye is limited. Once the “near point” has been reached, further accommodation is impossible when the object moves even closer. The near point is therefore the point at which the object can be seen in sharp focus at the greatest possible visual angle (σ ).

The distance of the near point from the eye varies from person to person. An average value of 250 mm is generally accepted. This is known as the “least distance of distinct vision” or “conventional visual distance”. 

The least distance of distinct vision serves as a reference value for calculating the magnification of optical instruments for visual use. Therefore, a magnifying optical instrument enables an object to be observed at a larger visual angle than would be possible for observing the same object from a distance of 250 mm. 

If you look at the above diagram of the visual angle, you will notice that the retina image becomes larger as the visual angle increases. It is easy to see that doubling the tangent of the visual angle doubles the size of the retina image. The magnification V of a visually used optical aid is therefore calculated from the quotient:

tan(σ') tan(σ)

tan(σ'): tangent of the visual angle with optical aid
tan(
σ): tangent of the visual angle without an optical aid with the least distance of distinct vision of 250 mm

The easiest way of observing objects at a larger visual angle is to use a magnifier. The following shows how the visual angle is increased when a magnifier is used. Basically, magnifiers are simple converging lenses. If you put an object between the front focal plane of a converging lens and the lens itself, the rays emerging from an object point are still divergent behind the lens.  Under these circumstances, a real image is not formed with a converging lens either. Instead, a virtual image is created, similar to the case of the diverging lens.

Figure 11: Virtual image by a converging lens. In the case of the virtual image, the light rays are refracted in such a way that they emanate from a different point of origin from the perspective on the right of the lens. This causes an apparent change in the position and size of the object (= “virtual image”).
Figure 12a: Determining the magnification of a magnifier. Situation without magnifier.
Figure 12b: Determining the magnification of a magnifier. Situation with magnifier.

A magnifier with the magnification 10x therefore has a focal length of  250mm/10 = 25mm. The above diagram shows that the magnification also depends on the distance between the eye and the magnifier. However, this does not apply if the observed object is exactly in the front focal plane of the magnifier, when the magnification of the magnifier is always 250/f as shown here. This is the value that is generally given for the magnification of a magnifier.

Magnifiers with a short focal length and generally with a stand, specimen stage and sometimes an illumination unit used to be called "simple microscopes". Most of them had a magnification of about 15-60x. This simple design predominated in the early days of microscopy, as the more complex instruments were more prone to image aberrations.

 

Antoni van Leeuwenhoek (1632-1723) from Delft in southern Holland must have acquired special skills in making lenses of extremely short focal length. He managed to make simple microscopes with a magnification of at least 250x (corresponding to a focal length of about 1mm). The practically spherical lenses he employed were naturally tiny, and it is not known how he made them. He was also the first person to clearly describe and draw bacteria with their basic shapes (spherical, cylindrical and spiral).

The objective

Converging lenses always generate a real image of an object when it is situated in front of the focal plane on the side of the object. Systems producing an image in this way are called “objective”. The real image can be captured on a projection surface (e.g. a camera chip). As the size ratios of the object and its image can be directly measured, we do not speak of “magnification” here as in the case of the magnifier, but of “reproduction scale” to represent the size ratio of the object image and the actual object. For example, a reproduction scale of 40:1 means that distances in the image are 40 times larger than on the object itself.

Figure 13: Using a converging lens as an objective
You can see that the nearer the object is to the object-side focal plane, the larger the image. The image moves further and further away from the objective into the image space.
Object space: Reproduction scale M = f/z
z = distance between object and object-side focal plane
Image space: Reproduction scale M = z’/f’
z’ = distance between image and image-side focal plane

Example: A converging lens with a focal length of 15 mm is to be used as an objective and provide a reproduction scale of 10:1. At what distance must an object be placed in front of the lens (more precisely: in front of its principal plane) and at what distance behind the lens is the image formed?

The formula M = f/z is reformulated as z = f/M.
For the given case, this results in z = 15mm/10 = 1.5mm.
An object placed at a distance of 16.5mm in front of the principal plane of the lens will be imaged with a reproduction scale of 10:1.
The calculation for the image space is:

z' = f' = 10 * 15 = 150mm

So the image is formed at a distance of 165mm from the principal plane of the lens.

The "compound microscope"

To obtain a larger view of objects, it is possible to use a magnifier. However, higher magnifications require such small focal lengths that such magnifiers are extremely difficult to produce and use.

It is much easier to use a magnifier to observe a real and magnified image that has already been produced by an objective. This takes us to be the principle of the two-stage “compound microscope”. Here, the objective generates a magnified and real intermediate image that is viewed through a magnifier called an “eyepiece”. All commonly used light microscopes work on this important basic two-stage imaging principle.

Figure 14: The compound microscope.

Incidentally, the intermediate image is not captured on a projection surface, but viewed directly with the eyepiece as a so-called “aerial image”. Unlike the above illustration, the intermediate image in a compound microscope is normally directly in the focal plane of the eyepiece. This means that the rays emanating from an image point are parallel with each other when they reach the eye of the observer and come from a virtual image at a distance . This allows relaxed, fatigue-free viewing of the microscope specimen. However, some microscopists initially have problems with correct eye accommodation, as they intuitively try to focus the eye on a particularly near point.  This can be counteracted by focusing one’s eyes on a far-distant object immediately before looking through the microscope with this “visual setting”. 

When a microscope is used visually, its method of functioning is based on a magnification of the visual angle. Therefore the term magnification is used in the same way as for the magnifier. The magnification is obtained by multiplying the objective scale with the eyepiece magnification.

 

Vmicroscope = Mobjective * Veyepiece

 

Example: A 40:1 objective is combined with a 10x eyepiece in a compound microscope. The magnification of the microscope is then 40 * 10 = 400x.

Now that we understand the working principle of the compound microscope, we can theoretically build one ourselves out of two converging lenses.

Example: The aim is to build a microscope with 100x magnification from two identical converging lenses each with a focal length of 25 mm. 

One of the two converging lenses is used as an eyepiece. According to the magnifier magnification formula (V=250/f), the eyepiece has the magnification 10x.

The second lens is used as an objective and has to produce the reproduction scale M of 10:1 for the real intermediate image.
According to the formula M=z'/f' for the reproduction scale of an objective, it is clear that this intermediate image has to be formed at a distance of 250 mm + 25 mm = 275 mm behind the lens (more precisely: behind its principal plane). As we want the intermediate image to be in the front focal plane of the eyepiece, we therefore have to design the microscope so that the distance between the principal planes of the objective and eyepiece is exactly 300 mm. 
The specimen, on the other hand, must be placed at a distance of 25 mm + 2.5 mm in front of the principal plane of the objective in accordance with the formula M=f/z so that it can be imaged in the real intermediate image.

The following diagram shows the microscope we have designed with a magnification of 100x:

Figure 15: Compound microscope 100x comprising two converging lenses with a focal length of 25 mm.

The focal length of a compound microscope can be calculated with the formula:

fMicroscope = fObjective * fEyepiece / Optical tube length

 

The focal length of the microscope we have designed is therefore 25 mm * 25 mm/250 mm = 2.5 mm. As the overall microscope produces a virtual image like a magnifier, the overall magnification of the microscope is 250 mm / 2.5 mm = 100x.

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