# Microscope Magnification

The magnification of a microscope can be determined by using geometrical optics.

1. Magnification of a Finite Tube Length Microscope

Let us first consider the simplest case of a microscope as shown in Figure 24. This is a finite tube length microscope that uses a single lens. Only the objective. The sample kept at a distance between 'f' and '2f'. As discussed earlier, for a single lens if an object is placed between f and 2f a real and magnified, but an inverted image is formed on the other side of the lens at a distance between 2f and infinity. Again for the reasons discussed before, we will consider an object shaped like an arrow.

Figure 24: Magnification of a Finite Tube Length microscope using camera as a detector

Lets us now look at what magnification is? Simply put, it is the ratio of : (Size of the Image/Size of the Object).

Let us take a closer look at Figure 24.

(Size of the Image/Size of the Object)=(O'A'/OA)

Now let us consider the triangles COA and CO'A'. It is easy to see that these are similar triangles having the same angles.

Hene by property of similar triangles.

(O'A'/OA)=(CO'/CO):------------------------------------------------------(1)

This Implies:

Magnification of Finite Tube Length microscope=

(Distance of the image from the center of the Objective/Distance of the object from the center of the Objective)-----(1)

Now let us consider a hypothetical scenario where the inverted image of the arrow is moved or translated across the principal axis to the center of the lens as shown in Figure 25.

Figure 25: Magnification of a Finite Tube Length microscope using the camera as a detector. Image translated to the center of the objective.

A line that is drawn from the top of the object (arrow) and passing through the focal point (F) of the objective will intersect the top of the inverted image of the object.

Lets now look at the similar triangles AOF and FCA''.

Again from the property of similar triangles (CA''/AO) = Magnification= (FC/FO)=(f/r)------------------------------- (2)

This implies:

Magnification of finite tube length microscope=

(Focal Length of the Objective/Distance of the object from the focal point)-----------------------------------(2)

As we had noted earlier for an object placed at the front of the objective a magnified image is formed at the back of the objective only if the object is kept between 'f' and '2f'.

As the object moves towards f from equation (2) it is obvious that the magnification increases and from equation (1) it turns out that the image will be formed further away from the lens.

It can also be seen from equation (2) that as the object approaches 'f' the magnification approaches infinity which simply means that the beam comes out collimated and the image is formed at infinity. This is also true from equation (1) as the distance of the image from the objective approaches infinity, magnification approaches infinity.

It is now clear that for a finite tube lens microscope the magnification is not a constant, it changes with the distance of the object from the focal point.

Please note that when a manufacturer specifies a magnification for an objective for finite tube length configuration it is matched to a certain tube length. The tube length specifies the distance from the objective where the specified magnification is achieved. If a detector is fixed at this distance, the sample will come into focus only at a fixed distance away from the focal point of the objective.

If the tube length is increased, magnification increases and if the tube length is decreased, magnification decreases.

2.Magnification of an Infinity Corrected Microscope

Now let us consider an infinity corrected microscope as shown in Figure 26. The objective has a focal length of 'fo' and the tube lens has a focal length of 'fi'. Working of the infinity corrected microscope is described in detail in an earlier article.

Figure 26: Magnification of an infinity corrected microscope using the camera as a detector.

The object is kept at a focal distance ('fo') away from the objective and the image is formed at a focal distance ('fi') from the tube lens. An infinity path exists between the objective and the tube lens.

For ease of ray tracing, the tube lens is kept focal distance ('fi') away from the objective lens.

From equation (1), magnification=(distance of image from lens/distance of object from the lens)

In an infinity corrected configuration image is formed a focal distance ('fi') away from the tube lens and the object is kept focal distance ('fo') away from the objective.

It then turns out that-

Magnification of Infinity Corrected Microscope=(fi/fo)-----------------------(3)

=(Focal length of tube lens/Focal length of the objective)-------------------(3)

To have a positive magnification the tube lens focal length ('fi') has to be greater than the objective focal length ('fo'). This is mostly the case with most microscopes. They have an objective focal length of few mm and the tube lens focal length typically > 100mm. Please donot confuse the working distance of the objective and focal length of the objective here. The focal length will be much larger than the working distance of an objective.

So, now if magnification depends both on the objective focal length and the tube lens focal length what does it mean to say an objective has a magnification of say 60X. This, in fact, means that the objective will provide a magnification of 60X when coupled with a tube lens of a standard focal length supplied by the manufacturer for microscopes for which that particular 60X objective was designed.

If you mate the objective with a microscope from another manufacturer or a different model by the same manufacturer for which this objective was not designed for you will not get 60X magnification: because the tube lens focal length is different. This may also lead to other problems like with aberration corrections which may not work well as they are matched to a certain tube lens.

In case you are building your own microscope and using one of the commercially available objectives, make sure you know for what tube lens focal length this objective was designed for.

# Köhler Illumination

In order to achieve optimal image clarity, contrast and resolution, microscopes require proper illumination. This is provided by the Illumination system of the microscope consisting of a light source/s and a set of optics. This illumination system should satisfy the following conditions:

1. Light from the source should be concentrated onto the sample plane where the specimen is placed
2. The system should create a uniform and grainless illumination across the Field of View (FOV)

OK, we need to concentrate the light from the source onto the sample. This is more difficult than we think. Intuitively, we might imagine focusing the light from the source onto the sample by using a lens or a combination of lenses. But this presents a problem.  This can be easily understood from Figure 12.

Figure 12: Light from a lamp filament focused by a pair of lenses

It can be seen that when you focus the light from an extended source, it creates an image of the source. In this case, the image of the lamp filament is formed on the sample plane.  Of course, if your light source is an arc lamp an image of the arc gap is created and if a LED source is used an image of the LED die is created. An extended source has a finite dimension unlike a point source (infinitesimally small). So any attempt at focusing the light from such a source will result in an image of the physical form source being formed.

This focusing and the resulting image formation creates two problems:

1. The field of view of the sample is not uniformly illuminated.
2. The image of the filament is superimposed on the image of the sample or in other words the observer sees the lamp filament in the background of the sample image when viewed through the eyepiece or a camera.

So why not use a point source of light? In its real literal sense, a point source cannot be created. We could try using a light source that is very small (smaller than the resolution limit) but then the light emitted would be so little to be of any practical use. Our best bet is to emulate a point source using a coherent source of light like a LASER. But it has its own problems. Widefield illumination from a coherent source will give rise to speckle patterns in the Field of Illumination (FOI). The intensity distribution in the FOI is Gaussian (intense at the center and progressively dimmer towards the periphery). Hene it is not really uniform. The biggest problem -they are expensive.

We hence need a way of getting an extended source to work as an illumination source. Ideally,  a way of concentrating the light without the need to focus the source. This can be achieved by what is called as condensing of light. Condensing is different from focusing. This is what Köhler Illumination is all about.

What do we need to do to achieve Köhler Illumination conditions?

We need a set of three lenses:

1. Collector Lens with a focal length fc
2. Field Lens with a focal length ff
3. Condenser Lens with a focal length fn

These lenses should be arranged in the following way:

1. The collector lens placed at a distance 'fc' from the light source.
2. The field lens kept a distance 'fc+ff' from the collector lens.
3. The condenser lens placed at a distance 'ff+fn' from the field lens.
4. The sample is kept at a distance 'fn' from the condenser lens.

The resulting configuration of Köhler Illumination is shown in Figure 13.

Figure 13: Schematic of Köhler Illumination

In order to understand how Köhler Illumination works, let's first consider a single point on the lamp filament, at exactly its center as shown in Figure 14.  Here the center of the filament is aligned with the center of the collector lens. This single point acts as a point source of light. Since this point source is kept at a focal distance ('fc') away from the collector lens, the light coming out of the collector lens will be collimated (parallel beams of light), see Figure 2

Figure 14: Köhler Illumination-Light from a single point on the lamp filament located at the center of the filament.

Since a collimated beam of light enters the field lens, it focuses this light onto its focal point (at a distance 'ff'). An image of our point (at the center of the lamp filament) is hence formed here. The separation between the field lens and the condenser lens is 'ff+fn'. This means this image is sitting at the focal distance ('fn') away from the condenser lens. In short, the central point of the lamp filament is imaged to the back focal point of the condenser.

The condenser lens now creates a collimated/parallel beam of light that traverses through the sample. This collimated beam of light is called a 'Beam Pencil'. No focusing happens. Hence, an image of the lamp filament point is not created on the sample plane. We have now managed not to focus the light from the source or create an image on the sample plane. This, however, does not explain how the light from the lamp is concentrated or condensed.

Let us consider a second point on the filament, this time at the upper portion of the filament as shown in Figure 15. Light from this second point, again, since it is a focal distance away from the collector lens, exits out as a collimated beam. Please note, however, that this beam itself is not parallel to the beam created by the first point (at the center of the filament). Please also note that that different colors have been used only for the sake of clarity and do not represent different wavelengths. The two sets of parallel beams (one created by a point on the center of the filament and the other created by a point at the upper portion of the filament) are diverging away from each other.

Figure 15: Köhler Illumination-Light from two points (center and top) on the lamp filament.

After transmitting through the field lens this second beam is focused at distance 'fn' from the condenser lens. However, this focal point is created at a small distance below the focal point created by the light from the first point (at the center of the filament). After exiting the condenser lens this second point also creates a parallel beam of light, a 'Beam Pencil'. This second Beam Pencil is deflected up by the condenser and intersects the first Beam Pencil at a distance 'fn'  from the condenser lens (focal distance away from the condenser lens).

Now consider a third point as well on the filament, this time at the lower portion as shown in Figure 16. Light from this point after exiting the field lens gets focused to a point above the focal point created by the light from the first point.  After exiting the condenser lens it also creates a collimated beam of light, a 'Beam Pencil' like before. This Beam Pencil is now deflected down by the condenser and intersects the Beam Pencil created by the light from the first point at a distance 'fn' (focal distance away from the condenser lens). This is similar to what happened with the second point.

Figure 16: Köhler Illumination. Light from two points (center and bottom) on the lamp filament.

Now let us consider all the three points on the filament together, a point at the center, a point on the upper portion and a point o the lower portion of the filament as shown in Figure 17. Light from all of these three points after exiting the field lens is focused onto different points but on the same plane. This plane is focal distance 'ff' away from the field lens and focal distance 'fn' away from the condenser lens.

Figure 17: Köhler Illumination. Light from three points (center, top, and bottom) on the lamp filament.

The three points on the lamp filament are imaged onto three distinct points on the condenser back focal plane. Each of these after exiting the condenser lens creates their own Beam Pencils. Each Beam Pencil is a collimated shaft of light, hence does not create an image of the point from which it was generated. However, the beams themselves are not parallel to each other instead intersect or converge at a focal distance away ('fn' away) from the condenser.

While we have considered only three points, it is not difficult to imagine that every point (in fact an infinite number of them) on the lamp filament creates its own Beam Pencil. These Beam Pencils are bent in such a way that they intersect at a certain distance away from the condenser. This intersection point is the focal point of the condenser. This where we place the sample.

This process, involving an infinite number of  Beam Pencils, create a solid cone of light with the apex of the cone at the focal point of the condenser. This is what concentrates or condenses the light onto the sample plane. It is clear that this condensing happens without focusing the light from the filament as each Beam Pencil in the cone is a collimated beam shaft. So no image of the filament is created on the sample plane.

A closer look reveals that the cone of light has an apex that is not a point but a spot (a circle with a certain diameter), resembling a cone with a portion of the top cut off creating a plateau. The diameter of this plateau defines the field of illumination. This is more or less defined by the diameter of a single beam pencil. This, in turn, is determined by the focal lengths of field and condenser lenses used in the setup. Longer the focal length of the condenser lens, larger will be the diameter of a Beam Pencil and hence the Field of Illumination but then smaller will be the condensing effect.

For an illumination system aligned for Köhler Illumination, if you move the condenser up and down, it gives the impression of light coming in and out of focus. This is just the apex of the light cone moving above and below the sample plane. No image of the filament will come into focus at any point unless the condenser is moved way closer to the field lens.

The image of the filament is in fact created at the condenser back focal plane as can be seen from Figure 18. This is also the front focal plane of the field lens. The magnification created by the collector lens + field lens combination is ('ff/fc'). Since fc is usually less than ff, a magnified image of the filament is created at the back aperture of the condenser. This magnification factor is important as we will discuss later and determines the choice of fc and ff.

Figure 18: Köhler Illumination-The light from the lamp filament is focused and an image created at the back focal plane of the condenser

Condenser Numerical Aperture

The numerical aperture of the condenser determines the angle subtended by the illumination cone at the sample plane. The larger this cone greater is the resolution. Hence it is desirable to have a greater NA for the condenser. However, in practice, greater NA decreases the focal length of the condenser and can introduce practical difficulties in mounting the sample etc. So typically NA of about 0.5 is used for microscope condensers. However high NA oil immersion condensers with NA upto 1.4 are available commercially with very short focal lengths.

It should also be noted that the full NA of the condenser can only be utilized if the image of the lamp filament fills the condenser back aperture. Hence the magnification of the collector lens + field lens combination should be designed so as to slightly overfill the condenser NA. Too high a magnification over and above what is required to fill the condenser NA will result in wasted optical power.

Field Diaphram and Aperture Diaphragm

Two other important elements of the Köhler illumination design are the field diaphragm and the aperture diaphragm.

The field diaphragm is kept at a focal distance away from the back aperture of the field lens as shown in Figure 19. As a result, the light from the edges of the field diaphragm gets focused on the sample plane. Image of the field diaphragm is created on the sample plane. Since the diaphragm only allows light to pass through its opening, the field diaphragm helps control the field of illumination on the sample plane.

The size of the field diaphragm image falling on the sample plane is determined by the magnification of the Field Lens + Condenser Lens combination ('fc/ff'). Since 'fc/ff' < 1 (condenser focal length 'fc' < field lens focal length 'ff'), a demagnified image of the field diaphragm is formed. This demagnified field aperture image is especially useful when controlling FOI of high magnification objectives with very small fields of view.

Figure 19: Köhler Illumination-The image of the field diaphragm is projected on sample plane. Controls field of illumination.

It is important to match the field of view and the field of illumination. Even the best of the cameras have a chip size much smaller than the field of projection of the objective. Hence by making the FOI equal to the FOV, we make sure that the region of the sample that is not in the view of the camera is not unnecessarily illuminated. This is even more important in biological applications with live cells or tissue samples.

The Aperture Diaphragm is placed at a focal distance away from the condenser-condenser back focal plane- as shown in Figure 20. It sits on the plane where the image of the lamp filament is formed.

Figure 20: Köhler Illumination-Aperture Diaphragm is placed at the condenser back focal plane. Controls the Numerical Aperture of the condenser.

As the condenser aperture is closed Beam Pencils that intersect the sample at steeper angles are progressively cut off decreasing the angle of light cone that falls on the sample. This has two effects:

1. Decreases the amount of light falling on the samples
2. Decreases the effective Numerical Aperture of the condenser and thus decreasing resolution

In fact the Aperture Diaphragm is very similar to the Aperture Stop used in photography. The only difference is that in photographic cameras the Aperture Stop sits in the imaging path while in microscopy it sits in the illumination path.

Like in photography we can define a field number or f/N ratio (ratio of focal length of the lens and effective diameter of the aperture) for the illumination optics of the microscope.

Diffuser, Heat Blocking Filter and Green Interference Filter (GIF)

There are three more components that are useful to have in an illumination system of the microscope.

Diffuser: If the Beam Pencils that create the light cone all have equal intensity then the sample plane is uniformly illuminated. But this is not always the case as there are variations in the light coming out of various points on the lamp filament. A diffuser can be placed in the illumination system after the light source. The diffuser acts like an optical shaker that mixes up light from different points on the filament through multiple scattering events and creates a homogeneous distribution of light. This ensures a uniform illumination on the sample plane. When we introduce the diffuser we no longer see the image of the lamp filament at the back focal plane of the condenser. The intensity variations are washed out.

Heat Blocking Filter: An incandescent lamp produces a large amount of power in the NIR and IR. These wavelengths are not useful for routine imaging but can significantly heat up the sample. A heat blocking filter can be used to block these wavelengths.

Green Interference Filter (GIF): Introducing a GIF can significantly improve image quality due to the following reasons:

1. Makes the light monochromatic and decreases chromatic aberrations
2. Most lenses including the ones with fewer aberration corrections are corrected for spherical aberration in the green
3. Phase rings and other contrast enhancing elements work best with monochromatic light, most commonly green.
4. Most detectors including the human eye are most sensitive in the green end of the spectrum.

GIF will make a significant difference while using contrast enhancing techniques like phase contrast microscopy especially with very small samples like Bacteria.

The diffuser, heat blocking filter, and the GIF are usually placed between the collector lens and field lens before the field diaphragm. This makes sure that any dust or other imperfections are not imaged onto the sample plane.

A complete optical layout of a microscope that implements Köhler Illumination is shown in Figure 21. This shows both Illumination and Imaging optics. The names of various components used will become obvious when looking at this figure

1. Collector Lens: Collects light from the light source
2. Field Aperture: Controls the Field of Illumination on the sample
3. Filed Lens: Acts as an objective for the field aperture
4. Aperture Diaphragm: Controls the Numerical Aperture of the Condenser Lens
5. Condenser Lens: Condenses the light on the sample plane

Figure 21: Schematic of a microscope implementing Köhler Illumination. The illumination path is shown in green and imaging path is shown in orange

Conjugate focal planes

For a microscope in the Köhler Illumination configuration, we can define two sets of conjugate focal planes. One for the light source and the other for the sample plane.

Conjugate focal planes for the light source are all those planes in the microscope optical train where the light source is brought into focus. This can easily be identified by tracing light coming out of a single point on the light source and noting all the planes in the optical train where this light is focused. All the planes where the light is collimated will not form an image of the source. This is shown in Figure 22. Please note that the image of the filament can only be observed if the diffuser is removed out of the optical path.

Figure 22: Köhler Illumination-Conjugate focal planes of the lamp filament

From Figure 22 we can easily identify the following conjugate planes for the llight source:

1. Back focal plane of Condenser
2. Back focal plane of Objective
3. Front focal plane of the Eyepiece

It may be noted that in addition to not allowing an image of the light source being formed on the sample plane, Köhler Illumination also ensures that the image of the light source is not formed on the camera image plane or the Retina of the eye. This means the observer will not see the lamp filament in the background.

It may also be noted that in phase contrast microscopy the phase annulus is kept at the back focal plane of the condenser and the phase ring is kept at the objective back focal plane, both of which are conjugate planes for the light source. This brings both of components' image onto one optical plane and at the same time makes sure their images are not projected onto the sample plane. So the observer sees neither the phase ring or the phase annulus.

Conjugate focal planes for sample plane are all those planes in the microscope optical train where the sample plane comes into focus. Like in the case of the lamp filament, this can be identified by tracing the light coming out of a single point on the sample plane and observing all the planes in the optical train where this light comes into focus. This is shown in Figure 23.

Figure 23: Köhler Illumination-Conjugate focal planes of the sample plane

From Figure 23 we can identify the following conjugate planes for the sample plane.

1. Camera sensor plane
2. Retina of the eye
3. Back focal plane of the field lens

Since the objective has a shorter focal length than the tube lens, a magnified image of the sample is projected at the camera plane. This image is further magnified by the eyepiece and eye combination and projected onto the retina.

A magnified image of the sample is also created at the field lens back focal plane. Here the condenser acts as the objective to capture the image. The image is magnified by the ratio ('ff/fn'). Usually, it is difficult to make use of this image as it becomes difficult to decouple the forward traveling illumination light and backward traveling reflected light from the sample. There are however ways to do this decoupling; the easiest being the use of a partially reflecting mirror placed just behind the field lens.

Adjusting/aligning a microscope for proper Köhler Illumination

It is a good practice to check for Köhler Illumination alignment at least once before the each imaging session. Each microscope company has a slightly different way of doing it. The basic idea, however, is the same and can be summarized in the following steps:

1. Mount a sample and bring it into focus using the desired objective.
2. Close the Field Diaphragm and the Aperture Diaphragm to their minimum positions. The closed field diaphragm will decrease the Field of Illumination (FOI) below the Filed of View (FOV). This helps us visualize the edges of the field diaphragm.
3. As discussed before, if properly aligned, a sharp image of the Field Diaphragm is formed on the sample plane. Move the condenser in the Z axis to make sure this happens by observing the edges of the field diaphragm image. Most microscopes have the condenser mounted on a Rack and Pinion arrangement to enable motion in the Z direction.
4. Once the image of the Field Diaphragm is in focus, translate the condenser lens in X and Y so that the image of the closed field diaphragm sits at the center of the field of view. Condensers on most microscopes are mounted on a X-Y translation stage for this purpose.
5. Once the condenser is aligned in X,Y and Z, the field diaphragm can be opened sufficiently as to fully illuminated the desired FOV and the aperture diaphragm can be opened sufficiently for the required contrast while adjusting the intensity of the light source.

# Microscope Fundamentals

Now that we understand how lenses work lets try to understand how these lenses can be put together to build a microscope. Basic microscopes can be divided into two broad categories:

a. Finite Tube Length Microscope

b. Infinite Tube Length Microscope

### Finite Tube Length Microscope

Finite tube length microscopes represent the earliest designs of light microscopy. It primarily uses two lenses:

a. Objective Lens: So called because it is placed close to the object or the specimen

b. Eye Piece Lens: So called because it is placed close to the eye of the observer

As can be seen from Figure 9 if an object is placed between the focal length (f)  and twice the focal length (2f) then an inverted an magnified image is formed between the 2f and infinity on the other side of the lens. the magnification increases as the object get away from 2f and closer to f. This effect is exploited in this configuration.

It would then look sufficient to have only one lens to make a microscope. This is true if we are to capture this image on a screen, a photographic film or the sensor chip of a camera. However, we need to note that if an observer needs to see the magnified image of the sample through the microscope having only one lens is not sufficient. This is because the eye has a lens of its own. The eye requires parallel rays to enter it so that an image could be formed on the retina.

This problem is solved using the eyepiece which collects the light from the image created by the objective and presents a parallel beam of light to the eye of the observer so as to form an image on the retina.

The construction of a finite tube length microscope is shown in Figure 10. Here fo is the objective focal length and fe is the eye piece focal length.

For this arrangement to work the specimen is placed between f0 and 2fo. This creates a magnified and inverted image beyond 2fo on the other side of the objective lens. The image created by the objective now acts as the object for eyepiece lens. The eyepiece lens needs to be placed so that the image formed by the objective is eyepiece focal length distance (fe) away from it. As can be seen from Figure 5 if an object is placed at focal distance light rays emerge parallel. When the observer places her/his eye in this path the parallel rays are converged by the lens of the eye to create an image on the retina.

It can be appreciated that the distance between objective and eyepiece is fixed and cannot be altered at will. Traditionally these two lenses were mounted on opposite sides of a tube and the tube length was designed to correspond to this fixed distance. Hence the name finite tube length microscope.

### Figure 10

Finite Tube Length Microscope

### Infinite Tube Length Microscope

Modern microscopes use an infinite tube length configuration. This is because of a number of practical problems in a finite tube lengths configuration. Microscope designers wanted more flexibility in the length of the microscope tube and wanted the sample to be placed at a focal distance away from the objective. This gave rise to infinite tube length configuration.

The construction of an infinite tube length microscope is shown in Figure 11. Here a third lens called the intermediate lens or tube lens is introduced between the objective and the tube lens. In Figure 11  'fo' is the focal length of the tube lens, 'fi' the focal length of the intermediate lens and 'fe' the focal length of the eyepiece lens. The following arrangement is used:

a. The object is placed at a focal distance (fo) away from the objective. This makes the light from a given point on the object come out collimated on the other side of the objective.

b. Since the light out of the objective is collimated the intermediate lens or tube lens can be placed ideally at any distance from the objective. This in principle creates an infinite space between the objective and the intermediate lens. Microscope designers could then chose a tube length of their choice independent of the focal lengths of the lenses used in the microscope. The intermediate lens creates a magnified and inverted image on the other side at the intermediate lens focal length (fi) distance from it. This image is called the intermediate image.

c. The eyepiece is placed similarly to that in a finite tube length configuration. The image created by the intermediate lens acts as the object for the eyepiece. The eyepiece lens is placed so that the image formed by the intermediate lens is eyepiece focal length (fe) distance away from it. This makes the light from a given point on the intermediate image come out collimated on the other side of the eyepiece. When the observer places her/his eye in this path the parallel rays are converged by the lens of the eye to create the final image on the retina.

### Figure 11

Infinite Tube Length Microscope