Basic Optics

Understanding basic microscopy starts with understanding basic optics.

To do that we will need to know a little bit about light.

Light, as we know, is an electromagnetic wave. An electric field varying in space and time creates a magnetic field that varies in space and time. A magnetic field that varies in space and time creates an electric field that varies in space and time. These varying electric and magnetic fields can sustain each other. Such waves do not require a medium to travel. This is a very simple way of looking at electromagnetic waves such as light. Waves interact with matter in a complex way. For electromagnetic waves, the laws governing their interaction is described by Maxwell's Equations.

We don't need Maxwell's Equations for a very basic yet intuitive understanding of optics. We can do with what is called as Ray Optics or geometrical optics.

Ray optics considers a light beam as a bundle of rays. These rays interact with a medium and undergo reflection or refraction. Reflection and Refraction obey the Laws of Reflection and the Laws of Refraction respectively which can be figured out using simple concepts in geometry. While the rays themselves are considered to be travelling in straight lines, the collective behavior of these rays in a beam changes as they interact with  optical elements. This can change the shape of the beam.

Let us try to use ray optics to understand how lenses work.

Lenses can be classified into two primary categories:

1. Convex Lenses (Lenses that are thicker at the center and thinner at the periphery)

2. Concave Lenses (Lenses that are thinner at the center and thicker at the periphery)

Most of what is used in microscopes are convex lenses. Hence we will primarily discuss how convex lenses work:

Lets us first consider a collimated beam of light passing through a convex lens. A collimated beam is characterized by rays constituting the beam traveling parallel to each other.

A collimated beam of light after passing through a convex lens gets focused at what is called as the focal point of the lens. The distance of the focal point from the center of the lens is called the focal length of the lens. This is shown in Figure 1. Here f is the focal point and fL is the focal length. The arrows on the light beam indicate the direction of travel of the light rays.

After intersecting at the focal point the rays diverge and the cone of light inverts. 

Figure 1

Picture2Parallel rays passing through a convex lens get focused at the lens focal point

Now let us consider the inverse scenario. Let us consider a point source of light, ie, a source with physical dimensions that are infinitesimally small. Let us consider that this point source is placed at the focal point of a convex lens. The source radiates isotropically. A cone of this isotropically radiated light is intercepted by the lens. It can be observed that after passing through the lens the rays of light emerge parallel to each other. This is shown in Figure 2. Here the orange dot at the focal point represents the point source of light.

Figure 2

lens2
Light emitted by a point source placed at the focal point of a convex lens after passing through the lens travels collimated

Theory of image formation

Now let us consider how images are formed. 

Lets us consider an object, lets say a sphere as shown in Figure 3. In-order to create an image of this object we need to use some kind of probes which can be made to impinge on the object. The probes used for imaging are ions, photons, electrons etc. In principle any particle can be used as long as they are much smaller than the object being imaged and available in large quantities.

Figure 3

imaging
Image formation

Any object obviously would have an infinite number of points on it. Ideally, we require an infinite number of probes to impinge on all of these infinite points on the object (even though practically this is not possible and actually not really required). The probes are scattered by the points on the object and collected by a collector.

The collector's job is to make sure that all the probes scattered by a single point on the object are focused onto a single point onto the detector. The detector has the job of measuring the number of probes falling at any given point on the detector and converting this information to some form that can be easily understood by us.

Imaging hence essentially is a mapping process. The collector maps the points on the object to points on the detector/screen. This is done as mentioned by collecting the probes from a given point on the object by the collector and focusing them to a single point on the screen. 

In doing so two facts have to be considered.
1. It is not really necessary that the separation between two points on the object is exactly maintained between the two corresponding points mapped onto the detector.
2. However, it is very important that if we consider any three arbitrary points on the object, the ratio of distances between them is also maintained between the correspondingly mapped points on the detector. 

This means that while the shape of the image is exactly made similar to the object, the size of the image need not necessarily be same. The image can be magnified or diminished with respect to the size of the object.

It is however not practically or even theoretically possible to create an image exactly as the object. The image is distorted to some extent or as we call it there is image blur.

Image blur is created due to the following reasons:
1. It is not always possible to make sure that probes falling on a particular point on the detector originated from a single point on the object.
2. The probes that originate from a point (infinitesimally small spot)  on the object don't always fall onto as single point on the detector. The probes kind of spread out and overlap with other spots.
3. Detectors themselves are limited by how small a spot they can collect information from independently of other regions on the detector.

Optical Imaging

In optical imaging the probes used are photons, the collector is a lens or group of lenses and the detector is some photosensitive device like eye, CCD, PMT etc.

Photons have a number of advantages when used as probes for imaging:
1. Are very small
2. Are practically mass-less
3. Can be made to travel any distance
4. Large quantities of them can be easily generated are made to propagate in any given direction.
5. Since they are Bosons, they don't obey Pauli's exclusion principle like Fermions and hence any number of them can be made to fall on a single spot.
6. Can be easily detected

However, image formation using photons cannot be fully understood by using ray optics alone. We need wave optics which considers light as a wave and not just as a bundle of rays. This is because diffraction which is a wave phenomenon plays an important role in determining how images are formed. However, ray optics can still give us some insights as we will see below.

Image formation by a lens

Let us consider how images are formed by a single convex lens. A convex lens as we have seen before will converge a collimated beam of light at the focal point. Now depending on the distance of the object from the lens, the image formed has different attributes. 

Let us consider a single convex lens as shown in Figure  4. The line passing through the center of the lens and perpendicular to the plane/face of the lens is called the principal axis. This is denoted by a black line in Figure 4. The focal point is on the principal axis and is focal length away from the face of the lens. Since a lens as we discussed before is a reciprocal system we have focal points on both sides of the lens. This is indicated by 'f' on both sides on the principal axis.

We can also see the points denoted as '2f' on the principal axis. These are the points that are twice the focal length away from the center of the lens.

We have considered the object in the form of a single-headed arrow. This has the advantage of being very simple and providing a sense of orientation for the image formed. This can be done by looking at the orientation of the arrow head in the image with respect to the object.

Like any other object the arrow has infinite number of points on it. Ideally infinite number of photons should strike each of these points and then get scattered off the object. Each of these scattered photons can be considered as rays emanating from the various points on the object. To understand how an image is formed, these infinite number of rays from each of the infinite number of points on the object should be traced through the lens and focused at the focal plane. The collection of these points on the focal plane in principle creates an image. However, considering all these infinite number of rays is not practically possible. We will do some thing simpler.

We will consider only 2 points on the Object.
1. The point at the top of the arrow head
2. A point at the bottom of the foot of the arrow touching the principal axis.

From these two points we will consider only two scattered rays.
1. One travelling parallel to the principle axis. After passing through the lens this ray will travel through the focus as does any ray that is parallel to the principal axis. 
2. One travelling though the center of the lens. Such a ray travels undeviated through the lens.

So we have simplified the problem to just 4 rays. Two from the topmost point and two from the bottom-most point of the object.

The point on the image plane where the two rays from the topmost point of the arrow meet after tracing it through the lens is the location of that point on the image. The same holds true for the point on the bottom of the arrow. Once we know where the topmost and bottom-most points on the image are we can simply fill the gap in-between with a straight line since the object we have chosen is an arrow.

We will however see that the two rays that are considered for the bottom-most point will lie on the principal axis. One that is parallel to the principle axis and the one that travels straight through undeviated both lie on the principal axis. So we can simply avoid them. We now end up considering only two rays form the top-most point of the arrow. Once we know the location of this point on the image plane, we can simply draw a line form this point to intersect the principal axis perpendicular to it. This is because we know that the bottom-most point on the image plane should lie on the principal axis.

In Figure 4 we have considered the object (arrow) to be located between 2f and infinity distance from the lens. After tracing the two rays we realize that they meet at a point below the principal axis between the points f and 2f. Once we connect this point by a straight line to the principal axis we realize the following:
1. Image is formed between f and 2f
2. Image is inverted. 
3. Image is smaller than the object (diminished)

Note that the image has been shown with dotted lines to distinguish it form the object. 

Figure 4

Object_bt_2f_infinityImage formation by a lens. Object between 2f and infinity 

In Figure 5 we have considered the object (arrow) to be located at 2f distance from the lens. After tracing the two rays we realize that they meet at a point below the principle axis at the point 2f on the other side. Once we connect this point by a straight line to the principle axis we realize the following:

1. Image is formed at 2f
2. Image is inverted
3. Image is same size as that of the object

 

Figure 5


Object_at_2f
Image formation by a lens. Object at 2f

In Figure 6 we have considered the object (arrow) to be located between 2f and f from the lens. After tracing the two rays we realize that they meet at a point below the principal axis between 2f and infinity on the other side. Once we connect this point by a straight line to the principal axis we realize the following:

1. Image is formed between 2f and infinity
2. Image is inverted
3. Image is larger than the object (magnified)

 

Figure 6


Object_bt_2f_fImage formation by a lens. Object between 2f and f

In Figure 7 we have considered the object (arrow) to be located at a distance 'f' from the lens. After tracing the two rays we realize that they go parallel and donot meet.We realize the following:

1. Image is not formed as the light rays donot meet, they travel parallel to each other.

The results are similar to that in Figure 2. A lens cannot form and image if an object is placed at the focal point. The rays emanate out parallel to each other and cannot meet to form an image.

 

 

Figure 7


Object_at_focusImage formation by a lens. Object at f

In Figure 8 we have considered the object (arrow) to be located between f and the lens itself. After tracing the two rays we realize that they diverge after exiting the lens and cannot meet.We realize the following:

1. Image is not formed as the light rays donot meet, they diverge after exiting the lens. In this configuration we cannot caputre a real image of the object. Real image is an image that can be captured on a screen when on the side opposite to the side of the lens that has the object.
2. As we will see later an object placed between f and the lens however can create a virtual image on the same side as that of the object. 

Figure 8


Object_bt_f_lensImage formation by a lens. Object between f and lens

Let us now consider a more complex as shown in Figure 9. This object not only extends above and below the principal axis but also has 4 distinct features on it. The object is placed between f and 2f. We have considered two rays like before this time from each of the four distinct features. For clarity the rays from each distinct feature is shown in different colors. The distinct features on the image are formed at the points where the rays from the corresponding points on the object meet. The space between the distinct features are filled with straight lines. It can easily be appreciated how an inverted and magnified image of the object is formed on the image plane.

Figure 9


Complex_ObjectImage formation by a lens. A complex object placed between f and 2f

Posted in Basic Optics.

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