Chapter 1
Introduction to Optics

Rainer Heintzmann1,3 and Ulrich Kubitscheck2

1Friedrich Schiller-Universität, Institut für Physikalische Chemie und Abbe Center of Photonics, Helmholtzweg 4, 07743 Jena, Germany

2Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Physikalische & Theoretische Chemie, Wegelerstr. 12, 53115 Bonn, Germany

3Leibniz Institute of Photonic Technology, Albert-Einstein Str. 9, 07745 Jena, Germany

In this chapter, we introduce the wave nature of light by discussing interference, which is then used to explain the laws of refraction, reflection, and diffraction. We then discuss light propagation in the form of rays, which leads to the laws of lenses and ray diagrams of= optical systems. Finally, the working principles of the most common optical elements are outlined.

1.1 A Short History of Theories about Light

For a long time, scientists like Pierre Gassendi (1592-1655) and Sir Isaac Newton (1643-1727) believed that light consisted of particles named corpuscles traveling along straight lines, the so-called light rays. This concept explains brightness and darkness, and effects such as shadows or even the fuzzy boundary of shadows due to the extent of the sun in the sky. However, in the sixteenth century, it was discovered that light can sometimes "bend" around sharp edges by a phenomenon called diffraction. This phenomenon was not compatible with the predictions of the ray theory, but rather, light must be described as a wave. In the beginning, the wave theory of light - based on Christiaan Huygens' (1629-1695) work and expanded later by Augustin Jean Fresnel (1788-1827) - was not accepted. Siméon Denis Poisson, one of the judges for evaluating Fresnel's work in a science competition, tried to ridicule it by showing that Fresnel's theory would predict a bright spot in the center of the dark shadow of a round, illuminated obstacle. Poisson considered this to be nonsense. Another judge, Arago, however, then demonstrated that this spot does indeed exist and can be observed when measurements are done very carefully. This was a phenomenal success of the wave description of light, and the spot was named Poisson's spot.

Only at the beginning of the twentieth century quantum mechanics fused both theories and suggested that light has a dual character: a wave nature and a particle nature. Since then, the description of light has maintained this dual view. When it is interacting with matter, one often has to consider the quantum or particle nature of light. However, the propagation of these particles is described by equations written down by James Clerk Maxwell (1831-1879). The famous Maxwell wave equations of electrodynamics identify oscillating electromagnetic fields as the waves responsible for what we call light and still serve today as an extremely precise description of most aspects of light.

The wave concept is required to understand the behavior of light in the context of microscopy. Light propagation in the form of rays can explain refraction, reflection, and even aberrations, but fails to explain diffraction and interference. Therefore, we start out by introducing interference, an effect that is observed only when experiments are designed very carefully.

Figure 1.1 Interference experiment. In a simple arrangement of mirrors M1 and M2 and 50/50 beam splitters BS1 and BS2, an incoming light beam is split by BS1 and after reflection at M1 or M2 passes through BS2. Constructive interference occurs in direction of exit 1 if the optical path lengths of the two beams are exactly equal. But the light in exit 2 cancels by destructive interference.

1.2 Properties of Light Waves

1.2.1 An Experiment on Interference

Suppose that we perform the following experiment: We construct the instrument shown in Figure 1.1 consisting of only two ordinary mirrors and two 50/50 beam splitters. These beam splitters reflect 50% of the incoming light and transmit 50%. Now we use a laser of any color to illuminate beam splitter BS1. It is absolutely crucial that the distances along the two different light paths between the two beam splitters are exactly equal with a precision better than 1/10 000 of a millimeter. Then we observe that something surprising is happening: the light entering the device will leave only through exit 1. Exit 2 will be completely dark. This is very strange, as one would expect 50% of the light exiting on either side. Even more surprising is what happens if one blocks one of the two light paths inside the instrument. Now, 25% of the light will emerge from both exits.

The explanation for this effect of interference lies in the wave nature of light. According to the wave description, light is an electromagnetic field oscillating with time. If two electromagnetic waves with identical amplitudes spatially superimpose but oscillate in exactly opposite directions, they will completely cancel each other (Figure 1.2a). Exactly this happens at exit 2 of our instrument. At exit 1, constructive interference of the two beams coming along light path 1 and 2 occurs. In constructive interference, the two waves oscillate in phase and therefore add up (Figure 1.2b). That is why all light is exiting at exit 1. Of course, the exact reasons for the asymmetric interference processes are not immediately obvious, since they are a consequence of subtle asymmetries in light path 1 and 2 for the two beams reaching the different exits. The effect will be explained later in detail after we covered some basic features of light (see Box 1.4).

Figure 1.2 Destructive and constructive interference. (a) Two waves cancel each other if the electric fields of two interfering light waves E1(t) and E2(t) are always of opposite value, that is, they have a phase shift of p. (b) Constructive interference occurs if the two waves are completely in phase.

The explanation for the second part of the experiment - the effect of blocking one of the light paths - is simple. Blocking one light path will prevent interference and yield 25% brightness at either exit, as expected when splitting 50% of the total input light again in two equal amounts.

The discussed instrument is called a Mach-Zehnder interferometer. Such interferometers are extremely sensitive instruments capable of detecting extremely small path length differences in the two arms of the interferometer.

1.2.2 Physical Description of Light Waves

In wave-optical terms, a light ray is an oscillating and propagating electromagnetic field. What does that mean?

Figure 1.3 provides a graphical representation of such a light wave in the simplest case - a plane wave in vacuum. The wave comprises an electric and a magnetic field component. Both oscillate perpendicular to each other and also to the propagation direction of the wave. Therefore, light is called a transverse wave. Since most of the effects of light on matter are caused by its electric field, we often neglect for simplicity the magnetic field altogether. In the figure, several important parameters describing such waves are indicated. The wavelength, ?, describes the spatial distance between two electric field maxima. The direction and strength of the oscillating electrical field is given by the vector , and the propagation direction is characterized by the vector . The oscillation direction of the electric field in Figure 1.3 is constant in space. We call this the polarization direction of the light wave and such a wave linearly polarized. The polarization direction is not necessarily constant, for example, it may also rotate around the propagation direction. Such waves are called elliptically or circular polarized (see Box 1.1 for details).

Figure 1.3 Sketch of a linearly polarized electromagnetic wave. (a) Wave with electric and magnetic field components, and . (b) Temporal oscillation at a fixed place in space. (c) Still image of the wave.

Figure 1.4 Electromagnetic spectrum. Different types of radiation are essentially electromagnetic waves with oscillation frequencies or vacuum wavelengths ranging over many orders of magnitude. English version of a graphic by Horst Frank (https://de.wikipedia.org/wiki/Elektromagnetisches_Spektrum, https://en.wikipedia.org/wiki/GNU_Free_Documentation_License).

The oscillation of the electric field as a function of time at a specific position in space is shown in Figure 1.3b. The time period until the electric field assumes again an identical profile is designated as oscillation duration T. The frequency ? = 1/T at which the electric field oscillates at a given position defines the color of the light wave. Electromagnetic waves exist over a vast range of frequencies, out of which only a very small range is perceived as "light" and detected by our eyes or cameras (Figure 1.4). Yet, the same wave theory of light governs all wavelength ranges of the electromagnetic spectrum, from cosmic waves to gamma rays. However, light microscopy uses only the visible and near-infrared range. Blue light has a higher frequency ? and higher energy E = h? per photon than green, yellow,...