Wave Optics

What is Light?

Light is a name for a range of electromagnetic radiation that can be detected by the human eye. What is electromagnetic radiation?Electromagnetic radiation has a dual nature as both particles and waves. One way to look at it is as changing electric and magnetic fields which propagate through space, forming an electromagnetic wave. This wave has amplitude, which is the brightness of the light, wavelength, which is the color of the light, and direction at which it is vibrating, called polarization. This is the classical interpretation, described by Maxwell's Equations.

What is Light?

Than Planck, Einstein and others came along with quantum theory. In terms of the modern quantum theory, electromagnetic radiation consists of particles called photons, which are packets ("quanta") of energy which move at the speed of light. In this particle view of light, the brightness of the light is the number of photons, the color of the light is the energy contained in each photon, and four quantum numbers determine the polarization.

What is Light?

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Which interpretation is correct? Both of them, actually. It turns out electromagnetic radiation can have both wave-like and particle-like properties. In this exploration of light, we will primarily take the wave viewpoint as it is a more useful description of the everyday properties of light, but keep in mind that both viewpoints are valid, and sometimes we will use the quantum viewpoint too.Light ranges from wavelengths of 7x10-5 cm (red) to 4x10-5 cm (violet) and (like all electromagnetic radiation) travels at the speed of light, 299,792,458 meters per second.The frequency (number of wavelengths per second) of a light wave may be calculated using the equation

In quantum theory, a photon has energy equal to

Physical optics

Physical optics, or wave optics, is the branch of optics which studies interference, diffraction, polarization, and other phenomena for which the ray approximation of geometric optics is not valid.Geometric optics is an approximation of physical optics which ignores wavelength , and consequently wave effects.

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Huygens' principleHuygens considered light to be a wave. He envisioned a wave crest advancing by imagining each point along the wave crest to be source point for small, circular, expanding wavelets, which expand with the speed of the wave. The surface tangent to these wavelets determines the contour of the advancing wave. Figure illustrates Huygens' construction for a plane wave (a) and for a spherical wave (b)

Conditions for Interference

Coherent waves maintain constant phase in time for each point of space

For interference of light waves the sources must be coherent that is they maintain constant phase with respect to each other.To get coherence, use a monochromatic source shining through 2 slits, or use laser, which is monochromatic, and send the beam through 2 different paths (can use slits or other ways)

Don’t normally see interference in light, because:Light waves have such small wavelengthsLight waves are usually not coherent Ordinary sources produce well-phased light for only about 10-8 sec, effectively incoherentOrdinary sources have random changing relative phase - incoherent

Interference

Because light is a wave, the superposition principle is valid to determine the constructive and destructive interferences for light waves. Interference in light waves is not easy to observe because the wavelengths are so short. For constructive interference, two waves must have the two contributing crests and the two troughs arriving at the same time. For destructive interference, a crest from one wave and a trough from the other must arrive at a given point at the same time.

Young’s Double-Slit ExperimentThomas Young first demonstrated interference from light waves with a double slit. The schematic diagram for this experiment is shown in Figure.The single light source is located at S0 (the light waves must have identical frequency and phase. The light beam is also considered to be of one color), and the light goes through two very narrow openings at S1 and S2. Each of the slits act as a source for circular expanding waves. The points of intersection of two crests, one from each slit, are points of constructive interference. The point of intersection of a crest from one slit and a trough from the other slit is a point of destructive interference. Therefore, the interference pattern called fringes, consisting of alternating light and dark bars, will be seen on the screen.

Figure illustrates the rays coming through two slits that are directed to the point P on the screen.

The difference in path length of the two rays is given by d sin θ = l2 = l1. If the path difference is a whole number of wavelengths, then constructive interference takes place. If the paths differ by a half number of wave lengths, destructive interference occurs. Using n to represent any integer, the two cases may be written

Diffraction of light through a single slit.Fresnel diffraction

A single slit yields an interference pattern due to diffraction and interference. Imagine that the slit is wide enough to allow a number of wavelets. The rays from A and B interfere at P on a distant screen. As shown, AP exceeds BP by half a wavelength; therefore, the represented waves destructively interfere. Also for every wave originating between A and B, there is another point between B and C with a wavelet that will destructively interfere. The wavelets cancel in pairs; thus, point P is a minimum or dark point on the screen. Whenever the path difference between AP and CP is a whole number of wavelengths, a dark fringe will be produced on the screen because the wavelets can be seen to completely cancel in pairs.

Figure illustrates the light rays traveling to another point on the screen.

The region of wavelets is divided into three. Again, the waves through two regions cancel in pairs, but now the waves from one region constructively interfere to produce a bright point on the screen. This is partial reinforcement.

The positions of the light and dark fringes formed by a single slit are summarized in the intensity versus angle sketch shown in Figure. The center region of the pattern will be the brightest band because the wavelets completely, constructively interfere in the middle.

Fraunhofer diffraction

If you replace a circular hole with a circular disk of the same size, you get essentially the same pattern, even with a bright spot in the center.

The Diffraction Grating

Polarization of light

Light is a transverse wave, meaning that it oscillates in a direction perpendicular to the direction in which it is traveling. However, a wave is free to oscillate right and left or up and down or at any angle between the vertical and horizontal.

Natural light is generally unpolarized, all planes of propagation being equally probable.

Classification of Polarization. Linear polarizationLight in the form of a plane wave in space is said to be linearly polarized. The transverse electric field wave is accompanied by a magnetic field wave as illustrated.

Circular Polarization

Circularly polarized light consists of two perpendicular electromagnetic plane waves of equal amplitude and 90° difference in phase. The light illustrated is right- circularly polarized.

Elliptical Polarization

Elliptically polarized light consists of two perpendicular waves of unequal amplitude which differ in phase by 90°. The illustration shows right- elliptically polarized light.

Methods for achieving Polarization of lightPolarization may be achieved by reflection, scattering, absorption or by the use of Birefringent Materials in Nicol prism.Polarization by AbsorptionSome kinds of crystals have a special property of polarizing light, meaning that they force light to oscillate only in the direction in which the crystals are aligned. They absorb more light in one incident plane than another, so that light progressing through the material become more and more polarized as they proceed. This anisotropy in absorption is called dichroism. There are several naturally occurring dichroic materials, and the commercial material polaroid also polarizes by selective absorption.

The scattering of light off air molecules produces linearly polarized light in the plane perpendicular to the incident light.

Polarization can be analized by crossed polarizers. Law of Malus.

Since the transmitted intensity is proportional to the amplitude squared, the intensity is given by:

The Law of Malus gives the transmitted intensity through two ideal polarizers. Note that it gives zero intensity for crossed polarizers.