Wave phenomena

    1.Wave phenomena

    The properties of light we set off to describe - reflection, refraction, diffraction, and interference - can all be explained, both qualitatively and quantitatively, in terms of light viewed as a wave. The success of these descriptions of the properties of light was a triumph of the wave picture, and by the 1850s this model of light was the generally accepted one. 

    As a wave it have such properties: Refraction, Dispersion, Interference of Waves,

    Diffraction

    2. Interference of Waves

    In contrast to refraction, interference can only be naturally explained by thinking of light as waves. Consider two waves meeting as shown in the Figure below:   

     
For constructive interference, the waves meet in phase, i.e. so that the crests of each wave coincide. Indestructive interference, the waves meet out of phase, so that the crest of one wave coincides with a trough of the other wave, and they cancel each other out. This readily explains the double slit interference pattern of Fig. The two light rays start off in phase since they come from the same source. By the time they reach the screen after having been diffracted through the slits, they have travelled different distances, so that the crests of the ray that travels further (the bottom ray in Fig. lag a little behind those of the top ray. If they lag behind by a half a wave length, the crest of one ray will meet the trough of the other at the screen, and destructive interference will naturally result. On the other hand if one ray lags behind by a full wavelength (or two or three full wavelengths, etc.) a crest will still meet a crest at the screen, and a bright spot will appear due to constructive interference. 

    3. Polarization 

    Polarization is a property of certain types of waves that describes the orientation of their oscillations. Electromagnetic waves, such as light exhibit polarization, sound waves do not have polarization because the direction of vibration and direction of propagation are the same.

    Polarization of light is described by specifying the orientation of the wave's electric field. Usually the polarization is perpendicular to the wave's direction of travel. In this case, the electric field may be oriented in a single direction (linear polarization), or it may rotate as the wave travels (circular or elliptical polarization). In much complicated cases, the oscillations can rotate both. The type of rotation in a given wave is called the wave's chirality or handedness.

    Unpolarized or natural light is the set of electromagnetic waves with all possible planes of oscillation of vectors E and H 

    Light is considered to be "linearly polarized" when it contains waves that only fluctuate in one specific plane. A polarizer is a material that allows only light with a specific angle of vibration to pass through. The direction of fluctuation passed by the polarizer is called the "easy" axis. 

    4. Diffraction is a phenomenon that appears when a wave meets an obstacle. In classical physics, the diffraction phenomenon is described as the apparent bending of waves around small obstacles and the spreading out of waves past small openings. Similar effects occur when light waves travel through a medium with a varying refractive index. Diffraction occurs with all waves, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. Its effects are generally most pronounced for waves where the wavelength is roughly similar to the dimensions of the diffracting objects.

    Single-slit diffraction

    A long slit of infinitesimal width which is illuminated by light diffracts the light into a series of circular waves which emerges from with uniform intensity.

    A slit which is wider than a wavelength produces interference effects in the space after the slit. These can be explained by assuming that the slit behaves as though it has a large number of point sources spaced evenly across the width of the slit. If the light is monochromatic, these sources all have the same phase. So we may obtain minima and maxima in the diffracted light. (Newton’s rings)

    We can find the angle of first minimum in the diffracted light by the following reasoning. The light from a source located at the top edge of the slit interferes destructively with a source located at the middle of the slit, when the path difference between them is equal to λ/2. Similarly, the source just below the top of the slit will interfere destructively with the source located just below the middle of the slit at the same angle. We can continue this reasoning along the entire height of the slit to conclude that the condition for destructive interference for the entire slit is the same as the condition for destructive interference between two narrow slits a distance apart that is half the width of the slit. The path difference is given by   so that the minimum intensity occurs at an angle θ_min given by

    

    where

• d is the width of the slit,
• θ_min is the angle of incidence at which the minimum intensity occurs, and
• λ is the wavelength of the light

    A similar argument can be used to show that if we imagine the slit to be divided into four, six, eight parts, etc., minima are obtained at angles θ_n given by

    

    where

• n is an integer other than zero.

    5. Photon

    Under the photon theory of light, a photon is a discrete bundle (or quantum) of electromagnetic (or light) energy. Photons are always in motion and, in a vacuum, have a constant speed of light to all observers, at the vacuum speed of light (more commonly just called the speed of light) of c = 2.998 x 10⁸ m/s.

    Basic Properties of Photons According to the photon theory of light, photons . . .

• move at a constant velocity, c = 2.9979 x 10⁸ m/s (i.e. "the speed of light"), in free space
• have zero mass and rest energy.
• carry energy and momentum, which are also related to the frequency nu and wavelength lamdba of the electromagnetic wave by E = h nu and p = h / lambda.
• can be destroyed/created when radiation is absorbed/emitted.
• can have particle-like interactions (i.e. collisions) with electrons and other particles, such as in the Compton effect.

    Let us see some of the properties of photons.

• The energy of a photon related to an electromagnetic wave whose frequency is   is given by 
• We know light propagates with the speed of light. Therefore the speed of photons is c.
• The rest mass   of a photon is 0.

    We learned earlier that objects with nonzero rest mass cannot propagate with the speed of light. This comes from the fact, that if you calculate the dynamical massfor   greater than 0 and v=c, you get nonsense, a diverging expression. It looks as if the dynamical mass became infinite, which means the energy of the particle became infinite. However, if the rest mass is zero, you get 0/0, which in mathematics means an arbitrary number. By using mathematical tricks (limits), one can still extract some information from such limits and get a meaningful value for the dynamical mass.

• The dynamical mass of the photon is

    

• The momentum of a photon is

      (just as in classical physics, momentum equals mass times speed) which gives us 

    6. Momentum of photon

    We should quickly note the case where the rest mass of an object is zero (such is the case for a photon--a particle of light). Given the equation for the energy in the form of Equation 1:8 (E = gamma*m*c^2), one might at first glance think that the energy was zero when m = 0. However, note that massless particles like the photon travel at the speed of light. Since gamma goes to infinity as the velocity of an object goes to c, the equation E = gamma*m*c^2 involves one part which goes to zero (m) and one part which goes to infinity (gamma). Thus, it is not obvious what the energy would be. However, if we use the energy equation in the form of Equation 1:7 
(E^2 = p^2*c^2 + m^2*c^4), then we can see that when m = 0 then the energy is given by E = p*c).  Now, a photon has a momentum (it can "slam" into particles and change their motion, for example) which is determined by its wavelength (lambda) in the equation p = h/lambda (where h = 6.626E(-34) Joules is called Planck's constant). A photon of wavelength lambda thus has an energy given by E = p*c= h*c/lambda, even though it has no rest mass.

    7. Photon mass

    Photons are traditionally said to be massless.  This is a figure of speech that physicists use to describe something about how a photon's particle-like properties are described by the language of special relativity.

    The logic can be constructed in many ways, and the following is one such.  Take an isolated system (called a "particle") and accelerate it to some velocity v (a vector).  Newton defined the "momentum" p of this particle (also a vector), such that p behaves in a simple way when the particle is accelerated, or when it's involved in a collision. For this simple behaviour to hold, it turns out that p must be proportional to v. The proportionality constant is called the particle's "mass" m, so that p = mv.

    In special relativity, it turns out that we are still able to define a particle's momentum p such that it behaves in well-defined ways that are an extension of the newtonian case.  Although p and v still point in the same direction, it turns out that they are no longer proportional; the best we can do is relate them via the particle's "relativistic mass" m_rel.  Thus           

    p = m_relv .

    When the particle is at rest, its relativistic mass has a minimum value called the "rest mass" m_rest.  The rest mass is always the same for the same type of particle.  For example, all protons, electrons, and neutrons have the same rest mass; it's something that can be looked up in a table.  As the particle is accelerated to ever higher speeds, its relativistic mass increases without limit.

    It also turns out that in special relativity, we are able to define the concept of "energy" E, such that E has simple and well-defined properties just like those it has in newtonian mechanics.  When a particle has been accelerated so that it has some momentum p (the length of the vector p) and relativistic mass m_rel, then its energy E turns out to be given by           

    E = m_relc² ,   and also    E² = p²c² + m²_restc⁴ .           (1)

    There are two interesting cases of this last equation:

1. If the particle is at rest, then p = 0, and E = m_restc².
2. If we set the rest mass equal to zero (regardless of whether or not that's a reasonable thing to do), then E = pc.

    In classical electromagnetic theory, light turns out to have energy E and momentum p, and these happen to be related by E = pc.  Quantum mechanics introduces the idea that light can be viewed as a collection of "particles": photons.  Even though these photons cannot be brought to rest, and so the idea of rest mass doesn't really apply to them, we can certainly bring these "particles" of light into the fold of equation (1) by just considering them to have no rest mass.  That way, equation (1) gives the correct expression for light, E = pc, and no harm has been done.  Equation (1) is now able to be applied to particles of matter and "particles" of light.  It can now be used as a fully general equation, and that makes it very useful.

    8. Photoelectric effect

    The photoelectric effect posed a significant challenge to the study of optics in the latter portion of the 1800s. It challenged theclassical wave theory of light, which was the prevailing theory of the time. It was the solution to this physics dilemma that catapulted Einstein into prominence in the physics community, ultimately earning him the 1921 Nobel Prize.

    When a light source (or, more generally, electromagnetic radiation) is incident upon a metallic surface, the surface can emit electrons. Electrons emitted in this fashion are calledphotoelectrons (although they are still just electrons). This is depicted in the image to the right.

    By administering a negative voltage potential (the black box in the picture) to the collector, it takes more energy for the electrons to complete the journey and initiate the current. The point at which no electrons make it to the collector is called the stopping potential V_s, and can be used to determine the maximum kinetic energy K_max of the electrons (which have electronic charge e) by using the following equation:

    K_max = eV_s

    It is significant to note that not all of the electrons will have this energy, but will be emitted with a range of energies based upon the properties of the metal being used. The above equation allows us to calculate the maximum kinetic energy or, in other words, the energy of the particles knocked free of the metal surface with the greatest speed, which will be the trait that is most useful in the rest of this analysis.

    The Classical Wave Explanation

    In classical wave theory, the energy of electromagnetic radiation is carried within the wave itself. As the electromagnetic wave (of intensity I) collides with the surface, the electron absorbs the energy from the wave until it exceeds the binding energy, releasing the electron from the metal. The minimum energy needed to remove the electron is the work function phi of the material. (Phi is in the range of a few electron-volts for most common photoelectric materials.)

    Three main predictions come from this classical explanation:

1. The intensity of the radiation should have a proportional relationship with the resulting maximum kinetic energy.
2. The photoelectric effect should occur for any light, regardless of frequency or wavelength.
3. There should be a delay on the order of seconds between the radiation’s contact with the metal and the initial release of photoelectrons.

    By 1902, the properties of the photoelectric effect were well documented. Experiment showed that:

1. The intensity of the light source had no effect on the maximum kinetic energy of the photoelectrons.
2. Below a certain frequency, the photoelectric effect does not occur at all.
3. There is no significant delay (less than 10^-9 s) between the light source activation and the emission of the first photoelectrons.

    As you can tell, these three results are the exact opposite of the wave theory predictions. Not only that, but they are all three completely counter-intuitive. Why would low-frequency light not trigger the photoelectric effect, since it still carries energy? How do the photoelectrons release so quickly? The photon's energy would be associated with its frequency (nu), through a proportionality constant known as Planck's constant (h), or alternately, using the wavelength (lambda) and the speed of light

    E = h nu = hc / lambda

    or the momentum equation: p = h / lambda

    In Einstein's theory, a photoelectron releases as a result of an interaction with a single photon, rather than an interaction with the wave as a whole. The energy from that photon transfers instantaneously to a single electron, knocking it free from the metal if the energy (which is, recall, proportional to the frequency nu) is high enough to overcome the work function (phi) of the metal. If the energy (or frequency) is too low, no electrons are knocked free.

    If, however, there is excess energy, beyond phi, in the photon, the excess energy is converted into the kinetic energy of the electron:

    K_max = h nu - phi

    The maximum kinetic energy results when the least-tightly-bound electrons break free, but what about the most-tightly-bound ones; The ones in which there is just enough energy in the photon to knock it loose, but the kinetic energy that results in zero? Setting K_max equal to zero for this cutoff frequency (nu_c), we get:

    nu_c = phi / h

    or the cutoff wavelength: lambda_c = hc / phi

    These equations indicate why a low-frequency light source would be unable to free electrons from the metal, and thus would produce no photoelectrons. 

    9. Double refraction

    Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certainanisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientistRasmus Bartholin in 1669, who saw it in calcite.^[1] The effect is now known to also occur in certain plastics, magnetic materials, various noncrystalline materials, and liquid crystals.^[2]