12 Optical Properties, Lecture 1
K Asokan
Learning Objectives:
From this module students may get to know about the following:
1. Behavior of a light wave in a homogenous medium.
2. Refractive index behavior in a medium (direction dependent).
3. Dispersion – Refractive index-wave length behavior in a medium, inter dependency on each other.
4. Snell law and total internal reflection in a medium.
Light or visible light is the portion of electromagnetic radiation that is visible to the human eye, responsible for the sense of sight. Visible light has a wavelength in a range from about 380 or 400 nanometres to about 760 or 780 nm, with a frequency range of about 405 THz to 790 THz. In physics, the term light often comprises the adjacent radiate on regions of infrared (at lower frequencies) and ultraviolet (at higher), not visible to the human eye.
Primary properties of light are intensity, propagation direction, frequency or wavelength spectrum, and polarization, while its speed, about 300,000,000 meters per second (300,000 kilometres per second) in vacuum, is one of the fundamental constants of nature.
Light, which is emitted and absorbed in tiny “packets” called photons, exhibits properties of both waves and particles. This property is referred to as the wave–particle duality. The study of light, known as optics, is an important research area in modern physics. Wave Optics Wavefront.
(a) A wavefront is the locus of all the points in space which receive light waves from a source in phase. If the source of light is a point source and the medium is homogeneous and isotropic, the wavefront will be spherical in shape. However, at very large distance from the point source, the shape of the wavefront changes from spherical to a plane wavefront.
(b) Shape of the wavefront may also change due to its passage through a refracting medium such as a lens.
(c) A wavefront is always normal to the light rays.
(d) A wavefront does not propagate in the backward direction.
Cylindrical wavefront. When the source of light is linear in shape, cylindrical wavefront is formed.
Plane wavefront. A small part of a spherical or cylindrical wavefront originating from a distant source can be considered as a plane wavefront
Huygens’ Principle.
(i) Each point on a given primary wavefront acts as a source of secondary wavelets, sending out disturbances (waves) in all directions in a similar manner as the original source of light does.
(ii) The new position of the wavefront at any instant (secondary wavefront) is given by the forward envelope to the secondary wavelets at that instant. Huygens’ construction (See Fig.)
Reflection on the Basis of Wave Theory.
The angle between the reflected ray and the normal is called angle of reflection.
The two laws of reflection are : (i) Angle of incidence is equal to angle of reflection. (ii) The incident ray, the reflected ray and the normal at the point of incidence all lie in the same plane.
LIGHT WAVES IN A HOMOGENEOUS MEDIUM:
We know from well-established experiments that light exhibits typical wave-like properties such as interference and diffraction. We can treat light as an EM wave with time-varying electric and magnetic fields and , respectively, which propagate through space in such a way that they are always perpendicular to each other and the direction of propagation is as depicted in Figure 1. The simplest traveling wave is a sinusoidal wave, which, for propagation along ?, has the general mathematical form,
Figure1: An electromagnetic wave is a traveling wave that has time-varying electric and magnetic fields that are perpendicular to each other and the direction of propagation z.
Equation (1) describes a monochromatic plane wave of infinite extent traveling in the positive direction as depicted in Figure 2. In any plane perpendicular to the direction of propagation (along z) the phase of the wave, according to Equation 1, is constant which means that the field in this plane is also constant. A surface over which the phase of a wave is constant is referred to as a wave front. A wave front of a plane wave is obviously a plane perpendicular to the direction of propagation as shown in Figure 2.
We know from electromagnetism that time-varying magnetic fields result in time-varying electric fields (Faraday’s law) and vice versa. A time-varying electric field would set up a time-varying magnetic field with the same frequency. According to electromagnetic principles, a traveling electric field as represented by Equation (1) would always be accompanied by a traveling magnetic field with the same wave frequency and propagation constant ( and k) but the directions of the two fields would be orthogonal as in Figure 1. Thus, there is a similar traveling wave equation for the magnetic field component . We generally describe the interaction of a light wave with a nonconducting matter (conductivity, = 0) through the electric field component rather than by because it is the electric field that displaces the electrons in molecules or ions in the crystal and thereby gives rise to the polarization of matter. However, the two fields are linked, as in Figure1, and there is an intimate relationship between the two fields. The optical field refers to the electric field.
Figure 2: A plane EM wave traveling along z, has the same (or ) at any point in a given xy plane. All electric field vectors in a given xy plane are therefore in phase. The xy planes are of infinite extent in the x and y directions.
We can also represent a traveling wave using the exponential notation since cos = [exp( )] where Re refers to the real part. We then need to take the real part of any complex result at the end of calculations. Thus, we can write Equation 1 as
planes as indicated in Figure 2. When the EM wave is propagating along some arbitrary direction k, as indicated in Figure 3, then the electric field E(r,t) at a point r on a plane perpendicular to k is
?(?,?)=?0 cos(??−??+?0) …………(3)
Because the dot product ?.? is along the direction of propagation similar to ??. The dot product is the product of ? and the projection of r onto k which is r’ in Figure 3, So ?.?=??′. Indeed, if propagation is along z, k • r becomes ??. In general, if k has components ??,?? and ?? along the ?,? and ? directions, then from the definition of the dot product, ?.?=???+???+??? .
Figure 3: A traveling plane EM wave along a direction k.
The time and space evolution of a given phase , for example, the phase corresponding to a maximum field, according to Equation 1 is described by
?=??−??+?0=????????
During a time interval , this constant phase (and hence the maximum field) moves a distance . The phase velocity of this wave is therefore . Thus the phase velocity is
where ? is the frequency ?=2??.
We are frequently interested in the phase difference Δ? at a given time between two points on a wave (Figure 1) that are separated by a certain distance. If the wave is traveling along z with a wavevector ?, as in Equation 1, then the phase difference between two points separated by Δ? is simply (? Δ?) since ?? is the same for each point. If this phase difference is 0 or multiples of 2?, then the two points are in phase. Thus, the phase difference Δ? can be expressed as ? Δ? or
REFRACTIVE INDEX:
When an EM wave is traveling in a dielectric medium, the oscillating electric field polarizes the molecules of the medium at the frequency of the wave. Intuitively, the EM wave propagation can be considered to be the propagation of this polarization in the medium. The field and the induced molecular dipoles become coupled. The net effect is that the polarization mechanism delays the propagation of the EM wave. The stronger the interaction between the field and the dipoles, the slower is the propagation of the wave. The relative permittivity measures the ease with which the medium becomes polarized, and hence it indicates the extent of interaction between the field and the induced dipoles. For an EM wave traveling in a nonmagnetic dielectric medium of relative permittivity , the phase velocity v is given by
Equation (6) is in agreement with our intuition that light propagates more slowly in a denser medium which has a higher refractive index. We should note that the frequency remains the same. The refractive index of a medium is not necessarily the same in all directions. In noncrystalline materials such as glasses and liquids, the material structure is the same in all directions and n does not depend on the direction. The refractive index is then isotropic. In crystals, however, the atomic arrangements and interatomic bonding are different along different directions. Crystals, in general, have nonisotropic, or anisotropic, properties. Depending on the crystal structure, the relative permittivity is different along different crystal directions. This means that, in general, the refractive index n seen by a propagating EM wave in a crystal will depend on the value of along the direction of the oscillating electric field (that is, along the direction of polarization). For example, suppose that the wave in Figure 1 is traveling along the direction in a particular crystal with its electric field oscillating along the x direction. If the relative permittivity along this x direction is, then √ . The wave therefore propagates with a phase velocity that is . The variation of n with direction of propagation and the direction of the electric field depends on the particular crystal structure. With the exception of cubic crystals (such as diamond) all crystals exhibit a degree of optical anisotropy which leads to a number of important applications. Typically noncrystalline solids, such as glasses and liquids, and cubic crystals are optically isotropic; they possess only one refractive index for all directions.
DISPERSION: REFRACTIVE -INDEX-WAVELENGTH BEHAVIOUR:
The refractive index of materials in general depends on the frequency, or the wavelength. This wavelength dependence follows directly from the frequency dependence of the relative permittivity . Figure 4 shows what happens to an atom in the presence of an oscillating electric field E which is due to a light wave passing through this location; it may also be due to an applied external field.
Figure 4: Electronic polarization of an atom. In the presence of a field in the +x direction, the electrons are displaced in the –x direction (from O), and the restoring force is in the +x direction.
In the absence of an electric field and in equilibrium, the center of mass C of the orbital motions of the electrons coincides with the positively charged nucleus at O and the net electric dipole moment is zero as indicated in Figure 4a. Suppose that the atom has Z number of electrons orbiting the nucleus and all the electrons are contained within a given shell. In the presence of the electric field E, however, the light electrons become displaced in the opposite direction to the field, so their center of mass C is shifted by some distance x with respect to the nucleus O which we take to be the origin as shown in Figure 4b. As the electrons are “pushed” away by the applied field, the Coulombic attraction between the electrons and nuclear charge “pulls in” the electrons. The force on the electrons, due to E trying to separate them away from the nuclear charge is ZeE. The restoring force Fr which is the Coulombic attractive force between the electrons and the nucleus, can be taken to be proportional to the displacement x provided that the latter is small. The reason is that Fr = Fr(x) can be expanded in powers of x, and for small x only the linear term matters. The restoring force Fr is obviously zero when C coincides with O (x = 0). We can write Fr =− where a constant and the negative sign is indicates that Fr is always directed toward the nucleus O.
First consider applying a dc field. In equilibrium, the net force on the negative charge is zero or ZeE = from which x is known. Therefore the magnitude of the induced electronic dipole moment is given by
As expected induced is proportional to the applied field. The electronic dipole moment in Equation (8) is valid under static conditions, i.e., when the electric field is a dc field. Suppose that we suddenly remove the applied electric field polarizing the atom. There is then only the restoring force – , which always acts to pull the electrons toward the nucleus O. The equation of motion of the negative charge center is then (force = mass x acceleration)
By solving this differential equation we can show that the displacement at any time is a simple harmonic motion, that is,
In essence, this is the oscillation frequency of the center of mass of the electron cloud about the nucleus and x0 is the displacement before the removal of the field. After the removal of the field, the electronic charge cloud executes simple harmonic motion about the nucleus with a natural frequency 0 determined by Equation 9; 0is also called the resonance frequency. The oscillations, of course, die out with time because there is an inevitable loss of energy from an oscillating charge cloud. An oscillating electron is like an oscillating current and loses energy by radiating EM waves; all accelerating charges emit radiation.
Consider now the presence of an oscillating electric field due to an EM wave passing through the location of this atom as in Figure 4b. The applied field oscillates harmonically in the +x and -x directions, that is, E = E0exp( ). This field will drive and oscillate the electrons about the nucleus. There is again a restoring force Fr acting on the displaced electrons trying to bring back the electron shell to its equilibrium placement around the nucleus. For simplicity we will again neglect energy losses. Newton’s second law for Ze electrons with mass Zme driven by E is given by
The induced electronic dipole moment is then simply given by Pinduced = – (Ze)x. The negative sign is needed because normally x is measured from negative to positive charge whereas in Figure 4b it is measured from the nucleus. By definition, the electronic polarizability is the induced dipole moment per unit electric field,
Thus, the displacement x and hence electronic polarizability increase as increases. Both become very large when approaches the natural frequency 0. In practice, charge separation x and hence polarizability ae do not become infinite at = 0 because two factors impose a limit. First, at large , the system is no longer linear and this analysis is not valid. Secondly, there is always some energy loss.
Given that the polarizability is frequency dependent as in Equation 9.12, the effect on the refractive index n is easy to predict. The simplest (and a very rough) relationship between the relative permittivity and polarizability is
where N is the number of atoms per unit volume. Given that the refractive index n is related to by n2 = , it is clear that n must be frequency dependent, i.e.
This type of relationship between n and the frequency co, or wavelength x, is called the dispersion relation.
SNELL’S LAW AND TOTAL INTERNAL REFLECTION (TIR):
We have so far discussed the propagation of electromagnetic wave in an isotropic, homogeneous, dielectric medium, such as in air or vacuum. In this lecture, we would discuss what happens when a plane electromagnetic wave is incident at the interface between two dielectric media. For being specific, we will take one of the medium to be air or vacuum and the other to be a dielectric such as glass.
Let us choose the interface to be the xy plane (z=0). The angles of incidence, reflection and refraction are the angles made by the respective propagation vectors with the common normal at the interface.
We have indicated the propation vectors in the appropriate medium by capital letters I, R and T so as not to confuse with the notation for the position vector and time t.
The principle that we use to establish the laws of reflection and refraction is the continuity of the tangential components of the electric field at the interface, as discussed extensively during the course of these lectures. Let us represent the component of the electric field parallel to the interface by the superscript ∥. We then have,
This equation must remain valid at all points in the interface and at all times. That is obviously possible if the exponential factors is the same for all the three terms or if they differe at best by a constant phase factor. Considering, the incident and the reflection terms, we have,
Here n is the refractive index of the second medium with respect to the incident medium.
This is Snell’s law which relates the angles of incidence and refraction to the refractive indices of the media.
When ( n1 > n2), then obviously the transmitted angle is greater than the incidence angle When the refraction angle reaches 90°, the incidence angle is called the critical angle which is given by
When the incidence angle , exceeds , then there is no transmitted wave but only a reflected wave. The latter phenomenon is called total internal reflection (TIR). The effect of increasing the incidence angle is shown in Figure 6. It is the TIR phenomenon that leads to the propagation of waves in a dielectric medium surrounded by a medium of smaller refractive index as in optical waveguides (e.g., optical fibers).
Figure 6: Light wave traveling in a denser medium strikes a less dense medium. Depending on the incidence angle with respect to determined by the ratio of the refractive indices, the wave may be transmitted (refracted) or reflected, (a) ??<?? (b) ??=?? (c) ??>?? and total internal reflection (TIR).
Summary:
- Behaviour of a light in a homogenous medium is explained.
- Direction dependent behavior of the refractive index is explained.
- Snell’s law and total internal reflection is also explained.
- Dispersion – Refractive index-wave length behavior in a medium,interdependency on each other is also described.
| you can view video on Optical Properties,Lecture 1 | |
References:
1. A.J. Dekker (1957). Solid state physics, Prentce-Hall, Inc .
2. Leonid A. Azaroff (1960), Introduction to solids, McGRAW-HILL BOOK COMPANY INC.
3 Kasap, S. (2006). Principles of electronic materials and devices (3rd ed.). Boston: McGraw-Hill
4. Frederick Wooten (1972) ACADEMIC PRESS INC
5. Harald Ibach, Hans Luth (2009) Springer series on material science
6. Ashcroft & Mermin (1976) Harcourt College Publisher.
7. Philip Hofmann (2008) , Solid State Physics An Introduction, WILEY-VCH Verlag GmbH & Co. KGaA.
References and Suggestive Readings
- James Patterson, Bernard Bailey (2010) Solid State Physics, Second Edition , Springier series on material science.
- S.L. Kakani (2004), Material Science , New age international (P) Limited, publishers
- Harald lbach & Hans Luth (1995) , Solid state physics- An introduction to material science , Springier
- Donald A. Neamen (2003), Semiconductor physics and devices, Third edition, Mc-Graw Hill Higher Education
Web Links
1. http://www.physics.usyd.edu.au/super/life_sciences/AN/AN4.pdf
2. http://hyperphysics.phy-astr.gsu.edu/hbase/mod3.html
3. http://www.ehs.utoronto.ca/services/radiation/radtraining/module3.html
4. http://www.tesec-int.org/TechHaz-site%2008/Radiation-interaction.pdf
5. https://www.youtube.com/watch?v=FzfR_IwfMW0
6. https://www.youtube.com/watch?v=424QV3tD4PE
Additional Topics to be studied
Interaction of Radiation with Matter
The emitted light usually has a longer wavelength, and therefore lower energy, than the absorbed radiation. Fluorescence occurs when an orbital electron of a molecule or atom relaxes to its ground state by emitting a photon of light after being excited to a higher quantum state by some type of energy. In a phosphorescence, excitation of electrons to a higher state is accompanied with the change of a spin state. Relaxation is a slow process since it involves energy state transitions “forbidden” in quantum mechanics.
Molecules have energy levels determined by the molecular orbitals that hold the molecule bound together. In the case of atoms it is the atomic orbitals what determines the energy levels of the electrons. In this section we will concern ourselves with molecular photoluminescence.
Photoluminescence • Molecules that have an electronic excitation are excited and that have a vibrational excitation are hot. With light absorption, molecules may become hot and excited. Physical process that leads to excited molecules can be physical (e.g. absorption of light), mechanical (e.g. friction), or chemical (e.g. reactions). When excited molecular states decay back to the ground state, resulting in the emission of light, they are undergoing a luminescence process. Generation of excited molecules by light absorption, that then decay emitting visible light, is photoluminescence. Photoluminescence processes are divided into 2 classes: –Fluorescence and Phosphorescence
Fluorescence:
Property of some atoms or molecules to absorb light at a particular wavelength and then emit light at a longer wavelength (lower frequency) than the incident light. Absorption process occurs over short time interval ( 15 10 s) and does not change the direction of the e spin. Vibrational relaxation (emission of IR while lowering vibrational state) occurs in ~ 12 10 s.
De-excitation to electronic ground state with emission of lower frequency light and IR occurs in 10 9 s. Because vibrational relaxation occurs ~1000 times faster than de-excitation, most molecules return to a low-vibrational state before the de-excitation takes place – Emitted wavelengths nearly independent of incident radiation. Shift in wavelength between absorption and emission spectra is the Stokes shift.
A quantitative expression of the efficiency of fluorescence is the fluorescent quantum yield,Φf , which is the fraction of excited molecules returning to the ground state by fluorescence.
Phosphorescence: In the fluorescence process, the electron did not change its spin direction but under the appropriate conditions, a spin-flip can occur.
Spin flip can occur during absorption, or afterwards. The situation where no spin flip occurs, the molecule is in a singlet state. When the electron undergoes a spin-flip, a triplet state is created.
The light emission process must wait until electron undergoes a spin-flip to revert back to it original state
- May take 3 10 – 2 10 s.
- Light emission is delayed long enough so that materials “glow in the dark” after exposure to light
- Because of the lower energy of triplet state, lower energy photon emission than incident or fluorescence photons
Molecular triplet states are more often involved in photo-chemical and photo-biological reactions than singlet states in molecules because of the long lifetime.
X- and Gamma Ray
The interaction of photons (γ-quantums) with matter involves several distinct processes. The relative importance and efficiency of each process is strongly dependent upon the energy of the photons and upon the density and atomic number of the absorbing medium. We shall first consider the general case of photon attenuation and then discuss some of the important processes separately.
Rayleigh Scattering
When a photon interacts with atom, it may or may not impart some energy to it. The photon may be deflected with no energy transfer. This process is called Rayleigh scattering and is most probable for very low-energy photons
Compton Effect
The Compton effect is usually the predominant type of interaction for medium energy photons (0.3 to 3 MeV). In this process the photon interacts with an atomic electron sufficiently to eject it from orbit, the photon retains a portion of its original energy and continues moving in a new direction. Thus, the Compton effect has an absorption component and scattering component. The amount of energy lost by the photon can be related to the angle at which the scattered photon travels relative to the original direction of travel. The scattered photon will interact again, but since its energy has decreased, it becomes more probable that it will enter into a photoelectric or Rayleigh interaction. The free electron produced by the Compton process may be quite energetic and behave like a beta particle of similar energy, producing secondary ionization and excitation before coming to rest.
Photoelectric Absorption
The most probable fate of a photon having energy slightly higher than the binding energy of atomic electrons is photoelectric absorption. In this process, the photon transfers all of its energy to the electron and its own existence terminates. The electron will escape its orbit with a kinetic energy equal to the difference between the photon energy and its own binding energy. Photoelectric absorption is most important for photons below 0.1 MeV if the absorbing medium is water or biological tissue. However, in high Z (atomic mass number) materials such as lead, this process is relatively important for photons up to about 1 MeV. As with ionization produced by any process, secondary radiation are initiated, in this case, by the photoelectron which may have sufficient energy to produce additional ionization and excitation of orbital electrons. Also, filling of the electron vacancy left by the photoelectron results in characteristic X-rays.
Pair Production
Photons with energy greater than 1.024 MeV, under the influence of the electromagnetic field of a nucleus, may be converted into electron and positron. At least 1.024 MeV of photons energy are required for pair production, because the energy equivalent of the rest mass of the electron and positron is 0.51 MeV each. Pair production is not very probable, however, until the photon energy exceeds about 5 MeV. The available kinetic energy to be shared by the electron and the positron is the photon energy minus 1.02 MeV, or that energy needed to create the pair. The probability of pair production increases with Z of the absorber and with the photon energy.
Relative Importance of Photon Attenuation Processes
The various processes of photon attenuation can now be considered by examining the effects of photon energy and atomic mass number of the absorber on their relative importance (Figure 1). The lines in the figure indicate the values of the photon energy and Z where the probabilities of occurrence of two major processes are equal.
