 an electromagnetic wave impinges on an atom or a material containing atomic species. That implies that it could be an ion or a metallically bonded material, the electron cloud is set into oscillation. This situation is like a dipole oscillator and therefore, this dipole oscillator radiates emission all around it in this having the same frequency and this is we can for a first approximate assume is not direction dependent. This process of this is the basically the process which we called scattering. In other words, it is the electron cloud or an atom which is responsible in combination of ion cores. Initially consider the ion core is fixed and the electrons are one which are oscillating and this dipole oscillator sends out radiation in all directions and this is the process of scattering. In other words scattering is not merely some kind of reflection, it is actually a dipole oscillator at work and similar to absorption scattering is also frequency dependent. So, in other words the strength of scattering depends on the frequency. Now, there are two possible ways in which transmission can take place, where earlier pointed out there are three possibilities when light falls on a medium or light passes array passes through a medium. One is reflection, other is refraction and third is absorption and when you are talking about refraction we are essentially assuming that the ray is going into the second medium, which we otherwise we call transmission. Of course, if there as we shall see soon if the refractive index is close to one in that case the refraction angle means the deviation in angle is very small. The two possibilities are for transmission. Number one, if the medium is extremely sparse like you are talking about interstellar space, wherein the atomic density or the matter density is extremely small, then the wave actually pass between the particles. So, schematically I can draw it like this. So, I have my medium, which is extremely sparse and my wave is actually passing between the particles. So, essentially there is no interaction and this is what I mean what you can call the trivial case of transmission. But the more common mechanism of transmission and this is we are talking through a denser media is a phenomenon of forward scattering. In other words what we normally call transmission like suppose light is coming from the left of this room and going towards the right of this room, we call it transmission. It looks to us as if it is just motion of light through the medium without any interaction at all, but in reality it is dominated by forward scattering. And when we are talking about molecules in the atmosphere for instance, the blue color is coming because of the frequency dependence of scattering. In other words this frequency goes as omega power 4 and that means that the higher the frequency more will be the scattering. And that is where the blue light is scattered and therefore, we see our sky is blue and this phenomenon is known as Raleigh scattering. The unscattered light which is the longer wavelength red radiation which goes through straight is what is responsible for the red redishness of the sky in the evening because in the evening what happens is scattered light is lost. The unimpeded or the unscattered light the forward scattered light which goes through is the one you which meets our eye and therefore, the evening sky looks red while the daytime sky actually looks blue. So, this is the phenomenon of Raleigh scattering and therefore, we have to note that what we call transmission in common usage may turn out to be what we might call forward scattering. So, this has to be remembered that means the forward wave is reinforced and if there is no lateral scattering at all that means the all the other waves are destructively interfering and therefore, we essentially have a wave transmission in the forward direction, but as we saw in the case of the scattering through the atmosphere often there is a lateral scattering though the amount may be small, but this is what is responsible for the color of the sky. When a wave is being transmitted from one medium to another say vacuum to a glass for instance to take an example the frequency remains constant. So, what is constant when a wave actually propagates from a rarer medium to a denser medium for instance we take the example of vacuum to glass it is the frequency which remains constant, but the velocity of the wave is altered. In fact, the velocity of the wave is decreased and the ratio of the velocities of the wave in vacuum which is a fundamental constant C divided by the velocity in the medium. So, the denominator is the velocity in the medium this velocity this is what we call in common language the refractive index. Therefore, the wave enters the medium its velocity decreases and this ratio we call the refractive index though we will not be talking too much about refractive index per say of nano structures material then suppose I take a glass sphere and keep on reducing its size to the nano scale what happens to its refractive index. So, I will mention it in the passing that often as we note here that the refractive index is a function of the can be written in terms of the permittivity and permeability of the medium with respect to the permittivity and permeability of free space. In other words they depend on what we call the relative permittivity and the relative permeability of the medium and because these constants or these characteristics of the medium the permeability and permittivity are not sensitive functions of the size. Therefore, often you will find that the even when you approach the nano scale the refractive index does not change much, but however there is expected to be some change. So, we have this concept called a refractive index which is basically the ratio of the velocity of light in vacuum vis a vis the velocity in a given medium. Now, it in the previous relation the K m the relative permeability or the relative magnetic permeability is actually close to unity and therefore, K e is what is going to determine the refractive index essentially and this K e is a function of the frequency of the electromagnetic wave and this is what is responsible for the phenomena of dispersion. Dispersion essentially implies we are referring to some kind of a Newton's experiment where we in sand in white light into a prism and we see that it actually spreads into the what you might call the Wibgier colors and this is coming from dispersion the phenomena of dispersion which essentially tells you that now my refractive index n is a function of the frequency of the material or frequency of the electromagnetic wave and this dependence of n is coming from K e because K m is essentially close to unity. Therefore, because K e depends on the frequency we see that the refractive index of a material actually depends on the frequency. Physically if you look where is this dependence of n coming from physically you note that the dependence of frequency dependence of n is coming from actually four factors if include three of them are the main factors usually cited, but there are four factors here. So, let me write down this as four totally and these four factors are what you might call orientation polarization, electronic polarization, ionic polarization and space charge polarization. And from these terms it is obvious that I am referring to for instance a medium like a dielectric medium where in polarization is possible. And therefore, I am talking about the propagation of an electromagnetic wave like light and let me for now assume that light is passing through some medium like quartz or some other crystalline medium and all these polarizations are responsible for what you might call the frequency dependence of n. That means that these polarizations are activated at selective frequency regimes and therefore, I mean in other words these are not active at all frequency regimes. And therefore, there is a dependence of n on the dependence of n on the frequency of the electromagnetic wave. In electronic polarization essentially you see that because the presence of an electric field of course, in the case of a wave this electric field you know and the magnetic field are oscillating quantities. Because you know that an electromagnetic wave is nothing but a coupled electric and magnetic oscillations it propagating through space. Now, because for now we will assume showing it as if it is a fixed e, but we have to remember that is actually we are talking about and time dependent e and correspondingly time dependent h. Due to this electric field what happens is that the center of mass of the electron cloud is shifted with respect to the center of mass of the nucleus. And essentially it is the electron cloud which shifts and this implies that there is a net dipole created in the material. So, this dipole which is instantaneously created at the atomic level is what is responsible for this what you might call electronic polarization. So, this is actually point number two which is electronic polarization. The other possibility is what you might call ionic polarization in a and this can obviously, happen only in a dielectric medium having ions. In that case what happens is the center of mass of the negative charges shifts with respect to the center of mass of the positive charges which means again a dipole is created. And of course, this dipole will oscillate if the electric field e oscillates. And therefore, again there is a dipole being created at the atomic level and this dipole is responsible for the for the frequency dependence. Wherever and it is clear that suppose I am trying to shift the electron cloud the electron cloud is much less massive compared to an ion. That means that the electron code shifting is going to take can be operative at higher frequencies as compared to the ionic polarization which may be active at lower frequencies. Third example is of course, the third point is orientation polarization. And we have already talked about this orientation polarization when we talked about the case of microwave heating of water. When you put an and we know for schematically show the water molecule for instance by this wherein essentially the oxygen molecule is slightly negative the hydrogens are slightly positive and you have these molecules oriented randomly in a medium like water. Now, what happens is that when you apply an electric field there is a tendency for the dipoles to align themselves such that the of course, the positive charge. So, therefore, there is a dipole alignment along this direction all the individual atomic dipoles or molecular dipoles align themselves. And this is responsible for what you might call orientation polarization. Here obviously when you are talking about entire molecules have to rotate and this implies that this is going to kick in at even lower frequencies as compared to a phenomena like electronic polarization or ionic polarization. And this and we also pointed out while heating a water that when the electric field is actually oscillating that implies that these molecules are constantly trying to rotate and align themselves parallel to the field. And this when dissipatively operates you can see that actually it is to heating of water in your microwave oven. The fourth of these is the space charge polarization wherein at any positive Kelvin temperature you may have actually dissociation of the positive and negative charges in a medium and this could typically occur at a interface for instance. And when you apply an electric field these mobile charges are specially separated of course, these may not be mobile to a long distance may be be short distance mobile. And therefore, again you get a net dipole moment in a material and this dipole moment for instance as I pointed out could be localized to an interface. In other words there are underlying atomic mechanisms which are responsible for this frequency dependence of n and these are the four we have just now cited. Now a few more words about the refractive index usually the refractive index is greater than 1. And that means that velocity of light in a medium is going to be greater than is going to be smaller than and this is velocity of light. Of course, this is true for any electromagnetic radiation, but I am just writing it for light is going to be less than c which is what we expect because we know c is the upper limit for velocities in nature. Now there are materials in which n is less than 1 and still we are talking about a positive kind of an n. And this is very surprising because if n is less than 1 it implies that light is travelling faster than the speed of light which is an apparent contradiction to the theory of relativity. And the word apparent is used because in reality it is not in contradiction to the theory of relativity. Because wherever the n is less than 1 the velocity one needs to consider in this context is not the phase velocity, but the group velocity. And of course, in certain other cases where group velocity even cannot resolve the case you actually consider something known as a signal velocity. And whatever if you either consider the group velocity of the signal velocity you will notice that the signal itself is not propagating faster than the speed of light. In other words causality is not being violated which is implies that the Einstein's postulate is correct that c is the fundamental constant of nature beyond which or equal to which we cannot send any signal. So, even though we may have a refractive index less than 1 which implies a velocity of speed greater than or the speed of electromagnetic wave in a medium greater than that of speed of light. This does not violate in any sense in the principle of causality which implies that the signal or the information cannot be transmitted faster than the speed of light. So, this is important to keep in mind, but we will briefly consider in these set of lectures not n is less than 1 materials, but materials wherein and n is less than 1 materials have been found for instance in Bose Einstein condensates wherein light can I mean there are special materials which have been created where light has been slowed down or light has been speeded up, but we will not talk about those materials, but we will briefly take up what is known as negative refractive index materials. In reality these are not really materials and you should note that they are actually typically structures. Of course, we know the difference between materials and structures when you refer to the word material we imply that there is no geometry, there is no other parameter, it is kind of an infinitum and no point in the material is different from any other point in the material. Of course, the properties may be direction dependent, but the points in the material themselves are all identical. In a structure there is some geometry, suppose I talk of material like copper I assume that it is a big block of copper wherein two points are identical, but suppose I am talking about a copper plate then there is a geometry involved in it because there is a length, there is a blade, there is a thickness to this plate and therefore, such a thing would also have a characteristic of a structure. Now, in these negative refractive index materials the refracted beam will be on the other side of the normal. So, this is a very strange kind of a phenomena wherein and these kind of materials sometimes are called meta materials in which the incident train comes normally like this and the normal refracted beam is like this. And if you are talking about an optically denser medium this blue material then you would notice that if this is the theta incident and this is my theta refracted then theta refracted would be less than theta incident we know this. But in the case of negative refraction actually the refracted beam lies on the other side of the normal. So, this is a very special kind of refraction which is known as negative refraction this was theoretically postulate quite sometime back, but then it is physically realized especially for radio waves and other kind of electromagnetic radiation in quite recent times the last couple of decades. And it is actually a region area of very active interest because people are talking about making cloaking devices based on negative refractive index materials. Now, we take about of talk about a few more other aspects wherein we are talking about interaction of electromagnetic waves with what you might call a matter. And for now we will restrict ourselves to what you might call crystalline form of matter. In other words there is an periodic array of atoms in the material and of course, I have just shown one row of atoms and this could be actually a series of rows like in a three dimensional crystal. And I assume now for this spacing between the atoms is a which is now the inter atomic spacing or the lattice parameter. And I have three positions three situations shown here case one that the wavelength of the radiation lambda is much larger than the inter atomic spacing. The second position is that the wavelength of the incoming radiation is of the order of the inter atomic spacing. And the third case is when the wavelength is much much smaller than the inter atomic spacing. The easiest of these three to visualize is the fact that the when wavelength is very small then essentially transmission dominates which is very easy to see like it is the case like we drew here that when you have these point scatterers very much far apart then the radiation essentially can go unimpeded through space. But the remaining two are the more interesting cases well one example light to site of transmission for instance suppose I am sending light through a material then the wavelength is for instance I take copper and send light then you would think that the wavelength of light is much larger than the inter atomic spacing. But suppose I send what you might call an electron beam which is what you do in a transmission electron microscope then the wavelength of that electromagnetic what you may call the electrons which is basically because of the debroy relationship we have a wave particle duality. And this implies that the wavelength of that electron beam is actually much smaller it is of the order of picometers. And this depends on of course on the voltage of the accelerating voltage. But this picometer sized or the wavelength radiation can actually pass through matter as if there existed none. So, there was this could be a situation very close to transmission when the lattice parameter say of copper is of the order of Ewangstrom's. Now, the other two situations the case of the scatterer spacing being of the order of the wavelength here or much smaller than the wavelength are of interest to us you know the wavelength is much larger than the scattering spacing are of interest to us. Because the first case leads to essentially what you may call reflection. So, most of the wave when the lambda is much larger than a is actually reflective. And of course, a little bit of the radiation will still leak through and this will be a small amount. And this amount which leaks through is given is frequency dependent obviously of the wavelength dependent and is given by the Bethes relationship. Now, the important a nice example of this would be again we go back to our microwave oven when you look at the microwave oven you would notice that in front of the microwave oven actually you have a mesh wire mesh. So, it is typically a small there are a lot of small holes in the mesh in other words I schematically draw this wire mesh is a mesh like this. So, this is my front window I just assume it schematically to be my front window and there are these small holes in the front window. So, you might assume that the electromagnetic radiation in the case microwave is and these microwaves have a wavelength of the order of lambda of the order of 3 centimeters would actually come through these holes come out through these holes and actually I mean can affect you in other words can heat you up or can cause damage to your tissue. But, this does not happen because this is the case we are now considering in which case the wavelength is much smaller than this whole dimensions or in other words scattering spacing. So, because the wavelength is much larger most of the electromagnetic waves is actually going to be reflected in other words the microwave is not much of the microwave is going to be reflected from those meshes and nothing is actually going to be or very little is going to be actually transmitted through that kind of a whole kind of a structure in a microwave oven. Another example could be for instance when you are using radio waves to collect you typically have a radio wave antenna like this. And you might have noticed that many of these radio wave antennas actually have a parabolic or a kind of a shape and they have these what you may call rod like structures or they may even have a mesh like structure. And if you look from the side you will notice that this dish has a collector here. In other words electromagnetic waves are reflected and there is a detector here a transponder or transducer which converts this energy into electrical signal. Now, the question which may be asked is that again if this such a holy structure is most of it is actually a rods or wires here then would not the electromagnetic waves go through it and not very little reflected here again the answer lies because this scattering spacing of the holes are actually much smaller than the wave length of the radio waves which implies that you are actually even though it looks like a structure which has practically holes in it you would actually see that most of it is reflection dominant. In other words this can out like act like almost like a mirror we use in a normal telescope a mirror reflecting telescope has these kind of large mirrors and these large mirrors actually reflect. So, for what is for light those mirrors is for radio waves these kind of dish antennas. Now, equally interesting to this case is the case where in the wave length of the lambda is of the same order of the inter atomic spacing and now we are we have said we are going to consider actually point scatters we will have to for now assume this is slightly different from these cases because here we have finite thickness scatters, but for now for simplicity we will assume that these are point scatters. So, when the lambda is much of the order of a then we note that actually that you have a phenomenon of diffraction in diffraction what happens is that not only you have a transmitted beam, but the energy is redistributed in between a transmitted beam and many of the diffracted beams and of course, that distribution is not equal most of the energy goes into the transmitted beam very little of it can be found actually in the diffracted beams, but if it is a crystalline material then the diffracted beams given by Bragg's equation would occur at very specific angles. And of course, we can conduct a diffraction experiment using light in photonic crystals we can conduct and we will have to use laser on photonic crystals to get a diffraction or a optical grating as you call it you can use an optical grating and laser to get a diffraction pattern you can use x rays and you can use a crystal like copper to get diffraction patterns you can use microwaves and you can use a crystal made up of ball bearing balls in other words I can take large macroscopic crystals in which there are ball bearing balls and I use microwaves like we just now pointed out like in a microwave oven and I can do a diffraction experiment. So, these three possible outcomes have to be kept in our mind and for now we have been talking about electromagnetic waves, but we have to remember diffraction and transmission are more common or more universal than just for electromagnetic waves. And we know that for instance an x double slit experiment can be performed in a ripple tank in other words you can use water waves and you can use what you may call wooden blocks as scatterers to actually get scattering and interference in a water tank that means we are using mechanical waves transverse waves on the surface of water and those will show the phenomena like diffraction interference etcetera. So, these are universal phenomena and we have three regimes in which we can talk about before we go to the origin of color, let us briefly summarize what we have been talking about so far. In optical properties we noted that there are three phenomena which can take place, one is refraction another is reflection other than third is absorption. And we said that this is basically occurring when you have electromagnetic radiation falling on a medium which we assume to be denser than the outer medium in which it is immersed, but we said from a fundamental perspective there are only two kinds of phenomena one is known as scattering and the other is absorption. And we also said that if the absorption is possible if this external electromagnetic wave sets up some kind of an electronic vibrational or rotational resonance within the material. And of course once the material has been excited using this kind of an external electromagnetic radiation the material may relax back and in the process actually re emit electromagnetic radiation. This re emission of course we noted can take place immediately which is why when it is called scattering basically in other words the material behaves like a electron atoms in the material act like dipole oscillators. And it is nearly of course with a small time lag re scattered back in which case you call it an oscillator and this is not called an excitation. And we also said that the velocity of light in a dense optically denser medium is smaller than the opt velocity of light in a rare medium like air or vacuum and which implies that there is a quantity called as refractive index which is the ratio of the velocities. We also pointed out that this refractive index actually has a frequency dependence which gives rise to the familiar phenomena known as dispersion which means that white light is actually spread by a prism into its component colors which is known as the wibbier colors. And we said that there are mechanisms like orientation electronic ionic and space charge polarization which lies at the heart of this kind of frequency dependence of n or the refractive index. And we also said when we talk about refraction there are interesting materials known as negative refractive index materials wherein the refracted beam actually lies on the same side of the normal as the incoming beam or the incident ray. And we also said that this when you have an electromagnetic wave interacting with a crystal which is a periodic array of point scatterers. Then there are three phenomena possible one is reflection other is the interesting case of diffraction and finally, transmission is also is possible. And of course, depending on various other parameters all these three might be simultaneously happening it is not that we are talking about dominant regimes but this not mean preclude that only one will happen even in case where reflection is taking place then some amount of matter is actually transmitted and this is has a some kind of a lambda power 4 dependence. So, the next topic we take up is the origin of color. Color can arise from various mechanisms as outlined below the most common what we are accustomed to is the absorption or emission colors that means that some part of the electromagnetic spectrum is absorbed the remaining is transmitted. And therefore, you see a certain color or some part of the electromagnetic spectrum is actually absorbed and the remaining is reflected. And since you see what is called reflection colors of course, this absorption may be followed by emission and this emission can give rise to colors as well. And we will take up especially this topic of absorption and emission colors in nanoparticles. Color again if whatever is not absorbed or reflected is transmitted and if some part of the electromagnetic spectrum is actually absorbed some other part is reflected. But then if you are looking at a transparent material if you are standing on the back side of the transparent material then what you see is a transmitted color which could be which is a complementary in some sense if you are talking about a material which only reflects and absorbs. So, if you are one side of the partially absorbing partially transmitting system you will observe one color, but the complementary color would be observed on the other side which is transmitted. Now, you could also have scattering colors as we pointed out like in this case of the blue sky or the red sky in the evening which is basically coming from the scattering phenomena. And we will take up one such example of colloidal gold wherein you see colors because of scattering. There would we also know the dispersion itself like the case of the prism can give rise to colors in which case a white light which is being split and therefore, you have colors. Also interesting are the cases of interference colors like we know the case of oil films on water they have a certain color and this color changes when you viewing angle is changed. Your colors on a CD a compact disc you see that on the back side of a compact disc typically you see some kind of iridescent colors which is coming from interference. The colors of a butterfly wing for instance many of these monarch butterflies etcetera have a beautiful iridescent color which is coming from interference. And typically many of the blues in nature are actually interference colors many of the birds which also have blue color comes from this interference colors. And we have already noted that if this colors could come from atomic electronic transitions like for instance if you look at the yellow color of sodium in a flame test. We have for instance the three a state and two closely space three p states and when relaxation takes place from these excited states we have the two famous sodium vapor lines which are the in the yellow region. And therefore, this is coming from emission. So, this is a nice example of an emission color. So, this can be the for instance the sodium color sodium in a flame test or a sodium vapor lamp. Now, in other words there can be multiple origins of color it can come from absorption or emission reflection or transmission or a combination of all these factors like absorption part of the spectrum is absorbed part of it is reflected part of it is transmitted. And therefore, you may have depending on which side of the medium you are you may observe different kind of colors. Therefore, and there could also be scattering colors as we just noted the case of the blue sky or the red sky in the evening there could be colors because of dispersion. And more interestingly there could be iridescent colors becoming because of in the phenomena of interference that is white light falls on a for instance a film of oil on top of water. For instance you have an oil in a water. So, this is my oil on water then typically what happens is that some part of the electromagnetic spectrum this thickness is destructively interfering. And some part of the electromagnetic spectrum this delta happens to be integral multiples of lambda that means that is going to be constructive interference. And therefore, you will see some colors being removed from the electromagnetic spectrum while others are enhanced. And therefore, you have colors in this coming rise purely from interference. So, we will have a little more to talk about this origin of color in nano materials in some of the coming slides. What happens when an electromagnetic radiation falls on a metal? Now, this picture we are taking up because we said that when you have a series of scatterers and you have an electromagnetic radiation falling on it there are three possible scenarios. But this is assuming that these point scatterers are virtually inert point scatterers, but then we know that we could actually be talking about these point scatterers like ion cores ion cores in a metal residing in an electron gas. That means we are talking about metallic bonding where there are free electrons available in the material. In such a material obviously the behavior is more complicated and what really does this electromagnetic wave do to this material is that it actually excites what are known as plasmon. Like you have phonons for lattice vibration plasmon or collective oscillations of free electrons. So, when you have an electromagnetic radiation impinging on a metallic surface it excite plasmon waves or oscillations inside the material. Typically what we call bulk plasmon or longitudinal oscillations of electrons with respect to the ion cores. So, the plasmon which is excited in the bulk of the material is has a longitudinal character like we have sound waves traveling through a medium like air in this room which are longitudinal oscillations. On the other hand suppose a pluck a gita string we know that is a transverse oscillation. Now, because the incoming electromagnetic wave sets up plasmon oscillations this implies that this material there is a resonance happening. That means an external electromagnetic field wave is going to be absorbed by the medium because now it is going to set up these electromagnetic or oscillations of the free electrons which we call plasmon plasmon. Therefore, because of this plasmon resonance what is happening is that you typically find metals are opaque metals do not transmit light. We will see that this is true for the visible radiation. We will have take a more detailed picture that what kind of a wavelength how it interacts with the metal in a coming slides. But this is true for visible radiation typically we find that metals are opaque to the visible radiation. Again we will take up one more interesting case later on wherein we make thin films of metallic material like gold. So, thin films of metallic material can actually become transparent and this we when we pointed out about how the properties of nano materials come about in those lectures we talked about a phenomena that when there is insufficient material. Then you would have certain effects coming like for instance transmission through gold and this is not really a nano phenomena. This is not something new physics coming in just there is insufficient material. Suppose I did a Bragg's equation on a material on a point scatterers like this. If I had an infinite array a three dimensional crystal and I throw electromagnetic radiation then I would obtain sharp delta peaks in reciprocal space. But suppose now I had a finite crystal there will be a peak broadening and this peak broadening is essentially coming from insufficient number of atomic planes. And this is no new physics out there. Similarly, you will notice that when you make thin gold foils they actually can become transparent to some parts of the electromagnetic spectrum. But we will take up the these cases in detail very soon. So, we are seeing that when you have an incoming electromagnetic radiation we have to worry about something more than just the presence of the ion codes or scatterers. We have to worry about the free electron clouds surrounding these ion codes. These free electron codes are set into collective oscillations and these collective quantized oscillations are called plasmon's and these plasmon oscillations or resonance can actually give rise to absorption in a metal. In other words metals are not transparent as we know. Now, what is the regime of transparency and opaquency of a metal? So, when we talk about visible radiation we have already pointed out what really happens and that is of course for a typical metal like silver for instance. But what is the scenario when you have for instance the entire electromagnetic radiation starting from gamma rays with very high frequency about 10 power 23 in the circular frequency or about 10 power 22 3 into 10 power 22 in the linear frequency then and on the other hand of the frequency regime you have the radio waves which have very long wavelengths of the order of about 10 power 4 meters. That means you have wavelengths of the order of kilometers. You have these extremely short wavelength 10 power minus 14 meter gamma rays on one end of the spectrum and you have wavelengths of the order of kilometers. Obviously, all these different regions of the electromagnetic spectrum are not going to interact identically with a given metal. So, what are the factors which come into play? So, we can actually divide the electromagnetic spectrum into three frequency regimes and these three frequency regimes two of them come in the range of the high frequency regime and one of them is the low frequency regime. How do I determine my high or low frequency regime? I multiply my frequency or of course omega being the circular frequency with the mean free collision time which we had encountered before. And typically we had noted that the mean free collision time is a collision time between of electrons between in other words it accelerates or maintains and accelerates increases velocity between two collisions. But then there are it may then interact with the phonon or an impurity and gets scattered and the time between the two is called the mean free time more casually or the average collision time which happens to be of the order of about 10 power minus 14 seconds for most metals. So, I multiply my circular frequency with my mean free time and note if it is much less than if it is less than 1 or it is much greater than 1. So, let us take an example for instance of the case of copper and when you are talking about a frequency like 10 power 7 hertz. And we see that this electromagnetic radiation is actually absorbed and of course some of it will be reflected. Now, in making these discussions as I pointed out we have to invoke the concept of plasmon's and plasmon's have a certain characteristic frequency depend on the material. And this plasmon frequency typically lies in the range of about 10 power 15 hertz. So, suppose I am talking about the plasmon frequency which is given which is a function of the what you might call the electron density and of course, also depends on the free electron mass in the permittivity of the medium. This plasmon frequency lies in the range of about 10 power 15 hertz for gold for instance is 2.18 into 10 power 15 hertz. So, what is happening now is that in the low frequency regime the beam penetrates the metal for a short distance which is called the skin depth and then it is absorbed and some of it is reflected. And when I am talking about the low frequency regime which is a reflecting and absorbing regime I am talking about the regime on the right hand side. So, this is my low frequency regime. So, here in reflection and absorption is being dominated and what is my switch over dominant or the switch over kind of frequency which I have to monitor. This is the plasmon frequency and I noted that this plasmon frequency is of the order of 10 power 15 hertz. In other words in the omega scale this is my 10 power 15 here. So, my plasmon frequencies are approximately in this regime and my visible lies to the right of the plasmon frequencies. That means that in the visible region metals are reflecting and absorbing they do not transmit any of the radiation. So, bulk metals typically we know that we already have this experience that they do not transmit any of the radiation. In the high frequency regime where in the omega tau is greater than 1 and we see that this is of course coming in the visible and the ultraviolet region. There are two possibilities when omega is less than omega p that means this region where the omega is actually less than omega p this region to the right of it the visible in the infrared regime. So, you have the to the right of the plasmon frequency you have the visible and the infrared which is actually the low frequency regime given in the of course the high frequency regime where in omega tau is much greater than 1. In other words I have the low frequency regime where I know that metals are not going to transmit then I take up the high frequency regime, but then subdivided into two parts that part where the omega is less than omega plasmon and the case where omega is greater than omega plasmon. Noting that the omega plasmon is of the order of 10 power 15 hertz which lies somewhere in the frequency spectrum from here. So, the visible and infrared are actually larger frequencies compared to the what you might call the omega plasmon. So, in this high frequency regime you see that visible and ultraviolet radiation is actually reflected because this regime lines because the plasmon frequency is less than the or greater than the omega of the radiation. But suppose you go to a regime where omega is much greater than p then the metal becomes a non absorbing transparent dielectric. So, if you are in this regime wherein the omega is much larger than omega p. So, you are somewhere here then you would notice that for radiations like gamma rays and x rays a metal can actually become transparent. So, you have three kind of interactions possible of radiation with a metal one regime in which is absorbed and reflected essentially other regime wherein the visible and ultraviolet regime where there is reflection and there is another regime wherein you can actually see that the metal actually becomes and some kind of a transparent dielectric which happens only in very high frequency regimes. Now, having this broad picture we that means metals have to be treated slightly separately from semiconductors and other dielectrics. So, we see when we talk about a dielectric like glass we know or as a Si O 2 crystal we know that I do not have to worry about plasmon at all. That means I can talk about purely in terms of the normal language of a transparent dielectric except close to those resonances wherein the dielectric starts to absorb and we already noted the mechanisms of those resonances. But when you are talking about a metal the predominant behavior of metal to electromagnetic radiation is coming from these free electrons and of course, when these free electrons stop to respond in the case when the omega is much greater than omega p then you note that actually the metal can also become transmitting to an electromagnetic radiation as it may happen to high gamma rays etcetera. The optical properties of a semiconductor as you would expect is going to be very different from that of a metal and this is going to be obviously dominated by the phenomena of the band gap. So, we will take up briefly here what are the issues of the band gap and how is it going to affect my absorption and emission from a semiconductor and we are talking about absorption and emission of electromagnetic waves. Now and we will take up in somewhat detail the fact that what happens when I make a semiconductor particle like CDSE or gallium arsenide in a very nano form that means I have particle size of the order of 10 or 20 or 5 nanometers. Ionic crystals show strong absorption and reflection in infrared region due to interaction of light with the optical phonons. Because now the absorption is not because of plasma because there are no free electrons in the system this is a semiconductor. And this ionic material which is dielectric tends to absorb in the infrared region because now you are setting up resonances in the optical phonon spectrum. Compound semiconductors have a partial ionic character and also exhibit absorption and reflection in the infrared. Like when we are talking about compound semiconductor it can be the direct band gaps gallium arsenide or gallium forstite. And these have a certain partial ionic character which implies that they also have a partial behavior which is similar to ionic crystals. Now if the energy of the incoming photon is greater than the band gap then the photon is absorbed. So, you have the balance band in a semiconductor which is schematically shown here. You have the conduction band and you know in a semiconductor at 0 Kelvin the balance band is full and the conduction band is empty. And if you have you are at a finite Kelvin temperature you know that thermal excitation of electrons will put some electrons into the conduction band and we know that this fraction is given by the Fermi Dirac statistics. Now what is happening is that if I am sending electromagnetic radiation and the measure of the energy of the electromagnetic radiation is frequency. And if the frequency is much smaller than the critical frequency given by E G by H then there is no absorption and like that of a metal. Because they are always free electrons which can be oscillated. But now if you keep on increasing the frequency then the energy in the incoming photon can exceed that of E G. In other words an electron can be excited from the balance band into the conduction band and you leave a hole in the balance band. A hole is nothing but the fact that there are actually now n minus 1 electrons in the conduction valence band. I can either deal with the motion of these n minus 1 electrons in opposition to the electric field in a direction opposite to the electric field or I can talk about the motion of one hole in the direction of the electric field. So, this is the convenience I use the language of hole is for convenience. So, I do not have to deal with n minus 1 electron, but I can just talk about one hole moving in the valence band. Therefore, I have would have a strong absorption when my frequency exceeds this critical frequency of the absorption edge. And we will therefore absorb the electro the semiconductor will absorb the electromagnetic radiation. Now, as the wave vector of a photon in optical region is very small and only constant momentum transfers are allowed across the band gap vertical transition in case space are allowed. That means that if I have a direct band gap semiconductor I can ignore the wave vector of the photon incoming photon, because it is very small and I can only talk about the energy transition. So, that I need to know not worry about the what you call the momentum of the photon, but on the other hand suppose I am talking about the indirect band gap semiconductor in which case the conduction band is actually shifted in case space. That means the conduction band minima lies at a certain fixed value of k here compared to the maxima in the valence band in energy. This implies that when I want to excite an electron to the from the valence band to the conduction band then not only have I to supply the energy e g, but additionally I will have to supply if I will just add and substitute call a k g kind of a momentum. Now, as I pointed out the photon itself does not have this momentum. So, this would imply that purely by a phononic photon excitation I cannot excite an electron from the valence band to the conduction band, but luckily a phonon comes to our aid here. That implies that in indirect band gap semiconductors like germanium both a photon and phonon together give rise to the excitation. And in this case of the phonon I ignore its energy like for the photon I ignore its momentum, because it was small for the phonon I can effectively ignore its energy, because its energy is small. And I can think of the phonon only contributing only the momentum to the electron. So, I have now two aspects coming in when I am trying to excite an electron in the indirect band gap semiconductor. One is the energy coming from the photon the other is momentum coming from the phonon and put together they supply the required momentum and the energy to excite a electron from the valence band to the conduction band. Typically the band gap of semiconductors of the order of about e v's a few e v's that implies that the fundamental edge actually is in the infrared. And therefore, semiconductors though cannot be used as lenses in the visible region, because you know semiconductors are not transparent in the visible region. But you can make infrared when you make infrared cameras lenses can be made out of these semiconductors, because below the absorption edge they actually would transmit. And therefore, I can make a lens out of a semiconductor. Now, therefore, to summarize the optical properties of semiconductor the optical properties of semiconductor are dominated by two factors one of which is given in the slide. The other one we will take up in the next slide is the fact of the band gap. That means any energy or frequency which is lower than the critical energy or the critical frequency is not going to be absorbed. On the other hand any other higher energy than the band gap is going to be absorbed by the semiconductor. And later on we will have a lot to say about how this band gap actually changes with confinement how do you make it when you make the material smaller how the band gap changes. And we said that in the context of absorption in semiconductors not only have I to worry about this band gap energy e g. I also have to worry about the relative position of the conduction band minima vis a vis with respect to the valence band maxima. So, this could be a direct band gap semiconductor in which the minima lies right above the maxima or it could be indirect band gap semiconductor in which case there is a shift of the minima with respect to the maxima of the valence band in the case space in the momentum space which implies that I have to supply the energy coming from a photon ignoring its momentum. And the momentum coming from a phonon ignoring its energy and put together the phonon photon actually give rise to an excitation of an electron from the valence band to a conduction band. When I said that two factors come into play when I am talking about absorption in semiconductors the second factor is what is called the exciton. Now, having excited an hole I mean excited electron to the conduction band you are left with an electron in the conduction band and a hole in the valence band. So, you have these two factors you have an hole in the valence band and you have an electron in the conduction band. But in principle of course, you might assume that these hole would move in its valence band independently in response to an electric field of course, dictated by its mobility in the valence band which would be different from the mobility of an electron in the conduction band. And you would assume that this electron would move freely in the conduction band this situation may or may not be. So, in fact there would be an Coulombic attraction between the electron and a hole and you may form what is called a bound state between the electron and the hole which is called an exciton. So, an exciton is a bound state of an electron and a hole this binding is due to electrostatic or Coulombic attraction and the exciton has a lower energy than the unbound electron plus and hole. So, obviously, because there is a lowering of energy that this binding is taking place and this implies a very important thing that brings the energy levels closer to the conduction band. So, what is exactly happening is that. So, you have the conduction bandage and you have a certain E g that means if I do not supply my E g then I cannot excite an electron from the valence band to the conduction band. But, because of the Coulombic attraction between the hole and the electron you can have a bound system and this bound system has a lowering of energy with respect to the conduction band and this is now my exciton binding energy. So, this small number here and this exciton binding energy is typically very small it is of the order of 0.01 E v or about 10 milli electron volts this is small binding energy. Nevertheless, this energy is lower than the conduction bandage energy. This is important consequences to the absorption of a semiconductor as we shall see now. But, so this exciton forms and what is this exciton this is a bound state and typically we will see that this exciton can be thought of as an electron and a hole moving around that means they are spatially confined and they are moving around each other center of mass. And this it can be calculated this radius around which they move around is can be called as the Bohr exciton radius which is given by some of the fundamental constants of an electron. And this Bohr exciton radius is of the order of about 11.8 nanometer. So, it is in the nano scale it is not like the normal Bohr radius of a hydrogen atom which is much smaller as you know therefore, you can have an electron and hole in a bound state and therefore, they are spatially localized. A few more factors we will see how when this localization can break down, but we have introduced a new concept called the concept of an exciton which is now spatially bound and spatially bound means that it is spatially bound to a radius of the Bohr radius we just now pointed out they move around the center you can think of them as moving around a common center of mass. And this give rise to an energy level the bound level of an exciton in the band gap. So, this is very very important and this exciton is an electrically neutral quasi particle because now there is a hole in electron together they form a neutral quasi particle that can exist in insulators and semiconductors and we will talk about the relevance of excitons only in semiconductors here. The exciton can be thought of as an elementary excitation in materials which can transport energy without transporting electric charge because this some total of the electron hole is a neutral particle this is when it moves through the crystal as in the bound form. This is can be thought of as transporting energy without transporting electric charge this excited state can travel through a lattice without transfer of charge. So, this is very important and for now we will restrict ourselves to what is known as a free exciton or otherwise given the name as a motvenor exciton which can move through the crystal there can be other forms of excitons which are known as bound excitons which we will not talk about here. The exciton can itself we thought to have as an effective mass if an effective reduced mass which depends on the effective masses of the electrons and holes as given in this formula. And if you look at the effective mass of an exciton you can see that and for example, I am taking gallium arsenate here it is got a mass of a 0.059 Me in other words it is much smaller than the mass of an electron. So, exciton can be ascribed an exciton bore radius similar to the concept of an bore radius in a hydrogen atom. An exciton can be given a mass which is the effective mass or mu star of an exciton which is of course, as we just now saw it has a much smaller value than the electron mass that means a much lighter particle quasi particle as we should be more careful with this terminology because it is a quasi particle as compared to an electron. Now, we have seen that the photon absorption by semiconductor can lead to the formation of an exciton. The exciton binding energy for most semiconductors is of the order of few to few tens of milli electron volts. So, this energy we have already seen that it is much smaller the exciton binding energy is much smaller than the typical band gap energy of semiconductor this is very important note. For instance the exciton energy in gallium arsenate is 4.6 milli electron volts for cadmium sulphide it is of the order of 28 milli electron volts. So, this is the range of these binding energies of this exciton and if you want for a comparison k t at room temperature is about 40 milli electron volts. This tells you directly that since k t at room temperature and now talking k t at room temperature and k b is the Boltzmann constant is greater than the exciton binding energy. This implies that at room temperature in bulk semiconductors the exciton will be unbound. The thermal energy will overcome the binding energy of the exciton and implies that the exciton actually will be dissociated at room temperature. So, that we implies we take a room temperature and do the electromagnetic radiation absorption experiment then you would note that there will be no peak corresponding to the exciton. Even and if you want to take a even higher energy scale you will note that the binding energy of an hydrogen atom is about 13.6 electron volts much larger than the exciton binding energy. So, we note that we have an important concept like exciton, but it becomes important only at low temperatures and absorption of semiconductors. One of the reasons we are taking up this discussion of excitons in details is that often when we read literature regarding nanoparticles semiconductor nanoparticles and their absorption the word excitonic absorption is used. And this sometimes can be misleading because the actual effect is coming from confinement effect and not from excitonic absorption because and therefore, we have to be little careful with the terminology and the physics behind it. And if you took at a semiconductor low temperature then the excitonic absorption spectrum is a sharp line just below the fundamental edge of the semiconductor. And this is observed at low temperature where thermal energy is lower than the binding energy. So, if you go to low temperature the KTB contribution goes down the exciton binding energy is larger than KTB which implies that I would observe an excitonic absorption. And for now I am just showing one excitonic level a simplified diagram where in I would have an absorption corresponding to the exciton just below the fundamental edge of the. So, I am plotting H nu minus E g in electron volts as a function of the absorption coefficient. And for a normal semiconductor without the exciton you would note that the absorption coefficient would go increase like this this theoretical plot. But in the presence of the exciton you will observe a strong peak here which is now my excitonic peak which can be observed only at low temperatures. So, because now even though the energy level is only slightly below the conduction bandage, but statistically this implies that many more electrons are going to be excited to the exciton level and which gives you a strong absorption excitonic absorption. Now, an important point we already saw the case of the exciton radius is the fact that the exciton radius itself is a nano scale sub entity within the semiconductor. So, it is a critical number within the material which is the exciton binding radius which is of the order of nanometers. For instance for gallium arsenide the with an exciton energy of about 4.6 e v m e v sorry milli electron volts the exciton radius is of the order of 11.8 nanometers. The bulk band gap is of the order of 1.43 e v which is much larger than the m e v quantities. For c d s e the exciton radius about 5 nanometers for c d s it is about the order of 3 or 4 nanometers. And if you look at a more slightly more detailed picture of the exciton you can actually see that you can ascribe quantum numbers to the excitons. In other words the lowest and this is of course, an exaggerated picture because we know the exciton energy is very small and you have the compare to the band gap energy this is out of scale diagrams. So, we should note that this is like the previous diagram I showed you and you can see that in this you can have multiple quantum numbers. So, this could be the n equal to 1 state and n equal to infinity state is nothing but the conduction band gap edge the conduction edge itself becomes a infinity state. So, if you can have multiple levels excitonic levels and therefore, you can have multiple excitonic absorptions and they will suppose you look at a spectrum like this the other excitonic absorptions would also show up in such a spectrum and you have to go to low temperatures for this. But then these excitonic levels can also be thought of as quantized levels having quantum numbers of its own. And we can note that if the crystal becomes very small of the order of the excitonic radius then there are important things which we need to consider.