WEBVTT Kind: captions Language: en 00:00:31.200 --> 00:00:38.700 Hello students, welcome to the 10th lecture of the online course of Nanophotronics, Plasmonics 00:00:38.700 --> 00:00:44.940 and Metamaterials. Today we will be discussing Matrix Theory of Dielectric Layered Media. 00:00:46.260 --> 00:00:52.320 So here is the lecture outline, we will have a quick recap of the Fresnel's equation and then 00:00:52.320 --> 00:00:58.500 we will introduce a very effective theoretical tool called transfer matrix or T matrix method 00:00:59.220 --> 00:01:05.040 that can be used for calculation of reflection and transmission across any multilayered medium. 00:01:05.940 --> 00:01:13.260 So in doing that we will first learn how to obtain T matrix for an interface, then we will look for 00:01:13.260 --> 00:01:20.820 T matrix for a layer and then how to obtain a overall T matrix for a multilayered system. We 00:01:20.820 --> 00:01:26.580 will try to correlate T matrix to reflection and transmission coefficients so that we are 00:01:26.580 --> 00:01:32.160 able to calculate reflection and transmission of any multilayered system from T matrix that 00:01:32.160 --> 00:01:36.240 we will see. And then we will take an example of a very popular optical 00:01:36.240 --> 00:01:42.480 device called Fabry-Perot interferometer and we will obtain the T matrix for that device. 00:01:43.080 --> 00:01:46.620 We will show you how to do that calculation and also, 00:01:46.620 --> 00:01:51.960 we will obtain reflection and transmission coefficients and other important parameters 00:01:51.960 --> 00:01:58.380 for a Fabry-Perot cavity that will be finesse and Q factor. So here is a quick recap of Fresnel's 00:01:58.380 --> 00:02:04.020 equation. So if you remember from the last few lectures we have discussed about light 00:02:04.680 --> 00:02:09.420 falling on a particular interface between two different media. In that case some 00:02:09.420 --> 00:02:14.640 portion of the light is getting reflected and some portion is getting transmitted. 00:02:14.640 --> 00:02:20.580 Now depending on the polarization state of the incident light the amount of reflection 00:02:20.580 --> 00:02:29.160 or transmission will vary. So in the case of S polarized light, so if you remember that S 00:02:29.160 --> 00:02:35.160 polarization is basically so the polarization where the electric field basically strikes out 00:02:35.160 --> 00:02:40.680 or it is perpendicular to the plane of incidence. Now when I say what is plane of incidence? Plane 00:02:40.680 --> 00:02:46.740 of incidence is basically that plane where the incident ray, the normal to the interface and the 00:02:46.740 --> 00:02:54.240 reflected ray they all lie in one plane. So if you consider the incident wave propagating this way 00:02:54.780 --> 00:03:01.560 you can consider the electric field to be either striking out that is S polarization or it can 00:03:01.560 --> 00:03:09.600 be parallel to the plane of polarization. So in case for S polarized light which is also known as 00:03:11.580 --> 00:03:17.700 TE polarized light you can find out what is the reflection coefficient 00:03:17.700 --> 00:03:22.080 and transmission coefficient that is R perpendicular and T perpendicular. 00:03:22.620 --> 00:03:27.480 On the other hand for T polarization light or polarized light which is also known as 00:03:29.160 --> 00:03:36.960 parallel polarization or TM polarization. So you can find out what is R parallel and T parallel. 00:03:37.500 --> 00:03:46.920 And we have seen that in the case of normal incidence that is when these equations basically 00:03:47.640 --> 00:03:54.840 become 1. So, R parallel will become equal to R perpendicular and that boils down to a very 00:03:54.840 --> 00:04:03.180 simple equation that is . What is n1 n2 ? So n1 is the refractive index of the first medium n2 00:04:03.180 --> 00:04:08.760 is the refractive index of the second medium. We also can see that the transmission coefficient 00:04:08.760 --> 00:04:14.940 T parallel and T perpendicular will also become same and they will become 2 n1 over n1 plus n2. 00:04:15.720 --> 00:04:21.660 So with this understanding we have to now approach with real life problems where 00:04:21.660 --> 00:04:30.480 there are multiple layered medium that we will encounter. In that case how to use Fresnel's 00:04:30.480 --> 00:04:37.320 equation and that method is known as Transfer matrix method. Always remember here Fresnel 00:04:37.320 --> 00:04:42.780 equations actually tell you the reflection and transmission from one particular interface. 00:04:44.100 --> 00:04:48.780 Now let us take a general problem. So the general problem is of a wave 00:04:48.780 --> 00:04:54.120 propagation in and through a multilayer structure. So the multilayer structure 00:04:54.120 --> 00:04:59.160 can be regarded as an optical system where there is one input port and one output port. 00:04:59.700 --> 00:05:05.940 And the Transfer matrix method will allow you to calculate the overall reflection 00:05:05.940 --> 00:05:12.180 from this multilayered system and what is finally coming out of this. So in all this 00:05:12.180 --> 00:05:16.920 multilayered there are many many interfaces that you will be encountering like here. 00:05:17.520 --> 00:05:23.760 So if you consider this is your incident wave that is coming down with a wave vector of k 00:05:24.360 --> 00:05:31.080 and hitting this multilayered structure as you can see this 6 particular layers. 00:05:31.080 --> 00:05:38.220 So every layer will have a different interface because this one let us assume 00:05:38.220 --> 00:05:44.100 this structure is in air. So you have air and this particular material. So you have this as 00:05:44.100 --> 00:05:49.740 one interface. Then you have this material and this material. So this is another interface. 00:05:49.740 --> 00:05:54.900 Similarly this is one interface. This is one interface. This is one interface and so 00:05:54.900 --> 00:06:02.160 on. So there are many many interfaces which are coming in one after another. Also there 00:06:02.160 --> 00:06:05.220 are layers of different thickness different material property. 00:06:05.220 --> 00:06:11.820 So how light wave is basically going through each of this interface and then layer interface 00:06:11.820 --> 00:06:17.820 and then layer all these things can be taken care of by the T matrix or transfer matrix. 00:06:19.380 --> 00:06:27.660 So let us see how it works. So let the normal to the interface plane So this is the interface plane 00:06:27.660 --> 00:06:34.020 as you can see. We assume that the normal to the all interface plane is basically X. 00:06:35.520 --> 00:06:41.820 And in that case the interface planes are basically parallel to YZ that are that 00:06:41.820 --> 00:06:49.200 those are basically vertically out of this. You can see. And we have also defined different 00:06:50.220 --> 00:06:55.140 refractive index for different region. It means there are different material present 00:06:55.140 --> 00:07:00.540 at different position. That is how the multilayer structure has been made. So 00:07:00.540 --> 00:07:10.080 if you write that nx equals n0 that is basically for the region when x is less than x0. It means 00:07:10.080 --> 00:07:20.040 before x0 the material is always n0 or n0. Then you can see that between x0 and x1 means 00:07:20.040 --> 00:07:27.060 in this region the material property is different. It is having a refractive index of n1. So this is 00:07:27.060 --> 00:07:33.420 basically nx. It means the permittivity of this multilayer structure is a function of x and it 00:07:33.420 --> 00:07:40.020 is changing. And this is changing discretely in different steps and this is how it is written. 00:07:40.020 --> 00:07:48.960 So between x1 and x2 as you can see here it is given as n2. And then between x(N-1) and x 00:07:51.000 --> 00:07:58.200 that is this one you are basically having xN. Or you can say that when 00:07:58.860 --> 00:08:06.900 x is greater than x(N-1) that is when you have crossed the multilayer structure or the light 00:08:06.900 --> 00:08:12.300 wave has crossed the multilayer structure what is the refractive index it is able to see that is 00:08:13.380 --> 00:08:24.600 nN. So this is how we have defined all the refractive index for each of this medium. 00:08:25.200 --> 00:08:29.280 So you have to remember that there are two possibilities of the incident wave. 00:08:29.280 --> 00:08:36.120 It can be either a S wave that is a TE wave. So in that case the electric field is either 00:08:36.120 --> 00:08:42.900 striking out or it is perpendicularly going in as it is shown here. This is the wave factor. And 00:08:42.900 --> 00:08:49.920 it can be also P wave that is parallel wave or TM wave. In that case the electric field is actually 00:08:50.880 --> 00:08:55.200 parallel to this plane. So there are these two possibilities. 00:08:55.860 --> 00:09:01.320 So we can actually define the thickness of each of these layers. So you can call this 00:09:01.320 --> 00:09:12.180 thickness as a parameter d. So you can call di is basically xi plus 1 minus xi. So that is a 00:09:12.180 --> 00:09:16.560 general formula that will tell you about the thickness of each of these layers. 00:09:16.560 --> 00:09:25.440 And you can refer to each of these interfaces in a general form using say i and i plus 1. So 00:09:25.440 --> 00:09:34.020 one interface is formed between i and i plus 1. So that way you can generalize them. Now what happens 00:09:34.020 --> 00:09:41.340 in this case that the normal to the interface plane and the wave factor they define a particular 00:09:41.340 --> 00:09:45.600 plane that is the incident plane. So what we have seen here. 00:09:47.400 --> 00:09:51.900 So when so that is in this case the incident plane is basically this screen 00:09:53.460 --> 00:10:01.140 because that is the interface. The interface is basically perpendicular to this particular 00:10:01.140 --> 00:10:08.760 screen. So you will see that the wave vector is basically along this plane. We are assuming that. 00:10:09.720 --> 00:10:16.020 And in doing so we are also considering that ky component is 0. 00:10:16.020 --> 00:10:22.320 It means the wave vector will not have any Y component. That is an assumption. Even 00:10:22.320 --> 00:10:27.960 if they have it we can use transfer matrix method but the calculation becomes little 00:10:27.960 --> 00:10:35.220 complicated. So let us assume in this case that the wave is particularly lying in that 00:10:35.220 --> 00:10:42.540 plane. So the wave vector is basically in xz plane that is this particular plane. 00:10:43.140 --> 00:10:52.440 Now we will consider each of these cases TE and TM and we will be able to describe 00:10:52.440 --> 00:10:58.860 the electric field in terms of its amplitude which is E and E is the function of x y and z. 00:11:00.060 --> 00:11:06.720 Now in this structure each layer will have the contribution of forward propagating wave. 00:11:07.500 --> 00:11:15.120 It means it is towards the increasing x and it will also have contributions from some reflection 00:11:15.120 --> 00:11:21.000 from the interface. Similarly it will also have backward propagating wave as I told you the 00:11:21.000 --> 00:11:29.460 reflections are considered as backward propagating wave. So the total field that you can see in layer 00:11:29.460 --> 00:11:37.380 i can be given as so this is the total field. So total field is basically AF that is the 00:11:37.380 --> 00:11:43.140 amplitude of the forward moving wave. So the forward moving wave is moving towards plus X 00:11:43.140 --> 00:11:52.560 direction and it has also got a component along Z. So it is AF e to the power minus 00:11:53.160 --> 00:12:05.520 j kx i x plus kz z plus AB. AB is the backward propagating wave. So you can write e to the 00:12:05.520 --> 00:12:10.920 power minus j so backward propagating wave will propagate along minus x direction. 00:12:10.920 --> 00:12:19.980 So it will be minus kx i times x plus kz z okay and then when you break this 00:12:21.480 --> 00:12:27.540 exponential into two components and then you if you club this two together you can write this as 00:12:27.540 --> 00:12:33.720 the forward electric field. So this is the forward electric field this is the backward electric field 00:12:33.720 --> 00:12:40.080 or you can say forward propagating electric field or backward propagating electric field okay. And 00:12:40.080 --> 00:12:48.240 what are these components? So kx i is basically square root of ni k0 whole square minus kz square 00:12:49.140 --> 00:12:56.640 okay. So in this case i is nothing but all those different layers that you will be encountering. 00:12:56.640 --> 00:13:03.000 So starting from 0 because if you remember here the first layer the incident medium is 0 00:13:03.000 --> 00:13:08.340 and then the transmission medium is n. So until that you will have all different 00:13:08.340 --> 00:13:18.000 wave vector in each of these layers okay. So I will range from 0 1 2 up to n. What is k0 that 00:13:18.000 --> 00:13:23.100 is basically the free space wave vector given by omega by C. Now do you understand this one? 00:13:23.100 --> 00:13:33.000 This is nothing but that the wave vector k0 will get modified in the in every layer which has got 00:13:33.000 --> 00:13:42.360 a refractive index of ni. So it will become ni k0 and ni k0 is nothing but so you can write in 00:13:42.360 --> 00:13:49.380 vector form that ni k0 square is equal to the x component square plus Y component square okay. 00:13:49.380 --> 00:13:57.720 So that way you actually get this. So from that only you can obtain what is the X component. 00:13:58.680 --> 00:14:05.340 Now here is the pictorial representation of what I was describing. So in this case let 00:14:05.340 --> 00:14:12.360 us take that you are talking about interface between ni and nj. So there are two layers 00:14:12.360 --> 00:14:17.460 okay. So right now let us find out what is the transform matrix for one interface. 00:14:17.460 --> 00:14:24.480 So these are two layers ni and nj. So there is a interface in between okay clear. So this interface 00:14:24.480 --> 00:14:36.900 it is at a position of xi that is the position of the interface. Now let us assume that we will be 00:14:36.900 --> 00:14:44.340 writing the forward propagating waves and backward propagating waves in the two region. So this is xi 00:14:44.340 --> 00:14:52.200 so anything before xi we will write as xi minus and anything after this xi boundary we will write 00:14:52.200 --> 00:14:58.620 as xi plus. So right now what we are doing we are trying to correlate this okay. 00:14:59.280 --> 00:15:05.340 So let us first write we will not look into the matrix equation first let us write what 00:15:05.340 --> 00:15:14.220 is EF xi plus that is so whenever there is some incident wave okay that is forward propagating 00:15:14.220 --> 00:15:21.840 the forward direction we will write it as EF and then to mention the location we are saying it is 00:15:21.840 --> 00:15:27.360 xi minus. Now at this interface there will be some reflection. So reflection is a backward 00:15:27.360 --> 00:15:34.260 propagating wave so we write it as EB and what is the location it is again behind the interface 00:15:34.260 --> 00:15:41.580 so we will put X I minus. Now there is some transmission so that transmission part we write as 00:15:42.480 --> 00:15:49.260 EF xi plus why xi plus because it is on the other side of the interface. 00:15:49.260 --> 00:15:56.100 There could be another ray if there is like another source on the other side but it is not the 00:15:56.100 --> 00:16:02.040 case we are assuming that the source is only from one side so this this line can be disregarded. 00:16:02.580 --> 00:16:09.660 So in that case what you can write okay but if this is there let us assume if it is there what 00:16:09.660 --> 00:16:16.260 will be this called this is basically a backward propagating electric field the location is xi plus 00:16:17.220 --> 00:16:24.300 clear. So now let us try to write down the equation for EF xi plus that is the forward 00:16:24.300 --> 00:16:31.920 propagating wave that is in this region. So this is nothing but you take this one the 00:16:31.920 --> 00:16:36.720 forward propagating wave and multiply with the transmittance. So whatever is the transmittance 00:16:36.720 --> 00:16:46.320 across this interface we name it as t ij. So t ij times this one okay so that is one component. 00:16:46.320 --> 00:16:52.200 Another thing would be if there is a backward propagating wave here some part will get reflected 00:16:52.200 --> 00:16:58.980 and that will also come here right. So this two rays will be on the same direction so the other 00:16:58.980 --> 00:17:07.740 component will be r ji why ji not ij because here the light is actually in region j and it 00:17:07.740 --> 00:17:14.880 is going towards i so it is r ji reflection from the interface while going from j to i so it is 00:17:14.880 --> 00:17:23.160 r ji and then we multiply EB xi plus clear. So this is how you get this equation equals t ij 00:17:23.160 --> 00:17:34.200 EF xi minus plus r ji EB xi plus okay. Again if you try to write what is this one that is this 00:17:34.200 --> 00:17:40.140 particular wave so how is this particular wave coming? So this particular wave is basically a 00:17:40.140 --> 00:17:48.420 reflection of this forward propagating wave or it could be transmission of this particular backward 00:17:48.420 --> 00:17:58.200 propagating wave on the other side. So EB xi minus can be written as r ij EF xi minus this one plus 00:17:58.200 --> 00:18:05.520 t ji that is this one the transmission across this times EB xi plus clear. 00:18:05.520 --> 00:18:13.200 So this is how you are able to make a relationship between the forward and backward propagating waves 00:18:13.200 --> 00:18:21.120 on both sides. Now as you can see they are bit you know xi plus xi minus they are mixed up so if you 00:18:21.120 --> 00:18:32.340 try to do some algebra and try to rearrange the terms in such a way that you get all the left side 00:18:33.060 --> 00:18:40.680 fields in terms of the right side fields that is how you can rearrange them. In that case so 00:18:40.680 --> 00:18:45.300 if you have the right side fields you multiply with a particular matrix you will be able to 00:18:45.300 --> 00:18:53.340 get the left side field so that way this becomes the transfer matrix of this particular interface. 00:18:54.420 --> 00:18:59.760 So how do you obtain this? This is very simple you actually start with this and try to do the 00:18:59.760 --> 00:19:07.140 algebra to rearrange the terms okay nothing else and here you also can use some other you 00:19:07.140 --> 00:19:15.000 know symmetry relation from the Fresnel equation something like r ij will be equal to minus r ji 00:19:15.000 --> 00:19:27.000 and there is another equality that is t ij times t ji minus r ij times r ji will be equal to 1. 00:19:27.000 --> 00:19:32.100 So if you use these two relations which are coming from Fresnel equation you can easily 00:19:32.760 --> 00:19:39.900 verify this by yourselves if you go to the first slide where I have shown the Fresnel equations 00:19:39.900 --> 00:19:47.400 if you actually put the values here you will be able to prove those relationships okay fine. 00:19:48.060 --> 00:19:55.020 So after you do that you can actually simplify this further and this will get a very nice look 00:19:55.020 --> 00:20:03.240 it looks like this so all the left side fields okay the forward and the backward field they look 00:20:03.240 --> 00:20:13.680 like 1 over t ij times 1 r ij, r ij 1 this is the matrix and then these are the right side fields so 00:20:13.680 --> 00:20:19.320 forward and backward propagating both the fields are there, got it so this is how you obtain 00:20:21.180 --> 00:20:32.040 the transfer matrix for an interface. Now if you see here that what is r ij? 00:20:32.700 --> 00:20:42.900 r ij is nothing but the reflection coefficient right between the interface i and j so if you see 00:20:42.900 --> 00:20:48.540 here what is the reflection coefficient this is basically the ratio of this field over this field 00:20:48.540 --> 00:20:55.560 so you can write this over this right and r ij is already known to you from Fresnel coefficients so 00:20:55.560 --> 00:21:02.640 you can plug in those values here which is ni cos theta i minus nj cos theta j over ni cos theta i 00:21:02.640 --> 00:21:11.220 plus nj cos theta j okay. Similarly you can also write what is t ij? t ij will be EF xi plus that 00:21:11.220 --> 00:21:17.040 is a transmitted electric field over the incident electric field is this one so you can also 00:21:17.640 --> 00:21:22.680 see that this is nothing but 1 plus r ij and that takes this particular 00:21:22.680 --> 00:21:29.220 term when you put this value of r ij okay. Similarly you can also do this exercise for 00:21:29.820 --> 00:21:37.020 p polarization or TM polarization they are the value of r ij the ratio is still same that 00:21:37.020 --> 00:21:43.500 only determines but then the value here will be different similarly t ij will also be different. 00:21:44.220 --> 00:21:52.620 So this is finally the thing that you can obtain because you have to talk in terms of r ij and t ij 00:21:52.620 --> 00:22:00.060 parameters so when you write the transfer matrix capital t ij for the wave propagation through an 00:22:00.060 --> 00:22:06.420 interface between layer i and j this is how it looks like so this is the transfer matrix T ij 00:22:06.420 --> 00:22:14.700 will be 1 over t ij this is the transmission coefficient and 1 r ij and then r ij 1. 00:22:14.700 --> 00:22:19.980 So which value to choose depending on which polarization of the light you are incidenting 00:22:19.980 --> 00:22:25.740 you can choose that value and that will allow you to compute this clear. Now this is what 00:22:25.740 --> 00:22:31.860 is happening in the interface and after the light has actually crossed the interface it is actually 00:22:31.860 --> 00:22:39.120 inside a particular layer now depending on the material and the depth of the layer there will 00:22:39.120 --> 00:22:46.860 be some changes in the phase being accumulated. So let us see how do we actually make a matrix 00:22:46.860 --> 00:22:52.680 of a layer. So the next step is to build the transfer matrix for wave propagation through a 00:22:52.680 --> 00:23:01.560 layer so let us consider that layer is i in that case the refractive index can be written as ni. 00:23:03.060 --> 00:23:09.180 So what are the waves here so this is the wave that has entered so it has basically come from 00:23:09.180 --> 00:23:18.060 this interface right so we will call it as EF xi minus 1 but it is on the other side so it is plus 00:23:18.720 --> 00:23:27.360 fine so it goes like this and this wave only will become the forward propagating 00:23:27.360 --> 00:23:36.420 wave of xi minus because now if you see this is another interface this is the forward electric 00:23:36.420 --> 00:23:42.180 field but this is on the left side of that interface so you can write EF xi minus. 00:23:42.960 --> 00:23:46.200 Now from here what happens there is there will be some reflection 00:23:47.520 --> 00:23:52.860 we do not forget about the transmission here because that will give you the interface 00:23:53.580 --> 00:23:58.140 transfer matrix of the interface but we are interested about the what is happening inside 00:23:58.140 --> 00:24:05.280 a particular layer. So when you move this and hit this particular interface you get some reflection 00:24:06.120 --> 00:24:13.260 how do you name this reflection EB xi minus backward propagating and it is xi minus 00:24:13.260 --> 00:24:20.040 and this field when it will travel and reach here it will be called as EB xi minus 1 plus. 00:24:20.880 --> 00:24:24.360 Now these two electric field do you think they will be exactly same 00:24:25.020 --> 00:24:33.000 or there will be a phase delay because of the path they are travelling from this point to this point. 00:24:33.540 --> 00:24:40.080 So this is what will be the contribution of a particular layer. So let us try to write them in 00:24:40.080 --> 00:24:49.080 terms of equation so you can write this equation that EF xi minus is nothing but EF xi minus 1 plus 00:24:50.160 --> 00:24:52.920 then whatever the phase it has accumulated here. 00:24:53.640 --> 00:25:01.680 So you have to look for the x component so you will e to the power minus j kxi di 00:25:03.780 --> 00:25:11.460 and then this one also can be correlated that this field EB xi minus 1 plus 00:25:12.300 --> 00:25:23.100 can be EB xi minus times this much amount of phase that is added e to the power minus j kxi di. 00:25:24.240 --> 00:25:29.940 Now what is di? di is as I mentioned before di is nothing but the thickness of this layer 00:25:29.940 --> 00:25:38.640 of material ni so this is di this thickness Now we can write it in short form that the transfer 00:25:38.640 --> 00:25:44.520 matrix so you can actually again do the same kind of maths and find out that the transfer matrix 00:25:44.520 --> 00:25:51.780 that correlates the you know left sides field with the right side field will be something like 00:25:51.780 --> 00:26:01.140 this one. So T i will be e to the power j phi i 0 and 0 e to the power minus j phi i. Now what is 00:26:01.140 --> 00:26:09.300 phi i is nothing but kxi di that is the amount of phase so it is a very generic quantity or general 00:26:10.500 --> 00:26:18.000 like it is a in general a complex quantity I must say and in the case of a lossless medium 00:26:18.000 --> 00:26:24.660 and in the absence of total internal reflection you will see that this phase is a real quantity 00:26:25.320 --> 00:26:32.100 and how do you calculate this phase as I told phi i is nothing but kxi d i so k can be kxi 00:26:32.100 --> 00:26:39.420 can be written as 2 pi over lambda naught times ni right and then you have to also look for the 00:26:39.960 --> 00:26:46.740 x component it is a kx right. So what is the kx if this is the direction of wave 00:26:46.740 --> 00:26:53.640 propagation what is this component if this angle is theta i it is cos theta i right so 00:26:53.640 --> 00:27:01.200 that is how cos theta i has actually come and distance is di so this is how you obtain it. 00:27:01.920 --> 00:27:10.020 So if you look into this particular equation that also tells you one more important thing that phi i 00:27:10.020 --> 00:27:19.860 is basically a phase change but this phase change is not related to the amount of the path length 00:27:19.860 --> 00:27:29.940 AC okay. So it is not about from here to here, because of the angle you have you have seen that 00:27:29.940 --> 00:27:39.180 you are basically considering a 90 degree angle here and this is a plane this is another plane So 00:27:39.180 --> 00:27:45.000 this phase difference is basically coming from the distance between these two planes and this 00:27:45.000 --> 00:27:52.920 perpendicular plane to the wave factor is nothing but the phase front. So d or the phase change that 00:27:52.920 --> 00:28:00.960 you are calculating is basically coming from the difference in the phase front that is AB prime. 00:28:01.500 --> 00:28:07.800 So that is the difference between the two phase fronts and phi corresponds to that particular 00:28:08.460 --> 00:28:16.200 value of a b prime clear. So now we have got both the things ready we know what is the 00:28:17.460 --> 00:28:23.760 transfer matrix for one interface between two different material and we also know what is 00:28:23.760 --> 00:28:33.840 the transfer matrix for one particular layer. And then if you put them together you can multiply 00:28:33.840 --> 00:28:40.860 not add them up you have to multiply them okay and that will give you an overall transfer matrix. So 00:28:40.860 --> 00:28:47.160 if you write the left side with respect to the right side quantities then the transfer matrix 00:28:47.160 --> 00:28:55.740 is called T0 to N if you remember the first layer is 0 the last layer was N. So this matrix will 00:28:55.740 --> 00:29:04.920 also have four components T 11 or four elements T 12, T 21 and T 22 and how do you obtain this? 00:29:04.920 --> 00:29:11.760 You have to start with the first layer. So if you remember if you go back you will see 00:29:12.300 --> 00:29:18.840 that yeah. So this is one layer so you will get a transfer matrix for this one, 00:29:18.840 --> 00:29:26.280 but this is the first this is the layer where the wave is already there the wave is propagating in 00:29:26.280 --> 00:29:31.140 that medium so you do not do anything for that. So what you first encounter is the interface 00:29:31.860 --> 00:29:39.900 between n0 and n1 so you have to got a transfer matrix for this one okay. Then you have this 00:29:39.900 --> 00:29:47.700 particular layer then you have this interface then again you have this layer and so on. Finally 00:29:47.700 --> 00:29:54.120 you will also have this interface and then this particular layer. So that many transfer matrices 00:29:54.120 --> 00:30:00.780 you have to put together and multiply and that is how you will be able to get the overall 00:30:00.780 --> 00:30:10.080 transfer matrix which is given as T0 to N. So it looks bit cumbersome and lengthy process but you 00:30:10.080 --> 00:30:14.820 can use MATLAB codes for writing this transfer matrix and you can do the matrix multiplication 00:30:14.820 --> 00:30:21.180 very easily and that will give you the overall transfer matrix of a multi-layered system. 00:30:22.140 --> 00:30:27.840 So now let us look into how to obtain the overall transfer matrix as I mentioned that this 00:30:27.840 --> 00:30:35.580 expression can be used for solving a variety of wave propagation problem. So first thing you will 00:30:35.580 --> 00:30:44.400 start with transfer matrix for that plane or that interface where the plane wave was propagating 00:30:45.180 --> 00:30:52.680 and then it will encounter the first layer so you have to put the transfer matrix for that interface 00:30:53.400 --> 00:31:00.840 okay. So first interface then layer then interface then layer and finally you will 00:31:00.840 --> 00:31:06.420 get another interface and then it will enter into another layer and that is it that will be 00:31:06.420 --> 00:31:13.800 the overall transfer matrix. Now as we mentioned earlier also that we can assume that there is no 00:31:13.800 --> 00:31:21.300 field that is coming from the right side that means EB n minus xN minus 1 plus is 00:31:21.300 --> 00:31:28.680 always 0 that is no electric field from the other direction. So once you do that you can simplify 00:31:28.680 --> 00:31:37.920 the equation like this EF x0 minus EB x0 minus can be correlated by the transfer matrix this 00:31:37.920 --> 00:31:44.880 is the overall transfer matrix that is why look at the superscript 0 to N 0 to N and these are 00:31:44.880 --> 00:31:52.140 the elements of the matrix 11 12 21 and 22 and here we have put that this term is 0 okay. 00:31:52.800 --> 00:32:04.680 EB xN minus 1 plus is 0 clear. So this is the final transfer matrix. Now will you be able 00:32:04.680 --> 00:32:11.340 to correlate the reflection and transmission from this transfer matrix the answer is yes. 00:32:11.940 --> 00:32:17.400 Now if you see this is the forward propagating wave that was basically the incident wave 00:32:17.400 --> 00:32:23.820 what is EB x0 minus that is the reflected wave from the very first interface. So if you want to 00:32:23.820 --> 00:32:31.020 get reflection coefficient it will be nothing but the ratio of EB x0 minus over EF x0 minus 00:32:32.040 --> 00:32:37.020 and if you talk in terms of the matrix elements which you have just computed 00:32:37.020 --> 00:32:40.020 you will compute this matrix rate by multiplying all the matrices 00:32:40.740 --> 00:32:48.420 you can actually get this ratio by doing t 0 to N you have to look for the 21 element this one and 00:32:48.420 --> 00:32:54.960 divide it by the 11 element then this will gives you this will give you the reflection coefficient. 00:32:55.920 --> 00:33:03.060 Whatever transmission the transmission coefficient is nothing but this over the incident one. 00:33:03.060 --> 00:33:13.800 So EF xN minus 1 plus divided by EF x0 minus okay and that can be simplified and you can 00:33:13.800 --> 00:33:22.860 from the equation you can see that it is simply 1 over t 11 okay. So the main important part is 00:33:22.860 --> 00:33:28.260 to calculate this transfer matrix once you are done with the calculation you have got the 4 00:33:28.260 --> 00:33:33.240 elements you can find out what is the reflection coefficient what is the transmission coefficient 00:33:34.380 --> 00:33:40.320 and once you know the reflection and transmission coefficient, reflectance and transmittance 00:33:40.320 --> 00:33:46.860 are very easy to find reflectance will be nothing but modulus square of your reflection 00:33:46.860 --> 00:33:55.680 coefficient transmittance will be nothing but real of n N cos theta N over real of n0 00:33:56.280 --> 00:34:03.060 that is the incident media times cos theta 0 this ratio will be multiplying the modulus of 00:34:03.060 --> 00:34:09.420 transmission coefficient square very simple and always remember if it is a non-absorbing media 00:34:09.960 --> 00:34:17.940 capital R plus capital T should be equal to 1 okay. So that completes the discussion on transfer 00:34:17.940 --> 00:34:25.620 matrix. So this transfer matrix will allow you to find out you know reflection or transmission from 00:34:25.620 --> 00:34:33.180 any multi-layered system and it is a very very effective tool lot of researchers even today use 00:34:33.180 --> 00:34:38.580 transfer matrix method for calculating reflection and transmission from multi-layered systems. 00:34:39.480 --> 00:34:44.880 One such multi-layered system is Fabry-Perot interferometer cavity or also called as etalon. 00:34:44.880 --> 00:34:48.300 So you might have heard of this Fabry-Perot interferometer before. 00:34:49.020 --> 00:34:56.160 So it actually tells you that is an optical resonator. So you have optical or you can say 00:34:56.160 --> 00:35:04.320 light rays entering here ok and they will there will be some reflection obviously. 00:35:04.980 --> 00:35:11.040 So say this is the light ray incidenting then you have got some reflection 00:35:11.580 --> 00:35:19.980 it has entered after being refracted into this medium. So this is n prime this medium is n and 00:35:19.980 --> 00:35:25.680 again this is n prime the same old medium. So you can take air glass air to easily 00:35:25.680 --> 00:35:34.860 understand. So this wave has entered hit this interface some part got reflected then remaining 00:35:34.860 --> 00:35:40.020 part got transmitted. This transmitted light when it hits this particular interface again 00:35:40.020 --> 00:35:50.520 some part got reflected ok and remaining will get transmitted. Out of this reflection when it 00:35:50.520 --> 00:35:56.940 will hit this particular interface again some part will get reflected remaining will get transmitted 00:35:56.940 --> 00:36:08.580 and it will go on happening okay. So in that case you what we will see if all these lights which are 00:36:08.580 --> 00:36:15.660 basically coming out of this they constructively interfere you will get a particular color. 00:36:16.320 --> 00:36:22.020 Similarly so these are all you can say because these are all back in the same direction you can 00:36:22.020 --> 00:36:27.360 say all these are basically reflected light. All these are on the other side of the incident 00:36:27.360 --> 00:36:33.120 wave you can say these are all transmitted light okay. So you can say that when all of them will 00:36:33.120 --> 00:36:40.380 interfere constructively you will get a color out there. So this is how typically etalon or 00:36:40.380 --> 00:36:45.960 Fabry-Perot interferometer works. So this is again a multi-layer structure one dielectric 00:36:45.960 --> 00:36:50.700 another dielectric and then third dielectric. So this is a three layer system one two and three. 00:36:50.700 --> 00:36:56.760 How many interfaces are there? Two. This is the interface between first and second layer 00:36:57.660 --> 00:37:01.260 this is the interface between second and third layer, this is the interface 00:37:02.340 --> 00:37:09.060 right. So it is a three layer system. So now we have to first write what is the transfer matrix 00:37:09.060 --> 00:37:18.060 for T12 this particular interface. So this is the formula we can bring from those lessons 00:37:18.060 --> 00:37:23.880 we have learned today that for a particular interface this is how it looks like ok. 00:37:23.880 --> 00:37:33.240 Now R12 R12 you can write as R okay. T12 you can calculate and find out what is where you 00:37:33.240 --> 00:37:38.160 will get this from again these are the Fresnel coefficient okay. So you can simply do the Fresnel 00:37:38.160 --> 00:37:44.460 coefficients and calculate and put them here. What about capital T23 that is basically the transfer 00:37:44.460 --> 00:37:52.260 matrix for this particular interface that is given as this. So here R and here this will be minus R 00:37:52.260 --> 00:37:58.260 because these are the same media just opposite okay. So it was air to glass here it is glass 00:37:58.260 --> 00:38:07.020 to air so you can put minus r minus r here. So as it is also shown that r is nothing but r12 00:38:07.020 --> 00:38:14.940 which is minus r23 or if you don't like this kind of short simplified relations see blindly take 00:38:14.940 --> 00:38:20.100 the Fresnel equation and put the parameters you will get the equations. So you have got the two 00:38:20.100 --> 00:38:28.680 transfer matrices. What is left the only thing left is the transfer matrix in layer 2 that we 00:38:28.680 --> 00:38:38.820 name as T2 ok fine. So see it is only written 2 not ij format layer 2 so it is only 2. 00:38:38.820 --> 00:38:46.320 So e to the power j phi, 0, 0, e to the power minus j phi. Now your job is to put the value of 00:38:46.320 --> 00:38:55.140 phi here. So you can also take this one common and make it look like this. So phi is nothing but 2 pi 00:38:56.400 --> 00:39:02.940 lambda 0 that is k0 times n that is the refractive index times l that 00:39:02.940 --> 00:39:07.920 is the thickness and it is basically the x component so cos theta. So this is the value. 00:39:08.580 --> 00:39:15.840 So now you have got all the three matrices T12 T23 and also T2 this one this is the layer 1. 00:39:15.840 --> 00:39:25.620 So the overall transfer matrix is nothing but you can name it as 13 it is like 0 N okay. So that is 00:39:25.620 --> 00:39:31.560 nothing but T and that is multiplication of all this transfer matrix. So you put all the elements 00:39:31.560 --> 00:39:37.980 and multiply them and after you multiply them you will get the four components. Once you got the 00:39:37.980 --> 00:39:44.220 four components your reflection coefficient will be again you go back to this formula 00:39:45.480 --> 00:39:52.140 reflection coefficient will be nothing but 21 element divided by 11 element and transmission 00:39:52.140 --> 00:40:00.420 coefficient will be 1 over the 11 element, that is what so that is very simple. So let us see 00:40:01.140 --> 00:40:08.280 yeah T21 over T12 these are the two elements so from this matrix after multiplication whatever 00:40:08.280 --> 00:40:14.700 you have got you pick those two 21 and 11 you have put them here so this becomes your 00:40:14.700 --> 00:40:22.860 reflection coefficient and T will be like this 1 over T11 so you put all these values 00:40:22.860 --> 00:40:28.680 here this is what you get very simple. So this is how you can get the reflection 00:40:28.680 --> 00:40:37.620 and transmission coefficient. Obviously it is easy to find out what is the reflectance and 00:40:37.620 --> 00:40:44.100 transmittance as I mentioned so it will be square of the reflection coefficient also square of the 00:40:44.100 --> 00:40:52.440 transmission coefficient here the final medium and initial medium are same so that ratio will be 1 ok 00:40:53.640 --> 00:41:01.500 clear. Now why this is important this particular Fabry-Perot interferometer has got a lot of 00:41:01.500 --> 00:41:06.900 application even in laser cavity you have Fabry-Perot interferometer. So here what 00:41:06.900 --> 00:41:12.720 is happening so if the incident intensity is taken as unity the first transmitted intensity 00:41:12.720 --> 00:41:24.360 this one will have you know the values of t12 that is the transmission here and then whatever 00:41:24.360 --> 00:41:31.020 you are getting here is T12 times whatever is the transmission coefficient of this interface that is 00:41:31.020 --> 00:41:42.960 t23 so this value becomes t12 times t23. Similarly that if you look for the reflection condition the 00:41:42.960 --> 00:41:50.280 second one this one will get twice reflected so it is getting a reflection from here so it will have 00:41:50.280 --> 00:42:00.240 r23 also it will have multiplied by r12 on top of that you will have the coefficients of t12 and 00:42:01.200 --> 00:42:07.440 t23 to find out what is coming out here. So they will become you know progressively 00:42:07.440 --> 00:42:14.280 weaker as you can understand and if you are able to maintain the phase difference 00:42:15.180 --> 00:42:20.760 ok in that case they will add up constructively and you will get a particular color. 00:42:20.760 --> 00:42:27.480 So the phase difference is nothing but delta phi sorry phase difference is given by delta that 00:42:27.480 --> 00:42:33.900 is taken as round trip phase difference so it is given as 2 phi. So phi you already know how what 00:42:33.900 --> 00:42:40.920 is the formula you put 2 phi here and then you can understand that if this round trip phase 00:42:40.920 --> 00:42:48.900 difference is an integral multiple of phi so if it is equal to m phi then you are able to get a 00:42:48.900 --> 00:42:55.680 constructive interference and your transmission will be maximum at those particular points okay 00:42:55.680 --> 00:43:02.520 or you can say at those particular angle theta. So in that case you can also find out what will 00:43:02.520 --> 00:43:07.920 be the maximum transmission but in the other case if you take when the round trip phase difference 00:43:07.920 --> 00:43:17.220 is basically odd multiple of or you can say 2m minus 1 times phi ok in that case you can find out 00:43:17.880 --> 00:43:26.940 oh it is odd multiple of phi by 2 in that case it will give a minima in the transmission because in 00:43:26.940 --> 00:43:34.200 that case all these waves will destructively interfere ok. So that that is how you are able 00:43:34.200 --> 00:43:40.140 to Fabry-perot is able to do the filtering so only at certain wavelength you will get 00:43:40.140 --> 00:43:47.940 very sharp transmission other wavelengths will be blocked okay. So these are the conditions 00:43:47.940 --> 00:43:54.600 and if you try to relate it to wavelength so lambda is nothing but c by f okay. 00:43:54.600 --> 00:44:02.340 So you can find out the condition for maximal transmission as nu M, M is an integer nu or f 00:44:03.000 --> 00:44:14.160 so it will be c over 2nl cos theta. So, this will tell you that theta n and l will tell you which 00:44:14.160 --> 00:44:22.080 frequency you will be letting let out of this particular interferometer. So it actually becomes 00:44:22.080 --> 00:44:30.240 like a filter, Febry-Perot cavity based filter okay. So with that you can also find out that 00:44:31.200 --> 00:44:39.660 all transmitted contributions at resonance they will interfere constructively but in off 00:44:39.660 --> 00:44:46.680 resonance the contributions do not go longer they will no longer interfere constructively, 00:44:47.700 --> 00:44:54.840 in that case the two neighboring resonance frequencies will be separated by so called 00:44:54.840 --> 00:45:00.660 spectral range free spectral range FSR. So that is given as delta nu which is nothing 00:45:00.660 --> 00:45:08.700 but nu m plus 1 minus nu m. So if you do the calculation you see it turns out to be c over 2nl 00:45:08.700 --> 00:45:14.580 cos theta that means how much the two resonant frequencies will be different to each other 00:45:15.180 --> 00:45:23.100 ok. So that also tells you that how much channel spacing you can obtain from that particular 00:45:23.100 --> 00:45:31.140 filter. There is another parameter that is called finness or you can say fineness how fine it is ok. 00:45:31.740 --> 00:45:38.040 So it is directly related to the mirrors reflection at each interface why mirror because 00:45:38.040 --> 00:45:43.080 every reflecting surface is a mirror right. So in eta learn also you have seen that there are two 00:45:43.860 --> 00:45:50.700 ideally fabry-perot cavity that is used in laser they will have very good reflector in 00:45:50.700 --> 00:45:58.260 one end and in another one it will be like 99. 95 kind of reflectance. So they are very very good 00:45:58.260 --> 00:46:04.380 mirror to give you very good spectral selectivity. So this mirror reflection plays a important role 00:46:04.380 --> 00:46:11.880 so the finness factor is given as pi square root of R over 1 minus R. So you can see the higher 00:46:11.880 --> 00:46:18.420 the mirrors reflection at each interface higher is the finness it means the sharper will be the 00:46:18.420 --> 00:46:26.880 transmission peak. So sharper peak means you will get very good quality output filtered output. 00:46:27.420 --> 00:46:33.840 Another important thing that will also correlate so I already gave out the hint that when it is 00:46:33.840 --> 00:46:41.340 very fine you also will get very high quality factor and quality factor Q is defined as omega 00:46:41.340 --> 00:46:47.760 R that is the resonant frequency divided by the spread in frequency okay that is delta omega. 00:46:47.760 --> 00:46:54.000 So delta omega is nothing but FWHM full width half maxima ok. So it actually gives you a measure of 00:46:54.000 --> 00:47:03.600 the resonance in the spectrum fine. So this is how it looks like so if you take R equals 30 so 00:47:03.600 --> 00:47:10.200 you will get this solid line so you see they are not that fine not that sharp so you still have 00:47:10.200 --> 00:47:18.240 transmission but it is not that selective. So on the other side so this is basically transmission 00:47:18.240 --> 00:47:28.680 spectrum so at this particular cases where the phase is 4 times pi or 5 times pi or 6 times 00:47:28.680 --> 00:47:37.080 pi it means every m pi okay, you see there is a resonance fine and this is transmission resonance 00:47:37.080 --> 00:47:44.640 it means if you look into the reflection spectrum or capital R which is nothing but modulus of 00:47:44.640 --> 00:47:51.960 small r square you will see that there is drop. So reflection based it is basically band stop filters 00:47:51.960 --> 00:48:00.840 but in the case of transmission you can actually get band pass filters based on Fabry-Perot cavity. 00:48:01.560 --> 00:48:11.040 So here the cavity length is actually taken care of by this one so 5 pi that actually 00:48:11.040 --> 00:48:16.620 gives you an idea of what will be the cavity length because this value is changing fine. 00:48:16.620 --> 00:48:22.920 So you can actually correlate with that but always remember that the resonating peaks will appear at 00:48:22.920 --> 00:48:37.740 m pi locations where M is an integer fine. So with that we will conclude our discussion today on T 00:48:37.740 --> 00:48:43.740 matrix as I mentioned it is a very very useful tool for theoretical calculation of reflection 00:48:43.740 --> 00:48:50.520 and transmission across any multi-layer system. This one is this one was an example of multi-layer 00:48:50.520 --> 00:48:56.400 system it is called Fabry-Perot interferometer or etalon ok. It is a three layer system you can do 00:48:56.400 --> 00:49:02.220 it for any layer system 4 5 6 7 8 9 10 11 or even after 200 layer system does not matter. 00:49:02.220 --> 00:49:07.800 The same formula of using the transfer matrix for the interface layer interface layer 00:49:07.800 --> 00:49:12.600 multiply everything you will get a transfer matrix with only four elements okay. 00:49:12.600 --> 00:49:19.500 And then if you take the ratio of 21 over 11 you get reflection coefficient if you take the 00:49:19.500 --> 00:49:27.060 inverse of the 11 element you get transmission coefficient once you have both this coefficient 00:49:27.060 --> 00:49:32.040 you take the square of it ok modulus square of it and for transmittance you have to add 00:49:32.040 --> 00:49:39.240 that factor you get reflection and transmission that is it very simple. So with that we stop here 00:49:39.240 --> 00:49:46.080 today and in the next lecture we will start the discussion of 1D photonic crystals. Thank you.