WEBVTT Kind: captions Language: en 00:00:02.580 --> 00:00:07.020 Hello 00:00:31.860 --> 00:00:37.800 students, welcome to lecture 12 of the online course on Nanophotonics, Plasmonics and 00:00:37.800 --> 00:00:44.220 Metamaterials. In this lecture we will be covering dispersion relation and photonic band structure. 00:00:45.480 --> 00:00:49.680 So, here is the lecture outline. So, we will introduce the eigenvalue problem 00:00:50.220 --> 00:00:57.180 on dispersion relation and bloch modes. And we will discuss the matrix 00:00:57.900 --> 00:01:05.010 optics approach for solving you know eigenvalue problem and obtaining the bloch modes. We 00:01:05.010 --> 00:01:10.500 will also see how to obtain the dispersion relation calculating photonic band structure 00:01:10.500 --> 00:01:15.840 and obtaining phase and group velocities. So, eigenvalue problem and dispersion relation. 00:01:15.840 --> 00:01:22.320 If you remember from the previous lecture that till now we have established the mathematical 00:01:22.320 --> 00:01:28.200 form of the bloch modes as imposed by the translational symmetry of the periodic medium. 00:01:28.980 --> 00:01:36.960 So, we were considering periodic medium where the refractive index periodically alters 00:01:37.620 --> 00:01:44.700 and these are the bloch modes, ok. And this is the typical 1D periodic medium 00:01:44.700 --> 00:01:51.240 we have discussed in the previous lecture. We have also seen that you know for this 1D 00:01:51.240 --> 00:01:58.620 periodic medium we are able to find dispersion relation. Dispersion relation in brief we can 00:01:58.620 --> 00:02:06.540 say it is basically omega k relation and in this particular case we were also able to see that 00:02:06.540 --> 00:02:14.280 certain you know frequencies were not allowed to propagate. So, those actually gave you band gap 1, 00:02:14.280 --> 00:02:22.200 band gap 2 and so on. So, our objective here is to know how do we go to this particular dispersion 00:02:22.200 --> 00:02:30.300 relation starting from a 1D periodic medium. So, our objective here would be to solve the 00:02:30.300 --> 00:02:37.440 eigenvalue problem described by the generalized Helmholtz equation for this 1D periodic system. 00:02:38.160 --> 00:02:43.740 So, for this there are two approaches one is Fourier optics another is matrix optics. 00:02:45.600 --> 00:02:50.160 So, the first approach let us have a quick look that is called Fourier optics. 00:02:50.880 --> 00:02:59.640 Now this approach is based on expanding the periodic function say eta of z of the medium, ok 00:02:59.640 --> 00:03:06.840 or you can say n z eta is the impedance or you can also talk in terms of refractive index, ok. And 00:03:06.840 --> 00:03:13.920 the periodic function p k z of the block mode. So, you have to expand this in Fourier series, ok. 00:03:14.640 --> 00:03:21.660 And then you convert the Helmholtz differential equation into a set of algebraic equation cast 00:03:21.660 --> 00:03:28.740 in the form of matrix eigenvalue problem and this you have to solve numerically. So, that's 00:03:29.340 --> 00:03:35.640 what is known as the Fourier optics method. The other method is called matrix optics. 00:03:36.660 --> 00:03:43.080 So, in this case this particular method is applicable to layered 00:03:43.620 --> 00:03:49.020 which are basically piecewise homogeneous like the previous example you have taken periodic 00:03:49.860 --> 00:03:57.900 alteration of refractive index a dielectric 1 2 1 2 1 2 and so on, ok. And if you have those 00:03:57.900 --> 00:04:05.460 kind of layered media with planar boundaries you are able to use matrix optics method. 00:04:05.460 --> 00:04:09.180 Now, in this particular method instead of solving the Helmholtz equation 00:04:09.900 --> 00:04:15.780 we make direct use of the laws of propagation reflection and refraction that is more or less 00:04:15.780 --> 00:04:21.780 you know the transfer matrix formulation that you have studied couple of lectures back. So, 00:04:21.780 --> 00:04:27.840 you can use those you know laws of propagation, reflection, refraction at the boundaries, ok, 00:04:27.840 --> 00:04:34.860 which are basically the known of the Maxwell's equation. And then you can use the matrix methods 00:04:34.860 --> 00:04:41.340 developed for multilayer media, right. So, when you apply matrix optics method finally, 00:04:41.340 --> 00:04:48.720 you will get a 2 by 2 matrix eigenvalue problem from which you can obtain the dispersion relation 00:04:48.720 --> 00:04:55.260 and the bloch modes. So, we will be mainly covering this particular matrix optics approach 00:04:55.260 --> 00:05:00.480 in this lecture and this course of course. So, let's look into matrix optics 00:05:00.480 --> 00:05:05.400 approach. The complex amplitude of the forward and backward waves 00:05:05.940 --> 00:05:14.400 through the boundaries of multilayered medium is facilitated by the use of matrix method something 00:05:14.400 --> 00:05:23.340 like this. So, here you can take an example of multilayered media, medium 1 2 3 4 and so on. So, 00:05:23.340 --> 00:05:28.680 just to make you understand how complex the system could be. So, you start with one particular wave 00:05:29.280 --> 00:05:34.860 that is partially getting reflected, some part is getting refracted or transmitted. 00:05:35.460 --> 00:05:43.560 Now this light when it is in this particular medium, when it hits this particular interface 00:05:43.560 --> 00:05:49.980 between this medium and this medium, some part of this light is getting partially reflected, 00:05:49.980 --> 00:05:56.340 remaining is getting transmitted. And again this transmitted light when it encounters 00:05:56.340 --> 00:06:00.960 this particular interface, some part is getting reflected back, some is getting transmitted. 00:06:01.500 --> 00:06:07.140 Now what happens to this reflection? This basically a backward propagating light or wave. 00:06:07.140 --> 00:06:13.020 So, here again at this interface it has got two options, one is to transmit, 00:06:13.020 --> 00:06:20.700 one is to reflect and so on. So, this is how things happen now in a multilayered medium. 00:06:20.700 --> 00:06:25.020 So, you start with the single wave, but because of these boundaries you 00:06:25.020 --> 00:06:31.080 end up with getting you know numerous transmitted and reflected light beams. 00:06:32.400 --> 00:06:40.260 Now in part b this particular figure what is shown is that in each layer the forward moving 00:06:40.260 --> 00:06:50.040 waves can be named as plus, ok. And backward ones can be denoted using this minus symbol. So, 00:06:50.040 --> 00:06:58.200 when you are in say medium 1 you can say that it has got a you know all the forward moving waves 00:06:58.200 --> 00:07:04.980 can be summed up together that can be called as U1 plus and all the backward propagating waves 00:07:04.980 --> 00:07:10.020 means all these reflections they can be summed up together or collected together and you can 00:07:10.020 --> 00:07:16.260 call them as U 1 minus. Same in layer 2 you can have U 2 plus and U 2 minus. 00:07:16.260 --> 00:07:21.300 Here you can have this is the incident medium and this is the final transmitted medium. So, 00:07:21.300 --> 00:07:27.300 we are just these are the two you know layers that actually form this multilayered system 00:07:27.300 --> 00:07:40.500 in this case. Now there is a way this U 1 plus and U 1 minus are kind of correlated. 00:07:41.280 --> 00:07:48.900 So, the amplitudes of this forward and backward collected waves they can be represented using a 00:07:48.900 --> 00:07:56.100 column matrix. So, if you actually consider this particular layer by a matrix M. 00:07:56.100 --> 00:08:03.900 So, what you see here is that there is incoming wave, there is a backward propagating wave and 00:08:03.900 --> 00:08:08.100 there is a forward and again backward propagating wave. So, you can actually 00:08:09.660 --> 00:08:19.260 represent this interface using this matrix M. So, this is the column matrix. So, 00:08:19.260 --> 00:08:26.340 let's correlate the coefficients. So, you can have on the right side you have U 2 plus U 2 minus 00:08:27.300 --> 00:08:35.700 and this M matrix can have four elements A B C D that are correlating this amplitude to the 00:08:36.240 --> 00:08:41.160 amplitudes on the left hand side that is U 1 plus and U 1 minus. 00:08:42.600 --> 00:08:52.620 So, the matrix M whose elements are A B C D this is called the wave transfer matrix and it depends 00:08:52.620 --> 00:09:00.960 between depends on the optical properties of the layered medium between the two planes. Now, 00:09:00.960 --> 00:09:07.380 how do you apply this for a periodic media? So, in a periodic media you can actually see that this 00:09:08.580 --> 00:09:15.000 a unit cell is basically repeated right. So, if you have two alternating dielectric 00:09:15.000 --> 00:09:21.240 material say one is having high and another is having relatively low refractive index. So, 00:09:21.240 --> 00:09:27.240 the periodic media will be something like high low high low high low and so on. So, you can actually 00:09:27.240 --> 00:09:34.320 represent each unit cell using one matrix and then you can repeat it like this ok. 00:09:34.320 --> 00:09:38.640 So, this is the wave transfer matrix representation of a periodic medium 00:09:39.180 --> 00:09:45.420 right. So, you can see that you know the periodicity is basically capital lambda. So, 00:09:45.420 --> 00:09:51.660 here you can say this is m lambda and this is m plus 1 lambda. What will be this one? This is m 00:09:51.660 --> 00:09:57.540 minus 1 lambda like from here to here it is the period ok. That is given by capital lambda. 00:09:58.740 --> 00:10:06.720 So, between one period to the other there is a matrix that is correlating 00:10:07.320 --> 00:10:17.580 the parameters of U m plus U m minus to U m plus 1 plus and U m plus 1 minus. So, you are 00:10:17.580 --> 00:10:23.940 basically correlating the forward propagating waves and the backward propagating waves. So, 00:10:23.940 --> 00:10:31.740 as you can see here a 1D periodic medium comprises of this identical segments like M O ok. They are 00:10:31.740 --> 00:10:37.140 called unit cells. They are repeated along one direction in this case we have considered Z is 00:10:37.140 --> 00:10:43.620 the direction of periodicity ok and they are separated by period capital lambda. 00:10:45.300 --> 00:10:52.500 And unit cells contain repetition of lossless dielectric layers or you can say these are 00:10:52.500 --> 00:10:58.380 partially reflective mirrors.Why they are called partially reflective mirrors? If you remember 00:10:58.380 --> 00:11:05.220 that any interface wherever there is a difference between the refractive indices ok there will be 00:11:05.220 --> 00:11:12.300 light reflection. This can also be now discussed in terms of impedance mismatch. So, if you 00:11:12.300 --> 00:11:16.800 take the impedance of the two different layers across the interface you will see that there is 00:11:16.800 --> 00:11:22.380 some difference in the or there is a mismatch in the impedance and that is why you will get some 00:11:22.380 --> 00:11:27.660 reflection of the incident wave from that interface. Now forming a symmetric system 00:11:27.660 --> 00:11:35.820 generated by a generic wave transfer matrix. So, you can actually make a generic wave transfer 00:11:35.820 --> 00:11:42.600 matrix which looks like this. What are the elements here? 1 by T conjugate R over T 00:11:42.600 --> 00:11:50.700 R conjugate over T conjugate and 1 over T. So, this is how you are able to write a generic wave 00:11:50.700 --> 00:11:59.700 transfer matrix where R and T are basically the reflectance and transmittance. This you 00:11:59.700 --> 00:12:05.460 have already seen from the Fresnel equation you know what is reflectance and transmittance. So, 00:12:05.460 --> 00:12:11.940 if you want to calculate what is or you can this these are basically amplitude transmittance. 00:12:11.940 --> 00:12:17.400 So, small t can be called as transmission coefficient or you can call them as 00:12:17.400 --> 00:12:26.400 amplitude transmittance, small r can be called as reflection coefficient or amplitude reflectance. 00:12:26.400 --> 00:12:32.520 So, correspondingly you can find out what is the intensity transmittance and intensity reflectance. 00:12:32.520 --> 00:12:41.160 So, this is how you can calculate ok.T equals modulus T square R equals modulus small r 00:12:41.160 --> 00:12:46.920 square. So, the electromagnetic wave traveling through the medium they will undergo numerous 00:12:46.920 --> 00:12:55.080 transmissions and reflections that we have seen in the previous slide and that will actually give you 00:12:55.080 --> 00:13:02.640 one particular you know forward and one particular backward moving wave at each plane ok. 00:13:03.360 --> 00:13:09.480 And the transfer matrix this particular matrix method not transfer matrix this is called matrix 00:13:09.480 --> 00:13:18.660 optics method this can be used to determine the block modes. So, let's assume U M plus minus as 00:13:18.660 --> 00:13:24.960 the complex amplitudes. So, plus 1 correspond to the forward and minus corresponds to the 00:13:24.960 --> 00:13:34.380 backward wave at any initial position z equals m lambda. So, here this particular one. So, 00:13:34.380 --> 00:13:41.280 what is m? m is the number of the unit cell ok. So, the amplitudes elsewhere within the cell can 00:13:41.280 --> 00:13:47.520 be determined by a straightforward application of the appropriate wave transfer matrices as 00:13:47.520 --> 00:13:53.640 discussed in the previous lecture. So, we have seen this already that you know if you know at 00:13:53.640 --> 00:14:00.360 one position you can add that phase and you can get the amplitude at any other location. 00:14:02.400 --> 00:14:11.340 Now the dynamics of the amplitude varies from one cell to another which is described by the 00:14:11.340 --> 00:14:18.240 recurrence relation.That means the amplitude U m plus and U m minus they will vary from one cell 00:14:18.240 --> 00:14:25.140 to another, but in a repeated pattern. So, what is the pattern? That is kind of like this. 00:14:25.740 --> 00:14:33.840 So, this is the initial amplitude and you multiply it by this particular unit cell matrix that is M 00:14:33.840 --> 00:14:40.140 0 you will get the next set of amplitude of the forward and backward moving wave. 00:14:40.740 --> 00:14:47.160 So, this relations are used to compute the complex amplitude at any particular cell if the amplitude 00:14:47.160 --> 00:14:52.440 of the previous cell are known. Make sense? These are the amplitude of the previous cell, 00:14:52.440 --> 00:14:59.340 this is the cell matrix. So, when you multiply this you get the amplitude of the current cell. 00:15:00.120 --> 00:15:04.560 Now let us see how do you obtain eigenvalue problem and block modes from this. 00:15:05.160 --> 00:15:09.540 So, by definition the modes of the periodic medium 00:15:10.140 --> 00:15:17.340 are self reproducing and why so because they actually maintain a particular phase relation. 00:15:18.120 --> 00:15:25.260 So, you can say that if you take the amplitude of the forward and backward moving waves for mth 00:15:25.260 --> 00:15:33.180 cell where m is 1 or 2 or 3 or so dot dot dot anything. In that case if you multiply this by 00:15:33.180 --> 00:15:39.180 e to the power minus j phi that is the amount of phase accumulated while crossing this unit 00:15:39.180 --> 00:15:47.640 cell you can actually get the amplitude of the forward and backward propagating waves of the next 00:15:47.640 --> 00:15:57.060 unit cell ok. So, here what is important that you know this phi is basically the phase accumulated 00:15:57.780 --> 00:16:04.260 over the distance of the period and the period is nothing, but capital lambda. 00:16:05.460 --> 00:16:13.860 So, there is a name to this phase. So, this phase are basically altered by a common shift phi and 00:16:13.860 --> 00:16:23.400 this phi is called as block phase ok. So, there is a corresponding block wave number which is defined 00:16:23.400 --> 00:16:32.280 as K capital K that is given by phi over capital lambda. So, obviously what is that then phi which 00:16:32.280 --> 00:16:39.720 is block phase phi turns out to be K capital lambda ok. So, this is nothing, but block phase. 00:16:40.980 --> 00:16:47.580 So, finding the complex amplitudes that is Um plus minus and the phase phi which is 00:16:47.580 --> 00:16:54.360 defined as K capital lambda from the following equation which satisfy the self reproduction 00:16:55.740 --> 00:17:02.160 condition can be cast as an eigenvalue problem. So, let us see how it looks like. So, if you take 00:17:02.160 --> 00:17:10.020 this particular problem where you know you already seen this equation that this is the 00:17:10.020 --> 00:17:14.940 phase relationship between the amplitude of the next cell and the previous cell 00:17:15.660 --> 00:17:28.890 and here if you put m equals 0 you get U 0 plus U 0 minus ok and what you will have here basically U 00:17:28.890 --> 00:17:41.760 1 plus and U 1 minus right and U 1 plus and U 1 minus you can go back and from here you can get 00:17:41.760 --> 00:17:50.520 that U 1 plus and U 1 minus can be written as m 0 u 0 plus and u 0 minus right. So, if you bring 00:17:50.520 --> 00:18:00.600 this equation here. So, on the left hand side you get M0 U0 plus U 0 minus is equal to e to 00:18:00.600 --> 00:18:12.240 the power minus J phi U0 plus U0 minus. So, this is an eigenvalue problem of this 2 by 2 unit cell 00:18:12.240 --> 00:18:20.340 matrix M0 right. So, here if you look into this particular equation this is your eigenvalue. 00:18:20.880 --> 00:18:31.560 So, the factor e to the power minus J phi is the eigenvalue ok and the vector with components U 00:18:31.560 --> 00:18:38.700 0 plus and U0 minus are basically the eigenvector right. So, how do you obtain the eigenvalues? The 00:18:38.700 --> 00:18:46.080 eigenvalues are basically obtained by equating the determinant of the matrix that is M0 minus e 00:18:46.080 --> 00:18:54.720 to the power minus J phi times identity matrix. If you take this determinant and equate it to 0 you 00:18:54.720 --> 00:19:01.560 will get the values at which you will get the those solutions are basically the eigenvalues 00:19:01.560 --> 00:19:09.600 ok. Now we already know that you know the reflect these are non-absorbing material. So, 00:19:10.620 --> 00:19:17.160 amplitude of transmission amplitude transmission coefficient square plus 00:19:18.000 --> 00:19:24.780 you know square of the amplitude reflection coefficient square is equal to 1. 00:19:24.780 --> 00:19:31.320 In that case you can actually find out the values which is e to the power minus J phi 00:19:31.320 --> 00:19:38.580 ok can be given as this quantity. So, you have this transmission coefficient also its 00:19:38.580 --> 00:19:46.440 conjugate and this is the value that you obtain. And from this you can write that you know if you 00:19:47.700 --> 00:19:54.000 separate it out to the real and imaginary part on both sides you can find that cos 00:19:54.000 --> 00:20:04.440 phi can be written as real of 1 over the amplitude transmission coefficient. So, real of 1 over T. 00:20:05.640 --> 00:20:11.880 So, keep this equation in mind. So, now let us try to obtain what is the dispersion relationship. 00:20:11.880 --> 00:20:19.200 Always remember dispersion relationship is basically the relationship between the bloch 00:20:19.200 --> 00:20:28.200 wave number K and the angular frequency omega that is we are looking for omega k relationship ok. So, 00:20:28.200 --> 00:20:37.080 the previous equation that you have seen this one this equation. So, this equation provides 00:20:37.080 --> 00:20:45.000 the eigenvalues which is exponential minus J phi of the unit cell matrix. And this is 00:20:45.000 --> 00:20:50.880 basically the progenerator or the source of the dispersion relation for the 1D periodic medium. 00:20:51.480 --> 00:20:59.880 So, how it works? So, we already know that phi that is the phase can be given as capital K that 00:20:59.880 --> 00:21:05.220 is the bloch wave number times the period. So, phi phase is basically proportional to 00:21:06.000 --> 00:21:15.420 k ok and T the transmission is also associated with frequency at different different wavelength 00:21:15.420 --> 00:21:22.560 of frequency will have different transmission. So, T can be written as T omega right. So, these two 00:21:22.560 --> 00:21:28.140 are related through the phase delay associated with the propagation through the unit cell. 00:21:28.980 --> 00:21:39.960 So, you can actually write that you know cos 2 pi K over g times phi is nothing but real of 1 over 00:21:40.800 --> 00:21:45.900 T which is a function of omega. So, this one directly correlates 00:21:46.500 --> 00:21:54.360 your K and omega and hence it can be named as dispersion relation. Now, the question arises 00:21:54.360 --> 00:22:01.740 what is g here? g is basically the fundamental spatial frequency of the periodic medium. So, 00:22:01.740 --> 00:22:07.260 what is the period? Period is capital lambda. So, g will be 2 pi over capital lambda 00:22:08.340 --> 00:22:14.940 right. So, this particular function that you see here on the left side cos 2 pi 00:22:14.940 --> 00:22:22.980 k over g is nothing but a periodic function of this k which has got a period of g ok. 00:22:23.760 --> 00:22:30.300 So, g is nothing, but 2 pi over capital lambda and this gives multiple solutions 00:22:31.260 --> 00:22:39.360 for you know this equation for any given omega ok. And that is how you are able to obtain 00:22:39.360 --> 00:22:45.600 that dispersion relation which is typically shown in as in the photonic band diagram. 00:22:46.500 --> 00:22:54.300 Now, but the solutions separated by the period g they are not independent. 00:22:54.840 --> 00:23:02.940 They basically lead to identical block modes. So, the domain of the dispersion relation is 00:23:02.940 --> 00:23:12.000 typically limited with the values of K ranging from interval of minus g by 2 to g by 2. 00:23:12.000 --> 00:23:19.260 That means, it is basically ranging from minus pi by capital lambda to plus pi by capital lambda 00:23:19.260 --> 00:23:24.360 which is nothing, but the Brillouin zone. So, that is where the concept of Brillouin zone comes on. 00:23:25.200 --> 00:23:34.080 And that allows your phase ok phi to be limited to an interval of minus pi to pi. So, once you 00:23:34.080 --> 00:23:40.860 know the phase variation from minus pi to pi you are basically covering the entire 2 pi right. 00:23:40.860 --> 00:23:46.440 So, after that it is just a repetition. So, there is no point computing those ok. So, 00:23:46.440 --> 00:23:54.660 this range this interval can give you the interval of phase starting from minus pi to pi. Also, 00:23:54.660 --> 00:23:59.640 we need to keep in mind that this cos function is an even function of 00:24:00.240 --> 00:24:07.740 K. So, for each value of omega there are 2 possible values of k ok 2 independent 00:24:07.740 --> 00:24:15.420 values and they could be equal in magnitude, but opposite in sign within the same Brillouin zone 00:24:16.080 --> 00:24:20.700 ok or within the Brillouin zone. So, Brillouin zone is from minus 00:24:22.140 --> 00:24:29.520 pi to pi in terms of phase or you can say it is from minus g by 2 to g by 2 in terms of K. 00:24:30.240 --> 00:24:37.440 So, this actually gives us that they are independent bloch waves. So, one solution 00:24:37.440 --> 00:24:46.080 is for you know forward propagating wave another solution is for the backward waves. So, dispersion 00:24:46.080 --> 00:24:53.700 relation gives you the photonic band structure. So, dispersion relation will also tell you 00:24:53.700 --> 00:25:01.500 the multiple spectral bands which can be typically classified into 2 regions or 2 regimes. 00:25:02.100 --> 00:25:10.260 So, one is propagation regime. So, spectral band within which capital K that is the block 00:25:10.260 --> 00:25:19.560 wave number is real those are the propagating modes. So, in those cases the real part of 1 over 00:25:20.400 --> 00:25:29.040 T which is a function of omega ok that is less than 1 and these bands can be numbered as 1, 00:25:29.040 --> 00:25:38.220 2 and so on starting from the lowest, make sense. And there could be other cases where in 00:25:38.220 --> 00:25:47.700 some spectral bands this K is complex. That means, they correspond to evanescent waves ok. It means 00:25:47.700 --> 00:25:52.740 these waves will get rapidly attenuated and they cannot propagate within that periodic medium. 00:25:53.340 --> 00:26:02.100 So, in this case if you see they will give you ignore this particular sign it's only modulus of 00:26:02.700 --> 00:26:06.720 real 1 over T and that will come out to be greater than 1. 00:26:08.220 --> 00:26:14.340 And these bands behave as stop bands of that diffraction grating. 00:26:15.240 --> 00:26:22.440 So, they are also called as photonic band gap PVG or forbidden band gap since no existing 00:26:22.440 --> 00:26:30.360 propagation mode are possible in this particular case. So, now let us look into the calculation of 00:26:30.360 --> 00:26:38.820 photonic band structure by taking an example of periodic stack of partially reflective mirrors. 00:26:39.420 --> 00:26:47.280 So, here is a stack of periodic stack you should say or partially reflective mirror 00:26:48.060 --> 00:26:54.660 and the wave travelling along the axis of the periodic stack is in the direction of z. 00:26:55.260 --> 00:27:03.240 What is the period here? Capital lambda. Now, let us see how we actually characterize this. 00:27:03.240 --> 00:27:10.380 So, the dispersion relation for a wave travelling along the axis of the periodic stack of identical 00:27:10.380 --> 00:27:16.620 that is very important identical partially reflective lossless mirrors which are separated by 00:27:16.620 --> 00:27:25.380 capital lambda. So, in this case the power reflectance is modulus of r square 00:27:26.100 --> 00:27:34.020 and intensity transmittance is nothing but what is not reflected is getting transmitted because these 00:27:34.020 --> 00:27:39.540 are non absorbing case. So, you can say modulus of t square that is transmittance is nothing, 00:27:39.540 --> 00:27:46.500 but 1 minus modulus of small r square. Now, let us use the matrix optics approach 00:27:47.040 --> 00:27:53.340 to derive explicit expression for elements of the scattering matrix of the composite system 00:27:53.940 --> 00:28:01.440 in terms of the elements of the scattering matrix of the constituent system that is we will take the 00:28:01.980 --> 00:28:11.040 elements of the unit cell and we will try to get the matrix elements for the overall system. 00:28:11.040 --> 00:28:20.520 So, the matrix M whose elements are say A B C D you can call them as wave transfer matrix 00:28:21.240 --> 00:28:27.600 which we have seen in this particular equation. So, they depend on the optical properties of the 00:28:27.600 --> 00:28:35.940 layered media between the 2 planes, right. So, we have already seen this particular case that you 00:28:35.940 --> 00:28:42.660 can obtain those equations or those elements from Fresnel reflection coefficients. An alternative 00:28:42.660 --> 00:28:52.080 to the wave transfer matrix that relates the 4 complex amplitude of the at the 2 edges of layered 00:28:52.080 --> 00:28:59.160 medium is a scattering matrix S matrix. So, you can also have S matrix and S matrix are more 00:28:59.160 --> 00:29:07.380 popularly used in describing transmission lines microwave circuits and scattering systems ok. 00:29:07.920 --> 00:29:13.980 So, in this case the outgoing waves are basically expressed in terms of 00:29:15.060 --> 00:29:24.300 incoming waves something like this. So, S matrix is used to describe transmission lines microwave 00:29:24.300 --> 00:29:30.180 circuits and scattering systems. So, in this case the outgoing waves are basically expressed 00:29:30.180 --> 00:29:37.140 in terms of the incoming waves. So, here is a schematic representation of S matrix. So, 00:29:37.140 --> 00:29:46.380 you see that you have the incoming wave U1 plus and you have one outgoing wave that is U2 plus. 00:29:47.400 --> 00:29:56.400 Now in this case this is a reflection, but you are actually trying to represent it in terms of 00:29:56.940 --> 00:30:00.060 outgoing wave because the reflection is also outgoing. 00:30:00.840 --> 00:30:10.740 So, you put it on the right side. So, you call it as U1 minus and the reflection from the other side 00:30:11.820 --> 00:30:17.400 becomes kind of incoming wave. So, you can actually take that U2 minus 00:30:18.600 --> 00:30:30.540 as a incoming one. So, in that case you can simply see that what are the 2 outgoing waves from this 00:30:30.540 --> 00:30:38.880 particular system that is U2 plus and U1 minus. So, U2 plus and U1 minus are the outgoing and what 00:30:38.880 --> 00:30:49.800 are the incoming U1 plus and U2 minus. So, U1 plus and U2 minus and you are trying to correlate this 00:30:50.340 --> 00:30:58.560 outgoing set of waves with the incoming set of waves. So, what are the coefficients? So, 00:30:58.560 --> 00:31:08.820 U2 plus as you know U2 plus will be nothing, but U1 plus times the transmission 00:31:10.200 --> 00:31:19.680 that is t 12. So, t 12 times U1 plus also it will have another component coming from this one. 00:31:20.220 --> 00:31:28.740 So, whatever is this wave whatever is getting reflected that will also contribute to U2 plus. 00:31:29.280 --> 00:31:36.780 So, you can have this is 2 this is 1. So, you can this reflection coefficient will be called r 21. 00:31:37.620 --> 00:31:46.260 So, you will have r 21 times U2 minus. Is it clear? So, you will have U2 plus that is given 00:31:46.260 --> 00:31:55.500 as t 12 times U1 plus. So, t 12 times U1 plus this one. So, there is also one contribution 00:31:55.500 --> 00:32:02.760 coming from this one some part of it will get reflected and add up to this outgoing wave. 00:32:03.300 --> 00:32:14.400 That will be r 21 times U2 minus.The other one also you can easily make it. So, this one 00:32:15.780 --> 00:32:24.720 U 1 minus is nothing, but r 12 U 1 plus. So, whatever is incidenting some part is getting 00:32:24.720 --> 00:32:32.640 reflected. So, that reflection is this one r 1 2 u 1 plus and then whatever you are putting 00:32:32.640 --> 00:32:40.380 here some part is getting transmitted and that also comes back as u 1 minus. So, that is t 21 00:32:40.380 --> 00:32:52.740 times U2 minus. So, this equation U 1 minus is nothing, but r 12 U 1 plus plus t 21 U 2 minus. 00:32:53.460 --> 00:33:01.980 Clear? So, unlike the wave transfer matrix this elements here in scattering matrix they have 00:33:01.980 --> 00:33:09.660 direct physical significance. Something like you know if you take r 1 2 and r 2 1 they are 00:33:09.660 --> 00:33:16.320 basically the forward amplitude transmittance and reflection. That is they are basically the 00:33:16.320 --> 00:33:24.000 transmittance and reflection coefficient of the wave incident from the left side. On the 00:33:24.000 --> 00:33:32.040 other hand if you see t 21 and t1 t 21 and r 21 they are basically amplitude transmittance and 00:33:33.240 --> 00:33:39.480 reflectance in the backward direction that is for wave that is coming from the right side.So, 00:33:40.080 --> 00:33:47.040 it is easy to you know correlate physically what is happening in the case of S matrix. 00:33:47.040 --> 00:33:56.100 Now, for a homogeneous layer of width d. So, this interface we have seen that what happens with the 00:33:56.100 --> 00:34:03.600 incoming and outgoing incoming and outgoing or you can say what is falling getting reflected and 00:34:03.600 --> 00:34:13.080 so on. So, for a homogeneous layer of width d and refractive index n that is shown here the complex 00:34:13.080 --> 00:34:19.620 amplitudes of the collected waves at the planes indicated by the arrow. So, if you are looking 00:34:19.620 --> 00:34:28.500 about the complex amplitude at this particular planes. So, you can call this as U1 ok. So, 00:34:28.500 --> 00:34:38.280 this is U1 forward one will be U1 plus backward one will be U1 minus this will be U2 ok, the 00:34:38.280 --> 00:34:45.420 amplitudes here will be U2 the forward one will be U2 plus and the reverse one will be U2 minus. 00:34:45.420 --> 00:34:52.200 So, how they are related? They are propagating or they are travelling this particular distance. So, 00:34:52.200 --> 00:34:59.880 they will add up a phase. So, U 2 plus will be simply U1 plus times e to the power minus j phi. 00:34:59.880 --> 00:35:08.760 What is phi? It will be n k naught small k naught and d ok, n is a refractive index k naught is the 00:35:09.600 --> 00:35:16.140 free space wave factor or wave number and d is the thickness of that layer. 00:35:16.140 --> 00:35:21.780 So, that is similarly you can also correlate what is u 1 minus and u 2 minus. 00:35:23.040 --> 00:35:31.020 So, that is how you can obtain the wave transfer matrix as well as scattering matrix for this 00:35:31.020 --> 00:35:38.520 particular case. So, if you see that wave transfer matrix M will look like exponential minus j phi, 00:35:38.520 --> 00:35:45.720 0, 0, exponential plus j phi. Whereas, the scattering matrix because scattering 00:35:45.720 --> 00:35:51.180 matrix will try to represent all outgoing in terms of incoming not left and right ok. So, 00:35:51.180 --> 00:35:59.280 it will look like e to the power minus j phi, 0, 0, e to the power minus j phi ok. So, that's the 00:35:59.280 --> 00:36:05.940 only difference between the wave matrix where wave transfer matrix and scattering matrix. 00:36:06.480 --> 00:36:13.260 So, now let us consider a wave transmitted through a system which is described by 00:36:13.860 --> 00:36:22.980 S matrix which has got this elements t 12, t 21, r 12, r 21. So, it these are 00:36:22.980 --> 00:36:27.960 easy to handle because we already know this transmission and reflection coefficient from 00:36:27.960 --> 00:36:37.080 the Fresnel equation. So, let's assume that you know this system has got two such separate systems 00:36:37.080 --> 00:36:43.020 ok. And these are the S matrix for these two separate systems. 00:36:43.620 --> 00:36:52.320 So, by multiplying the two associated M matrix. So, you can convert this into M matrix this one 00:36:52.320 --> 00:36:57.240 into M matrix you can multiply the M matrix and then convert it back to the scattering 00:36:57.240 --> 00:37:03.780 matrices ok. And you will be able to obtain the overall transmittance and reflection. 00:37:05.400 --> 00:37:12.600 So, overall transmittance in this case will be t 13 which is given by t 12, 00:37:12.600 --> 00:37:21.180 t 23 ok over 1 plus r 21, r 23. So, this is how you will be analyzing the multilayer system ok. 00:37:21.180 --> 00:37:26.160 You can also find out what is the reflection coefficient for this overall system 00:37:27.060 --> 00:37:33.960 ok. So, one important thing is that the relationship between M and S matrix in this 00:37:33.960 --> 00:37:42.060 case. So, as I mentioned M matrix are having four elements a b c d and they are not directly the 00:37:42.060 --> 00:37:49.500 reflection and transmission coefficient whereas, the S matrix are directly the reflection and 00:37:49.500 --> 00:37:57.660 transmission coefficient.So, sometimes it is easy to deal with S matrices. Now in this particular 00:37:57.660 --> 00:38:03.240 case the transmission of a plane wave through a cascade of two separate system that we have seen 00:38:03.240 --> 00:38:12.060 which are separated by a distance of d ok. Now we have assumed that if the two cascaded systems 00:38:12.720 --> 00:38:18.720 are mediated by propagation through a homogeneous medium it means the medium in 00:38:18.720 --> 00:38:25.200 between is a homogeneous medium then the overall transmittance and reflectance will also have this 00:38:25.200 --> 00:38:35.640 extra factor adding up that is exponential minus j phi and phi is nothing but n k naught d ok. 00:38:35.640 --> 00:38:39.600 So, that way the equations will also get slightly modified 00:38:40.680 --> 00:38:48.240 ok. So, as I mentioned here the phase phi is nothing, but n k naught d and d is the propagation 00:38:48.240 --> 00:38:54.240 distance n is nothing but the refractive index of this particular medium inside. So, 00:38:54.240 --> 00:39:01.740 with that what we learnt is that we understood the overall reflectance and transmittance 00:39:01.740 --> 00:39:10.560 and for this periodic stack of identical partially reflective lossless mirrors. So, using the 00:39:10.560 --> 00:39:19.620 equations this and this you can obtain what is the dispersion relation or you can find out that 00:39:21.120 --> 00:39:28.800 cos of 2 pi k by g is nothing, but 1 over mod t cos omega over omega B. 00:39:28.800 --> 00:39:35.880 So, here a new term omega B has come. So, omega b. So, g you already know g is 2 pi by capital 00:39:35.880 --> 00:39:45.540 lambda that is the special frequency spatial space related. So, spatial frequency and that you have 00:39:45.540 --> 00:39:54.540 omega b which is c pi by capital lambda. So, this is particularly a plot of the dispersion relation 00:39:55.380 --> 00:40:03.000 for a set of periodic mirrors. So, here certain values have been assumed like 00:40:03.000 --> 00:40:10.920 modulus t square has been taken as 0. 5 and they have been considered to have a 00:40:10.920 --> 00:40:20.880 separation of capital lambda. Omega B is c pi by capital lambda, g is 2 pi by capital lambda those 00:40:20.880 --> 00:40:27.600 are all fine. So, only important thing is the value of t is already assumed here and you can see 00:40:27.600 --> 00:40:34.800 this red dotted straight lines they are basically the approximation of a homogeneous medium. So, 00:40:34.800 --> 00:40:42.540 if you assume the entire medium to be homogeneous in which omega by K equals c or you can 00:40:43.560 --> 00:40:53.040 carefully work this out and see that omega by K will be omega B times g by 2 that also comes out 00:40:53.040 --> 00:40:59.280 to be c. So, you will have this straight lines. So, these are basically the homogeneous medium 00:40:59.280 --> 00:41:04.800 approximation. So, what it tells you that you know this graph tells you that because 00:41:04.800 --> 00:41:12.180 of the periodicity how much the dispersion relation deviates from the homogeneous medium 00:41:12.780 --> 00:41:20.580 and in homogeneous medium you see there is no band cap also. So, all the bands are allowed all 00:41:20.580 --> 00:41:28.620 the bands are allowed means all the frequencies have some k vector. It means at all frequencies 00:41:28.620 --> 00:41:37.440 you have solution for waves which has got real propagation constants, ok. But here in this case 00:41:37.440 --> 00:41:42.900 you can see it starts with a band gap then there is some band which is allowed then again there is 00:41:42.900 --> 00:41:48.600 a band gap then again there is some band where the propagation is allowed then again there is a band 00:41:48.600 --> 00:41:57.600 gap and so on ok. Now, yeah this is what I have already discussed that here the photonic band gaps 00:41:58.740 --> 00:42:01.620 there is no real solution. So, you do not have anything 00:42:02.640 --> 00:42:06.240 and all the band gap frequencies are basically centered around 00:42:06.900 --> 00:42:17.160 omega omega or you say omega B, 2 omega B and so on 3 omega B. So, the frequencies they this 00:42:17.160 --> 00:42:24.720 particular frequencies they do not permit any propagating mode rather in that case if the wave 00:42:24.720 --> 00:42:30.000 is not allowed to propagate inside the periodic medium what will happen? In terms of reflectance 00:42:30.000 --> 00:42:35.040 you will see that they have unity reflectance and this is a particular system where you also 00:42:35.040 --> 00:42:42.060 see that you know the lowest photonic band gap is at omega equals 0. So, if you take a real 00:42:42.060 --> 00:42:51.060 example with some values like n 1 equals 1. 5 and n 2 equals 3.5 and keep the thickness of 00:42:51.060 --> 00:42:56.100 the 2 layer similar. So, this is one layer this is another layer and then you are repeating this 00:42:56.100 --> 00:43:03.540 unit cell. So, this is the period ok period of the unit cell. So, if you take this and 00:43:03.540 --> 00:43:11.220 you try to calculate the dispersion relation you will see that you know the photonic band 00:43:11.220 --> 00:43:19.320 gaps have center frequencies at omega B here also omega B and its multiples like omega B, 00:43:19.320 --> 00:43:27.600 2B, 3B and so on. And they occur at either the brilliant zone center that is K equals 0 or at 00:43:27.600 --> 00:43:34.620 the edges that is K equals plus minus g by 2. So, this is the range of the brilliant zone. So, 00:43:34.620 --> 00:43:40.800 K value starts from you know minus g by 2 to g by 2 as we discussed before. So, here you see 00:43:40.800 --> 00:43:47.580 that initially all frequencies are permitted at omega B you have a particular band gap again at 2 00:43:47.580 --> 00:43:54.420 omega b you have a band gap and so on ok. And this is how it deviates from the homogeneous medium 00:43:54.420 --> 00:44:01.740 approximation. So, these are the values associated with this particular band gap. Now in this setup 00:44:01.740 --> 00:44:09.360 of partially reflective mirrors the frequency region surrounding to omega equals 0 does not fall 00:44:09.360 --> 00:44:13.680 in the band gap it has got some solution. So, its good in this case 00:44:15.000 --> 00:44:23.820 there are some propagating modes possible here ok. Now dielectric materials with lower contrast 00:44:24.420 --> 00:44:31.260 they will have band gaps of smaller width. Now here you see the contrast is really good. 00:44:31.980 --> 00:44:38.520 So, n1 is 1.5, n2 is 3.5. Now if you take 2 material where the difference between 00:44:39.180 --> 00:44:48.480 n1 and n2 like you can say deltan is less this band gap will also become very narrow ok. 00:44:48.480 --> 00:44:53.580 So, if you want the larger band gap you choose 2 materials which have higher contrast between them 00:44:54.240 --> 00:45:01.620 ok. And this red straight lines we already mentioned that this is how light would have 00:45:01.620 --> 00:45:09.600 behaved if we have a homogeneous medium with refractive index of the mean of n1 and n2 fine. 00:45:09.600 --> 00:45:14.820 So, from this you also can derive the information about the phase and group velocities. 00:45:14.820 --> 00:45:23.940 So, the propagation constant capital K it correlates to the phase velocity as well. So, 00:45:23.940 --> 00:45:28.860 phase velocity will be omega over capital K. So, once you know the phase velocity 00:45:28.860 --> 00:45:34.980 you can also find out what is the effective refractive index that is small n effective 00:45:34.980 --> 00:45:43.380 ok that is c naught over the phase velocity. So, you will get c naught capital K over omega fine. 00:45:43.380 --> 00:45:51.120 So, this is we are taking only up to this one. So, here you can see that clearly see what is the 00:45:51.120 --> 00:46:05.100 photonic band gap ok. And this is the plot of effective refractive index that is c naught K over 00:46:06.300 --> 00:46:13.620 omega ok. So, that is basically the effective refractive index. You can also find out what 00:46:13.620 --> 00:46:21.300 is the group velocity that is v equals d omega by d K ok which corresponds to the pulse propagation 00:46:22.140 --> 00:46:27.000 in the medium. So, any pulse will have a you know frequency 00:46:28.260 --> 00:46:33.840 spread ok means it will not be monochromatic it will have certain frequencies. 00:46:34.380 --> 00:46:40.020 So, you should calculate the group velocity in that case. So, group velocity should be 00:46:40.020 --> 00:46:45.900 obtained by d omega by d K. So, accordingly you can also find out what is the effective 00:46:45.900 --> 00:46:52.800 index seen by that group or you can call it effective group index that is also defined as 00:46:52.800 --> 00:46:57.360 capital N effective and that can be given as c naught over 00:46:57.960 --> 00:47:08.880 this v. So, you get c naught d omega dK by d omega ok. Now, these velocities can be calculated at any 00:47:08.880 --> 00:47:16.920 point of the dispersion relation curve by deriving the slope d omega by dK and you can also take the 00:47:16.920 --> 00:47:22.980 ratio of omega by K. So, as shown here you can also you can calculate what is small n effective, 00:47:22.980 --> 00:47:29.700 what is your capital N effective, what is n bar n bar this is the mean value ok, 00:47:31.140 --> 00:47:34.740 mean refractive index. So, the figure here 00:47:35.340 --> 00:47:42.780 the first one shows the dispersion relation of a long rotating layer of dielectric medium and this 00:47:42.780 --> 00:47:53.220 two shows the effective index of one particular frequency and this is the group index ok. So, 00:47:53.220 --> 00:47:58.740 here what you see you are able to clearly see two frequency bands where the propagation is possible 00:47:58.740 --> 00:48:07.260 and there is a definite photonic band gap ok. And for lower frequencies within the first photonic 00:48:07.260 --> 00:48:15.960 band you can see that this is how the effective index is. So, you can say that n effective is 00:48:15.960 --> 00:48:22.200 very close to the average effective index. So, initially this dotted line and this blue 00:48:22.200 --> 00:48:28.740 line they are very much overlapping ok. So, here also you see they are very much overlapping, 00:48:28.740 --> 00:48:40.020 but as you keep on you know going further with omega or you can say with wavelength 00:48:40.020 --> 00:48:49.380 it is expected that at longer wavelength the material becomes homogeneous. So, K is reducing 00:48:50.760 --> 00:48:57.840 means wavelength will be increasing. So, this is the case where you see more 00:48:58.560 --> 00:49:05.400 homogenized picture of your periodic medium, but as K is increasing your wavelength is basically 00:49:05.400 --> 00:49:14.280 reducing. So, you will be able to see the definite structures and that is where your you will be 00:49:14.280 --> 00:49:22.680 deviating from the line this particular red dotted line which represents a homogenized medium ok. 00:49:22.680 --> 00:49:28.860 So, here also you can see with frequency increase. So, this way the frequency is increasing 00:49:28.860 --> 00:49:35.040 ok this way the frequency is increasing. So, it is better to correlate with frequency and wavelength 00:49:35.040 --> 00:49:44.400 and this is the spatial period ok. So, you can correlate with the frequency here that at lower 00:49:44.400 --> 00:49:53.160 frequency wavelengths are high. So, you are seeing much homogeneous picture whereas, when you go for 00:49:53.160 --> 00:49:59.220 higher frequency you have lower wavelength you start deviating from the mean refractive index ok 00:49:59.220 --> 00:50:08.520 that is the crux of this particular thing. And at the second at the bottom of the second band that 00:50:08.520 --> 00:50:18.060 is here you will see that you know the n effective is much smaller than the mean refractive index. 00:50:18.060 --> 00:50:27.960 So, that is how it works that you know with the frequency increase initially n effective goes 00:50:28.560 --> 00:50:36.960 way above the mean value, but then suddenly it encounters a band gap and after the band gap 00:50:36.960 --> 00:50:43.920 at the bottom of the second band you will see that the n effective starts from a value which 00:50:43.920 --> 00:50:51.480 is much slower or much smaller than the mean refractive index. So, n effective increases at 00:50:51.480 --> 00:51:01.020 higher frequencies and with approaching to n bar which is the mean value at the middle of the band, 00:51:02.100 --> 00:51:11.220 understood. Now this drop of n effective from a value above average which is just below the band 00:51:11.220 --> 00:51:19.080 gap to a value which is below average just above the band gap is due to the significantly different 00:51:19.080 --> 00:51:25.440 spatial distribution of the corresponding block modes. So, there is a band gap because of which 00:51:25.440 --> 00:51:31.320 the bloch modes which are propagating here and here are significantly different. So, we do 00:51:31.320 --> 00:51:36.540 not expect them to have you know similar kind of feature and that is why there is a drastic change 00:51:36.540 --> 00:51:43.320 in this effective refractive index as well. And this bloch modes are orthogonal to each other. 00:51:44.280 --> 00:51:51.000 So, there is no similarity basically between this bloch modes. Now, the bloch modes at the 00:51:51.000 --> 00:51:58.320 top of the lower band has greater energy in the dielectric layers with higher refractive index. 00:51:58.920 --> 00:52:05.880 So, that the effective index is basically greater than the average. And if you look into the block 00:52:05.880 --> 00:52:13.200 modes at the bottom of the upper band it will be reverse. It means in that case greater energy is 00:52:13.200 --> 00:52:19.140 localized in the layers which are having lower refractive index and that is why the overall 00:52:19.140 --> 00:52:24.240 effective index is lower than the average. So, these are like two different mode 00:52:24.240 --> 00:52:32.340 configuration for the two different bloch modes which is present here and here fine. 00:52:32.940 --> 00:52:39.420 Lastly let us also look into the frequency dependence of the capital N effective which 00:52:39.420 --> 00:52:45.180 is the group effective index. And you see the group effective index increases at the 00:52:45.180 --> 00:52:56.940 edges of the band gap either from below or above ok. In both cases it is behaving the same way and 00:52:58.140 --> 00:53:03.960 that means the group velocity is much smaller. So, when index is 00:53:03.960 --> 00:53:10.140 larger the group velocity is smaller. It means when you are approaching a band 00:53:10.140 --> 00:53:19.440 gap you will see that the waves are much slower. So, the optical pulses are significantly slow near 00:53:19.440 --> 00:53:26.460 band gaps edge. So, that way you can actually make different devices based on this particular 00:53:26.460 --> 00:53:33.060 concept. So, with that we will stop here today. Thank you. Any questions you can drop an email 00:53:33.060 --> 00:53:38.787 to this particular email address and we will see you in the next lecture. Bye. Thank you.