WEBVTT Kind: captions Language: en 00:00:02.580 --> 00:00:31.200 Hello students. Welcome to lecture 29 00:00:31.200 --> 00:00:38.220 of the online course on Nanophotronics, Plasmonics and Metamaterials. Today's 00:00:38.220 --> 00:00:43.800 lecture will be on guided mode resonance. So, here is the lecture outline. We will give a 00:00:43.800 --> 00:00:49.620 quick introduction to guided mode resonance. We will go through the definition, the basic 00:00:49.620 --> 00:00:54.420 concepts and theory, some of the polarization properties of the guided mode resonance. 00:00:54.420 --> 00:01:00.300 And then we will see the filter design based on guided mode resonance. The filter spectral 00:01:00.300 --> 00:01:07.620 response and resonance regime of GMR and then we will provide a summary of this particular topic. 00:01:08.220 --> 00:01:12.960 So, the first question comes to mind is what is guided mode resonance? So, 00:01:12.960 --> 00:01:19.500 we have seen resonance phenomena in photonics which actually allow for strong localization of 00:01:19.500 --> 00:01:26.820 electromagnetic waves. And that has got numerous applications something like narrowband filtering, 00:01:26.820 --> 00:01:31.560 chemical and biological sensing, lasing, harmonic generation, 00:01:31.560 --> 00:01:38.760 Raman scattering, photovoltaics etcetera. Now, the important parameters that describe 00:01:38.760 --> 00:01:45.600 a resonance feature are its intensity and the spectral line width. That will actually decide 00:01:45.600 --> 00:01:51.240 the Q factor, the quality factor of the resonance. In most practical applications, 00:01:52.380 --> 00:01:58.200 resonances with strong intensity and narrow line width are desirable. As you can understand 00:01:58.200 --> 00:02:04.380 high Q resonances are always desirable in most practical applications. There are applications 00:02:04.380 --> 00:02:08.700 where you actually look for broadband absorption or broadband resonance. 00:02:08.700 --> 00:02:14.760 Those cases will come in the subsequent lectures. Today we are looking for very very narrow line 00:02:14.760 --> 00:02:22.380 width filters ok. So, high resonance intensity what are the benefits that can give you better 00:02:22.380 --> 00:02:29.340 signal to noise ratio. And when you have narrow line width that actually gives you very very 00:02:29.340 --> 00:02:37.080 strong field confinement. So, you are actually focusing your beam at a very very narrow spot. 00:02:37.080 --> 00:02:45.780 So, this is what is the thing you can relate to narrow line width. So, it can give you 00:02:45.780 --> 00:02:52.020 larger field confinement. However, most of the resonances are constrained by a kind of 00:02:52.020 --> 00:02:58.260 tradeoff between these two important parameters resonance intensity and line width. And this 00:02:58.260 --> 00:03:05.820 limits the possibility of independent tailoring of the resonant features at will. So, you are not 00:03:05.820 --> 00:03:12.060 able to kind of tune intensity and line width independently based on your requirement. 00:03:12.060 --> 00:03:18.660 So, that that lecture is not there. Now, when you think of guided mode resonance this is where the 00:03:18.660 --> 00:03:24.840 guided mode resonance becomes very important. So, it can provide a tailorable resonance intensity 00:03:24.840 --> 00:03:32.160 with narrow line width through geometrical design and selection of material. So, this kind 00:03:32.160 --> 00:03:38.580 of filters the filters design based on guided mode resonance can give you very high quality 00:03:38.580 --> 00:03:45.780 factor. So, due to this versatile nature of GMRs they are found in wide range of applications. 00:03:45.780 --> 00:03:52.860 One application we can see here is that high Q filter. So, this is how typically 00:03:52.860 --> 00:03:59.340 a GMR will look like. So, it is a guided mode resonance there is a grating and then there is 00:03:59.340 --> 00:04:05.700 a waveguide below this. So, as you can see there is light incident some part is getting reflected 00:04:05.700 --> 00:04:12.660 and the remaining is getting transmitted. So, the refractive index here if this is air. 00:04:12.660 --> 00:04:18.360 So, you can take refractive index to be 1 this is silica. So, it is 1.4 we are 00:04:18.360 --> 00:04:25.620 considering the periodicity the lattice period to be 6.91 that is a grating period. You can 00:04:25.620 --> 00:04:30.240 think of high low high low and these are made of germanium and selenium. 00:04:30.240 --> 00:04:35.040 So, that is n equal 4 and 2.64. And when you actually 00:04:35.820 --> 00:04:41.820 look into the grating thickness this is particular thickness of the grating that is 3.8 micrometer. 00:04:41.820 --> 00:04:48.960 And the fill factor of the high material is 0. 42. So, 42 percent of the grating period is 00:04:48.960 --> 00:04:55.320 basically germanium the remaining is Se ok. So, with that when you see the 00:04:55.320 --> 00:05:01.140 diffraction efficiency we will see that over the wavelength if this is the wavelength thing. So, 00:05:01.140 --> 00:05:05.460 this actually gives the spectrum. So, the blue one tells you the transmission peak 00:05:05.460 --> 00:05:11.160 it is a very narrow peak as you can see ok. And it is giving also a reflection dip. 00:05:11.160 --> 00:05:18.900 So, that particular wavelength only one particular wavelength is able to escape and remaining all are 00:05:18.900 --> 00:05:28.800 reflected. You can also look into the electric field profile at the peak wavelength of T0 and 00:05:28.800 --> 00:05:35.040 that corresponds to the T polarization state. And what is this dashed line that is basically 00:05:35.040 --> 00:05:41.520 the effective homogeneous layer. So, here you see you do not actually see any kind of resonance. So, 00:05:41.520 --> 00:05:45.480 the resonance comes from the grating and the grating does something which 00:05:45.480 --> 00:05:51.900 because of which you are able to only allow one particular wavelength to pass through it. 00:05:51.900 --> 00:05:56.700 Remaining all are getting reflected. So, there is something very very interesting about this 00:05:56.700 --> 00:06:03.480 particular phenomena. So, as you see the GMR devices primarily consist of a diffractive grating 00:06:03.480 --> 00:06:09.360 and then you have a in plane waveguide. The grating will diffract the in fact incident light 00:06:10.200 --> 00:06:16.920 and some some diffracted wave will basically coupled to the waveguide and it propagates as 00:06:16.920 --> 00:06:25.380 the guided mode. And this guided mode is designed to be leaky and it leaks out when it interferes 00:06:25.380 --> 00:06:32.520 with the free space propagating electromagnetic wave and that gives that resonance GMR ok. 00:06:32.520 --> 00:06:38.040 So, through the selection of material grating design dielectric layer thickness angle of 00:06:38.040 --> 00:06:44.520 incidence the GMR can provide a wide range of spectral feature. So, you can actually tune the 00:06:44.520 --> 00:06:51.660 transmission or resonance peak or deep as you can see here depending on all these parameters 00:06:51.660 --> 00:07:00.240 which are very easily tunable. So, that makes this GMR filters very very interesting. So, there are 00:07:00.240 --> 00:07:09.780 other names of this particular device it is also called resonant waveguide gratings RWG or we have 00:07:09.780 --> 00:07:16.680 already seen this name GMR grating or you can also call them waveguide mode resonant gratings 00:07:17.280 --> 00:07:22.680 ok. So, this RWGs as you can see they are basically dielectric structures where the 00:07:22.680 --> 00:07:28.980 resonant diffractive element they actually benefit from the leaky guided modes ok. 00:07:28.980 --> 00:07:35.940 And they can be tuned from UV to microwave and other frequencies also ok for different different 00:07:35.940 --> 00:07:43.080 configuration. Mainly they are used in this particular range and they are they can also be 00:07:43.080 --> 00:07:51.360 tuned to optical and infrared ok. So, a resonant waveguide grating can be defined can be defined 00:07:51.360 --> 00:08:00.840 as a thin wave guiding film in optical contact ok or it is merged with a grating as you can see here 00:08:01.380 --> 00:08:07.800 ok. So, this particular one actually shows a grating and then there is this 00:08:07.800 --> 00:08:14.640 waveguide below it ok. So, this particular figure shows a 4 port configuration. 00:08:14.640 --> 00:08:21.120 So, 1, 2, 3, 4 there are 4 ports and the white arrows are basically the inputs ok 00:08:21.120 --> 00:08:27.360 and the blue ones are basically the outputs ok. Light is considered to be incident from the free 00:08:27.360 --> 00:08:35.340 space which is coupled into a waveguide mode. And then that can out couple resonantly in a specular 00:08:35.340 --> 00:08:41.340 reflection or transmission and that is how it works. And here the substrate and super straight 00:08:41.340 --> 00:08:47.580 they are not shown, but they actually act as a cladding. So, when you think of this wave guiding 00:08:47.580 --> 00:08:53.760 film ok it operates usually by having a higher refractive index than the surrounding media. 00:08:53.760 --> 00:08:58.920 Because that is how wave guiding will take place. Wave guiding typically happens based 00:08:58.920 --> 00:09:04.800 on your total internal reflection for which the core, this will be the core in that case, 00:09:04.800 --> 00:09:11.040 waveguide core has to have a higher refractive index than the cladding. The thin film supports 00:09:11.040 --> 00:09:15.840 a discrete number of guided modes because of its thin dimension. So, the finite number of 00:09:15.840 --> 00:09:21.720 modes are allowed and the waveguide modes can be limited to the fundamental mode that is a zeroth 00:09:21.720 --> 00:09:30.480 order mode in case this is very very thin. Or it can actually go up to few modes if they are bit 00:09:30.480 --> 00:09:36.180 slightly thicker and for TE and TM polarization you can have different mode indices. 00:09:38.040 --> 00:09:45.240 Now depending on the wavelength this leads to a very high reflection or transmission giving rise 00:09:45.240 --> 00:09:53.040 to zeroth order reflection as you can see here ok. So, this is how the grating is made on top of 00:09:54.900 --> 00:10:00.660 thin waveguide and you see this the incident plane wave. Some part is getting reflected some 00:10:00.660 --> 00:10:06.540 is some is getting deflected and when this deflected mode couples with the leaky mode 00:10:06.540 --> 00:10:14.700 ok it actually. So, this is how it travels ok and it leaks out as well. So, this is how you 00:10:14.700 --> 00:10:19.680 actually get the transmission ok. So, in this particular example it has 00:10:19.680 --> 00:10:24.540 been designed to have a reflection peak and a transmission dip. So, 00:10:24.540 --> 00:10:32.040 you can actually design the filters or the guided mode resonance filters like that. 00:10:32.940 --> 00:10:40.140 So, those efficient resonances that you have seen here can be as narrow as 0.1 nanometer 00:10:40.140 --> 00:10:45.960 line width and they are very very sensitive to the incident angle and the wavelength. 00:10:46.500 --> 00:10:53.010 And you can see a typical angular to spectral line width ratio is like 0. 00:10:53.010 --> 00:10:58.260 1 degree nanometer inverse. So, you can actually see they are very very sharp. So, 00:10:58.260 --> 00:11:04.200 they can give rise to very high Q resonance and they are also tunable. Now, depending on 00:11:05.640 --> 00:11:10.680 the length and the phase delay that is accumulated during the propagation in the 00:11:10.680 --> 00:11:17.580 waveguide the destructive interference can occur either in reflection or in transmission. Now, this 00:11:17.580 --> 00:11:23.400 is why in the previous case we saw a transmission peak in this case we saw a transmission dip. So, 00:11:23.400 --> 00:11:28.920 you can actually so the modes that are coming the waves that are coming out if the destructive 00:11:28.920 --> 00:11:33.900 interfere they will cancel that particular transmission at that particular wavelength. 00:11:33.900 --> 00:11:38.520 If they constructively interfere they will get a peak. So, you can actually design this 00:11:38.520 --> 00:11:45.480 particular phase delay and that is accumulated along the propagation of the waveguide. So, 00:11:45.480 --> 00:11:52.800 the length of this device also plays a important role. Now, for a given polarization and wavelength 00:11:52.800 --> 00:11:58.320 an RWG can support various guided modes as you can understand having different mode index and 00:11:58.320 --> 00:12:04.320 therefore, transverse polarization different transverse propagation speed and momentum. So, 00:12:04.320 --> 00:12:09.240 light can be coupled to the waveguide modes by different grating orders. 00:12:09.240 --> 00:12:15.240 So, diffraction grating if you remember not only the zeroth order there will be plus 1 plus plus 1 00:12:15.240 --> 00:12:21.420 minus 1 plus 2 minus 2 plus 3 minus 3 and so on. So, all the diffraction orders are possible. So, 00:12:21.420 --> 00:12:29.760 particular which angle that actually meet that that that is more than the critical angle at 00:12:29.760 --> 00:12:34.800 this interface that will be totally internally reflected and that will continue to propagate in 00:12:34.800 --> 00:12:40.140 this waveguide right. And some part of it will leak out and that that is how the leaky waves 00:12:40.140 --> 00:12:47.280 are also coming. So, in this particular case you can see that some of this guided wave is 00:12:47.280 --> 00:12:55.680 diffracted out of the waveguide while propagating coupling back to the radiation and interferes 00:12:55.680 --> 00:13:02.160 with the non coupled reflected lights. So, here the reflected ones are basically 00:13:02.160 --> 00:13:10.020 ok the reflected ones this should be blue the reflected ones should be ok there is there is a 00:13:10.020 --> 00:13:17.400 mistake here. So, the reflected ones are basically magenta and transmitted ones are blue ok. So, yeah 00:13:18.600 --> 00:13:23.460 this this should be corrected. So, this is how it it works ok. So, 00:13:23.460 --> 00:13:28.440 here you can actually see this light yellowish kind of wave propagation inside. 00:13:28.440 --> 00:13:34.980 So, when a complete destructive interference happens in the transmission that happens at the 00:13:34.980 --> 00:13:41.400 specific angle and the specific wavelength you can actually get a narrow band reflection. So, 00:13:41.400 --> 00:13:47.160 that is how you can actually get this particular feature as you have seen here. 00:13:47.700 --> 00:13:57.420 So, it is clear that this RWGs or GMRs are very good at filtering ok and for especially for 00:13:57.420 --> 00:14:05.100 collimated light they are very good filters and they have extremely efficient diffraction elements 00:14:05.100 --> 00:14:13.860 they can be designed to be ok. And the structures usually consist of all dielectric materials. So, 00:14:13.860 --> 00:14:20.280 they are highly transparent and they can be used either in transmission or reflection mode. 00:14:20.280 --> 00:14:24.300 There is nothing absorbing here there is no metallic component. So, 00:14:24.300 --> 00:14:31.440 these structures do not suffer from thermal heating and they can be used for very high 00:14:31.440 --> 00:14:36.780 optical power applications such as mirrors or other diffractive elements. So, these are safe 00:14:36.780 --> 00:14:44.100 to use they will not heat up. Now, depending on the geometry there are different types of 00:14:45.360 --> 00:14:52.500 RWGs ok. So, each ridge and groove corrugating the waveguide layer that 00:14:52.500 --> 00:14:56.880 you see this one this is how you are corrugating the waveguide layer ok. 00:14:56.880 --> 00:15:04.020 You can actually design different kind of RWGs. So, this one as you see this is a single sided 00:15:04.020 --> 00:15:10.680 rectangular corrugation of the wave guiding layer, but this one is a double sided corrugation ok. 00:15:11.580 --> 00:15:17.820 This one is a wave guiding layer corrugated to its full thickness. So, it is basically you have just 00:15:19.860 --> 00:15:26.280 etched the entire thing ok and this is actually giving you discrete ribbon kind of structure. 00:15:26.280 --> 00:15:32.220 And this is the same thing, but you are having this ribbons on top of a wave guiding layer. 00:15:32.220 --> 00:15:38.160 This is a single sided sinusoidal corrugation of the wave guiding layer whereas, this one 00:15:38.160 --> 00:15:44.100 is a double sided sinusoidal corrugation. So, these are different kind of corrugations that 00:15:44.100 --> 00:15:52.980 you can make ok. And they actually decide the pattern of your quasi guided or leaky modes and 00:15:52.980 --> 00:16:00.180 also tell what kind of mode can propagate. So, RWGs can therefore, be considered as temporal 00:16:00.180 --> 00:16:06.420 or spatial optical integrators as well as they can be used to enhance the local electromagnetic field 00:16:07.500 --> 00:16:14.040 as example for sensing or non-linear optics applications. Now, let us have 00:16:14.040 --> 00:16:22.140 a look at the basic concepts of GMR ok. So, this is a grating structure and this is 00:16:22.140 --> 00:16:28.560 the waveguide that you have seen. Now, in most elementary structure in this case is a planar 00:16:29.100 --> 00:16:35.880 unslanted grating ok in a asymmetric waveguide geometry. Why we are calling asymmetric? Because 00:16:35.880 --> 00:16:43.560 epsilon 1 and epsilon 3 are not same here that we take them as different. And the relative 00:16:43.560 --> 00:16:49.980 permittivity or the dielectric constant of this region 2 can be specially modulated like this. 00:16:49.980 --> 00:16:56.820 So, this is x direction this is z. So, epsilon x can be varied like this epsilon g is basically 00:16:56.820 --> 00:17:06.660 the average hm relative permittivity. And delta epsilon is the modulation amplitude and it is 00:17:06.660 --> 00:17:13.980 having cosine variation, where k is basically 2 pi by lambda where lambda capital lambda is the 00:17:13.980 --> 00:17:21.480 grating period. Now, we understand that the waveguide grating ok this is the waveguide 00:17:21.480 --> 00:17:27.300 grating. So, it should have a permittivity which is larger than both epsilon 1 and epsilon 3 ok. 00:17:27.300 --> 00:17:35.520 So, this is where the waveguide grating is ok. So, this is also called as guided mode resonance 00:17:35.520 --> 00:17:41.880 filter or GMRF. Now, for TE polarization you can consider the electric field vectors to be 00:17:41.880 --> 00:17:48.660 normal to the plane of incidence in this figure ok. The coupled mode equations which govern the 00:17:48.660 --> 00:17:54.780 wave propagation in this waveguide grating can be written in this form. I will not go into 00:17:54.780 --> 00:18:00.480 the details of this, but I will just highlight the few important factors that is this S cap is 00:18:00.480 --> 00:18:06.120 basically the amplitude of the inhomogeneous plane wave of the ith space harmonic. So, 00:18:06.120 --> 00:18:12.600 you have got this i index, k is basically the free space wavelength and theta is the internal 00:18:12.600 --> 00:18:19.080 angle of reflection and that is this one ok. So, you start with the coupled mode equations 00:18:19.080 --> 00:18:27.000 and when you put delta phi equals 0 in this equation ok you actually make this 00:18:27.540 --> 00:18:33.900 to be a unmodulated dielectric waveguide it simply becomes a normal waveguide. So, 00:18:33.900 --> 00:18:41.040 in that case the equation also simplifies and it looks like a normal wave equation of this 00:18:41.040 --> 00:18:49.260 form where beta is the propagation constant. Now, a guided wave can be excited if the average or if 00:18:49.260 --> 00:18:57.900 the effective waveguide index of refraction N that is given by beta by k is in this range. So, 00:18:57.900 --> 00:19:06.000 it has to be the modulus of n should be lower than square root of epsilon g, but it has to be greater 00:19:06.000 --> 00:19:11.880 than or equal to the maximum of the refractive index of this or this whichever is maximum. 00:19:11.880 --> 00:19:16.920 So, this is how you can ensure that the mode will be propagating guided ok. 00:19:16.920 --> 00:19:22.980 So, this is very this is a very common kind of requirement that comes from the 00:19:22.980 --> 00:19:30.540 waveguides that you can study in any other course ok. But one important thing is that when you put 00:19:30.540 --> 00:19:37.740 this delta epsilon to be 0 in that equation and when you compare that with the equation 2 that we 00:19:37.740 --> 00:19:45.000 have seen here ok you can find out that beta can be written as this ok. So, that is basically the 00:19:45.000 --> 00:19:51.780 effective propagation constant in the waveguide grating and this also corresponds to a effective 00:19:52.740 --> 00:19:59.220 refractive index, but there it corresponds to the ith mode ok. So, n i can be written as beta i over 00:19:59.220 --> 00:20:06.600 k. Now, the propagation constant beta i of the waveguiding in the limit that delta epsilon is 00:20:06.600 --> 00:20:14.520 0 or tending to 0 ok is thus given in the terms of the basic parameters something like basic wave 00:20:14.520 --> 00:20:21.420 guiding grating parameter that is capital lambda epsilon g theta lambda i and all these things. 00:20:21.420 --> 00:20:27.540 So, this is how you can actually see these 2 equations and you can put the same arguments 00:20:27.540 --> 00:20:36.840 here also for the TM case and you will see that this equations are valid this conditions are valid 00:20:36.840 --> 00:20:46.920 TE and TM polarization ok. So, here let us refer to the eigen mode equations of this unmodulated 00:20:46.920 --> 00:20:53.400 slab waveguide ok. So, this will become slab waveguide when delta epsilon tends to 0 as you can 00:20:53.400 --> 00:21:02.700 understand the corresponding eigen value equation of the modulated waveguide can be written as this. 00:21:03.300 --> 00:21:12.420 So, you can actually find out that tan delta tan kappa i d will be given by this one. So, 00:21:12.420 --> 00:21:18.660 kappa i is related to the propagation constant in the different regions ok. 00:21:19.740 --> 00:21:25.860 So, this these are again all coming from the waveguide theory. So, we will just have a quick 00:21:25.860 --> 00:21:31.320 look we will not go into very much details of this just I am just showing you the formula here that 00:21:31.320 --> 00:21:38.100 will give you some idea that how this the GMR effect has come. So, for TM polarization the 00:21:38.100 --> 00:21:43.200 eigen value equations looks like this. So, the previous one was for TE this one is for TM. 00:21:43.200 --> 00:21:51.120 So, in the limit of delta epsilon tending to 0 ok. So, the equation that you have seen for TE or 00:21:51.120 --> 00:22:02.760 TM and the range that is equation 3 this one they are all holding good. So, they all govern the mode 00:22:02.760 --> 00:22:11.640 coupling. So, it all tell you that which all modes are possible ok this that equation 3 and this also 00:22:11.640 --> 00:22:17.880 governs the resonant behavior of this waveguide grating filter with the modified propagation 00:22:17.880 --> 00:22:26.340 constant beta i ok. So, here in this case this will contain the grating parameters explicitly. 00:22:26.340 --> 00:22:31.380 What is d? You remember d is basically the thickness of this waveguide grating. 00:22:31.380 --> 00:22:38.820 So, these are the conditions that you have seen ok and for any parameter varied. In fact, 00:22:38.820 --> 00:22:46.800 the resonance free parametric range in the limit of epsilon tending to 0 can be found from 00:22:46.800 --> 00:22:52.020 these two equations. So, these are the eigen mode equations that tell you the modes which 00:22:52.020 --> 00:23:00.300 are allowed to propagate in this particular grating and this you can find because all 00:23:00.300 --> 00:23:08.700 these coefficients gamma i delta i and kappa i they depend on epsilon g theta lambda this is 00:23:08.700 --> 00:23:14.040 the wavelength of light this is the grating period and this is the index ok. So, the resonance free 00:23:14.040 --> 00:23:20.040 region especially represents a separation between the two modes make sense. So, every 00:23:20.040 --> 00:23:27.360 resonant mode is there and there is also a region between those where different modes can propagate 00:23:28.080 --> 00:23:36.660 in a equivalent unmodulated slab waveguide that corresponds to themwaveguide grating. 00:23:36.660 --> 00:23:42.360 So, whenever there is grating there are some specific modes. So, in between there will be some 00:23:42.360 --> 00:23:48.480 resonance free region. So, we will also look into those how to identify those particular region. 00:23:48.480 --> 00:23:54.420 Now, if the i th diffracted wave corresponds to the guided mode then the resonance free 00:23:54.420 --> 00:24:01.980 range in the thickness in thickness for both TE and TM polarization can be given as delta 00:24:01.980 --> 00:24:09.120 d which is pi over kappa i and you can write this as expand this in this particular form. 00:24:10.200 --> 00:24:14.700 So, let us take one example. So, if you are looking at the TE 00:24:15.540 --> 00:24:21.780 eigenvalue equation. So, the TE eigenvalue equation was written as this 00:24:21.780 --> 00:24:29.040 ok and solve when you solve for lambda that gives a free spectral range that can be written 00:24:29.040 --> 00:24:37.440 as delta lambda FSR nu and that is basically the resonance at nu plus 1 minus the resonance 00:24:37.440 --> 00:24:43.560 wavelength at nu. So, nu is basically your 0 1 2 and so on these are the integers which are used 00:24:43.560 --> 00:24:52.320 for labeling the waveguide modes. Now, since the eigenvalue expressions for TE and TM modes 00:24:52.320 --> 00:24:59.520 are different and the resonance occur at different parametric locations ok. So, that that is that is 00:25:00.360 --> 00:25:05.580 easily understood that the eigenvalue equations for TE and TM modes are different. So, 00:25:05.580 --> 00:25:13.680 the resonance for TE and TM mode will occur at different parametric location So, these are the 00:25:13.680 --> 00:25:20.880 two equations for your quick reference. Now, you can consider this as parameters ok 00:25:20.880 --> 00:25:27.360 and that that will allow you to calculate. These are like some parameters that you 00:25:27.360 --> 00:25:34.080 can take and you can calculate what will be the positions of the TE and TM resonance modes. Now, 00:25:34.080 --> 00:25:42.600 to quantify the TE, TM polarization separation of the filters you can apply this equations to 00:25:42.600 --> 00:25:50.100 produce this particular graph. So, this is how you are plotting the TE0 and this TM0 ok. 00:25:50.100 --> 00:25:58.080 This is TE1, TM1 and this is TE2 and TM2. So, what are these? These are basically showing you the TE, 00:25:58.080 --> 00:26:06.240 TM polarization separation and this all these are normalized. So, this is basically the grating 00:26:06.240 --> 00:26:12.060 thickness normalized by its period this is the lambda normalized to the grating period. So, 00:26:12.060 --> 00:26:20.820 these are normalized one this also tells you about the separation between the modes ok and it also 00:26:20.820 --> 00:26:25.440 tells you the resonance free regions. So, these are the regions which are free of resonances. 00:26:26.460 --> 00:26:33.960 So, for a given normalized filter thickness ok d lambda the normalized 00:26:33.960 --> 00:26:38.880 wavelength separation between TE and TM mode you can find from here ok. 00:26:40.440 --> 00:26:49.200 So, for a particular value you can find out what is the lambda separation for the zeroth 00:26:49.200 --> 00:26:54.240 order TM mode and zeroth order TE mode you can use this graph and find it out. 00:26:54.840 --> 00:27:06.180 So, if you assume that d equals say 1000 nanometer ok and you can see that the fundamental the mode 00:27:06.180 --> 00:27:12.600 that is nu equals 0 and if you consider the separation between the TE0 and TM0 you will find 00:27:12.600 --> 00:27:20.280 that the resonances are separated by 24 nanometer. So, that you can find from this particular graph 00:27:21.180 --> 00:27:27.120 ok. Other other kind of separations like what is the separation between TE0 and 00:27:28.500 --> 00:27:34.980 like this particular when you take this value you can actually have TE1 mode as well as TE0 00:27:34.980 --> 00:27:41.760 mode you can actually find out what is the lambda corresponding to this two resonances. So, this 00:27:41.760 --> 00:27:48.120 particular figure is very important because it gives you all this resonance positions and also it 00:27:48.120 --> 00:27:55.260 tells you about the resonance free regions ok. Now as we have considered normal incidence in this 00:27:55.260 --> 00:28:02.340 example. So, here what are the parameters if you see the parameters this will be the parameters 00:28:02.340 --> 00:28:07.680 considered ok. So, we are considering normal incidence and other parameters are also given. 00:28:08.280 --> 00:28:15.180 So, the inequality actually boils down to this. So, this is the condition when you put epsilon 00:28:15.180 --> 00:28:21.960 1 and epsilon 2 and other parameters you will see this is the condition that will actually give the 00:28:21.960 --> 00:28:30.060 normalized wavelength range that you see. So, this particular range 1.47 to 1.73. So, 00:28:30.060 --> 00:28:36.780 this is the range that you have plotted here. So, this has basically come from this one ok. So, 00:28:36.780 --> 00:28:45.900 this actually tells you. So, you have actually put i equals plus minus 1 in this equation 00:28:45.900 --> 00:28:50.640 ok and you have obtained epsilon 1 epsilon 3 are the values that you have shown there. 00:28:50.640 --> 00:28:56.160 So, once you put that you will get this particular range. So, this is why this particular range 00:28:56.160 --> 00:29:02.220 is considered because in this configuration or the material choice this will be the normalized 00:29:03.120 --> 00:29:08.880 wavelength that you have to consider. Now the figure also additionally can be used to find 00:29:08.880 --> 00:29:16.020 out the resonance free region as I told you. So, you can actually see this regions ok in filter 00:29:16.020 --> 00:29:24.540 wavelength and thickness in both parameter which region is resonance free you can find out. So, 00:29:24.540 --> 00:29:30.900 you can actually take d by lambda equals constant like this and obtain those region where resonance 00:29:30.900 --> 00:29:37.800 is not there or you can consider the thickness and see whichever portion has no resonance. 00:29:38.940 --> 00:29:44.340 So, the analytical expression you can obtain for resonance free range. 00:29:45.000 --> 00:29:53.100 So, when you calculate. So, you will see the delta d that is the in thickness what 00:29:53.100 --> 00:29:58.860 is that thickness where there is no resonance you can obtain it like this. I am not going to 00:29:59.400 --> 00:30:04.980 the details of this equations there are complicated equations even you can obtain 00:30:04.980 --> 00:30:12.840 them from this particular paper, but this actually tells you how this separations are basically 00:30:12.840 --> 00:30:20.880 obtained. And when you combine this inequality of this equation and this one you will actually get 00:30:20.880 --> 00:30:27.840 this particular equation because the relationship between beta by k is basically your N. 00:30:27.840 --> 00:30:33.120 So, you can put this guy here and you will get this particular expression. So, 00:30:33.120 --> 00:30:42.180 here you can also see that if you want to make epsilon square root of epsilon 1 that is n 1 sin 00:30:42.180 --> 00:30:49.500 theta 1 equals the refractive index of this one that is square root of epsilon g sin theta ok. So, 00:30:49.500 --> 00:30:59.880 that way you will be able to make a range which are the allowed angles for your particular device. 00:31:00.540 --> 00:31:06.480 So, with that you can identify the resonance regime ok. So, this expression allows you 00:31:07.920 --> 00:31:12.660 to define the parametric regime where the guided mode resonance actually occurs. 00:31:13.260 --> 00:31:21.420 So, this is particularly a plot that shows those resonance regime ok. So, on the x axis you have 00:31:21.420 --> 00:31:31.080 the normalized wavelength and on the y axis you have angle theta okay. So, this actually gives you 00:31:31.080 --> 00:31:37.200 the selected values of the average permittivities with the deflection order i as a parameter. So, 00:31:37.200 --> 00:31:42.900 here you see this is for i equals plus 1, this is i equals minus 1, this is i equals plus 2, this 00:31:42.900 --> 00:31:49.980 is i equals minus 2 ok. And again the parameters that you have considered here is epsilon 1 is 1, 00:31:49.980 --> 00:31:57.000 epsilon g is 3 and epsilon 3 is 2. 161. So, the solid curves that you see here on the 00:31:57.000 --> 00:32:04.380 left side of the inequality ok. So, here the left side of the inequality is actually coming from 00:32:04.380 --> 00:32:13.620 this solid line ok and the right side is shown by this dashed line that is coming from this one. 00:32:14.160 --> 00:32:20.520 So, the right side inequality gives you this one for a particular i ok and the left side in 00:32:20.520 --> 00:32:26.580 equality gives you this solid line ok for that particular i. And the shaded region between 00:32:26.580 --> 00:32:32.940 the two lines are basically those parameter values for which resonance can takes place. So, 00:32:32.940 --> 00:32:41.760 this can be called as resonance regime ok and these are the boundaries for i equals1 right. So, 00:32:41.760 --> 00:32:50.040 the solid boundaries correspond to the deflected order i at the grazing angle ok at which the 00:32:50.040 --> 00:32:55.440 classic Rayleigh anomaly is associated. So, we will see what what is that Rayleigh 00:32:55.440 --> 00:33:05.220 anomaly in the next slides. So, here one important thing is to note that at the intersection of the 00:33:05.220 --> 00:33:13.320 two solid curves a double Rayleigh anomaly takes place ok something like this. Now when 00:33:13.320 --> 00:33:18.360 you say about Rayleigh anomaly this is nothing, but a anomalous or abnormal behaviour of light 00:33:18.960 --> 00:33:24.660 that is being reflected from a periodically corrugated surface or diffraction grating 00:33:25.200 --> 00:33:30.480 that was observed in the form of rapid variation of intensity of the diffraction 00:33:30.480 --> 00:33:39.780 orders as a function of the wavelength. So, it happens like this a sharp intensity variation 00:33:39.780 --> 00:33:46.560 and related exclusively to the emergence of diffracted beams when the incident beam 00:33:46.560 --> 00:33:54.960 is at glazing angle and you will also get beams which are parallel to the surface of the grating. 00:33:56.700 --> 00:34:01.680 So, Rayleigh was the first to represent this particular 00:34:01.680 --> 00:34:07.200 effect and he was able to explain this. So, that is why the anomaly is named after 00:34:07.200 --> 00:34:13.140 him. It was pointed out that the anomaly in reflection occurs at wavelengths for which 00:34:13.140 --> 00:34:19.380 one of the diffracted order becomes parallel to the main plane of the grating like this ok 00:34:19.380 --> 00:34:26.220 and then eventually it will vanish at greater wavelengths because after that you cannot actually 00:34:27.660 --> 00:34:32.400 like the reflected one cannot go inside. So, that has to vanish 00:34:32.400 --> 00:34:38.640 ok. So, this changes the power distribution among the remaining diffraction orders including 00:34:38.640 --> 00:34:47.700 the specular reflection or the zeroth order reflection right. So, if one mode is not allowed 00:34:47.700 --> 00:34:54.840 or one particular order is vanishing. So, the power has to be distributed among 00:34:54.840 --> 00:35:00.840 the remaining. So, that is how the anomaly disturbs the system. Another cause of the 00:35:01.500 --> 00:35:08.100 anomalous behavior takes into account the possible existence of leaky modes located at the surface. 00:35:08.700 --> 00:35:15.420 So, which are coupled to the incident wave such as in the case of surface plus one polar atoms. 00:35:15.420 --> 00:35:21.420 So, there also you can see in the case of grating it is mainly for those diffracted 00:35:21.420 --> 00:35:26.580 beam orders which are basically parallel to the grating. Now, in the case of the 00:35:26.580 --> 00:35:33.780 flat surface the wave guides of such surface waves are greater than the incident waves. 00:35:33.780 --> 00:35:40.680 So, that they do not interact, but when you look for periodic structures the wave factors 00:35:40.680 --> 00:35:48.060 of the grating come into the play. So, because of that the grating or because of that the coupling 00:35:48.060 --> 00:35:54.240 becomes possible and as a result at certain frequency the incident wave will generate the 00:35:54.240 --> 00:36:02.520 leaky surface waves and in that case the reflected power will decrease. So, this effect becomes 00:36:02.520 --> 00:36:10.440 resonant and because of that the reflection curve will get a narrow dip very narrow dip and that 00:36:10.440 --> 00:36:18.360 will also give you a transmission maximum. So, for the light diffraction at the periodic structure 00:36:18.360 --> 00:36:25.200 which has got a grating number of g okay. G can be written as 2 pi by the lattice period 00:36:25.200 --> 00:36:33.900 ok. You can say that Rayleigh anomaly will occur if k x plus n times G will be equal to the 00:36:33.900 --> 00:36:40.140 incident wave factor. So, if this matches so n is basically 1, 2, 3 this is happening because of the 00:36:40.140 --> 00:36:48.480 grating when there is a match ok. So, a Rayleigh anomaly will take place the wave will leak in ok 00:36:49.260 --> 00:36:55.980 and you will get a reflection dip. So, here k is the wave vector or wave number of the incident 00:36:55.980 --> 00:37:00.600 light and k x basically is the component of the component that is parallel to the 00:37:00.600 --> 00:37:09.780 grating plane of the wave vector right So, the 2 symmetric double Rayleigh anomalies can be seen 00:37:09.780 --> 00:37:18.060 here when the incident angle is 0 degree. You can see it for i plus minus 1 there is a 00:37:18.060 --> 00:37:25.080 degeneracy the other one is for i plus minus 2. So, these are basically symmetric double 00:37:25.080 --> 00:37:31.860 Rayleigh anomalies. There is also one asymmetric double Rayleigh anomaly that happens at theta 00:37:31.860 --> 00:37:40.020 equals 30 degree when i equals plus 2 and i equals minus 1 ok. Now, during the interval 00:37:41.040 --> 00:37:51.120 designated as this resonance regime the order i can actually correspond to the guided mode. So, 00:37:51.120 --> 00:37:56.100 you can say this is the first order, second order and so on and this is the equation that 00:37:56.100 --> 00:38:02.820 you have already seen. So, this theta is the incident angle and this limit tells you that 00:38:02.820 --> 00:38:09.420 which all modes including the order of the mode that can be allowed to propagate. 00:38:09.420 --> 00:38:17.520 Now if you look into the dashed curve ok with the increasing wavelength now beyond the dashed curve 00:38:17.520 --> 00:38:25.560 that is we are looking for this one the dashed curves ok. The order i is neither propagating 00:38:25.560 --> 00:38:32.520 nor it is possible to beyond this it is not able to propagate or it is not even guided. 00:38:33.180 --> 00:38:39.660 So, in order to actually strike a resonance with this particular regime the eigenvalue 00:38:39.660 --> 00:38:46.560 equations need to be satisfied. So, these are the equations that need to be satisfied to fall 00:38:46.560 --> 00:38:54.840 within this particular regime right. So, these are the supported modes which basically can propagate 00:38:54.840 --> 00:39:00.960 and leak out and give you that resonance. Now their solutions fall within the resonance 00:39:00.960 --> 00:39:09.180 regime with the location additionally dependent on the thickness of the filter that is the parameter 00:39:09.180 --> 00:39:21.240 d. So, here you can see at 0 degree the resonance of plus minus 1 is indistinguishable because they 00:39:21.240 --> 00:39:28.800 are overlapping here and at nonzero theta prime that is at nonzero degeneracy. So, you can put 00:39:28.800 --> 00:39:36.060 plus minus 1 and you can actually see that they actually do not have the degeneracy anymore. 00:39:37.200 --> 00:39:44.100 And the solid curve that you see here that is leveled as lambda r over capital lambda 00:39:44.100 --> 00:39:49.500 this indicates the last propagating higher order diffracted mode that is here 00:39:49.500 --> 00:39:57.180 in this case it is i equals plus 1 ok. And it gets cut off at grazing angle okay, 00:39:58.260 --> 00:40:04.260 as the wavelength is permitted to increase. So, as you keep on increasing it actually does 00:40:04.260 --> 00:40:11.160 not increase further it will actually get cut off here. So, this R is basically the Rayleigh 00:40:11.160 --> 00:40:17.340 wavelength. So, you can actually call this as the Rayleigh wavelength ok and when lambda is 00:40:17.340 --> 00:40:26.880 greater than lambda R ok you can say only 0th order wave can propagate ok. So, in this regime 00:40:27.480 --> 00:40:35.580 only the 0th order mode will be propagating not the higher order modes. So, the diagram such as 00:40:35.580 --> 00:40:41.880 this figure are also useful in visualizing the resonance properties of diffraction gratings. 00:40:42.660 --> 00:40:49.560 Now let us look at the effect of modulation amplitude ok. So, this particular figure 00:40:49.560 --> 00:40:57.660 calculates an example of the spectral behavior of the symmetric high symmetric filter or 00:40:57.660 --> 00:41:03.480 symmetric waveguide symmetric means epsilon 1 is considered to be same as epsilon 3 ok. And 00:41:04.740 --> 00:41:09.180 it has high spatial frequency. So, lambda is considered to be greater 00:41:09.180 --> 00:41:16.380 than the grating period ok. And what are the parameters epsilon 1 epsilon 3 to be taken 00:41:16.380 --> 00:41:23.160 as 2.5 epsilon g is 3 this normal incidence these are the parameters we have considered 00:41:23.160 --> 00:41:31.080 the center free space wavelength to be 1. 669 and the line width we got is 0.01 nanometer 00:41:31.080 --> 00:41:36.300 and this DE10 these are basically the diffraction efficiencies ok. 00:41:37.680 --> 00:41:47.160 So, 1 0 is reflected 3 0 is the transmitted one. So, you can only see that the zero forward 00:41:47.160 --> 00:41:54.540 and backward diffracted waves propagate and in this case all other modes are basically cut off 00:41:55.200 --> 00:42:01.080 right. And the diffraction efficiency represents the intensity of the various diffracted wave. So, 00:42:01.080 --> 00:42:07.080 here you can see that this is 3 0 the transmitted one is getting a dip and 00:42:07.080 --> 00:42:13.200 the reflection 10 is getting a peak ok. And there is a 100 percent energy exchange 00:42:13.200 --> 00:42:20.760 between these 2 modes and the smooth lines can be obtained. And what is the good thing about 00:42:20.760 --> 00:42:25.740 this kind of symmetric filter the symmetric waveguide grating filter they also produce 00:42:25.740 --> 00:42:30.840 very symmetric spectral response. So, you have 0 nulls on both side of the peak and that is a 00:42:30.840 --> 00:42:35.880 perfectly symmetrical filter which is usually not the case with any other resonant kind of 00:42:35.880 --> 00:42:45.900 structure. And if you see the transmission you get a very beautiful notch filter ok. And if you are 00:42:45.900 --> 00:42:51.960 allowing the higher order waves to propagate you will see that pure nulls may not be obtained. 00:42:51.960 --> 00:42:57.840 So, one of these legs will be kind of asymmetrical and it will be different. 00:42:58.560 --> 00:43:05.760 So, if you look into asymmetrical filter where you have epsilon 1 not equal to epsilon 3 and 00:43:05.760 --> 00:43:11.880 these are the parameters. So, in that case surely your spectral response is not symmetrical and you 00:43:11.880 --> 00:43:18.300 do not have null on both sides perfect null on both sides. Let us also look into the effect 00:43:18.300 --> 00:43:30.420 of amplitude modulation ok. So, here it shows the normalized filter line width as a function of the 00:43:30.420 --> 00:43:38.220 modulation amplitude of the waveguide grating. So, these are all kind of normalized. So, 00:43:38.220 --> 00:43:44.340 delta epsilon is the modulation index and delta lambda is basically the line width and if you 00:43:44.340 --> 00:43:52.440 take these are the parameters. So, this is in a symmetric parameter and you can see that if you 00:43:52.440 --> 00:43:59.220 consider grating period to be 1 micrometer the line width that you calculate falls in 00:43:59.220 --> 00:44:06.840 the range of 20 femtometer to 200 picometer. So, that is basically the range. So, you can 00:44:06.840 --> 00:44:13.500 understand how narrow these resonance filters are and they are very very high quality resonance. 00:44:13.500 --> 00:44:18.600 So, this is a graph that will tell you that for what kind of modulation you should have what kind 00:44:18.600 --> 00:44:26.460 of line width. You can also think about the mode confinement from this particular graph it allows 00:44:26.460 --> 00:44:31.920 you to see the spectral line width can also be controlled by the degree of confinement of the 00:44:31.920 --> 00:44:40.620 mode in an associated unmodulated waveguide. So, if you do not have modulation and you can think of 00:44:42.120 --> 00:44:47.880 an effective unmodulated waveguide. So, this particular figure shows the numerically calculated 00:44:47.880 --> 00:44:56.520 plot of normalized line width. So, here you are taking a symmetric case, but you are considering 00:44:56.520 --> 00:45:05.040 the difference in the refractive index to be your x axis and you are varying the line width. 00:45:05.040 --> 00:45:14.760 So, here you will see that you are varying the parameters from 1 to 2.95 right. This is 00:45:14.760 --> 00:45:23.820 the epsilon 1 that you are varying and this is how you actually get this variation okay. So, strongly 00:45:23.820 --> 00:45:32.640 confined states something like delta n will be tending to 0 exhibit the largest linewidth. You 00:45:32.640 --> 00:45:39.300 can also use this for electro optic switching in some applications something like this particular 00:45:39.300 --> 00:45:46.380 filter illustrates the resonance line under the variation of the average grating permittivity 00:45:46.380 --> 00:45:53.580 of an asymmetric filter or asymmetric waveguide grating structure. So, here you can see there are 00:45:53.580 --> 00:46:00.660 many other modes which are getting diffracted. So, here in this particular example the break grating 00:46:00.660 --> 00:46:07.860 is satisfied, but this is not a condition requirement for the resonance to occur. 00:46:07.860 --> 00:46:14.580 So, the break grating condition lambda over capital lambda equals 2 sin theta 00:46:14.580 --> 00:46:21.840 is satisfied here and these are the parameters that we considered and you see that 1 0, 1 1, 00:46:21.840 --> 00:46:29.220 3 0 3 1 these modes are also present ok and that is why you you are not seeing perfect 00:46:29.220 --> 00:46:38.160 null in this resonance. The point is that here the resonance can also occur under break condition and 00:46:38.160 --> 00:46:45.660 with non-zero diffracted wave also propagating as you can see here in this inset pure nulls are not 00:46:46.620 --> 00:46:52.560 obtained next to the resonance and a variation in the relative average relative permittivity 00:46:52.560 --> 00:47:00.840 can then introduce a resonance. So, in that case a electro optic effect can be used to switch energy 00:47:00.840 --> 00:47:08.160 from the transmitted to the refracted because you can switch change slightly the effective 00:47:09.240 --> 00:47:15.600 relative permittivity of this grating and that can introduce the resonance or it can move out 00:47:15.600 --> 00:47:22.320 the resonance. So, you can actually use a electro optic effect for getting this resonance or not 00:47:22.320 --> 00:47:29.760 ok. This particular figure shows the calculated TE filter reflectivity as a function of 00:47:29.760 --> 00:47:36.780 wavelength with different incident angle. So, you have got theta equals 0 plus 10 minus 00:47:36.780 --> 00:47:43.800 10 plus 20 minus 20 plus 30 minus 30 and so on. So, the peak efficiencies they are all dependent 00:47:43.800 --> 00:47:50.520 on the incident angle as well. So, here are the parameters which are used for calculating this 00:47:51.720 --> 00:47:58.320 and for this particular case you have also obtained what are the resonance regime as we 00:47:58.320 --> 00:48:05.040 have seen. So, here you can actually see that we have calculated for i equals plus 1 and minus 1 00:48:05.040 --> 00:48:13.440 ok and there is a large peak corresponds to i equals minus 1 with smaller peaks happening at 00:48:13.440 --> 00:48:23.280 i equals plus 2 ok because there is a case here corresponding to 30 ok. So, here you can see that 00:48:24.240 --> 00:48:36.840 at yeah this is the case for theta prime equals 30 degree which corresponds to your plus 2 ok. 00:48:36.840 --> 00:48:44.460 The large peak at theta prime equals 30 degree corresponds to i equals minus 1. So, 00:48:44.460 --> 00:48:51.720 this one with a smaller peak caused by the wave with i equals plus 2. So, i equals 00:48:51.720 --> 00:48:59.160 plus 2 gives this particular smaller width ok. So, this is basically seen from this one. So, 00:48:59.160 --> 00:49:05.100 the same results can also be demonstrated for that other polarization case TM polarization case. So, 00:49:05.100 --> 00:49:11.160 in this case you can see the resonances are even more narrower ok and with that we will 00:49:11.160 --> 00:49:18.060 conclude what we understood in the GMR. So, RWG or GMR basically they use the 00:49:18.060 --> 00:49:23.340 periodicity of a grating to couple light into a thin waveguide. They have been therefore, used 00:49:23.340 --> 00:49:29.220 very extensively as waveguide couplers for optical communication and signal processing for both in 00:49:29.220 --> 00:49:34.920 coupling and out coupling of thin waveguide modes with strong wavelength polarization and 00:49:34.920 --> 00:49:41.700 angular dependence. Their in coupled quasi guided mode can interfere dramatically with the incident 00:49:41.700 --> 00:49:48.000 illumination depending on the phase delay that is accumulated in the in coupling of the 00:49:48.000 --> 00:49:54.480 waveguide which can create anomalous reflection or transmission features creating unique zeroth 00:49:54.480 --> 00:50:00.660 order properties. So, you can actually get those very sharp reflection or transmission peak or dip. 00:50:00.660 --> 00:50:06.660 This mechanism makes them highly efficient narrow band or broadband reflectors as well 00:50:06.660 --> 00:50:13.500 as transmission filters with applications as laser mirror, advanced detection systems, spectrometers 00:50:14.220 --> 00:50:21.660 etc. So, in other direction you can think of cost efficient fabrication processes and unique 00:50:21.660 --> 00:50:26.520 appearance have enabled their applications in optical authentication and document security. 00:50:26.520 --> 00:50:32.280 Their polarization dependent behavior can help them to be used as polarizer, 00:50:32.280 --> 00:50:39.900 polarization rotator, wave plates etc. The control of the optical near field 00:50:40.440 --> 00:50:45.600 has found widespread application in biological refractive index sensing, fluorescence sensing, 00:50:45.600 --> 00:50:52.380 non-linear effects, optical switching etc as well as enhancement of the solar light harvesting. So, 00:50:52.380 --> 00:50:58.440 in the next lecture we will discuss some of these devices based on this effect. So, with 00:50:58.440 --> 00:51:03.900 that we conclude. Thank you for your attention. In the next lecture we will consider and discuss 00:51:03.900 --> 00:51:09.900 the applications of matter surfaces and this GMR based devices. If you have got any queries 00:51:09.900 --> 00:51:15.000 you can drop an email to this email address mentioning MOOC in the subject line. Thank you.