 communication ray theory 2. So, in the previous lecture we have studied about the total internal reflection and critical angle here. So, here these are the learning outcomes. At end of this session students will be able to describe the acceptance angle, numerical aperture and the skew rays here. So, before starting to this session just recall what is the critical angle and total internal reflection here because those two concepts are required for this session. So, now before starting to the numerical aperture, so we will see the different rays now how the rays will travel inside the core of an fiber optical cable here. So, as shown in this figure there are three different types of rays can be transmitted through the fiber optical fiber here. So, first one is the skew rays, meridional rays and axial rays here. So, the difference we will see now. So, this is the end view here. So, how it will look like in the end view. So, the skew rays will never passes through the center of the core here as we have seen in the previous lectures. So, this is the core and this is the cladding and above this one mechanical jacket will be there for the to protect from the damage and the moisture of the environment here. So, the skew rays will never passes through this center of the core here as shown in this figure. So, it will be touching to the boundaries of the core and cladding surface here and it will be look like in the this manner when we see at the end view here. So, these are the skew rays. Meridional rays the ray passes through the center of the core and it is reflected back from the parallel surface of the core and passes through the center here. So, it will look like this. So, it will look like a straight line when we see in the end view here and from the side view it will be look like this here. So, this is the core and this is the dotted line is the center here. So, it passes through the center of the core here. So, one more is there excel. So, excel is the dot as shown in this end view because the excel ray travels straight through the center of the core it never touches to the parallel surface of the core and the cladding here. So, these are the three different rays are there. So, which are going to be transmitted inside the optical fiber here skew rays, meridional rays and the excel rays here. So, now we will see the acceptance angle here that is the theta A. So, the acceptance angle is nothing but that is the theta A is the maximum angle over which the light rays enters the fiber guiding along its core here. So, as shown in this figure this is the core and this is the cladding here. So, now this is the acceptance cone as shown in this figure here. So, this is canonical half angle here that is the theta A here. So, the light ray will incident on this core here and it will get passed through this fiber optical here. So, what is the rule is there? So, we should incident this angle greater than the critical angle so that this ray will remain inside the core only. So, that is why this core inside this core will be following if the angle is incident less than that critical angle like the ray B here. So, it will get surpassed through the core and it will get into the cladding where the data will be lost here. So, that is why it is written here eventually lost by the radiation here. So, that is the most important thing at which angle you are going to incident the ray here. So, that is the theta A is the maximum angle over which the light rays entering the fiber guided along its core here. So, now we will see the what is numerical aperture here. So, the numerical aperture is nothing but it is the acceptance angle is normally measured in the terms of NA that is numerical aperture. Numerical aperture is measure of the light gathering capability of an fiber here. So, when you are going to incident the light that is the acceptance angles so the maximum light gathering capability will be there here. So, at which angle you are going to incident here. So, that is why it is shown in this figure here this is the inside the core and this is the cladding here. Now, we will see the derivation how to calculate an numerical aperture that is an NA here. But before that we will straight away see that how it is calculated that is the optical system NA that is a numerical aperture is being calculated in terms of N0 into sin of theta A. So, theta A is nothing but the acceptance angle here. Where N0 is the index of refraction of the surrounding medium surrounding medium here. So, we have seen in the previous slide that is theta A is nothing but angle of maximum cone of a light that can enter through the optical system so that the light remains inside the core only here. So, this is regarding about the numerical aperture how it is calculated it is calculated by NA is equals to N0 sin of theta A here. So, now we will see how it is calculated here. So, if you see applying the snail law so in the previous video lecture what we have seen the snail's law that is N0 sin of theta I is equals to N1 sin theta R. So, this is the normal snail's law which has to be written here. So, from figure C using trigonometry we can write this theta R can be written as a sin of 90 minus a phi that is theta C here that is in critical angle here. So, I shown in this figure so this is the incident angle here at which the acceptance cone is there that is the theta max here. So, it will get travel here and if you draw the normal here if you draw the normal here. So, if you take this as an phi means theta C so it will be automatically by trigonometry rule it will be 90 minus a theta C here. So, automatically it will be also in corresponding angles will be an theta C here to follow the total internal reflection just recall that diagram which was used in the previous lecture here that is phi that is the theta C into theta C here. So, using that relation we are going to understand that. So, theta R is equals to sin 90 minus theta C where cos is equals to cos theta C here. So, 90 minus theta C can be written as an cos theta C here. Now, putting the value of in theta R into the snails law so we are going to put this cos theta C into this equation that is the snails law that will be come to be an n 0 upon n 1 into sin theta I which is equals to theta cos C this is the equation number 4 here. Now, squaring on both the side that is n square 0 upon n 1 square into sin theta okay sin square theta I is which is equals to cos square theta C. So, which can be written as an 1 minus sin square theta C here. So, this is by using the mathematical expression that is cos square theta C can be written as 1 minus sin square theta C which is known as an can be calculated as an sin square theta C that is critical angle how it is calculated theta C is equals to sin inverse of n 2 upon n 1 here. So, the same thing we have to replace here now that is 1 minus n square n 2 square upon n 1 square here okay this will be the equation 5. So, thus we know that n 0 sin theta I is equals to n 1 square minus n 2 square here half that is the equation number 6 okay. Now, from the equation 1 and 6. So, we can write that is n is equals to n 1 square minus n 2 square of half where n 1 and n 2 are known nothing, but they are the refractive indexes of the core and the cladding respectively. So, the equation 1 is nothing, but so this is the equation 1 that is the n 0 sin theta A here okay. So, an equation 6 is this n 0 sin theta I is equals to n 1 square minus n 2 square half okay. So, this are n A can be written as an n 1 square minus n 2 square here okay. So, this is the n A also depends on the refractive index of the material which is used that is core and cladding here okay. So, the n A may be also may be given in terms of relative refractive index here that is the difference delta between the core and cladding which is defined as okay. So, delta is equals to n 1 square minus n 2 square upon 2 n 1 square here okay. So, that can be written as an where n 1 upon n 1 minus n 2 upon n 1 for delta less than 1. If the delta that is the relative refractive index of n 1 and n 2 is less than 1. So, you can write directly this equation here okay. Now, the companion equation this 7 and 9 we can have n A is equals to n 1 bracket 2 delta raise to half here because so n A can be calculated. So, while solving some examples in some examples they will give the values of an refractive index n 1 and n 2 or they will give the relative refractive index here. So, students must know how to calculate and which formula is going to be used for that particular example to solving here. So, when they give the relative refractive index n and that is delta or n 1. So, n A is nothing but n A is nothing but it is the measure of light gathering capability of an fiber here okay. So, by this we can understand that to calculate the numerical aperture. So, we should understand what is the critical angle and the total internal reflection here okay. So, these are my references. Thank you.