Lecture -- The Grating Equation

[@homoduran]

tahnks again. also at 2:48, for longitudinal wave vector from dispersion relation, where is the equation from? Sorry I didn't take the previous semester course, and couldn't find this online either. thank you

[@empossible1577]

To learn about the dispersion relation, checkout Lecture 6c in the Electromagnetic Field Theory website: empossible.net/academics/emp3302/ It is essentially just a rule for how the wave vector relates to the frequency and refractive index for a wave in some medium. Hope this helps!

[@okeokomo9613]

Thank you for your lectures, in minute 4:29, How do we get to the fact that the magnitude of the Kx vector is K0navg? The derivation of this is shown in previous lessons? What is this K0?

[@empossible1577]

k0 is the free space wave number. The magnitude of any wave vector inside of a medium with refractive index navg is |k| = k0*navg A wave vector at angle theta in the xz plane is written as k = k0*navg*[ sin(theta)*ax + cos(theta)*az] where ax and ay are unit vectors in the x and y directions respectively. The x component of this is kx = k0*navg*sin(theta)

[@okeokomo9613]

@@empossible1577 Thank you!

[@homoduran]

Thanks for your lecture. I am a bit confused at 7:53. what does n-ref mean (or n_average in the previous slide). let's say the light incidents from n_air, grating material has n-grating. does n_average (n ref) here stands for just n-air？ or the effective refractive index of the gratings (average from the grating material and air)? Many thanks

[@empossible1577]

n_ref is the refractive index in the reflection region. Yes, n_ref = n_inc. You can sort of think of n_ref and n_trn as the refractive index wherever you may be observing the diffraction orders. If you are observing the diffraction orders in the middle of the grating, then the refractive index where you see n_ref and n_trn would be n_avg and would the average refractive index. Hope this helps!

[@homoduran]

@@empossible1577 thank you very much, Professor. One more question, do we have the grating equation for a more generalized case. For example, we have all kx_inc, ky_inc and kz_inc components. (In the slides ky is not considered as it’s on the plane of xoz) many thanks

[@empossible1577]

@@homoduran First, let me point you to the official course website. You can download the notes, get links to the latest version of the videos, and get access to other learning resources. empossible.net/academics/emp6303/ The grating equation sort of is the most general form. For crossed gratings, you have diffraction along both of the grating vectors that describe the symmetry of the grating. You will write the grating equation twice and this will give you an array of diffraction orders kx(m,n) and ky(m,n). From there, you calculate kz(m,n) from the dispersion relation.

[@okeokomo]

Hello, Thank you for all the content in your channel and the available pdfs on your website. I have a few questions: 1. Why do we have to set those boundary conditions for phase matching in slide 4, and what do we mean by "be continuos"? 2. In slide 6, why we only consider the tangential components? 3. On slide 21, the condition of having a theta angle real, is that the value of the magnitude is less than one. But for the m(max) it looks that we made the value of the magnitude equal to one. Is this correct?

[@empossible1577]

Thank you!! 1. The phase matching condition is what ensures the field is continuous across the interface. When there is a grating, the phase matching condition is modified to account for all of the sum and difference terms that arise. 2. It is only the tangential component that determines what the field looks like at the interface so it is only this term that has to be matched across the interface. Remember the field varies according to kx*x+ky*y+kz*z, but at the interface z=0 leaving only the tangential terms to describe the field on both sides. 3. Correct. This is to approximate what maximum value of m makes that argument equal to 1. Smaller values of m will keep the value less than one. m must be an integer so you may have to round down after this calculation.

[@okeokomo]

@@empossible1577 Thank you. Following 1: When we say across the interface we are talking about being continuos through the grated material once they crossed the interface and splitting into specific modes? and the sum and difference terms(k) we are talking about are the same we got from this lesson: kzbin.info/www/bejne/nWWwfamkbqmSga8 ?

[@empossible1577]

@@okeokomo It is not correct to say anything about "through" the grated material when it comes to phase matching. Phase matching only applies at the interface over a distance of zero, not through the material. Yes, the sum and difference terms are the same as from the video you pointed to.

[@okeokomo]

@@empossible1577 I see, thank you again

[@okeokomo9613]

@@empossible1577 I have a few confusions with question 2 here: 1.I understand that we do not use kz as z = 0, but why we do not consider ky if we are considering kx? I believe both are at the interface, is it for simplicity? 2. I also have some doubts about using z=0, in the image we see that the transmitted light has components on z and x, and later we obtain the grating equation that applies to all thetha(m). What is confusing for me is that we obtain the grating equation only using kx, and the grating equation itself describes properties of light with x and z components. I hope I was able to explain myself haha.

[@rabiulislamsikder344]

Dear Prof. I have one question: In Bragg grating it is said that it couples power between counter propagating waves. What is mean by that? In the slide we see only the reflected wave of the incident beam.

[@empossible1577]

Take a look at Slide 5 in Lecture 4j here: empossible.net/academics/emp6303/ Here, I divide gratings into short period, diffraction, and long period gratings. There are also subwavelength gratings that have periods every shorter than short period gratings. This all comes down to the grating vector K being equal to the difference between the phase constants of the waves the grating couples. K = 2*pi/L = | k1 - k2 | Short period gratings, or Bragg gratings, couple energy between waves propagating in parallel but opposite directions. Since the waves are propagating in opposite directions, the difference in the wave vectors k1 and k2 is maximum, therefore the period L of the grating is minimum. Next are diffraction gratings where the difference between k1 and k2 is smaller due to the angles. Therefore, the period is a bit longer. Next are long period gratings. Since the waves are propagating in the same direction, the difference between their wave vectors is very small. This leads to periods that are very large. Does this make sense?

[@rabiulislamsikder344]

@@empossible1577 Thanks

[@mishuk2008]

Dear Professor, based on the lecture I had a question in mind. Let's say I have the following structure. A medium with index n_1, a grating region with index n_avg, and a medium with index n_2. Light is coming from medium 1, hits the interface between n_1 and n_avg. The grating equation gives me the angular distribution of the transmitted waves within the material n_avg. If I want to find the angular distribution of the spectrum in the material n_2, should I modify the directions I got from the grating equation with Snell's law that would be at play when light is crossing the interface between n_avg and n_2. Thank you in advance!

[@empossible1577]

Just replace n_avg with n_2. That is it! Think of n_avg in the grating equation as the refractive index where the diffraction orders are being observed. Sometimes I even write the grating equation as n_obs to convey this point. Hope this helps!

[@ngavu4997]

What happens that the wave transmits through grating region if we have a uniform dielectric after that? Do they still remain “chopped wave” or become back like applied wave?

[@empossible1577]

Yes, the wave remains periodic with the same period as the grating. The diffraction orders will refract and alter their direction. This is already accounted for in the grating equation presented in this lecture. This is in a refractive index of the medium where the diffraction orders are being observed.

[@imadboukhallot8476]

Dear Prof. I have one question my project is fabrication of transmission diffraction grating How do I choose the depth of diffraction grating