she's been teaching English at Madison High School, Armist Brooks has been able to get along with almost all the members of the faculty. Her relationship with her principal, however, might best be described as catchest catch can. That's about it, and Mr. Conklin catches me whenever he can. But our principal's eccentricities seemed mild indeed, compared to the recent project instigated by the head of the Board of Education. It seems the Board had hired a psychologist to determine the mental and emotional fitness of the teachers in our district. However, last Thursday morning, I had something else on my mind. Mr. Boynton said he had a surprise for me, and had invited me along on his customary early morning walk in the park. It was beautiful at that time, and as we strolled along with the sun barely visible above the treetops, I sensed an element of romance in the air. Suddenly he stopped, tenderly took my hand in his, and in a voice tense with emotion said, Mr. Brooks, look! A perfect specimen of the burrowing spadefoot toad! Mr. Boynton, is that all you wanted to say to me? Well, no indeed, Mr. Brooks. When first unearthed, the spadefoot or hermit toad, as he is sometimes known, looks like a small ball of brown earth. But then it expelled the air from its lungs, and the deflated sides collapsed like a punctured balloon. Don't go y'all. Mr. Boynton, I... This is truly a remarkable reptile. But you said you had a surprise for me, Mr. Boynton. I have, Mr. Brooks. I've written a play. What? It's for the kids at Madison. They needed one for their dramatic festival next month, so I decided to write it. What's the play about? Well, when I decided to write it, I asked myself one question. What was the one thing nearest and dearest to my heart? Yes, Mr. Boynton. There was only one answer. So I wrote a play about animals. The play entirely concerned with animal life is rather offbeat, Ms. Brooks. Rather, but with the student body at Madison, you shouldn't have any trouble casting it. I just hope I'll still be around to see it. What do you mean? Well, these psychological tests the Board of Education is giving teachers have produced some rather drastic results. Surely you're not worried because a couple of teachers at Jefferson High are being transferred. Some of the Clay City faculty members have been removed, too. And if I'm not mistaken, Madison is next on the list. Well, why worry about that now? I don't believe in crossing my bridges till I come to them. Why, we're on the subject of bridges. Have you ever thought of burning a few behind you? How was your walk with Mr. Boynton, dear? What was his surprise? Was it what you expected? Did he propose? Did he set the date? Not bad. Nothing much. No, no, and uh-huh. What have we ever had? Well, better luck next time. Did you ask him in for a bite? Oh, if he wouldn't bite in the park, he'd never... Breakfast? No, I didn't know if you'd have enough. Oh, I've made enough breakfast before. Walter Denton's in the dinette already. Well, luckily I'm not very hungry. What's Walter doing here so early? He said he wanted to talk to you. Go on in, Connie. I'll be along in a few minutes. All right. Green, me friend of Tarzan. For my part, Mr. Boynton's play. Did I really frighten you much? Oh, that's all right. Who needs five years' growth? Animal heads, too? Yes, ma'am. Oh, you'd swear Harriet Conklin was a living Tigris and stretch snodgrass as a donkey. I know. What's he playing in the show? Well, you'd better take that ape's head off before Mrs. Davis comes in. She's liable to pass out from the shot. Well, it's... Aren't pretty tight, Miss Brooks, but I'll try it. Ooh, too late. Here's your breakfast, Connie. I kept your eggs nice and warm. Thanks, Mrs. Davis. And how was your breakfast, Walter, dear? Fine, thanks. Well, don't you notice anything different about Walter today, Mrs. Davis? Come to think of it, I do. For a boy his age, he's got a pretty heavy beard. You do seem rather sunburned. Now, I've got an ape's head on me, Mrs. Davis. Well, don't worry. Your marks will improve. Ape's head mess, Mrs. Davis. It's for a play Mr. Boynton's written for the school. Well, isn't that interesting? But I'm afraid we're all out of bananas, dear. I'll go get you another glass of milk, though. Thanks, Mrs. Davis. What did you want to speak to me about, Walter? Well, the reason I called for you early was to get you to school in plenty of time to ask Mr. Conklin for a free period between two and three. That's the only time the entire cast can get together. I see. Well, that should help prepare me for my new job. What do you mean? I'll be getting some practical experience with a real, live lion. So, Mr. Conklin, we'll adopt the same procedure at Madison that I have used at other schools. That is, I shall test each teacher individually to judge his emotional and mental adequacy for his job. Very well, Mr. Gidney, but frankly, I'm not at all certain that I see the value in these tests. Mr. Stone, the head of the Board of Education personally engaged me for this project, Mr. Conklin. These tests are invaluable. However, I've been a bit concerned by rumors that you recommended several teachers at Clay City for transfer. They're more than rumors, Mr. Conklin. I am simply trying to fit the square pegs into the square holes and the round pegs into the round holes. In other words, get rid of the dead wood. I see. But in getting rid of these wooden pegs, the square hole, the dead square. Dead wood. Yes. Don't you think mistakes can be made? Mr. Conklin, I've made a life study of testing emotional and mental capabilities adequately and fairly. Oh, I have no doubt of that, Mr. Gidney. I don't like to brag, but I can often tell almost at one glance what a person is and what he does. Really? I'd like to see you try that sometime, Mr. Gidney. Well, I'll do it for you right now, Mr. Conklin. We'll use whoever that is at your door. Oh, Splendid. Splendid. Come in. Good morning, Mr. Conklin. Good morning. Good morning. This is Mr. Gidney, but I'm not going to tell him who you are. He wants to try a little experiment. Experiment? Mr. Gidney contends that he can tell who you are and what you do just by looking at you. Well, now isn't that nice? Go right ahead, sir. Hmm? You see, I would say that this woman is a happily married housewife. Why, you dear man. She is also the mother of a couple of young children. An amazing revelation. What am I doing so far? So far, you owe me a husband and two children. I'm an English teacher. An English teacher? Well, that certainly is surprising. It's a constant source of surprise to me. I've never been this far off in my calculations. No doubt it's due to the fact that I've been working too hard lately. Well, once I give the tests in this district, I can take a little vacation. The tests? What tests? Mr. Gidney is the psychologist. Mr. Stone engaged. The psychologist? Mr. Stone engaged? Mr. Conklin and I are simply trying to fit the round pegs and the round holes. Get rid of the dead wood, so to speak. And what better bit of... And what better person? I'm going to start the tests than Miss Brooks here. Oh, now let's not rush this thing. That is, I've got to rush along now, sir. What's your hurry, Miss Brooks? You haven't even told me why you came in here yet. I know, but right now I have something much more important to do. More important? Yes, sir. I have some information to bring to several round, square, and oblong pegs. Do you remember a tune called One Kisser? The Board of Education, in a mood to accept at face value the reports furnished by this psychologist, Mr. Gidney, I was more than a little concerned. By the time I joined Mr. Boynton for lunch in the school cafeteria, a number of the teachers had been tested, and for Mr. Boynton's behavior, I had a feeling he had been among them. I know you're upset about something, Mr. Boynton. Why don't you tell me about it? Upset? Who's upset? What makes you think I'm upset? You paid for my salad. Tell me what happened. Did you take the test? Well, yes. What were the results? Mr. Gidney thinks I'm in the wrong profession. The wrong profession? He says I should be doing something that brings me into closer contact with the public. He suggested some branch of show business. Oh, that's just dandy. Maybe you could do a vaudeville act with your frog. Boynton and MacDougall. Fong's dances and funny cloaks. Oh, and this is fantastic. It's pretty unbelievable to me, too, but he says on the basis of his test, he's going to suggest a transfer for me. Well, he can't get away with it. He simply doesn't know what he's talking about. Maybe not, but he's done it to several teachers at other schools. Gosh, isn't going to be easy to leave Madison after all these years? There must be something we can do. Well, I don't know what, Miss Brooks. His power is very extensive. But he seems more unstable than any of the people he's testing. Hello, Miss Brooks. Hi, Mr. Boynton. Oh, hello, Walter. What's the latest? After taking her psychological test, what was the best music teachers we've ever had? What did Mr. Gidney recommend for her? That she try her hand at welding. Welding? Mr. Gidney is definitely operating with a minimum of marbles. I'm another of the teachers he recommended for transfer, Walter. You, Mr. Boynton? Oh, but he can't do this. He's taking the cream of another wise, dupious crop. That's his present company. That's it, of course, Mr. Brown. Hello, everyone. Well, how do you like this tiger's head on me? Oh, it's extremely lifelike, Harriet. Yeah, it's for a minute. I thought you were Miss Enright. Now, you might as well forget about that tiger's head, Harriet. In fact, it looks like what I forget about the whole play. But why? I've taken Mr. Gidney's tests and I've been recommended for transfer. What? Now, that's an outrage. You couldn't get along without you, Mr. Boynton. You're not just chomping your whiskers, Tabby. Well, there's no telling where this can end. Golly, Miss Brooks' daddy said you're to be in his office at three for your test. Well, I'm not packing my bags yet. I just got an idea that may cancel all the effects of this psychological brainstorm right now. What kind of an idea? Well, first tell me, how many of those animal masks have you on hand, Mr. Boynton? Wolves, apes, tigress and lion's head. Well, all we're going to need is the pig, wolf and tigress. But, Miss Brooks, I don't get it. I'll explain. By his own admission, Mr. Gidney's been working very hard lately and he's been under a great strain. Now, if I were taking the test and three people in succession were to walk into Mr. Conklin's office looking like a pig, a wolf and a tigress, and I didn't acknowledge anything unusual about their appearance, you're a great light is beginning to dawn. You want Mr. Gidney to think he's got bats in his bell freeze. Crazy mixed-up bats. If we can convince Mr. Conklin that Mr. Gidney's having upper-story trouble, none of his reports will hold water. Miss Brooks, suppose Mr. Conklin discovers us and we're thrown out of school. Walder, if that happens, I know there's an opening for all three of you in the San Diego Zoo. A surgical tester already under a great strain, I had every intention of straining him right out of Madison High School. My plan was to convince Mr. Conklin that Mr. Gidney should be taking tests instead of giving them. At three that afternoon, the time set for my test, I was seated in our principal's office intent on separating a few more marbles for Mr. Gidney's already depleted stock. Now then, we'll begin by filling out this form. Your name, of course, is Constance Brooks. Now, your age? 25 and some months. How many months? About 50. Parents' names, please. Mommy and daddy. I mean, what did they call each other? Oh, Snooki. Maybe that isn't important. Were you the only child? The only child who what? The only child in your parents' home. Well, yes, until I was 15. And then there was another child. No, I grew up. The past few months have been extremely trying for me. Please try to cooperate. Oh, of course, Mr. Gidney. Thank you. Thank you. Now I'm going to start you a psychological test which is an excellent indication to me of one's emotional stability. No doubt you've heard of word associations. Yes, sir, but I'm not much of a joiner. Although if the dues aren't too high, I don't mind joining. Please. I'm referring to a psychopathological test or to try to put it simply a compulsory action indicator. You didn't try hard enough. Well, never mind. I'm ready if you are. Fine. Fine.  Ready? Black. All right. Black. Black. I thought you'd mention another color. We just met. Why should I start an argument? There's no argument at all. It's a simple test. If I said, boy, you'd say, girl, now do you understand? Oh, I think so. All right. All right. Chair. Thanks. I've got one. Here's the word. You're supposed to say another word. I'm sorry. I'll try again, daughter. Okay. Try it again. I am trying it again. Listen, daughter. I am listening, Pa. What word does it suggest to you? Sun. At last. Now you've grasped the fundamental. Oh, hey. Hello. Good. Bad. That isn't the word. William Byrne. What is his connection with Table? At the last party we were on, he wound up under one. Maybe these past few weeks have been too much for me. However, I'll try one more approach. This is the cycle quiz used by many of my contemporaries. Alice Brooks, have you had your choice? Which of all of the animals in the world would you rather be? Animals. Let's see. Oh, that's easy. I'd rather be a turtle. Turtle? Why? Because I like to dance. I'd rather be a turtle because you like to dance. Well, I don't want you to think I'm eccentric, Mr. Gidney, so I'd like to make a qualifying statement with that answer. What is it? I like to dance underwater. So much for the cycle quiz. Oh, come in, Mr. Gidney. That's something... Well, why... Oh, oh, I see now. He's wearing a pig's head. That's no way to talk about the editor of our school paper. Well, then meet Mr. Gidney. Glad to know you, sir.  Only when he eats. So you're feeling all right, Mr. Gidney? I thought so too now. You're positive that you don't see a pig's head on this fellow. Are you positive you do see one? Well, it could be a transitory illusion, I suppose. Oh, I certainly hope so, but if it is... Come in. Hello, Miss Brooks. Well, Mr. Boynton, how nice to see you. Now, these interruptions have got... Do you remember Mr. Boynton? Mr. Boynton, but he looks like a wolf. Boy, have you got the wrong number. Perfectly normal to me.  Walter, do you see a wolf's head on Mr. Boynton?  How nutty can you get? I don't know, but Mr. Gidney is nudging a fruitcake. I'm afraid the strain you've been under is affecting your nervous system, sir. But I've never had any symptoms like this before. Fall in. Come in. Hi, folks. Conklin. Is nothing the matter again, Mr. Gidney?  What do you see this time? This time, you go first. Oh, Harry, it looks perfectly fine to me. I know. Why don't you ask her what animal she'd like to be? From my point of view, that's fairly apparent. She already has the head of a Tigris. Hasn't she? A Tigris? That's a hard one. It's no laughing matter. I must be cracking up. I shouldn't have taken on this assignment. What I need is a good long rest. But what about your reports to the board of education? In my condition, I'd better notify the board to overlook them, or why it would be... Good afternoon, everyone. Mr. Conklin. What's going on here?  Then you see them, too, Mr. Conklin. Oh, certainly, I see them. You're both nervous, Rex. We never really thought you were crazy, Mr. Gidney. Crazy?  It's quite simple, Mr. Conklin. These precious associates of yours almost succeeded in making me believe that I was out of my mind. What? They wanted me to think that I was incapable of giving these tests, but now that I realize that I'm not having a nervous breakdown, I'll send in my reports immediately. And you will be the first one transferred, Mr. Conklin. I'll be transferred? But why? What were the results of my tests? You will make an excellent brush salesman. Put me down for three whiffs, Froome. Boy, this is preposterous. This is...this is... Answer the phone, Ms. Broke. The phone, sir? The phone? Yes. Don't you hear it ringing? I don't hear a thing. Well, then I'll input myself. Hello? Who? What number do you want? Oh, no, I'm sorry. You must have the wrong number. You...you heard the phone? Oh, he hears it constantly. He also hears doorbells. Oh, there goes that phone again. Well, this time I'm just going to let it ring. What?  We are on the same boat. I'm getting out of here. I need at least a year's vacation. But, Mr. Gidney, what about your report? Channel F was broken up with the paper they're written on. Well, that's bad. Thanks for your quick thinking, Mr. Conklin. You've saved all our jobs. Never mind the soft soap, Ms. Broke. For your part in perpetrating this thoroughly nefarious scheme without first consulting me, you shall be punished and punished more. Oh, pardon me, sir. I think I hear the doorbell. I'd better hurry and answer it. Doorbell, I... Doorbell, I... Oh, it's over at my house. Oh, it's... Sometimes I think I'd rather be a spade-foot toad and sink right out of sight, leaving such headaches as Mr. Gidney far behind. Mr. Conklin was played by Gail Gordon. Others in tonight's cast were Jane Morgan, Dick Crenna, Bob Rockwell, Gloria McMillan, and Bob Sweeney.


Okay, this is where we stopped in the last module, we are talking about second order nonlinear phenomena and we have arrived at an expression for the second order polarization and we see that it consists of two terms, one is a constant polarization which is called DC effect and the second one is the term that is responsible for second harmonic generation and the reason why we say that this is the term that is responsible for second harmonic generation is that it is associated with an angular frequency 2 omega 1 which is double the frequency of what we had started with, okay. Now let us go ahead and see how to actually get the second harmonic generation but before that we have unfinished business from the last module, we have to talk a little bit about this k vector, now to do that I want to show you the expression of something that I think we have discussed earlier phase velocity, phase velocity V photon is given by omega by k which is equal to C by N, what is N, number of photons of frequency omega, no, what is N, we hint N also goes by the symbol mu, refractive index, refractive index nothing else, why do I write refractive index and then I write omega n bracket because refractive index is color dependent, frequency dependent, remember this, this is something that we are going to build upon in the subsequent discussion, okay. So this is the expression for phase velocity and it is going to come handy in a minute. Now the reason why we are even talking about the k vector again without going into too much of discussion of optics as such is that the k vector tells you what the momentum of photon is, momentum of photon is P equal to k h cross, see h cross is the unit of everything in quantum mechanics is not it, what is the angular momentum, if you ask that question hydrogen atom let us say, angular momentum is root over L into L plus 1 into h cross, what is the z component of angular momentum, M into h cross, right even when you talk about NMR spectroscopy you get this h cross everywhere, this is a unit. So what we see is that this k then is a number that gives you an idea of momentum of photon, okay, right. Now comes the important discussion again more or less axiomatically without going into the math, it is important that we know this that phase matching condition, condition that has to be satisfied and we will see why it has to be satisfied for getting second harmonic generation is that the momentum well here this expression is written in the in a little more general term where we are saying omega 1 and omega 2, two different kinds of photons are combining to give you some frequency generation, okay. So for some frequency generation the phase matching condition is k 3 which is the k vector of the some frequency photon has to be equal to k 1 plus k 2 where k 1 and k 2 are the characteristic k vectors for photons of angular frequency omega 1 and omega 2, what is the meaning of this, what is the meaning of this, to understand the meaning of this first of all let us say we are working with collinear beams, do we use collinear beams for some frequency generation or second harmonic generation, for second harmonic generation in any case it is collinear, right it is one beam that goes in for some frequency you can have a collinear geometry, you can have a non collinear geometry. So for now let us only talk about collinear geometry, so if it is collinear geometry then things become a little easier, why does it become easier because these k vectors all have the same direction, we do not really have to worry about what is the angle, what is the direction so on and so forth, collinear means all k vectors are aligned. So for a collinear beam what will happen, what is k from this phase velocity definition k multiplied by c is equal to omega multiplied by n, yeah omega multiplied by n of omega and I am not worried about the direction because it is collinear, so what can I write instead of k3, instead of k3 I can write omega 3 multiplied by n for omega 3 that is the left hand side, what will I have on the right hand side, n1 multiplied by omega 1 plus n2 multiplied by omega 2, okay, okay, but even well before going further what is the implication of phase matching, k vector of the sum frequency, sum frequency light is equal to sum of the k vectors of the combining frequencies, what is the implication of that, once again the answer is there in front of you in the projection, what does k stand for, what does k define which property, momentum, so are we not saying that momentum has to be conserved in some frequency generation that is the physical meaning, all we are saying is that when 2 photons combine they should combine in such a way that the momentum well momentum is conserved, momentum of the sum frequency photon should be equal to sum of momentum of the combining photons that is all we are saying and then it boils down to for collinear beams it boils down to something very simple n3 omega 3 is equal to n1 omega 1 plus n2 omega 2 and it becomes even more simple if you talk about second harmonic generation, in second harmonic generation omega 1 equal to omega 2, yeah, so n1 equal to n2 what does the right hand side become second harmonic generation omega 1 equal to omega 2 equal to say omega and n1 equal to n2 equal to say n1, n1 omega 1 right, what is the right hand side then, 2 multiplied by n1 omega 1 what is the left hand side what is omega 3 second harmonic generation, second harmonic generation 2 omega 1 right, so left hand side is 2 n3 omega 1, right hand side is 2 n1 omega 1, so 2 and 2 cancel and you are left with this condition that diffractive index for 2 omega 1 sum frequency is equal to diffractive index of the incident frequency, okay, but a little while ago Sharma sister told us that diffractive index is a function of omega, so do we expect in a crystal solid medium do we expect the diffractive index to be the same for omega 1 and 2 omega 1, so see where we have got, we are saying that you can get second harmonic generation only when diffractive index for omega 1 is equal to that for 2 omega 1 on one hand. On the other hand we know for ourselves that in a medium usually the diffractive index is a function of frequency, so diffractive index of omega 1 for omega 1 should not be equal to diffractive index for 2 omega 1 right, which means that second harmonic generation cannot be done and we can go home to get the point, what we have shown here is that an essential condition for second harmonic generation is that the diffractive index indices have to be the same, diffractive index has to be the same for incident frequency and second harmonic and on the other hand from our conventional knowledge we are saying that it is not possible, which means it is not possible to get second harmonic generation unless we have material in which somehow your diffractive index is the same for omega 1 and 2 omega 1 under some conditions ok, that is why not everything can give you second harmonic generation. So, as you are going to discuss it is possible to achieve this condition only for birefringent crystals, what is the meaning of birefringent crystals, what is the meaning of birefringent, y is 2 then birefringent yes, so it has a simpler sounding name actually birefringent goes by another name, which you actually just said almost double refraction ok, we will come back to birefringence shortly, in fact we might have forgotten but all chemists have read about birefringence, if not in 11 and 12 definitely in first year BSc, that too not in physical chemistry, organic chemistry yeah, tartaric acid, what does tartaric acid do or you might have even done some experiment, polarization remember rotation of polarization and all that, polarimeter what do you use, what is there inside a polarimeter other than lenses and all, don't you have some birefringent crystal, what is it calcite calcite cube cut diagonally and joined by Canada balsam yeah, so that is where we have all encountered birefringence, we are not saying anything new, it is as if we might not have used that term, so we might have forgotten ok, so we will come back to see how elegantly birefringence can be used to get second harmonic generation and we will come back to see why it is that we take that birefringent crystal, what is the magic that we turn it like this and all of a sudden we get second harmonic, we will get to get to that, before that for the sake of completion of this discussion, let me show you the expression, we will not derive this, if you are interested there is plenty of resources are available, two things since this is for the sake of completeness, the psi of psi which is a function of x and t equal to a cos kx minus omega t plus phi is just a little more elaborate way of writing the expression we have written already, instead of e we have written psi here, this is another way of writing, so if you read some literature where suddenly you see psi do not get scared, sometimes we get scared the moment we seek the greek alphabet, there is no reason ok, second thing is what is phi, yes we did not write it earlier, we assume that phi 1 equal to phi 2 that is not very difficult to achieve, but look at the intensity expression for intensity of the second harmonic, intensity of second harmonic is equal to some constant k multiplied by square of the second order non-linear susceptibility multiplied by L square where L is the length of the crystal multiplied by our old friend, sin delta k L by 2 divided by delta k L by 2 square, where have we encountered this, this function does it ring a bell sin square theta by theta square, where have we encountered this everywhere, we have encountered it when we talked about not in this course, when we talked about interaction of radiation with matter, this is what appears in the probability of transition from one state to the other and in this course we encountered this kind of expression, sin square theta by theta square when we talked about the shape of the pulse, modulocking remember, modulocking we encountered sin square theta by theta square, so what does this function look like, sin square theta by theta square where does it have a maximum at 0, but then there is a problem right, because at 0 theta equal to 0 of course theta equal to 0 and sin theta is also 0, so you have 0 by 0 which is not defined, then why are you saying that it has a maximum at theta equal to 0, because the shape is as you approach 0 from both sides this is the shape, yeah so you can close as you can as close as you want to 0, just for that one point where theta equal to 0 it will be undefined, but you can keep going close close close close as much as you want and the limiting value will be 1 sin square theta by theta square, so this is called a removable discontinuity and you can take the limit of 1 for that one point where theta equal to 0, okay, now remember what it looks like, it starts from 0 maximum goes down and then it has some ripples that slowly die off, remember that shape something like this, so where will this be maximum, okay we will come back to that, then that is multiplied by I0 square of omega, square of intensity of the incident light, so let us take factor by factor, so first of all in order to get I2 omega you should have a high value of second order non-linear susceptibility, point number 1 and then it is well known that if there is centrosymmetry then that second order non-linear susceptibility is actually equal to 0, so first thing you look for is that you must have a crystal which does not have centrosymmetry, point number 1, point number 2 and then of course it has to be high also, whether it will be high or not that I cannot really discuss so qualitatively, second thing is what is the second factor, L, so L should be large, but then it cannot be too large also, I think it is better to use a big crystal, so I try to use a one foot long crystal, that is not a good idea, first of all if you are using pulse laser it is going to get broadened, secondly it is not so easy to grow a one foot long crystal anyway, millimeter long crystal is extremely difficult to grow that is why they cost something like 50,000 rupees, 1 lakh rupees, 2 lakh rupees something like that and thirdly there is a problem, there is something called a coherence length, coherence length roughly means for which length omega and 2 omega are in constructive interference, see problems you have 2 kinds of wavelengths, omega and 2 omega, initially they are in phase, as they go out then you will have destructive interference, so if you go beyond a certain length and in fact there is something odd multiple of coherence length or even multiple of coherence length it becomes 0, destructive interference is there, so one has to take care of coherence length as well, then we are going to discuss later that we can do angle tuning to get phase matching, so delta K has to be equal to 0, this is what we have discussed already, we said that condition for second harmonic generation is that delta K must be equal to 0, momentum must be conserved, this is why momentum has to be conserved, because when delta K equal to 0 that is when the sin square theta by theta square function has a maximum value of 1, that is why delta K must be equal to 0, what about this I0, if I0 is large will intensity of second harmonic be large or small, large definitely but then like everything else in laser spectroscopy too good is no good, if your intensity is too large then your optics will burn, so it has to be large but not so large that the crystal will burn anyway, okay and then one more thing that we are going to see why this happens, polarization of the second harmonic is going to be perpendicular to that of fundamental, this is not really included in this expression but we will see shortly in the next 10-12 minutes why that is going to happen, in the next module we are going to take it from there, great. Now let us go a little bit slower and see why we need birefringent crystals and how you can get phase matching using a birefringent crystal, remember it is not enough to be birefringent right, it should be birefringent and it should have a high value of second order non-linear susceptibility then only it will work, so there are very few materials that fulfill all the conditions, earlier it used to be KTP, KDP all these things, now it is BVO all the way by and large, okay great. So let us say that this rectangle is the crystal and let us say this arrow here is the optic axis Z, what is the optic axis? So let us hold off our angle of incidence is actually correct, well I would say direction of incidence rather than angle at the moment but it has something to do with direction of incidence but let us wait a little bit we will come back to this and then let us say this is the incident ray, omega 1 okay, now see this is a birefringent crystal, what is the meaning of birefringent? The moment this incident light enters the crystal it breaks into 2 parts, 2 rays, does anybody remember what these rays are called? Lantian Organic Chemistry or in 11-12 physics, physical optics, ore is an E-rays does that ring a bell, ore and E-ray, I am not calling anybody these are names, ore means ordinary ray and ordinary ray is the component of the incident light that continues in the same direction okay, if there is an angle of course it will be refracted that is a different issue, E-ray is the part of the light that goes in a different direction, this is called extraordinary ray, why ordinary, why extraordinary we will see in a moment but for now who did this experiment, does it look like, so I will tell you hygens and somebody else I forgot the second name, hygens, heard of hygens, hygens was one of the pioneers of physical optics long ago he did all these experiments that have to do with wave nature of light and all, so hygens did this experiment, so I do not exactly remember which century it was but it was a long long time ago, definitely not 20th century, much before that I think 19th century maybe okay, so hygens and scope workers are actually shown that what happens is this ordinary ray if you use polarized light, ordinary ray has the same polarization as the incident ray, however the extraordinary ray has perpendicular polarization, also the refractive index of the ordinary ray and now comes why it is ordinary, why it is extraordinary and this is the most crucial part of the discussion, refractive index of the ordinary ray is the same no matter what the angle of incidence is, of course refraction will be there because the light is passing from air to a denser medium but it is the same so refractive index of the ore is independent of the incident angle, however refractive index of the e-ray is dependent on the incident angle okay, I am just stating facts we are not proving anything, this is something that has been known for 150 years or more that if you are birefringent crystal then incident light divides into two parts in the crystal, one has polarization same as the same as that of the incident light, the other one has perpendicular polarization, first one is ordinary ray, second one is extraordinary ray and refractive index of ordinary ray does not depend on the angle of incidence, refractive index of the extraordinary ray does depend on the angle of incidence, we take a break here and complete the discussion in hopefully significantly shorter module that is to follow.