I'm back and you're probably sick and tired of me by now, because I was on yesterday, I'm on today, yeah, you are sick of me, and I just have to put a bit of lip gloss on, no, I can't find it, I can't find anything quickly at the moment, it seems to have all gone, okay, so you know that's just how it is, so I promised, when I was doing the readings earlier, I promised that I would come on and do some affirmations, so I do have an hour where I can come on and do this, because then I've got to do a reading, so yeah, that's what I thought, I've got lipstick on my teeth, like you do, here's my coffee done, yeah, so cheers everybody, cheers, we're going to be doing affirmations for 2023 today, that's what I thought we would do, Greg has got his football Sunday on at the moment, he makes nice food and he wears his chief's outfit, his t-shirt, he has a nice Sunday, he watches his football and totally relaxes, which is good, we all need that, otherwise we just work right through, especially if he's self-employed, you just do it, so hi Sarah, hello Sarah, good to see you, cheers, drinking coffee, I've got to try and stay away, I did fall asleep earlier, I finished the live readings and then I went, sorry, fell asleep, so good job I didn't sleep through this, or the reading that I've got in an hour, that would have been awful, you don't plan to fall asleep, do you sometimes, and you just wake up and think, oops, okay, so I'll wait and see if there's any more people come in, did you see I created a guided journal, no I didn't Sarah, well done you though, well done you, what does that entail, I'd like to see that, yeah, so how is your little one Sarah, obviously I just spoke to Shannon, who you connected with now, you've got a lot in common haven't you, you and Sarah, sorry, is it Shannon, you and Shannon, you've got a lot in common, she's lovely, she's very lovely, all in 5D guys are lovely, you guys have just welcomed me, it's just made my heart sing and I'm so happy, it's called passion and purpose, self-discovery journey journal for women now available on Amazon, Sarah that is amazing, put it in the link here please, that's really well done, I think, wow you're so clever, I mean, Sarah, if you don't know what Sarah does, Sarah, she created, we did a shout out at one point because in 5D was really struggling and we called upon people to save in 5D basically and because obviously Greg is working on it, he does everything he can, morning, noon and night he does everything he can, but because there's a shadow banning it's just hard work and so he's got loads of followers and he's got everything you need for a successful YouTube channel, but if you can't reach the people because they ban you, because they don't let your content get out there, then you're not going to get much creation going, so yeah it's hard, so Sarah came and she did a whole business plan thing and it was amazing, hi sticker, I found Shannon on Facebook, good, there we go, so if anybody wants to have a look at Sarah's book there, a journal, yeah, well it helped us, it helped us no end, we're still working through some of the ideas that she put forward, we're still working on them, we still want to create a monthly raffle, we still want to do that, so yeah, we'll get there, we'll get through it all, we will. Okay, so today I thought we would do some affirmations, some positive affirmations, now for those of you who are watching this, probably not many of you will not understand this but I've got to do it for the few that might not know, an affirmation is a positive statement that is yet to come true, but because you're saying it, because you're believing it and because you're pushing it out there into the universe in a positive light hearted way, it then manifests and then starts to come to you. So, for instance, if you wanted a new car, you would say thank you universe for my new car, it's perfect for me in every way, it's very affordable and it runs like a dream and I love driving it, so you basically put as many positive statements about that car as if it's already happened, if you've already got it, because what that does is it speeds up that affirmation. So if you guys learn about Law of Attraction via The Secret, and you know, what is it, Abraham Hicks, so follow any of those on Facebook and maybe the book The Secret, it will explain it a lot better than I have, but that's the gist of it really, but it is amazing, it is. So hi Seeker, it's snowing in Seattle, oh Seeker, no, oh so hi Janet, oh hi Janet. Yeah, so who is up for in the chat, who is up for creating some of these positive statements for the new year, who's up for that, who would like to have a go? Oh, so Sherry and Deborah bought one, so excited, yay, oh that's good, we do have a wonderful in 5D family, we are very blessed. And we can feel your love and anybody who's just joining in, welcome, welcome, welcome, family can only get bigger and brighter, can't it? So yeah, so if any of you would like to share this, we would appreciate it. If anybody would like to, you know, press the like button, that would be amazing as well, and we thank you for that. So all the people at the bottom of the screen, we've got, thank you to all the Patreons, I thought I'd put the wrong one up there. And thank you to our newest Patreons, Sonya, Sonya L, thank you. We've got Vera, Sharon P, Renee R, Lisa L, Lisa S, Kelly M, Julia P, Vaughn, Terry G, Sarah A, Sherry S, Stacy J, Stephanie S, Marejo, Linda W, Julie S, Julie E, Elaine C, Deb T, Rita G, Gracie W, Shannon H, Phil C, Doreen H, Debra P, and Debra B. Thank you, thank you, thank you, thank you for supporting me. It means that I can come on and do these things. And you know, it's just helping us get, helping people who are creating videos to get by, basically by supporting them. And there's five different tiers for me. Is it five or six or five, I think? And it's $1.11, which works at about 96p. Then the next one up is $5.55. The next one up from that is $11.11. And then it's $22.22. And then $33.33. So yeah, it's pretty awesome. And I'm so grateful for Patreon's means a lot to me. Okay, so let's start off with doing some affirmations. Hi Vera. Is it possible to do affirmations for income? 100%.   Anything, anything, new house, new car, a family, you know, whatever your heart desires. And the affirmations are there. You can do it for health as well. Everything is in any statement that you can make that is a positive.  So the way I can only do show you what I do with affirmations. And this is how I manifested my soulmate, my perfect partner. Okay, this is how I manifested this. I manifested this and it took 10 years for this to come about. But nowadays it doesn't take that long. Nowadays what happens is, I wasn't ready 10 years ago. So the reason why it took 10 years was because it just wasn't ready. I was still kind of in a lot of emotional pain. I still wasn't kind of like, I cleared, I hadn't cleared out my stuff.  I hadn't done any of that back then and probably no at he. And so we have to get rid of all of that so that when you meet the other half to you, you're not going to miss it out by being dysfunctional. You don't want to be dysfunctional, do you? And so I visited him in his dream. I'm going to talk you through how I did it because it works. It jolly well works, okay, affirmations. So where do I start? Now affirmations, affirmations, prayers, cosmic ordering, law of attraction, even spell, good spell work, all of that is the same kind of thing. It's all about the intention that makes it good or bad.  Even if that was, if you're capable of that, it would be so wrong on so many different levels. Yeah, you just don't do that. It's not right. But what you can do is to, with the right energy, you can have that, that the right person come to you. You can have abundance, financial abundance, stability. You can have good health. I'm still working on that one. Now, everything that you want, there has to be the cosmic, sorry, the karmic order of things. And so some things, if you were in this lifetime supposed to be, you know, skimped, not having much money. Okay, no matter how much you do positive affirmations, if you're in the middle of that lesson, you have to continue learning that lesson. Okay, you have to. It is what it is. But the secret of it is how do you change things? How do you change? It's like, how do I learn that lesson then so that I can then move on? Listen to the patterns in your life. Listen to the patterns. Listen to what you're thinking. Listen to what you're feeling. So say if you're feeling anxious and panicky and sad, angry, nervous, what energy is that pushing out to the universe? Is it going to work if you're pushing those energies out there? No.  Yes. In these circumstances, yes, because everybody can do a bit of acting. Okay, you have to do it for 30 seconds at least a day to be able to fake it. Okay, so you might be sitting there thinking, no, that's not right. It's all about vibrational energy. Okay, so for instance, if I wanted, say, if I wanted abundance, if I wanted me to be self-sufficient, okay, if I wanted that, which I'm not, if I wanted that, which I do, I would say thank you universe for the abundance financial, for the financial abundance that you are pouring towards me. Thank you universe for bringing me lots of wonderful people who would like readings with me. Thank you universe for bringing me, you know, finances to fill up my bank account and to help me with my future endeavors. Thank you universe for giving me peace of mind with my finances, with my abundance. Okay, so all of these things you might not have yet. So for that 30 seconds, you are kind of closing your eyes and you're visualising what it would feel like to be self-sufficient financially, to not have any worries at all regarding money. So let's sit there and just think about that. Automatically, we're feeling happier, we're in this space. So you hold that space as long as possible. You might only be able to do it for 10 seconds to start with, which is fine. Just for 10 seconds, you're visualising that everything is perfect financially for you. Okay, so if you wanted a perfect partner, then you would say thank you universe for my perfect partner. You have to put in we, we love each other unconditionally and we are so happy together. My family, this is important to me, my family love my partner as well and we all get on so well. Okay, so you have to add on all the things that would make your life happy. Okay, all that. So all of these things. Yeah, so it's all about the wording and the wording is very important. The wording can also mess up as well. And I'll tell you about that in a minute. So let me have a little look. I see what people are saying. Okay, so Vera says also as affirmation to live a life without fear, whatever is needed to get your life back on track. Yeah. Yeah, definitely. Affirming your safety and keeping your field clear. Yes. The one I made up is what yours is yours. What's mine is mine. Clear and transmute. Divine. Try not to make it too complicated because the universe is like a child. Okay, it's like a child. So the simpler, the better because it couldn't be misconstrued. Some things, some things that you say could be misconstrued and then you're in trouble. Okay. My guides gave me I'm sure it's been a while. Yeah, so yeah, you have to kind of be direct, be direct with these guides. Okay, be direct and simple. Yeah. So it's amazing though how it all comes about. It really is. Okay, so we're going to create some today. Like I said, I've only got a little bit of time which I'm gutted about. But yeah, I went and fell asleep, didn't I? Sorry. Okay, so you can do affirmations with herbs and make it a spell or without herbs. If you're uncomfortable with mixing a few herbs together and calling it a spell or an affirmation, then that's okay. Ignore this part of it and do it just without. Okay, that is entirely up to you and your preference. Okay. So let's get writing them down. Greg's calling now. Hold on. Hello, darling. I'm live at the moment. Sorry, my darling. I'm sorry. You've been watching football? Yes. I love you. I've got a reading after this. Okay, I love you. I love you too, baby. Bye. Bye, darling. Bye. I'm sorry. So yeah, I feel absolutely blessed to have our vibrations move together in this lifetime. We're definitely twins, twin flames. You can see it the way we even look alike. There you go. We look alike. Okay. So yeah. And so we're going to create some now. Okay. All right. So I've got a blank bit of paper that I'm going to write out my stuff. Rough. Okay. I'm going to write it just on a rough bit of paper. And then I'm going to transfer it into this. Now you can get some of this creative paper. Okay. So it's kind of all these different sheets and look, they're all pretty. It's all pretty. You see? And so I can write it out neater for when I've got it. You then need to get yourself a candle. So you can get just a nice candle holder if you want to. This is my circle of friends. My neighbours bought me this for Christmas, which I love last Christmas. So, and you can just replace the candle. Okay. So you can do that. Or you can buy a nice candle. The secret is to only really use this when you are pushing out the energy, the positive energies. Okay. So that's the best time to use your particular candle. Or you buy a special candle and that has positive energies to it. Okay. Okay. So the first thing that you have to do with affirmations is to be grateful for what you have. Okay. Ah, there's my love. I love you. I was just talking about you and then you rang. So, um, so, yeah, so the first thing we need to do is to think about our gratitude. Okay. So being grateful for what it is that we do have. So you light your candle once a day. You light your candle once a day and you're basically saying thank you for, and I, this is me. I would say thank you for my grandchildren. Thank you for my, my children. Thank you for my mum. Thank you for my husband. Thank you for it doesn't have to be in any particular order. It doesn't make any difference. It's just being grateful. And you are, when you're, when you're saying these people, um, you are letting your heart sing. You're letting your emotions rise to the surface in a, in a wonderful way. Okay. Um, you know, you would say to, um, you know, thank you for the roof over my head. Thank you for the food that I have. Thank you for, um, you know, my job. Um, thank, thank you for, so you're basically going through and going through everything and thanking the universe. Okay. You're thanking the universe for it. Okay. So, um, on that note, so I would put the first thing I would put is, um, I am grateful for, and you don't have to write anything. Okay. On that bit, you could just leave that to whatever you want to say at the time because things might change. Okay. I'm grateful for. And then you start your affirmations. So your affirmations need to start with I am it is thank you for, um, I say thank you universe, but you could say anything. If you wanted to add God, you can add God. If you wanted to add anything, it doesn't matter. It's got to be just on that high vibrational energy. That is the key. Okay. So I'm grateful for and I would do dot dot dot. Um, and then the first thing is, um, my, my twin flame relationship goes from strength to strength. Make sure you keep it simple. Make sure the universe can't get mixed up. Okay. With some of the wording, it has to be, you know, the, the correct words, strength to strength. We love each other unconditionally. So you can put your own. So if you haven't got a partner, you could write, um, thank you universe for bringing me my perfect partner. Should you put this? We love each other unconditionally. Um, I remember I did this affirmation and I put, I love my partner unconditionally. My partner loves me unconditionally. Well, what happened was I had ex-boyfriends coming at me, um, you know, claiming to love me and come, want to come back. And then, then I was, I was in love with, or thought I was in love with someone else. And I was aiming it at them, but they weren't reciprocating. So, um, that, so I didn't get the unconditional love back from the person I was intending to. Um, and I had, so it didn't sync up. Does that make sense? I didn't say we. I didn't say we. Did I? So, um, wouldn't have mattered now. Um, because I was meant to, I was meant to keep myself for, for Greg anyway. So that's that. Uh, excuse me. Gotta look good for my man. He's watching. So yeah, we love each other unconditionally. So, and that's always a good thing. Okay. So I have financial, you don't have to copy mine. Abundance. So if you've got a good one, if you've got a good affirmation, write it in here and maybe other people could, could maybe use it and change it about a bit to make it theirs. I financial abundance. I am, I'm rich beyond belief. Don't be shy. The universe doesn't care is only energy. Money is energy. What would you do with your riches? What would you do with your abundance? It's not greedy. We are taught that we don't, we, to be greedy, financial greedy is bad thing. What would you do with all of that abundance? When you're sorted out, when your kids are sorted out, when your family is sorted out and everybody is secure, you would then reach out, wouldn't you? And see what your community needed. You would then reach out and see what you could do globally. You know, so don't shy away from it. Okay. Don't shy away. So I have financial abundance. I'm rich beyond belief. Okay. My family, happy, healthy and comfortable. I don't like the word comfortable. So this is why you have to do it on a scrap paper. So my family are happy, healthy and abundance is all around them. Okay. So you could put financial abundances all around them. You could put that. Okay. So you get my point, don't you? So you have to say it as if it's already happened, that then speeds up the process. Okay. It speeds up the process of the universe bringing this to you. If for whatever reason you aren't, this doesn't work and you're doing it religiously and everything else. It could be that it's a big wish, which means they've got to work hard in the background to make it happen. Try and watch the film. What's it called? Adjustment Bureau. Okay. Adjustment Bureau. It is, it tells it perfectly. Adjustment Bureau. It tells the story. Okay. About it. Adjustment Bureau. In fact, I want to watch that with Greg. And what it is, it's like this man, I think it's not Matt Damon. Might be Matt Damon. Anyway, I'm not sure. And Rachel someone anyway. So these two were destined to meet, but they were trying to keep him from meeting her because he was supposed to be the president. Okay. The next president, he was supposed to be running for presidency and he was supposed to win. And the people in the background like the million black in the background were trying to prevent them from meeting because if he'd have met her, he wouldn't have been interested in politics. He'd have just fallen in love and that would have been it. And so this is, this was the idea of the film. And in the end, they had to give in and allow this to happen.  And I think he got into politics anyway. So yeah, it's a, it's good. So you've written down all your affirmations. Okay. You've written down all those. Okay. And you might then write them out on a nice bit of paper, a bit of creative paper, or maybe you bought yourself a lovely card. Okay. And inside this card is a, is a card for you. Okay. And these are affirmations for 2023, but start them now. Definitely. But you're aiming for that. Okay. So how I did mine was, I got a little royal bag, like one of these little bags. Okay. That's what I got. Right. I then had a little spell kit. Okay. I won it in, in a raffle where I had, I was going to a natural magic course. And the lady, Linda Rowan, she was running it and I won the raffle. I couldn't believe it, but it changed my life. It really did. And that was all meant to be as well. So I've now created some of these raffle as some of these spell boxes. Okay. Hi Tracy R. Okay. So I've created these little boxes. Okay. This one is 20 pounds. Okay. So, and that is full up with all things that you can use to create a positive affirmation with. You don't have to use any herbs. For some people, you can just do this. Okay. You do all your writing. You have yourself a nice candle. Okay. That is linked with this. So once a day, you would light your candle just in case people are switching off because I mentioned spells and everybody thinks it's like, Oh God. That's really evil. It's not evil. The only thing evil is the intention behind these things. Intention. If your intention is love, harmony and energy and a positive level, there isn't anything evil in that. It's just how it's always been. It's always been. What do you think you're doing when you're cooking and all the ingredients is going in? You're creating love.  So no different. So yeah, you would, so I'll tell you what to do with these words in a minute. Okay. But so yeah, that's all you would do on that score. So you would then end that and then I'll do the end bit. If you aren't scared. I've kind of catered for everybody. If you're not scared and you understand kind of the law of attraction.  You don't have to have this particular box. Okay. You might have some just herbs in the cupboard. Okay. You might just have some of those. So in, in this little box. Okay. In my little box, there is, I've got a candle. In fact, there's usually more candles than that. But anyway, so there's usually a group of candles. Each candle has a meaning for it. There's a little bit of a sage kind of smudging kit there. There's some quartz crystals in there. There's a lucky pebble in there that you can keep.  Well, it's this. So it's, you have them at parties. It's like a party thing. And that's, so you get ready for success. Okay. With that. This is sandalwood. Okay. Sandalwood is good for love and protection. And I've written on each one. So you would put a pinch of this, a pinch of that in. Okay. Inside here, you've got a leaf that you could blow into the wind. And to start your spell off is another, there's another few little leaves in there. This one is agrimony, element of, eliminate negative energy and cleansing.  There's some spare little bottles here that you would put your spells in as well. You could do this, pinch of this and pinch of that.  Where's the foil bag gone? There it is. Okay. So then you would put that bottle, you'd fill the bottle up with a few herbs and then you placed it in there and hang it in the sunniest part of your house. Okay. Your affirmation, just so you can remember it is, this is just a key ring on a bit of wood and you could write down their positivity or, you know, I wish or I believe in myself or something positive. So every time you see it, it sets that positive energy out. So there's lots of money. I've got little tissues and I cut little tissues up, 20 per note and things. This one stops negativity, brings good fortune, protection. So each one of these little bottles of herbs has got something else in it. So rose petals for love. Okay. So you would do a pinch of this to the pinch of that. Okay. In your royal bag. Maybe in here. Okay. Now I had, I wanted a partner for life. Okay. I just didn't want any part. I wanted a partner for life. So what I did, you laugh at all the things I've got in my pill pot. Okay. Let's find it. There it is. Okay. So this is what I put in as well. It's a silver swan. Okay. Now swans make for life. Do they not? So swans make for life. So I added that into my little pot. I had some feathers. I added that in. I had some rose petals. I added that in.  I added that in. You know, anything that I could think of that would, would prompt me to feel positive about this, this creation that I was making really. You could have a little photograph of you and hearts and things, but you can't put your photograph of your partner in here because you can't make someone fall in love with you. That is wrong. But it's the intention of love. Absolutely. In that kit, there's also some little hearts. Okay. So yeah, you could put two hearts in there, which then represents hearts coming together or people coming together. Okay. So you then put this and hang this in the, in the sunniest place in your home or it could be on your altar. You know, if you've got an altar or a dressing table or something. So you're putting that in there. You can activate the ingredients by maybe pouring a little bit of the herbs onto your hand on a windy day, and then you then let it go into the wind or you blow it into the wind with the intention of love and happiness again. Okay. You could pour a little bit of the herbs into a plant pot and you know, you could then, you know, water it and you could have a sunflower seed and you could plant a couple of sunflower seeds and every time you do affirmation, you could water it. You know, it doesn't have to be sunflower. It could be anything.  Everything has to be done with love whenever it comes to spell work. Okay.  You've got it all there. Okay. So this is your prescription. Okay.  Okay. So my time is usually the evening. Okay. So I would light my candle. Okay. So I would light my candle. Okay. And I would then slowly, now you can use the spell candles. Okay. You can use those if you want to, you know, depending on what it is that you want. So if you wanted, I mean, oranges, you can ask me, okay, to send you bits and pieces of information about which candles mean what? Okay. There's nothing stopping you, lighting three of them. You know, tie the three together and light them, light each week. It doesn't make sense. So they're equally burning anyway. So much to this. I'm trying to rush it a little bit because I've got to go in a bit. But so, yes, you light your candle. Okay. So I've got my candle lit and I then go into my, my wording. So I'm going to say it, how I would normally say it. Okay. And you can do the same guys, that if you're following this and doing your affirmations, then you can follow it with me. I am grateful for, and then I'm going to think about people that I'm grateful for, my wonderful husband, grateful for my mama, my children, I'm grateful for the house I'm living in, my grandchildren, the job that I do, the people that I meet in 5D, you know, so I'm going through everything and it doesn't have to be in an order. Okay. It doesn't have to be. It's just what you're feeling at the time. Okay.  And when you're on that high vibrational energy, that's when you start these words. Okay. So, my twin flame relationship goes from strength to strength. We love each other unconditionally. I'm feeling every word. I'm feeling it and the slow, do it slowly so that you can feel every word. And I would say that again, my twin flame relationship goes from strength to strength. We love each other unconditionally. And once again, my twin flame relationship goes from strength to strength. We love each other unconditionally. Then you go to the next sentence. I have financial abundance. I'm visualizing myself with financial abundance. As I'm saying it, I am rich beyond belief. I'm feeling myself rich beyond belief. I have financial abundance. I'm rich beyond belief. I have financial abundance. I'm rich beyond belief, but obviously slow.  Okay. So you could then say about a car if you wanted a car. You could then say about a house if you wanted a house. It can all be on there. It doesn't have to be just one thing. Okay. But equally you have to give them all as much energy. And then the most important thing that you say is and let this be done that it harm no one. Most important because you don't want to come into riches when someone else is going through something bad. Doesn't make sense. So you don't want that. Okay. So and let this be done that it harm no one and let this be done that it harm no one. And so it is. Okay. And so it is.  You blow your candle out. You thank the universe. Done. Once a day. Once a day. I'm telling you you will see results. You will see results. Okay. You know you you know it could be that the relationship side of things really kicks off quickly. But the financial side is a little bit slower or the opposite way round. You know your perfect job you can wish for kind of anything you your heart desires. Use the law of attraction guys. Law of attraction is amazing. Okay. It's absolutely amazing. And it really works. Really, really works. So this is my my own personal spell kit. It's good. It's got little drawers in there and there's little kind of things inside. So they're all just herbs. Just little herbs. And yeah. I love it. I believe. Yes, Tracy. I believe. So let's see the still still here listening. No fear here. Intention is what's all about. Yes. It's a wonderful jewelry box. Yeah. You could use a joy box as well. It really means a lot to show us what it is to take away the fear and how beautiful the silver swan for life is beautiful. You're adorable. Thank you.  Me too. Before I forget I have sent links to those spam profiles on OK, darling. I've got my eyes. I've got to go. My eyes are getting sleepy. OK, Vera. Great ideas. Thank you. Gratefulness and compassion are the key. It is. It's great for us in fashion. But do you not get into this? I'm I don't deserve or other people need things more than me because a stewardess when they are doing when they are reading you all the things on the airplane. They're saying you put your oxygen mask on first. You then help others because you have to be OK. You have to be you have to be grounded and OK first before you can help others. OK, that's that's how it has to be. All right, so so I'm hoping this helped you and you know, take care and I will see you all soon. I'm going to go and do a wonderful reading now and I feel absolutely blessed I'm able to do what I adore doing in my life. So I am lucky. OK, take care everybody and have a positive week.


Yeah, or secret key. So yeah, we have some private data that we are wondering like, okay, what to do to protect it, do we make up a security solution for it, or do we get some other party to protect it, because it is difficult to do security. So we got like two consultants, actually two cryptographers to consult us, let's call them Alice and Bo. So Alice called us, well, why don't you use like this secret sharing, which is really like, it's great, because you don't have to, well, first Alice called us, don't do it by yourself, because security is hard, just let some other people do it, like outsource it to someone. But it says, okay, it's better not to trust with like a single party, just maybe just secret share your secret and use a few parties, and that will be better. We thought, okay, yeah, that sounds nice. But the problem was, okay, we still have like, okay, now instead of one secret, you have like n secrets, so we still have a problem, because okay, what do we do with this share? Okay, we have to find someone who will say, okay, would you be able to hold my share, or would you, so it's still a problem. So instead of one problem, it's how to protect the secret, we have like n problems how to protect the share. So that was a problem for us, so we went to Bob and said, hey, Bob, what do we do? Because Alice has this great idea, but okay, we are not convinced about it. So Bob said, wait a minute, your secret is valuable, right? So, well, if your secret is valuable, then why don't you just, instead of buying space or buying some service or to protect your share, why don't you just sell your shares? Because that, I mean, it is actually a commission business case study, okay, because it's some valuable share or some secret, so we thought, yeah, that's interesting. So we packaged all this story together, we went to the investors and said, well, yeah, so we have these shares and okay, we'd like to sell them and so on and so on. But then it turns out that, okay, the investors, they have some very interesting requirements. Like there would be interesting, interested to know, for example, they don't just want to give money and just stay in some share there, which doesn't, and in particular, they're interested in growth potential, so they want to, okay, I'm gonna give money, then the share is gonna be, you know, will it grow? Well, in this particular case, literally, will your share grow? So we thought, okay, we don't know about that. Will it grow? So we went back to Alice and said, well, would it be possible to make this share grow, like in some sense, grow in some sense? Alice says, I don't think it can grow, it's not supposed to grow, but I don't know what I can do for you. Alice says, okay, tell us. Well, Alice says, well, I can give you a proof that it cannot grow. Alice says, okay, a proof that it cannot grow. What can I do with a proof that a share cannot grow? Can the proof grow? Alice says, okay, the proof cannot grow, but I can give you as many proof as you want, so that's actually interesting, right? Well, you said, okay, if you give me as many proofs as I want, then I can do something with that, right? So we went, so this is a very nice story, because if we can get it for a share, as many different proofs as you want, and they are all different, you can repackage them, you can install them, and so on. So it's very cool. So, yeah, so it turned out, this is very interesting, and so the investors were happy, the investors were happy, and said, yeah, okay, this is very interesting, I think I'm interested in your shares, actually, I want them all, yeah? So we said, oh, he wants all our shares, that's fine, that's fine, that's brilliant. So we were happy, we sell them all for the shares, the investors were happy. Alice was not happy. I mean, Alice, we went back to Alice, she said, okay, it's crazy, because, well, if you get into the talk, I think this is like, means the entropy of the secret is zero, just to use the vocabulary. So the problem was that, so Alice was not happy, but she was even less happy when she found out that actually the contract, we couldn't even tear down the contract because the contract was already on the blockchain, so we couldn't tear down the contract, we had to tear down the blockchain. So the question is, well, actually, do we tear down the blockchain? Actually, well, we will be happy just to remove our data from the blockchain because the contract is on the blockchain, the shares on the blockchain, so everything. And so I'm almost out of time, I don't have to tell you how we did that, but yeah, we did some research and I can tell you it's, we did find some solutions, but maybe it will be for next time, okay?


Welcome to the 20th lecture in the course Engineering Electromagnetics. Like the last few lectures, we continue with the phenomenon of reflection and refraction of waves and the topics that we plan to discuss today are the following. We discussed the phenomenon known as Brewster angle and then consider the phenomenon of total internal reflection. Considering the first topic that is the Brewster angle you would recall that for the two cases of perpendicular and parallel polarization we have written down the expressions for the reflection coefficient and the transmission coefficient considering an interface between two perfect dielectrics. You would recall that we consider the following situation. This is the interface and this is the normal to the interface and we consider that these two media are perfect dielectrics with permittivities epsilon 1 and epsilon 2 and a uniform plane wave represented by a ray let us say like this incident at an angle theta 1 with respect to the normal at the interface is partly in general is partly reflected back at the same angle and is partly transmitted into the second medium and let us say this is the way the transmitted wave looks like and this is the angle of refraction. Now considering the expression for the reflection coefficient for the case of the parallel polarization you would recall that we wrote E r by E i equal to epsilon 2 by epsilon 1 cos theta 1 minus whole square root of epsilon 2 by epsilon 1 minus sin square theta 1 divided by a denominator consisting of the same terms, but with the change of sin. So, that it is epsilon 2 by epsilon 1 cos theta 1 plus epsilon 2 by epsilon 1 minus sin square theta 1 and we said that in general part of the incident wave is reflected back into the same medium and part is transmitted. Now one can raise a question now is it possible for some angle of incidence that there is nothing which is reflected back and the transmission is complete that would be a very interesting possibility. So, let us see if that is indicated by this expression we just apply the condition that the reflection coefficient should be 0 and try to find out the corresponding angle of incidence if it exists. And from that point of view let us say that for E r by E i to be equal to 0 we should have epsilon 2 by epsilon 1 cos theta 1 equal to square root of epsilon 2 by epsilon 1 minus sin square theta 1 and we square up both sides and write cosine theta 1 as 1 minus sin square theta 1. So, that we get epsilon 2 by epsilon 1 whole square minus epsilon 2 by epsilon 1 whole square into sin square theta 1. Let us say that this happens when the angle of incidence theta 1 is equal to some angle say theta b. So, that we can change these 2 theta b this is going to happen if it does at some particular value of the angle therefore, we specialize the angle of incidence theta 1 to theta b which should be equal to epsilon 2 by epsilon 1 minus sin square theta b. Now, one can collect the terms and one can see that sin square theta b is going to be equal to epsilon 2 by epsilon 1 whole square minus epsilon 2 by epsilon 1 on one hand and 1 minus epsilon 2 by epsilon 1 whole square minus 1 on the other side which can be simplified further and you see that we are going to have the final result as epsilon 2 by epsilon 1 plus epsilon 2 which makes sense and we see that there is an angle of incidence equal to theta b where the reflection coefficient for this polarization can be 0. One can process it further. So, that let us say cos square theta b that is 1 minus sin square theta b is going to be epsilon 1 upon epsilon 1 plus epsilon 2 and therefore, one can say that the tan of this angle theta b is going to be epsilon 2 by epsilon 1 whole square root or this theta b is going to be equal to tan inverse of epsilon 2 by epsilon 1 which angle is known as Brewster angle, angle is theta 1 equal to theta b. What is going to happen at this angle of incidence? If the incident wave is parallel polarized that is it has an electric field vector which is in the plane of incidence then the wave is going to be completely transmitted. If there is no reflection then the entire wave is going to be completely transmitted which can have very interesting applications. But before we discuss the applications perhaps it could be a question what about the other polarization the perpendicular polarization. Let us look at that for a short while for the case of perpendicular polarization continuing with our simplifying assumptions of perfect dielectric media. We have E r by E i equal to cos theta 1 minus epsilon 2 by epsilon 1 minus sin square theta 1 divided by cos theta 1 plus epsilon 2 by epsilon 1 minus sin square theta 1. And now looking for a possible Brewster angle for this case we will apply the condition that the reflection coefficient is 0 which will give us the condition that cosine of theta 1 is equal to whole square root of epsilon 2 by epsilon 1 minus sin square theta 1. Let us say this happens at theta 1 equal to theta b. Now squaring up and writing cosine in terms of sin we will have 1 minus sin square theta b equal to epsilon 2 by epsilon 1 minus sin square theta b leading to the condition that epsilon 1 is equal to epsilon 2 which is a trivial situation just indicating that there is really no interface there are no two different media it is just one single media which case is perfectly alright. Otherwise in the case of perpendicular polarization we do not get an angle corresponding to the Brewster angle as we have got for the case of parallel polarization. One can think of interesting applications of this phenomenon related to Brewster angle. For example if the incident wave is circularly polarized it can be considered to be composed of two components of perpendicular polarized component and a parallel polarized component. If the angle of incidence is equal to the Brewster angle then the parallel component will not be reflected back it will be completely transmitted. Only the perpendicular component will be reflected back and therefore a circularly polarized wave can be converted into a linearly polarized wave or in general an elliptically polarized wave can be converted into a linearly polarized wave. We consider another interesting application of this phenomenon known as Brewster angle and for that we consider we go to the overhead projector. This is the schematic of what is known as a gas laser. Now laser is short into prominence many years ago because of its some unique properties. This is a device which can be used for generating laser radiation. It consists of a gas discharge tube which is filled with a suitable gas such as a mixture of helium neon at low pressure. It is a very low pressure almost close to vacuum and then a discharge is arranged in this gas discharge tube. As a result of this there are there is a complex series of phenomena that is initiated. Electrons are knocked off their shells they are raised to higher energy levels then they come back to lower energy levels and in that process emitting light. Now this light is to be coupled out of this gas laser in a suitable manner without allowing the vacuum or low pressure to leak and therefore this gas discharge tube is closed with what are called glass windows which have to be made of a low loss glass at the wavelength which this device is generating and quartz is a popular material for that purpose. So let us say these are quartz windows which would be fixed so that they seal the gas discharge tube properly. This entire assembly is then placed between two mirrors as shown here. So these are the mirrors. One of these mirrors has a very small pinhole. Now the light which is generated or radiated by the atoms and molecules inside the gas discharge tube can pass through these quartz windows and is going to strike these mirrors and in general is going to be reflected between these two mirrors and only the light which is exactly along this axis and travelling in this direction is going to be let out of the system. And before this happens a ray which comes out like this may have got reflected back and forth back and forth several thousand times and of course that requires that these two mirrors have a very good alignment, alignment to within a few seconds of parallelism. So that is one thing and that explains how the laser light generated through this kind of devices say gas lasers is highly collimated. It will have very small angular divergence and therefore it can propagate over very long distances. Where is the Brewster angle phenomenon coming in here? We would like that the reflection and reflection that takes place at these interfaces. These are now two different media. So there is an interface between this medium and this medium. Let us say this is very close to free space and this is that of quartz and once again from quartz to another medium say free space. So these interfaces are going to be encountered by the light rays as they reflect back and forth between these two mirrors and we would like that the transmission through these quartz windows is as efficient as possible. One way of doing that is that we arrange these windows at an angle to the axis shown here which is the Brewster angle. In such a case what is going to happen? We consider let us say a ray of light which is incident in this direction along the axis on this interface. Now we draw the normal at the interface. This is the incident ray and depending on the Snell's law it will make some angle with the normal here and let us say it will come out in this direction. Since this medium has a greater dielectric constant than this the angle of refraction will be less than the angle of incidence and for this kind of a symmetric interface this medium and this medium have identical properties. This ray is again going to come out in a direction which is parallel to the direction of the incident ray or the axis of the system. And this is going to be reflected here and then once again the same path is going to be traversed back and similarly it will pass through these and then we reflected back at this mirror. If this angle is the Brewster angle then the parallel polarized component the light that is generated here has a random polarization and in general all these randomly polarized emissions or radiations can be broken up into two components perpendicular and parallel. The parallel polarized component is going to be perfectly transmitted completely transmission since this is Brewster angle but the perpendicular polarized component is not going to be completely transmitted. Partly it will be reflected back and partly it will be transmitted. This part transmission will take place at this interface for a ray travelling in this direction at this interface and as it is reflected back once again at this interface and at this interface and so on. And since there are going to be several thousand reflections before a ray really comes out through this narrow pin hole one can make out that it is the transmissions of the parallel polarized component which will be complete which will be efficient and transmissions of the perpendicular polarized components are not going to be efficient. Then if we consider a transmission coefficient for the perpendicular polarized component say to be 0.9 which is a very good value of transmission coefficient. A single pass for a ray at this point travelling to this direction and then coming back at this point will require the field amplitude to drop by a factor of 0.9 to the power 4. Since there are 4 transmissions taking place at these 4 interfaces. So if we start with an amplitude of 1 we will end up with a reflected field which is of the order of 0.65. So one pass reduces the value of the perpendicular polarized component by about 35 percent and this way one can make out that what comes out here is essentially going to be of parallel polarization and the parallel polarized component will not suffer this kind of partial transmission. So that is a very interesting application of the phenomenon that we have called related to the Brewster angle. If there are any questions here we can consider those otherwise we proceed further. The second topic that we have listed for this lecture is the total internal reflection and that is what we will like to deal with next. Let us consider the diagram that we have made representing an interface between two perfect dielectrics. Now the angle of incidence and the angle of refraction are related through the Snell's law. So that we have sin theta 1 upon sin theta 2 equal to square root of epsilon 2 by epsilon 1. Let us consider the somewhat restricted category of situations where epsilon 1 is greater than epsilon 2. Then what happens? Then theta 2 is going to be greater than theta 1 and there will come some particular value of theta 1 where theta 2 is 90 degrees. What will be that particular value? That will be sin and let us say that particular value is theta c sin theta c equal to square root of epsilon 2 by epsilon 1 beyond which value of the angle if theta 1 exceeds theta c then the incident ray is going to be completely reflected back into the same medium. This is how we have been introduced to the phenomenon of total internal reflection and therefore, we say that if sin theta 1 is greater than epsilon 2 by epsilon 1 then the incident wave is totally internally reflected back when epsilon 1 is greater than epsilon 2. Let us see how this phenomenon corresponds to or is consistent with the detailed expressions that we have derived based on field theory. For this case let us consider the second term that is appearing in both the expressions for the reflection coefficient both for perpendicular polarization and parallel polarization. And we see that if we consider epsilon 2 by epsilon 1 minus sin squared theta 1. Now, sin squared theta 1 for these situations is going to be greater than epsilon 2 by epsilon 1 and therefore, this term is going to be the argument is going to be negative and the square root is going to be imaginary. And therefore, we can write this as going to let us say plus minus j sin squared theta 1 minus epsilon 2 by epsilon 1 which is what we can write for the reflection coefficient for both polarizations.  If you notice we have interchanged the order of sin compared to this expression it should have been minus plus according to this, but we are writing plus minus the reason for this will become evident later. However, now we have made sure that the quantity within the square root sin is positive and therefore, we can write this as let us say A 1 plus minus j B 1 and A 1 minus plus j B 1. Of course, the choice of sin in both numerator and denominator will go together it will be either upper sin or lower sin which will go together. What is the magnitude of this complex expression? Magnitude is 1, the phase could be anything depending on B 1 and A 1 and therefore, let us say that this has a value which is e to the power j phi perpendicular. Since this corresponds to the perpendicular polarization, phi perpendicular will be the phase shift between the incident wave and the reflected wave electric fields. And since the magnitude is coming out 1 it means that the entire electric field is reflected back completely and which is what is meant by the phenomenon of total internal reflection. And we see that whatever we had learnt earlier in simple terms is coming out from these expressions also. Let us see what happens for the parallel polarized component. We are going to have in a similar manner this equal to epsilon 2 by epsilon 1 cos theta 1 and then plus minus sin square theta 1 minus epsilon 2 by epsilon 1 divided by epsilon 2 by epsilon 1 cos theta 1 minus plus sin square theta 1 minus epsilon 2 by epsilon 1 which also can be written as A 2 plus minus j B 1 and A 2 minus plus j B 1 which again when these conditions are satisfied implies that the magnitude of the reflection coefficient is 1 and they could be a phase difference between the reflected wave electric field and the incident wave electric field which is represented as e to the power j phi parallel. So, in this case for the phenomenon of total internal reflection there is no so to say discrimination between the two polarizations for both types of polarizations total internal reflection can take place. And the condition for total internal reflection remains the same for both polarizations. However, the phase shift that occurs for the two polarizations is different which again can form the basis of some very interesting applications. For example, an elliptically polarized wave where the magnitudes of the two polarization components are equal, but the phase differences are not proper for circular polarization could be converted into a circularly polarized wave by an appropriate design by choosing a suitable angle of incidence and choosing proper materials. So, this can become a basis of very interesting design for this purpose changing the state of polarization. This phenomenon of total internal reflection is used very frequently in optics work. In fact, if you recall the optical fibers support propagation of light through the phenomenon of total internal reflection through very thin glass fibers consisting of a core and a cladding where the core index, core refractive index which can be related to the permittivity is greater than the cladding refractive index. So, that light remains guided between the core in this manner which I have mentioned earlier. Which optical fibers are revolutionizing the entire field of communication. In fact, optical fibers are just one kind of optical waveguide. They can be waveguides of many different types. Another type of waveguides occurs in the field of what is called integrated optics where one makes the optical waveguides on the surface of a substrate, a suitable substrate which could be say lithium niobate or even glass or silicon or gallium arsenide depending on the final application and there are various special features of each of these. So, there what one would be doing is that on a substrate let us say like this in a narrow portion one would be increasing the refractive index through some means. And then this becomes a waveguide where through total internal reflection between this core and the rest of the material which acts as cladding light rays can be guided into this waveguide. And if anyone of you takes up this kind of work he will see that these expressions utilizing these phase shifts which occur when total internal reflection takes place at different interfaces are very useful in that field. While one can use reflectors say metal coated reflectors for changing the path of light rays one can do the same thing using the phenomenon of total internal reflection also which concept I am sure you are familiar with. The change of the path of light rays using total internal reflection is supposed to be somewhat more efficient than accomplishing the same thing using reflectors. The loss that may be associated with the reflecting material is avoided in the total internal reflection based light ray path changing schemes. Now, we go a little further this seems to indicate this may be an impression that we may form that when total internal reflection is taking place the entire electromagnetic phenomenon is restricted to the region above the interface or just at the interface that impression is not correct. Why is it not correct? One simple argument that can be given is we consider the total tangential field let us say total tangential electric field at the interface in region 1 it will be E i plus E r or combination of these two depending on which polarization we are talking about. If that is the total tangential field in medium 1 at the interface there must be a corresponding tangential electric field in medium 2 as well. So, that the boundary conditions are satisfied therefore, there is field at the interface in medium 2 it is required from the continuity of tangential electric field components. If there is field in medium 2 even when total internal reflection is taking place in medium 1 what is the nature of this field? That is what we like to investigate next. Now, you would recall that we were writing the expression for the field related to the different waves the incident wave the reflected wave and the transmitted wave in the following manner. So, in medium 2 we write the field as let us say E t into E to the power minus j beta 2 beta 2 being the phase shift constant for a uniform plane wave in medium 2. And then let us say these are the y and the z directions then for the angles shown we have here y sin theta 2 minus z theta 2 minus z theta 2 minus cos theta 2. If there is a transmitted wave this is what we would write. We separate the y portion and the z portion our experience tells that these two components behave somewhat differently. So, we consider the y part here and let us say the z part here and the y part would be written as ignoring the amplitude factor E to the power minus j beta 2 y into sin theta 2. If time permits we can comment on the nature of theta 2 later on, but right now we are not sure about the nature of the role of theta 2. We are talking about the situation when total internal reflection is taking place. So, let us write theta 2 in terms of theta 1 using snails law that is we have sin theta 2 equal to epsilon 1 by epsilon 2 into sin theta 1. Theta 1 we are sure of that is the angle of incidence and the angle of reflection and therefore, this becomes E to the power minus j beta 2 y and instead of sin theta 2 we write root of epsilon 1 by epsilon 2 sin theta 1 which indicates or can be interpreted as a wave which is travelling along the y direction. This is not a subscript this is beta 2 times y with the phase shift constant beta y equal to beta 2 into square root of epsilon 1 by epsilon 2 into sin theta 1. It indicates a wave travelling in the y direction with this phase shift constant that is with the corresponding phase velocity v y equal to omega by beta y equal to omega by beta 2 square root of epsilon 1 by epsilon 2 times sin theta 1. Mathematical what is the nature of the factor multiplying beta 2 that is under the conditions of total internal reflection. Total internal reflection means that sin theta 1 is greater than epsilon 2 by epsilon 1 square root and therefore, this factor is greater than 1 under total internal reflection conditions implying that this phase velocity is less than omega by beta 2 or the velocity of light in this infinite medium 2. Otherwise it travels along in the y direction with this phase shift constant and with this phase velocity. What is more important from the practical point of view is the z variation that the wave this is the behavior of the combined wave also in medium 1 travels along the y direction is quite alright. Let us see what is the z behavior and there we get e to the power plus j beta 2 z times cos theta 2 which would simplify to e to the power j beta 2 z and then cos theta 2 we write in terms of sin theta 2 which is further going to be written in terms of sin theta 1 and therefore, we have cos theta 2 as 1 minus sin square theta 2 or e to the power j beta 2 z and then we have here sin square theta 2 is now written in terms of sin theta 1. So, that it is 1 minus epsilon 1 by epsilon 2 sin square theta 1. Under the conditions of total internal reflection the second term under the square root sign is going to be less than 1. Since, sin theta 1 is greater than root epsilon 2 by epsilon 1. Therefore, to make the argument real we take the negative sign out and that will give us e to the power j beta 2 z and then we have plus minus j and then we have sin square theta 1 times epsilon 1 epsilon 2 minus 1. Now, we have ensured that the quantity under the square root sign is positive this simplifies to e to the power minus plus beta 2 z into epsilon 1 by epsilon 2 sin square theta 1. Now, comes the question of choice of sign mathematically both signs are admissible. This wave is propagating in which direction in medium to in the negative z direction minus z cos theta 2 is what we use and since it is a passive medium with no source of energy the wave amplitude could decay in the direction of propagation it could not build up. Since, other quantities are positive here we must choose the lower sign that is the positive sign. So, that for decreasing values of z or increasingly negative values of z the wave amplitude goes down exponentially. And therefore, we choose a sign such that it is e to the power plus beta 2 z then epsilon 1 by epsilon 2 sin square theta 1 minus 1 anticipating this choice of lower sign in the square root signs we had reversed the order of signs in the earlier expressions. So, what is the behavior of the field in the z direction in medium to it is an exponentially decaying field it is not a wave it is not propagating in the z direction it does not have the characteristics of a propagating wave in the z direction for a propagating wave it is this kind of exponent which is required with a j sign. So, this is just a decaying wave exponentially decaying wave in the z direction. In fact, this is considered to be an example of a non uniform plane wave because in the plane normal to the direction of propagation the amplitude is not uniform. So, this is considered to be an example of a non uniform plane wave there is a very interesting consequence of this observation what is the observation the observation we have made is that even under total internal reflection conditions there is a field in the second medium total internal reflection is taking place in medium one, but there exists a field in the second medium which is exponentially decay. And therefore, it is necessary for efficient total internal reflection that this surface in medium two is kept absolutely clean and free of dust particles and dirt etcetera. If that is not so then the total internal reflection would not be 100 percent efficient which otherwise is indicated theoretically. With this observation which is of practical importance we can stop here unless there are some questions. Sir, can you clarify why root of epsilon by epsilon 2 sin theta 1 is greater than 1. That is quite straight forward you see for this situation and above angles greater than theta c only total internal reflection takes place otherwise it is the normal situation of partial reflection and partial transmission. So, since theta 1 is going to be greater than theta c for total internal reflection sin theta 1 is going to be greater than this quantity. So, what happens to the power flow? There is no power flow in the second medium perpendicular to the interface and whatever power was incident normally on the interface is reflected back. Since, the reflected wave electric field magnitude is the same as that of the incident wave electric field magnitude. So, it will form a standing wave in the z direction and in the y direction the entire composite wave a standing wave in this direction exponentially decaying wave in this direction will travel in the y direction. So, I suppose we can stop here.