As discussed before, I feel like I have been very busy - I have a lot to balance: learning french, doing research, socializing with the other people in my program, getting settled in my living area and studying for the GRE.
Just to summarize a few notable experiences,
Yesterday, I went with two other people on a "pasta-venture." My goal was to get a pan, pasta, and some toppings. We took the train and bus to a very large supermarket (called a hypermarche/). Carrefour is basically the French Walmart. It was a lot of fun to go around and see the all of the different food sections. There was a huge cheese section, a fish section, etc. Another cool area was the french book area. There were a lot of children's books. I spent about 15 minutes reading Winnie the Pooh (or Winnie l'Ourson). It was fun to read but I did not recognize some of the grammar structures. It was 10 euros so I didn't buy it but I do have fond memories of Winnie the Pooh from when I was younger! Another fun feature of the store is that they had moving walkways to go upstairs so you could take your shopping cart. The surrounding shopping center had these as well.
When I went to the checkout, apparently I almost bought a 50 euro pot. I also didn't weigh my bananas. So I was able to use my french to say no thanks. Apparently I misread the label for the pot and it actually cost 20 euros only if I bought 50 vignettes whatever that means. I was able to get another pot.
Another thing that I have been doing is that I have been trying different French cheeses. I have always eaten a lot of cheese and so it has been fun to explore different kinds. At my lunch, I have been getting a little cheese with each meal. I also bought some cheese from the store. So far, I have had Camembert, Brie, and Roquefort. The last cheese there is one of the moldy cheeses that is supposed to be quite strong but I thought it was tasty. Tomorrow, I hope to try Munster. I also bought "mimolette jeune". It looks like cheddar but I haven't tasted it yet. I hope to continue to try different french cheeses.
Tonight, I was successful in using my pot to cook pasta and I put some cheese on it (emmental). After looking this up, it is just what we refer to as Swiss cheese. I also got two other pasta sauces. One is a tomato sauce with Ricotta, and another is tomato sauce "basquaise" I just chose it randomly when I was there. Apparently it is a sauce with tomatoes and hot peppers. All in all, I am looking forward to making more pasta!
I have been trying to spend less money lately in general so I think that I will be doing well now that I am settled. For breakfast, I have a few cereal bars (they were quite cheap at that store), for lunch, it is subsidized so that is cheap, and dinner will be cheap with the pasta. I will go out occasionally but not everyday like I was doing at the start of the trip.
Also I am not sure if I have talked about the lunch at work. So basically, around 12, the whole lab moseys over to the lunch area. There is typically a long line so we talk a bit. When you are in line, you pick up a tray and put various dishes on your plate. A typical lunch for me includes: a hot plate with a meat and rice or pasta, a small salad, some fruit, a dessert (I have gotten a variety of things such as fruit with pudding or a piece of pie), some cheese, and a roll or two of bread. I enjoy lunch very much and the food is subsidized for students so that all costs around 2.50 euros! It is cool how balanced the meals are here. You have a lot of small plates instead of just one large dish of food.
So thats about that. I haven't been sleeping too well so I should be getting to bed now. Yesterday, I stayed up late doing some GRE studying since I haven't been keeping up with my studying goals. I really felt it at work so I hope to get at least 7 hours of sleep tonight.
Tuesday, June 7, 2011
Collecting thoughts on Research
My first attempt to calculate the phase from the simulated reflectivity data did not seem to work so I am going back and reading as much as I can. I am looking for ways to improve my calculation of phase.
I attempted to implement a method in matlab that integrated reflectivity data to get the phase but the results are not very good. (I took reflectivity data for every 0.1 eV (apparently this is typical for real data), created a function that interpolated those points into a smooth curve (I used a cubic spline), then I numerically integrated that using an adaptive simpson's rule routine. In order to handle the principal part integral, I removed a symmetric neighborhood of the singularity). For extrapolation, I assumed that the reflectivity was constant.
One problem: I am getting high reflectivity in regions where we wouldn't normally measure the reflectivity. I am looking at an aperiodic mirror from "Aperiodic multilayer mirrors for efficient broadband reflection
in the extreme ultraviolet". So there is reflectivity of about 0.15 in a wide region (30-80 eV) where the mirror was optimized, but there is also another peak near 10 to 20 eV. I am not sure if this is due to bad optical constants, or if this is just how their calculation worked out. Anyways this may be causing problems in my integration (and my way of extrapolating the reflectivity to low frequencies is not going to work)
So I looked at a lot of papers today. There is a ton of work that has been done in the area of using the KK relations. Here is a summary of things to do / things that I am thinking about:
1. learn more optics
a. how is the reflection coefficient derived from known constants for a multi-layer (maybe this would help me understand where the zero reflectivity points are)
b. does the reflectivity coefficient have zeros in the complex frequency plane (how can I predict their position... eg. the dielectric function for a semi-infinite substrate has its only zeros on the imaginary axis)
c. qualitatively, what is the reflectivity of a multilayer for high frequency/low frequency (so I can extrapolate correctly)
d. what effect does polarization, angle of incidence have on the calculations that I am doing?
e. is there a relationship between the complex reflectivity coefficient and the complex dielectric function
2. Kramers-Kronig relationship
a. which representation should I be using
i. real and imaginary part of a boundary condition for an analytic function are related by the Hilbert transform
ii. the transformation can be done as two Fourier transforms
b. zeros of reflectivity (for complex frequency) give rise to singularities in the integral above and result in extra terms being needed in the transformation
c. how is this derived? how can one use causality to show that reflectivity as a function of a complex variable is analytic?
3. Using matlab to do the hilbert transform numerically (there are MANY papers on this subject)
a. use of gaussian quadrature method (write integral as INT w(x) p(x) where w(x) is the weight function log x^(-1)... this may require a different program as you need high precision arithmetic for large numbers of sample points
b. write the transform of a sequence of fourier transforms and approximate as a discrete fourier transform (cooley-turkey algorithm?)
c. look into algorithms that allow one to isolate the zeros of the reflectivity as a function of a complex variable (eg. find the blaschke factors that are needed to recover a function that has singularities when it is fourier transformed...one paper suggests using transmission data?)
d. look into methods that acceleration of convergence of the integral by knowing the phase (eg. multiply subtractive KK method)
Anyways, I feel like there is SO much that I can do at the moment.
I attempted to implement a method in matlab that integrated reflectivity data to get the phase but the results are not very good. (I took reflectivity data for every 0.1 eV (apparently this is typical for real data), created a function that interpolated those points into a smooth curve (I used a cubic spline), then I numerically integrated that using an adaptive simpson's rule routine. In order to handle the principal part integral, I removed a symmetric neighborhood of the singularity). For extrapolation, I assumed that the reflectivity was constant.
One problem: I am getting high reflectivity in regions where we wouldn't normally measure the reflectivity. I am looking at an aperiodic mirror from "Aperiodic multilayer mirrors for efficient broadband reflection
in the extreme ultraviolet". So there is reflectivity of about 0.15 in a wide region (30-80 eV) where the mirror was optimized, but there is also another peak near 10 to 20 eV. I am not sure if this is due to bad optical constants, or if this is just how their calculation worked out. Anyways this may be causing problems in my integration (and my way of extrapolating the reflectivity to low frequencies is not going to work)
So I looked at a lot of papers today. There is a ton of work that has been done in the area of using the KK relations. Here is a summary of things to do / things that I am thinking about:
1. learn more optics
a. how is the reflection coefficient derived from known constants for a multi-layer (maybe this would help me understand where the zero reflectivity points are)
b. does the reflectivity coefficient have zeros in the complex frequency plane (how can I predict their position... eg. the dielectric function for a semi-infinite substrate has its only zeros on the imaginary axis)
c. qualitatively, what is the reflectivity of a multilayer for high frequency/low frequency (so I can extrapolate correctly)
d. what effect does polarization, angle of incidence have on the calculations that I am doing?
e. is there a relationship between the complex reflectivity coefficient and the complex dielectric function
2. Kramers-Kronig relationship
a. which representation should I be using
i. real and imaginary part of a boundary condition for an analytic function are related by the Hilbert transform
ii. the transformation can be done as two Fourier transforms
b. zeros of reflectivity (for complex frequency) give rise to singularities in the integral above and result in extra terms being needed in the transformation
c. how is this derived? how can one use causality to show that reflectivity as a function of a complex variable is analytic?
3. Using matlab to do the hilbert transform numerically (there are MANY papers on this subject)
a. use of gaussian quadrature method (write integral as INT w(x) p(x) where w(x) is the weight function log x^(-1)... this may require a different program as you need high precision arithmetic for large numbers of sample points
b. write the transform of a sequence of fourier transforms and approximate as a discrete fourier transform (cooley-turkey algorithm?)
c. look into algorithms that allow one to isolate the zeros of the reflectivity as a function of a complex variable (eg. find the blaschke factors that are needed to recover a function that has singularities when it is fourier transformed...one paper suggests using transmission data?)
d. look into methods that acceleration of convergence of the integral by knowing the phase (eg. multiply subtractive KK method)
Anyways, I feel like there is SO much that I can do at the moment.
Monday, June 6, 2011
matlab fatlab?
So I am at work again and I am going to write a few thoughts: Phase can be calculated from reflectivity using the the integral
phi(E)= -1/pi int 0 to infinity ln R(X/E) / (X^2 -1) dX (this is understood to be a principal part integral)
So the main issues for me are as follows:
1. We only know the function R for a limited range so we need to extrapolate it for other values of energy (frequency). For the moment, I am just assuming that R is constant outside that range. I do not know if this a reasonable assumption or how to go about saying what the reflectivity is for large frequencies.
For large frequency, I get the error as Log(R_infinity)/(2 Pi) Log | ( Emax-E)/(Emax+E) |
For small frequency, Log(R_0)/(2 Pi) Log | (E-Emin)/(E+Emin)|
R_infinity = R(E_max)
R_0=R(E_min)
2. I have to find an appropriate method of discretizing the integral that takes into account that we need to take a principal part integral.
2a. For the moment, I am using a right-side riemann sum. Perhaps I will implement simpson's rule. The disadvantage of the latter method is that it assumes equal spacing so I cannot see if the large spacing is the issue.
2b. Say we want to estimate the phase for a frequency omega. Then we need to know the reflectivity very precisely near omega. However, from what I hear from the lab, the reflectivity is measured every 0.2 eV, which I do not think is accurate enough.
Anyways, so I spent most of today trying to get the phase to predict more accurately. I had an idea that would let me integrate over the singularity. Basically, I just wanted to subtract the singularity out around a small symmetric neighborhood. This contributes nothing to the principal part integral but removes the singularity. This seemed fine but when I went to try and integrate this new function, I spent probably 3 hours trying to figure out how to get matlab to integrate this. This was pretty frustrating for me because I could have done this in 15 mins in mathematica but my lab does not use that program. I understand that this is just something that I will learn in time but I don't like having work go so slowly like this. I eventually asked one of the guys that I work with a few questions that I was able to figure it out. The main idea is that many functions need to be able to handle a vector input. I also didn't realize that when you write equations, you need to allow for vectors to work as well.
phi(E)= -1/pi int 0 to infinity ln R(X/E) / (X^2 -1) dX (this is understood to be a principal part integral)
So the main issues for me are as follows:
1. We only know the function R for a limited range so we need to extrapolate it for other values of energy (frequency). For the moment, I am just assuming that R is constant outside that range. I do not know if this a reasonable assumption or how to go about saying what the reflectivity is for large frequencies.
For large frequency, I get the error as Log(R_infinity)/(2 Pi) Log | ( Emax-E)/(Emax+E) |
For small frequency, Log(R_0)/(2 Pi) Log | (E-Emin)/(E+Emin)|
R_infinity = R(E_max)
R_0=R(E_min)
2. I have to find an appropriate method of discretizing the integral that takes into account that we need to take a principal part integral.
2a. For the moment, I am using a right-side riemann sum. Perhaps I will implement simpson's rule. The disadvantage of the latter method is that it assumes equal spacing so I cannot see if the large spacing is the issue.
2b. Say we want to estimate the phase for a frequency omega. Then we need to know the reflectivity very precisely near omega. However, from what I hear from the lab, the reflectivity is measured every 0.2 eV, which I do not think is accurate enough.
Anyways, so I spent most of today trying to get the phase to predict more accurately. I had an idea that would let me integrate over the singularity. Basically, I just wanted to subtract the singularity out around a small symmetric neighborhood. This contributes nothing to the principal part integral but removes the singularity. This seemed fine but when I went to try and integrate this new function, I spent probably 3 hours trying to figure out how to get matlab to integrate this. This was pretty frustrating for me because I could have done this in 15 mins in mathematica but my lab does not use that program. I understand that this is just something that I will learn in time but I don't like having work go so slowly like this. I eventually asked one of the guys that I work with a few questions that I was able to figure it out. The main idea is that many functions need to be able to handle a vector input. I also didn't realize that when you write equations, you need to allow for vectors to work as well.
Sunday, June 5, 2011
Visiting the Louvre and More Wanderings
The Louvre was AMAZING!! There is a seemingly endless collection of treasures. Further, the building that houses these masterpieces is equally as amazing. I wish that I know much more about the history. After going to the Centre Pompidou, it is so clear in my mind that modern art is nothing compared to the art here. There were so many sculptures and paintings that captured intensity and emotion. One of my favorite sculptures depicted a lion biting a man's leg. You could see the ferocity of the lion it his eyes and the fear in the man whose leg was getting devoured. Their entangled bodies captured the intensity of the fight. It is hard for me to see anything close to that in the art that I saw at the modern art museum.
It was impossible to appreciate everything at the Louvre. I hope to return sometime. Each tiny piece of art is the result of many hours or even years of careful craftsmanship. Further, there is a huge history behind all of the art. I was disappointed that all of the descriptions of the art were in French that was too complicated for me to understand.
This visit also makes me feel like it is somewhat pointless to visit a whole bunch of countries. There is so much to do in Paris. Seeing the details in the French culture is much more satisfying to me than rushing through a bunch of tourist attractions in other countries.
We decided to go to lunch outside the louvre to save some money. We ended up going to another street that was even move expensive. We looked at a BMW store and Louis Vuitton. The prices were so outrageous there. There was a not so fantastic looking handbag that was 25,000 euros!! I am really disgusted with anybody who buys things like that. There are so many other things that could use that kind of money. I tried to take pictures but they did not like that too much. After that, we came back to the Louvre but the line to get back in literally stretched a fourth of a mile. So we just wandered around for a bit.
It is hard to keep up with the blog because so much is going on. In other news, yesterday, I spent most of the day studying while the others went out. I felt pretty good about this because I got a fair amount done.
Also here are some pictures that I took at centre pompidou (modern art) and of cite universitaire (the place that I am living now).
Centre Pompidou and Cite Universitaire
http://www.facebook.com/media/set/?set=a.2159398021917.2101909.1155513641&l=c959a8756b
It was impossible to appreciate everything at the Louvre. I hope to return sometime. Each tiny piece of art is the result of many hours or even years of careful craftsmanship. Further, there is a huge history behind all of the art. I was disappointed that all of the descriptions of the art were in French that was too complicated for me to understand.
This visit also makes me feel like it is somewhat pointless to visit a whole bunch of countries. There is so much to do in Paris. Seeing the details in the French culture is much more satisfying to me than rushing through a bunch of tourist attractions in other countries.
Louvre (see picture descriptions for more detail about my day):
http://www.facebook.com/media/set/?set=a.2159452143270.2101910.1155513641&l=f431ca771c
We decided to go to lunch outside the louvre to save some money. We ended up going to another street that was even move expensive. We looked at a BMW store and Louis Vuitton. The prices were so outrageous there. There was a not so fantastic looking handbag that was 25,000 euros!! I am really disgusted with anybody who buys things like that. There are so many other things that could use that kind of money. I tried to take pictures but they did not like that too much. After that, we came back to the Louvre but the line to get back in literally stretched a fourth of a mile. So we just wandered around for a bit.
It is hard to keep up with the blog because so much is going on. In other news, yesterday, I spent most of the day studying while the others went out. I felt pretty good about this because I got a fair amount done.
Also here are some pictures that I took at centre pompidou (modern art) and of cite universitaire (the place that I am living now).
Centre Pompidou and Cite Universitaire
http://www.facebook.com/media/set/?set=a.2159398021917.2101909.1155513641&l=c959a8756b
Friday, June 3, 2011
Some feelings while in Paris
One thing that I wanted to blog a bit about was my mood. When we started the program, they showed us the happiness curve. It starts out erratic, then you go through a period of happiness due to being in new surroundings. Then there is a low when you start missing home and eventually, you get acclimated to the new country and it is similar to other places.
Anyways, I would say that I am in the erratic stage now. For the most part, I have been having a great time with seeing all that there is to see in France. I enjoy seeing the city but more importantly I have enjoyed interacting with French people. Too bad that I don't have time to do some type of French class. I feel like I am solidifying my French knowledge but I am not learning that many new things. But one thing that has been getting to me a bit is that I do feel like is that my social situation is unstable and that I have to put a lot of effort into that.
One example is that when we are in a group, I feel like I need to be loud otherwise my voice gets drowned out. In general, I am a person who likes to talk to people one on one. I feel like I have had good small group conversations with people but in the larger group I am sometimes quiet. This is not that different from other parts of my life but I do wish that I was more dynamic in a group setting at times. I remember one particular instance where some of us went out to rue de mouffetard and I was just quiet and I didn't feel like I had a good way to break into the conversation. For some reason, I was not in a very good mood.
One thing that is somewhat related is that I am not sleeping particularly well (I am averaging around 7 hours). I anticipated this earlier in the trip but now that it is happening, I am not sure that I am going to do much about it. When I am tired, I am less likely to speak up.
Looking at how I've been feeling, it is a bit strange that I have been so concerned with these small social issues. In the past, these things have not been on my mind. One aspect of it for me is that I feel like my relationships with the people on this trip are changing quickly (I know a few people that I have changed the way that I am acting around them a lot since the start of the trip). The main insecurity for me is that I don't want to feel like I don't have anybody to go around with.
In a related note, I have to be very careful with how I manage my time. I have been devoting a lot of time to social activities. I am taking the GRE general test on the 27th so I really do need to study for that more than I have been doing. I also feel that I haven't quite been doing as much as I could be on my research. One of the things that I have noticed is that it takes us a really long time to get dinner and I do not get that much done during the evenings. I think that until the 27th, I need to not be going out most of the time. I have some reservations about doing this because of the social issues that I mentioned above. I am glad that I chose that date though because I can then just be done with all of that and enjoy the rest of the summer.
In a bit unrelated note, I have noticed that I really do enjoy a bit of alone time. Tonight, I just sifted through my emails and watched a few starcraft videos. This was relaxing for me and I do like to have time with my thoughts. We are always running around and I think that it is great to have some personal relaxing time.
As always, there is way too much for me to talk about. Next time, I will talk more about my travels in the past few days and some things about research. Hopefully I get to do a practice GRE tomorrow morning...
Anyways, I would say that I am in the erratic stage now. For the most part, I have been having a great time with seeing all that there is to see in France. I enjoy seeing the city but more importantly I have enjoyed interacting with French people. Too bad that I don't have time to do some type of French class. I feel like I am solidifying my French knowledge but I am not learning that many new things. But one thing that has been getting to me a bit is that I do feel like is that my social situation is unstable and that I have to put a lot of effort into that.
One example is that when we are in a group, I feel like I need to be loud otherwise my voice gets drowned out. In general, I am a person who likes to talk to people one on one. I feel like I have had good small group conversations with people but in the larger group I am sometimes quiet. This is not that different from other parts of my life but I do wish that I was more dynamic in a group setting at times. I remember one particular instance where some of us went out to rue de mouffetard and I was just quiet and I didn't feel like I had a good way to break into the conversation. For some reason, I was not in a very good mood.
One thing that is somewhat related is that I am not sleeping particularly well (I am averaging around 7 hours). I anticipated this earlier in the trip but now that it is happening, I am not sure that I am going to do much about it. When I am tired, I am less likely to speak up.
Looking at how I've been feeling, it is a bit strange that I have been so concerned with these small social issues. In the past, these things have not been on my mind. One aspect of it for me is that I feel like my relationships with the people on this trip are changing quickly (I know a few people that I have changed the way that I am acting around them a lot since the start of the trip). The main insecurity for me is that I don't want to feel like I don't have anybody to go around with.
In a related note, I have to be very careful with how I manage my time. I have been devoting a lot of time to social activities. I am taking the GRE general test on the 27th so I really do need to study for that more than I have been doing. I also feel that I haven't quite been doing as much as I could be on my research. One of the things that I have noticed is that it takes us a really long time to get dinner and I do not get that much done during the evenings. I think that until the 27th, I need to not be going out most of the time. I have some reservations about doing this because of the social issues that I mentioned above. I am glad that I chose that date though because I can then just be done with all of that and enjoy the rest of the summer.
In a bit unrelated note, I have noticed that I really do enjoy a bit of alone time. Tonight, I just sifted through my emails and watched a few starcraft videos. This was relaxing for me and I do like to have time with my thoughts. We are always running around and I think that it is great to have some personal relaxing time.
As always, there is way too much for me to talk about. Next time, I will talk more about my travels in the past few days and some things about research. Hopefully I get to do a practice GRE tomorrow morning...
More Paris
So there are just way too many things to talk about so I will try to be brief. We moved into our new living area, "Cite universitaire". The goal of this place is to house international students studying in Paris. Most of the major countries have a house. We live in the German house. The room is somewhat bare bones. We have to walk down the hallway to go to the bathroom and shower. I have never lived in dorm like this but it will not be that big a deal. Honestly, I hope to spend as little time as possible in the room. The cite universitaire campus is really nice. I have taken a lot of pictures of the different buildings. There is also a lot of open space for relaxing or playing sports.
Yesterday, we went to centre pompidou. I am not a big fan of modern art but it was fun to see. One trend that seemed new to me is that the modern art pieces included sound as well as visuals. One of my favorite pieces involved a bunch of mirrors hanging down from the wall and the way that they were situated allowed for you to see multiple images of yourself. I enjoyed reading some of the descriptions of the art work. One of the funniest was something along the lines of: this artist's trademark is to paint alternating white and colored stripes that are approximately 8.7 cm. Then the piece of art was just a wall painted in stripes. But I suppose if this guy can tell a good enough story to sell his art, good for him.
Another cool thing that I did was that I played some frisbee with french people at Cite. They were very nice to me and allowed me to play. They didn't speak very much english but we got along fine. They invited me to play with them again (they play every thursday evening). In the near future, I hope to go around and talk to more people at Cite. The people there are especially friendly.
Another interesting thing is that Thursday was a holiday so most French people take Friday off as well. There are not that many people in the lab and most of the eateries in the area where I work were closed.
Anyways, onto research:
I feel a bit intimidated by my project because I feel that there has been a lot of work done on it already. However, I know that isn't the correct attitude so I will continue to read more papers. Even if I don't get any results, I think that I will learn how to make progress on a difficult problem. For now, I am playing around with matlab because I want to use some simulated data. I prefer using mathematica but knowing how to use matlab would be a good skill for me to have. I have spent most of the day wrestling with matlab. At the moment, my goal is to write matlab code that allows me to convert simulated reflectivity data into phase data. I can compare this phase data with simulated phase data. So far, my results are not that good but that is not much of a surprise.
Yesterday, we went to centre pompidou. I am not a big fan of modern art but it was fun to see. One trend that seemed new to me is that the modern art pieces included sound as well as visuals. One of my favorite pieces involved a bunch of mirrors hanging down from the wall and the way that they were situated allowed for you to see multiple images of yourself. I enjoyed reading some of the descriptions of the art work. One of the funniest was something along the lines of: this artist's trademark is to paint alternating white and colored stripes that are approximately 8.7 cm. Then the piece of art was just a wall painted in stripes. But I suppose if this guy can tell a good enough story to sell his art, good for him.
Another cool thing that I did was that I played some frisbee with french people at Cite. They were very nice to me and allowed me to play. They didn't speak very much english but we got along fine. They invited me to play with them again (they play every thursday evening). In the near future, I hope to go around and talk to more people at Cite. The people there are especially friendly.
Another interesting thing is that Thursday was a holiday so most French people take Friday off as well. There are not that many people in the lab and most of the eateries in the area where I work were closed.
Anyways, onto research:
I feel a bit intimidated by my project because I feel that there has been a lot of work done on it already. However, I know that isn't the correct attitude so I will continue to read more papers. Even if I don't get any results, I think that I will learn how to make progress on a difficult problem. For now, I am playing around with matlab because I want to use some simulated data. I prefer using mathematica but knowing how to use matlab would be a good skill for me to have. I have spent most of the day wrestling with matlab. At the moment, my goal is to write matlab code that allows me to convert simulated reflectivity data into phase data. I can compare this phase data with simulated phase data. So far, my results are not that good but that is not much of a surprise.
Wednesday, June 1, 2011
Another Day of Research
So I am really getting started on my research. I am pleased that I will be doing a challenging project. To review, there is a mathematical relationship between the reflectivity and the delay (phase delay) of a mirror. Strictly speaking, this relationship requires that we know the reflectivity for all frequencies in order to determine the phase for a particular frequency. As it is not practical to measure the reflectivity for all phases, I am looking to approximate that relation so that the reflectivity only needs to be measured over a band. Also our lab builds mirrors by making layers of different metals. The lab uses a matlab program to optimize the depths of the layers. I am interested in learning how this program works.
Things to do:
1. continue to try to understand papers given to me and look at other papers that I have found on this subject
2. understand the matlab code used to simulate the reflectivity (this is more of a personal goal)
2a. learn about the simulated annealing algorithm
3. use simulated reflectivity data to estimate phase
3a. understand better what typical reflectivity data looks like for multilayer mirrors and see where the major contributions come from in terms of the
Today, I continued to read over the papers that one of the professors gave me. Now I am going to be more specific. So given a mirror, we can define a complex number called reflectivity, r(omega). This is a function of frequency. The magnitude of this complex number is the amplitude of the reflected wave divided by the input amplitude. The phase of this complex number is equal to the phase delay of the pulse after it is reflected. So it turns out that defining a complex reflectivity in this way has a number of nice properties. This is somewhat obvious, but if we write the input pulse in complex number form, then to transform that pulse (in the frequency domain), we just multiply by the reflectivity. So if we have a pulse in the time domain, we just Fourier transform it, multiply by the reflectivity, and then inverse fourier transform.
The other property that is really nice is that r(omega) is an analytic function (we extend omega to allow for complex frequencies) except at isolated points for Im(omega)>0. I am not quite sure why this is so. It may be arguable due to "causality arguments" but I am not sure. Anyways, experimentally, it is relatively easy to calculate R(omega) := |r|^2. So we know the magnitude but not the phase. We do a bit of a trick, we note that f(w) = ln r(w) = (1/2) ln R(w)+i phi(w) where phi(w) denotes the phase as a function of frequency. Since r is an analytic function, so is its logarithm. Now we know that the real an imaginary parts of a complex analytic function are related. This relation involves an integral from w=0 to infinity.
The formula that lets us calculate the imaginary from the real part requires that the real part be well behaved. In general, this is not a valid assumption so we need to do some normalization procedures in order to make the integral converge. As far as I can see, doing this requires some general knowledge of the function. Ex. for a semi infinite medium, the reflectivity for large frequencies is a real constant. Here, the integral diverges as ln(constant not equal to 1) is nonzero. So we divide out this constant and use that normalized reflectance to calculate the phase. I will write more about this when I get to it.
Things to do:
1. continue to try to understand papers given to me and look at other papers that I have found on this subject
2. understand the matlab code used to simulate the reflectivity (this is more of a personal goal)
2a. learn about the simulated annealing algorithm
3. use simulated reflectivity data to estimate phase
3a. understand better what typical reflectivity data looks like for multilayer mirrors and see where the major contributions come from in terms of the
Today, I continued to read over the papers that one of the professors gave me. Now I am going to be more specific. So given a mirror, we can define a complex number called reflectivity, r(omega). This is a function of frequency. The magnitude of this complex number is the amplitude of the reflected wave divided by the input amplitude. The phase of this complex number is equal to the phase delay of the pulse after it is reflected. So it turns out that defining a complex reflectivity in this way has a number of nice properties. This is somewhat obvious, but if we write the input pulse in complex number form, then to transform that pulse (in the frequency domain), we just multiply by the reflectivity. So if we have a pulse in the time domain, we just Fourier transform it, multiply by the reflectivity, and then inverse fourier transform.
The other property that is really nice is that r(omega) is an analytic function (we extend omega to allow for complex frequencies) except at isolated points for Im(omega)>0. I am not quite sure why this is so. It may be arguable due to "causality arguments" but I am not sure. Anyways, experimentally, it is relatively easy to calculate R(omega) := |r|^2. So we know the magnitude but not the phase. We do a bit of a trick, we note that f(w) = ln r(w) = (1/2) ln R(w)+i phi(w) where phi(w) denotes the phase as a function of frequency. Since r is an analytic function, so is its logarithm. Now we know that the real an imaginary parts of a complex analytic function are related. This relation involves an integral from w=0 to infinity.
The formula that lets us calculate the imaginary from the real part requires that the real part be well behaved. In general, this is not a valid assumption so we need to do some normalization procedures in order to make the integral converge. As far as I can see, doing this requires some general knowledge of the function. Ex. for a semi infinite medium, the reflectivity for large frequencies is a real constant. Here, the integral diverges as ln(constant not equal to 1) is nonzero. So we divide out this constant and use that normalized reflectance to calculate the phase. I will write more about this when I get to it.
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