Homework Statement
I need to solve this integral for a>0:
\int _0^{\infty }\frac{\text{Sin}[x]}{x}\frac{1}{x^2+a^2}dx
The Attempt at a Solution
Using wolfram mathematica, I get that this integral is:
\frac{\pi -e^{-x} \pi }{2x^2}=\frac{\pi (1-\text{Cosh}[a]+\text{Sinh}[a])}{2...
Homework Statement
I am attempting to manipulate this equation into a form that, presumably, has 3 Sinc terms. I am attempting to do this because my professor has written "reduce the solution to terms having sinc functions", and I am assuming there are 3, because plotting this equation in...
Here's a quote from my notes:
"For a given 'Illumination Aperture Stop', the degree of coherence depends on the size of the Pupil Plane Aperture (i.e. resolution)."
and
COHERENT ILLUMINATION (in context of a microscope with Koehler Illumination)
Remember the size of the Pupil determines the...
Yes Andy, I suppose it does. It is essentially a confirmation of my intuition, but I am still uncertain about what my professors mean when they ask me about the coherence of a system. Perhaps this is a question for them.
So, what role does my imaging system play in the coherence of my image, in...
This is a very vague question, but I hope by asking it, we can pin down an answer eventually.
In optics, there are such things as "Coherent Imaging" and "Incoherent Imaging." From what I understand, the degree of coherence has to do with the size of your Point Spread Function (PSF), a.k.a...
if I am understanding the problem correctly, which is not an assumption I would bet any substantial amount of money on, I thought that the point was to show that f_m is the best substitution for \gamma_m, e.g. show that f_m=\gamma_m reduces E better than any other substitution for \gamma_m...
Homework Statement
So I'm supposed to show that a finite Fourier approximation is the optimal approximation for a given function.
I am to suppose we have a given set of functions \phi _k(x),k=1,2,\text{...}N defined on a\leq x\leq b.
They are orthogonal \int _a^b\phi _m(x)\phi _n(x)dx=0...
wow...never even crossed my mind to use trig substitution. In fact, I completely forgot that as a method. And its so useful! Thank you! I guess I ought to review calc 2 stuff from HS...
Thanks again!
Homework Statement
\int \frac{1}{\left(x^2+z^2\right)^{3/2}} \, dx
I have been trying various u-substitutions for about 2 hours now, but I cannot seem to find a way to solve this by hand! I used mathematica to solve the problem. I feel like it will be fairly straightforward once I figure...
This is simply in the context of the Geometrical Optics course I'm taking. I'm just getting into optics in school, and I am just trying to gain some insight into this method for evaluating optical systems.
I use Excel to automate the process. The 20 lens system was just hypothetical.
Moving...
I have heard those terms briefly. My professor at the institute of optics at the university of rochester probably just wanted to shorten the terms, so he calls them a and b rays.
Anyways, I'm just trying to get a good grasp of how to evaluate a given optical system using those rays, why you...
An a-ray is just an axial ray that goes through the aperture stop. Generally, you find the angle in the object plane that corresponds to when the a-ray just kisses the edge of the aperture stop, AKA the Y height at that point is the radius of the aperture stop.
A b-ray is a ray that goes...
Is anyone familiar with the "y n u" ray trace method for optical systems, using paraxial rays where Snell's law is approximated using:
n1sinx1 ≈ n1tanx1
≈ n1u1 = n2u2
If anyone is familiar with this, I'd be interested to discuss the finer points of "a-ray" and "b-ray" tracing, as well...
I had the same thought. Should it be defined to mean the number of "n" terms? Maybe a large N? Or are you saying there is something more fundamentally wrong with it. Thanks!