Differential Equations (Honors)

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MATH235 - Differential Equations with Honors - Spring2008

Course Information

Professor: Dr. Chip Durfee

Office: Meyer Hall 330

Meeting Times: MWF 9:00-9:50 AM

Location: Berthoud Hall 106 Go out west entrance of Meyer Hall, cross directly over the street to Berthoud. Room is on the first floor, halfway down on the left.

Announcements

9 December 2007

In problem 4a; Treat the gradient-index window as a phase mask rather than propagating the beam through as we did with the derivation of the Fresnel equations. You may also ignore any amplitude changes that may result from the passage of the beam through the gradient-index window.

In problem 4d, you may assume the aperture is one-dimensional. That is, you can ignore the dimension in and out of the page. In this case, you're diffraction integral is only across the same dimension in which the refractive index of the window varies.


8 December 2007

There was a problem with numbering - problem 4 was labeled as problem 3. There are 4 problems, and you need to do all 4. All homework solutions and the midterm 2 solutions are in the forum section.

6 December 2007

Take-home final is posted below in the homework area. Due next tuesday at 5pm.

3 December 2007

Comments/hints on HW11:

Problem 1, part b: For parameters, choose ω0 = 100,t0 = 1 and vary the delay τ so that the interference fringes are apparent.

Problem 3: So that all output angles would be visible, imagine the observation screen covers a semicircle around the grating. When I say that you can "just see" the +/- 2 orders, that means that those orders are going out at 90 degrees. (Actually the wavelengths within the visible range that are diffracted to the largest angle are visible for those orders.)

Problem 5: There is a typo in this problem. The finesse is (Δλ)fsr / (Δλ)res, where (Δλ)res = λ / Rres is the minimum resolvable wavelength (the Δλ in equation 11.92) and Rres is the resolving power defined in that same equation.

Problem 6: Here you are going to expand the transmission function for a Fabry-Perot around one of the maxima (where the sine function is near zero). You should find that you get a Lorentzian function, and you're looking for the width of that.

Problem 8: The main point/conclusion from the notes from monday was that a lens places at the focal plane the far-field (Fraunhofer) pattern that would normally be at infinity. So you use the far-field integrals with R substituted by the focal length f. That is, you're taking the Fourier transform of a Gaussian. At the end, evaluate the intensity and look for the 1 / e2 radius of that pattern.

Problem 9: The output of the waveguide is the standing wave pattern of the mode, truncated with a rect function to represent the edges of the waveguide. You then take the Fourier transform of this to get the far field.

1 December 2007

Midterm 2 takehomes are graded. Generally good: avg was 32.7/40 or about 80%. Grad avg was about 36/40 or 90%. I'll bring them to class Monday.

Look in "syllabus and reading list" section for fourier transform sheets. Old ones had been there all semester, but these are slightly tweaked to fix a couple of things.

25 November 2007

The final homework is posted below. It is somewhat longer than a single hw, less than 2. Due the last day of class.

19 November 2007

I have decided to give you a little extra time to turn in the take home. It is now due at 12noon on Tuesday 20 Nov.

Link to polarization control lecture 10 fixed. A couple of clarifications (also made in the document of the MT):

Problem 1d: you may treat the gas as ideal, and at room temperature as the pressure is raised. Problem 3a: In this problem, make the assumption that the path of the electron is unaffected by the energy lost to radiation during the first several cycles of motion in the potential.

17 November 2007

Problem 1: clarification. The molecules have one long axis, and two, equal length short axes. Picture them as cigar-shaped ellipsoids. When aligned (as they are throughout this problem), they are symmetric around the x-axis. In part (c), ny − 1 should be 20% less than nx − 1. Here, you may also assume beta is much smaller than omega or the resonance omegas.

Problem 3: The electron starts off at the origin.

16 November 2007 Solutions to most all problems for HW6-9 are posted. Lecture notes are all up to date through dispersion.

15 November 2007 Midterm II take home is posted below in the Homework area. Due by 5 pm in my office next monday. Open book and notes, but no communication with other people. In the (hopefully) unlikely event that clarifications need to be made, they will be posted here.

13 November 2007

HW 10: for the first problem, you can use the dipole moment of water which is p0 = 6.3 10^-30 coulombs-meters

11 November 2007

HW#7 (50 points) has been graded. Undergrad. Avg. = 83.85%. Grad. Avg. = 95.00%.
HW#8 (30 points) has been graded. Undergrad. Avg. = 78.24%. Grad. Avg. = 97.50%. As a matter of principle, one should work with <P> (average power) in problem 4, i.e., one should maximize/minimize <P> and NOT P.
--AJY

8 November 2007 HW 10 posted below.

6 November 2007 Homework 9: I've posted a short section below on the Maxwell-Boltzmann distribution function that should help you with the Doppler broadening problem. For that problem, you may consider the natural linewidth to be very narrow, so that if the input frequency matches up with the Doppler-shifted frequency, it gets absorbed. Thus there is a direct mapping of the velocity to the absorbed frequency. I am interested in the lineshape only, not the normalization factors.

2 November 2007 Midterms are graded and in the envelope outside my office door. Thank you for your patience.

Avg: undergrad 71% (39/55), grad 77% (42.6/55)

31 October 2007 HW 9 posted below: due 1 week from today, 7 Nov.

28 October 2007 Grad HW posted below.

22 October 2007 HW 8 posted below. Due by 5pm next monday (29 Oct).

HW#6 (50 points) has been graded. Undergrad. Avg. = 76.78%. Grad. Avg. = 92.50%.
--AJY

13 October 2007

HW7 is posted below.

8 October 2007

Review session: Tuesday 11am in MH220. Although I used the smartboard during the review, I didn't get it saved on the right computer, so I can't post the notes. To summarize: using the stress-tensor and the Einstein index notation for vector ID's won't be on the midterm. We discussed calculating intensity from the fields and worked with total internal reflection in the case where the index is gradually decreasing. This latter problem is on one of the posted midterms, I believe.

As I said in class, you can bring in a single-sided sheet with equations if you like. Calculator shouldn't be necessary.


HW#5 (45 points) has been graded. Undergrad. Avg. = 92.89%. Grad. Avg. = 98.33%. You can pick these up outside MH330.
--AJY


6 October 2007

Some grad HW problems posted below.

5 October 2007

In the forum section, I've posted some old midterms with solutions to help you study for next wednesday's test. The approach and pace in this course has varied over the years, so you may see problems on material we haven't done yet. HW 7 will be due two weeks from today. I'll try to get at least part of it posted by tomorrow morning for those of you who want to get started early.

4 October 2007

I've moved all scans from book sections over to the Forum section where they are supposed to be. For 462, this is the reading on plasmas and polarization, for 507, reading on multipoles and multilayer films. Lecture notes for 507 on waves in magnetized plasmas and matrix techniques for multilayer films are posted below.

3 October 2007

graded HW 3, 4 are outside my office.

HW5 solutions are up. Still waiting on the grader to bring in the graded HW 3 and 4.

Solutions for HW 1 through 4 are posted in the forum section. The math files are in 5.2 format, but I don't think much change will be required to read in 6.0 except to delete semicolons after plot commands. Later today HW1 solutions will updated to include problems 1 and 2 after I scan them. There are still several of you that need to register for the Forum (or maybe have user names I can't recognize). Thanks to Eliot Grafil and Kenton Larson for working to get the Forum up and running, and thanks to you for your patience.

HW#4 has been graded. Undergrad. Avg. = 76.42%. Grad. Avg. = 82.33%
HW#3 has been graded. Undergrad. Avg. = 79.49%. Grad. Avg. = 87.86%

--AJY

2 October 2007

I've gone through the list of people already registered for the forum, and added to the member list those names I could recognize. Missing: Brewer, Buehrle, Coulson, Justis, Meehan, Mogge, Parmentier, Patton, Przekwas, Reitz, Rister, Schubert, Studer, Welch, Strong, Wall. If you're on this list, email me with your user name after you are registered.

1 October 2007

Forum for the course is partially up. See the link and instructions below. You must register for the Forum and email me with your user name. I'm still having trouble posting, but I'm getting closer.

29 Sept 2007

HW 6 posted below.

I'm targeting the first midterm to be during the week of 8-12 Oct. Let me know if there are dates to avoid.

27 Sept 2007 HW5 tips: (you can turn this in at the end of the day (5pm) in my box.

Problem 1: look in class notes for information about Fermat's principle. You may assume the light travels in a straight line to a point on the interface, then from that point to the final point.

Problem 2: the Fresnel equations I use are most easily related to the first expressions in 6.26 and 6.27 (and 6.32 and 6.33). The expressions that don't explicitly have the refractive index are tricky to use in the limit of zero angle of incidence. Both numerator and denominator go to zero, and you have to take the limit.

Problem 3: brush up on your geometry to relate the two internal angles. Be sure that you are showing that the deviation angle is minimized when the internal angle is parallel to the base. You may show this graphically/numerically if you must. Analytically, try representing the deviation angle in terms of one of the internal angles.

Problem 4: first connect the rhomb angle shown to the angle of incidence on the first internal surface (more geometry). Then use the notebook to evaluate the net phase difference. You will have the most luck if you take the ratio of the two output fields to get the phase difference. Otherwise, you can have some discontinuities in the phase function (that don't really mean anything).

24 Sept 2007 Still working on getting the Forum set up for posting solutions. I appreciate your patience. (CGD)

HW#2 has been graded. Undergrad. Avg. = 82.56%. Grad. Avg. = 92.14%

--AJY

These are posted outside my office (CGD)

21 Sept 2007

HW5 posted, second HW posted for 507 students, 462 lecture notes are up to date.

20 Sept 2007 HW#1 has been graded. Undergrad. Avg. = 60.37%. Grad. Avg. = 81.36%
--AJY


19 Sept 2007

Problem 2: The integral to determine the force is best done numerically. You are trying to figure the sign of the force, so you may set I0=1, c=1, radius a=1, n=1.5 and alpha=0.1. I would suggest making a plot of the integrand from -a to a to help you see the contribution of the different ray heights. This way you can see how it all cancels when the intensity profile is uniform.

Problem 4 (HM6.3): the condition for zero reflection should relate the vacuum wavelength of the incoming light to the index and thickness of the thin layer.

14 Sept 2007

HW 4 is posted below.

13 Sept 2007 Notes for all lectures to date are scanned in. The treatment of polarization I did in class is slightly different from the notes.

HW3: I made a figure for problem 2 that may help you visualize the coordinates. Reading my lecture notes will probably help you with some of the terminology.

31 Aug 2007

HW 2 is posted below. Hopefully you will find this more straightforward.

29 Aug 2007

> Notes for the early lectures are posted below. Remember that there are often more complete derivations in there, so you might consider them as an additional reading. resource.

>HW1 HM problem 4.12 In this problem, solve for Q(t) first. The Maxwell equations you are using are given in equations 4.21-4.24. The second set on that page are for microscopic EM, where there are no bound charges or dielectric media. So in our context, there is no free current, only the displacement current.

24 Aug 2007:

> I've posted a couple of links below to give some more documentation on vector ID's. If any of you find anything useful, please post it. Many pages I found were more specific to the use of the notation in relativity, which is more involved than what we are interested in at this time.

> I will be at the SPIE conference in San Diego to give a talk. I'll be leaving after class on Monday, returning Tuesday night. No change in class schedule, but I'll move tuesday's office hour. Next week: office hours W1-3, Th1-4

> During the week of 3-7 Sept, I'm in Santa Fe to give a talk and poster. Dr Kowalski will be teaching in my place.

> I've posted HW1 below. Also look at the pdf from an intro plasma physics book by Chen. Later, I'll post copies like that in the forum section. --Ayuffa 06:21, 24 September 2007 (MDT)--Ayuffa 06:21, 24 September 2007 (MDT)

Office hours

Times - Tuesday 3-4p, Wednesday 2-3, Thursday 1-4p

Let me know if you can't make these - we can make an appointment.

Course Forum: Supplementary Readings, Homework Solutions

open to class members only:

If you are not already registered for the Forum, go through the registration steps (this is separate from logging into the Wiki).

In any case, email me to let me know what your user name is.

Forum for 462/507 http://ticc.mines.edu/csm/forum/index.php?f=83

You may want to open this in a new tab or browser window to maintain access to the wiki page.

Course Material

Syllabus and Reading List

Pdf.png These downloads require Adobe Acrobat Reader
Syllabus for PHGN-507
Syllabus for PHGN-462
List of topics we covered for Midterm 1
Optics FAQ: summary of basic concepts from this course
Fourier transform sheet (time and frequency variables)
Fourier transform sheet (space and spatial variables)

The material below is from 2006.

Pdf.png These downloads require Adobe Acrobat Reader
equation sheet used in 2006 for exams

Homework Assignments: all students

Pdf.png These downloads require Adobe Acrobat Reader
Homework 1: due in class 31 Aug
Homework 2: due in class 7 Sept
Homework 3: due in class 14 Sept. Revised 13 Sept to add a figure for problem 2
Homework 4: due in class 21 Sept.
Homework 5: due 28 Sept.
Fresnel equations.nb notebook (came from version 5.2)
Homework 6: due 5 Oct.
Homework 7: due 19 Oct.
Homework 8: due 29 Oct.
Homework 9: due 7 Nov.
a short summary of the Maxwell-Boltzmann distribution
Homework 10: due 14 Nov.
Midterm II take-home: due 10 Nov.
Homework 11 (last homework): due 5 Dec.
Final take-home: due Tuesday 11 Dec 5pm.


Homework Assignments: 507

14 Sept 2007: For the 507 group: do problems 3.4, 3.5 and 3.6 in the handout below. You are welcome to use Mathematica where it helps. You can turn this in along with HW 4 on 21 Sept.

Pdf.png These downloads require Adobe Acrobat Reader
hw 2
hw 3
hw 4

Lecture Notes

Pdf.png These downloads require Adobe Acrobat Reader
lec1
lec2
lec3
lec4
lec5
lec6 (EM Plane Waves)
lec7 (Vector Properties of EM Waves)
lec8 (Polarization States)
lec9 (Intensity, Momentum of Waves)
lec10 (polarization control)
lec11 (waves in conductors)
lec12 (skin effect, reflection normal incidence)
lec13 (refraction, Fermat's principle, Fresnel equations)
lec14 (TIR and reflection from metals)
lec15 (guided waves: scalar view)
lec16 metal waveguides
lec17 dielectric waveguides, includes a section on cylindrical geometry
retarded potentials and fields from rho and J
Lienard-weichert potentials, field of a moving charge
Radiation from accelerated charges
Antennas: intro, radiation damping
Antenna arrays
Intro to scattering and dispersion
Plasma refractive index
refractive index of gases and solids
fourier propagation of non-monochromatic light
group vel and pulse spreading
convolution and linear systems
michelson interferometer
coherence - diffraction grating
double beam interference
fabry perot
diffraction essentials
FT with lens-spatial frequencies
circ apertures-Gaussian beams

Lecture Notes for 507 Only

Pdf.png These downloads require Adobe Acrobat Reader
waves in magnetized plasmas
multilayer films

Mathematica Demos

These aren't actually pdf's. Do a "save link as" to save these to your computer, then open with Mathematica.

Pdf.png These downloads require Adobe Acrobat Reader
Stress tensor setup in Mathematica shown in class

Course Links

Students: when you post links, mark your name and the date of the post

Einstein summation convention and vector ID's: Brief discussion of use of Einstein notation to prove vector ID's

A more general discussion of Vector calculus, some reference to the summation convention

Heald-Marion text: Errata for the Heald/Marion textbook

Online Demo's JavaOptics: a nice collection of optics-related demonstrations

Lectures from GaTech (Rick Trebino): a very useful collection of graphics and worked examples in optics

Fabry-Perot demonstration

diffraction by slits

propagating waves, including waveguides (demo in class)

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