PHYSICAL OPTICS
Excel Documentation
Excel Tables
(Frefl1.xls)
By using the electromagnetic approach of propagation of light at an
interface, a relationship between the electric field and magnetic field
of the incident, reflected, and refracted light can be obtained. These
relationships are obtained by using the continuity of the wave at the boundary.
Interpretation of the wave in a linear, isotropic, and homogenous media
leads to the determination of amplitude, reflection, and transmission coefficents.
These coefficients are different for different polarization of the wave.
The spreadsheet program shows the variation of the coefficients at an air-glass
interface. Animation of the reflection coefficients for different refractive
indicies is used to show the variation of the Brewster angle.
1. Using a spreadsheet, show how the reflection and transmission coefficients
change when light passes form glass to air.
2. Using the spread sheet determine the Brewster angle when light passes
from glass to air.
Going Further:
3. Create an animation that shows the change in the reflection coefficients
due to different indicies of refraction.
|
Reflection and transmission coefficients at an interface |
|||||
|
|
|
|
|
|
|
|
|
|
Air-Glass Interface |
|
|
|
|
|
|
|
|
|
|
|
|
n1 = |
1 |
n2 = |
1.5 |
|
|
|
|
Pi = |
3.1415927 |
|
|
|
|
|
|
|
|
|
|
|
|
R (Per) |
T(Per) |
R(Par) |
T(Par) |
|
1 |
0.0174533 |
0.0400162 |
0.9599838 |
0.0399838 |
0.9600162 |
|
4 |
0.0698132 |
0.0402608 |
0.9597392 |
0.03974 |
0.96026 |
|
7 |
0.122173 |
0.0408044 |
0.9591956 |
0.039203 |
0.960797 |
|
10 |
0.1745329 |
0.0416595 |
0.9583405 |
0.0383715 |
0.9616285 |
|
13 |
0.2268928 |
0.0428462 |
0.9571538 |
0.0372437 |
0.9627563 |
|
16 |
0.2792527 |
0.0443927 |
0.9556073 |
0.035818 |
0.964182 |
|
19 |
0.3316126 |
0.0463366 |
0.9536634 |
0.034093 |
0.965907 |
|
22 |
0.3839724 |
0.0487264 |
0.9512736 |
0.0320691 |
0.9679309 |
|
25 |
0.4363323 |
0.0516235 |
0.9483765 |
0.0297493 |
0.9702507 |
|
28 |
0.4886922 |
0.0551046 |
0.9448954 |
0.0271412 |
0.9728588 |
|
31 |
0.5410521 |
0.0592653 |
0.9407347 |
0.0242594 |
0.9757406 |
|
34 |
0.5934119 |
0.0642241 |
0.9357759 |
0.0211292 |
0.9788708 |
|
37 |
0.6457718 |
0.0701276 |
0.9298724 |
0.0177915 |
0.9822085 |
|
40 |
0.6981317 |
0.0771577 |
0.9228423 |
0.0143095 |
0.9856905 |
|
43 |
0.7504916 |
0.0855395 |
0.9144605 |
0.0107792 |
0.9892208 |
|
46 |
0.8028515 |
0.0955518 |
0.9044482 |
0.0073425 |
0.9926575 |
|
49 |
0.8552113 |
0.1075411 |
0.8924589 |
0.004207 |
0.995793 |
|
52 |
0.9075712 |
0.1219371 |
0.8780629 |
0.0016741 |
0.9983259 |
|
55 |
0.9599311 |
0.1392735 |
0.8607265 |
0.0001778 |
0.9998222 |
|
58 |
1.012291 |
0.1602126 |
0.8397874 |
0.0003422 |
0.9996578 |
|
61 |
1.0646508 |
0.1855756 |
0.8144244 |
0.0030623 |
0.9969377 |
|
64 |
1.1170107 |
0.216378 |
0.783622 |
0.0096236 |
0.9903764 |
|
67 |
1.1693706 |
0.253872 |
0.746128 |
0.0218757 |
0.9781243 |
|
70 |
1.2217305 |
0.2995947 |
0.7004053 |
0.0424904 |
0.9575096 |
|
73 |
1.2740904 |
0.3554213 |
0.6445787 |
0.0753494 |
0.9246506 |
|
76 |
1.3264502 |
0.4236234 |
0.5763766 |
0.126135 |
0.873865 |
|
79 |
1.3788101 |
0.5069272 |
0.4930728 |
0.2032463 |
0.7967537 |
|
82 |
1.43117 |
0.6085697 |
0.3914303 |
0.3192525 |
0.6807475 |
|
85 |
1.4835299 |
0.7323455 |
0.2676545 |
0.4932538 |
0.5067462 |
|
88 |
1.5358897 |
0.8826381 |
0.1173619 |
0.754831 |
0.245169 |
|
89 |
1.553343 |
0.9394722 |
0.0605278 |
0.8688977 |
0.1311023 |
|
90 |
1.5707963 |
1 |
-7.338E-10 |
1 |
-1.651E-09 |
|
|
Identifying Brewster's angle |
|
|||
|
|
|
|
|
|
|
|
48 |
0.837758 |
0.1033004 |
0.8966996 |
0.0052024 |
0.9947976 |
|
49 |
0.8552113 |
0.1075411 |
0.8924589 |
0.004207 |
0.995793 |
|
50 |
0.8726646 |
0.1120484 |
0.8879516 |
0.0032775 |
0.9967225 |
|
51 |
0.8901179 |
0.1168405 |
0.8831595 |
0.0024279 |
0.9975721 |
|
52 |
0.9075712 |
0.1219371 |
0.8780629 |
0.0016741 |
0.9983259 |
|
53 |
0.9250245 |
0.1273594 |
0.8726406 |
0.0010339 |
0.9989661 |
|
54 |
0.9424778 |
0.13313 |
0.86687 |
0.0005276 |
0.9994724 |
|
55 |
0.9599311 |
0.1392735 |
0.8607265 |
0.0001778 |
0.9998222 |
|
56 |
0.9773844 |
0.1458159 |
0.8541841 |
1.044E-05 |
0.9999896 |
|
57 |
0.9948377 |
0.1527856 |
0.8472144 |
5.433E-05 |
0.9999457 |
|
58 |
1.012291 |
0.1602126 |
0.8397874 |
0.0003422 |
0.9996578 |
|
59 |
1.0297443 |
0.1681296 |
0.8318704 |
0.0009109 |
0.9990891 |
|
60 |
1.0471976 |
0.1765715 |
0.8234285 |
0.0018019 |
0.9981981 |
|
61 |
1.0646508 |
0.1855756 |
0.8144244 |
0.0030623 |
0.9969377 |
|
|
The cells below are for a slide show which demonstrates the effects of a changing index of refraction. |
|
|
|
||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
1.3 |
1.35 |
1.4 |
1.45 |
1.5 |
1.55 |
1.6 |
1.65 |
1.7 |
1.75 |
1.8 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
45 |
0.7853982 |
0.0020755 |
0.0032348 |
0.004688 |
0.0064341 |
0.0084665 |
0.0107746 |
0.0133454 |
0.0161642 |
0.0192155 |
0.0224832 |
0.0259515 |
|
46 |
0.8028515 |
0.0016305 |
0.0026349 |
0.0039212 |
0.0054917 |
0.0073425 |
0.0094654 |
0.011849 |
0.0144803 |
0.0173449 |
0.0204276 |
0.0237135 |
|
47 |
0.8203047 |
0.0012204 |
0.0020712 |
0.0031904 |
0.0045841 |
0.0062512 |
0.008186 |
0.010379 |
0.0128188 |
0.0154921 |
0.0183851 |
0.0214835 |
|
48 |
0.837758 |
0.0008531 |
0.001552 |
0.0025047 |
0.0037208 |
0.0052024 |
0.0069464 |
0.0089454 |
0.0111896 |
0.0136674 |
0.0163658 |
0.0192715 |
|
49 |
0.8552113 |
0.0005375 |
0.0010872 |
0.0018743 |
0.0029124 |
0.004207 |
0.0057578 |
0.0075596 |
0.0096044 |
0.011882 |
0.014381 |
0.0170887 |
|
50 |
0.8726646 |
0.0002837 |
0.0006877 |
0.0013108 |
0.002171 |
0.0032775 |
0.0046328 |
0.0062343 |
0.0080759 |
0.0101491 |
0.0124436 |
0.014948 |
|
51 |
0.8901179 |
0.0001035 |
0.000366 |
0.0008274 |
0.0015103 |
0.0024279 |
0.0035858 |
0.004984 |
0.0066187 |
0.008483 |
0.0105682 |
0.0128639 |
|
52 |
0.9075712 |
9.892E-06 |
0.0001363 |
0.000439 |
0.0009458 |
0.0016741 |
0.0026329 |
0.0038251 |
0.0052492 |
0.0069003 |
0.0087711 |
0.0108527 |
|
53 |
0.9250245 |
1.81E-05 |
1.47E-05 |
0.0001625 |
0.000495 |
0.0010339 |
0.0017922 |
0.0027759 |
0.003986 |
0.0054195 |
0.007071 |
0.0089328 |
|
54 |
0.9424778 |
0.0001452 |
1.948E-05 |
1.706E-05 |
0.0001775 |
0.0005276 |
0.0010843 |
0.0018572 |
0.0028498 |
0.0040615 |
0.0054885 |
0.007125 |
|
55 |
0.9599311 |
0.0004108 |
0.0001714 |
2.425E-05 |
1.562E-05 |
0.0001778 |
0.0005324 |
0.0010922 |
0.0018641 |
0.0028497 |
0.0040473 |
0.0054526 |
|
56 |
0.9773844 |
0.000837 |
0.0004939 |
0.0002086 |
3.462E-05 |
1.044E-05 |
0.0001624 |
0.0005074 |
0.0010553 |
0.0018106 |
0.0027738 |
0.0039421 |
|
57 |
0.9948377 |
0.0014492 |
0.0010137 |
0.0005977 |
0.0002629 |
5.433E-05 |
3.646E-06 |
0.0001322 |
0.0004532 |
0.0009742 |
0.0016977 |
0.0026231 |
|
58 |
1.012291 |
0.0022762 |
0.0017611 |
0.001223 |
0.0007326 |
0.0003422 |
8.933E-05 |
4.058E-09 |
9.137E-05 |
0.0003738 |
0.0008527 |
0.0015293 |
|
59 |
1.0297443 |
0.0033508 |
0.0027704 |
0.0021198 |
0.0014799 |
0.0009109 |
0.0004567 |
0.0001485 |
7.478E-06 |
4.751E-05 |
0.0002766 |
0.0006983 |
|
60 |
1.0471976 |
0.0047104 |
0.0040807 |
0.0033284 |
0.0025458 |
0.0018019 |
0.0011477 |
0.0006198 |
0.0002441 |
3.778E-05 |
1.197E-05 |
0.0001727 |
|
61 |
1.0646508 |
0.0063975 |
0.0057359 |
0.0048941 |
0.0039766 |
0.0030623 |
0.0022098 |
0.0014618 |
0.000849 |
0.0003927 |
0.000107 |
6.037E-07 |
|
62 |
1.0821041 |
0.0084607 |
0.0077865 |
0.0068683 |
0.0058246 |
0.004745 |
0.0036964 |
0.0027282 |
0.0018764 |
0.0011664 |
0.0006158 |
0.0002362 |
|
63 |
1.0995574 |
0.0109551 |
0.0102893 |
0.0093092 |
0.0081487 |
0.0069096 |
0.0056677 |
0.0044796 |
0.003387 |
0.0024199 |
0.0015995 |
0.0009405 |
|
64 |
1.1170107 |
0.0139439 |
0.013309 |
0.0122826 |
0.011016 |
0.0096236 |
0.0081917 |
0.0067844 |
0.0054496 |
0.0042221 |
0.0031272 |
0.0021827 |
|
65 |
1.134464 |
0.0174987 |
0.0169192 |
0.0158633 |
0.0145017 |
0.0129634 |
0.0113452 |
0.0097198 |
0.0081416 |
0.0066507 |
0.0052768 |
0.0040409 |
Mirrors can be one of the simplest optical tools that we use every day but they can also be the msot complicted tools. A common example is a plane mirror, where as the side mirrors in an automobile are more complex. These are mirrors used in the headlight of a car which are carefully designed. The shape of the mirror and its size dictate the specific application of the mirror. The maple/mathematica exercises allow the students to experience in a simple way one of the problems in the imaging property of spherical and parabolic mirrors. The abberation the students will learn about is spherical abberation.
Exercises:
1. How far away do two rays, one close to the optical axis (paraxial ray)
and the other, at quite a distance from the optic axis (marginal ray),
that are parallel to the optical axis, intersect the optical axis for a
spherical mirror.
2. What happens if the mirror is parabolic in shape?
|
Reflection from a spherical mirror |
|||||||
|---|---|---|---|---|---|---|---|
|
Radius of curvature 'R' |
|||||||
|
Distance from the optic axis 'y' |
|||||||
|
Angle of reflection is theta |
|||||||
|
x' indicates variation in focal length |
|||||||
|
|
|
|
|
|
|
|
|
|
y |
R |
DC |
R' |
q |
q1 |
Dq |
x' |
|
0 |
100 |
0 |
100 |
0 |
1.5707963 |
1.5707963 |
0 |
|
1.5 |
100 |
0.0112506 |
99.988749 |
0.0150023 |
1.5557941 |
1.5407918 |
49.977496 |
|
3 |
100 |
0.0450101 |
99.95499 |
0.030018 |
1.5407783 |
1.5107603 |
49.909939 |
|
4.5 |
100 |
0.1013013 |
99.898699 |
0.0450609 |
1.5257354 |
1.4806746 |
49.797192 |
|
6 |
100 |
0.1801623 |
99.819838 |
0.0601445 |
1.5106518 |
1.4505072 |
49.639023 |
|
7.5 |
100 |
0.2816466 |
99.718353 |
0.0752829 |
1.4955134 |
1.4202305 |
49.435109 |
|
9 |
100 |
0.4058235 |
99.594177 |
0.0904902 |
1.4803062 |
1.389816 |
49.185025 |
|
10.5 |
100 |
0.5527778 |
99.447222 |
0.1057808 |
1.4650155 |
1.3592347 |
48.888247 |
|
12 |
100 |
0.7226108 |
99.277389 |
0.1211697 |
1.4496266 |
1.3284569 |
48.544143 |
|
13.5 |
100 |
0.9154402 |
99.08456 |
0.1366724 |
1.434124 |
1.2974516 |
48.151966 |
|
15 |
100 |
1.1314003 |
98.8686 |
0.1523047 |
1.4184917 |
1.266187 |
47.710851 |
|
16.5 |
100 |
1.3706433 |
98.629357 |
0.1680833 |
1.402713 |
1.2346297 |
47.219802 |
|
18 |
100 |
1.633339 |
98.366661 |
0.1840258 |
1.3867706 |
1.2027448 |
46.677685 |
|
19.5 |
100 |
1.9196758 |
98.080324 |
0.2001503 |
1.370646 |
1.1704957 |
46.08321 |
|
21 |
100 |
2.2298614 |
97.770139 |
0.2164763 |
1.35432 |
1.1378437 |
45.434924 |
|
22.5 |
100 |
2.5641237 |
97.435876 |
0.2330243 |
1.3377721 |
1.1047478 |
44.731186 |
|
24 |
100 |
2.9227112 |
97.077289 |
0.249816 |
1.3209803 |
1.0711643 |
43.970153 |
|
25.5 |
100 |
3.3058947 |
96.694105 |
0.2668749 |
1.3039214 |
1.0370465 |
43.149749 |
|
27 |
100 |
3.7139678 |
96.286032 |
0.2842259 |
1.2865704 |
1.0023445 |
42.267644 |
|
28.5 |
100 |
4.1472483 |
95.852752 |
0.3018961 |
1.2689003 |
0.9670042 |
41.321216 |
|
30 |
100 |
4.6060799 |
95.39392 |
0.3199146 |
1.2508818 |
0.9309672 |
40.307507 |
|
31.5 |
100 |
5.0908329 |
94.909167 |
0.3383131 |
1.2324832 |
0.8941701 |
39.223182 |
|
33 |
100 |
5.6019068 |
94.398093 |
0.3571263 |
1.21367 |
0.8565436 |
38.064463 |
|
34.5 |
100 |
6.1397315 |
93.860268 |
0.3763922 |
1.1944041 |
0.8180118 |
36.827063 |
|
36 |
100 |
6.7047697 |
93.29523 |
0.3961526 |
1.1746437 |
0.778491 |
35.506091 |
|
37.5 |
100 |
7.2975189 |
92.702481 |
0.4164538 |
1.1543425 |
0.7378887 |
34.095952 |
|
39 |
100 |
7.9185143 |
92.081486 |
0.4373473 |
1.1334491 |
0.6961018 |
32.590207 |
|
40.5 |
100 |
8.5683315 |
91.431668 |
0.4588906 |
1.1119057 |
0.6530151 |
30.981405 |
|
42 |
100 |
9.2475896 |
90.75241 |
0.4811485 |
1.0896478 |
0.6084992 |
29.260868 |
|
43.5 |
100 |
9.9569547 |
90.043045 |
0.5041944 |
1.0666019 |
0.5624075 |
27.418413 |
|
45 |
100 |
10.697145 |
89.302855 |
0.5281118 |
1.0426846 |
0.5145728 |
25.441996 |
|
46.5 |
100 |
11.468932 |
88.531068 |
0.5529964 |
1.0178 |
0.4648036 |
23.317232 |
|
48 |
100 |
12.273151 |
87.726849 |
0.578959 |
0.9918373 |
0.4128783 |
21.02676 |
|
49.5 |
100 |
13.110703 |
86.889297 |
0.6061291 |
0.9646672 |
0.3585381 |
18.549369 |
|
51 |
100 |
13.98256 |
86.01744 |
0.6346591 |
0.9361372 |
0.3014782 |
15.858786 |
|
52.5 |
100 |
14.889777 |
85.110223 |
0.6647307 |
0.9060657 |
0.241335 |
12.921935 |
|
54 |
100 |
15.833498 |
84.166502 |
0.6965634 |
0.8742329 |
0.1776696 |
9.696399 |
|
55.5 |
100 |
16.814965 |
83.185035 |
0.7304264 |
0.8403699 |
0.1099435 |
6.126571 |
|
57 |
100 |
17.835531 |
82.164469 |
0.7666558 |
0.8041405 |
0.0374847 |
2.1376296 |
|
58.5 |
100 |
18.896671 |
81.103329 |
0.8056804 |
0.7651159 |
-0.0405645 |
-2.374326 |
|
60 |
100 |
20 |
80 |
0.8480621 |
0.7227342 |
-0.1253278 |
-7.5592895 |
Back
to Physical Optics