Fraunhoffer Diffraction of a Circular Aperture

Matthew Kenworthy, November 2006

Using the formulae from Born and Wolf, I have generated the first 100 Airy ring radii and dark ring radii for Fraunhoffer diffraction of a circular aperture. The spacing between adjacent bright rings tends towards pi for large ring number, and this applies to dark ring spacing too.

Also included is the energy enclosed within a given dark ring, the energy within a given bright ring, and the energy density within a given Airy ring.

KEY:

Ring # r h Mag(h)E(ring)EdEnclEResidE
001 dark 1.220π= 3.832 ------------ 0.8377849 1.622e-01
001 1.635π= 5.136 1.750e-02 4.3937.215e-023.661e-02------
002 dark 2.233π= 7.016 ------------ 0.9099305 9.007e-02
002 2.679π= 8.417 4.158e-03 5.9532.772e-028.950e-03------
003 dark 3.238π= 10.173 ------------ 0.9376475 6.235e-02
003 3.699π= 11.620 1.601e-03 6.9891.467e-023.474e-03------
004 dark 4.241π= 13.324 ------------ 0.9523192 4.768e-02
004 4.710π= 14.796 7.794e-04 7.7719.082e-031.698e-03------
005 dark 5.243π= 16.471 ------------ 0.9614014 3.860e-02
005 5.717π= 17.960 4.370e-04 8.3996.176e-039.537e-04------
006 dark 6.244π= 19.616 ------------ 0.9675772 3.242e-02
006 6.722π= 21.117 2.693e-04 8.9244.472e-035.883e-04------
007 dark 7.245π= 22.760 ------------ 0.9720493 2.795e-02
007 7.725π= 24.270 1.776e-04 9.3773.388e-033.882e-04------
008 dark 8.245π= 25.904 ------------ 0.9754373 2.456e-02
008 8.728π= 27.421 1.232e-04 9.7732.655e-032.695e-04------
009 dark 9.246π= 29.047 ------------ 0.9780927 2.191e-02
009 9.730π= 30.569 8.896e-05 10.1272.137e-031.946e-04------
010 dark 10.246π= 32.190 ------------ 0.9802300 1.977e-02
010 10.732π= 33.717 6.633e-05 10.4461.757e-031.452e-04------
011 dark 11.247π= 35.332 ------------ 0.9819873 1.801e-02
011 11.734π= 36.863 5.077e-05 10.7361.470e-031.111e-04------
012 dark 12.247π= 38.475 ------------ 0.9834578 1.654e-02
012 12.735π= 40.008 3.972e-05 11.0031.248e-038.695e-05------
013 dark 13.247π= 41.617 ------------ 0.9847062 1.529e-02
013 13.736π= 43.153 3.166e-05 11.2491.073e-036.931e-05------
014 dark 14.247π= 44.759 ------------ 0.9857795 1.422e-02
014 14.737π= 46.298 2.564e-05 11.4789.325e-045.614e-05------
015 dark 15.248π= 47.901 ------------ 0.9867120 1.329e-02
015 15.738π= 49.442 2.105e-05 11.6928.177e-044.610e-05------
016 dark 16.248π= 51.044 ------------ 0.9875297 1.247e-02
016 16.739π= 52.586 1.750e-05 11.8927.229e-043.832e-05------
017 dark 17.248π= 54.186 ------------ 0.9882526 1.175e-02
017 17.739π= 55.730 1.470e-05 12.0816.437e-043.220e-05------
018 dark 18.248π= 57.328 ------------ 0.9888963 1.110e-02
018 18.740π= 58.873 1.247e-05 12.2605.768e-042.732e-05------
019 dark 19.248π= 60.469 ------------ 0.9894731 1.053e-02
019 19.740π= 62.016 1.067e-05 12.4295.198e-042.337e-05------
020 dark 20.248π= 63.611 ------------ 0.9899930 1.001e-02
020 20.741π= 65.159 9.201e-06 12.5904.709e-042.015e-05------
021 dark 21.248π= 66.753 ------------ 0.9904639 9.536e-03
021 21.741π= 68.302 7.988e-06 12.7444.286e-041.750e-05------
022 dark 22.248π= 69.895 ------------ 0.9908925 9.108e-03
022 22.742π= 71.445 6.980e-06 12.8903.917e-041.529e-05------
023 dark 23.248π= 73.037 ------------ 0.9912842 8.716e-03
023 23.742π= 74.588 6.135e-06 13.0313.594e-041.344e-05------
024 dark 24.248π= 76.179 ------------ 0.9916436 8.356e-03
024 24.742π= 77.730 5.420e-06 13.1653.309e-041.187e-05------
025 dark 25.249π= 79.320 ------------ 0.9919746 8.025e-03
025 25.743π= 80.873 4.813e-06 13.2943.057e-041.054e-05------
026 dark 26.249π= 82.462 ------------ 0.9922803 7.720e-03
026 26.743π= 84.015 4.293e-06 13.4182.833e-049.404e-06------
027 dark 27.249π= 85.604 ------------ 0.9925636 7.436e-03
027 27.743π= 87.158 3.845e-06 13.5382.632e-048.423e-06------
028 dark 28.249π= 88.746 ------------ 0.9928268 7.173e-03
028 28.743π= 90.300 3.458e-06 13.6532.452e-047.574e-06------
029 dark 29.249π= 91.888 ------------ 0.9930721 6.928e-03
029 29.744π= 93.442 3.120e-06 13.7642.290e-046.836e-06------
030 dark 30.249π= 95.029 ------------ 0.9933011 6.699e-03
030 30.744π= 96.585 2.826e-06 13.8722.144e-046.190e-06------
031 dark 31.249π= 98.171 ------------ 0.9935154 6.485e-03
031 31.744π= 99.727 2.567e-06 13.9762.011e-045.623e-06------
032 dark 32.249π=101.313 ------------ 0.9937165 6.283e-03
032 32.744π=102.869 2.339e-06 14.0771.890e-045.124e-06------
033 dark 33.249π=104.454 ------------ 0.9939055 6.095e-03
033 33.744π=106.011 2.137e-06 14.1751.779e-044.682e-06------
034 dark 34.249π=107.596 ------------ 0.9940834 5.917e-03
034 34.745π=109.153 1.958e-06 14.2711.678e-044.289e-06------
035 dark 35.249π=110.738 ------------ 0.9942513 5.749e-03
035 35.745π=112.295 1.798e-06 14.3631.586e-043.939e-06------
036 dark 36.249π=113.879 ------------ 0.9944099 5.590e-03
036 36.745π=115.437 1.655e-06 14.4531.501e-043.626e-06------
037 dark 37.249π=117.021 ------------ 0.9945599 5.440e-03
037 37.745π=118.579 1.527e-06 14.5401.422e-043.345e-06------
038 dark 38.249π=120.163 ------------ 0.9947022 5.298e-03
038 38.745π=121.721 1.412e-06 14.6261.350e-043.093e-06------
039 dark 39.249π=123.304 ------------ 0.9948371 5.163e-03
039 39.745π=124.863 1.308e-06 14.7091.283e-042.865e-06------
040 dark 40.249π=126.446 ------------ 0.9949654 5.035e-03
040 40.745π=128.005 1.214e-06 14.7891.221e-042.660e-06------
041 dark 41.249π=129.588 ------------ 0.9950875 4.913e-03
041 41.745π=131.147 1.129e-06 14.8681.163e-042.473e-06------
042 dark 42.249π=132.729 ------------ 0.9952037 4.796e-03
042 42.746π=134.289 1.051e-06 14.9461.109e-042.303e-06------
043 dark 43.249π=135.871 ------------ 0.9953146 4.685e-03
043 43.746π=137.431 9.809e-07 15.0211.059e-042.149e-06------
044 dark 44.249π=139.013 ------------ 0.9954205 4.579e-03
044 44.746π=140.573 9.166e-07 15.0951.012e-042.008e-06------
045 dark 45.249π=142.154 ------------ 0.9955217 4.478e-03
045 45.746π=143.715 8.578e-07 15.1679.683e-051.879e-06------
046 dark 46.249π=145.296 ------------ 0.9956185 4.381e-03
046 46.746π=146.857 8.039e-07 15.2379.273e-051.761e-06------
047 dark 47.249π=148.438 ------------ 0.9957113 4.289e-03
047 47.746π=149.999 7.545e-07 15.3068.889e-051.653e-06------
048 dark 48.249π=151.579 ------------ 0.9958002 4.200e-03
048 48.746π=153.140 7.090e-07 15.3738.528e-051.553e-06------
049 dark 49.249π=154.721 ------------ 0.9958854 4.115e-03
049 49.746π=156.282 6.671e-07 15.4408.188e-051.461e-06------
050 dark 50.249π=157.863 ------------ 0.9959673 4.033e-03
050 50.746π=159.424 6.284e-07 15.5047.869e-051.377e-06------
051 dark 51.249π=161.004 ------------ 0.9960460 3.954e-03
051 51.746π=162.566 5.927e-07 15.5687.567e-051.298e-06------
052 dark 52.249π=164.146 ------------ 0.9961217 3.878e-03
052 52.746π=165.708 5.596e-07 15.6307.283e-051.226e-06------
053 dark 53.249π=167.288 ------------ 0.9961945 3.805e-03
053 53.747π=168.850 5.289e-07 15.6917.015e-051.159e-06------
054 dark 54.249π=170.429 ------------ 0.9962647 3.735e-03
054 54.747π=171.991 5.005e-07 15.7526.761e-051.096e-06------
055 dark 55.249π=173.571 ------------ 0.9963323 3.668e-03
055 55.747π=175.133 4.740e-07 15.8106.520e-051.039e-06------
056 dark 56.249π=176.712 ------------ 0.9963975 3.603e-03
056 56.747π=178.275 4.494e-07 15.8686.293e-059.846e-07------
057 dark 57.249π=179.854 ------------ 0.9964604 3.540e-03
057 57.747π=181.417 4.265e-07 15.9256.077e-059.343e-07------
058 dark 58.249π=182.996 ------------ 0.9965212 3.479e-03
058 58.747π=184.558 4.051e-07 15.9815.871e-058.874e-07------
059 dark 59.249π=186.137 ------------ 0.9965799 3.420e-03
059 59.747π=187.700 3.851e-07 16.0365.677e-058.436e-07------
060 dark 60.249π=189.279 ------------ 0.9966366 3.363e-03
060 60.747π=190.842 3.664e-07 16.0905.491e-058.026e-07------
061 dark 61.249π=192.421 ------------ 0.9966916 3.308e-03
061 61.747π=193.984 3.488e-07 16.1435.315e-057.643e-07------
062 dark 62.249π=195.562 ------------ 0.9967447 3.255e-03
062 62.747π=197.125 3.324e-07 16.1965.147e-057.283e-07------
063 dark 63.249π=198.704 ------------ 0.9967962 3.204e-03
063 63.747π=200.267 3.170e-07 16.2474.987e-056.946e-07------
064 dark 64.249π=201.845 ------------ 0.9968460 3.154e-03
064 64.747π=203.409 3.026e-07 16.2984.834e-056.629e-07------
065 dark 65.249π=204.987 ------------ 0.9968944 3.106e-03
065 65.747π=206.551 2.890e-07 16.3484.688e-056.331e-07------
066 dark 66.249π=208.129 ------------ 0.9969412 3.059e-03
066 66.747π=209.692 2.762e-07 16.3974.548e-056.050e-07------
067 dark 67.249π=211.270 ------------ 0.9969867 3.013e-03
067 67.747π=212.834 2.641e-07 16.4464.415e-055.786e-07------
068 dark 68.250π=214.412 ------------ 0.9970309 2.969e-03
068 68.747π=215.976 2.528e-07 16.4934.288e-055.538e-07------
069 dark 69.250π=217.554 ------------ 0.9970738 2.926e-03
069 69.747π=219.118 2.420e-07 16.5404.165e-055.303e-07------
070 dark 70.250π=220.695 ------------ 0.9971154 2.885e-03
070 70.747π=222.259 2.319e-07 16.5874.049e-055.081e-07------
071 dark 71.250π=223.837 ------------ 0.9971559 2.844e-03
071 71.747π=225.401 2.224e-07 16.6323.936e-054.872e-07------
072 dark 72.250π=226.978 ------------ 0.9971953 2.805e-03
072 72.747π=228.543 2.133e-07 16.6773.829e-054.673e-07------
073 dark 73.250π=230.120 ------------ 0.9972336 2.766e-03
073 73.747π=231.684 2.048e-07 16.7223.726e-054.486e-07------
074 dark 74.250π=233.262 ------------ 0.9972708 2.729e-03
074 74.748π=234.826 1.966e-07 16.7663.627e-054.308e-07------
075 dark 75.250π=236.403 ------------ 0.9973071 2.693e-03
075 75.748π=237.968 1.890e-07 16.8093.532e-054.140e-07------
076 dark 76.250π=239.545 ------------ 0.9973424 2.658e-03
076 76.748π=241.109 1.817e-07 16.8523.440e-053.980e-07------
077 dark 77.250π=242.686 ------------ 0.9973768 2.623e-03
077 77.748π=244.251 1.747e-07 16.8943.352e-053.829e-07------
078 dark 78.250π=245.828 ------------ 0.9974103 2.590e-03
078 78.748π=247.393 1.682e-07 16.9363.268e-053.685e-07------
079 dark 79.250π=248.970 ------------ 0.9974430 2.557e-03
079 79.748π=250.535 1.619e-07 16.9773.186e-053.548e-07------
080 dark 80.250π=252.111 ------------ 0.9974749 2.525e-03
080 80.748π=253.676 1.560e-07 17.0173.108e-053.417e-07------
081 dark 81.250π=255.253 ------------ 0.9975059 2.494e-03
081 81.748π=256.818 1.503e-07 17.0573.032e-053.294e-07------
082 dark 82.250π=258.395 ------------ 0.9975363 2.464e-03
082 82.748π=259.960 1.449e-07 17.0972.959e-053.176e-07------
083 dark 83.250π=261.536 ------------ 0.9975659 2.434e-03
083 83.748π=263.101 1.398e-07 17.1362.889e-053.063e-07------
084 dark 84.250π=264.678 ------------ 0.9975947 2.405e-03
084 84.748π=266.243 1.349e-07 17.1752.821e-052.956e-07------
085 dark 85.250π=267.819 ------------ 0.9976230 2.377e-03
085 85.748π=269.385 1.303e-07 17.2132.756e-052.854e-07------
086 dark 86.250π=270.961 ------------ 0.9976505 2.349e-03
086 86.748π=272.526 1.258e-07 17.2512.693e-052.756e-07------
087 dark 87.250π=274.103 ------------ 0.9976775 2.323e-03
087 87.748π=275.668 1.216e-07 17.2882.632e-052.663e-07------
088 dark 88.250π=277.244 ------------ 0.9977038 2.296e-03
088 88.748π=278.810 1.175e-07 17.3252.573e-052.574e-07------
089 dark 89.250π=280.386 ------------ 0.9977295 2.271e-03
089 89.748π=281.951 1.136e-07 17.3612.516e-052.489e-07------
090 dark 90.250π=283.527 ------------ 0.9977547 2.245e-03
090 90.748π=285.093 1.099e-07 17.3982.461e-052.408e-07------
091 dark 91.250π=286.669 ------------ 0.9977793 2.221e-03
091 91.748π=288.235 1.063e-07 17.4332.407e-052.330e-07------
092 dark 92.250π=289.811 ------------ 0.9978033 2.197e-03
092 92.748π=291.376 1.029e-07 17.4692.356e-052.255e-07------
093 dark 93.250π=292.952 ------------ 0.9978269 2.173e-03
093 93.748π=294.518 9.968e-08 17.5042.306e-052.184e-07------
094 dark 94.250π=296.094 ------------ 0.9978499 2.150e-03
094 94.748π=297.660 9.655e-08 17.5382.257e-052.115e-07------
095 dark 95.250π=299.235 ------------ 0.9978725 2.127e-03
095 95.748π=300.801 9.356e-08 17.5722.210e-052.050e-07------
096 dark 96.250π=302.377 ------------ 0.9978946 2.105e-03
096 96.748π=303.943 9.069e-08 17.6062.165e-051.987e-07------
097 dark 97.250π=305.519 ------------ 0.9979163 2.084e-03
097 97.748π=307.085 8.793e-08 17.6402.121e-051.927e-07------
098 dark 98.250π=308.660 ------------ 0.9979375 2.063e-03
098 98.748π=310.226 8.529e-08 17.6732.078e-051.869e-07------
099 dark 99.250π=311.802 ------------ 0.9979583 2.042e-03
099 99.748π=313.368 8.275e-08 17.7062.037e-051.813e-07------
100 dark 100.250π=314.943 ------------ 0.9979786 2.021e-03
100 100.748π=316.510 8.031e-08 17.7381.996e-051.760e-07------

Last updated: Wednesday January 02, 2008

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