Resolution and the Rayleigh criterion
Key idea: Diffraction spreads each point source into a pattern and limits resolution. Rayleigh's criterion places one source's central maximum at the other's first minimum for just-resolved sources.
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Guided lessons · coming soon
Guided lesson
Guided steps
- Explain how diffraction blurs nearby sources.
- Compare the individual patterns using the Rayleigh criterion.
- Find the limiting angle and minimum separation.
- Calculate both angles and decide whether the lamps are resolved.
- Explain what happens when wavelength and aperture width both halve.
- Correct the claim about the combined pattern.
- Explain how aperture width and brightness affect resolution.
- Find the minimum aperture width in millimetres.
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ContinueThe two headlights of a distant car look like a single light; as the car comes closer, they separate into two. Light from each headlamp diffracts as it enters your eye, so each forms a small blurred spot rather than a sharp point, and when the spots overlap too much you cannot tell them apart. This lesson uses diffraction to explain this limit and the Rayleigh criterion, θ ≈ λ/b, which sets the smallest angle at which two sources can be told apart.
Why diffraction limits resolution
Light from a point source passing through an aperture of width b, such as a lens or the pupil of an eye, is diffracted. Its image is not a point but a central maximum with fainter fringes around it. For a single slit, the central maximum extends to the first minimum at sin θ = λ/b. Other aperture shapes change the numerical factor; θ ≈ λ/b gives the H2 estimate of the diffraction angle.
Two sources close together give two of these patterns, side by side. If the patterns overlap a lot, they merge into one blur. The ability to show two close objects as separate is called resolution.
The Rayleigh criterion
The Rayleigh criterion states that two sources are just resolved when the central maximum of the diffraction pattern of one falls on the first minimum of the diffraction pattern of the other.
Individual images at the Rayleigh limit
Two equal-strength single-source slit intensity profiles centred one slit-resolution limit apart, with their summed intensity.
Scroll across the figure to read all labels.
View figure data
| Image angle / slit limit, θ/(λ/b) (unitless) | Left source only | Right source only | Sum of intensities |
|---|---|---|---|
| -2.00000 | 4.503e-02 | 1.621e-02 | 6.124e-02 |
| -1.98333 | 4.592e-02 | 1.638e-02 | 6.231e-02 |
| -1.96667 | 4.659e-02 | 1.647e-02 | 6.306e-02 |
| -1.95000 | 4.701e-02 | 1.647e-02 | 6.348e-02 |
| -1.93333 | 4.719e-02 | 1.637e-02 | 6.356e-02 |
| -1.91667 | 4.710e-02 | 1.619e-02 | 6.329e-02 |
| -1.90000 | 4.676e-02 | 1.591e-02 | 6.267e-02 |
| -1.88333 | 4.615e-02 | 1.555e-02 | 6.169e-02 |
| -1.86667 | 4.527e-02 | 1.510e-02 | 6.037e-02 |
| -1.85000 | 4.414e-02 | 1.457e-02 | 5.870e-02 |
| -1.83333 | 4.274e-02 | 1.396e-02 | 5.670e-02 |
| -1.81667 | 4.111e-02 | 1.328e-02 | 5.439e-02 |
| -1.80000 | 3.924e-02 | 1.254e-02 | 5.178e-02 |
| -1.78333 | 3.716e-02 | 1.174e-02 | 4.889e-02 |
| -1.76667 | 3.488e-02 | 1.089e-02 | 4.577e-02 |
| -1.75000 | 3.242e-02 | 1.001e-02 | 4.243e-02 |
| -1.73333 | 2.982e-02 | 9.095e-03 | 3.892e-02 |
| -1.71667 | 2.711e-02 | 8.167e-03 | 3.527e-02 |
| -1.70000 | 2.431e-02 | 7.233e-03 | 3.154e-02 |
| -1.68333 | 2.146e-02 | 6.305e-03 | 2.777e-02 |
| -1.66667 | 1.861e-02 | 5.396e-03 | 2.401e-02 |
| -1.65000 | 1.579e-02 | 4.518e-03 | 2.031e-02 |
| -1.63333 | 1.305e-02 | 3.683e-03 | 1.673e-02 |
| -1.61667 | 1.044e-02 | 2.904e-03 | 1.334e-02 |
| -1.60000 | 7.996e-03 | 2.194e-03 | 1.019e-02 |
| -1.58333 | 5.783e-03 | 1.564e-03 | 7.347e-03 |
| -1.56667 | 3.849e-03 | 1.025e-03 | 4.875e-03 |
| -1.55000 | 2.249e-03 | 5.900e-04 | 2.839e-03 |
| -1.53333 | 1.037e-03 | 2.678e-04 | 1.305e-03 |
| -1.51667 | 2.685e-04 | 6.824e-05 | 3.367e-04 |
| -1.50000 | 1.520e-33 | 1.520e-33 | 3.039e-33 |
| -1.48333 | 2.870e-04 | 7.055e-05 | 3.576e-04 |
| -1.46667 | 1.185e-03 | 2.862e-04 | 1.471e-03 |
| -1.45000 | 2.747e-03 | 6.521e-04 | 3.399e-03 |
| -1.43333 | 5.028e-03 | 1.172e-03 | 6.200e-03 |
| -1.41667 | 8.077e-03 | 1.848e-03 | 9.925e-03 |
| -1.40000 | 1.194e-02 | 2.680e-03 | 1.462e-02 |
| -1.38333 | 1.668e-02 | 3.669e-03 | 2.035e-02 |
| -1.36667 | 2.232e-02 | 4.811e-03 | 2.713e-02 |
| -1.35000 | 2.890e-02 | 6.102e-03 | 3.501e-02 |
| -1.33333 | 3.648e-02 | 7.536e-03 | 4.401e-02 |
| -1.31667 | 4.506e-02 | 9.107e-03 | 5.417e-02 |
| -1.30000 | 5.470e-02 | 1.080e-02 | 6.550e-02 |
| -1.28333 | 6.540e-02 | 1.262e-02 | 7.801e-02 |
| -1.26667 | 7.718e-02 | 1.453e-02 | 9.172e-02 |
| -1.25000 | 9.006e-02 | 1.654e-02 | 1.066e-01 |
| -1.23333 | 1.041e-01 | 1.862e-02 | 1.227e-01 |
| -1.21667 | 1.191e-01 | 2.077e-02 | 1.399e-01 |
| -1.20000 | 1.353e-01 | 2.295e-02 | 1.583e-01 |
| -1.18333 | 1.526e-01 | 2.515e-02 | 1.778e-01 |
| -1.16667 | 1.710e-01 | 2.736e-02 | 1.983e-01 |
| -1.15000 | 1.904e-01 | 2.955e-02 | 2.199e-01 |
| -1.13333 | 2.108e-01 | 3.170e-02 | 2.425e-01 |
| -1.11667 | 2.322e-01 | 3.379e-02 | 2.660e-01 |
| -1.10000 | 2.546e-01 | 3.580e-02 | 2.904e-01 |
| -1.08333 | 2.778e-01 | 3.771e-02 | 3.155e-01 |
| -1.06667 | 3.019e-01 | 3.950e-02 | 3.414e-01 |
| -1.05000 | 3.267e-01 | 4.114e-02 | 3.679e-01 |
| -1.03333 | 3.523e-01 | 4.262e-02 | 3.949e-01 |
| -1.01667 | 3.785e-01 | 4.393e-02 | 4.224e-01 |
| -1.00000 | 4.053e-01 | 4.503e-02 | 4.503e-01 |
| -0.98333 | 4.325e-01 | 4.592e-02 | 4.785e-01 |
| -0.96667 | 4.602e-01 | 4.659e-02 | 5.068e-01 |
| -0.95000 | 4.881e-01 | 4.701e-02 | 5.351e-01 |
| -0.93333 | 5.163e-01 | 4.719e-02 | 5.634e-01 |
| -0.91667 | 5.445e-01 | 4.710e-02 | 5.916e-01 |
| -0.90000 | 5.728e-01 | 4.676e-02 | 6.195e-01 |
| -0.88333 | 6.010e-01 | 4.615e-02 | 6.471e-01 |
| -0.86667 | 6.290e-01 | 4.527e-02 | 6.742e-01 |
| -0.85000 | 6.566e-01 | 4.414e-02 | 7.008e-01 |
| -0.83333 | 6.839e-01 | 4.274e-02 | 7.267e-01 |
| -0.81667 | 7.107e-01 | 4.111e-02 | 7.518e-01 |
| -0.80000 | 7.368e-01 | 3.924e-02 | 7.761e-01 |
| -0.78333 | 7.623e-01 | 3.716e-02 | 7.994e-01 |
| -0.76667 | 7.869e-01 | 3.488e-02 | 8.218e-01 |
| -0.75000 | 8.106e-01 | 3.242e-02 | 8.430e-01 |
| -0.73333 | 8.332e-01 | 2.982e-02 | 8.631e-01 |
| -0.71667 | 8.548e-01 | 2.711e-02 | 8.819e-01 |
| -0.70000 | 8.751e-01 | 2.431e-02 | 8.994e-01 |
| -0.68333 | 8.942e-01 | 2.146e-02 | 9.157e-01 |
| -0.66667 | 9.119e-01 | 1.861e-02 | 9.305e-01 |
| -0.65000 | 9.281e-01 | 1.579e-02 | 9.439e-01 |
| -0.63333 | 9.429e-01 | 1.305e-02 | 9.559e-01 |
| -0.61667 | 9.560e-01 | 1.044e-02 | 9.665e-01 |
| -0.60000 | 9.675e-01 | 7.996e-03 | 9.755e-01 |
| -0.58333 | 9.774e-01 | 5.783e-03 | 9.831e-01 |
| -0.56667 | 9.855e-01 | 3.849e-03 | 9.893e-01 |
| -0.55000 | 9.918e-01 | 2.249e-03 | 9.941e-01 |
| -0.53333 | 9.963e-01 | 1.037e-03 | 9.974e-01 |
| -0.51667 | 9.991e-01 | 2.685e-04 | 9.994e-01 |
| -0.50000 | 1.000e+00 | 1.520e-33 | 1.000e+00 |
| -0.48333 | 9.991e-01 | 2.870e-04 | 9.994e-01 |
| -0.46667 | 9.963e-01 | 1.185e-03 | 9.975e-01 |
| -0.45000 | 9.918e-01 | 2.747e-03 | 9.945e-01 |
| -0.43333 | 9.855e-01 | 5.028e-03 | 9.905e-01 |
| -0.41667 | 9.774e-01 | 8.077e-03 | 9.854e-01 |
| -0.40000 | 9.675e-01 | 1.194e-02 | 9.795e-01 |
| -0.38333 | 9.560e-01 | 1.668e-02 | 9.727e-01 |
| -0.36667 | 9.429e-01 | 2.232e-02 | 9.652e-01 |
| -0.35000 | 9.281e-01 | 2.890e-02 | 9.570e-01 |
| -0.33333 | 9.119e-01 | 3.648e-02 | 9.484e-01 |
| -0.31667 | 8.942e-01 | 4.506e-02 | 9.393e-01 |
| -0.30000 | 8.751e-01 | 5.470e-02 | 9.298e-01 |
| -0.28333 | 8.548e-01 | 6.540e-02 | 9.202e-01 |
| -0.26667 | 8.332e-01 | 7.718e-02 | 9.104e-01 |
| -0.25000 | 8.106e-01 | 9.006e-02 | 9.006e-01 |
| -0.23333 | 7.869e-01 | 1.041e-01 | 8.909e-01 |
| -0.21667 | 7.623e-01 | 1.191e-01 | 8.814e-01 |
| -0.20000 | 7.368e-01 | 1.353e-01 | 8.722e-01 |
| -0.18333 | 7.107e-01 | 1.526e-01 | 8.633e-01 |
| -0.16667 | 6.839e-01 | 1.710e-01 | 8.549e-01 |
| -0.15000 | 6.566e-01 | 1.904e-01 | 8.470e-01 |
| -0.13333 | 6.290e-01 | 2.108e-01 | 8.398e-01 |
| -0.11667 | 6.010e-01 | 2.322e-01 | 8.332e-01 |
| -0.10000 | 5.728e-01 | 2.546e-01 | 8.274e-01 |
| -0.08333 | 5.445e-01 | 2.778e-01 | 8.223e-01 |
| -0.06667 | 5.163e-01 | 3.019e-01 | 8.181e-01 |
| -0.05000 | 4.881e-01 | 3.267e-01 | 8.149e-01 |
| -0.03333 | 4.602e-01 | 3.523e-01 | 8.125e-01 |
| -0.01667 | 4.325e-01 | 3.785e-01 | 8.110e-01 |
| 0.00000 | 4.053e-01 | 4.053e-01 | 8.106e-01 |
| 0.01667 | 3.785e-01 | 4.325e-01 | 8.110e-01 |
| 0.03333 | 3.523e-01 | 4.602e-01 | 8.125e-01 |
| 0.05000 | 3.267e-01 | 4.881e-01 | 8.149e-01 |
| 0.06667 | 3.019e-01 | 5.163e-01 | 8.181e-01 |
| 0.08333 | 2.778e-01 | 5.445e-01 | 8.223e-01 |
| 0.10000 | 2.546e-01 | 5.728e-01 | 8.274e-01 |
| 0.11667 | 2.322e-01 | 6.010e-01 | 8.332e-01 |
| 0.13333 | 2.108e-01 | 6.290e-01 | 8.398e-01 |
| 0.15000 | 1.904e-01 | 6.566e-01 | 8.470e-01 |
| 0.16667 | 1.710e-01 | 6.839e-01 | 8.549e-01 |
| 0.18333 | 1.526e-01 | 7.107e-01 | 8.633e-01 |
| 0.20000 | 1.353e-01 | 7.368e-01 | 8.722e-01 |
| 0.21667 | 1.191e-01 | 7.623e-01 | 8.814e-01 |
| 0.23333 | 1.041e-01 | 7.869e-01 | 8.909e-01 |
| 0.25000 | 9.006e-02 | 8.106e-01 | 9.006e-01 |
| 0.26667 | 7.718e-02 | 8.332e-01 | 9.104e-01 |
| 0.28333 | 6.540e-02 | 8.548e-01 | 9.202e-01 |
| 0.30000 | 5.470e-02 | 8.751e-01 | 9.298e-01 |
| 0.31667 | 4.506e-02 | 8.942e-01 | 9.393e-01 |
| 0.33333 | 3.648e-02 | 9.119e-01 | 9.484e-01 |
| 0.35000 | 2.890e-02 | 9.281e-01 | 9.570e-01 |
| 0.36667 | 2.232e-02 | 9.429e-01 | 9.652e-01 |
| 0.38333 | 1.668e-02 | 9.560e-01 | 9.727e-01 |
| 0.40000 | 1.194e-02 | 9.675e-01 | 9.795e-01 |
| 0.41667 | 8.077e-03 | 9.774e-01 | 9.854e-01 |
| 0.43333 | 5.028e-03 | 9.855e-01 | 9.905e-01 |
| 0.45000 | 2.747e-03 | 9.918e-01 | 9.945e-01 |
| 0.46667 | 1.185e-03 | 9.963e-01 | 9.975e-01 |
| 0.48333 | 2.870e-04 | 9.991e-01 | 9.994e-01 |
| 0.50000 | 1.520e-33 | 1.000e+00 | 1.000e+00 |
| 0.51667 | 2.685e-04 | 9.991e-01 | 9.994e-01 |
| 0.53333 | 1.037e-03 | 9.963e-01 | 9.974e-01 |
| 0.55000 | 2.249e-03 | 9.918e-01 | 9.941e-01 |
| 0.56667 | 3.849e-03 | 9.855e-01 | 9.893e-01 |
| 0.58333 | 5.783e-03 | 9.774e-01 | 9.831e-01 |
| 0.60000 | 7.996e-03 | 9.675e-01 | 9.755e-01 |
| 0.61667 | 1.044e-02 | 9.560e-01 | 9.665e-01 |
| 0.63333 | 1.305e-02 | 9.429e-01 | 9.559e-01 |
| 0.65000 | 1.579e-02 | 9.281e-01 | 9.439e-01 |
| 0.66667 | 1.861e-02 | 9.119e-01 | 9.305e-01 |
| 0.68333 | 2.146e-02 | 8.942e-01 | 9.157e-01 |
| 0.70000 | 2.431e-02 | 8.751e-01 | 8.994e-01 |
| 0.71667 | 2.711e-02 | 8.548e-01 | 8.819e-01 |
| 0.73333 | 2.982e-02 | 8.332e-01 | 8.631e-01 |
| 0.75000 | 3.242e-02 | 8.106e-01 | 8.430e-01 |
| 0.76667 | 3.488e-02 | 7.869e-01 | 8.218e-01 |
| 0.78333 | 3.716e-02 | 7.623e-01 | 7.994e-01 |
| 0.80000 | 3.924e-02 | 7.368e-01 | 7.761e-01 |
| 0.81667 | 4.111e-02 | 7.107e-01 | 7.518e-01 |
| 0.83333 | 4.274e-02 | 6.839e-01 | 7.267e-01 |
| 0.85000 | 4.414e-02 | 6.566e-01 | 7.008e-01 |
| 0.86667 | 4.527e-02 | 6.290e-01 | 6.742e-01 |
| 0.88333 | 4.615e-02 | 6.010e-01 | 6.471e-01 |
| 0.90000 | 4.676e-02 | 5.728e-01 | 6.195e-01 |
| 0.91667 | 4.710e-02 | 5.445e-01 | 5.916e-01 |
| 0.93333 | 4.719e-02 | 5.163e-01 | 5.634e-01 |
| 0.95000 | 4.701e-02 | 4.881e-01 | 5.351e-01 |
| 0.96667 | 4.659e-02 | 4.602e-01 | 5.068e-01 |
| 0.98333 | 4.592e-02 | 4.325e-01 | 4.785e-01 |
| 1.00000 | 4.503e-02 | 4.053e-01 | 4.503e-01 |
| 1.01667 | 4.393e-02 | 3.785e-01 | 4.224e-01 |
| 1.03333 | 4.262e-02 | 3.523e-01 | 3.949e-01 |
| 1.05000 | 4.114e-02 | 3.267e-01 | 3.679e-01 |
| 1.06667 | 3.950e-02 | 3.019e-01 | 3.414e-01 |
| 1.08333 | 3.771e-02 | 2.778e-01 | 3.155e-01 |
| 1.10000 | 3.580e-02 | 2.546e-01 | 2.904e-01 |
| 1.11667 | 3.379e-02 | 2.322e-01 | 2.660e-01 |
| 1.13333 | 3.170e-02 | 2.108e-01 | 2.425e-01 |
| 1.15000 | 2.955e-02 | 1.904e-01 | 2.199e-01 |
| 1.16667 | 2.736e-02 | 1.710e-01 | 1.983e-01 |
| 1.18333 | 2.515e-02 | 1.526e-01 | 1.778e-01 |
| 1.20000 | 2.295e-02 | 1.353e-01 | 1.583e-01 |
| 1.21667 | 2.077e-02 | 1.191e-01 | 1.399e-01 |
| 1.23333 | 1.862e-02 | 1.041e-01 | 1.227e-01 |
| 1.25000 | 1.654e-02 | 9.006e-02 | 1.066e-01 |
| 1.26667 | 1.453e-02 | 7.718e-02 | 9.172e-02 |
| 1.28333 | 1.262e-02 | 6.540e-02 | 7.801e-02 |
| 1.30000 | 1.080e-02 | 5.470e-02 | 6.550e-02 |
| 1.31667 | 9.107e-03 | 4.506e-02 | 5.417e-02 |
| 1.33333 | 7.536e-03 | 3.648e-02 | 4.401e-02 |
| 1.35000 | 6.102e-03 | 2.890e-02 | 3.501e-02 |
| 1.36667 | 4.811e-03 | 2.232e-02 | 2.713e-02 |
| 1.38333 | 3.669e-03 | 1.668e-02 | 2.035e-02 |
| 1.40000 | 2.680e-03 | 1.194e-02 | 1.462e-02 |
| 1.41667 | 1.848e-03 | 8.077e-03 | 9.925e-03 |
| 1.43333 | 1.172e-03 | 5.028e-03 | 6.200e-03 |
| 1.45000 | 6.521e-04 | 2.747e-03 | 3.399e-03 |
| 1.46667 | 2.862e-04 | 1.185e-03 | 1.471e-03 |
| 1.48333 | 7.055e-05 | 2.870e-04 | 3.576e-04 |
| 1.50000 | 1.520e-33 | 1.520e-33 | 3.039e-33 |
| 1.51667 | 6.824e-05 | 2.685e-04 | 3.367e-04 |
| 1.53333 | 2.678e-04 | 1.037e-03 | 1.305e-03 |
| 1.55000 | 5.900e-04 | 2.249e-03 | 2.839e-03 |
| 1.56667 | 1.025e-03 | 3.849e-03 | 4.875e-03 |
| 1.58333 | 1.564e-03 | 5.783e-03 | 7.347e-03 |
| 1.60000 | 2.194e-03 | 7.996e-03 | 1.019e-02 |
| 1.61667 | 2.904e-03 | 1.044e-02 | 1.334e-02 |
| 1.63333 | 3.683e-03 | 1.305e-02 | 1.673e-02 |
| 1.65000 | 4.518e-03 | 1.579e-02 | 2.031e-02 |
| 1.66667 | 5.396e-03 | 1.861e-02 | 2.401e-02 |
| 1.68333 | 6.305e-03 | 2.146e-02 | 2.777e-02 |
| 1.70000 | 7.233e-03 | 2.431e-02 | 3.154e-02 |
| 1.71667 | 8.167e-03 | 2.711e-02 | 3.527e-02 |
| 1.73333 | 9.095e-03 | 2.982e-02 | 3.892e-02 |
| 1.75000 | 1.001e-02 | 3.242e-02 | 4.243e-02 |
| 1.76667 | 1.089e-02 | 3.488e-02 | 4.577e-02 |
| 1.78333 | 1.174e-02 | 3.716e-02 | 4.889e-02 |
| 1.80000 | 1.254e-02 | 3.924e-02 | 5.178e-02 |
| 1.81667 | 1.328e-02 | 4.111e-02 | 5.439e-02 |
| 1.83333 | 1.396e-02 | 4.274e-02 | 5.670e-02 |
| 1.85000 | 1.457e-02 | 4.414e-02 | 5.870e-02 |
| 1.86667 | 1.510e-02 | 4.527e-02 | 6.037e-02 |
| 1.88333 | 1.555e-02 | 4.615e-02 | 6.169e-02 |
| 1.90000 | 1.591e-02 | 4.676e-02 | 6.267e-02 |
| 1.91667 | 1.619e-02 | 4.710e-02 | 6.329e-02 |
| 1.93333 | 1.637e-02 | 4.719e-02 | 6.356e-02 |
| 1.95000 | 1.647e-02 | 4.701e-02 | 6.348e-02 |
| 1.96667 | 1.647e-02 | 4.659e-02 | 6.306e-02 |
| 1.98333 | 1.638e-02 | 4.592e-02 | 6.231e-02 |
| 2.00000 | 1.621e-02 | 4.503e-02 | 6.124e-02 |
For the slit model, the first minimum is at sin θ = λ/b. An aperture many wavelengths wide gives a small diffraction angle, so sin θ ≈ θ. Use the H2 approximation for the numerical comparisons below. The smallest angle between two sources that can be resolved is
with θ in radians. If the angle between the sources is greater than this, they are resolved; if it is smaller, they are not.
Two source images: below, at and above the Rayleigh limit
At the Rayleigh separation Δθ/(λ/b) = 1, one individual maximum coincides with the other individual first minimum. The combined curve has two overlapping peaks. At half that separation it has one broad central peak.
Scroll across the figure to read all labels.
View figure data
| Image angle / slit limit, θ/(λ/b) (unitless) | Below limit: Δθ/(λ/b) = 0.5 | Rayleigh limit: Δθ/(λ/b) = 1 | Above limit: Δθ/(λ/b) = 1.5 |
|---|---|---|---|
| -3.00000 | 0.01150 | 2.448e-02 | 0.013610 |
| -2.98333 | 0.01183 | 2.471e-02 | 0.013110 |
| -2.96667 | 0.01217 | 2.481e-02 | 0.012597 |
| -2.95000 | 0.01252 | 2.477e-02 | 0.012077 |
| -2.93333 | 0.01286 | 2.460e-02 | 0.011558 |
| -2.91667 | 0.01321 | 2.428e-02 | 0.011048 |
| -2.90000 | 0.01356 | 2.384e-02 | 0.010555 |
| -2.88333 | 0.01390 | 2.326e-02 | 0.010089 |
| -2.86667 | 0.01424 | 2.256e-02 | 0.009656 |
| -2.85000 | 0.01456 | 2.173e-02 | 0.009265 |
| -2.83333 | 0.01488 | 2.080e-02 | 0.008926 |
| -2.81667 | 0.01518 | 1.976e-02 | 0.008646 |
| -2.80000 | 0.01547 | 1.863e-02 | 0.008433 |
| -2.78333 | 0.01574 | 1.741e-02 | 0.008295 |
| -2.76667 | 0.01598 | 1.613e-02 | 0.008239 |
| -2.75000 | 0.01621 | 1.480e-02 | 0.008271 |
| -2.73333 | 0.01642 | 1.343e-02 | 0.008398 |
| -2.71667 | 0.01660 | 1.204e-02 | 0.008625 |
| -2.70000 | 0.01675 | 1.065e-02 | 0.008956 |
| -2.68333 | 0.01688 | 9.271e-03 | 0.009396 |
| -2.66667 | 0.01698 | 7.922e-03 | 0.009946 |
| -2.65000 | 0.01706 | 6.622e-03 | 0.010608 |
| -2.63333 | 0.01711 | 5.390e-03 | 0.011383 |
| -2.61667 | 0.01714 | 4.244e-03 | 0.012271 |
| -2.60000 | 0.01714 | 3.201e-03 | 0.013269 |
| -2.58333 | 0.01711 | 2.278e-03 | 0.014375 |
| -2.56667 | 0.01707 | 1.491e-03 | 0.015585 |
| -2.55000 | 0.01700 | 8.565e-04 | 0.016894 |
| -2.53333 | 0.01692 | 3.881e-04 | 0.018294 |
| -2.51667 | 0.01682 | 9.874e-05 | 0.019779 |
| -2.50000 | 0.01671 | 3.039e-33 | 0.021339 |
| -2.48333 | 0.01658 | 1.017e-04 | 0.022964 |
| -2.46667 | 0.01646 | 4.120e-04 | 0.024643 |
| -2.45000 | 0.01633 | 9.370e-04 | 0.026365 |
| -2.43333 | 0.01620 | 1.681e-03 | 0.028116 |
| -2.41667 | 0.01608 | 2.645e-03 | 0.029883 |
| -2.40000 | 0.01597 | 3.831e-03 | 0.031650 |
| -2.38333 | 0.01588 | 5.234e-03 | 0.033404 |
| -2.36667 | 0.01580 | 6.850e-03 | 0.035128 |
| -2.35000 | 0.01575 | 8.673e-03 | 0.036806 |
| -2.33333 | 0.01573 | 1.069e-02 | 0.038423 |
| -2.31667 | 0.01574 | 1.290e-02 | 0.039962 |
| -2.30000 | 0.01579 | 1.527e-02 | 0.041408 |
| -2.28333 | 0.01588 | 1.780e-02 | 0.042744 |
| -2.26667 | 0.01602 | 2.046e-02 | 0.043957 |
| -2.25000 | 0.01621 | 2.324e-02 | 0.045032 |
| -2.23333 | 0.01646 | 2.611e-02 | 0.045954 |
| -2.21667 | 0.01676 | 2.906e-02 | 0.046713 |
| -2.20000 | 0.01712 | 3.204e-02 | 0.047296 |
| -2.18333 | 0.01754 | 3.505e-02 | 0.047695 |
| -2.16667 | 0.01803 | 3.804e-02 | 0.047901 |
| -2.15000 | 0.01859 | 4.100e-02 | 0.047909 |
| -2.13333 | 0.01922 | 4.389e-02 | 0.047713 |
| -2.11667 | 0.01991 | 4.669e-02 | 0.047312 |
| -2.10000 | 0.02067 | 4.936e-02 | 0.046707 |
| -2.08333 | 0.02149 | 5.187e-02 | 0.045900 |
| -2.06667 | 0.02239 | 5.421e-02 | 0.044897 |
| -2.05000 | 0.02334 | 5.634e-02 | 0.043705 |
| -2.03333 | 0.02435 | 5.824e-02 | 0.042336 |
| -2.01667 | 0.02543 | 5.988e-02 | 0.040802 |
| -2.00000 | 0.02655 | 6.124e-02 | 0.039122 |
| -1.98333 | 0.02772 | 6.231e-02 | 0.037313 |
| -1.96667 | 0.02893 | 6.306e-02 | 0.035400 |
| -1.95000 | 0.03018 | 6.348e-02 | 0.033406 |
| -1.93333 | 0.03146 | 6.356e-02 | 0.031361 |
| -1.91667 | 0.03275 | 6.329e-02 | 0.029296 |
| -1.90000 | 0.03406 | 6.267e-02 | 0.027245 |
| -1.88333 | 0.03538 | 6.169e-02 | 0.025244 |
| -1.86667 | 0.03669 | 6.037e-02 | 0.023333 |
| -1.85000 | 0.03799 | 5.870e-02 | 0.021553 |
| -1.83333 | 0.03927 | 5.670e-02 | 0.019949 |
| -1.81667 | 0.04052 | 5.439e-02 | 0.018565 |
| -1.80000 | 0.04173 | 5.178e-02 | 0.017450 |
| -1.78333 | 0.04289 | 4.889e-02 | 0.016652 |
| -1.76667 | 0.04399 | 4.577e-02 | 0.016222 |
| -1.75000 | 0.04503 | 4.243e-02 | 0.016211 |
| -1.73333 | 0.04599 | 3.892e-02 | 0.016672 |
| -1.71667 | 0.04687 | 3.527e-02 | 0.017655 |
| -1.70000 | 0.04766 | 3.154e-02 | 0.019214 |
| -1.68333 | 0.04836 | 2.777e-02 | 0.021400 |
| -1.66667 | 0.04895 | 2.401e-02 | 0.024264 |
| -1.65000 | 0.04944 | 2.031e-02 | 0.027856 |
| -1.63333 | 0.04982 | 1.673e-02 | 0.032223 |
| -1.61667 | 0.05008 | 1.334e-02 | 0.037413 |
| -1.60000 | 0.05024 | 1.019e-02 | 0.043469 |
| -1.58333 | 0.05028 | 7.347e-03 | 0.050433 |
| -1.56667 | 0.05022 | 4.875e-03 | 0.058343 |
| -1.55000 | 0.05004 | 2.839e-03 | 0.067232 |
| -1.53333 | 0.04977 | 1.305e-03 | 0.077133 |
| -1.51667 | 0.04941 | 3.367e-04 | 0.088072 |
| -1.50000 | 0.04897 | 3.039e-33 | 0.100070 |
| -1.48333 | 0.04845 | 3.576e-04 | 0.113146 |
| -1.46667 | 0.04787 | 1.471e-03 | 0.127310 |
| -1.45000 | 0.04726 | 3.399e-03 | 0.142570 |
| -1.43333 | 0.04661 | 6.200e-03 | 0.158927 |
| -1.41667 | 0.04597 | 9.925e-03 | 0.176375 |
| -1.40000 | 0.04534 | 1.462e-02 | 0.194904 |
| -1.38333 | 0.04475 | 2.035e-02 | 0.214495 |
| -1.36667 | 0.04422 | 2.713e-02 | 0.235126 |
| -1.35000 | 0.04380 | 3.501e-02 | 0.256766 |
| -1.33333 | 0.04349 | 4.401e-02 | 0.279378 |
| -1.31667 | 0.04335 | 5.417e-02 | 0.302919 |
| -1.30000 | 0.04339 | 6.550e-02 | 0.327339 |
| -1.28333 | 0.04366 | 7.801e-02 | 0.352583 |
| -1.26667 | 0.04420 | 9.172e-02 | 0.378588 |
| -1.25000 | 0.04503 | 1.066e-01 | 0.405285 |
| -1.23333 | 0.04621 | 1.227e-01 | 0.432600 |
| -1.21667 | 0.04777 | 1.399e-01 | 0.460453 |
| -1.20000 | 0.04976 | 1.583e-01 | 0.488759 |
| -1.18333 | 0.05221 | 1.778e-01 | 0.517427 |
| -1.16667 | 0.05518 | 1.983e-01 | 0.546363 |
| -1.15000 | 0.05870 | 2.199e-01 | 0.575467 |
| -1.13333 | 0.06282 | 2.425e-01 | 0.604636 |
| -1.11667 | 0.06759 | 2.660e-01 | 0.633763 |
| -1.10000 | 0.07304 | 2.904e-01 | 0.662740 |
| -1.08333 | 0.07922 | 3.155e-01 | 0.691454 |
| -1.06667 | 0.08617 | 3.414e-01 | 0.719794 |
| -1.05000 | 0.09394 | 3.679e-01 | 0.747644 |
| -1.03333 | 0.10255 | 3.949e-01 | 0.774890 |
| -1.01667 | 0.11206 | 4.224e-01 | 0.801417 |
| -1.00000 | 0.12249 | 4.503e-01 | 0.827112 |
| -0.98333 | 0.13387 | 4.785e-01 | 0.851861 |
| -0.96667 | 0.14625 | 5.068e-01 | 0.875556 |
| -0.95000 | 0.15965 | 5.351e-01 | 0.898087 |
| -0.93333 | 0.17409 | 5.634e-01 | 0.919351 |
| -0.91667 | 0.18959 | 5.916e-01 | 0.939247 |
| -0.90000 | 0.20618 | 6.195e-01 | 0.957681 |
| -0.88333 | 0.22386 | 6.471e-01 | 0.974561 |
| -0.86667 | 0.24266 | 6.742e-01 | 0.989804 |
| -0.85000 | 0.26257 | 7.008e-01 | 1.003330 |
| -0.83333 | 0.28360 | 7.267e-01 | 1.015070 |
| -0.81667 | 0.30574 | 7.518e-01 | 1.024960 |
| -0.80000 | 0.32900 | 7.761e-01 | 1.032944 |
| -0.78333 | 0.35335 | 7.994e-01 | 1.038974 |
| -0.76667 | 0.37879 | 8.218e-01 | 1.043013 |
| -0.75000 | 0.40528 | 8.430e-01 | 1.045032 |
| -0.73333 | 0.43282 | 8.631e-01 | 1.045010 |
| -0.71667 | 0.46135 | 8.819e-01 | 1.042937 |
| -0.70000 | 0.49085 | 8.994e-01 | 1.038814 |
| -0.68333 | 0.52128 | 9.157e-01 | 1.032650 |
| -0.66667 | 0.55259 | 9.305e-01 | 1.024465 |
| -0.65000 | 0.58473 | 9.439e-01 | 1.014289 |
| -0.63333 | 0.61764 | 9.559e-01 | 1.002163 |
| -0.61667 | 0.65127 | 9.665e-01 | 0.988137 |
| -0.60000 | 0.68554 | 9.755e-01 | 0.972271 |
| -0.58333 | 0.72039 | 9.831e-01 | 0.954636 |
| -0.56667 | 0.75575 | 9.893e-01 | 0.935309 |
| -0.55000 | 0.79154 | 9.941e-01 | 0.914380 |
| -0.53333 | 0.82767 | 9.974e-01 | 0.891946 |
| -0.51667 | 0.86406 | 9.994e-01 | 0.868113 |
| -0.50000 | 0.90063 | 1.000e+00 | 0.842992 |
| -0.48333 | 0.93729 | 9.994e-01 | 0.816706 |
| -0.46667 | 0.97393 | 9.975e-01 | 0.789380 |
| -0.45000 | 1.01048 | 9.945e-01 | 0.761149 |
| -0.43333 | 1.04682 | 9.905e-01 | 0.732151 |
| -0.41667 | 1.08287 | 9.854e-01 | 0.702528 |
| -0.40000 | 1.11852 | 9.795e-01 | 0.672428 |
| -0.38333 | 1.15368 | 9.727e-01 | 0.642002 |
| -0.36667 | 1.18824 | 9.652e-01 | 0.611402 |
| -0.35000 | 1.22210 | 9.570e-01 | 0.580783 |
| -0.33333 | 1.25518 | 9.484e-01 | 0.550299 |
| -0.31667 | 1.28736 | 9.393e-01 | 0.520105 |
| -0.30000 | 1.31855 | 9.298e-01 | 0.490356 |
| -0.28333 | 1.34867 | 9.202e-01 | 0.461204 |
| -0.26667 | 1.37761 | 9.104e-01 | 0.432798 |
| -0.25000 | 1.40528 | 9.006e-01 | 0.405285 |
| -0.23333 | 1.43162 | 8.909e-01 | 0.378806 |
| -0.21667 | 1.45652 | 8.814e-01 | 0.353500 |
| -0.20000 | 1.47991 | 8.722e-01 | 0.329497 |
| -0.18333 | 1.50172 | 8.633e-01 | 0.306921 |
| -0.16667 | 1.52188 | 8.549e-01 | 0.285891 |
| -0.15000 | 1.54032 | 8.470e-01 | 0.266517 |
| -0.13333 | 1.55698 | 8.398e-01 | 0.248898 |
| -0.11667 | 1.57182 | 8.332e-01 | 0.233129 |
| -0.10000 | 1.58477 | 8.274e-01 | 0.219290 |
| -0.08333 | 1.59581 | 8.223e-01 | 0.207455 |
| -0.06667 | 1.60489 | 8.181e-01 | 0.197686 |
| -0.05000 | 1.61198 | 8.149e-01 | 0.190034 |
| -0.03333 | 1.61706 | 8.125e-01 | 0.184540 |
| -0.01667 | 1.62012 | 8.110e-01 | 0.181231 |
| 0.00000 | 1.62114 | 8.106e-01 | 0.180127 |
| 0.01667 | 1.62012 | 8.110e-01 | 0.181231 |
| 0.03333 | 1.61706 | 8.125e-01 | 0.184540 |
| 0.05000 | 1.61198 | 8.149e-01 | 0.190034 |
| 0.06667 | 1.60489 | 8.181e-01 | 0.197686 |
| 0.08333 | 1.59581 | 8.223e-01 | 0.207455 |
| 0.10000 | 1.58477 | 8.274e-01 | 0.219290 |
| 0.11667 | 1.57182 | 8.332e-01 | 0.233129 |
| 0.13333 | 1.55698 | 8.398e-01 | 0.248898 |
| 0.15000 | 1.54032 | 8.470e-01 | 0.266517 |
| 0.16667 | 1.52188 | 8.549e-01 | 0.285891 |
| 0.18333 | 1.50172 | 8.633e-01 | 0.306921 |
| 0.20000 | 1.47991 | 8.722e-01 | 0.329497 |
| 0.21667 | 1.45652 | 8.814e-01 | 0.353500 |
| 0.23333 | 1.43162 | 8.909e-01 | 0.378806 |
| 0.25000 | 1.40528 | 9.006e-01 | 0.405285 |
| 0.26667 | 1.37761 | 9.104e-01 | 0.432798 |
| 0.28333 | 1.34867 | 9.202e-01 | 0.461204 |
| 0.30000 | 1.31855 | 9.298e-01 | 0.490356 |
| 0.31667 | 1.28736 | 9.393e-01 | 0.520105 |
| 0.33333 | 1.25518 | 9.484e-01 | 0.550299 |
| 0.35000 | 1.22210 | 9.570e-01 | 0.580783 |
| 0.36667 | 1.18824 | 9.652e-01 | 0.611402 |
| 0.38333 | 1.15368 | 9.727e-01 | 0.642002 |
| 0.40000 | 1.11852 | 9.795e-01 | 0.672428 |
| 0.41667 | 1.08287 | 9.854e-01 | 0.702528 |
| 0.43333 | 1.04682 | 9.905e-01 | 0.732151 |
| 0.45000 | 1.01048 | 9.945e-01 | 0.761149 |
| 0.46667 | 0.97393 | 9.975e-01 | 0.789380 |
| 0.48333 | 0.93729 | 9.994e-01 | 0.816706 |
| 0.50000 | 0.90063 | 1.000e+00 | 0.842992 |
| 0.51667 | 0.86406 | 9.994e-01 | 0.868113 |
| 0.53333 | 0.82767 | 9.974e-01 | 0.891946 |
| 0.55000 | 0.79154 | 9.941e-01 | 0.914380 |
| 0.56667 | 0.75575 | 9.893e-01 | 0.935309 |
| 0.58333 | 0.72039 | 9.831e-01 | 0.954636 |
| 0.60000 | 0.68554 | 9.755e-01 | 0.972271 |
| 0.61667 | 0.65127 | 9.665e-01 | 0.988137 |
| 0.63333 | 0.61764 | 9.559e-01 | 1.002163 |
| 0.65000 | 0.58473 | 9.439e-01 | 1.014289 |
| 0.66667 | 0.55259 | 9.305e-01 | 1.024465 |
| 0.68333 | 0.52128 | 9.157e-01 | 1.032650 |
| 0.70000 | 0.49085 | 8.994e-01 | 1.038814 |
| 0.71667 | 0.46135 | 8.819e-01 | 1.042937 |
| 0.73333 | 0.43282 | 8.631e-01 | 1.045010 |
| 0.75000 | 0.40528 | 8.430e-01 | 1.045032 |
| 0.76667 | 0.37879 | 8.218e-01 | 1.043013 |
| 0.78333 | 0.35335 | 7.994e-01 | 1.038974 |
| 0.80000 | 0.32900 | 7.761e-01 | 1.032944 |
| 0.81667 | 0.30574 | 7.518e-01 | 1.024960 |
| 0.83333 | 0.28360 | 7.267e-01 | 1.015070 |
| 0.85000 | 0.26257 | 7.008e-01 | 1.003330 |
| 0.86667 | 0.24266 | 6.742e-01 | 0.989804 |
| 0.88333 | 0.22386 | 6.471e-01 | 0.974561 |
| 0.90000 | 0.20618 | 6.195e-01 | 0.957681 |
| 0.91667 | 0.18959 | 5.916e-01 | 0.939247 |
| 0.93333 | 0.17409 | 5.634e-01 | 0.919351 |
| 0.95000 | 0.15965 | 5.351e-01 | 0.898087 |
| 0.96667 | 0.14625 | 5.068e-01 | 0.875556 |
| 0.98333 | 0.13387 | 4.785e-01 | 0.851861 |
| 1.00000 | 0.12249 | 4.503e-01 | 0.827112 |
| 1.01667 | 0.11206 | 4.224e-01 | 0.801417 |
| 1.03333 | 0.10255 | 3.949e-01 | 0.774890 |
| 1.05000 | 0.09394 | 3.679e-01 | 0.747644 |
| 1.06667 | 0.08617 | 3.414e-01 | 0.719794 |
| 1.08333 | 0.07922 | 3.155e-01 | 0.691454 |
| 1.10000 | 0.07304 | 2.904e-01 | 0.662740 |
| 1.11667 | 0.06759 | 2.660e-01 | 0.633763 |
| 1.13333 | 0.06282 | 2.425e-01 | 0.604636 |
| 1.15000 | 0.05870 | 2.199e-01 | 0.575467 |
| 1.16667 | 0.05518 | 1.983e-01 | 0.546363 |
| 1.18333 | 0.05221 | 1.778e-01 | 0.517427 |
| 1.20000 | 0.04976 | 1.583e-01 | 0.488759 |
| 1.21667 | 0.04777 | 1.399e-01 | 0.460453 |
| 1.23333 | 0.04621 | 1.227e-01 | 0.432600 |
| 1.25000 | 0.04503 | 1.066e-01 | 0.405285 |
| 1.26667 | 0.04420 | 9.172e-02 | 0.378588 |
| 1.28333 | 0.04366 | 7.801e-02 | 0.352583 |
| 1.30000 | 0.04339 | 6.550e-02 | 0.327339 |
| 1.31667 | 0.04335 | 5.417e-02 | 0.302919 |
| 1.33333 | 0.04349 | 4.401e-02 | 0.279378 |
| 1.35000 | 0.04380 | 3.501e-02 | 0.256766 |
| 1.36667 | 0.04422 | 2.713e-02 | 0.235126 |
| 1.38333 | 0.04475 | 2.035e-02 | 0.214495 |
| 1.40000 | 0.04534 | 1.462e-02 | 0.194904 |
| 1.41667 | 0.04597 | 9.925e-03 | 0.176375 |
| 1.43333 | 0.04661 | 6.200e-03 | 0.158927 |
| 1.45000 | 0.04726 | 3.399e-03 | 0.142570 |
| 1.46667 | 0.04787 | 1.471e-03 | 0.127310 |
| 1.48333 | 0.04845 | 3.576e-04 | 0.113146 |
| 1.50000 | 0.04897 | 3.039e-33 | 0.100070 |
| 1.51667 | 0.04941 | 3.367e-04 | 0.088072 |
| 1.53333 | 0.04977 | 1.305e-03 | 0.077133 |
| 1.55000 | 0.05004 | 2.839e-03 | 0.067232 |
| 1.56667 | 0.05022 | 4.875e-03 | 0.058343 |
| 1.58333 | 0.05028 | 7.347e-03 | 0.050433 |
| 1.60000 | 0.05024 | 1.019e-02 | 0.043469 |
| 1.61667 | 0.05008 | 1.334e-02 | 0.037413 |
| 1.63333 | 0.04982 | 1.673e-02 | 0.032223 |
| 1.65000 | 0.04944 | 2.031e-02 | 0.027856 |
| 1.66667 | 0.04895 | 2.401e-02 | 0.024264 |
| 1.68333 | 0.04836 | 2.777e-02 | 0.021400 |
| 1.70000 | 0.04766 | 3.154e-02 | 0.019214 |
| 1.71667 | 0.04687 | 3.527e-02 | 0.017655 |
| 1.73333 | 0.04599 | 3.892e-02 | 0.016672 |
| 1.75000 | 0.04503 | 4.243e-02 | 0.016211 |
| 1.76667 | 0.04399 | 4.577e-02 | 0.016222 |
| 1.78333 | 0.04289 | 4.889e-02 | 0.016652 |
| 1.80000 | 0.04173 | 5.178e-02 | 0.017450 |
| 1.81667 | 0.04052 | 5.439e-02 | 0.018565 |
| 1.83333 | 0.03927 | 5.670e-02 | 0.019949 |
| 1.85000 | 0.03799 | 5.870e-02 | 0.021553 |
| 1.86667 | 0.03669 | 6.037e-02 | 0.023333 |
| 1.88333 | 0.03538 | 6.169e-02 | 0.025244 |
| 1.90000 | 0.03406 | 6.267e-02 | 0.027245 |
| 1.91667 | 0.03275 | 6.329e-02 | 0.029296 |
| 1.93333 | 0.03146 | 6.356e-02 | 0.031361 |
| 1.95000 | 0.03018 | 6.348e-02 | 0.033406 |
| 1.96667 | 0.02893 | 6.306e-02 | 0.035400 |
| 1.98333 | 0.02772 | 6.231e-02 | 0.037313 |
| 2.00000 | 0.02655 | 6.124e-02 | 0.039122 |
| 2.01667 | 0.02543 | 5.988e-02 | 0.040802 |
| 2.03333 | 0.02435 | 5.824e-02 | 0.042336 |
| 2.05000 | 0.02334 | 5.634e-02 | 0.043705 |
| 2.06667 | 0.02239 | 5.421e-02 | 0.044897 |
| 2.08333 | 0.02149 | 5.187e-02 | 0.045900 |
| 2.10000 | 0.02067 | 4.936e-02 | 0.046707 |
| 2.11667 | 0.01991 | 4.669e-02 | 0.047312 |
| 2.13333 | 0.01922 | 4.389e-02 | 0.047713 |
| 2.15000 | 0.01859 | 4.100e-02 | 0.047909 |
| 2.16667 | 0.01803 | 3.804e-02 | 0.047901 |
| 2.18333 | 0.01754 | 3.505e-02 | 0.047695 |
| 2.20000 | 0.01712 | 3.204e-02 | 0.047296 |
| 2.21667 | 0.01676 | 2.906e-02 | 0.046713 |
| 2.23333 | 0.01646 | 2.611e-02 | 0.045954 |
| 2.25000 | 0.01621 | 2.324e-02 | 0.045032 |
| 2.26667 | 0.01602 | 2.046e-02 | 0.043957 |
| 2.28333 | 0.01588 | 1.780e-02 | 0.042744 |
| 2.30000 | 0.01579 | 1.527e-02 | 0.041408 |
| 2.31667 | 0.01574 | 1.290e-02 | 0.039962 |
| 2.33333 | 0.01573 | 1.069e-02 | 0.038423 |
| 2.35000 | 0.01575 | 8.673e-03 | 0.036806 |
| 2.36667 | 0.01580 | 6.850e-03 | 0.035128 |
| 2.38333 | 0.01588 | 5.234e-03 | 0.033404 |
| 2.40000 | 0.01597 | 3.831e-03 | 0.031650 |
| 2.41667 | 0.01608 | 2.645e-03 | 0.029883 |
| 2.43333 | 0.01620 | 1.681e-03 | 0.028116 |
| 2.45000 | 0.01633 | 9.370e-04 | 0.026365 |
| 2.46667 | 0.01646 | 4.120e-04 | 0.024643 |
| 2.48333 | 0.01658 | 1.017e-04 | 0.022964 |
| 2.50000 | 0.01671 | 3.039e-33 | 0.021339 |
| 2.51667 | 0.01682 | 9.874e-05 | 0.019779 |
| 2.53333 | 0.01692 | 3.881e-04 | 0.018294 |
| 2.55000 | 0.01700 | 8.565e-04 | 0.016894 |
| 2.56667 | 0.01707 | 1.491e-03 | 0.015585 |
| 2.58333 | 0.01711 | 2.278e-03 | 0.014375 |
| 2.60000 | 0.01714 | 3.201e-03 | 0.013269 |
| 2.61667 | 0.01714 | 4.244e-03 | 0.012271 |
| 2.63333 | 0.01711 | 5.390e-03 | 0.011383 |
| 2.65000 | 0.01706 | 6.622e-03 | 0.010608 |
| 2.66667 | 0.01698 | 7.922e-03 | 0.009946 |
| 2.68333 | 0.01688 | 9.271e-03 | 0.009396 |
| 2.70000 | 0.01675 | 1.065e-02 | 0.008956 |
| 2.71667 | 0.01660 | 1.204e-02 | 0.008625 |
| 2.73333 | 0.01642 | 1.343e-02 | 0.008398 |
| 2.75000 | 0.01621 | 1.480e-02 | 0.008271 |
| 2.76667 | 0.01598 | 1.613e-02 | 0.008239 |
| 2.78333 | 0.01574 | 1.741e-02 | 0.008295 |
| 2.80000 | 0.01547 | 1.863e-02 | 0.008433 |
| 2.81667 | 0.01518 | 1.976e-02 | 0.008646 |
| 2.83333 | 0.01488 | 2.080e-02 | 0.008926 |
| 2.85000 | 0.01456 | 2.173e-02 | 0.009265 |
| 2.86667 | 0.01424 | 2.256e-02 | 0.009656 |
| 2.88333 | 0.01390 | 2.326e-02 | 0.010089 |
| 2.90000 | 0.01356 | 2.384e-02 | 0.010555 |
| 2.91667 | 0.01321 | 2.428e-02 | 0.011048 |
| 2.93333 | 0.01286 | 2.460e-02 | 0.011558 |
| 2.95000 | 0.01252 | 2.477e-02 | 0.012077 |
| 2.96667 | 0.01217 | 2.481e-02 | 0.012597 |
| 2.98333 | 0.01183 | 2.471e-02 | 0.013110 |
| 3.00000 | 0.01150 | 2.448e-02 | 0.013610 |
The graph shows what you would see at three separations. The sources are independent, so the brightness of their two patterns simply adds. At the Rayleigh separation the combined pattern has a clear dip between two peaks. Closer together, the dip disappears and the two images look like one.
A smaller limiting angle means better resolution. So a wider aperture, or a shorter wavelength, resolves finer detail. This is why large telescopes have wide mirrors.
To decide whether a particular pair of sources is resolved, compare the angle between them with the limit. Two sources a distance s apart, at a distance R away, are separated by an angle θ ≈ s/R.
Worked example 1
From the limiting angle to a separation
Problem
A camera has an aperture of width 0.120 m and is used with light of wavelength 5.50 × 10² nm. Using the Rayleigh criterion, find the smallest angle it can resolve, and the smallest separation of two objects 2.00 km away that it can resolve.
Show full solution
Find the limiting angle
Method
Use θ ≈ λ/b with lengths in metres.Reason
This is the angle at which the two diffraction patterns just meet the criterion.Working
Answer
θ ≈ (5.50 × 10⁻⁷)/0.120 = 4.58 × 10⁻⁶ rad
Convert to a separation
Working
Answer
s ≈ Rθ = (2.00 × 10³)(5.50 × 10⁻⁷/0.120) = 9.17 × 10⁻³ m, about 9 mm.
Quick check
The limiting angle belongs to the camera. The smallest separation it can resolve also depends on how far away the objects are.
Guided practice 2
Are the two sources resolved?
Problem
Two small lamps 0.0200 m apart are viewed from 1.00 km away through an aperture 40.0 mm wide. The light has wavelength 6.00 × 10² nm. Are the lamps resolved?
Compare the two angles
Show the steps Optional working
Show the steps Optional working
Hints
Hint 1
two angles
Which two angles must you compare to decide whether the lamps are resolved?
Show solution step by step
Find both angles
Reason
Resolution depends on how the real separation compares with the limit.Working
Answer
Actual angle: 0.0200/1000 = 2.00 × 10⁻⁵ rad. Limit: (6.00 × 10⁻⁷)/0.0400 = 1.50 × 10⁻⁵ rad.
Compare
Working
Answer
The lamps are separated by more than the limiting angle, so they are resolved.
Check your understanding 1
An instrument just resolves two sources. Its aperture is then halved, and the wavelength is halved too. Are the sources still resolved?
Show answer
Yes, just. The new limit is (λ/2)/(b/2) = λ/b, unchanged, so the sources are still just resolved. The narrower aperture lets in less light, but brightness does not enter the criterion.
Spot the mistake 3
Which patterns does the criterion compare?
Learner claim
A learner looks at the combined image of two sources and says they are just resolved when the two peaks of the combined pattern are a first-minimum angle apart. What does the Rayleigh criterion actually compare?
Diagnose the claim
Show solution step by step
Separate the two patterns
Method
Draw each source’s diffraction pattern on its own.Reason
The criterion is about the individual patterns, not their sum.Working
Answer
Find the central maximum of one source’s pattern and the first minimum of the other’s.
State the condition
Working
Answer
The sources are just resolved when the central maximum of one pattern falls on the first minimum of the other.
Common mistake
The combined pattern is what you see, which is why the claim is tempting. But its peaks are not at the positions of the individual maxima, because the two patterns overlap. The criterion is defined by the individual patterns.
Exam-style question 1
Sharper, not brighter
Examination question
Explain how the resolution of a telescope could be improved without changing the wavelength of the light it receives. Explain why making the objects brighter would not improve the resolution. [3 marks]
Write your answer before viewing the mark scheme
Show solution step by step
Widen the aperture
1 markMethod
Increase the aperture width b.Reason
A smaller λ/b resolves sources that are closer together.Working
Answer
A wider aperture makes the limiting angle θ ≈ λ/b smaller.
Brightness and resolution
2 marksWorking
Answer
Brighter objects make each image brighter, but the width of each diffraction pattern, set by λ/b, is unchanged. So the patterns overlap just as much, and the limiting angle is the same.
Self-mark with the mark scheme
Tick each mark point that your answer includes.
Tick each point only if your answer states it clearly.
Check your understanding 2
Two sources are 0.0150 m apart at a range of 1.50 km. Their light has wavelength 600 nm. Using θ ≈ λ/b, find the minimum aperture width needed to just resolve them. Give the width in millimetres.
Show the steps Optional working
Show answer
The actual angle is 0.0150/1500 = 1.00 × 10⁻⁵ rad. Set λ/b = s/R, giving b = λ R/s = 0.0600 m = 60.0 mm. A wider aperture lowers the limiting angle further, whereas a narrower aperture fails the criterion.
Common mistakes
- Thinking a larger limiting angle is better. A smaller limiting angle resolves finer detail.
- Comparing the limiting angle with a separation in metres instead of the angle s/R.
- Using degrees in θ ≈ λ/b. The result is in radians.
- Thinking a brighter image is a sharper one.
Before you move on
Check your understanding 3
Without looking back: state the Rayleigh criterion. How do you decide whether two particular sources are resolved? Name two ways to improve resolution.
Show answer
Two sources are just resolved when the central maximum of one diffraction pattern falls on the first minimum of the other. Compare the angle between them, about s/R, with the limit λ/b: they are resolved if their angle is larger. Use a wider aperture or a shorter wavelength.
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Syllabus and review details
- GCE A-Level H2 Physics 2027 · 2027
Content Overview, PDF pages 9–10; Subject Content, PDF pages 11–30
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