Diffraction
Diffraction is the spreading and interference produced when a wave passes through a finite opening or around an obstacle. A narrower aperture does not merely transmit a narrower geometric beam. Different parts of the opening contribute different phases, so the outgoing field forms an angular pattern.
Diffraction becomes important when an aperture or feature is not enormously larger than the wavelength. It places a fundamental wave limit on imaging resolution even when the lens and detector are otherwise perfect.
Huygens–Fresnel viewpoint
Every point across a wavefront can be treated as contributing a secondary wave. At an observation point, those contributions arrive with phases set by their path lengths. The combined amplitude is their superposition.
For an aperture coordinate \(x'\) and observation point \(x\), a scalar diffraction calculation has the schematic form
\[U(x)\propto \int_{\mathrm{aperture}} A(x')e^{ikR(x,x')}\,W(x,x')\,dx'\]The aperture function \(A(x')\) can contain amplitude and phase. The distance is \(R\), and \(W\) depends on the chosen approximation. Intensity is proportional to \(|U|^2\).
Fresnel and Fraunhofer regimes
Near-field, or Fresnel, diffraction retains curvature in the propagation paths. Far-field, or Fraunhofer, diffraction uses an approximation in which observation direction determines a nearly linear phase across the aperture. A lens can form a Fraunhofer pattern in its focal plane even when the screen is not physically very far away.
A useful scale is the Fresnel number
\[N_F=\frac{a^2}{\lambda L}\]Here \(a\) is an aperture scale and \(L\) is the observation distance. A value much smaller than one suggests a far-field regime, while values near or above one require care. Exact thresholds depend on geometry and desired accuracy.
Single-slit far-field pattern
For a uniformly illuminated slit of width \(a\), the Fraunhofer intensity is
\[I(\theta)=I(0) \left(\frac{\sin\beta}{\beta}\right)^2, \qquad \beta=\frac{\pi a\sin\theta}{\lambda}\]At the center, the ratio has the limit one even though direct substitution gives zero over zero:
\[\lim_{\beta\to0}\frac{\sin\beta}{\beta}=1\]The minima occur when
\[a\sin\theta_m=m\lambda, \qquad m=\pm1,\pm2,\ldots\]The central maximum lies between the first minima on either side and is twice the angular width of the neighboring maxima in the small-angle approximation.
Central-maximum width
For small angles, the first minima satisfy \(\theta_1\approx\lambda/a\). On a distant screen at distance \(L\), their positions are approximately
\[y_1\approx\frac{\lambda L}{a}\]so the central width between negative and positive first minima is
\[w_{\mathrm{central}} \approx\frac{2\lambda L}{a}\]Figure 1 compares the same wavelength and screen distance for two slit widths. Read the central-maximum width between the first zeros, not across the whole visible pattern: the upper bracket is labeled W and the lower one 2W. Halving the slit width doubles that span. Increasing wavelength also broadens the pattern, contrary to a purely geometric “narrow opening makes a narrow beam” intuition.
Both curves in Figure 1 are normalized to their own central intensities. Their equal drawn peak heights do not mean equal transmitted power or equal absolute brightness.
Worked slit example
Let a slit have width \(a=0.20\ \mathrm{mm}\) and use light of wavelength \(\lambda=500\ \mathrm{nm}\). The small first-minimum angle is
\[\theta_1\approx\frac{500\times10^{-9}\ \mathrm m} {0.20\times10^{-3}\ \mathrm m} =2.5\times10^{-3}\ \mathrm{rad}\]At a screen two metres away, the first minimum is about five millimetres from the center, so the central maximum is about ten millimetres wide.
This is the upper-row width W in Figure 1. If the slit is narrowed to 0.10 millimetres with wavelength and screen distance unchanged, the first minima move to about ten millimetres on either side and the central maximum becomes twenty millimetres wide, matching the lower-row 2W comparison.
Circular aperture and Airy pattern
A uniformly illuminated circular aperture produces an Airy pattern rather than a one- dimensional sinc-squared pattern. The first dark ring occurs at approximately
\[\sin\theta_1=1.22\frac{\lambda}{D}\]where \(D\) is the diameter. For small angles, the radius in a focal plane of focal length \(f\) is
\[r_1\approx1.22\frac{\lambda f}{D}\]A larger aperture makes the central spot narrower and improves diffraction-limited angular resolution.
Rayleigh resolution criterion
Two equally bright point sources imaged through a circular aperture are conventionally called just resolved when the central maximum of one Airy pattern falls at the first minimum of the other. The Rayleigh angular criterion is
\[\theta_R\approx1.22\frac{\lambda}{D}\]This is a convention for distinguishability, not a sharp law of information. Signal-to- noise ratio, detector sampling, prior knowledge, and image processing affect practical resolution, but none removes the physical diffraction pattern.
Diffraction grating
For many equally spaced narrow slits with separation \(d\), principal maxima satisfy
\[d\sin\theta_m=m\lambda\]More illuminated slits make principal maxima narrower, improving separation of nearby wavelengths. The highest possible order must satisfy \(|m\lambda/d|\le1\). Finite slit width supplies a diffraction envelope that can suppress some orders.
The resolving power of an ideal grating is
\[\mathcal R=\frac{\lambda}{\Delta\lambda}=mN\]where \(N\) is the number of illuminated grooves and \(m\) is the order used.
Aperture as a Fourier transform
In the Fraunhofer approximation, the angular field is proportional to the Fourier transform of the complex aperture function. A rectangular opening transforms to sinc- like structure; a circular opening transforms to the Airy pattern. Phase variation across an aperture changes the far-field pattern even when transmitted intensity across the aperture is unchanged.
import numpy as np
wavelength, width = 500e-9, 0.1e-3
aperture_x = np.linspace(-width / 2, width / 2, 1001)
angles = np.linspace(-0.02, 0.02, 201)
intensities = []
for angle in angles:
phase = 2 * np.pi / wavelength * aperture_x * np.sin(angle)
amplitude = np.trapezoid(np.exp(1j * phase), aperture_x)
intensities.append(abs(amplitude / width) ** 2)
reference = np.sinc(width * np.sin(angles) / wavelength) ** 2
print("Largest quadrature error:", np.max(abs(np.array(intensities) - reference)))
Refine the aperture sampling and observation-angle sampling separately. A smooth plotted curve can still have incorrectly located minima if the aperture is under-resolved or the transform convention is wrong.
Babinet's principle
Under scalar diffraction conditions, an opaque object and a complementary aperture have related diffraction fields away from the direct unblocked beam. A narrow wire can therefore produce a diffraction pattern related to that of a slit of the same width. Babinet's principle is useful for reasoning about obstacles, but it does not make the fields identical everywhere, particularly in the direct forward direction.
Diffraction is not ordinary geometric blur
Geometric blur and diffraction can both broaden an image, but they respond differently to aperture size. In an imperfect lens, reducing the aperture often blocks highly aberrated outer rays and sharpens the geometric image. Diffraction moves the other way: reducing the aperture broadens the central wave pattern. Real optical systems therefore have an aperture range where the combined blur is smallest rather than improving without limit as the opening closes.
Defocus also differs from diffraction. Defocus moves the detector away from the image plane and spreads rays according to geometry. A diffraction pattern remains even at perfect focus because a finite pupil transmits only a limited range of propagation directions. A measured spot can include both effects, so fitting it with an Airy pattern without verifying focus and aberrations can give a misleading aperture estimate.
Detector sampling introduces another limit. If pixels are too widely spaced to resolve the diffraction pattern, the recorded image can alias fine structure or make the central spot appear artificially narrow or wide. Increasing the numerical sampling of a plotted formula does not recover information that a physical detector never measured. Optical resolution, detector resolution, and computational grid resolution are separate entries in the uncertainty budget.
The aperture illumination also matters. The standard sinc and Airy formulas assume uniform amplitude and phase across the opening. A tapered amplitude can lower side lobes while broadening the central maximum. A phase error can move energy away from the ideal central spot. This tradeoff is used intentionally in antenna and optical pupil design and occurs unintentionally through aberration or imperfect surfaces.
Verification and limits
The single-slit pattern must be symmetric about zero angle for uniform normal illumination. Its central limit must be finite. The first minimum must move outward when wavelength increases or slit width decreases. The circular-aperture result must scale inversely with diameter. Numerical aperture integration should converge in vector or complex amplitude, not only in a normalized picture.
These formulas assume coherent monochromatic scalar waves, ideal apertures, and stated near- or far-field approximations. Real systems include polarization, aperture thickness, edge roughness, lens aberrations, material response, partial coherence, spectral bandwidth, detector pixel size, and noise. When features approach wavelength and vector boundary conditions matter, scalar diffraction can be insufficient.
A worked Python notebook
Resolve the far-field pattern of a 0.1 millimetre slit illuminated at 500 nanometres. The question for this calculation is precise: how does normalised single-slit intensity change when we change observation angle, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.
The positive-angle interval avoids the removable zero-over-zero expression at the central maximum. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0.0001 to 0.02 rad. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.
Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.
State the relationship before choosing a graph
In this small example, q stands for observation angle, expressed in rad. The output is
normalised single-slit intensity, expressed in dimensionless. The equation below uses the numerical values in those
units. Any fixed coefficients therefore carry the units needed to make the relation
consistent. A value supplied in another unit must be converted before it enters this
expression.
This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.
At 0.004 rad, the expression gives 0.0546976 dimensionless. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.
import numpy as np
import sympy as sp
import matplotlib.pyplot as plt
q = sp.Symbol("q", real=True)
expression = sp.sympify(
"(sin(pi*0.0001*sin(q)/500e-9)/(pi*0.0001*sin(q)/500e-9))**2", locals={"q": q}
)
reference_input = 0.004
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "rad")
print("Predicted normalised single-slit intensity:", reference_output, "dimensionless")
sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement
to that symbol. subs then replaces the symbol with our chosen input, and float
converts the resulting symbolic number into a standard Python numerical value.
Keeping these steps separate lets us inspect the equation before numerical substitution.
It also prevents a later plot from becoming the only surviving record of the model.
Compare a few cases you can explain
Before drawing hundreds of points, calculate three. The lower endpoint gives 0.998685 dimensionless. The midpoint, at 0.01005 rad, gives 2.45774e-05 dimensionless. The upper endpoint gives 4.44486e-09 dimensionless. Read these together with the assumptions: they describe different choices of observation angle within the same example, with the other inputs held fixed.
These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.
evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0001, 0.01005, 0.02])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
print(f"{input_value:.6g} rad -> {output_value:.6g} dimensionless")
assert np.all(np.isfinite(outputs))
lambdify turns the symbolic expression into a numerical function. We explicitly select
NumPy as its numerical backend so it can accept an array. The same expression now has
two useful forms: one for algebra and one for calculating many cases. The assertion
checks that the chosen inputs produce finite outputs. It would catch an infinity or an
undefined numerical result, but a finite value can still be physically inappropriate.
Make a graph that answers the question
The horizontal axis shows observation angle; the vertical axis shows normalised single-slit intensity. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.
input_grid = np.linspace(0.0001, 0.02, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Observation angle (rad)")
ax.set_ylabel("Normalised single-slit intensity (dimensionless)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()
Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.
Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.
Measure the local sensitivity
A derivative asks a more specific question than the whole graph: how quickly does normalised single-slit intensity change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.004 rad is -121.952 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.
\[f'(q) = \frac{0.01 \sin{\left(200 \pi \sin{\left(q \right)} \right)} \cos{\left(q \right)} \cos{\left(200 \pi \sin{\left(q \right)} \right)}}{\pi \sin^{2}{\left(q \right)}} - \frac{5.0 \cdot 10^{-5} \sin^{2}{\left(200 \pi \sin{\left(q \right)} \right)} \cos{\left(q \right)}}{\pi^{2} \sin^{3}{\left(q \right)}}\]For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.
slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.0040995
linear_prediction = reference_output + slope_at_reference * (
changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "dimensionless")
print("Full expression:", full_prediction, "dimensionless")
print("Approximation difference:", full_prediction - linear_prediction, "dimensionless")
Here the changed input is 0.0040995 rad. The tangent estimate is 0.0425633 dimensionless, while the full expression gives 0.0433181 dimensionless. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.
This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.
Check the derivative with numerical differences
We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.
The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.
input_span = 0.0199
for fraction in [1e-2, 1e-3, 1e-4]:
step = input_span * fraction
upper = float(evaluate(reference_input + step))
lower = float(evaluate(reference_input - step))
numerical_slope = (upper - lower) / (2 * step)
difference = numerical_slope - slope_at_reference
print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")
For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.
An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.
Change the example and preserve the reasoning
Make one copy of the reference calculation before editing it. Choose a second value of observation angle inside the stated interval, predict whether normalised single-slit intensity will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.
Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.
Finally, return to the physical limit stated at the start: The positive-angle interval avoids the removable zero-over-zero expression at the central maximum. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.
Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.
Recover the output change by integrating the slope
There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of dimensionless: the input unit in the integration cancels the input unit in the denominator of the slope.
This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.
slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(0.02) - evaluate(0.0001))
for points in [101, 201, 401]:
grid = np.linspace(0.0001, 0.02, points)
slopes = np.asarray(slope_function(grid), dtype=float)
slopes = np.broadcast_to(slopes, grid.shape)
integrated_change = np.trapezoid(slopes, grid)
print(
points,
"integrated change:",
integrated_change,
"endpoint change:",
endpoint_change,
)
The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.
Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.
Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.