SCIENCE · INVESTIGATE AND EXPLAIN

Explain the broad central band from a single slit

Compare slit width, wavelength and the dark minima of a diffraction pattern.

Learning outcome: Predict single-slit minima and central width while distinguishing width from two-slit separation.

Starting recap

Different points across one illuminated opening contribute waves. Their amplitudes can reinforce or cancel.

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Try

Can one opening produce dark bands without a second slit?

Open the experiment

Predict

At width 0.040 mm and wavelength 500 nm, predict how the central band changes when width doubles.

Your prediction and reason

Experiment

Capture widths 0.040 and 0.080 mm at 500 nm. Compare the first dark minimum and central width. Then use 0.020 mm and 700 nm to inspect features outside the plotted window.

Resolve diffraction from one finite slit

Worked teaching model · synthetic values · One uniformly illuminated slit of finite width; far-field sinc-squared intensity from interference across that width. Classical monochromatic scalar waves, normal incidence and ideal far-field angular interference at a screen 2 m away. Screen coordinate x uses sin(theta) = x / sqrt(L² + x²); retaining this screen geometry does not make the wave model an exact near-field calculation. Fixed screen range −80 to +80 mm; normalized intensity 0–1 with central intensity defined as one independently for each setting. No absolute power comparison, photon detection, noise, polarization mismatch or incoherence. The strip is a display encoding of model values, not a photograph or measured brightness. Features outside the finite plot remain listed.

Single slit width: 0.04 mm Wavelength: 500 nm

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Positive-side feature positions and path differences
Feature	Path difference / wavelength	Position (mm)	Within plot?
Dark order 1	1	25.001953	Yes
Dark order 2	2	50.015632	Yes
Dark order 3	3	75.052790	Yes

Path difference is between the edges of the single slit. Nonzero integer orders give dark bands; order zero is the bright center, not a minimum. Negative-side features mirror the positive side. The finite sample grid may not land exactly on an analytic maximum or minimum.

Pattern scale and approximation
Quantity	Value	Meaning
Single slit width	0.040 mm	Width of one opening
Wavelength	500 nm	Same wavelength for all interfering contributions
First positive feature	25.001953 mm	First dark minimum
Small-angle estimate	25.000000 mm	L × wavelength / width
Relative approximation difference	0.007813%	Relative to retained screen geometry; not experimental uncertainty
Central maximum width	50.003907 mm	Distance between the first two dark minima

Every setting has its own central normalization I₀. Changing slit width or separation and seeing the same peak value does not establish equal total power. The two-slit model deliberately omits the finite-width envelope that would modulate real double-slit fringes.

First positive minimum at 25.001953 mm; central maximum width 50.003907 mm.

Small-angle estimate for the first positive feature differs by 0.007813% from retained screen geometry. Intensity is normalized separately for each scenario; peak height is not transmitted power.

Observation to attach to a capture

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Understand

One opening contains many wave contributions

A finite-width opening does not transmit only one ray. In this model a uniform incident wave illuminates every part of a slit in phase. Contributions from different points across that width travel different distances toward an off-axis screen location. They can cancel even though there is only one slit.

At width a = 0.040 mm and wavelength λ = 500 nm, the first positive dark minimum lies at 25.001953 mm. The negative partner is at −25.001953 mm. The bright central band’s width, measured between these two minima, is therefore 50.003907 mm. This definition is not a full-width-at-half-maximum measurement and is not the physical width of the opening.

Pair contributions at the first minimum

Let θ be the screen angle from the central direction. At a sin θ = λ, contributions separated across half the slit width have a path difference (a/2) sin θ = λ/2. Every contribution in one half has an opposite-phase partner in the other half, so the resulting amplitude cancels.

More generally, dark minima satisfy a sin θ = mλ, with nonzero integer m. The bright center is m = 0 in the path-difference expression, but it is not a dark minimum: all contributions there are in phase. Do not mechanically substitute zero into the dark-order list.

Compare this with two-slit interference. Integer-wavelength differences between two slit centers give bright fringes, whereas nonzero integer edge-to-edge differences across one uniformly illuminated slit give minima. The geometry and number of contributing points differ. The symbol a here is one opening’s width; d in the two-slit lesson was center separation.

Predict pattern width before moving the slider

Double slit width from 0.040 to 0.080 mm at 500 nm. The first minimum moves inward to 12.500244 mm, giving central width 25.000488 mm. A wider slit produces a narrower central angular pattern. Narrowing the opening broadens the normalized pattern; that does not establish an increase in transmitted power.

Near the axis, first-minimum position is approximately x₁ ≈ λL/a, so central width is approximately 2λL/a. The model holds L = 2 m. Increasing wavelength broadens the central band at fixed width. The plot retains the same horizontal scale in every case, avoiding an automatic zoom that could hide this change.

At width 0.020 mm and wavelength 700 nm, the first minimum is 70.042914 mm and the central width is 140.085829 mm. Higher positive minima lie outside +80 mm. The table labels these as outside the plot; absence from the visible curve is not evidence that they do not exist.

Read intensity without mistaking a limit for division by zero

The model’s normalized intensity is I/I₀ = [sin β/β]², where β = πa sin θ/λ is dimensionless. This expression is often called sinc-squared. At the center β = 0, the limiting ratio is one; the implementation handles that center explicitly rather than producing 0/0.

Side bands are weaker than the central peak. Their maxima are not generally halfway between adjacent dark minima. The browser lists analytic minima, not invented equally spaced side-peak positions. The same 0–1 intensity scale is used throughout, so weak side bands remain weak rather than being individually rescaled.

I₀ is defined separately for each setting. A central value of one in every plot does not mean a narrower slit transmits the same total power. The finite plot is not an energy integral. A real double-slit pattern combines an interference pattern with a finite-width diffraction envelope; the separate two-point model intentionally leaves that envelope out.

Transfer to an unfamiliar opening

For a = 0.050 mm, λ = 600 nm and L = 1.5 m, estimate the first minimum and central width using the small-angle rule. Then predict what happens if width halves while wavelength and L stay fixed.

Check the diffraction transfer

x₁ ≈ (600 × 10⁻⁹ × 1.5)/(0.050 × 10⁻³) = 0.018 m = 18 mm. Central width is approximately 36 mm. Halving a doubles both values to approximately 36 mm and 72 mm. The estimate concerns pattern width, not total transmitted power.

OpenStax: single-slit diffraction develops the cancellation condition, and single-slit intensity gives the angular intensity expression. These are ideal far-field predictions, not a near-field solver or photon-counting simulation.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Explain why order zero is bright, why narrower slits produce a broader normalized central band, and why this does not imply more transmitted power.

Your explanation and limitations

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Previous: Infer slit separation from a fringe pattern Next: Predict attraction and repulsion between charges

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