Separate interference order from finite-width brightness
A principal interference direction is a phase condition, not a guarantee of a bright observed spot. Equal coherent slits separated by d have constructive path differences d sinθ=mλ and destructive differences (m+1/2)λ. The angle is measured from the central axis. A screen perpendicular to that axis at distance L records y=L tanθ, whereas y≈Lmλ/d adds a small-angle approximation.
Learning outcome: Compute order1 and4 locations for two wavelengths, retaining the approximation error separately from the envelope factor. Predict the first two-slit dark direction and explain why it differs from a single-slit first zero. Propose which geometry change would move a missing order.
Starting recap
Use normal-based angles, signed conjugate distances and explicit optical-path/reflection phases. Separate geometric direction, amplitude and energy; preserve focal and critical boundaries.
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At λ=600 nm, order1 has sinθ=.03 and exact screen position on a2 m screen is2(.03)/√(1−.03²)≈.0600270 m. The small-angle value is .0600000 m. Order4 has sinθ=.12 but w sinθ=600 nm, so the ideal envelope vanishes there.
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With center spacing20 µm and slit width5 µm, why is principal order4 missing?
Experiment
Compute order1 and4 locations for two wavelengths, retaining the approximation error separately from the envelope factor. Predict the first two-slit dark direction and explain why it differs from a single-slit first zero. Propose which geometry change would move a missing order.
Separate interference orders from the diffraction envelope
Worked teaching model · synthetic values · Normal coherent illumination of equal slits with center spacing20 µm and width5 µm, screen perpendicular at2 m. Principal orders satisfy sin(theta)=m lambda/d. Exact planar-screen position is L tan(theta); finite-width envelope can suppress an allowed order. Far-field scalar approximation, no laser activity.
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Exact and approximate order locations
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: exact planar-screen geometry (solid).2: small-angle geometry (dashed). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Exact and approximate order locations
exact planar-screen geometry
| principal order number | screen displacement (m) |
|---|---|
| 0 | 0 |
| 1 | 0.060027 |
| 2 | 0.120217 |
| 3 | 0.180733 |
| 4 | 0.241747 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
small-angle geometry
| principal order number | screen displacement (m) |
|---|---|
| 0 | 0 |
| 1 | 0.06 |
| 2 | 0.12 |
| 3 | 0.18 |
| 4 | 0.24 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Principal-order angle | 1.71913 | degrees |
| Exact screen displacement | 0.060027 | m |
| Small-angle screen displacement | 0.06 | m |
| Small-angle minus exact displacement | -0.0000270182 | m |
| Single-slit envelope factor at principal order | 0.810569 | ratio |
| Missing principal order | 0 | flag |
| First two-slit dark angle | 0.859469 | degrees |
An interference condition can coincide with a diffraction zero; an allowed geometric order is not necessarily bright.
Equal coherent two-slit maxima have d sin(theta)=m lambda, and minima have (m+1/2)lambda. A finite slit width supplies a squared-sinc envelope with zeros w sin(theta)=p lambda. Since d/w=4, principal order4 coincides with the first envelope zero. Increasing coherent slit count sharpens principal peaks under the ideal grating model but does not remove this same finite-width zero. L tan(theta) is exact screen geometry within the far-field angular model, not a replacement for its approximation assumptions.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
An interference condition can coincide with a diffraction zero; an allowed geometric order is not necessarily bright.
Equal coherent two-slit maxima have d sin(theta)=m lambda, and minima have (m+1/2)lambda. A finite slit width supplies a squared-sinc envelope with zeros w sin(theta)=p lambda. Since d/w=4, principal order4 coincides with the first envelope zero. Increasing coherent slit count sharpens principal peaks under the ideal grating model but does not remove this same finite-width zero. L tan(theta) is exact screen geometry within the far-field angular model, not a replacement for its approximation assumptions.
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