SCIENCE · INVESTIGATE AND EXPLAIN

Connect optical path difference to bright and dark fringes

Predict two-slit interference from phase and compare amplitude with intensity.

Learning outcome: Relate integer and half-integer path differences to interference and explain the model’s coherence assumptions.

Starting recap

Wavelength is the distance for one cycle. Phase describes position within a cycle; a half-cycle offset reverses an oscillation.

All lessons are open directly. Your saved experiments remain when you leave and return.

Try

Why can two illuminated openings produce a dark band?

Open the experiment

Predict

At separation 0.20 mm and wavelength 500 nm, predict whether path differences 0, 250 and 500 nm produce bright or dark bands.

Your prediction and reason

Experiment

Inspect the first bright and dark feature rows at the default setting. Change wavelength to 700 nm and compare the feature positions while keeping separation fixed.

Compare path difference and interference fringes

Worked teaching model · synthetic values · Two equally strong coherent in-phase pointlike slit contributions; individual slit width and its diffraction envelope are neglected. 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.

Slit center separation: 0.2 mm Wavelength: 500 nm

Reset keeps your prediction, captures and practice history.

Positive-side feature positions and path differences
Feature	Path difference / wavelength	Position (mm)	Normalized intensity	Within plot?
Bright order 1	1	5.000016	1	Yes
Bright order 2	2	10.000125	1	Yes
Bright order 3	3	15.000422	1	Yes
Dark band 1	0.5	2.500002	0	Yes
Dark band 2	1.5	7.500053	0	Yes
Dark band 3	2.5	12.500244	0	Yes

Path difference is between the two slit centers. Bright orders have integer wavelength differences; dark bands have half-integer differences. 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
Slit center separation	0.200 mm	Distance between two narrow openings
Wavelength	500 nm	Same wavelength for all interfering contributions
First positive feature	5.000016 mm	Bright order one
Small-angle estimate	5.000000 mm	L × wavelength / separation
Relative approximation difference	0.000313%	Relative to retained screen geometry; not experimental uncertainty

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 bright fringe at 5.000016 mm; small-angle spacing estimate 5.000000 mm.

Small-angle estimate for the first positive feature differs by 0.000313% from retained screen geometry. Intensity is normalized separately for each scenario; peak height is not transmitted power.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Two contributions can cancel at one place

The model uses two equally strong, narrow slit contributions illuminated in phase. They have the same wavelength and a stable phase relationship: they are coherent. At the center of the screen both paths are equal, so the contributions reinforce. Away from the center their path lengths differ.

At wavelength 500 nm, a path difference of 500 nm is one full cycle, so the waves arrive in phase again. A path difference of 250 nm is half a cycle: one contribution is opposite to the other and equal amplitudes cancel. Read these conditions in the feature table before changing either control.

For the default separation 0.20 mm and screen distance 2 m, the first positive bright fringe is at 5.000016 mm. The first dark band is about 2.500002 mm from the center. Symmetric partners occur at negative positions. The first dark band is not at the first integer-wavelength path difference.

Turn path difference into phase

Let δ be path difference and λ wavelength, in the same length unit. The fraction δ/λ counts how many cycles one path adds. Phase difference is φ = 2πδ/λ radians: one cycle is 2π radians, and half a cycle is π radians. A radian is an angle unit; the phase angle describes an oscillation, not a ray’s direction on the diagram.

For integer m, δ = mλ gives bright fringes. For integer m, δ = (m + 1/2)λ gives dark bands in this equal-amplitude, in-phase-source model. At m = 0 the bright condition is the center; the first positive dark condition has m = 0 in the half-integer expression.

The far-field path-difference model is δ = d sin θ, where d is slit-center separation and θ is screen angle from the central direction. The screen is L = 2 m away. For horizontal screen coordinate x, sin θ = x/√(L² + x²). Convert x, L and d to compatible units before combining them. Keeping this geometric sine does not remove the far-field approximation used for the wave contributions.

Add amplitudes before finding intensity

Equal waves with amplitude E can sum to amplitude 2E when in phase, or zero when opposite. Intensity is proportional to the square of the resulting wave amplitude. In this ideal case the normalized pattern is I/I₀ = cos²(φ/2), where I₀ is the central intensity. At phase zero this is one; at phase π it is zero; at phase π/2 it is one-half.

A dark location does not mean the waves destroy energy everywhere. The pattern redistributes the intensity over positions. This finite, separately normalized plot is not a total-power accounting calculation. Its central value is fixed at one independently for every setting, so equal peak heights across different settings do not establish equal optical power.

The plotted curve is the numerical intensity ratio. The strip below encodes those values as display brightness on black; it is not a photograph, an exposure simulation or a prediction of human brightness perception. Use the graph and numerical table for comparisons.

Identify what was simplified

The two slits are treated as pointlike contributions. Their individual finite widths and resulting diffraction envelope are omitted, so this model’s bright fringes do not fade under a finite-width envelope. Real finite slits combine both effects. Unequal amplitudes or loss of coherence would also alter the contrast; neither is a control here.

This is a classical wave-intensity calculation. It does not simulate individual photon detections or establish conclusions about a quantum measurement experiment. A ray-only image calculation cannot produce the alternating intensity pattern shown here.

Transfer a phase claim

For wavelength 600 nm, classify path differences 300, 600 and 900 nm. Then find the normalized intensity for a 150 nm path difference in the ideal equal-amplitude model.

Check the interference transfer

300 nm and 900 nm are half-integer wavelength differences and are dark. 600 nm is one full wavelength and is bright. At 150 nm, δ/λ = 1/4 and φ = π/2, so I/I₀ = cos²(π/4) = 1/2.

OpenStax: Young’s double-slit interference introduces the coherent-source model. Continue to fringe spacing and inference to connect the pattern with the slit separation.

Apply

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

Explain amplitude cancellation, coherence and why normalized intensity does not measure total transmitted power.

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Estimate focal length from repeated image measurements Next: Infer slit separation from a fringe pattern

Save a science backup or import one

Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.

A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.

Choose a science backup (maximum 36 MB)

Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.

Complete file sets and course backups

A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.

Export scope

The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.

Choose the manifest and all parts together

0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.

Science storage: loading. No account or remote computation is required.