Include reflection phase in the optical path ledger
Thin-film interference combines a propagation difference with possible phase changes on reflection. At normal incidence, the beam that travels into and back out of a film accumulates optical path2nt, where t is physical thickness and n is the film index. Using vacuum wavelength λ0, the propagation phase is4πnt/λ0. Physical distance alone is not the optical path inside a material.
Learning outcome: Predict which thickness condition applies before counting wavelengths. Compare zero thickness and the first constructive thickness, then change the assumed substrate index and explain which relative reflection phase changes. Keep a phase-condition result separate from a calibrated reflectance claim.
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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For λ0=560 nm and n=1.4, the first constructive thickness is λ0/(4n)=100 nm. Destructive thicknesses include0 and200 nm, and constructive thickness spacing is200 nm. The interactive wavelength grid can compare nearby550 and600 nm; the worked560 nm packet is a separate exact arithmetic case.
Open the experimentPredict
With one relative reflection phase reversal, the first positive constructive reflected thickness is:
Experiment
Predict which thickness condition applies before counting wavelengths. Compare zero thickness and the first constructive thickness, then change the assumed substrate index and explain which relative reflection phase changes. Keep a phase-condition result separate from a calibrated reflectance claim.
Include the reflection phase change in a film ledger
Worked teaching model · synthetic values · Normal incidence on index1.4 film with air on both sides. One reflected contribution has a pi phase reversal. Supplied equal-amplitude two-beam comparison reports normalized resultant amplitude squared, neglecting multiple reflections and actual interface amplitude differences; not calibrated film reflectance or perceived color.
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Two-beam phase comparison versus thickness
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: one reflection phase reversal (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Two-beam phase comparison versus thickness
one reflection phase reversal
| film thickness (nm) | normalized amplitude squared |
|---|---|
| 0 | 0 |
| 7.5 | 0.0572782 |
| 15 | 0.225832 |
| 22.5 | 0.496007 |
| 30 | 0.852327 |
| 37.5 | 1.27438 |
| 45 | 1.738 |
| 52.5 | 2.21663 |
| 60 | 2.68285 |
| 67.5 | 3.10995 |
| 75 | 3.47348 |
| 82.5 | 3.75261 |
| 90 | 3.93136 |
| 97.5 | 3.99948 |
| 105 | 3.95307 |
| 112.5 | 3.7948 |
| 120 | 3.53372 |
| 127.5 | 3.18479 |
| 135 | 2.768 |
| 142.5 | 2.30723 |
| 150 | 1.82885 |
| 157.5 | 1.36028 |
| 165 | 0.928346 |
| 172.5 | 0.557798 |
| 180 | 0.269856 |
| 187.5 | 0.0810141 |
| 195 | 0.00208775 |
| 202.5 | 0.0375982 |
| 210 | 0.185511 |
| 217.5 | 0.437355 |
| 225 | 0.778704 |
| 232.5 | 1.19001 |
| 240 | 1.6477 |
| 247.5 | 2.12558 |
| 255 | 2.59626 |
| 262.5 | 3.03279 |
| 270 | 3.41017 |
| 277.5 | 3.70677 |
| 285 | 3.90561 |
| 292.5 | 3.9953 |
| 300 | 3.97071 |
| 307.5 | 3.83323 |
| 315 | 3.59075 |
| 322.5 | 3.25716 |
| 330 | 2.85156 |
| 337.5 | 2.39718 |
| 345 | 1.92005 |
| 352.5 | 1.44751 |
| 360 | 1.0066 |
| 367.5 | 0.6226 |
| 375 | 0.317493 |
| 382.5 | 0.108756 |
| 390 | 0.00834665 |
| 397.5 | 0.0220152 |
| 405 | 0.148979 |
| 412.5 | 0.381966 |
| 420 | 0.707631 |
| 427.5 | 1.10732 |
| 435 | 1.55814 |
| 442.5 | 2.03427 |
| 450 | 2.50844 |
| 457.5 | 2.95348 |
| 465 | 3.34391 |
| 472.5 | 3.65737 |
| 480 | 3.87589 |
| 487.5 | 3.98696 |
| 495 | 3.98423 |
| 502.5 | 3.86784 |
| 510 | 3.64447 |
| 517.5 | 3.3269 |
| 525 | 2.93333 |
| 532.5 | 2.48631 |
| 540 | 2.01142 |
| 547.5 | 1.53589 |
| 555 | 1.08693 |
| 562.5 | 0.690279 |
| 570 | 0.368642 |
| 577.5 | 0.140447 |
| 585 | 0.0187636 |
| 592.5 | 0.0105618 |
| 600 | 0.116311 |
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 |
|---|---|---|
| Round-trip optical path | 280 | nm |
| Propagation phase in cycles | 0.509091 | cycles |
| Total relative phase including reflection | 6.34031 | rad |
| Normalized equal-beam resultant amplitude squared | 3.99674 | ratio |
| First constructive thickness | 98.2143 | nm |
| Constructive thickness spacing | 196.429 | nm |
One relative reflection reversal exchanges the constructive and destructive thickness conditions.
At normal incidence optical path difference is2nt and propagation phase4pi nt/lambda0. Reflection from air toward the higher-index film reverses phase; reflection from film toward air does not. Thus constructive reflected interference requires2nt=(m+1/2)lambda0 and destructive requires2nt=m lambda0. If both reflections reverse or neither does, those conditions exchange. Equal-amplitude two-beam normalization is an explicitly supplied comparison, not an exact Fresnel reflectance or a perceived-color prediction.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
One relative reflection reversal exchanges the constructive and destructive thickness conditions.
At normal incidence optical path difference is2nt and propagation phase4pi nt/lambda0. Reflection from air toward the higher-index film reverses phase; reflection from film toward air does not. Thus constructive reflected interference requires2nt=(m+1/2)lambda0 and destructive requires2nt=m lambda0. If both reflections reverse or neither does, those conditions exchange. Equal-amplitude two-beam normalization is an explicitly supplied comparison, not an exact Fresnel reflectance or a perceived-color prediction.
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