Apply each polarizer to the light reaching it
Polarization describes the direction of a transverse wave field. An ideal first linear polarizer transmits half the intensity of unpolarized input. Once the light is linearly polarized, a second ideal filter transmits a fraction cos²θ, where θ is the angle between its axis and the incoming polarization. That angle must be updated after each filter rather than repeatedly measured from the original source.
Learning outcome: Compare θ=0°,45°,90° with β=90°. Retain each intermediate intensity and its relative projection angle. Then place infrared and ultraviolet on opposite sides of visible light and predict which has the larger vacuum wavelength without assigning a health threshold.
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 θ=45° and β=90°, the successive intensities are5,2.5,1.25 W/m². Directly crossed first and third filters would give0. At θ=90°, the second filter already removes all ideal input and no third orientation restores it. Violet has higher frequency and shorter vacuum wavelength than red under the stated ordering.
Open the experimentPredict
Why can a45° intermediate polarizer allow light through crossed first and final axes?
Experiment
Compare θ=0°,45°,90° with β=90°. Retain each intermediate intensity and its relative projection angle. Then place infrared and ultraviolet on opposite sides of visible light and predict which has the larger vacuum wavelength without assigning a health threshold.
Track each projection relative to the preceding polarizer
Worked teaching model · synthetic values · Unpolarized input intensity10 W/m². First ideal polarizer passes half; second and third ideal axes are specified relative to first. Malus projections use angle differences, not all absolute angles. No optical exposure, real filter loss or biological vision model.
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Intermediate-axis dependence with fixed final axis
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: three-polarizer chain (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Intermediate-axis dependence with fixed final axis
three-polarizer chain
| second axis from first (degrees) | final intensity (W/m²) |
|---|---|
| 0 | 0 |
| 1.125 | 0.00192667 |
| 2.25 | 0.00769479 |
| 3.375 | 0.0172688 |
| 4.5 | 0.0305897 |
| 5.625 | 0.0475753 |
| 6.75 | 0.0681209 |
| 7.875 | 0.0920999 |
| 9 | 0.119364 |
| 10.125 | 0.149746 |
| 11.25 | 0.183058 |
| 12.375 | 0.219095 |
| 13.5 | 0.257634 |
| 14.625 | 0.298438 |
| 15.75 | 0.341256 |
| 16.875 | 0.385823 |
| 18 | 0.431864 |
| 19.125 | 0.479097 |
| 20.25 | 0.527228 |
| 21.375 | 0.575963 |
| 22.5 | 0.625 |
| 23.625 | 0.674037 |
| 24.75 | 0.722772 |
| 25.875 | 0.770903 |
| 27 | 0.818136 |
| 28.125 | 0.864177 |
| 29.25 | 0.908744 |
| 30.375 | 0.951562 |
| 31.5 | 0.992366 |
| 32.625 | 1.03091 |
| 33.75 | 1.06694 |
| 34.875 | 1.10025 |
| 36 | 1.13064 |
| 37.125 | 1.1579 |
| 38.25 | 1.18188 |
| 39.375 | 1.20242 |
| 40.5 | 1.21941 |
| 41.625 | 1.23273 |
| 42.75 | 1.24231 |
| 43.875 | 1.24807 |
| 45 | 1.25 |
| 46.125 | 1.24807 |
| 47.25 | 1.24231 |
| 48.375 | 1.23273 |
| 49.5 | 1.21941 |
| 50.625 | 1.20242 |
| 51.75 | 1.18188 |
| 52.875 | 1.1579 |
| 54 | 1.13064 |
| 55.125 | 1.10025 |
| 56.25 | 1.06694 |
| 57.375 | 1.03091 |
| 58.5 | 0.992366 |
| 59.625 | 0.951562 |
| 60.75 | 0.908744 |
| 61.875 | 0.864177 |
| 63 | 0.818136 |
| 64.125 | 0.770903 |
| 65.25 | 0.722772 |
| 66.375 | 0.674037 |
| 67.5 | 0.625 |
| 68.625 | 0.575963 |
| 69.75 | 0.527228 |
| 70.875 | 0.479097 |
| 72 | 0.431864 |
| 73.125 | 0.385823 |
| 74.25 | 0.341256 |
| 75.375 | 0.298438 |
| 76.5 | 0.257634 |
| 77.625 | 0.219095 |
| 78.75 | 0.183058 |
| 79.875 | 0.149746 |
| 81 | 0.119364 |
| 82.125 | 0.0920999 |
| 83.25 | 0.0681209 |
| 84.375 | 0.0475753 |
| 85.5 | 0.0305897 |
| 86.625 | 0.0172688 |
| 87.75 | 0.00769479 |
| 88.875 | 0.00192667 |
| 90 | 0 |
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 |
|---|---|---|
| After first ideal polarizer | 5 | W/m² |
| After second polarizer | 2.5 | W/m² |
| After third polarizer | 1.25 | W/m² |
| Without the intermediate second polarizer | 0 | W/m² |
| Total rejected or absorbed intensity | 8.75 | W/m² |
Each filter projects the polarization incident on it; inserted projection can enable nonzero final transmission without creating energy.
Unpolarized intensity halves at the first ideal linear polarizer. The next projection uses theta and the final projection beta−theta. At crossed first/last axes beta90, an intermediate theta45 gives1.25 W/m² while directly crossed filters give zero. Every stage transmits no more than its own input. Polarization is evidence of transverse electromagnetic waves; an ideal linear-polarizer model is not a description of ordinary longitudinal sound in a fluid.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Each filter projects the polarization incident on it; inserted projection can enable nonzero final transmission without creating energy.
Unpolarized intensity halves at the first ideal linear polarizer. The next projection uses theta and the final projection beta−theta. At crossed first/last axes beta90, an intermediate theta45 gives1.25 W/m² while directly crossed filters give zero. Every stage transmits no more than its own input. Polarization is evidence of transverse electromagnetic waves; an ideal linear-polarizer model is not a description of ordinary longitudinal sound in a fluid.
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