University Physics III · topic UPIII-05.04
The Michelson interferometer and optical metrology
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
The Michelson Interferometer & Optical Metrology: predict a finite change
A Michelson interferometer divides a beam's amplitude at a beam splitter, returns the two halves from separate mirrors and reads the path difference 2d between the arms as a fringe pattern that shifts by one fringe for every λ/2 of mirror travel. Use the supplied topic model Y_out = lam^2/(1000*dl), where Y_out is COHERENCE LENGTH λ²/Δλ (μm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. dl is Spectral width Δλ, measured in nm. Keep lam (Centre wavelength λ) = 633 nm; d (Mirror displacement d) = 0.95 μm fixed. For this exercise use only 32.7687 nm ≤ dl ≤ 268.231 nm on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at dl = 84.5705 nm and 117.535 nm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δdl across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
COHERENCE LENGTH λ²/Δλ against dl
Predictions from the supplied The Michelson Interferometer & Optical Metrology model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): COHERENCE LENGTH λ²/Δλ / μm
Horizontal axis (x): dl: Spectral width Δλ / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| dl: Spectral width Δλ / nm | COHERENCE LENGTH λ²/Δλ / μm |
|---|---|
| 32.76875 | 12.227777989700552 |
| 91.63437499999999 | 4.372693107799339 |
| 150.5 | 2.6623853820598007 |
| 209.36562499999997 | 1.9138242010836308 |
| 268.23125 | 1.4938192324719808 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of dl separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At dl = 84.5705 nm, substitute into Y_out = lam^2/(1000*dl): Y₀ = 4.73793 μm.
- At dl = 117.535 nm, the same fixed-parameter model gives Y₁ = 3.4091 μm.
- Signed change: ΔY_out = Y₁ − Y₀ = -1.32883 μm; the input change is Δdl = 32.9647 nm.
- Average sensitivity is ΔY_out/Δdl = -0.0403107 μm per nm. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 4.73793 μm, Y₁ = 3.4091 μm; ΔY_out = -1.32883 μm; average sensitivity = -0.0403107 μm per nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
The Michelson Interferometer & Optical Metrology: infer an input with uncertainty
A Michelson interferometer divides a beam's amplitude at a beam splitter, returns the two halves from separate mirrors and reads the path difference 2d between the arms as a fringe pattern that shifts by one fringe for every λ/2 of mirror travel. Use the supplied topic model Y_out = lam^2/(1000*dl), where Y_out is COHERENCE LENGTH λ²/Δλ (μm). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. dl is Spectral width Δλ, measured in nm. Keep lam (Centre wavelength λ) = 633 nm; d (Mirror displacement d) = 0.95 μm fixed. For this exercise use only 32.7687 nm ≤ dl ≤ 268.231 nm on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 2.2714683 μm, with a bounded reading uncertainty ±0.11665 μm. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown dl by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0235462 nm, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
COHERENCE LENGTH λ²/Δλ against dl
Predictions from the supplied The Michelson Interferometer & Optical Metrology model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): COHERENCE LENGTH λ²/Δλ / μm
Horizontal axis (x): dl: Spectral width Δλ / nm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| dl: Spectral width Δλ / nm | COHERENCE LENGTH λ²/Δλ / μm |
|---|---|
| 32.76875 | 12.227777989700552 |
| 91.63437499999999 | 4.372693107799339 |
| 150.5 | 2.6623853820598007 |
| 209.36562499999997 | 1.9138242010836308 |
| 268.23125 | 1.4938192324719808 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(dl) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(32.7687) = 12.2278 μm and Y_out(268.231) = 1.49382 μm. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives dl_line = 32.7687 + (2.27147 − 12.2278)(268.231 − 32.7687)/(1.49382 − 12.2278) = 251.173 nm. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives dl = 176.401 nm; substituting back gives Y_out = 2.2714683 μm.
- The interpolation error in the input is |dl_line − dl| = 74.7717 nm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 2.1548183 μm and 2.3881183 μm. Sort the roots even if the response decreases: 167.784 nm ≤ dl ≤ 185.95 nm.
- The lower and upper input deviations are 8.61648 nm and 9.54937 nm. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer dl ≈ 176.401 nm; compatible interval [167.784, 185.95] nm. Endpoint interpolation gives 251.173 nm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.