University Physics II · topic UPII-14.08
Physical-optics synthesis
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.
Physical-Optics Synthesis: predict a finite change
Physical-optics synthesis treats an aperture as a continuum of coherent strips, adds their fields as phasors with the far-field phase (2π/λ) x sinθ, and squares the total amplitude to get the intensity at angle θ. Use the supplied topic model Y_out = 1000*lam/d, where Y_out is FRINGE SPACING λ/d (mrad). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Slit separation d, measured in μm. Keep a (Slit width a) = 10 μm; lam (Wavelength λ) = 0.6 μm fixed. For this exercise use only 29.7813 μm ≤ d ≤ 65.2188 μm 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 d = 37.5775 μm and 42.5388 μm. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δd across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
FRINGE SPACING λ/d against d
Predictions from the supplied Physical-Optics Synthesis 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): FRINGE SPACING λ/d / mrad
Horizontal axis (x): d: Slit separation d / μm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Slit separation d / μm | FRINGE SPACING λ/d / mrad |
|---|---|
| 29.78125 | 20.146904512067156 |
| 38.640625 | 15.527699150828953 |
| 47.5 | 12.631578947368421 |
| 56.359375 | 10.645966176878293 |
| 65.21875 | 9.199808337326306 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of d separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At d = 37.5775 μm, substitute into Y_out = 1000*lam/d: Y₀ = 15.967 mrad.
- At d = 42.5388 μm, the same fixed-parameter model gives Y₁ = 14.1048 mrad.
- Signed change: ΔY_out = Y₁ − Y₀ = -1.86221 mrad; the input change is Δd = 4.96125 μm.
- Average sensitivity is ΔY_out/Δd = -0.375352 mrad per μm. 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 = 15.967 mrad, Y₁ = 14.1048 mrad; ΔY_out = -1.86221 mrad; average sensitivity = -0.375352 mrad per μm.
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.
Physical-Optics Synthesis: infer an input with uncertainty
Physical-optics synthesis treats an aperture as a continuum of coherent strips, adds their fields as phasors with the far-field phase (2π/λ) x sinθ, and squares the total amplitude to get the intensity at angle θ. Use the supplied topic model Y_out = 1000*lam/d, where Y_out is FRINGE SPACING λ/d (mrad). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. d is Slit separation d, measured in μm. Keep a (Slit width a) = 10 μm; lam (Wavelength λ) = 0.6 μm fixed. For this exercise use only 29.7813 μm ≤ d ≤ 65.2188 μm 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 = 11.673578 mrad, with a bounded reading uncertainty ±0.37107 mrad. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown d by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00354375 μm, 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.
FRINGE SPACING λ/d against d
Predictions from the supplied Physical-Optics Synthesis 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): FRINGE SPACING λ/d / mrad
Horizontal axis (x): d: Slit separation d / μm
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| d: Slit separation d / μm | FRINGE SPACING λ/d / mrad |
|---|---|
| 29.78125 | 20.146904512067156 |
| 38.640625 | 15.527699150828953 |
| 47.5 | 12.631578947368421 |
| 56.359375 | 10.645966176878293 |
| 65.21875 | 9.199808337326306 |
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(d) = 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(29.7813) = 20.1469 mrad and Y_out(65.2188) = 9.19981 mrad. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives d_line = 29.7813 + (11.6736 − 20.1469)(65.2188 − 29.7813)/(9.19981 − 20.1469) = 57.2108 μm. 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 d = 51.3981 μm; substituting back gives Y_out = 11.673578 mrad.
- The interpolation error in the input is |d_line − d| = 5.81264 μm. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 11.302508 mrad and 12.044648 mrad. Sort the roots even if the response decreases: 49.8147 μm ≤ d ≤ 53.0856 μm.
- The lower and upper input deviations are 1.58347 μm and 1.68744 μm. 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 d ≈ 51.3981 μm; compatible interval [49.8147, 53.0856] μm. Endpoint interpolation gives 57.2108 μm.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.