Double-Slit Fringe Loss with an Extended LED Source
A teaching lab compares two green sources in a double-slit experiment. The slit mask has nominal slit separation d = 0.250 mm and slit width a = 0.040 mm, and the screen is L = 1.50 m from the mask. Source A is a frequency-doubled Nd:YAG laser at 532 nm with a linewidth below 0.1 nm. Source B is a green LED centered at 530 nm with a 25 nm FWHM bandpass and an emitting area of 1.0 mm × 1.0 mm, placed 200 mm in front of the slit mask. The laser produces clean fringes. The LED produces a broad, almost uniform patch with no visible fringes, even though the calculated fringe spacing is a few millimetres. A student proposes adding a 10 nm bandpass filter to the LED and also covering one slit with a 5.0 mm thick BK7 plate (n = 1.517) to prove that the LED is 'coherent enough'.
1. Calculate the expected fringe spacing and the spatial coherence width at the slit plane for the LED. Use these numbers to explain the missing fringes.
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The geometrical fringe spacing is y = λL/d = (530 × 10⁻⁹ m)(1.50 m)/(0.250 × 10⁻³ m) = 3.18 mm. That is easily visible in principle.
The relevant limitation is spatial coherence. For a source of linear size s at distance z, the transverse coherence width at the slits is approximately w_c ≈ λz/s. Here z = 0.200 m and s = 1.0 mm, so w_c ≈ (530 × 10⁻⁹)(0.200)/(1.0 × 10⁻³) = 1.06 × 10⁻⁴ m = 0.106 mm.
Because d = 0.250 mm is larger than w_c, the fields at the two slits are substantially uncorrelated. The two slit contributions add in intensity rather than producing stable interference, so visibility is low. The single-slit envelope is not the problem: its first minimum is at y = λL/a = (530 × 10⁻⁹)(1.50)/(0.040 × 10⁻³) = 19.9 mm, so several fringes would fit inside the central envelope if the light were spatially coherent. The laser has a much smaller effective source etendue, so its coherence width at the mask is much larger than 0.25 mm.
2. Will a 10 nm bandpass filter restore fringes? Calculate the temporal coherence length for the LED before and after filtering, and state what must actually change.
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The filter improves temporal coherence, not spatial coherence. Using L_c ≈ λ²/Δλ:
- Unfiltered LED: L_c ≈ (530 × 10⁻⁹)²/(25 × 10⁻⁹) = 1.12 × 10⁻⁵ m = 11.2 µm.
- With a 10 nm filter: L_c ≈ (530 × 10⁻⁹)²/(10 × 10⁻⁹) = 2.81 × 10⁻⁵ m = 28.1 µm.
At zero path difference the temporal coherence length is not the limiting factor; the LED already fails because the slit separation exceeds the spatial coherence width. A narrower bandpass therefore does not restore fringes.
To recover fringes the lab must increase w_c or decrease d. Options include: reducing the slit separation below about 0.1 mm; moving the LED farther away, since w_c scales with z; or placing a small pinhole in front of the LED so the effective source size is much smaller. For example, moving the LED to z ≈ 0.47 m would give w_c ≈ 0.25 mm for the same 1 mm source. Increasing the slit width alone would not help and would reduce fringe visibility through the single-slit envelope.
3. For the 5.0 mm BK7 plate over one slit, compute the optical path difference. Predict the fringe visibility for both sources, and explain why this test does not diagnose the LED problem.
Read the worked answer
The plate adds an optical path difference OPD = (n − 1)t = (1.517 − 1)(5.0 mm) = 2.585 mm.
For the LED, L_c is only about 11 µm unfiltered or 28 µm with the 10 nm filter, so an OPD of 2.585 mm is roughly 100–200 coherence lengths. The two slit fields are effectively incoherent and no fringes should appear.
For the laser, L_c ≈ λ²/Δλ. With Δλ = 0.1 nm, L_c ≈ (532 × 10⁻⁹)²/(0.1 × 10⁻⁹) = 2.83 mm. The plate OPD is about 0.91 L_c, so the fringe visibility should be noticeably reduced and may be weak, depending on the actual lineshape and linewidth. If the laser linewidth is substantially below 0.1 nm, the visibility will be better.
The test diagnoses temporal coherence, not spatial coherence. The LED already fails at zero OPD because its spatial coherence width is too small at the slit plane. Adding a plate therefore cannot show that the LED is 'coherent enough' for the original double slit; it only confirms that the LED has a short coherence length.