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Experiment 5 — Diffraction and the Resolution Limit

Objectives

By the end of this experiment you should be able to:

Textbook connection

Read §5.1–5.6. The Rayleigh criterion introduced there is the reason Chapter 7’s electron microscope is worth building, and the grating equation is the instrument that Experiments 10, 11, and 12 all depend on. Treat this week as calibration for the second half of the course.

Theory

The single slit

A slit of width aa produces minima where

asin⁡θ=mλ,m=±1,±2,…a\sin\theta = m\lambda, \qquad m = \pm1, \pm2, \ldots
(5.1)

(note: m=0m = 0 is not a minimum — it is the central maximum), and an intensity profile

I(θ)=I0[sin⁡ββ]2,β=πasin⁡θλ.I(\theta) = I_0\left[\frac{\sin\beta}{\beta}\right]^{2}, \qquad \beta = \frac{\pi a \sin\theta}{\lambda} .
(5.2)

The central maximum is twice as wide as the others and contains about 90% of the transmitted power. Narrowing the slit widens the pattern — the inverse relationship between aperture and angular spread that, translated into momentum language in Chapter 7, becomes the uncertainty principle.

The circular aperture

For a circular aperture of diameter DD the pattern is the Airy disc, and the first zero falls at

sin⁡θ1=1.22 λD.\sin\theta_1 = 1.22\,\frac{\lambda}{D} .
(5.3)

The 1.22 is the first positive zero of J1J_1 divided by π\pi, for an ideal, uniformly illuminated circular aperture. It differs from the single-slit result, so use the right aperture model when inferring DD.

The Rayleigh criterion

Two incoherent point sources are said to be just resolved when the first zero of one Airy pattern falls on the central maximum of the other:

θmin⁡=1.22 λD.\theta_{\min} = 1.22\,\frac{\lambda}{D} .
(5.4)

This is a convention for two equally bright, incoherent point sources in an ideal circular-aperture image. Contrast, noise, aberrations, detector sampling, and the source itself affect a real threshold. A known point-spread function can support more precise localization under suitable conditions, but the Rayleigh angle remains a useful diffraction scale. Electron microscopes gain from much shorter electron wavelengths, while their lenses impose other limits.

Two circular-aperture Airy point-spread profiles separated by one first-zero radius overlap; their sum has a central dip about 26.5 percent below the peaks.

Figure 5.1:Just resolved under the Rayleigh convention. Each source alone (dashed) has an Airy profile; separated by θmin⁡\theta_{\min}, their summed intensity (solid) has a central dip about 26.5% below the peaks. The figure is a one-dimensional cross section of the circular pattern.

The grating

NN equally spaced slits of pitch dd give maxima at

dsin⁡θ=mλ,d\sin\theta = m\lambda ,
(5.5)

the same condition as two slits — but the maxima are far narrower, with angular width ∝1/N\propto 1/N. That sharpening, not the position of the maxima, is what makes a grating a spectroscopic instrument. Its resolving power is

(λΔλ)grating=mN,\left(\frac{\lambda}{\Delta\lambda}\right)_{\mathrm{grating}} = mN ,
(5.6)

so an ideal 1000 lines/mm1000\ \text{lines/mm} grating with a measured 5 mm5\ \text{mm} illuminated width has N=5000N=5000 and a first-order grating limit of Δλ≈0.13 nm\Delta\lambda\approx0.13\ \text{nm} at 650 nm650\ \text{nm}. A narrow laser beam may illuminate fewer grooves, and a spectrometer’s entrance slit, optics, alignment, and detector can make its actual resolution worse. See the Feynman grating derivation and this spectrograph laboratory guide.

Optical discs as gratings

The data tracks of a CD or DVD form a reflection grating. Nominal pitches are 1.6 μm1.6\ \mu\text{m} for CD, 0.74 μm0.74\ \mu\text{m} for DVD, and 0.32 μm0.32\ \mu\text{m} for Blu-ray. For a DVD, λ/d=650/740=0.88\lambda/d = 650/740 = 0.88, so first order appears at 62°62° — a large angle where the small-angle approximation fails completely and you must measure the actual angle. For Blu-ray at 650 nm650\ \text{nm}, λ/d>1\lambda/d > 1 and no first order can propagate in air at normal incidence in the simple grating model. Failure to see an order alone is not proof of this inequality: weak efficiency, disc cover layers, alignment, and limited angular coverage can also hide one. The nominal pitches are tabulated in this Stony Brook optical-disc teaching project.

Pre-lab

Apparatus

A laser beam crosses a transmission slit, aperture, or grating on a mount, and diffracted light fans out to a screen or detector on the far side.

Figure 5.2:The transmission-diffraction bench for the slit, aperture, and grating. Measure large angles with a rotating detector or protractor. Optical discs require a separate reflection layout, with the diffracted orders detected on the incident side.

Procedure

Before you take diffraction data

Part A — Single slit

  1. Set L≈2 mL \approx 2\ \text{m} and illuminate a slit of known width.

  2. Record the positions of visible minima on both sides of the center, through m=±4m=\pm4 if the field and dynamic range permit. The separation of the m=+4m=+4 and m=−4m=-4 minima spans eight orders; divide by eight in the small-angle limit, or fit the exact-angle relation below.

  3. Take a quantitative profile with the camera or the scanning photodiode, as in Interference of Light. Watch for saturation.

  4. Repeat with two more slit widths, and with the unknown.

[ ] Checkpoint 1. Confirm with the instructor that your widest and narrowest slits give patterns of the expected relative widths. If narrowing the slit does not widen the pattern, check alignment, source illumination, detector field, and whether you located the minima.

Part B — Circular aperture and the Airy disc

  1. Replace the slit with a circular aperture. Photograph the Airy pattern with a long enough exposure to see the first dark ring and with a shorter exposure to keep the center unsaturated. Use each image for the feature it resolves; retain its exposure and scale rather than merging intensities without a calibrated response.

  2. Measure the radius of the first dark ring, and extract DD from (5.3). Compare with a direct measurement of the aperture under a traveling microscope if one is available.

Part C — The grating

  1. Mount the grating on the rotation stage with the beam at normal incidence. Check the grating normal using the reflected zero order or the mount’s reference, then compare the m=+1m=+1 and m=−1m=-1 angles as a symmetry check. A mismatch can indicate tilt or an angle-zero error.

  2. Measure θm\theta_m for every visible order, on both sides.

  3. Using the specified line density, compute λ\lambda; then use the independently measured λ\lambda from Experiment 1 to infer line density. Compare with uncertainties, including the specification’s tolerance if given.

Part D — Optical discs

  1. Mount the CD in reflection near normal incidence. Block the specular return beam and locate first orders on the incident side using a rotating detector or screen. Measure signed angles from the reflected zero-order direction, or measure spot position and screen distance and compute θ=arctan⁡(y/L)\theta=\arctan(y/L). Check +1/−1+1/-1 symmetry before using the normal-incidence formula.

  2. Repeat for the DVD. Note that the angle is large: do not use the small-angle approximation, and measure the geometry carefully enough that you know θ\theta to a degree or better.

  3. If a Blu-ray disc is available, scan the accessible incident-side angles for a first order and record the angular coverage and detection limit. An absent spot is consistent with d<λd<\lambda at normal incidence but alone does not establish a pitch bound.

Part E — Resolution

  1. Set up two small pinholes with measured separation and diameter, back-illuminated by a diffuse lamp at measured distance. Avoid a coherent laser source, which would produce a different two-source pattern.

  2. View them through a calibrated variable iris near the eye. Record the iris diameter at which they merge while keeping ambient light and viewing distance fixed. Note the actual eye pupil and pinhole sizes as practical limitations.

  3. Repeat with both observers. Report trial-to-trial and observer variation, plus iris calibration and geometry uncertainty; a visual threshold is not a direct measurement of the ideal Rayleigh boundary.

Analysis

Slit width from minima

The minima positions satisfy ym=mLλ/ay_m = mL\lambda/a in the small-angle limit, but do not use small angles if you do not have to. Fit

sin⁡θm=ymym2+L2=mλa\sin\theta_m = \frac{y_m}{\sqrt{y_m^2 + L^2}} = \frac{m\lambda}{a}
(5.7)

as a straight line in signed order mm; its slope is λ/a\lambda/a. Allow a small intercept to diagnose a mislocated pattern center.

import numpy as np
from scipy.optimize import curve_fit

m  = np.array([-4, -3, -2, -1, 1, 2, 3, 4])
y  = np.array([...])            # minima positions relative to center, meters
sy = np.array([...])
sin_th  = y / np.sqrt(y**2 + L**2)
ssin_th = sy * L**2 / (y**2 + L**2)**1.5      # independent position errors

popt, pcov = curve_fit(lambda m, b, s: b + s*m, m, sin_th,
                       sigma=ssin_th, absolute_sigma=True)
slope = popt[1]
a  = lam / slope
sa = a * np.sqrt((np.sqrt(pcov[1,1])/slope)**2 + (slam/lam)**2)

Here y, sy, L, lam, and slam are your measured positions, their standard uncertainties, screen distance, wavelength, and wavelength uncertainty in SI units. Refit at plausible LL and center-position limits to include their shared systematic effects; do not put one common LL error into every independent sigma entry. Inspect the intercept and residuals. Use absolute_sigma=True only for defensible absolute standard uncertainties. Then fit the full profile with (5.2) if the detector response is linear enough, and compare the two values of aa with their shared wavelength and distance errors. Saturation or camera processing can distort an intensity fit.

Grating and discs

For a near-normal transmission grating, fit signed sin⁡θm\sin\theta_m against signed mm with an intercept; its slope is λ/d\lambda/d. A nonzero intercept or differing +m/−m+m/-m estimates may indicate angle-zero or incidence error. Common wavelength and angle calibrations correlate the inferred pitches, so do not treat all orders as independent measurements when quoting uncertainty. For discs, use the reflected zero-order direction and the separately drawn reflection geometry. The same simple dsin⁡θ=mλd\sin\theta=m\lambda form applies only after normal incidence and an appropriate diffraction order have been established.

Resolution

Compare your measured threshold iris diameter with the prediction from (5.4), using the measured pinhole separation and distance. Report the ratio measured/predicted with the uncertainties you can support. Discuss pinhole size, eye optics, pupil diameter, brightness, and observer judgment before interpreting disagreement with the ideal criterion.

Post-lab questions

Going further

Experiments
4. Interference of Light
Experiments
6. Planck's Constant from LEDs