Problem 1 IYPT 2027: 1. Invent yourself: Pinhole sunglasses

Problem Description & Analysis

Official Problem Statement

Investigate the optimal arrangement of pinholes in an opaque film to produce sunglasses that reduce light intensity and help correct short-sightedness.

Key Physics Concepts

Geometric Optics — Depth of Field

Each pinhole acts as a tiny aperture stop. As the hole diameter decreases, the cone of light passing through narrows, reducing the circle of confusion on the retina — allowing even myopic or hyperopic eyes to see a relatively sharper image.

d_blur = D × |1/f − 1/v − 1/u|

where:
d_blur : diameter of the blur (circle of confusion) on the retina — in metres (m)
D : pinhole diameter — in metres (m)
f : focal length of the eye’s lens — in metres (m) ≈ 0.017 m
v : image distance (lens to retina) — in metres (m) ≈ 0.017 m
u : object distance (object to lens) — in metres (m)

As D decreases, the blur circle shrinks and depth of field increases.

Wave Optics — Diffraction

Reducing the aperture too much introduces a wave-optics limit: when D approaches the wavelength of light, diffraction dominates and the Airy pattern blurs the image. Rayleigh’s resolution limit:

θ_min = 1.22 × λ / D

where:
θ_min : minimum resolvable angle (Rayleigh criterion) — in radians (rad)
λ : wavelength of light — in metres (m); green light: λ ≈ 550 nm = 550×10⁻⁹ m
D : pinhole diameter — in metres (m)
1.22 : Rayleigh constant (dimensionless, from Bessel function zero)

Conclusion: there is an optimal diameter that balances geometric blur (large D) and diffraction blur (small D).

Optimal Pinhole Diameter

At the point where geometric blur equals diffraction blur, the total blur is minimised. For image distance v (approximately the focal length of the eye, ≈ 17 mm) and green light (λ ≈ 550 nm):

D_opt ≈ √(2.44 × λ × v) ≈ 0.15 mm

where:
D_opt : optimal pinhole diameter — in metres (m)
λ : wavelength of light — in metres (m); green light: λ ≈ 550 nm = 550×10⁻⁹ m
v : retinal distance from lens — in metres (m); v ≈ 17 mm = 0.017 m
2.44 : Rayleigh diffraction constant (= 2 × 1.22) — dimensionless

This value matches commercially available pinhole glasses (0.1–0.3 mm holes).

Parameters to Investigate

Hole diameter D — directly controls depth of field and diffraction intensity
Hole spacing (pitch) — if too close, diffraction patterns from adjacent holes interfere
Arrangement pattern — square lattice vs. hexagonal lattice
Film thickness — thicker film = smaller acceptance angle = narrower field of view
Hole density — more holes = brighter image but higher interference risk

Suggested Experimental Procedure

Fabricate several samples with different diameters (0.1–0.5 mm)
Measure visual acuity for each sample using a Snellen chart
Measure transmitted light intensity with a lux meter
Determine the useful field of view by gradually rotating the glasses
Compare results with the theoretical model above

Related Images

pinholes

Interactive Simulation (Python)

This simulation runs directly in your browser — no Python installation needed. Initial loading may take a few seconds.

Hole diameter D 0.15 mm

Wavelength λ 550 nm

Refractive error 2 D

Hole spacing (pitch) 1 mm

Press the button below to start.

(a) Blur circle diameter on the retina vs. hole diameter — shows that total blur reaches a minimum at optimal D_opt; diffraction dominates for smaller diameters, geometric blur for larger ones.
(b) Light transmission (%) vs. hole diameter — shows the quadratic dependence of transmitted intensity on the D/pitch ratio.
(c) Comparison of diffraction, geometric, and total blur values (in µm) — displayed for the user-selected diameter.

Python Code Lab

Edit this starter code, enter your own experimental data, and run it directly in the browser — no Python installation required.

Python — Editable

# ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
#  Code Lab: Pinhole Sunglasses (IYPT 2027 — Problem 1)
#  Edit this code, enter your own data, and run in browser
# ━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━

import numpy as np
import matplotlib.pyplot as plt

# ─── 1. Your experimental data ───────────────────────────
D_mm = [0.10, 0.15, 0.20, 0.30, 0.40, 0.50]   # <-- replace with your values

# Visual acuity measured with a Snellen chart (1.0 = perfect)
acuity_measured = [0.72, 0.88, 0.82, 0.65, 0.48, 0.34]  # <-- your measurements

# Light transmission measured with a lux meter (% of direct light)
light_pct = [0.9, 2.1, 3.8, 8.5, 15.1, 23.6]   # <-- your measurements

# ─── 2. Theoretical model (simple pinhole) ───────────────
# Geometric blur = hole diameter (image of a point source through the hole)
# Diffraction blur = Airy disk diameter at the retina
v_mm   = 17.0         # retinal distance (mm)
lam_mm = 550e-6       # wavelength of green light (mm)

D_theory = np.linspace(0.01, 0.80, 400)
blur_geo  = D_theory                           # mm — geometric blur = hole size
blur_diff = 2.44 * lam_mm * v_mm / D_theory   # mm — Airy disk
blur_tot  = np.sqrt(blur_geo**2 + blur_diff**2)

# Optimal D: where geo = diff → D_opt = √(2.44 × λ × v)
D_opt = np.sqrt(2.44 * lam_mm * v_mm)

# Approximate visual acuity from blur
blur_ref      = blur_tot.min()
acuity_theory = np.clip(blur_ref / blur_tot, 0, 1)

# ─── 3. Plots ────────────────────────────────────────────
fig, axes = plt.subplots(1, 3, figsize=(13, 4.5))
fig.suptitle('Pinhole Sunglasses — Measured vs Theory', fontsize=12, fontweight='bold')

# Plot a: Visual acuity
ax = axes[0]
ax.plot(D_theory, acuity_theory, '-', color='#1a73e8', lw=2, label=r'Theory:  propto b_{min}/b_{tot}$')
ax.scatter(D_mm, acuity_measured, color='#e65100', s=80, zorder=5, label='Measured')
ax.axvline(D_opt, color='#6a0dad', ls=':', lw=1.5, label=f'D_opt = {D_opt*1000:.0f} um')
ax.text(0.35, float(np.interp(0.35, D_theory, acuity_theory)) + 0.04,
        r' propto b_{min}/b_{tot}$', color='#1a73e8', fontsize=8, fontstyle='italic')
ax.set_xlabel('Hole diameter D (mm)')
ax.set_ylabel('Relative visual acuity')
ax.set_title('(a) Visual Acuity vs D')
ax.legend(fontsize=8.5)
ax.set_xlim(0, 0.85); ax.set_ylim(0, 1.1)
ax.grid(True, alpha=0.3, ls='--')

# Plot b: Light transmission
ax = axes[1]
pitch_mm  = 1.0
D_t2      = np.linspace(0.01, pitch_mm * 0.98, 200)
theory_lt = (D_t2 / pitch_mm) ** 2 * 100
ax.plot(D_t2, theory_lt, '-', color='#1a73e8', lw=2, label=r'Theory: =(D/P)^2	imes100%$')
ax.scatter(D_mm, light_pct, color='#e65100', s=80, zorder=5, label='Measured')
ax.text(0.60, float((0.60/pitch_mm)**2 * 100) + 3,
        r'=(D/P)^2 	imes 100%$', color='#1a73e8', fontsize=8, fontstyle='italic')
ax.set_xlabel('Hole diameter D (mm)')
ax.set_ylabel('Light transmission (%)')
ax.set_title('(b) Light Transmission vs D')
ax.legend(fontsize=8.5); ax.set_ylim(0, 110)
ax.grid(True, alpha=0.3, ls='--')

# Plot c: Blur components
ax = axes[2]
ax.plot(D_theory, blur_diff * 1000, '--', color='#1a73e8', lw=1.8, label=r'Diffraction: =2.44lambda v/D$')
ax.plot(D_theory, blur_geo  * 1000, '--', color='#e65100', lw=1.8, label=r'Geometric: =D$')
ax.plot(D_theory, blur_tot  * 1000, '-',  color='#2d6a4f', lw=2.5, label=r'Total: =sqrt{b_g^2+b_d^2}$')
ax.axvline(D_opt, color='#6a0dad', ls=':', lw=1.5)
idx_label = 250
ax.text(D_theory[idx_label] + 0.01, blur_diff[idx_label]*1000 + 15,
        'Diffraction', color='#1a73e8', fontsize=7.5)
ax.text(D_theory[idx_label] + 0.01, blur_geo[idx_label]*1000 + 15,
        'Geometric',   color='#e65100', fontsize=7.5)
ax.text(D_theory[idx_label] + 0.01, blur_tot[idx_label]*1000 + 15,
        'Total',       color='#2d6a4f', fontsize=7.5)
ax.set_xlabel('Hole diameter D (mm)')
ax.set_ylabel('Blur on retina (um)')
ax.set_title('(c) Blur Components')
ax.set_xlim(0, 0.85)
ax.legend(fontsize=7.5, loc='upper right')
ax.grid(True, alpha=0.3, ls='--')


# ── Watermark ────────────────────────────────────────────────────────────────
for _wx in [0.18, 0.50, 0.82]:
    for _wy in [0.22, 0.52, 0.80]:
        fig.text(_wx, _wy, 'physicsme.ir',
                 fontsize=16, color='#b0b8cc', alpha=0.22,
                 ha='center', va='center', rotation=30,
                 transform=fig.transFigure, fontweight='bold', style='italic')
        fig.text(_wx, _wy - 0.055, 'man_e fizikee',
                 fontsize=10, color='#b0b8cc', alpha=0.22,
                 ha='center', va='center', rotation=30,
                 transform=fig.transFigure, fontweight='bold', style='italic')
fig.text(0.98, 0.012, '© physicsme.ir  ·  man_e fizikee',
         fontsize=8, color='#888', alpha=0.75,
         ha='right', va='bottom', transform=fig.transFigure)
plt.tight_layout()

# ─── 4. Numerical summary ─────────────────────────────────
print(f"Optimal D (theory):   {D_opt*1000:.1f} um  ({D_opt:.3f} mm)")
print(f"Minimum blur:          {blur_tot.min()*1000:.2f} um")
print(f"At D=0.15mm → blur:   {np.interp(0.15, D_theory, blur_tot)*1000:.1f} um")

Ready to run

Output