Refraction Through Glass Slab

When a ray of light passes through a glass slab with parallel faces, it undergoes refraction twice — once entering the glass and once leaving it. At the first surface, light bends towards the normal because it slows down entering a denser medium. At the second surface, it bends away from the normal as it speeds back up into air. Because the two surfaces are perfectly parallel, the exit ray emerges travelling in exactly the same direction as the incident ray — but it has been shifted sideways. This sideways shift is called lateral displacement. Lateral displacement occurs because, although the direction of the ray is restored upon exiting, the path through the glass has physically moved the ray to one side. Think of it like a pedestrian crossing a road diagonally, then walking diagonally back at the same angle on the other side — they end up travelling in their original direction but have drifted along the road. The thicker the slab, or the greater the angle of incidence, the larger the lateral displacement. This is a direct, measurable consequence of Snell's Law applied twice at parallel surfaces. Lateral displacement has real-world significance. Optical instruments, camera lenses, and even simple window glass all introduce small lateral displacements that must be accounted for in precision optics. Understanding this phenomenon connects refraction theory to practical instrument design and explains why objects viewed through a thick glass pane appear slightly shifted from their true positions.

Key takeaways
  • Setting Up the Geometry
  • Applying Snell's Law at Both Surfaces
  • Deriving the Lateral Displacement Formula
  • Physical Interpretation

Setting Up the Geometry

Consider a glass slab of thickness t (measured perpendicular to its faces) and refractive index n. A ray hits the top surface at angle of incidence i (measured from the normal). Inside the glass, it travels at angle of refraction r, given by Snell's Law: sin i = n sin r. The ray travels through the slab diagonally, covering a path length of t/cos r inside the glass.

Applying Snell's Law at Both Surfaces

At the first surface: n₁ sin i = n₂ sin r, where n₁ = 1 (air) and n₂ = n (glass), giving sin i = n sin r. At the second surface, the ray hits the parallel face at the same angle r (geometry of parallel lines). Applying Snell's Law again: n sin r = 1 × sin e, so e = i. The emergent ray is parallel to the incident ray — direction is fully restored, confirming that only a lateral shift remains.

Deriving the Lateral Displacement Formula

The perpendicular distance between the incident ray (extended) and the emergent ray defines the lateral displacement d. Inside the slab, the ray travels a path length PQ = t/cos r. The angle between this internal ray and the incident ray's extension is (i − r). Therefore: d = PQ × sin(i − r) = (t/cos r) × sin(i − r). This compact formula shows d increases with slab thickness t and with larger angles (i − r).

Physical Interpretation

When i = 0° (normal incidence), r = 0°, so d = 0 — no lateral displacement. As i increases towards 90°, d grows, approaching a maximum. For a given angle, a higher refractive index means a smaller r (light bends more), which increases (i − r) and thus increases d. Counterintuitively, a higher refractive index also shortens the effective path in some contexts, so the formula must be evaluated carefully for each scenario.

The Key Equation

d=tcos⁡rsin⁡(i−r)d = \frac{t}{\cos r} \sin(i - r)
  • d — Lateral displacement (metres (m)): The perpendicular distance between the original incident ray direction and the emergent ray.
  • t — Thickness of the slab (metres (m)): The perpendicular distance between the two parallel faces of the glass slab.
  • i — Angle of incidence (degrees (°) or radians): The angle between the incident ray and the normal to the first surface of the slab.
  • r — Angle of refraction (degrees (°) or radians): The angle between the refracted ray inside the glass and the normal, found using Snell's Law.

Worked Example

Problem. A ray of light strikes a glass slab of thickness 4.0 cm and refractive index 1.5 at an angle of incidence of 45°. Calculate the lateral displacement of the emergent ray.

d≈1.32 cmd \approx 1.32 \text{ cm}

Answer. The lateral displacement is approximately 1.32 cm.

Figure
Diagram
Glass Slab (n = 1.5)dIncident rayRefracted rayEmergent rayirre=iSurface 1Surface 2t
Origins · 1 minWillebrord Snellius and how this was discovered
WS
Willebrord Snellius
1580–1626 · Dutch

Willebrord Snellius (Snell) discovered the mathematical law of refraction in 1621, now universally known as Snell's Law: n₁ sin θ₁ = n₂ sin θ₂. This law is the absolute foundation of calculating angles of refraction at any interface, and it underpins the derivation of the lateral displacement formula for a glass slab. Without Snell's Law, we could not quantitatively predict how much light bends or shifts as it passes through optical media. Snell never published his discovery during his lifetime — it was Christiaan Huygens who found and published Snell's manuscript decades after his death, saving the law from being lost to history entirely.

  • Snell's Law — Snell's Law is applied twice at the two surfaces of the glass slab and is essential for deriving the lateral displacement formula.
  • Refractive Index — The refractive index of the glass determines how much the ray bends at each surface and therefore directly controls the magnitude of the lateral displacement.
  • Reflection of Light — Understanding the normal to a surface and angle conventions is shared between reflection and refraction, forming the geometric foundation of ray optics.
  • Total Internal Reflection — Total internal reflection is the extreme consequence of refraction at a glass–air boundary, directly related to the same Snell's Law geometry used in the glass slab analysis.