Refraction in a Glass Slab

Refraction in a glass slab is a standard case of light passing through a parallel-sided medium. When a ray travels from air into glass, it slows down and bends toward the normal; when it exits from glass back into air, it speeds up and bends away from the normal.2 Because the two faces of the slab are parallel, the total angular deviation cancels out, so the emergent ray is parallel to the incident ray, although shifted sideways.2

This sideways separation is called lateral displacement or lateral shift.2 The core law governing the phenomenon is Snell's law:

$$n_1 \sin i = n_2 \sin r$$

where $n_1$ and $n_2$ are refractive indices of the two media, $i$ is the angle of incidence, and $r$ is the angle of refraction.2

For a rectangular glass slab in air:

• the ray bends toward the normal at the first surface because glass has higher refractive index than air,2
• the ray bends away from the normal at the second surface,2
• the angle of emergence equals the angle of incidence, $e=i$,
• the incident and emergent rays are parallel,2
• but they are separated by a finite lateral shift if $i\neq 0$.2

Footnotes

1. Refractive index | Definition & Equation | Britannica - Defines refractive index, gives $n=c/v$ and typical values for air and glass. ↩ ↩² ↩³

2. Refraction | Definition, Examples, & Facts | Britannica - Explains why light bends when crossing media and how direction depends on optical density. ↩ ↩² ↩³ ↩⁴

3. Snell's Law Lab | Lab - Edubirdie - Describes the rectangular glass slab experiment, angle relations, and lateral displacement. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶

4. Refraction through glass slab (video) | Khan Academy - Explains parallel emergence and the concept of lateral shift in a parallel-sided medium. ↩ ↩² ↩³ ↩⁴

Physical basis of refraction in a slab

The refractive index of a material is defined by

$$n=\frac{c}{v}$$

where $c$ is the speed of light in vacuum and $v$ is its speed in the medium. Since light travels more slowly in glass than in air, ordinary visible light entering glass bends toward the normal.2 Typical values for yellow light are approximately $n \approx 1.0003$ for air and $n \approx 1.517$ for crown glass.

At the first boundary:

$$n_{\text{air}}\sin i = n_{\text{glass}}\sin r$$

Since $n_{\text{glass}} > n_{\text{air}}$, for nonzero $i$ we get $r < i$, meaning the refracted ray lies closer to the normal.2

At the second boundary:

$$n_{\text{glass}}\sin r = n_{\text{air}}\sin e$$

Combining both equations gives

$$\sin e = \sin i$$

and for ordinary geometrical conditions this yields

$$e=i$$

which explains why the emergent ray is parallel to the incident ray.2

A useful geometric expression for lateral displacement $d$ of a ray traversing a slab of thickness $t$ is

$$d = t\frac{\sin(i-r)}{\cos r}$$

which shows that shift depends on slab thickness, angle of incidence, and the refractive index through $r$. From this relation and ray-diagram analysis, the lateral shift becomes zero for normal incidence $i=0$ and increases with thickness for fixed optical conditions.2

Footnotes

1. Refractive index | Definition & Equation | Britannica - Defines refractive index, gives $n=c/v$ and typical values for air and glass. ↩ ↩² ↩³ ↩⁴

2. Refraction | Definition, Examples, & Facts | Britannica - Explains why light bends when crossing media and how direction depends on optical density. ↩ ↩²

3. Snell's Law Lab | Lab - Edubirdie - Describes the rectangular glass slab experiment, angle relations, and lateral displacement. ↩

4. Refraction through glass slab (video) | Khan Academy - Explains parallel emergence and the concept of lateral shift in a parallel-sided medium. ↩ ↩²

5. Calculating light's lateral shift in a glass slab - Physics Stack Exchange - Presents and derives the standard formula $d=t\\frac{\\sin(i-r)}{\\cos r}$ for lateral shift. ↩ ↩²

Lateral displacement: interpretation and dependence

The thickness of the slab strongly influences the lateral shift. From

$$d = t\frac{\sin(i-r)}{\cos r}$$

we see that for fixed $i$ and material, $d$ is directly proportional to $t$. Thus a thicker slab produces a larger sideways shift.2

The angle of incidence also matters. If $i=0^\circ$, then the ray enters along the normal and no bending occurs; therefore:

$$d=0$$

This is why a ray incident normally passes undeviated and unshifted through the slab.2

The refractive index of glass affects the amount of bending. A larger refractive index generally causes a smaller internal angle $r$ for the same incident angle $i$, which changes $\sin(i-r)$ and therefore the lateral shift.2 This is why different optical glasses can produce different displacement values under the same geometry.

A common confusion is to mix up lateral shift with apparent depth. Apparent depth usually refers to viewing an object across a single plane interface, such as a coin under water. In contrast, refraction through a glass slab with two parallel faces mainly highlights the emergent parallel ray and lateral displacement.2

Footnotes

1. Calculating light's lateral shift in a glass slab - Physics Stack Exchange - Presents and derives the standard formula $d=t\\frac{\\sin(i-r)}{\\cos r}$ for lateral shift. ↩ ↩² ↩³ ↩⁴

2. Refraction through glass slab (video) | Khan Academy - Explains parallel emergence and the concept of lateral shift in a parallel-sided medium. ↩ ↩²

3. Snell's Law Lab | Lab - Edubirdie - Describes the rectangular glass slab experiment, angle relations, and lateral displacement. ↩ ↩²

4. Refractive index | Definition & Equation | Britannica - Defines refractive index, gives $n=c/v$ and typical values for air and glass. ↩ ↩²

5. apparent depth - BYJU'S - Explains apparent depth and distinguishes it from lateral or normal shift phenomena. ↩

Laboratory verification and ray-diagram method

A standard school experiment uses a rectangular glass slab, a sheet of paper, optical pins, and a protractor to verify slab refraction and Snell's law. The incident ray is marked, the slab outline is traced, and pins are aligned so that the path of the emergent ray can be reconstructed. After removing the slab, the traced rays show that the emergent ray is parallel to the incident ray.

The experiment also demonstrates:

1. $i>r$ when light goes from air to glass,
2. $e=i$ for a rectangular slab in air,
3. the emergent ray is laterally displaced,2
4. a plot of $\sin i$ versus $\sin r$ is approximately linear, consistent with Snell's law.

This makes the glass slab an excellent model for connecting ray optics with measurable geometry. It links optical density to directional bending and shows how symmetrical boundaries can preserve direction while changing position.3

Footnotes

1. Snell's Law Lab | Lab - Edubirdie - Describes the rectangular glass slab experiment, angle relations, and lateral displacement. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸

2. Calculating light's lateral shift in a glass slab - Physics Stack Exchange - Presents and derives the standard formula $d=t\\frac{\\sin(i-r)}{\\cos r}$ for lateral shift. ↩

3. Refractive index | Definition & Equation | Britannica - Defines refractive index, gives $n=c/v$ and typical values for air and glass. ↩

4. Refraction | Definition, Examples, & Facts | Britannica - Explains why light bends when crossing media and how direction depends on optical density. ↩

Mathematical summary

For refraction through a slab surrounded by the same medium on both sides:

$$n_{\text{air}}\sin i = n_{\text{glass}}\sin r$$ $$n_{\text{glass}}\sin r = n_{\text{air}}\sin e$$

Therefore,

$$e=i$$

and the lateral displacement is

$$d = t\frac{\sin(i-r)}{\cos r}$$

Special cases:

• If $i=0^\circ$, then $r=0^\circ$, $e=0^\circ$, and $d=0$.2
• If slab thickness $t$ increases, then $d$ increases proportionally for fixed geometry.
• If refractive index increases, the internal angle $r$ changes according to Snell's law, modifying the shift.2

These relations explain everyday observations such as the slight sideways apparent shift of objects viewed through flat glass windows or plates at an angle.

Footnotes

1. Snell's Law Lab | Lab - Edubirdie - Describes the rectangular glass slab experiment, angle relations, and lateral displacement. ↩

2. Calculating light's lateral shift in a glass slab - Physics Stack Exchange - Presents and derives the standard formula $d=t\\frac{\\sin(i-r)}{\\cos r}$ for lateral shift. ↩ ↩² ↩³

3. Refractive index | Definition & Equation | Britannica - Defines refractive index, gives $n=c/v$ and typical values for air and glass. ↩

4. Refraction through glass slab (video) | Khan Academy - Explains parallel emergence and the concept of lateral shift in a parallel-sided medium. ↩