Understanding Lateral Displacement of Light

Lateral displacement of light is a key concept in ray optics, particularly when studying how a light ray passes through a parallel-sided transparent medium such as a glass slab. It describes the measure of sideways shift experienced by the ray, which is significant for understanding optical instruments, experimental setups, and exam problems in JEE and board physics.

Definition of Lateral Displacement of Light

Lateral displacement refers to the perpendicular distance between the path of the incident ray and the emergent ray after a light ray passes obliquely through a parallel-sided transparent slab. The incident and emergent rays are always parallel but not collinear due to refraction at both slab surfaces.

Cause and Nature of Lateral Displacement

When a ray of light enters a glass slab at an angle other than 90 degrees, it bends towards the normal due to a change in speed. Upon leaving the slab, it bends away from the normal, emerging parallel to the incident direction but shifted sideways. This shift is called lateral displacement.

Ray Diagram for Lateral Displacement

A standard ray diagram illustrates lateral displacement. A glass slab is represented with two parallel sides. An incident ray strikes the first surface at an angle, refracts towards the normal, travels inside the slab, refracts again at the second surface, and emerges parallel to the incident ray, but shifted by a fixed distance.

Mathematical Expression and Symbols

The magnitude of lateral displacement ($d$) for a slab of thickness $t$, angle of incidence $i$, and angle of refraction $r$ is given by:

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

Where $t$ is the perpendicular thickness of the slab, $i$ is the angle the incident ray makes with the normal, and $r$ is the angle of the refracted ray within the slab.

Factors Affecting Lateral Displacement

Lateral displacement depends on multiple factors related to the physical and optical properties of the system. Each parameter influences the shift as follows.

• Greater slab thickness increases displacement
• Larger angle of incidence increases displacement
• Higher refractive index increases displacement
• Displacement is zero for normal incidence

Comparison with Refraction and Diffraction

Lateral displacement arises from refraction through a parallel-sided slab. It differs from deviation or bending of rays at a single interface, and from the spreading of light due to diffraction at apertures or edges. For further clarity between these concepts, refer to the Difference Between Reflection And Refraction page.

Sample Calculation for Lateral Displacement

If a glass slab is $2\, \text{cm}$ thick, the angle of incidence $i = 45^\circ$, and the refracted angle $r \approx 28^\circ$ with refractive index $\mu = 1.5$, then:

$d = 2.0 \times \dfrac{\sin(45^\circ - 28^\circ)}{\cos 28^\circ}$

With $\sin 17^\circ \approx 0.292$ and $\cos 28^\circ \approx 0.882$, the displacement $d \approx 0.66\, \text{cm}$.

Key Properties and Experimental Conditions

Lateral displacement is only observed when the incident ray meets the slab at an angle other than 90 degrees. For normal incidence, the emergent and incident rays are collinear, resulting in zero displacement. This concept is also examined in optical experiments to measure the refractive index using a glass slab.

Applications of Lateral Displacement

Lateral displacement is relevant in the design of optical instruments, laboratory experiments, and in calculating the refractive index. It is crucial for correcting image alignment and understanding the behavior of light in beam alignment systems.

Dependence on Wavelength and Slab Geometry

The lateral shift is very weakly dependent on the wavelength, as the refractive index of the medium varies slightly for different colors of light. However, in curved or non-parallel slabs, light does not maintain a parallel emergent path, so uniform lateral displacement does not occur.

Further Study and Related Topics

For a deeper understanding of sign conventions in optics, slab ray path analysis, or to compare with prism refraction, consult the pages on Sign Convention In Optics, Path Of Ray Of Light Through Prism, and Refraction Of Light Through Prism.

Summary of Lateral Displacement of Light

The lateral displacement of light through a glass slab is the parallel but sideways shift between the incident and emergent rays. It depends on slab thickness, incident angle, and refractive index, but is zero for normal incidence. This topic is essential in ray optics for JEE and board level exams. For detailed explanations and additional solved problems on this topic, refer to the Lateral Displacement resource.