Show that in the paraxial domain the magnification produced by a single spherical interface between two continuous media, as shown in Fig., is given by Use the small-angle approximation for Snell’s Law and approximate the angles by their tangents.

Answer

Magnification at a Spherical Interface Using the Paraxial Approximation

In geometrical optics, when a ray of light travels from one medium to another across a spherical interface, it refracts according to Snell’s Law. In the paraxial (small-angle) approximation, this behavior simplifies, and we can derive an expression for the transverse magnification produced by the interface.

Step-by-Step Derivation:

1. Geometry and Parameters

• n₁: Refractive index of the first medium (left of interface)
• n₂: Refractive index of the second medium (right of interface)
• s₀: Object distance from the pole of the surface
• s_i: Image distance from the pole
• y₀, y_i: Heights of the object and image
• θ_i, θ_t: Angles of incidence and refraction

2. Use the Small Angle Approximation

In the paraxial domain, small angles are approximated using their tangents:

From the geometry of the diagram:

• tan(θ_i) ≈ y₀ / s₀
• tan(θ_t) ≈ y_i / s_i

3. Apply Snell’s Law in Small-Angle Form

Substitute the angle approximations:

4. Rearrange to Solve for Magnification

Transverse magnification MT is defined as:

Using the previous relation:

Therefore:

The negative sign indicates an image inversion relative to the object.

Conclusion

This formula is fundamental in optics for determining image characteristics at a spherical interface between different media. It assumes the object and image lie along the optical axis and that all rays make small angles with the axis (paraxial approximation).