Ray Optics and Optical Instruments — Notes

New Cartesian sign convention: distances from the pole; along the incident light +, against it −; heights above the axis +.

• Concave: f < 0; real image (v < 0, m < 0) unless the object is inside F (virtual, erect, enlarged).
• Convex: f > 0; image always virtual, erect, diminished; wide field of view (rear-view mirrors).

Snell’s law: $n_1\sin i = n_2\sin r$; $n = c/v$. Towards the normal into a denser medium. Apparent depth = real depth / n (near-normal viewing). Frequency unchanged across a boundary.

Total internal reflection: from denser to rarer, at the critical angle C the refracted ray grazes the surface: $\sin C = n_2/n_1$ (into air: $1/n$). For $i > C$ all light is reflected. Conditions: denser → rarer and i > C.

Critical angles: glass (1.5) ≈ 41.8°, water (1.33) ≈ 48.8°, diamond (2.42) ≈ 24.4°.

Applications: optical fibres (core of higher n than cladding; guided by repeated TIR; communication, endoscopy), sparkle of diamonds, totally reflecting prisms (periscopes, binoculars), mirages.

One spherical surface: $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ (paraxial rays).

Lens maker’s formula (thin lens of index n in air): $\dfrac{1}{f} = (n - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)$. In a medium of index $n_m$ use $n/n_m$.

Thin lens formula: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$, $m = \dfrac{v}{u}$. Convex f > 0; concave f < 0.

Power: $P = 1/f$ (f in m), dioptre (D). Thin lenses in contact: $P = P_1 + P_2 + \dots$, $m = m_1m_2$.

For a prism of angle A: $r_1 + r_2 = A$; deviation $\delta = i + e - A$.

As i increases, δ decreases to a minimum $\delta_m$ and then increases. At minimum deviation $i = e$, $r_1 = r_2 = A/2$, the ray inside is parallel to the base, and $n = \dfrac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ For a 60° prism of n = 1.5: $i = 48.6°$, $\delta_m \approx 37.2°$. (Thin prism: $\delta \approx (n - 1)A$.)

Magnifying power = angle subtended at the eye by the image / angle subtended by the object at the near point (D = 25 cm).

• Simple microscope: $m = 1 + D/f$ (image at D); $m = D/f$ (image at infinity).
• Compound microscope: objective (very short $f_o$) gives a real, inverted, enlarged image just inside the focus of the eyepiece; $m = m_om_e \approx \dfrac{L}{f_o}\cdot\dfrac{D}{f_e}$ (infinity) or $\dfrac{L}{f_o}\left(1 + \dfrac{D}{f_e}\right)$ (at D), L = tube length.
• Astronomical telescope: long $f_o$, large aperture; normal adjustment $m = f_o/f_e$, tube length $f_o + f_e$; final image inverted.
• Reflecting telescope (Cassegrain): concave primary, convex secondary, light through a hole in the primary. Advantages: no chromatic aberration, paraboloidal mirror removes spherical aberration, large mirrors can be supported from behind, cheaper and lighter.