A compound microscope uses two converging lenses to achieve high magnification. Objective lens ($f_o$ short, a few mm): placed just outside its focal point. Creates a real, inverted, magnified intermediate image at distance $L$ from its back focal point. Eyepiece ($f_e$ longer): acts as a simple magnifying glass on the intermediate image. Total magnification: $M = m_o \times m_e = \dfrac{L}{f_o} \times \dfrac{D}{f_e}$ (image at near point, $D=25$ cm). For final image at infinity: $M = \dfrac{L}{f_o} \times \dfrac{D}{f_e}$ (same formula here).

Objective: large focal length $f_o$, large aperture (to collect more light from distant objects). Eyepiece: short focal length $f_e$ (for high magnification). Magnification: $M = f_o/f_e$ (for relaxed eye, final image at infinity). Length of telescope: $L = f_o + f_e$. Resolving power: $\theta_{min} = 1.22\lambda/D$ (Rayleigh criterion). Larger $D$ → better resolution. Reflecting telescope: uses concave mirror as objective (no chromatic aberration, easier to make large). Refracting: uses lens (chromatic aberration problem).

Single convex lens used as magnifying glass. Object placed inside focal length. Virtual, erect, magnified image. Magnification: $M = 1 + D/f$ (image at near point $D=25$ cm). For relaxed eye: $M = D/f$. For this problem, eyepiece magnification $= D/f_e = 25/4 = 6.25$. Compound microscope achieves much higher magnification by multiplying: $125 = 20 \times 6.25$. Maximum useful magnification of optical microscope: $\sim 1500\times$ (limited by diffraction at visible wavelengths).

$$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ $n$ = refractive index of lens material. $R_1$, $R_2$ = radii of curvature of two surfaces (sign convention: positive if centre of curvature on transmission side). Thin lens formula: $1/v - 1/u = 1/f$. Power $P = 1/f$ (D). Lenses in contact: $P = P_1 + P_2$. For minimum spherical aberration: plano-convex lens oriented with flat side toward image.

$$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$$ $n_1$ = refractive index of incident medium, $n_2$ = refractive index of refraction medium, $R$ = radius of curvature. Thin lens formula derived from applying this twice (at two surfaces). Refraction through prism: deviation $\delta = (\mu - 1)A$ for thin prism. Minimum deviation: $\sin[(A+\delta_m)/2] = \mu\sin(A/2)$.

Dispersion: different wavelengths refract differently ($\mu$ varies with $\lambda$). Cauchy formula: $\mu = A + B/\lambda^2$. Violet bends most, red least (in glass). Rainbow: (1) refraction at entry + (2) TIR inside drop + (3) refraction at exit. Primary bow: $42°$, colours red outside-violet inside. Secondary bow: $51°$ (extra TIR), colours inverted. Scattering: Rayleigh scattering $\propto 1/\lambda^4$ → blue sky (blue scatters most), red/orange sunset (blue scattered away leaving red/orange).

Myopia (short-sight): image forms in front of retina (eyeball too long or lens too powerful). Far point closer than infinity. Corrected by diverging lens (negative power). Hyperopia (long-sight): image behind retina. Near point farther than 25 cm. Corrected by converging lens. Astigmatism: different power in different meridians. Corrected by cylindrical lens. Presbyopia: loss of accommodation with age (lens hardens). Needs reading glasses (converging). Cataract: lens becomes opaque. Treated by lens replacement (IOL surgery).

Young double slit: $\beta = \lambda D/d$ (fringe width). Condition: coherent sources. Thin film interference: constructive if $2nt = (m+1/2)\lambda$ (phase change at one surface). Anti-reflection coating: $t = \lambda/(4n)$. Single slit diffraction: first minimum at $\sin\theta = \lambda/a$. Diffraction grating: $d\sin\theta = m\lambda$. Resolving power of grating $= mN$ ($N$ = number of slits). Limit of resolution (Rayleigh criterion): $\theta_{min} = 1.22\lambda/D$.