Optics

Laws of Reflection: (i) Angle of incidence = angle of reflection. (ii) Incident ray, reflected ray, normal lie in same plane.

Sign Convention (Cartesian): Distances measured from pole, in direction of incident light positive, against negative. Heights above principal axis positive, below negative.

Mirror Equation: $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$, where $f$ = focal length, $u$ = object distance, $v$ = image distance.

Magnification: $m = -v/u = h'/h$. Positive: virtual & erect; negative: real & inverted.

Relation: $f = R/2$ for spherical mirrors ($R$ = radius of curvature).

Concave mirror ($f < 0$ in convention): produces real, inverted images for object beyond $F$; virtual, erect, magnified for object between $F$ and $P$.

Convex mirror ($f > 0$): always virtual, erect, diminished image (used as rear-view mirror).

Image Formation by Concave Mirror:

For convex mirror: image is always virtual, erect, diminished, between $P$ and $F$.

Laws of Refraction (Snell's): $n_1\sin\theta_1 = n_2\sin\theta_2$. Refractive index: $n = c/v$ where $v$ = speed of light in medium.

Total Internal Reflection (TIR): Occurs when light travels from denser to rarer medium at angle $> \theta_c$ (critical angle): $$\sin\theta_c = \frac{1}{n} = \frac{n_2}{n_1} \quad (n_1 > n_2)$$ Examples: mirage, optical fibre, brilliance of diamond ($n = 2.42$, $\theta_c \approx 24°$).

Refraction at Spherical Surface: $$\frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}$$

Lens Maker's Formula: $$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$ $R_1, R_2$ with sign convention.

Thin Lens Formula: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$. Magnification: $m = v/u$. Power: $P = 1/f(\text{in m})$, units diopter (D).

Lens Combinations (in contact): $P = P_1 + P_2$, $1/f = 1/f_1 + 1/f_2$.

Prism:

• Deviation: $\delta = (i_1 + i_2) - A$ where $A$ = prism angle
• Minimum deviation when $i_1 = i_2$ ($r_1 = r_2 = A/2$)
• $n = \dfrac{\sin[(A+\delta_m)/2]}{\sin(A/2)}$
• For small prism: $\delta = (n-1)A$

Dispersion: Splitting of white light into colors by prism. Dispersive power: $\omega = (\mu_v - \mu_r)/(\mu - 1)$.

Achromatic combination of prisms: $\omega_1 A_1 + \omega_2 A_2 = 0$ (chosen so dispersions cancel but deviation remains).

Huygens' Principle: Every point on a wavefront is a source of secondary wavelets; the new wavefront is the envelope of these.

Coherent Sources: Sources of same frequency, fixed phase relation, comparable amplitude. Required for sustained interference. Typically derived from same source (Young's slits, Lloyd's mirror, Fresnel biprism).

Superposition / Interference: Resultant intensity at point with phase difference $\phi$: $$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$$

For $I_1 = I_2 = I_0$: $I = 4I_0\cos^2(\phi/2)$.

• Constructive interference: $\phi = 2n\pi$ (path difference $= n\lambda$); $I_{\max} = (\sqrt{I_1}+\sqrt{I_2})^2$
• Destructive interference: $\phi = (2n+1)\pi$ (path difference $= (n+1/2)\lambda$); $I_{\min} = (\sqrt{I_1}-\sqrt{I_2})^2$

Young's Double Slit Experiment (YDSE): Two slits separated by $d$, screen at distance $D$.

Fringe width independent of fringe order. Decreases if $d$ increases or $D$ decreases.

In medium of refractive index $n$: $\lambda' = \lambda/n$, so fringe width decreases by factor $n$.

Effect of inserting glass slab of thickness $t$ and index $n$ in one path: extra path = $(n-1)t$ → pattern shifts by $(n-1)tD/d$.

Diffraction: Bending of light around obstacles or through narrow openings. Significant when slit width $\sim \lambda$.

Single Slit Diffraction:

• Position of dark fringes: $d\sin\theta = n\lambda$, $n = \pm 1, \pm 2, \ldots$
• Position of secondary maxima: $d\sin\theta \approx (n + 1/2)\lambda$
• Width of central maximum: $2\lambda D/d$ (twice the width of secondary maxima)

Resolving Power:

• Microscope: $R = 2n\sin\theta/(1.22\lambda)$
• Telescope: $R = D/(1.22\lambda)$, where $D$ = aperture

Larger aperture / shorter wavelength → better resolution.

Polarization: EM waves are transverse → can be polarized. Unpolarized light has $\vec{E}$ vibrating in all directions; polarized has it in one plane.

Malus' Law: $I = I_0\cos^2\theta$, where $\theta$ = angle between polarizer axis and $\vec{E}$.

Brewster's Law: When light strikes a dielectric at $\theta_B$ such that reflected and refracted rays are perpendicular, the reflected ray is fully polarized. $$\tan\theta_B = n$$ (Brewster's angle).

Optical Instruments:

For telescope: large $f_o$ and small $f_e$ for high magnification.