Shortcuts & Tricks for Numerical Problems in Optics

Diffraction:

• Single Slit:
  • For the central maximum:

$$width = 2\lambda L / d$$

• For the first secondary maxima (on either side of central maxima): $$ \theta = \sin^{-1} \frac{\lambda}{d} $$

where (\lambda) is the wavelength, (L) is the distance between the slit and the screen, and (d) is the slit width.

• Multiple Slits:
• Use $$ d\sin \theta = m \lambda$$ to determine the number of slits (N):

$$N = \frac{width\ of \ central \ maxima}{width \ of \ individual\ slits} $$

Interference:

• Double-Slit:

• Path difference for constructive interference: $$d \sin\theta = m\lambda$$

• Fringe spacing: $$ \Delta y = \lambda L/d $$

where (\Delta y) is the distance between adjacent bright fringes, (L) is the distance to the screen, and (\theta) is the angle from the central maximum.

• Thin Films:

-For constructive interference: $$2nt=m\lambda, \ \ m=0, 1, 2, 3…$$

Polarization:

• Brewster’s Angle:

$$ \theta_B = \tan^{-1} \left( \frac{n_2}{n_1} \right)$$

where (\theta_B) is Brewster’s angle, and (n_1) and (n_2) are the refractive indices of the two media.

• Intensity of Transmitted Light: $$I_t= I_0\cos^2\theta$$

where (I_t) is the intensity of transmitted light, (I_0) is the intensity of incident light, and (θ) is the angle between the polarization direction of the incident light and the transmission axis of the polarizer.

Note: These are just a few general shortcuts and tricks for numerical problems in optics. Depending on the specific question, you may need to use a combination of these methods and apply other relevant equations and concepts from optics.