Shortcut Methods and Tricks to Solve Numerical Problems on Interference of Light

Double-Slit Interference

• Fringe width: $$\beta = \frac{\lambda D}{d},$$ where (\beta) is the fringe width, (\lambda) is the wavelength of light, (D) is the distance from the slits to the screen, and (d) is the slit separation.
• Separation of bright fringes: $$\Delta x = \frac{\lambda D}{d}$$
• Intensity distribution: $$I = I_0 \cos^2 \left(\frac{\pi d}{\lambda D}x\right),$$ where (I_0) is the maximum intensity and (x) is the distance from the central fringe.

Michelson Interferometer

• Wavelength of light $$\lambda = \frac{2D}{N},$$ where (D) is the path difference and (N) is the number of fringes observed.
• Coherence Length $$l_c = \frac{\lambda}{2(\Delta \lambda)},$$ where (l_c) is the coherence length and (\Delta \lambda) is the spectral bandwidth of the light source.

Young’s Double-Slit Experiment

• Fringe spacing $$x = \frac{\lambda D}{d},$$ where (x) is the fringe spacing, (\lambda) is the wavelength of light, (D) is the distance to the screen, and (d) is the slit separation.
• Fringe width $$\beta = \frac{2\lambda D}{d},$$ where (\beta) is the fringe width.
• Total Number of fringes: $$N = \frac{D}{\beta}=\frac{d}{2\lambda}$$

Coherence and Incoherence

• Coherent sources: Emit waves with the same frequency, constant phase difference, and a definite phase relation.
• Incoherent sources: Emit waves with random phase differences and no definite phase relation.

Thin Film Interference

• Condition for constructive interference: $$2tn = m\lambda, \quad m=0, 1, 2, 3,…$$
• Condition for destructive interference: $$2tn = (m+\frac{1}{2})\lambda, \quad m=0, 1, 2, 3,…$$ where (t) is the film thickness, (n) is the refractive index of the film, (\lambda) is the wavelength of light, and (m) is the order of interference.