Diffraction and interference

Jan. 30, 2018

Diffraction: resulting from the wave nature of light.
Tendency for a small beam to spread.
Smaller is the hole, biggest is the difference between `D_{act}` and `D_{exp}`, i.e. `D_{act} \prop 1/D`; `D` beam diameter.
Similar to Bandwidth (`Delta nu`) relates to pulse width (`Delta T`) by `Delta nu prop 1/{Delta T}`.
• Fourier optics: field `U( k_x , k_y )` far away from the source relates to field `u(x,y)` near the source by the 2D Fourier transform, i.e. `U(k_x, k_y ) = int_{-oo}^{oo} int_{-oo}^{oo} u(x,y) e^{-j 2 pi (k_x x+k_y y)} dx dy` where `k_x = x_1/(lambda d)` and `k_y= y_1/(lambda d)` are spatial freq relating to the coordinates `(x_1 , y_1 )` in far field and a distance `d` from the source.
• Observations: Any sharp change in intensity generates ripples after propagation in space.
Consider a `D_x times D_y` rectangular slit, this slit increases the divergence angle `theta_x = lambda / D_x` & `theta_y = lambda / D_y`
Consider an aperture with radius `rho = D / 2`, divergence angle defined as `theta = 1.22 lambda /D`
Ref. 2 Sec. 4.3, 4.5

Gaussian beam: `E ( rho , z ) = E_o exp (-{rho / w } ^2)` where beam radius (waist) `w = w_o sqrt( 1 + (z / z_o)^2)`, `w_o` is the initial beam radius at focus, `z_o = {pi w_o^2}/lambda` is the diffraction length.
From these, divergence angle `theta = {2 lambda} /{pi w_o}`

Ref. 2 Sec. 3.1


• Interference: `E_1 = e^{-j k_1 r_1}` and `E_2 = e^{-j k_2 r_2}`
Result of interference depends on phase difference `Delta phi = k_1 r_1 - k_2 r_2`
Destructive when `Delta phi = p pi` where `p` is an odd integer.
Constructive when `Delta phi = q pi` where `q` is an even integer.
Observation `- Delta phi` is controlled by refractive index and/or distance.
Applications - interferometer (filter) (e.g. Mach-Zehnder, Michelson, Sagnac), antireflective coatings.

Ref. 2 Sec. 2.5


• Grating: a structure that causes periodic modulation of amplitude or phase, e.g. Blazed grating that has maximum reflection for wave with a certain wavelengthλB traveling at a blaze angle with respect to the normal of the grating `alpha = sin ^ (-1) (lambda_B / {2a})` where `a` is the grating period. (read pp. 118-122 2nd Ed. or pp. 124-128 3rd Ed. of the text)
Results from interference.

Holography: interfere of object wave `U_o` and reference wave `U_r` `=> |U_o + U_r|^2 = I_o + I_r+U_r^{**}U_o + U_rU_o^{**}`
Notice that `U_r^{**}U_o` & `U_rU_o^{**}` `prop cos Delta phi` where `Delta phi = phi_r - phi_o`
The interference pattern is recorded on film.
Playback output `sqrt(I_r) [ I_o + I_r + sqrt(I_r) U_o + sqrt(I_r) U_o^{**} ]`
Similar technique is used to fabricate grating onto optical fibers.

Ref. 2 Sec. 4.3, 4.5


Last Modified: January 29, 2018
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