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}

• 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.

• Grating: a structure that causes periodic modulation of amplitude or phase, e.g. Blazed grating that has maximum reflection for wave with a certain wavelength$\text{ }{\lambda }_{B}$ 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