Diffraction and interference |
Jan. 30, 2018 |
• :
resulting from the wave
nature of light.
Tendency for a small beam to spread.
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`
,
this slit increases the divergence
angle `theta_x = lambda / D_x`
& `theta_y = lambda / D_y`
Consider an
with radius `rho = D / 2`, divergence angle defined
as `theta = 1.22 lambda /D`
• :
`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.
,
Michelson,
),
antireflective coatings.
• Grating: a structure that causes periodic modulation
of amplitude or phase, e.g.
that has maximum
reflection for wave with a certain
wavelength$\text{\hspace{0.5em}}{\lambda}_{B}$
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.
• : 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.
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.