Download Overview from last week

Overview from last week
• Optical systems act as linear shift-invariant
(LSI) filters (we have not yet seen why)
• Analysis tool for LSI filters: Fourier transform
– decompose arbitrary 2D functions into
superpositions of 2D sinusoids (Fourier
transform)
– use the transfer function to determine what
happens to each 2D sinusoid as it is transmitted
through the system (filtering)
– recompose the filtered 2D sinusoids to
determine the output 2D function (Fourier
integral, aka inverse Fourier transform)
MIT 2.71/2.710 Optics
11/01/04 wk9-a-1
Today
• Wave description of optical systems
• Diffraction
– very short distances: near field, we skip
– intermediate distances: Fresnel diffraction
expressed as a convolution
– long distances (∞): Fraunhofer diffraction
expressed as a Fourier transform
MIT 2.71/2.710 Optics
11/01/04 wk9-a-2
Space and spatial frequency
representations
SPACE DOMAIN
2D Fourier transform
SPATIAL FREQUENCY
DOMAIN
MIT 2.71/2.710 Optics
11/01/04 wk9-a-3
2D Fourier integral
Aka
inverse 2D Fourier transform
2D linear shift invariant systems
input
output
transform
multiplication with transfer function
MIT 2.71/2.710 Optics
11/01/04 wk9-a-4
inverse Fourier
transform
Fourier
convolution with impulse response
Wave description of optical
imaging systems
MIT 2.71/2.710 Optics
11/01/04 wk9-a-5
coherent
illumination:
plane
wave
Thin transparencies
Transmission function:
Field before transparence:
Field after transparence:
assumptions: transparence at z=0
transparency thickness can be ignored
MIT 2.71/2.710 Optics
11/01/04 wk9-a-6
Diffraction: Huygens principle
incident
plane
Wave
Field at distance d:
contains contributions
from all spherical waves
emitted at the transparency,
summed coherently
Field after transparence:
MIT 2.71/2.710 Optics
11/01/04 wk9-a-7
Huygens principle: one point
source
spherical
wave
incoming
plane wave
opaque
screen
MIT 2.71/2.710 Optics
11/01/04 wk9-a-8
Simple interference: two point
sources
intensity
incoming
plane wave
MIT 2.71/2.710 Optics
11/01/04 wk9-a-9
opaque
screen
Two point sources interfering:
math…
(paraxial approximation)
Amplitude:
intensity
MIT 2.71/2.710 Optics
11/01/04 wk9-a-10
Diffraction: many point sources
many spherical waves
tightly packed
incoming
plane wave
opaque
screen
MIT 2.71/2.710 Optics
11/01/04 wk9-a-11
Diffraction: many point
sources,attenuated & phase-delayed
incoming
plane wave
thin
transparency
MIT 2.71/2.710 Optics
11/01/04 wk9-a-12
Diffraction: many point sources
attenuated & phase-delayed, math
Thin transparency
Transmission function
wave
incoming
plane wave
field
MIT 2.71/2.710 Optics
11/01/04 wk9-a-13
continuous
limit
Fresnel diffraction
The diffracted field is the convolution of the
transparency with a spherical wave
CONSTANT:
NOT interesting
amplitude distribution transparency
transmission
at output plane
function
(complex teiφ)
MIT 2.71/2.710 Optics
11/01/04 wk9-a-14
FUNCTION OF LATERAL COORDINATES:
Interesting!!!
spherical wave
@z=l
(aka Green’s function)
Example: circular aperture
input field
MIT 2.71/2.710 Optics
11/01/04 wk9-a-15
Example: circular aperture
input field
MIT 2.71/2.710 Optics
11/01/04 wk9-a-16
Example: circular aperture
input field
MIT 2.71/2.710 Optics
11/01/04 wk9-a-17
Example: circular aperture
Image removed due to copyright concerns
input field
MIT 2.71/2.710 Optics
11/01/04 wk9-a-18
(from Hecht, Optics, 4thedition, page 494)
Fraunhofer diffraction
propagation distance lis “very large”
approximaton valid if
MIT 2.71/2.710 Optics
11/01/04 wk9-a-19
Fraunhofer diffraction
The“far-field” (i.e. the diffraction pattern at a large
longitudinal distance l equals the Fourier transform
of the original transparency
calculated at spatial frequencies
MIT 2.71/2.710 Optics
11/01/04 wk9-a-20
Fraunhofer diffraction
spherical wave
originating at x
MIT 2.71/2.710 Optics
11/01/04 wk9-a-21
plane wave propagating
at angle –x/l
⇔ spatial frequency –x/(λl)
Fraunhofer diffraction
superposition of
spherical waves
originating at
various points
along
MIT 2.71/2.710 Optics
11/01/04 wk9-a-22
superposition of
plane waves propagating
at corresponding angles –x/l
⇔spatial frequencies –x/(λl)
Example: rectangular apertureinput
free space
propagation by
input field
MIT 2.71/2.710 Optics
11/01/04 wk9-a-23
far field
Example: circular aperture
free space
propagation by
far field
input field
also known as
Airy pattern, or
MIT 2.71/2.710 Optics
11/01/04 wk9-a-24