Download incoherent

Today
• Review of spatial filtering with coherent coherent illumination
• Derivation of the lens law using wave optics
• Point-spread function of a system with incoherent incoherent
illumination
• The Modulation Transfer Function (MTF) and Optical
Transfer Function (OTF)
• Comparison of coherent and incoherent imaging
• Resolution and image quality
– The meaning of resolution
– Rayleigh criterion and image quality
MIT 2.71/2.710 Optics
11/10/04 wk10-b-1
Coherent imaging
as a linear, shift-invariant system
Thin transparency
output
amplitude
impulse response
convolution
illumi
nation
Fourier
transform
Fourier
transform
transfer function
(≡plane wave
spectrum)
multiplication
transfer function
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aka pupil function
The 4F system with FP aperture
object plane
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Fourier plane: aperture-limited
Image plane: blurred
(i.e. low-pass filtered)
Single-lens imaging condition
object
lens
image
Imaging condition
(akaLens Law)
Derivation using
wave optics ?!?
lateral
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Magnification
Single-lens imaging system
object
lens
image
spatial
“LSI” system“
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Single-lens imaging system
Impulse response (PSF)
spatial
“LSI” system“
Ideal PSF:
Diffraction-Limited
PSF:
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Imaging with incoherent light
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Two types of incoherence
temporal
incoherence
point
source
Michelson interferometer
poly-chromaticlight
(=multi-color, broadband)
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spatial
incoherence
matched
paths
Young interferometer
mono-chromaticlight
(= single color, narrowband)
Two types of incoherence
temporal
incoherence
point
source
waves from unequal paths
do not interfere
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spatial
incoherence
matched
paths
waves with equal paths
but from different points
on the wavefront
do not
interfere
Coherent vs incoherent beams
Mutually coherent: superposition field amplitude
is described by sum of complex amplitudes
Mutually incoherent: superposition field intensity
is described by sum of intensities
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(the phases of the individual beams vary
randomly with respect to each other;
hence,
we would need statistical formulation to
describe them properly —statistical optics)
Imaging with spatially incoherent light
simple object: two point sources
narrowband, mutually incoherent
(input field is spatially incoherent)
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Imaging with spatially incoherent light
incoherent: adding in intensity ⇒
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Imaging with spatially incoherent light
Generalizing:
thin transparency with
sp. incoherent illumination
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intensity at the output
of the imaging system
Incoherent imaging
as a linear, shift-invariant system
Thin transparency
incoherent
impulse response
illumi
nation
convolution
Incoherent imaging is linear in intensity
with incoherent impulse response (iPSF)
where h(x,y) is the coherent impulse response (cPSF)
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output
intensity
Incoherent imaging
as a linear, shift-invariant system
Thin transparency
incoherent
impulse response
convolution
illumi
nation
Fourier
transform
Fourier
transform
(≡plane wave
spectrum)
transfer function of incoherent system:
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output
intensity
transfer function
multiplication
optical transfer function (OTF)
The Optical Transfer Function
normalized to 1
real
real
max
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max
max
max
some terminology ...
Amplitude transfer function
(coherent)
Optical Transfer Function (OTF)
(incoherent)
Modulation Transfer Function (MTF)
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MTF of circular aperture
physical aperture
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filter shape (MTF)
MTF of rectangular aperture
physical aperture
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filter shape (MTF)
Incoherent low–pass filtering
MTF
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Intensity @ image plane
Incoherent low–pass filtering
MTF
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Intensity @ image plane
Incoherent low–pass filtering
MTF
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Intensity @ image plane
Diffraction-limited vs aberrated MTF
real
ideal thin lens,
finite aperturez
realistic lens
finite aperture
& aberrations
max
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max
Imaging with polychromatic light
Monochromatic, spatially incoherent response
at wavelength λ0:
Polychromatic (temporally and spatially incoherent)
response:
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Comments on coherent vs incoherent
• Incoherent generally gives better image quality:
– no ringing artifacts
– no speckle
– higher bandwidth (even though higher frequencies are
attenuated because of the MTF roll-off)
• However, incoherent imaging is insensitive to phas
objects
• Polychromatic imaging introduces further blurring due to
chromatic aberration (dependence of the MTF on
wavelength)
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Resolution
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Connection between PSF and NA
Monochromatic
coherent on-axis
illumination
object plane
impulse
Fourier plane
circ-aperture
image plane
observed field
(PSF)
Fourier
transform
radial coordinate
@ Fourier plane
radial coordinate
@ image plane
(unit magnification)
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Connection between PSF and NA
Monochromatic
coherent on-axis
illumination
NA: angle
of acceptance
for on–axis
point object
Fourier plane
circ-aperture
Numerical Aperture (NA)
by definition:
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image plane
Numerical Aperture and Speed (or F–
Number)
medium of
refr. index n
half-angle subtended by the
imaging system from
an axial object
Numerical Aperture
Speed(f/#)=1/2(NA)
pronounced f-number, e.g.
f/8 means (f/#)=8.
Aperture stop
the physical element which
limits the angle of acceptance of
the imaging system
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Connection between PSF and NA
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Connection between PSF and NA
lobe width
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NA in unit–mag imaging systems
Monochromatic
coherent on-axis
illumination
in both cases,
Monochromatic
coherent on-axis
illumination
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The incoherent case:
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The two–point resolution problem
Imaging
system
object: two point sources,
mutually incoherent
(e.g. two stars in the night sky;
two fluorescent beads in a solution)
intensity
pattern
observed
(e.g. with
digital
camera)
The resolution question [Rayleigh, 1879]: when do we cease
to be able to resolve the two point sources (i.e., tell them apart)
due to the blurring introduced in the image by the finite
(NA)?
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The meaning of “resolution”
[from the New Merriam-Webster Dictionary, 1989 ed.]:
resolve v: 1to break up into constituent parts: ANALYZE;
2to find an answer to : SOLVE; 3DETERMINE, DECIDE;
4to make or pass a formal resolution
resolution n: 1the act or process of resolving 2the action
of solving, also: SOLUTION; 3the quality of being resolute :
FIRMNESS, DETERMINATION; 4a formal statement
expressing the opinion, will or, intent of a body of persons
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Resolution in optical system
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Resolution in optical system
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Resolution in optical system
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Resolution in optical system
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Resolution in optical system
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Resolution in optical system
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Resolution in noisy optical
systems
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“Safe” resolution in optical
system
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Diffraction–limited resolution (safe)
Two point objects are “just resolvable” (limited by diffraction only)
if they are separated by:
Two–dimensional systems
(rotationally symmetric PSF)
One–dimensional systems
(e.g. slit–like aperture)
Safe definition:
(one–lobe spacing)
Pushy definition:
(1/2–lobe spacing)
You will see different authors giving different definitions.
Rayleigh in his original paper (1879) noted the issue of noise
and warned that the definition of “just–resolvable”
points
MIT 2.71/2.710 Optics
is system–or application –dependent
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