What the Eye Can See
Human eye limited in:
• Range of EM wavelengths detectable, ~400 – 700 nm (best for green, ~560 nm)
• Signal intensity needed to trigger “recognition” in brain
(~100 photons per “pixel”)
• Angular separation of image details that eye can resolve
• Integration time over which an image is recorded by eye (0.1 s)
MSE 421/521 Structural Characterization
Lenses
Used to focus/magnify image
M = S2/S1 = v/u
Focal
plane
Object size,
S1
Plane of
lens
Principal
image
plane
Newton’s Lens Equation
Rays passing through
center are unchanged.
Focal
point
Image size,
f
Direction
Information
f = focal length
MSE 421/521 Structural Characterization
S2
v
u
1/f = 1/u + 1/v
Position
Information
Rays parallel to axis pass
through focal point.
Parallel rays leaving
object are focused in
focal plane (diffraction
pattern).
Rays leaving object at same
point are focused in image
plane (image)
Maxwell’s Conditions
for a Perfect Lens
All rays leaving one point on the object must meet at
one point in the image - no aberrations.
If points on the object lie on a plane perpendicular to
the lens axis then points in the image must also lie in a
plane perpendicular to the lens axis - no field/image
curvature.
Ratio of distances in object must be the same as those
in the image - no distortion.
MSE 421/521 Structural Characterization
Real vs Virtual Images
f
v
Real Image
u
Virtual Image
0<u<f
Virtual
Not inverted
Magnified (M < 0, v < 0)
u=f
--
--
Image at infinity – no image
f < u ≤ 2f
Real
Inverted
Magnified (M ≥ 1)
u > 2f
Real
Inverted
Demagnified (0 < M < 1)
MSE 421/521 Structural Characterization
f
Diffraction Barrier
or Diffraction Limit
Resolved
Unresolved
d1/2
d1
d1
Airy disc
George Biddell Airy (1801 – 1892)
MSE 421/521 Structural Characterization
Resolution
The Abbe Criterion
θ
θ
θ
θ
Diffraction Limit:
If grating is too narrow, θ is
too large and only zero-order
beams contribute to image.
object
aperture
d1/2 = rd =
0.5λ
µ sinβ
β
lens
Ernst Karl Abbe (1840 - 1905)
MSE 421/521 Structural Characterization
µ = refractive index
β = semi-angle/half-angle
µ sinβ = numerical aperture
Resolution
The Rayleigh Criterion
Resolved
Rayleigh
Criterion
d1/2
d1
Unresolved
15% drop in
intensity
d1
Diffraction Limit:
d1/2 = rd = δ =
aperture
object
0.61λ
µsinβ
µ = refractive index
β
β = semi-angle/half-angle
µ sinβ = numerical aperture
lens
Lord Ravleigh (John William Strutt) (1877 - 1919)
MSE 421/521 Structural Characterization
Resolution
Rayleigh: d1/2 = rd = δ = 0.61λ
µsinβ
Abbe:
Sparrow:
d1/2 = rd = δ =
0.5λ
µ sinβ
0.47λ
d1/2 = rd = δ =
µsinβ
The distance at which the first maximum in
one Airy disc coincides with the first minimum
of the adjacent Airy disc.
The finest spacing that can be imaged with
both the undiffracted and first-order
diffracted waves.
The distance at which there is no longer a
dip in peak intensity between two adjacent
Airy discs, but rather constant brightness
across the region between the peaks.
http://www.youtube.com/watch?v=0ULRY7RscOY
MSE 421/521 Structural Characterization
The Eye as Microscope?
- cornea: n = 1.376
- vitreous humor ~ water: n = 1:33
- lens: n = 1:39 - 1:41
index varies: high in center, low at
edges
- iris: variable aperture stop
diameter: 2-8 mm
- retina: detector
- resolution…?
MSE 421/521 Structural Characterization
The Eye as Microscope?
Sensitivity:
Fully expanded pupil can see I ≤ 10-10 W/m2
from point source
Power = IA = (10-10 W/m2)[π(4mm)2] ~ 10-14 W
Maximum irradiance (sunlight) ~ 250 mW/m2
Pupil area ~ π(1mm)2
So maximum power ~ 1mW
(damage threshold for laser)
Dynamic range: 10-14 to 10-6 W
(8 orders of magnitude)
(instantaneous range: ~ 5 orders of
magnitude)
Best artificial detectors:
photographic film & high-end CCDs
dynamic range 4 orders of magnitude
(10 worse than eye!)
MSE 421/521 Structural Characterization
- cornea: n = 1.376
- vitreous humor ~ water: n = 1:33
- lens: n = 1:39 - 1:41
index varies: high in center, low at
edges
- iris: variable aperture stop
diameter: 2-8 mm
- retina: detector
- resolution…?
Astigmatism
Differences in optical properties of lens from point to point →
Rays in the horizontal plane are focused at a different point from
those traveling in the vertical plane.
Disc of
least
confusion
lens
rast = β∆f
β must be minimised (unlike for rd)
Astigmatism can also be observed with a non-astigmatic
lens for object points far off the optic axis
MSE 421/521 Structural Characterization
Chromatic Aberration
Rays of different wavelength are refracted by different amounts
in the lens and so are brought to focus at different places.
Shorter wavelengths refract more.
blue green
red
This effect is caused by an energy spread in the electron beam (voltage instabilities)
or inelastic scattering in the sample.
rc = Ccβ (δE/E0)
Cc and β must be minimised (unlike for rd)
Cc (JEOL 2100) = 1.4 mm δE ~ 20 eV
MSE 421/521 Structural Characterization
Problem increases with thickness
Spherical Aberration
Rays passing near the centre of the lens are focused at
a different point from rays passing through the edges.
Marginal
focus
Axial
focus
β
Disc of
least confusion,
r = ¼rs
Gaussian image plane,
r = rs
rs = Csβ 3
(position of image
when very small
apertures are used
- paraxial rays)
Cs and β (strong dependence) must be minimised (unlike for rd)
Cs (JEOL 2100) = 1 mm
MSE 421/521 Structural Characterization
Distortion
If the magnification of the lens varies from the centre
to the edge, the result is a distorted image.
Rectangular object
Barrel distortion
Pin-cushion distortion
The overall effect of aberrations on resolution
In the electron microscope it is very difficult to correct for achromatic aberrations, especially Cs.
Net resolution: ropt = rd2 + rs2 then:
dopt (JEOL 2100) = 2.3 Å
W&C says 0.77
βopt = 0.67λ1/4Cs-1/4
ropt = 1.21λ3/4Cs1/4
MSE 421/521 Structural Characterization
0.7 ≤ x ≤ 1.21
W&C says 0.91
Depth of Field/Focus
Range of object/image distances over which the object can be placed/image
can be viewed without the eye detecting any change in sharpness
MSE 421/521 Structural Characterization
Depth of Field/Focus
Range of object/image distances over which the object can be placed/image
can be viewed without the eye detecting any change in sharpness
aperture
lens
In EM, the small β
necessary to reduce
spherical aberration
results in a large
depth of field/focus
by default (SEM)
despite the small λ.
β
d1
d1
α
Dim
Depth of field : Dob =
0.61λ
µ sinβtanβ
≈
0.61λ
β2
increase (but increases rd)
decrease (but increases rd)
by inserting a smaller aperture
(For EM, µ = 1 and sinβ ≈ tanβ ≈ β)
Depth of focus : Dim = DobM2
Optical microscopy is a compromise between resolution and depth of field/focus,
but in EM, reducing β can improve resolution by reducing aberrations.
MSE 421/521 Structural Characterization