§§ INTRODUCTION
Microscopy with light or electron
Light optics
{ Electron optics
§1 Light optics
1.22 F 1.22 1.22 0.61



d
d
2

F
Airy’s Disk
(occupy 84% of the light)
Ⅰ. The Diffraction Barrier
Intensity distribution in image of
point object formed by perfect lens
→ Rayleight Criterion
Rayleight Criterion
→ 1873, Ernst Abbe proposed that
0.61
… (1)
n sin 
d0 
n sin   NA (numerical aperature) of the objective lens
Where d 0 = the minimum resolvable separation in the object
= Resolution
 = wavelength of the illuminant
 = angular aperature of the objective lens
n = refraction index in the space between object and objective lens
… Properties of wave of the light, as shown in Fig.1 and Fig.2.
Airy’s Disk … Fig.3 and Fig.4
The radius of Airy’s Disk is D   d 0   
0.61
… (2)
n sin 
Where  = magnification

e.g.  =5,500  ,

sin  = 0.94, n = 1.5
→ d 0  2400 
if  = 100, then D  2.4  10 3 cm … almost no effect in light microscopy.
Rayleight Criterion … d 0  resolution
→ Magnification and resolution:
useful magnification …
empty magnification …
} Fig.5 … if
s  d 0 ,  Airy’s disks are overlapped,  It is an
empty magnification no matter how large of the  .
 for increasing magnification, resolution of the objective len s should be improved, i.e.
( i )increasing n … Fig.6
( ii )  → 90  … e.g. shorten the working distance … Fig.6
( iii )decreasing  … e.g., replacing the light by the electron beam.
Ⅱ. Aberration
Spherical aberration … Fig.7
( i )Spherical aberration … Fig.7
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( ii )Comma: Fig.7
( iii )An off-axis point will not get a point image. (L is called as the circle of least confusion)
…Fig.7
( iv )Distorsion … pincushion or barrel … Fig.7
Astigmatism
Fig.8 … due to the transval & longitudinal focus lengths are different.
… different wavelength having different n value,  the focus length is also different.
→ Depth of field and depth of focus.
Fig.9(a): definition of the depth of focus.
Fig.9(b): the depth of focus is increased by reducing the angular aperature of the lens.
Fig.9(c)(d): definition of the depth of field. D 
2d 0

… (3) … will be further discussed in eq.(22).
§2 Electron optics

h2
…………… (4)
2m0 eV
where h  6.626  10 34 Joul-sec
m0  9.1096  10 31 kg
V  volt
e  1.602  10 19 Coulomb
 
12.3
V
…………… (5)

where  in 
For voltage  50 KV ,
m
m0
…………… (6)
V
1  ( )2
c
where c  2.988  10 8 m /sec
m = electron mass in motion
 
h
2eV
eV 2
m0 c
(
)
2
m0 c
m0 c 2
…………… (7)
For example:
2

For 100KV,   0.037 

200KV,   0.0251 
{
→ Magnetic lenses,
Electrostatic lens … out of date
Electromagnetic lens … up to date
Ⅰ. Electron in Magnetic Field B
F  q  V  B …………… (8)
where F is Lrenz force, q in Coublomb, B in Maxwell/m2 = 104 Gauss,
V in m/sec
 F  F  q V B sin  …………… (9)
The relationship between F , B and V can be shown as :
( i )when   0  or 180  , F = 0
 shown in Fig.10
( ii )when   90  , then F  q V B …………… as shown in Fig.11
electron moves on the plane of F and V , the radius of motion r is
r  mV / q B
…………… (10)
and time period of the motion is
T  m / q B  2r / V …………… (11)
( iii )when   0  , 90  or 180  . Then electron will move in helical path.
Fig.12 … infinite magnetic field
Fig.13 … finite magnetic field (such as in electron microscope, it has the
focusing action + rotation)
Fig.14 …
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Ⅱ.Eelectron in Electric Field
 v

2eV
m
where v = velocity of electron under V volt
V1 sin  1  V2 sin  2 …………… (13)
called Snell rule in geometric electron optics
Ⅲ. Electrostatic lens
˙Equi-potential surface
˙From conservation of energy:
change in potential = change in kinetic
Fig.15 …
Fig.16 …
Fig.17 …
˙Some kinds of electrostatic lens
( i ) Aperture lens … Fig.18
1  2 '1 '
…………… (14)

f
4 0
where  2 ' = field strength of image space
 1 ' = field strength of image space
 0 : electric potential of center of aperture
( ii ) Immersion lens (two aperture lens)
… Fig.19
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
1 3 2 2
 [( )  1](1  1 ) …………… (15)
f 8l 1
2
( iii )Cylinder lens … use them in the multistage accelerating tube of super high voltage type
electron microscope.
… Fig.20
( iv )Electrostatic unipotential lens
… composed from three common-axis aperture electrodes, as shown in Fig.21.
… used as the short-focal lens
( v )Cathode lens
… Fig.22 … used as the lens electron gun for emission type EM, e.g. W-filament.
˙Advantage and disadvantage of electrostatic lens:
Advantage: ( i ) rotation-free imaging.
( ii ) the ability to work with simpler and less highly stablised voltage supplies.
( iii ) Cs is higher.
Disadvantage: ( i ) require a high precision in construction.
( ii ) require a high precision in alignment.
( iii ) require extreme cleanliness in operation to avoid voltage breakdown and
formation of surface insulating films.
… has had only a limited success commercially.
－ seldom used as part of the magnifying lens system in EM.
－ the illuminating systems invariably contain an electrostatic lens (i.e., cathode lens) to extract
and accelerate the electrons.
Ⅳ. Electromagnetic lens
→ Progrssion of electromagnetic lens configuration. … Fig.23
A. 為最原始的電磁透鏡，它只是一個中空的線圈，由於磁場越強，改變電子速度方向的程度
亦越大，即焦距越短，因此有了如 B，C 和 D 的改進。
B. 線圈外側加一層鐵殼，如此和電磁鐵的原理一樣，產生的磁場比沒有鐵殼時增加 10 到 100
倍以上。
C. 後來再加上內殼，使其有磁場的範圍縮小磁場，即磁力密度也增加。
D. 鐵殼上加漏斗狀極片，成為現代的電磁透鏡。電磁透鏡之好壞是電子顯微鏡中決定性能的
主要因素，一般來說，極片的洞(即電子線通過的地方)越短小越精密則愈好(此洞必須成為
正確的圓柱形)，任何偏離圓柱形之偏差，便會引起各種像差而限制了放大率。
E. 同 C。
F. High resolution objective lens … top-entry specimen stage.
G. High resolution objective lens … side-entry specimen stage.
H. Mini-lens … use very high current in a water-cooled coil.
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→ Action of an inhomogeneous (magnetic) field
… In all lens configurations encounted in an EM, the magnetic field distribution is inhomogeneous,
i.e., varying in space.
… Field distribution (z ) and (r ) in a symmetrical magnetic lens.
… Bell-shaped Field
0
 ( z) 
…………… (16)
z 2
1 ( )
a
2
2
… Fig.11-6 and 11-7 of Chen’s book.
… Fig.24 shows the image formation with a magnetic lens depicting electron rotation about the
z -axis.
→ Aberration of electromagnetic lens
including:
( i ) Consider the magnetic distribution to 3rd order (also called 3rd-order aberration):
˙spherical aberration*
˙comma
˙curvature field aberration
˙astigmatism*
˙distorsion
( ii )Consider the defects in magnetic lens, called as “anisotropic aberration”.
˙anisotropic distorsion
˙anisotropic astigmatism
˙anisotropic comma
( iii ) Other factor induced aberration
˙chromatic aberration*
˙rotational chromatic aberration
˙diffraction aberration
˙axial astigmatism
( * : more important )
→ 另一個影響“resolution”的是 diffraction in the objective lens → 即“Rayleight criterion”，由於
“astigmatism 像差”可由 astigmator 校正過來。而 chromatic 像差，則因 TEM 試片很薄，其 ΔE
之 改 變 可以忽略， 故 chromatic 像差也 可忽略 ，故一般由球 面係差及 diff.兩者來 決定
“resolution”。
→ The two main factors determining the resolving power of a microscope system are
aberration and
diffraction in the objective lens (also called 1st lens).
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spherical
r 
ri
…………… (17)
M
where M is magnification
ri is the disc radius in Gaussian image plane
r is the equivalent disc in object plane.
From eqs of motion of electron in magnetic lens, one can get
rs  C s 3 …………… (18)
where C s is the spherical aberration coefficient and C s is a constant
 , in radians, is dep. on the aperture size.
→From the “Rayleight Criterion” in diffraction:
Two points are just “resolved” when the separation of the peak equals the radius of the Airy disc.
 rd 
0.61
for small  …………… (19)

→ Combine effects
 rtotal  rs  rd
 C s 3 
1
  opt  (0.61) 4 3
1
4
3
1
 4Cs 
1
3
3
and rtotal ) min  (0.61) 4 3
4
4
(
 4 Cs
1
0.61 14
)
3C s

…………… (20)
}
…… (21)
3
4
0.61
 0.3028 4 Cs
1
4
Here rtotal ) min is the resolving power d 0 in eq.(1).
e.g. 100KV, C s  focal length of object lens  3.5mm
  opt  5 10 3 radians (0.29 degree)

d 0  0.45 nm  4.5 
( i ) Discussion of eqs (18) and (19):
 value in spherical aberration of EM is in the order of 10 2 ~ 10 3 radians, but in light
microscope  value is about 0.5 radians.
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…
 It has a factor of (10 2 ) 3 times better in eq.(18),
but has a factor of (10 2 ) times worse in eq.(19).
 still leave us 10 3~ 4 better off in resolving power.
( ii ) Discussion the small  in EM.
From Fig.9(b), (c) and (d), we know that for small  in EM, the depth of field and depth of
focus will be greater than that of the light objective at the same resolution.
( iii )depth of field, D
… This is the distance in object space over which you can focus and still get a good image.
r 
 D
D
D
tan   
2
2
2 r
for small 
…………… (22)


e.g. EM, r  5 
(resolution)
  5  1 03 radians

 D  2000 

 The thickness of TEM specimen is usually below 2000  ,
 The instrument is in focus over the whole specimen at once.
( iv ) depth of focus ～ depth of field but now in image space.

v
 magnification
u
1 1 1
 
u v f
For fixed f,
 u v

0
u2 v2
 v 
v2
u   2 u   2 D
u2
 depth of focus   2  (depth of field) …………… (23)

e.g. For   10 4 , D  2000 
then depth of focus  2000cm  20m
 Depth of focus can be looked as effectively infinite in image space. Mean that
screen and camera need not be coplanar.
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→ Chromatic aberration (C.A.) … in EM, it is caused from the electron energy difference.
→ In electron optics, the equivalent of colour is electron energy.
→ C.A. is the lens defect which degrades the image whenever electrons in the beam cease to be
monoenergetic.
Causes: (1) electron starts with a spread of energies
(2) the accelerating voltage fluctuating with time
→ From calculation, rc  C c


…………… (24)
where rc is the radius of the disc caused by C.A.
C c is the C.A. constant.
(slightly small than the focal length)
 is the deviation of the electron energy from it mean value E.

e.g. The C c =3.4mm, let rc = 0.45nm  4.5  = resolving power
  5  1 03 rad.    2.6 in 10 5 V … the order of stability required.
If we allow for some fluctuation in lens current, then  =1 in 10 5 V for pactial
required stability.
→ Astigmatism
→ caused by the electron-deflecting fields not being perfectly symmetrical about the len axis, so
that the lens has a different focus length in different orientations. (may be due to the pole-pieces
of a magnetic lens are not machined carefully, or slightly inhomogeneity in the iron used for the
pole-pieces.)
→ can be corrected by astigmatism corrector or stigmator in at least one lens or sometime in more.
An astigmatic lens can be considered as a combination of an axially symmetrical lens and a
weak cylindrical lens. (Fig.25 upper)
The stigmator introduces a balancing cylindrical lens field perpendicular to the unwanted one
and hence cancels out it effect. (Fig.25 lower)
→ TEM & SEM pictures on astigmatism
Fig.26 … SEM, Fig.27 … TEM
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