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Ideal lenses and the Scherzer theorem: A supplement
Ultrarrucroscopy 26 (1988) 385-358 North-Holland. A m s t e r d a m
385
IDEAL LENSES AND THE SCHERZER THEOREM: A SUPPLEMENT
Zhifeng S H A O and A.V. C R E W E Department of Physics and Enrico Fermi Institute. Untcersttv of Chzcago. 5640 S. Elhs .4ce.. Chicago. Ilhnots 6063". USA Received 22 April 1988
In this article, v,e discuss the form of an ideal magxmtic field ~ h i c h vdll be free from spherical aberration. The result is that a divergent point is needed in the m a ~ e t i c field distribution, so that an ideal lens tn this sense cannot be achie,,ed. Ho~e,.er. one may be able to approach the limit in a practical design by a p p r o x i m a t i n g the ideal field.
we can write down the ray equation up to the third order [2]:
1. Introduction
Since Scherzer published his famous theorem [1] about the spherical aberration of rotationally symmetric systems, man.', attempts have been made to find an ideal field distribution which provides adequate focusing without suffering f r o m spherical aberration. These efforts have failed for many reasons, but on the other hand the', m a y have helped us to u n d e r s t a n d the Scherzer theorem a little better because if w-e know the exact form of the ideal field distribution, we can alv,-avs tr-, to m a k e a real lens ver~ close to the ideal one. In this short paper, we intend to supplement a previous paper [2] bv one of the authors (A.V.C.) on this subject a n d put the pre,dous results on m o r e rigorous m a t h e m a t i c a l basis. We hope our result will benefit the understanding of the basic principles of electron optics and provide some hints for the design of a better lens.
r" =
4R z
rS,
4 R z -g-
t
B]"~
2R: }
3 1
- r " { rB] - r'B,., )I - " " " J
!4)
v,here R is the mag.netic riNdity, and B,., is the a.vial field and is a function of - only. B~. inspection. we see that in order to set all third-order terms to zero. v,e should require r'=-rB,=
4R:.
~5t
~r:{B('-B,~,2R:}-r'{rB,;-r'Bo}=O.
(6)
If w-e now substitute the expression for r from eq. (5) into eq. {61. v,e have
-
rr'a,;c~ ( r'B.._
r':B,:,
B, I
2. Field distributions
It has been shown by many authors that. from the equations of t~-iiOtiOn t o l ,.t tt0tattitotialt~, s~,-mmetric s',stem with its optical axis along the _--axis. F-to-=
l (.~io .._
--
---B:ro. r:o)
e B.rg ,
D7
.
e ( B.i"
11) B,-)
(') ( a)
,,vhich is a very convenient i n t e g a t e directly to obtain
2r'B
form since v,e can
= r( -4 - B.' ).
~S
v, here A is a function of r onb.. N o ~ ~f v,e differentiate this equation vdth respect to - again and substitute into the first-order ra,, equation, we have the general condition "B.-,( B,~' - .B) 2 R : ) - B,~(4.4 - 3B/) = . 4 : .
0304-3991,;88<$03.50:2 Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Phvsics Publishing Division)
19 ~,
Z. Shao, A. V. C r e w e / I d e a l lenses a n d t h e S c h e m e r t h e o r e m
386
which we can use to determine an appropriate function for B 0. This is a complicated second-order differential equation, but we can simplify the problem if we take the special case A = 0, in which case the above equation can be written as 2Bo( B o -. Bao12R 2 ) - 3Bo 2 = 0.
(10)
One special solution of this equation has been pointed out previously as being [2]
Bo(z)=n/(b-z
).
(11)
However, there are also other solutions. By rewriting the condition for A --0, we have ,, B°
Bo3 2R2
3 Bo2 = 0. 2 Bo
(12)
Since B o is a function of z only. we can write 3 2B °
p(So)=
and
q(B0)=
2R 2,
except near the origin. We can see readily now that the Glaser field (bell-shaped field),
B o = A / ( z : + b"),
(19)
in the region far away from origin (z >> b) is very close to our ideal field and this probably indicates why a bell-shaped field provides such a good performance. Since the ideal field diverges at z = +_2 R / c 1, we see that in the Glaser field, it is the region close to origin which produces spherical aberration because when z--, 0, the Glaser field approaches a constant instead of infinity. H o w ever, this ideal field c a n n o t be produced in laboratory. and therefore we concentrate on the other class of solutions which is easier to handle, namely, those corresponding to c I = 0. For the case of cl = 0, we have
=,,(s,, <,,,o)
(13)
1/2
dBo +
¢2'
and the equation becomes
B o" + p( Bo)Bo 2 + q ( B o ) = 0 .
(14)
This is a standard second-order differential equation. It has the following general solution [3]
~20) which reduces to Bo= R / ( c 2 - ").
d o)
(21)
(15)
which is just the field obtained before. Furthermore, we can set c z = 0 since this again corresponds to moving the origin. Note also that there are no arbitral 3, constants. The exact value for B is specified at all points.
(16)
3. T h e z - 1 field and the ray equation
z= f [-2fqexp(2SPdBo)dBo+c,l't:dB° + c 2.
/
The soludon corresponding to c~ 4= 0 is Bo(z) =
4R
R
ci (c z - z ) 2 - 4 R 2 / c 2
We can absorb c 2 into z without losing generality (it is equivalent to changing the origin of the coordinates), and then the ideal field distribution would be Bo =
4R
R , c1 z ~ _ 4R2/c 2
(17)
Now for c, >> 2R, we can approximate the field as
Bo--
4R R C1
2 2 "
If we consider the z - i field obtained from last section, we have the following ray equation r
,,
=
~
,
r
/_2 ,. ,
[k '~) ' -) !~
and the general solution can then be written as r = ( ~ ( C , + C: In z).
(23)
By introducing the initial condition: z = l, r ' = 0 and r = ro, we can evaluate the constants C 1 and C 2. This gives "-z--
(18)
r " ~ro)/ l ( 2 - ' n l ).
(24)
Z. Shao. A. $: Crewe / Ideal lenses and the Schcrzer theorem
If we now use the fact that lim ¢7 in z = 0,
(25)
g ---*0
we see that r = 0 at z = 0. This indicates that a parallel ray pencil at r - - ! will converge at the point z = 0, i n d e p e n d e n t of the value of r 0, so that this is a focal point a n d the focal length associated with this focal p o i n t is 0. This is u n d e r s t a n d a b l e because at z = 0, the field goes to infinity and the force on electron also diverges. The peculiar behavior is that for a n y given ray, we can trace it to the zero point f r o m " o u t s i d e " , but we c a n n o t trace it back from " i n s i d e " . The point z = 0 is a very special point, but we also see that z = l e : ( e = 2 . 7 1 8 2 8 . . . ), r = 0. This means that there is another focal point. Physically, a point source at - = le z will be imaged at z = 0 without spherical aberration passing through the position z - l as a pencil o f rays parallel to the axis. It also should be clear that for any arbitrary l, a source at le 2 will be focused at z = 0. In fact, the general solution itself guarantees that at z = 0, r = 0 and r ' = z¢, ind e p e n d e n t of the values of C 1 and Cz. This is an intrinsic property of this special field. Since the focal point is at the diverging point of the field, it does not have much practical value and so we now consider a more general field (26)
Bo(z)=kR/z,
where k is a constant and may be smaller or larger than one. The ray equation can be written as r
"" =
--
¼kEr/
(27)
. . ' ~".
F o r a weak field k 2 < 1, we suppose a solution r = z" and then we have a(a-1)=-lk2,
ct=½+~Vq-k
2.
(28)
~0 that #_. is real. If we now let 1 - k 2-= A,8 2, we have "~ = I _~ fl ~,,,a r = C1-1/2+/~ +
C2ZI/'e-IJ-'VI-z-( C12B + C2/2fl ) , (29)
fl_< ~. where 4/3" = 1 - k ' > 0- _a n d k~ > 0 s o t h a t By using the same initial condition, we have r=
1-2f14fl ro ( ).i / 2 + a . 1 + 1+2/34____~r° (.)l,2--B_l_
(30)
3~7
Since fl _< ,1, we see that - = 0 is a focal point. The conclusion is that for a weak field, only the divergent point can be a focal point which is then independent of the electron energy. A n o t h e r focal poi,qt is located at ( z 0 ) 2t~ 1 + 2fl 7 - l-2fl"
(31)
d e p e n d i n g on the field strength and the electron energy through B. N o w let us consider a strong field with k 2 > 1. To m a k e the solution straightforward, we use a n o t h e r way to obtain a solution. If we substitute z = e ~' into the ray equation we have d?r
dr
dq~:
d~
(32)
ik'r.
Suppose r = e "~, w e have 0t2 -
a
+
~k z = 0,
(33)
which gives us the following solution t r = ,1 + . ,i
ivy.,
--1
.
(34)
Again, let k - " - 1 = 4fl 2. we have a = ,~ _+ ifl. The solution can now be written as r = C, exp[qs(:' + ifl)] + C:
exp[
q,(
I, - i 13)] . (35)
or m o r e conventionally, r = e x p ( ,1 q5 )( C1 sin fl~, + C, cos ~
).
(36)
R e m e m b e r i n g that ~ = in -, we have the following solution which is written in a form equivalent to the one given in the previous paper r=~-'(C:
sin/3 In - + C z c o s . / 3
In - t .
By taking tile parallel ray condition and r ' = 0. we have ,=
(
i( '
- = 1. r = r,,
t
1
+ c o s f l In / + - ~ s i n
(37)
'
fl in I
•t c o s f l
in - .
1
13S)
388
Z Shao, ,4. V. Crewe / Ideal lenses and the Scherzer theorem
Assuming tan(fl In 1 ) = 1 / 2 f l , we have r=r 0
1+
cosfllnz
.
(39)
It should be noted that if we let k 2 = 1 + t then we get exactly the previous solution
r = ro
+ - cos(½~f~c In z ). E
which gives us : = 1 e x p ( - 2 r r / ~ ) . Since the explicit form of ray solution is obtained, one can directly use Scherzer integral to evaluate spherical aberration coefficient. It has been shown in the previous paper that for c - 0.2, C J f is less than 1%. For more details, one is referred to ref. [2].
(40)
4. Conclusion
From this result, we have two focal points !V~'2 In z = +½~r,_
(41)
that is z,, 2 = exp( +_ rr/V/~c),
(42)
where z I and z 2 are on opposite sides of the point z = l. If we have a point source at zl then we should have an image at z 2. The focal lengths at z~ and z2 are
For the minus sign, at c --* 0, we have f ~ 0 which agrees with the previous result. If we take the following condition z = l, r = 0 and r ' = ro, then r= --~
sin fl In 7 .
(44)
In conclusion, we see that the Glaser field is a good approximation to the ideal field distribution in the region far away from divergent point but differs significantly from the ideal field in the region near z = 0. To achieve zero aberration with adequate focusing, a divergence of the field seems necessary. This limitation seems to rule out the possibility of building an aberration-free lens. However, one can always approach this limit in many ways in practical design.
Acknowledgement This study was supported by grants from the Department of Energy and the National Science F o u n d a t i o n of the United States.
Since fl = ½v/~", it can be written as ,,
=
fZf
0V 7
z
sin ½v~ ln-f
References (45)
Clearly, the closest focal point is at
½¢r~ I n ( z / l ) =
-or,
(46)
[11 O. Scherzer, Z. Physik 101 (1936) 593. |2l A.V. Crewe, Ultramicroscopy 2 (1977) 281. [3] I. Gradshteyn and !. Ryzhik, Tables of Integrals, Series and Products (Academic Press. New York, 1980).
385
IDEAL LENSES AND THE SCHERZER THEOREM: A SUPPLEMENT
Zhifeng S H A O and A.V. C R E W E Department of Physics and Enrico Fermi Institute. Untcersttv of Chzcago. 5640 S. Elhs .4ce.. Chicago. Ilhnots 6063". USA Received 22 April 1988
In this article, v,e discuss the form of an ideal magxmtic field ~ h i c h vdll be free from spherical aberration. The result is that a divergent point is needed in the m a ~ e t i c field distribution, so that an ideal lens tn this sense cannot be achie,,ed. Ho~e,.er. one may be able to approach the limit in a practical design by a p p r o x i m a t i n g the ideal field.
we can write down the ray equation up to the third order [2]:
1. Introduction
Since Scherzer published his famous theorem [1] about the spherical aberration of rotationally symmetric systems, man.', attempts have been made to find an ideal field distribution which provides adequate focusing without suffering f r o m spherical aberration. These efforts have failed for many reasons, but on the other hand the', m a y have helped us to u n d e r s t a n d the Scherzer theorem a little better because if w-e know the exact form of the ideal field distribution, we can alv,-avs tr-, to m a k e a real lens ver~ close to the ideal one. In this short paper, we intend to supplement a previous paper [2] bv one of the authors (A.V.C.) on this subject a n d put the pre,dous results on m o r e rigorous m a t h e m a t i c a l basis. We hope our result will benefit the understanding of the basic principles of electron optics and provide some hints for the design of a better lens.
r" =
4R z
rS,
4 R z -g-
t
B]"~
2R: }
3 1
- r " { rB] - r'B,., )I - " " " J
!4)
v,here R is the mag.netic riNdity, and B,., is the a.vial field and is a function of - only. B~. inspection. we see that in order to set all third-order terms to zero. v,e should require r'=-rB,=
4R:.
~5t
~r:{B('-B,~,2R:}-r'{rB,;-r'Bo}=O.
(6)
If w-e now substitute the expression for r from eq. (5) into eq. {61. v,e have
-
rr'a,;c~ ( r'B.._
r':B,:,
B, I
2. Field distributions
It has been shown by many authors that. from the equations of t~-iiOtiOn t o l ,.t tt0tattitotialt~, s~,-mmetric s',stem with its optical axis along the _--axis. F-to-=
l (.~io .._
--
---B:ro. r:o)
e B.rg ,
D7
.
e ( B.i"
11) B,-)
(') ( a)
,,vhich is a very convenient i n t e g a t e directly to obtain
2r'B
form since v,e can
= r( -4 - B.' ).
~S
v, here A is a function of r onb.. N o ~ ~f v,e differentiate this equation vdth respect to - again and substitute into the first-order ra,, equation, we have the general condition "B.-,( B,~' - .B) 2 R : ) - B,~(4.4 - 3B/) = . 4 : .
0304-3991,;88<$03.50:2 Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Phvsics Publishing Division)
19 ~,
Z. Shao, A. V. C r e w e / I d e a l lenses a n d t h e S c h e m e r t h e o r e m
386
which we can use to determine an appropriate function for B 0. This is a complicated second-order differential equation, but we can simplify the problem if we take the special case A = 0, in which case the above equation can be written as 2Bo( B o -. Bao12R 2 ) - 3Bo 2 = 0.
(10)
One special solution of this equation has been pointed out previously as being [2]
Bo(z)=n/(b-z
).
(11)
However, there are also other solutions. By rewriting the condition for A --0, we have ,, B°
Bo3 2R2
3 Bo2 = 0. 2 Bo
(12)
Since B o is a function of z only. we can write 3 2B °
p(So)=
and
q(B0)=
2R 2,
except near the origin. We can see readily now that the Glaser field (bell-shaped field),
B o = A / ( z : + b"),
(19)
in the region far away from origin (z >> b) is very close to our ideal field and this probably indicates why a bell-shaped field provides such a good performance. Since the ideal field diverges at z = +_2 R / c 1, we see that in the Glaser field, it is the region close to origin which produces spherical aberration because when z--, 0, the Glaser field approaches a constant instead of infinity. H o w ever, this ideal field c a n n o t be produced in laboratory. and therefore we concentrate on the other class of solutions which is easier to handle, namely, those corresponding to c I = 0. For the case of cl = 0, we have
=,,(s,, <,,,o)
(13)
1/2
dBo +
¢2'
and the equation becomes
B o" + p( Bo)Bo 2 + q ( B o ) = 0 .
(14)
This is a standard second-order differential equation. It has the following general solution [3]
~20) which reduces to Bo= R / ( c 2 - ").
d o)
(21)
(15)
which is just the field obtained before. Furthermore, we can set c z = 0 since this again corresponds to moving the origin. Note also that there are no arbitral 3, constants. The exact value for B is specified at all points.
(16)
3. T h e z - 1 field and the ray equation
z= f [-2fqexp(2SPdBo)dBo+c,l't:dB° + c 2.
/
The soludon corresponding to c~ 4= 0 is Bo(z) =
4R
R
ci (c z - z ) 2 - 4 R 2 / c 2
We can absorb c 2 into z without losing generality (it is equivalent to changing the origin of the coordinates), and then the ideal field distribution would be Bo =
4R
R , c1 z ~ _ 4R2/c 2
(17)
Now for c, >> 2R, we can approximate the field as
Bo--
4R R C1
2 2 "
If we consider the z - i field obtained from last section, we have the following ray equation r
,,
=
~
,
r
/_2 ,. ,
[k '~) ' -) !~
and the general solution can then be written as r = ( ~ ( C , + C: In z).
(23)
By introducing the initial condition: z = l, r ' = 0 and r = ro, we can evaluate the constants C 1 and C 2. This gives "-z--
(18)
r " ~ro)/ l ( 2 - ' n l ).
(24)
Z. Shao. A. $: Crewe / Ideal lenses and the Schcrzer theorem
If we now use the fact that lim ¢7 in z = 0,
(25)
g ---*0
we see that r = 0 at z = 0. This indicates that a parallel ray pencil at r - - ! will converge at the point z = 0, i n d e p e n d e n t of the value of r 0, so that this is a focal point a n d the focal length associated with this focal p o i n t is 0. This is u n d e r s t a n d a b l e because at z = 0, the field goes to infinity and the force on electron also diverges. The peculiar behavior is that for a n y given ray, we can trace it to the zero point f r o m " o u t s i d e " , but we c a n n o t trace it back from " i n s i d e " . The point z = 0 is a very special point, but we also see that z = l e : ( e = 2 . 7 1 8 2 8 . . . ), r = 0. This means that there is another focal point. Physically, a point source at - = le z will be imaged at z = 0 without spherical aberration passing through the position z - l as a pencil o f rays parallel to the axis. It also should be clear that for any arbitrary l, a source at le 2 will be focused at z = 0. In fact, the general solution itself guarantees that at z = 0, r = 0 and r ' = z¢, ind e p e n d e n t of the values of C 1 and Cz. This is an intrinsic property of this special field. Since the focal point is at the diverging point of the field, it does not have much practical value and so we now consider a more general field (26)
Bo(z)=kR/z,
where k is a constant and may be smaller or larger than one. The ray equation can be written as r
"" =
--
¼kEr/
(27)
. . ' ~".
F o r a weak field k 2 < 1, we suppose a solution r = z" and then we have a(a-1)=-lk2,
ct=½+~Vq-k
2.
(28)
~0 that #_. is real. If we now let 1 - k 2-= A,8 2, we have "~ = I _~ fl ~,,,a r = C1-1/2+/~ +
C2ZI/'e-IJ-'VI-z-( C12B + C2/2fl ) , (29)
fl_< ~. where 4/3" = 1 - k ' > 0- _a n d k~ > 0 s o t h a t By using the same initial condition, we have r=
1-2f14fl ro ( ).i / 2 + a . 1 + 1+2/34____~r° (.)l,2--B_l_
(30)
3~7
Since fl _< ,1, we see that - = 0 is a focal point. The conclusion is that for a weak field, only the divergent point can be a focal point which is then independent of the electron energy. A n o t h e r focal poi,qt is located at ( z 0 ) 2t~ 1 + 2fl 7 - l-2fl"
(31)
d e p e n d i n g on the field strength and the electron energy through B. N o w let us consider a strong field with k 2 > 1. To m a k e the solution straightforward, we use a n o t h e r way to obtain a solution. If we substitute z = e ~' into the ray equation we have d?r
dr
dq~:
d~
(32)
ik'r.
Suppose r = e "~, w e have 0t2 -
a
+
~k z = 0,
(33)
which gives us the following solution t r = ,1 + . ,i
ivy.,
--1
.
(34)
Again, let k - " - 1 = 4fl 2. we have a = ,~ _+ ifl. The solution can now be written as r = C, exp[qs(:' + ifl)] + C:
exp[
q,(
I, - i 13)] . (35)
or m o r e conventionally, r = e x p ( ,1 q5 )( C1 sin fl~, + C, cos ~
).
(36)
R e m e m b e r i n g that ~ = in -, we have the following solution which is written in a form equivalent to the one given in the previous paper r=~-'(C:
sin/3 In - + C z c o s . / 3
In - t .
By taking tile parallel ray condition and r ' = 0. we have ,=
(
i( '
- = 1. r = r,,
t
1
+ c o s f l In / + - ~ s i n
(37)
'
fl in I
•t c o s f l
in - .
1
13S)
388
Z Shao, ,4. V. Crewe / Ideal lenses and the Scherzer theorem
Assuming tan(fl In 1 ) = 1 / 2 f l , we have r=r 0
1+
cosfllnz
.
(39)
It should be noted that if we let k 2 = 1 + t then we get exactly the previous solution
r = ro
+ - cos(½~f~c In z ). E
which gives us : = 1 e x p ( - 2 r r / ~ ) . Since the explicit form of ray solution is obtained, one can directly use Scherzer integral to evaluate spherical aberration coefficient. It has been shown in the previous paper that for c - 0.2, C J f is less than 1%. For more details, one is referred to ref. [2].
(40)
4. Conclusion
From this result, we have two focal points !V~'2 In z = +½~r,_
(41)
that is z,, 2 = exp( +_ rr/V/~c),
(42)
where z I and z 2 are on opposite sides of the point z = l. If we have a point source at zl then we should have an image at z 2. The focal lengths at z~ and z2 are
For the minus sign, at c --* 0, we have f ~ 0 which agrees with the previous result. If we take the following condition z = l, r = 0 and r ' = ro, then r= --~
sin fl In 7 .
(44)
In conclusion, we see that the Glaser field is a good approximation to the ideal field distribution in the region far away from divergent point but differs significantly from the ideal field in the region near z = 0. To achieve zero aberration with adequate focusing, a divergence of the field seems necessary. This limitation seems to rule out the possibility of building an aberration-free lens. However, one can always approach this limit in many ways in practical design.
Acknowledgement This study was supported by grants from the Department of Energy and the National Science F o u n d a t i o n of the United States.
Since fl = ½v/~", it can be written as ,,
=
fZf
0V 7
z
sin ½v~ ln-f
References (45)
Clearly, the closest focal point is at
½¢r~ I n ( z / l ) =
-or,
(46)
[11 O. Scherzer, Z. Physik 101 (1936) 593. |2l A.V. Crewe, Ultramicroscopy 2 (1977) 281. [3] I. Gradshteyn and !. Ryzhik, Tables of Integrals, Series and Products (Academic Press. New York, 1980).