.EF _ CTION
OF LGHT -4 FOiI
_
_ .
IMUTI- T
YLR FILT
by
ivionarch L. Cutler
Submitted inr P3artial Fulfillment of the
Requirements for the Degree of
BACHELOR OF SCENCE
from the
Massachusetts Institute of Technology
1939
Signatu're of Author
.
. . -..
Department of Physics, MIy 1,
1939
Signature of Professor
in
Charge of Research
.
-
Signature of Head of Department
i1
I:
.
..
-.
.
PREPFACE
This thesis is the result of work leading to the
degree
o
achelor
o
Science in Physics.
Recently Dr. Hawley C. Cartwright
ad Dr. Arthur F.
Turner proposed the placing of thin films of varying
indices on
lass to
roduce color filters through the
interference of light by these films.
This paper is a
theoretical study of the reflection and transmission
characteristics of these proposed filters.
In Part I,
equations are derived to enable the calculation of the
reflection and transmission curves of multi-layer films.
In Part II, these equclations are applied to the special
case of the color filters.
In the Appendix are several
tables and charts that were conmputed in the early stages
of this work.
They
ere not found to be as useful as
expected and are included here only because it is
believed that similar material is
not available at
present in other sources.
Grateful acknowledment is ext ended to Professor
Arthur C. Hardy, my tesis
advisor, to Dr.
Cartwright
and Dr. Turner for information concerning their work
in the production of
he color filters,
Fey3mmn
connection with Part I.
for his aid in
and to Richard P.
i~onarch
L.
Cutler
Camibridge, hivass.,
Mlay, 1939.
Sk.'M
..
~G
TABLE OF CONTETS
Page
Part I:
General Solution for the Determination
of Reflection from Multi-la er Films.
The Vector Method of Analysis .
...
.
Reflection and Transmission from Two
Surfaces.
.......
....
.
.
Case I:
.
The Single Film
.. .5.
....
Normalization of the Equations
. .
m + 1 Films
Case II:
Case III:
.
6
8
.
. . . . . .
m + 1 + n Films
Use of the Eauations
3
.
.
.
.
9
10
..
Part II: Conditions Necessarcf to Produce Color
Filters From mult-layer FlmsP
.s
.
Conditions for High Reflectivity
Conditions for a Sharp
. .
Special Situation .
.
.
.
.
Maximum Reflection
.
.
.
. .
.
.
.
.
.
"Latitude'" of the Maximum ......
..
Secondary Maxima
Varying nd cos r
.
Summary
.
.
..
.
.....
14
.
.
.
.
.
. .
.
....
Appendix.
Charts and Tables of Angles of Refraction
and Reflection Factors for Various Indices.
II""'"I"'""~-~L-
--PIII------------
-------
--
21
21
........
.
18
19
.
. .
12
14aximmn
14
.
.
.
.
22
77 7':
;i
if
~PART
GE1IERAI
SOLJTIO
I
FOGR THE DTP'B'lThATTIOJ OF
R FL CTI ON FROM " MUT-LAYER F'TMS
Relection from a thin film is
basically a
problem in the interference of light.
Essentially,
the amplitude of the incident vibrating light is
divided into two sections by a combination of
reflection and refraction at the first surface of
the film.
These two sections are later reunited,
this time by reflection from the lower surface, to
produce interference.
THE VECTOR iETHOD OF AITALYSIS
If a plane wave is incident on the surface 1-2
of the film in figure 1,
the wave reflected at this
surface will interfere with the one reflected at
surface 34 in the plane ab (or ab').
The difference
in the phase of these two waves is 2.
2
where:
= 4nd cos r/noAo,
(1)
d = thickness of film,
r = angle of refraction,
no= index of first
miedium,
n = index of reflecting medium,
A/= wavelength of the light in the first
Id
j1
--- --
---- ----
i·
c
E
P1IPY·-·IP-r--UI
~us*wr~-n--i~o~i~
~
)_
e__by
vA
L.
Z
__
(_
l--_Yi-jL*iiilY%-I-1Yi.i_
si·---r~l·.-~---I~--
.I·-C :·
------
...
1;
Figaure 1.
.... . .. .
-
.1 I . .
I
..
,. .
..
I
--
I.
~~~~~~
~ ~ ~
I -.
. .. ..
.. ..
axi
.
~
~ ~
.
~
.I . .
~
----,
-
~ ~
---~~~?s'
m~~u-·rr-nurrn-r
of imainmriess
I
1
.
axi s of
re als
..
\
.----
F igu re 2.
i(lOll---------
-·I--
-
I-
_
___
A:
J
-2-
Since the light waves under consideration are
amlitbud.es,
fi gre 2,
they mayr be treated as vectors.
A represents the incident vector of unit
amplitude incident on a single surface.
t is
In
An amount
transmitted and travels throu-gh one thickness
of the f ilm, thus producing a phase change,
.
This
phase change is represented on the vector diagram as
a rotation of t through an angle
is the complex
quantity teJi.
; the vector notation
The reflected vector,
r, encounters a phase change of re jv,
and its notation is
which is simply -r; on the vector diagram, it
therefore points in the negative direction.
If 'te
transmitted light is desgnated by T t and
the reflected light by R', where both are complex
quantities, then to obtain the intensity of each, T'
and R' mrst be multiplied by their respective complex
conjugates (notated by double priming), so that
It = TT" and Ir =
Rt1"
If the example in figure 2 is used,
j
It = (tei)
I
Since I r + I t
.^ ______1__1_
(teie)
= (rej w ) (re'J)
2
2
= 1, r +t
=1.
r
_I
_1_1
t
2
= r2 .
__
RE=F
'CTIONA3,
In
TRAT-',irSSTO
FROuI
T0 STRFACES.
determining the reflection and transmission
from two surfaces,
sidered.
the interreflections must be con-
Figure 3 shows the values of the amplitudes
of several interreflected plane waves.
The arrow
indicates the direction in which the waves are
traveling, while the value of the amplitudes are
taken at the points market with a small circle.
The
solution evolved is for any thickness of film and for
any combination of indices, therefore it is assumaed that
there is no phase change on reflection.
This assumption
is justified since in the actual use of the final
equations,
a + or - sign is placed before the reflection
factor r or s, depending upon which of the mediums used
is the denser.
For notation in this discussion and that which
follows, r or R'
indicates that the reflection is to
the left, while s or S' indicates that the reflection
is to the right; u or U
is taken for transmission to
the left and t or T', to the right.
The small letters
are used for reflection factors from a single boundary
surface, while the capitals are the comple: reflection
factors for one or more films.
__·_1_
111
1
The light changes
______
/
i
surfpe
:p
I
-·r-·-;
-
*-
..
16
.
rIo
t ....
-
·
-·--P·
-
/<
.... e6
'.
---
tI
l
I
2)
1
'·
.
t I 10,
I
I
`
r
i
I.
2
.
.
·
2
uI2 - - . - .
t
TI1
/r
surface
I1
tlte
.,
I: n
-1.1
t. I
-
-
T,
.
i
L
Is
c
I
t
t ituul rS2r1'
.2 .S
I
3
.1
i
i
t
2
t tzr
t 2 rs.
t~
ir
s'I
1 '~
12if
1
2 3
' 2
Q'
--
--°
12K-
et :
eera
tc
Light Incident from Right
_-
..z.
..
_
.
t.3
.
....................................................
surIf
..
e
,
,
.>.
.s.
.+#v^
.
1/1
--...
.
surface
i
~~~~~~~~
2
b-
-
A !: 1
.t-·v
-7,-U------
V
o
SI
Y I
r
r
If,,rc
;
L-_;rs
4
I--
.
-
Ir
e
:k
sl·_
l__r__
S
..
. iL2
,
et cetera
Li
z
2r2rH J
tgu2si
ht Incident from Left
Figure 3.
-----
- -
--
i
-4-
in passing through one thick-
phase by the angle
ness of the film.
For convenience, eje is represented
by .
The total transmitted amuplitude is obtained by
1
igure 3.
22
+ tlt 2r s
2
2e
2
2
1 +
+ r2 s2- "-).
adding the vectors on the right in
T' = tt
+
2
=tlt 2
L
tt 12r2
(1 + r2s 1
.....
Since the expression in parenthesis is a simple power
series, it can be written as
t t
T'
tlt2eJ
1 -
(2)
slr 2 e J
The reflected amplitude is obtained in the same
manner
*
= rl+tlulr 2 2
R'
(lr 2s
l
J2
+ r2 2 1 2
+ .....
)
2
tlulr2 eji
1 - r 2 ls ei 2 e
(3)
If the amplitude, A = 1,
is incident from the
right in figure 3, rather than from the left, the total
transmitted wave is
j'
r:j~~~~~~-
= Uluu 2e
(
r2sleJ
and the reflected amplitude is
"_~~~~r·~1
S' = s2 + t2u2sleJ2e
r
1
(5)
I
1I
: ) :· :
LI II I
_'k-
···
single
film
ri, Il1
w_
SI
tl
Sl,
,
..
.-
.
,
.
.
. t. .
.
",
nd cos r '
r 1 =-S
1
&Clj-LLII
U1
r 2 , 'Ua
=' t 1
(a)
R
.
I
. e_
m-
singl.e
film
l' ,
ii
mT
m
nd cO
r
-
i_
SZ
t
-
film.J
films
fnM
.I
s inile
Case I:
i·iU -,·
uZ'
t2
Ii
.....
II
r
( b)
Case I:
m
filmrns
R,' U'r
I
i
I
S
i
i
,,
tz
ilms.
n
fi'lms
i
T'
&
nd cos r
ir
I
iI
i
i
'Figure 4:
m + 1
sLingle
fillm
i·
(c)
, uz Sz
IR',
Case r1.1
U
n
S irn, T'
Tf n
m + 1 + n films.
Cases to which the multi-layer film
equations apply,
"11
I
. I. I
.~
1.....
- .1 I
- .
, ,
..
- %
-
. -1 -_~
1 I , -..
I.. .1
~
-
.-
I
.
i
.1 I
.1
-5Equations (2), (3), (4), and (5) are the general
solutions for any two surfaces.
In the case of a single
thin film, the quantities r, s, t, and u are simple
owever, when multiple films are used,
reflectAfactors.
Figure 4
these values become complex themraselves.
illustrates the three possible cases under which the
above eau.ations may be applied.
TkT
CAS1' I:
S TM GLE FILI.
The reflection and transmission of a single film is
readily determined by setting s = -r and u = t, which facts
,rue in such a case, so that by using
are necessarily
2
2
equations (2), (3), (4), and (5) become
= 1
+ t
r
R'
= r + r2 ej 2 e
l+rlr 2 e
j 2e
ei j(6b)
r
_?
l+rrl
T' = U' =
(sa)
e2 3 j
tl t 2 .e
(6c)
ltrlr2eJ 2
As has been stated previously,
these vectors must be
multiplied by their complex conjugates, R't , S", and T",
respectively, to obtain the intensity, so that
t1 2 2
It = T'T" =
(lr12r
_-
_
---~~_~~~~~_
2
2
) +2rlr 2 cos 23
,
(7a)
-6 -
Ir
= R"=
(312+
2
)+2r
es
CO
(7b)
(l+rl2r2 2 ) +2rlr 2 cos 2e
Tr = IR = R
r
s
S 'S".
If equations (7) are added it will be seen that
their sum is unity.
NONI'.R
}IZATION OF THEEO'- J ATIOTS .
Equations ()
and their conjugates may be cumber-
some to handle when the action of several films is
Since they are complex numbers, they may
considered.
be expressed by the form Ke
ix
The normalized eauations are
R'
= Re
j u
and R" = Re-ju;
(8a)
S' = sejV, and S" = Se - jv;
(8b)
T' = TelW, and T" = Te-JW;
(8c)
Since, I t = T
R
4jr
-S
=
,
T " =
T =
t.
Likewise,
, so that the only difference between
the reflection factors R' and S' is in the normalization
angles, v and w.
If a vector diagram, similar to
figure 1, is set u-p, it will be seen that u, v, and w
are merely the phases of the vectors of magnitudes
R',
S,
and T', respectively.
In order to determine tdhe value of u, ecuations
(8a) are expanded into
,?
i
!I2
..
-7-
= R (cos u + j sin u)
R
R" = R (cos u -
and
j sin u);
adding the above:
R' + R"
= 2R cos u;
(9)
subtracting:
(0)
R' - R" = 2Rj sin u.
If equation (10) is divided by ecuation (9), the result
is
tan u = R' - R" .
j
(ila)
Rt + Rt"
By doing the same operations with (b)
and (8c)
the
following are obtained:
j
tan v =S' - S
j
tan wT'
T'
T"
T11
(11b)
(lc)
.
laTen the equations for R t, S',
and T
and their
conjugates are placed in equations (11), the
t's cancel
out and the real angles u, v, and w may be determined
that
by remembering
e i
and
+ e ix
eJx _- -x
= 2 cos x
= 2j sin x.
In the case of a single film, equations (6) are
placed in equations (11) and the results tlhus obtained
are
__
.-1.1-_1_-^.-·11
111·-·1·11··-·---
1
-8-
tlulr 2 sin 28
tan u
r(i+r2
(12a)
)+r2 (l+r
)cos
e
t 2 u 2 rlsin 2e
tan v =
(12b)
r 2 (l
2
tan w =
(
1 l
+ r l 2 ) +r l
1
l-rlr 2
+r2 2 )
2
cos 28
tan 8
(12c)
l+rlr 2
CASE II:
m + 1 FILMSf.
If the reflection and trsnsmission for m films have
been determined and it is desirous to find them for one
more film, or m + 1 films, the process is analagous to
that just performed for a single film.
(3), (4), and (5) can be used if ,
Equations (2),
sl, t,
and u
are
replaced by the complex reflection and transmission
factors (R'm,
S't,
m~
T'm ~ and~ U'm)
of m films.
m~
When these
operations are completed, the results With reference
to
iLure 4b are
R'm+l
R
m
+ T
U'mr2 e
j2
=
s 2+
j2e
t2u2-S 'rme2
j
1-S'mr2e
(a)
e
1-Sf mr2e
S m+
e
2e
(
+
T
I
(13d)
=
U'Jm+
1
^ T2 TTT ·
A
'sJL.~
.'l..L
l.
±T
*
e
e2 3
'mr2
!-S
i
(13c)
2J
t
I
'
gl
.LJT-L
TTkitc
LL LLJ
In some cases it may be desirous to calculate the
reflection and transmission of m films and of n films
and then determine these characteristics if the two
sets are put together with a single film between
them (i. e. m + 1 + n films) as shown in fugure 4c.
The forms of equations (2), (3), (4), and (5) in
this case are
R'
:R'
m
m+l+n
+
e
T' U
m
2
1-S lne
mR
m+l+n
=
n,, ~ TnU'
S nI
(14b)
S'mRnej2 e
1-S
U'm+l+n =
m
Sm nn
e
j2e
U'U e j2e
1-S 'mRt e,
^----lc-r*··)-m·nn_.
____
_
(14a)
j
·
.,
~ ~~.
~I
1 1.5.... .I
·
..
i
: : , :s : i::
: ;z:!{ ~? i ::o
!~'
~ivfb
: ! !i: :0 :'
.
}-;'
.;
.-H~
.
.-4
-C4
-
.~
.,~-
'C
;
,
-1e
. -
_.
.-i
.1-11
p~~~
ulS
C)
r-4'
,
.r -4
·
40
I
. ,o
4
,
,,4
-
u~~~~
t
W
%:%~~~~~~~~
/
C
.X
s_~~~~~.
Y
IJ
t
~~n
~
,
--
. . 4
sC
. .
<
s,
-
.-.....................
- ' m-0£
0
H-
/i
0;2
. . . ---. ,,
. _
-
1
I
;4
w
~
/~~
W
.
--
I '
~~~
I·
to
d
5'
0 -
--
c .:
·
po,
U)
pq~~~~~~~~~~~~~~~~~~~~~~~~~~~~:
. t M.l'
¢
. :iiwS
i'. < :s - if t
{;
:
.
0
H~
-
.
.~
.
SC)~~~It
~
+>
X
&,
-Q
~
r|
..,t
~tQ
'.
.
11~~~~
~
~ ~ ~~11
(1;as
;
-~~~~~~~~~~~~+
II VBcC
,,,
:~~-.*H
4
v-
-~i
C)b
'
t
-
"-
44 --.
I-C3
SIQ
k
~
cr;~
FC9
~U
dikJ>:;~~~
.
---
.
.r . - 1,
,CQ
i
,
ItC
19
U
W
:
4
+r -
.
.
r
........
p.j
.
k
-k
.W
-10-
'1
~ -
I
Wdhen an attempt
I
--
- - I -1 _ I - _ _
s mace to proceea Lurizer,
in the case of a single
.
as
ilrm, with the generalized
forms of equations (13) and (14), the results become
rather complicated.
A
For this reason it is
:
best to con-
tinue from here to whatever special case may be under
consideration; a special case often
has enough
idiosyncrasies to help reduce the formulae.
USE OF THE EQUJATIONS.
If the equations are studied it will be seen that
:
-[
it is far easier to calculate the transmission than
the reflection.
throughout.-
This procedure is followed
Since, as will be seen later, the calculations are
quite tedious, it is desirous to use these formulae in
such a manner that the least amount of computation need
be done.
The
ost straight forward method'
-
'
of determining
the transmission wold be to calculate it for a single
film by euations
(6)
face and use ecq4,uations
and then
eep a.ding oe
(13) thereafter.
Such a proceduro
is illustrated diagrisnatically in figure 5a.
method is
is
illustrated in
that if
all the h-h
figure 5b.
the H films
Another
The advant age here-
indices and all the
are respectively the same,
more sur-
1ow
indices
the values calculated for
ould be the samie and thus alleviate
uch
unnecessar-y work re(qired in usinLg the previous ly
'F
MaL
r-·
i"'
'"~~~~~~~~~~~~~~~~~~~~
,;
1
I
i
I
II*
-------"-- -1-·c- ----------- ·-----·~`"p"P*'`'LJ.U;
le·iiL-·LUU·Y-DiCirr;b-;·
;
:
-
:::
I3
I
i
I
IoI
.Y-.l
Sw
LOICLi·III1_
ar<.-:_ffie
..
I
~~a
f
t
if
s
.1i
I
I
z
I
.
'
'
Ii.
I_.
_wE;i>-S_-
··
., K ,_.
W
I
!
i-
'§leo<~,p~r.
.sF-~--yrod
w
rc-
-.
_r
,.11-
1-
-'*r-O
t-.
wrw -wa -.
~--·
i
i 7--
--- -i
i
i
il
¥Ik
I
I
t
i
I
v'
1.
Yg
,e~
'igre
61
A
ew ways in which the m-alt -layer film eqtUtins
Oay be applied to a series of sevehn fiims.
-I-
mentioned successive surface nmeth.od.
T'h-is -m!ethod may
be called the alter-nate l.ayer method. and entails the
use of equa.tions
(14),
h.ich are for rm + 1 + n films.
There are a large number of
combine these two mrethods.
It
is
ossible ways to
that with
es tiate.
seven films (eight bloundarry surf ces)
there are w-ell
over fifty possible methods of approach to the final
solution of the reflection or trr.nlsml ssio n
few of these
_
___II________II__I___I__
curve-
A
re show.n diagramatically in fi~gre 6.
-C----·--
·-.
I-
-12;
._~,."
?:PT II
I
PiOL .. JLTI-LYWR FT¥h,}S
.-
Tlhe ideal color filter
would be one thai
lar g e reflection (or transmission)
l ength band and a ssha,.rp
;wvelengths of te
gave a
for a narrow wave-
rop to around zero
sectr-ai.
or other
The conditions necessary
to -.roduce such a response from a series o
may be stAdied with
in
Part I.
fi
ters
CO-I>TITOPS
tlil
t.lin films
aid of the equations
This discussion deals with 4e
e.ol ved
reflection
OR 13
ICR FI LiSCTTTy7.
If we have a film, as in fiLure 7, with an inex
higher than Ltht of
lass
and of infinite thinnes3,
then the reflection factors r
inter-reflectionls)
and r 2 (neglecting
from surfaces ab and cd, respectively,
will point in opposite directions since rl
is
negative
and r 2 is positive.
The ilmA
f
m,-st theremore be O
that the vector r 2, on passing fromr;
-A;[i
l
rad-ans;
the 2
.
of euaton (1) euals
¶r
a
1iickness
ab to cd and back
a ain, will be rotated through
or so that
This condition, it
can be s een, occurs when nd cos r = kA0 ,/4,where k is
'an odd integer.
}?/i.[i
sch
'i
··· ;
''
';'
: ''': ': ",;'
~'
l'
:)~~~~~~'i
~-.--, -.-..' ~|:--.
.-'-''':'
.,,r
' ~':
0:''''','0'',08
..... ' ' ''
.'~~~~~~~~~~~~~~~~~~~~~.
. .'
,:
'='
·-;-:--
--
-lbhW1I
: .: --7-1--l-r
Z_
':-
bX
":
r·
. . '·: · ··):?:. ·.
:·; .''·.L
·;L.'
:-: I· 'Wi;;lL:;-:l-s.,.:i-,s_,i.-.;;;.r
:::::;-:i
___
_______________
,,
. .1 I ., a
__
. -;·Ue~jt~;l
t%0:
air
/
b
-
.-..
.- :-'
'·
i
~.w
E,>tDVjt
'd
)
..
,
..-
.
.
''..
.
.
.
.....
'
.:.=-,.
(b)
d ia-n.
.,
-4
I
h i gh' -nd ex
l
r a'r
-et 3t
teeet or- d Iarm
7.
.
-
.
- -
--.-1-
.
-I-ri1 r
ror r 4
r5
r6
ht
i
hi
.r
rl
;··
.
r
·f71 c SS
(a)
Schematic dia ram
i
(b)
Vector diaram:
Filt,~ infiniely thin
Fifu re 8 m_
_
_
>_
_
_Ki;
-x'
'^' ;IUIW--O"PUYCWLWW;_·YY··*·i;UMhii*
I
.
-. ...
·iiiLL\ei_ii:-i-iLSliLieil)llRL
` _1_1_4
-3--
If several filams are used, it is desir.ble that
actors rl,
the reflection
,r
r3, etc.
e high.
condition can be ffilled by alternt'nl
so tha
This
the films
t e upper one has a high index, the next a
low-; index, then high., and so on throuGhut the f ilter,
as in figure
.
The reflection factors may also be
increased by (designing the filter for use at hi -,h
angles of incid.ence.
If each.
ilm is
of infinite thinness, the
vectors point in the directions shov.
To make r,
r4, and r 6 pomnt in the sme rd'rectionas
vectors,
it
is
necessary to make nd cos r eal
od --quarter wavelength for each film.
rr radians;
rotated
the other
r,
2£Fradians; r,
to an
Then r
3T
2
is
radisns
and so on, until all the vectors point in the same
-di.rcction as rl, which of course cannot be changed
since it
does
ot
pass throug-h the films.
WUith these vectors all
direction,
att
-i nable
they
ad
ointin2 in the se
up to give the
reflection
b- Just vary
ing the thicirkness of the fil:ls.
For other wavelengths nd cos r does not eccal oneone-quarter wavelenot'lh and. the vectors
point in the sam.e direction
ill not all
and thus the refelection
~~~~~~~~~~~~~~~~~~
I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~-
I·
'·
·
.`
ru
·
Ci-·----i--·i
..I
-
.
.
i
-·
.
'v--
l.,
i
1 ,:5>
=
I.,
.
--·--- i-:·_---;
-
n
=-
-·..;:-·
-;;.Ur-:--
,·
r
L .J
:---·;i--- ··
·
. L
.
.C)
7J7
"0
-
; - -I
- 4l
-
-
'''
4
0
·- ·
I --. .
. -
2
42
.
_
-
-r~
2
7
rC
angle of incider.ce
nd cos r
(a)
in
450
fAm/4
for each film
escripton o f te
tic Cv.te m from
.Liruire.Ot10
c co -Out e d.
vir
7
1.e 55
>2
trnlch
the curves
air
! r
Js
I/
l-al
(b)
ivgrrrmetic i:llustration of
toe rmulti-layr film equt'-tiz'xr
pute th~e eiarvcsin
igd{re 10.
"
- . ...- ~~
I --1,~~~
.···--.
-- -LI--'-----
I~.1.
. . .
}siamrcr inr ;hici
'crc: t'ic-ed to com-
ihe,
-~-
- I I - I . I - 1-.- I --I
------ ------- --------I--
a
Pi gure 9-
-9
--. -111193111·1*·111411ILslllrr(·------·
·
.·_ __
_
I-I
_··
.
·
_
*- - -
br;-·l
':
-1
,L
'
i
I·
Other L-mailma a.r,-,arat
on each side.
if
cuarter-wave
C
T"iTrIT'iTS FOR A s'fH-R.
films
to make the
as sharp
..a;imml
rom sevr;al lrayers of thin
case is
a special
Rath.er than build.ing several films and
deteimning teir
S
etc.,
To obtain an idea of the sharpness of
films, a quan titative aalysis o
phot;roeter,
3Ao, S5A,
T
K ITlT:J
the maxium obtainable
necesssry.
values
are used throu-llhcul.
roble-m is
The ne xt
as possible.
and cdecrcasingn
a:imnuM at ,
curve will show a
chara,-ct eristics on a spectro-
the eua.tions evolved in
Prt
I m-3ay be
C. ·
SPPOiiCT.
S.TUl
~
,ON.
The series of£ fil
S
that will be discu ssed is
illustrated in figsre 9 with the values of the indices
and the reflection and transmission factors.
lr.Ls ae
r-anged
to act asa
The
reflection filter
with
t an angle of 45 degrees.
the light to be incident
The light must, of course, be parallel to have the
filter
The reflection factors
finction as desi-red.
of the bouncary surf. ce
are calculted from Presnel's
f orrnml a
R
= -
sin
(i
-
r)
,
I:
.jl
L
1
_i ----ir-r
lll--IllC5 --
-._
-
·-
''-==----
---
I
1·
- % 1"1
1
- E.
_
.-
1:i
'
--
Reflectdion curves,
rrAr iB
and-I
for the system c
in figuum
4-I
i'T'
S)
.1 I
H
i.is.t.1-il -Li
w@
---
la.-sis
·
r
,,,
- .
w:;·
.r
'"r- I
T-lwvml
~I ~
. .t1-'
t
9 __
i
t
------
.T- : r7
Itl.-T
'
I1 I
1WHH
¢flJX
rt771-i
I j7T
4-'
-t'I tH
"I
t
-!h-4
11'f
'-t- I
I
at-4 I
+
1-'
"lit·t-:
ThT h-~1-
.
. 4,4_,-
.1
,
' 4j-j J
lj2dH2
_-
40
I_ij
,
iFI~
',4_
I.S,
f iJi
I f
*
.,I
q441i ~.~J'
;
¥~
~IZ
I
-Hj+
_! . ! {
III',
f
-:?4-
H-H
.-".
t- T I
1'q44
-.
;·.·
ij
i
kt4-!
¶4
1+
- i
1 _
Ie
-H'
;·
..:..:
I
1:1- _
:·-
I;
' !- ,
z
";
" "
'`
78.if- ,
H- f ee s, .
i:i'+ /i 's.r I
l l I
,t.4+1
I: _1.i
ff
k
Hih
i7 t
-i
4-
;'11-111±1
,
4-
_D t 1_
A.,
14111TV
-- t -rc--tn*
m f 1Lt4P
iFi-Hr
t
. It1_::
7H_ f-i'
.zI-
ihii
_,
j -i H4
i"i
E
.j
+i
,."
-.
I,{I
t5!%_:-
I-
i..
!':-iY!
mt
g
TI
ti
i!:.r4;1 2_
~;j
,4,1-4
,l-
.4 i--
t--t4~
I; I l I I ' ';
(rI:( I tI:_I
f
LE
14T
-
_P~t±+_gj
ffIi2I
li' .!i_"JT!
·
-L - '.v .
Li
i;
A
- 't1,
'I ' t-l
+P`+H_
-+
_4
1 d:,L
I
:J.L'_,
2t
.i
Ii]
414·
-H.-lf
YT4
f.91''
II
i
i
4
Vf+' .LF-f£+M
1
-,
-
_iH
rt'
'ti r~-- T-"I1tT.
3 .
L--t+..
JuI
g Tlle
i1 - 1
?74
TTT
IE!- IM
T-j- i I.
?ft
.[ZT-F-'~'.
Huff~t-t
;. .-.
IT 11'W111-11:
4T44TRIP-11tW.
tt
tH'14:
T
l iTT.
-:
T l4J1-44l
-I
a
I,'1'Al
ig
_i-il]
p-
-+
I-s' Sleti+'
:!t
fil ms i 'l ust rated
,.
.
W-4X
TX i
-
4
'v" '
..
:~~~~"
"
i%·
.
_L.4-
?^
pTIL--, I
.
i-
I'"~_
3()
ii
·i·
,·
;;
Pq
20
10
TTTTTT1,
_I U.
.-1
7- I
_.4t
1
;7Ft-
'_;"J't
-J--1
t·
I4
;SL,__-T
. T-,
Pt-t
4 'tE,~-
.;r4
:i - t' ! i {:
_114- '
4 I.
' -
:-
i.-i-
-i l
,
ar
.....
L
.
,
.
S
0Q
2.0
1
t.9
Z
1.8
_44_
-1
.
i
.....
,
1.9
I-r
.
1+'I
I ] T r
.
1.6
1.5
4
1-a-
:
17;n
,,..
_
.7
.5
.3 .4
I- L:,,J. i ; L:;.
'i k
-4.L'4I-A.
_`_
IL-iI I
'Y"
.' renU;-i· _
,
FRMi 3 T
,.A-M
;
U
_L tL
',-- -
-,T Fhfw-.H
1.4 'l
.
..
_.-L.~ L-~I
Al LI½LAL
!d
,4-
,t114:;t
T- -t-·
j-.IMIT
.i I LL
1-r-T-f-T
:
,
,6
7
r
liltI
_
":
..
.
1,
t
.
-,
i·IIIi ii 1: III
f
.
t-i-
i
.
.
.9
1.1
I
I0
.
.,
I
.. .
A/Ar
I
10
Figure 10.
.........
~},'
t',
":' .
.
. ,.
' ,
.' :'
. ..
I
/j
i~
so that ani additional recuirement is
be
olarized in a
that
the light
I
i
i
lane parallel to its trace upon
i
i
I
the plane of incidence.
The value of nd cos r
fillm, xwhere \
is
s m/4 for each
t.ken
is
the wavelength
have the Baximumm. reflection.
(in
Since 2
air) that is
to
= 4rndcs
r/
rT/AA , the curves can be plotted a, ainst the
X =
Am/
raction
I
I
II
i
I
i
i
I!
i1
1!
I
=7r
, or 2
Figure 10 shows the curves calculat>sd to illustrate
the effect of building up a series of films.
is
the reflection from the first
fil ,
um.nder the cnditions shown in f
shows the result whener the first
Curve A
of index 1.92,
ure 10; curve B,
two films are con-
sidered; and curve C, for the frst
three films.
The forula used to determine curve A is deduced
from elquation (7a):
2
1
A
(! -r
!
2
2
2 2
r 2 ) -2rr
The only variable here is 2
2
cos28
= XTr, so that this cl-
culation is relatively simple and. can be reduced to
.765
ItA
1 after the values
.109 cos X r
iven
in
filure 9 have been inserted.
I
t;i :
I·
I
-- ·-----
--------------
-16-
Before determining curve B,
one more value Ylmst
be found:
tan v
t 2 rsin29
=
A
r2 (l+r
i
-
2 )+r,(l+r 2 )cos2Q
2
.392
sin 20
_m
- I -
.574
Sin 2@
ItB is obtained from equation (13c) by multiplying
7 =
complex congugate and remembering that TT"
it
by its
I
and S'
t
Se j3v and S-R,
It
see pge 6).
t22
0
ItB
B
(l+r2 IrA) -2r 2 RACOS (20+vA)
ii
In a similar manner
2
ii
It
illii
i
Itc
:li
t2
(l+rz2IrB)+2r2 RBcos (2Q+vB)
Ir;
where .
S t
tan VB
-
iin·,
-St
B
IB
-
B
B
ii
which reduce
to
t2RA sin (20+v A )
tan v B
'!··if.
i
I;
i.
3
I
W1
"""' _s
T|
r
-r IrAcos
tt(2@+vA).
r22
Ai-
RAoSA)
ttIr
A
*
)-.-r
?
.l~~
~ ~,-17,
To calculate tan v
a rather td
for around twenty values would
be
ious ,.ask for an indi-,_vi>il since there
four variables in
re
he- ecuation
For thi s reason it
is bst
to revert from the
successive surface met;od of using the forulae to the
alternatte
ayer
et:!od:, shon
sm
d agrmat.it:ca1ly in f iure
8b. Such a system would require the detiermination of
ItH but it would also eliminate the calculation of
ItB and of tan v . Actusnlly the curve or IB is not
required and is shown here only
or the purpose of
illustrating the effect of building up a series of
films.
I
e
t2
....b
i ItH(
.903
=
(1+r 2 4 )-2r2 cos24
Me
(14c),
.
1-.096cos 2
ormula for ItC -maybe resolved from equation
-whichis for m +
+ n films.
ItAItH
1+I AIr
-
2R AR
*
c
u
o s ( 2 + v ·IrA +f ;s )
where
tan v
,...,-·
W
-. 3592 sin 2e
A
-
-
-
~
1- . 574cos2(
J
-·-------------
~~~~~~~~----
.
w.i
I
.i
W
·
·
·
·
.
-·
l
I
·
I
I,
I
·
·
'!.:
E-I! -,4.
-.
'1-4
'
FiHi
4-4t4H4l1M-l
RFj--E;4+4+ti4-t4-Hi lt1t t!~ itt~·~~~~·.t--
t-4_1j:q1
-i--q
i4.i,
FtHi i-i
-,
~ :If
~ -~YX!~:
-.. : '--r
itti-l
+!-H!-I
-J iri
:7 1
:i
-TTTI
IFg: -, 14 I~-rI'
-1,,~'!I!
L1J--L-"
4 --,
ff
I~~~~-q
·
4:-t'
-i'.~,
!-H- ;.:+--.
hi
ii4
.i
fi
-'--;-''--'t',:'"-.-!-:-,-,,'7-:,,-f';:,?'_t~
'?-''
TLI
ixIt_
d i.- _- _tIltt-N
__,*4-t4H-Hi itnUfl .I1'==
;0; ~-t 7: + -- '5x-t~-$ ''qm-u?~?
~i
'*k
~,
:i~~
~*. .i
~-S;:T
iS~.t:'~~'
i-Hltt'J~J~.,
f i I-e
lEt ,';~.:-r
_?i
rrr
.riE'l¥
,
J
4
-4.,
;-4.',4+Wq~i- 44---,.- 4;-'.4
. ;q
,I-~-~
~.~,*~~~~~~"~ a:..-¢~t
:..-~,.
i...q~
·.
1
00~~~~~~~~~~
44+
S:
,
-s n-i
-_.- -
ii~~" t.20
J'2-;-;-u'l_Yii_i
0
.': ,.
I-.
I]
,I
t
_ rq--.-H,-t l.
.'.L.
r -' ,
k.
e444--?-'/f
H
,I;.-k- ; t
r
-l~~~~~~~~~~~~~I : .I I, tIJ i ! ! +i
t
ti-i."
4 -vM"'+ I
1,144
iI
:1 14 fill] 2'lijjjL
EI
L
i;4"i7-TCFf
ttt't¢=4irII'M
a-,-
4;-
.'j
ti+
t'- -
-t-i ti
-- - i i
. 1.
-
l
!-,
.
I
r
-k-',4f
1.11-
.1-
.
',' 'i~·i--:;-.,:~i?;:_t
:,r:-:--4::1~~~,-:_ ' _
~
60~~L
t4~-~'.. 4 ~
- --t- 'i--i
'~.,,r.;i.,; - :'-?Mq -4-.4-.-
*,-";
'.~
2~t .~.-I ~,
~
:~'Tt,-,,~_L
'"'~!T
llJ.'T!-,L
O
T
---.
-: -.-
&i
~
,_ , i ] i ,'
ii!f''.'
]S
i
Z.T--''[t!.~q]
q
::-,- .--
"-.r
.:
i-'-T
'--.
~
F~
t
-4--.
tS.-C-
: - i$-.-F---i
5A0
P,40
I
W
:" ~'-
!~i,,t' -F:Z..+--i
'
i
:--4''
i
i-
i"
;
!q4-.
~~,j
r-'
'-:
R:1-'$.
. . HI
. 4qH--iI
;~~~~~~~~~~~~~~~~~~~~~~~7
_Tlk-' -
-"+q'
;;
'~j- + '
i.L:~~~~~~~~~~~~~~~]
;A
......
r
!--H
-F:
:·
-:
F~~Tfi
44
I- .i-.
..
-''f''i-.N
.. H
yt
-1...
TiI
4
;iS:
i.~---~-'.,4.-'
~:-- . '+-.
i4-.
. ...
-
,.
,~..i--41H...,..
, ,,,
LI
Number Of films on Al-ass
1'-- ~
t .I--,!
-'
.
~ .
i.~-I:
--:-~-- -";t'
-,-+.-.,.~~-,--~.~~
,
..
I
:
,'
,!·
~
]~~.-t
...
4- H-!i--:~,..
,, t.
-? ~4i+-f-!,
-+- -~" ~''~,"xi.''N-.-hr r,['
,i--1--,i.',k~
'-lf -Ht--.'---l.~-H I.' i''t~"
t'?
-Tc-'-',v'''-rt ;10,Li-'q·
~~--4'-.;-.-
;
;n: n1' ..
Z ~ iF
e,--.';i,;!-;~
L :N
m.T"":q1 t'::
{," q4F ;!' !!
r :/!;i! ::
,i+ , 'dS 4f~~'rti---,'
l-~-~';-i-ih,
on.1-F~.,eE
~ltumber of films
Figare 11.
--
·---
--
I
·,
-
-
--
ILt
-18-
as previously deterrmined, and
t 2 2 r 2 sin 2@
tan .r =.
-0.0D sin 2e
r 2 (l+r 2 2 )(1-cs 2,
It
1 - cos
cn be readily seen that cor^utation is
reduced by eli-'inatins
,!IIM.UivU
29
appreciabl.
t n v
-
REFLECTION.
It is evident from figure 10 that as more films
are added, th6 intensity of
A
increases.
The nxt
step is to determine the nixnber of films necessary to
produce a relatively high reflection atA .
were made with one to five
Calculations
ilms on glass and with the
high index varying from 2.0 to 2.8; the low index was
kept at 1.35.
The results are reproduced in figure 11.
From examining the chart it should be obvious that five
layers are sufficient to
roduce a satisfactory high
reflection (between 85 and 100 percent), even with the
high index at 2.0.
The multi-layer film
equations reduce to a very
simple form for reflection at Am.
The general reflection
equation becomes
R tn+r
fn +
1 + R nr
n
where only the mgnitudes of R'n and r are placed in
--
-b·
I"-"ls-
70
T +I
.I
i
I
Hi
;
!·i
4
....
i-
-
' 'T
, --*..
T41
t
:
.... ,..
50
t~~~~~~~~
~~~~
~'
..
...
'- ' cL -141* t - '. ~-,
"'' ·
.'-~r
·- '-
3...·
-
- '
'
9
- .
-..
...
-l'~-F.kr
..
f{~p-,'~
pl.t- .. ~ .. ~ :
...
I ....
.[L i~I'"l!:-'
40
---i,
ll-----
?-
~:
...
:~----~.
ri
-?.
'
-
_..+4.
'
"
-
.
f
+i
i~
~
-
,
' ~.c&
II~~~~~~~~~~I
.. ,'
-~r- .-~
.,,,
,_,.T;:
-
-'i
.
.
1 i!
.
I
'
..
[~'--"
~~"
"',t'~"
.~...I
:
~
- .- ~ '
~
t '-C
"
v
-tIir-'~t~'':t~t-7
'- ~ .r:/- ~i' Li.'1i
,
' ,'4--:.'~
~
I;t':-i" 'Z ,.L'".: .11.._:.:~-+[4..?. : ~.;r',,"r· _,'~1 r~7
f-t
:.iJ-i
. .....
...........
.. [ t1, ,,..-:2
..........E
~-w~-,1
.... t .......
.........................
~* ..........+.
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20
4-
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10
.
2
3
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5
6
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1 7
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Figure 12.-
· · ·
~~~~~~~~~~~~~~~~~~~~~I
Il110
7
8
-1 .3 1v2
.9
1.0
1.1
1.0
ii
·i
-19the equation since the + or - signs of these
The above
have already been taken into account.
applies to the special case of an
actors
rrangement of
alternately high and low index film-s (figure
wfhich nd cos r of each film is
length.
a),
in
an odd quarter wave-
film, R'n is merely the
For the first
reflection factor of the air-film boundary surface.
LA TiTU"
ITHEr
OF
-FMAXIOiIH.
.
The latitude of the maximaum is defined as one
minus the value of A
7A
at the last point of zero slope,
before the curve begins to rise toward the maximum at
Am
.
_
This latitude may also be termed the half-breadth
of the maximum.
In figure 10, the latitude for curve IC
-
IrA,
.52 = .48; for curve IrB
1 -
.37r
= .3;
is
for curve
it is unity, since the only low turning point on the
curve is at
ero.
A small latitude is desired in order that a filter
have a sharp maximum.
latitude decreases.
As more films are added, the
However, the rate of decrease is
not constant, but decreases as the latitude decreases.
These facts may be seen from figure 10.
It has been
estimated fromn somre rough calculations (not indicated
here) that an addition of two more layers to the three
already considered will reduce the latitude to no less
than between .
-~-- -I-"
and .40; a further increase in
-n-'------------
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70
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Figurfe 13.
- __
·I_
--
I
1.i
9
1.e2 I
....___
1
1 iC
-20the number of films causes a very small reduction.
I-i
A very good filter
.01.
Such a condition will give a spread of the
peak in te
Am
should have a latitude of about
reflection curve from
is 475 mn-.
if
70 mai to 480 m
An arrangement of films to
this sharpness has not been found b
rocduce
the theoretical
study made in this paper.
However,
if
the films are increased in thickness,
a reasonably small half-breadth may be obtained.
It
vill be retmembered that the nd cos r of each film must
be an odd quarter-wavelength to produc;e a maximum
reflection atA
considered .
.
if, in figsure 9,
increased to 3
12 results; i
occurs.
So far only a value of
m/4 has been
nd cos r of each film is
m/4, the reflection curve for I
5
/4 is
used, the curve of figure
In the former the latitude is
while by using 5Am/4,
it
;
reduced to .16,
can be made as low as .096.
The half-breadth for k
m/4 divided by k.
in f iglre
,/4
is
smply that for
rom a purely theoretical stand-
point, the value of k may be made large and thus produce
a very sharp filter.
II
In doing so, the films are made
*1
so thick that serious effects caused by scatterinc and
absorption occur.
Also, it is evident from figures 10, 12, and 13 that
as the thickness of the
ilms increases, the number of
II
Iz?
70
it
y'.
i%:
;i -l
.'
6o
-ji;1i t~i ~t"iI ~
i'
1
7
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: t
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20
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.
-1171-71
IT
' 71 ,-·rL---LLU-i-LL
Thl'i' I , 4;
-I .__- -1 . i_
-7
c---ii;-·
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iijt_4111:'- - 1:;i- -·.-
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f.
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2;
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0
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.4
.5
.
.7
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1.
2~0
1.9
1. 8
1,.7
1. 6
1 .5
1,4
t. -3
1,g2
1.
I t.0
Pigu.re 14,-
-II
--
-
-21-
airimu', reflecti on
peaks of
ras.
if a filter for
the visible region (400 mp to 700 m)
is desired, the
5j\ m/4 flms can be used only whenAm is between 470 m.
and 530 mi,
otherwise two peaks of maximum reflection
or the 3m/4 films
will occur in the visible region.
the 1imits are from 335 m
two
to 610 m.
eaks occur in the visible
Above 610 m
rn
art of the sectrm.uv
A7:-lA'
-::y~(i
,....A~.
MAXI',A
(DARY
SEC
It is believed that the small secondary mrraxima in
';
the reflection curves will decrease in intensity as the
number of films increase, but no special calculations
have been made to prove this point.
7TARVuINTG nd cos r.
Another interesting
in a series of films.
ossibility is varying nd cos r
In the previous discussion all of
a sirngle series have been 1/4 or 3/4 or 5/4 wavelength
films.
It is possible to makce a filter with the nd cos r
of the first
on.
film, 3
m/4,' of second film,)m/4, and so
Figures 14 and 15 represent the reflection curves
of Ir
for the system sketched in figure 9, but with
the films of different thicknesses.
first and third films are
3
3
m/
In figsre 14, the
m/4, while second
iim is
; in figure 15, the first and third filrms are
X4,
while the second is
a
Xm/4 film.
m~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.
.~~~~~~~
,.
-
:
-
-
!
70
t......
:<i., | -L- t - S i
60
!
4-~~~~~~~~~~~~~~.
!'iL~.
.~!
I~i
;
-!1
,~;
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j
;
1
T.,i;!4|.....0Xt 1"
it
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m
r, 0
:L-!'i":'~-.,'
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m.-:..L
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l
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<
: ....::'- 7
:::
1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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30
is
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i'.':
:4~~~~~~~~.
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j
|
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1
[ ..j: .-. *;
~:T '- !
wt~~~~~~~~~~~
t, [~_'[i
--..
~' ~
t
I ~ ;~'~
~:'"
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W-Wj~~~
I
:... ::
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.j4;,-...Ii
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!.[~~~~~~~~~~~~~
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tF
.---~~*---~-~-J.....L-~-f--:: ..
t L : .~i~,:
"'' i: ' ~' ' ~ ; :' i
I
'~ J::,~~~~~~~~~~~~~~~
:.:i
!77
j;~~~~~~~~~~~~~~~~~~~~~~~~~~
;,
10
[~~~~~~~~1
0
.1
2.0
1.9
.2
e3
1.8 1.7
c4
e5
o6
1.6
1t5
1.4
s7
Figure 15.
I-----*
_
-1.3
e8
ii9
1c
1.2_
1.1
1.0
-2
)
The seconSry raxim.s arc ,uite
cases and it
Is. probable that the-
icsh in these two
will be d-ifficult to
reduce with the addition of more films.
it
Consequentl,
probably undesirable to make the filters with
is
varying values of nd cos r.
STJ- mARY .
From the foregoing discussion the following limitations
in d esignin
a filter by using thin films have been men-
tioned:
A Factors determining the number of films.
1.
The intensity of
m.
2.
Scattering and absorption.
3· The size of the latitude (sharpness of the
filter).
B
actors determining the thictmess of the films.
1.
The size of the latitude (sharpness of the
filter).
2.
The number of peaks of maxirmum reflection
occuring
3.
Sc~4ttering and absorption.
4.
The intensity and numiber of the secondary maxima
nd cos r of the films in a filter
wh en
ried.
is v --
.. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~---
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400
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BOO
560
600
Wavelength in millimicrons
1;
f
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6,
.~~~~~~~~,
700
Fisuire 16 shows how a filter
react with,
iin the blue region.
o
three films would
The reflection curve
was detenine. from figure 13.
i
-------
-- i--·-·--n--*p·--·a----p------------·rrr-·--·^··cl-ac·--c·-----··
This appendix cons~sts of tables and
charts of angles of refraction and
reflection factors for various indices
and at angles of incidence for 0
to
80 ° .
::;
ill
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~~~~~~~~
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1.4
1.3
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i
i
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z
TArBLE II
RTEFL]CTION PICTOB'S
Rs =
- sin (i- - : r)
tan (i
ip
'~
$
O
r)
rY
z.1
*,
i}
-
tan? (i
1.2
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R,
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1.3
R
1476
+1476
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i
477
476
911
5
485
463
10
477
15
Rs
±1909
o1304
+°1304
909
1304
1304
914
903
1317
1297
453
939
886
1336
1273
506
445
961
855
1377
1234
20
532
419
°1006
S01
1433
1175
25
570
384
1071
749
1515
1095
30
618
336
1149
663
1616
989
35
680
272
1244
562
1752
854
40
713
189
1387
414
1939
679
45
875
78
1568
246
2
2133
456
95
2408
171
2
50
O 1047
- 79
1788
55
1232
1 ,1
281
2083
+ 300
2747
-1207
60
1518
573
2483
714
3190
696
35
1917
989
3002
01277
3760
9326
70
2501
01605
3722
2051
4483
2172
75
3409
2535
4669
3134
5413
3317
8o0
4774
4002
5954
4659
6597
4864
-`^I-"I-'-----~-----
-
TABLE II
(cont.)
REFLECTION FACTORS
\n
-------------
- R
IiIp x,~-,
1.4
1.5
Rs
16
Rp
RS
Rp
0
-°1667
+01667
-02000
+°2000
-02303
+02303
1
1668
1666
2003
1999
2307
2300
5
1674
1657
2011
1990
2318
2296
10
1707
1630
2041
1959
2355
2263
15
1748
1582
2096
1906
2408
2205
20
1821
1514
2168
1829
2490
2122
25
1909
1420
2261
1725
2601
2014
30
2033
1297'
2396
L589
2743
1852
35
2179
1139
2566
1478
2917
1680
40
2382
1939
2773
1196
3135
1447
45
2618
685
3038
999
3410
1155
50
2910
364
3367
574
3722
1790
55
3289
50
3733
133
4110
329
60
3752
1572.
4197
-1430
4578
65
4329
-0o1267
4776
01138
5145
984
70
5037
2154
5472
2064
5821
o 1934
75
5934
3337
6317
3274
6630
3170
80
7030
4806
7341
4870
7578
4770
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