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On the exact calculation of the reflectance of glass, coated with an arbitrary number of non-absorbing layers
P h y s i c a X l I I , no 6 - - 7
Augustus 194"7
ON THE EXACT CALCULATION OF THE REFLECTANCE OF GLASS, COATED WITH AN ARBITRARY NUMBER OF NON-ABSORBING LAYERS by B. S. BLAISSE and J. J. VAN D E R SANDE N.V. Optische industrie ,,De Oude Delft", Delft
Up till now a general method of calculating exactly the reflectance of glass, coated with an arbitrary number of layers has been lacking in the published literature. The one-layer case however, has been treated exactly by B 1 o d g e t t 1) ; the solution of the two-layer case for normal.incidence has been given by M o o n e y 2) and that of a layer with a continuously varying index of refraction by M e ys i n g a). The latter authors both treated the problem starting from M a x w e 1 l's equations. The electromagnetic theory however allows also the solving of the general problem for glass, coated with an arbitrary number of layers, at all angles of incidence. This can be done for the case of light polarized perpendicular as well as parallel to the plane of incidence. General expressions obtained in this way for an arbit.rary number of layers are given in this note. We take a glass coated with N - - 2 layers 2, 3 . . . . . N - - 1 . Medium 1 represents the glass and medium N is air or vacuum. Thus there are N media; the thicknesses of the different layers are a 2 , a 3 . . . . a N - - l ; the refractive indices are n t , n 2 . . . , n N (see Fig.). A Plane light wave with an angle of incidence of 9 N is falling from the left side on the outside layer N - - 1. In each layer l the vector potentials A z and A s' are introduced describing respectively the ingoing lightwaves moving to the glass medium and the outgoing waves moving in the opposite direction. In the glass itself only the vector potential A 1 exists, as in this medium there is no reflected light, whereas in the medium N the two - -
413
- -
414
B. S. BLAISSE AND J. J. VAN DER SANDE
potentials denote respectively the incident wave AN and the totality of all reflected waves AN. The reflectance RN thus is given by the square of the quotient of the moduli of amplitudes A ~ and AN t
A,N ~~ ,N, , ANAN
RN =
the asterisk indicating the complex conjugate quantity. Now the problem is to find RN if the indices of refraction of all media, the thicknesses of the layers, the angle of incidence and the polarisation are given. This can be accomplished by considering the boundary conditions between the electric and magnetic vectors
at each interface giving relations between the A's of adjacent media. In this manner it can be shown that the quotient A;+I/AI+ 1 of the complex amplitudes of the ingoing and outgoing waves in the medium l + t can be expressed as a function of A~/A l in the medium l by the recursion relation A ' t+1
At+ 1
--
(st+l--st) exp (iqt)+(s/+l +st) exp (--iqt) . A t / A t I (st+t+ s,) exp (iq,)+(st+l--s,) exp (--iqt).As/At
(1)
In this expression qt is equal to . ql = 2"unbar cos 9~/~
(2)
where 9~ is the angle of incidence in the yh medium and I is the wave length of the light in vacuo. The quantities st are defined as SlJ. ~-
~1 COS ~t
s~u ~
--
(3)
1
nl
COS ~o~
REFLECTANCE
OF GLASS,
COATED
WITH
NON-ABSORBING
LAYERS
4r5
in the cases of light polarized perpendicular or parallel to the plane of incidence respectively. ¢ As in the medium 1 there is no reflected wave, A~ = 0 and we have t
A~
S2
A,2
s2 + sl
-
S1
-
(4)
giving for the reflectance of one interface the well-known F r e sn e 1 formula
R2 = ( S 2 - - S I I 2 \s2 + s2/
(S)
The value of A'N/AN for an arbitrary n u m b e r of layers can be calculated from (1) by complete induction using (4). B y squaring the modulus of this expression, the reflectance R N can be written as G RN =
+
s,s s
....
sN
(6)
in which G = 27 27 . . . . 27 Ck,, k,..., kA-_1 COS 2 (k2q2+k3q3+ . . . . kN--t qN--t) k~ ha
(7)
kN-- 1
The quantities C~, . . . . kN_t are functions of s I . . . . s N and the s u m m a t i o n 27 is to be carried out over the values ks= + 1,0 and - - 1 . ks For the case t h a t each layer has an optical thickness of a quarter wave length at the considered angle of incidence, each q~ has the value n/2, giving the following simple expression for the reflectance RN (v--w'12 R1v = \v + w / (8) The quantities v and w are given, in the case t h a t N is even, b y
v = sts]s ] Z~) ~
.
.
.
.
~2 - 2 ^2 .~2.~4.~6 . . . .
s2,_,
(9a)
S2
N__2SN
whereas for N odd v =
s,s]s~
_2 ^2 ^2
....
Ze) ~-- 52.~4.~6 . . . .
s~_2s;v
(9b)
S2
N--I
For light at normal incidence each ~0t is zero, the quantities s t
416
R E F L E C T A N C E OF GLASS, C O A T E D W I T H N O N - A B S O R B I N G L A Y E R S
being replaced by the refractive indices nv For the one and two layer case one gets the expressions allready found by B 1 o dg e t t x) and M o o n e y ~ ) . Details of the method will be published soon in "Applied Scientific Research". We wish to express our thanks to Prof, J. d e B o e r and Prof. A. v a n H e e 1 for valuable help and discussion.
Note added in proo]. After this note had gone to press, we became acquainted with an article by Miss D. C a b a 11 e r o 4) published in the March number of the Journal of the Optical Society of America, in which she gives a recursion formula for the case of normal incidence equivalent with (1) but deduced by the method of multiple reflections. Received April 10th, 1947.
REFERENCES 1) 2) 3) 4)
K. B l o d g e t t , Phys. Rev. 57, 921,1940. R . L . M o o n e y, J. opt. Soc. Am. 35,574, 1946. N.J. Meysing, P h y s i c a S , 687, 1941. D. C a b a l l e r o J. opt. Soc. Am. 37, 176, 1947.
Augustus 194"7
ON THE EXACT CALCULATION OF THE REFLECTANCE OF GLASS, COATED WITH AN ARBITRARY NUMBER OF NON-ABSORBING LAYERS by B. S. BLAISSE and J. J. VAN D E R SANDE N.V. Optische industrie ,,De Oude Delft", Delft
Up till now a general method of calculating exactly the reflectance of glass, coated with an arbitrary number of layers has been lacking in the published literature. The one-layer case however, has been treated exactly by B 1 o d g e t t 1) ; the solution of the two-layer case for normal.incidence has been given by M o o n e y 2) and that of a layer with a continuously varying index of refraction by M e ys i n g a). The latter authors both treated the problem starting from M a x w e 1 l's equations. The electromagnetic theory however allows also the solving of the general problem for glass, coated with an arbitrary number of layers, at all angles of incidence. This can be done for the case of light polarized perpendicular as well as parallel to the plane of incidence. General expressions obtained in this way for an arbit.rary number of layers are given in this note. We take a glass coated with N - - 2 layers 2, 3 . . . . . N - - 1 . Medium 1 represents the glass and medium N is air or vacuum. Thus there are N media; the thicknesses of the different layers are a 2 , a 3 . . . . a N - - l ; the refractive indices are n t , n 2 . . . , n N (see Fig.). A Plane light wave with an angle of incidence of 9 N is falling from the left side on the outside layer N - - 1. In each layer l the vector potentials A z and A s' are introduced describing respectively the ingoing lightwaves moving to the glass medium and the outgoing waves moving in the opposite direction. In the glass itself only the vector potential A 1 exists, as in this medium there is no reflected light, whereas in the medium N the two - -
413
- -
414
B. S. BLAISSE AND J. J. VAN DER SANDE
potentials denote respectively the incident wave AN and the totality of all reflected waves AN. The reflectance RN thus is given by the square of the quotient of the moduli of amplitudes A ~ and AN t
A,N ~~ ,N, , ANAN
RN =
the asterisk indicating the complex conjugate quantity. Now the problem is to find RN if the indices of refraction of all media, the thicknesses of the layers, the angle of incidence and the polarisation are given. This can be accomplished by considering the boundary conditions between the electric and magnetic vectors
at each interface giving relations between the A's of adjacent media. In this manner it can be shown that the quotient A;+I/AI+ 1 of the complex amplitudes of the ingoing and outgoing waves in the medium l + t can be expressed as a function of A~/A l in the medium l by the recursion relation A ' t+1
At+ 1
--
(st+l--st) exp (iqt)+(s/+l +st) exp (--iqt) . A t / A t I (st+t+ s,) exp (iq,)+(st+l--s,) exp (--iqt).As/At
(1)
In this expression qt is equal to . ql = 2"unbar cos 9~/~
(2)
where 9~ is the angle of incidence in the yh medium and I is the wave length of the light in vacuo. The quantities st are defined as SlJ. ~-
~1 COS ~t
s~u ~
--
(3)
1
nl
COS ~o~
REFLECTANCE
OF GLASS,
COATED
WITH
NON-ABSORBING
LAYERS
4r5
in the cases of light polarized perpendicular or parallel to the plane of incidence respectively. ¢ As in the medium 1 there is no reflected wave, A~ = 0 and we have t
A~
S2
A,2
s2 + sl
-
S1
-
(4)
giving for the reflectance of one interface the well-known F r e sn e 1 formula
R2 = ( S 2 - - S I I 2 \s2 + s2/
(S)
The value of A'N/AN for an arbitrary n u m b e r of layers can be calculated from (1) by complete induction using (4). B y squaring the modulus of this expression, the reflectance R N can be written as G RN =
+
s,s s
....
sN
(6)
in which G = 27 27 . . . . 27 Ck,, k,..., kA-_1 COS 2 (k2q2+k3q3+ . . . . kN--t qN--t) k~ ha
(7)
kN-- 1
The quantities C~, . . . . kN_t are functions of s I . . . . s N and the s u m m a t i o n 27 is to be carried out over the values ks= + 1,0 and - - 1 . ks For the case t h a t each layer has an optical thickness of a quarter wave length at the considered angle of incidence, each q~ has the value n/2, giving the following simple expression for the reflectance RN (v--w'12 R1v = \v + w / (8) The quantities v and w are given, in the case t h a t N is even, b y
v = sts]s ] Z~) ~
.
.
.
.
~2 - 2 ^2 .~2.~4.~6 . . . .
s2,_,
(9a)
S2
N__2SN
whereas for N odd v =
s,s]s~
_2 ^2 ^2
....
Ze) ~-- 52.~4.~6 . . . .
s~_2s;v
(9b)
S2
N--I
For light at normal incidence each ~0t is zero, the quantities s t
416
R E F L E C T A N C E OF GLASS, C O A T E D W I T H N O N - A B S O R B I N G L A Y E R S
being replaced by the refractive indices nv For the one and two layer case one gets the expressions allready found by B 1 o dg e t t x) and M o o n e y ~ ) . Details of the method will be published soon in "Applied Scientific Research". We wish to express our thanks to Prof, J. d e B o e r and Prof. A. v a n H e e 1 for valuable help and discussion.
Note added in proo]. After this note had gone to press, we became acquainted with an article by Miss D. C a b a 11 e r o 4) published in the March number of the Journal of the Optical Society of America, in which she gives a recursion formula for the case of normal incidence equivalent with (1) but deduced by the method of multiple reflections. Received April 10th, 1947.
REFERENCES 1) 2) 3) 4)
K. B l o d g e t t , Phys. Rev. 57, 921,1940. R . L . M o o n e y, J. opt. Soc. Am. 35,574, 1946. N.J. Meysing, P h y s i c a S , 687, 1941. D. C a b a l l e r o J. opt. Soc. Am. 37, 176, 1947.