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Reflectance of multilayer systems with randomly rough boundaries
Volume 7 1, number
OPTICS
6
REFLECTANCE
OF MULTILAYER
15 June 1989
COMMUNICATIONS
SYSTEMS
WITH
RANDOMLY
ROUGH
BOUNDARIES
Ivan OHLiDAL Department of solid State Physics, Faculty of Science, Purkyne University Brno, Koriihkci 2, 61137 Brno, Czechoslovakia Received
25 November
1988
In this theoretical paper new formulae for calculating the specular reflectance of multilayer systems with randomly rough boundaries are presented. These formulae are derived within the scalar theory of diffraction. They are suitable for investigating the multilayer systems with slightly rough boundaries. The theoretical results are illustrated through the spectral dependence of the reflectance characterizing a multilayer system formed by seven nonabsorbing films.
The influence of roughness of boundaries on the optical properties of multilayer systems has been studied within different approximate theoretical approaches [ l-6 1. The most exact way for evaluating the reflectances, transmittance and scatter losses of the multilayer systems with randomly rough boundaries was presented in Eastman’s paper [ 5 1. Eastman’s approximate approach is based on a matrix formalism for deriving the formulae of the Fresnel coefficients characterizing the systems mentioned [ 51. However, in general this approach is rather complicated from the mathematical point of view so that one must often use algebraic manipulation programs for evaluating the Fresnel coefficients and reflectances and transmittance corresponding to the systems of interest. In this paper approximate formulae for calculating the specular reflectance of the multilayer system with the randomly rough boundaries formed by the nonidentical thin films will be derived. These formulae are very simple ones compared with those presented by Eastman [ 5 1. The physical model of the system under consideration is specified by the following assumptions. 1. Materials forming the ambient, the substrate and the thin films are isotropic and homogeneous from the optical point of views. 2. Boundaries of the system are locally smooth (see, e.g. ref. [ 7 ] ). 3. The slopes of the irregularities on all boundaries are small enough that the shadowing and multiple 0 030-4018/89/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
reflections among these irregularities can be neglected (the rms value of angles corresponding to the slopes is much smaller than unity. 4. The rms value ITof the height irregularities on all boundaries is considerably smaller than the wavelength of incident light (o< ;1 or cr<;l). 5. The mean levels of all boundaries are formed by planes. 6. The rough boundaries of the system are generated by stationary gaussian random processes. 7. The dimensions of the illuminated surface 2X and 2 Y are much larger than the wavelength I of incident light. 8. All boundaries of the multilayer system are mutually independent from the statistical point of view, i.e. the system is formed by the nonidentical thin films known as the general thin films [ 81. Further, let us assume that the parallel beam of monochromatic light falls on the mean planes of boundaries at nearly normal incidence. Moreover, let conditions of the Fraunhofer approximation be fulfilled when light flux is detected by a detector. We shall assume that the detector is situated in a focal plane of a lens placed in front of the system investigated. Then the electric field g at the point P(xf, yfj of the focal plane of the lens is given by the following integral [ 91
B.V.
323
Volume 7 1, number 6
x
&Xf, Yf) =R
OPTICS COMMUNICATIONS
I5 June 1989
Y
IS ax> Y)
(4)
-X-Y
xexp[-i(2~lV)(xx,+~~r)l
dxdy,
(1)
where
where v= - (4z/A)n,. Owing to the assumptions 1-5 the Fresnel coefficient l? can be expressed by recursion formulae implied by those presented in refs. [ 121 and [ 13 1, i.e. rl
rT=
fis focal length of the lens, R. denotes the distance between the lens and the upper boundary of the multilayer system under investigation, xr and y, are coordinates of the points laying in the focal plane of the lens and 0(x, y) denotes the electric field corresponding to the mean plane of the wavefronts leaving the upper boundary of the system (i.e. 0(x, y) is the near-field complex amplitude; see, e.g. ref. [ lo] ). Using assumption 2 we can write Qx,~)=R&exp]--i(4~/~)~7i(x,~)l,
[ill) R,= (4XYA;)-’ m cc IS
(&x‘,Y~))
(J%-GY~)
b,dy,,
(3)
-CC --oo where (I?(xr, yr)) is the statistical mean value of g( xf, yr), (I? ( xf, yf) ) represents the complex conjugate value of (g( xr, yr) ) and 4XYAi corresponds to the light flux of incident light. Eq. (3) implies that
324
1
(5)
’
where F
=
2
r2 +k3exp(ix2) 1 +r2k3 exp(ix,)
’
ew(kk) Fk= rk +Fkkfl 1 +T$~+,
exp(ix,)
’
(2)
where A0 is the amplitude of a monochromatic plane wave representing the beam of incident light, R denotes the complex Fresnel coefficient of the multilayer system, q, (x, y ) is a random function describing the upper boundary and no denotes the refractive index of the ambient. Below we are only interested in the reflectance R, corresponding to light scattered coherently from the system under consideration (i.e. under the assumptions outlined the reflectance R, is practically identical with the specular reflectance of the system). Ifthe relations X&%Aff/2 and YJ7r%Af/ 2 (Xr and Vf denote maximal values of xf and yf determined by an aperture of the detector) are valid the reflectance R, is given as follows (see, e.g. ref.
X
+k2ew(h
l+r,F,exp(ix,)
rN+rN+l
rN=1 +rNrN+, rk =
nk--l
-nk
nk-l
+nk
eXP(kN)
exp(ixN) )
k=l,2
(N denotes the number tem considered ),
’ ,...) TV+1 of thin films forming the sys-
o’= 1,2, ...) N), nN+, = n being the refractive index of the substrate, n, and/or 6, denotes the refractive index and/or the mean thickness of the jth film (a, is identical with the distance between the mean planes of thejth and the u+ 1 )th boundaries) and q,(x, y) and/or q,+, (x, y) is a random function describing the roughness of thejth and/or G+ 1 )th boundary. Eqs. (4) and (5) show that the reflectance R, cannot be calculated in an exact way. However, if the values of o, are sufficiently smaller than A (see assumption 4) we are satisfied to use an approximation expressed by the following equation
Volume 7 1.number
OPTICS
6
COMMUNICATIONS
15 June 1989
= (r,exp[itnl,(x,y)l+F;exp(i[x~+~~(x,Y)l))
(l+r,F*exp(ix,))
(6) The mean values of the quantities ~11, i2 ewG[x~+m(x,
Y)II
r, exp[iq,
and rlk2 exp(b)
(x, can
also be calculated in the manner expressed by eq. ( 6 ) . In this way one can proceed at calculating other mean values of random quantities needed for evaluating the mean values of the quantities mentioned above. Thus, the following recursion formulae are obtained for (Rexp[iw,(x, y)]), i.e.
ffocl+&,o
(~exp[iw(x,y)l)=
I;r
I
R
I.1
T
(7)
where ffo,,=r,(exp[ivrl~(x,y)l),
(8)
~;;1,0=(~2exp(i[x,+ly,(x,y)lj)
&=(t+,
I?, , +$, lir R ’ ,
g~,k +kj+ =
2 2.2
exp[i(x,+x,_,
(11)
’
(exp[i(x,+x,_,
gj3k=r,+I
(10)
+...+x~)])
I,k
1+rj+IR,+r,,+1
+...+x~)]),
(12)
j=2, 3, .... N,‘k=j, j- 1, .... 0, &,,, =fN,k and l?N,N =&QJ. One can easily show that the following equation is valid eXp[i(xj+xj_1
R,.k=r,+, Xexp
(
-i
where
+...+xk)]
$
-QpQp-,0;-QpQp+,a;+,
=r:[(I~212)-I(~2exp(~,))121,
(13)
(14)
where D{q denotes the variance of the complex random quantity f, the symbol 121 denotes the absolute 2 and quantity complex value of the (9) = 1 + r, ( r2 exp (ix, ) ) . Assuming the relation D{j>/ I (9) (2-x 1 we can easily prove that the following equation is true, 6R, = iz 2R,mf
(9)
ff,,,=(FZexp(ix,))=
and a0 =o,. This means that eqs. (4) and (7)-( 13) enable us to calculate the reflectance R, within the approximation specified above. An accuracy of evaluating the values of R, using the equations mentioned can be estimated as follows. Let us assume that 9~ 1 +r,f2exp(ix,). Then it holds that
( (1) (,
(15)
where dR, denotes the error of R, calculated using eqs. (4) and ( 7)-( 13) under assumption that the absolute value of 3 lies into the interval limited by the values of the real quantities I ( j) I -m and see eqs. (4) and (6)). Equations I(m+Jm ( for calculating the quantities D{a and (9) are similar to those expressing (R exp [iv, (x, y) ] ) . These equations will be published elsewhere. If the relation ) 6R,) G \ AR,) (AR, denotes an experimental error of R,) is fulfilled the equation for R, outlined here can exactly be used for interpreting experimental values of R,. In fig. 1 the spectral dependence of R, characterizing the multilayer system formed by seven nonabsorbing thin films calculated using eq. (4) is plotted together with that representing the same system with ideally smooth boundaries. In conclusion of this paper it should be noted that the formulae for calculating the coherent reflectance of the nonabsorbing multilayer systems presented here can be applied by employing simple calculators. It is evident that the formulae can be generalized for absorbing multilayer systems in an simple way. In a forthcoming paper equations for evaluating the incoherent reflectance of the system considered will be presented. Moreover, equations allowing to calculate the coherent transmittance, incoherent transmittance and scatter losses will be introduced as well. 325
Volume 7 1, number
OPTICS
6
COMMUNICATIONS
15 June 1989
References [ 1] R. Blazey, Appl. Optics 6 ( 1967) 83 1.
4m
500
6m
700
800
900
1000 h tna3
Fig. 1. The spectral dependence of R, for the system formed by seven nonabsorbing films with randomly rough boundaries placed on a glass substrate (p) and the spectral dependence of the reflectance of the same system with ideally smooth boundaries (---): n,=l, n=1.5, n,=n,=ns=n,=1.46, n2=n4=n6=2.3, d,=dl=d5=d,=300 nm, d,=d,=d,=200 nm for both the dependences. gI =r~~=...=cr~= 10 nm for R,.
The author wishes to thank Dr. J. MusilovA for her help with the preparation of the program used for the numerical calculations.
326
[2] K.H. Guenther, H.L. Gruber and H.K. Pulker, Thin Solid Films 34 (1976) 363. [ 31 H.K. Pulker, Thin Solid Films 34 ( 1976) 343. [4] 0. Arnon, Appl. Optics 16 (1977) 2147. [5] J.M. Eastman, Phys. Thin Films 10 (1978) 167. [ 61 J. Ebert, H. Panhorst, H. Kilster and H. Welling, Appl. Optics 18 (1979) 818. [7] P. Beckmann and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces (Pergamon, Oxford, 1963). [ 81 I. Ohlidal, K. Navrikil and F. LukeS, Optics Comm. 3 (1971) 40. [9] J.W. Goodman, Introduction to Fourier optics (McGrawHill, San Francisco 1968). [IO] P.J. Chandley and W.T. Welford, Opt. Quantum Electron. 7 (1975) 393. [ 11 ] I. Ohlidal, F. Luke: and K. Navratil, J. Phys. Paris 38, CS77 ( 1977) (Colloque C-5, Supplement au No I1 ). [ 121 A. VaSiEek, Optics of thin films (North-Holland, 1960). [ 131 Z. Knittl, Optics of thin films (Wiley, London, 1976).
OPTICS
6
REFLECTANCE
OF MULTILAYER
15 June 1989
COMMUNICATIONS
SYSTEMS
WITH
RANDOMLY
ROUGH
BOUNDARIES
Ivan OHLiDAL Department of solid State Physics, Faculty of Science, Purkyne University Brno, Koriihkci 2, 61137 Brno, Czechoslovakia Received
25 November
1988
In this theoretical paper new formulae for calculating the specular reflectance of multilayer systems with randomly rough boundaries are presented. These formulae are derived within the scalar theory of diffraction. They are suitable for investigating the multilayer systems with slightly rough boundaries. The theoretical results are illustrated through the spectral dependence of the reflectance characterizing a multilayer system formed by seven nonabsorbing films.
The influence of roughness of boundaries on the optical properties of multilayer systems has been studied within different approximate theoretical approaches [ l-6 1. The most exact way for evaluating the reflectances, transmittance and scatter losses of the multilayer systems with randomly rough boundaries was presented in Eastman’s paper [ 5 1. Eastman’s approximate approach is based on a matrix formalism for deriving the formulae of the Fresnel coefficients characterizing the systems mentioned [ 51. However, in general this approach is rather complicated from the mathematical point of view so that one must often use algebraic manipulation programs for evaluating the Fresnel coefficients and reflectances and transmittance corresponding to the systems of interest. In this paper approximate formulae for calculating the specular reflectance of the multilayer system with the randomly rough boundaries formed by the nonidentical thin films will be derived. These formulae are very simple ones compared with those presented by Eastman [ 5 1. The physical model of the system under consideration is specified by the following assumptions. 1. Materials forming the ambient, the substrate and the thin films are isotropic and homogeneous from the optical point of views. 2. Boundaries of the system are locally smooth (see, e.g. ref. [ 7 ] ). 3. The slopes of the irregularities on all boundaries are small enough that the shadowing and multiple 0 030-4018/89/$03.50 0 Elsevier Science Publishers ( North-Holland Physics Publishing Division )
reflections among these irregularities can be neglected (the rms value of angles corresponding to the slopes is much smaller than unity. 4. The rms value ITof the height irregularities on all boundaries is considerably smaller than the wavelength of incident light (o< ;1 or cr<;l). 5. The mean levels of all boundaries are formed by planes. 6. The rough boundaries of the system are generated by stationary gaussian random processes. 7. The dimensions of the illuminated surface 2X and 2 Y are much larger than the wavelength I of incident light. 8. All boundaries of the multilayer system are mutually independent from the statistical point of view, i.e. the system is formed by the nonidentical thin films known as the general thin films [ 81. Further, let us assume that the parallel beam of monochromatic light falls on the mean planes of boundaries at nearly normal incidence. Moreover, let conditions of the Fraunhofer approximation be fulfilled when light flux is detected by a detector. We shall assume that the detector is situated in a focal plane of a lens placed in front of the system investigated. Then the electric field g at the point P(xf, yfj of the focal plane of the lens is given by the following integral [ 91
B.V.
323
Volume 7 1, number 6
x
&Xf, Yf) =R
OPTICS COMMUNICATIONS
I5 June 1989
Y
IS ax> Y)
(4)
-X-Y
xexp[-i(2~lV)(xx,+~~r)l
dxdy,
(1)
where
where v= - (4z/A)n,. Owing to the assumptions 1-5 the Fresnel coefficient l? can be expressed by recursion formulae implied by those presented in refs. [ 121 and [ 13 1, i.e. rl
rT=
fis focal length of the lens, R. denotes the distance between the lens and the upper boundary of the multilayer system under investigation, xr and y, are coordinates of the points laying in the focal plane of the lens and 0(x, y) denotes the electric field corresponding to the mean plane of the wavefronts leaving the upper boundary of the system (i.e. 0(x, y) is the near-field complex amplitude; see, e.g. ref. [ lo] ). Using assumption 2 we can write Qx,~)=R&exp]--i(4~/~)~7i(x,~)l,
[ill) R,= (4XYA;)-’ m cc IS
(&x‘,Y~))
(J%-GY~)
b,dy,,
(3)
-CC --oo where (I?(xr, yr)) is the statistical mean value of g( xf, yr), (I? ( xf, yf) ) represents the complex conjugate value of (g( xr, yr) ) and 4XYAi corresponds to the light flux of incident light. Eq. (3) implies that
324
1
(5)
’
where F
=
2
r2 +k3exp(ix2) 1 +r2k3 exp(ix,)
’
ew(kk) Fk= rk +Fkkfl 1 +T$~+,
exp(ix,)
’
(2)
where A0 is the amplitude of a monochromatic plane wave representing the beam of incident light, R denotes the complex Fresnel coefficient of the multilayer system, q, (x, y ) is a random function describing the upper boundary and no denotes the refractive index of the ambient. Below we are only interested in the reflectance R, corresponding to light scattered coherently from the system under consideration (i.e. under the assumptions outlined the reflectance R, is practically identical with the specular reflectance of the system). Ifthe relations X&%Aff/2 and YJ7r%Af/ 2 (Xr and Vf denote maximal values of xf and yf determined by an aperture of the detector) are valid the reflectance R, is given as follows (see, e.g. ref.
X
+k2ew(h
l+r,F,exp(ix,)
rN+rN+l
rN=1 +rNrN+, rk =
nk--l
-nk
nk-l
+nk
eXP(kN)
exp(ixN) )
k=l,2
(N denotes the number tem considered ),
’ ,...) TV+1 of thin films forming the sys-
o’= 1,2, ...) N), nN+, = n being the refractive index of the substrate, n, and/or 6, denotes the refractive index and/or the mean thickness of the jth film (a, is identical with the distance between the mean planes of thejth and the u+ 1 )th boundaries) and q,(x, y) and/or q,+, (x, y) is a random function describing the roughness of thejth and/or G+ 1 )th boundary. Eqs. (4) and (5) show that the reflectance R, cannot be calculated in an exact way. However, if the values of o, are sufficiently smaller than A (see assumption 4) we are satisfied to use an approximation expressed by the following equation
Volume 7 1.number
OPTICS
6
COMMUNICATIONS
15 June 1989
(l+r,F*exp(ix,))
(6) The mean values of the quantities ~11, i2 ewG[x~+m(x,
Y)II
r, exp[iq,
and rlk2 exp(b)
(x, can
also be calculated in the manner expressed by eq. ( 6 ) . In this way one can proceed at calculating other mean values of random quantities needed for evaluating the mean values of the quantities mentioned above. Thus, the following recursion formulae are obtained for (Rexp[iw,(x, y)]), i.e.
ffocl+&,o
(~exp[iw(x,y)l)=
I;r
I
R
I.1
T
(7)
where ffo,,=r,(exp[ivrl~(x,y)l),
(8)
~;;1,0=(~2exp(i[x,+ly,(x,y)lj)
&=(t+,
I?, , +$, lir R ’ ,
g~,k +kj+ =
2 2.2
exp[i(x,+x,_,
(11)
’
(exp[i(x,+x,_,
gj3k=r,+I
(10)
+...+x~)])
I,k
1+rj+IR,+r,,+1
+...+x~)]),
(12)
j=2, 3, .... N,‘k=j, j- 1, .... 0, &,,, =fN,k and l?N,N =&QJ. One can easily show that the following equation is valid eXp[i(xj+xj_1
R,.k=r,+, Xexp
(
-i
where
+...+xk)]
$
-QpQp-,0;-QpQp+,a;+,
=r:[(I~212)-I(~2exp(~,))121,
(13)
(14)
where D{q denotes the variance of the complex random quantity f, the symbol 121 denotes the absolute 2 and quantity complex value of the (9) = 1 + r, ( r2 exp (ix, ) ) . Assuming the relation D{j>/ I (9) (2-x 1 we can easily prove that the following equation is true, 6R, = iz 2R,mf
(9)
ff,,,=(FZexp(ix,))=
and a0 =o,. This means that eqs. (4) and (7)-( 13) enable us to calculate the reflectance R, within the approximation specified above. An accuracy of evaluating the values of R, using the equations mentioned can be estimated as follows. Let us assume that 9~ 1 +r,f2exp(ix,). Then it holds that
( (1) (,
(15)
where dR, denotes the error of R, calculated using eqs. (4) and ( 7)-( 13) under assumption that the absolute value of 3 lies into the interval limited by the values of the real quantities I ( j) I -m and see eqs. (4) and (6)). Equations I(m+Jm ( for calculating the quantities D{a and (9) are similar to those expressing (R exp [iv, (x, y) ] ) . These equations will be published elsewhere. If the relation ) 6R,) G \ AR,) (AR, denotes an experimental error of R,) is fulfilled the equation for R, outlined here can exactly be used for interpreting experimental values of R,. In fig. 1 the spectral dependence of R, characterizing the multilayer system formed by seven nonabsorbing thin films calculated using eq. (4) is plotted together with that representing the same system with ideally smooth boundaries. In conclusion of this paper it should be noted that the formulae for calculating the coherent reflectance of the nonabsorbing multilayer systems presented here can be applied by employing simple calculators. It is evident that the formulae can be generalized for absorbing multilayer systems in an simple way. In a forthcoming paper equations for evaluating the incoherent reflectance of the system considered will be presented. Moreover, equations allowing to calculate the coherent transmittance, incoherent transmittance and scatter losses will be introduced as well. 325
Volume 7 1, number
OPTICS
6
COMMUNICATIONS
15 June 1989
References [ 1] R. Blazey, Appl. Optics 6 ( 1967) 83 1.
4m
500
6m
700
800
900
1000 h tna3
Fig. 1. The spectral dependence of R, for the system formed by seven nonabsorbing films with randomly rough boundaries placed on a glass substrate (p) and the spectral dependence of the reflectance of the same system with ideally smooth boundaries (---): n,=l, n=1.5, n,=n,=ns=n,=1.46, n2=n4=n6=2.3, d,=dl=d5=d,=300 nm, d,=d,=d,=200 nm for both the dependences. gI =r~~=...=cr~= 10 nm for R,.
The author wishes to thank Dr. J. MusilovA for her help with the preparation of the program used for the numerical calculations.
326
[2] K.H. Guenther, H.L. Gruber and H.K. Pulker, Thin Solid Films 34 (1976) 363. [ 31 H.K. Pulker, Thin Solid Films 34 ( 1976) 343. [4] 0. Arnon, Appl. Optics 16 (1977) 2147. [5] J.M. Eastman, Phys. Thin Films 10 (1978) 167. [ 61 J. Ebert, H. Panhorst, H. Kilster and H. Welling, Appl. Optics 18 (1979) 818. [7] P. Beckmann and A. Spizzichino, The scattering of electromagnetic waves from rough surfaces (Pergamon, Oxford, 1963). [ 81 I. Ohlidal, K. Navrikil and F. LukeS, Optics Comm. 3 (1971) 40. [9] J.W. Goodman, Introduction to Fourier optics (McGrawHill, San Francisco 1968). [IO] P.J. Chandley and W.T. Welford, Opt. Quantum Electron. 7 (1975) 393. [ 11 ] I. Ohlidal, F. Luke: and K. Navratil, J. Phys. Paris 38, CS77 ( 1977) (Colloque C-5, Supplement au No I1 ). [ 121 A. VaSiEek, Optics of thin films (North-Holland, 1960). [ 131 Z. Knittl, Optics of thin films (Wiley, London, 1976).