The paper proposes new approach top the design of wide-band interference filters with suppression adjacent stop-bands. The method of research of the multi-layered interference system, allowing to find the necessary of refractive indices of layers, is presented. The results described in communication open new possibilities for construction and calculation of devices based on thin film materials.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3095

Bilozertseva, V.I.

Gaman, D.A.

Khlyap, H.M.

Nhu Tranh, Duong Thi

Ovcharenko, A.P.

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 5 No 1, 01010(3pp) (2013) Том 5 № 1, 01010(3cc) (2013) 
 
 
2077-6772/2013/5(1)01010(3) 01010-1  2013 Sumy State University 
Design of Broadband Optical Interference Filters Based on Six-Layer Period 
 
A.P. Ovcharenko1,*, V.I. Bilozertseva2, D.A. Gaman2, H.M. Khlyap3, Duong Thi Nhu Tranh1 
 
1 V.N. Karazin Kharkov National University, 4, Svobody Sq., 61022 Kharkiv. Ukraine 
2 National Technical University “KPI”, 21, Frunze Str., 61002 Kharkov, Ukraine 
3 University of Technology, Fachbereich Physik, 56, E.-Schrödinger Str., D-67663 Kaiserslautern, Germany 
 
(Received 12 December 2012; published online 28 March 2013) 
 
The paper proposes new approach top the design of wide-band interference filters with suppression ad-
jacent stop-bands. The method of research of the multi-layered interference system, allowing to find the 
necessary of refractive indices of layers, is presented. The results described in communication open new 
possibilities for construction and calculation of devices based on thin film materials. 
 
Keywords: Thin film, Wideband interference coating, Narrow-band interference coating, Refractive indi-
cies. 
 
 PACS numbers: 42.25.Yz, 42.79.Ci 
 
 
                                                        
* apo@univer.kharkov.ua 
The article was reported at the 2nd International Conference «Nanomaterials: Applications  Properties-2012» 
1. INTRODUCTION 
 
Many applications need non-absorbed coatings with 
high coefficient of reflection or transmission in the wide 
spectral range. There are mirrors for re-tuned lasers, 
cut-off and heat-reflected filters, polarizers, etc. How-
ever, the existing film compounds make it possible to 
manufacture coatings with relatively low width of the 
high reflectivity band, especially for UV- and visible 
spectral ranges. The known constructions of coatings 
allowing to expand the range of low transmission are 
characterized by completeness of their practical reali-
zation due to using layers (or groups of layers) with not 
multiple optical thicknesses. The system of two groups 
of layers with periodical alignment (AB) and (2AB), 
where A, B are quarter-wave (0.25 0) layers with re-
fraction indices nA, nB. Nevertheless, this coating has 
satisfactory characteristic only if the ratio of the refrac-
tion coefficients of the layers A and B is larger than 2, 
what can be realized in the IR spectral range. 
 
2. RESULTS AND DISCUSSION 
 
Authors have proposed [1, 2] coatings containing 
two groups of the layers with periods (А2ВАВ2АВ) and 
(АВ). For these periods the wide-band systems can be 
realized at the ratio of the layers’ refractive indices 
equal 1.67 and more. Compounds forming thin films 
which provide fulfillment of this condition in visible 
and near-UV spectral range, are existing, what makes 
possible to manufacture required wide-band coatings.  
The spectral dependencies of the transmission coef-
ficient T of the multilayer dielectric coating are defined 
by the elements of its interference matrix Ai,k (this ma-
trix is a product of the matrices describing the layers as 
components of the coating) and the refraction indices 
n0, nS of environmental media are written as follows 
[3, 4]: 
 
 0
2 2
11 0 22 12 0 21
4
( ) ( )
S
S S
n n
T
A n A n A n n A

  
 (1) 
The interference matrix of the j-layer is 
 
 
1
,
cos sin
( )
sin cos
j j j
k l j
j j
in
m
in
 
 
 
 
 
 
 (2) 
 
Where j  2njtj /   0.5vj is the phase thickness of 
the layer, and nj, tj are the refraction index and the 
geometrical thickness of the layer, respectively, j is a 
relative layer thickness in the units 0/4, 0/,  is a 
wavelength.  
Figures (1-3) present calculated transmittance of 
narrowband filters with period SN(AB)2AN(BA)S,  bi-
lateral cut filters with period SN(2AB)2AS (Fig. 1, 2)  
 
 
 
 
Fig. 1 – Spectral transmittance dependencies of 17-layer nar-
rowband filter (solid line) and cut filter based on 2AB-period 
(dotted line) for nA = 2.7, nB = 1.32 (a) and resultant curve (b) 
 
 
and with period SN(A2BAB2AB)A2BAS (Fig. 3). The 
resultant transmittance is also shown. Here N is a 
number of matched periods, nS = 1.52, 0 is doubled 
optical thickness of basic narrowband filter period (AB).  
 
 A.P. OVCHARENKO, V.I. BILOZERTSEVA, D.A. GAMAN, ET AL. J. NANO- ELECTRON. PHYS. 5, 01010 (2013) 
 
 
01010-2 
As one can see, bilateral cut filter based on (2AB)-
period can be used on condition that nA / nB ≥ 2. But if 
nA / nB < 2 it is better to use cut filter based on period 
(A2BAB2AB). 
 
 
 
 
 
Fig. 2 – Spectral transmittance dependencies of 19-layer nar-
rowband filter (solid line) and cut filter based on 2AB-period 
(dotted line) for nA = 2.2, nB = 1.32 (a) and resultant curve (b) 
 
 
 
 
 
Fig. 3 – Spectral transmittance dependencies of 19-layer nar-
rowband filter (solid line) and 21-layer cut filter based on 
A2BAB2AB-period (dotted line) for nA = 2.2, nB = 1.32 (a) and 
resultant curve (b) 
 
On the other hand, if the six-fold period (А2ВАВ2АВ) is 
considered as a principal one, and the value 0 is de-
termined as its double optical thickness, we obtain the 
transmission spectra with only uneven suppressed fre-
quency bands, and the function T() has a period equal 
8. In this case the corresponding change of the refrac-
tion indices of the layers included in the filter structure 
leads to removing some bands. Then one can obtain a 
wide band filter with suppressed adjacent frequency 
bands. For this purpose we will consider the six-layer 
period consisting of two symmetrical three-layer struc-
tures: (A2XA)(B2YB) (*). 
The homogeneous layer with equivalent refraction 
index NE can be associated with each of these layers 
[4]. For the left part of the period (*): 
 
 
cos
cos
A X
A X
E A
A X
A X
n n
n n
N n
n n
n n





 



 (3) 
 
The equivalent thickness ГЕ is determined as 
follows: 
 
 
2
2( )cos 1 sin
2
A X
E
A X
n n
n n


   
 
 (4) 
 
For the second subsystem from (*) the expressions 
are analogous. Here  is a phase thickness of the center 
layer.  
As one sees from the expression (4), at the same 
phase thicknesses of the center layers the spectral 
dependencies of the equivalent thicknesses for tri-layer 
structures will be identical under conditions:  
 
 A B X Yn n n n    (5) 
 
or 
 
 A Y X Bn n n n    (6) 
 
Now we require that for some  the equivalent 
thicknesses of every tri-layer structure from (*) 
satisfied the condition cos(ГЕ) = 0, and for the same  
the refraction indices of the tri-layers were also equal: 
 
 NE(A2XA) = NE(B2YB) (7) 
 
Then, for the corresponding  all the period (*) may 
be considered as a half-wave layer. At the same time, 
the conditions (6), (7) give physically unreliable 
solutions, and (5) and (7) produce two solutions for 
 
 
2 2 2
1
( ) ( ) ( 6 )
4
A B A B A A B B
X
B
n n n n n n n n
n
n
     
  (8) 
 
and 
 
 
2 2 2
2
( ) ( ) ( 6 )
4
A B A B A A B B
X
B
n n n n n n n n
n
n
     
  (9) 
 
The expression (5) gives the corresponding values 
for nY. 
Figures (4) and (5) show spectral functions of the 
transmission T() and the envelope curve of stray 
extremums E() of wide-band filters 
 
 SN(A2XAB2YB)S (10) 
 
for the solutions of (8) and (9), respectively. Here N is a 
number of periods.  
As one can see from these figures, for refraction 
indices from (5) and (8) there are only suppressed 
frequency bands 1 and 7 in the spectra (at the period 
8), and for the solution (5) and (9) there are only bands 
3 and 5.  
 DESING OF BROADBAND OPTICAL INTERFERENCE FILTERS… J. NANO- ELECTRON. PHYS. 5, 01010 (2013) 
 
 
 
01010-3 
 
 
Fig. 4 - Spectral dependencies T() and E() of the filter (8) at 
nA = 2.3, nX = 2.89, nB = 1.67, nY = 1.338, nS = 1.5, N = 10 
 
 
 
Fig. 5 - Spectral dependencies T() and E() of the filter (8) at 
nA = 2.3, nX = 1.83, nB = 1.67, nY = 2.1, nS = 1.5, N = 10 
The spectral dependencies of T() are similar with 
those described in [4] for a period (ABCDDCBA), which 
can be represented analogically to (*): (B2ABC2DC). 
However, authors [4] do not use the joint solution (3) 
and (4), the condition (5) is rather postulated, but is not 
proved. Such a postulation leads to a large equation 
binding parameters nB / nC and nA / nB. 
 
3. CONCLUSIONS 
 
Thereby the cyclic rearrangement of the layers in 
the period (10) does not lead to change of position and 
number of suppressed frequency bands, but at the same 
time, the depth of extremum in the ranges of 
transparency can be drastically changed. This depth 
also strongly depends on the refractive index nS of the 
environmental media. 
 
 
 
 
 
 
REFERENCES 
 
1. A.P. Ovcharenko, E.A. Lupashko, Intern. Conf. of 
OPTDIM'95, 202 (Kiev: 1995).  
2. A.P. Ovcharenko, E.A. Lupashko, International workshop 
on advanced technologies of multicomponent solid films 
and structures and their application in photonics, 77 
(Uzhgorod: 1996). 
3. H.A. Macleod, Thin-Film Optical Filters (Bristol: Adam 
Hilger Ltd.: 1986). 
4. A. Thelen, Design of Optical Interference Coatings (New 
York: McGraw-Hill: 1988). 
 


Design of Broadband Optical Interference Filters Based on Six-Layer Period

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 5 No 1, 01010(3pp) (2013) Том 5 № 1, 01010(3cc) (2013)   2077-6772/2013/5(1)01010(3) 01010-1  2013 Sumy State University Design of Broadband Optical Interference Filters Based on Six-Layer Period  A.P. Ovcharenko1,*, V.I. Bilozertseva2, D.A. Gaman2, H.M. Khlyap3, Duong Thi Nhu Tranh1  1 V.N. Karazin Kharkov National University, 4, Svobody Sq., 61022 Kharkiv. Ukraine 2 National Technical University “KPI”, 21, Frunze Str., 61002 Kharkov, Ukraine 3 University of Technology, Fachbereich Physik, 56, E.-Schrödinger Str., D-67663 Kaiserslautern, Germany  (Received 12 December 2012; published online 28 March 2013)  The paper proposes new approach top the design of wide-band interference filters with suppression ad-jacent stop-bands. The method of research of the multi-layered interference system, allowing to find the necessary of refractive indices of layers, is presented. The results described in communication open new possibilities for construction and calculation of devices based on thin film materials.  Keywords: Thin film, Wideband interference coating, Narrow-band interference coating, Refractive indi-cies.   PACS numbers: 42.25.Yz, 42.79.Ci                                                           * apo@univer.kharkov.ua The article was reported at the 2nd International Conference «Nanomaterials: Applications  Properties-2012» 1. INTRODUCTION  Many applications need non-absorbed coatings with high coefficient of reflection or transmission in the wide spectral range. There are mirrors for re-tuned lasers, cut-off and heat-reflected filters, polarizers, etc. How-ever, the existing film compounds make it possible to manufacture coatings with relatively low width of the high reflectivity band, especially for UV- and visible spectral ranges. The known constructions of coatings allowing to expand the range of low transmission are characterized by completeness of their practical reali-zation due to using layers (or groups of layers) with not multiple optical thicknesses. The system of two groups of layers with periodical alignment (AB) and (2AB), where A, B are quarter-wave (0.25 0) layers with re-fraction indices nA, nB. Nevertheless, this coating has satisfactory characteristic only if the ratio of the refrac-tion coefficients of the layers A and B is larger than 2, what can be realized in the IR spectral range.  2. RESULTS AND DISCUSSION  Authors have proposed [1, 2] coatings containing two groups of the layers with periods (А2ВАВ2АВ) and (АВ). For these periods the wide-band systems can be realized at the ratio of the layers’ refractive indices equal 1.67 and more. Compounds forming thin films which provide fulfillment of this condition in visible and near-UV spectral range, are existing, what makes possible to manufacture required wide-band coatings.  The spectral dependencies of the transmission coef-ficient T of the multilayer dielectric coating are defined by the elements of its interference matrix Ai,k (this ma-trix is a product of the matrices describing the layers as components of the coating) and the refraction indices n0, nS of environmental media are written as follows [3, 4]:   02 211 0 22 12 0 214( ) ( )SS Sn nTA n A n A n n A   (1) The interference matrix of the j-layer is   1,cos sin( )sin cosj j jk l jj jinmin       (2)  Where j  2njtj /   0.5vj is the phase thickness of the layer, and nj, tj are the refraction index and the geometrical thickness of the layer, respectively, j is a relative layer thickness in the units 0/4, 0/,  is a wavelength.  Figures (1-3) present calculated transmittance of narrowband filters with period SN(AB)2AN(BA)S,  bi-lateral cut filters with period SN(2AB)2AS (Fig. 1, 2)      Fig. 1 – Spectral transmittance dependencies of 17-layer nar-rowband filter (solid line) and cut filter based on 2AB-period (dotted line) for nA = 2.7, nB = 1.32 (a) and resultant curve (b)   and with period SN(A2BAB2AB)A2BAS (Fig. 3). The resultant transmittance is also shown. Here N is a number of matched periods, nS = 1.52, 0 is doubled optical thickness of basic narrowband filter period (AB).    A.P. OVCHARENKO, V.I. BILOZERTSEVA, D.A. GAMAN, ET AL. J. NANO- ELECTRON. PHYS. 5, 01010 (2013)   01010-2 As one can see, bilateral cut filter based on (2AB)-period can be used on condition that nA / nB ≥ 2. But if nA / nB < 2 it is better to use cut filter based on period (A2BAB2AB).      Fig. 2 – Spectral transmittance dependencies of 19-layer nar-rowband filter (solid line) and cut filter based on 2AB-period (dotted line) for nA = 2.2, nB = 1.32 (a) and resultant curve (b)      Fig. 3 – Spectral transmittance dependencies of 19-layer nar-rowband filter (solid line) and 21-layer cut filter based on A2BAB2AB-period (dotted line) for nA = 2.2, nB = 1.32 (a) and resultant curve (b)  On the other hand, if the six-fold period (А2ВАВ2АВ) is considered as a principal one, and the value 0 is de-termined as its double optical thickness, we obtain the transmission spectra with only uneven suppressed fre-quency bands, and the function T() has a period equal 8. In this case the corresponding change of the refrac-tion indices of the layers included in the filter structure leads to removing some bands. Then one can obtain a wide band filter with suppressed adjacent frequency bands. For this purpose we will consider the six-layer period consisting of two symmetrical three-layer struc-tures: (A2XA)(B2YB) (*). The homogeneous layer with equivalent refraction index NE can be associated with each of these layers [4]. For the left part of the period (*):   coscosA XA XE AA XA Xn nn nN nn nn n  (3)  The equivalent thickness ГЕ is determined as follows:   22( )cos 1 sin2A XEA Xn nn n     (4)  For the second subsystem from (*) the expressions are analogous. Here  is a phase thickness of the center layer.  As one sees from the expression (4), at the same phase thicknesses of the center layers the spectral dependencies of the equivalent thicknesses for tri-layer structures will be identical under conditions:    A B X Yn n n n    (5)  or   A Y X Bn n n n    (6)  Now we require that for some  the equivalent thicknesses of every tri-layer structure from (*) satisfied the condition cos(ГЕ) = 0, and for the same  the refraction indices of the tri-layers were also equal:   NE(A2XA) = NE(B2YB) (7)  Then, for the corresponding  all the period (*) may be considered as a half-wave layer. At the same time, the conditions (6), (7) give physically unreliable solutions, and (5) and (7) produce two solutions for   2 2 21( ) ( ) ( 6 )4A B A B A A B BXBn n n n n n n nnn       (8)  and   2 2 22( ) ( ) ( 6 )4A B A B A A B BXBn n n n n n n nnn       (9)  The expression (5) gives the corresponding values for nY. Figures (4) and (5) show spectral functions of the transmission T() and the envelope curve of stray extremums E() of wide-band filters   SN(A2XAB2YB)S (10)  for the solutions of (8) and (9), respectively. Here N is a number of periods.  As one can see from these figures, for refraction indices from (5) and (8) there are only suppressed frequency bands 1 and 7 in the spectra (at the period 8), and for the solution (5) and (9) there are only bands 3 and 5.   DESING OF BROADBAND OPTICAL INTERFERENCE FILTERS… J. NANO- ELECTRON. PHYS. 5, 01010 (2013)    01010-3   Fig. 4 - Spectral dependencies T() and E() of the filter (8) at nA = 2.3, nX = 2.89, nB = 1.67, nY = 1.338, nS = 1.5, N = 10    Fig. 5 - Spectral dependencies T() and E() of the filter (8) at nA = 2.3, nX = 1.83, nB = 1.67, nY = 2.1, nS = 1.5, N = 10 The spectral dependencies of T() are similar with those described in [4] for a period (ABCDDCBA), which can be represented analogically to (*): (B2ABC2DC). However, authors [4] do not use the joint solution (3) and (4), the condition (5) is rather postulated, but is not proved. Such a postulation leads to a large equation binding parameters nB / nC and nA / nB.  3. CONCLUSIONS  Thereby the cyclic rearrangement of the layers in the period (10) does not lead to change of position and number of suppressed frequency bands, but at the same time, the depth of extremum in the ranges of transparency can be drastically changed. This depth also strongly depends on the refractive index nS of the environmental media.       REFERENCES  1. A.P. Ovcharenko, E.A. Lupashko, Intern. Conf. of OPTDIM'95, 202 (Kiev: 1995).  2. A.P. Ovcharenko, E.A. Lupashko, International workshop on advanced technologies of multicomponent solid films and structures and their application in photonics, 77 (Uzhgorod: 1996). 3. H.A. Macleod, Thin-Film Optical Filters (Bristol: Adam Hilger Ltd.: 1986). 4. A. Thelen, Design of Optical Interference Coatings (New York: McGraw-Hill: 1988).  

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Design of Broadband Optical Interference Filters Based on Six-Layer Period

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