- Email: [email protected]
High reflectance bands of some simple two-material dielectric multilayers
Thin Solid Films, 167 (1988) 33-43
33
ELECTRONICS AND OPTICS
HIGH REFLECTANCE BANDS OF SOME SIMPLE TWO-MATERIAL DIELECTRIC MULTILAYERS Z. LJd AND L. JANKOWIAK Institute of Physics, Jageilonian University, Reymonta 4,30-059 Krakdw (Poland)
(Received August 6,1987; revised February 23,1988; accepted May 27,1988)
The high reflectance bands and other optical properties of the two structures [(HL)“H]” and [(HL)P(LHfl’ are discussed and compared with those of the structures (HL)’ and [(H/2)L(H/2)‘.
1. INTRODUCTION
Four thin film structures are commonly used when dielectric two-material multilayers are being designed: structure A: (HL)k = HLH.. . HL
(1)
structure B: [(H/2)L(H/2)]’
(2)
= (H/2)LH.. . L(H/2)
structure C: (HL)mH = HLH . . . LH
(3)
structure D: (HL)P(LH)P = HLH . . . HLLH . . . LH
(4)
Here H denotes a high index film which is one-quarter of a wavelength thick at 1, and L denotes an equally thick low index film, and k, m and p are integers. There is of course a second version of each of the above structures in which the H and L layers are interchanged. The optical properties of the first two structures are much easier to calculate than those of the latter two and have been known for quite a long time. Their description can be found for example in the three famous monographs by Macleod’, KnittlZ and Liddel13. Structure B in particular has proved to be very useful in thin film theory, as shown in the pioneering papers by Epstein4 and Thelen’. Structures C and D are also inherent in many thin film assemblies but they are much more difficult to treat mathematically. Some aspects of the properties of structures C were first utilized in the design of multilayer filters by Thelens, whose work has been extended recently by Jacobs6 and by Pelletier and Macleod’. In this paper we want to describe some other aspects of these structures. We shall be mainly concerned with structures C and D but we shall discuss them together with structures A and B to be able to point out some similarities and differences between all of them. ocw%6090/88/$3.50
0 Elsevier Sequoia/Printed in The Netherlands
34
Z. LEi, L. JANKOWIAK
2. SOMEGENERAL PROPERTIES OF STRUCTURES
A-D
(1) Let us start with a short account of some general optical properties of a multilayer (they can be found, for example, in the monographs of Macleod’ and Knittl’). (a) The spectral characteristics reflectivity R(IZ)and/or transmittance T(I) of any multilayer are entirely defined by its characteristic unimodular matrix (5) which is the product of the characteristic and also unimodular matrices of the individual layers. In the case when the light is incident normally on the strata, the matrix for a homogeneous non-absorbing layer of refractive index nj and phase thickness 6j takes the form mj =
COS
(i/n3 sin aj
6j
[ injSin 6j
COS dj
1
(6)
where i* = - 1 and 6j = 2m,dj/A, d, being the geometrical thickness of the jth layer and 1 the vacuum wavelength. It is also possible to express the phase thickness 6j in terms of the wavelength &, at which the given layer becomes a quarter-wavelength layer: njdj = A,/4 and thus 6j = (n/2)nej/n. Therefore the characteristic matrix of a layer can be described as a function of the quantity g = &,/A. For I = lo we have g = 1 and 6 = x/2. (b) Any multilayer consisting of at least two layers possesses so-called high reflectance bands (HRBs) that can be defined as follows. If we consider a multilayer to be a fundamental period and start piling up identical fundamental periods one onto another, the reflectance of the periodic multilayer being produced will increase monotonically within the HRBs. The HRBs exist at those values of g = 1,/1 for which the condition I~,,+~221/2
3
1
(7)
is fulfilled, where M, 1 and Mz2 are the matrix elements from eqn. (5) referring to the fundamental period. The HRBs are also called the stop or rejection bands as the increasing reflectance causes automatically the decreasing transmittance. All the other spectral regions are called pass bands. (2) Structures A-D have in pairs or in groups of three certain features in common, which can be easily seen from eqns. (l)-(4) defining these structures. (a) Structures A and B are periodic. The fundamental period of structure A is two-layer HL and that of structure B is three-layer (H/2)L(H/2). Neither structure C nor structure D is periodic. In order to build periodic structures out of them, we have to repeat them at least once more while the numbers m and p are kept constct. (b) Structures B-D are symmetrical, whereas structure A is not. For any symmetrical multilayer matrix eqn. (5) has M,, = Mz2. This means that such a multilayer can be described as a single layer matrix (eqn. (6)) with its so-called equivalent index n, and its equivalent phase thickness 6,. It follows from eqns. (5)
HIGH REFLECTANCE
BANDS OF TWO-MATERIAL
DIELECTRICS
35
and (6) that n, = (M21/M12Y2
@a)
6, = arccos M 11
@W
The manipulation of equivalent parameters is very advantageous, especially when dealing with structures that are both symmetrical and periodic, such as structure B, because the equivalent index of the whole system is equal to that of the fundamental period and the equivalent phase thickness of the whole system is equal to the product of the equivalent phase thickness of the fundamental period and the number of periods. Naturally, as the number of layers that constitutes the fundamental period increases, the calculations of n, and 6, to be performed become more elaborate. For symmetrical structures, the condition (7) describing HRBs takes the form IMlll 2 1
(9)
It can be seen easily that the equality in expression (9), which gives the positions of the edges of HRBs, is equivalent to two conditions: Ml,=0
or
M,,=O
(10)
Within the HRBs, the equivalent index n, is imaginary, which follows from the unimodularity of matrix (5) as well as eqns. (8a) and (9). (c) At certain I = & (i.e. g = 1) each layer of structures A and C is one-quarter of a wavelength thick. This is why such systems are called tuned or centred or quarter-wavelength systems. It is very simple to calculate the reflectance of such systems at 1 = 1, by means of the so-called effective index of refraction’ nerf. We then have Rk = I) = {Cv- G~)/(Y + n,,,‘)>’
(1 Ia)
where y is a function of the indices of the surrounding media only and n
cff -
n1n3n5..
.
n2n,n6..
.
(1lb)
nl, n2,. . . being
the refractive indices of successive layers. For structure A (even number of layers, equal to 2k) we have
Y=
no/n,
(114
4ff
=
UN
hM
and for structure C (odd number of layers, equal to 2m + 1) y=
nOng
n eff
-
h&dk)m
(114 (llf)
The symbols no and n, are the refractive indices of the incident medium and the substrate respectively. The notion of effective index can actually be applied also to stacks in which
36
Z. Lk,
L. JANKOWIAK
some or all layers are multiples of a given quarter-wave. The example is structure D and it follows from eqn. (11b) that ncrf = 1, which agrees with the fact that at 1= lo the whole structure becomes latent. Because structure B is detuned, its reflectivity cannot be expressed in the form of eqn. (1 la) and so the effective index is not defined for it. (d) Structures A and B have one more characteristic in common, which is that they have exactly the same HRBs. This follows from the fact that, applying condition (7) to the fundamental period of structure A and condition (9) or (10) to that of structure B, we obtain the same expression: cosa,,,
= *(n”-nL)/(%r+nr,)
(12)
where 6, and a2 are the HRB edges in units of phase. As far as structures C and D are concerned, some information on the HRBs of structures C can be found in refs. 5-7 but, to our knowlege, there does not exist in the literature any detailed account of the subject. We try to fill in this gap in this paper. As we shall show, the HRBs of some pairs of structures C and D are very much alike, but they always differ essentially from those of structures A and B. All our computations were performed for normal incidence of the light and for non-absorbing materials, with nH = 2.35 (ZnS), nL = 1.35 (cryolite) and no = n, = 1. The abscissa in all figures is the quantity g = lo/12 referring to that layer of a stack which at Iz = L, becomes one quarter-wave thick (H or L). 3.
HIGH REFLECTANCE BANDS OF THE
(HL)mH
STRUCTURES
(1) Figure 1 gives the reflectance characteristics R(g) of the simplest representatives of structures A-C. The characteristic R(g)of the LH structure has been omitted because it is the same as that of the HL structure. The straight horizontal lines below the abscissae show the HRBs. They were computed from condition (7) or (9), the lines situated slightly higher than the others belonging to the value - 1, and those l.O0.5. O-Y, 1.0.
10. c
14' 0.5. On
-
Fig. 1. Reflectance R(g) of the HL (curve a), (H/2)L(H/2) (curve b), (L/2)H(L/2) (curve c), HLH (curve d) and LHL (curve e) structures, with indicated HRBs.
HIGH REFLECTANCE
BANDS OF TWO-MATERIAL
DIELECTRICS
37
slightly lower to the value + 1. The latter appear only for structures C. The range [0,4] ofg comprises one order of the reflectance curve R(g) for structures B (curves b and c), and two such orders for structures A and C (curves a, d and e). Curves a-c show that, although the characteristics R(g) are very different from each other, the HRBs are identical. This corresponds to the fact that in all three cases the HRBs are described by the same equation (12). In contrast, the HRBs of structures C (curves d and e) differ from those of curves a-c fundamentally because they reveal gaps at g = 1 and g = 3. Analytical calculations of the HRBs for structures C, with the help of condition (9), would be very tedious even for such an easy example as the HLH or LHL structure because one encounters an equation cubic in cos 6. It appears, however, that for those simple structures the analytical expressions for the edges of the HRBs can be obtained very easily by means of conditions (10). We then obtain cos&,,
= Ifi&.r/(n,+h,)
cos 83.4 = * IZJ(nn + k)
(13)
where d1-6* give the four edges of the two HRBs (in units of phase) in the region g = [0,2], which are in agreement with the numerical calculations (curves d and e). Figure 2 gives the first-order reflectance characteristics R(g) for structures (HLH)” with n = 1-5. In accordance with the definition of the HRBs, within their limits the reflectance increases with the number of the repetitions of the fundamental period, whereas in the gap between the two HRBs more and more local minima and maxima occur. For even n it is always found that R(g = 1) = 0, and for odd n R(g = 1) reaches a local maximum, which follow from the fact that the whole system can be reduced to one half-wave layer or to the fundamental period respectively. (2) The numerical computations of the HRBs of systems (HL)“H with M 2 2 show that, as well as the two principal HRBs which are the broadest and the nearest to the g = 1 point, additional much narrower HRBs occur, so that each (HL)‘“H system reveals 2m HRBs altogether. Curves a-d in Fig. 3 give the characteristics R(g)
Fig. 2. Reflectance R(g) of the (HLH)” structures with n = l-5 (curves a-e respectively).
2. LE6, L. JANKOWIAK
38
of the [(HL)“H]” systems with m = 2 and n = l-4. It appears that with the increase in m the edges of the principal HRBs rapidly approach the point g = 1. Nevertheless, in the region including this point the gap between the two HRBs always remains, producing a pass band in which, when the fundamental period is repeated, a transmittance region occurs, for n = 2 without any structure and for n 2 3 with a structure the same as for m = 2 (Fig. 2) and m = 3 (Fig. 3, curves a-d). It is just these local maxima of T(g) (i.e. the local minima of R(g)) that were utilized in ref. 6 for designing interference filters with multiple peaks.
20 m 15
10
5
0I 0.5
0.6
0.7
0.9
0.9 p=&.lX
l.0
Fig. 3. Reflectance R(g) of the [(HL)‘H]” structures with n = l-4 (curves a-d respectively) and the [(HL)*(LH)‘]’ structure (curve e). Fig. 4. The positions and widths of one branch of the principal HRBs of the (HL)“H systems with m = l-20 as function of g = 1,/k
Figure 4 shows the positions and widths of one branch of the principal HRBs of the (HL)“H systems with m = l-20. It appears that the g values of both edges of these HRBs fit quite well with the experimental curves, the constants of which depend on the nH/nL ratio. Thus on a logarithmic scale we would obtain two nearly straight lines. In the first five columns of Table I the numerical values of the above HRBs for m = l-4 and m = 20 are given. One can see that with increasing m the widths of these HRBs at first slightly increase, reaching a maximum for m = 3, and then decrease, whereas the ratios g,/g, of the HRB edges, which are independent of the value of A,, decrease monotonically all the time. (3) The approximate values of those principal edges of the (Hl)mH structures that are near the point g = 1 (i.e. g, in Table I) can be found from Thelen’s formula for the half-width of the pass band of these structures’. This formula is valid only for g not too far removed from g = 1 and has been obtained with the help of the equality (9) and the approximations given by Seely’. For the purpose of this paper we shall write it in the form gz” = 1 -(l/n)arcsin{2(n,/n,-
l)/(n,/n,)m+ ‘>
(14)
HIGH REFLECTANCE BANDS OF TWO-MATERIAL
39
DIELECTRICS
TABLE I NUMERICAL
(HL)“H
g VALUES
AND
OF TIE
(HL)p(LH)p
EDGES OF ONE BRANCH
OF THJ3 PRINCIPAL
HIGH REFLECTANCE
HRB edges
OF THE
(HL)p(LH)p
(HL)“H m
BANDS
STRUCI-URJS
gz-g1
g,fg,
g?
g1
gz
1 2 3 4
0.562 0.682 0.733 0.760
0.762 0.894 0.945 0.970
0.200 0.211 0.212 0.209
1.356 1.311 1.290 1.276
0.837 0.909 0.948 0.970
20
0.820
1.000
0.180
1.226
1.000
HRB edges
P
gz-g,
g,fg,
g1
gz
1 2 3 4
0.415 0.638 0.712 0.750
0.587 0.844 0.924 0.959
0.172 0.206 0.212 0.209
1.414 1.323 1.300 1.279
20
0.820
1.000
0.180
1.226
The values ofgih calculated with nu = 2.35, nL = 1.35, and m = l-4 and m = 20 are given in the middle column of Table I. We can see that the agreement with the accurate values of g, (third column) is rather good even for small values of m and improves rapidly with increasing m. One obtains nearly the same values of g;” after the sign of the sine term in eqn. (14) has been omitted (the discrepancy amounts to. only 0.007 even for m = 1). 4.
HIGH REFLECTANCE BANDS OF THE
(HL)P(LH)P
STRUCTURES
(1) The simplest representative of structure D (eqn. (4)) is the HLLH system. Its characteristic R(g) is the same as that for the (H/2)L(H/2) system given in Fig. 1, curve b, provided that we change the g scale to g’ = g/2. This means that, for example, R = 0 appearing at g = 2 in Fig. 1, curve b, will be situated at g’ = 1 for the HLLH system. In consequence the range g’ = [0,4] will cover two orders of the reflectance curves of the HLLH structure, the same as for structures A and C in Fig. 1, curves a, d and e. In contrast, &cause the ratio gJg, of the HRB edges is independent of the choice of lo, it has the same value for both systems (H/2)L(H/2) and HLLH, which can be calculated from eqn. (12). All the above relations are of course also valid for the multiple structures, i.e. [(H/2)L(H/2)lk and (HLLH)’ with k = 1. For the easy structure HLLH it is also possible to obtain simple analytical expressions for the edges of the HRBs. They are cos6,,,
= &{nu/(n,+nL))‘iz
cos a,,, = f {nL/(n, + nL)} ‘/’
(15)
We can see that these expressions are the square roots of those for the HLH structure (eqn. (13)). (2) The (HL)P (LH)P structures with p > 2 do not have anything in common with the simple and well-known structure (H/2)L(H/2). From the numerical calculations it appears, however, that the spectral characteristics of [(HL)P(LH)“]r structures can be very similar to those of the [(HL)mH]” structures if the pairs of numbers m and p as weli as n and I are matched properly. Thus the condition m = p
40
Z.
Lti,
L. JANKO-K
secures the same number of HRBs, whereas the same number of half-wave films makes both structures have similar patterns of the R(g) characteristics in the pass band including the point g = 1. Because all the [(HLP)p (LHM’ systems have always an odd number of half-wave films, which causes R(g= 1) = 0, the number n has to be even. The simplest example of such a matched pair of structures C and D are the systems (HLH)’ and HLLH, whose characteristics R(g) are given in Figs. 2, curve b, and 1, curve b, respectively, provided that the g scale in Fig. 1, curve b, has been changed to g’ = g/2 (Section 4(l)). As another illustrative example, let us consider curves d and e in Fig. 3, where the spectral characteristics R(g)of the structures [(HL)2H]4 and [(HL)2(LH)2]2 are given. It will be seen that in both of the above examples the spectral characteristics as well as the HRBs of the two components of each pair are really very similar. To compare the HRBs of the two structures being discussed it is sufficient to deal with their fundamental periods only. The right-hand part of Table I also shows the values concerning one branch of the principal HRBs of the (HL)‘(LH)p structures with p = l-4 and p = 20. When comparing them with those of the (HL)mH structures (the first five columns of this table) one can see that they are very similar and the existing differences diminish with the increase in m and p. 5. EQUIVALENT AND EFFECTIVE INDICES OF THE STRUCTURES UNDER DISCUSSION
(1) Figure 5 gives the equivalent index n, of the structures HLH and LHL (curves 1 and 2 respectively) as a function of g = A,,/& and Fig. 6 gives the same characteristics n,(g) of the structures (HL)2H and (LH)‘L. The equivalent indices of all these systems are real at g = 1, in agreement with the fact that there is no HRB there (Fig. 1, curves d and e, and Fig. 3). This is quite different from the more often drawn curves n,(g) of structures B (e.g. refs. 1 and 2) whose equivalent indices are imaginary at g = 1 as they have the HRB at this point (Fig. 1, curves b and c). It can be seen in Figs. 5 and 6 that structures C with m = 1 have two regions with imaginary indices (the regions in which no values are given) and those with m = 2 four such -
6
Fig. 5. The equivalent index n,(g) of the HLH (curves 1) and LHL (curves 2) structures. Fig. 6. The equivalent index n,(g) of the (HL)‘H (curves 1) and (LH)2L (curves 2) structures.
HIGH REFLECTANCE
BANDS OF TWO-MATERIAL DIELECTRICS
41
regions, in accordance with the number of HRBs of all these structures (e.g. Figs. 2, curve a, and 3, curve a). As far as the structures D are concerned, the n,(g) graphs for the HLLH and LHHL structures are the same as those very well known for the (H/2)L(H/2) and (L/2)H(L/2) structures (see for example refs. l-4), provided that the g scales have been changed tog = g/2 (Section 4(l)). However, the general properties of n,(g) of the (HL)P(LH)P structures with any value of p will be very similar to those of the proper (HL)“‘H structures, which is clear from the suitably chosen spectral characteristics R(g) (e.g. Fig. 3, curves d and e). (2) Some features of curves n,(g) for structures C and D can be found analytically. (a) Limiting values n,(g --t 0) of any symmetrical structure can be found easily with the help of the general statement by Thelen (Theorem (V) in his pape?). For the (HL)‘“H structure we obtain 112 (m+ l)n, + rn?rL n&+0) = (m + l)n, + mnHnHnL}
1
It appears that for the (HL)P (LH)P structures n,(g + 0) is independent of the number of layers as we have n,(g --) 0) = (?iH?rL)l’2 This formula is of course the same as that for the (H/2)L(H/2) structure. (b) For the simplest structures C, i.e. HLH and LHL, the values ofg at which n, becomes zero or infinite (the edges of the HRBs) are given by eqns. (13), whereas for the simplest structures D, i.e. HLLH and LHHL, they are given by eqns. (15). The approximate values of such points, but only of those nearest the g = 1 point, can be found for any (HL)“H structure from eqn. (14). (c) It is also possible to find general formulae for the values of the equivalent indices at g = 1 for structures C and D. For structures C we obtain %(s = 1) =
dh-ll~L)m
(16)
and for structures D we obtain n,(g = 1) = rIu(?ru/nL)p-1’2
(17)
These two formulae can be reduced to one if we consider the structures [(HL)“H]’ and (HL)P(LH)P, because only such structures can have similar spectral characteristics (Section 4(2)). We then obtain n,(g = 1) = uH{(nH/nL)N’2-1}1’2
(18a)
where N equals the number of quarter-wave layers in a stack, i.e. N = 2(2m+ 1) or N = 4p (the half-wave layers being counted as two quarter-wave layers) or n&g = 1) = n,{(n,/nL)(K- l)‘2}l’2
(18b)
where K = N - 1 equals the number of layers in each of the stacks. Naturally, the n,(g = 1) values of the structures [(HL)‘“H]” do not depend on n, and we have taken into account the [(HL)“H12 structures only to be able to derive eqns. (18).
Z. LEi, L. JANKOWIAK
42
It follows from eqns.(l6)-(18b) that, with increasing number of layers, the values of n,(g = 1) of structures C and D rapidly increase and, after having interchanged the H and L layers, rapidly decrease. The first few values of n,(g = 1) for both these types of structures C and D are shown in Table II. TABLE II THEn,(g = 1) STRUCTURES,
N
P
4 6 8 10 12 14 16 18 20
1
VALUES
OF THE
EACH STRUCTURE
m
1 2 2 3 3 4 4 5
(HL)mH ANJJ (HL)p(LH)P STRUCTURES WITH N QUARTER-WAVE LAYERS (H.L)
n,(g=i).
ANDTHE
(LH)“L
AND
(LH)P(HL)P
n,(g = I), (L,H)
3.10 4.09 5.40 7.12 9.40 12.39 16.35 21.57 28.48
1.35 0.78 0.59 0.45 0.34 0.26 0.19 0.15 0.11
(3) Since structures C are tuned, they also have an effective index n,rr, defined at g = 1 only (Section 2(2)(c)). Comparing eqns. (Ill) and (16), we see that for these structures neff = n,(g = 1). It is easy to see that this is also valid for the [(HL)“H]” structures but only when n is an odd number. It follows from eqn. (1 lb) that for even it we always have n,rr = 1. neff = 1 is also always found for the [(HL)P(LHfl’ structures with any value of p and 1(structures D and their multiples). All these relations have been gathered in Table III. The first two rows refer to structures A and B. Structure A being asymmetric, n, is not defined, whereas structure B does not exhibit a value of neff because it is detuned. TABLE III VALUES
OF n,(g
=
1) AND
Multilayer
ncff FOR VARIOUS
MULTILAYERS
n,(g=I)
4,
Imaginary
-
ndn&~)m nH(nH/nL)p-‘I2
6.
(n”/d n odd: n,(n,/n,)m
{n even:
1
1
FINAL REMARKS
(1) We would like to stress once more the point that the HRBs of structures C and D differ essentially from those of structures A and B. A priori one could believe that in particular structures A and C should have identical or at least similar HRBs and such suggestions can be found in the literature. Two possible reasons for this are as follows: firstly, that the spectral characteristics R(g) of both of these structures
HIGH REFLECTANCE BANDS OF TWO-MATERIAL
DIELECTRICS
43
(with the exception of the LHL system that has R = Rmin at g = 1) are exactly of the same type and, secondly, that the R(g = 1) values of all the members of structure A (with various k) as well as those of structures C (with various m) increase monotonically with k and m. (2) The structures [(HL)“H]” have proved very useful when designing various multilayer filters5. Table11 shows that they can be replaced by suitable [(HL)P(LHM’ structures, which may sometimes be very advantageous because of the different values of n,(g = 1) for the same values of nu and nL. REFERENCES
H. A. Macleod, Thin-Film Optical Filters, Hilger, London, 1969. Z. Knittl, Optics of Thin Films, Wiley, London, 1976. H. M. Liddell, Computer-aided Techniquesfor the Design of Multilayer Filters, Hilger, Bristol, 1981. L. I. Epstein, J. Opt. Sot. Am., 42 (1952) 806. 5 A. Thelen,J. Opt. Sot. Am., 50(1966) 1533. 6 C. Jacobs, Appl. Opt., 20(1981) 1039. 7 E. Pelletier and H. A. Macleod, J. Opt. Sot. Am., 72 (1982) 683. 8 J. S. Seely, J. Opt. Sot. Am., 51(1964) 342.
1 2 3 4
33
ELECTRONICS AND OPTICS
HIGH REFLECTANCE BANDS OF SOME SIMPLE TWO-MATERIAL DIELECTRIC MULTILAYERS Z. LJd AND L. JANKOWIAK Institute of Physics, Jageilonian University, Reymonta 4,30-059 Krakdw (Poland)
(Received August 6,1987; revised February 23,1988; accepted May 27,1988)
The high reflectance bands and other optical properties of the two structures [(HL)“H]” and [(HL)P(LHfl’ are discussed and compared with those of the structures (HL)’ and [(H/2)L(H/2)‘.
1. INTRODUCTION
Four thin film structures are commonly used when dielectric two-material multilayers are being designed: structure A: (HL)k = HLH.. . HL
(1)
structure B: [(H/2)L(H/2)]’
(2)
= (H/2)LH.. . L(H/2)
structure C: (HL)mH = HLH . . . LH
(3)
structure D: (HL)P(LH)P = HLH . . . HLLH . . . LH
(4)
Here H denotes a high index film which is one-quarter of a wavelength thick at 1, and L denotes an equally thick low index film, and k, m and p are integers. There is of course a second version of each of the above structures in which the H and L layers are interchanged. The optical properties of the first two structures are much easier to calculate than those of the latter two and have been known for quite a long time. Their description can be found for example in the three famous monographs by Macleod’, KnittlZ and Liddel13. Structure B in particular has proved to be very useful in thin film theory, as shown in the pioneering papers by Epstein4 and Thelen’. Structures C and D are also inherent in many thin film assemblies but they are much more difficult to treat mathematically. Some aspects of the properties of structures C were first utilized in the design of multilayer filters by Thelens, whose work has been extended recently by Jacobs6 and by Pelletier and Macleod’. In this paper we want to describe some other aspects of these structures. We shall be mainly concerned with structures C and D but we shall discuss them together with structures A and B to be able to point out some similarities and differences between all of them. ocw%6090/88/$3.50
0 Elsevier Sequoia/Printed in The Netherlands
34
Z. LEi, L. JANKOWIAK
2. SOMEGENERAL PROPERTIES OF STRUCTURES
A-D
(1) Let us start with a short account of some general optical properties of a multilayer (they can be found, for example, in the monographs of Macleod’ and Knittl’). (a) The spectral characteristics reflectivity R(IZ)and/or transmittance T(I) of any multilayer are entirely defined by its characteristic unimodular matrix (5) which is the product of the characteristic and also unimodular matrices of the individual layers. In the case when the light is incident normally on the strata, the matrix for a homogeneous non-absorbing layer of refractive index nj and phase thickness 6j takes the form mj =
COS
(i/n3 sin aj
6j
[ injSin 6j
COS dj
1
(6)
where i* = - 1 and 6j = 2m,dj/A, d, being the geometrical thickness of the jth layer and 1 the vacuum wavelength. It is also possible to express the phase thickness 6j in terms of the wavelength &, at which the given layer becomes a quarter-wavelength layer: njdj = A,/4 and thus 6j = (n/2)nej/n. Therefore the characteristic matrix of a layer can be described as a function of the quantity g = &,/A. For I = lo we have g = 1 and 6 = x/2. (b) Any multilayer consisting of at least two layers possesses so-called high reflectance bands (HRBs) that can be defined as follows. If we consider a multilayer to be a fundamental period and start piling up identical fundamental periods one onto another, the reflectance of the periodic multilayer being produced will increase monotonically within the HRBs. The HRBs exist at those values of g = 1,/1 for which the condition I~,,+~221/2
3
1
(7)
is fulfilled, where M, 1 and Mz2 are the matrix elements from eqn. (5) referring to the fundamental period. The HRBs are also called the stop or rejection bands as the increasing reflectance causes automatically the decreasing transmittance. All the other spectral regions are called pass bands. (2) Structures A-D have in pairs or in groups of three certain features in common, which can be easily seen from eqns. (l)-(4) defining these structures. (a) Structures A and B are periodic. The fundamental period of structure A is two-layer HL and that of structure B is three-layer (H/2)L(H/2). Neither structure C nor structure D is periodic. In order to build periodic structures out of them, we have to repeat them at least once more while the numbers m and p are kept constct. (b) Structures B-D are symmetrical, whereas structure A is not. For any symmetrical multilayer matrix eqn. (5) has M,, = Mz2. This means that such a multilayer can be described as a single layer matrix (eqn. (6)) with its so-called equivalent index n, and its equivalent phase thickness 6,. It follows from eqns. (5)
HIGH REFLECTANCE
BANDS OF TWO-MATERIAL
DIELECTRICS
35
and (6) that n, = (M21/M12Y2
@a)
6, = arccos M 11
@W
The manipulation of equivalent parameters is very advantageous, especially when dealing with structures that are both symmetrical and periodic, such as structure B, because the equivalent index of the whole system is equal to that of the fundamental period and the equivalent phase thickness of the whole system is equal to the product of the equivalent phase thickness of the fundamental period and the number of periods. Naturally, as the number of layers that constitutes the fundamental period increases, the calculations of n, and 6, to be performed become more elaborate. For symmetrical structures, the condition (7) describing HRBs takes the form IMlll 2 1
(9)
It can be seen easily that the equality in expression (9), which gives the positions of the edges of HRBs, is equivalent to two conditions: Ml,=0
or
M,,=O
(10)
Within the HRBs, the equivalent index n, is imaginary, which follows from the unimodularity of matrix (5) as well as eqns. (8a) and (9). (c) At certain I = & (i.e. g = 1) each layer of structures A and C is one-quarter of a wavelength thick. This is why such systems are called tuned or centred or quarter-wavelength systems. It is very simple to calculate the reflectance of such systems at 1 = 1, by means of the so-called effective index of refraction’ nerf. We then have Rk = I) = {Cv- G~)/(Y + n,,,‘)>’
(1 Ia)
where y is a function of the indices of the surrounding media only and n
cff -
n1n3n5..
.
n2n,n6..
.
(1lb)
nl, n2,. . . being
the refractive indices of successive layers. For structure A (even number of layers, equal to 2k) we have
Y=
no/n,
(114
4ff
=
UN
hM
and for structure C (odd number of layers, equal to 2m + 1) y=
nOng
n eff
-
h&dk)m
(114 (llf)
The symbols no and n, are the refractive indices of the incident medium and the substrate respectively. The notion of effective index can actually be applied also to stacks in which
36
Z. Lk,
L. JANKOWIAK
some or all layers are multiples of a given quarter-wave. The example is structure D and it follows from eqn. (11b) that ncrf = 1, which agrees with the fact that at 1= lo the whole structure becomes latent. Because structure B is detuned, its reflectivity cannot be expressed in the form of eqn. (1 la) and so the effective index is not defined for it. (d) Structures A and B have one more characteristic in common, which is that they have exactly the same HRBs. This follows from the fact that, applying condition (7) to the fundamental period of structure A and condition (9) or (10) to that of structure B, we obtain the same expression: cosa,,,
= *(n”-nL)/(%r+nr,)
(12)
where 6, and a2 are the HRB edges in units of phase. As far as structures C and D are concerned, some information on the HRBs of structures C can be found in refs. 5-7 but, to our knowlege, there does not exist in the literature any detailed account of the subject. We try to fill in this gap in this paper. As we shall show, the HRBs of some pairs of structures C and D are very much alike, but they always differ essentially from those of structures A and B. All our computations were performed for normal incidence of the light and for non-absorbing materials, with nH = 2.35 (ZnS), nL = 1.35 (cryolite) and no = n, = 1. The abscissa in all figures is the quantity g = lo/12 referring to that layer of a stack which at Iz = L, becomes one quarter-wave thick (H or L). 3.
HIGH REFLECTANCE BANDS OF THE
(HL)mH
STRUCTURES
(1) Figure 1 gives the reflectance characteristics R(g) of the simplest representatives of structures A-C. The characteristic R(g)of the LH structure has been omitted because it is the same as that of the HL structure. The straight horizontal lines below the abscissae show the HRBs. They were computed from condition (7) or (9), the lines situated slightly higher than the others belonging to the value - 1, and those l.O0.5. O-Y, 1.0.
10. c
14' 0.5. On
-
Fig. 1. Reflectance R(g) of the HL (curve a), (H/2)L(H/2) (curve b), (L/2)H(L/2) (curve c), HLH (curve d) and LHL (curve e) structures, with indicated HRBs.
HIGH REFLECTANCE
BANDS OF TWO-MATERIAL
DIELECTRICS
37
slightly lower to the value + 1. The latter appear only for structures C. The range [0,4] ofg comprises one order of the reflectance curve R(g) for structures B (curves b and c), and two such orders for structures A and C (curves a, d and e). Curves a-c show that, although the characteristics R(g) are very different from each other, the HRBs are identical. This corresponds to the fact that in all three cases the HRBs are described by the same equation (12). In contrast, the HRBs of structures C (curves d and e) differ from those of curves a-c fundamentally because they reveal gaps at g = 1 and g = 3. Analytical calculations of the HRBs for structures C, with the help of condition (9), would be very tedious even for such an easy example as the HLH or LHL structure because one encounters an equation cubic in cos 6. It appears, however, that for those simple structures the analytical expressions for the edges of the HRBs can be obtained very easily by means of conditions (10). We then obtain cos&,,
= Ifi&.r/(n,+h,)
cos 83.4 = * IZJ(nn + k)
(13)
where d1-6* give the four edges of the two HRBs (in units of phase) in the region g = [0,2], which are in agreement with the numerical calculations (curves d and e). Figure 2 gives the first-order reflectance characteristics R(g) for structures (HLH)” with n = 1-5. In accordance with the definition of the HRBs, within their limits the reflectance increases with the number of the repetitions of the fundamental period, whereas in the gap between the two HRBs more and more local minima and maxima occur. For even n it is always found that R(g = 1) = 0, and for odd n R(g = 1) reaches a local maximum, which follow from the fact that the whole system can be reduced to one half-wave layer or to the fundamental period respectively. (2) The numerical computations of the HRBs of systems (HL)“H with M 2 2 show that, as well as the two principal HRBs which are the broadest and the nearest to the g = 1 point, additional much narrower HRBs occur, so that each (HL)‘“H system reveals 2m HRBs altogether. Curves a-d in Fig. 3 give the characteristics R(g)
Fig. 2. Reflectance R(g) of the (HLH)” structures with n = l-5 (curves a-e respectively).
2. LE6, L. JANKOWIAK
38
of the [(HL)“H]” systems with m = 2 and n = l-4. It appears that with the increase in m the edges of the principal HRBs rapidly approach the point g = 1. Nevertheless, in the region including this point the gap between the two HRBs always remains, producing a pass band in which, when the fundamental period is repeated, a transmittance region occurs, for n = 2 without any structure and for n 2 3 with a structure the same as for m = 2 (Fig. 2) and m = 3 (Fig. 3, curves a-d). It is just these local maxima of T(g) (i.e. the local minima of R(g)) that were utilized in ref. 6 for designing interference filters with multiple peaks.
20 m 15
10
5
0I 0.5
0.6
0.7
0.9
0.9 p=&.lX
l.0
Fig. 3. Reflectance R(g) of the [(HL)‘H]” structures with n = l-4 (curves a-d respectively) and the [(HL)*(LH)‘]’ structure (curve e). Fig. 4. The positions and widths of one branch of the principal HRBs of the (HL)“H systems with m = l-20 as function of g = 1,/k
Figure 4 shows the positions and widths of one branch of the principal HRBs of the (HL)“H systems with m = l-20. It appears that the g values of both edges of these HRBs fit quite well with the experimental curves, the constants of which depend on the nH/nL ratio. Thus on a logarithmic scale we would obtain two nearly straight lines. In the first five columns of Table I the numerical values of the above HRBs for m = l-4 and m = 20 are given. One can see that with increasing m the widths of these HRBs at first slightly increase, reaching a maximum for m = 3, and then decrease, whereas the ratios g,/g, of the HRB edges, which are independent of the value of A,, decrease monotonically all the time. (3) The approximate values of those principal edges of the (Hl)mH structures that are near the point g = 1 (i.e. g, in Table I) can be found from Thelen’s formula for the half-width of the pass band of these structures’. This formula is valid only for g not too far removed from g = 1 and has been obtained with the help of the equality (9) and the approximations given by Seely’. For the purpose of this paper we shall write it in the form gz” = 1 -(l/n)arcsin{2(n,/n,-
l)/(n,/n,)m+ ‘>
(14)
HIGH REFLECTANCE BANDS OF TWO-MATERIAL
39
DIELECTRICS
TABLE I NUMERICAL
(HL)“H
g VALUES
AND
OF TIE
(HL)p(LH)p
EDGES OF ONE BRANCH
OF THJ3 PRINCIPAL
HIGH REFLECTANCE
HRB edges
OF THE
(HL)p(LH)p
(HL)“H m
BANDS
STRUCI-URJS
gz-g1
g,fg,
g?
g1
gz
1 2 3 4
0.562 0.682 0.733 0.760
0.762 0.894 0.945 0.970
0.200 0.211 0.212 0.209
1.356 1.311 1.290 1.276
0.837 0.909 0.948 0.970
20
0.820
1.000
0.180
1.226
1.000
HRB edges
P
gz-g,
g,fg,
g1
gz
1 2 3 4
0.415 0.638 0.712 0.750
0.587 0.844 0.924 0.959
0.172 0.206 0.212 0.209
1.414 1.323 1.300 1.279
20
0.820
1.000
0.180
1.226
The values ofgih calculated with nu = 2.35, nL = 1.35, and m = l-4 and m = 20 are given in the middle column of Table I. We can see that the agreement with the accurate values of g, (third column) is rather good even for small values of m and improves rapidly with increasing m. One obtains nearly the same values of g;” after the sign of the sine term in eqn. (14) has been omitted (the discrepancy amounts to. only 0.007 even for m = 1). 4.
HIGH REFLECTANCE BANDS OF THE
(HL)P(LH)P
STRUCTURES
(1) The simplest representative of structure D (eqn. (4)) is the HLLH system. Its characteristic R(g) is the same as that for the (H/2)L(H/2) system given in Fig. 1, curve b, provided that we change the g scale to g’ = g/2. This means that, for example, R = 0 appearing at g = 2 in Fig. 1, curve b, will be situated at g’ = 1 for the HLLH system. In consequence the range g’ = [0,4] will cover two orders of the reflectance curves of the HLLH structure, the same as for structures A and C in Fig. 1, curves a, d and e. In contrast, &cause the ratio gJg, of the HRB edges is independent of the choice of lo, it has the same value for both systems (H/2)L(H/2) and HLLH, which can be calculated from eqn. (12). All the above relations are of course also valid for the multiple structures, i.e. [(H/2)L(H/2)lk and (HLLH)’ with k = 1. For the easy structure HLLH it is also possible to obtain simple analytical expressions for the edges of the HRBs. They are cos6,,,
= &{nu/(n,+nL))‘iz
cos a,,, = f {nL/(n, + nL)} ‘/’
(15)
We can see that these expressions are the square roots of those for the HLH structure (eqn. (13)). (2) The (HL)P (LH)P structures with p > 2 do not have anything in common with the simple and well-known structure (H/2)L(H/2). From the numerical calculations it appears, however, that the spectral characteristics of [(HL)P(LH)“]r structures can be very similar to those of the [(HL)mH]” structures if the pairs of numbers m and p as weli as n and I are matched properly. Thus the condition m = p
40
Z.
Lti,
L. JANKO-K
secures the same number of HRBs, whereas the same number of half-wave films makes both structures have similar patterns of the R(g) characteristics in the pass band including the point g = 1. Because all the [(HLP)p (LHM’ systems have always an odd number of half-wave films, which causes R(g= 1) = 0, the number n has to be even. The simplest example of such a matched pair of structures C and D are the systems (HLH)’ and HLLH, whose characteristics R(g) are given in Figs. 2, curve b, and 1, curve b, respectively, provided that the g scale in Fig. 1, curve b, has been changed to g’ = g/2 (Section 4(l)). As another illustrative example, let us consider curves d and e in Fig. 3, where the spectral characteristics R(g)of the structures [(HL)2H]4 and [(HL)2(LH)2]2 are given. It will be seen that in both of the above examples the spectral characteristics as well as the HRBs of the two components of each pair are really very similar. To compare the HRBs of the two structures being discussed it is sufficient to deal with their fundamental periods only. The right-hand part of Table I also shows the values concerning one branch of the principal HRBs of the (HL)‘(LH)p structures with p = l-4 and p = 20. When comparing them with those of the (HL)mH structures (the first five columns of this table) one can see that they are very similar and the existing differences diminish with the increase in m and p. 5. EQUIVALENT AND EFFECTIVE INDICES OF THE STRUCTURES UNDER DISCUSSION
(1) Figure 5 gives the equivalent index n, of the structures HLH and LHL (curves 1 and 2 respectively) as a function of g = A,,/& and Fig. 6 gives the same characteristics n,(g) of the structures (HL)2H and (LH)‘L. The equivalent indices of all these systems are real at g = 1, in agreement with the fact that there is no HRB there (Fig. 1, curves d and e, and Fig. 3). This is quite different from the more often drawn curves n,(g) of structures B (e.g. refs. 1 and 2) whose equivalent indices are imaginary at g = 1 as they have the HRB at this point (Fig. 1, curves b and c). It can be seen in Figs. 5 and 6 that structures C with m = 1 have two regions with imaginary indices (the regions in which no values are given) and those with m = 2 four such -
6
Fig. 5. The equivalent index n,(g) of the HLH (curves 1) and LHL (curves 2) structures. Fig. 6. The equivalent index n,(g) of the (HL)‘H (curves 1) and (LH)2L (curves 2) structures.
HIGH REFLECTANCE
BANDS OF TWO-MATERIAL DIELECTRICS
41
regions, in accordance with the number of HRBs of all these structures (e.g. Figs. 2, curve a, and 3, curve a). As far as the structures D are concerned, the n,(g) graphs for the HLLH and LHHL structures are the same as those very well known for the (H/2)L(H/2) and (L/2)H(L/2) structures (see for example refs. l-4), provided that the g scales have been changed tog = g/2 (Section 4(l)). However, the general properties of n,(g) of the (HL)P(LH)P structures with any value of p will be very similar to those of the proper (HL)“‘H structures, which is clear from the suitably chosen spectral characteristics R(g) (e.g. Fig. 3, curves d and e). (2) Some features of curves n,(g) for structures C and D can be found analytically. (a) Limiting values n,(g --t 0) of any symmetrical structure can be found easily with the help of the general statement by Thelen (Theorem (V) in his pape?). For the (HL)‘“H structure we obtain 112 (m+ l)n, + rn?rL n&+0) = (m + l)n, + mnHnHnL}
1
It appears that for the (HL)P (LH)P structures n,(g + 0) is independent of the number of layers as we have n,(g --) 0) = (?iH?rL)l’2 This formula is of course the same as that for the (H/2)L(H/2) structure. (b) For the simplest structures C, i.e. HLH and LHL, the values ofg at which n, becomes zero or infinite (the edges of the HRBs) are given by eqns. (13), whereas for the simplest structures D, i.e. HLLH and LHHL, they are given by eqns. (15). The approximate values of such points, but only of those nearest the g = 1 point, can be found for any (HL)“H structure from eqn. (14). (c) It is also possible to find general formulae for the values of the equivalent indices at g = 1 for structures C and D. For structures C we obtain %(s = 1) =
dh-ll~L)m
(16)
and for structures D we obtain n,(g = 1) = rIu(?ru/nL)p-1’2
(17)
These two formulae can be reduced to one if we consider the structures [(HL)“H]’ and (HL)P(LH)P, because only such structures can have similar spectral characteristics (Section 4(2)). We then obtain n,(g = 1) = uH{(nH/nL)N’2-1}1’2
(18a)
where N equals the number of quarter-wave layers in a stack, i.e. N = 2(2m+ 1) or N = 4p (the half-wave layers being counted as two quarter-wave layers) or n&g = 1) = n,{(n,/nL)(K- l)‘2}l’2
(18b)
where K = N - 1 equals the number of layers in each of the stacks. Naturally, the n,(g = 1) values of the structures [(HL)‘“H]” do not depend on n, and we have taken into account the [(HL)“H12 structures only to be able to derive eqns. (18).
Z. LEi, L. JANKOWIAK
42
It follows from eqns.(l6)-(18b) that, with increasing number of layers, the values of n,(g = 1) of structures C and D rapidly increase and, after having interchanged the H and L layers, rapidly decrease. The first few values of n,(g = 1) for both these types of structures C and D are shown in Table II. TABLE II THEn,(g = 1) STRUCTURES,
N
P
4 6 8 10 12 14 16 18 20
1
VALUES
OF THE
EACH STRUCTURE
m
1 2 2 3 3 4 4 5
(HL)mH ANJJ (HL)p(LH)P STRUCTURES WITH N QUARTER-WAVE LAYERS (H.L)
n,(g=i).
ANDTHE
(LH)“L
AND
(LH)P(HL)P
n,(g = I), (L,H)
3.10 4.09 5.40 7.12 9.40 12.39 16.35 21.57 28.48
1.35 0.78 0.59 0.45 0.34 0.26 0.19 0.15 0.11
(3) Since structures C are tuned, they also have an effective index n,rr, defined at g = 1 only (Section 2(2)(c)). Comparing eqns. (Ill) and (16), we see that for these structures neff = n,(g = 1). It is easy to see that this is also valid for the [(HL)“H]” structures but only when n is an odd number. It follows from eqn. (1 lb) that for even it we always have n,rr = 1. neff = 1 is also always found for the [(HL)P(LHfl’ structures with any value of p and 1(structures D and their multiples). All these relations have been gathered in Table III. The first two rows refer to structures A and B. Structure A being asymmetric, n, is not defined, whereas structure B does not exhibit a value of neff because it is detuned. TABLE III VALUES
OF n,(g
=
1) AND
Multilayer
ncff FOR VARIOUS
MULTILAYERS
n,(g=I)
4,
Imaginary
-
ndn&~)m nH(nH/nL)p-‘I2
6.
(n”/d n odd: n,(n,/n,)m
{n even:
1
1
FINAL REMARKS
(1) We would like to stress once more the point that the HRBs of structures C and D differ essentially from those of structures A and B. A priori one could believe that in particular structures A and C should have identical or at least similar HRBs and such suggestions can be found in the literature. Two possible reasons for this are as follows: firstly, that the spectral characteristics R(g) of both of these structures
HIGH REFLECTANCE BANDS OF TWO-MATERIAL
DIELECTRICS
43
(with the exception of the LHL system that has R = Rmin at g = 1) are exactly of the same type and, secondly, that the R(g = 1) values of all the members of structure A (with various k) as well as those of structures C (with various m) increase monotonically with k and m. (2) The structures [(HL)“H]” have proved very useful when designing various multilayer filters5. Table11 shows that they can be replaced by suitable [(HL)P(LHM’ structures, which may sometimes be very advantageous because of the different values of n,(g = 1) for the same values of nu and nL. REFERENCES
H. A. Macleod, Thin-Film Optical Filters, Hilger, London, 1969. Z. Knittl, Optics of Thin Films, Wiley, London, 1976. H. M. Liddell, Computer-aided Techniquesfor the Design of Multilayer Filters, Hilger, Bristol, 1981. L. I. Epstein, J. Opt. Sot. Am., 42 (1952) 806. 5 A. Thelen,J. Opt. Sot. Am., 50(1966) 1533. 6 C. Jacobs, Appl. Opt., 20(1981) 1039. 7 E. Pelletier and H. A. Macleod, J. Opt. Sot. Am., 72 (1982) 683. 8 J. S. Seely, J. Opt. Sot. Am., 51(1964) 342.
1 2 3 4