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Polarized light transmission through generalized Fibonacci multilayers: I. Dynamical maps approach and II. Numerical results
Optik 115, No. 6 (2004) 257–276 http://www.elsevier.de/ijleo
International Journal for Light and Electron Optics
Polarized light transmission through generalized Fibonacci multilayers: I. Dynamical maps approach Agnieszka Klauzer-Kruszyna, Włodzimierz Salejda, Michał H. Tyc Institute of Physics, Wrocław University of Technology, Wybrzez˙e Wyspian´skiego 27, 50-370 Wroclaw, Poland
Abstract: The theory of polarized light propagation in dielectric generalized Fibonacci multilayers is developed. The matrix formulation and dynamical maps technique is used. New objects: diagonal antitraces, symmetric and antisymmetric nondiagonal antitraces of characteristic matrices are introduced. Dynamical maps for these objects are derived. Transmittance for s- and p-type polarized light of the studied aperiodic multilayers placed between two homogenous media is expressed in terms of traces and antitraces of characteristic matrices. Three interesting physical situations are considered, allowing to study the influence of surrounding media on light transmission properties of Fibonacci-type multilayers. Key words: Fibonaccian superlattices – polarized light transmittance – multilayer systems
1. Introduction Since Shechtman et al. [1] had discovered quasicrystals, aperiodic crystals and quasi-one-dimensional aperiodic structures have been investigated experimentally and theoretically [2–11]. The quasi-one-dimensional aperiodic systems are multilayer structures composed of at least two different substrates (homogenous layers) A and B distributed along given direction. The A and B layers alignment is precisely defined by the substitution rules or inflation scheme [2, 3, 12, 13]. In the past, the main research efforts were focused on electron spectra properties and electron wave functions [3–5], electric conduction [6, 7], phonon spectra [8, 13], magnetic properties [9], acoustic [10] and electromagnetic waves transmission [11, 12], [14–31]. In their pioneering work [14], Kohmoto, Sutherland and Iguchi considered light propagation through an optical Fibonaccian multilayer, using dynamical maps of traces of unimodular real transfer matrices to calcu-
Received 17 February 2004; accepted 10 May 2004. Correspondence to: W. Salejda Fax: ++48-71-3283696 E-mail: [email protected]
late transmittance. The dynamical maps formalism has been introduced firstly by Kohmoto, Kadanoff and Tang [15]. Transmittance of generalized Fibonaccian superlattices, introduced by Gumbs and Ali [5], was calculated using the transfer matrix trace maps technique developed by Kola´rˇ and Ali [16] and then Iguchi [17] and applied in the papers [18–23]. This technique was extended to unimodular complex matrices [7] and modified by Dulea, Severin and Riklund [24], who introduced transfer matrix antitraces and derived socalled antitrace maps. Mathematical properties of generalized Fibonacci superlattice trace maps and some of their physical applications were presented in review papers [4, 25]. The main reason for studying dielectric two-substrate periodic, aperiodic and random multilayer system [26, 27] was the possibility of practical application in, e.g., Cantor type [28] or Fibbonaci type [29] FabryPerot resonators, optical filters and optical memories [20, 27]. In the most of published papers on the electromagnetic wave propagation through aperiodic multilayer media, it was assumed that: 1. incident wave has s polarization, 2. optical thicknesses of A and B layers are equal, 3. neither reflection nor refraction occurs on the interfaces between the system and external media. In the present paper we study, in the framework of dynamical maps technique, polarized light transmission properties of dielectric Fibonaccian superlattices placed between two dielectric media. For these structures, called Optical Generalized Fibonacci Superlattices (OGFS), we calculate in particular the transmittance as a function of light wavelength and incidence angle q. Unlike in previous published papers, we consider the propagation of the s- and p-wave incident on the structure at arbitrary angle. We give and prove, in the explicit form, dynamical trace and antitrace maps for OGFS characteristic matrices. We derive formulas for polarised light transmittance of the studied system placed between two homogenous media. We express the transmittance in terms of trace and antitraces of 0030-4026/04/115/06-257 $ 30.00/0
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A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
characteristic matrices. We discuss the following cases: 1. both external media are A-type, 2. both external media have arbitrary but identical refractive index, 3. external media have arbitrary refractive indices. The paper is organized as follows. In section 2 we recall briefly the matrix formulation of light propagation for isotropic layered media. The OGFS construction rule and derived formulas for transmittance expressed as a function of trace and antitraces maps are given in section 3. Section 4 contains final conclusions and remarks. Appendix includes basic definitions and derivation of formulas from section 2 and 3.
2. Matrix formulation of light propagation for isotropic layered media We describe the light propagation through a dielectric multilayer medium (fig. 1) consisting of J isotropic and homogeneous layers with refractive indices nj and thicknesses dj ¼ xj xj1 distributed along x-axis. The subscript j denotes refraction indices of media, j ¼ 0; 1; 2; . . . ; J; J þ 1, where j ¼ 0 corresponds to incident medium ‘‘in” and j ¼ J þ 1 – substrate medium ‘‘out”, onto which the subsequent layers are deposited. We assume that an electromagnetic plane wave of length l falls at angle qin upon the interface between the media with refractive indices nin and n1 . The amplitudes of electric field vector for incident ðþÞ ðÞ ðþÞ Ein , reflected Ein and transmitted Fout light wave through dielectric multilayer media (fig. 1) are related by formula (within the framework of transfer matrix formalism) [30, 31] " ðþÞ # " # " ðþÞ # " ðþÞ # J Ein Q Fout F R j Dj; jþ1 ¼ Din;1 ¼ G out ; ðÞ ðÞ ðÞ j¼1 Fout Fout Ein ð1Þ ðÞ
where Fout ¼ 0, and G is the characteristic matrix of layered system. Matrix P j , called jth medium propagation matrix, can be written as ij 2p 0 e j Pj ¼ cos qj ; ð2Þ ; jj ¼ dj nj 0 eijj l
y
z
Fig. 1. Dielectric multilayer structure placed between homogeneous dielectric media with refractive indices nin and nout
Fig. 2. Electric E and magnetic H field vectors of s and p waves at the interface between jth and (j + 1)st media.
and the matrix Dj; jþ1 , called transmission matrix from jth to ð j þ 1Þst medium, takes the form 1 1 rj;jþ1 Dj; jþ1 ¼ ; ð3Þ 1 tj; jþ1 rj; jþ1 where the matrix elements depend on light polarization (see fig. 2), and Fresnel transmision and reflection coefficients are given by [30, 31] 8 2nj cos qj > > for s wave ; > < nj cos qj þ njþ1 cos qjþ1 ð4Þ tj; jþ1 ¼ > 2nj cos qj > > for p wave ; : nj cos qjþ1 þ njþ1 cos qj 8 nj cos qj njþ1 cos qjþ1 > > for s wave ; > < n cos q þ n cos q j j jþ1 jþ1 ð5Þ rj;jþ1 ¼ > n cos qjþ1 njþ1 cos qj > > j for p wave : : nj cos qjþ1 þ njþ1 cos qj The characteristic matrix G " # N Q G 11 G 12 P j Dj; jþ1 G¼ ¼ Din;1 G 21 G 22 j¼1
ð6Þ
represents, as we can see, light wave propagation from the incident medium ‘‘in”, through the considered structure, to the substrate medium ‘‘out” and it is a product of all transmission and propagation matrices of individual layers. Transmittance T and reflectance R of the whole multilayer structure can be expressed in terms of the G elements as [30, 31]: nout cos qout 1 2 ; ð7Þ TG ¼ nin cos qin G 11 2 G 21 ð8Þ RG ¼ : G 11 For nin ¼ nout , the matrix G is unimodular (the definition and selected properties of unimodular matrix are given in Appendix A). In this case, the transmittance can be written in the following form (see Appendix D): 2 1 4 T G ¼ ¼ ; ð9Þ 2 G 11 js G j þ jtG j2
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers Table 1.
259
Generalized Fibonacci chains of types (M = 1, N = 1), (M = 2, N = 1) and (M = 1, N = 2) for the five initial generations.
L
SLþ1 ¼ SL SL1
L
SLþ1 ¼ S2L SL1
L
SLþ1 ¼ SL S2L1
0 1 2 3 4
B A AB ABA ABAAB
0 1 2 3 4
B A AAB AABAABA AABAABAAABAABAAAB
0 1 2 3 4
B A ABB ABBAA ABBAAABBABB
where the unimodular matrix
where tðGÞ ¼ tG ¼ G 11 þ G 22
ð10Þ
is the trace of G matrix and sðGÞ ¼ s G ¼ G 11 G 22
ð11Þ
will be called diagonal antitrace. We introduce additional quantities, essential in our formalism: 1. Antisymmetric nondiagonal antitrace VðGÞ ¼ VG ¼ G 21 G 12 :
ð12Þ
This quantity has been defined for the first time by Dulea, Severin and Riklund [24] and called antitrace. 2. Symmetric nondiagonal antitrace hðGÞ ¼ hG ¼ G 21 þ G 12 :
ð13Þ
Definitions and basic properties of antitraces are presented in Appendix A.
ð14Þ
where S0 ¼ B, S1 ¼ A, M is the number of Lth chain repetitions, N – the number of ðL 1Þst chain repetitions, L – generation index, with M; N 2 N, L 2 N [ f0g. The formula (14) is equivalent to the substitution rule A 7! AM BN :
ð15Þ
Table 1 shows five initial generations of selected generalized Fibonacci chains. The OGFS transfer matrix G Lþ1 of ðL þ 1Þst generation can be written in the form G Lþ1 ¼ Din; A QLþ1 DA; out ;
ð17Þ
is the transfer matrix of the generalized Fibonacci multilayer structure placed between two A media, with Q0 ¼ DAB P B DBA and Q1 ¼ P A . To simplify the formulas, from here on we will use the following notation: T Lþ1 ¼ T ðQLþ1 Þ ;
RLþ1 ¼ RðQLþ1 Þ ;
tLþ1 ¼ tðQLþ1 Þ ;
sLþ1 ¼ sðQLþ1 Þ ;
VLþ1 ¼ VðQLþ1 Þ ;
hLþ1 ¼ hðQLþ1 Þ :
3.1. OGFS immersed in A-type medium We show in Appendix D, that the transmittance T Lþ1 of OGFS placed between two A media can be expressed as T ðG Lþ1 Þ ¼ T Lþ1 ¼
tLþ1 ¼
In this paper we discuss aperiodic optical generalized Fibonacci superlattices. These multilayer media consist of two different types of dielectric thin films: A and B layers characterized respectively by the refractive indices nA and nB as well as the film thicknesses dA and dB . The order of A and B layers along given direction is defined by the following concatenation rule [5]
B 7! A ;
L2
4 jsLþ1 j2 þ jtLþ1 j2
:
ð18Þ
The values of trace tLþ1 and diagonal antitrace sLþ1 fulfill the nonlinear dynamical maps
3. Transmittance of optical generalized Fibonacci superlattices
N SLþ1 ¼ SM L SL1 ;
N QLþ1 ¼ QM L QL1 ;
ð16Þ
uM ðtL Þ uN ðtL1 Þ uM ðtL1 Þ ½uMþ1 ðtL1 Þ tL uNþ1 ðtL2 Þ þ uN1 ðtL2 Þ ½uMþ1 ðtL Þ uN1 ðtL1 Þ þ uM1 ðtL Þ uNþ1 ðtL1 Þ ;
s Lþ1
(19) uMþ1 ðtL1 Þ uN ðtL1 Þ ¼ uM ðtL Þ uNþ1 ðtL1 Þ sL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ s L1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ sL2 ; uM ðtL1 Þ
ð20Þ
where uj ðyÞ is modified Chebyshev polynomial (see Appendix B). The trace map (19) has been derived by Kola´rˇ and Ali [16]. The formula (20) is a new result (see Appendix C). We note that in order to calculate tLþ1 we must know traces of matrices of three previous generations OGFS: tL , tL1 , tL2 . To calculate the diagonal antitrace sLþ1, we additionally need sL , s L1 , sL2 . Explicit formulas for three initial traces and diagonal antitraces are shown in tables 2 and 3.
260
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
Table 2. Traces of matrices Q0 = DABPBDBA, Q1 = PA and N Q2 = QM 1 Q0 . L
tL
0
2 cos jB
1
2 cos jA 1 ½2 cos ðjA þ jB Þ tAB tBA 2 2 cos ðjA jB Þ uM1 ðt1 Þ uNþ1 ðt0 Þ rAB uMþ1 ðt1 Þ uN1 ðt0 Þ
uM ðt1 Þ uN ðt0 Þ 2
Table 3. Diagonal antitraces of matrices Q0 = DABPBDBA, N Q1 = PA and Q2 = QM 1 Q0 . L
sL
0
2 2ð1 þ rAB Þ i sin jB tAB tBA
1
2i sin jA
L
VL
0
i
1
0
2
V0 ½uM ðt1 Þ uN ðt0 Þ cos ðjA Þ uM1 ðt1 Þ uN ðt0 Þ
2 4rAB sin jB tAB tBA
The formulas for initial nondiagonal antisymmetric antitraces are presented in table 4.
3.3. OGFS placed between two homogenous media with arbitrary refractive indices In this case the OGFS transmittance can be expressed in the following form:
2i ½sin ðjA þ jB Þ tAB tBA 2 sin ðjA jB Þ uM ðt1 Þ uN1 ðt0 Þ s1 rAB uM1 ðt1 Þ uN ðt0 Þ s 0
uM ðt1 Þ uN ðt0 Þ 2
Table 4. Nondiagonal antisymmetric antitraces of matrices N Q0 = DABPBDBA, Q1 = PA and Q2 = QM 1 Q0 .
T ðG Lþ1 Þ ¼ T ðW Lþ1 Þ ¼
4 jsðW Lþ1 Þj2 þ jtðW Lþ1 Þj2
;
ð25Þ where W ¼ ðdet G Lþ1 Þ1=2 G Lþ1 ;
3.2. OGFS immersed in arbitrary medium The transmittance T ðG Lþ1 Þ of the generalized Fibonacci multilayer structure placed between two identical media with the refractive index nin ¼ nout ¼ n0 6¼ nA , can be expressed as 4 T ðG Lþ1 Þ ¼ ; ð21Þ 2 jsðG Lþ1 Þj þ jtðG Lþ1 Þj2 where the trace tðG Lþ1 Þ of the unimodular transfer matrix G Lþ1 is given by ð22Þ tðG Lþ1 Þ ¼ tLþ1 ; and the diagonal antitrace sðG Lþ1 Þ of the transfer matrix G Lþ1 2 1 þ r0A 2r0A sðG Lþ1 Þ ¼ s Lþ1 þ V : ð23Þ t0A tA0 t0A tA0 Lþ1 The derivation of (22) and (23) is presented in Appendix E. The diagonal antitrace sðG Lþ1 Þ (23) depends on diagonal antitrace sLþ1 of the matrix QLþ1 and nondiagonal antisymmetric antitrace VLþ1 of the matrix QLþ1 . The trace tLþ1 and the diagonal antitrace s Lþ1, occuring in (21)–(23), satisfy dynamical maps (19) and (20). It can be shown (Appendix C) that VLþ1 also fulfills nonlinear map uMþ1 ðtL1 Þ uN ðtL1 Þ VLþ1 ¼ uM ðtL Þ uNþ1 ðtL1 Þ VL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ VL1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ VL2 : uM ðtL1 Þ
ð24Þ
ð26Þ
is unimodular matrix with the determinant det G ¼
nout cos qout : nin cos qin
It can be shown that the trace tðW Lþ1 Þ and diagonal antitrace sðW Lþ1 Þ of matrix W Lþ1 are given by (see Appendix F): tðW Lþ1 Þ ¼
ð1 þ rin; A rA; out Þ tLþ1 þ ðrin; A þ rA; out Þ hLþ1 pffiffiffiffiffiffiffiffiffiffiffiffi ; tin; A tA; out det G ð27Þ
sðW Lþ1 Þ ¼
ð1 rin; A rA; out Þ sLþ1 þ ðrin; A rA; out Þ VLþ1 pffiffiffiffiffiffiffiffiffiffiffiffi : tin; A tA; out det G ð28Þ
The quantities tLþ1, sLþ1 and VLþ1 satisfy nonlinear dynamical maps (19), (20) and (24) presented in previous subsections. Symmetric nondiagonal antitrace hLþ1 also fulfills dynamical map uMþ1 ðtL1 Þ uN ðtL1 Þ hLþ1 ¼ uM ðtL Þ uNþ1 ðtL1 Þ hL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ hL1 uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ hL2 : þ uM ðtL1 Þ
ð29Þ
Tables 2–5 contain explicit formulas for initial traces and antitraces.
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers Nondiagonal symmetric antitrace of matrices Q0, Q1
Table 5. and Q2. L
hL
0
0
1
0
2
uM ðt1 Þ uN ðt0 Þ
4rAB sin ðjA Þ sin ðjB Þ tAB tBA
Let us note that the trace (10) and introduced antitraces (11)–(13) of unimodular matrix are not independent quantities. They fulfill indentity (see subsection A.5) t2Lþ1 h2Lþ1 þ V2Lþ1 s2Lþ1 ¼ 4 :
ð30Þ
Using (30) we can express, for example, the symmetric nondiagonal antitrace hLþ1 as a function of tLþ1, sLþ1 and VLþ1 : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð31Þ hLþ1 ¼ V2Lþ1 s2Lþ1 þ t2Lþ1 4 : Unfortunately, the ambiguity of the square root does not allow to incorporate the identity (30) into the developed formalism. Therefore we are forced to use a set of four nonlinear dynamical maps given by (19), (20), (24) and (29). We point out that all considered antitraces (20), (24) and (29) have the same mathematical properties (Appendix A). Namely, they satisfy dynamical maps of the identical form uMþ1 ðtL1 Þ uN ðtL1 Þ aLþ1 ¼ uM ðtL Þ uNþ1 ðtL1 Þ aL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ aL1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 ; uM ðtL1 Þ
ð32Þ
where aj 2 fs j ; Vj ; hj g. The only differences exist in the initial values a0, a1 and a2 of the antitraces shown in tables 3–5.
Three interesting physical situations, described in section 1, have been considered. The proposed approach can be applied in design of optical devices or instruments like filters, optic switches or resonators. In the next paper we apply the presented formalism and show that it is useful in optical engineering. Finally let us note that we have limited our considerations to incidence angles q < qcrit, where qcrit is the critical angle at which total internal reflection occurs. However, the presented formalism can also be applied to analysis of light tunneling phenomenon in ultrathin dielectric multilayers (d l). We have neglected light dispersion [32], but this phenomenon can be easily incorporated into our approach. This is of great importance for composite materials with negative refractive indices, called also lefthanded materials with negative permittivity and permeability [33]. It should be noted, that the discussed structures can be treated as quasi-one-dimensional aperiodic photonic crystals [34]. In this context, a very interesting problem is the photonic band structure of these multilayered systems [35].
Appendix This Appendix contains basic mathematical definitions and theorems with proofs as well as derivations of the dynamical maps.
A. Selected properties of 2 2 matrices Here we present definitions and discuss mathematical properties of 2 2 complex matrices, which are of great importance for our studies. A.1. Unimodular matrix Definition A.1: Nonsingular complex matrix W of dimension 2 2 we define as unimodular matrix if det W ¼ 1 :
4. Conclusions and final remarks In conclusion, we have proposed and developed –– in the framework of matrix formulation and dynamical maps technique –– a new approach to polarized light propagation in optical generalized Fibonacci superlattices being one-dimensional aperiodic structures. We have introduced auxiliary objects: diagonal antitraces, symmetric and antisymmetric nondiagonal antitraces of characteristic matrices, and investigated their mathematical properties. The explicit forms of dynamical maps for these objects have been derived. We have shown that all three kinds of antitraces satisfy identical dynamical maps. Transmittance of the studied aperiodic dielectric multilayers placed between two homogenous media has been expressed in terms of traces and antitraces of characteristic matrices.
261
ð33Þ
Corollary A.2: Product of unimodular matrices is also the unimodular matrix, which is a simple consequence of the definition. Theorem A.3: Suppose that C is non-singular complex matrix of dimension 2 2. Then W ¼ ðdet CÞ1=2 C
ð34Þ
is a unimodular matrix. Relation (34) can be verified by direct calculations. A.2. Diagonal antitrace of matrix Definition A.4: Assume that C is nonsingular complex matrix of dimension 2 2. The function sðCÞ ¼ s C ¼ C11 C22 : is called diagonal antitrace sðCÞ of the matrix C.
ð35Þ
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A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
Theorem A.5: Suppose that B and C are nonsingular complex 2 2 matrices. Then the following relations are satisfied: (i)
sðBCÞ þ sðCBÞ ¼ sB tC þ sC tB ;
(ii) sðaB þ bCÞ ¼ as B þ bs C
ð36Þ
for a; b 2 R ; ð37Þ
1
(iii) sðB Þ ¼ sB =det B :
ð38Þ
Theorem A.9: Assume that B and C are nonsingular complex 2 2 matrices. Then the following relations are fulfilled: (i)
hðBCÞ þ hðCBÞ ¼ hB tC þ hC tB ;
ð44Þ
for a; b 2 R ; ð45Þ
(ii) hðaB þ bCÞ ¼ ahB þ bhC 1
(iii) hðB Þ ¼ hB =det B :
ð46Þ
Proof. Analogous to theorem A.5.
Proofs. (i) If we take matrices B and C in the form B11 B12 C11 C12 B¼ ; C¼ B21 B22 C21 C22
A.5. Identity Lemma A.10: The trace tC , diagonal antitrace sC , antisymmetric nondiagonal antitrace VC and symmetric nondiagonal antitrace hC of unimodular matrix C fulfill the relation
then sðBCÞ þ sðCBÞ ¼ B11 C11 þ B12 C21 B22 C22 B21 C12 þ C11 B11
t2C h2C þ V2C s2C ¼ 4 :
þ C12 B21 C22 B22 C21 B12
ð47Þ
The identity (47) can be proved using definitions (10)– (13) and unimodularity of matrix C.
¼ 2B11 C11 2B22 C22 ¼ B11 C11 þ B22 C11 B11 C22 B22 C22 þ B11 C11 B22 C11 þ B11 C22 B22 C22
B. Modified Chebyshev polynomial
¼ ðB11 þ B22 ÞðC11 C22 Þ þ ðC11 þ C22 ÞðB11 B22 Þ ¼ tB s C þ tC s B : (ii) This property is obvious, as s is a linear combination of matrix elements. (iii) The inverse matrix B1 has the form 1 B22 B12 1 B ¼ det B B21 B11 and sðB1 Þ ¼
ðB22 B11 Þ sB ¼ : det B det B
Definition A.6: Suppose that C is nonsingular complex 2 2 matrix. Antisymmetric nondiagonal antitrace VðCÞ of the matrix C is the function ð39Þ
Theorem A.7: Assume that B and C are nonsingular complex 2 2 matrices. Then the following relations are fulfilled: ð40Þ (i) VðBCÞ þ VðCBÞ ¼ VB tC þ VC tB ; (ii) VðaB þ bCÞ ¼ aVB þ bVC for a; b 2 R ; ð41Þ (iii) VðB1 Þ ¼ VB =det B :
where m is an integer. The modified Chebyshev polynomials uj ðxÞ are related to Chebyshev polynomials of the first Sj ðyÞ and second kind Uj ðyÞ as follows [16, 36]: um ðxÞ ¼ Sm1 ðxÞ ¼ Um1 ðx=2Þ for m > 1 :
A.3. Antisymmetric nondiagonal antitrace of matrix
VðCÞ ¼ VC ¼ C21 C12 :
Definition B.1: Polynomial of the complex argument x is called modified Chebyshev polynomial if 8 0 for m < 0 ; < 1 for m ¼ 0 ; ð48Þ umþ1 ðxÞ ¼ : xum ðxÞ um1 ðxÞ for m > 0 ;
ð42Þ
Theorem B.2: The modified Chebyshev polynomials um ðxÞ satisfy the identity [16] u2m ðxÞ umþ1 ðxÞ um1 ðxÞ ¼ 1
for m ¼ 1; 2; 3; . . . ð49Þ
which can be proven by mathematical induction using (48). Theorem B.3: (Abele´s Theorem) [30] Let the matrix W of dimension 2 2 is unimodular. Then W m ¼ um ðtW Þ W um1 ðtW Þ I ;
ð50Þ
where tW is trace of matrix W and I – unit matrix.
The way of proving this theorem is analogous to one presented for theorem A.5.
C. Dynamical maps of antitraces for OGFS
A.4. Symmetric nondiagonal antitrace
Let aj represents one of three antitraces introduced in section 2, i.e.,
Definition A.8: Suppose that C is nonsingular complex matrix of dimension 2 2. Symmetric nondiagonal antitrace hðCÞ of the matrix C is the function: hðCÞ ¼ hC ¼ C21 þ C12 :
ð43Þ
aj 2 fs j ; Vj ; hj g : Below we present the explicit derivation of dynamical maps for aj .
263
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
From Abele´s theorem QM L ¼ uM ðtL Þ QL uM1 ðt L Þ I ; QN L1 ¼ uN ðt L1 Þ QL1 uN1 ðt L1 Þ
We further insert to (53) the above expression and obtain I:
aðQL QL1 Þ ¼ aL tL1 þ aL1 tL
After substituting the above into N aLþ1 ¼ aðQLþ1 Þ ¼ aðQM L QL1 Þ ; we obtain aLþ1 ¼ uM ðtL Þ uN ðtL1 Þ aðQL QL1 Þ
ð51Þ
where we have taken into acount that aj ðIÞ ¼ 0. Using the properties (36), (40) and (44) we find out aðQL QL1 Þ ¼ aL tL1 þ aL1 tL aðQL1 QL Þ ; ð53Þ and then we calculate
¼ uM ðtL Þ uN ðtL1 Þ aL tL1 þ aL1 tL
þ uM ðtL Þ uN ðtL1 Þ aL1 tL
uMþ1 ðtL1 Þ uN1 ðtL2 Þ aL1 ð54Þ
as well as aL ¼ uM ðtL1 Þ uN ðtL2 Þ aðQL1 QL2 Þ
uM1 ðtL Þ uN ðtL1 Þ aL1 þ uM ðtL Þ uN ðtL1 Þ uMþ1 ðtL1 Þ uN ðtL2 Þ aL þ aL2 uM ðtL1 Þ uM ðtL1 Þ ¼ uM ðtL Þ ½uN ðtL1 Þ tL1 uN1 ðtL1 Þ aL
uM ðtL1 Þ uN1 ðtL2 Þ aL1 ð55Þ
We obtain aðQL1 QL2 Þ from (55) and insert into (54)
þ uN ðtL1 Þ ½uM ðtL Þ tL uM1 ðtL Þ aL1
uMþ1 ðtL1 Þ uM ðtL Þ uN ðtL1 Þ aL uM ðtL1 Þ
þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 uM ðtL1 Þ
aðQL1 QL Þ
þ uM1 ðtL1 Þ uN ðtL2 Þ aL2 uMþ1 ðtL1 ÞuN1 ðtL2 ÞaL1 uM ðtL1 ÞuN ðtL2 ÞaL2 1 ½uMþ1 ðtL1 Þ aL ¼ uM ðtL1 Þ þ uMþ1 ðtL1 Þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 þ uMþ1 ðtL1 Þ uM1 ðtL1 Þ uN ðtL2 Þ aL2 uMþ1 ðtL1 Þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 u2M ðtL1 Þ uN ðtL2 Þ aL2 1 ½uMþ1 ðtL1 Þ aL ¼ uM ðtL1 Þ þ uMþ1 ðtL1 Þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 uMþ1 ðtL1 Þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 ½uMþ1 ðtL1 Þ uM1 ðtL1 Þ uN ðtL2 Þ aL2 þ u2M ðtL1 Þ uM ðtL1 Þ uMþ1 ðtL1 ÞaL uN ðtL2 ÞaL2 ; ¼ uM ðtL1 Þ where the indentity (49) has been applied.
¼ uM ðtL Þ uN ðtL1 Þ aL tL1 uM ðtL Þ uN1 ðtL1 Þ aL
¼ uMþ1 ðtL1 Þ uN ðtL2 Þ aðQL1 QL2 Þ
uMþ1 ðtL1 Þ ½aL þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 uM ðtL1 Þ
uMþ1 ðtL1 Þ uN ðtL2 Þ aL þ aL2 uM ðtL1 Þ uM ðtL1 Þ
uM ðtL Þ uN1 ðtL1 Þ aL uM1 ðtL Þ uN ðtL1 Þ aL1
N aðQL1 QL Þ ¼ aðQMþ1 L1 QL2 Þ
¼
ð56Þ
aLþ1
uM1 ðtL Þ uN ðtL1 Þ aL1 ;
uM1 ðtL1 Þ uN ðtL2 Þ aL2 :
uN ðtL2 Þ aL2 : uM ðtL1 Þ
After substituting (56) into (52) we find out, using (48), the final form of dynamical map:
uM ðtL Þ uN1 ðtL1 Þ aL
uM ðtL1 Þ uN ðtL2 Þ aL2 ;
þ
uMþ1 ðtL1 Þ aL uM ðtL1 Þ
¼ uM ðtL Þ uNþ1 ðtL1 Þ aL þ uMþ1 ðtL Þ uN ðtL1 Þ aL1
uMþ1 ðtL1 Þ uM ðtL Þ uN ðtL1 Þ aL uM ðtL1 Þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 uM ðtL1 Þ uMþ1 ðtL1 Þ uN ðtL1 Þ ¼ uM ðtL Þ uNþ1 ðtL1 Þ aL uM ðtL1 Þ þ
þ uMþ1 ðtL Þ uN ðtL1 Þ aL1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 : uM ðtL1 Þ
D. Transmittance of mutilayer medium with unimodular characteristic matrix Theorem D.1: Suppose that characteristic matrix W of the system is unimodular. Then the transmittance of multilayer system can be expressed in the form TW ¼
4 2
jtW j þ jsW j2
:
ð57Þ
264
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
Finally, applying the definition (35), we obtain diagonal antitrace
Proof. From equation (9) we obtain TW ¼
1 W11 W22
sG ¼ G 11 G 22 1 2 ¼ ðQ11 þ r0A Q21 r0A Q12 r0A Q22 t0A tA0 2 þ r0A Q11 þ r0A Q21 r0A Q12 Q22 Þ 1 2 ½ðQ11 Q22 Þ þ r0A ðQ11 Q22 Þ ¼ t0A tA0 þ 2r0A ðQ21 Q12 Þ 1 2 ½ð1 þ r0A Þ sQ þ 2r0A VQ ; ¼ t0A tA0
1 ¼1 4 ½2<ðW11 Þ þ 2i=ðW11 Þ ½2<ðW11 Þ 2i=ðW11 Þ 4 4 ¼ ; ¼ 2 2 2 j2<ðW11 Þj þ j2i=ðW11 Þj jtW j þ jsW j2 where the following relations, fulfilled by elements of the characteristic matrix, have been taken into account [30, 31]: W21 ¼ W *12 ;
W22 ¼ W *11 :
We note that the trace tW and diagonal antitrace s W of the matrix W are equal respectively to tW ¼ W11 þ W22 ¼ W11 þ W *11 ¼ 2<ðW11 Þ ;
F. Non-unimodular characteristic matrix G
sW ¼ W11 W22 ¼ W11 W *11 ¼ 2i=ðW11 Þ :
E. Characteristic matrix G of multilayer system immersed in medium with arbitrary refractive index Transmission of multilayer system placed between two media with arbitrary refractive index n0 is given by TG ¼
4
because the characteristic matrix G is unimodular. The matrix G can be written as Q12 Q22
1
tA0
1
rA0
rA0
1
;ð58Þ
¼ tðD1 0A D0A QÞ ¼ t Q : In order to calculate the diagonal antitrace of matrix G we have to derive its elements: 1 2 ðQ11 þ r0A Q21 r0A Q12 r0A Q22 Þ; t0A tA0 ð59Þ 1 2 ¼ ðr0A Q11 r0A Q21 þ Q12 þ r0A Q22 Þ ; t0A tA0 ð60Þ 1 2 ¼ ðr0A Q11 þ Q21 r0A Q12 þ rA0 Q22 Þ ; t0A tA0 ð61Þ
G 11 ¼
G 22 ¼
ð63Þ
with nout cos qout : nin cos qin
G ¼ ðdet GÞ1=2 W ;
tG ¼ tðD0A QDA0 Þ ¼ tðDA0 D0A QÞ
G 21
W ¼ ðdet GÞ1=2 G ;
Since
where Q represents multilayer system placed between two A-type medium. The trace tG is equal to the trace of matrix Q
G 12
Non-unimodular characteristic matrix G can be transformed into unimodular one W with the help of theorem (A.3)
det G ¼
js G j2 þ jtG j2
G ¼ D0A QDA0 Q11 1 r0A 1 ¼ t0A r0A 1 Q21
which corresponds to formula (23) used in section 3.2.
1 2 ðr0A Q11 rA0 Q21 þ r0A Q12 þ Q22 Þ : t0A tA0 ð62Þ
ð64Þ
the transmittance of OFGS can be written in the following form 1 2 : ð65Þ TG ¼TW ¼ W11 Proof. 2 1 T G ¼ det G G 11 2 1 2 1 ¼TW: ¼ det G ¼ ðdet GÞ1=2 W11 W11 As a consequence of unimodularity of matrix W , the transmittance is given by TW ¼
4 2
jtW j þ jsW j2
:
ð66Þ
Below we present the derivation of tW and sW . The characteristic matrix has the form 1 rin; A 1 G ¼ Din; A QDA; out ¼ tin; A rin; A 1 1 rA; out Q11 Q12 1 ; ð67Þ 1 Q21 Q22 tA; out rA; out
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
where unimodular matrix Q represents multilayer system placed between two A-type media. Then elements of matrix W are given by W11 ¼ W12
[7]
rA; out Q11 þ rin; A rA; out Q21 þ Q12 þ rin; A Q22 pffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ tin; A tA; out det G
W21 ¼ W22
Q11 þ rin; A Q21 þ rA; out Q12 þ rin; A rA; out Q22 pffiffiffiffiffiffiffiffiffiffiffiffi ; tin; A tA; out det G
rin; A Q11 þ Q21 þ rin; A rA; out Q12 þ rA; out Q22 pffiffiffiffiffiffiffiffiffiffiffiffi ; tin; A tA; out det G
[8]
rin; A rA; out Q11 þ rA; out Q21 þ rin; A Q12 þ Q22 pffiffiffiffiffiffiffiffiffiffiffiffi : ¼ tin; A tA; out det G
Therefore the trace and diagonal antitrace of the matrix W can be written as tW ¼
ð1 þ rin; A rA; out Þ tQ þ ðrin; A þ rA; out Þ hQ pffiffiffiffiffiffiffiffiffiffiffiffi ; tin; A tA; out det G
ð68Þ
sW ¼
ð1 rin; A rA; out Þ s Q þ ðrin; A rA; out Þ VQ pffiffiffiffiffiffiffiffiffiffiffiffi : tin; A tA; out det G
ð69Þ
[9]
Acknowledgement. This work was supported in part by European Network of Excellence for Micro-Optics NEMO, contract No.: 003897.
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Polarized light transmission through generalized Fibonacci multilayers. II. Numerical results Agnieszka Klauzer-Kruszyna, Włodzimierz Salejda, Michał H. Tyc Institute of Physics, Wrocław University of Technology, Wybrzez˙e Wyspian´skiego 27, 50-370 Wroclaw, Poland
Abstract: The theory of polarized light propagation in generalized Fibonacci multilayers, recently developed in the framework of matrix formulation and dynamical maps technique, is applied. Transmittance of studied systems is numerically calculated and presented in gray scale figures. The main tendencies in dependences of transmittance on model parameters are presented and discussed. We find a strong dependence of transmittance on refractive indices of surrounding media. We show that the proposed approach can be useful in optical engineering. Key words: Fibonaccian superlattices – polarized light transmittance – multilayer systems
1. Introduction In the previous paper [1], the new approach to propagation of polarized light in dielectric generalized Fibonacci multilayers has been proposed. By means of the matrix formulation [2] and dynamical maps technique [3], the transmittance of generalized Fibonacci multilayers placed between two homogeneous media has been expressed in terms of traces and antitraces of characteristic matrices. The dynamical maps for diagonal antitraces, symmetric and antisymmetric nondiagonal antitraces of characteristic matrices have been derived. In this paper, we apply the developed approach [1] to study the dependence of transmittance of optical generalized Fibonacci superlattices on model parameters: polarization type, wavelength, incidence angle, refractive indices of layers and surrounding media, thicknesses of layers, as well as the type of Fibonacci superlattice described by generation number L and concatenation multiplicities M, N. We show that transmittance characteristics strongly depend on refractive indices of external media. We also discuss how aperiodicity of multilayers, i.e., neither disordered nor periodic distribution of two different dielectric films along given direction, influences polarized light transmission properties of studied systems. We hope that intentional aperiodicity – a new degree of freedom in the studied systems – enables design of new multilayer optical devices with functionality not available to conventional periodic systems.
The paper is organized as follows. In the next section, the explicit formulas for polarized light transmittance of studied systems are given. Section 3 contains numerical results. Final conclusions and remarks are presented in section 4. The Appendix contains formulas for trace and antitraces in case of a periodic structure.
2. Transmittance of generalized Fibonacci multilayer placed between two homogeneous media An optical generalized Fibonacci multilayer (OGFM) is build from dielectric thin films of two different types, denoted by A and B, according to the following inflation scheme: N SLþ1 ¼ SM L SL1 ;
ð1Þ
where S0 ¼ B, S1 ¼ A, M and N are concatenation multiplicities, with M; N 2 N; L 2 N [ f0g defines the generation number of multilayer. The light propagation through a Fibonaccian multilayer structure placed between two arbitrary homogenous media ‘in’ and ‘out’ is described, in the framework of transfer matrix approach, by the characteristic matrix [1] G Lþ1 ¼ Din; A QLþ1 DA; out ;
ð2Þ
where the unimodular matrix N QLþ1 ¼ QM L QL1
ðL 2Þ
ð3Þ
is the characteristic matrix of OGFM placed between two A media, with Q0 ¼ DAB P B DBA and Q1 ¼ P A . We have shown [1] that transmittance T of the ðL þ 1Þst generation of OGFM can be written in the following forms: 1. for nin ¼ nout ¼ nA , T ðG Lþ1 Þ ¼ T ðQLþ1 Þ ¼
4 2
jsLþ1 j þ jtLþ1 j2
;
ð4Þ
where tLþ1 and s Lþ1 are the trace and diagonal antitrace of matrix QLþ1 , respectively;
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A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
2. for nin ¼ nout ¼ n0, T ðG Lþ1 Þ ¼
4 jsðG Lþ1 Þj2 þ jtðG Lþ1 Þj2
;
tðG Lþ1 Þ ¼ tLþ1 ; sðG Lþ1 Þ ¼
ð5Þ ð6Þ
2 1 þ r0A 2r0A s Lþ1 þ V ; t0A tA0 t0A tA0 Lþ1
ð7Þ
where tðG Lþ1 Þ and sðG Lþ1 Þ are trace and diagonal antitrace of characteristic matrix (2), respectively, and VLþ1 is antisymmetric nondiagonal antitrace of matrix (3); 3. for nin 6¼ nout, TG ¼TW ¼
4 js W j2 þ jtW j2
tW ¼ ðdet GÞ1=2
;
1 ½ð1 þ rin; A rA; out Þ tLþ1 tin; A tA; out
þ ðrin; A þ rA; out Þ hLþ1 ; sW ¼ ðdet GÞ1=2
ð8Þ
ð9Þ
1 ½ð1 rin; A rA; out Þ s Lþ1 tin; A tA; out
þ ðrin; A rA; out Þ VLþ1 ;
ð10Þ
nout cos qout ; nin cos qin is symmetric nondiagonal antitrace of matrix
where W ¼ ðdet GÞ1=2 G with det G ¼ hLþ1 (3).
The quantities tLþ1 and aLþ1 , where aLþ1 2 s Lþ1 ; VLþ1 ; hLþ1 , fulfill the nonlinear dynamical maps tLþ1 ¼
uM ðtL Þ uN ðtL1 Þ ½uMþ1 ðtL1 Þ tL uNþ1 ðtL2 Þ uM ðtL1 Þ þ uN1 ðtL2 Þ ½uMþ1 ðtL Þ uN1 ðtL1 Þ
aLþ1
ð11Þ þ uM1 ðtL Þ uNþ1 ðtL1 Þ ; uMþ1 ðtL1 Þ uN ðtL1 Þ ¼ uM ðtL Þ uNþ1 ðtL1 Þ aL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ aL1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 ; uM ðtL1 Þ
ð12Þ
here uj ðyÞ denotes the modified Chebyshev polynomial [1].
3. Numerical results We have performed extensive numerical studies of propagation of polarized light through OGFM in order to find out dependence of transmittance on model parameters: 1. light polarization (s and p), 2. wavelength l 2 ð300 nm; 700 nmÞ, 3. incidence angle q 2 ½0; p=2Þ, 4. layer thicknesses dA ; dB 2 ð250 nm; 100 nmÞ,
5. concatenation multiplicities M, N and generation number L, 6. refractive indices of layers nA ; nB 2 ð1:0; 3:0Þ, 7. refractive indices of surrounding media nin and nout (all refractive indices are chosen to guarantee that total internal refraction does not occur on any interfaces), 8. dispersion curves. In our numerical simulations, we have taken the optical phase length in the following form: 2p cos ql ; ð13Þ jl ¼ d~l nl l~ where l~ ¼ l=l0 and d~l ¼ dl =l0 with l0 ¼ 100 nm. It is straightforward to show that the initial values of traces tL, diagonal antitraces sL, antisymmetric nondiagonal antitrace VL and symmetric nondiagonal antitrace hL are given by expressions t0 ¼ 2 cos jB ; t2 ¼
t1 ¼ 2 cos jA ;
uM ðt1 Þ uN ðt0 Þ ½2 cos ðjA þ jB Þ tAB tBA 2 cos ðjA jB Þ 2rAB uM1 ðt1 Þ uNþ1 ðt0 Þ uMþ1 ðt1 Þ uN1 ðt0 Þ;
s0 ¼ i
2 2ð1 þ rAB Þ sin jB ; tAB tBA
s1 ¼ 2i sin jA ;
2uM ðt1 Þ uN ðt0 Þ ½sin ðjA þ jB Þ tAB tBA 2 sin ðjA jB Þ uM ðt1 Þ uN1 ðt0 Þ s 1 rAB uM1 ðt1 Þ uN ðt0 Þ s0 ;
s2 ¼ i
V0 ¼ i
4rAB sin jB ; tAB tBA
V1 ¼ 0 ;
V2 ¼ V0 ½uM ðt1 Þ uN ðt0 Þ cos ðjA Þ uM1 ðt1 Þ uN ðt0 Þ; h0 ¼ 0 ;
h1 ¼ 0 ;
h2 ¼ uM ðt1 Þ uN ðt0 Þ
4rAB sin ðjA Þ sin ðjB Þ : tAB tBA
The numerically obtained transmittances T are presented in gray scale maps (figs. 2, 3, 5–8, 10–18), where white pixels correspond to transmittance equal to 1 and black ones – to 0. Figs. 2 and 3 present polarized light transmittance maps T ðl~; qÞ for periodic (I ¼ 11) and Fibonaccian multilayers with M ¼ N ¼ 1 and L ¼ 7 at indicated model parameters. Cross-sections of these maps for fixed angle q ¼ p=12 are presented in fig. 4. The transmittance maps T ðl~; qÞ of polarized light for OGFM with ðM ¼ 2; N ¼ 1Þ and ðM ¼ 1; N ¼ 2Þ are depicted in figs. 5 and 6, respectively. Figs. 7 and 8 present the dependence of the transmittance T ðl~; qÞ on the number of unit cells I for periodic multilayers and on the generation number L for Fibonaccian ones, respectively (the compared structures have been selected so that they are built from similar
269
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers OGFM
OGFM
Fig. 1. Transmittance maps T (~ l; d~A Þ OGFM as well as periodic structure for fixed parameters: nA ¼ 1:43, nB ¼ 2:3, d~B ¼ 0:45 and q ¼ 0.
OGFM
Fig. 3. Transmittance maps T (~ l; q) of 22-layer periodic structure and 21-layer OGFM for: p-type polarization, refraction indices nA ¼ 1:43 and nB ¼ 2:3, as well as given thicknesses d~A and d~B .
number of layers). Cross-sections of these maps for q ¼ p=12 are put together in fig. 9. In the maps for p-type of polarization (figs. 3, 5–8), we can observe white horizontal stripes corresponding to maximum transmittance at Brewster angle. Figs. 10 and 11 show transmittance as a function of layer thicknesses d~A and d~B , respectively, for selected periodic and aperiodic multilayers. In fig. 10, a few vertical lines of maximum transmittance can be seen. They correspond to constructive interference of light in single thin film with refractive index nB and Table 1. Interference maxima of a single thin film with thick~ B and refraction index nB = 2.3. The first row correness D sponds to the upper and middle maps in fig. 10 and the second row – to the bottom maps. Fig. 2. Transmittance maps T (~ l; q) of 22-layer periodic structure and 21-layer OGFM for: s-type polarization, refraction indices nA ¼ 1:43 and nB ¼ 2:3, as well as given thicknesses d~A and d~B .
~ B ¼ d~B D
2 j ~ B =j 6.90 l~j ¼ 2nB D
~ B ¼ 2d~B D
4 5 6 7 8 9 j ~ B =j 6.90 5.52 4.60 3.94 3.45 3.07 l~j ¼ 2nB D
3 4.60
4 3.45
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A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
λλ
λλ
Fig. 4. Cross-sections of transmittance maps T (l~; q) (fig. 2) for Fibonaccian (top graph) and periodic (bottom graph) structures with fixed q ¼ p=12.
~ B immersed in A medium. The transmitthickness D tance maxima are observed at wavelengths given by formula ~ B =j l~j ¼ 2nB D and shown in table 1. The same phenomena occur on the transmittance maps T ðl~; nA Þ, which we do not present. In fig. 11 we observe vertical lines of high transmittance (along these lines transmittance is not equal to 1, but oscillates just below 1). Another kind of straight lines with T ¼ 1 can be seen as well; they are described by formula j ~ l; d~ðl~Þ ¼ 2nB where j ¼ 1; 2; . . . numbers the interference maxima. For example, in the upper right corner of fig. 10, we can see lines from j ¼ 2 to j ¼ 15. Let us note that these lines vanish for sufficiently small d~B , what is illustrated in fig. 11.
Fig. 5. Transmittance maps T (~ l; q) of OGFM with M ¼ 2, N ¼ 1 for: L ¼ 7, nA ¼ 1:43, nB ¼ 2:3 and given thicknesses ~ ~ dA and dB .
Transmittance as a function of l~ and nB or nA for periodic structures and OGFM is presented in fig. 12. The maps were calculated for s-type of polarization and non-zero incidence angle q. The influence of external media refractive indices on light transmittance is shown in fig. 13–18. Fig. 19 presents transmittance for OGFM (M ¼ 1, N ¼ 1, L ¼ 7), with dispersion taken into account. In this example we assume that the refractive index of B layer depends on wavelength as C2 nB ðl~Þ ¼ C1 þ 2 ; l~ where C1 and C2 are constants.
ð14Þ
Fig. 6. Transmittance maps T (l~; q) of OGFM with M ¼ 1, N ¼ 2 for: L ¼ 7, nA ¼ 1:43, nB ¼ 2:3, and given thicknesses d~A and d~B . Fig. 7. Transmittance maps T (~ l; q) of periodic structures for: nA ¼ 1:43 and nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 and given numbers of unit cells I. Fig. 8. Transmittance maps T (~ l; q) of OGFM with M ¼ 1, N ¼ 1 for: nA ¼ 1:43 and nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 and given values of genaration parameters L.
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A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
λλ
λλ
Fig. 9. Cross-sections of maps (figs. 7 and 8) for fixed q ¼ p=12. Calculations were performed for Fibonaccian (left graph) and periodic (right graph) structures for s-type of polarization.
OGFM
Fig. 10. Transmittance maps T (~ l; d~A ) of selected periodic structure and OGFM for: nA ¼ 1:43 and nB ¼ 2:3, d~B ¼ 3:0 and incidence angle q ¼ 0.
OGFM
Fig. 11. Transmittance maps T (~ l; d~B ) of selected periodic structure and OGFM for: nA ¼ 1:3 and nB ¼ 2:3, d~A ¼ 3:0 and incidence angle q ¼ 0.
OGFM
Periodic structure
OGFM
OGFM
Fig. 12. Transmittance maps T (l~; nB ) and T (~ l; nA ) of s-type of polarization 22-layer periodic structure and 21-layer OGFM, with d~A ¼ d~B ¼ 3:0, and given incidence angles. Fig. 13. Transmittance maps T (~ l; q) of periodic structure and OGFM for: s-type of polarization, d~A ¼ d~B ¼ 3:0, nA ¼ 1:43, nB ¼ 2:3 and given values of external refraction index n0 . Fig. 14. Transmittance maps T (~ l; q) of periodic structure and OGFM for: p-type of polarization, d~A ¼ d~B ¼ 3:0, nA ¼ 1:43, nB ¼ 2:3 and given values of external refraction index n0 .
OGFM
OGFM
OGFM
Fig. 15. Transmittance maps T (~ l; q) of periodic structure and OGFM for: s-type of polarization, nA ¼ 1:43, nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 as well as given external refractive indices nin and nout . Fig. 16. Transmittance maps T (~ l; q) of periodic structure and OGFM for: p-type of polarization, nA ¼ 1:43, nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 as well as given external refractive indices nin and nout . Fig. 17. Transmittance maps T (~ l; q) of 22-layer periodic structure and 21-layer OGFM for: s-type of polarization, nA ¼ 1:43, nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 as well as nin ¼ nA and given values nout .
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
275
4. Summary and final remarks
OGFM
In the present paper we have applied the new formalism [1] for studied properties of polarized light propagation in aperiodic Fibonacci-type dielectric multilayers. The result of numerical investigations can be summarized as follows:
Fig. 18. Transmittance maps T (~ l; q) of 22-layer periodic structure and 21-layer OGFM for: p-type of polarization, nA ¼ 1:43, nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 as well as nin ¼ nA and given values nout .
λ/λ 0
λ /λ 0
Fig. 19. Dispersion influence. Transmittance of OGFM for: M ¼ 1, N ¼ 1, L ¼ 7, d~A ¼ d~B ¼ 3:0, q ¼ 0, nA ¼ 1:43 as C2 lÞ ¼ C1 þ 2 , where well as nB ¼ 2:3 (black curve) and nB ð~ ~ l C1 ¼ 2:2775 and C2 ¼ 1:1025. Dispersion curve nB ð~ lÞ is shown in the figure as inset.
1. Transmittance maps for polarized light, presented in gray scale, characterize completely the filtering properties of studied systems. 2. Transmission properties of discussed structures depend on the layers arrangement. The filtering properties of periodic structures can be modified by changing thicknesses, refractive indices or number of layers (figs. 2, 3, 7). Optical generalized Fibonaccian multilayers give us new possibilities of modifying filter properties by using the special kind of aperiodicity defined by the concatenation parameters M and N. It can be seen that with increasing M the transmission channels narrow down and split up. With increasing N, the number of wavelength intervals containing transmission channels grows (figs. 5, 6). 3. For both periodic and aperiodic structures, we observe that with increasing layer thicknesses dA and dB (fig. 2, 3) or the total number of layers J (figs. 7, 8), the number of transmittance channels grows and their widths significantly drop. Remarkable differences in transmission maps of periodic and aperiodic structures appear in the hierarchy of transmittance channels and band gaps. In the case of periodic structures, the regularly spaced high transmittance channels stay in the same spectral regions, independent of the number of layers (figs. 7, 9). In the case of OGFM, with increasing thicknesses dA or dB or generation number L, the transmittance channels become more fragmented and exhibit self-similar character (figs 2, 3, 8, 9). Additional spectral regions with high transmittance channels shift with increasing dA , dB or J (figs. 4, 9). It is interesting that transmittance channels exist also in the spectral region inaccesible in the case of periodic structures (fig. 9). 4. Multilayer system built from B-layer with constant thickness ordered arbitrarily in A medium will be always perfectly transparent at wavelengths corresponding to interference maxima of the single thin film (fig. 10). This kind of transmission channels exists in the spectra of all structures in which all layers of one type (here it is B) have the same thickness. 5. Modifying external media refractive index causes remarkable changes in the filtering properties of investigated structures (figs. 13–16). The following effects can be seen in the transmission maps (if nin 6¼ nA 6¼ nout ): (i) The distribution of transmission channels and band gaps changes. In case of periodic structures (figs. 13 and 15), considerable effects ocp cur for incidence angles q > , and in case of 4
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OGFM they can be observed for arbitrary q (figs. 14 and 16). Transmission characteristics of aperiodic Fibonacci-type structures are more sensitive to boundary conditions than of periodic ones. (ii) The total transmittance decreases. (iii) The transmittance maximum at Brewster angle (p-polarization only) decays. The case when nin ¼ nA and nout is high (figs. 17, 18) is also interesting. In a periodic structure, the transmission channels decay with increasing nout . In an aperiodic structure, this effect occurs for most transmission channels, but not for all: some channels just narrow down, keeping their maximum transmittance and can be denoted as transmission windows. In this model an arbitrary dispersion relation can be taken into account. We performed calculations for weak dispersion, typical for dielectric materials in visible light range (fig. 19), and did not observed any qualitative changes in the transmittance spectrum. The developed formalism and numerical algorithm applied in a computational enviroment (available from us) can be useful in design of optical devices with desirable filtering properties (filters [4], resonators [5], included asymetric ones [6]).
Appendix Periodic structure Suppose that the unit cell of periodic structure consists of L layers: A1 A2 . . . AL . The characteristic matrix of the unit cell can be expressed as G ¼ Din; A1 Qu DA1 ; out ; where Qu ¼ P A1 DA1 ; A2 P A2 DA2 ; A3 . . . DAj1 ; Aj P Aj DAj ;Ajþ1 . . . DAL1 ; AL P AL DAL ; A1 : The unimodularity of Qu allows to express transmittance of periodic structure as a function of trace and antitraces.
The characteristic matrix of a structure composed of I unit cells is given by G ¼ Din; A1 QIu DA1 ; out : The trace and antitraces of periodic structure can be expressed in the following form: tðQI Þ ¼ uI ðtðQu ÞÞ tðQu Þ 2uI1 tðQu Þ ;
ð15Þ
VðQI Þ ¼ uI ðtðQu ÞÞ VðQu Þ ;
ð16Þ
sðQI Þ ¼ uI ðtðQu ÞÞ sðQu Þ ;
ð17Þ
hðQI Þ ¼ uI ðtðQu ÞÞ hðQu Þ :
ð18Þ
Acknowledgement. This work was supported in part by European Network of Excellence for Micro-Optics NEMO, contract No.: 003897.
References [1] Klauzer-Kruszyna A, Salejda W, Tyc MH: Polarized light transmission through generalized Fibonacci multilayers. I. Dynamical maps approach. Optik 115 (2004) 253–262 [2] Born M, Wolf E: Principles of optics. Pergamon Press, London 1968; Yeh P: Optical waves in layered media. John Wiley & Sons, New York 1988 [3] Kohmoto M, Sutherland B, Iguchi K: Localization in optics: Quasiperiodic media. Phys. Rev. Lett. 58 (1987) 2436– 2438; Dulea M, Severin M, Riklund R: Transmission of light through deterministic aperiodic non-Fibonaccian multilayers. Phys. Rev. B 42 (1990) 3680–3689; Avishai Y, Berend D, Tkachenko V: Trace maps. Int. J. Mod. Phys. B 11 (1997) 3525–3542; Yang XB, Liu YY, Fu XJ: Transmission properties of light trough the Fibonacci-class multilayers. Phys. Rev. B 59 (1999) 4545–4548; Wang X, Grimm U, Schreiber M: Trace and antitrace maps for aperiodic sequences, their extensions and applications. Phys. Rev. B 62 (2000) 14020–14031, available as arXiv: cond-mat/0005463; Wang X, Pan S, Yang G: Antitrace maps and light transmission coefficients for a generalized Fibonacci multilayers. arXiv: cond-mat/0106378 (2001) [4] Zhang Q, Li X, Shen J, Wu G, Wang J, Chen L: ZrO2 thin films and ZrO2 =SiO2 optical reflection filters deposited by sol-gel method. Materials Letters 45 (2000) 311–314; Sullivan BT, Dobrowolski JA, Clarke G, Akiyama T, Osborne N, Ranger M, Howe L, Matsumoto A, Song Y, Kikuchi K: Manufacture of complex optical multilayer filters using an automated deposition system. Vacuum 51 (1998) 647–654; Vlastou R, Fokitis E, Maltezos S, Kalliabakos G, Kokkoris M, Kossionides E: Characterization of optical UV filters using Rutherford backscattering spectroscopy. Nucl. Instr. and Meth. B 161–163 (2000) 590–594 [5] Szerling A, Wawer D, Hejduk K, Piwoski T, Wo´jcik A, Mroziewicz B, Bugajski M: Reflectance study of Si2 =Si3 N4 dielectric Bragg reflectors. Opp. Appl. 32 (2002) 523–527 [6] Nolte DD, Melloch MR, Ding Y, Dinu M, Kwolek KM, Lahiri I: Photorefractive Semiconductor Nanostructures. In Nalwa HS (Ed.): Nanostructured Materials and Nanotechnology, pp. 484–562. Academic Press, New York 2002
International Journal for Light and Electron Optics
Polarized light transmission through generalized Fibonacci multilayers: I. Dynamical maps approach Agnieszka Klauzer-Kruszyna, Włodzimierz Salejda, Michał H. Tyc Institute of Physics, Wrocław University of Technology, Wybrzez˙e Wyspian´skiego 27, 50-370 Wroclaw, Poland
Abstract: The theory of polarized light propagation in dielectric generalized Fibonacci multilayers is developed. The matrix formulation and dynamical maps technique is used. New objects: diagonal antitraces, symmetric and antisymmetric nondiagonal antitraces of characteristic matrices are introduced. Dynamical maps for these objects are derived. Transmittance for s- and p-type polarized light of the studied aperiodic multilayers placed between two homogenous media is expressed in terms of traces and antitraces of characteristic matrices. Three interesting physical situations are considered, allowing to study the influence of surrounding media on light transmission properties of Fibonacci-type multilayers. Key words: Fibonaccian superlattices – polarized light transmittance – multilayer systems
1. Introduction Since Shechtman et al. [1] had discovered quasicrystals, aperiodic crystals and quasi-one-dimensional aperiodic structures have been investigated experimentally and theoretically [2–11]. The quasi-one-dimensional aperiodic systems are multilayer structures composed of at least two different substrates (homogenous layers) A and B distributed along given direction. The A and B layers alignment is precisely defined by the substitution rules or inflation scheme [2, 3, 12, 13]. In the past, the main research efforts were focused on electron spectra properties and electron wave functions [3–5], electric conduction [6, 7], phonon spectra [8, 13], magnetic properties [9], acoustic [10] and electromagnetic waves transmission [11, 12], [14–31]. In their pioneering work [14], Kohmoto, Sutherland and Iguchi considered light propagation through an optical Fibonaccian multilayer, using dynamical maps of traces of unimodular real transfer matrices to calcu-
Received 17 February 2004; accepted 10 May 2004. Correspondence to: W. Salejda Fax: ++48-71-3283696 E-mail: [email protected]
late transmittance. The dynamical maps formalism has been introduced firstly by Kohmoto, Kadanoff and Tang [15]. Transmittance of generalized Fibonaccian superlattices, introduced by Gumbs and Ali [5], was calculated using the transfer matrix trace maps technique developed by Kola´rˇ and Ali [16] and then Iguchi [17] and applied in the papers [18–23]. This technique was extended to unimodular complex matrices [7] and modified by Dulea, Severin and Riklund [24], who introduced transfer matrix antitraces and derived socalled antitrace maps. Mathematical properties of generalized Fibonacci superlattice trace maps and some of their physical applications were presented in review papers [4, 25]. The main reason for studying dielectric two-substrate periodic, aperiodic and random multilayer system [26, 27] was the possibility of practical application in, e.g., Cantor type [28] or Fibbonaci type [29] FabryPerot resonators, optical filters and optical memories [20, 27]. In the most of published papers on the electromagnetic wave propagation through aperiodic multilayer media, it was assumed that: 1. incident wave has s polarization, 2. optical thicknesses of A and B layers are equal, 3. neither reflection nor refraction occurs on the interfaces between the system and external media. In the present paper we study, in the framework of dynamical maps technique, polarized light transmission properties of dielectric Fibonaccian superlattices placed between two dielectric media. For these structures, called Optical Generalized Fibonacci Superlattices (OGFS), we calculate in particular the transmittance as a function of light wavelength and incidence angle q. Unlike in previous published papers, we consider the propagation of the s- and p-wave incident on the structure at arbitrary angle. We give and prove, in the explicit form, dynamical trace and antitrace maps for OGFS characteristic matrices. We derive formulas for polarised light transmittance of the studied system placed between two homogenous media. We express the transmittance in terms of trace and antitraces of 0030-4026/04/115/06-257 $ 30.00/0
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A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
characteristic matrices. We discuss the following cases: 1. both external media are A-type, 2. both external media have arbitrary but identical refractive index, 3. external media have arbitrary refractive indices. The paper is organized as follows. In section 2 we recall briefly the matrix formulation of light propagation for isotropic layered media. The OGFS construction rule and derived formulas for transmittance expressed as a function of trace and antitraces maps are given in section 3. Section 4 contains final conclusions and remarks. Appendix includes basic definitions and derivation of formulas from section 2 and 3.
2. Matrix formulation of light propagation for isotropic layered media We describe the light propagation through a dielectric multilayer medium (fig. 1) consisting of J isotropic and homogeneous layers with refractive indices nj and thicknesses dj ¼ xj xj1 distributed along x-axis. The subscript j denotes refraction indices of media, j ¼ 0; 1; 2; . . . ; J; J þ 1, where j ¼ 0 corresponds to incident medium ‘‘in” and j ¼ J þ 1 – substrate medium ‘‘out”, onto which the subsequent layers are deposited. We assume that an electromagnetic plane wave of length l falls at angle qin upon the interface between the media with refractive indices nin and n1 . The amplitudes of electric field vector for incident ðþÞ ðÞ ðþÞ Ein , reflected Ein and transmitted Fout light wave through dielectric multilayer media (fig. 1) are related by formula (within the framework of transfer matrix formalism) [30, 31] " ðþÞ # " # " ðþÞ # " ðþÞ # J Ein Q Fout F R j Dj; jþ1 ¼ Din;1 ¼ G out ; ðÞ ðÞ ðÞ j¼1 Fout Fout Ein ð1Þ ðÞ
where Fout ¼ 0, and G is the characteristic matrix of layered system. Matrix P j , called jth medium propagation matrix, can be written as ij 2p 0 e j Pj ¼ cos qj ; ð2Þ ; jj ¼ dj nj 0 eijj l
y
z
Fig. 1. Dielectric multilayer structure placed between homogeneous dielectric media with refractive indices nin and nout
Fig. 2. Electric E and magnetic H field vectors of s and p waves at the interface between jth and (j + 1)st media.
and the matrix Dj; jþ1 , called transmission matrix from jth to ð j þ 1Þst medium, takes the form 1 1 rj;jþ1 Dj; jþ1 ¼ ; ð3Þ 1 tj; jþ1 rj; jþ1 where the matrix elements depend on light polarization (see fig. 2), and Fresnel transmision and reflection coefficients are given by [30, 31] 8 2nj cos qj > > for s wave ; > < nj cos qj þ njþ1 cos qjþ1 ð4Þ tj; jþ1 ¼ > 2nj cos qj > > for p wave ; : nj cos qjþ1 þ njþ1 cos qj 8 nj cos qj njþ1 cos qjþ1 > > for s wave ; > < n cos q þ n cos q j j jþ1 jþ1 ð5Þ rj;jþ1 ¼ > n cos qjþ1 njþ1 cos qj > > j for p wave : : nj cos qjþ1 þ njþ1 cos qj The characteristic matrix G " # N Q G 11 G 12 P j Dj; jþ1 G¼ ¼ Din;1 G 21 G 22 j¼1
ð6Þ
represents, as we can see, light wave propagation from the incident medium ‘‘in”, through the considered structure, to the substrate medium ‘‘out” and it is a product of all transmission and propagation matrices of individual layers. Transmittance T and reflectance R of the whole multilayer structure can be expressed in terms of the G elements as [30, 31]: nout cos qout 1 2 ; ð7Þ TG ¼ nin cos qin G 11 2 G 21 ð8Þ RG ¼ : G 11 For nin ¼ nout , the matrix G is unimodular (the definition and selected properties of unimodular matrix are given in Appendix A). In this case, the transmittance can be written in the following form (see Appendix D): 2 1 4 T G ¼ ¼ ; ð9Þ 2 G 11 js G j þ jtG j2
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers Table 1.
259
Generalized Fibonacci chains of types (M = 1, N = 1), (M = 2, N = 1) and (M = 1, N = 2) for the five initial generations.
L
SLþ1 ¼ SL SL1
L
SLþ1 ¼ S2L SL1
L
SLþ1 ¼ SL S2L1
0 1 2 3 4
B A AB ABA ABAAB
0 1 2 3 4
B A AAB AABAABA AABAABAAABAABAAAB
0 1 2 3 4
B A ABB ABBAA ABBAAABBABB
where the unimodular matrix
where tðGÞ ¼ tG ¼ G 11 þ G 22
ð10Þ
is the trace of G matrix and sðGÞ ¼ s G ¼ G 11 G 22
ð11Þ
will be called diagonal antitrace. We introduce additional quantities, essential in our formalism: 1. Antisymmetric nondiagonal antitrace VðGÞ ¼ VG ¼ G 21 G 12 :
ð12Þ
This quantity has been defined for the first time by Dulea, Severin and Riklund [24] and called antitrace. 2. Symmetric nondiagonal antitrace hðGÞ ¼ hG ¼ G 21 þ G 12 :
ð13Þ
Definitions and basic properties of antitraces are presented in Appendix A.
ð14Þ
where S0 ¼ B, S1 ¼ A, M is the number of Lth chain repetitions, N – the number of ðL 1Þst chain repetitions, L – generation index, with M; N 2 N, L 2 N [ f0g. The formula (14) is equivalent to the substitution rule A 7! AM BN :
ð15Þ
Table 1 shows five initial generations of selected generalized Fibonacci chains. The OGFS transfer matrix G Lþ1 of ðL þ 1Þst generation can be written in the form G Lþ1 ¼ Din; A QLþ1 DA; out ;
ð17Þ
is the transfer matrix of the generalized Fibonacci multilayer structure placed between two A media, with Q0 ¼ DAB P B DBA and Q1 ¼ P A . To simplify the formulas, from here on we will use the following notation: T Lþ1 ¼ T ðQLþ1 Þ ;
RLþ1 ¼ RðQLþ1 Þ ;
tLþ1 ¼ tðQLþ1 Þ ;
sLþ1 ¼ sðQLþ1 Þ ;
VLþ1 ¼ VðQLþ1 Þ ;
hLþ1 ¼ hðQLþ1 Þ :
3.1. OGFS immersed in A-type medium We show in Appendix D, that the transmittance T Lþ1 of OGFS placed between two A media can be expressed as T ðG Lþ1 Þ ¼ T Lþ1 ¼
tLþ1 ¼
In this paper we discuss aperiodic optical generalized Fibonacci superlattices. These multilayer media consist of two different types of dielectric thin films: A and B layers characterized respectively by the refractive indices nA and nB as well as the film thicknesses dA and dB . The order of A and B layers along given direction is defined by the following concatenation rule [5]
B 7! A ;
L2
4 jsLþ1 j2 þ jtLþ1 j2
:
ð18Þ
The values of trace tLþ1 and diagonal antitrace sLþ1 fulfill the nonlinear dynamical maps
3. Transmittance of optical generalized Fibonacci superlattices
N SLþ1 ¼ SM L SL1 ;
N QLþ1 ¼ QM L QL1 ;
ð16Þ
uM ðtL Þ uN ðtL1 Þ uM ðtL1 Þ ½uMþ1 ðtL1 Þ tL uNþ1 ðtL2 Þ þ uN1 ðtL2 Þ ½uMþ1 ðtL Þ uN1 ðtL1 Þ þ uM1 ðtL Þ uNþ1 ðtL1 Þ ;
s Lþ1
(19) uMþ1 ðtL1 Þ uN ðtL1 Þ ¼ uM ðtL Þ uNþ1 ðtL1 Þ sL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ s L1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ sL2 ; uM ðtL1 Þ
ð20Þ
where uj ðyÞ is modified Chebyshev polynomial (see Appendix B). The trace map (19) has been derived by Kola´rˇ and Ali [16]. The formula (20) is a new result (see Appendix C). We note that in order to calculate tLþ1 we must know traces of matrices of three previous generations OGFS: tL , tL1 , tL2 . To calculate the diagonal antitrace sLþ1, we additionally need sL , s L1 , sL2 . Explicit formulas for three initial traces and diagonal antitraces are shown in tables 2 and 3.
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A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
Table 2. Traces of matrices Q0 = DABPBDBA, Q1 = PA and N Q2 = QM 1 Q0 . L
tL
0
2 cos jB
1
2 cos jA 1 ½2 cos ðjA þ jB Þ tAB tBA 2 2 cos ðjA jB Þ uM1 ðt1 Þ uNþ1 ðt0 Þ rAB uMþ1 ðt1 Þ uN1 ðt0 Þ
uM ðt1 Þ uN ðt0 Þ 2
Table 3. Diagonal antitraces of matrices Q0 = DABPBDBA, N Q1 = PA and Q2 = QM 1 Q0 . L
sL
0
2 2ð1 þ rAB Þ i sin jB tAB tBA
1
2i sin jA
L
VL
0
i
1
0
2
V0 ½uM ðt1 Þ uN ðt0 Þ cos ðjA Þ uM1 ðt1 Þ uN ðt0 Þ
2 4rAB sin jB tAB tBA
The formulas for initial nondiagonal antisymmetric antitraces are presented in table 4.
3.3. OGFS placed between two homogenous media with arbitrary refractive indices In this case the OGFS transmittance can be expressed in the following form:
2i ½sin ðjA þ jB Þ tAB tBA 2 sin ðjA jB Þ uM ðt1 Þ uN1 ðt0 Þ s1 rAB uM1 ðt1 Þ uN ðt0 Þ s 0
uM ðt1 Þ uN ðt0 Þ 2
Table 4. Nondiagonal antisymmetric antitraces of matrices N Q0 = DABPBDBA, Q1 = PA and Q2 = QM 1 Q0 .
T ðG Lþ1 Þ ¼ T ðW Lþ1 Þ ¼
4 jsðW Lþ1 Þj2 þ jtðW Lþ1 Þj2
;
ð25Þ where W ¼ ðdet G Lþ1 Þ1=2 G Lþ1 ;
3.2. OGFS immersed in arbitrary medium The transmittance T ðG Lþ1 Þ of the generalized Fibonacci multilayer structure placed between two identical media with the refractive index nin ¼ nout ¼ n0 6¼ nA , can be expressed as 4 T ðG Lþ1 Þ ¼ ; ð21Þ 2 jsðG Lþ1 Þj þ jtðG Lþ1 Þj2 where the trace tðG Lþ1 Þ of the unimodular transfer matrix G Lþ1 is given by ð22Þ tðG Lþ1 Þ ¼ tLþ1 ; and the diagonal antitrace sðG Lþ1 Þ of the transfer matrix G Lþ1 2 1 þ r0A 2r0A sðG Lþ1 Þ ¼ s Lþ1 þ V : ð23Þ t0A tA0 t0A tA0 Lþ1 The derivation of (22) and (23) is presented in Appendix E. The diagonal antitrace sðG Lþ1 Þ (23) depends on diagonal antitrace sLþ1 of the matrix QLþ1 and nondiagonal antisymmetric antitrace VLþ1 of the matrix QLþ1 . The trace tLþ1 and the diagonal antitrace s Lþ1, occuring in (21)–(23), satisfy dynamical maps (19) and (20). It can be shown (Appendix C) that VLþ1 also fulfills nonlinear map uMþ1 ðtL1 Þ uN ðtL1 Þ VLþ1 ¼ uM ðtL Þ uNþ1 ðtL1 Þ VL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ VL1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ VL2 : uM ðtL1 Þ
ð24Þ
ð26Þ
is unimodular matrix with the determinant det G ¼
nout cos qout : nin cos qin
It can be shown that the trace tðW Lþ1 Þ and diagonal antitrace sðW Lþ1 Þ of matrix W Lþ1 are given by (see Appendix F): tðW Lþ1 Þ ¼
ð1 þ rin; A rA; out Þ tLþ1 þ ðrin; A þ rA; out Þ hLþ1 pffiffiffiffiffiffiffiffiffiffiffiffi ; tin; A tA; out det G ð27Þ
sðW Lþ1 Þ ¼
ð1 rin; A rA; out Þ sLþ1 þ ðrin; A rA; out Þ VLþ1 pffiffiffiffiffiffiffiffiffiffiffiffi : tin; A tA; out det G ð28Þ
The quantities tLþ1, sLþ1 and VLþ1 satisfy nonlinear dynamical maps (19), (20) and (24) presented in previous subsections. Symmetric nondiagonal antitrace hLþ1 also fulfills dynamical map uMþ1 ðtL1 Þ uN ðtL1 Þ hLþ1 ¼ uM ðtL Þ uNþ1 ðtL1 Þ hL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ hL1 uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ hL2 : þ uM ðtL1 Þ
ð29Þ
Tables 2–5 contain explicit formulas for initial traces and antitraces.
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers Nondiagonal symmetric antitrace of matrices Q0, Q1
Table 5. and Q2. L
hL
0
0
1
0
2
uM ðt1 Þ uN ðt0 Þ
4rAB sin ðjA Þ sin ðjB Þ tAB tBA
Let us note that the trace (10) and introduced antitraces (11)–(13) of unimodular matrix are not independent quantities. They fulfill indentity (see subsection A.5) t2Lþ1 h2Lþ1 þ V2Lþ1 s2Lþ1 ¼ 4 :
ð30Þ
Using (30) we can express, for example, the symmetric nondiagonal antitrace hLþ1 as a function of tLþ1, sLþ1 and VLþ1 : qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð31Þ hLþ1 ¼ V2Lþ1 s2Lþ1 þ t2Lþ1 4 : Unfortunately, the ambiguity of the square root does not allow to incorporate the identity (30) into the developed formalism. Therefore we are forced to use a set of four nonlinear dynamical maps given by (19), (20), (24) and (29). We point out that all considered antitraces (20), (24) and (29) have the same mathematical properties (Appendix A). Namely, they satisfy dynamical maps of the identical form uMþ1 ðtL1 Þ uN ðtL1 Þ aLþ1 ¼ uM ðtL Þ uNþ1 ðtL1 Þ aL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ aL1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 ; uM ðtL1 Þ
ð32Þ
where aj 2 fs j ; Vj ; hj g. The only differences exist in the initial values a0, a1 and a2 of the antitraces shown in tables 3–5.
Three interesting physical situations, described in section 1, have been considered. The proposed approach can be applied in design of optical devices or instruments like filters, optic switches or resonators. In the next paper we apply the presented formalism and show that it is useful in optical engineering. Finally let us note that we have limited our considerations to incidence angles q < qcrit, where qcrit is the critical angle at which total internal reflection occurs. However, the presented formalism can also be applied to analysis of light tunneling phenomenon in ultrathin dielectric multilayers (d l). We have neglected light dispersion [32], but this phenomenon can be easily incorporated into our approach. This is of great importance for composite materials with negative refractive indices, called also lefthanded materials with negative permittivity and permeability [33]. It should be noted, that the discussed structures can be treated as quasi-one-dimensional aperiodic photonic crystals [34]. In this context, a very interesting problem is the photonic band structure of these multilayered systems [35].
Appendix This Appendix contains basic mathematical definitions and theorems with proofs as well as derivations of the dynamical maps.
A. Selected properties of 2 2 matrices Here we present definitions and discuss mathematical properties of 2 2 complex matrices, which are of great importance for our studies. A.1. Unimodular matrix Definition A.1: Nonsingular complex matrix W of dimension 2 2 we define as unimodular matrix if det W ¼ 1 :
4. Conclusions and final remarks In conclusion, we have proposed and developed –– in the framework of matrix formulation and dynamical maps technique –– a new approach to polarized light propagation in optical generalized Fibonacci superlattices being one-dimensional aperiodic structures. We have introduced auxiliary objects: diagonal antitraces, symmetric and antisymmetric nondiagonal antitraces of characteristic matrices, and investigated their mathematical properties. The explicit forms of dynamical maps for these objects have been derived. We have shown that all three kinds of antitraces satisfy identical dynamical maps. Transmittance of the studied aperiodic dielectric multilayers placed between two homogenous media has been expressed in terms of traces and antitraces of characteristic matrices.
261
ð33Þ
Corollary A.2: Product of unimodular matrices is also the unimodular matrix, which is a simple consequence of the definition. Theorem A.3: Suppose that C is non-singular complex matrix of dimension 2 2. Then W ¼ ðdet CÞ1=2 C
ð34Þ
is a unimodular matrix. Relation (34) can be verified by direct calculations. A.2. Diagonal antitrace of matrix Definition A.4: Assume that C is nonsingular complex matrix of dimension 2 2. The function sðCÞ ¼ s C ¼ C11 C22 : is called diagonal antitrace sðCÞ of the matrix C.
ð35Þ
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A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
Theorem A.5: Suppose that B and C are nonsingular complex 2 2 matrices. Then the following relations are satisfied: (i)
sðBCÞ þ sðCBÞ ¼ sB tC þ sC tB ;
(ii) sðaB þ bCÞ ¼ as B þ bs C
ð36Þ
for a; b 2 R ; ð37Þ
1
(iii) sðB Þ ¼ sB =det B :
ð38Þ
Theorem A.9: Assume that B and C are nonsingular complex 2 2 matrices. Then the following relations are fulfilled: (i)
hðBCÞ þ hðCBÞ ¼ hB tC þ hC tB ;
ð44Þ
for a; b 2 R ; ð45Þ
(ii) hðaB þ bCÞ ¼ ahB þ bhC 1
(iii) hðB Þ ¼ hB =det B :
ð46Þ
Proof. Analogous to theorem A.5.
Proofs. (i) If we take matrices B and C in the form B11 B12 C11 C12 B¼ ; C¼ B21 B22 C21 C22
A.5. Identity Lemma A.10: The trace tC , diagonal antitrace sC , antisymmetric nondiagonal antitrace VC and symmetric nondiagonal antitrace hC of unimodular matrix C fulfill the relation
then sðBCÞ þ sðCBÞ ¼ B11 C11 þ B12 C21 B22 C22 B21 C12 þ C11 B11
t2C h2C þ V2C s2C ¼ 4 :
þ C12 B21 C22 B22 C21 B12
ð47Þ
The identity (47) can be proved using definitions (10)– (13) and unimodularity of matrix C.
¼ 2B11 C11 2B22 C22 ¼ B11 C11 þ B22 C11 B11 C22 B22 C22 þ B11 C11 B22 C11 þ B11 C22 B22 C22
B. Modified Chebyshev polynomial
¼ ðB11 þ B22 ÞðC11 C22 Þ þ ðC11 þ C22 ÞðB11 B22 Þ ¼ tB s C þ tC s B : (ii) This property is obvious, as s is a linear combination of matrix elements. (iii) The inverse matrix B1 has the form 1 B22 B12 1 B ¼ det B B21 B11 and sðB1 Þ ¼
ðB22 B11 Þ sB ¼ : det B det B
Definition A.6: Suppose that C is nonsingular complex 2 2 matrix. Antisymmetric nondiagonal antitrace VðCÞ of the matrix C is the function ð39Þ
Theorem A.7: Assume that B and C are nonsingular complex 2 2 matrices. Then the following relations are fulfilled: ð40Þ (i) VðBCÞ þ VðCBÞ ¼ VB tC þ VC tB ; (ii) VðaB þ bCÞ ¼ aVB þ bVC for a; b 2 R ; ð41Þ (iii) VðB1 Þ ¼ VB =det B :
where m is an integer. The modified Chebyshev polynomials uj ðxÞ are related to Chebyshev polynomials of the first Sj ðyÞ and second kind Uj ðyÞ as follows [16, 36]: um ðxÞ ¼ Sm1 ðxÞ ¼ Um1 ðx=2Þ for m > 1 :
A.3. Antisymmetric nondiagonal antitrace of matrix
VðCÞ ¼ VC ¼ C21 C12 :
Definition B.1: Polynomial of the complex argument x is called modified Chebyshev polynomial if 8 0 for m < 0 ; < 1 for m ¼ 0 ; ð48Þ umþ1 ðxÞ ¼ : xum ðxÞ um1 ðxÞ for m > 0 ;
ð42Þ
Theorem B.2: The modified Chebyshev polynomials um ðxÞ satisfy the identity [16] u2m ðxÞ umþ1 ðxÞ um1 ðxÞ ¼ 1
for m ¼ 1; 2; 3; . . . ð49Þ
which can be proven by mathematical induction using (48). Theorem B.3: (Abele´s Theorem) [30] Let the matrix W of dimension 2 2 is unimodular. Then W m ¼ um ðtW Þ W um1 ðtW Þ I ;
ð50Þ
where tW is trace of matrix W and I – unit matrix.
The way of proving this theorem is analogous to one presented for theorem A.5.
C. Dynamical maps of antitraces for OGFS
A.4. Symmetric nondiagonal antitrace
Let aj represents one of three antitraces introduced in section 2, i.e.,
Definition A.8: Suppose that C is nonsingular complex matrix of dimension 2 2. Symmetric nondiagonal antitrace hðCÞ of the matrix C is the function: hðCÞ ¼ hC ¼ C21 þ C12 :
ð43Þ
aj 2 fs j ; Vj ; hj g : Below we present the explicit derivation of dynamical maps for aj .
263
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
From Abele´s theorem QM L ¼ uM ðtL Þ QL uM1 ðt L Þ I ; QN L1 ¼ uN ðt L1 Þ QL1 uN1 ðt L1 Þ
We further insert to (53) the above expression and obtain I:
aðQL QL1 Þ ¼ aL tL1 þ aL1 tL
After substituting the above into N aLþ1 ¼ aðQLþ1 Þ ¼ aðQM L QL1 Þ ; we obtain aLþ1 ¼ uM ðtL Þ uN ðtL1 Þ aðQL QL1 Þ
ð51Þ
where we have taken into acount that aj ðIÞ ¼ 0. Using the properties (36), (40) and (44) we find out aðQL QL1 Þ ¼ aL tL1 þ aL1 tL aðQL1 QL Þ ; ð53Þ and then we calculate
¼ uM ðtL Þ uN ðtL1 Þ aL tL1 þ aL1 tL
þ uM ðtL Þ uN ðtL1 Þ aL1 tL
uMþ1 ðtL1 Þ uN1 ðtL2 Þ aL1 ð54Þ
as well as aL ¼ uM ðtL1 Þ uN ðtL2 Þ aðQL1 QL2 Þ
uM1 ðtL Þ uN ðtL1 Þ aL1 þ uM ðtL Þ uN ðtL1 Þ uMþ1 ðtL1 Þ uN ðtL2 Þ aL þ aL2 uM ðtL1 Þ uM ðtL1 Þ ¼ uM ðtL Þ ½uN ðtL1 Þ tL1 uN1 ðtL1 Þ aL
uM ðtL1 Þ uN1 ðtL2 Þ aL1 ð55Þ
We obtain aðQL1 QL2 Þ from (55) and insert into (54)
þ uN ðtL1 Þ ½uM ðtL Þ tL uM1 ðtL Þ aL1
uMþ1 ðtL1 Þ uM ðtL Þ uN ðtL1 Þ aL uM ðtL1 Þ
þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 uM ðtL1 Þ
aðQL1 QL Þ
þ uM1 ðtL1 Þ uN ðtL2 Þ aL2 uMþ1 ðtL1 ÞuN1 ðtL2 ÞaL1 uM ðtL1 ÞuN ðtL2 ÞaL2 1 ½uMþ1 ðtL1 Þ aL ¼ uM ðtL1 Þ þ uMþ1 ðtL1 Þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 þ uMþ1 ðtL1 Þ uM1 ðtL1 Þ uN ðtL2 Þ aL2 uMþ1 ðtL1 Þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 u2M ðtL1 Þ uN ðtL2 Þ aL2 1 ½uMþ1 ðtL1 Þ aL ¼ uM ðtL1 Þ þ uMþ1 ðtL1 Þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 uMþ1 ðtL1 Þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 ½uMþ1 ðtL1 Þ uM1 ðtL1 Þ uN ðtL2 Þ aL2 þ u2M ðtL1 Þ uM ðtL1 Þ uMþ1 ðtL1 ÞaL uN ðtL2 ÞaL2 ; ¼ uM ðtL1 Þ where the indentity (49) has been applied.
¼ uM ðtL Þ uN ðtL1 Þ aL tL1 uM ðtL Þ uN1 ðtL1 Þ aL
¼ uMþ1 ðtL1 Þ uN ðtL2 Þ aðQL1 QL2 Þ
uMþ1 ðtL1 Þ ½aL þ uM ðtL1 Þ uN1 ðtL2 Þ aL1 uM ðtL1 Þ
uMþ1 ðtL1 Þ uN ðtL2 Þ aL þ aL2 uM ðtL1 Þ uM ðtL1 Þ
uM ðtL Þ uN1 ðtL1 Þ aL uM1 ðtL Þ uN ðtL1 Þ aL1
N aðQL1 QL Þ ¼ aðQMþ1 L1 QL2 Þ
¼
ð56Þ
aLþ1
uM1 ðtL Þ uN ðtL1 Þ aL1 ;
uM1 ðtL1 Þ uN ðtL2 Þ aL2 :
uN ðtL2 Þ aL2 : uM ðtL1 Þ
After substituting (56) into (52) we find out, using (48), the final form of dynamical map:
uM ðtL Þ uN1 ðtL1 Þ aL
uM ðtL1 Þ uN ðtL2 Þ aL2 ;
þ
uMþ1 ðtL1 Þ aL uM ðtL1 Þ
¼ uM ðtL Þ uNþ1 ðtL1 Þ aL þ uMþ1 ðtL Þ uN ðtL1 Þ aL1
uMþ1 ðtL1 Þ uM ðtL Þ uN ðtL1 Þ aL uM ðtL1 Þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 uM ðtL1 Þ uMþ1 ðtL1 Þ uN ðtL1 Þ ¼ uM ðtL Þ uNþ1 ðtL1 Þ aL uM ðtL1 Þ þ
þ uMþ1 ðtL Þ uN ðtL1 Þ aL1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 : uM ðtL1 Þ
D. Transmittance of mutilayer medium with unimodular characteristic matrix Theorem D.1: Suppose that characteristic matrix W of the system is unimodular. Then the transmittance of multilayer system can be expressed in the form TW ¼
4 2
jtW j þ jsW j2
:
ð57Þ
264
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
Finally, applying the definition (35), we obtain diagonal antitrace
Proof. From equation (9) we obtain TW ¼
1 W11 W22
sG ¼ G 11 G 22 1 2 ¼ ðQ11 þ r0A Q21 r0A Q12 r0A Q22 t0A tA0 2 þ r0A Q11 þ r0A Q21 r0A Q12 Q22 Þ 1 2 ½ðQ11 Q22 Þ þ r0A ðQ11 Q22 Þ ¼ t0A tA0 þ 2r0A ðQ21 Q12 Þ 1 2 ½ð1 þ r0A Þ sQ þ 2r0A VQ ; ¼ t0A tA0
1 ¼1 4 ½2<ðW11 Þ þ 2i=ðW11 Þ ½2<ðW11 Þ 2i=ðW11 Þ 4 4 ¼ ; ¼ 2 2 2 j2<ðW11 Þj þ j2i=ðW11 Þj jtW j þ jsW j2 where the following relations, fulfilled by elements of the characteristic matrix, have been taken into account [30, 31]: W21 ¼ W *12 ;
W22 ¼ W *11 :
We note that the trace tW and diagonal antitrace s W of the matrix W are equal respectively to tW ¼ W11 þ W22 ¼ W11 þ W *11 ¼ 2<ðW11 Þ ;
F. Non-unimodular characteristic matrix G
sW ¼ W11 W22 ¼ W11 W *11 ¼ 2i=ðW11 Þ :
E. Characteristic matrix G of multilayer system immersed in medium with arbitrary refractive index Transmission of multilayer system placed between two media with arbitrary refractive index n0 is given by TG ¼
4
because the characteristic matrix G is unimodular. The matrix G can be written as Q12 Q22
1
tA0
1
rA0
rA0
1
;ð58Þ
¼ tðD1 0A D0A QÞ ¼ t Q : In order to calculate the diagonal antitrace of matrix G we have to derive its elements: 1 2 ðQ11 þ r0A Q21 r0A Q12 r0A Q22 Þ; t0A tA0 ð59Þ 1 2 ¼ ðr0A Q11 r0A Q21 þ Q12 þ r0A Q22 Þ ; t0A tA0 ð60Þ 1 2 ¼ ðr0A Q11 þ Q21 r0A Q12 þ rA0 Q22 Þ ; t0A tA0 ð61Þ
G 11 ¼
G 22 ¼
ð63Þ
with nout cos qout : nin cos qin
G ¼ ðdet GÞ1=2 W ;
tG ¼ tðD0A QDA0 Þ ¼ tðDA0 D0A QÞ
G 21
W ¼ ðdet GÞ1=2 G ;
Since
where Q represents multilayer system placed between two A-type medium. The trace tG is equal to the trace of matrix Q
G 12
Non-unimodular characteristic matrix G can be transformed into unimodular one W with the help of theorem (A.3)
det G ¼
js G j2 þ jtG j2
G ¼ D0A QDA0 Q11 1 r0A 1 ¼ t0A r0A 1 Q21
which corresponds to formula (23) used in section 3.2.
1 2 ðr0A Q11 rA0 Q21 þ r0A Q12 þ Q22 Þ : t0A tA0 ð62Þ
ð64Þ
the transmittance of OFGS can be written in the following form 1 2 : ð65Þ TG ¼TW ¼ W11 Proof. 2 1 T G ¼ det G G 11 2 1 2 1 ¼TW: ¼ det G ¼ ðdet GÞ1=2 W11 W11 As a consequence of unimodularity of matrix W , the transmittance is given by TW ¼
4 2
jtW j þ jsW j2
:
ð66Þ
Below we present the derivation of tW and sW . The characteristic matrix has the form 1 rin; A 1 G ¼ Din; A QDA; out ¼ tin; A rin; A 1 1 rA; out Q11 Q12 1 ; ð67Þ 1 Q21 Q22 tA; out rA; out
A. Klauzer-Kruszyna et al., Polarized light transmission through generalized Fibonacci multilayers
where unimodular matrix Q represents multilayer system placed between two A-type media. Then elements of matrix W are given by W11 ¼ W12
[7]
rA; out Q11 þ rin; A rA; out Q21 þ Q12 þ rin; A Q22 pffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ tin; A tA; out det G
W21 ¼ W22
Q11 þ rin; A Q21 þ rA; out Q12 þ rin; A rA; out Q22 pffiffiffiffiffiffiffiffiffiffiffiffi ; tin; A tA; out det G
rin; A Q11 þ Q21 þ rin; A rA; out Q12 þ rA; out Q22 pffiffiffiffiffiffiffiffiffiffiffiffi ; tin; A tA; out det G
[8]
rin; A rA; out Q11 þ rA; out Q21 þ rin; A Q12 þ Q22 pffiffiffiffiffiffiffiffiffiffiffiffi : ¼ tin; A tA; out det G
Therefore the trace and diagonal antitrace of the matrix W can be written as tW ¼
ð1 þ rin; A rA; out Þ tQ þ ðrin; A þ rA; out Þ hQ pffiffiffiffiffiffiffiffiffiffiffiffi ; tin; A tA; out det G
ð68Þ
sW ¼
ð1 rin; A rA; out Þ s Q þ ðrin; A rA; out Þ VQ pffiffiffiffiffiffiffiffiffiffiffiffi : tin; A tA; out det G
ð69Þ
[9]
Acknowledgement. This work was supported in part by European Network of Excellence for Micro-Optics NEMO, contract No.: 003897.
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Polarized light transmission through generalized Fibonacci multilayers. II. Numerical results Agnieszka Klauzer-Kruszyna, Włodzimierz Salejda, Michał H. Tyc Institute of Physics, Wrocław University of Technology, Wybrzez˙e Wyspian´skiego 27, 50-370 Wroclaw, Poland
Abstract: The theory of polarized light propagation in generalized Fibonacci multilayers, recently developed in the framework of matrix formulation and dynamical maps technique, is applied. Transmittance of studied systems is numerically calculated and presented in gray scale figures. The main tendencies in dependences of transmittance on model parameters are presented and discussed. We find a strong dependence of transmittance on refractive indices of surrounding media. We show that the proposed approach can be useful in optical engineering. Key words: Fibonaccian superlattices – polarized light transmittance – multilayer systems
1. Introduction In the previous paper [1], the new approach to propagation of polarized light in dielectric generalized Fibonacci multilayers has been proposed. By means of the matrix formulation [2] and dynamical maps technique [3], the transmittance of generalized Fibonacci multilayers placed between two homogeneous media has been expressed in terms of traces and antitraces of characteristic matrices. The dynamical maps for diagonal antitraces, symmetric and antisymmetric nondiagonal antitraces of characteristic matrices have been derived. In this paper, we apply the developed approach [1] to study the dependence of transmittance of optical generalized Fibonacci superlattices on model parameters: polarization type, wavelength, incidence angle, refractive indices of layers and surrounding media, thicknesses of layers, as well as the type of Fibonacci superlattice described by generation number L and concatenation multiplicities M, N. We show that transmittance characteristics strongly depend on refractive indices of external media. We also discuss how aperiodicity of multilayers, i.e., neither disordered nor periodic distribution of two different dielectric films along given direction, influences polarized light transmission properties of studied systems. We hope that intentional aperiodicity – a new degree of freedom in the studied systems – enables design of new multilayer optical devices with functionality not available to conventional periodic systems.
The paper is organized as follows. In the next section, the explicit formulas for polarized light transmittance of studied systems are given. Section 3 contains numerical results. Final conclusions and remarks are presented in section 4. The Appendix contains formulas for trace and antitraces in case of a periodic structure.
2. Transmittance of generalized Fibonacci multilayer placed between two homogeneous media An optical generalized Fibonacci multilayer (OGFM) is build from dielectric thin films of two different types, denoted by A and B, according to the following inflation scheme: N SLþ1 ¼ SM L SL1 ;
ð1Þ
where S0 ¼ B, S1 ¼ A, M and N are concatenation multiplicities, with M; N 2 N; L 2 N [ f0g defines the generation number of multilayer. The light propagation through a Fibonaccian multilayer structure placed between two arbitrary homogenous media ‘in’ and ‘out’ is described, in the framework of transfer matrix approach, by the characteristic matrix [1] G Lþ1 ¼ Din; A QLþ1 DA; out ;
ð2Þ
where the unimodular matrix N QLþ1 ¼ QM L QL1
ðL 2Þ
ð3Þ
is the characteristic matrix of OGFM placed between two A media, with Q0 ¼ DAB P B DBA and Q1 ¼ P A . We have shown [1] that transmittance T of the ðL þ 1Þst generation of OGFM can be written in the following forms: 1. for nin ¼ nout ¼ nA , T ðG Lþ1 Þ ¼ T ðQLþ1 Þ ¼
4 2
jsLþ1 j þ jtLþ1 j2
;
ð4Þ
where tLþ1 and s Lþ1 are the trace and diagonal antitrace of matrix QLþ1 , respectively;
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2. for nin ¼ nout ¼ n0, T ðG Lþ1 Þ ¼
4 jsðG Lþ1 Þj2 þ jtðG Lþ1 Þj2
;
tðG Lþ1 Þ ¼ tLþ1 ; sðG Lþ1 Þ ¼
ð5Þ ð6Þ
2 1 þ r0A 2r0A s Lþ1 þ V ; t0A tA0 t0A tA0 Lþ1
ð7Þ
where tðG Lþ1 Þ and sðG Lþ1 Þ are trace and diagonal antitrace of characteristic matrix (2), respectively, and VLþ1 is antisymmetric nondiagonal antitrace of matrix (3); 3. for nin 6¼ nout, TG ¼TW ¼
4 js W j2 þ jtW j2
tW ¼ ðdet GÞ1=2
;
1 ½ð1 þ rin; A rA; out Þ tLþ1 tin; A tA; out
þ ðrin; A þ rA; out Þ hLþ1 ; sW ¼ ðdet GÞ1=2
ð8Þ
ð9Þ
1 ½ð1 rin; A rA; out Þ s Lþ1 tin; A tA; out
þ ðrin; A rA; out Þ VLþ1 ;
ð10Þ
nout cos qout ; nin cos qin is symmetric nondiagonal antitrace of matrix
where W ¼ ðdet GÞ1=2 G with det G ¼ hLþ1 (3).
The quantities tLþ1 and aLþ1 , where aLþ1 2 s Lþ1 ; VLþ1 ; hLþ1 , fulfill the nonlinear dynamical maps tLþ1 ¼
uM ðtL Þ uN ðtL1 Þ ½uMþ1 ðtL1 Þ tL uNþ1 ðtL2 Þ uM ðtL1 Þ þ uN1 ðtL2 Þ ½uMþ1 ðtL Þ uN1 ðtL1 Þ
aLþ1
ð11Þ þ uM1 ðtL Þ uNþ1 ðtL1 Þ ; uMþ1 ðtL1 Þ uN ðtL1 Þ ¼ uM ðtL Þ uNþ1 ðtL1 Þ aL uM ðtL1 Þ þ uMþ1 ðtL Þ uN ðtL1 Þ aL1 þ
uM ðtL Þ uN ðtL1 Þ uN ðtL2 Þ aL2 ; uM ðtL1 Þ
ð12Þ
here uj ðyÞ denotes the modified Chebyshev polynomial [1].
3. Numerical results We have performed extensive numerical studies of propagation of polarized light through OGFM in order to find out dependence of transmittance on model parameters: 1. light polarization (s and p), 2. wavelength l 2 ð300 nm; 700 nmÞ, 3. incidence angle q 2 ½0; p=2Þ, 4. layer thicknesses dA ; dB 2 ð250 nm; 100 nmÞ,
5. concatenation multiplicities M, N and generation number L, 6. refractive indices of layers nA ; nB 2 ð1:0; 3:0Þ, 7. refractive indices of surrounding media nin and nout (all refractive indices are chosen to guarantee that total internal refraction does not occur on any interfaces), 8. dispersion curves. In our numerical simulations, we have taken the optical phase length in the following form: 2p cos ql ; ð13Þ jl ¼ d~l nl l~ where l~ ¼ l=l0 and d~l ¼ dl =l0 with l0 ¼ 100 nm. It is straightforward to show that the initial values of traces tL, diagonal antitraces sL, antisymmetric nondiagonal antitrace VL and symmetric nondiagonal antitrace hL are given by expressions t0 ¼ 2 cos jB ; t2 ¼
t1 ¼ 2 cos jA ;
uM ðt1 Þ uN ðt0 Þ ½2 cos ðjA þ jB Þ tAB tBA 2 cos ðjA jB Þ 2rAB uM1 ðt1 Þ uNþ1 ðt0 Þ uMþ1 ðt1 Þ uN1 ðt0 Þ;
s0 ¼ i
2 2ð1 þ rAB Þ sin jB ; tAB tBA
s1 ¼ 2i sin jA ;
2uM ðt1 Þ uN ðt0 Þ ½sin ðjA þ jB Þ tAB tBA 2 sin ðjA jB Þ uM ðt1 Þ uN1 ðt0 Þ s 1 rAB uM1 ðt1 Þ uN ðt0 Þ s0 ;
s2 ¼ i
V0 ¼ i
4rAB sin jB ; tAB tBA
V1 ¼ 0 ;
V2 ¼ V0 ½uM ðt1 Þ uN ðt0 Þ cos ðjA Þ uM1 ðt1 Þ uN ðt0 Þ; h0 ¼ 0 ;
h1 ¼ 0 ;
h2 ¼ uM ðt1 Þ uN ðt0 Þ
4rAB sin ðjA Þ sin ðjB Þ : tAB tBA
The numerically obtained transmittances T are presented in gray scale maps (figs. 2, 3, 5–8, 10–18), where white pixels correspond to transmittance equal to 1 and black ones – to 0. Figs. 2 and 3 present polarized light transmittance maps T ðl~; qÞ for periodic (I ¼ 11) and Fibonaccian multilayers with M ¼ N ¼ 1 and L ¼ 7 at indicated model parameters. Cross-sections of these maps for fixed angle q ¼ p=12 are presented in fig. 4. The transmittance maps T ðl~; qÞ of polarized light for OGFM with ðM ¼ 2; N ¼ 1Þ and ðM ¼ 1; N ¼ 2Þ are depicted in figs. 5 and 6, respectively. Figs. 7 and 8 present the dependence of the transmittance T ðl~; qÞ on the number of unit cells I for periodic multilayers and on the generation number L for Fibonaccian ones, respectively (the compared structures have been selected so that they are built from similar
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OGFM
Fig. 1. Transmittance maps T (~ l; d~A Þ OGFM as well as periodic structure for fixed parameters: nA ¼ 1:43, nB ¼ 2:3, d~B ¼ 0:45 and q ¼ 0.
OGFM
Fig. 3. Transmittance maps T (~ l; q) of 22-layer periodic structure and 21-layer OGFM for: p-type polarization, refraction indices nA ¼ 1:43 and nB ¼ 2:3, as well as given thicknesses d~A and d~B .
number of layers). Cross-sections of these maps for q ¼ p=12 are put together in fig. 9. In the maps for p-type of polarization (figs. 3, 5–8), we can observe white horizontal stripes corresponding to maximum transmittance at Brewster angle. Figs. 10 and 11 show transmittance as a function of layer thicknesses d~A and d~B , respectively, for selected periodic and aperiodic multilayers. In fig. 10, a few vertical lines of maximum transmittance can be seen. They correspond to constructive interference of light in single thin film with refractive index nB and Table 1. Interference maxima of a single thin film with thick~ B and refraction index nB = 2.3. The first row correness D sponds to the upper and middle maps in fig. 10 and the second row – to the bottom maps. Fig. 2. Transmittance maps T (~ l; q) of 22-layer periodic structure and 21-layer OGFM for: s-type polarization, refraction indices nA ¼ 1:43 and nB ¼ 2:3, as well as given thicknesses d~A and d~B .
~ B ¼ d~B D
2 j ~ B =j 6.90 l~j ¼ 2nB D
~ B ¼ 2d~B D
4 5 6 7 8 9 j ~ B =j 6.90 5.52 4.60 3.94 3.45 3.07 l~j ¼ 2nB D
3 4.60
4 3.45
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λλ
λλ
Fig. 4. Cross-sections of transmittance maps T (l~; q) (fig. 2) for Fibonaccian (top graph) and periodic (bottom graph) structures with fixed q ¼ p=12.
~ B immersed in A medium. The transmitthickness D tance maxima are observed at wavelengths given by formula ~ B =j l~j ¼ 2nB D and shown in table 1. The same phenomena occur on the transmittance maps T ðl~; nA Þ, which we do not present. In fig. 11 we observe vertical lines of high transmittance (along these lines transmittance is not equal to 1, but oscillates just below 1). Another kind of straight lines with T ¼ 1 can be seen as well; they are described by formula j ~ l; d~ðl~Þ ¼ 2nB where j ¼ 1; 2; . . . numbers the interference maxima. For example, in the upper right corner of fig. 10, we can see lines from j ¼ 2 to j ¼ 15. Let us note that these lines vanish for sufficiently small d~B , what is illustrated in fig. 11.
Fig. 5. Transmittance maps T (~ l; q) of OGFM with M ¼ 2, N ¼ 1 for: L ¼ 7, nA ¼ 1:43, nB ¼ 2:3 and given thicknesses ~ ~ dA and dB .
Transmittance as a function of l~ and nB or nA for periodic structures and OGFM is presented in fig. 12. The maps were calculated for s-type of polarization and non-zero incidence angle q. The influence of external media refractive indices on light transmittance is shown in fig. 13–18. Fig. 19 presents transmittance for OGFM (M ¼ 1, N ¼ 1, L ¼ 7), with dispersion taken into account. In this example we assume that the refractive index of B layer depends on wavelength as C2 nB ðl~Þ ¼ C1 þ 2 ; l~ where C1 and C2 are constants.
ð14Þ
Fig. 6. Transmittance maps T (l~; q) of OGFM with M ¼ 1, N ¼ 2 for: L ¼ 7, nA ¼ 1:43, nB ¼ 2:3, and given thicknesses d~A and d~B . Fig. 7. Transmittance maps T (~ l; q) of periodic structures for: nA ¼ 1:43 and nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 and given numbers of unit cells I. Fig. 8. Transmittance maps T (~ l; q) of OGFM with M ¼ 1, N ¼ 1 for: nA ¼ 1:43 and nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 and given values of genaration parameters L.
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λλ
λλ
Fig. 9. Cross-sections of maps (figs. 7 and 8) for fixed q ¼ p=12. Calculations were performed for Fibonaccian (left graph) and periodic (right graph) structures for s-type of polarization.
OGFM
Fig. 10. Transmittance maps T (~ l; d~A ) of selected periodic structure and OGFM for: nA ¼ 1:43 and nB ¼ 2:3, d~B ¼ 3:0 and incidence angle q ¼ 0.
OGFM
Fig. 11. Transmittance maps T (~ l; d~B ) of selected periodic structure and OGFM for: nA ¼ 1:3 and nB ¼ 2:3, d~A ¼ 3:0 and incidence angle q ¼ 0.
OGFM
Periodic structure
OGFM
OGFM
Fig. 12. Transmittance maps T (l~; nB ) and T (~ l; nA ) of s-type of polarization 22-layer periodic structure and 21-layer OGFM, with d~A ¼ d~B ¼ 3:0, and given incidence angles. Fig. 13. Transmittance maps T (~ l; q) of periodic structure and OGFM for: s-type of polarization, d~A ¼ d~B ¼ 3:0, nA ¼ 1:43, nB ¼ 2:3 and given values of external refraction index n0 . Fig. 14. Transmittance maps T (~ l; q) of periodic structure and OGFM for: p-type of polarization, d~A ¼ d~B ¼ 3:0, nA ¼ 1:43, nB ¼ 2:3 and given values of external refraction index n0 .
OGFM
OGFM
OGFM
Fig. 15. Transmittance maps T (~ l; q) of periodic structure and OGFM for: s-type of polarization, nA ¼ 1:43, nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 as well as given external refractive indices nin and nout . Fig. 16. Transmittance maps T (~ l; q) of periodic structure and OGFM for: p-type of polarization, nA ¼ 1:43, nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 as well as given external refractive indices nin and nout . Fig. 17. Transmittance maps T (~ l; q) of 22-layer periodic structure and 21-layer OGFM for: s-type of polarization, nA ¼ 1:43, nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 as well as nin ¼ nA and given values nout .
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4. Summary and final remarks
OGFM
In the present paper we have applied the new formalism [1] for studied properties of polarized light propagation in aperiodic Fibonacci-type dielectric multilayers. The result of numerical investigations can be summarized as follows:
Fig. 18. Transmittance maps T (~ l; q) of 22-layer periodic structure and 21-layer OGFM for: p-type of polarization, nA ¼ 1:43, nB ¼ 2:3, d~A ¼ d~B ¼ 3:0 as well as nin ¼ nA and given values nout .
λ/λ 0
λ /λ 0
Fig. 19. Dispersion influence. Transmittance of OGFM for: M ¼ 1, N ¼ 1, L ¼ 7, d~A ¼ d~B ¼ 3:0, q ¼ 0, nA ¼ 1:43 as C2 lÞ ¼ C1 þ 2 , where well as nB ¼ 2:3 (black curve) and nB ð~ ~ l C1 ¼ 2:2775 and C2 ¼ 1:1025. Dispersion curve nB ð~ lÞ is shown in the figure as inset.
1. Transmittance maps for polarized light, presented in gray scale, characterize completely the filtering properties of studied systems. 2. Transmission properties of discussed structures depend on the layers arrangement. The filtering properties of periodic structures can be modified by changing thicknesses, refractive indices or number of layers (figs. 2, 3, 7). Optical generalized Fibonaccian multilayers give us new possibilities of modifying filter properties by using the special kind of aperiodicity defined by the concatenation parameters M and N. It can be seen that with increasing M the transmission channels narrow down and split up. With increasing N, the number of wavelength intervals containing transmission channels grows (figs. 5, 6). 3. For both periodic and aperiodic structures, we observe that with increasing layer thicknesses dA and dB (fig. 2, 3) or the total number of layers J (figs. 7, 8), the number of transmittance channels grows and their widths significantly drop. Remarkable differences in transmission maps of periodic and aperiodic structures appear in the hierarchy of transmittance channels and band gaps. In the case of periodic structures, the regularly spaced high transmittance channels stay in the same spectral regions, independent of the number of layers (figs. 7, 9). In the case of OGFM, with increasing thicknesses dA or dB or generation number L, the transmittance channels become more fragmented and exhibit self-similar character (figs 2, 3, 8, 9). Additional spectral regions with high transmittance channels shift with increasing dA , dB or J (figs. 4, 9). It is interesting that transmittance channels exist also in the spectral region inaccesible in the case of periodic structures (fig. 9). 4. Multilayer system built from B-layer with constant thickness ordered arbitrarily in A medium will be always perfectly transparent at wavelengths corresponding to interference maxima of the single thin film (fig. 10). This kind of transmission channels exists in the spectra of all structures in which all layers of one type (here it is B) have the same thickness. 5. Modifying external media refractive index causes remarkable changes in the filtering properties of investigated structures (figs. 13–16). The following effects can be seen in the transmission maps (if nin 6¼ nA 6¼ nout ): (i) The distribution of transmission channels and band gaps changes. In case of periodic structures (figs. 13 and 15), considerable effects ocp cur for incidence angles q > , and in case of 4
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OGFM they can be observed for arbitrary q (figs. 14 and 16). Transmission characteristics of aperiodic Fibonacci-type structures are more sensitive to boundary conditions than of periodic ones. (ii) The total transmittance decreases. (iii) The transmittance maximum at Brewster angle (p-polarization only) decays. The case when nin ¼ nA and nout is high (figs. 17, 18) is also interesting. In a periodic structure, the transmission channels decay with increasing nout . In an aperiodic structure, this effect occurs for most transmission channels, but not for all: some channels just narrow down, keeping their maximum transmittance and can be denoted as transmission windows. In this model an arbitrary dispersion relation can be taken into account. We performed calculations for weak dispersion, typical for dielectric materials in visible light range (fig. 19), and did not observed any qualitative changes in the transmittance spectrum. The developed formalism and numerical algorithm applied in a computational enviroment (available from us) can be useful in design of optical devices with desirable filtering properties (filters [4], resonators [5], included asymetric ones [6]).
Appendix Periodic structure Suppose that the unit cell of periodic structure consists of L layers: A1 A2 . . . AL . The characteristic matrix of the unit cell can be expressed as G ¼ Din; A1 Qu DA1 ; out ; where Qu ¼ P A1 DA1 ; A2 P A2 DA2 ; A3 . . . DAj1 ; Aj P Aj DAj ;Ajþ1 . . . DAL1 ; AL P AL DAL ; A1 : The unimodularity of Qu allows to express transmittance of periodic structure as a function of trace and antitraces.
The characteristic matrix of a structure composed of I unit cells is given by G ¼ Din; A1 QIu DA1 ; out : The trace and antitraces of periodic structure can be expressed in the following form: tðQI Þ ¼ uI ðtðQu ÞÞ tðQu Þ 2uI1 tðQu Þ ;
ð15Þ
VðQI Þ ¼ uI ðtðQu ÞÞ VðQu Þ ;
ð16Þ
sðQI Þ ¼ uI ðtðQu ÞÞ sðQu Þ ;
ð17Þ
hðQI Þ ¼ uI ðtðQu ÞÞ hðQu Þ :
ð18Þ
Acknowledgement. This work was supported in part by European Network of Excellence for Micro-Optics NEMO, contract No.: 003897.
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