Transmission properties of light through the Fibonacci-class multilayers

PHYSICAL REVIEW B

VOLUME 59, NUMBER 7

15 FEBRUARY 1999-I

Transmission properties of light through the Fibonacci-class multilayers Xiangbo Yang Department of Physics, South China University of Technology, Guangzhou 510641, China and Physics Group, Department of Mathematics and Physics, Guangdong University of Technology, Guangzhou 510090, China

Youyan Liu CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China and Department of Physics, South China University of Technology, Guangzhou 510641, China

Xiujun Fu Department of Physics, South China University of Technology, Guangzhou 510641, China ~Received 28 August 1997; revised manuscript received 22 July 1998! We study the transmission properties of light through the Fibonacci-class quasiperiodic multilayers. The trace map of the propagation matrices, its invariant of motion, and the expression for the transmission coefficient are obtained. For the normal incidence of light, the transmission coefficient exhibits a switchlike property in a family of the Fibonacci-class multilayers, while a six-cycle feature in another family is found. @S0163-1829~99!11603-5#

I. INTRODUCTION

Since the fundamental discovery of the diffraction pattern with fivefold symmetry,1 the quasilattices have attracted much attention from both mathematicians and physicists. In particular, the Fibonacci quasilattice has been extensively studied2–9 and a lot of experimental work has been concerned with the propagation of electrons or other classical waves in one-dimensional ~1D! quasiperiodic superlattices or dielectric multilayers. Merlin, Bajema, and Clarke10 have grown a Fibonacci lattice made of GaAs and AlAs for the first time and have studied its x-ray diffraction and Raman scattering properties. Following this work, several interesting experimental studies have been reported. An incomplete list includes transmission of bulk acoustic phonons,11 third sound along superfluid helium thin film,12 surface acoustic wave,13 photonic dispersion relation,14 localization of light waves,15 photorefractive properties,16 and second17 and third18 harmonic generation in a quasiperiodic optical superlattice of LiTaO3. These experimental works not only confirm the present theoretical results, but also open some new subjects to study. To exhibit the localization of classical waves, Kohmoto, Sutherland, and Iguchi7 have suggested an ideal experiment. They studied the transmission of light through dielectric multilayers consisting of two kinds of layers, which are arranged following the 1D Fibonacci sequence and their theoretical results are confirmed by the dielectric multilayer experiment.15 Later, the transmission of light through the multilayers arranged by nonFibonaccian sequences was theoretically discussed by Riklund and Severin19 and Dulea, Severin, and Riklund.20 On the other hand, Schwartz21 suggested the possible application of quasiperiodic multilayers as optical switches and memories. Huang, Lui, and Mo22 proposed a so-called intergrowth quasiperiodic model and found an interesting switchlike property in the light transmission efficient. 0163-1829/99/59~7!/4545~4!/$15.00

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In this paper, we study the transmission properties of light through the quasiperiodic multilayers arranged by the Fibonacci-class @ FC(n) # sequences.23 The FC(n) sequences are a class of quasilattices generated by the substitution rules B→B n21 A, A→B n21 AB, where n is a positive integer. A specific n corresponds to a certain quasilattice that is denoted as FC(n). Starting with a B, the first-three generations of FC(n) are S 1 5B, S 2 5B n21 A, ~1!

] which show the following recursion relation S j 5S nj21 S j22

~ j>3 ! .

~2!

The FC(n) sequence can also be obtained by the projection method from a two-dimensional square lattice.23 It has been proved that for all the Fibonacci-class lattices the electronic spectra are perfect self-similar, and there is a universal gap-labeling theorem.23 Naturally, a question comes to us: are there any other universal physical properties in the FC quasilattices? In this paper, we will investigate the light transmission of FC(n) multilayers. Based on the characteristics of the construction of FC(n) sequences, we first deduce the trace map of the propagation matrices and look for the invariant of motion. Then we study the transmission properties of light through FC(n) multilayers. Analytical results show that the transmission coefficient for the FC(2m) multilayers (m51,2,3, . . . ), displays a switchlike property ~on-off-on-off-¯!, but for the FC(2m 4545

©1999 The American Physical Society

4546

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BRIEF REPORTS 2m12

1

(

n51

~ 2m11 ! a 2m11 ~ n ! x n2 z j21 j21

~ j>5,m>1 ! ,

~4!

a 2m ~ n ! 5 ~ 21 ! m a 2m21 ~ n21 ! ,

~5!

where

n21

a 2m11 ~ n ! 5 ~ 21 ! m

(

j52m21

a 2m ~ j ! ,

~6!

z j 5Tr~ M j M j21 ! 5x j x j21 2x n22 j21 z j21 FIG. 1. Propagation of light on the reflective and the transmission surface of FC(n) multilayer, where E I , E R , and E O are the input, reflective, and output electromagnetic fields, respectively.

2m21

(

1

a 2m ~ n ! x n22m j21 x j22

n51 2m11

11) ones (m51,2,3, . . . ), there exist a six-cycle property around u 5 p (l6 21 ), which is similar to the Fibonacci model24 (m50). We organize this paper as follows. Section II is devoted to study the trace map of the propagation matrices and the constant of motion. In Sec. III, we investigate the transmission properties of light propagating normally through FC(n) multilayers. And a brief summary is given in Sec. IV.

1

M j 5M nj21 M j22

~ j>3 ! .

~3!

n22 n23 5x n21 j21 z j21 2x j21 x j22 2 ~ n22 ! x j21 z j21

5x 24 1x 23 1z 24 2x 4 x 3 z 4

(

C 2m x n27 j21 z j21

n26

1

(

m52

As shown in Fig. 1, for the general incidence the transmission coefficient T is T5

(

4 ~ m 11m 222m 12m 21! 2 ~ m 111m 22! 2 1 ~ m 122m 21! 2

,

~9!

where m i j (i, j51,2) is the element of the matrix M j . As M j being the unimodular transfer matrix, the transmission coefficient T can be written as T5

4 21m 2111m 2121m 2211m 222

.

~10!

Now, we study a special case, i.e., the light vertically transmits the multilayer, we can expect to obtain some interesting analytical results. In this case, we denote R5

nB , nA

~11!

where n i is the index of refractive of media i. Furthermore, we choose appropriate thickness of the layers d A and d B in order to make the following condition be satisfied: n A d A 5n B d B ;

~12!

u i 5n i d i k[ u ,

~13!

then we have

2m

n51

~8!

III. PROPERTIES OF TRANSMISSION COEFFICIENT

C 2m x n28 j21 x j22 1¯

n21 ! z j21 1 5x ~j21

~ j>4 ! .

This invariant I is always positive and represents the strength of the effect of quasiperiodicity.

n25 m52

~7!

5x 2j 1x 2j21 1z 2j 2x j x j21 z j

2 n25 1 ~ n23 ! x n24 j21 x j22 1C n23 x j21 z j21

2C 2n24 x n26 j21 x j22 2

~ j>2 ! .

I5x 2j11 1x 2j 1z 2j11 2x j11 x j z j11

Because the propagation matrices of FC(n) multilayer are all unimodular,22 after a straightforward calculation we obtain a general form of the trace map, x j 5Tr~ M j ! 5Tr~ M nj21 M j22 !

~ 2m11 ! a 2m11 ~ n ! x n2 z j21 j21

We have also obtained the invariant, or constant of motion,2,25 as follows:

II. PROPERTIES OF THE PROPAGATION MATRICES

Now let us consider the matrices of light propagating through the FC(n) multilayer of jth generation S j , which is sandwiched by two media of material of type B, shown in Fig. 1. By the use of the recursion relation ~2!, it is easy to get the propagation matrices of light through the multilayers of the first and second generations: M 1 5T B , M 2 5T n21 B T BA T A T AB , where T AB (T BA ) stands for the propagation matrix from layer A (B) to layer B (A) and T A (T B ) is the propagation matrix through a single layer A (B). Then for the multilayer of the jth generation, the following recursion relation holds

(

n51

a 2m ~ n ! x n22m j21 x j22

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BRIEF REPORTS

where k is the wave number in vacuum, and u is the phase difference between the ends of a layer, A or B. For the special phases

u 5 ~ l6 21 ! p ,

~14!

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To study the transmission properties, we classify the FC(n) quasilattices to two families: the even family with n 52m and the odd one with n52m11, m50,1,2, . . . . By a straightforward calculation, for the systems of the even family FC(2m) we can prove that

the accordingly propagation matrices will be T AB 5 T BA 5

S

S D 1 0

0 5T 21 BA , R

1 0

T A 5T B 5

~15!

D

0 5T 21 AB , R 21

~16!

S

~17!

M 2 j ~ 2m ! 5

D S D

21 , 0

0 1

1 T AA 5T BB 5 0

0 . 1

~18!

T q ~ n ! 5T q ~ 2m ! 5

5

q52 j11, q52 j,

where C 1 is a constant, and ‘‘off’’ and ‘‘on’’ display the switchlike character of FC(2m) multilayers. The result means that under the special condition given above the system of even family FC(2m) multilayers shows a switchlike property with ‘‘off’’ for the odd generation and ‘‘on’’ for the even generation. As for the systems of odd family FC(2m11), the propagation matrices possess a standard six-cycle property: M 6 j11 ~ 2m11 ! 5M 1 ~ 2m11 ! 5

S

D

0 1

M 6 j12 ~ 2m11 ! 5M 2 ~ 2m11 ! 5

S

M 6 j13 ~ 2m11 ! 5M 3 ~ 2m11 ! 5

S

2R 21 0

M 6 j14 ~ 2m11 ! 5M 4 ~ 2m11 ! 5

S

~ 21 !

R

~22!

D

~23!

~25! M 6 j15 ~ 2m11 ! 5M 5 ~ 2m11 ! 5

S

0 R

2m11

0 R

22m j

~ 21 ! m j R 0

D

2R 2m j , 0

~19!

D

0 . ~ 21 ! m j R 21

~20!

Applying the above results to Eq. ~9!, we obtain the transmission coefficient T of FC(2m) multilayers for the qth generation,

21R

j→1` 4 24m j ——→ 0 1R

4m j

4 5C 1 2 21R 1R 22

~ off!

~21! ~ on! ,

M 6 j ~ 2m11 ! 5M 6 ~ 2m11 ! 5

S

R 0

D

0 . R 21

~27!

The corresponding transmission coefficient T is obtained as

D

D

2m12

S

S

0 , ~24! 2R

~ 21 ! m R 22m22 , 0

0 m11

21 , 0

~ 21 ! m11 R 21 , 0

0 ~ 21 ! m R

M 2 j11 ~ 2m ! 5

D

2R 22m21 , 0 ~26!

~28! where C 1 and C 2 are all constants, and ‘‘on’’ and ‘‘off’’ display the corresponding ‘‘switch’’ property. This analytical calculation shows that the transmission coefficient T of FC(2m11) system possess a six-cycle property under the special cases given above. To end this section, it should be noted that the Integrowth model proposed by Huang, Liu, and Mo22 is in fact a mem-

BRIEF REPORTS

4548

ber of the FC(n) sequences with n52, and our present analysis is general, which contains all the results obtained there. IV. SUMMARY

We have investigated the transmission properties of light through the Fibonacci-class quasiperiodic multilayers and found some interesting results. The trace map of propagation matrices is deduced and the invariant of motion is found. We obtained the expression of the coefficient T analytically for

1

D. S. Shechtman, I. Blech, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 ~1984!. 2 M. Kohmoto, L. P. Kadanoff, and C. Tang, Phys. Rev. Lett. 50, 1870 ~1983!. 3 J. M. Luck and D. Petrities, J. Stat. Phys. 42, 289 ~1986!. 4 Q. Niu and F. Nori, Phys. Rev. Lett. 57, 2070 ~1986!. 5 Y. Liu and R. Riklund, Phys. Rev. B 35, 6034 ~1987!. 6 M. Kohmoto, B. Sutherland, and C. Tang, Phys. Rev. Lett. 35, 1020 ~1987!. 7 M. Kohmoto, B. Sutherland, and K. Iguchi, Phys. Rev. Lett. 58, 2436 ~1987!. 8 Y. Liu, X. Fu, W. Deng, and S. Wang, Phys. Rev. B 46, 9216 ~1992!. 9 T. Odagak and M. Kaneko, J. Phys. A 27, 1683 ~1994!. 10 R. Merlin, K. Bajema, and R. Clarke, Phys. Rev. Lett. 55, 1768 ~1985!. 11 D. C. Hurley, S. Tamura, J. P. Wolfe, K. Ploog, and J. Nagle, Phys. Rev. B 37, 8829 ~1988!. 12 D. T. Smith, C. P. Lorenson, and R. B. Hallock, Phys. Rev. B 40, 6634 ~1989!. 13 L. Maeon, J. P. Desieleri, and D. Sornette, Phys. Rev. B 44, 6755

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general incidences and normal one. When u 5(l6 21 ) p the transmission coefficient for FC(2m) multilayers displays a switchlike property ~on-off-on-off-¯!, but for FC(2m11) systems, the transmission coefficient possesses a six-cycle feature. ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China, Grant No.: 19874021, and Advanced Education Foundation of Guangdong Province ~960001!.

~1991!. T. Hattori, N. Tsurumachi, S. Kawato, and H. Nakatsuka, Phys. Rev. B 50, 4220 ~1994!. 15 W. Gellermann, M. Kohmoto, B. Sutherland, and P. C. Taylor, Phys. Rev. Lett. 72, 633 ~1994!. 16 M. Dinu, M. R. Melloch, and D. D. Nolte, J. Appl. Phys. 79, 3787 ~1996!. 17 S. Zhu, Y. Zhu, Y. Qin, H. Wang, C. Ge, and N. Ming, Phys. Rev. Lett. 78, 2752 ~1997!. 18 S. Zhu, Y. Zhu, and N. Ming, Science 278, 843 ~1997!. 19 R. Riklund and M. Severin, J. Phys. C 21, 3217 ~1988!. 20 M. Dulea, M. Severin, and R. Riklund, Phys. Rev. B 42, 80 ~1990!. 21 C. Schwartz, Appl. Opt. 27, 1232 ~1988!. 22 X. Huang, Y. Liu, and D. Mo, Solid State Commun. 87, 601 ~1993!. 23 X. Fu, Y. Liu, P. Zhou, and W. Sritrakool, Phys. Rev. B 55, 2882 ~1997!. 24 M. Kohmoto and J. R. Banavar, Phys. Rev. B 34, 563 ~1986!. 25 G. Y. Oh and M. H. Lee, Phys. Rev. B 48, 12 465 ~1993!. 14