Photonic tunneling between two wires

Progress in Surface Science 67 (2001) 347±354 www.elsevier.com/locate/progsurf

Photonic tunneling between two wires L. Dobrzynski a,*, B. Djafari-Rouhani a, A. Akjouj a, J.O. Vasseur a, J. Zemmouri b a b

UFR de Physique, Universit e de Lille 1, ESA CNRS 8024, 59655 Villeneuve d'Ascq Cedex, France UFR de Physique, Universit e de Lille 1, UMR CNRS 8523, 59655 Villeneuve d'Ascq Cedex, France

Abstract The tunneling of a well-de®ned propagating state from one wire to another, leaving all other neighbor states una€ected, is of great importance for photons, electrons and phonons. We present a simple coupling structure enabling such a transfer to be realized, for any given state energy and state width. We illustrate the analytic study of this structure by an application to the transmission of a telecommunication signal. We discuss then how such a structure could be realized with the help of current surface technologies. Ó 2001 Elsevier Science Ltd. All rights reserved. PACS: 42.79.Sz Keywords: Surface; Wire; Interface; Photon; Tunneling; Telecommunication; Nanometric; Multiplexer; Electron; Phonon

1. Introduction Although surface physics is still a young science, it passed through many evolutions in the last decades. Many surface properties are now well characterized. Their deeper understanding will still keep surface physicists busy. Nevertheless, some of them have already started to dream about man-made surface devices at the nanometer scale. Such a dream presents a very important challenge at the beginning of this century. As a contribution to this dream, the present paper will focus on a simple photonic device. In today's telecommunications, the wavelengths of the signals are 1550 nm and have a width 0.5 nm. Several such signals are traveling together in optical ®bers *

Corresponding author. Fax: +33-3-20-43-40-84. E-mail address: [email protected] (L. Dobrzynski).

0079-6816/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 7 9 - 6 8 1 6 ( 0 1 ) 0 0 0 3 5 - 1

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across our planet. When emitted, they are multiplexed (associated together) and when received de-multiplexed (de-associated). For these purposes, one needs devices able to transfer from one ®ber to another, a well-de®ned signal, leaving all other neighbor signals una€ected. Present devices rely on the opening of small gaps in modulated optical ®bers (see, for example [1]) and work by re¯ection. Investigations of alternative multiplexing photonic devices started recently [2,3]. The aim is to realize a multiplexing system as a planar waveguide surface device. Di€erent surface technologies enable monomode wires to be created within 2D photonic crystals having a gap in a wavelength domain around 1550 nm. By monomode wires, we mean wires whose width enables the propagation of a single mode. This is currently realized by the removal of one line of rods of about 0.2 lm of diameter in a surface layer of a composite material made out of such cylindrical holes in a thin semiconductor layer. Fan et al. [2] proposed building the ®rst multiplexing device by adjusting the diameters and the dielectric constants of a few rods. They showed that complete transfer could occur between two wires by creating, in this way, resonant states of di€erent symmetry, and by forcing an accidental degeneracy between them. We proposed di€erent solutions [3] for this problem using monomode wires and stubs. A simple experimental setup, made out of coaxial wires, con®rmed our solution in the MHz frequency range. In the present paper, we propose the simplest of these solutions and show, for the ®rst time, how it enables the width (even to 0) of the transferred signal to be tuned. We then give all the parameters of such structure for the signals used in modern telecommunications, namely, for a wavelength of 1550 nm and a signal width of 0.5 nm.

2. Theory Let us consider the system presented in Fig 1, which has the symmetry of two mirror planes. The two continua are the two in®nite lines passing by, respectively, points …1; 2† and …3; 4†. The distance d0 between points 1 and 2 is the same as that between points 3 and 4. Four identical monomode structures are branched between points …1; 5†, …5; 4†, …2; 6† and …6; 3†. These structures have one side loop of length d2

Fig. 1. Photonic system under study. d2 and d4 are lengths of corresponding side loops. System has symmetries of two orthogonal planes.

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branched in the middle of the lines of length 2d1 situated between the above-cited points. Such structures enable transmission gaps to be opened in the desired frequency range in the same manner as similar stub structures [4,5]. Between points 5 and 6 is ®xed a waveguide of length 2d3 with a side loop of length d4 in its middle, which acts as a resonant cavity with a localized mode in the above-mentioned gap. More details about such localized modes are given in [5]. Using loops rather than stubs, in such systems, avoids the diculties of determining the boundary conditions at the free extremity of each stub. In order to study the transmission and re¯ection of a propagating state U …x†, incoming from 1 to the site 1, it is helpful to use the following form for the Green function g…DD† of the structure depicted in Fig. 1: g…DD† ˆ G…DD† ‡ G…DM†T …MM†G…MD†;

…2:1†

where G…DD† is the reference Green's function (GF) formed out of truncated parts of the corresponding bulk GF; D stands for the complete space of the composite system and M for the interface space formed by the four points …1; 2; 3; 4†. The transmitted and re¯ected waves u…D† can be obtained from: u…D† ˆ U …D†

G…DM†‰G…MM†Š 1 U …M† ‡ G…DM†‰G…MM†Š 1 g…MM† 1

 ‰G…MM†Š U …M†:

…2:2†

It is then straightforward to ®nd the re¯ection and transmission coecients in the function of the elements of g…MM†, namely R ˆ j1 ‡ 2iag…11†j2

…2:3a†

and T1j ˆ j2iag…1j†j2 ;

j ˆ 2; 3; 4:

…2:3b†

In the above equations, p the parameter a is de®ned, for monomode electromagnetic waves, as a ˆ 2p e=k, where k is the wavelength,  the relative dielectric p constant of all the photonic waveguides, and i ˆ 1. The inverse of g…MM† can be obtained from the linear superposition of the inverse of the appropriate surface gs …MM† of the di€erent constituent pieces of the composite system of Fig. 1. Let us recall [6,7] that: (a) for the semi-in®nite wire ‰gs …00†Š

1

ˆ

ia;

(b) for a ®nite wire of length dm ,   Am Bm 1 ‰gs …MM†Š ˆ a ; Bm Am

…2:4†

…2:5a†

where Am ˆ

1 tan…adm †

…2:5b†

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and Bm ˆ

1 ; sin…adm †

…2:5c†

with m ˆ 0, 1 or 3 (see Fig. 1). One then easily obtains for the structure grafted between points 1 and 5,   A5 B5 1 ‰gs …MM†Š ˆ a ; …2:6a† B5 A5 with A5 ˆ A1 ‡ B5 ; B5 ˆ

…2:6b†

B21 ; 2A1 ‡ A2

where



A2 ˆ 2 tan

…2:6c†

 ad2 ; 2

…2:6d†

d2 being the length of the side loops. The structure situated between points 5 and 6 is characterized by the same expression as (2.6a), but with A6 and B6 instead of A5 and B5 de®ned as A6 ˆ A3 ‡ B6 ; B6 ˆ

…2:7a†

B23 ; 2A3 ‡ A4

where



A4 ˆ 2 tan

…2:7b†

 ad4 ; 2

…2:7c†

d4 being the length of the corresponding side loop. By linear superposition of the above results (2.4),(2.5a)±(2.5c), (2.6a)±(2.6d), 1 (2.7a)±(2.7c), it is straightforward [6,7] to obtain the ‰g…M 0 M 0 †Š with 0 M ˆ …1; 2; 3; 4; 5; 6† of the composite system under study. One takes then advantage of the two orthogonal planes of symmetry of the system in order to obtain: g…11† ˆ Z1 ‡ Z2 ‡ Z3 ‡ Z4 ;

…2:8a†

g…12† ˆ Z1 ‡ Z2 Z3 g…13† ˆ Z1 Z2 ‡ Z3

Z4 ; Z4 ;

…2:8b† …2:8c†

Z3 ‡ Z4 ;

…2:8d†

g…14† ˆ Z1

Z2

where Zn ˆ ‰4a…yn

i†Š 1 ;

n ˆ 1; 2; 3; 4

…2:8e†

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and 2B25 =…2A5 ‡ A6 ‡ B6 †;   ad0 y2 ˆ tan ‡ A5 ; 2 1 ‡ A5 ; y3 ˆ tan…ad0 =2†

y1 ˆ y2

y4 ˆ y3

2B25 =…2A5 ‡ A6

B6 †:

…2:8f† …2:8g† …2:8h† …2:8i†

In (2.8a)±(2.8i), once absorption is neglected, the yn are purely real quantities determined by the ®nite structure contained between the points …1; 2; 3; 4†. One then ®nds that, in order to obtain a complete transfer of a propagating state of wavelength k0 from site 1 to site 3 (namely T13 ˆ 1 and R ˆ T12 ˆ T14 ˆ 0), one must ful®ll the following conditions: y1 ˆ y3 ˆ

y2 1 ˆ

y4 1 :

…2:9†

If one wants the wavelength k0 to lie in the middle of a wavelength domain for which only direct transmission exists (namely, T12 ˆ 1 and R ˆ T13 ˆ T14 ˆ 0), one must ful®ll, in this domain, the following conditions: y1 ˆ y2 ˆ

y3 1 ˆ

y4 1 :

…2:10†

In other words, conditions (2.10) require that the system which couples the two continua, must have a gap in the above wavelength domain. Condition (2.9) implies that the system must also have one resonant state k0 inside this gap. These conclusions are equivalent to the ones obtained by Fan et al. [2] for their type of multiplexer. Conditions (2.9) and (2.10) enable us to completely determine from closed-form expressions the di€erent wire lengths of our system. First, the condition y2 y3 ˆ 1 is satis®ed for tan…2a0 d1 † ˆ

1 : tan…a0 d2 =2†

…2:11†

Then the condition y1 ˆ y3 and y2 ˆ y4 provides us with the following relations: tan…2a0 d3 † ˆ

1 ; tan…a0 d4 =2†

…2:12†

and tan…a0 d3 † ˆ tan2 …a0 d1 †:

…2:13†

When studying the variations with k of the yi around k0 , one sees that the transferred signal has a symmetrical shape, only for p a0 d0 ˆ ‡ 2n0 p; n0 ˆ 0; 1; 2; . . . …2:14† 2 One may also notice that T13  1=2, when y4 diverges and y1 vanishes for the same wavelength k0 , namely, when A3 …k0 † vanishes. It is then helpful to de®ne an approximated quality factor

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qˆ

2…k

0

k0 k0 †

:

…2:15†

It is possible to choose d3 such that this quality factor becomes as large as one wishes, namely d3 ˆ

rp 1 ‡ 2n3 ; 2a0 1 ‡ …2q† 1

n3 ˆ 0; 1; 2; . . .

…2:16†

So, as described above, our system enables to determine completely and in closedform all the wire lengths for a complete channel transfer, once the wavelength k0 and the order of magnitude of the quality factor q are chosen. Relations (2.14) and (2.16) give us the values of d0 and d3 . Let us note that several solutions are possible due to the di€erent values of the integers n3 and n0 . Knowing d3 , the lengths d1 ; d2 and d4 are obtained from (2.13), (2.11) and (2.12), respectively. Let us note that these three lengths can be such that di ˆ …di †min ‡ ni p;

i ˆ 1; 2 and 4;

…2:17†

where …di †min stands for the minimal value of di . The choice of the values of the integers n0 ; n1 ; n2 ; n3 ; n4 enables the size of the system to be increased. It also in¯uences the width of the region for which the direct transmission (T12 ˆ 1) is not a€ected by the multiplexer. The width of this region is important for the determination of the number of signals of a given width, one may multiplex with such devices. 3. Applications and discussions Following the above theory, we present one application. We have chosen k0  1550 nm,  ˆ 11:56, n3 ˆ 3 and q ˆ 1600. Eq. (2.16) provides us with d3 ˆ 797:5 nm. Then with n0 ˆ 3 and n1 ˆ n2 ˆ n4 ˆ 1 and (2.11)±(2.15), we obtain d0 ˆ 1481:6 nm, d1 ˆ 337:6 nm, d2 ˆ 244:9 nm and d4 ˆ 228:9 nm. Fig. 2 shows the transmission factors with a precision of the order of the nanometer for all distances, namely

Fig. 2. Variation of intensity of transmitted signal from site 1 to site 2 (solid lines), and of transferred signal T13 (long dashed lines), in structure shown in Fig. 1 versus the wavelength k. Dots represent signal intensity T14 in backward direction. Theoretical results were obtained for d0 ˆ 1482 nm, d1 ˆ 338 nm, d2 ˆ 245 nm, d3 ˆ 797 nm, d4 ˆ 229 nm and  ˆ 11:56.

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d0 ˆ 1482 nm, d1 ˆ 338 nm, d2 ˆ 245 nm, d3 ˆ 797 nm and d4 ˆ 229 nm. The re¯ection coecient R (not represented in this ®gure) and T14 are almost 0. These approximated values of the distances neither modify nor shift the transmitted signals. However, a precision 10 nm was found insucient for maintaining good stability of the demultiplexer. This is, of course, consistent with the required precision chosen for k0 and the width (about 0.5 nm) of the transferred peak. The ®nal value obtained for the quality factor Q was found to be 4000. It is better than the input value q ˆ 1600, as (2.16) was approximated. This example shows that one may produce multiplexers with dimensions 1 lm and even 0.5 lm for n0 ˆ n1 ˆ n3 ˆ 0 and n2 ˆ n4 ˆ 1, working for wavelengths of 1.55 lm with signals widths of 0.5 nm. The diculty of producing such devices lies mostly in the precision of about 1 nm for all the wire lengths. Higher quality factors Q can also be obtained, but the precision on the distances was found to be related to the width of the transferred peak. We also checked that the solutions for higher integer values ni and greater values of distances di were good solutions. However, the required precision on the distances was found to remain correlated with the width of the telecommunication signals, namely, approximately a few nanometers for signal widths 0.5 nm. The small size of such devices is highly desirable for the future development of telecommunications. Present devices, using optical ®bers with modulated dielectric constants and working by re¯ection, have much bigger dimensions. Present technologies can produce [8,9], within an adsorbed layer of a semiconductor, photonic materials made out of periodic vacuum rods. For telecommunication purposes, one uses such photonic materials with a gap around 1550 nm. Removal of rams of rods within such photonic materials is the usual manner of obtaining monomode wires without transmission problems at wire connections. This is the expected manner for manufacturing devices of the type represented in Fig. 1. The diculty, which remains, lies mostly in the nanometric precision required, as seen above. Nevertheless, with the help of modern technologies enabling one atom at a time to be deposited on surfaces, we may hope that such devices will appear in the next decade. The small size of such devices will also minimize the losses, although they can already be overcome by erbium doping of such wires. Such doping even enables the signals to be ampli®ed. Finally, let us mention that similar structures may prove to be helpful for transferring simple electrons from one wire to another [10,11] and also for selecting a well-de®ned acoustic phonon from one slender tube to another [12]. Acknowledgements We thank D. Lippens and O. Vanbesien for stimulating discussions and the Centre National de la Recherche Scienti®que, Programme Telecommunications for support.

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