WO2006120162A1

WO2006120162A1 - Elasto-optical dilatational devices and method of establishing same

Info

Publication number: WO2006120162A1
Application number: PCT/EP2006/062077
Authority: WO
Inventors: Alexander Boris Movchan; Sebastien Guenneau
Original assignee: The University Of Liverpool
Priority date: 2005-05-06
Filing date: 2006-05-04
Publication date: 2006-11-16
Also published as: GB2425845A; EP1877857A1; GB0509262D0

Abstract

A method and apparatus for excitation of acoustic waves in a photonic crystal are shown wherein high-frequency localised dilatational modes are established in micro-structured photo-elastic material (10) by the application thereto of elastic vibrations. The multi-core photo-elastic material (10) contains a multi-core channel within a n array of voids (12) . A generator (14) of elastic vibrations is coupled to the elastic material (10) for establishing localised dilatational deformations in the material (10) to cause changes in the optical properties of the multi-cored channel .

Description

DESCRIPTION

RLASTO-QPTICAL DILATATIONAL DEVICES AND

METHOD OF ESTABLISHING SAME

The present invention is concerned with the modelling and construction of elasto-

optical devices for use, for example, as optical switches.

A new branch of physics has emerged in the past five years, following the

discovery by John Pendry of new structures that lead to negative refraction of electromagnetic waves.

These ideas have been used in the design of modern resonance imaging devices

of subwavelength resolution. There is a serious need of mathematical modelling of such

composite structures, analysis of their interaction with physical fields of different nature, including electromagnetic, acoustic, elastic waves as well as coupled fields.

Also, over the past ten years there has been much interest in the fabrication and modelling of so-called photonic crystal fibres (PCFs). These fibres guide light in a

unique way. Instead of channelling the light by using a stepped refractive index gradient,

the fibre itself contains inhomogeneities which are used to "trap^"' light in a central core.

Materials of this kind are often referred to as photonic band-gap materials, since they suppress the propagation of light over a small band of frequencies, and are the optical

analogue of semiconductors.

A related question involves the propagation of elastic waves through

inhomogeneous materials. Such ^*'phononic band-gap materials^" would use the spacing of inhomogeneities in the materials to suppress mechanical vibrations (such as elastic

waves; in one or all directions.

A major problem is how to investigate the interaction between elastic and

electromagnetic waves within Photonic Crystal Fibre structures. An acoustic band-gap

could be designed and tuned for vibration- free operation of high-precision machine

systems. In addition, it is suggested that materials with an acoustic band-gap in a

suitable frequency range might be used to build passive mass dampers as well as elasto-

optical switches. Modes which are forbidden to propagate within a band-gap will tend to cluster around a defect in the material, an effect known as localisation.

There are a number of challenges involved in studying the motion of elastic waves through an inhomogenous material. The first is that the spacing of the inhomogeneities must not be of the same order as the wavelength of the vibration; for

this reason the material cannot, in general be homogenised and the field equations must be solved in full. Also, there is more structure to the elastic problem than in the

electromagnetic case. For propagating light, two polarizations of light are possible,

whereas the equations of isotropic elastodynamics allow for three types of vibration: two transverse and one longitudinal.

In accordance with one aspect of the present invention, we use a micro-structured opto-elastic fibre designed to transmit a signal within an acoustic or optical range of

frequencies and which contains a multi-core channel within an array of circular voids.

We found that such a fibre can be used efficient!} for transmitting elastic pressure waves, and for certain frequencies it possesses localised standing waves, which are either

symmetric or skew-symmetric (even in both x and>-, or odd in x and >>, respectively). For

photo-elastic materials, the localised pressure wave will affect/enhance the propagation

of light within the multi-core channel. The type of symmetry of the localised elastic mode within the photo-elastic fibre can be clearly recognised in the transmitted optical

signal, and hence this property is useful in the design of optical switches.

With no elastic vibrations applied to the fibre it is designed to transmit the optical

signal within a certain frequency range [ω,, ω₂] corresponding to a stop band for a certain

spectral problem for the Maxwell system in a doubly-periodic array of circular voids.

Simultaneously, the same structure can be considered for equations of linear elasticity,

and we have found that it supports localised standing pressure waves concentrated within

the multi-core channel. The frequencies ω and ω_skew of localised elastic modes

correspond to symmetric and skew-symmetric states as shown in accompanying Fig. 2.

All frequencies are normalised as ωd/v_b, where d is the distance between centres of

neighbouring voids, and v_b is the shear wave speed. For applications in photonic crystal

fibres, d is typically 0.5μm and ω is within the TH_Z range.

The micro-structured opto-elastic fibre is attached to a generator of elastic

vibrations tuned to frequencies ω_sγm or ω_ikew. Due to elastic vibrations, the corresponding

region of the micro-structured fibre is deformed as shown in accompanying Fig. 3 (with

a symmetric mode corresponding to the frequency ω_SV7n) or Fig. 2 (with a skew-symmetric

mode corresponding to the frequency ω_skew). Both localised states correspond to dilatational deformations. For photo-elastic materials, these localised deformations yield

changes in optical properties of the multi-core channel. As a result, one can observe

changes in the transmitted optical signal. This is used as the basis of an optical switch:

the optical signal is changed due to symmetric or skew-symmetric dilatational localised

elastic vibrations initiated at frequencies ω and ω_skew respectively.

In accordance with a second aspect of the present invention, an optical switch is

obtained by the establishment of high-frequency dilatational localised modes in a micro-

structured photo-elastic material. An important feature of such an arrangement is the

instant response of the optical signal to the mechanical vibration frequency

corresponding to the localised defect mode.

The invention is described further hereinafter, by way of example only, with

reference to the accompanying drawings, in which: -

Fig. 1 illustrates the band structure for a square array of circular holes in fused

silica, for a normalised radius r_cd= 0.4 and inter-hole space d= 1;

Fig. 2 shows two transverse sections through an opto-elastic fibre having a square

array of 47 air holes running longitudinally through it, illustrating two different

dilatational modes associated to the normalised eigenfrequency 7.71;

Fig. 3 shows the dilatational mode in the array to the normalised eigenfrequency

7.81: and

Fig 4 shows the phononic band structure for a square array of circular holes in

fused silica. Brillouin predicted in 1922 the diffraction of light by an acoustically perturbed

medium: this is the so-called Brillouin scattering. When an elastic wave propagates in

a medium, there is an associated strain field which results in a change of the index of

refraction. This is referred to as the photoelastic effect. The photoelastic effect in a

material couples the mechanical strain tensor S_k, to the optical index of refraction n. This

effect is described by the change in the so-called optical impermeability tensor:

^Δ (^ϊ) ^ PijkiSki . i, j, k. l = l , .., 2 ^Kn ' ^ (1)

where p is the rank four strain-optic tensor.

Thanks to the symmetry of the optical impermeability and strain tensors, one can

rewrite (1) in contracted form:

A (^)₁ = PyS, , Z₁ J = I, ..^ (2) where (S₁)³ _{J = x} = (ε_u, ε₂₂, ε₁₂)^τ.

In the case of 2D isotropic photoelastic material, p is given by

It is noted that if p., = p_P then the contribution to shear strain is cancelled and only

pressure modes are important. As an example, one can consider glass (Ge₂₁Se₅₅As₁-)

where p_n = p₁₂ = 0.21 for the wavelength λ = l .Oόαtn. On the other hand, for fused

silica P₁, = 0.121. p.~ = 0.270 for the wavelength λ = 0.63 um.. and in this case the optical properties would change within the region of localised strain, both in pressure and

shear modes.

In the numerical model for the analysis of phononic crystal fibres, in-plane

harmonic Bioch waves within an isotropic elastic medium 10 of density p and Lame coefficients λ and μ can be considered. The elastic structure consists of a periodic

assembly of cylindrical inclusions 12 of circular cross-section (invariant along the z axis)

in a homogeneous isotropic medium.

The vector of amplitudes u in the elastic material is described by the Navier system of equations

(λ + 2μ) W ^• u(r) - μV x V x u(r) + pω²u(r) = 0, r e IR³ (4) where r = (x,y) is the position vector and ω is the radian frequency of vibration.

The boundary conditions come from the traction free boundary condition on the

surface of each circular void. Expressed in cylindrical coordinates, the components of

the stress tensor O_n., σ_rθ and O_n, must vanish. As the cylinders are considered to be

infinitely long, the problem is Inherently two-dimensional, and u = u(x,y). In this case,

on the boundary of each void:

σ_*r = μ~- = 0 ^)

G¹Ur

where U_x, u_θ and u_z denote the three components of u in cylindrical coordinates.

In the mathematical model it is assumed that the medium is periodic, with the

macro-cell consisting of 47 air holes 12. as shown in Figs. 2 and 3. The periodicity of

the problem implies that u must satisfy the Floquet-Bloch condition:

u(r + R₁₃) = u(r) exp(ik_moch -R_p), r e!R³. (8)

where R_p lies entirely in thex - y plane and points at the center of thepth cavity, whereas the Bloch vector k_Btoch belongs to the reciprocal plane. For a given k_Bloch one finds a set

of real positive eigenvalues ω of the system that correspond to propagating elastic modes within the lattice.

First, we consider a doubly-periodic array of circular voids 12, with an elementary square cell containing a single circular void: the corresponding dispersion diagram is shown in Fig. I .

Fig. 1 shows the phononic band structure for the square array of circular air-holes

in fused-siϋca (normalised radius r_cd = 0.4, inter-hole spacing d — 1). The

skewsymmetric and symmetric localised modes (respectively at normalised radian

frequency ω_skew = 7.71 and ω = 7.81, the horizontal straight lines) depicted on Fig. 2

and Fig. 3 sit well above the sonic band-gap, which extends from 4.4 to 5.2. The inset

shows the irreducible segment of the first Brillouin zone [of vertices Y = (0.0),M = (0.

T, d\. K = (π'd. rudjl described by the Bloch vector.

Fig. 2 shorn s dilatational modes associated to the normalised eigenfrequency

7.71. Trie's are odd in x and even in v ( left) and even in x and odd in γ (right). The scale is in arbitrary units.

Fig. 3 shows a ciiiatational mode associated to the normalised eigenfrequeiicy

7.81. Il is even in x and JΛ The scale is in arbitrary units.

Fig. 4 shows the phononic band structure for the square array of circular air holes in fased-siiica (normalised radius r_Ld~ 0.4, inter-hole spacing d = 1). Four holes have

been removed to create a multi-core defect. ω_skew corresponds to the mode of Fig. 2 whereas ω_sym corresponds to the mode of Fig. 3. We note that the first Brill ouiii zone is

defined by the three vertices F = (0,0), M = (0, πlld), K = (π/7d, %!ld) (7 x 7 macrocell).

We select a contour FMK within an irreducible Brillouin zone (see Fig. 1) and

take the Bloch vector k_B]och as a position vector of a point along FMK. On the horizontal

axis of diagram in Fig. 1 we plot the modulus of the Bloch vector k_Bloch and along the vertical axis we plot the radian frequency ω. It is shown that within a certain frequency

range there is a band-gap corresponding to pressure and in-plane shear waves.

Next, we consider doubly-periodic structures with defects. It is modelled as a

doubly-periodic array of 7 by 7 macro cells 12 containing a multi-core defect in the

centre, as shown in Fig. 4. Of course, one can construct the dispersion diagram for such

a structure. The number of dispersion curves will dramatically increase compared to Fig.

1. We concentrate on the analysis of localised dilatational modes associated with the

presence of a defect. Here, we show the dispersion diagram corresponding to a certain

range of frequencies (see fig. 4). We emphasise that the localised pressure modes occur

at freoueiicies above the shaded stop band shown on the diagram of Fig. 1. The diagram created for the macro-cell with the multi-core defect exhibits additional high-frequency

band-gaps associated with the presence of the multi-core defect. The dispersion curve

corresponding to the frequency ω_cym is associated with the symmetric localised mode shown in Fig. 3, whereas the dispersion curves corresponding to frequencies around ω,_kevy

are associated with the skew symmetric localised modes shown in Fig. 2. If the multi-

core region is used for transmitting optical signals, as in photonic crystal fibres, then the

optical properties of the material will change when vibrations of frequencies ω or ω_sfcew are initiated by means of a generator 14. shown diagrammatically in Fig. 3.

In practice, the generator 14 may be positioned as appropriate in relation to the fibre 10 and may, for example, embrace the fibre or lie to the side thereof.

Although a 7 x 7 array of voids 12 is shown in the drawings, with a square sectioned fibre, in practice, the number of voids and sectional shape can be varied widely

within the scope of the invention.

Claims

IOCLAIMS

1. A method of forming an optical switch wherein high-frequency localised

dilatational modes are established in a micro-structured photo-elastic material by the

application thereto of elastic vibrations.

2. An elasto-optical dilatational switch device comprising a micro-structured

opto-elastic material arranged to transmit a signal within an acoustic or optical range of frequencies and which contains a multi-core channel within an array of voids, and a

generator of elastic vibrations coupled to the opto-elastic material for establishing

localised dilatational deformations in the material to cause changes in the optical properties of the multi-cored channel.

3. An elasto-optical dilatational switch as claimed in claim 2, wherein the

generator is arranged to transmit elastic pressure waves such as to establish localised

standing waves in the material which are either symmetric or skewsymmetric, on-off

conditions of the switch corresponding respectively to the establishment of symmetric

or skewsymmetric dilatational localised elastic vibrations initiated at frequencies ω_svro and ω_skew, respectively.

4. An elasto-optical dilatational switch as claimed in claim 2 or 3, wherein the

micro-structured opto-elastic material comprises a fibre containing an array of air-holes

extending therealong.

5. An elasto-optical dilatational switch as claimed in claim 4. wherein the micro-

structured opto-elastic fibre contains an array of n x n air-holes, e.g. 7 x 7.

6. An elasto-optical dilatational switch as claimed in any of claims 2 to 5,

wherein the micro-structured opto-elastic material satisfies the constitutive law of

photoelastic material (1) - (3), eg fused silica.