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Properties and implementation of acousto-optic superlattice tunable filters in fibers Supriyo Sinha and Karel “Double K” Edward Urbanek Ginzton Labs, Stanford University, Stanford, CA 94305 I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. Introduction Physics of the Acousto-optic Effect Introduction to Optical Fiber Properties Fiber Bragg Grating Modelling Superlattice Theory Modeling of AOSTFs Simulation Past Experimental Results Comparison with Other Filter Designs Future Directions Conclusions Acknowledgements/References I. Introduction The attraction of acousto-optic tunable filters in fiber lies in the ability to avoid coupling losses of a comparable waveguide or external device. It also makes sense to create a device of this sort in a fiber due to the fact that the efficiency is significantly improved with smaller crosssection. There has been little success in fabricating an all-fiber acousto-optic filter with narrow linewidth due to the fact that the linewidth is inversely proportional to the length of the device, and the acoustic power is attenuated along the fiber. We will investigate the physical limits of these devices and show the promise of a superlattice acousto-optic modulator. periodic perturbation. For resonant coupling to occur, the beat length of the two optical wavelengths in the fiber must match the acoustic wavelength. The photoelastic effect in glass couples the mechanical strain caused by an acoustic wave to the optical index of refraction as illustrated in the following equation 1 ij 2 pijkl Skl p n ij Examining the coefficients of the strain optic tensor, pijkl, for the material in question yields insight into the magnitude of the photoelastic effect. Isotropic fused silica has the following strain-optic tensor: 0 0 0 0.121 0.270 0.270 0.270 0.121 0.270 0 0 0 0.270 0.270 0.121 0 0 0 0 0 0.12895 0 0 0 0 0 0 0 0.12895 0 0 0 0 0 0 0.12895 Skl(p) is the strain field from an acoustic wave and is given by Skl ( p) s(ks z st ) so cos ks z st II. Physics of the Acousto-optic Effect An acoustic wave in the fiber core periodically strains the material, which creates a periodic change in refractive index. This, of course, can act like a diffraction grating. If the Bragg condition is satisfied for a given set of optical wavelengths and acoustic wavelengths, i.e.: 2 k m l then coupling occurs between the two optical waves. Here, the propagation constants of the kth and mth modes are given by k and m respectively, is the acoustic wavelength in the guide, and l is the Fourier component of the where (ksz-st) is the optical phase, ks is the acoustic wave vector, is the acoustic wavelength, s is the angular frequency and so is the peak strain value. The peak strain caused by a guided acoustic wave with area, A, power Ps, in a medium with Young’s modulus E and acoustic group velocity vgs is solved as s0 2 Ps / EA gs 1/ 2 Using equations (??) and (??), and solving for the index of refraction for light propagating along the fiber (in the z-direction) gives: 1 nx n0 n03 0.270 S sin st K s z 2 x C The acousto-optic grating propagates through the fiber at speed of /K. A z t e i t z d 0 where ξμt and ξρt are radial transverse field distributions. All of the modes, both guided and radiation, are orthogonal. The transverse component of the magnetic field is given as Ht 0 r 0 ez z t The electric and magnetic fields must satisfy the wave equation and are subject to the boundary conditions imposed by the waveguide. The solution is given by (ref) r cos x C J a sin H y neff 0 x 0 in the core where Jμ is the J-Bessel function In the cladding, the fields take the form 2 a 2 a 2 2 ncore nclad 2 n 2 ncore clad 2 2 2 n nclad neff nclad b core nclad 1 l i t z A z t e c.c. 2 1 0 0 x where Kμ is the K-Bessel function. Several normalized parameters have been introduced above which are defined below. A basic understanding of the modes in an optical fiber is beneficial to understanding the superlattice effect. The modes in an optical fiber consist of guided modes and radiation modes. An arbitrary electrical field can be decomposed into a sum of transverse guided mode amplitudes, Aμ(z), and a continuum of radiation mode amplitudes Aρ(z) with propagation constants of βμ and βρ respectively. The decomposition is represented below Et r cos K K a sin H y neff (here we will show some “back of the envelope” calculations for various fiber sizes and acoustic powers) III. Introduction to Optical Fiber Properties J b 1 2 2 The normalization constant Cμ can be solved for using Poynting vector relationship which states that the power carried by the mode is |Aμ|2. 1/ 2 0 / 0 2w C av neff e J 1 J 1 where eμ= 2 for the fundamental mode (μ=0), otherwise eμ = 1. Matching the fields at the core and cladding yields the following eigenvalue equation which can be solved to yield the propagation constants of the desired mode. J 1 u J u w K 1 w K w IV. Fiber Bragg Grating Modeling Fiber Bragg gratings (FBGs) have been an enabling technology for several applications in fiber optics such as distributed feedback lasers (ref), wavelength filters (ref), pulse compression (ref) and optical sensors (ref). In the most general sense, fiber Bragg gratings are perturbations in the dielectric tensor of the fiber waveguide that possess some degree of periodicity. Their fabrication is an extensively studied area but one that is not overly relevant to the understanding of the physics of the superlattices. The interested reader is referred to ??guys?? (ref) for a comprehensive discussion of the material science and fabrication behind FBGs. Several methods have been developed in the late 20th century to accurately describe and model electromagnetic wave propagation across periodic structures. The most popular approach is coupled mode theory (ref), in which the coupling between the eigenmodes of the unperturbed waveguide is represented by a set of differential equations. This method has the advantage that grating characteristics such as reflection efficiency and bandwidth can easily be solved. However, since coupled mode theory assumes that the eigenmodes do not change in the presence of the grating, the solution is an approximation. A second method uses Bloch wave analysis on the grating structure allowing for exact solutions, since Block waves are the natural modes of periodic media. The Bloch treatment is more invovled than coupled mode theory, but it is more appropriate for more complex structures such as superlattices or when insight into dispersion or microstructures characteristics is required. Bloch wave analysis can be extended to aperiodic structures using numerical techniques. In addition to coupled mode theory and Bloch mode theory, other purely numerical techniques based on matrix methods (Ref,ref) are often employed in computer algorithms that break the grating down into suitably small pieces, model each representative piece with it’s own matrix and then multiply them all together to obtain the transfer function of the entire structure. More physics oriented approaches also exist such as the WKB (ref), Hamiltonian (ref) and variational (ref), but they are not commonly used. In this paper, we will discuss only coupled mode theory and Bloch wave theory. The former will be used throughout the paper to model the basic characteristics of a simple periodic Bragg grating. The latter will be used to analyze how this Bragg grating behaves when modulated by a coarse grating such as an acoustic wave. This hybrid approach provides the easiest path to understanding the physics of superlattices in fibers. Coupled Wave Theory In section ??, we have derived the field distributions in the fiber. The effects of the dielectric tensor perturbation on the amplitudes and phases of the eigenmodes must now be included. We begin once again with the wave equation. 2 E 2 P E 0 0 2 2 t t The response of the polarization in the medium can be decomposed into an unperturbed term and a grating term. P Punpert P grating where Punpert E . 1 Substituting the eigenmodes of the system solved for in section ?? in the above equation results in an extremely complicated equation. Neglecting coupling to radiation modes in the structure and using the slowly varying envelope approximation (SVEA) on the field allows the wave equation to be simplified to A 1 z i t e i t z 2 c.c. 0 2 Pgating ,t t Multiplying by the conjugate of the eigenmode and then using the orthogonality property of eigenmodes, we arrive at A it z e c.c. 2i0 z 1 can be transferred between the two eigenmodes. The strength of the interaction is given by the mode overlap between the refractive index profile and the mode field which is represented in eqn ?? as the transverse integral on the right hand side. This requires that for non-zero coupling between a symmetric mode and an antisymmetric mode requires a transverse refractive index profile that is asymmetric. The second important quantity results from the principle of conservation of momentum. This dictates that for efficient energy transfer, the phase constants must be same between the two coupling modes on either side of the equation. The resulting detuning constant, ??delta_b?, given by l 0 2 * Pgrating ,t t dxdy t 2 This is a general wave propagation equation which can be used to describe coupling between all the guided modes of the fiber, both codirectional and counter-propagating (the signs of the ??beta?? will be opposite for the eigenmodes for counter-propagating. Next, the more specific case of a periodic modulation of the index of refraction is examined. The total polarization is given by P r 1 z E 2 N is zero for maximum energy transfer. Assuming the perturbation of the index of refraction is small, we can write Now, we have the tools to examine the coupling between two identical counterpropagating modes. This is the situation that we will be examining in detail in this paper. In this case (4.2.10) reduces to z 2n n z where B v n z n 1 ei [ 2 N / z z ] c.c. 2 z Substituting 4.2.17 and 4.2.16 into 4.2.14, we obtain n i [ 2 N / z z ] Ppert 2n 0 n e c.c. 2 E i z z i dc B i ac A e where we have introduced some new notation to yield some more physical insight. The derivative of ??phi versus distance defines a chirp in the grating. The DC coupling constant ??Kdc?? arises from any change in the average refractive index of the mode and is given by dc n 0 n t t *dxdy We can now substitute 4.2.20 into 4.2.13 to obtain an expression that describes the coupling between the various eigenmodes in the presence of a periodic refractive index modulation. l 2i 1 A 0 z e i t z 2 c.c. 0 2 {2n 0 n t n i [ 2 N / z z ] e c.c. 2 E t * }dxdy There are two important quantities in this expression that determine how efficiently energy Absorption, gain or scatter loss is incorporated into the magnitude and sign of the imaginary part of ??Kdc. The ac coupling constant is ac n 0 n t t *dxdy dc 2 2 The amplitude of the driving mode can be also written in a similar manner to obtain the second of the two coupled differential equations. A i z z * i dc A i ac B e z Solving the coupled equations is a simple eigenvalue problem. After using the boundary conditions that there is no driving field incident from the z = L and no driven field at the input incident at z = 0 (where L is the size of the grating), we obtain an amplitude reflection coefficient ρ given as ac sinh L S (0) R(0) sinh L i cosh L where d z 1 dc 2 dz max B 1 n n where the Bragg wavelength is given the phase matching condition. neff , neff , Coupled mode theory can also yield the grating’s bandwidth. The bandwidth, defined as the size of the central lobe of the reflectivity spectrum, (and found by setting the argument αL in (4.3.11) equal to π) is solved straightforwardly as 2 2 neff L ( ac L) 2 2 ac 2 2 The power reflection co-efficient is simply the magnitude squared of 4.3.11 ac 2 sinh 2 L 2 ac cosh 2 L 2 2 When |κac| < δ, amplitude and power reflection co-efficients become ac sin( L) i cos( L) sin( L) ac 2 2 sin 2 L 2 ac cos2 L 2 For a uniform grating there is no chirp and the peak reflectivity occurs when δ = 0 and has a value of tanh 2 ac L Bloch Wave Analysis The coupled mode approach is a one step approach to solving the grating problem. By contrast, the Bloch method uses two stages. First, the dispersion relation is solved to obtain the eigenmodes in the periodic structure. Second, the eigenmodes that are excited by the plane wave spectrum are identified. We begin with Floquet’s theorem which states that for a chosen forward wave vector kf, only wave vectors spaced by integral multiples of the grating vector are permitted as diffracted wave vectors. For full accuracy, partial waves from all wave vectors should be included in any analysis of a periodic medium. However, it can be assumed that only two of these vectors are resonantly coupled, those at kf and kf – K. This assumption breaks down at small Bragg angles when higher Fourier harmonics can no longer be neglected, but this case is not considered in this paper. 2 As we are dealing with identical counterpropagating modes, the transverse integral in 4.3.5 can be assumed be unity, yielding a peak reflectivity wavelength of A Bloch wave in a grating structure can be written as E ( z, t ) 1 ik z i ( K k f ) z V f e f Vbe eit c.c. 2 where Vf and Vb are the field amplitudes for the forward and backward waves respectively. The partial wave with the larger amplitude is referred to as the dominant wave vector. Once again we are analyzing a linear grating with a permittivity modulation given by r z r r cos Kz for z0 where εr is the relative permittivity of the unperturbed fiber. Maxwell’s equations are used with (??2) and (??above equation) to yield a homogeneous system of equations with eigenvalues kf and eigenvectors {Vf, Vb}. Using assumption that kf and ko are close enough in value to justify following approximation k 2f k02 2k0 k f k0 we have 2k0 K Thus the solution of (??5) produces independent Bloch waves for each ξ. eigenvalues can either be real or complex. traveling waves, the eigenvalues are real given by V f 0 Vb 0 where coupling constant κ is given as Mk0 / 4 M m / 1 0 1 1 where the kf+ eigenvalue corresponds to a traveling wave with a group velocity in the –z direction. The evanescent eigenvalues are given by K / 2 j 1 2 where kf+ is the wave decaying in the –z direction. We have introduced a parameter δ, defined as / 2 4 1 B / / M which distinguishes between the traveling wave and evanescent wave regimes. When δ2 > 1, traveling waves are obtained; when δ2 > 1, evanescent waves are obtained. Below is the dispersion map for the grating. and dephasing parameter, ξ, is given as two The For and k f k0 K / 2 ( / 2)(1 1/ 2 )1/ 2 (kf 2n0 4 n B 2 0 B c B Figure 1(a) When δ > 1, the dominant wave-vector is shorter than ko, the Bloch wave is called a fast wave and this case appears on the blue-shifted branch of the Bragg condition. In this case, the energy in the optical Bloch wave is redistributed in regions of low refractive index. Similarly, slow waves in which dominant wave-vector is larger than ko appear on the red-shifted branch. For slow waves δ < 1, and the energy is primarily distributed in regions of high refractive index. Solving for the group velocity yields an interesting physical insight into the Bloch wave. If we approximate the change in coupling constant with respect to frequency as zero, we solve for the group velocity as 1 k g f regions in the grating with zero optical energy density. Consequently, no power flow is possible implying that the group velocity must tend to zero at stop band and the grating traps the optical energy. Physically, it is clear why the real component of kf remains unchanged in the stop band. In the stop band, there is no dominant wavevector, thus both would have to increase or decrease equally for any index change experienced by the Bloch wave. Floquet’s theorem requires that neither wave-vector’s real component may change in the stop band, thus only the imaginary part may change. Further properties given by Bloch wave analysis such as higher-order dispersion will not be discussed here. See (ref??) for a more thorough treatment of Bloch theory applied to periodic structures. c / n0 1 1/ 2 1/ 2 This implies that the two-waves forming the Bloch wave are bound together by the grating, establishing a coherent group of discrete waves possessing the same group velocity. The intensity profile across grating is obtained for the evanescent and propagating case as I e 1 cos Kz arccos , 2 1 I p 1 1/ cos Kz , 2 1 respectively. Clearly in the stop band, the fringe visibility is unity. This means that there are Superlattice reflection sideband: efficiency s tanh 2 LJ m into mth 2 Ps EA gs Figure shows zero coupling in unperturbed Fiber Bragg grating in (a). In (b), an acoustic wave allows coupling between forward and backwards traveling Bloch waves. If we look at the dispersion relation, we see the Bragg condition relates the momentum of the counter propagating optical waves to the acoustic wave. Figure. Frequency wave-vector diagram for a Bragg grating. In A, a forward traveling acoustic wave vector couples a forward-traveling Bloch wave to a downshifted backwards traveling Bloch wave. In case B, a backwards traveling Bloch wave couples a forward-traveling Bloch wave into an anamalously downshifted backward Bloch wave.