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Project no: NMP-CT-2003-505699 Project acronym: SPP Project title: Surface Plasmon Photonics Instrument: STREP Thematic Priority: NMP D14 Report assessing theoretical models for density of photonic states in periodically textured metallic systems Due date of deliverable: Month 24 Actual submission date: Month 24 Start date of project: January 1st 2004 Duration: 3 Years Organisation name of lead contractor for this deliverable: UAM _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 1 of 11 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Deliverable 14: Report assessing theoretical models for density of photonic states in periodically textured metallic systems. (Part of WP3 – Surface Plasmon Devices). Due: Month 24. Partners: UNEXE, KFUG, ULP, UAM, UZ, IC Contents. 1 Introduction/Overview ....................................................................................................... 3 2 Previous work (before project commenced) ...................................................................... 4 3 Results from project. .......................................................................................................... 6 4 Overall summary ................................................................................................................ 9 Project co-funded by the European Commission within the Sixth Framework Programme (2002-2006) Dissemination Level PU Public X PP Restricted to other programme participants (including the Commission Services) RE Restricted to a group specified by the consortium (including the Commission Services) CO Confidential, only for members of the consortium (including the Commission Services) _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 2 of 11 Deliverable 14 Report assessing theoretical models for density of photonic states in periodically textured metallic systems. ___________________________________________________________________________ 1 Introduction/Overview The photonic density of states is a quantitative measure of the extent to which a structure (surface, particle etc..) modifies the vacuum fluctuations at optical frequencies from their free space values. This concept was first articulated by Purcell in a short note published in 1946 [1], and elaborated in a series of articles in the mid 1970s [2-5]. To give an example of where the density of photonic states plays a role, the extent to which the photonic mode density deviates from the value 1 indicates to what extend a process such as fluorescence might be modified [6]. In the context of the present project the density of photonic states was anticipated at the project design stage as being of potential importance in achieving our objectives. However, there are two reasons why this has become less important to us than we anticipated. Firstly, very considerable progress has been made by others – see for example the beautiful optical corral demonstration by Dereux et al. [7] – and very ably reviewed by Girard [8]. Second, such work has been developed primarily in the context of scanning near-field optical microscopy (NSOM). Although we have made extensive use of this technique in the present project to map optical fields, we have not had occasion to need the capacity to predict the photonic density of states. Nonetheless, it is possible that we may need to make use of such calculations in the final year of the project and for this reason we have retained activity so as to be able to establish a numerical code quickly should the need arise. In what follows we assess the techniques that have been employed by others and indicate why there is really only way to provide anything that is better than that which has gone before, that is a technique specifically targeted at calculating the photonic density of states near nanostructured metals – a technique based on the finite difference time domain approach. _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 3 of 11 2 Previous work (before project commenced) The local density of photonic states (LDOPS), r), at location r and for a frequency of light is defined through the imaginary part of the trace of the retarded dyadic Green’s function, GR(,r,r) (a 6 x 6 matrix), 6 1 ( , r ) Im G R ( , r , r ) (1) jj j 1 G R (, r, r ) measures how a perturbation (an electromagnetic wave in this case) generated jj at position r' and at a frequency propagates to another point r. It is a 6 x 6 matrix because there are potentially six field components to the perturbations E x , E y , Ez , Hx , H y , Hz . For the calculation of the photonic density of states we need to calculate this quantity for r = r' and the trace of the 6 x 6 matrix. In equation 1, the first three terms in the j-sum of equation (1) run over the three components of the E-field and the last three terms run over the three components of the H-field. In principle, a theoretical formalism working in the frequency domain such as the Transfer Matrix method (TM) might be ideally suited to calculate Green’s functions, and has certainly been used to good effect in many situations before [9]. However, we know that the TM approach does not give accurate results for 3D periodically textured metallic systems that are the typical structures we intend to analyze within this project (see D5 “Report providing specification of computational techniques appropriate for modelling fields associated with SPs”). Consequently the best numerical candidate in order to treat this type of system is the FDTD (Finite Difference Time Domain) theoretical framework. Some years ago the team of John Pendry at Imperial College developed FDTD numerical code for calculating Green’s functions for periodically structure dielectrics (photonic crystals). Details of this numerical framework can be found in references [10, 11]. Here we provide a very brief summary. Within FDTD formalism, Green’s functions are obtained in time-domain, GR(t,r,r). The relation between time and frequency domains is just a Fourier transform integral: _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 4 of 11 G ( , r , r ) G R (t , r , r )eit dt R jj jj (2) 0 In the usual way, the dyadic GR(t,r,r), is just the solution of the quasi-homogeneous equation involving Maxwell equations: 1 I M G R (t , r , r' ) I (t ) (r r' ) (3) c t where c is the velocity of light, I is the identity matrix and the spatial operator M can be written (in Gaussian unit system) as, 1 0 - x (r ) M (4) 1 ( r ) x 0 with (r) being the local relative electric permeability and (r) the corresponding relative magnetic permittivity. The physical meaning of equation 3 is as follows, The (i,j) component of tensor (matrix) GR(t,r,r’) measures the magnitude of the i-component of the electromagnetic (EM) field at the point r and at time t after the EM propagation of the j- component of the EM-field located at point r’ at t = 0. Within the FDTD framework, the time-derivatives in equation 3 are approximated in the usual by, that is, F(r , t ) F ( r , t t ) F( r , t ) (5) t whereas for the spatial coordinates finite differences are also taken in place of derivatives, F(r , t ) F ( r a , t ) F( r , t ) (6) x Therefore, in order to calculate the local density of photonic states r) at point r for a given structure we have to run six numerical time propagations of equation 3 with six different seeds (field components) located at r at t = 0 by setting the initial fields to be delta functions in space for each one of the six EM-field components, whilst the other five are set to zero. _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 5 of 11 3 Results from project. As explained above, the numerical code developed at Imperial College was able to treat only dielectric media, in which the dielectric permeability does not depend on time. Clearly such restrictions will be over at best very limited value in computing the photonic density of states associated with metallic nanostructures. What we have done within the project is to extend this FDTD algorithm so that we are able to treat dispersive materials such as metals. Due to time invariance, the relationship between the displacement field, D and the electric field E is very simple in the frequency domain, it is, D() = E However, as in the original FDTD implementation we work in time domain, the relation between the two fields (at a given point r) then involves a time integral, t D(t ) ( ) E (t )d (7) 0 where (is the Fourier transform of 1 ( )e i ( ) (8) 2 From this it would appear that in order to update the displacement field we need to know the electric field at all previous times. There are several ways to deal with this problem that keep the most important characteristic of FDTD - the equations are almost local in time, so that the fields can be updated efficiently. For developing our numerical code, we have chosen to work within the Piecewise-Linear Recursive Convolution (PLRC) scheme whose technical details can be found in [12]. Here we give the details on how this method treats the time derivatives of the displacement field that is just the new ingredient needed when dealing with dispersive materials such as metals. The rest of the theoretical framework and the corresponding numerical code remain unaltered with respect to the case of dielectrics. The basic idea within PLRC is to approximate the integral of equation 7 by evaluating at discrete time the values of D and E. The displacement field D at time t=nt (n being an integer) can be written as: n 1 Dn ( m ) E n m ( m ) ( E n-m -1 E n m ) (9) m 0 where, ( m 1) t ( m) ( )d (10) m t _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 6 of 11 and, 1 ( m1) t t m t ( m) ( )(m t )d (11) In the above integration the continuous time function E(t) was approximated, at each time interval, by the linear interpolation of the values of E(t) at the beginning and end of the time interval. Therefore, in the discrete time evolution it is possible to relate Dn+1 with Dn simply by, Dn 1 Dn ( (0) (0) ) E n 1 (0) E n n (12) where, n 1 n ( m ) En m ( m ) ( En-m -1 En m ) (13) m 0 with, ( m) ( m) ( m1) (14) ( m) ( m) ( m1) (15) Equation 11 gives us the clue about how to update the displacement field. We just have to keep track of n during the time evolution of the EM-fields. The different magnitudes appearing in equations 11-14, in particular the set [ have very simple expressions when the metal can be treated within the Drude formula, i.e. when the relative permittivity of the metal can be written as, p2 ( ) 1 (16) ( i ) For this particular case, p2 ( ) ( ) (1 e )( ) (17) being the Delta function and the Heaviside time-step function. _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 7 of 11 During the last months we have been checking the above-described new algorithm to treat dispersive materials by comparing the new algorithm against experimental data. We have performed this verification by undertaking some numerical simulations on 2D square arrays of rectangular holes perforated on a thick silver film. We have studied the transmission properties of these arrays under the condition of normally incident plane wave light and we have compared our theoretical findings with experimental results obtained by us recently. In figures 1 and 2 we present the main results of this study. The data show good agreement between numerical simulations and experimental data, giving us confidence in the suitability of the FDTD numerical code to treat in a realistic way the optical properties of periodically textured metallic surfaces. Figure 1. FDTD simulations of the normalized-to-area transmittance as a function of wavelength (in nm) for a normal incident plane wavelength impinging at a 2D square hole array (different periods of the array ranging from P = 325 nm to P = 775 nm) made of rectangular holes of dimensions d x = 260 nm and dy = 200 nm. The thickness of the silver film is 400nm. The polarization of the incident wave is such that the E-field is pointing along the short side of the rectangle. In the inset are shown the corresponding experimental data. _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 8 of 11 Figure 2. As in Figure 1, but in this case the polarization of the normal incident plane wave is such that the E- field is pointing along the long side of the rectangle. Again, in the inset we show the experimental results obtained by the ULP team. 4 Overall summary The key step we have made is to identify the most practical technique for determining the photonic density of states associated with metallic nanostructures – an extension of the FDTD technique that has been so successfully used in the field of photonic band gap materials. The natural alternative is a Transfer Matrix type approach, but as we discussed in D5, such an approach is not able to represent the structured metallic systems we require well enough. We have implemented this modified version of FDTD and verified it by comparison with experimental data. Should the need to perform photonic density of states calculations arise we are well placed to undertake them. However, as noted in the introduction, we have thus _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 9 of 11 far had far greater need of capabilities to compute field strengths (something that is very closely related to the density of photonic states) and have accordingly devoted more effort to this. Previous results Results from SPP project Conclusions Previous results Finite Difference Time Although we have not yet No capacity within project to Domain (FDTD) capability needed this capability it calculate density of states. now established capable of exists in case we do need it in giving density of states. WP3. 5 References 1 "Spontaneous emission probabilities at radio frequencies" E. M. Purcell Physical Review, (1946), 69, pp 681 2 "Quantum electrodynamics in the presence of dielectrics and conductors. I. Electromagnetic-field response functions and black-body fluctuations in finite geometries" G. S. Agarwal Physical Review A, (1975), 11, pp 230-242 3 "Quantum electrodynamics in the presence of dielectrics and conductors. II. Theory of dispersion forces" G. S. Agarwal Physical Review A, (1975), 11, pp 243-252 4 "Quantum electrodynamics in the presence of dielectrics and conductors. III. Relations among one-photon transition probabilities in stationary and nonstationary fields, density of states, the field-correlation functions, and surface-dependent response functions" G. S. Agarwal Physical Review A, (1975), 11, pp 253-264 5 "Quantum electrodynamics in the presence of dielectrics and conductors. IV. General theory for spontaneous emission in finite geometries" G. S. Agarwal Physical Review A, (1975), 12, pp 1475-1497 6 "Fluorescence near interfaces: the role of photonic mode density" W. L. Barnes Journal of Modern Optics, (1998), 45, pp 661-699 7 "Subwavelength mapping of surface photonic states" A. Dereux, C. Girard, C. Chicanne, G. Colas des Francs, T. David, E. Bourillot, Y. Lacroute and J.-C. Weeber Nanotechnology, (2003), 14, pp 935-938 8 "Near fields in nanostructures" C. Girard Reports on Progress in Physics, (2005), 68, pp 1883-1933 9 "Green's functions for Maxwell's equations: Application to spontaneous emission" _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 10 of 11 F. Wijnands, J. B. Pendry, F. J. GarciaVidal, P. M. Bell, P. J. Roberts and L. M. Moreno Optical and Quantum Electronics, (1997), 29, pp 199-216 10 "Calculating photonic Green's functions using a nonorthogonal finite-difference time- domain method" A. J. Ward and J. B. Pendry Physical Review B, (1998), 58, pp 7252-7259 11 "A program for calculating photonic band structures, Green’s functions and transmission/reflection coefficients using a non-orthogonal FDTD method" A. J. Ward and J. B. Pendry Computer Physics Communications, (2000), 128, pp 590-621 12 "Computational Electrodynamics: the finite-difference time-domain method" A. Taflove Artech House, (1995). _________________________________________________________________________________________ Deliverable 14 Surface Plasmon Photonics (SPP) NMP-CT-2003-505699 Page 11 of 11