Surface Plasmon Photonics by 4T2tc7

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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




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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)


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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.




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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:




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                                                      
                                  G ( , r , r )   G R (t , r , r )eit 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.



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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=nt (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

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and,
                                              1 ( m1) 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)   ( m1)                  (14)


                                               ( m)   ( m)   ( m1)                  (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.

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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.



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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

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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"

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       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).




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