Qabs for gold in IR
Maxim Yurkin
research, which I think will fit in the framework of this group.
We had a discussion with Baptiste some time ago on accuracy of simulations of scattering cross section for gold
particles using DDA (ADDA in particular). Then we concluded that it can be done, quite easily. Recently David (I hope he
will also join this discussion) was trying to do some simulations of _absorption_ cross sections for gold spheres and he
had large errors for large wavelength (say 0.8 um). I had done some quick simulations on my laptop (using grid sizes
from 16 to 64) for gold spheres with diameter D=30 nm and wavelength 0.82 um and compared it with Mie. I have used
standard (LDR) formulation and the FCD (see manual for description). The errors in Qsca are indeed satisfactory (about
5%), but for Qabs they are huge (about 100%)! I believe using grid sizes as large as 256 may bring Qabs to satisfactory
accuracy as well, but that means much large computational complexity (supercomputer, etc.). Moreover, I have seen
similar effects when modelling other "extreme" refractive indices using LDR and FCD.
The questions are:
Do people (i.e. in published literature) use absorption cross section of gold in IR in any applications? Light heating
is one option, but I do not know the typical wavelengths.
If they do, how do they compute it (for non-spherical shapes)?
If they use DDA, are they aware of its real (in)accuracy in this regime?
Is there any published accuracy results for Qabs using DDA in IR?
In summary, how to compute absorption of small gold particles in IR?
Maxim.
Alfons Hoekstra
surprising, isn't it? How was Qabs computed, from Qext or directly?
Bye,
Alfons
Maxim Yurkin wrote:
Alfons G. Hoekstra
Section Computational Science
University of Amsterdam
+3120 5257543
http://www.science.uva.nl/research/scs
http://www.science.uva.nl/~alfons
International Master Computational Science
http://www.studeren.uva.nl/msc_computational_science
DaviddeKanter
Hi Alfons and Maxim,
Qext is independently computed isn't it ?
I'll make more simulations with higher grids even for next week, and
see what the result is and post it later.
But we must bear in mind, that we have a situation in contrary to the
situation of the HWBC's we computed earlier, we have very many dipoles
per wavelength; so it becomes very much a computational problem.
It would be logical, physically speaking , to use at most 50 dipoles
per wavelength; however I have seen earlier, that the error is very
large, but it is challenging how DDA works in this regime, although
physically speaking it is in my opinion a somewhat artificial
situation, completely different from the large HWPC's
bye, David.
On Apr 9, 11:40 am, Alfons Hoekstra <A.G.Hoeks...@uva.nl> wrote:
> Hi Maxim,
> surprising, isn't it? How was Qabs computed, from Qext or directly?
> Bye,
> Alfons
> Maxim Yurkin wrote:Baptiste, I have noticed that you have joined this group. And it happens that I have a question related to your research, which I think will fit in the framework of this group. We had a discussion with Baptiste some time ago on accuracy of simulations of scattering cross section for gold particles using DDA (ADDA in particular). Then we concluded that it can be done, quite easily. Recently David (I hope he will also join this discussion) was trying to do some simulations of _absorption_ cross sections for gold spheres and he had large errors for large wavelength (say 0.8 um). I had done some quick simulations on my laptop (using grid sizes from 16 to 64) for gold spheres with diameter D=30 nm and wavelength 0.82 um and compared it with Mie. I have used standard (LDR) formulation and the FCD (see manual for description). The errors in Qsca are indeed satisfactory (about 5%), but for Qabs they are huge (about 100%)! I believe using grid sizes as large as 256 may bring Qabs to satisfactory accuracy as well, but that means much large computational complexity (supercomputer, etc.). Moreover, I have seen similar effects when modelling other "extreme" refractive indices using LDR and FCD. The questions are: Do people (i.e. in published literature) use absorption cross section of gold in IR in any applications? Light heating is one option, but I do not know the typical wavelengths. If they do, how do they compute it (for non-spherical shapes)? If they use DDA, are they aware of its real (in)accuracy in this regime? Is there any published accuracy results for Qabs using DDA in IR? In summary, how to compute absorption of small gold particles in IR? Maxim.
>
> Section Computational Science
> University of Amsterdam
> +3120 5257543http://www.science.uva.nl/research/scshttp://www.science.uva.nl/~alfons
> International Master Computational Sciencehttp://www.studeren.uva.nl/msc_computational_science
>
>
>
> 53KViewDownload
>
> image_png_part
> 51KViewDownload
Maxim Yurkin
inaccurate, while their difference _is_ accurate. Qsca can be also computed by integration, but this will make no
difference (see Section 11.3 of the manual). As for the physical reasons, we can think something like: Qabs is more
dependent on the boundary dipoles than Qsca, hence it should be less accurate (the boundary dipoles are least accurate
described by DDA) - but this is too vague. For small particles both Qsca and Qabs is determined by the total
polarizability of the particle (alpha): Qsca~|alpha|^2, Qabs~Im(alpha). So it seems, that absolute value of alpha is
determined much more accurately than its phase - I have no idea why.
Even more complicating is the fact, that in some sense even D=30nm can be not completely in Rayleigh regime (when such
extreme refractive indices are involved).
About large dpl values - it is better just to forget about them. They may well become say a million for sufficiently
small particles. So the grid value (e.g. size along one chosen dimension, n_x for ADDA) is the only relevant value. And
this value increases with refractive index (more precisely, with refractive index becoming more extreme). It is hard to
justify how large or small it should be based on some common sense. If we think in terms of discretizing integral
equation, then necessary discretization depends on the derivatives of the functions involved - and they may become
large. The situation, most probably, can be improved by using more "clever" discretization, such as different basis
functions or other tricks, but this is an open research question.
Maxim.
DaviddeKanter
well , interesting; I have some simulations on LISA running up
to grid 128, good Easter days, David
baptiste
Thanks Maxim for bringing my attention to this interesting topic, I
forgot to check this group after I subscribed (i'm also very busy
completing my thesis).
I won't be of much help as I've mostly used Thomas Wriedt's T-matrix
code to model the scattering and extinction spectra of non-spherical
gold particles in the visible and IR. I have obtained several spectra
for ellipsoids that I trust using Wriedt's code (I compared it with
the separation of variables method, A. Moroz' improvement to
Mishchenko's code, Mie theory, etc.). ADDA was however just too slow
in comparison for the simple purpose of calculating far-field cross-
sections, so I never went far in the comparison.
I do have a few comments and several references at the end (ask me if
you can't access them), but bear with my little knowledge of
scattering theory,
> For small particles both Qsca and Qabs is determined by the total
> polarizability of the particle (alpha): Qsca~|alpha|^2, Qabs~Im(alpha). So it seems, that absolute value of alpha is
> determined much more accurately than its phase - I have no idea why.
particles has been proposed in the literature [1,2,3] that accounts
for the radiative damping and dynamic depolarisation. I believe this
last effect is the reason why the phase is more sensitive in gold
particles. Perhaps a good improvement in the DDA scheme would be to
start the calculation not from the first Born approximation for the
internal polarisation but using this approximation following Meier and
Wokaun[1]. Also, because of the excitation of the surface charge
density on a thickness of order the skin depth, it may be that the
surface dipoles require a special consideration [12].
> this value increases with refractive index (more precisely, with refractive index becoming more extreme).
(*)), but below unity and with a strong dispersion and absorption. The
permittivity is large in absolute value. The energy density of the
field in dispersive media depends on the slope of the dielectric
function with respect to frequency, which may be another hint as to
why the calculation is behaving like this.
> Do people (i.e. in published literature) use absorption cross section of gold in IR in any applications?
analysed where the absorption dominates over scattering: a measurement
of extinction (quite common) is therefore almost identical to
absorption measurements [9]. Some studies have considered the melting
of nanorods due to Joule heating upon resonant excitation of a
localised plasmon mode [10]. In this case absorption is indeed the
important observable. Further, as far as plasmonics is concerned, the
quality factor of the resonance is a subject of much research interest
(quite low, generally around 10) and this is partly dictated by the
energy loss per cycle [8].
>If they do, how do they compute it (for non-spherical shapes)?
commercial packages), DDA (I think DDSCAT is the most common
implementation), and T-matrix codes (BEM). [4--7]
>If they use DDA, are they aware of its real (in)accuracy in this regime?
recent version of ADDA introduced a bug, but you fixed it before i
could worry about it).
>Is there any published accuracy results for Qabs using DDA in IR?
in the past a modified version of DDSCAT to model gold and silver
particles.
Bye for now,
baptiste
1. Meier and Wokaun. Enhanced fields on large metal particles: dynamic
depolarization. Optics Letters (1983) vol. 8
2. Moroz. Depolarization field of spheroidal particles. (submitted
2009)
3. Kuwata-Gonokami et al. Resonant light scattering from metal
nanoparticles: Practical analysis beyond Rayleigh approximation. App.
Phys. Lett. (2003) vol. 83 (22) pp. 4625--4627
#DDA for metal particles
4. Jensen et al. Electrodynamics of Noble Metal Nanoparticles and
Nanoparticle Clusters. Journal of Cluster Science (1999) vol. V10 (2)
pp. 295--317
5. Kelly et al. The Optical Properties of Metal Nanoparticles: The
Influence of Size, Shape, and Dielectric Environment. Journal of
Physical Chemistry B (2003) vol. 107 (3) pp. 668--677
6. Pecharromán et al. Redshift of surface plasmon modes of small gold
rods due to their atomic roughness and end-cap geometry. Physical
Review B (2008) vol. 77 (3)
# Modelling of gold particles
7. Myroshnychenko et al. Modelling the optical response of gold
nanoparticles. Chem Soc Rev (2008) vol. 37 (9) pp. 1792-1805
8. Wang and Shen. General Properties of Local Plasmons in Metal
Nanostructures. Physical Review Letters (2006) vol. 97 (20)
#Measurements
9. Sonnichsen et al. Drastic Reduction of Plasmon Damping in Gold
Nanorods. Phys. Rev. Lett. (2002) vol. 88 (7) pp. 077402
10. Mulvaney. On the temperature stability of gold nanorods:
comparison between thermal and ultrafast laser-induced heating .
(2006)
#Absorption
11. Teperik and Popov. Void plasmons and total absorption of light in
nanoporous metallic films. Physical Review B (2005) vol. 71 (8)
#Surface dipoles
12. Kooij and Poelsema. Shape and size effects in the optical
properties of metallic nanorods. Physical Chemistry Chemical Physics
(2006) vol. 8 pp. 3349--3357
13. Collinge and Draine. Discrete-dipole approximation with
polarizabilities that account for both finite wavelength and target
geometry. J Opt Soc Am A (2004) vol. 21 (10) pp. 2023-2028
(*) http://picasaweb.google.com/lh/photo/1agX2VXhVR1pl9eHSmHPEw?feat=directlink
Maxim Yurkin
quick comments:
>> this value increases with refractive index (more precisely, with refractive index becoming more extreme).
> The refractive index for gold is actually not large per se (plot here (*)), but below unity and with a strong
> dispersion and absorption.
Yes, that is what I mean by "extreme" - those values that are far from usual range (1--2 + 0--1i) in complex plane. So
both large absolute values (like 10+10i) and small real part (like 0.1 + 2i). Actually, I studied both with respect to
FCD formulation, and it seems that the latter in some respect are even worse for DDA than the former.
>> If they do, how do they compute it (for non-spherical shapes)?
>
> Most of the literature seems to use either FDTD, FEM, (mostly commercial packages), DDA (I think DDSCAT is the most
> common implementation), and T-matrix codes (BEM). [4--7]
>
>> If they use DDA, are they aware of its real (in)accuracy in this regime?
>
> I have never heard of such large discrepancy before (except when a recent version of ADDA introduced a bug, but you
> fixed it before i could worry about it).
Then it seems that a quick comparison with DDSCAT can add some confidence in that effect. I believe that ADDA do not
have any particular problems in this case, but additional verification is always good.
>> Is there any published accuracy results for Qabs using DDA in IR?
>
> You may want to contact George Schatz (Northwestern Uni), he has used in the past a modified version of DDSCAT to
> model gold and silver particles.
I will contact him.
Maxim.
Maxim Yurkin
following:
When dealing with metallic nanoparticles researchers are interested in spectrum and not in cross section values at a
particular wavelength. If you shift a resonance peak a little bit relative to the exact one, these two spectra will seem
still similar. However, at a particular wavelength the value can easily differ by a factor of two. In a few papers I
have found statement like: "We choose number of dipoles so that further increase of it do not significantly improve the
accuracy", but without any quantitative criteria for the accuracy. In one paper:
I.O. Sosa, C. Noguez, and R.G. Barrera, “Optical properties of metal nanoparticles with arbitrary shapes,” The Journal
of Physical Chemistry B, vol. 107, Jul. 2003, pp. 6269-6275.
the researchers had some ripples in their spectra, which they say is most probably due to the convergence problems of
DDA, but again they believe their simulations are accurate enough (without any numbers given for the accuracy).
Maxim.
DaviddeKanter
Thanks for all the information I did not know, and the references
Baptiste. for ADDA nQabs and nQscat I have examined the full spectrum
for grid 128, which in in quite good accordance with MIE,
I will publish this plot next week tuesday. , it is interesting what
happens at the resonance peak; Alfons will be back from a trip next
week; I will discuss with him and talk to him what I can do next,
David.
baptiste
stumbled upon this article that seems quite relevant,
Phys. Rev. E 70, 036606 (2004)
Coupled dipole method for scatterers with large permittivity
http://link.aps.org/doi/10.1103/PhysRevE.70.036606
baptiste
Maxim Yurkin
> relevant,
>
> Phys. Rev. E 70, 036606 (2004) Coupled dipole method for scatterers with large permittivity
> http://link.aps.org/doi/10.1103/PhysRevE.70.036606
I am aware of this paper. The idea presented there for scatterers with large permittivity is indeed interesting.
However, it involves integration of highly oscillatory functions, which is hard to implement numerically (to be robust
and relatively fast). That's why it is not yet implemented in ADDA, and it is not possible to produce a lot of benchmark
results (so we can only use those presented in this paper). And they show results only in positive quadrant of
permittivity (both Re and Im parts are positive), while our "extreme" values of interest lies near the negative real axis.
By the way, I have also contacted George Schatz and he said, that researchers are aware of some limitations to DDA
accuracy, but they believe that effects of not perfectly known size, shape and dielectric constants (in range 1-2 um)
are more significant. But that probably applies to similarity of the spectra, not to accuracy of simulation for a single
wavelength.
Maxim.
DaviddeKanter
Hi, I dont know whether it will be needed to implement this, I will
discuss this with Alfons,
in the meantime II have made plots of nQabs and nQsca for ADDA grids
up to 128, for the worst case, for a particle value of 0.03 micron,
but also nQsca for 0.03 and 0.001 micron, where in the last case one
can see that nQsca is not that accurate with respect to MIE, I will
discuss with Alfons if this is accurate enough for our purpose,
bye David.
PS I dont know how to attach the plots now
DaviddeKanter
Maxim Yurkin
http://dx.doi.org/10.1016/j.jqsrt.2009.02.020
They studied DDA accuracy for water particles smaller than wavelength for radar frequencies (refractive index 8.3+2.3i)
and also noted large errors for Qabs in certain cases. But that is another class of extreme refractive indices.
Maxim.
vlad
I did a validation at 500 nm for Au particles from 10 nm to 100 nm in
diameter.
at that point, n = 0.6 + 1.9 i. I had agreement with Mie for all
sizes to better than 5% for both sca and abs.
I was using grid sizes of 30. Did I just get lucky with these n, k
values?
vlad
guess helps to keep the index contrast lower. Au particles rarely
float in vacuum, you know!
Maxim Yurkin
> agreement with Mie for all sizes to better than 5% for both sca and abs. I was using grid sizes of 30. Did I just get
> lucky with these n, k values?
I think that this value of refractive index is a moderate one, so good agreement (with grid size 30) is not surprising.
And placing gold in some medium also helps, of course. However, as noted in this discussion before, a systematic study
of DDA accuracy for gold (especially in IR range) is still lacking (to the best of our knowledge). So it is hard to make
any predictions before actually testing the accuracy for a particular range.
Maxim.
vlad
student time-- it may happen in the next couple of months. I will be
happy to post progress..
baptiste auguie
instead of doing a thorough comparison of (a)dda for gold at specific
wavelengths --- and learn some figure of confidence for the code when
applied to this material, would it make sense to :
1) identify the most likely parameter that makes the computation so
inaccurate with gold as opposed to glass at this wavelength (say, the
phase shift across the particle, or the field gradient, or some more
complicated derived variable). I'm guessing that the paper I suggested
last time (*) has some hints into the origin of the problem.
(*) Phys. Rev. E 70, 036606 (2004) Chaumet et al.
permittivity artificially to span the space of this parameter (go
smoothly from a 'good' situation to a 'bad' one) and quantify the
evolution of the accuracy of (a)dda with respect to this parameter
alone.
That way, we could obtain: 1) a better understanding of the physical
origin for the failure of DDA in predicting absorption in gold
nanoparticles; 2) a generic result that could apply to other materials
(say, silver, or silicon at THz frequencies, etc...).
Now the tricky (but interesting) question is really to find what this
relevant parameter could be. I saw that the latest version of DDSCAT
can model infinite 2D planar sheets. Perhaps it could be a good
exercise to see the accuracy of the calculated absorption in such a
minimalistic system against a rigorous calculation using Fresnel
coefficients as a function of skin depth for a fixed wavelength? If my
guess is correct, there would be some sort of scaling law of the
required density of dipoles inversely proportional to the skin depth.
Does this make sense?
Best,
baptiste
_____________________________
Baptiste Auguié
School of Physics
University of Exeter
Stocker Road,
Exeter, Devon,
EX4 4QL, UK
Phone: +44 1392 264187
http://newton.ex.ac.uk/research/emag
______________________________
Maxim Yurkin
understand (at least empirically) the dependence of the accuracy on all the relevant parameters (size parameter,
refractive index, shape, dipole size) with a predictive power. Such a dependence may be used to set dipole size (grid
size) for a particular scattering problem a priori, and to estimate realistic computational time based on a fixed
requirement for the accuracy.
The problem in reaching this goal is basically the number of free parameters involved, as well as lack for some definite
theoretical hint for the "grand parameter". The obvious choice is to study the whole range of complex refractive index
(the one studied in the paper by Chaumet et al plus the domain of metallic refractive indices - negative real part of
permittivity) and then try to catch some trends from the results.
One more problem is that the conclusions may depend on the scattering quantity, which is simulated. And it is hard to
choose a universal one. What you propose with 2D planar sheets is just another possible scattering quantity. Shape is
also an important factor. Accuracy study can be done easily for (coated) spheres and probably a few other axisymmetric
shapes (comparing to T-matrix method). But some other shapes, like cubes or rough particles, which may have different
behavior of errors, are not that trivial to study due to the uncertainty and computational complexity of the reference
methods.
Having said all the above, I think that you proposed the correct development direction. And limiting oneself to the gold
is just a first step, which is significantly simpler and has some practical value by itself.
Maxim.
baptiste auguie
I suggested looking at a planar system to get rid off the dependency
on the shape description and simplify as much as possible the field
distribution. The observable should be the absorption as it is the
main subject of our discussion. I think that the wavelength is not so
relevant (scaling rule), so the only free parameters left are the
index and density of dipoles. I get the feeling that spanning the
complex plane is not as enlightening (understand easy to draw
conclusions on) than having an intermediate variable such as the decay
length but that's just an educated guess. Another option is to take a
physical model for the permittivity (say, Drude model) and vary the
plasma frequency and damping parameter.
But, perhaps this is too naive and the influence of shape and material
are correlated (sort of non-local effects I guess), is that so?
baptiste
Maxim Yurkin
> possible the field distribution. The observable should be the absorption as it is the main subject of our discussion.
> I think that the wavelength is not so relevant (scaling rule), so the only free parameters left are the index and
> density of dipoles.
layer additionally to what you specified). However, there are two possible caveats: how accurate is the formulation of
DDA for 2D periodic arrays, when the period is equal to particle size (that is required to model a homogeneous layer).
As I know some kind of Fourier transform is involved, which may have poor convergence in such a particular range
(although it may be perfectly OK). The other, more important, issue is that currently DDSCAT do not include FCD (which
is most probably should be included in the testing), and implementing FCD _together_ with 2D periodic arrays seem to be
not trivial at all. The FCD modifies the expression for Green's tensor, which should be summed (with some Fourier
harmonics) anew to get the formulae required for 2D periodic arrays.
> I get the feeling that spanning the complex plane is not as enlightening (understand easy to draw
> conclusions on) than having an intermediate variable such as the decay length but that's just an educated guess.
> Another option is to take a physical model for the permittivity (say, Drude model) and vary the plasma frequency and
> damping parameter.
Let's think in terms of number of parameters. To describe the refractive index two real parameters are required. If you
choose any two other parameters (like Drude parameters) that map one-to-one into the original Re and Im parts, then it
is perfectly fine. But such a mapping can also be done afterwards - i.e. you may first compute accuracy in complex plane
of refractive index and then perform any mapping to get dependence of accuracy on a different pair of parameters. This
way you may try to "guess" a pair in which one is a "strong" parameter and one is "weak".
If you choose a "strong" parameter (like decay length) a priori based on some physical insight, you will cut down the
simulation burden significantly (one parameter instead of two). But you will have less confidence in your conclusions
(based on the validity of the physical insight). And, strictly speaking, to get enough confidence you have only one
option - to study the dependence of accuracy on the "hidden" parameter (complementary to the strong one), which is
equivalent to covering the whole complex plane of refractive index.
> But, perhaps this is too naive and the influence of shape and material are correlated (sort of non-local effects I
> guess), is that so?
Of course, both shapes and materials are important. But whether it is strongly correlated or their influence can be
(approximately) separated is unclear. However, I can describe two examples:
1) When I was developing Second Order (SO) formulation for dipole polarizability (implemented in ADDA as "so", but not
yet published) a couple of years ago, I was testing it on a homogeneous infinite medium. I was choosing a large cube of
dipoles and calculating influence of all dipoles on the central one, and then compared it with the exact answer for a
propagating plane wave. In some respect it is similar to the analytical derivation of LDR formulation. SO was much more
better than all other formulations for this artificial case actually reaching fourth order convergence (in dipole size).
But it did not perform that well on usual particles compared to wavelength being comparable to other formulations.
Basically, that is the reason I haven't published SO yet.
2) I saw in many accuracy studies that error behavior for spheres and cubes can be significantly different. This is
probably because shape errors (caused by imperfect description of particle shape by cubical subvolumes) have different
dependence on problem parameters than inherent discretization errors (due to finite dipole size). There is also an
opinion that spheres are different from all (or at least a majority of) other shapes in this respect.
So I guess sooner or later we will need to study different shapes to make definite conclusions.
Maxim.
Rick
Sorry to join the discussion so late. I have just recently started
doing computational work again after a long experimental period.
I have spent a lot of time studying the issue of DDA calculations for
gold nanorods in the near infrared. We recently published some work
using some of these results [1]. I was aware of the problems
associated with calculating Qabs, and I tried to get around it by
using massive number of dipoles (up to 256 across the nanorod
diameter!), which is a very brute force approach. I have plots of the
convergence for nanosphere cross section values with increasing number
of dipoles that I can send if you are interested. In this case, using
gold spheres with diametrs ranging from 20 - 150 nm, I found residual
errors in the Qabs calculation of 15 - 25% even for very large numbers
of dipoleswhen compared with Mie calculations. The values for Qsca did
agree, though.
For our applications (cell imaging and ablation using plasmonic
enhancement), we are interested in the absolute magnitude of the three
cross-sections, and also the near fields. I have implemented a method
for calculating near fields and Poynting vectors from dipole
polarizations, but as of now it is very, very slow in Matlab. I
benchmarked my ADDA calculations with near-field calculations with Mie
theory with acceptable errors (I have some images for this too).
I also always use size (and damping)-corrected values for the
refractive indices (using the methods in [2] and [3]), but this
doesn't really matter for comparisons between the Mie and DDA
calculations. I am also always trying to find nice experimental works
where the Cabs for single particles can be extracted. Recently, I used
very high resolution SEM images to fit the shape of my gold nanorods
and was able to achieve relatively good agreement between the
extinction spectra.
Since our research group is largely interested in biomedical studies,
most of my simulations have been in a water medium. I have also begun
work on comparing near fields around gold nanorods in FDTD and DDA,
but I still have work to do in that area.
Anyways, I can't comment on the interesting suggestions for important
parameters and planar systems, but I did want to say what I had done
so far.
Regards,
Rick Harrison
Ph.D. Student
Ben-Yakar Research Group
Mechanical Engineering
University of Texas at Austin
References:
1) Ekici, O., R. K. Harrison, N. J. Durr, D. S. Eversole, M. Lee, and
A. Ben-Yakar. “Thermal analysis of gold nanorods heated with
femtosecond laser pulses.” Journal of Physics D: Applied Physics 41,
no. 18 (2008): 185501.
2) Coronado, E. A., and G. C. Schatz. “Surface plasmon broadening for
arbitrary shape nanoparticles: a geometrical probability approach.”
The Journal of Chemical Physics 119 (2003): 3926.
3) Scaffardi, L. B., and J. O. Tocho. “Size dependence of refractive
index of gold nanoparticles.” Nanotechnology 17, no. 5 (2006):
1309-1315.
Maxim Yurkin
found and joined this discussion yourself. First, I want to note that I am glad that one more person is not ashamed of
using brute force DDA simulations with a huge amount of dipoles, at least as a last resort.
Second, it would be great if you can post your convergence plots here for sphere, both for Qabs and Qsca. It is also
interesting whether the convergence of these observables for nanorods is similar to the spheres. Of course, for nanorods
you probably do not have a rigorous reference solution (although T-matrix method can be an option). But even without
reference it may be possible to see whether there is a difference between these two shapes.
Maxim.
Rick
Sorry for the delay. I was rounding up some old files from different
machines.
When the new version came out (adda 0.78.2), I also tested the FCD
implementation and found considerably better convergence. Except for
where specifically listed, all polarization and interactions are
computed with the default method (LDR). In the comparison of different
diameter spheres and nanorods, the refractive indices are slightly
different in each case to account for size damping. In all cases, the
simulations are at lambda = 780 nm in water (n = 1.333). The
refractive indices are near n = 0.2 and k = 3.7 in all cases. I
didn't use orientation averaging in any of these cases. The nanorods
were modeled as spherically capped cylinders.
I find that as the number of dipoles becomes massive, the accuracy in
Qabs improves. FCD seems to accelerate this process for large
refractive indices, as expected. Qsca approaches reasonable accuracy
at much smaller number of dipoles. I think the shape modeling (fitting
a curve with a set nuber of small cubes) may have something to do with
the accuracy as well, but this source of error should die away as the
number of dipoles grows. In my simulations, as the size of the
nanosphere decreases, the DDA solution converges more rapidly. The
convergence for nanorods is quite different than for spheres. The
values don't decrease to a set value, but rather stay close to one
value. Now I am especially curious about looking at some sort of exact
solution for nanorod geometry. I'd also appreciate any comments about
my methods or results.
The graphs will be in the following email.
Rick
Ph.D. Student
Ben-Yakar Research Group
Mechanical Engineering
University of Texas at Austin
vlad
I am trying to understand the concept of the "large index values"
Your simulations are in water. Therefore, effective n and k are
reduced accordingly.
Yet, you still have Qabs problems. Can you make some estimates at what
n and k values this becomes a problem?
I know that this is not the most general approach, but I would like to
know where the "safe index territory" is, at least
for the case of gold spheres.
VL
>
> - Show quoted text -
Maxim Yurkin
follow below.
> I think the shape modeling (fitting a curve with a set nuber of small cubes) may have something to do with
> the accuracy as well, but this source of error should die away as the number of dipoles grows.
Actually, all sources of error vanish when size of the dipoles goes to zero. Which of them decreases most slowly is an
open question. We have tried to separate shape and discretization errors in the paper
http://dx.doi.org/10.1364/JOSAA.23.002592 . But recently I noticed that part of discretization errors are related to
surface as well (even if it is perfectly described by the sphere) so simulations for the cube are not always more
accurate than for the sphere. So more research is required to make definite conclusions (possibly using the methodology
described in above cited paper).
> In my simulations, as the size of the nanosphere decreases, the DDA solution converges more rapidly.
It is interesting that with decreasing size of the sphere (keeping grid fixed), i.e. going to Rayleigh limit, accuracy
of Qabs improves while accuracy of Qsca deteriorates. Although the latter is still always better than the former.
> The convergence for nanorods is quite different than for spheres. The values don't decrease to a set value, but rather stay close to one value. Now I am
> especially curious about looking at some sort of exact solution for nanorod geometry.
The most striking here is results for Qsca for nanorods. It seems that LDR and FCD converge to a different
(significantly different) values. So at least one of this formulations should be quite inaccurate. I think, that capped
cylinders can be handled by some of T-matrix codes, but I can't point to any specific one. Maybe a new site by Wriedt et
al. (http://www.scattport.org/) may help to find a suitable code.
Another possibility is to use DDA itself as an "exact" solution. For that you need to increase the grid size as much as
possible and additionally use extrapolation technique described in the paper cited above. Briefly, the idea is to
compute the result for different grid sizes and then extrapolate to the infinite grid size. This can be done for two
formulations (LDR and FCD), and for each of them the procedure will give some number and error estimate. If two
formulations agree within their error estimates that is one more proof that one is moving in the right direction.
However, this extrapolation technique is an empiric one (though it has some theoretical foundation) and hence its
accuracy is not guaranteed (so it should be checked at least for a few cases against a different reference results).
Rick, I have the question about your experience with FCD. Have you noticed any difference in number of iterations of the
iterative solver (hence, simulation time) between LDR and FCD? I am asking because I have noticed up to 6 times
acceleration of the convergence using FCD in certain cases.
Finally, about Vlad's question on "safe territory" I think it is hard to specify any sharp boundary. If you fix a
certain accuracy, then computational resources for a given particle increase with wavelength (due to refractive index
becoming more extreme). So the boundary is determined by the available computational resources. It would be nice to
deduce some kind of scaling law between refractive index, size of the particle, grid size, and accuracy. Similar
attempts have been made in paper http://dx.doi.org/10.1016/j.jqsrt.2006.06.006 , but there range of gold refractive
indices was not covered. The problem, of course, is that such a scaling would probably be rather complex even if we
limit ourselves to gold only.
Maxim.
baptiste auguie
I have reached similar conclusions working with DDSCAT recently. I
modeled a 50nm radius gold sphere in water, and the error in
absorption is indeed very important at long wavelengths where gold
behaves like a good Drude metal. I used the default settings of DDSCAT
version 7.1.0.
The attached plot shows the error against Mie theory as a function of
number of dipoles for scattering, extinction and absorption
cross-sections at three particular wavelengths. In the region of
interband transitions and also around the localised plasmon peak the
convergence of Cabs is rather good --better than Cscat--, however
towards the infra-red the convergence of Cabs is atrocious even with
half a million dipoles. Looking at the extremely slow convergence of
Cabs in the bottom panel, I doubt that an extrapolation would be of
much help.
Best regards,
baptiste
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Maxim Yurkin
same results (so that is not some weird peculiarity of ADDA).
Coming back to this discussion, David, Alfons, and I have performed some benchmark studies last autumn and recently
published them in Journal of Nanophotonics: http://dx.doi.org/10.1117/1.3335329. The PDF file can also be downloaded
from my homepage:
http://sites.google.com/site/yurkin/publications/papers/Yurkinetal.-2010-Accuracyofthediscretedipoleapproximationfor.pdf
The main message is that DDA and gold may go well together but only with proper caution. For instance, the required
number of dipoles for a sphere to keep accuracy of Qabs within 10% can be as large as 10 million. That is probably not
very practical for many applications.
I also want to point out that in this paper we concentrated on gold in vacuum (to choose just one possible case), so
there is still space for further work in this field (e.g. silver and/or water medium), especially if driven by a
particular application.
Maxim.
Maxim
http://groups.google.com/group/adda-discuss/msg/6b045dd5c41ce937 ), so
I first repeat it here for archival purposes:
===============================
Dear Maxim,
Thank you for mentioning your recent paper on the subject, I was not
aware of it. The conclusions are rather bad for the use of DDA in the
field of nano-plasmonics, at least until the reason for the poor
convergence of Qabs is known (and perhaps overcome). It would seem
that T-matrix methods (and EBCM, NFM with DS, ...) are more suited to
this class of particles. The cause for the low accuracy of DDA in this
regime is a very intriguing question though. Could some insight be
gained from comparing the instantaneous internal polarisation with the
internal field calculated using, e.g., Mie theory for a sphere? Such a
comparison might illustrate the phase error that appears to undermine
the estimation of the absorption loss.
Best regards,
baptiste
=================================
The following is my answer. About the use of the DDA in general, I
think that it is better to replace "bad" by "should be used with
care", because the conclusions very much depend on the particle under
consideration, wavelength and scattering quantity of interest. But it
is true that the DDA in this field is not as universally suitable
method as was considered by at least some of researchers. The T-matrix
methods probably can claim such suitability, but not for all particle
shapes. So I am not sure if any universally suitable "black-box"
solver exist for the whole field of nanoplasmonics.
About improving the DDA - this definitely should be done. There exist
a number of ideas in the literature, e.g. weighted discretization to
decrease shape errors, as well as several unpublished ideas. However,
they will not help a lot until they are implemented in a publicly
available DDA code. So there is a lot of space for coding and testing.
About internal fields - this is definitely worth trying. We wanted to
do it in the paper, which I mentioned in the previous message, but
decided to leave it for the future research. David has done some
simulations, but I do not know if any conclusions can be based on
them. Concerning the published results, there was a paper by Hoekstra
et al. ( http://dx.doi.org/10.1364/AO.37.008482 ) devoted to this
subject. Its main conclusion is that errors in the internal field are
mostly concentrated near the particle surface. But it is not clear,
how this can be connected with errors in Qabs.
Maxim.