First of all, please use correct physical terms: not "density" but "volume fraction" of particles, not "photon density" but "irradiation"... Your question is really very simple and does not depend on the type of particles. You can find an answer in my recent book: http://www.begellhouse.com/books/6d17e856430c0b8d.html

Thanks a lot, Leonid! I'll look at the book. In the mean time, perhaps you can point to some of your representative publications dealing with this problem, please? It just might be a while until I get my hands on the book.

Serge,

I think you can consider this system as an inhomogeneous distribution of absorbers with known sources of light.

Depending on the level of scattering and dimensions, you may find that you can approximate the radiative transport equation for light using a diffusion equation. There are a variety of tools developed to describe such a system; these tools have subsequently been applied to benchtop and clinical experiements. What you've described above is termed the 'forward problem' in diffuse optical tomography.

Durduran et al., Reports in Progress in Physics, 2010 provides a complete review and many references.

Regards,

David

Hi Aaron! Yes, it helps. Perhaps, I should add more clarifications since the questions was too vague indeed. I was thinking in terms of more simplified (?) conditions when the individual nanoparticle can be considered as a perfect absorber and then would become "the black body" nanoparticle. Or, even better: a local density of "realistic" absorbers (i.e., nanoparticles) is so high that all the radiation (photons) is absorbed. Example: you see such saturation effects when attempting to measure highly absorbing samples in a spectroscopic cuvette by observing a flattened (transmission) spectrum near the limit of the device. Normally, one would go with thinner cuvettes if the NPs concentration for some reasons has to be fixed and then the true extinction will be obtained. Alternatively, one could increase the power of the light source (the flux) and again, the true extinction would be measured.

So, I'm really interested in such "dense" inclusions of NPs that allow achieving local black body behavior in a turbid medium with fixed illumination levels (flux). Adjusting the concentration would correspond to a transition from flat black-body type absorption to the one characteristic for the the particular type of NP (i.e., one would see the absorption spectrum close to the real one). I'm curious if such conditions would simplify or constrain the problem and allow predicting the local photon density from local "saturated" absorption measurements.

Since I'm coming from an experimental standpoint, the variable is the concentration of NP and the measured quantity is spectrally resolved absorption. So, I was hoping for some phenomenological model to account for such a transition.

David, yes, DA would be the first choice. However, having a strong localized absorber embedded in the turbid medium may undermine the applicability of DA. Similar to the case of non-scattering void inclusion which requires adding radiosity model to describe propagation of light inside such inclusions.

I'd like to see some intuitive approach but perhaps I want too much:)

I'm not sure if I understand the question correctly, but if you want to relate optical absorption (extinction cross section) to nanoparticle concentration this is definitely possible by relatively simple analytical models. You need to know the size and shape of the particles though. Is that information available to you?

Hi Andreas. No, this is not what I want to get:( I know how to do that. Sorry for not being very clear - mea culpa! What I want is to be able to say that since X number of nanoparticles absorb ALL incoming photons in a selected location inside a turbid medium and I can measure the absorbance (extinction) in this location, then the local density of photons must be the number Y and this number can be obtained from given information (i.e., X and saturated absorption condition). Is it more clear?

Sergei,

Ideally, there is no way to absorb all photons, if the density of photon states is non-zero. Perhaps, the quantum optics is not what you want to study. Moreover, there is no way to absorb all photons by increasing the density of metal nanoparticles because you will end up with a metal mirror instead. On the other hand, if local fields of nanoparticles are disturbed by their neighbors, there will be "hot" spots in between them. And the absorption rate will be enhanced in proportion to the field strength. That is why the absorption in tightly aggregated ensembles of nanoparticles can be very high. For example, light attenuation in the fcc lattice of closely-packed metal-dielectric core-shells (shells from Au nanoparticles) can be as high as 80 dB/micrometer (Zhuromskyy et al, 2011)

Sergei, thanks - it's getting warmer! Yes, quantum optics is not the framework I want to treat this problem indeed. I also argue that one can absorb all photons and one doesn't have to have a lot of nanoparticles. Just very few photons to start. In the example above (with spectroscopic measurements in a cuvette) a transmitted light intensity is (nearly, of course) going to zero either because the density of NPs is very high in the cuvette or incoming flux is too low. However, the result is obvious - instead of a characteristic plasmonic absorption spectrum one measures a truncated or flattened spectral profile with the fixed OD value. Obviously, this not what usually researchers want so they go for thinner cuvettes, lower concentrations while increasing incident light power is hardly an option in most of spectrophotometers...The point is that you can absorb (nearly) most of light and record spectral manifestations of it. Since there are so many particles and so few photons, NPs act as they were the black body particles in the limited spectral range (say, 500 - 800 nm, it all depends on the plasmon resonance ).

Now, imagine you put the same density of NP in small volume somewhere in a biological tissue. And it is not difficult to provide illumination conditions that ensure that the flux or density of diffuse migrating photons in a selected point is so low that once you insert the inclusion with NP, all photons will be absorbed the NP. So, one can implement such "truncated" spectrum condition inside biological tissues. I do see this in experiments.

I agree with you that interparticle distance comes to play in such experiments because if one manages to measure extinction spectrum at high concentrations, the characteristic plasmonic spectrum is usually broadened. But I posit that you don't have to have local hot spots to see such effect - just high density of NPs in the entire volume of interest. Agreed?

Serge,

Am I right that „truncated“ transmission spectrum is the effect of a limited sensitivity of a spectrophotometer? if yes, there is no physics behind such observation.

Is the black body a radiating system in thermal equilibrium with environment or a name for the absolute absorber? If the latter, this is a poor model for plasmonic NPs, which not only absorb, but scatter (re-emit) the radiation.

I am afraid, I do not understand the aim of experiment.

If the aim is to detect the presence of NP in a tissue, the truncation of the transmission spectrum is not a criterion.

If the aim is to suppress the light propagation, the intensity will follow the Beer-Lambert law at a low volume fraction of NPs (effective medium approx. should work and hot spots are disregarded), i.e., the intensity of incident light is irrelevant.

If the aim is to find, where NPs are localized in a tissue and the concentration of NPs, then one have to combine Z-scan of the focus of laser beam with measurements of, e.g., the second harmonic intensity.

More sophisticated experiment, that can relate the position of a NP cloud, concentration of NPs and their interaction might be, perhaps, Z-scan of the second harmonic of excitation wavelength in combination with time-resolved spectroscopy of the second harmonic in order to estimate the dynamics of absorption. With cw experiments it would be difficult to say how fast the population of photons decays, because the corresponding homogenous spectrum broadening might be lost on the inhomogeneously broad background.

Another issue is the transparency window of a tissue that is usually the near-IR range, i.e., it sits well above the plasmon resonance of a small NP. Naomi Halas used Au nanoshells to shift the plasmon resonance down to this window.

Sergei, thanks for a number of excellent points! I consider NPs that have absorption dominating over scattering. Such particles are available and their properties can be verified both theoretically and experimentally. So, for the purpose of the current discussion the assumption of having absorbing (only) NPs is justified.

You've suggested a number of good ways to treat particular problems.

My aim is an attempt to extract quantitative information about local photon densities in turbid media from a particular experiment using Au NPs as a local probe.

Localized absorber in this case would act a probe indeed because photons normally present in this location would be removed after introducing the absorber. By translating the inclusion (aka point absorber) from point to point inside a turbid medium and measuring the reduction in the optical signal, one can recreate the distribution of photons and get, essentially the map of photon distributions. Such concepts were illustrated in the following articles:

S. C. Feng, F. A. Zeng, and B. Chance, "Photon migration in the presence of a single defect - a perturbation analysis," Applied Optics 34, 3826-3837 (1995) and

S. B. Colak, D. G. Papaioannou, G. W. tHooft, M. B. vanderMark, H. Schomberg, J. C. J. Paasschens, J. B. M. Melissen, and N. vanAsten, "Tomographic image reconstruction from optical projections in light-diffusing media," Applied Optics 36, 180-213 (1997).

The function of the point absorber is simple - to remove photons from the subsequent migration and detect their absence. One is getting, essentially a "negative" image of the photon density from such measurements.

Next, the concept of the perfect or ideal absorber is a good concept indeed. But can one achieve it in practice? From the theoretical standpoint, the perfect absorber works for any illumination conditions, the value of the flux is not important. ...and the wavelength either - that's why we see the flat black body absorption.

I know the next step may appear trivial with "no physics involved" but bear with me and these concepts:) Now, we start introducing a localized inclusion composed of absorbing Au NPs and and we can vary the concentration of NPs in this very small localized volume. Such inclusion is placed in a particular location inside the turbid medium and there are also the light source and the detector embedded into the medium as well and the detector is connected to a spectrometer. At low concentration of Au NPs in the medium the detector connected to a spectrometer will measure a reduction in the flux a and give you some number in count per second (cps). If you compare such spectrum with the original one that was obtained without the inclusion present, you would see the very characteristic extinction spectrum of Au NPs (i.e., with the plasmon resonance) measured inside the turbid medium. Going up the ladder of concentrations would bring you gradually to a flat line spectrum because the number of nanoparticle is so high and density of photons in that location is so low that high absorption at wavelengths other then the plasmon resonance would smear the resonance and make the spectrum look almost flat.

Can we say that this is because of low sensitivity of our measurement system as in the case of the experiment with a spectrometer and cuvette? Perhaps, even though I think this might be not the best way to describe it. However, while there is not much (interesting, useful etc.) physics in the experiment with the cuvette and spectrometer, there is something cool once you do it inside the turbid medium. By varying the concentration of NPs and keeping the flux fixed one can observe the onset of total absorption of local photons AND register a very characteristic spectral transformation at his point. What I want to do is to relate the concentration of Au NPs which leads to the total absorption to the local density of photons...

I PROMISE NOT TO WRITE SUCH LONG REPLIES IN FUTURE!

Serge,

You want to make a kind of optical tomography for random scattering medium? For this you want to use cluster of NPs as the local probe for the number of photons. The light source is placed in front of the sample and the detector - behind it and measures the total flux? Then you are going to move the NP cluster through the tissue, while keeping the detector in the same position? The changes of detector readings then should be interpreted as the measure of the photon numbers that passed through the position of NP cluster? By changing the NP concentration the rate of absorption change is the measure of number of photons at the NP location? Is this what you mean? Or, perhaps, you want to excite the light source locally near the position of NP cluster? Let us assume that al these things are technically possible.

Then, I see the following constrains:

1. This is a concern about non-invasive measurements, except for the negligible refractive index contrast between scatterers and environment. However, the latter case is trivial, because it corresponds to homogeneous flux distribution. The reason is the interference of scattered in a tissue light. You can see it as a speckle, if such medium is illuminated by a laser beam. If part of photon trajectories is eliminated by absorption, the interference pattern will be changed and photon flux will be redistributed. The higher the absorption, the lager the difference with flux distribution in undisturbed tissue. More than this, the interference pattern will depend on particular spatial distribution of NPs in a cloud, if the interparticle distance will exceed 1/4-wavelength.

2. With white light source the different interference patterns will be formed at each wavelength. Hence, the same NP configuration will change interference pattern in individual way for each wavelength.

3. You can see through turbid media at any particular wavelength by using multichannel LCD screen, which changes the phase of the propagating light at each pixel by applying a computerized feedback with a detector placed behind the sample. This also applies to biological tissues. I am not working in this field and cannot say about the latest progress in terms of tomography applications. To some degree, your aim to achieve complete blocking for different locations by changing concentration is another reincarnation of the idea to look through turbid medium. But it seems less tractable because of invasive sort of measurements. Complete absorption here is just an arbitrarily chosen criterion. It can be 10 times decrease of transmission instead.

4. How to make the black body absorber out of the plasmonic particle? One needs to change the balance between real and imaginary parts (scattering and absorption, respectively) of the dielectric function of metal NP. This can be achieved by nanostructuring in a manner of metamaterials. As far as I know, only in the narrow spectral range of the LC-like (electrical circuit) resonance the imaginary part can be much larger than the real one. Well, this is all about stealth technology that is renowned for relatively narrowband functionality.

Perhaps, these thoughts are not the ultimate truth but might be useful.

Sergei,

I'm not trying to re-invent optical tomography for highly-scattering media. Such technique, known as diffuse optical tomography (DOT) has been used widely for detecting a location and optical properties of various optical inhomogeneities deeply embedded in turbid media. This technique requires arrays of sources and detectors positioned usually on a surface of the object of interest. By varying positions of illumination and detecting points and applying various back-projection algorithms one can achieve pretty good results in seeing-through tissue as was illustrated in the article by Durduran (2010) mentioned by David. There are some issues to overcome in DOT but I don't want to go in there.

I'm not really trying to get image the inclusion per se. I want to stick with single detector (fiber based) and single broad-band light source ( say, few cm apart), measure changes in detector readings and spectrally resolving them and compare readings of two systems, with and without the inclusion. A ratio of the two data sets produces a spectral signature of the inclusion that you use a localized probe. Say, if it is Au NPs, then you would see a spectrum that is very close to the characteristic spectrum of Au NPs measured in the conventional spectrophotometer. If one does more elaborate measurements by having angular-sensitive detector, then one can extract information about how positioning the absorptive inclusion at a particular angle (relative to the source-detector direction) affects density of photons in the entire 360-degree range. Due to the diffuse nature of light and multiple scattering in turbid media, there is a reduction in the signal (photon density) for ALL directions (i.e., angles) but the highest reduction occurs in the direction of the inclusion. Multiple scattering also broadens the angle(s) at which one "see" the inclusion.

The ratio of two measurements bears information about the angular location of the inclusion, the spectroscopic signature of the inclusion in the turbid media, absorption of the inclusion and local photon density in the location of the inclusion. The speckle effect doesn't distort the spectroscopic response but it is modified of course by spectrally-dependent optical properties of the medium. One can also probably say that the speckle effect is responsible for spectrally-dependent angular broadening of the angular signature of the inclusion and I'd agree with this.

I agree with you that interpretation of the observed reduction in the signal is not trivial. As Aaron correctly pointed out, measuring absorbance ( let's just call it that way even though the measured quantity is proportional to it but not equal) does not tell you what was the flux in that location since some photons pass though the inclusion. Even extracting the absorption coefficient of the inclusion is far from trivial, because as oppose classical absorbance measurements one doesn't have knowledge of incident and transmitted intensity of light but only a fraction of photons that originate from the light source, pass through the region occupied by the inclusion and are detected by the detector.

High concentration of Au NPs placed in a small thin capillary tube does look like black ink and behave like black ink in optical measurements. So, my hope is by saturating absorption at high concentrations arrive to the absolute fraction of photons that were present and ALL absorbed by the inclusion. Since no light passes though and there is no scattering, I'd know that I'm measuring the actual density in this location and the absorbance measured at this condition would give me the density in this point. Perhaps, I'm starting to repeat myself - better stop at this point...

Serge,

1. If you completely absorb the light, there will no the second signal to compare.

2. If you change the speckle, you change the density of photon states (this is not a density of photons), then you will compare samples with different light distributions. The calibration of absorption of NPs by concentration will not help.

3. High absorption of NP powder is a collective effect. It might not be realized inside the tissue, hence dilution will bring it back to normal localized plasmon behaviour.

Sergei,

"Yes" to #1 and #2, "no" to #3. It's actually doable and observable inside the tissue. BTW, what would be characteristic distances at which the speckle from an incoherent white light source would be observed deep in a turbid medium?

To be clear, aren't we talking about the same thing? When one inserts a spherical diffuser acting as a light source inside the turbid medium, it provides uniform illumination that decreases with distance from the light source. Multiple scattering homogenizes photons so that they form a light cloud inside the medium where photons visit multiple locations multiple times because of random and isotropic scattering. If one places detectors around the light source at various distances and angles, a steady pattern of light distribution is observed. Would you call some random fluctuations on top of this pattern the random speckles?

I can see how one can observe light speckles in OCT using low coherent broad band source at distances characteristic for OCT but not with using an incandescent lamp which has virtually non-existing coherence length and at several cm distances from the light source...

Perhaps, I'm missing something here.

Serge

You are talking about diffuse regime of light propagation. In this case your approach may work with or without total absorption. To my opinion, one can measure the light intensity with better precision then the concentration of NPs in random medium.

The major point, however, that your approach completely disregards the light localization in randomly inhomogeneous medium (other names are coherent backscattering, speckle, etc.). It will work correctly only in the case of negligibly small contribution of the interference to light propagation in such medium. Since the localization is a fundamental phenomenon, with each new sample you will have to prove the validity of such omission.

Simple example. The transmission of the Fabry-Perot resonator with two highly reflecting mirror is equal to unity at the resonance and zero away from the resonance frequency. If the thin sheet of absorber will block the light propagation at the resonance frequency, there will be no standing wave in the resonator, i.e. your measurement will make absolutely different distribution of electromagnetic density inside the resonator. Then, instead of the sequence of Fabry-Perot resonances in the spectrum, you will detect zero transmission at any frequency.

This is the error introduced by your measurement scheme, which kills light interference..

Sergei,

In general, I agree with you that interference if present may introduce an error to the measurements. Perhaps, to be more specific about its effects more concrete and detailed cases should be considered. Thanks for bringing it to my attention - I completely ignored it.

From Coherence Theory and the Planck Equation Minimum Intensity of zero implies zero temperature, zero energy and zero distance and zero frequency.

David, thanks for another good point - no black body is perfect in practical implementations:)

We've found some way to relate the density of nanoparticles to the density of photons incident on a target and then, to the optical properties of the target composed of Au NPs:

https://www.researchgate.net/publication/272350657_Interstitial_diffuse_radiance_spectroscopy_of_gold_nanocages_and_nanorods_in_bulk_muscle_tissues?ev=prf_pub

Article Interstitial diffuse radiance spectroscopy of gold nanocages...

Bernoulli Statistics still apply. The E-field is non-coherent, the energy density is still 1.0000(00). Homogenous distribution is an assumption, the Lagrange invariant still applies. An average Coherence length should still apply.