Physically meaningful cross sections in complex environments

[myurkin]

Extinction, absorption, and scattering cross sections (or, more generally, powers - making it applicable to arbitrary incident field) are ubiquitously used for scattering in a non-absorbing homogeneous medium (e.g., in vacuum). The corresponding powers have well-defined meanings based on integrals of the Poynting vector over closed surfaces and/or potential detector readings.

When more complicated environment is considered, the physical meaning of cross sections can be questioned. This cases include planar substrate (potentially, multi-layered - #235) or absorbing homogeneous medium (#219). Moreover, it makes sense to calculate more quantities instead of three basic ones. The following summarizes some ideas, which should guide the implementation of various new features (complex environments) in ADDA.

• Scattering power (Wsca) is more or less clear - it is an integral of the scattered-field Poynting vector over the far-field sphere. In homogeneous absorbing sphere this will decay with sphere radius, thus only some normalized Wsca can be defined (by integrating, e.g, the distance-normalized phase function). For planar substrate (normal along the z-axis), we may have up to three components: upper hemisphere, lower hemisphere, and guided or surface-plasmon waves (equivalent to horizontal scattering). Each component may decay differently with distance - hence, need to be normalized differently. (Actually, these three components should also be separately described in mueller and decay constants can be given in log). Thus, adding them into a total Wsca is possible, but makes sense only for fully transparent layers.
• Absorption power (Wabs) is also fundamental as the integral of the total Poynting vector over the particle surface, which is equivalent to total power absorbed inside (and heating the particle). However, complex environment can also absorb radiation. Total absorbed power is then often infinite, but some meaningful quantities can be obtained if we remove the contribution of the incident field. For instance, it is definitely meaningful when the particle is placed on the strongly-absorbing substrate - then Wsca only in the upper hemisphere is meaningful, while the scattering into the substrate contributes only to absorption inside it.
• Extinction power (Wext) is the most problematic, since there are two fundamentally different approaches to consider it:
  • through interference with the incident field in the forward direction, which is naturally related to the detector reading with and without particle present (although even this is not trivial for a single fixed particle). This is discussed in a recent paper about SMUTHI [1] - that for a planar media there are two parts of the incident field (transmitted and reflected), each of which has a corresponding Wext. One may also add guided-wave contribution to this list. Similar to scattering, each component may scale with distance according to layer absorption.
  • based on some optical theorem (sum of scattering and absorption) or as an integral of some cross-term of the Poynting vector over some closed surface around the particle. While in the free space the surface can be deformed freely, making it equivalent to the previous viewpoint, there are limitations in complex environment. If the host medium (or part of it) is absorbing, the surface integral will scale with sphere radius and depend on its shape (to some extent).
  • More importantly, the whole optical theorem (as some equality between different flow integrals over the same surface) will be valid for any surface, but the physical meaning of its constituents may depend on the surface. For instance, Wabs has a clear meaning for particle surface, while Wext (and Wsca) - for far-field sphere. For a two-layer system, where the upper layer (with particle) is not absorbing, some generality (similar to the free-space case) can be retained if the integration surface stays completely in the upper layer - see [2]. This is additionally complicated by Optical theorem for advanced DDA formulations #148.
• Another issue is related to volume-integral representation of Wext (which is very convenient for the DDA). For a plane wave it is fully equivalent to the far-field-interference expressions - that is already discussed inside ADDA code for a case of semi-infinite substrate (surface mode). But if other incident field is used, the volume integral expression stays valid (equal to flow integral of the cross-term Poynting vector over the particle volume), but it no more has direct analogy with some far-field interference. Moreover, it is not clear how this volume integral can be decomposed into two parts, corresponding to the extinction of the transmitted and reflected incident waves. However, in this case the whole description of far-field extinction (keeping the reading of some detector in mind) becomes shaky; hence, a rigorous theoretical analysis need to be performed first.

Specific actions that I see at this point

• In the case of homogeneous absorbing host medium, this should be addressed in Complex frequency (wavelength) #219.
• Current surface mode can be improved, by computing more quantities (e.g. three different scattering and extinction components) and absorption power inside the substrate. Relations between them should be described in the manual. This is related to Incorrect calculation of Csca for a limited angular range #195 and Calculate as many cross section as possible when it is fast  #133.
• After the previous one is solved, implementation of multi-layered substrates (Multi-layered substrate #235) should be done similarly.

References

1. Section 4.7 of Egel A., Czajkowski K.M., Theobald D., Ladutenko K., Kuznetsov A.S., and Pattelli L. SMUTHI: A python package for the simulation of light scattering by multiple particles near or between planar interfaces, J. Quant. Spectrosc. Radiat. Transfer 273, 107846 (2021).
2. Section IX of Moskalensky A.E. and Yurkin M.A. Energy budget and optical theorem for scattering of source-induced fields, Phys. Rev. A 99, 053824 (2019).

/cc @Sunmosk @AmosEgel

[AmosEgel]

Dear Maxim!

As you know, I also had to find a way how to treat cross sections for systems involving planarly layered media in a meaningful way during the development of the SMUTHI code.Your above summary seems already quite well developed to me, and I think there is not much new that I can add, but I'll still share what we came up with, maybe some of the details are interesting.

Two or more far field components

• In SMUTHI, far field quantities (such as the scattered power) are defined as a pair for top and bottom hemisphere. E.g., the total scattered power is the integral over the top hemisphere far field plus the integral over the bottom hemisphere far field.
• If the top or the bottom layer are absorbing, no far field quantity is computed for this hemisphere. I can see that there may be a possibility to define a far field by correcting the bulk absorption from the decay of the scattered power. I haven't investigated in that direction and just skipped those figures.
• I can see that a third (and actually also fourth, fifth, ...) scattered far field contribution could be defined for the individual waveguide modes (including SPP). A different kind of far field would have to be defined, in the sense of dP/d\phi rather than dP/d\Omega (i.e., power-per-azimuth-direction rather than power-per-solid-angle), because guided modes are parameterized by a single angle. Accordingly, the power law for flux decay with distance is different from that in free space. For the calculation of the waveguide mode refractive index (as a complex number, where the imaginary part indicates the decay length), I have made good experiences with the mode-finding algorithm described in [1]. It is based on numerical contour integrals around closed loops in the complex plane which yield zero if no singularity (i.e., guide mode resonance) is included and non-zero otherwise. By iteratively sub-dividing the complex plane of effective refractive index for the guided modes, you can precisely get the resonance location which then also gives you the decay length (from the imaginary part of the resonance). If you are interested in more details, I'll be happy to share more description and/or the code that I used (if I can still find it). Let me add that for SMUTHI, we have not yet implemented this kind of analysis (i.e., we do no attribution of scattered power into the individual waveguide modes). Instead, power that is neither scattered into the top nor bottom hemisphere far field is not accounted for in the total scattered power. This way, we currently don't discriminate between power absorbed by the particles, power absorbed by the layer system and power scattered into the guided modes.

Absorption

• In SMUTHI, we don't calculate particle or layer absorption, which is not a trivial thing in the T-matrix formalism for particles near planar interfaces. However, I guess that DDA is ideally suited to evaluate particle absorption, as you have the near field and the internal field available. So I guess that the evaluation of particle absorption should be not more difficult for particles near planar interfaces than for particles in free space.
• Regarding layer absorption: Yes, the presence of particles may alter the power absorbed by layers. It may be possible to express the change in layer absorption through a suitable cross section quantity. However, this portion may interfer with the guided mode absorption, and it may be necessary to distinguish between power absorbed by the layered medium from guided modes from that absorbed by the layered medium from propagating modes. I think there is plenty of freedom how to define such a quantity, one only needs to take care with the interpretation in terms of conservation of energy. By that I mean statements such as "extinction = scattering + absorption". It might be possible to define extinction, scattering and absorption cross sections in a way such that this optical theorem holds. To this end, one would need to be careful to avoid double-counting of, say, layer system absorption and guided mode contribution (which will also be eventually absorbed).

Extinction

In general, I prefer to define quantities with regard to measurable observables in contrast to terms of mathematical expressions (e.g., extinction as some cross term in a specific mathematical expression). But I think this is a matter of taste.

As you already wrote, in SMUTHI we follow a picture of extinction in the sense of "power taken away from the specular (reflected/transmitted) components". This notion is related to the cross term (i.e., interference) between incident plane wave and scattered field. The approach yields two extinction quantities, one for reflection and one for transmission. Again, we haven't pushed that to the case of absorbing bottom or top layer. In such a case, we just return a single extinction figure (namely the extinction in reflection from, e.g., a particle on a metal substrate).

So far, we have only considered extinction for plane wave excitation. I am not aware of how to physically (i.e., in terms of measurable quantities) define extinction for other initial fied types. For that reason, the extinction signal can only have a guided wave contribution if the initial field is a guided wave, too. This is currently not supported by SMUTHI, but may be in a future version.

Conclusion

To me, this is a very interesting topic. I am curious to learn what concept you and your colleagues will finally implement in ADDA. I acknowledge that it is not at all trivial to identify a consistent picture in which the naive notion of "extinction = scattering + absorption" still makes sense for scenarios with absorbing and/or planarly layered media. Wish you all the best with that quest :)

References

[1] B. Hu and W. C. Chew, “Fast inhomogeneous plane wave algorithm for electromagnetic solutions in layered medium structures: Two-dimensional case,” Radio Science 35, 31–43 (2000).

[AmosEgel]

P.S.: I forgot to mention that with the concept of extinction as implemented in SMUTHI, negative extinction can occur when the layer absorption is reduced by a particle. For example, imagine a reflecting particle on a dark substrate. Without the particle, less power is reflected into the specular direction than with the particle. So, the particle takes away a negative amount of power from the reflected signal.

[myurkin]

Amos, thanks a lot for valuable comments. Especially, now I better understand guided mode contribution - this seems to require a separate design of output. It seems that for each mode we have a separate phase function (versus phi) and a separate set of cross sections (scattering and, maybe, extinction and absoption). And the description of these guiding modes (including effective refractive indices) requires a separate output as well.

Also, it seems that some cross sections (or similar quantities) are easier to compute with one code (e.g. DDA-based), while others - with T-matrix based codes. So even after writing the whole "energy budget" with a lot of components, efficient evaluation of all these components may require combination of several codes.

Concerning the negative extinction. If one looks at it as a difference, it is natural to have it of any sign. The common expectation of the extinction to be positive spans, in my opinion, from the optical theorem in the simplest free-space case (Wext=Wabs+Wsca). Since both its two constituents are positive (or non-negative), so is the extinction. In more complicated environments, it is this simple optical theorem that first breaks down, although Wabs and Wsca are still non-negative (at least, when some of the possible definitions are employed).