Do nano-objects have color?

I've been reading Jim Pivarski's blog Coffeeshop Physics for some time, and I always find the topics interesting and the perspective refreshing. However, I think that his latest post "Viruses have no color" contains a number of fundamental errors, beyond the imprecisions inherent in a simplified account.

Pivarski's stated point is that objects smaller than the wavelength of light have no color, and he explains this by the uncertainty principle. Instead, he illustrates that small objects scatter less light than large ones, using a "geometrical" point of view that ignores the composition of the objects and sees them simply as opaque to the incoming light. Of course, in this approximation even large objects are colorless, since their scattering properties will not change much over the visible spectrum¹.

The relevant parameter when discussing the color of an object is not the wavelength but the frequency of the incoming light. For instance, gold nanoparticles a few tens of nanometers in diameter both absorb and scatter green light more effectively than at other visible frequencies because in this range the electromagnetic field couples very effectively with the oscillation modes (plasmons) of the conduction electrons in the particle. Dispersions of such particles are therefore green when seen in reflection and red in transmission, as illustrated by the Lycurgus cup. Even atoms can be said to "have color" if we think of their characteristic transition lines (for instance, sodium lamps glow yellow).

The uncertainty principle² only tells us that the image of the nanoparticles cannot be sharper than the wavelength used to look at them, not that this image is colorless (see such colored images here and here).

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^{1. I neglect here the λ4 dependence in Thompson scattering, leading to the "blue-sky effect".↩}

^{2. I preserve here the author's terminology, although "the uncertainty principle" is generally associated with quantum mechanics. Here the reasoning is completely classical, so we might as well call the result "the Abbe resolution limit".↩}

Infrared dichroism of gold nanorods controlled using a magnetically addressable mesophase

Our paper just appeared in J. Mater. Chem. C !

Saint-Venant's principle and its relatives - part 2

In part 1 of this discussion we had concluded that the Laplacian was a "zero-sum game", i.e. that a static modulation along one space dimension was exactly matched by a decay in the perpendicular dimension: \(q_x^2 + q_z^2 = 0\).

What happens for a time-dependent field distribution? For simplicity, let us assume a purely harmonic dependence: \(f(x,z,t) = F(x,z) \exp(i \omega t)\), with translation invariance along \(y\), where \(f\) stands for a component of the electric or magnetic field (in the scalar wave approximation).

The field now obeys:

\begin{equation}
\label{eq:dalemb}
\Box \, f = \frac{1}{c^2} \partial _t^2 f(x,z,t) - \underbrace {( \partial _x^2 + \partial _z^2)}_{\Delta} f(x,z,t)= 0
\end{equation}

where \(c\) is the speed of light and \(\Delta\) is the Laplacian operator discussed in part 1. The wave operator \(\Box \) is often called d'Alembertian.

The Carvallo paradox - part II

I mentioned the Carvallo paradox in a previous post. Here, I will give a simpler version and some comments. Consider that, instead of using a spectrometer, we only use the dispersive element (e.g. a prism). After refraction, the signal components (the various colors) are projected onto a perfectly absorbing screen:

From Wikimedia; under CC-SA 1.0 License.

Even though the incoming signal \(s(t)\) is of finite length (and thus energy), each component (of infinite length) \(f_i(t)\), where \(s(t)=\sum_i f_i(t)\), has finite and constant power, meaning that over a period of time larger than the duration of the original signal the screen will receive a higher amount of energy (which can in fact be made arbitrarily large). The Carvallo paradox can then be restated as follows:

1) A finite-length signal is a sum of infinite-length components,
2) which can be separated and manipulated individually.

There is obviously a problem with 1) or 2) (or both!), since accepting them would break both causality and energy conservation. Note that invoking quantum mechanics does not solve the paradox: we can envision a similar setup using for instance sound waves.

Whether 2) is valid or not, the "separation" step is non-trivial, as one can see from the power spectrum of the signal: \(P(\omega) = |\tilde{S}(\omega)|^2 = \left | \sum_i \tilde{F}(\omega)\right |^2\), where the uppercase and tilde combination denote the Fourier transform. On the other hand, the power spectrum after separation (e.g. that absorbed by the screen) is: \(P'(\omega) = \sum_i  \left | \tilde{F}(\omega)\right |^2\). What I called separation thus amounts to decorrelating the various signal components.

Coherence 4 - Space coherence

The Michelson setup is an amplitude-splitting interferometer, and hence it is well adapted for time coherence measurements since the two beams originate in the same point \(P_1 = P_2\) and the time shift \(\tau\) is simply given by the difference in arm length. To measure the space coherence, we need a wavefront-splitting interferometer, with the two beams originating in different points \(P_1 \neq P_2\) but at the same time: \(\tau = 0\). The simplest example is the Young interferometer.

Coherence 3 - The Michelson interferometer

The Michelson interferometer (shown below) is an excellent setup for illustrating the concept of longitudinal coherence.  We will simplify the formalism of the previous post down to three essential ingredients:

• The beam consists of wave pulses of length l (corresponding to ξ_l in post 2).
• They are "split in half"; each half goes through one of the arms.
• The "twin" pulses arrive at the detector with a shift ΔL.

Coherence 2 - Time coherence

Time coherence is directly linked to the spectral width. Indeed, a perfectly monochromatic signal is coherent at all times: a maximum now ensures a maximum \(N\) periods later. Not so for a polychromatic source: two oscillations of slightly different frequency are in phase at \(t=0\) and become out of phase \(N\) periods later, as shown below.

Image from [BW].

This defines the coherence time (and the longitudinal coherence length \(\xi _l\)), via:
\begin{equation}
\label{shift}
\xi _l = N \lambda = (N - \frac{1}{2})(\lambda + \Delta \lambda) \Rightarrow N \simeq \frac{\lambda}{2 \Delta \lambda}\Rightarrow \xi _l \simeq \frac{\lambda^2}{2 \Delta \lambda}
\end{equation}

In conclusion, the larger the spectral width \(\Delta \lambda\), the shorter the longitudinal correlation length \(\xi _l\). Once again, the descriptions in the time and frequency domains are complementary.

In the next installment, we will look at time coherence in more detail, using the Michelson interferometer.