A dark mode is simply by definition a mode that cannot be excited by a plane wave. The physical explanation is symmetry. As an example think of a nanoparticle dimer: the bonding mode, with a total dipole moment p can be excited by a plane wave, but the antibonding mode, where charges with the same sign face at the dimer gap, cannot of course be excited by a plane wave. A dipole source breaks these symmetry constraints imposed by a plane wave, and therefore allows the excitation of dark modes.

It is possible if you only excite part of metallic structure.

Thank you very much. Yes, it is. I wander is there any physical explaination about this fact, i.e. the dipole can excite the dark modes, but the plane wave cannot.

A dark mode is simply by definition a mode that cannot be excited by a plane wave. The physical explanation is symmetry. As an example think of a nanoparticle dimer: the bonding mode, with a total dipole moment p can be excited by a plane wave, but the antibonding mode, where charges with the same sign face at the dimer gap, cannot of course be excited by a plane wave. A dipole source breaks these symmetry constraints imposed by a plane wave, and therefore allows the excitation of dark modes.

Thank your for your answer. You mean that the excitation way of the modes is limited to its symmetry. I think it is quite resonable. Can we find some mathematical foundation for this explanation? 

A number of papers come to mind. I would say:

• All the Nordlander and Prodan papers regarding their hybridization method: just look for the authors on google and the papers will pop up
• "The benefits of darkness" and references therein:

Best

Giovanni

I'll read your recommended papers. Thank you very much.

To comment on Giovanni's answer - you can actually excite anti-bonding modes with a non-normal incidence of light. Thus, the "darkness" is in my opinion not a very well defined property... For a simple picture of dipole coupling to "dark" modes in nanorods, I suggest the following paper (pardon the shameful self-promotion!)

Article Control of Single Emitter Radiation by Polarization- and Pos...

Thank you for your comment. I've read your OL paper based on numerical simulation with BEM. From your paper I know that using a dipole emitter to excite dark modes isn't always work. Can you offer some physical explainations about this result?

I'd say it's a geometric problem really. Consider the electric near field of dipolar emitter in the plane perpendicular to the dipole. You'll see the it has a point symmetry, right (point is given by the dipole axis crossing the plane)? Now, if the dipole approaches the nanorod, and it's normal to the nanorod's long axis, this electric field will be driving charges in opposite directions along this axis. That is already something that normally incident planewave cannot do, as it has a homogemeous electric field - thus, it drives all the charges in the same direction.

There is a very nice paper, though a bit technical, from Tim Taminiau, that also discusses this problem and puts forward a formalism.

If it's still unclear, let me know!

Article Optical Nanorod Antennas Modeled as Cavities for Dipolar Emi...

Lets assume that we have nano metallic nanoobjects coupling to a field E with rates g1 and g2.

roughly we write the coupling as g1 E a1 + g2 E a2 + H.c.

So if g1=g2, the field equally couples to the plasmon modes a1 and a2 in two nanoobjects. Therefore we have g E (a1 + a2). The field only couples to the bright (symmetric) mode. 

If g1 not equal to g2, then we have (g1 + g2)/2 E( a1 + a2) + (g1 - g2)/2 E( a1 - a2) . The second term drives the dark (antisymmetric) mode.

I hope my answer is useful for you.

I think the definition of "dark" modes is ambiguous. It only describes how effectively the system radiates in a relative sense. Except for electric monopoles, no modes are perfectly dark. In terms of excitation, since the collective excitation of the system, say quadrupoles and other multipolar modes can be generally expressed in the form of a group of dipoles (electric or magnetic by nature), they can be excited locally using a dipolar source.

Thank you, Fang. I agree with you that the defenition of "dark mode" is somewhat vague. What's the relation between the dipole group property of the multipolar modes and the excitability by dipole source? Why can't the plane wave?

Quote Xia: "Lets assume that we have nano metallic nanoobjects coupling to a field E with rates g1 and g2."

I was quite confused with the coupling rates. Why there are two coupling rates? what does them mean? How to calculate them?

Just a couple of questions to Giovanni Pellegrini:

Why are you sure that the antibonding mode can be excited by a dipole source?

Did you calculate such excitation yourself or your knowledge is based on another source of information?

I disagree that in practice, plane wave cannot excite the multipolar modes. My logic is based on reciprocity. If the system can radiate to the far field, then what you receive at a long distance and within a small solid angle can be approximated as a plane wave. If you agree on that point, then by reciprocity when we send a beam in the same direction but flip the arrows of light, then you are going to excite the multipolar system. By the same argument a dark mode cannot be be dark if you are allowed to illuminate the system at all solid angles!

However, since most of our experiments and simulation are conducted at finite distance and within limited solid angles, we tend to neglect some features that are allowed by the system.  For example, if we study an object of perfect mirror symmetry and our beam is at normal incidence, then we have excluded a polarization state of the object that is anti -symmetric with respect to the mirror plane. On the other hand, if you assume the object is made of many discrete dipoles that are assembled together, then it is not hard to find out how each dipoles can be excited when a dipole source is placed in their neirborhood. We can then use the symmetric groups or methods of multipole expansion, to distinguish the collective effect of the group of the excited dipoles and call them quadrupoles, magnetic dipoles, etc. You might find a nice example from my colleague, Prof Kin Hung Fung's paper:

Kin Hung Fung, Anil Kumar, and Nicholas X. Fang, "Electron-photon scattering mediated by localized plasmons: A quantitative analysis by eigen-response theory", Physics Review B, 89, 045408(2014).

Thank you, Fang. I'll read your paper.

My answer to Professor Tishchenko regarding excitation of antibonding modes by a dipole:

- Yes we did calculate the excitation using the method shown in the above paper (Kin Hung Fung et al, PRB, 2014). 

Thank you Professor Fang, I am intriguing by this story.

I hope that Dr. Pellegrini will also bring some light.

Very interesting discussion. Perhaps I may add another viewpoint: 

Some electromagnetic modes derive their darkness (as explained above by Pellegrini)  from global symmetry, encompassing the configuration of the source, scattering object, and detector. In contrast, a dipole source is localized. Therefore, it probes the local electromagnetic field and it is insensitive to global phase differences determining the far-field extinction.

In PRL 109, 166803 (2012)  we showed that a spectral-angular region of far-field induced transparency also hosts enhanced local fields at certain positions, where we placed emitters. Therefore, the emission is enhanced and the extinction is suppressed.

In another paper [numerical study only, Opt. Lett. 38, 1238 (2013)], we showed the existence of collective "dark" modes, i.e. quadrupolar Bloch waves   with enhanced local fields but suppressed far-field extinction.

Hi

Let me clarify my answer:

For the rest I would invite everyone to read the answer of Prof. Fang, which I find quite illuminating.

Best

Giovanni

I think that the "dark plasmon modes" can be excited by a planewave. In the momentum space, there is a wide momentum distribution when a planewave interacts with a metallic nanoobject. Therefore, the "dark plasmon modes" supported by a metallic nanoobject can be excited. However, the generated efficiency of the "dark plasmon modes" should be very low when the excitation is a planewave. 

Thank you. Dr Chang. Is there a quantitative expression for the efficiency you mentioned?

The generated efficiency of the "dark plasmon modes" depends on the coupling strength and the loss term of the "dark plasmon modes". The coupling strength is related to the generated efficiency of surface plasmon polariton (SPP). You are suggested to read the reference "IEEE Journal of Selected Topics in Quantum Electronics, vol. 14, No. 6, pp. 1522-1529 (2008)". However, it is not easy to have an analytical solution for calculating the coupling strength between planewave and SPP. Numerical methods (such as FDTD and FEM) are widely used to solve the interaction between electromagnetic waves and metallic nanoobjects.  You can read my past report "IEEE Photonics Technology Letters. 23(22):1727-1729 (2010)" for understanding the FDTD simulation.

Thank you. Dr. Chang. You mention that it is not easy to have an analytical expression. But what's the trouble on the way?