Probing the dynamical behavior of surface dipoles through energy-absorption interferometry

Spatial interferometry, based on the measurement of total absorbed power, can be used to determine the state of coherence of the electromagnetic field to which any energy-absorbing structure is sensitive. The measured coherence tensor can be diagonalized to give the amplitude, phase, polarization patterns, and responsivities of the individual electromagnetic modes through which the structure can absorb energy. Because the electromagnetic modes are intimately related to dynamical modes of the system, information about collective excitations can be found. We present simulations, based on the discrete dipole approximation (DDA), showing how the dynamical modes of systems of surface dipoles can be recovered. Interactions are taken into consideration, leading to long-range coherent phenomena, which are revealed by the method. The use of DDA enables the interferometric response of a wide variety of objects to be modeled, from patterned photonic films to biological macromolecules.

Figure 1

A collection of particles on a surface. The individual particles have dipole moments ${\mathbf{p}}_i$. The system is illuminated by a pair of phase-locked sources, which can have any polarization and position. The experiment measures the complex visibility of the absorbed power as the differential phase between the sources is varied.

Figure 2

Total power absorbed (arbitrary units) as a function of frequency for two parallel dipoles, 0.1 mm apart, each having ${\omega }_o=300$ GHz, and $\Gamma =20$ GHz. The lines correspond to polarizabilities of $\alpha /{\epsilon }_0=0.0005$ mm${}^3$ (solid line), $\alpha /{\epsilon }_0=0.005$ mm${}^3$ (dotted line), and $\alpha /{\epsilon }_0=0.01$ mm${}^3$ (dashed line).

Figure 3

Total power absorbed (arbitrary units) as a single source is swept in the $x$ direction at a distance $y=5$ mm from two parallel dipoles, 0.1 mm apart, each having ${\omega }_o=300$ GHz, $\Gamma =20$ GHz, and $\alpha /{\epsilon }_0=0.005$ mm${}^3$. The source frequencies were 240 GHz (dashed line), 300 GHz (solid line), and 340 GHz (dotted line).

Figure 4

Total power absorbed (arbitrary units) as a function of frequency for a variety of ringlike configurations: three dipoles having $\alpha /{\epsilon }_0=0.005$ mm${}^3$ placed 0.1 mm apart at the corners of a triangle (solid line); four dipoles having $\alpha /{\epsilon }_0=0.003$ mm${}^3$ placed 0.1 mm apart at the corners of a square (dotted line); and eight dipoles having $\alpha /{\epsilon }_0=0.0004$ mm${}^3$ placed 0.0383 mm apart at the corners of an octagon (dashed line). The power for the eight-dipole case was scaled by a factor of 10 before plotting.

Figure 5

(a) Amplitude and (b) phase of the continuous fringe visibility of the triangular system at 240 GHz (solid line), the square system at 289 GHz (dotted line), and the octagonal system at 390 GHz (dashed line). The sources were swept around in a circle, 5 mm from the center. The reference angle was 90${}^{\circ }$.

Figure 6

Modal forms of the 413 GHz resonance of octagonal systems of dipoles with the resonant frequency of one dipole increased to 400 GHz. The reduced-mass dipole was moved around the chain: none (solid line), position 1 (dashed line), position 2 (dotted line), and position 3 (dot-dashed line).

Figure 7

(a) Amplitude and (b) phase of the recorded fringe visibility at 413 GHz of octagonal systems, with the resonant frequency of one dipole changed to 400 GHz. The lines correspond to none (solid line), position 1 (corresponding to the 90${}^{\circ }$ reference angle) (dashed line), position 2 (dotted line), and position 3 (dot-dashed line).

Figure 8

Total power absorbed (arbitrary units) as a function of frequency for a linear chain of five parallel dipoles, 0.1 mm apart, each having ${\omega }_o=300$ GHz and $\Gamma =20$ GHz. The different lines correspond to polarizabilities $\alpha /{\epsilon }_0=0.0005$ mm${}^3$ (solid line), $\alpha /{\epsilon }_0=0.005$ mm${}^3$ (dotted line), and $\alpha /{\epsilon }_0=0.01$ mm${}^3$ (dashed line).

Figure 9

Modal forms of the resonances at 239 GHz (solid line), 293 GHz (dashed line), and 332 GHz (dotted line) in Fig. 8. All dipoles are either perfectly in phase or out of phase. The lines are drawn to guide the eye only.

Figure 10

Total power absorbed (arbitrary units) as a function of frequency for 11 parallel dipoles, 0.1 mm apart, each having ${\omega }_o=300$ GHz and $\Gamma =20$ GHz. All of the dipoles except the central one had a polarizability of $\alpha /{\epsilon }_0=0.0005$ mm${}^3$, whereas the central one had $\alpha /{\epsilon }_0=0.001$ mm${}^3$ (solid line), $\alpha /{\epsilon }_0=0.003$ mm${}^3$ (dotted line), and $\alpha /{\epsilon }_0=0.005$ mm${}^3$ (dashed line).

Figure 11

Modal forms of the strongest absorptive modes at 276 GHz (solid line), 323 GHz (dashed line), and 300 GHz (dotted line) of the dashed spectrum shown in Fig. 10. All dipoles are either perfectly in phase or perfectly out of phase, and the lines are drawn to guide the eye only.

Figure 12

(a) Amplitude and (b) phase of the recorded fringe visibility at frequencies of 276 GHz (solid line), 323 GHz (dashed line), and 300 GHz (dotted line). The sources were swept along a line, 1 mm from the chain of dipoles, and the visibilities are referenced to the central position.

Figure 13

(a) Amplitude and (b) phase of the recorded fringe visibility at 300 GHz. A highly polarizable dipole was placed at $x=0.0$ mm (solid line), $x=0.1$ mm (dashed line), and $x=0.2$ mm (dotted line). The sources were swept along a line, 1 mm from the chain of dipoles, and the visibilities are referenced to the central position.

Figure 14

Modal forms of the strongest absorptive modes at 270 GHz (solid line), 330 GHz (dashed line), and 284 GHz (dotted line) of a system comprising 11 dipoles. All of the dipoles had a polarizability of $\alpha /{\epsilon }_0=0.0005$ mm${}^3$, except the ones at $x=-0.2$ mm and $x=+0.2$ mm (dipoles 4 and 8), which had a polarizability of $\alpha /{\epsilon }_0=0.005$ mm${}^3$.

Figure 15

Total power absorbed (arbitrary units) as a single probe is scanned along a line parallel to a linear chain of 21 dipoles placed 0.2 mm apart, having ${\omega }_0=300$ GHz, $\Gamma =20$ GHz, and $\alpha /{\epsilon }_0=0.5$ mm${}^3$ The scan distance was 1 mm. The lines correspond to frequencies of 42 GHz (dotted line), 205 GHz (dashed line), and 500 GHz (solid line): the powers have been divided by $5\times {10}^8$, $1\times {10}^4$, and $1\times {10}^7$, respectively, before plotting.

Figure 16

Amplitude of the continuous fringe visibility for an array of 21 dipoles placed 0.2 mm apart, ${\omega }_0=300$ GHz, and $\Gamma =20$ GHz. (a) 42 GHz, (b) 205 GHz, and (c) 510 GHz. The reference positions are 0 mm (solid line), $-2.5$ (dashed line), and $-5.0$ mm (dotted line) in all cases. The ends of the chain are at $\pm 2$ mm.

Figure 17

Modal forms of the strongest absorptive modes at 42 GHz (dotted line), 205 GHz (dashed line), and 500 GHz (solid line) of a linear chain of 21 dipoles placed 0.2 mm apart, having ${\omega }_0=300$ GHz, $\Gamma =20$ GHz, and $\alpha /{\epsilon }_0=0.5$ mm${}^3$. The lines are drawn to guide the eye only.