^{1}, M. Michelle Easter

^{2}, L. David Wellems

^{1}, Henry Mozer

^{1}, Godfrey Gumbs

^{2}, D. A. Cardimona

^{1}and A. A. Maradudin

^{3}

### Abstract

Both dynamic and static approaches are proposed and investigated for controlling the optical phase of a p-polarized light wave guided through a surface-patterned metallic structure with subwavelength features. For dynamic control, the important role of photo-excited electrons in a slit-embedded atomic system with field-induced transparency (FIT) is discovered within a narrow frequency window for modulating the intensity of focused transmitted light in the near-field region. This is facilitated by electromagnetic coupling to surface plasmons between the two FIT-atom embedded slits. The near-field distribution can be adjusted by employing a symmetric (or asymmetric) slit configuration and by a small (or large) slit separation. In addition, the cross-transmission of a light beam is also predicted as a result of this strong coupling between optical transitions in embedded FIT atoms and surface plasmons. For static control, the role of surface curvature is found for focused transmitted light passing through a Gaussian-shaped metallic microlens embedded with a linear array of slits. A negative light-refraction pattern, which is associated with higher-order diffraction modes, was also found for large angles of incidence in the near-field region. This anomalous negative refraction can be suppressed when higher-order waveguide modes of light leak through a very thin film. In addition, this negative refraction can also be suppressed with a reinforced reflection at the left foothill of a Gaussian-shaped slit array of the forward-propagating surface-plasmon wave at large angles of incidence. A prediction is given of near-field focusing of light with its sharpness dynamically controlled by the frequency of the light in a very narrow window. Moreover, a different scheme based on Green's second integral identity is proposed for overcoming a difficulty in calculating the near-field distribution very close to a metallic surface by means of a finite-difference-time-domain method.

This research was supported by the Air Force Office of Scientific Research (AFOSR). M.E. and H.M. also would like to thank the support from the AFRL Phillips Scholars Program.

I. INTRODUCTION

II. MODEL SYSTEMS FOR OPTICAL-PHASE CONTROL

A. Dynamic approach

B. Static approach

III. NUMERICAL RESULTS AND DISCUSSIONS

A. Quantum-interference effect in dynamic control

B. Curvature effect in static control

IV. CONCLUSIONS

### Key Topics

- Surface plasmons
- 36.0
- Metallic thin films
- 20.0
- Surface patterning
- 13.0
- Dielectric thin films
- 11.0
- Electron optics
- 11.0

## Figures

Schematic illustration for a z aligned slit array (blue), which extends in the y direction. In our notation, zj and denote the center position and the slit width of the jth slit, respectively, with . The regions on the left- and right-hand sides of the slits are indicated as regions I and III, respectively, with real dielectric constants and . The region forthe slit array is denoted as region II, and slits are filled with medium (orange) having dielectric constant (real) for . Randomly distributed atoms (red dots) are embedded inside each slit-filled dielectric medium with a concentration cj for and described by an effective, frequency-dependent, and complex dielectric function. The depth of slits in the x direction is 2d, and represents the dielectric function of the metal film containing slits. A Gaussian beam is incident on the slit array from the left-hand side with the angle of incidence and at a center position . The incident wave number is and is the incident wave vector along the z direction.

Schematic illustration for a z aligned slit array (blue), which extends in the y direction. In our notation, zj and denote the center position and the slit width of the jth slit, respectively, with . The regions on the left- and right-hand sides of the slits are indicated as regions I and III, respectively, with real dielectric constants and . The region forthe slit array is denoted as region II, and slits are filled with medium (orange) having dielectric constant (real) for . Randomly distributed atoms (red dots) are embedded inside each slit-filled dielectric medium with a concentration cj for and described by an effective, frequency-dependent, and complex dielectric function. The depth of slits in the x direction is 2d, and represents the dielectric function of the metal film containing slits. A Gaussian beam is incident on the slit array from the left-hand side with the angle of incidence and at a center position . The incident wave number is and is the incident wave vector along the z direction.

Diagram representing a shaped metallic lens. A Gaussian-overlapped profile with an embedded slit array is chosen for the surface on the entry side. A flat surface is assumed for the exit side. A p-polarized plane-wave light with out-of-plane magnetic-field component is incident with angle of incidence where the dielectric constant is . The substrate dielectric constant is on the exit side. A complex dynamical dielectric function is employed for metal with optical loss included, where represents the incident photon energy. In addition, the metal-film thickness is ( with being the incident wavelength), and D 0 represents the lens aperture size.

Diagram representing a shaped metallic lens. A Gaussian-overlapped profile with an embedded slit array is chosen for the surface on the entry side. A flat surface is assumed for the exit side. A p-polarized plane-wave light with out-of-plane magnetic-field component is incident with angle of incidence where the dielectric constant is . The substrate dielectric constant is on the exit side. A complex dynamical dielectric function is employed for metal with optical loss included, where represents the incident photon energy. In addition, the metal-film thickness is ( with being the incident wavelength), and D 0 represents the lens aperture size.

Schematic illustration for the effect due to off-diagonal radiative-decay coupling in the bare-atom picture [in (a)] and the scaled , defined through Eq. (1) , as functions of the detuning in the other panels (b)–(d), where is the atom concentration, with and for j = 1, 2, 3 stand for the energies of three labeled states in (a). The notations , and in (a) represent different energy states in the bare-atom picture, is the probe-field frequency and are nonzero optical dipole moments, and denotes the real part of both the diagonal ( and ) and off-diagonal ( and ) radiative-decay rates. In (b)–(d), both the real (index of refraction) and the imaginary (absorption) parts of have been presented for the pure atomic system in (a) with three different given values for the probe-field amplitude , i.e., (weak), 0.2 (medium), and 0.67 (strong), where , where d 13 = d 12 and d 23 = 0 are assumed.

Schematic illustration for the effect due to off-diagonal radiative-decay coupling in the bare-atom picture [in (a)] and the scaled , defined through Eq. (1) , as functions of the detuning in the other panels (b)–(d), where is the atom concentration, with and for j = 1, 2, 3 stand for the energies of three labeled states in (a). The notations , and in (a) represent different energy states in the bare-atom picture, is the probe-field frequency and are nonzero optical dipole moments, and denotes the real part of both the diagonal ( and ) and off-diagonal ( and ) radiative-decay rates. In (b)–(d), both the real (index of refraction) and the imaginary (absorption) parts of have been presented for the pure atomic system in (a) with three different given values for the probe-field amplitude , i.e., (weak), 0.2 (medium), and 0.67 (strong), where , where d 13 = d 12 and d 23 = 0 are assumed.

Transmission coefficient calculated self-consistently as a function of δ in the upper-left panel. Also, shown are color maps for scaled with a single slit. The slit is filled with a dielectric material ( ), which is embedded with randomly distributed atoms in the rest of the panels with (upper-right), 0.25 (lower-left), and 0.5 (lower-right), respectively. The width and position of the slit are and . Here, the incident-light wavelength is and the p-polarized plane-wave amplitude is . For the three color maps of in this figure, a pair of vertical arrows has been used to indicate the positions of two surfaces of a metallic film.

Transmission coefficient calculated self-consistently as a function of δ in the upper-left panel. Also, shown are color maps for scaled with a single slit. The slit is filled with a dielectric material ( ), which is embedded with randomly distributed atoms in the rest of the panels with (upper-right), 0.25 (lower-left), and 0.5 (lower-right), respectively. The width and position of the slit are and . Here, the incident-light wavelength is and the p-polarized plane-wave amplitude is . For the three color maps of in this figure, a pair of vertical arrows has been used to indicate the positions of two surfaces of a metallic film.

Self-consistently determined double-slit color maps for scaled filled with dielectric material ( ), which are both embedded with randomly distributed atoms. The upper panels correspond to a symmetry configuration having the same embedded atoms in both slits, i.e., (upper left) and (upper-right). The lower panels are for an asymmetric configuration with different types of embedded atoms in two slits, i.e., and for the lower and upper slits, respectively. In addition, for the two lower panels, we display the results for both coupled (lower-left with a small slit separation) and uncoupled (lower-right with a large slit separation) slits. The widths of two slits are . For the upper two and lower-left panels, the positions for these slits are and , while and for the lower-right panel. Here, the incident-light wavelength is and the p-polarized plane-wave amplitude is . For all the color maps of , a pair of vertical arrows has been added to show the positions of two surfaces of a metallic film.

Self-consistently determined double-slit color maps for scaled filled with dielectric material ( ), which are both embedded with randomly distributed atoms. The upper panels correspond to a symmetry configuration having the same embedded atoms in both slits, i.e., (upper left) and (upper-right). The lower panels are for an asymmetric configuration with different types of embedded atoms in two slits, i.e., and for the lower and upper slits, respectively. In addition, for the two lower panels, we display the results for both coupled (lower-left with a small slit separation) and uncoupled (lower-right with a large slit separation) slits. The widths of two slits are . For the upper two and lower-left panels, the positions for these slits are and , while and for the lower-right panel. Here, the incident-light wavelength is and the p-polarized plane-wave amplitude is . For all the color maps of , a pair of vertical arrows has been added to show the positions of two surfaces of a metallic film.

Self-consistent calculations of direct (bottom) and cross (middle andtop) triple-slit transmissions as functions of δ in the upper-left ( ) and lower-left ( ) panels. Also plotted are their corresponding color maps for scaled with triple slits. All slits are filled with the same dielectric material ( ), but only the top slit is embedded with randomly distributed atoms having for two right panels. The widths for these three slits are and , while the positions for these slits are , and . In addition, only the lowest slit is illuminated with light of a Gaussian beam, the incident-light wavelength is (upper panels) or (lower panels), and the peak amplitude of a p-polarized Gaussian-beam is . For the two color maps of in this figure, a pair of vertical arrows has been put on to indicate the positions of two surfaces of a metallic film.

Self-consistent calculations of direct (bottom) and cross (middle andtop) triple-slit transmissions as functions of δ in the upper-left ( ) and lower-left ( ) panels. Also plotted are their corresponding color maps for scaled with triple slits. All slits are filled with the same dielectric material ( ), but only the top slit is embedded with randomly distributed atoms having for two right panels. The widths for these three slits are and , while the positions for these slits are , and . In addition, only the lowest slit is illuminated with light of a Gaussian beam, the incident-light wavelength is (upper panels) or (lower panels), and the peak amplitude of a p-polarized Gaussian-beam is . For the two color maps of in this figure, a pair of vertical arrows has been put on to indicate the positions of two surfaces of a metallic film.

Color plots of scaled for transmitted light through a flat and slit-embedded metallic film when . Various angles of incidence are chosen. The color scales are from 0 (blue) to 4.5 (red) for , 0 to 5 for and 0 to 4 for .

Color plots of scaled for transmitted light through a flat and slit-embedded metallic film when . Various angles of incidence are chosen. The color scales are from 0 (blue) to 4.5 (red) for , 0 to 5 for and 0 to 4 for .

Density plots of scaled for transmitted light through a shaped and slit-embedded metallic film when for various angles of incidence . The color scale ranges from 0 to 5 for , 0 to 6 for and 0 to 7 for .

Density plots of scaled for transmitted light through a shaped and slit-embedded metallic film when for various angles of incidence . The color scale ranges from 0 to 5 for , 0 to 6 for and 0 to 7 for .

Density plots of scaled for light transmitted through a shaped and slit-embedded metallic film. We chose with the half values for and Q 0 used in Fig. 7 and various angles of incidence . The color scale ranges from 0 to 4 for , 0 to 5 for and 0 to 3 for .

Density plots of scaled for light transmitted through a shaped and slit-embedded metallic film. We chose with the half values for and Q 0 used in Fig. 7 and various angles of incidence . The color scale ranges from 0 to 4 for , 0 to 5 for and 0 to 3 for .

Density plots of scaled for transmitted light through a shaped and slit-embedded metallic film when for various angles of incidence . The color scale is from 0 (blue) to 5 (red) for , 0 to 6 for and 0 to 7 for .

Density plots of scaled for transmitted light through a shaped and slit-embedded metallic film when for various angles of incidence . The color scale is from 0 (blue) to 5 (red) for , 0 to 6 for and 0 to 7 for .

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