Remarkable Radiation Efficiency through Leakage Modes in Two

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Remarkable Radiation Efficiency through Leakage Modes in Two-Dimensional Silver Nanoparticle Arrays Wenfang Hu and Shengli Zou* Department of Chemistry, University of Central Florida, 4000 Central Florida Boulevard, Orlando, Florida 32816-2366, United States ABSTRACT: Using the coupled dipole method, we investigated radiation efficiency through leakage modes in two-dimensional silver nanoparticle arrays. The leakage modes are defined as radiations along directions other than reflection and transmission directions. The leakage efficiency depends on the particle size, lattice spacing, the number of particles in the array, and the angle of incidence. There is also a strong orientation dependence of leakage efficiency at different diffractive coupling orders among particles. We found that a total leakage efficiency of over 75% can be obtained due to the improved coherent coupling between metallic nanoparticles and optimized orientation of the induced dipoles relative to the reflection, transmission, as well as leakage directions. For a specified direction on the same side of the surface normal relative to the incident light, a leakage radiation efficiency as high as over 50% can be obtained. The simulations from the coupled dipole method were also compared to the results from the discrete dipole approximation method, which includes high order excitations of metal nanoparticles, and the differences between the coupled dipole and the discrete dipole approximation methods are explained.

’ INTRODUCTION Metal nanoparticles have been found useful in applications of various disciplines including sensing,1
11 waveguide,12,13 metamaterials,14,15 and solar cells.16,17 Those applications are closely associated with their optical properties due to excited surface plasmons of the metallic particles. For particles arranged in oneor two-dimensional arrays, when the scattered light from induced dipoles of particles in the array is coherent (in phase) with each other, the coherent coupling between particles leads to extremely narrow resonance peaks, which had been theoretically predicted and experimentally observed.18
27 Most of the reported works focus on reflection, extinction, or transmission spectra of metallic nanoparticle and particle arrays, or other properties related to the enhanced electric field of metal nanoparticles. Leakage radiation due to high order diffractive interference of induced dipoles of particles and propagating along directions other than the transmission and reflection directions attracts less attention. Using a couple dipole method, we found that light may propagate along a specific direction, other than the conventional transmission and refection directions, with an efficiency of over 50% at optimized conditions. The results can be of interest for controllable light propagation along a specific direction in a device of reduced size.

light. After the induced dipoles are obtained, the absorption cross section and the orientation-dependent scattering cross section of the array can be calculated using equations:
 4πk 3N 2 3
1    Im ½Pj 3 ðRj Þ Pj 
k Pj 3 Pj ð1Þ Cabs ¼ 3 jEinc j2 j ¼ 1

’ THEORY Using the coupled dipole method, we investigated leakage modes of two-dimensional silver nanoparticle arrays. The leakage modes are defined as radiation along directions other than those of reflection and transmission. The coupled dipole method has been discussed in detail in the previous literature.18,22 In brief, for an array of N particles, 3N linear equations need to be solved to obtain the induced dipoles of particles by considering the interaction between induced dipoles of particles and the incident

’ RESULTS AND DISCUSSION We start from arrays illuminated with light of normal incidence. For an array of particles, the allowed orientations of diffractive radiation need to satisfy the phase match conditions between all induced dipoles. The reflection and transmission

r 2011 American Chemical Society

∑

Csca

k4 ¼ jEinc j2

Z

 2  3N    dΩ ½Pj
^nð^n 3 Pj Þ expð
ik^n 3 rj Þ  j¼1 

∑

ð2Þ where k = 2π/λ is the incident wave vector at wavelength λ, Einc represents the magnitude of the incident light, Pj and Rj are the induced dipole and polarizability of particle j along the x, y, or z axis, respectively, rj is the coordinate vector of particle j, and ^n represents the unit vector along the scattering direction. The dielectric function of silver was taken from Palik’s handbook.28 All of the efficiencies of the spectra are calculated by dividing the calculated cross section over the physical area of the array plane. The incident light is included along the transmission direction to calculate transmission spectra. The software used is the one currently developed in our group.

Received: June 7, 2011 Revised: August 4, 2011 Published: August 08, 2011 17328

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Figure 1. Sketch of particle arrays for (a) top view, (b) side view. (c) Reflection, absorption, transmission, and sum (=reflection + absorption + transmission) efficiencies for 50 nm radius silver particles arranged in a square array of 500 nm spacing.

Figure 2. Contour plot of the scattering efficiency of 50 nm particles arranged in a square array of 500 nm spacing at different wavelengths and polar angles (θ) for a given azimuth angle (ϕ). (a) ϕ = 0°, (b) ϕ = 22.5°, (c) ϕ = 45°, (d) ϕ = 90°.

direction are the same as those defined in a smooth surface where the transmission is the radiation through the particle plane and propagating along the same direction of the incident light, and the reflection is the radiation whose angle relative to the surface normal is the same as that of the incident light. The reflected light, transmitted light, surface normal, and the incident light are in the same plane (plane of incidence). Besides the reflection and transmission direction, which is perpendicular to the particle plane for the normal incidence, radiations are also allowed along directions when the phase difference between two adjacent particles equals n  2π (light path= n  wavelength, where n is an integer). A schematic of the target array is illustrated in Figure 1a (top view) and b (side view). The particles are arranged in a square lattice. Incident light propagates along the z axis, and the incident polarization is along the x axis. Figure 1c shows the reflection, absorption, and transmission efficiency spectra of an array of 50 nm radius particles and 500 nm lattice spacing. The total leakage efficiency, including leakage radiation of all directions except those of transmission and reflection, is defined as 1
sum (sum = reflection + absorption + transmission) efficiency. The sum efficiency spectrum is also included in Figure 1c, which shows a sharp dip at 500 nm wavelength and a broad dip at about 400 nm. In Figure 1c, the sharp dip of the sum efficiency spectrum at 500 nm wavelength is due to the (1,0) order diffractive coupling between particles, which is the diffractive coupling between

particles arranged perpendicular to the incident polarization direction. The broad dip at around 400 nm wavelength is due to the excited local mode of the metal nanoparticles. There is a weak peak in the sum efficiency spectrum at 350 nm wavelength due to the (1,1) order diffractive coupling between particles. As sketched in Figure 1b, when the phase difference between scattered light of adjacent particles equals n  2π, the radiation is allowed along that direction. Figure 2a
d shows the contour plot of scattered light at different wavelengths and different polar angles (θ = 0
180°) for a given azimuth angle (ϕ) of (a) ϕ = 0°, (b) ϕ = 22.5°, (c) ϕ = 45°, and (d) ϕ = 90°. We only included scattered light in Figure 2 and excluded incident light along the transmission direction. It is worth noting that narrow bands at θ = 0° and 180° are not clearly shown in Figure 2. Figure 2a shows a curve symmetric about the θ = 90° line, which satisfies λ = d sin(θ), where d is the lattice spacing of the particle array and θ represents the polar angle between the leakage light propagation direction and the z axis. Those spectra have a narrow line width of less than 10 nm. Because the incident polarization is along the x axis, the diffracted light disappears when the polar angle θ is close to 90° and ϕ = 0°. The leakage scattering intensity is high at around 400 nm wavelength, which corresponds to the dip at 400 nm of the sum efficiency spectrum in Figure 1c. Figure 2b shows no leakage radiation other than the backward reflection and forward transmission directions when the azimuth angle ϕ = 22.5°. Figure 2c displays the leakage radiation intensity 17329

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Figure 3. (a) Sum efficiency and (b) reflection spectra of 50 nm particle arrays of different spacing; (c) sum efficiency spectra of 100 nm particle arrays of different spacing; and (d) sum efficiency spectra of 100 nm particles arrays of 800 nm spacing for different numbers of particles, n, included in the calculations: black, n = 100; red, n = 1000; blue, n = 1500; green, in the limit of an infinite array via the discrete dipole approximation (DDA) method.

when ϕ √ = 45° and the wavelength matches the equation of λ = d sin(θ)/ 2, which is due to the (1,1) diffractive coupling among particles. Figure 2d shows leakage radiation along the direction of ϕ = 90°, which is parallel to the y axis and perpendicular to the incident polarization direction. Following the energy distribution of an oscillating dipole, Figure 2d shows a similar feature but much stronger intensity in comparison to Figure 2a when ϕ = 0°. Because a finite number of particles is used in the calculations, a strong leakage radiation along the array plane is also obtained at about 500 nm wavelength. The distance dependence of leakage modes for arrays of 50 nm radius particles is shown in Figure 3a. The corresponding reflection spectra of the arrays are shown in Figure 3b. For arrays of 500 and 600 nm spacing, sum efficiency (Figure 3a) and reflection (Figure 3b) spectra show similar features. The resonance dip wavelengths in Figure 3a match the resonance peak wavelengths in Figure 3b. For the array of 400 nm spacing, the similarity between sum efficiency (Figure 3a) and the reflection (Figure 3b) spectra is not so great. The difference is due to the strong absorption efficiency of particles at around 420 nm wavelength. The leakage radiation for arrays of 400 nm spacing at 420 nm wavelength, which propagates in the particle plane, is absorbed by the particles in the array, and the sum efficiency spectrum in Figure 3a only shows a small dip at 420 nm wavelength where a strong reflection peak appears in the reflection spectrum (Figure 3b). The particle size dependence of the leakage modes is examined by simulating arrays of 100 nm radius particles. Figure 3c shows the sum efficiency spectra for arrays of 400, 600, and 800 nm spacing for particles of 100 nm radius. The spectra show features and trends similar to those in Figure 3a. More high order diffractive couplings, whose number increases with increasing particle spacing and decreasing incident wavelength, were obtained. The maximum total leakage efficiency is also increased to

over 40% (1
sum efficiency of 60%) at around 800 nm wavelength when the lattice spacing is 800 nm. Figure 3d shows the particle number dependence of the sum efficiency, which shows that the sum efficiency is converged at most wavelengths when only 100 particles are included. When the particle number is increased from 1000 to 1500, the two spectra are very similar, indicating the convergence of the calculations. To confirm the reliability of the coupled dipole method, we also calculated the sum efficiency spectra of 100 nm radius particles arranged in a square lattice of 800 nm spacing using the discrete dipole approximation (DDA) method.29 The absorption, reflection, as well as transmission spectra of particles array can be calculated using the DDA method by applying periodic boundary conditions.24,29 Figure 3d shows that the coupled dipole method caught most of the features in the sum efficiency spectrum in comparison to the DDA method, which includes high order excitations of the metal nanoparticles. The difference between the DDA method and the coupled dipole method is due to that periodic boundary conditions are applied in the DDA method and the leakage mode in the particle plane is forbidden. For the coupled dipole method used in this article, only up to 1500 particles are used, and significant radiation energy in the particle plane can still be obtained, which was also demonstrated in the contour plot in Figure 2d. In consequence, the leakage efficiencies calculated using the coupled dipole method are more than those from the DDA method. In the DDA method, due to the periodic boundary conditions, most of the features become sharper, for example, the Fano line shape30,31 at around 580 nm wavelength. Please note that the observed Fano line shape is different in mechanism from the Fano resonance in his original paper.30,31 The line shape is due to the interference between the radiated light rays of different magnitudes and phases from dipoles of different orientations or arranged along different directions.32,33 The significant difference between the results from the coupled 17330

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The Journal of Physical Chemistry C dipole and the DDA methods is the feature at around 800 nm wavelength. In the coupled dipole method, a sharp dip in the sum efficiency spectrum is obtained at around 800 nm due to the leakage radiation along the array plane. In the DDA method, because the applied periodic boundary conditions forbid leakage radiation in the particle plane, the sharp dip in the sum efficiency spectrum obtained in the coupled dipole method at around 800 nm wavelength is not obtained in the DDA method. It is worth noting that a narrow reflection peak and transmission dip at around 800 nm can still be obtained in the DDA method. We also examined the angle of incidence dependence of leakage radiations. When the angle of incidence is changed, the induced dipole direction of the particles will be varied accordingly. Because of the different coupling strengths of the induced in-plane and out-of-plane (perpendicular to the particle plane) dipoles, the induced dipole direction of particles may not be the same as the incident polarization direction. As sketched in Figure 4, when the induced dipole direction is parallel to the reflection direction, which is defined as the same of a mirror reflection direction, the reflection efficiency will be significantly reduced due to the nature of an oscillating dipole. The transmission efficiency will be quenched due to the phase delay (destructive interference) of the

Figure 4. Schematic of leakage radiation along different orientations relative to the incident wave and induced polarization directions.

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scattered light of metal nanoparticles along the transmission direction and the incident light. Consequently, the sum efficiency will decrease dramatically, and the total leakage efficiency will be amplified. There are many leakage radiations along different directions; we take only two leakage modes as an example to discuss their angle of incidence dependence. Both of the two discussed leakage modes are due to the 2π phase difference between radiated light rays of two neighboring columns and along the same side of the surface normal as that of the incident light (“negative” reflection and refraction). The strong leakage mode is along the direction approximately perpendicular to the induced polarization direction, and the weak leakage mode is along the direction approximately parallel to the induced polarization direction. In the simulations, the particle radius is taken to be 100 nm, and they are arranged in a square array of 600 nm spacing. Figure 5a shows that the lowest sum efficiency (the highest total leakage efficiency) at around 600 nm wavelength drops when the angle of incidence is changed from 10° to 30° and grows with further increasing angle of incidence. When the angle of incidence is 30°, the sum efficiency at 600 nm wavelength can be as low as 25%, indicating a total of 75% leakage efficiency. Figure 5b shows the reflection spectra of particle arrays of different angles of incidence. The reflection spectra show a consistent trend as that of sum efficiency. The highest reflection efficiency drops with increasing angle of incidence from 10° to 30° degrees and grows when the angle is further changed from 30° to 50°. There is a dip in the reflection spectrum at 600 nm wavelength when the angle of incidence is 30°, which had been similarly shown in previous experimental reports.26,34 Figure 5c shows the leakage radiation efficiency along the direction of θ = 119°, 150°, and 164° when the angle of incidence is changed from 10° to 30° and 50°, respectively. Those radiation angles are selected for the highest leakage efficiency at around a resonance wavelength of 600 nm. Figure 5c indicates that a leakage radiation efficiency of over 50% can be obtained along the direction of θ = 150° when the angle of incidence is 30°. Figure 5d displays the

Figure 5. (a) The lowest sum efficiency at around 600 nm wavelength, (b) reflection, (c) strong leakage, and (d) weak leakage efficiency spectra for 100 nm radius particle arrays of 600 nm spacing at different angles of incidence. Sum efficiency is the sum of absorption, reflection, and transmission efficiencies. The strong and weak leakages refer to the radiations along two directions sketched in Figure 4. 17331

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Figure 6. Spectra of particles of different polarizabilities in and out of (perpendicular) the particle plane at the angle of incidence of 30°. The particles are arranged in a square array of 600 nm spacing. (a) Reflection, absorption, and transmission spectra of 100 nm spherical particles. (b) Reflection spectra for particles of effective in-plane radius 100 nm and out-of-plane radius zero nm and vice versa. (c) Reflection spectra for particles of a 100 nm effective radius in the particle plane and different out-of-plane effective radii ranging from 80 to 150 nm. (d) The same spectra as those of (c) for wavelengths between 860 and 960 nm.

radiation efficiency for weak leakage radiation along the direction of θ = 61°, 30°, and 16° when the angle of incidence is at 10° to 30° and 50°, respectively. The spectra in Figure 5d show a trend and feature similar to those of reflection spectra. A dip is also obtained at around 600 nm wavelength along the direction of θ = 30° when the incident angle is 30°. The mechanism leading to the dip at reflection and leakage mode spectra at around 600 nm wavelength when the angle of incidence is 30° was further understood by using particles of different effective radii (or polarizabilities) along directions in the particle plane (x and y) and out of (perpendicular to) the particle plane (z). Figure 6a shows the reflection, absorption, and transmission spectra of 100 nm spherical particles when the angle of incidence is 30°. As shown in Figure 6a, both scattering and transmission spectra show a dip at 600 nm wavelength, while an absorption peak is observed at the same wavelength. The peak in the absorption spectra indicates that the coupling between particles is constructive in the particle plane. As has been discussed in the previous paragraph, the orientation of the induced dipole depends on the relative magnitudes of the induced in-plane and out-of-plane dipoles. A dip can be obtained in the reflection spectrum when the induced dipole is approximately parallel to the reflection direction. Figure 6b shows the reflection spectrum of particle arrays when the effective in-plane radius is taken as 100 nm and out-of-plane radius is set to be 0 nm and vice versa. Setting 0 nm radius is not physically applicable but will reveal a better physical pictures for mechanism study. Figure 6b indicates that one broad peak is obtained when only the in-plane polarization is included and the out-of-plane polarizability (or radius) is set to be 0. When only the out-of-plane polarization is included, there are three peaks of interest (the one at 600 nm is very clear while the other two are quite weak). One is

at 600 nm wavelength, which is due to coupling of particles arranged along the perpendicular direction (y) to the polarization direction. For the coupling of particles arranged in the plane of incidence (xz plane), two weak resonance peaks at wavelengths of λ = d(1 ( sin(Θ)), where d is the distance between particles and Θ represents the angle of incidence, and ( accounts for the phase delay of the incident radiation between neighboring particles, can be obtained.35 When the incident angle is at 30°, the two weak peaks are located at 300 and 900 nm wavelengths. We also calculated reflection spectra for particles of a fixed effective in-plane radius of 100 nm and different effective outof-plane radii ranging from 80 to 150 nm. Figure 6c shows that the dip in the reflection spectrum shifts to red with increasing effective out-of-plane radius of the particles, which is consistent with the previous reports about the size effect of particle arrays.22,23 Figure 6d shows the same reflection spectra as those of Figure 6c but between 860 and 960 nm wavelengths. Figure 6d shows a similar red shift of the resonance dip with increasing effective out-of-plane radius of the particles. When the effective out-of-plane radius is increased to 150 nm, the resonance due to the coupling between the induced out-of-plane dipoles becomes dominant relative to that of the in-plane polarization, and the resonance dip disappears.

’ CONCLUSION In summary, we investigated leakage radiation in two-dimensional silver nanoparticle arrays. The effects of particle size, spacing, number of particles, and the angle of incidence to the leakage efficiency were discussed. We found that improved coherent coupling between particles could enhance leakage efficiency, and an optimized leakage radiation efficiency over 17332

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The Journal of Physical Chemistry C 50% along a specified direction can be achieved by controlling the orientation of the induced dipoles relative to the reflection and transmission directions. The comparison between the results from the discrete dipole approximation method of periodic boundary conditions and the coupled dipole method including finite number of particles showed that in-plane leakage modes were blocked due to the applied periodic boundary conditions in the discrete dipole approximation method.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This work was supported by the ACS Petroleum Research Fund No. 48268-G6, NSF CBET 0827725, and ONR N00014-01-1118.

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