Polarized light emitting diodes using silver nanoellipsoids ¨ Sepsi,* Tibor G´al, and P´al Koppa Ors Budapest University of Technology and Economics, Department of Atomic Physics, Budapest, 1111, Hungary *
[email protected]
Abstract: We investigate the polarizing properties of periodic array of silver nanoellipsoids placed on top of a planar LED structure. The response of the particles is calculated with the periodic layered Green’s tensor in the electrostatic limit with dynamic depolarization and radiation damping corrections. We investigate the degree of polarization and the total extracted power spectra depending on parameters like lattice period, axial ratio and particle size. The proposed model is applicable over a wide range of parameters and appropriate to optimize the given structure. The optimization procedure shows that particles in the size range of 100 nm are optimal to reach 50% degree of polarization and less than 15% absorbance for an uncollimated and unpolarized dipole source. © 2014 Optical Society of America OCIS codes: (050.6624) Subwavelength structures; (230.3670) Light-emitting diodes; (240.5440) Polarization-selective devices.
References and links 1. R. Otte, L. P. de Joung, and A. H. M. van Roermund, Low-power Wireless Infrared Communications (Kluwer Academic Publishers, 1999). 2. J. C. Ramella-Roman, K. Lee, S. A. Prahl, and S. L. Jacques, “Polarized light imaging with a handheld camera,” Proc. SPIE 5068, 284–293 (2003). 3. J. S. Baba, S. S. Gleason, J. S. Goddard, and J. M. Paulus, “Application of polarization for optical motionregistered SPECT functional imaging of tumors in mice,” Proc. SPIE 5702, 97–103 (2005). 4. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley, Canada, 1999). 5. M. F. Schubert, S. Chhajed, J. K. Kim, E. F. Schubert, and J. Cho, “Polarization of light emission by 460 nm GaInN/GaN light-emitting diodes grown on (0001) oriented sapphire substrates,” Appl. Phys. Lett. 91, 051117 (2007). 6. J. H. Oh, S. J. Yang, and Y. R. Do, “Polarized white light from LEDs using remote-phosphor layer sandwiched between reflective polarizer and light-recycling dichroic filter,” Opt. Express 21, A765–A773 (2013). ¨ Sepsi, I. Szanda, and P. Koppa, “Investigation of polarized light emitting diodes with integrated wire grid 7. O. polarizer,” Opt. Express 18, 14547–14552 (2010). 8. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998). 9. S. G. Moiseev, “Thin-film polarizer made of heterogeneous medium with uniformly oriented silver nanoparticles,” Appl. Phys. A 103, 775–777 (2011). 10. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). 11. B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010). 12. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000). 13. U. Kreibig, “Electronic properties of small silver particles, the optical constant and their temperature dependence,” J. Phys. F: Metal Phys 4, 999 (1974). 14. K. Kambe, “Theory of Electron Diffraction by Crystals. Green’s function and integral equation,” Z. Naturforschg. 22a, 422–431 (1967).
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1190
15. N. A. Sanford, A. Munkholm, M. R. Krames, A. Shapiro, I. Levin, A. V. Davydov, S. Sayan, L. S. Wielunski, and T. E. Madey, “Refractive index and birefringence of InxGa1-xN films grown by MOCVD,” phys. stat. sol. (C) 2, 2783–2786 (2005). 16. L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonance wavelength and width,” J. Phys. Chem. B 107, 7343–7350 (2003).
1.
Introduction
Light emitting diodes (LEDs) became in the past decade indispensable part of our life. They are integral part of many applications ranging from general lighting to optoelectronic devices. Many of these applications can benefit from polarized light, such as optical communications [1], high contrast imaging [2, 3] or liquid crystal displays [4]. There exist inherently polarized LEDs [5], but they represent the minority of current technology and usually external polarizers are used for these applications. However in this case more than half of the light is lost due to absorption and reflection. As stated in recent studies [6, 7], recycling the reflected waves enhances the extracted power of the device and thus compact and efficient polarized LEDs can be engineered. Metallic nanoparticles attract great attention nowadays because of their peculiar optical responses. It is well known that the response of such particles will depend on the polarization of the exciting light if we introduce anisotropy into their shape [8]. Recently ellipsoidal metallic nanoparticles embedded in a transparent host were suggested for ultrathin polarizers [9]. We use this idea in this work to obtain polarized light extraction from conventional LED chips. In order to characterize quantitatively the device we carry out electromagnetic calculations of the interface between the semiconductor and packaging. We focus our attention on the modeling of the polarizer structure on top of the LED chip. The whole LED chip is described with a homogeneous refractive index, but the model provided can be easily extended to a multilayer structure. The schematic of the investigated geometry is shown in Fig. 1. The ellipsoidal nanoparticles are assumed to be oriented in the same direction and are placed into a two-dimensional square lattice. The maximal lattice period was chosen to be in the order of 100 nm to eliminate diffraction effects. The nanoparticles are modeled as dipole scatterers with their dipole moment calculated using the modified long wavelength approximation (MLWA) [10]. To calculate the scattering of the nanoparticle lattice at the interface we have used the periodic layered Green’s tensor formalism [11, 12]. The LED source was modeled as a single unpolarized dipole emitter by the incoherent summation of three dipole sources with orthogonal polarizations. The computational cost of this model is much lower than that of the rigorous methods like FDTD or finite elements, thus suitable optimization can be carried out in order to find the optimal parameters for the desired application. n2=1.5 n1=2.45
Fig. 1. Geometry of the investigated planar LED structure with silver nanoellipsoids on top.
The aim of the present work is to investigate the degree of polarization (DOP), transmittance, reflectance and absorbance of the interface shown in Fig. 1 as the function of the geometrical parameters of the nanoparticle lattice. Moreover we demonstrate that using the suggested model #208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1191
it is possible to optimize a suitable geometry which yields high enough DOP and very low absorbance. The polarizer should work in ”reflective mode”, i.e. the waves with the wrong polarization should be reflected rather than absorbed. This makes the nanostructure a potential candidate for using with polarization recycling layers in combination. In the second section we introduce the electromagnetic model of the nanoparticle lattice in a layered medium. In the third section we calculate the size, period and axial ratio dependence of the DOP, transmittance, reflectance and absorbance spectra and seek for optimal geometrical parameters for a single wavelength. 2.
Electromagnetic model
If the size of the particles are much smaller than the wavelength (λ ), the scattered field of a single particle is of dipolar nature and the quasistatic approximation can be used. The particle is described with an oscillating dipole placed in the center (denoted by r0 ) of the particle. The dipole moment of the particle is calculated as p = εh α E(r0 ),
(1)
where εh is the absolute electric permittivity of the host medium, α is the dipolar polarizability tensor and E(r0 ) is the electric field at the r0 position. This electric field does not contain the direct emission of the dipole, but it contains the electric field emitted by the dipole and reflected back by the environment. The dipolar polarizability tensor of an ellipsoid is calculated in the quasistatic approximation as 4π lx ly lz ε p − εh αij = δi j , (2) 3 εh + Li (ε p − εh ) where lx , ly and lz are the length of the semi-principal axes, ε p is the electric permittivity of the particle and δi j is the Kronecker delta. The Li parameters are called shape factors [8]. With increasing nanoparticle size the quasistatic approximation becomes more and more inaccurate. Radiation damping and dynamic depolarization begin to play an important role even in the case when the scattered fields are still of dipolar nature. Introducing the modified long wavelength approximation (MLWA) into the dipolar polarizability the validity of the dipolar approximation can be ensured for particles as large as the tenth of the wavelength: −1 kh2 1 3 α˜ i j = α i j 1 − α − i kh α i j , (3) 4π li i j 6π where kh is the wavenumber in the host material [10]. For nanoparticles of size smaller than mean free path of the conductive electrons (≈ 50 nm in metals like gold and silver), the dielectric function of the particles is modified to take the nonlocality into account [13]. The total electric field can be written as the sum of the incident and scattered electric field. The incident electric field is defined in the absence of the particles but the single interface is taken into account in the calculation. In the case of multiple particles the scattered field is the coherent sum of the individual scatterers. The electric field can be calculated using the layered Green’s tensor formalism [12]: E(r) = Einc (r) + ω 2 µ0 ∑ G(r, r j )p j ,
(4)
j
where Einc denotes the incident electric field, ω denotes the frequency of the wave, G denotes the Green’s tensor, and p j denotes the dipole moment associated with a particle placed in position r j . Applying Eq. (4) at the lattice points and inserting the dipole moments defined in #208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1192
Eq. (1) into the total electric field, we get a system of linear equations for the unknown dipole moments. Using incident plane waves and the Bloch-condition the derivation of the unknown p j dipole moments results in −1 ˜ 2 µ0 Glat (r0 ) p j = eiK(r j −r0 ) I − εh αω εh α˜ Einc (r0 ),
(5)
where K is the projection of the incident wave vector to the xy-plane, I is the identity tensor, r j is a lattice point, and εh is the permittivity of the medium containing the scatterers. Glat is called the lattice sum and is defined as Glat (r0 ) = ∑ G(r0 , r j )eiK(r j −r0 ) . ′
(6)
j
The slash in the superscript of the summation sign denotes the omission of the direct dipole emission from the r0 position. During the evaluation of the sum the direct emission part and the reflected part of the Green’s tensor is separated. The reflected part is a smooth function in the spatial domain and hence the sum converges quickly in the Fourier domain. The sum of direct part converges very slowly. In this case Ewald summation [11, 14] can be used to improve the convergence. Knowing the dipole moments of the scatterers the total electromagnetic field can be calculated using Eq. (4). Transmittance and reflectance of the structure is calculated using the electric field from Eq. (4) below and above the structure. Technically we use the spectral representation of the field and take only the zeroth order because of the small period of the nanoparticle lattice. 3.
Results and discussion
We have calculated the response of an “unpolarized” and uncollimated dipole source placed into the sample structure depicted by Fig. 1. The refractive index of the substrate and the superstrate was taken to be 2.45 and 1.5 respectively. These values correspond to an InGaN [15] LED structure. The ellipsoidal scatterers were placed into square lattice in order to eliminate the effect of lattice anisotropy. The distance between the surface of the particles and the interface was set to 5 nm. 3.1. Dependence of the spectra on the parameters of the nanoparticle layer A parameter sweep was used to investigate the effect of the parameters of the structure on four quantities: the degree of polarization (DOP), transmittance, reflectance and absorbance. The DOP is defined by (I⊥ − Ik )/(I⊥ + Ik), where I⊥ and Ik are the intensities transmitted through a linear polarizer placed perpendicular to and parallel with the major axis of the ellipsoids respectively. The length of the semi-major axis (lx ) of the ellipsoid varied between 2.5 nm and 40 nm. The axial ratio (AR= lx /ly ) varied between 1.1 and 5.0. The lattice constant (p) was chosen so that the separation between the particles was at least 1 nm and at most 15 nm. Figure 2 shows the investigated four quantities for different particle sizes at fixed axial ratio value of 2.0 while the separation between the particles was kept at constant 5 nm. The DOP curves show that the nanoellipsoid lattice achieves polarization selection as expected. With increasing particle size the maximum of the DOP increases and as we reach larger particle sizes the increase slows down and the maximum value approaches 50%. For the smallest investigated particle size the peak maximum is around 1.5%. The width of the peak first narrows with increasing particle size then broadens above 15 nm semi-major axis size. For small particle sizes the DOP curve is broad due to the reduced mean free path of the conducting electrons. With increasing particle sizes the size effect diminishes thus the width of the DOP
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1193
lx=2.5 nm lx=5.0 nm
lx=10.0 nm lx=15.0 nm
60 40 0
T [%]
DOP [%]
20 −20 −40 −60 450
500
550 600 wl [nm]
650
700
50
90
40
80
30
70
R [%]
A [%]
−80 400
20 18 16 14 12 10 8 6 4 2 400
20 10 0 400
lx=25.0 nm lx=40.0 nm
450
500
550 600 wl [nm]
650
700
450
500
550 600 wl [nm]
650
700
60 50
450
500
550 600 wl [nm]
650
700
40 400
Fig. 2. Degree of polarization (top left), transmittance (top right), absorbance (bottom left) and reflectance (bottom right) spectra of periodic array of silver nanospheroids placed on an LED chip. The major semi-axis values are presented in the legend of the figure, the AR is constant 2 and the period of the lattice is 2lx + 5 nm.
peak decreases. However above 20 nm radiation damping becomes important and causes again broadening in the DOP peak. Moreover a redshift is visible in the DOP spectra which can be attributed to the dynamic depolarization at larger particle sizes. The transmittance of the interface decreases as the particle size increases. Obviously waves with not allowed polarization will be either reflected or absorbed by the nanoparticles. It is more important however that the waves that are not transmitted should not be absorbed by the particles but rather reflected. Using a proper polarization recycler the reflected waves have another chance to escape from the device. It can be seen that with increasing size the absorbance decreases at the peak position and the reflectance increases to about 80%. Figure 3 shows the DOP and the absorbance for different lattice constants at lx = 20 nm and AR = 2.0. By decreasing the distance between the particles redshift and broadening of the DOP peak can be observed. This redshift and broadening is a sign of electrostatic dipolar interactions between the particles [16]. By reducing the distance between the particles the absorbance of the nanoparticle array decreases. This leads to higher reflection due to the strong dipolar interaction of the nanoparticles. We have investigated the effect of the axial ratio on the DOP and absorbance spectra too. We have found that AR values between 1.5 and 2.5 yield the best DOP (about 50%) for the visible wavelengths and in this range the axial ratio only influences the peak position.
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1194
45
60
35
20 A [%]
DOP [%]
40
0 −20
p=51.0 p=55.0 p=60.0
−40 −60 400
p=51.0 p=55.0 p=60.0
40
450
500
550 600 wl [nm]
650
30 25 20 15 10
700
5 400
450
500
550 600 wl [nm]
650
700
Fig. 3. Degree of polarization (left) and absorbance (right) spectra of periodic array of silver nanospheroids placed on an LED chip. The major semi-axis was lx = 25 nm, the axial ratio was equal to 2. The lattice period is presented in the legend of the figure.
3.2. Optimal parameters for single wavelength In order to obtain the optimal parameters we have carried out another parameter sweep with finer resolution at a single wavelength of 620 nm. The parameter sweep showed that for each particle size the arrangement calculated with the smallest period and with AR = 2.2 yielded the highest DOP and lowest absorbance values. Figure 4 shows the DOP, absorbance, transmittance and reflectance curves as a function of the particle semi-major axis at 620 nm wavelength, AR = 2.2 and p/lx = 2.1. The DOP values increase while the absorbance values decrease with increasing particle size. The curves approach a specific limit but do not reach in this size regime the extreme values. Note however that at the margin of the investigated interval the change of the curves is minor.
DOP, T, R, A [%]
100
DOP A
80
T R
60 40 20 0
0
10
20
30 lx [nm]
40
50
Fig. 4. DOP, absorbance, transmittance and reflectance at 620 nm as function of particle semi-major axis. The axial ratio was equal to 2.2, the period was p = 2.1lx .
The best configuration yields about 50% DOP, and less than 15% absorbance. The transmittance of the interface is about 10% while the reflectance reaches 75%.
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1195
4.
Conclusion
In this study we have investigated the polarizing properties of silver nanoellipsoid array on a conventional LED surface. We have calculated the response of the system with the layered periodic Green’s tensor and the nanoparticles were represented as polarizable dipoles. The method was extended to larger particle sizes with the use of correction factors for dynamic depolarization and radiation damping in the particle polarizability. The finite size effect of the particles was too taken into account in the dielectric function of the silver. The investigated curves show strong size, shape, and lattice period dependence. We have demonstrated that optimization of the structure is possible with the available method at a single wavelength. The best results were obtained at particle major axis of 100 nm, axial ratio of 2.2 and period of 210 nm. The results indicate that particle sizes above 100 nm can be subject of future investigation in order to achieve even higher DOP values and lower absorbance. For the best configurations the DOP values are near 50%, while the absorbance is below 15% for the unpolarized dipole emitter. The transmittance and reflectance is about 10% and 75% respectively. Using a proper polarization recycler to recover the reflected waves, the extracted power can largely surpass the LED + external polarizer configuration. Acknowledgments ´ This work has been supported by the grants TAMOP – 4.2.2.B-10/1–2010-0009 and VKSZ12-1-2013-80.
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1196
[email protected]
Abstract: We investigate the polarizing properties of periodic array of silver nanoellipsoids placed on top of a planar LED structure. The response of the particles is calculated with the periodic layered Green’s tensor in the electrostatic limit with dynamic depolarization and radiation damping corrections. We investigate the degree of polarization and the total extracted power spectra depending on parameters like lattice period, axial ratio and particle size. The proposed model is applicable over a wide range of parameters and appropriate to optimize the given structure. The optimization procedure shows that particles in the size range of 100 nm are optimal to reach 50% degree of polarization and less than 15% absorbance for an uncollimated and unpolarized dipole source. © 2014 Optical Society of America OCIS codes: (050.6624) Subwavelength structures; (230.3670) Light-emitting diodes; (240.5440) Polarization-selective devices.
References and links 1. R. Otte, L. P. de Joung, and A. H. M. van Roermund, Low-power Wireless Infrared Communications (Kluwer Academic Publishers, 1999). 2. J. C. Ramella-Roman, K. Lee, S. A. Prahl, and S. L. Jacques, “Polarized light imaging with a handheld camera,” Proc. SPIE 5068, 284–293 (2003). 3. J. S. Baba, S. S. Gleason, J. S. Goddard, and J. M. Paulus, “Application of polarization for optical motionregistered SPECT functional imaging of tumors in mice,” Proc. SPIE 5702, 97–103 (2005). 4. P. Yeh and C. Gu, Optics of Liquid Crystal Displays (John Wiley, Canada, 1999). 5. M. F. Schubert, S. Chhajed, J. K. Kim, E. F. Schubert, and J. Cho, “Polarization of light emission by 460 nm GaInN/GaN light-emitting diodes grown on (0001) oriented sapphire substrates,” Appl. Phys. Lett. 91, 051117 (2007). 6. J. H. Oh, S. J. Yang, and Y. R. Do, “Polarized white light from LEDs using remote-phosphor layer sandwiched between reflective polarizer and light-recycling dichroic filter,” Opt. Express 21, A765–A773 (2013). ¨ Sepsi, I. Szanda, and P. Koppa, “Investigation of polarized light emitting diodes with integrated wire grid 7. O. polarizer,” Opt. Express 18, 14547–14552 (2010). 8. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998). 9. S. G. Moiseev, “Thin-film polarizer made of heterogeneous medium with uniformly oriented silver nanoparticles,” Appl. Phys. A 103, 775–777 (2011). 10. K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003). 11. B. Gallinet, A. M. Kern, and O. J. F. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010). 12. M. Paulus, P. Gay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green’s tensor for stratified media,” Phys. Rev. E 62, 5797–5807 (2000). 13. U. Kreibig, “Electronic properties of small silver particles, the optical constant and their temperature dependence,” J. Phys. F: Metal Phys 4, 999 (1974). 14. K. Kambe, “Theory of Electron Diffraction by Crystals. Green’s function and integral equation,” Z. Naturforschg. 22a, 422–431 (1967).
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1190
15. N. A. Sanford, A. Munkholm, M. R. Krames, A. Shapiro, I. Levin, A. V. Davydov, S. Sayan, L. S. Wielunski, and T. E. Madey, “Refractive index and birefringence of InxGa1-xN films grown by MOCVD,” phys. stat. sol. (C) 2, 2783–2786 (2005). 16. L. Zhao, K. L. Kelly, and G. C. Schatz, “The extinction spectra of silver nanoparticle arrays: influence of array structure on plasmon resonance wavelength and width,” J. Phys. Chem. B 107, 7343–7350 (2003).
1.
Introduction
Light emitting diodes (LEDs) became in the past decade indispensable part of our life. They are integral part of many applications ranging from general lighting to optoelectronic devices. Many of these applications can benefit from polarized light, such as optical communications [1], high contrast imaging [2, 3] or liquid crystal displays [4]. There exist inherently polarized LEDs [5], but they represent the minority of current technology and usually external polarizers are used for these applications. However in this case more than half of the light is lost due to absorption and reflection. As stated in recent studies [6, 7], recycling the reflected waves enhances the extracted power of the device and thus compact and efficient polarized LEDs can be engineered. Metallic nanoparticles attract great attention nowadays because of their peculiar optical responses. It is well known that the response of such particles will depend on the polarization of the exciting light if we introduce anisotropy into their shape [8]. Recently ellipsoidal metallic nanoparticles embedded in a transparent host were suggested for ultrathin polarizers [9]. We use this idea in this work to obtain polarized light extraction from conventional LED chips. In order to characterize quantitatively the device we carry out electromagnetic calculations of the interface between the semiconductor and packaging. We focus our attention on the modeling of the polarizer structure on top of the LED chip. The whole LED chip is described with a homogeneous refractive index, but the model provided can be easily extended to a multilayer structure. The schematic of the investigated geometry is shown in Fig. 1. The ellipsoidal nanoparticles are assumed to be oriented in the same direction and are placed into a two-dimensional square lattice. The maximal lattice period was chosen to be in the order of 100 nm to eliminate diffraction effects. The nanoparticles are modeled as dipole scatterers with their dipole moment calculated using the modified long wavelength approximation (MLWA) [10]. To calculate the scattering of the nanoparticle lattice at the interface we have used the periodic layered Green’s tensor formalism [11, 12]. The LED source was modeled as a single unpolarized dipole emitter by the incoherent summation of three dipole sources with orthogonal polarizations. The computational cost of this model is much lower than that of the rigorous methods like FDTD or finite elements, thus suitable optimization can be carried out in order to find the optimal parameters for the desired application. n2=1.5 n1=2.45
Fig. 1. Geometry of the investigated planar LED structure with silver nanoellipsoids on top.
The aim of the present work is to investigate the degree of polarization (DOP), transmittance, reflectance and absorbance of the interface shown in Fig. 1 as the function of the geometrical parameters of the nanoparticle lattice. Moreover we demonstrate that using the suggested model #208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1191
it is possible to optimize a suitable geometry which yields high enough DOP and very low absorbance. The polarizer should work in ”reflective mode”, i.e. the waves with the wrong polarization should be reflected rather than absorbed. This makes the nanostructure a potential candidate for using with polarization recycling layers in combination. In the second section we introduce the electromagnetic model of the nanoparticle lattice in a layered medium. In the third section we calculate the size, period and axial ratio dependence of the DOP, transmittance, reflectance and absorbance spectra and seek for optimal geometrical parameters for a single wavelength. 2.
Electromagnetic model
If the size of the particles are much smaller than the wavelength (λ ), the scattered field of a single particle is of dipolar nature and the quasistatic approximation can be used. The particle is described with an oscillating dipole placed in the center (denoted by r0 ) of the particle. The dipole moment of the particle is calculated as p = εh α E(r0 ),
(1)
where εh is the absolute electric permittivity of the host medium, α is the dipolar polarizability tensor and E(r0 ) is the electric field at the r0 position. This electric field does not contain the direct emission of the dipole, but it contains the electric field emitted by the dipole and reflected back by the environment. The dipolar polarizability tensor of an ellipsoid is calculated in the quasistatic approximation as 4π lx ly lz ε p − εh αij = δi j , (2) 3 εh + Li (ε p − εh ) where lx , ly and lz are the length of the semi-principal axes, ε p is the electric permittivity of the particle and δi j is the Kronecker delta. The Li parameters are called shape factors [8]. With increasing nanoparticle size the quasistatic approximation becomes more and more inaccurate. Radiation damping and dynamic depolarization begin to play an important role even in the case when the scattered fields are still of dipolar nature. Introducing the modified long wavelength approximation (MLWA) into the dipolar polarizability the validity of the dipolar approximation can be ensured for particles as large as the tenth of the wavelength: −1 kh2 1 3 α˜ i j = α i j 1 − α − i kh α i j , (3) 4π li i j 6π where kh is the wavenumber in the host material [10]. For nanoparticles of size smaller than mean free path of the conductive electrons (≈ 50 nm in metals like gold and silver), the dielectric function of the particles is modified to take the nonlocality into account [13]. The total electric field can be written as the sum of the incident and scattered electric field. The incident electric field is defined in the absence of the particles but the single interface is taken into account in the calculation. In the case of multiple particles the scattered field is the coherent sum of the individual scatterers. The electric field can be calculated using the layered Green’s tensor formalism [12]: E(r) = Einc (r) + ω 2 µ0 ∑ G(r, r j )p j ,
(4)
j
where Einc denotes the incident electric field, ω denotes the frequency of the wave, G denotes the Green’s tensor, and p j denotes the dipole moment associated with a particle placed in position r j . Applying Eq. (4) at the lattice points and inserting the dipole moments defined in #208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1192
Eq. (1) into the total electric field, we get a system of linear equations for the unknown dipole moments. Using incident plane waves and the Bloch-condition the derivation of the unknown p j dipole moments results in −1 ˜ 2 µ0 Glat (r0 ) p j = eiK(r j −r0 ) I − εh αω εh α˜ Einc (r0 ),
(5)
where K is the projection of the incident wave vector to the xy-plane, I is the identity tensor, r j is a lattice point, and εh is the permittivity of the medium containing the scatterers. Glat is called the lattice sum and is defined as Glat (r0 ) = ∑ G(r0 , r j )eiK(r j −r0 ) . ′
(6)
j
The slash in the superscript of the summation sign denotes the omission of the direct dipole emission from the r0 position. During the evaluation of the sum the direct emission part and the reflected part of the Green’s tensor is separated. The reflected part is a smooth function in the spatial domain and hence the sum converges quickly in the Fourier domain. The sum of direct part converges very slowly. In this case Ewald summation [11, 14] can be used to improve the convergence. Knowing the dipole moments of the scatterers the total electromagnetic field can be calculated using Eq. (4). Transmittance and reflectance of the structure is calculated using the electric field from Eq. (4) below and above the structure. Technically we use the spectral representation of the field and take only the zeroth order because of the small period of the nanoparticle lattice. 3.
Results and discussion
We have calculated the response of an “unpolarized” and uncollimated dipole source placed into the sample structure depicted by Fig. 1. The refractive index of the substrate and the superstrate was taken to be 2.45 and 1.5 respectively. These values correspond to an InGaN [15] LED structure. The ellipsoidal scatterers were placed into square lattice in order to eliminate the effect of lattice anisotropy. The distance between the surface of the particles and the interface was set to 5 nm. 3.1. Dependence of the spectra on the parameters of the nanoparticle layer A parameter sweep was used to investigate the effect of the parameters of the structure on four quantities: the degree of polarization (DOP), transmittance, reflectance and absorbance. The DOP is defined by (I⊥ − Ik )/(I⊥ + Ik), where I⊥ and Ik are the intensities transmitted through a linear polarizer placed perpendicular to and parallel with the major axis of the ellipsoids respectively. The length of the semi-major axis (lx ) of the ellipsoid varied between 2.5 nm and 40 nm. The axial ratio (AR= lx /ly ) varied between 1.1 and 5.0. The lattice constant (p) was chosen so that the separation between the particles was at least 1 nm and at most 15 nm. Figure 2 shows the investigated four quantities for different particle sizes at fixed axial ratio value of 2.0 while the separation between the particles was kept at constant 5 nm. The DOP curves show that the nanoellipsoid lattice achieves polarization selection as expected. With increasing particle size the maximum of the DOP increases and as we reach larger particle sizes the increase slows down and the maximum value approaches 50%. For the smallest investigated particle size the peak maximum is around 1.5%. The width of the peak first narrows with increasing particle size then broadens above 15 nm semi-major axis size. For small particle sizes the DOP curve is broad due to the reduced mean free path of the conducting electrons. With increasing particle sizes the size effect diminishes thus the width of the DOP
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1193
lx=2.5 nm lx=5.0 nm
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60 40 0
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Fig. 2. Degree of polarization (top left), transmittance (top right), absorbance (bottom left) and reflectance (bottom right) spectra of periodic array of silver nanospheroids placed on an LED chip. The major semi-axis values are presented in the legend of the figure, the AR is constant 2 and the period of the lattice is 2lx + 5 nm.
peak decreases. However above 20 nm radiation damping becomes important and causes again broadening in the DOP peak. Moreover a redshift is visible in the DOP spectra which can be attributed to the dynamic depolarization at larger particle sizes. The transmittance of the interface decreases as the particle size increases. Obviously waves with not allowed polarization will be either reflected or absorbed by the nanoparticles. It is more important however that the waves that are not transmitted should not be absorbed by the particles but rather reflected. Using a proper polarization recycler the reflected waves have another chance to escape from the device. It can be seen that with increasing size the absorbance decreases at the peak position and the reflectance increases to about 80%. Figure 3 shows the DOP and the absorbance for different lattice constants at lx = 20 nm and AR = 2.0. By decreasing the distance between the particles redshift and broadening of the DOP peak can be observed. This redshift and broadening is a sign of electrostatic dipolar interactions between the particles [16]. By reducing the distance between the particles the absorbance of the nanoparticle array decreases. This leads to higher reflection due to the strong dipolar interaction of the nanoparticles. We have investigated the effect of the axial ratio on the DOP and absorbance spectra too. We have found that AR values between 1.5 and 2.5 yield the best DOP (about 50%) for the visible wavelengths and in this range the axial ratio only influences the peak position.
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1194
45
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Fig. 3. Degree of polarization (left) and absorbance (right) spectra of periodic array of silver nanospheroids placed on an LED chip. The major semi-axis was lx = 25 nm, the axial ratio was equal to 2. The lattice period is presented in the legend of the figure.
3.2. Optimal parameters for single wavelength In order to obtain the optimal parameters we have carried out another parameter sweep with finer resolution at a single wavelength of 620 nm. The parameter sweep showed that for each particle size the arrangement calculated with the smallest period and with AR = 2.2 yielded the highest DOP and lowest absorbance values. Figure 4 shows the DOP, absorbance, transmittance and reflectance curves as a function of the particle semi-major axis at 620 nm wavelength, AR = 2.2 and p/lx = 2.1. The DOP values increase while the absorbance values decrease with increasing particle size. The curves approach a specific limit but do not reach in this size regime the extreme values. Note however that at the margin of the investigated interval the change of the curves is minor.
DOP, T, R, A [%]
100
DOP A
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T R
60 40 20 0
0
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30 lx [nm]
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Fig. 4. DOP, absorbance, transmittance and reflectance at 620 nm as function of particle semi-major axis. The axial ratio was equal to 2.2, the period was p = 2.1lx .
The best configuration yields about 50% DOP, and less than 15% absorbance. The transmittance of the interface is about 10% while the reflectance reaches 75%.
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1195
4.
Conclusion
In this study we have investigated the polarizing properties of silver nanoellipsoid array on a conventional LED surface. We have calculated the response of the system with the layered periodic Green’s tensor and the nanoparticles were represented as polarizable dipoles. The method was extended to larger particle sizes with the use of correction factors for dynamic depolarization and radiation damping in the particle polarizability. The finite size effect of the particles was too taken into account in the dielectric function of the silver. The investigated curves show strong size, shape, and lattice period dependence. We have demonstrated that optimization of the structure is possible with the available method at a single wavelength. The best results were obtained at particle major axis of 100 nm, axial ratio of 2.2 and period of 210 nm. The results indicate that particle sizes above 100 nm can be subject of future investigation in order to achieve even higher DOP values and lower absorbance. For the best configurations the DOP values are near 50%, while the absorbance is below 15% for the unpolarized dipole emitter. The transmittance and reflectance is about 10% and 75% respectively. Using a proper polarization recycler to recover the reflected waves, the extracted power can largely surpass the LED + external polarizer configuration. Acknowledgments ´ This work has been supported by the grants TAMOP – 4.2.2.B-10/1–2010-0009 and VKSZ12-1-2013-80.
#208613 - $15.00 USD Received 19 Mar 2014; revised 10 May 2014; accepted 12 May 2014; published 19 Jun 2014 (C) 2014 OSA 30 June 2014 | Vol. 22, No. S4 | DOI:10.1364/OE.22.0A1190 | OPTICS EXPRESS A1196
