|
1.IntroductionPhotonic crystals (PCs) are periodic structures of dielectric material that can reflect light with high efficiency due to optical interference effects; this allows PCs to bend and control light with high precision. PCs can be designed to have forbidden optical frequency bands that are known as photonic band gaps (PBGs). In addition to commercially available low-loss PC fibers, other applications of PCs include low-threshold lasers,1 single-mode light emitting diodes,2,3 Bragg mirrors,4–6 optical filters,7–9 and efficient planar antennas.10 PCs two-dimensional (2-D) periodicity can guide light in a plane. The propagated light inside the PC structure is affected mainly by material properties (refractive index) and the period size of the crystal structure. In addition, the direction of light polarization and wavelength has a significant effect on reflectivity of a PC waveguide.11–13 Various numerical computation methods exist for studying the optical properties of PCs. Transfer matrix methods,14,15 finite-element methods (FEMs),16–20 finite difference time domain,21–28 finite difference frequency domain (FDFD) methods,29–32 and plane-wave expansion methods33–36 have been developed over the past few decades for this purpose. When modeling 2-D PCs, infinitely periodic boundary conditions are often used in both dimensions,26,30,33,37 or the models can be finite in both directions.38–40 Yet, it is impossible to fabricate a PC with an infinite number of periods, and it is challenging and expensive to fabricate highly uniform PCs with many periods. Therefore, investigating the effects of varying the number of periods in a PC is important for optimally fabricating them. The model introduced here can help precisely determine how many periods are needed to achieve the desired reflectivity for a given incident wavelength. Others have made use of semi-infinite models to study PC properties, but none of these specifically investigates the precise effect on the number of periods relative to reflection spectra.41–43 This work introduces a useful model to calculate the reflectivity spectrum for a 2-D PC with a finite number of periods in one dimension and an infinite periodicity in the perpendicular direction. The effects of varying the situational parameters (polarization, material properties, number of periods, and so on) can be easily investigated, as demonstrated in this work. Incorporation of wavelength-dependent dielectric functions is a key benefit of the model. The results are compared to the photonic band structure calculated through a FDFD model to compare the reflectivity spectrum with the photonic band structure. 2.Reflection ModelReflection spectra from various PC configurations were calculated using a FEM model (geometry created and discretized through COMSOL Multiphysics). The model, shown in Fig. 1(a), is a semiperiodic lattice of circles (cylinders, if the infinite -dimension is considered) of one material within a matrix of another material. The -dimension has a finite number of periods, whereas in the -direction, it is modeled to be infinitely periodic by assigning periodic boundary conditions. This model has been developed to study the effects of variable geometric and material parameters, especially the number of periods and the period size, on the reflection spectra of light incident on the structure in the negative -direction. The materials chosen were GaAs due to its relatively high refractive index in the visible range, to 5, and air with .44 The large ratio improves the overall reflectivity of the PC.45 The model consists of a vertical unit cell of a square crystal lattice with periodic boundary conditions in one dimension ( directions). The number of periods in the -direction, , is varied, as shown in Fig. 1(a). Light is incident in the negative -direction. Materials studied in this work include a GaAs, Si, or slab with air holes, and GaAs posts surrounded by air (the model extends infinitely in the -direction). The thickness of the simulation space is a function of the number of periods inside the unit cell so that the distance between the incident light port and the PC is always four times the period (). In this model, the incident light is polarized in either the transverse electric (TE) or transverse magnetic (TM) direction, with respect to the -axis. Figure 1(b) shows how the reflection spectrum changes as a function of the number of periods, , in the unit cell. The results were calculated for air holes in GaAs with a constant period (center-to-center distance) of . Incident light was polarized in the -direction, which is also known as the TM mode for PCs.46 As was increased, the reflection peak was found to increase for PBG wavelengths, as expected. The inset plots the maximum reflection at the PBG as a function of . At 1255 nm, the reflection magnitude for is 16.4%, and for , it reaches 99.9%. The reflectivity reaches a value at and exceeds 99% when . The work by Abdulhalim47 demonstrates the same trend of increasing reflection for an increasing number of periods for stacked anisotropic wave plate layers. Figure 2 shows the polarization dependence of the PC reflection spectrum. The constant parameters for this test were and with air holes in GaAs. Only the polarization was changed, with the incident light being polarized in (TE mode) and (TM mode), as shown in Figs. 2(a) and 2(b), respectively. The width of the PBG was found to be greater for the TE mode compared to the TM mode. The PBG wavelength range was for the TE mode and 300 nm for the TM mode; for TE polarization, the reflectivity is near 100% across the entire width of the PBG. The effects of varying material properties on the PBG were also investigated, as seen in Fig. 3. GaAs posts in air [Fig. 3(a)] exhibit a wider PBG than air holes in GaAs [Fig. 3(b)]. The width of the PBG wavelength range for GaAs posts was , while for air holes, it was . Air holes in GaAs show higher reflectivity () than the peak value for GaAs posts in air () when both unit cells contain periods. For GaAs posts, the PBG has a center wavelength near 725 nm, and for air holes, the PBG center wavelength was near 1255 nm. This model was lastly used to plot the PBG peak shift as a function of the PC period. The resulting reflection spectra are shown in Fig. 4 in a waterfall plot, where varies from 200 to 300 nm. As is increased, the PBG central wavelength is redshifted. This trend is plotted in the inset of Fig. 4 for the wavelengths of peak reflection, ; it shows that increases linearly with , where . A small PBG can be seen for to 300 nm. The gap shifts to lower wavelengths as a function of , similar to the behavior of the main higher-wavelength PBG region. Once it shifts to wavelengths , the reflectivity drops to zero because at this point, the photon energy exceeds the bandgap of GaAs, permitting light to be absorbed into the GaAs. The presence of this phenomenon helps to confirm the validity of the results. Using a material with a larger bandgap can extend the PBG to lower wavelengths; however, the bandgap width is typically inversely proportional to the refractive index of a given material.48 This reduces the reflectivity, as increases with ; therefore, a larger period number, , is required to maintain sufficient reflectivity.45 Models such as the one described in this work may prove beneficial for the determination of optimal parameters prior to fabrication or experimental characterization of PC structures for optical applications. 3.Finite Difference Frequency Domain ModelNext, the FDFD method was used to get a complete plot of the photonic band structure for a 2-D square lattice of dielectric cylinders in a matrix with a different dielectric constant and to compare these results with the above model. As is often done for the TM mode in PC calculations, we begin with Eq. (1) (directly derived from Maxwell’s equations), as it is dependent only on the electric field.32,46,49 The solutions of this eigenvalue equation provide the photonic band structures for the desired crystal and light parameters: Equation (1) is an eigenvalue problem that was solved using the FDFD method.49 This was accomplished by first setting up a mesh along the desired geometry and calculating the relevant -vectors along the geometry’s irreducible Brillouin zone.To begin, Eq. (1) is simplified using the curl of curl identity: We assume that there are no free charges, which demands that Equation (2) now simplifies to Expressing Eq. (3) in terms of second order central finite differences gives the following: where and are the electric field vectors along and , and and are the physical distances between the mesh elements along the and axis, respectively. Equation (4) is fully decoupled; that is, changes in happen only along (-axis), and changes in happen only along (-axis). This allows it to be rewritten as To solve this eigenvalue problem, a mesh is generated along a unit cell of the PC, with values corresponding to mesh elements. Figure 5 shows an example of the mesh in which the electric field at each position is labeled . In the finite differences equation, the field at a given position, , corresponds to one position, . The and terms correspond to adjacent mesh cells in the lattice. Elements within the unit cell are shown as bounded by solid lines and those within adjacent cells are represented by dashed lines. The Appendix further discusses the mesh elements of the unit cell, as well as the required mesh size at which calculated bands converge to a common value.The matrix multiplying the electric field eigenvector on the left side of Eq. (5) takes into account the connections between each mesh element and its neighboring elements. Thus, in Fig. 5, connects to and within the same unit cell and connects to and in adjacent unit cells. Periodic boundary conditions are therefore applied to the resulting matrix in the appropriate positions to account for these intercell connections between the edge mesh elements. This requires the use of imaginary exponential terms,, for periodic Bloch waves in a periodic crystal. Bloch’s theorem maintains that all of the waves capable of propagating within the crystal will do so periodically such that they will exist for -values within the first Brillouin zone of the reciprocal lattice. Appending this exponential to the necessary terms conforms to this formulation and is necessary since only one unit cell is contained in the model geometry. The term within the exponential refers to the component of the wave vector () along the direction of the basis vector () corresponding to the direction of the connection between mesh points. In the square case, is along the direction and is along the direction. These correspond to - and -directions, respectively. Once the eigenvalues are calculated, they are multiplied by and square-rooted to isolate the eigenmodes, , which are then normalized. The 2-D photonic band structure was calculated for the TM mode of the square lattice GaAs model and is plotted in Figs. 6(a) and 6(b) next to the reflection spectra solution [Fig. 6(c)]. In both the FDFD and reflection models in Fig. 6, the dielectric constant was set to one for air and 11.8 for GaAs. Figure 6(a) also shows the real and reciprocal space lattices used in the model. The photonic band structure result was found to closely match the reflection spectrum as well as agreeing with existing results.46 Last, due to the complexity of the FDFD model, constant material properties were implemented. Thus, the dielectric function of the materials in the model was constant instead of being wavelength dependent. As shown in Fig. 6, the FDFD band gap results match very well with the reflection spectrum from the other model in which constant material properties were also used. In physical reality, instead of being constant, the dielectric function depends on the wavelength of the incident light. For this reason, reflection spectra were calculated for a dielectric constant, , and for a dielectric as a function of wavelength, .44 The resulting spectra are shown in Fig. 7(a) for the wavelength range plotted in previous figures with the constant permittivity shown in blue and variable dielectric function in red. The values of the real () and imaginary () parts of the dielectric function were plotted versus wavelength as well, as shown in Figs. 7(b) and 7(c), respectively. For a wavelength-dependent dielectric function, continues to vary, albeit close to the constant value of 11.8, as demonstrated in Fig. 7(b). The imaginary part is zero for a constant permittivity and quickly approaches zero for . The larger discrepancy between the reflection spectra for wavelengths from 800 to 1000 nm, as compared to the 1000 to 2000 nm range, is due to the difference between values of for and . See the Appendix for results calculated for air holes in Si and surrounding media. 4.Conclusion and DiscussionA compact model was designed to evaluate the reflection of a 2-D square lattice PC with a finite number of periods, , under different conditions. The results showed precisely how the reflectivity changes with . With the reflection model, the effects of using both a constant and variable permittivity were studied. Results reveal the potential errors resulting from using a constant dielectric function in pursuit of the most accurate calculations of photonic band structures. A FDFD model was used to calculate the full TM mode photonic band structure for comparison with the reflection model. The FDFD calculated photonic bandgaps align well with the reflection bands determined through the reflection model. Variations of the model introduced here can also be used to simulate a hexagonal lattice, calculate transmission spectra, and investigate various incident angles. Utilizing this type of model will help facilitate advances in the development of PC technologies. AppendicesAppendixAdditional materials, Si and , were studied with the reflection model as well. Here, models were established to calculate the reflection spectra for air holes in surrounding media of each material for the TM mode with , and wavelength-dependent dielectric functions were implemented.50,51 Figures 8(a) and 8(b) show the results of varying for each material. As was the case for air holes in GaAs, , the wavelength of the reflection peak is redshifted for increasing . A comparison between the peak shifts for each material is shown in Fig. 8(c). The plot shows the values for versus for each material. Comparing the peak widths in Figs. 8(a) and 8(b), the peaks are clearly narrower than those calculated for Si. In order to illustrate the convergence of the FDFD model for a sufficient number of mesh elements in the unit cell, the plot shown in Fig. 9 was created. The plot was created by taking the wavelength values of the first and second bands at the point in reciprocal space as a function of mesh elements in one row of the unit cell. This was done for with GaAs (). The plot demonstrates convergence for a unit row containing 25 or more mesh elements. Thus, in the interest of accuracy, a unit cell of mesh elements was utilized for the FDFD calculations performed to create Figs. 6(a) and 6(b). AcknowledgmentsAuthors would like to acknowledge the help of Salvador Barraza-Lopez through many useful conversations and discussions. Also, we want to recognize Paul Millett for helping to inspire the motivation for this research project. S.B. has received support from the SPIE Optics and Photonics Education Scholarship and the Doctoral Academy Fellowship. J.M. has received support through the George D. Lingelbach Memorial Scholarship and the Student Undergraduate Research Fellowship (SURF) provided by the Arkansas Department of Higher Education. J.H. has received support from the Arkansas Biosciences Institute and from the Provost’s Collaborative Research Grant funded by the Office of the Provost at University of Arkansas. ReferencesL. Marko, Y. Tomoyuki and S. Axel,
“Low-threshold photonic crystal laser,”
Appl. Phys. Lett., 81
(15), 2680
(2002). http://dx.doi.org/10.1063/1.1511538 APPLAB 0003-6951 Google Scholar
I. Schnitzer et al.,
“The single-mode light-emitting-diode,”
Nano ASI, 308 369
–378
(1993). http://dx.doi.org/10.1007/978-1-4899-1606-8_28 Google Scholar
G. Shambat et al.,
“Ultrafast direct modulation of a single-mode photonic crystal nanocavity light-emitting diode,”
Nat. Commun., 2 539
(2011). http://dx.doi.org/10.1038/ncomms1543 NCAOBW 2041-1723 Google Scholar
V. Berger,
“Nonlinear photonic crystals,”
Phys. Rev. Lett., 81 4136
–4139
(1998). http://dx.doi.org/10.1103/PhysRevLett.81.4136 PRLTAO 0031-9007 Google Scholar
A. Kavokin et al.,
“Photonic Bloch oscillations in laterally confined Bragg mirrors,”
Phys. Rev. B, 61 4413
–4416
(2000). http://dx.doi.org/10.1103/PhysRevB.61.4413 Google Scholar
B. Bakir et al.,
“Surface-emitting microlaser combining two-dimensional photonic crystal membrane and vertical Bragg mirror,”
Appl. Phys. Lett., 88
(8),
(2006). http://dx.doi.org/10.1063/1.2172730 APPLAB 0003-6951 Google Scholar
M. Qiu et al.,
“Photonic crystal optical filter based on contra-directional waveguide coupling,”
Appl. Phys. Lett., 83
(5121),
(2003). http://dx.doi.org/10.1063/1.1634373 APPLAB 0003-6951 Google Scholar
M. Qiu,
“Ultra-compact optical filter in two-dimensional photonic crystal,”
Electron. Lett., 40
(9), 539
–540
(2004). http://dx.doi.org/10.1049/el:20040362 ELLEAK 0013-5194 Google Scholar
G. Shambat et al.,
“Photonic crystal filters for multi-band optical filtering on a monolithic substrate,”
J. Nanophotonics, 3 031506
(2009). http://dx.doi.org/10.1117/1.3110223 1934-2608 Google Scholar
E. R. Brown and E. R. Parker,
“Radiation properties of a planar antenna on a photonic crystal,”
Opt. Soc., 10
(2), 404
(1993). http://dx.doi.org/10.1364/JOSAB.10.000404 Google Scholar
T. Liu et al.,
“Design of a compact photonic-crystal-based polarizing beam splitter,”
IEEE Photonics Technol. Lett., 17
(7), 1435
–1437
(2005). http://dx.doi.org/10.1109/LPT.2005.848278 IPTLEL 1041-1135 Google Scholar
S. G. Johnson et al.,
“Guided modes in photonic crystal slabs,”
Phys. Rev. B, 60
(8), 5751
–5758
(1999). http://dx.doi.org/10.1103/PhysRevB.60.5751 Google Scholar
T. R. Woliński et al.,
“Polarization effects in photonic liquid crystal fibers,”
Meas. Sci. Technol., 18
(10), 3061
–3069
(2007). http://dx.doi.org/10.1088/0957-0233/18/10/S03 MSTCEP 0957-0233 Google Scholar
P. M. Bell et al.,
“A program for calculating photonic band structures and transmission coefficients of complex structures,”
Comput. Phys. Commun., 85
(2), 306
–322
(1995). http://dx.doi.org/10.1016/0010-4655(94)00131-K CPHCBZ 0010-4655 Google Scholar
M. M. Sigalas et al.,
“Theoretical investigation of off-plane propagation of electromagnetic waves in two-dimensional photonic crystals,”
Phys. Rev. B, 58
(11), 6791
–6794
(1998). http://dx.doi.org/10.1103/PhysRevB.58.6791 Google Scholar
B. P. Hiett et al.,
“Application of finite element methods to photonic crystal modelling,”
IEE Proc.—Sci. Meas. Technol., 149
(5), 293
–296
(2002). http://dx.doi.org/10.1049/ip-smt:20020642 Google Scholar
W. Axmann and P. Kuchment,
“An efficient finite element method for computing spectra of photonic and acoustic band-gap materials,”
J. Comput. Phys., 150
(2), 468
–481
(1999). http://dx.doi.org/10.1006/jcph.1999.6188 JCTPAH 0021-9991 Google Scholar
A. Peyrilloux et al.,
“Comparison between the finite element method, the localized function method and a novel equivalent averaged index method for modelling photonic crystal fibres,”
J. Opt. Pure Appl. Opt., 4
(3), 257
–262
(2002). http://dx.doi.org/10.1088/1464-4258/4/3/307 Google Scholar
D. C. Dobson,
“An efficient method for band structure calculations in 2D photonic crystals,”
J. Comput. Phys., 149
(2), 363
–376
(1999). http://dx.doi.org/10.1006/jcph.1998.6157 JCTPAH 0021-9991 Google Scholar
D. C. Dobson, J. Gopalakrishnan and J. E. Pasciak,
“An efficient method for band structure calculations in 3D photonic crystals,”
J. Comput. Phys., 161
(2), 668
–679
(2000). http://dx.doi.org/10.1006/jcph.2000.6521 JCTPAH 0021-9991 Google Scholar
T. Baba et al.,
“Light propagation characteristics of straight single-line-defect waveguides in photonic crystal slabs fabricated into a silicon-on-insulator substrate,”
IEEE J. Quantum Electron., 38
(7), 743
–752
(2002). http://dx.doi.org/10.1109/JQE.2002.1017584 IEJQA7 0018-9197 Google Scholar
A. Chutinan and S. Noda,
“Waveguides and waveguide bends in two-dimensional photonic crystal slabs,”
Phys. Rev. B, 62
(7), 4488
–4492
(2000). http://dx.doi.org/10.1103/PhysRevB.62.4488 Google Scholar
M. Qiu and S. He,
“A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,”
J. Appl. Phys., 87
(12), 8268
–8275
(2000). http://dx.doi.org/10.1063/1.373537 JAPIAU 0021-8979 Google Scholar
M. Qiu and S. He,
“FDTD algorithm for computing the off-plane band structure in a two-dimensional photonic crystal with dielectric or metallic inclusions,”
Phys. Lett. A, 278
(6), 348
–354
(2001). http://dx.doi.org/10.1016/S0375-9601(00)00795-7 PYLAAG 0375-9601 Google Scholar
L. Wu and S. He,
“Revised finite-difference time-domain algorithm in a nonorthogonal coordinate system and its application to the computation of the band structure of a photonic crystal,”
J. Appl. Phys., 91
(10), 6499
–6506
(2002). http://dx.doi.org/10.1063/1.1473676 JAPIAU 0021-8979 Google Scholar
Y. Cao, Z. Hou and Y. Liu,
“Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals,”
Solid State Commun., 132
(8), 539
–543
(2004). http://dx.doi.org/10.1016/j.ssc.2004.09.003 SSCOA4 0038-1098 Google Scholar
M. Qiu,
“Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,”
Microwave Opt. Technol. Lett., 30
(5), 327
–330
(2001). http://dx.doi.org/10.1002/(ISSN)1098-2760 MOTLEO 0895-2477 Google Scholar
J.-H. Sun and T.-T. Wu,
“Propagation of surface acoustic waves through sharply bent two-dimensional phononic crystal waveguides using a finite-difference time-domain method,”
Phys. Rev. B, 74
(17), 174305
(2006). http://dx.doi.org/10.1103/PhysRevB.74.174305 Google Scholar
H. Y. D. Yang,
“Finite difference analysis of 2-D photonic crystals,”
IEEE Trans. Microwave Theory Tech., 44
(12), 2688
–2695
(1996). http://dx.doi.org/10.1109/22.554631 IETMAB 0018-9480 Google Scholar
C. Yu and H. Chang,
“Compact finite-difference frequency-domain method for the analysis of two-dimensional photonic crystals,”
Opt. Express, 12
(7), 1397
(2004). http://dx.doi.org/10.1364/OPEX.12.001397 OPEXFF 1094-4087 Google Scholar
S. Guo et al.,
“Photonic band gap analysis using finite-difference frequency-domain method,”
Opt. Express, 12
(8), 1741
(2004). http://dx.doi.org/10.1364/OPEX.12.001741 OPEXFF 1094-4087 Google Scholar
C. Van Der Mee, S. Seatzu and P. Contu,
“Fast and effective finite difference method for 2D photonic crystals,”
Commun. Appl. Ind. Math., 2
(2), 1
–18
(2011). http://dx.doi.org/10.1685/journal.caim.374 Google Scholar
R. D. Meade et al.,
“Accurate theoretical analysis of photonic band-gap materials,”
Phys. Rev. B, 48
(11), 8434
–8437
(1993). http://dx.doi.org/10.1103/PhysRevB.48.8434 Google Scholar
K. M. Ho, C. T. Chan and C. M. Soukoulis,
“Existence of a photonic gap in periodic dielectric structures,”
Phys. Rev. Lett., 65
(25), 3152
–3155
(1990). http://dx.doi.org/10.1103/PhysRevLett.65.3152 PRLTAO 0031-9007 Google Scholar
C. T. Chan, Q. L. Yu and K. M. Ho,
“Order-N spectral method for electromagnetic waves,”
Phys. Rev. B, 51
(23), 16635
–16642
(1995). http://dx.doi.org/10.1103/PhysRevB.51.16635 Google Scholar
S. Xiao, L. Shen and S. He,
“A plane-wave expansion method based on the effective medium theory for calculating the band structure of a two-dimensional photonic crystal,”
Phys. Lett. A, 313
(1–2), 132
–138
(2003). http://dx.doi.org/10.1016/S0375-9601(03)00690-X PYLAAG 0375-9601 Google Scholar
A. G. Hanif, T. Arima and T. Uno,
“Finite-Difference frequency-domain algorithm for band-diagram calculation of 2-D photonic crystals composed of debye-type dispersive materials,”
IEEE Antennas Wireless Propag. Lett., 11 41
–44
(2012). http://dx.doi.org/10.1109/LAWP.2011.2182173 Google Scholar
A. Mekis et al.,
“High transmission through sharp bends in photonic crystal waveguides,”
Phys. Rev. Lett., 77
(18), 3787
–3790
(1996). http://dx.doi.org/10.1103/PhysRevLett.77.3787 PRLTAO 0031-9007 Google Scholar
H.-Y. Ryu, M. Notomi and Y.-H. Lee,
“Finite-difference time-domain investigation of band-edge resonant modes in finite-size two-dimensional photonic crystal slab,”
Phys. Rev. B, 68
(4), 45209
(2003). http://dx.doi.org/10.1103/PhysRevB.68.045209 Google Scholar
S. Yamada et al.,
“Simulation of optical pulse propagation in a two-dimensional photonic crystal waveguide using a high accuracy finite-difference time-domain algorithm,”
J. Appl. Phys., 93
(4), 1859
–1864
(2003). http://dx.doi.org/10.1063/1.1539542 JAPIAU 0021-8979 Google Scholar
Z.-Y. Li and K.-M. Ho,
“Light propagation in semi-infinite photonic crystals and related waveguide structures,”
Phys. Rev. B, 68
(15), 155101
(2003). http://dx.doi.org/10.1103/PhysRevB.68.155101 Google Scholar
Y.-C. Hsue and T.-J. Yang,
“Applying a modified plane-wave expansion method to the calculations of transmittivity and reflectivity of a semi-infinite photonic crystal,”
Phys. Rev. E, 70
(1), 16706
(2004). http://dx.doi.org/10.1103/PhysRevE.70.016706 Google Scholar
W. J. Kim and J. D. O’Brien,
“Optimization of a two-dimensional photonic-crystal waveguide branch by simulated annealing and the finite-element method,”
J. Opt. Soc. Am. B, 21
(2), 289
(2004). http://dx.doi.org/10.1364/JOSAB.21.000289 JOBPDE 0740-3224 Google Scholar
J. B. Theeten, D. E. Aspnes and R. P. H. Chang,
“A new resonant ellipsometric technique for characterizing the interface between GaAs and its plasma-grown oxide,”
J. Appl. Phys., 49
(12), 6097
–6102
(1978). http://dx.doi.org/10.1063/1.324529 JAPIAU 0021-8979 Google Scholar
F. L. Pedrotti and L. S. Pedrotti, Introduction to Optics, 2nd ed.Prentice Hall, New Jersey
(1993). Google Scholar
J. D. Joannopoulos et al., Photonic Crystals: Molding the Flow of Light, 2nd ed.Princeton University Press, Princeton
(2008). Google Scholar
I. Abdulhalim,
“Effect of the number of sublayers on axial optics of anisotropic helical structures,”
Appl. Opt., 47
(16), 3002
(2008). http://dx.doi.org/10.1364/AO.47.003002 APOPAI 0003-6935 Google Scholar
S. K. Tripathy,
“Refractive indices of semiconductors from energy gaps,”
Opt. Mater., 46 240
–246
(2015). http://dx.doi.org/10.1016/j.optmat.2015.04.026 OMATET 0925-3467 Google Scholar
M. Sarollahi et al.,
“The significance of the number of periods and period size in 2D photonic crystal waveguides,”
Proc. SPIE, 9556 95561B
(2015). http://dx.doi.org/10.1117/12.2188321 PSISDG 0277-786X Google Scholar
M. A. Green and M. J. Keevers,
“Optical properties of intrinsic silicon at 300 K,”
Prog. Photovoltaic Res. Appl., 3
(3), 189
–192
(1995). http://dx.doi.org/10.1002/(ISSN)1099-159X PPHOED 1062-7995 Google Scholar
L. Gao, F. Lemarchand and M. Lequime,
“Refractive index determination of SiO2 layer in the UV/Vis/NIR range: spectrophotometric reverse engineering on single and bi-layer designs,”
J. Eur. Opt. Soc. Rapid Publ., 8 130101
(2013). http://dx.doi.org/10.2971/jeos.2013.13010 Google Scholar
BiographyMirsaeid Sarollahi is a PhD student in electrical engineering at the University of Arkansas studying solar cell efficiency. He received his bachelor of science degree in physics from Zanjan University in Iran and his master’s in physics from Azad University (Iran). He worked with fiber optic design at the Telecommunication Company of Zanjan, Iran. He performed the present work with Dr. Herzog at the University of Arkansas. Stephen J. Bauman is currently a PhD candidate in the microelectronics and photonics graduate program at the University of Arkansas. He studies experimental and theoretical plasmonics and other areas of nano-optics, such as photonic crystals. He received his dual BS degrees from Southeast Missouri State University in physics and engineering physics and received his MS in microelectronics and photonics from the University of Arkansas. He is actively involved with SPIE. Jonathan Mishler joined the Herzog Lab in 2013. He graduated from the University of Arkansas with a bachelor’s in physics. He then became a postbaccalaureate at the National Institutes of Health in Bethesda, Maryland. In fall 2016, he began graduate school at the University of Washington (Seattle) in the Department of Bioengineering. Joseph B. Herzog is an assistant professor in the Department of Physics at the University of Arkansas. He received his BS degree from Louisiana State University and his MS and PhD degrees from the University of Notre Dame. He was a Welch postdoctoral research associate at Rice University. His research focuses on nano-optics including plasmonics and photonic crystals. He is an active member of SPIE, serving on various committees. |