Rough 1D photonic crystals: A transfer matrix approach

Ortiz, Guillermo P.; Ricci, María Luz Martínez; Toranzos, V.; Mochán, W. Luis

doi:10.1016/j.optmat.2020.110012

Cited by 17 publications

(14 citation statements)

References 39 publications

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“…Thus, as our GRIN sample is not macroscopically homogeneous and its properties have a gradient, the experimental results have contributions from regions with slightly different porosities and thicknesses, partially averaging out the interference maxima and minima. We expect microscopic inhomogeneities such as interface roughness might also reduce the contrast through scattering and its corresponding apparent dissipation [50]. The loss of contrast is most notable for the thicker regions towards the left side of Fig.…”

Section: Reflectancementioning

confidence: 92%

Analytical Model for the Current Density in the Electrochemical Synthesis of Porous Silicon Structures with a Lateral Gradient

Ospina-Delacruz,

Agarwal,

Mochán

2021

Preprint

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Layered optical devices with a lateral gradient can be fabricated through electrochemical synthesis of porous silicon (PS) using a position dependent etching current density ( ‖ ). Predicting the local value of ( ‖ ) and the corresponding porosity ( ‖ ) and etching rate ( ‖ ) is desirable for their systematic design. We develop a simple analytical model for the calculation of ( ‖ ) within a prism shaped cell. Graded single layer PS samples were synthesized and their local calibration curves vs and vs were obtained from our model and their reflectance spectra. The agreement found between the calibration curves from different samples shows that from one sample we could obtain full calibration curves which may be used to predict, design, and fabricate more complex non-homogeneous multilayered devices with lateral gradients for manifold applications.

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Section: Reflectancementioning

confidence: 92%

Analytical Model for the Current Density in the Electrochemical Synthesis of Porous Silicon Structures with a Lateral Gradient

Ospina-Delacruz,

Agarwal,

Mochán

2021

Preprint

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“…The measured reflectance is lower than the calculated one, more so at small wavelengths and large angles. This difference may be partially attributed to the scattering of light due to the roughness present at the sample [43,30,29] that we have not taken into account in our theory. Due to the limitations of our spectrophotometer, we couldn't perform measurements for > 60 • and thus check the ODB.…”

Section: Chirped-type Bragg Mirrorsmentioning

confidence: 93%

“…Many equivalent formulations have been proposed to obtain and use [28,29,30]. Here we use a recently developed formalism for the numerically stable calculation of the reflectance of large multilayer systems, as proposed by Puente-Díaz, et al [31], summarized in the supplementary information.…”

Section: Theorymentioning

confidence: 99%

Optimization of wide-band quasi-omnidirectional 1-D photonic structures

Castillo-Gallardo,

Puente-Díaz,

Ariza-Flores

et al. 2021

Preprint

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We have designed, optimized, fabricated and characterized highly reflective quasi-omnidirectional (angular range of 0 − 60 • ) multilayered structures with a wide spectral range. Two techniques, chirping (a continuous change in thicknesses) and stacking of Bragg-type sub-structures, have been used to enhance the reflectance with minimum thickness for a given pair of refractive indices. Numerical calculations were carried out employing the transfer matrix method and we optimized the design parameters to obtain maximal reflectance averaged over different spectral ranges for all angles. We fabricated some of the optimized structures with porous silicon dielectric multilayers with low refractive contrast and compared their measured optical properties with the calculations. Two chirped structures with thicknesses 21.6 m and 60.4 m, resulting in quasi omnidirectional mirrors with bandwidths of 360 nm and 1800 nm, centered at 1160 nm and 1925 nm respectively have been shown. In addition, we fabricated a stacked sub-Bragg mirror structure with a quasi omnidirectional bandwidth of 1800 nm (centered at 1850 nm) and a thickness of 41.5 m, which is almost two third (in thickness) of the chirped structure. Thus, our techniques allowed us to obtain relatively thin quasi-omnidirectional mirrors with wide bands over different wavelength ranges. Our analysis techniques can be used for the optimization of the reflectance not only in multilayered PS systems with different refractive index contrasts but also for systems with other types of materials with low refractive index contrast. The present study could also be useful for obtaining omnidirectional dielectric mirrors in large spectral regions using different materials, flat focusing reflectors, thermal regulators, or, if defects are included, as filters or remote chemical/biosensors with a wide angular independent response.

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“…It is considered a dispersive, absorbent, isotropic and nonmagnetic material with refractive index

. Each interface is characterized by an admittance matrix

and phase matrix

given by [ 30 , 31 ]:

where

and

are the modified Fresnel coefficients that introduce the interface rms roughness (

) (assuming

) in order to deal with incoherent propagation due to thick substrates,

is the polarization factor,

is the layer thickness, and

is the incidence angle [ 4 , 30 ]. Thus, for the structure Air/Psi/Si, the total transfer matrix is given by:

and the reflectance can be determined from S components as:

…”

Section: Theoretical Modelsmentioning

confidence: 99%

In Situ Photoacoustic Study of Optical Properties of P-Type (111) Porous Silicon Thin Films

et al. 2021

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Porous silicon (PSi) on p++-type (111) silicon substrate has been fabricated by electronically etching method in hydrofluoric acid (HF) media from 5 to 110 mA/cm2 of anodizing current density. The problem of determining the optical properties of (111) PSi is board through implementing a photoacoustic (PA) technique coupled to an electrochemical cell for real-time monitoring of the formation of porous silicon thin films. PA amplitude allows the calculation of the real part of the films refractive index and porosity using the reflectance self-modulation due to the interference effect between the PSi film and the substrate that produces a periodic PA amplitude. The optical properties are studied from specular reflectance measurements fitted through genetic algorithms, transfer matrix method (TMM), and the effective medium theory, where the Maxwell Garnett (MG), Bruggeman (BR), and Looyenga (LLL) models were tested to determine the most suitable for pore geometry and compared with the in situ PA method. It was found that (111) PSi exhibit a branched pore geometry producing optical anisotropy and high scattering films.

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Rough 1D photonic crystals: A transfer matrix approach

Cited by 17 publications

References 39 publications

Analytical Model for the Current Density in the Electrochemical Synthesis of Porous Silicon Structures with a Lateral Gradient

Analytical Model for the Current Density in the Electrochemical Synthesis of Porous Silicon Structures with a Lateral Gradient

Optimization of wide-band quasi-omnidirectional 1-D photonic structures

In Situ Photoacoustic Study of Optical Properties of P-Type (111) Porous Silicon Thin Films

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