# Optical Effects Accompanying the Dynamical Bragg Diffraction in Linear 1D Photonic Crystals Based on Porous Silicon

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## Abstract

**:**

## 1. Introduction

## 2. Dynamical Bragg Diffraction in Linear 1D PhC: The Main Approach

**Figure 1.**(

**a**) Scheme of the dynamical Bragg diffraction and of the temporal pulse splitting in the direct and diffraction directions; (

**b**) SEM image of the cross-section of the porous silicon based PhC.

_{0}± Ω], where ω

_{0}is the central frequency, Ω << ω

_{0}, and with the amplitudes that are the functions of the coordinates z, x and of the time t; the coordinate system is shown in Figure 1a as well. In the frame of the two-wave approximation and under the fulfillment of the Bragg diffraction condition, sin ϑ

_{B}= λ/2d, d being the PC period, the p-polarized electric field within the bulk of the photonic crystal is determined by the expression [16]:

**E**

_{0}(x, z, t) and

**E**

_{h}(x, z, t) are the amplitudes of the transmitted and diffracted waves, respectively; the wave vector k = ω/c = k

_{0x}+ K, k

_{0x}= k

_{0}sinθ is the lateral projection of the wave vector, θ is the angle of incidence. In other words, a coherent superposition of the incident, E

_{0}, and of the diffracted, E

_{h}, pulse fields determine the field in every point within the PhC structure. This two-wave approximation is valid in particular due to an achievable large modulation of the permittivity (of the refraction index) of the layers that compose the photonic crystal. The permittivity of the PhC in this approximation is:

_{0}, ε

_{h}, ε

_{−}

_{h}are the spatial Fourier components of the permittivity, the exact equations that determine their dependence on the PhC material and geometrical parameters of a particular structure are described in [16]. Importantly, the amplitudes of the confined fields

**E**

_{0}(x, z, t) and

**E**

_{h}(x, z, t) are given by integrating over the frequency-angular distribution of the propagating pulses, thus giving the dependence of the output field on both the dielectric parameters of the photonic crystal, as well as on the duration, phase, amplitude and propagation direction of the incident pulsed radiation.

## 3. Composition of Multilayer Porous Silicon Based 1D Photonic Crystals

_{2}O:ethanol in the ratio 2:4:3 for the HF concentration of 21% (w/w)), which results in the formation of pores oriented along the [100] direction, i.e., perpendicularly to the surface of the Si(001) plate. During this procedure, the silicon plate serves as the bottom electrode, while the platinum wire is the second electrode in a two-electrode electrochemical cell. It was checked that the pores’ diameter and thus the porosity of the porous silicon layer are proportional to the electrochemical current density, while the etching time determines the depth of the pores. Thus, the periodic modulation of the current density results in the formation of spatially periodic porous structure, the layers of the constant porosity being oriented parallel to the Si(001) surface. The average pores’ diameter in our conditions were varied in the interval 10–60 nm [18], which is much smaller that the optical wavelength, so that such porous layers may be treated as a homogeneous media. As a result, the PC made of several hundreds of layers with low porosity (high refractive index, n

_{1}) and of high porosity (smaller refractive index, n

_{2}) was made, its average thickness being about 300 microns.

_{1}d

_{1}= n

_{2}d

_{2}= λ/4, where λ is the central wavelength of the photonic band gap and determines the Bragg condition and is close to 800 nm, which is the fundamental wavelength of our laser systems. As a result, a high periodic structure is formed; the cross-section is shown in Figure 1b.

## 4. Experimental Results

#### 4.1. Temporal Splitting of the Femtosecond Laser Pulses

_{1}= 1.45 and n

_{2}= 1.35. The radiation of a Ti-sapphire laser at the wavelength of 800 nm and the pulse duration of 30 fs was used, which was focused on a cut-off of a PhC structure into a spot of approximately 30 µm in diameter. The diffraction Bragg condition was realized at the angle of incidence of 31°.

**Figure 2.**Finite difference time domain (FDTD) calculation of the propagation of a 30 fs laser pulse through a PhC in the Laue scheme of the dynamical Bragg diffraction; the PhC layers are parallel to the z-axis (shown in panel a) and perpendicular to the plane of the figure. (

**a**) Spatial distribution of the pulse intensity after the laser pulse travels for 200 fs, 400 fs, 600 fs and 800 fs inside a PhC; (

**b**) cross-sections of the panel a made for the layers of low porosity (upper red curve) and of high porosity (bottom blue curve).

_{CC}(τ) ∞ ${\mathrm{\int}}_{-\mathrm{\infty}}^{\mathrm{\infty}}$ I

_{S}(t) I

_{R}(t + τ)dt, where I

_{S}(t) and I

_{R}(t + τ) are the pulse intensities in the signal and the reference channels.

**Figure 3.**(

**a**) Cross-correlation functions measured for the p- and s-polarized fundamental radiation and (

**b**) as a function of the angle between the polarization plane of the fundamental radiation and the plane of incidence.

_{0}, ε

_{h}are the zero and the first spatial Fourier components of the permittivity, and c is the speed of light.

_{1,0}= 1.35 ± 0.01; n

_{1,e}= 1.32 ± 0.01 and n

_{2,0}= 1.46 ± 0.01; n

_{2,e}= 1.45 ± 0.01. The performed calculations show a good correlation with the experimental dependencies if both types of anisotropy are considered.

#### 4.2. Dynamical Bragg Diffraction of Chirped Femtosecond Laser Pulses: The Effect of Selective Pulse Compression

_{0}, τ are the pulse duration prior to the compression and after it, correspondingly, ϕ" is the second phase derivative, that is the group delay dispersion, ∆ω is the pulse spectral width. The compressed signal pulse was focused on the PhC facet into a spot of 20 µm in diameter. The pulse duration in the reference channel was kept to be 30 fs.

**Figure 4.**(

**a**) Experimental and (

**b**) calculated dependencies of the values of the duration of the Borrmann, anti-Borrmann and input chirped pulses on the chirp parameter β.

#### 4.3. Pendulum Effect in the Laue Diffraction Scheme in 1D Porous Silicon PhC

**Figure 5.**(

**a**) Intensity spectrum of the diffracted beam passed through the porous quartz PhC of 0.2 mm long in the Laue geometry; (

**b**) spectra of the transmitted (red symbols) and of diffracted (black symbols) beams under measured under the similar conditions.

^{(s,p)}is the extinction length and q

_{z}

_{1}

^{(s,p)}, q

_{z}

_{2}

^{(s,p)}are the z-projections of the wave vectors of the Borrmann and of the anti-Borrmann p- or s-polarized waves within the photonic crystal. Figure 5 demonstrates the pendulum effect induced by the variation of the wavelength of light incident on the photonic crystal, while qualitatively similar dependencies can be realized for say when changing continuously the length of the PhC structure [28].

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Maydykovskiy, A.; Novikov, V.; Svyakhovskiy, S.; Murzina, T.
Optical Effects Accompanying the Dynamical Bragg Diffraction in Linear 1D Photonic Crystals Based on Porous Silicon. *Crystals* **2014**, *4*, 427-438.
https://doi.org/10.3390/cryst4040427

**AMA Style**

Maydykovskiy A, Novikov V, Svyakhovskiy S, Murzina T.
Optical Effects Accompanying the Dynamical Bragg Diffraction in Linear 1D Photonic Crystals Based on Porous Silicon. *Crystals*. 2014; 4(4):427-438.
https://doi.org/10.3390/cryst4040427

**Chicago/Turabian Style**

Maydykovskiy, Anton, Vladimir Novikov, Sergey Svyakhovskiy, and Tatiana Murzina.
2014. "Optical Effects Accompanying the Dynamical Bragg Diffraction in Linear 1D Photonic Crystals Based on Porous Silicon" *Crystals* 4, no. 4: 427-438.
https://doi.org/10.3390/cryst4040427