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A single Gaussian pulse propagation through a two-dimensional photonic crystal is considered. The spectrum of photonic crystal corresponds to the both permitted and forbidden frequency ranges for the passage of radiation. The possibility of radiation localization within the photonic crystal with finite dimensions is shown.
The photonic crystals (PC) are the class of artificial objects actively researched in recent years and representing periodic structures with the specific scale of electrophysical parameters change comparable to radiation wavelength. It causes the existence of the band structure of a photonic crystal passing spectrum including also the frequencies ranges forbidden for radiation propagation within which radiation propagation in a crystal is completely suppressed that, in particular, makes it possible to light velocity in it [1]. Dispersing properties of photonic crystal result in the unusual reflection and refraction pattern on boundaries of such structures even for harmonic electromagnetic oscillations [2,3]. Interaction of pulse radiations with such objects should cause more interesting effects. Let us illustrate the observed information, having considered the following task.
A plane electromagnetic wave formed as a Gaussian pulse is incident on the border of two-dimensional periodical insulating crystal (Fig.1) formed from tubular elements of circular cross-section having dielectric capacity of 13 (gallium-arsenide):
u ( t ) = exp ( –( t – t0 )2 / a2 ) cos ( ω t – k r ),
where t0 is temporary pulse movement corresponding to zero reading, a is pulse width, ω is a circular frequency, k is a wave number in interelement field, r is a coordinate along the pulse of propagation direction. E polarization of radiation is considered.
Let us suppose that photonic crystal phase is 1 mm, elements radius is 0.15 mm. For these parameters the calculation of field level for photonic crystal is performed by the FDTD method (Fig. 2) which showed the existence of a single complete band gap (in the plane, perpendicular to axes of elements) in a range of photonic crystal proper state which is located in the range of 100–140 GHz.
Let us clarify the features of propagation through concerned photonic crystal of a pulse radiation the spectral content of which corresponds to different sections of received frequency response. Let us use Gaussian pulse having following parameters: t0 = 25 ps and a = 10 ps. With such signal duration the width of its range on e–1/2 level from maximum level is 140 GHz. Oscillation frequency that creates the pulse should be accepted as 110 GHz, i. e. corresponding to band gap.
The simulation result received outside the photonic crystal at points 1 and 2 are given in Fig. 3 (see Fig. 1). Data received from the irradiated side of structure are easily interpreted. As an essential part of a signal range falls within the forbidden frequency band, a single reflected signal of considerable amplitude and duration, close to duration of the pulse falling on photonic crystal is considered. This means that radiation penetrates into photonic crystal slightly, having met the reflection practically close to its border. The action of the temporal relation received on the other side of a crystal is not so obvious. The passed signal has low amplitude and reaches measuring point at the point of time corresponding to its propagation with light velocity in the free space. However time chart concerns that only low frequency components of initial radiation corresponding to the allowed bands of photonic crystal reach the measuring point by 100 ps, and the main last transmitted signal begins to be formed only at approximately 250 ps. It is the subsequence of bell-shaped pulses having exponential envelope. In general this image resembles beating of harmonic process similar in frequency dying out at a time.
Let us consider similar time charts but those received at points 3 and 4 (see Fig. 1), i. e. in structure directly near its borders for specification of a physical pattern of pulse interaction with photonic crystal.
The main aspect caused by these relations is the coincidence of oscillations mode outside the photonic crystal and within it, but there is the essential distinction in amplitude characteristics. So, at photonic crystal output the field amplitude in a signal maximum within the structure is 5 times more than appropriate level beyond its limits. This means that the crystal is mismatched strongly with external environment, and accordingly, the wave entered into the structure is reflected effectively by its borders and has been concentrated in it for extended periods of time. At the same time spatially constrained photonic crystal represents a peculiar resonator of Fabry-Perot resonator. Besides, comparing the provided temporal relations, it is obvious that at radiation movement, for example, in the opposite direction on photonic crystal its amplitude changes very slightly (points of 540 ps in Fig. 4a and 295ps in Fig. 4b). It is indistinctive for wave processes action the frequency of which is strictly laid down in borders of the band gaps of photonic crystal, and as is well-known, in this case field’s exponential decrease is considered in process of passing through a crystal, physically connected to Bragg reflection of waves. In the considered situation such loosening also happens, but only in case of pulse penetration into photonic crystal, and then study level doesn’t decrease exponentially any more, but only slowly decreases by means of energy effluence through borders. It is important to note that the similar amplitude signals characteristics should be considered both in less and in more extensive structures.
In conclusion we should mark the following information. Time of pulse propagation through photon crystal determined on moments at which its maximum radiation level was considered near borders or at appearance of a signal, reflected by one of crystal edges and escaped through another one in appropriate time charts considerably exceeds the time which is required for wave to pass the same distances in the free space. This allows basically referring to reduction of group radiation velocity considered earlier in detail in photonic crystal with the linear defects [4]. At the same time it is incorrectly in this case to consider it as some medium with active refraction index differed from the correspondent feature of external environment [5]. First, it is shown above that separate signal components move with light velocity in a vacuum. Secondly, the concept of group velocity generally doesn’t make a sense (or it is necessary to consider its proximity to zero) for the spectral components of a signal corresponding to the band gap, as radiation propagation in photonic crystal in this range of frequencies doesn’t appear. Therefore it is incorrect in our case to treat results of simulation and as the reduction of speed of a total wavepackage which is, in essence, Gaussian pulse because rather broadband signal is considered, and only a part of a range of which is within borders of the band gap. In our opinion, it is more correctly to describe observed results as localization of Gaussian pulse with driving frequency, appropriate to complete band gaps, in spatially limited photonic crystal.
The work is performed with financial support of the Russian Federal Property Fund within the scientific project No. 15-47-04315.
A plane electromagnetic wave formed as a Gaussian pulse is incident on the border of two-dimensional periodical insulating crystal (Fig.1) formed from tubular elements of circular cross-section having dielectric capacity of 13 (gallium-arsenide):
u ( t ) = exp ( –( t – t0 )2 / a2 ) cos ( ω t – k r ),
where t0 is temporary pulse movement corresponding to zero reading, a is pulse width, ω is a circular frequency, k is a wave number in interelement field, r is a coordinate along the pulse of propagation direction. E polarization of radiation is considered.
Let us suppose that photonic crystal phase is 1 mm, elements radius is 0.15 mm. For these parameters the calculation of field level for photonic crystal is performed by the FDTD method (Fig. 2) which showed the existence of a single complete band gap (in the plane, perpendicular to axes of elements) in a range of photonic crystal proper state which is located in the range of 100–140 GHz.
Let us clarify the features of propagation through concerned photonic crystal of a pulse radiation the spectral content of which corresponds to different sections of received frequency response. Let us use Gaussian pulse having following parameters: t0 = 25 ps and a = 10 ps. With such signal duration the width of its range on e–1/2 level from maximum level is 140 GHz. Oscillation frequency that creates the pulse should be accepted as 110 GHz, i. e. corresponding to band gap.
The simulation result received outside the photonic crystal at points 1 and 2 are given in Fig. 3 (see Fig. 1). Data received from the irradiated side of structure are easily interpreted. As an essential part of a signal range falls within the forbidden frequency band, a single reflected signal of considerable amplitude and duration, close to duration of the pulse falling on photonic crystal is considered. This means that radiation penetrates into photonic crystal slightly, having met the reflection practically close to its border. The action of the temporal relation received on the other side of a crystal is not so obvious. The passed signal has low amplitude and reaches measuring point at the point of time corresponding to its propagation with light velocity in the free space. However time chart concerns that only low frequency components of initial radiation corresponding to the allowed bands of photonic crystal reach the measuring point by 100 ps, and the main last transmitted signal begins to be formed only at approximately 250 ps. It is the subsequence of bell-shaped pulses having exponential envelope. In general this image resembles beating of harmonic process similar in frequency dying out at a time.
Let us consider similar time charts but those received at points 3 and 4 (see Fig. 1), i. e. in structure directly near its borders for specification of a physical pattern of pulse interaction with photonic crystal.
The main aspect caused by these relations is the coincidence of oscillations mode outside the photonic crystal and within it, but there is the essential distinction in amplitude characteristics. So, at photonic crystal output the field amplitude in a signal maximum within the structure is 5 times more than appropriate level beyond its limits. This means that the crystal is mismatched strongly with external environment, and accordingly, the wave entered into the structure is reflected effectively by its borders and has been concentrated in it for extended periods of time. At the same time spatially constrained photonic crystal represents a peculiar resonator of Fabry-Perot resonator. Besides, comparing the provided temporal relations, it is obvious that at radiation movement, for example, in the opposite direction on photonic crystal its amplitude changes very slightly (points of 540 ps in Fig. 4a and 295ps in Fig. 4b). It is indistinctive for wave processes action the frequency of which is strictly laid down in borders of the band gaps of photonic crystal, and as is well-known, in this case field’s exponential decrease is considered in process of passing through a crystal, physically connected to Bragg reflection of waves. In the considered situation such loosening also happens, but only in case of pulse penetration into photonic crystal, and then study level doesn’t decrease exponentially any more, but only slowly decreases by means of energy effluence through borders. It is important to note that the similar amplitude signals characteristics should be considered both in less and in more extensive structures.
In conclusion we should mark the following information. Time of pulse propagation through photon crystal determined on moments at which its maximum radiation level was considered near borders or at appearance of a signal, reflected by one of crystal edges and escaped through another one in appropriate time charts considerably exceeds the time which is required for wave to pass the same distances in the free space. This allows basically referring to reduction of group radiation velocity considered earlier in detail in photonic crystal with the linear defects [4]. At the same time it is incorrectly in this case to consider it as some medium with active refraction index differed from the correspondent feature of external environment [5]. First, it is shown above that separate signal components move with light velocity in a vacuum. Secondly, the concept of group velocity generally doesn’t make a sense (or it is necessary to consider its proximity to zero) for the spectral components of a signal corresponding to the band gap, as radiation propagation in photonic crystal in this range of frequencies doesn’t appear. Therefore it is incorrect in our case to treat results of simulation and as the reduction of speed of a total wavepackage which is, in essence, Gaussian pulse because rather broadband signal is considered, and only a part of a range of which is within borders of the band gap. In our opinion, it is more correctly to describe observed results as localization of Gaussian pulse with driving frequency, appropriate to complete band gaps, in spatially limited photonic crystal.
The work is performed with financial support of the Russian Federal Property Fund within the scientific project No. 15-47-04315.
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