The time delay, space shift and widening of wave packet transmitted and reflected by structures with Bragg mirrors are investigated. The specific structures such as Bragg mirrors, resonators, and structures with chirp variation of thickness of the “period” are also considered. The calculation is carried out under the conditions that carrier frequency and incidence angle are in the vicinity of the Bragg resonance. Integral (mass center) and differential (group) estimates of the delay time and space shift are then compared. The conditions for the appearance of anomalous (negative) mass center delay or mass center shift (Goos-Hänchen shift) of the reflected wave packet are determined. The shape transformations of the wave packet illuminating periodic and quasiperiodic apodized Bragg reflectors are then considered. Spatial apodization of permittivity contrast yields much smaller shape deformation of the transmitted wave packet upon incidence at angles and carrier frequency near the edges of reflection band, as well in the Bragg reflection band, in comparison with the phenomena in similar periodic structures. The values of group delay for layered structures with a small chirp variation of optical (electrical) thickness of the period along longitudinal coordinates are experimentally obtained in microwave range.

(a) |
(b) |

(c) |
(d) |

*The reflection of the wave packet from Bragg resonators: (a) is the envelope of incident wave packet; (b) is the wave packet envelope reflected by an asymmetric Bragg resonator with far location of the resonance layer; (c) is the wave packet envelope reflected by an asymmetric Bragg resonator with near location of the resonance layer; (d) is the wave packet envelope reflected by symmetric Bragg resonator.*

Negative values of group delay and group shift for packet are inherent to asymmetric Bragg resonators and observed under the condition that a resonance thickness layer is located closer to the end of a structure. In the case of large wave packet durations and extents, integral and deferential packet estimates take approximately equal values. Center mass delay and center mass shift have anomalous values via packet distortions: widening, variation of asymmetry, coefficient of kurtosis.

Significant relative reduction of the side maximum level can be achieved by spatial permeability contrast apodization of multilayer structures. It provides a much smaller distortion of the reflected or transmitted wave packet with the angle of incidence and carrier frequency near the edge of the Bragg band reflection in comparison with similar periodic structures.

For the pulses with insufficient frequency localization, neither group delay nor center mass delay can be used as a unique estimate of delay time of the transmitted and reflected pulses under Bragg structures irradiation.

**Languages:** Matlab

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― Alan J. Perlis
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“FORTRAN is not a flower but a weed — it is hardy, occasionally blooms, and grows in every computer.”