Scattering of the optical pulses on multiple micro-resonator band-stop filters

Pulses’ envelopes scattered on several Microresonator band-stop filters were examined in the optical transmission line. The most widespread optical pulses have been studied. The possibilities of pulses' time manipulation with various carrier frequencies are showed. New approaches to the indistinguishable pulses' separation have been proposed by means of transformation in to observable ones distinguished by amplitudes or time. A compression possibility has been shown for the chirped pulses. Reference 8, figures 8.


Introduction
Today various 3-d optical elements for envelops are applied in order to provide management of optical pulses.They include prisms, diffraction grating pairs, chirped mirrors and and some others [1][2][3].The filters on the dielectric micro-resonators can be used for changing of the optical pulses too [4,5].In contrast to the traditional devices, the optical filters are widely used for integral processing of the optical circuits' manufacturing, however their capabilities of scattered pulses shaping are currently insufficiently investigated.
Compression and time manipulation of the pulses in the optical fibers can be used according to the application of the several band-stop filters on micro-resonators with whispering gallery oscillations.Such technique allows realizing quasi-onedimensional structures with comparatively small dimensions.
The goal of the current article is the investigation of optical pulses manipulation by using several band-stop filters.A few band-stop filters with nonoverlapping frequency bands are considered.It has been demonstrated that such structures are characterized by pulses' compression and time manipulation as well as extracting discernible signals from indistinguishable pulses.

Statement of the problem
Let's supposes that we have a few (in the general case N ) band-stop filters, situated in the optical transmission line (Fig. 1, a).Every band-stop filter consists of S ring micro-resonators and have a frequency band is not coinciding with others.
The goal of the current article is the investigation of optical pulses scattered on this multielement system of band-stop filters.

Green's functions of the band-stop filter systems
For the purpose of optical pulses shapes calculation in the time region, it's necessary to make a Green's functions of band-stop filter system.According to [6], in case of mismatched frequency bands, the transmission coefficient T and the reflection coefficient R of the system can be presented in form: Where Γ -is the longitudinal wave number of the transmission line; ∆z is the distance between adjacent filters.
define the contribution of scattering coefficients of v-th filter [5]: The amplitudes ( ) The received above relations (1 -4) were used for the temporal Green's functions' calculation for the waves are reflected in the direction of the source: where, from [5], the following expression is true for any filter:

Pulse envelops' calculation
The presented relations (5,6) are used in order to find the general analytic expressions for pulse shapes (t) out E − scattered on band-stop filter's system in the incident direction: where is the pulse envelop is reflected from v -th band-stop filter calculated in [5].The fig. 2 demonstrates the result of scattering on two band-stop filters for various shapes of pulses at fig. 1: 1) For the case of the rectangular envelop of the incident pulse (fig.2, a where θ( ) t -is the Heaviside step function [8].Where the parameter m stands for front steepness and σ -is the width of the pulse envelop [7]; 3) For the Gaussian pulse envelop (fig.2, c-d):    Mutual influence of a few pulses scattered on two band-stop filters are shown in figure 4.This result demonstrates well distinguished pulses after scattering.
Preliminarily chirped pulses can be good separated too after scattering on band-stop filters systems, even if these pulses were indistinguishable before.The pulses, prepared in a special manner, even having coinciding envelopes, can be separated, if they have different frequency carriers.Let a pulse of complicated form, namely:   Hence, the pulses can be separated in time (fig.7, 8, b) scattered toward source direction from band-stop filters, distributed in space by the transmission line (fig.1, a) as in a case of two gratings.

Conclusions
By using a sequentially allocated in the transmission line band-stop filter systems with nonoverlapping frequency bands the management of optical pulses' parameters , namely: delay, separation and compression, can be provided.In contrast to 2d and 3-d structures, the proposed devices can be easier realized in optical integral circuits in the form of quasi-one-dimensional structures.

Fig. 1 .
Fig. 1.System of the N band-stop filters on S ring micro-resonators with whispering-gallery oscillations in the optical waveguide (a).Frequency response of the transmission coefficient (b); the reflection coefficient (c) -of two of 10-section filters the dielectric loss tangent of the micro-resonator`s material; λ vs -is the eigenvalue of the coupling operator: v K [4], ( 1,2,...,S s = ); ω -is the current angular frequency; ω 0 v -is the free oscillation's angular frequency of each insulated micro-resonator in the line of the vth filter.
vs ω -is the complex angular frequency of s-th coupling oscillation of the v -th filter of the microresonator system in line;/ of the Maxwell's equations' solutions is proposed to be equal to + ω exp( ) i t .The results of the S-matrix frequency responses' tions are showed at fig. 1, (b), (c) according to the expressions (1 -4) for two band-stop filters on 10 ring micro-resonators.The frequency of the filter isolated in the line of micro-resonators is supposed to be equal to 01 192,5 f = THz and 02 207,5 f = THz.The central frequency of the structure is 0 200 f = THz (fig. 1, b, c).The dielectric quality factor of the resonator's material constitutes between adjacent micro-resonators' centers of the filter was supposed to be 21 / (2 ) π Γ .The mutual coupling coefficients for non-propagating waves are considered to be unequal to zero only between adjacent micro-resonators: 0,015 sn k = ,as well as the coupling coefficients between nonadjacent microresonators for propagating waves of the transmission line.The coupling coefficients on the propa-

(
The constant C defines a chirp as a magni- tude of frequency modulation of the pulse carrying[7]);

Fig. 4 . 4 )
Fig. 4. Mutual influence of the rectangular pulses reflected from two micro-resonator filters (The duration of each incident pulse is equal to 0,2 ps; the temporary interval between adjacent pulse centers is δ = 0,4 t ps)

(
Where parameters i N and σ denote the pulse du- ration); 7) For the exponential pulse (fig.2, h):

Fig. 5
Fig. 5 shows two types of the scattering of the indistinguishable pulses, namely without and with chirping.The same result was obtained for four pulses.As it can be seen from fig. 6 the pulses are well distinguished.Eventually the scattered pulses are narrower as compared with incident.The pulses, prepared in a special manner, even having coinciding envelopes, can be separated, if they have different frequency carriers.Let a pulse of complicated form, namely: = π is the central frequency of the n -th filter.

Fig. 8 .
Fig. 8. Three Gaussian pulses with coincident envelops (a).Intensities of the pulses (a) after their scattering on three filters (b)