CC BY
2
УДК 621.372
Scattering of Optical Pulses by Add-Drop Filters on Dielectric Microresonators
Ъ-иЫп A. A.
National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv, Ukraine
E-mail: atrubin&ukrpoet.net
We consider a system of ring microresonators with whispering gallery oscillations of ult.raliigh-Q, which are widely used to construct various integrated filters of the optical wavelength range. Using the perturbation theory, an elect.rodynamic model has been developed that describes a complex system of coupled microresonators with doubly degenerate types of natural oscillations, as well as located between two different transmission lines. General analytical expressions are obtained for describing the non-mutual characteristics of the scattering of the eigenwaves of a line on a system of optical microresonators that form a channel splitter. The frequency dependences of the scattering matrix of optical filter couplers with several communication channels are calculated. Based on the constructed analytical model, the time Green's functions are calculated for filters with serial coupling between microresonators, filters with microresonators coupled along the side wall and two transmission lines, as well as filters built on a double lattice of microresonators coupled along two transmission lines. The envelopes of optical pulses scattered by filters into various channels are considered. The envelopes of a rectangular and Gaussian single pulses scattered by 10-cavity filters of various designs are studied. The mutual influence of several rectangular as well as Gaussian pulses during their scattering by multilink optical splitters is investigated. Based on a comparison of the data obtained for the three types of structures, it is concluded that filters with laterally coupled microresonators are preferred. The obtained practical simulation results can significantly reduce the computation time and optimize complex multi-resonator structures of optical communication systems that simultaneously perform the functions of separation, or combination of channels.
Key words: dielectric microresonator: scattering: pulse: Add-Drop filter: Double-channel SCISSOR: Twisted Double-channel SCISSOR
DOI: 10.20535/RADAP. 2020.83.29-35
Introduction
Optical filters built on the basis of microresonators are widely used in various devices of the optical and infrared wavelength ranges fl 14]. The most promising is the use of dielectric microresonators in optical communication systems for the separation and combination of channels in WDM technologies. To solve this problem, it is proposed to use the so-called Add-Drop filters. Of greatest interest are filters built using planar structures of microresonators. The better part studied Add-Drop filters are divided into filters with sequential arrangement of microresonators and filters with lateral connection with optical transmission lines. Each of these types of filters has its own advantages and disadvantages. However, no comparative analysis of the filters was carried out. To compare the characteristics of the filters, it is desirable to study the scattering of optical pulses. Currently, only frequency response of scattering matrices of Add-Drop filters is considered. The scattering of optical pulses by such filters has not been fully investigated.
1 Statement of the problem
The purpose of this article is analysis of envelopes of optical pulses scattered by different Add-Drop filters: research of the best type of filters for transmitting optical pulses with minimal distortion. An analytical solution to the pulse scattering problem can be obtained by calculating and applying the temporal Green's functions [10]. In this case, we will only be interested in the pulse envelopes in the input and output transmission lines of the filter. As a result, the solution to the problem is obtained in a simplified form of an integral of the time Green's function and the envelope of the incident pulse.
2 Calculation Green' functions of the Add-Drop filters
Suppose we have a complex system of the N coupled microresonators. which is located in open space and a part of it also coupled with two different
transmission linos (Fig. 1-3.a). Suppose that each of the microresonators has a doubly degenerate type of natural oscillations on the frequency = 2-kf0, each of which is characterized by a given symmetry with respect to the selected plane: even (e%, hf), or odd (e°, h°) [15]. Let a wave falls on 1 port of
the system via regular transmission line (Fig. 1-3.a).
The eigenoscillation field of the system of N dielectric resonators we represented as a superposition of isolated fields of the resonators of even and odd types:
N
N
+ E b°n
N
N
(1)
hs = E hnn°° + E .
n=1
n=1
The matrix B
B
'bf bf bf bf
b°1 b°2 N N
bfN b?N
bT
(2)
E:
2N
E + + Eases
s=1 2N
(3)
H «H + +E'
Tv(w) = SV2 —
Q
D
det B
S=1Qs(w) '
(4)
where Qs(w) = + 2iQD(w/w0 — 1 — Xs/2);
QD
=
Q
- 1,...,4). An expression of the matrix BV is
presented below ((9). (11). (13)) for a specific kind of filter topology.
The time-domain Green's function of the Add-Drop filter, as follow from [10]. can be obtain from (4) taking into account the principle of causality:
9v (t)--
wlioro
S(T)6v2 +
1+2i QD
0, T < 0
N
E AV
s=1
0
av = _qd det BV As = Q det B
(5)
(6)
The field E°ut (t) in the plane of "first" mi-moment t is a integral on falling pulse E\n(t):
1
E°°ut(t)= f gv(t — t')E\n(t')df .
(7)
of the amplitudes of coupled microresonator oscillations bet'°'s should satisfy the equation system as an eigenvector of coupling operator K [ ]. The found eigenvalue A = 2(50+io")/oQ] (So = Re(w— oq); o" = Im(w)) and eigenvectors of the matrix K we used for solving the scattering problem of the wave (E + ,H +) on a system of coupled microresonators of the filter. For the solution of the problem we represented decomposition:
3
Calculating pulses scattering by Add-Drop filters
The design of the simplest Add-Drop filter is shown in Fig. l.a. In this case, the coupling matrix of the microresonators we represented in the form:
K = ||i (kfôs 1 + k°Ss2 + k%Ss{2N-1) + + kN$s(2 N ) + kOS ) Ssn + Ksn(1 — Ssn )
(8)
where as are the unknown amplitudes (s = 1,2,..., 2N) and (es,hs) are the s-th eigenoscillation field of the coupled microresonator system (1) with complex frequency os.
As a result, the transfer coefficient between the 1
where kf - is the coupling coefficient of the 1st microresonator with the transmission line 1-2 (Fig. l.a) on an even (or odd) mode oscillation; kt° - is the
N
the transmission line 3-4 also on an even (odd) oscillation; kos - is the coupling coefficient of the microresonator with open space; Ksn = = kt'° - is the mutual coupling coefficient between microresonators. if |s — n| = 1 and KSn = 0 in other cas es; Sst - is the
The matrix, defining a scattering on the Add-Drop filter:
s
sEh s
B s=
'bf
°
° N
b f~k I + b° sk ° bfk e°-+ + b° sk
1 1
bfN'
°2 N
°2 N N
bs-
bf °
° N
Le sk e e+ + + L° sk ° e++ K21 + K21 le sk e °++ 1 L° sk ° °++ °1 K21 + °1 K21
0
bfN' bfN
°2 N N
(9)
0
bs
'bf bf
°1 N
Les kee++ xL°s k°e+ +
N^31
les ke°++ + h°sk°°++
N 31
bfN' bfN
°2 N N
Bf
'bf bf
°1 N
Le s k ee—+ + 1° s k ° e—+ UN-1^41 + UN^41 Les ke°—+ + h°sk°°—+
bfN' bfN
°N2 N
0
0
In this case, index s is determined by the column number in matrix B. Here
I a b++ = ira+rb+*
mn \ m n
/(<№) =(fcm6n)o e-4 r( "
L a b--+ _
mn
<?~ cb+*
n
/(^o»n) =(Lmbn)o^4 r(
c°n± - is the expansion coefficient of the n-th microresonator field with a - mode on the propagating wave of the transmission line [ ]; wn - is the energy-stored in the dielectric of microresonator; r - is the longitudinal wave number of the transmission line: zn - is the longitudinal coordinate of the n-th microresonator center.
In Fig. l.b-e shows the results, based on (2 9). of the dependences on the frequency of the ¿"-matrix (S^i =20lg |) of a 10-section Add-Drop filter. As can be seen from the above data, the filter has minimal attenuation between ports 1-4.
The calculation results of the envelopes of rectangular and Gaussian pulses reflected and transmitted through the filter, calculated using (5 9). are shown in Fig. l.f-i. As follows from the data obtained. the passage of pulses through a chain of microresonators is accompanied by a noticeable delay-in time. The magnitude of the delay increases with increasing number of resonators. The envelopes of the transmitted pulses with not wide enough filter pass bands tend to a Gaussian-like distribution. With the simultaneous scattering of several pulses (Fig. l.h.i). their distinguishability is determined by the passband of the filter, the width and the relative distance between them. It should also be noted that the relative amplitudes of the initially identical pulses change as they pass through the filter.
Fig. 1. A - laterally coupled microresonator filter, ¿"-matrix responses of the 10-section bandpass filter as functions of frequency (b-e). Coupling coefficients of the first and last microresonators with transmission lines: =3,4 • 10-4 for even oscillations; knm = 3,48 • 10 4 for odd oscillations. Open Space microresonator
coupling coefficients: kos = 1 • 10 7. Mutual coupling coefficients of the microresonators for even oscillations:
k f2 = k*21 = 2,4 • 10 4; for odd oscillât ions k °2 = i = -2,4 • 10 4. Frequency of free microresonators oscillations
4
io = 200 THz; Qd
106. The envelops of the rectangular (f, h); Gaussian (g, i) pulses, scattered by 10-section
bandpass filter.
4 Calculating pulses scattering by Double-channel SCISSORs
The sketch of parallel coupled microresonator Add-Drop filter is shown in Fig. 2, a. In this case, each microresonator is coupled simultaneously with two transmission lines.
The ^-matrix of SCISSOR (side-coupled integrated spaced sequence of resonators) has the form:
K = (12)S (34) s + ¡¿OS )SSn + «Sn(1 -SSn) . (10)
Formally, the filter transfer coefficient is also (4), but for the SCISSOR structure shown on Fig. 2, a:
B{
Bs3
bf
bf
h°1
bf
bf
°N
2N
V^ H kae—+
u=1 2N
£ K<°—+ ..
u= 2N
V^ hska°—+
u= 2 N
£ KkUi++ ..
u= 2 N
£ K~kau°++ ..
u=
2 N £b:
u=
s ka°++
KuN
bfN
°2 N
°N2 N
bfN
°2 N
°N2 N
Bl
Bl
bf
°
° N
bf
°
° N
2 N
V hs kae++
u u
u= 2 N
£ Kkau°++
u=
2 N
£
u=
l,s ka°++ u u N
2 N
V hs kae—+
u= 2 N
£ K~kau2—+
u= 2 N
V^ hs ka°—+
°uKuN
u=
bfN
°2 N
°2 N N
bfN
°2 N
°2 N N
(ID
Here k^v)s = is the coupling coefficient of the s-th microresonator with the w-th transmission line; Ksn = ksn + iksn±± is the complex mutual coupling coefficient of the microresonators [ ]. Here a, b takes values even or odd depending on the type of w-th microresonator oscillations.
As known. Donble-channel SCISSOR is characterized by a transfer function with minimal attenuation
to the port 3 (Fig. 2.b.c). The delay in the pulses during transmission through a filter is minimal, but is characterized by their visible distortions (Fig. 2.d-g) as well as uneven frequency response (Fig. 2.b.c). Apparently this is duo to beats that occur between the microresonators of this structure duo to the mutual connection along the propagating waves of both lines.
Fig. 2. Parallel coupled microresonator Add-Drop filter on Double-channel SCISSOR (a), ¿"-matrix responses of the 10-section Add-Drop filter as functions of frequency (b. c). Coupling coefficients of the microresonators with transmission lines: k^m = k°°m = 1,2 • 10—44. Open Space microresonator coupling coefficients: kos = 1 • 10—7. Mutual coupling coefficients of the microresonators for even oscillations: kf2 = = 3 • 10—5; for odd oscillations k°2 = k° = —3 • 10— The envelops of the rectangular (d,f); Gaussian (e, g) pulses, scattered by 10-section
Add-Drop filter.
5 Calculating pulses scattering The sketch of Twisted Double-channel SCISSOR i
by Twisted Double-channel shown in Fig a.
17 lhe coupling and Bt matrices of the Double-
SCISSORs n
IS
channel SCISSOR has the form:
K = i(k( +kos)Ssn + «sn(1 - Ssn) , (12)
To increase out-of-band attenuation, the use of so-called Twisted Double-channel SCISSORs is proposed, where k( = k(l12)s if s<^d k( = k(l34)s if s > N.
Fig. 3. Optical Add-Drop filter on a Twisted double-channel SCISSOR (a), ¿-matrix responses of the 10-section Add-Drop filter as functions of frequency (b.c). Coupling coefficients of the microresonators with transmission lines: knm = knm = 2 • 10-4. Open Space microresonator coupling coefficients: kos = 1 • 10-7 . Mutual coupling coefficients of the microresonators for even oscillations: kf2 = k%1 =2 • 10-6; for odd oscillât ions =
-2 • 10-6; hfy =4 • 10-5; k°v = -4 • 10-5; f0 = 200THz; QD = 109. The envelops of the rectangular(d,f);
Gaussian (e. g) pulses, scattered by 10-section filter.
Bt =
bf
bf
loi . 2 N
4 N
E bsukau1-+
U=i
4 N
E &UkUO-+
u=1
0
6f4N
o4 N
bOiN.
Bt =
bf
o
o
. 2 N
4 N
E bUukaut++
u=1 4 N
E &UkUO++
u=1
bfN
5o4N
bOiN.
(13)
B3
bf
o
loi °2N
4N
E &UkU!++
U=i
4 N
E &UkUO++
U=1
6f4N
bofN
b°2%N.
; B4
bf
o
o
2 N
4 N
E &UkUr+
U=1
4 N
E 6UkU2-+
U=1
6f4N
o4 N
bONN.
0
0
0
Here also Ksn = ksn + iii^n the complex coupling coefficient [12]. For different microresonators with the same longitudinal coordinates Ksn = ksn = ky, where ky is the mutual coupling coefficient of "vertically conplod" microcavities (Fig. 3,a).
Twisted Donble-channel SCISSOR is characterized by a minimal attenuation to port 4 (Fig. 3. b.c).
The transmission of pnlsos through such a filter is characterized by their large distortions (Fig. 3,d,o), and in addition, by the phenomena of elevated mutual interference (Fig. 3,f, g).
6 Discussion and Conclusion
As follows from our calculations, the use of parallel structures of microresonators. such as Donble-channel SCISSOR and Twisted Donble-channel SCISSOR, leads to an increase in ont-of-band attenuation, but is accompanied by processes of destructive interference of pulses which is unacceptable for nso in separation of communication channels with increased speeds. The best scattering parameters are possessed by filters with laterally conplod microresonators.
In this paper, we have extended the scattering theory of the electromagnetic pnlsos on Add-Drop filters of receiving and transmitting optical communication systems. The proposed theory can be used to model and optimizing wide class of multiplexors, of the optical communication systems.
References
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Розсиовання оптичних 1мпуль-с!в фшьтрами-розгалужувачами на д!електричних м1крорезонаторах
Трубгн О. О.
Розглядаеться система шльцевих м!крорезопатор1в з коливаппями шепочучей галере! падвисоко! добротность як! широко використовуються для побудови р!зпо-маштпих штегралышх ф!льтр1в оптнчпого д!апазопу довжш1 хвнль. За допомогою теорп збурень розроблепа електродипам1чпа модель, яка описуе складну систему зв'язапих м!ж собою м!крорезопатор1в з двократпо ви-роджмшмн тршамн влаишх колтшань. а також розташо-вапнх м!ж двома р1зпими лш!ямн передачи. Отрпмап! загальш апал1тпчп1 впразп для оппсу не взаемпих характеристик розс1юваш1я влаишх хвиль лшп па систем! ОПТИЧ1ШХ м1крорезопатор!в. як! утворюють ф!льтр-розгалужувач. Розраховап! частотш залежпост! матриц! розсиоваппя оптичпих ф!льтр1в-розгалужувач1в з де-к!лькома каналами зв'язку. На основ! побудовапо! апа-л1тичио1 модел!. розраховаш часов! фупкц!! Гр1па для
ф!льтр1в з поондовнпм зв'язком м!ж мшрорезонатора-мп, ф!льтр!в з мшрорезонаторамп зв'язаними по бо-ковш станц!, та двом лш!ям передачи, а також фшь-тр!в, побудованих на подвшнш репптщ мшрорезона-тор!в, зв'язаних з двома л!шями передачи. За допо-могою використання знайдених часових функцш Грь на, розраховаш огинаюч! оптичних ¡мпульав, розаюва-пих фшьтрами-розгалужувачамп в р!зш капали зв'язку. Розглянута огинаюч! прямокутпого та Гауссовского оди-почних ¡миульав, розаюваних на 10-резонаторних фшь-трах р!зних конструкцШ. Дослщжено взаемний вплив дешлькох прямокутних, а також Гауссовских ¡мпуль-с!в при 1х розаюванш на багатоланкових оптичних ф!льтрах-розгалужувачах. На основ! пор!вняння отри-маних даних для трьох найб!льш поширених вид!в кон-струкцш, зроблено висновок про перевагу застосування ф!льтр!в з поондовно зв'язаними мшрорезонаторами. Отримаш практичш результата моделювання дозволя-ють суттево скоротити час розрахуншв та оптим1зувати складш багаторезонаторш структури оптичних систем зв'язку, одночасно виконуючих функцп розподшу, або об'еднання канал!в.
Ключовг слова: д!електричний мшрорезопатор; роз-аюваппя; ¡мпульс; ф1льтр-розгалужувач; двоканальний ф!льтр на оптичних мшрорезонаторах, зв'язаних по бо-ковш станц!
Рассеяние оптических импульсов фильтрами-разветвите л ями на диэлектрических микрорезонаторах
Трубин А. А.
Рассматривается система кольцевых микрорезонаторов с колебаниями шепчущей галереи сверхвысокой добротности, широко применяемых для построения различных интегральных фильтров оптического диапазона длин волн. С помощью теории возмущений разработана электродинамическая модель, описывающая
сложную систему связанных между собой микрорезонаторов с двукратно вырожденными типами собственных колебаний, а также расположенных между двумя разными линиями передачи. Получены общие аналитические выражения для описания не взаимных характеристик рассеяния собственных волн линии на системе оптических микрорезонаторов, которые образуют фильтр-разветвитель каналов. Рассчитаны частотные зависимости матрицы рассеяния оптических фильтров-разветвителей с несколькими каналами связи. На основе построенной аналитической модели, рассчитаны временные функции Грина для фильтров с последовательной связью между микрорезонаторами, фильтров с микрорезонаторами, связанными по боковой стенке и двум линиям передачи, а также фильтров, построенных на двойной решетке микрорезонаторов, связанных по двум линиям передачи. Рассмотрены огибающие оптических импульсов, рассеиваемых фильтрами-разветвителями в различные каналы. Рассмотрены огибающие прямоугольного и Гауссовского одиночного импульсов, рассеиваемых на 10-резонаторных фильтрах различной конструкции. Исследовано взаимное влияние нескольких прямоугольных, а также Гауссовских импульсов при их рассеянии на многозвенных оптических фильтрах-разветвителях. На основе сравнения полученных данных для трех видов конструкций, сделан вывод о предпочтительности применения фильтров с последовательно связанными микрорезонаторами. Полученные практические результаты моделирования позволяют значительно сократить время вычислений и оптимизировать сложные многорезонаторные структуры оптических систем связи, одновременно выполняющие функции разделения, или объединения каналов.
Ключевые слова: диэлектрический микрорезонатор; рассеяние; импульс; фильтр-разветвитель; двухканаль-ный фильтр на оптических микрорезонаторах, связанных по боковой стенке
