analysis of double-parallel

7/28/2019 Analysis of Double-parallel

1/7

Analysis of double-parallelamplified recirculating optical-delay lines

M. C. Vazquez, R. Civera, M. Lpez-Amo, andM. A. Muriel

A novel method of analysis of double-paral lel amplified recirculating optical-delay lines (DPAROD) ispresented. The location of the maxima and the minima of the trans fer function for this configuration iscalculated and experimentally demonstrated. The influence of different parameters, such as th ecoupling coefficients, gains, lengths of the fiber loops and frac tiona l losses of the directional couplers , on

the shape of the tra nsfer function are analyzed. Different measurements have been taken to verify thismodel. The potenti al application of these inte rconnected delay loops as filters is a reason for developingthis method.

lntroductionSingle, amplified recirculating optical-delay ines havebeen widely analyzed,ly2 showing their capability forthe development of signal-processing functions suchas filtering, pulse-train generation, conv~lut ion ,~ ndmemo^-y.4 A higher degree of design flexibility can

be achieved with more complex structures, such as

the double-parallel amplified recirculating optical-delay line (DPAROD),l which is depicted in Fig. 1.

New features obtained with DPAROD allow one toreshape the ir frequency response. For instance, onecan obtain a predetermined relative amplitude be-tween the main and secondary maxima and minimaof the frequency response of DPAROD. This perfor-mance may be fairly useful for filtering, when discrimi-nation is made on the basis not only of the signalfrequency, but also on its signal-to-noise ratio.Another application of the DPAROD may be in thefield of fiber-optic sensors, owing to the dependence ofthe DPAROD response on the gain of the fiber loops.If the results to be shown are extrapolated to inte-

grated optics, large free spectral ranges should beachieved with DPAROD, as compared with single-structure configurations. ~r ev io us eports have beenmade in the coherent analysis of a three-coupler

The auth ors a re with the Departamento de Tecnologa Fotnica,Escuela Tcnica Superior de Ingenieros Telecomunicacin, Univer-sidad Politcnica de Madrid, Ciudad Universitaria, E-28040 Madrid,Spain.

Received 9 February 1993; revision reical So.

configuration,5 while with our DPAROD only twocouplers are needed.

An incoherent analysis (i.e., we consider intens itiesra ther t han fields) of the DPAROD has been carriedout. Our main interest lies in predicting the fre-quency-domain location of the maxima-minima andthe shape of the t ransfe r function, we have developeda new model to do so. Some experimental results a re

also shown.

Nove l Metho d for Analysis of a Double-A mplifiedRecirculating Optical-Delay LineTo describe the signal in fiber-optic recirculatingdelay lines one usually utilizes the Z t r a n ~ f o r m . ~This usage is possible a s long as a basic time delay Tcan be defined such that the other relevant systemdelays are integer multiples of th is basic delay, andthe system can be considered discrete in time. Suchis the case in the single-structure configuration,ly2 orin Nth-order fiber-optic lattice structure s, where thedelay T s the same for al1 se ~ ti o ns .~ his is not truefor the general DPAROD analyzed in this paper:different loop lengths, the use of which implies differ-ent time delays, are used. These time delays do notnecessarily have to be integer multiples of a basictime delay, so the Z transform cannot be applied.A novel method will be used.

The DPAROD is a linear system with respect tointensity, so the calculation of t he transf er functionH ( 0 ) with 11+ s the input and 12+ s the output, seeFig. 1) s carried out with the intensity equations ofthe directional couplers and the recirculating condi-tions imposed by each loop. The following expres-

1


2/7

II3

Fig. 1. Double-parallel amplified recirculating optical-delay line(DPAROD): Ii%e the optical-field intensities at the input and

'1 1

output ports ~ - f he directional couplers, while yj and kj are thefractional losses and th e coupling ratio of th e j directional coupler.

OPTICAL-

OURCE

sion is obtained:

where

- .

lC 1

The definitions of the utilized parameters are asfollows: yj, aj, and kj are th e coupler fractionallosses, the fiber attenuation , and the coupling ratio ofthe directional couplers, respectively, as indicated inFig. 1, where j = 1 efers to coupler 1 and j = 2 refersto coupler 2. G1, G2, and G3 are the gains introducedin each of the fiber loop lines, and 0 is the frequencymodulating the optical carrier. lpi. orresponds to theunamplified fiber length, while T~ 1s the propagation-time delay introduced by each loop line i. Thesedelays are given by th e expression

2 'OPTICALDETECTOR

where 1 is the length of the arnplified fiber, c is thelight speed in vacuum, and n is th e refractive index ofthe unamplified fiber (because the refractive-indexdifference between the amplified and th e unamplifiedfiber is no greater t han 0.02, only one refractive indexhas been used).

The time delays given by Eq. (4) do not have to beinteger multiples of a basic delay, so as explainedabove, the Z transform cannot be applied.

The new method of analyzing DPAROD7s consistsof splitting the system into three single rings: The

first is made of the fiber sections of length l l and 12, asshown in Fig. 2(a). This loop is closed throughcoupler 2, giving the consequent factor (1 - k2) n theoverall gain of the loop [see Eq. (6)]. The second ringcontains the fiber of length l3 [Fig. 2(b)]. The entiretransfer function is multiplied by the factor k12G1G2because the first and the last optical field intensitiescross through coupler 1 during each circulation.

The third r ing comprises al1 three loop lengths-11,12, and 13-and a factor of (k2)2 rom crossing throughcoupler 2 is added [Fig. 2(c)] o the overall gain of theloop [see Eq. (8)].

Adding together the different results of the equa-tions for the three rings yields an approximation ofthe ou tput of the system. This method considerscertain common terms, such as the direct pass throughcoupler 1 or rings 1, 2, and 3 or the direct pass acrosscoupler 2 for rings 1 and 2, only once.

The expression of the transfer function for eachsingle ring is given by

where M = G*Z-l, with G* = lO-(*lp/lO) x G( l - Y),

Fig. 2. Novel separation of the double structure into threedifferent rings: (a) ing 1 comprises l l and 12; (b) ing 2 comprisesZ3; and (c) ring3 compriscsZI, Z2,and 13 .

2


3/7

Z = ejn7 and, for rings 1 ,2 , and 3, respectively,

The parameters that have not been identified asecond time have t he same meaning as in th e double-ring configuration.

For each single-ring configuration, it can be in-ferred from Eq. (5) that the frequencies f of themaxima-minima of the transfer function are locatedat

Here, N, M, and L are integer numbers. Thesefrequencies correspond to those of th e laser-modula-tion frequency.

So al1 possible maximum and minimum frequen-cies belong to the set of values

f m a x = + 72)UM/73UN/(71 + 7 2 + 73 ) (12>

fmin = (1/2 + L)/(71+ 72)U(1/2 + M)/73U(1/2 + N )/(TI + 7 2 + 73). (13)

From Eqs. (12) and (13) t can be inferred tha t th eonly parameters that influence the location of themaxima and th e minima are 11, 12, 13, and n. Therelative amplitude of the different maxima and minimawould be fixed through th e remaining paramete rs kl,k2, Gl, G2, and G3.

Periodicity of the Transfer Function

Knowing tha t H ( a ) s a periodic function, it would beinteresting to determine the margin of frequenciesthat represent one period of H(0 ). Considering again

the division into thr ee rings , the period of eachseparate ring would be l/fm,, as shown in Eqs.(9)-(11). If rings 2 and 3 are considered simulta-neously, the period of H ( a ) of th e compound struc-tu re would be obtained by se lecting a certain pair ofthe smallest integers N and M, which follow

so MFP = M ( ~ / T ~ ) , here M FP is the margin offrequencies of a period, which means the range infrequency between two consecutive main maxima.

Equation (14) is also valid when including ring 1,

because if we operate with i t

Other Terms Affecting the Model

The general expression for the delay associated withthi s double configuration are a s follows:

So the transfer function would have a maximum a t

This equation could imply tha t Eq. (12) was incom-plete, but th at conclusion would be wrong because thenew terms added in Eq. (15) would never con tributeto generate a maximum frequency. Let us consider,for example, the initial second-order term, which hasbeen neglected: M = 1 and N = 2, indicating one triparound ring 3 and two trips around ring 1. It can beseen tha t, although completion of th e total path cancause in-phase interference, other interference a t the12+ erminal have occurred during the intermediatesteps (see Fig. l), and the interference may havecanceled the effects a t 1 2 + . he same reasoning canbe applied to t he minima frequencies.

Overlap Effect

For a certain set of values (kl, k2, Gl, G2, and G3), nwhich the influence of the thr ee rings is fairly similar,an overlap effect would appear such t ha t th e frequen-cies of the secondary maxima would suffer a shiftwith respect to the habitual case, where overlap is notpresent. This overlap usually occurs when th e cou-pling coefficients kl and k2 are approximately 0.5,which we do not consider useful for our purposes.An example where this effect appears follows: Theparameters of a structure where overlap could occurarekl = 0.72,k2 = 0.63, G1 = G2 = G3 = l , n = 1.44,l l = 1 m , 1 2 = 2 m , 1 3 = 3 m , a = 0 , a n d y = 0 . Tablellists the frequencies of the maxima determined withthe simu1ation.l The second column gives the re-sul ts calculated with Eq. (12), as if no overlap effectexisted.

Table 1. Calculated and Simulated Frequenciesa

Simulated mm ( M H z ) Calculatedb ,, ( M H z )

Frequencieso o

45.37 41.6684.42 83.33

123.48 125163.11 166.66208.33 208.33--

aParameters are k l = 0.72, k2 = 0.63, Gi = G2 = G 3 = 1, n = 1.44,11= 1 m , 1 2 = 2 m , 1 3 = 3 m , ~ = 0 , a n d ~ = 0 .

bFrequencies calculated with Eq. (12).

3


4/7

Table 2. Calculated and Measured Frequencies orMaxima and Minimaa

Table 3. Prevailing Rings as k, and k, Are Varieda

MeasuredCalculated fmaxb Calculated fmin FrequenciesC

Contributing Contributing f m a x f m i nRing MHz Ring MHz (MHz) (MHz)

Prevailing Ring

Conditions k l * 1 k 1 0

aAll other DPAROD parameters are kept constant.bThe nfluence of ring 2 increases as k 1approaches unity.

Experime ntal Confirmation

aFrequencies where maxima and minima can be located whenL1 = L2 = l 3 = 2 m a n d n = 1.44.

bCalculated f determined with Eq. (12); calculated fmin eter-mined with Eq. (13).

CMaxima nd minima measured when l l = 12 = l 3 = 2 m, a l =a2 = a3 = 0.0003dB/m, y1 = y2 = 0.05, k l = 0.958, k 2 = 0.912,G l = G 3 = 0.5, and G 2 = 0.25.

A cw laser-diode output of 1.5 pm was sinusoidallymodulated in frequency by using the tracking genera-tor of an rf spectrum analyzer at the desired frequen-cies. This modulated signal was the input to therecirculating delay line. The set up was constructedwith two 2 x 2, k-variable, polarization-preservingfiber couplers with 2 m of pigtails that constitu te thedifferent recirculating delay lines of the DPAROD,that isll = l2 = l3 = 2 m. AccordingtoEqs. (12) and

(13), it is possible for maxima and minima to belocated at the frequencies listed in the second andthird columns of Table 2. It is important to notetha t these are the frequencies where the maxima canbe located, but choosing values of some parameters(e.g., kl , k2, G1, G2, and G3), permits the amplitude ofthe optical field to be low enough that there is not amaximum at some of those freqencies. This can bethe case with the minimum frequencies. In Table 2,the first and fourth columns list the ring or rings thatcontribute to the maximum and the minimum fre-

Fig. 3. Transfer function of the DPAROD (experimental resul ts) for l l = 12 = l 3 = 2 m, a l = a:, = a g = 0.0003 dB/m, y1 = y2 = 0.05, n =1.44,hl = 0.958,h2 = 0.912,G1 = G3 = 0.5,ilndG2 = 0.25.

4


5/7

Fig. 4. Transfer function of the DPAROD (experimental results) for l l = l2 = l3 = 2 m, al = a 2 = a3 = 0.0003 dB/m, y1 = y2 = 0.05, n =1 . 4 4 , G 1 = G 3 = 0 . 5 , G 2 = 0 . 2 5 , a n d k 2 0 . 8 6 : a,k1=0.95;andb,kl=0.98.

quencies, respectively, and the fifth and sixth col- the maximum frequency (103.5 MHz). The otherumns list the measured frequencies for k l = 0.958, frequencies correspond to the minima of ring 3 (a tk 2 = 0.912, G l = G3 = 0.5, G2 = 0.24, a = 0.03, and 120.9 and 190.2 MHz) or to the maxima of the samey = 0.05 (see Fig. 3) . Through this example, which ring. It can be seen tha t, for those frequenciesshows the periodicity of H ( f l ) ,we can analyze one where the maxima and the minima of different ringsMFP of the transfer function. The absolute maxima overlap ( mml = fmin2,, = 52 MHz) because of the greatare those where the three rings contribute to achieve influence of rings 2 and 3 (see the next section and

5


6/7

frsquency, Hz x l 06

Fig. 5. Transfer function of the DPAROD (calculated) for k l = 0.95, k z 0.1, L1 = 1 m, Lz L3 2 m, n = 1.44, a = y = 0, and G1 G2 1:a, G3 1; b, G3 1.1.

Table 3), the measured amplitude corresponds to aminimum. If other lengths were selected, the FSRwould increase, as in Ref. 5. With respect to theMFP, by applying Eq. (15) we find that 104.l.M =52.08L, so L/M = 2, which means tha t for L = 2 andM = 1, the margin frequency = 104.1 MHz, as can beseen in Fig. 3 and Table 2. There is a perfectagreement between the measured and predicted val-ues, so the model is demonstrated to be valid fordetermining the maxima and the minima.

lnfluence of the Different Parameters on the Frequency

ResponseThroughout this analysis, the parameters that arenot being tested will remain constant at Gi = 1, ai = 0,and y i = 0.

lnfluence of the Refractive lndex and of the Loop LengthsThe refractive index n and the lengths of the fiberloops 11, L2, and l3 are the parameters tha t determinethe location of th e maxima. The effect of varyingany of them would be similar. A significant case iswhen L3 = ll + 12, SO the maxima of rings 1 and 2 aresuperimposed, and a secondary maximum will appearonly at the half of the main one. If any of the looplengths are changed, more secondary maxima willappear.

lnfluence of k l and k2

As a first step it is necessary to delimit the ranges ofkl and k2 to those where the response is governed bya single ring. By comparing any two rings (e.g., iand j), he ratio H(CI),;Ui/H(CI),,j is calculated. Bychanging kl and k2, this ratio can be increased ordecreased, showing the prevailing ring or rings ineach case. If only one ring dominates, the analysis ofthe structure will be equivalent to that of a singlering2 witll aii equivalent gain, wliich would be de-

scribed by the expressions in Eqs. (6)-(8). The ex-pressions of th e different ratios are omitted and onlythe conclusions are given in Table 3. This analysis isalso applicable to t he minima.

I t can be seen in Fig. 4 how, a t 86.0 and 120.9 MHz,there are two minima whose levels increase whenk2 = 0.86 and kl changes from 0.95 [Fig. 4(a)] o 0.98[Fig. 4(b)]. As kl changes, the influence of ring 2increases with respect to ring 3, as predicted in Table3 (see conditions for kl

*

0, and kl - 1 whenk2

*

1). The same effects occur at 155.2 and 52.0MHz, which decrease because they correspond to theminima of ring 2. The rest of the parameters of thestructure have been kept unaltered, i.e., G1 = G3 =0.5, G, = 0.25, a = 0.03 dB/km, y = 0.05, L1 = l2 =Z3 = 2 m. This example helps us to understand theutility of Table 3 and it s possible interpretation.

lnfluence of G1, G2, and G3Following the same steps as in the preceding sectionand taking into account the gains Gi of each loop, theresults shown in Table 3 still prevail, but moredegrees of freedom (G1 x G2, G3) have been added tothe system. Now the new gains will be determinedby Eqs. (6)-(8) by allowing the values of G1, G2, and G3to change.

Special attention is given to the influence of G3when k2 = O. From Table 2 we determine that ifk2

*O and kl - 1, the influence of both ring 1 and

ring 2 appears. But changes in G3 affect only ring 2.This is confirmed in Fig. 5, where kl = 0.95 and k2 =O.l(ll = 1m,12 = 2m,13 = 2 m , n = 1 . 4 4 , ~ ~ y = 0,G1 = G2 = 1) and Gg changes from 1 to 1.1 (linearscale) so the amplitude of the maximum at 104.2 MHz(related to ring 2) increases while the maximum ofring 1 emains the same. So only the amplitude for afixed frequency has been altered, and t he rest of thetrans fer fuilctionhas not been modified. This behav-

6


7/7

ior is different from that of a typical tuner , where al1elements in the tr ansfer function are shifted.

lnfluence of the Attenuation of the Fiber a

Any change in a s equivalent to a change in the gainloop, because it can be defined as

lnfluence of Fractional Losses at the Couplers

Losses at t he couplers can be taken into account whenconsidering the factor (1 - yl) related to each couplerand should be included into the overall gain:

SummaryDetermining the locations of the maximum and mini-mum frequencies in double-parallel amplified recircu-lating optical delay lines (DPAROD), while essentialfor their application as filters, cannot be obtainedusing the Z transform techniques, as can be done forsingle amplified recirculating delay lines. In th iswork, we have presented a new model for calculationof these max&um and minimum frequencies. Themodel consists of spli tting the whole system intothree independent rings. Experimental results havebeen reported to help to understand it s meaning.

The versatility of DPAROD's has been shown, andthe model has been applied to the analysis of theinfluence of different parameters on the shape of the

transfer function. The model permits one to delimitthe intervals for the values of the different pararn-eters that should be interesting to work in filtering(or demultiplexing) applications.

This work was supported by Spanish ComunidadAutnoma de Madrid-contract PRI (C059/90), Comi-sion Inteerministerial de Ciencia y Tecnologia TIC92-0052-C02 and 89-21 1, and Universidad Politcnica deMadrid (Acciones Concertadas). We thank B. Vizosoand ATT Espaa S.A. for their help.

References

1. M. C. Vzquez, B. Vizoso, M. Lpez-Amo, and M. A. Muriel,"Single and double recirculating delay lines as fiber-opticfilters," Electron. Lett. 28,1017-1019 (1992).

2. B. Mosheli and J. W. Goodman, "Novel amplified fiber-opticrecirculating delay line processor," J . Lightwave Technol. 10,1142-1 147 (1992).

3. K. P. Jackson, S. A. Newton, B. Moslehi, M. Tur, C. ChapinCutler, J. W. Goodman, and H. J. Shaw, "Optical fiber delay-line signal processing," IEEE Trans. Microwave Theory Tech.33,193-209 (1985).

4. T. J . Soukup, R. J. Feuerstein, andV. P. Heuring. "Implementa-tion of a fiber-optic delay-line memory," Appl. Opt. 31, 3233-3240 (1992).

5. K. Oda, N. Takato, and H. Toba, "A wide FSR waveguide doublering resonator for optical FDM transmission systems," J .Lightwave Technol. 9,728-736 (1991).

6. B. Mosheli, J. W. Goodman, M. Tu r, and H. J . Shaw, "Fiber-optic lattice signal processing," IEEE Proc. Natl. Aerosp.Electron. Conf. 72,909-930 (1984).

7

analysis of double-parallel

Documents