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Optics Communications 274 (2007) 124–129

Dispersion compensation using mismatched multicavityetalon all-pass filter

Li Wei a,*, Zhi Huang b, John W.Y. Lit a

a Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ont., Canada N2L 3C5b LightMachinery Inc., NRC Building M-50, 1200 Montreal Road, Ottawa, Ont., Canada K1A 0R6

Received 23 August 2006; received in revised form 12 February 2007; accepted 14 February 2007

Abstract

In this paper, we propose a novel mismatched multicavity etalon (MME) all-pass filter for fixed-value dispersion compensation anddispersion slope compensation. It is a multicavity Gires-Tournois (GT) etalon filter with unequal cavity lengths between adjacent cav-ities. A theoretical study is presented. Analytical expressions for the group delay and chromatic dispersion are derived to allow devicedesigns to be made. The simulation results show that nearly linear group delay or quasi-flat chromatic dispersion response can beobtained by suitably choosing the reflectances of the reflectors and the cavity mismatch length. The chromatic dispersions and the band-width of the quasi-flat-dispersion band can be tailored by changing the reflectances of the reflectors and the mismatched cavity length.The dispersion slope compensation can be obtained by slightly modulating the reflectances of the reflectors. Increasing the number ofcavities can enhance the performance and the design flexibility of the dispersion compensator.� 2007 Elsevier B.V. All rights reserved.

Indexing terms: All-pass filters; Dispersion compensator; Dispersion slope compensator; Fiber dispersion compensator; Multicavity etalon; Mismatchedcavity; Gires-Tournois etalon

1. Introduction

Chromatic dispersion is one of the most fundamentalcharacteristics of optical fiber. Fiber chromatic dispersionhas critically limited the network system performance, inparticular, with the increase of the number of DWDMchannels and the data rate of each channel, because theimpact of the chromatic dispersion rises sharply as thesquare of the increase of data rate. Chromatic dispersioncompensation is necessary over a fiber link with a bit rateof 10 Gbit/s or higher in network systems. The main partof the dispersion in the fiber links is usually compensatedwith a fixed-value dispersion compensator while the resid-ual dispersion caused by environmental changing, re-rout-ing, and power variation is dynamically compensated with

0030-4018/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2007.02.017

* Corresponding author. Tel.: +1 519 884 1970x2903; fax: +1 519 7460677.

E-mail address: lwei@wlu.ca (L. Wei).

a tunable compensator. For fixed-value dispersion compen-sation, dispersion compensating fibers (DCFs) [1,2] andchirped fiber Bragg gratings (CFBGs) [3] are commonlyused. Although DCF is presently the most deployed solu-tion, it suffers from high insertion loss and nonlineareffects. CFBG, the other widely used approach, tradition-ally suffers from high group delay ripples, although it hasnow been improved somewhat with new writingtechniques.

Other compensating techniques have been reported,including all-pass ring resonator filters [4,5], birefringentequalizing filters [6], and Gires-Tournois (GT) etalons[7–13]. All-pass filters with fiber Bragg grating distributedGT etalons [7,8] and thin-film-based GT etalons [9–13]have recently attracted much attention because they aretheoretically lossless, and are phase dispersive devices withperiodic responses which make them ideal for applicationsin multi-channel dense wavelength division multiplexing(DWDM) systems. Several approaches have been

L. Wei et al. / Optics Communications 274 (2007) 124–129 125

developed to exploit the advantages of GT etalons for dis-persion compensation. One of the approaches is to use twosets of uniform (equally spaced) GT etalons [9,10]; theslopes of the two sets of GT etalons have to be designedwith equal magnitude but opposite sign, so as to achievetunable dispersion compensation by tuning the centerwavelength of one of the GT etalons. Another is to cascadeseveral single-cavity GT etalons [11–13]. However, in orderto realize a constant dispersion response, tailoring the cav-ity lengths and reflectivities [12], or actively controlling thetemperature of each of the multiple single-cavity GT eta-lons [13] is required, which are very complicated and costly.Moreover, the wavelength alignment (centering or off-cen-tering) of each of the multiple single-cavity etalons wouldintroduce additional manufacturing complexity.

Generally speaking, to design GT etalon-based disper-sion compensators, a flat dispersion band with wide band-width for each channel is highly desired, since this can easethe tight tolerance on the stability of center frequency. Inthis paper, we propose a novel mismatched multicavity eta-lon (MME) dispersion compensating filter that is compact,low loss and potentially much less expensive to implement.We derive the analytical formulae for the group delay andthe chromatic dispersion of the MME filter. Our resultsshow that the novel MME all-pass filter can offer nearlylinear group delay, i.e., quasi-flat-dispersion response bysuitably choosing the reflectances of the reflectors and themismatched cavity length. We discuss the compensator per-formance in terms of the dispersion value and bandwidthof the quasi-flat-dispersion band. The dispersion slopecompensation can be obtained by modulating the reflec-tance in terms of wavelength. In addition, by increasingthe number of cavities, the performance of the dispersioncompensator and design flexibility can be enhanced.

2. MME filter and theory

The proposed MME filter, formed by N + 1 reflectors(R0,R1,R2, . . . ,Rn), is a GT all-pass filter, as shown inFig. 1. The lengths of the cavities are d1, d2, d3, and dn,which are slightly and successively mismatched with a con-

M0

d1d2d3

…...

dn

M1MM3Mn-1Mn M0

d1d2d3

…...

dn

M1M2M3Mn-1Mn

Fig. 1. Schematic illustration of the mismatched multicavity Gires-Tournois (MME) filter.

stant cavity mismatch length D, i.e., dn = dn�1 + D. Notethat the MME filter may be implemented by using eitherdielectric thin-film mirrors or Bragg gratings (fiber orwaveguide). The reflection coefficients and reflectances ofthe N + 1 reflectors are r0, r1, r2, . . . , rn, and R0,R1,R2, . . . ,Rn, respectively. The reflectance R0 of the rearreflector is assumed to be unity so theoretically such a filteris a lossless filter with 100% reflection, though the resultantphase is dispersive. To find the reflection coefficient r of theMME filter, firstly, one can start with the last rear cavity(with reflection coefficients of r1, r0) and calculate its resul-tant reflection coefficient r10, which may be written as [14]

r10 ¼r1 þ r0 e�i2d1

1þ r1r0 e�i2d1; ð1Þ

where d1 ¼ 2pnd1=k is the phase shift of the rear cavity, n isthe refractive index of the cavity, and k is the wavelength invacuum. Given the unity reflectance of the rear reflector,Eq. (1) can be simplified to

r10 ¼ �ei2/1 ð2Þ

with /1 = �tan�1(a1tand1) and a1 = (1 + r1)/(1 � r1).Obviously, the resultant reflectance of the effective reflectorR10 (formed by R1 and R0) is also unity, but the resultantphase 2/1 is dispersive and is dependent on the reflectioncoefficient. Next, we can find the resultant reflection coeffi-cient r20 of the two coupled cavities R20 (formed by R2, R1,and R0) by considering it as a single cavity formed by R2

and R10 to obtain the following equation:

r20 ¼r2 þ r10 e�i2d2

1þ r2r10 e�i2d2¼ �ei2/2 ; ð3Þ

where /2 ¼ � tan�1ða2 tanðd2 � /1ÞÞ, a2 ¼ ð1þ r2Þ=ð1� r2Þ, and d2 ¼ 2pnd2=k is the phase shift of the rear cav-ity (formed by R2 and R1). Similarly, by repeating theabove process, the general expression of the resultantreflection coefficient for the MME filter with N + 1 reflec-tors can be obtained as:

rn0 ¼rn þ rðn�1Þ0 e�i2dn

1þ rnrðn�1Þ0 e�i2dn¼ �ei2/n ; ð4Þ

where r(n�1)0 is the reflection coefficient of an MME filterwith N reflector. Note that the formula in Eq. (4) is recur-sive. The general expression of the resultant phase 2/n isalso recursive:

2/n ¼ �2 tan�1½an tanðdn � /n�1Þ� ð5Þwith

an ¼ ð1þ rnÞ=ð1� rnÞ; ð6Þdn ¼ 2pndn=k: ð7Þ

By using Eqs. (5)–(7), the group delays of an MME filterwith different number cavities can be derived as

GD1 ¼ 2s1c1 ðone cavityÞ; ð8ÞGD2 ¼ 2s2c2ð1þ c1d12Þ ðtwo cavitiesÞ; ð9Þ

Fig. 2. (a) Group delay and group delay ripple. (b) Chromatic dispersionresponses of a two-cavity MME filter.

126 L. Wei et al. / Optics Communications 274 (2007) 124–129

GD3 ¼ 2s3c3ð1þ c2d23ð1þ c1d12ÞÞ ðthree cavitiesÞ; ð10Þ

GDn ¼ 2sncn 1þ dðn�1Þn

2sn�1

GDn�1

� �ðN cavitiesÞ ð11Þ

with

cn ¼an

1þ ðan � 1Þ2 sin2ðdn � /n�1Þ; ð12Þ

where dnm ¼ dn=dm and sn ¼ ndn=c is a unit time delay of asingle cavity. Based on Eqs. (5)–(11), the correspondingchromatic dispersions of the MME filter can be obtained as

CD1 ¼ 2h1s1g1 ðone cavityÞ; ð13ÞCD2 ¼ 2h2s2½g2ð1þ c1d12Þ2 þ g1c2d12� ðtwo cavitiesÞ;

ð14ÞCD3 ¼ 2h3s3½g3ð1þ c2d23ð1þ c1d12ÞÞ2

þ g2ð1þ c1d12Þ2c3d223 þ g1c3c2d2

13� ðthree cavitiesÞ:ð15Þ

Assuming a very small variation of the cavity length d,we may assume dnm ffi 1. The chromatic dispersions of anN-cavity MME filter can be simplified to the recursiverelation:

CDn ¼ 2hnsn gnGDn

2sncn

� �2

þ cnCDn�1

2hn�1sn�1

� �" #ðN cavitiesÞ

ð16Þwith

hn ¼ 2pndn=k2; ð17Þ

gn ¼ða2

n � 1Þan sin 2ðdn � /n�1Þ

1þ ða2n � 1Þ2 sin2ðdn � /n�1Þ

h i2: ð18Þ

It is very interesting to conclude the following pointsabout GT all-pass filters from above equations:

1. From Eqs. (8) and (12), the group delay of a single-cav-ity GT etalon has an Airy function, similar to the trans-mitted intensity of a Fabry–Perot etalon [15]. The groupdelay has a periodic response [16] with a maximum valueof 2s1a1 and a minimum value of 2s1a1=ð1þ ða2

1 � 1Þ2Þ.2. From Eq. (16), the chromatic dispersion is proportional

to ðndnÞ2. This means for an MME filter, if the FSR ishalved, the chromatic dispersion will be increased by fourtimes because FSR is inversely proportional to ðndnÞ2.

3. The resultant reflection coefficient, resultant phase, thegroup delay and chromatic dispersion of the N-cavityMME filter are all recursive functions. It is the uniquerecursive relations that can provide such a dispersioncompensating filter with flexible and powerful designs,as will be seen in the next section.

3. Results and discussion

The MME filter we proposed has slightly mismatchedthe cavities, and thus the free spectral range (FSR) of each

of the single cavities is also slightly mismatched. Because ofthis special feature, the group delay response may beadjusted to have a nearly linear response, which would givea quasi-flat chromatic dispersion response within a certainbandwidth. In this section, we present the results and dis-cussions. The MME filters with two cavities and three cav-ities are used for specific illustrations.

3.1. Two-cavity MME filter

Fig. 2 shows the calculated group delay, group delay rip-ple (GDR), and the chromatic dispersion responses of atwo-cavity MME filter with an FSR of 50 GHz. TheGDR is obtained by subtracting a linear fit of the groupdelay over a nearly linear group delay region. One cansee clearly in Fig. 2a that there is a nearly linear regionin the group delay response and a quasi-flat GDR region,which gives rise to a quasi-flat chromatic dispersionresponse, as shown in Fig. 2b. The optimum reflectancesof R1 and R2 are 29% and 1.1%, respectively. The cavitymismatch length D is 28.7 nm and the refractive index of1.44408 is used. For fixed-value dispersion compensation,

Fig. 4. Relations of R2 and cavity mismatch length with R1 for theoptimum design.

L. Wei et al. / Optics Communications 274 (2007) 124–129 127

it is highly desirable to have a large dispersion valuetogether with a wide quasi-flat-dispersion bandwidth. Herein this paper the bandwidth is determined at the pointwhere the value of the dispersion deviates by ±2% fromthe fixed dispersion value. Fig. 3 displays the chromatic dis-persion and the bandwidth of quasi-flat-dispersion band asfunctions of reflectance R1 for an optimized design. It canbe seen that when the reflectance R1 increases, the magni-tude of the chromatic dispersion increases, but the band-width of the quasi-flat-dispersion band decreases. Thissuggests that the MME filter could provide a large disper-sion value, with a tradeoff between the dispersion and thequasi-flat-dispersion bandwidth. The adjustable parame-ters for an optimum design (i.e., nearly linear groupresponse) are the reflectances R1, R2, and the cavity mis-match length D. Fig. 4 shows the relations of reflectanceR2 and cavity mismatch length D with reflectance R1 forthe optimum design in Fig. 3. From Figs. 3 and 4, onecan see that the reflectance of R2 increases with R1 and lar-ger dispersions can be obtained by having larger reflec-tances and smaller cavity mismatch lengths.

For applications in fiber dispersion compensation, it ishighly desirable to have the dispersion slope to be compen-sated together with fixed-value dispersion compensationbecause the standard Corning SMF-28 fiber has a positiveslope. The dispersion curve of the SMF-28 fiber is nearly astraight line between 1200 and 1600 nm, and the dispersioncan be analytically expressed as DðkÞ � S0ðk� k4

0=k3Þ=

4 ps=ðnm kmÞ, where S0 is the zero dispersion slope andk0 is the zero dispersion wavelength. For the dispersionslope compensation of an MME filter, we may obtain thedesired slope compensation by slightly modulating thereflectance of the reflectors relative to wavelength, as thedispersion varies with the reflectance as shown in Fig. 3.Fig. 5 shows the chromatic dispersion and GDR responsesfor the shortest wavelength and the longest wavelength ofC band with modulated R1 and R2. One can see that theamplitudes of dispersions for the two wavelengths are

Fig. 3. Chromatic dispersion and bandwidth of quasi-flat-dispersion bandas functions of the reflectance R1 for the two-cavity MME filter.

slightly modulated. By using four passes of the design inFig. 5, we can completely compensate a 120-km SMF-28fiber to reach the amplitude of dispersion of 1932 ps/nmat the shortest wavelength, and that of 2141 ps/nm at thelongest wavelength, as shown in Fig. 6 which displayedthe dispersion and dispersion slope compensation with 74channels in the whole C band.

3.2. Three-cavity MME filter

In Section 2, we have seen that all the parameters of anMME filter, such as phase and group delay, are recursivefunctions. With additional cavities, the group delayresponse may be more finely adjusted by tailoring morecavity parameters. This suggests that by increasing thenumber of cavities, the design flexibility and the perfor-mance of the compensator may be improved. This makessuch a filter a very convenient and powerful device.

Fig. 5. Chromatic dispersion and GDR responses for (a) shortestwavelength and (b) longest wavelength of C band.

Fig. 6. Chromatic dispersion of the two-cavity MME filter in the whole C

band.

Fig. 8. (a) Bandwidth and (b) dispersion of quasi-flat-dispersion band asfunctions of reflectance R2 for different reflectance R1 for the three-cavityMME filter.

128 L. Wei et al. / Optics Communications 274 (2007) 124–129

The results above are all based on a two-cavity MMEfilter. As an example, we now consider a three-cavityMME filter to illustrate the increased design flexibilityand also to show how the performance may be enhanced.In Fig. 7, we have plotted the dispersion and group delayresponses for a two-cavity and a three-cavity MME filter.For comparison, the three-cavity MME filter has beendesigned to have the same dispersion value as the two-cav-ity MME filter. It is obvious that the three-cavity MME fil-ter gives a wider quasi-flat-dispersion bandwidth with thesame dispersion value. If the three-cavity MME filter isdesigned to have the same bandwidth as the two-cavityMME filter, then the dispersion value would be larger.Not only the performance of the dispersion compensatoris improved, but also the flexibility is greatly increased,which can be seen from Fig. 8. Figs. 8a and b show thebandwidth and dispersion of quasi-flat-dispersion band asfunctions of the reflectance R2 for different fixed reflec-

Fig. 7. Group delay and chromatic dispersion responses of a two-cavityand a three-cavity MME filter.

tances R1 for a three-cavity MME filter. For each fixedR1, there is a distinct separate curve; this demonstratesthe design flexibility. The figures also show that for a fixedreflectance R1, there is an optimum reflectance R2 that willgive a maximum bandwidth that increases with decrease ofR2. In Fig. 9, we have plotted the reflectance R3 and thecavity mismatch length D in terms of the reflectance R2

for different fixed R1 for the optimum design in Fig. 8. Itis obvious that the reflectance R3 increases with R2, whichin turn increases with R1; the cavity mismatch length Dincreases with decrease of R2, which in turn decreases withR1. Comparing Figs. 3 and 4 for a two-cavity MME filterwith Figs. 8 and 9 for a three-cavity MME filter, one canclearly see that the same trends occur in both cases, which

Fig. 9. (a) Reflectance R3 (b) Cavity mismatch length D as functions ofreflectance R2 for different reflectance R1 for the three-cavity MME filter.

L. Wei et al. / Optics Communications 274 (2007) 124–129 129

follow from the fact that larger reflectances and smallercavity mismatch length would give higher dispersions,and the quasi-flat-dispersion bandwidth and amplitude ofthe dispersion have opposite effects relative to all theparameters, such as reflectances and cavity mismatchlength.

It is worthy to note that using an MME filter to design adispersion compensator, the rule of thumb to choose thereflectance is to set R1 � R2 � � � � � Rn. This can be seenfrom Fig. 4 which shows that the optimum reflectance R2

is almost one order of magnitude lower than R1. Fig. 9shows that the optimum reflectance R3 is almost one orderof magnitude lower than R2, which in turn is almost oneorder of magnitude lower than R1. This can be attributedto the unique recursive relations of the MME filters. Infact, it is this feature that limits the number of the cavitiesthat can be employed in practice as with the increase of thenumber of cavities, the optimal reflectivity of the frontreflector could be very small, which presents difficulty inmaking it if the reflectors are implemented by dielectricthin-film mirrors.

Finally, to connect our theoretical model to a practicaldevice, more attention shall be paid on bandwidth and dis-persion, as they are the key specifications for such a device.The bandwidth of the two-cavity MME filter shown inFig. 3 varies between about 7 and 10 GHz, which may beinadequate for applications at 10 Gb/s. However, it isdelighted to see that the bandwidth requirements havedecreased substantially in the past five years with the useof advanced modulation formats. On the other hand, largerbandwidths may be obtained by using a three-cavity MMEfilter (see Fig. 7) and by choosing smaller reflectivities. Asmentioned earlier, there is a tradeoff between the quasi-flat-dispersion bandwidth and dispersion, i.e., while thebandwidth is enhanced, the dispersion will be dropped.Fortunately, this may be relieved by using the strategy ofconcatenation of multiple MME filters to increase the dis-persion to useful ranges.

4. Conclusions

In this paper, we have proposed to use a novel mis-matched multicavity etalon (MME) all-pass filter forfixed-value dispersion compensation and dispersion slopecompensation. A theoretical analysis of the MME filter ispresented. The MME is a multicavity all-pass GT etalonwith each of the cavities equally mismatched by a constant

small amount. With such a design, by suitably choosing thereflectances of the reflectors and the mismatched cavitylength, one may obtain a nearly linear group delay, i.e.,quasi-flat-dispersion response. We have derived the generalanalytical expressions for the group delay and chromaticdispersion for the MME filters with N cavities. Specialcases with two and three cavities are presented. The simu-lation results show that for a two-cavity MME filter, higherdispersion values can be obtained by increasing the reflec-tance R1, but there is a tradeoff with the bandwidth ofthe quasi-flat-dispersion band. The fixed fiber slope com-pensation can also be obtained by modulating the reflec-tance of the reflector in terms of wavelength. The resultsalso show that by increasing the number of cavities, theflexibility and the performance of the compensator maybe improved. Such a filter with a simple structure (andpotentially low manufacturing cost) should find usefulapplications in network systems with high data rates.

Acknowledgment

This work is supported by the Natural Sciences andEngineering Research Council of Canada (NSERC).

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