is slightly different from the desirable value due to the inaccuratelength of the central period. Furthermore, we can tailor the trans-mission spectrum by changing the applied pressure. Figure 4(b)shows the evolution of a 3�/2 phase-shifted LPFG under differentpressures. This provides another degree of freedom for tailoringthe transmission spectra.

Finally, a grating with a period of 630 �m and a total of 79periods was used to flatten the spectral output of an Er-dopedsuperfluorescent fiber source. The result is illustrated in Figure 5.The variation in the spectral output has been flattened to less than2 dB over 35 nm, compared to 9.2 dB without the grating. Ofcourse, the amount of flattening can be improved by placinganother segment of fiber under the same plate in order to cascadethe gratings to flatten the small spectral variations.

CONCLUSION

In this paper, we have presented a novel method to fabricateband-rejection and bandpass filters based on mechanically inducedLPFGs with periodically wrapped wires. This method is simpleand reconfigurable and offers great tunability by changing theexternal pressures or introducing a phase-shift in the grating.Based on this type of grating, we have successfully flattened anEr-doped superfluorescent fiber source, obtaining a flatness lessthan 2 dB over a bandwidth of 35 nm. This type of grating isflexible in spectra shaping and will be useful for special filterdesigns.

ACKNOWLEDGMENT

This work is supported by the National Natural Science Founda-tion of China under grant no. 60137010.

REFERENCES

1. A.M. Vengsarkar, J.R. Pedrazzani, J.B. Judkins, P.J. Lemaire, N.S.Bergano, and C.R. Davidson, Long-period fiber-grating-based gainequalizers, Opt Lett 21 (1996), 336–338.

2. A.M. Vengsarkar, P.J. Lemaire, J.B. Judkins, V. Bhatia, T. Erdogan,and J.E. Sipe, Long-period fiber gratings as band-rejection filters, JLightwave Technol 14 (1996), 58–65.

3. V. Bhatia and A.M. Vengsarkar, Optical fiber long-period gratingsensors, Opt Lett 21 (1996), 692–694.

4. X. Li, C. Yue, L. Xia, X. Chen, and X. Xie, Novel technique for longperiod gratings fabrication using broad spectrum ultraviolet source,Microwave Opt Technol Lett 33 (2002), 368–370.

5. Y. Lin and L.A. Wang, A wavelength- and loss-tunable band-rejection

filter based on corrugated long-period fiber grating, IEEE Photon Tech-nol Lett 13 (2001), 332–334.

6. S. Savin, M.J.F. Digonnet, G.S. Kino, and H.J. Shaw, Tunable mechan-ically induced long-period fiber gratings, Opt Lett 25 (2000), 710–712.

7. T.W. MacDougall, S. Pilevar, C.W. Haggans, and M.A. Jackson, Gen-eralized expression for the growth of long period gratings, IEEE PhotonTechnol Lett 10 (1998), 1449–1451.

© 2004 Wiley Periodicals, Inc.

PERIODIC BANDPASS FILTER WITHNEAR-IDEAL FILTER RESPONSEBASED ON SAGNAC LOOPINTERFEROMETER

Li Wei,1 Qinghe Mao,1 and John W. Y. Lit1,2

1 Department of Physics and Computer ScienceWilfrid Laurier UniversityWaterloo, ON, N2L 3C5, Canada2 Department of PhysicsUniversity of WaterlooWaterloo, ON, N2L 3G1, Canada

Received 16 December 2003

ABSTRACT: In this paper, an analytical analysis is made on a peri-odic bandpass filter formed by a two-cavity etalon incorporated into aSagnac loop interferometer. The general optimum conditions for a quasi-flat-top spectrum and a ripple-free group-delay response are derived. Ourresults show that by properly choosing the reflectance of the mirrors, thefilter can be designed with a quasi-flat-top spectral response that is alsofree from group-delay ripples. © 2004 Wiley Periodicals, Inc. MicrowaveOpt Technol Lett 42: 17–21, 2004; Published online in Wiley Inter-Science (www.interscience.wiley.com). DOI 10.1002/mop.20194

Key words: optical filters; Sagnac loop; wavelength division multiplex-ing (WDM); group delay

INTRODUCTION

Optical filters are essential components in dense wavelength-division-multiplexing (DWDM) systems and networks for realiz-ing various fundamental network functions, including add-drop,multiplexing/demultiplexing, and interleaving. For such applica-tions, the filters will perform best if they have flat-top spectralresponses. Such flat-top filters can be made with various elements,including fiber Bragg gratings (FBG) [1, 2], planar lightwavecircuits (PLCs) [3, 4], optical-fiber delay lines [5], and variousinterferometers based on bulk optics [6, 7]. Recently, a ripple-freeoptical filter using a Michelson interferometer (MI) with a Gires–Tournois resonator (GTR) inserted into one arm was proposed andtheoretically analyzed [6]. The performance of the filters was thenimproved when both arms of the interferometers had a GTR builtinto them [7]. However, as bulk-optic elements are used, the lightbeams have to be collimated and the system has high losses. On theother hand, the Sagnac loop interferometer (SLI) in fiber form ismuch more robust by comparison. It has the advantages of low lossand low cost. More significantly, it is inherently compatible withfiber systems. A periodic filter [8, 9] formed by an SLI with anFBG in the fiber loop has been proposed and demonstrated. How-ever, because only one reflector (the FBG) is used, the spectralresponse of the filter has a Lorentzianlike shape. In order toachieve flat-top spectral response, more reflectors may have to beused in a Sagnac loop. Thus, it is very important to investigate andoptimize the functions of an SLI with multiple reflectors with

Figure 5 Spectra of Er-doped superfluorescent source with and withoutflattening

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 1, July 5 2004 17

respect to its filtering properties, particularly, the group-delaybehaviors [10] in high-speed systems.

In this paper, we study a periodic filter based on an SLI withmultiple reflectors, in this case, a two-cavity etalon in a fiber loopis used to obtain flat-top spectral response. The two-cavity etalonis used to form two GTRs. The general optimum conditions foroptimum design to obtain flat-top spectral response and ripple-freegroup-delay response are derived. The result shows that the trans-mission spectrum and group-delay behavior of such a filter aresignificantly controlled by the reflectance of the mirrors. Thedependences of the passband bandwidth and the steepness on thereflectance of mirrors are investigated. When properly designed,such a filter can have a quasi-flat-top spectrum that is also freefrom group-delay ripples. It is noted that the two-cavity etalon maybe replaced by FBGs in order to turn the filter into an all-fiberdevice.

DESIGN AND FORMULATION

Figure 1 shows the schematic diagram of our periodic filter, whichis based on an SLI with an etalon inserted into the fiber loop. Theetalon consists of three mirrors, M1, M2, and M3, which may beformed by two glass spacers with three dielectric thin film coat-ings. The two glass spacers have the same thickness d. Thereflectances of M1, M2, and M3 are r1, r2, and r3, respectively.The coupler has a coupling coefficient k. The port of the coupleris numbered in a clockwise manner. The lengths of the fiber fromM1 to port 2 of the coupler and from M2 to port 3 of the couplerare L1 and L2, respectively. We have chosen r3 to be unity suchthat the central mirror M3 may be shared by M1 and M2, thusproducing two cavities—a left cavity formed by M1 and M3 and aright one formed by M2 and M3, respectively. With this design,each of the two cavities may serve as a GTR. We assume that thecoupler and fiber are polarization maintaining; therefore, the com-plex fields can be treated as scalars. We also assume that thecoupler is a point device, that is, it has zero length. The complexelectric fields at ports i (i � 1, 2, 3, 4) of the coupler are denotedby Ei

t and Eia, where the superscripts t and a indicate the directions

of the electric field toward and away from the ports, respectively.Referring to Figure 1, the complex electric fields traveling

away from the four ports of the coupler can be written as

E1,2a � �1 � � ��1 � k E2,1

t � j�k E3,4t �, (1)

E3,4a � �1 � � � j�k E1,2

t � �1 � k E4,3t �, (2)

where � is the intensity loss of the coupler. After propagation inthe loop, the electric fields traveling toward the coupler are givenby

E2,3t � E2,3

a exp� j�1,2 � j2�fL1,2�, (3)

with

�1,2 � �2 arctan�a1,2tan��sd��, (4)

a1,2 � �1 � r1,2�/�1 � r1,2�, (5)

�f,s � 2�nf,s/, (6)

where �1 and �2 are the phase shifts of the two GTRs [11], a1 anda2 are defined as the reflective factors, �f and �s are the respectivepropagation constants in the fiber and the glass substrate, nf and ns

are the respective refractive indices of the fiber and glass substrate,and is the wavelength of the light in a vacuum.

We assume that port 1 is the input port and the input at port 4is zero; therefore,

E1t � Ein, E4

t � 0. (7)

By combining Eqs. (1)–(7), one can obtain the transmissioncoefficient at port 4 as

t �E4

a

Ein� j�1 � ��2��k�1 � k� �exp� j��1 � 2�fL1��

� exp� j��2 � 2�fL2���. (8)

Considering that the coupler has 3-dB lossless coupling, that is,k � 0.5 and � � 0, Eq. (8) can be simplified as follows:

t � j cos��� � 2�fL�/2�exp� j���1 � �2� � 2�f�L1 � L2��/2�,

(9)

where � � �1 � �2 is the phase difference between the two GTRsand L � L1 � L2 is the difference of the two fiber lengths. Inorder to obtain a quasi-flat-top spectral response, the phase of thesingle cavity � has to be twice that generated by the lengthdifference of the fiber sections, that is,

� � �sd � 2�fL. (10)

CHARACTERIZATION

General Condition for the Quasi-Flat-Top SpectrumThe normalized transmission intensity at port 4 can be obtainedfrom Eqs. (4)–(10) as follows:

T � �1 � cos��2�arctan�a1tan �� � arctan�a2tan ��� � ���/2

(11)

From Eq. (11), we can see that the transmission spectrum isperiodic with a free spectral range (FSR) determined by c/ 2nsd,which highly depends on the reflective factors a1 and a2. If the

Figure 1 Sagnac loop interferometer with a two-cavity etalon in the fiberloop

18 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 1, July 5 2004

transmission intensity is unity, the following equation must betrue:

�2�arctan�a1tan �� � arctan�a2tan ��� � � � 2m�. (12)

At resonance, that is, � � �sd � 2m�, from Eq. (12) we canobtain the following equation:

a2 � a1 �1

2. (13)

Therefore, Eq. (13) is the condition for unit transmission.Moreover, it is the general condition for the quasi-flat-top spec-trum, as will be shown below. After substituting Eq. (5) into Eq.(13), one can obtain the optimum reflectance for the mirrors. Thedependence of the optimum value for r2 on r1 is plotted in Figure2. It can be seen that r2 changes with r1 monotonously. It shouldbe noted from Eq. (13) that when r1 is equal to �1, the optimumvalue of r2 is �1/3. This is a special case; it means that such a filtermay be made with a loop and one single-cavity etalon. Theinterference is between waves and comes from the 100% mirrorwith one GTR instead of the two GTRs. The flat-top condition atr2 � �1/3 is in agreement with that given in [6]. Obviously, withtwo GTRs, the filter has a variable optimum reflectance, as shownin Figure 2, whereas with one GTR, the optimum reflectance hasa constant value at �1/3. For example, if r1 � �0.6, the optimumvalue for r2 is �0.143 according to Eq. (13). Figure 3(a) shows thespectra for a few values of r2 with r1 equal to �0.6. The details ofthe passband spectra are shown in Figure 3(b). From Figure 3 wecan see clearly what happens if r2 deviates from the optimumvalue of �0.143. When r2 is less than the optimum value, thespectrum has three peaks with ripples and side lobes. When r2 isgreater than the optimum value, the spectrum has a roundedmaximum instead of a flat top. When r2 has the optimum value of�0.143, the spectrum shows quite a broad flat-top passband. It isworthwhile to mention that the negative sign of the reflectancemeans that the coating phase of the mirror should be �.

To evaluate the performance of an optical filter, the steepnessand the bandwidth are two important parameters. An ideal filterwould have large steepness and large bandwidth, which will givehigh isolation between adjacent channels and low losses within thepassband. Since all pairs of r1, r2 that satisfy Eq. (13) give the

quasi-flat-top spectrum, it would be very interesting to see howdifferent sets of r1, r2 affect the bandwidth of the flat portion andthe steepness of the filter. Here we define the range of the flat topas the passband bandwidth with ripples less than 0.005 dB. Steep-ness is defined as the slope of the normalized transmission inten-sity with wavelength, where the normalized intensity is 1/2, that is,3 dB [1, 6]. In Eq. (11), if � � (2m 1/ 2)�, then T � 1/ 2 forall values of a1 and a2. This is a unique characteristic for the filter.This means that all spectra will pass through the point T � 1/ 2 at� � (2m 1/ 2)�, independent of the reflectance of the mirrors.According to the above definition and analysis, the steepness canbe derived as

S �dT

d�

T�3 dB

� � 2

a2�

2

a1� 1� � �nsd/2. (14)

Figure 4 plots the range of the flat-top passband and thesteepness as functions of the reflectance of r1, where r2 has theoptimum value given by Eq. (13). From Figure 4 one can see thatthe bandwidth has maximum value when r1 is around �0.64.Because reflectivity is the square of reflectance, it is clear from

Figure 2 Reflectance of mirrors for the quasi-flat-top spectrum

Figure 3 Transmission spectra for different r2 with fixed r1 at �0.6(dashed, solid, and dotted lines correspond to the values of r2 at �0.043,�0.143, and �0.243, respectively)

Figure 4 Steepness and range of the quasi-flat-top passband as a func-tion of reflectance of M1

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 1, July 5 2004 19

Figure 4 that steepness increases with increase of the reflectivity ofM1. Note that in Figure 4 we consider the spectrum to have aquasi-flat response. In practice, filters with small ripples are alwaysallowable. If small ripples are allowed, one could expect that thebandwidth may be enhanced. Figure 5 shows passband bandwidthand reflectance of M2 as functions of the reflectance of M1 fordifferent amount of ripples, where the reflectance of M2 is opti-mized in terms of passband bandwidth. From Figure 5, we can seeclearly that, with increasing ripples, the bandwidth increases andthe peak of optimum bandwidth shifts to the left. In other words,the optimum value of reflectivity of M1 increases.

General Condition for Ripple-Free Group-Delay ResponseGroup delay is another important characteristic parameter of pe-riodic transmission filters when such filters are employed in high-speed DWDM systems. Filters with an in-band constant group-delay response [9, 10, 12] are preferred for the system design inorder to avoid signal distortions. From Eq. (9), we may derive thegroup delay and chromatic dispersion of the filter as

GD�� � �c� a1

cos2� � a12sin2�

�a2

cos2� � a22sin2�

�nf�L1 � L2�

nsd�, (15)

CD�� � ��csin 2�� a1�a12 � 1�

�cos2� � a12sin2��2

�a2�a2

2 � 1�

�cos2� � a22sin2��2� � �2�nsd

2 �, (16)

where �c � nsd/c is a unit delay of the single cavity. In Eq. (16),we can see clearly that chromatic dispersion arises from the twosets of phase dispersive GTRs.

At resonance, in order to have zero chromatic dispersion, thefollowing equation has to be satisfied:

a1�a12 � 1� � a2�a2

2 � 1� � 0. (17)

Eq. (17) is the condition for ripple-free group delay response. Bysubstituting Eq. (5) into Eq. (17), one can obtain the optimum

reflectance for the mirrors by numerical calculation. The relation-ship between r1 and r2 for ripple-free group delay is shown as asolid line in Figure 6, where the optimum condition for the flat-topspectrum is depicted by a dotted line. From Figure 6 we can seethat r1 changes very slowly with r2 for ripple-free group delay,while for the flat-top spectrum r1 changes almost linearly with r2.It is noted that the two curves have an intersection point P. At thispoint, one may expect that both ripple-free group delay and flat-topspectrum may be achieved simultaneously. Figure 7 shows thegroup delay response for a few values of r2, with r1 equal to �0.4.One can see clearly that when r2 deviates from its optimum valueof 0.067 (see Fig. 6), the group delay loses its flat response. Whenit is larger than its optimum value, its response has two extremevalues. When it is smaller than its optimum value, it has a roundedresponse. When r2 has the optimum value, the group delay re-sponse has a wide ripple-free flat region. This is because the groupdelay of the filter comes from the sum of two phase shifts of theGTRs, where the two GTRs have an opposite chromatic dispersion

Figure 5 Reflectance of M2 and passband bandwidth as a function ofreflectance of M1 for different ripples (solid, dashed, and dotted linescorrespond to the ripples at 0.05 dB, 0.15 dB, and 0.25 dB, respectively) Figure 6 Reflectance of mirrors at optimum conditions (solid line:

ripple-free group delay; dotted line: flat-top spectral response)

Figure 7 Group-delay responses for different r2 with fixed r1 at �0.4(dashed, solid, and dotted lines correspond to the values of r2 at 0.16,0.067, and 0.03, respectively)

20 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 1, July 5 2004

slope over a certain passband region; the second GTR with a smallreflectance serves as a chromatic dispersion compensator [13],giving rise to a ripple-free region. It should be noted that thebandwidth of the ripple-free region may be improved by using aGTR with more cavities, say, three or four.

CONCLUSION

We have presented analytic expressions for the transmission andgroup-delay responses of an SLI incorporating two GTRs formedby a two-cavity etalon. The two-cavity etalon may be replaced byFBGs to form all-fiber optical devices. The general optimumconditions for a quasi-flat-top spectrum are derived. The optimumreflectance of the mirrors for the flat-top spectrum is variable foran SLI with two GTRs, compared to that with one GTR whichoccurs at �1/3. If the reflectivity of M1 increases, the steepnessincreases and there is optimum reflectivity, which will give amaximum bandwidth at the flat-top condition. Relaxing the pass-band ripple will enhance passband bandwidth. As the ripple in-creases, the optimum reflectivity increases. We have also derivedthe general ripple-free condition for the group-delay response. Thenumerical results show that by properly choosing the reflectance ofthe mirrors, filters with the desirable characteristics of flat-topspectral response and ripple-free group-delay response can bedesigned. Such filters may be very useful for applications inhigh-data-rate DWDM systems.

ACKNOWLEDGMENT

This work is supported by the Natural Sciences and EngineeringResearch Council of Canada.

REFERENCES

1. L. Wei and J.W.Y. Lit, Phase-shifted Bragg-grating filters with sym-metrical structures, J Lightwave Technol 15 (1997), 1433–1437.

2. L.R. Chen, Design of flat-top bandpass filters based on symmetricphase-shifted long-period fiber gratings, Opt Comm 205 (2002), 271–276.

3. M. Oguma, K. Jinguji, T. Kitoh, T. Shibata, and A. Himeno, Flat-passband interleave filter with 200 GHz channel spacing based onplanar lightwave circuit-type lattice structure, Electron Lett 36 (2000),1299–1300.

4. M. Oguma, T. Kitoh, K. Jinguji, T. Shibata, A. Himeno, and Y.Hibino, Passband-width broadening design for WDM filter with lat-tice-form interleave filter and arrayed-waveguide gratings, IEEE Pho-ton Technol Lett 14 (2002), 328–331.

5. Y. Lai, W. Zhang, and J.A.R. Williams, All-fibre multichannel flattopfilter based on coherent fibre delay line structure, Electron Lett 38(2002), 473–475.

6. B.B. Dingel and T. Aruga, Properties of a novel noncascaed type,easy-to-design, ripple-free optical bandpass filter, J Lightwave Tech-nol 17 (1999), 1461–1469.

7. C. Hsieh, R. Wang, Z.J. Wen, I. McMichael, P. Yeh, C. Lee, and W.Cheng, Flat-top interleavers using two Gires–Tournois etalons asphase-dispersive mirrors in a Michelson interferometer, IEEE PhotonTechnol Lett 15 (2003), 242–244.

8. X. Shu, L. Yu, D. Zhao, L. Zhang, K. Sugden, and I. Bennion,Transmission characteristics of Sagnac interferometers based on fiberBragg grating, J Opt Soc Am B 19 (2002), 2770–2779.

9. X. Shu, D. Zhao, L. Zhang, K. Sugden, and I. Bennion, Near-constanttime delay response of fiber grating Sagnac loop filters, IEEE PhotonTechnol Lett 14 (2002), 1127–1129.

10. Q. Zeng, Delay behaviors of phase-shifted Bragg grating filters, Elec-tron Lett 34 (1998), 1098–1100.

11. P. Yeh, Optical waves in layered media, Wiley, New York, 1988,p 150.

12. R. Slavik and S. LaRochelle, Large-band periodic filters for DWDMusing multiple superimposed fiber Bragg gratings, IEEE Photon Tech-nol Lett 14 (2002), 1704–1706.

13. D.J. Moss, M. Lamont, S. McLaughlin, G. Randall, P. Colbourne, S.Kiran, and C.A. Hulse, Tunable dispersion and dispersion slope com-pensators for 10 Gb/s using all-pass multicavity etalons, IEEE PhotonTechnol Lett 15 (2003), 730–732.

© 2004 Wiley Periodicals, Inc.

ANTENNA FAR-FIELD PATTERNRECONSTRUCTION USINGEQUIVALENT CURRENTS ANDGENETIC ALGORITHMS

J. R. Perez and J. BasterrecheaDepartment of Communications EngineeringETSIITUniversity of CantabriaAvda. de Los Castros s/n39005 Santander, Spain

Received 13 December 2003

ABSTRACT: A method for computing the far-field pattern of an an-tenna under test (AUT) from near-field data is presented. Based on theequivalence principle, the AUT is replaced by equivalent magnetic sur-face currents, whose components are optimized using microgenetic algo-rithms. Results of reconstructed far-field patterns from synthetic near-field samples, a comparison to the approach using classical methods,and an analysis of the method’s performance and limitations are in-cluded. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 42:21–25, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20195

Key words: genetic algorithms; near-field; far-field; equivalence princi-ple

1. INTRODUCTION

Near-field antenna measurement techniques are a reliable alterna-tive to far-field testing. The near-field test system measures theenergy in the radiating near-field region of the antenna under test(AUT) and converts those measurements into the far-field resultusing a suitable near-field to far-field (NF-FF) transformationmethod [1]. Among the most commonly used near-field test setupconfigurations (planar, cylindrical, and spherical), near-field mea-surements on planar surfaces have attracted the most attention, dueto the degree of mathematical and computational simplicity [2].Focusing on the planar geometry, the classical fast Fourier trans-form (FFT)-based approach is the most widely used NF-FF trans-formation method [3]. However, the accuracy of the far-fieldresults depends on the planar grid size and spacing, and on theAUT directivity, thus making it necessary to look for more versa-tile solutions.

In this paper, the application of the second equivalence princi-ple along with a genetic algorithm (GA)-based optimization pro-cess proves to be an efficient technique when applied to NF-FFtransformation issues. Basically, the AUT is replaced by equiva-lent magnetic currents (EMCs) placed on a fictitious surface en-closing the antenna aperture and each component of the EMC isoptimized using a binary-nature micro-genetic algorithm (MGA).Antenna far fields can then be easily computed from the optimizedEMC. This approach has no severe restrictions with regard to thenear-field data, sampling spacing, or near-field geometry, thus

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 42, No. 1, July 5 2004 21