APPLICATION OF WINDOW THEORY TOOPTIMIZATION OF FIBER BRAGGGRATINGS FOR WAVELENGTH-SELECTABLE ANTENNABEAM STEERINGAlex V. Petrov,1 Shizhuo Yin,1 and Francis T. S. Yu11 Department of Electrical EngineeringPennsylvania State UniversityUniversity Park, Pennsylvania 16802

Recei ed 23 September 1997

( )ABSTRACT: The profile of fiber Bragg gratings FBGs utilized in fibergrating prism and other wa¨elength-selecti e phased-array beamformerscan be optimized using the results of window theory. In this paper, thewindow formalism is adopted to quantify the relationship between theFBG profile and its spectral selecti ity. It is shown how application ofoptimum and suboptimum windows affects the reflecti ity, bandwidth,and sidelobe le¨el of FBG. Also, the impact of taper on the group-delaynonuniformity, an essential issue for true time delay processors, isdiscussed. Q 1998 John Wiley & Sons, Inc. Microwave Opt TechnolLett 17: 161]164, 1998.

Key words: optical fiber; Bragg gratings; antenna beam steering; phasedarrays

1. INTRODUCTION

Over the past few years, the concept of fiber grating prismŽ .FGP has evolved as a hardware-compressive approach to

Ž . w xtime steering of phased-array antennas PAAs 1]3 . InFGP, fiber delay lines supply the PAA subarrays with pro-gressively delayed signal replicas. At the subarrays, the de-layed signals are phase shifted and applied to the radiatingelements.

Wavelength-selective fiber delay lines are central to theFGP architecture. For a transmit PAA, the delay lines in-clude a tunable laser source, external modulator, opticalcirculators, single-mode fibers with permanent fiber Bragg

Ž . Ž .gratings FBGs , and photodetectors Fig. 1 . The RF signal isimposed onto an optical carrier with an electro-optic modula-tor and directed toward the FBG array. The delay is insertedby tuning the laser wavelength to have the light reflect fromthe appropriately located FBG. Finally, the reflected signal is

Figure 1 Wavelength-selective fiber delay line for a subarray oftransmit PAA

routed to the PAA subarray. The described layout can bemodified to transmit and receive multiple antenna beamsw x2, 3 .

Insertion loss and crosstalk incurred by the FGP beam-former depend on the spectral selectivity of FBG. In thispaper, the theory of optimum windows is applied to relate theFBG performance to its taper. Of particular interest are thefollowing FBG parameters:

Ž .1. Reflectivity, which affects the insertion loss IL of theFGP optics:

Ž 2 . Ž .IL s 10 log 1 y ht t 1mod circ

where h is FBG reflectivity; t ( 0.15 ??? 0.25 andmodt ( 0.8 are transmittances of the modulator andcirc

w xcirculator, respectively 2 .Ž .2. Full bandwidth at half maximum FWHM , which must

be commensurate with the bandwidth of the modulatedRF waveform.

Ž .3. Sidelobe level SLL , which is proportional to the ex-tent of optical crosstalk and RF signal distortion. SLLis defined as the maximum reflectivity outside the FBGchannel bandwidth:

Ž � Ž . Ž .4.SLL s y10 log max h b rh b ,0

< < Ž .b y b G Db 20 ch

where b is the propagation constant, b is the reflec-0Ž .tivity peak, Db is the channel separation, and h b isch

the FBG reflectivity at arbitrary wavelength.

Along with these parameters, the group-delay distortionhas to be analyzed for true time-delay steering. The distor-tion is caused by mode depletion in FBG, which skews thedistribution of the incident wave and makes the phase of thereflected signal nonlinear.

In the proximity of the reflectivity peak, group delay canbe decomposed as follows:

Ž . Ž . Ž .Ž . Ž .t v s t v q c 0 v v y v 3g g 0 0 0

where v and v are angular frequencies of interest and at0Ž . Ž .the reflectivity peak; c v and t v are the phase andg

Ž .group delay of reflected signal. From 3 , the upper bound onthe delay error is

Ž .dt F dt FWHMr2 4g g1

� Ž .4 < <where dt s max c 0 v over v y v F FWHMr2 is theg1 0maximum distortion of group delay per unit bandwidth. Forsquint-free operation of PAA, the following condition musthold:

Ž .dt D f < 1rf 5g1 RF c

where f and D f are, respectively, the central frequency ofc RFthe RF signal and its bandwidth.

2. FBG AND FORMALISM OF WINDOW THEORY

Ž .FBG is similar to a finite-impulse response FIR filter in thata grating period is a tap in the transmission line, and areflected wave is thus a series of the delayed and weightedreplicas of the input signal. Therefore, window theory pro-

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 17, No. 3, February 20 1998 161

vides an analytical tool to optimize the FBG taper under thedesign constraints. The minimum length condition has beenselected as the optimization criterion. It results in highertemporal resolution, easier fabrication, and less stringentconstraints on the laser coherent length.

Distribution of the refractive index within FBG is given by

Ž . Ž . Ž . Ž .n z s n q n w z cos 2p zrL 60 1

where n and n are the refractive index of the fiber core and0 1Ž .its maximum perturbation, respectively, w z is the FBG

taper, and L is the FBG period.Distributed reflection due to impedance mismatch over

the infinitesimal segment dz is

Ž . Ž . Ž . Ž .dr z s n9 z dzr2n z . 7

Ž . y1With current FBG fabrication techniques, w9 z < L ,Ž . Ž .and Eqs. 6 ] 7 reduce to

p n1Ž . Ž . Ž . Ž .dr z s y sin 2p zrL w z dz. 8n L0

A specific requirement on the FBGs for PAA applicationsŽ .is high SLL in excess of 20 dB to avoid crosstalk between

the delays. Hence, in the sidelobe regions, depletion of theincident wave is insignificant, and its magnitude is close touniform over the FBG length. Also, since n F 10y3n , the1 0

w xsecondary reflections can be neglected 4 . So, for a sidelobe,the FBG reflection coefficient is an integral of phase-retarded elemental waves of the generic form

Ž . Ž Ž .. Ž .dr z exp yj b z q Lr2 . 9

Ž . Ž .Substituting 8 for 9 and integrating over the FBGlength yield the following result:

p n Lr21Ž . Ž . Ž . Ž .r b s j exp yj bLr2 w z exp yj2Db z dzH4L n yLr20

Ž .10

where Db s b y prL is Bragg mismatch.Ž .From 10 , the problem of FBG optimization has been

Ž .reduced to finding the shortest window w z such that itsŽ . Ž .Fourier transform W n meets the SLL specification 2

Ž � Ž . Ž .4. < <y10 log max W 2Db rW 0 G SLL, Db G Db .ch

Ž .11

The optimum solution has been found for FIR filterw xapplications, and is known as a Kaiser window 6 :

2'Ž . Ž . Ž . < < Ž .w z s I a 1 y 2 zrL rI a , z F Lr2 12k 0 0ž /Ž .where I a is a zero-order modified Bessel function, and a0

is the shape parameter calculated as follows:

Ž .0.1102 SLL y 8.7 , SLL ) 50 dB¡0.4Ž . Ž .0.5842 SLL y 21 q 0.07886 SLL y 21 ,~ Ž .a s 13

21 - SLL F 50 dB¢0, SLL F 21 dB.

Ž .Since the profile given by 12 may be difficult to imple-ment, it can be approximated with some distribution thatwould be compatible with the FBG fabrication technique. Forexample, an FBG array can be fabricated with the transverse

w xholographic technique 6, 7 , that is, using interference ofcrossed Gaussian beams inside the photosensitive core ofGe-doped fiber. In this case, the Kaiser window can beclosely approximated with the truncated Gaussian windowŽ .TGW :

22Ž . Ž Ž . . < < Ž .w z s exp yz r2 s L , z F Lr2 14g

where s is the ratio of standard deviation and FBG length. Ifcollinear beams are used for FBG recording, the raised-cosine

w xlaw is a more suitable approximation 8 .

3. NUMERICAL COMPUTATIONS

To characterize the performance of tapered FBGs, coupled-w xmode equations 9 were solved numerically for Kaiser and

corresponding TGW profiles with the shape parameters asgiven in Table 1.

The two windows exhibited close performance in terms ofreflectivity, FWHM, SLL, and group-delay distortion. For thisreason, only the results for TGW are shown in Figures 2]5.Notice that SLL values are less than the prespecified ones inTable 1. For small values kL, this discrepancy is due to SLL

Žnormalization by reflectivity the frequency response of a.digital filter is unit at its maximum . Also, as kL increases,

the secondary reflections become more significant and affectthe sidelobes. However, the SLL curves are nearly horizontal,so that the discrepancy can be accounted for in advance.

From the delay nonuniformity curves, phase nonlinearityis insignificant for applications with bandwidth up to 5]10GHz depending on the tolerance on the delay ripple. Forhigher frequencies, tapering can be considered as a means toimprove the group-delay uniformity.

TABLE 1 Window Parameters

Specified Kaiser Window TGWŽ .SLL Value dB Parameter a Parameter s

a21 0.000 4 130 2.166 0.37640 3.395 0.27750 4.551 0.235

a Special case of rectangular window.

Figure 2 FBG reflectivity as a function of kL

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 17, No. 3, February 20 1998162

Figure 3 FBG SLL as a function of kL

Figure 4 FBG FWHM as a function of kL

Figure 5 Group-delay distortion as a function of kL

The FBG profile can be synthesized as follows. First, themaximum value of kL given FWHM is estimated usingFigure 4. Then, the shape parameter is found from the SLLspecification and Figure 3. Finally, reflectivity and groupdelay are compared with the specified values. If the specifica-tions are not met, kL or s should be modified, and anotheriteration is run.

4. EXAMPLE OF BEAMFORMER DESIGN

In order to illustrate the FBG design procedure, let usconsider the design of a beamforming network for a 1-D40-element PAA with parameters as given in Table 2. Atemporal resolution of "1r6 f had been found to be suffi-c

TABLE 2 PAA Parameters and FBG Specifications

Parameter Symbol Value Unit

Central frequency f 5 GHzcBandwidth D f 2.5 GHzSteering angle a 60 deg

Ž .Number of subarrays time steered 20 NrAŽ .Number of elements phase shifted 2 per subarray NrA

Reflectivity h 0.80 NrASidelobe level SLL 40 dB

cient to maintain a quality antenna beam. Also, the spacingŽbetween the PAA elements was chosen to be 2.4 cm half the

.minimum RF wavelength to avoid grating lobes. It can bew xshown that such array calls for 78 FBGs spaced by 6.7 mm 3 .

In order to accommodate a double-sideband amplitude-modulated RF signal, an optical bandwidth of at least 2 f qcD f s 12.5 GHz must be allocated for each FBG. To allow forcertain wavelength instability, a greater value of FWHM s 30GHz was assumed. Then, according to Figure 4, the couplingconstant must satisfy 0 - k F 2.8rL. With that, to meet theSLL specification with the highest reflectivity, the shapeparameter should be s s 0.22. From the solution tocoupled-mode equations, it was found that s s 0.22 wouldresult in SLL s 39 dB. With a slight correction, s s 0.217yielded the desired performance of h s 81.4%, FWHM s40.4 GHz, and SLL s 40.1 dB.

For a feasible value of index perturbation of ; 10y4 andwavelength 1.3 mm, the coupling constant would equal k s0.242 mmy1. Thus, the FBG length should be kLrk s2.8r0.242 s 11.6 mm, which would imply an overlap betweenthe adjacent gratings. However, the overlap is under half theFBG length, and overall index perturbation does not exceedthe value at the FBG center, so that such an FBG array canbe practically implemented.

From the solution to coupled-mode equations, it wasfound that channel separation should be Dl s 0.87 nm.Hence, to drive a fiber delay line, a laser with tunable rangeDl = 78 s 70 nm, coherent length of about 2 L s 22.4 mm,and modulation bandwidth 10 GHz is required. Such lasersare expected to soon become available as the technology of

w xsuperstructure grating DBR lasers matures 11, 12 .The FGP irregularities were factored into the PAA model.

First, since group-delay variation over the RF bandwidth isdt D f s 0.4 ? 2.5 s 1 ps < 1rf , phase nonlinearity is notg1 RF ccritical in this example. Therefore, two major error sources,inherent to FGP, are delay quantization and optical crosstalk.The quantization error for each subarray was assumed to bea random variable with uniform distribution over "1r6 f .cThe magnitude error caused by optical crosstalk was simu-lated as a random variable distributed uniformly within

Ž"0.78%. 0.78% is the worst case value for 40 dB crosstalk of78 .78 gratings since 1-0.9999 s 0.0078.

Figure 6 shows the calculated radiation pattern for a PAAdriven with an FGP. For comparison, the pattern for amonochromatic phase-shifted RF signal is shown as an idealtemplate to assess the beam quality. As seen in Figure 6, thedelay-line irregularities resulted in 1 dB lower directivity and; 18 of residual squint. The increased sidelobe level of thebroadband PAA at 88 is attributed to the frequency-dependent null of the subarray pattern, which is caused byphase steering of the subarray elements rather than delay-lineimperfection. Overall, the degradation due to wavelength-

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 17, No. 3, February 20 1998 163

Figure 6 Radiation patterns for broadband and monochromaticsignals

Žselective fiber delay lines is relatively small compared to the.width of the main beam , and can be reduced by enhancing

the temporal resolution of FGP.

5. CONCLUSION

Application of window theory to FBG profile synthesis fol-lows the generic similarity between FBG and FIR filters, andaids in minimizing the FBG length given SLL. Compared tothe case of uniform grating, up to 20 dB improvement in SLLcan be traded for 10]20% downfall in reflectivity by varyingthe window parameters. Also, tapering mitigates the problemof group-delay distortion, and thus aids in the design of truetime delays for antennas in the K -band.a

The use of window theory to optimize FBG is not limitedto FGP processors, and can be applied to wavelength-

w xmultiplexed beamforming networks 12 , distributed sensing,DBR lasers, and FIR filtering of RF signals.

REFERENCES

1. R. A. Soref, ‘‘Fiber Grating Prism for True Time Delay Beam-steering,’’ Fiber Integr. Opt., Vol. 15, 1996, pp. 325]330.

2. H. Zmuda, R. Soref, P. Payson, S. Johns, and E. Toughlian,‘‘Photonic Beamformer for Phased Array Antennas Using aFiber Grating Prism,’’ IEEE Photon. Technol. Lett., Vol. 9, 1997,pp. 241]243.

3. L. J. Lembo, T. Holcomb, M. Wickham, P. Wisseman, and J. C.Brock, ‘‘Low-Loss Fiber Optic Time-Delay Element for Phased-Array Antennas,’’ Proc. SPIE, Vol. 2155, 1994, pp. 13]23.

4. D. M. Pozar, Microwa e Engineering, Addison-Wesley, Reading,MA, 1990.

5. J. F. Kaiser, ‘‘Nonrecursive Digital Filter Design Using theI -sinh Window Function,’’ Proc. IEEE Int. Symp. Circuits Syst.,01974, pp. 20]23.

6. H. Patrick and S. L. Gilbert, ‘‘Growth of Bragg Gratings Pro-duced by Continuous-Wave Ultraviolet Light in Optical Fiber,’’Opt. Lett., Vol. 18, 1993, pp. 1484]1486.

7. C. G. Askins, M. A. Putnam, G. M. Williams, and E. J. Friebele,‘‘Stepped-Wavelength Optical-Fiber Bragg Grating Arrays Fabri-cated in Line on a Draw Tower,’’ Opt. Lett., Vol. 19, 1994,pp. 147]149.

8. S. Yin, B. D. Guenther, and F. T. S. Yu, ‘‘Narrow-Band FilterUsing Multiple Photorefractive Gratings,’’ Proc. SPIE, Vol. 2529,1995, pp. 196]205.

9. A. Yariv, ‘‘Coupled-Mode Theory for Guided-Wave Optics,’’IEEE J. Quantum Electron., Vol. QE-9, 1973, pp. 919]933.

10. M. Y. Frankel, R. D. Esman, and J. F. Weller, ‘‘Rapid Continu-ous Tuning of a Single-Polarization Fiber Ring Laser,’’ IEEEPhoton. Technol. Lett., Vol. 6, 1994, pp. 591]593.

11. H. Ishii, H. Tanobe, F. Kano, Y. Tohmori, Y. Kondo, and Y.Ž .Yoshikuni, ‘‘Broad-Range Wavelength Coverage 62.4 nm with

Superstructure-Grating DBR Laser,’’ Electron. Lett., Vol. 32,1997, pp. 454]455.

12. R. A. Minasian, K. E. Alameh, and N. Fourikis, ‘‘Wavelength-Multiplexed Photonic Beam-Former for Microwave Phased Ar-rays,’’ Microwa e Opt. Technol. Lett., Vol. 10, 1995, pp. 84]86.

Q 1998 John Wiley & Sons, Inc.CCC 0895-2477r98

APPLICATION OF THEMATERIAL-INDEPENDENT PMLABSORBERS TO THE FDTD ANALYSISOF ELECTROMAGNETIC WAVES INNONLINEAR MEDIAAn Ping Zhao1*1 Electronics LaboratoryNokia Research CenterP.O. Box 407FIN-00045 Nokia Group, Finland

Recei ed 26 August 1997

ABSTRACT: To more accurately and effecti ely simulate wa¨es propa-( )gating in nonlinear media with the finite-difference time-domain FDTD

( )method, a material-independent perfectly matched layer MIPML ab-sorber is proposed. Within this absorber, electric displacement D andmagnetic field H are directly absorbed, whereas electric field E is simulta-neously absorbed through the relation of D and E. It is shown that, withthe help of this MIPML absorber, Berenger’s PML can be simply andeffecti ely extended to nonlinear media. Q 1998 John Wiley & Sons, Inc.Microwave Opt Technol Lett 17: 164]168, 1998.

Key words: FDTD; nonlinear media; MIPML absorbers

1. INTRODUCTION

Recently, application of the finite-difference time-domainŽ .method to nonlinear andror dispersive media has been

w xinvestigated 1]4 . It is understood that, to obtain moreaccurate results for open nonlinear problems, developing a

Ž .better absorbing boundary condition ABC is one of themost important issues for these problems. Among the re-

w xported works 1]3 for nonlinear cases, not much attentionwas paid to this issue. Although the extension of Berenger’s

w x w xPML 5 to nonlinear media was investigated 4 for a 1-Dnonlinear problem, more investigations on this topic are still

w x .needed due to the following drawbacks in 4 . 1 The NL-PMLŽmatching conditions i.e., the relation between conductivities

E H . w xs and s used in 4 are directly related to the permittiv-w xity « of the nonlinear material. Even though this does not

Žproduce any difficulties for linear isotropic or diagonal.anisotropic cases, it is rather difficult to derive a satisfactory

PML matching condition for nonlinear cases because, inw xthese cases, the permittivity « varies with time, power.intensity, as well as space. 2 Furthermore, the approach

w xproposed in 4 has difficulties when applied to 2-D andror3-D nonlinear guiding structures since the NL-PML cornersregions cannot be easily constructed when the NL-PML is

w xconstructed in the way reported in 4 . To overcome theabove drawbacks, the material-independent PML absorber,which was successfully applied to arbitrary anisotropic dielec-

* A. P. Zhao was with the Radio Laboratory Department of Electrical andCommunications Engineering, Helsinki University of Technology, FIN-02150 Espoo, Finland.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 17, No. 3, February 20 1998164