Figure 5 Measured radiation pattern at 20 GHz. }} copolariza-tion, - - - cross polarization

France TelecomrCNET La Turbie for providing experimen-tal and computer resources.

REFERENCES

1. C. Dourthe, C. Pichot, and J.-Y. Dauvignac, ‘‘MicrowaveDiffraction Tomography Algorithm for Buried Objects with Arbi-trary Space and Time Near-Field Illumination,’’ PIERS’97, Cam-bridge, MA, 1997, p. 735.

2. E. Gazit, ‘‘Improved Design of the Vivaldi Antenna,’’ Proc. Inst.Elect. Eng., Vol. 135, Pt. H, No. 2, 1988, pp. 89]92.

3. J. D. S. Langley, P. S. Hall, and P. Newham, ‘‘Novel Ultrawide-Bandwidth Vivaldi Antenna and Low Crosspolarisation,’’ Elec-tron. Lett., Vol. 29, No. 23, 1993, pp. 2004]2005.

4. P. Ratajczak, P. Brachat, and J.-L. Guiraud, ‘‘Rigorous Analysisof Three Dimensional Structures Incorporating Dielectrics,’’IEEE Trans. Antennas Propagat., Vol. 42, Aug. 1994, pp.1077]1088.

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

PERFORMANCE OF S-BENDS FORINTEGRATED-OPTIC WAVEGUIDESApurva Kumar1 and Sheel Aditya11 Department of Electrical EngineeringIndian Institute of Technology, DelhiHauz Khas, New Delhi 110016, India

Recei ed 9 April 1998

ABSTRACT: The performance of S-bends, which are used for connect-ing wa¨eguides offset with respect to each other, is ¨ery important inintegrated-optic wa¨eguide circuits. Se¨eral approaches for the design ofS-bends ha¨e been reported in the literature. These include, in particular,designs based on simple sinusoidal cur es as well as relati ely compli-cated, but more general, polynomial cur es. This paper reports a compar-ison of the calculated pure bend loss of such S-bends for a typicalsilica-based wa¨eguide operating at a wa¨elength of 1550 nm. It is shownthat the simple cosine S-bands yield a performance which is close to thatobtained from the general polynomial cur e-based S-bends. Design cur esare also presented for these two types of S-bends. The calculations arebased on the analytical expression for bend loss proposed by Marcuse.Q 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 19:289]292, 1998.

Key words: S-bends; integrated-optic wa¨eguides

INTRODUCTION

In integrated-optic circuits such as directional couplers,switches, multiplexers, and power splitters, it is often re-quired to increase the separation between adjacent wave-guide ports to accommodate standard optical fibers for thepurpose of pigtailing. This requires a connection betweenstraight waveguides which are offset with respect to eachother. It is desirable that such a connection, typically calledan S-bend, introduce low additional loss and only moderateincrease in device length. Therefore, the design and perfor-mance of S-bends is important in determining the cost andperformance of the overall circuit.

For connecting offset integrated-optic waveguides, cornerbends and multisection corner bends have been studied inw x w x1]4 . In 3 , it was shown that, for large lateral offsets, anS-bend based on circular arcs performs better than the onecomposed of straight sections with sharp corners. A furtherimprovement in bend transmission was achieved by removingdiscontinuities in the first- and second-order derivations tominimize loss due to abrupt curvature changes at thestraight-curved transitions and due to curvature reversals at

w xthe midpoint of the S-bend 5]6 . This led to simple designsbased on sine and cosine curves. More general designs, whichcan be optimized for bend loss or radiation loss, have also

w xbeen considered in 7]8 .The use of lateral offsets for reducing the transition loss

due to abrupt curvature changes for S-bends based on circu-w xlar arcs has been suggested in 9 . A number of other studies

on bent waveguides have been reported in the recent litera-w xture, e.g., 10]11 , and it remains an active area of research.

Typically, the transition loss is calculated using the overlapintegral, and the bend loss is calculated using the expressions

w x w xdeveloped by Marcuse 12 or Marcatilli 13 for the attenua-tion constant of curved waveguides.

Recently, a general approach to optical waveguide pathw x w xdesign has been reported in 14 . In 14 , a general polynomial

Ž .family of curves P-curves has been suggested to link twoarbitrary points on the optical circuit, respecting the given

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 19, No. 4, November 1998 289

slope and curvature at both ends to minimize transition loss,and still offering one free parameter enabling a bending lossminimization. Further, a loss minimization procedure hasbeen introduced which, in conjunction with this general fam-ily of curves, can, in certain cases, produce a substantial lossreduction compared with the concatenation of line segmentsand arcs of circles. In general, since the curves used have acontinuously changing curvature, this procedure provides fora continuous change in the waveguide width to effect acancellation of the infinitesimal mode offsets due to theinfinitesimal difference in curvature.

While the above method is very general, for the simplecase of connection between offset, parallel straight wave-guides, a comparison between the results obtained using thismethod and the S-bends based on sine-cosine functions wouldappear to be of interest since the latter designs, employingconstant width, are very simple and can be easily automated.Such a comparison does not appear to be reported in theliterature. This paper reports on such a comparison of theo-retical bend loss calculated for a typical silica-based wave-guide operating at 1550 nm. Design curves are also presentedfor the cosine S-bends and for the optimum S-bends basedon P-curves.

METHOD OF CALCULATION

Ž .The waveguide geometry considered is shown in Figure 1 aw xand is the same as in 14 . For the purpose of calculations,

the channel waveguide is converted into a step-index slabw xwaveguide by using the effective index method 15 . The

Ž .equivalent slab waveguide is shown in Figure 1 b for wave-lengths of 1310 and 1500 nm. The waveguide half width is r,the core index is n , and the cladding index is n . Theco cl

Ž .Figure 1 a Geometry and other parameters of the partially etchedŽ .ribbed waveguide. b Equivalent symmetric slab waveguide; numbers

in parentheses denote the wavelength in microns

Figure 2 S-bend connecting two straight waveguides with a lateraloffset of X and longitudinal offset of Zs s

general shape of the S-bend considered is shown in Figure 2,with the lateral offset X and the longitudinal offset Zs sbetween the input and output ports.

The expressions for the S-bends based on sine and cosinefunctions are, respectively,

X X 2ps sŽ . Ž .x z s z y sin z 1ž /Z 2p Zz s

X psŽ . Ž .x z s 1 y cos z . 2ž /2 Zs

For an input power P , assuming that the power remaining atoa given point is

sŽ . Ž . Ž .P s s P exp y a s9 ds9 , 3H0 ½ 5

0

the integrated attenuation constant for a curve of total arclength L is

L Ž . Ž .I s a s9 ds9. 4H0

The bend loss in decibels is obtained as

10 IŽ . Ž .bend loss dB s . 5

ln 10

w xFollowing 12 , for weak-guidance simplification, the attenua-tion constant a , representing pure bend loss or radiationloss, is given by

1 U 2W 2 4 W 3D Rc2W Ž .a s e exp y 62 2 2ž /Ž . 3 rkn r 1 q 2W V Veff

where the various symbols have the following expressions:

n2 y n2co cl Ž .D s 722nco

' Ž .V s krn 2D 8co

2 2 Ž .'U s kr n y n 9co eff

2 2 Ž .'W s kr n y n . 10eff cl

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 19, No. 4, November 1998290

The effective index n for the fundamental TE mode iseffcalculated from the eigenvalue equation

Ž .W s U tan U. 11

Ž . Ž .In 8 , k is the free-space wavenumber, and R in 6 is thecradius of curvature at different points.

For estimating the bend loss of optimum S-bends basedŽ .on polynomial curves, the integrated attenuation I in 4 is

Ž .treated as I L where L is the free parameter available foroptimization. L is determined subject to the condition of

w xminimum loss 14 :

Ž .dI LŽ .s 0. 12

dL

Calculations are carried out for both constant width andvariable width curves. For all calculations reported here, thetransition loss between the straight and the curved waveguidehas been neglected. The parent straight waveguide is alsoconsidered to be lossless. Also, for determining the free

Ž .parameter L, in addition to 12 , manual fine-tuning as re-w xported in 14 has not been done.

RESULTS AND DISCUSSION

Table 1 lists the value of total attenuation or bend loss indecibels for a lateral offset of 250 mm, for different values oflongitudinal offset Z . The S-bend designs considered aresbased on sin, cos, and P-curves. The results have beencalculated for l s 1550 nm. The table shows that the sinS-bends have the highest loss. The cosine S-bends have a lossperformance which is midway between the two types ofP-curve-based S-bends; while the constant-width P curvesŽ .P show a higher loss than that for cosine S-bends, thec

Ž .variable-width P curves P show a lower loss. Similarwcalculations have been carried out for different values oflateral offset X , as well as for other wavelengths such ass1310 nm. The general behavior mentioned above is repeatedin all of these results. Table 1 also shows that, as thelongitudinal offset increases, all types of S-bends show arapid decreasing loss. We have also observed that for l s1310 nm, other parameters being the same, the loss is less ascompared to that for l s 1550 nm. These are well-knownaspects of bend loss in curved waveguides.

( )TABLE 1 Bend Loss dB for Different Values of LongitudinalOffset Z , for Lateral Offset X = 250 mm and l = 1550 nm,s sfor S-Bends Based on Sine, Cosine, P-Curves with

( )Constant Width P , and P-Curves with Variablec( )Width Pw

LongitudinalOffsetŽ .Z mm z-sin z cos z P Ps c w

2800 5.0156 2.2191 3.1417 1.69773000 3.3870 1.2725 2.0503 0.90433200 2.2182 0.7356 1.2906 0.48093400 1.4093 0.4095 0.7844 0.25373600 0.8690 0.2196 0.4607 0.13213800 0.5202 0.1135 0.2616 0.06764000 0.3023 0.0565 0.1437 0.03394200 0.1706 0.0271 0.0764 0.01674400 0.0935 0.0125 0.0393 0.00804600 0.0498 0.0056 0.0195 0.0037

Figure 3 Variation of bend loss as a function of longitudinal offsetZ for S-bends based on cosine curve and variable-width P-curvesŽ .P for different lateral offsetsw

For X s 250 mm and Z s 4000 mm, the loss for P-s scurve-based S-bends in Table 1 is less than that reported inw x14, Table V for the same parameters. The latter result hasbeen obtained using the beam propagation method. Further,this difference may also occur because the transition lossbetween then straight and curved waveguides is ignored here.

Figure 3 presents the loss curves for cosine S-bends forl s 1550 nm, for values of lateral offset 125, 250, 500, and1000 mm, as a function of the longitudinal offset. Similar

Ž .information for variable-width P-curve-based S-bends P iswalso presented in the same figure. For the same bend loss, thecosine bends are approximately 5% longer. These resultsshould be useful in the design of a number of integrated-opticdevices. Similar results can be calculated for the TM mode by

Ž .replacing 1 by the corresponding eigenvalue equation.

CONCLUSION

Calculated results have been presented to compare the bendloss of well-known sine- and cosine-based S-bends with thatof newly proposed S-bends based on P-curves. It is shownthat the cosine S-bends have a performance which is betterthan that of the constant-width P-curve S-bends. On theother hand, the variable-width P-curve S-bends indicate aperformance which is better than that of cosine S-bends. Inaddition to the above results, design curves have been pre-sented for two types of S-bends for a typical silica basedwaveguide operating at 1550 nm.

ACKNOWLEDGMENT

The assistance of Mr. Mohammad Zaffar in some of thecalculations is gratefully acknowledged.

REFERENCES

1. H. F. Taylor, ‘‘Power Loss at Directional Changes in DielectricWaveguides,’’ Appl. Opt., Vol. 13, Mar. 1974, pp. 642]647.

2. D. Marcuse, ‘‘Length Optimization of an S-Shaped TransitionBetween Offset Optical Waveguides,’’ Appl. Opt., Vol. 17, Mar.1978, pp. 763]768.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 19, No. 4, November 1998 291

3. L. D. Hutcheson, I. A. White, and J. J. Burke, ‘‘Comparison ofBending Losses in Integrated Optical Circuits,’’ Opt. Lett., Vol. 5,June 1980, pp. 276]278.

4. L. M. Johnson and F. J. Leonberger, ‘‘Low-Loss LiNbO Wave-3guide Bends with Coherent Coupling,’’ Opt. Lett., Vol. 8, Feb.1983, pp. 111]113.

5. V. Ramaswamy and M. D. Divino, ‘‘Low-Loss Bends for Inte-grated Optics,’’ Proc. Conf. on Lasers and Electrooptics, Washing-ton DC, 1981, paper THP1.

6. W. J. Minford, S. K. Korotky, and R. C. Alferness, ‘‘Low-LossTi:LiNbO Bends at s 1.3 m,’’ IEEE J. Quantum Electron., Vol.3QE-18, Oct. 1982, pp. 1802]1806.

7. R. Baets and P. E. Lagasse, ‘‘Loss Calculation and Design andArbitrarily Curved Integrated-Optic Waveguides,’’ J. Opt. Soc.Amer., Vol. 73, Feb. 1983, pp. 177]182.

8. F. J. Mustieles, E. Ballesteros, and P. Baquero, ‘‘TheoreticalS-Bend Profile of Optimization of Optical Waveguide RadiationLosses,’’ IEEE Photon. Technol. Lett., Vol. 5, May 1993, pp.551]553.

9. T. Kitoh, N. Takato, M. Yasu, and M. Kawachi, ‘‘Bending LossReduction in Silica-Based Waveguides by Using Lateral Offsets,’’J. Lightwa¨e Technol., Vol. 13, Apr. 1995, pp. 555]562.

10. C. Seo and J. C. Chen, ‘‘Low Transactions Losses in Bent RibWaveguides,’’ J. Lightwa¨e Technol., Vol. 14, Oct. 1996, pp.2255]2259.

11. M. Heinbach, M. Schienle, A. Schmid, B. Acklin, and G. Muller,‘‘Low-Loss Bent Connections for Optical Switches,’’ J. Lightwa¨eTechnol., vol. 15, May 1997, pp. 833]837.

12. D. Marcuse, Light Transmission Optics, Van Nostrand Reinhold,New York, 1973.

13. E. A. J. Marcatilli, ‘‘Bends in Optical Dielectric Guides,’’ BellSyst. Tech. J., Vol. 48, Sept. 1969, pp. 2103]2132.

14. F. Ladouceur and P. Labeye, ‘‘A New General Approach toOptical Waveguide Path Design,’’ J. Lightwa¨e Technol., Vol. 13,Mar. 1995, pp. 481]492.

15. G. B. Hocker and W. K. Burns, ‘‘Mode Dispersion in DiffusedChannel Waveguides by the Effective Index Method,’’ Appl. Opt.,Vol. 16, Jan. 1977, pp. 113]118.

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

RADIATION OF SMITH – PURCELLTYPE BY A PERIODIC STRIPGRATING OVER A GROUNDEDDIELECTRIC SLABCheol Hoon Lee,1 Jong Ig Lee,1 Ung Hee Cho,1 andYoung Ki Cho11 Department of Electronics EngineeringKyungpook National UniversityTaegu, 702-701, Korea

Recei ed 24 March 1998

ABSTRACT: The radiation problem of the Smith]Purcell type is consid-ered for the case where a line charge mo¨es, at a constant speed, parallelto a periodic strip grating o¨er a grounded dielectric slab. Numericalresults pertaining to the radiation intensities and magnitudes of spaceharmonics for ¨arious cases of geometrical parameters and dielectricpermitti ities are presented, based upon which some discussions on theoccurrence of the sharp maximum radiation intensity of the n s y1space harmonic and the characteristic beha¨ior of neighboring spaceharmonic magnitudes under such a condition of maximum radiation aregi en. The relationship between the k d y bd diagram and the0Smith]Purcell radiation are also briefly discussed. Q 1998 John Wiley& Sons, Inc. Microwave Opt Technol Lett 19: 292]296, 1998.

Key words: reflection grating; Smith]Purcell radiation; leaky wa¨e

1. INTRODUCTION

Ž . w xSince the phenomenon of SP Smith]Purcell radiation 1was observed in 1953, many researchers have contributed to

w xunderstanding the nature of this radiation, as listed in 2]4 .w xAmong them, Palocz and Oliner 2, 3 pointed out a strong

similarity between SP radiation and leaky wave antennas, byuse of which they solved the problem of SP radiation for asheet electron beam of finite thickness passing between two

w xparallel planar strip transmission gratings, and Hessel 4dealt with the problem of radiation from an electric current

Žsheath above an SMRS sinusoidally modulated reactance.surface which was used as a model of an optical reflection

w xgrating. For more details on other prior works, refer to 2]4 .However, because the SMRS is a kind of an oversimplified

w xmodel as the author 4 mentioned, the study of such a modelis not expected to provide much practically useful informa-

Žtion although it sheds lights on the properties common to a.large class of grating to some extent . So the motivation of

this work is the expectation that the study of the practicalw x Ž .reflection grating 5 as shown in Fig. 1 , which also has been

widely used as a leaky-wave antenna structure, can give morepractical results than that of the SMRS model.

The purpose of this work is to examine the radiationcharacteristics of the SP type of the present geometry inFigure 1 in connection with the radiation characteristics ofthe leaky-wave antenna problem.

2. FORMULATION

w xConsider a line charge density of q coulombrm moving at aw xconstant velocity ¨ mrs in the y-direction above the strip0

grating plane over a grounded dielectric slab as shown inFigure 1, where d, a, and h denote a period, slot width, anddielectric thickness, respectively, and the other symbols formaterial constants follow the usual meanings.

The current density in the frequency domain is J syyjk Žcr ¨ 0 . y Ž .qe d z y z where d is the Dirac delta function,0

k s v m e s 2prl, and c is the velocity of light in free'0 0 0space. The time factor e jv t is suppressed throughout.

In each region, the total magnetic fields are expressed as

1ŽI . y2 jg z y jg Ž zyz . y j b y01 0 01 0 0Ž .H s 1 y e e e x02

`yjg z y j b yn1 n Ž .q A e e x , z ) z 1Ý n 0 0

nsy`

ŽII. y jg 01 z0 Ž . yj b 0 yH s yqe cos g z e x01 0

`yjg z y j b yn1 n Ž .q A e e x , z ) z ) 0 2Ý n 0 0

nsy`

Figure 1 Radiation geometry

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 19, No. 4, November 1998292