REFERENCES 1. W. Khor, M. Bialkowski, A. Abbosh, N. Seman, and S. Crozier, An ultra wideband microwave imaging system for breast cancer detec- tion, IEICE Trans Commun E-90B (2007), 2376–2381. 2. M. Chiappe and G. Gragnani, Vivaldi antennas for microwave imag- ing: Theoretical analysis and design considerations, IEEE Trans Ins- trum Meas 55 (2006), 1885–1891. 3. A. Abbosh, H. Kan, and M. Bialkowski, Compact ultra-wideband planar tapered slot antenna for use in a microwave imaging system, Microwave Opt Technol Lett 48 (2006), 2212–2216. 4. A. Abbosh and M. Bialkowski, A UWB directional antenna for microwave imaging applications, In IEEE Antennas and Propagation Symposium, APS2007, USA, 2007. 5. A. Abbosh and M. Bialkowski, Design of ultra wideband planar monopole antennas of circular and elliptical shape, IEEE Trans Antennas Propag 56 (2008), 17–23. 6. A. Abbosh, M. Bialkowski, and S. Crozier, Investigations into opti- mum characteristics for the coupling medium in UWB breast cancer imaging systems, In IEEE Antennas and Propagation Symposium, San Diego, USA, 2008. 7. S. Davis, H. Tandradinata, S. Hagness, and B. Van Veen, Ultra wideband microwave breast cancer detection: A detection-theoretic approach using the generalized likelihood ratio test, IEEE Trans Biomed Eng 52 (2005), 1237–1250. V C 2009 Wiley Periodicals, Inc. MINIMUM LOSS CONDITION OF A BENT RECTANGULAR HOLLOW WAVEGUIDE J. Yamauchi, S. Harada, and H. Nakano Faculty of Engineering, Hosei University, 3-7-2 Kajino-cho, Koganei, Tokyo 184-8584, Japan; Corresponding author: j.yma@k.hosei.ac.jp Received 11 March 2009 ABSTRACT: The leakage loss of a hollow dielectric waveguide is analyzed analytically and numerically. A minimum loss condition of a bent rectangular hollow waveguide is derived in terms of the refractive index of the cladding using the perturbation method. The validity of the derived minimum loss condition is confirmed by the beam-propagation method. V C 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 2901–2902, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24761 Key words: beam-propagation method; perturbation method; leakage loss; bend loss 1. INTRODUCTION A hollow dielectric waveguide has attracted much attention owing to its unique propagation characteristics [1–6]. The loss of a straight and a bent slab hollow waveguide have already been investigated in detail using the perturbation method [2, 3]. It has also been indicated that the loss of a straight rectangular hollow waveguide is approximated by the sum of the two losses for the TE and TM modes in a two-dimensional (2D) slab hol- low waveguide [3, 4]. These facts motivate us to apply the loss formulae derived by the perturbation method for the slab hollow waveguide to a bent rectangular hollow waveguide. In this article, we explicitly show the loss formula of a bent rectangular hollow waveguide by the perturbation method. Tak- ing advantage of the analytical technique, we will derive a mini- mum loss condition in terms of the refractive index of the clad- ding. We also analyze a straight and a bent rectangular hollow waveguide by the three-dimensional imaginary distance beam- propagation method (3D-BPM), which demonstrates the validity of the derived minimum loss condition. 2. FORMULATIONS We study a rectangular hollow waveguide composed of a low- refractive index material surrounded by a high-refractive index material, as shown in Figure 1(a). The refractive indices of the core and cladding are designated as n co and n cl , and the core width and height are as 2w and 2h, respectively. The hollow waveguide is bent with a radius R in the x–z plane. It has been found that the leakage loss in a straight rectangular hollow waveguide can be estimated by the sum of two losses obtained from straight slab hollow waveguides in the TE and TM modes [3, 4]. It is, therefore, expected that the loss in a bent rectangu- lar hollow waveguide can be approximated by the sum of two losses obtained from the perturbation method [2, 3] for the straight and the bent slab configurations shown in Figures 1(b) and 1(c), respectively. The sum of the two losses leads to the following attenuation constant for the E x mn mode: a x mn ¼ A þ BC ðn co k 0 Þ 2 ffiffiffiffiffiffiffiffiffiffi ffi C À 1 p (1) where k 0 is the free-space wavenumber, A ¼ u 2 n /h 3 , B ¼ u 2 m  c/ w 3 , C ¼ (n cl /n co ) 2 , u m ¼ mp/2, and u n ¼ np/2, in which m and n are mode numbers. For the E y mn mode, we can obtain a y mn by interchanging A and B in Eq. (1). In this article, the order of the fundamental mode is defined as one. The coefficient c includes the effect of bending loss, and becomes unity for a straight waveguide. Depending on the bending radius R, c is evaluated by [2, 3], c L ¼ 1 À 2 3 1 À 15 4u 2 m   D 2 E 2 À 5 9 1 À 105 2u 2 m þ 495 4u 4 m   D 4 E 4 for large R ð2Þ c S ¼ DE for small R (3) where D ¼ (n co k 0 w/u m ) 2 and E ¼ w/R. In the derivation of Eq. (1), it is assumed that the core width and height are sufficiently large compared with the wavelength. It is interesting to note that for a straight (c ¼ 1) rectangular hollow waveguide, Eq. (1) is substantially the same as that derived by Isaac and Khalil using a ray-optics approach [6]. In other words, Eq. (1) can be regarded as an equation extended to a bent rectangular hollow waveguide. To find the location of the local extremum of the loss, we differentiate Eq. (1) with respect to the refractive index of the cladding, that is, qa mn /qn cl ¼ 0. As a result, we obtain the fol- lowing minimum loss condition: n min cl ¼ n co ffiffiffiffiffiffiffiffiffiffiffi 2 þ F p (4) where F ¼ (n/m) 2 (w/h) 3 /c ¼ f for the E x mode and F ¼ f À1 for the E y mode. For a straight square (h ¼ w) waveguide with m ¼ n, the minimum loss condition of both E x and E y modes is simply expressed as n min cl ¼ ffiffi 3 p n co (5) which serves as a rough guideline for determination of n cl , as can be seen in Figures 2 and 3. DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 12, December 2009 2901