一個使用毫米波位於兩平行板之間的緊湊型繞射儀被研製出來。其使用可單獨裝在一個可旋轉結構上的微加工製作人工晶體。和以前的工作相比，因為我們將多重散射效應最小化，所以實驗結果相當符合布拉格的預測。我們也對一些影響分辨率和信號強度的因素進行了分析，如散射體數目，圓柱體的半徑，探測器和人工晶體之間的距離。
在毫米波布拉格繞射中，可以觀察到角度分布因素的影響，其為影響散射波的振幅。另一個會影響散射波的振幅重要因素，是多重散射的效應。在這裡，我們測量了許多二維金屬短圓柱結構的晶格類型。此外，我們研發了電腦自動控制的旋轉平台，用於加快測量。繞射波的強度多樣性不能簡單地歸納來預測，但散射波的強度可以通過這兩種效應去預測。使得我們的計算結果相當吻合實驗結果。我們對這兩種效應有了更好地理解，使得對角度偏移布拉格預測值的現象也有進一步的解釋。
該儀器還可以使用於測量光子帶隙。基於模式分析理論，一個解決短圓柱金屬光子帶隙問題的理論方法被提出來。毫米波繞射系統在二維人工介質隱形斗篷也具有潛在的應用。

A compact diffraction apparatus is developed with millimeter wave propagation between two parallel plates. Micro-fabricated model crystals are individually mounted on a rotatable structure. In contrast to previous work, the experimental results agree well with Bragg's predictions because multiple scattering is minimized in this configuration. Factors that affect the resolution and signal strength, such as the number of scatterers, cylinder radius, and the distance between the detector and model crystal, are analyzed.
The effects of (atomic) form factor can be observed in the millimeter-wave Bragg diffraction, which affects the amplitude of scattering wave. Another important factor that affects the amplitude of the scattering wave is from the multiple scattering. Here we used many two-dimensional lattice types of metal-stub structures. Furthermore, a computer-automatically-controlled rotary platform was used to speed up the measurements. The diversification of intensity can not be briefly summarized, but the intensity of scattering wave can be predicted by these two effects. The calculations also matched well with the experimental results. With a better understanding of these two effects, the phenomena of angle offset to Bragg’s law perdition also have further explanations.
The apparatus can also be using to measure the photonic band gap. A theoretical method is presented to solve metal-stub photonic-band-gap (PBG) problems based on modal analysis. The automatic millimeter-wave diffraction system has also potential applications in two-dimensional meta-material electromagnetic cloak.

[1] W. H. Bragg and W. L. Bragg, “The reflection of x-rays by crystals,” Proc. R. Soc. London, Ser. A 88, 428–438 (1913).
[2] Roland A. Allen, “Verification of Bragg’s law by the use of microwaves,” Am. J. Phys. 23, 297–298 (1955).
[3] Robert G. Marcley, “Apparatus drawings project #6: Bragg diffraction apparatus,” Am. J. Phys. 28, 415–417 (1960).
[4] Bailey L. Donnally, George Bradley, and Jacob Dewitt, “Models for microwave analogs of Bragg scattering,” Am. J. Phys. 36, 920 (1968).
[5] T. G. Bullen, “An improved mounting for the Welch–Bragg diffraction apparatus,” Am. J. Phys. 37, 333 (1969).
[6] Thomas D. Rossing, Rodney Stadum, and Douglas Lang, “Bragg diffraction of microwaves,” Am. J. Phys. 41, 129–130 (1973).
[7] William H. Murray, “Microwave diffraction techniques from macroscopic crystal models,” Am. J. Phys. 42, 137–140 (1974).
[8] M. T. Cornick and S. B. Field, “Microwave Bragg diffraction in a model crystal lattice for the undergraduate laboratory,” Am. J. Phys. 72, 154–158 (2004).
[9] Joseph C. Amato and Roger E. Williams, “Rotating crystal microwave Bragg diffraction apparatus,” Am. J. Phys. 77, 942–945 (2009).
[10] Ansoft HFSS version 10.1, Ansoft Corporation, PA, USA.
[11] J. D. Williams and W. J. Wang, “Study on the postbaking process and the effects on UV lithography of high aspect ratio SU-8 microstructures,” J. Microlithogr. Microfabr. Microsyst. 3, 563–568 (2004).
[12] C. P. Yuan, S. Y. Lin, T. H. Chang, and B. Y. Shew, “Millimeter-wave Bragg diffraction of microfabricated crystal structures,” Am. J. Phys. 79, 619–623 (2011).
[13] C. P. Yuan and T. H. Chang, “Modal analysis of metal-stub photonic band gap structures in a parallel-plate waveguide,” Prog. Electromagn. Res. 119, 345–361 (2011).
[14] R. W. P. King and T. T. Wu, The Scattering and Diffraction of Waves (Harvard University Press, Cambridge, MA, 1959), p. 69.
[15] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, San Diego, 2000), p. 933.
[16] A. Sommerfeld, Partial Differential Equations in Physics (Academic Press, New York, 1949), p. 193.
[17] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed. (Dover, New York, 1972), p. 364.
[18] C. Kittel, Introduction to Solid State Physics, 8th ed. (John Wiley & Sons, New York, 2005), p. 39.
[19] P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge University Press, Cambridge, New York, 2006), pp. 1–4.
[20] E. Hecht, Optics, 4th ed. (Addison-Wesley, San Francisco, 2002), pp. 452–457.
[21] J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, New York, 1998), pp. 358–359, 479.
[22] E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed. (Springer-Verlag, Berlin, 2006), pp. 11–12.
[23] Reference 15, p. 940.
[24] Reference 15, p. 660.
[25] Reference 20, pp. 447–448.
[26] L.-M. Li and Z.-Q. Zhang, “Multiple-scattering approach to finite-sized photonic band-gap materials,” Phys. Rev. B 58, No. 15, 9587–9590 (1998).
[27] D. M. Pozar, Microwave Engineering, 3rd ed. (Addison-Wesley, New York, 2005), pp. 197–203.
[28] H. Y. Yao and T. H. Chang, “Effect of high-order modes on tunneling characteristics,” Prog. Electromagn. Res. 101, 291–306 (2010).
[29] A. Wexler, “Solution of waveguide discontinuities by modal analysis,” IEEE Tran. Microwave Theory Tech. 15, No. 9, 509–517 (1967).
[30] N. Marcuvitz, Waveguide Handbook, (McGraw-Hill, New York, 1951).
[31] Reference 21, p. 359.
[32] Reference 21, p. 36.
[33] Reference 21, p. 358.
[34] S. Arscott, F. Gatet, P. Mounaix, L. Duvillatet, J.-L. Coutaz, and D. Lippens, “Terahertz time-domain spectroscopy of films fabricated from SU-8,” Electron. Lett. 35, 243–244 (1999).
[35] R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. 26, No. 11, 846–848 (2001).
[36] R. Mendis and D. Gischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microw. Wirel. Compon. Lett. 11, No. 11, 444–446 (2001).
[37] S. Coleman and D. Grischkowsky, “Parallel plate THz transmitter,” Appl. Phys. Lett. 84, No. 5, 654–656 (2004).
[38] M. Nagel, P. H. Bolivar, and H. Kurz, “Modular parallel-plate THz components for cost-efficient biosensing systems,” Semicond. Sci. Technol. 20, 281–285 (2005).
[39] R. Mendis, “Nature of subpicosecond terahertz pulse propagation in practical dielectic-filled parallel-plate waveguides,” Opt. Lett. 31, No. 17, 2643–2645 (2006).
[40] D. G. Cooke and P. U. Jepsen, “Optical modulation of terahertz pulses in a parallel plate waveguide,” Opt. Express 16, No. 19, 15123–15129 (2008).
[41] R. Mendis and D. M. Mittleman, “An investigation of the lowest-order transverse-electric (TE1) mode of the parallel-plate waveguide for THz pulse propagation,” J. Opt. Soc. Am. B 26, No. 9, A6–A13 (2009).
[42] G. Guida, A. de Lustrac, and A. Priou, “An introduction to photonic band gap (PBG) materials,” Prog. Electromagn. Res. 41, 1–20 (2003).
[43] C. Lin, C. Chen, G. J. Schneider, P. Yao, S. Shi, A. Sharkawy, and D. W. Prather, “Wavelength scale terahertz two-dimensional photonic crystal waveguides,” Opt. Express 12, No. 23, 5723–5728 (2004).
[44] A. L. Bingham and D. R. Grischkowsky, “Terahertz 2-D photonic crystal waveguides,” IEEE Microwave Wireless Compon. Lett. 18, No. 7, 428–430 (2008).
[45] Y. Zhao and D. Grischkowsky, “Terahertz demonstrations of effectively two-dimensional photonic bandgap structures,” Opt. Lett. 31, No. 10, 1534–1536, (2006).
[46] A. Bingham, Y. Zhao, and D. Grischkowsky, “THz parallel plate photonic waveguides,” Appl. Phys. Lett. 87, 051101 (2005).
[47] A-C. Tarot , S. Collardey, and K. Mahdjoubi, “Numerical studies of metallic PBG structures,” Prog. Electromagn. Res. 41, 133–157 (2003).
[48] H. Benisty, D. Labilloy, C. Weisbuch, C. J. M. Smith, T. F. Krauss, D. Cassagne, A. Beraud, and C. Jouanin, “Radiation losses of waveguide-based two-dimensional photonic crystals: Positive role of the substrate,” Appl. Phys. Lett. 76, No. 5, 532–534 (2000).
[49] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals-Molding the Flow of Light, (Princeton University Press, Princeton, NJ, 1995).
[50] K. Inoue and K. Ohtaka, Photonic Crystals: Physics, Fabrication and Applications, Chap. 3, (Springer-Verlag, New York, 2004).
[51] J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices, Chap. 1, (Springer-Verlag, Berlin, 2005).
[52] J. D. Callen, Fundamentals of Plasma Physics, Appendix D, (Draft, University of Wisconsin, Madison, 2006).
[53] U. Lenonhardt and T. G. Philbin, “General relativity in electrical engineering,” New J. Phys. 8, 247 (2006).
[54] J. B. Pendry, D. Schuring, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780–1782 (2006).
[55] U. Leonhardt, “Optical conformal mapping”, Science 312, 1777–1780 (2006).
[56] U. Leonhardt, “Notes on Conformal Invisibility Devices”, New. J. Phys. 8, 118 (2006).
[57] A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the Scattering Tomography for Central Potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[58] D. Schuring, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial Electromagnetic Cloak at Microwave Frequencies,” Science 314, 977–910 (2006).
[59] A. Nicolet and F. Zolla, “Cloaking with Curved Spaces,” Science 323, 46–47, (2009).
[60] S. A. Cummer and D. Schurig, “One path to acoustic cloaking,” New J. Phys. 9, 45 (2007).
[61] Z. Ruan, M. Yan, C. W. Neff, and M. Qiu, “Ideal cylindrical cloak: perfect but sensitive to tiny perturbations,” Phys. Rev. Lett. 99, 113903 (2007).
[62] Reference 21, pp. 258–267.