## Terahertz scattering by subwavelength cylindrical arrays |

Optics Express, Vol. 19, Issue 11, pp. 10138-10152 (2011)

http://dx.doi.org/10.1364/OE.19.010138

Acrobat PDF (1750 KB)

### Abstract

We demonstrate the use of a full-wave electromagnetic field simulator to verify terahertz (THz) transmission-mode spectroscopic measurements of periodic arrays containing subwavelength cylindrical scatterers. Many existing THz scattering studies utilize analytical solutions, which were developed for a single scatterer. For multiple scatterers, a scaling factor equal to the number of scatterers is applied, accounting for interference between far-field radiative contributions from those scatterers but not their near-field mutual coupling. Consequently, analytical solutions do not accurately verify measurements. Conversely, results from the full-wave electromagnetic field simulator elucidate our measurements well, and provide an important insight into how the scattering behavior of cylindrical scatterers is influenced by test conditions.

© 2011 OSA

## 1. Introduction

*l*≫ diameter

*d*,

*l*≫ wavelength

*λ*, and is orientated orthogonal to the direction of propagation of an incident electromagnetic beam. Many naturally occurring organisms and minerals, such as some viruses and asbestos, can be modeled accurately by infinitely long circular cylinders [1]. Equally ubiquitous in everyday life are synthetic objects that too can be represented as infinitely long cylinders, such as fabric fibers or hairline scratches on hard surfaces. Arrays of infinitely long cylindrical objects can display distinct scattering and transmission characteristics, such as birefringence or dichroism, which allow for their detection and identification (e.g. birefringent proteins are luminous when dyed appropriately). The prevalence of infinitely long circular cylinders make them interesting and useful objects to study. A study of scattering from arrays of infinitely long circular cylinders in the terahertz frequency range (THz or T-ray, frequency

*ν*= 0.1–10 THz, wavelength

*λ*= 3 mm–30

*μ*m) forms the basis of this paper.

2. L. M. Zurk, B. Orlowski, D. P. Winebrenner, E. I. Thorsos, M. R. Leahy-Hoppa, and L. M. Hayden, “Terahertz scattering from granular material,” J. Opt. Soc. Am. B **24**, 2238–2243 (2007). [CrossRef]

5. M. Franz, B. M. Fischer, and M. Walther, “The Christiansen effect in terahertz time-domain spectra of coarse-grained powders,” Appl. Phys. Lett. **92**, 021107 (2008). [CrossRef]

6. J. E. Bjarnason, T. L. J. Chan, A. W. M. Lee, M. A. Celis, and E. R. Brown, “Millimeter-wave, terahertz, and mid-infrared transmission through common clothing,” Appl. Phys. Lett. **85**, 519–521 (2004). [CrossRef]

8. J. R. Fletcher, G. P. Swift, D. C. Dai, J. A. Levitt, and J. M. Chamberlain, “Propagation of terahertz radiation through random structures: An alternative theoretical approach and experimental validation,” J. Appl. Phys. **101**, 013102 (2007). [CrossRef]

9. Y. L. Hor, J. F. Federici, and R. L. Wample, “Nondestructive evaluation of cork enclosures using terahertz/millimeter wave spectroscopy and imaging,” Appl. Opt. **47**, 72–78 (2008). [CrossRef]

10. P. Y. Han, G. C. Cho, and X.-C. Zhang, “Time-domain transillumination of biological tissues with terahertz pulses,” Opt. Lett. **25**, 242–244 (2000). [CrossRef]

11. M. Reid and R. Fedosejevs, “Terahertz birefringence and attenuation properties of wood and paper,” Appl. Opt. **45**, 2766–2772 (2006). [CrossRef] [PubMed]

12. J. Beckmann, H. Richter, U. Zscherpel, U. Ewert, J. Weinzierl, L.-P. Schmidt, F. Rutz, M. Koch, H. Richter, and H.-W. Hübers, “Imaging capability of terahertz and millimeter-wave instrumentations for NDT of polymer materials,” in Proceedings of 9th European Conference on Nondestructive Testing (NDT), R. Diederichs, ed. (The e-Journal of Nondestructive Testing, 2006).

13. K. Naito, Y. Kagawa, S. Utsuno, T. Naganuma, and K. Kurihara, “Dielectric properties of eight-harness-stain fabric glass fiber reinforced polyimide matrix composite in the THz frequency range,” NDT & E Int. **42**, 441–445 (2009). [CrossRef]

14. K. Naito, Y. Kagawa, S. Utsuno, T. Naganuma, and K. Kurihara, “Dielectric properties of woven fabric glass fiber reinforced polymer matrix composites in the THz frequency range,” Compos. Sci. Technol. **69**, 2027–2029 (2009). [CrossRef]

15. X.-L. Tang, Y.-W. Shi, Y. Matsuura, K. Iwai, and M. Miyagi, “Transmission characteristics of terahertz hollow fiber with an absorptive dielectric inner-coating film,” Opt. Lett. **34**, 2231–2233 (2009). [CrossRef] [PubMed]

16. Y. Dikmelik, J. B. Spicer, M. J. Fitch, and R. Osiander, “Effects of surface roughness on reflection spectra obtained by terahertz time-domain spectroscopy,” Opt. Lett. **31**, 3653–3655 (2006). [CrossRef] [PubMed]

18. R. Piesiewicz, C. Jansen, D. Mittleman, T. Kleine-Ostmann, M. Koch, and T. Kürner, “Scattering analysis for the modeling of THz communication systems,” IEEE Trans. Antennas Propag . **55**, 3002–3009 (2007). [CrossRef]

19. J. Pearce, Z. Jian, and D. M. Mittleman, “Statistics of multiply scattered broadband terahertz pulses,” Phys. Rev. Lett. **91**, 043903 (2003). [CrossRef] [PubMed]

22. S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, “Experimental and numerical investigation of terahertz transmission through strongly scattering sub-wavelength size spheres,” Appl. Phys. Lett. **85**, 6284–6286 (2004). [CrossRef]

8. J. R. Fletcher, G. P. Swift, D. C. Dai, J. A. Levitt, and J. M. Chamberlain, “Propagation of terahertz radiation through random structures: An alternative theoretical approach and experimental validation,” J. Appl. Phys. **101**, 013102 (2007). [CrossRef]

23. X. J. Zhong, T. J. Cui, Z. Li, Y. B. Tao, and H. Lin, “Terahertz-wave scattering by perfectly electrical conducting objects,” J. Electromagn. Waves Appl. **21**, 2331–2340 (2007). [CrossRef]

8. J. R. Fletcher, G. P. Swift, D. C. Dai, J. A. Levitt, and J. M. Chamberlain, “Propagation of terahertz radiation through random structures: An alternative theoretical approach and experimental validation,” J. Appl. Phys. **101**, 013102 (2007). [CrossRef]

25. G. M. Png, R. J. Falconer, B. M. Fischer, H. A. Zakaria, S. P. Mickan, A. P. J. Middelberg, and D. Abbott, “Terahertz spectroscopic differentiation of microstructures in protein gels,” Opt. Express **17**, 13102–13115 (2009). [CrossRef] [PubMed]

## 2. Fibrillar samples and experimental results

^{2}) as shown in Fig. 1(a).

*μ*m in diameter (

*λ*

_{min}/3,

*λ*

_{max}/300) and is 15 mm long (500

*λ*

_{min}, 5

*λ*

_{max}). Since the length of each strand is 1500 times larger than the diameter, then each strand can be approximated by an infinite cylinder according to the definition given in Section 1. The strands are aligned as a

*near-periodic*array as shown in Fig. 1(b). The term “near-periodic” is used as it is acknowledged that the strands’ small cross-sectional dimensions make handling difficult, resulting in imperfections in the array structure. These imperfections are taken into consideration later in Section 5.

*extinction coefficient*(with symbolic notation

*α*) is used instead of

*absorption coefficient*to account for scattering. Each THz measurement is averaged between 16 to 25 scans. Results are verified using two different THz-TDS systems (at the University of Adelaide, and at the University of Leeds).

*n*are fairly flat, varying only slightly over the frequencies of interest (sample 1:

*n*

_{⊥}= 1.71 at 0.5 THz; 1.63 at 2 THz).

*n*profiles that vary more over frequency (sample 1:

*n*

_{||}= 2.13 at 0.5 THz; 1.74 at 2 THz), indicating dispersion in the fiberglass strands. The difference in refractive index between the parallel and perpendicular orientations suggests that the strands are birefringent, and that the measured refractive indices are

*effective*values that depend in geometry rather than solely

*bulk*values. Furthermore, given the more significant increase in extinction for the parallel samples, the strands may also be dichroic. Alternatively, the increase in

*α*could be due to strong THz scattering in this orientation. In order to elucidate if scattering causes the increase in

*α*for the parallel orientation, and to investigate why the two parallel samples have differing transmission profiles, Mie scattering is investigated in the next Section.

## 3. Analytical solutions: Mie and Rayleigh scattering

*λ*—can be found in numerous pieces of literature [1, 28, 29], hence they will not be elaborated on here. However a summary of salient equations is provided as follows. Referring to Fig. 1(e), when an electromagnetic wave

**E**

_{inc}propagating in a medium with complex refractive index

*n̂*

*is incident normal (*

_{i}*θ*= 90°) to the long (

*z*) axis of an infinite cylinder with complex refractive index

*n̂*

_{t}, some of the power from the incident wave is scattered back by the target object (“backscatter”). The measure of this backscattered power is the scattering cross section

*C*

_{sca}(units: m

^{2}), which is analogous to the monostatic radar cross section (or echo area) [30]. The backscattering cross sections for the parallel and perpendicular cases are defined in Eqs. (1) and (2): where wavenumber

*k*= 2

*π/λ*,

*r*= cylinder’s radius,

*l*= cylinder’s length,

*n̂*=

*n – iκ*,

*κ*= imaginary component of the refractive index or absorption index =

*αλ/*(4

*π*),

*m*=

*n̂*

_{t}

*/n̂*

_{i},

*J*

*(*

_{s}*p*) and

*Y*

*(*

_{s}*p*) are the Bessel functions of the first and second kind respectively with integral order

*s*∈ ℝ,

*J*

*(*

_{s}*p*), and

## 4. Comparison between experimental and analytical results

*α*with frequency as shown in Section 2, the scattering cross sections

*C*

_{sca, ||}and

*C*

_{sca, ⊥}in Eqs. (1) and (2) are investigated. However, knowledge of

*n̂*is needed for

*m*in Eqs. (3)–(5). Since the measured effective material parameters

*n*and

*α*of the fiberglass strands (displayed in Fig. 2) are dependent on orientation, they are not suitable for use in

*n̂*. One alternative is to substitute the

*n*and

*α*values of the fiberglass strands with those of bulk glass reported in literature. A second alternative is to use

*n*and

*α*of bulk fiberglass cloth.

*C*

_{sca, ||}and

*C*

_{sca, ⊥}plots on a logarithmic scale when

**E**

_{inc}is polarized parallel to the

*z*-axis in Fig. 1. For the frequencies under investigation

*C*

_{sca, ⊥}≪

*C*

_{sca, ||}. This means that a fiber oriented perpendicular to the incident

*E*field has a much smaller scattering area than one that is oriented parallel to

*E*. In accordance with the definition of backscattering cross section in Section 3, a smaller scattering area implies that a weaker signal is scattered back to the illumination source, resulting in a stronger transmitted signal. This trend is consistent with those in Figs. 2(a) and 2(c).

*α*from 2 THz onwards for the parallel case is an artifact. The relative difference between

*C*

_{sca, ||}and

*C*

_{sca, ⊥}is also exaggerated in the analytical model. These gaps in the analytical model motivate the exploration of other modeling techniques. One possible technique is to account for constructive and destructive interference from the array elements by using an array factor; this is elaborated on in the next Subsection. Another technique is to use a numerical electromagnetic simulation tool as described in Section 5.

### 4.1. Modeling an array with analytical solutions

3. A. Bandyopadhyay, A. Sengupta, R. B. Barat, D. E. Gary, J. F. Federici, M. Chen, and D. B. Tanner, “Effects of scattering on THz spectra of granular solids,” Int. J. Infrared Millim. Waves **28**, 969–978 (2007). [CrossRef]

25. G. M. Png, R. J. Falconer, B. M. Fischer, H. A. Zakaria, S. P. Mickan, A. P. J. Middelberg, and D. Abbott, “Terahertz spectroscopic differentiation of microstructures in protein gels,” Opt. Express **17**, 13102–13115 (2009). [CrossRef] [PubMed]

*N*isotropic sources [33]: where

*k*= 2

*π*

*ν*(

*ε*

_{t}

*μ*

_{t})

^{1/2}= 2

*π*

*νn*

*̂*

_{t}(

*ε*

_{0}

*μ*

_{0}

*μ*

_{r})

^{1/2},

*d*is the distance between two adjacent sources,

*β*is the difference in phase excitation between adjacent sources,

*ϕ*is the azimuth (with −90° being the forward scatter direction, and 90° being the back scatter direction),

*ε*

_{0}is the electric permittivity

*in vacuo*,

*μ*

_{0}is the magnetic permeability

*in vacuo*, and

*μ*

_{r}= 1 is the relative magnetic permeability of fiberglass. Assuming that all the fiberglass strands are excited simultaneously by the incident THz plane wave, then

*β*= 0. The plot of the total scattering cross section

*C*

_{sca, total}=

*C*

_{sca}× AF when diameter

*d*= 1

*μ*m ≈

*λ*

_{min}/100 is presented in Fig. 3(b), where the maximum frequency

*ν*

_{max}= 2.788 THz.

*C*

_{sca}by AF becomes evident when Figs. 3(a) and 3(b) are compared, showing that AF scales

*C*

_{sca}up by 2 orders of magnitude. The inclusion of AF for normal incidence is therefore no different from multiplying the Mie scattering equations by a scaling factor equal to the number of scatterers. This consideration is inadequate for modeling THz scattering for the test scenarios presented in this paper. In the next Section, an alternative model based on full-wave electromagnetic field simulation is presented. The full-wave solver produces results that are a better match with the experimental observations.

## 5. Full-wave electromagnetic field solver

*μ*m long strands (each 10

*μ*m in diameter) seated between periodic master/slave side boundaries. To create the infinitely long cylinders (with respect to wavelength) along the

*z*-axis, symmetrical boundaries are applied to the top and bottom airbox walls that touch the ends of the cylinders (symmetrical

**E**boundaries for the parallel case; symmetrical

**H**boundaries for the perpendicular case). Given that HFSS utilizes a 3D solver, the symmetrical boundaries also serve the dual purpose of reducing the problem space into a 2D one, simplifying and speeding up the simulation. It is noted here that other electromagnetic field simulators may be more suited for the 2D problem presented in this study, e.g. COMSOL (by COMSOL AB, also based on FEM but includes a 2D implementation) and the Multiple Multipole Program [36]. The latter simulator is most suited for 2D problems, particularly periodic surfaces.

*y*-

*z*plane are assigned perfectly matched layers (PML) as absorbing boundaries in order to simulate the extension of the problem space into infinity. Since the far-field response is of interest in this study, the incident THz beam is modeled in HFSS as a plane wave as described in Fig. 1(e) instead of as a Gaussian beam.

*x*-axis) airbox seated tightly inside the first airbox. This airbox is similar in shape to the first, except the walls which are parallel to the

*y*-

*z*plane are at a shorter distance (

*λ*

_{max}/2.05) from the cylinders, meaning they do not touch the PML boundaries. None of the walls of the second inner airbox are assigned specific material boundary types, but they are used as Huygen’s surface for near-field to far-field computation. The 0.05 factor used in the length of the second airbox (i.e. the difference in the denominators of

*λ*

_{max}/2.05 and

*λ*

_{max}/2) is chosen because it is a small number. If another small number is used, such as 0.01, similar results are generated.

*S*

_{11}and

*S*

_{21}are the reflection and transmission coefficients respectively. The S-parameters, which are usually expressed in decibels, are defined in [33] either in terms of the electric field

*E*or magnetic field

*H*of a plane wave transmitting from medium 1 into medium 2: where

*η*

_{1}= (

*μ*

_{1}

*/ε*

_{1})

^{1/2}, and

*η*

_{2}= (

*μ*

_{2}

*/ε*

_{2})

^{1/2}. Values of

*S*

_{11}describe backscattering, whereas

*S*

_{21}describe transmission;

*S*

_{21}therefore provides a means for elucidating the behavior of the THz transmission magnitude plots in Fig. 2(a). If

*S*

_{21}= 1 (or

*S*

_{21}= 0 dB), then the incident THz signal is 100% transmitted, implying that the transmission magnitude → 0 dB. If

*S*

_{11}= 1 (or

*S*

_{11}= 0 dB), then the signal is 100% reflected, implying that the transmission magnitude → ∞. In an ideal condition with no losses,

*|S*

_{21}

*|*

^{2}+

*|S*

_{11}

*|*

^{2}= 1. The next Section presents the S-parameters from the model in Fig. 4 and compares them with the measured THz transmission magnitudes.

## 6. Comparison between experimental and numerical results

*S*

_{21}results obtained with the double-cylinder, periodic model shown in Fig. 4. Compared with the measured THz transmission magnitude in Fig. 2(a), the following observations are made with regard to their qualitative and quantitative match. There is a good qualitative match over all frequencies between the measurements and the

*S*

_{21}results for both the parallel and perpendicular cases. This match is superior to the poor match between the scattering cross section

*C*

_{sca, ||}and

*C*

_{sca, ⊥}plots in Fig. 3(a) with the measured extinction coefficients

*α*in Fig. 2(c). Quantitatively, the match is poor between the measured THz transmission magnitude in Fig. 2(a) and the

*S*

_{21}results in Fig. 5(a), raising the question whether the single layered HFSS model used is adequate for modeling the actual array, which may contain more than one layer due to imperfections. This question is answered in Section 6.1.

*S*

_{21}plot also matches the measured THz transmission magnitudes more accurately at the lower THz frequencies. However, the

*S*

_{21}plot for the parallel case still does not explain if the upwards-turning curve after 2 THz as seen in Fig. 2(a) for the parallel sample 1 is an artifact. To ascertain if this feature is an artifact, we explore the effect of changing the space between the cylinders in our HFSS simulation. This scenario is realistic, as the fiberglass strands are not secured very tightly to the polyethylene frame in order to prevent them from snapping. The strands may therefore have moved out of place prior to measurements, creating spaces between the strands that are wider than anticipated.

*μ*m, 10

*μ*m and 15

*μ*m, were arbitrarily chosen to examine the effect of varying the space between adjacent strands. The

*S*

_{21}plots generated from these new scenarios are presented together with the 1

*μ*m case in Fig. 5(b). Comparing only the perpendicular cases, the

*S*

_{21}plots shift upwards as the strand spacing increases, with similar profiles except a change in gradient; the changes in gradients are less significant between the 5, 10 and 15

*μ*m cases than between the 1

*μ*m and 5

*μ*m cases. These observations for the perpendicular case suggests that the the fiberglass arrays in the perpendicular orientation are fairly robust to small changes to strand spacing, possibly explaining why no difference in THz transmission magnitude is measured in Fig. 2(a) for the two perpendicular samples.

*S*

_{21}plots for the parallel case also shift upwards as the strand spacing increases. The extent of shifting is greater than for the perpendicular case, suggesting that the fiberglass arrays in the parallel orientation are more sensitive to small changes to strand spacing, thus the difference in the measured THz transmission magnitudes of the two parallel samples shown in Fig. 2(a). However, there is no evidence in the

*S*

_{21}plots for the parallel case of the upwards-turning characteristic present in Fig. 2(a).

*S*

_{21}plots suggests that

*(i)*the fiberglass arrays in the parallel orientation are particularly sensitive to small changes in the strand spacing, manifesting as different THz transmission magnitudes as reported in Section 2 for the two parallel samples; and

*(ii)*the upwards-turn seen in Fig. 2(a) for the parallel sample 1 cannot be reproduced in the HFSS model, hence it can be concluded that the curve beyond ≈ 2.2 THz in Fig. 2(a) is an artifact. However, the lower THz transmission magnitude of the parallel sample 1 when compared to the parallel sample 2 at 2 THz is genuine, and could be due to the aforementioned sensitivity to strand spacing. Alternatively, there could be more than one layer of cylinders in the path of the THz beam. The effect of multiple cylinder layers is investigated in the next Subsection.

### 6.1. Influence of multiple cylinder layers in the array

*S*

_{21}value of the perpendicular sample in Fig. 5(a) is ≈ −0.5 dB. This difference may be due to multiple layers of cylinders in the arrays used in the THz measurements. Referring to Fig. 1(b), there is a strong likelihood that multiple layers of fiber threads are present in the path of the THz beam.

*S*

_{21}plots of the models containing multiple cylinder layers for both the parallel and perpendicular orientations. The normalized THz transmission magnitudes of the samples as presented previously in Fig. 2(a) are included for comparison.

*S*

_{21}value by ≈ 1 dB for the parallel orientation, and by ≈ 0.5 dB for the perpendicular orientation. This results in a distinct decrease in the

*S*

_{21}values for the parallel case when compared with the perpendicular case, indicating less transmission for the parallel case that is consistent with the results shown in Fig. 2(a). The

*S*

_{21}plots of the 24-cylinder model for both orientations are in good quantitative agreement with the normalized transmission magnitudes in Fig. 2(a) (with the exception of parallel sample 1), indicating that the transmission ratio between both orientations (parallel vs. perpendicular) has been very closely found. Therefore, the use of multiple cylinder layers is a more accurate means of modeling the arrays used in this study.

*S*

_{21}value by ≈ 1 dB, then it is also possible that sample 1 contains more layers than sample 2.

### 6.2. Verifying consistency in periodic models

*i*), which is similar to Fig. 6(a), contains only one cylinder enclosed inside similar periodic and symmetric boundaries as Fig. 4(d). The second model (model

*ii*) contains two cylinders at the chosen spacing, and therefore considers two periods of model

*i*. The third model (model

*iii*), which is similar to Fig. 4(d), introduces a small offset between the cylinders. This offset breaks the periodicity, and introduces some randomness in the cylinder arrangement.

*i*is structurally simple as it contains only one cylinder, reducing any errors that may arise from the presence of multiple cylinders in the problem space. Model

*ii*is equivalent to model

*i*, and is included to check the consistency of the periodic computational model. Therefore the S-parameters of models

*i*and

*ii*should be similar with only small differences that arise from meshing. If the presence of multiple cylinders causes any inconsistency in the model, then the two sets of S-parameters will differ noticeably.

*S*

_{21}plots of the single- (model

*i*) and the double-cylinder with no offset (model

*ii*) periodic models overlap when juxtaposed in Fig. 7(b), indicating that both models are indeed equivalent. Furthermore, the model in Fig. 4(d) (model

*iii*) also overlaps the model

*i*and model

*ii*plots. This means that artifacts have not been introduced from the presence of multiple cylinders in the problem space, nor by the offset between cylinders. It can therefore be concluded that the numerical results generated by HFSS are not corrupted by meshing inconsistencies.

## 7. Conclusion

## Acknowledgments

## References and links

1. | C. F. Bohren and D. R. Huffman, |

2. | L. M. Zurk, B. Orlowski, D. P. Winebrenner, E. I. Thorsos, M. R. Leahy-Hoppa, and L. M. Hayden, “Terahertz scattering from granular material,” J. Opt. Soc. Am. B |

3. | A. Bandyopadhyay, A. Sengupta, R. B. Barat, D. E. Gary, J. F. Federici, M. Chen, and D. B. Tanner, “Effects of scattering on THz spectra of granular solids,” Int. J. Infrared Millim. Waves |

4. | Y.-C. Shen, P. F. Taday, and M. Pepper, “Elimination of scattering effects in spectral measurement of granulated materials using terahertz pulsed spectroscopy,” Appl. Phys. Lett. |

5. | M. Franz, B. M. Fischer, and M. Walther, “The Christiansen effect in terahertz time-domain spectra of coarse-grained powders,” Appl. Phys. Lett. |

6. | J. E. Bjarnason, T. L. J. Chan, A. W. M. Lee, M. A. Celis, and E. R. Brown, “Millimeter-wave, terahertz, and mid-infrared transmission through common clothing,” Appl. Phys. Lett. |

7. | M. A. Kaliteevski, D. M. Beggs, S. Brand, R. A. Abram, J. R. Fletcher, G. P. Swift, and J. M. Chamberlain, “Propagation of electromagnetic waves through a system of randomly placed cylinders: The partial scattering wave resonance,” J. Mod. Opt. |

8. | J. R. Fletcher, G. P. Swift, D. C. Dai, J. A. Levitt, and J. M. Chamberlain, “Propagation of terahertz radiation through random structures: An alternative theoretical approach and experimental validation,” J. Appl. Phys. |

9. | Y. L. Hor, J. F. Federici, and R. L. Wample, “Nondestructive evaluation of cork enclosures using terahertz/millimeter wave spectroscopy and imaging,” Appl. Opt. |

10. | P. Y. Han, G. C. Cho, and X.-C. Zhang, “Time-domain transillumination of biological tissues with terahertz pulses,” Opt. Lett. |

11. | M. Reid and R. Fedosejevs, “Terahertz birefringence and attenuation properties of wood and paper,” Appl. Opt. |

12. | J. Beckmann, H. Richter, U. Zscherpel, U. Ewert, J. Weinzierl, L.-P. Schmidt, F. Rutz, M. Koch, H. Richter, and H.-W. Hübers, “Imaging capability of terahertz and millimeter-wave instrumentations for NDT of polymer materials,” in Proceedings of 9th European Conference on Nondestructive Testing (NDT), R. Diederichs, ed. (The e-Journal of Nondestructive Testing, 2006). |

13. | K. Naito, Y. Kagawa, S. Utsuno, T. Naganuma, and K. Kurihara, “Dielectric properties of eight-harness-stain fabric glass fiber reinforced polyimide matrix composite in the THz frequency range,” NDT & E Int. |

14. | K. Naito, Y. Kagawa, S. Utsuno, T. Naganuma, and K. Kurihara, “Dielectric properties of woven fabric glass fiber reinforced polymer matrix composites in the THz frequency range,” Compos. Sci. Technol. |

15. | X.-L. Tang, Y.-W. Shi, Y. Matsuura, K. Iwai, and M. Miyagi, “Transmission characteristics of terahertz hollow fiber with an absorptive dielectric inner-coating film,” Opt. Lett. |

16. | Y. Dikmelik, J. B. Spicer, M. J. Fitch, and R. Osiander, “Effects of surface roughness on reflection spectra obtained by terahertz time-domain spectroscopy,” Opt. Lett. |

17. | T. Kleine-Ostmann, C. Jansen, R. Piesiewicz, D. Mittleman, M. Koch, and T. Kürner, “Propagation modeling based on measurements and simulations of surface scattering in specular direction,” in Proceedings of Joint 32nd International Conference on Infrared and Millimetre Waves, and 15th International Conference on Terahertz Electronics, M. J. Griffin, P. C. Hargrave, T. J. Parker, and K. P. Wood, eds., vol. 1, pp. 408–410 (2007). |

18. | R. Piesiewicz, C. Jansen, D. Mittleman, T. Kleine-Ostmann, M. Koch, and T. Kürner, “Scattering analysis for the modeling of THz communication systems,” IEEE Trans. Antennas Propag . |

19. | J. Pearce, Z. Jian, and D. M. Mittleman, “Statistics of multiply scattered broadband terahertz pulses,” Phys. Rev. Lett. |

20. | J. Pearce and D. M. Mittleman, “Using terahertz pulses to study light scattering,” Phys. B |

21. | Z. Jian, J. Pearce, and D. M. Mittleman, “Characterizing individual scattering events by measuring the amplitude and phase of the electric field diffusing through a random medium,” Phys. Rev. Lett. |

22. | S. Mujumdar, K. J. Chau, and A. Y. Elezzabi, “Experimental and numerical investigation of terahertz transmission through strongly scattering sub-wavelength size spheres,” Appl. Phys. Lett. |

23. | X. J. Zhong, T. J. Cui, Z. Li, Y. B. Tao, and H. Lin, “Terahertz-wave scattering by perfectly electrical conducting objects,” J. Electromagn. Waves Appl. |

24. | R. A. Cheville, M. T. Reiten, R. McGowan, and D. R. Grischkowsky, |

25. | G. M. Png, R. J. Falconer, B. M. Fischer, H. A. Zakaria, S. P. Mickan, A. P. J. Middelberg, and D. Abbott, “Terahertz spectroscopic differentiation of microstructures in protein gels,” Opt. Express |

26. | C. A. Balanis, |

27. | E. Hecht, |

28. | H. C. van de Hulst, |

29. | M. Kerker, |

30. | E. F. Knott, J. F. Shaeffer, and M. T. Tuley, Radar Cross Section (SciTech Publishing Inc., 2004). |

31. | R. Piesiewicz, T. Kleine-Ostmann, N. Krumbholz, D. Mittleman, M. Koch, and T. Kürner, “Terahertz characterisation of building materials,” Electron. Lett. |

32. | M. Naftaly and R. E. Miles, “Terahertz time-domain spectroscopy of silicate glasses and the relationship to material properties,” J. Appl. Phys. |

33. | C. A. Balanis, |

34. | G. Chattopadhyay, J. Glenn, J. J. Bock, B. K. Rownd, M. Caldwell, and M. J. Griffin, “Feed horn coupled bolometer arrays for SPIRE—design, simulations, and measurements,” IEEE Trans. Microwave Theory Tech . |

35. | L. Liu, S. M. Matitsine, Y. B. Gan, and K. N. Rozanov, “Effective permittivity of planar composites with randomly or periodically distributed conducting fibers,” J. Appl. Phys. |

36. | C. Hafner, |

**OCIS Codes**

(120.4290) Instrumentation, measurement, and metrology : Nondestructive testing

(290.5820) Scattering : Scattering measurements

(050.1755) Diffraction and gratings : Computational electromagnetic methods

(300.6495) Spectroscopy : Spectroscopy, teraherz

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: March 16, 2011

Revised Manuscript: April 25, 2011

Manuscript Accepted: April 26, 2011

Published: May 9, 2011

**Virtual Issues**

Vol. 6, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Gretel M. Png, Christophe Fumeaux, Mark R. Stringer, Robert E. Miles, and Derek Abbott, "Terahertz scattering by subwavelength cylindrical arrays," Opt. Express **19**, 10138-10152 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10138

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