## Coupling of terahertz waves to a two-wire waveguide |

Optics Express, Vol. 18, Issue 22, pp. 22614-22624 (2010)

http://dx.doi.org/10.1364/OE.18.022614

Acrobat PDF (1237 KB)

### Abstract

We calculate theoretically the coupling of a terahertz wave from a dipole into a two-wire waveguide. The field transmission and reflection are obtained using a Single Mode Matching (SMM) technique at the input port of the two-wire waveguide. The results show more than 70 percent coupling efficiency for the waveguide using 500μm radii wires with 2mm center-to-center separation and the exciting field cross section of 1mm × 1mm. The results also show good agreement with the full-wave numerical simulations using the Finite Element Method (FEM).

© 2010 OSA

## 1. Introduction

1. M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. **39**(6), 910–916 (1991). [CrossRef]

2. C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B **17**(5), 851–863 (2000). [CrossRef]

3. S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. **76**(15), 1987–1989 (2000). [CrossRef]

4. R. Mendis and D. Grischkowsky, “Plastic ribbon thz waveguides,” J. Appl. Phys. **88**(7), 4449–4451 (2000). [CrossRef]

5. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature **432**(7015), 376–379 (2004). [CrossRef] [PubMed]

6. J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express **14**(1), 279–290 (2006). [CrossRef] [PubMed]

6. J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express **14**(1), 279–290 (2006). [CrossRef] [PubMed]

8. M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. **95**(23), 233506 (2009). [CrossRef]

9. H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express **18**(7), 7415–7420 (2010). [CrossRef] [PubMed]

## 2. Approach in analytic framework

10. S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. **70**(5), 559‒561 (1997). [CrossRef]

13. S. Matsuura and H. Ito, “Generation of CW terahertz radiation with photomixing,” Top. Appl. Phys. **97**, 157–202 (2005). [CrossRef]

14. W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. **67**(12), 1607–1614 (1977). [CrossRef]

19. P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. **20**(8), 807–809 (1995). [CrossRef] [PubMed]

19. P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. **20**(8), 807–809 (1995). [CrossRef] [PubMed]

19. P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. **20**(8), 807–809 (1995). [CrossRef] [PubMed]

21. P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B **13**(11), 2424–2436 (1996). [CrossRef]

*w × d*and with 1W power) corresponding to a plane wave, impinges on the input port of a two-wire waveguide. The coupling efficiency is the ratio of the transmitted power to the incident power [2

2. C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B **17**(5), 851–863 (2000). [CrossRef]

## 3. The mode-matching technique to calculate the coupling

22. R. Gordon, “Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons,” Phys. Rev. B **74**, 153417 (2006). URL http://link.aps.org/abstract/PRB/v74/e153417.

*E*,

*H*are the electric and magnetic field,

*a*,

_{m}*b*are the field coefficients for forward and backward propagating waves,

_{m}*e*,

_{m}*h*are the normalized electric and magnetic fields of the

_{m}*m*th mode, and subscripts

*1*,

*2*are for the parallel-plate waveguide and the two-wire waveguide, respectively. The orthonormalization of the modes can be written [25,26] where

*l*,

*m*are the mode subscripts, and

*δ*is the Kronecker delta. The boundary conditions at the interface of the two waveguides require: where the subscript

_{lm}*t*indicates the transverse components of the fields. The single lowest-order TEM mode with 1W power is incident from the left side. Continuity of the transverse electric field expanded in terms of the orthogonal modes on both sides gives:where

*r*,

*t*are the reflection and transmission coefficients of the fields for the TEM modes, respectively. Multiplying both sides of Eq. (6) by

*1/2(h*, integrating over the transverse plane (XY-plane), and using orthoganality relations for the parallel-plate waveguide's modes yieldswhere,

_{1})_{TEM}**1/2(e*, integrating over the transverse plane, and using the orthogonality relations for the two-wire waveguide's modes yields

_{2})_{TEM}**|t|*is a measure of the coupling coefficient. Therefore, the coupling coefficient can be obtained by calculating

^{2}*κ*. The amplitude of

*κ*is a number between 0 and 1 and indicates how well the field distributions of the two TEM modes are matched, 1 corresponds to the perfectly-matched case and 0 when the modes are not matched at all.

## 4. Calculation of the *κ*

*κ*calls for knowing the electric and magnetic fields of the TEM mode for the two waveguides. The TEM mode for the parallel-plate waveguide is [24]: where

*η*is the intrinsic impedance of free space and

_{0}*A*is the normalization constant that is chosen so that the TEM mode carries 1W power:where

_{1}*w*are the width of the plates, and

*d*is the distance between the plates. The TEM mode for the two-wire waveguide is [9

9. H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express **18**(7), 7415–7420 (2010). [CrossRef] [PubMed]

*C*and

_{1}*C*in Eq. (18), (19) depend on the radii of the wires (

_{2}*R*), and the distance between them (

*D*), by

*A*is the normalization constant that can be obtained from the equation below,that yieldsSubstituting the fields in Eq. (8) yieldswhere

_{2}*S*is the surface specified with the pattern in Fig. 2 . Explained in the Appendix, integration over

_{1}*S*using Green's theorem gives

_{1}## 5. Results and discussion

*|t|*,

^{2}*|r|*) versus

^{2}*R*, obtained from the proposed method for the same situation. They show the expected behavior, consistent with the intuitive conclusion.

*w*×

*d*= 1mm × 0.5mm (equal to the parallel-plate dimensions) for the parallel-plate case, and

*w*×

*d*= 1mm × 1mm (equal to the cross section of the box) for the dipole one. At the lower limit, when the radii of the wires go to zero, there is no waveguide to support the TEM mode; thus, the coupling should go to zero. Coupling should also go to zero at the higher limit, when the radii of the wires become equal to

*D/2*(the center-to-center distance of the wires,

*D*, is constant and equal to 400μm) because the edges of the two wires touch each other and the waveguide cannot support the TEM mode. Therefore, there should be an optimum value for the radius corresponding to the peak value of the coupling between the limits. Figure 4 shows that the results are consistent with this expectation. The results from the theory and the simulations with the parallel-plate waveguide, depicted in Fig. 4(a), show good agreement. Figure 4(b) also illustrates the results from the theory and the simulations with the dipole source. They show the same overall behavior even though there are some discrepancies due to neglecting the excitation of radiating and higher-order modes at the input port in the theoretical calculations.

*w*×

*d*= 1mm × 1mm impinges a two-wire waveguide with constant radii of the wires, 500μm, for different center-to-center distances. Figure 5 shows the coupling efficiency for this case.

*D*starts from its minimum value,

*2R*, where the two wires touch each other and the coupling is zero. As

*D*increases, the waveguide starts to support the TEM mode that overlaps with the field of the plane wave, enhancing the coupling. But when the edges of the wires are too close, the TEM mode supported by the waveguide is mostly concentrated in a very small area between the wires, resulting in small coupling. However, as

*D*becomes larger the field is more distributed on the surface of the wires [9

9. H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express **18**(7), 7415–7420 (2010). [CrossRef] [PubMed]

8. M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. **95**(23), 233506 (2009). [CrossRef]

**18**(7), 7415–7420 (2010). [CrossRef] [PubMed]

*R*and

*D*. But, roughly speaking, the optimum point always happens when the size of the aperture is close to the edge-to-edge distance of the wires, like the case shown in Fig. 6(a). Note that the proposed method is only accurate when a single TEM mode dominates the behavior of the waveguide. Avoiding higher-order modes limits the dimensions of the waveguide. However, the fact that the linearly-polarized incident field is well-matched with the TEM mode supported by the waveguide alleviates that concern to some extent.

## 6. Conclusion

## Appendix

*F*and

_{1}*F*be continuous differentiable functions on a simply connected domain

_{2}*S*, and let Γ be a positively oriented simple closed contour around

*S*. Then Eq. (26) is valid:In [9

**18**(7), 7415–7420 (2010). [CrossRef] [PubMed]

*x*component of the electric field in Eq. (8) and using Green's theorem yieldswhere the surface

*S*and the contour Γ are shown in Fig. 7 . The integrations on the routes 1 and 12 are zero because

_{1}*y*is constant. So are the integrations on the routes 3 to 10 because the integrand is the even function of

*y*. Thus, the calculation of

*κ*reduces to the integrations on the routes 2, 11 and 13 that yield

*y*is constant. So the calculation of

*κ*reduces to the integrations on routes 1 and 3 that result in the same equation as Eq. (30). However, the case shown in Fig. 8(a) results in a different relationship for

*κ*. The integrations over 1 and 6 are zero because

*y*is constant, and so are the integrations on routes 3, 4, 8, and 9 because the integrand is the even function of

*y*. Therefore, the calculation of

*κ*reduces to the integrations on the routes 2, 5, 7, and 10 that yield

## Acknowledgement

## References and links

1. | M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. |

2. | C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B |

3. | S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. |

4. | R. Mendis and D. Grischkowsky, “Plastic ribbon thz waveguides,” J. Appl. Phys. |

5. | K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature |

6. | J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express |

7. | M. K. Mbonye, V. Astley, W. L. Chan, J. A. Deibel, and D. M. Mittleman, “A terahertz dual wire waveguide,” in |

8. | M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. |

9. | H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express |

10. | S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. |

11. | D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Elsevier, Progress in Quantum Electronics |

12. | S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. |

13. | S. Matsuura and H. Ito, “Generation of CW terahertz radiation with photomixing,” Top. Appl. Phys. |

14. | W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. |

15. | W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. |

16. | W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. |

17. | J. Y. Courtois, J. M. Courty, and J. C. Mertz, “Internal dynamics of multilevel atoms near a vacuum-dielectric interface,” Phys. Rev. A |

18. | L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” J. Phys. |

19. | P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. |

20. | C. Fattinger and D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. |

21. | P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B |

22. | R. Gordon, “Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons,” Phys. Rev. B |

23. | R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B |

24. | D. M. Pozar, |

25. | A. Yariv, and P. Yeh, |

26. | J. D. Jackson, |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(230.7370) Optical devices : Waveguides

**ToC Category:**

Optical Devices

**History**

Original Manuscript: July 21, 2010

Revised Manuscript: September 17, 2010

Manuscript Accepted: October 4, 2010

Published: October 11, 2010

**Citation**

Hamid Pahlevaninezhad and Thomas E. Darcie, "Coupling of terahertz waves to a two-wire waveguide," Opt. Express **18**, 22614-22624 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-22614

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### References

- M. Y. Frankel, S. Gupta, J. A. Valdmanis, and G. A. Mourou, “Terahertz attenuation and dispersion characteristics of coplanar transmission lines,” IEEE Trans. Microw. Theory Tech. 39(6), 910–916 (1991). [CrossRef]
- C. G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B 17(5), 851–863 (2000). [CrossRef]
- S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fibers,” Appl. Phys. Lett. 76(15), 1987–1989 (2000). [CrossRef]
- R. Mendis and D. Grischkowsky, “Plastic ribbon thz waveguides,” J. Appl. Phys. 88(7), 4449–4451 (2000). [CrossRef]
- K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature 432(7015), 376–379 (2004). [CrossRef] [PubMed]
- J. A. Deibel, K. Wang, M. D. Escarra, and D. M. Mittleman, “Enhanced coupling of terahertz radiation to cylindrical wire waveguides,” Opt. Express 14(1), 279–290 (2006). [CrossRef] [PubMed]
- M. K. Mbonye, V. Astley, W. L. Chan, J. A. Deibel, and D. M. Mittleman, “A terahertz dual wire waveguide,” in Lasers and Electro-Optics Conference, Optical Society of America, 2007, paper CThLL1.
- M. K. Mbonye, R. Mendis, and D. M. Mittleman, “A terahertz two-wire waveguide with low bending loss,” Appl. Phys. Lett. 95(23), 233506 (2009). [CrossRef]
- H. Pahlevaninezhad, T. E. Darcie, and B. Heshmat, “Two-wire waveguide for terahertz,” Opt. Express 18(7), 7415–7420 (2010). [CrossRef] [PubMed]
- S. Matsuura, M. Tani, and K. Sakai, “Generation of coherent terahertz radiation by photomixing in dipole photoconductive antennas,” Appl. Phys. Lett. 70(5), 559‒561 (1997). [CrossRef]
- D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Elsevier, Progress in Quantum Electronics 28(1), 1–66 (2004), doi:. [CrossRef]
- S. M. Duffy, S. Verghese, A. McIntosh, A. Jackson, A. C. Gossard, and S. Matsuura, “Accurate modeling of dual dipole and slot elements used with photomixers for coherent terahertz output power,” IEEE Trans. Microw. Theory Tech. 49(6), 1032–1038 (2001). [CrossRef]
- S. Matsuura and H. Ito, “Generation of CW terahertz radiation with photomixing,” Top. Appl. Phys. 97, 157–202 (2005). [CrossRef]
- W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane interface. I. Total radiated power,” J. Opt. Soc. Am. 67(12), 1607–1614 (1977). [CrossRef]
- W. Lukosz and R. E. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67(12), 1615–1619 (1977). [CrossRef]
- W. Lukosz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. III. Radiation patterns of dipoles with arbitrary orientation,” J. Opt. Soc. Am. 69(11), 1495–1502 (1979). [CrossRef]
- J. Y. Courtois, J. M. Courty, and J. C. Mertz, “Internal dynamics of multilevel atoms near a vacuum-dielectric interface,” Phys. Rev. A 53(3), 1862–1878 (1996). [CrossRef] [PubMed]
- L. Luan, P. R. Sievert, and J. B. Ketterson, “Near-field and far-field electric dipole radiation in the vicinity of a planar dielectric half space,” J. Phys. 8, 264 (2006), doi:.
- P. U. Jepsen and S. R. Keiding, “Radiation patterns from lens-coupled terahertz antennas,” Opt. Lett. 20(8), 807–809 (1995). [CrossRef] [PubMed]
- C. Fattinger and D. Grischkowsky, “Terahertz beams,” Appl. Phys. Lett. 54(6), 490 (1989). [CrossRef]
- P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B 13(11), 2424–2436 (1996). [CrossRef]
- R. Gordon, “Vectorial method for calculating the Fresnel reflection of surface plasmon polaritons,” Phys. Rev. B 74, 153417 (2006). URL http://link.aps.org/abstract/PRB/v74/e153417 .
- R. Gordon, “Light in a subwavelength slit in a metal: Propagation and reflection,” Phys. Rev. B 73, 153405 (2006). URL http://link.aps.org/abstract/PRB/v73/e153405 .
- D. M. Pozar, Microwave engineering: 3rd Ed. (John Wiley & Sons, 2005), Chap.4.
- A. Yariv, and P. Yeh, Optical waves in crystals: propagation and control of laser radiation (John Wiley & Sons, 1984), Chap.11.
- J. D. Jackson, Classical electrodynamics 3rd Ed. (John Wiley & Sons,1999), pp. 390–394.

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