Abstract

The process of terahertz generation through optical rectification in a nonlinear crystal is modeled using discretized equivalent current sources. The equivalent terahertz sources are distributed in the active volume and computed based on a separately modeled near-infrared pump beam. This approach can be used to define an appropriate excitation for full-wave electromagnetic numerical simulations of the generated terahertz radiation. This enables predictive modeling of the near-field interactions of the terahertz beam with micro-structured samples, e.g. in a near-field time-resolved microscopy system. The distributed source model is described in detail, and an implementation in a particular full-wave simulation tool is presented. The numerical results are then validated through a series of measurements on square apertures. The general principle can be applied to other nonlinear processes with possible implementation in any full-wave numerical electromagnetic solver.

©2012 Optical Society of America

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  1. A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from <110> zinc-blende crystals,” Appl. Phys. Lett. 64(11), 1324–1326 (1994).
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  26. D. H. Auston, “Subpicosecond electro-optic shock waves,” Appl. Phys. Lett. 43(8), 713–715 (1983).
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    [Crossref]
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    [Crossref]
  31. C. Fumeaux, K. Sankaran, and R. Vahldieck, “Spherical perfectly matched absorber for finite-volume 3-D domain truncation,” IEEE Trans. Microw. Theory Tech. 55(12), 2773–2781 (2007).
    [Crossref]
  32. D. Baumann, C. Fumeaux, C. Hafner, and E. P. Li, “A modular implementation of dispersive materials for time-domain simulations with application to gold nanospheres at optical frequencies,” Opt. Express 17(17), 15186–15200 (2009).
    [Crossref] [PubMed]
  33. M. C. Hoffmann, K.-L. Yeh, J. Hebling, and K. A. Nelson, “Efficient terahertz generation by optical rectification at 1035 nm,” Opt. Express 15(18), 11706–11713 (2007).
    [Crossref] [PubMed]
  34. E. K. Rahani and T. Kundu, “Electromagnetic THz radiation modeling by DPSM,” Int. J. Infrared Millim. Waves 33(3), 376–390 (2012).
    [Crossref]
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    [Crossref]
  37. A. J. L. Adam, J. M. Brok, M. A. Seo, K. J. Ahn, D. S. Kim, J. H. Kang, Q. H. Park, M. Nagel, and P. C. Planken, “Advanced terahertz electric near-field measurements at sub-wavelength diameter metallic apertures,” Opt. Express 16(10), 7407–7417 (2008).
    [Crossref] [PubMed]
  38. M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express 16(25), 20484–20489 (2008).
    [Crossref] [PubMed]
  39. A. Bitzer and M. Walther, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92(23), 231101 (2008).
    [Crossref]
  40. K. Serita, S. Mizuno, H. Murakami, I. Kawayama, Y. Takahashi, M. Yoshimura, Y. Mori, J. Darmo, and M. Tonouchi, “Scanning laser terahertz near-field imaging system,” Opt. Express 20(12), 12959–12965 (2012).
    [Crossref] [PubMed]
  41. D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Čherenkov radiation from sfemtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53(16), 1555–1558 (1984).
    [Crossref]
  42. P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30(1), 29–31 (2005).
    [Crossref] [PubMed]
  43. W. Withayachumnankul, H. Lin, K. Serita, C. M. Shah, S. Sriram, M. Bhaskaran, M. Tonouchi, C. Fumeaux, and D. Abbott, “Sub-diffraction thin-film sensing with planar terahertz metamaterials,” Opt. Express 20(3), 3345–3352 (2012).
    [Crossref] [PubMed]

2012 (3)

2011 (2)

2010 (2)

2009 (2)

2008 (7)

R. Lecaque, S. Grésillon, and C. Boccara, “THz emission microscopy with sub-wavelength broadband source,” Opt. Express 16(7), 4731–4738 (2008).
[Crossref] [PubMed]

A. J. L. Adam, J. M. Brok, M. A. Seo, K. J. Ahn, D. S. Kim, J. H. Kang, Q. H. Park, M. Nagel, and P. C. Planken, “Advanced terahertz electric near-field measurements at sub-wavelength diameter metallic apertures,” Opt. Express 16(10), 7407–7417 (2008).
[Crossref] [PubMed]

M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express 16(25), 20484–20489 (2008).
[Crossref] [PubMed]

J. B. Khurgin, M. W. Pruessner, T. H. Stievater, and W. S. Rabinovich, “Suspended AlGaAs waveguides for tunable difference frequency generation in mid-infrared,” Opt. Lett. 33(24), 2904–2906 (2008).
[Crossref] [PubMed]

A. Bitzer and M. Walther, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92(23), 231101 (2008).
[Crossref]

M. Neshat, D. Saeedkia, and S. Safavi-Naeini, “Semi-analytical calculation of terahertz signal generated from photocurrent radiation in traveling-wave photonic mixers,” Int. J. Infrared Millim. Waves 29(9), 809–822 (2008).
[Crossref]

D. Li and G. Ma, “Pump-wavelength dependence of terahertz radiation via optical rectification in (110)-oriented ZnTe crystal,” J. Appl. Phys. 103(12), 123101 (2008).
[Crossref]

2007 (3)

W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95(8), 1528–1558 (2007).
[Crossref]

C. Fumeaux, K. Sankaran, and R. Vahldieck, “Spherical perfectly matched absorber for finite-volume 3-D domain truncation,” IEEE Trans. Microw. Theory Tech. 55(12), 2773–2781 (2007).
[Crossref]

M. C. Hoffmann, K.-L. Yeh, J. Hebling, and K. A. Nelson, “Efficient terahertz generation by optical rectification at 1035 nm,” Opt. Express 15(18), 11706–11713 (2007).
[Crossref] [PubMed]

2005 (2)

2004 (2)

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, “Antenna effects in terahertz apertureless near-field optical microscopy,” Appl. Phys. Lett. 85(14), 2715–2717 (2004).
[Crossref]

C. Fumeaux, D. Baumann, P. Leuchtmann, and R. Vahldieck, “A generalized local time-step scheme for efficient FVTD simulations in strongly inhomogeneous meshes,” IEEE Trans. Microw. Theory Tech. 52(3), 1067–1076 (2004).
[Crossref]

2003 (2)

2002 (1)

2000 (1)

1994 (1)

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from <110> zinc-blende crystals,” Appl. Phys. Lett. 64(11), 1324–1326 (1994).
[Crossref]

1990 (2)

N. K. Madsen and R. W. Ziolkowski, “A three-dimensional modified finite volume technique for Maxwell’s equations,” Electromagnetics 10(1-2), 147–161 (1990).
[Crossref]

V. Shankar, A. H. Mohammadian, and W. F. Hall, “A time-domain, finite-volume treatment for the Maxwell equations,” Electromagnetics 10(1-2), 127–145 (1990).
[Crossref]

1984 (2)

D. A. Kleinman and D. H. Auston, “Theory of electrooptic shock radiation in nonlinear optical media,” IEEE J. Quantum Electron. 20(8), 964–970 (1984).
[Crossref]

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Čherenkov radiation from sfemtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53(16), 1555–1558 (1984).
[Crossref]

1983 (1)

D. H. Auston, “Subpicosecond electro-optic shock waves,” Appl. Phys. Lett. 43(8), 713–715 (1983).
[Crossref]

1977 (1)

T. Weiland, “A discretization method for the solution of Maxwell’s equations for six-component fields,” AEU, Int. J. Electron. Commun. 31(3), 116–120 (1977).

Abbott, D.

Adam, A. J. L.

Agrawal, G. P.

Ahmed, I.

Ahn, K. J.

Alexander, M.

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from <110> zinc-blende crystals,” Appl. Phys. Lett. 64(11), 1324–1326 (1994).
[Crossref]

Atakaramians, S.

W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95(8), 1528–1558 (2007).
[Crossref]

Auston, D. H.

D. A. Kleinman and D. H. Auston, “Theory of electrooptic shock radiation in nonlinear optical media,” IEEE J. Quantum Electron. 20(8), 964–970 (1984).
[Crossref]

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Čherenkov radiation from sfemtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53(16), 1555–1558 (1984).
[Crossref]

D. H. Auston, “Subpicosecond electro-optic shock waves,” Appl. Phys. Lett. 43(8), 713–715 (1983).
[Crossref]

Balakrishnan, J.

W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95(8), 1528–1558 (2007).
[Crossref]

Baumann, D.

D. Baumann, C. Fumeaux, C. Hafner, and E. P. Li, “A modular implementation of dispersive materials for time-domain simulations with application to gold nanospheres at optical frequencies,” Opt. Express 17(17), 15186–15200 (2009).
[Crossref] [PubMed]

C. Fumeaux, D. Baumann, P. Leuchtmann, and R. Vahldieck, “A generalized local time-step scheme for efficient FVTD simulations in strongly inhomogeneous meshes,” IEEE Trans. Microw. Theory Tech. 52(3), 1067–1076 (2004).
[Crossref]

Bhaskaran, M.

Bitzer, A.

A. Bitzer and M. Walther, “Terahertz near-field imaging of metallic subwavelength holes and hole arrays,” Appl. Phys. Lett. 92(23), 231101 (2008).
[Crossref]

Bliss, D.

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from <110> zinc-blende crystals,” Appl. Phys. Lett. 64(11), 1324–1326 (1994).
[Crossref]

Boccara, C.

Brok, J. M.

Chen, Q.

Cheung, K. P.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Čherenkov radiation from sfemtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53(16), 1555–1558 (1984).
[Crossref]

Côté, D.

Dakovski, G.

Darmo, J.

Dissanayake, C. M.

Fan, S.

Fejer, M. M.

Ferguson, B.

W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95(8), 1528–1558 (2007).
[Crossref]

Fischer, B. M.

H. Lin, C. Fumeaux, B. M. Fischer, and D. Abbott, “Modelling of sub-wavelength THz sources as Gaussian apertures,” Opt. Express 18(17), 17672–17683 (2010).
[Crossref] [PubMed]

W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95(8), 1528–1558 (2007).
[Crossref]

P. U. Jepsen and B. M. Fischer, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett. 30(1), 29–31 (2005).
[Crossref] [PubMed]

Fumeaux, C.

Grésillon, S.

Hafner, C.

Hall, W. F.

V. Shankar, A. H. Mohammadian, and W. F. Hall, “A time-domain, finite-volume treatment for the Maxwell equations,” Electromagnetics 10(1-2), 127–145 (1990).
[Crossref]

Hebling, J.

Hoffmann, M. C.

Jepsen, P. U.

Jiang, Z.

Jin, Y.

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from <110> zinc-blende crystals,” Appl. Phys. Lett. 64(11), 1324–1326 (1994).
[Crossref]

Jones, I.

W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95(8), 1528–1558 (2007).
[Crossref]

Kang, J. H.

Kawase, K.

Kawayama, I.

Khoo, E. H.

Khurgin, J. B.

Kim, D. S.

Kiwa, T.

Kleinman, D. A.

D. H. Auston, K. P. Cheung, J. A. Valdmanis, and D. A. Kleinman, “Čherenkov radiation from sfemtosecond optical pulses in electro-optic media,” Phys. Rev. Lett. 53(16), 1555–1558 (1984).
[Crossref]

D. A. Kleinman and D. H. Auston, “Theory of electrooptic shock radiation in nonlinear optical media,” IEEE J. Quantum Electron. 20(8), 964–970 (1984).
[Crossref]

Kubera, B.

Kundu, T.

E. K. Rahani and T. Kundu, “Electromagnetic THz radiation modeling by DPSM,” Int. J. Infrared Millim. Waves 33(3), 376–390 (2012).
[Crossref]

Kurniawan, O.

Larkin, J.

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from <110> zinc-blende crystals,” Appl. Phys. Lett. 64(11), 1324–1326 (1994).
[Crossref]

Lecaque, R.

Lee, J. W.

Leuchtmann, P.

C. Fumeaux, D. Baumann, P. Leuchtmann, and R. Vahldieck, “A generalized local time-step scheme for efficient FVTD simulations in strongly inhomogeneous meshes,” IEEE Trans. Microw. Theory Tech. 52(3), 1067–1076 (2004).
[Crossref]

Li, D.

D. Li and G. Ma, “Pump-wavelength dependence of terahertz radiation via optical rectification in (110)-oriented ZnTe crystal,” J. Appl. Phys. 103(12), 123101 (2008).
[Crossref]

Li, E. P.

Lin, H.

Ma, G.

D. Li and G. Ma, “Pump-wavelength dependence of terahertz radiation via optical rectification in (110)-oriented ZnTe crystal,” J. Appl. Phys. 103(12), 123101 (2008).
[Crossref]

Ma, X. F.

A. Rice, Y. Jin, X. F. Ma, X.-C. Zhang, D. Bliss, J. Larkin, and M. Alexander, “Terahertz optical rectification from <110> zinc-blende crystals,” Appl. Phys. Lett. 64(11), 1324–1326 (1994).
[Crossref]

Madsen, N. K.

N. K. Madsen and R. W. Ziolkowski, “A three-dimensional modified finite volume technique for Maxwell’s equations,” Electromagnetics 10(1-2), 147–161 (1990).
[Crossref]

Mickan, S. P.

W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95(8), 1528–1558 (2007).
[Crossref]

Mittleman, D. M.

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, “Antenna effects in terahertz apertureless near-field optical microscopy,” Appl. Phys. Lett. 85(14), 2715–2717 (2004).
[Crossref]

Mizuno, S.

Mohammadian, A. H.

V. Shankar, A. H. Mohammadian, and W. F. Hall, “A time-domain, finite-volume treatment for the Maxwell equations,” Electromagnetics 10(1-2), 127–145 (1990).
[Crossref]

Mori, Y.

Murakami, H.

Nagel, M.

Nelson, K. A.

Neshat, M.

M. Neshat, D. Saeedkia, and S. Safavi-Naeini, “Semi-analytical calculation of terahertz signal generated from photocurrent radiation in traveling-wave photonic mixers,” Int. J. Infrared Millim. Waves 29(9), 809–822 (2008).
[Crossref]

Ng, B. W.-H.

W. Withayachumnankul, G. M. Png, X. Yin, S. Atakaramians, I. Jones, H. Lin, B. S. Y. Ung, J. Balakrishnan, B. W.-H. Ng, B. Ferguson, S. P. Mickan, B. M. Fischer, and D. Abbott, “T-ray sensing and imaging,” Proc. IEEE 95(8), 1528–1558 (2007).
[Crossref]

Park, Q. H.

Planken, P. C.

Planken, P. C. M.

M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, K. J. Ahn, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Near field imaging of terahertz focusing onto rectangular apertures,” Opt. Express 16(25), 20484–20489 (2008).
[Crossref] [PubMed]

K. Wang, D. M. Mittleman, N. C. J. van der Valk, and P. C. M. Planken, “Antenna effects in terahertz apertureless near-field optical microscopy,” Appl. Phys. Lett. 85(14), 2715–2717 (2004).
[Crossref]

Png, G. M.

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E. K. Rahani and T. Kundu, “Electromagnetic THz radiation modeling by DPSM,” Int. J. Infrared Millim. Waves 33(3), 376–390 (2012).
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Supplementary Material (1)

» Media 1: MOV (2428 KB)     

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Figures (10)

Fig. 1
Fig. 1 Illustration of the paraxial modeling of the pump beam in the GaAs crystal, including first reflection. (a) Schematic of the spatial beam profile w inside the crystal, showing an exaggerated divergence with the incident beam profile in solid blue line and the reflected beam profile in red dashed line. (b) Schematic of the transient pulse propagation showing envelope attenuation and modeled normalized pump beam power profile at three different points in time.

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Fig. 2
Fig. 2 Schematic principle of the distributed source modeling for nonlinear rectification. (a) On-axis discretization of the nonlinear crystal in slices modeled as radiating apertures, with transverse discretization of sources schematically depicted as dots. The dots are evenly distributed for representation, but can also be arranged on an unstructured grid. (b) Equivalence of spatial source amplitude distribution to instantaneous IR power density. (c) Time-domain linear transfer function from the IR pulse to the generated terahertz pulse.

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Fig. 3
Fig. 3 Full-wave simulation set-up. (a) General view of interfaces in the FVTD model, including spherical absorbing boundary conditions (ABC), as well as perfect electric conductor (PEC) and perfect magnetic conductor (PMC) symmetries. The materials are indicated in bold fonts. (b) Partial view of one discretized source aperture inside the crystal, the outline of other source planes being shown as thick lines. The tetrahedral volume discretization is not shown for the sake of clarity. (c) Schematic of one triangular element on a source aperture, with indication of the typical dimension, as well as the orientation of normal vector and current source vectors.

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Fig. 4
Fig. 4 Schematic of far-field detection process, (a) far-field radiation pattern and acceptance angle by the first parabolic mirror or lens, (b) example of far-field amplitude pattern at 2 THz for the case considered, and illustration of the acceptance angle inside which the detected signal is integrated.

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Fig. 5
Fig. 5 Probing apertures on a metallized GaAs crystal. (a) Photographs of manufactured apertures with three different sizes. (b) Triangular surface discretization of the FVTD model including the three possible apertures. The size of the aperture can be selected by selecting the surface properties either as metal or transparent GaAs-Air interface.

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Fig. 6
Fig. 6 (a) Instantaneous images of the E-field magnitude at different times, showing the terahertz pulse amplification and propagation through the nonlinear crystal. Only one half of the symmetric field distribution (for x > 0) is shown, with a PEC symmetry at x = 0. The excitation pulse is a sine-modulated Gaussian pulse as described in Sec. 2.3. (Media 1). (b) Zoom on the (rescaled) simulated THz pulse in the GaAs illustrating the Čherenkov cone with an angle close to 20.42 degrees, arising from the velocity mismatch between the IR and terahertz beam (c) Simulated pulse detected at a sensor located 400 μm in front of the aperture.

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Fig. 7
Fig. 7 Comparison of measured and simulated normalized transmission E-field spectrum for the three apertures considered (25, 50 and 75 μm) when the pump beam is focused on the crystal input surface with a waist of 25 μm. (a) Spectrum including the initial terahertz pulse only. (b) Spectrum including the first secondary pulse.

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Fig. 8
Fig. 8 Normalized transmission E-field spectrum in two independent measurements, compared to the simulation. The curves (a) correspond to the top curves of Fig. 7(b).

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Fig. 9
Fig. 9 Comparison of measured and simulated normalized transmission E-field spectrum for the three apertures considered (25, 50 and 75 μm) for the pump beam defocused, with a beam radius of around 50 μm at the input surface. (a) Spectrum including the initial terahertz pulse only. (b) Spectrum including the first secondary pulse.

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Fig. 10
Fig. 10 Illustrative results from the source simulations. (a) Amplitude growth in the crystal. In those graphs, the velocity mismatch between pump and terahertz radiation is the same as in the previous validation example. The standing wave pattern is due to reflections. (b) Field amplitude at 2 THz for different distances (0.5, 25 and 50 μm) from the output surface. There is a localized field enhancement in the small aperture, and therefore, the scale for the top right plot is capped to about 1/3 of the achieved value.

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Equations (7)

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J → THz (t, r → )= n ^ × H → THz (t, r → ) M → THz (t, r → )=− n ^ × E → THz (t, r → ) ,
E THz (t, r → )= P IR (t, r → )*T(t) H THz (t, r → )= P IR (t, r → )*T(t)/ Z ε r ,THz  ,
J → THz (t,x,y,z)= ∑ m refl =0 M refl η optrect P IR,max ︸ (a) ⋅ e − ( t D /2 σ G ) 2 ⋅sin(2π f c t D ) ︸ (b)  ⋅ e − α IR z p ︸ (c) ⋅ R m refl ︸ (d) ⋅ e −2( x 2 + y 2 )/ w 2 ( z p ) ︸ (e) ⋅ ( ρ x ρ y 0 ) ︸ (f)
M → THz (t,x,y,z)= ∑ m refl =0 M refl ( | J → THz (t, r → ) |⋅ Z ε r ,THz ) m refl ︸ (g) ⋅ (−1) m refl ⋅( − ρ y ρ x 0 ) ︸ (h) ,
z p ={ z+ m refl ⋅D       for m refl even (D−z)+ m refl ⋅D for m refl odd ,
t D =t−3 σ G − z p c 0 / n IR .
α C ⌣ =arccos( ε r,IR ε r,THz ).

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