Abstract

We apply Kirchhoff’s heat equation to model the influence of a CW terahertz beam on a sample of water, which is assumed to be static. We develop a generalized model, which easily can be applied to other liquids and solids by changing the material constants. If the terahertz light source is focused down to a spot with a diameter of 0.5 mm, we find that the steady-state temperature increase per milliwatt of transmitted power is 1.8°C/mW. A quantum cascade laser can produce a CW beam in the order of several milliwatts and this motivates the need to estimate the effect of beam power on the sample temperature. For THz time domain systems, we indicate how to use our model as a worst-case approximation based on the beam average power. It turns out that THz pulses created from photoconductive antennas give a negligible increase in temperature. As biotissue contains a high water content, this leads to a discussion of worst-case predictions for THz heating of the human body in order to motivate future detailed study. An open source Matlab implementation of our model is freely available for use at www.eleceng.adelaide.edu.au/thz.

© 2010 Optical Society of America

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    [CrossRef]

2009 (2)

2008 (3)

G. M. Png, J.W. Choi, B.W.-H. Ng, S. P. Mickan, D. Abbott, and X.-C. Zhang, "The impact of hydration changes in fresh bio-tissue on THz spectroscopic measurements," Phys. Med. Biol. 53, 3501-3517 (2008).
[CrossRef] [PubMed]

N. P. Bondar, I. L. Kovalenko, D. F. Avgustinovich, A. G. Khamoyan, and N. N. Kudryavtseva, "Behavioral effect of terahertz waves in male mice," B. Exp. Biol. Med. 145,401-405 (2008).
[CrossRef]

A. Beneduci, "Which is the effective time scale of the fast Debye relaxation process in water?" J. Mol. Liq. 138, 55-60 (2008).
[CrossRef]

2007 (4)

R. Appleby and H. B. Wallace, "Standoff detection of weapons and contraband in the 100 GHz to 1 THz region," IEEE Trans. Antennas Propag. 55,2944-2956 (2007).
[CrossRef]

P. U. Jepsen, U. Moeller, and H. Merbold, "Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy," Opt. Express 15,14717-14737 (2007).
[CrossRef] [PubMed]

J. R. Knab, J.-Y. Chen, Y. He, and A. G. Markelz, "Terahertz measurements of protein relaxational dynamics," Proc. IEEE 95,1605-1610 (2007).
[CrossRef]

W. J. Ellison, "Permittivity of pure water, at standard atmospheric pressure, over the frequency range 0-25 THz and the temperature range 0-100◦C," J. Phys. Chem. Ref. Data 36,1-18 (2007).
[CrossRef]

2006 (2)

2004 (1)

J. K. Vij, D. R. J. Simpson, and O. E. Panarina, "Far infrared spectroscopy of water at different temperatures: GHz to THz dielectric spectroscopy of water," J. Mol. Liq. 112,125-135 (2004).
[CrossRef]

2003 (1)

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

2002 (5)

B. M. Fischer, M. Walther, and P. U. Jepsen, "Far-infrared vibrational modes of DNA components studied by terahertz time-domain spectroscopy," Phys. Med. Biol. 47,3807-3814 (2002).
[CrossRef] [PubMed]

S. P. Mickan, A, Menikh, H. B. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, and X. C. Zhang, "Labelfree bioaffinity detection using terahertz technology," Phys. Med. Biol. 47,3789-3795 (2002).
[CrossRef] [PubMed]

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, "High-power terahertz radiation from relativistic electrons," Nature 420,153-156 (2002).
[CrossRef] [PubMed]

M. Sherwin, "Terahertz power," Nature 420,131-132 (2002).
[CrossRef] [PubMed]

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

1998 (1)

D. Abbott, B. Davis, B. Gonzalez, A. Hernandez, and K. Eshraghian, "Modelling of low power CW laser beam heating effects on a GaAs substrate," Solid-State Electron. 42,809-816 (1998).
[CrossRef]

1996 (1)

J. T. Kindt and C. A. Schmuttenmaer, "Far-infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy," J. Phys. Chem. 100,10373-10379 (1996).
[CrossRef]

1995 (3)

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, "THz reflection spectroscopy of liquid water," Chem. Phys. Lett. 240,330-333 (1995).
[CrossRef]

F. Bonani and G. Ghione, "On the application of the Kirchhoff transformation to the steady-state thermal analysis of semiconductor devices with temperature-dependent and piecewise inhomogeneous thermal conductivity," Solid-State Electron. 38,1409-1412 (1995).
[CrossRef]

M. L. V. Ramires, C. A. Nieto de Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W. A. Wakeham, "Standard reference data for the thermal conductivity of water," J. Phys. Chem. Ref. Data 24,1377-1381 (1995).
[CrossRef]

1992 (1)

W. Grundler and F. Kaiser, "Experimental evidence for coherent excitations correlated with cell growth," Nanobiology 1,163-176 (1992).

1988 (1)

R. R. Warner,M. C. Myers, and D. A. Taylor, "Electron probe analysis of human skin: Determination of the water concentration profile," J. Invest. Dermatol. 90,218-224 (1988).
[CrossRef] [PubMed]

1980 (1)

H. Fröhlich, "The biological effects of microwaves and related questions," Adv. Electron. El. Phys. 53,85-152 (1980).
[CrossRef]

1973 (1)

R. W. Guynn and R. L. Veech, "The equilibrium constants of the adenosine triphosphate hydrolysis and the adenosine triphosphate citrate lyase reactions," J. Biol. Chem. 248,6966-6972 (1973).
[PubMed]

Abbott, D.

J. Balakrishnan, B. M. Fischer, and D. Abbott, "Fixed dual-thickness terahertz liquid spectroscopy using a spinning sample technique," IEEE Photon. J. 1,88-98 (2009).
[CrossRef]

G. M. Png, J.W. Choi, B.W.-H. Ng, S. P. Mickan, D. Abbott, and X.-C. Zhang, "The impact of hydration changes in fresh bio-tissue on THz spectroscopic measurements," Phys. Med. Biol. 53, 3501-3517 (2008).
[CrossRef] [PubMed]

D. Abbott, B. Davis, B. Gonzalez, A. Hernandez, and K. Eshraghian, "Modelling of low power CW laser beam heating effects on a GaAs substrate," Solid-State Electron. 42,809-816 (1998).
[CrossRef]

Allen, S. J.

J. Xu, K. W. Plaxco, and S. J. Allen, "Absorption spectra of liquid water and aqueous buffers between 0.3 and 3.72 THz," J. Chem. Phys. 124,1-3 (2006).
[CrossRef]

Alton, J.

Appleby, R.

R. Appleby and H. B. Wallace, "Standoff detection of weapons and contraband in the 100 GHz to 1 THz region," IEEE Trans. Antennas Propag. 55,2944-2956 (2007).
[CrossRef]

Arnone, D. D.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

Ashworth, P. C.

Assael, M. J.

M. L. V. Ramires, C. A. Nieto de Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W. A. Wakeham, "Standard reference data for the thermal conductivity of water," J. Phys. Chem. Ref. Data 24,1377-1381 (1995).
[CrossRef]

Avgustinovich, D. F.

N. P. Bondar, I. L. Kovalenko, D. F. Avgustinovich, A. G. Khamoyan, and N. N. Kudryavtseva, "Behavioral effect of terahertz waves in male mice," B. Exp. Biol. Med. 145,401-405 (2008).
[CrossRef]

Balakrishnan, J.

J. Balakrishnan, B. M. Fischer, and D. Abbott, "Fixed dual-thickness terahertz liquid spectroscopy using a spinning sample technique," IEEE Photon. J. 1,88-98 (2009).
[CrossRef]

Barbieri, S.

Beere, H. E.

Beneduci, A.

A. Beneduci, "Which is the effective time scale of the fast Debye relaxation process in water?" J. Mol. Liq. 138, 55-60 (2008).
[CrossRef]

Berry, E.

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

Bonani, F.

F. Bonani and G. Ghione, "On the application of the Kirchhoff transformation to the steady-state thermal analysis of semiconductor devices with temperature-dependent and piecewise inhomogeneous thermal conductivity," Solid-State Electron. 38,1409-1412 (1995).
[CrossRef]

Bondar, N. P.

N. P. Bondar, I. L. Kovalenko, D. F. Avgustinovich, A. G. Khamoyan, and N. N. Kudryavtseva, "Behavioral effect of terahertz waves in male mice," B. Exp. Biol. Med. 145,401-405 (2008).
[CrossRef]

Carr, G. L.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, "High-power terahertz radiation from relativistic electrons," Nature 420,153-156 (2002).
[CrossRef] [PubMed]

Chamberlain, M.

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

Chen, J.-Y.

J. R. Knab, J.-Y. Chen, Y. He, and A. G. Markelz, "Terahertz measurements of protein relaxational dynamics," Proc. IEEE 95,1605-1610 (2007).
[CrossRef]

Choi, J.W.

G. M. Png, J.W. Choi, B.W.-H. Ng, S. P. Mickan, D. Abbott, and X.-C. Zhang, "The impact of hydration changes in fresh bio-tissue on THz spectroscopic measurements," Phys. Med. Biol. 53, 3501-3517 (2008).
[CrossRef] [PubMed]

Cole, B. E.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

Davis, B.

D. Abbott, B. Davis, B. Gonzalez, A. Hernandez, and K. Eshraghian, "Modelling of low power CW laser beam heating effects on a GaAs substrate," Solid-State Electron. 42,809-816 (1998).
[CrossRef]

Ellison, W. J.

W. J. Ellison, "Permittivity of pure water, at standard atmospheric pressure, over the frequency range 0-25 THz and the temperature range 0-100◦C," J. Phys. Chem. Ref. Data 36,1-18 (2007).
[CrossRef]

Eshraghian, K.

D. Abbott, B. Davis, B. Gonzalez, A. Hernandez, and K. Eshraghian, "Modelling of low power CW laser beam heating effects on a GaAs substrate," Solid-State Electron. 42,809-816 (1998).
[CrossRef]

Fischer, B. M.

J. Balakrishnan, B. M. Fischer, and D. Abbott, "Fixed dual-thickness terahertz liquid spectroscopy using a spinning sample technique," IEEE Photon. J. 1,88-98 (2009).
[CrossRef]

B. M. Fischer, M. Walther, and P. U. Jepsen, "Far-infrared vibrational modes of DNA components studied by terahertz time-domain spectroscopy," Phys. Med. Biol. 47,3807-3814 (2002).
[CrossRef] [PubMed]

Fitzgerald, A. J.

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

Fröhlich, H.

H. Fröhlich, "The biological effects of microwaves and related questions," Adv. Electron. El. Phys. 53,85-152 (1980).
[CrossRef]

Ghione, G.

F. Bonani and G. Ghione, "On the application of the Kirchhoff transformation to the steady-state thermal analysis of semiconductor devices with temperature-dependent and piecewise inhomogeneous thermal conductivity," Solid-State Electron. 38,1409-1412 (1995).
[CrossRef]

Gonzalez, B.

D. Abbott, B. Davis, B. Gonzalez, A. Hernandez, and K. Eshraghian, "Modelling of low power CW laser beam heating effects on a GaAs substrate," Solid-State Electron. 42,809-816 (1998).
[CrossRef]

Grundler, W.

W. Grundler and F. Kaiser, "Experimental evidence for coherent excitations correlated with cell growth," Nanobiology 1,163-176 (1992).

Guynn, R. W.

R. W. Guynn and R. L. Veech, "The equilibrium constants of the adenosine triphosphate hydrolysis and the adenosine triphosphate citrate lyase reactions," J. Biol. Chem. 248,6966-6972 (1973).
[PubMed]

He, Y.

J. R. Knab, J.-Y. Chen, Y. He, and A. G. Markelz, "Terahertz measurements of protein relaxational dynamics," Proc. IEEE 95,1605-1610 (2007).
[CrossRef]

Hernandez, A.

D. Abbott, B. Davis, B. Gonzalez, A. Hernandez, and K. Eshraghian, "Modelling of low power CW laser beam heating effects on a GaAs substrate," Solid-State Electron. 42,809-816 (1998).
[CrossRef]

Houghton, M.

Jacobsen, R. H.

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, "THz reflection spectroscopy of liquid water," Chem. Phys. Lett. 240,330-333 (1995).
[CrossRef]

Jepsen, P. U.

P. U. Jepsen, U. Moeller, and H. Merbold, "Investigation of aqueous alcohol and sugar solutions with reflection terahertz time-domain spectroscopy," Opt. Express 15,14717-14737 (2007).
[CrossRef] [PubMed]

B. M. Fischer, M. Walther, and P. U. Jepsen, "Far-infrared vibrational modes of DNA components studied by terahertz time-domain spectroscopy," Phys. Med. Biol. 47,3807-3814 (2002).
[CrossRef] [PubMed]

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, "THz reflection spectroscopy of liquid water," Chem. Phys. Lett. 240,330-333 (1995).
[CrossRef]

Jordan, K.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, "High-power terahertz radiation from relativistic electrons," Nature 420,153-156 (2002).
[CrossRef] [PubMed]

Kaiser, F.

W. Grundler and F. Kaiser, "Experimental evidence for coherent excitations correlated with cell growth," Nanobiology 1,163-176 (1992).

Keiding, S. R.

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, "THz reflection spectroscopy of liquid water," Chem. Phys. Lett. 240,330-333 (1995).
[CrossRef]

Khamoyan, A. G.

N. P. Bondar, I. L. Kovalenko, D. F. Avgustinovich, A. G. Khamoyan, and N. N. Kudryavtseva, "Behavioral effect of terahertz waves in male mice," B. Exp. Biol. Med. 145,401-405 (2008).
[CrossRef]

Kindt, J. T.

J. T. Kindt and C. A. Schmuttenmaer, "Far-infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy," J. Phys. Chem. 100,10373-10379 (1996).
[CrossRef]

Knab, J. R.

J. R. Knab, J.-Y. Chen, Y. He, and A. G. Markelz, "Terahertz measurements of protein relaxational dynamics," Proc. IEEE 95,1605-1610 (2007).
[CrossRef]

Kovalenko, I. L.

N. P. Bondar, I. L. Kovalenko, D. F. Avgustinovich, A. G. Khamoyan, and N. N. Kudryavtseva, "Behavioral effect of terahertz waves in male mice," B. Exp. Biol. Med. 145,401-405 (2008).
[CrossRef]

Kudryavtseva, N. N.

N. P. Bondar, I. L. Kovalenko, D. F. Avgustinovich, A. G. Khamoyan, and N. N. Kudryavtseva, "Behavioral effect of terahertz waves in male mice," B. Exp. Biol. Med. 145,401-405 (2008).
[CrossRef]

Linfield, E. H.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

Markelz, A. G.

J. R. Knab, J.-Y. Chen, Y. He, and A. G. Markelz, "Terahertz measurements of protein relaxational dynamics," Proc. IEEE 95,1605-1610 (2007).
[CrossRef]

Martin, M. C.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, "High-power terahertz radiation from relativistic electrons," Nature 420,153-156 (2002).
[CrossRef] [PubMed]

McKinney, W. R.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, "High-power terahertz radiation from relativistic electrons," Nature 420,153-156 (2002).
[CrossRef] [PubMed]

Merbold, H.

Mickan, S. P.

G. M. Png, J.W. Choi, B.W.-H. Ng, S. P. Mickan, D. Abbott, and X.-C. Zhang, "The impact of hydration changes in fresh bio-tissue on THz spectroscopic measurements," Phys. Med. Biol. 53, 3501-3517 (2008).
[CrossRef] [PubMed]

S. P. Mickan, A, Menikh, H. B. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, and X. C. Zhang, "Labelfree bioaffinity detection using terahertz technology," Phys. Med. Biol. 47,3789-3795 (2002).
[CrossRef] [PubMed]

Miles, R. E.

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

Moeller, U.

Myers, M. C.

R. R. Warner,M. C. Myers, and D. A. Taylor, "Electron probe analysis of human skin: Determination of the water concentration profile," J. Invest. Dermatol. 90,218-224 (1988).
[CrossRef] [PubMed]

Nagasaka, Y.

M. L. V. Ramires, C. A. Nieto de Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W. A. Wakeham, "Standard reference data for the thermal conductivity of water," J. Phys. Chem. Ref. Data 24,1377-1381 (1995).
[CrossRef]

Nagashima, A.

M. L. V. Ramires, C. A. Nieto de Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W. A. Wakeham, "Standard reference data for the thermal conductivity of water," J. Phys. Chem. Ref. Data 24,1377-1381 (1995).
[CrossRef]

Neil, G. R.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, "High-power terahertz radiation from relativistic electrons," Nature 420,153-156 (2002).
[CrossRef] [PubMed]

Ng, B.W.-H.

G. M. Png, J.W. Choi, B.W.-H. Ng, S. P. Mickan, D. Abbott, and X.-C. Zhang, "The impact of hydration changes in fresh bio-tissue on THz spectroscopic measurements," Phys. Med. Biol. 53, 3501-3517 (2008).
[CrossRef] [PubMed]

Nieto de Castro, C. A.

M. L. V. Ramires, C. A. Nieto de Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W. A. Wakeham, "Standard reference data for the thermal conductivity of water," J. Phys. Chem. Ref. Data 24,1377-1381 (1995).
[CrossRef]

Panarina, O. E.

J. K. Vij, D. R. J. Simpson, and O. E. Panarina, "Far infrared spectroscopy of water at different temperatures: GHz to THz dielectric spectroscopy of water," J. Mol. Liq. 112,125-135 (2004).
[CrossRef]

Pepper, M.

P. C. Ashworth, E. Pickwell-MacPherson, E. Provenzano, S. E. Pinder, A. D. Purushotham, M. Pepper, and V. P. Wallace, "Terahertz pulsed spectroscopy of freshly excised human breast cancer," Opt. Express 17, 12444-12454 (2009).
[CrossRef] [PubMed]

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

Pickwell-MacPherson, E.

Pinder, S. E.

Plaxco, K. W.

J. Xu, K. W. Plaxco, and S. J. Allen, "Absorption spectra of liquid water and aqueous buffers between 0.3 and 3.72 THz," J. Chem. Phys. 124,1-3 (2006).
[CrossRef]

Png, G. M.

G. M. Png, J.W. Choi, B.W.-H. Ng, S. P. Mickan, D. Abbott, and X.-C. Zhang, "The impact of hydration changes in fresh bio-tissue on THz spectroscopic measurements," Phys. Med. Biol. 53, 3501-3517 (2008).
[CrossRef] [PubMed]

Provenzano, E.

Purushotham, A. D.

Pye, R. J.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

Ramires, M. L. V.

M. L. V. Ramires, C. A. Nieto de Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W. A. Wakeham, "Standard reference data for the thermal conductivity of water," J. Phys. Chem. Ref. Data 24,1377-1381 (1995).
[CrossRef]

Ritchie, D.

Schmuttenmaer, C. A.

J. T. Kindt and C. A. Schmuttenmaer, "Far-infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy," J. Phys. Chem. 100,10373-10379 (1996).
[CrossRef]

Sherwin, M.

M. Sherwin, "Terahertz power," Nature 420,131-132 (2002).
[CrossRef] [PubMed]

Simpson, D. R. J.

J. K. Vij, D. R. J. Simpson, and O. E. Panarina, "Far infrared spectroscopy of water at different temperatures: GHz to THz dielectric spectroscopy of water," J. Mol. Liq. 112,125-135 (2004).
[CrossRef]

Sirtori, C.

Smith, M. A.

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

Smye, S. W.

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

Taylor, D. A.

R. R. Warner,M. C. Myers, and D. A. Taylor, "Electron probe analysis of human skin: Determination of the water concentration profile," J. Invest. Dermatol. 90,218-224 (1988).
[CrossRef] [PubMed]

Thrane, L.

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, "THz reflection spectroscopy of liquid water," Chem. Phys. Lett. 240,330-333 (1995).
[CrossRef]

Veech, R. L.

R. W. Guynn and R. L. Veech, "The equilibrium constants of the adenosine triphosphate hydrolysis and the adenosine triphosphate citrate lyase reactions," J. Biol. Chem. 248,6966-6972 (1973).
[PubMed]

Vij, J. K.

J. K. Vij, D. R. J. Simpson, and O. E. Panarina, "Far infrared spectroscopy of water at different temperatures: GHz to THz dielectric spectroscopy of water," J. Mol. Liq. 112,125-135 (2004).
[CrossRef]

Wakeham, W. A.

M. L. V. Ramires, C. A. Nieto de Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W. A. Wakeham, "Standard reference data for the thermal conductivity of water," J. Phys. Chem. Ref. Data 24,1377-1381 (1995).
[CrossRef]

Walker, G. C.

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

Wallace, H. B.

R. Appleby and H. B. Wallace, "Standoff detection of weapons and contraband in the 100 GHz to 1 THz region," IEEE Trans. Antennas Propag. 55,2944-2956 (2007).
[CrossRef]

Wallace, V. P.

P. C. Ashworth, E. Pickwell-MacPherson, E. Provenzano, S. E. Pinder, A. D. Purushotham, M. Pepper, and V. P. Wallace, "Terahertz pulsed spectroscopy of freshly excised human breast cancer," Opt. Express 17, 12444-12454 (2009).
[CrossRef] [PubMed]

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

Walther, M.

B. M. Fischer, M. Walther, and P. U. Jepsen, "Far-infrared vibrational modes of DNA components studied by terahertz time-domain spectroscopy," Phys. Med. Biol. 47,3807-3814 (2002).
[CrossRef] [PubMed]

Warner, R. R.

R. R. Warner,M. C. Myers, and D. A. Taylor, "Electron probe analysis of human skin: Determination of the water concentration profile," J. Invest. Dermatol. 90,218-224 (1988).
[CrossRef] [PubMed]

Williams, G. P.

G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, "High-power terahertz radiation from relativistic electrons," Nature 420,153-156 (2002).
[CrossRef] [PubMed]

Woodward, R. M.

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

Worral, C.

Xu, J.

J. Xu, K. W. Plaxco, and S. J. Allen, "Absorption spectra of liquid water and aqueous buffers between 0.3 and 3.72 THz," J. Chem. Phys. 124,1-3 (2006).
[CrossRef]

Zhang, X.-C.

G. M. Png, J.W. Choi, B.W.-H. Ng, S. P. Mickan, D. Abbott, and X.-C. Zhang, "The impact of hydration changes in fresh bio-tissue on THz spectroscopic measurements," Phys. Med. Biol. 53, 3501-3517 (2008).
[CrossRef] [PubMed]

Zinov’ev, N. N.

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

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H. Fröhlich, "The biological effects of microwaves and related questions," Adv. Electron. El. Phys. 53,85-152 (1980).
[CrossRef]

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N. P. Bondar, I. L. Kovalenko, D. F. Avgustinovich, A. G. Khamoyan, and N. N. Kudryavtseva, "Behavioral effect of terahertz waves in male mice," B. Exp. Biol. Med. 145,401-405 (2008).
[CrossRef]

Chem. Phys. Lett. (1)

L. Thrane, R. H. Jacobsen, P. U. Jepsen, and S. R. Keiding, "THz reflection spectroscopy of liquid water," Chem. Phys. Lett. 240,330-333 (1995).
[CrossRef]

IEEE Photon. J. (1)

J. Balakrishnan, B. M. Fischer, and D. Abbott, "Fixed dual-thickness terahertz liquid spectroscopy using a spinning sample technique," IEEE Photon. J. 1,88-98 (2009).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

R. Appleby and H. B. Wallace, "Standoff detection of weapons and contraband in the 100 GHz to 1 THz region," IEEE Trans. Antennas Propag. 55,2944-2956 (2007).
[CrossRef]

J. Biol. Chem. (1)

R. W. Guynn and R. L. Veech, "The equilibrium constants of the adenosine triphosphate hydrolysis and the adenosine triphosphate citrate lyase reactions," J. Biol. Chem. 248,6966-6972 (1973).
[PubMed]

J. Chem. Phys. (1)

J. Xu, K. W. Plaxco, and S. J. Allen, "Absorption spectra of liquid water and aqueous buffers between 0.3 and 3.72 THz," J. Chem. Phys. 124,1-3 (2006).
[CrossRef]

J. Invest. Dermatol. (1)

R. R. Warner,M. C. Myers, and D. A. Taylor, "Electron probe analysis of human skin: Determination of the water concentration profile," J. Invest. Dermatol. 90,218-224 (1988).
[CrossRef] [PubMed]

J. Laser Appl. (1)

E. Berry, G. C. Walker, A. J. Fitzgerald, N. N. Zinov’ev, M. Chamberlain, S. W. Smye, R. E. Miles, and M. A. Smith, "Do in vivo terahertz imaging systems comply with safety guidelines?" J. Laser Appl. 15,192-198 (2003).
[CrossRef]

J. Mol. Liq. (2)

A. Beneduci, "Which is the effective time scale of the fast Debye relaxation process in water?" J. Mol. Liq. 138, 55-60 (2008).
[CrossRef]

J. K. Vij, D. R. J. Simpson, and O. E. Panarina, "Far infrared spectroscopy of water at different temperatures: GHz to THz dielectric spectroscopy of water," J. Mol. Liq. 112,125-135 (2004).
[CrossRef]

J. Phys. Chem. (1)

J. T. Kindt and C. A. Schmuttenmaer, "Far-infrared dielectric properties of polar liquids probed by femtosecond terahertz pulse spectroscopy," J. Phys. Chem. 100,10373-10379 (1996).
[CrossRef]

J. Phys. Chem. Ref. Data (2)

M. L. V. Ramires, C. A. Nieto de Castro, Y. Nagasaka, A. Nagashima, M. J. Assael, and W. A. Wakeham, "Standard reference data for the thermal conductivity of water," J. Phys. Chem. Ref. Data 24,1377-1381 (1995).
[CrossRef]

W. J. Ellison, "Permittivity of pure water, at standard atmospheric pressure, over the frequency range 0-25 THz and the temperature range 0-100◦C," J. Phys. Chem. Ref. Data 36,1-18 (2007).
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G. L. Carr, M. C. Martin, W. R. McKinney, K. Jordan, G. R. Neil, and G. P. Williams, "High-power terahertz radiation from relativistic electrons," Nature 420,153-156 (2002).
[CrossRef] [PubMed]

M. Sherwin, "Terahertz power," Nature 420,131-132 (2002).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Med. Biol. (4)

B. M. Fischer, M. Walther, and P. U. Jepsen, "Far-infrared vibrational modes of DNA components studied by terahertz time-domain spectroscopy," Phys. Med. Biol. 47,3807-3814 (2002).
[CrossRef] [PubMed]

S. P. Mickan, A, Menikh, H. B. Liu, C. A. Mannella, R. MacColl, D. Abbott, J. Munch, and X. C. Zhang, "Labelfree bioaffinity detection using terahertz technology," Phys. Med. Biol. 47,3789-3795 (2002).
[CrossRef] [PubMed]

G. M. Png, J.W. Choi, B.W.-H. Ng, S. P. Mickan, D. Abbott, and X.-C. Zhang, "The impact of hydration changes in fresh bio-tissue on THz spectroscopic measurements," Phys. Med. Biol. 53, 3501-3517 (2008).
[CrossRef] [PubMed]

R. M. Woodward, B. E. Cole, V. P. Wallace, R. J. Pye, D. D. Arnone, E. H. Linfield, and M. Pepper, "Terahertz pulse imaging in reflection geometry of human skin cancer and skin tissue," Phys. Med. Biol. 47,3853-3863 (2002).
[CrossRef] [PubMed]

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B. S. Alexandrov, V. Gelev, A. R. Bishop, A. Usheva, and K. O. Rasmussen, "DNA breathing dynamics in the presence of a terahertz field," arXiv:0910.5294v1 (2009).

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Figures (7)

Fig. 1.
Fig. 1.

Illustration of the cross-section of the disc of water, which is used in the calculations. Here, b and d are the radius and the thickness of the disc, respectively, a is the radius of the THz beam and [a], [b], [c] and [d] denote the chosen boundary conditions for the transformed temperature, U (r, z). The calculations are performed in cylindrical coordinates, which is why only half of the cross-section is needed by argument from symmetry.

Fig. 2.
Fig. 2.

The temperature change per milliwatt, Ũ(r,z), (a) at z = 0 as a function of the radial distance, r, and (b) at r = 0 as a function of the thickness, z. It is seen that the maximal heating occurs just in the center of the beam and that the result has an exponential tail, but this may vary depending on the choice of dissipated power.

Fig. 3.
Fig. 3.

(a) The temperature change per milliwatt at (r,z) = (0,0) as a function of the number of terms used in the sum. After 400 terms it appears that the value of Ũ(r,z) is saturated - but to increase the accuracy, 1000 terms will be used in all further calculations. (b) The temperature change at (r,z) = (0,0) as a function of the sample radius, b, where the dashed red line indicates the chosen initial value. The result is shown for the three cases where boundary conditions b.1 and b.2 are used and when Cn is neglected. The last case makes the result independent of which boundary condition is used. It is seen that the dependence of Ũ(r,z) on the radius of the disc is negligible and that the choice of boundary conditions at the side of the disc is unimportant when b > 25 mm. For smaller b the three results are very different, which makes the result less reliable and therefore we choose b = 50mm.

Fig. 4.
Fig. 4.

The dashed red line on both graphs indicates the chosen initial values. (a) The temperature change per milliwatt at (r,z) = (0,0) as a function of the sample thickness, d. If the chosen thickness of the water disc is too small, the profile along the z-axis loses its exponential behavior, which is unphysical. Thus we have chosen a thickness of 15 mm for the sake of example. (b) The temperature change vs. the beam radius, a, at (r,z) = (0,0). It is hard to focus a THz beam with a center frequency of 1 THz down to a spot radius smaller than 0.25 mm [20] due to the diffraction limit, and most often the radius will be around 0.5 mm. Note that this quantity is investigated even further in Fig. 5.

Fig. 5.
Fig. 5.

(a)-(f) Contour plots, with the normalized radius, r/a, on the x-axis and the thickness, z on the y-axis showing the heating caused by the THz beam for six different beam radii a = 0.05mm, a = 0.25mm, a = 0.75mm, a = 2.50mm, a = 5.00mm and a = 10.00mm. The colors indicate the level of heating and the color-bar above each plot shows the conversion (in Kelvin per milliwatt), the vertical dashed lines show where the beam hits the water. (g)-(h) The normalized temperature change, Ũ(r,z)/Ũmax, as a function of the normalized radius, r/a, and the thickness z, respectively. The six curves represent the six different beam radii as shown on the contour plots.

Fig. 6.
Fig. 6.

(a) The temperature increase at (r,z) = (0,0) of the initial temperature, T0. (b) The temperature increase at (r,z) = (0,0) as a function of the beam frequency, ν. The red dashed line on each plot indicates the chosen initial value.

Fig. 7. (a)
Fig. 7. (a)

Contour plot of the sample near the center, the colors indicate the temperature change per milliwatt, Ũ(r,z), and the vertical lines show where the terahertz beam hits the disc. (b) The power needed to create a temperature change of 1 K as a function of the beam radius. The red vertical dashed line indicates the chosen initial value for the beam radius and the horizontal line shows the power corresponding to this value (0.56 mW). Note that P and U are directly proportional (P = U/Ũ(r,z)), i.e with a beam radius of 0.25 mm the amount of transmitted power needed to heat the water for instance 2°C is 2 × 0.56mW = 1.2mW. The initial conditions given in Table 2 is used in both figures.

Tables (2)

Tables Icon

Table 1. Definition of the Boolean variables χ and θ

Tables Icon

Table 2. Initial values for the calculations.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

( k ( T ) T ) = g ( z ) ,
U ( T ) = 1 k 0 T 0 T k ( T ) d T ,
2 U r 2 + 1 r U r + 2 U z 2 = g ,
P ̃ = P P 0 U ̃ ( r , z ) = U ( r , z ) P 0 ,
U ( r , z ) = P U ̃ ( r , z ) .
U ̃ ( r , z ) = n = 1 cos ( p n z ) g n p n 2 { χ ̂ A n [ B n ( r ) + C n ( r ) ] } ,
p n = ( 2 n 1 ) π 2 d
g n = 2 α P ̃ π d a 2 k 0 ( α ( 1 ) n p n e α d α 2 + p n 2 )
A n = 1 I 1 ( p n a ) K 0 ( p n a ) + I 0 ( p n a ) K 1 ( p n a )
B n ( r ) = ( 1 ) χ K χ ̂ ( χ ̂ p n a + χ p n r ) I χ ( χ p n a + χ ̂ p n r )
C n ( r ) = ( 1 ) θ I 1 ( p n a ) K θ ( p n b ) I θ ( p n b ) I 0 ( χ p n r ) ,
k 0 ( T 0 ) = 0.6065 ( 1.48445 + 4.12292 298.15 T 0 1.63866 298.15 2 T 0 2 ) .
α ( ν , T 0 ) = 4 πν c 0 ( ε ( ν , T 0 ) 2 + ε ( ν , T 0 ) 2 2 ε ( ν , T 0 ) 2 ) 1 2 ,
B THz = π ν min ν max d ν 2 h ν 3 c 0 2 [ exp ( h ν k B T ) 1 ] 1 ,
= 4 n ̂ sample ( 1 + n ̂ sample ) 2 .
n ̂ sample ( ν , T 0 ) = ( ε ' + ( ε ) 2 + ( ε ) 2 2 ) 1 2 + i ( ε ' + ( ε ) 2 + ( ε ) 2 2 ) 1 2

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