Abstract

We have measured terahertz near-field spectra of cesium iodide crystals as small as ∼10 μm in diameter, which were deposited on single, sub-wavelength-sized apertures created in thin gold films on a substrate. The advantage of using small apertures for terahertz microspectroscopy is that only terahertz light that has interacted with the cesium iodide is observed. We find that around the transverse optical phonon frequency of cesium iodide, the amplitude transmission is as much influenced by the refractive index as by the absorption. We show that the ability to measure in the near-field of the apertures, where signals are relatively strong, allows us to measure on sample volumes as small as ∼5×10−16 m3.

© 2013 OSA

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  1. P. U. Jepsen and B. M. Fisher, “Dynamic range in terahertz time-domain transmission and reflection spectroscopy,” Opt. Lett.30, 29–31 (2005).
    [CrossRef] [PubMed]
  2. M. Walther, B. Fischer, and P. U. Jepsen, “Noncovalent intermolecular forces in polycrystalline and amorphous saccharides in the far infrared,” Chem. Phys.288, 261–268 (2003).
    [CrossRef]
  3. Y. C. Shen, P. C. Upadhya, E. H. Linfield, and A. G. Davies, “Temperature-dependent low-frequency vibrational spectra of purine and adenine,” Appl. Phys. Lett.82, 2350–2352 (2003).
    [CrossRef]
  4. M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
    [CrossRef]
  5. R. Chakkittakandy, J. A. W. M. Corver, and P. C. M. Planken, “Terahertz spectroscopy to identify the polymorphs in freeze-dried mannitol,” J. Pharm. Sci.99, 932–940 (2010).
  6. A. J. L. Adam, “Review of near-field terahertz measurement methods and their applications,” J. Infrared Milli. Terahz. Waves32, 976–1019 (2011).
    [CrossRef]
  7. P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, “Opto-electronic pulsed THz systems,” Semicon. Sci. Techn.20, s121–s127 (2005).
    [CrossRef]
  8. A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett.8, 3766–3770 (2008).
    [CrossRef] [PubMed]
  9. H. -T. Chen, R. Kersting, and G. C. Cho, “Terahertz imaging with nanometer resolution,” Appl. Phys. Lett.83, 3009–3011 (2003).
    [CrossRef]
  10. A. Bitzer and M. Walther, “Terahertz near-field imaging of metallic sub-wavelength holes and hole arrays,” Appl. Phys. Lett.92, 231101 (2008).
    [CrossRef]
  11. A. Bitzer, H. Merbold, A. Thoman, T. Feurer, H. Helm, and M. Walther, “Terahertz near-field imaging of electric and magnetic resonances of a planar metamaterial,” Opt. Express17, 3826–3834 (2009).
    [CrossRef] [PubMed]
  12. M. Wächter, M. Nagel, and H. Kurz, “Tapered photoconductive field probe with subwavelength spatial resolution”, Appl. Phys. Lett.95, 041112 (2009).
    [CrossRef]
  13. M. Nagel, A. Michalski, T. Botzem, and H. Kurz, “Near-field investigation of THz surface-wave emission from optically excited graphite flakes,” Opt. Express19, 4667–4672 (2011).
    [CrossRef] [PubMed]
  14. M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, S. C. Jeoung, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Fourier-transform terahertz near-field imaging of one-dimensional slit arrays: mapping of electric-field-, magnetic-field-, and Poynting vectors,” Opt. Express15, 11781–11789 (2007).
    [CrossRef] [PubMed]
  15. 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. M. Planken, “Advanced terahertz electric near-field measurements at sub-wavelength diameter metallic apertures,” Opt. Express16, 7407–7417 (2008).
    [CrossRef] [PubMed]
  16. J. R. Knab, A. J. L. Adam, M. Nagel, E. Shaner, M. A. Seo, D. S. Kim, and P. C. M. Planken, “Terahertz near-Field vectorial imaging of subwavelength apertures and aperture arrays,” Opt. Express17, 15072–15086 (2009).
    [CrossRef] [PubMed]
  17. N. Kumar, A. C. Strikwerda, K. Fan, X. Zhang, R. D. Averitt, P. C. M. Planken, and A. J. L. Adam, “THz near-field Faraday imaging in hybrid metamaterials,” Opt. Express20, 11277–11287 (2012).
    [CrossRef] [PubMed]
  18. J. R. Knab, A. J. L. Adam, R. Chakkittakandy, and P. C. M. Planken, “Terahertz near-field microspectroscopy,” Appl. Phys. Lett.97, 031115 (2010).
    [CrossRef]
  19. L. Guestin, A. J. L. Adam, J. R. Knab, M. Nagel, and P. C. M. Planken, “Influence of the dielectric substrate on the terahertz electric near-field of a hole in a metal,” Opt. Express17, 17412–17425 (2009).
    [CrossRef] [PubMed]
  20. Y. Rong and K. A. Zaki, “Characteristics of generalized rectangular and circular ridge waveguides,” IEEE Trans. Microwave Theory Tech.48, 258–265 (2000).
    [CrossRef]
  21. A. Agrawal, H. Cao, and A. Nahata, “Time-domain analysis of enhanced transmission through a single subwavelength aperture,” Opt. Express13, 3535–3542 (2005).
    [CrossRef] [PubMed]
  22. G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, “Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,” Rev. Sci. Instrum.73, 1715–1719 (2002).
    [CrossRef]
  23. O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
    [CrossRef]
  24. O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
    [CrossRef]
  25. O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
    [CrossRef]
  26. R. Chakkittakandy, A. W. M. Corver, and P. C. M. Planken, “Quasi-near field terahertz generation and detection,” Opt. Express16, 12794–12805 (2008).
    [CrossRef] [PubMed]
  27. M. Yamaguchi, F. Miyamaru, K. Yamamoto, M. Tani, and M. Hangyo, “Terahertz absorption spectra of L-, D, and DL-alanine and their application to determination of enantiometric composition,” Appl. Phys. Lett.86, 053903 (2005).
    [CrossRef]
  28. E. R. Brown, J. E. Bjarnason, A. M. Fedor, and T. M. Korter, “On the strong and narrow absorption feature in lactose at 0.53 THz,” Appl. Phys. Lett.90, 061908 (2007).
    [CrossRef]
  29. C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).
  30. C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep.5, 401–422 (1950).

2012 (1)

2011 (2)

A. J. L. Adam, “Review of near-field terahertz measurement methods and their applications,” J. Infrared Milli. Terahz. Waves32, 976–1019 (2011).
[CrossRef]

M. Nagel, A. Michalski, T. Botzem, and H. Kurz, “Near-field investigation of THz surface-wave emission from optically excited graphite flakes,” Opt. Express19, 4667–4672 (2011).
[CrossRef] [PubMed]

2010 (2)

R. Chakkittakandy, J. A. W. M. Corver, and P. C. M. Planken, “Terahertz spectroscopy to identify the polymorphs in freeze-dried mannitol,” J. Pharm. Sci.99, 932–940 (2010).

J. R. Knab, A. J. L. Adam, R. Chakkittakandy, and P. C. M. Planken, “Terahertz near-field microspectroscopy,” Appl. Phys. Lett.97, 031115 (2010).
[CrossRef]

2009 (4)

2008 (4)

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. M. Planken, “Advanced terahertz electric near-field measurements at sub-wavelength diameter metallic apertures,” Opt. Express16, 7407–7417 (2008).
[CrossRef] [PubMed]

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

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett.8, 3766–3770 (2008).
[CrossRef] [PubMed]

R. Chakkittakandy, A. W. M. Corver, and P. C. M. Planken, “Quasi-near field terahertz generation and detection,” Opt. Express16, 12794–12805 (2008).
[CrossRef] [PubMed]

2007 (2)

2005 (4)

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, “Opto-electronic pulsed THz systems,” Semicon. Sci. Techn.20, s121–s127 (2005).
[CrossRef]

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

M. Yamaguchi, F. Miyamaru, K. Yamamoto, M. Tani, and M. Hangyo, “Terahertz absorption spectra of L-, D, and DL-alanine and their application to determination of enantiometric composition,” Appl. Phys. Lett.86, 053903 (2005).
[CrossRef]

A. Agrawal, H. Cao, and A. Nahata, “Time-domain analysis of enhanced transmission through a single subwavelength aperture,” Opt. Express13, 3535–3542 (2005).
[CrossRef] [PubMed]

2003 (4)

M. Walther, B. Fischer, and P. U. Jepsen, “Noncovalent intermolecular forces in polycrystalline and amorphous saccharides in the far infrared,” Chem. Phys.288, 261–268 (2003).
[CrossRef]

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and A. G. Davies, “Temperature-dependent low-frequency vibrational spectra of purine and adenine,” Appl. Phys. Lett.82, 2350–2352 (2003).
[CrossRef]

M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
[CrossRef]

H. -T. Chen, R. Kersting, and G. C. Cho, “Terahertz imaging with nanometer resolution,” Appl. Phys. Lett.83, 3009–3011 (2003).
[CrossRef]

2002 (1)

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, “Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,” Rev. Sci. Instrum.73, 1715–1719 (2002).
[CrossRef]

2001 (3)

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

2000 (1)

Y. Rong and K. A. Zaki, “Characteristics of generalized rectangular and circular ridge waveguides,” IEEE Trans. Microwave Theory Tech.48, 258–265 (2000).
[CrossRef]

1950 (2)

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep.5, 401–422 (1950).

Adam, A. J. L.

N. Kumar, A. C. Strikwerda, K. Fan, X. Zhang, R. D. Averitt, P. C. M. Planken, and A. J. L. Adam, “THz near-field Faraday imaging in hybrid metamaterials,” Opt. Express20, 11277–11287 (2012).
[CrossRef] [PubMed]

A. J. L. Adam, “Review of near-field terahertz measurement methods and their applications,” J. Infrared Milli. Terahz. Waves32, 976–1019 (2011).
[CrossRef]

J. R. Knab, A. J. L. Adam, R. Chakkittakandy, and P. C. M. Planken, “Terahertz near-field microspectroscopy,” Appl. Phys. Lett.97, 031115 (2010).
[CrossRef]

J. R. Knab, A. J. L. Adam, M. Nagel, E. Shaner, M. A. Seo, D. S. Kim, and P. C. M. Planken, “Terahertz near-Field vectorial imaging of subwavelength apertures and aperture arrays,” Opt. Express17, 15072–15086 (2009).
[CrossRef] [PubMed]

L. Guestin, A. J. L. Adam, J. R. Knab, M. Nagel, and P. C. M. Planken, “Influence of the dielectric substrate on the terahertz electric near-field of a hole in a metal,” Opt. Express17, 17412–17425 (2009).
[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. M. Planken, “Advanced terahertz electric near-field measurements at sub-wavelength diameter metallic apertures,” Opt. Express16, 7407–7417 (2008).
[CrossRef] [PubMed]

M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, S. C. Jeoung, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Fourier-transform terahertz near-field imaging of one-dimensional slit arrays: mapping of electric-field-, magnetic-field-, and Poynting vectors,” Opt. Express15, 11781–11789 (2007).
[CrossRef] [PubMed]

Agrawal, A.

Ahn, K. J.

Aizpurua, J.

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett.8, 3766–3770 (2008).
[CrossRef] [PubMed]

Averitt, R. D.

Bitzer, A.

Bjarnason, J. E.

E. R. Brown, J. E. Bjarnason, A. M. Fedor, and T. M. Korter, “On the strong and narrow absorption feature in lactose at 0.53 THz,” Appl. Phys. Lett.90, 061908 (2007).
[CrossRef]

Botzem, T.

Bouwkamp, C. J.

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep.5, 401–422 (1950).

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).

Brok, J. M.

Brown, E. R.

E. R. Brown, J. E. Bjarnason, A. M. Fedor, and T. M. Korter, “On the strong and narrow absorption feature in lactose at 0.53 THz,” Appl. Phys. Lett.90, 061908 (2007).
[CrossRef]

Cao, H.

Chakkittakandy, R.

J. R. Knab, A. J. L. Adam, R. Chakkittakandy, and P. C. M. Planken, “Terahertz near-field microspectroscopy,” Appl. Phys. Lett.97, 031115 (2010).
[CrossRef]

R. Chakkittakandy, J. A. W. M. Corver, and P. C. M. Planken, “Terahertz spectroscopy to identify the polymorphs in freeze-dried mannitol,” J. Pharm. Sci.99, 932–940 (2010).

R. Chakkittakandy, A. W. M. Corver, and P. C. M. Planken, “Quasi-near field terahertz generation and detection,” Opt. Express16, 12794–12805 (2008).
[CrossRef] [PubMed]

Chen, H. -T.

H. -T. Chen, R. Kersting, and G. C. Cho, “Terahertz imaging with nanometer resolution,” Appl. Phys. Lett.83, 3009–3011 (2003).
[CrossRef]

Cho, G. C.

H. -T. Chen, R. Kersting, and G. C. Cho, “Terahertz imaging with nanometer resolution,” Appl. Phys. Lett.83, 3009–3011 (2003).
[CrossRef]

Corver, A. W. M.

Corver, J. A. W. M.

R. Chakkittakandy, J. A. W. M. Corver, and P. C. M. Planken, “Terahertz spectroscopy to identify the polymorphs in freeze-dried mannitol,” J. Pharm. Sci.99, 932–940 (2010).

Davies, A. G.

M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
[CrossRef]

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and A. G. Davies, “Temperature-dependent low-frequency vibrational spectra of purine and adenine,” Appl. Phys. Lett.82, 2350–2352 (2003).
[CrossRef]

Fan, K.

Federici, J.

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

Federici, J. F.

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

Fedor, A. M.

E. R. Brown, J. E. Bjarnason, A. M. Fedor, and T. M. Korter, “On the strong and narrow absorption feature in lactose at 0.53 THz,” Appl. Phys. Lett.90, 061908 (2007).
[CrossRef]

Feurer, T.

Fischer, B.

M. Walther, B. Fischer, and P. U. Jepsen, “Noncovalent intermolecular forces in polycrystalline and amorphous saccharides in the far infrared,” Chem. Phys.288, 261–268 (2003).
[CrossRef]

Fisher, B. M.

Guestin, L.

Hangyo, M.

M. Yamaguchi, F. Miyamaru, K. Yamamoto, M. Tani, and M. Hangyo, “Terahertz absorption spectra of L-, D, and DL-alanine and their application to determination of enantiometric composition,” Appl. Phys. Lett.86, 053903 (2005).
[CrossRef]

Harel, R.

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

Helm, H.

Herz, L. M.

M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
[CrossRef]

Hillenbrand, R.

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett.8, 3766–3770 (2008).
[CrossRef] [PubMed]

Hsu, J. W. P.

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

Huber, A. J.

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett.8, 3766–3770 (2008).
[CrossRef] [PubMed]

Jeoung, S. C.

Jepsen, P. U.

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

M. Walther, B. Fischer, and P. U. Jepsen, “Noncovalent intermolecular forces in polycrystalline and amorphous saccharides in the far infrared,” Chem. Phys.288, 261–268 (2003).
[CrossRef]

Johnston, M. B.

M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
[CrossRef]

Kang, J. H.

Keilmann, F.

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett.8, 3766–3770 (2008).
[CrossRef] [PubMed]

Kersting, R.

H. -T. Chen, R. Kersting, and G. C. Cho, “Terahertz imaging with nanometer resolution,” Appl. Phys. Lett.83, 3009–3011 (2003).
[CrossRef]

Khan, A.

M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
[CrossRef]

Kim, D. S.

Knab, J. R.

Köhler, A.

M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
[CrossRef]

Korter, T. M.

E. R. Brown, J. E. Bjarnason, A. M. Fedor, and T. M. Korter, “On the strong and narrow absorption feature in lactose at 0.53 THz,” Appl. Phys. Lett.90, 061908 (2007).
[CrossRef]

Kumar, N.

Kurz, H.

M. Nagel, A. Michalski, T. Botzem, and H. Kurz, “Near-field investigation of THz surface-wave emission from optically excited graphite flakes,” Opt. Express19, 4667–4672 (2011).
[CrossRef] [PubMed]

M. Wächter, M. Nagel, and H. Kurz, “Tapered photoconductive field probe with subwavelength spatial resolution”, Appl. Phys. Lett.95, 041112 (2009).
[CrossRef]

Lee, J. W.

Lee, M.

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

Linfield, E. H.

M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
[CrossRef]

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and A. G. Davies, “Temperature-dependent low-frequency vibrational spectra of purine and adenine,” Appl. Phys. Lett.82, 2350–2352 (2003).
[CrossRef]

Merbold, H.

Michalski, A.

Mitrofanov, O.

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

Miyamaru, F.

M. Yamaguchi, F. Miyamaru, K. Yamamoto, M. Tani, and M. Hangyo, “Terahertz absorption spectra of L-, D, and DL-alanine and their application to determination of enantiometric composition,” Appl. Phys. Lett.86, 053903 (2005).
[CrossRef]

Nagel, M.

Nahata, A.

Park, Q. H.

Pfeiffer, L. N.

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

Planken, P. C. M.

N. Kumar, A. C. Strikwerda, K. Fan, X. Zhang, R. D. Averitt, P. C. M. Planken, and A. J. L. Adam, “THz near-field Faraday imaging in hybrid metamaterials,” Opt. Express20, 11277–11287 (2012).
[CrossRef] [PubMed]

J. R. Knab, A. J. L. Adam, R. Chakkittakandy, and P. C. M. Planken, “Terahertz near-field microspectroscopy,” Appl. Phys. Lett.97, 031115 (2010).
[CrossRef]

R. Chakkittakandy, J. A. W. M. Corver, and P. C. M. Planken, “Terahertz spectroscopy to identify the polymorphs in freeze-dried mannitol,” J. Pharm. Sci.99, 932–940 (2010).

L. Guestin, A. J. L. Adam, J. R. Knab, M. Nagel, and P. C. M. Planken, “Influence of the dielectric substrate on the terahertz electric near-field of a hole in a metal,” Opt. Express17, 17412–17425 (2009).
[CrossRef] [PubMed]

J. R. Knab, A. J. L. Adam, M. Nagel, E. Shaner, M. A. Seo, D. S. Kim, and P. C. M. Planken, “Terahertz near-Field vectorial imaging of subwavelength apertures and aperture arrays,” Opt. Express17, 15072–15086 (2009).
[CrossRef] [PubMed]

R. Chakkittakandy, A. W. M. Corver, and P. C. M. Planken, “Quasi-near field terahertz generation and detection,” Opt. Express16, 12794–12805 (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. M. Planken, “Advanced terahertz electric near-field measurements at sub-wavelength diameter metallic apertures,” Opt. Express16, 7407–7417 (2008).
[CrossRef] [PubMed]

M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, S. C. Jeoung, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Fourier-transform terahertz near-field imaging of one-dimensional slit arrays: mapping of electric-field-, magnetic-field-, and Poynting vectors,” Opt. Express15, 11781–11789 (2007).
[CrossRef] [PubMed]

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, “Opto-electronic pulsed THz systems,” Semicon. Sci. Techn.20, s121–s127 (2005).
[CrossRef]

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, “Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,” Rev. Sci. Instrum.73, 1715–1719 (2002).
[CrossRef]

Rong, Y.

Y. Rong and K. A. Zaki, “Characteristics of generalized rectangular and circular ridge waveguides,” IEEE Trans. Microwave Theory Tech.48, 258–265 (2000).
[CrossRef]

Schouten, R. N.

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, “Opto-electronic pulsed THz systems,” Semicon. Sci. Techn.20, s121–s127 (2005).
[CrossRef]

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, “Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,” Rev. Sci. Instrum.73, 1715–1719 (2002).
[CrossRef]

Seo, M. A.

Shaner, E.

Shen, Y. C.

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and A. G. Davies, “Temperature-dependent low-frequency vibrational spectra of purine and adenine,” Appl. Phys. Lett.82, 2350–2352 (2003).
[CrossRef]

Strikwerda, A. C.

Tani, M.

M. Yamaguchi, F. Miyamaru, K. Yamamoto, M. Tani, and M. Hangyo, “Terahertz absorption spectra of L-, D, and DL-alanine and their application to determination of enantiometric composition,” Appl. Phys. Lett.86, 053903 (2005).
[CrossRef]

Thoman, A.

Upadhya, P. C.

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and A. G. Davies, “Temperature-dependent low-frequency vibrational spectra of purine and adenine,” Appl. Phys. Lett.82, 2350–2352 (2003).
[CrossRef]

van der Valk, N.

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, “Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,” Rev. Sci. Instrum.73, 1715–1719 (2002).
[CrossRef]

van Rijmenam, C. E. W. M.

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, “Opto-electronic pulsed THz systems,” Semicon. Sci. Techn.20, s121–s127 (2005).
[CrossRef]

Wächter, M.

M. Wächter, M. Nagel, and H. Kurz, “Tapered photoconductive field probe with subwavelength spatial resolution”, Appl. Phys. Lett.95, 041112 (2009).
[CrossRef]

Walther, M.

A. Bitzer, H. Merbold, A. Thoman, T. Feurer, H. Helm, and M. Walther, “Terahertz near-field imaging of electric and magnetic resonances of a planar metamaterial,” Opt. Express17, 3826–3834 (2009).
[CrossRef] [PubMed]

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

M. Walther, B. Fischer, and P. U. Jepsen, “Noncovalent intermolecular forces in polycrystalline and amorphous saccharides in the far infrared,” Chem. Phys.288, 261–268 (2003).
[CrossRef]

Wenckebach, W. Th.

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, “Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,” Rev. Sci. Instrum.73, 1715–1719 (2002).
[CrossRef]

West, K. W.

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

Wittborn, J.

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett.8, 3766–3770 (2008).
[CrossRef] [PubMed]

Wynn, J. D.

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

Yamaguchi, M.

M. Yamaguchi, F. Miyamaru, K. Yamamoto, M. Tani, and M. Hangyo, “Terahertz absorption spectra of L-, D, and DL-alanine and their application to determination of enantiometric composition,” Appl. Phys. Lett.86, 053903 (2005).
[CrossRef]

Yamamoto, K.

M. Yamaguchi, F. Miyamaru, K. Yamamoto, M. Tani, and M. Hangyo, “Terahertz absorption spectra of L-, D, and DL-alanine and their application to determination of enantiometric composition,” Appl. Phys. Lett.86, 053903 (2005).
[CrossRef]

Zaki, K. A.

Y. Rong and K. A. Zaki, “Characteristics of generalized rectangular and circular ridge waveguides,” IEEE Trans. Microwave Theory Tech.48, 258–265 (2000).
[CrossRef]

Zhang, X.

Zhao, G.

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, “Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,” Rev. Sci. Instrum.73, 1715–1719 (2002).
[CrossRef]

Appl. Phys. Lett. (9)

Y. C. Shen, P. C. Upadhya, E. H. Linfield, and A. G. Davies, “Temperature-dependent low-frequency vibrational spectra of purine and adenine,” Appl. Phys. Lett.82, 2350–2352 (2003).
[CrossRef]

H. -T. Chen, R. Kersting, and G. C. Cho, “Terahertz imaging with nanometer resolution,” Appl. Phys. Lett.83, 3009–3011 (2003).
[CrossRef]

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

M. Wächter, M. Nagel, and H. Kurz, “Tapered photoconductive field probe with subwavelength spatial resolution”, Appl. Phys. Lett.95, 041112 (2009).
[CrossRef]

J. R. Knab, A. J. L. Adam, R. Chakkittakandy, and P. C. M. Planken, “Terahertz near-field microspectroscopy,” Appl. Phys. Lett.97, 031115 (2010).
[CrossRef]

O. Mitrofanov, M. Lee, J. W. P. Hsu, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. F. Federici, “Terahertz pulse propagation through small apertures,” Appl. Phys. Lett.79, 907–909 (2001).
[CrossRef]

O. Mitrofanov, R. Harel, M. Lee, L. N. Pfeiffer, K. W. West, J. D. Wynn, and J. Federici, “Study of single-cycle pulse propagation inside a terahertz near-field probe,” Appl. Phys. Lett.78, 252–254 (2001).
[CrossRef]

M. Yamaguchi, F. Miyamaru, K. Yamamoto, M. Tani, and M. Hangyo, “Terahertz absorption spectra of L-, D, and DL-alanine and their application to determination of enantiometric composition,” Appl. Phys. Lett.86, 053903 (2005).
[CrossRef]

E. R. Brown, J. E. Bjarnason, A. M. Fedor, and T. M. Korter, “On the strong and narrow absorption feature in lactose at 0.53 THz,” Appl. Phys. Lett.90, 061908 (2007).
[CrossRef]

Chem. Phys. (1)

M. Walther, B. Fischer, and P. U. Jepsen, “Noncovalent intermolecular forces in polycrystalline and amorphous saccharides in the far infrared,” Chem. Phys.288, 261–268 (2003).
[CrossRef]

Chem. Phys. Lett. (1)

M. B. Johnston, L. M. Herz, A. Khan, A. Köhler, A. G. Davies, and E. H. Linfield, “Low-energy vibrational modes in phenylene oligomers studied by THz time-domain spectroscopy,” Chem. Phys. Lett.377, 256262 (2003).
[CrossRef]

IEEE J. Sel. Top. Quant. Electron. (1)

O. Mitrofanov, M. Lee, J. W. P. Hsu, R. Harel, J. F. Federici, J. D. Wynn, L. N. Pfeiffer, and K. W. West, “Collection-mode near-field imaging With 0.5-THz Pulses,” IEEE J. Sel. Top. Quant. Electron.7, 600–607 (2001).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

Y. Rong and K. A. Zaki, “Characteristics of generalized rectangular and circular ridge waveguides,” IEEE Trans. Microwave Theory Tech.48, 258–265 (2000).
[CrossRef]

J. Infrared Milli. Terahz. Waves (1)

A. J. L. Adam, “Review of near-field terahertz measurement methods and their applications,” J. Infrared Milli. Terahz. Waves32, 976–1019 (2011).
[CrossRef]

J. Pharm. Sci. (1)

R. Chakkittakandy, J. A. W. M. Corver, and P. C. M. Planken, “Terahertz spectroscopy to identify the polymorphs in freeze-dried mannitol,” J. Pharm. Sci.99, 932–940 (2010).

Nano Lett. (1)

A. J. Huber, F. Keilmann, J. Wittborn, J. Aizpurua, and R. Hillenbrand, “Terahertz near-field nanoscopy of mobile carriers in single semiconductor nanodevices,” Nano Lett.8, 3766–3770 (2008).
[CrossRef] [PubMed]

Opt. Express (9)

R. Chakkittakandy, A. W. M. Corver, and P. C. M. Planken, “Quasi-near field terahertz generation and detection,” Opt. Express16, 12794–12805 (2008).
[CrossRef] [PubMed]

A. Bitzer, H. Merbold, A. Thoman, T. Feurer, H. Helm, and M. Walther, “Terahertz near-field imaging of electric and magnetic resonances of a planar metamaterial,” Opt. Express17, 3826–3834 (2009).
[CrossRef] [PubMed]

A. Agrawal, H. Cao, and A. Nahata, “Time-domain analysis of enhanced transmission through a single subwavelength aperture,” Opt. Express13, 3535–3542 (2005).
[CrossRef] [PubMed]

L. Guestin, A. J. L. Adam, J. R. Knab, M. Nagel, and P. C. M. Planken, “Influence of the dielectric substrate on the terahertz electric near-field of a hole in a metal,” Opt. Express17, 17412–17425 (2009).
[CrossRef] [PubMed]

M. Nagel, A. Michalski, T. Botzem, and H. Kurz, “Near-field investigation of THz surface-wave emission from optically excited graphite flakes,” Opt. Express19, 4667–4672 (2011).
[CrossRef] [PubMed]

M. A. Seo, A. J. L. Adam, J. H. Kang, J. W. Lee, S. C. Jeoung, Q. H. Park, P. C. M. Planken, and D. S. Kim, “Fourier-transform terahertz near-field imaging of one-dimensional slit arrays: mapping of electric-field-, magnetic-field-, and Poynting vectors,” Opt. Express15, 11781–11789 (2007).
[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. M. Planken, “Advanced terahertz electric near-field measurements at sub-wavelength diameter metallic apertures,” Opt. Express16, 7407–7417 (2008).
[CrossRef] [PubMed]

J. R. Knab, A. J. L. Adam, M. Nagel, E. Shaner, M. A. Seo, D. S. Kim, and P. C. M. Planken, “Terahertz near-Field vectorial imaging of subwavelength apertures and aperture arrays,” Opt. Express17, 15072–15086 (2009).
[CrossRef] [PubMed]

N. Kumar, A. C. Strikwerda, K. Fan, X. Zhang, R. D. Averitt, P. C. M. Planken, and A. J. L. Adam, “THz near-field Faraday imaging in hybrid metamaterials,” Opt. Express20, 11277–11287 (2012).
[CrossRef] [PubMed]

Opt. Lett. (1)

Philips Res. Rep. (2)

C. J. Bouwkamp, “On Bethe’s theory of diffraction by small holes,” Philips Res. Rep.5, 321–332 (1950).

C. J. Bouwkamp, “On the diffraction of electromagnetic waves by small circular disks and holes,” Philips Res. Rep.5, 401–422 (1950).

Rev. Sci. Instrum. (1)

G. Zhao, R. N. Schouten, N. van der Valk, W. Th. Wenckebach, and P. C. M. Planken, “Design and performance of a THz emission and detection setup based on a semi-insulating GaAs emitter,” Rev. Sci. Instrum.73, 1715–1719 (2002).
[CrossRef]

Semicon. Sci. Techn. (1)

P. C. M. Planken, C. E. W. M. van Rijmenam, and R. N. Schouten, “Opto-electronic pulsed THz systems,” Semicon. Sci. Techn.20, s121–s127 (2005).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic drawing of part of the experimental setup. A 500 nm-thick gold film with a lithographically fabricated aperture is deposited onto a (111) oriented GaP electro-optic crystal. From above, a THz pulse is focused onto the aperture. From below, femtosecond probe laser pulses are focused onto the front surface to an approximately 5 μm diameter spot size. The probe pulses sample the THz electric near-field immediately behind the aperture. On the right is a photograph of a CsI crystal deposited onto a 20 μm diameter aperture, taken with a conventional optical microscope.

Fig. 2
Fig. 2

(a) Measured electric field as a function of time of the THz pulse incident on the aperture. (b) Electric near-field as a function of time measured behind an unfilled 10 μm aperture. (c) Electric near-field as a function of time measured behind a 10 μm aperture covered with a small CsI crystal.

Fig. 3
Fig. 3

(a)–(c) Electric near-field amplitude as a function of frequency, measured behind empty apertures of 10, 20 and 40 μm diameter. (d)–(f) Electric near-field amplitude as a function of frequency, measured behind apertures of 10, 20 and 40 μm diameter, filled with CsI. For each of the measurements, the data were normalized with respect to the spectrum of the incident THz pulse, obtained using a bare part of the same (111) oriented GaP crystal on which the CsI samples were deposited.(g)–(i) Electric near-field amplitude as a function of frequency, calculated behind apertures of 10, 20, and 40 μm diameter, filled with CsI.

Fig. 4
Fig. 4

(a) Refractive index, n, and (b) absorption coefficient, α, as a function of frequency, calculated using Eq. (1), with parameters adapted from ref. [1]. We have used ωTO =2π×1.8×1012 rad/s, ΓTO =0.3×1012 rad/s, ε =3.17, and εstat =6.03.

Fig. 5
Fig. 5

Normalized electric near-field as a function of aperture diameter, calculated for four different distances, z, behind the center of the aperture: z =0, 5, 10, 50 μm. The data points represent numerical calculations at a frequency of 0.5 THz using CST Microwave Studio. The solid lines are calculated using Eq. (2).

Fig. 6
Fig. 6

(a) Normalized electric near-field behind a 10 μm diameter aperture, calculated for three different positions of a 10 μm diameter, 5 μm thick CsI crystal placed on top of the aperture. The three positions correspond to a crystal centered at the aperture, and shifted 5 and 10 μm to the left, respectively. The illustration above the graph shows the polarization direction of the incident field and a CsI sample displaced to the left of the center of the aperture in the gold film. (b) Normalized electric near-field behind a 10 μm diameter aperture, calculated for three different positions of a 10 μm diameter, 5 μm thick CsI crystal. The three positions correspond to a crystal centered at the aperture, and shifted 5 and 10 μm upwards, respectively. (c) Normalized electric near-field behind a 10 μm diameter aperture, calculated for three different crystal diameters of 10, 20 and 40 μm respectively. The crystal thickness is 5 μm. As in Figs. 3(g)–3(i), the fields are calculated in the center of the aperture, 10 μm below the surface of the GaP crystal.

Equations (4)

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

ε ( ω ) = ε + ( ε stat ε ) ω T O 2 ( ω T O 2 ω 2 i ω Γ T O )
E x ( x , y , z ) = E 0 [ i k z 2 π i k a u ( 1 + v arctan ( v ) + 1 3 1 u 2 + v 2 + x 2 y 2 3 a 2 ( u 2 + v 2 ) ( 1 + v 2 ) 2 ) ]
u 2 = 1 2 [ 1 x 2 + y 2 + z 2 a 2 ] + 1 2 [ x 2 + y 2 + z 2 a 2 1 ] 2 + 4 z 2 a 2
v 2 = z 2 a 2 u 2 .

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