Abstract

We report on data processing for continuous wave (CW) terahertz (THz) spectroscopy measurements based on a Hilbert spectral analysis to achieve MHz resolution. As an example we investigate the spectral properties of a whispering gallery mode (WGM) THz bubble resonator at critical coupling. The experimental verification clearly demonstrates the significant advantages in relative frequency resolution and required acquisition time of the proposed method over the traditional data analysis. An effective frequency resolution, only limited by the precision and stability of the laser beat signal, can be achieved without complex extensions to a standard commercially available CW THz spectrometer.

© 2017 Optical Society of America

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References

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  1. A.J. Deninger, A. Roggenbuck, S. Schindler, and S. Preu, “2.75 THz tuning with a triple-DFB laser system at 1550 nm and InGaAs photomixers,” J. Infrared Millim. Te. 36(3), 269–277 (2015)
    [Crossref]
  2. D. Mittleman, Sensing with terahertz radiation (Springer, 2013)
  3. A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
    [Crossref]
  4. G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
    [Crossref]
  5. A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
    [Crossref]
  6. D. W. Vogt and R. Leonhardt, Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, The University of Auckland, Private Bag 92019, Auckland, New Zealand are preparing a manuscript to be called “Low order terahertz whispering gallery mode bubble resonators.”
  7. E. Alexander and D. Poularikas, The Handbook of Formulas and Tables for Signal Processing (CRC Press, 1998)
  8. Y. W. Liu, Hilbert Transform and Applications (INTECH Open Access Publisher, 2012)
  9. E. Jones, T. Oliphant, P. Peterson, and et al., “SciPy: Open source scientific tools for Python,” http://www.scipy.org/
  10. S. Van De Walt, S.C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” Comput. Sci. Eng. 13(2), 22–30 (2011)
    [Crossref]
  11. W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
    [Crossref]

2015 (1)

A.J. Deninger, A. Roggenbuck, S. Schindler, and S. Preu, “2.75 THz tuning with a triple-DFB laser system at 1550 nm and InGaAs photomixers,” J. Infrared Millim. Te. 36(3), 269–277 (2015)
[Crossref]

2012 (2)

A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
[Crossref]

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

2011 (1)

S. Van De Walt, S.C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” Comput. Sci. Eng. 13(2), 22–30 (2011)
[Crossref]

2010 (1)

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

2007 (1)

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Alexander, E.

E. Alexander and D. Poularikas, The Handbook of Formulas and Tables for Signal Processing (CRC Press, 1998)

Baets, R.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Bienstman, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Bigourd, D.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Blary, K.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Bocquet, R.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Bogaerts, W.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Claes, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Colbert, S.C.

S. Van De Walt, S.C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” Comput. Sci. Eng. 13(2), 22–30 (2011)
[Crossref]

Cuisset, A.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

De Heyn, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

De Vos, K.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Deninger, A.

A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
[Crossref]

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

Deninger, A.J.

A.J. Deninger, A. Roggenbuck, S. Schindler, and S. Preu, “2.75 THz tuning with a triple-DFB laser system at 1550 nm and InGaAs photomixers,” J. Infrared Millim. Te. 36(3), 269–277 (2015)
[Crossref]

Dumon, P.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Grüninger, M.

A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
[Crossref]

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

Güsten, R.

A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
[Crossref]

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

Hemberger, J.

A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
[Crossref]

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

Hindle, F.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Kumar Selvaraja, S.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Lampin, J.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Leonhardt, R.

D. W. Vogt and R. Leonhardt, Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, The University of Auckland, Private Bag 92019, Auckland, New Zealand are preparing a manuscript to be called “Low order terahertz whispering gallery mode bubble resonators.”

Lippens, D.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Liu, Y. W.

Y. W. Liu, Hilbert Transform and Applications (INTECH Open Access Publisher, 2012)

Marx, J.

Matton, S.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Mayorga, I.C.

A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
[Crossref]

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

Mittleman, D.

D. Mittleman, Sensing with terahertz radiation (Springer, 2013)

Mouret, G.

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Poularikas, D.

E. Alexander and D. Poularikas, The Handbook of Formulas and Tables for Signal Processing (CRC Press, 1998)

Preu, S.

A.J. Deninger, A. Roggenbuck, S. Schindler, and S. Preu, “2.75 THz tuning with a triple-DFB laser system at 1550 nm and InGaAs photomixers,” J. Infrared Millim. Te. 36(3), 269–277 (2015)
[Crossref]

Roggenbuck, A.

A.J. Deninger, A. Roggenbuck, S. Schindler, and S. Preu, “2.75 THz tuning with a triple-DFB laser system at 1550 nm and InGaAs photomixers,” J. Infrared Millim. Te. 36(3), 269–277 (2015)
[Crossref]

A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
[Crossref]

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

Schindler, S.

A.J. Deninger, A. Roggenbuck, S. Schindler, and S. Preu, “2.75 THz tuning with a triple-DFB laser system at 1550 nm and InGaAs photomixers,” J. Infrared Millim. Te. 36(3), 269–277 (2015)
[Crossref]

Schmitz, H.

A. Roggenbuck, K. Thirunavukkuarasu, H. Schmitz, J. Marx, A. Deninger, I.C. Mayorga, R. Güsten, J. Hemberger, and M. Grüninger, “Using a fiber stretcher as a fast phase modulator in a continuous wave terahertz spectrometer,” J. Opt. Soc. Am. B 29(4), 614–620 (2012)
[Crossref]

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

Thirunavukkuarasu, K.

Van De Walt, S.

S. Van De Walt, S.C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” Comput. Sci. Eng. 13(2), 22–30 (2011)
[Crossref]

Van Thourhout, D.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Van Vaerenbergh, T.

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

Varoquaux, G.

S. Van De Walt, S.C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” Comput. Sci. Eng. 13(2), 22–30 (2011)
[Crossref]

Vogt, D. W.

D. W. Vogt and R. Leonhardt, Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, The University of Auckland, Private Bag 92019, Auckland, New Zealand are preparing a manuscript to be called “Low order terahertz whispering gallery mode bubble resonators.”

Appl. Phys. B (1)

G. Mouret, S. Matton, R. Bocquet, D. Bigourd, F. Hindle, A. Cuisset, J. Lampin, K. Blary, and D. Lippens, “THz media characterization by means of coherent homodyne detection, results and potential applications,” Appl. Phys. B 89(2), 395–399 (2007)
[Crossref]

Comput. Sci. Eng. (1)

S. Van De Walt, S.C. Colbert, and G. Varoquaux, “The NumPy array: a structure for efficient numerical computation,” Comput. Sci. Eng. 13(2), 22–30 (2011)
[Crossref]

J. Infrared Millim. Te. (1)

A.J. Deninger, A. Roggenbuck, S. Schindler, and S. Preu, “2.75 THz tuning with a triple-DFB laser system at 1550 nm and InGaAs photomixers,” J. Infrared Millim. Te. 36(3), 269–277 (2015)
[Crossref]

J. Opt. Soc. Am. B (1)

Laser Photon Rev. (1)

W. Bogaerts, P. De Heyn, T. Van Vaerenbergh, K. De Vos, S. Kumar Selvaraja, T. Claes, P. Dumon, P. Bienstman, D. Van Thourhout, and R. Baets, “Silicon microring resonators,” Laser Photon Rev. 6(1), 47–73 (2012)
[Crossref]

New J. Phys. (1)

A. Roggenbuck, H. Schmitz, A. Deninger, I.C. Mayorga, J. Hemberger, R. Güsten, and M. Grüninger, “Coherent broadband continuous-wave terahertz spectroscopy on solid-state samples,” New J. Phys. 12(4), 043,017 (2010)
[Crossref]

Other (5)

D. Mittleman, Sensing with terahertz radiation (Springer, 2013)

D. W. Vogt and R. Leonhardt, Dodd-Walls Centre for Photonic and Quantum Technologies, Department of Physics, The University of Auckland, Private Bag 92019, Auckland, New Zealand are preparing a manuscript to be called “Low order terahertz whispering gallery mode bubble resonators.”

E. Alexander and D. Poularikas, The Handbook of Formulas and Tables for Signal Processing (CRC Press, 1998)

Y. W. Liu, Hilbert Transform and Applications (INTECH Open Access Publisher, 2012)

E. Jones, T. Oliphant, P. Peterson, and et al., “SciPy: Open source scientific tools for Python,” http://www.scipy.org/

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Figures (3)

Fig. 1
Fig. 1 Schematic diagram of the experimental setup with the Toptica TeraScan 1550nm system. The THz path length is about 0.8 m. The inset shows a microscope picture of the silica fiber and the THz bubble resonator. The sub-wavelength fiber has a diameter of 200 μm. The bubble resonator made of quartz glass has a diameter of 6.3 mm and a wall thickness of 78 μm.
Fig. 2
Fig. 2 (a) Detected photocurrent in the frequency range from 0.4642 THz to 0.4672 THz for the sample (red dots) and reference (black dots) scans, respectively. The frequency step size is 1 MHz and the data is averaged over 5 scans. The red and black solid lines show the corresponding envelopes retrieved from the analytical signal and (b) shows the phase of both sample and reference scan obtained from the data processing and (c) shows the amplitude ratio of the sample and reference envelope shown in (a). The absorption due to the resonance of the THz bubble is clearly visible and (d) presents the relative phase shift of the WGM. A very steep transition of π at the resonance frequency of 0.46574 THz is clearly visible.
Fig. 3
Fig. 3 Relative phase shift (black dots) of a WGM of a THz bubble resonator at critical coupling. The measurements are averaged over seven scans and the error bars show the standard deviation of the calculated relative phase shift. The measurements are taken with a 1 MHz step size and are analyzed with Hilbert spectral analysis. The frequency axis is centered around the resonance frequency of the WGM. The experimental results are fitted (cyan solid line) with the analytical equation for a 2D ring resonator for critical coupling, convoluted with a Gaussian beam. The fitted FWHM of the Gaussian beam is ∼ 4 MHz. For comparison, the red solid line shows the phase profile for critical coupling of the analytical model without the convolution.

Equations (5)

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

I ph ( Δ φ ) ~ E THz cos ( Δ φ ) = E THz cos ( 2 π Δ L f c ) ,
I a ( Δ φ ) = 1 { [ I ph ( Δ φ ) ] 2 𝒰 } = I ph ( Δ φ ) + i [ I ph ( Δ φ ) ] ,
I a ( Δ φ ) = Λ ( Δ φ ) e i Ψ ( Δ φ ) ,
t = Λ s ( Δ φ r ) Λ r ( Δ φ r ) ,
Φ = Ψ s ( Δ φ s ) Ψ r ( Δ φ r ) + 2 π n ,

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