Causality-based method for determining the time origin in terahertz emission spectroscopy |
Optics Express, Vol. 19, Issue 13, pp. 12759-12765 (2011)
http://dx.doi.org/10.1364/OE.19.012759
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Abstract
We propose a method for determining the time origin on the basis of causality in terahertz (THz) emission spectroscopy. The method is formulated in terms of the singly subtractive Kramers-Kronig relation, which is useful for the situation where not only the amplitude spectrum but also partial phase information is available within the measurement frequency range. Numerical analysis of several simulated and observed THz emission data shows that the misplacement of the time origin in THz waveforms can be detected by the method with an accuracy that is an order of magnitude higher than the given temporal resolutions.
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1. Introduction
1. R. de L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. 12(6), 547–557 (1926). [CrossRef]
4. R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1(2), 818–831 (1970). [CrossRef]
7. V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4–6), 409–414 (2003). [CrossRef]
8. V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B 72(12), 125107 (2005). [CrossRef]
9. T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010). [CrossRef] [PubMed]
2. Causality principle in transient THz emission
7. V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4–6), 409–414 (2003). [CrossRef]
3. Results and discussion
3.1. Application to trial THz waveforms
10. T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B 81(12), 125329 (2010). [CrossRef]
11. N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). For simplicity, the parameter c of the Drude-Smith model is set to –1 in the present paper. [CrossRef]
3.2. Application to actual THz waveforms
9. T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010). [CrossRef] [PubMed]
10. T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B 81(12), 125329 (2010). [CrossRef]
10. T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B 81(12), 125329 (2010). [CrossRef]
9. T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010). [CrossRef] [PubMed]
3.3. Remarks
9. T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010). [CrossRef] [PubMed]
4. Summary
Acknowledgments
References and links
1. | R. de L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. 12(6), 547–557 (1926). [CrossRef] |
2. | H. A. Kramers, “La diffusion de la lumière par les atomes,” in Atti del Congresso Internazionale dei Fisici, Como (Zanichelli, 1927), Vol. 2, pp. 545–557. |
3. | V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005). |
4. | R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1(2), 818–831 (1970). [CrossRef] |
5. | R. K. Ahrenkiel, “Modified Kramers-Kronig analysis of optical spectra,” J. Opt. Soc. Am. 61(12), 1651–1655 (1971). [CrossRef] |
6. | K. F. Palmer, M. Z. Williams, and B. A. Budde, “Multiply subtractive Kramers- Kronig analysis of optical data,” Appl. Opt. 37(13), 2660–2673 (1998). [CrossRef] |
7. | V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4–6), 409–414 (2003). [CrossRef] |
8. | V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B 72(12), 125107 (2005). [CrossRef] |
9. | T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010). [CrossRef] [PubMed] |
10. | T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B 81(12), 125329 (2010). [CrossRef] |
11. | N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). For simplicity, the parameter c of the Drude-Smith model is set to –1 in the present paper. [CrossRef] |
12. | Numerical results for K(δt) were insensitive to the way of cutting off the integration frequency range in Eq. (7). |
13. | K(δt) becomes a periodic function of δt with a period equal to the original temporal length of E_{THz}(t). |
14. | Even if E_{THz}(t) is recorded with non-flat sensitivity, causality should hold and thus our approach will also work. However, non-flat sensitivity will require a more complicated interpretation of THz signals. |
OCIS Codes
(000.3860) General : Mathematical methods in physics
(300.2140) Spectroscopy : Emission
(300.6495) Spectroscopy : Spectroscopy, teraherz
ToC Category:
Spectroscopy
History
Original Manuscript: April 26, 2011
Revised Manuscript: June 5, 2011
Manuscript Accepted: June 6, 2011
Published: June 16, 2011
Citation
Takeya Unuma, Yusuke Ino, Kai-Erik Peiponen, Erik M. Vartiainen, Makoto Kuwata-Gonokami, and Kazuhiko Hirakawa, "Causality-based method for determining the time origin in terahertz emission spectroscopy," Opt. Express 19, 12759-12765 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12759
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References
- R. de L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. 12(6), 547–557 (1926). [CrossRef]
- H. A. Kramers, “La diffusion de la lumière par les atomes,” in Atti del Congresso Internazionale dei Fisici, Como (Zanichelli, 1927), Vol. 2, pp. 545–557.
- V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, 2005).
- R. Z. Bachrach and F. C. Brown, “Exciton-optical properties of TlBr and TlCl,” Phys. Rev. B 1(2), 818–831 (1970). [CrossRef]
- R. K. Ahrenkiel, “Modified Kramers-Kronig analysis of optical spectra,” J. Opt. Soc. Am. 61(12), 1651–1655 (1971). [CrossRef]
- K. F. Palmer, M. Z. Williams, and B. A. Budde, “Multiply subtractive Kramers- Kronig analysis of optical data,” Appl. Opt. 37(13), 2660–2673 (1998). [CrossRef]
- V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, “Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibilities,” Opt. Commun. 218(4–6), 409–414 (2003). [CrossRef]
- V. Lucarini, Y. Ino, K.-E. Peiponen, and M. Kuwata-Gonokami, “Detection and correction of the misplacement error in terahertz spectroscopy by application of singly subtractive Kramers-Kronig relations,” Phys. Rev. B 72(12), 125107 (2005). [CrossRef]
- T. Unuma, Y. Ino, M. Kuwata-Gonokami, E. M. Vartiainen, K.-E. Peiponen, and K. Hirakawa, “Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy,” Opt. Express 18(15), 15853–15858 (2010). [CrossRef] [PubMed]
- T. Unuma, Y. Ino, M. Kuwata-Gonokami, G. Bastard, and K. Hirakawa, “Transient Bloch oscillation with the symmetry-governed phase in semiconductor superlattices,” Phys. Rev. B 81(12), 125329 (2010). [CrossRef]
- N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). For simplicity, the parameter c of the Drude-Smith model is set to –1 in the present paper. [CrossRef]
- Numerical results for K(δt) were insensitive to the way of cutting off the integration frequency range in Eq. (7).
- K(δt) becomes a periodic function of δt with a period equal to the original temporal length of ETHz(t).
- Even if ETHz(t) is recorded with non-flat sensitivity, causality should hold and thus our approach will also work. However, non-flat sensitivity will require a more complicated interpretation of THz signals.
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