## Determination of the time origin by the maximum entropy method in time-domain terahertz emission spectroscopy |

Optics Express, Vol. 18, Issue 15, pp. 15853-15858 (2010)

http://dx.doi.org/10.1364/OE.18.015853

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### Abstract

We have developed a scheme for determining the time origin by the maximum entropy method (MEM) in time-domain terahertz (THz) emission spectroscopy. By applying the MEM to trial damped sinusoidal waveforms, we confirmed that the MEM gives true phase shifts across the resonance features and that its inherent uncertainty in determining the time origin is ±15 fs for 100-fs-class excitation/sampling optical pulses. Furthermore, when the MEM was applied to a THz waveform recorded experimentally with a finite sampling interval for the Bloch oscillation in a semiconductor superlattice, a misplacement of the time origin was indeed detected with an accuracy limited by the worse of the MEM inherent uncertainty and the sampling interval.

© 2010 OSA

## 1. Introduction

1. Q. Wu, M. Litz, and X.-C. Zhang, “Broadband detection capability of ZnTe electro-optic field detectors,” Appl. Phys. Lett. **68**(21), 2924 (1996). [CrossRef]

2. A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory,” Appl. Phys. Lett. **74**(11), 1516 (1999). [CrossRef]

4. M. Abe, S. Madhavi, Y. Shimada, Y. Otsuka, K. Hirakawa, and K. Tomizawa, “Transient carrier velocities in bulk GaAs: Quantitative comparison between terahertz data and ensemble Monte Carlo calculations,” Appl. Phys. Lett. **81**(4), 679 (2002). [CrossRef]

5. E. Hendry, M. Koeberg, J. M. Schins, L. D. A. Siebbeles, and M. Bonn, “Ultrafast charge generation in a semiconducting polymer studied with THz emission spectroscopy,” Phys. Rev. B **70**(3), 033202 (2004). [CrossRef]

6. N. Sekine and K. Hirakawa, “Dispersive terahertz gain of a nonclassical oscillator: Bloch oscillation in semiconductor superlattices,” Phys. Rev. Lett. **94**(5), 057408 (2005). [CrossRef] [PubMed]

7. E. Beaurepaire, G. M. Turner, S. M. Harrel, M. C. Beard, J.-Y. Bigot, and C. A. Schmuttenmaer, “Coherent terahertz emission from ferromagnetic films excited by femtosecond laser pulses,” Appl. Phys. Lett. **84**(18), 3465 (2004). [CrossRef]

*E*

_{THz}(

*t*), i.e., the moment when carriers excited by the peak of femtosecond laser pulses start emitting THz radiation. A possible misplacement

*δt*of the time origin in

*E*

_{THz}(

*t*) gives rise to an artificial phase shift by −

*ωδt*in its complex Fourier transform defined byand deforms both the real and imaginary parts of the spectrum. This is indeed very troublesome particularly when complex ac conductivities, Re

*σ*(

*ω*) and Im

*σ*(

*ω*), are derived from the spectral shapes of emitted THz electric fields, Re

*Ẽ*

_{THz}(

*ω*) and Im

*Ẽ*

_{THz}(

*ω*), respectively [6

6. N. Sekine and K. Hirakawa, “Dispersive terahertz gain of a nonclassical oscillator: Bloch oscillation in semiconductor superlattices,” Phys. Rev. Lett. **94**(5), 057408 (2005). [CrossRef] [PubMed]

8. Y. Shimada, K. Hirakawa, M. Odnoblioudov, and K. A. Chao, “Terahertz conductivity and possible Bloch gain in semiconductor superlattices,” Phys. Rev. Lett. **90**(4), 046806 (2003). [CrossRef] [PubMed]

*E*

_{THz}(

*t*) is crucial in making the most of its unique information. The conventional way of removing

*δt*is based on an empirical search for the −

*ωδt*component in the phase spectrum arg

*Ẽ*

_{THz}(

*ω*) [6

6. N. Sekine and K. Hirakawa, “Dispersive terahertz gain of a nonclassical oscillator: Bloch oscillation in semiconductor superlattices,” Phys. Rev. Lett. **94**(5), 057408 (2005). [CrossRef] [PubMed]

*ωδt*is comparable to or smaller than true phase shifts across resonance features, the empirical method does not work very accurately for such a spectral shape of arg

*Ẽ*

_{THz}(

*ω*).

*Ẽ*

_{THz}(

*ω*)|

^{2}and to separate it from the −

*ωδt*component in the phase spectrum arg

*Ẽ*

_{THz}(

*ω*). By testing the MEM on THz waveforms simulated for two types of damped sinusoidal oscillation current, we confirmed that the MEM properly derives true phase shifts across the resonance features and that its inherent uncertainty in determining the time origin is ±15 fs for 100-fs-class excitation/sampling optical pulses. Furthermore, when the MEM was applied to a THz waveform recorded experimentally with an arbitrary time origin and a finite sampling interval for the Bloch oscillation in a GaAs-based superlattice, a misplacement of the time origin was detected with a practically sufficient accuracy that is comparable to the sampling interval of the data points.

## 2. Scheme for determining the time origin

*E*

_{THz}(

*t*) consists of signal and small random noise, the fluctuation of

*E*

_{THz}(

*t*) does not have any particular polarity. In terms of the information theory, this situation can be formulated into the condition that the entropy for

*E*

_{THz}(

*t*) should be maximized [9]. The entropy for

*E*

_{THz}(

*t*) is known to be a functional of the power spectrum |

*Ẽ*

_{THz}(

*ω*)|

^{2}[9]. By maximizing the entropy, we obtain an analytical expression for |

*Ẽ*

_{THz}(

*ω*)|

^{2}:Here, Δ

*t*is the sampling interval of the data points in

*E*

_{THz}(

*t*), and 2

*M*is the number of the data points in |

*Ẽ*

_{THz}(

*ω*)|

^{2}in a frequency range used for the MEM analysis.

*a*(

_{m}*m*= 1, 2, …,

*M*) and

*b*are parameters called the MEM coefficients, which can be determined in such a way that Eq. (2) fits the measured power spectrum [9–11].

*Ẽ*

_{THz}(

*ω*) towith a phase factor exp[

*iφ*

_{err}(

*ω*)]. We obtain an important message from Eq. (3) that the phase spectrum arg

*Ẽ*

_{THz}(

*ω*) can be separated into two parts:withHere,

*ψ*(

_{M}*ω*) is the true phase shift, called the MEM phase, that is derived from the spectral feature of the power spectrum |

*Ẽ*

_{THz}(

*ω*)|

^{2}.

*φ*

_{err}(

*ω*) is the remaining part, called the error phase [10

10. E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Phase retrieval in optical spectroscopy: Resolving optical constants from power spectra,” Appl. Spec. **50**(10), 1283 (1996). [CrossRef]

*ωδt*due to a misplacement

*δt*of the time origin. It should be noted that, since |

*Ẽ*

_{THz}(

*ω*)|

^{2}does not depend on the position of

*t*= 0, the misplacement of the time origin does not affect

*ψ*(

_{M}*ω*) and appears only in

*φ*

_{err}(

*ω*) as the −

*ωδt*component.

*φ*

_{err}(

*ω*) has a simpler spectral shape than the original phase spectrum arg

*Ẽ*

_{THz}(

*ω*), which consists of both

*ψ*(

_{M}*ω*) and

*φ*

_{err}(

*ω*). Thus, the MEM will allow us to detect

*δt*in the slope of

*φ*

_{err}(

*ω*) more accurately than the empirical method does in the slope of arg

*Ẽ*

_{THz}(

*ω*). The scheme described above for removing

*δt*in THz emission spectroscopy is similar to that developed for removing the spatial misplacement between sample and reference in THz reflection spectroscopy [12

12. E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, “Numerical phase correction method for terahertz time-domain reflection spectroscopy,” J. Appl. Phys. **96**(8), 4171 (2004). [CrossRef]

13. Y. Ino, B. Héroux, T. Mukaiyama, and M. Kuwata-Gonokami, “Reflection-type pulsed terahertz imaging with a phase retrieval algorithm,” Appl. Phys. Lett. **88**(4), 041114 (2006). [CrossRef]

## 3. Results and discussion

### 3.1. Application to trial THz waveforms

*Ẽ*

_{THz}(

*ω*)|

^{2}, we first applied the MEM to trial THz waveforms

*E*

_{THz}(

*t*) simulated for two types of damped sinusoidal oscillation current

*J*(

*t*) =

*J*

_{0}Θ(

*t*)exp(

*−γt*)sin

*ω*

_{0}

*t*and

*J*

_{0}Θ(

*t*)exp(

*−γt*)cos

*ω*

_{0}

*t*. Here,

*J*

_{0}is the magnitude of the current, Θ(

*t*) the unit step function,

*γ*the dephasing rate, and

*ω*

_{0}the eigenfrequency. In this simulation, where

*E*

_{THz}(

*t*) has the exact time origin (i.e.,

*δt*= 0) and does not include random noise, the difference between the exact phase spectrum arg

*Ẽ*

_{THz}(

*ω*) and the MEM phase spectrum

*ψ*(

_{M}*ω*) is attributed to numerical errors in the MEM algorithm itself. We can thus estimate its inherent uncertainty in detecting

*δt*from the slope of the error phase spectrum

*φ*

_{err}(

*ω*) = arg

*Ẽ*

_{THz}(

*ω*)−

*ψ*(

_{M}*ω*). Numerical data on trial THz waveforms were prepared by convolving ∂

*J*(

*t*)/∂

*t*with a system response function that characterizes a temporal resolution

*τ*

_{res}.

*E*

_{THz}(

*t*) simulated for the damped sine and cosine currents, respectively, with parameters

*ω*

_{0}/2

*π*= 1.5 THz,

*γ*= 1.1 THz, and

*τ*

_{res}= 300 fs. Here, vertical dashed lines denote the exact position of

*t*= 0 used for the simulations. As seen in the figures, the two THz waveforms have rather different features: First, peak A is at

*t*= 0 in Fig. 1(a), whereas peak A' is at

*t*< 0 in Fig. 2(a). Second, peak A is of similar height to peak C and much higher than peak D in Fig. 1(a), whereas peak A' is much lower than peak C' and of similar height to peak D' in Fig. 2(a). Third, peak B is significantly higher than peak C in Fig. 1(a), whereas peak B' is of similar height to peak C' in Fig. 2(a).

*Ẽ*

_{THz}(

*ω*)| (solid curve) and the phase spectrum arg

*Ẽ*

_{THz}(

*ω*) (dash-dotted curve) calculated from the THz waveform

*E*

_{THz}(

*t*) of Fig. 1(a). As seen in the figure, |

*Ẽ*

_{THz}(

*ω*)| has a resonance peak at

*ω*/2

*π*= 1.48 THz and arg

*Ẽ*

_{THz}(

*ω*) displays a true phase shift from −

*π*/2 to

*π*/2 across the resonance feature. We then fitted the power spectrum |

*Ẽ*

_{THz}(

*ω*)|

^{2}with Eq. (2) to determine MEM coefficients

*a*(

_{m}*m*= 1, 2, …,

*M*) and

*b*, which can be numerically done in a Toeplitz matrix equation [12

12. E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, “Numerical phase correction method for terahertz time-domain reflection spectroscopy,” J. Appl. Phys. **96**(8), 4171 (2004). [CrossRef]

*ψ*(

_{M}*ω*) computed using Eq. (5) with the obtained set of

*a*in three different frequency ranges: 0.14–2.99 THz (blue), 0.20–2.94 THz (red), and 0.29–2.90 THz (green) [14]. As seen in the figure,

_{m}*ψ*(

_{M}*ω*) changes across the resonance feature and curls up near the edges of the respective frequency ranges. The difference between arg

*Ẽ*

_{THz}(

*ω*) and

*ψ*(

_{M}*ω*), i.e., the error phase spectrum

*φ*

_{err}(

*ω*), is shown in Fig. 1(d) by solid curves in the corresponding colors. As seen in the figure,

*φ*

_{err}(

*ω*) exhibits small but finite values, indicating numerical errors inherent to the MEM algorithm. We performed a least-squares fitting of

*φ*

_{err}(

*ω*) with a straight line in the respective frequency ranges (dashed lines). From different slopes of these three linear fits to

*φ*

_{err}(

*ω*), we find that a typical error in detecting

*δt*by the MEM is

*δt*

_{err}= ±15 fs for the given temporal resolution of

*τ*

_{res}= 300 fs.

*Ẽ*

_{THz}(

*ω*)| (solid curve) and arg

*Ẽ*

_{THz}(

*ω*) (dash-dotted curve),

*ψ*(

_{M}*ω*), and

*φ*

_{err}(

*ω*) for the THz waveform

*E*

_{THz}(

*t*) of Fig. 2(a) are shown in Figs. 2(b)–(d), respectively. As seen in the figures, the same analysis works for the THz waveform of Fig. 2(a), except that the obtained error phase spectrum

*φ*

_{err}(

*ω*) has an offset by −

*π*/2 due to the initial phase of the oscillatory waveform. It should be noted that, since

*ω*

_{0}

*δt*

_{err}is calculated to be ±0.14 and an order of magnitude smaller than

*π*/2, the MEM can definitely distinguish the two THz waveforms shown in Figs. 1(a) and 2(a).

### 3.2. Application to actual THz waveforms

*F*was measured in the time domain using a ZnTe electro-optic sensor that has flat sensitivity up to 3.5 THz. More experimental details are presented in Refs [6

**94**(5), 057408 (2005). [CrossRef] [PubMed]

15. T. Unuma, N. Sekine, and K. Hirakawa, “Dephasing of Bloch oscillating electrons in GaAs-based superlattices due to interface roughness scattering,” Appl. Phys. Lett. **89**(16), 161913 (2006). [CrossRef]

*E*

_{THz}(

*t*) observed for a GaAs(7.5 nm)/AlAs(0.8 nm) superlattice biased at

*F*= 8.2 kV/cm. Here, the time origin is placed at a tentative position; a possible misplacement

*δt*of the time origin exists in

*E*

_{THz}(

*t*). The sampling interval of the data points in

*E*

_{THz}(

*t*) is Δ

*t*= 30 fs. Figure 3(b) shows the amplitude spectrum |

*Ẽ*

_{THz}(

*ω*)| (solid curve) and the phase spectrum arg

*Ẽ*

_{THz}(

*ω*) (dash-dotted curve) calculated from the THz waveform

*E*

_{THz}(

*t*) of Fig. 3(a). As seen in the figure, |

*Ẽ*

_{THz}(

*ω*)| has a resonance peak at

*ω*/2

*π*= 1.47 THz, nearly equal to the expected Bloch frequency

*eFd/h*(

*d*: superlattice period) [15

15. T. Unuma, N. Sekine, and K. Hirakawa, “Dephasing of Bloch oscillating electrons in GaAs-based superlattices due to interface roughness scattering,” Appl. Phys. Lett. **89**(16), 161913 (2006). [CrossRef]

*Ẽ*

_{THz}(

*ω*) exhibits no clear linear phase shift at this stage because the −

*ωδt*component was already partially minimized by the empirical method.

*a*(

_{m}*m*= 1, 2, …,

*M*) and

*b*for the measured power spectrum |

*Ẽ*

_{THz}(

*ω*)|

^{2}, following the same procedure described in the previous subsection. Figure 3(c) shows the MEM phase spectrum

*ψ*(

_{M}*ω*) computed using Eq. (5) with the obtained set of

*a*in three different frequency ranges: 0.22–2.88 THz (blue), 0.28–2.97 THz (red), and 0.38–2.58 THz (green) [14]. The error phase spectrum

_{m}*φ*

_{err}(

*ω*) is shown in Fig. 3(d) by solid curves in the corresponding colors. As seen in the figures,

*ψ*(

_{M}*ω*) displays a phase shift by ~

*π*across the resonance feature in |

*Ẽ*

_{THz}(

*ω*)|, and

*φ*

_{err}(

*ω*) exhibits a nearly linear phase shift due to misplacement of the time origin. By fitting the obtained spectral shapes of

*φ*

_{err}(

*ω*) with straight lines (dashed lines) and estimating their slopes, we determined

*δt*to be 180 ± 20 fs, the uncertainty of which is comparable to the sampling interval of the original data points (Δ

*t*= 30 fs) shown in Fig. 3(a). This means that

*δt*has been detected with a practically sufficient accuracy and that the time origin should be shifted to the position indicated by a red vertical line in Fig. 3(a).

*ωδt*component is not easy to empirically find in the original phase spectrum arg

*Ẽ*

_{THz}(

*ω*) of Fig. 3(b): the −

*ωδt*phase shift is indeed obscured by the true phase shift [see Figs. 3(c) and 3(d)] in arg

*Ẽ*

_{THz}(

*ω*). This gives the implication that, when the MEM is applied to different spectral shapes [16], the MEM will excel the empirical method more greatly at detecting

*δt*in THz emission with larger or more complicated true phase shifts.

### 3.3. Note

17. For example, seeA. Lisauskas, M. M. Dignam, N. V. Demarina, E. Mohler, and H. G. Roskos, “Examining the terahertz signal from a photoexcited biased semiconductor superlattice for evidence of gain,” Appl. Phys. Lett. **93**(2), 021122 (2008). [CrossRef]

*J*(

*t*) starts from its maximum as damped cos

*ω*

_{0}

*t*at

*t*= 0. Here, we would like to emphasize that the MEM can make a clear distinction between the two. Very recently, it has been reported that the observed phase of the Bloch oscillation is ascribed to a capacitive response of electrons distributed on the Wannier-Stark ladder with translational symmetry [18

18. 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]

## 4. Summary

## Acknowledgments

## References and links

1. | Q. Wu, M. Litz, and X.-C. Zhang, “Broadband detection capability of ZnTe electro-optic field detectors,” Appl. Phys. Lett. |

2. | A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory,” Appl. Phys. Lett. |

3. | A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Femtosecond charge transport in polar semiconductors,” Phys. Rev. Lett. |

4. | M. Abe, S. Madhavi, Y. Shimada, Y. Otsuka, K. Hirakawa, and K. Tomizawa, “Transient carrier velocities in bulk GaAs: Quantitative comparison between terahertz data and ensemble Monte Carlo calculations,” Appl. Phys. Lett. |

5. | E. Hendry, M. Koeberg, J. M. Schins, L. D. A. Siebbeles, and M. Bonn, “Ultrafast charge generation in a semiconducting polymer studied with THz emission spectroscopy,” Phys. Rev. B |

6. | N. Sekine and K. Hirakawa, “Dispersive terahertz gain of a nonclassical oscillator: Bloch oscillation in semiconductor superlattices,” Phys. Rev. Lett. |

7. | E. Beaurepaire, G. M. Turner, S. M. Harrel, M. C. Beard, J.-Y. Bigot, and C. A. Schmuttenmaer, “Coherent terahertz emission from ferromagnetic films excited by femtosecond laser pulses,” Appl. Phys. Lett. |

8. | Y. Shimada, K. Hirakawa, M. Odnoblioudov, and K. A. Chao, “Terahertz conductivity and possible Bloch gain in semiconductor superlattices,” Phys. Rev. Lett. |

9. | S. Haykin, |

10. | E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Phase retrieval in optical spectroscopy: Resolving optical constants from power spectra,” Appl. Spec. |

11. | K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, |

12. | E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, “Numerical phase correction method for terahertz time-domain reflection spectroscopy,” J. Appl. Phys. |

13. | Y. Ino, B. Héroux, T. Mukaiyama, and M. Kuwata-Gonokami, “Reflection-type pulsed terahertz imaging with a phase retrieval algorithm,” Appl. Phys. Lett. |

14. | We chose these frequency ranges by considering a tradeoff: the resonance feature is more accurately captured in wider frequency ranges, while the numerical error in solving the Toeplitz matrix equation becomes less in narrower frequency ranges. |

15. | T. Unuma, N. Sekine, and K. Hirakawa, “Dephasing of Bloch oscillating electrons in GaAs-based superlattices due to interface roughness scattering,” Appl. Phys. Lett. |

16. | V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, |

17. | For example, seeA. Lisauskas, M. M. Dignam, N. V. Demarina, E. Mohler, and H. G. Roskos, “Examining the terahertz signal from a photoexcited biased semiconductor superlattice for evidence of gain,” Appl. Phys. Lett. |

18. | 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 |

**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 23, 2010

Revised Manuscript: June 7, 2010

Manuscript Accepted: June 7, 2010

Published: July 12, 2010

**Citation**

Takeya Unuma, Yusuke Ino, Makoto Kuwata-Gonokami, Erik M. Vartiainen, Kai-Erik Peiponen, and Kazuhiko Hirakawa, "Determination of the time origin by
the maximum entropy method in
time-domain terahertz emission spectroscopy," Opt. Express **18**, 15853-15858 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-15-15853

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### References

- Q. Wu, M. Litz, and X.-C. Zhang, “Broadband detection capability of ZnTe electro-optic field detectors,” Appl. Phys. Lett. 68(21), 2924 (1996). [CrossRef]
- A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Detectors and sources for ultrabroadband electro-optic sampling: Experiment and theory,” Appl. Phys. Lett. 74(11), 1516 (1999). [CrossRef]
- A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Nuss, and W. H. Knox, “Femtosecond charge transport in polar semiconductors,” Phys. Rev. Lett. 82 (25), 5140 (1999); “Femtosecond high-field transport in compound semiconductors,” Phys. Rev. B 61(24), 16642–16652 (2000).
- M. Abe, S. Madhavi, Y. Shimada, Y. Otsuka, K. Hirakawa, and K. Tomizawa, “Transient carrier velocities in bulk GaAs: Quantitative comparison between terahertz data and ensemble Monte Carlo calculations,” Appl. Phys. Lett. 81(4), 679 (2002). [CrossRef]
- E. Hendry, M. Koeberg, J. M. Schins, L. D. A. Siebbeles, and M. Bonn, “Ultrafast charge generation in a semiconducting polymer studied with THz emission spectroscopy,” Phys. Rev. B 70(3), 033202 (2004). [CrossRef]
- N. Sekine and K. Hirakawa, “Dispersive terahertz gain of a nonclassical oscillator: Bloch oscillation in semiconductor superlattices,” Phys. Rev. Lett. 94(5), 057408 (2005). [CrossRef] [PubMed]
- E. Beaurepaire, G. M. Turner, S. M. Harrel, M. C. Beard, J.-Y. Bigot, and C. A. Schmuttenmaer, “Coherent terahertz emission from ferromagnetic films excited by femtosecond laser pulses,” Appl. Phys. Lett. 84(18), 3465 (2004). [CrossRef]
- Y. Shimada, K. Hirakawa, M. Odnoblioudov, and K. A. Chao, “Terahertz conductivity and possible Bloch gain in semiconductor superlattices,” Phys. Rev. Lett. 90(4), 046806 (2003). [CrossRef] [PubMed]
- S. Haykin, Nonlinear Methods of Spectral Analysis (Springer, Berlin, 1983), Chap. 2.
- E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Phase retrieval in optical spectroscopy: Resolving optical constants from power spectra,” Appl. Spec. 50(10), 1283 (1996). [CrossRef]
- K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, Heidelberg, 1999), Chap. 5.
- E. M. Vartiainen, Y. Ino, R. Shimano, M. Kuwata-Gonokami, Y. P. Svirko, and K.-E. Peiponen, “Numerical phase correction method for terahertz time-domain reflection spectroscopy,” J. Appl. Phys. 96(8), 4171 (2004). [CrossRef]
- Y. Ino, B. Héroux, T. Mukaiyama, and M. Kuwata-Gonokami, “Reflection-type pulsed terahertz imaging with a phase retrieval algorithm,” Appl. Phys. Lett. 88(4), 041114 (2006). [CrossRef]
- We chose these frequency ranges by considering a tradeoff: the resonance feature is more accurately captured in wider frequency ranges, while the numerical error in solving the Toeplitz matrix equation becomes less in narrower frequency ranges.
- T. Unuma, N. Sekine, and K. Hirakawa, “Dephasing of Bloch oscillating electrons in GaAs-based superlattices due to interface roughness scattering,” Appl. Phys. Lett. 89(16), 161913 (2006). [CrossRef]
- V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer, Berlin, 2005), Chap. 10.
- For example, seeA. Lisauskas, M. M. Dignam, N. V. Demarina, E. Mohler, and H. G. Roskos, “Examining the terahertz signal from a photoexcited biased semiconductor superlattice for evidence of gain,” Appl. Phys. Lett. 93(2), 021122 (2008). [CrossRef]
- 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]

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