- Email: [email protected]
Frequency aspects of the THz photomixer
Optics Communications 285 (2012) 1308–1313
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Frequency aspects of the THz photomixer Przemysław P. Jarzab, Kacper Nowak ⁎, Edward F. Plinski Department of Electronics, Wroclaw University of Technology, Wroclaw 50–370, Poland
a r t i c l e
i n f o
Article history: Received 27 April 2011 Received in revised form 28 July 2011 Accepted 14 September 2011 Available online 6 October 2011
a b s t r a c t In the paper methods of the terahertz photomixer operational frequency estimation are considered. Three methods are investigated — estimation of the frequency via direct measurements of the laser heterodyne compounds, via calculations from datasheets, and via calculations from raw collected data obtained from the photomixer system. © 2011 Elsevier B.V. All rights reserved.
Keywords: Terahertz wave THz photomixer Homodyne detection Photomixer frequency Terahertz spectroscopy
1. Introduction In the recent years the terahertz radiation is the subject of many worldwide commercial and scientific projects and conferences [1–5]. Terahertz radiation represents a so called “forgotten region” in the radio wave and light spectrum at the frequency range between 100 GHz and 30 THz. Frequencies from the “terahertz gap” are used to recognize the spectral properties of several substances and biological materials with no destruction during the process of investigations. There are many ways to generate terahertz radiation for spectroscopy purposes. The most popular is the THz-TDS [6] (Terahertz Time Domain Spectroscopy) which includes a laser system for ultra short pulses generation. Most of the THz-TDS uses photoconductive technique with an optical delay line to achieve generation and detection of the submillimeter waves in the same setup. The results are obtained from Fast Fourier Transform (FFT) applied to the measured time-domain signal. Another technique is the oldest THz source — the molecular far infrared (FIR) laser [7] which can produce frequencies between 300 GHz and 10 THz at relatively high output power. Unfortunately the FIR laser setup is complicated and its emission can occur only at fixed frequencies. Quantum Cascade Lasers (QCL) development setups are very promising [8] but they usually require cryogenic temperatures in a terahertz region and cannot be widely tunable. Among continuous wave sources there are BWO tubes (Backward-Wave Oscillators) [9] and FTIR (Fourier Transform Infra-Red) [10]. The BWO devices are inconvenient because of magnets size and weight. On the other hand, the FTIR
⁎ Corresponding author. E-mail addresses: [email protected] (K. Nowak), [email protected] (E.F. Plinski). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.09.053
apparatus is equipped in complicated automatic adjusting system and sophisticated filaments. The solution which can be competitive to above systems is a CW terahertz spectrometer which contains two lasers (a laser heterodyne) working in the system using photomixing technique. The frequency of the terahertz signal is equal to the frequency of the laser heterodyne [11,12]. CW spectrometer parameters can be similar to the TDS in terms of the ease of operation and measurements frequency range. This solution can be cheaper than femtosecond systems but requires more time to setup and adjust the arrangement. A crucial problem in the photomixer is correct estimation of the operation frequency. In this paper we consider three possible methods to find reliable solutions for the problem. 2. Photomixing The photomixing [13] is a process of the optical heterodyne conversion, when two waves at the f1 and f2 frequencies are in a beating (Fig. 1). For this reason, interference of two optical beams (Fig. 2) can be used for generation of electromagnetic waves with gigahertz and terahertz frequencies. The interference of two optical collinear polarized beams can be expressed as: ω þ ω ω −ω 2 2 Eðt Þ ¼ E0 cosðω1 t Þ þ E0 cosðω2 t Þ ¼ 2E0 cos 1 t cos 1 ; 2 2 ð1Þ where: E0 — electric field amplitude equal for each beam, ω1 = 2πf1, and f1 — laser 1 frequency, ω2 = 2πf2, and f2 — laser 2 frequency.
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
1309
3. THz operating frequency: method 1 — laser frequency direct measurements The terahertz frequency can be calculated as: fTHz ¼ f2 −f1 ¼
c c − ; λ2 λ1
ð3Þ
Fig. 1. Photomixer setup: f1–f2 differential frequency between Laser1 and Laser2.
The incident optical power Pi(t) is proportional to the square of the electric field E(t):
Pi ðt Þ ¼ E0
2
1 1 1 þ cos ðω1 þ ω2 Þt þ cos ðω1 −ω2 Þt þ cos ð2ω1 t Þ þ cos ð2ω2 t Þ : 2 2
ð2Þ
Four waves at the frequencies f1 − f2, 2f1, 2f2 and f1 + f2 are the mathematical result of photomixing but only the frequency difference f1 − f2 could be achieved in a real photomixer system based on a low-temperature grown GaAs (LT-GaAs) medium, which is used in our experiment. The other result frequencies are much higher than electron–hole recombination time therefore they can be omitted. The optical beating periodically modulates the photoconductance of the antenna deposited on the LT-GaAs. Under the presence of an external electric field (bias), the AC component will be flowing through the antenna and thus emit electromagnetic radiation. There are several ways to detect THz radiation including incoherent (thermal-based sensors like bolometers, Golay cells) and coherent detection. Each has their advantages and disadvantages. In our experiment, we use a coherent detection method [14] which is non-thermal-based and does not require low temperature. In addition the coherent detection provides phase information. In order to perform the coherent detection, a second detecting photoconductive antenna has to be illuminated with the beam including frequency beat used for the generation of THz radiation. In contrast to the emitting antenna, there is no bias applied to the detector antenna. The optical beating periodically modulates the photoconductance of the detector antenna. The induced current is proportional to the conductivity modulated by the optical power incident on the detector, however only an average current over time can be measured. The phases of the optical and terahertz signal have to be matched to give a maximum signal which can be measured by a lock-in amplifier. Precise measurements/calculations of the photomixer frequency are crucial for correct results obtained with the system. Below several methods are discussed.
where: fTHz — terahertz wave frequency,λ1, f1 — wavelength/frequency of the laser 1,λ2, f2 — wavelength/frequency of the laser 2. The dependency above shows that accuracy of the terahertz wave frequency measurement depends on the accuracy of laser wavelength/ frequency measurements. Error caused by a wavelength measurement can be calculated using a differential method. The frequency measurements can be performed in two ways: measuring the differential frequency between two laser sources with an optical spectrum analyzer or measuring each laser frequency separately with an optical wavelength meter. 3.1. Optical spectrum analyzer In this method both laser frequencies are measured by an optical spectrum analyzer (OSA) — see Fig. 3. This means that both measurements of the wavelengths λ1, λ2 have the same measurement error Δλ1 = Δλ2 = Δλe. The measurement error Δλe consists of a common part for both wavelength measurements (a systematic error) and a part specific for each wavelengths separately. The systematic error will subtract to 0, because both lasers are measured in one measurement. Equation below shows the worst case measurement error (a real error is smaller): ΔfTHz ¼
c c c c ⋅Δλe þ ⋅Δλe ¼ Δλe ⋅ þ ; 2 2 2 2 ðλ2 Þ ðλ1 Þ ðλ1 Þ ðλ2 Þ
ð4Þ
where: Δλe — measurement error of the optical spectrum analyzer. A spectrum analyzer is characterized by several parameters. For terahertz measurements it is important to measure a central wavelength of the laser. For this reason a frequency resolution is the most important parameter of the OSA device in THz measurements. It determines how fine the OSA slices the optical signal, which limits accuracy of the laser beam central frequency determination. The frequency resolution typically ranges from 0.1 nm to 0.01 nm. (E.g. in frequency domain the frequency resolution of 0.01 nm at a wavelength of 850 nm is equal to 4 GHz). The other two parameters that should be considered are the wavelength accuracy and range. The wavelength accuracy limits a minimal difference between λ1 and λ2. The typical wavelength accuracy of the OSA is better than 0.1 nm. At the wavelength of 850 nm it determines a minimal distance of 41 GHz. A minimal frequency distance that can be recognized by OSA is around 80 GHz [15]. A typical value of the OSA wavelength range is 400–1700 nm. For the LT-GaAs
Fig. 2. Time and frequency domain result of the beam interference.
1310
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
difference caused by temperature change ΔT,λ0 — diode center wavelength in temperature T0. Because ΔT1 = T1−T0 and f ¼ λc then: fTHz ¼
Fig. 3. Optical spectrum analyzer (OSA) measurement setup. λ1, λ2-wavelengths of Laser1 and Laser2, respectively.
c c c⋅ðλ1 −λ2 Þ c ðT2 −T1 Þ − ¼ ¼ ⋅ ; λ2 λ1 λ1 ⋅λ2 k T2 ⋅T1
where: T1 and T2 are absolute temperatures of the diodes 1 and 2, respectively. When the temperature coefficient of wavelength is used to calculate the frequency fTHz, then the measurement error from a differential method is given by a formula: ΔfTHz ¼
terahertz chip used in our experiment, a high absorbency factor is for the wavelength around 850 nm, and it falls in this range. 3.2. Optical wavelength meter The optical wavelength meters are devices that typically use Michelson or Fizeau interferometers to measure the wavelength of the laser beam. Manufacturers provide a measurement error in wavelength units (for vacuum) and/or in frequency units. The wavelength meter (WM1, WM2) is a sensitive, high-resolution device that can be used to measure laser 1 and 2 DFB diode characteristics. Absolute accuracy of this device for wavelength of 850 nm can be around Δf = 60 MHz [16]. In this method, two wavelength meters are used (Fig. 4), therefore frequency measurement error ΔfTHz can be defined as: ΔfTHz ¼ Δf1 þ Δf2 ;
ð5Þ
where: Δf1 — measurement error of the first wavelength meter,Δf2 — measurement error of the second wavelength meter. 4. THz operating frequency: method 2 — calculations from datasheets A main factor determining the laser 1 and laser 2 DFB diode wavelengths is diode temperature. For this reason, this dependency is being used to control the laser diode wavelength. Assuming that constant current is applied, it is possible to tune the wavelength by setting the temperature precisely. Usually the current coefficient of wavelength, taken from datasheet, is around 10 times lower than the temperature coefficient, so under constant current it is possible to calculate the diode wavelength λ using an approximation: λ≈λ0 þ Δλ ¼ λ0 þ k⋅ΔT;
ð6Þ
where: k — temperature coefficient of wavelength,Δλ — wavelength
ð7Þ
c ðT2 −T1 Þ c 1 1 ⋅Δk þ ⋅ΔT2 þ ⋅ΔT1 : 2 T ⋅T k T2 T1 k 2 1
ð8Þ
where: Δk — error of the temperature coefficient of wavelength k parameter,ΔT1 — measurement error of the temperature of the laser 1,ΔT2 — measurement error of the temperature of the laser 2. Some producers provide the temperature tuning coefficient of a frequency parameter and the formula (6) can be rewritten as follow: fTHz ¼ kf ⋅ðT2 −T1 Þ;
ð9Þ
where: kf — temperature coefficient of frequency. When the temperature coefficient of frequency is used to calculate the frequency fTHz, then the measurement error from a differential method is given by a formula: ΔfTHz ¼ ðT2 −T1 Þ⋅Δkf þ kf ðΔT2 þ ΔT1 Þ
ð10Þ
where: Δkf — error of the temperature coefficient of frequency kf parameter. Parameters λ0 and k can be appointed from OSA measurements, but also they are usually given in a laser diode datasheet. Unfortunately if OSA is not available, datasheet k and kf parameters are usually not very precise which is shown in a further part of this paper. In that way, an error can be introduced in the frequency axis scale of the spectroscopic characteristics. The method based on values of temperature of frequency/wavelength coefficients taken from datasheets is very simple and does not need any extra equipment. Laser temperature can be read from a laser diode controller. Another advantage is that every next measurement point has frequency higher than previous measurement point, therefore frequency axis on spectrum chart is always ascending, which makes it very easy to plot. Wavelength changes are not linear to temperature changes. The wavelength changes are not monotonic and usually “steps” can be observed in the temperature vs. wavelength characteristics. That is why application of this method involves a risk — wrong frequencies can be assigned to several measurement points. Most of the time it is not noticeable, but in some worst case scenarios false positive spectrum peak can be observed. 5. The operating frequency: method 3 — via coherent detection
Fig. 4. Measurement setup. WM1, WM2 — wavelength meters.
As it is known, in a classical optical interferometer two optical signals of the same frequency meet together at the surface of the screen (or detector). When the optical length of one of the interferometer arm is changed, then we can observe moving of interferometric fringes or a periodical current/voltage signal at the detector. In the coherent detection method the picture is more complicated. The terahertz wave induces a voltage signal at the receiving antenna technologically fit to the surface of the semiconductor. The same laser beam which is used as a pumping beam at the emitting antenna to release a THz wave from alternative semiconductor chip is applied as a probing beam to key the terahertz signal at the receiving antenna. The setup — a dipole antenna and semiconductor are often called a
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
1311
Fig. 5. Lock-in amplitude signals for fTHz = 400, 500 and 600 GHz photomixer operation frequency (a), a typical lock-in phase signal (b).
photoconductive antenna or a terahertz chip [17]. Depending on the phase relations between terahertz and probing beams some constant average signal is received. When one of the arms of the “interferometer” is changed we can observe a periodical signal like in a classical interferometer (Fig. 5a). The obtained signal is used in the method as a source for calculation of the terahertz wave frequency. Additionally, the calibration of the delay line is needed for calculations. The frequency of the terahertz wave can be calculated from the period of the signal in Fig. 5a scaled in length of the delay line shift [18]. Another method used in this paper is FFT (Fast Fourier Transform) applied for a measured periodical signal. It gives directly the operation frequency of the photomixer. The measurement procedure is performed in three steps. First — the photomixer is fixed at some chosen THz frequency; the frequency is set with temperature controllers of the laser diodes. Second — at the fixed frequency the delay line is tuned in its maximum possible range to obtain the periodic signal as long as possible. Third — FFT of the signal is calculated and the frequency axis is scaled using a known dependency linking speed of the light c, frequency f and wavelength λ: c = λ ⋅ f. The maximum of the FFT result determines operating frequency of the photomixer. Additionally, the average amplitude of the detected signal can be received. Next, the procedure is repeated for other temperature controller settings linearly changing the difference in temperatures in all possible range of laser diodes used. The third step needs more explanation. FFT is calculated from the signal shown at Fig. 5a. The signal is obtained from a raw lock-in signal — an absolute value of the amplitude. The necessary signal for FFT is calculated monitoring the phase signal changes. A final amplitude
Fig. 6. FFT of the lock-in amplitude signal obtained directly from measurements for fTHz = 400, 500 and 600 GHz.
signal is a source for FFT procedure. The result is shown in Fig. 6. To obtain a spectrum of the investigated sample the information about amplitude Am of the signal in Fig. 5a. is necessary as well. The amplitude is calculated from a known formula: Am ¼
2A max ; NF
ð11Þ
where: Amax — amplitude of the FFT signal (Fig. 5a),NF — number of the FFT points. Higher resolution of the FFT signal can be obtained artificially adding zeros to the measured signal vector. The efficiency of the FFT can be increased when the length of the signal vector is a power of two (e.g. … 1024, 2048, 4096 …). The frequency of the terahertz radiation, necessary to scale the xaxis of the spectroscopic characteristics, can be calculated from the lock-in phase signal only (Fig. 5b). Anyway, the amplitude of the signal, necessary to scale the y-axis of spectroscopic characteristics, has to be calculated from the formula (11). 6. Experimental setup and results The experiment was arranged to compare the second and third methods. The first method measures directly the frequency of the heterodyne components — lasers. The method is not interesting for us because it measures the frequencies before introducing the laser beams to the measuring setup. It means, the method does not take into account many factors influencing the measuring process. As it is known, the arrangement of the terahertz photomixer is, in its essence, the interferometric setup which is sensitive to all possible turbulences. It is why we can expect influences of the incorrect laser beam overlapping, acoustical and mechanical vibrations and temperature changes.
Fig. 7. Experimental setup. LH — laser heterodyne, SG — signal generator, BS — cubic beam splitter, DL — delay line, EA — emitting antenna, RA — receiving antenna, PM — parabolic mirrors.
1312
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
The source of the signal in the measuring system is a setup of the laser heterodyne LH (usually CW single mode diode lasers). The heterodyne signal is split into two arms with a cubic beam splitter BS (Fig. 7). The photomixer consists of two main optical arms: the first arm with a pumping beam from the heterodyne plus a terahertz beam conducted with parabolic mirrors PM, the second arm with a detecting beam. Both of them are adjusted to the same length using an optical delay line DL. For the measuring process, it is useful when the perturbation signal (split into both arms) reaches the receiving antenna RA at the same time. In our setup, we used two 852 nm DFB laser diodes and two identical LT-GaAs chips with the 200 μm dipoles and 5 μm gap [19]. A coherent detection method is applied. The output signal is measured by a lock-in amplifier LA at the receiving antenna RA. The emitting antenna EA is gated with a signal generator SG. The SG signal is used as a reference for the lock-in amplifier. Laser diodes are tunable to the differential frequency at the value up to 900 GHz. The step dl of the delay line DL was set at 16 μm. In the experiment we moved the delay line of the distance lP = 2000 μm (see Fig. 5a), that results in number of NP = 125 samples being collected. To enhance the resolution of the FFT algorithm a number of samples was increased to NF = 4096 by adding zeros to the measured signal. This operation artificially increases the distance lPto a value lF = dl ⋅ NF = 65.54 mm. As a consequence, the calculated minimum differential frequency change (the frequency accuracy) was: ΔfTHz ¼ c=lF ¼ 4:6GHz;
ð12Þ
The accuracy of the delay line translation was measured with an optical interferometer and it was Δdl = 1 μm. This value is a negligible percentage ((Δdl ⋅ NP/lF) ⋅ 100% ≈ 0.2%) of the lF = 65.54 mm length used in Fourier Transform and therefore can be omitted. The repeatability of the delay line used was estimated as 10 μm. In our experiment absolute value of the delay line position does not play any role. Only a relative move must be accurate to ensure equal steps along the delay line path during the measurement. Additionally, every move was always performed in one direction with removal of the initial mechanical backlash. The experiment was performed in a closed box at saturated water vapor atmosphere. The well known in literature water spectrum is used in our experiment as a reference [20]. We performed spectroscopy measurements using the methods two and three. The results are shown in Fig. 8. The black line shows the results using the method 2 (from datasheets). The method 3 (via coherent detection) is illustrated by red and blue lines for amplitude and phase signals, respectively. As seen, the amplitude of the signal drops due to decreasing efficiency of the photomixing process with frequency. The second factor determining attenuation of the
Fig. 8. Spectroscopy measurements of the saturated water vapor (in the range of 400– 800 GHz). The photomixer operation frequencies f are calculated using method 2 and method 3 (see the main text). A, B — recognized absorption lines (method 3); a, b — recognized absorption lines (method 2).
signal is an impedance matching which varies when the operating frequency of the photomixer is changed. The impedance matching depends directly on a THz antenna used (LT-GaAs chip with a Hertz dipole). The effective matching is obtained for a resonance frequency fm for a Hertz dipole of length deff using the formula [21]: fm ≈
c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; deff ⋅2 ðεr þ 1Þ=2
ð13Þ
where: εr = 12.85. For deff = 100 μm used in our experiment it gives appr. 570 GHz, which is close to the center of our investigated band. We recognized two absorption lines of water vapor at 560 GHz and 756 GHz indicated with A and B, respectively (blue and red lines in Fig. 8). We achieved a measurement error of ±4.6 GHz. It is a result of a number of the measurement points and stability of the photomixer setup. The not overlapping characteristics (blue and red) at the absorption line B fall in the range of the measurement error. The result of the frequency calculations taken from the DFB diode datasheet is also shown in Fig. 8 — black line. The frequency temperature coefficient factor kf (see formula (9)) was taken from the datasheet of the laser system used in our experiment. As seen in Fig. 8, the water spectral lines (indicated as a and b in the figure) vary significantly from lines A and B, respectively. The explanation of the effect is clear: for calculations we used the laser diodes frequency factor corresponding to the diode temperature as 25 GHz/1 °C given by a producer. Our careful investigations have shown that the overlapping is obtained at the frequency factor value of 23.8 GHz/1 °C. 7. Conclusion The most important parameter of the CW terahertz photomixer setup is its frequency. In this work, three methods of the terahertz wave frequency estimation, necessary for x-axis scaling for spectroscopic applications, have been considered. Accuracy in the frequency determination in all three methods has been discussed. Principle of method 1 is direct measurements of the laser heterodyne frequency. Two variants are shown: a direct measurement of both laser beams at once using an optical spectrum analyzer and a direct measurement of both lasers separately using two wavelength meters. The method 2 bases on the data provided by a laser head or laser diode manufacturer. In this method, either a temperature coefficient of the wavelength or a temperature coefficient of the frequency parameter is known and used for calculations of the differential frequency. On the other hand, the proposed method 3 is also based on calculations, but does not depend on any datasheet parameters of the laser diode. In principle, all calculations are based on the results obtained from the lock-in amplifier. Therefore, a real terahertz wave frequency of the measured signal is estimated. In this study, the accuracy of all three methods is discussed. The best accuracy is achieved using wavelength meters (the method 1) at the level of ± 120 MHz. Some devices provide a Proportional Integral Derivative (PID) controller. It can use a feedback signal to fine tuning of the lasers over the full frequency range. In this case, both the frequency and laser power are stabilized. The Optical Spectrum Analyzer (OSA) variant in principle is the same as Wavelength Meter (WM) variant, but the WM variant cannot be used for the laser PID stabilization and the accuracy of the method 1 is much lower — at the level of ±4 GHz. The disadvantage of method 1 (in both variants) is an additional cost of the device and the modification of the setup is necessary to place the device into the photomixer arrangement. The method 2 basis on the datasheet parameters of the laser diodes and its accuracy depends on those parameters. We have shown that the water vapor absorption lines are shifted approximately of 45 GHz in relation to literature data when the data from
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
the manufacturer datasheet is used. The method 3 basis on the raw data is obtained from the lock-in amplifier. In this method, the appropriate frequency of the terahertz wave is calculated. In the method, the data is obtained from the spectrum measurements. In that way, no additional measurement procedures are necessary, and no arrangement changes or additional devices are needed. All calculations are performed during the data analysis. Method 3 is much more reliable than method 2 and does not depend on any laser parameters. Method 3 depends on the delay line accuracy. The accuracy achieved in our photomixer setup was at the level of ±4.6 GHz and it is comparable to the OSA measurements. Complementary considerations a reader can find in [22] where signal to noise ratio aspects of the photomixer are discussed. In [23] the authors claim that a THz spectrometer based on the frequency comb is able to achieve an accuracy better than 50 kHz. Acknowledgments We acknowledge Dr. J. S. Witkowski for his helpful discussions, and Dr. G. Dudzik for his contribution to the experiment part of the work. References [1] E.R. Brown, F.W. Smith, K.A. McIntosh, Journal of Applied Physics 73 (3) (1993) 1480. [2] D. Mittleman (Ed.), Sensing with Terahertz Radiation, Springer-Verlag, New York, 2002. [3] X.-C. Zhang, Xu Jingzhou, Intrudction to THz Wave Photonics, Springer, New York, 2010.
1313
[4] K. Sakai, Terahertz Optoelectronics, Springer-Verlag, Berlin-Heidelberg, 2005. [5] Yun-Shik Lee, Principle of Terahertz Science and Technology, Springer, New York, 2009. [6] B. Fischer, M. Hoffmann, H. Helm, R. Wilk, F. Rutz, T. Kleine-Ostmann, M. Koch, P. Jepsen, Optics Express 13 (14) (2005) 5205. [7] K.M. Abramski, E.F. Plinski, in: M. Endo, R.F. Walter (Eds.), gas lasers, CRC, Press, New York, 2007, p. 497. [8] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Science 264 (5158) (1994) 553. [9] B. Gorshunov, A. Volkov, I. Spektor, A. Prokhorov, A. Mukhin, M. Dressel, S. Uchida, A. Loidl, International Journal of Infrared and Millimeter Waves 26 (9) (2005) 1217. [10] P. Hamm, M.H. Lim, R.M. Hochstrasser, The Journal of Physical Chemistry. B 102 (31) (1998) 6123. [11] D. Saeedkia, R.R. Mansour, S. Safavi-Naeini, IEEE Journal of Quantum Electronics 41 (9) (2005) 1188. [12] K.A. McIntosh, E.R. Brown, K.B. Nichols, O.B. McMahon, W.F. DiNatale, T.M. Lyszczarz, Applied Physics Letters 67 (26) (1995) 3844. [13] E.F. Plinski, Bulletin PAS-Technology Science 58 (4) (2010) 463. [14] J.T. Nahata, Yardley, T.F. Heinz, Applied Physics Letters 75 (17) (1999) 2524. [15] R. Wilk, “Switchable THz Reflectors”, PhD. Dissertation, Technischen Universitat Carolo-Wilhelmina zu Braunschweig 2007. [16] A.J. Deninger, T. Göbel, D. Schönherr, T. Kinder, A. Roggenbuck, M. Köberle, F. Lison, T. Müller-Wirts, P. Meissner, The Review of Scientific Instruments 79 (2008) 044702. [17] M. Mikulics, M. Marso, S. Wu, A. Fox, M. Lepsa, D. Grützmacher, R. Sobolewski, P. Kordoš, IEEE Photonics Technology Letters 20 (12) (2008) 1054. [18] R. Wilk, Bulletin PAS-Technology Science 58 (4) (2010) 713. [19] R. Wilk, F. Breitfeld, M. Mikulics, M. Koch, Applied Optics 47 (16) (2008) 3023. [20] M. Van Exter, C.H. Fattinger, D. Grischkowsky, Optics Letters 14 (20) (1989) 1128. [21] G.M. Rebeiz, Proceedings of the IEEE 80 (11) (1992) 1748. [22] J.E. Bjarnason, E.R. Brown, Applied Physics Letters 87 (2005) 134105. [23] G. Mouret, F. Hindle, A. Cuisset, Ch. Yang, R. Bocquet, M. Lours, D. Rovera, Optics Express 17 (24) (2009) 22031.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Frequency aspects of the THz photomixer Przemysław P. Jarzab, Kacper Nowak ⁎, Edward F. Plinski Department of Electronics, Wroclaw University of Technology, Wroclaw 50–370, Poland
a r t i c l e
i n f o
Article history: Received 27 April 2011 Received in revised form 28 July 2011 Accepted 14 September 2011 Available online 6 October 2011
a b s t r a c t In the paper methods of the terahertz photomixer operational frequency estimation are considered. Three methods are investigated — estimation of the frequency via direct measurements of the laser heterodyne compounds, via calculations from datasheets, and via calculations from raw collected data obtained from the photomixer system. © 2011 Elsevier B.V. All rights reserved.
Keywords: Terahertz wave THz photomixer Homodyne detection Photomixer frequency Terahertz spectroscopy
1. Introduction In the recent years the terahertz radiation is the subject of many worldwide commercial and scientific projects and conferences [1–5]. Terahertz radiation represents a so called “forgotten region” in the radio wave and light spectrum at the frequency range between 100 GHz and 30 THz. Frequencies from the “terahertz gap” are used to recognize the spectral properties of several substances and biological materials with no destruction during the process of investigations. There are many ways to generate terahertz radiation for spectroscopy purposes. The most popular is the THz-TDS [6] (Terahertz Time Domain Spectroscopy) which includes a laser system for ultra short pulses generation. Most of the THz-TDS uses photoconductive technique with an optical delay line to achieve generation and detection of the submillimeter waves in the same setup. The results are obtained from Fast Fourier Transform (FFT) applied to the measured time-domain signal. Another technique is the oldest THz source — the molecular far infrared (FIR) laser [7] which can produce frequencies between 300 GHz and 10 THz at relatively high output power. Unfortunately the FIR laser setup is complicated and its emission can occur only at fixed frequencies. Quantum Cascade Lasers (QCL) development setups are very promising [8] but they usually require cryogenic temperatures in a terahertz region and cannot be widely tunable. Among continuous wave sources there are BWO tubes (Backward-Wave Oscillators) [9] and FTIR (Fourier Transform Infra-Red) [10]. The BWO devices are inconvenient because of magnets size and weight. On the other hand, the FTIR
⁎ Corresponding author. E-mail addresses: [email protected] (K. Nowak), [email protected] (E.F. Plinski). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.09.053
apparatus is equipped in complicated automatic adjusting system and sophisticated filaments. The solution which can be competitive to above systems is a CW terahertz spectrometer which contains two lasers (a laser heterodyne) working in the system using photomixing technique. The frequency of the terahertz signal is equal to the frequency of the laser heterodyne [11,12]. CW spectrometer parameters can be similar to the TDS in terms of the ease of operation and measurements frequency range. This solution can be cheaper than femtosecond systems but requires more time to setup and adjust the arrangement. A crucial problem in the photomixer is correct estimation of the operation frequency. In this paper we consider three possible methods to find reliable solutions for the problem. 2. Photomixing The photomixing [13] is a process of the optical heterodyne conversion, when two waves at the f1 and f2 frequencies are in a beating (Fig. 1). For this reason, interference of two optical beams (Fig. 2) can be used for generation of electromagnetic waves with gigahertz and terahertz frequencies. The interference of two optical collinear polarized beams can be expressed as: ω þ ω ω −ω 2 2 Eðt Þ ¼ E0 cosðω1 t Þ þ E0 cosðω2 t Þ ¼ 2E0 cos 1 t cos 1 ; 2 2 ð1Þ where: E0 — electric field amplitude equal for each beam, ω1 = 2πf1, and f1 — laser 1 frequency, ω2 = 2πf2, and f2 — laser 2 frequency.
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
1309
3. THz operating frequency: method 1 — laser frequency direct measurements The terahertz frequency can be calculated as: fTHz ¼ f2 −f1 ¼
c c − ; λ2 λ1
ð3Þ
Fig. 1. Photomixer setup: f1–f2 differential frequency between Laser1 and Laser2.
The incident optical power Pi(t) is proportional to the square of the electric field E(t):
Pi ðt Þ ¼ E0
2
1 1 1 þ cos ðω1 þ ω2 Þt þ cos ðω1 −ω2 Þt þ cos ð2ω1 t Þ þ cos ð2ω2 t Þ : 2 2
ð2Þ
Four waves at the frequencies f1 − f2, 2f1, 2f2 and f1 + f2 are the mathematical result of photomixing but only the frequency difference f1 − f2 could be achieved in a real photomixer system based on a low-temperature grown GaAs (LT-GaAs) medium, which is used in our experiment. The other result frequencies are much higher than electron–hole recombination time therefore they can be omitted. The optical beating periodically modulates the photoconductance of the antenna deposited on the LT-GaAs. Under the presence of an external electric field (bias), the AC component will be flowing through the antenna and thus emit electromagnetic radiation. There are several ways to detect THz radiation including incoherent (thermal-based sensors like bolometers, Golay cells) and coherent detection. Each has their advantages and disadvantages. In our experiment, we use a coherent detection method [14] which is non-thermal-based and does not require low temperature. In addition the coherent detection provides phase information. In order to perform the coherent detection, a second detecting photoconductive antenna has to be illuminated with the beam including frequency beat used for the generation of THz radiation. In contrast to the emitting antenna, there is no bias applied to the detector antenna. The optical beating periodically modulates the photoconductance of the detector antenna. The induced current is proportional to the conductivity modulated by the optical power incident on the detector, however only an average current over time can be measured. The phases of the optical and terahertz signal have to be matched to give a maximum signal which can be measured by a lock-in amplifier. Precise measurements/calculations of the photomixer frequency are crucial for correct results obtained with the system. Below several methods are discussed.
where: fTHz — terahertz wave frequency,λ1, f1 — wavelength/frequency of the laser 1,λ2, f2 — wavelength/frequency of the laser 2. The dependency above shows that accuracy of the terahertz wave frequency measurement depends on the accuracy of laser wavelength/ frequency measurements. Error caused by a wavelength measurement can be calculated using a differential method. The frequency measurements can be performed in two ways: measuring the differential frequency between two laser sources with an optical spectrum analyzer or measuring each laser frequency separately with an optical wavelength meter. 3.1. Optical spectrum analyzer In this method both laser frequencies are measured by an optical spectrum analyzer (OSA) — see Fig. 3. This means that both measurements of the wavelengths λ1, λ2 have the same measurement error Δλ1 = Δλ2 = Δλe. The measurement error Δλe consists of a common part for both wavelength measurements (a systematic error) and a part specific for each wavelengths separately. The systematic error will subtract to 0, because both lasers are measured in one measurement. Equation below shows the worst case measurement error (a real error is smaller): ΔfTHz ¼
c c c c ⋅Δλe þ ⋅Δλe ¼ Δλe ⋅ þ ; 2 2 2 2 ðλ2 Þ ðλ1 Þ ðλ1 Þ ðλ2 Þ
ð4Þ
where: Δλe — measurement error of the optical spectrum analyzer. A spectrum analyzer is characterized by several parameters. For terahertz measurements it is important to measure a central wavelength of the laser. For this reason a frequency resolution is the most important parameter of the OSA device in THz measurements. It determines how fine the OSA slices the optical signal, which limits accuracy of the laser beam central frequency determination. The frequency resolution typically ranges from 0.1 nm to 0.01 nm. (E.g. in frequency domain the frequency resolution of 0.01 nm at a wavelength of 850 nm is equal to 4 GHz). The other two parameters that should be considered are the wavelength accuracy and range. The wavelength accuracy limits a minimal difference between λ1 and λ2. The typical wavelength accuracy of the OSA is better than 0.1 nm. At the wavelength of 850 nm it determines a minimal distance of 41 GHz. A minimal frequency distance that can be recognized by OSA is around 80 GHz [15]. A typical value of the OSA wavelength range is 400–1700 nm. For the LT-GaAs
Fig. 2. Time and frequency domain result of the beam interference.
1310
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
difference caused by temperature change ΔT,λ0 — diode center wavelength in temperature T0. Because ΔT1 = T1−T0 and f ¼ λc then: fTHz ¼
Fig. 3. Optical spectrum analyzer (OSA) measurement setup. λ1, λ2-wavelengths of Laser1 and Laser2, respectively.
c c c⋅ðλ1 −λ2 Þ c ðT2 −T1 Þ − ¼ ¼ ⋅ ; λ2 λ1 λ1 ⋅λ2 k T2 ⋅T1
where: T1 and T2 are absolute temperatures of the diodes 1 and 2, respectively. When the temperature coefficient of wavelength is used to calculate the frequency fTHz, then the measurement error from a differential method is given by a formula: ΔfTHz ¼
terahertz chip used in our experiment, a high absorbency factor is for the wavelength around 850 nm, and it falls in this range. 3.2. Optical wavelength meter The optical wavelength meters are devices that typically use Michelson or Fizeau interferometers to measure the wavelength of the laser beam. Manufacturers provide a measurement error in wavelength units (for vacuum) and/or in frequency units. The wavelength meter (WM1, WM2) is a sensitive, high-resolution device that can be used to measure laser 1 and 2 DFB diode characteristics. Absolute accuracy of this device for wavelength of 850 nm can be around Δf = 60 MHz [16]. In this method, two wavelength meters are used (Fig. 4), therefore frequency measurement error ΔfTHz can be defined as: ΔfTHz ¼ Δf1 þ Δf2 ;
ð5Þ
where: Δf1 — measurement error of the first wavelength meter,Δf2 — measurement error of the second wavelength meter. 4. THz operating frequency: method 2 — calculations from datasheets A main factor determining the laser 1 and laser 2 DFB diode wavelengths is diode temperature. For this reason, this dependency is being used to control the laser diode wavelength. Assuming that constant current is applied, it is possible to tune the wavelength by setting the temperature precisely. Usually the current coefficient of wavelength, taken from datasheet, is around 10 times lower than the temperature coefficient, so under constant current it is possible to calculate the diode wavelength λ using an approximation: λ≈λ0 þ Δλ ¼ λ0 þ k⋅ΔT;
ð6Þ
where: k — temperature coefficient of wavelength,Δλ — wavelength
ð7Þ
c ðT2 −T1 Þ c 1 1 ⋅Δk þ ⋅ΔT2 þ ⋅ΔT1 : 2 T ⋅T k T2 T1 k 2 1
ð8Þ
where: Δk — error of the temperature coefficient of wavelength k parameter,ΔT1 — measurement error of the temperature of the laser 1,ΔT2 — measurement error of the temperature of the laser 2. Some producers provide the temperature tuning coefficient of a frequency parameter and the formula (6) can be rewritten as follow: fTHz ¼ kf ⋅ðT2 −T1 Þ;
ð9Þ
where: kf — temperature coefficient of frequency. When the temperature coefficient of frequency is used to calculate the frequency fTHz, then the measurement error from a differential method is given by a formula: ΔfTHz ¼ ðT2 −T1 Þ⋅Δkf þ kf ðΔT2 þ ΔT1 Þ
ð10Þ
where: Δkf — error of the temperature coefficient of frequency kf parameter. Parameters λ0 and k can be appointed from OSA measurements, but also they are usually given in a laser diode datasheet. Unfortunately if OSA is not available, datasheet k and kf parameters are usually not very precise which is shown in a further part of this paper. In that way, an error can be introduced in the frequency axis scale of the spectroscopic characteristics. The method based on values of temperature of frequency/wavelength coefficients taken from datasheets is very simple and does not need any extra equipment. Laser temperature can be read from a laser diode controller. Another advantage is that every next measurement point has frequency higher than previous measurement point, therefore frequency axis on spectrum chart is always ascending, which makes it very easy to plot. Wavelength changes are not linear to temperature changes. The wavelength changes are not monotonic and usually “steps” can be observed in the temperature vs. wavelength characteristics. That is why application of this method involves a risk — wrong frequencies can be assigned to several measurement points. Most of the time it is not noticeable, but in some worst case scenarios false positive spectrum peak can be observed. 5. The operating frequency: method 3 — via coherent detection
Fig. 4. Measurement setup. WM1, WM2 — wavelength meters.
As it is known, in a classical optical interferometer two optical signals of the same frequency meet together at the surface of the screen (or detector). When the optical length of one of the interferometer arm is changed, then we can observe moving of interferometric fringes or a periodical current/voltage signal at the detector. In the coherent detection method the picture is more complicated. The terahertz wave induces a voltage signal at the receiving antenna technologically fit to the surface of the semiconductor. The same laser beam which is used as a pumping beam at the emitting antenna to release a THz wave from alternative semiconductor chip is applied as a probing beam to key the terahertz signal at the receiving antenna. The setup — a dipole antenna and semiconductor are often called a
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
1311
Fig. 5. Lock-in amplitude signals for fTHz = 400, 500 and 600 GHz photomixer operation frequency (a), a typical lock-in phase signal (b).
photoconductive antenna or a terahertz chip [17]. Depending on the phase relations between terahertz and probing beams some constant average signal is received. When one of the arms of the “interferometer” is changed we can observe a periodical signal like in a classical interferometer (Fig. 5a). The obtained signal is used in the method as a source for calculation of the terahertz wave frequency. Additionally, the calibration of the delay line is needed for calculations. The frequency of the terahertz wave can be calculated from the period of the signal in Fig. 5a scaled in length of the delay line shift [18]. Another method used in this paper is FFT (Fast Fourier Transform) applied for a measured periodical signal. It gives directly the operation frequency of the photomixer. The measurement procedure is performed in three steps. First — the photomixer is fixed at some chosen THz frequency; the frequency is set with temperature controllers of the laser diodes. Second — at the fixed frequency the delay line is tuned in its maximum possible range to obtain the periodic signal as long as possible. Third — FFT of the signal is calculated and the frequency axis is scaled using a known dependency linking speed of the light c, frequency f and wavelength λ: c = λ ⋅ f. The maximum of the FFT result determines operating frequency of the photomixer. Additionally, the average amplitude of the detected signal can be received. Next, the procedure is repeated for other temperature controller settings linearly changing the difference in temperatures in all possible range of laser diodes used. The third step needs more explanation. FFT is calculated from the signal shown at Fig. 5a. The signal is obtained from a raw lock-in signal — an absolute value of the amplitude. The necessary signal for FFT is calculated monitoring the phase signal changes. A final amplitude
Fig. 6. FFT of the lock-in amplitude signal obtained directly from measurements for fTHz = 400, 500 and 600 GHz.
signal is a source for FFT procedure. The result is shown in Fig. 6. To obtain a spectrum of the investigated sample the information about amplitude Am of the signal in Fig. 5a. is necessary as well. The amplitude is calculated from a known formula: Am ¼
2A max ; NF
ð11Þ
where: Amax — amplitude of the FFT signal (Fig. 5a),NF — number of the FFT points. Higher resolution of the FFT signal can be obtained artificially adding zeros to the measured signal vector. The efficiency of the FFT can be increased when the length of the signal vector is a power of two (e.g. … 1024, 2048, 4096 …). The frequency of the terahertz radiation, necessary to scale the xaxis of the spectroscopic characteristics, can be calculated from the lock-in phase signal only (Fig. 5b). Anyway, the amplitude of the signal, necessary to scale the y-axis of spectroscopic characteristics, has to be calculated from the formula (11). 6. Experimental setup and results The experiment was arranged to compare the second and third methods. The first method measures directly the frequency of the heterodyne components — lasers. The method is not interesting for us because it measures the frequencies before introducing the laser beams to the measuring setup. It means, the method does not take into account many factors influencing the measuring process. As it is known, the arrangement of the terahertz photomixer is, in its essence, the interferometric setup which is sensitive to all possible turbulences. It is why we can expect influences of the incorrect laser beam overlapping, acoustical and mechanical vibrations and temperature changes.
Fig. 7. Experimental setup. LH — laser heterodyne, SG — signal generator, BS — cubic beam splitter, DL — delay line, EA — emitting antenna, RA — receiving antenna, PM — parabolic mirrors.
1312
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
The source of the signal in the measuring system is a setup of the laser heterodyne LH (usually CW single mode diode lasers). The heterodyne signal is split into two arms with a cubic beam splitter BS (Fig. 7). The photomixer consists of two main optical arms: the first arm with a pumping beam from the heterodyne plus a terahertz beam conducted with parabolic mirrors PM, the second arm with a detecting beam. Both of them are adjusted to the same length using an optical delay line DL. For the measuring process, it is useful when the perturbation signal (split into both arms) reaches the receiving antenna RA at the same time. In our setup, we used two 852 nm DFB laser diodes and two identical LT-GaAs chips with the 200 μm dipoles and 5 μm gap [19]. A coherent detection method is applied. The output signal is measured by a lock-in amplifier LA at the receiving antenna RA. The emitting antenna EA is gated with a signal generator SG. The SG signal is used as a reference for the lock-in amplifier. Laser diodes are tunable to the differential frequency at the value up to 900 GHz. The step dl of the delay line DL was set at 16 μm. In the experiment we moved the delay line of the distance lP = 2000 μm (see Fig. 5a), that results in number of NP = 125 samples being collected. To enhance the resolution of the FFT algorithm a number of samples was increased to NF = 4096 by adding zeros to the measured signal. This operation artificially increases the distance lPto a value lF = dl ⋅ NF = 65.54 mm. As a consequence, the calculated minimum differential frequency change (the frequency accuracy) was: ΔfTHz ¼ c=lF ¼ 4:6GHz;
ð12Þ
The accuracy of the delay line translation was measured with an optical interferometer and it was Δdl = 1 μm. This value is a negligible percentage ((Δdl ⋅ NP/lF) ⋅ 100% ≈ 0.2%) of the lF = 65.54 mm length used in Fourier Transform and therefore can be omitted. The repeatability of the delay line used was estimated as 10 μm. In our experiment absolute value of the delay line position does not play any role. Only a relative move must be accurate to ensure equal steps along the delay line path during the measurement. Additionally, every move was always performed in one direction with removal of the initial mechanical backlash. The experiment was performed in a closed box at saturated water vapor atmosphere. The well known in literature water spectrum is used in our experiment as a reference [20]. We performed spectroscopy measurements using the methods two and three. The results are shown in Fig. 8. The black line shows the results using the method 2 (from datasheets). The method 3 (via coherent detection) is illustrated by red and blue lines for amplitude and phase signals, respectively. As seen, the amplitude of the signal drops due to decreasing efficiency of the photomixing process with frequency. The second factor determining attenuation of the
Fig. 8. Spectroscopy measurements of the saturated water vapor (in the range of 400– 800 GHz). The photomixer operation frequencies f are calculated using method 2 and method 3 (see the main text). A, B — recognized absorption lines (method 3); a, b — recognized absorption lines (method 2).
signal is an impedance matching which varies when the operating frequency of the photomixer is changed. The impedance matching depends directly on a THz antenna used (LT-GaAs chip with a Hertz dipole). The effective matching is obtained for a resonance frequency fm for a Hertz dipole of length deff using the formula [21]: fm ≈
c pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; deff ⋅2 ðεr þ 1Þ=2
ð13Þ
where: εr = 12.85. For deff = 100 μm used in our experiment it gives appr. 570 GHz, which is close to the center of our investigated band. We recognized two absorption lines of water vapor at 560 GHz and 756 GHz indicated with A and B, respectively (blue and red lines in Fig. 8). We achieved a measurement error of ±4.6 GHz. It is a result of a number of the measurement points and stability of the photomixer setup. The not overlapping characteristics (blue and red) at the absorption line B fall in the range of the measurement error. The result of the frequency calculations taken from the DFB diode datasheet is also shown in Fig. 8 — black line. The frequency temperature coefficient factor kf (see formula (9)) was taken from the datasheet of the laser system used in our experiment. As seen in Fig. 8, the water spectral lines (indicated as a and b in the figure) vary significantly from lines A and B, respectively. The explanation of the effect is clear: for calculations we used the laser diodes frequency factor corresponding to the diode temperature as 25 GHz/1 °C given by a producer. Our careful investigations have shown that the overlapping is obtained at the frequency factor value of 23.8 GHz/1 °C. 7. Conclusion The most important parameter of the CW terahertz photomixer setup is its frequency. In this work, three methods of the terahertz wave frequency estimation, necessary for x-axis scaling for spectroscopic applications, have been considered. Accuracy in the frequency determination in all three methods has been discussed. Principle of method 1 is direct measurements of the laser heterodyne frequency. Two variants are shown: a direct measurement of both laser beams at once using an optical spectrum analyzer and a direct measurement of both lasers separately using two wavelength meters. The method 2 bases on the data provided by a laser head or laser diode manufacturer. In this method, either a temperature coefficient of the wavelength or a temperature coefficient of the frequency parameter is known and used for calculations of the differential frequency. On the other hand, the proposed method 3 is also based on calculations, but does not depend on any datasheet parameters of the laser diode. In principle, all calculations are based on the results obtained from the lock-in amplifier. Therefore, a real terahertz wave frequency of the measured signal is estimated. In this study, the accuracy of all three methods is discussed. The best accuracy is achieved using wavelength meters (the method 1) at the level of ± 120 MHz. Some devices provide a Proportional Integral Derivative (PID) controller. It can use a feedback signal to fine tuning of the lasers over the full frequency range. In this case, both the frequency and laser power are stabilized. The Optical Spectrum Analyzer (OSA) variant in principle is the same as Wavelength Meter (WM) variant, but the WM variant cannot be used for the laser PID stabilization and the accuracy of the method 1 is much lower — at the level of ±4 GHz. The disadvantage of method 1 (in both variants) is an additional cost of the device and the modification of the setup is necessary to place the device into the photomixer arrangement. The method 2 basis on the datasheet parameters of the laser diodes and its accuracy depends on those parameters. We have shown that the water vapor absorption lines are shifted approximately of 45 GHz in relation to literature data when the data from
P.łP. Jarzab et al. / Optics Communications 285 (2012) 1308–1313
the manufacturer datasheet is used. The method 3 basis on the raw data is obtained from the lock-in amplifier. In this method, the appropriate frequency of the terahertz wave is calculated. In the method, the data is obtained from the spectrum measurements. In that way, no additional measurement procedures are necessary, and no arrangement changes or additional devices are needed. All calculations are performed during the data analysis. Method 3 is much more reliable than method 2 and does not depend on any laser parameters. Method 3 depends on the delay line accuracy. The accuracy achieved in our photomixer setup was at the level of ±4.6 GHz and it is comparable to the OSA measurements. Complementary considerations a reader can find in [22] where signal to noise ratio aspects of the photomixer are discussed. In [23] the authors claim that a THz spectrometer based on the frequency comb is able to achieve an accuracy better than 50 kHz. Acknowledgments We acknowledge Dr. J. S. Witkowski for his helpful discussions, and Dr. G. Dudzik for his contribution to the experiment part of the work. References [1] E.R. Brown, F.W. Smith, K.A. McIntosh, Journal of Applied Physics 73 (3) (1993) 1480. [2] D. Mittleman (Ed.), Sensing with Terahertz Radiation, Springer-Verlag, New York, 2002. [3] X.-C. Zhang, Xu Jingzhou, Intrudction to THz Wave Photonics, Springer, New York, 2010.
1313
[4] K. Sakai, Terahertz Optoelectronics, Springer-Verlag, Berlin-Heidelberg, 2005. [5] Yun-Shik Lee, Principle of Terahertz Science and Technology, Springer, New York, 2009. [6] B. Fischer, M. Hoffmann, H. Helm, R. Wilk, F. Rutz, T. Kleine-Ostmann, M. Koch, P. Jepsen, Optics Express 13 (14) (2005) 5205. [7] K.M. Abramski, E.F. Plinski, in: M. Endo, R.F. Walter (Eds.), gas lasers, CRC, Press, New York, 2007, p. 497. [8] J. Faist, F. Capasso, D.L. Sivco, C. Sirtori, A.L. Hutchinson, A.Y. Cho, Science 264 (5158) (1994) 553. [9] B. Gorshunov, A. Volkov, I. Spektor, A. Prokhorov, A. Mukhin, M. Dressel, S. Uchida, A. Loidl, International Journal of Infrared and Millimeter Waves 26 (9) (2005) 1217. [10] P. Hamm, M.H. Lim, R.M. Hochstrasser, The Journal of Physical Chemistry. B 102 (31) (1998) 6123. [11] D. Saeedkia, R.R. Mansour, S. Safavi-Naeini, IEEE Journal of Quantum Electronics 41 (9) (2005) 1188. [12] K.A. McIntosh, E.R. Brown, K.B. Nichols, O.B. McMahon, W.F. DiNatale, T.M. Lyszczarz, Applied Physics Letters 67 (26) (1995) 3844. [13] E.F. Plinski, Bulletin PAS-Technology Science 58 (4) (2010) 463. [14] J.T. Nahata, Yardley, T.F. Heinz, Applied Physics Letters 75 (17) (1999) 2524. [15] R. Wilk, “Switchable THz Reflectors”, PhD. Dissertation, Technischen Universitat Carolo-Wilhelmina zu Braunschweig 2007. [16] A.J. Deninger, T. Göbel, D. Schönherr, T. Kinder, A. Roggenbuck, M. Köberle, F. Lison, T. Müller-Wirts, P. Meissner, The Review of Scientific Instruments 79 (2008) 044702. [17] M. Mikulics, M. Marso, S. Wu, A. Fox, M. Lepsa, D. Grützmacher, R. Sobolewski, P. Kordoš, IEEE Photonics Technology Letters 20 (12) (2008) 1054. [18] R. Wilk, Bulletin PAS-Technology Science 58 (4) (2010) 713. [19] R. Wilk, F. Breitfeld, M. Mikulics, M. Koch, Applied Optics 47 (16) (2008) 3023. [20] M. Van Exter, C.H. Fattinger, D. Grischkowsky, Optics Letters 14 (20) (1989) 1128. [21] G.M. Rebeiz, Proceedings of the IEEE 80 (11) (1992) 1748. [22] J.E. Bjarnason, E.R. Brown, Applied Physics Letters 87 (2005) 134105. [23] G. Mouret, F. Hindle, A. Cuisset, Ch. Yang, R. Bocquet, M. Lours, D. Rovera, Optics Express 17 (24) (2009) 22031.