It is shown theoretically and experimentally that for the specific case of an equidistant frequency spacing of semiconductor laser modes, signals similar to terahertz (THz) time domain spectroscopy (TDS) can be detected in a standard photomixer setup. This quasi TDS system approach enables for both, time and frequency domain data processing. Measurements with a THz system which is based on a low cost multimode laser diode are presented. The system exhibits a bandwidth of 600 GHz and can be applied to the classical THz TDS application scenarios.
©2009 Optical Society of America
The development of systems working in the terahertz (THz) frequency range is a subject of active research due to the plethora of potential application fields for THz technology. Applications include non-destructive test of industrial processes and products [1–4], of cultural heritage [5–7], and the detection of hazardous materials [8,9] and THz near-field imaging [10–13]. Two different system approaches exist. Firstly, THz time domain spectroscopy (TDS), where femtosecond lasers are employed to generate broadband THz pulses  (see  for a comprehensive review). The overall advantage of TDS systems is that a single measurement contains the information on the complete THz frequency window, only restricted by the systems bandwidth. Due to the enormous amount of information obtained by every measurement, TDS systems are discussed for industrial and real world applications. Yet, their core component, the femtosecond laser, is still rather expensive and often not reliable and stable enough for long term industrial operation. Hence, photomixing systems are investigated for their practical applicability due to their potentially lower price. Here, two continuous wave (cw) single mode lasers with slightly different center wavelength are heterodyned onto a photomixer to generate cw THz radiation [16–20]. But the measurements of this system type provide only a limited amount of information since only a single frequency component is detected. Furthermore, the thickness determination is complicated because of the 2π ambiguity of the detected phase . In addition a precise and sophisticated frequency stabilization of the laser diodes is essential to preserve the phase information .
In this paper, we present an alternative system approach which provides quasi TDS (QTDS) like signals. It is based upon a low cost commercial multimode laser diode. Thus, the proposed system approach bears the potential to be used in real world applications of THz technology.
While multimode laser diodes with three or more oscillating modes have been employed to drive THz spectrometers [23–25], a comprehensive analytical discussion of this generation and detection scheme is essential to fully explore the potential of multi-frequency THz systems and especially of the QTDS approach  which we introduce here. The paper is organized as follows: First, the theory of cw multi-frequency THz systems is discussed in general. This includes the most promising case of QTDS. Afterwards, the theoretical deductions are validated by measurements obtained with a QTDS system. Finally, the system is utilized for typical THz application scenarios to highlight its practical applicability.
While the theory of continuous wave THz systems based on photoconductive antennas gated with two laser modes has been deeply discussed e.g. in , we will analyze here the system properties for the general case of multi-frequency excitation. After calculating the emitted THz field we derive the resulting photocurrent in the detector antenna. We will conclude by discussing the specific case of equidistant mode spacing.
If a laser beam, consisting of multiple frequency components, is focused onto the emitter antenna, the incident photons excite free carriers in the semiconductor material. These carriers are accelerated by the bias voltage which leads to a photocurrent. Due to the fact that the carrier generation is proportional to the laser power, a frequency mixing of the differing laser modes occurs and the photocurrent is modulated by the various mixing products.
However, the free carrier lifetime of the semiconductor material in the range of hundreds of femtoseconds induces a low pass characteristic and hence, the effective current modulation just consists of the difference frequencies between the laser modes. These difference frequencies lie in the THz range. According to  the emitted steady state THz field is proportional to time derivative of the photocurrent IE, which itself depends linearly on the antennas conductance GE and thus on the amount of optically excited free carriers nE, which is a function of the optical excitation power POpt,E.
The density of the photo carriers can be determined as the solution of the differential equation Eq. (2) in the case of a sinusoidal excitation with the angular frequency ω is given by
Yet, it can be concluded from the equation, that the carrier lifetime induces a low pass characteristics. Due to the fact that the optical power POpt,E is given by the squared sum of the electrical field of the M different laser modes, each of them oscillating with the angular frequency ωi:
For clarity, we will first specify the problem to the case of three laser modes (M = 3). Then the optical power at the emitter antenna is:
Here we account for the amplitude Ei and the time varying phase ϕi of the different laser modes, oscillating with the angular frequencies ωi. By employing Eq. (1) and 3, the THz field can be calculated as:
Since the same laser modes also modulate the carriers and thus the conductance in the detector antenna GD(t), a photocurrent can be detected, which is proportional to the time dependent conductance and the momentarily present electrical THz field at the position of the detector antenna. Due to the quasi DC measurement, the detected photocurrent is the result of a temporal average over these fast oscillations  and is given by:27]. As can be seen from Eq. (9), the phase of the detected THz frequency components is not related to the random phase between the multimode laser lines. However, due to the complex antenna impedance and dispersive materials in the THz path, the phase of the components may be slightly altered. After introducing the constant phase values ϕij, Eq. (9) becomes
Analogue to the deductions above, the more general case of M laser mode mixing can be calculated. Here, the photocurrent is given by:
The last point that will be generally discussed concerns the power efficiency. As can be seen from Eq. (11), the detected signal is a function of the individual power Pi of the laser modes. Thus by assuming an equal power spreading over the M laser modes, i.e. Pi = P/M, with P being the total power of the laser, the amplitude of the resulting THz frequency components ID(ω) are proportional to
The theoretical waveforms for the case of three and multiple modes and identical total power are shown Fig. 1 : An equidistant frequency spacing of the laser modes leads to TDS-like pulses, which occur with a repetition rate determined by the mode spacing. A frequency difference of 25GHz for instance induces an effective pulse repetition time of 40ps. This is especially interesting, since here the elaborate signal processing known from THz TDS can be utilized to analyze the measured data. Consequently, this specific case will be referred in the following as quasi time domain spectroscopy (QTDS).
The equidistant mode spacing Δf induces a constructive enhancement of the individual frequency components, which is illustrated in the Fig. 2 . Therefore, in the case of a QTDS system, the detected signal is given by
As a consequence, the signals peak-to-peak amplitude is theoretically only weakly dependent on the number of oscillating modes. On the one hand the lower frequency components are enhanced by the constructive superposition of the modes with the factor (M-m). On the other hand the signal is also proportional to the factor (2πmΔf ). Thus, the higher frequencies are amplified as well. Yet, the low pass characteristics induced by the free carriers, which is considered in the spectral efficiency of the system A(2πmΔf), effectively lower the pulses amplitude for higher bandwidths, since the higher frequency components have smaller amplitudes than the lower ones. The Fig. 3 shows simulated signals based on Eq. (13) for a different number of laser modes M, and identical total power and a resonance free antenna structure.
However, the metallization structures of the utilized photoconductive antennas affect the resulting waveform. Consequently, the antennas resonances can be optimized with respect to the employed laser diodes emission bandwidth.
3. System & Experiment
To experimentally verify the above conclusions, we employ a commercial multi mode laser diode operating at a center wavelength of 660nm. Instead of using a custom-tailored laser diode, we utilized an inexpensive device that was produced for the consumer electronic industry. The diode emits multiple laser lines, spaced by a constant difference frequency Δf of about 24GHz. The diodes spectrum which is shown in Fig. 4 exhibits an emission bandwidth of several hundreds of GHz. The total output power is 100mW with an electrical power consumption of 400mW. The laser beam is collimated by a standard low cost polymeric lens. The overall laser device including the driver electronics consumes the space of a laser pointer.
The laser beam is spitted by a beam splitter into two paths with equal powers. Afterwards, one of the beams is focused onto the emitter antenna. The other one is focused onto the detector antenna after being guided over a linear stage which is utilized as optical delay line. The antennas comprise a 100µm dipole and a 5µm gap without interdigital structures . They are made of standard low temperature (LT)-gallium arsenide (GaAs). This material provides fast recombination times of the optically exited free carriers and is consequently suitable for cw THz systems  as well as for the QTDS system discussed there. The emitter antenna is biased with 35V at 4.6kHz and a lock-in amplifier is utilized to enhance the signal-to-noise ratio. The integration time is set to 50ms. Four parabolic off-axis parabolic mirrors are employed for collimating and focusing the THz beam. The Fig. 5 shows a schematic of the setup.
In the beginning of this section, we will discuss the experimentally obtained THz waveform and its Fourier spectrum. Afterwards, we will highlight the broad applicability of the QTDS approach by demonstrating its suitability for a few practical application scenarios.
The signal obtained with the system is shown in Fig. 6 . As can be seen from the figure, the resulting pulse repetition time is 41ps, which is given by the inverse mode spacing of the utilized diode laser. Please note that this time constant can be optimized for the specific application scenario by choosing a laser diode with the preferred mode spacing. The Fig. 7 shows the corresponding THz spectra for a time window of 82ps and for one of only 41ps. For the latter, the second pulse is not considered within the time window of the measurement. This leads to a smooth spectrum. However, the frequency resolution is determined by the oscillation laser modes. In this case the mode spacing is 24.1GHz. As can be further seen from the spectra, the signal-to-noise ratio of this demonstrator system is in the range of 50 dB and the bandwidth is on the order of 600GHz. The latter is determined by the restricted spectral bandwidth of the employed laser diode and the antenna characteristic, which exhibits its constructive resonance at 500GHz. The second maximum in the pulse shape originates from the back reflections from the dipole ends . The ringing of the ground level is caused by water absorption lines and by back reflections from the contact pads.
In analogy to THz TDS measurements, a sample inserted into the THz path induces a delay of the pulses and reduces their amplitude due to reflection and absorption losses. Furthermore, Fabry Pérot (FP) echo pulses occur caused by the multiple reflections within the sample [31,32]. By comparing the reference and the sample measurement, the optical parameters of the sample can be determined.
To demonstrate the applicability of the QTDS approach for material characterization as well as thickness determination, we conduct measurements on a high resistive silicon wafer. The results are presented in Fig. 8 . The figure shows the delayed pulse and two clearly observable echo pulses. This signal can be exploited to analyze the sample by a time domain algorithm recently presented in . By numerically shifting the time axis and by amplitude scaling of the reference pulse to optimally overlap with the samples pulse, the refractive index and the absolute thickness of the sample under investigation can be determined simultaneously.
Figure 9 shows on the left the measured waveform compared to the results of the time domain algorithm. A very good agreement is observed. The derived spectrally averaged refractive index of 3.418 and the thickness of 531µm are in accordance to the literate value  and the values obtained by a micrometer screw, respectively. Additionally, a frequency domain based algorithm, proposed in , was employed to extract the refractive index of the sample from a single measurement over a broad frequency range. The extracted data are shown on the right side of Fig. 9.
These results clearly demonstrate that the QTDS approach is suitable to determine the thickness and the dielectric THz properties of samples analogous to TDS systems with both, time domain and frequency domain parameter extraction.
A second potential application of THz systems is the monitoring of the water status of plants [35,36]. A leaf of coffea arabica was detached from the plant and afterwards measured in the system for 6 hours. With time the leaf looses water by evaporation. Consequently, the thickness of the leaf decreases and its THz transmittance increases. Figure 10 shows on the left hand side the amplitude of the transmitted THz pulse (100% corresponds to the pulse amplitude without the leaf in the beam path) which rises with time as the water evaporates. On the right hand side we plot the temporal evolution of the position of the pulse maximum. Both measurands can be exploited to monitor the plant water status.
A third promising application of THz spectroscopy systems is the orientation analysis of birefringent samples such as fiber enforced polymers, liquid crystal polymers, paper tissue or optical crystals [37–40]. As one example, a 4mm thick LCP sample is measured at different orientation angles. This sample exhibits a distinctive birefringence as shown by the measured refractive index presented on the left side of Fig. 11 . This birefringence induces an extended pulse delay if the THz wave is polarized along the slow axis of the sample. The waveforms for the signals parallel and perpendicular to the slow axis are shown in Fig. 11.
After the samples THz properties have been determined using the procedure described in , we use the algorithm presented in  to analyze its orientation angle with respect to the slow axis. Figure 12 shows the waveforms for two arbitrarily adjusted orientations together with the simulated signals. A very good agreement allows for a precise orientation analysis.
Overall, the results of theses selective examples suggest that the QTDS approach combines the practically applicability of THz TDS with an unrivaled cost-effectiveness.
A new approach for THz spectroscopy systems based on inexpensive, commercially available multimode laser diodes has been proposed. This approach provides signals similar to that of THz TDS systems and is consequently referred to quasi time domain spectroscopy (QTDS). A demonstration system was presented, which exhibits a bandwidth of 600GHz and a signal-to-noise ratio of about 50 dB. The potential of QTDS was demonstrated by means of three examples.
While the presented system performance already allows for a broad applicability, the THz bandwidth could be extended by employing antenna structures with different resonance behavior and laser diodes with broader spectral emission. In conclusion, the QTDS approach carries the potential to foster real world applications of THz technology.
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