System-theoretical modeling of terahertz time-domain spectroscopy with ultra-high repetition rate mode-locked lasers

Kevin Kolpatzeck; Xuan Liu; Kai-Henning Tybussek; Kai-Henning Tybussek; Lars Häring; Marlene Zander; Wolfgang Rehbein; Martin Moehrle; Andreas Czylwik; Jan C. Balzer

doi:10.1364/OE.389632

1. Introduction

Since the first demonstration of terahertz time-domain spectroscopy (THz-TDS) in 1989 [1], the interest in this specific frequency range steadily increases [2]. While the first systems were driven by dye lasers, the advent of ultrafast solid-state lasers, in particular the titanium-sapphire laser, made terahertz technology more accessible and enabled first THz images [3]. The breakthrough for terahertz applications outside of a laboratory environment was the use of an ultrafast fiber laser with suitable photoconductive antennas [4]. This led to many industry-related applications [5]. However, ultrafast fiber lasers make still up to 50% of the total system costs [6]. A promising alternative for compact and inexpensive light sources are monolithic edge-emitting semiconductor lasers. In 2000, Morikawa et al. presented a terahertz system driven by a multi-mode laser diode and called this technique cross-correlation spectroscopy (CCS) [7]. In 2009, Scheller et al. adapted this approach and named it quasi time-domain spectroscopy (QTDS) [8]. Another demonstration of CCS was shown by Molter et al., where an ultrabroadband superluminescent diode was used [9]. The first demonstration of THz-TDS based on a research-grade semiconductor laser in 2017 by Merghem et al. exhibited a bandwidth of 0.8 THz [10]. Two years later, Tybussek et al. employed a commercially available laser diode and achieved a bandwidth of up to 1.3 THz [11]. Summarizing, there are four main regimes in which terahertz spectroscopy systems using semiconductor-based antennas have been demonstrated:

• Frequency-domain spectroscopy (FDS): Continuous-wave operation using two tunable single-mode lasers or a single dual-mode laser.
• Time-domain spectroscopy (TDS): Pulsed operation using a femtosecond pulse laser.
• Cross-correlation spectroscopy (CCS) or quasi time-domain spectroscopy (QTDS): Chaotic operation using a multi-mode laser diode or a superluminescent diode.
• Time-domain spectroscopy with ultra-high repetition rate: Pulsed operation using a mode-locked laser diode (MLLD).

FDS is thoroughly described in [12,13]. The classical TDS approach using a femtosecond pulse laser is comprehensively modeled in [12,14]. The CCS or QTDS approach using a multi-mode laser diode is described in detail in [7,8,15]. While the models using a multi-mode laser diode describe a mode structure similar to that found in MLLDs, they do not take the spectral phase of the laser signal into account, as there is no stable phase relationship between the laser modes without mode locking. And yet, the terahertz spectrum can be related to the optical amplitude spectrum because the random fluctuations of the modes are averaged out in the detection process. In the case of TDS using a MLLD as the light source, the amplitudes of the equidistant laser modes are stable and there is a distinct phase relationship between the modes. The optical signal is thus periodic in the time domain. For certain phase relationships, it has the shape of a train of sub-ps pulses whose repetition rate is equal to the round-trip time of the laser cavity.

In this paper, we provide a system-theoretical model that accurately describes the detected terahertz spectrum as a function of the amplitude and phase spectra of the MLLD and the transfer function of the terahertz path. We furthermore find a relationship between the intensity autocorrelation of the MLLD and the detected terahertz spectrum. We discuss the consequences of these findings for linearly chirped MLLDs and prove the validity of the model with detailed experimental results.

The paper is structured as follows. In Section 2, the THz-TDS system is described and the system-theoretical model is given. In Section 3, the effect of pulse chirp on the detected terahertz spectrum is theoretically analyzed. Finally, in Section 4, the model is verified experimentally for a single-section and a two-section MLLD.

2. System-theoretical modeling

A simplified block diagram of a conventional fiber-coupled THz-TDS system is depicted in Fig. 1. The optical output signal of the light source is distributed to a terahertz emitter module and, through a variable delay line, to a terahertz detector module. The emitter is realized as a biased antenna-integrated photodiode or a biased photoconductive antenna, whereas the detector uses an unbiased photoconductive antenna. The terahertz signal radiated by the emitter is transmitted through a terahertz path (e.g. optics, samples,…) and received by the detector. A block diagram of the system model including all relevant quantities both in the time and the frequency domain is depicted in Fig. 2.

Fig. 1. Block diagram of a fiber-coupled THz-TDS setup. The optical output signal of the mode-locked laser source is distributed to a terahertz emitter (THz Tx) and, through a variable delay line, to a terahertz detector (THz Rx). The terahertz radiation generated by the emitter is transmitted through a sample and focused into the detector.

Download Full Size | PDF

Fig. 2. System model of a THz-TDS system. Quantities in the time domain are represented by lower-case letters, wheras quantities in the frequency domain are represented by capital letters. The input signal $e_\textrm {opt}(t)$ of the terahertz spectrometer is the optical output signal of the light source. The output signal $i_\textrm {det}(\tau )$ is the time-averaged photocurrent at the output of the detector as a function of the delay $\tau$ of the variabel delay line in the setup.

Download Full Size | PDF

At the emitter, the optical signal generates free carriers that are accelerated by the applied bias voltage. The accelerated carriers create a photocurrent $i_\textrm {ph}(t)$ in the antenna. For sufficiently low optical peak powers and at considerably low frequencies, where RC effects and lifetime or carrier acceleration effects play a negligible role, it can be assumed that the opto-electrical conversion is linear and thus the photocurrent $i_\textrm {ph}(t)$ is proportional to the instantaneous power $p_\textrm {opt}(t)$ of the incident optical signal. In the far-field, the radiated electrical field $e_\textrm {THz}(t)$ is proportional to the time-derivative of the current $i_\textrm {ph}(t)$ flowing in the antenna [12]. The above-mentioned lifetime and RC effects at high frequencies as well as the antenna frequency response can be accounted for by a complex transfer function $H_\textrm {Tx}(\omega )$ [16]. The spectrum of the emitted electrical field is weighted by the transfer function $H_\textrm {path}(\omega )$ of the terahertz path between emitter and detector. This results in a modification of the amplitude and phase of the emitted spectrum.

At the detector, the delayed optical signal generates free carriers and thus modulates the instantaneous conductivity of the photoconductor. Under the assumption that the opto-electrical conversion is linear, the instantaneous conductivity $g(t)$ is proportional to the instantaneous power $p_\textrm {opt}(t-\tau )$ of the incident optical signal, where $\tau$ is the delay introduced by the variable delay line. Simultaneously, the electrical field $e_\textrm {THz}'(t)$ received by the antenna induces a voltage $v(t)$ at the terminals of the photoconductor. The resulting photocurrent can be described as the product $g(t) \cdot v(t)$. Due to the low-pass behavior of the detector, the detected current $i_\textrm {det}(\tau )$ is proportional to the time average of the product $g(t) \cdot v(t) \propto p_\textrm {opt}(t-\tau ) \cdot e_\textrm {THz}'(t)$. The frequency dependence of the detector can be accounted for by a complex transfer function $H_\textrm {Rx}(\omega )$. Due to the linearity assumption of the opto-electronic conversion, the quantity that describes both the terahertz generation and the terahertz detection process is the instantaneous optical power $p_\textrm {opt}(t)$.

The remainder of this section is structured as follows. In Section 2.1, a description of the complex optical spectrum of the MLLD is given. In Section 2.2, the instantaneous optical power $p_\textrm {opt}(t)$ is calculated from the complex optical spectrum. Based on the instantaneous optical power, a formula for the detected photocurrent $i_\textrm {det}(\tau )$ as a function of the complex optical spectrum and the transfer function of the terahertz system is derived in Section 2.3. In Section 2.4, the optical intensity autocorrelation $R_{pp}(\tau )$ is calculated from the complex optical spectrum and compared to the detected photocurrent $i_\textrm {det}(\tau )$.

2.1 Mathematical description of the mode-locked laser

The optical spectrum of a mode-locked laser consists of several discrete modes whose spacing in the frequency domain is given by the round-trip time, i.e. free spectral range, of the laser cavity. In stable mode-locked operation, the amplitudes and phase differences between the modes are considered to be constant. Thus, each mode is characterized by its number $k$, amplitude $E_k$, and phase $\varphi _k$. The complex electrical field at the output of the light source is described by the sum

(1)$$e_\textrm{opt}(t) = \sum_{k=0}^{N-1} E_k \cdot \textrm{e}^{\textrm{j}\left[ \left(\omega_0 + k\Omega \right) \cdot t + \varphi_k \right]} ~,~ \Omega = 2\pi\cdot F ~,$$

where $N$ is the number of significant modes of the laser, $\omega _0$ denotes the angular frequency of the first significant mode, and $F$ is the free spectral range of the laser.

2.2 Instantaneous optical power

Both terahertz generation and detection are based on the incident instantaneous optical power $p_\textrm {opt}(t)$. In the time domain, the instantaneous optical power is proportional to the squared magnitude of the complex electrical field, i.e.

(2)$$\begin{aligned}p_\textrm{opt}(t) \propto \left|e_\textrm{opt}(t)\right|^2 &= \left( \sum_{k=0}^{N-1} E_k \cdot \textrm{e}^{\textrm{j}\left[\left(\omega_0 + k\Omega \right) \cdot t + \varphi_k \right]} \right) \cdot \left( \sum_{l=0}^{N-1} E_l \cdot \textrm{e}^{-\textrm{j}\left[ \left(\omega_0 + l\Omega \right) \cdot t + \varphi_l \right]}\right)\\ &= \left( \sum_{k=0}^{N-1} E_k \cdot \textrm{e}^{\textrm{j}\left[ k\Omega t + \varphi_k \right]} \right) \cdot \left( \sum_{l=0}^{N-1} E_l \cdot \textrm{e}^{-\textrm{j}\left[ l\Omega t + \varphi_l \right]}\right) ~. \end{aligned}$$

Rearranging the resulting product terms results in the simplified expression

(3)$$p_\textrm{opt}(t) \propto \sum_{k=0}^{N-1} E_k^2 ~+~ 2 \cdot \sum_{m=1}^{N-1} \sum_{k=m}^{N-1} E_k E_{k-m} \cdot \cos \left( m \Omega t + \varphi_k - \varphi_{k-m} \right) ~.$$

This notation is similar to that found in [17]. By Fourier transform, the spectrum of the instantaneous optical power

(4)$$P_\textrm{opt}(\omega) \propto 2\pi \cdot \sum_{m=0}^{N-1} \sum_{k=m}^{N-1} E_k E_{k-m} \cdot \left[\delta\left(\omega - m\Omega \right) \cdot \textrm{e}^{-\textrm{j}\left(\varphi_k - \varphi_{k-m} \right)} + \delta\left(\omega + m\Omega \right) \cdot \textrm{e}^{\textrm{j}\left(\varphi_k - \varphi_{k-m} \right)}\right] ~,$$

where $\delta (\omega )$ is the Dirac delta function, is obtained. The instantaneous optical power – and thus both the transmitted and detected terahertz signal – contains discrete spectral components at frequencies that are integer multiples of the free spectral range of the mode-locked laser. Each spectral component is the superposition of all mixing products of pairs of laser modes whose frequency difference is the frequency of the respective spectral component. It can be seen that the angular frequency of the first mode $\omega _0$ is a common-mode term that cancels in the mixing process. The average power (i.e. the zero-frequency component of the instantaneous optical power) is given by the self-mixing of all laser modes and is thus equal to the sum of the powers of all laser modes. The amplitude and phase of all other spectral components at the angular frequencies $m\Omega$ depends on the amplitudes and phases of the laser modes $m$ through $N-1$. Depending on the phase relationship between those modes, the superposition of the mixing products can be anywhere from perfectly constructive to perfectly destructive.

The shape of the instantaneous optical power and thus the shape of the radiated terahertz pulse depend both on the amplitude and the phase spectrum of the mode-locked laser. It can be seen that perfect constructive interference occurs for the case of linear phase, i.e. an unchirped laser pulse, as in this case the term $\varphi _k - \varphi _{k-m}$ is constant for a given value of $m$. In the unchirped case, the pulses are minimally short and maximally high in peak amplitude. It should be noted that in this context a constant phase can be considered as linear with zero slope. A non-zero slope simply represents a delay of the signal.

2.3 Detected photocurrent

To calculate the detected photocurrent, both the emitter and the detector side of the system need to be considered. As described above, in the far field the terahertz electric field radiated by the emitter antenna is proportional to the time derivative of the instantaneous optical power at the emitter. The frequency dependence of the emitter is accounted for by the complex transfer function $H_\textrm {Tx}(\omega )$. Thus, in the frequency domain the electric field radiated by the emitter is

(5)$$\begin{aligned}E_\textrm{THz}(\omega) &\propto \textrm{j}\omega \cdot H_\textrm{Tx}(\omega) \cdot P_\textrm{opt}(\omega) \\ &\propto \textrm{j}\omega \cdot H_\textrm{Tx}(\omega)\\ & ~ \cdot 2\pi \cdot \sum_{m=0}^{N-1} \sum_{k=m}^{N-1} E_k E_{k-m} \cdot \left[\delta\left(\omega - m\Omega \right) \cdot \textrm{e}^{-\textrm{j}\left(\varphi_k - \varphi_{k-m} \right)} + \delta\left(\omega + m\Omega \right) \cdot \textrm{e}^{\textrm{j}\left(\varphi_k - \varphi_{k-m} \right)}\right] ~, \end{aligned}$$

where the term $\textrm {j}\omega$ represents the first derivative with respect to time in the frequency domain. As described above, the electric field at the detector is the product of the terahertz path’s transfer function and the electric field radiated by the emitter:

(6)$$\begin{aligned} E_\textrm{THz}'(\omega) &= H_\textrm{path}(\omega) \cdot E_\textrm{THz}(\omega) \\ &\propto \textrm{j}\omega \cdot H_\textrm{Tx}(\omega) \cdot H_\textrm{path}(\omega)\\ & ~ \cdot 2\pi \cdot \sum_{m=0}^{N-1} \sum_{k=m}^{N-1} E_k \cdot E_{k-m} \cdot \left[\delta\left(\omega - m\Omega \right) \cdot \textrm{e}^{-\textrm{j}\left(\varphi_k - \varphi_{k-m} \right)} + \delta\left(\omega + m\Omega \right) \cdot \textrm{e}^{\textrm{j}\left(\varphi_k - \varphi_{k-m} \right)}\right]~. \end{aligned}$$

At the detector, the delayed instantaneous optical power is multiplied in the time domain with the received electrical field weighted with the complex transfer function $H_\textrm {Rx}(\omega )$ of the detector. By time averaging, the detected current is the zero-frequency component of this product, i.e.

(7)$$\begin{aligned} i_\textrm{det}(\tau) &\propto \mathcal{F}^{{-}1} \left( \left. \mathcal{F} \left\lbrace p_\textrm{opt}(t-\tau) \cdot \mathcal{F}^{{-}1} \left[ H_\textrm{Rx}(\omega) \cdot E_\textrm{THz}'(\omega) \right] (t) \right\rbrace (\omega) \right|_{\omega = 0} \right) \\ &= \mathcal{F}^{{-}1} \left\lbrace\frac{1}{2\pi} \cdot \left. \left[ P_\textrm{opt}(\omega) \cdot \textrm{e}^{-\textrm{j}\omega\tau} \right] * \left[ H_\textrm{Rx}(\omega) \cdot E_\textrm{THz}'(\omega) \right] \right|_{\omega=0} \right\rbrace ~. \end{aligned}$$

Inserting Eqs. (4) and (6) into Eq. (7) results in the convolution of two double sums, yielding a plethora of terms containing delta functions. However, only a small number of these are non-zero for $\omega =0$, allowing the result to be significantly reduced to

(8)$$\begin{aligned}i_\textrm{det}(\tau) &\propto 2 \cdot \sum_{m=1}^{N-1} {\bigg \lbrace} \left| H_\textrm{THz}(m\Omega) \right| \\ &\, \cdot \sum_{k=m}^{N-1} \sum_{l=m}^{N-1} E_k E_{k-m} E_l E_{l-m} \cdot \sin \left[m\Omega\tau + \angle{H_\textrm{THz}(m\Omega)} + \left(\varphi_k - \varphi_{k-m} \right) - \left(\varphi_l - \varphi_{l-m} \right)\right] {\bigg \rbrace} ~, \end{aligned}$$

where

(9)$$H_\textrm{THz}(m\Omega) = m\Omega \cdot H_\textrm{Tx}(m\Omega) \cdot H_\textrm{path}(m\Omega) \cdot H_\textrm{Rx}(m\Omega) ~,~ m = 1 \,\ldots\, N-1 ~,$$

describes the combined transfer function of the terahertz system. Further simplification is possible by using the relation $\sin (\alpha + \beta ) = \sin (\alpha )\cos (\beta ) + \cos (\alpha )\sin (\beta )$ and by taking into account that the indices $k$ and $l$ run over the same range of values:

(10)$$\begin{aligned}i_\textrm{det}(\tau) &\propto 2 \cdot \sum_{m=1}^{N-1} {\bigg \lbrace} \left| H_\textrm{THz}(m\Omega) \right| \cdot \sin \left[m\Omega\tau + \angle{H_\textrm{THz}(m\Omega)} \right] \\ &\quad\quad \cdot \sum_{k=m}^{N-1} \sum_{l=m}^{N-1} E_k E_{k-m} E_l E_{l-m} \cdot \cos \left[\left(\varphi_k - \varphi_{k-m} \right) - \left(\varphi_l - \varphi_{l-m} \right)\right] {\bigg \rbrace} ~. \end{aligned}$$

The term

(11)$$A_m = \sum_{k=m}^{N-1} \sum_{l=m}^{N-1} E_k E_{k-m} E_l E_{l-m} \cdot \cos \left[\left(\varphi_k - \varphi_{k-m} \right) - \left(\varphi_l - \varphi_{l-m} \right)\right] ~,~ m = 0 \,\ldots\, N-1~,$$

describes an amplitude factor for each spectral component of the detected photocurrent that is determined by the complex optical spectrum. Using Eq. (11) results in the straightforward expression for the detected photocurrent

(12)$$i_\textrm{det}(\tau) \propto 2 \cdot \sum_{m=1}^{N-1} \left| H_\textrm{THz}(m\Omega) \right| \cdot \sin \left[m\Omega\tau + \angle{H_\textrm{THz}(m\Omega)} \right] \cdot A_m ~.$$

It can be seen that the time-average of the detector output current consists of spectral components at frequencies $\nu =mF, m=1\,\ldots \,N-1$, that are integer multiples of the mode-locked laser’s free spectral range. The amplitude of each spectral component is the product of the magnitude of the combined transfer function $\left | H_\textrm {THz}(m\Omega ) \right |$ and the amplitude factor $A_m$ given by the complex optical spectrum of the mode-locked laser. Given that the laser is stable, the factors $A_m$ are static and depend only on the amplitudes and phases of the laser modes. The phase of each spectral component in the detected terahertz spectrum is equal to the phase of the combined transfer function $H_\textrm {THz}(m\Omega )$ and does not depend on the phases of the laser modes. These results confirm the experimental observation that the amplitude and phase differences between two terahertz spectra (e.g. with and without a sample) measured with the same THz-TDS system depend only on the transfer functions of the terahertz system during these measurements. The amplitude and phase spectrum of the light source only affects the maximum amplitude obtainable at each frequency. Unlike the well-described CCS/QTDS case using a multi-mode laser without mode locking [7,8,15], the detection mechanism does not depend on temporal averaging over random fluctuations of the laser modes.

It should be noted that the given model allows the prediction of the detected terahertz spectrum from the complex optical spectrum and the transfer function of the terahertz system. This can be useful to develop light sources with optical spectra that are optimized for use in a THz-TDS system.

2.4 Intensity autocorrelation

A common way to characterize a mode-locked laser is the intensity autocorrelation. It is used to determine the laser’s pulse width. The intensity autocorrelation $R_{pp}(\tau )$, measured for example by a second-harmonic generation (SHG) autocorrelator, is the autocorrelation of the instantaneous optical power

(13)$$R_{pp}(\tau) = p_\textrm{opt}(\tau) * p_\textrm{opt}(-\tau) ~.$$

For convenience, the calculation is carried out in the frequency domain. The power spectral density of the instantaneous optical power is

(14)$$S_{pp}(2\pi\nu) = P_\textrm{opt}(2\pi\nu) \cdot P^*_\textrm{opt}(2\pi\nu) ~,$$

where $\nu$ represents frequency with respect to delay $\tau$. Inserting Eq. (4) into Eq. (14) results in products of delta functions that are mathematically ill-defined. However, in the given context the delta functions represent narrow spectral lines with non-zero width, so it is appropriate to adapt the sifting property in the form of

(15)$$\delta(x-x_0) \cdot \delta(x-x_1) \propto \left\lbrace \begin{array}{cc} \delta(x-x_0) & ,~x_0 = x_1 \\ 0 & ,~x_0 \neq x_1 \end{array}\right. ~,$$

resulting in the expression:

(16)$$\begin{aligned}S_{pp}&(2\pi\nu) \propto (2\pi)^2 \cdot \sum_{m=0}^{N-1}\sum_{k=m}^{N-1}\sum_{l=m}^{N-1} E_k E_{k-m} E_l E_{l-m} \\ &\quad\cdot \left\lbrace \delta(2\pi\nu-m\Omega) \cdot \textrm{e}^{-\textrm{j}\left[(\varphi_k - \varphi_{k-m})-(\varphi_l - \varphi_{l-m})\right]} + \delta(2\pi\nu+m\Omega) \cdot \textrm{e}^{\textrm{j}\left[(\varphi_k - \varphi_{k-m})-(\varphi_l - \varphi_{l-m})\right]} \right\rbrace ~. \end{aligned}$$

The intensity autocorrelation is then obtained by inverse Fourier transform:

(17)$$R_{pp}(\tau) \propto \sum_{m=0}^{N-1}\sum_{k=m}^{N-1} \sum_{l=m}^{N-1} E_k E_{k-m} E_l E_{l-m} \cdot \cos \left[m\Omega\tau + \left(\varphi_k - \varphi_{k-m} \right) - \left(\varphi_l - \varphi_{l-m} \right)\right] ~.$$

Using the relation $\cos (\alpha + \beta ) = \cos (\alpha )\cos (\beta ) - \sin (\alpha )\sin (\beta )$ and by taking into account that the indices $k$ and $l$ run over the same range of values, this can be simplified to

(18)$$R_{pp}(\tau) \propto \sum_{m=0}^{N-1} \left\lbrace\cos \left(m\Omega\tau \right) \cdot \sum_{k=m}^{N-1} \sum_{l=m}^{N-1} E_k E_{k-m} E_l E_{l-m} \cdot \cos \left[\left(\varphi_k - \varphi_{k-m} \right) - \left(\varphi_l - \varphi_{l-m} \right)\right]\right\rbrace ~.$$

Using the amplitude factors $A_m$ defined in Eq. (11), the intensity autocorrelation can be written as

(19)$$R_{pp}(\tau) \propto \sum_{m=0}^{N-1} \cos \left(m\Omega\tau \right) \cdot A_m ~.$$

This result is notable. Firstly, it indicates that for a mode-locked laser the intensity autocorrelation contains spectral components at frequencies $\nu =mF, m=0,\,\ldots \,,N-1$, that are integer multiples of the laser’s free spectral range. All of these spectral components have the same phase. Their amplitudes are determined by the complex optical spectrum of the laser through the factors $A_m$. Measured SHG autocorrelations always have the shape of symmetrical pulses whose exact shape – in particular amplitude and width – is determined both by the amplitude and phase spectrum of the laser.

Secondly, by comparing Eq. (12) and Eq. (19), it can be seen that there is a close relationship between the intensity autocorrelation and the detected photocurrent in a THz-TDS system. Effectively, the spectrum of the detected photocurrent is the spectrum of the intensity autocorrelation weighted with the transfer function of the terahertz system. Thus, the intensity autocorrelation is a meaningful measure for a light source’s performance in a THz-TDS system. Its bandwidth determines the maximum bandwidth achievable. This is particularly useful, as the intensity autocorrelation can be measured directly.

3. Theoretical results for linearly chirped lasers

In previous work it has been shown that the bandwidth of THz-TDS systems using MLLDs can be increased by adding a suitable section of single-mode fiber between the output of the laser and the rest of the THz-TDS system [10,11]. Since MLLDs have been shown to exhibit a linear chirp, i.e. their phase spectra have a square profile, this observed increase in bandwidth can be attributed to the pulse compression by the anomalous dispersion of the single-mode fiber [17–19]. In the case of linear chirp, the phase of mode $k$ is

(20)$$\varphi_k = \frac{1}{2} \cdot B \cdot \left[ 2 \pi \cdot \left(f_k - f_0 \right) \right]^2 = \frac{1}{2} \cdot B \cdot \left( 2 \pi \cdot k \cdot F \right)^2 = \frac{1}{2} \cdot b \cdot \left( 2 \pi k \right)^2 ~,$$

where $b = B \cdot F^2$ is the dispersion coefficient $B$ normalized to the free spectral range $F$ of the laser. For the instructive case of a rectangular laser spectrum

(21)$$E_k = E_0 ~,~ m = 0 \,\ldots\, N-1 ~,$$

the resulting amplitudes according to Eq. (11) are

(22)$$A_m = \sum_{k=m}^{N-1} \sum_{l=m}^{N-1} E_0^4 \cdot \cos \left[b\cdot \left(2\pi \right)^2 \cdot m\cdot \left(k-l\right)\right] ~,~ m = 0 \,\ldots\, N-1~.$$

By using the Dirichlet kernel, the closed expression

(23)$$A_m = E_0^4 \cdot \left\lbrace\frac{\sin \left[\dfrac{1}{2} \cdot b\cdot \left(2\pi \right)^2 \cdot m \cdot \left(m-N\right) \right]}{\sin \left[\dfrac{1}{2} \cdot b\cdot \left(2\pi \right)^2 \cdot m\right]}\right\rbrace^2 ~,~ m = 0 \,\ldots\, N-1~,$$

is obtained. The highest frequency component, $A_{N-1} = E_0^4$, is independent of the chirp, as only two modes are involved in the mixing process. For all other spectral components, the highest amplitudes $A_m = E_0^4 \cdot \left (m-N\right )^2$ are achieved in the unchirped case, $b=0$.

It has been shown experimentally that the dispersion coefficient of single-section MLLDs can be estimated according to [20]

(24)$$B_\textrm{MLL} = \left(2\pi \cdot f_{10\,\textrm{dB}}\cdot F\right)^{{-}1} ~,$$

where $f_{10\,\textrm {dB}}$ is the optical 10 dB bandwidth of the laser. For a rectangular spectrum, all modes lie within the 10 dB bandwidth, thus

(25)$$B_\textrm{MLL} = \left(2\pi \cdot N\cdot F^2\right)^{{-}1}~.$$

For analyzing the effect of chirp on the amplitudes $A_m$, the normalized dispersion coefficient

(26)$$b_\textrm{MLL} = \left( 2\pi N \right)^{{-}1}$$

is used as the upper boundary for laser chirp that can be expected at the output of the light source. Figure 3 depicts the amplitudes $A_m$ for different amounts of chirp

(27)$$\varphi_k = \frac{1}{2} \cdot c \cdot b_\textrm{MLL} \cdot \left( 2 \pi k \right)^2 ~,~ c = 0\,\ldots\,1 ~,$$

for a laser with $N=101$ modes, where $c=0$ describes the case of no chirp and $c=1$ describes the chirp directly at the output of the laser diode.

Fig. 3. Resulting amplitudes $A_m$ for a MLLD with 101 modes with equal amplitude. The parameter $c=0$ describes the case of no chirp and $c=1$ describes the chirp that can be expected directly at the output of the laser diode. Subfigure (a) shows the frequency dependence of the amplitudes $A_m$ for different amounts of chirp, whereas (b) shows the chirp dependence of the amplitudes $A_m$ for different frequencies.

Download Full Size | PDF

The unchirped case, $c=0$, in Fig. 3(a) shows the predicted $\left (m-N\right )^2$ roll-off. Small amounts of chirp ($c<0.1$) lead to a strong degradation towards higher frequencies with little effect on the spectral components at low frequencies. Larger amounts of chirp ($c>0.1$) have a strong negative effect on both low- and high-frequency components. Poorly compensated chirp can lead to a reduction of the detected terahertz amplitude by several tens of decibels. Figure 3(b) highlights the effect of linear chirp on select spectral components. It can be seen that increasing chirp generally leads to a decrease in the amplitude of all spectral components within the detected terahertz spectrum. For certain amounts of chirp it leads to nulls in the spectrum. These nulls are close together for spectral components in the middle of the spectrum and far apart at the edges of the spectrum. The highest frequency component is independent of the chirp.

4. Measurements and comparison to the model

To verify the model proposed in Section 2 and the investigations in Section 3, we conduct measurements with two different MLLDs. The first one is a low-cost and commercially available single-section Fabry-Perot laser diode (Thorlabs FPL1009P) that has been shown to exhibit mode-locking behavior and be suitable for THz-TDS applications [11,21]. The second one is a two-section buried heterostructure (BH) quantum dot laser diode from the Fraunhofer Heinrich Hertz Institute (HHI), Berlin. The active layers are comprised of 7x stacked layers of InAs/InP QDs. The total Fabry-Perot cavity length is 840 $\mu$m to give a mode spacing of approximately 50 GHz. The saturable absorber has a length of 50 $\mu$m [22,23].

We investigate different amounts of chirp by using the anomalous dispersion of a variable-length section of single-mode fiber between the output of the respective laser and the terahertz spectrometer. For each combination of laser and single-mode fiber length we measure

• the complex optical spectrum using an optical spectrum analyzer and a stepped-heterodyne measurement approach,
• the intensity autocorrelation using a SHG autocorrelator, and
• the resulting terahertz spectrum using a conventional THz-TDS setup.

Furthermore, we measure the transfer function of the terahertz system using a frequency-domain spectroscopy system. These measurements allow us to

• compare the intensity autocorrelation calculated from the measured complex optical spectrum with the measured intensity autocorrelation and
• compare the detected terahertz spectrum calculated from
- - the measured complex optical spectrum or
- - the intensity autocorrelation

and the transfer function of the terahertz system with the detected terahertz spectrum measured with the THz-TDS setup.

In Section 4.1, the measurement setup for measuring the optical amplitude and phase spectra is explained and the measured spectra for both lasers are shown. In Section 4.2, the measurement setup for measuring the intensity autocorrelation is explained and the measured autocorrelations are compared to the autocorrelations calculated from the complex optical spectra. Finally, in Section 4.4, the THz-TDS setup is explained and the measured terahertz spectra are compared with the spectra calculated from either the complex optical spectrum or the intensity autocorrelation.

4.1 Complex optical spectrum

The complex optical spectrum consists of an amplitude spectrum and a phase spectrum. While the amplitude spectrum depends essentially on the laser temperature and the injection current, the phase spectrum also changes with the additional single-mode fiber length between the laser and the terahertz system. The fiber introduces mainly linear chirp. We measure the amplitude spectrum with an Anritsu MS9740A optical spectrum analyzer. The amplitude spectra of the Thorlabs FPL1009P and the HHI QD laser are depicted in Figs. 4(a) and (b), respectively. The FPL1009P exhibits a 10 dB-bandwidth of 1.5 THz and a 40 dB-bandwidth of 3.2 THz, whereas the HHI QD laser exhibits a 10 dB-bandwidth of 2 THz and a 40 dB-bandwidth of 3.1 THz.

Fig. 4. Amplitude spectra of (a) the Thorlabs FPL1009P and (b) the HHI QD laser. The spectra measured with the optical spectrum analyzer are plotted in solid blue lines and the modes within the 40 dB-bandwidth are highlighted with red crosses.

Download Full Size | PDF

To get the phase spectrum, we use the stepped-heterodyne approach proposed in [24]. The measurement setup is depicted in Fig. 5. The output of the laser first passes a fiber isolator and a polarization controller before entering a variable-length section of single-mode fiber. The polarization at the output of the single-mode fiber is aligned to the slow axis of a polarization maintaining fiber at one output of a polarization beam splitter (PBS) by minimizing the power at the other output of the PBS. The signal from the MLLD under test is combined with the output of a Pure Photonics PPCL300 tunable low-noise external-cavity laser and fed into an IPHOBAC-NG 70 GHz photodiode.

Fig. 5. Measurement setup for the stepped-heterodyne measurement of the optical phase spectrum and the intensity autocorrelation for different single-mode fiber lengths. Red lines indicate single-mode fibers, blue lines indicate polarization maintaining fibers, and black lines indicate electrical connections.

Download Full Size | PDF

We use a Keysight DSA-Z 634A 63 GHz real-time oscilloscope for digitizing. We step the frequency of the tunable single-mode laser through the spectrum of the MLLD, making sure that it is asymmetrically located between two adjacent modes, and record the resulting photocurrent at the photodiode over a period of 5 $\mu$s with a sampling-rate of 160 GSamples/s. The phase spectrum is calculated offline from all steps using the algorithm described in [24]. The resulting phase spectra of the FPL1009P and the HHI QD laser for different fiber lengths are depicted in Figs. 6(a) and (b), respectively. Cubic fits are used to approximate the phase of the laser modes at the edges of the spectra whose power is too low to be measured with the stepped-heterodyne technique. For the FPL1009P and the HHI QD laser, the smallest square components, i.e. the best chirp compensation, are achieved with fiber lengths of $L=70\,\textrm {m}$ and $L=50\,\textrm {m}$, respectively.

Fig. 6. Phase spectra of (a) the Thorlabs FPL1009P and (b) the HHI QD laser measured with the stepped-heterodyne technique for different single-mode fiber lengths.

Download Full Size | PDF

4.2 Intensity autocorrelation

As a next step, we use an APE pulseCheck autocorrelator to measure the intensity autocorrelation for both lasers for the same fiber lengths as in Section 4.1. The measured autocorrelations are compared to the autocorrelations calculated from the complex optical spectra according to Eq. (18) for the FPL1009P and the HHI QD laser in Figs. 7(a) and (b), respectively. The measured and calculated intensity autocorrelations agree for both lasers for all fiber lengths. The traces in the time domain (left column) without normalization show that the shortest pulses with the highest amplitudes are obtained in the cases with the smallest chirp with $L=70\,\textrm {m}$ respectively $L=50\,\textrm {m}$. In the normalized traces in the frequency domain (right column), it can be seen that – in agreement with the investigations in Section 3 – the cases with the smallest chirp also exhibit the highest bandwidth. It should be noted that the apparent variation of the noise floor of the measured autocorrelation is due to the individual normalization of the frequency domain traces.

Fig. 7. Measured (blue) and calculated (red) intensity autocorrelations of (a) the Thorlabs FPL1009P and (b) the HHI QD laser for different single-mode fiber lengths. The dashed black line represents the noise floor of the measured autocorrelation.

Download Full Size | PDF

4.3 Terahertz spectrum from frequency-domain spectroscopy

To compare the detected terahertz spectrum calculated from the measured complex optical spectrum with the detected terahertz spectrum measured with the THz-TDS setup, we need to know the frequency-discrete transfer function $H_\textrm {THz}(m\Omega )$ of the terahertz system. We measure the transfer function from 20 to 1300 GHz with a frequency resolution of 10 MHz using a Toptica Terascan 1550 frequency-domain spectroscopy system. The alignment of the terahertz setup is not changed between this measurement and the THz-TDS measurements. The magnitude of the transfer function $\left | H_\textrm {THz}(f) \right |$ is depicted in Fig. 8. Due to the high frequency-resolution of the measured transfer function, the values at the discrete frequencies $mF$ are accurately obtained for both lasers.

Fig. 8. Terahertz transfer function measured by frequency-domain spectroscopy.

Download Full Size | PDF

4.4 Terahertz spectra from time-domain spectroscopy

We measure the detected terahertz spectra using a standard THz-TDS setup for both lasers and all the fiber lengths discussed in Sections 4.1 and 4.2. The measurement setup is depicted in Fig. 9. We use InGaAs terahertz emitter and detector modules intended for continuous-wave operation from Fraunhofer HHI (Toptica #EK-000724 and #EK-000725), the current input of a Zurich Instrument MFLI lock-in amplifier, and an OZ Optics ODL-650 motorized variable delay line with a maximum delay of 330 ps. We operate the delay line in stepped mode with a time step of 0.1 ps. The integration time of the lock-in amplifier is 300 ms. The bias voltage at the emitter is -0.9 V. The average optical power at the emitter and detector is just below 30 mW for the FPL1009P and around 5 mW for the HHI QD laser. According to the measured optical spectra and Eq. (3) this corresponds to peak optical powers of about 1.3 W for the FPL1009P and 220 mW for the HHI QD laser.

Fig. 9. Measurement setup for THz-TDS.

Download Full Size | PDF

The divergent radiation of the terahertz emitter is collimated and focused into the terahertz receiver with two 2” off-axis parabolic mirrors. A piece of polarization-maintaining fiber is used between the beam splitter and the emitter module to compensate for the additional fiber length of the delay line. By ensuring that the total fiber lengths from the MLLD to both the emitter and the detector module are equal to the fiber lengths to the photodiode respectively the autocorrelator in Sections 4.1 and 4.2, the measured complex optical spectra and intensity autocorrelations precisely match those in the THz-TDS setup. The measured terahertz spectra for the FPL1009P and the HHI QD laser are depicted for all fiber lengths as solid blue lines in Figs. 10(a) and (b), respectively. The amplitudes of the discrete spectral components in the measured spectra are highlighted with red crosses. The terahertz spectra measured with both lasers exhibit comparable bandwidths with detectable spectral components up to about 1.3 THz. This high-frequency limit can be attributed to the fact that we could optimize the alignment of the terahertz path only up to a frequency of 1.3 THz. The HHI QD laser offers a more gentle roll-off with a 40 dB-bandwidth of about 950 GHz compared to the 40 dB-bandwidth of about 870 GHz offered by the FPL1009P.

Fig. 10. Calculated and measured detected terahertz spectra acquired (a) with the Thorlabs FPL1009P and (b) with the HHI QD laser for different single-mode fiber lengths. Blue traces: Measured spectra. Red crosses: Amplitudes of the discrete spectral components in the measured spectra. Yellow circles: Amplitudes calculated from the complex optical spectra (COS). Purple squares: Amplitudes calculated from the intensity autocorrelation function (ACF). Both the amplitudes calculated from the COS and the ACF are normalized to the amplitudes of the measured spectra.

Download Full Size | PDF

Overlaid with the measured spectra are the amplitudes calculated from the complex optical spectra in yellow and the amplitudes calculated from the intensity autocorrelation in purple. The measured and the calculated spectra agree for both lasers and all investigated fiber lengths. This is proof that according to Eqs. (11), (12), and (19) the detected photocurrent can be calculated from the complex optical spectrum or the intensity autocorrelation and the terahertz transfer function. That means that the intensity autocorrelation of a MLLD accurately determines its performance in a THz-TDS system and gives a useful upper bound for the bandwidth achievable with the system. Furthermore, this confirms that for the employed components and the occurring peak optical powers the linearity assumption about the photomixers is valid.

5. Conclusion

In this work, we have developed a system-theoretical model of a THz-TDS system using a MLLD as the light source. We have derived a simple formula that accurately describes the detected terahertz spectrum as a function of the complex optical spectrum and the transfer function of the terahertz system. Furthermore, we have provided a relationship between the intensity autocorrelation of the MLLD and the detected terahertz spectrum. These findings improve the understanding of THz-TDS with ultra-high repetition rate lasers and allow the engineering of optimized optical spectra. Based on the derived formula, we have investigated the deterioration of the detected terahertz spectrum due to pulse chirp. We have concluded our investigation with a thorough experimental comparison of the model with measurement results for a single-section and a two-section MLLD. While these results remarkably show the validity of the proposed model for the demonstrated combination of MLLDs, terahertz emitter, and detector, it would be of high interest to extend these investigations to light sources with higher peak optical powers and other types of terahertz emitters and detectors.

Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 287022738 - TRR 196.

Acknowledgments

We acknowledge support by the Open Access Publication Fund of the University of Duisburg-Essen.

Disclosures

The authors declare no conflicts of interest.

References

1. M. van Exter, C. Fattinger, and D. Grischkowsky, “Terahertz time-domain spectroscopy of water vapor,” Opt. Lett. 14(20), 1128–1130 (1989). [CrossRef]

2. M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1(2), 97–105 (2007). [CrossRef]

3. B. B. Hu and M. C. Nuss, “Imaging with terahertz waves,” Opt. Lett. 20(16), 1716–1718 (1995). [CrossRef]

4. B. Sartorius, H. Roehle, H. Künzel, J. Böttcher, M. Schlak, D. Stanze, H. Venghaus, and M. Schell, “All-fiber terahertz time-domain spectrometer operating at 1.5 µm telecom wavelengths,” Opt. Express 16(13), 9565–9570 (2008). [CrossRef]

5. M. Naftaly, N. Vieweg, and A. Deninger, “Industrial applications of terahertz sensing: State of play,” Sensors 19(19), 4203 (2019). [CrossRef]

6. T. Hochrein, “Markets, availability, notice, and technical performance of terahertz systems: Historic development, present, and trends,” J. Infrared, Millimeter, Terahertz Waves 36(3), 235–254 (2015). [CrossRef]

7. O. Morikawa, M. Tonouchi, and M. Hangyo, “A cross-correlation spectroscopy in subterahertz region using an incoherent light source,” Appl. Phys. Lett. 76(12), 1519–1521 (2000). [CrossRef]

8. M. Scheller and M. Koch, “Terahertz quasi time domain spectroscopy,” Opt. Express 17(20), 17723–17733 (2009). [CrossRef]

9. D. Molter, M. Kolano, and G. von Freymann, “Terahertz cross-correlation spectroscopy driven by incoherent light from a superluminescent diode,” Opt. Express 27(9), 12659–12665 (2019). [CrossRef]

10. K. Merghem, S. F. Busch, F. Lelarge, M. Koch, A. Ramdane, and J. C. Balzer, “Terahertz time-domain spectroscopy system driven by a monolithic semiconductor laser,” J. Infrared, Millimeter, Terahertz Waves 38(8), 958–962 (2017). [CrossRef]

11. K.-H. Tybussek, K. Kolpatzeck, F. Faridi, S. Preu, and J. C. Balzer, “Terahertz time-domain spectroscopy based on commercially available 1550 nm fabry-perot laser diode and eras:in(al)gaas photoconductors,” Appl. Sci. 9(13), 2704 (2019). [CrossRef]

12. P. Jepsen, D. Cooke, and M. Koch, “Terahertz spectroscopy and imaging - modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011). [CrossRef]

13. S. Preu, G. H. Döhler, S. Malzer, L. J. Wang, and A. C. Gossard, “Tunable, continuous-wave terahertz photomixer sources and applications,” J. Appl. Phys. 109(6), 061301 (2011). [CrossRef]

14. K. Wynne and J. J. Carey, “An integrated description of terahertz generation through optical rectification, charge transfer, and current surge,” Opt. Commun. 256(4-6), 400–413 (2005). [CrossRef]

15. M. Tani, S. Matsuura, K. Sakai, and M. Hangyo, “Multiple-frequency generation of sub-terahertz radiation by multimode ld excitation of photoconductive antenna,” IEEE Microw. Guid. Wave Lett. 7(9), 282–284 (1997). [CrossRef]

16. S. Preu, “A unified derivation of the terahertz spectra generated by photoconductors and diodes,” J. Infrared, Millimeter, Terahertz Waves 35(12), 998–1010 (2014). [CrossRef]

17. R. Rosales, S. G. Murdoch, R. Watts, K. Merghem, A. Martinez, F. Lelarge, A. Accard, L. P. Barry, and A. Ramdane, “High performance mode locking characteristics of single section quantum dash lasers,” Opt. Express 20(8), 8649–8657 (2012). [CrossRef]

18. X. Tang, A. S. Karar, J. C. Cartledge, A. Shen, and G. Duan, “Characterization of a mode-locked quantum-dash fabry-perot laser based on measurement of the complex optical spectrum,” in 2009 35th European Conference on Optical Communication, (2009), pp. 1–2

19. S. G. Murdoch, R. T. Watts, Y. Q. Xu, R. Maldonado-Basilio, J. Parra-Cetina, S. Latkowski, P. Landais, and L. P. Barry, “Spectral amplitude and phase measurement of a 40 GHz free-running quantum-dash modelocked laser diode,” Opt. Express 19(14), 13628–13635 (2011). [CrossRef]

20. S. P. O. Duill, S. G. Murdoch, R. T. Watts, R. Rosales, A. Ramdane, P. Landais, and L. P. Barry, “Simple dispersion estimate for single-section quantum-dash and quantum-dot mode-locked laser diodes,” Opt. Lett. 41(24), 5676–5679 (2016). [CrossRef]

21. S. C. Tonder, K. Kolpatzeck, X. Liu, S. Rumpza, A. Czylwik, and J. C. Balzer, “A compact THz quasi TDS system for mobile scenarios,” in 2019 Second International Workshop on Mobile Terahertz Systems (IWMTS), (2019), pp. 1–5.

22. M. Zander, W. Rehbein, M. Moehrle, S. Breuer, D. Franke, and D. Bimberg, InAs/InP QD and InGaAsP/InP QW comb lasers for >1 Tb/s transmission, in 2019 Compound Semiconductor Week (CSW), (2019), p. 1.

23. M. Zander, W. Rehbein, M. Moehrle, S. Breuer, D. Franke, M. Schell, K. Kolpatzeck, and J. C. Balzer, “High performance bh inas/inp qd and ingaasp/inp qw mode-locked lasers as comb and pulse sources,” Optical Fiber Communication Conference (OFC) 2020, (Optical Society of America, 2020), p. T3C.4.

24. D. A. Reid, S. G. Murdoch, and L. P. Barry, “Stepped-heterodyne optical complex spectrum analyzer,” Opt. Express 18(19), 19724–19731 (2010). [CrossRef]

System-theoretical modeling of terahertz time-domain spectroscopy with ultra-high repetition rate mode-locked lasers

Abstract

1. Introduction

2. System-theoretical modeling

2.1 Mathematical description of the mode-locked laser

2.2 Instantaneous optical power

2.3 Detected photocurrent

2.4 Intensity autocorrelation

3. Theoretical results for linearly chirped lasers

4. Measurements and comparison to the model

4.1 Complex optical spectrum

4.2 Intensity autocorrelation

4.3 Terahertz spectrum from frequency-domain spectroscopy

4.4 Terahertz spectra from time-domain spectroscopy

5. Conclusion

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (10)

Equations (27)

Optics Express