1. Introduction

Autocorrelation (AC) is one of the key technologies to characterize ultrashort pulses. Autocorrelation based on second harmonic generation (SHG) has been a great success during the last few decades [1–6]. Moreover, technologies developed from SHG including frequency-resolved optical gating (FROG) [7–10] and spectral phase interferometry for directelectric-field reconstruction (SPIDER) [11–18], have enabled retrieval of pulse amplitude and phase information simultaneously. However, there are also some limitations caused either by technological or engineering issues [4, 5, 19]. For example, light at different wavelength may require phase matching by beam alignment to optimize the SHG efficiency in the SHG crystal. SHG crystal is difficult to be integrated with current optical integration platforms (Si, Si₃N₄, III-V materials, etc.). The fabrication of high-quality SHG crystal may also be costly.

Another autocorrelation technology based on two-photon absorption (TPA) effect in semiconductors has also been widely investigated [1, 20–30]. TPA effect allows very compact autocorrelator design [24, 25] and high detection sensitivity [21, 27]. TPA effect usually has picosecond decay time in bulk semiconductors which may lead to limitation on the measurement of pulse width [27–30]. Hybrid integration between semiconductors and insulators (e.g., Si₃N₄ or SiO₂) is still challenging if the insulator waveguides are preferred to reduce the propagation loss of the pulses.

On the other hand, low dimensional materials (LDMs) including one-dimensional (1D) carbon nanotubes (CNTs) and various two-dimensional (2D) materials have shown abundant photonic and optoelectronic properties. Their saturable absorption(SA), i.e., intensity dependent nonlinear transmission, has been widely used for mode locking operation in pulsed lasers including carbon nanotube in [31, 32], graphene in [33–35], topological insulators in [36–39], transition metal dichalcogenides in [40–46], black phosphorus in [47–50] and MXene in [51], etc. Saturable absorption also enables the materials to be a nonlinear medium for pulse interactions, which is a key requirement for pulsewidth measurement technologies such as AC. Compared with TPA, most LDMs exhibit femtosecond-level decay time after photonic excitation which is promising for measuring ultrashort pulses with minimum distortion [30, 52–54]. The atomic level thickness of the LDMs also minimizes the potential pulse broadening caused by the optical dispersion in the bulk materials, which is another advantage to achieve ultrashort pulse measurement [54–57]. Moreover, hybrid integration between LDMs and planar waveguides or between LDMs and optical fibers has been developed with simple procedures [54, 56–59]. Therefore, pulsewidth measurement based on the saturable absorption of LDMs may provide a new technical route for pulse characterization, which is potentially capable of measuring ultrashort pulses and compatible with various integrated or non-integrated photonic platforms.

Meanwhile, it is known that an ideal intensity AC requires the nonlinear medium to have a response proportional to the square of pulse intensity, which is unfortunately not the case of saturable absorption in LDMs. However, if there is still a nearly linear relation between the input pulse width and the pulse width of measured trace, and the pulse width measurement error is controlled within a reasonable range, e.g., less than 10%, we can still treat this LDM based technology as a quasi-AC measurement considering its advantages mentioned above.

In this paper, we propose and demonstrate a proof-of-concept quasi-autocorrelation technology based on the saturable absorption of carbon nanotubes, called saturable absorption based pulsewidth measurement(SAPM). Low saturation intensity of CNTs allows low input pulse intensity and relatively large modulation depth of CNTs allows better measurement result. A minimum measurable pulse energy of 75 fJ is obtained limited by the system loss. The corresponding average-power-peak-power product ($P_{av}\cdot P_{pk}$) is $5.44\times {10}^{-7}$ W² and the measurement time resolution is 9.0 fs. By further optimizing the experimental setup, it is estimated that the specifications can be improved to 10 fJ for minimum pulse energy and $1.3\times {10}^{-9}$ W² for $P_{av}\cdot P_{pk}$. The measurement error of the pulse width is less than 6% due to the non-ideal AC measurement. We believe this work may pave the way to a new type of pulsewidth measurement technology which is capable of achieving pulse width measurement down to few-femtosecond level and is easy to be integrated to various photonic platforms.

2. Principle and simulation

An ideal AC measurement requires the nonlinear medium to have a response proportional to the square of input pulse intensity. On the other hand, it is known that an ideal AC measurement does not reflect the actual input pulse shape [60]. So from a more practical point of view, the most important function of an AC measurement is to obtain the input pulse width based on a linear relation given by the following equation:

In the following part of this section, we will discuss the preparation of LDM carbon nanotubes, the characterization of their saturable absorption, the experimental setup of a CNT based SAPM system, and the modelling and simulation of such a SAPM system, respectively.

2.1. CNT preparation and characterization

A high-quality CNT thin film is the key component of our system. The single wall CNTs are synthesized by the catalytic chemical vapor decomposition method and the diameters are from 1 nm to 1.5 nm. Then CNTs are mixed with polyvinyl alcohol (PVA) to form thin films with the following steps. Firstly, 0.5 mg$\cdot$mL⁻¹ CNT dispersion is prepared by dispersing filiform CNTs in deionized water using an ultrasonic cleaner for five hours. The dispersing agent used in the experiment is sodium dodecyl benzenesulfonate. Secondly, the CNT dispersion is centrifuged at 12000 g for several hours, and upper supernatant is collected to reduce unwanted scattering losses from large-scale agglomeration. Thirdly, 10 wt% aqueous PVA solution and CNT dispersion are mixed at a volume ratio of 1 : 2 by a magnetic stirrer for three hours. Finally, the obtained CNT-PVA mixture is dropped on a Petri dish for a two-day evaporation and a CNT-PVA thin film is obtained. Figure 1(a) shows the transmission electron microscopy (TEM) image of CNTs and Fig. 1(b) shows the image of the fabricated CNT-PVA thin film. The CNT-PVA thin film is then cut into small pieces and sandwiched between two fiber connectors to form a saturable absorber as shown in Fig. 1(c).

 

Fig. 1 (a) Transmission electron microscopy image of CNTs. (b) CNT thin film. (c) CNT thin film transferred onto a fiber end. (d) Experimental setup of the CNT saturable absorption measurement. (e) Nonlinear saturable absorption characteristics of the CNT saturable absorber (data and fit). (f) Transmission spectrum of the CNT.

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The nonlinear saturable absorption property of the CNT is characterized by a standard two-arm measurement as shown in Fig. 1(d). A passively Erbium-doped mode-locked laser (MLL) with a repetition rate of 37 MHz and a pulse width of 560 fs is used as an optical source with variable output power. After passing through a protective isolator (ISO), the pulses are divided by a 99:1 coupler. 1 % power is directly measured by a power meter (denoted as power meter 1) as reference and 99 % output power propagates through the CNT saturable absorber and is detected by another power meter (denoted as power meter 2). A 10 dB fixed attenuator (ATT) is added between the CNT saturable absorber and the power meter 2 to maintain a suitable detection power range. Figure 1(e) shows the measured optical transmittance of the CNT saturable absorber under different incident pulse intensity. The red curve in Fig. 1(e) is the fitting curve using the following saturable absorption formula [40]:

2.2. Experimental setup, theory and simulation

The experimental setup of the proposed system with the CNT saturable absorber is shown in Fig. 2(a). The all-fiber configuration is chosen to allow an alignment free system. A homemade passively mode-locked laser with a repetition of 100 MHz and a center wavelength of 1560 nm is used as the optical pulsed source. An optical pulse from the MLL is divided by a polarization beam splitter (PBS). One of the split pulse is modulated by an electro-optic modulator (EOM) with a 1 kHz sinusoidal signal (denoted as $R\left(t\right)$) and the other pulse propagates through a variable time delay line (DL, General Photonics MDL-002). A tunable attenuator (ATT) is added to balance the power of two paths. Two optical pulses are recombined by another PBS. Polarization-maintaining optical fiber devices are used to avoid the interference between two paths. The output of the PBS (denoted as $P\left(t\right)$) passes through the CNT saturable absorber. The output from the CNT saturable absorber (denoted as $P_1\left(t\right)$) is detected by a photodetector (PD) and measured by a Lock-in Amplifier (LIA, Signal Recovery 7280) system. The setup is very similar to a conventional intensity AC system except that the nonlinear medium is CNT instead of SHG crystal.

 

Fig. 2 (a) Experimental setup of the CNT based SAPM. (b) Simulated transmittance change of CNT saturable absorber with respect to different delay time.

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The principle of the proposed SAPM can be explained as follows. When two split pulses combine at PBS, the peak power of the combined output pulse is dependent on the time delay between two split pulses. The transmission of the CNT saturable absorber is dependent on the peak power of the incident pulse. Therefore, the output power from the CNT saturable absorber is monotonically determined by the time delay of two split pulses. By measuring the average output photocurrent from the PD, the SAPM pulse profile can be obtained. The LIA and EOM are utilized to improve the measurement precision. Figure 2(b) shows an example of simulation result of the dependence between the CNT transmission and the delay time with an input pulse full width at half maximum (FWHM)of 1 ps.

The theoretical analysis of the proposed system is performed as follows. The temporal pulse power profile from the MLL is assumed Gaussian and given by:

As mentioned previously, saturable absorption does not provide an ideal AC function, so it is important to understand how our quasi-AC SAPM technology is different from an ideal AC. To compare, we choose four different input pulse types, i.e., Gaussian pulse, soliton pulse (sech² shape) and pulse with tale ringing caused by third order dispersion. The simulation results are summarized in Fig. 3 shown below. All the pulses have a pulse width (FWHM) of 1 ps and a normalized peak power P_n of 1. Here the normalized peak power P_n is defined as the ratio of pulse peak power over saturation power of CNT saturable absorber. And the saturation power equals to $I_{sat}\cdot$80 μm² = 16.55 W for our CNT saturable absorber.

In Figs. 3(a)–3(c), the SAPM traces are shown in red dashed lines for Gaussian pulse, soliton pulse and pulse with tale ringing, respectively. The insets are the pulse shapes of the input pulses. For comparison, we also calculate the ideal AC traces generated by SHG based AC, shown in black solid lines. It can be seen that in all three cases, the traces generated by our SAPM technology match well with the AC traces by SHG based AC. In the pulse peak and wing region of the traces, there do exist slight differences in three cases, shown in Figs. 3(d)–3(f). These differences are caused by the non-ideal AC nature of saturable absorption.

 

Fig. 3 Simulated CNT based SAPM traces (red) and SHG AC traces (black) for an input of (a) Gaussian pulse, (b) soliton pulse, and (c) pulse with tale ringing. Insets in (a)-(c): Input pulse shapes. (d)-(f) Zoomed views of wing region of the traces in (a)-(c). Insets in (d)-(f): Zoomed views of peak region of the traces.

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Moreover, it is known that Gaussian pulse and soliton pulse input can be separated by comparing the fitting with Gaussian and soliton functions in the SHG AC trace. This can also be achieved by our technology. In Fig. 3(e), we also fit the soliton SAPM traces with a Gaussian function (green). It can be clearly seen that Gaussian fitting is different from both two traces in the wing region.

In Fig. 3, both SAPM traces and SHG AC traces are normalized by the corresponding $C_{con}$ (in Eq. (1)) in time axis because $C_{con}$ for SAPM traces is different from $C_{con}$ for SHG AC traces as discussed above. For example, for SHG AC trace with Gaussian/soliton pulse input, the corresponding $C_{con}$ = 1.414/1.543. For SAPM trace with same Gaussian/soliton pulse input, the corresponding $C_{con}$ = 1.365/1.779. This treatment is to allow a direct comparison of the trace shapes. The calculation of $C_{con}$ in the SAPM technology will be discussed later.

A pulse pair input is also simulated to compare with SHG AC, shown in Fig. 4. The pulse width is 1 ps and the pulse spacing is 4 ps. Time axis is not normalized by $C_{con}$ this time because the pulse separation will be changed. It can be observed that 4 ps pulse spacing can be clearly observed and the ratio between the main peak and side peak is 2, which confirms the capability of our SAPM technology to detect pulse pair.

 

Fig. 4 Simulation of pulse pair input for (a) SHG based AC and (b) CNT based SAPM. Inset: input pulse pair.

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Then we investigate the conversion coefficient $C_{con}$ in Eq. (1). Figure 5(a) shows the simulation result of the relation between the input pulse width and the pulse width of the SAPM trace. The pulse shape is Gaussian and the normalized peak power of input pulse is fixed to 1. A nearly linear relation confirms our previous assumption that Eq. (1) is still held in our CNT based SAPM except for a different $C_{con}$. The corresponding $C_{con}$ =1.365.

 

Fig. 5 Simulation of (a) SAPM pulse width with respect to the input pulse width and (b) SAPM pulse width with respect to the input pulse peak power. (c) Three-dimensional simulation.

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The situation when input pulses have different peak power is also studied. The simulation result of the relation between normalized input peak power and SAPM pulse width is shown in Fig. 5(b). The pulse shape is Gaussian and the input pulse width is fixedto 1. The normalized peak power is from 0.001 to 2 because the system is working on the monotonic region of CNT saturable absorber to prevent data ambiguity (P₀<32W or $P_n<1.93$) as shown in Fig. 1(e).

It can be seen that, due to the non-ideal AC nature of saturable absorption, the SAPM pulse width increases with the increase of input peak power. This means for different input pulse peak power, there will be a different conversion coefficient $C_{con}$, which is not desired. However, it is noted that the maximum change of $C_{con}$ is 0.17 when the normalized input peak power changes from 0.001 to 2. If we choose $C_{con}$ = 1.365 as a standard value (normalized peak power P_n = 1), the pulse measurement error induced by $C_{con}$ is less than 6% which is within a reasonable range.

A more complete three-dimensional simulation result is shown in Fig. 5(c) to indicate the dependence of SAPM pulse width on input pulse width and normalized peak power. The curves corresponding to Figs. 5(a) and 5(b) are also denoted.

Lastly, it is worth mentioning that the value of $C_{con}$ is dependent on the nonlinear transmission property of CNT (and other LDMs to be used). So a calibration procedure is needed to determine the value of $C_{con}$ in the system.

3. Experimental results

3.1. SAPM pulse profile

A typical SAPM experimental result is shown as red circles in Fig. 6(a). The horizontal axis is the time delay between two pulses and the vertical axis is the measured LIA signal amplitude. Each data point is averaged by 100 times to reduce the noise from the environmental perturbation. The input pulse has a bandwidth of ∼ 57.2 nm, a repetition rate of 100 MHz and a center wavelength of 1560 nm. The red curve is the fitted SAPM pulse profile using a Gaussian shape. The pulse width (FWHM) of the SAPM trace is ∼ 11.94 ps based on the Gaussian fit. The pulse is broadened by the fiber dispersion of the system. The calibration of the system will be discussed in the next paragraph. The inset of Fig. 6(a) shows an AC trace measured by a standard SHG autocorrelator (Femtochrome FR-103 XL) with a pulse width (FWHM) of 1.03 ps. The pulse is not transform limited due to the ∼ 1-meter long fiber pigtail from the laser. The pulse peak power at the input port of the system is 23.635 W and the peak power entering the CNT saturable absorber is 0.345 W (zero delay between two arms) due to dispersion induced pulse broadening and the system loss of 7.71 dB. The corresponding normalized peak power is 0.02. The Fig. 6(b) shows the SAPM trace of a soliton pulse with hyperbolic secant profile and the inset of Fig. 6(b) is the corresponding standard SHG AC trace.

 

Fig. 6 (a) SAPM trace (purple circles) and Gaussian fit (red line), inset: AC trace with standard SHG autocorrelator, (b) SAPM trace (purple circles), hyperbolic secant fit (red line) and Gaussian fit (blue dashed line), inset: AC trace with standard SHG autocorrelator.

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3.2. Influence of pulse width and peak power

The investigation on the influence of pulse width and peak power is also a calibration process for the system. The value of conversion coefficient $C_{con}$ is dependent on the saturable absorption of the nanomaterials. The all fiber setup allows alignment free but also induces pulse broadening by the dispersion, i.e., the group delay dispersion (GDD) with the units of ps/nm. So the calibration process is to determine $C_{con}$ and GDD of the SAPM system. A pulse train with fixed peak power and tunable pulse width is first generated. This pulse train is obtained by sending a pulse train from a mode-locked laser to the single mode fiber with different length. Fiber dispersion will broaden the pulse width. Meanwhile, the average power of the pulse train is adjusted so that the peak power is fixed. Then the pulse train is measured both by a standard SHG autocorrelator and by our SAPM system. Figure 7(a) shows four typical measured SAPM traces and their fitting curves for input pulses with four different pulse widths denoted on the figure. The traces are vertically offset to provide a clear view. Figure 7(b) summarizes the relation between the input pulse width and measured SAPM pulse width using our system. A linear fit is then plotted in red in Fig. 7(b). The relation between the measured SAPM pulse widths ${\tau }_{SAPM}$ and the input pulse widths ${\tau }_{in}$ is given by:

 

Fig. 7 (a) SAPM traces with different input pulse width. (b) Relationship of the measured SAPM pulse width and the input pulse width. (c) SAPM traces with different input pulse power. (d) SAPM pulse width with respect to the normalized peak power.

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The influence of pulse peak power is also investigated when the input pulse peak power is increased from 46.6 W to 127.3 W at system input port corresponding to a normalized peak power P_n from 0.15 to 0.41 at the CNT saturable absorber. Four typical SAPM traces are shown in Fig. 7(c) with P_n denoted. Figure 7(d) summarizes the relation between measured pulse width and normalized peak power. It can be seen that when the normalized peak power is increased from 0.15 to 0.41 at the CNT saturable absorber, the measured SAPM pulse width is increased by $\sim 2\%$. This pulse broadening is due to the nonlinear transmission of the CNT. The red curve is the linear fit which is consistent with the CNT saturable absorption described in Eq. (2) and the simulation in Fig. 5(b).

To briefly summarize, there are 2 steps for the system calibration: Firstly, determining the conversion coefficient $C_{con}$ by comparing the input pulse widths (using SHG AC) and measured pulse widths of our SAPM traces using a certain pulse train. This step is necessary because $C_{con}$ is dependent on the nonlinear transmission property of LDM used. Secondly, determining the pulse broadening effect, i.e. group delay dispersion (ps/nm), caused by the fiber dispersion. This step can be removed in the future by applying a free-space system.

3.3. Stability and sensitivity

The stability of the system output is also investigated as shown in Fig. 8. Figure 8(a) shows a long-time SAPM traces (LIA signal train) obtained by measuring a known pulse 30 times which indicates that the measurement results are stable under environmental disturbance. Four typical measurement results with fitted pulse profiles are shown in Fig. 8(b). The data are vertically offset to give a clear view.

Figure 8(c) summarizes the measured pulse width deviation for each measurement similar to [1]. Blue circles represent the measured actual pulse width errors using SAPM technology according to Eq. (6) for 30 measurements with reference to the pulse width measured by the SHG autocorrelator. Red square (9.0 fs) represents the root-mean-square (rms) error of the measured pulse width. The error bar denotes the upper (21.6 fs) and lower (-17.6 fs) bound of the errors. The error histogram is shown in Fig. 8(d). The measurement error is mainly limited by the loss variation (0.6 dB) and the delay accuracy (∼ 10 fs) of the tunable delay line.

 

Fig. 8 (a) Long-time stability test of the SAPM. (b) Four typical SAPM traces and their fit from (a). (c) Measurement error of the pulse widths. (d) Histogram of the error.

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Fig. 9 SAPM measurement with a pulse energy of 75fJ (-21.26 dBm average power).

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The sensitivity of our system is also studied. A variable optical attenuator with negligible excess dispersion is inserted before the input port to adjust the input average power. Figure 9 shows a typical SAPM trace with a minimum detectable input power of -21.26 dBm which in fact has exceeded the minimum measurable power of our SHG autocorrelator. The measured FWHM of SAPM pulse is 11.78 ps corresponding to an input pulse width (FWHM) of 895 fs according to Eq. (6). This calculated result has acceptable deviation ($13.5\%$) in this low power condition compared with the actual input pulse width data in Fig. 6(a) (1029 fs (FWHM) measured at high power with SHG autocorrelator) and the corresponding single pulse energy is 75 fJ with a $P_{av}\cdot P_{pk}$ of $5.44\times {10}^{-7}$ W² for the system.

4. Discussion

In this section, we compare the properties of two current AC technologies SHG and TPA with our quasi-AC SAPM technology, shown in Table I.

Table 1. Property comparison among SHG, TPA and LDM based SAPM technologies

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Firstly, SHG technology requires phase matching whereas TPA and SAPM technologies don’t which gives more flexibility in the beam alignment. Secondly, for integration, it is still challenging to integrate SHG crystal with current integration platform. TPA materials are compatible with semiconductor platform (e.g., Si or InP [24, 25]) but hybrid integration with insulator platform (e.g. Si₃N₄ or SiO₂) is challenging and lossy. LDMs used in SAPM are relatively easy to be integrated to various platforms [54–57]. Thirdly, the recovery time of the nonlinear material is an important measure of the minimum measurable pulse width. The recovery time is nearly instant forSHG but from tens of picoseconds to nanosecond for TPA [61–63] (e.g. 25ps in [22] and 25ns in [64]) and from tens of femtoseconds to sub-picosecond for LDMs [52–54] (e.g. <30 fs for graphene [65], 280fs for CNT [66]). We simulate the influence of the recovery time for LDM saturable absorbers by using the slow saturable absorber model as follows [67]:

 

Fig. 10 (a) Four typical simulated SAPM traces with different recovery time and their fit. (b) Relationship of the measurement error and NRT, inset: measurement error against NRT when error is less than 10%. (c) Simulated SAPM trace (purple circles) and Gaussian fit (red line) with NRT equals 10. (d) Simulated SAPM trace (purple circles) and Gaussian fit (red line) with NRT equals 0.3.

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Lastly the material thickness is also important because a thinner material induces less dispersion and thus less pulse broadening. The typical thickness of these nonlinear materials is tens of microns to few millimeters for SHG crystals, sub-micron to few microns for TPA materials [30] and sub-nanometer to few nanometers for LDMs. Therefore, LDM based SAPM technology has advantages in beam alignment, integration and thickness. Its material recovery time is between SHG crystals and TPA materials.

Table II summarizes the specifications achieved by current SHG and TPA AC technologies as well as our LDM based SAPM technology. There are also other newly demonstrated pulse width measurement technologies including time-lens temporal imaging [68, 69], FWM-X-SPIDER [16], transverse second harmonic generation (TSHG) in nanowires [4], third-harmonic generation (THG) in photonic crystal waveguide [70] and graphene photodetector [71, 72]. Their specifications are also provided in Table II.

Table 2. Different methods for pulse width measurement

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Here we analyze the potential performance of our LDM based SAPM system. For operation bandwidth, a broadband LDM can be chosen to cover a wide wavelength range [54, 56, 57]. For minimum pulse width, the nanometer or sub-nanometer thickness and the ultrafast decay process of LDMs can support the measurement of pulse width down to few femtoseconds [52–54]. For the average-power-peak-power product and minimum measurable pulse energy, current system is limited by the loss (7.71 dB) mainly introduced by the EOM. If a better EOM with low insertion loss is chosen, or even the LIA is removed if the whole system is well isolated from the noise, the allowable single pulse energy sensitivity can be $\sim 10$ fJ and the $P_{av}\cdot P_{pk}$ can be improved to $1.3\times {10}^{-9}$ W² (corresponding to an average power of -21.26 dBm - 7.71 dB = -28.97 dBm). Choosing a saturable absorber material with even lower saturation intensity and higher modulation depth may further improve the sensitivity of the system to several ∼ fJ. For pulse width resolution, a better tunable delay line with less loss variation and better delay accuracy can potentially minimize the measurement error on the pulse width to sub-femtosecond level [1]. Free-space system may also be considered to avoid the dispersion effect.

5. Conclusions

In conclusion, we have proposed and demonstrated a SAPM technology by using low dimensional nanomaterials based saturable absorber. In a proof-of-concept experiment with carbon nanotube, the SAPM system has a minimum measurable single pulse energy of 75 fJ, an average-power-peak-power product $P_{av}\cdot P_{pk}$ of $5.44\times {10}^{-7}$ W², and a time resolution of 9.0 fs. By further optimizing the experimental setup, it is estimated that the system can potentially support fJ level pulse energy with fs pulse width. We believe this work may pave a new way for pulsewidth measurement which is potentially capable of measuring ultrashort pulses with high sensitivity and resolution. Our technology is also compatible to various integration platforms and can benefit the realization of a high-performance fully integrated pulsewidth measurement device.