## Abstract

The technique of multiform time derivatives of pulse has been shown necessary to achieve various time-space metrology goals with a precision at or beyond the standard quantum limit. However, the efficient generation of the desired time derivatives remains challenging. In this paper, we report on the efficient realization of multiform time derivatives with a programmable 4-f pulse shaping system. The first-order time derivative of the pulse electric field has been achieved with a generation efficiency of 72.12%, which is more than 20 times higher than that of previous methods. Moreover, the first- and second-order time derivatives of the pulse envelope have been achieved with the generation efficiencies being 11.10% and 3.53%, respectively. In comparison, these efficiencies are three times higher than those for previously reported methods. Meanwhile, the measured fidelities of the three time-derived pulses are reasonably high, with values of 99.53%, 98.37% and 97.32% respectively.

© 2017 Optical Society of America

## 1. Introduction

Along with the remarkable development of atomic clock technology, the accuracy of time-frequency signals has reached the E-18 order [1]. With the help of accurate space-time positioning technologies, these high-precision time-frequencies have played an incredibly important role in areas ranging from fundamental physics to engineering and technological applications. Most breakthroughs in these areas are dependent on the ability to accurately measure both time and frequency. Therefore, many attempts have been made to improve the sensitivity of the time delay measurements [2].

Traditionally, two methods are used to measure the time delays: incoherent time-of-flight (TOF) [3] and coherent phase (PH) measurements [4,5]. For a light pulse of central frequency ${\omega}_{0}$and bandwidth$\Delta {\omega}_{}$, the achievable best accuracy of the measured time delay in these two schemes is governed by the so-called standard quantum limits (SQL), which are given by${(\Delta u)}_{SQL}^{tof}=1/(2\Delta \omega \sqrt{N})$ for the TOF and${(\Delta u)}_{SQL}^{ph}=1/(2{\omega}_{0}\sqrt{N})$for the PH respectively, and$N$denotes the measured mean number of photons during the detection time. In order to further improve the measurement precision, Lamine et al. proposed a quantum-improved time transfer scheme that implements homodyne detection and a first-order time derivative on the local oscillator (LO) pulse electric field [6]. With this scheme, a new SQL ${(\Delta u)}_{SQL}=1/(2\sqrt{{\omega}_{0}^{2}+\Delta {\omega}^{2}}\sqrt{N})$ can be achieved, which implies a sensitivity that potentially reaches the yoctosecond range (10^{−21}–10^{−24} s). The latest research shows that by applying different time derivatives on the LO pulse, various quantum metrology goals [7] can be achieved. For example, the TOF of the signal pulse can be measured at the SQL by using the first-order envelope-derived pulse as the LO. Moreover, one can make the measurement sensitivity immune to atmospheric fluctuations by introducing the second-order envelope-derivative on the LO pulse. Thus, the ability to precisely shape the LO determines the accuracy of the time delay measurement for different quantum metrology purposes.

For a LO with a center frequency of ${\omega}_{0}$ and an envelope of $A(t)$, the complex electric field of which can be written as $E(t)=A(t)\mathrm{exp}(-i{\omega}_{0}t)$. Consider a Fourier transform limited Gaussian pulse of frequency spread $\Delta \omega $, its envelope can be expressed as:$A(t)={(\Delta \omega )}^{1/2}{(2/\pi )}^{1/4}\mathrm{exp}[-{(\Delta \omega t)}^{2}]$. To achieve the desired measurement of phase delay, group delay and group velocity dispersion delay with a precision immune from environmental parameters fluctuations, the respective temporally shaped LO pulses should have the forms of the first-order time derivative of the electric field, the first-order time derivative of the envelope, and the modified second-order derivatives of the envelope respectively, which can be expressed as:

where the subscript*e*denotes the derivative of the pulse envelope. Through Fourier transformation, the time derivatives can be given by [7]:

Several methods have been reported to realize these time derivatives of the pulse. For example, Miguel et al. used a fiber Bragg grating to generate the first-order derivative of the pulse envelope [8]. This method requires manufacturing different fiber gratings for different input pulses, and is limited by the spectral width and input energy. Park et al. used a Michelson interferometer to obtain the first- and second-order time derivatives of the pulse envelope [9]. Labroille et al. used a Babinet-Soleil-Bravais (BSB) compensator to realize the time derivative of both the pulse electric field and envelope [10]. These methods are almost compatible with any bandwidth of the input spectrum. However, the generation efficiencies are low. In addition, if higher-order derivatives are to be implemented, the derivative manipulation process needs to be performed more than once, which seriously complicates the experimental facilities. The method of using a programmable spatial light modulator (SLM) in a pulse shaper was firstly proposed by Weiner et al. [11,12], and has since been widely used in a variety of pulse shaping applications due to its versatility [13–15]. However, to the best of our knowledge, obtaining the pulse derivative using a Fourier pulse shaping system has not been reported.

In this paper, we demonstrate the efficient generation of the required first-order time derivative of the pulse electric field as well as the first- and modified second-order time derivatives of the pulse envelope using a programmable SLM in a reflective configured 4-f Fourier pulse shaping system. The results show that the generation efficiency can be greatly improved while maintaining a reasonably high fidelity.

## 2. Experimental setup

The experimental setup is shown in Fig. 1. The input pulse is generated by a commercial Ti: Sapphire laser (Fusion 100-1200, FemtoLasers) centered at 813.3 nm with a bandwidth of 6 nm and a repetition rate of 75 MHz. After being collimated by the pair of lenses (L1 and L2), the beam is passed into the pulse shaper by the combination of the 45° highly-reflective mirror (M) and the D-shaped pickoff mirror (D-M). The optical configuration of the Diffractive Pulse Shaper (MIIPS-HD, Biophotonic) is plotted and enclosed by a dash-dotted frame. The half-wave plate before the pulse shaper (HWP1) is used to rotate the polarization of the input electric field so that most of input energy can be diffracted by the polarization-dependent grating G (PC2100NIR from SPECTROGON), which has a maximum diffraction efficiency of about 92.5%. In the pulse shaping system, the different frequency components of the input pulse are spatially dispersed along the x-axis by the grating. A reflection type liquid crystal on silicon-spatial light modulator (LCOS-SLM, X10468-2, Hamamatsu), with 792 pixels (the pixel size is 19.6 μm with a 0.4 μm gap between the pixels) and light utilization efficiency of 98%, is placed on the Fourier plane. When the spatially dispersed light pulse enters the LCOS-SLM, it is phase-modulated along the x-axis and amplitude-modulated based on the diffraction-based method [16] along the y-axis as needed and then reflected. In our experiment, the spatial frequency was set as 2.5 lp/mm, which corresponds to a diffraction efficiency of 88% [17]. Thus the output efficiency of the pulse shaping system without amplitude modulation is 73.86%. The reflected light pulse has a vertical offset of approximately 2.4 cm relative to the input light pulse at the D-M position and can be departed from the input pulse. Afterwards, an aperture behind the D-M is used to block the undesired zero-order diffraction light. It should be noted that the pixel gap cannot modulate the spectrum. Consequently, approximately 2% of the spectrum remains unmodulated, which may decrease the fidelity. By applying the appropriate amplitude and phase modulation to each pixel, the pulse is shaped into the desired form at the cost of efficiency decreasing.

A portion of the output light is reflected by the PBS2 and sent to SM1 to record the spectral intensities of $\text{|}{E}_{0}(\omega ){\text{|}}^{2}$,$\text{|}{E}_{1}(\omega ){\text{|}}^{2}$,$\text{|}{E}_{e1}(\omega ){\text{|}}^{2}$ and$\text{|}{E}_{e2}(\omega ){\text{|}}^{2}$. The transmission portion is focused on a beta barium borate (BBO) crystal (10 μm thickness) and the second-harmonic generation (SHG) is then measured by SM2.

To measure the phase of the shaped pulse, the Fourier-transform spectral interferometry (FTSI) method is used [18,19]. A portion of input light reflected by PBS1 acts as the reference pulse. After the delay line, the reference pulse is combined with the shaped pulse at PBS4. The HWP6 is used to change the polarization of the combined pulse. The interference spectrum is recorded by SM3.

## 3. Results and analysis

#### 3.1 Dispersion compensation

In principle, the Fourier transform of an electromagnetic pulse spectrum includes its temporal information; however, it is only accurate when the pulse is transform-limited and the different frequency components have the same phase. Unfortunately, the input light pulse in the experiment is not always transform-limited. Thus, it is necessary to compensate for the dispersion before adding modulation to the SLM. In our experiment, we used the multiphoton intrapulse interference phase scan (MIIPS) method [20] to accomplish the dispersion compensation. In the MIIPS method, an initial reference phase$f(\omega ,{p}_{0}^{})$ was added to the input pulse by the modulation applied onto the SLM, where$p={p}_{0}$is an adjustable parameter used to change the phase. The total phase of output pulse is$\Phi (\omega )=f(\omega ,{p}_{0})+\varphi (\omega )$, where $\varphi (\omega )$ is the unknown phase of the input pulse. The output pulse was focused on a BBO crystal (10 μm thickness) to generate the second-harmonic generation (SHG) signal, and parameter $p$ was dependent on the SHG signal intensity. When the maximum SHG signal was observed at a particular${p}_{opt}$, the total phase$\Phi (\omega )$was close to zero and the inverse of the corresponding reference phase$\text{-}f(\omega ,{p}_{opt})$could be used as the phase compensation function. In our experiment, the time-bandwidth product was reduced from 1.58 to 1.05, and the pulse duration was compressed from 183.1 fs (blue dotted line) to 121.6 fs (red solid line) after dispersion compensation, as shown in Fig. 2(a). In addition, the phase of the pulse electric field changed from a triple curve (blue dotted line) to a constant close to zero (red solid line), as shown in Fig. 2(b). Thereby, a nearly transform-limited pulse corresponding to$\text{E}(\omega )$was generated and subsequently used for implementing the desired time derivatives.

#### 3.2 Time derivative of the pulse electric field and envelope

The phase and amplitude modulation functions needed to realize the first-order time derivative of the pulse electric field, and the first- and second-order time derivatives of the pulse envelope are plotted in Figs. 3(a)-3(c), respectively. As illustrated by the black solid line in Fig. 3(a), the phase function for the first-order time derivative of the pulse electric field should have a constant phase shift of $-\pi /2$ across the entire spectral range. Also, as shown by the black solid lines in Figs. 3(b)-3(c), a $\pi $phase shift of the phase function is expected at the center wavelength of 813.3 nm for the generated first-order derivative of the pulse envelope, and the $\pi $phase shifts at both 810.6 nm and 816.0 nm are necessary for the generated second-order derivative of the pulse envelope. By including the phase function for dispersion compensation, the applied phase functions are shown by the red solid lines in Figs. 3(a)-3(c). As the amplitude modulation is applied on the spectral intensity of the pulse, Figs. 3(a)-3(c) show the intensity modulation functions with the blue dashed lines, whose theoretical energy conversation efficiency are 97.75%, 14.52% and 4.23% respectively. Combining with the above experimental parameters, the expected total generation efficiencies of the three forms of shaped pulses are 72.13%, 10.71% and 3.12% respectively. For the first-order time derivative of the pulse electric field, the intensity modulation function is a monotonic quadratic function of the wavelength; however, as$\Delta {\omega}_{}$is much smaller than ${\omega}_{0}$in our case, it appears to be a linear function. For the first-order derivative of the pulse envelope, the intensity modulation function is quadratic with a zero point at 813.3 nm. For the second-order derivative of the pulse envelope, the intensity modulation function is a quartic function with two zero points at the two phase shift points (810.6 nm and 816.0 nm). It should be noted that, although the phase and intensity modulation functions shown in Fig. 3 are continuous, the actual functions in the experiment are discrete and each wavelength component corresponds to a pixel in the SLM. By adding these phase and intensity modulations to the SLM in a properly configured 4-f Fourier system, the input pulse can be transformed into the desired pulse.

Based on the above modulation functions, the first-order time derivative of the pulse electric field, and the first- and second-order time derivatives of the pulse envelope are implemented. The spectra of the obtained time-derived pulses are then measured and shown by the square root of the ratios to that of the fundamental mode. Figure 4(a) shows the result for ${\text{|E}}_{1}^{}\left(\lambda \right){\text{|/|E}}_{0}\left(\lambda \right)\text{|}$. Due to the pulse-to-pulse noise and small $\Delta \omega /{\omega}_{0}$, apparent fluctuations are observed. Even so, the comparison between the experimental results and theory gives a fidelity of 99.53%. By measuring the total power of the input and output pulses, which were 23.80 mW and 17.14 mW, respectively, a generation efficiency of 72.12% was achieved. In comparison, this efficiency is more than twenty times better than that for previously reported results [10]. However, we should note that the generation efficiency of the first-order time derivative of the pulse electric field is dependent on the input spectrum bandwidth. With a pulse bandwidth as broad as that in Ref [10], the generation efficiency will be reduced to 47.27%, which is still 12 times better. As the spectral resolution of the SLM will be decreased with broader spectrum, the achievable fidelity may decrease as well. This may be solved by choosing a SLM with more pixels.

Similarly, the measured spectrum of the generated first-order derivative of the pulse envelope as compared to that of the fundamental mode in the form of the square root of the ratios, ${\text{|E}}_{e1}^{}\left(\lambda \right){\text{|/|E}}_{0}\left(\lambda \right)\text{|}$, is shown in Fig. 4(b). The zero amplitude point can be seen at the center wavelength of 813.3 nm, and a fidelity of 98.37% was obtained. With 23.80 mW input, the output was measured to be 2.64 mW, which corresponds to a generation efficiency of 11.10%. This efficiency is about three times higher than the previously reported efficiency [9]. The resultant spectrum of the second-order derivative of the pulse envelope in the form of ${\text{|E}}_{e2}^{}\left(\lambda \right){\text{|/|E}}_{0}\left(\lambda \right)\text{|}$ and the corresponding theoretical results are plotted in Fig. 4(c). As estimated by Fig. 3(c), two zero amplitude points are observed at the phase shift wavelengths of 810.6 nm and 816.0 nm. By comparing the input and output powers, the generation efficiency was determined to be 3.53%, which is about three times higher than that reported in [9]. The fidelity was measured to be 97.32%. Note that the achieved fidelities based on our method were slightly lower than those for the previously reported methods. This is due to the effects of the discrete sampling and pixel gaps [21].

#### 3.3 Phase measurement

In order to fully evaluate the resultant time-derived pulses, it is also necessary to measure the field phase in addition to the ratios of the field amplitudes. Common methods for this are FTSI and polarization spectral interferometry [22], which are able to accurately measure abrupt phase changes. Here, we used FTSI to retrieve the phases of the above generated time-derived pulses. It should be pointed out that the reference pulse in FTSI is not transform-limited, in order to give the phase of differentiated pulse, the phase difference ${\phi}_{d}(\omega )$between the dispersion compensated pulse and the reference pulse is measured and subtracted from the phase difference ${\phi}_{r}(\omega )$between the differentiated pulse and the reference pulse. Figures 5(a)-5(c) show the retrieved spectral phases${\phi}_{r}(\omega )\text{-}{\phi}_{d}(\omega )$of the generated first-order derivative of the pulse electric field, and the first- and second-order derivatives of the pulse envelope, respectively. Figure 5(a) shows that the first-order derivative pulse of the electric field has a flat phase shape that takes the value of$-\pi /2$. In Fig. 5(b), the generated first-order envelope derivative pulse clearly shows the expected$\pi $phase shift at the center wavelength of 813.3 nm. In Fig. 5(c), the generated second-order envelope derivative pulse clearly shows the expected $\pi $phase shifts at 810.6 nm and 816.0 nm. The agreement between the results and theoretical phases confirm that we did indeed obtain the desired pulse given by Eqs. (4)-(6). The residual discrepancies between the theory and experimental results are mainly due to the pixel gap and limited phase modulation precision of the SLM. Additionally, as the FTSI method is an indirect measurement, some error may also arise from the Fourier transform process [19].

## 4. Conclusion

Based on a 4-f pulse shaping system, we demonstrated that the first-order time derivative of the pulse electric field, and the first and second-order time derivatives of the pulse envelope can be efficiently realized by modulating the amplitude and phase of the input pulse with an SLM. The generation efficiencies of the time derivatives based on our method are all three times higher while the obtained fidelities maintain comparable to previously reported methods. By simply manipulating the phase and amplitude of the SLM, the higher-order field or envelope derivatives of the pulse can also be readily obtained. Therefore, our method is highly efficient and convenient to implement multiform time derivatives of pulse for various needs of quantum time metrology.

## Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 91336108, Y133ZK1101, 11504292, 11273024), the “Young Top-Notch Talents” Program of Organization Department of the CPC Central Committee, China (Grant No. [2013]33), CAS Frontier Science Key Research Project (Grant No. QYZDB-SSW-SLH007) and Natural Science Basic Research Plan in Shaanxi Province of China (Grant No. 2016JQ1036).

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