## Abstract

Discrete Fourier transform (DFT) plays an important role in digital signal processing. In this paper, we present a novel optical real-time discrete Fourier transformer with complex-valued outputs, which is enabled by the inverse temporal Talbot effect. In the system, an input pulse train is first quadratically phase-modulated as in an inverse temporal Talbot system and then split into two channels. In the first channel, the pulse train is further amplitude-modulated pulse-by-pulse by a discrete data sequence to be transformed. In the second channel, a reference signal modulates the pulse train, which is for removing the residual quadratic phase profile in the output pulse train. The pulse trains in the two channels propagate through a shared dispersion medium with a proper dispersion value determined by the inverse temporal Talbot effect. A 90-degree optical hybrid and two balanced photodetectors are employed to retrieve the real and imaginary parts of the DFT results. In this scheme, the pulse repetition rate of the output pulse train is equal to the input one. In addition, we present a full theoretical framework to explain exactly the DFT relationship. We also demonstrate that the input data sequence can be complex-valued with the help of an I/Q modulator.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The self-imaging phenomenon is a near-field diffraction effect, which is also known as the Talbot effect in honor of its discoverer [1]. When a plane wave propagates through a periodic grating, the same pattern of grating appears at specific distances from the grating. The Talbot effect also occurs in the time domain due to the space-time duality between the paraxial diffraction of beams in space and the chromatic dispersion of pulses in time [2–3]. The temporal Talbot phenomenon can be observed when the period ${T_\textrm{0}}$ of an input pulse train and the dispersion amount $\ddot{\Phi }$ satisfy $|\ddot{\Phi }|= (p/m)T_0^2/(2\pi )$, where *p* and *m* are co-prime numbers [4–6]. When $m = 1$, an integer temporal Talbot phenomenon occurs, in which an exact replica of the input pulse pattern is observed at the output. When $m > 1$, corresponding to the fractional temporal Talbot phenomenon, the repetition rate of the pulse train is multiplied by a factor of *m* while the waveform of each pulse is unchanged if the pulse duration is less than ${T_0}/m$ [7]. In an integer-order pulse pattern, the temporal pulse-to-pulse phase profile is linear or uniform. However, in a fractional-order pulse pattern, there is a quadratic pulse-to-pulse phase profile within each group of *m* pulses in a periodic way despite of a uniform amplitude envelope among all pulses [8]. The fractional temporal Talbot effect provides a potential solution to the generation of optical pulse trains with ultra-high repetition rate [9–11]. It has been reported by Maram et al. that passive amplification of optical pulses can be realized by dividing the pulse repetition rate via a system based on the inverse temporal Talbot effect (ITTE), in which the input pulse train is quadratically phase-modulated within each group of pulses prior to experiencing dispersion [12–14].

In recent years, two interesting optical methods for computing discrete Fourier transform (DFT) based on the temporal Talbot effect were proposed and studied in [15–19]. In the first approach [15–16], an input pulse train is firstly quadratically phase-modulated as in an ITTE-based passive amplification system and then further amplitude-modulated pulse-by-pulse by a discrete data with a length of *m* in a periodic way. The modulated pulse train then propagates through a dispersion medium with a dispersion value defined by ITTE. The output pulse train, whose pulse repetition rate is the same as the input one, is repetitive every *m* pulses and the amplitudes of the pulses in each period are the DFT results of the input discrete data. The second approach [17] is based on the fractional temporal Talbot effect, in which the stage of quadratic phase modulation is removed, the discrete data sequence under test is directly modulated upon the input pulse train, and the dispersion value should satisfy the fractional temporal Talbot condition. In this approach, the output pulse train is also periodic, but its repetition rate is multiplied by a factor of *m*. It was approved that the amplitudes of the pulses in a period are equal to the *m*-point DFT of the input data sequence. This approach provides a potential way to generate pulse trains with ultra-high repetition rate as well as arbitrary and programmable envelope, as demonstrated in [18–19]. However, the multiplication of repetition rate also brings challenges when we apply it to the field of optical computing, especially in the case of large DFT size, since the multiplication factor *m* equals the length of data sequence. In addition, in order to avoid pulse overlapping at the output, the pulse duration of the input pulse train should be less than 1/*m* of the pulse period, also owing to the multiplication of pulse repetition rate [17].

The above schemes based on the temporal Talbot effect provide potential solutions for fast DFT computing. However, the phase information of the DFT results is lost due to the square-law detection of photodetectors, which may be unacceptable since the spectral phase is of great importance in many applications, such as image processing and speech perception [20]. In this paper, we present a novel optical real-time discrete Fourier transformer with complex-valued outputs. In the system, there are two optical channels for realizing ITTE-based DFT. One is for the data input and the other is as the reference for removing the residual quadratic phase profile in the output pulse train. The repetition rate of the output pulse train remains the same as the input one thanks to the quadratic phase modulation in advance determined by ITTE, which is highly desired in the applications with large DFT size. We use a 90-degree optical hybrid and two balanced detectors (BPDs) to retrieve the real and imaginary parts of the DFT results. A concise theoretical framework is presented, which exactly explains the DFT relationship between the input and output pulse trains. We present some numerical results to verify the proposed discrete Fourier transformer with real- and imaginary-part outputs. It is also demonstrated that the input data sequence can be complex-valued with the help of an I/Q modulator.

## 2. Principle

A schematic diagram of the ITTE-based pulse-repetition-rate division system is shown in Fig. 1(a). In the system, the repetition rate of the input pulse train is decreased while the shape of each pulse is maintained, which results in the effect of pulse passive amplification [12–13]. To achieve ITTE, the input pulse train with a period of ${T_\textrm{0}}$ should be quadratically phase-modulated pulse-by-pulse by a phase signal ${\varphi _k} = [(m - 1)/m]\pi {k^2}$, where *m* is a positive integer and ${\varphi _k}$ is the phase modulated upon the *k*-th pulse [21]. Note that in the quadratic phase signal ${\varphi _k} = [(m - 1)/m]\pi {k^2}$, the factor *m*-1 in the numerator is only one of many possible solutions. The advantage of using *m*-1 is that this minimizes the required dispersion amount, and therefore simplifies the experimental setup [21]. After propagating through a dispersion medium with a dispersion value of $mT_0^2/(2\pi )$, a pulse train with a decreased repetition rate (with a factor of $1/m$) and an amplified pulse intensity can be obtained. In this approach, the phase modulation on the pulse train is to produce a periodic waveform at a fractional Talbot distance and the dispersive propagation further converts the waveform at the fractional Talbot distance into the one at an integer Talbot distance. Pulse repetition rate division and passive amplification of pulse intensity are thus realized. The proposed system enabled by ITTE for realizing optical real-time DFT with complex-valued outputs is shown in Fig. 1(b). In the system, an input pulse train is firstly phase-modulated quadratically; and then the phase-modulated pulse train is split into two channels via a coupler. In the first channel, the pulse train is further amplitude-modulated by a discrete data sequence $\{ {c_0},\textrm{ }{c_1},\textrm{ } \cdot{\cdot} \cdot ,\textrm{ }{c_{m - 1}}\} $ with a length of *m* pulse-by-pulse repetitively; while in the second channel, it is modulated by the data sequence $\{ 1,\textrm{ }0,\textrm{ }0,\textrm{ } \cdot{\cdot} \cdot \} $ (also with a length of *m*) in a periodic way. Both the pulse trains propagate through a dispersion medium with a dispersion amount of $mT_0^2/(2\pi )$, which is shared by the two channels with the help of two circulators. The output pulse trains are connected to the two input ports of a 90-degree optical hybrid. The four output signals from the hybrid are sent to two balanced detectors (BPDs). The output electrical signals from the two BPDs are proportional to the real and imaginary parts of the DFT of $\{ {c_0},\textrm{ }{c_1},\textrm{ } \cdot{\cdot} \cdot ,\textrm{ }{c_{m - 1}}\} $, respectively, which will be discussed in the following paragraphs. In this system, the repetition rate of the output pulse train is the same as the input one, i.e. $1/{T_\textrm{0}}$, and complex-valued DFT results can be obtained, which is different from the previous reports [15–19].

We assume the input pulse train with a repetition rate of $1/{T_\textrm{0}}$ has equal amplitude and phase, the complex amplitude of which is represented by

where $g(t)$ denotes the individual pulse in a period. The input pulse train is firstly modulated by a discrete phase signal $\exp (j{\varphi _l})$, where ${\varphi _l} = [(m - 1)/m]\pi {l^2}$ is assumed constant within the duration of each pulse. Note that $\exp (j{\varphi _l})$ is periodic with period*m*. The phase-modulated pulse train is expressed as

*m*. The pulse train ${e_M}(t)$ with both the phase and amplitude modulation can be written as

Then, the modulated pulse train propagates through the dispersion medium with a dispersion value $\ddot{\Phi }$ which satisfies the condition $|\ddot{\Phi }|= mT_0^2/(2\pi )$. The frequency response of the dispersion medium, with ignoring the constant and linear phase terms, is given by $H(\omega ) = \exp ( - j\ddot{\Phi }{\omega ^2}/2)$. Therefore, the spectrum of the output signal can be expressed as

By observing (6), ${X_1}(\omega )$ can be regarded as the output spectrum of a system, where an input pulse train $\sum\limits_{k ={-} \infty }^\infty {g(t - km{T_0})} $ with a period of $m{T_0}$ propagates through a dispersion medium with a dispersion value $\ddot{\Phi }$. It is found that the pulse period $m{T_0}$ and dispersion amount $mT_0^2/(2\pi )$ exactly satisfy the condition of the fractional temporal Talbot effect with an order of $1/m$ [4,8]. According to the knowledge on the Talbot effect, the pulse repetition rate is multiplied by a factor of *m* and there is a pulse-to-pulse quadratic phase profile within each pulse group with *m* pulses. Therefore, the time-domain signal ${x_1}(t)$ can be expressed as

*s*is decided by the parameter $1/m$ of the fractional Talbot effect, that is [8]

*m*, ${x_1}(t)$ is periodic with period $m{T_0}$.

The other time-domain signal ${x_2}(t)$ can be obtained by applying the inverse Fourier transform to ${X_2}(\omega )$ in (7), which is

In the convolution operation between ${x_1}(t)$ and ${x_2}(t)$, the term $\delta (t - l{T_0})$ in ${x_2}(t)$ means applying a time shift of $l{T_0}$ to ${x_2}(t)$. Let us first consider the case of even *m*. By noticing the periodicity of ${x_1}(t)$, the equation $\exp [j{(m - q)^2}\pi /m] = \exp (j{q^2}\pi /m)$ holds true for any integer *q* if *m* is even, the output signal ${e_D}(t)$, as the result of the convolution operation, can be expressed as

For odd *m*, since we have the equations $\exp [j(m + 1){l^2}\pi /m] = {( - 1)^l}\exp (j{l^2}\pi /m)$ and $\exp [j{(m - l)^2}\pi /m] ={-} \exp (j{l^2}\pi /m)$, the convolution result can be expressed by the following formula,

*m*is odd.

So far, we have proved that, under the condition of quadratic phase modulation and dispersion value required in the ITTE system, the amplitude of the output pulse train is the *m*-point DFT of the input sequence, while the pulse repetition rate keeps unchanged. However, according to (11) and (12), there is an additional phase profile $\exp (j{k^2}\pi /m)$ besides the amplitude profile ${\hat{C}_k}$ or ${( - 1)^k}{\hat{C}_k}$. In order to eliminate the influence of this residual phase profile, we modulate a reference data sequence $\{ {r_k}\} = \{ 1,0,0,0 \cdot{\cdot} \cdot{\cdot} \}$ (also with a length of *m*) periodically on the phase-modulated pulse train ${e_P}(t)$ in the second channel. Since the DFT of $\{ {r_k}\}$ is $\{ 1,\textrm{ }1,\textrm{ }1, \cdot{\cdot} \cdot{\cdot} \}$, the output pulse train after passing the dispersion medium can be obtained according to (12), which is

*m*is odd. The optical signals at the four output ports of the optical hybrid are as ${e_D}(t) + {e_R}(t)$, ${e_D}(t) - {e_R}(t)$, ${e_D}(t) + j{e_R}(t)$ and ${e_D}(t) - j{e_R}(t)$, respectively. The electrical signals from the two BPDs can be expressed as

*m*.

## 3. Results and discussions

Now we present numerical examples to demonstrate the real-time DFT with complex-valued outputs and verify the above theoretical results. In the simulations, the period of the input pulse train is set to be 0.03 ns and each pulse is Gaussian-shaped as $\exp [ - {t^2}/(2\tau _0^2)]$, where the pulse width ${\tau _0}$ is set to 3.3 ps. The phase modulation upon pulses for realizing the ITTE is set according to ${\varphi _k} = [(m - 1)/m]\pi {k^2}$ and the dispersion amount is set according to $\ddot{\Phi } = mT_0^2/(2\pi )$. In the first example, the parameter *m* is set to be 7. The input data sequence $\{ {c_l}\} $ is set to be [1, 0.5, 0.9, 0.4, 0.8, 0.3, 0.7] and the reference data sequence is [1, 0, 0, 0, 0, 0, 0]. The input pulse train in amplitude is shown in Fig. 2(a). We give the both output electrical signals from the two BPDs by simulations, i.e. the real and imaginary parts, in Figs. 2(b) and 2(c), respectively. The results by direct DFT calculation are also shown. It is seen that the simulation and calculated results match with each other perfectly. In the second example, the parameter *m* is chosen to be an even number, i.e. *m*=8. The input data sequence $\{ {c_l}\} $ with 8 data is set to be [1, 0.9, -0.8, -0.7, -0.6, -0.5, -0.4, 0.3] and the reference signal is [1, 0, 0, 0, 0, 0, 0, 0] in this case. Note that the negative modulation coefficient can be realized by biasing the external modulator at its minimum transmission point [19]. The input pulse train is shown in Fig. 2(d) and the output real and imaginary parts are shown in Figs. 2(e) and 2(f), respectively. Again, the numerical and theoretical results match perfectly, which verifies the correctness of the proposed scheme for real-time DFT with complex-valued outputs. In both the examples, the repetition rate of the input and output pulse trains is the same as expected from the theoretical results.

In a real-time DFT system without the reference channel and with a single-ended photodetector [15–16], the output signal contains only the magnitude information of the DFT results. In this case, if the input data sequence is restricted to be real-valued numbers, the envelope of the output pulse train is always symmetrical due to the square-law detection, which is decided by the property of Fourier transform [19,22–23]. In our proposed system, the complex-valued information of the DFT results, with the real and imaginary parts, can be obtained thanks to the employment of the reference channel, the optical hybrid as well as the BPDs. In fact, the input data sequence $\{ {c_l}\} $ could be complex-valued signals, as we did not restrict $\{ {c_l}\} $ to be real in the above analysis. To modulate complex signals on a pulse train, we may use an I/Q modulator or the cascade of phase and amplitude modulators [17–19]. Here we present two examples to demonstrate the complex signal modulation. Figure 3(a) shows the pulse train modulated by a complex-valued data sequence, where *m*=7. The input complex data sequence $\{ {c_l}\} $ is [1, -0.3+*j*0.3, -0.4-*j*0.4, -0.5-*j*0.5, -0.5+*j*0.5, -0.4-*j*0.4, 0.3-*j*0.3] and the reference data sequence is [1, 0, 0, 0, 0, 0, 0]. Figures 3(b) and 3(c) give the real and imaginary parts of the output complex-valued signal, respectively, where both the simulation and calculation results are given. Figures 3(d), 3(e) and 3(f) correspond to the case of *m*=8, where $\{ {c_l}\} $ is set to be [1, -0.5-*j*0.5, 0.5-*j*0.5, -0.5+*j*0.5, 0, 0, 0.6+*j*0.8, 0]. Again, there is also no visible difference between the output waveforms by simulation and predicted by DFT calculation.

In the above simulations, the input pulses are all assumed Gaussian-shaped. We find that the real-time DFT function keeps unchanged if the pulses are not Gaussian, such as the pulse shape modeled by the Dirichlet kernel function. On the experimental demonstration of the proposed scheme, there are two major challenges. One is the quadratic phase modulation and the other is the dispersion medium for different *m*. The quadratic phase signal ${\varphi _k} = [(m - 1)/m]\pi {k^2}$ can be generated by a high-speed arbitrary waveform generator, as in [12]. As for the dispersion medium, it is not practical to use different dispersion medium for different length of data. We may determine a maximum *m* for a given system with a fixed dispersion value. Any input data shorter than *m* should be extended to the length of *m* by zero padding. It can avoid the use of adjustable dispersion medium.

Since the DFT is fundamental in many applications with digital signal processing, the proposed optical real-time DFT may play an important role in optical computing and optical signal processing. The major advantage of the given approach to realizing DFT lies in its capability of complex-valued signal in and out, while the pulse repetition rate keeps unchanged. With lower repetition rate, the output pulses would contain more energy with respect to the approaches, where the repetition rate was multiplied in the output [17–19]. In addition, the pulse repetition rate multiplication would make the output pulse repetition rate extremely high if the DFT size is in magnitude of hundreds or above. It is interesting to compare the discussed real-time DFT with the phenomenon of frequency-to-time mapping (FTTM) based on dispersive propagation, which also realizes the real-time Fourier transform between the input and output waveforms [24–26]. FTTM realizes the scaled Fourier transform of continuous signals with a finite duration, while the real-time DFT discussed here realizes the DFT of any discrete data sequence with arbitrary length.

## 4. Conclusion

We have proposed a real-time discrete Fourier transformer with complex-valued outputs. A concise analytical model with both the time- and frequency-domain analysis is presented to exactly explain the DFT relationship. Owing to the employment of ITTE, the periodic modulation of data signal does not multiply the pulse repetition rate, which is highly desired in the DFT computing with large data size. In the system, we use a reference channel to remove the residual quadratic phase profile in the output pulse train. A 90-degree optical hybrid and two BPDs are employed to obtain the real and imaginary parts of the DFT results. Some numerical results have been presented to verify the proposed transformer. It is shown that the input data can be either real or complex numbers. The proposed discrete Fourier transformer is a potential powerful candidate for optical computing as it preserves the complete complex information of the DFT results.

## Funding

National Key Research and Development Program of China (2019YFB2203204); National Natural Science Foundation of China (61975048, 61901148, 41905024); Natural Science Foundation of Zhejiang Province (LZ20F010003, LQ18F050002); Department of Education of Zhejiang Province (Y201635688).

## Disclosures

The authors declare no conflicts of interest.

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