Investigation of temporal Talbot operation in a conventional optical tapped delay line structure

Jianqi Hu; Simon J. Fabbri; Chen-Bin Huang; Camille-Sophie Brès

doi:10.1364/OE.27.007922

1. Introduction

High repetition-rate optical pulse trains have been instrumental to a variety of applications. For instance, they are widely used in high speed optical time-division-multiplexing (OTDM) systems as data carriers, and similarly at the demultiplexer end to all-optically down-sample the OTDM signals [1]. Pulse trains of high rate distribute data to base stations in radio-over-fiber (RoF) links, and the data carried by optical pulses is translated to the electrical carrier after photodetection [2]. Moreover, they bridge optical frequency and radiofrequency (RF) domains [3–7], opening new pathways for RF signal generation of ultrahigh frequency [4], arbitrary waveform [5], and rapid reconfigurability [6], as well as filtering [7], ranging from millimeter-wave (MMW) to THz.

In particular, multiplying the repetition-rate of optical pulse trains outside their laser cavity is widely demanded. This is attractive when the normal mode-locking techniques or external modulation are not readily available for the target rate [8]. Spectrally, periodic optical pulse train corresponds to a series of equally spaced frequency components. Thus, repetition-rate multiplication (RRM) can be achieved through either one of the following approaches: spectral amplitude filtering [9–11], spectral phase-only filtering [12–16], or a combination of both [17]. In the case of spectral amplitude filtering, RRM results from the increased spectral spacing, generally through amplitude shaping in pulse shaper or various delay line structures [9–11]. For spectral phase-only filtering, or equivalently temporal fractional Talbot effect, the pulse energy is redistributed equally into its temporal self-images inside each period, with the spectral shape of the initial pulse train unaltered. This is in principle an energy preserving process, unlike the spectral amplitude filtering. The temporal Talbot effect has been utilized to generate THz wave beating in an ultrahigh frequency photodiode [4], and could also mitigate the self-phase modulation of an ultrashort pulse delivery along optical fibers [18]. Furthermore, temporal Talbot effect has also been utilized to demonstrate exotic phenomena, such as noiseless pulse amplification [19], and recently spectral cloaking [20].

Typical methods to implement temporal Talbot effect are through a spool of optical fiber, or linearly chirped fiber Bragg grating (LCFBG) [12], based on the first-order dispersion. Optical line-by-line pulse shaper [13,14], instead of propagation, carries out the task in the frequency domain by directly associating frequency components with a predefined Talbot phase pattern. Ring-resonator-based filter structures of various types [15,16] have also been proposed to achieve temporal self-imaging in a compact configuration, with stringent control of both coupling coefficient and round-trip phase of each resonator.

The optical tapped delay line structure with fast modulator embedded in each arm has been widely used as transmitter architecture for OTDM systems [21]. Here, we show that it is possible to implement temporal Talbot effect in a conventional optical tapped delay line structure, same as the one used for spectral amplitude filtering. To the best of our knowledge, temporal Talbot effect has not been investigated in the past in such a simple architecture. We need to point out here the Talbot effect in the proposed scheme possesses intrinsic loss as spectral amplitude filtering, unlike the other Talbot demonstrations mentioned above. This is because the temporal Talbot achieved in tapped delay line structure is based on the coherent interference of multiple delayed signals. However, multiple Talbot pulse trains featuring lossless operation can be implemented simultaneously in such a structure. Additionally, different combination of temporal Talbot effect (spectral phase filtering) and spectral amplitude filtering is also demonstrated by simulation, all eventually resulting in the same multiplication factor. We theoretically derive and numerically simulate the working principle of the concept. We also confirm the proposed approach through a proof-of-concept experiment with a commercial Mach-Zehnder delay line interferometer (DLI), which is equivalent to the simplest version of the proposed structure. We address the efficiency, as well as the potential degradation of the temporal Talbot arising from the structure’s power imbalance, delay line length inaccuracy, and the applied phase error. We conclude by comparing the proposed scheme to a conventional 4-f pulse shaper and LCFBG in terms of performing temporal Talbot effect.

2. Principle of operation

2.1 Optical tapped delay line structure

Figure 1 depicts the optical tapped delay line structure proposed for temporal Talbot effect. It consists of N parallel delay lines with incremental delay T₀ from 0 to (N-1)T₀, each embedded with a phase tuning element of applied phase ϕ_n (n = 0, 1, 2,…, N-1). Here the delay T₀ is chosen to be 1/N of the input optical pulse period. We suppose that the splitter used at the input and combiner at the output equally splits and collects light with no additional phase relation between the arms: the additional phase in each arm can be compensated through the corresponding phase tuning element without loss of generality. This architecture is well-known and amenable to photonic integration, and is indeed a simplified version of the photonic integrated circuits demonstrated in [9,22], albeit without optical power control of each arm. Note that phase tuning elements are required in the proposed structure in order to shape both the waveform and spectrum of the input pulse train. This makes it different from the case without phase control between arms [23], when only the temporal RRM is the target. The transfer function of the proposed structure is given by,

H (f) = \frac{1}{N} \sum_{n = 0}^{N - 1} \exp (- i 2 π n f T_{0} - i ϕ_{n})

It can be seen that the filter response is periodic with a free spectral range (FSR) equal to 1/T₀. Before discussing the temporal Talbot operation of the proposed structure, we first review the well-known spectral amplitude filtering as a comparison.

Fig. 1 Schematic diagram of the optical tapped delay line structure. T₀ is the unit delay time, and ϕ_n (n = 0, 1, 2,…, N-1) are the phase applied to phase tuning elements.

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2.2 Spectral amplitude filtering

When all the phase shifters are set in-phase, for example ϕ_n = 0 (n = 0, 1, 2,…, N-1), the filter response becomes a periodic sinc function [9,21]. This was employed in all-optical demultiplexing of orthogonal frequency division multiplexing (OFDM) systems [10]. For instance, the specific case N = 8 is shown in Fig. 2(a) with transmission at the top and phase response at the bottom. It can be seen that the filter suppresses all the frequency components at every 1/N (grey markers) of FSR, except maximum transmission at every integer FSR (red markers). Note that the overshoot spikes in the phase response of the filter are artifacts at zero amplitudes. Therefore, if a pulsed source with repetition-rate FSR/N is sent through the filter such that comb line locations of the corresponding spectrum are aligned to the filter FSR, the comb spacing will be increased N-times resulting in an N-times multiplication of the repetition-rate. This is shown in Fig. 2(b). We take a periodic Gaussian pulse train with full-width-half-maximum (FWHM) that corresponds to 1/25 of its period as the input. The pulse repetition-rate at the output of the filter is multiplied 8-times due to the increased comb spacing.

Fig. 2 (a) Power (top) and phase (bottom) response of the spectral amplitude filtering, when the phase shifters are configured all in-phase, [0,0,0,0,0,0,0,0]. (b) Optical spectra (top) and temporal waveforms (bottom) at the input and output the filter.

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2.3 Spectral phase-only filtering

Spectral phase-only filtering, or temporal Talbot effect, is characterized by RRM in the time domain while keeping the spectral shape. Consider that the phase sequence applied to the phase tuning elements follows the Talbot phase relation, one of which is given by [15]:

ϕ_{n} = {\begin{matrix} \frac{π n^{2}}{N} & if N \equiv 0 (\mod 2) \\ \frac{2 π n^{2}}{N} & if N \equiv 1 (\mod 2) \end{matrix}

where the actual phases applied are residues of ϕ_n (n = 0, 1, 2,…, N-1) after modulo 2π. As such, identical transmission self-images inside each FSR of the filter transfer function are created. In this case, the spectral comb structure of an input pulse train is maintained, when the comb lines are aligned with the filter FSR. By substituting Eq. (2) to Eq. (1), this can be formulated by,

\begin{matrix} H (f_{k} = \frac{k}{N T_{0}}) = \frac{1}{N} \sum_{n = 0}^{N - 1} \exp (- i \frac{2 π n k}{N} - i ϕ_{n}) = \frac{1}{\sqrt{N}} \exp (- i θ + i s ϕ_{k}) \\ = {\begin{matrix} \frac{1}{\sqrt{N}} \exp (- i \frac{π}{4} + i ϕ_{k}) & if N \equiv 0 (\mod 2) \\ \frac{1}{\sqrt{N}} \exp (i \frac{3 N + 1}{4} ϕ_{k}) & if N \equiv 1 (\mod 4) \\ \frac{1}{\sqrt{N}} \exp (- i \frac{π}{2} + i \frac{N + 1}{4} ϕ_{k}) & if N \equiv 3 (\mod 4) \end{matrix} \end{matrix}

where f_k = k/(NT₀) are the frequency locations of the input comb line (k is the comb index with respect to the center carrier). θ and s are respectively the constant phase and an multiplication factor of the Talbot phase defined in Eq. (2). They correspond to π/4 and 1 if N is an even number, 0 and (3N + 1)/4 if N≡1 (mod 4), π/2 and (N + 1)/4 if N≡3 (mod 4). Note the multiplication factor (3N + 1)/4 when N≡1 (mod 4) or (N + 1)/4 when N≡3 (mod 4) is integer, and it is easy to confirm that in both situations coprime with N. The proof of the second and third equality is given in the Appendix of the paper. This is related to extended quadratic Gauss sum [24,25], and is of similar mathematical foundation as temporal Talbot array illuminator [26]. Indeed, this can also be traced back to the phase relation of Talbot self-images in multimode interference (MMI) couplers [27]. Here the varying part of the filter phase response, sϕ_k, complies again the Talbot phase (N is even) or multiple of Talbot phase (N is odd) given in Eq. (2). As long as the multiplication factor s is coprime with N, sϕ_k are also the phase sequence fulfilling the Talbot condition [24]. Thus, in addition to the same transmission for all the input frequency components, the relative phase response of the filter at each comb line results in temporal Talbot effect of input pulse train, leading to N-times RRM.

A more straightforward explanation is from the time domain. As discussed in section 2.1, the tapped delay line structure with phase shifters tuned all in-phase enlarges the spectral spacing by N-times. Then if we configure the phase shifter corresponding to the Talbot phase, the spectral Talbot effect would occur, converting the spectrum with increased spacing back to the original shape as the input. This is similar to spectral self-imaging achieved by periodic multilevel Talbot phase modulation of pulse train, through arbitrary waveform generator (AWG) [19,28,29]. Here the phase modulation of pulse train is implemented through phase tuning element associated to each copy of initial pulse train. Figure 3 shows again the case for N = 8 in this regard, and we take the same pulse train as input to the proposed structure. The phases applied to the phase shifters in this case are Talbot phase of N = 8, i.e. [0,π/8,π/2,9π/8,0,9π/8,π/2,π/8]. As shown in Fig. 3(a), the phase response of the filter (red markers) is again of periodic [0,π/8,π/2,9π/8,0,9π/8,π/2,π/8], exactly matching the phase required for temporal Talbot effect, with equal amplitude transmission (red markers). Note that θ is omitted for illustration purpose here and afterwards, as a constant phase will not change the waveform. The current result is completely different from spectral amplitude filtering. It can be clearly seen in Fig. 3(b) that though the repetition-rate of the pulse train is 8-times multiplied as for spectral amplitude filtering, the spectral shape remains identical to the input spectrum.

Fig. 3 (a) Power (top) and phase (bottom) response of the spectral phase-only filtering, when the phase shifters are configured by Talbot phase of N = 8, [0,π/8,π/2,9π/8,0,9π/8,π/2,π/8]. (b) Optical spectra (top) and temporal waveforms (bottom) at the input and output the filter.

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2.4 Combination of spectral amplitude and phase filtering

In addition, the combination of spectral amplitude and phase filtering [17], in between pure amplitude and pure phase-only filtering, can also be realized in the proposed structure enabling the same N-times RRM. As an illustration, 8-times RRM can be synthetized by combining either 2-times amplitude filtering with 4-times Talbot effect or 4-times amplitude filtering with 2-times Talbot effect, as shown in Figs. 4(a) and (b), respectively. Figure 4(a) illustrates the case when the phase shifters are configured according to the repeated Talbot phase of N = 4, i.e. [0,π/4,π,π/4,0,π/4,π,π/4], 2-times amplitude filtering is realized due to 2-times repetition of applied phase to the phase shifters. This is characterized both from the transmission profile (red/grey markers) of the filter response and the doubled comb spacing at the filter output. Moreover, the phase response at the remaining comb line follows the periodic Talbot phase relation [0,π/4,π,π/4], so that 4-times temporal Talbot effect is simultaneously achieved. Therefore, 8-times RRM is observed at the output of the filter while the spectrum appears as a frequency comb with comb line separation that is only doubled. Similar condition applies to Fig. 4(b) for 8-times RRM, when the phase shifters are configured to repeated Talbot phase sequence of N = 2, i.e. [0,π/2,0,π/2,0,π/2,0,π/2]. In general, N-times RRM can be synthesized in the proposed architecture through arbitrary integer factor of N by spectral amplitude filtering, together with the complementary factor by temporal Talbot effect. The strategy for the phase applied is the same as the example herein.

Fig. 4 Left: power (top) and phase (bottom) response of the spectral phase and amplitude filtering; Right: optical spectra (top) and temporal waveforms (bottom) at the output the filter. The phase shifters are configured by repeated Talbot phase of (a) N = 4, [0,π/4,π,π/4,0,π/4,π,π/4] and (b) N = 2, [0,π/2,0,π/2, 0,π/2,0,π/2].

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3. Results and analysis

3.1 Proof-of-concept test with delay line interferometer

A proof-of-concept test is carried out with a commercial Mach-Zehnder DLI (KYLIA mint-1x2-U-10GHz), which is functionally equivalent to the proposed structure of N = 2. The delay time between the two arms of interferometer is fixed at 100 ps, thereby is generally used as demodulator for 10 GHz differential-phase-shift-keying (DPSK) signal. A piezo phase shifter is embedded in one arm to control the phase relation between the two arms of the interferometer. Here we demonstrate that a 10 GHz pulse train can be generated using the DLI from a 5 GHz pulse train through 2-times temporal Talbot effect.

Figure 5(a) is the experimental setup. A 5 GHz repetition-rate, 21-line comb source of rectangular shape is used as the input light source to the DLI, as shown in Fig. 5(b). It is synthesized from two modulators in cascade [21]. The first MZM is driven by synchronized multi-harmonic 5 GHz, 10 GHz, and 15 GHz RF signals [30] to generate 7-line comb, while the second modulator is driven by synchronized 35 GHz RF clock to triple the comb line number to 21. The spectra and waveforms are measured by an optical spectrum analyzer (OSA) and an optical sampling oscilloscope (OSO), respectively. The embedded phase shifter is tuned so that the output spectrum is identical to the input spectrum. We can see that for both two output ports of DLI, the pulse repetition-rates are doubled in the time domain, while the spectra maintain the original shape. This is temporal Talbot effect, confirming the state of operation proposed in the simple case of N = 2. The asymmetrical pulse behavior in the time domain is mainly attributed to the power imbalance of the test device, or equivalently extinction ratio of the DLI (nominal 18dB). Besides, the power discrepancy between input and each of the two outputs is roughly 5 dB, due to a 2 dB nominal insertion loss of the DLI and 3 dB because of two outputs with equal power.

Fig. 5 (a) Experimental setup. MZM: Mach-Zehnder modulator; φ: electrical phase shifter; DLI: delay line interferometer; OSA: optical spectrum analyzer; OSO: optical sampling oscilloscope. (b) Optical spectra (top) and temporal waveforms (bottom) measured at the input and two output ports of the DLI.

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3.2 Efficiency

The efficiency of performing temporal Talbot effect in the proposed structure, however, is not a lossless process due to interference. In this section, we investigate the efficiency of Talbot implementation in this condition. Suppose the input signal is a comb source of K-lines spaced by a repetition-rate of f_m, the optical field is given by,

E_{i n} (f) = \frac{E_{0}}{\sqrt{K}} \sum_{k = 0}^{K - 1} A (k) δ (f - k f_{m})

where A(k) is the relative amplitude of the k-th frequency component. Thus, the total power of the input source is

\frac{{| E_{0} |}^{2}}{K} \sum_{k = 0}^{K - 1} {| A (k) |}^{2}

. When the comb lines are aligned with filter FSR, the output field from the proposed structure in the Talbot condition can be calculated,

E_{o u t} (f) = E_{i n} (f) H (f) = \frac{E_{0}}{\sqrt{K N}} \sum_{k = 0}^{K - 1} A (k) \exp (- i θ + i s ϕ_{k}) δ (f - k f_{m})

where ϕ_k, θ and s are respectively the Talbot phase, constant phase and integer multiplication factor as previously defined. The total power at the output of the structure is therefore

\frac{{| E_{0} |}^{2}}{K N} \sum_{k = 0}^{K - 1} {| A (k) |}^{2}

, which corresponds to 1/N of the input power. Indeed, the power of each frequency component is reduced by N-times, while the overall shape is maintained. The loss inevitably scales up for large number RRM by temporal Talbot effect. When N is small (e.g. N≤4), however, the efficiency of proposed method is comparable to 4-f pulse shaper technique [13,14,17]. Nevertheless, this efficiency is similar to performing spectral amplitude filtering in the same structure, where every one out of N frequency components are selected.

Moreover, if N × N discrete Fourier transform (DFT) network [31] is installed as the output coupler, N temporal Talbot RRM pulse trains instead of one can be accessed simultaneously. In fact, the N × N DFT network has been utilized in spectral amplitude filtering to direct different frequency spectra to different ports, hence outputs N optical pulse trains with N-times RRM. Similarly, the incremental linear phase imposed by the DFT network for each output port would incrementally shift the Talbot pulse train by 1/T₀ in time domain. Thus, all the N output pulse trains are time translated replica of initial Talbot pulse train. When all these Talbot pulse trains are considered as a whole, the optical power is conserved. This is experimentally shown above in the case of N = 2. Both outputs of the DLI exhibit 2-times Talbot pulse train, making the overall structure in principle lossless if insertion loss is subtracted.

3.3 Performance analysis

As seen in the preliminary test result with a DLI, the output signal deviates from ideal temporal Talbot effect. To assess the performance of the proposed temporal Talbot multiplier, we evaluate the possible degradation due to the power imbalance and delay line length inaccuracy of the structure, together with the phase error applied to the phase shifters. Here we consider tapped delay line structure in integrated platform. The deviation of all these factors are investigated separately based on empirical values of Triplex technology [9,32,33]. As for other platforms like silicon or InP, the analysis can be carried out in the same way.

The power imbalance arises from both input and output couplers, as well as the unequal loss in each arm due to the delay line length difference. Take 8-parallel delay line as an example. We assume here 1% standard deviation of power imbalance for 50/50 coupler, which is used as building block for both input and output couplers. The power imbalance induced from delay length difference depends on the unit delay length. Obviously it scales up for long delay line length. Here we target for temporal Talbot RRM of 10 GHz pulse train into 80 GHz. Since the group refractive index of Si₃N₄/SiO₂ waveguide at 1550 nm is 1.71 with propagation loss of 0.15 dB/cm [9,32,33], the incremental loss between adjacent arms can be estimated to be around 0.033 dB. Note that the delay line length inaccuracy is not considered in this part as it is negligible compared to delay line length difference. Figure 6(a) illustrates 200 superimposed traces with deviation defined above. As previous examples, the same input pulse train is launched to the proposed structure. The multiplied pulse train deviates from unitary intensity by up to 32.6%, indicated by both the superimposed pink markers in filter response, and pink traces in waveform and spectrum plots. We confirm here the non-uniform pulse intensity generated by the DLI is mainly due to the power imbalance of the structure. Additional tunable Mach-Zehnder couplers [9] or variable in-line attenuators can be adopted in the structure to tackle the imbalance. This would be necessary to improve the performance of temporal Talbot effect for sub-GHz pulse train, as power imbalance increases with the delay line length difference.

Fig. 6 Left to right: Power and phase response of the proposed filter, output optical spectra and temporal waveforms in the presence of (a) power imbalance (b) delay line length inaccuracy, and (c) phase error. The pink markers and traces correspond to 200 superimposed variations with predefined standard deviation.

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Another factor that would degrade the performance is the delay line length inaccuracy. The filter response is no longer periodic in this scenario. Note that here the delay line length inaccuracy can be minimized by iteratively optimizing the design of each delay line. We assume here the 8 delay lines possess 1% standard deviation of the unit delay line length. The corresponding filter response, together with the output pulse and spectrum variation are shown in Fig. 6(b). It can be clearly seen that the filter response, here plotted over multiple FSR for better illustration, is non-periodic and varies more significantly away from the center frequency of the input signal, when the applied phase compensates the length inaccuracy induced phase for the center frequency [34]. Though the filter response diverges significantly from its ideal shape, the 200 superimposed output waveforms only show up to 3.3% variation. However, the output spectra are altered non-uniformly thereby deviating from ideal temporal Talbot condition. Since most of the power lies in the center part of the filter, the waveform shape is well-maintained. Nevertheless, this would inevitably pose a hurdle for pulse train with broad bandwidth. For non-Gaussian shaped pulse, for example, sinc pulse with similar pulse width, the deviation is 15.3%, larger than the Gaussian pulse. This is due to the fact that the two-ends of the rectangular spectrum possess the same amplitude as the center frequency components, hence suffering quite a lot variation.

The error of applied phase to the phase tuning elements can also degrade the implementation of the temporal Talbot effect. Note the phase error here has nothing to do with the structure itself, but the applied phase variation. We take into account here standard deviation of 2 degrees from the ideal Talbot phase. Figure 6(c) shows the filter response, multiplied pulse train and spectrum of 200 superimposed traces. As we can see, the filter response is indeed varying as indicated by the pink markers, however the pulse train is repetition-rate multiplied with equal amplitude. Thus, RRM of pulse train is unaffected by the presence of applied phase error, once the initial pulse width is narrow enough so that will not overlap each other after RRM. This can be understood from the nature of tapped delay line structure: shifting and recombining the optical pulse in time domain. However, this is not exactly the temporal Talbot effect as the spectrum is reshaped by the filter response. Also note that when the input pulse is a sinc pulse with similar pulse width, 4.3% deviation is observed unlike zero in the case of Gaussian pulse. This is because the sinc pulse possesses sidelobes causing interference, while Gaussian pulse is sidelobe-free.

4. Conclusions

In conclusion, a novel concept of performing temporal Talbot effect is proposed. The scheme is based on an optical tapped delay line structure, which is conventionally used for spectral amplitude filtering. We demonstrate that, temporal Talbot effect and combined amplitude and phase filtering can be synthesized in such an architecture, leading to the same RRM factor. The working principle is theoretically derived and numerically simulated, also confirmed by a proof-of-concept experiment. In addition, we evaluate the efficiency and potential performance degradation of proposed scheme.

The proposed approach enables a new regime for temporal Talbot effect and versatile regimes of optical pulse train RRM. Compared to its spatial light modulator (SLM) based pulse shaper counterpart, the method features several advantages in terms of temporal Talbot implementation. The proposed scheme benefits from the periodic nature of its transfer function, thereby eliminating the use of SLM to apply periodic Talbot phase. And it is particularly appealing to multiply relatively low repetition-rate (sub-GHz) pulse train, which is not easily accessible by 4-f pulse shaper of limited spectral resolution. Most importantly, while 4-f pulse shaper is generally bulky, the structure is ready to be integrated to all kinds of platforms in a simple and compact manner. Compared to the LCFBG, the tapped delay line structure is able to implement complex spectral amplitude and phase filtering functions. Also the working bandwidth of the proposed structure is generally larger than the LCFBG. Besides, the performance of temporal Talbot effect achieved by LCFBG is vulnerable to higher-order dispersion. Nevertheless, loss is intrinsic to the proposed scheme as well as all the deviation factors discussed in section 3.3. Overall, it provides a complementary approach for temporal Talbot effect which is amenable to photonic integration.

5 Appendix

In order to prove the second/third equality of Eq. (3), we first introduce the generalized Landsberg-Schaar identity [35, Eq. (2.8)],

\sum_{n = 0}^{N - 1} \exp (- i π \frac{n^{2} l + n m}{N}) = \sqrt{\frac{N}{l}} \exp (i π \frac{m^{2} - N l}{4 N l}) \sum_{n = 0}^{l - 1} \exp (i π \frac{n^{2} N + n m}{l})

where m is an integer, N and l are positive integers satisfying Nl + m is an even number. We divide the proof of Eq. (3) into three cases: 1) N is an even number; 2) N≡1 (mod 4); 3) N≡3 (mod 4).

1) N is an even number
When N is an even number, substitute m = 0, l = 1 (satisfies Nl + m is an even number) into Eq. (6), we can derive the following identity,
$\sum_{n = 0}^{N - 1} \exp (- i \frac{π n^{2}}{N}) = \sqrt{N} \exp (- i \frac{π}{4})$
This can be inserted to Eq. (3) so that it becomes,
$\begin{matrix} \sum_{n = 0}^{N - 1} \exp (- i \frac{2 π n k + π n^{2}}{N}) = \exp (i \frac{π k^{2}}{N}) \sum_{n = 0}^{N - 1} \exp (- i \frac{π {(n + k)}^{2}}{N}) \\ = \sqrt{N} \exp (- i \frac{π}{4} + i \frac{π k^{2}}{N}) \end{matrix}$
Note that the quadratic sum of n and (n + k) is the same in Eq. (8). Therefore, we prove Eq. (3) in the condition of an even number N.
2) N≡1 (mod 4)
When N is an odd number, let m = 2k (k is an integer), l = 2 (also satisfies Nl + m is an even number) in Eq. (6), we derive that,
$\sum_{n = 0}^{N - 1} \exp (- i π \frac{2 n^{2} + 2 n k}{N}) = \sqrt{\frac{N}{2}} \exp (i \frac{π k^{2}}{2 N} - i \frac{π}{4}) (1 + \exp (i π \frac{N + 2 k}{2}))$
Here Eq. (9) is valid whenever N is an odd number, so it holds for both N≡1 (mod 4) and N≡3 (mod 4) cases. When N≡1 (mod 4),
$1 + \exp (i π \frac{N + 2 k}{2}) = {\begin{matrix} \sqrt{2} \exp (i \frac{π}{4}) & if k \equiv 0 (\mod 2) \\ \sqrt{2} \exp (- i \frac{π}{4}) & if k \equiv 1 (\mod 2) \end{matrix}$
Note that
$\exp (i π \frac{k^{2}}{2 N}) = {\begin{matrix} \exp (i π \frac{k^{2}}{2 N} + i \frac{3 π}{2} k^{2}) & if k \equiv 0 (\mod 2) \\ \exp (i π \frac{k^{2}}{2 N} + i \frac{3 π}{2} (k^{2} - 1)) & if k \equiv 1 (\mod 2) \end{matrix}$
We use the fact that k² is divisible by 4 if k is an even number, or (k²-1) is divisible by 4 if k is an odd number in Eq. (11). Substitute Eq. (10) and Eq. (11) into Eq. (9), we can degenerate the two conditions regardless k is even or odd,
$\begin{matrix} \sum_{n = 0}^{N - 1} \exp (- i π \frac{2 n^{2} + 2 n k}{N}) = \sqrt{N} \exp (i π \frac{k^{2}}{2 N} + i \frac{3 π}{2} k^{2}) \\ = \sqrt{N} \exp (i \frac{3 N + 1}{4} \frac{2 π k^{2}}{N}) \end{matrix}$
In this case, (3N + 1)/4 is also an integer number, and is coprime with N. Therefore, Eq. (3) is proved in the condition of N≡1 (mod 4).
3) N≡3 (mod 4)
Note that Eq. (9) is also valid for N≡3 (mod 4). In this case,
$1 + \exp (i π \frac{N + 2 k}{2}) = {\begin{matrix} \sqrt{2} \exp (- i \frac{π}{4}) & if k \equiv 0 (\mod 2) \\ \sqrt{2} \exp (i \frac{π}{4}) & if k \equiv 1 (\mod 2) \end{matrix}$
Similar expression as Eq. (11) for N≡3 (mod 4) is found,
$\exp (i π \frac{k^{2}}{2 N}) = {\begin{matrix} \exp (i π \frac{k^{2}}{2 N} + i \frac{π}{2} k^{2}) & if k \equiv 0 (\mod 2) \\ \exp (i π \frac{k^{2}}{2 N} + i \frac{π}{2} (k^{2} - 1)) & if k \equiv 1 (\mod 2) \end{matrix}$
Therefore, the odd and even condition of k can also be merged into one equality,
$\begin{matrix} \sum_{n = 0}^{N - 1} \exp (- i π \frac{2 n^{2} + 2 n k}{N}) = \sqrt{N} \exp (- i \frac{π}{2} + i π \frac{k^{2}}{2 N} + i \frac{π}{2} k^{2}) \\ = \sqrt{N} \exp (- i \frac{π}{2} + i \frac{N + 1}{4} \frac{2 π k^{2}}{N}) \end{matrix}$
In Eq. (15), (N + 1)/4 is also an integer and is easy to show it is coprime with N. Here we derive Eq. (3) in the condition of N≡3 (mod 4). Overall, Eq. (3) is proved incorporating all the parts above.

Funding

Swiss National Science Foundation (159 897).

References

1. M. Nakazawa, E. Yoshida, T. Yamamoto, E. Yamada, and A. Sahara, “TDM single channel 640Gbit/s transmission experiment over 60 km using 400fs pulse train and walk-off free, dispersion flattened nonlinear optical loop mirror,” Electron. Lett. 34(9), 907–908 (1998). [CrossRef]

2. F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011). [CrossRef]

3. V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radiofrequency photonics,” Laser Photonics Rev. 8(3), 368–393 (2014). [CrossRef]

4. J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016). [CrossRef]

5. J. P. Yao, “Photonic generation of microwave arbitrary waveforms,” Opt. Commun. 284(15), 3723–3736 (2011). [CrossRef]

6. V. Torres-Company, A. J. Metcalf, D. Leaird, and A. M. Weiner, “Multichannel radio-frequency arbitrary waveform generation based on multiwavelength comb switching and 2-D line-by-line pulse shaping,” IEEE Photonics Technol. Lett. 24(11), 891–893 (2012). [CrossRef]

7. V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012). [CrossRef]

8. R. Maram, L. R. Cortés, and J. Azaña, “Programmable fiber-optics pulse repetition-rate multiplier,” J. Lightwave Technol. 34(2), 448–455 (2016). [CrossRef]

9. Y. Xie, L. Zhuang, and A. J. Lowery, “Picosecond optical pulse processing using a terahertz-bandwidth reconfigurable photonic integrated circuit,” Nanophotonics 7(5), 837–852 (2018). [CrossRef]

10. D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010). [CrossRef] [PubMed]

11. Z. Geng, Y. Xie, L. Zhuang, M. Burla, M. Hoekman, C. G. H. Roeloffzen, and A. J. Lowery, “Photonic integrated circuit implementation of a sub-GHz-selectivity frequency comb filter for optical clock multiplication,” Opt. Express 25(22), 27635–27645 (2017). [CrossRef] [PubMed]

12. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001). [CrossRef]

13. J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. 32(6), 716–718 (2007). [CrossRef] [PubMed]

14. C.-C. Chen, I.-C. Hsieh, S.-D. Yang, and C.-B. Huang, “Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect,” Opt. Express 20(24), 27062–27070 (2012). [CrossRef] [PubMed]

15. C.-B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photonics Technol. Lett. 12(2), 167–169 (2000). [CrossRef]

16. M. A. Preciado and M. A. Muriel, “All-pass optical structures for repetition rate multiplication,” Opt. Express 16(15), 11162–11168 (2008). [CrossRef] [PubMed]

17. H.-P. Chuang and C.-B. Huang, “Generation and delivery of 1-ps optical pulses with ultrahigh repetition-rates over 25-km single mode fiber by a spectral line-by-line pulse shaper,” Opt. Express 18(23), 24003–24011 (2010). [CrossRef] [PubMed]

18. M. Seghilani, R. Maram, and J. Azaña, “Mitigating nonlinear propagation impairments of ultrashort pulses by fractional temporal self-imaging,” Opt. Lett. 42(4), 879–882 (2017). [CrossRef] [PubMed]

19. R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014). [CrossRef] [PubMed]

20. L. R. Cortés, M. Seghilani, R. Maram, and J. Azaña, “Full-field broadband invisibility through reversible wave frequency-spectrum control,” Optica 5(7), 779–786 (2018). [CrossRef]

21. M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013). [CrossRef] [PubMed]

22. S. Liao, Y. Ding, J. Dong, T. Yang, X. Chen, D. Gao, and X. Zhang, “Arbitrary waveform generator and differentiator employing an integrated optical pulse shaper,” Opt. Express 23(9), 12161–12173 (2015). [CrossRef] [PubMed]

23. S. Min, Y. Zhao, and S. Fleming, “Repetition rate multiplication in figure-eight fibre laser with 3 dB couplers,” Opt. Commun. 277(2), 411–413 (2007). [CrossRef]

24. L. R. Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self-imaging effects on periodic objects,” Opt. Lett. 41(2), 340–343 (2016). [CrossRef] [PubMed]

25. C. R. Fernández-Pousa, “On the structure of quadratic Gauss sums in the Talbot effect,” J. Opt. Soc. Am. A 34(5), 732–742 (2017). [CrossRef] [PubMed]

26. C. R. Fernández-Pousa, R. Maram, and J. Azaña, “CW-to-pulse conversion using temporal Talbot array illuminators,” Opt. Lett. 42(13), 2427–2430 (2017). [CrossRef] [PubMed]

27. M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33(18), 3905–3911 (1994). [CrossRef] [PubMed]

28. J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. 36(6), 858–860 (2011). [CrossRef] [PubMed]

29. A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21(4), 4139–4144 (2013). [CrossRef] [PubMed]

30. J. Hu, S. Fabbri, and C.-S. Brès, “Flexible width Nyquist pulse based on a single Mach-Zehnder modulator,” in CLEO: Science and Innovations (Optical Society of America, 2018), paper SF3N.6. [CrossRef]

31. M. E. Marhic, “Discrete Fourier transforms by single-mode star networks,” Opt. Lett. 12(1), 63–65 (1987). [CrossRef] [PubMed]

32. K. Wörhoff, R. G. Heideman, A. Leinse, and M. Hoekman, “TriPleX: a versatile dielectric photonic platform,” Adv. Opt. Technol. 4(2), 189–207 (2015).

33. Y. Li, J. Li, H. Yu, H. Yu, H. Chen, S. Yang, and M. Chen, “On-chip photonic microsystem for optical signal processing based on silicon and silicon nitride platforms,” Adv. Opt. Technol. 7(1-2), 81–101 (2018). [CrossRef]

34. Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express 16(7), 4713–4718 (2008). [CrossRef] [PubMed]

35. B. C. Berndt and R. J. Evans, “The determination of Gauss sums,” Bull. Am. Math. Soc. (N.S.) 5(2), 107–129 (1981). [CrossRef]

Investigation of temporal Talbot operation in a conventional optical tapped delay line structure

Abstract

1. Introduction

2. Principle of operation

2.1 Optical tapped delay line structure

2.2 Spectral amplitude filtering

2.3 Spectral phase-only filtering

2.4 Combination of spectral amplitude and phase filtering

3. Results and analysis

3.1 Proof-of-concept test with delay line interferometer

3.2 Efficiency

3.3 Performance analysis

4. Conclusions

5 Appendix

Funding

References

Cited By

Figures (6)

Equations (15)

Optics Express