## Abstract

Microwave photonic arbitrary waveform generation based on incoherent frequency-to-time-mapping (FTTM) accompanied by intersymbol interference, so called crossed FTTM (CFTTM). The pulse shape can be defined and tuned by properly adjusting the spectrum shaper (symbol shape) and the degree of intersymbol interference. UWB-, triangular-, rectangle-, comb- and user-defined pulse shapes are experimentally obtained.

©2013 Optical Society of America

## 1. Introduction

Photonic-assisted arbitrary waveform generation (AWG) offers a promising solution to many applications in ultra-wideband (UWB) fiber-wireless communication systems, pulsed radar, radio frequency communications and sensor networks etc [1,2]. Compared with the electronic ones, photonic approaches exhibit superior advantages with light weight, small size (or compact design), large tunability and immunity to electromagnetic interference (EMI) [3]. So far Fourier-transform optical pulse shaping is prevailing and widely-used technology to generate arbitrarily shaped pulses [4,5]. It is enabled by combining the spectral shaping with the frequency-to-time mapping (FTTM) technique. Such method allows the amplitude and phase of discrete optical spectral lines to be independently controlled and enables arbitrary waveform to be synthesized in a time aperture, which is also called the spectral line-by-line pulse shaping [6]. On the other hand, the incoherent pulse shaping is demonstrated recently with intrinsic low cost and potential applications in optoelectronic measurements. Note that the desired intensity profile based on incoherent FTTM could be achieved only through statistical averaging [7,8].

Nowadays, there are many photonic approaches of arbitrary waveform generation based on Fourier-transform optical pulse shaping, such as the combination of diffraction gratings and a liquid crystal spatial light modulator (SLM) or the scheme utilizing a dense wavelength division multiplexing (D-WDM) channel controller and an amplified spontaneous emission (ASE) source [9–11]. Such demonstrations show good performance or ability in generating arbitrary radio frequency waveforms, while the insertion loss, complicity in the setup and alignment control are still quite challenging for further optimization [12,13]. In order to find alternate approaches in addition to the spatial ones, several schemes of fiber-optic Fourier-transform pulse shaping based on the FTTM process have been reported recently, such as UWBs, triangle-shaped, rectangle-shaped and arbitrary-shaped pulses generator [14–16]. However, mostly those approaches rely on the spectrum shaper to offer the tunability of the pulse shaping. The reconfigurability of pulse generators is still limited.

In this paper, we propose an all-fiber pulse shaping approach based on incoherent crossed frequency to time mapping (CFTTM) process, i.e. by introducing the intersymbol interference (ISI) into conventional FTTM process. It is important to note that the ISI here isn’t equivalent to the normal coherent interference, but merely the overlap of temporal intensities due to the wavelength separation. The output pulse shape could be defined and tuned by adjusting the spectrum shape (the symbol shape) and the degree of ISI. Through theoretical analyses, we derive the relationship between the output pulse shape and the ISI, and experimentally demonstrate arbitrary waveform generation based on the CFTTM process. The advantages of our approach compared with conventional FTTM process are illustrated by reconfigurable generation of various pulses, including UWB-shaped, triangular-shaped and user-defined ones.

## 2. Principle of crossed frequency to time mapping

Figure 1 is the typical scheme of photonic arbitrary waveform generation based on incoherent frequency to time mapping. It consists of a time-gating broadband light source, a spectrum shaper, a dispersion element and a photodetector. The optical spectrum of the light source is firstly shaped using a tunable filter module (i.e. the spectrum shaper) and then launched into a linear dispersive element. The generated temporal pulse in average is the counterpart of the spectrum shape after the conventional frequency to time mapping as shown in Fig. 1(a) [17]. In order to realize accurate mapping between the optical spectrum and the stretched temporal pulse, a sufficient amount of dispersion must be introduced on the input pulse as [18–20]

where*Ф*is the total group delay dispersion (GDD) parameter(ps/nm),

_{2}*σ*and

_{ω}*Δω*is the bandwidth of the spectrally incoherent light source and the external modulator for time-gating function respectively .

The average output of incoherent light intensity after time-gating and a linear dispersion element can be expressed as

*S(ω)*is the energy spectrum of the incoherent source (e.g. ASE) centered at the baseband,

*F(ω)*denotes the complex transfer function of the TFM and

*M(ω)*is the spectral complex transfer function of the external modulator for time-gating. When Eq. (1) is satisfied, the output intensity can be written as

Subsequently the temporal pulse becomes a scaled replica of its optical spectrum, i.e. the incoherent frequency to time mapping takes place. The full width of the temporal pulse (*ΔT*) is proportional to the frequency bandwidth (*Δλ*) of the spectral shape as

Meanwhile, the effect of dispersion on the smallest period of the pulse source (*1/B)* has to be limited by using the criterion *BΔT<1* (similar to the concept of bit-rate capacity). The dispersion coefficient *Ф _{2}* has to satisfy

By satisfying Eq. (5), signal generation based on conventional FTTM could be achieved. On the other hand, when the above criterion is broken, i.e. *BΔT>1*, we have

The intersymbol interference would happen as illustrated in Fig. 1(b). The frequency to time mapping accompanying by the intersymbol interference is called the crossed frequency to time mapping (CFTTM) in the paper. The degree of ISI is proportional to the value of *BΔT*. Note that the value of *ΔT* is fixed once the spectrum shape and the dispersion parameter is pre-defined, and the degree of ISI can be determined by the parameter of *B*. Obviously, when the parameter *B* of the FTTM is set as *B _{1}>B_{2}*, the output pulse shape would be quite different. The output pulse shape of the CFTTM is the superposition of several symbol shapes (spectrum shape of the input). Meanwhile the shape of an arbitrary signal can be regarded as the synthesis of certain basic shapes with different combinations, therefore photonic arbitrary waveform generation based on the CFTTM could be achieved with simplified spectrum manipulation in conventional FTTM-based schemes.

## 3. Waveform synthesis

According to the above principle, the output pulse shape of our proposed arbitrary waveform generation based on CFTTM can be defined by the symbol shape (the spectrum shape) and the degree of ISI (the value of *BΔT*). Figure 2
show typical configurations to tune the pulse shape. The solid lines of Fig. 2(a) are the synthetic waveforms of two triangle-shaped symbols with different degree of ISI (*B _{1}ΔT, 2/B_{1}ΔT* and

*3/B*). Therefore the output pulse shape can be tuned by adjusting the period of the pulse source (i.e.

_{1}ΔT*1/B*). Assuming that the shape of the symbol is merely changed into notch-shape, we could obtain totally different synthetic waveforms as shown Fig. 2(b). Therefore, by adjusting the spectrum shaper to filter out different symbol shape, the output pulse shape could be tuned. Note that the bandwidth of the spectrum (

*Δλ)*could affect the degree of ISI through the parameter of

*ΔT*. Moreover, if

*B*as illustrated in Fig. 2(c)

_{1}≠B_{2}_{,}the non-uniform degree of ISI could be introduced to facilitate the arbitrary pulse shaping. The synthesis of three triangle-shaped symbols with uniform (left) and nonuniform (right) distributions of ISI could end with two different pulse shapes. Therefore the period of pulse (

*1/B*) might also be set as unequal to introduce the non-uniform ISI for the pulse shaping. Such approaches could be utilized either individually or by combinations for arbitrary waveform generation in accordance with specific environment requirements.

## 4. Experiment

Figure 3
is our experimental setup of the proposed arbitrary waveform generation based on CFTTM. It consists of an ASE source with a time-gating control, a symbol (spectrum) shaper, a spool of 10-km SMF and a high-speed PD. The control of time-gating is realized by setting the pattern and the bit-rate of the RF-driving signal to adjust the value of *1/B*. The symbol shaper is composed of an array of FBGs with different characteristics (e.g. the central-wavelength of 1555.3, 1556.2 nm; the bandwidth of 0.1, 0.2 nm and the reflectivity of 85%, 70%, respectively) and an optical tunable bandpass filter (TBF). By varying the parameters of the TBF (i.e. the bandwidth and central wavelength), the symbol shape could be tuned. Here, a 10-km SMF is used as the first-order linear dispersive element with the total GDD (*Ф _{2}*) value of 220 ps/nm

_{.}In our experiment, the light from the ASE source is intensity-modulated by a Mach-Zehnder modulator (MZM) driven by a user-defined RF signal, and then launched into the tunable spectrum shaper to filter out the desired spectrum shape. After the CFTTM process in the SMF, an arbitrary-shaped waveform can be generated at the output of PD. A sampling oscilloscope (Agilent 86100C)with eight-time averaging is used to measure the generated waveforms. In addition, an EDFA is used to compensate the power loss of the system.

First we verify the requirements for the conventional incoherent FTTM and CFTTM as indicated by Eqs. (5) and (6). As an example, the driving RF signal of MZM is set as 10-Gbit/s with a fixed pattern of “00010001” (*1/B = 400 ps*) and the TBF with its central wavelength of ~1556.3 nm and its bandwidth of ~1.3 nm is chosen. Figure 4(a)
is obtained symbol shape (left) and the corresponding averaged notch-shaped electrical pulse (right). It can be observed that the generated temporal pulse shape is similar to the spectrum shape and its full width (*ΔT*) after a 10-km SMF is ~286 ps. Thus, the corresponding value of *BΔT* is ~0.71. The requirement of CFTTM (*BΔT >1*) can’t be met, therefore the obtained optical pulse in average is a scaled copy of the spectrum based on the conventional FTTM process.

However, if we change the pattern of driving RF signal to be “01010101” (*1/B = 200 ps)*, Fig. 4(b) shows the measured optical spectrum after the spectrum shaper (left) and the generated triangle-shaped electrical pulse with averaged (right). As indicated in the insets of the left figure, the corresponding degree of ISI (*BΔT)* is about 1.43, meeting the requirements for CFTTM, therefore the output pulse shape is not only determined by the spectrum shape, but also the degree of ISI. In addition to the results in Fig. 4(b), if we slightly adjust the central wavelength of the TBF to left or right (e.g. ~0.2 nm), we could easily obtain electrical averaged sawtooth-shaped pulses with positive ramp and negative ramp as shown in Figs. 5(a)
and 5(b), respectively.

Furthermore, the tunability of the symbol shape with the same degree of ISI is illustrated in Fig. 6
, where the positive [Fig. 6(a)] and negative [Fig. 6(b)] monocycle-shaped and doublet-shaped [Fig. 6(c)] UWB signals are obtained in average. The degree of ISI (*BΔT*) for all cases is ~1.1 using different combinations of the period of the pulse source (*1/B)* and the bandwidth of the TBF (i.e. *Δλ,* which has a fixed relationship with the pulse width *ΔT*): (i) For the monocycle-shaped pulse generation, we adjust the full-bandwidth of the TBF to be ~2 nm and the central wavelength to be ~1556.5 nm (positive) and ~1555.8 nm (negative). The period of the pulse source (*1/B)* is ~400 ps while the symbol shapes are different. (ii) For the doublet-shaped pulse generation, we adjust the bandwidth of symbol to be ~2.2 nm (the corresponding value of *ΔT* is ~440 ps), and tune the bit-rate of the driving RF signal to be 9 Gbit/s (the corresponding period of the pulse source *1/B* is 484 ps).

Meanwhile, the tunability of the degree of ISI (i.e. *1/B*) with the same symbol shape (spectrum shape) for pulse shaping is verified. Figure 7
is the measured spectrum shape with full-bandwidth of ~1.9 nm for all cases (*ΔT≈418 ps*). Three output signals with different shapes in average are generated: (i) if we fix the driving RF signal of MZM at 10 Gbit/s with the pattern of “100100”, a rectangle-shaped electrical pulse train could be obtained [Fig. 8(a)
], here *1/B≈300 ps and BΔT≈1.4*; (ii) if the RF pattern is adjust to be “101010” (*1/B≈200 ps, BΔT≈2.1)*, the shape of generated pulse is changed into the comb one as shown in Fig. 8(b). Note that both cases are still under a uniform ISI distribution. (iii) If we set the driving RF signal at 10-Gbit/s with a fixed pattern of “10011000”, there will be a relatively weak ISI (*1/B _{1}≈200 ps*,

*B*) for the first two pulses and a strong ISI

_{1}ΔT≈2.1*(*1/B

*for the last two pulses. i.e. the ISI distribution is not uniform any more. The generated electrical pulses are illustrated in Fig. 8(c). Therefore, it is feasible to introduce the non-uniform distribution of ISI for the arbitrary waveform generation by proper selection of the ISI distribution and the driving signals.*

_{1}≈100 ps,B_{2}ΔT≈4.2)## 5. Analysis of key parameters

The pulse width and the repetition rate are two important parameters for the pulse shaping, therefore we further analyze those parameters with examples. The pulse width of obtained pulses based on the CFTTM is determined by the value of *ΔT* (*ΔT = χΔλ*) and the ISI (*BΔT*). Once the pulse shape is defined, the values of *Δλ* and *BΔT* are fixed. In order to tune the pulse width, we have to properly adjust the dispersion element (GDD) and the *1/B* value at the same time. The adjustment of the GDD value is for the pulse width, and varying the 1/B value is for the shape preserving (i.e. fixed *BΔT*). For example, under the same experimental parameters in Figs. 7 and 8 except that the pattern is set as “100010000000000”, the pulse width of generated doublet-like pulse is *2ΔT-1/B* with the ISI (*BΔT≈2*) fixed (see the inserts of Fig. 9
). By properly adjusting the values of the total GDD amount and the 1/B period, we could demonstrate the tunability of the pulse width for the doublet-like pulse as shown in Fig. 9. The inserts of Fig. 9 are waveforms and electrical spectra of two generated doublet-like pulses with the pulse-width of ~780(a-1, a-2) and ~1100 (b-1, b-2) ps, respectively. Due to the limitation of modulated RF signal (the maximum bit-rate of 12.5 Gbit/s) and the bandwidth of the MZM, the available maximum RF bandwidth is ~10 GHz [17].

The repetition rate of generated pulses could also be tuned as illustrated in Fig. 10
. First, we can directly tune the pattern of the MZM-modulated signal to change the repetition rate. For example, the repetition rate of Fig. 10(a) is ~500 MHz (up) and ~600 MHz (down), with the corresponding pattern of “100010000000000” and “1000100000”, respectively. Here the bit sequence of “100010000” corresponds to the doublet-like pulse while the repetition rate can be tuned by modifying the number of unoccupied “0”-bit followed. On the other hand, we could also tune the bit-rate of MZM-modulated signal when all bits are occupied for the pulse shaping. Note that the period of *1/B* is related to the bit-rate, so we need to change the pattern (modifying the number of “0”-bit between “1”-bits, i.e. *1/B*) or the dispersion amount (i.e. *ΔT)* with the fixed ISI at same time. Figure 10(b) illustrates generated triangle-shaped pulses with the repetition rate of 5 (up) and 6.25 (down) GHz. Here the bit-rate of modulated signals with same pattern of “1010” is set to be 10 Gbit/s (*1/B≈200 ps*) and 12.5 Gbit/s (*1/B≈160 ps*), respectively. For the upper case (i.e. 5 GHz), we obtain the signal by utilizing similar parameters as Fig. 4(b) (i.e. *BΔT≈1.47*). For the lower case (i.e. 6.25 GHz), the GDD value is merely adjusted to be ~180 ps/nm (*ΔT≈180 × 1.3 = 235 ps*) and the ISI is fixed (i.e. *BΔT≈1.47*). Incidentally, we can’t directly change the pattern from “1010” (1/B≈160 ps) into “100100” (1/B≈240 ps), i.e. adding a “0”-bit for compensating the different of 1/B for the 12.5 Gbit/s case.

## 6. Discussion

Although we have demonstrated a photonic approach to generate arbitrary waveform generation based on incoherent CFTTM, there are some points worthwhile to be addressed: i) similar to other proposals, the claim of “arbitrary” has to be further investigated to justify whether “fully” arbitrary or “conditional” arbitrary”; ii) it’s possible to realize the overlap of two or more shapes based on the CFTTM process by means of parallel paths; the limit of how many pulses that could be crossed before seeing coherent effects is determined by the number of pulsed of precise time-gating control during *ΔT*, as well as the bandwidth of the light source and the dispersion amount; iii) although we demonstrate such approach based on incoherent light source due to the advantages of simplicity and low cost [8], it is still interesting to further investigate the performance of signal generation based on coherent CFTTM. The contributions of optical noises and the coherent length of the light source should be taken into account.

## 7. Conclusion

We have proposed a photonic approach to generate arbitrary shaped waveforms based on the CFTTM process. The requirements of CFTTM are theoretically analyzed compared with conventional FTTM and the three approaches of arbitrary-shaped waveform synthesis are offered and proved with experiments in the paper. The output pulse shape can be defined by the value of *BΔT* (the degree of intersymbol interference) and the spectrum shape (the shape of symbol).

## Acknowledgments

The research is supported by the National Basic Research Program of China (2012CB315704), the Natural Science Foundation of China (No. 61275068) and the Key Grant Project of Chinese Ministry of Education (No.313049).

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