## Abstract

An optical orthogonal frequency division multiplexing (OFDM) scheme with Fourier transform in optical domain using time lenses both at the transmitter and at the receiver is analyzed. The comparison of performance between this scheme with the optical OFDM scheme that utilizes fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) in electrical domain is made. The nonlinear effects induced by Mach-Zehnder modulator (MZM) as well as by the fiber are investigated for both schemes. Results show that the coherent OFDM using time lenses has almost the same performance as that using FFT when the electrical driving message signal voltages are low so that MZM operates in the linear region. The nonlinearity of MZM deteriorates the conventional coherent OFDM based on FFT when the power of electrical driving signal increases significantly, but only has negligible impairment on the coherent OFDM using time lenses. Details of the time lens set up are provided and a novel scheme to implement the time lens without requiring the quadratic dependence of the driving voltage is presented.

©2009 Optical Society of America

## 1. Introduction

Orthogonal frequency division multiplexing (OFDM) has drawn significant research interest owing to its large dispersion tolerance and high spectral efficiency [1–7]. Recently, Refs. [5]. and [6] have proposed to realize Fourier transform (FT) and inverse Fourier transform (IFT) in optical domain. In Ref. [5], the discrete Fourier transform (DFT) circuit is designed using combinations of optical delays and phase shifters. In Ref. [6], FT and IFT are realized using a time lens which is nothing but a cascade of dispersive element (such as optical fiber or fiber grating), phase modulator and a dispersive element. It is well known that for a conventional lens, the optical field distribution at the back focal plane is Fourier transform of the field distribution at the front focal plane. Replacing the diffraction by dispersion and spatial chirp of a conventional lens by a temporal chirp introduced by a phase modulator, Fourier transformation of a temporal signal by a time lens is discussed in Ref. [8]. The time-domain Fourier transformation of a single Gaussian pulse is experimentally demonstrated in Ref. [9]. The advantage of the optical domain realization of FT is that high speed digital signal processing needed for FFT and IFFT implementation is now replaced by optical signal processing by time lenses which have inherently high bandwidth. Reference [6]. has demonstrated the feasibility of 320 Gb/s transmission over a 400 km fiber-optic link using numerical simulations. However, fiber nonlinearity effects are ignored in the study of Ref. [6]. Due to inherently high peak-to-average power ratio (PAPR) of OFDM signal, fiber nonlinearity causes serious performance degradation. Therefore, the impact of fiber nonlinearity in optical OFDM systems and its mitigation have drawn significant attention, recently [10–20]. The effects of MZM nonlinearity on optical OFDM systems both for coherent detection [21,22] and for direct detection [23,24] have also been investigated and digital clipping and pre-distortion processes were applied to compensate for the nonlinear impairments introduced by in-phase and quadrature MZMs in Ref. [22].

In this paper, we study the nonlinear tolerance of a coherent optical OFDM system using time lenses. The nonlinear effects induced by MZM are also investigated and compared with the OFDM scheme based on FFT. Results show that the nonlinearity of MZM significantly degrades the performance of the conventional coherent OFDM using FFT as the power of the driving message signal increases, while it only has minor impairment on the coherent OFDM using time lenses. This is because MZM is placed after IFFT for the conventional OFDM, and the output signal of IFFT has the property of high peak to average power ratio (PAPR), such that some part of the input signal to MZM with high instant peak power falls into the nonlinear region of MZM, and therefore, the output of MZM is distorted by the nonlinear response of MZM. At the receiver end, the received OFDM sub-channels are no longer orthogonal and therefore, after FFT the demodulated signal cannot be recovered without errors. In contrast, for the coherent OFDM using time lenses, the MZM is placed before FT, so that, in the absence of fiber dispersion, nonlinearity and ASE noise, the signal after FT at the receiver end is identical to the output signal of MZM at the transmitter end. This implies that the nonlinear distortion induced by MZM is applied directly to the message signal while it does not affect the OFDM modulation and demodulation. The simulation results show that the nonlinearity of MZM does not have significant impairment on the coherent optical OFDM system using time lenses.

One of the drawbacks of the scheme proposed in Ref. [6]. is that to realize FT and IFT, one would need fibers with anomalous as well as normal dispersion and therefore, the dispersion tolerance of such a scheme becomes questionable. Instead, in this paper, we consider a scheme in which FTs are introduced both at the transmitter and at the receiver so that the SMF can be used as a dispersive element of the time-lens setup. As a result, the received sequence gets time reversed within a frame. But it can be easily corrected using the digital signal processing at the receiver.

To realize the time lens, a quadratic phase chirp should be introduced using a phase modulator. As the frame time increases, one would expect that the driving voltage increases quadratically with time and at the edge of the time frame, the required driving voltage becomes unreasonably large. But, in this paper, we find that it is possible to realize the time lens even with low driving voltage by making use of the periodic property of the sinusoidal transfer function of the phase modulator.

## 2. System modeling and time-lens setup

Figure 1 shows a block diagram of fiber-optic communication system based on optical OFDM, in which the FT blocks are implemented in optical domain using the Fourier transforming property of time lenses. In this scheme, the in-phase and quadrature components of the message signals from various channels are combined using electrical time division multiplexing (ETDM) units ETDM-I and ETDM-Q, respectively. The outputs of ETDM-I and ETDM-Q drive the in-phase and quadrature Mach-Zehnder modulators (MZM), MZM-I and MZM-Q, respectively. In contrast, in the case of conventional OFDM [1–4], message signals from various channels modulate the sub-carriers through IFFT. The outputs of the MZMs are combined and pass through the FT block. The output of the FT block is launched to the fiber-optic link and then to another FT block at the receiver. Since the Fourier transform of a Fourier transform leads to time reversal within an OFDM frame, the transmitted signal can be recovered by introducing time reversal using digital signal processing. In the case of coherent optical/electrical OFDM, a phase chirp is introduced across the frame due to fiber dispersive effects which can be cancelled using equalization algorithms [4]. The optical field envelope at the MZM-I and MZM-Q outputs can be written as [25]

The input of the Fourier transformer under these conditions is

whereHere, we define the Laser power, ${P}_{c}$, in-phase/quadrature message signal average power, ${P}_{m}$, and average input power launched to fiber, ${P}_{in}$ as

Next, let us consider the implementation of the Fourier transformation using time lenses. Let the input optical field envelope be ${u}_{in}(t)$. The FT pairs are defined as

A time-lens-based Fourier transformer is shown in Fig. 2
. ${\beta}_{2}^{F}$ and *F* are the second-order dispersion of the standard single-mode fiber (SMF) and the length of the fiber, respectively. The accumulated second-order dispersion of the dispersive fiber of the time lens is defined as ${S}_{1}={\beta}_{2}^{F}F$. Propagation of an input optical field envelope in a 2*F* system (see Fig. 2) consisting of dispersive elements (fibers) and a phase modulator (time lens) results in either FT or IFT of the input signal at the output of 2*F* system [8], depending on the signs of second-order dispersion coefficients and chirp factors of the phase modulator [26]. The phase modulator multiplies the incident optical field envelope by a function

The output of FT at the transmitter is given by

Then, ${u}_{out-tx}^{FT}(t)$ is transmitted over the fiber-optic link, and the output of the fiber-optic link, ${u}_{out}^{fiber}(t)$ passes through the Fourier transformer at the receiver whose output is

Consider the optical OFDM signal ${u}_{out-tx}^{FT}(t)$ propagating in a fiber whose linear transfer function is

where*L*is the fiber length, and ${\beta}_{n}$ is the

*n*th-order dispersion coefficient. In the absence of fiber nonlinearity and noise, the fiber output iswhere ⊗ denotes convolution and ${h}_{F}(t)={\mathcal{F}}^{-1}\left[{H}_{F}(f)\right]$ is the fiber impulse response function. Convolution in the time domain becomes a product in the frequency domain and therefore, after FT at the receiver, the output signal given in Eq. (13) can be rewritten as

Using Eqs. (12) and (14) in Eq. (17), we obtain

From Eq. (18), we see that the effect of fiber dispersion is a mere phase shift in the time-lens-based OFDM system and therefore, dispersive effects completely disappear if direct detection is used since the photocurrent is directly proportional to the incident optical power. In the case of coherent detection, the phase shift introduced by the fiber dispersion is cancelled using electrical equalizers. Note that the received signal is time-reversed in both detection schemes which can be easily undone in electrical domain.

Since the input signal to the time-lens-based system is processed in block-by-block basis, periodic time lenses with finite aperture should be introduced [6,26,27]. Suppose that the time frame of Fourier transform (FT) is ${T}_{FT}$, the OFDM bandwidth is ${f}_{s}$, the number of sub-channels is ${n}_{sc}$, and $\mathrm{\Delta}{f}_{s}$ is the frequency space between OFDM sub-channels, then we have

The phase modulator multiplies the incident optical field envelope by a function

*n-*th block. Consider the input signal ${u}_{n}(t)$, which is limited to the interval $\left[-{T}_{FT}/2,{T}_{FT}/2\right]$. To see the spectrum of the OFDM signal, taking the FT of Eq. (12) and using Eq. (24), we obtainwhere

Equation (26) implies that the desirable OFDM spectrum, ${\tilde{u}}_{out-tx}^{FT}(f)$ can be obtained by appropriately shaping the input message signal ${u}_{n}(t)$ in time domain and the time axis, *t* is related to the frequency axis, *f* by Eq. (27). For the given frequency space, $\mathrm{\Delta}{f}_{s}$ between the sub-channels, the corresponding time spacing for the time-lens-based system is

In other words, the samples of the message signals corresponding to adjacent subcarriers should be separated by $\mathrm{\Delta}{t}_{s}.$ For the OFDM signal with bandwidth of ${f}_{s}$, the frequency *f* in Eq. (26) should be confined to $\left[-{f}_{s}/2,{f}_{s}/2\right]$and the time frame boundary ${T}_{FT}/2$ should correspond to the maximum frequency, ${f}_{s}/2$. Using Eq. (27), we obtain

Substituting Eq. (29) in Eq. (28) and using Eq. (20), we obtain

The driving voltage of the phase modulator in the time lens set up could be very high if the duration of OFDM symbol is getting large, which can be seen from Eq. (23). Using Eqs. (11), (19), (20), (23) and (29), the maximum phase shift introduced by the phase modulator in frame-by-frame basis is given by

Because of the periodic property, the driving voltage needs not increase quadratically with time. The broken line in Fig. 3 shows the normalized driving voltage increasing quadratically with time and the solid line in Fig. 3 shows the driving voltage that provides the same amount of phase shift. For example, when$t={T}_{FT}/2=25.6$ns, the driving voltage with quadratic dependence is $256{V}_{\pi}$, whereas the voltage shown in the solid line of Fig. 3 does not increase beyond$2{V}_{\pi}$. Because of the periodic property, the maximum driving voltage becomes independent of ${T}_{FT}$. The required driving voltage shown in the solid line of Fig. 3 can be obtained using arbitrary waveform generator (AWG). Typically, the commercially available AWG has a bandwidth less than 10 GHz. For the example shown in Fig. 3, the required bandwidth is around 2 GHz.

For the proper operation of the proposed scheme, the driving voltage of the phase modulator should be synchronized with OFDM frames. This can be achieved by extracting a clock at the frame rate which would be used at the AWG electronics to synchronize the driving voltage with the OFDM frame. Similar techniques have been used to reduce the timing jitter of solitons using synchronous amplitude modulators [28]. Since the frame rate is much lower than the symbol rate, this technique is not too hard to implement. Alternatively, the periodic time lenses could be free running without synchronizing them to the OFDM frames. Let the timing shifts between the OFDM frame and the driving voltage of the phase modulators at the transmitter and receiver be *t*
_{1} and *t*
_{2}. The parameters *t*
_{1} and *t*
_{2} can be estimated using digital signal processing at the receiver.

## 3. Simulation and results

To compare the performance of the optical OFDM using time lenses with the conventional OFDM using FFT, the bit error rate (BER) is calculated at the receiver as a function of the launch power. Fiber nonlinearity and amplified spontaneous emission (ASE) noise are both taken into account. The simulation parameters are given in Table 1
for OFDM setup and Table 2
for the transmission link setup, respectively. The data rate is 40 Gb/s, and we use 4-QAM format as the message data. The OFDM bandwidth,${f}_{s}$ is 20 GHz. The number of sub-channels is 1024 and the space between sub-channels, $\mathrm{\Delta}{f}_{s}$ is 19.53 MHz. 64 OFDM symbols are simulated such that the total number of bits is ${2}^{17}$. A De Bruijin sequence of length ${2}^{17}$ is used in the Monte-Carlo simulation. The accumulated second-order dispersion ${S}_{1}$ in the time-lens-based system is 0.407 ns^{2}. The bandwidth of optical filter used in our simulation is 10 GHz.

First, the amplitudes of ${m}_{I}(t)$ and ${m}_{Q}(t)$ are carefully chosen such that they operate in the linear region of MZM. We choose ${P}_{m}$ (as defined in Eq. (7)) to be $0.12$ mW. The disadvantage of choosing such low values is that the significant laser power is lost after introducing MZMs. Figure 4 shows the BER as a function of launch power, ${P}_{in}$ (as defined in Eq. (7)), at ${P}_{m}=0.12$ mW. The launch power to the fiber-optic link can be varied by changing the laser power ${P}_{c}$ (as defined in Eq. (7)). The broken lines with squares and diamonds show the back-to-back BER for the case of FFT-based and time-lens-based OFDM,

respectively. Back-to-back BER is calculated by introducing a noise source that is equivalent to the cascade of all in-line amplifiers. The solid line with crosses stands for the conventional coherent OFDM using FFT and the circles stand for the coherent OFDM using time lenses, respectively. It is found that both of these two schemes of coherent OFDM systems have almost the same performance. This is because the nature of both conventional OFDM based on FFT and optical OFDM using time lenses is essentially same and the only difference is that for the conventional OFDM, the IFT/FT is implemented in electrical domain but for the optical OFDM, that is done in the optical domain using Fourier transforming property of time lenses. However, the advantage of optical OFDM is that all the signal processing needed to obtain OFDM signal can be accomplished in the optical domain. This implies that intrinsically higher available bandwidth in optical domain can be utilized and thereby, high speed digital signal processing (DSP) to implement FFT/IFFT can be eliminated. From Fig. 4, it is can be seen that when the launch power is small, BER decreases as the launch power increases. This is the linear regime as shown on the left-hand side of Fig. 4. When the launch power becomes large, the fiber nonlinearity dominates over the ASE noise, such that the BER increases with the launch power. This is the nonlinear regime as depicted on the right-hand side of Fig. 4.

Figure 5a shows the normalized in-phase component of input $m(t)$ of the first 32 bits in the first OFDM frame for a coherent optical OFDM system using time lenses. The solid and dotted lines in Fig. 5b show the normalized output current after the coherent detector, but before the time-reversing circuit at the average optical launch power of $-13$ dBm and $-4$ dBm, respectively. The significant nonlinear impairment can be seen for the case of ${P}_{in}=-4\text{dBm}$, while for the case of ${P}_{in}=-13\text{dBm}$, the effect of fiber nonlinearity is negligible. Note that the output bit sequence shown in Fig. 5b is time-reversed within a frame. Figures 6a and 6b show the spectra of FFT-based and time-lens-based OFDM. As can be seen, they are identical and therefore, the coherent OFDM using time lenses has the same spectral efficiency as the OFDM using FFTs.

The above simulation is done under the assumption that MZM is working as a linear modulator. However, when the power of driving message signal launched to MZM increases, this assumption would not hold any more. Next, we study the impact of MZM nonlinearity as well as fiber nonlinearity on the coherent OFDM system based on FFT and time lenses. First, to investigate the nonlinear effect induced by MZM alone, the fiber nonlinearity is turned off and the ASE noise is adjusted to generate the OSNR required to obtain the BER of $2\times {10}^{-3}$.

Figure 7 shows the required OSNR at BER of $2\times {10}^{-3}$ varying as the average power of the driving message signal. Figure 7a shows the MZM nonlinear effect on coherent OFDM systems without fiber nonlinearity. In Fig. 7b, the fiber nonlinearity is taken into account and the power launched to the transmission fiber is at $-10$ dBm. In both cases, it is shown that the required OSNR at BER of $2\times {10}^{-3}$ for coherent optical OFDM based on FFT increases drastically when ${P}_{m}>200\text{mW}$, whereas for the coherent optical OFDM using time lenses, the required OSNR at BER of $2\times {10}^{-3}$ does not change significantly. For the coherent OFDM using FFT, the MZM is placed between IFFT block at the transmitter and FFT block at the receiver, so MZM nonlinearity destroys the orthogonality of sub-channels of OFDM signal, which leads to the significant impairment on the coherent OFDM using FFT. In contrast, for the coherent OFDM using time lenses, MZM is placed before IFT, so that the OFDM signal can be demodulated correctly after passing through FT. Therefore, the MZM nonlinearity only has minor impairment on the coherent OFDM using time lenses as can be seen in Fig. 7. When the fiber nonlinearity is present (Fig. 7b), the required OSNR at BER of $2\times {10}^{-3}$ increases by $1~2\text{dB}$ for both coherent OFDM schemes, compared to that shown in Fig. 7a.

This implies that the fiber nonlinearity further worsens the system performance along with the MZM nonlinearity. Figure 8 shows the BER as a function of the launch power, ${P}_{in}$, when a larger average power of electrical driving message signal, ${P}_{m}$ is chosen. The parameters used for Fig. 8 is same as that of Fig. 4 except that ${P}_{m}=0.12$ mW in Fig. 4 and ${P}_{m}=500$ mW in Fig. 8. In Fig. 4, we found that the performance of the OFDM based on FFT is almost same as that based on time lenses. In contrast, when ${P}_{m}$ is large, OFDM based on time lenses has a superior performance as can be seen from Fig. 8.

## 4. Conclusions

A coherent optical OFDM scheme utilizing time lenses for implementing Fourier transforms both at the transmitter and at the receiver is analyzed. Impact of MZM nonlinearity as well as the fiber nonlinearity on the performance of the optical OFDM system is studied. The results are compared with the optical OFDM using FFTs. It is found that when the driving voltage of the MZM is large, MZM nonlinearity destroys orthogonality of sub-channels of OFDM signal for coherent OFDM using FFT, and therefore degrades its performance significantly. Whereas for the coherent OFDM using time lenses, MZM nonlinearity only has minor impairment because the Fourier transform block is introduced after the MZM. When the MZM operates in the linear regime, almost same performance is obtained for both schemes. In addition, the setup of time-lens-based Fourier transformer is discussed and a novel scheme to obtain the quadratic phase chirp without requiring the quadratic driving voltage is proposed.

## Acknowledgments

The authors like to thank Natural Sciences and Engineering Research Council of Canada (NSERC) foundation for research grant.

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