## Abstract

We introduce an all-optical arithmetic unit operating a weighted addition and subtraction between multiple phase-and-amplitude coded signals. The scheme corresponds to calculating the field dot-product of frequency channels with a static vector of coefficients. The system is reconfigurable and format transparent. It is based on Fourier-domain processing and multiple simultaneous four-wave mixing processes inside a single nonlinear element. We demonstrate the device with up to three channels at 40 Gb/s and evaluate its efficiency by measuring the bit-error-rate of a distortion compensation operation between two signals.

© 2014 Optical Society of America

## 1. Introduction

Advanced modulation formats producing multilevel phase-and-amplitude encoded signals are now widely used in optical communication systems [1]. Presently, very few mathematical operations can be carried all-optically on such signals. Their processing relies on coherent detection and digital signal processing (DSP). For example, fast Fourier transform (FFT), turbo codes and pattern recognition are essential functions in most optical information networks for encoding, error correction, impairments mitigation and routing [2, 3].

State-of-the-art DSP logic provides extremely versatile and powerful solutions to most computational needs. However, these come at the cost of increased latency, power consumption and limited bandwidth since analog signals are converted to the electrical domain and processed by algorithms at the bandwidth of digital circuits [4]. In order to alleviate these limitations, the photonics community has been looking at harnessing ultrafast optical effects to carry critical functions. All-optical and electro-optical matrix operations have been demonstrated in the past using free space linear optics, spatial light modulators [5, 6, 7] and nonlinear media [8]. Ultrafast arithmetic and logic circuits have also been developed using guided-optics components that are better suited for high speed fibre telecommunication applications [9, 10, 11, 12]. However, they remain relatively rudimentary and are typically limited to binary signals. Generally these techniques attempt to respond to the needs of particular application sectors that require extremely low latency such as data centres and financial networks which are willing to reduce or even remove the load on DSP in order to minimize delaying of the information [13, 14].

In this paper we introduce a novel all-optical scheme for calculating the dot-product between optical fields based on a combination of linear and nonlinear optical effects by Fourier-domain optical processing (FD-OP) and four-wave mixing (FWM). The method is applicable to any encoding format and potentially enables operation at ultra high baudrate thanks to the fast response of the Kerr processes involved. We believe that such high speed optical arithmetic operation has potential applications in the field of all-optical signal processing as it corresponds to the calculation of a complex linear form in linear algebra. In particular, summation of frequency components with harmonic coefficients can be the base of an optical fast Fourier transform. In a broader context, an all-optical dot-product implementation is a valuable addition to ongoing efforts for developing an optical signal processing toolbox [15, 16]. Such a toolbox is desirable in particular to enable ultra-high bandwidth processing systems with low latency [17, 18, 19]. We demonstrate an all-optical field dot-product at 40 Gbaud with up to three intensity modulated signals. The error-free operation of the scheme is demonstrated for a particular case with two signals. Moreover, the accuracy of the dot-product operation is quantified by configuring it to remove a signal component introduced intentionally and measuring the bit-error-rate improvement.

## 2. Principle

This work aims at calculating a dot-product between a time dependent vector of values **E**(*t*) and a static vector of coefficients **a**. Mathematically, this operation is a linear form in the complex space resulting in a scalar *E*_{Itot} (*t*) = ∑_{j}*a _{j}E_{j}*(

*t*) where

*E*(

_{j}*t*) and

*a*are the components of the two vectors. In optical terms, this would correspond to a weighted sum between orthogonal light fields

_{j}*E*(

_{j}*t*), for example signals encoded on separate WDM channels.

The method consists of a combination of linear and nonlinear optical processing. Linear FD-OP first applies attenuation and phase values corresponding to the modulus and argument of the coefficients *a _{j}* to the fields. Mixing of the modified fields with pump fields in a Kerr medium operates frequency-conversion of the WDM channels to a unique idler frequency where they add or subtract coherently. We show for up to three signals that this scheme implements the dot-product operation of the analogue values contained in the complex optical fields of phasors

*E*

_{S}_{1},

*E*

_{S}_{2}and

*E*

_{S}_{3}with a vector of complex coefficients (

*a*

_{1},

*a*

_{2},

*a*

_{3}) normalized so that |

*a*≤ 1 ∀

_{j}|*j*= 1, 2, 3. The following equations are written for the case of three signals, but can be extended to any number of channels. With the coefficients expressed with their modules |

*a*| and arguments

_{j}*ϕ*, this corresponds to the phasor in Eq. (2).

_{j}The working principle of the device is shown in Fig. 1. The wavelength division multiplexed (WDM) signals to be processed are encoded onto a flat frequency comb with a frequency separation Δ*ν*. The dot-product device sets them with custom attenuation levels and relative phases before combining them with pump signals having half their spectral separation
$\frac{\mathrm{\Delta}\nu}{2}$. Subsequent nonlinear mixing inside a Kerr medium generates frequency converted idlers via degenerate FWM. Due to the choice of the spectral separations of the pumps and signals, the idlers fall on the same frequency and add or subtract coherently according to their phase relations.

The incoming WDM signals **E**_{S}_{1}(*t*), **E**_{S}_{2}(*t*) and **E**_{S}_{3}(*t*) are phase locked due to the fact that they originate from the same frequency comb source. The pump signals **E**_{P}_{1}(*t*), **E**_{P}_{2}(*t*) and **E**_{P}_{3}(*t*) are also phase locked together, however they do not need to be locked to the signals. Their electric field is expressed in Eqs. (3) to (8).

*E*

_{S}_{1},

*E*

_{S}_{2},

*E*

_{S}_{3},

*E*

_{P}_{1},

*E*

_{P}_{2}and

*E*

_{P}_{3}are the phasors of the fields. In the following development, we consider that their amplitudes, defined by

*A*= |

_{Sj}*E*| and

_{Sj}*A*= |

_{Pj}*E*|, are respectively

_{Pj}*A*

_{S}_{1},

*A*

_{S}_{2},

*A*

_{S}_{3},

*A*

_{P}_{1}=

*A*

_{P}_{2}=

*A*

_{P}_{3}=

*A*and wave vectors

_{P}*k*

_{S}_{1},

*k*

_{S}_{2},

*k*

_{S}_{3},

*k*

_{P}_{1},

*k*

_{P}_{2}and

*k*

_{P}_{3}. The first order FWM idlers generated from these waves have phasors

*E*

_{I}_{1}=

*A*

_{I}_{1}

*e*

^{iϕI1},

*E*

_{I}_{2}=

*A*

_{I}_{2}

*e*

^{iϕI2}and

*E*

_{I}_{3}=

*A*

_{I}_{3}

*e*

^{iϕI3}. The wave vectors of the generated idlers are determined by the phase matching condition of the FWM processes [20]: From these relations and considering that signals are phase locked and pumps are phase locked, the explicit phase relations of the idlers can be written [21]: where the phases of the signals are

*ϕ*

_{S}_{1},

*ϕ*

_{S}_{2}and

*ϕ*

_{S}_{3}and the pump phase is set so that

*ϕ*

_{P}_{1}=

*ϕ*

_{P}_{2}=

*ϕ*

_{P}_{3}=

*ϕ*. We consider the phase of the incoming phase-locked signals as a reference and shift their relative phases respectively by

_{P}*ϕ*→

_{Sj}*ϕ*+

_{Sj}*ϕ*∀

_{j}*j*= 1, 2, 3 in order to implement the argument of the coefficients in the dot-product. In case that (

*a*

_{1},

*a*

_{2},

*a*

_{3}) are real, these phases are 0 (addition) or

*π*(subtraction). The coherent idlers add or subtract accordingly due to interference. In Fig. 1, we consider real coefficients (

*a*

_{1},

*a*

_{2},

*a*

_{3}). Signals with opposite phases are symbolized by arrows pointing in opposite directions.

In the undepleted pump approximation of degenerate FWM, the amplitude of the idler is proportional to the amplitude of the signal. As a consequence, the relative amplitudes of the idler waves can be controlled by power attenuation coefficients (|*a*_{1}|^{2}, |*a*_{2}|^{2}, |*a*_{3}|^{2}), which are applied to the signal waves. In terms of field amplitude, this results in Eqs. (15) to (17).

The relations above allow writing the phasor of the total generated idler *E*_{Itot} as a function of the characteristics of the incoming fields and the attenuation and phase shifts applied.

*ϕ*results in an absolute phase offset of the total idler, which is not relevant, and hence no phase relation is required between signals and pumps. Although other first-order cross products are generated from the pumps and signals, these appear at other frequencies than the idler of interest and can be removed by bandpass filtering. The method can be generalized to a dot-product between more than three channels by using more wavelength channels and pump waves.

_{P}In practice, the attenuation |*a _{j}*|

^{2}and phase shift |

*ϕ*| are applied via attenuation and phase control using a FD-POP device schematically represented in the inset of Fig. 1.

_{j}## 3. Experiment

Figure 2 describes our experimental implementation of the scheme. A pulse train with 2.2 ps pulse width from a 40 GHz modelocked (ML) fibre laser centered at 1550 nm was spectrally broadened by self phase modulation in two successive highly nonlinear fibre (HNLF) broadening stages. The resulting flat frequency comb was split into a pump pulse train and a data signal modulated with a 2^{31} – 1 pseudo random bit sequence (PRBS) via a Mach-Zehnder modulator (M-Z).

Three independent data channels were derived from this signal by spectral slicing and generation of a one bit-delay between each of them with a Finisar Waveshaper FD-POP [22]. The device allows for delays up to 60 ps, with increasing insertion loss for long delays. The two channels were separated with the shortest possible delay (1 bit) in order to reduce OSNR impairments.

The dot-product device was based on a second FD-POP controlling the phase and magnitude of the signals involved in the operation, an optical amplifier and a 30 m section of HNLF. The square root of the intensity attenuation applied to the channels translated the absolute value of the coefficients in Eq. (20). The relative phase between the signals controlled the argument of the coefficients, or their sign in the case where they were real values. The idler wavelength was then extracted using a narrow (70 GHz) bandpass filter and measured via a sampling oscilloscope with 65 GHz bandwidth. Note that only the intensity of the output signal was observed, given that an intensity detector was used.

The optical spectrum after the HNLF is shown in Fig. 3. The dashed and dotted traces correspond to configurations with only *S*_{1} and *S*_{2} or *S*_{2} and *S*_{3} that were used for calibration of the relative phases. When the pair of signals had the same bit pattern and were set with opposite phases, a dip appeared in the spectrum of the idler as a signature of the destructive interference. The power contrast due to interference between identical signals in the cases of addition and subtraction exceeded 20 dB. The solid trace features the dot-product operating with all three signals with a given set of phase and amplitude coefficients.

## 4. Results

#### 4.1. Dot-product with 3 signals

Figure 4 presents the time trace of the idler intensity measured with the sampling scope, corresponding to the modulus squared of the total field. The initial data stream *S* was a 64 bits on-off keying (OOK) pattern. *S*_{1}, *S*_{2} and *S*_{3} were obtained by delaying *S* respectively by −1, 0 and 1 bit-delay (25 ps). The results are given for four different configurations of coefficients (*a*_{1}, *a*_{2}, *a*_{3}). From left to right and top to bottom, these take the values (1, −0.4, 0), (0.3, −1, 0.7), (0.3, −1, 0.3) and (0.5, −1, −0.5).

Although the signals generated in our experiment were limited to OOK data, the linear combination can be operated between channels encoded with complex modulation formats without modification of the system. The field dot-product of multilevel phase-and-amplitude coded signals, for example QAM data, would produce an idler according to Eq. (20). The intensity trace resulting from direct detection could have more levels than those obtained with OOK.

On the figure, the theoretically expected envelope (blue trace) of the output signal is overlayed with the data measured. The agreement of the traces gives a good confirmation that the system is working as a dot-product of the signals.

#### 4.2. BER performance with 2 signals

In order to provide a quantitative evaluation of the function, we designed an experimental configuration of the dot-product where the expected output sequence was a known binary pattern, to allow for direct bit-error rate (BER) measurement. The principle of the method is summarized in Fig. 5. Two initial 40 Gb/s OOK data streams were voluntarily distorted according to a linear transformation represented by a matrix **X**. Basic algebra shows that this distortion can be undone by multiplication with the corresponding inverse matrix **X**^{−1}. Reverting the transformation was applied to one channel via the all-optical dot-product device. After cancellation of the impairment, the resulting recovered channel was analyzed through BER and eye diagram measurement.

*Distortion generation:* The linear transformation of the two signals by multiplication with the matrix **X** was operated inside the first FD-POP represented on Fig. 2. The operation can be represented mathematically by a matrix product **D** = **XS** where **S** is an array of the signals before modification (for example optical fields), **D** represents the distorted signals, and **X** is the transformation matrix. Distortion can therefore be canceled by applying the inverse transfer matrix **X**^{−1} to the degraded signals **S** = **X**^{−1}**D**, supposing that the adjacent channels are known. Here the matrices are normalized to verify conservation of energy and expressed with their coefficients. The expression of **X**^{−1} is given for the particular case of 2-by-2 matrices.

**X**is similar to the introduction of linear crosstalk between the two signals. Such linear crosstalk can appear in optical networks due to imperfect filtering in optical add-drop multiplexers (OADM) combining WDM channels from different origins [23, 24]. However, the comparison to this effect is limited, as discussed later.

The matrix **X** was applied by an interferometric technique inside the FD-POP. The method emulates the effect without applying an actual power transfer between the channels. This is allowed by the manner used to generate the incoming signals by delaying a single initial bit-pattern. Considering that both channels are identical replicas delayed by one bit-length Δ*L*, the extra-diagonal coefficient *b* corresponding to a leak from channel 2 into channel 1 can be synthesized by adding a fraction *b* of each symbol into the next one in the manner of a delay line interferometer (DLI) illustrated in Fig. 6 (right). Power and phase transfer functions of such DLI are derived for a signal of phasor *E _{S}*

_{1}=

*A*

_{S}_{1}

*e*

^{iϕS1}. Constructive interference after recombination of the arms

*a*and

*b*results in a total field equivalent to

*E*

_{S}_{1}distorted by a leak from

*E*

_{S}_{2}=

*A*

_{S}_{2}

*e*

^{iϕS2}. The latter signal can be written as a function of

*S*1 thanks to the link that exists between the two pseudo-independent channels: ${E}_{S2}={A}_{S2}{e}^{i({\varphi}_{S2})}=\underset{={E}_{S1}}{\underbrace{{A}_{S1}{e}^{i{\varphi}_{S1}}}}{e}^{ik\mathrm{\Delta}L}$. Eq. (23) expresses the distorted field.

*ν*is the frequency.

**X**.

*Distortion compensation:* The dot-product device was used on the distorted signal to undo the linear impairment. The attenuation level of the channels translated the absolute value of the matrix coefficients of **X**^{−1}, while their relative phase (either 0 or *π*) set the signs. The FD-POP is configured so that both channels are *π* out of phase and the channel to be subtracted is attenuated by *b*^{2}. The idler wavelength is then filtered using a bandpass filter and measured using a 40 Gb/s bit-error rate tester and a sampling oscilloscope.

The optical spectrum after the HNLF is shown on Fig. 7. The blue and black traces correspond to signal 1 and signal 2 being identical and undistorted, respectively with equal and opposite phases. When phases are opposite, a dip appears on the blue trace at the idler wavelength with 20 dB of contrast, which is the signature of the destructive interference between idlers. The green trace features both signals affected by a mutual crosstalk of 30 % and the compensator set to cancel this distortion. Signal 2 is attenuated by a factor *b*^{2} = 0.185 = −7.3*dB* and flipped by a *π* phase shift. Note that the spectral shape of the signals changes slightly when the distortion is applied, due to the FD-POP spectral shaping operation.

The method is assessed for coefficients of 0.2 and 0.3 with symmetric matrices

Figure 8 presents BER measurements for the back-to-back case (compensator bypassed) and with the compensator active. Curves are shown in the case of non distorted signals (blue), crosstalk with 20 percent leak (green) and 30 percent leak (red). A significant improvement of the power penalty was observed in both cases of crosstalk coefficients of 0.3 and 0.2. The wavelength conversion operation on an undistorted signal was causing a 1 dB power penalty regarding the back-to-back operation (B2B clean) at error free level.

## 5. Discussion

These measurements confirm the ability of the system to operate a dot-product operation. Comparison of the BER curves for the clean converted idler and the back-to-back cases shows that the operation introduces a minimum increase of the power penalty of about 1 dB at error free level. This added penalty is due to introduction of noise in the reamplification of the idler product. Moreover, the significant decrease in BER due to the action of the dot-product on distorted signals shows that the scheme reliably carries the operation. The drop in signal quality of the recovered channel compared to a clean channel is mostly due to the optical signal-to-noise ratio (OSNR) degradation caused by the coherent subtraction. Indeed, configuration of the system so as to subtract the signals leads to a total idler of lower power.

This OSNR issue limits the range of coefficients practically accessible by our method, according to the configuration implemented. Indeed, choices of coefficients leading to subtraction of signals of similar magnitudes excessively reduce the OSNR.

The coherent dot-product operation requires the incoming signals to be phase-locked. This condition is automatically satisfied provided that the signals originate from the same transmitter, such as a comb source encoded with monolithically integrated modulators. Pump channels must also be phase-locked, however no phase relation is required between signals and pumps. In our experiment pumps and signals both originated from the same source, although this is not required for the technique to function properly. Real world implementation would rely on either a distinct pump source driven by a clock recovery module or copropagation of a clock signal along with the data.

When comparing our distortion technique to crosstalk, it is important to note that this situation does not correspond to realistic interchannel crosstalk. Typical crosstalk profiles usually present strong leakage on the side of the channels and weak in the centre. Moreover, the phase-locking requirement is not fulfilled in case of signals originating from different sources and mixed through an OADM.

## 6. Conclusion

We demonstrated a novel all-optical dot-product scheme between WDM channels and a fixed vector of coefficients using a combination of linear and nonlinear optical processing. We applied the concept to two and three OOK data streams at 40 Gb/s. Observation of the measured sampling scope traces and BER confirms the operation of the system as a field dot-product for the various configurations tested.

## Acknowledgments

The authors acknowledge the Australian Research Council (ARC) Centres of Excellence Program (Project CE110001018), Laureate Fellowship (Project FL120100029), Discovery Early Career Researcher Award (DECRA) (Project DE120101329) and an ARC Linkage grant with Finisar Australia (Project LP120100661).

## References and links

**1. **P. J. Winzer and R.-J. Essiambre, “Advanced modulation formats for high-capacity optical transport networks,” J. Lightwave Technol. **24**, 4711–4728 (2006). [CrossRef]

**2. **K. Azadet, E. Haratsch, and H. Kim, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid State Circuits **37**, 317–327 (2002). [CrossRef]

**3. **L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express **18**, 17075–17088 (2010). [CrossRef] [PubMed]

**4. **P. Bower and I. Dedic, “High speed converters and DSP for 100G and beyond,” Opt. Fiber Technol. **17**, 464–471 (2011). [CrossRef]

**5. **R. A. Athale, “Optical matrix algebraic processors: a survey,” in *10th International Optical Computing Conference*,S. Horvitz, ed. (International Society for Optics and Photonics, 1983), pp. 24–31. [CrossRef]

**6. **X. Wang, J. Peng, M. Li, Z. Shen, and O. Shan, “Carry-free vector-matrix multiplication on a dynamically reconfigurable optical platform,” Appl. Opt. **49**, 2352–2362 (2010). [CrossRef] [PubMed]

**7. **D. E. Tamir, N. T. Shaked, P. J. Wilson, and S. Dolev, “High-speed and low-power electro-optical DSP coprocessor,” J. Opt. Soc. Am. A **26**, 11–20 (2009). [CrossRef]

**8. **P. Yeh and A. E. T. Chiou, “Optical matrix-vector multiplication through four-wave mixing in photorefractive media,” Opt. Lett. **12**, 138–140 (1987). [CrossRef] [PubMed]

**9. **H. Ishikawa, *Ultrafast All-Optical Signal Processing Devices* (John Wiley, 2008). [CrossRef]

**10. **G. Berrettini, A. Simi, A. Malacarne, A. Bogoni, and L. Potí, “Ultrafast integrable and reconfigurable XNOR, AND, NOR, and NOT photonic logic gate,” IEEE Photonics Technol. Lett. **18**, 917–919 (2006). [CrossRef]

**11. **S. Singh and L. Lovkesh, “Ultrahigh speed optical signal processing logic based on an SOA-MZI,” IEEE J. Sel. Top. Quantum Electron. **18**, 970–977 (2012). [CrossRef]

**12. **D. Hillerkuss, R. Schmogrow, T. Schellinger, M. Jordan, M. Winter, G. Huber, T. Vallaitis, R. Bonk, P. Kleinow, F. Frey, M. Roeger, S. Koenig, A. Ludwig, A. Marculescu, J. Li, M. Hoh, M. Dreschmann, J. Meyer, S. Ben Ezra, N. Narkiss, B. Nebendahl, F. Parmigiani, P. Petropoulos, B. Resan, A. Oehler, K. Weingarten, T. Ellermeyer, J. Lutz, M. Moeller, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “26 Tbit/s line-rate super-channel transmission utilizing all-optical fast Fourier transform processing,” Nat. Photonics **5**, 364–371 (2011). [CrossRef]

**13. **M. Freiberger, D. Templeton, and E. Mercado, “Low latency optical services,” in National Fiber Optic Engineers Conference(2012), paper NTu2E.1. [CrossRef]

**14. **D. Schneider, “Trading at the speed of light,” IEEE Spectrum **48**, 11–12 (2011). [CrossRef]

**15. **L. Yan, A. Willner, X. Wu, and A. Yi, “All-optical signal processing for ultrahigh speed optical systems and networks,” J. Lightwave Technol. **30**, 3760–3770 (2012). [CrossRef]

**16. **J. Kurumida and S. J. B. Yoo, “Nonlinear optical signal processing in optical packet switching systems,” IEEE J. Sel. Top. Quantum Electron. **18**, 978–987 (2012). [CrossRef]

**17. **H. J. Caulfield and S. Dolev, “Why future supercomputing requires optics,” Nat. Photonics **4**, 261–263 (2010). [CrossRef]

**18. **A. Willner, O. Yilmaz, J. Wang, X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. **17**, 320–332 (2011). [CrossRef]

**19. **Y. Paquot, J. Schröder, M. D. Pelusi, and B. J. Eggleton, “All-optical hash code generation and verification for low latency communications,” Opt. Express **21**, 23873–23884 (2013). [CrossRef] [PubMed]

**20. **G. P. Agrawal, *Nonlinear Fiber Optics* (Academic, 2001).

**21. **J. Wang, Q. Sun, and J. Sun, “All-optical 40 Gbit/s CSRZ-DPSK logic XOR gate and format conversion using four-wave mixing,” Opt. Express **17**, 12555–12563 (2009). [CrossRef] [PubMed]

**22. **J. Schröder, M. A. F. Roelens, L. B. Du, A. J. Lowery, S. Frisken, and B. J. Eggleton, “An optical FPGA: reconfigurable simultaneous multi-output spectral pulse-shaping for linear optical processing,” Opt. Express **21**, 690–697 (2013). [CrossRef] [PubMed]

**23. **A. Hill and D. Payne, “Linear crosstalk in wavelength-division-multiplexed optical-fiber transmission systems,” J. Lightwave Technol. **3**, 643–651 (1985). [CrossRef]

**24. **J. Zhou, M. O’Mahony, and S. Walker, “Analysis of optical crosstalk effects in multi-wavelength switched networks,” IEEE Photonics Technol. Lett. **6**, 302–305 (1994). [CrossRef]