2.1. All optical OFDM transmitter and receiver designs
When modulated carriers are packed too closely to each other, so that carrier spectra overlap partially, it is critical to demultiplex individual carriers while avoiding crosstalk. An OFDM receiver demultiplexes closely packed carriers, eliminating the intercarrier interference (ICI) at certain time positions. In order to investigate the principle of operation, let us consider a conceptual OFDM demultiplexer model at a receiver, as illustrated in Fig. 2a with 4 carriers (N = 4). An all-optical OFDM receiver implements two main functions: a delay array for serial-to-parallel conversion and a DFT processor [20, 25]. Denoted by and , respectively, the delay and DFT transfer functions of the mth circuit path can be combined to produce the frequency-domain transfer function for the nth carrier as follows:
where ω is the angular frequency, and τ is the chip period. The OFDM symbol period is T = Nτ. Accordingly, the carrier spacing in the angular frequency is defined by . Neglecting the constant time delay factor of , we obtain
Figure 2. (online color at: www.lpr-journal.org) Schematic illustrations of the principle of operation of all-optical OFDM demultiplexer consisting of delay array and DFT circuitry for the output port n = 2, (a), as an example, and its application for a receiver, (b). In (a), the sinusoidal curves indicate two carriers that are delayed and phase shifted.
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In order to gain an insight into the demultiplexing process, Fig. 2 shows two input carriers ω1 = Δ and ω2 = 2Δ, modulated by a rectangle function of duration of 2 × 4τ (two symbols in the same modulation state) arriving at the demultiplexer, as an example. From inspection of Fig. 2, it is evident that the delaying and phasing of the replicas of the carrier ω1 create destructive interference, whereas all the replicas of ω2 sum up constructively. It is also worth observing that complete orthogonality between the two carriers holds only for a certain period, where all the replicas for each carrier interfere destructively. Figure 2b is the corresponding receiver system model in which N optical detectors can be incorporated.
In principle, orthogonality is preserved with any kind of modulation format as long as i) each modulation state lasts for To = Nτ or longer, where the cyclic prefix T ′ = Kτ is defined by temporal duration in excess of Nτ, ii) the rise and fall times of modulation are short enough compared with τ, and iii) the modulation is applied to all carriers with the same modulator clock phases so that all the carriers arrive at the receiver at the same time, in order to arrange ICI from all other carriers such that it is located on the symbol boundaries in the time domain. The resulting symbol configurations are illustrated in Fig. 3, in which the ICI-free region is indicated by the white space in the time domain; Figs. 3a and b depict how ICI free periods are formed by all-optic DFT processes due to destructive interference. This property allows the use of advanced modulation formats such as M-ary phase shift key (PSK), QAM, and even another layer of optical OFDM, as well as simple nonreturn-to-zero on–off keying (NRZ-OOK) for intensity-modulation/direct-detection (IM/DD) systems and differential PSK (DPSK). In this sense, AO-OFDM is modulation-format transparent.
Figure 3. (online color at: www.lpr-journal.org) A schematic illustration of an OFDM symbol structure shown in the time domain; τ is the OFDM sampling space, and To = Nτ and T’ = kτ are the minimum symbol and cyclic prefix periods, respectively, in an N = 4 carrier OFDM system. (a) and (b) represent optical carriers produced by short optical pulses; (c) and (d) by a cw source. The color codes indicate phase rotation of the carrier with respect to a reference carrer frequency, e.g. the first carrier. (b) and (d) compare how cyclic prefixes can be applied in pulsed and cw mode OFDM symbol structures. Symbol values are dectected only in the ICI-free regions.
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Figure 4 shows a more precise numerical analysis to illustrate the orthogonality behavior in a 4-carrier (N = 4) transmission system with a carrier spacing of 10 GHz, corresponding to a symbol period of To = Nτ = 100 ps. In order to show that ICI occurs only when a carrier is modulated, i.e. only on the boundary between symbols, unit step modulation is applied to a carrier at 195.00 THz in the system model shown in Fig. 4a. The corresponding demultiplexed optical waveforms are plotted in Fig. 4b: The target signal waveform at the 195.00-THz carrier frequency is demultiplexed to form a slowly increasing optical power waveform, and ICI waveforms for other carrier output ports of other frequencies are shown in the plots for ICIs. All ICI waveforms last only for a period of To, and hence one can infer that if carriers are modulated at every To, an interference-free time position can be found at every To, as indicated with red arrows in the plots. This behavior in two-carrier experiments was first demonstrated by Sanjoh et al.  and its 30 × 100 Gb/s transmission is achieved by Sano . Because the interference-free interval is narrow compared to a symbol period, as shown in Fig. 3, the demultiplexed symbol should be sampled optically at each output port of an AO-OFDM demultiplexer or electrically after optical-to-electrical conversion in conjunction with digital signal processing. As depicted in Fig. 3d , the modulation symbol period T can be made longer than To, keeping the rise and fall times unchanged, to increase the interference-free range. This is equivalent to adding a cyclic prefix (Fig. 3b) used in conventional OFDM transmission systems [25, 26]. The cyclic prefix is also used to mitigate time-domain impairments in fading radio channels, for example. The same application can mitigate chromatic dispersion effects of the optical-fiber medium [27, 34].
Figure 4. (online color at: www.lpr-journal.org) A numerical example of AO-OFDM demultiplexer characteristics for a 4–10 GHz OFDM system, (a) system model where the laser is modulated to a ‘on’ state at t = 100 ps, (b) demultiplexed signal and ICIs into other carrier ports after demultiplex. Red arrows indicate the time-domain sampling positions for data recovery at the center of each symbol period that corresponds to a 10 Gbit/s data rate.
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An OFDM transmitter is required to produce frequency-locked optical carriers that exactly match the optical frequencies of the OFDM demultiplexer of the receiver. There are several transmitter designs that can generate such frequency arrangements. A comb spectrum source, produced by a modelocked laser or by a continuous wave (cw) laser modulated by an over-driven phase modulator, is the most widely adopted method. A comb spectrum is used to generate optical OFDM carriers through aggregation with a pulse-mode AO-OFDM multiplexer or a cw-mode AO carrier multiplexer.
The most comprehensive transmitter model that can present insights into AO-OFDM communication is a pulse-mode transmitter model with a pulsed optical source as shown in Fig. 5, which gives an immediate analogy with electrically generated OFDM symbols . A short pulse train with a pulse width narrower or equal to τ generated by a modelocked laser diode at a repetition rate B is split into N replicas, and each of them is individually modulated and fed into each input port of the AO-OFDM multiplexer (MUX). An input pulse at an input port i is split into N replicas for all different output ports k = 0, 1, . . N–1. Pulses shown in the boxes of the figure indicate the time positions of optical signal with respect to the OFDM symbol slots. The iDFT block introduces phase shifts between input port i and output port k according to the following expression:
Figure 5. (online color at: www.lpr-journal.org) Conceptual schematic of all-optical ODFM multiplexer of a pulse-mode transmitter design. The spectrum illustrations on the bottom shows the modulation broadening.
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This is the same as the discrete Fourier transform (DFT) used in the electrical OFDM communication. At the outputs of the iDFT block, a set of optical delays are introduced as shown in Fig. 5. This delay array has delays of multiples of τ and N delayed signals are combined by a standard N × 1 coupler, providing parallel-to-serial conversion. The corresponding output of this OFDM symbol includes multiple carriers with designated frequencies. Applying the same analysis used to obtain Eq. (2), we find the OFDM multiplexer function for the ith carrier input as
At the receiver, the same principle of operation is applied, following the system model of Eq. (1). The symbols are delayed by a matched set of an optical delay array, followed by the DFT, output signals filtering, and photodetection to recover the input data.
In order to construct the mathematical model for pulse-mode OFDM signal multiplexing and demultiplexing, let us consider single-pulse propagation through an input port i and output port n at an AO-OFDM multiplexer and demultiplexer, respectively. The frequency-domain amplitude representation of a single pulse for the ith carrier can be defined as , where and denote the data modulation and the modulation spectrum of the pulse, respectively. Here, the pulse width is assumed to be the same as τ. The corresponding OFDM signal is represented as
Then an optical pulse sequentially propagating through the system from the ith port of an AO-OFDM multiplexer to the nth port of a demultiplexer is obtained by
The corresponding time-domain representation of the received and demultiplexed signal can be obtained by a Fourier transform:
Using the Fourier transform , where is the time-domain representation of , and , we obtain the optical field of the demultiplexed signal as
The evaluation of Eq. (7) for is zero at for integer s as long as for . For the case of i = n, i.e. for the target carrier being demultiplexed, we find
When N carriers are multiplexed with data modulation of each carrier , where ais is the data sequence of the ith carrier, the output at the demultiplexer port is given by
It is noted that in a pulse-mode AO-OFDM transmitter, the performance of the optical data modulator can be eased because orthogonality depends on the performance of the AO-OFDM multiplexer and the shape of the pulse.
A cw-mode AO-OFDM transmitter generates a multiplexed OFDM signal either from a comb spectrum source or by a set of frequency-controlled cw sources. Figure 6a shows the key principles of AO-OFDM superchannel generation that can be demultiplexed and demodulated by digital coherent carrier multiplexing [1, 8, 22, 27], which has been widely exploited in high-speed AO-OFDM transmission experiments. At the transmitter, a multitone generator can be used as an optical multicarrier source with phase-locked optical tones at integer multiples of the driving source clock frequency B. Such a multicarrier source can be realized using a mode-locked laser synchronized at its fundamental frequency , concatenated overdriven optical phase modulators [6, 35], or a recirculating loop where a new tone is generated at each pass through the loop . In the case of multicarrier tone generation via wideband frequency modulation, the optical powers of the generated carriers will follow an nth-order Bessel function of the first kind, such that intercarrier power variation can be large. A flexible band WSS  can then be employed with an all-optical carrier demultiplexer to equalize the intercarrier powers, while an optical carrier demultiplexer, such as an AWG, can be used to separate the orthogonal frequency components.
Figure 6. (online color at: www.lpr-journal.org) Conceptual schematic of cw-mode all-optical ODFM transmitter designs by the use of (a) phase-locked tone generation for superchannel generation, and (b) individual free-running cw carrier sources with precise wavelength control. The AO carrier MUXs in both designs can be achieved by various design options. Tones and carriers are generated with exact carrier spacing B. The modulators can use advanced coherent modulation formats at the B baud rate.
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A comprehensive understanding of orthogonality can be extended from the fact that orthogonality is independent of carrier–carrier phase correlation, as inferred from Fig. 4. As shown in Fig. 6b, a set of free-running cw lasers can produce AO-OFDM carriers as long as the laser frequencies are precisely tuned to the receiver AO-OFDM demultiplexer frequencies and the modulation clock phases are aligned to produce orthogonality. In this configuration, however, the instantaneous power waveform of an OFDM symbol is not deterministic because the carriers are not phase-locked. Subsequently, when coherent detection with electrical adaptive equalization by digital signal processing is used at a receiver, the impact of ICI may not be eliminated as it manifests fast beat noise fluctuation due to fast phase rotations of free-running sources. In addition, fiber nonlinearity impairments may not be deterministic, making nonlinear penalty equalization by receiver digital signal processing difficult. Nonetheless, this scheme can also be modulation-format transparent.
A generalized mathematical model for cw-mode AO-OFDM can be obtained by modification of Eqs. (5) and (6). A modulated carrier i with a modulation spectrum Si(ω) = aiΦ(ω−iΔ), where Φ(ω) is the modulation spectrum due to optical data modulation, propagating through the OFDM system to an AO-OFDM demultiplexer at the receiver through the nth port is denoted by
2.2. Subrate OFDM Transmission
In an optical OFDM system, transmission impairments can cause two-fold penalties of intersymbol interference (ISI) and intercarrier interferences (ICI). The ISI penalty in an OFDM system is the same as that of a single-carrier system. On the other hand, the ICI degrades the orthogonal property of an OFDM system, and hence can be more critical than the ISI penalty. As an example, orthogonality degradation due to fiber chromatic dispersion (CD) penalty is more severe if the OFDM symbol bandwidth exceeds 100 GHz. In order to model an example of CD penalty in an AO-OFDM demultiplexing, Eq. (10) can be modified as
Here, the fiber CD transfer function is used, where ; D, L, λ, and c, are the fiber CD coefficient, fiber length, wavelength, and speed of light, respectively. Using the system model of Fig. 4a with a fiber inserted in front of the demultiplexer, the ISI and ICI behaviors are evaluated by a numerical Fourier transform of Eq. (13). This data shows a 4-carrier example where the carrier at 195.00-THz turns on at t = 100 ps, and the corresponding ISI and ICIs are observed at the receiver AO-OFDM demultiplexer ports as shown in Fig. 7. From the data plots, one can see the ISI and ICI are spread in the time domain, making the orthogonal property poor. In order to mitigate such interference penalties, the OFDM carrier baud rate BC can be lowered below B so that detection points can be sampled at a lower rate as can be seen in comparison of Fig. 4b and Fig. 7. Such subrate OFDM transmission can be achieved by cyclic prefix insertion and subrate (BC <B) modulation in the pulse-mode [27, 34] and cw-mode [18, 25, 26] AO-OFDM transmission systems, respectively. The use of subrate modulation is also equivalent to the insertion of a temporal guard band between symbols.
Figure 7. (online color at: www.lpr-journal.org) AO-OFDM symbol propagation in dispersive fiber observed at different output ports of the AO-OFDM demultiplexer with 20 km of SMF28 fiber inserted in the setup shown in Fig. 4a, for comparison with the waveforms of Fig. 4b. Red arrows indicate a reasonable choice of the subrate sampling positions for data recovery.
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2.3. Superchannel digital coherent demultiplexing
A superchannel can be regarded as a bank of AO-OFDM carriers that can be generated by the scheme discussed in Fig. 6a of Section 'All optical OFDM transmitter and receiver designs'. Figure 8 illustrates the key principle of digital coherent carrier demultiplexing of AO-OFDM superchannel, which has been widely exploited in high-speed AO-OFDM transmission experiments. In this scheme, optical carriers are modulated in parallel using a bank of optical modulators (Fig. 6a). As aforementioned, the typical modulation format used for each optical carrier can be arbitrary; specifically, either OFDM or single-carrier (SC) modulation in the form of M-ary PSK (M-PSK) or M-ary quadrature amplitude modulation (M-QAM) can be used on each optical carrier, without loss of generality. In the case of SC transmission per optical carrier, the baud rate per carrier should be less than or equal to the frequency spacing of the all-optical carriers B to ensure digital carrier demultiplexing can be performed with low distortion, as aforementioned. For the special case of zero-guard-interval OFDM, . It is also noted that, apart from the all-optical tone generator, the AO-OFDM approach with SC per-carrier modulation can use identical transmitters and receivers as those developed for 100 Gb/s coherent systems, which is attractive from the system-upgrade point of view. The aggregate data rate of the AO-OFDM superchannel can then be scaled by simply increasing the number of optically generated carriers and SC transceivers.
Figure 8. (online color at: www.lpr-journal.org) Digital coherent AO-OFDM carrier demultiplexing. LO: local oscillator; PD: photodetector; ADC: analog-to-digital converter; FDE: frequency domain equalizer; TDE: time-domain equalizer; CR: carrier recovery; SD: symbol detection.
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At the receiver side, to detect an all-optical carrier of interest, a cw local oscillator (LO) laser can be tuned near the carrier center frequency, essentially downconverting the target optical signal to the electrical baseband by an optical hybrid. Following such LO-based downconversion, a standard coherent receiver front-end can be used for digital coherent carrier demultiplexing . If the bandwidth of the coherent receiver front-end is sufficiently wide, it is also possible to detect multiple carriers per transceiver by placing the LO midway between the target carriers and using DSP to accomplish any residual baseband shifting prior to digital impairment compensation.
To process the target AO-OFDM carriers with low distortion, the analog-digital converter (ADC) bandwidth and oversampling rate must be high enough to capture sufficient energy of main and side lobes (Fig. 8). It is also noted that the ADC bandwidth and oversampling rate requirements also increase with constellation size. Following digitization, residual CD must be first compensated using a frequency-domain equalizer (FDE)  to restore orthogonality between the carriers. For a DSP sampling rate that is M times the carrier symbol baud rate, the carrier of interest can be demultiplexed digitally by initializing an adaptive time-domain equalizer (TDE)  with coefficients that are equal to the first column of an M-point FFT matrix. In other words, the TDE coefficients are initialized so as to have a frequency response of an M-point FFT, where M < N is equivalent to the number of AO-OFDM carriers being jointly processed in DSP. This is done in order to have a sufficiently high oversampling rate so as to perform the partial digital coherent carrier demultiplexing without incurring too much penalty. It is noted that this differs from conventional OFDM, wherein the entire OFDM band is captured and a single-joint FFT is used for carrier demultiplexing. It is also noted that the TDE both demultiplexes the carrier(s) of interest and equalizes the residual impulse response of the channel. The TDE coefficients can subsequently be adapted using any commonly used algorithm for SC systems, such as common modulus or decision-directed algorithms . Finally, carrier recovery (CR) and symbol detection (SD) are performed. Consequently, notwithstanding potentially higher oversampling requirements and the need for proper initialization of the adaptive TDE, digital carrier demultiplexing for AO-OFDM can be performed largely using the same DSP algorithms as those exploited in current-generation 100 Gb/s digital coherent transmission systems.