High or extended data rate optical fourier transform method and apparatus

Abstract

A method of operating an optical Fourier transform 'OFT' apparatus, the OFT apparatus comprising N input channels and K output channels, the method comprising: quantising the light at the input channels to form a quantised output at each output channel.

Description

High or Extended Data Rate Optical Fourier Transform Method and ApparatusFieldThe present disclosure relates to an optical Fourier transform apparatus and a method of operating an optical Fourier transform apparatus.BackgroundOptical and photonic computing approaches promise to perform mathematical operations such as Fourier transforms at much higher speed whilst consuming much lower energy compared to performing similar operations using electronic circuits. Optical Fourier transform (OFT) is traditionally performed using free-space optics and, more recently, integrated photonics.Free-space OFT systems are well known and comprise a Fourier transform lens that is illuminated with beams of coherent light, either arranged in one or two dimensions. The output from the lens is OFT of the input beams.A photonic integrated circuit (PIC) for OFT comprises an integrated photonic device capable of performing one-dimensional OFT. The input to the OFT device is an array of input waveguides or light channels, which carry light.Inside the OFT device the light from the input waveguides or light channels undergoes diffraction, the properties of which are governed by the geometries of the OFT device. The OFT device geometries, including the placement of the input and output waveguide arrays, are so selected such that diffracted light forms an interference pattern within the OFT device. The diffracted light illuminates the output waveguide array. In other words, an output optical field at the output waveguide array is the OFT of the optical field incident to the OFT device that was inserted at the input waveguide array.OFT is an analog process, the output of which is dependent on the number of inputs and the input values themselves. In a system with a large number of inputs and with a large dynamic input range, the output of an OFT is quasi-continuous over a large dynamic range.Whilst the analog nature of the OFT lends itself to its large bandwidth, it imposes considerable challenges in designing a suitable electronic transmitter (or encoder) and receiver (or decoder) to fulfil the capabilities of the OFT device or PIC.As such, OFT devices or PICs require high precision analog to digital converters (ADCs) for digital processing of the OFT data.The large dynamic range of OFT devices and PICs and the need for high precision ADCs limit the operating frequency of the system, because they impose restrictions on the speed of the modulation of the inputs and sampling of the outputs. An increase in the operating speed will increase the noise at the receiver, and will therefore require a higher optical signal at the inputs to the OFT in order to achieve the same signal to noise ratio.Due to these constraints, the speed of modulation of the input functions to the OFT device is almost an order of magnitude lower than the speed at which the photonic devices delivering the input functions can be driven.SummaryAn invention is defined in the appended claims.Brief description of the drawingsThe following description of the disclosure will be better understood when read in conjunction with the appended drawings. It should be understood, however, that the disclosure is not limited to the precise OFT apparatus arrangements, the methods of operation thereof nor the precise logic or data formats shown. In the drawings:Figure la shows a schematic of an optical Fourier transform apparatus;Figure lb shows a schematic of an optical Fourier transform apparatus showing an arrangement of input and output ports;Figure 2 shows an optical circuit diagram of a photonic integrated circuit including an optical Fourier transform apparatus;Figure 3 shows a pair of Argand diagrams showing unlimited continuous possible values across the real and imaginary space for both the input states and output states of an optical Fourier transform apparatus;Figure 4 shows pair of Argand diagrams showing a limited range of discrete or quantised values across the real and imaginary space for both the input states and output states of an optical Fourier transform apparatus;Figure 5 shows a pair of Argand diagrams showing a limited range of more heavily quantised values across the real and imaginary space for both the input states and output states of an optical Fourier transform apparatus;Figure 6 shows a pair Argand diagrams depicting possible input and output states of a deterministic OFT system for a quad-binary optical Fourier transform device;Figure 7 shows four pairs of Argand diagrams, each diagram depicting possible input or output states at a particular input or output port among four input ports and four output ports of a deterministic OFT system;Figure 8 shows sixteen diagrams depicting the real parts of output states of an OFT system having four input ports and four output ports;Figure 9 shows sixteen diagrams depicting the imaginary parts of output states of an OFT system having four input ports and four output ports; andFigure 10 shows a truth table mapping input vectors to expected output vectors for a quadbinary OFT system.Figure 11a is a block diagram of an example optical Fourier transform system.Figure lib is an exploded view of the part of the block diagram that depicts the host and its various components of the example optical Fourier transform system depicted in Figure 11a. Figure 12 is a flow diagram of an example method of processing an output signal from an optical Fourier transform apparatus, which can be implemented in the optical Fourier transform system of Figure 11a.Figure 13 is a flow diagram of an example method of obtaining a full complex optical Fourier transform result from an optical Fourier transform apparatus, which can be implemented in the optical Fourier transform system of Figure 11a.Figure 14 is a flow diagram of an example method of processing data using a Fourier transform apparatus, which can be implemented in the optical Fourier transform system of Figure 11a. Figure 15 is a flow diagram of an example method of processing data using a Fourier transform apparatus, which can be implemented in the optical Fourier transform system of Figure 11a.Figure 16 is a flow diagram of an example method of operating an optical Fourier transform apparatus or system comprising N input channels and K output channels, for example the optical Fourier transform system of Figure 11a.Figure 17 is a truth table showing example input-output combinations for an optical Fourier transform apparatus, such as the optical Fourier transform apparatus of Figure 11a, wherein raw output data is modified by a carry add method to form a PAM4 signal and the raw output data is subsequently recovered from the PAM4 signal.Figure 18 is a truth table showing example input-output combinations for an optical Fourier transform apparatus, such as the optical Fourier transform apparatus of Figure 11a, wherein raw output data is modified by a first saturation method to form a PAM4 signal and the raw output data is subsequently recovered from the PAM4 signal.Figure 19 is a truth table showing example input-output combinations for an optical Fourier transform apparatus, such as the optical Fourier transform apparatus of Figure 11a, wherein raw output data is modified by a second saturation method to form a PAM4 signal and the raw output data is subsequently recovered from the PAM4 signal.Figure 20 is a truth table showing example input-output combinations for an optical Fourier transform apparatus, such as the optical Fourier transform apparatus of Figure 11a, wherein a first subset of outputs is derived from a second subset of outputs based on a known relationship between them.OverviewThe present disclosure seeks to address the above problems and related problems with OFT apparatuses and methods for operating such apparatuses. This overview is not intended to be limiting and various modifications and variants of the described methods and apparatuses are encompassed in addition to those described.Deterministic, quantised OFT apparatus and methodRather than providing a continuous input to the waveguides or input ports, in embodiments the input to the OFT is quantised or discretised at each input channel so that the input optical field has only a finite number of allowable values and a finite number of combinations of values at the finite set of input ports. The discretisation (or quantisation) of the input channels causes a discretisation (or quantisation) of the output optical field at the output channels. The output is also deterministic and if the input values are known, the output values can be predicted in advance.Predictable output values existing only in discrete levels or bands allows a very accurate and reliable determination of the output OFT without the need for high precision receivers / decoders and associated ADCs. Similarly, discretisation of the input allows the input optical field to be created without the need for high precision transmitters / encoders and associated DACs. A receiver with a lower effective number of bits (ENoB) can therefore be used compared with analogue OFT devices. Lower ENoB allows faster throughput because the receivers can operate at higher frequency, increasing bandwidth and data throughput. Furthermore, the predictability of the output for a giveninput can allow for reliable integrated testing and calibration of the OFT apparatus to provide a basis for more reliable and robust data processing. Further still, when the output is restricted to a low number of output values (or levels or states), data can be packaged and sent more easily over widely- used telecommunications links.That is, in an analogue OFT apparatus, the possible states or values transmitted from (or encoded at) each input channel at the input ports are continuous and can have any value within the dynamic range of the emitters in the input channels. Consequently, the possible states or values received (or decoded) at the output channels is not discretised (also continuous) and can have any values dictated by the OFT of the input optical field within the dynamic range of the OFT device.In order to determine the OFT of such an analogue optical function, the receivers (or detectors or decoders) in the output channels must be capable of determining the optical signal to a high degree of accuracy and precision. In other words, the receivers must have a high resolution and produce repeatable readings for the OFT apparatus to be able to process accurately over many frames.The inventors have recognised that a quantised output signal can be obtained at each output channel by controlling the optical signal at the input channels such that each input channel produces a quantised signal with a limited number of states or levels. In this way, the output becomes (more easily) deterministic because the output can be mapped by predicting the signal state at each of the output channels based on the each possible pattern of input states. In particular, the real and imaginary values at the output channels can be detected separately to determine the full complex values at each output channel. In this way, the output OFT of the input function is determined for a single frame.Furthermore, because the output is quantised, the receivers or decoders at each output channel can determine the correct output states faster or more accurately (or both), or a lower resolution or lower fidelity receiver or decoder can be used. Hence, a more accurate, higher frequency and / or cheaper OFT apparatus is obtainable.That is, the possible input values (or input states) to an OFT device or PIC is traditionally not bounded or restrictive, and can have any value. However, in the OFT apparatus and methods described herein, the input and output values in an OFT are optimised such that the output of the OFT :• Is not a continuum of numbers but is discretised or quantised. This means that when noise is introduced at either the transmitter or receiver, as long as the separation between quantised levels is greater than the noise level, the result can still be sorted into the quantised states and is deterministic.• Can be represented by a finite number of output quantised states with a uniform difference between quantised states.• Can allow for an offset or adjustment of the quantised states in multiples of the difference between quantised states. This can allow reusable receiver designs with a pre-determined or known offset.• Can be constructed as a truth table as a function of the input states.• Can have a separation in quantisation states proportional to the separation of the input states (inherits the separation between states from the input, scaled down by any losses in the OFT device)• Can have a larger dynamic range than the inputs, with the maximum dynamic range of any output channel scaling with the number of inputs.An advantage can include that the electronic drivers to the OFT can be constructed using an identical transmitter or encoder electronic circuit for each input channel in the OFT, with offsets or adjustments, and / or using an identical receiver or decoder electronic circuit for each output channel in the OFT, with simple offsets or adjustments, though the present disclosure is not limited thereto.The method of controlling the OFT apparatus include any one or a combination of the following:• Restricting the number of inputs to the OFT, whilst keeping the number of inputs to be the same as the number of outputs from the OFT device.• Restricting the dynamic range of the inputs and in each input channel.• Introducing asymmetry in the dynamic range and discretisation between the inputs and the outputs.• Offsetting or adjusting the output results for some or all output channels such that the dynamic range for all channels is a subset of the dynamic range of the entire OFT outputs.The OFT apparatus may include:• same transmitter or encoder electronic driver for each input channel.• the same receiver or decoder electronic driver for each output channel.• a transmitter or encoder, and receiver or decoder electronic drivers such that they permit highest operating frequency, for example by controlling the ENoB of the receiver in any of the ways described herein.There is therefore provided a method of operating an optical Fourier transform 'OFT' apparatus, the OFT apparatus comprising N input channels and K output channels, the method comprising: quantising the light at the input channels to form a quantised output at each output channel.There is also provided an optical Fourier transform apparatus comprising: N input channels and K output channels. The input channels each comprise a modulator, transmitter or encoder arranged to cause the input channels to emit a quantised input signal. The output channels each comprise a receiver arranged to detect a quantised state at the output channelThe OFT apparatus (or OFT device) can be any optical apparatus capable of performing an optical Fourier transform of an input optical field delivered via two or more input channels and received at a corresponding number of output channels. This can take the form of a waveguide array including N input channels, the output of which is split into N mixing channels, nth mixing channels from each input channel are then joined to form K output channels. However, a more efficient arrangement is a free-space OFT apparatus in which N input channels deliver light into a free space region which allows the light from the input channels to diffract and create an interference pattern at K output channels. N is an integer of at least 2, optionally an integer multiple of 2 or 4, optionally 4. K is at least equal to N. Examples of an OFT apparatus are described with reference to Figures la, lb and 2, though the present disclosure is not limited to these examples.For an input vector comprising the frame of input states simultaneously present at the input channels, the full discrete optical Fourier transform of the frame of input states is derivable from the output states detected at all output channels. In this sense, the OFT apparatus is arranged to perform a discrete Fourier transform. The discrete Fourier transform satisfies equation 1:(Equation 1) where k is the index of output channel, Xk is the channel k output, n is the index of input channel, xnis the channel n input, N is the number of input channels.Quantising the light at the input channels is to limit the possible input states of the light at each input channel so that each state is separated. In other words, quantising the light at the input channels comprises limiting the light emitted by each input channel to a finite number of possible input states. In an OFT system which suffers from noise, the possible states are separated by an appropriately-sized interval relative to the noise so that they are reliably discernible from each other by any detector or monitor connected to the input channels. That is, an appropriate signal to noise ratio is obtained by selecting an appropriate quantisation interval. Examples of quantisation of input states leading to quantisation of output states, as represented on an Argand diagram, are described with reference to Figures 4 and 5, though the present disclosure is not limited thereto.Limiting to two possible input states per input channel allows a binary input to each input channel so that data in the form of a single bit can be delivered to each input channel at any one time. Alternatively or in addition, the input channels are controlled so that the possible input states for one input channel are identical to those for every other input channel. These measures separately and in combination provide a simplified interface with an electronic system controlling the input channel states and a simplified data flow through the OFT apparatus. They also maximally restrict the number of possible output states received at the output channels, leading to simplification of the data extraction from the OFT apparatus. Examples of maximally restricted (i.e. binary) input states and the effect on output states are described with reference to the Argand diagrams of Figures 6 and 7. However, the possible input states are not limited to binary. For example, 3 or even 4 possible input states is possible while still ensuring useful quantisation of the output.Limitation of the possible input states can be through configuration of one or more transmitters in each input channel and / or the electronics driving the one or more transmitters so that they can only input a limited number of input states. Alternatively, or in addition, the data sent to the one or more transmitters can be limited so as to limit the possible input states.The possible input states are limited in phase and / or amplitude of the light carried by the input channels. For example, the possible input states comprise or consist of: a first input state having a first amplitude and a second input state having a second amplitude different from the first amplitude, wherein the first input state is in phase with the second input state. In other words the input states are limited by amplitude shift keying (ASK). In another example, the possible input states comprise or consist of: a first input state having a first phase and a second input state having a second phase out of phase with the first phase, wherein the first input state has the sameamplitude as the second input state. In other words, the input states are limited by phase shift keying (PSK). In yet another example, the possible input states comprise or consist of: a first input state having a first phase and a first amplitude and a second input state having a second phase out of phase with the first phase and a second amplitude different from the first amplitude. In other words, the input states are limited by binary phase shift keying (BPSK).Increasing the separation of the possible input states can improve the signal to noise ratio in the system. This can be achieved through the configuration of the one or more transmitters or the electronics controlling them, or can be controlled via the data sent to the one or more transmitters. For example, in any of the ASK, PSK or BPSK examples, the first state and second state are separated by two or more discernible states, the discernible states determined by the signal to noise ratio and resolution of the one or more transmitters (and / or the associated control electronics).For maximal effect, the first input state is a lowest possible input state of the input channel and the second input state is a highest possible input state of the input channel. Otherwise, the first input state has a maximal possible separation from the second input state - for example, the first phase is 90 or 180 degrees out of phase with the second phase.The possible input states can be limited in any of the ways described so as to ensure that the input states are unsigned real or unsigned imaginary states.Limiting the number of possible input states for each input channel restricts the number of possible input vectors. The input vector is defined as the representation of the input optical signal applied to the OFT apparatus by the full set of input channels. The possible input vectors are limited by the combinations of possible input states of the input channels. The number of possible input states can be limited such that the set of all input channels can display only MNdistinct input vectors, where each combination is in turn a N logjM-bit combination, wherein each input vector is represented by the set of input states applied at the input channels in a single frame. For example, for a quadbinary system (four input ports and binary input states) the possible input vectors may be represented as

[0000] to

[1111] and the fourteen vectors in between as described with reference to Figure 10. M is the number of levels in an input and is a positive integer. A frame is a snapshot in time or a sample period during which the input vector is constant. The input vector can change from frame to frame, causing an output vector received at the set of output channels to change from frame to frame.The output of the optical Fourier transform stage is deterministic such that the finite number of possible input states gives rise to a finite number of possible output states at each output channel. In this way, the limitation of possible input vectors limits the number of possible output vectors. The output from each channel in the OFT device or PIC can be represented by a subset of the quantised values that represents the full result set of the OFT device or PIC. The output can be offset or adjusted by any multiples of the difference between quantised states to form the output values of OFT. Examples of how the output vectors are limited for a binary input system having four input and four output ports are described with reference to Figures 8 and 9, though the present disclosure is not limited thereto.By limiting the number of possible output vectors (via limiting the number of possible input vectors), the output receivers need not have a high resolution. It is still possible to use high resolution receivers, but these may be replaced by those of lower resolution to provide improvements in bandwidth. Lower resolution receivers may be capable of receiving only discernible output states, the discernible states determined by the signal to noise ratio and resolution of the one or more receivers (and / or the associated control electronics). The number of discernible output states of the receiver can be described by the ENoB of the receiver. By limiting the receiver ENoB, the data throughput of the system or the operating frequency of the receiver (and hence the OFT apparatus) can be increased.That is, the receiver ENoB need not be limited, however to increase the maximum operating frequency of the OFT apparatus it can be useful to do so. For maximum operating frequency, the ENoB can be limited so that the number of levels of each output channel receiver is equal to the maximum number of levels of the output signal generatable by the quantised input. To this effect, the effective number of bits 'ENoB' for a receiver in each of the output channels is equal to log2(P+Q), wherein P is the total number of possible output states at the respective output channel given the possible input states, wherein Q is the number of spare levels (equal to 0 for minimum ENoB). Additional levels over and above the minimum required to read the output signal can be added. In other words, Q can be greater than zero, for example 1, 2, 3, 4, 6, 8 or 10. This can reduce the need to offset the receiver to encompass all levels of the output signal or can allow the OFT apparatus to be still usable if the number of possible input states is increased. Thereby, increasing Q can improve the ease of configuration or increase the flexibility of use of the OFT apparatus.The ENoB does not necessarily need to be restricted to such a degree. A lower ENoB can of 12 or less, optionally 10 or less, optionally 8 or less, optionally 6 or less, optionally 4 or less, optionally 3 or less, optionally 2 or less can provide increase in operation frequency compared to more continuous or pseudo-continuous receivers.The spacing of levels in the receiver matches the spacing of the possible output states. For ENoB values greater than the minimum (i.e. when Q > 0) the spacing of receiver levels either matches the spacing of the possible output states (leaving spare levels above and / or below) or the spacing of receiver levels is a fraction (with rational denominator) of the number of possible output states.The possible input states are controlled such that there is a uniform difference between: a first output state and a second output state of the possible output states, and between the first output state and a third output state of the possible output states. Alternatively, or in addition, there is a uniform difference between: a first output state and a second output state of the possible output states; and between a third output state of the possible output states other than the first and second output states and a fourth output state of the possible output states other than the first, second and third output states. The output receiver configuration can be simplified if there is a uniform difference between all adjacent output states of the possible output states; that is, if the output of the OFT is represented by a finite number of output states with a uniform interval between output states. The adjacent output states are real output states or imaginary output states.The difference between adjacent quantised output states for each output channel is proportional to the difference between adjacent quantised input states for each input channel. The constant of proportionality is proportional to losses in the OFT device or an allied system or component (for example, DAC or ADC systems, amplifiers, filters or analog equalisers) connected to the OFT apparatus.OFT data packingThe present disclosure includes a method of handling the raw output data from an optical Fourier transform apparatus so that it can be processed more easily and efficiently and without losing any information relating to the result of the optical Fourier transform. The method involves modifying the raw output data so that it is PAM4 compliant. In other words, the present disclosure includes a method of processing an output signal from an optical Fourier transform apparatus comprising:applying a input vector to the optical Fourier transform apparatus, obtaining an output vector representing the optical Fourier transform of the input vector; modifying a channel (i.e. level or value) of the output vector to form a PAM4 signal.Depending on the configuration of the OFT apparatus, the raw output data may not be naturally PAM4 compliant. The raw output data may be in in another format other than PAM4 due to the number of possible output states receivable at an output channel of the OFT apparatus. For example, if the number of possible output states receivable is five, then the raw output data may be said to be (or include) a PAM5 signal (one having 5 possible states or levels or values). That is, the number of levels in the channel can be greater than four, optionally wherein the channel carries a PAM5 signal (one having five possible levels). Modification of the channel includes monitoring the fifth (or a higher than fourth) level to produce the PAM4 signal.The raw output data that is modified may have some channels which do not need to be modified to become PAM4 compliant. Other channels of the raw output data are not PAM4 compliant and it is these channels which can be modified according to the methods described herein. A channel of the output data (or output signal) may be understood to be the data that is received at a physical output channel of the OFT apparatus.The raw output signal includes the full complex result of the optical Fourier transform from the optical Fourier transform stage. The channel can represent the real component of the optical Fourier transform. In a quad-binary OFT apparatus, the first or third output channel of the optical Fourier transform apparatus represent the real component of the Fourier transform result. Due to the range of the possible output vectors in such a system, the real component at the first and third channel comprises a PAM5 signal. That is, for some input vectors, a signal at the fifth level is received at the first or third channels. This is illustrated in e.g. the truth table of Figure 10 for the input vector (1010) and (1111), for which the output vectors are (2020) and (0040) respectively (note that output channel Outl in this table has five levels ranging from -2 to 2 and output channel Out2 has five levels ranging from 0 to 4).Modifying the raw output data is carried out on a frame by frame basis. In each frame, the raw output data of the OFT apparatus includes an input vector which defines the input state at each input channel as a function of input channel position and an output vector which defines output state at each output channel as a function of channel position Thus a vector is a matrix, wherein theposition in the matrix corresponds to a physical channel position in the OFT apparatus (which can have a circular offset). The output states are quantised as described throughout the present disclosure. For a one-dimensional OFT device having N input ports the vector can take the vector form (output state at channel 1 .... output state at channel N). Examples of vectors are provided in the truth table of Figure 10 if each row of output data is represented in the aforementioned vector form (e.g. (1030)).The raw output data can be modified via any one of a number of methods. Methods include acting on the non-compliant channel of the raw output data to remove one or more levels so that the number of levels is reduced to four or on the compliant channel of the raw output data to add one or more levels. When there are more than four levels, modifying the channel includes detecting the highest level(s) of the channel and removing the highest level(s) from the channel. When there are fewer than four levels, modifying the channel includes adding one or more levels. One example is if the channel of the raw output data is a PAM5 signal, the method includes removing the highest (fifth) level, or removing the lowest first level and offsetting the remaining levels by minus 1.Another example is if the channel of the raw output data is a PAM3 signal, the method includes adding one level to the channel to form a PAM4 signal. Adding one level to the PAM3 signal may involve increasing the value by more than one if the value on that channel is on the first or second level.Removing the highest level(s) from the channel comprises: saturating or clipping the highest level(s) to the fourth level, or subtracting one level or more levels to form the fourth level. In the PAM5 example, the fifth level can be removed by simply subtracting one level from the fifth level of the signal, but can also be removed by, for example, saturating the fifth level to make it the fourth level. In more general terms, the higher levels (fifth and greater) of the non-compliant channel are saturated. Saturating the higher levels can be done on the receiver itself by reducing the dynamic range of the receiver in an output channel of the OFT apparatus so that the higher levels are clipped, or by post processing the raw output data, for example by amplifying and / or applying a limiter to achieve saturation.Different channels of the raw output signal may have different numbers of levels, some more than four, some fewer than four. In such cases, a carry add logic can be applied to a pair of channels - a first channel having more than four levels and a second channel having less than four levels. One or more (or all) of the excess levels over four can be removed (using subtraction or saturation) from thefirst channel and then added to the second channel so that both channels are PAM4 compliant. The carry add logic can be applied so that it creates a unique output vector - one in which the values across the channels is a unique combination among all the possible output vectors obtainable by the OFT apparatus, given all the possible input vectors.Once the channel has been modified to form the PAM4 signal, the PAM4 signal can then be sent over a PAM4 link. In this sense, the signal has become PAM4 compliant. This opens possibilities to use the optical Fourier transform apparatus as part of a communications channel, allowing data to be processed inline in the communication channel.A detailed example of a data packing method is described with reference to Figure 12.Data unpackingThe raw output data can be recovered from the PAM4 signal by reversing the modification of the output signal, for example by reversing the modification of the channel of the output signal. In other words, the method can include retrieving the correct full complex OFT data from the PAM4 signal by reversing the modification of the channel.Where one or more levels were originally removed from the raw output data to form the PAM4 signal, the correct full complex OFT data comprises deriving the one or more removed levels from the PAM4 signal. For example, where the fifth level was originally removed from the raw output data to form the PAM4 signal, the correct full complex OFT data comprises deriving a fifth level of a PAM5 signal from the PAM4 signal. This can be achieved via a logic operation on the PAM4 signal. For example in the case of a carry add operation on the channel to form a unique output vector including the PAM4 signal, the raw output data can be recovered with the knowledge of the relationship between the unique output vector and the carry add logic applied. Reversing the carry add logic provides the raw output data.Through knowledge of methods for modifying the output data to form a PAM4 signal in the ways described throughout this disclosure, deriving the raw data from the PAM4 signal can be carried out by an entity receiving data, for example over a PAM4 link, even if that entity was not that which carried out the optical Fourier transform. To this end, the present disclosure also includes a method of processing an output signal from an optical Fourier transform apparatus, the method comprising: receiving the output signal from the optical Fourier transform apparatus in a form comprising aPAM4 signal; retrieving the full complex output of the optical Fourier transform apparatus from the output signal. This can be achieved by decoding the PAM4 signal, for example by deriving a signal with a number of layers greater than four from the PAM4 signal, optionally deriving a PAM5 signal from the PAM4 signal. Alternatively decoding the PAM4 signal includes deriving a signal with a number of layers fewer than four from the PAM4 signal, optionally deriving a PAM3 signal from the PAM4 signal. The full complex output of the optical Fourier transform apparatus retrieved from the PAM4 signal can be defined at least in part by the PAM5 and / or PAM3 signals.A detailed example of data unpacking to retrieve the raw output signal from a signal including a PAM4 signal is described with reference to Figure 15.Logical derivation of outputsThe present disclosure also includes determining the Fourier transform of some input vectors in a data stream using optical means and other input vectors in the same data stream by other (non- optical) means, in other words without carrying out an optical Fourier transform of the other input vectors. The inventors have recognised that although in general an optical Fourier transform can process input vectors at a much higher frequency than electronic means, overall data throughput for OFT data can be increased yet further by selectively avoiding performing an optical Fourier transform for a subset of input vectors in the data stream and using other means instead.There is therefore provided a method of processing data using an Fourier transform apparatus, the method comprising applying an input data stream to the Fourier transform apparatus, wherein the input data stream is applied in a series of sequential frames, each frame comprising an input vector. The method further comprises producing an output data stream from the input data stream by determining an output vector for each frame, wherein each output vector is the Fourier transform of the corresponding input vector in the input data stream. For a first input vector having a first value, a corresponding first output vector is determined via an optical Fourier transform of the first input vector. For a second input vector having a second value different from the first value, a corresponding second output vector is determined other than via an optical Fourier transform of the second input vector.For the second input vector, the corresponding second output vector can be determined via a logic operation applied to the second input vector. For example, if the second input vector has an overall value of zero, the corresponding second output vector is determined by copying the second inputvector to produce the second output vector or by determining that the second input vector also has an overall value of zero. That is, the particular input vector for which the Fourier transform is performed other than by optical means can have an overall zero value. In such a case the resulting output will have an overall value of zero too, in which case the input vector is simply copied to form the output vector or otherwise the output vector is determined logically, digitally or electronically from the input vector to have an overall value of zero.Alternatively, for the second input vector, the corresponding second output vector is determined via the optical Fourier transform of the first input vector. In this case, the second input vector is codified in the input data stream as a repeated input of the first input vector and the second output vector is determined logically from the optical Fourier transform of the first input vector. The second input vectors can be those which have closely matched corresponding output vector with the first input vector that has an output vector which is, by contrast, determined optically using the OFT apparatus. In this case, the second output vector can contain the same corresponding vector values as the first output vector except for one differing vector value, wherein the second output vector is determined logically from the first vector via a known relationship between the differing vector values. The known relationship can be an inverted sign.A detailed example of logical derivation of outputs is described with reference to Figure 14.Optimisation of receiversThe overall receiver configuration can be simplified by determining the form of all the possible output states at each output channel. The form of all the possible output states can be determined theoretically given the number of input and output channel and all the possible input states at each input channel. This may be carried out mathematically from a discrete Fourier transform equation such as Equation 1 or electronically. The output vectors for each input vector may be stored in a lookup table or other reference database. The lookup table can contain all possible output vectors. With knowledge of all possible output vectors, the inventors have recognised that it is possible to simplify the receivers in the output channels by recognising redundancies in the output or to make use of only one receiver per output channel and yet still determine the real and imaginary components of output states at every output channel. In addition, the inventors have recognised that independent detection of the imaginary values and real values at the output can also simplify the receiver configuration.For example, each output channel has a receiver which differs from the receiver of at least one other output channel. Alternatively, or in addition, at least one of the output channels includes a receiver for recovering real data only and another of the output channels includes a receiver for recovering imaginary data only, for example three of the output channels include a receiver for recovering real data only and another one of the output channels includes a receiver for recovering imaginary data only. Alternatively, or in addition, the number of levels detectible by the receivers may differ. For example, the ENoB of the receiver of one of the output channels differs from the ENoB of the receiver of another one of the output channels. Optionally, each output channel has only a single receiver. The receivers compared are those for detecting the OFT result, rather than for any other purpose such as monitoring.There is therefore provided a method of processing obtaining a full complex optical Fourier transform result from an optical Fourier transform apparatus, the optical Fourier transform apparatus comprising first to fourth output channels. The method comprises determining the real components of a channel output signal in each of the first, second and third output channels, and detecting the imaginary component of a channel output signal in the fourth output channel. The fourth output channel is different from the first, second and third output channel. The method further comprises determining the full complex result by obtaining the real and imaginary components of all four output channels using only the detected real components from the first second and third output channels and the detected imaginary component from the fourth output channel.The operations applied to the fourth and second channels can be swapped so that the method comprises determining the real components of a channel output signal in each of the first, fourth and third output channels, and detecting the imaginary component of a channel output signal in the second output channel. The second output channel is different from the first, fourth and third output channel. The method further comprises determining the full complex result by obtaining the real and imaginary components of all four output channels using only the detected real components from the first, fourth and third output channels and the detected imaginary component from the second output channel.Optionally, obtaining the imaginary component of the channel output signal in the second output channel can include applying a logic operation to the imaginary component detected in the fourth output channel. Or, in an alternative arrangement, obtaining the imaginary component of thechannel output signal in the fourth output channel can include applying a logic operation to the imaginary component detected in the second output channel.Alternatively, or in addition, obtaining the real component of the channel output signal in the fourth output channel includes applying a logic operation to the real component of the channel output signal detected in the second output channel. Or, in the alternative arrangement, obtaining the real component of the channel output signal in the second output channel includes applying a logic operation to the real component of the channel output signal detected in the fourth output channel.Alternatively, or in addition, obtaining the imaginary component of the channel output signal in the first and third output channels includes determining that the imaginary component is zero. Alternatively, or in addition, obtaining the real component of the channel output signal in any of the first to third output channels and / or obtaining the imaginary component of the channel output signal in either of the second or fourth channel comprises transforming detected optical output signal levels into signed numbers or their digital equivalents.More generally, there is provided a method of processing obtaining a full complex optical Fourier transform result from an optical Fourier transform apparatus, the optical Fourier transform apparatus comprising four output channels, the method comprising: determining the real components of a channel output signal in each of three of the four output channels, and detecting the imaginary component of a channel output signal in the remaining output channel. The method further includes determining the full complex result by obtaining the real and imaginary components of all four output channels using only the detected real components from the three output channels and the detected imaginary component from the remaining output channel.Obtaining the imaginary component of the channel output signal in one of the three channels can include applying a logic operation to the imaginary component detected in the remaining output channel. Alternatively, or in addition, the real component of the channel output signal in the remaining output channel comprises applying a logic operation to the real component of the channel output signal detected in the one of the three output channels. Alternatively, or in addition, obtaining the imaginary component of the channel output signal in the two of the three output channels comprises determining that the imaginary component is zero. Alternatively, or in addition, the real component of the channel output signal in any of the three output channels and / or obtaining the imaginary component of the channel output signal in either the one of the three orremaining channel comprises transforming detected optical output signal levels into signed numbers or their digital equivalents.More specific examples of variation of receivers between different output channels include any combination of the following:• at least one of the output channels includes a receiver for recovering real data having only three possible real output states, optionally wherein the three possible real output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.• at least one of the output channels includes a receiver for recovering imaginary data having only three possible imaginary output states, optionally wherein the three possible imaginary output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.• two of the output channels include a receiver for recovering real data having only three possible real output states, optionally wherein the three possible real output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.• two of the output channels include a receiver for recovering imaginary data having only three possible imaginary output states, optionally wherein the three possible imaginary output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.• at least one of the output channels includes a receiver for recovering real data having only five possible real output states.• two of the output channels include a receiver for recovering real data having only five possible real output states.A detailed example of optimising the configuration of receivers is described with reference to Figure 13.Limiting the receivers in any of the ways described herein can lead to simplification of the construction of the OFT apparatus, reducing manufacturing and component costs and / or can lead to higher receiver operation frequencies.It may be understood that one or more of the methods of data packing, data unpacking, optimisation of receivers and logical derivation of outputs described in this overview can be combined together to form a combined method.Description of the drawingsThis description is intended to be read in the context of the overview and is not intended to be limiting.Deterministic, quantised OFT device and methodA ID OFT device (such as an integrated photonics OFT device) can be realised in a two-dimensional structure, where the light is confined in one dimension and diffracts freely in the other two. In an example of such an OFT device, the input and output of the OFT device are arrays of input and output ports serving as the exit or entrance of waveguides or light channels. The light from each input port is diffracted into a wavefront wide enough to cover all output ports. At each output port, the contribution from each input port has the same intensity when the optical power at all the input ports is the same. The angle of the wavefront at the output array is determined by the angle of the input port, and sets the phase delay at each output port.One way to achieve this is to have the input and output arrays arranged on the arc of a different circle, on which lies the centre of the other circle. The Fourier transform of the input array forms on the line of the second circle. For a given radius of circle and number of input ports the angular spacing of the input and output arrays can be calculated so that the output ports sample a single order of the Fourier transform of the input data.The device will have some tolerance to how close the input and output ports need to be to the arc, with small changes in radial distance or lateral distance (but not angle) able to be compensated. If the radius is large enough, the arc can be approximated to a straight line. 'Large enough' means the distance between the arc and its tangent at the edge of the zero order FT is much less than a wavelength.Figure la shows a known ID OFT device (or Fourier transform waveguide) 100 of this type. In brief summary, Figure 1 shows a Fourier Transform slab waveguide 100 where the light is constrained in one (cartesian) dimension (z) and diffracts freely in the othertwo dimensions (x and y). The OFT device includes a first array 110 of input ports 111, a second array 120 of output ports 121. The second arraybe understood by the skilled person, the Fourier plane may be a plane or a curved surface at which the Fourier transformation of the light at the input ports is formed. An interference region 130 is defined in the space or volume between the first array and second array, and may include an input coupling region connecting the first input array and the diffraction region and an output coupling region connecting the diffraction region and the second output array. The interference region 130 is surrounded in the x and y plane by a boundary 141. The boundary 141 may be directly adjacent the interference region 130 or there may be a spatial margin between at least some edges (perimeters) of the interference region 130 and the boundary 141.The interference region and any surrounding volume in which the light from the input ports propagates can be formed by etching or machining a block, slab or wafer of material to form the interference region and any surrounding volume within the boundary 141 (e.g. as a relief therein or as a standalone block or an embossed structure). The boundary 141 surrounding or adjacent the interference region 130 therefore includes sides (as in the minor faces of a three dimensional slab). The interference region is constrained in the z-direction (as in the thickness direction of the slab) by faces (as in the major faces of the slab). The faces of the boundary surrounding or adjacent to the interference region 130 are exposed to the light emitted from the input ports.The slab waveguide is any photonic waveguide and may be realised using any of the following technology or combination of:• A slab waveguide in silicon-on-insulator (SOI)• A photonic crystal (PhC) with or without periodic defects or sub-wavelength holes• A slab waveguide in silicon nitride (SiN)• A waveguide on a compound photonic technology (lll-V or ll-VI)• Any plasmonic waveguide• A metasurface• Or another photonic technologyIn the interference region, there may be a void or, alternatively, a (for example, homogenous) medium, and / or a medium with uniform refractive index or engineered to have a uniform refractive index.The first array 110 is arranged on (or along) a first arc 110a of a first circle 110c and the second array 120 is arranged on (or along) a second arc 120a of a second circle 120c offset from the first circle. Preferably, the first circle 110c has a centre which lies on the second arc 120a and the second circle120c has a centre which lies on the first arc 110a. Preferably, the centre of the first circle 110c lies at or near the centre of the second array 120 and the centre of the second circle 120c lies at or near the centre of the first array 110.The first arc 110a and second arc 120a respectively define a first portion 131a and second portion 131b of the perimeter 131 of the interference region 130. A third portion 131c and fourth portion 131d of the perimeter 131 join respective ends (or edges) of the first portion 131a and second portion 131b. That is, the third portion 131c is a virtual face which extends in a plane between a first end of the first portion 131a and first end of the second portion 131b, and the fourth portion 131d is a virtual face which extends in a straight line (or a plane) between a second end of the first portion 131a and second end of the second portion 131b. The first to fourth portions 131a-d constitute the whole of the perimeter 131 in the x-y plane (the plane in which the input and output ports are arranged).The input ports 111 are the exits (e.g. exit pupils or exit apertures) of input channels 101, which can also be described as waveguides or coupling structures. That is, the input ports can be connected to any of the following, or combination of:• Waveguides manufactured using the same technology, methods and / or materials as the free space Fourier Transform slab waveguide region• Coupling structures such as grating couplers or edge couplers, that couple in light from an external source• Coupling structures such as tapered couplers, that couple in light between vertically displaced waveguides (i.e. waveguides in a silicon layer coupling light into a SiN Fourier Transform slab waveguide region).• Impedance matching structures that reduce the reflection between the free space Fourier Transform slab waveguide and any of the above.Embodiments can be used with existing photonic technology. For example, both the phase and amplitude of light in input channels 101 connected to the input ports 111 can be controlled or modulated using a transmitter in the input channel. The term transmitter when used throughout this disclosure can be any photonic modulator. It can encompass a light source or emitter (provided that it is coherent and each transmitter has some detectable or controllable phase relationship to other transmitters in the OFT apparatus); a transmitter arranged to control the transmission of (coherent) light from a light source; a modulator arranged to modulate (coherent) light from a light source; or an encoder arranged to encode a signal onto a (coherent) optical signal. That is, the transmitter can bean optical light transmitter, optical encoder, photoemitter or photomodulator, including, by way of non-limiting example, the following or a combination of:• Mach Zehnder Modulators• PN modulators• Ring modulators• Thermal modulators• Another photonic integrated circuit arranged to modulate an optical input signal.The term transmitter also encompasses any of the optical encoders described with reference to Figures 2, 3a, 3b, 4a-g, 5a-f, 6a-e, 7a and 7b of W02023170405(A1), the disclosure of which is hereby incorporated by reference. Protection is sought for an OFT apparatus as described herein and including such optical encoders, though the present disclosure is not limited to these types of transmitters.The output ports 121 can be placed to sample one or more orders of the Fourier transform of the input data either as a fast Fourier transform or otherwise. Generally, the output ports are positioned to capture or sample the zero-order Fourier transform; however, in some embodiments, output ports may alternatively or in addition be positioned to capture or sample higher order Fourier transform modes, such as the first or second order.Embodiments are directed to sampling a Fast Fourier transform. If sampling a fast Fourier transform, the number of output ports used to extract data must be at least the same as the number of input ports. The maximum number of data points is the same as the number of input ports i.e. additional ports can be used to extract other orders of the Fourier transform but the extra ports will contain copies of the zero order data and no additional information. The number of input ports 111 can be two or more up to sixteen. As will be described with reference to Figure lb, the number of input ports in a particularly advantageous arrangement is four.In operation, the Fourier Transform slab waveguide is illuminated with modulated or unmodulated light at one or more of the input ports 111.In more detail, a light source, e.g. a solid-state semiconductor laser source (not shown), provides coherent light. The light source can be housed off-chip, in which case the light is coupled into the OFT device using a fibre and coupling said fibre via grating couplers, or edge couplers using ferrules, or V- shaped grooves. Alternatively, the light source can be packaged in the same carrier substrate with theOFT device and light coupled into the OFT device through edge couplers and photonic wire bonds. Alternatively, the light source can be integrated with the OFT device and light coupled into the OFT device through edge couplers and tapers with or without photonic wire bonds.Coherent light from the light source is amplified using an optical amplifier (not shown) before being split into the plurality of input channels 101. This is achieved by either splitting coherent light from the light source off-chip and coupling in light to fibre bundles which feed the input channels or act as the input channels themselves, or using a fibre splitter, or by coupling in the light to a single channel initially, and splitting the light into input streams using cascaded MMIs orY-branches. If using cascaded MMIs or Y-branches the input streams stem from the output of the last stacked or cascaded 1x2 splitters. The input channels 101 carry the input streams of light split from the light source to the input ports 111.The inserted modulated or unmodulated light undergoes diffraction within the interference region 130. The output ports 121 are placed on the Fourier plane, i.e. where the diffracted light forms the optical Fourier transform of the light pattern from the input ports 111.The input ports 111 bring light into the interference region 130. The light at an input port 111 has information encoded into the phase and / or amplitude of the field. The size and shape of the optical field contained within each input port 111 to the free space Fourier transform slab waveguide will define the envelope function of the Fourier transform according to the convolution theorem.The output ports 121 are placed in the Fourier plane (which lies along the second arc 120a). Light arrives at the output ports 121 as a set of waves at a range of angles which match the relative positions of the input ports 111. The waves from all input ports 111 sum (e.g., coherently) to form an analog Fourier transform. The Fourier transform in the Fourier (or output) plane (which lies along the second arc 120a) contains the full Fourier transform of the light pattern at the input plane (which lies along the first arc 110a), including the input mode shape and input envelope function.The FT is sampled for detection by the output ports 121, and then the detection method will determine if the solution to the Fourier transform is analog or digital. Light collected by the output ports 121 is detected typically using at least one receiver.The term 'receiver' when used throughout the present disclosure includes any component arranged to detect the parameters of an optical signal, in particular relative or absolute amplitude (intensity), relative or absolute phase, or both. That is, the term receiver means an optical receiver. The receiver can be analogue (connected to an ADC) or digital. A receiver can include one or more (e.g. a pair of) photodiodes and said light is either detected on a single photodiode per channel or the signal is mixed with a reference beam for homodyne detection. The receiver can be a coherent detector with 909optical hybrid for detecting the phase of the optical signal. PINs or Avalanche Photo Diodes may be used with trans impedance amplifiers. The receivers that are used to extract data for detection may preserve the amplitude of the Fourier transform, and the phase if the phase is being detected. The term receiver also encompasses any of the balanced detectors or combinations thereof described with reference to Figure 8a or 8b of W02023170405(A1), the disclosure of which is hereby incorporated by reference. Protection is sought for an OFT apparatus as described herein and including such balanced detector arrangements, and the associated methods of operating such an OFT apparatus, though the present disclosure is not limited to these types of receivers.In addition to the first arc 110a and the second arc 120a, the waveguide shown in Figure la comprises boundaries adjacent to, and outside of, the third portion 131c and fourth portion 131d of the perimeter 131 interference region 130. In particular, the waveguide of Figure 1 comprises a first waveguide boundary 141a and a second waveguide boundary 141b. The first waveguide boundary 141a is a planar surface extending between a first end of the first arc 110a and a first end of the second arc 120a. The second waveguide boundary 141a is a planar surface extending between a second end of the first arc 110a and a second end of the second arc 120a.The interference region 130 is positioned between the input ports 111 (i.e. at the input plane of the OFT device) and the output ports 121 (i.e. at the output or Fourier plane of the OFT device). In the interference region 130, the light from the individual input ports interferes to produce a diffraction pattern. The output ports 121 are positioned to capture the diffraction pattern at the Fourier plane (though a diffraction pattern will be present throughout the interference region).Each output port 121 is connected via an output channel 102 to an optical amplifier (not shown) and then on to a detector (which may also be described as a receiver or decoder). Each optical amplifier is arranged to amplify the respective output stream. The optical amplifiers may be semiconductor optical amplifiers (SOAs). However, embodiments are not limited thereto and other types of optical amplifier may be used. Alternatively, no optical amplifiers are used.The value of a first complex element encoded onto each of the input streams emerging from the input ports 111 can differ between input streams so that the value of the input function varies with the positions of the input ports in the first array due to the spatial variations in the input function.Two variables can be used to define or approximate each of the input and output functions: (i) the relative position of the ports within the array; and (ii) the value of the complex number encoded onto the streams of light passing through (e.g. entering or exiting) the ports. It may therefore be understood that the input and output functions are each sampled versions of a continuous function, wherein the sampling resolution is determined by the aperture size of the ports and / or the port spacing or pitch.Temporal variation in the input streams can also be applied by varying the value of the first complex element encoded onto each input stream over time. The input streams may be continuous (always on, or on for multiple cycles of a clock signal). Alternatively, the input streams may also be pulsed (intermittently on and off, optionally in sync with a clock signal). The value of the first complex elements encoded onto the input streams may change with each clock cycle so that multiple optical Fourier transforms can be carried out consecutively, frame-by-frame.The detector (or receiver or decoder) is arranged to detect (or decode) a second complex element encoded onto each output stream collected by the output ports 121.The decoder is arranged to decode a second complex element from each of the output streams based on at least one characteristic of the respective output stream. A value of the at least one characteristic is detected by the decoder and translated into a form which is representative of the second complex element. In embodiments, the at least one characteristic is a phase and / or amplitude of the output stream and the value of said phase and / or amplitude is equal to or correlates with the value of the second complex element. The phase may be a phase relative to the phase of the input streams at the input ports 111.Figure lb shows a special form of the ID OFT device described with reference to Figure la.In Figure lb, each of the input ports are arranged on a first circular arc and each of the output ports are arranged on a second circular arc. The distance between the first and second circular arcs is equal to the radius of the first circular arc and also equal to the radius of the second circular arc. The input ports are arranged at angles 0n from a zeroth position on the second arc and the output portsare arranged at angles 0n from a zeroth position on the first arc. The zeroth position on the first arc is directly opposite the zeroth position on the second arc with respect to a virtual line bisecting the common radius of the first arc and second arc. That is, the zeroth position on the first arc and the zeroth position on the second arc lie at opposite ends of the common radius of the first arc and second arc. The angles 0n are defined by equation 2: (Equation 2)wherein n is the port number, N is the total number of input ports or output ports, R is the radius and neff is the effective index of the waveguide mode. In Figure lb, the port number n starts at - (NZ2)+1 if N is even and -((N-1))Z2 if N is odd at the lowest port, and increases in integers up to the top. The corresponding port numbers in Figure 10 are input 1 corresponding to the uppermost port in the Figure and input 4 corresponding to the lowermost (with the same convention for the output ports). The effective index of the waveguide mode neff describes the characteristics of guiding mode propagation in the OFT device and can be determined using methods known to those skilled in the art from the optical properties of the materials used to construct the waveguide and the dimensions of the waveguide.The OFT device of embodiments can also take other forms other than that described with reference to Figure lb. Embodiments include any OFT device capable of preforming a discrete Fourier transform optically. Such OFT devices satisfy the discrete Fourier transform equation (equation 1).Figure 2 shows an optical circuit diagram for an OFT apparatus such as those described with reference to Figures la or lb. The OFT apparatus includes an OFT stage 210 having input channels 201 and output channels 202, a grating coupler 203, a source channel 211, splitters 220, combiners 230, transmitters 240, input channel phase controls 242, monitor channel receivers 260, monitor channel phase controls 265, output channel receivers 270, reference channel phase controls 280, reference channels 281, a reference source channel 282, and a reference channel amplifier 290.The grating coupler 203 provides an inlet for a light source (e.g. laser) such as that described with reference to Figure la. The input channels 201 are branched off from a single source channel 211 using a series of cascaded splitters 220 and extend to the OFT apparatus. A transmitter (or encoder) 240 is provided in-line in each input channel.A splitter 220 is provided to separate a portion of the optical input signal from the transmitter 240 via a monitor channel 261 before the transmitter signal reaches the OFT device 210. A combiner 230combines the optical input signal from the monitor channel 261 with an optical perturbation signal provided via a perturbation channel 262 and feeds the combined signal to the monitor channel receiver 260. The perturbation channel 262 includes one of the monitor phase controls 265.Each output channel 202 is connected between the OFT stage 210 and a combiner 230 which combines the optical output signal from a corresponding output channel 202 with an optical reference signal from a corresponding reference channel 281 and feeds the combined optical signal into a corresponding output channel receiver 270. Each reference channel 281 includes a reference channel phase control 280. The reference channels 281 are connected via series of cascaded splitters 220 to the reference source channel 282.Each transmitter 240 includes a modulator 241 and an input channel phase control 242. In the example of Figure 2 the modulator 241 is a modulator pair and the phase control 242 is a phase control pair on separate parallel branches of the input channel 201. In other examples, the input channel phase control 242 is a single phase control provided in the input channel downstream of the transmitter.In operation, the modulator 241 in each input channel 201 modulates light from the light source to provide the optical input signal having an input state to the OFT stage 210. The phase control 242 adjusts the input state.The phase of the light in the perturbation channel 262 is controlled by the monitor phase control 265 to produce the optical perturbation signal. The monitor channel receiver 260 detects the state of the combined optical input signal and optical perturbation signal, or just the input state if there is no perturbation light provided in the perturbation channel.The output channels each carry the optical output signal in a corresponding output state from the OFT stage 210.The phase of the light in the reference channels 281 is controlled by the reference channel phase controls 280 to produce the optical reference signal. The reference channel amplifier 290 controls the amplitude of the optical reference signal. The output channel receiver 270 detects the state of the combined optical output signal and optical reference signal, or just the output state if the optical reference signal is switched off by the reference channel amplifier 290.The term 'state' (e.g. input state or output state) as used in the present disclosure refers to a parameter or parameters of an optical signal, or the representation of the parameter(s) in digital or numerical form. The state may be understood to be or represent a value or a level. The parameter may include amplitude or intensity of the signal or may be phase or relative phase of the signal. The digital or numerical form may be a complex number, or a signed or unsigned real or imaginary number, or a (bit) representation thereof.Figure 3 shows a first Argand diagram 301 and a second Argand diagram 302, each comprising a real axis 311 and an imaginary axis 312 which bisect each other at right angles and an origin 313 that lies at the point of intersection of the real axis 311 and the imaginary axis 312. Figure 3 is an illustration of possible values at input and output ports in an ideal system. Each port or channel, whether input or output, can have any value. The first Argand diagram 301 depicts possible input states (e.g. from an input port 111 or an array 110 of input ports) to an OFT system, such as that shown in Figure la. The second Argand diagram 302 depicts possible output states (e.g. at an output port 121 or an array 120 of output ports) of the OFT system. Notably, in Figure 3, the set of possible input and / or output states includes every complex number. That is, such OFT systems have (possibly unbounded) continuous inputs and continuous outputs. In practice, there is an upper bound on both the number of possible input states to and the number of possible output states from an OFT system. For example, the number of possible input states may be determined by the amplitude of light in input channels 101 connected to the input ports 111, and the number of possible output states may be determined by the sensitivity (range) of the receiver (or detector or decoder).The values represented on the Argand diagrams in the drawings represent mathematical inputs to the OFT apparatus and not the magnitude or phase of light at either input or output channels or ports. The mathematical values in the inputs can be set using the transmitter. The mathematical values in the outputs can be read using the receiver.In the OFT apparatus, the magnitude (or amplitude or intensity) of light at the input channel can be any value between 0 and a limit set by the combination of the light source and transmitter. The magnitude (or amplitude or intensity) of light at the output can be any value between 0 and a limit set by the sensitivity of the receiver. The phase of the light at either input or output can be any value between 0 to 2n or multiples thereof, and can be detected using the receiver, for example a coherent detector with 909optical hybrid.Likewise, Figure 4 shows two Argand diagrams 401, 402 each comprising a real axis 411 and an imaginary axis 412 which bisect each other at right angles and an origin 413 that lies at the point of intersection of the real axis 411 and the imaginary axis 412. Figure 4 is an illustration of possible values at input and output ports in a real system. Each channel or port, whether input or output, can have a value that is discretised the quantisation of which is set by the system that inserts or sets the input values (i.e. the transmitter and its associated systems), and by those that read the output values (i.e. the receiver and its associated systems).In contrast to Figure 3, the set of possible input and / or output states depicted in the diagrams of Figure 4 is discrete. That is to say, the space of input and / or output complex numbers is discretised by creating discrete intervals 421, 422 (e.g. corresponding to the divisions of a grid overlaid on the complex plane) the union of which spans the complex plane (or a possibly finite subplane thereof). Each discrete interval 421, 422 is a finite subplane of the complex plane, the dimensions of which are determined by the quantisation set by the system that inserts or inputs the input states and / or by the system that reads the output states. The dimensions of the discrete intervals 421, 422 may be smaller or larger depending on the quantisation of the system. A larger quantisation (i.e. a discretisation yielding larger discrete intervals of the complex plane) may be set by the electronics in and / or by the noise of the system. In the case of Figure 4 (and Figure 5, discussed below), there is a one-to-one mapping between the quantised input / output states and the intersection points or divisions of the grid.Figure 5 also shows two Argand diagrams 501, 502. The quantisation of the input and / or output states depicted in Figure 5 is greater or more pronounced than that depicted in Figure 4. That is, Figure 5 illustrates possible values at input and output ports in a system with a large quantisation (difference between different values or state levels), representative of a real-world analog system where large quantisation is set by the sensitivity of the electronics and noise in the system. That is, each discrete interval 521, 522 of Figure 5 spans a larger area of the complex plane than each discrete interval 421, 422 of Figure 4. Thus, for example, a larger discretisation of a given finite subplane of the complex plane yields fewer discrete intervals than for a smaller discretisation of the same subplane.With reference to Figures 4 and 5, although all discrete intervals are depicted as having the same dimensions, this is not necessarily the case. For example, the quantisation of the input states may differ from the quantisation of the output values such that the dimensions of a discrete interval of thespace of possible input values may be different from those of a discrete interval of the space of possible output states. Similarly, the span of possible input states (i.e. the width of each discrete interval yielded by the quantisation of the input states) may differ from the span of possible output states (i.e. the width of each discrete interval yielded by the quantisation of the output states). That is, the quantisation, dynamic range and span of the input and output channel may be the same or different. The quantisation, dynamic range and span of each input channel may be different from another input channel. The quantisation, dynamic range and span of each output channel may be different from another output channel.Figure 6 similarly shows two Argand diagrams depicting possible input and output states of a deterministic OFT system. Figure 6 illustrates input and output states of a deterministic optical Fourier transform device with a small number of input and output ports, for e.g. a one-dimensional OFT device with four input ports and four output ports such as that shown in Figures lb or 2. In this case the set of possible input states is restricted to the real axis, and the quantisation of the input is such that the discrete intervals 621 are of width 6in. In particular, there is exactly one discrete interval 621 (since the size of the set of possible input states coincides exactly with the width of a discrete interval 6in) and therefore there are exactly two quantised input channel states 631, 632, which may be denoted in binary form as real values of 0 and 1. Due to the restriction on the possible input states, the set of possible output states is also restricted. In particular, in Figure 6, there are 27 possible output states: nine of which have imaginary part +bi for some constant b, nine of which have imaginary part Or, and nine of which have imaginary part —bi. The real part of each of the 27 possible output states takes on one of nine values. The output of the OFT system is therefore discretised over nine levels 641 on the real axis and over three levels 642 on the imaginary axis; thus, there are 27 possible quantised output states. The quantisation of the output states is such that the discrete intervals 622 are of width 6out-Figure 7 shows four pairs 701, 702, 703, 704 of Argand diagrams, each diagram depicting possible input or output states at a particular input or output port among four input ports and four output ports of a deterministic OFT system. Figure 7 differs from Figure 6 in that it breaks down the possible input states and output states for each of the input and output channels in the OFT apparatus. The output states are restricted such that each output channel receives only a smaller subset of the overall dynamic range of the OFT output shown in Figure 6; and that all channels have the same quantisation steps (dout). This way the demand on receiver resolution is reduced and the same receiver circuit is used in all ports with a suitable offset or adjustment applied. These outputpatterns are possible by restricting the input states to only a few (in this case two) levels (with a quantisation din), and also restricting the inputs to be unsigned real or unsigned imaginary.The first pair 701 of Argand diagrams depicts possible input and output states at a first input port and a first output port respectively; the second pair 702 depicts possible input and output states at a second input port and a second output port respectively; and so on for the third and fourth pair 703, 704. The set of possible input states in each of the four cases is restricted to the real axis, and consequently the corresponding set of possible output values is also restricted. In particular, in each of the four cases, the size of the set of possible input values coincides exactly with the width of a discrete interval 6in. Quantisation therefore yields exactly one discrete interval, and hence there are exactly two quantised input channel states 731, 732 for each of the four input ports, which may be denoted in binary form as real values of 0 and 1.The output of the first and third output ports is discretised over five levels 741, 742 on the real axis (where the five levels 741 for the first output port overlap with but differ from the five levels 742 for the third output port) and over the same one level of the imaginary axis. The output of the second and fourth output ports is discretised over the same three levels 743 of the real axis and over the same three levels 744 of the imaginary axis. Output from the four output ports is therefore discretised over seven unique levels of the real axis and three unique levels 744 of the imaginary axis.Figure 8 shows sixteen diagrams depicting the real parts of output states of an OFT system having four input ports and four output ports. Figure 8 illustrates the input and output states at each port in an exemplar deterministic OFT system with four inputs and four outputs, where the input is restricted to be binary unsigned real. The input and output states are obtained from a simulation of a onedimensional OFT device, where each of the four input ports is sequentially illuminated with a binary unsigned real input. The result from the real part of the output is compared with the result expected from FFT algorithms (with a circular shift) for all 16 input combinations (4-bit).In Figure 8, the set of possible input states is restricted to the real axis and in particular wherein the size of the set of possible inputs coincides exactly with the width of a discrete interval (and therefore there are exactly two quantised input channel states, denoted in binary form as real values of 0 or 1), as in the system described in relation to Figure 7. For instance, the first diagram 801 shows that the real parts of the output states at the four output ports are 0, 0, 0 and 0 respectively on input 0, 0, 0 and 0 at the four input ports respectively. As another example, the sixteenth diagram 802 shows thatthe real parts of the output states at the four output ports are 0, 0, 4 and 0 respectively on input 1, 1, 1 and 1 at the four input ports respectively. An offset is required only for port 3.Figure 9 similarly shows sixteen diagrams depicting the imaginary parts of output values of an OFT system having four input ports and four output ports, wherein the set of possible input states is restricted to the real axis and in particular wherein the size of the set of possible inputs coincides exactly with the width of a discrete interval (and therefore there are exactly two quantised input channel states, denoted in binary form as real values of 0 or 1), as in the system described in relation to Figure 7. The imaginary part of the output is compared with the result expected from FFT algorithms (with a circular shift) for all 16 input combinations (4-bit). For instance, the first diagram 901 shows that the imaginary parts of the output states at the four output ports are Or, Or, Or and Or respectively on input 0, 0, 0 and 0 at the four input ports respectively. As another example, the thirteenth diagram 902 shows that the imaginary parts of the output states at the four output ports are Or, — 11, 01 and +1 i respectively on input 1, 1, 0 and 0 at the four input ports respectively.In Figures 8 and 9, the quantised states are shown in dashed lines and are consistent for all input ports and for all input combinations. The quantisation to recover the real and imaginary part of the OFT output is the same for all input combinations and for all output ports.Figures 6, 7, 8 and 9 show that for a four input channel / four output channel OFT apparatus for which the input states are binary, the output of the OFT is discretised over seven levels on the real axis and three levels on the imaginary axis. A receiver capable of discerning only five levels is sufficient to recover all OFT data (both real and imaginary data). Further performance improvement is possible by simplifying the receiver arrangement as follows:• using two three-level receivers to recover real data;• using two three-level receivers to recover imaginary data;• using two five-level receivers to recover real data; and• removing two receivers that would have otherwise been used to recover imaginary data from output channels 1 and 3 (since they are not needed to measure an imaginary value which is always 0 for all driving conditions).Figure 10 shows a table depicting all possible input-output combinations of an OFT system having four input ports and four output ports, wherein each input port has exactly two quantised input states (which may be denoted as real values of 0 or 1), as in the systems described in relations to Figures 7, 8 and 9. For instance, the first row 1001 shows that inputting (real values) 0, 0, 0 and 0 at the fourinput ports yields real parts 0, 0, 0 and 0 at the four output ports and imaginary parts 01, 01, 01 and 01 at the four output ports. As another example, the thirteenth row 1002 shows that inputting (real values) 1, 1, 0 and 0 at the four input ports yields real parts 0, -1, 2 and -1 at the four output ports and imaginary parts 01, —11, 01 and +1 i at the four output ports.Figure 11a illustrates the components of an optical Fourier transform 'OFT' apparatus or system 1100. The OFT system 1100 comprises a host 1101, a data and control interface 1111, a calibration and monitor controller 1121, an input digital-to-analog converter 1131, an input analog signal conditioner 1141, a photonic system 1151, an output analog signal conditioner 1161, and an output analog-to- digital converter 1171.The application logic 1101a of the host 1101 generates data which is processed by the OFT system 1100. The data is stored in transmitter ('TX') data frame memory 1101b, which is then sent to the transmitter data packing logic 1101c. The transmitter data packing logic 1101c may determine whether or not to allow certain data to be processed further by the OFT system 1100. For instance, the transmitter data packing logic 1101c may determine that an input data frame consisting of all zeroes should not be processed further by the OFT system 1100. As another example, the transmitter data packing logic 1101c may determine that an input data frame consisting of either of the bits 0101 or 1010 should not be processed further by the OFT system 1100. The transmitter data packing logic 1101c may also serialise incoming data before transmitting it further through the OFT system 1100. That is to say, the transmitter data packing logic 1101c may transform, for example, a 64-bit input into 64 individual 1-bit inputs. Once the data has been processed by the transmitter data packing logic 1101c, it is sent to TX queue memory llOld, from where it then exits the host 1101 and enters the data and control interface 1111.The control system 1111a of the data and control interface 1111 determines whether or not input data should be transmitted further through the OFT system 1100 or whether further transmission of the input data should be delayed, depending on a result obtained from the calibration and monitor controller 1121. Input data that is selected to be transmitted further through the OFT system 1100 then leaves the data and control interface 1111 and enters the input digital-to-analog converter 1131.The input digital-to-analog converter 1131 comprises a digital equaliser 1131a, a data encoder 1131b, a pre-emphasis stage 1131c and a digital-to-analog converter 1131d, as is common in the state of the art. The digital equaliser 1131a corrects any errors that may have accumulated in the input dataduring its transmission from the host 1101 to the input digital-to-analog converter 1131. The data encoder 1131b may encode the input data into a format that is recognisable by the OFT system 1100. The data is then converted from digital data to analog data by the digital-to-analog converter 1131d, and then leaves the input digital-to-analog converter 1131 and enters the input analog signal conditioner 1141.The input analog signal conditioner 1141 comprises a pre-driver 1141a, a driver 1141b and an amplifier chain 1141c, as is common in the state of the art. The input analog signal conditioner 1141 amplifies the analog signal received from the input digital-to-analog converter 1131 in such a way that the signal is compliant with a photonic encoder 1151a of the photonic system 1151. The amplifier chain 1141c may comprise for example gain blocks or amplifier(s). The amplified or conditioned signal leaves the amplifier chain 1141c and enters the photonic encoder 1151a of the photonic system 1151.The photonic encoder 1151a of the photonic system 1151 converts the received electrical analog signal into a photonic signal. An optical Fourier transform stage 1151b is applied to the photonic signal, and the resulting photonic signal is received by the photonic receiver 1151c. The photonic receiver 1151c transmits a corresponding photocurrent to a trans-impedance amplifier 1161a of the output analog signal conditioner 1161.The output analog signal conditioner 1161 comprises a trans-impedance amplifier 1161a, an amplifier chain 1161b and a filter stage 1161c. The trans-impedance amplifier 1161a amplifies the received photocurrent and converts it into a voltage. The amplifier chain 1161b and the filter stage 1161c conditions the signal in preparation for digitisation by the output analog-to-digital converter 1171. The output analog signal conditioner 1161 transmits the conditioned signal from the filter stage 1161c to an analog equaliser 1171a of the output analog-to-digital converter 1171.The output analog-to-digital converter 1171 comprises an analog equaliser 1171a, an aligner 1171b, a sampler 1171c and an analog-to-digital converter 1171d, as is common in the state of the art. The purpose of the output analog-to-digital converter 1171 is to digitise the incoming raw voltage from the output analog signal conditioner 1161. The analog equaliser 1171a may apply for example continuous time linear equalisation (CTLE). The analog equaliser 1171a modifies the incoming raw voltage from the output analog signal conditioner 1161 in such a way that it is compliant with other components of the output analog-to-digital converter 1171, such as the aligner 1171b or the analog- to-digital converter 1171d. The aligner 1171b may introduce a delay or some other fine temporalcharacteristic into the signal such that the signal aligns with a particular point. This point is then sampled or extracted by the sampler 1171c and then transmitted to the analog-to-digital converter 1171d to be digitised. Consequently, the output analog-to-digital converter does not necessarily digitise the entirety of the incoming raw voltage, but instead may digitise only a point or points selected by the aligner 1171b and / or sampler 1171c. The digitised signal is then transmitted from the analog-to-digital converter 1171d to an output data receiver 1111b of the data and control interface 1111.The output data receiver 1111b of the data and control interface 1111 transmits the received data to the host 1101, where it enters a receiver queue memory llOle. The receiver detection and recovery logic (or 'deserialiser') llOlf of the host 1101 may apply an unpacking method to data received from the receiver queue memory llOle in order to, for example, recover full complex data from PAM4 data. The receiver detection and recovery logic llOlf is applied to (possibly many) data frames, the output of which is then transmitted to receiver data frame memory 1101g. Finally, data from receiver data frame memory 1101g is consumed by the application logic 1101a.Figure lib shows a magnified view 1102 of the host 1101 of an OFT system 1100. As in Figure 11a, the host 1101 comprises transmitter data frame memory 1101b, receiver data frame memory 1101g, transmitter data packing logic 1101c, receiver detection and recovery logic llOlf, transmitter queue memory llOld and receiver queue memory llOle.Data is stored in transmitter data frame memory 1101b, which is then sent to the transmitter data packing logic 1101c. The transmitter data packing logic 1101c may determine whether or not to allow certain data to be processed further by the OFT system 1100. For instance, in the case of Figure lib, the transmitter data packing logic 1101c checks at step S1103 whether data received from transmitter data frame memory 1101b consists of all zero bits; if so, the input data is copied and then transmitted directly to receiver data frame memory 1101g. That is, in the case that the input data consists of all zero bits, the data is simply copied to receiver data frame memory 1101g without further processing by other components of the OFT system 1100, such as input and output analog signal conditioners 1141, 1161 or the photonic system 1151. In the event that the input data does not consist of all zero bits, step S1104 then checks whether the signal pattern of the data is of the form 1010, in which case at step S1105 the input data is inverted and duplicated, and then sent to transmitter queue memory llOld. Thus, input (1010) is transformed into (

[0101]

[0101] ), i.e. two distinct input frames each comprising a 4-bit input vector, at step S1105. If step S1104 determines that the input signal patternof the data is not of the form (1010), then the data is transmitted directly to transmitter queue memory llOld. In any case, the (possibly transformed) data is then sent to the data and control interface 1111, from where it is processed by other components of the OFT system 1100, and then returns to the data and control interface 1111 and is sent to receiver queue memory llOle. In particular, data that does not satisfy any of the checks of steps S1103 and S1104, i.e. data that does not consist of all zero bits and whose signal pattern is not of the form (1010), proceeds to transmitter queue memory llOld without modification.When data is received into receiver queue memory llOle from the data and control interface 1111, it is sent for processing by the receiver detection and recovery logic llOlf. In the case of Figure lib, each data frame from receiver queue memory llOle comprises four channels, each channel being in one of four possible levels or states. Steps S1106 and S1107 determine whether an output data frame corresponds to input data that was subject to the inversion and duplication by steps S1104 and S1105 of the transmitter data packing logic 1101c. If this is determined to be the case, a known relationship between the output of the OFT system 1100 yielded by two different inputs (e.g. inputs (0101) and (1010) is used to determine the output for one of the two different inputs.At step S1106, each received data frame is checked to determine whether a first channel of the data frame is at a first level, a second channel of the data frame is at a second level, a third channel of the data frame is at a third level and a fourth channel of the data frame is at a second level. If it is determined that this is not the case for a given data frame, then the data frame is transmitted directly for processing by receiver detection and recovery logic llOlf. Otherwise (i.e. if the channels of a data frame are determined to be at the above-specified levels), the data frame may be modified in accordance with a known relationship between the outputs of the OFT system 1100 given two different inputs. In particular, at step S1107, the contents of four channels stored in a temporary storage 1102a are checked to determine whether they are at the same levels as above (i.e. a first channel, second channel, third channel and fourth channel are at a first level, second level, third level and second level respectively). If so, then the data received from the receiver queue memory llOle is modified as follows: the first channel is set to the fourth level, and the remaining three channels remain unchanged. If the check of step S1107 fails, then the data received from the receiver queue memory llOle is unmodified. In either case, the (modified or unmodified) data is transmitted for processing by the receiver detection and recovery logic llOlf, and from there it is sent to receiver data frame memory 1101g.OFT data packingFigure 12 is a flow diagram of an example method 1200 of processing an output signal from an optical Fourier transform apparatus. The method 1200 can be implemented in the optical Fourier transform system 1100 of Figure 11a or in another suitable system or apparatus. The method 1200 can be implemented as a set of computer-readable instructions stored on a transitory or non-transitory computer-readable medium and executable by one or more processors, for example.The method 1200 begins at step S1202, where an input vector is applied to an optical Fourier transform apparatus (e.g. to the optical Fourier transform system 1100 of Figure 11a). At step S1204, an output vector representing the optical Fourier transform of the input vector is obtained. The output vector representing the optical Fourier transform of the input vector may be obtained for example by the photonic system 1151 of the optical Fourier transform system 1100 of Figure 11a on which the method 1200 is implemented.At step S1206, a channel of the output vector is modified to form a PAM4 signal. This channel of the output vector may comprise more than four levels. For example, the channel may carry a PAM5 signal. In the case that this channel comprises more than four levels, the modification of this channel at step S1206 comprises reducing the number of levels in the channel to form a PAM4 signal. For example, if the channel carries a PAM5 signal, then the number of levels in the channel is reduced by one to form a PAM4 signal. The number of levels in the channel may be reduced by detecting the highest level(s) of the channel and removing the highest level(s) from the channel. For example, the fifth level of a channel carrying a PAM5 signal may be removed to form a PAM4 signal. Removing a level or levels from the channel may be performed by saturating or clipping the highest level(s) to the fourth level, or by subtracting one or more levels to form the fourth level.Alternatively, the channel may comprise fewer than four levels. For example, the channel may carry a PAM3 signal. In the event that the channel comprises fewer than four levels, the modification of this channel at step S1206 comprises adding one or more levels to the channel to form the PAM4 signal. For example, one or two levels may be added to form a fourth level.Once the PAM4 signal has been formed at step S1206, the PAM4 signal may optionally be transmitted over a PAM4 link at step S1208. For instance, the PAM4 signal may be transmitted over a PAM4 link from the host 1101 to the data and control interface 1111 of the OFT system 1100 of Figure 11a in which the method 1200 may be implemented. At step S1210, correct full complex OFT data isretrieved from the PAM4 signal by reversing the modification of the channel. For example, correct full complex OFT data may be retrieved from the PAM4 signal by deriving a fifth level of a PAM5 signal from the PAM4 signal. Steps S1208 and S1210 are both optional steps, i.e. only one may be performed, both may be performed, or neither may be performed. Thus, for example, the method 1200 may end immediately after performing step S1206 (in which case neither of steps S1208 and S1210 is performed); or the method 1200 may perform step S1208 after step S1206 and then end (in which case step S1208 is performed but step S1210 is not); or the method 1200 may perform step S1210 after step 1206 and then end (in which case step S1210 is performed but step S1208 is not); or the method 1200 may perform step S1206 followed by step S1208 followed by step S1210 and then end.Optimisation of receiversFigure 13 is a flow diagram of an example method 1300 of obtaining a full complex optical Fourier transform result from an optical Fourier transform apparatus. The method 1300 can be implemented in the optical Fourier transform system 1100 of Figure 11a or in another suitable system or apparatus. The method 1300 can be implemented as a set of computer-readable instructions stored on a transitory or non-transitory computer-readable medium and executable by one or more processors, for example.The method 1300 begins at step S1302, where the real components of a channel output signal in each of a first, second and third output channel of the optical Fourier transform apparatus are determined. For instance, the method 1300 may determine at step S1302 that the first, second and third output channels are at a first level, a second level and a third level respectively, in which case the corresponding real components may be the real numbers -2, 0 and 2 respectively. The real numbers corresponding to the levels of the first output channel may range from -2 to +2, the real numbers corresponding to the levels of the second output channel may range from -1 to +1, and the real numbers corresponding to the levels of the third output channel may range from 0 to 4.At step S1304, the method 1300 detects the imaginary component of a channel output signal in a fourth output channel of the optical Fourier transform apparatus, where the fourth output channel is different from the first, second and third output channels. For example, the method 1300 may determine at step S1304 that the fourth output channel is at one of a first level, a second level or a third level, in which case the corresponding imaginary component may be the imaginary number — 1 i, 01 or -Fir respectively.At step S1306, the method determines a full complex result by obtaining the real and imaginary components of all four output channels using only the detected real components from the first, second and third output channels and the detected imaginary component from the fourth output channel. For instance, obtaining the imaginary component of the channel output signal in the second output channel may involve applying a logic operation to the imaginary component detected in the fourth output channel. Similarly, obtaining the real component of the channel output signal in the fourth output channel may involve applying a logic operation to the real component of the channel output signal detected in the second output channel. As another example, obtaining the imaginary component of the channel output signal in the first and third output channels may involve determining that the imaginary component is zero. As a further example, obtaining the real component of the channel output signal in any of the first to third output channels and / or obtaining the imaginary component of the channel output signal in either of the second or fourth channel may involve transforming detected optical output signal levels into signed numbers or their digital equivalents. In particular, for example, a first detected optical output signal level may be transformed into the signed number 1, a second detected optical output signal level may be transformed into the signed number 2, and so on.Logical derivation of outputsFigure 14 is a flow diagram of an example method 1400 of processing data using a Fourier transform apparatus. The method 1400 can be implemented in the optical Fourier transform system 1100 of Figure 11a or in another suitable system or apparatus. The method 1400 can be implemented as a set of computer-readable instructions stored on a transitory or non-transitory computer-readable medium and executable by one or more processors, for example.The method 1400 begins at step S1402, where an input data stream is applied to a Fourier transform apparatus. The input data stream is applied in a series of sequential frames and each frame comprises an input vector. For instance, an input vector of a first frame may be of the form 0001 and an input vector of a next frame in the series of sequential frames may be of the form 0010. The input data stream and / or the series of sequential frames of the input data stream may be stored for example in transmitter data frame memory 1101b of the optical Fourier transform system 1100 of Figure 11a. At step S1404, an output data stream is produced from the input data stream by determining an output vector for each frame, where each output vector is the Fourier transform of the corresponding input vector in the input data stream; for example, for the input vector (0010) the output vector may be the Fourier transform of the input vector, namely (1111). At step S1406, for a first input vector having afirst value, a corresponding first output vector is determined by taking an optical Fourier transform of the first input vector. At step S1408, for a second input vector having a second value different from the first value, a corresponding second output vector is determined by means other than taking a Fourier transform of the second input vector. For instance, the corresponding second output vector may be determined by applying a logic operation to the second input vector. If the second input vector has an overall value of zero, then the second output vector may be determined by copying the second input vector. The corresponding second output vector may additionally or alternatively be determined by taking an optical Fourier transform of the first input vector, or from the first output vector via a known relationship (e.g. an inverted sign) between differing vector values.Data unpackingFigure 15 is a flow diagram of an example method 1500 of processing data using a Fourier transform apparatus. The method 1500 can be implemented in the optical Fourier transform system 1100 of Figure 11a or in another suitable system or apparatus. The method 1500 can be implemented as a set of computer-readable instructions stored on a transitory or non-transitory computer-readable medium and executable by one or more processors, for example. The method 1500 may be seen as the inverse of the method 1200 of Figure 12. That is to say, the method 1200 of Figure 12 is a data packing method in which the output or a Fourier transform apparatus is modified in order to be compliant with a PAM4 data link and the method 1500 of Figure 15 comprises unpacking the PAM4 data to recover the full Fourier transform result.The method 1500 begins at step S1502, where an output signal comprising a PAM4 signal is received from an optical Fourier transform apparatus. At step S1504, a full complex output of the optical Fourier transform apparatus is retrieved from the output signal. The retrieval may comprise decoding the PAM4 signal, which in turn may comprise deriving a PAM5 signal from the PAM4 signal or deriving a PAM3 signal from the PAM4 signal.Figure 16 is a flow diagram of an example method 1600 of operating an optical Fourier transform apparatus or system comprising N input channels and K output channels. The method 1600 can be implemented in the optical Fourier transform system 1100 of Figure 11a or in another suitable system or apparatus. The method 1600 can be implemented as a set of computer-readable instructions stored on a transitory or non-transitory computer-readable medium and executable by one or more processors, for example.The method 1600 begins at step S1602, where light arriving at the input channels of the optical Fourier transform apparatus or system is quantised to form a quantised output at each output channel. For example, the light emitted by each input channel may be limited to a finite number of possible input states (e.g. to two, three or four possible input states). Then, at step S1604, any combination of at least one of the methods 1200, 1300, 1400 and 1500 of Figures 12, 13, 14 and 15 respectively is applied.Truth tablesFigure 17 is a truth table 1700 showing example input-output combinations for an optical Fourier transform apparatus (e.g. the optical Fourier transform apparatus 1100 of Figure 11a). The truth table 1700 comprises six sets of columns 1710, 1711a, 1711b, 1712, 1713a, 1713b, each set comprising four columns.The real components of all possible 4-bit input combinations are shown in the first four columns 1710 of the truth table 1700, where each bit of each possible input corresponds to a respective input channel of the optical Fourier transform apparatus. The next four columns 1711a show the real components of the raw or unprocessed output data of the optical Fourier transform apparatus when said system is given as input the corresponding 4-bits from the first four columns 1710. For example, the truth table 1700 shows that on input

[0000] the (real components of the) raw or unprocessed output of the optical Fourier transform apparatus is

[3212] , The output represented in the second set 1711a of columns may be obtained for example by an optical Fourier transform stage 1151b of the optical Fourier transform apparatus. Likewise, the third set of four columns 1711b shows the imaginary components of the raw or unprocessed output of the optical Fourier transform apparatus when said system is given as input the corresponding four bits from the first four columns 1710. For instance, the truth table 1700 shows that on input

[0000] the (imaginary components of the) raw or unprocessed output of the optical Fourier transform apparatus is at a second level at both a second and a fourth output channel. The output represented in the third set 1711b of columns may be obtained for example by an optical Fourier transform stage 1151b of the optical Fourier transform apparatus.Depending on the configuration of the optical Fourier transform apparatus, the raw output data may not naturally be PAM4 compliant. The raw output data may be in another format other than PAM4 due to the number of possible output states receivable at an output channel of the OFT apparatus. For example, if the number of possible output states receivable is five, then the raw output data may be said to be (or include) a PAM5 signal (one having 5 possible states or levels or values). The opticalFourier transform apparatus may therefore modify the raw output data so that it is PAM4 compliant. The fourth set 1712 of columns of the truth table 1700 depicts the results of such a modification of the raw output data of the second and third sets 1711a, 1711b of columns. In general, such a modification may be performed as follows:• Detect the fifth level of the PAM5 signal for a first output channel and a third output channel.• Subtract the fifth level from the PAM5 signal, yielding a PAM4 signal.• Add two levels to the PAM3 signal for a second output channel.As an example, the eleventh row 1720 of the truth table 1700 shows that on input

[1010] the optical Fourier transform apparatus outputs real component levels 5, 2, 3 and 2 at the first, second, third and fourth output channel respectively and outputs imaginary component level 2 at both the second and fourth output channels. Thus, the fifth level 1708 of the PAM5 signal is detected for the first output channel, and therefore this fifth level 1708 is subtracted from the PAM5 signal, yielding a PAM4 signal 1701, and two levels are added to the PAM3 signal 1702 for the second output channel.The optical Fourier transform apparatus may recover the raw output data from the PAM4 signal by reversing the above-described modification of the output signal. In particular, therefore, the optical Fourier transform apparatus may retrieve the correct full complex OFT data from the PAM4 signal by reversing the modification of the output signal. Where one or more levels were originally removed from the raw output data to form the PAM4 signal, the correct full complex OFT data comprises deriving the one or more removed levels from the PAM4 signal. For example, where a fifth level was originally removed from the raw output data to form the PAM4 signal, the correct full complex OFT data comprises deriving a fifth level of a PAM5 signal from the PAM4 signal.The fifth and sixth sets 1713a, 1713b of columns of the truth table 1700 represent the results of retrieving the correct full complex OFT data from the PAM4 signal that is represented in the fourth set 1712 of columns. Notably, no information is lost during the formation of the PAM4 signal or recovery of data from the PAM4 signal, in the sense that the retrieved data ("PROCESSED OUTPUT" in the figure) is identical to the corresponding raw output data. In general, the real components of the raw output data (e.g. the correct full complex OFT data) may be retrieved from the PAM4 signal as follows:• Adjust the real component of the first output channel of the PAM4 signal to be equal to: o a third level, if the second output channel and the third output channel of the PAM4 signal are both equal to a fourth level; oro its current level plus an additional level, if the second output channel of the PAM4 signal is equal to a fourth level and the third output channel of the PAM4 signal is not equal to the fourth level; or otherwise o its current level.• Adjust the real component of the second output channel of the PAM4 signal to be equal to: o a second level, if the second output channel of the PAM4 signal is equal to a fourth level; or otherwise o its current level.• Adjust the real component of the third output channel of the PAM4 signal to be equal to: o a third level, if the second output channel and the first output channel of the PAM4 signal are both equal to a fourth level; or o its current level plus an additional level, if the second output channel of the PAM4 signal is equal to a fourth level; or otherwise o its current level.• The real component of the fourth output channel of the data recovered from the PAM4 signal is equal to the real component of the fourth output channel of the raw output data.The imaginary components of the raw output data (e.g. the correct full complex OFT data) may be retrieved from the PAM4 signal as follows:• The imaginary component of the first output channel of the data recovered from the PAM4 signal is equal to the second level.• The imaginary component of the second output channel of the data recovered from the PAM4 signal is equal to: o a third level, if the imaginary component of the fourth output channel of the PAM4 signal is equal to the first level; or o a first level, if the imaginary component of the fourth output channel of the PAM4 signal is equal to the third level; or otherwise is equal to o the second level.• The imaginary component of the third output channel of the data recovered from the PAM4 signal is equal to the second level.• The imaginary component of the fourth output channel of the data recovered from the PAM4 signal is equal to: o a first level, if the imaginary component of the fourth output channel of the PAM4 signal is equal to the first level; oro a third level, if the imaginary component of the fourth output channel of the PAM4 signal is equal to the third level; or otherwise is equal to o the second level.The imaginary components of the first and third output channels are always zero (i.e. level 2 among levels 1, 2, 3).For example, the eleventh row 1720 of the truth table 1700 shows that on input

[1010] the optical Fourier transform apparatus outputs real component levels five 1708, two, three and two at the first, second, third and fourth output channel respectively and outputs imaginary component level two at both the second and fourth output channels. The corresponding PAM4 signal, formed in accordance with the above description, comprises first 1701, second 1702, third 1703 and fourth output channels at a fourth, fourth, third and second level respectively.The following is an example, restricted to the eleventh row 1720 of the truth table 1700, of retrieving the raw output data (e.g. the correct full complex OFT data) from the PAM4 signal. Since the real component of the second output channel 1702 of the PAM4 signal is equal to a fourth level and the real component of the third output channel 1703 of the PAM4 signal is not equal to the fourth level, the real component of the first output channel 1704 of the PAM4 signal is adjusted so that it is equal to its current level plus an additional level (as per the second sub-bullet of the first bullet above). Thus, in particular, the real component of the first output channel 1704 of the PAM4 signal is adjusted so that it is equal to a fifth level. Since the real component of the second output channel 1702 of the PAM4 signal is equal to the fourth level, the real component of the second output channel 1702 of the PAM4 signal is adjusted so that it is equal to the second level (as per the first sub-bullet of the second bullet above). The real component of the third output channel 1703 of the PAM4 signal is not adjusted in this particular case (as per the third sub-bullet of the third bullet above), and thus remains equal to the third level. The real component of the fourth output channel of the data recovered from the PAM4 signal is equal to the real component of the fourth output channel of the raw output data (as per the fourth bullet above), and in particular is therefore equal to the second level. As for the retrieval of the imaginary components of the raw output data, the imaginary components of a first and third output channel of the data recovered from the PAM4 signal is equal to the second level (as per the first and third bullets respectively, above). Since the imaginary component of the fourth output channel of the PAM4 signal is equal to neither the first level nor the third level, the imaginary component of each of the second and fourth output channels of the data recovered from the PAM4signal is equal to the second level (as per the third sub-bullet of the second bullet and the third subbullet of the fourth bullet, respectively).Figure 18 likewise is a truth table 1800 showing example input-output combinations for an optical Fourier transform apparatus (e.g. the optical Fourier transform apparatus 1100 of Figure 11a). The truth table 1800 comprises six sets of columns 1810, 1811a, 1811b, 1812, 1813a, 1813b, each set comprising four columns.As in the truth table 1700 of Figure 17, the real components of all possible 4-bit input combinations are shown in the first four columns 1810 of the truth table 1800 of Figure 18, where each bit of each possible input corresponds to a respective input channel of the optical Fourier transform apparatus. The next four columns 1811a show the real components of the raw or unprocessed output data of the optical Fourier transform apparatus when said system is given as input the corresponding 4-bits from the first four columns 1810. Likewise, the third set of four columns 1811b shows the imaginary components of the raw or unprocessed output of the optical Fourier transform apparatus when said system is given as input the corresponding 4-bits from the first four columns 1810.As described in relation to Figure 17, the optical Fourier transform apparatus may modify the raw output data so that it is PAM4 compliant. The fourth set 1812 of columns of the truth table 1800 depicts the results of such a modification of the raw output data of the second and third sets 1181a, 1811b of columns. In contrast to the modification of raw output data described in relation to Figure 17, the modification of raw output data in the context of Figure 18 is performed as follows:• Saturate the highest signal level (e.g. a fifth level) of the raw output data to the second highest signal level (e.g. a fourth level) for both the first and third output channels.The fifth and sixth sets 1813a, 1813b of columns of the truth table 1800 represent the results of retrieving the correct full complex OFT data from the PAM4 signal that is represented in the fourth set 1812 of columns. Notably, no information is lost during the formation of the PAM4 signal or recovery of data from the PAM4 signal, in the sense that the retrieved data ("PROCESSED OUTPUT" in the figure) is identical to the corresponding raw output data. As a consequence of the saturation described above, the real components of the raw output data (e.g. the correct full complex OFT data) may be retrieved from the PAM4 signal more simply than as described in relation to Figure 17, and in particular as follows:• Adjust the real component of the first output channel of the PAM4 signal to be equal to:o the fifth level, if the real component of the first output channel of the PAM4 signal is equal to the fourth level and the real component of the third output channel of the PAM4 signal is equal to the third level; or otherwise o its current level.• The real component of the second output channel of the data recovered from the PAM4 signal is equal to its current level.• Adjust the real component of the third output channel of the PAM4 signal to be equal to: o the fifth level, if the real component of the first output channel of the PAM4 signal is equal to the third level and the real component of the third output channel of the PAM4 signal is equal to the fourth level; or otherwise o its current level.• The real component of the fourth output channel of the data recovered from the PAM4 signal is equal to the real component of the fourth output channel of the raw output data.The imaginary components of the raw output data (e.g. the correct full complex OFT data) may be retrieved from the PAM4 signal in the same way as described in relation to Figure 17.Figure 19 likewise is a truth table 1900 showing example input-output combinations for an optical Fourier transform apparatus (e.g. the optical Fourier transform apparatus 1100 of Figure 11a). The truth table 1900 comprises six sets of columns 1910, 1911a, 1911b, 1912, 1913a, 1913b, each set comprising four columns.As in the truth tables of Figures 17 and 18, the real components of possible 4-bit input combinations are shown in the first four columns 1910 of the truth table 1900 of Figure 19, where each bit of each possible input corresponds to a respective input channel of the optical Fourier transform apparatus. The next four columns 1911a show the real components of the raw or unprocessed output data of the optical Fourier transform apparatus when said system is given as input the corresponding 4-bits from the first four columns 1910. Likewise, the third set of four columns 1911b shows the imaginary components of the raw or unprocessed output of the optical Fourier transform apparatus when said system is given as input the corresponding 4-bits from the first four columns 1910.As described in relation to Figures 17 and 18, the optical Fourier transform apparatus may modify the raw output data so that it is PAM4 compliant. The fourth set 1912 of columns of the truth table 1900 depicts the results of such a modification of the raw output data of the second and third sets 1191a,1911b of columns. In contrast to the modification of raw output data described in relation to Figures 17 and 18, the modification of raw output data in the context of Figure 19 is performed as follows:• Saturate the highest signal level (e.g. a fifth level) of the raw output data to the second highest signal level (e.g. a fourth level) for the first output channel.Notably, in contrast to Figure 18, in which context the signal levels for both the first and third output channels are saturated, in the context of Figure 19 signal levels for only the first output channel are saturated.The fifth and sixth sets 1913a, 1913b of columns of the truth table 1900 represent the results of retrieving the correct full complex OFT data from the PAM4 signal that is represented in the fourth set 1912 of columns. Notably, no information is lost during the formation of the PAM4 signal or recovery of data from the PAM4 signal, in the sense that the retrieved data ("PROCESSED OUTPUT" in the figure) is identical to the corresponding raw output data. As a consequence of the saturation described above, the real components of the raw output data (e.g. the correct full complex OFT data) may be retrieved from the PAM4 signal more simply than as described in relation to either of Figures 17 and 18, and in particular as follows:• Adjust the real component of the first output channel of the PAM4 signal to be equal to: o the fifth level, if the real component of the first channel output of the PAM4 signal is equal to the fourth level and the real component of the third channel output of the PAM4 signal is equal to the third level; or otherwise o its current level.• The real components of the second, third and fourth output channels of the data recovered from the PAM4 signal are equal to the real components of the second, third and fourth output channels of the raw output data respectively.The imaginary components of the raw output data (e.g. the correct full complex OFT data) may be retrieved from the PAM4 signal in exactly the same way as described in relation to Figures 17 and 18.Figure 20 likewise is a truth table 2000 showing example input-output combinations for an optical Fourier transform apparatus (e.g. the optical Fourier transform apparatus 1100 of Figure 11a). The truth table 2000 comprises six sets of columns 2010, 2011a, 2011b, 2012, 2013a, 2013b, each set comprising four columns.As in the truth tables of Figures 17, 18 and 19, the real components of possible 4-bit input combinations are shown in the first four columns 2010 of the truth table 2000 of Figure 20, where each bit of each possible input corresponds to a respective input channel of the optical Fourier transform apparatus. The next four columns 2011a show the real components of the raw or unprocessed output data of the optical Fourier transform apparatus when said system is given as input the corresponding 4-bits from the first four columns 2010. Likewise, the third set of four columns 2011b shows the imaginary components of the raw or unprocessed output of the optical Fourier transform apparatus when said system is given as input the corresponding 4-bits from the first four columns 2010.The optical Fourier transform apparatus may modify the raw output data so that it is PAM4 compliant in any of the ways described in relation to any of Figures 17, 18 and 19. Similarly, the real and imaginary components of the raw output data (e.g. the correct full complex OFT data) may be retrieved from the PAM4 signal in any of the ways described in relation to any of Figures 17, 18 and 19.Two of the sixteen rows, namely the first row 2001 and the eleventh row 2003, of the truth table 2000 of Figure 20 are omitted or blacked out, which depicts a form of input truncation that may be formed by the optical Fourier transform apparatus. The first row 2001 corresponds to the input vector

[0000] and the eleventh row 2003 corresponds to the input vector

[1010] , The optical Fourier transform apparatus always outputs

[0000] on input

[0000] , and therefore an optical Fourier transform stage (e.g. the optical Fourier transform stage 1151b of the optical Fourier transform apparatus 1100 of Figure 11a) and any other associated processing need not be performed. Rather, the output vector

[0000] may simply be copied from the input vector

[0000] , For instance, the input vector

[0000] may be directly copied into the receiver data frame memory 1101g of the optical Fourier transform apparatus 1100 of Figure 11a. The eleventh row 2003 corresponds to the input vector

[1010] , There is a known relationship between the outputs of the optical Fourier transform apparatus on inputs

[1010] and

[0101] , In particular, the output at each output channel besides the first output channel is the same for both inputs

[1010] and

[0101] , The outputs at each output channel besides the first output channel on input

[1010] may therefore be copied from the corresponding output channels of the optical Fourier transform apparatus on input

[0101] , Similarly, the outputs at each output channel besides the first output channel on input

[0101] may therefore be copied from the corresponding output channels of the optical Fourier transform apparatus on input

[1010] , The output at the first output channel on input

[1010] may be derived by inverting the output at the first output channel ofthe optical Fourier transform apparatus on input

[0101] , For example, if the output at the first output channel on input

[0101] is equal to a first level, then the output at the first output channel on input

[1010] may be determined to be equal to a fifth level. Similarly, if the output at the first output channel on input

[0101] is equal to a second level, then the output at the first output channel on input

[1010] may be determined to be equal to a fourth level. Likewise, the output at the first output channel on input

[0101] may be derived from the output of the first output channel of the optical Fourier transform apparatus on input

[1010] ,Advantages of the OFT apparatuses and methods described herein can include:• Simplification of dynamic range of the receiver or decoder electronic circuits.• Allowing discretisation of the OFT results by permitting only a subset of all possible input and output combinations to be processed by the driver circuits.• Allowing for construction of truth tables describing the operating combinations of the OFT device or PIC.• Allowing for efficient calibration routines to be designed and constructed that ensures accurate OFT calculation over the lifetime of the device.• Removal of extensive startup routines, easing of system auto-calibration routines and simplifying of stability circuits.• Simplification of receiver hardware, configuration or use, thereby reducing cost and complexity of manufacture and / or reducing complexity of operation and / or increasing data throughput of an optical Fourier transform device.• Use of OFT apparatuses (e.g. quad-Binary OFT devices) with PAM4 compliant transceiver ports thereby allowing the OFT devices to, for example, be used more easily or efficiently in line in data transport systems, data centre links, transceiver ports, communication links or computing systems or any hybrids thereof.• Compatibility of OFT apparatuses (e.g. quad-Binary OFT devices) and systems with existing backplane infrastructure, communication backends, ports and legacy technologies employing PAM4 signals.• Increased throughput of data or increased bandwidth in a Fourier transform apparatus or system.Any of the methods described in the present disclosure may be implemented on or executed on one or more controllers (e.g. one or more data controllers), one or more processors, computer system,data transport system or hybrid combination thereof connected with and configured to control the OFT apparatus.There is provided a system (e.g. a computing system, or data transport system, or a hybrid combination thereof) comprising: an optical Fourier transform apparatus comprising N input channels and K output channels; and one or more processors, wherein the one or more processors are configured to perform any of the methods described throughout the present disclosure, or alternatively wherein the system comprises one or more transitory or non-transitory computer- readable media storing instructions that when executed by the one or more processors cause the one or more processors to perform operations including any of the methods described throughout the present disclosure.There is also provided one or more tangible or non-tangible, transitory or non-transitory computer- readable media storing computer-readable instructions that when executed by one or more processors or controllers, cause the one or more processors or controllers to perform operations including any of the methods described throughout the present disclosure.It will be appreciated by those skilled in the art that various modifications and alterations could be made to disclosure above without departing from the broad inventive concepts thereof. Some of these have been discussed above and others will be apparent to those skilled in the art. It is understood, therefore, that this invention is not limited to the particular embodiments disclosed, but it is intended to cover modifications within the spirit and scope of the present disclosure, as set forth in the appended claims.Also described herein are the following numbered items:Item 1. An optical Fourier transform apparatus comprising:N input channels;K output channels; wherein the input channels each comprise a modulator, transmitter or encoder arranged to cause the input channels to emit a quantised input signal, wherein the output channels each comprise a receiver arranged to detect a quantised state at the output channel.Item 2. The optical Fourier transform apparatus of any preceding item, wherein N is an integer multiple of 1, optionally an integer multiple of 4, optionally 4.Item 3. The optical Fourier transform apparatus of any preceding item, wherein K is equal to N.Item 4. The optical Fourier transform apparatus of any preceding item, wherein the receiver in each of the output channels has an effective number of bits 'ENoB' of 12 or less, optionally 10 or less, optionally 8 or less, optionally 6 or less, optionally 4 or less, optionally 3 or less, optionally 2 or less.Item 5. The optical Fourier transform apparatus of any preceding item, wherein the receiver in each of the output channels has an effective number of bits 'ENoB' equal to log2(P+Q), wherein P is the total number of possible output states at the respective output channel given the possible input states, wherein Q. is 0, 1, 2, 3, 4, 6, 8 or 10.Item 6. The optical Fourier transform apparatus of any preceding item, wherein each output channel has a receiver which differs from the receiver of at least one other output channel.Item 7. The optical Fourier transform apparatus of any preceding item, wherein at least one of the output channels includes a receiver for recovering real data only and another of the output channels includes a receiver for recovering imaginary data only.Item 8. The optical Fourier transform apparatus of any preceding item, wherein three of the output channels include a receiver for recovering real data only and another one of the output channels includes a receiver for recovering imaginary data only.Item 9. The optical Fourier transform apparatus of any preceding item, wherein the ENoB of the receiver of one of the output channels differs from the ENoB of the receiver of another one of the output channels.Item 10. The optical Fourier transform apparatus of any preceding item, wherein each output channel has only a single receiver.Item 11. The optical Fourier transform apparatus of any preceding item, wherein at least one of the output channels includes a receiver for recovering real data having only three possible real outputstates, optionally wherein the three possible real output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.Item 12. The optical Fourier transform apparatus of any preceding item, wherein at least one of the output channels includes a receiver for recovering imaginary data having only three possible imaginary output states, optionally wherein the three possible imaginary output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.Item 13. The optical Fourier transform apparatus of any preceding item, wherein two of the output channels include a receiver for recovering real data having only three possible real output states, optionally wherein the three possible real output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.Item 14. The optical Fourier transform apparatus of any preceding item, wherein two of the output channels include a receiver for recovering imaginary data having only three possible imaginary output states, optionally wherein the three possible imaginary output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.Item 15. The optical Fourier transform apparatus of any preceding item, wherein at least one of the output channels includes a receiver for recovering real data having only five possible real output states.Item 16. The optical Fourier transform apparatus of any preceding item, wherein two of the output channels include a receiver for recovering real data having only five possible real output states.Item 17. The optical Fourier transform apparatus of any preceding item, wherein the quantised input signal comprises a finite number of possible input states.Item 18. The optical Fourier transform apparatus of any preceding item, wherein the number of possible input states for each input channel is two, three or four.Item 19. The optical Fourier transform apparatus of any preceding item, wherein the possible input states for one input channel are identical to those for every other input channel.Item 20. The optical Fourier transform apparatus of any preceding item, wherein the possible input states comprise or consist of: a first input state having a first amplitude and a second input state having a second amplitude different from the first amplitude, wherein the first input state is in phase with the second input state, or a first input state having a first phase and a second input state having a second phase out of phase with the first phase, wherein the first input state has the same amplitude as the second input state, or a first input state having a first phase and a first amplitude and a second input state having a second phase out of phase with the first phase and a second amplitude different from the first amplitude, optionally wherein the first input state is a lowest possible input state of the input channel and the second input state is a highest possible input state of the input channel, and / or wherein the first phase is 180 degrees out of phase with the second phase.Item 21. The optical Fourier transform apparatus of any preceding item, wherein the number of possible input states is such that the set of all input channels can display only MNdistinct N-bit input vectors, where each combination is in turn a N logjM-bit combination, wherein each input vector is represented by the set of input states applied at the input channels in a single frame, wherein M is the number of levels in an input and is a positive integer.Item 22. The optical Fourier transform apparatus of any preceding item, wherein each of input ports at the exit of each input channel are arranged on a first circular arc and each of output ports at the entrance of each output channel are arranged on a second circular arc where the distance between the first and second circular arcs is equal to the radius of the first circular arc and equal to the radius of the second circular arc.Item 23. The optical Fourier transform apparatus of item 22, wherein the input ports are arranged at angles 0n from a zeroth position on the second arc.Item 24. The optical Fourier transform apparatus of item 22 or 23, wherein the output ports are arranged at angles 0n from a zeroth position on the first arc.Item 25. The optical Fourier transform apparatus of item 24, wherein the zeroth position on the first arc is directly opposite the zeroth position on the second arc with respect to a virtual line bisecting the common radius of the first arc and second arc.Item 26. The optical Fourier transform apparatus of item 24, wherein the zeroth position on the first arc is and the zeroth position on the second arc lie at opposite ends of the common radius of the first arc and second arc.Item 27. The optical Fourier transform apparatus of any preceding item, wherein the angles 0n are defined by:wherein n is the port number, N is the total number of input ports or output ports, R is the radius and neff is the effective index of the waveguide mode, wherein the port number n starts at -N / 2 if N is even and -((N-1))Z2 if N is odd, and increases in integers.Also described herein are the following numbered clauses:DATA PACKINGClause 1. A method of processing an output signal from an optical Fourier transform apparatus comprising: applying a input vector to the optical Fourier transform apparatus, obtaining an output vector representing the optical Fourier transform of the input vector; modifying a channel of the output vector to form a PAM4 signal.Clause 2. The method of clause 1, wherein the number of levels in the channel is greater than four, optionally wherein the channel carries a PAM5 signal.Clause 3. The method of clause 2, wherein the channel represents the real component of the optical Fourier transform, optionally wherein the channel corresponds to a first or third output channel of the optical Fourier transform apparatus.Clause 4. The method of clause 2 or 3, wherein modifying the channel comprises reducing the number of levels in the channel to form the PAM4 signal, optionally wherein the channel carries a PAM5 signal and the number of levels is reduced to four.Clause 5. The method of clause 1, 3 or 4, wherein modifying the channel comprises detecting the highest level(s) of the channel and removing the highest level(s) from the channel, optionally wherein the channel carries a PAM5 signal and modifying the channel comprises removing the fifth level from the PAM5 signal.Clause 6. The method of clause 5, wherein removing the highest level(s) from the channel comprises: saturating or clipping the highest level(s) to the fourth level, or subtracting one level or more levels to form the fourth level.Clause 7. The method of any preceding clause, wherein the channel carries a signal with fewer than four levels, optionally a PAM3 signal.Clause 8. The method of clause 7, wherein modifying the channel comprises adding one or more levels to the channel to form the PAM4 signal.Clause 9. The method of clause 7 or 8, wherein the channel carries the real or imaginary component and wherein the channel corresponds to a second or fourth output channel of the optical Fourier transform stage.Clause 10. The method of any of clauses 7-9, wherein modifying the channel comprises forming the PAM4 signal by adding one or two levels to create a fourth level.Clause 11. The method of any preceding clause, wherein the output vector comprises the full complex result of the optical Fourier transform from the optical Fourier transform stage.Clause 12. The method of any preceding clause, further comprising transmitting the PAM4 signal over a PAM4 link.Clause 13. The method of any preceding clause, further comprising retrieving the correct full complex OFT data from the PAM4 signal by reversing the modification of the channel.Clause 14. The method of clause 13, wherein retrieving the correct full complex OFT data comprises deriving the one or more levels removed from the raw output data from the PAM4 signal.OPTIMISATION OF RECEIVERSClause 15. A method of processing obtaining a full complex optical Fourier transform result from an optical Fourier transform apparatus, the optical Fourier transform apparatus comprising four output channels, the method comprising: determining the real components of a channel output signal in each of three of the four output channels, and detecting the imaginary component of a channel output signal in the remaining output channel, determining the full complex result by obtaining the real and imaginary components of all four output channels using only the detected real components from the three output channels and the detected imaginary component from the remaining output channel.Clause 16. The method of clause 15, wherein obtaining the imaginary component of the channel output signal in one of the three channels comprises applying a logic operation to the imaginary component detected in the remaining output channel.Clause 17. The method of clause 16, wherein obtaining the real component of the channel output signal in the remaining output channel comprises applying a logic operation to the real component of the channel output signal detected in the one of the three output channels.Clause 18. The method of any of clauses 15-17, wherein obtaining the imaginary component of the channel output signal in the two of the three output channels comprises determining that the imaginary component is zero.Clause 19. The method of any of clauses 15-18, wherein obtaining the real component of the channel output signal in any of the three output channels and / or obtaining the imaginary component of the channel output signal in either the one of the three or remaining channelcomprises transforming detected optical output signal levels into signed numbers or their digital equivalents.LOGICAL DERIVATION OF OUTPUTSClause 20. A method of processing data using an Fourier transform apparatus, the method comprising: applying an input data stream to the Fourier transform apparatus, wherein the input data stream is applied in a series of sequential frames, each frame comprising an input vector, and producing an output data stream from the input data stream by determining an output vector for each frame, wherein each output vector is the Fourier transform of the corresponding input vector in the input data stream, wherein for a first input vector having a first value, a corresponding first output vector is determined via an optical Fourier transform of the first input vector, for a second input vector having a second value different from the first value, a corresponding second output vector is determined other than via an optical Fourier transform of the second input vector.Clause 21. The method of clause 20, wherein for the second input vector, the corresponding second output vector is determined via a logic operation applied to the second input vector, optionally wherein the second input vector has an overall value of zero and the corresponding second output vector is determined by copying the second input vector to produce the second output vector or by determining that the second input vector also has an overall value of zero.Clause 22. The method of clause 20 or 21, wherein for the second input vector, the corresponding second output vector is determined via the optical Fourier transform of the first input vector.Clause 23. The method of clause 22, wherein the second input vector is codified in the input data stream as a repeated input of the first input vector, optionally wherein the second output vector is determined logically from the optical Fourier transform of the first input vector.Clause 24. The method of any of clauses 20-23, wherein the second output vector contains the same corresponding vector values as the first output vector except for one differing vector value, wherein the second output vector is determined logically from the first vector via a known relationship between the differing vector values.Clause 25. The method of clause 24, wherein the known relationship is an inverted sign.DATA UNPACKINGClause 26. A method of processing an output signal from an optical Fourier transform apparatus, the method comprising: receiving the output signal from the optical Fourier transform apparatus in a form comprising a PAM4 signal; retrieving the full complex output of the optical Fourier transform apparatus from the output signal.Clause 27. The method of clause 26, wherein retrieving the full complex output of the optical Fourier transform apparatus from the output signal comprises decoding the PAM4 signal.Clause 28. The method of clause 27, wherein decoding the PAM4 signal comprises deriving a signal with a number of layers greater than four from the PAM4 signal, optionally deriving a PAM5 signal from the PAM4 signal.Clause 29. The method of clause 27 or 28, wherein decoding the PAM4 signal comprises deriving a signal with a number of layers fewer than four from the PAM4 signal, optionally deriving a PAM3 signal from the PAM4 signal.Clause 30. The method of clause 28 or clause 29 when dependent on clause 28, wherein the full complex output of the optical Fourier transform apparatus retrieved from the PAM4 signal is defined at least in part by the PAM3 and / or PAM5 signals.Clause 31. A method of operating an optical Fourier transform 'OFT' apparatus, the OFT apparatus comprising N input channels and K output channels, the method comprising:quantising the light at the input channels to form a quantised output at each output channel, and the method of any of clauses 1-14; and / or the method of any of clauses 15-19; and / or the method of any of clauses 20-25; and / or the method of any of clauses 26-30.Clause 32. The method of clause 26, wherein quantising the light at the input channels comprises limiting the light emitted by each input channel to a finite number of possible input states.Clause 33. The method of clause 27, wherein the number of possible input states for each input channel is two, three or four.Clause 34. The method of clause 27 or 28, wherein the possible input states for one input channel are identical to those for every other input channel.Clause 35. The method of any of clauses 27-29, wherein the possible input states comprise or consist of: a first input state having a first amplitude and a second input state having a second amplitude different from the first amplitude, wherein the first input state is in phase with the second input state, or a first input state having a first phase and a second input state having a second phase out of phase with the first phase, wherein the first input state has the same amplitude as the second input state, or a first input state having a first phase and a first amplitude and a second input state having a second phase out of phase with the first phase and a second amplitude different from the first amplitude, optionally wherein the first input state is a lowest possible input state of the input channel and the second input state is a highest possible input state of the input channel, and / or wherein the first phase is 180 degrees out of phase with the second phase.Clause 36. The method of any of clauses 32-35, wherein the number of possible input states is such that the set of all input channels can display only MNdistinct N-bit input vectors, where eachcombination is in turn a NMIogj-bit combination, wherein each input vector is represented by the set of input states applied at the input channels in a single frame, wherein M is a positive integer.Clause 37. The method of any of clauses 32-36, wherein the output of the optical Fourier transform apparatus is deterministic such that the finite number of possible input states gives rise to a finite number of possible output states at each output channel.Clause 38. The method of clause 37, wherein an effective number of bits 'ENoB' for a receiver in each of the output channels is equal to log2(P+Q), wherein P is the total number of possible output states at the respective output channel given the possible input states, wherein Q. is 0, 1, 2, 3, 4, 6, 8 or 10.Clause 39. The method of any of clauses 32-38, wherein there is a uniform difference between: a first output state and a second output state of the possible output states, and between the first output state and a third output state of the possible output states; or wherein there is a uniform difference between: a first output state and a second output state of the possible output states; and between a third output state of the possible output states other than the first and second output states and a fourth output state of the possible output states other than the first, second and third output states; or wherein there is a uniform difference between all adjacent output states of the possible output states; or wherein the output of the OFT is represented by a finite number of output states with a uniform interval between output states.Clause 40. The method of any preceding clause, wherein a receiver in each of the output channels has an effective number of bits 'ENoB' of 12 or less, optionally 10 or less, optionally 8 or less, optionally 6 or less, optionally 4 or less, optionally 3 or less, optionally 2 or less.Clause 41. The method of any preceding clause, wherein N is an integer multiple of 2, optionally an integer multiple of 4, optionally 4.Clause 42. The method of any preceding clause, wherein K is equal to N.Clause 43. The method of any preceding clause, wherein each output channel has a receiver which differs from the receiver of at least one other output channel.Clause 44. The method of any preceding clause, wherein at least one of the output channels includes a receiver for recovering real data only and another of the output channels includes a receiver for recovering imaginary data only.Clause 45. The method of any preceding clause, wherein three of the output channels include a receiver for recovering real data only and another one of the output channels includes a receiver for recovering imaginary data only.Clause 46. The method of any preceding clause, wherein the ENoB of the receiver of one of the output channels differs from the ENoB of the receiver of another one of the output channels.Clause 47. The method of any preceding clause, wherein each output channel has only a single receiver.Clause 48. The method of any preceding clause, wherein at least one of the output channels includes a receiver for recovering real data having only three possible real output states, optionally wherein the three possible real output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.Clause 49. The method of any preceding clause, wherein at least one of the output channels includes a receiver for recovering imaginary data having only three possible imaginary output states, optionally wherein the three possible imaginary output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.Clause 50. The method of any preceding clause, wherein two of the output channels include a receiver for recovering real data having only three possible real output states, optionally wherein the three possible real output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.Clause 51. The method of any preceding clause, wherein two of the output channels include a receiver for recovering imaginary data having only three possible imaginary output states, optionallywherein the three possible imaginary output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.Clause 52. The method of any preceding clause, wherein at least one of the output channels includes a receiver for recovering real data having only five possible real output states, or wherein two of the output channels include a receiver for recovering real data having only five possible real output states.Clause 53. A method comprising: the method of any of clauses 1-14; and / or the method of any of clauses 15-19; and / or the method of any of clauses 20-25; and / or the method of any of clauses 26-30; and / or the method of any of clauses 31-52.Clause 54. A computing system comprising: an optical Fourier transform apparatus; one or more processors, wherein the one or more processors are configured to perform the method of any of clauses 1- 53, or wherein the computing system comprises one or more transitory or non-transitory computer- readable media storing instructions that when executed by the one or more processors cause the one or more processors to perform operations including the method of any of clauses 1-53.Clause 55. One or more tangible or non-tangible, transitory or non-transitory computer-readable media storing computer-readable instructions that when executed by one or more processors, cause the one or more processors to perform operations including the method of any of clauses 1-53.

Claims

CLAIMS1. A method of operating an optical Fourier transform 'OFT' apparatus, the OFT apparatus comprising N input channels and K output channels, the method comprising: quantising the light at the input channels to form a quantised output at each output channel.

2. The method of claim 1, wherein quantising the light at the input channels comprises limiting the light emitted by each input channel to a finite number of possible input states.

3. The method of claim 2, wherein the number of possible input states for each input channel is two, three or four.

4. The method of claim 2 or 3, wherein the possible input states for one input channel are identical to those for every other input channel.

5. The method of any of claims 2-4, wherein the possible input states comprise or consist of: a first input state having a first amplitude and a second input state having a second amplitude different from the first amplitude, wherein the first input state is in phase with the second input state, or a first input state having a first phase and a second input state having a second phase out of phase with the first phase, wherein the first input state has the same amplitude as the second input state, or a first input state having a first phase and a first amplitude and a second input state having a second phase out of phase with the first phase and a second amplitude different from the first amplitude, optionally wherein the first input state is a lowest possible input state of the input channel and the second input state is a highest possible input state of the input channel, and / or wherein the first phase is 180 degrees out of phase with the second phase.

6. The method of any of claims 2-5, wherein the number of possible input states is such that the set of all input channels can display only MNdistinct N-bit input vectors, where each combination is in turn a N logjM-bit combination, wherein each input vector is represented by the set of input states applied at the input channels in a single frame, wherein M is the number of levels in an input and is a positive integer.

7. The method of any of claims 2-6, wherein the output of the optical Fourier transform stage is deterministic such that the finite number of possible input states gives rise to a finite number of possible output states at each output channel.

8. The method of claim 7 , wherein an effective number of bits 'ENoB' for a receiver in each of the output channels is equal to log2(P+Q), wherein P is the total number of possible output states at the respective output channel given the possible input states, wherein Q. is 0, 1, 2, 3, 4, 6, 8 or 10.

9. The method of any of claims 2-8, wherein there is a uniform difference between: a first output state and a second output state of the possible output states, and between the first output state and a third output state of the possible output states; or wherein there is a uniform difference between: a first output state and a second output state of the possible output states; and between a third output state of the possible output states other than the first and second output states and a fourth output state of the possible output states other than the first, second and third output states; or wherein there is a uniform difference between all adjacent output states of the possible output states; or wherein the output of the OFT is represented by a finite number of output states with a uniform interval between output states.

10. The method of any preceding claim, wherein a receiver in each of the output channels has an effective number of bits 'ENoB' of 12 or less, optionally 10 or less, optionally 8 or less, optionally 6 or less, optionally 4 or less, optionally 3 or less, optionally 2 or less.

11. The method of any preceding claim, wherein N is an integer multiple of 2, optionally an integer multiple of 4, optionally 4.

12. The method of any preceding claim, wherein K is equal to N.

13. The method of any preceding claim, wherein each output channel has a receiver which differs from the receiver of at least one other output channel.

14. The method of any preceding claim, wherein at least one of the output channels includes a receiver for recovering real data only and another of the output channels includes a receiver for recovering imaginary data only.

15. The method of any preceding claim, wherein three of the output channels include a receiver for recovering real data only and another one of the output channels includes a receiver for recovering imaginary data only.

16. The method of any preceding claim, wherein the ENoB of the receiver of one of the output channels differs from the ENoB of the receiver of another one of the output channels.

17. The method of any preceding claim, wherein each output channel has only a single receiver.

18. The method of any preceding claim, wherein at least one of the output channels includes a receiver for recovering real data having only three possible real output states, optionally wherein the three possible real output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.

19. The method of any preceding claim, wherein at least one of the output channels includes a receiver for recovering imaginary data having only three possible imaginary output states, optionally wherein the three possible imaginary output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.

20. The method of any preceding claim, wherein two of the output channels include a receiver for recovering real data having only three possible real output states, optionally wherein the three possible real output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.

21. The method of any preceding claim, wherein two of the output channels include a receiver for recovering imaginary data having only three possible imaginary output states, optionally wherein the three possible imaginary output states are zero, a positive value and a negative value, wherein the positive value has a same magnitude as the negative value.

22. The method of any preceding claim, wherein at least one of the output channels includes a receiver for recovering real data having only five possible real output states.

23. The method of any preceding claim, wherein two of the output channels include a receiver for recovering real data having only five possible real output states.

24. A system comprising: an optical Fourier transform apparatus comprising N input channels and K output channels; one or more processors or controllers, wherein the one or more processors or controllers are configured to perform the method of any of claims 1-23, or wherein the computing system comprises one or more transitory or non-transitory computer- readable media storing instructions that, when executed by the one or more processors or controllers, cause the one or more processors to perform operations including the method any of claims 1-23.

25. One or more tangible or non-tangible, transitory or non-transitory computer-readable media storing computer-readable instructions that when executed by one or more processors, cause the one or more processors to perform operations including the method any of claims 1-23.

Images