Patent Publication Number: US-2022229340-A1

Title: Optical modulator and method of use

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     The present application is a continuation-in-part of U.S. patent application Ser. No. 16/494,675, filed Sep. 16, 2019, which is a national phase application of International Application No. PCT/EP2018/056540, filed Mar. 15, 2018, which claims priority to and the benefit of U.S. Provisional Application No. 62/473,011, filed Mar. 17, 2017; the entire contents of all of the documents identified in this paragraph are incorporated herein by reference. 
    
    
     FIELD 
     Embodiments of the present invention relate to optoelectronic devices, and particularly to optoelectronic devices for quadrature-amplitude modulation and a method of modulating an optical signal in accordance with a quadrature-amplitude modulation scheme. 
     BACKGROUND 
     Quadrature-amplitude modulation (QAM) is a candidate modulation format for future optical interconnect modules, and may achieve 200 Gb/s per wavelength. This can be particularly suitable for inter-datacentre applications. In this modulation scheme, the information is encoded in the amplitude and the phase of the optical signal. In QAM-16, there are 16 points available in the in-phase-quadrature-phase complex plane. The points  101  are generally equally spaced, as shown in  FIG. 1 . Four specific bits can be associated to each symbol/point. The constellation size is chosen arbitrarily. 
     A series of pulses each with the characteristics of one of the 16 constellation points, representing four specific bits, can be sent to communicate information. According to different applications, the spacing, relative orientation, and the position of the centre point  102 , would be subject to change. 
     Conventionally, optical QAM schemes are achieved by using a nested Mach-Zehnder in-phase modulators (MZ IQ modulators). In these configurations, a cascade of Y-junctions or 3-dB couplers are used to split or combine the light. A disadvantage to such configurations is the inherent 3-dB loss each time two optical branches are combined. In addition, increasing the number of couplers will result in an increase in the total optical loss which can be significant. This is because each 3-dB coupler has a certain insertion loss (typical compact structures have around 0.1 dB of insertion loss). 
     The use of ring resonator modulations has also been demonstrated in implementing a QAM-16 modulation scheme. However, they have a low practicality for a commercial product due to their high sensitivity to environmental conditions and fabrication tolerances. They would therefore require power intensive stabilization circuits. 
     There is a desire to provide a device suitable for QAM-16 modulation which does not suffer the above drawbacks. 
     Pulse-amplitude modulation (PAM) is a modulation format selected by an IEEE standards committee for achieving 50 Gb/s per lambda for next generation optical interconnects, where information comprising a message is encoded in the amplitude of a series of pulses comprising the signal. In PAM-4 modulation, 2 2  (=4) discrete pulse amplitudes are available, which are generally equally spaced in the linear regime as shown in  FIG. 11 . 
     1 st  Summary 
     Accordingly, in a first aspect embodiments of the invention provide an optoelectronic device for quadrature-amplitude modulation (QAM), comprising:
         an input waveguide;   two intermediate waveguides, each coupled to the input waveguide via an input coupler; and   an output waveguide, coupled to each of the intermediate waveguides via an output coupler;   wherein each intermediate waveguide includes a modulating component connected in series with a phase shifting component, and each modulating component is connected to a respective electronic driver, the electronic drivers together being operable to produce a QAM-N modulated output from light entering the device from the input waveguide.       

     Such a device enables QAM-N, and preferably QAM-16, modulation in a compact (high integration density) transmitter. The device also requires a lower power consumption as compared to conventional devices, and the driver signals can be simpler. In some examples, the electronic drivers provide identical driver signals to each of the modulating components. There is no requirement for a digital-to-analogue-converter (DAC) in a device as per embodiments of the present invention. Therefore, in some embodiments, there is no DAC provided in the device. By electronic driver, it may be meant that the electronic driver is an electronic driver chip. 
     By modulating component, it may be meant each intermediate waveguide includes either: a phase modulator, an amplitude modulator, or a phase modulator and an amplitude modulator. The electronic drivers may be operable to produce a QAM-16 modulated output. 
     Optional features of embodiments of the invention will now be set out. These are applicable singly or in any combination with any aspect of embodiments of the invention. 
     The phase shifting component within each intermediate waveguide may be a DC phase shifter. By DC phase shifter, it may be meant that a component provides a specific static phase shift that does not need to be changed frequently and significantly (i.e. does not need to be modulated, and should generally be regulated to a certain value). 
     Each electronic driver may be a PAM-4 electronic driver, such that the respective modulating component can operate in four modulation states. One of these modulation states may be where the modulating component does not alter the properties of the light passing therethrough, i.e. the characteristics of the light are not actively modified. However, for clarity, it is referred to as one of the four modulation states that may be provided. 
     The input coupler may be configured to equally split input light between the two intermediate waveguides. In this example, the modulating component may be an amplitude modulator. For example, the modulating component may be an electro-absorption modulator. Alternatively, the modulating component may be a ring modulator, or an amplitude modulating using polarization modulators. 
     The device may further comprise a DC phase shifting intermediate waveguide, coupled to the input waveguide and the output waveguide via the respective couplers, the DC phase shifting intermediate waveguide being configured to re-centre a constellation corresponding to the QAM-N outputs available from the device. This DC phase shifting intermediate waveguide may contain, asides from the waveguide itself, only a DC phase shifter. With the introduction of this intermediate waveguide, the light may be split between the two intermediate waveguides and the DC phase shifting waveguide at a ratio of 1:x:1, such that the power is split equally between the two intermediate waveguides and some fraction x is provided to the DC phase shifting waveguide. The value of x may be defined by the characteristics of the modulating components within the two intermediate waveguides. Since the characteristics of the modulator (i.e. extinction ratio, chirp and insertion loss for an EAM, or insertion loss in a phase modulator) will define the exact shape of the constellation diagram (e.g. size, shift from origin, and relative rotation), the power portion “x” in the extra arm should be such that it brings back the centre point of the constellation to the origin. 
     In further detail, if each intermediate waveguide is considered to have a representative vector in the IQ plane (see, e.g.  FIG. 1 ) the vector corresponding to the DC phase shifter intermediate waveguide should have an amplitude proportional to the distance of the centre of the constellation to the origin, and a phase of 180° away from the phase of the centre-point vector, so that when it is combined with the vectors of the two intermediate waveguides it brings the centre point of the constellation back to the origin (0,0). 
     A static relative phase difference between the two intermediate waveguides may be set at 90°. This static relative phase difference may be set by the phase shifting component (e.g. DC phase shifter) within each intermediate waveguide. 
     The input coupler may be configured to split input light between the two intermediate waveguides at a ratio of 2:1 such that one of the waveguides receives twice the optical power of the other. In this example, the modulating component may be a phase modulator. For example, the phase modulator may be a heater, a pn junction, a p-i-n phase modulator, or a metal-oxide-semiconductor (MOS) based phase modulator. In this example, a static phase difference between the two intermediate waveguides may be set to 0°. 
     The two intermediate waveguides may be first and second intermediate waveguides, and the device may further comprise a third intermediate waveguide and a fourth intermediate waveguide, each coupled to the input waveguide via the input coupler and coupled to the output waveguide via the output coupler. The third and fourth intermediate waveguides may each include a modulating component connected in series with a phase shifting component and each modulating component may be connected to an electronic driver, the electronic drivers of each of the first and second intermediate waveguides and third and fourth intermediate waveguides may be operable to produce a QAM-N modulated output from light entering the device from the input waveguide. The phase shifting component within the third and fourth intermediate waveguides may be a DC phase shifter. Each driver may be a binary non-return-to-zero driver, such that the respective modulating component can be operated in two modulation states. The input coupler may be configured to split input light between the intermediate waveguides at a ratio of 2:1:2:1, such that two of the waveguides receive twice the optical power of the other two. Where a DC phase shifting intermediate waveguide is also present, the input light may be split at a ratio of 2:1:x:2:1, where x is the amount provided to the DC phase shifting intermediate waveguide and defined by the modulation components characteristics. A static phase difference between the first intermediate waveguide and the second intermediate waveguide may be 180°, and a static phase difference between the third intermediate waveguide and the fourth intermediate waveguide may be 180°. A static phase difference between the first intermediate waveguide and the third intermediate waveguide may be 90°, and a static phase difference between the second intermediate waveguide and the fourth intermediate waveguide may be 90°. Each modulating component may be a phase modulator (for example those discussed above), or each modulating component may be an amplitude modulator (for example those discussed above). The modulating components in all of the first to fourth intermediate waveguides may have substantially identical characteristics. For example, if the modulating components are all electro-absorption modulators they may all have the same length. 
     Where the device includes first, second, third, and fourth intermediate waveguides and also includes a DC phase shifting intermediate waveguide, the light may be split between the five intermediate waveguides at a ratio of 2:1:x:2:1 where ‘x’ is (as above) defined by the modulators characteristics. 
     Each intermediate waveguide may include two modulating components connected in series: a phase modulator; and an amplitude modulator. 
     Either or both of the input coupler and the output coupler may comprise at least one phase correcting channel, said phase correcting channel having a width, as measured in a direction perpendicular to a guiding direction of the channel, which varies along a direction parallel to the guiding direction of the channel. By channel, it may be meant that a path exists for an optical signal passing through the coupler. For example, if the coupler is a 1×5 coupler, it may comprise at least 5 channels at the output of the coupler. 
     In all examples of the embodiments of invention, the modulating components within each of the intermediate waveguides may have the same modulating characteristics. For example, where the modulating component is an electro-absorption modulator, the electro-absorption modulators present in each of the intermediate waveguides should have the same length and other characteristics that affect their modulation. 
     In a second aspect, the embodiments of invention provide a method of modulating an optical signal according to a quadrature-amplitude modulation (QAM) scheme, comprising:
         providing an optical signal at an input waveguide;   splitting the optical signal into two intermediate waveguides, each coupled to the input waveguide via an input coupler;   modulating the amplitude and phase of the optical signal present in each of the intermediate waveguides, according to a QAM scheme and by respective modulating components connected in series with respective phase shifting components, said modulating components being respectively connected to and driven by an electronic driver; and   recombining the optical signal into an output waveguide, coupled to each of the intermediate waveguides via an output coupler, wherein the recombined optical signal.       

     Optional features of embodiments of the invention will now be set out. These are applicable singly or in any combination with any aspect of embodiments of the invention. 
     The phase shifting component may be a DC phase shifter. 
     Each modulating component may be driven by a PAM-4 electronic driver so as to be operable in four modulation states. 
     The input coupler may equally split the optical signal between the two intermediate waveguides. The modulating component may be an amplitude modulator, for example an electro-absorption modulator, and may modulate the amplitude of the optical signal. 
     A DC phase shifting intermediate waveguide may be coupled to the input waveguide and the output waveguide via the respective couplers, and may re-centre a constellation corresponding to the QAM-N outputs. 
     A static phase difference between the two intermediate waveguides may be set at 90°. 
     The input coupler may split the optical signal between the two intermediate waveguides at a ratio of 2:1, such that one of the waveguides receives twice the optical power of the other. The modulating component may be a phase modulator. A static phase difference between the two intermediate waveguides may be set at 0°. 
     The two intermediate waveguides may be first and second intermediate waveguides, and the optical signal may be split between the first and second intermediate waveguide and also a third and fourth intermediate waveguide, the third and fourth intermediate waveguides may each be coupled to the input waveguide via the input coupler and coupled to the output waveguide via the output coupler. The amplitude and phase of the optical signal present in each of the intermediate waveguides may be modulated according to a QAM scheme by respective modulating components connected in series with respective phase shifting components. The phase shifting components may be DC phase shifters. Each modulating component may be driven by a non-return-to-zero electronic driver so as to be operable in two modulating states. The input coupler may split the optical signal between the intermediate waveguides at a ratio of 2:1:2:1, such that two of the waveguides receive twice the optical power of the other two. A static phase difference between the first intermediate waveguide and the second intermediate waveguide may be 180° and a static phase difference between the third intermediate waveguide and the fourth intermediate waveguide may be 180°. A static phase difference between the first intermediate waveguide and the third intermediate waveguide may be 90°, and a static phase difference between the second intermediate waveguide and the fourth intermediate waveguide may be 90°. Each modulating component may be a phase modulator, or each modulating component may be an amplitude modulator. 
     Each intermediate waveguide may include two modulating components connected in series: a phase modulating component; and an amplitude modulating component. Each modulating component may be driven by a non-return-to-zero electronic driver. 
     Either or both of the input coupler and the output coupler may comprise at least one phase correcting channel, said phase correcting channel having a width, as measured in a direction perpendicular to a guiding direction of the channel, which varies along a direction parallel to the guiding direction of the channel. By channel, it may be meant that a path exists for an optical signal passing through the coupler. For example, if the coupler is a 1×5 coupler, it may comprise at least 5 channels at the output of the coupler. 
     The method may be performed on an optoelectronic device according to the first aspect. 
     Aspects and embodiments of the present invention will now be discussed with reference to the accompanying figures. Further aspects and embodiments will be apparent to those skilled in the art. All documents mentioned in this text are incorporated herein by reference 
     2 nd  Summary 
     Accordingly, in a first aspect, embodiments of the invention provide an optical device, operable to provide a PAM-N modulated output, comprising:
         an input waveguide, configured to receive light;   a first electro-absorption modulator, coupled to receive light from the input waveguide, and operable to produce a first output or a second output, wherein the second output has a lower amplitude than the first output;   a second electro-absorption modulator, coupled to receive light from the input waveguide, and operable to produce a third output or a fourth output, wherein the fourth output has a lower amplitude than the third output; and   an output waveguide, coupled to receive light from the first electro-absorption modulator and the second electro-absorption modulator, and output a combined signal comprising an output of the first electro-absorption modulator and an output of the second electro-absorption modulator;   wherein the first electro-absorption modulator and the second electro-absorption modulator are disposed in parallel.       

     In a second aspect, embodiments of the invention provide a Mach-Zehnder interferometer, comprising:
         an input waveguide;   an output waveguide;   a first arm, optically connecting the input waveguide and the output waveguide;   a second arm, optically connecting the input waveguide and the output waveguide, the second arm being disposed in parallel with the first arm; and   at least two electro-absorption modulators, disposed respectively in the first arm and the second arm, and operable to provide PAM-N modulation.       

     Optional features of embodiments of the invention will now be set out. These are applicable singly or in any combination with any aspect of embodiments of the invention. 
     The optical device may further comprise:
         a variable input coupler, disposed between the input waveguide and both of the first electro-absorption modulator and the second electro-absorption modulator.       

     The input coupler may be configured to modify the light transmitted to the first electro-absorption modulator and the second electro-absorption modulator such that adjacent PAM-N modulated outputs generated by the device are linearly spaced. The input coupler may be configured to unequally split input light between the first electro-absorption modulator and the second electro-absorption modulator. The input coupler may be a Mach-Zehnder interferometer, comprising an input coupler coupled to the input waveguide, two parallel arms and an output coupler. At least one of the parallel arms may include a phase-shifter. 
     Disposed between the output waveguide and the first electro-absorption modulator and the second electro-absorption modulator may be a variable output coupler. The output coupler may be configured to modify the received light from the first electro-absorption modulator and the second electro-absorption modulator such that adjacent PAM-N modulated outputs are linearly spaced. The output power of light which can be outputted by the coupler may be described by: 
     
       
         
           
             
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     where P OUT  is the output power, k is a coupling coefficient P in1  is the input power from the first electro-absorption modulator, θ 1  is the phase of light entering the coupler from the first electro-absorption modulator, P in2  is the input power from the second electro-absorption modulator, and θ 2  is the phase of light entering the coupler from the second electro-absorption modulator. k may be less than 0.44. The variable output coupler may a Mach-Zehnder interferometer, comprising an input coupler coupled to the first electro-absorption modulator and the second electro-absorption modulator, two parallel arms and an output coupler coupled to the output waveguide. At least one of the parallel arms may include a phase-shifter. 
     The optical device may further comprise:
         a first analogue driver, said first driver configured to provide a first driver signal to the first electro-absorption modulator; and   a second analogue driver, said second driver configured to provide a second driver signal to the second electro-absorption modulator;   wherein, when driven by the respective driver signal, the first electro-absorption modulator produces the second output and the second electro-absorption modulator produces the fourth output. The first driver signal and the second driver signal are identical.       

     The two analogue drivers may be two driver circuits occupying a single driver chip. The drivers may independently provide voltages V 1  and V 2 , which may have the same value or different values. V 1  and V 2  may both equal 2V. By using this configuration, it is possible to implement the conversion of a 2-bit symbol into 4 analogue optical states without the use of a digital-to-analogue-converter. One of the drivers can correspond to the least-significant bit, and the other may correspond to the most-significant bit. 
     The optical device may include a Mach-Zehnder interferometer, and the first electro-absorption modulator is disposed within a first arm of the Mach-Zehnder interferometer and the second electro-absorption modulator is disposed within a second arm of the Mach-Zehnder interferometer. 
     The device may include a phase-shifter associated with at least one of the electro-absorption modulators, said phase-shifter being operable to modify a phase shift associated with the respective electro-absorption modulator. The phase-shifter may be disposed within an arm of the Mach-Zehnder interferometer. The phase-shifter may be any one or more of: a heater; a PIN phase shifter; or a PN phase shifter. 
     The device may include a directional coupler coupled to the output waveguide, configured to redirect a portion of the output power for use to bias and stabilize the phase shifter. The portion of the output power may be less than 5%. 
     According to an embodiment of the present disclosure, there is provided an optoelectronic device for modulation, including: an input waveguide; a first intermediate waveguide and a second intermediate waveguide, each coupled to the input waveguide via an input coupler; and an output waveguide, coupled to each of the intermediate waveguides via an output coupler, wherein each intermediate waveguide includes a modulating component connected in series with a phase shifting component, and each modulating component is connected to a respective electronic driver, the electronic drivers together being operable to produce a modulated output from light entering the optoelectronic device from the input waveguide, and wherein: the input coupler is a variable input coupler; the output coupler is a variable output coupler; or the input coupler is a variable input coupler and the output coupler is a variable output coupler. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Embodiments of the invention will now be described by way of example with reference to the accompanying drawings in which: 
         FIG. 1  shows a constellation diagram illustrating QAM-16 modulation; 
         FIG. 2A  shows a device for QAM-16 modulation; 
         FIG. 2B  shows a constellation diagram for the device in  FIG. 2A ; 
         FIG. 3A  shows a variant device for QAM-16 modulation; 
         FIG. 3B  shows a constellation diagram for the device in  FIG. 3A ; 
         FIG. 4A  shows a variant device for QAM-16 modulation; 
         FIG. 4B  shows a constellation diagram for the device in  FIG. 4A ; 
         FIG. 5  shows a constellation diagram illustrating a non-ideality when using non-return-to-zero drivers; 
         FIG. 6A  shows a variant device for QAM-16 modulation; 
         FIG. 6B  shows a constellation diagram for the device in  FIG. 6A ; 
         FIG. 7  is a plot showing the variation of in total input power vs modulator length; 
         FIG. 8A  shows a variant device for QAM-16 modulation; 
         FIG. 8B  shows a constellation diagram for the device in  FIG. 8A ; 
         FIG. 9  shows a top-down plan view of a chip layout including a device for QAM-16 modulation; 
         FIG. 10  shows a schematic view of an optimised coupler; 
         FIG. 11  shows a plot of signal (in normalised units) against time; 
         FIG. 12  shows an optical device; 
         FIG. 13  shows a plot of signal against time for the optical device shown in  FIG. 12 ; 
         FIGS. 14A and 14B  show, respectively, the behaviour of a coupler of the optical device of  FIG. 12 ; 
         FIG. 15  shows a plot of PAM-4 transmission, together with Gaussian noise; 
         FIG. 16  shows a plot of power against coupling factor k; 
         FIG. 17A  shows a plot of EAM output power against coupling factor k; 
         FIG. 17B  shows a plot of EMA input power against coupling factor k; 
         FIG. 18  shows a variant optical device; 
         FIG. 19  shows a further variant optical device; 
         FIGS. 20A and 20B  show a transmission system including the optical device of  FIG. 12 ,  FIG. 18 , or  FIG. 19 ; 
         FIG. 21  shows a graph of extinction ratio vs. input power; 
         FIG. 22  shows a graph of extinction ratio vs. input power; 
         FIG. 23  shows a diagram of input power vectors and a diagram of output power vectors; 
         FIG. 24  shows a power level diagram; 
         FIG. 25  shows a diagram of input power vectors and a diagram of output power vectors; 
         FIG. 26  shows a power level diagram; 
         FIG. 27  shows three vector diagrams; 
         FIG. 28  shows an optical device; and 
         FIG. 29  shows an optical device. 
     
    
    
     DETAILED DESCRIPTION AND FURTHER OPTIONAL FEATURES 
       FIG. 2A  shows a schematic view of an optoelectronic device according to some embodiments of the present invention. An input waveguide  201 , provides an optical signal with power P in  to an input multi-mode interference coupler  202 . The optical signal is split by the coupler into a first intermediate waveguide  203  and a second intermediate waveguide  204 . The coupler is configured to split the optical signal at a ratio of 1:1 between the waveguides, and so each receives an optical signal with a power of P in /2. 
     The first intermediate waveguide  203  includes, disposed along its optical path, a DC phase shifter  205   a  and an electro-absorption modulator  206   a . The DC phase shifter is operated to apply a static phase shift to the optical signal as it is transmitted through the intermediate waveguide. The DC phase shifter may be implemented as a heater, a p-i-n junction, or a pn junction. The electro-absorption modulator  206   a  is driven by a PAM-4 driver  207   a  able to provide four operating voltages: V 0  to V 3 . Each of these operating voltages corresponds to a modulating state of the electro-absorption modulator. In this way, the electro-absorption modulator  206   a  can be considered to provide a PAM-4 modulated output from the optical signal provided into the intermediate waveguide. In a 66 μm long elector-absorption modulator operating at a wavelength of 1543 nm, V 0  to V 3  may take values of: 0V, 0.4V, 1V, and 2V. These voltages may provide 4 equally spaced field outputs (and not intensity outputs). It should be noted that the voltages may not be equally spaced due to the non-linear behaviour of the EAM in this example. 
     Similarly, the second intermediate waveguide  204  includes, disposed along its optical path, a DC phase shifter  205   b  and an electro-absorption modulator  206   b . The DC phase shifter is operated to apply a phase shift to the optical signal as it is transmitted through the intermediate waveguide. As with the previous electro-absorption modulator, this modulator  206   b  is also driven by a PAM-4 driver  207   b  able to provide four operating voltages: V 0  to V 3 . Each of these operating voltages corresponds to a modulating state of the electro-absorption modulator. Both modulators may operable using the Franz-Keldysh effect, or the Quantum-confined Stark effect if the modulators contain quantum wells. In this way, the electro-absorption modulator  206   b  can be considered to provide a PAM-4 modulated output from the optical signal provided into the intermediate waveguide. The modulators may be silicon germanium based electro-absorption modulators. 
     An output multi-mode interference coupler  208  is connected to both the first and second intermediate waveguides. This coupler  208  recombines the optical signals transmitted through each intermediate waveguide, and provides an output signal with power P out . As the optical signals provided by each intermediate waveguide are different (or can be different) in phase, the output signal can be modulated according to a QAM-16 scheme. The output coupler  208  is connected to an output waveguide  209 , which allows the modulated signal to exit the device. Coupled between the input coupler  202  and the output coupler  202  is a DC phase shifter  210 . This DC phase shifter  210  may be contained within a DC phase shifting intermediate waveguide, and may be used to bring the centre point of a constellation corresponding to the outputs of the device to the origin. 
     A constellation diagram, showing the possible outputs, is shown in  FIG. 2B . Sixteen points, shown in area  211 , are available for transmitting information, and can be utilized by the combination of the optical signal modulated in each intermediate waveguide. Four possible modulation states are available from the modulator  206   a  in the first intermediate waveguide:  212   a   1 ,  212   a   2 ,  212   a   3 , and  212   a   4 . Modulation state  212   a   1  is achieved by not actively modulating the light passing through the amplitude modulator  206   a . The remaining modulation states are achieved by applying increasing degrees of amplitude modulation. In this example, the four modulation states vary in amplitude and phase. The variation in phase is caused by the chirp of the modulator, and leads to the sixteen points in area  211  being arranged in an array which has been rotated. Similarly, four available modulation states are available from the modulator  206   b :  212   b   1 ,  212   b   2 ,  212   b   3 , and  212   b   4 . Again, modulation state  212   b   1  is achieved by not actively modulating the light passing through the amplitude modulator  206   b . The remaining modulation states are achieved by applying increasing degrees of amplitude modulation. It should be noted that in this example the modulators  206   a  and  206   b  are identical with respect to their modulating characteristics, e.g. length, extinction ratio, and chirp (i.e. an unintentional change in phase caused by the amplitude modulator). 
       FIG. 3A  shows a variant optoelectronic device according to an embodiment of the invention. It shares a number of features with the device shown in  FIG. 2A , and so like reference numerals are used for like features. A notable difference between this device and that shown in  FIG. 2A , is that the modulating components  301   a  and  301   b  here are phase modulators. The phase modulators may operate by the Pockels effect. A further difference is that the optical signal is not split at a ratio of 1:1 as in the previous device. Instead, the input coupler  301  splits the optical signal at a ratio of 2:1 between the first and second intermediate waveguides. Therefore the first intermediate waveguide receives an optical signal with power P in,1  and the second intermediate waveguide receives an optical signal with power P in,2 . The static relative phase difference in this example should be set to 0°. 
     Again, the phase modulators  301   a  and  301   b  are driven by respective PAM-4 drivers  207   a  and  207   b  such that each is operable in four modulating states. As shown in  FIG. 3B , the modulation states  310   b   1 - 310   b   4  indicate the states which may be provided by whichever of the intermediate waveguides that received an optical signal with the lowest power. Modulation states  310   a   1 - 310   a   4  may be provided by whichever of the intermediate waveguides that received an optical signal with the highest power. Each of the sixteen points of the constellation diagram can be used by a set combination of outputs from the first and second intermediate waveguide. It should be noted that, in this example, the centre point of the constellation should remain at the origin regardless of non-idealities in the modulators, drivers, or couplers, and therefore it should not be necessary to include a DC phase shifting intermediate waveguide. 
       FIG. 4A  shows a further variant device according to some embodiments of the present invention. Here, the input waveguide  201  is connected to a 1×4 star coupler  401 . It should be noted that the star coupler could be replaced by a multi-mode interference coupler, which is configured to produce a 1:2:1:2 splitting or coupling ratio. The star coupler  401  splits the incoming optical signal between four intermediate waveguides: first intermediate waveguide  402 ; second intermediate waveguide  403 ; third intermediate waveguide  404 ; and fourth intermediate waveguide  405 . Each of the intermediate waveguides includes a DC phase shifter as per previous examples. However, in this instance, phase modulators  406   a - 406   d  are connected to non-return-to-zero drivers  407   a - 407   b  which are substantially identical. Therefore each modulator is operable in two modulating states, and can modulate the phase of the light passing therethrough between either 0° or 180°. 
     The star coupler  401  splits or couples the incoming optical signal between the four intermediate waveguides according to the ratio: 1:2:2:1, such that the second  403  and third  404  intermediate waveguides each receive twice the optical power as compared to the signal received by the first  402  and fourth  405  intermediate waveguides. A static relative phase difference between the intermediate waveguides should be set as: 180°, 0°, 90°, and 270° such that the first and second intermediate waveguide are separated by 180° of phase and the third and fourth intermediate waveguides are separated by 180° of phase. However the first and third intermediate waveguides are separated by 90° of phase, and the second and fourth intermediate waveguides are also separated by 90° of phase. 
     A device as shown in  FIG. 4A  is able to produce the modulation states  440  shown in  FIG. 4B . These are obtained by combining the modulation states of each of the four intermediate waveguides. The first intermediate waveguide  402 , provides modulation states  410   b   1  and  410   b   2  which are generally the same in magnitude but with a 180° phase change. The second intermediate waveguide  403  provides modulation states  410   a   1  and  410   a   2  which are, again, generally the same in magnitude but with a 180° phase change. The third intermediate waveguide  404  provides modulation states  410   c   1  and  410   c   2 , and the fourth intermediate waveguide  405  provides modulation states  410   d   1  and  410   d   2 . 
     The centre point of the constellation, as shown in  FIG. 4B , should remain at the origin when using the device shown in  FIG. 4A . However, non-idealities in the phase modulator characteristics, electronic drivers, or the input/output couplers will result in a shift of the centre point and an additional intermediate waveguide may be necessary to bring the centre point of the constellation back to the centre (as has been discussed previously). In this case, the power and the phase of the additional intermediate waveguide would need to be adjusted accordingly and the input/output couplers would need to be modified to a 1×5 configuration with a corresponding adjustment to the splitting/coupling ratio. 
       FIG. 5  illustrates a constellation diagram corresponding to a device such as that shown in  FIG. 4A , but where the modulators or drivers possess a non-ideality. In this example, the non-ideality has led to a phase modulation of less than 180° in each intermediate waveguide. Therefore the 16 constellation points have shrunk in total size relative to the example shown in  FIG. 4B , and rotated around the inner-upper-right black point in a clockwise fashion. 
       FIG. 6A  shows a further variant device according to some embodiments of the present invention. The input waveguide  201  provides an optical signal with power P in  to a 1×5 input star coupler  601 . It should be noted that the star coupler could be replaced by a multi-mode interference coupler, which is configured to produce a 1:2:2:1 splitting or coupling ratio. The input coupler in this example splits the light into four intermediate waveguides: a first  602 , a second  603 , a third  604 , and a fourth  605 . The input power is split between the intermediate waveguides at a ratio of 1:2:2:1, such that the optical power in two of the intermediate waveguides is twice the optical power in the other two intermediate waveguides. In this example, the input power into the second and third intermediate waveguides is twice the input power into the first and fourth intermediate waveguides. A static relative phase difference between the intermediate waveguides should be as described above, namely: 180°, 00, 90°, and 270°. 
     Each of the intermediate waveguides of the device shown in  FIG. 6A  include DC phase shifters  606   a - 606   d  as described previously. They also all include an electro-absorption modulator  608   a - 608   d  respectively. The modulators should be substantially identical, and so have the same modulation characteristics e.g. length, extinction ratio, and chirp. Each modulator is driven by one of non-return-to-zero drivers  607   a - 607   d . These drivers allow the respective modulator to operate in one of two modulation states. The star output coupler  609  recombines the optical signals from each of the intermediate waveguides and provides a combined modulated output from the device. 
     A device as shown in  FIG. 6A , but without the DC phase shifting intermediate waveguide  210 , is able to produce the modulation states  660  shown in  FIG. 6B . These are obtained by combining the modulation states of each of the four intermediate waveguides. The first intermediate waveguide  602 , provides modulation states  610   c   1  and  610   c   2 . Modulation state  610   c , corresponds to a 0V signal applied by the driver, e.g. a modulation state where the amplitude of the signal present in the first intermediate waveguide is not modified. Modulation state  610   c   2  corresponds to a signal greater than 0V being applied by the driver, and so the amplitude of the modulation state is less than that of modulation state  610   c   1 . Moreover, as the modulator  608   a  will unintentionally apply a phase shift (known as chirp), modulation state  610   c   2  is not aligned with modulation state  610   c   1 . Similarly, second intermediate waveguide  603  is operable to provide modulation states  610   d   1  and  610   d   2 . Third intermediate waveguide  604  is operable to provide modulation states  610   a   1  and  610   a   2 , and fourth intermediate waveguide  605  is operable to provide modulation states  610   b   1  and  610   b   2 . 
     As has been discussed previously, the constellation diagram has rotated due to the chirp introduced by the electro-absorption modulators. By using the DC phase shifting intermediate waveguide discussed it is possible to shift the constellation centre point back to the origin (0,0). 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 discusses two examples of electro-absorption modulators 
               
               
                 usable with some embodiments of the present invention: 
               
            
           
           
               
               
               
               
            
               
                   
                 Version 
                 Standard 
                 Aggressive 
               
               
                   
                   
               
               
                   
                 Modulator Type 
                 EAM 
                 EAM 
               
               
                   
                 EAM Length 
                 41.5 μm 
                 78 μm 
               
               
                   
                 EAM extinction ratio 
                 4.5 dB 
                 8.48 dB 
               
               
                   
                 EAM Δϕ 
                 0° , 14°  
                 0° , 26.3°  
               
               
                   
                 EAM Insertion Loss 
                 2.98 dB 
                 5.6 dB 
               
               
                   
                 Power Splitting Ratio 
                 1:2:0.78:2:1 
                 1:2:0.5:2:1 
               
               
                   
                 Driver Type 
                 4 × Binary NRZ 
                 4 × Binary NRZ 
               
               
                   
                 Total Driver Power 
                 Approx. 140 mW 
                 Approx. 260 mW 
               
               
                   
                 consumption (50 Gbaud) 
                   
                   
               
               
                   
                 Optical Power Penalty 
                 23.27 dB 
                 21.62 dB 
               
               
                   
                   
               
            
           
         
       
     
       FIG. 7  is a plot of representative input power against electro-absorption modulator length. It shows, for example, that when providing an EAM with a length of 41.5 μm an extinction ratio of 4.5 is possible (when centralization is included). Similarly, when an EAM with a length of 78 μm is used an extinction ratio of 8.48 is possible. It can be seen from the plot how much power is required for a given target constellation size (chosen based on the requirements of the device), using an EAM with a certain length (in the x axis). For example, in an illustrative example, to achieve the targeted constellation in  FIG. 6B  using an EAM with length 41.5 μm (and each EAM having an extinction ratio of 4.5 dB when modulated), will require around 50 mW of input power into the system. Similarly, it can be seen that an optimal length of the EAMs is around 78 μm, giving an extinction ratio of 8.48 dB each, which would require 37 mW of input power. 
     In general, increasing the EAM length improves the system since the extinction ratio associated with the EAM increases. However, the insertion loss also increases with length and there may be a trade-off that results in an optimum point which minimises the required input power. 
       FIG. 8A  shows a further variant device according to some embodiments of the present invention. It shares a number of features with the device shown in  FIG. 2A , and so like features are indicated by like reference numerals. 
     The first intermediate waveguide  801  of this device includes, disposed along its length, a DC phase shifter  803   a , followed by an electro-absorption modulator  805   a , and finally a phase modulator  807   a . The electro-absorption modulator and phase modulator are driven by respective non-return-to-zero (NRZ) drivers  804   a  and  808   a . Therefore, an optical signal provided to the first intermediate waveguide  801  via the coupler  202  may be modulated in both amplitude and phase by a single intermediate waveguide. The second intermediate waveguide  802  similarly includes a DC phase shiftier  803   b , electro-absorption modulator  805   b , and phase modulator  807   b . Whilst these examples show the EAM and phase modulators arranged with the phase modulator after the EAM, it is of course possible to reverse this order. 
     The phase modulators  807   a  and  807   b  should be operable to modulate the phase of light passing therethrough between the states of 0° and 180° relative phase difference. The EAMs  805   a  and  805   b  should be operable to generate an extinction ratio of 20×log(3/1)=9.54 dB with zero chirp so as to provide equally spaced constellation points. The right length for the EAM should be chosen so as to avoid high insertion losses. One method for compensating for EAM chirp is to segment the respective phase modulator into a main segment and a smaller secondary segment, and use the smaller secondary segment to correct the phase change induced by the EAM. This would require a second smaller electronic binary NRZ driver for the smaller secondary segment of the phase modulator. 
     The input coupler  201  and output coupler  208  may be a 3-dB coupler. A static relative phase difference between the two intermediate waveguides should be set at 90°. As noted before, non-idealities in the phase modulators or EAMs may cause the centre point of a constellation corresponding to the output of the device to shift and rotate. If so, an additional DC phase shifting intermediate waveguide may be introduced as discussed previously. If such an intermediate waveguide is introduced, the input and output couplers should be modified to a 1×3 configuration with the appropriate splitting/coupling ratios. 
       FIG. 8B  shows a constellation diagram indicating the possible modulation states of one of the intermediate waveguides. Modulation state  810   a   1  is achieved by modulating neither amplitude nor phase. Modulation state  810   a   2  is achieved by modulating only the amplitude of the light passing through the respective intermediate waveguide. Modulation state  810   a   3  is achieved by modulating only the phase of the light passing therethrough, and modulating state  810   a   4  is achieved by modulating both the amplitude and the phase of the light. It should be noted in this diagram, that the chirp incurred by the EAMs is not corrected for, and so modulation states  810   a   2  and  810   a   4  are not aligned with  810   a   1  and  810   a   3  respectively. 
     Similar behaviour occurs in the other of the two intermediate waveguides, whilst statically rotated by 90°, to thereby provide 16 constellation points at the output coupler. In this example, an extinction ratio of 9.54 dB and an arbitrary chirp of 14° is assumed for the EAMs. 
       FIG. 9  shows an example chip layout incorporating a device as shown in  FIG. 6A . Like features are indicated by like reference numerals. A pitch  902 , between adjacent waveguides containing modulating components is around 2 mm and a distance  903  of each modulating component from an edge of the chip is around 271 μm and may be around 254 μm. The lower portion  901  of the chip is given over to testing structures. 
       FIG. 10  shows an optimized coupler which forms a part of this disclosure. The coupler  1000  shown may be provided as an implementation of either the input or output coupler (or both) where the device has more than two intermediate waveguides. In an example where the optimized coupler is an input coupler, it is connected at one end to the input waveguide  201  discussed above. The input waveguide  201  connects to a 1×7 multi-mode interference portion  1100 , which divides light between seven channels. Two pairs of the channels:  1002  and  1003 , and  1005  and  1006 , are combined so that the coupler is a 1×5 coupler. However, the portions of light arriving at channels  1002  and  1003  are out of phase by 
     
       
         
           
             
               π 
               7 
             
             , 
           
         
       
     
     as they have respectively experienced relative phase shifts of: 
     
       
         
           
             
               
                 6 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 π 
               
               7 
             
             ⁢ 
             
                 
             
             ⁢ 
             and 
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 
                   5 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                 
                 7 
               
               . 
             
           
         
       
     
     Similarly, the portions of light arriving channels  1005  and  1006  are also out of phase by 
     
       
         
           
             
               π 
               7 
             
             , 
           
         
       
     
     as they have respectively experienced relative phase shifts of 
     
       
         
           
             
               
                 5 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 π 
               
               7 
             
             ⁢ 
             
                 
             
             ⁢ 
             and 
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 
                   6 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                 
                 7 
               
               . 
             
           
         
       
     
     Therefore, if left unmodified, when the portions of light recombined in 2×1 MMIs  1008  and  1009 , the phase mismatch would cause significant losses. In order to rectify this, channels  1002 ,  1003 ,  1005 , and  1006  have widths which taper in a direction parallel to the guiding direction of the channel. In channels  1002  and  1006 , the width of the channel initially decreases before increasing after a midpoint in the channel. Whereas, the width of channels  1003  and  1005  initially increase before decreasing after a midpoint of the channel. Tapered channels such as these can be engineered in order to remove any relative phase shift between respective pairs of channels. Each pair of channels connects to a 2×1 MMI, which provides output channels  1010  and  1012  of the coupler. 
     The remaining three channels:  1001 ,  1004 , and  1007  are not combined and so there are no issues regarding phase mismatch. These channels therefore correspond to further output channels of the coupler. Each of channels  1001 ,  1010 ,  1013 ,  1012 , and  1007  are connected to a respective intermediate waveguide. In channel  1004 , a custom 1×2 MMI  1011  is provided, an output channel  1013  is provided from the splitter and in this example is connected to the DC phase shifting intermediate waveguide. A channel  1014  is connected to the MMI but receives no output from the splitter. In examples where the EAM length is 41.5 μm, a final splitting/coupling ration of 1:2:0.78:2:1 may be needed. 
     The output of the first 1×7 MMI is 1:1:1:1:1:1:1 and after phase correction and combinations of the inner waveguides, before the last custom 1×2 MMI, the splitting ratio may be 1:2:1:2:1. The middle arm may be decreased from 1 to 0.78 and that is the role of the last custom 1×2 MMI. This last 1×2 MMI may be custom designed to provide a 0.78 portion of the input light. The other output of it (0.22 portion which corresponds to the output number  1014 ) is not used in the system (i.e. it is “thrown away” when the whole thing is used as a splitter or has a zero input light to it when used as a coupler). In the case of the splitter when the 0.22 portion is discarded, it is ensured that this portion will not reflect back to the splitter to distort the performance. 
     Other techniques can be used to achieve a 1:2:0.78:2:1 (or in general 1:2:x:2:1) splitting/coupling ratios, such as a custom designed 1×5 star coupler by engineering the output waveguide widths and positions. However, MMI couplers may have clean outputs and perform better and more robust through process variations. 
     If the output coupler is implemented according to the optimized coupler above, the ordering is reversed i.e. it will receive light into each of channels  1001 ,  1010 ,  1013 ,  1012 , and  1007  and provide light out of output waveguide  209 . 
     While the invention has been described in conjunction with the exemplary embodiments described above, many equivalent modifications and variations will be apparent to those skilled in the art when given this disclosure. Accordingly, the exemplary embodiments of the invention set forth above are considered to be illustrative and not limiting. Various changes to the described embodiments may be made without departing from the spirit and scope of the invention. 
     All references referred to above are hereby incorporated by reference. 
     LIST OF FEATURES 
     
         
           101  Constellation point 
           102  Centre of constellation 
           201  Input waveguide 
           202  MMI input coupler 
           203  First intermediate waveguide 
           204  Second intermediate waveguide 
           205  DC phase shifter 
           206  EAM 
           207  PAM-4 driver 
           208  MMI output coupler 
           209  Output waveguide 
           210  DC phase shifter 
           211  16-point constellation 
           212  Outputs from first intermediate waveguide 
           213  Outputs from second intermediate waveguide 
           301  MMI input coupler 
           302  Phase modulator 
           310  Outputs from first intermediate waveguide 
           311  Outputs from second intermediate waveguide 
           401  Star input coupler 
           402  First intermediate waveguide 
           403  Second intermediate waveguide 
           404  Third intermediate waveguide 
           405  Fourth intermediate waveguide 
           406   a - d  DC phase shifters 
           407   a - d  NRZ drivers 
           408   a - d  Phase modulators 
           409  Star output coupler 
           410   n   i  Modulation state 
           440  Constellation points 
           601  Star input coupler 
           602  First intermediate waveguide 
           603  Second intermediate waveguide 
           604  Third intermediate waveguide 
           605  Fourth intermediate waveguide 
           606   a - d  DC phase shifters 
           607   a - d  NRZ drivers 
           608   a - d  Electro-absorption modulators 
           609  Star output coupler 
           610   n   i  Modulation state 
           660  Constellation points 
           801  First intermediate waveguide 
           802  Second intermediate waveguide 
           803   a/b  DC Phase shifter 
           804   a/b  NRZ driver for EAM 
           805   a/b  Electro-absorption modulator 
           808   a/b  NRZ driver for phase modulator 
           807   a/b  Phase modulator 
           810   n   i  Modulation state 
           901  Test area 
           902  Pitch 
           903  Distance from edge of chip 
           1000  Optimized coupler 
           1001 - 14  Channels 
           1100  1×7 MMI 
           1008 ,  1009  2×1 MMI 
           1011  Customised 1×2 MMI 
       
    
       FIG. 12  shows an optical device  9200 , which broadly comprises: an input waveguide  9201 , a first electro-absorption modulator  9202 , a second electro-absorption modulator  9203 , and an output waveguide  9204 . Light enters the input waveguide, and has a power P in . This is then divided between the first and second electro-absorption modulators, as they are disposed in different optical paths of the device. The first electro-absorption modulator receives P in,a , and the second electro-absorption modulator receives P in,b . The modulators are driven, respectively, by AC sources  9205  and  9206  at voltages V 1 , V 2 , V 3 , or V 4 . Light then exits via output waveguide  9204 , having an associated power P out . 
     The first electro-absorption modulator  9202  is operable to produce an output with either a first P a  or second P a ′ power level. As shown in  FIG. 12 , the second power level is less than the first power level. Similarly, the second electro-absorption modulator is operable to produce an output with either a third P b  or second P b ′ power level. This allows PAM-4 modulation, as at least four amplitudes are available from the device. 
       FIG. 13  shows the outputs available from a device as discussed above. The lowest output in power output, P 0  is formed by a combination of P a  and P b . The next output, increasing in power, P 1 , is formed by a combination of P a  and P b . Next, P 2  is formed by a combination of P a  and P b . Finally, P 3 , with the highest output power is formed by a combination of P a  and P b . The difference, ΔP (0,3) , between P 0  and P 3  is around 3.9 dBm (or 2.45 mW). Whereas a difference ΔP 1,2 , between P 1  and P 2  is around −0.87 dBm (or 818 μW). 
     The combined output power, as well as individual powers generated by each of the first and second electro-absorption modulators, can be calculated as follows: 
     
       
         
           
             
               
                 
                   
                     OMA 
                     
                       t 
                       ⁢ 
                       o 
                       ⁢ 
                       t 
                       ⁢ 
                       a 
                       ⁢ 
                       l 
                     
                   
                   = 
                   
                     
                       
                         P 
                         3 
                       
                       - 
                       
                         P 
                         0 
                       
                     
                     = 
                     
                       
                         2 
                         . 
                         4 
                       
                       ⁢ 
                       5 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       mW 
                     
                   
                 
               
               
                 
                     
                 
               
               
                 
                   
                     P 
                     3 
                   
                   = 
                   
                     3.58 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     mW 
                   
                 
               
             
             
               
                 
                   ER 
                   = 
                   
                     
                       10 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         log 
                         ⁡ 
                         
                           ( 
                           
                             
                               P 
                               3 
                             
                             
                               P 
                               0 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     5 
                   
                 
               
               
                 ⇒ 
               
               
                 
                   
                     P 
                     2 
                   
                   = 
                   
                     2.77 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     mW 
                   
                 
               
             
             
               
                 
                   
                     
                       P 
                       
                         i 
                         + 
                         1 
                       
                     
                     - 
                     
                       P 
                       i 
                     
                   
                   = 
                   
                     
                       
                         O 
                         ⁢ 
                         M 
                         ⁢ 
                         
                           A 
                           
                             t 
                             ⁢ 
                             o 
                             ⁢ 
                             t 
                             ⁢ 
                             a 
                             ⁢ 
                             l 
                           
                         
                       
                       3 
                     
                     = 
                     
                       817 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       µW 
                     
                   
                 
               
               
                 
                     
                 
               
               
                 
                   
                     P 
                     1 
                   
                   = 
                   
                     1.95 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     mW 
                   
                 
               
             
             
               
                 
                     
                 
               
               
                 
                     
                 
               
               
                 
                   
                     P 
                     0 
                   
                   = 
                   
                     
                       1 
                       . 
                       1 
                     
                     ⁢ 
                     3 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     mW 
                   
                 
               
             
           
         
       
     
     where OMA total  is the total optical modulation amplitude of the PAM-4 signal and ER is the extinction ratio. These four output power levels, P 3 , P 2 , P 1 , and P 0  should be generated by the power combination of the two arms of the device (i.e. the two electro-absorption modulators). However, it may be important to consider whether the output powers of the electro-absorption modulators add linearly to generate the final output power levels. 
     The field vectors of the each output power may be summed to provide the total output power. Therefore, for a symmetric coupler, the total output power may be described by: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       out 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               1 
                               
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 P 
                                 
                                   in 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                             
                           
                           + 
                           
                             
                               1 
                               
                                 2 
                               
                             
                             ⁢ 
                             
                               
                                 P 
                                 
                                   in 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                             
                           
                         
                          
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   OR 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     P 
                     out 
                   
                   = 
                   
                     
                        
                       
                         
                           
                             1 
                             
                               2 
                             
                           
                           ⁢ 
                           
                             
                               P 
                               
                                 in 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                         
                         + 
                         
                           
                             1 
                             
                               2 
                             
                           
                           ⁢ 
                           
                             
                               P 
                               
                                 in 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                         
                       
                        
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     Equation (1) is used where there is a phase difference between the light output from the first electro-absorption modulator and the light output from the second electro-absorption modulator. Equation (2) is used where there is no phase difference. 
     It is noted that, in this example, even when there is zero phase difference the output power of the device is not a linear addition of the respective output powers of the electro-absorption modulators. 
     The total output power may be described by: 
     
       
         
           
             
               
                 
                   
                     
                       P 
                       out 
                     
                     = 
                     
                       
                          
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     k 
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   P 
                                   
                                     in 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   1 
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                                 ⁢ 
                                 
                                   P 
                                   
                                     in 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 j 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   θ 
                                   2 
                                 
                               
                             
                           
                         
                          
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   OR 
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     P 
                     out 
                   
                   = 
                   
                     
                        
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 k 
                               
                               ) 
                             
                             ⁢ 
                             
                               P 
                               
                                 in 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               ( 
                               k 
                               ) 
                             
                             ⁢ 
                             
                               P 
                               
                                 in 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                           
                         
                       
                        
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Where k is the coupler coefficient. 
     Equation (3) is used where there is a phase difference between the light output from the first electro-absorption modulator and the light output from the second electro-absorption modulator. Equation (4) is used where there is no phase difference. Equations (3) and (4) are generally true for any value of k where 0&lt;k&lt;1. In examples where the output power P out  is equal to P in1 +P in2 , k may be set to equal 
     
       
         
           
             
               P 
               
                 in 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 2 
               
             
             
               
                 P 
                 
                   in 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
               + 
               
                 P 
                 
                   in 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     In this example. the equations describing the generation of the 4 output levels of the device would be as follows: 
       √{square root over ((1− k ) P   a )}+√{square root over ( kP   b )}=√{square root over ( P   3 )}
 
       √{square root over ((1− k ) P   a )}+√{square root over ( kP   b ′)}=√{square root over ( P   2 )}
 
       √{square root over ((1− k ) P   a ′)}+√{square root over ( kP   b )}=√{square root over ( P   1 )}
 
       √{square root over ((1− k ) P   a ′)}+√{square root over ( kP   b ′)}=√{square root over ( P   0 )}
 
     This can be considered as a set of 4 non-linear equations, with 5 unknowns: P a , P a ′, P b , P b ′, and k. However, the equations can be manipulated and rewritten in the following form: 
         a+b =√{square root over ( P   3 )}
 
         a+b ′=√{square root over ( P   2 )}
 
         a′+b =√{square root over ( P   1 )}
 
         a′+b ′=√{square root over ( P   0 )}
 
     This forms a set of 4 linear equations with 4 unknowns. As the left-hand side of the this set of linear equations are dependent (i.e. a linear combination of 3 chosen equations can build the 4 th ), an associated matrix A to this system is singular (i.e. if the system above is written in the form Ax=b where x is the vector of unknowns, the 4-by-4 matrix A would have a rank of 3 rather than 4). 
     For this set of equations to have any solutions, it should be required that: 
       √{square root over ( P   1 )}+√{square root over ( P   2 )}=√{square root over ( P   0 )}+√{square root over ( P   3 )}
 
     And as it is preferred that the output power levels be equally spaced, it should be stated that: 
     
       
      
       P 
       1 
       +P 
       2 
       =P 
       0 
       +P 
       3  
      
     
     It is preferred to calculate a set of values for a, b, a′, and b′ which minimises an error of the system described above. This can be formulated as a convex optimization problem, with several factors to consider minimising in this system. 
     Therefore, set out below is a derivation of an equation for an unequally spaced PAM-4 bit-error-rate. The derivation assumes: that symbols are equally likely to be transmitted; any noise is Gaussian in form; and that the noise and its power is independent from the optical signal.  FIG. 15  is a demonstration of a PAM-4 transmission which includes Gaussian noise. 
     The symbol error rate may be stated as: 
     
       
         
           
             
               P 
               s 
             
             = 
             
               
                 1 
                 2 
               
               ⁡ 
               
                 [ 
                 
                   
                     Q 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                               P 
                               1 
                             
                           
                           
                             2 
                             ⁢ 
                             
                               N 
                               0 
                             
                           
                         
                       
                       ) 
                     
                   
                   + 
                   
                     Q 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                               P 
                               2 
                             
                           
                           
                             2 
                             ⁢ 
                             
                               N 
                               0 
                             
                           
                         
                       
                       ) 
                     
                   
                   + 
                   
                     Q 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             Δ 
                             ⁢ 
                             
                               P 
                               3 
                             
                           
                           
                             2 
                             ⁢ 
                             
                               N 
                               0 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     where 
     
       
         
           
             
               
                 Q 
                 ⁡ 
                 
                   ( 
                   x 
                   ) 
                 
               
               = 
               
                 
                   1 
                   
                     2 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                   
                 
                 ⁢ 
                 
                   
                     ∫ 
                     x 
                     ∞ 
                   
                   ⁢ 
                   
                     
                       e 
                       
                         
                           t 
                           2 
                         
                         2 
                       
                     
                     ⁢ 
                     d 
                     ⁢ 
                     t 
                   
                 
               
             
             , 
           
         
       
     
     t is time, Q is a Q function, and N 0  is the Gaussian noise power in mW/Hz. 
     It can be noted that the dominant factor in affecting the bit-error-rate will be the smallest output level (i.e. P 0 ), as the changes in the Q function are steep as the argument goes beyond 7 to achieve a bit-error-rate of the order −12. 
     Therefore, the constraints that should be used in the optimizer result in solving the following problem: 
     
       
         
           
             
               
                 
                   System 
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   Ax 
                   = 
                   b 
                 
               
             
             
               
                 Optimization 
               
               
                 
                   min 
                   ⁡ 
                   
                     ( 
                     
                       
                          
                         
                           
                             eye 
                             3 
                           
                           - 
                           
                             eye 
                             2 
                           
                         
                          
                       
                       2 
                     
                     ) 
                   
                 
               
             
             
               
                 
                   parameters 
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   min 
                   ⁡ 
                   
                     ( 
                     
                       
                          
                         
                           
                             eye 
                             2 
                           
                           - 
                           
                             eye 
                             1 
                           
                         
                          
                       
                       2 
                     
                     ) 
                   
                 
               
             
             
               
                 
                     
                 
               
               
                 
                   min 
                   ⁡ 
                   
                     ( 
                     
                       
                          
                         
                           
                             eye 
                             1 
                           
                           - 
                           
                             eye 
                             3 
                           
                         
                          
                       
                       2 
                     
                     ) 
                   
                 
               
             
             
               
                 
                   Constraints 
                   ⁢ 
                   
                     : 
                   
                 
               
               
                 
                   
                     
                       
                         eye 
                         1 
                       
                       ≥ 
                       
                         
                           
                             OMA 
                             outer 
                           
                           3 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         for 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         i 
                       
                     
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   3 
                 
               
             
           
         
       
       
         
           
             
               x 
               = 
               
                 [ 
                 
                   
                     
                       a 
                     
                   
                   
                     
                       
                         a 
                         ′ 
                       
                     
                   
                   
                     
                       b 
                     
                   
                   
                     
                       
                         b 
                         ′ 
                       
                     
                   
                 
                 ] 
               
             
             ⁢ 
             
                 
             
             , 
             
               b 
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           P 
                           3 
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           2 
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
                           P 
                           0 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
             
               A 
               = 
               
                 [ 
                 
                   
                     
                       1 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       0 
                     
                   
                   
                     
                       1 
                     
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       1 
                     
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                     
                       1 
                     
                     
                       0 
                     
                     
                       1 
                     
                   
                 
                 ] 
               
             
           
         
       
     
     Where eye refers to the output level for a particular level of PAM-4 modulation. 
     The results of this optimization are shown in  FIG. 16 . The levels are unequally spaced, however the minimum output level is still larger than the minimum eye closure requirement. 
     The extinction ratios for each of the first and second electro-absorption modulators can therefore be calculated as: 
     
       
         
           
             
               
                 
                   a 
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           k 
                         
                         ) 
                       
                       ⁢ 
                       
                         P 
                         a 
                       
                     
                   
                 
               
               
                 
                     
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     a 
                     ′ 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           k 
                         
                         ) 
                       
                       ⁢ 
                       
                         P 
                         a 
                         ′ 
                       
                     
                   
                 
               
               
                 ⇒ 
               
               
                 
                   
                     ER 
                     1 
                   
                   = 
                   
                     
                       10 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         log 
                         ⁡ 
                         
                           ( 
                           
                             
                               ( 
                               
                                 a 
                                 
                                   a 
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         6 
                         . 
                         3 
                       
                       ⁢ 
                       3 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       dB 
                     
                   
                 
               
             
             
               
                 
                   b 
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           k 
                         
                         ) 
                       
                       ⁢ 
                       
                         P 
                         b 
                       
                     
                   
                 
               
               
                 
                     
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     b 
                     ′ 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           1 
                           - 
                           k 
                         
                         ) 
                       
                       ⁢ 
                       
                         P 
                         b 
                         ′ 
                       
                     
                   
                 
               
               
                 ⇒ 
               
               
                 
                   
                     ER 
                     2 
                   
                   = 
                   
                     
                       10 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         log 
                         ⁡ 
                         
                           ( 
                           
                             
                               ( 
                               
                                 b 
                                 
                                   b 
                                   ′ 
                                 
                               
                               ) 
                             
                             2 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         3 
                         . 
                         4 
                       
                       ⁢ 
                       9 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       dB 
                     
                   
                 
               
             
           
         
       
     
     It may be useful to determine the optimum coupling coefficient k to use in order to minimize the necessary input power to the device. The plot in  FIG. 17A  shows that by sweeping over k, it is possible to achieve different combinations of the first and second electro-absorption modulator output powers necessary for operation. From this data, and by considering the insertion loss from each modulator, the necessary input power to the first and second electro-absorption modulators and their sum (which is generally referred to as a measure for the total input power) can be derived as shown in  FIG. 17B . 
     One limit for input power may be that the input power to each of the first and second electro-absorption modulator respectively should not exceed +8 dBm (the line  9701  indicated in  FIG. 17B ). This limit results in the optimum value fork being around 0.39 (which would result in a minimisation of the necessary total input power). However, due to the saturation input power constraint, a value of k=0.3 is also considered. 
     When using multimode interference couplers for output coupler and input coupler, and assuming an excess loss of 0.1 dB for each, a proposed circuit is shown in  FIG. 18 . The specifications for this circuit are discussed in tables 1 and 2 below: 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 (where k = 0.39) 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                   
                 EAM 
                   
                   
                   
                   
                 Insertion 
                 Link 
               
               
                 Sec- 
                 Lengths 
                 ER 
                 P off   
                 P on   
                 P in   
                 Loss 
                 penalty 
               
               
                 tions 
                 (μm) 
                 (dB) 
                 (mW) 
                 (mW) 
                 (mW) 
                 (dB) 
                 (dB) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Arm 1 
                 66 
                 6.33 
                 2.18 
                 0.51 
                 7.45 
                 5.34 
                 6.49 
               
               
                 Arm 2 
                 48 
                 3.49 
                 2.12 
                 0.95 
                 5.00 
                 3.73 
                 6.3 
               
               
                 Total 
                 — 
                 4.95 
                 4.26 
                 1.36 
                 12.46 
                 4.66 
                 6.33 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 (where k = 0.3) 
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                   
                 EAM 
                   
                   
                   
                   
                 Insertion 
                 Link 
               
               
                 Sec- 
                 Lengths 
                 ER 
                 P off   
                 P on   
                 P in   
                 Loss 
                 penalty 
               
               
                 tions 
                 (μm) 
                 (dB) 
                 (mW) 
                 (mW) 
                 (mW) 
                 (dB) 
                 (dB) 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 Arm 1 
                 66 
                 6.33 
                 1.9 
                 0.44 
                 6.49 
                 5.34 
                 6.49 
               
               
                 Arm 2 
                 48 
                 3.49 
                 2.74 
                 1.23 
                 6.44 
                 3.72 
                 6.3 
               
               
                 Total 
                 — 
                 4.95 
                 4.26 
                 1.36 
                 12.94 
                 4.82 
                 6.49 
               
               
                   
               
            
           
         
       
     
     Of particular note is that the devices discussed above require no digital to analogue converters in order to operate as PAM-N modulating devices. All driver signals on the electro-absorption modulators may be the same, and the laser power into the device can be increased by 3 dB relative to a transmitter implemented with a single electro-absorption modulator or a series of modulators in a single waveguide. 
     Moreover, variable couplers or tunable Mach-Zehnder interferometers can be used at the input and output of the device (i.e. coupled with the input waveguide and output waveguide respectively) to adjust the input and output split ratios. This can compensate for non-perfectly linearly separated layers generated at the output of the device due to the absorption of the electro-absorption modulators and/or the phase responses being different from different modulators. 
     This is illustrated in  FIG. 19 , where a further variant device is illustrated. Broadly, the device includes: an input waveguide  9901 , a variable input coupler  9902 , a first electro-absorption modulator  9903  and a second electro-absorption modulator  9904 , a variable output coupler  9905 , and an output waveguide  9906 . 
     The variable input coupler  9902  is formed of a first Mach-Zehnder interferometer, having arms b 1  and b 2 . A 1×2 multimode interference coupler splits the light received from the input waveguide  9901  into each of the arms. The first arm b 1  includes a phase shifter  9907   a , and the second arm b 2  includes a phase shifter  9907   b . The resulting light is combined in a further multimode interference coupler  9908 , before being divided again into the first electro-absorption modulator  9903  and the second electro-absorption modulator  9904 . The first and second electro-absorption modulators are disposed within arms of a second Mach-Zehnder interferometer, and have associated phase shifters  9909   a  and  9909   b.    
     The light is then transmitted into the variable output coupler  9905 , which is formed of a third Mach-Zehnder interferometer having arms b 5  and b 6 . Each of the arms b 5  and b 6  include respective phase shifters  9910   a  and  9910   b . The light in the variable output coupler  9905  is then recombined in a 2×1 multimode interference coupler (MMI), before exiting the device via output waveguide  9906 . 
     In all of the Mach-Zehnder interferometers, there may be more than one phase shifter. The arms of the Mach-Zehnder interferometers are generally parallel. A tap  9911  is present in the variable output coupler that allows a feedback signal to be obtained to bias and stabilize the phase shifters present in the device. 
       FIG. 29  shows a device similar to that of  FIG. 19 . In the device of  FIG. 29 , a 2×2 MMI  9912  is used at the output instead of the combination, in  FIG. 19 , of a 2×1 MMI and the tap  9911 . The 2×2 MMI  9912  may have a (power) coupling ratio of between 0.1% and 20.0%, i.e., of the optical power received by the 2×2 MMI  9912  from the arms b 5  and b 6 , between 0.1% and 20.0% may be transmitted to the coupled output  9913  of the 2×2 MMI  9912 , and the remainder, less any optical loss incurred in the 2×2 MMI  9912 , may be transmitted to the output waveguide  9906 . 
     In the second Mach-Zehnder interferometer, each of the phase shifters  9909   a ,  9909   b  may be thermally coupled to a respective heater of two heaters, a first heater  9914   a  and a second heater  9914   b . Each heater may include, e.g., a resistive metal (e.g., tungsten) or semiconductor layer on or adjacent to a waveguide. Current driven through the resistive layer may cause the temperature of the heater, and the temperature of the waveguide, to increase, which may cause the phase delay of light propagating through the waveguide to change according to the thermo-optic coefficient of the material of the waveguide. 
     Each of the heaters  9914   a ,  9914   b  may also be thermally coupled to a respective one of the first electro-absorption modulator  9903  and the second electro-absorption modulator  9904 , as a result, for example, of the proximity of each of the first electro-absorption modulator  9903  and the second electro-absorption modulator  9904  to a respective one of the phase shifters  9909   a ,  9909   b  and to a respective one of the heaters  9914   a ,  9914   b . For example, some of the heat dissipated in the first heater  9914   a  may be conducted, (e.g., through the waveguide connecting the first heater  9914   a  to the first modulator  9903 ) to the first electro-absorption modulator  9903 , causing its temperature to increase. The heating coefficient of each heater for the respective electro-absorption modulator may be between 0.01 and 2.0 of the heating coefficient of each heater for the respective phase shifter. As used herein, the “heating coefficient” of a heater for a component is the rate of change of temperature of the component with respect to power dissipated in the heater, under normal operating conditions (e.g., with the device mounted on a suitable heat sink at a normal operating temperature). 
     Each of the first electro-absorption modulator  9903  and the second electro-absorption modulator  9904  may have an extinction ratio (ER) and an insertion loss (IL), each of which is a function of both wavelength and temperature. As such, it may be advantageous to adjust the temperature of the first electro-absorption modulator  9903  and the second electro-absorption modulator  9904  such that the peak, as a function of wavelength, of the normalized extinction ratio is at, or near, the operating wavelength of the system (where the operating wavelength is, e.g., the wavelength of light generated by a laser supplying light to the device). As used herein, the “normalized extinction ratio” is the ratio of the extinction ratio to the insertion loss. 
     It may also be advantageous to adjust the temperature of each of the phase shifters  9909   a ,  9909   b  such that the phase difference in the two arms of the second Mach-Zehnder interferometer is at or near the correct operating point. The correct operating point may be one at which the phase difference is zero degrees (resulting in constructive interference at the output waveguide  9906 ) or within an offset angle of zero degrees (where the offset angle may be between 1 degree and 15 degrees (e.g., 5 degrees or 10 degrees) and selected to compensate for the phase modulation the EAMs  9903 ,  9904  may produce). 
     In some embodiments, the differential heater current (i.e., the difference between the current driven through the first heater  9914   a  and the current driven through the second heater  9914   b ) is adjusted so as to achieve the desired phase difference, and the common mode heater current (e.g., the average of the current (or the total current) driven through the first heater  9914   a  and the current driven through the second heater  9914   b ) is adjusted to so that the peak (as a function of wavelength) of the normalized extinction ratio of each of the first electro-absorption modulator  9903  and the second electro-absorption modulator  9904  is at or near the operating wavelength. 
     In some embodiments, the first electro-absorption modulator  9903  and the second electro-absorption modulator  9904  may be sufficiently different (e.g., having different lengths) that when the temperature is set such that the peak, as a function of wavelength, of the normalized extinction ratio of one of the electro-absorption modulators  9903 ,  9904  (which may be referred to simply as the peak of the modulator) is at the operating wavelength, the peak, as a function of wavelength, of the normalized extinction ratio of the other modulator is at a wavelength differing from the operating wavelength. In such a case, the common mode heater current (and the corresponding temperature) may be selected, e.g., such that (i) the peak of each modulator differs from the operating wavelength by less than a certain maximum offset, or (ii) the operating wavelength is between the peak of the first modulator and the peak of the second modulator. The maximum offset may be (i) between 0.1 nm and 10 nm or (ii) a constant times the separation between the peak of the first modulator and the peak of the second modulator, the constant being between 0.5 and 5.0. 
     In some embodiments, one or more additional heaters may be present (e.g., a heater may be on the first electro-absorption modulator  9903  or on the second electro-absorption modulator  9904 ); such a configuration may facilitate the independent temperature control of the first electro-absorption modulator  9903  and the second electro-absorption modulator  9904 . Each of the phase shifters  9907   a ,  9907   b ,  9910   a ,  9910   b  in the first Mach-Zehnder interferometer and in the third Mach-Zehnder interferometer may also be controlled by a respective heater. 
     A control circuit  9916  (which may be or may include a processing circuit) may control the heaters, based on a feedback signal derived from the tap  9911  ( FIG. 19 ) or derived from the coupled output of the 2×2 MMI  9912  ( FIG. 29 ). The heater currents may be selected, for example, if four heater currents are to be controlled (one for each of two heaters  9914   a ,  9914   b  in the second Mach-Zehnder interferometer, one for a heater on one of the phase shifters  9907   a ,  9907   b  in the first Mach-Zehnder interferometer, and one for a heater on one of the phase shifters  9910   a ,  9910   b  in the third Mach-Zehnder interferometer), by performing an exhaustive search over a (4-dimensional) grid of heater current values (each point in the grid corresponding to a respective set of four respective heater current values for the four heaters), at each point testing the extent to which the modulation produced differs from ideal PAM-4 modulation (with evenly spaced levels). The set of heater current values that results in the best performance may then be selected as the operating point. In some embodiments another method (e.g., a search that is a log search along one or more of the dimensions, or a gradient descent method) may be employed instead of an exhaustive search. 
       FIGS. 20A and 20B  show respective parts of a transmission system using the optical device described above and their desired properties. 
     The system starts from an optical source, in this case a laser  9801 , producing light with a signal amplitude of around +11.9 dBm. The laser is coupled to a fibre  9802 , which couples the light into a device  9803  according to the above description. The fibre should preferably incur no more than 2 dB loss at the attach points, and so the signal received by the device  9803  should have a signal amplitude of approximately +9.9 dBm. The device  9803  itself will incur a loss equivalent to the OMA link penalty, which may be approximately 6 dB. The result is a modulated signal with an OMA of approximately +3.9 dBm. This modulated signal is fed into a multiplexer  9804  ‘MUX’ which may incur a loss of 1.5 dB, and so the resulting signal output OMA from the multiplexer is approximately +2.4 dBm. The light then passes through a Tx fibre  9805 , which may incur a loss of around 1 dB at the attach point. Further losses are illustrated by box  9806  and  9807 , which illustrate respectively the dispersion and transmission signal-to-noise ratio penalties (2.4 dB) and PAM-4 signal-to-noise ratio penalty (4.8 dB). The result is that a signal with an OMA of approximately −5.8 dBm is received by link fibre  9808 . The link fibre  9808  can incur a loss of around 4 dB. The light then passes into an RX fibre  9809 . The signal received by the Rx fibre should have an amplitude of approximately −9.8 dBm. The Rx fibre attach transmits the light into a demultiplexer  9810  ‘DeMUX’, incurring a loss of around 1 dB at the attach point. The received signal by the demultiplexer may have an amplitude of approximately −10.8 dBm. Finally, the signal is received by receiver  9811 , having incurred a loss in the demultiplexer of around 3 dB its amplitude may be approximately −13.3 dBm. 
     All of the devices described above may be used in a photonic integrated circuit, e.g., in a photonic integrated circuit fabricated on a silicon-on-insulator (SOI) wafer or on a double-silicon-on-insulator (DSOI) wafer. 
     The term “processing circuit” is used herein to mean any combination of hardware, firmware, and software, employed to process data or digital signals. Processing circuit hardware may include, for example, application specific integrated circuits (ASICs), general purpose or special purpose central processing units (CPUs), digital signal processors (DSPs), graphics processing units (GPUs), and programmable logic devices such as field programmable gate arrays (FPGAs). In a processing circuit, as used herein, each function is performed either by hardware configured, i.e., hard-wired, to perform that function, or by more general-purpose hardware, such as a CPU, configured to execute instructions stored in a non-transitory storage medium. A processing circuit may be fabricated on a single printed circuit board (PCB) or distributed over several interconnected PCBs. A processing circuit may contain other processing circuits; for example, a processing circuit may include two processing circuits, an FPGA and a CPU, interconnected on a PCB. 
     As used herein, the word “or” is inclusive, so that, for example, “A or B” means any one of (i) A, (ii) B, and (iii) A and B. Any numerical range recited herein is intended to include all subranges of the same numerical precision subsumed within the recited range. For example, a range of “1.0 to 10.0” or “between 1.0 and 10.0” is intended to include all subranges between (and including) the recited minimum value of 1.0 and the recited maximum value of 10.0, that is, having a minimum value equal to or greater than 1.0 and a maximum value equal to or less than 10.0, such as, for example, 2.4 to 7.6. Similarly, a range described as “within 35% of 10” is intended to include all subranges between (and including) the recited minimum value of 6.5 (i.e., (1-35/100) times 10) and the recited maximum value of 13.5 (i.e., (1+35/100) times 10), that is, having a minimum value equal to or greater than 6.5 and a maximum value equal to or less than 13.5, such as, for example, 7.4 to 10.6. Any maximum numerical limitation recited herein is intended to include all lower numerical limitations subsumed therein and any minimum numerical limitation recited in this specification is intended to include all higher numerical limitations subsumed therein. 
     While the invention has been described in conjunction with the exemplary embodiments described above, many equivalent modifications and variations will be apparent to those skilled in the art when given this disclosure. Accordingly, the exemplary embodiments of the invention set forth above are considered to be illustrative and not limiting. Various changes to the described embodiments may be made without departing from the spirit and scope of the invention. 
     All references referred to above are hereby incorporated by reference. 
     The following numbered clauses contain statements of broad combinations of technical features in accordance with various aspects of the device and methods disclosed herein: 
     Clause 
     1. An optical device, operable to provide a PAM-N modulated output, comprising:
         an input waveguide, configured to receive light;   a first electro-absorption modulator, coupled to receive light from the input waveguide, and operable to produce a first output or a second output, wherein the second output has a lower amplitude than the first output;   a second electro-absorption modulator, coupled to receive light from the input waveguide, and operable to produce a third output or a fourth output, wherein the fourth output has a lower amplitude than the third output; and   an output waveguide, coupled to receive light from the first electro-absorption modulator and the second electro-absorption modulator, and output a combined signal comprising an output of the first electro-absorption modulator and an output of the second electro-absorption modulator;   wherein the first electro-absorption modulator and the second electro-absorption modulator are disposed in parallel.       

     2. The optical device of clause 1 further comprising:
         a variable input coupler, disposed between the input waveguide and both of the first electro-absorption modulator and the second electro-absorption modulator.       

     3. The optical device of clause 2, wherein the input coupler is configured to modify the light transmitted to the first electro-absorption modulator and the second electro-absorption modulator such that adjacent PAM-N modulated outputs generated by the device are linearly spaced. 
     4. The optical device of either clause 2 or clause 3, wherein the input coupler is configured to unequally split input light between the first electro-absorption modulator and the second electro-absorption modulator. 
     5. The optical device of any of clauses 2-4, wherein the input coupler comprises a Mach-Zehnder interferometer, comprising an input coupler coupled to the input waveguide, two parallel arms and an output coupler. 
     6. The optical device of clause 5, wherein at least one of the parallel arms includes a phase-shifter. 
     7. The optical device of any of clauses 1-6, wherein, disposed between the output waveguide and the first electro-absorption modulator and the second electro-absorption modulator is a variable output coupler. 
     8. The optical device of clause 7, wherein the output coupler is configured to modify the received light from the first electro-absorption modulator and the second electro-absorption modulator such that adjacent PAM-N modulated outputs are linearly spaced. 
     9. The optical device of either clause 7 or clause 8, wherein the output power of light which can be outputted by the coupler is described by: 
     
       
         
           
             
               P 
               OUT 
             
             = 
             
               
                  
                 
                   
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             k 
                           
                           ) 
                         
                         ⁢ 
                         
                           P 
                           
                             in 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         j 
                         ⁢ 
                         
                           θ 
                           1 
                         
                       
                     
                   
                   + 
                   
                     
                       
                         
                           ( 
                           k 
                           ) 
                         
                         ⁢ 
                         
                           P 
                           
                             in 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                     
                     ⁢ 
                     
                       e 
                       
                         j 
                         ⁢ 
                         
                           θ 
                           2 
                         
                       
                     
                   
                 
                  
               
               2 
             
           
         
       
     
     where P OUT  is the output power, k is a coupling coefficient P in1  is the input power from the first electro-absorption modulator, θ 1  is the phase of light entering the coupler from the first electro-absorption modulator, P in2  is the input power from the second electro-absorption modulator, and 82 is the phase of light entering the coupler from the second electro-absorption modulator. 
     10. The optical device of clause 9, wherein k is less than 0.44. 
     11. The optical device of any of clauses 7-10, wherein the variable output coupler is a Mach-Zehnder interferometer, comprising an input coupler coupled to the first electro-absorption modulator and the second electro-absorption modulator, two parallel arms and an output coupler coupled to the output waveguide. 
     12. The optical device of clause 11, wherein at least one of the parallel arms includes a phase-shifter. 
     13. The optical device of any of clauses 1-12, further comprising:
         a first analogue driver, said first driver configured to provide a first driver signal to the first electro-absorption modulator; and   a second analogue driver, said second driver configured to provide a second driver signal to the second electro-absorption modulator;   wherein, when driven by the respective driver signal, the first electro-absorption modulator produces the second output and the second electro-absorption modulator produces the fourth output.       

     14. The optical device of clause 13, wherein the first driver signal and the second driver signal are identical. 
     15. The optical device of any of clauses 1-14, wherein the optical device includes a Mach-Zehnder interferometer, and the first electro-absorption modulator is disposed within a first arm of the Mach-Zehnder interferometer and the second electro-absorption modulator is disposed within a second arm of the Mach-Zehnder interferometer. 
     16. The optical device of any of clauses 1-15, wherein the device includes a phase-shifter associated with at least one of the electro-absorption modulators, said phase-shifter being operable to modify a phase shift associated with the respective electro-absorption modulator. 
     17. The optical device of clause 16 as dependent on clause 15, wherein the phase-shifter is disposed within an arm of the Mach-Zehnder interferometer. 
     18. The optical device of either clause 16 or 17, wherein the phase-shifter is any one or more of: a heater; a PIN phase shifter; or a PN phase shifter. 
     19. The optical device of any of clauses 15-18, further including a directional coupler coupled to the output waveguide, configured to redirect a portion of the output power for use to bias and stabilize the phase shifter. 
     20. The optical device of clause 19, wherein the portion of the output power is less than 5%. 
     21. A Mach-Zehnder interferometer, comprising:
         an input waveguide;   an output waveguide;   a first arm, optically connecting the input waveguide and the output waveguide;   a second arm, optically connecting the input waveguide and the output waveguide, the second arm being disposed in parallel with the first arm; and   at least two electro-absorption modulators, disposed respectively in the first arm and the second arm, and operable to provide PAM-N modulation.