Once we have these beams, we then feed each waveguide into its own pair of MZIs where we perfo

If the phase of light travelling along one arm changes relative to that of the light travelling down the other arm, recombining the two fields will create a field with different phase and amplitude to the input.
The paths themselves are cut into the silicon and can’t change in length, so we have to use a different method of altering the relative phases. Instead of changing the distance, we can achieve the same effect by changing the local speed of light along each arm by altering the refractive index of the silicon.
The phase of an optical field travelling through a material is given by
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Where φ is the phase of the optical field, λ is the wavelength,​ l is the length of the path through the material, and n is the refractive index. Change n, and we introduce a proportional change in the phase.
There are a number of ways in which we can change n, and the method we choose can have a dramatic impact on the performance of the system. We’ll cover this subject in another article, but for now one of the simpler ways is by changing the temperature of the silicon waveguide.
This gives us a means of tuning the phase. From a mathematical point of view, if we say that the light entering one arm of an MZI has an amplitude equal to 1, changing the phase is equivalent to multiplying the input amplitude by a complex number with modulus 1​. If you’re familiar with complex numbers, this can be written as multiplication with the exponential exp(i φ), which is usually written as e with the argument in superscript​.
We can represent this schematically as:
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with the input on the left and output on the right.
We can make a Mach Zehnder interferometer that uses only a single heater on one arm of the MZI, but it would not be able to change the modulus and phase independently. Instead, by using two heaters (one on each arm in what is known as a push-pull configuration), we only need to use half of the maximum temperature to cover the same phase range.
A single Mach-Zehnder interferometer performs the following tasks:
A single input beam of light is separated into two by splitting the waveguide. The amplitude of the beam in each arm is given by the amplitude of the input beam divided by √2.
The field in the upper branch is multiplied by ​the exponential of an imaginary number, adding φ₁​ to its phase.
The field in the lower branch is multiplied by ​the exponential of another imaginary number, adding φ₂​ to its phase.
At the recombination point, the fields are summed with their amplitudes divided by √2.
The amplitude of the recombined field is therefore:
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Let x​ be a complex number that we want to encode into a beam of light. If ​its modulus is no larger than 1, we can encode x into the beam by choosing the phases such that​
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with the associated complex argument
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The notation ‘mod​’ indicates that the maximum range of the complex argument arg(x) can only lie between 0 and 2π​. This is because a complex number is not changed by adding 2π​​ to its phase, which amounts to a rotation by 360°​ in the complex plane.
The above gives us a method of linking numerical values with adjustments to the properties of light, allowing us to make the conversion from a digital representation of a number into an optical one. By using two arrays of multiple Mach Zehnder interferometers in series, we can perform these operations on many beams of light at once.
Optical Multiplication
However, we aren’t done quite yet. In our first article, we described how the Fourier transform can be used to simplify certain problems. Another very useful mathematical tool, the “convolution” of two functions, can be performed efficiently in Fourier space through the equation
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The convolution equation; I and K are 2-dimensional arrays of data and the curved F represents the Fourier transform. Raising F to the negative power indicates the inverse Fourier transform.
If we already know the Fourier transforms of I and K (which we can rapidly calculate using our device) then we can perform the element-wise multiplication in this equation optically by using two arrays of MZIs in series, like so:
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Here the numbers φ₁, φ₂, ψ₁, and ψ₂ are phase values which can be chosen at will by adjusting n. Passing the output of this multiplication into the free space optics will perform the inverse Fourier transform on the data, completing the calculation of the convolution of I and K.
If we don’t want to perform a multiplication (if, for instance, we are calculating the initial Fourier transform) then we are not obliged to do so; we can simply set the second MZI such that it does not alter the phase.
The convolution operation is a major part of executing a Convolutional Neural Network, or CNN. CNNs are well known as cutting-edge techniques in machine vision, but they have also been applied to a wider range of problems including network security analysis, the protein folding problem, and the task of predicting electron density parameters in complex quantum systems.
Not only is the optical Fourier transform naturally much faster than the electronic variant (both in terms of the number of operations and the maximum operating speed of an optical system), but the ability to simultaneously perform both the multiplication and inverse Fourier transformation required for the convolution in a single optical pass makes Fourier-optical computing a natural fit for high speed processing in machine vision tasks.
Putting it all together
Silicon photonics gives us everything we need to be able to reliably control light, but there’s still more work to do. So far, we have focused on how we can write digital information into light, but how do we get light into the system in the first place, and what do we do with it once we’ve finished altering it?
We want to be able to process as much data as possible in a single operation so we need multiple beams, as many as is practical. To get light into our chip, we start with a single source of laser light. This can be from an integrated laser built into the surrounding control electronics, or (as in the case of our prototype system) it can be generated by a desktop laser source.
We use a fibre-optic cable to convey light from the laser source to an input waveguide on the silicon photonic chip. We can then divide this single source of light in two by splitting this input waveguide into two paths, just as we do when making an MZI.
However, rather than recombining the two beams, we can split each beam again using the same method until we have as many individual beams as we like. This is called a branching-tree multiplexer. Of course, every time we split the waveguides we decrease the total optical power present in each beam, so there is a limit to how many beams we can make from a single laser source.
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Once we have these beams, we then feed each waveguide into its own pair of MZIs where we perform the encoding and multiplication operations described above.
After encoding the light, we then want to create a 2-dimensional optical field that can be processed by a lens. In our system, each output waveguide carrying light processed by the MZI arrays feeds into a 2-dimensional grid of light-emitting grating couplers that lie just under the surface of the silicon photonic chip. These are structures cut into the waveguides which cause the light to reflect upwards.
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