Question 5 of 7: Y-Branch Mach–Zehnder Interferometer — Transfer Function, Modulator, Switch and QAM

Question 5: Y-Branch Mach–Zehnder Interferometer — Transfer Function, Modulator, Switch and QAM (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A symmetric Y-branch Mach–Zehnder interferometer (MZI) with a phase modulator (e.g. electro-optic, on one arm) — no numeric data supplied; the answer is a general derivation and device description.

Find. (a) structure/operation, (b) intensity transfer function (derived), (c)–(e) device applications, (f) speed-limiting factors.

Approach. Split the input field equally at the first Y-junction, apply a relative phase shift $\Delta\phi$ on one arm, and recombine at the second Y-junction; interference between the two arms converts phase modulation into intensity modulation.

1. Part (a) — structure and operation.
   Y-branch Mach-Zehnder interferometer: split, differential phase shift, recombine.
   Light enters a single input waveguide, splits equally at a Y-junction into two parallel arms of equal nominal length, and recombines at a second Y-junction into a single output waveguide. One arm (or, in a push-pull design, both arms differentially) carries an electro-optic phase shifter — a region of the waveguide, typically LiNbO$_3$, with electrodes that change the local refractive index via the Pockels effect in proportion to an applied voltage. Because the two arms recombine coherently, whatever relative optical phase $\Delta\phi$ builds up between them during propagation controls how much of the recombined light couples back into the guided output mode versus radiating away into the substrate as an unguided mode.
2. Part (b) — intensity transfer function.
   Intensity transfer function T = cos^2(Delta phi / 2).
   Let the input field $E_{\text{in}}$ split equally into two arms, each carrying amplitude $E_{\text{in}}/\sqrt2$. After propagation, arm 2 carries an extra phase $\Delta\phi$ relative to arm 1. The recombined field at the output Y-junction is the coherent sum of the two (each again divided by $\sqrt2$ at the junction): $$E_{\text{out}}=\frac{E_{\text{in}}}{2}\left(1+e^{j\Delta\phi}\right).$$ The output intensity is $I_{\text{out}}=|E_{\text{out}}|^2$, and using $1+e^{j\Delta\phi}=2\cos(\Delta\phi/2)\,e^{j\Delta\phi/2}$, $$\frac{I_{\text{out}}}{I_{\text{in}}}=\boxed{\cos^2\!\left(\frac{\Delta\phi}{2}\right)}.$$ This is a raised-cosine transfer function: full transmission at $\Delta\phi=0,2\pi,\ldots$ (constructive recombination back into the guided mode) and complete extinction at $\Delta\phi=\pi,3\pi,\ldots$ (destructive interference — the light that cannot couple back into the single output mode radiates into the substrate).
3. Part (c) — amplitude modulator. Biasing the device at the quadrature point $\Delta\phi=\pi/2$ (the steepest, most linear part of the $\cos^2(\Delta\phi/2)$ curve) and superimposing a small time-varying drive voltage on the phase-shifter electrodes linearly modulates $\Delta\phi(t)$ about that bias point, which in turn linearly modulates the output intensity — an analog or digital (on–off keyed) amplitude modulator whose extinction ratio is set by how close to $\Delta\phi=0/\pi$ the drive voltage can swing.
4. Part (d) — optical switch. Driving $\Delta\phi$ digitally between exactly $0$ (bar/"on" state, full transmission) and exactly $\pi$ (full extinction) turns the same device into a fast, non-mechanical 1×1 optical on–off switch (or, with a second output branch instead of a single recombining Y, a 1×2 routing switch), with switching speed set by how quickly the drive voltage can traverse that half-wave voltage $V_\pi$.
5. Part (e) — 4-QAM (quadrature) modulator. Two MZIs are nested inside one larger Mach–Zehnder structure: an outer Y-splitter feeds an in-phase (I) MZI and a quadrature (Q) MZI in its two arms, the Q-arm carries an extra fixed $\pi/2$ bias, and an outer Y-combiner recombines them. Each inner MZI is driven at its own $\pm V_\pi/2$ points to produce a bipolar $\pm1$ amplitude (BPSK) on its own I or Q axis; combined through the 90°-offset outer stage, the four combinations of (I,Q) $=(\pm1,\pm1)$ synthesize the four constellation points of 4-QAM (equivalently QPSK) directly in the optical domain.
6. Part (f) — speed-limiting factors. (i) The electrode/drive-circuit $RC$ time constant and microwave loss of the traveling-wave electrode at high drive frequencies; (ii) velocity mismatch between the microwave drive signal propagating along the electrode and the optical group velocity in the waveguide, which walks the two out of sync over a long device and limits the usable modulation bandwidth; (iii) the electro-optic material's own response time (fast, sub-picosecond, for Pockels-effect LiNbO$_3$, so rarely the bottleneck); and (iv) how large a drive voltage swing ($V_\pi$) is needed — a longer, lower-$V_\pi$ device trades bandwidth for drive-voltage efficiency because of the velocity-mismatch penalty in (ii).