Homodyne detection

Introduction

Optical homodyne detection is a method for detecting messages transmitted in optical signals, where a frequency or phase modulated signal is compared to what is misleadingly called the "local oscillator" (LO) signal, which is generated from the same source but not modulated with the message. In order to probe quantum effects, it is important to bring the noise of the detector down to the shot-noise limit, where the only fluctuations observed arise from the discrete nature of photons, which can be theoretically modelled as the vacuum-state fluctuations of the quantised electromagnetic field. This project's first objective is to build a homodyne detector from scratch.

Lab Location: S11-02-04 (Optics Lab)

Application: Continuous-variable QKD with Gaussian modulation and coherent states

While the first protocols for quantum key distribution (QKD) involved discrete variables (DV) in finite, small dimensions, QKD can also be done using continuous variables (CV) in infinite dimensions, i.e. the state of an electromagnetic field. Using Gaussian modulation and coherent states makes the QKD system relatively easy to implement and analyse, although getting positive key rates is a different matter. The homodyne detector is an essential component of this setup, but if we have the capacity, we can try to develop other parts of the system, such as the implementation of the QKD protocol itself in software. We want to pay particular attention to the design of the amplifier for the homodyne detector, for which there are stringent requirements and difficult tradeoffs to make.

Reality

What we actually ended up working towards is a simple version of the system proposed above. Instead of building a system capable of sending useful information, we aimed for the simpler objective of just being able to produce some kind of phase modulation. This is a stepping stone towards a full-blown optical communication system, where digital data is modulated onto the laser by a computer and read off from the results of the homodyne detection.

Background Reading

Our primary source of background information is Lasers and Electro-Optics, available for download via NUS Library. The relevant chapters are:

• Chapter 5 Laser Radiation, discussing the basic background on lasers
• Chapter 18 The Electro-optic and Acousto-optic Effects and Modulation of Light Beams, discussing how modulation can be achieved
• Chapter 21 Detection of Optical Radiation, discusses noise in detectors and the process of homodyne detection

For DV-QKD theorists who stumbled into this (like me), here's some background reading on CV-QKD:

• Review of Gaussian quantum-information processing
• Self-contained tutorial on theory of Gaussian-modulated CV-QKD

Tragically, we were nowhere close to being able to use the material on CV-QKD, but we leave the references here for posterity.

Theory

Phase Modulation

The next question is how to vary $\delta$ in an easily controllable manner. TODO

Talk about LiNbO3 structure, birefringence, electro-optic effect

Noise

Finally, we have to control for the effect of noise in our experiment. There are four primary sources of noise in semiconductor photodetectors:

• Shot noise, or quantum noise, arising from the quantisation of light: because the light must arrive at the detector one photon at a time, there will be some fluctuation in the signal read at any point in time.
• Nyquist-Johnson noise or thermal noise, arising from thermal fluctuations in the electronics.
• $1/f$ noise, which is apparently a very mysterious and universal phenomenon caused by various different factors, but has a power spectral density proportional to $1/f$, hence the name.
• Generation-recombination (GR) noise, caused by holes and electrons randomly being generated and recombining in the semiconductor.

$\Delta f$ is the sampling bandwidth, related to how often we take electrical measurements. By Nyquist's theorem, we must have a sampling period $T=1/\Delta f\leq 1/2f$ to reconstruct a modulated signal of maximum frequency component $f$: that is, we must have $\Delta f\geq 2f$.

Setup & Methodology

Full setup of Michelson interferometer with $\mathrm {LiNbO_{3}}$ crystal.

Homodyne Detection Setup

The overview for the initially intended experiment is:

1. Split the laser beam into two, one component will be LO and another will be the signal
2. Frequency-shift the signal beam using an acousto-optical modulator (AOM)
3. Phase-modulate the signal beam using an electro-optical modulator (EOM). This is typically a $\mathrm {LiNbO_{3}}$ crystal, whose birefringence is controlled by the voltage applied to it. For more advanced applications, we can use a transformer to amplify a small change in voltage into a large difference, producing a large change in the birefringence.
4. Recombine the signal and the LO in a 50:50 beam splitter
5. Send the two output beams to two reverse-biased photodiodes, and connect the junction between the photodiodes to a current detector to convert it to a voltage. The signal will be modulated according to the beat frequency.

The noise in the photodiodes tends to be low frequency, so aiming for a signal of frequency $10~\mathrm {MHz}$ or so will make it easier to remove the noise without affecting the signal.

Actual Setup

As the theory section indicates, we are constructing a Michelson interferometer with a $\mathrm {LiNbO_{3}}$ crystal on one arm, serving as a voltage-controlled phase modulator. The voltage across the crystal controls the refractive index, which changes the path difference $\delta$ and therefore the phase difference. If the fringes are circular, the total intensity on the photodiode will vary significantly as the phase changes. This can be read off by using the photodiode as a current source and measuring the voltage it induces across a resistor.

A high-level overview is as follows:

1. Modify and wire up all battery-powered equipment (detectors and laser pointer) to use benchtop power supplies instead, and mount them on optical posts
2. Align the laser to ensure it is parallel to the table and its grid
3. Align the beam splitter to the grid of the table and the laser
4. Align the mirrors to produce circular fringes
5. Place and align the crystal so the beam passes through it
6. Align the lens to magnify the interference pattern, and align the photodiode so that it coincides with the pattern

Results & Discussion

Equipment Parameters

We are using a Hamamatsu S5106 Si PIN photodiode as our detector, with a resistive load of $1~\mathrm {M\Omega }$. Measurements are taken with a Kenwood CS-5270 oscilloscope with $\Delta f=100~\mathrm {MHz}$.