E S E LO ∝ √— I 2 + Q 2 and ∆Φ = tan – 1( Q— I)
OPTICAL heterodyNE detection
PHOTONICS INSIGHTS provides a photocurrent which is proportional to the squared modulus of the total electric field, thus introducing the mandatory nonlinearity for heterodyne detection:
I PD ∝ | E S + E LO | 2
The resulting photocurrent can therefore be written as:
I PD = | E S | 2 + | E LO | 2 + 2E S E LO { cos( ∆ωt) cos( ∆Φ)– sin( ∆ωt) sin( ∆Φ)}
With ∆Φ = φ S – φ LO the relative phase shift between the signal and the L O and where ∆ω = ω S – ω LO lies within the electrical bandwidth of the detector and subsequent electronics. Thus, the interference term produces a measurable beat note at the difference frequency. This beat signal retains the amplitude and phase of the optical signal relative to the local oscillator. This provides information that would otherwise be inaccessible through direct intensity detection alone. The beat note can then be demodulated using a lock-in amplifier, an analogue I / Q demodulator, or digital signal processing. Digital lock-in technology from companies such as Zurich Instruments and Liquid Instruments removes drift from analogue electronics and enables faster signal processing and demodulation of multiple frequency components simultaneously. Taking ∆ω as the reference frequency for demodulation, this yields an in-phase component, I ∝ E S E LO cos( ∆Φ) and a quadrature component, Q ∝ E S E LO sin( ∆Φ), so that the signal amplitude and relative phase can then be reconstructed independently from I and Q:
E S E LO ∝ √— I 2 + Q 2 and ∆Φ = tan – 1( Q— I)
The local oscillator fulfils two complementary functions. Firstly, it provides a stable phase and frequency reference against which the unknown optical field is measured. Secondly, when its optical power is much greater than that of the signal, it increases the amplitude of the interference term, which scales as E S E LO. While the local oscillator does not amplify the signal optically, it converts a weak signal field into a comparatively large electrical beat note. This heterodyne gain makes it possible to detect extremely weak optical fields, and under the right conditions, operation can occur close to the shot-noise limit. In many implementations, the signal and local oscillator are derived from the same laser at a frequency of ω 0 in order to maximise their mutual coherence and minimize the impact of laser-frequency noise. Before recombination, one arm is shifted in frequency, most commonly using an acousto-optic modulator. The traveling acoustic wave inside the modulator acts as a moving diffraction grating and transfers its radio-frequency drive frequency, ∆ω = ω AOM, to the diffracted optical beam. The shifted field therefore oscillates at ω 0 ±
ω AOM depending on the diffraction order(± 1) and geometry, while the unshifted arm remains at ω 0. Their interference produces a stable beat note at the frequency difference
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