Optical Heterodyne Detection

Optical heterodyne detection mixes a weak signal field with a frequency-shifted local oscillator (LO). Their optical carriers are much too fast for a camera to follow, but their interference contains a difference-frequency term that can fall inside the camera bandwidth.

This lesson first treats the fields themselves, then introduces the square-law detector that turns their interference into a measurable intensity.

Two fields and their beat

Consider two real, co-polarized fields at one point on the detector:

\[ s_\mathrm{sig}(t)=A_\mathrm{sig}\cos(2\pi f_\mathrm{sig}t+\phi_\mathrm{sig}), \qquad s_\mathrm{LO}(t)=A_\mathrm{LO}\cos(2\pi f_\mathrm{LO}t+\phi_\mathrm{LO}). \]

In the animations, s_beat denotes their summed optical field:

\[ s_\mathrm{beat}(t)=s_\mathrm{sig}(t)+s_\mathrm{LO}(t), \qquad \Delta f=f_\mathrm{LO}-f_\mathrm{sig}. \]

Using the sum-to-product identity

\[ \cos a+\cos b = 2\cos\!\left(\frac{a-b}{2}\right) \cos\!\left(\frac{a+b}{2}\right), \]

the equal-amplitude, equal-phase sum can be factored as

\[ s_\mathrm{beat}(t)=2A \cos\!\left(\pi\Delta f\,t\right) \cos\!\left(2\pi\frac{f_\mathrm{sig}+f_\mathrm{LO}}{2}t\right). \]

The rapidly oscillating carrier therefore sits inside a slowly changing envelope.

Equal field amplitudes with a fixed 20 kHz detuning. The carrier frequencies are compressed for display.

Detuning sets the beat frequency

Changing \(\Delta f\) changes the rate at which the two fields move into and out of phase. Small detuning produces a slow envelope; larger detuning produces more envelope cycles in the same observation time.

The next animation holds both amplitudes fixed while sweeping the displayed LO detuning between 5 and 35 kHz. The phase evolves continuously throughout the loop.

Increasing the detuning makes the beat envelope evolve more rapidly.

What the local oscillator amplifies

Increasing \(A_\mathrm{LO}\) increases the amplitude of the combined field, but it does not amplify the original signal field. Heterodyne gain appears when a square-law detector measures the irradiance

\[ I_\mathrm{raw}(t)\propto \left[s_\mathrm{sig}(t)+s_\mathrm{LO}(t)\right]^2. \]

A detector that averages over many optical periods removes terms near twice the optical carrier. The remaining signal is

\[ \overline I(t)= \frac{A_\mathrm{sig}^{2}+A_\mathrm{LO}^{2}}{2} +A_\mathrm{sig}A_\mathrm{LO} \cos\!\left(2\pi\Delta f\,t+\Delta\phi\right), \]

where \(\Delta\phi=\phi_\mathrm{LO}-\phi_\mathrm{sig}\). The useful heterodyne cross-term has amplitude \(A_\mathrm{sig}A_\mathrm{LO}\): for a fixed weak signal, a stronger LO produces a larger absolute modulation at the detector. The LO also contributes a DC term proportional to \(A_\mathrm{LO}^2\), so more LO power is not an unlimited improvement—detector saturation, shot noise, and technical noise still matter.

The LO field amplitude varies while the signal and 20 kHz detuning remain fixed. The annotation tracks the square-law cross-term amplitude.

From an optical carrier to camera samples

At 852 nm, the optical carrier frequency is approximately

\[ f_0=\frac{c}{\lambda}\approx 351.87\ \mathrm{THz}. \]

No conventional camera samples that field oscillation directly. A pixel integrates irradiance during an exposure of duration \(T_\mathrm{exp}\). For a rectangular exposure centered on \(t\), the beat term is attenuated by the exposure transfer function:

\[ \overline I_{T_\mathrm{exp}}(t)= \frac{A_\mathrm{sig}^{2}+A_\mathrm{LO}^{2}}{2} +A_\mathrm{sig}A_\mathrm{LO} \operatorname{sinc}(\Delta f T_\mathrm{exp}) \cos\!\left(2\pi\Delta f\,t+\Delta\phi\right), \]

with \(\operatorname{sinc}(x)=\sin(\pi x)/(\pi x)\). The animation uses \(\Delta f=20\ \mathrm{kHz}\), a 50 kfps camera, and a full-frame \(20\ \mu\mathrm{s}\) exposure. The beat attenuation is therefore \(\operatorname{sinc}(0.4)\approx0.757\). Its 25 kHz Nyquist frequency remains above the 20 kHz beat.

The displayed 2 MHz carrier is still an enormous compression of the physical 351.87 THz carrier. It exists only to make the raw oscillations drawable; the averaged curve is calculated from the analytic exposure model above.

Finite exposure suppresses the fast carrier terms and leaves a clean, sampled difference-frequency modulation.

What if the camera runs at 37 kfps?

A 37 kfps camera has a Nyquist frequency of 18.5 kHz. A 20 kHz beat is above that limit and aliases to \(|20-37|=17\ \mathrm{kHz}\). Exposure averaging still rejects the optical carrier, but an unambiguous 20 kHz measurement requires a higher frame rate or a lower detuning.

Takeaways

• The sum of two nearby-frequency fields has a beat envelope governed by \(|\Delta f|\).
• A square-law detector creates a low-frequency cross-term proportional to \(A_\mathrm{sig}A_\mathrm{LO}\).
• Exposure integration rejects optical-frequency oscillations but attenuates the beat according to a sinc response.
• The camera frame rate must still satisfy the sampling requirements of the difference-frequency signal.