## homodyne versus heterodyne

The merits of homodyne versus heterodyne detection is somewhat of a crossover topic, I guess, since it applies to both optical communication and interferometry. We will try to keep the discussion neutral and point out differences where they occur.

#### Basics

Ignoring polarization effects in this post, we start with a scalar harmonic plane wave whose magnitude and phase vary in time, albeit very slowly compared to a period of the carrier (i.e. the wave is almost monochromatic):

\begin{aligned} E_\mathrm{S}(t) &= \frac{1}{2} \Bigl( A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr) + \mathrm{c.c.} \Bigr)\\ &= \Re \Bigl\lbrace A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr) \Bigr\rbrace\\ &= \Bigl|A_\mathrm{S}(t)\Bigr| \cos \Bigl( \omega_\mathrm{S} t + \mathrm{arg}\bigl[A_\mathrm{S}(t)\bigr]\Bigr) \tag{1} \end{aligned}

Here, $E$ is the (real-valued) electric field, $A_\mathrm{S}$ is the complex amplitude which contains the actual information signal, $\omega_\mathrm{S}$ is the carrier frequency, and “c.c.” stands for the complex conjugate of everything that precedes it. We have used

$$\Re \bigl\lbrace x \bigr\rbrace = \frac{1}{2}\bigl( x + x^* \bigr) \quad \text{and} \quad \Im \bigl\lbrace x \bigr\rbrace = \frac{1}{2i}\bigl( x – x^* \bigr)$$

to define $E_\mathrm{S}$ in (1) as the real part of $A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr)$. As long as our operations on the field are linear, we may operate directly on the complex amplitude $A_\mathrm{S}(t)$ instead of its real part (i.e. the field). However, for nonlinear operations we must always first calculate the real part.

In coherent detection, the passband signal around the optical frequency $\omega_\mathrm{S}$ is mixed down into regions of the electromagnetic spectrum that electronics can handle. Mixing is always a nonlinear process, and for mixing at light frequencies we can use the nonlinear characteristic of a photodiode (more about that in a minute). The oscillator which generates the mixing frequency in this case is a laser with

$$E_\mathrm{LO}(t) = \frac{1}{2} \Bigl( A_\mathrm{LO}(t) \exp\bigl(i \omega_\mathrm{LO} t\bigr) + \mathrm{c.c.} \Bigr)\tag{2}$$

where $A_\mathrm{LO}$ describes the amplitude this local oscillator. This amplitude would ideally be constant, but e.g. laser phase and intensity noise cause it to be time-varying. Depending on its oscillation frequency $\omega_\mathrm{LO}$ we distinguish between homodyne ($\omega_\mathrm{LO} = \omega_\mathrm{S}$, homo is the Greek prefix for “same”) and heterodyne ($\omega_\mathrm{LO} \ne \omega_\mathrm{S}$, hetero being the Greek prefix for “different”). Usually, in heterodyne reception the frequency difference is larger than the bandwidth of the signal; otherwise the label intradyne has been introduced in optical communication ($\omega_\mathrm{LO} \approx \omega_\mathrm{S}$, intra is rooted in Latin and means “within”).

#### Heterodyning

In heterodyne detection, both waves (the signal and the local oscillator) must be combined using e.g. a directional coupler (or 180° hybrid) in fiber optics or a beam splitter in free-space optics before being mixed in the photodiode. Both devices have the same scattering matrix for the complex amplitudes,$^1$

$$\mathbf{S}_{180} = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\ 1 & -1\end{pmatrix} \tag{3}$$

With $A_\mathrm{S}$ on input 1 and the local oscillator $A_\mathrm{LO}$ on input 2, there are two signals $A_1$ and $A_2$ available for detection:

$$\begin{pmatrix} A_1(t) \\ A_2(t) \end{pmatrix} = \mathbf{S}_{180} \begin{pmatrix} A_\mathrm{S}(t) \\ A_\mathrm{LO}(t) \end{pmatrix}$$

with

\begin{aligned} A_1(t) &= \frac{1}{\sqrt{2}} \Bigl[ A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr) + A_\mathrm{LO}(t) \exp\bigl(i \omega_\mathrm{LO} t\bigr) \Bigr] \\ A_2(t) &= \frac{1}{\sqrt{2}} \Bigl[ A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr) – A_\mathrm{LO}(t) \exp\bigl(i \omega_\mathrm{LO} t\bigr) \Bigr] \end{aligned}\tag{3}

The second terms on the right side are 180° out of phase, giving the 180° hybrid its name. Figure 1 illustrates some examples of such a hybrid.

Figure 1: 180° hybrids: a) fiber-optic implementation, b) free-space optics (beam splitter) implementation, c) symbol.

The photodetector converts the optical field into a photocurrent according to$^2$

$$I(t) = R \cdot E^2\tag{4}$$

with a photodiode-dependent coefficient $R$ (responsivity) which we set to unity for the present discussion. Since this operation is a nonlinear one, we must not work with the complex amplitudes but with the real-valued fields $E_1$ and $E_2$ instead:

\begin{aligned} I_1(t) &\propto \frac{1}{2} E_\mathrm{S}^2(t) + \frac{1}{2} E_\mathrm{LO}^2(t) + E_\mathrm{S}(t) E_\mathrm{LO}(t)\\ I_2(t) &\propto \frac{1}{2} E_\mathrm{S}^2(t) + \frac{1}{2} E_\mathrm{LO}^2(t) – E_\mathrm{S}(t) E_\mathrm{LO}(t) \end{aligned}\tag{5}

We can get rid of the constant terms and double the usable signal by subtracting both currents (which is usually called balanced or differential detection)

\begin{aligned} I_{1-2}(t) &\propto 2 E_\mathrm{S}(t) E_\mathrm{LO}(t)\\ &= \frac{1}{2} \Bigl[ A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\Bigl(i \bigl[\omega_\mathrm{S} – \omega_\mathrm{LO} \bigr] t\Bigr) \\ &\quad\quad \color{gray}{ + A_\mathrm{S}(t) A_\mathrm{LO}(t) \exp\Bigl(i \bigl[\omega_\mathrm{S} + \omega_\mathrm{LO} \bigr] t\Bigr)} + \mathrm{c.c.} \Bigr]\\ &= \Re \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\Bigl(i \bigl[\omega_\mathrm{S} – \omega_\mathrm{LO} \bigr] t\Bigr) \\ &\quad\quad \color{gray}{ + A_\mathrm{S}(t) A_\mathrm{LO}(t) \exp\Bigl(i \bigl[\omega_\mathrm{S} + \omega_\mathrm{LO} \bigr] t\Bigr)} \Bigr\rbrace \end{aligned}\tag{6}

From the above equation, we get a current component at the difference (or intermediate) frequency

$$\omega_\mathrm{IF} = \omega_\mathrm{S} – \omega_\mathrm{LO}$$

and one at the sum frequency. However, the sum frequency term is never observable. The external current is the integral of all current densities inside the detector. As soon as the photodetector is longer than maybe half a wavelength of the sum-frequency signal, the corresponding current densities created in various positions along the detector add and only the average current remains. To achieve any meaningful value of conversion efficiency, photodetectors are usually many wavelengths long and the observable current is then the average of the instantaneous power of this number of oscillations – one must take care though that the detector is short enough to not affect signals at $\omega_\mathrm{IF}$. The second term in the equation above can therefore be safely dropped.

Since there is a factor $A_\mathrm{LO}$ in the photocurrent, the local laser intensity can be used to amplify the incoming signal directly during the detection process without further hardware, which can be very useful for weak signals.

At this point it is also very important that $\omega_\mathrm{IF}$ is sufficiently high so that the two constituent parts of the real part do not overlap spectrally. Otherwise, an intradyne receiver is needed to fully recover the information in $A_\mathrm{S}$ (see below).

We can now either digitize the signal at the intermediate frequency $\omega_\mathrm{IF}$ and mix it into the baseband digitally or use an electrical mixer to downconvert the signal into the baseband and then digitize it for further processing. The first method is more expensive but involves fewer RF components. Once the signal is digital, we extract the complex modulation signal by removing the lower sideband – the “c.c.” in (6) – and then multiplying by $\exp\bigl(-i \omega_\mathrm{IF} t\bigr)$:

\begin{aligned} B_\mathrm{digital}(t) &= \frac{1}{2} A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\bigl(i \omega_\mathrm{IF} t\bigr) \cdot \exp\bigl(- i \omega_\mathrm{IF} t\bigr) \\ &= \frac{1}{2} A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \end{aligned}\tag{7}

In an analog implementation, since the desired output signal $A_\mathrm{S}$ is complex, we need separate circuits giving us a signal proportional to its real and its imaginary part (there exists no such thing as a complex current). We can get such signals by multiplying with $A_\mathrm{IF} \cos \bigl(\omega_\mathrm{IF} t\bigr)$ and $-A_\mathrm{IF} \sin \bigl(\omega_\mathrm{IF} t\bigr)$, where $A_\mathrm{IF} A_\mathrm{IF}^*$ is the total output power of a mixing oscillator. Since the current $I_{1-2}$ must be split into both circuits, we have another factor $1/\sqrt{2}$ due to the law of conservation of energy:

\begin{aligned} I_\mathrm{I}(t) &= A_\mathrm{IF} \cos \bigl(\omega_\mathrm{IF} t\bigr) \cdot \frac{1}{\sqrt{2}}\Re \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\bigl(i \omega_\mathrm{IF} t\bigr) \Bigr\rbrace \\ &= \frac{A_\mathrm{IF}}{2\sqrt{2}} \Re \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \color{gray}{ + A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\bigl(i 2 \omega_\mathrm{IF} t\bigr)} \Bigr\rbrace\\ I_\mathrm{Q}(t) &= – A_\mathrm{IF} \sin \bigl(\omega_\mathrm{IF} t\bigr) \cdot \frac{1}{\sqrt{2}} \Re \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\bigl(i \omega_\mathrm{IF} t\bigr) \Bigr\rbrace \\ &= \frac{A_\mathrm{IF}}{2\sqrt{2}} \Im \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \color{gray}{ – A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\bigl(i 2 \omega_\mathrm{IF} t\bigr)} \Bigr\rbrace\\ \end{aligned}\tag{8}

where we simply expand $\Re \lbrace x \rbrace$, $\cos x$ and $\sin x$ into their exponential forms before doing the multiplication, or alternatively write the real part in its cosine form as in (1) and then apply product-to-sum trigonometric identities. The terms at $2\omega_\mathrm{IF}$ are removed by low-pass filtering, the signals are digitized and then used to build the complex output

$$B_\mathrm{analog}(t) = I_\mathrm{I}(t) + i I_\mathrm{Q}(t) = \frac{A_\mathrm{IF}}{2\sqrt{2}} A_\mathrm{S}(t) A_\mathrm{LO}^*(t)\tag{9}$$

which is the same result as in digital processing except for a lower signal amplitude.

#### Homodyning

In homodyne detection, we have $\omega_\mathrm{S} = \omega_\mathrm{LO}$. In optical communications this can be a problem since the local oscillator laser cannot be the same device as the signal laser and it is difficult to synchronize two lasers. In interferometry it is quite possible to use one laser source for both tasks.

The homodyne case is very similar to the downconversion from $\omega_\mathrm{IF}$ to the baseband in heterodyne detection, except that $\omega_\mathrm{IF}$ here corresponds to $\omega_\mathrm{LO}$. Also, instead of simple multiplication with $\cos \bigl(\omega_\mathrm{LO} t\bigr)$ and $-\sin \bigl(\omega_\mathrm{LO} t\bigr)$ we need to employ a combination of a six-port device called 90° hybrid to combine the signals and photodiodes to perform the mixing. The scattering matrix of this hybrid is

$$\mathbf{S}_{90} = \frac{1}{2} \begin{pmatrix} 1 & 1 \\ 1 & -1 \\ 1 & -i \\ 1 & i \end{pmatrix}\tag{10}$$

with which we can again write down the output amplitudes

\begin{aligned} A_1(t) &= \frac{1}{2} \Bigl[ A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr) + A_\mathrm{LO}(t) \exp\bigl(i \omega_\mathrm{LO} t\bigr) \Bigr]\\ A_2(t) &= \frac{1}{2} \Bigl[ A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr) – A_\mathrm{LO}(t) \exp\bigl(i \omega_\mathrm{LO} t\bigr) \Bigr]\\ A_3(t) &= \frac{1}{2} \Bigl[ A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr) – i A_\mathrm{LO}(t) \exp\bigl(i \omega_\mathrm{LO} t\bigr) \Bigr]\\ A_4(t) &= \frac{1}{2} \Bigl[ A_\mathrm{S}(t) \exp\bigl(i \omega_\mathrm{S} t\bigr) + i A_\mathrm{LO}(t) \exp\bigl(i \omega_\mathrm{LO} t\bigr) \Bigr]\\ \end{aligned}\tag{11}

and with these we obtain the differential photocurrents after a lot of algebra (also compare (6) for the real part):

\begin{aligned} I_{1-2}(t) &\propto \frac{1}{2} \Re \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \\ &\quad\quad \color{gray}{ + A_\mathrm{S}(t) A_\mathrm{LO}(t) \exp\bigl(i 2\omega_\mathrm{S} t\bigr)} \Bigr\rbrace\\ I_{4-3}(t) &\propto \frac{1}{2} \Im \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \\ &\quad\quad \color{gray}{ – A_\mathrm{S}(t) A_\mathrm{LO}(t) \exp\bigl(i 2\omega_\mathrm{S} t\bigr)} \Bigr\rbrace \end{aligned}\tag{12}

We neglect the unobservable currents at $2\omega_\mathrm{S}$, digitize and combine both signals to obtain

$$C(t) = I_{1-2}(t) + i I_{4-3}(t) = \frac{1}{2} A_\mathrm{S}(t) A_\mathrm{LO}^*(t)\tag{13}$$

which is the same output signal as in the heterodyne case except for a 3dB higher electrical signal power. This is due to the heterodyne terms at $2\omega_\mathrm{IF}$ which are discarded during downconversion into the baseband, but contain half the electrical signal power.

Figure 2: 90° hybrids: a) fiber-optic implementation, b) free-space optics implementation.

In the case that different lasers are used to generate $E_\mathrm{S}$ and $E_\mathrm{LO}$, it is difficult to achieve $\omega_\mathrm{S} = \omega_\mathrm{LO}$ (unless one has a working optical phase-locked loop). However, the equations governing homodyne detection apply equally here, except for a residual frequency modulation:

\begin{aligned} I_{1-2}(t) &\propto \frac{1}{2} \Re \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\bigl(i \Delta\omega t\bigr) \Bigr\rbrace\\ I_{4-3}(t) &\propto \frac{1}{2} \Im \Bigl\lbrace A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\bigl(i \Delta\omega t\bigr) \Bigr\rbrace \end{aligned}\tag{14}

with $\Delta\omega = \omega_\mathrm{S} – \omega_\mathrm{LO}$ being small (smaller than half the bandwidth of the signal). We neglected the sum-frequency terms in writing (14). After analog-to-digital conversion we can combine both signals and obtain

$$D(t) = I_{1-2}(t) + i I_{4-3}(t) = \frac{1}{2} A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \exp\bigl(i \Delta\omega t\bigr)\tag{15}$$

It is then just a matter of digitally multiplying the signal with $\exp\bigl(-i \Delta\omega t\bigr)$ ($\Delta\omega$ must be known) to obtain the original modulation. The bandwidth required for the digitizers is a bit higher than for homodyne, but there is no need for a phase-locked loop.

If we used a 180° degree hybrid and tried to detect such a signal in a heterodyne manner, our usable signal would be given by $I_{1-2}$ in (14), or

$$I_{1-2}(t) \propto \frac{1}{2} \bigl|A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \bigr| \cos\Bigl(\Delta\omega t + \arg \bigl[ A_\mathrm{S}(t) A_\mathrm{LO}^*(t) \bigr]\Bigr)$$

Due to the symmetry / ambiguity of the cosine, we would not be able to determine the complex signal correctly – $\Delta\omega t$ changes slower than the phase of $A_\mathrm{S}$ per definition of intradyne (otherwise it would be heterodyne).

In the next post we’ll take a look at noise performance.

1 The scattering matrix for the 3dB coupler (symmetric power splitter) is not unique. The magnitude of its elements must be $1/\sqrt{2}$, but their phase is arbitrary as long as the matrix remains unitary. A different description which is often used due to its symmetry is

$$\mathbf{S}_{180} = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & i \\ i & 1\end{pmatrix}$$

2 The photocurrent density is proportional to the energy density of the field, given by $S = \frac{\varepsilon}{4\pi} E^2$. The observable photocurrent is then the integral of the photocurrent density over the thickness of the active layer of the photodiode.