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.