# Homodyne Detection and Heterodyne Detection of Light

In this chapter we introduce the concept of homodyne detection and heterodyne detection of light in the framework of electromagnetic wave theory of radiation, including Maxwell’s continuous picture and Einstein’s quantized granularity picture of light. Optical homodyne detection and heterodyne detection are both adapted from radio frequency modulation technology. Unlike standard photodetection, homodyne detection and heterodyne detection measure the signal radiation, which may have modulated complex amplitude, by mixing with radiation of a reference frequency, which is usually generated by a local oscillator. Roughly speaking, in homodyne detection the reference frequency equals that of the input signal radiation; in heterodyne detection, the reference light is frequency-shifted.

## Optical Homodyne and Heterodyne Detection

Figure 13.1.1 schematically shows an optical homodyne or heterodyne detection setup. The signal field and the reference field are mixed at a 50% : 50% beamsplitter and superposed at photodetectors *D*i and To simplify the notation, the following analysis will be in 1-D

by focusing on one of the polarizations of the signal and reference radiation. The intensities

FIGURE 13.1.1 Schematic setup of a homodyne or heterodyne detection experiment.

of *1**1** (t)* and /■>(/) are the results of the superposition between the input signal field *E _{s}* and the reference field

*E*

_{r},

The common cross term *E _{s}(t)E*(t)* is formally written as

where *v = oj*_{s} — uj_{s}о is the detuning frequency and w_{s}o the central frequency of the input signal radiation, r_{s} = *t — z _{s}/c,* r

_{r}=

*t — z*In Eq. (13.1.2), we have assumed a single-mode reference radiation field

_{r}jc.and a general multi-subfields and multi-Fourier-mode field of input signal with certain distribution of complex amplitude,

which is formally written as a wavepacket with carrier frequency *u _{s}o* and modulated “envelope” corresponding to the Fourier transform of the complex amplitude.

For homodyne detection, *u> _{r} = lj*

_{s}o = u>o, Eq. (13.1.2) can be formally written as the Fourier transform of the spectrum

where *Aj(i/)* = «_,((/) *a _{r}.* indicating an amplified amplitude of

*aj(v)*in the case of a strong local oscillator. The cross term

*E*represents the interference between the input signal radiation

_{s}(t)E*(t)*E*and the reference field

_{s}(t)*E*As we have studied earlier, the relative phases

_{r}(t).*= constant, and will have null contribution if the relative phase (//) —* —

*tp*will play an important role in determining the measured values of /) and

_{r}*1-2*

*■*The interference term will contribute to the measured values of

*I*and /2 significantly when

*4*

*?j(v)*—

r

*(p*takes all possible random values from 0 to 2тг. The optical path difference

_{r}*z*is another factor in determining the contribution of the interference term in the case of

_{s}— z_{r } 1/) —

= constant. The value of iaJo(z

_{s}

*— z*determines the constructive-destructive property of the interference and consequently determines the magnitude for each and for all of the Fourier amplitudes. It is interesting to see the relative phase

_{r})/c*and the relative phase delay wo(^s*r

^{—}«

_{r})/c between the input field and the local oscillator are both included in the Fourier transform. A spectrum analyzer can retrieve this important information for certain observations. This property has been widely adapted in the studies of squeezed state and other coherent and statistical properties of light.

For heterodyne detection, taking *ш _{г} ф u>_{s}*о, Eq. (13.1.2) can be formally written as

where oj,i = — *u*>_{s}о is the frequency of beats. Ecp (13.1.4) is recognized as a modulated

harmonic oscillation of frequency uy = w_{s}o ^{—} w_{r}. The Fourier transform of the spectrum is the modulation function that modulates the harmonic oscillation.

## Balanced Homodyne and Heterodyne Detection

Figure 13.2.1 schematically shows a balanced homodyne or heterodyne detection setup. The input signal field *E _{s}(t)* and the reference field

*E*are mixed at a 50% : 50% beamsplitter. The output fields are directed and superposed at photodetectors

_{r}(t)*D*and

_{x}*D*The photocurrent

_{2}.*i{t)*and

*i*

_{2}*{t)*are subtracted from each other in an electronic circuit. A standard spectrum analyzer follows to select, amplify and rectify a certain bandwidth of the Fourier spectral composition in the waveform of

*i*— electronically. The observed

_{x}(t)output of the spectrum analyzer is a measure of the amplitude of the chosen Fourier spectral composition.

FIGURE 13.2.1 Schematic setup of a balanced homodyne and heterodyne detection experiment. In homodyne detection the reference frequency equals the central frequency of the input signal radiation, *u) _{r}* = w

_{s}о = wo-

Based on the experimental setup of Fig. 13.2.1 we now calculate the expected output from the spectrum analyzer. We start from calculating *i _{x}(t_)—*г

_{2}(£_) ос

*I*/2(^2); where

_{x}(t_{x}) —*t*

_{a}= t_-ri^{e}a =*1*

*,*

*2*

*,*with

*Ta^*the electronic delay in the cables and the electronic circuits associated with a-th photodetector, and f_ is the time for photocurrent “subtraction”. To simplify the mathematics, we choose

*т*r

_{х}^{е}^ =_{2}

^{<}'

^{1}to achieve

*t*

_{x}= t*-2*

*= t.*It is easy to see

FIGURE 13.2.2 Simplified block diagram of a classic spectrum analyzer.

from Eq. (13.1.1), Eq. (13.1.3), and Eq. (13.1.4) for balanced homodvne detection, and

for balanced heterodyne detection. We may consider homodyne detection a special case of heterodyne detection when taking *u>d* = 0. There is no significant difference in the modulation function, except a trivial phase factor of *u>o(z _{s}* —

*z*It is the spectrum of the modulation function that will be analyzed by the spectrum analyzer. To simplify the discussion, we will focus our attention on the balanced homodyne detection in the following analysis. We will show how a spectrum analyzer works in determining the spectrum of the modulation function with the coherence and path information of the measured light. Before exploring the working mechanism of the spectrum analyzer, we estimate the expected value of its input current *i(f) —

_{r})/c.*i*

*2*

*{t)*oc

*Ii(t)*—

*I-*

*2*

*(t).*

The expectation value of *{Ii(t)* — /^(f)) is easy to calculate by taking into account all possible values of *ipj(v) — *

r

within the superposition. For single-mode reference field*E*the coherent behavior of the input signal, which is mainly determined by the phases of the sub-fields 9j(^), will determine the expectation. We discuss two extreme cases:

_{r}(t),Case (1): Random

It is easy to see that the only surviving terms in the superposition are the terms with *=* i*p _{r}* when taking into account all possible values of Obviously, the chances of

having (//) =

r

are quite small. The expectation value of*(I (t)*—

*12*

*(t*)) is thus effectively zero in this case. In a real measurement, however, the superposition may not take all possible values of the random phases and the interference cancelation may not be complete. These non-canceled terms of

*h(t)*— /^(f) will be analyzed and displayed by the spectrum analyzer in terms of the Fourier composition of

*1*

*/,*which is effectively the beats frequency

*ш*

*—*

_{3}*ш*in the homodyne detection measurement,

_{г}where ^_{surv} represents the sum of the non-canceled surviving terms in the superposition. These surviving terms are traditionally treated as the noise or fluctuations of the radiation. The spectrum analyzer is thus considered of measuring the spectrum of the noise or the fluctuations of the radiation.

Case (2): —

r

= constant.In the case of *tpj* (i/) —

r = *I(t)* — *h(t)* oc *Re [E _{s}(t)E*(t*)], i.e.,

without interference cancelation will be received by the spectrum analyzer. The spectrum analyzer no longer measures the noise or the fluctuation of the surviving input signal, instead, it receives and analyzes the entire interference term of *E _{s}{t)E*[t).*

The design and working mechanism of specific spectrum analyzers can be quite different from each other. Nevertheless, the output reading of modern spectrum analyzers can be roughly divided into two categories: linear normal spectrum and nonlinear power spectrum. Linear normal spectrum simply presents the spectral amplitude of the input signal as a function of frequency. The nonlinear power spectrum provides much more detail than the normal spectrum. A power spectrum includes not only the Fourier composition of the input current *i{t) — iiit)-* but also their beats and sum-frequencies that fall within the passband of the chosen spectral filter in the heterodyne circuit of the spectrum analyzer, such as the IF filter shown in Fig. 13.2.2.

A simplified block diagram of a classic spectrum analyzer is illustrated in Fig. 13.2.2. The input signal of *i{t)* — *h{t*), which is either proportional to Eq. (13.2.3) or Eq. (13.2.4), is mixed with a sinusoidal reference current of tunable RF frequency *иц„* in an electronic mixer. The RF current of w/_{0} is generated from a local oscillator. The mixer has a nonlinear response to the inputs. Taking account of the first-order and the second-order response of the mixer to a good approximation, the output of the mixer contains the input signal *i{t)* — *h{t),* the reference oscillation of *oj( _{0},* their second-harmonics, and a cross term

for the measurement of thermal light, and

for the measurement of coherent light, where

s,z_{r}) = co_{s}o(z_{s}

*z*and we have assumed a simple harmonic local oscillator of frequency w;

_{r})/c+ vz_{s}/c,_{0},

*ii*

_{0}*(t) = Ai*

_{0}*cosu>i*

_{0}*t.*Eqs. (13.2.5) and (13.2.6) indicate that the nonlinear response of the mixer produces a down-converted Fourier composition ojif = w/

_{0}— // and an up-converted Fourier composition lojf = W(o +

*v*in terms of each Fourier composition of the input signal. An electronic spectral filter follows after the mixer to select a narrowband RF current of frequency

*wjp*from either the down-converted set or the up-converted set of the Fourier-modes. uiif is technically called the intermediate frequency. To simplify the mathematics, we assume the bandwidth of

*ujp*much narrower than that of the input signal so that the selected Fourier composition of ojif can be treated as single-mode. The selected single-mode Fourier composition of u>if is then amplified by a linear amplifier and rectified by a nonlinear envelope detector, resulting in an output that is proportional to the power spectrum

for the measurement of incoherent thermal light, and

for the measurement of coherent light.

We thus have the following results for the above two extreme cases.

(1): Measurement of incoherent thermal light.

Thermal light is statistically stationary and ergodic, by choosing an appropriate time parameter (integration time) of the spectrum analyzer, we may treat the measurement as an ensemble average

where (...) denotes, again, an ensemble average by means of *taking into account all possible realizations of the field.* As we have discussed earlier, when taking into account all possible values of *p)j(is),* the stochastic superposition results in a non-zero value from the first cosine term, which includes all the surviving diagonal terms of *j = k,* and a zero value from the second cosine term of Eq. (13.2.9). The expected power spectrum of thermal light is thus a simple sum of the squared amplitudes

In reality, the radiation field may not take all possible realization within the time integral of the spectrum analyzer, the incomplete interference cancellation may still cause a random fluctuation in the neighborhood of (P(*y))* from time to time.

Case (2): Measurement of coherent light.

Taking *(u) — tp _{r} = constant, Eq. (13.2.9) becomes*

It is interesting to find from Eq. (13.2.11) the power spectrum of coherent light is a sinusoidal function of *z _{r}) = *(lo

_{s}z

_{s}

*—*

*u*~ w

_{r}z_{r})/c_{s}o(^

_{s}—

*z*The change of the relative optical path between the signal field and the reference field, which can be realized by adjusting the position of the mirror

_{r})/c.*M*in Fig. 13.2.1, will produce an interference pattern as a function of

*z*similar to that of the interference between two individual but synchronized laser beams.

_{s}— z_{r},