22. Lecture, 16 November 1999

Transcription

1 Astronomy 3/43, all 999 Lecture, 6 Novemer 999 Coherent detection Another way to detect light using photodetectors is to use the same method your radio uses: coherent, or linear, detection n this method the wave properties of light are used explicitly, and the measurements amount to the determination of the amplitude and phase of the electric and magnetic fields in the radiation emitted y the distant source Coherent detection itself comes in two forms that turn out to have the same sensitivity in ideal systems: coherent preamplification, in which incident light is passed through a medium that can impart gain, amplifying the wave amplitudes directly The output of such a preamplifier is usually detected y a heterodyne receiver (see elow), though an incoherent detector could e used instead; the idea is for the gain to e so large that any detector could e used susequently without affecting the signal-tonoise ratio The paradigm for astronomical coherent preamplifiers is the maser, and the asic principles involved are those that apply to the (y now) more familiar oscillator forms of masers and lasers We will discuss these devices superficially elow n the past decade the highest frequencies at which transistors can e used as coherent preamplifiers have crept up to tens of Gz (wavelengths down to cm or so) with the development of EMTs (see 88); these components are currently used in most radio-astronomical coherent preamplifiers heterodyne detection, in which the signal one wishes to measure is mixed with coherent light (constant frequency, phase and amplitude) efore shining on the detector The additional coherent light, which comprises requency and phase reference, is called the local oscillator () The detector used here has to have a response time short enough that currents can exist in it at the frequency difference etween the signal and : that is, at the frequency of eats etween the signal and waves Amplitudes and frequencies of the eats can e measured y the normal techniques of low-frequency electronics n what follows we will use ν as the symol for signal and frequencies (light), and f for the lower, eat, frequencies (currents) eterodyne detection is used commonly practically universally y astronomers at wavelengths longer than aout mm, so we will discuss this technique in detail ensitivity of heterodyne detection Consider a photodetector used in heterodyne mode in a telescope system similar to that used in (see igure ) uppose that the eam of light is matched to the signal eam (that is, has the same eam waist size and wavefront curvature) and is injected into the signal path y use of a diplexer that attenuates the signal and ackground power negligily n most applications the power availale from the local oscillator is many orders of magnitude larger than the signal and ackground power, so the diplexer can consist simply of a thin dielectric eamsplitter that transmits virtually all of the incident light ut still reflects enough ; we will assume that this is the case in the following Note, however, that there are many instruments in which more complicated schemes such as Michelson interferometers or folded ary-erot interferometers are used to comine the eams At the surface of the photodetector (z ), the field from the is E E e i ω t, () 999 University of Rochester All rights reserved

2 Astronomy 3/43, all 999 Diplexer ( ε) + B ( ε) + B τ( ε) + + τb + τ ignal Warm optics, transmission - ε Local oscillator Cold optics, transmission τ hotodetector, quantum efficiency η ignal (ν) (ν ) Detector + R DC C AC amplifier Beats (f ν ν ) R power detector igure : signal chain for calculation of sensitivity of a heterodyne receiver where ω πν The signal and ackground are not generally monochromatic or characterized y constant phase; however, we can consider for now one frequency component and phase of the signal, E i ωt+ φ Ee g, () at the detector s surface We will consider the field amplitudes to e real (or simplicity we can leave the ackground power out for now; it will return in a little while) The photocurrent induced y these two radiation fields is simply η Gq ω, (3) where + τ ε + τ is given in terms of the fields at the detector s surface y z c E + E da, (4) A and where A is the detector s area in the focal plane, and cgs units are used uppose furthermore that the detector is uniformly illuminated y oth signal and ; then ca E + E ca E E e i t i e i L t + NM ω ω φ ω ω φ E E g E E g ca E + E + E E c ω ω t φh cos O Q (5) Assume that the same polarization if used for signal and, and define 999 University of Rochester All rights reserved

3 Astronomy 3/43, all 999 ca ca E E ; (6) then we can write + + cos ω ω t φ c h (7) The first two terms in Equation 7 give rise to DC photocurrents, and the third term is the eat etween signal and (see igure ), which oscillates at the intermediate frequency (), f ν ν Usually the detector is followed y an amplifier that works only on the component of this current, g c h ηgq i ω, t cos ω ω t φ, ω (8) or an associated voltage iω, tg R, as shown in igure The power detected at the output of the amplifier is proportional to the electrical power dissipated in the resistor R, or 4η G q R ir cos ω ω t φ ω c h (9) ince it oscillates periodically, the average of this power over a large numer of eat oscillation periods is the same as the average over a single period, π/ ω ω g: [ E( ω t) ] Re, ( ωt φ) i + Re E e [ E( ω t) ] ω ( E e ) Re, Re i t ( ω, ) ( ω, ) ( ω, ) ( ω, ) E t E t + E t E t Time igure : eats 999 University of Rochester 3 All rights reserved

4 Astronomy 3/43, all 999 π/ ω ω 4η G q R ir ω ω cos ω π η G q R z g c ω ω t φ dt h () n terms of power at the system input rather than at the detector surface, this is (see igure ): () ετ ηg q R ir This time, we consider our electrical signal to e a power, rather than a current g g, requency f in the photocurrent corresponds to Note that ecause cos ω ω t cos ω ω t the detection of two signal frequencies, ν ν ± f, that therefore cannot e told apart simply from the signal requencies of detected light greater than that of the are called the upper sideand, and lower frequencies are called the lower sideand eparation of the two sideands generally requires additional, interferometric optics to transmit one or the other, and this is desirale if, for instance, a spectral line is oserved in one sideand, and one would like to avoid the additional noise from detection of the other sideand n the following we will restrict our attention to heterodyne systems that detect oth sideands, and are called doule-sideand receivers Now we shall deal with the noise We assume again that the amplifier is designed to render Johnson noise negligile compared to shot noise, so the noise power, at the input of the amplifier, is, from Equation 7, ir NR β GqR, () K sn where is the average total current in the detector, and the lael sn just stands for shot noise ere follows the sutle trick of heterodyne detection: suppose that the power on the detector is y far the largest component of the total power: Gq τη hν τηgq so ir βgq R K sn hν, (3) (4) Let us assume that the signal and frequencies are very similar ω ω << ωg Then the (doule sideand) signal-to-noise ratio is G N K J K, ir ir ετ ηg q R hν βτηg q R sn ετη β hν (5) 999 University of Rochester 4 All rights reserved

5 Astronomy 3/43, all 999 independent of the power (!) Turn the power up high enough, increasing the photocurrent shot noise all the way, and eventually the signal-to-noise ratio doesn t depend upon power or this noise The form of Equation 4 is a good illustration of the workings of heterodyne detection, and turns out to e correct at the shortest wavelengths at which the technique is used y astronomers t is, however, incomplete; it turns out that in using simply the shot noise we have omitted a noise process that is important at longer wavelengths, where heterodyne detection is used most often 3 Background radiation and its fluctuations in heterodyne detection We have only dealt so far with signal and power, since we had the << limit in mind all along t is not much troule to account also for detection of ackground power, ecause its eats with the would have exactly the same form as those of the signal one would therefore expect to repeat the derivation of Equation, changing for B and ecause the eats etween signal and ackground would e negligily small compared to the eats etween either with the much more powerful Thus Equation ecomes (see igure ) τ η G q R ir ε + B (6) Just as is the case for direct detection, the form of this expression shows that we have to make two measurements, in practice, to determine for a celestial oject: one with the telescope pointing at the oject, which leads to power at the input to the amplifier given y τ η G q R ir ε + K B, (7) and one with the telescope pointing at lank sky: G q R ir K τ η B, (8) so that the difference etween the two measured powers is proportional to the quantity we re actually trying to measure, : (9) G q R ir ir K ετ η K Thus the ackground can e separated from the signal owever, it is essentially always the case that B >> (as well as >> B ), since interesting astronomical ojects are faint This results in a contriution to the noise y ackground radiation, and this cannot e sutracted off That the power in lackody radiation must fluctuate was really shown aove ( 3), when we discussed the limiting cases of the photon proaility distriution; we need merely flesh out this claim here uppose a single-mode eam is used, and a single polarization (the s) is selected; then 999 University of Rochester 5 All rights reserved

6 Astronomy 3/43, all h B B εbνatf νaω ε ν N νλ εhν νn, () c where as usual N e h ν/ e kt j The average value of B is, analogously, and the variance of the ackground power is, () B εhν ν N d i d i, () B εhν ν N εhν ν N N + Bεhν ν N + where we have used Equation 34 in the last step At sumillimeter wavelengths and longer λ 35µ mg, and common amient temperatures (T ~ 3 K), N is consideraly greater than unity, so, (3) B εhν ν N B or B B g rms ; (4) that is, the ackground power at these wavelengths follows Gaussian statistics, and the rms fluctuations, far from eing small, are as large as the average ackground power itself 4 The quantum limit to heterodyne detection f our heterodyne detector can detect the ackground, then it can detect the ackground power fluctuations characterized y Equation 4 as well The detected fluctuations are another form of noise, and need to e added to the shot noise power (Equation 3) in order to otain a correct form for the signal-to-noise ratio This time, however, the noise is not simply due to the finite charge on the electron; as we ll se elow ( 3), it is due to the uncertainty principle rom Equation 6 we see that the ackground power fluctuations detected y our heterodyne receiver give rise to electrical power, referred to the input of the amplifier, of g (5) τ η G q R τ η G q R ir B K rms B f We should add this term to Equation 3 to get the total noise power, referred to the amplifier imput: K K + K i R i R i R sn f βτηg q R τ η G q R + B hν (6) Note that since the frequency of the is fixed, the signal andwidth ν is equal to the andwidth for a single sideand, or for doule-sideand response We will continue to assume that a doule-sideand receiver is used, for which Equations and therefore give 999 University of Rochester 6 All rights reserved

7 Astronomy 3/43, all 999 G q R G q R i R 4 h N K βτη h + τ η ε ν ν h G K J G βν K J + τη G τ η G q R KJ τη N (7) The signal power is given y Equation 9, and the doule -sideand signal-to-noise ratio is τ η G q R ε i R h i R G q R h f N G h KJ ν τ η G βν K J τη + ν τη G εg τη β hν ν τη + N G N K J g KJ (8) The only difference etween this expression and the incomplete Equation 5 is the last factor Note that since N >>, as it is at long wavelengths and common amient temperatures, this factor can reduce the signal-to-noise ratio significantly if τη / β and ε are large enough Rememer that the factor of two accompanying the εn factor is from the assumption that ackground fluctuations were detected in oth sideands, and that ν ; this factor goes away, and ν, for single sieand receivers Normally in radio astronomy the andwidth is on the order of ν / Mz, corresponding to integration time on the order of t / 5 5µ s As you might imagine, one normally averages for much longer than a small fraction of a second As we have seen repeatedly, fluctuations integrate down ecome smaller in proportion with the square root of the exposure time Thus if the signal and noise are averaged over an exposure time t >> t, the signal-to-noise ratio increases y the factor t t, which gives N t t G N K J t g (9) + ε τη β hν τη N This is the complete expression for the signal-to-noise ratio of an ideal, quantum-noise limited doulesideand heterodyne receiver As we did for incoherent detection, we can define a noise equivalent power for the ideal heterodyne receiver This would e the value of / f that corresponds to /N, with / t: or g NE ε τη β hν τη + N NE hν g ε τη β G τη + N KJ ν, (3) (3) The same factor-of-two differences etween doule-sideand (used here) and single-sideand response that we noted in connection with Equation 8 also apply here 999 University of Rochester 7 All rights reserved