DESIGNCONSIDERATIONS FOR OPTICAL HETERODYNERECEIVERS: A RFXIEW
John J. Degnan Instrument Electro-optics Branch NASA GoddardSpace Flight Center Greenbelt,Maryland 20771
ABSTRACT
By its verynature, an optical heterodyne receiver is both a receiver and anantenna. Certain fundamental antenna properties ofheterodyne receivers are describedwhich set theoretical limits onthe receiver sensitivity for the detectionof coherent point sources, scattered light, and thermal radiation. Inorder to approachthese limiting sensitivities, the geometry of the optical antenna-heterodyne receiver configurationmust be carefully tailored to the intendedapplication. The geometric factors which affect system sensitivity includethe local osciliator (LO) amplitudedistribution, mismatches between the signaland LO phasefronts,central obscurations of the optical antenna, and nonuniformmixer quantum efficiencies. The current state of knowledge in this area, which rests heavilyon modern concepts of partial coherence, is reviewed.
Following a discussion of noiseprocesses in the heterodyne receiver and the manner in whichsensitivity is increasedthrough time integrationof the detectedsignal, we derivean expression for the mean squaresignal current obtained by mixing a coherent local oscillator with a partially coherent, quasi- monochromaticsource. We thendemonstrate the manner in which the IF signal calculationcan be transferred to anyconvenient plane in the optical front end ofthe receiver. Using these techniques, we obtain a relativelysimple equation forthe coherently detected signal from anextended incoherent source and apply it to theheterodyne detection of an extended thermal source and to theback- scatter lidarproblem where the antenna patterns of both the transmitter beam andheterodyne receiver mustbe taken into account. Finally, we considerthe detectionof a coherentsource and, in particular, a distantpoint source such as a star or laser transmitter in a longrange heterodyne communications system.
461 1. INTRODUCTION
Heterodyne or coherent detection can beadvantageous in a variety of applications.Heterodyne receivers have at least two features which are quali- tativelydifferent from incoherent(or direct detection) receivers (ref. 1). First of all, the receiving bandwidth is determined by the IF bandwidthwhich, in principle, canbe varied at will to give very high spectral resolution. Secondly,information related to the phase of the radiation signal is retained in the IF output and theoutputs of two or more receivers can be correlated to make coherence measurements comparable to the aperturesynthesis techniques of radioastronomy.
To achieve high spectral radiation with incoherent or direct detection systems,radiation filters or spectrometers mustbe utilized and thecombination ofvery narrow bandwidth and high sensitivity (low loss) is usually difficult to realize. In general, a heterodynereceiver will be more sensitivethan a direct detectionreceiver with an equivalentnoise equivalent power (NEP) forspectral resolutions below a cutoff bandwidth which depends on the NEP and the infrared wavelength (refs. 1, 2). The sub-Doppler spectralresolution of heterodyne receivers can be exploited to study the molecular constituents and kinematics ofremote sources yielding specific information such as altitude profiles of absoluteabundance of the species, vertical temperature profiles, and wind velocities(ref. 3). In detectingextraterrestrial thermal sources, the infor- mation is gathered by passiveheterodyne spectrometers whereas, in our own atmo- sphereor in planetaryatmospheres visited by spacecraft, active backscatter lidars canbe employed. In contrastto the above applications where theradia- tionsignal is totallyincoherent or only partially coherent, the signal from thelaser transmitter in a heterodyne communicationsystem (ref. 4) is coherent exceptas modified by atmosphericeffects (ref. 5). Thisarticle attempts to present a unified theory of heterodynereceivers which addresses the optical design considerations for all of these applications.
A representativeheterodyne receiver is illustrated in Figure 1. Signal radiation is collected by an optical antenna and focused,along with a local oscillator beam, onto a square-lawfrequency mixer operating at the radiation frequency. The latter beams have centerfrequencies Vs and VL andpowers Ps and PL. Thetwo frequencies mix togive an outputspectrum centered at the intermediatefrequency VIF = VS - VL where VIF is much smallerthan the infraredfrequencies VS and VL and typically on theorder of a GHz orless. The resultingsignal current is amplified by an IF amplifier ofbandwidth BIF and rectified by a nominallysquare-law detector to give a current output pro- portionalto the power in the IF. This is usuallyinput to a low-frequency filter or integrating circuit to further enhancethe spectral resolution and/or sensitivity and is thenrecorded.
Although the present article will address most factorsinfluencing the per- formanceof the receiver in Figure 1, it will emphasizethe design of theoptica front end of thereceiver for a variety of applicationsand, in particular, the manner in which the optical antenna geometry and local oscillator distribution affect system sensitivity. In Section 2 of thispaper, we reviewthe noise processesrelevant to the IF signal and discussthe system signal-to-noise in the IF in termsof an asyet undefined mean squaresignal current. Section 3
462 brieflyoutlines the sensitivity improvementachieved by time integrationtech- ’ niques.In Section 4, we addressthe calculation of the mean squaresignal current in the mixer plane for a general,partially coherent, quasi-monochromatic sourceand, in Section 5, demonstratethe manner in which the IF signalcalcula- tion can be transferred to any convenient plane in the optical front end of the receiver. InSection 6, we applythe general result to thespecific problem of coherentlydetecting an extended incoherent source. The resultsof that section are thenapplied to theheterodyne detection of an extended thermal source in Section 7 and to the backscatter lidar problem in Section 8 and some useful designguidelines are generated.In Section 9, we applythe results of Sec- tion 4 to the detection of a spatiallycoherent source such as a laser trans- mitter in a heterodynecommunications system or a distantpoint source such as a star.
2. THE SIGNAL-TO-NOISE RATIO OF A HETERODYNE RECEIVER
The power signal-to-noise ratio of a heterodynereceiver is a measure of its sensitivity since setting the ratio equal to onepermits calculation of the noiseequivalent power (NEP). It is given,in most cases ofinterest, by (ref. 1)
We will leave thecalculation of the mean squaresignal current
The localoscillator induced shot noise, or quantum noise,
2- -
- where iDc is the DC currentgenerated by the LO, e is theelectronic charge, BIF is theintermediate frequency bandwidth, and hV is thephoton energy. The integrandcontains the detector quantum efficiency nQ and the LO intensity IL which are assumed tovary over the plane of thedetector defined by the two- -+ dimensionalcoordinate rD. The parameter B equals 1 forphotoemissive mixers
463 while,for photoconductors, it equals 2 dueto fluctuations in the generation andrecombination of charge carriers as described by Levinstein(ref. 6).
One can rewrite Equation(2.2) in the more familiarform
- if we definean average quantum efficiency nQ by
and PL is the local oscillator power incidenton the detector.
Radiationfrom a thermalsource contained within the receiver field of view and thereceiver bandwidth BIF will becoherently detected and subjectto so-called"heterodyne amplification." In some experiments,such as inpassive heterodynespectrometry, this thermal source is theobject of study,while in others it correspondsto unwanted backgroundnoise. We will show in later sections that it can be described bythe equation
where nT is 'an overallefficiency whichdepends inpart on thedesign of the opticalfront end.
Fluctuationsin background radiation, which spectrally is outsidethe receiverbandwidth but within the infrared response band of the mixer, will also producenoise currents, given by
Two otherimportant sources of noise are Johnson orthermal noise asso- ciatedwith the mixer and the IF amplifier. The mixer noise is given by
464
Clearly, other sources of noiseexist. "Excess noise" is common in receivers which employ diode laser local oscillators and generally arises from multimode effects or other non-ideal behavior in theLO. Noise can also be introduced at the electrical contacts to the mixer elementor by temperature fluctuations in the mixer. These sources are unique to specific systems and will not be considered further here.
With sufficientLO power, mostof the above noise sources can be made negligible relative to the quantum noise
where Ps is the received signal power and qHET is an as yet undefined heterodyne receiver efficiency, then, under strongLO illumination, the signal- to-noise ratio tends to
Setting the latter ratio equal to one and solvingfor PS/BIF yields the noise equivalent power per unit bandwidth;i.e.,
NEP (W/Hz)= -hV { B + QT [exp (hV/KT) - (2.10) ~HET 13 "> where nHET and nT bothdepend on the optical frontend geometry. In the quantumnoise limit (hV >> KT) , Equation(2 .lo) reduces to
BhV NEP(W/Hz) = - (2.11) ~HET
whereas, inthe thermal limit (hV << KT), it becomes nnl NEP (W/Hz) = KT (2.12) ~HET
If we includemixer and amplifier Johnsonnoise, we can write for a general photoconductor
2hV + nThv K(TM + TA) NEP (W/Hz) = - + (2.13) %ET [exp(hw/KT) - 11 G 'HET
where G is the"conversion gain" defined by Arams et al. (ref. 8)
3. DETECTION AND TIMEINTEGRATION
If the powersignal-to-noise ratio in the IF is less thanunity, the signal can be detected by integrating the detector outputover a sufficiently long period of time. The voltagesignal-to-noise ratio at the filter outputin Figure 1 is linearly related to the power S/N bythe equation (ref. 1)
The latter equation assumes that the IF amplifier has a rectangularbandpass spectrum(double sideband), the rectifying detector is anideal square-law device,the final output filter has a noisebandwidth Bo much less than BIF andthe power S/N is much less thanunity. Smith (ref. 9) hasconsidered the more general case wherethe IF amplifier is notstrictly square-law and does nothave a rectangularbandpass spectrum. He has also considered power S/N ratios much greaterthan unity. If theoutput filter is a singlestage RC circuitsuch that Bo = -c0/4 = RC/4, Equation(3.1) becomes
466 4. COHERENT DETECTIONOF A GENERAL QUASI-MONOCHROMATIC SOURCE 2 We turn now to the calculationof the mean square signal current
where W = 2TVS and the complex signal field envelope E, t) at the point +- S rD in the detector plane varies slowly in time relative to the exponential exp(iwSt). The time dependence of the envelope might reflect the modulated output of a transmitter laser ina heterodyne communications system, the ampli- tude and phase fluctuations inherent in the signal from an incoherent thermal source or backscatter lidar, or even the effects of atmospheric turbulence on the signal. The envelope, through its dependence on the detector coordinate -+ rD, also contains spatially dependent amplitude and phasefront information.
If we represent the LO fieldby a similar expression, the current out of the square-law mixeris given by
where wL is the LO center frequency and the integral is over the active detector area. Upon performing the quadratic multiplication of fields in Equation (4.2), we obtain both sum and difference frequencies. High-frequency sum terms varying as exp(+2iwst), exp(f2iwLt), exp(fi(W +W ) t) , lie outside the $L bandwidth of the mixer and hence can be ignored.The dlfference terms produce two "DC" currents corresponding to the average signal and local oscillator induced currents and an additional mixing term given by
467
L" . where the IF frequencyWIF - Ws - WL. Squaring Equation (4.3) yields
If we average the above expression over a time intervalT short compared to the coherence timesof the signal and local oscillator field (Ts and TL) but long compared to the IF beat period, TIF, we may write
2 dt iM (t)
since the field envelopes can be viewed as effectively constant over this time interval and hence the terms varying as exp(k2iwIFt) in Equation (4.4) average to zero over an IF beat period.In certain applications,such as passive heterodyne spectrometry of a thermal source, the integrationtime can be arbitrarily long. The limit of Equation (4.5) as T approaches infinity is then
468 where we have invoked thefact that the signal and local oscillator fields are statistically independent and hence the fourth-order correlation function +
Under the assumptionof cross spectral purity (refs.12, 13) , the spatial and time variables are separable leading to
++ where g(0) = 1 and Js (rlrr2) is the "mutual intensity function" (MIF) of the signal field. From Equations (4.7) and (4.8) we note that *-+a + +I -++
where Js(;D,ZD') and JL(ZD' ,;?,) are the mutual intensity functions of the signal and local oscillator fields in the detector plane. Calculationof the mean square mixingcurrent by means of Equation+ (4.9) is not always a simple task due to the difficultyin computing Js (ZDrrD') for many sourcesof practical interest. In ensuing sections, we will demonstrate how the calcula- tion can be carried out in optical planes other than the detector plane and the enormous simplifications that often result. Beforeclosing this section, it is worthwhile to note two useful properties ofthe mutual intensity function; i.e.,
(4.10)
and
(4.11)
+ where Is(?D) is the time averagedsignal intensity at thepoint rD.
5. PROPAGATION OF THE MUTUAL INTENSITYFUNCTION
Considerthe signal electric fieldpropagating from the antenna plane in Figure 2 tothe detector plane. Small angle scalar diffractiontheory (ref. 12) givesthe electric field in the detector plane; i.e.,
where k = 2~r/A, PA(ZA) is theantenna pupil function and-+ the term inbrackets correspondsto a Huygen'swavelet emanating from a point rA inthe antenna planeand traveling a distance rlU! to a point -frD inthe mixer plane. Then, from thedefinition of the mutual Intensity function (MIF), it is clear that
For a stationaryprocessl the time origin is of no consequenceand therefore
(ESkA.t - -)I-'AD C ES*(ZA' ,t - %)>C = (Es(sAlt)E, rA ,t - *[. I C -
(5.3)
470 NOW, if the-transverse dimensions of the antenna and detector pupil are small compared tothe coherence length of thesignal radiation defined by 1 = c/AVs, the variation of the signal electric field over a time interval t = - rm)/c is negligible andEquation (5.3) is effectivelythe signal MIF inthe antenna plane. Equation (5.2) then becomes thepropagation law forthe MIF as first derived by Zernike(refs. 12, 14); i.e.,
If we substituteEquation (4.5) in (4.9) €or the mean squaresignal current and reversethe order of integration, we obtain
where PD (;D) is the mixer pupilfunction. If we now definean effective local oscillator field given by
thecorresponding effective MIF is thenequal to
SubstitutingEquation (5.7) into (5.5) andcomparing theresulting expressionwith the MIF propagation law (5.4), we notethat the bracketed term inEquation (5.5) is simplythe MIF ofthe effective local oscillator back- propagated to theantenna plane. We may thereforewrite for the mean square mixing current
471 The physicalsignificance of Equation (5.8) is thatthe calculation of mean square IF signal current can be carried out in anyconvenient optical plane as firstpointed out by Rye (ref. 10). Thishas practical importancesince it is usually easier, for example, to compute thebackpropagation of a coherent LO electric field through an optical system than to propagate the MIF of an incoherentsource in a forwarddirection through the system to the mixer. This fact will be well illustrated in later sections.
Although we have consideredonly free space propagation in the present derivation,the approach is equally valid when interveningoptical elements such aslenses, mirrors, and aperturesare present. The simple Huygens wavelet inEquation (5.1) is thenreplaced by an appropriatetransmission function €or theoptical system (refs. 10, 12).
6. HETERODYNE DETECTION OF AN EXTENDED INCOHERJ3NT SOURCE
The expressionsderived up to this point have assumed a general, partially coherent,quasi-monochromatic source. We consider now animportant practical application in which thesignal radiation emanates froman extendedincoherent source and propagatesto the antenna plane as in Figure 3. The propagationof the MIF proceeds in preciselythe same fashionas in theprevious section exceptthat there is no coherencebetween the Huygens waveletsemanating from -+ -+ theinfinitesimal sources located at rS and rs'. Thus thesecond-order correlation function in the source-antenna plane version of Equation (5.3) becomes
(6.1) -+ -+ where Is(rs) is thetime averaged radiation intensity at the point rS in the -f -+ sourceplane and 6(rs - rs') is thetwo-dimensional Dirac delta function. It canbe shown thatsubstitution of Equation(6.1) into the source-antenna plane -+ version ofEquation (5.2) and performing the double integral over rs' yields thepropagation law forthe MIF of an incoherentsource (ref. 13); i.e.,
where theintegral is overthe finite dimensions of thesource. We maynow substituteEquation (6.2) into (5.8) and reversethe order of integrationto obtainfor the mean square IF signalcurrent
47 2 2
Throughuse of the MIF propagation law givenby Equation (5.4), we recognize thebracketed term inEquation (6.3) as themutual intensity function of the backpropagatedeffective local oscillator (BPELO) evaluated at thepoints -+ -+ -++ rS = rs' But,since JE(rs ,rS) = IE(zs) , the time averagedintensity of the BPELO inthe source plane, Equation (6.3) reduces to the relatively simple expression
Thus we havethe very useful result that the mean squareIF signal current is proportional to theoverlap integral of the extended incoherent source intensity withthe backpropagated effective LO intensity.In the next two sections, we will applythis result to the detection of thermal radiation and to the back- scatter lidar problem.
7. THERMAL SOURCEDETECTION
The total power AP radiatedinto a hemisphere,within the IF bandwidth BIF, from a small area AA on a blackbody is
2n hwBIF AP = - AA A2 [eXp (hV/KT) - 11
Onlythe power emitted in the direction of the receiver contributes to the signal MIF inthe antenna plane. Thus, if the receiver is in a directionnormal to the plane of theblackbody, we mustmultiply the above expression by a factor l/n corresponding to the power emitted per steradianin the normal direction. We must also multiplyby 1/2 to account for the fact thatthe heterodyne receiver detects onlyone polarization component. Thus, the intensity to be substituted intoEquation (6.4) is givenby
-1 hvBIF X2 [exp (hV/KT) - 13
473 andEquation (6.4) becomes
(7.3)
where the integral is simplythe total backpropagated effective LO powersub- tended by thesource.
If the dominant noise mechanism is the LO-induced shotnoise given by Equation(2.3), the IF signal-to-noise ratio is
where qT is theoverall heterodyne receiver efficiency for thermal source detectionintroduced in Equation (2.5) and defined by
- where qQ is theaverage mixer quantum efficiency defined byEquation (2.4) and PL is the LO power incidenton the detector. If the mixer quantum effi- ciency is uniform,Equation (7.5) reduces to
-+ where we haveused Equations (5.7) and (4.11) . The quantity I~(rs)is the intensity of theactual backpropagated LO ratherthan the effective LO. The quantity ?IT replacesthe mixer efficiency in the corresponding equations in theclassic paper bySiegman (ref.15).
If the source is so largethat the backpropagated LO is containedentirely within its diskradius, the integral in Equation (7.6) is simplythe total LO power inthe source plane. Except for an atmospheric transmission factor qA, the latter is equal to thebackpropagated LO power exiting fromthe antenna. Thus, theoverall heterodyne efficiency (7.6) can be broken down intoseveral components; i.e.,
where qo takesinto account routine optical losses due to reflections and scatteringwhile rlR is a geometricefficiency which takes into account vignetting,central obstructions, LO phasefrontcurvature, etc. inthe optical
474 antenna.Numerically, qR is equalto the fraction of the original LO power which exits from the antennaduring backpropagation.
where W is thegaussian spot radius in theantenna plane and we have defined two parameters(ref. 16) a = a/w and y = b/a. The geometricefficiency has been plottedas a function of c1 and y in Figure 5.
The importantthing to note in Figure 5 is that, for a givennonzero value of the linearobscuration ratio y = b/a,the optimum efficiency is lessthan what one would expectbased on simpleblockage of the incoming radiation by the centralobscuration. For example, y = 0.5 would implyan arealobscuration efficiencyof 1 - y2 or 75%. The peak efficiency in Figure 5 , however, would only beabout 47% if one were to choosean optimum gaussianspot radius corre- sponding to c1 = 1.3. Nonoptimum choicesclearly result in significantly worse performance.
Clearly, to maximize the efficiency of coherentdetection of a thermal source which fills thereceiver field of view,one wishes to choosean optical geometrywhich allowsthe effective backpropagated LO to exit from thetelescope with near-unityefficiency. Although this is most easily accomplished with off-axisreflective telescope geometries which eliminatethe central obscuration problem,one is notlimited to such geometries in general. For example, if we useappropriate masks in the LO beam to create a local oscillator distribution in the mixerplane which matchesthe Airy pattern of the centrally obscured Cassegraintelescope in Figure 4, thebackpropagated LO will forman annular disk in theantenna plane which matches theantenna pupil function and provides unity transmission. This result assumes,of course, thatthe mixer quantum efficiency is reasonablyuniform. The transmission loss ofthe beam splitter in Figure 4 is included in theoptical efficiency Qo.
Forsuch large sources, the efficiency is notsensitive to the wavefront curvature of the LO beam except to the extent that it modifiesthe LO trans- missionthrough the antenna pupil. For example, if one considers two systems, projectingthe same gaussianspot size in theantenna plane of Figure 4 but having two different radii ofcurvature for the LO phasefronts, the fractional transmission and hence thereceiver efficiency will be the same.The system with thewider backpropagated LO divergence will detect point sources near the optic axis with less sensitivity but this will becompensated for by the
47 5 detection of additional point sources which are beyond the field ofview of the receiverwith the smaller backpropagated LO divergence. On theother hand, if the source is of limited spatial extent, maximum detection efficiency dictates that the backpropagated LO be contained totally within the source pupil function and hence LO phasefrontcurvature effects will play a more importantrole. For small thermalsources in the near field ofthe receiver, as in a laboratory experiment, this canbe accomplished by choosing an optical systemwhich effec- tivelyfocuses the backpropagated LO ontothe target source and providesnear- unitytransmission efficiency for the backpropagated LO.
8. INCOHERENT BACKSCATTER LIDAR
Considerthe lidar system in Figure 6. An outgoingpulse of temporal width 6 is transmittedthrough the atmosphere illuminating the aerosol scatterers in its path. The mixer current at time t is due toradiation scattered at a time t - R/c from a volume defined by thelength c6/2 within the receiver field ofview asdetermined by the backpropagatedeffective LO intensity. Although theaerosol scatterers are randomlyspaced and typically many wavelengthsapart, the return is notstrictly incoherent since the scatterers within the volume of interest are "frozen" in their positions during thepassage of a shortlaser pulse, thereby producing a coherentor "speckle1' component in thereturn. Thus, based on a single return, onecannot perform thelong time average necessary to progress from Equation (4.5) to (4.6) in our derivationof the mean squaremixing current
With theadditional argumentgiven above, we can applyEquation (6.4) to thepulsed backscatter lidar problem. The sourceintensity function Is which is now a functionof range (Z coordinate)as well as the transverse coordinates, is given by
+ where IT(R,rS) is the intensity of thecoherent transmitter beam atthe -+ range R and transversecoordinate rs, dO(T)/dR is thedifferential scatter- ingcross section in the-+ backward direction, c6/2 is thelength of the scattering volume, p(R,rS) is thedensity distribution of scatterers, and p is a factorof order unity or less which takesinto account depolarization effects. The product[da(r)/dR] IT(R,?s) is the power scattered in the back- ward direction per steradian by a single scatterer located at the coordinates
47 6 -+ (R,rS) whilethe product p (R,;s) (c6/2) is the number of scatterers perunit cross-sectional area inthe source volume. SubstitutingEquation (8.1) into (6.4) gives
which yields the important result that the mean square signal current is pro- portional to the overlap integral of three quantities - the coherent transmitter intensity, the backpropagated effective LO intensity, and the density distribu- tionof scatterers. It is usefulto note that we have not made theassumption that the transmitted and local oscillator beams are coaxial in deriving Equa- tion (8.2).In fact, the equation canbe used forbistatic lidar systems pro- vided the transmitter and receiver optical axes are nearly parallel andan appropriate offset between transmitter and LO beams is includedbefore computing theintegral. If thetransverse separation between transmitter and receiver is smallrelative to the spot sizes of thetransmitter and BPELO at the range R, the bistatic system can be treated as coaxial to a good approximation.
As a simplenumerical example, we now considerthe case of gaussiantrans- mitter and local oscillator beams described by
and
where PT and PL arethe transmitter and localoscillator output powers and uT(R) and uL(R) arethe corresponding guassian radii at the range R. Sub- stitution ofEquations (8.3) and (8.4) into (8.2) yields
where we have assumed a uniformscattering density p(R) and a uniformmixer efficiency ne. Clearly, < iM2> increaseswith decreasing uT and uL imply- ing that the signal level will be maximized in a laboratory scattering experi- ment by focusingthe transmitter andbackpropagated LO into the sample.
If the scattering volume in the lidar systemof Figure 6 lies in the far field of the transmitter and LO beam waists, we canuse the approximations
477 which exhibitsthe familiar R-2 dependencefor the lidar equation. Equa- tions(8.3) and (8.4) suggest the definition of an effective area forthe 2 2 gaussian beam waists given by AT = TUTO /2 and AL = noLo /2. Furtherdefin- ingan average antenna area A = (AT + AL)/2 and letting AL = EA and + = (2 - €)x, Equation(8.6) becomes
whichhas a maximum for & = 1 given by
Thus, we havedemonstrated that, if we constrainthe sum ofthe transmitter and receiver areas tothe value 2A, we obtain a maximum signal when E = 1 or AL = AT, i.e., when theantenna areas are matched. To includeoptical and atmospherictransmission losses, Equation (8.3) should be multiplied by qAqT0 andEquation (8.4) by rlAqRo where n, is theatmospheric transmission for therange R and qTO and 'lRo are theefficiencies of the transmitter and receiveroptical systems.
It shouldbe clear that, just as in the case ofthermal source detection, any LO power falling on the mixer that cannot be backpropagatedthrough the receiveroptics to the source will contribute to the shot noise but not to the signalcurrent and therefore represents a reductionin system signal-to-noise. Thus,vignetting, central obscurations, and phasefront errors can have a major impacton the lidar efficiency by (1) reducingthe transmission of the back propagated LO and (2) influencingthe antenna pattern of the backpropagated effective LO inEquation (8.2). The antennapatterns of vignetted, centrally obscured,and decollimated gaussian beams havebeen computed by Klein and Degnan (ref. 16).
478 9. COHERENT SOURCE DETECTION
For a spatiallycoherent source such as a laser or distant star, we can write €or the mutual intensity function at the mixer
where ES and @s are real functionswhich describe the signal amplitude dis- tribution andphasefront in the mixer plane. A similar expressioncan be written€or the laser LO. SubstitutingEquation (9.1) and the LO equivalent intoour general expression €or
In the trivial case where themixer efficiency and thesignal and LO beams are uniformover the mixer of area AD, Equation(9.2) reduces to the familiar form
where Ps = E~~A~.Inthe most general case, we canuse Equation (2.8) to define a coherentheterodyne efficiency given by
479 plane is containedin the central lobe of the signal Airy pattern. Degnan and Klein (ref. 18) considered several illuminationprofiles for the LO including uniform, gaussian, and an Airy pattern matched to the signal Airy pattern. Theirresults are summarized inFigure 7. Optimum detectionefficiency is achieved when the mixer captures the entire signal Airy pattern and a matched Lo is used.In this instance, the receiver efficiency is simply 1 - y2 (where y is theobscuration ratio defined previously for the Cassegrain antennain Figure 4) correspondingto the areal obscurationloss andrepre- sentedby the "matched" LO curvein Figure 7. The differencebetween the ideal or "matched" LO curveand the uniform or gaussian curves corresponds to the heterodynedetection efficiency oHET.
If the mixer is illuminated by a uniform LO, the optimum Airy disk radius (tothe first null) is found to be RA 2 1.35RDwhere RD is themixer radius. It should be noted that the Airy disk radius varies with the obscuration ratio foran optical antenna havirlg a given f-number (ref. 18). The optimum effi- ciency omT is approximately 83% for no obscurationand falls rapidly as the obscuration ratio is increasedeven if one chooses an optimum signal spot size. An optimizedgaussian LO with waist radius w = 0.64RA and a centralAiry signaldisk which matches the mixer radius RD yieldsgreater sensitivity com- paredto the uniform LO since it more closely matches the intensity distribution ofthe central Airy disk for the signal. Thepower containedin the outer rings ofthe Airy pattern is lost, however,and thisaccounts for the major difference between the"ideal" matched LO andgaussian LO curvesin Figure 7. For a more detailed discussion, and for more generalplots of non-optimized geometries, thereader is referred to theoriginal paper by Degnan andKlein (ref. 18).
It is a simple matter to compute theeffects of misalignment between the signal and LO beams or of a mismatchbetween phasefront curvatures using the generalexpression (9.5). For example, if the two wavefronts are misaligned by anangle 8 inthe yD directionillustrated in Figure 2, theexponential argument inEquation (9.5) is
-t -b where kS and k, are thepropagation vectors for the signal and LO beams, lkSl 2 lkLl =: k = 2~rr/X, and yD is the y-component ofthe vector rD. For cylindrically symmetric fields,Equation (9.5) reduces to a specialcase previouslyderived by Cohen (ref. 19); i.e.,
(9.6) [LrodrD rD oQ(rD)EL2 (rD)] [lmdrD rD EL2(rD)3
48 0 where r is the radius of the mixer, and we have used yD = rD cos @D and 0 the integral expression €or the Bessel function Jo(z),i.e.,
Cohen (ref. 19) has generated plotsof qHET for a variety of source-LO illumination function combinations suchas uniform-uniform, Airy-uniform, matched Airy-Airy, uniform-gaussian, and Airy-gaussian. He considered the tolerance of the various combinations to misalignment and allowed for a quad- ratically varying mixer quantum efficiency.The sensitivity to misalignment for the various combinations varied less than 15% relative to the most sensitive uniform-uniform case givenby
(kr, sin 8) kro sin 8 I’ Thus, qHET = nQ for no misalignment and oHET = nQ/2 for 8 = 0.5X/(2rO) corresponding to a half-wavelength phase difference over the mixer diameter2r0. For a wavelength of10 ym and a mixer diameterof 200 ym, the misalignment angle at which the detection efficiencyis reduced by a factor of 2 is 8 = 1.4O.
For a mismatch in phasefront curvatures, the exponential argumentin Equation (9.5) is
where Cs and CL are the curvatures of the signal and LO phasefronts at the mixer plane. For cylindrically symmetric beams, Equation (9.5) reduces to
481 For the uniform-uniformcase,
~HET (9.10)
and nHET = nQ for A($) = ($ - 6)= 0 while nHET = 0 for A - = 2 A/ro where ro is themixer radius. Thus, if thelocal oscillator beamC) has a planarphasefront (C, = ") , the signal beam phasefrontcurvature must satisfy CS >> rO2/2~.
It should be noted in closing that we have arbitrarily chosen to perform the above calculations in the mixerplane. For a particularantenna or LO geometry, it may be more convenient to perform the computation in some other optical plane as noted previously in Section 5.
10. CONCLUDING REMARKS
This article has attempted to present a unifiedapproach to the calculation ofsignal-to-noise ratios in opticalheterodyne receivers for a varietyof importantapplications. No attempthas been made togive an exhaustivereview ofthe existing literature. The referencescited are those which, in the author'sopinion, either lend themselves particularly well to the development of the general theory of optical heterodyne receivers given here or have pre- sentednumerical results having widespread application. There are, for example, various uncited articles which present calculations of signal-to-noise for very specificincoherent source or backscatter lidar geometries. Thesehave usually employed bruteforce computational methods thatgive little insightinto the generalapproach for optimizing system sensitivity. While theseprovide excellent tests of the general theory, the articles were deemed to be too specializedto be included in thepresent review.
Clearly, no attention has been paid to the effects of the atmosphere on coherent wave propagation.Although the amplitude and phasefluctuations pro- duced by theatmosphere are inherently included in the complex electric field envelopesintroduced in Section 4, no attempthas been made hereto give a quantitativeassessment of their impact. In theapproach taken here, the atmo- spherecan be viewed assimply another optical element through which thecoher- entbackpropagated effective LO must pass to reach the signal source or vice versa. In thethermal source detection and backscatterlidar problem, the atmo- spherepresumably modifies the backpropagated effective LO intensity distribu- tionthereby influencing the overlap integral in Equation (6.4). A number of papers in this area haveappeared since the early work of Fried(ref. 5) including a ratherextensive recent report by Capron et al. (ref. 20) appli- cableto coherent optical radar. 1. T.G. Blaney, Space Science Reviews,=, 691 (1975). 2. J.H. McElroy, Applied Optics,~,1619(1972). 3. M.J. Mumma, T. Kostiuk, and D. Buhl, Optical Engineering, -17,50(1978). 4. J.H. McElroy, N. McAvoy, E.H. Johnson, J.J. Degnan, F.E. Goodwin, D.M. Henderson, T. A. Nussmeier, L.S. Stokes. B.J. Peyton, and T. Flattau, Proc. IEEE,=,221(1977). 5. D.L. Fried, Proc. IEEE,55,57(1967). 6. H. Levinstein,Applied Osics,4,639( 1965). 7. T.P. McLean and E.H..Putley,RRE Journa1,=,5(1965). 8. F.R. Arams, E.W. Sard, B.J. Peyton, and F.P. Pace, IEEE JQE, h-3,11(1967). 9. R.A. Smith, Proc. IEEE,98,43(1951). 10.B.J. Rye,Applied OpticsFx, 1390( 1979). 11.D. McGuire,Optics Letters,z,73(1980). 12.M.Born and E.Wolf, "Principles of Optics", 5th Ed., Chapt. 10 (Pergamon,New York, 1975). 13.L.Mandel and E.Wolf,Rev. Mod. Phys.,37,231(1965).- 14.F.Zernike,Physika,2,791r1938). 15.A.E. Siegman,Applied Optics,z,1588(1966). 16.B.J. Klein and J.J. Degnan,Applied Optics,s,2134(1974). 17.A.E. Siegman,8'An Introduction to Lasers and Masers", Chapter 8(McGraw-Hill,New York,1971). 18.J.J. Degnan and B.J. Klein,Applied Optics,s,2397(1974); Erratum,s,2762(1974). 19.S.C. Cohen,Applied Optics,z,1953(1975). 20.B.A.Capron,R.C. Harney, and J.H. Shapiro,"Turbulence Effects on the Receiver Operating Characteristics of a Heterodyne Reception Optical Radar", Project Report TsT-33,Lincoln Laboratory. Massachusetts Institute uf Technology( 1979).
483 BANDWIDTH BANDWIDTH 'IF Bo
OUTPUT MIXER AMPLIFIER/ FILTER/ "IF , DETECTOR ' VQ FILTER INTEGRATOR "IF "D.C." I
YIFi us- YL 1 8,; - OPTICAL 'IF << "S 4r ANTENNA "IF << "L " " LOCAL OSCILLATOR "L' PL
Figure 1.- Block diagram of a representative heterodyne receiver.
J HUYGENS WAVELETS
Figure 2.- Huygen's wavelet model for propagation of themutual intensity function between the antenna and mixer plane.
ANTENNA PUPIL
= 'AD /XA
'A' D'
HUYGEN'S WAVELETS Figure 3.- Huygen's wavelet model for propagation of themutual intensity from an extended incoherent source. BACK PROPAGATED LO
Figure 4.- Backpropagation of a gaussian local oscillatorbeam through a Cassegrain telescope.
1 .o
0.9
0.0
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0 1 2 3
Figure 5.- Geometric receiver efficiency for a large thermal source viewed through a centrally obscured telescopeby a mixer illuminated by a gaussian local oscillatorbeam.
485 INSTANTANEOUS SIGNAL AT TIME T
Figure 6.- Functional diagram of a heterodyne incoherent backscatter lidar system.
OPTIMIZEDGAUSSIAN
LO (RD == RA, (U = 0.64 Rp
-0 LL
a: w SIGNALSPOT (RA z= 1.35 Ro) z - -3.0 iU I Y i bfa 3 RA : AIRY SPOTRADIUS 2 + RD DETECTORfMIXER RADIUS a 0 -4.0 - w GAUSSAN LO RADIUS
0.0 0.1 0.2 0.3 0.4 0.5
OBSCURATIONRATIO, y
Figure 7.- Maximum receiver efficiency factors in dB for detection of a distant point source by a heterodyne receiver consisting of a general centrally obscured telescope (primary radiusa, secondary radius b) asa function of linear obscuration ratioy = b/a and several optimized LO distributions (uniform, gaussian, and matched Airy) .
486