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Coherent Receivers Principles Downconversion Heterodyne receivers mix signals of different frequency; if two such signals are added together, they beat against each other. The resulting signal contains frequencies only from the original two signals, but its amplitude is modulated at the difference, or beat, frequency; this downconverted beat signal is used for the detection. Unlike incoherent detectors, this signal encodes the spectrum of the incoming signal over a range of input frequencies and also retains information about the phase of the incoming wavefront. This “extra” information allows efficient spectral multiplexing (many spectral elements observed simultaneously with a single receiver) and very flexible use of arrays of telescopes and receivers for interferometry. For heterodyne operation, the mixed field must be passed through a nonlinear circuit element or mixer that converts power from the original frequencies to the beat frequency. In the submillimeter- and millimeter- wave (and radio) region this element is a diode or other nonlinear electrical circuit component. For visible and infrared operation, the nonlinear element is a photon detector, sometimes also termed a photomixer.

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Page 1: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

Coherent Receivers• Principles

DownconversionHeterodyne receivers mix signals of different frequency; if two suchsignals are added together, they beat against each other. The resultingsignal contains frequencies only from the original two signals, but itsamplitude is modulated at the difference, or beat, frequency; thisdownconverted beat signal is used for the detection. Unlike incoherentdetectors, this signal encodes the spectrum of the incoming signal over arange of input frequencies and also retains information about the phase ofthe incoming wavefront. This “extra” information allows efficient spectralmultiplexing (many spectral elements observed simultaneously with asingle receiver) and very flexible use of arrays of telescopes and receiversfor interferometry.

For heterodyne operation, the mixed field must be passed through anonlinear circuit element or mixer that converts power from the originalfrequencies to the beat frequency. In the submillimeter- and millimeter-wave (and radio) region this element is a diode or other nonlinearelectrical circuit component. For visible and infrared operation, thenonlinear element is a photon detector, sometimes also termed aphotomixer.

Page 2: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

If the mixer has a linear I-V curve, then the conversion efficiency is zero. Similarly, any mixer having a characteristic curve that is an odd function ofvoltage around the origin will have zero conversion efficiency if operated atzero bias. Much greater efficiency is achieved, however, with acharacteristic curve that is an even function of voltage; if I ∝ V 2, the outputcurrent is proportional to the square of the signal amplitude. I ∝ V 2 ∝ ξ2 ∝P, where ξ is the strength of the electric field; hence, we prefer to usesquare law devices as fundamental mixers because their output is linearwith input power.

Assume that we are using a mixer illuminated with a mixture of twosources of power, one a signal at frequency ωS and the other at ωLO. Letthe power at ωLO originate from the local oscillator (LO) within theinstrument. We also specify that ωS > ωLO. Then the mixed signal will beamplitude modulated at the intermediate frequency ωIF = |ωS - ωLO|. Thesignal at ωIF contains spectral and phase information about the signal atωS. The signal has been downconverted to a much lower frequency thanωS or ωLO, in the frequency range where it can be processed by

Page 3: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

conventional electronics. SidebandsThere is no way of telling in the mixed signal whether ωS > ωLO or ωLO >ωS. Because we have lost the initial information regarding the relativevalues of ωS and ωLO, many of the derivations of receiver performance willassume that the input signal contains two components of equal strength,one above and the other below the LO frequency ωLO. Since the signal atωIF can arise from a combination of true inputs at ωLO + ωIF and ωLO - ωIF,it is referred to as a double sideband signal. When observing continuumsources, the ambiguity in the frequency of the input signal is a minorinconvenience. When observing spectral lines, the image frequencysignal at the off-line sideband results in complications.

• Heterodyne Receivers MixerAt the highest frequencies at which heterodyne receivers are used, acontinuous wave (CW) laser is used as the local oscillator. The laser lightand the signal are combined by a beam splitter, sometimes called adiplexer. The output is mixed in a photon detector -- a photoconductor,photodiode, photomultiplier, or bolometer. Because a photomixerresponds to power, or field strength squared, it is a square law mixer bydefinition. However, such heterodyne receivers are uncommon. Thetechnique comes into its own in the submm and at lower frequencies,where the mixer is a specialized electrical element onto which the energy

Page 4: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

is concentrated by antennae, waveguides, and other non-focussing optics.

The picture above is a SIS mixer block. The local oscillator and the inputsignal are coupled into the mixers by the twin slot antenna. The IF is takenout to the right, while the volume to the left helps tune the response forefficiency (but at a specific frequency). SIS stands for "superconductor-insulator-superconductor. A sandwich of these materials produces the"diode or other nonlinear device" described above as the heart of a mixer.Its operation is illustrated below.

(a) shows the device without a bias voltage. The superconductors areshown in the band diagram approximation; the band gap is a few meV. Abias has been applied in (b). When it becomes big enough to align the"valence" band on the left with the "conduction" one on the right, suddenlyCooper pairs tunnel through the insulator efficiently, causing a current.See the plot in (c) for the overall behavior. Because the inflection in thebias curve is so sharp at 2∆/q, relatively little local oscillator power isneeded to get a good IF signal. However, the tuning required for goodsignal coupling generally restricts the operation to a relatively narrowspectral range (10-20%).

The full receiver is shown below:

Page 5: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

Using the photodetector case as an example, the mixer need not respondfast enough to track the frequency of the two input signals, so thosesignals just produce constant photocurrents. In addition, the photocurrentcontains a component oscillating at the intermediate frequency, ωIF = |ωS -ωL|. The IF current is the heterodyne signal and has a mean-square-amplitude of

where IL is the current in the detector from the LO signal and IS is that fromthe source.

It is important to note that the signal strength in equation (1) depends onthe LO power. As a result, many forms of noise can be overcome byincreasing the output of the local oscillator. The ability to provide anincrease in power while downconverting the input signal frequency ischaracteristic of quantum mixers, such as the photomixers discussedhere. The conversion gain is defined as the IF output power that can bedelivered by the mixer to the next stage of electronics divided by the inputsignal power. Classical mixers do the downconversion without gain - butthe use of very low noise electronics in the GHz range of the IF signalmakes it useful to carry out this operation even without gain.

Post mixer ElectronicsThe heterodyne signal derived above is a low level, high frequency ACcurrent; it needs to be amplified and converted into a slowly varying

)1(,22SILIIFI =⟩⟨

Page 6: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

voltage that is proportional to the time-averaged input signal power. Thefirst step in this process is amplification. We want to preserve as high an IFfrequency range as possible, since the spectral band over which thereceiver works is just 2 ∆fIF (assuming we use it double sideband).

The best performance for the IF amplifier is obtained with high electronmobility transistors (HEMTs) built on GaAs. The HEMT is based on themetal- semiconductor field effect transistor (MESFET). It consists of asubstrate of GaAs with an n-doped layer grown on it to form the channel,with contacts for the source and drain and a gate formed as a Schottkydiode between them on this layer. The electron flow between source anddrain in this channel is regulated by the reverse bias on the gate; as withthe JFET, with an adequately large reverse bias the depletion regiongrows to the semi-insulating layer and pinches off the current. Becausethis structure is very simple, MESFETs can be made extremely small,which reduces the electron transit time between the source and drain andincreases the response speed.

Conduction band Conduction band

GaAlAs GaAs

Valence bandValence band

Page 7: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

In the HEMT, the MESFET performance is further improved by using aheterojunction (junction between two different semiconductors withdifferent bandgaps) so the electrons flow in undoped GaAs.

To achieve this result, the MESFET is grown on heavily doped GaAlAs,whose Fermi level lies above the bottom of the conduction band in theundoped GaAs layer. Thus, the conduction electrons collect in the GaAsand flow through it under the influence of the MESFET fields. The veryhigh mobility in undoped GaAs makes for very fast response, to ~ 1011 Hz.

Page 8: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

Detector StageThe conversion to a slowly varying output can be done by a detector stagethat rectifies the signal and sends it through a low-pass filter. We wouldlike the circuit to act as a square law detector because <IIF2> isproportional to IS (see equation (1)), which in turn is proportional to thepower in the incoming signal.

To demonstrate how the detector stage achieves this goal, we solve thediode equation for voltage and expand in I/I0:

)2(.......4

041

3

031

2

021

001ln

+−+−≈+=

II

II

II

II

qkT

II

qkTV

The first and third terms in the expansion will have zero or smallconversion efficiency, and the 4th and higher terms will be small if I << I0.Thus, the detector stage does act as a square law device.

Sometimes it is desirable to carry out a variety of operations with the IFsignal itself before smoothing it. For example, imagine that theheterodyne signal is sent to a bank of narrow bandpass electronic filterswith a smoothing circuit on the output of each filter. The frequenciespresent at the input to the filter bank are limited to the bandwidth of the IFamplifier or of the photomixer (typically ~1GHz). The filters can be tunedto divide this IF signal into components at a range of frequencies over theIF stage bandpass. Because the intermediate frequency goes as |ωS -ωL|, the frequencies in the heterodyne signal correspond to a similar rangeof frequencies in the source centered on the signal and imagefrequencies. The filter bank therefore provides the spectrum of the sourceunder study.

Local Oscillator

Page 9: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

At the high frequencies of the infrared and visible regions, the only localoscillator with reasonably high power output is a continuous wave laser. Inthe submm and lower frequencies, tunable LOs are available - usually bystarting with a lower frequency oscillator and multiplying it up to theoperating frequency. In any case LO power is a critical asset.

• Fundamental limits BandwidthThe spectral bandwidth of a heterodyne receiver is determined by theachievable bandwidth at the intermediate frequency.

ThroughputThe signal photons cannot be concentrated onto the mixer in a parallelbeam; even for a point source, they will strike it over a range of angles. The requirement that interference occurs at the mixer between the laserand the signal photons sets a requirement on the useful range ofacceptance angle for the heterodyne receiver. This condition can beexpressed as

where D is the diameter of the telescope aperture and Φ is the angulardiameter of the field of view on the sky. Thus, a coherent receiver mustoperate at the diffraction limit of the telescope.

)3(,D

λ≈Φ

Page 10: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

A second restriction is that the interference that produces a heterodynesignal only occurs for components of the source photon electric fieldvector that are parallel to the electric field vector of the laser power; i.e.,only a single polarization of the source emission produces any signal. Therelation between wavelength and etendue and the constraints onpolarization are manifestations of the antenna theorem, applicable to allheterodyne receivers.

Signal to Noise and Detection LimitsThe detection limits of a heterodyne receiver are considered differentlythan for incoherent detectors. We need to distinguish noise: (1) that isindependent of the LO-generated current, IL, and (2) that depends on IL. In principle, the first category can be eliminated by using a local oscillatorwith sufficient power to raise the signal strength out of the noise. Thus,the second category alone contains the fundamental noise limits. Twotypes of fundamental noise are: (1) quantum noise in the mixer from thegeneration of charge carriers by the LO power; and (2) noise from thermalbackground detected by the system. The division between the tworegimes can often be simplified by stating that for hν >> kTB, the quantumlimit holds, while for hν << kTB we get the thermal limit. Here, TB is thetemperature associated with the background power.

Noise temperatureTo describe the performance on continuum sources, a thermal source isintroduced through a noise temperature, TN, defined such that a matchedblackbody at the receiver input at a temperature TN produces S/N = 1. The lower TN, the fainter a source gives S/N = 1, and the better is theperformance of the receiver (so the behavior is again like NEP – thesmaller, the better).

In the thermal limit, if the effective source emissivity ε = 1, then bydefinition TN = TB. For the ideal double sideband case, the quantum limit is

The quantum limit expressed in equation (4) can be justified in terms ofthe Heisenberg uncertainty principle, which states that the uncertainty, ∆P,in a measurement of power will be

. )4(. k

h ν≈NT

Page 11: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

where ∆t is the observation time. Starting with the equation for Johnsonnoise in a resistor within a frequency bandwidth df and converting topower noise within a time interval ∆t,

Setting ∆P ≈ <P>, we obtain

It is often convenient to express the flux from a source as an antennatemperature, TS, in analogy with the noise temperatures. This concept isparticularly useful in the millimeter and submillimeter (and radio) regions,where the observations are virtually always at frequencies that are in theRayleigh Jeans regime (hv << kT). In this case, the antenna temperatureis linearly related to the input flux density:

where ∆ν is the frequency bandwidth. To maintain the simple formalism interms of noise and antenna temperatures, it is conventional to use aRayleigh Jeans equivalent temperature such that equation (7) holds bydefinition whether the RayleighJeans approximation is valid or not.

The achievable signal-to-noise ratio for a coherent receiver is given interms of antenna and system noise temperatures by the Dicke radiometerequation:

where ∆t is the integration time of the observation and K is a constant oforder one.

)5(,t h = P∆

∆ ν

)6(.NT k = t >P< ∆

. k

h ν=NT

)7(,Tk 2 = SPSν∆

)8(, ) 2/1 t f( T

T =

c∆∆Κ

IFS

N

S NS

Page 12: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

Comparisons with incoherent detectorsEquations (7) and (8) give us the means to compare the performance ofcoherent and incoherent detectors, as long as we also keep in mind thebandwidth and single mode detection restrictions that we have alreadydiscussed. From equation (7) and the definition of NEP, the signal-to-noise ratio with an incoherent detector system operating at the diffractionlimit is

Therefore, using equation (8), we obtain the ratios of signal to noiseachievable with the two types of system under the same measurementconditions:

Suppose a bolometer is operating background limited and we compareits signal to noise on a continuum source with a heterodyne receiveroperating at the quantum limit. We set the bolometer field of view at thediffraction limit, AΩ = λ2 and assume that the background is in theRayleigh Jeans regime (e.g., thermal background at 270K observednear 1mm). The background limited NEP is:

The photon incidence rate, ϕ, can be shown to be

If we assume the bolometer is operated at 25% spectral bandwidth, ∆ν= 0.25ν, then

We have taken the IF bandwidth to be 3 x 109 Hz, a typical value.

)9(..NEP

) 2/1t( Tk 2 =

i

∆∆

νS NS

)10(. k 2

) 2/1 f( NEP =

)i (S/Nc

ν∆

SNT

IF) (S/N

)11(.2/12

= ηϕ

λhcNEP

)12(.2

ν

νηϕ

hBkT ∆

=

( )( )

)13(.11106.2~2/1)4(

6104.2~//

νν×∆×

IFfiNScNS

Page 13: Coherent Receivers Principles - University of Arizonaircamera.as.arizona.edu/Astr_518/hetdyne.pdf · Coherent Receivers • Principles ¾ Downconversion Heterodyne receivers mix signals

Thus, the incoherent detector becomes more sensitive near 2.6 x 1011Hz,or at a wavelength just longer than 1mm. Actually, this comparison isslightly unfair to it (since, for example, it does not have to work at thediffraction limit), so it is the detector of choice for continuum detections towavelengths of 2 to 3mm. Hence, the development of large scalebolometer cameras for mm-wave and submm telescopes.

Of course, the coherent detectors are preferred for high resolutionspectroscopy and for interferometry.