Heterodyne Detection with Superconducting Tunnel Diodes - CaltechTHESIS

results of any type of power measurement accurately. For this case, our theory is a complete theory for

heterodyne measurement. For larger output frequencies, the details of the output measuring system must
be considered.
If ω0 is small, we can also set J⅞- to zero. Then the υo- term in (2.4.10) does not contribute to

the mixer output power, so (2.4.11) becomes

F,0+ = G⅛B

Zomzζm, (∏Γjmm∙ + HEmm1}

for Po- = θ-

(2.4.18)

mml

Most of Tucker’s expressions are derived under these assumptions. Tucker’s mixer noise expressions
involve sums like (2.4.18) over an H matrix which is equivalent to Hd. As we have defined He in
(2.4.15), it accounts for mixer response to signal power, but as we show below, it also includes an

additional noise term which completes tunnel junction mixer noise theory.
Eqn. (2.4.10) is close to the formalism used by Caves (1982) for consideration of lower limits on

noise in amplifiers. Caves’ eqn. (3.5) relates the output signal photon operator to input signal photon

operators and a noise operator Hq through

αo+

= Σ (*⅛'mα∣ωτnJ- ■ Lom α,

.) + Ho

(2.4.19)

m∈5

where S is the set of sidebands which carry signal. This is equivalent to (2.4.10) if

i√⅛⅛Fr!m

fc"->0

Λ⅜n =
otherwise
f√½⅛

for ωm < 0
(2.4.20)

otherwise
^∑√⅛e¾^÷∑
τn

+ ∙kθmα∣ωm∣-)

τn%S

The Γgm for positive ωm correspond to Caves’ phase-preserving gain terms Mom, while negative ωm
produce phase-conjugated gain Lom- Caves refers to the modes which are summed over to produce

77o as internal modes. As he points out just before his conclusion, sideband modes which do not carry

signal must be included with the noise-adding internal modes of the device.

40
Since our formalism is equivalent to Caves’, it is consistent with the quantum limits he has derived

for linear amplifiers. Although we are not going to pursue the interesting field of quantum non-demolition
measurements in this paper, it is clear that (2.4.10) can be used to handle correlated photon input states

(squeezed states) to heterodyne mixers.

2.5 Mixer Gain and Noise
In a classical theory, voltages can be measured instantaneously and exactly, so classical receiver

theories use voltage and signal interchangeably. Quantum mechanically, the measurement of voltage is
distinguished from voltage itself, so signal must be identified with the expectation value of the voltage
waveform. The simplest use of a mixer is to determine the input voltage phasor in one sideband by

measuring the output voltage phasor. For convenience we choose the m = 1 sideband to carry the signal.
The input and output signals are

½n = {vr-}
(2.5.1)
VθUT = (t>0+) = Γoι½N

where Γ0ι in (2.4.10) is now identified as the voltage gain of the mixer. It is common to express mixer
performance in terms of powers. The signal powers are
Ffrj = ⅛G,ι ∣½n∣2

(2.5.2)
Poor = i1G⅛ ∣Vbur∣2

and they are linearly related by the conversion gain of the mixer (2.4.17)

Pout = (7oiPin∙

(2.5.3)

Noise in the voltage measurement is characterized by the expected mean square deviation of a set of

voltage measurements from the true expectation value. The voltage phasor operator is not an observable
operator, so noise is calculated in the time domain using (2.2.13),
ι⅛√(‰(i)-i⅛(i)}]2)

= I (([‰+'υi+]+) - ‰M,

Pn = GqV⅛

(2.5.4)

41
The minimum detectable signal power is defined to be the signal power which τesults in an output signal
to noise ratio (SNR) of unity

⅛t = P1v∕‰

(2-5.5)

The minimum detectable power is generally proportional to the bandwidth B, so the noise figure of merit

for a mixer is a detectable noise power per bandwidth

Em = Pdet/B.

(2.5.6)

Traditionally, this is presented in units of temperature

Tm = EM/kB

(2-5.7)

but it can also be expressed in units of photons

Nm - ΕΜ/Κωι.

(2.5.8)

We now work out the noise for this measurement. Assume that the signal state ∣jS1-) is an

eigenstate of v‡_ with eigenvalue 1⅛. Since vf_ is proportional to ai_, this eigenstate is a Glauber
coherent state. This state carries power (2.2.13)
Pi_ = ≤i(‰∣ [<,i1.]+∣‰)

= ⅝ (¼N∣ vi-«L ∣½n) +fiω1B∕2

(2.5.9)

= ∙γ ∣½N∣2 +Λω1B∕2
= 7⅛t +fiωιB∕2
where (2.2.11) was used to expand the anti-commutator. Comparison with (2.2.13) shows that photons

counted by the number operator

account exactly for signal power in this state, but power flux exceeds

signal power by an unavoidable half photon which we attribute to the vacuum. To calculate minimum
mixer noise, assume that all other incoming radiation states are in their vacuum states. Subtracting

output signal power (2.5.3) from the total output power (2.4.16), we have the output noise power

Pn = G⅛B

Zom'
mm,

m m, + B

Qom - hω m ■

(2.5.10)

42

The first term above is the noise term which Tucker derives for tunnel diode mixers. The second term is

due to quantization of the external circuit, and it represents the minimum noise possible for the signal in
one sideband only, since (nm~} > 0. For this mixer, the vacuum half photon at every sideband appears

to be frequency converted into noise power at the mixer output.

The non-signal sidebands may be in thermal equilibrium photon states. The mth sideband photon

state in thermal equilibrium at temperature Tm has
)71m = θ
(2.5.11)
+ 2)τm ” 5 COth(7i⅛∕2⅛2jTm)

(Robinson, 1974) which has limiting forms
{τim- ÷ Ï/ — ) t∏Tπ
v fi)ωm∣

for Tm = 0;

fθr k∕}Tm i>∕i ]u,m∣.

(2.5.12)

These limits are the vacuum state and the Rayleigh-Jeans limit, respectively. The noise in (2.5.10)

represents the case that all unused sidebands are terminated at 0 K. We can include the effect of finite
temperature termination of non-signal sidebands by replacing the ∣ in the second term on the right of

(2.5.10) with the second line in (2.5.11).
Tucker suggests in his eqn. (7.8) that the Planck formula,
'ιτ~

exp^ωm∕⅛Tm)-1’

(2'5'13)

be used to describe thermal noise. This is the traditional formula for the available photons, it does
not include the unavailable one-half photon in (2.5.11) that is attributed to the vacuum. This relation

between the Planck formula and (2.5.11) serves to emphasize the identification of the second term on
the right of (2.5.10) as the quantum noise term. Amplifier gain and noise measurements are often made

using black bodies at various known temperatures as calibrated power sources. Since one-half photon of
(2.5.11) must always be identified with unavoidable noise, the Planck formula should be used to describe
thermal “signals”.

2.6 Photodiode Noise in SIS Mixers
We have developed the theory above very generally so it can serve as a framework for a quantum

mixer theory of any device. The inclusion of the external circuit noise from He is not device dependent

43
at all, nor is the fact that the Y and Z matrices relate voltage and current operators in the external
circuit. However, our formalism has been developed so that it will fit with the formalism developed by

Tucker (1979) for SIS diodes, and complete the noise theory of these devices. We now show that this

noise theory applied to perfect SIS mixers reduces to photodiode mixer theory.
In theory, a perfect SIS, dc biased just below its turn-on voltage Vg, has the responsivity of a

perfect photodiode (Tucker, 1979). The SIS photo-current responsivity is e∕hv, i.e., one electron joins
the photo-current for each photon that is absorbed. It might be expected that the noise properties of

such an SIS would be identical to the noise properties of a photodiode. Theories for the noise in mixers

based on ideal photodiodes are described in textbooks by Kingston (1978) and Marcuse (1980).
For a perfect SIS operated at low LO powers, the expressions that Tucker gives for the Y and ¾>

matrices can be simplified, see for example Tucker and Feldman, (1985) and Feldman, (1986). The
mixer noise (2.5.8) calculated for such an SIS, including the noise term He introduced in the previous
section, is
Nm = ∖∣η

(2.6.1)

where η has the form of a power coupling coefficient between two admittances,

4∣Ylo + JΛ ∣2

(2.6.2)

Ylo is the admittance presented by the SIS to the sinusoidal LO,

Ylo =

J-lo

(2.6.3)

7lo can be calculated from Tucker’s eqn. (5.2). ¾ is the signal source admittance. Guo and Gι are the
real parts of J⅛o and ¾ respectively.

Note that the optimum source admittance is the admittance which the device presents to the LO and
not the admittance it presents at the signal frequency. The load admittance which the SIS mixer presents

to the input radiation at m = 1 can be calculated from the Y matrix and the terminating admittances ym.

In general, that admittance is not equal to 1>'lo, which is completely independent of all ym. R. McGrath

has suggested to us that the m = 1 mixer load admittance can be derived from an active circuit with a

44
feedback loop. T’lû is apparently the passive admittance in this circuit. It is therefore this admittance
to which Callen and Welton’s (1951) fluctuation-dissipation theorem applies, and this admittance which
the external circuit must match to minimize measurement noise.

This equation for mixer temperature is independent of image termination, JLι.
optimum mixer with Nm = 1 at ¾ = y£o·

Consider the

the image is also optimally coupled to the mixer, then

>’_i = 37lo and all of the mixer noise is algebraically due to the vacuum photon term on the right of
(2.5.10). If, however, JLj is purely reactive, then half of Nm comes from the vacuum photons, and

half comes from the device term involving Ho. We are seeing the detailed workings of the fluctuationdissipation theorem. When the y_i is well coupled to the mixer, its dissipation enforces the quantum

limit, but decoupling J¼ from the mixer does not lower the quantum limit, instead dissipation in the
device provides the necessary noise.

Equation (2.6.1) is the same as that obtained for a perfect photodiode. For photodiode mixers,
η is defined as the fraction of radiation incident on the front of the photodiode which manages to get

absorbed in the photodiode. Eqn. (2.6.2) is the circuit analog for this quantity. When (2.6.1) is derived
for a photodiode mixer, it is calculated from the statistics associated with LO photon absorption. The
fact that we derive this same amount of noise for an ideal SIS mixer leads us to conclude that, as in

a photodiode mixer, noise in an ideal SIS mixer can ultimately be identified with the statistics of the

photon absorption process. In fact, Devyatov et al., (1986) provide expressions for SIS current which
contain terms proportional to ata and ααt. We conclude that not only is an ideal SIS tunnel junction
like a perfect photodiode in its response, producing one electron in its output current for each photon

absorbed as Tucker has shown, but that it is like a perfect photodiode in its noise properties as well.
Real SIS mixers are not operated at low LO powers. At a higher LO power, SIS response to the LO
saturates; not all absorbed photons result in the conduction of electrons across the device. While (2.6.1)
is no longer exact for this case, we show that it is still a good approximation. Computer calculations
of the mixer noise have been done for a nearly ideal SIS with the realistic values ‰> = ∕L>lo = e½j

and a variety of conditions on the image termination y_ 1. This value of LO voltage represents more

absorbed LO power than we typically find necessary in our actual mixers (Wengler et al., 1985; Woody,

45

-+->

23
CL

co
∙÷-'
-C
CL

<υ

G1 X 50 Ω
Figure 2.2 Mixer noise Nm vs. real source admittance Gι for a nearly ideal SIS with large
LO power. The a) dashed, b) dotted and c) dash-dot lines are for image termination equal to
a) signal termination, b) open circuit, and c) short circuit. The solid line is a plot of (2.6.1)
for comparison. The x-axis is the zero reactance slice through a standard Smith chart.
Miller and Wengler, 1985). These results are shown in fig. 2.2, along with a calculation of (2.6.1) for
comparison. The image termination JG1 slightly affects mixer noise for this high LO power, but simple

photodiode theory still gives a good approximation to the mixer noise.
In this section, we have shown that the full noise theory for SIS mixers developed in previous

sections is consistent with the conceptually simpler theory for noise in photodiode mixers. Since noise

in photodiode mixers is completely derived from the statistics of the photon absorption process, we

conclude that this is true for the ideal SIS mixer as well, even though the complex algebra of the full
theory seems to obscure this relation. Finally, we have shown by calculations from the full noise theory,

that photodiode theory describes, to reasonable accuracy, the noise in a good quality SIS mixer at
realistic LO power levels. Of course, for poorer quality tunnel devices, noise sources other than photon

noise are important, for these (2.5.10) from the full quantum mixer theory must be used to predict noise.

2.7 Summary
We have produced a fully quantum mixer theory which includes all noise effects in a heterodyne
measurement. For the standard use of a mixer, we present expressions for mixer gain and noise. These

46

are derived from the expectation value of an output voltage operator and its square. Photodiode mixer
theory is shown to be sufficient for describing noise in high quality SIS mixers. The operator formalism

presented here can be easily extended to make predictions of SIS response to correlated photon states,
which are the subject of much current research.
Our output operator expressions are similar to Caves’ (1982), so his limits on amplifier performance
should apply to any mixer analyzed with our theory. We have shown that, theoretically, SIS mixers can

reach this limit. In fact, we have shown that for a good quality SIS, mixer noise is predicted by the
simple expressions of photodiode mixer theory, and is simply related to how well radiation is coupled

to the SIS.

47

Chapter 3 — Numerical Results from SIS Mixer Theory

“Looks an awful lot like a game of Empire to me.”
— Jeff, in Moby Mike or Captain Phillips and
the Great White Wengler, by Erich Grossman and Jeff Stem.

3.1 Introduction
Tucker’s theory for mixing in tunnel diodes (Tucker, 1979), combined with the quantum heterodyne

theory presented in chapter 2, lend themselves readily to use in computer programs. I have written
an extensive set of such programs to aid in setting engineering specifications for SIS mixers, and to
determine theoretical limits to SIS mixer performance. In this chapter, a brief outline of the information

available from these programs is presented. Other numerical results from Tucker’s theory have been

published, (Danchi and Sutton, 1986; Devyatov et al., 1986; Face, 1987; Feldman et al., 1983; Feldman

and Rudner, 1983; Hartfuß and Hitter, 1983 & 1984; Wengler and Woody in Phillips and Dolan, 1982;
Phillips and Woody, 1982; Richards and Shen, 1980; Shen and Richards, 1981; Shen et al., 1980).

Many of these are reviewed by Tucker and Feldman (1985).
The engineering parameters considered here are 1) SIS junction quality, as measured by their I-V

curves, 2) dc and LO bias 3) the admittances of the signal and image sources, yi and >_i, and the IF
load, 3⅛. The theoretical limits to SIS mixer performance are investigated by determining optimized

mixer performance as a function of signal frequency, for four different tunnel diode I-V curves.
Mixer performance parameters are discussed in section 1.5. The mixing performance parameters
presented in this chapter are mixer gain and mixer noise. Mixer input and output powers are related

through

Po = G(Ps + Pn)

(3.1.1)

where Po is the power coupled out of the mixer, Ps is the incident signal power, Pn is the mixer noise
power referred to the mixer input, and Q is the mixer gain. The gain reported in this section is the

“available” gain, that which occurs for a properly matched IF amplifier. Mixer noise is reported in terms

48

of “mixer noise photons at the input,”

Nm =

"JV

⅛ploB

(3.1.2)

where B is the bandwidth in which the mixer output signal is measured. This is related to the more

usual “mixer input noise temperature” by

r*'=⅞⅛∙

(3.1.3)

Nm, as described in chapter 3, has a quantum lower limit of unity for heterodyne detection. This
quantum lower limit in terms of mixer noise temperature is huuo∕kB-, which is 4.8 K at 100 GHz.
A complete receiver consists of a mixer followed by an IF amplification chain. Especially when

the mixer gain is small, SNR is degraded because of noise in the IF amplifiers. As for the mixer, noise

in the IF amplifier chain is characterized by its noise temperature referred to its input, 7⅛. Total noise
from the mixer and IF amplifier chain combination is also characterized by a receiver noise temperature,

Tr = Tm + G~xT↑r

(3.1.4)

From this relationship, it can be seen that high mixer gain is important insofar as 7⅛ is large. In the

following sections, G and Tm are reported. For a given 7⅛, the overall figure of merit, 7⅛, can be
estimated from (3.1.4).
Until very recently, the state of the art for 500 MHz bandwidth amplifiers in the 1-2 GHz range

was 2⅛ ~ 10 K, achieved with cooled GaAsFETs (Weinreb, Fenstermacher, and Harris, 1982). New
transistors called HEMTs give early evidence of achieving 7⅛ ~ 3 K in this same range. Use of these

amplifiers lower the importance of high mixer gain to overall receiver performance. Weinreb (1987)
discusses a radically different approach to SIS-HEMT mixer-amplifier design.

3.2 Digitized I-V and Kronig-Kramers Transform
In his theory for SIS heterodyne response, Tucker (1979) considers current flow due to tunneling
of quasiparticles, or single-electrons. Cooper pairs of electron can also tunnel, the currents due to

this process are called Josephson currents (Iosephson, 1965). For the mixer theory, the tunnel diode’s

49

electronic structure is specified in terms of its complex response function, j(x) = u(x) + iv(χ), which
is discussed in section 1.4. The portion of the device I-V which is due only to single-electron currents,
∕dc(¼>). fully specifies the imaginary part of j,

v(x) = 7dc(Vo)

(x = eVo∕ħ) .

(3.2.1)

The real part of j is related to v through a Kronig-Kramers transform (1.4.2).
For use in our programs, a measured υ(r) is treated as though it were piecewise linear between

M+1 digitized points. For all devices that we consider, v(-x) - ~v(χ) so it is only necessary to digitize
the positive x side of v. Fig. 1.3b shows a measured SIS I-V curve with some of its interesting features

noted. Most of the I-V is due to single electron currents, but the supercurrent, Iq, and the dropback

voltage, Vo, are results of the interaction of the Josephson currents and the quasiparticle currents. The
finite current Ic which flows with no dc voltage across the SIS consists of pairs. No quasiparticle
current flows if there is no dc voltage, so xo = ι>o = 0 is the first digitized point for all SIS I-Vs.

Stable bias is not possible for voltages between zero and V∏, due to chaos associated with ac Josephson
currents. The second digitized point is x↑ = eVf>∕ħ, υι = Idc(Vf>), and the quasiparticle Iic is assumed

linear between zero volts and this point. The SIS I-V asymptotically approaches an ohmic response,

jrdc ≈ GnVq, as Vo gets large. We take a number of points Vi = ⅛(⅛t), Xi ≈ eVoi∕ħ in ascending order,
until ∕dc(⅛Λi) = GnVqm to within the accuracy which we care about. We then calculate slope, b., and

intercept, ai, for the straight line connecting (xi, vi) and (zf+ι, vi+↑). For voltages above Vom , the ohmic

values b1^ =IiGn∕r and aM = 0 are used. The current at any positive x = eVo∕ħ is given by

fdc(⅛) =

α, + b{X
bM x

for xi < x < X{+↑
for x > xm∙

(3.2.2)

We refer to the collection of ai and bi as the “linefit” parameters of v(x). W& use linefit parameters to

represent other curves in our programs.
To get the full complex response function j(x) that corresponds to the measured I<=c, the Kronig-

Kramers transform of v(x) must be done numerically. From the relation (1.4.2), it can be seen that
⅛(⅛) = — Idc(-⅛) implies j(χ') = j*(-κ), and (1.4.2) can be rewritten as

(3.2.3)

50
All physical quantities in the mixer theory depend on differences u(χ↑) — u(xq), or on u'(x), so we can

arbitrarily set u(0) = 0, in which case

u(x)

_2 Γ∞
X √0

x2
i v(xr)

-dx'.
{x,i - x2^)
x'{x'

(3.2.4)

By setting u(0) = 0, we have made the integral finite when performed along the real axis, thus removing

the requirement of taking a Cauchy principal value.

To evaluate this numerically, we write the integral as a sum over the linear bits of υ. First we note
that the transform of v(x) is equal to the transform of v(x) - b>r∣ι. This allows the elimination of the

integral from xm to x = oo. The integral from 0 to xm is the sum of its bits,

, Λf-1

u(χo)=- yι a*in
∙7Γ
7r

-ti

(⅛i - xθ)xi

(χi

- xq)xIλ

,,L-f>

jin

mv

~ x>÷l×aι0 + χj)

(χo + *i+ι)(zo - χi) '

(3.2.5)

For each junction Lc, we calculate this transform at 128 points and store its linefit parameters for use

in further programs.
The first derivatives of u and v are used for finding elements of the Y matrix. We pretend that 6,is the value of v, halfway through the interval xi to aτi+ι. Linefit parameters for v' are then generated

and stored for use by other programs. The same procedure is used on the linefit parameters of u'. Our
avoidance of subtler splines for interpolation of the digitized data is made up for by our willingness to
digitize many points wherever the curve has an interesting shape.

Each Zdc we digitize can be used for making mixer predictions. Through the years, we have built up
a library of Zdc curves representing different quality junctions cooled to different physical temperatures.

We store the linefit parameters for j and j' for each junction, and so any calculation we write a program
for can be done for all these different sorts of junctions. In fig. 3.1 are shown I-V curves and their

Kronig-Kramers transforms for four tunnel diodes. Table 3.1 summarizes some of the parameters of
these junctions. The gap frequency,
MjAP =

elzGAP
h ’

(3.2.6)

is a characteristic frequency for each junction. The current scales of these junctions have been scaled

so that all four have Gn = (50 Ω)^^1. Junction 1 is a PbBi alloy SIS cooled to 2.5 K (Dynes et al.,

51

mv.

Figure 3.1 Complex response function, j, for four tunnel diodes, a) Im j is the measured I-V
of each diode, b) Re j is the numerically determined Kronig-Kramers transform of Im j.

1978). Its I-V is nearly ideal in the sense that their is virtually no current flowing below Vgap, and the
current turns on very suddenly at Vzgap∙ Junction 2 is also cooled to around 2.5 K. It is a PblnAu alloy

SIS that was used in early mixer experiments at Caltech. Junction 3 is also a PblnAu alloy SIS, cooled
to 4.5 K. This junction is the one used for the first bowtie mixer experiments (Wengler et al., 1985),
which are also discussed in chapter 4 of this thesis. Finally, junction 4 is a lead SIN tunnel junction

cooled to 1.6 K (Giaever, 1960). An SIN is a superconductor-insulator-normal metal tunnel diode. Its

I-V is much less non-linear than the SIS curves. SINs have been proposed for receivers because there
is no Joseρhson effect in them, the Josephson effect is responsible for excess noise in some SIS mixers
(Tucker and Feldman, 1985).

Mixer results for these four junctions are presented below. The three SIS ∕-Vs cover the full range

of quality for which mixing experiments are reported. Hence, the effect of I-V quality can be determined

52

Table 3.1 - Junction data for Figure 3.1

Junction

Vgap> mV

rzGAPj GHz

Physical Temperature, K

1 (SIS)
2 (SIS)
3 (SIS)
4 (SIN)

3.06
2.70
2.13
1.75

740.
653.
516.
423.

2.5
2.5
4.5
1.6

from the results below. The inclusion of the SIN curve in calculations shows what performance must

be sacrificed if these are used in mixers instead of SISs.

3.3 Assumptions for Numerical Calculations
A number of assumptions are made so that numerical calculations of mixer performance can be

carried out. These assumptions are collected here, and the conditions under which they are valid are
discussed.

For the mixer we investigate the IF or output frequency ωo is assumed small enough to be treated
classically. Tucker (1979) presents the criterion that frequencies ωo < δx can be treated classically,
where δx is the smallest a;-axis scale on which the complex response function, j(x), has structure. The

output frequencies typically used with SIS heterodyne receivers for radioastronomy are 1 to 2 GHz. All
of the I-Vs shown in fig. 3.1 are smooth on the voltage scale, about 8 μV, which corresponds to 2 GHz,

so all the calculations presented in this chapter are valid for astronomical heterodyne receivers.
For a mixer, the LO is physically due to a monochromatic power source at frequency ⅛o = wlo∕2tγ.
We assume this results in a sinusoidal voltage VLo cos ω‰oi across the device. For this voltage, the Wn

(1.4.19) are real, and
Wn=Jn(a}

forα = ^≥
nωijo

(3.3.1)

where Jn are Bessel functions of the first kind. A more general theory would allow for the possibility of

harmonic generation at the device by including harmonic components Vm cos (mωιχ√ + φm) for m > 2

in the LO waveform. These will be small in SIS millimeter wave mixers for two reasons. First, the

junction capacitance will tend to short out these high frequency harmonics. Second, as usually operated,
an SIS mixer does not have much harmonic content in its LO waveform. As will be seen below, a < 1

53

for efficient fundamental mixing in a good quality tunnel junction. Using (1.4.18) for a = 1, the second
harmonic current is about one-fifth the amplitude of the fundamental LO frequency current, ⅛ ≈ .2Ii.
Unless it is deliberately arranged to be otherwise, the circuit to which the junction is connected presents
similar admittances ∣>2∣ ~ ∣>ι ∣ at the LO fundamental and first harmonic. The harmonic voltage is then

∣½∣ ~ ∙2 ∣VLo∣. and ∣V2∣ is even smaller if junction capacitance is taken into account. If the harmonic
voltage is treated as an LO voltage, it corresponds to a a Bessel function argument α2 = e½∕β2ωLo 5,.1.
A value of α2 this small means that V2 corresponds to less than one-fifth the optimum LO voltage, or

less than 4% of the optimum LO power for mixing at 2wlo. The effects of this low harmonic content

should be negligible by comparison to effects from the LO fundamental.
All mixer calculations are done using 3 by 3 matrices for Y, Z, and H∏. Equations for these

matrices were developed by Tucker (1979), and are reviewed in section 1.6 of this thesis. The elements

used are the central nine,
ÇY,Z,HD)mm,

for m,m' = -1,0,1,

(3.3.2)

which relate voltages and currents at the signal, image and output frequencies. Matrix elements outside
these central nine describe mixing between harmonics of the LO and the higher order sidebands, ωm

(1.5.6) for ∣m∣ ≥ 2. Harmonic mixing calculations will depend strongly on harmonic content of the LO
voltage waveform. The assumption made above that there is no harmonic content is therefore consistent
with the use of only the central 3 by 3 terms from the Y, Z, and Hd matrices.
The expressions for currents, (1.4.18), and for the mixer matrix elements, (1.6.8) thru (1.6.10),

involve summations over an infinite number of terms. For the numerical calculations discussed below,
these summations are made finite by assuming

Wn = Jn(a) = 0

for ∣n∣ > 50.

(3.3.3)

The high order Bessel functions are given approximately by

Jn ≈ ~
n!

for α ≪ n.

(3.3.4)

As examples, J5o(lO) ~ 10-15 and J50(l) ~ 10^^65. The lower order Bessel functions are, by comparison,
of order unity for this range of a. Very litde is lost by terminating summations at ∣n∣ = 50.

54
The calculations presented here are made assuming the ambient temperature of the junction is 0 K

in (1.6.11), so that (1.6.12) is used for 1%. A finite temperature in (1.6.11) allows the inclusion of
Johnson noise in the predicted mixer noise. Using T = 0 K means we calculate only shot noise. The

highest temperature at which the SIS mixers we describe are operated is less than 5 K. Voltage bias

values are typically ~ 2 mV. For ⅛ = 1-5 mV and T = 5 K, the coth term in (1.6.11) is equal to 1.06,
while for T = 0 K it is equal to 1. This implies that Johnson noise accounts for at most a few percent

of the noise in SIS mixers, the rest is the shot noise, which we calculate.

Many of the calculations presented will be for the so-called double-sideband (DSB) mixer. This
refers to the condition that the external circuit presents the same admittance to the mixer at both the

signal and image frequencies. This is achieved by requiring

>1=V11.

(3.3.5)

The complex conjugation is necessary because of the definitions of ym in (2.4.3). Under this condition,
the conversion efficiencies for the signal and image are the same, <7oι = (7o-ι> hence the name double­

sideband. The physical motivation for condition (3.3.5) is that the signal and image frequencies usually
differ by only a few percent. Thus a circuit with a Q ≤ 10 will present very similar admittances at
these two frequencies. Mixers like the bowtie mixer described in chapter 4 almost certainly satisfy

(3.3.5). However, it is often the case for waveguide based mixers that different conversion efficiencies

are measured for the different sidebands. We present some calculations for non-DSB mixers which may
be relevant to these experiments.

3.4 Performance vs. dc and LO Bias
The dc bias voltage Vq and the LO voltage amplitude ‰ are referred to collectively as the voltage

bias. Mixer gain and noise are calculated numerically as a function of the voltage bias to determine the
best bias for an SIS mixer.

In fig. 3.2 are shown calculations of conversion gain vs. bias voltage for three situations. Calcula­
tions of mixer noise show the same structure as the gain, and so they are not shown. Fig. 3.2a shows
the calculation for the nearly ideal I-V of junction 1 shown in fig. 3.2, assuming that it is being operated

55

oZ ×≈
Figure 3.2 Conversion gain vs. de and LO voltage bias, a) Junction 1 from fig. 3.1 operated
at frequency .025z∕gap = 37 GHz. Gain is clipped at 2. b) Junction 2 operated at frequency
.025∕√gap = 32.6 GHz. The gain scale in this case is 5 times lower than for a) or c). c)
Junction 2 at a higher frequency, . 1^gap = 130.5 GHz. Gain is clipped at 2.
at a frequency .025pgap = 37 GHz. Fig. 3.2b shows results for junction 2 with the same normalized

frequency of .025m□ap, which corresponds to 32.6 GHz because Vgap is lower for this junction. Finally,
fig. 3.2c shows results for junction 2 at a higher frequency, . Hgλp = 130.5 GHz.
Figs. 3.2a and c show definite photon structure in the mixer response. In each of these cases, peaks

56

of conversion efficiency fall at dc bias voltages near the middle of each photon step,

⅛ ≈ Vgap - (n + ⅛)

for n = 0,1,2...

(3.4.1)

These plots also show the presence of high conversion gains at dc voltages near to Vgap∙ In this figure,
conversion efficiencies greater than 2 are plotted as equal to 2. The flat-topped regions in these two

plots actually include regions of infinite conversion efficiency, a non-classical phenomenon discussed

above.
A comparison of figs. 3.2a and b shows the effect of ∕-V curve sharpness on mixer conversion

efficiency. The photon steps which are so obvious in a are barely discernible as a slight ripple in
conversion efficiency in b. The weaker I-V non-linearity limits the maximum conversion efficiency in b

to .37, as compared to an infinite gain prediction for a.
For calculations of the gain vs. (Vo, Vlo), a mixer with external signal and image admittances

>’i = >'_i = Gjv is assumed. The IF load admittance is chosen to maximize mixer conversion efficiency
as discussed above. Calculations made assuming other values for Vi and V-ι show essentially the same

results described above. Therefore, the best bias point (Vo, VLo) can be considered to be independent of
the external circuit admittances chosen.

3.5 Performance vs. Signal and Image Source Admittance
The behavior of an SIS mixer as the external circuit admittance is varied is now determined. For

this calculation, a junction at a particular bias point and frequency is characterized by its Y and Ho
matrices. Calculations of mixer gain and noise, and receiver noise are calculated as Vi and/or y↑, 1 are
varied, results are presented on a Smith chart normalized to Gχ. The optimum operating point, found
as described above, is held constant for this calculation. Whichever of y↑ or >2ι represents the signal

sideband source admittance is labeled ys, and the other, which is then the source admittance of the
image sideband, is labelled V∕∙ For the low output frequency limit which is assumed here, results are

mathematically identical whether the signal is in the upper or lower sideband.

Calculations of mixer noise expressed in photons, Nm (2.5.8), vs. ys in the DSB case are shown

in fig. 3.3. These results are at the optimum operating point for junction 1 at f = .1pgap corresponding

57

Figure 3.3 Mixer noise vs. ys. Mixer noise is shown as contours of constant Nm- The
calculations are done for junction 2 at frequency .1^gap = 130.5 GHz, assuming ys = yι.
Minimum Nm is shown with a +, Flo is shown with a ∙.

to fig. 3.2b. Nm is minimum for ys ≈ Ylo, consistent with the results presented in section 2.6. This
minimum is fairly broad, a significant area of the Smith chart results in Nm < 2∙

Mixer conversion gain, shown in fig. 3.4, varies in a way which is quite different from the mixer

temperature. For most situations examined with these programs, infinite gain is available for values of

ys which appear in the lower left half of the Smith chart. This corresponds to source admittances with
a small to zero real conductance (high real impedance), and a small negative imaginary conductance. (A

negative imaginary conductance corresponds to inductance, positive imaginary conductance corresponds
to capacitance.)

The available gain from the mixer vs. ys is dominated by the change in the output admittance,
Gif, that the SIS mixer presents to the following IF amplifiers. Fig. 3.5 shows how G'tf varies with ys.

Especially at higher gain, the contours of constant gain are seen to be nearly parallel to the contours of

constant Gif- Infinite gain is available for Gif < 0 as discussed in section 1.5.
Given this effect, it would seem that an SIS mixer should be designed to have an output admittance

which is in the neighborhood of zero mhos, and a following amplifier with as high an input admittance,

58

Figure 3.4 Mixer gain vs. ys for the DSB mixer. Gain is shown in contours of constant
expressed in dB. The calculations are done for junction 2 at frequency .1^gap = 130.5 GHz,
assuming ys =y∕∙ The lower left half of the Smith chart, with no contours of gain in it, is
the region in which infinite gain is available from the mixer.

Figure 3.5 Mixer IF output admittance vs. ys ■ The contours are labeled with the normalized
output admittance, G↑f∕Gn. Calculations are done using junction 2 at frequency . 1z∕gλp =
130.5 GHz, assuming ys =yl.

59
Go, as can be achieved. Two effects must be considered in the construction of a mixer along these lines,
saturation and stray capacitance. Saturation of the SIS mixer can occur when the signal power becomes

comparable to the LO pump power, just as it can in classical mixers such as those based on Schottky
diodes.
However, a more important saturation problem has been identified by Smith and Richards (1982)

for SIS mixers with high conversion efficiency. This saturation results from the high rms voltages, Vbuτ,

developed across an SIS which is delivering a large output power to a following amplifier. For the case

of negative match, where Go = -<¾, Vbuτ is infinite. For Go matched to G⅛ > 0, Vzouτ oc G⅞1.
While Vbuτ accounts for the IF output power, it can also be considered as a random variation in the dc
bias voltage, ⅛. In section 3.3, the SIS mixer is seen to have a gain maximum which falls off if the

dc bias voltage is changed by a significant fraction of a “photon step” voltage, ΛωLo∕e. When Vbuτ is
a significant fraction of this voltage, the SIS mixer spends only some fraction of time biased near the

high gain maximum, for the rest of the time it is at a bias voltage corresponding to a lower conversion

efficiency. This results in a gain compression as an increasing Fouτ results in an increasing amount of
time spent in a low gain state. One way to avoid this kind of saturation when Gif gets small is to keep
Go at some reasonably large value. This results in an IF mismatch, lowering the mixer gain. However,
by turning down the gain, higher tolerance to saturation is achieved.

Another problem with high output impedances followed by high input impedance amplifiers is stray
capacitance. There will exist some amount of capacitance between the IF output line and ground. This
capacitance shunts Gif and Go. It causes degradation of output power when

Gpωo ~ Gif + Go-

(3.5.1)

As Gif gets small, and Go matches it, the amount of parasitic capacitance Cp that can be tolerated also

shrinks. Even though Cp can be tuned out by careful circuit design, the small amount of capacitance
that will ruin a low admittance IF circuit makes it very difficult to tune Cp out simultaneously over the

entire output bandwidth. This problem, like saturation, is mitigated by keeping Go reasonably large, at
the expense of a lower gain due to mismatch.

60

Figure 3.6 Mixer performance vs. y/, for ys fixed. The calculations are done using junction 2
at frequency .1pgap = 130.5 GHz. a) Available gain in dB vs. >’/. Infinite gain is available
for yI in the lower left half of the Smith Chart. The minimum gain point of -2.8 dB is shown
by a ∙. b) Mixer noise in photons, Nm, vs. 3-7- Minimum noise occurs at the point shown
with a +, maximum noise occurs at the point labeled with a ∙.

The results shown above are for the DSB mixer, where >⅛ and y1 are varied simultaneously. In

principal, a mixer could have ys and yj set to unequal optimum values. When J⅛ is held fixed, and
ys is allowed to vary alone, the pictures of mixer temperature and gain generated are not remarkably

different from the DSB case presented. This is true because the value of y$ has more effect on mixer

performance than the value of yl. When ys is held fixed, mixer performance vs yf shows optimum
performance for y1 ^ys, generally.

Performance found by varying y1 while ys is held near its optimum value are shown in fig. 3.6.
yI has a much more significant effect on available mixer gain than it does on mixer noise. Over all

possible values of ¾, the predicted mixer temperature ranges from 9.9 K (1.6 photons) to 13.6 K (2.2
photons). However, the available conversion gain varies from -2.8 dB to infinite available gain as the

image is tuned. As in the DSB case, this gain variation is essentially due to a change in G⅛ as >∕ is

changed.
The best and worst gain occurs at values of 3,∕ which are 1) completely reactive, lying on the outer
edge of the Smith chart, and 2) almost diametrically opposed on the Smith chart. While a poor choice

of y1 can reduce conversion gain, it cannot reduce it by very much. Similarly, an optimum choice of
yI cannot improve mixer performance by any significant amount over the DSB tuning results.

61

Figure 3.7 Gain vs. ys for two different junctions. Gain contours are labeled in dB. a) Junction
1 at frequency .1pgap = 148.0 GHz. Infinite gain is available for ys in the lower half of the
Smith chart, b) Junction 3 at frequency . 1pgap = 103.2 GHz. Large but finite gain is available
for ys in only a very small area at the lower left of the Smith chart.

There is a practical difficulty in a mixer with ys and

chosen to be at their different optima.

The output admittance of the mixer is actually, in general, complex,

Yif = Gif + ⅛

(3.5.2)

In principle, Btf can be tuned out by proper design of the IF circuit. Further, for the DSB mixer, ⅛ = 0

anyway due to the symmetries of the Y matrix. With the DSB constraint removed, Yif = (- .2 — .2z) Gn
when y1 is chosen to minimize Tm- For match, the IF circuit must present an approximately constant

reactive admittance of +.2iGχ over the entire IF output bandwidth. It is rather difficult to design a
circuit with a non-zero imaginary admittance that remains constant over any significant frequency range.
All the results shown so far are calculated using junction 2 from fig. 3.1. Fig. 3.7 shows the

DSB gain predictions for junctions 1 and 3. Junction 1 has an essentially ideal I-V, while junction 3 is
significantly less ideal than junction 2. The less ideal junction 3 does not quite achieve infinite available

gain, but gain greater than +10 dB is calculated. Again, it occurs for ys in the lower left of the Smith
chart, but the area of the Smith chart over which high gain is predicted is much less than it was with

junction 2. The effect of the nearly ideal I-V of junction 1 is to make infinite gain available over almost

all of the Smith chart. Mixer temperature for this junction is at the quantum limit when ys = Ylo as
discussed in chapter 2.

62

Figure 3.8 Gain vs. ys at a lower frequency. Available gain, labeled in dB, is calculated for
junction 2 at a frequency .025pgap = 32.6 GHz, keeping >’s = ÎV/. The maximum available
gain point is shown by a +.
All results so far have been at the same normalized frequency of .1^gap, which ranges from 103 GHz

for junction 3 through 130 GHz for junction 2 to 145 GHz for junction 1, due to the differences in Vgap∙
For normalized frequencies from .1 to about .8, the results are not very different from those shown. At

lower frequencies, the quantum mixer feature of available conversion gain disappears. Fig. 3.8 shows
the gain available from junction 2 at a frequency of .025pgap, or about 32 GHz. These contours show

that the highest conversion efficiencies occur for ys somewhere in the middle of the Smith chart, which
is what usually is expected in a classical diode mixer theory. It should be noted that the same calculation
for the nearly ideal junction 1 shows infinite available gain at most ys. The quantum mixer phenomenon
of infinite gain has a lower frequency limit which depends on the sharpness of the I-V. Junction 1 is so

sharp that even at a frequency of .025^gap> its behavior is still quantum.

3.6 Performance vs. Frequency
In this section, fully optimized mixer performance is presented as a function of LO frequency. The
calculations are done for the four junction I-Vs shown in fig. 3.1. For each junction, at each frequency,

63

Figure 3.9 Gain vs. frequency for four tunnel junctions. The available gains for the four
junctions of fig. 3.2 are shown as LO frequency is varied. At low frequencies, the available
gains for junctions 1 and 2 are infinite, and therefore do not appear in the figure.
the values of ys = F∕, ⅛, and ‰∣ are found which minimize the receiver noise temperature, 7⅛ in

(3.1.4), assuming an IF amplifier chain noise of 7⅛ = 10 K. The IF amplifier chain input admittance, >'o,

is assumed matched to the mixer output admittance, ⅛, for this optimization. By finding the optimum

mixer in this way, high mixer gain and low mixer noise are traded off against each other just as they
would be in a real receiver.

Fig. 3.9 shows the maximum available gain. The nearly ideal I-V of junction 1 has infinite available
gain up to its gap frequency, pgap∙ The good, but noticeably imperfect I-V of junction 2 also shows high
gain up to its gap frequency. The very non-ideal I-V of junction 3 provides non-classical conversion

efficiencies of > —3 dB up to about 300 GHz. Finally, junction 4, the SIN, is seen to provide at most

—6 dB gain, and this falls off faster than any of the SIS gains, since j^gap for the SIN is much less than
the SIS gap frequencies.

The mixer noise, shown in fig. 3.10, again favors the SISs over the SIN. The best SISs are well
within a factor of 2 of quantum limited noise, Nm = 1. over most of the frequency range up to z∕gap∙

The SIN is able to outperform junction 3 below 500 GHz. SIS junctions 1 and 2 are both at physically

higher temperatures than the SIN, and yet both the SISs have much lower noise at all frequencies.

64

Figure 3.10 Mixer noise, Nm> ys∙ frequency for the four junctions of fig. 3.1.

In both the gain and the mixer noise, their is an upper frequency limit to efficient mixer performance

of i^gap∙ The reason for this can be seen from the semiconductor model shown in fig. 1.7. When the

dc bias ⅛ is just below the diode turn on voltage of Vgap, absorption of a photon of small energy will
result in an extra electron in the diode current, and this photon is thus detected. However, there are
two ways to absorb a photon of energy ,⅛a-gap- It can result in an electron in the diode current, just as

occurs for low energy photons. However, its energy is sufficiently high that it can also cause an electron

to tunnel backwards across the diode, producing a current of the opposite sign. On average, there will
be more electrons tunneling forward than backward, so there will still be a net photocurrent. However,

the responsivity does drop to a fraction of what it was before this backward tunneling became possible,

resulting in a sharp drop off of mixer response at pgap∙
In the mixer noise, a feature can be seen in all three of the SIS predictions at a frequency of z×gap∕2.

This is a result of electron tunneling assisted by the simultaneous absorption of two photons. Below
the frequency j/gap/2, it is always possible to bias the junction so that even the two photon process can

only cause forward conduction of tunneling electrons. Above pgap∕2, any dc bias ⅛ < Vgap results in
a two photon absorption process which can result in electrons tunneling in either direction across the

65
diode. Just as above for the single photon process, this results in a sudden drop in diode photo-response.
Under optimum bias conditions, the single photon process is much more important than the two photon

process, so the degradation in performance from the loss of the two photon photocurrent is not nearly
as severe as the sudden fall off in performance at pgap∙

The dc and LO voltage bias points for optimum performance are shown in ög. 3.11. These are
the values which minimize the receiver noise temperature, T⅛, assuming an IF amplifier chain noise
temperature of 3⅛ = 10 K. The optimum dc voltage bias is always less than V'gap∙ The LO voltage is

shown in terms of the Bessel function argument, a = eVw∕hωχβ. Its optimum value is in the range .4

to .7 for the high frequency range, and it is nearly the same for all four junctions. At low frequencies,
this is not true. While junction 1 has a < 1 at the lowest frequency calculated, junctions 2 through 4
require higher LO drive levels at low frequencies.
This is due to the transition from quantum to classical behavior at low frequencies for these

junctions. In the expressions of Tucker’s theory, the Jn(α) Bessel function term can be identified with
the physical process where n photons are absorbed simultaneously to cause an electron to tunnel. Indeed,
the probability of the n-photon process is proportional to Jn(α). The Bessel functions have the property

that Jn(a) is very small for a ≤ n. Finding the optimum value of a to be in the range .4 to .7 over a

large range of frequencies indicates that the most efficient mixing occurs when the single photon process
dominates the ∏ = 2 and higher processes. This result holds for junction 1 right down to the lowest

frequency for which the calculations are done. However, junctions 2 through 4 do not have sharp ∕-Vs on
the scale of a single photon step at the low frequencies. Hence, it is necessary to absorb a few photons

simultaneously to get efficient photo-detection, and a must rise so that these multi photon processes

have reasonable probability.
The optimum signal source admittance 3⅛ = Gs + iBs is shown in fig. 3.12. The optimum Gs is
close to Gpf for all junctions over the higher part of the frequency range. However, junction 1 has an
optimum Gs which rises quickly at low frequencies. This is true for the low values of a which were

found to be optimum for this junction. Junction 1 is also capable of producing good mixer performance
when >5 ~ Gn- For this lower value of source admittance, a higher value of a optimizes the mixer.

66

Figure 3.11 Optimum bias vs. frequency. The dc and LO voltages which optimize 7⅛ for the
four junctions of fig. 3.1 are plotted, a) The dc bias voltage, b) The LO voltage, shown in
terms of the Bessel function argument a = e‰∕∕jωLo.

The imaginary part of the optimum source admittance is also shown in this figure.

It is not

tremendously different from zero for any of these junctions at any of the frequencies investigated. This

suggests in mixer design that it will be important to use very low capacitance SISs for higher frequencies,

or else to design a circuit in which SIS capacitance is carefully tuned out in some way.

3.7 Summary
In this chapter, an overview of results from numerical calculations has been presented. The method

67

Figure 3.12 Optimum signal source admittance vs. frequency. The real and imaginary parts of
Ts ≈ Gs + iBs, which result in minimum 7⅛ are plotted for the four junctions of fig. 3.1. a)
The real part, normalized to Gw. b) The imaginary part, similarly normalized,
for digitizing 7-Vs for use in the calculations was described. Mixing performance as a function of dc

and LO bias showed the transition from classical to quantum behavior as LO frequency was increased.
The effects on mixer performance of signal and image source admittances were discussed. While

highest mixer gain is usually achieved for very low source admittances, low mixer noise occurs for
y,s ≈ ‰>, which is typically close in value to Gn∙ The minima in mixer noise were fairly broad,

suggesting that a mixer circuit designed to present a source admittance ys & Gn should provide near
optimum mixer performance.

68

The optimum theoretical performance for various tunnel diodes was calculated as a function of
frequency. Reasonable SIS diodes were found to give mixer noises less than Nm = 2 up to a frequency
proportional to their gap voltage, z,gap = 2e½jAp∕Λ∙ SIN performance was predicted to be reasonable

at low frequencies, but its lower z√gap resulted in very poor performance above 600 GHz.

For the lead alloy junctions shown, the high frequency limit, ioap, is about 1500 GHz. Junctions

made from niobium-nitride are now being developed for mixer applications (LeDuc et al., 1986). These
junctions have higher superconducting transition temperatures than lead alloys (13 K instead of 7 K),

and correspondingly higher gap voltages (5.5 mV as opposed to 3 mV). With NbN SISs, z∕gap may

eventually be as high as about 3000 GHz.

69

Chapter 4 — SIS-Bowtie Mixer

“This new outfit will save us from Ahab’s wrath. You
see, this bowtie allows me to see to higher frequen­
cies. We’ll surely spot the quantum limit now.”
— Ishmael, in Moby Mike or Captain Phillips
and the Great White Wengler, by Erich Grossman and Jeff Stem.

4.1 Design Overview

This chapter will discuss the design, construction, and testing of a prototype mixer which provides
very low noise detection for millimeter and submillimeter wavelengths.

SIS tunnel diodes are the

detector for this mixer. Radiation is coupled to the tunnel diode through a scheme which has been used
for imaging arrays (Rutledge, Neikirk, and Kasilingam, 1984). The optical arrangement of this mixer

is shown in fig. 4.1. The SIS is fabricated at the center of a superconducting bowtie antenna. The
integrated diode and antenna are placed on the back of a truncated quartz sphere (hyperhemisphere).

Radiation is focussed into the bowtie by the hyperhemisphere, and a plastic lens in front of it. The lower

frequency limit (about 100 GHz) of this mixer is set by the size of the lenses and the bowtie antenna,

while the upper frequency limit of this mixer (unknown now, but greater than 761 GHz) is probably
limited by the capacitance of the SIS junction.
Low noise mixers in this spectral range are usually based on waveguide structures. Fig. 4.2 shows
a cross section of the 3 mm wavelength mixers (Woody, Miller, and Wengler, 1985) used at the Owens

Valley Radio Observatory (OVRO). Over about a 30% frequency range, scalar feedhoms give a nearly

perfect coupling between the waveguide mode and the Gaussian free space mode for which the telescope
optics are usually designed. This overall coupling is very important to the ultimate signal to noise ratio

(SNR) of astronomical observations. Poor mode match between the receiver and the telescope results

in a loss of telescope signal coupled to the receiver. Furthermore, the part of the receiver mode which
is not coupled to the telescope will probably be filled with thermal radiation from the receiver room or
the landscape surrounding the telescope. This results in an increase in the noise coupled to the receiver.

Coupling inefficiency therefore hurts SNR in two ways, it lowers signal and it raises noise.

70

Figure 4.1 Bowtie mixer optics. An SIS junction is fabricated integrally with a planar bowtie
antenna, on a thin quartz substrate. This is placed on the flat surface of a quartz hyperhemi­
sphere. Radiation incident on the plastic lens is focussed into the quartz hyperhemisphere,
which further focuses the beam into the center of the bowtie antenna. The radiation is thus
coupled to the SIS, which is at the center of that bowtie.

Figure 4.2 Cutaway view of the waveguide-feedhom mixer block in use at 0VR0. The SIS is
mounted across a circular waveguide. The waveguide diameter is 2.4 mm, about equal to the
free space wavelength of the radiation it will detect. A movable short in the waveguide behind
the SIS improves the coupling efficiency between the waveguide and the SIS. A feedhom
efficiently transforms the waveguide radiation to a Gaussian profile free space beam. (Woody,
Miller and Wengler, 1985).

The bowtie antenna shown in fig. 4.1 will have a coupling efficiency to the telescope of less than

80% at best, and probably less than 50%, but it will not change much over a very large frequency
range. Waveguide structures like the one shown in fig. 4.2, have efficiencies of well over 90% over their

frequency ranges of about 30%.
Why then, would one build a low noise receiver in a bowtie antenna? Because the theoretical

71
work reported previously in this thesis shows that the intrinsic noise of SIS detectors is 50 to 150

times lower than the best receiver noises which are achieved by existing technology. Even with a low
coupling efficiency, the SIS-bowtie could still be expected to have much better sensitivity than is actually

achieved with other technologies. In addition, the bowtie mixer has two large advantages over waveguide

structures. First, the bowtie mixer is cheaper and simpler to build. Second, a single bowtie mixer covers
a few octaves of the near-millimeter band, while a waveguide mixer covers a 30 % bandwidth. Even
if outperformed by a waveguide mixer in some narrow band, the multi-octave coverage of the bowtie

would still make it worthwhile for many applications. As will be seen, however, it does as well as or

better than other technologies which have been used in the 300 to 760 GHz range.

4.2 SIS Junctions
The SIS junction is the most critical element of an SIS mixer. The SIS junctions used for the mixer

described here are fabricated in R. E. Miller’s lab at AT&T Bell Laboratories in Murray Hill, New
Jersey. In the last five years, I have spent one or two months each summer developing and fabricating
junctions with Miller. Two different observatories have relied on junctions from this lab, to the exclusion

of all other possible technologies. The AT&T Bell Labs 7 m telescope at Crawford Hill, New Jersey has
used these junctions for the past six years for all of its 3 mm band observations. At Caltech’s Owens
Valley Radio Observatory (OVRO), junctions from this lab have been used over for the past six years in
receivers for 3 mm to 1 mm (Woody, Miller and Wengler, 1985; Sutton, 1983). For the past three years,

three SIS receivers have been run nearly constantly at OVRO as part of a three telescope interferometer
for 3 mm observations. Other observatories have used SIS receivers for particular experiments, but

only AT&T and Caltech have relied on them exclusively. As a result, a major fraction of the world’s
experience with SIS receivers on telescopes is with junctions from Ron Miller’s lab. In this section, the
desired properties of SISs for astronomy are presented. The fabrication procedures and experiences at

Bell Labs are discussed.
Except for recent results with niobium junctions at the National Radio Astronomy Observatory’s

Kitt Peak telescope (Pan et al., 1987), all SIS junctions used on radiotelescopes have been based on

72
lead alloys. In addition to the AT&T and Caltech efforts, lead alloy SIS receivers have been used on
telescopes at Onsala in Sweden (Olsson et al., 1983), at the NASA-Goddard Institute for Space Studies

(Pan et al., 1983), and at the IRAM telescope in Pico Veleta, Spain (Blundell et al., 1983; Ibruegger et
al., 1984).

The original lead alloy junction fabrication technology was developed at IBM (Greiner et al., 1980)
as part of a superconducting logic research effort. All of the labs cited above have developed their
processes from the IBM process. The fabrication process at Bell Labs was developed by Dolan, who
made the junctions for the first SIS quasiparticle mixers (Dolan, Phillips, and Woody, 1979).

Our SIS diodes are shown schematically in fig. 4.3. The base and counter electrodes are thermally
evaporated onto the quartz substrate in a vacuum system at pressures of about 10~6 torr. The electrodes

are alloys of lead including bismuth, indium and/or gold. Electrode thicknesses range from .2 to .3 μm.
One method used to form an electrode of a particular alloy is to make a bulk quantity of the proper alloy,

and then evaporate small samples of that alloy onto the substrate. Another method is to sequentially

evaporate lead, bismuth, indium and/or gold in the proper ratio to achieve the alloy desired. For the alloys
we have used, we have not observed differences in results achieved by these two very different methods.
For the sequential evaporation method, the order of evaporation of the elemental components does not

seem to matter. Since we do not see differences as we vary this part of the process, we conclude that,

for the limited range of alloys we have tried, the various components of the alloy interdiffuse readily.

After the base electrode is evaporated, the vacuum chamber is filled with dry oxygen gas to a
pressure of about 5 x 10~2 torr. A dc discharge is run through this gas in the vicinity of the substrate
for about five minutes. This produces a layer of oxide on the base electrode, which serves as the

insulating tunnel barrier for the eventual SIS. After oxidation, the chamber is pumped down once again
to 10~6 torr, and the counterelectrode is evaporated onto the substrate.

The base electrode alloys always contain 5% to 8% indium by weight The counter electrode never
contains any indium. This results in consistently high current density junctions. Researchers at IBM
claim that the electrons in this system are tunneling through a Schottky potential barrier (Baker and

Magerlein, 1983). They have determined that the predominant oxide in junctions made this way is

73

.5 μm

^^^∣^∖S2

S1
Insulator (Oxide)

«____________ ;_______________

S2

Quartz Substrate

2 μm

Figure 4.3 Schematic view of SIS junction. At the top is shown the view looking down onto
the junction. Below is shown a cross-section through a junction (not to scale). The current
path is from SI, through the insulator, to S2.
In2O3. This oxide is a semiconductor, as opposed to an insulator. In these junctions, the indium oxide

is degenerately doped by excess indium which diffuses into it from the base electrode. The counter
electrode, an alloy with no indium content, is on top of the doped indium oxide. Excess indium diffuses

out of the top few angstroms of the oxide layer, into the top electrode. The top few angstroms of

the In2O3 are therefore undoped. The result is a Schottky barrier between the oxide layer and the top

electrode.
The tunnel barriers in other SIS technologies usually consist of a good quality oxide of 1 to 2 nm
thickness. The oxide is an insulator, so it presents a potential barrier to electrons which is as wide
as the oxide layer is thick. As a result, tunnehng currents vary exponentially with oxide thickness in

these junctions. To fabricate junctions like this with consistently high current densities, it is necessary

to fabricate extremely thin oxides. The thinner an oxide, the more probability that there is a pinhole
flaw, which results in a short-circuited junction. The indium oxide layer, on the other hand, consists of

a physically thick oxide with a consistently thin Schottky tunnel barrier. As a result, consistently high
current densities are achieved with these junctions.
For making small area junctions, we use a method invented by Dolan of AT&T Bell Labs (Dolan,
1977) which is based on the tri-level stencil technique (Dunkleberger, 1978). In this method, the base

and counter electrodes are both deposited through the same photoresist stencil. In most other techniques,

Ί4

Figure 4.4 Micrograph of an SIS-Bowtie. A scanning electron micrograph of an SIS-Bowtie
shows the overlapping superconductors which form the SIS, and the small scale part of the
bowtie antenna. The bowtie shape is accurately maintained down to about 1 μm scale size.
substrate processing is required between the evaporations of the base and counter electrodes. This means

that the base electrode surface is contaminated by whatever processing steps are required to pattern the
counter electrode. With Dolan’s technique, however, the base electrode is deposited, oxidized, and the

counter electrode is evaporated without the vacuum system being opened up to air. It is also possible
with Dolan’s technique to make junctions with overlap dimensions of about .5 μm even when the

original photolithographic technique used to make the tri-level stencil has a resolution of 1 μm,. Using
this method we are able to achieve junction areas of ~ .3 (μrn)2 with tolerable consistency. An electron

micrograph of one of these junctions is shown in fig. 4.4.
Two different alloys have been used for junction fabrication. The first is Pb97Au03. The supercon­
ducting energy gap voltage (∆∕e) of this alloy is about 1.2 mV at 4.2 K, and it transition temperature,

Tc, is about 7.2 K. The gold is present in this alloy to avoid the formation of “hillocks” in the lead film
as the junction is cycled in temperature between room temperature and 4.2 K. The second alloy we have
tried is Pb71Bi29. This alloy has been used with good results by another lab (Gundlach et al., 1982).

PbBi alloy has a superconducting gap voltage of about 1.5 mV at 4.2 K, Tc is about 9 K. The presence

75

Table 4.1 - Desirable Characteristics for SIS diode
Desirable electronic characteristics:

1. Good quality I-V. As discussed in section 2.6, an SIS with a perfect I-V responds
perfectly to absorbed radiation.

2. Low capacitance. The lower the junction’s capacitance, the higher the frequency
of radiation that can be coupled to it from a given rf circuit.
3. High current density. A small area junction must carry a high current density
so that its radiation impedance is low enough.
4. High critical temperature (Tc). For a given operating temperature, an SIS made
from higher Tc superconductors has a better I-V.

Desirable physical characteristics:
1. Small size. Junction capacitance is proportional to junction area.

2. Thermal cyclability. Cycling between cryogenic operating temperatures and
room temperature should not degrade or destroy the SIS.
3. Shelf-life. The longer a good junction can be stored for later use, the better.

4. Yield/Repeatability. It should be possible to fabricate high quality junctions on
demand.

of ~ 8% indium in the base electrode does not seem to affect the superconducting characteristics of

either of these alloys.
We now discuss the quality of the junctions we make. The desirable characteritics of an SIS junction

for use in radioastronomy can be divided into its desirable electronic characteristics, and its desirable
physical characteristics. Table 4.1 lists these desirable characteristics. Our results with lead junctions

can be generally summed up by saying that their electronic characteristics are good to excellent, but that
their physical characteristics range from fair to just barely tolerable.
An I-V for one of our PbAu alloy junctions cooled to 4.5 K is shown in Fig. 1.3b. The curve
deviates from the ideal SIS curve shown in fig. 1.3a. The real SIS requires a finite dc voltage range

in which to switch from its off state to its on state. In its off state, the real SIS carries some current,
whereas the ideal SIS does not. There are discontinuities and features present which are due to tunneling

of Cooper pairs of electrons. One such feature is the existence at zero voltage of a finite current This is
the dc Josephson effect, also referred to as the supercurrent (Josephson, 1965). It is not possible to bias

76
the junction at dc voltages between zero and the dropback voltage (Vd). This is indirectly due to the

ac Josephson effect. If a finite dc voltage ⅛ exists across an SIS, there is a sinusoidal current flowing

across the junction at the Josephson frequency (1.2.3). If the junction has a large capacitance, or is

otherwise shorted, then these currents can flow without generating much voltage across the junction.
However, if the junction capacitance is small and the junction is not otherwise shorted, these
currents will generate relatively large voltages. These voltages will in turn change the Josephson current

being generated. Detailed solutions of the current and voltage waveforms depend on details of the
circuit external to the junction through which these currents must flow. These solutions include period
doubling, quadrupling and chaotic forms. The driving force in this system is the ac Josephson current

which flows at a frequency which is proportional to the dc voltage across the junction. Above a certain
voltage, Vd , this frequency is high enough that the junction capacitance Cj guarantees small rf voltages
which do not result in chaotic behavior. Below Vo, the Josephson frequency is low enough so that rf

voltages are large, and chaotic solutions keep the junction from having a stable dc voltage on it.
In fig. 3.1, junction I-Vs are shown for different alloy junctions cooled to different temperatures.
The best of these I-Vs are good enough so that at frequencies above 100 GHz, their mixer performance

should be only slightly worse than ideal. Therefore, in terms of I-V quality, lead alloy junctions are
good.

A real SIS can be thought of as an ideal “point” junction, shunted by a capacitance, Cj. This

capacitance is due to the fact that, except for the tunneling currents, an SIS is a superconducting parallel

plate capacitor. The capacitance between the base and counter electrodes is

where A is the overlap area of the junction, and d and e are the effective thickness and dielectric constant
for the tunnel barrier. For indium oxide Schottky barriers, the values of d and e yield a capacitance of

about

Cj = 50 fF

(μm)2

(4.2.2)

for high current density junctions (Baker and Magerlein, 1983). The smallest area junctions we make

are about 1/4 (pm)2. The minimum Cj is therefore about 13 fF.

77
The current densities in our junctions are about as high as any reported for SISs with good quality
Ι-Vs. We can fabricate a Rn = 25 Ω junction of area .25 (μm)2, which corresponds to supercurrent

densities of a few times 104 A /cm2. With a capacitance Cj = 13 fF, these junctions have a characteristic

RC roll off frequency of about 500 GHz. Our mixer results actually show good performance up to
about 750 GHz, verifying that we do indeed have very fast junctions.

The critical temperatures for the alloy films used in these junctions are about 7 K for the PbAu

alloys and about 8.5 K for the PbBi. The receivers at OVRO are cooled by closed cycle refrigerators

which achieve a minimum temperature of about 4.5 K. At this ambient temperature, the slightly higher
Tc of the PbBi junctions results in a significant difference in mixer performance. For mixers cooled

to 4.2 K by direct exposure to liquid helium, the PbAu I-Vs improve sufficiently so that there is not a
large difference between PbBi and PbAu junction performance.

The thermal cyclability of our junctions is more than adequate. Over 90% of our junctions will

survive their first cool down and warm up. Once a junction has survived two thermal cycles, it is
extremely rare that further cycling causes it to fail.

It is in terms of shelf-life that our junctions receive a barely tolerable rating. For the PbAu junctions,
we find that a usable percentage of junction batches will survive for a few months after fabrication. If
stored in liquid nitrogen, storage time may exceed one year, for some batches. Unfortunately, the higher

transition temperature PbBi alloy junctions have a very small survival rate when stored. Gundlach et
al., (1982) experience no such difficulties with their junctions. Because of the short shelf-life of PbBi

SISs, all mixer results presented in this thesis are with PbAu alloy junctions.
The fabrication technique we use for junctions is fairly reliable in the sense that a good batch of

junctions can be fabricated with only a few days notice. This is usually accomplished, however, by
fabricating three to six batches, of which at least one will be high quality. Many of those batches will,

however, be unacceptable. The most common failures are batches with excess current below the turn
on voltage Vqap-> and batches in which almost all junctions are short circuited.

In conclusion, lead alloy junctions from AT&T Bell Labs are capable of producing low noise
receivers for near-millimeter wavelengths. The reliability and shelf-life of these junctions is not very

78

high. It is sufficient for radioastronomy and laboratory research, as long as new batches from Bell Labs
can be fabricated on relatively short notice.

43 RF optics
A major difficulty in receiver design at mm and sub-mm wavelengths is to provide efficient coupling

of radiation to the detector over a reasonable spectral range. For mm waves, this is most effectively

accomplished by placing the detector in a waveguide which has one or more tuning stubs. Radiation is
coupled into the waveguide with feedhoms. Fig. 4.2 shows the design of the 3 mm wavelength receivers

used at Owens Valley Radio Observatory (Woody, Miller and Wengler, 1985).

Waveguide-based systems have a number of disadvantages at higher frequencies.

In the sub-

mm band, waveguide dimensions become difficult to work with. For example, a 500 GHz circular

waveguide of the same design as shown in fig. 4.2 would have a waveguide diameter of .5 mm. The
metal structures for mounting the SIS in the waveguide must be small compared to this diameter. An
additional difficulty, waveguide loss increases rapidly with frequency due to surface roughness of the

waveguide walls. Practically, this becomes important for frequencies above 500 GHz, even when great
care is taken to minimize surface roughness. Finally, waveguide-feedhom structures have only a one-half

octave bandwidth, so many different mixers must be built to cover the near-millimeter band.

For Schottky diode-based mixers, an alternative to waveguide mounting is the quasioptical comer

reflector and long wire antenna. These are the only broadband heterodyne receivers currently in use for
submillimeter astronomy (Röser et al., 1984; Betz and Zmuidzinas, 1984; Zmuidzinas, 1987). Unlike

waveguide structures, a single mixer covers 3 or more octaves in the submillimeter. The comer reflector,
long wire antenna radiation coupling scheme is particularly appropriate for Schottky diodes. It would be
possible to build an SIS receiver using this scheme. However, the thin-film-on-substrate SIS fabrication
technology suggest a planar antenna approach for SIS coupling.

The optics of our mixer were derived from the work of Dean Neikirk and David Rutledge of Caltech.

Similar optics have been used for an imaging system at a wavelength of 120 μm (Neikirk et al., 1982).
An excellent and extensive review of integrated circuit antennas, including the bowtie scheme discussed

79
here, has been presented by Rutledge, Neikirk and Kasiliπgam (1984). The intended operation of this

scheme for our mixer is illustrated in fig. 4.1. A 30° wide converging beam of radiation is focussed into
a 120° converging beam by the plastic lens and the curved surface of the quartz hyperhemisphere. This

beam is coupled, with some unknown efficiency, to a detector at the center of a planar bowtie antenna,

which is placed on the flat surface of the hyperhemisphere.
Above some lower frequency limit, the bowtie-on-quartz antenna properties do not change with

frequency, suggesting its use in extremely broadband systems. Its frequency independent properties are

best presented in the language of antenna design. They are: 1) the radiation pattem of the antenna is

not symmetric front to back, 80% of the beam is coupled into the quartz, only 20% goes backwards
into the vacuum; 2) the antenna impedance is purely real; 3) the SIS and the antenna form a monolithic
structure which minimizes parasitic reactances; and 4) choking of the IF line to avoid RF propagation
is achieved automatically by the bowtie antenna. These results are derived and experimentally verified

(Rutledge, Neikirk, and Kasilingam, 1984). A complete theory for the bowtie, which compares well
with measurements, has been presented (Compton et al., 1987).

The operation of the bowtie antenna can be understood by considering it to be a lossy transmission
line. An infinitely long bowtie in vacuum is a lossless radial transmission line. This transmission line
has some characteristic impedance Zu which is independent of frequency, and radiation travels along

this line at the speed of light, c. This same transmission line embedded in a dielectric of refractive
index n would have a characteristic impedance Z⅛∣n and radiation would travel along this line at c∕n.

If this transmission line is now placed on the planar boundary between a half-space of dielectric on
the one side and vacuum on the other, it is no longer lossless. As with any wave-guiding structure on

a dielectric-vacuum interface, there is a radiative loss which is predominantly into the dielectric. This
radiative loss occurs because purely guided wave solutions have different wave speeds in the dielectric
and in the vacuum, c/n and c respectively. These solutions do not match at the interface between the

two media. What actually happens is that radiation propagates along the bowtie at some intermediate
speed. The fields on the bowtie make a reasonable match with radiation modes that can propagate freely

through the dielectric, but they do not match well with any propagating modes in the vacuum. Hence,

80
a wave propagating along the bowtie is lost predominantly into the dielectric.

The antenna impedance for the bowtie is accurately calculated from the transmission line model

(Rutledge and Muha, 1982),
^ΛNΓ ~ Zü/tlm

(4.3.1)

where nm = 5∕(1 + n2)∕2. The index nm is intermediate between the vacuum and dielectric values, it
is interpreted as the effective index for the bowtie transmission line. For a 90°-wide bowtie on quartz

(n = 2), Zant is about 120 Ω.

The bowtie is frequency independent above some lower frequency limit. Consider an infinite bowtie
on a dielectric-vacuum interface. This system must have frequency independent performance since it is
self-similar on all length scales. Consider an observer at the center of the bowtie. Looking down the

bowtie is like looking into mist, since waves propagating on the bowtie get lost into the dielectric. The
distance at which the bowtie is lost in the mist must be some number of free-space wavelengths, say xλ0.

Now consider each arm of the bowtie truncated to some finite length L. When examined at wavelengths

L jx and shorter, the truncation is lost in the mist. Hence, the bowtie is effectively self-similar for length
scales L/x and shorter.
There are actually two important length scales for the bowtie. The antenna impedance of the

bowtie-on-quartz is frequency independent for Ao ≤ ∏L. However, the beam pattern is much more
sensitive to the length of the bowtie, and it becomes only approximately frequency independent for
Ao ≤ .2nL (Compton et al., 1987).
A final property of the bowtie which is particularly nice for millimeter and submillimeter detection

is that it automatically isolates the rf frequency from the low frequency leads. Low frequency electrical
connections to the antenna can be made at a distance of a few λ∕n or more from the center of the

bowtie. There will be no rf loss due to propagation of the rf into these connections, since the rf has

already jumped off the antenna into the dielectric by this point With a bowtie on dielectric, it is not
necessary to include rf filters on low frequency leads.

The ideal bowtie has no upper frequency limit. The real bowtie will deviate from ideal performance

if 1) the bowtie is not accurate on a scale of ~ . 1 A, or 2) the dielectric become lossy, or 3) the conducting

81

(b)

Figure 4.5 Bowtie antenna patterns, a) Pattern measured at 10 GHz on a scale model of an
imaging array (Rutledge and Muha, 1982). b) Pattem measured for a single bowtie at 94 GHz
(Compton et al., , 1987). The zero of the γ and S axes show degrees away from the normal
in the E-plane and H-plane of the bowtie respectively. Only one quadrant of the pattern is
shown, since the other three are simply related to this by the symmetries of the antenna.

material of the bowtie becomes lossy. The SIS that detects the radiation is fabricated at the center of
the bowtie as shown in Fig. 4.4. Deviations from bowtie shape occur on the scale of microns, which

is much shorter than the wavelength at which loss in the fused quartz will be important (~ 200 μm).
The electrical conductivity of the superconducting antenna metal is extremely high for frequencies up

to ∕ ~ 2Δ∕.⅞ which is about 500 GHz for the lead alloys used here. As the frequency rises above

this value, the lead alloy conductivity falls towards its normal state value (Tinkham, 1975). At what
frequency this conductivity falls enough to affect performance is not yet known.
The bowtie used in in this mixer has 90° wide bowtie aims, which have a measured antenna

impedance of 120 Ω on a dielectric with index n = 2 (Rutledge and Muha, 1982). The length of the

bowtie arms are 1.5 mm. For bowties with only 60° wide electrodes, beam patterns have been measured
at 10 GHz (Rutledge and Muha, 1982) and at 94 GHz (Compton et al., 1987). These measurements are
shown in fig 4.5. It has been suggested that the beam of a 90° bowtie should be similar to that of a
60° bowtie, but with a slightly wider E-plane pattern and a slightly narrower H-plane (Rutledge, private

communication).

The 10 GHz measurements show a pattern with most of the radiation included in a 60° half-angle

82
beam into the quartz, which is what the mixer optics were designed to illuminate. Additionally, these

measurements show only very small amounts of power in the beam propagating into the vacuum side of
the bowtie. The E-plane pattern shows large lobes at 30° away from a line normal to the bowtie plane.

For optimum coupling using the optics of fig. 4.1, the beam pattern should be maximum at 0°, but these
optics illuminate a large enough angle in the quartz that these 30° lobes should be well illuminated.

The recently reported theory and measurements at 94 GHz (Compton et al., 1987) show beam

patterns which are much worse than the earlier 10 GHz measurements around which our mixer was
designed. In the measurements at 94 GHz, shown in fig 4.5b, the E-plane pattern has large lobes which

are 60° away from the normal to the bowtie. These lobes, carrying most of the power in the bowtie
pattern, are just on the edge of the angle which the optics in fig. 4.1 are designed to illuminate. The

important difference between the two patterns in fig 4.5 is due to the different length L of the bowtie arms

in the two measurements. At 10 GHz, the bowtie arm length was L ~ .14λo∙ The 10 GHz pattern would
correspond to a frequency of about 29 GHz for the L = 1.5 mm bowtie arm in our mixer. At 94 GHz,

the arm length L was 2Λo, which corresponds to about 400 GHz for our bowtie. Discussions with Rick

Compton about his bowtie theory suggest that the lobes in the E-plane pattern increase in angle as the

bowtie is made longer. Contrary to the initial expectations of frequency independent beam patterns,
the bowtie pattern in our mixer is probably changing significantly throughout the entire frequency range

over which we have measured response.

The beam from the bowtie is focussed by a fused-quartz hyperhemisphere. The optical behavior

of a hyperhemisphere is discussed by Rutledge, Neikirk and Kasilingam (1984). It is used because it is
capable of aplanatic focussing, i.e., it can focus a beam with no aberrations. Our hyperhemispheres are

cut from spheres of fused quartz which have a radius r = 3.18 mm. Assuming that the quartz has an

index of refraction ∏q = 2, the sphere is truncated to a height of r(l + 1∕uq) = 4.76 mm for aplanatic
focussing.
Further focussing of the beam is provided by a plano-convex plastic lens with a design focal length

of 2.4 cm. The lens in Bowtie 1 is machined on a lathe from Teflon. In Bowtie 2, The lens is made
by pressing polyethylene, which has been heated to about 120 C, in a vacuum mold. A plano-convex

83

Figure 4.6 Bowtie mixer E-plane beam patterns. The solid lines show the measured E-plane
patterns of Bowtie 2. The dashed lines show Gaussian profiles which match the patterns at
their half power points, a) Measured at 115 GHz. The full angle between half power points
is 8.6°. b) Measured at 230 GHz. The full angle between half power points is 6.7°.
lens shape is used to minimize spherical aberration for the anticipated beam shapes (Jenkins and White,

1957). The curved front surface of the polyethylene lens has a radius of curvature of 1.27 cm, the

lens thickness at its center is .38 cm. Polyethylene lens design is based on an index of refraction at
submillimeter wavelengths of np = 1.52 (Smith and Loewenstein, 1975).
Relatively careful measurements of the E-plane beam pattern of our mixer are shown for two

frequencies in fig. 4.6. At 115 GHz, the beam is 8.6° FWHM (full width, half maximum). At 230 GHz
it is 6.7° FWHM. Low resolution measurements show an H-plane pattern with about the same angular

extent as the E-plane. Gaussian profiles that match the measurements at the half power points are shown

in the figure. Both beams show reasonably good profiles. They are not as good as scalar feedhom
beams, which would deviate only slightly from gaussian shapes if plotted on this same scale. However,
the power is reasonably well contained in a single forward lobe, and we can expect, therefore, to achieve

reasonably efficient coupling of that beam to a radiotelescope.
The patterns for the mixer shown in fig. 4.6 are much cleaner and simpler than the patterns for the
bowtie antenna shown in fig. 4.5. The reason for this is diffraction. For extremely short wavelengths, the
beam pattern of the mixer would have the same structures in it as are shown in fig. 4.5, except the angular

scale would be compressed by a magnification factor of four, due to focussing by the hyperhemisphere

84
and plastic lenses. For instance, lobes at ±30° in the bowtie pattern would be focussed to ±7.5° in the
mixer pattern. However, both the hyperhemisphere and the plastic lens are small enough compared to the

wavelength of radiation being focussed, that the sharp structures in the bowtie pattern must be smeared
out by diffraction. The small sized lenses act as low pass filters to high spatial-frequency components

of the patterns shown in fig. 4.5. Up to 230 GHz, these low pass filters work extremely well, as there
is very little high spatial-frequency content in the mixer beams, fig. 4.6. Diffraction effects become less
important as wavelengths become shorter, so above some high frequency, the mixer beam pattern will
degrade.

Shape is not the only important characteristic of the mixer beam. The overall coupling of the bowtie
antenna pattem to the mixer pattem must be considered. Power in the high spatial-frequency components

of the antenna beam does not show up in the main Gaussian lobe of the mixer beam. The overall coupling
efficiency can be inferred only approximately from the mixer measurements presented below, it is on the
order of 20% to 50%. This compares to numbers well over 90% for feedhom-waveguide based mixers.

In this section, the optics of the bowtie mixer have been described, and measured beam patterns

have been shown. Some of the relevant theoretical descriptions of bowtie operation (Rutledge, Neikirk,
and Kasilingam, 1984) have been briefly described. The bowtie mixer is shown to have a beam pattern
which is reasonably good at 115 and 230 GHz.

4.4 IF Circuit
The output port (IF) circuit of the bowtie mixer is designed to improve mixer gain, short out

voltages above the output frequency, and aid cooling of the SIS junction. The arrangement is shown in

fig. 4.7. The capacitor is formed by pressing the IF connection wire close to the mixer block, using a
.25 mm thick piece of crystal quartz as an insulator. The inductance is achieved by slightly coiling the

IF connection wire on its way to the mixer output SMA connector.

The improvement in mixer gain comes about because the IF circuit acts as a matching transformer

between the SIS and the following IF amplifiers. In chapter 3, it was seen that a properly operating
SIS has an IF output impedance whose magnitude is 100 Ω or higher.

This is mismatched from

85

(b)

Figure 4.7 Output (IF) circuit, a) The left side of the SIS is shorted to ground. The right side
is connected to the output (IF) port. The flat output wire is capacitively coupled to ground
as close to the SIS as possible. The wire going to the coaxial connector on the right has a
loop in it to provide inductance, b) The lumped-component equivalent circuit for analyzing
the output circuit The output circuit will transform an IF amplifier input impedance of Ra
on the right up to tRA at the SIS.

following amplifiers which usually have 50 Ω input impedances. The IF output circuit transforms the
50 Ω following amplifier impedance to 100 Ω at the SIS. For the case that the SIS output impedance
is 100 Ω, this transformation improves mixer gain by a factor of 9/8 (.5 dB). For higher SIS output

impedances, the improvement is more substantial. In the extreme case that the SIS output has an infinite

impedance, the mixer gain is improved by a factor of 4 (6 dB). The transformer will improve mixer

gain for all SIS output impedances above about 71 Ω.

The simple LC circuit shown in fig. 4.7 works as a 2:1 transformer over the output frequency range

of 1.1-1.9 GHz. To transform an amplifier input impedance Ra up by a factor of t at angular frequency
ω, the desired capacitance is
C = ÆI
idtRA

(4.4.1)

86

Figure 4.8 Reflection coefficient of IF circuit. The transformation properties of the IF circuit
are verified by measuring reflection return loss from the mixer with a 100 Ω chip resistor in
place of the SIS. The design IF band is centered at 1.5 GHz.

The inductance is chosen so that there is no net reactance at the design frequency,

L = tCR⅛.

(4.4.2)

The IF amplifer-isolator combination used in these experiments has Ra = 50 Ω. This is transformed

by t = 2 to 100 Ω at the SIS. Designed for an IF center frequency of ω∕2π = 1.5 GHz, the desired

capacitance is C = 1.1 pF and the inductance is L = 5.3 ∏H.
The circuit built to do this transformation is physically about .7 cm long, which is about .35λ at

the design frequency. The lumped circuit description of the circuit given above can be expected to be
inexact. The actual circuit is tuned using a network analyzer. A 100 Ω chip resistor is put in place of the
SIS while this circuit is being built. Transformer operation is then verified by looking at the mixer block
power reflection coefficient with the network analyzer attached to the mixer output If the transformer
is operating correctly, the 100 Ω chip resistance appears to be 50 Ω to the network analyzer, and there
is no power reflected from the mixer port The capacitance and inductance can be adjusted to achieve
the proper transformation ratio at the proper IF center frequency. The power reflection of the finished

transformer is shown in fig. 4.8. At 1.1 GHz, the power reflection is -14.1 dB, this is the worst point in
the IF band. At 1.58 GHz, the minimum reflection of -33.1 dB is achieved.

87
At frequencies immediately above the IF, the transformer appears as a capacitive short to the SIS.

Since the capacitor is only 2 mm away from the SIS, it will present a short circuit to frequencies as high

as about 15 GHz at the SIS. Smith and Richards (1982) show that an SIS mixer is susceptible to saturation

if voltages at or near the output frequency have a root mean square (rms) value of more than about
Ahvw/e. By presenting a short circuit at out of IF band frequencies, total rms voltage is minimized.

Recently, Weinreb (1987) has suggested that low impedance following amplifiers can reduce saturation
problems without reducing receiver sensitivity, even though mixer conversion efficiency is lowered.

Broad output bandwidths from a mixer such as Weinreb proposes could only be achieved through a
close integration of the SIS mixer diode and the transistor which does the first stage of IF amplification.

The capacitor in the mixer output circuit also aids in cooling the SIS diode. The IF line is made

to have the proper capacitance to ground by bolting it tightly in a sandwich of mixer block-insulator-IF
line-insulator-copper. The mixer block is extremely well heat sunk to a liquid helium bath (or to a closed
cycle refrigerator when used at OVRO). Most electrical insulators are fairly good thermal insulators as

well. However, crystals such as quartz have high thermal conductivity. By using crystalline quartz for
the insulator in this capacitor, the IF line is well heat-sunk to the mixer block. As a result, it serves to

help cool the SIS to which it is electrically connected.

4.5 Mixer Block

Two bowtie mixer blocks have been fabricated and tested. Results from the first one have been
previously reported (Wengler et al., 1985a, 1985b). In this section, the physical design of the mixer
blocks is presented. The primary difficulty of the bowtie mixer blocks has been a proper heat sinking

of the SIS junction to the mixer block. A less fundamental consideration dictating design choices is that
the bowtie mixer block be usable in cryogenic systems designed for waveguide-feedhom mixer blocks.

As with cryogenic mixers of many different sorts, the metal used for mixer block construction

is oxygen free copper, known usually as either OFC or OFHC. This material is chosen due to its

extremely high thermal and electrical conductivities at cryogenic temperatures, and its relatively easy
machinablility. An OFC mixer block bolted directly to the cold plate of a helium cryostat or closed-cycle

88

SIS-Bowtie
→------------1 cm-------------«ή
Figure 4.9 Cross section through Bowtie 1. The first mixer block is made so that the quartz
hyperhemisphere is located in a hole drilled in the copper mixer block, and held in place by
the plastic lens, which is attached to the front of the mixer block. IF connections to the SIS
are not shown in this figure.
refrigerator may be considered to be perfectly heat sunk, at least for temperatures (1-5 K) and thermal

power levels (milliwatts) involved in mixer design.
A cross-section through the first mixer block (Bowtie 1) constructed is shown in fig. 4.9.

photograph of Bowtie 1 is shown in fig 4.10. The block is basically two-sided. Radiation is incident
from the front, so the front side of the block serves to mount the lenses for the mixer. For Bowtie 1,

the quartz hyperhemisphere is placed in a hole which is drilled nearly through the mixer block. The
hyperhemisphere is held in the block by direct contact with a plastic (Teflon or polyethylene) lens which

is mounted with spring-loaded bolts to the mixer block. The hole in the mixer block must be made
slightly oversize since copper will shrink more on cooling than quartz will. In Bowtie 1, indium was

pressed around the base of the hyperhemisphere to mechanically immobilize it, and to improve thermal

conduction to it.

Mounted on the back of the mixer are the IF circuit and the SIS itself. The IF circuit described
in the previous section is used, except that the insulator for making the capacitor in Bowtie 1 is mylar.

(The use of crystal quartz for that insulator is an improvement made in Bowtie 2.) The SIS, which is
on a 3 × 3 × .25 mm fused quartz substrate, is laid on the back of the quartz hyperhemsiphere through
a hole of sufficient diameter drilled from the back of the block. By making this hole just large enough

89

Figure 4.10 Photographs of Bowtie 1. (a) ]⅞om the RF side, the plastic lens is shown out
of place so that the SIS-bowtie can be seen through the fused quartz hyperhemisphere, (b)
Rom the other side, the ground and IF connections to the SIS-bowtie can be seen. These
connections are made from springy metal to hold the SIS against the hyperhemisphere.

to clear the SIS-on-quartz, it can be used for locating the SIS relative to the hyperhemisphere with
reasonable accuracy. The SIS is held against the hyperhemisphere by two electrodes made from springy

beryllium-copper shim stock. One electrode is attached directly to the block to provide a ground to one

side of the SIS. This electrode should also present a high thermal conductivity path to the mixer block
for cooling the SIS. The other electrode is insulated from the block by a sheet of mylar. It is trimmed

to size to provide about 1 pF capacitance, as required for the IF circuit.
Cooling of the junction to the temperature of the mixer block was achieved only when indium

was painstakingly pushed into all the gaps between the quartz hyperhemisphere and the mixer block.
It was also found necessary to place a sheet of black polyethylene between the hyperhemisphere and

90

LSIS-Bowtie

Quartz Flat
-*------------1 cm------------ *j
Figure 4.11 Cross section through Bowtie 2. The hyperhemisphere in this mixer is glued to a
thin quartz flat, which is indium-soldered to the copper mixer block.
the Teflon front lens which held the hyperhemisphere into the block. The tentative conclusion is that

thermal radiation from the approximately 100 K radiation shield in the test Dewar is sufficient to heat
the fused quartz hyperhemisphere above mixer block temperature unless that radiation is blocked by

black polyethylene on the cold stage of the cryostat.

The second mixer block (Bowtie 2) differs from the first primarily in the way in which the quartz

hyperhemisphere is attached to the mixer. As shown in fig. 4.11, the hyperhemisphere is glued to a flat
piece of quartz, which is soldered to the copper mixer block. Indium is used for the quartz-copper solder
joint, lead-tin solder does not work. The hyperhemisphere is glued to the flat with superglue (Eastman

910 adhesive).
Their are a number of advantages to this mounting scheme over that used in Bowtie 1. The metal
sides of the hole in which the hyperhemisphere fits in Bowtie 1 are absent in Bowtie 2. The beam pattern

of Bowtie 1 may have been affected by reflection or diffraction from the walls of this hole. It is also

possible in Bowtie 2 to change the size of the hyperhemisphere. In Bowtie 1, a larger hyperhemisphere
would not fit in the hole, while a smaller one would not have been held in place by the front lens. It

had initially been expected that the mounting scheme in Bowtie 2 would make for very good junction
cooling.

It was intended that the flat piece of quartz to which the hyperhemisphere is glued could be made

91
from crystalline quartz, which has high thermal conductivity. Indium soldering that quartz to the mixer

block would provide a high thermal conductivity connection between the mixer block and the quartz flat.
Thus the SIS-on-quartz would be laid flat against a well heat-sunk piece of crystal quartz, and radiation
falling on the hyperhemisphere could not heat up the SIS. However, it was discovered that fused quartz

and crystalline quartz shrink by different amounts when cooled. When the fused quartz hyperhemisphere
is glued to the crystal flat, the quartz hyperhemisphere is broken by differential contraction on cooling.

Thus, it has been necessary to use fused quartz flats in Bowtie 2, since all of the hyperhemispheres are
made from fused quartz. To achieve efficient cooling of the SIS, it is necessary, as in Bowtie 1, to place
a sheet of black polyethylene between the plastic front lens and the hyperhemisphere.

Both bowtie mixers have been built with hyperhemispheres with radii of about 3.2 mm. Referring
to fig. 4.1, this should result in a beam with a half width of about 7 mm at the front of the plastic lens.
This matches reasonably well to the beams which exit the feedhoms of waveguide-feedhom mixers that

are used at OVRO (Sutton, 1983; Woody, Miller and Wengler, 1985). As a result, it has been possible
to use test Dewars originally constructed for use with these other mixers, but more importandy, it has
simplified the testing of the bowtie mixers at OVRO.

4.6 Bowtie Receiver Lab Measurements
The true test of a low-noise receiver for radioastronomy is to put it on a telescope and look at
spectral lines from known astronomical sources. The signal to noise ratio of those measurements can be
determined, and a system noise temperature can be assigned to the whole telescope-receiver-atmosphere

system. However, the first step in testing a receiver is to measure some of its properties in the lab. The
receiver does not typically go straight to a telescope because 1) time on a telescope is at a premium, and
therefore nearly completely given over to already proven receivers, and 2) performance on the telescope
will be much better if the receiver is properly coupled to it, which requires knowing a lot about the
receiver. The SIS receiver is more difficult to test in the lab than Schottky diode receivers. This is
primarily due to noise and instabilities associated with the Josephson effect in the SIS.

The receiver set-up for laboratory measurements is shown in fig. 4.12. LO is injected into the signal

92

Mylar
Beamsplitter

Figure 4.12 Receiver test set-up. The LO is combined with the test signal by partial reflection
from a 25 μm thick mylar sheet. The SIS-bowtie mixer and the first IF amplifier are in the
receiver cryostat. The mixer output is further amplified and filtered outside the cryostat, and
then measured with a power meter.
path by reflection off a 25 μm thick mylar beam-splitter. This reflects ~ 1% of the power from the LO

source into the cryostat. A thicker beam-splitter is used when higher LO reflection is required, but this
increases the reflection loss which the signal suffers as it is transmitted through the mylar. The LO and

signal are mixed in the cryostat, and the first stage of IF amplification is done in the cryostat. Outside
the cryostat, the IF is further amplified, and passed through a filter to remove out of IF band power.

Finally, the IF power is detected in a power meter.
The receiver cryostat layout is shown in fig 4.13. The RF radiation passes through a mylar vacuum
window on its way into the cryostat. The cryostat has a 100 K radiation shield, a single crystal quartz
window is mounted in this shield. The quartz blocks most of the 300 K radiation from outside the

cryostat since it is lossy at infrared frequencies, but it passes near-millimeter radiation with low loss.
The quartz is coated with black polyethylene which helps block infrared, and also acts as an imperfect
anti-reflection coating. The bowtie-SIS mixer is mounted on the cold plate, which is directly cooled

by liquid helium in the cryostat. The IF output of the mixer passes through an isolator, and is then
amplified by a GaAsFET amplifier, both of which are mounted on the cold plate. A superconducting
coil is mounted directly behind the mixer block. By passing a current through this coil, a magnetic field

93

Mylar
Vacuum
Window

RF-

∕j
Quartz
Window
1OO K
Figure 4.13 Receiver cryostat. RF passes through a mylar vacuum window and a Quartz
window at 100 K on its way to the bowtie mixer. The mixer, and the first stages of IF
amplification are mounted on the cold plate of this liquid helium cryostat

is produced which partially suppresses Josephson effect currents in the SIS (Tucker and Feldman, 1985;
Barone and Patemö, 1982).

The isolator-amplifier combination has performance which is completely characterized by a mini­
mum noise temperature, ‰, a maximum gain, Γifo, and an input impedance, 5⅛. The input impedance
of the combination is simply the transmission fine impedance of the isolator, 50 Ω for the standard ele­

ments used here. The minimum noise temperature and maximum gain for the combination occur when
the output impedance of the mixer, Zm , is matched to the isolator-amplifier input impedance of 50 Ω.
Isolator-amplifier response when Zλ∕ y 50 Ω is

I⅛ = Γifo

4∕2⅛f∕⅛

l¾ + ⅛Γ
3⅛ = 7⅛o

(4.6.1)

l⅞ + ⅞l
4∕i!jvi-¾F

Rm and fi⅛ are the real parts of Zm and Zif respectively. Both gain and noise degradation are due to

the mismatch between the mixer and the isolator. Without the isolator, gain would vary with mismatch

as in (4.6.1), but noise would have a more complicated behavior.
The isolator-amplifier gain and noise can be measured by using the SIS as a noise source (Woody,
Miller and Wengler, 1985). Consider the SIS dc biased at ⅛ > Vgap∙ The current flowing across the

SIS has shot-noise statistics just as the current in a Schottky diode or in a field-emission vacuum tube

94

Figure 4.14 I-V and IF power vs. de bias voltage. Below 3 mV, heterodyne response is seen,
this is for a waveguide mixer at 95 GHz (Woody, Miller and Wengler, 1985). Above 3 mV,
there is no heterodyne response, and the SIS shot noise is used to calibrate the IF amplifier
chain as described in the text.
does. IF power and the I-V curve are shown in fig. 4.14. The equation for the I-V above 3 mV is linear,
io = G(⅛ - V∕)

(4.6.2)

where G is the dynamic admittance of the I-V in the linear region, and Vj is non-zero because io does
not extrapolate to zero at Vo = 0. The standard expression for the shot noise ? in a current io is

(i2) = 2ei0B

(4.6.3)

where B is the bandwidth in which the fluctuations are measured. The standard expression for the
Johnson, or thermal noise coming from a resistor of admittance G in thermal equilibrium is
<⅞}=4⅛TGB.

(4.6.4)

95

The SIS shot noise source can be assigned a “shot temperature” 2shot by combining these two
expressions. Using (4.6.2) for Io, the biased SIS looks like a thermal source of temperature
Tshot =

(Vo — Vi)
5.8 K
(Vo - Vt)
1 mV

(4.6.5)

This result, based simply on the assumption that the SIS has shot noise in its dc current, can also be
derived from the expressions used by Tucker in his noise theory (Tucker, 1978; Tucker and Feldman,
1985).

The reading on the power meter at the end of the IF amplifier chain, Pm, is linearly related to the
noise power at the output of the mixer,

Pm = & (Tout + 7⅛)

(4.6.6)

where the mixer output power is ⅛g7buτB. Both 3⅛ and a can be determined through measurement

of Pm as a function of V0∙

This method of calibrating the IF amplifier chain does depend on 1) finding a sufficiently linear
portion of the I-V to use, and 2) the theory of current conduction in an SIS being correct. The first

requirement is generally met in our experiments, although other experimenters have claimed to get
unbelievable results when they apply this method. This may be because the isolator between the SIS

and the first IF amplifier makes our system less sensitive to small deviations from linearity in the I-V.

The second requirement has yet to be tested directly, and could easily be true for some SISs, but not
all. For our junctions, however, the 2⅛ calculated in this fashion typically agrees quite well with IF

amplifier noise measurements made with standard techniques.

This method, as described, is valid when a single SIS is used as the mixing element. For two SISs
in series, there is no consensus on whether the constant of proportionality, 5.8 K∕mV, in (4.6.5) should

be halved (Tucker, 1983), multiplied by √5 (van Kempen et al., , 1981), or left alone. Bowtie 2 was

tested with a detector which consists of two SISs in series. The characterization of the IF amplifier chain
for these measurements suffers from uncertainty about the correct way to deal with the shot noise from
these SISs in series.

96
The fundamental measurement of mixer/receiver performance is to measure the IF power as a
function of the input signal power. Blackbodies provide power ⅛b7⅛B at the mixer input, where B is

the bandwidth in which the radiation is measured, and
hv/ks
ehυ∣lcBTι, — 1

(4.6.7)

is the Planck-corrected “signal temperature” of a blackbody with physical temperature T¼, at frequency

V. Keeping terms to first order in v, the Rayleigh-Jeans blackbody signal temperature is

Tg = Tl — hv∕2kB

for hv

(4.6.8)

Neglecting the small correction proportional to hv will result in calculated receiver noise temperatures
which are low by an amount on the order of hv∕kβ, This amount is only a few percent of the receiver
temperatures reported here, but in some SIS receivers the total receiver noise temperature is within an
order of magnitude of this value, so it should not be neglected.
Measurements are made with a room temperature blackbody, assumed to have Tl = 290 K, and a

blackbody cooled with liquid nitrogen, assumed to have Tl = 80 K. These are called the “hot load”
and the “cold load” respectively. The receiver noise temperature depends on the ratio of the IF power

measurements with hot and cold loads at the mixer input,

Tr =

th-tc
y— ι

-Tc

(4.6.9)

where Y is the ratio between IF power measurements with hot and cold loads applied, and 7⅛ and

Tc are the signal temperatures (4.6.7) of the hot and cold loads. The calculation of receiver noise
temperature neither requires nor yields any information on the separate contributions to gain and noise
of the SIS mixer and the IF amplifier chain.

To derive the mixer gain and noise from the IF power measurements, (4.6.6) is used to determine

the mixer output temperature Tout as a function of the input signal, Ts- The mixer gain Γ⅛f and noise
temperature Tm are then determined by the linear relationship

Tom = Γ⅛f (Tm + Ts) ∙

(4.6.10)

97
The parameters in (4.6.6) are calculated based on what is really only a theoretical model of noise in

the SIS. It is more likely that mixer gain, Γju, is estimated improperly by this method than that mixer

noise is. The model prediction that the linear portion of the I-V should be associated with a linear SIS

noise power is verified in all mixer measurements, of which fig. 4.14 shows one example. It is difficult

to conceive of a physical system in which a noise power which is proportional to the current Io through
the SIS should extrapolate to any other value than zero when ∕0 is extrapolated to zero. However,

it is relatively easier to imagine that physical effects not included in the theory cause a constant of

proportionality other than 5.8 K/mV in (4.6.6). From this reasoning, it follows that the intercept, ½-, is

more reliably known than is the slope of the line in K/mV. Just as the receiver temperature Tr depends
only on the ratio of IF powers with hot and cold loads applied, Tm depends on the ratio of mixer output
temperatures. This ratio is unchanged if values other than 5.8 K/mV are used in (4.6.6). However,

calculated mixer gain, Vm, varies inversely as the assumed constant of proportionality. So, Tr is the

least model dependent of the parameters reported here, Tm relies on only one additional assumption
about SIS shot noise, and Γ⅛f depends on the accuracy of both slope and intercept in (4.6.6).

Measurements of the bowtie receiver in the lab are intrinsically double sideband (DSB). Mixer
IF output signal at the output frequency uq can be due to downconversion of RF input signal at two

different frequencies, plo + vo, the upper sideband (USB), and ⅛ro - w, the lower sideband (LSB). The
hot and cold biackbody signal sources used in the laboratory measurements provide equal amounts of

input power at each sideband. If the receiver responds equally to each sideband, then the gain and noise
measured with hot and cold loads are called (DSB) values. All gains and noises reported here are DSB

values unless otherwise noted. For a mixer with equal response in each sideband, the corresponding
single sideband (SSB) gain and noise are related to the DSB values,

ΓsλP = Γm∕2
(4.6.11)
γ7-tSSB

_ ozP
-i-MfR - ^-ιM,R∙

The bowtie mixers are almost certainly DSB, so (4.6.11) shows how to estimate their SSB performance
from the DSB measurements reported here.

To measure mixer and receiver performance at some frequency, an LO at that frequency must be

provided. The LO power causes a voltage ‰>, at frequency iio, across the SIS. For efficient mixing,

98
‰> must be about ⅛rκ>∕e. The LO power that is absorbed in the SIS when this voltage is produced

could be calculated from Tucker’s theory (Tucker, 1978). However, there is a simpler calculation which
yields the same result, and emphasizes the fact that SISs are photon counters. If the SIS is at some fixed

dc bias voltage on the first photon step below the gap, the dc current through the junction rises linearly
with applied LO power for small LO powers. When the current rises to about 1/4 of the value it would

have if the SIS was dc biased at just above Vgap, the photo-response of the SIS begins to saturate,
i.e., the current rises more slowly with additional applied power. Optimum mixing occurs when the

LO power is just above this saturation power. So the current that must flow through the SIS due to
absorbed LO photons is about GjvVgap∕3. This is about 15 juA for a 50 Ω lead alloy SIS. Tucker has

shown (Tucker, 1978) that an SIS has a current responsivity of “one electron per photon” as discussed
in chapter 2. The optimum absorbed LO power in an SIS is found by equating the electron flux acros
the SIS to the photon flux which produces it,

jdabs

<¾∙Vgap hv^o

rro
= 6 nW
100 GHz

(4.6.12)

1,

for a 50 Ω lead alloy SIS.

The required power scales linearly with Gw, so a 25 Ω junction must absorb twice as much LO power.
If two 25 Ω junctions in series are used for mixing, as is the case for the Bowtie 2 measurements

reported here, the necessary absorbed power will be four times the amount in (4.6.12).
For the Bowtie 1 experiments, LO power was generated by using a Klystron tuned to 116.5 GHz to

drive a Schottky-diode cross-guide frequency multiplier (Schneider, 1982). Sufficient power to drive a
single 50 Ω SIS was available up to the fourth harmonic, 466 GHz, with this system. For the Bowtie 2
experiments, a Gunn-diode oscillator at 75 GHz drove a Schottky-diode multiplier, to provide sufficient
LO power up to the fourth harmonic, 300 GHz. 420 GHz LO was supplied by multiplying a Klystron

oscillating at 140 GHz. LO at 525 and 761 GHz was supplied by a far infrared laser built by Dan

Watson.

An example of the raw data for Bowtie 1 measured at 466 GHz is shown in fig. 4.15. Results
are shown both with and without magnetic field applied to the SIS junction. When no magnetic field
is imposed on the SIS, the IF power has a steep slope in the dc bias range 1.9 mV < Vr0 < 2.2 mV.

99
The IF power with cold load applied shows sharp structure at Vo = 1.93 mV. It is impossible, from this
data, to say anything sensible about heterodyne performance. However, when magnetic field is applied

to the SIS, The IF power curve becomes flat over a region of .15 mV width, just below the structure at
Vo = 1.93 mV. One can quantitatively determine an IF power difference between hot and cold loads in

this case, and mixer and receiver performance can be calculated based on this difference.
The structure at 1.93 mV, and at other voltages indicated in the figure, are due to the ac Josephson
effect, described in section 1.2. The arrows labelled J2 and J3 in fig. 4.15 show the voltages at which

the Josephson frequency, (1.2.3), is exactly twice and three times the LO frequency. Structure in the
I-V or IF power curve at these voltages are due to mixing of the Josephson and LO oscillations.

The large rise in IF power below 1.9 mV with no magnetic field, and below 1.5 mV in the presence
of magnetic field is also attributed to Josephson effect currents. Its cause is the chaotic effects associated
with ac Josephson currents. In the dc I-V, fig. 1.3b, this chaos resulted in a region of forbidden dc bias
voltages, between 0 and Vp. With LO applied, this chaos produces a large output noise power from the

SIS. The difference between this chaotic power output when the mixer signal is a hot or cold load is
almost certainly not due to linear mixing of the LO and signal radiation.

Magnetic field is applied to the junction in an attempt to suppress Josephson currents. In larger

area SISs than those used here, it is possible to completely eliminate Josephson currents with a magnetic
field on the SIS (Barone and Patemö, 1982). However, in the SIS illustrated here, the Josephson currents

are lowered by only a factor of two with the application of field. Even with field applied, Josephson
structure in the I-V and IF are still apparent. However, the value of Vq at which the SIS is chaotic is
lowered by about .5 mV, allowing heterodyne response to be observed.
The SISs used in the testing of Bowtie 2 had much better quality I-Vs, and there is more effective

suppression of the Josephson effect than for the SIS in Bowtie 1. Fig. 4.16 shows raw data for Bowtie 2

at 225 GHz. The detector in Bowtie 2 consists of two SISs in series. If the two SISs were identical,
the series array would behave just like a single SIS, except the voltage scale in the problem would be
twice as large. This is essentially what is seen in the figure. This data shows a larger non-chaotic region

than the data in fig. 4.15. For this reason, the results from bowtie 2 are less likely to be contaminated

100

Figure 4.15 I-V and IF power at 466 GHz in Bowtie 1. a) With no magnetic field applied, the
IF power rises quickly as the dc bias is lowered, reliable heterodyne results seem unlikely, b)
With magnetic field applied, their is a flat region in the IF power curve in which heterodyne
response can be accurately measured. The voltages marked J2 and J3 are where the ac
Josephson frequency is respectively twice and three times the LO frequency.

101

Figure 4.16 I-V and IF power at 225 GHz in Bowtie 2. The IF power is measured with a small
magnetic field applied. A broad, flat region of heterodyne response is seen above the high IF
power region associated with chaotic noise in the SIS.

by non-heterodyne response. The results for Bowtie 2 are generally better than those for Bowtie 1,
presumably because the I-V for Bowtie 2 is more nearly ideal.

The mixer and receiver results for Bowties 1 and 2 are listed in tables 4.2 and 4.3. They are
plotted in fig. 4.17. The data is not corrected in any way for losses in the signal path between the
blackbody signal source and the SIS. Performance both with and without magnetic field applied are
shown. The major effect of applying magnetic field seems to be to lower mixer gain. The implication

is that Josephson currents are part of a useful gain mechanism for this mixer. Examination of the mixer

temperature results show that the Josephson currents do not necessarily produce extra mixer noise, as

has been postulated (Tucker and Feldman, 1985). The gain measurements for Bowtie 2 in table 4.3 are
difficult to estimate because two SISs in series are used as the detector in these experiments. The gains

reported here are calculated as though a single SIS detector had been used. These gains would go up
by a factor of -<∕2 (1.5 dB) or 2 (3 dB) if one or the other of the possibilities mentioned above for noise

102
Table 4.2 - Bowtie 1 Results (DSB)
LO, GHz

T⅛EC.K

7mix, K

Gain, dB

116
233
233
349
349
466

173
263
365
410
639
741

94
179
193
317
329
434

—6.6
—6.9
-9.4
-6.7
-11.9
-11.7

Mag. field
no
no
yes
no
yes
yes

Table 4.3 - Bowtie 2 1 Results (DSB)

LO, GHz

‰c. K

Tmx» K.

Gain*, dB

150
150
225
225
300
300
420
525
761

198
173
163
165
498
405
492
1160
1360

135
89
109
87
421
269
273

-4.2
-5.5
-3.6
-5.2
-5.1
-7.6
-9.6

Mag. field

no
yes
no
yes
no
yes
yes
yes
yes

* Gain is uncertain, see text.
in series arrays had been used.

Because of the possibility of non-heterodyne mechanisms for IF power modulation, a technique for
relative calibration of mixer gain was used to increase confidence in the performance numbers reported

above. The LO for most of the above experiments was an harmonic of a 75 to 140 Ghz oscillator.
The harmonic was produced by pumping a Schottky-diode with the fundamental frequency, the desired
harmonic is then coupled out through waveguide filters. For the purpose of gain calibration, a 1.5 GHz
signal is also imposed on the diode. Sidebands at ran ÷ 1.5 GHz are then produced on the Schottky.

The 1.5 GHz power level is kept low enough so that the sideband power is much less than the LO power
at ⅛o which is coupled out of the Schottky.

These sidebands can be used as a test signal of unknown absolute power. By monitoring the mixer

output on a spectrum analyzer, changes in mixer gain can be measured as dc bias voltage, ⅛, is changed.

The results of a measurement of Bowtie 1 at 350 GHz by this method are shown in fig. 4.18. Because
the calibration signal is reflected off the mylar beam splitter with the LO, it is possible to do these gain

measurements while illuminating the signal port of the mixer with hot (290 K) or cold (80 K) blackbody

103

Frequency, GHz.
Figure 4.17 Mixer and receiver double sideband performance vs. LO frequency. * - Bowtie 1
with magnetic field applied. ■ - Bowtie 1 with no magnetic field. Δ - Bowtie 2 with no
magnetic field, o - Bowtie 2 with magnetic field. These are the same data as in tables 4.2
and 4.3.
radiation. In this way, saturation of mixer gain can also be measured. Finally, it is possible to turn the

magnetic field on and off while observing the downconverted sidebands.
The curve of mixer gain vs. Vb inferred from measurements with hot and cold load can be plotted on
top of the relative gain vs. Vb curves measured with sidebands. The absolute gain scale for the sideband

measurements has been chosen so that the rms difference between sideband gain and hot/cold measured
gain is minimized for the data taken with magnetic field applied. As can be seen, gain measured by the

two methods has the same shape to within 1 dB. Hence, it is reasonable to conclude that the hot/cold

measurements are predominantly measuring heterodyne response.
Without magnetic field, the hot/cold measurements may overestimate actual mixer gain by as much

as about 3 dB. However, the mixer gain is significantly higher without magnetic field, even when

measured with sidebands. In fact, this effect of higher gain was seen at the three frequencies at which

104

-5

-10
CD

’σ

-15

-20

1.5

dc bias, mV
Figure 4.18 Gain measured with sidebands. The measurements were made using Bowtie 1 at
an LO frequency of 350 GHz. The dashed line is measured with cold load and magnetic field.
The dot-dash line, with hot load and magnetic field. Dotted line, with cold load, no magnetic
field. Dash-dot-dot-dot line, with hot load, no magnetic field. The lower and upper solid
curves are the gains inferred from hot/cold load measurements, with and without magnetic
field, respectively.

sideband measurements were done. Measured in Bowtie 1, mixer gains improved with removal of

magnetic field by 2.6 dB at 233 GHz, by 4 dB at 350 GHz, and by 6 dB at 466 GHz. The possible
positive effects of Josephson currents on submillimeter mixing results deserve more extensive study than

they have been given here.

Information about mixer saturation is also obtained from the sideband measurements. In general,
for signal power levels above some permissible maximum, heterodyne response is saturated, IF power
rises by less than 1 dB for each additional 1 dB of signal applied. Sideband power levels in this
experiment are deliberately chosen so that they are a small fraction of the thermal power load on the

mixer. By looking at the sideband gain as hot and cold loads illuminate the mixer, gain compression

due to these different thermal powers are observed. In fig. 4.18, the gain compression is less than .5 dB
for Vo > 1.2 mV. For the higher gains that occur when no magnetic field illuminates the mixer, gain

compression of about 1 dB is observed.

The receiver temperature results reported for Bowtie 2 are compared with the best results achieved

105

Frequency, GHz

Figure 4.19 Comparison of receiver noises for different technologies. Each waveguide point is
measured in a different receiver. The SIS bowtie numbers are from a single receiver (Bowtie
2). The Schottky comer cube numbers are from a single receiver (Betz, 1987; Zmuidzinas,
1987). Each Schottky waveguide datum is from a different receiver (Predmore et al., 1984;
Erickson, 1985 & 1987). Each SIS waveguide datum is from a different receiver (Räisänen
et al., , 1986; Pan et al., , 1983; Ibmegger et al., , 1984; Ellison and Miller, 1987; Theis
DeGrauw, private communication).
for Schottky diode and SIS low-noise receivers in fig. 4.19. The receiver temperatures are divided by
the quantum lower limit, so that as described in chapter 2, noise is quoted in photons instead of degrees

Kelvin. The theoretical limit of one noise photon is hardly even approached by real receivers. Even

ten times the quantum limit represents higher sensitivity than all but a few of these receivers. The

SIS receivers are seen to enjoy a slight advantage over Schottky receivers over most of the frequency
range. At millimeter wavelengths, waveguide-mounted detectors give the best results. At submillimeter

wavelengths, receivers are not as sensitive, even when that sensitivity is expressed in photons. This is
mainly due to the difficulties of building waveguide structures with sufficient accuracy for these very

short wavelengths. At the highest frequencies, “open structure” mixers such as the Schottky-comer-cube

and the SIS-bowtie are more sensitive.

106
4.7 Bowtie Performance on the Telescope
Both mixers, Bowtie 1 and Bowtie 2, have been tested on a 10.4 m diameter telescope at Caltech’s

Owens Valley Radio Observatory (OVRO). The maximum frequency of these tests, however, was only

260 GHz. Bowtie sensitivity up to this maximum frequency was good, but not as good as the SIS-

waveguide results obtained in this same frequency range (Sutton et al., 1985). Perhaps most importantly,
it was shown that the receiver can be calibrated by using hot and cold blackbodies as known signal

sources. It was also shown that the bowtie mixer beam pattern coupled to the telescope with about 60%
efficiency, as compared to about 90% for waveguide-feedhom mixers. Finally, the aperture efficiency

of the telescope when illuminated by the bowtie mixer beam was shown to be excellent at 115 GHz,
and probably as good as with waveguide mixers at 230 GHz.

The bowtie mixers were found to be usable at OVRO as drop-in replacements for the waveguide
based SIS mixers (Woody, Miller and Wengler, 1985) which are usually used there. Most importantly,
calibrating these receivers with hot and cold loads, as described above, was found to result in the correct
measurement of known line intensities at 115 and 230 GHz. Measurements of the line intensities of two

spectral lines of the CO molecule are shown in fig. 4.20. The measurements are of spectral intensity

in units of “antenna temperature”, Ta (Krauss, 1986). The bowtie mixer finds the same intensities for

these lines as other receivers at OVRO have found. The conclusion is that at 115 and 230 GHz, the
bowtie-SIS heterodyne response is correctly calibrated by use of hot and cold loads.
The mixer is coupled to a telescope at OVRO as shown in fig. 4.21. As is usual for millimeter

wavelength telescopes, there are a number of mirrors between the receiver and the primary mirror of
the telescope. The mirrors are designed assuming that Gaussian beams will be propagated through the
system. The beam patterns for Bowtie 2 are shown in fig. 4.6. It is apparent from that figure that these

patterns have a fairly large Gaussian component, so they should couple reasonably well to the telescope
optics.
The efficiency with which the bowtie mixer illuminates the telescope is measured by comparing

the receiver response to hot and cold loads placed at the Cassegrain focus (between the secondary and

tertiary mirrors) with response measured with hot and cold loads directly in front of the receiver. Hot

107

IF channel #
Figure 4.20 Two CO transitions in Orion measured with bowtie-SIS. The data are in 1024 chan­
nels spaced .5 MHz apart. The center channel corresponds to line frequencies of 230.531 GHz
for the J=2-l transition and 115.268 GHz for J=l-0. Signal was integrated for the times
shown.
and cold load illumination produces a change in IF power, ∂7⅛. The coupling efficiency, x, of the

receiver is the ratio of IF response at the Cassegrain focus to the IF response right at the receiver,
<57⅛ (Cass.)
<5⅛ (Rec.)

(4.7.1)

The waveguide receivers in use at OVRO (Woody, Miller, and Wengler, 1985), have x = .9. For the

bowtie receivers, x = .6 over the entire frequency range tested, (100-260 GHz).
Besides coupling efficiently in a spatial sense to the Cassegrain focus, it is also necessary that
the bowtie beam be focussable. In classical optics, things like spherical aberration and coma result in

beams which have minimum focal spots which are bigger than diffraction theory would predict. If the
beam leaving the bowtie mixer has non-spherical equiphase surfaces, it will not be possible to achieve

diffraction limited angular resolution with the OVRO telescope. The telescope on which the bowtie was
mounted is part of a three telescope interferometer. At 115 GHz, it is possible to make extremely high

108

Figure 4.21 Optics on 10.4 m telescope at OVRO. The extreme rays traced are for the -13 dB
contour of the Gaussian beam for which the optics are designed.

resolution maps of the beam pattern of one telescope in the interferometer by scanning that telescope

around while using the other two telescopes as a constant reference. With the bowtie mixer, the aperture
efficiency of the telescope was actually a little bit better than that measured with the waveguide receivers.
The two mixers illuminate the primary in slightly different ways, accounting for the small difference.
The important conclusion is that there is no evidence at all for any sort of aberation in the bowtie beam
at 115 GHz.

At higher frequencies, it was not possible to make accurate beam measurements on the telescope.
However, rough measurements at 230 GHz indicate a beamwidth that is no more than 40% wider than
the diffraction limit for a 10.4 m diameter telescope. A wider beam than the diffraction limit could be
due to inaccuracies in the primary dish of the antenna. It could also be due to a failure to illuminate the

entire primary reflector of the telescope, causing it to behave like a smaller telescope. In any case, there
is no strong evidence that the bowtie mixer beam cannot be focussed at 230 GHz, but the measurements
we made do not completely rule out problems either.

The bowtie mixer was found to have double sideband response, that is, it has equal gain to signals

109

in the upper and lower sidebands. The CO 2-1 spectral line at about 230 GHz was observed with two

different LO frequencies, so that the line was alternately in the upper and lower sidebands. The measured
line intensities under these two tunings differed by 1% (which is within the noise of the measurement).
As a result, it can be concluded that, at least at 230 GHz, the bowtie mixer has double sideband

response to high accuracy. This is expected from the bowtie, since there are no tuning structures to give
differential mixer response over a frequency range as small as the separation of the lower and upper

sidebands, about 3 GHz for the IF used at OVRO.
The bowtie mixer was checked for harmonic response, none was observed. In addition to response

at the upper and lower sidebands, rγjo ± r,o, some mixers respond to higher order mixing products. In
particular, signals at frequencies 2plo ± t¾ may ⅛,e downcoπverted into the IF at ιό. To test this, the
LO frequency was adjusted so that the CO 2-1 line would be at 2∕√lo + vq. The mixer was tuned in

the normal fashion to maximize its response to hot and cold loads. The resulting spectra showed no

sign of a spectral line, to an accuracy of about 2 K, or two parts in 40 for the CO 2-1 line in Orion.
We thus conclude that any harmonic response in this mixer is at least 13 dB below fundamental mixing

response. This is in a mixer tuned in the normal way for maximum fundamental mixing response. It is
quite likely that harmonic response could be achieved with SISs if it were specially tuned to maximize

that response.
In this section, we have discussed the performance of the SIS-bowtie mixer on a radiotelescope.

We have found that, for the millimeter wavelengths investigated, the SIS-bowtie presents no unfortunate
surprises to the radioastronomer. Its performance is generally about two-thirds as good as the SIS-

waveguide mixers which have been used at OVRO (Woody, Miller and Wengler, 1985; Sutton, 1983).
However, a single SIS-bowtie covers the whole 3 to 1 mm range, which requires at least two SIS-

waveguide mixers. In any case, the true strength of the SIS-bowtie should lie at shorter wavelengths,
where waveguide structures become increasingly inefficient and difficult to fabricate. The next step is

to test the SIS-bowtie on a telescope at submillimeter wavelengths.

110

4.8 Conclusion
In this chapter, we have described a prototype low noise mixer for near-millimeter wavelengths (100

to 1000 GHz). A lens-coupled bowtie antenna is used for the first time in a low-noise receiver. The
detectors are lead alloy SIS tunnel diodes of the same sort as are used in waveguide mixers at Caltech’s

Owens Valley Radio Observatory, and at AT&T Bell Labs’ Crawford Hill Observatory.

However,

they have been fabricated so as to maximize their switching speeds, which are estimated to be .3 ps,

corresponding to 500 GHz.

The bowtie mixer has demonstrated good low noise performance from 115 to 761 GHz, in laboratory
measurements. This multi-octave coverage of the near-millimeter is one of the advantages of the bowtie
optics. Above 300 GHz, the bowtie is approximately as good as a number of waveguide based mixers,

each of which covers about a half-octave.

Telescope tests, carried out to a maximum frequency of 260 GHz, show that the SIS-bowtie is

suitable for radioastronomy. At the frequencies tested, SIS-waveguide mixers are better than the SISbowtie. However, the real strength of the SIS-bowtie should be at submillimeter wavelengths. Telescope
tests at these short wavelengths are certainly the next step that should be taken.

Ill

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