Infrared Telescopes and Dyson Shells
Jansky Units, Microwave Telescopes, Infrared Telescopes
One Jansky is 1e26 W/m^{2}Hz. Hz implies bandwidth. The bandwidth of a radio telescope is obvious  the bandwidth in Hertz, probably after a chilled reflector into a chilled low noise amplifier, then through a filter. A wider filter picks up more power, and more thermal noise. So the power in Janksy units for a source is Jy = 1e26 * Power~Received / ( Area * ( f_2  f_1 ) ) with power in watts, Area of the receiving surface in square meters, and frequencies in Hertz.
Infrared telescopes have much wider bandwidth, and infrared is typically characterized with wavelength (usually micrometers) rather than Hertz. We can recast the formula as
Jy~=~1e26*(P/A)*c*(1/\lambda_21/\lambda_1)~=~1e26*(P/A)*c(\lambda_1\lambda_2)/(\lambda_1\lambda_2)
where c is the speed of light. Defining \Delta\lambda~=~\lambda_1\lambda_2 and center wavelength \lambda = \sqrt{\lambda_1\lambda_2}, this simplifies to:
Jy ~=~ 1e26*(P/A)*c~\Delta\lambda/\lambda^2 ~=~ 1e26*(P/A)*c~/( \lambda (\lambda / \Delta\lambda) )
So, how much is that? A parsec is 3.26 light years or 3.0857e16 meters, like a star that appears to move one arcsecond in the sky as the earth moves 1 AU across its orbit. So, a sphere 1 AU in diameter would appear one arcsecond across from 1 parsec away. From 200 parsecs away ( 6.1714e18 meters ), a sphere 100 AU in radius or 200 AU in diameter, and heated by the entire sun's output (3.86e26 W), would be about 60K, and deposit all that power on a 200 parsec diameter sphere with an area of 4.786e38 square meters, a power density of 8.07e13W/m^{2}. We still don't know enough to compute the Jansky units, because we don't know the bandwidth. That is a function of the filtering and imager on our telescope  and if the telescope is not in orbit, the passband and emissions of the atmosphere.
Dyson Shell Server Sky
A shell 100 AU in diameter at 60K enclosing the 3.84e26 Watt sun. Peak emissions at 48μm and a power flux of 0.1366 W/m^{2}. The inband fraction of total black body emissions as a function of the mean wavelength λ and the wavelength concentration λ/Δλ can be computed from this C program: irin01.c. The program expects three command line arguments  temperature T , λ, and λ/Δλ, then computes various parameters, including the fraction of total power inband. It is a bit of a kludge!
$ cc o irin01 irin01.c lm ; ./irin01 300.0 5.6 5.0 T = 3.00000e+02 K peak = 9.65924e+00 μm hckt = 4.79592e+01 wl = 5.60000e+00 μm wr = 5.00000e+00 μm ws = 1.12000e06 μm wl1 = 5.06793e+00 μm wl2 = 6.18793e+00 μm frac = 3.22066e02 frac = 3.22e02
James Webb Space Telescope
NASA's JWST Mid Infrared Instrument (MIRI), scheduled to deploy in 2018, will be the most powerful long distance infrared imager available, with a 6.5 meter mirror. The imager is a 1024x1024 pixel Raytheon Si:As sensor chip assembly (SCA) with 25 µm pixels. The pixels are 0.11 arcseconds.
This table gives the sensitivity, recomputed below,
 (detection limit, 10 sigma, 10,000 seconds)
Assuming 3.84e26 Watts, The point source detection distance is sqrt( fraction * 3.84e26 / 4 π 3.0857e16 ^{2} sensitivity ) parsecs
 = 1.79e4 sqrt( fraction / sensitivity ) parsecs.
 Earthlike planet, 152 PW of IR = 3.56E9 sqrt( fraction / sensitivity ) parsecs.

λ 
λ/Δλ 
sensitivity 
λ_{0} 
λ_{1} 
Δλ 
300K 
300K 
300K 
60K 
60K 









power 
384 YW 
152 PW 
power 
384 YW 

μm 

μJy 
W/m^{2} 
μm 
μm 
μm 
fraction 
parsec 
parsec 
fraction 
parsec 
F560W 
5.6 
5 
0.16 
1.7e20 
6.19 
5.07 
1.12 
3.22e02 
250000 
4.9 
1.43e13 
0.5 
F770W 
7.7 
3.5 
0.25 
2.8e20 
8.88 
6.68 
2.20 
1.30e01 
390000 
7.7 
6.32e09 
85 
F1000W 
10 
5 
0.54 
3.2e20 
11.05 
9.05 
2.00 
1.35e01 
370000 
7.3 
6.72e07 
820 
F1130W 
11.3 
16 
1.35 
2.2e20 
11.66 
10.95 
0.71 
4.54e02 
370000 
7.3 
1.24e06 
1300 
F1280W 
12.8 
5 
0.84 
3.9e20 
14.14 
11.58 
2.56 
1.46e01 
350000 
6.9 
3.74e05 
5500 
F1500W 
15 
5 
1.39 
5.6e20 
16.57 
13.57 
3.00 
1.36e01 
280000 
5.6 
2.79e04 
13000 
F1800W 
18 
6 
3.46 
9.6e20 
19.56 
16.56 
3.00 
9.66e02 
180000 
3.6 
1.44e03 
22000 
F2100W 
21 
4 
7.09 
2.5e19 
23.79 
18.54 
5.25 
1.19e01 
120000 
2.4 
7.98e03 
32000 
F2550W 
25.5 
6 
26.2 
5.1e19 
27.71 
23.46 
4.25 
5.78e02 
60000 
1.2 
1.69e02 
33000 
300K Sum 
0.898 
60K Sum 
0.027 

Using the F2100W and F2550W filters, JWST can detect a 60K 200 AU diameter Dyson shell as a point source almost anywhere in the galaxy. But a Dyson shell probably cannot be differentiated from a gas cloud unless it is close enough to show a disk, which it would at a distance of less than 200 AU / .11 (arc seconds) = 1800 parsecs, encompassing on the order of 100 million stars, perhaps a million G stars like our own.
JWST seems to be optimized for looking for 300K objects, perhaps nearby earthtemperature planets. Dyson shells emit such vast amounts of power that, even at 60K and 1/30th of the sensitivity, they will be visible across most of the galaxy.
We can expect to see "star" clusters of similar temperature (because of similar technology) Dyson shells in "small" regions a few hundred light years across, size dependent on the future discount rate for the source civilization.
If the JWST imager and mirror was colder, with longer wavelength Filters up to "F8000W" or so, it would be optimal for imaging 60K Dyson shells. With 3 times the wavelength, it would need 3 times larger optics to provide the same .11 arcsecond angular resolution, thus a 20 meter main mirror. In order to image a 200 AU diameter sphere at the far side of the galaxy, as a sphere, it would need to be 60 times larger, perhaps a 400 meter main mirror, or a few smaller mirrors arranged as a sparse interferometer.
JWST is beggaringly expensive with current launch technology and solid mirrors, but with welldeveloped thinsat and launchloop technology, far from the sun so the system is cool, we will "soon" be able to accurately characterize any Dyson shell in our galaxy  if there are any to be seen, and if they are warmer than 50K or so. A more computationally efficient Dyson shell might be colder and very much larger, resolvable only by kilometerscale telescopes.
Herschel
ESA's Herschel Space Telescope, active from 2009 to 2013, has a 3.5 meter mirror and a wavelength range of 55 to 672 μm. The Photodetector Array Camera and Spectrometer (PACS) uses two low resolution arrays of bolometers:
Red 
16x32 
6.4as 
Blue 
32x64 
3.2as 
These measure in 3 bands, either 70 alone or 100 and 160 together:
70 
60 to 85 μm 
5 mJy (5σ, 1h) 
100 
85 to 130 μm 
5 mJy (5σ, 1h) 
160 
130 to 210 μm 
10 mJy (5σ, 1h) 
Spitzer
IRAS