fluxcal

Monochromatic flux calibration of celestial sources.

If the goal is to perform a ground-based calibration of the monochromatic flux of celestial sources, then we need to make the following metrology chain work:

emitter calibrated relative to NIST diode
observe that emitter at a distance D1, full-aperture and almost collimated beam
observe celestial source
ratio of fluxes times calflux is the celestial flux.

This has historically been done "monochromatically", in reality done by measuring SED in some dlambda, then dividing by dlambda to get flux density.

Want a region where stellar spectrum is flat, and atmosphere is benign.

Since we want a monochromatic flux, dispersed imaging makes sense. Spectral resolution depends on structure in atmosphere and stellar spectrum.

Going to NIR helps with atmosphere, since aerosols are less of a problem than at historical wavelength of 555.5 nm (Megessier 1995)

Going faint helps us avoid secondary calibrators, and allows us to put the calibration source further away (where fraction distance errors are smaller).

For drone-based calibrator, getting the elevation right is easy, problem is azimuth. If we want part per thousand then cos^2 (one for Lambertian and one for projected area) < 10^-3 so cos < 0.032 ~ 2 degrees. Not so bad!

Can put a star tracker on the emitter and determine orientation after the fact!

Or, even better, put an out-of-band modulated source on the ground and point optical system at it. Now THAT would work!

tasks:

Devise closed-loop gimbaled emitted pointing scheme, decoupled from drone control system
Figure out narrowbanded dispersed imager, can use a standard spectrograph on Magellan, etc but it would be better to use a smaller telescope.
Find propitious wavelengths that are benign both in terms of the atmosphere and stellar spectra. Around 800 nm is good.
Do flux estimates
Figure out how far out of focus the point source would appear. Beam divergence is (mirror diameter)/(range), want this < 1 arcsec so < 5E-6. Smaller apertures are better, from this perspective. Also we can point them wherever we want...
Find out about position-stabilized drone performance, rms position error?
Think about flying one of those white light sources? Ans- no, spectrum is not really that flat.

Do a modulated-source, lock-in amplifier determination of angular dependence of emission from integrating sphere, rotating about output port plane. Is it really Lambertian? (Needs a source without a fiber)

Diameter	minimum range
1 mm	200 meters
1 inch	5 km (actually, 1.3 km to keep PSF below diffraction limit)
180 mm	36 km
1 meter	200 km
6 meters	1200 km

Think about systematics limitations, at part per thousand level

Systematic
Uncertainty in range to source	(dR/R)² < 1E-3, so dR < 0.5E-3 * R, so for 1m dR need R > 2 km	improves with increasing range, for fixed dR. Use differential GPS to get 1 m
Pointing uncertainty for emitter	cos²(phi) < 1E-3 so d(phi) < 1.8 degrees	pretty easy actually, with active pointing to ground.
Temperature dependence of flux calibration		measure and correct
Temperature dependence of wavelength		measure it on the ground, with spectrograph
Atmosphere.	need to know attenuation to 1E-3! Around 800 nm we have Rayleigh term dominating, assuming aerosols are negligible, and atmosphere is 95.3% transmissive. So let's say it's 5% scattering loss, we need to know that to a precision of 2%. Also need to worry about the horizontal component too.
Flux non-uniformity across input aperture	We want uniform surface brightness to 0.5 E-3 across the input aperture, for the local calibration source. Assume it falls off as cos^4(theta) < 0.9995 at edge of pupil. That means the half angle of the beam can't subtend an angle greater than theta = acosd(sqrt(sqrt(0.9995)) = 0.0158 radians, so full angle of beam can be 0.0316 radians, so a point source can be placed a distance of 32D where D is the telescope diameter. So a full-aperture Fresnel lens (with image quality better than an arcsec) can be used in conjunction with a point source to jointly satisfy the flux uniformity and the PSF conditions. In fact at f/32 we could perhaps even use a full-aperture Acrylic asphere rather than a Fresnel lens.
emitter isotropy	The emitter must broadcast light into a cone with opening angle of a couple of degrees with an angular uniformity no worse than 0.5E-3.
Depth of field	What displacement of the source, on-axis, produces a 1 arcsec displacement of the image? This is the displacement on axis that rotates the marginal ray by an arcsecond. Initial angle is atan(0.75/40) = 0.0187478 radians. Distance that rotates marginal ray by 1 arcsec satisfies theta+4.86E-6=atan(0.75/(D+x)) so that tan(theta+1 arcsec) = 0.75/(D+x) and so (D+x)=0.75/tan(theta+1arcsec), and x= 0.75/tan(theta + 1arcsec) - D = 0.01 m, i.e. 1 cm
Chromatic focal shift	400 nm to 700 nm changes focal length from 40 m to 39.5 meters.

from jt.dat, lists Rayleigh and molecular attenuation

nm Rayl mol product

795 0.954 0.998 0.952

796 0.954 0.998 0.951

797 0.953 1.000 0.953

798 0.953 1.001 0.954

799 0.953 1.001 0.954

800 0.953 0.999 0.952

801 0.953 0.998 0.951

802 0.953 0.999 0.952

803 0.953 1.000 0.952

804 0.953 1.001 0.953

805 0.953 1.003 0.955

806 0.953 1.003 0.955

807 0.953 1.002 0.954

808 0.952 1.000 0.953

809 0.952 1.000 0.952

810 0.952 0.998 0.951

Spectrum of Vega and molecular lines in atmosphere, in this regime:
Note that Rayleigh scattering at 555 nm has transmission of 0.876.

Conceptual design:

1) link NIST flux scale to instrument, using 1 cm diameter common aperture and a bright enough source, small enough that it's like a collimated point source at infinity. Diffraction limit for 1 cm aperture and 800 nm light is theta=0.8E-6/0.01 = 16 arcseconds. So a source can be 200 m away and have a diameter of 1.6 cm and match this. Measure integrating sphere internal flux with monitor diode there, I1. Do flux discrimination from background on photodiode in frequency domain (modulate at a few hundred Hz) and on main instrument in spatial domain.

2) remove small aperture and drop flux inside integrating sphere to get same flux on dispersed imager. Take ratio of photocurrents as scaling for obscured-full-aperture sensitivity. If it's a takahashi then cos^4 term is negligible for 1 cm source a distance 200 m away (i.e. < 1E-3). This approach keep horizontal atmosphere piece the same.

3) Now point at source of interest, and do extinction correction vs. airmass in the usual way.

4) Can do this at a variety of discrete wavelengths, avoiding the saturated water bands. Narrowband filter prevents any second order light issues.

an earlier version:

1) dispersed imager with ronchi grating in aperture, so that passband is defined on detector, in dispersion direction. Rotating aperture mask with small and large diameter apertures, or even better put 2 instruments on a sliding stage.

2) NIST photodiode mounted with optical axis that points at a common point 200 m away.

3) integrating sphere with really small output port, as Lambertian emitter. Modulated laser diode source at 808 nm.

4) measure intensity though small (~1mm) aperture on photodiode. Then rotate/swap aperture/instrument in front of dispersed imager to map NIST QE onto dispersed imager. This keeps PSF small. But integrating sphere aperture needs to be small as well. We could actually collimate the beam and swap instruments behind the aperture, keeping illumination the same. In that case the source does not need to be far away, and aperture can be larger, up to size of NIST diode. Question: presumably we do want to mask the NIST diode to avoid edge effects and stay in sweet spot.

Aperture that can be used with NIST diode is 1 cm diameter. Flux from Vega is, according to Bohlin et al, F-lambda = 3.4E-9 erg/cm^2/s/A at 555 nm. Each photon has hbar-omega = hc/lambda of energy so photon flux is Phi=F-lambda/(energy per photon), for zero optical bandwidth, which is equal to F-lambda*lambda/hc. The units are a mess, but mostly cgs. h=6.62E-27 erg-sec, and c=3E10 cm/sec so hc=1.98E-16. Lambda=0.555 microns = 0.555 E-4 cm. So putting all this together we end up with 961 photons per cm^2 per A per second from Vega. That's not very much! Even with QE=100% we get a photocurrent per cm^2 of about

i ~ 1000*1.6E-19 amps per cm^2 per A, so 0.16 fA per cm^2 per A. If we have a 10 nm wide interference filter then that's 100 A so we'd expect 16 fA at 100% QE, per cm^2, from Vega. It's going to be tough to just point a masked photodiode at Vega.

How about an artificial calibration source? Take a canonical laser diode power of 100 mW, narrowband, into 4 pi. The angle subtended by the photodiode aperture is pi R^2/(4 pi Range^2) = (R/2 Range)^2. Take R=0.5 cm, ie 10 mm diameter aperture, and 200 m distance. Fraction of flux that passes through the aperture is (0.05/200*2)^2 = 1.6E-8 so 1.6E-9 Watts. Number of photons per second is this power times lambda/hc=2.8E18 (MKS), and so photon flux is 4.5E9 photons/s * (P/100mW) * (D/1 cm)^2 * (200 m/R)^2, for an isotropic emitter, which at 100% QE corresponds to 0.72 nA. If we do any collimation at all, this only goes up.

5) rotate to larger aperture, so that phi_vega * (Dbig/Rvega)² ~ phi_calibrator * (Dcal/Rcal)²

6) Link flux seen on dispersed imager