Introduction

In its most basic form, an astrophotometer is just a telescope with a photodetector at the prime focus. The telescope serves both to define the field of view of the instrument and to gather more light than the photodetector would collect if it were simply masked to limit its field of view. The field of view is determined by the angle that the diameter of the detector subtends at the focal length of the objective. In the case of an instrument intended for measuring the brightness of an extended source, the flux captured by the detector is determined entirely by the f/number of the optical system and the brightness of the source. It is therefore a low f/number that gives a high sensitivity, rather than a large objective.

The drawing above shows the geometry of the photometer in relation to the sky. For the moment, consider the sky to be a uniformly glowing plane a distance R from the photometer. The idea is that the sky--even though it is really a three-dimensional volume of space--looks just like a glowing plane a finite distance away. In the drawing, the field of view of the detector is a patch of sky of area Σ. Each differential area element of this patch of sky radiates omnidirectionally, but only the fraction bounded by the solid angle Ω actually enters the lens and is focused onto the detector. If the sky has a surface intensity of L watts/m²-sr, then the total flux F landing on the detector is F = L Σ Ω.

Since Ω = π d² / 4 R² and Σ = R² a² / f², the total flux falling on the detector is

F = L (π/4) ( d² / f² ) a²

The distance to the sky, R, cancels out and therefore does not matter to the final result. The sky looks like a uniformly glowing surface an arbitrary distance from the photometer. As you would expect, the flux falling on the detector depends on the intrinsic brightness of the sky, L, and the sensitive area of the detector, a². It also depends on the ratio of the focal length to the objective diameter, called the aperture or f/number, n. The total captured flux is just

F = L (π/4) (a² / n²)

Mechanics and Optics

In order to keep the size and weight of the photometer manageable, the design uses a relatively small lens operating at about f/1. Since the imaging quality of the objective is of no consequence (all it has to do is concentrate light on the detector), the lens can be a simple plano-convex configuration. I used a plano-convex lens of 30mm diameter and 25mm focal length. After mounting (which steals a little of the diameter), the effective aperture of the optical system is just about f/1.

For a tube, I used a length of 1 1/2 inch PVC tubing salvaged from a sink drain tailpiece. (Given the size of the objective lens, it would have been better to use a piece of 1 1/4 inch tube.) To mount the lens, I made a pair of split rings from short pieces sawed from the tube. The overall length of the telescope tube was 4.1 inches including the flange at the end. This dimension is not at all critical. The only reason for making the tube extend past the objective is to provide shading from indirect sunlight.

In order to reduce dark current, and thereby reduce dark current noise, the detector should be cooled. Consequently I pressed the photodetector into a hole in a piece of 1x1/8 aluminum bar to serve as a cold finger. The telescope tube and electronic circuit board mount to the bar. In a final design, the cold finger will extend to a thermal radiator outside the top wall of the payload package. In the photo above, you can see that the cold finger is bent at a 45 degree angle so that when it is mounted vertically, the telescope points at a 45 degree angle.

The Photodetector

The choice of photodetector involved some tradeoffs, but was eventually driven by cost considerations. The clear choice for a detector is a silicon photodiode, since other devices are considerably noisier. Larger detector surface areas produce greater signal magnitudes, but also require larger optical systems to retain the desireable f/1 aperture and the target field of view of a couple of degrees. Instrumentation photodiodes typically have active element diameters from about 1 mm square to more than 10 mm square, and are quite expensive. One source of relatively low cost photodiodes of this sort is Pacific Silicon Sensor. They manufacture a line of suitable diodes--particularly their Series 6 devices. These units start at 1 square millimeter surface area and would be an excellent choice. They are available from Mouser.

Rather than invest in an expensive diode, however, I decided to see what could be done with a very inexpensive small-area device. I selected the Osram SFH229, available from Digi-Key. The active area of this device is 0.56 mm square (0.3 sq mm area), giving a field of view of 1.28 degrees (1.65 sq deg solid angle). This field of view is a little smaller than optimum, but should be quite suitable.

There are two advantages to the smaller diode area. First is that dark current is reduced, and second that noise is reduced. The disadvantage is lower signal strength, but that problem is easily overcome by the simple expedient of longer sampling times.

Electronic Front End

Signal to noise ratio sets the limit of sensitivity of any instrument. Consequently, for instruments that must detect very low level signals, the imperitive is to reduce noise and increase signal. The use of a low f/number optical system serves to maximize the signal placed on the detector, while careful design of the electronics keeps the noise as low as possible. You can find some useful guidance in the Burr-Brown application note Designing Photodiode Amplifier Circuits with OPA128, and the Hamamatsu website.

I used a straightforward integrator circuit to integrate the photocurrent from the photodiode. The critical elements are the choice of op amps, and the choice of integrating capacitor. For the integrator, I used the OPA129 from Texas Instruments. This device has an amazingly low input bias current of only 100 fA, so that the amplifier contributes an insignificant amount of current to be integrated. The integrator stage is isolated by a voltage follower constructed from an OPA27 op amp. The virtue of the OPA27 is extremely low noise.

The photocurrent being integrated is quite small--on the order of pA--so it is important to keep op-amp bias current and integrating capacitor leakage very small. I chose a 1000 pF Panasonic metallized polypropylene film capacitor (ECQ-P1H101JZ, available from Digi-Key.)

It is tempting to use a CMOS analog switch for the integrator reset switch, but leakage currents for these devices are typically much too large to be acceptable. The only practical alternative is a relay. I used a tiny DIP-packaged reed relay that operates on 5V TTL logic levels.

Considering the very small currents this circuit deals with, it is important to lay out the circuit board to maximize path lengths from the power supply to the integrator input terminals. After fabrication, it is important to clean the circuit thoroughly to eliminate any conductive surface contamination.

Knowing the temperature of the photodiode may turn out to be very useful for calibration purposes. The circuit diagram includes a simple temperature detector circuit using an LM335. This device acts like a Zener diode with an avalanche voltage proportional to absolute temperature. So with a simple bias circuit to allow about 50 μA to flow through the device, its terminal voltage follows temperature at 10 mV/K.

The LM335 comes in a TO-92 package that could easily be cemented to the cold finger or pressed into a hole. I did not implement this part of the system in the prototype, but doing so should be very straightforward.

Evaluation

To evaluate the design I mounted the photometer inside a light-tight box with the aperture of the telescope penetrating the wall. The telescope pointed straight up through the top of the box so as to detect sky brightness at the zenith. I connected the signal output to a 12-bit data acquisition system to record integrator output data.

At sunset the integrator saturates very quickly--on the order of a few milliseconds. As the sky darkens through twilight, the integration time gets longer and longer, until, under dark sky, on the order of 100 seconds of integration time is required to get a usable signal. The figure below shows the integrator output as a function of time for dark sky with a full moon a few degrees above the horizon.

Inverting the equation for flux on the detector, the sky brightness is

L = (4 / π) (n² / a²) F

Now the photodiode responds to a total flux F falling on it by producing a photocurrent I in proportion to the total power weighted by the spectral response. The constant of proportionality is K = 0.62 A/W. Consequently

L = (4 / π) (n² / a²) (I / K)

The photocurrent can be computed from the voltage rise ΔV of the integrator over time of integration Δt. The rate of integration is the capacitance C. Consequently,

I = C (ΔV / Δt )

and the sky brightness is

L = (4 / π) (n² / a²) (C / K) (ΔV / Δt )

From the graph below, the integrator voltage rose 1.81 V during in integration time of 200 sec. Using C = 1000 pF, the sky brightness is L = 59 μW/m²-sr.

This result, of course, assumes that the photodiode complies exactly with the datasheet, that the optics are precisely calibrated, and that all losses in the system are negligible. None of these assumptions is completely valid, but the result certainly seems plausible to within an order of magnitude.

But what about dark current? The data sheet for the photodiode indicates that the dark current at room temperature should be about 3 pA. Over 200 seconds, that current should integrate up ΔV = (I_D / C) Δt = 0.6 V, a significant fraction of the night sky brightness signal.

The plot below shows the output of the photometer with the telescope aperture covered. The integrator output is nearly constant, with no voltage rise due to dark current in evidence. Apparently the dark current and other stray currents nearly canceled under the specific conditions of the test--a remarkably fortuitous circumstance.

Conclusion

The questions of calibration, temperature dependence, and dark current suppression still remain to be addressed.