### Astronomy 512 -- Spring 2006 -- Assignment 4 -- Due Tuesday March 28

These problems are all meant to be worked individually. For 1) and 2) you are on your own (of course you can ask the staff questions...). For 3) you can consult heavily with the others in the class, but everyone should submit a unique solution to the problem.

1. Calculate the focal length at a wavelength at 4.0um of a biconvex Germanium lens at a temperature of 100K which has radii of curvature of 2000 and 300 millimeters. If the lens is 50mm in diameter what is the focal ratio when the lens is used without a stop? Suppose you imaged the sky with this lens onto a focal plane, what is the resulting platescale in arcseconds/millimeter?

2. Consider a 1 centimeter thick slab of Germanium at a temperature of 77K interposed in a f/3 beam of unaberrated monochromatic 8.00um light converging toward a focus.

• How much does the focus shift when the slab is introduced into the converging beam?
• Use Snell's law to determine how much transverse spherical aberration appears due to the slab if the beam is 10 cm in diameter when it encounters the slab.

3. Consider an experiment to conduct wide field imaging of the sky at thermal infrared wavelengths in the L' and M photometric bands. Use Oslo to design a refractive imaging system with the following properties.

• 8 cm diameter f/5 telescope objective (which forms the dewar "window" - a fundamentally bad idea (can you see why?) but for our purposes we'll let this slide.

• Reimaging (collimator/camera pair) optics to
• produce a pupil image to locate a cold Lyot stop and filters.
• produce a final platescale of 10 arcsec/pixel onto an array with 40um pixels.

• optimized for best image quality onto a detector with 256x256 elements (and thus 1 cm x 1 cm physical size).

a) Turn in your optimized solution lens file from Oslo.

b) Estimate the 5-sigma detection limit for point sources in 10 minutes of observation in the L'-band and M-band for two cases:

• ambient temperature 280K - a mid-latitude site
• ambient temperature 230K - South Pole winter.

The primary concern here is the emissivity of the objective - which will be at ambient temperature. Assume a good anti-reflection coating so that only bulk material transmission determines the absorption (and thus the emissivity of the front-end optics). If your bulk absorption is less than 1% then assume a minimum emissivity per warm optic of 3%. The other concern would be the emissivity of the sky itself, but we will assume that the temperature of the sky is much lower than that of the ambient environment and that you have chosen a bandpass that is restricted to significantly transparent regions of the atmosphere.

This problem could be pursued with an extensive set of lenses. Fortunately, Oslo comes to the rescue (at least the EDU edition) by only permitting you 10 surfaces). You can allocate them however you like, but let this Oslo restriction set the ultimate complexity of your design. You should use spherical surfaces, but, if you want to be ambitious I will allow one aspherical surface in the design. If you wish, instead of placing that asphere on one side of a lens you can use that asphere as a reflective component in the re-imager part of the system.

Oslo Reference Manual - 429 pages of fun.