Hi Bo, I have just recently started working with spectral/pseudospectral methods as well and I share your excitement about their capabilities.

Regarding your question I am currently dealing with a similar problem ergo. how to implement non-reflecting BC.

You can find a nice summary about PML for 2nd order wave equation here:

https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&ved=0CD0QFjAD&url=http%3A%2F%2Fwww.it.uu.se%2Fresearch%2Fpublications%2Flic%2F2010-004%2F2010-004.pdf&ei=0UXjU5XoHuHmyQOtyoHIDQ&usg=AFQjCNE6rGV9ASXE6PNkMW-HwFPFGGhQ4w&bvm=bv.72676100,d.bGQ&cad=rja

You will find many useful references in this work. This report is however not directly connected to spectral methods.

I don't know if you insist on implementing PML or you simply need some BC that would ensure no waves travelling back to your domain. In the second case I managed to implement absorbing boundary conditions for 2nd order wave equation and 2nd order elastic wave equation and a good starting point would be these papers:

https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCYQFjAA&url=http%3A%2F%2Fmtm.ufsc.br%2F~fermin%2Fabc_rkf.pdf&ei=J0fjU7yiDomfygP-3IKgAQ&usg=AFQjCNFPdwoa32SuoIsiWB4rtwd-pNc90A&bvm=bv.72676100,d.bGQ

https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0CCQQFjAA&url=https%3A%2F%2Fmath.la.asu.edu%2F~rosie%2Fmypapers%2Frenautfrojcp.pdf&ei=a0fjU6Ea4-7JA7LggrgG&usg=AFQjCNFAm3ec_AApZPKNsEAIX8ojAr8fTQ&bvm=bv.72676100,d.bGQ&cad=rja

https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCEQFjAA&url=http%3A%2F%2Fcomputation.llnl.gov%2Fcasc%2Fserpentine%2Fpubs%2FPS_Energy_09.pdf&ei=kUfjU6uLMoSaygPQn4DQCA&usg=AFQjCNGzH_YF5u0az0R2b8lYYABg5wT4wg&bvm=bv.72676100,d.bGQ

These papers provide, I would say, straightforward implementation guidelines. To get more theoretical knowledge behind these BC, you can look at one of the pioneering works from Enquist and Majda:

https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCEQFjAA&url=http%3A%2F%2Fwww.math.mcgill.ca%2F~gantumur%2Fdocs%2Fdown%2FEngquist77.pdf&ei=amnjU9PEL4LnyQOm04CwBw&usg=AFQjCNEqYYW6AFo6lt7bM4mcmU2bbD4wSQ&bvm=bv.72676100,d.bGQ

Please let me know if this helps you. As I am quite new to these methods as well I will be happy to hear any interesting remarks from you.  

These are great references! Thanks!

How  about using differentiable functions ( sigma(x) is differentiable) for the PML parameters ? Then make the equation only have one and second order  derivative about the unknown function.  The following implementations are the same as general differential equations except  that the coefficients of thedifferential eqution is complex.

Thanks!