1. My Life and Times with the Fourier Transform Spectroscope
Rebecca Dell
CARA Summer REU 2001
University of Chicago
Advisor: Prof. Stephan Meyer

2. The Item:

3. Detector
M2 Mirrors
M1 Mirrors
Dihedral Mirrors on Carriage
Black Body Radiator
Polarizers

4. The Magical Thing:
The Fourier transform of the interferogram is the spectrum of the light that you sent through the FTS in the first place.

5. Some Mathy Stuff
The fundamental concept of this coming mathematics is that the sum of an infinite number of cosine waves is exactly the Fourier Transform integral. It is demonstrating the magical principle so recently stated.

6. y (z) = (1/s) ∫0∞ a(s)cos(2psz)ds
For wavenumber s, position z, the power:
y (z, s) = a(s)cos(2psz)
For all s:
y (z) = (1/s) ∫0∞ a(s)cos(2psz)ds
where s is the average wavenumber.

7. ∫-∞∞ b(s)ei2pszds= ∫-∞0 b(s)ei2pszds + ∫0∞ b(s)ei2pszds
Fourier Integral:
∫-∞∞ b(s)ei2pszds= ∫-∞0 b(s)ei2pszds + ∫0∞ b(s)ei2pszds
= ∫0∞ b*(s)(ei2psz)*ds + ∫0∞ b(s)ei2pszds
=2 ∫0∞Re[b(s)ei2psz]ds
Now, use the handy Euler’s Fromula:
eix=cos(x) + isin(x)
∫-∞∞ b(s)ei2pszds = 2∫0∞ b(s)cos(2psz)ds
(½) ∫-∞∞ b(s)ei2pszds = ∫0∞ b(s)cos(2psz)ds
If b(s) = 2 a(s)/s:
y(z) = ∫-∞∞ b(s)ei2pszds

8. A more intuitive approach:
Monochromatic
a(s1)cos(2ps1z)
Dichromatic
a(s1)cos(2ps1z) +
a(s2)cos(2ps2z)
Broad Band
(1/s) ∫0∞ a(s)cos(2psz)ds

9. Advantages of the FTS
THROUGHPUT: all the light makes it through the instrument and is measured
MULTIPLEX: all frequencies are measured all the time
Works for any area of the E-M spectrum equally well (almost)
Inexpensive, small
Fast scanning time

10. What is the FTS good for?
Measuring the luminous ether, like Michelson and Morley
Measuring the CMBR, like COBE
Characterizing filters, like me
Any time one wants to characterize the nature of E-M radiation

11. Sample Interferogram

12. A little closer in:

13. Power Spectrum of the Black Body

14. Power Spectrum of Light Transmitted through the Filter

15. Divide the two to get the Filter Characterization

16. Black Body
FTS
Detector
What I did:
SPECTRA
Computer
Hard Drive
FFT