Fourier transform spectroscopy: an introduction David Naylor University of Lethbridge
Outline History Ideal vs real FTS Pros/cons Extension to ifts Examples: Sitelle, SPIRE, Safari
Michelson s original FTS design and measurements (1892)
. and measurements
Historical Background 1890 - Michelson invented the interferometer, discovered several multiplets and found the red cadmium line to be extremely narrow. He determined the wavelength to unprecedented accuracy 6438.4696 Angstroms. It remained the standard of length until 1960! 1911 - Rubens and Wood recorded the first interferogram. 1949 - Fellgett performed the first computation of Fourier transform and recognized the multiplex advantage of FTS. 1950 - Jacquinot advantage recognized. Area x solid angle (throughput, étendue, light grasp) of an FTS much higher than dispersive spectrometers 1950-1965 FTS used only by those who could not obtain their measurements by conventional spectroscopic techniques (Connes, Fellgett, Gebbie, Mertz) 1965 - Cooley-Tukey (re) invented Fast Fourier Transform algorithm (FFT). Time to compute Fourier transform reduced from days to minutes Over the last 50 years FTS have moved from the specialized domain of the physics laboratory and are now found as standard diagnostic tools in many branches of science and industry.
Lord Rayleigh s response (1892) All spectrometers operate under the principal of interference In all cases the maximum optical path difference between interfering beams determines the spectral resolution
Diffraction Grating A diffraction grating is a plate with a periodic surface modulation it creates multiple slit diffraction. Gratings can be designed for transmission, reflection, or phase operation. The diffraction peaks are wavelength sensitive so with a white light source the maxima are associated with particular wavelengths (colours). Schematic of a reflection diffraction grating Diffracted Beam Input Beam FN GN Constructive interference occurs when the optical path difference is an integer number of wavelengths: B d sin sin m Grating Equation c.f. Young slits formula asin m m B d
Fabry-Perot Interferometer A Fabry-Perot interferometer uses two highly reflecting plane parallel surfaces One of the plates is set on a translation stage so that the gap, d, can be tuned for a particular wavelength or scanned to cover a range of wavelengths. In reality several wavelengths are transmitted for a given gap, d: = 2d 0, 2d 0 /2, 2d 0 /3,. Filters are used to remove unwanted orders. This interference phenomena can be used to accurately measure distances by using a laser beam or in measuring the spectral nature of a source by scanning the separation so as to sequentially detect the intensity of different monochromatic components.
Michelson interferometer
Michelson interferometer
Michelson interferometer
Michelson interferometer
Michelson interferometer
I ( z ) B ( )exp( i ( ))exp( 2 i z ) d B(σ) = I(δ) cos 2πσδ dδ Spectrum, B, interferogram, I, wavenumber σ (cm -1 ) and optical path difference, δ (cm) I (δ) = B(σ) exp(iφ(σ))exp(i2πσδ) dσ φ total = φ zpd + φ electrical + φ optical + φ random
SPIRE Test FTS at Rutherford Appleton Laboratory
Advantages of Fourier spectroscopy Relatively simple opto-mechanical design High throughput (Jacquinot) Simultaneous measurements of all wavelengths (Felgett) Intrinsic Wavelength calibration (Connes) Best instrumental line shape function of any spectrometer Disadvantages of Fourier spectroscopy Sensitivity to fluctuations in source intensity Multiplex disadvantage under background limited conditions Complex math required for analysis
Key design considerations Beamsplitter Mirror Drive Metrology Dynamic Range Misaligned Mirrors Detector feed optics Channel Fringes
Instrument Line Shape (ILS) Consider a monochromatic spectral line at frequency σ 0 B( ) 2A cos(2 x)exp( i2x) dx 0 L B( ) L B( ) 2A cos2 ( ) xdx) 0 0 L sin 2 0 L 2A 2 0 L 0 cos2 ( 0 sin 2 2 0 L 0 ) xdx Rapidly decays Sinc ILS First zero of this function occurs at δσ = 1/2L In terms of FWHM δσ = 1.207/2L
Comparing Herschel SPIRE FTS ILS with theory
Line fitting Herschel SPIRE FTS spectra
But some people are never happy..
Optimal apodizing functions Naylor and Tahic, Apodizing functions for Fourier transform spectroscopy JOSA A (2007)
Optimal apodizing functions
Phase correction Goal: to correct for phase errors that arise from electrical, optical and sampling effects. Method: convolve interferogram with phase correction function (PCF) derived from phase information extracted from a short double-sided portion of the interferogram. PCF(δ) = exp(-iφ(σ))exp(i2πσδ) dσ I symmetrical (δ) = I(δ) * PCF(δ)
Raw asymmetrical interferogram
Real Imaginary Phase
Phase correction function
Raw asymmetrical interferogram
Phase corrected symmetrical interferogram
Real Imaginary Phase
Generic FTS Data Processing Pipeline Software Components Instrument Control Interferogram Processing Instrument diagnostics Observing diagnostics Inspect interferogram De-glitch interferogram Re-grid interferogram Phase correction Single Sided FT Double Sided FT Gain correction Wavelength correction Code FT in C or JAVA, optimize for speed Gain correction Wavelength correction Code FT in C or JAVA, optimize for speed Spectral Processing Flat fielding Quality control Spectral Math average, co-add, difference, etc. Archiving Design SQL database for spectra / interferograms / observational parameters Visualization Slicing, thresholding, template/pattern matching, 3-color mapping, profiling, averaging, etc.
The FTS is readily adapted to imaging spectroscopy - ifts
Imaging FTS Combine FTS with detector array Imaging + Spectroscopy 3D data product 2D spatial imaging 1D spectral
Imaging spectroscopy An imaging Fourier Transform Spectrometer (ifts)
The IFTS System
IFTS Graphical User Interface (GUI)
Imaging spectroscopy with a Mach-Zehnder ifts
d z S d z S d z RT S d z RT S d z S z I d z S d z S d z RT S d z RT S d z S z I BS BS B B A out BS BS B A A out ]} 2 )]sin[ ( RTsin[ { ]} )]cos[2 ( RTcos[ { ]} 2 )]sin[ ( 2 sin[ { ]} )]cos[2 ( cos[2 { ]} {RTcos[2 ) ( ]} 2 )]sin[ ( RTsin[ { ]} )]cos[2 ( RTcos[ { ]} cos[2 { ]} 2 )]sin[ ( 2 sin[ { ]} )]cos[2 ( {RTcos[2 ) ( 2 1 Even Odd
ifts data processing pipeline modules Inspect interferogram data cubes Deglitch cosmic ray events Phase correction Apodization Fourier transform time sampled interferograms Wavelength scale correction for off-axis pixels (the obliquity effect) Flat field array adjust gain for individual pixel responsivities Calibrate spectra in Jy or W m -2 Hz -1 Inspect spectral data cubes Merge spectral data cubes from two bands Spectral processing (average, difference, ratio, spectral/spatial integration)
ifts scan of a uniform white target
Imaging spectroscopy of something found in every well equipped physics research laboratory.. Smarties!
Now we have complete spectral information for each pixel FTS + Array Detector Hyperspectral Imaging
Or if we prefer we can fly through the spectral hypercube viewing a spatial image as a function of wavelength
KL S Bar
Poor S/N is not always a showstopper
Berlin, 22 Nov 1880 (Michelson to Simon Newcomb) With all due respect, however, I think differently, for if the apparatus is surrounded with melting ice, the temperature will be so nearly constant as possible. There is another and unexpected difficulty, which I fear will necessitate the postponement of the experiments indefinitely namely that the necessary funds do not seem to be forthcoming. Newcomb arranged for Alexander Graham Bell to provide the 100 to buy the optical components for Michelson s interferometer,and the rest is history
Funding agencies AAET CFI CMC Canadian Space Agency European Space Agency EU FP7 NSERC University of Lethbridge
And finally. the most important resource
Questions?