Fourier transform spectroscopy: an introduction · Integral wide field spectroscopy in astronomy...
-
Upload
nguyenliem -
Category
Documents
-
view
214 -
download
0
Transcript of Fourier transform spectroscopy: an introduction · Integral wide field spectroscopy in astronomy...
Fourier transform spectroscopy:
an introduction
David Naylor
University of Lethbridge
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 2
Outline
• History
• Ideal vs real FTS
• Pros/cons
• Extension to iFTS
• Examples
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 3
Suggested readings on Fourier transform spectroscopy
Papers provided:
1. SHIFTS: Simulator for the Herschel Imaging Fourier Transform Spectrometer,John Lindnder MSc thesis excerpt (2006)
2. Spectral Characterization of the Herschel SPIRE Photometer Locke Spencer MSc thesis excerpt (2005)
3. Astronomical imaging Fourier spectroscopy at far infrared wavelengths, Can. J. Phys., Naylor et al (2013)
4. Integral wide field spectroscopy in astronomy the iFTS solution, Experimental Astronomy, Maillard et al (2013)
Introductory Textbooks:
5. Eugene Hecht. Optics. Addison Wesley, fourth edition, (2002).
6. P. R. Griffths and J. A. Haseth. Fourier Transform Infrared Spectrometry. John Wiley and Sons, New York, (1986).
7. J. F. James, A students guide to Fourier transforms, Cambridge University Press (2002).
Advanced Textbooks:
8. R. J. Bell. Infrared Fourier Transform Spectroscopy. Academic Press, New York, (1972).
9. Sumner P. Davis, Mark C. Abrams, and James W. Brault. Fourier Transform Spectroscopy. Academic Press, (2001).
10. J. E. Chamberlain. The Principles of Interferometric Spectroscopy. John Wiley and Sons: Chichester, England (1979). Edited
by G. W. Chantry and N. W. B. Stone.
11. E. O. Brigham. The Fast Fourier Transform. Prentice-Hall Inc., (1974).
12. M. Born and E. Wolf. Principles of Optics, Cambridge University Press, (1980).
13. L. Mertz. Transformations in optics. New York: Wiley, (1965).
14. D. C. Champeney. Fourier Transforms and their Physical Applications. Academic Press: St. Louis, MO (1973).
15. R. M. Bracewell. Fourier Transforms and Its Applications. McGraw-Hill Book Co: New York, NY (1965).
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 4
Suggested readings on Fourier transform spectroscopy
Historical Papers:
16. A. A. Michelson, On the application of interference methods to spectroscopic measurements. Phil. Mag., 34:280, (1892).
17. H. Rubens & R. W. Wood, Focal isolation of long heat-waves. Philosophical Magazine, 21:249–261, (1911).
18. H. Nyquist, Certain topics in telegraph transmission theory. Trans. American Inst. of Elec.Engn, 47:617 – 644, (1928).
19. P. Fellgett.,A propos de la theorie du spectrometre interferentiel multiplex. Journal of Physics Radium, 19:187, (1958).
20. P. Jacquinot, New developments in interference spectroscopy. Rep. Prog. Phys., 23:267–312, (1960).
21. J Connes, Recherches sur la spectroscopie par transformation de Fourier, PhD Thesis, (1960).
22. J. W. Cooley & J. W. Tukey, An algorithm for the machine calculation of complex Fourier series. Math. Comp., 19:297, (1965).
23. M. L. Forman, Fast Fourier-Transform Technique and it’s Application to Fourier Spectroscopy. JOSA, 56(7):978, (1966).
24. M. L. Forman, W. Steel & G. A. Vanasse, Correction of Asymmetric Interferograms Obtained in Fourier Spectroscopy. JOSA,
56(1):59–64 (1966).
25. L. Mertz, Rapid scanning Fourier transform spectroscopy. J. Phys. Coll. C2, Suppl. 3-4, 28:88, (1967).
26. R. H. Norton & R. Beer, New apodizing functions for Fourier spectrometry. JOSA, 66:259 – 264, (1976).
27. F. J. Harris, On the use of windows for harmonic analysis with the discrete Fourier transform. IEEE, 66, 51–83, (1978).
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 5
Coherence LengthThe coherence length, L, the optical delay length over which interference
effects can be observed is related to the bandwidth, Δλ, and the mean
wavelength, λ0, of a source by:
L =C
Δf=
λ02
ΔλIn his search for the best optical emission line to be used as the standard
of length, Michelson looked for sources that had the largest coherence
length, using the visibility curve:
V(x) = Imax
−Imin
Imax
+Imin
Where I is the fringe intensity
Examples of coherence length of lasers
• Multimode He:Ne lasers have a typical coherence length of 20 cm
• Single mode He:Ne lasers are ~ km
• Singlemode fibre lasers can exceed 100 km.
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 6
Ok, some mental arithmetic:
What is coherence length of the visible light
emitted by an incandescent lamp?
What is the bandwidth of laser that could
produce a hologram of a small crater on the
moon from a terrestrial laboratory?
How challenging is this?
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 7
Michelson’s original FTS design and measurements (1892)
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 8
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy
Michelson’s visibility measurements…
9
Historical Background1890 - 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.
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 10
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
Constructive interference occurs when the optical
path difference is an integer number of wavelengths:
md sinsin Grating Equation
c.f. Young’ slits formula ma msin
GN
FN
Input BeamDiffracted
Beam
dB
B
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 11
Fabry-Perot Interferometer
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.
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: = 2d0, 2d0/2, 2d0/3,….
Filters are used to remove unwanted orders.
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 12
Michelson interferometer
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 13
Michelson interferometer
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 14
Michelson interferometer
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 15
Michelson interferometer
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 16
Michelson interferometer
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 17
dziiBzI )2exp())(exp()()(
dziiBzI )2exp())(exp()()(
I′(δ) = ∫ B(σ) exp(iφ(σ))exp(i2πσδ) dσ
B(σ) = ∫ I(δ) cos 2πσδ dδ
φtotal = φzpd + φelectrical + φoptical + φrandom
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy
Spectrum, B, interferogram, I, wavenumber σ (cm-1), optical path difference, δ (cm)
18
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 19
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy
SPIRE Test FTS at
Rutherford Appleton Laboratory
20
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 21
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
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 22
Key design considerations
• Beamsplitter
• Mirror Drive
• Metrology
• Dynamic Range
• Misaligned Mirrors
• Detector feed optics
• Channel Fringes
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 23
Instrument Line Shape (ILS)
0
0
0
0
2
2sin
2
2sin2)(
LLAB
dxxdxxABLL
)(2cos))(2cos2)(0
00
0
dxxixABL
L)2exp()2cos(2)( 0
First zero of this function occurs at δσ = 1/2L
In terms of FWHM δσ = 1.207/2L
Rapidly decays Sinc ILS
Consider a monochromatic spectral line at frequency σ0
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 24
Comparing Herschel SPIRE FTS ILS with theory
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 25
Line fitting Herschel SPIRE FTS spectra
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 26
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 27
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy
But some people are never happy…..
28
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 29
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 30
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy
Optimal apodizing functionsNaylor and Tahic, Apodizing functions for Fourier transform spectroscopy
JOSA A (2007)
31
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 32
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
Isymmetrical(δ) = I(δ) * PCF(δ)
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 33
Raw asymmetrical interferogram
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 34
Real Imaginary Phase
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 35
Phase correction function
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 36
Raw asymmetrical interferogram
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 37
Phase corrected symmetrical interferogram
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 38
Real Imaginary Phase
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 39
Generic FTS Data Processing
Pipeline
Instrument Control
• Instrument diagnostics
• Observing diagnostics
Inspect interferogram
De-glitch interferogram
• Re-grid interferogram
Interferogram Processing
• Flat fielding
• Quality control
• Spectral Math – average, co-add, difference, etc.
Spectral Processing
Design SQL database for spectra / interferograms / observational parametersArchiving
Slicing, thresholding, template/pattern matching, 3-color mapping, profiling, averaging, etc.
Visualization
• Gain correction
• Wavelength correction
Code FT in C or JAVA, optimize for speed
• Phase correction
• Gain correction
• Wavelength correction
Code FT in C or JAVA, optimize for speed
Single Sided
FT
DoubleSided
FT
Software Components
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 40
The FTS is readily adapted to imaging spectroscopy - iFTS
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 41
Imaging FTS
• Combine FTS with detector array
Imaging + Spectroscopy
• 3D data product
2D spatial imaging
1D spectral
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 42
Imaging spectroscopy
An imaging Fourier Transform Spectrometer (iFTS)
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 43
The IFTS System
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 44
IFTS Graphical User Interface (GUI)
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 45
Imaging spectroscopy with a Mach-Zehnder iFTS
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 46
dzS
dzS
dzRTS
dzRTS
dzSzI
dzS
dzS
dzRTS
dzRTS
dzSzI
BS
BS
B
B
Aout
BS
BS
B
A
Aout
]}2sin[)](sin[RT{
]}2cos[)](cos[RT{
]}2sin[)](2sin[{
]}2cos[])(cos[2{
]}2cos[RT{)(
]}2sin[)](sin[RT{
]}2cos[)](cos[RT{
]}2cos[{
]}2sin[)](2sin[{
]}2cos[])(RTcos[2{)(
2
1
Even
OddDunlap Institute Summer School 2015 Fourier Transform Spectroscopy 47
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)
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 48
iFTS scan of a uniform white target
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 49
Imaging spectroscopy of something found in every
well equipped physics research laboratory…..
Smarties!
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 50
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 51
FTS + Array Detector ⇒ Hyperspectral Imaging
Now we have complete spectral information for… each pixel
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 52
Or if we prefer we can fly through the spectral hypercube viewing a spatial image as a function of wavelength…
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 53
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 54
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 55
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 56
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 57
Bar
KL
S
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 58
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 59
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 60
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 61
Poor S/N is not always a showstopper…
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 62
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
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 63
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy
While visiting the Nautical Almanac Office in 1879 Michelson read a
letter from James Clerk Maxwell to the director in which he stated his
belief that no terrestrial experiment could be designed with sufficient
sensitivity to measure the motion of the earth through the aether.
(Nature, 21 314, (1880))
I always think of Michelson as the artist in Science. His greatest joy
seemed to come from the beauty of the experiment itself, and the
elegance of the method employed. … There is of course, no logical
way to the establishment of a theory, but only groping constructive
attempts controlled by careful consideration of factual knowledge.
(Nature, 171 101, (1953)) Albert Einstein
What his contemporaries thought
64
Writing about the Durham FTS Conference Sept 1983 Connes wrote:
(The conference) has just demonstrated that Fourier transform
spectroscopy is now one of the most versatile and widely useful tools in
the paraphernalia of modern optics. Will it become one of the
permanent features of the landscape, just like - let us say - the
Cathedral? Such a prediction would be risky of course; but any
participant who briefly stole away from one of the less thrilling sessions
and trespassed across the Close, surely discovered that all the great
surviving medieval cathedrals are first of all things of the mind:
destroyed, rebuilt, enlarged or mutilated, plastered all over and then
joyfully restored and rediscovered. What truly endures is oft more the
spirit than the stones. At the dissipation rate of all things modern, our
present interferometers (some of them anyhow) will be museum pieces
before the turn of the century-millenium, and likewise most of our
analysis or recording techniques. But the basic principles involved
might well exhibit far greater staying power, and earn a permanent
niche on the shelves of scientific methodology.
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 66
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 67
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 68
Questions?
Dunlap Institute Summer School 2015 Fourier Transform Spectroscopy 69