Fourier transform spectroscopy: an introduction · Integral wide field spectroscopy in astronomy the iFTS solution, ... A students guide to Fourier transforms, ... Rapid scanning

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


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).


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).


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.


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?


Michelson’s original FTS design and measurements (1892)


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.


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


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.


Michelson interferometer










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


Spectrum, B, interferogram, I, wavenumber σ (cm-1), optical path difference, δ (cm)

18



SPIRE Test FTS at

Rutherford Appleton Laboratory

20


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)

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


Comparing Herschel SPIRE FTS ILS with theory


Line fitting Herschel SPIRE FTS spectra




But some people are never happy…..

28




Optimal apodizing functionsNaylor and Tahic, Apodizing functions for Fourier transform spectroscopy

JOSA A (2007)

31


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)


Raw asymmetrical interferogram


Real Imaginary Phase


Phase correction function


Raw asymmetrical interferogram


Phase corrected symmetrical interferogram


Real Imaginary Phase


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


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


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)


iFTS scan of a uniform white target


Imaging spectroscopy of something found in every

well equipped physics research laboratory…..

Smarties!



FTS + Array Detector ⇒ Hyperspectral Imaging

Now we have complete spectral information for… each pixel


Or if we prefer we can fly through the spectral hypercube viewing a spatial image as a function of wavelength…






Bar

KL

S





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



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.




Questions?


Fourier transform spectroscopy: an introduction · Integral wide field spectroscopy in astronomy the iFTS solution, ... A students guide to Fourier transforms, ... Rapid scanning

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