Fourier Transform Spectrometry

Fourier Transform Spectrometry is of immediate use to those who use Fourier transform spectrometers in their research, or are considering their use. The authors' presentations enable readers to obtain a clear understanding of FTS, which is crucial to their studies and research. Due to the increasing complexity and commercialization of instrumentation, achieving optimum performance in research applications and automated usage can be challenging. For example, a thorough understanding of the instrument can dramatically affect the outcome of the experiment and the generation of reliable data in applications where conditions are not ideal and resulting signals are weak. This book provides a comprehensive discussion of FTS from the ground up, covering basic concepts, instrumentation, data-processing algorithms, and techniques for computerized spectral analysis.

Preface

xiii

Introduction

1

(16)

Spectra and Spectroscopic Measurement

5

(5)

The Classical Michelson Interferometer

10

(2)

Precision, Accuracy, and Dynamic Range

12

(1)

Units

13

(2)

A Glance Ahead and to the Side

15

(2)

Why Choose a Fourier Transform Spectrometer?

17

(12)

Quantity

18

(2)

Quality

20

(4)

Resolution and Line Shape

20

(1)

Fixed and Variable Quantities in Experiments

21

(3)

Cost

24

(1)

Summary

25

(4)

Theory of the Ideal Instrument

29

(12)

Equation for the Balanced Output

31

(1)

The Unbalanced Output

32

(1)

From Monochromatic to Broadband Light

33

(3)

Generalization to Polychromatic Light

35

(1)

Extension to Plus and Minus Infinity

35

(1)

The Fourier Transform Spectrometer as a Modulator

36

(1)

Summary

37

(4)

Fourier Analysis

41

(26)

Linear Systems and Superposition

42

(1)

A Practical Approach to Fourier's Theorem

43

(3)

Fourier Decomposition

46

(1)

Representations of Functions and Their Transforms

47

(2)

Practical Representations

49

(1)

Linear Systems and Convolution

49

(5)

Cross-Correlation

52

(1)

Definition of a Signal-to-Noise Ratio

53

(1)

Generalized Functions

54

(3)

The Dirac Delta Function

54

(1)

The Dirac Comb and Sampling of Continuous Functions

55

(2)

The Essential Theorems

57

(6)

Addition (Superposition)

57

(1)

Similarity (Stretching)

58

(1)

Shift

58

(1)

Modulation

58

(1)

Convolution (Folding Rule)

59

(1)

Autocorrelation

60

(1)

Derivative

61

(1)

Derivatives of a Convolution Integral

62

(1)

Rayleigh Theorem

62

(1)

Power

63

(1)

Zeroth Moment (Area)

63

(1)

Examples of Transform Pairs Fundamental to FTS

63

(4)

Nonideal (Real-World) Interferograms

67

(13)

Finite Path Difference

68

(1)

Finite Entrance Aperture Size

69

(4)

Finite Instrumental Function and Its Properties

73

(6)

Non-uniform or Non-symmetric Irradiation of the Aperture

79

(1)

Working with Digital Spectra and Fourier Transforms

80

(21)

Signals and Measurement

83

(3)

Discrete Transforms

86

(1)

Interpolation in the Spectrum

87

(6)

Apodization

93

(5)

Preparation for Making Transforms

98

(1)

Common Artifacts

98

(1)

Zero Mean Interferograms, End Effects, and Masking

98

(1)

Procedure for Reliable Transforms

98

(3)

Phase Corrections and Their Significance

101

(18)

The What and Why of Phase

101

(1)

Origins of Asymmetries in the Interferogram

102

(1)

From Asymmetries to Phase Errors

102

(5)

Asymmetric Truncation, Amertization, and Phase Errors

107

(5)

More on One-Sided vs. Symmetric Interferograms

112

(1)

Determining Phase Shifts

112

(4)

Low-Resolution Phase Determination

114

(2)

High-Resolution Phase Determination

116

(1)

Recommendations

116

(3)

Effects of Noise in Its Various Forms

119

(24)

Signal and Noise in the Two Domains

120

(2)

Noise Classifications

122

(11)

Detector Noise (k = 0): The Multiplex Advantage

122

(1)

Digitizing Noise (k = 0): Dynamic Range

123

(3)

Photon Noise (k = 0.5)

126

(5)

Source Noise (k = 1)

131

(2)

Unbalance, Misalignment, Modulation Efficiency, and Signal-to-Noise

133

(1)

Noise Limited Resolution

134

(1)

Localized Noise (Ghosts and Artifacts)

134

(6)

Periodic Position Error Ghosts

136

(1)

Coherent Ghosts

136

(3)

Ringing Ghosts

139

(1)

Summary

140

(3)

Line Positions, Line Profiles, and Fitting

143

(26)

Introduction

143

(1)

Line Finding and Line Shapes

144

(1)

Emission Lines

144

(1)

Absorption Lines

144

(1)

Derivative Line Finding

145

(9)

Kernel Functions

145

(3)

Line Position Accuracy

148

(1)

Useful Line-Finding Properties of First and Second Derivatives

149

(1)

Techniques of Derivative Line-Finding

150

(1)

Noise and Filtered Derivative Line-Finding

150

(2)

Resolution and Line-Finding

152

(2)

Least Square Fitting of Line Profiles to Voigtian Functions

154

(11)

Voigtian Functions

154

(1)

Fitting Considerations

155

(2)

Special Problems in Fitting Voigtian Functions

157

(1)

Fitting Errors from Sampling

158

(1)

Fitting Errors from Noise

158

(2)

Fitting Errors from the Sampling Grid

160

(1)

Fitting Underresolved Spectra

160

(3)

Rates of Convergence

163

(2)

Areas and Equivalent Widths

165

(1)

Wavenumber Calibration

166

(3)

Processing of Spectral Data

169

(10)

Emission Background Subtraction and Intensity Correction

170

(3)

Absorption Background and Intensity Corrections

173

(2)

Line Lists and Fitting

175

(4)

Discussions, Interventions, Digressions, and Obscurations

179

(34)

A Novel Approach to Sampling Systems

181

(5)

Introduction

181

(1)

Position Measurements

182

(1)

Sampling Systems

183

(1)

A New Approach to Sampling Systems

184

(1)

Practical Realization, Results, and Implications

185

(1)

The Imaging Fourier Transform Spectrometer (IFTS)

186

(3)

Characterization and Determination of Instrumental Line Shape Functions

189

(1)

Apodization

190

(9)

Introduction

190

(2)

Selected Apodization Functions/Windows

192

(3)

Evaluation Criteria and Relative Performance

195

(1)

Families of Functions

196

(1)

Method of Optimization

196

(1)

Some Useful Functions for Optimal Apodization

196

(2)

Discussion

198

(1)

To Apodize or Not to Apodize---Case Studies

198

(1)

The Quest for the Perfect Instrument

199

(5)

On Fourier Transform Spectrometry and Digital Signal Processing

204

(1)

From Echelle Spectrographs to Echelle Spectrographs---A 50-Year Circle

205

(6)

A Final Story

211

(2)

Chapter-by-Chapter Bibliography

213

(10)

Chronological Bibliography

223

(10)

Applications Bibliography

233

(10)

Author Bibliography

243

(16)

Index

259

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