Computation of a spectrum from a single-beam Fourier-transform infrared interferogram Avishai Ben-David and Agustin Ifarraguerri
A new high-accuracy method has been developed to transform asymmetric single-sided interferograms into spectra. We used a fraction 共short, double-sided兲 of the recorded interferogram and applied an iterative correction to the complete recorded interferogram for the linear part of the phase induced by the various optical elements. Iterative phase correction enhanced the symmetry in the recorded interferogram. We constructed a symmetric double-sided interferogram and followed the Mertz procedure 关Infrared Phys. 7, 17 共1967兲兴 but with symmetric apodization windows and with a nonlinear phase correction deduced from this double-sided interferogram. In comparing the solution spectrum with the source spectrum we applied the Rayleigh resolution criterion with a Gaussian instrument line shape. The accuracy of the solution is excellent, ranging from better than 0.1% for a blackbody spectrum to a few percent for a complicated atmospheric radiance spectrum. © 2002 Optical Society of America OCIS codes: 300.6300, 300.6340, 300.6190, 120.0280, 120.3180, 070.6020.
1. Introduction
Much research has been devoted to the subject of converting interferograms measured with singlebeam Fourier-transform infrared 共FTIR兲 instruments into spectra. FTIR spectrometers can measure simultaneously many of the thousands of gaseous chemicals that have absorption features in the infrared and are used routinely in air-pollution monitoring of toxic gases released by industrial processes and in atmospheric research in which the chemical composition of the atmosphere is of interest. Two issues that are important but are not well understood in converting an interferogram into a spectrum are phase-correction techniques and apodization. As the signal-to-noise ratio in the Fourier spectroscopy measurements increases, an accurate phase correction and high-quality apodization techniques become more important and may set the limit on the achievable accuracy of the deduced spectrum. The inter1– 6
A. Ben-David 共
[email protected]兲 is with the Science and Technology Corporation, Suite 206, 500 Edgewood Road, Edgewood, Maryland 21040. A. Ifarraguerri is with the Science Applications International Corporation, Suite 300, 4001 Fairfax Drive, Arlington, Virginia 22203; while this research was performed he was with the U.S. Army Edgewood Chemical Biological Center, Aberdeen Proving Ground, Maryland 21010-5424. Received 12 June 2001; revised manuscript received 14 September 2001. 0003-6935兾02兾061181-09$15.00兾0 © 2002 Optical Society of America
ferogram I共x兲 is the measured interference pattern for a displacement x 共in centimeters兲 between two interfering beams in a classic Michelson interferometer.1 The recorded interferogram 共Fig. 1兲 is a sequence of measurements that span the range ⫺xmin to ⫹xmax, where xmax is the maximum optical-path difference and is determined by the wave-number resolution 共inverse centimeters兲 requirement and xmin depends on the instrument design. In Fig. 1, only the modulated part 共ac兲 of the interferogram, which contains information on the spectrum, is shown, and the constant 共dc兲 component is omitted. The recorded interferogram is in general asymmetric. If xmax ⫽ xmin the interferogram is termed doublesided. In some instruments xmax ⬎⬎ xmin, and the interferogram is termed a single-sided asymmetric interferogram. The maximum of the interference pattern is located at an optical-path difference x0 and is called the center burst. At this location all the wavelengths of the light source that produced the 共ideal兲 interferogram are in phase, making this point in the recorded interferogram a unique reference point for phase-correction procedures. There are two components of phase in the recorded interferogram. The first is due to a phase shift in the incident radiance that results from the optical components such as beam splitter, lenses, and mirrors in the instrument, all of which have different wavelengthdependent refractive indices and optical thickness, and from the electronics 共e.g., amplifiers and filters兲, which exhibit a frequency-dependent time delay. 20 February 2002 兾 Vol. 41, No. 6 兾 APPLIED OPTICS
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Fig. 1. Single-sided asymmetric interferogram I共⫺xmin ⱕ x ⱕ xmax兲 computed for blackbody source B共500 ⬍ ⬍ 1500 cm⫺1兲 at a temperature of 300 K. Short, double-sided interferogram Is共x兲 ⫽ I共⫺xmin ⱕ x ⱕ xmin兲 and the location of center burst x0 are shown. Only the modulated part 共ac兲 of the interferogram that contains information on the spectrum is shown, and the constant 共dc兲 component is omitted.
The second part of the phase is due to the location of the true center burst in the recorded measurement sequence, which in general will fall between sample points. The two commonly used methods of converting an interferogram into a spectrum are the Mertz7 and the Forman8 methods 共see Ref. 1, Chap. 3, for a comprehensive discussion兲. In the Mertz method a phase estimate of the optical components 共i.e., the first part of the phase兲 is computed from a short region about the center burst of the interferogram 共usually denoted a short double-sided interferogram; Fig. 1兲. This phase estimate is deduced for only a limited number of frequencies and is then interpolated to all the frequencies that are contained within the measured interferogram. Then the recorded asymmetric interferogram is apodized by an asymmetric triangular window and rotated to zero out the phase shift introduced by the sampled location of the interferogram. One converts the interferogram into a spectrum by taking the Fourier transform; it is phase corrected by use of the phase estimate deduced from the short double-sided interferogram. This phase correction in a sense transfers residual energy from the imaginary frequency components of the Fourier transform operation back to the real frequency components of the spectrum. In the Forman method the phase correction is performed in the time domain 共i.e., the interferogram domain兲, whereas in the Mertz method the phase correction is done in the frequency domain 共i.e., the spectrum domain兲. In the Forman method no apodization is applied to the short double-sided interferogram, and no interpolation of the phase is performed. Chase2 stated that the Forman method is better than the Mertz method 1182
APPLIED OPTICS 兾 Vol. 41, No. 6 兾 20 February 2002
because the triangular apodization 共used in the Mertz method兲 has a discontinuity in its first derivative, and thus the phase estimate is less accurate, and suggested that no fewer than 100 data points 共preferably several hundreds兲 should be used in the phase estimate. In this paper we present a method that combines the Forman and Mertz methods of converting an interferogram into a spectrum. In our method we use the short double-sided interferogram to apply an iterative phase-correction procedure 共for the linear part of the phase induced by the various optical elements and for the linear phase delay that is due to the electronics兲 to an asymmetric single-sided recorded interferogram. From the linearly phase-corrected interferogram we construct a complete symmetric double-sided interferogram for the range ⫺xmax to ⫹xmax. We then compute the Fourier transform, using any user-selected apodizing window 共e.g., Hamming, Blackman, Chebyshev, and Kaiser兲. These windows have characteristics 共e.g., lower sidelobes兲 that are superior to those of the triangular apodizing window. A final phase correction for the nonlinear part of the phase induced by the various optical elements is performed on the derived Fourier transform to transfer energy from the imaginary frequency components to the real frequency components, as is done in the Mertz method. In our method the nonlinear phase is deduced directly from the constructed symmetric double-sided interferogram, and thus no interpolation on the nonlinear phase is needed. We discuss practical issues of numerical implementations, such as zero-fill interpolation and a deresolution algorithm for comparing the computed spectrum with the source spectrum that produced the interferogram, and show some examples of the method’s results. 2. Phase Correction of the Interferogram I(x)
The modulated 共ac兲 component of the recorded interferogram I共 x兲 共in volts兲 at n locations xi , ⫺xmin ⱕ x ⱕ xmax uniformly spaced every ␦ centimeters that results from a source B共兲 关W兾共cm2 sr cm⫺1兲兴, where is the wave number 共in inverse centimeters兲 is given1 by I共 x兲 ⫽
兰
2
K共兲0.5B共兲cos关2x ⫹ ⌽共兲兴d,
(1a)
1
⌽共兲 ⫽ ⌽ 1共兲 ⫹ ⌽ 2共兲, ⌽ 2共兲 ⫽ ⫺2x 0,
(1b) (1c)
where K共兲 关共V cm2 sr兲兾W兴 is the instrument constant that combines the responsivity 共volts per watt兲, the aperture area 共square centimeters兲 and the field of view 共steradians兲 of the instrument. For convenience we assume that K共兲 ⫽ 1 for 2 ⬍ ⬍ 1. The total phase ⌽共兲 is combined phase shift ⌽1共兲 from all optical elements and electronics components and phase shift ⌽2共兲 that is due to the shift x0 of the center burst in the recorded interferogram. The integration limits 1 and 2 are for a band-limited
source B共兲 or are due to a detector band-limited transfer function, and the modulation efficiency is assumed to be 1. For an ideal spectrometer for which ⌽共兲 ⫽ 0 and interferogram I共x兲 is sampled at all x values, I共x兲 and B共兲 are a Fourier pair given by I共 x兲 ⫽
兰
K共兲 B共兲cos共2x兲d ⫽ F ⫺1关K共兲 B共兲兴,
B共兲 ⫽ K共兲 ⫺1
兰
found from the maximum of the absolute value of I共 x兲, I共 xˆ0兲 ⫽ max关兩I共 x兲兩兴. A short double-sided interferogram Is共x兲 of length 共2n0 ⫺ 1兲 is formed, with I共xˆ0兲 at the center and 共n0 ⫺ 1兲 samples on each side 共n0 is the location, point number i for x ⫽ xˆ0兲. Short double-sided interferogram Is共x兲 is in general not symmetric about I共 xˆ0兲, and this departure from symmetry is due to the phase shift. Average departure ⌬ 共in centimeters兲 from symmetry is computed from 2n0⫺1
I共 x兲cos共2x兲dx ⫽ K共兲 ⫺1F关I共 x兲兴, ⌬ ⫽ ␦
(2)
where F共 兲 and F⫺1共 兲 denote the cosine Fouriertransform operation and the inverse cosine Fourier transform, respectively. In this research the difficulty in implementing the straightforward Fourier transform 关Eqs. 共2兲兴 because of phase ⌽共兲 and the finite sampling range ⫺xmin ⱕ x ⱕ xmax are addressed. We apply a phase correction to I共x兲 for the linear part of phase shift ⌽1共兲 and to the portion of phase 2共x0 ⫺ 关 x0兴兲 in ⌽2共兲, where 关 x0兴 is the rounded value of center-burst location x0 共关 x0兴 ⫽ k␦, where 1 ⬍ k ⬍ n is an integer sampling point number兲 in an iterative procedure that usually takes only a few iterations. Because of the finite number of sampling intervals, the center burst is not sampled exactly 共it usually falls between sampling points兲. Phase correction 2共x0 ⫺ 关 x0兴兲 shifts the center burst to the location 关 x0兴 that is one of the sampled xi locations. Phase shift 2关 x0兴 is a phantom phase 共owing to the arbitrary placement of the center-burst location in the interferogram vector兲 that is easily handled with a rotation process by rearrangement of the elements of I共x兲 to place the center burst at the first element of the vector. As our iterative procedure is a linear procedure, only the linear part of phase ⌽1共兲 is corrected. For a general function f 共x兲, phase ⌽f 共兲 is given by the arctangent of the ratio between the imaginary part and the real part of the Fourier transform of f 共x兲. If f 共x兲 is real and symmetric about the origin, phase ⌽f 共兲 is zero because the imaginary part of a complex Fourier transform of a real and symmetric function is zero. Because unknown source spectrum B共兲 is purely a real function, we would like to transform the real interferogram I共x兲 to be symmetric 共i.e., an even function兲 about x0; then, shifting the x axis 关i.e., forming the function I共x ⫺ x0兲兴 will yield a phase that has become zero. The general objective of our procedure is to estimate with high accuracy the location of center burst x0, using a short double-sided fraction of the recorded interferogram, and impose a time shift 共i.e., a shift of the x axis, by a resampling process兲 on the full recorded interferogram. When this is done, the symmetry of the interferogram tends to improve, resulting in a smaller phase component. The concept is applied iteratively; the correction improves following each iteration. A first estimate of center-burst location xˆ0 is easily
兺
i兩 I s共 x i 兲兩
兺
兩 I s共 x i 兲兩
⫺ n 0,
i⫽1 2n0⫺1
(3)
i⫽1
where n0 is the point number of center burst location I共 xˆ0兲, and ⌬兾␦ can be viewed as an estimate of the center of gravity of Is共x兲 or as the first moment of i 2n0⫺1 with 兩Is共xi 兲兩兾¥i⫽1 兩Is共xi 兲兩 taken as a weighing function. If Is共x兲 were exactly symmetric 关where Eq. 共3兲 is used as a measure of symmetry兴 about n0, then ⌬ would be zero. It is important to note that this procedure tends to improve the symmetry of the interferogram but does not guarantee complete symmetry, for which all odd moments of Is共x兲 are zero and Is共x0 ⫺ x兲 ⫽ Is共x0 ⫹ x兲. A common method of estimating ⌬ is to perform a polynomial fit about I共xˆ0兲 共where the proper polynomial order and the number of points for the fit affect ⌬ and should be predetermined兲 and computing the location of the maximum. In practice, because the signal-to-noise ratio in the measured interferogram decreases rapidly as x moves farther from the center-burst location, we use fewer points 共⬃200兲 and not all the 共2n0 ⫺ 1兲 points about the center burst to compute ⌬ in Eq. 共3兲. A linear phase-shift correction can be thought of as a convolution of I共x兲 by a shifted Dirac delta function ␦共x ⫺ ⌬兲, given by I共 x ⫺ ⌬兲 ⫽ ⫽
兰 兰
I共␣兲␦关␣ ⫺ 共 x ⫺ ⌬兲兴d␣ I共␣兲␦共 x ⫺ ⌬ ⫺ ␣兲d␣
(4)
关using the symmetry ␦共t兲 ⫽ ␦共⫺t兲兴 or with the Fouriertransform operation given as I共 x ⫺ ⌬兲 ⫽ F ⫺1兵F关I共 x兲兴F关␦共 x ⫺ ⌬兲兴其 ⫽ F ⫺1关F关I共 x兲兴exp共⫺j2⌬兲兴,
(5)
where j is the imaginary number 公⫺1. For a vector I共x兲 of length n uniformly spaced every ␦ cm, the n frequencies are 0, 共n␦兲⫺1, 2共n␦兲⫺1, 3共n␦兲⫺1, . . . , 共n ⫺ 1兲共n␦兲⫺1, and frequencies for Eq. 共5兲 are
再
ⱕ Nyquist , ⬎ Nyquist
(6)
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⫽
⫺共2 Nyquist ⫺ 兲
where Nyquist ⫽ 共2␦兲⫺1.
shift, which falls at point 750.25 on the x-axis grid 共i.e., between grid points兲. In Fig. 2 we show the function sinc共100x兲, the exact shifted function sinc关100共x ⫺ ⌬兲兴, and the approximation I共x ⫺ ⌬兲 computed with Eqs. 共5兲 and 共7兲. The residual ⑀共x兲 ⫽ I共x ⫺ ⌬兲 ⫺ sinc共x ⫺ ⌬兲 shows the improved accuracy with Eq. 共7兲. B.
Fig. 2. Sinc共100x兲 with ⫺xmin ⱕ x ⱕ xmax, exact shifted function sinc关100共x ⫺ ⌬兲兴, and approximations I共x ⫺ ⌬兲 from the shift theorem, where I共x ⫺ ⌬兲 computed with Eq. 共5兲 using single-sided and reconstructed double-sided I共⫺xmin ⱕ x ⱕ xmax兲 关Eq. 共7兲兴 is used. The residual ⑀共x兲 ⫽ I共x ⫺ ⌬兲 ⫺ sinc共x ⫺ ⌬兲 shows the improved accuracy when Eq. 共7兲 is used. The centerburst at x ⫽ 0 is not sampled on this x-axis grid and is located between two adjacent values of x. ⌬ ⫽ 0.075 ⫹ ␦兾4, where ␦ ⫽ 10⫺4 is the interval between adjacent points.
A.
Implementation of the Shift Theorem for I共x ⫺ ⌬兲
再
I共⫺x兲 I共 x兲
⫺x max ⱕ x ⱕ ⫺x min . ⫺x min ⬍ x ⱕ x max
(7)
A small discontinuity at ⫺xmin is introduced in Id共x兲. Then we take the new linearly phase-corrected region ⫺xmin ⬍ x ⬍ xmax to be I共x ⫺ ⌬兲. This improvement is modest when small shifts ⌬ are involved. In Fig. 2 we demonstrate the difference between the shift theorem 关Eq. 共5兲兴 for I共x兲 and 关Eq. 共7兲兴 for Id共x兲. For illustration of the significant difference we chose a sinc共100x兲 function for I共x兲 with ⫺xmin ⱕ x ⱕ xmax uniformly spaced from xmin ⫽ 0.1 ⫹ ␦兾2 to xmax ⫽ 0.25 ⫹ ␦兾2, where ␦ ⫽ 10⫺4 is the interval between adjacent points. The center burst at x ⫽ 0 is not sampled on this x-axis grid and is located between two adjacent values of x. A sinc function is the interferogram of rectangular B共兲 in Eqs. 共2兲. We chose a value ⌬ ⫽ 0.075 ⫹ ␦兾4 for the desired 1184
We implemented the linear phase correction for the sampled interferogram in an iterative process with which, from the first iteration phase-corrected interferogram, I共x ⫺ ⌬兲, we extracted a new short doublesided Is共x兲 and ⌬ 关Eq. 共3兲兴 and computed a second iteration phase-corrected interferogram with Eq. 共5兲. Usually no more than a few iterations are needed for a phase correction to be achieved with an accuracy of ⌬兾␦ ⬍ 0.001. At the end of the process we obtained interferogram I共x兲, where the center burst location was resampled with high accuracy and the linear part of the phase, ⌽共兲 in Eqs. 共1兲, was corrected to a large degree. In addition, the symmetry of shortdouble-sided region Is共x兲 was improved. For the rest of this paper we refer to I共 x ⫺ ⌬兲 in Eq. 共5兲 as phasecorrected asymmetric 共single-sided兲, ⫺xmin ⱕ x ⱕ xmax, interferogram I共x兲. 3. Apodizing Functions
The shift theorem implemented with a fast-Fouriertransform 共FFT兲 operation implies cyclical 共periodic兲 functions 共which are due to the FFT operation兲. But sampled interferogram I共x兲 is asymmetric and is sampled at ⫺xmin ⬍ x ⬍ xmax. Thus shift theorem I共 x ⫺ ⌬兲 implemented with a FFT moves 共clockwise rotation for ⌬ ⬎ 0兲 in regions near xmax to regions near xmin owing to the cyclical rotation. We solve this problem by creating a double-sided interferogram Id共x兲 for the complete range ⫺xmax ⱕ x ⱕ xmax by using the portion I共xmin ⱕ x ⱕ xmax兲 of the recorded interferogram for ⫺xmax ⱕ x ⱕ ⫺xmin with the rest of the recorded interferogram I共⫺xmin ⱕ x ⱕ xmax兲 before the shift operation with the FFT shift theorem 关Eq. 共5兲兴, where Id共x兲 is given by I d共 x兲 ⫽
Iteration Process for Linear Phase Correction
APPLIED OPTICS 兾 Vol. 41, No. 6 兾 20 February 2002
Spectrum B共兲 was obtained from I共 x兲 关Eqs. 共1兲兴 by a ˆ 共兲 ⫽ Fourier-transform operation 关Eqs. 共2兲兴, B F关I共x兲w共x兲兴, where w共x兲 is an apodizing function 共i.e., a window function兲 that is designed9 to minimize sidelobes and thereby to reduce the distortion of the ˆ 共兲 that results from derived estimated spectrum B the transfer of energy 共leakage兲 across frequencies , a leakage that can even produce negative values and therefore an unphysical spectrum. The design of the apodization function involved a trade-off between minimizing the sidelobes 共reducing energy leakage兲 of the power spectrum of the window at the expense of increasing the width of the main lobe, an increase that reduces the resolution 共i.e., a smearing effect兲. The resolution width is given in units of spectral bins, in which a bin size is the fundamental frequency resolution N⫺1, where N is the number of points in the window w共x兲. Usually the spectral bin widths of the apodization windows range from 1.2 to 3 spectral bins. Boxcar window w共x兲 has the narrowest main lobe 共1.21 spectral bins兲 and the highest sidelobes 共⫺13 dB, falling at a rate of ⫺6 dB兾octave兲. The traditionally used triangular window in a FTIR spectrometer has a resolution width of 1.78 spectral bins, a highest sidelobe level of ⫺27 dB, and a sidelobe falloff rate of ⫺12 dB兾octave, a decay rate that is usually too slow for accurate spectral analysis. A Chebychev window designed to have a constant level of sidelobes of ⫺80 dB has a resolution width of 2.31 spectral bins and for a constant level of sidelobes of ⫺50 dB has a resolution width of 1.85 spectral bins. With the phase correction, interferogram I共x兲 is single sided where the region about center-burst location x0 appears twice. Without proper weighing of
Fig. 3. Asymmetrical hybrid apodization windows. xmin ⫽ 5% of the window width 共100 points兲. 共a兲 Time domain where a ramp function is substituted for the range ⫺xmin ⱕ x ⱕ xmin. Triangle, Blackman, Kaiser, and Chebyshev apodization windows for the range xmin ⱕ x ⱕ xmax. 共b兲 Power spectral density magnitude 10 log10兵兩F关w共x兲兴兩2其 in the frequency domain where the Nyquist frequency is 1. Attenuation R ⫽ 100 dB for Chebyshev and Kaiser windows.
the interferogram, intensities of sharp absorption bands 共frequencies兲 in the deduced spectrum will be distorted. A proper weighing function w共x兲 such that w共x0 ⫺ x兲 ⫹ w共x0 ⫹ x兲 ⫽ 1 for the range ⫺xmin ⱕ x ⱕ xmin will correct this distortion. Mertz chose the weighing function w共x兲 to be a ramp, and Rahmelow and Hubner5 chose a fifth-order polynomial for ⫺xmin ⱕ x ⱕ xmin, w共x兲 ⫽ 0.5 ⫹ 1.25共x兾xmin兲3 ⫺ 0.75共x兾xmin兲5, which is smoother than a ramp function and thus is somewhat better for a Fouriertransform operation in which one wants to avoid sharp discontinuities. We tried to combine ramp function w共x兲 共or a fifthorder polynomial兲 for the range ⫺xmin ⱕ x ⱕ xmin with different windows w共x兲 for the range xmin ⱕ x ⱕ xmax to achieve the proper weighing of the interferogram in the short double-sided region and an improved apodizing window for the rest of the range. Examples of hybrid asymmetrical windows 共a ramp combined with a triangle, Blackman, Kaiser, and Chebyshev windows兲 are presented in Fig. 3共a兲 for xmin ⫽ 5% of the window width; in Fig. 3共b兲 the corresponding power spectral density magnitude, 10 log10关兩F共w共x兲兲兩2兴 共in decibels兲, of the windows w共x兲 is shown in the frequency domain. These asymmetric windows are not significantly better than the traditional Mertz asymmetric triangular window. The good properties 共i.e., narrow main lobe and low sidelobes兲 of the more-sophisticated windows 共e.g., Blackman, Kaiser, and Chebyshev兲 were lost owing to the asymmetry. The symmetrical apodization windows for triangle, Blackman, Kaiser, and Chebyshev functions are shown in Fig. 4 共R ⫽ 100 dB for Kaiser and Chebyshev windows兲, where parameter ␣ for a Kaiser
Fig. 4. Symmetrical apodization windows 共xmin ⫽ xmax兲 for which the peak location of the window 共in the time domain兲 is at the middle 共a兲 in the time domain and 共b兲 in the frequency domain. Attenuation R ⫽ 100 dB for Chebyshev and Kaiser windows.
window is given for attenuation R 共decibels兲 as ␣ ⫽ 0.5842共R ⫺ 21兲0.4 ⫹ 0.07886共R ⫺ 21兲 for 50 ⱖ R ⱖ 21 and ␣ ⫽ 0.1102共R ⫺ 8.7兲 for R ⬎ 50. The superiority of the symmetrical windows to the asymmetrical windows is evident. To take full advantage of the symmetrical apodization we constructed from the single-sided phase-corrected interferogram I共⫺xmin ⱕ x ⱕ xmax兲 a double-sided phase-corrected interferogram Id共x兲 关Eq. 共7兲兴. In this process we introduced a discontinuity at x ⫽ ⫺xmin, but this discontinuity is small because of the phase correction and will be reduced with the apodization window. In doublesided phase-corrected interferogram Id共x兲 the center burst location is in the middle, and all points in interferogram Id共x兲 appear twice 共therefore no special weighing is necessary兲, and thus a symmetrical apodization window 共Fig. 4兲 that is superior to the asymmetrical triangular window can be used. A demonstration of the superiority of a spectrum computed with the double-sided phase-corrected interferogram to a spectrum computed with a zero-fill vector for the missing data ⫺xmax ⱕ x ⱕ ⫺xmin is presented in Section 6 below. 4. Transforming the Recorded Interferogram I共x兲 to a Solution Spectrum S共兲
Given recorded interferogram I共x兲 sampled at n points, unknown spectrum B共兲 can be deduced at a set of discrete frequencies 共wave numbers兲 that are uniformly spaced between a zero frequency and a maximum frequency, the Nyquist frequency, max ⫽ Nyquist ⫽ 共2␦兲⫺1, where ␦, the distance between consecutive optical-path difference x values, is usually determined by an external clock 共e.g., a He–Ne laser兲. Padding the interferogram with m0 zeros helps to reduce the discontinuities that are inherent in a discrete Fourier-transform operation in which the n-point vector is treated as a periodic sequence of 20 February 2002 兾 Vol. 41, No. 6 兾 APPLIED OPTICS
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many n-point vectors. It is also sometimes of interest to obtain a representation of an estimated specˆ 共兲 at a different 共smaller兲 frequency interval trum B ⌬, an objective that can be achieved with appropriate zero padding. The number of available discrete frequencies n from ⫽ 0 to max for an n-point vector padded with m0 zeros is n ⫽
再
0.5共n ⫹ m 0兲 ⫹ 1 0.5共n ⫹ m 0 ⫹ 1兲
共n ⫹ m 0兲 even , 共n ⫹ m 0兲 odd
S共兲 ⫽ Re关B共兲兴
and the spacing 共intervals兲 between consecutive frequencies is ⌬ ⫽ 1兾关共n ⫹ m0兲␦兴 ⫽ 2共n ⫹ m0兲⫺1Nyquist. However, the minimum frequency 共i.e., ⫽ 0兲 and the maximum frequency remain unchanged with any zero padding. We apodized the phase-corrected double-sided interferogram 关Eq. 共7兲兴, Idw共x兲 ⫽ Id共x兲w共x兲, where the superscript w denotes the windowing 共apodizing兲 process. To obtain a specific interval ⌬ ⬍ 共2xmax兲⫺1 we padded Id共x兲 with m0 zeros, m0 ⫽ 共␦⌬兲⫺1 ⫺ nd, where nd ⫽ 2共n ⫺ n0兲 ⫹ 1 ⫽ 2xmax兾␦ ⫹ 1 is the number of points in Id共x兲. The phase shift ⌽2共兲 that is due to the location of the centerburst, at point n0d ⫽ 共n ⫺ n0 ⫹ 1兲 in double-sided phase-corrected interferogram Idw, is a phantom phase that is easily corrected by a simple rotation, as was demonstrated by Mertz,7 to place the center burst at the first element. ˆ 共兲 共a complex quanThe first estimate spectrum B tity兲 from Idw is given by ˆ 共兲 ⫽ 2 B
F共Idw兲兾K共兲 nd nd 2 Nyquist
⫽ 2␦F共Idw兲兾K共兲 ⬵
F共Idw兲兾K共兲 关W兾共cm2 sr cm⫺1兴, n d共2x max兲 ⫺1兾2
(8)
where the factor 2 compensates for the factor 0.5 in Eqs. 共1兲, nd in the denominator normalizes the length of 共unpadded兲 vector Id共x兲, factor 2nd⫺1Nyquist ⬵ 共2xmax兲⫺1 is the spectral bin size 共inverse centimeters兲 in the FFT operation to produce a spectrum per ˆ 共兲 wave number. For nd ⫹ m0 ⫽ odd, spectrum B has to be multiplied by 2 for all frequencies other than the zero frequency to account for the energy in the imaginary frequencies in the FFT operation. ˆ 共兲 has to be multiplied by 2 for For nd ⫹ m0 ⫽ even, B all frequencies other than the zero frequency and the Nyquist frequency, which appears once in the FFT vector. In the Mertz7 procedure, the residual nonlinear ˆ 1共兲 is estimated from a Fourier transform phase in ⌽ of the short double-sided interferogram for a limited number of frequencies and then interpolated 关assumˆ 1共兲 is a smooth function兴 to all frequencies ing that ⌽ ˆ 共兲. In our procedure, phase ⌽ ˆ 1共兲 that was not for B corrected in the iterative phase correction can be esˆ 共兲 and is timated directly for all frequencies from B ˆ 1共兲 ⫽ tan⫺1兵Im关B共兲兴兾关Re共B ˆ 共兲兴其, where given by ⌽ Im共 兲 denotes the imaginary part and special care is ˆ 共兲兴 and Re关B ˆ 共兲兴 to preserve given to the signs of Im关B 1186
ˆ 共兲兴 ⫽ ⫺1 and Re关B ˆ 共兲兴 ˆ 1共兲 兵e.g., for Im关B the sign of ⌽ ˆ 1共兲 ⫽ 45° in the first quadrant when the ⫽ ⫺1, ⌽ correct angle is ⫺135° in the third quadrant其. The ˆ 共兲 is corrected for the residual estimated spectrum B ˆ 1共兲 to give the sought-after nonlinear part of phase ⌽ spectrum B共兲 and is denoted solution spectrum S共兲:
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ˆ 共兲兴sin关⌽ ˆ 1共兲兴. ˆ 共兲兴cos关⌽ ˆ 1共兲兴 ⫹ Im关B ⫽ Re关B (9) With this phase correction we neglected the small imaginary part in the derived solution spectrum: ˆ 共兲兴sin关⌽ ˆ 1共兲兴 ⫹ Im关B ˆ 共兲兴cos关⌽ ˆ 1共兲兴. Im关S共兲兴 ⫽ ⫺Re关B
5. Comparison of Solution S共兲 and Source Spectrum B共兲
Solution S共兲 is a lower-resolution estimate 共sometimes also called a deresolved spectrum兲 of true input source spectrum B共兲. The source spectrum is degraded 共smeared兲 as a result of the instrument transfer function 关also called the instrument line shape 共ILS兲兴 and the maximum optical path difference xmax that was used in the interferogram measurements. Thus, to compare solution spectrum S共兲 derived with our method with B共兲 we must first degrade the resolution of B共兲 to the spectrum that would have been measured by the instrument 共i.e., without regard to the algorithm used to derive a spectrum from the recorded interferogram兲. The wavelength resolution of S共兲 is given by the Rayleigh resolution criterion1 as ⌬ ⫽ 2⌬ ⬵ 1兾xmax. With the Rayleigh resolution criterion, two monochromatic sources of a unit intensity at two frequencies, 1 and 2 ⫽ 1 ⫹ ⌬, are considered to be just resolved when they produce a combined spectrum 关for an instrument line shape in the form of a sinc2共 x兲 function兴 with a dip of 81% in the middle of the combined peak. For convenience 共it is easier to convolve Gaussian functions than sinc functions兲 we implement the Rayleigh resolution criterion for the ILS in the form of a Gaussian function given by
g共, 1兲 ⫽
exp关⫺0.5共 ⫺ 1兲 2兾共⌬兾2.638兲 2兲兴 , 共2兲 0.5共⌬兾2.638兲
(10)
for which the combined spectrum g共兲 ⫽ g共, 1兲 ⫹ g共, 1 ⫹ ⌬兲 is a good approximation of the Rayleigh criterion, as is shown in Fig. 5 for ⌬ ⫽ 2⌬ ⬵ 1兾xmax ⫽ 4 cm⫺1. The two Gaussians10 intersect at x ⫽ ⌬兾2, and each has a FWHM value of 0.8927⌬. When spectrum B共兲 is measured with a Gaussian ILS the resultant reduced resolution spectrum is denoted
Fig. 5. Rayleigh resolution criterion implemented with Gaussian functions. g共兲 ⫽ g共, 1兲 ⫹ g共, 1 ⫹ ⌬兲 is the combined spectrum of two monochromatic sources, g1 and g2, at frequencies 1 ⫽ 1000 cm⫺1 and 2 ⫽ 1 ⫹ ⌬ separated by ⌬ ⫽ 4 cm⫺1.
Fig. 7. Nonlinear phase error ⌽1共兲 rad 共obtained from a FTIR instrument兲 representing the dispersive optical elements in a spectrometer.
6. Results
L共兲 and is given by convolution of the ILS 关Eq. 共10兲兴 with B共兲, given by
L共兲 ⫽
兰 兰
⬁
B共t兲 g共t, 兲dt
0
⫽
⬁
0
B共t兲
exp关⫺0.5共t ⫺ 兲 2兾共⌬兾2.638兲 2兲兴 dt. 共2兲 0.5共⌬兾2.638兲 (11)
Fig. 6. Spectrum B共兲 W兾共cm2 sr cm⫺1兲. 共a兲 Planck function at a temperature of 300 K; 共b兲 Planck function at a temperature of 300 K with five Lorentzian absorption lines of 10-cm⫺1 FWHM and transmissions 0.1, 0.2, 0.3, 0.4, and 0.5 at frequencies 600, 800, 1000, 1200 and 1400 cm⫺1, respectively; 共c兲 atmospheric radiance 共1-cm⫺1 resolution兲 computed with the MODTRAN program for a 1976 U.S. Standard Atmosphere for an observer on the ground looking up. Wavelength range, 510 –1500 cm⫺1.
We simulated interferograms 关Eqs. 共1兲兴 for three types of source spectrum B共兲 shown in Fig. 6: a Planck function 共i.e., a blackbody兲, a Planck function with five Lorentzian1 absorption lines of 10-cm⫺1 FWHM and transmissions of 0.1 to 0.5, and atmospheric radiance 共1-cm⫺1 resolution兲 computed with the MODTRAN program11 for a 1976 U.S. Standard Atmosphere and an observer on the ground looking up. The wavelength range was 510 –1500 cm⫺1, and the temperature for the Planck blackbody function was 300 K. The maximum optical path difference was xmax ⫽ 0.25 cm, and there were n ⫽ 2048 sampled points from xmin ⫽ ⫺0.05 cm to xmax. The Nyquist frequency was 3411.7 cm⫺1. We tried to make the simulations as realistic as possible. Thus the location of the center burst 共x ⫽ 0兲 was not sampled in interferogram I共 x兲, and a nonlinear phase error, ⌽1共兲, obtained from a FTIR instrument and shown in Fig. 7, was used in simulating the interferograms. Derived spectrum S共兲 was constructed at equal intervals of ⌬ ⫽ 1.9995 cm⫺1 for zero padding, and a corresponding spectrum L共兲 was computed from B共兲 with ⌬ ⫽ xmax⫺1 ⫽ 4 cm⫺1. As noted above 共Section 3兲, the apodization function is a trade-off between minimizing the sidelobes 关and thereby minimizing the leakage from a nearby spectral features of B共 ⫽ 0兲 to derived spectrum S共0兲兴 at the expense of increasing the width of the main lobe and thus smearing and spreading S共0兲 共and thereby distorting its magnitude兲. Assuming that we would like to obtain a large dynamic range, of the order of 105, in derived spectra S共兲 to exploit the high signal-to-noise ratio 共SNR兲 in the interferogram measurements, an attenuation R 共in the Kaiser and Chebyshev apodization windows兲 of 50 –100 dB will be sufficient. The SNR for a shot-noise process 共owing to the statistical fluctuations in the incident photon flux兲 is the maximum attainable SNR. For a blackbody source B共500 ⬍ ⬍ 1500 cm⫺1兲 at a temperature of 300 K the SNR for 20 February 2002 兾 Vol. 41, No. 6 兾 APPLIED OPTICS
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Fig. 8. Radiance spectrum L共兲 W兾共cm2 sr cm⫺1兲 关Eq. 共11兲兴, with ⌬ ⫽ 4 cm⫺1 for B共兲, and solution spectrum S共兲 W兾共cm2 sr cm⫺1兲 共top兲. Residual percent difference 100兩L共兲 ⫺ S共兲兩兾L共兲 共middle兲. Absolute radiance difference 兩L共兲 ⫺ S共兲兩 W兾共cm2 sr cm⫺1兲 共bottom兲. Solution spectrum S共兲 is derived for Planck function B共兲 关Fig. 6共a兲兴 for a Chebyshev apodizing window with R ⫽ 100 dB.
a shot-noise process is of the order of 107, and the SNR that is due to detector noise and background photon flux in FTIR spectrometers is of the order of 103 to 104. It was also noted previously6 that Kaiser and Chebyshev apodization windows are good choices for use in retrievals. In Figs. 8 –10 we show spectra L共兲关W兾共cm2 sr cm⫺1兲兴 and solution spectrum S共兲关W兾共cm2 sr cm⫺1兴, the residual percentage difference 100兩L共兲 ⫺ S共兲兩兾
Fig. 9. Same as Fig. 8 but here source spectrum B共兲 关Fig. 6共b兲兴 is a Planck function with five Lorentzian absorption peaks 共10-cm⫺1 FWHM兲 with transmissions of 0.1, 0.2, 0.3, and 0.5 at frequencies 600, 800, 1000, 1200, and 1400 cm⫺1. Two spectra, S1共兲 and S2共兲 共computed with a Chebyshev window with R ⫽ 100 dB兲, are shown. For S2共兲 the double-sided phase-corrected interferogram was used, and for S1共兲 zeros were substituted for the missing data ⫺xmax ⱕ x ⱕ ⫺xmin in phase-corrected interferogram Id共x兲 关Eq. 共7兲兴. 1188
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Fig. 10. Same as Fig. 8 but here source spectrum B共兲 关Fig. 6共c兲兴 is atmospheric radiance 共1-cm⫺1 resolution兲 computed with MODTRAN and S共兲 is computed with a Kaiser window with R ⫽ 50 dB.
L共兲, and the absolute radiance difference 兩L共兲 ⫺ S共兲兩关W兾共cm2 sr cm⫺1兴. In Fig. 8, spectra S共兲 are shown for Planck function B共兲 关Fig. 6共a兲兴 for a Chebyshev apodizing window with R ⫽ 100 dB. The figure shows an error that is in general less than 0.1% and a radiance difference of less than 10⫺8 W兾共cm2 sr cm⫺1兲. The importance of constructing a double-sided phase-corrected interferogram 关expression 共8兲兴 is shown in Fig. 9, where two spectra, S1共兲 and S2共兲 共both computed with a Chebyshev window with R ⫽ 100 dB兲, are shown for radiance spectrum B共兲 of Fig. 6共b兲. For S1共兲, zeros were substituted for the missing data, ⫺xmax ⱕ x ⱕ ⫺xmin, in phase-corrected interferogram Id共x兲 关Eq. 共7兲兴, and for S2共兲 the doublesided phase-corrected interferogram was used. The figure shows the degradation in accuracy 共of the order of 5–10兲 for S1共兲, a degradation that increases as the range of missing data increases. Substituting zeros for the missing data and using symmetrical apodization functions yields an interferogram that is not weighted properly 共i.e., not all the fringes in the interferogram are counted equally兲. As a result, deduced spectrum S1共兲 is distorted. In general, fringes in the interferogram located farther from the center burst 共large values of x兲 contain spectral information on sharp spectral features of B共兲 共i.e., sharp absorption bands兲 and fringes closer to the center burst contain spectral information on smooth spectral features of source B共兲. In Fig. 10, the atmospheric radiance 共1-cm⫺1 resolution兲 of source spectrum B共兲 关Fig. 6共c兲兴 computed with the MODTRAN program in which there are many narrow absorption lines, and S共兲 computed with a Kaiser window with R ⫽ 50 dB, are shown. The accuracy of S共兲 is within a few percent, and radiance the difference is smaller than 10⫺7 W兾共cm2 sr cm⫺1兲. 7. Summary
A method for transforming symmetric single-sided interferograms into spectra that combines the For-
man and the Mertz methods has been developed. In this new method we use any symmetric apodizing windows 共e.g., triangular, Hamming, Hanning, Blackman, Chebyshev, and Kaiser兲. The measured interferogram is asymmetric and contains phase errors that are due to missampling of the interferogram center burst’s location and a phase error from dispersive optical elements and electronics in the spectrometer. We use a short double-sided fraction of the asymmetric interferogram and apply an iterative phase correction for the linear portion of the phase to the recorded single-sided asymmetric interferogram. The linear phase correction produces a phasecorrected single-sided interferogram and is implemented with an iterative method in which the location of the center burst is computed by estimation of the center of gravity of the short, double-sided portion of the measured interferogram, and the shift theorem is applied to a reconstructed double-sided interferogram. The iterative phase correction enhances the symmetry in the double-sided interferogram computed from the phase-corrected singlesided recorded interferogram. Because of the poor properties of the asymmetric apodization functions, we constructed a symmetric double-sided interferogram from the single-sided phase-corrected interferogram and followed the Mertz procedure to produce the solution spectrum but with symmetrical apodization windows and with nonlinear correction derived from the double-sided interferogram without the need to interpolate over frequencies as is done in the Mertz procedure. Examples of the accuracy of the derived spectrum have been shown for a Planck function of temperature of 300 K, for a Planck function of temperature of 300 K with five Lorentzian absorption lines of 10cm⫺1 FWHM and transmissions of 0.1 to 0.5, and for a moderate resolution radiance 共1 cm⫺1兲 computed with the MODTRAN program for a standard atmosphere for an observer on the ground looking up. A nonlinear phase error 共obtained from a FTIR spectrometer兲 was included in the simulated interferograms, and the location of the center burst 共x ⫽ 0兲 was not sampled so realistic simulated interferograms could be produced. In comparing the solution spectrum with the source spectrum we applied the Rayleigh resolution criterion with a Gaussian instrument line shape. The accuracy of the solution has been shown to be excellent, ranging from better than 0.1% for a Planck blackbody 关Fig. 6共a兲兴 and a radiance difference of less than 10⫺8 W兾共cm2 sr cm⫺1兲 to a few percent for a
complicated atmospheric radiance spectrum 关Fig. 6共c兲兴 and radiance difference smaller than 10⫺7 W兾共cm2 sr cm⫺1兲. This study was supported by the U.S. Army Soldier and Biological Chemical Command, Edgewood Chemical Biological Center 共ECBC; formerly Edgewood Research Development and Engineering Center兲 under contract DAAM01-94-C-0079. We thank Thomas Gruber and Rich Vanderbeek for many useful discussions and Bill Loerop of the ECBC for administrative support and encouragement. We are very grateful for the constructive and insightful suggestions of the anonymous reviewer, especially regarding the FFT operation and the shift theorem. References and Notes 1. P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectrometry 共Wiley, New York, 1986兲. 2. D. B. Chase, “Phase correction in FT-IR,” Appl. Spectrosc. 36, 240 –244 共1982兲. 3. H. E. Revercomb, H. Buijs, H. B. Howell, D. D. LaPorte, W. L. Smith, and L. A. Sromovsky, “Radiometric calibration of IR Fourier transform spectrometers: solution to a problem with the High-Resolution Interferometer Sounder,” Appl. Opt. 27, 3210 –3218 共1988兲. 4. R. C. M. Learner, A. P. Thorne, I. Wynne-Jones, J. W. Brault, and M. C. Abrams, “Phase correction of emission line Fourier transform spectra,” J. Opt. Soc. Am. A 12, 2165–2171 共1995兲. 5. K. Rahmelow and W. Hubner, “Phase correction in Fourier transform spectroscopy: subsequent displacement correction and error limit,” Appl. Opt. 36, 6678 – 6686 共1997兲. 6. C. D. Barnet, J. M. Blaisdell, and J. Susskind, “Practical method for rapid and accurate computation of interferometric spectra for remote sensing application,” IEEE Trans. Geosci. Remote Sens. 38, 169 –183 共2000兲. 7. L. Mertz, “Auxiliary computation for Fourier spectrometry,” Infrared Phys. 7, 17–23 共1967兲. 8. M. L. Forman, W. H. Steel, and G. A. Vanasse, “Correction of asymmetric interferograms obtained in Fourier spectroscopy,” J. Opt. Soc. Am. 56, 59 – 63 共1966兲. 9. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51– 83 共1978兲. 10. Based on probability theory, the probability of measuring a signal with a Gaussian probability function g共, 1 ⫽ ⌬兲 in the presence of zero-mean noise with probability-density function ⬁ g共, 1 ⫽ 0兲 is 兰⌬兾2 g共, ⌬兲d ⫽ 90% when a threshold of detection ⱖ ⌬兾2 is chosen. 11. A. Berk, G. P. Anderson, L. S. Bernstein, P. K. Acharya, H. Dothe, M. W. Matthew, S. M. Adler-Golden, J. H. Chetwynd, Jr., S. C. Richtsmeier, B. Pukall, C. L. Allred, L. S. Jeong, and M. L. Hoke, “MODTRAN4 radiative transfer modeling for atmospheric correction,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research III, A. M. Larar, ed., Proc. SPIE 3756, 348 –353 共1999兲.
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