Theoretical aspects of Fourier Transform Spectrometry and common path triangular interferometers

Recent investigations have induced relevant advancements of imaging interferometry, which is becoming a viable option for Earth remote sensing. Various research programs have chosen the Sagnac configuration for new imaging interferometers. Due to the growing diffusion of this technique, we have developed a self-contained theory for describing the signal produced by triangular FTSs and its optimal processing. We investigate the relevant disadvantages of multiplexing, and compare dispersive with FTS instruments. The paper addresses some methods for correcting the phase error, and the non-unitary transformation performed by a Sagnac interferometer. The effect of noise on spectral estimations is discussed. ©2010 Optical Society of America OCIS codes: (300.6300) Spectroscopy, Fourier Transforms; (300.6190) Spectroscopy, Spectrometers; (120.3180) Instrumentation, measurement; and metrology, Interferometry; (120.0280) Instrumentation, measurement; and metrology, Remote Sensing and sensors; (280.4991) Remote sensing and sensors, Passive remote sensing; Hyperspectral remote sensing; Sagnac interferometer; Instrument response compensation. References and links 1. M. J. Persky, “A review of spaceborne infrared Fourier transform spectrometers for remote sensing,” Rev. Sci. Instrum. 66(10), 4763–4797 (1995). 2. B. Harnisch, W. Posselt, K. Holota, H. O. Tittel, and M. Rost, “Compact Fourier-transform imaging spectrometer for small satellite missions,” Acta Astronaut. 52(9-12), 803–811 (2003). 3. M.-L. Junttila, “Stationary Fourier transform spectrometer,” Appl. Opt. 31(21), 4106–4112 (1992). 4. L. J. Otten, A. D. Meigs, B. A. Jones, P. 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Introduction
Recent investigations have induced relevant advancements of imaging interferometry, which is becoming a viable option for Earth remote sensing [1,2].Various research programs have focused the opportunity to adopt the common path configuration for new imaging interferometers under analysis.More specifically, the Sagnac triangular configuration has been adopted for the development of some sensor prototypes, as reported by [3][4][5][6].The Fourier Transform Hyperspectral Imager (FTHSI) was the first spaceborne imaging interferometer that operated on board of the AirForce Research Laboratory (AFRL) -U. S. Department of Defence (DoD) technological satellite MightySat II.1 (Sindri P99-1) after its launch in 2000 [4,7].Precursors of the FTHSI were the HyperCam and the IrCam developed by the Kestrel Corporation (USA) for airborne applications, the Spatially Modulated Fourier Transform Spectrometer (SMIFTS) developed by Hawaii University [5], the High Ètendue Imaging Fourier Transform Spectrometer (HEIFTS) by Science Application International Corporation (USA) [8], and the imaging interferometer developed by Applied Spectral Imaging (Israel) for laboratory applications [6].An additional scientific initiative that will exploit the use of a hyperspectral imaging interferometer (the ALISEO sensor [9] and [10]) is the MIOSAT mission of the Italian Space Agency (ASI).MIOSAT is a technological compact satellite (mass around 120 kg) that will host three payloads: a panchromatic camera, a MEMS Mach-Zehnder interferometer for atmospheric sounding, and the ALISEO sensor.Other remote sensing instruments adopting an interferometric approach have been reported in [11][12][13][14].
Some advantages are implicit in using imaging interferometers, due to their high signal (Jacquinot's effect), and the option to adjust the sampled spectral range and resolution by changing the sensor sampling step and the instrument Field-Of-View (FOV) [2][3][4][5][6]15,16].Critical points are instead connected with the high data-rate requested, the need to pre-filter the incoming radiation in order to avoid aliasing in the retrieved spectra, and the heavy data pre-processing for compensating instrument response and possible acquisition artefacts [17][18][19][20].In this connection achieving new insights on the performance of imaging interferometers is an important topic that should originate relevant contributions to remote sensing applications and spectroscopic studies.
In this work we propose a theoretical analysis of Fourier Transform Spectrometry (FTS) with specific relations to the common-path Sagnac configuration.We gives a brief description of the optical configuration (Sagnac, without input slit) of the new ALISEO sensor, and discusses the theory describing its signal and the spectral estimations derived by it.The remainder of this paper is organized as follow.In Sect. 2 the main optical characteristics of the developed instrument are outlined.Sect. 3 discusses the theory of Sagnac common-path interferometers and describes a self-contained model of interferogram and spectral estimation.In this section we demonstrate that multiplexing usually produces a severe radiometric disadvantage in FTS (as compared with dispersive spectrometry), while the absence of exit slit (here termed as Fellgett's advantage) is confirmed.We indicate some theoretical properties of the interferometric signal that can be useful for correcting the phase error originated by uncontrolled scanning offset.We also point out how this phase correction scheme interferes with experimental noise affecting the interferogram.We also find an analytical model describing the error introduced by the not unitary cosine-like transformation operated by the instrument, obtaining implicitly an algorithm for correcting its effects.Let us note that most of these findings are proved under very general assumptions, such that they represents as well the behaviour of almost each interferometer.Sect. 4 discusses the main consequences of our theoretical findings, while Sect. 5 summarizes the outcomes of our investigation, and sketches future work and open problems.Fig. 1.Layout of a stationary interferometer in the Sagnac (triangular common path) configuration.The light from the objective is first collimated by the input lens L1, and the semi-reflecting surface of the beam-splitter BS originates two rays (reflected and transmitted beam), which travel the instrument on the same triangular path limited by the two folding mirrors M1 and M2.On the exit port, light is focused onto a CCD plane by the lens L2.

The Sagnac interferometer
Sagnac is an optical configuration for a class of interferometers that have a triangular ray-path, common for the two rays of the interferometer.Due to this reason this type of instruments are also called common-path interferometers, as opposed to tilted interferometers like the Michelson-Morley and Mach Zehnder instruments.Figure 1 shows the instrument's optical layout, so describing the main characteristics of the Sagnac configuration.The light is first collimated by the lens L1, and travels the interferometer by means of a beam-splitter BS and two folding mirrors M1 and M2.Light emerging on the output port is then focused onto the output focal plane that holds an image detector by the lens L2.The instrument produces in its focal plane a stationary pattern of interference fringes of equal thickness that are localized at infinity.Let us note that interference fringes of a Sagnac instrument are sometimes termed as "equal inclination" fringes, as in Katzberg and Statham (1996) [12].However, equal inclination fringes are usually produced in Michelson-Morley interferometers, and are characterized by a non-uniform fringe distribution in the instrument focal plane, as discussed in [21].This feature is quite different from the behaviour of a narrow Field-Of-View (FOV) Sagnac instrument, in the focal plane of which fringe thickness is truly constant.
It can be shown that the BS provides part of phase-delay between the two interfering rays, the remaining part being originated outside the BS, and that the overall OPD linearly changes with varying the angle of the entering ray with respect to the instrumental optical axis [9,10].Due to the absence of entrance slit, the device acquires the image of an object superimposed to a fixed pattern of across-track interference fringes.Then, introducing a relative motion between the sensor and the object, each scene pixel exploits the entire interference pattern, hence its interferogram and spectrum can be assessed.We call this kind of optical layout "Leap frog" configuration.In remote sensing applications, each ground point is observed under several viewing angles while the sensor moves with respect to the target, so a 3-dim array of data (image stack) of varying phase offset is collected.This data-cube is first processed in order to extract the complete interferogram of every image pixel, then it is inverse cosine transformed to yield a wavenumber hyperspectral data-cube.
One of the main drawbacks of the Sagnac configuration is that the image at the output port is affected by relevant vignetting frequently.This phenomenon evidently influences the fringe visibility, and demands for complex data pre-processing necessary for restoring the correct radiometric level of the signal in the far wings of the interferogram.

Theory of a Sagnac imaging interferometer
In this Section an attempt is made to develop a comprehensive theory of an imaging interferometer operating in the common-path, triangular configuration.Nonetheless, the properties discussed in Sections from 3.1 until 3.6 have been deduced on such a quite general ground that they can be applied to almost any kind of Fourier Transform Spectrometer (FTS).
In a Fourier Transform Imaging Spectrometer (FTIS) the acquired physical information is the interferogram , that is the power of the interference pattern generated by the two rays at the position x in the focal plane of the instrument, as stated in the following relationship.
In this equation j is the imaginary unit, φ the initial phase of the impinging radiation field,

=
, and the integration over κ takes into account for a non-monochromatic light spectrum.Let us note that the OPD depends on the inclination ϑ of the entering ray, and that the corresponding focal plane position x is the projection of ϑ by the equivalent focal length of the focusing element of the instrument.
We note that both rays pass throughout the BS two times, being subject to two attenuations due to internal extinction of the BS.For this reason the electric field amplitude attenuation factors ( ) It is worth noting that the power internal transmittance ( ) κ τ i of the BS is measured over a 45° slant path in the typical Sagnac geometry.We also recall that the reflected ray experiences along its path two additional reflections more than the transmitted ray.Following these two supplementary reflections along the propagation direction of the light, the first reflection happens on an increasing step of the refraction index while the second one takes place on a to an overall π phase delay with respect to the transmitted ray for both the electrical field components (parallel and orthogonal).For sake of simplicity we also assume ( ) i τ κ to be a real quantity.It can easily be shown that the previous equations lead to the following fundamental law [9,10,27,32]: In this equation we have introduced the fringe visibility This characteristic is important since amplitude division is driven by ( ) κ ρ which varies with changing wavelength, usually abating the fringe visibility towards both sides of the exploited spectral interval.The relationship between OPD and the entering ray inclination ϑ over the instrument optical axis is linear for a Sagnac device, as long as the device FOV is small enough: the focal plane position conjugated to the entering direction ϑ , and f the effective focal length of the lens focusing the interference image.The maximum optical path difference max OPD observed by the interferometer depends on the width of the digitized FOV, the beam-splitter thickness, as well as its distance to folding mirrors.In Eq. ( 4) we suppose that the position of the x origin exactly corresponds to the interferogram centre ( ) 0 0 OPD = .The effect of an unknown offset of the interferogram centre will be addressed in Sect.3.2.As can be seen from Eq. (3), the interferogram is an oscillating function, which reaches a minimum at 0 OPD = at each wavelength in the typical Sagnac configuration.The interferogram also decays for large OPD s where useful information regarding the target spectrum is contained.
Actually, the complete raw interferogram of the energy coming from a certain pixel of the observed scene is convolved by the pixel dimension, and sampled with a (square) grid whose extension is limited by the detector size D .This is true for a system equipped with a (2 dim) array detector, like a CCD or a CMOS sensor.Therefore, the sampled interferogram ( ) x I S would be expressed as:: x is the centre of the sampling grid.Imposing a not null value for 0 0 ≠ x gives rise to sampling asymmetry, a condition in which the number of samples collected on the two sides of the interferogram is not balanced.Since the interferogram is a function having even symmetry, the instrument may be designed in order to collect interferogram samples only on one of its sides, i.e. on the left of the interferogram center.We take 0 x to be an integer multiple of the pixel pitch p , so assuming that a pixel exactly falls on the interferogram centre.In Sect.3.2 we will discuss the important effects introduced by a supplementary sampling offset in the form of a fraction of p .The symbol ( ) comb x p is coherent with the notation used by Goodman [22], and is proportional to the shah function adopted by Bracewell [23], and Walmsley et al [24].The ratio ( ) is the areal fill factor, a characteristic specific of the adopted detector.We define ) is given by: The parameter b takes into account the interferogram sampling asymmetry.For a null 0 x , b equals ½ and the interferogram sampling is symmetric having half samples on the left of the origin and half samples on its right.When 1 = b only one side of the interferogram is sampled.Let us note that in Eqs.(5) and (6) we suppose that the interferogram central sample always is measured, regardless of the actual b value.
Let us note that Eqs. ( 1) and (3) contain an integral transform that differs from the usual cosine transform, the difference being due to circumstance that the OPD also depends on the wavenumber κ .Therefore OPD and κ are not conjugate (canonical) variables and the integral transform performed by the instrument is not unitary.This point will be deeply addressed in Sect.

IT
to the Performing the inverse transform of the signal in Eq. ( 5) would produce a spectral estimation modulated by the cosine factor where As known, Eq. ( 9) also yields an important limit for the sampling frequency in order to avoid the aliasing phenomenon in the retrieved spectrum.As stated by Shannon's theorem for ideal sampling [9,25], the chosen sampling frequency s κ should be greater than the bandwidth max 2κ of the concerned source, otherwise aliases corresponding to adjacent m values overlap each other.In our case it should occur that: where, the bandwidth limit max κ may be due to either the system response ( ) ( ) S H κ κ or the observed source ( ) κ i .The above behaviour of the sampled interferogram has been thoroughly investigated in the past (e.g. in [24]), and Eqs from (5) to (10) are useful for introducing a clear nomenclature, and declaring the meaning of any symbols.Using Eq. ( 10), the minimum wavelength min λ for which the original spectrum can be estimated without aliasing errors is min 2 OPD λ δ = .In other words, the shortest observed wavelength (greatest observed wavenumber) gives rise to a monochromatic fringe pattern that has to be sampled not less than two times per cycle along the entire interferogram.All wavelengths longer than min λ will have their fringe cycles sampled by two detector elements at least and can be reconstructed.Supposing we haven't aliasing effects ( ) by low-pass filtering Eq. ( 9) using an ideal filter {} ⋅ F .This yields: It is worth noting that the utilization of an ideal filter is connected to the circumstance that its application happens at software level, by a data-processing procedure.More specifically, the low-pass filter in the wavenumber domain simply means that only wavelengths greater than Taking into consideration Eqs. ( 6), (10), and ( 12) it can be noticed that the parameters max OPD and OPD δ are related each other by means of the asymmetry factor b , so that the interferometer overall performance mainly depends on instrument parameters like OPD δ , M , and b .It is interesting to evaluate the interferometer performance while varying these parameters.Equations ( 12) can be rewritten as: This equation states that as long as the number of interferogram samples M is fixed a better spectral resolution (a lower λ δ ) can be obtained only after increasing the minimum wavelength min λ that can be reconstructed.In (quite asymmetric) shows the best interferometer performance but is unrealistic because symmetrisation requires a rough estimate of the interferogram centre, which can be assessed only relying on a sampling which overlaps both sides of the interferogram.Some samples have to be collected also on the "other" side of the interferogram making the parameter b less than unit.To this purpose it can be sufficient gathering two or three samples around the interferogram centre, so that b can approach the unit without reaching it.In this sense, the blue curve in the plot is anyway a realistic representation of the best interferometer performance.has contributions from the raypaths inside and outside the beam splitter, which is made up of some glass or crystal having a spectrally variable refractive index.In our specific configuration, the path outside the beam splitter is in vacuum or air (for the airborne and laboratory prototypes), the refractive index of which is assumed to be unitary at all wavelengths.In view of Eq. ( 4) and considering these characteristics we write the general equations below: where Air  8) would be degraded by some unknown error that must be carefully evaluated.

Spectral dependence of the optical path difference
Let us investigate the properties of the and how it departs from the ideal unitary operator: In this equation the x integral, representing the IT operator, can be rewritten as: and in view of Eqs. ( 14) one can write: .
In the above equations Air ϖ is a constant value that corresponds to the OPD contribution due the raypath outside the BS (in air), while .
In Eqs.(18) ( ) κ n′ represents the derivative of the refraction index with respect to the wavenumber, and the solution of the ξ integral demonstrates that the input -output relationship of a Sagnac interferometer is affected by a wavelength dependent distortion given by: The convolution with the spectrum.The argument of this sinc always is less than ½ for the entire wavelength range that can be reconstructed without aliasing.Therefore, this distortion is a smooth modulation superimposed on the source spectrum, and can be removed by calibrating the instrument response through a laboratory white reference standard.We point out that the factor equals the phase-to-group OPD ratio, as shown in the next equations: Here, g OPD is the group optical path difference as well as ( ) ( ) ( )κ κ κ κ is the group refraction index of the BS material [27].Let us note that the availability of an analytical formulation for correcting the effects of a spectrally dispersed OPD may be relevant to many interferometric applications [27].

Phase error due to sampling offset
Perfect interferogram sampling is an infrequent circumstance, and in real applications no sample is exactly collected on the interferogram center.We term this phenomenon as "sampling offset", and introduce the new symbol x ∆ that indicates an unknown x translation of the sampling grid.Obviously, the sampling offset has to be less than the sampling step x p ∆ ≤ , otherwise its integral part has to be included in the asymmetry parameter 0 x for a coherent analysis of sensor performance.In the past, a great effort has been devoted to mitigate or remove the phase error introduced by uncontrolled sampling offset [24,[28][29][30].As a consequence of offset, no sample is exactly matching the interferogram center, introducing a phase error ( )  in the source spectrum estimation computed as inverse complex interferometer transform The phase term disturbance can be removed taking the norm of the inverse complex transform , an operation that restores the result in the right hand-side of Eq. ( 20).This type of correction was devised by Connes [31], and it was investigated in depth by Walmsley et al [24].In the past, the main shortcoming of this correction scheme was its requirement for a significant computation burden that prevented its application.Some authors [32] have hypothesized that noise affecting the interferogram sampling may interfere with phase correcting procedures, however the effect of noise on the previous phase correction scheme has never been clarified.As a matter of fact, the interferogram measurement being processed is affected by electronic and quantization noise that without loss of generality may be assumed to be a stationary zero-mean Additive White Noise (AWN).The first problem is that the noise degrades the measured interferogram where we have introduced the ensemble average operator { } E .Equation (23) shows that the average of the inverse interferometer Fourier Transform is biased by noise, as reported in the second of Eqs.(23).This bias is due to the combination of the square modulus with the ensemble average operator that acts on the random component like an estimator of autocorrelation at zero-lag.As known, this zero-lag autocorrelation obtains the square modulus of the mean plus the variance.Therefore, the method of correcting the phase error by taking the modulus of the complex Fourier transform of the interferogram is well suited when the noise amplitude is negligible (low noise sensors), or whereas the noise standard deviation of the detector is calibrated with an independent measurement and removed from the final result.Let us note that only the noise components that are independent of the source (e.g.quantization error and electronic noise) can be reliably calibrated and removed.Moreover, photonic noise should directly affect the ( ) κ i symbol in Eq. ( 23), so the spectral radiance in these relationships should be considered as an additional stochastic field having mean ( ) κ i , and the standard deviation characteristic of photonic noise.Also for this more accurate modeling of ( ) κ i the combination of the square modulus with the ensemble average operator behaves like an estimator of autocorrelation at zero-lag.Hence, it is easy showing that this improved modeling of photonic and electronic noise would lead to a final result quite similar to that shown in the second of this Eq.(23).Indicating the standard deviation of photonic noise with the symbol where we have assumed that photonic noise contributions of adjacent wavenumbers are uncorrelated.Now, we are interested in evaluating the statistical difference between using the corrected estimator . 2 Similarly to the behavior of the inverse cosine transform, the standard estimator Re   3) and constitutes a basic property obeyed by any interferometers.This behavior is the main drawback of Fourier Transform Spectrometers.

Effects of the direct term
As shown in Eq. ( 3) the interferogram contains a constant contribution independent of x and proportional to the half power of the observed source.This contribution does not bring information about the source spectrum, and should be considered as a disturbance that may degrade the interferogram and the associated spectral estimations.The above reasoning is summarized in Fig. s (3) and (4).Figure (3) indicates the informative part of the interferometric signal (the small amplitude ripple of the orange curve) and the useless part of the interferogram: the dc term plotted as a green straight line.In Fig. 4 we show the inverse Fourier Transform (spectra) of the interferogram components already shown in Fig. 3.  Often, this direct term is removed before the inverse transformation that leads to the source spectrum estimation, although the presence in the sampled interferogram of a residual direct term does not affect the spectrum estimation significantly in many spectroscopic applications.In order to shows this property let us consider the presence of an additional constant term 0 I in the interferogram.Due to this constant Eq. ( 23) changes as stated by the following equation: Let us note that frequently max λ is hundreds (even thousands) times min λ for many sampled imaging interferometers where M is between 500 and 5000.The limits of the operating spectral interval due to the constant term 0 I are so wide to not affect at all the spectrum of  23) or (24).The sole exception is constituted by those cases in which the overall sensor response ( ) ( ) ( ) is not properly compensated, apodization is applied before inverse transformation, or uncontrolled vignetting introduces a broader bell-shaped profile whose convolution with the pulse in Eq. ( 26) gives rise to a disturbance that might interfere with spectral estimations (significantly lowering the max λ limit).

Fellgett's advantage of Fourier Transform Spectrometers
Fellgett's advantage should be connected to the circumstance that each interferogram sample (pixel) benefits of radiative contributions from any wavelengths of the observed source [34][35][36][37][38].In [38] Fellgett writes: "I recognize that a major inefficiency in spectrometry in the infrared region (...) is that the available observing time has to be shared among all the observed spectral elements.This inefficiency can be overcome by multiplexing all spectral elements through a single detector; that is to say, imposing mutually orthogonal modulations on the separate elements, and sorting out their individual contributions in the final output".In such a way the overall physical signal level outputted by an interferometer should be much higher than that achieved by a dispersive instrument (i.e., grating).Fellgett idea was that the multiplexing permitted the observation of all the spectral elements at once even using a single detector, as in [36], where he writes: "Multiplexing is specifically associated with orthogonal sets of functions, but it is not at all necessary that these should be the trigonometric functions as in Fourier spectrometry."This effect has received various interpretations in the past.As an instance the multiplexing advantage has been associated in [20] to the lack of output slit in interferometers, a rather different representation that may be also related to the property of Fourier Transform multiplexing of having poor and unimportant spectral dispersion in its output focal plane.Surely, the advent of 1-dim and 2-dim detectors (e.g.CCD or CMOS devices) permitted a similar advantage even for traditional dispersive spectrometers making Fellgett's advantage outdated.Other authors has pointed out that summation of independent interferogram samples implicit in performing the inverse cosine transform should reduce the effect of incoherent noise on the spectral estimations (e.g [12].).Often, Fellgett's advantage has been stated in term of an increased SNR available in interferometers, a phenomenon that should be more evident when the main noise source is uncorrelated and due to the detector [35].
Evidently, the multiplexing effect has been misinterpreted in the past frequently, also assuming that a higher spectral resolving power can be achieved by a Fourier Transform Spectrometer avoiding the effect typical of dispersive spectrometers where a higher spectral resolution implies the measurement of a fainter signal.It is easy to demonstrate that the above apparently accurate reasoning is very often untrue for a generic FTS that obeys Eq. ( 3).Hence, it is interesting stating on more firm basis (quantitatively) possible radiometric advantages of FTS, stemming from a physical and mathematical investigation of the signal produced by FTS instruments.
In the following we assume that Fellgett's advantage is associated to the absence of exit slit, while possible radiometric advantages connected with the typical FT multiplexing are termed multiplexing advantage.This definition of possible radiometric advantages of FTS help us to include in the following analysis new features of modern detectors that weren't available sixty years ago (e.g.array detectors).Let us note that in many works on interferometry the phrases "Fellgett advantage" and "multiplex advantage" are used synonymously.However, our purpose is not getting a better representation the original thinking of Fellgett or other authors.We arbitrarily assume the above definitions for sake of simplicity, aiming at obtaining a clear discussion of their implications.
In the above equation ( ) x I is as usual the interferogram, ( ) κ i the source spectrum, ( ) the additive zero-mean noise (e.g.AWN), and the Fourier Transform has been employed in lieu of the basic cosine transform.In this case we neglect the possible dependence of the OPD on the wavenumber κ , this characteristic being unnecessary for the following analysis.An unreal folk-lore piece frequently encountered when discussing the multiplexing advantage, is that white noise should combine incoherently in the inverse Fourier Transform which leads to spectral estimations.Apparently, some authors retain that spectral estimations should benefit from a reduced noise amplitude thanks to the

( ) { }
x I 1 − FT operator.As an example, Katzberg and Statham [12] write: "Fourier Transform Spectrometers, on the other hand, observe a linear sum of elements of all portions of the input spectrum.When reconstructed, the presumably uncorrelated noise samples combine incoherently, while the signal adds coherently".This claim is erroneous because it applies a property of the arithmetic addition of uncorrelated random samples to the FT of the same random sequence.As shown in Eq. ( 22), and in many textbooks on random fields and stochastic processes (e.g [33].) the FT of a random sequence obeys a different law, and the signal-to-noise ratio (SNR) is an invariant quantity with respect to Fourier transformation.For the interferogram of Eq. ( 23), the average variance We note that Eq. ( 29) represents a common form of noise amplitude estimator, adopted in many works.Interestingly, this equation holds true in strict sense if the involved noise term is at least a variance-ergodic field (process).The variance ergodicity simply requires an autouncorrelated normal field.Evidently, any form of white noise or of auto-uncorrelated normal noise is variance ergodic.The SNR in the interferogram domain is given by: .
With a similar reasoning, we obtain in the conjugated domain (spectrum) the following expression: , therefore the application of the inverse Fourier transform operator doesn't add radiometric benefit to the result.This outcome is a general characteristic of the FT operator, and is based on the FT property of being a complete representation of the original function.In a different wording, the FT operator exactly preserves the information content held in the processed data, and the original separation in its random and source (ideal) contributions.Let us note that Eqs. ( 31) and (32) show the average SNR of the interferometer physical signal in both domains, but that they do not represent the effective SNR of interferometric measurements.As shown in Section 3.3 the signal ( ) x I (and its

FT
) has a large amplitude due to the presence of a non-informative component, which beautifies the total SNR.

The constant term in the interferogram
The measured interferogram takes its high amplitude from a useless contribution represented by the first term in the right hand-side of Eq. ( 28).This term is not useful because it does not bring information concerning the source, with exception of its panchromatic energy.The truly source-informative signal in the interferogram is a tiny undulation

FT
. This behavior is evident when we consider the inverse FT of the constant term: it originates a pulse located 0 = κ that doesn't add information about the source spectrum, a circumstance that has been shown in Eq. (26).
It is worth noting that Plancherel theorem implies that the integrated power carried by the informative component of the interferogram equates a quarter of the integrated power carried by the source spectrum, ideally measured by a dispersive instrument.

( ) ( )
In comparison with traditional dispersive techniques, FTS can maintain some étendue advantage as long as no input slit is adopted (Jaquinot's advantage), or when the interferometer exhibits less optical losses of a grating spectrometer, but multiplexing doesn't involve any radiometric advantage for the effective signal.

The informative part of the interferogram
In this Section we will show that multiplexing by trigonometric orthonormal functions originates a serious radiometric disadvantage connected to the circumstance that the power carried by the effective signal in the interferogram is concentrated at lower OPD s.In other words, spectral information concerning the source at high resolution (the finest spectrum details) is held in the subtle tails of the interferogram [39].In order to elucidate this point with an example let us consider a radiation source ( ) It is evident that using a dispersive spectrometer at the generic spectral resolution κ δ we need a radiometric resolution

≅
. When considering a Fourier Transform Spectrometer (having the same scanning mechanism) the same spectral resolution κ δ can be achieved if the maximum OPD observed by the device is not less than . And this requires the observation of a tiny interferogram oscillation whose amplitude is promptly deduced by the sinc term in Eq. ( 34).

(
) ( ) Since the value is related to the requested spectral resolution κ δ , we can find a relationship between spectral resolution and minimal radiometric sensitivity FTS i δ for a FTS device when observing a source with rectangular spectrum: As can be seen, sensitivities DISP i δ and FTS i δ requested to dispersive and FT instruments yield the same numerical value apart from the π 1 factor, the ½ factor being due to the amplitude splitting operated by the beam-splitter plate and characteristic of any interferometers.
It is worth noting that the assumption of a rectangular source spectrum is rather unrealistic, and should be replaced by a generic function continuous and Lebesgue integrable with its first l derivatives.The requirement that the l th ( ) derivative is Lebesgue integrable and admits a finite and invertible Fourier Transform means that the interferogram ripple (i.e.: the Fourier transform of ( ) ) dies away for large enough OPD s at least as rapidly as . In such a case the radiometric sensitivity requested to a FTS device for reaching the spectral resolution κ δ is: Equation ( 37) is straightforwardly deduced applying the Reimann-Lebesgue lemma to the Fourier Transform of the l th ( ) κ i derivative, as shown in [23].Let us note that we have assumed the interferogram to be a function of the product ( ) x B OPD , a choice that conveniently accounts for the uncertainty principle.The FTS A term is an unknown factor of the asymptotic limit of the transform of the source spectrum . The above asymptotic behavior clearly shows that the previous estimation of FTS i δ held in Eq. ( 36) is the most favorable condition for an interferometric measurement, while for a realistic case we obtain from Eq. (37) The hypothesis of Lebesgue integrable spectra and interferograms is a sufficient condition for the existence of direct and inverse transforms but it is not necessary.The rect and sinc functions of Eq. ( 34) are examples of functions non-integrable in the sense of Lebesgue that, however, admit convergent Fourier integrals (or a generalized Fourier transform).Typical examples of source spectra that are Lebesgue integrable are the Gaussian and the Lorentz profiles.In summing up, Eqs.(37), and (38) mean that the radiometric accuracy requested to an interferometer for achieving the spectral resolution κ δ is usually higher than the one required to a dispersive spectrometer with the same performance.The above results can be condensed in the following theorem.Lemma 1.Given a FTS with a radiometric accuracy FTS i δ matched in the sense of Eq.
(38) to the desired spectral resolution κ δ , and assigned a dispersive spectrometer having a radiometric accuracy DISP i δ sufficient for obtaining the same spectral resolution κ δ , then a (high) spectral resolution limit 0 δ exists such that: In practice the FTS always will be disadvantageous with respect to a dispersive spectrometer for spectral resolutions fine enough.The demonstration of the above Lemma is a trivial consequence of the previous analysis (Eq. ( 38)).The threshold 0 δ can't be foreseen on theoretical basis since its value depends on the spectrum of the actual source, i.e. its bandwidth.

Interferogram quantization and noise effects
In this Section we analyses the possible benefits of using interferometric and dispersive techniques.Since we have shown that the average SNR is the same in the interferogram domain and in its spectral estimation, performance of FTS and dispersive instruments can be computed in the domain in which the measurement is executed (interferogram for the FTS and spectral domain for dispersive devices).The ratio is an important estimate of the radiometric accuracy required to the interferometer, in order to reach the spectral resolution κ δ .From Eq. ( 38) the following relationship is easily obtained: where we have introduced the minimal signal-to-noise ratio min SNR necessary for measuring the tiny amplitude on the top of the constant value , and the maximum number max K of in-band independent spectral channels that can be interpolated from such an interferometric measurement when observing a source  40) makes even more evident the possible disadvantage of using an interferometer for high spectral resolution measurements when 1 > l .Stemming from Eq. ( 40) it is possible to derive a simple relationship that states the minimal quantization accuracy (number of bits min Q ) requested to obtain the spectral resolution κ δ from interferometric observations: In the previous equation the maximum digitized signal value has been set to since the interferogram maximum is about twice its average; i.e. the dc term level.Moreover, the assumed quantum amplitude equals the effective signal amplitude In comparison, a dispersive spectrometer is not subject to severe restrictions in term of its quantization accuracy, provided that it is sufficient for measuring the source spectrum with the desired resolution  It is well known that noise affecting radiometric measurements comes from three main sources: quantization noise (the round-off error), photonic noise, and instrumental noise.Noise caused by quantization and electronic detector and circuitry is not strictly connected to the optical configuration of the spectrometer under examination, and we can suppose that they affect in the same way the SNR of dispersive and FTS measurements.Therefore, we will focus our attention mainly on the photonic noise when investigating the advantages or disadvantages of FTSs.Photonic noise instead originates effects that are specific of the interferometer optical configuration and further degrade the performance of FTS.The point here examined regards the informative component of the interferometer signal, the only component that contains source spectral information.While the constant term doesn't add contribution to the estimation of the source spectrum, the photonic noise originated by it strongly affects any spectral estimations.And this photonic noise contribution can't be separated from the informative component of the signal.Moreover, since the non-informative term holds most of the power of the measured signal, this photonic noise contribution might be overwhelming at high spectral resolution (large OPD s) with respect to the tiny informative component of the signal.
The photon flux ( ) experimented by an interferometer is related to the interferogram intensity that, according to Fig. , which summed over all the narrow spectral intervals gives rise to the photonic noise affecting the interferometer radiometric signal: In the above relationship we have performed a r.m.s.estimation of the overall photonic noise affecting the interferogram measurements.With simple mathematical steps this equation can be reformulated as follows: As can be seen the maximum effective signal-to-noise ratio ( )

FTS SNR
eff max for a FTS rapidly decreases with increasing the interferometer spectral resolution.The noise performance of an interferometer is worse than that obtained with the traditional dispersive spectrometer, which is reported for comparison in the next equation: Thus, the comparison of dispersive and interferometric techniques shows an important difference when the photonic noise performance is examined.In the limit for high spectral resolution (i.e. 1 << κ δ and large 1 OPD max >> ) the advantage of dispersive techniques is an exponential factor of 2 1 + l (figure of merit), when the maximum signal-to-noise ratio provided by the two techniques is compared.
Due to the above behavior, measurements performed in the visible spectral range where the photonic noise has a large impact can't be executed at high spectral resolution with a FTS.Nevertheless, in the infrared spectral range the photonic noise is mitigated by the lower photon energy, and high spectral resolution interferometric measurements are a viable alternative to dispersive instruments.

Lost samples during data collection or transmission
With the phrase lost samples we indicate measurements that, due to a transmission or acquisition error, assume a trivial (not informative) value, e.g.null.When such a circumstance occurs in the measurement performed with a dispersive spectrometer the concerned spectral channel has definitively gone.In general this lacking channel can't be in any way recovered by interpolation or other predictive modeling, unless a large error is tolerated.The important point is that the resulting error (information lack) is concentrated in a single spectral channel that becomes useless.
The corresponding situation with FTS devices is quite different and less disruptive.After the inverse transform procedure that leads to the spectral estimation, the effect of a missing interferogram sample (error) is spread as a small amplitude perturbation over all the interpolated spectral channels.Moreover, the corresponding spectral error is quite low in any spectral channels, and no channel is missing.This is an advantage typical of FTSs and all those spectrometric devices which impose mutually orthogonal modulations to the spectral radiance impinging on separate detector elements (multiplexing).

Discussion
The following list recaps the aspects relevant to the power efficiency of an generic FTS as compared to the traditional dispersive techniques.
1.The multiplexing measurement approach doesn't produce advantage in reducing the experimental noise, nor the final SNR is optimized by the inverse FT operation.
2. The interferometric physical signal has a power high above that of a grating instrument observing the same source in the same experimental conditions.Most of this power is carried by a non-informative component of the interferometric signal (a constant term); 3. Using the Plancherel theorem it can be shown that the power held in the informative part of the interferometer signal (the raw interferometer signal minus the constant factor) is on average the same as the power of the spectrally dispersed signal available for a dispersive spectrometer; spectrometers when high spectral resolution measurements are performed (large OPD s).
5. The interferometer has to measure a much more tiny signal as the dispersive spectrometer does, but this signal is found to be superimposed on a high-amplitude continuous radiation plateau.From a practical point of view, this feature represents a shortcoming since most of the digitalization accuracy of the employed detector is used to sense this useless (non informative) high signal.
6.This high-amplitude constant signal level originates a photonic noise of large standard deviation, so that the effective average SNR available in interferometric measurements is usually significantly lower than that obtained in dispersive observations with high spectral resolution.Decreasing is caused by both: the reduced level of effective signal amplitude at large OPD s (see lemma 1) and the noise increase due to the augmented photonic noise (large amplitude of the non-informative interferogram component).The above outcomes demonstrate that multiplexing is a radiometric disadvantage real for Fourier Transform Spectrometers.This characteristic was not fully recognized in many works about interferometry due to a misinterpretation of the noise role and the effect of the constant term.Fellgett's advantage is maintained in FTS as long as it is defined as the signal amplitude increase connected to the absence of exit slit.Today, this advantage no longer is a prerogative of FT spectrometers due to the availability of low-cost array detector.Similar conclusions has been drawn by Schumann and Lomheim [40].The following Table 1 resumes the main differences among the traditional dispersive spectrometers and the most recent triangular interferometers that have been examined in this paper.• Sources having a spectrum with almost rectangular profile can be observed by a FTS that would closely match the optimal radiometric-resolution / spectral resolution compromise achieved by dispersive instruments; • Sources observed in the Infrared spectral range may show a spectrum closely approximated by a negative exponential function of the wavenumber.It can be shown that the Fourier Transform of such kind of spectra approach a Lorentz function of the OPD .In such a case the radiometric accuracy required to interpolate the source spectrum with spectral resolution κ δ follows a rather favorable law that can be reliably measured with a FTS; • In the TIR spectral range the available signal contains radiative contributions from the optical elements along the ray path requiring cooling of the whole instrument.
Usually, an FT spectrometer has lesser optical losses than a dispersive instrument, and sometime a minor number of optical surfaces along the raypath.Hence, the source-informative signal may be higher in FT spectrometers and be degraded at a lesser extent with respect to a dispersive instrument.This difference may mitigate the needing for cooling in spaceborne devices, originating a possible advantage of FTS in the TIR.
• In the Visible spectral range, where due to the photonic noise the FTS performance is weakened, Fourier Transform Spectrometers can be advantageously employed for conducting low and medium spectral resolution observations of a broad-band source.
In fact our theoretical modeling of FTSs performance of Eq. ( 37) provides us with a prediction of the required radiometric accuracy only as an asymptotic approximation valid for large OPD s.Hence, in spectral applications where the maximum observed OPD is not huge, the FTS technique can be a viable tool.
• When observing a narrow-band source (e.g. a laser source with ), it is possible to perform high spectral resolution measurements while maintaining a mild radiometric resolution of the FTS even for applications covering the visible spectral range.Possible applications of multiplexing advantage potentially remain confined to instruments adopting a non trigonometric set of orthonormal functions for representing the observed spectrum.Our theoretical investigations only exclude that this advantage holds true for Fourier Transform Spectrometers.

Conclusions
In this paper a deep theoretical analysis has been performed about the principles of Fourier Spectrometry pertaining the Sagnac triangular configuration.A special effort has been devoted to study the cosine-like instrument transformation in order to take into account the circumstance that the Optical Path Difference function depends also on the wavenumber.This investigation led us to introduce the concept of group OPD , which expressly contains the spectral dispersion law of the material composing the beam-splitter (i.e., the essential element which originates by amplitude splitting the two interfering beams).It has been shown that the phase to group OPD ratio is the amplitude of the non-unitary transformation operated by the instrument.This theoretical result may be useful for correcting the spectral estimations obtained with a FTS or for other interferometric applications.
An analytical expression of the sampled interferogram has been found, which includes the effects of on-pixel integration, spatial sampling offset, truncation of the interferometric measurement, and additional instrumental effects such as fringe visibility.Attention has been devoted to the effects of a null-mean additive white noise affecting the interferometric measurement, and to signal degradation produced by the photonic noise.If the complex inverse (Fourier) transform estimator is adopted in lieu of the standard cosine-like operator (for an instance, in order to automatically remove the phase distortion in spectral estimations due to the sampling offset error), the retrieved spectrum is affected by a bias proportional to the noise standard deviation, and subject to an enhanced noise amplitude (40% relative increment).We have shown that this behavior is introduced by the fact that the basic cosinelike inverse transform selects only the even part of the noise record.
Finally, a new interpretation of the so called Fellgett's advantage has been discussed based on an analytical comparison of system performance (estimation of amplitude of the effective signal and Signal-to-Noise ratio) between the interferometric technique and the traditional dispersive spectrometers.This discussion has led to the conclusion that Fellgett's advantage has been misunderstood frequently in the past.We have proved that the informative tail of the interferogram to be resolved requires a radiometric resolution much finer (depending on the illumination source's bandwidth) than that is needed for a dispersive spectrometer operating at the same high spectral resolution.This circumstance also reveals the advantage of dispersive techniques of an exponential factor of 2 1 + l pertaining the maximum Signal-to-Noise ratio with respect to the interferometric technique.In the Visible spectral range, where due to the photonic noise the FTS performance is weakened, Fourier Transform Spectrometers can be advantageously employed for conducting low and medium spectral resolution observations of a broad-band source.On the other hand, when observing a Visible narrow-band or an Infrared broad-band source, it is possible to perform high spectral resolution measurements while maintaining a mild radiometric resolution of the interferometer.

#
124682 -$15.00USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11623 amplitude attenuation factors affecting the electric field amplitude ( ) κ E of the reflected and transmitted rays when passing from the input to the output port of the interferometer.The coefficient A relates the square modulus of the field amplitude of the radiation field), and OPD is the optical path difference introduced by the interferometer in the considered propagation direction.In Eq. (1), the wavenumber κ corresponds to the wavelength κ λ 1

V
and the overall response function of the interferometer ( ) H κ , which is modulated by the detector sensitivity, the transparency of optical components, and so on, that are collectively represented here with the notation ( ) κ S .For an ideal interferometer in the Sagnac configuration ( ) at any wavenumbers.If the first surface reflectance of the BS plate deviates significantly from the ideal value ( ) 2 1 = κ ρ the fringe visibility is substantially dimmed.

( 5 )
#124682 -$15.00USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSAwhere the symbol * indicates the standard convolution product, p is the distance between two adjacent pixels, d is the pixel size, M is the number of pixels of the detector, between adjacent pixels.With this naming convention, the maximum OPD in the sampled interferogram ( max OPD

3 . 1 .
In order to adopt a coherent notation we indicate here as follows the integral transform performed by the device with the symbol {} ⋅ IT , and reserve the {} ⋅ CT symbol for the standard cosine transform.The following equation shows the basic relationship linking the {} ⋅ −1

Fig. 2
this dependency is plotted for an interferogram formed of 1024 samples ( M ) for two values of b parameter (0.5 and 1

Fig. 2 .From
Fig. 2. Theoretical representation of the link between spectral resolution and minimum reconstructed wavelength computed at 500 nm (see Eq. (13)).The red curve represents the "standard" interferometer configuration sampled in a symmetric way with respect to the fringe pattern centre 0.5 b = , while blue curve is characteristic for an optimal configuration where only one side of the interferogram is sampled 1 b = .Let us note that this last case ( 1 b = ) is unrealistic because symmetrisation requires a rough estimate of the interferogram centre, that can be achieved gathering two or three samples around the interferogram centre.This makes b only slightly less than unit, thus the blue curve still is a realistic representation of the best interferometer performance.From a theoretical standpoint the

ϖ
are proportional to the geometric path differences obtained inside and outside the BS, and should be considered as constants, which are defined by the optical configuration of the Sagnac triangular raypath.Similar equations have been reported by Hilliard and Shepherd[26].The OPD depends not only on the position x (i.e. the input angle ϑ ), it is a function of the wavenumber κ too, hence κ and ( )OPD ,x κ cannot be considered a couple of conjugate variables.Due to this property, the integral transform performed by a Sagnac interferometer is not a true cosine transform and it might not be invertible, meaning that the composite operator unitary.In such a #124682 -$15.00USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSA case the source spectrum estimate in Eq. ( is proportional to the OPD originated inside the BS.The vanishing width of Dirac's pulse allows us to consider only ξ values approaching κ , and the second of Eqs.(17) can be fruitfully expressed using a power series of ( ) ξ n blocked to the first order: distortion in the retrieved source

σ
is the standard deviation of the interferogram noise, while ζ σ is the standard deviation of noise affecting the spectral estimation.Relying on this property we can assume that the effects of experimental noise is modeled simply adding a random term ( ) x z (zeromean stationary AWN) to the sampled interferogram, and its instrument Fourier Transform ( ) κ ζ on the right hand side of Eq. (20), as shown below: : #124682 -$15.00USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11634 which are random variables of the same type with the same standard deviation 2 z σ .The following relationships recap this circumstance:

Fig. 3 .
Fig. 3. Schematic of signals composing the measured interferogram (black curve).The interferogram is made up of two components, a dc term (green line) that holds the source halfenergy and the Fourier (cosine) Transform of the source spectrum (orange curve) that contains the desired spectral information.While the measured signal (interferogram) always is high and does not decreases with increasing the spectral resolution, the useful signal (orange curve) is tiny for high OPDs.The interferogram model here described arises from Eq. (3) and constitutes a basic property obeyed by any interferometers.This behavior is the main drawback of Fourier Transform Spectrometers.

Fig. 4 .
Fig.4.Schematic of signals composing the spectral estimate obtained stemming from a generic interferogram measurement.The interferogram is made up of two components, a constant dc term that inverse transforms in a pulse at zero wavenumber (green line), and the true source signal that inverse transforms in the desired spectrum (orange curve).The meaning of these two interferogram component in the conjugate OPD domain is discussed in the text and illustrated in Fig.3.This picture undoubtedly shows that the dc term, which is at the core of the multuplexing advantage, does not bring source information apart its energy.
) updating of Eq.(24) follows straightforwardly.The perturbation right hand-side of Eq. (26) has its first null.The condition for which the perturbation is negligible the operating spectral range of the interferometer, as stated by the following relationships: the average square interferogram is:

#
124682 -$15.00USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSA having bandwidth B .Equation ( .(40) and (41) implicitly assume that the standard deviation z σ of the overall noise affecting the interferogram measurement is less than the small amplitude ripple( ) { } 2 κ i FTto be observed.In a different wording, the maximum digitized OPD (the max OPD ) has to be less than the value n OPD where the envelope of the informative signal Interferogram samples collected at OPD s greater than n OPD are meaningless and don't add useful information but mainly noise to the spectral estimation.This elementary concept is illustrated in Fig.5where the blue curve represents the envelope of the informative component of the interferogram dotted line indicates the noise amplitude z σ .Remembering that the effective average SNR is the same in both domains for FTS, adding interferogram samples in which the signal amplitude is lower than that of noise results in abating the average SNR of the subsequent spectral estimations.#124682 -$15.00USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSA

Fig. 5 .
Fig. 5. Schematic of signals related to the Fourier Transform of the source spectrum (orange curve).The point where the envelope of the source Fourier Transform (blue curve) reaches the level of noise (black curve) defines the noise OPD limit OPD n .Interferogram samples gathered at OPDs greater than OPD n does not bring source information.
3 and Eq.(3), is mainly determined by the constant term flux is known to obey Poisson's statistics law, giving rise to a flux variability whose standard deviation is just the square root of the flux itself.The standard deviation flux impinging in a unitary time interval over a sensor having unitary equivalent surface area within the narrow spectral interval κ d around the generic wavenumber κ can be written as: $15.00 USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSA In the above equation c is the speed of light, and h Planck's constant.The above variability of the monochromatic photon flux produces the monochromatic radiometric variability 45)being av κ the spectral photo-center of the concerned source.Equation (45) allows us to write the maximum effective interferometer signal-to-noise ratio eff SNR max allowed by the photonic noise only: that the signal amplitude in the above equation has been taken equal to the norm of the effective signal.The obtained result yields an upper bound to the effective SNR achieved by an FTS, since it only contains contributions from photonic noise.Actually the effective SNR of an interferometer has to be less than the value OPD s (e.g., at high spectral resolution), resulting in a small ( )max eff SNRFTS .The situation is critical at large OPD s whose measurement is necessary to obtain high spectral resolution.In view of Eq. (38) we can write the asymptotic behaviour of the maximum signal- of a single spectral channel No spectral channel is lost -the lost information gives rise to a small error in any spectral channels Furthermore, we point out that: #124682 -$15.00USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11647 we have introduced the sampling frequency s Here as follows we will make use of continuous representations of signals involved in the interferogram and spectral estimations, avoiding the mathematical complexity of handling with sampled signals.Any results are easily extended to the case of a sampled signals.Let us write the single-sided interferogram signal model as: the following comparison: #124682 -$15.00USD Received 24 Feb 2010; revised 17 Mar 2010; accepted 19 Mar 2010; published 18 May 2010 (C) 2010 OSA 24 May 2010 / Vol. 18, No. 11 / OPTICS EXPRESS 11641