theoretical aspects of fourier transform spectrometry and ... aspects of fourier.pdf · (120.0280)...

Theoretical aspects of Fourier Transform

Spectrometry and common path triangular

interferometers

Alessandro Barducci*, Donatella Guzzi,

Cinzia Lastri, Paolo Marcoionni, Vanni Nardino, and Ivan Pippi

Consiglio Nazionale delle Ricerche – Istituto di Fisica Applicata “Nello Carrara”, Area di Ricerca di Firenze del

CNR, Via Madonna del Piano, 10, 50019 – Sesto Fiorentino, Italy

*[email protected]

Abstract: 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

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1. 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 [3–6].


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 [11–14].

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) [2–6,15,16]. 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 [17–20]. 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.

2. 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.

3. 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 )(xI , 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.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )2

0

exp exp 2 OPD , .r tI x A S t E j t E j j x dκ κ κ ϕ κ κ ϕ π κ κ κ+∞

= + +∫ (1)

In this equation j is the imaginary unit, φ the initial phase of the impinging radiation field,

( )κrt and ( )κtt are the overall amplitude attenuation factors affecting the electric field

amplitude ( )κE 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 ( )κE to the corresponding radiance ( )2

( )i A Eκ κ= (specific intensity 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 κλ 1= , 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 ( )κrt and ( )κtt can be expressed using the power reflectance ( )κρ and power

internal transmittance ( )κτ i of the BS:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

exp

1 1 1 .

r i i i

t i i i

t j

t

κ ρ κ τ κ ρ κ τ κ π ρ κ τ κ

κ ρ κ τ κ ρ κ τ κ ρ κ τ κ

= = −

= − − = − (2)

It is worth noting that the power internal transmittance ( )κτ 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


descent step (BS internal reflection). The composition of the additional reflections gives rise 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 ( )iτ 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]:

( ) ( ) ( ) ( ) ( ){ }

( ) ( ) ( ) ( )

( )( ) ( )( )

( ) ( )

0

22 2

22

( ) 1 cos 2 OPD ,

1

2 1.

1

i

I x S H i V x d

H

V

κ κ κ κ πκ κ π κ

κ τ κ ρ κ ρ κ

ρ κ ρ κκ

ρ κ ρ κ

+∞

= + +

= + −

−=

+ −

∫

(3)

In this equation we have introduced the fringe visibility ( ) 1 V κ κ≤ ∀ , 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 ( )κS . For an ideal interferometer in the Sagnac configuration ( ) 1 2H κ = and

( ) 1=κV at any wavenumbers. If the first surface reflectance of the BS plate deviates

significantly from the ideal value ( ) 21=κρ the fringe visibility is substantially dimmed.

This characteristic is important since amplitude division is driven by ( )κρ 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:

( ) ( )OPD , ,x xκ ϖ κ= (4)

Being ϑfx ≅ 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 maxOPD 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

( ) 00OPD = . 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 0OPD = 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 ( )xIS

would be expressed as::

( ) ( )( ) 01 1

OPD , rect comb rect

,

S

x xx xI x I x

d d p D D

D Mp

κ− = ∗

=

(5)


where 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,

( ) ( )combm

mx p p x mpδ

=+∞

=−∞= ⋅ −∑ is the sampling function, and 0x is the centre of the

sampling grid. Imposing a not null value for 00 ≠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 0x 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 ( )2pd is the areal fill factor, a characteristic specific of the adopted detector. We define

( )OPDd dϖ κ= as the ( )OPD ,x κ variation along the pixel area, and ( )OPD pδ ϖ κ= as the

smallest OPD variation between adjacent pixels. With this naming convention, the maximum OPD in the sampled interferogram ( maxOPD ) is given by:

( )( )

( )max 0

0

OPD / 2

2.

OPDx D Mb

x Db

D

ϖ κ δ= + =

+=

(6)

The parameter b takes into account the interferogram sampling asymmetry. For a null 0x , 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. 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 {}⋅−1IT to the {}⋅−1

CT operator:

( ){ } ( ) ( ){ }( )

1 1

S SI x I x

κϖ κϖ κ

− −=IT CT (7)

Performing the inverse transform of the signal in Eq. (5) would produce a spectral estimation

modulated by the cosine factor ( )( )0cos 2 OPD ,xπκ κ , due to the translation 0x . From a

mathematical standpoint this modulation can be related to the fact that interferogram samples

at lower OPD s enter two times the inverse interferometer transform {}⋅−1IT and their

contribution is overweighted. On the other hand, it is possible to eliminate this cosine

modulation simply applying the {}⋅−1IT operator to the symmetrised sampled interferogram

( )xISS . Symmetrization can be made duplicating samples at large OPD s on the opposite side


of the interferogram. Such operation clearly requires that at least a short sequence of samples is collected on both sides of the interferogram, in order to roughly know the position of the interferogram centre (with an accuracy better than one pixel

OPDδ ). In summary we have:

( ) ( )

( )

0 0

max 0

1 1rect comb rect

2 2

OPD2

1,

SS

S

S

OPD

x x xI x I x

d d p x D x D

Mb Dx ϖ κ

κ

κδ

= ∗ + +

= = +

=

(8)

where we have introduced the sampling frequency sκ . Supposing that the direct term, fringe

visibility ( )κV , and system response ( ) ( )S Hκ κ can be reliably removed from the sampled

data or temporary included in the source ( )κi (these options will be addressed in Sect. 3), the

inverse interferometer transform of the symmetrised sampled interferogram obeys the following law:

{ } ( ) ( )

( ) ( ){ }

1

1

2 ( ) sinc sinc

.

m

SS S S OPD

m S

MbI x i m m d

i iκ

κκ κ κ κ

κ

κ κ

=+∞−

=−∞

−

= − − ∗

=

∑IT

IT IT

ɶ

ɶ

(9)

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 sκ should be greater than the bandwidth

max2κ of the concerned source, otherwise aliases corresponding to adjacent m values

overlap each other. In our case it should occur that:

( ) ( ) ( )

max

max

12

0 ,

s

OPD

S H i

κ κδ

κ κ κ κ κ

= ≥

= ∀ ≥

(10)

where, the bandwidth limit maxκ may be due to either the system response ( ) ( )S Hκ κ or the

observed source ( )κi . 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 minλ for which the original spectrum can be estimated without aliasing errors is

min 2OPD

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 minλ will

have their fringe cycles sampled by two detector elements at least and can be reconstructed. Supposing we haven’t aliasing effects ( max2

sκ κ≥ ), it is possible to isolate the central alias

( 0=m ) by low-pass filtering Eq. (9) using an ideal filter {}⋅F . This yields:

( ){ }{ } ( ) ( ){ }1 2 sinc sinc .

SS OPD

S

MbI x i d

κκ κ

κ−

= ∗

F IT ɶ (11)


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

minλ are interpolated. Due to the convolution in Eq. (11), a monochromatic source

( )0( )i κ δ κ κ= − , i.e. a Dirac’s delta pulse, is estimated as a finite-width ( )maxsinc 2 OPDκ

function, so worsening the available spectral resolution. The best spectral resolution κδ ( λδ )

can be written as:

max

2

max

1

OPD

.OPD

κ

λ

δ

λδ

=

=

(12)

Taking into consideration Eqs. (6), (10), and (12) it can be noticed that the parameters

maxOPD and OPDδ are related each other by means of the asymmetry factor b , so that the

interferometer overall performance mainly depends on instrument parameters like OPDδ , M ,

and b . It is interesting to evaluate the interferometer performance while varying these parameters. Equations (12) can be rewritten as:

( )

( )min 2

2min

2

2 .OPD

OPD

bMbM

λλ

λ δλ

δ λ λλδ λ

δ

≥ ⇒ ≥

=

(13)

This equation states that as long as the number of interferogram samples M is fixed a better spectral resolution (a lower λδ ) can be obtained only after increasing the minimum

wavelength minλ that can be reconstructed. In Fig. 2 this dependency is plotted for an

interferogram formed of 1024 samples ( M ) for two values of b parameter (0.5 and 1), for nm 500=λ . The curve obtained for 1=b (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.


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 0.5b = , while blue curve is characteristic for an optimal configuration where only one side of the interferogram is sampled 1b = . Let us note that this last case ( 1b = ) 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 ( )κ,OPD x is related to the Geometrical Path Difference

( )xGPD through the material refraction index ( )κn . In the Sagnac configuration the

( )κ,OPD x 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:

( ) ( ) ( ) ( ) ( )

( ) ( )OPD , GPD GPD

,

BS Air BS Air

BS Air

x n x x n x x

n

κ κ κ ϖ ϖ

ϖ κ κ ϖ ϖ

= + = +

= + (14)

where Airϖ and BSϖ 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 {}{ }⋅−ITIT

1 could not be unitary. In such a


case the source spectrum estimate in Eq. (8) would be degraded by some unknown error that must be carefully evaluated.

3.1 Spectral dependence of the optical path difference

Let us investigate the properties of the {}{ }1− ⋅IT IT and how it departs from the ideal unitary

operator:

( ){ }{ } ( ) ( ) ( ){ } ( )

( )

1 Re exp 2 2

OPD .

i i j x j x d dx

d dx

κ ξ π ϖ ξ ξ π ϖ κ κ ξϖ κ

ϖ κ

+∞ +∞−

−∞ −∞

= − +

=

∫ ∫IT IT (15)

In this equation the x integral, representing the { }1−IT operator, can be rewritten as:

( ) ( ) ( ){ } ( ) ( ) ( )Re exp 2 ,jx dxϖ κ π ϖ ξ ξ ϖ κ κ ϖ κ δ ϖ ξ ξ ϖ κ κ+∞

−∞

− − = −

∫ (16)

and in view of Eqs. (14) one can write:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ){ }

( ) ( ) ( ) ( ){ }

Re exp 2

.

Air BS

Air BS

n n

jx dx

n n

ϖ ξ ξ ϖ κ κ ϖ ξ κ ϖ ξ ξ κ κ

ϖ κ π ϖ ξ ξ ϖ κ κ

ϖ κ δ ϖ ξ κ ϖ ξ ξ κ κ

+∞

−∞

− = − + −

− − =

= − + −

∫ (17)

In the above equations Airϖ is a constant value that corresponds to the OPD contribution due

the raypath outside the BS (in air), while ( )κϖ nBS 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 ( )ξn blocked to the first order:

( ) ( ) ( )( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ){ }{ } ( ) ( ) ( ) ( ) ( ) ( ){ }

2

1 .Air BS

n n n

n n n n o n n

i i n n d

ξ κ κ ξ κ

ξ ξ κ κ ξ κ κ ξ κ ξ κ ξ κ κ κ κ

κ ξ ϖ κ δ ϖ ξ κ ϖ ξ κ κ κ κ ξ+∞

−

−∞

′≅ + −

′ ′− ≅ − + + − ≅ − +

′≅ − + − + ∫IT IT

(18)

In Eqs. (18) ( )κn′ 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:

( ){ }{ } ( ) ( )( ) ( )

1 .i iϖ κ

κ κϖ κ ϖ κ κ

− =′+

IT IT (19)

Combining Eqs. (8) with the above result gives us the opportunity to write a thorough mathematical link connecting the true spectrum of the observed source with the filtered inverse interferometer transform of the sampled interferogram:


( ){ }{ } ( ) ( )( ) ( )

( ) ( )1maxsinc sinc 2 OPD .

SS OPDI x i d

ϖ κκ κ κ

ϖ κ ϖ κ κ−

= ∗ ′+ F IT (20)

The convolution with the ( )maxOPD 2sinc κ originates the finite interferometer spectral

resolution, while the factor ( )OPDdκsinc originates a spectral distortion in the retrieved source

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

( ) ( ) ( )( )κκϖκϖκϖ ′+ equals the phase-to-group OPD ratio, as shown in the next equations:

( ) ( ) ( )

( )( )

( ) ( ) ( )

( )( ) ( )

( )( )

OPD ,

OPD ,

OPD ,.

OPD ,

Air BS

g Air BS

g

x x n x

dx x x N x

d

x

x

κ ϖ κ ϖ ϖ κ

ϖ κ κκ ϖ κ ϖ κ κ ϖ ϖ κ

κϖ κ κ

ϖ κ ϖ κ κ κ

= = +

′= = + = +

=′+

(21)

Here, gOPD is the group optical path difference as well as ( ) ( ) ( )κκκκ nnN ′+= 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].

3.2 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

xp∆ ≤ , otherwise its integral part has to be included in the asymmetry parameter 0x 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 [24,28–30]. As a consequence of offset, no sample is exactly matching the interferogram center, introducing a

phase error ( )exp 2 j x jπ κϖ κ π+ in the source spectrum estimation computed as inverse

complex interferometer transform ( ){ }1SSI x

−IFT . The phase term disturbance can be

removed taking the norm of the inverse complex transform ( ){ }{ }1SS

I x−F IFT , 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 ( )SS

I x but we are


interested to its effects on the spectral estimation ( ){ }{ }1SS

I x−F IFT . A properties of

stochastic processes and random fields is that the Fourier Transform of a stationary white noise field (process) ( )xz again is a stationary white noise field (process) ( )κζ with the same

spectral power density (standard deviation) and null-mean [33]. Let ( )xz indicate the

measurement noise affecting the interferogram, and ( )κζ its Fourier transform. According to

[33], we can write:

( ) ( ){ }1

,z

x

ζ

κ

σ σ

−=

=

ζ IFT z (22)

where zσ is the standard deviation of the interferogram noise, while ζσ 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 ( )xz (zero-

mean stationary AWN) to the sampled interferogram, and its instrument Fourier Transform ( )κζ on the right hand side of Eq. (20), as shown below:

( ) ( ){ }{ }

( ) ( ) ( )( ) ( )

( ) ( ) ( )

( ) ( ){ }{ }{ }( ) ( )

( ) ( )( ) ( )

1

max

21

2

2max

exp 2 sinc sinc 2 OPD

sinc sinc 2 OPD ,

SS

OPD OPD

SS

OPD z

I x z x

i j d

I x z x

i d

ϖ κκ π κ κ κ κ

ϖ κ ϖ κ κ

ϖ κκ κ κ σ

ϖ κ ϖ κ κ

−

−

+ =

= − ∆ ∗ + ′+

+ =

= ∗ + ′+

F IFT

ζ

E F IFT (23)

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 ( )κi symbol in Eq. (23), so the spectral radiance in

these relationships should be considered as an additional stochastic field having mean ( )κi ,

and the standard deviation characteristic of photonic noise. Also for this more accurate modeling of ( )κi 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 ( )photzσ , it results:


( ) ( ){ }{ }{ }( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

( ) ( )

21

2

max

2

2 2max

sinc sinc 2 OPD

sinc sinc 2 OPD .

SS

OPD

z OPD z

I x z x

i d

phot d

ϖ κκ κ κ

ϖ κ ϖ κ κ

ϖ κσ κ κ σ

ϖ κ ϖ κ κ

− + =

+ ∗ + ′+

+ ∗ + ′+

E F IFT

(24)

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 ( ){ }{ }1SS

I x−F IFT and the basic one ( ){ }{ }1SS

I x−F IT , which is the

real part of the complex Fourier transform ( ){ }{ }{ }1ReSS

I x−F IFT . The noise superimposed

on the acquired signal (the interferogram) has non-trivial even ( )κEz and odd ( )κOz

components both of which are random variables of the same type with the same standard

deviation 2zσ . The following relationships recap this circumstance:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

22 2

2

2

2

.2E O

E O

E

O

z

z z

x xx

x xx

x xx

σσ σ

+=

+ −=

− −=

= =

z zz

z zz

z zz

(25)

Similarly to the behavior of the inverse cosine transform, the standard estimator

( ){ }{ } ( ){ }{ }{ }1 1ReSS SS

I x I x− −=F IT F IFT rejects the odd component of the noise that is

orthogonal to the cosine integral kernel. In such a way the basic estimation is affected by the

phase error modulation ( )OPD2cos ∆πκ and a random noise standard deviation of 2z

σ .

Conversely, the corrected estimator embeds in its spectral estimation the whole experimental random noise zσ , is free from phase errors, but its estimate is biased by a constant additive

radiometric error whose amplitude roughly equates the noise standard deviation. In a different

wording, random noise affecting the ( ){ }{ }1SS

I x−F IFT estimates is on average 40% higher

than that occurring with the basic estimation formula. We note that the previous modeling can be used with few changes even for a colored input noise ( )κz . In fact, ( )κζ still is white noise

(non stationary) as long as ( )κz is a general stationary field [33] (e.g. non-white noise).

Considering that the bias affecting spectral estimations can partially removed with a simple

calibration measurement, the corrected estimator ( ){ }{ }1SS

I x−F IFT is always preferable

because it prevents possible phase errors. It can be affirmed that the analyzed phase correction scheme provides reliable spectral estimations independent of possible sampling offset errors at the price of enhancing the weight of random noise and a possible residual estimation bias originated by photonic noise contribution (source dependent) that can’t be calibrated and removed.


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 half-energy 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.

3.3 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.


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.

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 0I

in the interferogram. Due to this constant Eq. (23) changes as stated by the following equation:

( ) ( ){ }{ }{ }( ) ( ) ( )

( ) ( )( ) ( )

12 21 2

0 max sinc sinc 2 OPD ;

SS z

OPD

I x z x

i I d

σ

ϖ κκ δ κ κ κ

ϖ κ ϖ κ κ

− + − =

= + ∗ ′+

E F IFT

(26)

updating of Eq. (24) follows straightforwardly. The perturbation ( )κδ0I is negligible for

wavenumbers κ greater than maxOPD5.0 , where the ( )maxOPD2sinc κ function on the right

hand-side of Eq. (26) has its first null. The condition for which the perturbation is negligible

maxOPD2≤λ defines an interval ( )maxmin ;λλ that evidently limits the operating spectral

range of the interferometer, as stated by the following relationships:

max max

min

2OPD 2

2 .OPD

OPD

Mbλ δ

λ δ

= =

= (27)

Let us note that frequently maxλ is hundreds (even thousands) times minλ for many sampled

imaging interferometers where M is between 500 and 5000. The limits of the operating spectral interval due to the constant term 0I are so wide to not affect at all the spectrum of


source observed by a Sagnac interferometer and estimated by Eqs. (23) or (24). The sole exception is constituted by those cases in which the overall sensor response ( ) ( ) ( )κκκ VHS 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 maxλ limit).

3.4 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 [34–38]. 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.


3.4.1 Noise and multiplexing advantage

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:

( ){ } { }

( )0max

( ) ( ) 0 OPD

20 .

xi i

x xI x

elsewhere

κ κ=

+ + ≤ ≤=

FT FTz (28)

In the above equation ( )xI is as usual the interferogram, ( )κi the source spectrum, ( )xz 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 ( ){ }xI

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 2zσ of an

auto-uncorrelated noise ( )xz can be estimated as:

( ) 22

max

1,

OPDz x dxσ+∞

−∞

= ∫ z (29)

while the average square interferogram is:

( ) ( )2 2

max

1.

OPDI x I x dx

+∞

−∞

= ∫ (30)

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 auto-uncorrelated 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:

( )

( )

( )

2

2

2

.I x

I x dx

SNR

z x dx

+∞

−∞+∞

−∞

=∫

∫ (31)

With a similar reasoning, we obtain in the conjugated domain (spectrum) the following expression:


( ){ }

( ){ }

( )1

21

2

2

,I x

I x d

SNR

d

κ

κ κ−

+∞−

−∞+∞

−∞

=∫

∫FT

FT

ζ

(32)

where ( )κζ is the inverse Fourier Transform of ( )xz , hence ( )κζ always is white noise.

Relying on the Plancherel theorem it is easy proving that the average SNR is the same in both

domains ( ){ } ( )1

2 2I xI x

SNR SNR− =FT

, 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 ( )xI (and its ( ){ }xI

1−FT ) has

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

3.4.2 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 ( ){ } 2κiFT around the

constant term ( ){ } 20=x

i κFT . 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.

( ) ( )2 21

.4informativeI x dx i dκ κ

+∞ +∞

−∞ −∞

=∫ ∫ (33)

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.

3.4.3 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 ( )κi having a rectangular spectrum with

bandwidth B as indicated in the following equation:


( )

( )( ) ( )( ) ( )( )

00

00

rect

OPD sinc OPD exp 2 OPD .2Informative

i IB

BII x B x j x

κ κκ

π κ

− =

= −

(34)

It is evident that using a dispersive spectrometer at the generic spectral resolution κδ we

need a radiometric resolution DISP

iδ finer (better) than the power transmitted by this frequency

range: κδδ 0IDISP

i ≅ . When considering a Fourier Transform Spectrometer (having the same

scanning mechanism) the same spectral resolution κδ can be achieved if the maximum OPD

observed by the device is not less than ( ) κδ1OPDOPD maxmax ≥= x . And this requires the

observation of a tiny interferogram oscillation whose amplitude is promptly deduced by the sinc term in Eq. (34).

( ) ( )0 0max 0 max

max

1sinc OPD exp 2 OPD .

2 2 OPDFTS

i

BI BIB j

Bδ π κ

π≅ − ≤ (35)

Since the ( )maxOPD x value is related to the requested spectral resolution κδ , we can find a

relationship between spectral resolution and minimal radiometric sensitivity FTS

iδ for a FTS

device when observing a source with rectangular spectrum:

0 0 .2 2

FTS

i

BI I

B

κ κδ δδ

π π≈ = (36)

As can be seen, sensitivities DISP

iδ and FTS

iδ requested to dispersive and FT instruments yield

the same numerical value apart from the π1 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 ( )κi derivative is Lebesgue integrable and admits

a finite and invertible Fourier Transform means that the interferogram ripple (i.e.: the Fourier transform of ( ) 2κi ) dies away for large enough OPD s at least as rapidly as

( ) ( )11 OPD

l

B x+

. In such a case the radiometric sensitivity requested to a FTS device for

reaching the spectral resolution κδ is:

( ){ }( ){ }

OPD 01

max

1 0.

2 2

FTSFTS

i l

A ii l

B

κδ κ =

+= ≈ ≥

FTFT

OPD (37)

Equation (37) is straightforwardly deduced applying the Reimann-Lebesgue lemma to the Fourier Transform of the l th ( )κi derivative, as shown in [23]. Let us note that we have

assumed the interferogram to be a function of the product ( )xB OPD , a choice that

conveniently accounts for the uncertainty principle. The FTSA term is an unknown factor of the

asymptotic limit of the transform of the source spectrum ( ){ }κiFT . The above asymptotic

behavior clearly shows that the previous estimation of FTS

iδ held in Eq. (36) is the most

favorable condition for an interferometric measurement, while for a realistic case we obtain from Eq. (37) the following comparison:


( ){ }

( )

1

OPD 0 0

2

.

l

FTS FTS

i

DISP

i

Ai l

B

i

κ

κ

δδ κ

δ κ δ

+

=

≈ ≥

≈

FT (38)

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 κδ 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 FTSiδ matched in the sense of Eq.

(38) to the desired spectral resolution κδ , and assigned a dispersive spectrometer having a

radiometric accuracy DISPiδ sufficient for obtaining the same spectral resolution κδ , then a

(high) spectral resolution limit 0δ exists such that:

( ) ( ) 0 .FTS DISP

i iκ κ κδ δ δ δ δ δ< ∀ < (39)

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.

3.5 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 ( ){ } 20OPD=

κiFT to FTS

iδ ratio is an important estimate of the radiometric accuracy

required to the interferometer, in order to reach the spectral resolution κδ . From Eq. (38) the

following relationship is easily obtained:

( ){ }( ) 1OPD 0

min max

max

1

2

,

l

FTS

FTSi

iSNR K

A

BK

κ

κ

δ

δ

+=≅ ≈

=

FT

(40)

where we have introduced the minimal signal-to-noise ratio min

SNR necessary for measuring

the tiny amplitude ( ){ } 2maxOPDOPD≈

κiFT on the top of the constant value

( ){ } 20OPD=

κiFT , and the maximum number maxK of in-band independent spectral channels

that can be interpolated from such an interferometric measurement when observing a source


having bandwidth B . Equation (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 minQ ) requested to obtain the spectral

resolution κδ from interferometric observations:

( ){ }

( ) ( ) ( )OPD 0min 2 2 max 2log 1 1 log log .

FTSFTS

i

iQ l K A

κ

δ=

≅ ≈ + + −

FT (41)

In the previous equation the maximum digitized signal value has been set to ( ){ }0OPD=

κiFT ,

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 ( ){ } 2κiFT around

maxOPD . 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 ( ) κδκδ iDISP

i ≈ .

Evidently, Eqs. (40) and (41) implicitly assume that the standard deviation zσ of the

overall noise affecting the interferogram measurement is less than the small amplitude ripple ( ){ } 2κiFT to be observed. In a different wording, the maximum digitized OPD (the maxOPD )

has to be less than the value nOPD where the envelope of the informative signal ( ){ } 2κiFT

equates the noise level zσ :

max .n

<OPD OPD (42)

Interferogram samples collected at OPD s greater than nOPD are meaningless and don’t

add useful information but mainly noise to the spectral estimation. This elementary concept is illustrated in Fig. 5 where the blue curve represents the envelope of the informative component of the interferogram ( ){ } 2κiFT and the black dotted line indicates the noise

amplitude zσ . 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.


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 OPDn does not bring source information.

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 ( )OPDΦ experimented by an interferometer is related to the

interferogram intensity that, according to Fig. 3 and Eq. (3), is mainly determined by the

constant term ( ){ }0OPD=

κiFT . The photon 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 ( )κσ ,OPDΦd of the photon flux impinging in a unitary time

interval over a sensor having unitary equivalent surface area within the narrow spectral interval κd around the generic wavenumber κ can be written as:

( ) ( ) ( ) ( ) ( ){ }OPD,

( )1 cos 2 OPD , .

id S H V x d

chκ

κσ κ κ κ πκ κ π κ

κΦ = + + (43)


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

( )κσ ,OPDId , which summed over all the narrow spectral intervals gives rise to the photonic

noise affecting the interferometer radiometric signal:

( ) ( ) ( ) ( ) ( ){ }OPD0

( ) 1 cos 2 OPD , .I

ch S H i V x dσ κ κ κ κ κ πκ κ π κ+∞

= + + ∫ (44)

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:

( )

( ){ } ( ){ }{ }

( ) ( ) ( ) ( ){ }

( ) ( ) ( ) ( ){ }

OPD 0OPD

0

0

2

( ) 1 cos 2 OPD ,

,

( ) 1 cos 2 OPD ,

avI

av

i ich

S H i V x d

S H i V x d

κ κσ κ

κ κ κ κ κ πκ κ π κκ

κ κ κ κ πκ κ π κ

=

+∞

+∞

+=

+ + =

+ +

∫

∫

FT FT

(45)

being avκ the spectral photo-center of the concerned source. Equation (45) allows us to write

the maximum effective interferometer signal-to-noise ratio effSNRmax allowed by the photonic

noise only:

( )( ){ }

( ){ } ( ){ }{ }max

OPD 0

2.

2

eff

av

iSNR FTS

i ich

κ

κ κκ =

=+

FT

FT FT

(46)

Let us note 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 ( )max

effSNR FTS . On the other hand, the

informative signal ( ){ } 2i κFT amplitude is far below the dc term ( ){ }OPD 0

2i κ=

FT for

large OPD s (e.g., at high spectral resolution), resulting in a small ( )maxeff

SNR FTS . 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-to-noise ( )max

effSNR FTS :

( )( ){ }

max

1

0max OPD

.2

l

eff OPD

FTS

av

iSNR FTS A

ch B

κκ δκ

+

= ≅

FT (47)

As can be seen the maximum effective signal-to-noise ratio ( )FTSSNReff

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:


( )1

2

max .eff

DISPSNR DISP AB

κδ ≅

(48)

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<<κδ and large 1OPDmax >> ) the advantage of dispersive techniques is an

exponential factor of 21+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.

3.6 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).

4. 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;

4. Unfortunately, the cosine-like transformation operated by the FTS concentrates this equal power level (in the informative component of the signal) at lower OPD s, strongly reducing the power efficiency of the FTS with respect to dispersive


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.

Table 1. Main differences between dispersive and interferometric technique about

throughput, multiplexing, radiometric accuracy and SNR issues.

Issue Dispersive spectrometer FTS device

Jacquinot advantage NO YES (some imaging FTSs adopt an input slit losing this advantage)

Fellgett’s advantage YES (for those devices that employ array detectors)

YES (for those devices that employ array detectors)

Multiplexing effect - YES

SNR improvement by the

{ }1−FT operator

- NO

Physical signal ( )κi ( ){ } ( ){ } 20OPD

+

=κκ ii FTFT

Effective signal ( )κi ( ){ } 2κiFT

Minimal radiometric Accuracy

( ) κδκi≈ ( ){ } 2

1

0OPD

+

=

≈

l

FTS

Bi

A κδκFT

Effective SNR (proportional to) ( )2

1

κδ

1+

l

B

κδ

Lost samples Lack 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:


• 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 κδ follows a rather favorable law

( ){ }2

OPD 0 ,

2FTS FTS

i

Ai

Bκδδ κ

=

≈

FT 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 1<<B ), 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.

5. 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 cosine-like 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 21+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.

Acknowledgements

This work was carried out with the support of the Italian Space Agency (ASI).


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