Figure  depicts the optical setup of an imaging spectrometer during solar calibration. The incoming sunlight is scattered by the diffuser. The scattering origin lies in the aperture plane with spatial coordinates g and h, which is perpendicular to the light's direction of propagation. The angular field distribution at the aperture plane is imaged to the slit plane with the coordinates x and y. The light is collimated onto a dispersive element (e.g., a diffraction grating), which splits it into its spectral components. The spatial information in the y direction of the slit is transformed into spectral information at the detector with coordinate b by imaging the diffracted beams of different wavelengths onto a 2D array detector. Beams of the same wavelength (within the spectral resolution) are assigned the same spectral detector coordinate b. The spatial information in the x direction is retained in the detector coordinate a. We relate the coordinates via the simplified linear spectrometer equations as follows: 1a=Mxx,2b=Myy+kλ, where Mx and My are the respective magnification factors in the x and y direction, k=db/dλ denotes the dispersion, and λ is the wavelength. For these simplified equations to hold the magnification factors and the dispersion, they are assumed to be independent of the wavelength and the spatial position (x,y). Also, the instrument point spread function (IPSF) is not accounted for. Sunlight is spatially coherent (), and we can assume collimated light is illuminating the diffuser. Additionally, the sunlight's temporal coherence is on the order of femtoseconds () and much smaller than the typical detector integration time of several hundred milliseconds of an optical instrument. As a consequence, cross-coherence terms of interfering fields of different wavelengths average out, and the net intensity distributions at the slit and the detector planes are well approximated by the superposition of monochromatic intensities. The Sun disk is comprised of many incoherent point sources, which should be considered for angular averaging contributions and are not part of this work as they will only account for a single-point source. For the purpose of the SFA measurement, the sequence of optical components is subdivided into two parts. The first part, ranging from the illuminated diffuser through the telescope to the entrance slit, is represented by the optical setup in the lab. The second part comprises the rest of the optical system from the slit plane to the instrument-detector plane. The data acquired in the first part are used as input for a numerical simulation of the optical setup after the slit plane. The setup layout is shown in Fig. . The Sun is mimicked as a single field point with a tunable laser source, which is spectrally stabilized by a wavemeter, and it illuminates the diffuser through a linear polarizer at normal incident with respect to the diffuser plane. The distance between the single-mode fiber output and the diffuser is chosen such that the divergent beam illuminates the diffuser homogeneously over the size of the apertures. The second aperture blocks any unwanted angular contributions. A power meter placed next to the diffuser records a fixed fraction of the emitted laser power. The telescope images the scattered light onto a 2D array detector positioned in the focal plane. The focal plane of the telescope represents the slit plane in Fig. . The diffuser plane is tilted by 10∘ with respect to aperture and slit plane. This ensures that only scattered light contributes to the measurement. The telescope is aligned perpendicular to the aperture and slit plane.

For a measurement, monochromatic speckle intensities are recorded subsequently over a wavelength range λ1…λN several times the spectral resolution λres of the real spectrometer that is being mimicked. An example of such an monochromatic speckle pattern is shown in Fig. a and d in the slit plane. The spectral tuning step size Δλ in between images needs to be sufficiently small, in order to properly sample the change of the speckle patterns. The intermediate result is a 3D data set Islit(x,y,λ) consisting of a spectrum of monochromatic speckle images, where x and y are the spatial coordinates in the slit plane and λ is the wavelength. Every speckle image is mapped to a certain position (a,b) at the focal plane, where all images are summed up in intensity. The summation on the intensity basis is justified as cross-coherence terms involving the interference of different wavelengths of actual sunlight, which will vanish for sufficiently long integration times. The summation procedure is detailed in and can be summarized as follows: 3Idet(a,b)=Δλλres∑λ=λ1λNIslitaMx,b-kλMy,λ×Θb-kλΘkλres-b+kλ, where slit coordinates are expressed in terms of the detector coordinates using Eqs. () and () and the Heaviside function with Θ(y)=0,y<0 and Θ(y)=1,y≥0. The result of the sum is a 2D intensity distribution in the focal plane of the instrument Idet(a,b), which is depicted in Fig. b and e. In a last step, Idet(a,b) is overlain with the actual instrument's detector pixel grid (ã,b̃), and intensities belonging to the same pixel are summed, which are shown in Fig. c and f. The SFA is calculated as the standard deviation of the mean normalized detector pixel intensity distribution Idet,binned(ã,b̃).

Measurements are conducted in the NIR regime around 777 nm and in the SWIR regime around 1570 nm, which represent wavelength bands with commonly monitored data products, such as water vapor, clouds, CO2, aerosols, or the O2 absorption, which are commonly used to calculate the effective path length and the air mass factor (see , , or ). The experimental setup is shown in Sect. . As light sources, they serve tunable monochromatic external cavity diode lasers with single-mode output and an integrated optical isolator. They are stabilized via a proportional–integral–derivative (PID) loop with feedback data from a wavemeter, which uses a Fizeau interferometer. Connecting fibers between the laser, wavemeter, and output serve single-mode (SM) fibers since the spectral tuning range will be narrow. Also, SM fibers will introduce no additional speckle in contrast to multi-mode fibers. A linear polarizer ensures polarization stability. The round diffuser plate has a diameter of 70 mm and a thickness of 3 mm. It is made of high-scattering fused silica HOD^®-500 material featuring inhomogeneities of 20 µm or less. The spatial coherence length of the Sun's light is around 60 µm in the NIR and 120 µm in the SWIR band, according to . Hence, the material, which was selected for the Sentinel-5/UVNS according to , is suited to the wavelengths in question and deemed a good choice for this study. The data collected with the power meter are used to normalize the acquired images. The round apertures are used to control the size of individual speckle correlation areas. The laser beam's uniformity at the diffuser plane was measured to be around 3 % in the NIR band and 6 % in the SWIR band over the size of the aperture. There were no additional speckle contribution by the fibers detected. The telescope has a focal length of ftel=1100mm. For the NIR, the laser source has a center wavelength of 780 nm and a nominal linewidth of 6×10-7nm. The Thorlabs 780HP SM fiber (SMF) type was used for all fiber connections in the NIR range. The charge-coupled device (CCD) detector features a 12.5 mm × 10.0 mm active area with 2750 × 2200 pixels of size 4.54 µm × 4.54 µm. The expected average 1D width of the speckle correlation area Sd in the slit plane is calculated using the following: 4Sd=2λftelDπ, which was derived by . The aperture's diameter is set to D=10mm, and, thus, the speckles are sampled by 19 pixels in one dimension, which is deemed sufficiently fine to ensure that the sampling with the detector does not alter the speckle pattern significantly. The laser wavelength was tuned over the range of 776.4–777.7 nm, with a step size of Δλ=1 pm. The step size is chosen so that there is a non-zero correlation between subsequent speckle images. For the SWIR measurement, the laser source and the detector and the fiber splitter are replaced, and the Thorlabs SMF-28 was used for the fiber connections. The SWIR laser source center wavelength is 1550 nm, with a single-mode output of nominal 2×10-6nm linewidth. The detector is a 640 × 512 pixel InGaAs camera with a pixel size of 15.5 µm × 15.5 µm. The tuning range was 1571–1577.5 nm, with a step size of Δλ=3.1pm. The aperture's diameter is set to D=13mm, which means a sampling of 10 pixels per speckle in one dimension. This is smaller than for the NIR configuration but is a trade off between sampling and SNR.

The laser source emits a single polarization state, which is ensured with the polarizer. The diffused light leaving the volume diffuser can be treated as depolarized due to multi-scattering p. 85. This corresponds to two orthogonal polarization configurations or two effective intensity distributions which cannot interfere. Therefore step, n=1 introduces two degrees of freedom Mpolarization=2 p. 49.

Step n=2 leads to spectral averaging at the detector. We recall the finding from Sect.  that the net intensities in the field planes (slit and detector) can be treated as the superposition of monochromatic intensities for integration times much greater than the coherence time. Let us consider the acquired speckle intensities In(x,y) and the underlying fields An(x,y), which are related by In=An2. They are recorded at various wavelengths λn. The magnitude of the statistical change of subsequent speckle intensities Im and In, with a wavelength difference Δλnm=λn-λm, can be described in terms of the first-order field correlation coefficient μmn, with the following: 9μmnλn,λm=〈AmAn∗〉〈Im〉〈In〉, where ∗ denotes the complex conjugate. The field correlation is influenced by two effects which, in our case, are both frequency dependent. The first effect is due to changing light paths through the diffuser medium. The second effect takes into account the spatial offset Δb=kΔλnm at the detector plane between individual speckle patterns In induced by the dispersion (see Eq. ). We start with the former contribution to the field correlation and follow the approach of , who presented an analytic equation for the angular correlation function of a slab geometry of a scattering media of thickness d, which can also be used for wavelength correlations as follows: 10Fλn,λm=d+2Bsinh⁡z0Q+BQcosh⁡z0Qz0+B1+B2Qsinh⁡dQ+2BQcosh⁡dQ, where we set Q=q2+α2 and q=i6π1λn-1λmβns/lt, with ns denoting the refractive index of the scattering material, lt being the transport mean free path used for anisotropic multi-scattering systems, and β being a constant factor taking into account the contribution of the tilted diffuser plane (e,f) with respect to the other planes and the specific geometry. The value z0 describes the average penetration depth after which the light is scattered for the first time. It does not have great influences in transmission geometry and is approximated with the transport mean free path lt. We set α=0, thereby ignoring absorption. The parameter for the boundary condition is given by B=lt2(1+R)3(1-R), where R is the reflection coefficient which is calculated using the Fresnel equations. It accounts for internal reflection due to the index of refraction mismatch at the boundaries.

The second contribution to the field correlation is due to changing spatial positions of speckle patterns which are distributed over the instrument detector in accordance with the spectral dispersion. This constitutes a spatial offset Δb between the speckle intensities In and Im at the detector plane (a,b). Keeping in mind Eq. () and () for the transformation from the slit to the detector plane, the correlation of speckle fields, which are separated spatially by Δb, can be expressed as follows: 11Ψ(Δa,Δb)Δa=0=∫-∞∞P(g,h)2e-i2πλ̃z(gΔa+hΔb)dgdh∫-∞∞P(g,h)2dgdhΔa=0, where P(g,h) is the aperture function of the imaging system and may be defined, e.g., for a circular aperture of diameter D, as P(g,h)=circ2g2+h2D, which is 1 inside the aperture and 0 otherwise. g and h are the a- and b-coordinate representations in the aperture plane, z is the distance between aperture and slit plane, and λ̃ is the mean wavelength p. 169. We set Δa=0 since we are only interested in the offset in the spectral direction. Combining the two effects, we can model the correlation between the speckle fields as follows: 12μmnλn,λm=Fλn,λmΨ(Δb). The accumulation of individual speckle patterns In with field correlations μmn at the detector can be interpreted as the summation of partially correlated speckle intensities as follows: 13Idet(a,b)=∑n=1N=λres/ΔλInaMx,b-kλMy,λn. The number of individual speckle intensities In contributing to the sum at arbitrary detector coordinates (a,b) is equal to the ratio of the spectral resolution λres with the step size Δλ. This also applies to the mean intensities, as follows: 14〈Idet(a,b)〉=∑n=1N=λres/ΔλInaMx,b-kλMy,λn. Using an established method by and , we define a coherency matrix with entries Jnm=〈AmAn∗〉 and use Eq. () to obtain the following: 15J=〈I1〉〈I1〉〈I2〉μ1,2⋯〈I1〉〈IN〉μ1,N〈I1〉〈I2〉μ1,2∗〈I2〉⋯〈I2〉〈IN〉μ2,N⋮⋮⋱⋮〈I1〉〈IN〉μ1,N∗〈I2〉〈IN〉μ2,N∗⋯〈IN〉. By the diagonalization of J with a unitary linear transformation L0, the ensemble of correlated speckle fields is transformed to a basis with no correlation between them. 16J′=L0JL0†=〈I1̃〉0⋯00〈I2̃〉⋯0⋮⋮⋱⋮00⋯〈IÑ〉, where † denotes the Hermitian transpose operation. The total mean intensity 〈Idet〉=∑n〈In〉=∑n〈Iñ〉 is conserved under this transformation but, in general, 〈In〉≠〈Iñ〉. The complex coherence factor μmn=|μmn|exp⁡(iΦnm) includes a phase Φnm. However, due to the specific construction of J, these phase terms can be omitted when calculating the eigenvalues Sect. 4.7.2. Finally, for the spectral degrees of freedom, we use the eigenvalues 〈Iñ〉 of the coherency matrix to obtain the following: 17Mspectral=〈Idet〉σdet2=∑n〈Iñ〉2∑n〈Iñ〉2. Note that changing Δλ and, thus, changing N will not change the result of Mspectral as long as Δλ is sufficiently small to sample the covariance μmn. The enabling property of the coherency matrix J is called Toeplitz, which implies an asymptotic behavior of its eigenvalues found by . gives a simplified proof in corollary 2.1 and 2.2 that both the numerator and denominator in Eq. () converge for large N.

In step n=3, an averaging due to the integration of the instrument-detector pixel takes place. We already established that the resultant intensity distribution at the detector Idet(a,b) is given by the summation in Eq. (). This effect impacts the speckle contrast if individual speckles are not sufficiently oversampled by the instrument detector pixel grid (ã,b̃). An analytical expression for the degrees of freedom Mdetector introduced by stationary speckles in one detector pixel with relative coordinates (Δa,Δb) is given by the following: 18Mdetector=1AD2∬-∞∞KD(Δa,Δb)μdet(Δa,Δb)2dΔadΔb-1, where AD is the area of a detector pixel, KD(Δa,Δb) is the autocorrelation function of the detector pixel, and μdet(Δa,Δb) is the field correlation at the detector plane p. 108. In order to accurately describe μdet, one needs to account for the evolution of the speckle size during the summation in Eq. (). Let us consider a single speckle correlation area I1(Sd/2≤x-x1,Sd/2≤y-y1,λ1) with a spatial extent denoted by Sd centered at (x1,y1) in the slit. Its correlation relative to this position is described by the following: 19Ψ(Δx,Δy)=∫-∞∞P(g,h)2e-i2πλ̃z(gΔx+hΔy)dgdh∫-∞∞P(g,h)2dgdh, with Δx=x-x1 and Δy=y-y1 being relative coordinates. The function Fλ1,λn, introduced previously, characterizes how the correlation area develops after n wavelength steps at the same position, In(Sd/2≤Δx,Sd/2≤Δy,λn), with n>1. In other words, it denotes the number of spectral steps after which a single speckle ceases to exist at a fixed position in the slit. The initial position of the speckle at the detector is (a1,b1)=(Mxx1,Myy1+kλ1). The subsequent contributions relative to the initial position are shifted in the b direction by kλ1-λn and have a magnitude denoted by Fλ1,λn. Therefore, the resultant speckle correlation function at the detector μdet(Δa,Δb)2 is a convolution of Ψ(Δa,Δb)2 with Fλn,λm2 as follows: 20μdet(Δa,Δb)2=Ψ(Δa,Δb)2Fλn,λm2. The symbol denotes the convolution.