A.

Particle contamination

A first source of diffuse straylight is contamination of the optical surfaces with dust particles. A synthetic review of this subject has been presented by Dittman [3]. The dust particles, of random shape and orientation, are traditionally considered spherical to simplify the modeling, although it can be remarked that for a mirror surface, in principle two particles should be considered, the particle and its mirror image. It is also assumed that the particles follow a log-log size distribution according to the MIL-1246B standard [4]. Dittman [3] discussed the possible use of two slope values (0.926 and 0.313) for respectively cleaned and uncleaned surfaces, as the cleaning process tends to remove preferably the largest particles. He also showed that Spyake and Wolfe [5] published BSDF curves are convolved with a finite detector aperture and thus should not be used as a starting point for a straylight model.

From the Dittman review, it appears that the most accurate way to get BSDFs is a numerical computation based on the Mie theory [6,7] and integration over the MIL-1246B size distribution. Scripts for Mie computation can be easily found on the web. It is also possible to write one’s own script, with the help of the Wiscombe criteria that tells when to stop the Mie infinite sums [8], online tools for validation [9], and some care for evaluating complex mathematical functions at very large orders. The Mie intensities i₁ and i₂ [6,7] evaluated for one specific particle size, are converted into a BSDF as follows. The scattered intensity I (square modulus of the electric field, proportional to an irradiance in W/m²) at a distance r from the particle is:

where I₀ is the incident intensity on the particle, k=2π/λ, θ the scattering angle and unpolarized light is considered. We assume σ particles per unit surface area, dS is a small area perpendicular to the incident beam over which we have σdS particles, and dA a small surface where the scattered light is measured. The BSDF is the ratio scattered radiance L(θ)/incident irradiance I₀:

With this formula we could recover the Dittman BSDF curves, compute them for a given distribution slope and several wavelengths on a regular grid. Our final particle contamination BSDF is obtained with a wavelength interpolation and scaling to reach the relevant contamination level. For a mirror, the forward scatter, after reflection adds up to the backscatter so it may be useful to sum up both contributions.

B.

Roughness of optical surfaces: Total Integrated Scatter (TIS)

The TIS for a rough surface is traditionally computed with:

where k=2π/λ, σ is the rms roughness and Δn the refractive index contrast at the interface (Δn =2 for a mirror). To better understand the underlying assumptions and establish the angular dependency, we show how this equation is derived from first principles. We consider a mirror surface with a roughness histogram p(h). This histogram is normalized (its integral is 1) and centered (its average is 0). The optical path difference (OPD) between the real ray reflected by the mirror at the height h, and the ideal ray reflected on the average flat surface for which h=0, is 2hcosθ. The decrease of specular reflection is obtained by interference between all the real reflected rays. We assume that the TIS is equal to 1 minus this specular reflection, by energy conservation:

On the right side we see the Fourier transform of the histogram. Under normal incidence (θ=0), with very small roughness (σ<<1), a Taylor expansion of the exponential allows to recover equation (3). If the angular dependency is kept, for very small roughness we recover the formula given by Stover [10]:

Finally, the usual formula with the exponential TIS_reflection = 1-exp(-4k²σ²cos²θ) is recovered with a gaussian histogram of arbitrary amplitude. In transmission, a similar reasoning allows to get the following TIS at arbitrary incidence and σ<<1:

where θ₁ and θ₂ are the incident and transmitted angles, respectively in media with indices n₁ and n₂. This equation is consistent with the predicted specular transmission established in the microwave remote sensing literature [11,12]. We note that, contrary to the case of reflection, it predicts an increase of TIS at large incidence angles.

The two TIS formulae (5) and (6) may be used to renormalize the considered roughness BSDFs. However, proper normalization is not easy to achieve as at non-normal incidence, the angular boundaries for computing a BSDF integral are shifted. Also, if the assumption TIS = 1-specular is reasonable in reflection on a metallic surface, it may be incorrect in transmission if the transmittance variations with angle are significant, for instance inside a material close to total internal reflection. In most cases, a simple BSDF normalization at normal incidence can be done with (3). If angles are large (e.g. folding mirrors with 45deg incidence) a more sophisticated normalization using (5) or (6) must be implemented.

C.

Roughness of optical surfaces: overview of existing BSDF models

A large choice of BSDF models is available in the literature to describe rough optical surfaces. Some of them are empirical, and care must be taken to select the parameters of these laws so that the BSDF integral is equal to the expected TIS. Others are physically based and proper normalization is automatically ensured. At normal incidence, proper normalization is achieved when:

The ABg [13], the 2-parameters [3] and 3-parameters [1,3] Harvey-Shack models are empirical. The Wein formula, as presented in [3], seems to be physically based but is actually an empirical modification of a rigorous model [14]. The exponent 3/2 was removed from equation (2.20) in [14], following the suggested empirical fitting law of equation (2.24). It can be easily checked that in the BSDF from reference [3], proper normalization is not automatically achieved. Finally, the K-correlation model [15, 16] is a physically-based BSDF obtained by inserting an analytical parametrization of the power spectral density (PSD) into the Rayleigh-Rice equation [10, 16]. As can be checked numerically, this model is inherently normalized to the correct TIS.

Dittman proposed to extend the K-correlation model to PSD functions with a slope s≤2 [15]. Such PSD laws are fitting well the experimental BSDFs, and indeed for s=2 one recovers the Wein equation [3]. Unfortunately, PSD laws with s≤2, if assumed mathematically correct over the entire frequency spectrum, result in an infinite roughness σ and cannot be associated to real physical surfaces. Further, the obtained BSDFs are not normalized, and if proper normalization factors are added the final BSDF expression does not follow anymore the Rayleigh-Rice theory. To resolve the issue, Dittman [15] makes the following reasoning: the TIS, given by the BSDF integral over one hemisphere, is also given by the PSD integral over a limited range of frequencies (limited by the hemisphere boundaries), and finally given by the traditional equation (3) with a band-limited roughness. This band-limited roughness is wavelength-dependent, with the consequences that it is no longer possible to associate a single roughness value to a surface, and the TIS is now determined by the surface PSD.

On the other hand, this concept of wavelength-dependent roughness seems to contain two inconsistencies: (a) it contradicts the basic result, demonstrated above in section II.B, that the TIS is fully determined by the roughness histogram (1^st order statistics) and does not depend on 2^nd order statistics, such as the PSD or autocorrelation. (b) the Rayleigh-Rice theory does not seem to explain what happens to the evanescent modes (light scattered at angles > 90 deg) and how their energy is redistributed in the propagating modes. Therefore, we can question its validity for PSDs with a significant energy in the high frequencies, such as the ones considered by Dittman for s≤2. In view of the confused situation, we decided to normalize the K-correlation (with s≤2) BSDFs with the usual TIS formula (3) and associate a single-valued roughness to each optical surface.

D.

Roughness of optical surfaces: expected BSDF dependencies

In this section, we try to clarify the dependencies that one should expect from a physically-based BSDF model, with respect to wave vector k=2π/λ, roughness σ, refractive index contrast Δn and angles (incidence angle θ₀, scattered angle θ and relative azimuth φ). In reflection, in the limit of very small roughness it can be shown that the Kirchhoff [11, 12], perturbation [12] and Rayleigh-Rice [10] rigorous theories all converge to a common law:

where no assumption (other than σ<<1 and isotropy) was done about the surface. We assume it is immersed in a medium of index n₁, with Δn=2n₁. R is a reflectance factor that also includes polarization projection terms, and F a general function that contains the angular variability. The obliquity factors (slow varying with the angles) have been neglected. For the transmission BSDF, very few references are available that provide rigorous equations. From [11,12] we find that the Kirchhoff and perturbation theories, in the limit σ<<1, converge to:

where light propagates from index n₁ to index n₂ so that Δn=n₂-n₁. Again, obliquity factors have been neglected, T is a generalized transmittance coefficient, and F the same function as for the reflection BSDF. In these two equations the TIS as given by (3) appears explicitly in front of the function F. If we exclude it, the remaining terms indicate that when the wavelength increases, the BSDF peak decreases and its curve becomes wider. We finally note that the transmission BSDF may be rewritten as a function of the scattered angle θ_out in medium n₁ (e.g. after it exits a glass component of index n₂), noting that the BSDF is a quantity per unit solid angle and per unit projected area:

The obtained expression, very similar to (8), justifies the practice of using the same BSDF models to describe scattering in reflection and transmission. As we will see in the next section, the angular dependency of the BSDFs in (8) and (11) is simplified to implement the Peterson formalism [1], by assuming normal incidence (θ₀=0) and small angles (sinθ~θ).

A.

Peterson equation

We now explain how the BSDFs of individual optical surfaces can be combined into a straylight kernel that describes the properties of the entire instrument. Equation (14) in reference [1] explains how the BSDFs of all surfaces must be summed with different weighting and stretching factors. In this equation, the term πa_ent² represents the entrance pupil area, so that πa_ent²E_ent is the incident flux that we assume equal to 1. The transmittance T affects equally the scattered and non-scattered light so it is disregarded and the final kernel expression is:

where the sum index j runs over all optical surfaces, (na) is the instrument numerical aperture, a_j the height of the marginal ray when it crosses surface j, and r the distance from the geometrical image measured in instrument focal plane, where the kernel is to be evaluated.

To establish this expression, Peterson only considered optical elements immersed into a medium of index n=1. We now derive a more general law by repeating Peterson’s demonstration and considering air/glass or glass/air interfaces. First, we note that equation (7) in [1] writes n_jθ_j = H/y_j in the general case, where n_j is the index of the medium where the scattered ray propagates. Then, we note that the power P and the product n²G, where n is the index and G the throughput, are conserved quantities through the instrument. Since P=LG where L is the radiance, we conclude that L/n² is also conserved. We then get the generalized Peterson law:

If we repeat the reasoning of section II.D and use the BSDF defined in the external medium n=1, we recover the original Peterson equation (12):

From this reasoning we can draw two conclusions. First, the Peterson equation, although derived with simplified assumptions, can be used for all types of interfaces in an optical system. Second, in that equation the BSDFs must be expressed in the external medium with n=1, which (as discussed in the previous section) means the same BSDF models can be used for the reflective and transmissive interfaces.

B.

Off-axis systems

Equation (12) was derived by Peterson under the assumption of a rotationally symmetric system. Its use for an off-axis instrument is not straightforward, as the height of the marginal ray a_j that appears in (12) provides information about the distance to the optical axis but has no link any more with the size of the beam footprint on surface j. We propose to redefine a_j as a function of the beam footprint area S_j on surface j:

To validate this approach we considered several simple configurations: (a) a parabolic mirror with a non-centered circular pupil, (b) a parabolic mirror with a non-centered elliptical pupil, (c) an Offner relay with 3 reflections on 2 concentric mirrors. The results are illustrated on figure 1, where the exact straylight kernel is computed with ASAP for a point source at infinity at the center of the field, and compared with the corresponding Peterson kernel computed with (12) and (15). The agreement is excellent.

Fig. 1.

Comparison between ASAP computed straylight kernel (blue) and Peterson kernel (red) for a parabolic mirror and non-centered circular pupil (left), parabolic mirror and non-centered elliptical pupil (middle), Offner relay (right).

C.

Diffuse straylight vignetting

In a complex optical system, the argument of the BSDF in equation (12), computed numerically, may take very large values sometimes larger than 90 degrees for some optical surfaces and a given r. This is of course a limitation of the approach, and the BSDF must be set to 0 for angles larger than 90 degrees. On the other hand, this feature highlights the fact that the Peterson approach allows BSDFs of individual surfaces to scatter light at arbitrary large angles. In practice, most of the scattered light will actually fall outside of the physical aperture of the next optical surface, or may be vignetted by another surface further away in the instrument. To evaluate the impact of this effect we compared three levels of approximation:

Fig. 2.

Computation of the vignetting angle for a refractive telescope. The observed object is at infinity and on axis (blue rays). At surface #11, scattered rays (in red) are emitted and vignetted at surface #18.

The comparison was performed with the spectrometers designed by the industrial primes Selex Finmeccanica (Italy) and Airbus (France) during the phase A-B1 study of the FLEX mission [17]. The impact of the different assumptions is reported in Table 1, for the peak of the straylight spectrum in the high resolution (HR) channel. We see that the implementation of a realistic vignetting angle reduces straylight by 10 to 15%, while implementing a vignetting function with a realistic shape has a small impact (around 2%). On this basis, we have implemented assumption (b) in our model.

Tab. 1.

Impact of the vignetting assumptions used on the peak straylight in the HR FLEX spectrometer.

A.

Principle

We now describe how the straylight kernel, obtained with the Peterson formalism described previously and assumed to be constant over the instrument field of view and spectral range, is used to compute straylight in an imaging spectrometer. Imagers are much simpler so the method we present can be easily adapted for these instruments. Fourier-transform spectrometers are more complex and will not be addressed.

Pushbroom imaging spectrometers usually consist of (a) a telescope that images a scene at infinity (b) a slit situated in the telescope focal plane (c) a collimator, with the slit placed at its focus (d) a disperser such as a grating, prism or grism (e) finally a camera that focuses rays coming from the disperser onto a detector array. The slit plays the role of a field stop, so that the detector records in its two dimensions, the spectral information and only one spatial dimension (along-slit). The remaining spatial dimension (across-slit) is acquired temporally while the instrument flies over the observed scene, the slit being perpendicular to the flight direction.

We assume that such an instrument observes a scene that has separated spatial and spectral dependencies. Such scene is typically constructed by aggregating spectra computed with a radiative transfer model for a spatially uniform scene L_j(λ), and giving them some spatial dependency g_j(x,y):

In this equation, x is measured along flight direction. The following two contributions are computed separately and will be discussed in the next sections:

B.

Telescope

To limit the model complexity, we will make two additional assumptions:

The nominal (straylight-free) illumination in the telescope focal plane becomes, ignoring spectral transmittance:

where V(x,y) is a function describing the vignetting that results from the telescope entrance baffle. It can be computed with a simple model, based on the convolution of the instrument pupil (oversized to get some margin) and the useful field, of rectangular shape, defined by the slit. The imaging PSFs before slit involve the telescope optical quality (diffraction, aberrations, multiple spots from a polarization scrambler) and potentially motion smear. The straylight is computed in the telescope focal plane by convolution with the telescope Peterson kernel K, and then propagated to the spectrometer focal plane:

If we neglect the imaging PSFs (replacing them with Dirac functions) we get:

This simple law has been implemented in our model with the following three steps: (a) computation of illumination on slit plane gj(y)V(x,y) (b) convolution with the Peterson kernel (c) implementation of slit opening (selection of the computed straylight for x=0) and multiplication with convolved spectra. Our model is programmed in Python, and makes use of the Numpy, Scipy and Matplotlib packages. The inputs and outputs are collected in Excel and the Python code is called by a Visual Basic macro. Some computation results are presented on figure 3 below to illustrate the method. A telescope with 17 optical surfaces, contamination levels ranging from 50 ppm to 400 ppm and roughness of 1nm rms was illuminated with a half-illuminated field. The simulation took less than 45 sec on a 2.80 GHz computer with 16 Gb RAM under Windows 7.

Fig. 3.

Computation steps for the telescope straylight. Upper left: half-illuminated scene observed on the telescope focal plane, limited by the telescope entrance baffle vignetting V(x,y). The array is zero-padded for the convolution. Upper right: resulting straylight after convolution with the Peterson kernel. Lower left: straylight observed on the spectrometer focal plane. Lower right: straylight spectra at 0, 10, 20, 40 and 80 spatial sampling distance (SSD, with 1SSD = 2 detector pixels) from the half field transition. The horizontal and vertical axes of the false color plots are in detector pixels.

C.

Spectrometer

To compute spectrometer straylight, we make the same two additional assumptions as for telescope straylight (slit illumination uniform in x, and separable PSFs). The nominal illumination on the instrument focal plane in the absence of straylight is:

where the imaging PSF includes all PSFs in the instrument (telescope, spectrometer, detector) along the y axis. The corresponding straylight is obtained through a slightly more complicated equation:

where T(λ) is the product between the instrument spectral transmittance and the detector quantum efficiency. When the convolution with the Peterson kernel is omitted, and if the variations of T(λ) are neglected across the width of the ISRF, equation (20) is recovered. If scenes with simple patterns of spatial contrast are used, and the straylight is evaluated a few pixels away from the transitions, the imaging PSF may be omitted.

Equation (21) is implemented in our model. Some computation results are presented on figure 4 below for a half-illuminated scene, a complex spectrometer with 40 optical surfaces, and similar assumptions as on figure 3. The simulation took less than 75 sec on a 2.80 GHz computer with 16 Gb RAM. It is interesting to note that the spectrometer straylight has a spectrally smooth signature, contrary to the telescope straylight.

Fig. 4.

Computation steps for spectrometer straylight. Left: half-illuminated scene observed on the instrument focal plane, multiplied by T(λ) along spectral axis, limited by slit length along spatial axis. The array is zero-padded for the convolution. Middle: resulting straylight after convolution with the kernel. Right: radiometrically calibrated straylight observed at 0, 10, 20, 40 and 80 spatial sampling distance (SSD, with 1SSD = 2pixels) from the half field transition. The horizontal and vertical axes of the false color plots are in pixels.

D.

Possible straylight definitions

The Peterson method allows to address various interpretations of the straylight definition. The straylight calculated as described in the previous sections includes all the photons that underwent a scattering event. Other definitions are possible: