3.1.

Deriving Thickness Function

The first step in removing the fringes is to determine the effective thickness function $T(x,y)$of the CCD sensor layer. Our approach was to apply a search procedure at each pixel to fit Eq. (2) to the flat-field intensity data. For each pixel, the intensity was extracted from the flat-field frames for the wavelength range from 820 to 940 nm (61 values), which corresponds to an approximately linearly sloped response region (Fig. 2). The data for each pixel were then processed using a high-pass filter to remove the slope from the flat-field intensity profile as shown in Fig. 2(b). It is worthy to note that because the thickness function is an argument in the cosine in Eq. (2), it is primarily the wavelength-dependent frequency and phase of the sinusoid signature in Fig. 2(b) that need to be fitted with Eq. (2) to find $T(x,y)$.

To apply Eq. (2) to a pixel at $(x,y)$, the term $\alpha (\lambda )$ was replaced with a constant value ${\alpha }_{\mathrm{rms}}(x,y)$, which is the root mean square (rms) value of the slope-removed flat-field signal for the particular pixel. For $n(\lambda )$, we used silicon refractive index values tabulated by Green¹⁶ for every 10 nm in the wavelength range of interest at room temperature (300 K). Linear interpolation was applied to find the refractive indices corresponding to the 61 wavelengths of interest.

Next, a set of test thickness values within a selected range was applied to Eq. (2). For each test value, the mean square error (MSE) between the computed intensity and the slope-removed flat-field intensity was calculated. The test value that produced the minimum MSE was chosen to be the thickness at the considered pixel. The example plot in Fig. 5 of the MSE versus thickness values for the pixel at (256, 256) shows that the MSE behaves in an oscillatory manner with the minimum near $12.6\mu \mathrm{m}$. The oscillation period is proportional to the finite wavelength range (820 to 940 nm) that is available from the flat-field data. Figure 6 shows a comparison of the slope-removed flat-field intensity ($I_{\text{flat}}$) and the intensity result from Eq. (2) for the thickness value obtained through the MSE search ($I_{\text{model}}$). Although the modeled intensity cannot account for all the detailed amplitude changes in the data, the model shows good agreement in terms of the frequency and phase, which is most relevant to the thickness fit.

Fig. 5

Behavior of the MSE between the modeled intensity and the slope-removed flat-field intensity for a range of thickness values at pixel $(x,y)$ = (256, 256). The red circle at $12.6\mu \mathrm{m}$ indicates the thickness value corresponding to the minimum MSE.

Fig. 6

The slope-removed flat-field intensity ($I_{\text{flat}}$) and the modeled intensity ($I_{\text{model}}$) at pixel $(x,y)$ = (256, 256) for the thickness value shown in Fig. 5. Only the match between fringing frequency and phase (not amplitude) is significant for determining the layer thickness.

It is worth noting that our choice of ${\alpha }_{\mathrm{rms}}(x,y)$ is not critical to the thickness results because the minimum MSE is primarily a function of the frequency and phase of the model signal rather than the amplitude. Other choices for $\alpha (x,y)$ produced the same results, but ${\alpha }_{\mathrm{rms}}(x,y)$ is convenient and provides a similar amplitude in the model result as in the flat-field data. With regard to variations in $n(\lambda )$, an Si detection layer in a CCD may contain dopants and other impurities,^1,2 and the refractive indices for Si presented in the literature may also lack accuracy.^1,16 Nevertheless, these differences are typically $<0.5\%$^1,16 and we found from simulation results that such variations can create up to 20-nm differences in the recovered thickness function but do not change the physical form of the function nor the level of final correction in the images.

The search approach described above was repeated for all pixels to obtain the 2-D thickness function $T(x,y)$. However, an issue with this approach is that the oscillatory nature of the MSE result (Fig. 5) combined with noise in the data can cause the minimum MSE metric to incidentally “jump” between two adjacent valleys. This leads to some ambiguous thickness jumps between pixels in the 2-D thickness profile, as noted in our previous work.¹⁵ Here, we assumed that large, discrete thickness jumps between adjacent pixels are extremely unlikely and developed an algorithm to exclude these features. We first computed the thickness for a starting pixel at the center of the frame and then limited the allowable change when computing the thickness for adjacent pixels. The thickness computation was typically started at the center pixel [$(x,y)$ = (256, 256)] because the signal intensity is relatively less noisy due to better AOTF sensitivity at the center. For this starting pixel, the search was conducted over a thickness range from 10 to $16\mu \mathrm{m}$ in 0.01-nm intervals. We investigated larger thickness ranges, for example 5 to $30\mu \mathrm{m}$, and found the global minimum does not change. The thickness search for each pixel is then stepped out toward the edges of the sensor. The allowable change for the thickness search in subsequent pixels was limited to $\pm 60\mathrm{nm}$ of the average thickness of the available adjacent pixels. The $\pm 60\mathrm{nm}$ change was chosen because the typical spurious amplitude step of the profile was $\sim 120\mathrm{nm}$. The algorithm was also tested with different starting pixels and we found the relative thickness profile was essentially unchanged, although, depending on the choice of the starting pixel, the mean thickness of the 2-D profile could change by $\pm 120\mathrm{nm}$. However, this difference has no effect on the final defringing results.

3.2.

Correcting Etaloning Effects

The next step for etaloning correction involves separately removing the fringes from each science frame and flat-field frame. Correcting the Jupiter and flat-field images separately compensates for the issue that the frames may have been formed with differing optical spectral content or other differences such as noise levels that affect the fringe contrast. Fringing is removed by dividing the frame of interest by a normalized synthetic fringe model frame (“fringe frame”) that is created by applying the derived 2-D thickness function along with an appropriate value of $\alpha (\lambda )$ in Eq. (2). It is not feasible with our data to recover $\alpha$-values for each pixel in a single spectral frame. Rather, a single value for $\alpha (\lambda )$ is found for each frame through an iterative process of minimizing the overall presence of the fringes in the frame.

Our metric for determining the amount of fringing present in a given frame is to compute the 2-D spatial power spectrum of the frame and examine a region of the power spectrum that contains the fringing signature. Figure 7(a) shows a grayscale representation of the power spectrum for a flat-field image where the tilted “bow-tie” feature is a signature of the fringing formed by our particular CCD. Normalized synthetic fringe frames for a range of $\alpha$-values were created with the 2-D thickness function applied to Eq. (2). The flat-field frame was divided by each synthetic frame trial and the resulting power spectra were monitored. The $\alpha$-value for the best correction was found by minimizing the integrated power within an appropriate spectral region. Multiple regions within the power spectrum were examined for each frame and the best-correction $\alpha$-values were averaged to reduce the error dependence on the region chosen. The standard deviation of the results for the multiple regions was also calculated to determine the sensitivity of the method to the region selection. An example set of regions for the analysis is shown by the green boxes in Fig. 7(a) where the areas were stepped by a factor of 2 while keeping the aspect ratio the same. Only boxes in one quadrant are used as the power spectrum of a real-valued input is symmetric about dc.

Fig. 7

Power spectrum of a flat-field at $\lambda =848\mathrm{nm}$ (a) before and (b) after defringing (i.e., divided by the synthetic fringe frame). Figures are shown in log scale for display purposes. The green boxes in (a) are example regions that are examined to minimize the fringe signature.

At every wavelength and for a considered region, the search for $\alpha$ was initially conducted in the range from $-0.03$ to 0.04 at intervals of ${10}^{-3}$. Once an initial solution was found in this range, another iteration of the search was conducted at $\pm 0.002$ around the initial solution in intervals of ${10}^{-6}$ to improve the accuracy of the solution. Although the contrast ($2\alpha$) is typically $<0.05$ (5%), the initial range for the $\alpha$ search was chosen to be somewhat larger to account for outlying values. The search range included negative values because the best-correction result could be slightly negative due to a response aspect of the model that is discussed in Sec. 4.3. The two-step search for $\alpha$-values made the search computationally efficient in terms of time and memory. Figure 7(b) shows the power spectrum result for the example flat-field frame after dividing by the synthetic fringe frame with the best-correction $\alpha$-value. This correction procedure was applied individually to all flat-field frames over the full wavelength range to get the best-correction $\alpha$-value (${\alpha }_{\mathrm{flat}}$) as a function of wavelength.

Next, the same steps were repeated for the Jupiter images to determine the best-correction $\alpha$-values (${\alpha }_{\mathrm{jupiter}}$) for the synthetic fringe frames as a function of wavelength, although more care needed to be taken when examining the power spectrum to avoid components associated with Jupiter’s structure and banding. A fringe-corrected Jupiter image was obtained by dividing by the appropriate synthetic fringe frame. The final science image was the fringe-corrected Jupiter image divided by the associated fringe-corrected flat-field image that is normalized by the respective mean.

4.1.

Thickness Function

Figure 8 shows the three-dimensional (3-D) projection and the 2-D profile of the thickness function obtained with our approach. The dish-shaped thickness between the sensor face and the internal layer is apparent in the 3-D projection of the thickness function and the 2-D profile clearly shows indications of polishing grooves on the top face that are consistent with the fringe characteristics seen in Fig. 3(a). Furthermore, the 3-D projection of the thickness function indicates different curvatures in $x$ and $y$ directions. These thickness features are also visible in one-dimensional (1-D) profiles of the thickness function in the $x$ and $y$ directions along the center of the CCD (Fig. 9). Notably, the 1-D profiles show no evidence of the discrete thickness jumps between adjacent pixels that can arise from noise affecting the MSE search.

Fig. 8

Thickness function $T(x,y)$ obtained by minimizing the MSE (a) 3-D projection and (b) 2-D profile in grayscale.

Fig. 9

The 1-D cross-section profiles of the thickness function $T(x,y)$ at (a) $x=256$ and (b) $y=256$.

As is evident from the thickness function figures, the mean thickness is about $12.67\mu \mathrm{m}$. However, depending on the starting pixel point for the thickness search and the noise present in the collected data, an uncertainty of $\sim \pm 0.120\mu \mathrm{m}$ can be expected for the computed thickness.

4.2.

Corrected Images

Figure 10 shows an example of a synthetic fringe frame obtained with the derived thickness function inserted in Eq. (2). The synthetic fringes are comparable to the fringes apparent in the flat-field frame of Fig. 3(a).

Fig. 10

Synthetic fringe pattern derived with thickness function at $\lambda =848\mathrm{nm}$ and best-correction $\alpha =0.0175$ [see Eqs. (1) and (2)].

Examples of fringe-corrected flat-field images for four different wavelengths are shown in Fig. 11. These examples represent the onset of fringes (726 nm), before and within one of the ${\mathrm{CH}}_4$ absorption regions of Jupiter (848 and 890 nm), and near the end of our data collection range (926 nm). The top row shows the frames with no correction and illustrates that the fringe contribution is minimal for 726 nm but becomes more pronounced as the wavelength increases. The bottom row shows the frames after correction and demonstrates the effectiveness of our fringe correction method. A few residual fringe contributions are apparent in the corrected and 926-nm frames. The horizontal banding and the spot-like artifact in the corrected frames are due to the AOTF transmission response and a contaminant on an optical surface. The contrast of these response-related artifacts can be as high as 30%. These features are corrected in the Jupiter images with the final division by the flat-field frames.

Fig. 11

Flat-field image correction examples, left to right in each row: $\lambda =726$, 848, 890, and 926 nm. (a)–(d) Original flat-field images and (e)–(h) defringed flat-field images. Image contrast is enhanced to emphasize features.

Figure 12 shows example results for the Jupiter images. The top row shows the final science frames where fringe and flat-field corrections have been completed. Even in uncorrected Jupiter frames, it can be difficult to visually identify fringing and transmission response artifacts because of the banding on the planet. Therefore, in the bottom row of Fig. 12, we show the percentage difference between the final science frames and the original uncorrected Jupiter frames. These difference maps illustrate the combination of AOTF/sensor transmission response and etaloning artifacts that are removed from the Jupiter images. The results illustrate that the response corrections tend to dominate but fringe structures are visible in the 848- and 926-nm difference frames. The lack of fringes in the 890-nm difference frame is discussed in Sec. 4.3.

Fig. 12

Jupiter image correction examples, left to right in each row: $\lambda =726$, 848, 890, and 926 nm. (a)–(d) Jupiter images corrected using synthetic fringe frames and divided by the defringed flat-fields. Image contrast is enhanced to emphasize features. (e)–(h) Percentage difference per pixel between original Jupiter images and the final corrected Jupiter images. The blue color intensity indicates negative differences (0% to $-35\%$) and yellow color intensity indicates positive difference (0% to $+35\%$).

By direct examination of several positions on the flat-field images, the contrast values ($2\alpha$) due to the fringes were measured. In the flat-field images before correction, the peak fringe contrast varies between $\sim 5\%$ and $\sim 2\%$ (i.e., 0.05 and 0.02) depending on the area of the sensor and the wavelength. However, at longer wavelengths ($>890\mathrm{nm}$), the overall fringe contrast in the uncorrected flat-fields tends to be close to or above $\sim 5\%$. After correction, the fringe contrast on average decreases to $\sim 2\%$ to 1%. This shows that our defringing method reduced the etaloning fringes by at least a factor of $\sim 2$; however, depending on position/wavelength, the correction can be as much as a factor of $\sim 8$.

4.3.

Best-Correction $\mathit{\alpha }$-Values Discussion

Plots of the best-correction $\alpha$-values for the flats (${\alpha }_{\mathrm{flat}}$) and Jupiter (${\alpha }_{\mathrm{Jupiter}}$) as a function of wavelength (Fig. 13) display features that allow further interpretations related to defringing correction, model behavior, and data signal levels. For $\lambda <700\mathrm{nm}$, there are no observable fringes in the flat-field or Jupiter images; therefore, the $\alpha$-values for both are near zero. As the wavelength increases ($\lambda >740\mathrm{nm}$), fringes appear as indicated by the increasing $\alpha$-values. The $\alpha$-values continue to rise with wavelength because the Si absorption decreases with wavelength providing more internally reflected intensity. When ${\alpha }_{\mathrm{flat}}$ and ${\alpha }_{\mathrm{Jupiter}}$ have comparable values for a particular wavelength, it indicates that the fringe patterns in both the flat-field and Jupiter frames have similar contrast levels and the correction of the science frames could likely be done with a conventional flat-field division.

Fig. 13

Best-correction $\alpha$-values [see Eqs. (1) and (2)] as a function of wavelength for the defringing results. The error bars indicate the sensitivity of the power spectrum search method as described in Sec. 3.2.

A curious feature in Fig. 13 is the slope and negative values for both curves in the 650- to 740-nm range. This corresponds to a contrast-reversed fringe correction. Through numerical simulation studies, we found that an $\alpha$-value oscillating with wavelength can arise if the single-layer model [Eqs. (1) and (2)] is applied to a volume with multiple internal reflecting surfaces. Specifically, if a second weak reflecting surface is near the principal reflecting surface, then a portion of a slowly oscillating response can appear like the negative feature in Fig. 13. In effect, this feature represents the response of our model to multiple layers. We believe this is the explanation for this feature and our simulation results show that the single-layer model with this response is still effective in the defringing process for multilayer situations where one surface reflection is dominant. It is straightforward to extend our approach to multiple layers for a higher fidelity model, but a new multivariable method would need to be developed to search for the thicknesses and contrast values.

Around the ${\mathrm{CH}}_4$ absorption region ($\lambda \sim 890\mathrm{nm}$) in Fig. 13, the ${\alpha }_{\mathrm{Jupiter}}$ curve dips significantly below ${\alpha }_{\mathrm{flat}}$ and drops to zero, or slightly negative, for a few points. A close examination of the Jupiter frames in this region showed that the absorption feature is deep enough that the intensity signal-to-noise ratio is poor and the fringe variations are buried in the noise, which explains the ${\alpha }_{\mathrm{Jupiter}}$ values dropping near zero. We found that applying an uncorrected flat-field introduced fringes back into these science frames. Thus, our approach of separate defringing of the flat-field frames is important for this wavelength region. The few negative points for ${\alpha }_{\mathrm{Jupiter}}$ in this region might again be influenced by the multilayer response effect mentioned in the previous paragraph.

It is not clear if the optical spectral energy distribution differences for the flat-field lamps and Jupiter also contribute to the difference in $\alpha$-values near $\lambda \sim 890\mathrm{nm}$ because it is difficult to separate this from other effects. QTH lamps mounted to the APO 3.5-m telescope structure were used for flat-field illumination and APO test calibration results for these lamps show a relatively smooth spectral irradiance profile that increases with wavelength throughout our range of interest. On the other hand, the ${\mathrm{CH}}_4$ absorption feature involves large gradients so it is possible there is a spectral energy difference contribution to the $\alpha$-value results. We have yet to find a definitive explanation for the peak and dip variation in the ${\alpha }_{\mathrm{flat}}$ curve for $\lambda >890\mathrm{nm}$; however, we saw a similar variation in other flat-field data collected for this CCD, so it may be device related.

4.4.

Disk-Averaged Spectrum with Defringed Jupiter Data

The corrected NAIC Jupiter images undergo a geometric and a photometric calibration prior to any spectral analysis. Application of these data in conjunction with the radiative transfer package NEMESIS (nonlinear optimal estimator for multivariate spectral analysis)¹⁷ for modeling of the structure and color of Jupiter’s uppermost cloud deck is ongoing. Nevertheless, as an initial verification of the defringing and calibration process, the disk-averaged NAIC Jupiter spectrum was compared with an established spectrum presented by Karkoschka.¹⁸ Figure 14 shows such a comparison with defringed and calibrated NAIC data that were collected on March 27, 2017, during the 5th perijove pass of Juno. The Karkoschka spectrum falls within the error margins estimated for the NAIC spectrum, which suggests the defringing process did not significantly perturb the spectrum. Some slight differences between the spectra, particularly in the absorption bands, can be attributed to limited NAIC spectral resolution, low signal-to-noise ratio, and a sensitive spectral leakage correction that accounts for filter transmission sidelobes.

Fig. 14

Disk-averaged spectrum of NAIC Jupiter data that were defringed, calibrated, and corrected for spectral leakage compared to a spectrum from Karkoschka.¹⁸ The estimation error envelope for the NAIC spectrum is marked in gray.