Fast Imaging Solar Spectrograph of the 1.6 Meter New Solar Telescope at Big Bear Solar Observatory

Abstract

For high resolution spectral observations of the Sun – particularly its chromosphere, we have developed a dual-band echelle spectrograph named Fast Imaging Solar Spectrograph (FISS), and installed it in a vertical optical table in the Coudé Lab of the 1.6 meter New Solar Telescope at Big Bear Solar Observatory. This instrument can cover any part of the visible and near-infrared spectrum, but it usually records the Hα band and the Ca ii 8542 Å band simultaneously using two CCD cameras, producing data well suited for the study of the structure and dynamics of the chromosphere and filaments/prominences. The instrument does imaging of high quality using a fast scan of the slit across the field of view with the aid of adaptive optics. We describe its design, specifics, and performance as well as data processing

Appendices

Appendix A: Flat Fielding

Let us denote the index of a pixel along the wavelength direction by i and that along the slit direction, by j. An averaged spectrogram, s _ij should depend on i only, but not on j if the spatial average is ideally done so that we may write s _ij =o _i where o _i is the one-dimensional spectrum of the average sun. Then the kth spectrogram $a_{ij}^{k}$, that is acquired by the camera with the spectrum being displaced by x _k from the reference spectrum along the wavelength direction can be mathematically modeled as:

$$ a_{ij}^k = c_k s_{i-x_k,j} f_{ij} = c_k o_{i-x_k} f_{ij}, $$

(3)

where f _ij is the flat pattern we intend to determine, and c _k is the relative level of intensity. We find it necessary to modify this equation a little bit, for the spectral lines in the acquired spectrograms are not vertically straight, but slightly curved. This distortion may be considered as a manifestation of horizontal displacement of spectrum depending on the position along the slit, which we denote by Δ _j . Taking into account this distortion, we modify the above equation into

$$ a_{ij}^k = c_k o_{i-\Delta_j-x_k} f_{ij} , $$

(4)

or into its logarithmic form,

$$ A_{ij}^k = C_k + O_{i-\Delta_j-x_k} + F_{ij} . $$

(5)

For given $A_{ij}^{k}$, x _k , and Δ _j , maximizing

(6)

and

$$ K = \sum_{ijk} \bigl( C_k + O_i + F_{i+\Delta_j+x_k, j} - A_{i+\Delta_k+x_k , j}^k \bigr)^2 w(i+\Delta_j+x_k) $$

(7)

we obtain the iterative formulas: for F _ij

$$ \Delta F_{ij} \approx- {\sum_k (C_k + O_{i-\Delta_j-x_k}+F_{ij} - A_{ij}^k ) w(i-\Delta_j-x_k) \over\sum _k w(i-\Delta_j-x_k) } , $$

(8)

for C _k

$$ \Delta C_k =- {\sum_{ij} (C_k + O_{i-\Delta_j -x_k} + F_{ij}-A_{ij}^k ) w(i-\Delta_j-x_k) \over\sum _{ij} w(i-\Delta_j-x_k) } , $$

(9)

and for O _i

$$ \Delta O_i \approx- {\sum_{jk} ( C_k + O_i + F_{i+\Delta_j+x_k, j} - A_{i+\Delta_k+x_k , j}^k ) \over \sum_{jk} w(i+\Delta_j+x_k) } . $$

(10)

Note that w(x) and v(y) are functions that have values of either unity or zero depending on whether x and y are inside the corresponding index ranges, respectively. These expressions for the iteration are similar to those given by Chae (2004) so we do not go into further details.

Appendix B: Compression

Generally speaking an arbitrary vector I _i of N elements (i=1,…,N) can be written as a linear combination of a set of N orthonormal vectors $s_{i}^{k}$ (k=1,…,N). Specifically we choose the form

$$ I_i = {\bar{I}} \sum_{k=1}^{N} c_k s_i^k, $$

(11)

where $\bar{I}$ is the average intensity. The coefficient c _k is determined from I _i :

$$ c_k = \sum_{i=1}^N {I_i \over\bar{I}} s_i^k . $$

(12)

If $s_{i}^{k}$ can be placed in descending order of statistical significance, we can reduce the number of coefficients to be kept to n<N while retaining most physical information:

$$ I_i \approx{\bar{I}} \sum_{k=1}^{n} c_k s_i^k . $$

(13)

Data are thus compressed by a factor of N/n.

The set of basis vectors $s_{i}^{k}$ is constructed from the database of model profiles, $J_{i}^{l}$ (l=1,…,M), that well represent spectral profiles to be analyzed. By referring to Rees et al. (2000), we define a symmetric covariance matrix

$$ A_{ij} = \sum_{l=1}^M {J_i^l \over\bar{J}^l} {J_j^l \over\bar{J}^l} $$

(14)

and identify its kth eigenvector with the kth basis vector $s_{i}^{k}$ while the eigenvectors are sorted in the descending order of the associated eigenvectors. Note that the statistical significance of $s_{i}^{k}$ is measured by its associated eigenvalue λ _k .

Appendix C: Stray Light Model

We model the effect of spatial stray (large-angle spread) light on the data using the equation

$$ {I_c^{\mathrm{obs}} (\mathbf {r}) \over I_c^{\mathrm{obs}} (0) } = (1-\epsilon) {I_c (\mathbf {r}) \over I_c (0)} + \epsilon{S(\mathbf {r})} $$

(15)

and that of spectral stray (large-wavelength spread) light using the equation

$$ {I_\lambda^{\mathrm{obs}} (\mathbf {r}) \over I_c^{\mathrm{obs}} (\mathbf {r}) } = (1-\zeta) {I_\lambda(\mathbf {r}) \over I_c ( \mathbf {r}) } + \zeta{ V_\lambda}, $$

(16)

where $I_{c}^{\mathrm{obs}}(\mathbf {r})$ and $I_{c}^{\mathrm{obs}} (0)$ are the observed intensities of the continuum at an arbitrary position r and the disk center, respectively, and $I_{\lambda}^{\mathrm{obs}}(\mathbf {r})$ and $I_{\lambda}^{\mathrm{obs}} (0)$ refer to those at an arbitrary wavelength λ. The intrinsic intensities I _c (r), I _c (0), I _λ (r), and I _λ (0) are defined in the same way.

In our model, the effects of spatial stray light and spectral stray light on a spectrogram are independent from each other; the spatial stray light affects the level of intensity equally over different wavelengths, and the spectral stray light affects the shape of each spectrum. The spatial stray light integral S(r) is a normalized function slowly varying over the position r, but may be approximated to be equal to a constant value, 1, near the disk center. In a similar way, the spectral stray light integral V _λ is a normalized function slowing varying over wavelength λ, but may be approximated to be equal to a constant value, 1. Therefore, the correction of stray light in disk observations can be done if two parameters, the fraction of spatial stray light ϵ and the fraction of spectral stray light ζ, are known.