本研究首先詳細說明雙橢圓曲面光柵（DEG）系統，包括光路徑函數及像差係數的分析，利用費馬原理與座標轉換求出雙橢圓曲面光柵中的像平面上之像差。但是因為固定曲率的關係，DEG系統的像差無法被調整。
將DES系統改為雙自由曲面光柵（DFSG）系統，利用調整自由曲面光柵參數與光柵條紋函數參數降低雙光柵曲面系統中的像平面上之像差。假設一光柵曲面為自由曲面，其中自由曲面光柵方程式包含自由曲面光柵參數，為可變參數，並利用費馬原理與座標轉換求出雙自由曲面光柵中的像平面上之像差。雙自由曲面光柵系統建立光柵結構參數後，可計算出像差，借由自由曲面光柵參數與光柵條紋函數參數改變光柵曲面並修正像差，解決無法調整像差的問題。

A double ellipsoidal grating (DEG) system with variable grating line spacing is presented, which includes the analysis of the optical path function and aberration coefficients. Fermat’s principle is applied to relate the coordinate systems between the two ellipsoidal gratings. Aberrations on the image plane can be calculated based on this analysis. However, the aberrations on the image plane for a DEG system is unable to be adjusted.
In order to reduce the aberrations on the image plane for a double grating system, we construct a double free-form-surface grating (DFSG) system with variable grating line spacing. The aberration coefficients of the optical path function of a DFSG system are compared to those of a DEG system in order to construct the coefficients of free-form grating surface profile and variable line spacing. By using Fermat’s principle, the coordinate transformation between the two gratings is obtained.
Aberrations on the image plane can be expressed as the grating surface profile coefficients and the grating line spacing coefficients. By adjusting these two sets of coefficients, we are able to reduce the aberrations on the image plane.

[1] G. Lu and B. Fei, “Medical hyperspectral imaging: a review,” J. Biomed. Opt., 19, 010901 (2014).
[2] J. S. Pearlman, P. S. Barry, C. C. Segal, John Shepanski, Debra Beiso and Stephen L. Carman, “Hyperion, a space-based imaging spectrometer,” IEEE Trans. Geosci. Remote Sens., 41, 1160-1173 (2003).
[3] A. Coradini, F. Capaccioni, P. Drossart, A. Semery, G. Arnold, U. Schade, F. Angrilli, M. A. Barucci, G. Bellucci, G. Bianchini, J. P. Bibring, A. Blanco, M. Blecka, D. Bockelee-Morvan, R. Bonsignori, M. Bouye, E. Bussoletti, M. T. Capria, R. Carlson, U. Carsenty, P. Cerroni, L. Colangeli, M. Combes, M. Combi, J. Crovisier, M. Dami, M. C. DeSanctis, A. M. DiLellis, E. Dotto, T. Encrenaz, E. Epifani, S.Erard, S. Espinasse, A. Fave, C. Federico, U. Fink, S. Fonti, V. Formisano, Y. Hello, H. Hirsch, G. Huntzinger, R. Knoll, D. Kouach, W. H. Ip, P. Irwin, J. Kachlicki, Y. Langevin, G. Magni, T. McCord, V. Mennella, H. Michaelis, G. Mondello, S. Mottola, G. Neukum, V. Orofino, R. Orosei, P. Palumbo, G. Peter, B. Pforte, G. Piccioni, J. M. Reess, E. Ress, B. Saggin, B. Schmitt, D. Stefanovitch, A. Stern, F. Taylor, D. Tiphene and G. Tozzi, “VIRTIS: an imaging spectrometer of the Rosetta mission,” Adv. Space Res., 24, 1095-1104(1999).
[4] S. Masui and T. Namioka, “Geometric aberration theory of double-element optical systems,” J. Opt. Soc. Am. A, 16, 2253-2268 (1999).
[5] 林琦焜，最小作用量原理，「數學傳播」，第三十五卷，第一期，第15-28頁 (2011)。
[6] T. Namioka, M. Koike and D. Content, “Geometric theory of the ellipsoidal grating,” Appl. Opt., 33, 7261–7274 (1994).
[7] T. Namioka and M. Koike, “Aspheric wave-front recording optics for holographic gratings,” Appl. Opt., 34, 2180–2186 (1995).
[8] H. Noda, T. Namioka and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. A, 64, 1031-1036 (1974).