一般的光學系統在成像時都會產生像差，影響成像品質，因此針對像差的研究便顯得格外重要，而其中最著名的研究莫過於Seidel提出的賽德像差與Buchdahl提出的計算賽德像差係數的方法，但此係數的計算方式會涉及複雜的複合函數運算，在計算上會非常吃力。因此本論文提出透過泰勒級數展開式，將原先複雜的複合函數運算簡化成較容易的多項式運算，大大簡化計算波前像差係數的難度。
本論文以物高h與入射瞳直角坐標 (Xa,Ya)作為獨立變數，將光程函數對光軸作泰勒展開到零至六階，再透過光程函數的對稱性質將其簡化及歸納，可整理出二、四及六階波前像差函數的公式，配合本研究推導出的光程函數之各階導數，即可計算二、四及六階波前像差之數值。最後將四階波前像差之結果與光學模擬軟體Zemax模擬之結果作比較，以驗證本研究之四階波前像差之正確性。由於Zemax內並無六階波前像差之數值，因此本文透過計算出的五階波前像差與有限差分法，以近似的方式得到六階波前像差的近似解，並將其與本研究得出的六階波前像差作比較，來驗證本研究理論之正確性。

Most published works of calculating the aberration coefficients showing lots of iterative operations are very complicated. Therefore, this study based on the well-known Taylor series expansions of a skew ray with respect to the object height and the Cartesian coordinates of entrance pupil provides another simpler way to calculate Seidel-aberrations from the viewpoint of wavefronts.
Regarding the object height and the Cartesian coordinates of entrance pupil as the independent variables, we expand the OPL function by Taylor series expansion with respect to . Then, the polynomial of the fourth- and sixth-order wavefront aberration can be obtained by the relation between OPD and the wavefront aberration. After we simplify both polynomials by the symmetric properties of the axis-symmetrical optical systems, we can derive the formulas of the fourth- and sixth-order wavefront aberrations.
Finally, substituting the value of aberration coefficients calculated by the numerical software our laboratory developed into the aberration formulas, we can solve the magnitude of variety aberrations and then compare the results with those from optical software. However, the software doesn’t provide the magnitude of sixth-order wavefront aberration. Therefore, we have fifth-order wavefront aberrations approximated to sixth-order wavefront aberrations by Finite-Difference methods, and then compare those to the results we calculated.
It’s found the calculated values of this thesis agree remarkably with the fourth-order results of software simulation and the sixth-order approximation respectively, which verifies the correctness of this theory.

[1.] Johnson, R. Barry. “Historical perspective on understanding optical aberrations.”Lens Design: A Critical Review. Vol. 10263. International Society for Optics and Photonics,1992.
[2.] Aiton,Eric John. “Johannes Kepler in the light of recent research.” History of
Science 14.2 (1976):77-100.
[3.] Descartes, René, and Et Gilson. Discours de la méthode. Vrin, 1987.
[4.] Swerdlow, Noel Mark. “Optical Profusion,” Chicago Journal, Isis, Vol.77, No.1,
pp.136-140, 1986.
[5.] Hamilton, William Rowan. “Theory of system of rays.” Transactions of the
Royal Irish Academy 15.1828 (1828):69-174.
[6.] Hamilton, William Rowan. “Supplement to an essay on the theory of systems of rays.” Transactions of the Royal Irish Academy 16.part 1 (1830):1-61.
[7.] Hamilton, William Rowan. “Second Supplement to an essay on the theory of
systems of rays.” Transactions of the Royal Irish Academy 16.part2 (1831):93-125.
[8.] Petzval, Joseph. Bericht über die Ergebnisse einiger dioptrischen Untersuchungen. Hartleben, 1843.
[9.] Seidel, Ludwig. “Zur Dioptrik. Über die Entwicklung der Glieder 3ter Ordnung
welche den Weg eines ausserhalb der Ebene der Axe gelegene Lichtstrahles
durch ein System brechender Medien bestimmen, vo Herrn Dr. L. Seidel.”Astronomische Nachrichten 43 (1856):289.
[10.] Buchdahl, Hans Adolph. Optical aberration coefficients. Dover
Publications,1968.93
[11.] Schwarzschild, Karl. Untersuchungen zur geometrischen Optik: Einleitung in
die Fehlertheorie optischer Instrumente auf Grund des Eikonalbegriffs. I. Vol. 1.
Druck der Dieterich'schen Univ.-Buchdruckerei (W. Fr. Kaestner), 1905.
[12.] Conrady, Alexander Eugen. Applied Optics and Optical Design, Part One.
Courier Corporation, 2013.
[13.] Kingslake, Rudolf, and R. Barry Johnson. Lens Design Fundamentals.
academic press, 2009.
[14.] 鄭子暘. “軸對稱光學系統的三階光線像差之研究.” 成功大學機械工程學
系學位論文 (2018):1-83.
[15.] 周冠宇. “軸對稱光學系統的四階波前像差與五階光線像差之研究.” 成功
大學機械工程學系學位論文 (2019):1-88.
[16.] Huygens, Christiaan. Traite de la lumiere: Où sont expliquées les causes de ce
qui luy arrive dans la reflexion, & dans la refraction. Et particulierement dans
l'etrange refraction du cristal d'Islande, par CHDZ Avec un Discours de la cause
de la pesanteur. Chez Pierre vander Aa marchand libraire, 1967.
[17.] Gauss, Carl Friedrich. Dioptrische Untersuchungen. Druck und Verlag der
Dieterichschen Buchhandlung, 1841.
[18.] Hopkins, Harold Horace. Wave Theory of Aberrations. Clarendon Press,1950.
[19.] Hopkins, Harold Horace. “Canonical Pupil Coordinates in Geometrical and
Diffraction Image Theory.” Japanese Journal of Applied Physics 3.S1 (1964): 31.
[20.] Mahajan, Virendra N. “Zernike annular polynomials for imaging systems with
annular pupils.” JOSA 71.1 (1981): 75-85.
[21.] Mahajan, Virendra N. ” Zernike Circle Polynomials and Optical Aberrations of
Systems with Circular Pupils.” Applied Optics 33.34 (1994): 8121-8124.
[22.] Mahajan, Virendra N. Optical Imaging and Aberrations: Ray Geometrical 94
Optics. Vol. 45. SPIE press, 1998.
[23.] Rayces, J.L. “Exact Relation between Wave Aberration and Ray Aberration.”
Optica Acta: International Journal of Optics 11.2 (1964): 85-88.
[24.] Lin, Psang Dain. Advanced geometrical optics. Springer Singapore, 2017.
[25.] Teodorescu, Petre, Stanescu, Nicolae-Doru, and Pandrea, Nicolae. Numerical
Analysis with Applications in Mechanics and Engineering. John Wiley & Sons,
2013.
[26.] Sadiku, Matthew NO. Numerical Techniques in Electromagnetics. CRC press,
2000.
[27.] Sasián, José. Introduction to Aberrations in Optical Imaging Systems.
Cambridge University Press, 2013