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J. Eur. Opt. Society-Rapid Publ. 21, 8( 2025)

Table 3. Transformed ray aberration coefficients and percentage errors for the system shown in Figure 2 with an object placed at infinity with b 0 = 4.377 °.

coordinate frame( xyz) attached to the entrance pupil by using the expressions x e = x e( x, y) andy e = y e( x, y), to obtain

P nx

P ny

R s n s

@ y @ y e axis

@ W ðh 0; x; yÞ =@ x @ W ðh 0; x; yÞ =@ y

: ð16Þ

Equation( 16) is obtained by assuming ox / oy e 0andoy / o x e 0, and setting oy / oy e = ox / ox e to be( oy / oy e) axis, which is evaluated along the optical axis of the axially symmetrical system. Again, equation( 16) is also valid for a system with an object at infinity, provided that the object height h 0 is replaced by the field angle b 0.

3 Conclusions

Many wavefront aberrations are determined using the Gaussian imaging equation. In other words, they are expressed in terms of the object height h 0 and coordinates in the image space as W( h 0, x e, y e), where( xyz) e is attached to the exit pupil. Thus, the classic relationship between the wavefront aberrations W( h 0, x e, y e) and ray aberrations( DP nx, DP ny) isconfined to coordinate frames in the image space. Alternatively, the primary wavefront aberrations can be determined without using the Gaussian imaging equation by using the Taylor series expansion of the optical path length to obtain the expression W( h 0, x, y), where( xyz) is the coordinate frame in the object space attached to the entrance pupil. To estimate the ray aberrations( DP nx, DP ny) fromthewavefrontaberrationsW( h 0, x, y) ofasystem with a Gaussian image plane, this study has converted the classic relationship equation between the ray and wavefront aberrations [ 6 ] using the chain rule based on q e = q e( q, /) and / e = / e( q, /). It has been shown that the percentage errors of the transformed ray aberrations relative to those derived in [ 14 ] and obtained by Zemax are negligible foranobjectplacedatP 0 ¼ð0; 17; �200Þ or at infinity with b 0 = 4.377 ° in the illustrative axially symmetrical system. Thus, the validity of the transformation equation is confirmed.

Funding National Science and Technology Council( 113-2221-E-006-103-).

Conflicts of interest The author declares no conflicts of interest.

Data availability statement

The author confirms that the data supporting the findings of this study are either available within the article and its supplementary materials or could be obtained from the authors upon reasonable request.

References

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( Cambridge University, 1999). 8 Malacara D, Malacara Z, Handbook of lens design( Marcel Dekker, New York, 1994).

9 Wyant JC, Creath K, Basic wavefront Aberration Theory for Optical Methodology. Applied Optics and Optical Engineering,