J. Eur. Opt. Society-Rapid Publ. 22, 42( 2026) 411
� d j ðk i Þ ¼ n
jðk i Þ� n j
�; ð2Þ n j � 1
with n j( k i) the refractive index of the optical glass of the jth lens for the three wavelengths k C = 656,281 nm, k d = 587,562 nm, and k F = 486,134 nm and n j the mean of the refractive indices across these three wavelengths.
This leads to two principal components, denoted d( j, PC 1) and d( j, PC 2). The principal component analysis, through the 83 optical glasses, transforms the basis of the d j diagram, resulting in d( j, PC 1) andd( j, PC 2) vectors which are combinations of the d j at different wavelengths and proportional to the 1st and 2nd order of colour aberrations respectively. These vectors are robust to changes in catalogues and wavelengths. The coefficients of the two principal components are normalized to a peak-to-valley deviation of 1 using the following equations:
� dðjPC 1 Þ ¼ �1: 077n
jðFÞþ0:282n j ðdÞþ0:795n j ðCÞ �; n j � 1 ð3Þ
Figure 6. Diagram of the reduced Schott Glass map after principal component analysis and the application of empirically selected criteria( 34 glasses).
� dðjPC 2 Þ ¼ 0: 274n
jðFÞ�1:000n j ðdÞþ0:726n j ðCÞ �: ð4Þ n j � 1
The axial chromatic aberration for first and second order axial colour for a system of k thin lenses in contact can therefore be defined as: dU PC i
¼ Xk j¼1Þ d j; PC i
U j; ð5Þ
with U j the partial refractive power of the lens j and dU PC i the ith axial chromatic aberration. The resulting diagram
for dU PC 1 = U anddU PC 2
= U is shown in Figure 5, with
U ¼ P k j¼1U j ¼ 1 = f where f is the total focal length. By tracing a straight line connecting two glasses and intersecting the DU PC 2 = U axis, i. e. DU PC 1
¼ 0, we can select a glass couple with first order axial colour corrected and
ðDU PC 2
= UÞ. f as secondary axial colour. The ratio of the distances between the selected glasses and the resulting doublet on the diagram reflect the individual refractive power. A larger separation between the glasses implies lower required refractive powers for each element. Lower refractive powers allow for the use of lenses with larger radii of curvature, thereby reducing spherical aberrations. Therefore, to design a well-corrected achromatic doublet, it is better to select glasses that are widely separated in the diagram and connected by a line oriented toward, or as close as possible to, the origin.
Under the specified conditions some glasses cannot be combined with others to design an achromat. This provides a good basis on which to further reduce the glass catalogue. Because we are designing an achromat, it is not necessary to achieve DU PC 2
= U ¼ 0. Instead, glass combinations are selected to satisfy empirically DU PC 2
= U values ranging from 0to�3.5 10 �4. This corresponds to a maximum allowable
Figure 7. Spot size in mm produced by the achromatic cemented doublet designed by the SPC method( yellow), A. Szulc method( Purple) [ 16 ] and the two off-the-shelf achromatic cemented doublets from Thorlabs( green) and Edmund Optics( blue).
secondary axial colour of f / 2857. A minimum separation of 0.012 was imposed between selected glasses on the map, to prevent the selection of materials with similar optical properties. Under these criteria, the glass map was reduced to a total of 34 glasses( see Fig. 6), tabulated in Table 4.
5 Results
5.1 Axial object field solutions on achromatic doublets
The SPC method, combined with the previous glass map, has led to the design of 68 achromatic doublets with a focal length of 100 mm and an f-number of 4. To assess the performance of achromatic cemented doublets, the spot sizes of the five best performing designed achromatic cemented doublets were compared with the five best achromatic cemented doublets designed by A. Szulc [ 16 ], as well as two off-theshelf achromatic cemented doublets: Thorlabs reference