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dispersion to compensate the later in the design process introduced lens thickness. However, there also should not only be negative dispersion present in the system. For the chosen s Ref = 1 mm, a lower limit of > �1 10 �3 is not advisable to be crossed. Applied to the in this article discussed two-lens examples, the resulting SF11, SF11 = 1 10 �4( comp. Appendix A. 2) and BK7, SF11 = �4.4 10 �5( see Fig. 5) accurately distinguish the two cases. The former does not provide sufficient negative dispersion and therefore, the paraxial calculations may not be trustworthy. The latter, fulfills the requirements and it can be assumed, that the paraxial calculations may form a sufficient accurate start system for further optimization. This is confirmed by the ray-tracing in Figure 5.
2.2 Design of a three-lens system
Based on the introduced equations and estimators, it can be derived, that a system of three refractive lenses has sufficient free parameters to achieve the primary goals( I) and( II). Following the systematic approach, it may be separated into a collimation subsystem consisting of a negative and a positive lens and a second focusing subsystem consisting of a single positive lens. In order to maximize demand( II) for the secondary spectrum, the subsystem should be chosen accordingly. As this cannot be simply derived from m i and p i of the individual lenses, a new quantity n 1, k to characterize the secondary dispersion of a system of k lenses shall be introduced: n 1; k ¼ UdC 1; k
: ð13Þ
U d 1; k
For single lens this equation simply represents the ratio./ m, just as used formerly in equation( 12). Atfirst, the two-lens collimator shall be designed with these paraxial estimations. The distance between the two lenses s 2 thereby must meet the collimation condition. Therefore, it can be estimated as:
s 2 ¼ 1
U d 1 Ud 2
� 2h 1
D 2 þ U d 1 þ Ud 2
¼� 2h 1f d 1 f d 2
D 2 þ f d 2 þ f d 1
ð14Þ
where h 1 = arcsin( NA F) is the aperture angle of the common fiber and D 2 the desired diameter of the collimated output beam. Using U d i for this estimation represents the compromise for exposure and measurement wavelength, k F and k C, respectively. Analogously, the first distance to the fiber facet can be derived. With insertion of equation( 14) follows:
s 1 ¼ 2h 1
� 1 f d
D 2 f2
d 1
D 2
2h 1
: ð15Þ
Thereby, D 2 = ð2h 1 Þ¼f1 d; 2 can be expressed as focal length of the collimator sub-system. s 1 and s 2 now control the parameter space of suitable focal lengths of the two lenses. For s 1, s 2 > 0, the two ranges to attain the combination f1 d < 0, f2 d > 0canbefoundas:
Figure 6. System dispersion of the first two-lens subsystem.
0 < f d 2 < f d 1; 2; ð16Þ
�1 = ½1 = f d 1; 2 � 1 = f d 2 Š < f d 1 < 0: ð17Þ
Opposite to f1 d > 0, f 2 d < 0, this case minimizes the total sub-system length s 1 + s 2. Using these equations, a collimator sub-system with a targeted D 2 = 4 for a common fiber NA F = 0.12 is directly fixed as f1 d; 2
¼ 16: 6 mm. The subsystem is estimated to consist of two stock lenses with f1 d ¼�9mm and f 2 d ¼ 12 mm, respectively. Given these, different material combinations can be analyzed to maximize the secondary system dispersion n 1, 2. Figure 6 shows these for the materials given in Table 1. For given available fiber-coupled laserdiodes, k F = 448 nm, k C1 = 642 nm, and k C2 = 662 nm are picked for the design. Combinations of strong dispersive with low dispersive glasses exhibit stronger dispersion for longer wavelengths. It is possible to achieve negative dispersion for combinations of low dispersion with high dispersion glasses. However, this might be reversed by the later added third focusing lens.
To judge whether achromatism for goal( I) is achievable, Figure 7 shows the from equation( 12) calculated reliabilities 1, 2. Greencolormarksasufficient negative dispersion for a safe estimation, while red values probably will not result in a viable solutions. It is confirmed, that the first negative lens should be of highly dispersive glass to ensure a safe estimation. Accordingly, the first lens should be made of SF11. For the second lens, all other glasses might be of interest for the whole system. These shall be further investigated and combined with the third focusing lens.
As the distances s 1 and s 2 are already fixed by the collimation criteria, the third lens must enforce the primary goal( I). For this purpose it is efficient, to use the recursive variant of equation( 10), similar to equation( 5):
U FC 1; 3 ¼ UFC 1; 2 þ UFC 3
� ðs 3 � t 2 Þ U F 1; 2 UF 3 � UC 1; 2 UC 3
: ð18Þ
Thereby U k
1; 2 ¼ Uk 1; 2 ðs 1; s 2 Þ is the two-lens collimator subsystem. Its t 2 is described by equation( 6). Inserting goal( I) and transforming for s 3, yields the searched distance for an achromatic paraxial result between k F and k C: